Research Collection. Algorithmic decision suport for train scheduling in a large and highly utilised railway network. Doctoral Thesis.

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1 Research Collection Doctoral Thesis Algorithmic decision suport for train scheduling in a large and highly utilised railway network Author(s): Caimi, Gabrio C. Publication Date: 2009 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library

2 Diss. ETH No Algorithmic decision support for train scheduling in a large and highly utilised railway network A dissertation submitted to the ETH ZURICH for the degree of DOCTOR OF SCIENCES presented by GABRIO CURZIO CAIMI Dipl. Math. ETH born 14 th June 1979 citizen of Ligornetto (TI) accepted on the recommendation of Prof. Dr. Hans-Jakob Lüthi, examiner Prof. Dr. Leo Kroon, co-examiner Prof. Dr. Ulrich Weidmann, co-examiner 2009

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4 Non c è successo se non c è amore in quello che fai

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6 Acknowledgments Since I was a kid, I have always been interested in railways. Being the son of a train driver, I had the opportunity to have an inside view of this world from the very beginning. This passion has remained with me as a private interest. Only after many years have I had the opportunity to combine my learned skills in mathematics with this early interest. For this, and for having helped me complete this thesis, I would like to thank many people. First of all, I am very grateful to Prof. Hans-Jakob Lüthi for having given me the possibility to stay at the institute after writing my diploma thesis so I could write this thesis on the railway topic, although two other colleagues were already working on it. Thank you for the trust and the freedom during these past few years. I have also to thank Prof. Ulrich Weidmann and Prof. Leo Kroon for accepting my request without hesitation to be co-examiners of this thesis. I have greatly appreciated the always very interesting discussions, where I could learn a lot and see things from another perspective. Your contribution for the successful outcome of the work was essential. A fundamental person all through the thesis work was my daily supervisor Dr. Marco Laumanns. You always had time for me even if you were involved in a thousand projects, you trusted me, you taught me how to conduct such a big project, how to structure a report and manage the time, how to take a step back and get the big picture, and many other substantial things that belongs to a PhD. A special thank goes also to Dr. Fabian Chudak, who supervised me in the beginning phase of the thesis. Some central ideas of the thesis arose already during discussions with you, very stimulating to learn new mathematical techniques not yet applied to railways problems. An applied project is really applied only if it is done in collaboration with an industrial partner. Swiss Federal Railways were this and much more. They continually brought interesting problems, participated in insightful discussions and listened very open-mindedly

7 iv to our proposed methods. Special thanks go to Dr. Felix Laube, Dr. Raimond Wüst, Samuel Roos, and Oskar Stalder who were particularly involved in the collaboration. This dissertation would not have been the same without the important contribution of my colleagues at IFOR also working on the railway project. I am deeply indebted to my former colleagues Dr. Dan Burkolter and Dr. Thomas Herrmann. You introduced me to the topic, supported me actively during the initial phase of my thesis and continued doing so also after having left the institute. An important role in this project was also played by my colleagues Martin Fuchsberger and Kaspar Schüpbach. They started doing their master thesis on this topic, and the attained positive results let them join the institute to continue working. Many many thanks, your contribution and implementation was decisive for the good results of this thesis. I wish you all the best for the continuation of you own thesis on railway projects and related topics. I am also very grateful to Stefan Wörner. The first ideas about partial periodicity was developed during his bachelor thesis and served as basis for the future development. I appreciated much your engagement and enthusiasm and I am sure that you will finish your master studies with success and I hope that you will not forget the railway world for your future career. Special thanks go to Marco Lüthi, who wrote his thesis in parallel to mine in collaboration with SBB. Even if the topics of our thesis were in its technical part relatively disjointed, you were always ready to give me an answer on technical question and always took time for me when necessary. I also have to thank Dr. Rico Zenklusen, Holger Flier, and Dr. Marc Nunkesser for their contribution in the theoretical part of Appendix B. The nature of the research topic required me to spend time in discussion with many people from both the railway industry and academics. For the always stimulating discussion I want to thank Heinz Egli, Falk Möser, Nicolas Regez, Dr. Stephan Thomann, Muriel Perron, Venkata Mahadevan, and Thomas Graffagnino. I also greatly appreciated the contact with international colleagues, in particular from Rotterdam, Utrecht, Delft, Trieste, Bologna, Berlin, Göttingen, and Paris. I would like to thank all my former and current colleagues at the Institute for Operations Research that made my time as a PhD student so enjoyable. A special thank go to Michael Guarisco, who started at the institute with me and was always there for discussions, support, or simply a good laugh. This time at ETH was also very enriching due to the various students that I had the opportunity to work with. I also thank them sincerely. I would like to thank Marco Laumanns, Revital, and Hilda Fritze-Vomvoris for proofreading this thesis and having contributed to improve the quality of the English. I am also grateful to all those that helped me in the translation of the abstract in the different

8 v languages: vielen Dank Kaspar, merci beaucoup Vania et Amparo, grazie mille Giorgio, grazia fitg Denise, hartelijk dank Diana. Un grande ringraziamento va anche a tutte le persone che mi hanno sostenuto in questi anni e hanno condiviso con me gioie e dolori. Un sentito grazie ai miei genitori per l incessante fiducia e sostegno di tutti questi anni, cosí come al resto della mia famiglia. Grazie Vido per il tuo continuo e contagiante entusiasmo; grazie a tutti gli amici ticinesi, zurighesi, e anche a quelli sparsi un po ovunque; grazie Grifone per ricordarmi che le vittorie si ottengono solo col sudore. E l ultimo ringraziamento va alla persona piú importante di tutte: mia moglie. Grazie Elisa! Per tutto, semplicemente.

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10 Abstract This thesis addresses the general problem of constructing train schedules, in particular for large and highly utilised railway networks. Commercial requirements for the timetable are assumed to be given, and the task is to provide detailed conflict-free track paths for each train that fulfil these requirements. In the thesis, a comprehensive approach from the commercial description of intended train services to a conflict-free detailed schedule for a whole day is developed. The methodology follows a divide-and-conquer strategy based on three description levels: the service intention, the macroscopic timetable, and the microscopic schedule. The levels are interfaced in such a way that planners have the possibility of intervening into the specifications on every level, and enabling a feedback loops for testing different alternative scenarios. Many models and algorithms for train scheduling have already been proposed in the literature, some of them with successful application in practice. However, they are either designed for large-scale problems, considering a simplified topology and safety system, or are detailed approaches, yet applicable only to a restricted area. This thesis combines both approaches for finding detailed schedules for large networks, partially relying known models and algorithms from the literature, adapted or extended, and partially developing totally new ideas and methods. The starting point of the approach is the construction of an appropriate structure for describing the intended train services, including periodicity information. This partial periodic Service Intention (ppsi) contains the commercial offer that a railway company would like to tender to the customers during a day. The purpose of the ppsi is to have a formal framework in which potential commercial offers can be developed and described systematically. An important property of modern railway timetables is their periodic pattern, so that they are easy for the customers to memorise. In addition to this regularity,

11 viii the introduction of irregular train services is necessary to cope with varying demand over the day. The developed ppsi can describe commercial railway offers with partial periodic structure in a compact form and can exploit these effectively in the train scheduling process. The basic idea to handle the partial periodicity of the service intention is an equivalent projection onto a single period time, resulting in an augmented periodic problem. This augmented periodic timetabling problem is then solved first globally on an aggregated topology with a simplified safety model (macroscopic level), and subsequently, locally refined by considering more details of the railway infrastructure and train dynamics (microscopic level). Finally, the generated periodic conflict-free train schedule is rolled out over the complete day to create a production plan that fulfils all requirements that were initially specified in the service intention. The macroscopic level focuses on global interdependencies over the entire network for generating the most important properties of the timetable and thus has to avoid dealing with large amounts of detailed information that is only locally relevant. According to the simplified topology model, safety constraints and train dynamics are also simplified. A well known model for this description level is the Periodic Event Scheduling Problem (PESP). In this thesis, an extension called Flexible Periodic Event Scheduling Problem (FPESP) is introduced and applied, allowing for time slots for each event instead of fixed times. Moreover, an extension of the FPESP model is proposed, the Flexbox model, which is a further generalisation of the FPESP that allows to make use of natural dependencies among events in the service intention. The resulting (flexible) macroscopic timetable is then used as input for the microscopic level. The existence of an operable production plan for a given draft timetable has to be checked on the microscopic level by taking into account detailed information that is important for ensuring the schedule to be conflict-free, but which are not relevant for the global structure of the timetable and have therefore been neglected on the macroscopic level. The safety model on the microscopic level follows the way the interlocking system works. It computes the blocking times on each infrastructure element based on the signalling system and ensures that these blocking time intervals do not overlap. Moreover, the computed track paths must specify a speed profile that the train driver can follow, given a reasonable tolerance band. As the microscopic scheduling problem might become infeasible, the event time slots of the FPESP solution give more freedom, which is particularly crucial in bottleneck regions with dense traffic, where the solution on the macroscopic level with fixed times is often too restrictive. The requested level of detail, together with dense traffic, leads to a enormous problem size. Therefore, a network separation approach is applied to divide the railway network into zones of manageable size by taking account of the network properties, distinguish-

12 ix ing condensation and compensation zones. Condensation zones are usually situated near main stations, where the track topology is complex and many different routes exist. As such an area is expected to have a high traffic density, it is also a capacity bottleneck and trains are required to travel through with maximum allowed speed and thus without time reserves. Collisions are avoided by exploiting the various routing possibilities in the station area. Conversely, a compensation zone connects two or more condensation zones and consists of a simpler topology and less traffic density. Here, time reserves should be introduced to improve timetable stability. The choice of an appropriate speed profile is the most important degree of freedom to exploit in these compensation zones. For both zones, a new model called Resource Tree Conflict Graph (RTCG) is developed for solving the microscopic scheduling problem. In this model, a set of scheduling alternatives for each train is computed first and stored in a compact way using a tree structure. In condensation zones, these alternatives are computed by looking at all routes through the topology as well as a discrete set of starting times for each train. In compensation zones, a reasonable set of alternative speed profiles for the few different routes is computed. Afterwards, constraints are derived that preclude conflicts between the alternatives on the involved resources. Based on a graph model, a resource-constrained integer multicommodity flow problem is formulated as an integer linear program. The model has considerably fewer and stronger constraints compared to previous formulations based on stable sets in conflict graphs, which leads to a much stronger LP relaxation and hence much shorter computation times. The RTCG model enables therefore large problems to be solved quickly. In the case of lack of feasibility in a zone on the microscopic level, a feedback strategy is applied to generate another macroscopic schedule according to the information obtained from the microscopic level. The proposed multi-level approach is validated through some real-world test cases from Switzerland. Computational results are presented for all steps of the timetable generation process, and are compared with previous methods for evaluating their improvements.

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14 Contents Acknowledgments Abstract iii vii 1 Introduction Motivation Goal of the thesis Main contributions Outline of the thesis Background Planning from the commercial offer to the conflict-free production plan Deregulation of railway market Planning stages Timetabling and periodicity The timetabling task Notion of periodicity A production-oriented approach Background and goal Planning and production methods Integrated real-time rescheduling Focus of this thesis Problem definition and two-level approach Related work

15 xii Contents 3.2 Basic idea of the two-level approach Macroscopic topology Purpose and properties Graph representation Macroscopic safety system using headway times Formal definition Microscopic topology Basics and goal Double vertex graph Resources Signals and blocking times Formal definition Train dynamics Microscopic train dynamics Macroscopic train dynamics Partial periodic service intention Macroscopic timetable and microscopic schedule Train scheduling problem and two-level algorithm Macroscopic timetabling Related work Periodic Event Scheduling Problem Classical PESP model Constraints Objective functions Cycle periodicity formulation Non-collision constraints Flexible PESP Motivation and basic idea Flexibility and robustness Properties FPESP model Objective functions Interaction with the microscopic level Flexbox model Motivation Definition Application examples

16 Contents xiii 4.5 Computational results Reference scenario Results for different objective functions Results for the Flexbox model Bi-objective analysis Summary and final remarks Microscopic scheduling: network decomposition approach Related work Network decomposition Condensation zones Compensation zones Interface between the zones Solving the micro scheduling problem Microscopic scheduling in condensation zones Problem formulation Related work Conflict graph model Other approaches Policies for train scheduling Time discretisation Route reduction The Resource Tree Conflict Graph model Resource trees Constraining flows: the Resource Tree Conflict Graph Objective function ILP Formulation Extension to departure time slots Robustness measure Dealing with infeasibility Variants of the RTCG model Tree Conflict Graph Resource Conflict Graph Computational results Scenarios and implementation Analysis of the scheduling policies Computational results and comparison between the models Results for the extended ILP

17 xiv Contents Robustness results Summary and final remarks Microscopic scheduling in compensation zones Related work Problem formulation and solution approach Generation of speed profiles for a single route Generating a set of β-profiles Restricting the speed profiles Quality measure Optimisation model Computational results Final remarks Train scheduling for partial periodicity Basic idea Procedure for partial periodic train scheduling Slot propagation Projection to a periodic instance Periodic timetable generation Rolling out the solution Equivalence between original and projected problem Choice of the period length T Computational results Final remarks Global results and added value of flexibility Results for the complete train scheduling procedure Added value of flexibility on the micro level Conclusions Critical appraisal of the results Outlook for future research Bibliography 213 A Scenarios used as test cases 231 A.1 Network in central Switzerland A.1.1 Macroscopic topology A.1.2 Microscopic topologies

18 Contents xv A.2 Condensation zone of Berne A.3 Macroscopic network in Ticino A.4 Overview on the usage of the scenarios B On stable sets and maximal cliques in intersection graphs 239 B.1 Motivation B.2 Intersection graphs and stable sets B.3 Interval graphs B.3.1 Performance proof of the algorithm B.4 Circular-arc graphs B.4.1 Non-degenerate circular-arc graphs B.5 Summary C Computation of track paths with blocking times 253 C.1 Different speed profiles for same route C.2 Platforms and connections C.3 Computation of blocking time start and end Zusammenfassung 261 Riassunto 265 Résumé 269 Riassunt 273 Samenvatting 277

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20 List of Figures 2.1 Perspective triangle Deregulated situation Planning stages Planning stages in time Planning stages in Rail Line plan Rolling stock circulation Production chain at SBB in Basic elements of the production chain according to PULS Current control process Future control process Construction of the macroscopic topology Headway and opposite headways Modeling train routes in networks Microscopic topology of the station of Berne in dvg format Double vertex graph with representation of the resources Calculation of a blocking time interval in a green wave policy with conventional safety system Calculation of the blocking time interval for a departing train Train run Example of ppsi Schematical representation of the addressed general train scheduling problem

21 xviii List of Figures 4.1 Overview on the principal methods in the literature for macroscopic timetabling A part of a PESP graph for two trains stopping in a minor station and then leaving in two different directions Case where the introduction of a non-collision constraint is necessary Flexibility for the events i and j Changed in upper and lower bounds when introducing event slots Example of Flexboxes Macro timetable with and without flexibility Flexible timetable for the line Muri Gotthard Flexible timetable for the line Baar Gotthard Pareto front for the scenario in central Switzerland Pareto front for the scenario in southern Switzerland Pareto front for the reference scenario for fixed train sequences Pareto front for the reference scenario with station priorities Pareto front for the reference scenario for fixed train sequences and station priorities Pareto front of the reference scenario with randomised weights for flexibilities Pareto front for the reference scenario for fixed train sequences and randomised weights for flexibilities Overview on the principal methods in the literature for microscopic scheduling Possible representation of the Swiss railway central part divided in condensation and compensation zones Illustration of Property Flexibility of the speed profile Procedure for the microscopic scheduling problem Overview on the principal methods in the literature for microscopic scheduling in station areas Example of a conflict graph Illustration of Policy Representation of a discretised timetable including track occupation versus the classical time-space diagram Dominated Path Similar Paths Resource Tree

22 List of Figures xix 6.8 RT with flow commodities Resource Tree Conflict Graph Allocation Schema Concurrent resource occupation Extension to departure time window Extended time slots Example of a Tree Conflict Graph Example of a Resource Conflict Graph Overview on the principal methods in the literature for generating conflict-free speed profiles on railway lines Two-step approach for micro scheduling in compensation zones Speed-location function of the α-profile Set of all generated speed profiles Speed profiles for the commuter train S1 Lucerne-Zug Example of slot propagation for a train run Illustration of the procedure for partial periodic train scheduling Necessity or not of introducing headway constraints in the projected version Flexibility against penalty on the micro level A.1 The region connecting the towns Zug Lucerne Arth Goldau A.2 The macro topology of the test scenario in central Switzerland A.3 A more precise macro topology of the network in central Switzerland A.4 Main station area Lucerne A.5 Micro topology of the core of the condensation zone Lucerne A.6 Micro topology of the railway line between the portals of Lucerne and Zug.235 A.7 Main station area Berne A.8 Switch region topology on the west side in front of Berne main station A.9 Switch region topology on the east side in front of Berne main station A.10 The third scenario region connecting the towns Chiasso Biasca Locarno Luino in southern Switzerland A.11 Macro topology of the region in southern Switzerland B.1 Intersection graph of an allocation schema B.2 Odd chordless cycles in circular-arc graphs B.3 Arc model A and corresponding circular-arc graph C.1 Different speed profiles for the same edge

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24 List of Tables 2.1 Example of mixture between periodicity and irregular services for the lines Zurich-Lucerne and Wil-St. Gallen in the 2009 timetable Example of mixture between periodicity and irregular services in Italy, France, and Germany in the 2009 timetable Considered type of information at the macro and micro level, as well as non considered information during the proposed train scheduling approach Data of the PESP graph and the MIP for the reference scenario Results of FPESP for different objective functions Effect of the limitation of δ i δ max Maximal flexibility for the different types of Flexboxes Conflict Matrix Impact of the interval length τ on problem size and computation time Impact of the route reduction on the conflict graph size and the computation time for 4 scenarios in Berne Statistics of the three models for three scenarios Conflict clique sizes in the RTCG model for the three scenarios Statistics of the three different ILP formulations of three scenarios, before and after the (standard) preprocessing step of CPLEX Processing times (in CPU seconds) for the various computation steps Solution quality and steps of the MIP solver Growth of CPU time depending on the number of allowed starting times. 165

25 xxii List of Tables 7.1 Statistics on speed profile generation Optimal values for the three objective functions depending on granularity Statistics of the generated PESP instance and solution time of the ILP optimisation for two different scenarios and some values of T Added value of flexibility A.1 Overview on the utilisation of the scenarios for the computations of this thesis B.1 Summary of the principal properties for describing stable sets in the three considered classes of intersection graphs

26 List of Algorithms 1 Approximated ε-constraint method for Pareto front Creation of conflict cliques for a resource r Binary search for maximising the static robustness of the schedule Checking necessity of headway constraints Projection of the ppsi to a fully periodic instance Adaptation of Algorithm 2 for partial periodicity Computation of the blocking time start Computation of the blocking time end

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28 Chapter 1 Introduction 1.1 Motivation Over the last few decades, railway traffic in Europe has increased considerably for both passenger and freight transportation, and this trend is expected to continue over a long term perspective [UVEK, 2006]. The Swiss Federal Railways (SBB), for instance, operate the densest railway network in Europe. On their 3000 track kilometers, approximately 9000 trains travel per day, yearly transporting a total of 300 millions passengers and 60 millions tons of freight. In some areas, the railway network is already at its capacity, while the demand for transportation services is still increasing. To face this increasing demand, an improvement of all aspects of the offer becomes necessary, e.g., faster connections, higher frequencies, and more comfortable rolling stock. Improving rail transportation services poses new challenges for the utilisation of the infrastructure and the production processes within railway companies. This situation is aggravated by the deregulation of the railway market currently underway in most European countries, which forces additional coordination effort between the different companies operating in the railway market with their unequal production concepts. One of the most crucial issues is to provide sufficient capacity for operating the desired services in a robust way. This target can be reached, for instance, with the construction of new railway infrastructure, such as the extension of a single line track to double track or the design of a new high-speed line to reduce the travel time. This solution, however, is very expensive, takes many years (or even decades) from its concept to its realisation, is often difficult to realise due to the many construction constraints, especially in city centers, and last but not least requires much landscape change raising also environmental concerns. Since this kind of investment has to be reduced to the minimum necessary, the existing

29 2 Chapter 1: Introduction infrastructure has to be used at its full capacity and other, less invasive measures must be considered first before building new infrastructure. Besides extension of the physical infrastructure, new technologies are being introduced for standardising the signalling system, improving safety, communication, and control in railway operations. This option is still quite costly, but considerably less expensive than new constructions and less invasive for the environment. Significant examples are the GSM-R and ETCS systems, which are part of the ERMTS (European Rail Traffic Management System) project, with the aim of creating a single Europe-wide standard for railway signalling and data transmission. Beside its main goal, ETCS also allows, in its levels 2 and 3, a significant reduction of the headway time between two trains. Depending on track and rolling stock, the headway can be reduced from 150 or more seconds down to 90 seconds, which increases the capacity of the furnished tracks substantially. Technological improvements are none the less still insufficient for the envisaged expansion of services. Improvements are also necessary with respect to internal planning and control processes. More precision is needed in both the planning and operation phase. A precise plan enables a smooth operation as long as no or only minor disruptions occur, e.g., avoiding stops in front of red signals that cause unnecessary capacity loss. Likewise, precise operation can reduce deviations from the plan, thus reducing delays and improving the stability and available capacity of the schedule. Current production processes still have many manual interfaces, and computers are mainly used for visualisation and data handling, with few automated decisions. This approach has so far produced good performance according to customer expectations, but its limits are being approached due to the increasing complexity of planning and operations. This necessity of more accuracy, together with higher traffic density, considerably increases the importance of automatic decision support during both planning and traffic management. Advanced computational methods can help to handle the increasing complexity and can therefore be of great support to the planners. Thus, many railway operators are investing in software systems to increase performance and reliability of their planning processes. The SBB, for example, have been allocating resources for the development of efficient methods for generating and operating railway schedules. This thesis is part of this endeavour and is concerned with the development of models and algorithms for the automatic generation of network-wide conflict-free train schedules. 1.2 Goal of the thesis The thesis contributes to the advancement of railway planning and control tools, in order to enable railway operators to achieve their goals of increasing train frequencies. In particular, the focus is on constructing conflict-free train schedules in large and highly utilised

30 1.2 Goal of the thesis 3 railway networks. To this end, a methodology is devised that allows train schedules to be automatically generated from a given train service intention, which is the description of the train services that passenger and freight rail companies would like to offer to the customers. A key requisite of the methodology is that is must deliver train schedules that are very detailed with respect to both the used track elements and the timing of the train runs. Thus, for each train, a track path consisting of a precise route, including passing times, through the railway topology has to be determined. Additionally, these track paths are required to be conflict-free, i.e., assuming no delays, all trains can run exactly as planned without creating safety conflicts. This feature is in contrast to today s timetables, which are typically not planned to be conflict-free and rely rather on on-line, manual resolution of resource conflicts as they appear, in real time. For this purpose, it is first necessary to develop and formally define, a framework for the description of the intended train services on the network from all concerned train companies. This train service intention contains all commercially relevant information such as travel times, connections, or periodicity, and is intended as a basis for negotiating the railway offer between the different railway companies without having to consider purely technical details. This allows planners to concentrate on what to offer as transportation services, whereas the question of how exactly this offer can be produced using the available resources is merely technical and shall be automatised using mathematical models and algorithms. Given this commercial train service intention input, the problem addressed in this thesis consists of finding a conflict-free schedule fulfilling the requirements described in the service intention. Because of the large size of the regarded railway network, it is impossible to consider each detail during one single optimisation step. Efficient and suitable models become necessary for the subdivision of the whole problem into treatable sub-problems and the solution of them exploiting their specific properties. The intended purpose of the proposed methodology is its usage as a flexible decision support system (DSS), and not just as a monolithic black box that is fed with the topology and train services and, after some computation time, delivers a complete production plan. Planners should still play an important role during the timetable generation process, with the possibility to interact with the tool, modifying decisions, testing alternatives, adding or relaxing some conditions. Furthermore, the approach should be suitable during all levels of the planning process, from long-term to short-term planning. This way, the planning process can be standardised and enables a better communication and feedback between the different actors involved in the planning of a railway operations. The technical approach of this thesis is twofold. On the one hand, the train scheduling problem is addressed with appropriate models that describe the reality accurately but

31 4 Chapter 1: Introduction remain mathematically tractable. On the other hand, suitable algorithms are developed to solve the different sub-problems efficiently without losing the target of obtaining a system-wide feasible schedule. 1.3 Main contributions This thesis presents a contribution to the field of train scheduling by presenting a complete, modular, and integrated multilevel approach for system-wide detailed conflict-free train scheduling in large and highly utilised railway networks, from the description of the commercial offer to the detailed production plan. The main achievements of the thesis are the following. A framework for the description of the commercial offer of rail services is introduced, the partial periodic service intention (ppsi). This object will be formally defined, discussed, and used as input for the subsequent goal of generating a train schedule that fulfills the properties specified in the service intention. The ppsi contains all information that are relevant for the customer and neglects all technical considerations that are only necessary to operate the schedule but have no commercial relevance. Periodicity of the offer is observed to be an important factor for passengers and is therefore included into the ppsi in an individual way for each train run, possibly resulting in commercial railway offers that are not completely periodic but contain some arbitrary periodic structure. The purpose of the ppsi description is to have a framework in which potential commercial offers can be developed and analysed. Thus, technical decisions, which should be automatised, are strictly separated from commercially relevant decisions, which should remain a decision made by human planners after negotiations between the involved companies and given as input. To the best of the author s knowledge, this is the first time that the input for the train scheduling problem directly addresses the notion of periodicity and formalises it in a general form that describes all mixed forms of periodic and non-periodic offers. A methodology for automatically generating train schedules in large networks is proposed and discussed. To cope with the enormous amount of data and detailed information, a two-level approach consisting of a macroscopic and microscopic description level is followed. This allows global decisions to be separated from only locally relevant decisions and therefore to efficiently handle large instances without abandoning the very important property of creating a detailed conflict-free schedule. This multi-level, modular planning approach is conceived as a decision support system for strategic, tactical, and operational planning horizons. It provides

32 1.3 Main contributions 5 an integrated and standardised tool that enables a continuous and smooth flow from long-term to short-term planning. Planners can intervene at each level by manually (slightly) adjusting the automatically generated schedule. On the macroscopic level, the FPESP model is proposed for the generation of simplified, network-wide railway timetables. To increase the chance that a generated macroscopic timetable could be extended into a conflict-free microscopic schedule, the well-known PESP model is augmented with the notion of flexibility. Instead of exact event times, the FPESP model yields time slots for the departure and arrival times of the trains with the property that any choice of event times inside the computed slots, which is now deferred to the microscopic level, is feasible from the macroscopic point of view. This additional freedom provided by the FPESP is important as the microscopic scheduling problem might become infeasible, especially in bottleneck regions with dense traffic, if the solution on the macro level is too restrictive. The FPESP is a generalisation of the known PESP model, and its formulation as a mixed integer program (MIP) also yields an extension to the efficient cycle periodicity formulation of the PESP, without having to introduce any new integer variable. As a further generalisation, the Flexbox model is introduced. The concept of Flexboxes takes advantage of the natural dependencies between certain events instead of considering them all as mutually independent. This way, flexibility can be placed where it is really necessary and can therefore be exploited more effectively on the microscopic level. Similarly to the FPESP, the Flexbox model can be formulated as a MIP with the same structure and integer variables of the corresponding PESP. Thus, the computational effort for both models is only moderately larger than for the PESP. Thus, it can make use of existing theoretical results for the PESP formulation and future improvements regarding the PESP are likely to carry over to the FPESP and the Flexbox model as well, still allowing for more flexibility and being therefore better suited for the purposes of the proposed multilevel approach. On the microscopic level, a network decomposition approach is proposed to divide the railway network into zones of manageable size by taking the network properties into account, distinguishing condensation and compensation zones. This approach seems a natural way to cope with the huge amount of data, and the separation into zones with distinct properties allows different degrees of freedom for planning, namely the route choice in condensation zones and the speed profiles in compensation zones. For microscopic scheduling in both condensation and compensation zones, the Resource Tree Conflict Graph (RTCG) model is developed and applied. The RTCG

33 6 Chapter 1: Introduction model represents the scheduling alternatives in a compact way and allows for a very efficient detection of the potential conflicts, which are described much more efficiently than in previous approaches. Thanks to the strong structure of this model, the resulting ILP formulation can be solved in seconds even for real instances of a large size, mainly because of its strong LP-relaxation, which often provides relaxed solutions that are already integer. The RTCG model is designed to deal with the flexible input delivered by the FPESP and Flexbox model from the macroscopic level and can effectively exploit the provided additional degrees of freedom. If the microscopic problem is still not feasible, the model delivers a feedback instead of just a negative response, by providing an alternative solution that fulfils the original requirements of the ppsi and violates the macro schedule in a minimal way. Furthermore, some variants of the RTCG model are also presented and discussed for their strengths and weaknesses. In particular, the alternative RCG model, without the tree structure for the alternatives, seems to be interesting for very large instances for which the RTCG model could develop memory problems, especially in long compensation zones. The presented methodology for periodic scheduling is extended to work also for service intentions that can have an arbitrary heterogeneous periodicity structure. The approach is based on a reduction (projection) of the ppsi to a periodic problem, to which the presented methods for periodic scheduling can be applied, in order to exploit the advancements in periodic scheduling also for partial periodic scheduling. The projection method exploits the existing periodic structure efficiently by eliminating all redundancy and creating one single decision variable for each event of a periodically repeated service. Under some realistic assumptions, it is possible to guarantee the equivalence between the original and the projected problem, which enables the optimality of the generated schedule or infeasibility of the input ppsi to be proven. The proposed projection procedure also permits an easy implementation within the current tools for train scheduling, as it basically consists of an add-on to the algorithms for periodic timetabling. 1.4 Outline of the thesis The thesis is structured as follows. Chapter 2 gives first an overview on the complete planning procedure of a railway system and continues discussing the specific properties of the timetabling stage, in particular the notion of periodicity. Furthermore, the project of Swiss Federal Railways that contextualise this thesis is presented and explained. The chapter ends describing the focus of

34 1.4 Outline of the thesis 7 this thesis into this context. Chapter 3 explains the general approach for solving the introduced train scheduling problem. The method is based on a two level approach, the macroscopic and the microscopic level. Both levels are introduced in this chapter, and the purpose and goal at each level is explained. The chapter formally describes all necessary input information and states clearly the form of the output at all levels. Based on these formalisations, the addressed train scheduling problem is precisely defined, and the procedure for solving it is finally depicted. Chapter 4 focuses on the periodic timetabling on the macroscopic level. It first introduces the well known Periodic Event Scheduling Problem (PESP). The main goal of the thesis on this level is then motivated, increasing the chance of finding a feasible solution on the microscopic level. Therefore, it proposes a generalisation of the PESP model to search for arrival and departure time intervals in lieu of exact event times, which are quite restrictive for the micro level and often lead to infeasibility. This additional flexibility for those events leads to the extended model developed in this chapter, the Flexible Periodic Event Scheduling Problem (FPESP). The FPESP model is further generalised with the so called Flexbox model. The added value of using this model is that it makes use of the natural dependencies between the events in order to additionally increase the chance of getting a feasible solution in the microscopic level. Chapter 5 focuses on the train scheduling problem on the microscopic level. This chapter does not present any algorithms for solving it, but it discusses the properties of the network and proposes a subdivision of the network in zones of treatable size, according to the different properties, called condensation and compensation zones. Finally, the procedure for dealing with the local zones and merging them into a global conflict-free train schedule is explained. Chapter 6 introduces models and algorithms for the micro scheduling problem in condensation zones, which are the bottlenecks of the network. It first presents a policy for scheduling in a condensation zone based on time discretisation and route reduction. Afterwards, the RTCG model is formally defined step by step, from the input to its final ILP formulation, and also some variants of the model are discussed. Finally, the chapter proposes a method for providing feedback to the macro level in case of infeasibility in a condensation zone. Chapter 7 introduces models and algorithms for the micro scheduling problem in compensation zones. It starts by discussing different approaches and models to generate speed profiles. The problem of conflict-free scheduling in compensation zones is then introduced formally and the proposed solution method is described. The generation of speed profiles is discussed and the applied optimisation problem is explained, based on an adapted version of the RTCG model from the previous chapter.

35 8 Chapter 1: Introduction Chapter 8 focuses on the train scheduling problem for a partial periodic input. This problem is solved by reducing the input to a periodic instance, where it is possible to apply the presented two-level approach for periodic scheduling. This reduction method for the partial periodic case is formally defined and discussed for its strengths and drawbacks, and its basic idea is given. The chapter continues introducing in detail the different steps of the procedure. Furthermore, the conditions for the equivalence of the reduced periodic and the original partial periodic problem are shown and discussed. Chapter 9 discusses aspects of the approach that are relevant for more than one of the steps presented in the previous chapters. It presents some results for a scenario in central Switzerland completely computed from its input description to the corresponding conflict-free micro schedule for the whole considered network. Moreover, the added value of flexibility that is generated on the macro level is evaluated for its impact on the condensation zones on the micro level. The main results of the thesis are discussed in Chapter 10, that also presents some reccommendations for future research. Appendix A illustrates the layout of the parts of the Swiss railway network that are used as test cases. Appendix B analyses from a graph theoretical point of view the structure of the allocation schemas for solving the micro scheduling problem in a local zone. After the motivation, the general notions of stable set polytopes in intersection graphs are introduced. The focus is then given first to the non-periodic case, where it is shown that the proposed algorithm is able to find all maximal cliques. The periodic case is then studied: it is first shown that the algorithm, even if always correct, does not always find all maximal cliques. Moreover, a realistic restriction to a subclass of these graphs is then introduced such that the algorithm remains able to find all maximal cliques. Finally, Appendix C describes additional technical details necessary for the computation of conflict-free track paths in a condensation zone. First, conditions for the correct modeling of speed profiles and platform consistency are presented. Then, the precise computation of the start and the end time of the blocking time for each track path is depicted.

36 Chapter 2 Background This thesis is written in the context of a joint project with the Infrastructure Division of the SBB for the purpose of improving the performance of the railway system. It addresses the problem of generating train schedules for large and highly utilised railway networks aligned with the conceptual basis of the project. In addition to being essential to the project, most of the proposed models and algorithms can nonetheless be applied to many other approaches out of the context of this project. A general introduction to the entire planning procedure in railway networks is given in Section 2.1, and Section 2.2 discusses the timetabling step in more detail, with particular attention to the notion of periodicity. The methodological approach of the joint project is explained in Section 2.3, and the specific role and focus of the thesis is outlined in Section Planning from the commercial offer to the conflict-free production plan The task of a railway system is to offer transportation services that are as reliable and as attractive for the customers as possible. But cost factors also play an important role, preventing the services from being too exaggerated and financially intolerable. In some situations, where some services are not profitable but are of social importance, e.g. services in isolated and low-populated areas, public authorities can intervene and subside the transportation company for offering this service. Thus, there are three main actors in the field of railway transportation: customers, railway companies, and the public authorities. Each actor has a different point of view and different expectations from the railway

37 10 Chapter 2: Background Figure 2.1: The three points of view of a railway system and its interactions. In the oval, the goals for each actor are described, and the arcs specify the interactions between them. system, as explained in detail in [Liebchen, 2006] and illustrated in Figure 2.1. For offering and operating train services, a railway company must first describe the commercial requirements (the so-called train service intention) that they would like to offer to the customers. The service intention then has to be converted into a production plan, i.e. a detailed schedule ready to be operated. If it turns out that the creation of such a schedule is infeasible, a new, less exigent train service intention has to be designed and then the procedure is repeated Deregulation of railway market The European Union (EU) started the deregulation of the railway market with the Directive 91/440 in Even though not a member of the EU, Switzerland also joined this initiative for an open and liberalised railway market. Deregulation was initially limited to international services, and the market was then opened up also for domestic freight transport. Full deregulation also for international passenger traffic is planned for the year 2010 [Hughes, 2008]. As an integral part of the deregulation, each country was required to separate provision of transport services from the management of the infrastructure, or at least to provide for an independent and fair track path allocation. This happened in most cases by splitting the former national railway companies into separate divisions or independent companies for infrastructure management and for passenger and freight transportation, the so-called train operating company (TOC).

38 2.1 Planning from the commercial offer to the conflict-free production plan 11 Figure 2.2: Division between infrastructure operator and TOCs in the deregulated railway market. Distribution of the tasks and interaction in Switzerland. Note that the independent track path allocation of the normal gauge network is outsourced from the infrastructure operators to the company trasse.ch [trasse.ch, 2009]. In other countries, the organisation can be slightly different. These two kinds of entities, the infrastructure managers on one side and the train operating companies on the other side, have different responsibilities and tasks, as illustrated in Figure 2.2. The infrastructure manager is not necessarily the owner of the railway infrastructure, which usually belongs to the state, but it has received the mandate to administer, maintain, and operate the infrastructure. At least in Switzerland, the train stations also belong to the infrastructure manager. The infrastructure manager has the task to provide fair access to the infrastructure to the different TOCs, which are their customers. On the other hand, the TOCs are responsible for offering train services to the passengers or freight forwarders, who are their customers. Based on their economic and operational strategy, they order and purchase track paths from the infrastructure managers in a market-oriented way. On these track paths they run trains with their own rolling stock and crew, and market their services to the customers. Timetable generation, as well as online traffic management, are tasks assigned to the infrastructure managers. For this task they in turn depend on information about future demand by TOCs to create useful track path catalogues. The general perception of this situation until now is that liberalisation takes place quite

39 12 Chapter 2: Background slowly [Heymann, 2006]. The railway sector is still behind other network industries, such as the telecommunication sector or the energy sector. Also, differences in the liberalisation progress between different countries are observed [Holvad et al., 2003]. From this point of view, Switzerland, in particular for the freight market, is in an advanced stage of deregulation. Many freight operators already actively compete in the Swiss railway network for international traffic. Deregulation also comes up against some difficulties. If an infrastructure manager is still associated with a national train operating company, it may hinder non-discriminatory access to the infrastructure for new TOCs. Additionally, there are technical barriers for new, in particular foreign companies, such as different traction voltage and safety systems. Currently, there is still one leading company for passenger transportation in almost every country in Europe, but this situation might change in the future. It is therefore important to define and clearly formalise the interfaces between the infrastructure manager and the TOCs. This thesis addresses the formalisation of the service intention, which is intended to be an interface for negotiations on the assignment of track paths to the TOCs. One possibility for the assignment is to conduct an auction, as prescribed by German law ([Borndörfer et al., 2005]). More in general, the assignment occurs after negotiation between the parties. Thus, it is particularly important for the infrastructure manager to ensure that the assigned track paths can result in an operable timetable. Once a track path has been sold, it is the responsibility of the infrastructure manager to ensure that the train can effectively run on the agreed track path. The generation of a conflict-free and robust train schedule fulfilling the given service intention is therefore of crucial importance for the infrastructure manager already in an early planning phase. This thesis intends to make a contribution on both levels: on the one hand the formalisation of the service intention as basis for negotiations and on the other hand, which is the most important contribution, the generation of a feasible schedule that fulfills a given train service intention Planning stages To reach the goal of a railway system which is to be reliable and to offer train services of good quality, accurate planning is paramount. As the railway infrastructure and resources need a long time to be extended, upgraded, or modified, the planning phase should start many years in advance. The deregulation of the railway market also changed the planning processes compared to the previous situation, requiring structural modification to enable the intended market structures, which is usually achieved by separating infrastructure managers and TOCs [Watson, 2001, Daduna, 2001]. As the production of the railway transportation service is a complex task that involves many different questions ranging from the strategic infrastructure extension to the details

40 2.1 Planning from the commercial offer to the conflict-free production plan 13 Figure 2.3: Classical hierarchical structure of the railway planning process. The infrastructure planning stage is usually the responsibility of the infrastructure manager in cooperation with the public authority. Rolling stock and crew scheduling are internal affairs of the TOC. Line planning and timetabling are usually the result of complicated negotiations between the infrastructure manager and the different TOCs. Note that also other hierarchical subdivisions in stages of the planning process are possible. of crew scheduling, it is usually divided hierarchically into stages, depending on the problem type to be considered. A good general overview of the different stages can be found in [Bussieck et al., 1997a]. In the literature, and also often in practice, the planning process is commonly hierarchically divided into six main stages (Figure 2.3). Each stage generates a result which is used as input for the next planning stage. This classical procedure also has some drawbacks, as for instance pointed out by [Laube, 2009]. In particular, in this method the timetabling step is based on a pre-defined infrastructure and not the other way round with the infrastructure based on timetable considerations, as it would make sense from a commercial point of view. The stages are however not independent of each other and cannot be considered in a purely sequential way. Everything is mutually influenced by everything else: for instance, a timetable which is more attractive to customers [Schittenhelm, 2008] may need additional rolling stock because of an inefficient vehicle turnaround. On the other hand, a slightly worse timetable in terms of customer attractiveness could improve the rolling stock circulation and therefore the cost-effectiveness of the railway offer. Introduction of feedback loops or the integration of several steps are possible ways to overcome these difficulties, but this increases the complexity of the problem that needs to be solved. These six planning stages are principally not meant to lead from long-term strategic planning to daily operation, but they are a hierarchical division into problems that need to be solved in sequence. Figure 2.4 illustrates the relevance of the six planning steps during strategic, tactical, and operational planning, as well as during real-time operations. Infrastructure changes need a very long time and this planning has to be done long time in advance, and line planning for passenger trains is fixed during the creation of the yearly basic schedule. Therefore, these two planning steps are not taken into consideration during operational planning and rescheduling. On the other hand, timetabling, rolling stock,

41 14 Chapter 2: Background Figure 2.4: Timely planning of the production process. During strategic planning, all steps are considered and the infrastructure changes are decided. In tactical planning, the line plan is fixed for the yearly basic plan. The timetabling, rolling stock, and crew planning stages are taken into account continuously until the online operation. and crew planning need planning on all strategic, tactical, and operational levels as well as during real-time operations. As indicated in the figure, focus and goals of the various levels are different, but these three stages have to be taken into account already during strategic planning for the necessary resources to be obtained, which usually takes many years. For example, in a press conference in April 2009, the SBB presented their order strategy for new rolling stock over a time horizon until 2030, for a total investment of 20 billion CHF. At the basis of the whole planning process is the demand estimation. The demand is defined as an estimation of the number of people wishing to travel from an origin to a destination over a certain period of time during the day, or the amount of goods to be transported. This can be conducted with passenger counts, interviews with current or potential customers, and through sales analysis. The results enable the creation of time-dependent origin-destination matrices (OD-matrices, [Zhao et al., 2007]) or of more aggregated data. Based on the demand estimation, the next step is the planning of the railway infrastructure. This is a crucial step in the planning procedure because of the long life and the very high investment costs of the railway infrastructure. The infrastructure can be extended, modified, or reduced, according to the passengers demand needs. Some relevant works on this topic can be found in [Niekerk and Voogd, 1999], [Middelkoop and Bouwman, 2000], [Kavicka and Klima, 2000], and [Romein et al., 2003]. This stage is the responsibility of

42 2.1 Planning from the commercial offer to the conflict-free production plan 15 Figure 2.5: Hierarchical structure of the planning process during the Rail 2000 project. The lines and the timetable are directly planned according to the demand, and the infrastructure is planned depending of the needs of the timetable. the infrastructure manager companies, in close collaboration with the public authorities. Ideally, decisions regarding the infrastructure should also be supported by generating a simplified timetable fulfilling the demand, and the infrastructure is then planned for supporting this generated timetable, as described for instance in [Tzieropoulos et al., 2008]. This approach was the pivotal idea for planning the infrastructure extension during the Rail2000 project [SBB, 2008a, Kräuchi and Stöckli, 2004]. The initial step has been a service intention which resulted in requirements for new infrastructure such as the high speed line (200 km/h) between Matstetten and Rothrist, as depicted in Figure 2.5. This is a notable difference to the classical approach, where infrastructure planning is only based on aggregated demand estimation [Stalder, 2006, Stalder, 2007, Laube, 2009]. After OD-matrices have been estimated and the infrastructure has been planned, a line plan has to be constructed. A line plan consists of a set of train lines that are direct connections offered to the customers between two terminal stations, with some intermediate stops, as illustrated in Figure 2.6. A train line also includes the specification of the type of vehicle used for this service, and its frequency, in case of regular periodic services. Note that on the same path in the railway network it is possible to have multiple train lines with different stop policies and rolling stock type. For instance, between Genève and Lausanne in Western Switzerland, the 2009 timetable [SBB, 2009] provides four types of regular hourly direct connections, taking 33, 39, 44, and 51 minutes with respectively 0, 2, 4, and 7 intermediate stops. Moreover, these services are offered with different kinds of rolling stock and are therefore defined as four distinct train lines. The line planning problem is then the problem of covering the railway network with train lines, such that traffic demand can be fulfilled and some objectives are optimised. Typical objectives are the maximisation of the customers having a direct connection as well as cost minimisation. The result of line planning can be formalised as the train service intention. The train service intention should be elaborated in collaboration between the infrastructure manager and the TOCs, as it concerns the service offer of the TOCs and the coordination of the railway network, which is task of the infrastructure manager. Significant work on line planning can be found in [Bussieck et al., 1997b], [Bussieck, 1998], [Bussieck et al., 2004], [Schöbel and Scholl, 2005], [Goossens et al., 2006], and [Schöbel and Schwarze, 2006]. Once a line plan has been specified and a train service intention is formalised, a

43 16 Chapter 2: Background Figure 2.6: Line plan for the commuter train system (S-Bahn) of Lucerne, Switzerland. Source: timetable has to be constructed which fulfils the requirements defined in the service intention and describes each individual train trip offered to the customers in detail. This is the last step that is published, whereas the next steps are only meant for company-internal planning and are not relevant for the customers. This planning step, called the timetabling problem or train scheduling problem, is a particularly critical step in the whole railway planning process because it has a direct impact on both customers and personnel of the railway companies [Liebchen, 2006]. Moreover, it touches the interest of both the TOCs, responsible for the train services, and the infrastructure manager, responsible for coordination of the traffic and marketing of track paths with all the TOCs on the entire network. Therefore, the timetabling step should be elaborated with the participation of both the infrastructure manager and the TOCs, each with their distinct and clear roles. For a more detailed description of the timetabling problem, which is the main focus of this thesis, the reader is referred to Section 3.8 on page 60, whereas the notion of periodicity, a key issue for timetabling, is discussed in detail in Section 2.2. A comprehensive literature review on railway timetabling is presented in Section 3.1 on page 31. There are also some very special cases where a railway service is offered without a timetable, but with the organisation of transport services ad hoc, depending on the current demand and the online situation [Malucelli et al., 2001, Cordeau, 2006, PostAuto, 2009]. After the timetabling stage, the next step of the planning process is rolling stock planning, also called vehicle scheduling. It consists of defining a set of trips operated in

44 2.1 Planning from the commercial offer to the conflict-free production plan km '46 '25 '39 '12 '46 '25 '39 '12 '46 '25 '39 '12 '46 '25 11:16 h Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf 13:39 h sh km '39 '12 '46 '25 '39 '12 '46 '25 '39 '12 '46 '25 '39 '12 '46 '25 12:49 h Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen 15:46 Hbf h sh km '40 '25 '39 '12 '46 '25 3:57 h Dustadt Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf 13:45 h sh km '39 '12 '46 '25 '39 '12 '46 '25 '39 '12 '46 '25 '39 '12 '46 '25 '39 '21 13:31 h Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Dustadt 16:42 Hbfh sh km '46 '25 '39 '12 '46 '25 '39 '12 '46 '25 '39 '12 '46 '25 11:16 h Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf Sieburg Hbf Wirhofen Hbf 14:39 h sh. Figure 2.7: Extract of a cyclic rolling stock schedule. Source: Extra module Umlaufplanung of Viriato, SMA und Partner AG 0.0 km 0:00 h sequence by the same rolling stock unit, usually on a cyclic basis, over a certain period. However, the same unit could then be assigned to a different set of trips in the next period, as illustrated in the example shown in Figure 2.7. The rolling stock plan is an internal task of the TOC, because it is not important for the customers to know which trip the rolling stock did before, as long as it is reliable, punctual, and a sufficient amount of seats are available. Rolling stock planning has only an impact on the internal processes and the costs of the TOC. Typical objectives are therefore the minimisation of the number of vehicles used, as well as minimising the number of empty trips and maintenance work, but also the maximisation of robustness against disruptions and of the probability that there are sufficient seats available. A comprehensive literature review over rolling stock planning can be found in [Maróti, 2006], and significant work is reported in [Peeters and Kroon, 2008], [Cacchiani et al., 2008a], [Alfieri et al., 2006], and [Fioole et al., 2006]. The last step in the planning process is crew management. It usually takes place in two stages: crew scheduling and crew rostering. Crew scheduling is the generation of a set of general work duties (also called shifts) covering all activities in the operations that require a certain type of personnel, such as train drivers or conductors. Crew rostering is then the assignment of personnel to the duties for a certain time horizon (weeks or months), taking into account holidays, crew member characteristics, (such as knowledge of the lines for train drivers) as well as laws and labor agreements. The rostering can be cyclic or noncyclic. Similarly to the rolling stock, crew management is an internal task of the TOC and therefore under its responsibility. A general literature overview about crew management can be found in [Ernst et al., 2004]. [Abbink et al., 2005] presents a model for crew scheduling used in practice for the planning at NS (Nederlandse Spoorwegen), the Dutch railways. [Caprara et al., 1998] states a model for crew rostering, [Hartog et al., 2009] describes an application for NS, and [Ernst et al., 2001] presents an integrated approach

45 18 Chapter 2: Background to train crew management. Finally, [Haase et al., 2001] and [Huisman, 2004] present two approaches for the integration of vehicle and crew scheduling in a single optimisation step. The last three steps of the planning process are subject to short term modifications, e.g. due to short-term track path requests, maintenance works, or the inavailability of some resources. It is therefore necessary to slightly modify the already created (and published) plans in order to match these changes. Afterwards, the production plan goes into operation. For different reasons, disruptions may occur so that rescheduling becomes necessary for an efficient realisation of the service. A good literature review over rescheduling approaches within these three steps is provided in [Jespersen-Groth et al., 2007] and focused on the timetabling stage in [Törnquist, 2006] and [D Ariano, 2008]. Approaches for rolling stock rescheduling is presented in [Nielsen et al., 2009], whereas crew rescheduling is addressed in [Huisman, 2007]. Besides visualisation tools, which are nowadays broadly used by almost all railways companies, there is also some commercially available software for automatic generation of a plan for one or more of the presented planning steps. Their utilisation, however, strongly depends on the stage. For infrastructure and line planning, there are only few tools available and not really broadly used. In these stages, the tools are still mainly used for visual representation, e.g. the software Viriato from SMA und Partner AG [SMA und Partner AG, 2009]. For the timetabling step, more involved tools are available, in particular solutions tailored for a specific transportation company, like the modules CADANS and STATIONS of the DONS project of Netherlands Railways [Odijk and Van den Berg, 1994, Kroon et al., 2009], or tools that originate directly from academic research work, like the TAKT tool of the Technical University in Dresden [Nachtigall and Opitz, 2008]. Some timetable optimisation modules are also available in the tool VISUM of PTV AG [PTV AG, 2009]. Finally, for more technical, companyinternal tasks such as vehicle and crew planning there are many tools available which are also broadly used in practice (MICROBUS, INTERPLAN, HASTUS, Carmen, TURNI, and others). Currently, the production process at SBB is characterised by a separation between the strategic, tactical, and operational planning. Data and information flow predominantly in one direction, while feedback loops are only partly formalised and only possible with manual effort, as illustrated in Figure 2.8. This is due to several technical and organisational reasons. As a consequence, it is difficult to exploit the optimisation potential in the system.

