Heuristic Model for Iberian Natural Gas Sources Location

Transcription

1 Heuristic Model for Iberian Natural Gas Sources Location TERESA NOGUEIRA 1, RUI MENDES 2, ZITA VALE 1 and JOSÉ C. CARDOSO 3 1 GECAD Knowledge Engineering and Decision Support Research Group Institute of Engineering, Polytechnique Institute of Porto Rua Dr. António Benardino de Almeida, Porto PORTUGAL tan@isep.ipp.pt, zav@isep.ipp.pt 2 EGP Porto School of Management University of Porto Rua de Salazares, Porto PORTUGAL RuiMendes@egp.up.pt 3 CETAV Centre for Technological, Environmental and Life Studies University of Trás-os-Montes and Alto Douro, Engineering Department Quinta de Prados Vila Real PORTUGAL jcardoso@utad.pt Abstract: - Because of the strong growth in Iberian Natural Gas industry, it is of crucial importance to find the most cost-effective way of locating the Gas Supply Units GSUs on primary network gas system. The right number of GSUs and their optimal location in gas network is a decision problem that can be formulated in order to minimize a cost function. The p-median problem is the classical model of locating P facilities on a network. In this paper we propose a lagrangean heuristic, an approach to solve the p-median problem. This heuristic is applied to the Iberian natural gas system, modelised with 65 demand nodes 18 Portuguese districts and 47 Spanish provinces. We present the results obtained for the location problem and the allocation of loads to gas sources, using a 2015 forecast natural gas demand scenario. Key-Words: NG natural gas, location model, gas supply units - GSUs, p-median problem, lagrangean relaxation, subgradiente optimization 1 Introduction Natural Gas has been the fastest growing component of world primary energy consumption among developing nations. To comply with the NG demand growth patterns and the Europe s import dependency, the Iberian Natural Gas industry needs to organize an efficient upstream infrastructure. The marine terminals, storage facilities infrastructures [1] and gas injection points, constitute the source points of the NG system, called in this work, Gas Supply Units GSUs. The right place to locate these facilities on a gas network is a location decisison-make issue. An important aspect of optimizing natural gas utilities operations is the art and science of locating GSUs to make the system more efficient and profitable. This involves large sums of capital resources and their economic effects are long term. The aim of this paper is the presentation of a location model applied to the Iberian high pressure transport gas, a system modelised with 65 demand nodes. For a growing demand scenario forecasted to 2015, we will study the optimal GSUs location on the NG network, minimizing expenses and maximizing throughput and security of supply.

2 In our model we consider that the best place to locate GSUs are nodes. Some of these nodes already contain an existing GSU which cannot be relocated. In literature, we can find different location models applied to different systems [2]. To deal with the realistic gas system, the mathematical models need to be extended, observing regulatory and technical restrictions. In case of natural gas systems, the choice of location problem is heavily dependent on the network size. For small networks, the fixed charge facility location problems can be used with satisfactory results [3]. For large networks, Beasley [4] describes very effective heuristics dealing with location problems. The search for p-median nodes on a network is a classical location problem that minimizes the sum of all distances from each demand point to its nearest facility. The p-median problem, due to its mathematical structure, is NP-hard and therefore cannot be solved in polynomial time. So, it is necessary to use software technologies or heuristics methods for large and realistic p-median problems. To apply this study to the Iberian network, we developed a lagrangean heuristic, using lagrangean relaxation and subgradient optimization to solve the dual problem [5]. With our location model, we intend to provide a decision support tool that helps to find the optimal number of GSUs and the right node to locate them on gas network. After this task we determine the most efficient allocation from demand points to gas sources, using an exact approach. In section two we explain how the Iberian gas network was modelised. Section three outlines an optimization-based approach to solve the p-median problem. The lagrangean relaxation applied to the p- median location model was inspired in Lorena works, [8] and [9]. The computational results are presented for the proposed location lagrangean