46 2.2 Timetabling and periodicity 19 Figure 2.8: Production chain at SBB in Source: SBB. 2.2 Timetabling and periodicity In this thesis, the timetabling step is seen as the key stage of the six-stage procedure and is therefore discussed in more detail The timetabling task Requirements and interests of the different TOCs are combined together to form a socalled train service intention, which describes the intended train services that the TOCs plan to offer to their customers. Therefore, the train service intention is a description of the commercial requirements that have to be fulfilled by the train schedule. There can also be other types of requests from the customers, or the public authorities, to the railway companies, such as the comfort and the number of seats in the rolling stock, different types of facilities at the train stations, and so on. These kinds of requirements are not considered further in this thesis, because they do not have a direct influence on the train schedule. Technical data, such as the length of the train and its dynamic properties, has to be included in the service intention because it has an impact on the production plan. The train service intention is the result of discussions and negotiations between customers, public authority, and railway companies (Figure 2.1), that can take place under different forms. For the rest of this thesis it is assumed that these negotiations have already occurred and that a train service intention is given. The task of the train scheduling step is then to determine the technical details for such a schedule by taking into account the optimisation objectives, like short trip times, good connections, and robustness against disruptions. This concept of a service intention can be seen, of course, more generally

47 20 Chapter 2: Background as the input of each work dealing with timetable generation. Nevertheless, only few authors have so far addressed the question of how to formalise a service intention in a way that represents the commercial requirements of the TOC well and is clear and exhaustive enough for starting the train scheduling procedure. Some preliminary work in this direction was done by [Erol et al., 2008], [Borndörfer and Schlechte, 2007], [Burkolter, 2005], and [Herrmann, 2005]. This thesis introduces a new formalisation of the train service intention, which is inspired by the current track path ordering process for the TOCs in Switzerland and by a parallel work of the SBB for an a posteriori description of the commercial services that can be found is an existing timetable, the Global Service Intention (GSI). The GSI contains the customer-relevant information for each service during a day and was intended as a reference for online rescheduling, where changes in the timetable should focus on reducing consequences for customers. The reader is referred to [Wüst et al., 2008, Laube and Mahadevan, 2008, Mahadevan, 2007] for more details about the GSI. The formalisation proposed in this thesis picks out the periodicity as a central theme and integrates it as part of the offer into the service intention. Definition 3.25 in Section 3.6 formalises this concept, and on this basis the addressed train scheduling problem is formally defined Notion of periodicity Many railway companies in Europe operate periodic timetables. In Switzerland, for instance, the periodicity of the train services is considered a main driver of the strong increase in passengers since its introduction in The periodicity makes it easier to remember the departure times so that the passengers can be much more spontaneous in taking a train, even without having a timetable at hand. Today, periodicity is considered a substantial part of the service offer and most of the trains are scheduled with a halfhour or one hour periodicity. The point of no return is by far already passed: the use of non-periodic timetables in Switzerland is simply no longer imaginable. The demand, however, is not distributed uniformly over the day, and also different days of the week can have different demands. Morning peak hours involve a large demand on trips from the suburbs to the city centers, and in late afternoon in opposite direction. Moreover, there is less demand in the evening than during the day and it is focused on direct connections rather than on short trip times. Exceptions in the periodicity of the timetable are therefore important and useful to cope with these irregularities of the demand. Examples of this situation are depicted in Table 2.1. For instance, in the 2009 timetable the IC train from Zurich to Berne is scheduled half-hourly with departures at the minutes 00 and 32. During the morning peak, the frequency is increased with two

48 2.2 Timetabling and periodicity 21 Zurich xx:04 xx:35 16:41 17:41 18:41 00:07 Lucerne xx:49 xx:25 17:39 18:39 19:39 01:07 Time 0:45 0:50 0:58 0:58 0:58 1:00 Wil xx:25 06:54 xx:54 22:54 St. Gallen xx:53 07:17 xx:15 23:17 stops Uzwil, Flawil, Gossau Uzwil, Gossau Gossau Uzwil, Gossau Time 0:28 0:23 0:21 0:23 Table 2.1: Example of mixture between periodicity and irregular services for the lines Zurich-Lucerne and Wil-St. Gallen in the 2009 timetable. additional trains at 6:47 and 7:47. Another example of irregularity is the hourly IC from Zurich to Lucerne, which is usually planned to have only two intermediate stops, but in the late evening stops at five additional stations with a 15 minutes longer trip time. These irregularities in the almost periodic timetable are not a specific Swiss situation. Also in other European countries the situation is similar, as illustrated in Table 2.2. In Italy, for instance, between Milan and Rome during the day one train per hour is scheduled leaving the Milan main station at minute 30 and taking 3 hours and 59 minutes. In the morning hours, because of different demand, there is a faster offer between 6:15 and 8:45 a that takes 3 hours and 30 minutes with a half-hourly periodicity. The overlap of these two offers leads to a periodicity of 15 minutes for the morning peak hours. This is on the one hand an adaptation to a larger (and different) demand in the morning hours and on the other hand the regularity of the offer is kept, because both morning and daily offer are periodic and matched together. Finding the right balance between the interest in a periodic timetable and the nonperiodic demand is a crucial issue for offering a good service to the passengers. Therefore, this issue should be supported by suitable planning tools. Unfortunately, this balance problem has not yet been addressed satisfactorily in the literature on timetable generation methods. Only for the extreme cases of purely periodic and non-periodic timetabling, many models and algorithms have been proposed, in some cases with successful realworld applications [Kroon et al., 2009, Liebchen, 2008]. Periodic timetables with a few non-periodic exceptions, however, have usually been treated as entirely periodic with some manual postprocessing. This procedure has the obvious drawback that only one part of the day, usually a typical hour with basic periodicity, or the hour with highest frequency, is optimised. The timetable for the rest of the day is then achieved by simply removing trains from the generated timetable, or by manually adapting or inserting extra trains. This sequential approach could lead to suboptimal

49 22 Chapter 2: Background Milan 07:15 07:45 08:15 08:30 08:45 xx:30 Rome 10:45 11:15 11:45 12:29 12:59 xx:29 Time 3:30 3:30 3:30 3:59 4:14 3:59 Paris 10:10 12:10 14:10 14:40 15:15 15:50 16:10 17:20 Bordeaux 13:11 15:24 17:14 17:38 18:48 18:52 19:17 20:23 Time 3:01 3:14 3:04 2:58 3:33 3:02 3:07 3:03 Stuttgart 12:07 14:07 14:40 16:07 16:42 18:07 18:40 Nuremberg 14:16 16:16 17:25 18:16 19:25 20:16 21:25 Time 2:09 2:09 2:45 2:09 2:43 2:09 2:45 Table 2.2: Example of mixture between periodicity and irregular services in Italy, France, and Germany in the 2009 timetable. solutions for the parts of the day which are not directly addressed by the optimisation algorithm, which could be avoided formulating a simultaneous problem that constructs the entire timetable in one step. Furthermore, manual postprocessing usually requires more time and effort by the planners that could be better utilised on a more strategic level. Approaching this problem of the balance between periodic offer and non-periodic demand with methods for generating non-periodic timetables is also not satisfactory. Although being able to deal well with irregular demand, these methods lose the important property of the periodic offer and usually result in instances of larger size, as they have to take into account a whole day-timetable. To overcome these drawbacks, this thesis introduces the notion of partial periodicity. Partial is meant in the sense that the periodicity is a property of each single train service, which can vary from service to service, even with some singular non-repeated trains. The union of these train runs results in a timetable that can neither be considered as periodic nor as non-periodic, but rather some heterogeneous mixture in-between. In this thesis, this intermediate situation is called partial periodicity, and it is actually the reality in almost all countries in Europe, while the proportion of periodic services varies from country to country. According to this concept, which is formalised in Definition 3.33, all operated timetables in the world can be considered partial periodic, as the purely periodic and non-periodic cases are simply special cases of this generalisation. The notion of partial periodicity will be introduced formally in Section 3.6. Based on this notion, the so-called Partial periodic service intention is then introduced to describe the intended commercial offer of partial periodic timetables.

50 2.3 A production-oriented approach A production-oriented approach This section describes a new approach for improving the performance of the railway system by following a production-oriented approach. This is the task of the project PULS 90 (German acronym for Produktorientierte Umsetzung der Leistungssteigerung: productoriented realisation of the improvement in performance) of the Infrastructure Division of SBB 1. First, the basic ideas of this approach are briefly described and then how the philosophy behind this approach influenced the methods described in this thesis is explained Background and goal Rail 2000 [SBB, 2008a, Kräuchi and Stöckli, 2004] (in German Bahn 2000, in French Rail 2000, in Italian Ferrovia 2000) is a large ongoing SBB project, which started in 1987 with the goal of improving the quality of the railway system. It involves measures for accelerating and intensifying existent connections between cities as well as modernising the rolling stock and is organised in stages. After the positive public vote in 1987, the first stage was completed in 2004 with around 130 construction projects and a budget of 5.9 billion CHF. Rail 2000 intended to extend the (already) dense periodic timetable in a demandoriented way. The project was mostly based on construction measures to increase the network capacity and to reduce the travel times between the main cities. Nowadays the network is at its capacity limit, and it is very difficult to accommodate additional demand. Financial resources are also more limited than in the past and thus new construction becomes increasingly difficult. The project PULS 90 was started in 2003 with the task to increase the performance (in terms of capacity, efficiency, reliability) of the currently available railway network without reducing quality (punctuality). The project uses a system-oriented approach. It should enable operation of denser timetables without needing to build new tracks, but rather by systematically reorganising the processes. Goal 2.1 (PULS 90) According to [Laube et al., 2007], the key goals of the PULS 90 project are: 1. Standardisation of the planning and production methods in an integrated system (see Section 2.3.2); 2. Reduction of reaction time and increase of production accuracy (from minutes to seconds, see Section 2.3.3); 1 with the support of ETH Zurich

51 24 Chapter 2: Background 3. Alignment of all processes to an optimal customer advantage. These goals are reached with the application of many different methods and the principal are resumed in the following. Method 2.2 The principal methods that are used to reach the goals of the PULS 90 project, described in Goal 2.1, are: 1. Dynamic integrated real-time rescheduling; 2. Permanent data flow and information chain; 3. Transition from sequential, event-oriented production to a time-oriented production (see Section 2.3.3); 4. High precision in the production; 5. Separation of the railway network in zones of tractable size for planning and control (see Chapter 5). The goals 2.1.1, 2, as well as the methods 2.2.1, 2.2.1, and are described briefly in the following sections. Point is described in ([Laube and Mahadevan, 2008, Wüst et al., 2008]). Finally, points and are the most relevant for this thesis, and will be described in detail in Chapter 5. For a more accurate description of the PULS 90 project in general, the reader is referred to [Laube et al., 2007, Schaffer et al., 2005, Lüthi and Stalder, 2007] Planning and production methods This project intends to resolve the problems created by the current production chain (Figure 2.8), in particular the explicit separation between the production levels, with the development of an integrated approach that enables a continuous feedback of information between the levels and therefore a better coordination and overview of the system, ultimately achieving the joint goal of a good service. This integrated process chain formalises the feedback loops, resulting in a closed control loop for the production processes. Figure 2.9 shows the basic elements of the future production chain according to PULS 90. This future approach for the production chain has the planning methods as its core element, which should be systematically used during the planning at all levels: strategic, tactical, operational, and also in real-time. Thereby, an instrumental amalgamation of different levels becomes possible. It is therefore important to keep in mind that the proposed methods should have the same structure for the different planning stages. It does not mean that the algorithms must be the same, because the levels have different goals

52 2.3 A production-oriented approach 25 Figure 2.9: Basic elements of the production chain according to PULS 90. Source: SBB. and properties, but rather that input and output are of the same form. This allows them to be combined in the production chain description as a unique element. Indeed, it can be possible that the algorithms for reaching the same output are different at each planning level, depending on the specific goal and characteristics of the level. For instance, during online operation the time needed for the computation is very critical and it is important to deliver an acceptable solution quickly. On the other hand, on the strategic or tactical level, more time is available but it is important to generate the best possible solution in order to increase customer satisfaction. In this thesis, the focus is primarily on the tactical level, where it is important to generate all details of the train schedule, while the time is not a constraining factor. Nevertheless, all levels of the timetable generation are kept in mind throughout thesis. On the strategic level, the purpose is basically to produce different input data, e.g. with some trains added or removed, or with some changes in the railway infrastructure, and then to create a detailed schedule using the same algorithms as in the tactical level. The so generated results can then be compared, and strategic decisions can be taken on the basis of precise analyses. On the rescheduling level, the main contribution of this thesis is given by trying to create algorithms which are as fast as possible, and CPU times for all computations are presented and discussed throughout the thesis. Stochasticity and dynamic aspects of a real-time system are not directly addressed. The methods presented in the thesis serve as a basis for the next logical step of specifically developing algorithms for real-time rescheduling. Rescheduling algorithms are a fundamental component of the integrated real-time rescheduling described in the next section, and their specific properties will be described there.

53 26 Chapter 2: Background Integrated real-time rescheduling Train control plays an essential role of the production chain for the operative stage. Figure 2.10 illustrates the current train control process. One can observe that the process is not formally closed and that the communication between the train and the infrastructure is only possible via signals. Dispatching decisions after a delay or a disruption are nowadays still taken manually. This happens in a heuristic and non-systematic way, depending to a large extent on the person who executes the dispatching. Figure 2.10: Control process at SBB in Source: [Lüthi, 2009]. As a great improvement for the operative service, it is intended to transform the operation of the trains and their dispatching into two superposed closed feedback control loops, where all steps are formalised and supported by automatic methods. This process is shown in Figure 2.11 and explained extensively in [Lüthi et al., 2007a, Lüthi et al., 2007c, Laube et al., 2007, Lüthi, 2009]. The first loop, the inner loop, is responsible for precise production. It assures that the give production plan is precisely followed within a given tolerance bandwidth and has to attenuate small disturbances. The second loop, the outer loop, supervises the train traffic and infrastructure state, and develops and transmits new production plans in case of disruptions and deviations from the current plan. This will al-

54 2.3 A production-oriented approach 27 Disturbance Published Timetable Data preparation Conditions & Constraints Production Plan Rescheduling Driver,Guard Passengers Infrastructure & Operator Train Threshold exceeded Current State Supervision Actual State Figure 2.11: Structure of the proposed integrated real-time rescheduling system. Source: [Lüthi, 2009]. low more precise driving and reduces the variance of the realised speed profiles. Thanks to these technical improvements, the tolerance band around a given track path can be set to plus or minus 7.5 seconds, whereas currently the variation is much larger. If a train varies from its plan by more than this threshold, a new production plan needs to be computed, where all trains travel again in their planned band. The potential of such a dynamic rescheduling tool can only be exploited if all actors of the production process (train driver, infrastructure manager, conductor, and so on) can execute the current production plan at any point in time and have the adequate instruments to do so. This approach for the improving the production precision can only work if there is a coordinated collaboration between infrastructure managers and train operating companies (TOC). The procedures need to be transparent and the responsibilities for each single process have to be clearly defined. This approach is called integrated real-time rescheduling (short irtr), or coproduction. With its implementation, there are three principal aspects that are essentially influenced and modified: Data acquisition Recording of train movements becomes denser, and available data for the computation will be more precise, updated, and of better quality. Precise prediction Prediction of future behaviour of the trains becomes more precise, giving a more stable environment for the computation of the new production plan. Information transmission The new generated production plan can be automatically transmitted to the involved actors very quickly with the support of appropriate devices. The realisation of these aspects also changes the boundary conditions for the decision making process, and hence the process itself. In particular, the irtr approach allows

55 28 Chapter 2: Background an earlier possible reaction, as soon as the the position of one or more trains deviates from the current production plan. This way, more time is available to produce a new conflict-free production plan according to the changed situation, instead of waiting for a conflict before starting to reschedule. It is therefore possible to avoid unnecessary stops in front of a red signal, leading to shorter travel times and, as a positive secondary effect, less energy consumption [Lüthi, 2008]. This continually updated schedule serves also as basis for an improved communication to the customers and the different TOCs involved. As a consequence, it is possible to reduce the planned reserves and thereby to offer more frequent trips to the customers, improving the attractiveness of the offer. This integrated real-time rescheduling is the main topic of the PhD thesis of Marco Lüthi [Lüthi, 2009] 2. Together with the Swiss Federal Railways, the author intensively works on both the theoretical and practical side for the development of the necessary instruments for an efficient implementation. 2.4 Focus of this thesis This thesis primarily deals with the timetabling step. It is assumed that the infrastructure and line planning stages are already finalised. Precise information about the track topology are therefore available. Furthermore, also a train service intention for a whole day, according to the notion of partial periodicity, is provided as input. The task is then to create a detailed conflict-free train schedule that fulfills the given requirements. If this train scheduling step can be done quickly enough, it enables a feedback loop to the previous step, and testing of different alternative infrastructure variants and service intentions becomes possible, which can lead to an improvement of the entire planning process. The main points of influence of the described production-oriented approach to this thesis are the following. First, the thesis focuses on the algorithmic part on the tactical level, without abandoning the project philosophy that the same structure for the computation should be kept from the strategic to the operational level, as well as for rescheduling during operation. However, the specific properties of the rescheduling level are not directly addressed, and it should be part of future work to develop algorithms that are suited for the specific rescheduling goals, in particular addressing the very restricted amount of computation time available, as well as the stochasticity and the dynamic aspects of the system. In particular, this thesis addresses the project goal and is mainly concerned with Method Goal is the standardisation of the planning and production methods in an integrated system. For this thesis this means that the developed methods and specifications should also be suited for an application to dynamic rescheduling. At each planning 2 written at the Institute for Transportation and Planning, ETH Zurich

56 2.4 Focus of this thesis 29 level, also on the strategic level, a conflict-free detailed schedule should be created to precisely evaluate different alternatives and to take the correct decisions. The interfaces (input and output) of the addressed problems are the same for each time horizon. In the remainder of the thesis, only the offline train scheduling problem is addressed. Method is the separation of the railway network into zones of tractable size. This network decomposition topic and the related specific policies and algorithms for each zone are described in detail in Chapter 5.

57

58 Chapter 3 Problem definition and two-level approach In this chapter, the train scheduling problem is formally defined and the general procedure for its solution is outlined. The solution procedure is based on a two-level approach, which distinguishes a macroscopic and a microscopic description level. Both planning levels are introduced in this chapter, and the purpose and goal of each level is explained. After a literature review on the related work in Section 3.1, an intuition about the proposed two-level approach is given in Section 3.2. Section formally describes all necessary input information, whereas Section 3.7 clearly states the form of the output at all levels. Based on these formalisations, in Section 3.8 the addressed train scheduling problem is precisely defined and the procedure for solving it is depicted. 3.1 Related work Railway problems in general and planning questions in particular have been intensively studied subjects over the last few decades. Many literature reviews can be found, from a very wide, general overview to specific reviews on specific topics. A comprehensive survey over different planning steps can be found in [Caprara et al., 2007b], [Huisman et al., 2005], and [Bussieck et al., 1997a]. [Assad, 1980] gives a survey with focus on network planning, [Maróti, 2006] on rolling stock scheduling, and [Ernst et al., 2004] on crew management. If the train scheduling step is considered (defined as Problem 3.35 or similarly), which is the main topic of this thesis, some dedicated surveys on this topic are also available in the literature. Application of Operations Research methods to the train scheduling prob-

59 32 Chapter 3: Problem definition and two-level approach lem are described in [Kroon et al., 2007b], who present a literature review on macroscopic and microscopic scheduling. [Vromans, 2005] focuses on works addressing the topic of reliability and robustness in timetables, and [Törnquist, 2006] reviews approaches for train rescheduling. Finally, [Cordeau et al., 1998] focuses on the routing and scheduling aspects of train scheduling. Train scheduling approaches can be basically grouped into few principal categories: Manual train scheduling Manual generation of train schedules is (still) the most common planning method used in practice by railway companies. Depending on the railway company, the scheduling approach can be very precise up to the creation of a conflict-free schedule, or more rough relying on traffic management for resolving conflicts. In the first case, manually generated schedules are often checked for feasibility using simulation tools such as OpenTrack [Nash and Hürlimann, 2004] or RailSys [Bendfeldt et al., 2000], which can reproduce the train dynamics and many practical constraints in detail, but without taking any decisions itself. Two-level approach The separation of complex systems with very large amount of data into two levels, a macroscopic level with coarse global view and one microscopic level with a detailed local view, is a common approach not only in railway planning. Besides the well known application in economics, also other contexts like mechanics [Tay et al., 2008] and politics [Huckfeldt, 2006] make use of the same idea. In the transportation field, a similar situation occurs also in road transportation: [Bullock et al., 2004] gives a literature review on microscopic and macroscopic simulation models for road transportation. Coming back to the railway world, only few projects address the train scheduling problem in large networks with a two-level approach. A project for the German railways based on the macro-micro approach is presented in [Sewcyk et al., 2007, Kettner et al., 2003, Radtke and Hauptmann, 2004]. It combines an estimation of traffic with the tool NEMO on the macroscopic level and a simulation with the tool RailSys [Bendfeldt et al., 2000] on the microscopic level, where only a very basic adaptation of a given macro timetable is allowed. So far, the most important and most successful project based on two-level scheduling is the Dutch project DONS [Odijk and Van den Berg, 1994, Hooghiemstra et al., 1999]. The goal of the project was the creation of a detailed train schedule for the entire railway network in the Netherlands. Using a macroscopic timetable generation model, a draft timetable is created that will subsequently be checked for feasibility on a micro-

60 3.1 Related work 33 scopic level (although not on the level of blocking times) only for the principal station areas of the country. The project also integrates rolling stock planning, which also plays a role in the quality and robustness of a train schedule. The approach was successfully implemented in practice and the generated schedule was introduced into operation in the Netherlands in December 2006, leading to an improvement in punctuality [Kroon et al., 2009, Kroon, 2008]. This project won the prestigious Franz Edelman Award of INFORMS [Kroon et al., 2009, Huisman and Wagelmans, 2008]. The approach developed in this thesis has some similarities to the DONS project, but with the ultimate goal of ensuring that the microscopic schedule is conflict-free on the blocking times level for all parts of the railway network, not only in main station areas like in DONS, even if these areas remain the most critical and difficult to schedule. Moreover, the interface between macro and micro level has been relaxed in order to be more suitable for the goals of each level. Finally, models and algorithms for the sub-problems have been adapted, extended, and improved compared to the methods presented in the DONS project. Macro timetabling Many works are only concerned with the automatic generation of macroscopic timetables, without determining feasibility on the microscopic level. Section 4.1 reviews approaches in the literature in detail for macroscopic scheduling. Micro scheduling Less intensively than for macro scheduling, some papers looked only at the generation of micro schedules, considering complete networks or only a local part of it. Section 5.1 reviews approaches for microscopic scheduling in general, while Section 6.2 presents a literature review for micro scheduling in main station areas. Finally, Section 7.1 gives an overview of micro scheduling on train lines connecting major stations and on corridors. Other approaches Capacity studies of a railway network, or part of it, have also been studied intensively in the literature. These works can then serve as basis for defining appropriate service intentions. [Burkolter, 2005, Herrmann, 2005] have studied a main station area from the points of view of capacity and robustness. [Burkolter, 2005, Burkolter et al., 2005] consider a given periodic service intention and measure the capacity of a main station area with respect to this service intention as the minimal period time that is necessary to periodically

61 34 Chapter 3: Problem definition and two-level approach repeat the services. This is done by applying meta-heuristics such as simulated annealing for finding the optimal train sequences and max-plus algebra and Petri net modeling [Akian et al., 1994, Goverde et al., 1998, Goverde, 2005] for finding the minimal period time. [Herrmann, 2005] presents models and algorithms for measuring the robustness of a given schedule and for optimising a schedule for a given service intention with respect to some static robustness criteria. The model uses a fixed point iteration heuristic for finding an initial feasible solution, whose robustness is then improved applying local search heuristics [Caimi et al., 2005]. [Powell and Wong, 2000] study the effect of the track layout on capacity, measured by the throughput rate of trains in a main station area. Analytic models based on queueing theory are presented in [Wakob, 1985] and [Wendler, 2007]. An integer program is formulated that determines optimal throughput rates, and the model is also used to analyse the capacity of some existing stations. 3.2 Basic idea of the two-level approach Considering all necessary details of the railway topology at once appears to be intractable due to the complexity of the problem and the very large amount of data. Therefore, an appropriate consideration of a subset of data at each time is necessary to overcome this difficulty. For this purpose, a two-level approach is proposed in this thesis, which first considers only a simplified version of the railway network with the most relevant information and then looks at the details necessary to generate guaranteed conflict-free train schedules locally. This procedures uses two levels of abstraction of the railway infrastructure: one aggregated and simplified level, called macroscopic (or simply macro) topology, and a detailed level, called microscopic (or simply micro) topology. Table 3.1 summarises these two levels of abstraction and shows the kind of information considered at each level. This method of modelling makes it possible to switch between different levels of detail according to the necessary accuracy for reaching the designated goal on each level [Gély et al., 2008]. On the upper level, the macroscopic topology for the complete railway network is considered at once. Hence, this level will be called macroscopic level, or simply macro level in the following. The purpose of this level is to provide a global view of the network for generating the most important properties of the timetable by avoiding getting stuck in large amounts of detailed, only locally relevant information. Parallel to the topology, also the information about train movements and their dynamics are simplified, basically by taking into consideration only the travel time between two consecutive stations. Because of this simplification, only a draft timetable can be created at the macro level. This draft still requires a check to determine whether the draft timetable can be implemented on the

62 3.2 Basic idea of the two-level approach 35 Macro level Micro level Not considered Track topology - Stations - Switches - Curves - Parallel tracks - Lengths - Steepness between stations - Signals Train dynamics Total travel time - Acceleration - Frictions - Braking - Weather conditions - Max speeds Conflict modeling Headway Blocking time - Shunting movements theory - Manual release Input Train Macro service intention timetable Output Macro Detailed timetable production plan Table 3.1: Considered type of information at the macro and micro level, as well as non considered information during the proposed train scheduling approach. microscopic topology. Section 3.3 presents the requirements and tasks at the macro level in detail. On the lower level, the microscopic topology of the railway network is considered. Hence, this level will be called microscopic level in the following, or simply micro level. The microscopic topology, however, will not be treated all at once, but will be considered only locally to cope with the enormous amount of data. A proper local decomposition of the network can then also help to exploit the different properties of different areas. On this level, the starting point is the macro timetable. In addition, a local view of all details of the railway topology is given, which are necessary for creating conflict-free schedules but are not relevant for the global structure of the timetable. Analogously, the information about the dynamic properties of the trains is also more accurate than on the macro level. However, details on the train dynamics are taken into account up to a certain degree. Overly precise calculation of the train dynamics, taking into account the necessary boundary conditions can be unnecessarily time consuming, as train drivers are unable to follow a predefined track path (see later Definition 3.9) with very high accuracy. Other factors, e.g. weather conditions, also influence the train dynamics. Section 3.4 presents the requirements and tasks at the micro level in detail, and Chapter 5 presents the approach to separate the railway network into parts of treatable size according to their different properties.

63 36 Chapter 3: Problem definition and two-level approach In the remainder of this thesis, the word timetable will generally refer to the macroscopic schedule, while the word schedule will refer to the microscopic schedule. 3.3 Macroscopic topology In this section, a formal definition of the macroscopic topology is introduced Purpose and properties The purpose of a macroscopic schedule is to provide a simplified description of the timetable, in the form of arrival and departure times of the trains in the principal stations. For this, aggregated information about the track topology and simplified train dynamics in the form of travel times between two stations are taken into account. The rationale of the macro level is that the information relevant for the passengers are considered at this level, whereas on the micro level only purely technical decisions are additionally needed for creating a conflict-free schedule, such as the exact route through a junction or the speed profile on the open track. A macroscopic schedule looks therefore quite similar to the published timetables that are available for the passengers. Some difference with the published timetable may arise due to the fact that minor stations, where the trains could stop, are considered on the macro level. On the other hand, passing times of the trains at relevant crossing points will be considered on the macro level, even if this information is not relevant from a commercial point of view, but necessary for the global structure of the timetable and the subsequent separation for micro scheduling. Components of a macroscopic topology are a list of stations or other important locations (junctions, crossing points, and so on) and their connections via lines, with single, double, or multiple tracks. Furthermore, information about (minimal) interconnection times in a station should be provided, as well as (minimal) travel times of each train type on each line connecting two stations. Minimal travel or interconnection times form technical information and belong to the information assigned to the macroscopic topology, as they do not depend on commercial considerations. On the contrary, maximal allowed times for travelling between two stations or changing trains in a station are commercial considerations and do not depend on the underlying track topology. These are assigned depending on acceptable times with respect to passenger patience and commercial interests of the operating company. As commercial requirements, these properties be provided in the train service intention, see Section 2.2.

64 3.3 Macroscopic topology Graph representation On the macroscopic level, the railway infrastructure can be represented as a graph G = (V,E) with nodes and (multiple) edges. A node v V represents a location in the network where trains may interact. Examples of nodes are stations where passengers can change trains or junctions where the infrastructure changes from double track to single track. Note that a stop station in a line with very short dwell time, without interconnection possibilities and without additional tracks does not need to be modeled as a node in the macroscopic topology, but can simply be treated as part of the travel on the track between two nodes. By viewing stations or junctions as nodes in the model, they are considered as a black box, and the information about their track topology is hidden. Train movements inside the nodes will be taken into account only on the microscopic level. However, [Peeters, 2003] presented some rough ideas to incorporate the node capacity already in the macroscopic level by limiting the number of trains that are simultaneously located in the node to a maximum number. The edges e E represent the tracks that connect the nodes. It is assumed that once the train enters an edge it will remain on it until it reaches its end, and thereby the adjacent node. Although this assumption might overlook some technically possible timetables, it is easily validated by common practice in reality. Effectively, this eventuality is only rarely used for letting a fast train overtake a slower train on the fly. This will not be allowed on the macroscopic level, where overtakings are only possible in stations, which are modelled as nodes. Later all these scheduling possibilities come again into consideration on the microscopic level. When multiple edges exist between a pair of nodes, each train is assumed to be assigned to one of the available tracks a priori. When two tracks are available for a line, each track is used for one travel direction. For more than two tracks, different train types are separated, according to their speed, to enable a higher train frequency and therefore a higher track utilisation [Peeters, 2003]. As a result, each train has its designated path in the graph representing the macroscopic topology. The aggregated representation of a railway track topology by the macroscopic topology is not straight-forward. The decision whether a certain part of the track topology should be represented by a node depends on the desired level of precision on the macroscopic level. A less precise macro topology of tractable size allows for faster computation times and thus can be used to consider larger networks, but it increases the risk of infeasibility on the micro level. This is well suited, for instance, for strategic decisions on large networks. On the other hand, the usage of a larger yet more precise macro topology is more advisable for the tactical timetable generation to be operated in practice, where it is important to ensure conflict-free track paths on the micro level. Figure 3.1 presents two

65 38 Chapter 3: Problem definition and two-level approach Figure 3.1: Construction of the macroscopic topology. At the top of the figure, the detailed topology of the region around Zug in central Switzerland is illustrated. On the bottom there are two possible macroscopic representations of the topology, which differ in the level of detail and therefore on the reliability of the resulting macro timetable. possible macroscopic representations of the same track topology for the region around Zug, in central Switzerland. A very similar graph representation of the topology was also used in many other macroscopic scheduling or timetabling approaches, e.g. [Peeters, 2003, Liebchen, 2006, Caprara et al., 2002, Fischer et al., 2008] Macroscopic safety system using headway times On the macroscopic level, the safety system is modeled in a simplified way. The precise safety system based on the blocking time theory (see later, Section 3.4) is not directly considered but is approximated by introducing the concept of headway time, or simply headway. Definition 3.1 (Headway) The headway h e (τ 1,τ 2 ) is the minimal time difference for passing the same point on a given track e E between two trains traveling in the same direction (of type τ 1 followed by type τ 2 ) so that the trains do not generate a conflict. For a given pair of trains, depending on the exact signal positions, this number can vary not only from track to track but also depending on the different positions on the same track. On the macro level, this is simplified by defining the headway of a track line as the

66 3.3 Macroscopic topology 39 Figure 3.2: Headway and opposite headway for the track line between stations A and B. Departure time of train 3 has to be at least h AB (τ 2,τ 3 ) later after train 2 leaves station A. h AB (τ 2,τ 3 ) is the headway associated to the track line A B for the pair of trains (2,3). Departure time of train 2 has to be at least h A AB (τ 1,τ 2 ) later after train 1 reaches station A. h A AB (τ 1,τ 2 ) is the opposite headway associated to station A and the track line A B for the pair of trains (1,2). Similar for h B AB. maximal (or sometimes average) value that it can take on the different segments of the track. This way, by ensuring a minimal time distance of the headway time between two trains over the entire track line, the detailed feasibility on the micro level should also not be at risk, even if this is not a guarantee. The headway is a property of a track line and a given pair of leading and following train type. Note that this relation is not necessarily symmetric. In this thesis, analogously to the headway concept, the notion of opposite headway is introduced for travelling on the same track in opposite directions. Definition 3.2 (Opposite headway) On a given track line e E, the opposite headway h i e(τ 1,τ 2 ) is the time that has to pass between a train of type τ 1 passing its end node i e and another train of type τ 2 entering e from the same node, traveling in the opposite direction, so that the trains do not generate a conflict. The opposite headway h i e(τ 1,τ 2 ) is a property of a track line e, its end node i e, and a given pair of incoming and successively outgoing train types (τ 1,τ 2 ). These two definitions formalise the requirement that train movements are not allowed to interfere on a track. Overtaking between a faster and a slower train that use the same track, for instance, has to take place in a node of macro topology. Figure 3.2 illustrates the concepts of headway and opposite headway for an easy example.

67 40 Chapter 3: Problem definition and two-level approach Formal definition The macroscopic topology of the considered railway network can now be formally defined. Definition 3.3 (Macroscopic topology) Let the (multi-)graph G = (V, E) represent a division of the railway network in nodes and edges. Additionally, let h : E Z Z R + be the headway function, h : V E Z Z R + the opposite headway function, t trip : E Z R + the minimal trip time in an edge, t dwell : V Z R + the minimal dwell time in a node, and θ : V Z Z R + the minimal interconnection time between two trains in a station. Then, a 7-tuple M = (V,E,h, h,t trip,t dwell,θ ) is called macroscopic topology. 3.4 Microscopic topology In this section, a formal definition of the microscopic topology is introduced, which will later serve as the basis for microscopic scheduling Basics and goal The purpose of a microscopic schedule is to provide a precise description of the train movements, in the form of complete routes along the train line and the corresponding passing times. For this, detailed information about the track topology and trains dynamic properties have to be taken into account. The rationale of this level of scheduling is that there is basically nothing more to decide for the dispatcher or the train driver, and they simply have to follow the plan laid out by the microscopic schedule. Components of a microscopic topology are the position of tracks, signals, switches and the safety system in use. With this information, together with the exact run of each train through the railway topology, it is possible to check whether the schedule is conflict-free, i.e., assuming no delays, the train can travel as planned without causing any conflicts. An essential part of microscopic scheduling is the blocking time theory, which permits the safety system in use to be modelled in a realistic and detailed way, in order to correctly compute whether the assigned track paths for two trains are in conflict or not. A detailed description of the blocking time theory is out of the scope of this thesis. Section introduces the relevant concepts for understanding this thesis. The safety systems considered in this thesis are based on fixed blocks, as it is currently the case in Switzerland on the entire railway network [Graffagnino, 2007]. Although the safety system is not unique, there are different system in use. For the vast majority, the safety system in use is based on conventional signals, where information is given by a

68 3.4 Microscopic topology 41 signal on the track. There are also few lines based on cab signalling, where the train driver reads the information directly from a screen in the cab. By 2008, only two lines are equipped with ETCS level 2 [SBB, 2008b, ERTMS, 2008], the cab signalling system currently in use in Switzerland: the new lines Matstetten-Rothrist (52 km between Berne and Olten) and the new Lötschberg base tunnel (34.5 km between the cantons Berne and Valais). All safety systems based on fixed blocks can be described correctly with the microscopic models developed in this thesis (conventional, ETCS level 1 and 2, or others), under the assumption of having all necessary information available. On the other hand, it is not possible to model moving blocks safety system in this way, like ETCS level 3, as they work with an entirely different concept. The fundamental property of a microscopic schedule is that it must be conflictfree. The idea is that one could run a detailed simulation of the computed microscopic schedule, e.g. [Hürlimann, 2002, Hürlimann, 2009, Nash and Hürlimann, 2004, Radtke and Hauptmann, 2004], without having to deviate from the plan under the assumption that no delays occur. Such simulation tools take into consideration much more details and are more precise in computing train track paths (see Definition 3.9). Some unavoidable imprecision in the computations should be covered by small reserves, generated by slightly underestimating the technical performance of the locomotive and by adding a few seconds to the blocking time of the train. Moreover, these reserves should also be helpful in case of weather difficulties, e.g. in rainy days, to permit a stable operation year round. Most of microscopic topology data used for the computations in this thesis are exported from the simulation tool OpenTrack [Nash and Hürlimann, 2004], which is based on the notion of double vertex graphs Double vertex graph Montigel proposed to describe the microscopic track topology with double vertex graphs (see [Montigel, 1992] and [Montigel, 1994]). A railway network, i.e. the track layout, can be represented by a special type of graph where each vertex is doubled, i. e., each vertex has a unique partner. Definition 3.4 (Double Vertex Graph) Let V be a finite set of vertices, E V V a finite set of edges between the vertices where E does not contain any loops or multiple edges. Moreover, let : V V be a mapping, the so called joining mapping, which satisfies (v) v and ( (v)) = v for all vertices v V and where v (v). Then a triple D = (V,E, ), where E contains no edges of the form (v,v ) is called a double vertex graph, or dvg in short. This definition of double vertex graphs must not be confused with another use of this term in the literature ([Alavi et al., 2002]), which does not have anything to do with this

69 42 Chapter 3: Problem definition and two-level approach work. In the special case where (v) = v for all vertices v V, i. e. the identity function, the double vertex graph is equivalent to an ordinary (simple) graph. The double vertex graph is intended to represent the railway topology in a realistic way. Therefore, the following properties will be assumed for the dvg in this thesis: Assumption 3.5 (Double vertex graph properties) i) No vertex has more than two outgoing edges (normally there are no switches with more than two outgoing tracks). ii) There are no loops or edges connecting two joined vertices (this special case cannot exist in reality). Edges of the dvg correspond to track segments, and they are the basic elements of the microscopic track topology model. All the additionally provided information about the track topology can be assigned to an edge, a node or a set of them in the double vertex graph. For instance, Figure 3.4 shows the microscopic track topology of the main station of Berne, Switzerland, represented in the dvg model. C A B D C C C o A B D o A A B B D D o o Figure 3.3: Modeling train routes in networks. In a railway network the route C B D is impossible (top), but would be a legal path in a conventional graph representation (lower left). However, with the double vertex graph and its rule always to pass both partner vertices, C B D becomes illegal (lower right). Double vertex graphs enable the correct description of feasible routes for the trains in the railway network: Definition 3.6 (Path in double vertex graph) A path p in the double vertex graph D = (V,E, ) is feasible if it visits both partners of each double vertex, with the exception of the

70 3.4 Microscopic topology 43 Figure 3.4: Microscopic topology of the station of Berne in dvg format. This screen shot is taken from the simulation tool OpenTrack [Hürlimann, 2009]. first one, leading to a sequence of the form vertex-edge-vertex-vertex-edge-vertex-vertex-...-vertex-vertex: p := {v 1,(v 1,v 2 ),v 2,v 2,(v 2,v 3 ),...,v n,v n} (3.1) An illustrative example is given in Figure 3.3. Note that the first vertex of the path determines the direction of the path. This formal definition of a path, however, is only necessary to enumerate all possible paths correctly. Always indicating both partner vertices is cumbersome and therefore, for simplicity, it is sufficient to describe the path in the double vertex graph in a simplified, yet unique, way. Remark 3.7 (Simplified Path Description) Each path in the double vertex graph is described by a sequence of single vertices and edges only, i. e. by p := v 1 (v 1,v 2 ) v 2... Equivalently, each path can be represented only by a sequence of edges p := (v 1,v 2 ) (v 2,v 3)... If not ambiguous (i.e. when it is unique), it is also possible to describe it as a list of vertices v 1 v 2 v 3... The definition of a train route can now be directly derived from the path description in the double vertex graph. Definition 3.8 (Train route) A feasible route through the railway topology of a train not changing the travel direction is represented as a path in the double vertex graph describing the microscopic topology, as already defined.

71 44 Chapter 3: Problem definition and two-level approach If, for technical reasons, it is not possible to choose a specific route that has a valid representation as a feasible path in the double vertex graph, this has to be additionally specified in the input of the microscopic scheduling step. A train route, augmented with temporal information, is called a track path, or track slot. Definition 3.9 (Track path) A track path, or track slot, is a feasible route augmented with passing times of the train head on all points represented as double nodes in the dvg. Direction change If a train changes direction during its trip, e.g. in terminal stations or during shunting movements, it is also not possible to represent its route properly in this form, but some special rules become necessary. Rule 3.10 (Direction change) If a train changes its traveling direction during the trip, its route should be described by one path for each drive maintaining the direction, and then connected while taking care of the train length. The trip is then split into as many paths as there are different directions. p := {v 1,(v 1,v 2 ),...,(v k 1 1,v k 1 ) d1, (v k1 +1,v k1 +2),...,v n,v n} (3.2) Here, (v k i 1,v k i ) di represent the edge and exact position of the train head before changing direction, i. e. the train head is d i meters from v k i 1. (v k i +1,v ki +2) is the edge of the location of the train head ready for the start in the opposite direction, i. e. the previous train end. The exact position is implicitly given by the exact position of train head before changing direction and the train length, by going back in the path for exactly the value of the train length. This rule for the direction change needs an assumption in order to be always feasible. Assumption 3.11 (Feasible direction change) Each train moves for at least its length before changing direction. Otherwise, situations can occur where a train route is not uniquely described by Rule A direct consequence of Assumption 3.11 is that the edge just after the direction change is equal to an edge in the path for the drive of the previous direction: (v ki +1,v ki +2) = (v l,v l+1) k i l < k i. Even if situations violating this Assumption 3.11 could be technically feasible, in this thesis only train drives fulfilling it will be considered. In particular, this special rule becomes necessary for trains changing direction in the station. This is the case, of course, for all trains stopping at a terminal station and some trains at other through stations, like the Zurich Berne Interlaken in the station of Berne.

72 3.4 Microscopic topology 45 Additional information to double vertex graph The double vertex graph as described above is still not sufficient to describe all necessary topological information on the microscopic level. In addition to that, it is necessary to know, among other things, the length of the edges, the maximally allowed speed v max (e) on these edges, the position of the signals, and the position of the platforms in stations. Furthermore, in order to correctly compute the blocking times and thus determine if the timetable is conflict-free, it is essential to have knowledge of the set of edges that correspond to a single resource and if they are therefore allocated simultaneously to a passing train. The definition of the double node graph can easily be augmented so that it contains all the needed information. All the information that is directly used or mentioned in this thesis is explicitly described. Some other technical data could be used implicitly for the computation of the speed profiles: these are directly exported from the tool Viriato of SMA und Partner AG and will also not be introduced in this thesis, as the author did not directly use them Resources A resource, also called track element according to the notation of [Montigel, 1992], is the basis of the railway safety system. It will be allocated and cleared as one single unit and can be blocked by at most one train at the same time, according to the blocking time theory. In this work, the mechanisms of the railway blocking time theory is described only roughly in order to understand the model. For a more accurate explanation of the blocking time theory for different safety systems, the reader is referred to [Hansen and Pachl, 2008, Pachl, 2002]. To define resources, the double vertex graph is partitioned into a subsets of its vertices. Definition 3.12 (Resource) Let the triple D = (V,E, ) be a double vertex graph. A resource R V is a set of vertices of the double vertex graph D. The nodes of the double vertex graph are partitioned in a disjoint set of resources R such that R V = R = R i. R R i=1 Let r : V R be the resource assignment function with the following properties: e = (v,w) E: r(v) = r(w). This means that an edge and their adjacent vertices belong to the same resource. Each edge e can therefore be associated with a unique resource r(e) := r(v). If r(v) = r(w) then there exists a sequence of vertices (v,u 1,...,u n,w) such that (v,u 1 ), (u j,u j+1 ) j = 1...n 1, and (u n,v)

73 46 Chapter 3: Problem definition and two-level approach are either an edge in E or a double vertex according to the function ( ). This means that all nodes of a resource are connected. This definition of resources uses assumptions (uniqueness, connectedness) that are not necessary from a mathematical point of view, but useful for a realistic description of the railway topology. Figure 3.5: Double vertex graph with representation of the resources. Each element of a resource is blocked by a train for the same time interval Signals and blocking times A track path blocks the resources corresponding to the occupied track elements for a certain amount of time. In order to determine the start and the end of the blocking time interval for each resource the position of the signals and the release (or clearing) points is needed. In Switzerland, similarly to many other countries, there are basically three types of signals: 1) Main signals, controlling the entrance of a block for each incoming direction; 2) Distant signals, indicating in advance if the corresponding box is free; 3) Combined signals, combining of both signals in the case of two-section signalling [Pachl, 2008]. This basic structure can have many variations and special cases that make the situation heterogeneous and complicated. For instance, some distant signals allow the possibility to keep driving with reduced speed, by giving a different combination of lights or with a table indicating the maximally allowed speed. Sometimes there are multi-aspect signals, which can give information about the next two (or more) blocks and where the maximally allowed speed also is not fixed but depends on the lights combination. For

74 3.4 Microscopic topology 47 a general overview of the signalling systems in use in Europe the reader is referred to [Bailey, 1995]. In this thesis, only the basic situation where each signals can give binary information (green or red, two binary information units for combined signals) is described. Nevertheless, all described methods for microscopic scheduling can be easily adapted by changing the way of computing the blocking times, whereas everything else can remain unchanged. For the purpose of computing the blocking times in the basic signalling situation, a signal function σ is introduced, which states the signal type or release point for each node in the double vertex graph. As some combinations of the signal types and/or the release point are possible, a coding of the different types will be introduced. The threedimensional binary code {0,1} 3 will describe which kind of signal or release point are present at the given vertex (in the dvg description). The first binary values states if there is a main signal, the second if there is a distant signal and the third one if this vertex is a release point in the network. For instance, the code (1,1,0) (or simply 110) means that in this node there is a combination of a main signal and a distant signal associated to the next main signal. 101 means that there is a main signal and at the same place there is a release point. Signals, at least conventional ones, are built for being operated in only one driving direction and are therefore not necessarily symmetric. As already observed, the relevant direction for the signal can be implicitly given by adding the signal information to the vertex of the double vertex looking at the driving direction, i.e. being the first vertex of all paths starting there and continuing in the desired direction. Definition 3.13 (Signal function) Let the triple D = (V,E, ) be a double vertex graph. A function σ : V {0,1} 3 is called signal function and describes the position of all signals and release points in the double vertex graph of the corresponding railway topology. Note that not every function σ corresponds to a meaningful or feasible signal assignment in the railway network. Depending on the safety system in use, the conditions that need to be fulfilled in order to have a realistic signal function are different. In conventional signal systems, each main signal must be provided with a distant signal for each direction that the main signal can be approached, in order to enable train braking at a red light in all cases. In order to avoid confusion and to remain controllable for the train driver, the distant signal must be located after the previous main signal on the train path (case 010) or together with the previous main signal, in this case called combination signal (case 110). Moreover, for each block section there should be at least one release point in order to take advantage of the signal that would otherwise be useless. In main station areas with many different routes of the trains it could be helpful to have many release points between two main signals to enable the release of partial routes. These release points could be located at the same place as the main signal (type 011 or 111) or later in an isolated position (type 001).

75 48 Chapter 3: Problem definition and two-level approach The distance between the main signal and the corresponding distant signal is directly correlated with the maximal allowed speed through the distant signal, as the train must be able to brake before reaching the main signal. Longer distances enable higher speeds but lead simultaneously to longer blocking times and therefore longer headway times, with a negative influence on the capacity utilisation of the track. On the other hand, short distances enforce lower speeds but allow shorter blocking times and a higher network utilisation. The choice where to place the distant signal should depend on the amount and the type of traffic that travels through this track as well as the properties of the railway network. Usually, during infrastructure planning the desired speed is first stated, and the distance between main and distant signals is then derived. Once the infrastructure is given, it is the task of the planners to define the adequate speed to cope with the requirements of maximising the capacity and to enable a smooth operation. For a given standard railway infrastructure, [Landex and Kaas, 2005] shows that the optimal travel speed to minimise the headway (and therefore maximise capacity) is between 60 and 100 kilometers per hour. In this thesis the railway topology is always considered as input and this question is not addressed here. In ETCS level 2, with cab signalling, there are detection points on the track that detect passing of the train. This information is then processed by a software system and the movement authority sends instructions to the train driver directly into the cab ([SBB, 2008b, ERTMS, 2008]). These detection points can be simultaneously interpreted as main and distant signals, as well as release points. In this case there will be only signals of the type 111, but the blocking times will be computed differently compared to conventional signalling system with all signals of this type. The computation of the release point follows basically the same rules as in the conventional system [Graffagnino, 2007]. Throughout this thesis a green wave situation is assumed and considered for. Definition 3.14 (Green wave) A green wave is a situation where the train always faces only green signals during the trip. However, also in the green wave the train can face a red signal during a planned stop. In a green wave, the train driver can continue the trip without braking at a red signal and then accelerating again. This has positive effects on energy consumption ([Lüthi, 2008, Albrecht, 2005, Albrecht, 2004]) and capacity utilisation ([Lüthi et al., 2007b, Lüthi et al., 2007c, Lüthi and Stalder, 2007, Mazzarello and Ottaviani, 2007]). In the green wave case, a resource is blocked from the time that the train head passes the distant signal of the corresponding main signal minus some technical time until the tail of the train passes the associated release point plus some minor release time [Graffagnino, 2007, Pachl, 2002]. Figure 3.6 illustrates the computation of these intervals. The only exception of this rule is when the train starts

76 3.4 Microscopic topology 49 Signal view point Main signal Distant signal Distance Head of train at begin of blocking time Clearing point Time to set the route Time between block signals Block section Signal watching time Approach time Head of train when train end passes clearing point Head of train at end of blocking time Clearing time Release time Time Figure 3.6: Calculation of a blocking time interval in a green wave policy with conventional safety system. Source: [Lüthi, 2009]. from a stop station, as shown in Figure 3.7. In this case the train has already passed the distant signal and stays at the station in front of the red signal without having the next resource allocated. The resource will then be blocked some seconds before the train starts moving [Graffagnino, 2007]. A detailed description of the computation of blocking time intervals is presented in Appendix C. In both cases, the start of the blocking time is the same for all used resources between two main signals, but the end time is not necessarily the same, as for each clearing point on the track the already passed resources can be released. This mechanism is known as partial route release, and is particularly important in station regions and complex track topologies, where often only a part of the route is used by two trains. In the planning phase, a pure green wave situation is sought. A similar green wave policy has been also implemented in the Netherlands [Badcock, 2003]. Other systems considering the possibility that the trains face a red signal are not addressed here and are treated as infeasible. During operation it may become necessary to abandon the green wave policy in order to reschedule trains in case of delay. This topic is not addressed here and the reader is further referred to [D Ariano, 2008, D Ariano and Albrecht, 2006, Wegele and Schnieder, 2004a]. A good evaluation of the added value of the green wave policy in real-time railway traffic management can be found in [Corman et al., 2008a] Formal definition It is now possible to formally define the microscopic topology of the considered railway network.

77 50 Chapter 3: Problem definition and two-level approach Main signal Distant signal Distance Platform Clearing point Block section Time to set the route Train departure process duration Approach time Time between block signals Head of train when train end passes clearing point Head of train at end of blocking time Clearing time Time Release time Figure 3.7: Calculation of the blocking time interval for a departing train. Source: [Lüthi, 2009]. Definition 3.15 (Microscopic topology) Let the triple D = (V, E, ) be a double vertex graph, R a feasible partition of V into resources and σ a signal function, feasible according to the safety system in use. Furthermore, let l : E R + be the function describing the edge lengths of D. Then, the 6-tuple T = (V, E,, R, σ, l) is called microscopic topology. The macroscopic and the microscopic topology are representations on two different levels of detail of the same railway topology in reality. Each element in the track topology has a representation on the macro and on the micro level. Thus, a function can be introduced a that maps the elements from the macro topology to the micro topology, and vice versa. Let m : M 2 T be the function that for each element in the macro topology returns the set of elements in the micro topology associated to this macro element (node or edge). On the other hand, let m : T M be the function that for each element of the micro topology (vertices or nodes of the dvg D) gives the corresponding macro element. These mapping functions are important in the interface between the macro and the micro scheduling to transmit the information correctly. 3.5 Train dynamics Information about the train dynamics are also necessary in order to compute feasible train schedules. The basis for its computation is a detailed speed profile computation, from which also the simplified version of train dynamics is derived.

78 3.5 Train dynamics Microscopic train dynamics On the micro level, not only the topology but also the train dynamics is precisely considered. This precision, however, has to be limited up to a certain level, which enables a certain reliability but remains computationally tractable. The task of microscopic train scheduling is to assign conflict-free track paths to all trains respecting their dynamic properties. In particular, only the passing times at the signal positions are relevant for the safety and therefore for the schedule to be conflict-free. More precisely, the crucial times are the passing time of the train s head through the distant signals (or the departure time in case of stop at a station) and the passing time of the train s tail through the release points. Nevertheless, the computed speed profile should be feasible to follow for the driver. Together with some small reserves, the imprecision of the calculations can be covered and the trajectory is reliable for the daily operations even in changing weather conditions. In order to compute the speed profiles of the trains, the following assumption about the driving behaviour is introduced. Assumption 3.16 (Micro train dynamics) The train dynamics of each train is assumed to be piecewise quadratic, i.e. it has piecewise constant acceleration rate, bounded by a maximal acceleration rate and a maximal braking rate (minimal acceleration). Moreover, the maximal speed limits on each track section are respected. Let s : Z R R,(z,t) s z (t) be the function describing the position of train z at time t in a given train route. This function fulfils the properties a) ṡ z (t) v max (e) track elements e D with s z (t) e; b) s z (t) is piecewise constant, i.e., it has a piecewise constant acceleration rate. These assumptions are made with the purpose of finding a compromise between efficient computation with a limited amount of necessary data and a practically useful result, but also any other method can fit into the model without any changes except its computation. Furthermore, the micro train dynamics is also the basis for computing the macro train dynamics, resulting therefore in a consistent description between the levels Macroscopic train dynamics In addition to the topology, the train dynamics is also simplified on the macroscopic level. Basically, only the trip time between two nodes in the macro topology is considered. This trip time is computed starting from the computation of the microscopic train dynamics, adding the desired time reserve and retaining finally only the total trip time between the nodes. This way, the exact speed profile on the edge is not directly taken into account. In particular, the acceleration and braking behaviour of the trains is neglected on this

79 52 Chapter 3: Problem definition and two-level approach level. The macroscopic safety system via headways performs satisfactorily if the dynamic behaviour of the two considered trains is similar. If this is not the case, either the headway should be increased or the speed profile needs to be adjusted on the micro level to find a conflict-free schedule. The first method is safer, but very conservative and could lead to capacity loss. The second method basically shifts the problem to the micro level and relies on good and flexible scheduling methods. Based on the macroscopic train dynamics, the minimal possible times for the trip on an edge or the stop in a node are defined. Definition 3.17 (Macro train dynamics) The macro train dynamics consist of giving the (minimal) times that a train needs to travel between two adjacent vertices of the macro topology, according to the micro train dynamics. Let ttrip : E Z R + be defined as the function for the minimal trip time on an edge. Furthermore, let tdwell : V Z R + the function for the minimal dwell time in a node. These times can also be increased, for commercial or technical reasons. An upper bound for these times is decided based on the maximal acceptable time for the passenger and will be part of the service intention. 3.6 Partial periodic service intention This section presents a method to describe the partially periodic structure of modern timetables. Chapter 8 then presents a method for effectively exploiting this structure for timetable generation. The concept of a partial periodic service intention (short ppsi) is introduced here as an interface between commercial offer and technical process planning. It consists of all services a railway company would like to offer during a day. Each train service is specified by its line, stopping stations, interconnection possibilities, periodicity, and the time frame.the ppsi is not the representation of a technical timetable, but it describes only the commercial offer and contains therefore only the customer-relevant information. The precise departure and arrival times as well as the detailed train routing will be decided during the timetable generation. Hence, the ppsi is an input to the timetabling problem. The purpose of the ppsi description is to have a framework in which potential commercial offers can be developed and evaluated. It serves as a starting point for process planning and analysis, e.g., timetable generation or rolling stock planning, as well as a level for discussions and negotiations between the infrastructure manager and the train operator companies. It enables the separation of technical and commercial consideration. The discussion between the companies can in this way take place based only on commercial requirements, whereas planners inside the infrastructure manager company can deal

80 3.6 Partial periodic service intention 53 with the technical problem of creating a conflict-free schedule, which may or may not be supported by automatic methods. The ppsi framework also allows to analyse the offer quality and to evaluate different alternative service intentions. However, this last point is out of the scope of this thesis. The basic element of the ppsi is the train run with the customer relevant information. The train run can be seen as the specification of its geographical information, the train line with the stop stations, and its temporal information, such as trip times, dwell times, time frame, and periodicity. Additionally, each train run is associated with a train type, which describes the type, and therefore the quality, of the rolling stock used for this service. The assignment of the train types to the train runs is a commercial decision, but has consequences to the dynamics of the train, in particular its running time, to be considered during scheduling. Definition 3.18 (Train type) A train type z describes a specific type of rolling stock, possibly with some small variations, that is used for the train run. It corresponds to a certain quality standard, and it has therefore to be part of the commercial offer, i.e. the service intention. All trains belonging to the train type have the same driving properties, or train dynamics, such that it is possible to compute correct running times and to assign feasible track paths with only this information. Some small variations will be tackled by taking into account the most conservative values. Let Z, called train type set, be the set of all considered train types. An example of train type could be the RABe 523 train (FLIRT) for commuter services in central Switzerland. This train type will be used by different train runs on various lines. Definition 3.19 (Train run) A train run z is defined as the run over K + 1 stations in the railway network, repeated R times with periodicity ρ minutes and is specified as z = ( z,(v k,tdwell k,tk+ dwell,tk trip,tk+ trip,ω k,ω+ k )K k=0,ρ,r), (3.3) where z Z is the train type used for the train run, v k V(M) is the macro vertex visited in the k-th step of the train run, t k /+ dwell the minimal and maximal dwell time of the train in the station node (a value of zero means that the train passes the station without stopping), [ttrip k,tk+ trip ] defines the allowed interval of the trip time between v k 1 and v k, and [ωk,ω+ k ] is the (optional) time slot for the departure event of the first train recurrence. The lower bound for the trip time is set to the minimum time needed for the train to run the distance, according to the macro train dynamics, plus a reserve of a few percent for ensuring a stable operation. The upper bound is the maximum acceptable time with respect to customer acceptance and track capacity usage. Dwell times should be long enough for boarding new passengers, coupling/decoupling wagons, loading/unloading goods, or

81 54 Chapter 3: Problem definition and two-level approach maintenance work on the train if necessary. It should not be much longer than necessary, however, as travelers would like to move on, and platform capacity within a station might be small. In both cases, the lower bounds depend mainly on technical considerations and less on commercial strategies. The decision on the upper bound for the trip and dwell times of a train is mainly a commercial requirement, and depends on the maximal acceptable time for the customers or other commercial considerations, but not on technical reasons. Each train run should have at least one associated time slot to approximately locate it temporally during the day. The time slots for all relevant events of the train run will then be derived from the time slots given in the train run, as described in detail later in Section They should not be too restrictive to enable flexibility in the timetable generation process but also not too large to avoid ambiguities, which could hamper the reduction to a periodic problem. Notice that this (compulsory) introduction of a time slot does not necessarily restrict the search space compared to the classical timetabling problem from scratch without time slots. Indeed, it is possible to choose the size of the time slot larger than or equal to the desired periodicity of the train run, such that each departure time of the train is still possible. Figure 3.8 illustrates an example of a train run. A train run describes a service which is repeated R times exactly in the same way. Irregularities in the commercial services, for instance additional stops in the late evening, are approached by introducing a train run for each different intended commercial service. Also a higher frequency in peak hours can be covered using two train runs: one for the peak hours and one for the rest of the day, or in an equivalent way one for the basic frequency all over the day and one for reinforcing it only during peak hours. Collecting all the train runs that are intended to be offered, it is possible to define the considered train set: Definition 3.20 (Train set) Let the set of all considered train runs z be defined as the train set Z. Abstracting form the temporal information and only keeping the geographical one, the train line for each train run can be specified: Definition 3.21 (Train line) A train line l z is a list of the K + 1 macro vertices that a specific train run z Z visit in sequence: l z := (v k ) K k=0. (3.4) Connections between trains also belong to a commercial description, as they permit the passenger to efficiently change trains to continue their trip. The minimum connection time depends on the infrastructure of the railway station, in particular on the distances

82 3.6 Partial periodic service intention 55 Figure 3.8: Example of a train run description, without the periodicity information. One can recognise the time slots [ωk,ω+ k ] for a subset of the stations (in blue), as well as lower (red) and upper (green) bound for trip and dwell times. passengers have to walk. Upper bounds are defined as the acceptable waiting times for the passengers. Definition 3.22 (Connection) A connection c is defined as the possibility for the passengers to change from train run z 1 to train run z 2 in the station vertex v V(M), c = (z 1,z 2,v,r 1,r 2,θ + ), (3.5) where both train runs z 1 and z 2 Z travel through station node v, and the connection takes place for the first time during the r 1 -th repetition of train run z 1 and the r 2 -th repetition of train run z 2. As a consequence, the connection has a periodicity of ρ := lcm(ρ 1,ρ 2 ) minutes and will be repeated R := min( ρ 1(R 1 r 1 + 1) ρ, ρ 2(R 2 r 2 + 1) )) (3.6) ρ times. The connection takes place within at most θ + minutes. A lower bound θ is given by the properties of the considered station, computed as the minimal time that passengers need to go from the arrival platform to the departure platform. Let C be the set of all connections. It is necessary to specify between which repetition of both train runs the connection takes place. This could be seen as a restriction for the scheduling algorithm, but it is necessary information for the used procedure and in reality it is usually decided in the

83 56 Chapter 3: Problem definition and two-level approach planning phase at which repetition the connection will occur. If a fixed prescribed connection constraint could be too limiting, it can be relaxed by choosing a large value for the upper bound θ + and introducing the connection in the objective function as element to minimise, like in [Liebchen, 2006]. Time dependencies between train events are also an important commercial requirement, as they separate two different train runs in time that are covering the same demand, at least partially. Definition 3.23 (Time dependency) A time dependency d is defined as a time constraint between any two departure or arrival events of the train runs, where d = (z 1,z 2,k 1,k 2,r 1,r 2,θ,θ + ). (3.7) The departure event of the k 1 -th node of train run z 1 should occur between θ and θ + minutes before the k 2 -th node of train run z 2. The dependency takes place for the first time during the r 1 -th repetition of train run z 1 and the r 2 -th repetition of train run z 2. As a consequence, the connection has a periodicity of ρ minutes and will be repeated R times, computed in the same way as in the connection case. Let D be the set of all time dependencies. Remark 3.24 A connection c C is a special case of time dependency, where the two events are the arrival of a train run and the departure of another train run at the same station. As connections have more structure and mean something specific, they are listed separately in the set C instead of in the list D of time dependencies. The difference is mostly used for the analysis of the service intention and the resulting timetable. However, they will be basically treated in the same way from a mathematical point of view. For instance, the half-hourly direct non-stop connection between Zurich and Berne is assured by the temporal separation of the hourly train runs St.Gallen Geneva and Romanshorn Brig. In this case a dependency entry can assure that the two trains together offer a half-hourly service on their common section Zurich Berne. This dependency can be restrictive, forcing an exact time difference of 30 minutes for the departure in Zurich and the arrival in Berne, or can just prevent trains from being too close in time, for instance setting the borders for the time difference to [θ,θ + ] = [20,40]. In the 2009 timetable, these trains are scheduled to depart at Zurich at the minutes 00 and 32, which fulfils the purpose of the time separation without forcing an exact 30 minute periodicity, as it concerns two train runs with a different remaining journey. It is now possible to formally define the partial periodic service intention, collecting all this information. An example of ppsi is depicted in Figure 3.9. The ppsi is included in a framework where the times can be absolute or cyclic. Absolute times mean that the

84 3.6 Partial periodic service intention 57 railway traffic starts and ends at a certain point in time. This is true if there is for instance a period without traffic during the night or it is a good approximation in case the night traffic is very low and can be easily adjusted by hand in a later phase. On the contrary, cyclic times are necessary when there is no point in time with no or very few intended railway traffic. In this case, the night train runs influence the trains of the morning again and the service intention has to be considered in a periodic way. This property belongs to the ppsi and is defined by the variable ρ. Definition 3.25 (Partial periodic service intention) A partial periodic service intention G (short ppsi) for a given railway network is defined as G = (Z,C,D, ρ), (3.8) where Z is the set of all train runs, C the set of all connections, D the set of all time dependencies, and ρ R + is defined as the considered period length. A value of ρ = means that the considered time frame of the ppsi is not cyclic and the times are interpreted in an absolute way. ρ < means a cyclic framework with period ρ. Figure 3.9: Example of a partial periodic service intention for a single line from A to B. The shadowed surfaces are the different train runs with their time slots, red arrows are connections, whereas blue arrows represent time dependencies. Note that in general time slots for train runs may overlap.

85 58 Chapter 3: Problem definition and two-level approach Remark 3.26 In case of a cyclic framework, i.e. ρ <, for all train runs z Z it needs to hold ρ R ρ (3.9) as well as all the time values have to be interpreted in a cyclic way, which is equivalent to setting 0 θ,θ +,ωk,ω+ k < ρ. This ppsi intends to describe a general situation, very common in reality, where the operated timetable is a mixture of trains with different periodicities and non-recurring trains. It is a general framework which generalises previous purely periodic service descriptions, and it is able to handle any periodic structure of the intended service, including the extremes of no periodicity and full periodicity. As described in detail later in Chapter 8, the approach for generating a train schedule fulfilling the ppsi is based on a reduction of the problem to a purely periodic instance for a certain time period T as small as possible, the application of methods for periodic timetable generation, and finally rolling it out again for getting a timetable for the originally required time period, usually one day. Of particular interest is the generation of timetables that balance objective values in different parts of the day. This thesis first presents a method for generating periodic train schedules, and afterwards explains how to extend this method for partial periodic timetabling, using the ppsi. Therefore, the class of purely periodic service intentions will also be introduced, and Chapters 4 to 7 focus on the creation of timetables for this type of service intention. Definition 3.27 (Periodic service intention) A periodic service intention G = (Z,C,D, ρ) (short psi) is a special case of ppsi where it exists a value T with the properties a) For all train runs z Z there exists a value k N with ρ(z) k = T. This means that the train run z is repeated exactly k times during the period. b) There is a time interval of length T where k repetitions of the train run are completely included, either with k full train runs or with k 1 full train runs, plus one leaving the time interval and one entering it in the same place. These two fractional train runs can be seen as one complete train run modulo T. In this case, the service intention G is said to have periodicity T. It could be described in an equivalent and efficient way using ρ = T. Remark 3.28 (ppsi) All ppsis are actually a psi if a sufficiently large value of T is chosen. In fact, one could say that a non-periodic timetable in the common sense is a periodic timetable with periodicity 1 day, or 1 week. Nevertheless, it makes sense to distinguish between ppsi and psi as the goal of the approach is to reduce the time period to be considered also for ppsi which are not psi, in order to make use of the

86 3.7 Macroscopic timetable and microscopic schedule 59 partially existing periodic structure within this potential large value of T, which could not be exploited if the ppsi were considered as a psi with very large periodicity. 3.7 Macroscopic timetable and microscopic schedule After having formalised all necessary inputs, this section defines the output of the different steps of the proposed train scheduling procedure. On the macroscopic level, both macro topology and train dynamics together contain the necessary information to define the commercial requirements of a timetable and to give a rough global structure of the train movements. On the basis of the macroscopic topology, it is possible to introduce the notion of macroscopic timetable. Definition 3.29 (Macroscopic timetable) A macroscopic timetable, or macroscopic schedule or simply timetable, is a set of departure and arrival times for each input train z Z at each visited vertex v V(M) such that the macro train dynamics and safety constraints are respected. In addition to the macroscopic timetable, the notion of flexible macroscopic timetable is also introduced, which is an augmented version where a time slot for each departure or arrival event is provided instead of exact times. Definition 3.30 (Flexible macroscopic timetable) A flexible macroscopic timetable, or flexible macro timetable or simply flexible timetable, is a set of time slots for the departure and arrival for each input train z Z at each visited node v V such that for any choice of the event times from these slots the macro train dynamics and safety constraints are respected. For more details, the reader is referred to Chapter 4 on macroscopic scheduling. Once a (flexible) macroscopic timetable has been generated, independently of the used method, it is used as input for the scheduling problem at the microscopic level. There, the micro topology and the train dynamics for each train type together contain all the necessary information to compute feasible track paths for each train. Definition 3.31 (Train schedule) A train schedule, or microscopic schedule, is a set of track paths through the considered microscopic topology T for each input train z Z respecting the micro train dynamics. Each track path is associated with a blocking time interval for each utilised resource, which is computed according to the green wave policy described in Definition This way, it is possible to generate a conflict-free train schedule.

87 60 Chapter 3: Problem definition and two-level approach Definition 3.32 (Conflict-free train schedule) A train schedule is conflict-free if the blocking time intervals of all trains do not overlap for all resources r R. The notion of conflict-free schedule uniquely refers to a schedule in the microscopic topology. A timetable, resp. train schedule, generated from an input ppsi will be also called partial periodic timetable resp. partial periodic train schedule, if it is necessary to point out its property of partial periodicity. Definition 3.33 (Partial periodic timetable/schedule) A partial periodic timetable resp. partial periodic schedule is a (macro) timetable resp. a (micro) train schedule that fulfils all the requirements described in the given partial periodic service intention G. As a consequence of Remark 3.28, also the periodic timetable (and schedule) can be defined accordingly. Definition 3.34 (Periodic timetable/schedule) A periodic timetable resp. periodic schedule is a (partial periodic) timetable that fulfils the requirements of a given periodic service intention G. The timetable (schedule) has periodicity T( G). 3.8 Train scheduling problem and two-level algorithm The previous sections have precisely defined the input and the output for train scheduling procedure. It remains therefore to formally define the main problem in this thesis, which is addressed in this section. Furthermore, the proposed two-level procedure for solving it is roughly presented. According to this approach, the sub-problems that need to be solved for generating conflict-free train schedules and therewith solving Problem 3.35 are defined in the following. Problem 3.35 (Train scheduling problem) Given are a macroscopic and microscopic description of the railway topology, information about macro and micro train dynamics, as well as a ppsi G. The train scheduling problem then consists of finding a conflict-free train schedule fulfilling the requirements described in G, respecting the safety system, and satisfying the train dynamics. Figure 3.10 illustrates the procedure for solving Problem 3.35 schematically. First, starting with the partial periodic service intention (Definition 3.25) an (equivalent) fully periodic instance is generated, which is given as input to the macro scheduling problem. A (flexible) macro timetable using an exact method is then generated, such that an infeasibility response is a sure indication that it is not possible to fulfil all the requirements

88 3.8 Train scheduling problem and two-level algorithm 61 ppsi Generate periodic input Macro scheduling Feedback Delete intention in ppsi no Feasible? yes Micro scheduling Conflict-free? no Detect conflict yes Conflict-free schedule, constraint report Appraisal, relaxation no Satisfies ppsi? yes Roll out, Production plan Figure 3.10: Schematical representation of the addressed general train scheduling problem. The green boxes are the core of this work, whereas the red box is out of the scope of this thesis. of the ppsi, and that this should be changed in order to generate a feasible solution. On the other hand, the generation of a feasible macro timetable is not yet a guarantee that a conflict-free train schedule exists on the microscopic level. If it is possible to generate a feasible solution on the macro level, this has to be augmented with all microscopic details by solving the micro scheduling problem for creating a conflict-free schedule. The micro scheduling problem will be considered only locally, and the boundary condition given by the macroscopic timetable guarantee that the micro schedule is conflict-free also across different local zones. If this is possible by fulfilling all ppsi requirements, a production plan for the whole day is obtained. If some intentions in the ppsi were deleted, the constructed schedule, together with a constraint report, is fed back to the planner for changing the input ppsi. This step is outside of the scope of the thesis and the reader is referred to [Laube and Mahadevan, 2008, Mahadevan, 2007] for some preliminary work in this topic. Finally, if the microscopic scheduling problem

89 62 Chapter 3: Problem definition and two-level approach is infeasible, a feedback loop to the macro scheduling is necessary in order to change the macro schedule. In this thesis, some ideas for the feedback loop will be given, but are not tested and implemented for the validation. In the following, the three subproblems for train scheduling are defined. Problem 3.36 (Periodic input generation) Given a ppsi G, the periodic input generation problem has the goal to find a fully periodic instance P with period T, which is equivalent to the original ppsi and where algorithms for periodic scheduling can be applied. It is equivalent in the sense that a conflict-free schedule for P can be rolled out to a conflictfree schedule fulfilling G and an infeasibility response for P is also a proof of infeasibility for G. Recall that in the ppsi the train line is given, i.e., the path through the macro topology is known for each train, including the information which edge (in case of multiple edges) connects two vertices. Problem 3.37 (Macroscopic timetabling) Given is a periodic instance P, consisting of a set of periodic train runs (including macro train dynamic), a set of connections and time dependencies between the train runs, and a set of train pairs needing headway constraints. Furthermore, the macroscopic topology M as well as information about macro train dynamics are given. The macroscopic timetabling problem has the goal to find a (flexible) macroscopic timetable m fulfilling all requirements described in P and respecting macro safety and train dynamics. Problem 3.38 (Microscopic scheduling problem) Given is a (flexible) macroscopic timetable m, the microscopic topology T := m(m), and micro train dynamics. The microscopic scheduling problem has the goal to find a conflict-free schedule s fulfilling the boundary conditions given by the macro timetable m and respecting micro safety and train dynamics.

90 3.8 Train scheduling problem and two-level algorithm 63 In the remainder of the thesis, models and algorithms for a fully periodic input P are presented first. Chapter 4 presents a method for solving the macro scheduling problem Chapter 5 describes the network decomposition approach for the micro timetabling problem Based on that, Chapter 6.2 focuses on condensation zones, which are the bottlenecks of the network and the most difficult regions to schedule. Chapter 7 focuses then on compensation zones that will be addressed after the condensation zones. Finally, the approach is extended to work also for partial periodic inputs. Chapter 8 presents a method for solving Problem 3.36, based on the projection of the partial periodic instance for a day to an (equivalent) fully periodic instance for a given time period.

91

92 Chapter 4 Macroscopic timetabling This chapter addresses the periodic timetabling problem on the macroscopic level. This problem can be modeled as a particular type of a Periodic Event Scheduling Problem (PESP). Its output (departure and arrival times) serves as the input for the micro level to check detailed feasibility. The goal of this contribution is to enlarge the solution space and thus reduce the risk of infeasibility on the microscopic level. This goal can be reached by generalising the PESP model to search for arrival and departure time intervals in lieu of exact times, which are quite restrictive for the micro level and often lead to infeasibility. This additional flexibility for those events leads to the extended model developed in this chapter, the Flexible Periodic Event Scheduling Problem (FPESP). The FPESP model is further generalised with the so-called Flexboxes. Flexboxes make use of the natural dependencies between the events in order to further increase the chance of finding a feasible solution on the microscopic level. This chapter assumes the input service intention to be fully periodic. The extension of macro scheduling for partial periodic instances is presented in Chapter 8. This chapter is organised as follows: Section 4.1 gives a literature review on relevant macroscopic approaches. In Section 4.2 the PESP is introduced as a powerful model to generate periodic macro timetables, which is the basis for the approach of this thesis. Section 4.3 explains the extension of the PESP model to the FPESP, with the introduction of flexibility for the events in the PESP model. In Section 4.4 the Flexbox model as a further generalisation of the flexibility idea for arbitrary sets of events by making use of natural dependencies in the timetable is presented. Section 4.5 presents computational results on scenarios in central and southern Switzerland, and finally Section 4.6 describes an investigation of the trade-off between the conflicting objectives of maximising the flexibility and minimising the trip times.

93 66 Chapter 4: Macroscopic timetabling This chapter is based on [Caimi et al., 2009d]. 4.1 Related work Macroscopic timetabling has been studied intensively in the literature. Each approach considers a different level of detail of the infrastructure. It can vary from very simplified to very precise, passing through diverse intermediate steps that are not always easily mapped into the level of topology described in this thesis. This literature review classifies the approaches that directly address the signalling system and the blocking times for the block sections as microscopic scheduling, whereas all other approaches, that simplify in one way or another the safety system are classified as macroscopic scheduling which are specifically reviewed in this section. The principal approaches are illustrated in Figure 4.1 and described in the following. Microscopic approaches are discussed in the later Chapters 5.1, 6.2, and 7.1, depending on their focus. Figure 4.1: Overview on the principal methods in the literature for macroscopic timetabling. As already introduced in Section 2.2, the approaches can be basically divided in two categories, periodic or non-periodic scheduling; no mixed versions are known to the author of the thesis. Non-periodic macro timetabling is broadly used in networks where freight traffic is predominant. This problem was already studied in the 1970s by [Szpigel, 1973], who proposes a mixed integer formulation, and became an intensively studied topic during the 1990s. Many special heuristic approaches were developed [Cai and Goh, 1994, Cai et al., 1998, Dorfman and Medanic, 2004], as well as general techniques like ant colony optimisation [Huang, 2006, Ghoseiri and Morshedsolouk, 2006], constraint pro-

94 4.1 Related work 67 gramming [Oliveira, 2001, Oliveira and Smith, 2001], or resource constrained formulation [Zhou and Zhong, 2007]. The most important approach for solving the macro timetabling problem, however, has remained integer programming. Many formulations have been proposed, ranging from nonlinear integer programming [Jovanovic and Harker, 1991, Higgins et al., 1996, Higgins and Kozan, 1997] to large scale integer linear programs, which are then solved for instance with Lagrangian relaxation [Brännlund et al., 1998, Fischer et al., 2008]. Integer programming based on network models have been established in literature, for both cases of periodic and non-periodic scheduling. [Ahuja et al., 2005] presents a comprehensive review of train scheduling methods based on network models. [Caprara et al., 2001, Caprara et al., 2002] introduce a graph theoretical formulation using a directed graph in which nodes correspond to departures/arrivals at a certain station at discretised times. Trains are modelled as one integer unit that has to flow from the source to its destination. The resulting multicommodity flow model is formulated as integer linear program and solved using Lagrangian relaxation. [Sahin et al., 2005] proposes heuristic algorithms, whereas [Cacchiani et al., 2008b] uses column generation for solving the train scheduling problem in a corridor; extensions for a network are introduced in [Cacchiani, 2007]. [Borndörfer and Schlechte, 2007, Borndörfer and Schlechte, 2008] design a similar network model and solve it using a column generation approach, yet starting from a different setting, an auction-based approach, where not all trains have to be scheduled but a schedule has to be found where an objective function is maximised. Some approaches have been also proposed specific for periodic scheduling. [Klemt and Stemme, 1988, Malucelli, 1996] models the macro scheduling problem as a quadratic semi-assignment problem. However, this method is not suitable for large instances, as reported in [Liebchen, 2006]. Results from practical application in the Netherlands [Schrijver and Steenbeek, 1994] and in Germany [Liebchen, 2006] suggest to use another model, the Periodic Event Scheduling Problem (short PESP). It is a powerful model for macroscopic scheduling introduced by [Serafini and Ukovich, 1989] and first applied to train scheduling by [Schrijver and Steenbeek, 1994]. The PESP has been intensively studied, among others, in many PhD theses [Odijk, 1997, Lindner, 2000, Peeters, 2003, Vromans, 2005, Liebchen, 2006]. Within the PESP framework also robustness aspects of periodic timetabling can be studied [Kroon et al., 2007a, Kroon et al., 2006, Liebchen and Stiller, 2009, Liebchen et al., 2009]. The PESP model was successfully applied as the core method for the generation of the 2005 timetable of the Berlin underground [Liebchen, 2008] and, as part of a larger framework, for the generation of the 2007 railway timetable in the Netherlands [Kroon et al., 2009]. Besides these implementations for a specific company, a software

95 68 Chapter 4: Macroscopic timetabling based on PESP, called TAKT, was developed [Nachtigall and Opitz, 2008]. Its particularity is that in case of infeasibility it delivers a feedback about a local conflict that prevents finding a feasible solution. 4.2 Periodic Event Scheduling Problem This section introduces the Periodic Event Scheduling Problem, which is the basis for the macroscopic scheduling approach proposed in this thesis. First, its basic elements are introduced in this section and then extensions will be built on it and are presented in the successive sections Classical PESP model Recall that according to Def a periodic railway timetable on the macroscopic level consists of a list of departure and arrival times at the nodes (stations) in the aggregated network for all train runs within a given time period T. Each departure or arrival of a train at a node is called an event i which takes place at a certain time π i. As the timetable is periodic with a time period T (often T = 60 min), the event i also takes place at times {...,π i T,π i,π i + T,π i + 2T,...}. Therefore, π i can be restricted without loss of generality to 0 π i < T, i.e., π i [0,T). The choice of event times π i depend on each other. For instance, two trains running on the same track cannot have the same departure times because they would generate a conflict. These dependencies are modeled as constraints in the PESP (see Problem 4.1). The constraints always concern two events i and j and specify minimum and maximum periodic time difference between the two. The constraint bounds are given as parameters of the model, and are used as bounds for the periodic time differences between events i, j V. The events and constraints constitute the elements of the Periodic Event Scheduling Problem. This can be represented as a directed (multi-)graph G = (V, A) whose node set V is the set of events and whose arcs A denote the constraints. With each arc a A connecting nodes i, j V, corresponding lower and upper bounds l a and u a are associated on the periodic time differences of events i, j V. Problem 4.1 (PESP, [Peeters, 2003]) Given is a set V of events, a set A of constraints, a periodic time T, and values [l a,u a ] for all elements a A. Let introduce the tail function s : E V and the head function t : E V for representing the two events i := s(a) and j := t(a) V connected by the arc a A. The Periodic Event Scheduling Problem PESP(G,T,l,u) is to find a periodic timetable π i [0,T), i V, satisfying (π j π i ) mod T [l a,u a ] a A,

96 4.2 Periodic Event Scheduling Problem 69 and optimising a given objective function f(π) : [0,T) V R, or to prove that no such schedule exists. For simplicity of notation, all through this chapter the tail s(a) of the arc a will be denoted by i := s(a) V and similarly the head t(a) by j := t(a) V. The PESP has been proven to be NP-complete by several authors, by reduction from the Hamiltonian Circuit Problem [Serafini and Ukovich, 1989], Graph K-colourability [Odijk, 1997], or the Linear Ordering Problem [Liebchen and Peeters, 2002]. Remark 4.2 (Multi-graph) Note that it is possible to have more than one parallel (or anti-parallel) arc between the same event nodes [Liebchen, 2006]. It is therefore necessary to use the notation of the arc a A instead of (i, j). Writing the constraints as linear inequalities, the PESP is then about finding event times π i for each event i that fulfil all constraints of the form l a π j π i + T p a u a, (4.1) or in short l a [π j π i ] T u a, where a A and p a Z. The integer variables p a allow the constraints to be fulfilled in the periodic sense. As an example, consider the InterRegio train running from Zurich to Zug in the 2009 timetable. Its hourly departure time in Zurich is at xx.35 and the trip duration is required to lie between 24 and 27 minutes. Plugging the data l a = 24, u a = 27, T = 60 into Eq. (4.1), and assuming the departure time π i = 35 as given, it results that 24 π j 35+60p a 27. The arrival time π j = 59 fulfills the constraint with p a = 0. Another solution is π j = 1 with p a = 1, where the integer variable p a makes the jump to the next time period possible. An example excerpt of a PESP graph is shown in Figure 4.2 for the arrival and departure events for two trains in the same station. This PESP can be solved by the corresponding mixed-integer linear program (MIP) formulation [Liebchen and Möhring, 2007, Peeters, 2003, Nachtigall, 1998]: Problem 4.3 (PESP, Original formulation) minimise f obj (π, p) (4.2) s. t. l a π j π i + T p a u a, a A (4.3) 0 π i < T, i V (4.4) p a Z, a A (4.5)

97 70 Chapter 4: Macroscopic timetabling Figure 4.2: A part of a PESP graph for two trains stopping in a minor station and then leaving in two different directions. Trip time constraints are drawn as full lines, dwell times as lines, headway as dashed lines, and connection as dotted lines. Eq. (4.3) imposes constraints for each arc in the graph such that the event time difference between two connected nodes is between the given lower and upper bound in the periodic sense. Eq. (4.4) requires all event times to be in the time period [0,T) and Eq. (4.5) requires the integrality of the period jump variables. To apply a MIP-solver, all constraints need to be in the form of inequalities, so Eq. (4.4) has to be transformed into 0 π i T. A solution of π i = T is then equivalent to π i = 0. Algorithms especially designed for the PESP have also been developed, e.g., constraint propagation [Schrijver and Steenbeek, 1994], genetic algorithms [Nachtigall and Voget, 1996], problem-specific branch-and-cut [Lindner, 2000], constraint generation [Odijk, 1996] or adapted backtracking algorithm [Serafini and Ukovich, 1989]. A discussion on different solution methods can be found in [Lindner, 2000]. These are specialised algorithms for finding feasible solutions quickly. However, for optimised solutions mostly MIP solvers are used [Liebchen, 2006] Constraints Many properties that a timetable on the macroscopic level should fulfil can be modeled via PESP constraints of the form (4.1): Trip time The trip time is the time needed for the train to run between two consecutive nodes. Trip times do not necessarily need to be fixed, but can also be variable, as reported in [Kroon and Peeters, 2003]. The values for lower and upper bounds are given in train run information of the service intention. The trip time bounds (l,u) form a constraint between the departure and arrival events of the same train on two consecutive stations in its line. Dwell time The dwell time is the duration of a train stop in a station. This constraint connects the arrival and departure event for the same station of a train run. Simi-

98 4.2 Periodic Event Scheduling Problem 71 larly to trip times, the values for lower and upper bounds are given in the train run information of the service intention. Connections Connection constraints relate the arrival event of one train to the departure event of another in order to enable passengers to change train. As the minimal and maximal connection time can have a major impact on the final structure of the timetable, they have to be chosen with care and could be subject to a planning model as well. In this case, the upper bounds are directly derived from the connection description of the input service intention, hence they are implicitly but uniquely defined by the commercial service specification on the upper planning level. Turnaround time The turnaround time is the time passing between the arrival of a train at its final destination and the departure of the next train in the opposite direction. A good choice of this value enables the usage of the same rolling stock for both trips and possibly also to use this time as a break time for the crew. This makes also an effective preparation for an efficient planning of the rolling stock and crew, usually done in a subsequent step of the planning process, as described in Section For more details the reader is referred to [Liebchen, 2006]. Headway The headway constraints separate two trains running on the same track e E(M) in time by at least the headway time h e (Def. 3.1) or by the opposite headway time h e (Def. 3.2), if the trains use the same track in opposite directions. The headway and the opposite headway are used in the PESP to avoid conflicts. For simplicity of notation, let h and h be the headways for the two sequences of the considered trains. The corresponding constraint (h,t h ) is introduced between each pair of events of the two trains running on the same track on both end nodes of the common edge. This guarantees that the departures and arrivals of the two trains on the same track have a safe temporal distance, without having to assume which train goes first, because it is a constraint in a periodic framework. To correctly introduce headway constraints, it is necessary to know a priori which train uses which edge on the macro topology. Assumption 4.4 (Train path) The path of a train through the macro topology is assumed to be fixed in advance. It can be given in the train run of the service intention, as specified in Definition 3.19, where commercial stops have to be given in any case. Alternatively, it can be given within previous planning steps, following simple and widely used policies like the traffic separation on the traffic on multiple parallel tracks according to their direction and subsequently speed or train type. This assumption is valid for long parallel tracks that are modeled in the macro topology. The situation is different in proximity to main stations, where many route

99 72 Chapter 4: Macroscopic timetabling Figure 4.3: Case where the headways constraints are not sufficient. At both stations trains are separated by at least the headway time, but they generate a conflict on the track in-between, which can only be avoided introducing a non-collision constraint, forcing the trains to not cross on the track. choices are possible and it is difficult and limiting to define a priori the track to enter the station. This particular case is handled in detail on the micro level, as discussed in Chapter 6. The headway time is only a simplification of the real safety system. More precise safety restrictions will be taken into account during micro scheduling, as outlined in Chapter 3 and tracked in detail in Chapter 6. The headway constraints do not prevent overtaking of trains during the run on the same track, which is, of course, impossible without a collision, as illustrated in Figure 4.3. The problem can be solved, for instance, by using more restrictive headway constraints [Kroon and Peeters, 2003]. The idea is to increase the headway times such that overtaking is impossible even for the largest possible trip time difference. For example, a fast train with trip time bounds (30,35) and a slow train with bounds (35,42) have a maximum trip time difference µ of 12 minutes. With a headway of h, the fast train would need to make up h minutes to catch up with the slow train and again h to restore the necessary headway before arrival at the destination. In the case of µ < 2h, collisions can be excluded. In the example, this would require a headway time of h > 6. If this condition is not fulfilled automatically, it can be enforced by lowering the trip time difference µ or by increasing the headway time h. Increasing headway should be avoided as it reduces the track capacity and flexibility. [Liebchen and Möhring, 2007] present an alternative approach to prevent overtaking without increasing the headway time h. They propose to subdivide an initial trip arc into

100 4.2 Periodic Event Scheduling Problem 73 new smaller ones such that u a l a < 2h for every new trip arc a. This ensures the safety constraint by reducing the trip time difference for each trip arc. The approach enables the constraints to remain inside the PESP framework and behaves well in practice. This method, however, does not fit for the purposes of this thesis, because it will artificially restrict the flexibility of each event as it makes use of the trip time differences, see later in Section 4.3. An alternative approach to cope with this problem that is suitable for our goal is therefore proposed in Section All the constraint types described above are of the form (4.1) and fit into the PESP model. Another constraint type will be introduced in Section 4.2.5, leading to an extended model. [Liebchen and Möhring, 2007] give a good overview on some further practical constraint that can be considered in macro timetabling and fit into the PESP framework Objective functions There are two classes of algorithms for solving the PESP: one looking for any feasible solution and the other looking for a solution that is optimal with respect to a certain quality criterion. Feasibility algorithms are often much faster, since they stop as soon as the first feasible solution is found. Optimal solutions can give a measure of the solution quality and guarantee that the output is a solution of maximum quality according to the considered objective. Because of time constraints, planners using manual methods are often satisfied when they find a timetable fulfilling the given requirements without comparing alternatives. Therefore, this guaranteed optimality is an added value of the computer-generated railway timetables compared to the human-made ones. An overview of possible optimisation goals in the PESP model can be found in [Peeters, 2003, pages 57-64]. Typical goals are minimisation of the total passenger travel time, minimisation of the required number of train units or maximisation of some measure of robustness. The objective functions used in this work are related to the flexible event slot concept that will be introduced in Section Cycle periodicity formulation The Cycle periodicity formulation (CPF) is an alternative MIP formulation for the PESP which turned out to be much more efficient in practice [Peeters and Kroon, 2001, Liebchen, 2006, Peeters, 2003]. Instead of solving for the event time variables π i, it solves for the time differences x a := π j π i +T p a. These time differences x a are called periodic tensions. and must obey the bounds l a x a u a for each constraint arc a A. In analogy to electric circuits, periodic tensions yields a periodic potential π i at each node if the sum

101 74 Chapter 4: Macroscopic timetabling of all tensions along each cycle C is equal to an integer multiple of T, hence a C + x a a C x a = T q C, (4.6) where q C is the integer number of period jumps along the cycle C, C + the set of the arcs in the cycle direction and C the set of arcs in the opposite direction. This becomes intuitive by looking at the back transformation from the CPF variables x a to the classical PESP variables π i. Starting by fixing any π 0, one can compute the neighbouring values π using the relation π j = π i + x a mod T, or π j = [π i + x a ] T. The same values for a π i result for every path one can take starting from π 0, if the sum of the x a along each cycle is an integer multiple of T. Consequently, the time differences x a have the same value for all parallel arcs. Thus, the following Cycle periodicity formulation (CPF) of the PESP is obtained: Problem 4.5 (PESP, Cycle periodicity formulation) minimise f obj (x,q) (4.7) s. t. l a x a u a, a A (4.8) a C + x a a C x a = T q C, C G (4.9) a C q C b C, C G (4.10) x a 0, a A (4.11) q C Z, C G (4.12) Eq. (4.10) uses upper and lower bounds b C resp. a C for the values q C by computing the smallest resp. the largest possible integer value that can be achieved along the cycle C [Odijk, 1996]. Eq. (4.9), (4.10), and (4.12) imposes constraints on each cycle in the graph. The number of cycles in a graph can be exponential in the number of nodes, but it can be shown that it is sufficient to require (4.9) to hold for an integral cycle basis B of G [Liebchen and Peeters, 2009, Liebchen and Rizzi, 2007]. Such a basis has the property that each cycle C in G can be expressed as a linear combination with integer coefficients of the cycles in B. Such an integral cycle basis B exists for every graph G and can be constructed by building a spanning tree Γ of G. When taking one chord a A/Γ together with Γ, a graph with exactly one cycle occurs. Adding one cycle per chord to the basis B gives an integral cycle basis of G. For a connected PESP graph with n nodes and m arcs, the cycle basis always contains B = m (n 1) cycles, as the spanning tree of G has n 1 arcs. The search space in the CPF formulation can be reduced considerably by using cutting planes (4.10) for the cycles in B [Odijk, 1996]. The cycle basis should be chosen such that it contains cycles with maximally restrictive cutting planes. The number of integer options for a q C is denoted by w C := b C a C + 1. Thus, the total number of

102 4.3 Flexible PESP 75 integer value combinations to potentially check equals C B w C and can be reduced significantly by using a good cycle basis. A theoretical discussion of minimal cycle bases can be found in [Liebchen, 2006], and many cycle basis construction heuristics are given in [Peeters, 2003]. In this thesis, the CPF formulation with an integer cycle basis generated by the minimum spanning tree (with respect to the arc spans as weights) approach is used, which is simple and gives good results in many cases. When using MIP solvers, it is important to find a good formulation to reach good performance. For the present case it is reported that the CPF formulation with a good cycle basis is more powerful than the original PESP [Peeters, 2003, Liebchen, 2006] Non-collision constraints The relation between periodic ordering of trains and the values q C of the cycle can be used as a constraint to avoid conflicts. Non-colliding trains have q C = 0 on the cycle consisting of the two trip time arcs and the two headway arcs (which must have the same direction, e.g., from train 1 to 2). This fact has been reported earlier [Peeters, 2003, Liebchen, 2006] and can also be adapted for non-collision constraints between trains of reversed direction on single tracks (with q C = 0 for the cycle consisting of the two trip time arcs having opposite directions and the two headway arcs with the same direction). The condition q C = 0 is a type of non-collision constraint that does not fit into the original PESP framework as it is not a proper PESP constraint of the form (4.1). However, it can easily be added to the MIP formulation of both original PESP (Problem 4.3) and CPF (Problem 4.5). The non-collision constraints q C = 0 fit directly into the CPF formulation by choosing a C = b C = 0 in Eq. (4.10). Ideally, these non-collision cycles are used for the cycle basis, as they have the smallest possible w C = Flexible PESP This section introduces the Flexible PESP model, a generalisation of the PESP model that is particularly well suited for the two-level approach presented in this thesis Motivation and basic idea The two-level approach couples the macroscopic scheduling problem with the microscopic scheduling by solving the PESP and passing the generated train departure and arrival times to the local scheduling algorithms to check feasibility on the micro level. This is the way the DONS project works in the interface between the two levels. How-

103 76 Chapter 4: Macroscopic timetabling ever, if no routing exists for these exact event times, it is necessary to go back to the macro level to adjust the macro timetable to find a conflict-free solution on the micro level. To avoid tedious and time-consuming iterations between micro and macro level in case of microscopic infeasibility, the idea is to improve the chance of finding a feasible solution by enlarging the solution space for the micro level. Some benefits of this approach were already observed in [Zwaneveld et al., 1996], where they allow for some minor deviation of the scheduled time in the micro level in order to overcome infeasibility of the problem. It is possible to reach this goal if the macro timetable does not impose exact event times π i but enables some freedom for choosing the event times π i. This flexibility can be added to the events π i by introducing lower and upper bounds π i and π i for π i as new decision variables instead of the event times π i. The final choice of the π i (π i,π i ) is left to the micro level, where exact times are chosen to construct conflict-free track paths for the trains. Property 4.6 (Independence of event flexibility) The choice of event times on the microscopic level shall be independent for each event not to endanger the global macroscopic feasibility of the timetable. This means that each value π j (π j,π j ) should fit with each value π i (π i,π i ) in the sense of Eq. (4.1). This temporal flexibility does not necessarily have to be added to all event nodes, but to which ones can be arbitrarily selected, for instance only to events in a main station area with high traffic density, where it is more difficult to schedule trains on the microscopic level. The micro scheduling algorithms proposed in this thesis are designed to cope with such flexible event time slots as input. Following this input of flexible event times the generalisation of the PESP for generating event slot timetables on a macroscopic level can be defined. This more general model is called Flexible Periodic Event Scheduling Problem, Flexible PESP, or just FPESP. Problem 4.7 (FPESP) Given is a set V of events, a set A of constraints, a periodic time T, and values [l a,u a ] for all elements a A. The Flexible Periodic Event Scheduling Problem FPESP(G,T,l,u) is the problem to find a time slot [π i,π i ] for each event i V, with 0 π i < T and π i π i < π i + T, such that (π j π i ) mod T [l a,u a ] a A, for all combinations of event times π i [π i,π i ], optimising a given objective function f(π,π) : [0,T) V [0,T) V R or to prove that no such schedule exists. Note that the input for the FPESP is exactly the same input as for the PESP. For simplifying the computations, the value for π i can be greater than the period time T. In this case, it has to be interpreted in the modulo sense, i.e., the periodic event time of i is actually π i mod T.

104 4.3 Flexible PESP Flexibility and robustness Flexible timetables with event time slots π i (π i,π i ) can also be used to overcome delay propagation in the network and thus make the timetable more robust. A train leaving a station at time π i at the latest will not start a delay cascade on following event times. If the published departure time π i is earlier than π i, the difference π i π i can be used to mitigate delays. The micro scheduling algorithm should therefore preferably choose event times π i (π i,π i ) that are close to π i so that the flexibility π i π i remaining for operation is maximised. Related approaches to incorporate robustness against delays in the PESP framework can be found in [Kroon et al., 2007a, Liebchen and Stiller, 2009, Liebchen et al., 2009, Fischetti et al., 2009]. In particular, [Liebchen and Stiller, 2009, Liebchen et al., 2009] introduce the notion of absorbing path, which is a path that absorbs a limited disturbance occurred at the first arc at least by the end of the path. This is achieved by adding time reserves to the lower bounds of the PESP formulation. The FPESP approach also restricts the feasible intervals on the arcs, but instead of associating these new variables directly with the arcs, it associates them with the events (nodes) of the network. Doing so, these variables serve as a measure for their events flexibility for microscopic scheduling and might lead to additional robustness on the operational level. The light robustness approach proposed in [Fischetti et al., 2009] has some similarities with the flexibility described in this thesis, in particular with the POSTOPT approach presented in Section Their idea is to improve the robustness of the macro schedule by changing the train departure/arrival times without altering the combinatorial structure of the train schedule. This is a special case of the flexibility described here, as any choice of the train departure/arrival times in the event time slots will not change the combinatorial structure. Another interesting approach is presented by [D Ariano et al., 2008b]. For improving timetable robustness, they introduce the concept of flexible timetable, yet defined in a different way compared to the flexibility concept of this thesis. Their principle is to plan less off-line and to rely on real-time traffic management to resolve conflicts during operations. The larger degree of freedom left to real-time management offers a better chance to recover from disturbances. According to this principle, they define the flexible timetable as (i) a set of feasible platform tracks for each train and for each station, (ii) time slot of minimum and maximum arrival/departure times (a larger time slot corresponding to having more flexibility), and (iii) a provisional (or even partial) order of trains at overtakes and junctions. The complete choice of these three elements is then postponed to real-time operations. The second point is similar to the flexible time slot concept of the FPESP. However,

105 78 Chapter 4: Macroscopic timetabling in contrast to Property 4.6 of the flexibility described in this thesis, they do not intend to guarantee any kind of (macro or micro) feasibility to the times in the time slot, but just aim to produce a set of alternative arrival/departure times to use in case of disruptions. The approach in this thesis is different. According to the project scheduling philosophy presented in Section 2.3, in each step of the planning process a conflict-free production plan is needed, and the goal of rescheduling is to produce a new conflict-free plan matching the changed situation. Property 4.6 will ensure that the changes, as long as the new times are inside the generated time slots, will not affect the rest of the schedule from a macroscopic point of view. Similar to the flexibility approach presented in this thesis, a drawback of their flexible timetable is that each travel time between two stations is increased by the size of the time slot. The fundamental tradeoff between these objectives is discussed and numerically investigated using the FPESP model later in Section Properties The ranges for the event time bounds are set to 0 π i < T for the lower bound and π i π i < π i + T for the upper bound. Definition 4.8 (Event flexibility) The flexibility δ i of an event i is defined as the size of its time slot δ i := π i π i. (4.13) Each constraint arc a A has a corresponding span γ a := u a l a. From any event time π i (π i,π i ) of event i V, any event time π j (π j,π j ) for an adjacent event j V must be reachable by fulfilling the constraint l a π j π i + p a T u a, as illustrated in Figure 4.4. Lemma 4.9 (Flexibility restriction) Let a A be an arc connecting event nodes i, j V and γ a be its arc span. Then the event flexibilities of i and j fulfil: δ i + δ j γ a. (4.14) Proof: Assume that arc a represents the only constraint that applies to π j. Then, π j = [π i +l a ] T is the first possible time for event j that fulfills constraint a for any π i (π i,π i ). Similarly, π j = [π i + u a ] T. The flexibility δ j is then given by δ j = π j π j = [(π i + u a ) (π i + l a )] T = γ a δ i. As other constraints can only restrict δ i further, the claim follows. Thus, each node flexibility of a feasible timetable is a non-negative value δ i 0, which is bounded above by the relation specified in Lemma 4.9. Moreover, Lemma 4.9 shows that the event flexibilities are dependent. Adding more flexibility at one node possibly restricts it at the neighbors, as illustrated in Figure 4.4. A weighted objective function

106 4.3 Flexible PESP 79 Figure 4.4: Flexibility for the events i and j. By increasing the flexibility δ i, the flexibility δ j for the adjacent node j will be reduced by the same amount such that the sum of both values is at most the arc span γ a. or a feedback strategy from the microscopic scheduling algorithm could help allocate flexibility where it is most useful. Remark 4.10 (Feasibility) Finding a set of non-negative values δ i fulfilling Eq. (4.14) does not guarantee a feasible macro timetable. For instance, when choosing all δ i = 0 in an infeasible PESP instance Eq. (4.14) is trivially satisfied but the problem remains infeasible FPESP model In this section, the model for introducing event flexibility into the PESP is presented by augmenting the constraints of the PESP. According to Property 4.6 for the Flexible PESP, event time slots require that the PESP constraints are fulfilled for any π i (π i,π i ), independently for each event. It is therefore possible to state the constraints that need to be fulfilled by the variables δ, π, and p for a feasible solution of the FPESP. Theorem 4.11 (FPESP constraint) Let (π, p) be the solution of a PESP instance and δ the event flexibility according to Property 4.6. Then for all arcs a A it holds that l a + δ i π j π i + T p a u a δ j. (4.15) Note that Eq corresponds to constraints in PESP form of type (4.1) for the variables π i, if the δ values are considered as fixed. Proof: The range of the time span x a = π j π i + T p a between two events i and j is given by π j π i + T p a π j π i + T p a π j π i + T p a. (4.16) Replacing the upper bounds π i for the event times with π i + δ i (Def. 4.8), the following inequalities result: π j (π i + δ i )+T p a π j π i + T p a (π j + δ j ) π i + T p a. (4.17)

107 80 Chapter 4: Macroscopic timetabling The PESP constraints (4.1) are satisfied for any combination of π i and π j (Property 4.6) if they are satisfied for the entire range of (π j π i + T p a ) in Eq. (4.17), i.e., l a π j (π i + δ i )+T p a π j π i + T p a (π j + δ j ) π i + T p a u a. (4.18) Considering the first and the last inequalities separately, constraints in PESP form for the variables π i are obtained as l a + δ i π j π i + T p a and π j π i + T p a u a δ j. (4.19) Putting these results together leads to l a + δ i π j π i + T p a u a δ j, (4.20) which was to be proven. Figure 4.5: Introducing an event slot of size δ i at event i leads to the adapted constraint bounds in the PESP graph, as stated in Theorem The upper bound of incoming constraints is reduced by δ i and the lower bound of outgoing arcs is increased by δ i. The necessary adaptation of the constraints for the FPESP model is illustrated in Figure 4.5. The constraints are more restrictive than in the original PESP, where γ a = (u a δ j ) (l a + δ i ) = γ a δ i δ j. As γ a must be non-negative, it follows again that δ i + δ j γ a as already stated in Lemma 4.9. Remark 4.12 (Original PESP) The original PESP without event slots is a special case of the FPESP where δ i = 0 for all i V. The Flexible PESP can now be solved for the decision variables π and δ (and p), while the input parameters remain the same as in the PESP formulation. Both the original and the CPF formulation are applicable. Following the original PESP formulation 4.3, Eq. (4.1) changes to l a + δ i π j π i + T p a u a δ j a A, (4.21) which leads to the following MIP formulation of the FPESP.

108 4.3 Flexible PESP 81 Problem 4.13 (FPESP, Original formulation) minimise f obj (π, p,δ) (4.22) s. t. l a + δ i π j π i + T p a u a δ j, a A (4.23) 0 π i < T, i V (4.24) 0 δ i < T, i V (4.25) p a Z, a A (4.26) In the CPF version, the change affects the bounds of Eq. (4.8), which become l a + δ i x a u a δ j a A. (4.27) Thus, the following MIP formulation is obtained for the FPESP, which generalises the CPF of the original PESP given in the MIP formulation 4.5 via the additional decision variables δ i for each i V. Problem 4.14 (FPESP, Cycle periodicity formulation) minimise f obj (x,δ,q) (4.28) s. t. l a + δ i x a u a δ j, a A (4.29) x a x a = T q C, a C + a C C B (4.30) a C q C b C, C B (4.31) x a 0, a A (4.32) δ i 0, i V (4.33) q C Z, C B (4.34) Remark 4.15 (No new integer variables) The FPESP model augments the PESP model by introducing the new continuous variables δ i. No new integer variables are added in both formulations Objective functions A good flexible timetable should (i) be good with respect to the classical PESP objectives, as briefly explained in Section 4.2.3; (ii) enable as much flexibility as possible, i.e., have as large event time slots as possible; (iii) contain homogeneously distributed event flexibilities, because small flexibilities everywhere in the network are more useful than large flexibilities for only few events.

109 82 Chapter 4: Macroscopic timetabling These goals are often conflicting, and the choice of an appropriate objective function is not obvious. The following list discusses some possible choices. A computational study of the interplay of the different objectives is presented later in the Sections 4.5 and 4.6. MINTRAVEL: This objective function measures a weighted sum of the passengerrelevant times, which is to minimise: minimise f tt (x,δ) t A T w t x t + d A D w d x d + c A C w c x c (4.35) where A T A is the set of trip arcs, A D A the set of dwell arcs and A C A the set of connection arcs. The weights can be chosen corresponding to the number of passengers using this activity or other priority criteria. MAXFLEX: This objective function measures a weighted sum of flexibilities, which is to maximise maximise f flex (x,δ) w i δ i (4.36) i V where V is the set of all events where flexibility is introduced. The weights can be chosen such that more flexibility is assigned to some parts of the graph, e.g., main station areas or network bottlenecks. MIXFLEX: An aggregated objective function allows a combination between the travel times and the time slots to be optimised. maximise f mixflex (x,δ) λ f f lex (1 λ) f tt (4.37) The timetable quality here is measured by a weighted sum, whose optimum constitutes a Pareto-optimal solution to the bi-objective problem of minimising travel time and maximising flexibility simultaneously. The weight λ (0, 1), balances the priorities of the two goals. CONTRAVEL: Instead of optimising a weighted sum of the objectives, the bi-objective problem can be addressed by constraining one objective and optimising the other. maximise f flex (x,δ) (4.38) s. t. f tt (x,δ) (1+ε) f tt (4.39) where f tt is the optimal value found for f tt in (4.35). By appropriate constraint values, any Pareto-optimal solution is reachable, and the quality of the final solution can be controlled more accurately than by means of a weighted sum. Here, the flexibility under a travel time constraint is optimised, where the minimum of f tt is used as a reference and allow a parametrised relative deviation of ε.

110 4.3 Flexible PESP 83 CONFLEX: This objective function is similar to CONTRAVEL, but reversing the role of the two objectives. Here, the travel time under a flexibility constraint is optimised, where the minimum of f flex is used as a reference and allows a parametrised relative deviation of ε: minimise f tt (x,δ) (4.40) s. t. f flex (x,δ) (1 ε) fflex (4.41) where fflex is the optimal value found for f flex in (4.36). POSTOPT: This approach is also based on two steps. In the first step, values for the integer variables q C are taken from (4.35). In the second step, all integer variables are fixed as q C = qc (i.e., all train sequences are fixed) and the resulting LP is optimised maximising the flexibility as in (4.36). This is a type of post optimisation, which is very fast, but has a limited solution space. The second step only shifts the event times while keeping the event order constant. A similar post-optimisation approach has been applied in [Kroon et al., 2007a] for finding an optimal distribution of time reserves along a train trip using stochastic optimisation. MAXMINFLEX: The idea here is to guarantee a minimum flexibility for a set of selected events i Ψ: maximise ϕ (4.42) s. t. ϕ δ i i Ψ. (4.43) In many cases, however, there are events that cannot have any flexibility. In such cases, ϕ will be zero and the approach will not give the desired result. The presented objectives, e.g. (4.36) for MAXFLEX, might lead to a few events with a lot of flexibility while all others with none. It is usually more desirable to have a balanced distribution of the flexibility among all events where event slots are introduced, possibly in a weighted version. By additionally constraining the maximum flexibility per node in the MIP formulations 4.13 and 4.14 with the constraints δ i δ max i V, (4.44) a more even distribution of flexibility can be obtained. Various choices for the value δ max are discussed in Section Interaction with the microscopic level When an optimal macro timetable is found, the event time slots are passed to the micro level, where each train in the macro timetable is augmented with exact track paths

111 84 Chapter 4: Macroscopic timetabling through the railway topology. The event slots enlarge the solution space of the micro scheduling problem, which can now choose from various routing possibilities, as well as from all event times π (π,π + δ) within the slots. The output of the macro level consists therefore of a list of trains Z with their π and δ values for the arrival and departure at each station of the train run. If no solution of the micro scheduling problem is found, a feedback loop can lead to a shift of the weights, w i in (4.36) and λ in (4.37), in the objective function of the macro scheduling. More flexibility can then be assigned to the event responsible for the micro infeasibility. 4.4 Flexbox model In this section the Flexbox model is developed, which is a generalisation of the flexibility concept just introduced in the last section. The Flexbox model makes use of natural dependencies between the events to increase the chance of finding a feasible solution in the microscopic level Motivation Recall the relation between the flexibilities stated in Lemma 4.9 as δ i + δ j γ a. This inequality imposes strong restrictions on the event flexibilities. It follows from δ j 0 that δ i max{γ a a A,t(a) = i h(a) = i}. (4.45) In the case of a small γ a, for instance in the dwell time between the arrival and departure event in a passing station, the corresponding flexibilities are also constrained to have a very small value. Consider the example shown in Figure 4.2: if few passengers leave or board the train, the dwell time can be as short as (0.5,1.5) minutes. It then holds that δ a1 + δ d1 γ (a1,d 1 ) = 1 and, similarly, δ a2 + δ d2 1, with the consequence that the flexibilities for each event are very limited. Thus, only a small part of the other constraint interval lengths γ a can be used for the event slots. To overcome this drawback, a generalisation of the event flexibility concept is introduced, the Flexbox model. The key idea behind the event flexibility introduced in the last section was to leave some freedom in scheduling this event to the micro level, as that the choice of the value in the event time slot can be taken independently. This independence is particularly important to decompose the micro scheduling problem by eliminating dependencies between different zones, see later in Chapter 5. Nevertheless, there are some natural dependencies between the events that should be kept, because scheduling these

112 4.4 Flexbox model 85 events on the micro level will be correlated in any case, like for instance the arrival at and the departure from the same station of one train. The idea of the Flexbox model is to give up the non-useful independence and thereby to increase the total flexibility of the resulting macro timetable, and keep only the independence between the events that we are interested in. This goal can be reached by introducing sets of events (called Flexboxes) with common slots for all events contained by the box instead of, or in addition to, individual time slots for single events. This way the independence of the events contained in the same box is given up in order to gain additional flexibility by the common time slot size of the box. Each box may then be shifted within its time slot independently of all other external events Definition The Flexbox concept outlined above can now be defined formally and interpreted into the FPESP model. Definition 4.16 (Flexbox) Let a Flexbox be defined as a subset F V of events in the PESP where it is intended to assign flexibility. Let the Flexbox set F 2 V be the set of all considered Flexboxes. Definition 4.17 (Box flexibility) Let F V be a Flexbox. The value δ F denotes the flexibility of the Flexbox F, i.e., the maximal time all events i F can be shifted jointly without affecting the other events i / F. The basic property of a Flexbox is that events inside the box F can be jointly postponed by at most δ F minutes without affecting the events outside the box. The effective event time π i of an event i F can be set as π i = π i + δ F, where δ F fulfills 0 δ F δ F and is the same for all events i F. Remark 4.18 (Multiple Flexboxes) There is no restriction on the choice of Flexboxes. In particular, they can be disjoint, overlap or be included in another Flexbox. Any choice of Flexbox set F 2 F is possible from a macroscopic point of view. However, the effective exploitation of the flexibility δ F on the micro level depends on the choice of appropriate Flexboxes. Particularly suitable Flexboxes are discussed later in Section If an event i is contained in several boxes, then the δ F values of each Flexbox F sum up as π i = π i + δ F. (4.46) F i The effective event times are then bounded by π i π i π i + δ F. (4.47) F i

113 86 Chapter 4: Macroscopic timetabling The choices of event times within the slots is now dependent on the other events included in the same Flexboxes, as the choice of the value δ F must be the same for all events inside a box. Problem 4.19 (Flexbox PESP) Given is a set V of events, a set A of constraints, a periodic time T, values [l a,u a ] for all elements a A, and a Flexbox set F 2 V. The Flexbox Periodic Event Scheduling Problem Flexbox(G, T, l, u, F) is to find periodic times π i [0,T) for each event i V and box flexibilities δ F for each Flexbox F, such that ( ) (π j + δ F) (π i + δ F) mod T [l a,u a ] a A, F j F i for all 0 δ F δ F, optimising a given objective function f(π,δ) : [0,T) V [0,T) F R or to prove that no such solution exists. By an adaptation of the PESP constraints, it can be ensured that any set of π i fulfilling Eq. (4.46) is a feasible solution of the original PESP if the set of π i is feasible for the adapted PESP. Theorem 4.20 (Flexbox constraint) Let (π, p) be the solution of a PESP instance and δ F the event flexibilities for all Flexboxes F F according to Definition Then the following inequalities are satisfied for all arcs a A with j = h(a) and i = t(a): l a + F i,f j δ F π j π i + T p a u a F i,f j δ F. (4.48) Note that, as for the FPESP model, Eq. (4.48) corresponds to constraints in PESP form of the type (4.1) for the variables π i if the δ values are considered as fixed. Proof: Recall that a set of π i is a solution of the original PESP if it fulfills the equation l a π j π i + p a T u a. The constraint bounds (l a,u a ) must be restricted such that any π i satisfies Eq. (4.46). The constraint corresponding to an arc pointing into a Flexbox F (i / F, j F) must fulfil l a (π j + δ F) π i + p a T u a, resulting in l a δ F π j π i + p a T u a δ F. This inequality holds for any 0 δ F δ F when l a π j π i + p a T u a δ F. Similar statements can be derived for constraints associated to box leaving arcs. The PESP constraints (l a,u a ) for arcs entering or leaving a box are then adapted in the following way. The lower bounds of arcs leaving a box F are increased by δ F and the upper bounds

114 4.4 Flexbox model 87 of arcs entering a box are decreased by δ F. If multiple boxes are entered or left, the restrictions cumulate, i.e., the new lower bound l a for the tension becomes l a = l a + F i,f j Similarly, for the new upper bound ũ a it holds that ũ a = u a F i,f j δ F. (4.49) δ F. (4.50) If a feasible solution (a set of π i ) can be found for the PESP with Flexboxes, then each set of π i fulfilling Eq. (4.46) is also a feasible solution. The values δ F can be chosen independently within the range 0 δ F δ F. This result can be used to enlarge the solution space of the micro scheduling problem, by deferring the final choice of the δ F to this stage. Box flexibility can also be seen as a robustness measure to overcome delay propagation: delays of at most δ F minutes will not have consequences outside the box, as a delayed event time π i = π i + δ F for 0 δ F δ F will be feasible and the choice of the δ F is independent for each box. Remark 4.21 (FPESP as special case) The FPESP model is a special case of the Flexbox model, where each event i is contained in a one-event Flexbox F i = {i} with the same flexibility value as in the FPESP model: δ Fi = δ i. Lemma 4.22 (Flexibility restriction for Flexbox) Let a A be an arc connecting event nodes i, j V and γ a be its arc span. Then it holds that F i,f j δ F + F i,f j δ F γ a. (4.51) Proof: The relation l a ũ a must still be satisfied for each constraint a after the introduction of the Flexboxes. It follows that l a + The claim then follows directly. F i,f j δ F u a F i,f j δ F. (4.52) Eq. (4.51) is the Flexbox equivalent to Eq. (4.14). It is less restrictive and gives more possibilities for assigning flexibility, as shown in Figure 4.6 (using the same example graph as in Figure 4.2). Remark 4.23 (Flexibility restriction) Lemma 4.9 is a special case of Lemma Its proof also follows directly from it having only the two Flexboxes F 1 = {i} and F 2 = { j}.

115 88 Chapter 4: Macroscopic timetabling To summarise, the model with an arbitrary set of Flexboxes F 2 V as input parameters leads to the following MIP formulation as a further generalisation of the FPESP model given in Section By considering the cycle periodicity formulation, the decision variables in the Flexbox model are the continuous tensions x and the integer variables q, as in the original formulation, as well as one variable δ F for each given Flexbox F F. Problem 4.24 (Flexbox PESP, Cycle periodicity formulation) way. minimise f obj (x,δ,q) (4.53) s. t. l a + F i,f j δ F x a, a A (4.54) x a u a F i,f j δ F, a A (4.55) a C + x a a C x a = T q C, C B (4.56) a C q C b C, C B (4.57) x a 0, a A (4.58) 0 δ F δ max, F F (4.59) q C Z, C B. (4.60) The original PESP formulation for the Flexbox model can be written in an analogous Observation 4.25 (No new integer variables) Similar to the FPESP model, in the Flexbox model no new integer variables are added in both formulations Application examples The Flexbox has been introduced in a general way for an arbitrary choice of event sets F. However, not all imaginable combinations of event sets in a Flexbox F can generate a box flexibility that is useful on the micro level. In particular, events that are not connected or, more generally, that are not dependent on each other should not be combined in a Flexbox, as the resulting flexibility would be useless. In this section, possible useful applications of the Flexbox model are discussed, in particular those suited for the two-level framework proposed in this thesis. DWELL BOX: A dwell arc connects the arrival and departure event of the same train at the same station. It typically has a short span, in particular for minor stations, and can therefore be the constraining factor for the flexibility. Moreover, independence of arrival and departure events of the same train at the same station is not needed, because their scheduling on the microscopic level is coordinated in any case. Therefore, one Flexbox for each dwell situation in the network can be introduced, and the generated flexibility can be exploited on the micro level.

116 4.4 Flexbox model 89 Definition 4.26 (Dwell box) A dwell box is a Flexbox containing the two events corresponding to the arrival and departure time of train t at a specific station s. F D ts := {i V i is arrival or departure of train t at station s}. The set of all dwell boxes is defined as F D := {F D ts } t T,s S. An example of a dwell box can be seen in Figure 4.6(b). STATION BOX: A station box contains all events that take place in a certain given region of the network, such as a station, a line, or a group of stations. Definition 4.27 (Station box) The station box for the station s (or a network area) is defined as the set of all events corresponding to the arrival or the departure of all trains at station s: F S s := {i V i is an event taking place at station s}. Consequently, the set of all station boxes is defined as F S := {F S s } s S. An example of a station box can be seen in Figure 4.6(c). In microscopic scheduling, as explained later in Chapter 5, the network is only considered locally and all events associated with the same local zone are scheduled simultaneously. Having one Flexbox containing all the events taking place in a certain region does not enlarge the search space for the micro scheduling inside the region, if this corresponds exactly to the local zone that will be scheduled on the micro level. However, the flexibility associated with this box allows all event times of the station to be shifted by a value of δ F, which gives a measure of delay absorption capability for the region. This will be especially important in large stations, where the schedule is more sensitive to small disruptions. BALANCE BOX: The periodic service intention could ask for a specific train to be offered with a higher frequency than the considered period T or have a time dependency giving a time restriction between two events for balancing them in time. In this case a constraint arc in the PESP is needed to force the departure/arrival time of these train services at the same station to be well distributed over the time period. If the constraint is very restrictive and forces the time distance to be exactly equal to the time period divided by the frequency (e. g., every 30 minutes for

117 90 Chapter 4: Macroscopic timetabling a double frequency in 1 hour), the resulting arc span is zero, and one can effectively contract the two events into one by a simple adaptation of the PESP graph [Liebchen and Möhring, 2007, Lindner, 2000]. This way the PESP graph becomes smaller. Sometimes, however, there are cases with a small non-zero span in the balance arc, which restricts the flexibility of these events. It is therefore meaningful to combine these two events in a Flexbox. Definition 4.28 (Balance box) A balance box is a Flexbox containing the two (or more) events corresponding to the events which are mutually connected by a balance (time-dependency) arc in the PESP: F B b := {i V i is an event connected by the balance arc (or set of arcs) b}. The set of all balance boxes is defined as F B := {F B b } b B, where B is the balance arc or the set of balance arcs that mutually connects dependent events. DWELL-BALANCE BOX: Dwell boxes and balance boxes can also be combined in order to simultaneously avoid the drawbacks of the short spans in the dwell and balance arcs. Definition 4.29 (Dwell-balance box) A dwell-balance box is a Flexbox containing the events corresponding to the arrival and departure time of all trains which are mutually connected by balance arcs b, i.e., F DB b := i b{i,d(i)}, where i b means that event i is either the head or tail of a balance arc in the set b and d(i) the opposite end of the dwell arc corresponding to i. Let F DB := {Fb DB } b B be defined as the set of all dwell-balance boxes. TRAIN PATH BOX: A train path is a sequence of departure and arrival events of the visited stations of a train run. A Flexbox with all these events will gain flexibility for the train to be shifted without influencing other trains. This can be interpreted as some freedom for the (manual or automatic) planner to schedule this train or a sort of absorption robustness before it starts affecting other trains.

118 4.4 Flexbox model 91 Definition 4.30 (Train path box) A Train path box is a Flexbox containing all events of a certain train t: F T P t := {i V i is an event of train t}. The set of all train path boxes is defined as F T P := {F T P t } t T, It is also imaginable to create a train (sub-)path box of only events of a sub-path, for instance only the sub-path corresponding to a certain microscopic region.

119 92 Chapter 4: Macroscopic timetabling (a) (b) (c) Figure 4.6: Example of Flexboxes. (a) The same example of Figure 4.2, with a Flexbox added for each single event. The same event flexibility results as described in Section 4.3. The flexibility value δ F for each box is indicated by a gray number. The black numbers represent the arc span. It can be easily verified that each arc span is larger than or equal to the flexibilities of the boxes it crosses and that Eq. (4.51) holds. (b) Dwell boxes containing two events. The box flexibilities are now larger while Eq. (4.51) still holds. The events π a1 and π d1 of the first train can now be shifted simultaneously by 1.5 minutes without affecting the event times of the second train, e.g. in the solution π a1 = 0,π d1 = 1,π a2 = 4,π d2 = 5. (c) Station box containing all events of a station. When used for delay absorption, the value δ station-box determines how many minutes delay may occur at one station without affecting the events of other stations. In this case, the absorption capability of the station is of 2.5 minutes.

120 4.5 Computational results Computational results The PESP, FPESP, and Flexbox models are all implemented using Matlab R with the MOSEK R [Mosek, 2007] and the ILOG CPLEX R solver for mixed-integer linear programs. The tests are run on a 2 GHz 64bit processor with 4GB RAM. All computations throughout this chapter are terminated when an optimality gap of 0.1% is reached Reference scenario Concepts and algorithms presented in this chapter are validated with computations on the macroscopic topology of central Switzerland, as presented in Appendix A.1. The reference scenario consists of 48 trains running on the described topology with a periodicity T = 60 min and headway time h = 2 min. Table 4.1 shows the sizes of the PESP graph and the MIP formulation for the reference scenario with and without flexibility. # # integer # δ # MIP # average (stdev) average variables variables variables constraints non-collision arc span [min] ω C (stdev) (4.8) 2.6 (1.1) Table 4.1: Data of the PESP graph and the MIP for the reference scenario in central Switzerland, with the variables δ for adding flexibility. The PESP graph of the reference scenario with 48 trains has 212 events and 647 arcs after resolving arcs with zero span. The average arc span and its standard deviation are computed excluding the headway constraints, which are 446 arcs with span 56 min (h = 2) Results for different objective functions The output of the timetable generation step, i.e. a macro timetable, is a list of all departure and arrival times of the trains at the nodes in the macro topology. This data can be visualised in the form of time-space diagrams for both cases of classical PESP and Flexible PESP, as illustrated in Figure 4.7 for the same service intention. One can notice that the introduction of flexibility on the one hand changes the combinatorial structure of the solution (i.e. train sequences) and on the other hand yields a better distribution of the trains over time. Figures 4.8 and 4.9 show an example of the resulting flexible timetable for the reference scenario on two lines: the first is the principal line for freight trains on the international north-south Gotthard route, the second is the main line for passenger trains from Zurich to Italy, with two single track sections.

121 94 Chapter 4: Macroscopic timetabling The timetable can be computed with both the original PESP formulation (Section 4.2.1) or with the CPF formulation (Section 4.2.4). First, the two formulations are compared, as well as the model with and without flexibility of the events. The objective function NOFLEX is used, i.e. MINTRAVEL without introducing the variables δ. The reference scenario takes more than 4000 sec to be solved with the original PESP formulation but only 14 sec (with CPLEX) with the CPF formulation. As often observed in the literature [Liebchen, 2006, Peeters and Kroon, 2001, Peeters, 2003], the CPF formulation seems more efficient and better suited for timetable generation. It is therefore used for all further tests throughout this section. If the values δ for the event flexibilities are introduced, the CPF formulation is considered, and the reference scenario is solved with the objective MINTRAVEL, a CPU time of 25 sec results, as reported in Table 4.2. Other tested scenarios give similarly increasing ratios of the CPU time when introducing event flexibility. Notice that an optimal solution of MINTRAVEL with all δ i = 0 corresponds to an optimal solution of the original PESP without flexibility. The CPLEX solver detects this structure and quickly delivers a solution with all δ i = 0. It is interesting that the MIP solver of MOSEK does not find this solution, but takes more time (210 sec) and provides a (more efficient) solution with δ i = 60. By solving CONTRAVEL with ε = 0, an optimal solution with δ i = 152 is obtained, which is the maximal amount of flexibility that one can get by keeping the optimal objective value for the travel time. This can be considered as the natural flexibility inherent in the considered scenario. Applying the classical PESP approach, this flexibility gets lost without having anything instead. If the travel time is regarded as extremely important by a planner, this option will allow in any case to get some flexibility for improving the quality of the scheduling on the micro level, without compromising on the travel times. By maximising the flexibility using the objective MAXFLEX, a CPU time of 336 sec (CPLEX) is reached. One can observe that the introduction of the additional variables δ, which more or less doubles the number of continuous variables in the MIP, increases the CPU time, but not too much as no additional integer variables are introduced. Furthermore, an appropriate choice of the objective function could help to improve the CPU time (see Table 4.2). Results on the reference scenario with event slots are displayed in Figure 4.7. Here, 48 trains are used, with a event slot sizes limitation to δ i 4. The limitation of the event flexibilities with inequality (4.44) (δ i δ max ) has several reasons. Large flexibilities are not very useful, neither for the micro scheduling nor for the delay management. On the contrary, events with large δ i restrict the δ j for other events because of Lemma 4.9 (δ j γ a δ i ). It seems more appropriate to have many small time slots instead of a few large ones. Table 4.3 shows the effect of the flexibility bounds for a slightly modified version of the reference scenario. A second drawback of large flexibilities is that travel times are increased, as the lower

122 4.5 Computational results 95 bound for the trip times increased, as stated in Theorem This is only acceptable if the increase is small and if it is compensated by additional timetable robustness. An analysis of the behaviour between these conflicting objectives is discussed in Section 4.6. name objective CPU time [s] δ i x t,d,c MOSEK CPLEX NOFLEX min x t,d,c MINTRAVEL min x t,d,c / MAXFLEX max δ i / 2241 MIXFLEX 1/2 max δ i x t,d,c MIXFLEX 2/3 max2 δ i x t,d,c MIXFLEX 9/10 max9 δ i x t,d,c POSTOPT max δ i for fixed q C CONTRAVEL max δ i s. t. x t,d,c 1.02 ftt CONFLEX min x t,d,c s. t. δ i 0.9 f flex Table 4.2: Results for the reference scenario with bounds for the flexibilities δ i 4. x t,d,c stands for the sum of all trip, dwell and connection times. NOFLEX means the original PESP solved with CPF formulation, without introducing the variables δ. Notice that for POSTOPT and CONTRAVEL a solution to NOFLEX is needed; the reported CPU time is without the time needed for NOFLEX. The entries 67/0 and 2251/2241 stand for different values provided by MOSEK 5 and CPLEX 11, because these objectives are not considered in the corresponding objective function. Generating a timetable with maximised (or large) flexibility needs more computation time compared to other objective functions. The increase can be explained by comparing the effects of the objective functions on the solver. An objective function that minimises the trip and connection times (MINTRAVEL, see Table 4.2) automatically saves the capacity of the tracks by trying to assign to each train the shortest track occupancy time possible. The objective function basically helps the solver to find a solution, as it is easier to find one when the trains use only little track capacity. MINTRAVEL has the additional advantage of offering passenger-friendly train schedules with low travel times. An objective function maximising the event slots (MAXFLEX) is the opposite extreme. An event with high flexibility also uses a lot of track capacity. This can be seen in Figure 4.7, where the flexible events occupy a band (filled in grey) instead of just a single line. With such an objective function, the solver starts looking for solutions with high

123 96 Chapter 4: Macroscopic timetabling δ max δ i number of events with δ i = 0 δ i = 1 δ i = 2 δ i = 3 δ i Table 4.3: This table shows the effect of the limitation δ i δ max when MAXFLEX is optimised in a slightly modified version of the reference scenario with one additional freight train slot. The choice of δ max has the conflicting goal of maximising δ i while minimising the number of events with zero flexibility. For the following tests, the flexibility bound δ max = 4 is used. flexibility, which are not likely to be feasible as they block a lot of track capacity. Hence, approaches that combine the advantages of both are desirable. The next section analyses this crucial tradeoff between the two objectives in more detail. The post-optimisation approach (POSTOPT) takes the NOFLEX solution, which is generated quickly, and adds flexibility in a second step while keeping the integer variables q C constant. The generated MIP is thus reduced to an LP for the second step and can therefore be solved almost instantaneously. It is interesting to see that the resulting flexibility is quite high, even compared to the maximally possible objective value in MAXFLEX. In particular, the generated solution is very close to be a Pareto-optimal solution, see later Figures 4.10 and It can be expected that the computation times of MAXFLEX grow faster than for MINTRAVEL with the problem size due to the capacity problem. This makes the POSTOPT concept attractive for larger instances. The objective CONTRAVEL works on a reduced search space that contains only the solutions with a maximum deviation from the optimum of MINTRAVEL of a ratio given by the value ε. It is interesting to see that the computation time of this approach depends on

124 4.5 Computational results 97 the input. For the reference scenario it provides good results but for some other instances it is quite time consuming. The reason might be that the travel time restrictions give different reduction of the search space of the integer variables. Similar to CONTRAVEL, the objective CONFLEX works on a reduced search space that contains only the solutions with a minimal flexibility of a factor 1 ε compared to the maximal one. Also in this case, the computation time is very sensitive to the choice of the scenario, the value ε, and also to the choice of the solver. In particular, small values of ε are quite difficult for the solver, probably because of the very constraining factor of the minimal flexibility. Larger values of ε become easier to solve, in particular when the solution of the first step is given as initial solution. Furthermore, in contrast to other objectives, it seems that MOSEK is better able to deal with the constrained flexibility than CPLEX in this case.

125 98 Chapter 4: Macroscopic timetabling (a) (b) Figure 4.7: (a) The generated timetable without event slots from the 2007 SBB service intention visualised in a time-space diagram. The horizontal axis represents the route from Arth-Goldau (GD) towards the alps (Erstfeld, ER), whereas the vertical axis represents the time. (b) Flexible timetable for the same service intention. When using event slots, each event gets an event time π i and a flexibility δ i. In this diagram, the earliest possible line and the latest possible line are filled in grey. Any choice for train trajectories in the grey zones are feasible from the macro scheduling point of view.

126 4.5 Computational results 99 Figure 4.8: Example of a flexible timetable for the reference scenario on the line Muri Rotkreuz Immensee Arth-Goldau Gotthard. Figure 4.9: Example of a flexible timetable for the reference scenario on the line Baar Zug Walchwil Arth-Goldau Gotthard. Notice the two single track lines Zug Walchwil and Walchwil Arth-Goldau, with possibility to cross in Walchwil.

127 100 Chapter 4: Macroscopic timetabling Results for the Flexbox model The benefits of the Flexbox model are demonstrated using the two topologies presented in Appendix A.1 and A.3. As scenarios, some variation of the service intentions for both topologies are presented. For the scenario Central Switzerland, the same reference scenario as Section is taken, which is introduced in Appendix A.1. It contains 3 track paths for freight trains per hour and direction on the north-south line. Furthermore, a variant with 4 track paths for freight trains is also considered. The reference scenario for the topology in southern Switzerland takes the service intention for passenger traffic, obtained by reverse-engineering the operated passenger timetable in 2007, and adds three track slots for freight traffic on the north-south axis per hour and direction. The second scenario adds an additional freight train per direction, whereas the third scenario tests an hypothetical frequency of 30 min on the commuter S3 line between Bellinzona and Luino. Special focus for the computation is put on the contribution of the different types of Flexboxes introduced in Section In each test, in addition to the different types of Flexboxes, the flexibility for each single event box is kept. The results are summarised in Table 4.4. scenario FPESP Station Dwell Balance Dwell-Balance Boxes Boxes Boxes Boxes Central Switzerland Central Switz. (4 freight) Southern Switzerland Southern Switz.(2x S3) Southern Switz. (4 freight) Table 4.4: Maximal flexibility for the different types of Flexboxes. In Table 4.4 one can observe that only in one case the station boxes lead to an improvement of the total flexibility. For most instances it seems difficult to guarantee absorption for any delay up to a certain amount in a station on the macroscopic level. Thus, an appropriate microscopic assignment (and online re-routing) appears necessary to absorb as much delay as possible in a station region and avoid delay propagation. Dwell and balance boxes show that natural dependencies can indeed be used to gain additional flexibility. In all tested scenarios, with the exception of one per box type, it was possible to improve the total flexibility, mostly significantly. Both of them have a similar average impact on the total flexibility, but this seems to be dependent on the scenario. For instance, the scenario in southern Switzerland exhibits good values from the dwell boxes

128 4.6 Bi-objective analysis 101 but not as much for the balance boxes. Changing the frequency of a commuter train from 1 hour to 30 min, which is a small change in the service intention, has a large impact of the performance of the two boxes. The additional balance boxes due to the additional local train S3 can increase their total flexibility, whereas the additional flexibility of the dwell boxes is decreased to zero even if the new train adds additional new dwell boxes. The combination of both Flexbox types seems to be even better: since the arc span of a dwell or separation arc is typically smaller than a trip or headway arc, the dwell-balance boxes lead to an improvement of the total flexibility for all scenarios, in most of the cases even very effectively compared with the other Flexbox types. As all events in a dwell-balance box correspond to the same station and the same train type, this additional flexibility can be exploited fully during microscopic scheduling without restricting the solution space and therefore to substantially improve the chance of getting a feasible solution of good quality. 4.6 Bi-objective analysis As seen from the results in the previous section, minimising trip time and maximising flexibility are conflicting objectives. In principle, it is desirable to set as much flexibility as necessary and simultaneously to minimise the trip times offered to the customers. It is, however, difficult to state which combination of both objectives is appropriate for a practical application without knowledge of the actual tradeoffs for the instance at hand. In fact, all efficient solutions could come into consideration as potential optimal solutions, depending on the specific input and the perception of the planners. An efficient solution is also called a Pareto-optimal solution. Definition 4.31 (Dominated solution) Let x 1, x 2 be two solutions of a bi-objective MIP max{(c T 1 x,ct 2 x) x X}. Let x 1 x 2 be the strict partial order meaning that c i (x 1 ) c i (x 2 ) for i = 1,2 and c j (x 1 ) < c j (x 2 ) for at least one objective j = {1,2}. x 1 x 2 is equivalent to x 2 x 1 and means that x 2 dominates x 1. An equivalent definition can be also stated if one or both functions have to be minimised instead of maximised, as it is the case here with maximising flexibility and minimising the total travel time. Definition 4.32 (Pareto-optimal solution) A feasible solution x is Pareto-optimal, or strongly Pareto-optimal, if there is no other feasible solution x such that x x. x + is called weakly Pareto-optimal, if there is no other feasible solution x such that c i (x + ) < c i (x) for both objectives. The set of all Pareto-optimal solutions is called the Pareto set X par : X par := {x X x is Pareto-optimal},

129 102 Chapter 4: Macroscopic timetabling and its image is defined as F par := {(c 1 (x ),c 2 (x )) x X par } and will be called Pareto front or efficient frontier. For a general introduction to bi-objective and multicriteria optimisation the reader is referred to [Ehrgott, 2000]. In general, as a MIP has both discrete and continuous variables, the Pareto front consists of piecewise continuous lines (due to the continuous variables) that could be disjoint (due to the integer variables). A method for exact computation of the Pareto set for mixed integer programs with an arbitrary number of objective functions based on a branch-and-bound approach is presented in [Mavrotas and Diakoulaki, 1998, Mavrotas and Diakoulaki, 2005]. [Laumanns et al., 2006] introduces an output sensitive iterative method for the same goal if there are only integer variables, whereas [Popović, 2009] presents an efficient specific developed method for computing exactly the Pareto front of a bi-objective MIP. These methods are able to find each element of the Pareto set, but are quite technical to implement and time consuming. The goal in this thesis is just to better understand the dependencies between the objectives and provide a representative collection of efficient candidate solutions, where the planners can choose from. Therefore, an ε-constraint approximation algorithm for the computation of the efficient frontier for this bi-objective problem is adopted. The Pareto front is sampled by applying the objective function CONTRAVEL for different values of ε, starting with ε = 0 and stopping when the maximum possible flexibility (computed with MAXFLEX) is reached, as described in Algorithm 1 for a FPESP instance. Computing the Pareto front for a Flexbox PESP instance is the same, with the only difference that at each step the MIP formulation 4.24 needs to be solved instead of MIP formulation Figure 4.10 shows the approximated efficient frontiers for the reference scenario and two variations. The first variation has two additional freight trains (one per direction) on the line Lenzburg Rotkreuz Immensee Arth-Goldau Gotthard, and flexibility was added only in the events occurring in the stations of Lucerne, Zug and Arth-Goldau, which are the larger ones in the network and the most difficult to schedule on the micro level. The second variation has the same trains as the first one, but flexibility was added to all events. To simplify reading of the picture, the sample points (the dots in the graphics) are connected by lines. In addition to the points computed with CONTRAVEL, the solution of NOFLEX is also displayed, with minimal trip time and no flexibility, as well as the solution of MAXFLEX, with maximal flexibility but not with the corresponding minimum trip time. Figure 4.11 presents approximate efficient frontiers for a (smaller) topology in southern Switzerland, introduced in Appendix A.3. The three considered variants of the service

130 4.6 Bi-objective analysis 103 Algorithm 1 Approximated ε-constraint method for Pareto front. Require: FPESP Instance FPESP(G,T,l,u), objective functions f tt (to minimise), f flex (to maximise), number of steps to compute n Ensure: Discretised Pareto front PS 1: Solve Problem 4.7 using objective MINTRAVEL. Get the optimal value f tt 2: Solve Problem 4.7 using objective MAXFLEX. Get the optimal value f flex 3: Solve Problem 4.7 using objective CONFLEX(ε = 0). Get the minimal travel time tt max f lex for the optimal value of flexibility fflex 4: Compute relative ratio ε := tt max f lex ftt 1 5: Set relative current constraint on travel time ε := 0 6: Set x old := 0 7: Initialise Pareto front PS := {(tt max f lex, f flex )} 8: for i = 0,...,n do 9: Solve Problem 4.7 using objective using objective CONTRAVEL(ε). Get solution x new with objective values ( f tt (x new ), f flex (x new )) 10: if f flex (x new ) > f flex (x old ) then 11: Update Pareto front PS := PS {( f tt (x new ), f flex (x new ))} 12: x old := x new 13: end if 14: ε := ε + ε n 15: Delete x new 16: end for 17: return Pareto front PS intention are the same as introduced in Section From all computations one can observe that even with minimal trip time the flexibility can be larger than zero. This means that there is some natural flexibility, not at all detected by NOFLEX and only partly by MINTRAVEL, which can be used without affecting the quality of the timetable (total trip time) offered to the passenger. In addition, there seems to be a wide region where the marginal rates of substitution (the slope of the efficient frontier) is equal to one, a particularity that is discussed in the following. Recall that the actual efficient frontier for such bi-objective MIPs consists of a set of piecewise continuous lines that could be disjoint. By considering any fixed feasible combination of the integer variables q C, i.e., the train sequences remain unchanged, an LP is obtained whose efficient frontier is a continuous, concave, and piecewise linear function [Ehrgott, 2000]. The maximum over these functions then represents the overall efficient frontier for the original MIP, which can then of course be non-concave and discontinuous. To give an idea of this superposition, Figure 4.12 displays the efficient frontiers for those

131 104 Chapter 4: Macroscopic timetabling Figure 4.10: Pareto front for the scenario in central Switzerland. q C combinations that are optimal for at least one considered value ε of CONTRAVEL during Algorithm 1. It can be observed that all curves also have a very long central part that is linear with slope one. By investigating the different optimal bases and their sensitivity, it was observed that within a large range there is mostly only one tension x a which increases and the corresponding flexibility δ i of the starting event also increases by the same amount. The cycle constraints are then fulfilled by adjusting other tensions on the cycles which are not in the objective function (e.g., headway or turn-around arcs). If all the trains have the same weight (priority), this implies a shadow price of one of the travel time constraint. When it is no longer possible to compensate only with tensions with weight zero, other tensions are needed, leading to a correspondingly smaller shadow price and slope. However, in all the tested cases it seems that this happens only close to the maximal flexibility. But even though the region with a shadow price of one is large, there is no single basis feasible for a long interval due to the enforced upper bounds on the flexibility values. Sensitivity analysis shows that each basis moves in a domain of 0-4 units (minutes) and then a basis change occurs, which means that the simultaneous increase of tension and corresponding flexibility takes place at a different arc. Additionally, the behaviour of the objective function according to their efficiency is discussed. Per construction, the objective function MIXFLEX with 0 < λ < 1, as well as CONTRAVEL and CONFLEX for ε not too large, generates a strongly Pareto-optimal solution. On the other hand, MAXFLEX, MINTRAVEL, and CONTRAVEL or CONFLEX

132 4.6 Bi-objective analysis Flexibility POSTOPT POSTOPT POSTOPT 3x freight 4x freight 3x freight and 2x S Trip time Figure 4.11: Pareto front for the scenario in southern Switzerland. for large ε generate weak Pareto-optimal solutions. In particular, the strongly Paretooptimal solution with minimal trip time can be computed with CONTRAVEL and ε = 0, providing for the reference scenario a solution with δ i = 152. The trip time value provided by the objective MAXFLEX shown in Table 4.2 is not the best possible. As the trip time is not in the objective function, different solvers could deliver solutions with different values. Both MOSEK and CPLEX find similar, not very efficient solutions, and these points are displayed as the right-most point of the efficient frontier on the figures. Similarly, the strongly Pareto-optimal solution with maximal flexibility can be computed with CONFLEX and ε = 0, providing a solution with x t,d,c = 2160 for the reference scenario. A combination of both objectives with MIXFLEX (0 < λ < 1)yields a strongly Paretooptimal solution. However, because of the observed precise linearity in the central part of the efficient frontier and the consequent degeneration of the optimal solution for λ = 2 1, all solutions with λ > 1 2 reside in the region close to the maximal flexibility, whereas all solutions with λ < 1 2 reside in the region close to the minimal trip time. The optimal solution with λ = 1 2 is degenerated, but neither CPLEX nor MOSEK provide solutions with a balance of both objectives, rather in one of the the extremes, as it could be expected. The heuristic approach POSTOPT performs quite well. The solutions of POSTOPT are indicated in Figures 4.10 and 4.11 on the head of the arrows. This quick method does not yield an efficient solution, but the solution is very close to efficiency for each

133 106 Chapter 4: Macroscopic timetabling Flexibility Flexibility Trip time (a) Trip time (b) Figure 4.12: (a) Pareto front for the reference scenario for fixed train sequences. (b) Zoom of the range x in [2080,2150] and y in [300, 350]. tested scenario and has the additional advantage that it provides a well balanced solution between the two objectives in some cases. As not all stations have the same size and the same difficulty to be scheduled, it is reasonable to give larger weights to events belonging to large stations and smaller (or even no flexibility) to events of minor stations. Figure 4.13 illustrates the approximated Pareto front for the reference scenario, where all events taking place in Lucerne are weighted with the value 3, events of Zug and Arth-Goldau with value 2, and all other events with value 1. One can observe that the long central line with slope 1 is now replaced by three parts with a relatively stable slope of respectively 3, 2, and 1 units. Effectively, by observing in detail the changes in the solution, each slope corresponds to a change in the solution of the flexibility and tension of an event with the corresponding priority. Figure 4.14 shows the complete Pareto front of those sequences that were Pareto-optimal in at least one computed point. Finally, priorities can be introduced for the different trains and stations (weights for the values x and δ). Figure 4.15 shows an example for an instance where the weights for the flexibilities were generated uniformly at random from the interval [1, 2]. One can notice that the long linear central part disappears, which is due to the fact that at each

134 4.6 Bi-objective analysis 107 Figure 4.13: Pareto front for the reference scenario with station priorities. basis change (at most every 4 units) the weights for the new basic variables δ i changes. Figure 4.16 shows the complete Pareto front of those sequences that were Pareto-optimal in at least one computed point.

135 108 Chapter 4: Macroscopic timetabling Figure 4.14: Pareto front for the reference scenario for fixed train sequences and station priorities Flexibility Trip time Figure 4.15: Pareto front of the reference scenario with randomised weights for flexibilities.

136 4.6 Bi-objective analysis Efficient frontier for the different sequences q Flexibility Trip time Figure 4.16: Pareto front for the reference scenario for fixed train sequences and randomised weights for flexibilities.

137 110 Chapter 4: Macroscopic timetabling 4.7 Summary and final remarks The classical PESP model with fixed event times could lead to a macro timetable that is infeasible at the microscopic level. Therefore, the Flexible Periodic Event Scheduling Problem (FPESP) is introduced, which allows time slots instead of exact event times to be generated. This enlarges the solution space, and hence the chance of finding a feasible solution, for the subsequent microscopic scheduling. Moreover, the concept of Flexboxes is developed as a further generalisation of the FPESP to make use of natural dependencies of events to further improve the total flexibility of the resulting macro schedule. The resulting problems can be formulated as MIP, with the same structure and integer variables of the corresponding PESP. Thus, the computational effort is only moderately larger than for the PESP, and future improvements regarding the PESP are likely to carry over to the FPESP as well. Computational results on real-world instances from the Swiss railway network show that it is possible to generate macro timetables with event slots and box flexibility for a scenario of medium size (48 trains in one hour) in a reasonable amount of time (2-7 minutes). It can be observed that the presented formulations (PESP, FPESP, and Flexbox) can be quite efficient also for large networks and many trains (like the complete IC network for Switzerland), but suffer particularly if the intended services become very dense compared to the available infrastructure, i.e., when it is near at the capacity limit. The introduction of the event slots and the Flexboxes does not seem to affect the computation time too much and should be more than compensated by the reduction of the necessary iterations between the macro and micro level within the presented two-level scheduling approach. An intuition for this is given in Section 9.2. However, this is so far only a conjecture which is not based on systematic computations. The FPESP yields a natural formulation as a bi-objective problem when considering the simultaneous minimisation of total travel time and maximisation of flexibility (or robustness). A detailed analysis of the trade-offs between these two conflicting objectives for different scenarios is given and the properties of the efficient frontier are discussed, as well as the behaviour of different solution approaches. The linearity of the efficient frontier within a wide range makes it difficult to find well-distributed efficient solutions by using a weighted sum of the two objectives. For this reason, solving a sequence of constrained single-objective problems seems more appropriate. On the other hand, the post-optimisation heuristic leads to solutions close to the efficient frontier with a good compromise between the two objectives. This very quick approach is promising for larger scenarios.

138 Chapter 5 Microscopic scheduling: network decomposition approach This chapter addresses the train scheduling problem on the microscopic level. On this level, the task is to find an operable production plan for a given macro timetable, in particular to determine exact track paths and their blocking times to ensure a conflictfree schedule. Therefore, micro scheduling considers all details of the railway topology that are important for describing local conflicts, but which are not relevant to the global structure of the timetable and were therefore neglected on the macro level. The considered safety system follows the way the interlocking system works. It computes the blocking times on each resource based on the signalling system and ensures that these blocking time intervals do not overlap. Finally, the computed track paths must specify a speed profile that a train driver can follow within a reasonable tolerance band. This chapter does not present any model or algorithm for solving the micro scheduling problem but discusses the properties of the network and develops a subdivision of the network into zones of treatable size, originally proposed at SBB [Laube et al., 2007]. The next Chapters 6 and 7 then focus on the methods for solving the microscopic scheduling problem in the different zones according to their different features. The chapter is organised as follows. Section 5.1 gives a literature review on relevant approaches for micro scheduling. Section 5.2 conceives a division of the network into different zones according to their specific properties, and Section 5.3 introduces the notion of a portal as the interface between the zones. Finally, Section 5.4 outlines the procedure for dealing with the local zones and merging their local solutions into a global conflictfree train schedule. This chapter is partially based on [Caimi et al., 2009e].

139 112 Chapter 5: Microscopic scheduling: network decomposition approach 5.1 Related work Efforts to find a suitable model and corresponding algorithms for conflict-free scheduling of a whole railway network have remained rare. Because of the size and complexity of the problem, the few existing approaches mostly resort to heuristic solution methods. An overview of the methods for microscopic train scheduling is illustrated in Figure 5.1. Figure 5.1: Overview on the principal methods in the literature for microscopic scheduling. [Carey, 1994, Carey and Crawford, 2007] consider a railway network composed of different lines with some non-trivial stations. The problem is solved by simultaneously considering the micro topology of the whole network and applying a heuristic based on step-wise fixing of the train sequences, similar to the manual methods in use in Great Britain. For medium size networks, the authors observe acceptable computation times. [Caprara et al., 2006] extend the (macroscopic) model introduced in [Caprara et al., 2002] by considering additional practical constraints that were neglected in previous papers. In particular, they address manual block signalling and station capacities, transforming this approach to micro scheduling, according to the classification explained in Section 4.1. They incorporate these additional constraints into the mathematical model for the basic version of the problem, and solve it again using a Lagrangian-based heuristic. In the DONS project [Hooghiemstra et al., 1999], also the micro scheduling step is addressed. However, the schedule is checked for feasibility only in station regions, while the simplified macro safety model using headways is considered sufficient for the lines. Even if not completely wrong from a feasibility point of view, this approach does not allow precise computation of conflict-free track paths, which results in imprecise train driving behaviour that can only be compensated by more generous time reserves and reliance on online traffic management. Even in station regions, the schedule is checked

140 5.1 Related work 113 only by considering the running times over the relevant track sections and the release times, but the blocking times are not inspected for conflicts in all resources. The model used in DONS for micro scheduling in station regions is described and discussed in detail in Section Some other interesting microscopic approaches are developed for real-time rescheduling. [D Ariano, 2008] develops a decision support system based on the so-called alternative graph formulation introduced by [Mascis and Pacciarelli, 2002]. The alternative graph model considers all possible scheduling alternatives for a given set of train routes and is able to efficiently model different signalling systems and therefore offer high reliability. For a given set of train routes, a branch-and-bound method is presented [Pranzo et al., 2005, D Ariano et al., 2007a] for finding optimal train sequences that minimise delay. This model is then augmented by also considering rerouting of trains, which is solved iteratively by computing an optimal train sequencing for given train routes, and then improving the solution by locally rerouting some trains [D Ariano et al., 2006, Corman et al., 2007, Corman et al., 2008b, D Ariano et al., 2008a]. During the optimisation, the authors also consider the computation of conflict-free and energy-efficient train speed profiles according to microscopic train dynamics [D Ariano et al., 2007b, D Ariano and Albrecht, 2006]. [Törnquist, 2006] describes a network-wide microscopic approach for train dispatching. The method follows an agent-based two-level approach. The upper level is represented by intelligent agents who simulate the behaviour of the traffic management and the operators and controls the lower level, which includes the physical network, the traffic flows, and the associated restrictions. Each agent can act independently and is tailored according to its specific task [Törnquist and Davidsson, 2002, Davidsson et al., 2005]. Each sub-problem is formulated as a mixed integer program (MIP), which is relaxed to a linear program (LP), and the integer variables are handled using heuristics (tabu search and simulated annealing, [Törnquist and Persson, 2005]). In a first step, a singletrack line with homogeneous traffic is considered [Törnquist and Persson, 2005] and is then extended to a network with a number of track lines and heterogeneous traffic [Törnquist and Persson, 2007]. Finally, in [Törnquist, 2007] the approach is augmented with heuristics for dealing with networks at their capacity limits, and extensive computational results are presented. [Wegele and Schnieder, 2004a, Wegele and Schnieder, 2004b, Wegele et al., 2007] present an optimisation method for train dispatching, which can also be used for offline train scheduling. The authors consider (simplified) block sections on the lines and ensure they do not overlap, but they only schedule trains to run at full speed and let them wait at stations. Based on a special coding for the timetable, they present an optimisation method with several levels, with the goal to first detect deadlocks and then to reduce the search

141 114 Chapter 5: Microscopic scheduling: network decomposition approach space. The objective is to minimise the delay propagation applying branch-and-bound for finding the first solution and improving it using genetic algorithms. Computational results for a railway network of medium size are presented. The monolithic approaches considering the whole micro topology at once, however, have resort to heuristic and relatively simple solution methods. Furthermore, they do not take into consideration the characteristics of the different parts of the railway networks. For specific, exact, and targeted approaches for a local part of the network, the reader is referred to Section 6.2 for a literature review on approaches for condensation zones and to Section 7.1 for compensation zones. 5.2 Network decomposition Micro scheduling cannot be simultaneously done on the whole network because of the enormous amount of data that would lead to both memory problems and prohibitively long computation times. This problem can be overcome by a targeted subdivision of the network into different zones of tractable size such that the detailed topology is considered only locally in regions of limited size. The railway network has been built according to mobility demand of the population and depends on population density and geographical properties. It grew over many decades according to the necessities, the budget, and the available techniques of the period. The result is a very heterogeneous railway network, with modern and dated tracks, various safety and signalling systems, and where parts of the network are used at full capacity while others are underutilised. This heterogeneity and the different characteristics of different parts the railway network are the basis for the network decomposition approach. This network decomposition should be done manually by experienced practitioners by taking into account the different properties of the railway network and the service intention. Policy 5.1 (Network decomposition) On the microscopic level, the railway topology is subdivided in a suitable, targeted, and systematic way exploiting the heterogeneous structure of the network. Two types of zones are distinguished: Condensation zones are the saturated zones of the network. The available unused capacity is scarce, thus they represent the bottlenecks of the network. Here, the network shall be used at the maximal possible capacity that allows stable operation. Therefore, the scheduling policy for these zones is such that no time reserves are introduced, and the specified track paths must be followed very precisely. Usually, but not necessarily, condensation zones are situated around major stations.

142 5.2 Network decomposition 115 Compensation zones are the zones that connect different condensation zones and mostly have additional capacity available. They serve as recovery zones for the trains with respect to punctuality and following of a determined track path. In compensation zones, trains will be controlled in order to enter the next condensation zone precisely at the predetermined time and speed. This way, the condensation zone can be passed without losing any capacity. Time reserves are introduced to reduce possible delays coming from previous condensation zones and thereby to improve timetable stability. Different policies for generating micro train schedules are then applied to the two zones according to their distinct properties. In particular, a clear difference of this approach compared to standard approaches is the strategy concerning the distribution of time reserves (also called running time supplements) along the train lines. In general, generous time reserves lead to a better punctuality of the railway services yet causing longer travel times. Therefore, a compromise must be found. According to UIC recommendations, time reserves are normally between 5 12% of the technical travel time [Lüthi, 2009]. In Switzerland, it is currently common practice to plan 7% running time supplement for passenger trains and 11% for cargo trains as rule of thumb [Lüthi, 2009, Kroon et al., 2006], without differentiating between compensation and condensation zones. In contrast to this standard approach, the policy described here applies calls for a different distribution of time reserves. In condensation zones, no running time reserves are introduced and the zone will be saturated up to the limit of a stable operation according to UIC recommendations, which gives a general guiding value of 15 25% of unplanned capacity during peak times [Union Internationale des Chemins de Fer (UIC), 2004]. However, some (small) time reserves will remain in the dwell and turn-around times at the platforms of the stations. In compensation zones, more generous time reserves are then introduced in order to compensate (hence the name of this zone) for the condensation zones. The saturation of these zones may not pass a certain amount to enable the necessary flexibility for steering trains so that they can enter the next condensation zone precisely on time. Figure 5.2 illustrates a suitable subdivision of the network into condensation and compensation zones for the central part of the Swiss railway network according to the operated service intention in Notice that a compensation zone does not necessarily need to be a line connecting two condensation zones, but can also can also be of more complex nature. In the figure, for instance, this is the case for the compensation zone between Basel, Brugg, and Liestal. Remark 5.2 (Network decomposition depending on the service intention) The network decomposition, distinguishing saturated from non-saturated zones, is not an intrin-

143 116 Chapter 5: Microscopic scheduling: network decomposition approach Figure 5.2: Possible representation of the Swiss railway central part divided in condensation and compensation zones. sic property of the network, but depends on the service intention. It is therefore possible that an adequate separation changes when the service intention is changed. The proposed network decomposition can be formalised by a function that labels the corresponding local zone and its type (condensation or compensation) for each element of the microscopic topology. Definition 5.3 (Network decomposition function) Let C := k 1 i=1 Ccond i k 2 j=1 C comp j be defined as the set of all local zones, where Ci cond denotes the i-th condensation zone and C comp j the j-th compensation zone. The network decomposition function Λ : V C indicates for each vertex v V of the considered micro topology T = (V,E,,R,σ,l) the zone it is associated to. Of course, not all possible functions Λ make sense, and it is out of the scope of this thesis to describe a general setting for all potentially useful network decomposition functions. Nevertheless, the given network decomposition should fulfil some basic properties. Property 5.4 (Network decomposition) The network decomposition, described by function Λ, must ensure that each local zone is connected Condensation zones According to Policy 5.1, condensation zones are the saturated zones in a network with little available capacity. The reason for this situation is a very high traffic density and a structural complexity of both the topology of the railway network and the layout of train lines. That is the case, usually, in urban regions close to main station areas.

144 5.2 Network decomposition 117 Property 5.5 (Condensation zone) A condensation zone contains the track resources of the recognised critical zone until the disjunction of the lines into different directions, plus a certain length to enable trains to accelerate to the desired speed independently from the chosen route. This property is illustrated in Figure 5.3. Figure 5.3: Illustration of Property 5.5. The condensation zone of the corresponding main station contain the tracks until a certain distance after the last disjunction. In urban regions, there are long-distance trains for the main directions, various commuter trains serving all parts of the conurbation with high frequency, and also some freight trains. The majority of these train services travel to the main station and cause heavy traffic in their proximity. There are many switches to enable connections in all directions so that the track layout gives rise to a large number of different possible routes connecting the borders of the condensation zone with the designated destination, for instance the platform in the main station or the exit of the zone. Thus, train overtakings and crossings are possible and quite common, but track paths blocking several switches have the obvious drawback of simultaneously disabling many other routes. Sometimes, this difficulty is overcome with the construction of fly-overs in front of main stations, like in Zurich, where several fly-overs are used to reduce the number of necessary crossing and thus enable the operation of a denser timetable. In other cases, like in Berne or Lucerne, fly-overs are not possible or at least very complicated and expensive, so that the entire operation must be conducted managing train crossings on the same level. This is the most critical situation and it also represents the most difficult condensation zones to operate. For this reason, the two main station areas of Lucerne and Berne are the two test cases for the computations of condensation zones presented in this thesis. The layout for the railway infrastructure in the main station regions of Lucerne and Berne is illustrated in Appendices A.1 and A.2, respectively.

145 118 Chapter 5: Microscopic scheduling: network decomposition approach As already mentioned, no running time reserves are included in track paths in compensation zones, which implies that trains run at their maximally allowed speed within the zone. Policy 5.6 (Scheduling in condensation zones) In condensation zones, each train is scheduled to travel at the maximally allowed speed on the designated route. This maximal speed profile can vary depending on the given route. Therefore, once the route is assigned, the speed profile is no longer free. It is thus sufficient to assign one passing time per train at a specific location (e.g., at the zone border or at the platform). All passing times within the condensation zone can then be derived. With the temporal component fixed, the most important degree of freedom to exploit in condensation zones is the assignment of exactly one route to each train, together with the suitable assignment of this (single) passing time. The model described in Chapter 6 makes use of exactly these properties for solving the micro scheduling problem in condensation zones. For the purpose of simplifying train scheduling, the layout properties of a typical condensation zone are exploited. Observation 5.7 (Layout of a condensation zone) Usually, condensation zones cover a relatively small area (radius up to 15 km). The typical layout consists of stretches of several relatively long parallel tracks (typically 1 3 km) without switches, leading to different directions. These stretches are connected by switch regions that allow all platforms and directions to be reached. The important fact is that the switch regions are short (usually up to 500 m) relative to the stretches and are, therefore, passed quickly by trains. For example, Figure A.7 on page 235 shows a rough layout of the track topology for the station region of Berne. The most complex switch regions lie east and west of the station just in front of the platforms. Their topologies are then shown in Figures A.8 and A.9, respectively Compensation zones Compensation zones connect two (or more) condensation zones. According to Policy 5.1, they are not saturated, and it is possible to introduce time reserves for recovering from delays. This happens because compensation zones have a lower traffic density and easier topologies with less necessary crossings. Because of the simple topology, the number of possible routes within a compensation zone is small. The route for each train is often known a priori by introducing simple and widely used policies, such as the utilisation of each track for only one direction or the separation of the tracks between freight and passenger traffic. Therefore, the choice

146 5.3 Interface between the zones 119 Figure 5.4: Flexibility of the speed profile in a compensation zone. The only constraints are the boundary conditions at the borders of the compensation zones: there, trains have a given passing time and speed. The shaded zone represents the flexible space for choosing the appropriate speed profile. of appropriate speed profiles is the most important degree of freedom to exploit in this zone, as illustrated in Figure 5.4. It can be freely chosen as long as it meets the boundary conditions at the borders with the condensation zones, where the exact passing time and speed are given. 5.3 Interface between the zones Besides scheduling the different zones individually, their coordination is necessary in order to construct a conflict-free train schedule for the entire railway network. For this purpose, it is important to define and coordinate clearly the interface between the zones. Therefore, the concept of a portal is introduced, which represents the boundary between condensation and compensation zones. Each portal comprises a certain number of parallel tracks. Definition 5.8 (Portal) A double vertex (v,v ) is called portal vertex, or portal node, iff Λ(v) Λ(v ). The portal between the condensation zone Ci cond and compensation zone is defined as the set of portal vertices between the two zones C comp j P i j := {(v,v ) (Λ(v) = C cond i Λ(v ) = C comp j )}. The boundary between a condensation and a neighbouring compensation zone is always located at a double vertex. This requires an additional restriction for a valid network division.

147 120 Chapter 5: Microscopic scheduling: network decomposition approach Property 5.9 (Network division function) A valid network division function Λ has to fulfill the property that Λ(v) = Λ(w) (v,w) E. Property 5.9 implies that a route connecting a zone with another one has to pass through one portal. Furthermore, in order to properly compute the blocking times, an assumption about the topology resources is necessary. Assumption 5.10 (Inseparability of resources) Let R be the set of resources of the microscopic topology. It is assumed that all resources R V in R belong to the same zone and that no resources are divided by the network decomposition approach, i.e., R R a zone c C : (Λ(v)) = c v R. For simplicity of notation, define Λ(s) := Λ(v) for one of the vertices v in each resource R R. Property 5.11 (Portals in both zones) The portal vertices (v,v ) are considered in the microscopic topology of both zones Λ(v) and Λ(v ), such that there is a proper double vertex graph in each local zone. Implicitly, a vertex v in a zone C i with the property that Λ(v) C i indicates the border of the considered zone with the neighbour one. A natural and good choice for the portals is usually to locate them in double vertices that are main signal vertices. This way, the blocking time computation for the resources is simplified. However, the signal position is often not symmetric, i.e, the main signals for the two travelling directions are not located in the two partner vertices of the double vertex graph. Therefore, as in at least one direction it would anyway not be the case, Assumption 5.10 is less restrictive and only requires the resources not to be separated. Once the boundary conditions (portal vertex, passing time, and speed) are fixed for each train and passed portal, the zones can indeed be treated independently with the property that the locally generated schedules will also be feasible globally. Theorem 5.12 (Zone independence) Let (s i ) i C be a set of local conflict-free train schedules for the corresponding zone i C with the property that for each train z Z and each passed portal P i j, it holds that i) The train path p z uses the same portal vertex (w,w ) in the adjacent zones, i.e., the last vertex of the train path p z i for the zone i is w and is the same as the first vertex of the train path p z j for the next zone j; ii) This portal vertex (w,w ) is passed at the same time ti z j from both track paths pz i and p z j ;

148 5.3 Interface between the zones 121 iii) The portal vertex (w,w ) is passed at the same speed v z i j from both track paths pz i and p z j. Then, the union i C s i of the locally conflict-free micro schedules s i forms a globally conflict-free micro schedule s. Proof: According to the properties of each train described in the theorem and Assumption 3.16, the union of the local track paths for a train z results in a (global) track path with a feasible micro train dynamics. Hence, the resulting union is a train schedule. Assumption 5.10 states that each resource r R will not be separated and therefore belongs completely either to the condensation or compensation zone q := Λ(r). As each local schedule s q is conflict-free, it means that in each resource r of the zone q the blocking times of the train using this resource do not overlap. As the local schedules s i are conflictfree for all zones i C and all resources r R belong to exactly one zone Λ(r), the result is that in all resources of the complete micro topology the blocking times of the trains do not overlap. The claim now follows directly. This theorem states that once the track, time, and speed at the portals are specified, each zone can be treated independently, and the union of the different locally conflict-free schedules provides a globally conflict-free train schedule. The sole assignment necessary on the macro level for guaranteeing independence of the zones are the described boundary conditions at the portals. Additional information or decisions from the macro level may be taken into consideration during local micro scheduling, but they are not necessary as also a change of these conditions does not prevent the union of the local schedules to be conflict-free, as long as they are locally conflict-free. Nevertheless, the computation of correct blocking times for one resource r belonging to the zone Λ(r) can depend on the track paths of the neighbouring zones, as described in Section During local micro scheduling, this problem is overcome by taking into consideration the possible speed profiles in the neighbouring zone and ensuring the local schedule to be conflict-free in relation to all these possibilities. More details about the computation of blocking times is given on the next chapters about local micro scheduling in the different of zones: Chapter 6 for condensation zones and Chapter 7 for compensation zones. Once the route is given, according to Policy 5.6 the speed profile in the condensation zone is fixed, and therefore also the train speed at the portal is fixed. To ensure the same speed at the portals for all routes inside the compensation zones it suffices to locate the portal far enough from the last switch coming from the condensation zone to enable, if necessary, acceleration of the train in order to reach the desired speed at the portal.

149 122 Chapter 5: Microscopic scheduling: network decomposition approach 5.4 Solving the micro scheduling problem The task on the microscopic level is to solve the micro scheduling problem stated in Problem The network decomposition approach already introduced is applied, and each zone is solved independently. However, as stated in Theorem 5.12, the union of the local schedules results in a global conflict-free schedule only if the boundary constraints at the portals are fulfilled. As the macroscopic level will deliver time slots and not exact times for each event, it will not be possible to schedule all zones independently and simultaneously. In this case there is the (well probable) risk that two neighbouring zones will set different times in the event slots at the portals for the trains passing through them, thus violating the assumptions of Theorem To cope with this situation, an intermediate version between simultaneous and sequential scheduling is used, in order to guarantee the independent local scheduling yet matching the requirements for the boundary conditions and ensuring therefore the generation of a global conflict-free schedule. First, all condensation zones are solved in parallel, taking as input the flexible schedule generated on the macro level and generating a locally conflict-free schedule. As each condensation zone only borders with compensation zones, at each portal the passing time is fixed only once and therefore cannot therefore create any violation of the theorem s assumption. After having solved all condensation zones, all tracks and passing times of the trains through the portals are fixed with the data from the local schedules. All the speeds at the portals are also already fixed by the application of Policy 5.6 during local scheduling in condensation zones. Subsequently, the micro scheduling problem is solved for all compensation zones in parallel, taking as input the already fixed boundary conditions at the portals. The so created conflict-free local schedules will automatically meet the boundary conditions, i.e. the assumptions of Theorem Finally, the so generated local schedules can be merged to form a global conflict-free schedule, which is the solution sought on the micro level. This approach for micro scheduling is schematically depicted in Figure 5.5. One can notice that the feedback loop to the macro level is displayed only after the scheduling in condensation zones. This is due to the fact that these are by far the most difficult zones to schedule, on the one hand because of the very dense and heterogeneous traffic and on the other hand because they are only badly modelled on the macro level, leading to a higher risk of being infeasible when scheduling them on the micro level. Conversely, headway times are a good approximation of the safety system in compensation zones. Therefore, there is usually no risk of creating an infeasible instance, and the introduction of a feedback loop is of minor importance. The flexible macro timetable can be seen as a collection of macro timetables, and in scheduling condensation zones, one of these will be fixed. In the very unlikely case of infeasibility in a compensation zone, this does not

150 5.4 Solving the micro scheduling problem 123 Figure 5.5: Procedure for the microscopic scheduling problem. The light green box corresponds to the micro scheduling box of Figure directly mean that the given flexible timetable is infeasible, but only that the boundary conditions set by the condensation zones is infeasible. A feedback loop to the concerned condensation zones becomes then also necessary. The application of the procedure described above is facilitated by the flexibility generated on the macro level. This way, the different condensation zones can be independently solved without having to fix portal tracks and passing time a priori, which would significantly reduce the freedom in the zone to find a good local schedule. The so generated local schedules will lead to travel times between two portals that are not known in advance, but are the result of the micro scheduling. As the portal times had to be chosen inside the given event slots, the travel time will certainly be between the lower and upper bounds given as input at the macro level. Under the assumption that a time reserve close to the recommended values satisfies the needs of stability and performance of the railway operators, and that the lower and upper bounds for the train travel on the macro level are set accordingly, this method would deliver exactly a schedule satisfying these requirements. Not giving a fixed value for the travel times leaves more flexibility for scheduling, resulting in solutions of better quality and, particularly important, a reduced risk that the macro schedule is infeasible on the micro level. Even though the microscopic scheduling procedure sketched in Figure 5.5 appears rea-

151 124 Chapter 5: Microscopic scheduling: network decomposition approach sonable, it can result in any case to be infeasible in at least one of the considered condensation zones. In this case, it is clear that the timetable generated on the macro level was infeasible. But this does not imply that the given service intention, which was used as the input for macro scheduling, is infeasible as well. It might well be possible that a different macro timetable, suboptimal from the macroscopic point of view, is feasible on the micro level, implying that the service intention is actually feasible. It is therefore necessary, in case of infeasibility on the micro level, that the method does not just give a negative answer but also some kind of feedback report that could help the macro scheduler to adjust the timetable in the right way. Section 6.5 presents a method for giving a feedback in the case of infeasibility in a condensation zone.

152 Chapter 6 Microscopic scheduling in condensation zones This chapter develops models and algorithms for the micro scheduling problem in condensation zones. Condensation zones are the capacity bottlenecks of the network for the given service intention. According to the procedure for solving the micro scheduling problem described in Section 5.4, condensation zones are solved first, in parallel and independently from each other. Thus, the problem addressed in this chapter is to generate (local) conflict-free train schedules for each considered zone individually respecting the boundary conditions given by the macro timetable. Following the scheduling policy for condensation zones, time reserves are excluded from the trip time of the trains to fully utilise the available capacity. Therefore, each possible route of a train within the zone yields exactly one possible speed profile: the one that runs at maximal allowed speed. Thus, the choice of an appropriate route for each train is the important degree of freedom to exploit in order to obtain an optimal and conflictfree schedule in the zone. Because the general procedure for solving the train scheduling problem requires a feedback to the macro level in case of infeasibility, an exact method for this (sub-)problem is necessary. Existing methods for microscopic scheduling can be roughly divided into heuristic and exact methods. Heuristic methods have the disadvantage that they are not able to prove problem infeasibility. The available exact methods based in formulations as integer linear programs have a weak structure and require very long computation times even for instances of moderate size. Therefore, a new exact approach is presented in this thesis that uses a model with substantially stronger structure by accounting for the context of the resource conflicts. In this model, called Resource Tree Conflict Graph (RTCG), the

153 126 Chapter 6: Microscopic scheduling in condensation zones scheduling alternatives for each train are represented by a tree structure. The place in the micro topology where a conflict occurs is explicitly considered, and for each resource all trains are taken into account simultaneously for analysing the temporal relation between conflicts. Encoding this information in a particular graph structure, a resource-constrained multicommodity flow problem can be formulated as an integer linear program with considerably fewer and stronger constraints compared to previous formulations. The resulting integer linear program for this formulation is then solved with a commercial solver. In tests with realistic, large-scale instances provided by SBB, the RTCG model generates solutions for major stations within seconds, even for cases where traditional approaches take very long time or even fail for memory reasons. Thus, this new model constitutes a great leap forward in the practical solvability of large-scale microscopic scheduling problems. This chapter is organised as follows. In Section 6.2 different approaches and models for train scheduling in complex station areas are discussed. In Section 6.1 the problem of conflict-free scheduling in condensation zones is introduced formally. Section 6.3 presents a policy for scheduling in condensation zones based on time discretisation, which reduces the problem size and thereby also enables manual dispatching based on the same conceptual model. In Section 6.4, the RTCG model is developed step by step, from the input to the final ILP formulation. Section 6.5 proposes a way of providing feedback to the macro level in case of infeasibility in a condensation zone. Computational results in Section 6.7 and an outlook in Section 6.8 conclude the chapter. This chapter is partially based on [Caimi et al., 2009a]. 6.1 Problem formulation The micro scheduling problem in condensation zones corresponds to the respective boxes in Figure 5.5, which describe the complete procedure for the micro scheduling problem over the entire network. A flexible macroscopic timetable m is given, and the first step of micro scheduling is to solve all condensation zones in parallel, with the goal of finding a microscopic schedule fulfilling the requirements given by the macro schedule. Problem 6.1 (Micro train scheduling in a condensation zone) Given are the following inputs: Microscopic topology for a condensation zone Ci cond portals; = (V,E,,R,σ,c), with its Microscopic train dynamics properties for each train type z Z, according to Assumption 3.16, as well as each train length l(z);

154 6.1 Problem formulation 127 A flexible macro timetable m, with a time slot at the entry/exit portal for each train z running through the condensation zone Ci cond, as well as Flexboxes with their respective flexibility value; The original periodic service intention G, in particular connection and time dependency requirements at the main station. The micro scheduling problem in the condensation zone then consists of finding a (local) conflict-free train schedule s i fulfilling the requirements of the input data, respecting the scheduling policy in the condensation zone (Policies 5.6, 6.3, and 6.4), and optimising a given objective function, see Section Note that the problem formulation requires a proof in case of infeasibility, in order to provide a feedback to the macro level, as illustrated in Figure This means that the problem must be addressed with exact methods, and a heuristic approach is not satisfactory. Thus, a model is proposed that leads to the formulation of an integer linear program, which can then be solved by a commercial solver. This makes it possible to provide a guaranteed optimal solution or a certificate for the infeasibility of the problem. The main goal of micro scheduling in condensation zones is to find a feasible solution, because the maximum speed policy in the zones makes the quality of each route similar from an operational point of view. Nevertheless, the introduction of quality criteria is still interesting, e.g. for maximising buffer times between the trains or for leaving some flexibility in case rescheduling becomes necessary. Some objectives for measuring the quality of the microscopic schedule are discussed in Section For the model developed in this chapter, the following concept of a train is introduced. Trains that travel to their platform in the main station and then continue out of the area are split and represented by two trains, an inbound train and an outbound train, as inbound and outbound routes can be set independently. It must be assured that the assigned platform is the same and that the time slots for each pair of inbound and outbound trains is temporally feasible. This will be ensured by adequate constraints, introduced in Appendix C.2. Trains entering and leaving the zone without stopping at the main station, e.g. freight trains, are modeled without splitting them into inbound and outbound trains. Accordingly, the notion of starting time of a train in a condensation zone is defined as follows. Definition 6.2 (Starting time) The starting time tz 0 of a train z traveling in the considered condensation zone is the passing time at the portal entering the zone for inbound trains and non-split trains or the departure time from the platform toward the exit portal for outbound trains.

155 128 Chapter 6: Microscopic scheduling in condensation zones 6.2 Related work Figure 6.1: Overview on the principal methods in the literature for microscopic scheduling in station areas. The micro scheduling problem for condensation zones, or main station areas, has been addressed in the literature many times. Often this problem is considered with a macro timetable as input. This special case is known in the literature as the train routing problem [Zwaneveld et al., 1996]. Furthermore, if the problem mainly focuses on the assignment of trains to platforms at the main station, with the routing inside the region a less relevant issue, it is also called train platforming problem [Cordeau et al., 1998]. The principal approaches for solving one of these variants are illustrated in Figure 6.1 and described in the following Conflict graph model The conflict graph model (short CG), also called node packing model, has become the standard approach in the literature for solving the micro scheduling (or train routing) problem in main station areas. It was first proposed by [Zwaneveld et al., 1996], with a very similar input to Problem 6.1, yet without flexibility in the given macro timetable. This model is used in the module STATIONS [Zwaneveld and Kroon, 1995], which is responsible for the micro scheduling part of the project DONS (see Section 3.1). The conflict graph model is very intuitive and represents all potential routes, which are computed in a preprocessing step, as vertices in a so-called conflict graph. If it is not possible to assign two routes simultaneously for violation of safety constraints, this pair

156 6.2 Related work 129 Figure 6.2: Example of a conflict graph. Each row represents the route alternatives for a train and at the end of the row the clique between nodes of the same train is illustrated separately for simplifying the understanding of the picture. The black nodes represent a feasible solution. of routes is called conflicting, and the two corresponding vertices are connected with an edge. In addition, all vertices representing different scheduling possibilities of the same train are mutually connected to form a clique. An example of conflict graph is illustrated in Figure 6.2. A conflict-free micro schedule then corresponds to choosing exactly one node per train such that the chosen nodes form a maximum stable set in this conflict graph. Formally, a set Z of n trains is given, each having a set S i = { f i1,..., f im(i) }, i = 1,...,n, of m(i) possible routes connecting its entry and exit portal within a condensation zone and passing through all minor stations where it is supposed to stop. Since the timetable is given, it is possible to calculate non-compatible routes of different trains, i.e., routes that block the some track segment at the same time. Two conflicting routes are denoted f pq f uv. Analogously, f pq f uv means that the routes q and v of trains p and u are compatible. Additionally, all routes of the same train are by definition in conflict, i.e., the corresponding nodes form a clique, since only one route for each train is needed. The conflict set is described by C = {( f ik, f jl ) i = j f ik f jl }. A feasible solution to the train routing problem is a set S of routes f pq such that each train receives a route, i.e., S = n, and the chosen routes are conflict-free, i.e., f pq f uv f pq, f uv S. Typical objectives are the maximisation of the preferences for routes (and also platforms if they are not fixed in advance), as well as the maximisation of buffer times between the routes. An instance of the CG can be visualised as a graph in which the node set is composed of the elements of S := S 1... S n and the edges correspond to the elements of C. The CG has a special structure since all routes of the same train form a clique in the graph. A stable set in this graph, i.e., a set of nodes such that no two chosen vertices are connected by an edge, corresponds to a conflict-free set of routes. Note that if the instance of the

157 130 Chapter 6: Microscopic scheduling in condensation zones routing problem is feasible, i.e., all trains can be routed, then a maximum stable set in the conflict graph includes exactly one node from each train clique. Finding a maximum stable set in an arbitrary graph is known to be NP-hard [Garey and Johnson, 1979]. The computational complexity of the specific structure of the conflict graph model is analysed in [Kroon et al., 1997], where it is shown that the train routing problem is NP-complete with respect to the infrastructure (topology) and the train service intention, i.e., the number of train routes through the station area. More precisely, it is shown that the train routing problem is NP-complete by a reduction from the satisfiability problem (SAT), if each train has at least three different routing possibilities. However, if each train has only two routing possibilities, then the problem can be solved in computation time polynomial in the number of trains. [Zwaneveld et al., 1996] propose to solve the train routing problem using the standard ILP formulation for the stable set problem, where each vertex f ik in the graph is assigned a binary variable x ik and for each conflict edge c C a constraint is inserted that forbids the simultaneous assignment of both variables to 1: x ik + x jl 1 ( f ik, f jl ) C. (6.1) This formulation is slightly improved by considering one fixed vertex and grouping all conflicts of other trains with this single vertex. The resulting ILP (see Formulation B.2 in Appendix B) is then solved with a commercial solver. However, the problem quickly becomes intractable even for moderate conflict graph sizes because of its weak linear relaxation. This general perception is verified by computational results later in Section 6.7. For solving larger instances, [Zwaneveld et al., 1996] propose some heuristic methods for finding a good initial solution, which is then improved by a branch-and-cut approach. [Zwaneveld et al., 2001] propose a pre-processing step for reducing the size of the conflict graph based on graph theoretical considerations. [Herrmann, 2005, Caimi et al., 2005] apply a fixed point iteration heuristic to find a feasible solution and improve its quality using a local search heuristic. In his dissertation, [Zwaneveld, 1997] proposes an extension of the routing problem to also consider some small deviations of planned arrival and departure times. [Caimi et al., 2009e] extends it further to work also for the micro scheduling problem addressed in this thesis, i.e., with a flexible macro timetable as input. These approaches help accelerating the computations, but the model remains hard to solve for large instances, in particular for extended departure time windows. [Billionet, 2003] also formulates an ILP for the train platforming problem based on the conflict graph model. [Caprara et al., 2007a] adopt the same conflict graph approach for the (non-periodic) train platforming problem, which is actually a generalisation of the train routing problem similar to the micro scheduling problem addressed in this thesis, but

158 6.2 Related work 131 where the platforming part has more relevance than the routing part because of the layout of the considered Italian stations. Starting from the conflict graph, the authors improve the conflict constraints by finding the cliques that contain vertices of two trains only, and consider a quadratic objective function which is then linearised. ILP-based heuristics are then applied for solving the problem. [Kroon and Maróti, 2008] aim for a robust solution for the train routing problem. They formulate an integer program with a quadratic objective function and linearise it with a sophisticated method based on a two-phase solution approach. The goal is to spread conflicting train routes in time, as much as possible. Finally, [Delorme, 2003, Delorme et al., 2006] use the conflict graph model for capacity studies at a junction. For this purpose, [Delorme et al., 2004] propose a GRASP (Greedy Randomized Adaptive Search Procedure) meta-heuristic and, alternatively, [Gandibleux et al., 2005] suggests a meta-heuristic based on ant colony optimisation Other approaches [Bourachot, 1986] introduces a quadratic non-convex integer program for solving the train routing problem, whereas [Carey and Carville, 2003] propose a heuristic approach similar to the manual methods currently in use in Great Britain. [Velasquez et al., 2005] proposes a set packing model for the train routing problem and uses column generation and a constraint branching technique to solve it. [Lusby et al., 2006, Lusby, 2006, Lusby, 2008] also model the problem with a set packing approach. They present a solution procedure which entails solving the dual of this formulation through the dynamic addition of violated cuts (primal variables), as well as an efficient pricing routine. The approach based on a set packing model seems very interesting, because it models conflicts by considering all trains simultaneously. The same principle is adopted in this thesis, yet the conflicts are detected much more efficiently. The method is described in Section [Rodriguez, 2007a] presents a job shop scheduling model which is then solved using constraint programming. Using this model, [Rodriguez, 2000, Rodriguez, 2007b] study the use of resources and traffic management for routing and scheduling trains. [Delorme et al., 2001] describes some heuristic methods for infrastructure saturation. [Takagi et al., 2006] proposes a method for optimal train control at a junction in a main line of the railway network using a new object-oriented signalling system model. A similar maximum speed policy to the one proposed in this thesis is applied in the Netherlands for the Schiphol tunnel (the railway tunnel serving the airport of Amsterdam-Schiphol), which is a bottleneck in the Dutch railway network. ProRail, the Dutch infrastructure manager, adopts scheduling policies that are specifically developed for the traffic management in the Schiphol tunnel [Schaafsma, 2001, Schaafsma, 2005,

159 132 Chapter 6: Microscopic scheduling in condensation zones Schaafsma and Bartholomeus, 2007]. The basic idea is to plan on a less detailed level during the offline planning (tactical and operational), and to postpone the resolution of possible conflicts among trains to the online traffic management [D Ariano et al., 2008b]. This way, more freedom is given to the real-time control by relaxing some of the timetable specifications. [Middelkoop and Hemelrijk, 2005] discuss its technical effects, [van den Top, 2005] focuses on the commercial consequences for the passengers, and [Middelkoop and Loeve, 2006] present a simulation tool for analysing traffic management methods. [De Luca Cardillo and Mione, 1998] proposes to solve the train platforming problem using sophisticated graph-coloring models and algorithms. 6.3 Policies for train scheduling With the intensification of the services in condensation zones, dispatchers will face more pressure during operation and will have to respond to disruptions more quickly. In [Herrmann, 2005] it was shown that for a potential future dense service intention, the dispatchers would have to react four times as often as today to resolve conflicts. Thus, the adoption of a new scheduling paradigm in condensation zones that can simplify the dispatchers work becomes necessary, so that the dispatchers have sufficient response time available. The maximum speed Policy 5.6 for scheduling condensation zones is meant for blocking as little capacity as possible in these bottlenecks, but also has the indirect scheduling (and rescheduling) advantage that once the route of a train and one single passing time have been fixed, the speed profile, i.e. all other passing times, can be directly derived, which simplifies the scheduling work. But this might not be sufficient. The policies to apply should enable planning (on all levels) as well as online rescheduling to be realised according to the same principles, according to goal 2.1.1, i.e., the standardisation of the planning and production methods in an integrated system. Moreover, the policies should be sufficiently simple and transparent to allow for manual operation following the same system. This way, a stepwise introduction of the methodology into practice becomes possible, which enables planners and dispatchers to familiarise themselves with the models, increasing acceptance and thereby improving the chances of a successful implementation. According to Observation 5.7, condensation zones can be divided into switch regions and stretches connecting those, and scheduling policies exploiting this structure are introduced. The principal one is based on a time discretisation, and is described in the next Section This scheduling concept was proposed first in [Wüst, 2006, Roos, 2006, Laube et al., 2007, Caimi et al., 2009e]. Another policy is the suitable reduction of the available routes for a train, as specified in Section

160 6.3 Policies for train scheduling Time discretisation The most relevant policy for scheduling trains in a condensation zone is the concept of time discretisation. A parameter τ(ci cond ), called pulse length, is used for time discretisation in a condensation zone Ci cond. This parameter will be simply called τ in the following and can be different for different condensation zones. The entry of a train in the zone at a portal or its departure from a platform will only be possible with a frequency (pulse) of τ seconds. Given a route, the set of available speed profiles for a train thus consists of the same profile simply shifted in time by a multiple of τ seconds. For each train type z and portal P i j of the condensation zone, a so-called phase Φ( z,p i j ) (or shorter Φ zi j ) is assigned, and this value states the exact starting time of one speed profile for each train of this type traveling through this portal. This phase value can be restricted without loss of generality to 0 Φ zi j < τ. This time discretisation policy is illustrated in Figure 6.3. Figure 6.3: Illustration of Policy 6.3. Φ gives the phase of the potential starting times (vertical lines), which are separated by τ seconds. Only the potential starting times included in the macro time slot (filled vertical lines) are feasible starting times and are included in the set T 0 z. Policy 6.3 (Time discretisation in condensation zone) Let τ be the given pulse length and Φ zi j the set of phases for each combination of train type and portal. Furthermore, let a train z have a time slot [π zi,π zi ] for its starting time in the condensation zone Ci cond and traveling through its designated portal P. Then, the time discretisation policy states that train z has a (discrete) set of possible starting times Tz 0 : T 0 z := {Φ zi j + kτ k Z, Φ zi j + kτ [π zi,π zi ]}. (6.2) The feasible starting times for a train are a set of points in the given (macro) time slot with distance τ and phase Φ zi j. If the set Tz 0 is empty, an adaptation of the phase for this train becomes necessary to ensure that at least one feasible starting time exists. Figure 6.4 shows a discretised timetable both in the classical space-time diagram and in a dedicated representation that shows the switch region and corresponding time discretisation, as well as the routes through the micro topology. In this view, each train is

161 134 Chapter 6: Microscopic scheduling in condensation zones Figure 6.4: Representation of a discretised timetable including track occupation (left) versus the classical time-space diagram (right). In the left picture, it is visible the assignment of a time slot for the traveled switch regions (columns) to each train. Source: [Laube et al., 2007]. assigned to a time slot for each traveled switch region (column). Together with information about the phase and the speed profile, exact passing times can be derived, making this representation a complete description of the track paths of each train. In the figure, τ is chosen equal to the headway such that it is possible in the majority of cases to use the same path in the switch region directly in the next time interval. Notice that each choice of the puls length τ > 0 is basically possible and that a very small value of τ (e.g., τ = 1 s) essentially matches a standard schedule without this restriction. With the time discretisation, the dispatchers gain better comprehensibility of the schedule and can therefore quickly decide in case of delays. However, an appropriate choice of the puls length τ and the values Φ zi j for the phases is crucial to allow for many combination possibilities for scheduling trains and thereby to guarantee a good quality of the discretised timetables. The values of Φ zi j can be chosen to minimise conflicts between the speed profiles of different train and/or directions, whereas the choice of τ should achieve a compromise between large values to avoid the explosion of computation time and small values to minimise the loss of potentially attractive track paths. Simplification of the dispatcher s and planner s work is a criterion that is very difficult to quantify. A potential indicator could be the number of distinct scheduling alternatives for each train during operations, which is approximately proportional to 1/τ. In this work, some different values for τ are tested, and the results shown in Section 6.7 suggest that a choice of τ equal to the headway time provides good performance. For Φ zi j, values provided by practitioners from SBB [Roos, 2006] are used for this case study, while models and algorithms for optimising these vales are presented in [Wörner, 2009].

162 6.3 Policies for train scheduling Route reduction A reduction of the set of allowed routes could be useful to provide a set of alternative routes of moderate size, which is transparent, keeps the variety of routes, and could also be set manually by dispatchers. Most importantly, a reduction helps to reduce the size of the considered model and therefore to accelerate computations. [Zwaneveld et al., 2001] propose to algorithmically delete the dominated nodes of the conflict graph, i.e., nodes whose set of conflict edges is a superset of the conflicts generated by another node. Computations showed that it was possible to reduce the conflict graph size by 90% without losing feasibility of the problem. However, this approach is very time consuming since it does not exploit the characteristics of the railway infrastructure. Furthermore, with this method the deleted routes are not known a priori and can change depending on the given train service intention. Therefore, it is not suitable for manual or semi-automatic train (re-)scheduling based on the same policies. A B s t D E C Switch region F Path AD BD s BD t BE CD CE CF AD BD s BD t BE CD CE CF Conflict Matrix Figure 6.5: Example of a dominated path. The link BD via s does not conflict with the link CE and dominates therefore the link via t. In this section, a different method based on an a priori reduction of the available routes by exploiting properties of the micro topology is proposed. The method applies a twostep reduction procedure to eliminate ineffective and very similar routing possibilities.

163 136 Chapter 6: Microscopic scheduling in condensation zones It enables very fast computation times by applying the following policy to reduce the possible routes. Policy 6.4 (Routes reduction) For each switch region, only one path connecting each pair of entry and exit track of the region is allowed. With this policy, it is possible to derive the exact route directly from the graphical interface illustrated in Figure 6.4. There, only the entry and exit tracks are depicted, and this is, under this policy, sufficient to know the exact path in the switch region. With the application of Policy 6.4, each feasible combination of parallel tracks between switch regions is still considered, and this method maintains therefore the global variety of routes but avoids the consideration of too many alternatives. To choose the most appropriate paths, a symmetric binary conflict matrix is introduced that shows whether two paths in the switch region are in conflict, i.e., whether it is possible to assign both paths simultaneously or not, similar to those described in [Pachl, 2002]. Each row of the matrix corresponds to a path in the switch region, and each entry of the matrix is set to 1 if the two paths are in conflict and otherwise to 0. Figures 6.5, 6.6 and Table 6.1 show three possible conflict matrices, where entries between two paths connecting the same entry and exit points are represented by a dash (corresponding to a 1) to improve readability. For applying Policy 6.4, it is first necessary to define the notion of dominance. Definition 6.5 (Dominated path) A path p 1 is called dominated if there exists a path p 2 with the same entry and exit points and the property that the row in the conflict matrix corresponding to p 1 is component-wise larger than or equal to the row corresponding to p 2. This dominance property can be easily detected. In a first step, each dominated path is removed. Figure 6.5 illustrates an example of a dominated path. Each train route using the path BD t as a sub-route will not be considered, and no trains will be scheduled on this sub-route. Paths that generate exactly the same row in the conflict matrix are equivalent and all but one (e.g. the shortest) can be removed. This first step reduces the number of itineraries to consider while preserving the feasibility of the original train scheduling problem. After deleting dominated paths, a second reduction step is applied where nondominated yet similar paths are removed as well. In some special cases, it could be meaningful to allow more than one path for the same entry and exit tracks of a switch region, e.g. in large switch regions two paths could have significant differences in generated conflicts. Let L αβ 0 be a parameter indicating the number of paths from α to β to remain after the second reduction step has been applied. If L αβ = 0 the route from α to β

164 6.3 Policies for train scheduling 137 D A E B s t F C G Switch region Path AD AE BD BE s BE t BF BG CE CF CG AD AE BD BE s BE t BF BG CE CF CG Conflict Matrix Figure 6.6: Example of non-dominated similar paths. The link BE via s generates other conflicts than the link via t. BE s is preferred as it has one conflict less. is not possible; otherwise a heuristic procedure is followed by taking the path that causes the least number of conflicts (1 in the matrix) with other routes. If more than one path is required, additional paths are chosen such that they offer the highest number of new alternatives, i.e., paths with a maximal number of zeros in the position where previous chosen paths had ones. Table 6.1 illustrates an example of a conflict matrix for an artificial switch region, where A and B are two parallel tracks in one direction, and C, D, and E are three parallel tracks in the other direction. After applying the two-step reduction procedure, dominated and similar routes are removed. From a theoretical point of view, this approach does not guarantee to maintain the feasibility of the problem. By decreasing the considered number of paths connecting the

165 138 Chapter 6: Microscopic scheduling in condensation zones Path AC 1 AC 2 AC 3 AE 1 AE 2 BC BD BE 1 BE 2 AC AC AC AE AE BC BD BE BE Table 6.1: Conflict Matrix. BE 2 is dominated by BE 1, whereas AC 2 is preferred to AC 1 or AC 3 because it causes less conflicts. As second path AC 3 is better than AC 1 since it offers more new alternatives. entry and exit tracks of the switch region, the risk of losing feasibility is increased but the size of the resulting problem and hence the solution time are reduced, which is important particularly for memory reasons in large condensation zones. In the computational experiments (see later see Section 6.7), feasibility was never lost even when Policy 6.4 is systematically applied in dense regions, i.e., only one path per entry and exit point was considered (all L αβ = 1 for feasible connections). 6.4 The Resource Tree Conflict Graph model This section introduces a new model for solving the micro scheduling problem in condensation zones (Problem 6.1). A flexible macro timetable is given, and conflict-free track paths for all trains have to be assigned. Recall that such an assignment is feasible (conflict-free) if each infrastructure resource (Def. 3.12) is blocked by at most one track path at any moment in time [Hansen and Pachl, 2008]. The proposed solution approach is based on a new resource constrained multicommodity flow model, which is formulated as an integer linear program (ILP) and solved with a commercial solver. As a first step, the model for the train routing sub-problem is presented. Section explains how to generate scheduling alternatives for each train separately, how to model them in so-called resource trees, and how a solution for a train would look like in these trees using a multicommodity flow description. In Section the resource trees are augmented to a Resource Tree Conflict Graph (RTCG), where the flows are constrained

166 6.4 The Resource Tree Conflict Graph model 139 by the capacities of infrastructure resources. Section discusses an objective function combining the measure of the schedule quality and the feedback needs in case of infeasibility. This issue is then deepened in Section 6.5. In Section 6.4.4, the introduced model concepts are assembled, and an integer linear programming formulation is derived. Section extends the model for the micro scheduling problem where the input macro timetable has flexible event slots. Finally, Section proposes a simple robustness measure to integrate into the model. This model, yet with another way for generating potential track paths according to the different scheduling policies, is also applied for solving the micro scheduling problem in compensation zones, which is elaborated in the next Chapter Resource trees The number of routes for a train with a given origin and destination can grow exponentially in the number of infrastructure resources [Herrmann, 2005]. To model routes compactly and without redundancies, the notion of a resource tree (short RT) is introduced, which represents all alternative routing possibilities for one train. The basic idea is that for each train a tree structure can be generated with all routes from the given origin to the designated destination, as illustrated in Figure 6.7. Definition 6.6 (Resource tree) Let z be an (inbound or outbound) train traveling from its origin s z V(T) to its destination t z V(T) in the micro topology T of the condensation zone. A resource tree rt z = (V z,e z ) is an acyclic graph that represent all routing possibilities for a train z. It has a special vertex rtz root V z, denoted as the root vertex, representing the origin s z, and all leaves lz i V z correspond to the destination node t z, uniquely identifying the complete path from the root to l i z. Consequently, the number of possible routes for the train z is equal to the number of leaves in its resource tree rt z, and is denoted by m z. Let rt := z Z rt z, called resource tree set, be the set of all considered resource trees for a condensation zone. Similarly, let ˆV := z Z V z and Ê := z Z E z be the set of all vertices resp. edges in the resource tree set. A vertex u V z matches a vertex dvg(u) of the dvg in the micro topology, and the (unique) path from the root vertex to it represents its previous (sub-)route. This vertex u implicitly groups all routes with a common sub-route from s z until this point dvg(u) in the micro topology. Accordingly, an edge e = (u,w) E z represents the edge dvg(e) = (dvg(u),dvg(w) ) in the double vertex graph linking the two corresponding vertices. To construct the RT of a train z intending to drive from an origin s z to a destination t z, a search tree with origin in s z = dvg(rtz root ) is built. This can be done, for instance, by breadth-first search [Knuth, 1997] in the double vertex representation of the micro topology. For every vertex v V(T) reachable from the vertex s z in the double node

167 140 Chapter 6: Microscopic scheduling in condensation zones (a) A double vertex graph with two trains running on it: train 1 drives from 0 to 10 and train 2 drives from 1 to 9. (b) Resource trees representing the routes of the two trains given in Figure 6.7a. Figure 6.7: Resource trees rt 1 = (V 1,E 1 ) and rt 2 = (V 2,E 2 ). The vertex u 0 V 1 is the root vertex rt root 1 and correspond to the node dvg(u 0 ) = 0 in the micro topology. Similarly, many vertices in the RTs correspond to the same vertex in the double vertex graph. These are drawn such that they are at the same level in the figure, with exception of the root and the leaves. For instance, dvg(u i ) = 6 for i = 6,16,25. graph, a child u with dvg(u) = v is appended to the root node rtz root in the resource tree. This step is repeated recursively for each child, and recursion for a branch terminates when one of the following cases occurs: i) the child node does not have any possible successor, i.e., it has reached the end of the considered micro topology; ii) an already visited vertex is reached, i.e., a cycle in the route is found and it does not make sense to continue; iii) the branch contains exactly the node sequence that was forbidden by the route reduction procedure, if this is applied; iv) the designated destination t z is reached. In this case, a feasible leaf l i z V z in the RT was found and the (unique) path from the root to this leaf corresponds to a possible

168 6.4 The Resource Tree Conflict Graph model 141 route for this train. In the cases (i)-(iii) it is not possible to reach the destination with this branch and the branch will be removed from the search tree up to a node in the RT where it is possible to continue with another branch. In the case (iv) a feasible leaf is reached. This branch represents a feasible route for the train. Alternatively to this recursive algorithm, one can first generate a set of all possible routes, e.g., using the algorithm described in [Herrmann, 2005] or taking them from a database, and then merging them into a RT. Remark 6.7 (Input for RT construction) A resource tree can be created from an arbitrary input set of routing alternatives for each train. In a second step, each resource tree is augmented with time information about the train trip along the routes. According to Policy 5.6, it is only necessary to assemble for each route its maximal allowed speed profile. Thus, it is sufficient to assign the passing time of the train to each node in the resource tree. With the spatial and temporal information of an RT, it is possible to compute the blocking times for each used resource. For each edge e = (u,w) E z in a RT, the corresponding resource r(e) R will be blocked. To correctly compute the start and end time of the blocking time interval for this resource in the case that the train z travels through the edge e in the RT, the utilisation of the blocking time theory for the safety system in use [Graffagnino, 2007, Hansen and Pachl, 2008] and the green wave policy 3.14 are necessary. In Appendix C, the details about the computation of the start and the end of the blocking time of each edge of the RT to each resource are presented. The RT, together with all blocking time intervals, contains all the necessary information for checking whether a chosen train schedule is conflict-free. Remark 6.8 (Periodic and non-periodic scheduling) The RT model can deal with periodic and non-periodic scheduling. The only difference in the model is that in non-periodic scheduling absolute times are considered for creating the blocking times, whereas in periodic scheduling all times have to be considered modulo T. A train driving through the micro topology can be represented by a unitary (non separable) network flow through the RT. Since several trains drive through the track topology and are not interchangeable, each train is assigned a unique, distinct, and distinguishable commodity. Consequently, each resource tree rt z is augmented by adding a source (s z V z ) and a sink (t z V z ) node for each commodity (train) z. Each source node of a commodity is connected with the root node rtz root of the corresponding RT by an edge. Similarly, each sink node of a commodity is connected to all the leaves (lz) i m z i=1 of the corresponding RT, as illustrated in Figure 6.8.

169 142 Chapter 6: Microscopic scheduling in condensation zones Figure 6.8: Flow commodities are introduced in the resource trees by adding source and sink nodes. The chosen route of a train is represented by a path through the resource tree and is indicated by indicator variables. As each train has to drive from its origin to its destination in one piece, each commodity has to flow from its source to its sink on a unique nonseparated unitary flow path. The chosen path is indicated by assigning binary decision variables x e to each edge e Ê in the RT, where x e = 1 if the corresponding edge belongs to the flow path, and x e = 0 otherwise Constraining flows: the Resource Tree Conflict Graph According to the blocking time theory, a resource cannot be blocked by more than one train route at any time in a conflict-free train schedule. Hence, the use of any resource at every time point must be restricted to at most one route. In [Velasquez et al., 2005, Lusby et al., 2006, Lusby, 2008], the time axis was discretised and constraints (in the ILP formulation) were added to force that every resources is used at most once at each time step. However, this approach has two significant drawbacks: On one hand, the discretisation might overlook conflicts if they occur between two discretised points. A very fine time discretisation, however, is very time consuming and generates a very large amount of constraints that have to be dealt with eventually by the ILP solver. On the other hand, most of the generated constraints are not necessary: either because no trains or only one train intends to allocate the resource (in this case no constraints are needed), or the constraints are redundant because the same conflict group occurs repeatedly over time.

170 6.4 The Resource Tree Conflict Graph model 143 Instead, a more compact representation is proposed in this thesis, where the allocations are constrained at a finite set of time points with the equivalent effect, but without discretising the time axis and without repetitions of the same conflict group in a resource. To this end, all potentially conflicting groups of resource assignments, the conflict cliques, are detected algorithmically. This model, where the flows are constrained by conflict cliques, is called Resource Tree Conflict Graph (RTCG). The RTCG can be represented graphically by connecting each node u of a RT with the corresponding resource r(u), as illustrated in Figure 6.9. Indeed, beside the representation of the RT as sequence of nodes of the micro topology, each RT can be also seen as a sequence of resources used by the corresponding path. Because of its tree structure, each vertex u V z \ rtz root has a unique incoming edge e = (w,u) E z, and the information related to edge e can therefore be associated with the corresponding vertex u. This allocation edge (u,r(u)) contains information about the blocking time of the resource r(u) by all sub-paths in the RT represented by the node u, and receives the same color in the figure as all other edges that intend to block the same resource at the same time. Each resource r R is in this way connected with all (sub-)paths in the RTs using it, and it is only necessary to consider these allocation edges to guarantee the schedule to be conflict-free. Figure 6.9: The Resource Tree Conflict Graph for the example introduced in Figure 6.7. Allocation edges of the same colour represent edges belonging to the same conflict clique I C r. The paths displayed as sequence of yellow nodes represent a possible conflict-free train schedule.

171 144 Chapter 6: Microscopic scheduling in condensation zones Figure 6.10: Allocation schema for one resource: the allocation of each track path is represented by an horizontal line. In order to correctly model the conflicts, the notion of allocation is formally introduced. Definition 6.9 (Allocation) An allocation a r by a track (sub-)path of train z Z represented by u V z contains information about the start and the end of the blocking time interval a r (u) := [bs r (u),be r (u)], (6.3) where bs r (u) and be r (u) are the start resp. the end of the blocking time interval of resource r caused by this track (sub-)path of train z. The allocation schema AS r is now defined as the union of all allocations for the same resource r. Definition 6.10 (Allocation schema) An allocation schema (short AS) AS r for resource r R is the collection of all possible allocations a r, i.e., AS r := a r (u). (6.4) z u V z :r(u)=r An allocation schema can be seen as a (graphical) representation of the resource blocking intervals by the potential track paths over time. The allocations of the different track paths are charted by a horizontal line with the start of the blocking time as the offset and the blocking duration as the length of the line. In periodic scheduling, time is considered as cyclic. For simplicity, the period length is anyways considered as a time interval between 0 and T. Therefore, allocation intervals could overlap the period length and continue at the beginning of the interval. In this case, allocations are split into two intervals by keeping track of the information that they belong to the same allocation. The first line starts at time 0 and ends at the original end time of the allocation modulo the period length. The second line starts at the original start time and ends at the period length T. An example of an allocation schema is given in Figure Two allocations are conflicting if they simultaneously require a resource. Their corresponding horizontal lines would intersect when projected to the time axis. These overlapping lines in the AS induce an intersection graph, where each vertex corresponds to one

172 6.4 The Resource Tree Conflict Graph model 145 allocation interval in the AS. Two nodes are then connected by an edge if the two corresponding allocations are conflicting, i.e., the horizontal lines intersect in the projection onto the time axis (see Figure B.1). Since a conflict in the AS is mapped to an edge in the intersection graph, no adjacent nodes in the intersection graph would be allowed in a feasible set of allocations. Therefore, every conflict-free set of blocking intervals in the allocation schemas corresponds to a stable (or independent) set in the corresponding intersection graph, and vice versa. Consequently, it is interesting to study the structure of stable sets in intersection graphs, and how to describe them. Appendix B presents an analysis and a review on the structure of intersection graphs for both cases of periodic and non-periodic scheduling. The problem of finding a stable set in an arbitrary graph can be formulated as an integer linear program, where for each edge the simultaneous assignment of both vertices to the stable set is forbidden. The ILP formulated using these constraints is in general not naturally integer, and its linear relaxation might be far away from an integer solution. This motivates one to look for stronger ILP formulations, which could significantly speed up the computation time to find an optimal solution. The generation of such a formulation, however, should not be too time consuming, otherwise the computational advantage of the strong formulation is already lost. Thus, it is important to search for the most efficient ILP formulation for the stable set problem that is applicable in practice. In this thesis, this compromise is achieved by looking for maximal cliques in the intersection graph derived from the RTCG model. Algorithm 2, which operates on the AS, is proposed for finding conflict cliques to generate constraints for an ILP formulation with a strong linear relaxation. The algorithm is quite intuitive and consists of two major steps: (1) sorting the start and end times of the blocking time intervals according to the (noncyclic) time; (2) walking through the times, keeping track of the currently open intervals in a list and constructing the set of conflict cliques among the currently open intervals at the end times. It is not necessary to form a clique at each end time, but only at end times where a new interval has been opened since the last iteration. Additionally, an open interval set containing one single interval will not be added because it would not be a necessary constraint. As an example, Figure 6.11 illustrates the result of the algorithm applied to an allocation schema. It is sufficient to consider the end times for the formation of conflict cliques, since at that point in time the blocking time interval is closed and the invariant that all concurrencies between already covered intervals are detected, is conserved. In Appendix B, the

173 146 Chapter 6: Microscopic scheduling in condensation zones Algorithm 2 Creation of conflict cliques for a resource r. Require: An allocation schema AS r for a resource r, consisting of allocation intervals { ar (u) = [bs r,be r ] } i=1,...,n Ensure: Set C r of conflict cliques 1: Create a list L of the 2n tuples (time R +, identifier {1,...,n}, Boolean is endtime) 2: Sort L according to the first key time 3: Initialise list of currently open intervals O := /0 4: Initialise new starttime := FALSE 5: for i = 1 to 2n do 6: if is endtime i = FALSE then 7: O := O {identifier i } 8: new starttime := T RUE 9: else 10: if (new starttime = T RUE)&&( O > 1) then 11: C r := C r O 12: new starttime := FALSE 13: end if 14: O = O \ {identifier i } 15: end if 16: end for 17: return C r proposed algorithm is studied for its behaviour of finding all maximal cliques. The result is well-known for the non-periodic case but new for the periodic setting. Theorem 6.11 (Finding maximal cliques) In the non-periodic setting, Algorithm 2 finds all maximal conflict cliques. For the periodic case, it might occur that the algorithm doesn t find all maximal cliques. However, with an additional assumption that will practically always be fulfilled in practice, the algorithm is also able to find all maximal cliques. The proof of this theorem is given in Appendix B.3 for the non-periodic case and Appendix B.4 for the periodic case. Remark 6.12 (Algorithm complexity) Algorithm 2 runs in O(nlogn) time, where n is the number of allocation intervals. In a first step the blocking intervals have to be sorted according to time, which takes O(n log n) time, see e.g. [Ottmann and Widmayer, 2002]. In a second step, a loop is executed 2n times, while each iteration takes constant time, which results in a running time complexity of O(n) for the loop and O(nlogn) for the whole algorithm.

174 6.4 The Resource Tree Conflict Graph model 147 Figure 6.11: Several track paths blocks a resource during different time intervals (horizontal bars). The dashed vertical bars illustrate conflict cliques among the blocking intervals formed by Algorithm 2. The restrictions to conflict-free allocations at each resource have to hold simultaneously over all resources. Formally, the intersection of the stable set polytopes (or the formulation by conflict cliques found by Algorithm 2) for each resource r R contains the incidence vectors of feasible simultaneous allocations of resources by trains in the network. With the described computation of conflicts, the resource trees are extended to a Resource Tree Conflict Graph where the flows of the trains are constrained for ensuring that a resource is never blocked by more than one train simultaneously. Definition 6.13 (Resource Tree Conflict Graph) Let rt = z rt z be the set of resource trees for all considered trains z. For each resource r R in the condensation zone, a resource vertex v r is introduced. Let R := (v r ) r R be the set of all these resource vertices. For each resource r, Algorithm 2 delivers a set C r of conflict cliques O ˆV. For each node u V z and each conflict clique O C r with u O, an allocation edge c = (u,r) is introduced, indicating with the set O c O C r to which conflict cliques it belongs: O c := {O C r c = (u,r), u O}. Let A := (u,r) u rt z,r R be defined as the set of all the allocation edges. Then, the triple RTCG := (rt,r,a) is called Resource Tree Conflict Graph (short RTCG). In a feasible solution of the RTCG model, i.e. a conflict-free schedule, for each conflict clique I C r at most one train z can use a vertex u I for its unitary flow from the source

175 148 Chapter 6: Microscopic scheduling in condensation zones to the destination. An illustrative example of a RTCG is given in Figure 6.9. The resource trees are extended by additional resource vertices v r (the squares in the figure) and sets of coloured allocation edges c = (u,r). In this simple RTCG, one can find a feasible solution by hand, e.g., the yellow nodes are one possible conflict-free solution since they are never connected by conflict edges of the same colour Objective function To formulate the micro scheduling problem as an optimisation problem, only the objective function is still missing. The goal is to schedule all input trains. Each train is represented by a unitary flow commodity, and the primary objective is therefore the maximisation of the total source output flow, constrained to at most one unit per resource tree, which corresponds to maximising the number of scheduled trains. In other words, the sum of the decision variables associated to the outgoing edges of the sources in the RTCG is maximised. As a subordinate objective, one could think of assigning a value measuring the quality of the corresponding route, for instance the time needed to travel through the condensation zone, the total length of the routes, or energy consumption. The sum of these values will be then minimised in order to assign the best set of routes to the trains. It is possible to consider both the primary and the subordinate objectives simultaneously and to incorporate a corresponding lexicographic ordering by scaling the values such that each solution that schedules s trains has a better objective value than any solution scheduling at most s 1 trains. For evaluating an entire route, weights are assigned only to edges which are directly connected to the sink node: at this node the route is completely known and a value describing its quality can be associated with this edge. All other edges are assigned a weight equal to zero. For each route l of train z, let the value q zl give a measure of its quality (to be maximised). Let q z be the smallest value and Q z the largest for train z. Let δ z := Q z q z be the maximal difference. Then all edges connected to the sink get the weight w zl := (q zl q z )+ Z k=1 δ k. (6.5) The worst possible solution scheduling s trains has an objective value of s n k=1 δ k, whereas the best one scheduling s 1 trains (in the set S ) has a value of (s 1) n k=1 δ k + k S δ k, which is always worse. It is also possible to consider a weighted version, where each train has a given priority and the goal is the maximisation of the weighted sum over the trains.

176 6.4 The Resource Tree Conflict Graph model ILP Formulation In the RTCG model, a train route is sought for each train is given, corresponding to a (non-separable) unit of flow through the RT, such the chosen set of routes is conflictfree with respect to the conflict cliques and the given objective function is maximized. This problem shall now be formulated as integer linear program based on the described multi-commodity flow problem. In addition to the flow conservation constraints, the flow has to avoid conflicts in each resource. To formulate these constraints in the ILP, the blocking time interval is mapped to the unique decision variable of the incoming edge of that node. Subsequently, the sum of the decision variables for each conflict clique is limited to 1 and thereby guarantees the solution to be conflict-free. Problem 6.14 (RTCG, ILP formulation) Given is the set of route weights w zl for each route l of train z, the set Ê of all edges in the RT, the set S z of edges connected to the source node of train z, the set T z of edges connected to the sink node of train z, the family C r of conflict cliques for the infrastructure resource r and the number of trains Z. The corresponding multi-commodity flow problem, formulated as an ILP, is then: max Z z=1 e T z w zl x e (6.6) s.t. x e 1 e S z z = 1,..., Z (6.7) x e 1 x e T z z = 1,..., Z (6.8) x e = x f z Z, e / f :e, f adjacent Z z=1 T z (6.9) x e 1 O C r, r (6.10) e=(u,w): w O x e {0,1} e Ê (6.11) This ILP maximises an objective function with the primarily goal of scheduling a maximal number of trains and the subordinate goal of optimising the quality of the train routes. Inequalities (6.7) and (6.8) constrain the induced flow at a source and the output flow at the sink to at most one, while (6.9) are the flow conservation constraints. For each node in a resource tree, the amount of in-flow x e should be equal to the amount of out-flow ( x f with e and f adjacent edges), which can only be equal to 1 (train uses this node) or 0 (node not used in the solution). Inequalities (6.10) are the conflict constraints according

177 150 Chapter 6: Microscopic scheduling in condensation zones to the gathered set C r of allocation cliques by Algorithm 2, and Eq. (6.11) restrict the flow variables to binary values Extension to departure time slots So far, it was assumed in the model that each train has an exact starting time given by a macroscopic draft timetable. In Chapter 4, an approach for generating a macro timetable with times slots instead of fixed times was presented. By applying the time discretisation Policy 6.3, it is possible to construct a finite set of possible starting times for each input time slot. This additional degree of freedom can be easily integrated into the RTCG model. [Zwaneveld et al., 1996] also propose a similar idea by allowing a discrete set of minor deviations from the scheduled starting times in the conflict graph model. As a consequence, each train z is associated with several resource trees one for each possible starting time (see Figure 6.12). The RTCG is extended by including these additional resource trees, connecting them with their corresponding sink and source, and calculating the conflict cliques in the same way as Algorithm 2 describes. The structure of the resulting associated ILP remains exactly the same, yet the problem size grows linearly with the granularity of the discretisation. Figure 6.12: A departure time slot [π i,π i ] of the macroscopic timetable is discretised according to Policy 6.3 and for each discrete starting time a resource tree is built and hooked into the corresponding source and sink Robustness measure An important property of a train schedule is its robustness. A train schedule is robust if it can absorb small disturbances in real-time operations [Hansen and Pachl, 2008, p. 142ff.]. A simple measure of robustness can be easily introduced into the model for evaluating the static robustness of the schedule, i.e., the amount of delay that can be tolerated without changing anything in the plan. This kind of robustness alone is

178 6.5 Dealing with infeasibility 151 by far not sufficient, but it is still useful in order to not force the dispatcher to react to very small disturbances. The introduction of similar robustness measures during algorithmic train scheduling on the planning stage has been also proposed by [Herrmann, 2005, Caimi et al., 2005, Fischetti et al., 2009], whereas other static measures are proposed by [Liebchen and Stiller, 2009, Liebchen et al., 2009, Caprara et al., 2009]. For measuring schedule robustness, the maximal time deviation that one single train may have (while all the other trains run as scheduled) is considered, such that it can be guaranteed that no conflict occurs. Recall that no conflict will occur if each blocking time interval of the deviating train has no overlaps with other allocation intervals of other scheduled routes. For this purpose, each blocking time interval is artificially extended by a value η. If a schedule with this value of η is found, each train can be delayed until a maximal value of η without affecting other trains. To find the maximal value of η, i.e., the schedule with the maximal static robustness, a binary search algorithm can be applied, as described in Algorithm 3. The optimal value of η guarantees that a delay up to η seconds does not create conflicts, but for some trains this value can also be greater. This lower bound is only tight for at least one train, therefore it is not possible to further increase the value η. Actually, a maximin problem is solved, where the minimal static delay tolerance over all trains is maximised. 6.5 Dealing with infeasibility The ILP formulation (6.6)-(6.11) is technically always feasible (e.g. x = 0 is a feasible solution), because it maximises the number of scheduled trains as primary objective and it does not force any train to be scheduled under hard constraints. However, if the optimal solution of the ILP does not schedule all trains under the assumed policies, it means that Problem 6.1 is infeasible, i.e., there is no conflict-free micro schedule in the considered condensation zone fulfilling the requirements of the input macro timetable. The ILP formulation 6.14 has the advantage that whenever not all trains can be scheduled, it is still possible to provide a schedule where the number of scheduled trains is maximised instead of just giving an infeasibility response. Using larger weights in the objective function, it is possible to schedule first the most important trains in case it is not possible to schedule all. This result could be used as a possible feedback to the macro level to generate another macro timetable with higher chance of being feasible on the micro level. However, this kind of feedback has the drawback that it does not provide any information about the trains that are not scheduled in the optimal solution of the ILP formulation. It is difficult, on the macro level, to generate another timetable with time slots for the trains so that it the chance of feasibility on the micro level in increased. Therefore, a different approach is proposed here, which allows trains to be scheduled outside the time

179 152 Chapter 6: Microscopic scheduling in condensation zones Algorithm 3 Binary search for maximising the static robustness of the schedule. Require: Start tolerance value ξ, accuracy ε, input of Problem 6.1 Ensure: A microscopic train schedule with a maximal static robustness η 1: Solve Problem 6.1 by applying the ILP formulation Define κ as the number of scheduled trains 2: Extend all blocking time intervals of the value ξ 3: Set last in f easible :=, last f easible := 0, current tolerance := ξ 4: while last in f easible last f easible < ε do 5: Extend all original blocking time intervals of the value current tolerance 6: Solve Problem 6.1 by applying the ILP formulation : if κ trains are scheduled then 8: S := x {best feasible schedule} 9: last f easible := current tolerance 10: if last in f easible = then 11: current tolerance = 2 current tolerance 12: else 13: current tolerance = 2 1 (last in f easible last f easible) 14: end if 15: else 16: last in f easible := current tolerance 17: current tolerance = 2 1 (last in f easible last f easible) 18: end if 19: end while 20: return S is the schedule with maximal static robustness η := current tolerance under accuracy ε slot given by the macro timetable, if it is not possible otherwise. Formally, let Tz 0 be the set of possible starting times inside the macro time slot [π zi,π zi ] for train z in the condensation zone Ci cond and traveling through the portal P i j. If the train gets assigned one starting time in Tz 0, it is feasible from a macro point of view and it corresponds to the case that the train is scheduled in Formulation On the other hand, if it is not possible to schedule train z in the given macro time slot, it is possible that the train can be scheduled with another starting time outside but close to the input time slot. This way, the resulting conflict-free schedule does not fulfil the macro requirements and is therefore infeasible. Nevertheless, such a schedule provides useful information to the macro level about how far the train is from its given time slots. This schedule can be seen as a suggestion to the macro level, how these time slots could be changed (enlarged or moved) to have a conflict-free schedule for this local part of the network. If it is

180 6.5 Dealing with infeasibility 153 Figure 6.13: Extended set of starting times. The green vertical lines are the feasible starting times Tz 0 of the train in the time slot of the flexible macro timetable and discretised according to Policy 6.3. The red vertical lines represent additional starting times T z 0 \ Tz 0 that are feasible according to the input psi but not feasible according to the current macro schedule, and will be therefore penalised in the objective function of the ILP formulation possible on the macro level to provide time slots that include all these assigned starting times (formerly still infeasible), then the micro schedule for this zone is guaranteed to be conflict-free. Even if this is not possible, this information could be used for generating a different timetable with more suitable time slots for finding a feasible solution. It does not make sense, however, to schedule a train outside of the given input time slot [ωk,ω+ k ] of the psi. If it is only possible to find schedules with at least one train with a starting time outside this time slot, it immediately means that the input psi is infeasible, i.e., a feedback loop to the macro level is meaningless. Therefore, an extension of the possible starting times for each train is proposed, which includes the time created according to Policy 6.3 in the time slot [ωk,ω+ k ]. Let the set of these points T z 0 be defined as T z 0 := {Φ zi j + kτ k Z, Φ zi j + kτ [ωk,ω+ k ]}. (6.12) Figure 6.13 illustrates this extension of the possible starting times to T z 0. As the primary objective remains the search for starting times in Tz 0 and starting times in T z 0 \Tz 0 are only taken into account as a second option, the objective values in the ILP formulation are accordingly chosen. For each starting time tz 0 Tz 0, all edges e connecting a leaf lz t0 associated with this starting time to the sink s z (i.e., all track paths l with starting time tz 0 ) get a weight q zl := 0 in the ILP formulation. For the starting times tz 0 T z 0 \ Tz 0 that are not in the macro time slot, the associated quality measure is set to its distance to the feasible time slot, meant as a penalty for not being in the macro time slot. Summarising, the quality measure q zl is defined as 0 if tz 0 Tz 0 q zl :=. (6.13) min{ t z 0 π zi, tz 0 π zi } if tz 0 / T 0 The ILP formulation is also adapted to provide a useful feedback in case of infeasibility. z

181 154 Chapter 6: Microscopic scheduling in condensation zones Problem 6.15 (RTCG, extended ILP formulation) For each train train z, the resource tree rt z is given, as well as the set of starting times T 0 z. Let w zl be the set of track path weights, computed according to Eq. 6.5 using the values for q zl of Eq Furthermore, given are the set Ê of all edges in rt, the set S z of edges connected to the source node of train z, the set T z of edges connected to the sink node of train z, the family C r of conflict cliques for the infrastructure resource r and the number of trains Z. The adapted multi-commodity flow problem, formulated as ILP, is the following: max Z z=1 e T z w zl x e (6.14) s.t. x e 1 e S z z = 1,..., Z (6.15) x e 1 e T z z = 1,..., Z (6.16) x e = x f z Z, e / f :e, f adjacent Z z=1 T z (6.17) x e 1 O C r, r (6.18) e=(u,w): w O x e {0,1} e Ê (6.19) This extended ILP has a similar structure as Formulation The only differences are the sets of available starting times ( T z 0 instead of Tz 0 ) and the quality measure q zl of its routes, which are defined here according to their starting times. The objective function of this formulation has the primary goal of scheduling a maximal number of trains and the subordinate goal of minimising the starting time penalty. As a consequence, if there is a solution that schedules all trains inside their macro time slots, this will yield a penalty value equal to 0 and is therefore an optimal solution. Otherwise, the optimal solution will schedule the maximal number of trains (primarily objective) with a minimal deviation from the macro time slots (subordinate objective). For instance, a solution where all trains are scheduled with a starting time in T z 0 \ Tz 0 has a better objective function and is therefore preferred to a solution where all trains except one gets a starting time in Tz 0 and one is not scheduled at all. If it is additionally of interest to still consider the quality of a route also in Formulation 6.15, this measure can be added as a subordinate objective of second order, i.e., Eq. 6.5 is applied twice for generating the weights for the ILP. Formulation 6.15 can become significantly larger than Formulation 6.14, because possible starting times have to be generated not only for the provided time slot from the macro level, but also for the entire time slots from the ppsi, which are usually as large

182 6.6 Variants of the RTCG model 155 as minutes or even larger compared to the usual macro time slots of 2-5 minutes. Computation times for both constructing the model and solving it grow but remain in the worst case around 1-2 minutes for the larger computed instances in Lucerne and Berne, see also results later in Section Variants of the RTCG model In this section two variants of the already introduced RTCG model are presented and discussed. Section introduces the tree conflict graph model, which considers the tree structure as in the RTCG but without grouping the conflicts in cliques. Section briefly explains the resource conflict graph model, which is the opposite consideration. It takes into account the grouping of the conflicting allocations into cliques, but without representing the routes with a tree structure Tree Conflict Graph The Tree Conflict Graph model (short TCG), pictured in Figure 6.14, is based on a multicommodity flow through resource trees, like in the RTCG model, with a localised conflict description in the RT node where the conflict actually occurs. It was first introduced by [Herrmann and Caimi, 2006]. Similar to the conflict graph model, the flow is restricted by interference edges between two vertices of two different RTs (pairwise consideration), as opposed to the RTCG where the flow is restricted by conflict cliques. The TCG model can also be formulated as an ILP, where the flow conservation constraints are described as in the RTCG model and the (pairwise) conflict constraints as in the CG model but assigned to the binary value corresponding to the exact location in the RT. This ILP formulation is equivalent to the formulation of the RTCG and the CG model. Its LP relaxation is guaranteed stronger than (or equal to) the CG formulation, because the localised pairwise conflicts correspond to a clique between the leaves in the two sub-trees of the RT. On the other hand, it is weaker than (or equal to) the RTCG formulation because of the pairwise considered conflicts instead of the cliques of all simultaneous allocations. This formulation is therefore only interesting from a didactical point of view, in order to better understand whether the improvements of the RTCG compared to the CG model are mainly due to the tree structure or due to the grouping of the simultaneous allocations into cliques. Section 6.7 therefore also contains computational results for the TCG model Resource Conflict Graph The Resource Conflict Graph model (short RCG) is also a combination of the CG and the RTCG model. The RCG is a graph G = (V R,E), in which each node v z l V corresponds

183 156 Chapter 6: Microscopic scheduling in condensation zones Figure 6.14: A Tree Conflict Graph where the multi-commodity flow is restricted by interference edges (red dashed lines). to a possible track path l of train z, and a node v r R corresponds to a resource r R. Edges in this graph correspond to resource allocations and always connect a node in V with a node in R, which makes the graph bipartite. Each assignment (track path) v z l is connected with all resource nodes that are used by this track path. Its exact blocking time interval for the resource r is calculated in the same way as for the RTCG model. For each resource r, Algorithm 2 can be applied for finding the conflict cliques. The gathering of the conflicting allocation is therefore made in the same way as in the RTCG model, but these allocations are directly assigned to a vertex representing a complete track path alternative, like in the CG model. Figure 6.15 shows an example of a RCG graph. The ILP formulation of the RCG model uses a binary variable x z l for each potential track path, like in the CG model, and a conflict constraint for each group of mutually conflicting track paths, like in the RTCG model. It is formulated as follows: Problem 6.16 (RCG, ILP formulation) max f(x) (6.20) s.t. xl = 1 l z Z (6.21) (l,z) O xl z 1 O Cr, r (6.22) x z l {0,1} (6.23) This ILP maximises the objective function (6.20), which can be defined similarly to

184 6.6 Variants of the RTCG model 157 Figure 6.15: Example of a Resource Conflict Graph. Edges of the same colour represent edges belonging to the same conflict clique I C r. The yellow nodes display a possible feasible solution. the objective function of the RTCG model (Section 6.4.3). Constraints (6.21) enforce the assignment of exactly one track path for each train. Constraints (6.22) enforce to assign at most one track path for each conflict clique O, where C r is the set of all conflict cliques for resource r. Finally, constraints (6.23) restrict the x-variables to binary values. The LP relaxation of ILP formulation 6.16 is stronger than (or equal to) the CG formulation because of the conflict clique constraints. On the other hand, the value of its LP relaxation is the same as for the RTCG formulation. This can be seen by taking an RTCG and using the flow conservation constraints 6.9 (or 6.17) for substituting variables until only the variables corresponding to leaves in the RT remain. At the end the formulation looks exactly like the RCG formulation. However, the RTCG model has the advantage compared to the RCG that these localised conflicts can be used by the commercial solver for creating stronger and more sophisticated cuts or by branching on a sub-tree in the branch-and-cut algorithm, which can lead to shorter computation times. On the other hand, the RCG model needs significantly less memory than the equivalent RTCG. It is therefore well suited for very large instances, where the RTCG could face memory problems. In this thesis, the RCG model is used for the micro scheduling problem in compensation zones described in Chapter 7, where the large amount of generated speed profiles for some instances made it impossible to use the RTCG model. Computational results are presented in Section 7.5.

185 158 Chapter 6: Microscopic scheduling in condensation zones 6.7 Computational results In this section computational results are presented for the condensation zones of Berne and Lucerne in Switzerland, starting from different macroscopic timetables. Results of the RTCG model are compared with the results of the CG and the TCG model. First, an outline of the test scenarios and the test environment in which the calculations took place are introduced in Section Then, Section analyses the scheduling policies in condensation zones, Section presents computational results and compares the different models, Section discusses results for the extended ILP formulation 6.15 and results on robustness are shown in Section Scenarios and implementation The different models are implemented and tested with real data provided by SBB for the condensation zones of Berne and Lucerne. Results for three representative scenarios are presented, which can roughly be classified as small, medium and large. Other scenarios for the condensation zones of Berne and Lucerne were computed and gave qualitatively similar results, which are not described in the thesis. The condensation zone of Lucerne is introduced in Appendix A.1.2. The macro timetable operated in 2006 is considered for the computational study, where 16 trains (8 Intercity trains and 8 local trains) drive through this area in half an hour during peak hours. This scenario is called Lucerne 2006 (the small scenario). The condensation zone of Berne is introduced in Appendix A.2. Two different macro timetables are considered, denoted Berne 2008 and Berne East. Berne 2008 (the large scenario) is a reverse-engineered draft timetable from the operated timetable in It consists of 67 in- or outbound trains (30 InterCity trains, 37 local trains) per hour driving through the condensation zone in peak hours. Berne East (the medium size scenario) is a condensed hypothetical timetable restricted to trains running through the eastern part of Berne, following the structure of the integrated fixed-interval [Lüthi, 2009, Liebchen, 2006] timetable with a periodicity of 15 min. In this scenario, a strict policy is applied where the platform track of each train in the main station is already specified. The existing buffer in the dwell and turn-around times in the main station (at least 1 minute for dwell and much more for turn-around) can be exploited to adjust departure times of trains to align the time discretisation in both sides of the main station. Therefore, it is possible to treat the west and the east side of the station separately, and then merge the so created optimal solutions into an optimal conflict-free schedule for the whole condensation zone. This scenario consists of 27 in- or outbound trains passing the zone within half an hour, and 16 in the most congested quarter of an hour.

186 6.7 Computational results 159 The computations are performed on a x86-64 processor (Dual-Core AMD Opteron R, 32 GB RAM). The code for constructing the models is written in C++ and the generated ILPs are solved with the commercial solver ILOG CPLEX Analysis of the scheduling policies Table 6.2 shows how the size of the conflict graph and the computation time increase by reducing the frequency τ of the time discretisation. Whereas the number of nodes grows linearly, the number of conflict edges and the computation time grow more or less quadratically. The choice of an interval length τ = 90 sec (equal to the headway time) seems to be reasonable because it is very fast while avoiding to waste capacity due to the discretisation. This value is also suitable for manual dispatching. Frequency # # Solution τ [s] nodes edges time [s] Table 6.2: Impact of the interval length τ on problem size and computation time. Results for Berne (West side, condensed hypothetical service intention, reduced routes). By applying the route reduction policy, the number of available routes is reduced approximately by a factor of 10, the number of (pairwise) conflicts by a factor of , and the computation time for solving the scheduling problem is reduced by a factor of compared to the resulting computation time applying only Policy 6.3. Note that in the resulting conflict graph, feasibility has never been lost, even if L K αβ = 1 for all entry/exit points α and β for all switch regions K. Table 6.3 shows the impact of the reduction on the problem size and the computation times on four scenarios in the condensation zones of Berne. For the route reduction the parameter value L K αβ = 1 was chosen for all existing paths αβ and all switch regions K.

187 160 Chapter 6: Microscopic scheduling in condensation zones Scenario Routes # trains # # Solution reduced in 30 min nodes edges time [s] West 2003 no West 2003 yes East 2003 no East 2003 yes Berne East no Berne East yes Berne West no Berne West yes Table 6.3: Impact of the route reduction on conflict graph size and computation time for 4 scenarios in Berne. Timetable without time slots, τ = 90 sec for the condensed timetables (Berne East and West) and τ = 150 sec for the 2003 timetables Computational results and comparison between the models Table 6.4 presents results for the three scenarios of different sizes. One can observe that the number of vertices and edges heavily depend on the model. The TCG and RTCG models are based on resource trees as opposed to the CG model and need more vertices. As the RTCG has additional resource vertices, it uses slightly more vertices than the TCG model. Both TCG and RTCG form a multi-commodity flow through their resource trees and therefore require flow edges. On the contrary, the CG model is based on a stable set approach and does not use flow edges. The differences in the number of conflict edges are explained by the different contexts in each model. In the CG, conflict edges are built between the vertices of two conflicting train routes and between the vertices of routes that belong to the same train. In the TCG, the interference edges are between conflicting resource allocations of two routes belonging to two different trains. In the RTCG, the allocation of a resource is represented by a conflict edge connecting the node in the RT with the corresponding resource node. For small scenarios, like Lucerne 2006, there are no significant differences in the number of conflict edges between the models. The TCG model seems to be slightly preferable. For medium scenarios like Berne East, TCG remains slightly better than the CG model. However, it can be observed that the number of conflict edges in the RTCG is much smaller than for the two other scenarios. This is due to the fact that for each conflict group there

188 6.7 Computational results 161 Scenario # Routes Model # Vertices # Conflict edges # Flow edges RTCG Lucerne TCG CG RTCG Berne East TCG CG RTCG Berne TCG CG Table 6.4: Statistics of the three models for three scenarios. Scenario Average Minimum Maximum Lucerne Berne East Berne Table 6.5: Conflict clique sizes in the RTCG model for the three scenarios. is a linear number of interference edges in the RTCG model but a quadratic number in the two others scenarios, due to the pairwise consideration of conflicts. For large scenarios like Berne 2008, the RTCG has by far the smallest number of conflict edges. It can be concluded that the denser the scenario, the larger is the benefit of the RTCG approach. Additionally, note that in Berne 2008 the number of conflicts in the TCG is larger than in the CG. This is due to the fact that many conflicts repeated several times, leading to redundancies that are not present in the CG, as all conflicts are represented only once between a pair of routes. However, after having resolved this redundancy, for example after the preprocessing of a commercial solver, one can observe that the number of conflicts in the TCG is smaller than in the CG, see Table 6.6. Table 6.5 displays the clique sizes of the RTCG model for all considered scenarios, which are on average quite large, particularly for dense scenarios. The maximum clique size ca be huge, meaning that for each scenario there is at least one resource and one point in time where many trains and routes could potentially make use of it, making it de facto a bottleneck in the network. Recall again that one single conflict clique replaces a quadratic number of interference edges in the TCG and even more in the CG model. Table 6.6 illustrates the effects of the models on the ILP sizes before and after the pre-

189 162 Chapter 6: Microscopic scheduling in condensation zones Scenario Model Before preprocessing After preprocessing Binary Conflict Binary Nonzero Density variables constraints variables constraints entries [%] RTCG Berne West TCG CG RTCG Lucerne 2006 TCG CG RTCG Berne East TCG CG RTCG Berne 2008 TCG CG Table 6.6: Statistics of the three different ILP formulations for the three scenarios, before and after the (standard) preprocessing of CPLEX. Berne West is an additional small scenario, with the same properties of Berne East for the western part of the condensation zone of Berne. processing step of the ILP solver CPLEX. The RTCG model has the fewest constraints, with beneficial effects on the memory requirements. This difference in the number of constraints needed by the models grows with the density of the models. As already observed, the TCG can group some constraints, resulting in a reduction compared to the CG, but it has some redundancy, thus also generating more constraints than the CG model. However, this redundancy can be efficiently eliminated during preprocessing, which finally leads to less constraints compared to the CG model, even for the largest scenario. Moreover, the effects of the preprocessing step of CPLEX on the problem size for the different models can be analysed. One can see that for the (small) scenario in Lucerne the preprocessing is very efficient for all models, leading to very small ILPs. For both Berne scenarios, the preprocessing is particularly efficient for the TCG model, where redundancy can be eliminated and both the number of variables and constraints can be drastically reduced. In contrast, preprocessing for the CG model is very weak, which indicates that the model lacks structure. The RTCG is already very effective in describing the conflicts so that the effect of preprocessing remains limited. The number of variables, however, is reduced considerably, probably by resolving many flow constraints of the ILP formulation.

190 6.7 Computational results 163 Scenario Model Model ILP Pre- Root- Branch & Total creation creation processing relaxation Cut time RTCG <<1 <<1 <<1 <<1 <<1 <<1 Berne West TCG <<1 <<1 <<1 <<1 <<1 <<1 CG <<1 <<1 <<1 <<1 <<1 1 RTCG <<1 <<1 <<1 <<1 <<1 <<1 Lucerne TCG 3 <<1 <<1 <<1 1 4 CG <<1 <<1 <<1 <<1 <<1 1 RTCG << Berne East TCG CG RTCG << Berne 2008 TCG CG Table 6.7: Processing times (in CPU seconds) for the various computation steps. Table 6.7 presents CPU times for the different computation steps from model creation up to computing the solution for the primary objective of maximising the number of scheduled trains. For small scenarios, there are basically no differences noticeable, as all models are very fast in all steps. For larger scenarios, one can observe that the RTCG model has enormously beneficial effects in both model creation and solution finding (up to 500 times faster) compared to both other approaches. This table well illustrates the impact of the constraints strength in the RTCG model: the linear relaxation is much tighter with enormous effects on the branch-and-cut procedure applied by the commercial solver, and therefore, ultimately, on the computation time. In the TCG model, the constraints are also helpful for a fast linear relaxation, but the construction is so time consuming that it does not pay off. Additionally, the TCG model is also the largest such that for large scenarios it also runs into memory problems on a typical workstation. Table 6.8 shows the quality of the generated solutions for the three considered scenarios. All models provide the same optimal value, as they are equivalent models. In Lucerne 2006, all models yield an optimal solution almost instantaneously, as the relaxed solution with preprocessing had already the optimal value and in the cases of RTCG and CG it was even directly an integer solution. In Bern East, the RTCG is the only model where the linear relaxation already delivered the optimal objective value, such that CPLEX was able to find an integer optimal solution using heuristics very quickly. For the two other models, some branch-and-bound nodes as well as many cuts were necessary to overcome the inte-

191 164 Chapter 6: Microscopic scheduling in condensation zones Scenario # trains Model Integrality gap [%] B&B before after preprocessing nodes Cuts RTCG Lucerne TCG CG RTCG Berne East 15 TCG CG RTCG Berne TCG CG Table 6.8: Solution quality and steps of the MIP solver. The symbol + after the number in the B&B columns denotes a solution generated by an heuristic process. grality gap of around 7% after the preprocessing phase, leading to quite long computation times. In Berne 2008, the linear relaxation of all models had an integrality gap. However, the gap for the RTCG was much smaller and CPLEX was still able to find the optimal solution during the heuristic pre-solve phase. This procedure failed for both TCG and CG, where branch-and-bound nodes and cuts were necessary to find the optimal solution, leading to longer CPU times. It is also instructive to investigate the difference between the solutions of the optimal integer solution and the pure linear relaxations, i.e., when all preprocessing of CPLEX is switched off. This gives a better intuition for the strengths of the different model foundations. For the small and medium size scenarios, the linear relaxation of the CG model is quite poor. The relaxation for the TCG model could reduce it to almost half, meaning that the structure could be improved significantly, yet remaining far from the natural integrality. Only the RTCG model was able to generate an integer solution directly, demonstrating the very strong structure of the RTCG model. For the large scenario, all models generate an integrality gap even after the CPLEX preprocessing phase. But also in this case, the RTCG model yields a much better linear relaxation compared to both other models Results for the extended ILP The additional freedom given by flexible time slots for the starting times instead of exact times can yield better schedules. This can occur by scheduling more trains, enabling a reduction in the penalty for starting times outside the macro time slot, or assigning track

192 6.7 Computational results 165 # Starting # # Clique size CPU time [s] times nodes constraints Min Average Max Constr. Solving Table 6.9: Growth of CPU time depending on the number of allowed starting times. RTCG model, Bern East scenario. paths of better quality. It cannot generate in any case a worse solution compared to the version with fixed times, as the solution space is enlarged. For example, in the scenario Berne East it was only possible to schedule 15 out of 16 trains using an input macro timetable with fixed starting times. By enlarging the time slot of the starting time of each train to an interval of 7.5 minutes, it became possible to schedule all 16 input trains. In the other two tested scenario it was already possible to schedule all trains with fixed times because the input times were directly reverse-engineered from the operated schedules of SBB. Table 6.9 presents the trade-off between the time slot size and the CPU time needed for solving the problem with the RTCG model. With the increase of the number of possible starting times, the number of vertices in the model grows linearly, as one could expect. Less easy to anticipate is that also the number of conflict constraints of the RTCG seems to grow linearly, whereas this number, for instance, grows quadratically in the CG model. Finally, particularly surprising is the fact that also the CPU time seems to grow linearly with the increase of the number of possible start times, indicating that the structure provided by the RTCG is sufficiently strong to enable efficient computation for realistic scenarios of this size. The CPU time as a function of the number of start times will probably explode at some point due to the complexity of the scheduling problems in main station areas, which was proven to be NP-hard in very similar settings [Kroon et al., 1997]. Fortunately, this point seems to be further away from the problem size that is needed to solve realistic instances of large and dense scenarios Robustness results The three scenarios have been investigated also for their static robustness, i.e., the minimal amount of time that a train can be delayed such that the current solution is still conflict-

193 166 Chapter 6: Microscopic scheduling in condensation zones free. The binary search described in Algorithm 3 was applied, with a starting tolerance value ξ = 30 sec and an accuracy of ε = 1 sec. For Lucerne 2006, an optimal value of η = 44 is obtained, reached by calling Problem times. This result demonstrates the importance of the precision in the driving behaviour of the driver, because the time interval to operate without creating delay propagation or forcing the dispatcher to intervene is less than one minute. For Berne East, an optimal value of η = 9 is obtained, also reached after 7 iterations. Such a results was to be expected, as this scenario was developed to test the limits of the system and is particularly dense and close to the capacity limits. Such a small delay tolerance makes the schedule difficult to operate in practice, as it will be very sensitive to delays and has to rely on an effective rescheduling mechanism controlling the propagation of delays. Finally, in Berne 2008, a schedule with an optimal value of η = 7 can be generated. This result is due to one particularly sensitive train, whereas other trains have larger values of delay tolerance. As with the current train control system it is impossible to conduct trains with this precision, the current timetable in the region Berne often requires the intervention of the dispatchers and breaking the full speed policy, which results in trains stopping in front of a red signal in the main station area. Nevertheless, the good work of the practitioners usually keeps the delays in Berne within an acceptable amount. 6.8 Summary and final remarks In this chapter, an exact method is proposed for solving the micro train scheduling problem in a condensation zone. To this end, the Resource Tree Conflict Graph model is developed, which describes all track path alternatives for each train in a compact way using a tree structure and conflict cliques. Conflicts are efficiently grouped without redundancies for each resource in a way that all potential simultaneous assignments are restricted to at most one. This results in an integer linear programming formulation that can be solved with a commercial solver. The model constitutes an immense improvement compared to previous existing models in the literature based on the conflict graph (e.g. [Zwaneveld et al., 1996, Herrmann, 2005]), where conflicts were described pairwise, leading to less efficient formulations with poor linear relaxations. Computational results show that even very large scenarios can be solved to optimality very quickly, and computation times improve dramatically compared to the classical ILP of the conflict graph approach. In particular, the RTCG model is able to capture the structure of the problem much better and generates much smaller integrality gaps. For the small and medium size scenarios, the relaxed solutions computed with RTCG were already integer so that the optimal solution was found very quickly. It is observed that

194 6.8 Summary and final remarks 167 large instances may face memory problems before longer CPU times become an issue. The RTCG model can also be applied for micro scheduling in a compensation zone. Therefore, Chapter 7 describes how to compute alternative scheduling possibilities following the policy of compensation zones of variable speed profiles, which can be captured in the same tree structure and solved with the same approach as presented in this chapter. Finally, the observed strong structure and the resulting very quick computation times encourage to apply the approach in an online rescheduling setting, where it is necessary to compute an alternative schedule quickly in case of deviations from the scheduled times. First steps in this direction using the RTCG model can be found in [Priewasser, 2009].

195

196 Chapter 7 Microscopic scheduling in compensation zones This chapter focuses on models and algorithms for the micro scheduling problem in compensation zones. According to the procedure for solving the micro scheduling problem proposed in Section 5.4, condensation zones are solved first. Based on their solution, tracks and passing times at the portals are fixed, and with these data as input, all compensation zones are solved in parallel afterwards. In compensation zones, the problem is to find a reliable track path for each train, such that the schedule is conflict-free and the requested boundary conditions at the portals are met. According to the scheduling policy introduced in Section 5.2, time reserves are incorporated in compensation zones to improve timetable stability. Therefore, many speed profiles are possible for each train to connect the fixed passing times at the portals. The choice of an appropriate speed profiles is the most important degree of freedom to exploit in order to obtain a conflict-free schedule. The goal is to find a conflict-free schedule that optimises a given objective function measuring energy consumption and the distribution of time reserves along the trip. To solve the micro scheduling problem in a compensation zone, the zone is first divided into a small number of segments. For each train trip, the total available reserve time is computed and discretised into a number of time units. Speed profiles are then created by the distribution of these time units over the segments, resulting in a variety of alternative speed profiles. Finally, each speed profile receives an objective value. A graph model is applied to find a set of conflict-free track paths that maximise the objective function. By building cliques of track paths which block track resources simultaneously, efficient restrictions for modeling conflicts in the ILP formulation are built, which is then solved with a commercial solver. Results show that track paths can be built and assigned to

197 170 Chapter 7: Microscopic scheduling in compensation zones several trains within less than a minute of computation time. This chapter is organised as follows: In Section 7.1 different approaches and models to generate speed profiles are discussed. In Section 7.2 the problem of conflict-free scheduling in compensation zones is introduced formally and the proposed solution method is described. The generation of speed profiles is discussed in Section 7.3, and the applied optimisation model is explained in Section 7.4. Computational results in Section 7.5 and some final remarks in Section 7.6 conclude the chapter. This chapter is based on [Caimi et al., 2009b]. 7.1 Related work Checking whether macro schedules are conflict-free on the micro level requires the computation of detailed train speed profiles according to the micro train dynamics. This is particularly important in compensation zones, which are usually track lines with easy topologies, because there the suitable speed profile is not clear a priori and it is in fact the most important degree of freedom to exploit. Therefore, methods for the generation and analysis of speed profiles are crucial for the development of micro schedules in compensation zones. Figure 7.1 depicts the principal approaches for speed profile generation both from the academic literature and from the railway praxis. These approaches are described in the following. Figure 7.1: Overview on the principal methods in the literature for generating conflictfree speed profiles on railway lines.

198 7.1 Related work 171 The problem of generating speed profiles has often been addressed in the literature. Some approaches assume a fixed (micro) train dynamics based on maximum speed, e.g. [Jacobs, 2004, Törnquist and Persson, 2007]. For the approach described in this thesis, this is the case only in condensation zones (Policy 5.6), but not in compensation zones. Other approaches dealing with variable speed profiles are concerned with the on-line generation of conflict-free speed profiles, with the goal of minimising the propagation of delays [D Ariano and Albrecht, 2006, D Ariano et al., 2007b]. These methods are often applied to rescheduling problems for recovering or avoiding delay propagation. Unfortunately, these approaches are not suitable for including time reserves in the planning process nor for finding speed profiles with minimal energy consumption. On the other hand, the offline generation of one detailed single speed profile for a given trip time optimising the energy consumption has been addressed several times, e.g. in [Franke et al., 2000, Khmelnitsky, 2000]. However, little attention has been paid to the simultaneous generation of speed profiles and the development of conflict-free schedules. [Lusby et al., 2006, Lusby, 2006] investigate the generation of many speed profiles for each train and eventually assign one appropriate profile per train for creating an optimal conflict-free assignment. They analyse a single railway junction and consider a combination of different routes and speed profiles. However, the method seems not to be applicable for large network topologies, as the number of generated speed profiles would be impossible to handle. [Ravizza, 2007] applies the same approach for a single line and comes to similar conclusions. [Gumy, 2005] also proposes to generate many speed profiles and to choose a conflict-free set using constraint propagation. Unfortunately, computational results are not presented. [Carey and Lockwood, 1995] sets out a model for the train pathing and timetabling problem for railway lines with separate tracks for trains in each direction. They propose solution heuristics and strategies analogous to those which are applied by expert planners to find a timetable for large-scale complex railway systems by traditional manual methods. Few papers address the problem of optimising the distribution of the time reserves over a trip. [Kroon et al., 2006] describes a stochastic optimisation model for allocating time reserves and buffer times in a given timetable in a way that the timetable becomes robust against stochastic disturbances. However, the model works on a macroscopic level and aims at distributing the amount of time reserves between the stations without generating a speed profile or checking feasibility on the blocking times. For an exhaustive literature review on speed profile generation the reader is referred to [D Ariano, 2008]. The Swiss Federal Railways developed a two-step method for generating a speed profile connecting the given fixed boundary conditions [Graffagnino and Möser, 2006]. First, the fastest possible speed profile between arbitrary given points A and B is generated given a

199 172 Chapter 7: Microscopic scheduling in compensation zones starting time and speed in A and a fixed route in the network. The resulting speed profile respects the microscopic train dynamics and speed restrictions on all sections of the route. This fastest speed profile is called α-profile, with which the train would arrive at time tb α in B. The difference t between the planned time t B and tb α is the time reserve on the train s journey from A to B. In a second step, this time surplus t is distributed by reducing the speed limits on chosen track sections. Criteria for choosing the track sections are energy consumption (e.g. less acceleration and braking) or the operators preference of having time reserves near the end of the journey. The procedure for reducing the maximum speed is iterated until all time reserves from the α-profile have been allocated to the trip. The resulting speed profile is called β-profile and fulfils the requirements on the desired journey time from A to B, i.e. the starting speed, train dynamics, and speed limits. This heuristic method generates good speed profiles concerning energy consumption and time reserve distribution, without directly addressing the feasibility question. In practice, the choice of the terminal points A and B can prevent train paths from being in conflict, i.e., potential conflict points are chosen to be terminal points. Conflict resolution between different trains therefore focuses on assigning the right values for arrival and departure times in A and B. The model presented in this chapter targets the same goals as this approach, but applies a more systematic approach and addresses the feasibility question directly and simultaneously. 7.2 Problem formulation and solution approach The micro scheduling problem in compensation zones is formulated as follows. problem corresponds to the respective boxes in Figure 5.5 that describe the complete procedure for the micro scheduling problem. Problem 7.1 (Micro train scheduling in a compensation zone) Given is a compensation zone C comp j, its micro topology T j, as well as fixed tracks, passing times, and speed at the start and end locations of each train. The micro scheduling problem in the compensation zone then consists of assigning a track path to each train within the compensation zone fulfilling safety requirements and the boundary condition given by the start and end of trains in the zone. The union of these track paths form a conflict-free micro schedule s j. From the point of view of a single compensation zone, a train starts and ends its journey at the portals of the zone, or at stations within the zone if the train does not traverse the compensation zone completely. A speed profile of a train can be seen as a time-location-function s(t) that assigns the location of the train for each point in time. The This function has to fulfil the following

200 7.2 Problem formulation and solution approach 173 Figure 7.2: Two-step approach for micro scheduling in compensation zones. First, a set of alternative track paths for each train is generated, and a conflictfree assignment of one track path per train is done optimising an objective function combining energy consumption and time reserve distribution. properties: 1. s(t A ) = A and s(t B ) = B for the start and end locations A and B of the train in the compensation zone, where t A and t B are fixed passing times given as input parameters; 2. ṡ(t A ) = v A and ṡ(t B ) = v B for the start and end locations A and B of the train in the compensation zone, where v A and v B are fixed speeds given as input parameter and ṡ(t) is the speed of the train at time t. The function s(t) must respect micro train dynamics (Assumption 3.16), i.e., piecewise constant acceleration (and braking) rates as well as the maximum allowed speed on each edge of the micro topology. The considered routes in the compensation zone consist of the meaningful track sequences connecting A and B. A train schedule in compensation zones has to meet two types of restrictions. On the one hand, it has to fulfil the time and speed boundary conditions. On the other hand, the track paths must be conflict-free. Whereas the first restriction mainly concerns the development of speed profiles for each train individually, the second restriction results in a feasibility problem affecting all trains. In addition to satisfying the restrictions, the goal is to optimise a given objective function based on reserve time distribution and energy consumption. For this purpose, a two step approach is applied, which consists of a track path generation step and an optimisation step covering the feasibility problem, as displayed in Figure 7.2. The first step comprises the generation of a set of possible scheduling alternatives for each train individually. This step itself contains two subtasks: the enumeration of meaningful route alternatives and the generation of viable speed profiles for each route according to a given objective function, which is the crucial element of the track path generation step. A wide variety of speed profiles helps finding conflict-free track paths in

201 174 Chapter 7: Microscopic scheduling in compensation zones the optimisation step. Moreover, since compensation zones are intended to be the areas where delays can be reduced, schedules can be compared by their time reserves distribution. Hence, a wide variety of track paths can improve the quality of the generated train schedule. This first step is finished by calculating weights for the generated track paths. These weights represent the quality of the individual track paths and consider energy consumption as well as the distribution of the designated time reserves. The details are described in Section 7.3. The second step is to solve a combinatorial optimisation problem in order to assign one track path to each train such that the resulting schedule is conflict-free and maximises the weights of the chosen track paths. For each train, the generated speed profiles of the first step are gathered in a resource tree, and then the resource conflict graph model is applied for finding an optimal solution for the micro scheduling problem. Section 7.4 describes this second step. This approach is modular and the two steps are not mutually linked. One could also consider generating speed profiles in this way and then applying different optimisation method in the second step, or to generate a set of speed profiles with other methods and then use the optimisation method proposed here. 7.3 Generation of speed profiles for a single route The β-profile generation method described in Section 7.1 does not provide different speed profiles due to its fixed procedure for allocating time reserves. However, the idea of first determining the α-profile and then allocating time reserves is a promising approach for the purpose of this thesis. By distributing the time reserves differently, it is possible to generate a set of diverse speed profile alternatives. Starting with the α-profile, the additional time reserve t is not added in a unique way to the α-profile, illustrated in Figure 7.3. Instead, the additional time reserve is divided into N pieces, which are then distributed in several different ways among the K track sections between A and B. The partitioning points for the sections are typically block section borders (signal locations) or stations. The parameters N and K are mainly introduced to control the computational complexity. Since time reserves of fractions of a second are not useful in the real world, there is no need to exaggerate time accuracy, which is determined by the choice of N. On the other hand, the parameter K influences the geographic accuracy of the reserve time placement. Having many track sections increases the possibilities of placing the time reserves and thus also the number of possible speed profiles. An appropriate choice of K is needed to limit the number of speed profiles (and thus track paths) to a manageable amount, without reducing the variety too much such that feasible solutions may still be found in the

202 7.3 Generation of speed profiles for a single route Example alpha speed profile for S1 Lucerne-Zug Velocity [m/s] Distance [m] Figure 7.3: Speed-location function of the α-profile for the commuter train S1 between Lucerne and Zug, in central Switzerland. optimisation step of the algorithm Generating a set of β-profiles By discretising the time reserves and partitioning the compensation zones into K sections, the β-profile generation algorithm works as follows. First, the α-profile, displayed in Figure 7.3, is computed by assuming a constant acceleration and deceleration rate, which results in a piecewise linear speed-time diagram. Then, the fastest possible track path for the train is computed that respects the speed limitations on the track sections as well as the speeds at the portals. Let N be the (train dependent) parameter controlling the time discretisation, i.e., the number of time units that need to be distributed among the K sections. The reserve time per unit is given by t N, where t is the difference between the time needed by the α-profile and the planned trip time. If t is small, the resulting time units may be too short and therefore not useful in practice, since train drivers might not be able to follow a speed profile that accurately. Therefore, time units must have a minimal length ( ) t u z = min 1 n N n t n u min (7.1) for any train z. Let m be the argument of u z in Eq Thus, instead of N time units, m time units have to be distributed among the K sections. This leads to ( K+m 1) m 1 different

203 176 Chapter 7: Microscopic scheduling in compensation zones speed profiles for a route i of a train z. A speed profile j for a route i of train z can be represented by a code c z i j, where each section s, 1 s K, is assigned an integer value ri z j s indicating the number of allocated time reserve units. Of course, the sum of all entries r z i j s must be equal to m: c z i j := (rz i j 1,...,r z i j K ), r z i j s N, K ri z j s = m. (7.2) s=1 For each code, the passing time R z i j s at the end of a section s of a speed profile is determined by the amount of all consumed time reserve units until section s and the time t z α is needed by the α-profile until the end of section s, i.e., R z i j s = t z α s is + l=1 r z i j l u z = s l=1 (ν z α i l + ri z j l u z ), where ν z α i l is the time a train needs to pass section l using the α-profile. This approach is similar to the stochastic optimisation model proposed by [Kroon et al., 2006], yet working on the micro instead of the macro level. Figure 7.4 illustrates the variety of all speed profiles for a train. Figure 7.4: Set of all generated speed profiles with K = 5 and N = 20 for the commuter train S1 between Lucerne and Zug. The magenta line displays the α-profile.

204 7.3 Generation of speed profiles for a single route 177 Each code c z i j determines the passing time of the corresponding speed profile at the control points. A precise calculation of the speed profile can be done by solving differential equations with constraints respecting micro train dynamics and the passing times as boundary conditions. In the computations presented in this thesis, this step is simplified by distributing the time reserves ri z j s uniformly over s in such a way that the speed limits and the micro train dynamics within each section are respected Restricting the speed profiles Enumerating all possible codes c z i j for large values of K and N may result in a very large number of alternatives. Many of these alternatives are too similar to be distinguishable with respect to the generated conflicts and their quality. Moreover, some codes might be undesirable as the corresponding speed profiles are poor regarding the given objective. Thus, the number of alternatives is reduced in advance by applying rules on the speed profiles. Rule 7.2 (Feasible speed profile code) A code c z i j describing a possible speed profile of train z is considered feasible and will be inserted in the list of possible speed profiles for the train if the following requirements to the codes are respected: (i) Assigning too many time reserve units u z to one section is avoided by limiting the number to a maximal value u max for all sections. (ii) A large difference in allocated time reserves between two neighbouring sections s 1 and s is not permitted, as this would imply strong acceleration or braking leading to unnecessary energy consumption. Therefore, r z i j s r z i j s 1 is restricted to m νz α i j s t z α i where γ N is an input parameter and t z α i z for traveling over route i. γ ri z j s ri z j s 1 m ν z α i j s t z +γ, (7.3) α i the needed time of the α-profile for train Quality measure The quality of a speed profile is measured according to its energy consumption and the distribution of time reserve along the route. From an operational point of view it is desirable to place time reserves towards the end of the trip, to retain re-scheduling flexibility. This target can be quantified using only the information provided by its code description c z i j with D z K ( ) i j := M + lz i s s=1 Li z ri z j s, (7.4)

205 178 Chapter 7: Microscopic scheduling in compensation zones where l z i s is the length of route i until section s, L z i the total length of route i and M is an input parameter, which determines the influence of the sections on the goal. For large values of M, the contribution of r z i j s is similar for all s. Choosing M to be small favours r z i j s over r z i j s 1. The parameter M thus serves as a calibration parameter. On the other hand, the energy consumption of the speed profile is also an important criterion for its quality. In general, the energy consumption of a speed profile is the time integral of the force F times the speed v: E := Fvdt (7.5) In this thesis, this integral is approximated by summing up the interpolated energy consumption of each topology element, resulting in the expression E z i j := E e=2 m z 4l z i e (v e + v e 1 ) 2 (v e v e 1 ) +, (7.6) where m z is the mass of the train, l z i e the length of topology element e, and v e the speed at the end of e. With this formula, energy recuperation is not considered, i.e., energy that is fed back into the overhead line by using the motor as a generator during a braking phase. Since a good track path should allocate the time reserves at the end and use little energy, the two objectives are combined, resulting in an aggregated weight for speed profile j of route i of train z as w z i j := λ D z i j (1 λ) Ẽ z i j, (7.7) where λ [0,1] and D respectively Ẽ are the normalised values over all speed profiles of the time reserve distribution and energy consumption. Notice that D and Ẽ are measured with the same unit, and some fine tuning for the choice of a suitable value for λ could become necessary. 7.4 Optimisation model Once a set of alternative track paths has been generated, an optimisation model for choosing one of these alternatives per train is applied. Basically, one could apply the conflict graph model (Section 6.2.1), the RTCG model (Section 6.4), or one of its variants described in Section 6.6. In this thesis, the resource conflict graph model (RCG) is proposed for solving Problem 7.1. The model is described in detail in Section Each feasible code c z i j representing a possible speed profile for train z will be assigned a node v z in the resource conflict graph, and the blocking times for the used resources are computed using the green wave policy 3.14, as in condensation zones. The optimisation problem is then represented using ILP formulation 6.16, which is solved to optimality with a commercial solver. Results are presented in the next Section 7.5.

206 7.5 Computational results Computational results The concepts and algorithms of the previous sections are validated on a scenario in central Switzerland. The effect of the speed profile granularity on the solution quality will also be analysed. The considered compensation zone is located between the portals of Lucerne and Zug and is described in Appendix A.1.2 and illustrated in Figure A.6. The designated traffic of the 2008 SBB timetable is used for the test runs. It consists of two intercity, one InterRegio and two commuter trains per hour and direction. Portal times used as input for our computations are directly re-engineered from the timetable operated in The code was written in Java, and the resulting ILP was then solved with the commercial solver ILOG CPLEX 11. The tests were conducted on a x86-64 machine (Dual-Core AMD Opteron R, 4 GB RAM). N \ K infeasible feasible feasible feasible infeasible feasible feasible feasible > infeasible feasible feasible out of memory > infeasible feasible feasible out of memory Table 7.1: Statistics on speed profile generation. Each box contains the total number of speed profiles that were generated for the corresponding number of time reserve units N and the number of control points K and whether the generated speed profiles resulted in a problem that was feasible, infeasible, or had too large memory requirements. Using up to K = 9 partitioning points and up to N = 20 time reserve units, the test produced the results shown in Table 7.1. The following input parameters have been used: u min = 10s,γ = 1,M = 1. Control points are located at each station on the line as well as at the beginning and end of the single track lines. Tests with fewer control points discarded some of these points, in a fashion that all lengths between two consecutive points remained similar. All computation times of the tested scenario that did not run into memory problems were smaller than 30 seconds. The time was mostly used for the generation of speed

207 180 Chapter 7: Microscopic scheduling in compensation zones N \ K * * Table 7.2: Optimal values for the three objective functions depending on granularity. The values in each box are the optimal objective values for λ = 1, 0, and 1 2. The values for time reserve distribution are here not normalised. profiles, while the construction of the ILP and the computation of the optimal solution by the ILP solver was very fast. Table 7.1 presents the number of speed profiles that were generated for a selection of values for N and K and lists, for each level of granularity, whether the problem was feasible, infeasible, or if the computer ran out of memory. First of all, one can observe that working with too few control points makes the problem infeasible, in contrast to the fact that even a small choice of N can still create feasible train paths. Obviously, the number of generated speed profiles is considerably larger for finer granularities, and the computation runs out of memory. Table 7.2 presents the optimal objective values for λ equal to 1, 0, and 2 1, i.e., considering either only maximisation of time reserve distribution, or only minimisation of energy consumption, or maximising a combination of both objectives. Figure 7.5 illustrates the three optimal speed profiles of a train for N = 20 and K = 5. One can first observe that with the growth of N, i.e. smaller time reserve units, all objective values improve. However, this effect weakens as N grows. On the other hand, increasing the number of control points does not always lead to an improvement of the objective function value. This becomes true only if N is sufficiently large, which can be explained by the fact that having many control points has little value if only few time reserve units can be distributed. This would lead to the allocation of relatively large time reserves to short sections, forcing

208 7.6 Final remarks Set of possible speedprofiles Time [s] Distance [m] Figure 7.5: Speed profiles for the commuter train S1 Lucerne-Zug in the optimal solution: minimisation of total energy consumption is displayed in green, maximisation of time reserve distribution desirability in red and the combination of both objectives in blue. the train to drive at full speed in some sections and slowly in others, which has negative effects on the speed profile quality. The main observations of these tests can be summarised as follows. The parameter N seems critical for improving the solution quality, while K appears important for obtaining feasibility of the problem. The parameter K also plays a role in improving solution quality if N is large. 7.6 Final remarks In this chapter, the generation of conflict-free train schedules in compensation zones was addressed. Using quality criteria for speed profiles in terms of energy consumption and time reserve distribution, an optimal track path allocation was sought. The problem was addressed with the the RCG model formulated as an ILP, which considers all trains simultaneously, whereas previous methods usually applied heuristic approaches that treat trains sequentially. As a special case, this approach could also be applied to insert additional trains into an existing timetable, where modifying relevant times of pre-existing trains would be prohibited.

209 182 Chapter 7: Microscopic scheduling in compensation zones Computational tests show that the generation of speed profiles is very sensitive to the choice of the input parameters, and some fine tuning of these values is necessary to get a reasonable amount of speed profile alternatives for obtaining solutions of good quality while avoiding too long computation times. Test results indicate that the number of time reserve units should be similar to or greater than the number of control points. These points should be located at the stations on the line or at the borders of the single track lines. In order to make the problem feasible, a sufficient number of track paths are needed, which also improves the solution quality. In future work, methods could be developed that connect the points given by the reserve time allocation smoothly to generate more realistic speed profiles, e.g. by applying dynamic programming or solving differential equations. Moreover, larger scenarios should be tested to help understand the limits of this model with respect to computation time. Furthermore, it would also be interesting to investigate the proposed procedure for rescheduling applications, in particular to see whether it is possible to generate speed profiles in real time depending on the actual situation, or whether an off-line pre-computation of scenario-dependent speed profiles is mandatory.

210 Chapter 8 Train scheduling for partial periodicity This chapter focuses on the train scheduling problem for a partially periodic input, as introduced in Problem So far, Chapters 4 to 7 have presented a method for generating a periodic timetable starting for a given periodic service intention. If the given partial periodic service intention G (Definition 3.25) does not fulfil the conditions of Definition 3.27, i.e., it is not a periodic service intention, it is not possible apply the proposed methods directly. This problem is overcome by reducing (projecting) the ppsi G to a fully periodic instance, to which it is possible to apply the presented two-level approach for periodic scheduling. This reduction procedure is formally defined in Problem 3.36 and described in detail in this chapter. Finally, the resulting reduced periodic schedule has to be transformed back into a conflict-free train schedule for the complete considered period fulfilling the original ppsi G. The chapter is organised as follows: Section 8.1 explains the basic ideas of the general procedure for solving the train scheduling problem in the partially periodic case, and gives a motivation for this choice. Section 8.2 describes the different steps of the procedure in detail. In Section 8.3, conditions for the equivalence of the projected periodic and the original partially periodic problem are discussed. Computational results are presented in Section 8.4, and Section 8.5 concludes the chapter with some final remarks and an outlook. This chapter is partially based on [Caimi et al., 2009f].

211 184 Chapter 8: Train scheduling for partial periodicity 8.1 Basic idea An input service intention G (Definition 3.25) is in general not fully periodic, and presented methods for periodic service intentions has to be extended to work for all G. The basic idea of this extension is to apply a projection algorithm for reformulating the problem to an augmented periodic problem over a certain time period. Let T R + be the considered time period, which together with the given ppsi forms the input for Problem The projection algorithm reduces the ppsi to a periodic instance over the period T, after which well-known methods for periodic timetabling can be applied. The projected fully periodic instance is then in a suitable form for applying the macro scheduling algorithm developed in this thesis. It is important to note that the projected instance does not represent an input to a periodic timetable, i.e., it is not a psi. The reason is that the projection will enable some trains to disregard the headway time, which is of course infeasible if this is directly interpreted as a periodic timetable. Therefore, this projection will be called fully periodic instance, as an extended version of the feasible inputs for periodic scheduling. Similarly, also the resulting periodic output is not a conflict-free periodic schedule for the period T, but has to be interpreted as the compact (projected) representation of the partially periodic schedule. A final roll out algorithm is necessary to obtain the desired conflict-free schedule fulfilling the commercial requirements described in G. A discussion for a good choice of the value T is given later in Section Current approaches for dealing with partial periodicity mainly rely on manual postprocessing after the generation of a basic periodic schedule. Contrarily, the method proposed in this chapter enables the consideration of all exceptions of the periodicity directly during the timetable optimisation step. This has the advantage that no postprocessing is needed to include these exceptions and that the resulting schedule can be optimised over the whole day and not only for a certain sub-period. Even though the approach is primarily developed for service intentions with a very strong periodic structure and few exceptions, it is able to cope with any input that can be described within the ppsi framework, e.g. also a completely non-periodic service intention or every scenario in-between. The reduction of the problem to a periodic instance allows theoretic and practical advances in the field of periodic timetabling to be taken into account. Furthermore, the method allows an easy implementation in practice, because it can be seen as an extension, or add-on, to the tools for periodic timetabling, without having to change tools, methods, or production processes for adopting it.

212 8.2 Procedure for partial periodic train scheduling Procedure for partial periodic train scheduling The generation of partially periodic timetables from a given ppsi is explained in the following. The procedure is divided into four steps: (i) Slot propagation, (ii) Projection to the periodic case, (iii) Solution of the periodic problem, (iv) Rolling-out. An overview of the procedure is given in Figure 8.1 (step i) and Figure 8.2 (steps ii to iv). The steps are explained in the upcoming sections with reference to the example of the Figure Slot propagation Slot propagation is a preprocessing step that generates time slots for each event, i.e., time bounds for the departure and arrival at each station for each train. Some events have prescribed time slots [ωk,ω+ k ] in the input, specified in the train run of the ppsi (see Definition 3.19). For the other events, the slots can be derived by propagating the lower and the upper bounds for the trip, dwell, and connection time. Figure 8.1 shows how the time slots are propagated within a train run. In Figure 8.1a, a train run from v 0 to v 5, with given time slots (green) for departure in v 0 and arrival in v 5 is depicted. For the events in-between, time slots can be computed using constraint propagation. Feasible event times at point v i are bounded below by the earliest possible departure time in v 0 plus the fastest allowed trip time from v 0 to v i. Similarly, the event times have upper bounds. The event times reachable from the given departure time slot are visualised in transparent blue. Similar bounds can be computed by propagating the trip time constraints from the v 5 slot backwards. This gives two upper and lower bounds for each v i and the feasible time slot in-between (dark blue area and blue intervals). Figure 8.1b shows a case with three given time slots. As each constraint is propagated forward and backward, three upper and three lower bounds can be computed for each v i. The feasible event times lie between the largest lower bound and the smallest upper bound. It may happen that the defined slots (green bars) are further restricted (blue intervals) through the propagation procedure. Similarly, slots are propagated between the trains, using connection and dependency constraints in both directions. The resulting time slots always consist of one contiguous interval. If intervals are empty, this directly implies infeasibility of the ppsi.

213 186 Chapter 8: Train scheduling for partial periodicity Figure 8.1: Two examples of slot propagation for a train run. The green intervals are the time slots given in the ppsi, whereas the blue ones are derived during slot propagation. For the length of the time slots after slot propagation, the following assumption is made. Assumption 8.1 (Maximum time slot length) After having applied the slot propagation algorithm, the sizes of the time slots fulfil the following inequality ω k + θ kl < T k,(k,l), (8.1) where ω k := ω k + ω k is the length of the time slot at event k and θ kl = θ kl + θ kl span of the constraint between i and j, in both directions. is the Note that the assumption together with θ kl 0 implies ω k < T. The assumption shall be satisfied for each event and each incoming and outgoing constraint. The necessity of Assumption 8.1 is explained in Section 8.3. If the assumption is violated, the procedure is stopped because it is not possible to ensure a unique representation during the subsequent projection algorithm. To continue, the ω k may be reduced by defining additional or stronger time constraints in the ppsi, or by choosing a larger T Projection to a periodic instance This section explains how a ppsi with partial periodicity can be transformed into an instance for periodic timetabling. In the example illustrated in Figure 8.2a, the morning hours of a ppsi are shown. Four train types (different colors) run on a track from A to B, visualised by the time slots obtained through constraint propagation. The projection step

214 8.2 Procedure for partial periodic train scheduling : 00 A B 0 A B 7 : 00 A B Projection 8 : 00 T (b) Timetable Generation 8 : 00 9 : 00 0 A B 9 : 00 Roll-out 10 : 00 (a) T (c) 10 : 00 (d) Figure 8.2: Illustration of the procedure for partial periodic train scheduling. leads to a periodic projection of the problem (Figure 8.2b) with period length T = 60 min. It contains the trains from the original problem with modulo T projected event slots. The projections of trains with periodicity T coincides, so one representative is sufficient. This reduces the number of trains in the periodic problem considerably. More formally, equivalence classes for train runs are introduced. Definition 8.2 (Equivalence class for train runs) An equivalence class for train runs z is an abstract train run in the projection with time period T. It has the same properties as the original train run of Definition 3.19, with the exception that it is not repeated and that the time slots ωk,ω+ k have to be seen as times modulo T. Two recurrences of a train run z (or in some cases two different train runs z 1 and z 2 ) belong to the same equivalence class z if their descriptions are identical (trivial for two recurrences of the same train run) except for a temporal shift of an integer multiple of T. For instance, all recurrences of a train run z have the same equivalence class if the train run has a periodicity ρ(z) = kt with integer k N. More generally, if a train run has a periodicity such that it is repeated λ times within T k (so ρ = T k λ ) then λ equivalence classes for this train run are needed (if the train is repeated at least λ times). If the expression ρ T = k k,λ N, (8.2) λ is rational and k and λ are chosen such that k/λ is irreducible, then λ is the number of equivalence classes required. For instance, if ρ = 40 min and T = 60 min, then λ = 3 and k = 2. The projected problem is then an instance for periodic scheduling on the time period T. It contains one train for each equivalence class, see Figure 8.2 for an example. The

215 188 Chapter 8: Train scheduling for partial periodicity green train runs exactly once per hour and is therefore represented once in the projection. The blue train runs once per hour but with additional half-hour periodicity in the morning rush hour, leading to two representatives in the projected periodic problem. The projected problem can be formulated in PESP form, as defined in Section 4.2. In this case, the PESP nodes are the events of the equivalence classes. The PESP constraints are the projected trip time, connection, and dependency constraints between two equivalence classes. The constraints will be introduced if they occur at least once between the corresponding equivalence classes in the original problem. For instance, if a connection between two hourly trains is only defined in the ppsi to happen every second hour, it is equivalent to require it for every hour as also one that is not required in the other hours will take place in any case. If the periodicity of a train run is smaller than T, multiple equivalence classes for this train become necessary. The required fixed time shift between them is enforced by using periodicity constraints. For example, T = 60 min and ρ = 30 min results in λ = 2 required equivalence classes. Introducing periodicity constraints asserts the temporal difference of exactly 30 min between the two. Somewhat more complicated is the case of headway constraints on the projected version. Headway constraints guarantee that two trains do not collide (on single-track with opposing directions), overtake, or get too close violating safety constraints. Headways between equivalence classes are necessary if their time slots overlap in the original problem and are always introduced for two trains on a common track section. Multiple trains are treated by considering each pair separately. It is therefore sufficient to discuss the possible cases for a pair of trains on one track section between event nodes. Notice that it may occur that two equivalence classes for train runs indeed overlap, but no headway constraint should be introduced. This is the case if the corresponding original train runs are sufficiently separated in time and only their projection via equivalence classes makes them appear too close. For each such train pair, one has to check whether the event time slots of the two trains get closer than the required headway distance. For this purpose, a safety margin with length of the required headway is added to each slot. If these slots with safety margins overlap, the trains may possibly get too close, and headway constraints need to be introduced. The procedure to determine the necessary headway constraints is specified in Algorithm 4. The possible cases for a pair of trains on a common track section are described here and are illustrated in Figure 8.3. In the trivial case where the slots of two projected equivalence classes do not overlap, there is no need for headway constraints. For any combination of trajectories inside the slots, the trains run at a safe distance. This is the case in the example of Figure 8.2.

216 8.2 Procedure for partial periodic train scheduling : 00 A B A B A B 8 : 00 9 : 00 Projection Projection Projection 0 A B A B A B T (a) (b) (c) Figure 8.3: Necessity or not of introducing headway constraints in the projected version. 3 out of 4 different possible cases. If the time slots of two equivalence classes overlap in both projected and original problem, headway constraints are necessary to avoid conflicts (Figure 8.3a). If the slots do overlap in the projected case but the corresponding slots in the original problem do not overlap, headway constraints should not be introduced (Figure 8.3b). As the slots are sufficiently separated in the original problem, no headway constraints are necessary here. This overlap does not represent a potential conflict but it is only an overlap of their representations. It is even better to omit headway constraints, as their introduction would cut off feasible solutions leading to an unnecessary limitation of solution possibilities. For instance, the green and blue trains may have the same periodic arrival time in B without creating any conflicts, which would be prohibited by a headway constraint. The projected case can have four overlaps, two at each end of the track section. In some cases it may happen that one overlap requires headway constraints while another one implies that they should not be introduced (Figure 8.3c). The safety aspect requires headway constraints to be introduced despite the possible loss of potentially feasible solutions. In this case, the equivalence between original problem and projected problem would be lost (see later Section 8.3 for details).

217 190 Chapter 8: Train scheduling for partial periodicity This procedure for the introduction of headway constraints between events of the equivalence classes is the reason why the resulting instance cannot directly be interpreted as a description of a periodic service intention. The resulting periodic timetable in the space of the equivalence classes fulfilling all requirements, including the headway constraints, will have conflicts. This makes sense only if it is seen as a compact formulation for a partially periodic timetable. The projection procedure can be summarised by the pseudo-code described in Algorithm 5. The running time of the algorithm is polynomial in the number of train services.

218 8.2 Procedure for partial periodic train scheduling 191 Algorithm 4 Checking necessity of headway constraints. Require: Set of equivalence classes Z, time period T Ensure: Set H of headway constraints 1: for each pair of equivalence classes in Z do 2: Determine common track sections 3: for each common track section do 4: Find overlapping time slots in the projected period. Slots are considered to be overlapping if they violate the minimum required headway distance 5: {maximum 4 overlaps possible, 2 per track section end} 6: headway necessary := f alse 7: headway ok := true 8: for each overlapping do 9: Check in the original problem if the overlapping also occurs there 10: if overlapping also in original problem then 11: headway necessary := true 12: else 13: headway ok := f alse 14: end if 15: end for 16: if headway necessary = true and headway ok = true then 17: Introduce headway constraints in H 18: else if headway necessary = f alse and headway ok = f alse then 19: No headway constraints 20: else if headway necessary = true and headway ok = f alse then 22: else 21: Introduce headway constraints in H {warning: equivalence assumption violated} 23: No headway constraints {headway constraints have no consequences here as 24: end if 25: end for 26: end for 27: return H trains cannot conflict}

219 192 Chapter 8: Train scheduling for partial periodicity Algorithm 5 Projection of the ppsi to a fully periodic instance. Require: ppsi G, period time T Ensure: Fully periodic instance Ḡ, list of headway constraints H 1: Initialisation: Z := /0, C := /0, D := /0, Ḡ := /0 2: for each train run z Z(G) do 3: Determine the corresponding equivalence classes z i, i = 1,...,n z according to Definition 8.2 4: Trip and dwell times remain unchanged, time slots are computed modulo T 5: for each z i do 6: Z := Z z 7: end for 8: for i = 1 to n z do 9: for each pair of events in z i and z i+1 associated to the same original arrival or departure event in v k do 10: Introduce a time dependency constraint d between these events of z i, z i+1, with θ = θ + = T λ {n z + 1 := 1} 11: D := D d 12: end for 13: end for 14: end for 15: for each connection c C(G) do 16: Introduce constraints d i between the arrival resp. departure events at station v for the connection of the corresponding equivalence classes for train runs z 1 and z 2, including periodicity 17: Lower and upper bound remain unchanged 18: for each c i do 19: C := C c i 20: end for 21: end for 22: for each time dependency d D(G) do 23: Introduce constraints d i between the events of the time dependency of the corresponding equivalence classes for train runs z 1 and z 2, including periodicity 24: Lower and upper bound remain unchanged 25: for each d i do 26: D := D d i 27: end for 28: end for 29: Ḡ := ( Z, C, D, T) {is the fully periodic instance} 30: Apply Algorithm 4 for determining the set H of necessary headway constraints 31: return Ḡ, H

220 8.2 Procedure for partial periodic train scheduling Periodic timetable generation The projected periodic problem (Ḡ,H) with trip time, connection, dependency, periodicity and headway constraints can then be solved using the PESP formulation, as stated in Section 4.2.1, resulting in event times (or event time slots) for the equivalence classes (see Figure 8.2c). In addition to the classical PESP constraints described in Section 4.2.2, which are constraints between two arrival or departure events stating lower and upper bounds for their relative time difference, the PESP formulation resulting from the projection Algorithm 5 also includes absolute time constraints. These lower and upper bounds for the absolute event times come from the time slots [ωk,ω+ k ] computed during the slot propagation step and the subsequent projection to absolute time constraints for the events of the equivalence classes. Remark 8.3 (Absolute times) It is not possible to model the absolute time slots constraints as a relative time difference between two existing events in the PESP. Therefore, it is necessary to introduce an artificial zero-node v 0 that takes place per definition at time π 0 = 0. Absolute time constraints can then be modeled as an arc between the zero-node and the concerned node, where lower and upper bounds are directly the start and end time of the time slot ω k and ω+ k. The objective function used for solving the PESP formulation of the projected problem can basically be the same as in an ordinary PESP, as described in Section 4.2.3, e.g. a linear function of the length of trip, dwell, and connection times. To ensure an equal optimisation over the whole considered period ρ, each tension x a needs to be augmented with a weight w a that accounts for the number of repetitions that this trip, dwell, or connection occurs in the original ppsi. Similarly to the original PESP, it is also possible to apply all extended FPESP and Flexbox formulations defined in Sections 4.3 and 4.4. Also in these cases, the only difference from the formulations presented for periodic scheduling are the introduction of the zero-node and the weights in the objective function. Here, of course, the weights for both functions f tt and f flex need to be adapted according to the frequency of the corresponding arc or event. The resulting periodic macro schedule should then be checked for feasibility on the micro level as before. This step can basically be done in two ways. One possibility is to first transform back the periodic schedule to the partially periodic timetable in the original space using the roll-out method described in next Section The micro scheduling algorithms are then applied to this rolled-out timetable by using the non-periodic version of the micro scheduling approach for both condensation and compensation zones, as noticed in Remark 6.8. Even if embedded in a (partial) periodic framework, it is possible

221 194 Chapter 8: Train scheduling for partial periodicity to apply the non-periodic micro scheduling because the passenger-relevant decisions are already taken on the macro level. On the micro level, only technical and locally relevant decisions are taken. If these are different for each repetition of a train run, the periodic structure that is perceived by the passenger is not affected. On the other hand, the micro scheduling problem that needs to be solved in this way is of particularly large size. This disadvantage can be overcome, for instance, by iteratively applying the scheduling algorithm in a rolling horizon fashion (which may lead to suboptimal solutions) exploiting the structure given by the service intention, which in Switzerland is often based on the integral fixed-interval timetable [Lüthi, 2009, Liebchen, 2006]. Another way to overcome the large size of the problem is to apply the micro scheduling algorithms before rolling out the schedule. This way there is the double advantage that the problem to solve is of a smaller size and the required periodicity is accomplished up to the finest level of detail, i.e., all repetitions of the same train run are assigned exactly the same routes and speed profiles, even if this is not commercially relevant. However, for doing so, the RTCG model for micro scheduling needs to be slightly adapted. More precisely, the computation of the conflict cliques of Algorithm 2 has to be changed to deal with partial periodicity. Similar to Algorithm 4 for checking the necessity of headway constraints, a conflict on the projected space of equivalence classes does not automatically imply a conflict in the original formulation. Conversely, if in the projected space there is no conflict, it follows that also the original train runs do not conflict. Therefore, only line 11 of Algorithm 2 must be adapted, as described in Algorithm 6. Instead of considering the clique O of all allocations blocking the resource r at a certain point in time in the projection, the original version has to be inspected for the group of allocations O i that really blocks the resource simultaneously for at least one repetition i. Finally, only the maximal ones are inserted into the list C r of conflict cliques for the resource r. Algorithm 6 Adaptation of Algorithm 2 for partial periodicity. Require: Conflict clique O of allocations in the projected space {Line 11 of Algorithm 4 states: C r := C r O. This line is replaced by the following code:} 1: for each repetition i of time period T in the original ppsi do 2: Compute clique O i O of allocations occurring in repetition i 3: if C C r such that O i C then 4: C r := C r O i 5: end if 6: end for 7: return Set of conflict cliques C r

222 8.3 Equivalence between original and projected problem Rolling out the solution The periodic schedule generated as a solution to the projected periodic problem must be finally transformed back (rolled out) to a partially periodic schedule for the entire planning period, e.g. one complete day, as illustrated in Figure 8.2d. This step can be performed after having applied only the macro or both the macro and the micro scheduling algorithms to the projected periodic instance. If the macro schedule is rolled out, the PESP solution consists of event times in the interval [0, T) for each equivalence class in the projected problem. These are representatives of a set of trains in the original problem, being identical except for a temporal shift of kt, k Z. It is therefore sufficient to re-transform the event times of an equivalence class to the corresponding times in the time slots for each repetition of the train run in the original problem. The result is a timetable consisting of event times for each train run described in the ppsi and fulfilling all required constraints. Remark 8.4 (Uniqueness of roll out) The roll out result is unique if Assumption 8.1 is fulfilled. This is because the assumption implies that the size of the time slot ω is strictly smaller than T for each event, making the correspondence between original and projected problem unambiguous. The roll out procedure of a microscopic schedule is basically the same. Instead of only event times, the passing times through each topology element of the micro topology need to be re-transformed into passing times for each repetition in the original problem in the same way. Equivalently to Remark 8.4, uniqueness is ensured also in this case if Assumption 8.1 holds. 8.3 Equivalence between original and projected problem In the last section, a procedure is presented to generate a partially periodic schedule from a ppsi by reducing the problem to periodic instance, to which existing methods for periodic scheduling can be applied. The question arises if the original ppsi and its reduced periodic instance are actually equivalent. For ensuring equivalence of the two problems an assumption on the result of Algorithm 4 becomes necessary. Assumption 8.5 (Headway special case) In Algorithm 4, the case (headway necessary = true) and (headway ok = false) does not occur. This situation is depicted in Figure 8.3c.

223 196 Chapter 8: Train scheduling for partial periodicity This assumption, together with Assumptions 8.1, is a sufficient condition for stating the equivalence of the original partial periodic and the projected periodic problem. Theorem 8.6 (Equivalence of original and projected problem) Under Assumptions 8.1 and 8.5, the problem of generating a partially periodic schedule for a given ppsi and the problem of generating a periodic schedule for the corresponding projected fully periodic instance are equivalent. The theorem states that an infeasibility response from the periodic instance implies infeasibility of the original ppsi. Similarly, finding the optimal periodic solution implies that the corresponding rolled-out partially periodic schedule is also optimal, according to the same objective function. Proof: Each event of the train run in the ppsi is uniquely represented by an event in an equivalence class. On the other hand, an equivalence class represents only train runs that are identical expect for being shifted in time. Therefore, a partially periodic schedule can be uniquely mapped to a periodic schedule in the projected space, and vice versa. Whereas the projected mapping is always possible unambiguously, the mapping via rollout is unique only if Assumption 8.1 holds. Note that projecting a schedule and rolling it out again will result in the schedule itself, without any changes. It is now necessary to show that a partially periodic schedule and its projection fulfil (or violate) exactly the same set of constraints. This will be proven here for the (original and projected) macroscopic schedules. It can be shown that the rolled out solution of the periodic problem does (or does not) fulfil the same trip time, connection, and time dependency constraints defined in the ppsi, because during the projection these time were kept unchanged. It is shown here that the PESP constraints (4.1) are equivalent to the ppsi constraints. Let ω i be the event time of a schedule in the original formulation. It then holds for each ppsi constraint (i, j) that θ ω j ω i θ +. (8.3) Eq. (4.1) can then be applied to the ppsi data l i j = θ, u i j = θ +, π i = ω i + k i T, π j = ω j k j T, with k i,k j Z. It results in θ ω j k j T ω i k i T + p i j T θ + (8.4) and, reformulated, θ ω j ω i +(p i j k i k j )T θ +, (8.5) which is equivalent to the ppsi constraint if p i j k i k j = 0. It remains to be shown that p i j k i k j = 0. From slot propagation the resulting bounds for the event times are ωi ω i ω i +. (8.6)

224 8.3 Equivalence between original and projected problem 197 It follows that for each arc (i, j) that This leads to the bounds and ω i + θ ω j ω j ω + j ω + i + θ +. (8.7) (p i j k i k j )T θ + ω j + ω i θ + (ω i + θ )+ω + i = ω i + θ i j (8.8) (p i j k i k j )T θ ω j + ω i θ (ω + i + θ + )+ω i = ( ω i + θ i j ). (8.9) Combining both inequalities results in ω i + θ i j T (p i j k i k j ) ω i + θ i j. (8.10) T If Assumption 8.1 holds, it follows, together with the integrality of the expression, that p i j k i k j = 0. Therefore, the equivalence of the PESP to the ppsi constraint is shown. The introduction of headway constraints in the projected version is explained in Algorithm 4 and discussed in Section Assumption 8.5 excludes the critical case where the headway constraints for the projected and for the original formulation do not correspond. In that case, equivalence cannot hold and solutions may be lost. If no assumptions are violated and a solution to the projected problem is found, then this solution fulfils all the constraints from the original problem. Conversely, if no solution for the PESP is found and the assumptions hold, the ppsi defines an infeasible problem instance. The equivalence of the microscopic solutions of the projected and the original problem can be shown in a similar way, considering passing times through each topology element instead of event times in macroscopic nodes. If violated, both assumptions can be satisfied by increasing the value of T, which is discussed next in Section If Assumption 8.5 does not hold, it is not necessary to interrupt the algorithm. In line 21 of Algorithm 4, a violation of Assumption 8.5 is detected and signalised with a warning, but the algorithm keeps going. Once the projection is finished, the periodic scheduling problem is solved. If an optimal solution is found, this is surely feasible also after being rolled out, because when detecting an equivalence violation, a more conservative constraint than the original is introduced. However, the corresponding rolled-out solution is not necessarily optimal because it can be possible that better solutions were cut off during the projection with the insertion of these more conservative constraints. For the same reason, an infeasibility response of the periodic scheduling algorithm does not imply infeasibility of the original problem, if the assumption is violated. One could also consider modifying line 21 of Algorithm 4 in such a way that if a violation of Assumption 8.5 is detected, no headway constraints are introduced. This way,

225 198 Chapter 8: Train scheduling for partial periodicity an optimal solution found on the projected space is no longer guaranteed to be feasible after being rolled out. On the other hand, if it is feasible then it is clear that it is also optimal, and an infeasibility response from the periodic scheduling algorithm is a certificate of infeasibility of the original problem. This alternative becomes interesting, for instance, for checking infeasibility of the original problem when the other variant, with the introduction of the headway constraints in line 21, results to be infeasible Choice of the period length T As the projection is possible for an arbitrary value of projection period T, the question arises which is the best choice of the parameter T. On one hand the value T has to satisfy the required assumptions, but on the other hand it should reduce of the problem size as much as possible. Assumption 8.1 depends directly on T and can easily be satisfied by increasing its value. Assumption 8.5 also depends on T, but in a more complicated way. Doubling the value from T to 2T certainly preserves the assumption. However, this is not generally true for values in-between. Increasing T generally leads to satisfying the assumptions. In particular, there can always be found a value T max large enough to contain all time slots of the original problem and fulfilling both assumptions. An easy and intuitive choice is T = ρ (see Definition 3.25). This leads to a projected problem with one equivalence class per train service. This is equivalent to a non-periodic train scheduling problem without any reduction of problem size. Also, a smaller value for T can be selected in order to ensure the fulfilment of the assumptions. Theorem 8.7 (Choice T for fulfilling the assumptions) Let the period time T be defined fulfilling T > 2 max i V ω i + max (i, j) A θ i j. (8.11) Then, Assumption 8.1 and Assumption 8.5 are satisfied by using time period T, and the projected problem is guaranteed to be equivalent to the original ppsi. Proof: It is sufficient to check that both assumptions cannot be violated with a choice of T = 2 max i V ω i + max (i, j) A θ i j + ε, where ε > 0 is an arbitrary (small) positive value. Assumption 8.1 states ω i + θ i j < T. Inserting the defined value for T this results in ω i + θ i j < 2 max i V ω i + max (i, j) A θ i j + ε, (8.12) which is trivially true for each event i V and arc (i, j) A.

226 8.4 Computational results 199 Assumption 8.5 can only be violated if line 21 in Algorithm 4 is reached, i.e., the time slots of the two considered events overlap twice in the periodic time modulo T. This is only possible if the sum of the two time slot sizes is at least T, i.e., However, ω i + ω j T. (8.13) T = 2 max i V ω i + max (i, j) A θ i j + ε 2 max i V ω i + ε > ω i + ω j (8.14) for each pair of events i and j, which makes the violation of the assumption impossible. The choice of T as small as possible is of course more effective. If the ppsi contains some periodic structure, the number of equivalence classes and therefore the size of the projected periodic problem shrinks if T divides the periodicities ρ z of many trains. A too small choice of T, however, could lead to a violation of the assumptions. If one assumption is violated, the reduction of the time slot size in the ppsi can be used to overcome the problem and to fulfil the assumption. Without touching the input ppsi, it is also possible to fulfil the assumptions by choosing a larger value for the time period T. Corollary 8.8 (Doubling T ) Let Assumption 8.1 be satisfied and Assumption 8.5 be violated with time period T. Then, by choosing the new time period T = 2T, both assumptions are fulfilled for T. Proof: As Assumption 8.1 is already satisfied, it follows directly that ω i < T (8.15) for all events i. Hence, ω i + ω j < 2 T = T (8.16) for each pair of events i and j. This condition is sufficient to ensure that Assumption 8.5 cannot be violated. Usually, the offer of a railway company has a basic periodicity of half an hour, one hour or two hours. It is the natural choice to use one of these values for T. Computational results for various choices of T are given in the next section. 8.4 Computational results The method presented in this chapter has been tested using the scenario in central Switzerland, explained in Appendix A.1. The underlying ppsi G is reverse-engineered from the

227 200 Chapter 8: Train scheduling for partial periodicity Scenario T [min] variables int. variables constraints CPU time [s] Partial periodic SI Regular trains Table 8.1: Statistics of the generated PESP instance and solution time of the ILP optimisation for two different service intentions in the central Switzerland scenario and some values of T. T = 1200 means actually that T = ρ is chosen and that no projection was done. The asterisk means that the equivalence Assumption 8.5 was violated, resulting in an infeasible instance timetable of SBB for a standard Wednesday. The code is written in Java, and the resulting ILP of the projected periodic instance is then solved with the commercial solver ILOG CPLEX 11. The tests were conducted on a x86-64 machine (Dual-Core AMD Opteron R, 4 GB RAM). The projection algorithm works almost instantaneously for all tested scenarios, as well as the roll-out procedure. In all computations an objective function that measures the sum over all passenger-relevant travel times is minimized, as already discussed in Section Table 8.1 presents the results for the ppsi G. Furthermore, in order to better compare the additional complexity of the partial periodic case, a version of the test case with only the regular trains is tested as well, omitting all the exceptions that occur. This version results in a fully periodic psi G and would have been the input of classical approaches for periodic timetabling. The resulting optimal timetable for G would then need some (manual or automatic) postprocessing to adapt it for each period in the day to fulfil the requirements of the ppsi. This adaptation can also have negative effects on the timetable quality, as the exceptions are not considered during the optimisation phase. This postprocessing phase is not considered here. The algorithm has been tested for two different values of the time period T, as well as without the projection (T = T max = ρ = 1200 min), considering the whole day as a single period. One can first observe that, depending on the chosen time period T for the projection, the generated (equivalent) PESP instances have different sizes. As most trains in the test scenario have a periodicity of one hour, the projection with T = 60 min results in a smaller PESP instance. On the other hand, the smaller the value T, the smaller the time slots for the events should be as well. For example, the initial time slots specified

228 8.5 Final remarks 201 in the ppsi are relatively large, and choosing T = 60 min leads to a violation of the equivalence Assumption 8.5, eventually resulting in an infeasible PESP instance. In the case of T = 120 min, the projection is possible by keeping the equivalence of the problem, and an optimal solution is computed in less than half a minute. For comparison, solving the same problem without making use of the projection and the equivalence classes (i.e., choosing T = ρ) produces a much larger ILP that will take much longer to solve. An optimal solution for this scenario is then rolled out for a complete day and results directly in a schedule fulfilling the ppsi with its partial periodicity. The fully periodic scenario G can be projected to a smaller PESP instance as all special trains are not considered here. The smaller T is, the larger the difference, because regular trains need less equivalence classes than special train runs occurring only once. In the non-projected version, the difference between the partial periodic and the fully periodic PESP instances is small, as periodic trains are often repeated, whereas special trains occur only few times. Fully periodic instances can be solved 2-10 times quicker, but need a postprocessing that can also be time consuming, while the solution quality might deteriorate considerably. The generally good computation times compared to the purely periodic case can be partially explained with the structure of the problem. Composing a ppsi requires planning that already determines some structure of the timetables, thus limiting the search space of the solver. However, the main complexity of timetabling remains unchanged, consisting, among other aspects, of finding feasible train orderings on the tracks. Summarising, the best value of T for solving Problem 3.35 depends on the ppsi but can be roughly determined as the smallest possible periodicity of the train runs where the equivalence assumption is not violated. Directly considering the ppsi with its partial periodicity for the optimisation problem leads to longer computation times. However, its explosion is avoided and is largely compensated by the fact that no more postprocessing is necessary and all requirements in the ppsi are taken into consideration for the optimisation, which results in timetables of better quality. 8.5 Final remarks This chapter presents a method for solving the train scheduling problem for a partially periodic input. The method is based on the projection of the intended train runs over equivalence classes and thereby reducing the ppsi to an equivalent periodic timetabling instance. This partially periodic structure can be exploited effectively, as each equivalence class is represented only once in the projected problem. It is then possible to use existing well-known models for periodic scheduling, and thereby also to take advantage of future improvements in this field. It is observed that the stronger the periodicity of the offer is,

229 202 Chapter 8: Train scheduling for partial periodicity the larger the reduction of the problem size and the shorter the computation times are. The proposed model and approach is therefore particularly well suited for offers with a strong periodicity but some irregularities, which could not be treated properly by the existing periodic timetable generation approaches. Existing methods for micro scheduling cannot be directly applied, because the equivalence classes are not the same as a single train run, and two trains could use the same resource simultaneously in the projected problem, yet remain feasible because they are temporally separated in the original problem. A slight adaptation of the micro scheduling algorithm is therefore necessary to cope with these new features coming from the projection procedure. In future work, the presented approach should be compared to classic approaches with manual postprocessing to analyse computation times and to compare the quality of the generated timetables. Serious discussions with planners are also necessary to make the ppsi an effective instrument for describing real service intentions in practice.

230 Chapter 9 Global results and added value of flexibility This chapter discusses some general aspects of the approach that are relevant for more than one of the steps presented in the previous chapters. Section 9.1 presents results for a scenario in central Switzerland, which has been completely computed from its ppsi description to the corresponding conflict-free micro schedule for the whole considered network. In Section 9.2, the added value of flexibility that is generated on the macro level is evaluated for its impact on condensation zones on the micro level in the scenario in central Switzerland. 9.1 Results for the complete train scheduling procedure So far, Chapters 4 to 8 have presented computations on real-world instances for each single method or algorithm presented in this thesis, with the main goal to compare alternative approaches and to validate the presented models. Besides this important aspect, a scenario was also tested through the entire methodology developed in this thesis and outlined in Section 3.8 for solving Problem 3.35, from its representation as ppsi to the detailed conflict-free schedule for the whole day. The complete procedure is applied to a test case scenario in the railway network of central Switzerland, introduced in Appendix A.1. The considered partial periodic service intention is constructed by reverse-engineering the operated timetable for a typical Wednesday in It contains 33 periodic train runs, which provide a total of 196 single

231 204 Chapter 9: Global results and added value of flexibility train services for a whole day. The time slots are created by setting a time slot of 20 minutes around the published departure time of the train in the first station and around the published arrival time of the train in the last station of the run. All other time slots are then computed applying slot propagation. The so generated ppsi is first projected on a reference time period of 1 hour, generating 48 equivalence classes. The subsequent PESP instance on the macro level has 212 events and 647 arcs, after resolving arcs with zero span. This instance results in a MIP formulation with 1083 variables (436 integer) and 1730 constraints. Choosing the objective function MIXFLEX 1/2, the solver delivered an optimal solution with a flexibility of 189 units (the maximum is 330) and a sum of the passenger relevant trip times of 1971 (minimum 1914) after 57 seconds. The resulting macro timetable is then fed as input for micro scheduling. As explained in Section 5.4, condensation zones are solved first. The most interesting one is around the main station of Lucerne, which is a terminal station with 12 platforms. The timetable obtained from the macro level gives rise to 1217 different routes in the condensation zone Lucerne, which results in an RTCG with nodes, blocking edges, and flow edges. The resulting MIP, using the extended formulation as presented in Problem 6.15, has binary variables, with only conflict constraints. The chosen objective function is hierarchically organised as described in Section 6.4.3, with the primary goal of maximising the number of scheduled trains, and the subordinate goal of minimising the penalty if trains are scheduled outside of the given macroscopic time slot. After 10 seconds of preprocessing, the MIP solver delivered a relaxed solution that was integer, scheduling all trains without penalties, i.e., the computed macroscopic schedule was feasible also on the micro level. Finally, the compensation zones are solved. The longest line is between Lucerne and Zug. 12 representative trains (equivalence classes) travel on this line over the projected time period T = 1 h. For these 12 trains a total of 873 speed profiles are generated, which took about 5 seconds of computation time. The construction of the RTCG model required 0.3 sec, and the MIP-solver could find the optimal solution for all three tested objective functions (energy consumption, time reserves distribution or a mixed version) almost instantaneously, as the relaxed solution was integer. The resulting production plan can also be compared with the timetable of 2008, which was based on the same service intention. The underlying structure is similar, with for instance the crossing of the commuter train and the EuroCity in Walchwil in both directions or the connections in Arth-Goldau. On the other hand, the departure times are almost always different, and sometimes also some train sequences on the same track are not equal. This differences have many reasons: first of all, these computations have only considered the scenario region in central Switzerland, whereas the operated timetable has of course to

232 9.2 Added value of flexibility on the micro level 205 take into account the entire network, which gives some additional boundary conditions to the region considered here. Furthermore, it is quite possible that the planners considered additional constraints, e.g. for rolling stock or shunting movements, which was neglected for the computations presented in this thesis. 9.2 Added value of flexibility on the micro level An important aspect that remains to be analysed is the effect of the flexibility on the micro level. As described in Chapter 4, different objectives can be applied, resulting in different solutions regarding attractiveness and flexibility. Of particular interest are the Pareto-optimal solutions, because they provide the maximum possible total flexibility for a timetable quality regarding travel time. In theory, it can happen that a solution with less flexibility is conflict-free on the micro level whereas one with larger total flexibility is not, because the feasibility strongly depends on where this flexibility is allocated. It is not clear a priori where flexibility is necessary and where not, otherwise the solution would already be known. Some indication can of course come from the intuition and the experience of the planners, as well as from previous tests, and can be incorporated into the objective function by giving more weight to the events of critical trains and/or stations. Therefore, Pareto-optimal solutions with respect to the given objective functions are the most attractive ones to analyse for the added value of flexibility for finding conflict-free solutions in condensation zones. The generated flexible macro timetables are then tested for their performance in condensation zones on the micro level, applying the RTCG model described in Chapter 6. More precisely, Formulation 6.15 for the extended problem can provide an alternative schedule minimising a penalty function in the case where the input time slots are not feasible. Remark 9.1 (Same ILP) The formulation 6.15 for solving a condensation zone is always the same and does not depend on the provided time slots from the FPESP, with the exception of the objective function. This is due to the fact that an alternative starting time is generated in any case for each discretised point inside the time slots provided by the ppsi, and these remain unchanged during macro scheduling, even if iterated. Also the discretisation depends on the values for the pulse length and the phases, which are given as input and do not change during the train scheduling algorithm. Only their weight in the objective function changes each time depending on the provided macro time slots. The impact of the macro flexibility is tested in the condensation zone of Lucerne, described in Appendix A.1.2. The macro scheduling step is applied for the test case of

233 206 Chapter 9: Global results and added value of flexibility Objective total total # trains total function travel time flexibility not in time slot penalty NOFLEX MIXFLEX 1/ MIXFLEX 2/ MIXFLEX 9/ POSTOPT Lucerne prio only Lucerne Lucerne topology Table 9.1: Added value of flexibility. In the scenario Lucerne prio3, all events corresponding to the station of Lucerne have become a weight of 3 in the objective function MIXFLEX 2/3. In only Lucerne, only the events of Lucerne are considered in the objective function. Finally, in Lucerne topology2 the macro topology of Figure A.2 is used instead of the more precise one. central Switzerland (Appendix A.1.1) using the same reference service intention introduced in Section for testing the FPESP model on the macro level. The topology used for macroscopic scheduling is the more precise one described in Figure A.3. Furthermore, each event of the train runs in the input ppsi had a time slot of around 15 to 20 minutes. The available routes are reduced according to Policy 6.4, and time discretisation is applied with τ = 60 sec and values for the phases provided by practitioners. In the condensation zone of Lucerne, 30 normal gauge trains run per hour. For a total of 705 available routes, the resulting RTCG has vertices and 7388 leaves. Applying Formulation 6.15 an ILP results with variables, constraints, including conflict constraints that contain 31 variables on average and a maximum of 252 variables. The creation of the model takes approximately 30 s, whereas the solution time varies from 70 to 160 seconds, depending on the objective function. All instances have an integrality gap ranging from 0 (sometimes even with the relaxed solution being integer) to a maximum of 0.5%. Recall that the computed macro schedule includes 48 trains, the minimal possible total travel time is 1914 minutes and the maximal reachable total flexibility is 330 sec. Table 9.1 and Figure 9.1 present the results of different macro timetables. The first general observation is that, even if not systematic, the more flexibility a macro timetable has, the smaller the penalties and the more likely a feasible solution according to the macro time slots can be found. For a sufficiently large amount of flexibility, which depends on

234 9.2 Added value of flexibility on the micro level Scheduling penalties against flexibility Equal weights everywhere Weight 3 for Lucerne Flexibility only in Lucerne 50 Penalty Flexibility Figure 9.1: Flexibility against penalty on the micro level for three different (macro) objective functions for the reference scenario in central Switzerland. The larger the generated flexibility, the smaller the resulting penalty value, until for a certain value of flexibility a conflict-free schedule can be found. the chosen objective function, a penalty value of 0 is reached, which means that a conflictfree schedule could be found. However, it is very difficult to state a priori the right amount of flexibility that is needed and the location where it should be allocated. This is very situation sensitive. It could be, for instance, that two equivalent macro schedules according to the macroscopic objective are very different from a microscopic point of view, but the macro scheduler just randomly gives back one of the two. Therefore, also some luck is necessary, as on the macro level the details of the microscopic network necessary to make the schedule feasible are not visible. As it can be seen in Table 9.1, the macro schedule generated using the POSTOPT, which is not Pareto optimal, performs better than the comparable Pareto-optimal solution generated with the objective MIXFLEX 2/3. It can be easily observed, however, that a higher concentration of flexibility in the larger or denser condensation zones makes sense, as they are the places where more crossings are needed to reach the designated platforms. The two other condensation zones of the test case, Zug and Arth-Goldau, are smaller and less dense than Lucerne. Indeed, in Arth- Goldau the micro schedule is always feasible even for fixed event times for all conducted computations. This is the case also in Zug, with the exception of one single case with fixed times that could be easily be overcome by adding smaller event slots to the critical

235 208 Chapter 9: Global results and added value of flexibility trains. Furthermore, the consideration of a more precise topology (Figure A.3) on the macro level clearly seems to perform better and offer more possibilities to be feasible than the use of the less precise topology (Figure A.2), which on its part is faster to compute. In particular, the neglected single track lines in condensation zones seem to be very difficult to allocate if this structure is not already considered in the macro topology. These considerations are just meant to give an impression on the added value of flexibility for the micro level. Some representative results are presented and confirm the intuition that macroscopic flexibility can be helpful on the microscopic level for finding a conflict-free solution. However, computations have not been not sufficiently exhaustive and it is not pretended that their implications are valid in general. The behaviour and the performance of flexibility may differ for a different topology or when a different service intention is taken into account. Further and more systematic test are necessary in order to arrive at a more significant conclusion about the added value of flexibility.

236 Chapter 10 Conclusions The work presented in this thesis provides formal methods for solving the train scheduling problems of large and highly utilised railway networks. This thesis is written in collaboration with a production-oriented project of the Swiss Federal Railways, Infrastructure Division, and follows in some points the philosophy of this project. However, most of the proposed models and algorithms can also be applied out of the context of this project. In some cases the applied policies could restrict the general problem, yet remain relevant for the general context. This concluding chapter first discusses the main results of the thesis. Some suggestions for further research on train scheduling and rescheduling are finally proposed Critical appraisal of the results This thesis has introduced a complete, modular, and integrated multilevel approach for system-wide detailed and conflict-free train scheduling in large and highly utilised railway networks. This multi-stage planning approach is principally conceived to be a decision support system for the strategic, tactical, and operational planning levels. Depending on specific needs, it is easily possible to use only a part of the proposed framework or to intervene with some manual changes when passing from one stage to the next. This way, planners can still play a role as a protagonist during the train scheduling procedure, and the algorithms only carry into effect his requirements. The presented method can also be used to automatically check feasibility of a given production plan. This (simpler) use of the model allows the automatic conflict detection at the microscopic level to be beneficial by keeping all decisions to be done manually. The procedure described in this thesis generates train schedules that respect the applied

237 210 Chapter 10: Conclusions policies: green wave, maximum speed in condensation zones (and at portals), a limited number of available routes in condensation zones and speed profiles in compensation zones, and the time discretisation policy. There could be a schedule that is conflict-free and fulfils all commercial requirements, but where at least one of the previous operational polices are not respected. Such a schedule would be impossible to create with the presented methodology. Nevertheless, it is possible to also check these schedules to be conflict-free, except when the green wave policy is violated. In this case, the blocking times would have to be computed in a different and more complicated way, where additional information such as the sight distance to the main signals become necessary. This is a limitation of the proposed procedure. The presented method also has other limitations. Various aspects that might also be important for the generation of efficient train schedules were neglected, such as the effect of the schedule on rolling stock or crew planning. These issues can be partially incorporated into the presented framework by introducing additional constraints and considering other objective functions [Liebchen, 2006]. Furthermore, the formalisation of the input ppsi can be rather restrictive. It presumes an already clear idea about the structure of the schedule in order to describe connections and time dependencies. In particular, exact prescription of the train runs connecting repetitions is required. This is certainly a condition that forces the planners to undertake some preparatory work before applying the proposed methodology and possibly run it even with some alternative service intentions. However, this is not a limitation exclusive to the ppsi formalisation of this thesis, but appears also in other descriptions of the periodic service intention, for instance when compiling the input for the standard PESP model. In fact, the ppsi can be seen as an aid for planners to structure an input specification process that is necessary in any case Outlook for future research All algorithms of this thesis are developed in a laboratory environment, with the given limitations of this situation. The results of this thesis have the potential to become the core of a support tool for planners. However, some work remains to be done for reaching the goal of practical implementation. First of all, a practical validation of the results is necessary. Some unavoidably neglected aspects have to be checked to determine if they can easily be integrated into the generated schedule. Furthermore, the algorithms should be accessible through a userfriendly interface, relying on an up-to-date and complete database to avoid the creation of out-dated, useless results. In particular, the intervention possibilities of the planners during the scheduling process need to be formalised and implemented. Besides these practical aspects, also in regards to the theoretical and academic part of

238 10.2 Outlook for future research 211 the work is not completely finished. Several interesting issues are left to be considered in detail and need further investigation. An important component of the train scheduling methodology is the development of a strategy on the macro level to deal with the feedback provided by the micro level. Critical trains that cannot be scheduled as desired are detected in each condensation zone, and an alternative, locally feasible schedule is suggested. These counter-proposals have to be analysed altogether to compute a new flexible macro timetable that is more likely to be feasible in all local regions. In this thesis, some simple criteria for improving the robustness of the schedule are introduced on both macro and micro level. They are based on a static concept, i.e. the capability of the schedule to absorb delays or at least avoid its propagation without having to actively adjust the schedule, e.g. to change train sequences or to break connections. In practice, this property is helpful only in a very limited way, because changes in the schedule are allowed with the goal of reducing delays. Dispatchers can intervene at any time, and a more important characteristic of a train schedule is its capability to offer simple and effective rescheduling measures to stabilise the schedule. Therefore, more advanced criteria of robustness should be considered during the train scheduling process, such as the recently developed concepts of recoverable robustness [Liebchen et al., 2007] or adaptive robustness [Caimi et al., 2009c]. An important aspect of the train scheduling process that was not addressed in this thesis is the appraisal and the relaxation of the ppsi if the finally generated production plan does not satisfy all requirements prescribed in the ppsi, as displayed in the bottom left box of Figure 3.10 on page 61. Here, commercial considerations dominate technical ones. The importance of each service has to be analysed, e.g., from its economical point of view, its impact on other services, or its fairness towards all train operating companies. Based on this, a less restrictive ppsi has to be generated and the train scheduling procedure started over again. Larger scenarios need to be tested. In particular, during strategic and tactical planning more time is available and the goal of considering all trains running in Switzerland should be the aim. Although some tests were already efficiently and successfully completed for the entire Swiss Intercity network, it remains an open question as to what extent the presented methodology is suitable for computing the daily timetable for the complete railway network in Switzerland, particularly because the density of the services is the decisive factor for the CPU time and therefore the practicality of the presented approach. It is observed that a more precise macro topology is of great assistance for the feasibility of the generated timetable on the micro level, yet it increases the computation time. A compromise is therefore necessary on the precision on the macro level between the computability of a single macro timetable and the reduction of the iterations between micro

239 212 Chapter 10: Conclusions and macro level with the goal of minimising the total time needed by the train scheduling approach. Moreover, if the considered network and/or the service intention are so large that the approach does not work in a reasonable amount of time (one could expect this, e.g., for the case of the German railway network or even smaller networks), a systematic subdivision of the macroscopic topology might become necessary. The train scheduling step is only one part of the complete planning procedure for railway system. The presented methodology should be integrated into the complete production chain and combined with other planning steps, in particular rolling stock and crew scheduling, as well as line planning. Input and output of each step have to be mutually arranged, and feedback has to be defined with the goal of improving the quality of the entire railway system. The milestone reached with this thesis is still only the beginning for a model-based coproduction of planning and operation. The real challenge comes with the introduction of a decision support system for dispatching. The topic of rescheduling, i.e., the continuous adaptation of the current production plan to the real-time situation, opens a very interesting research area, with the additional difficulty compared to the offline situation given by the intrinsic properties of the stochasticity of disruptions, incompleteness of available information, and system dynamics, that were not present in the already very challenging train scheduling problem. This will set new limits for the current operations research methods while requiring an even stronger collaboration between the involved actors in planning, operation and process design.

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258 Appendix A Scenarios used as test cases Three test cases are set up to validate the concepts and algorithms of this work. The first scenario, presented in Section A.1, is the most important one and describes a network in central Switzerland which is used for testing all steps of the train scheduling algorithm presented in this thesis. It is either used as complete network or just partially, for instance only one line or a station region. The second scenario is described in Section A.2 and is the condensation zone of Berne, which is only used for testing the microscopic scheduling approach in condensation zones presented in Chapter 6, in addition to the first scenario. This scenario has a larger condensation zone compared to the ones in the first scenario and it is therefore more interesting for testing the performance of the algorithms. The third scenario, introduced in Section A.3, pictures the macroscopic network of the Ticino region, in southern Switzerland. This scenario is only used for computations of the Flexbox model and the bi-objective analysis of the FPESP model presented in Sections resp A.1 Network in central Switzerland The first scenario includes the cities Lucerne, Zug and Arth-Goldau as the major nodes in the network. Figure A.1 shows the geographical region in central Switzerland where the network is located. This network scenario is not very large, as it has a radius of about 25 km and consists of 4 lines, partially overlapping, in the interior of the network, and 7 diverging lines. Nevertheless, it is quite interesting because it included many changes from double to single track and junctions where lines from different directions join. Moreover, there is a mixture of freight trains, long distance and local passenger trains running on this part of the Swiss network.

259 232 Appendix A: Scenarios used as test cases Figure A.1: The region connecting the towns Zug Lucerne Arth Goldau in central Switzerland. As a reference scenario, the periodic service intention obtained by reverse-engineering the 2007 SBB timetable is used. It contains Intercity trains running from Baar (Zurich) and Sursee (Basel) to Erstfeld and the Gotthard tunnel through the Alps. Additionally, there are InterRegio trains running from Lucerne to Baar and to Biberbrugg (St. Gallen). Regional trains run in the triangle Lucerne Zug Arth-Goldau with several stops in between, as well as on all other lines described in Figure A.2 (b). Several slots for freight trains are reserved every hour on the double-track line Lenzburg Rotkreuz Immensee Arth-Goldau Gotthard in both directions, which is the main freight line between Germany and Italy with nearly the entire freight traffic passing through this corridor. A.1.1 Macroscopic topology The macroscopic track topology is shown in Figure A.2 and Figure A.3. All the events in the PESP model will correspond to departure or arrival times at stations in these figures. They are two alternative possibilities to describe the railway topology in this region on the macro level, depending on the level of detail which is to be considered. Figure A.2 is less detailed but enables faster computations. It can therefore be useful for comparing alternative service intentions on the strategic level. Using this topology the resulting macro timetable will not contain crossings resp. overtakings in minor stations that are not explicitly represented. These can be then taken into account again on the micro level, making them optional to use only if necessary, but not in a systematic way. On the other hand, Figure A.3 is more detailed and contains all changes from single to double track as well as all stations where it is possible to cross and overtake. The macro timetable on this topology will make use of these possibilities each time it allows to improve the objective function. Using this more detailed topology is more suitable during tactical and short-term planning, although it requires longer computation times.

260 A.1 Network in central Switzerland 233 Figure A.2: The macro topology of the test scenario in central Switzerland. It is partly double track and partly single track and is used by regional and intercity trains as well as freight trains. Figure A.3: A more precise macro topology of the network in central Switzerland. This version can be used in alternative to the one illustrated in Figure A.2.

261 234 Appendix A: Scenarios used as test cases A.1.2 Microscopic topologies In this network, three condensation zones were considered on the micro level, the zones around the three stations Lucerne, Zug, and Arth-Goldau, which coincide with the corresponding nodes in the macro topology. It is also conceivable to consider the area around Rotkreuz as a fourth condensation zone, although not a particularly large one, in order to clearly separate the lines (compensation zones) connecting them. All computations in this thesis are done without considering Rotkreuz as a condensation zone. The major main station area of the considered network is the area around Lucerne, which is also the larger city in central Switzerland. The terminal station has 12 platforms, and the considered condensation zone has a radius of about 6 kilometers, consisting of roughly 40 switches and 60 main signals, while serving five directions. Figure A.4 sketches the condensation zone of Lucerne, and Figure A.5 illustrates the track topology of the core part of the condensation zone Lucerne in detail. In all calculations the narrow-gauge infrastructure and trains were excluded. 6 1 Entlebuch Sursee - Olten Luzern Lenzburg Rotkreuz Immensee Figure A.4: Sketch of the the condensation zone Lucerne, located in central Switzerland. From Lucerne normal gauge trains can drive into five directions. Figure A.5: Micro topology of the core of the condensation zone Lucerne with its terminal station. Source: [Lüthi et al., 2008]. The longest connection between two condensation zones is the compensation zone between the portals of Lucerne and Zug, illustrated in Figure A.6. The line is 25km long,

262 A.2 Condensation zone of Berne 235 Figure A.6: Micro topology of the railway line between the portals of Lucerne and Zug. with two single track sections of roughly 5km and 3km. Although the zone is not large, the complexity for scheduling is demanding since it changes between single and double track sections and is used by a mixture of different train types. A.2 Condensation zone of Berne The second scenario is the condensation zone of Berne, which is the Swiss capital, geographically located in the center of the Swiss plateau. Berne is one of the largest condensation zones and a major hub of the railway network of Switzerland. The main station is a through station and has twelve platforms. The considered condensation zone of Berne has a radius of about 10 kilometers, consists of roughly 600 switches and serves 6 directions. Figure A.7 roughly illustrates the layout of the condensation zone Berne, while Figure A.8 illustrates the track topology of the switch region in front of main station Berne on the west side in detail. The detailed topology of the switch regions in the west and east side in front of the main station is illustrated in Figure A.8 and Figure A.9. An in- or outbound train traveling through the condensation zone has an average of 300 possible routes (maximum 1433) if the route reduction policy is not applied. Figure A.7: Sketch of the condensation zone Berne. Approximate distances given in meters. Both East and West side serve three directions. Source: [Burkolter, 2005].

263 236 Appendix A: Scenarios used as test cases Neuchatel/Fribourg/Belp Sidings yard Figure A.8: Switch region topology in the west side in front of Berne main station. Source: [Burkolter, 2005]. Sidings Biel/Olten/Thun Figure A.9: Switch region topology in the east side in front of Berne main station. On the left side a schematical illustration of the tracks and on the right side its representation as double vertex graph. Source: [Burkolter, 2005] resp. [Hürlimann, 2009]. A.3 Macroscopic network in Ticino The third scenario is the railway network in Ticino, an Italian speaking region in southern Switzerland depicted in Figure A.10. It connects the towns of Chiasso, Biasca, Locarno, and Luino, which is actually in Italy but the line is operated by the Swiss Federal Railways. The Gotthard-line north of Biasca to central and northern Switzerland will not be considered in this scenario. The track topology is divided in a main north south double track line and two minor single track lines to Locarno and Luino. The double track line is used by regional and intercity trains as well as freight trains, whereas in the single track lines have mainly regional traffic with some InterRegio trains. This scenario is only used for testing the Flexbox model in Section and for the bi-objective analysis on the macro level for the conflicting objectives of short travel times and large flexibility, explained in Section 4.6. Therefore, only the macro topology of this network is considered and illustrated in Figure A.11. All the events in the PESP model correspond to departure or arrival times at stations in this figure.

264 A.4 Overview on the usage of the scenarios 237 Figure A.10: The third scenario region connecting the towns Chiasso Biasca Locarno Luino in southern Switzerland. A.4 Overview on the usage of the scenarios All computations throughout the thesis refer to one of the three track topologies described above. The corresponding service intentions used for the various computations are described directly in the section where the results are presented, because they are not always identical and depend on the computational study. For the purpose of testing general properties of the models and algorithms, also some variants of the actually operated timetable are considered. Table A.1 summarises all scenarios used within the computational studies of this thesis.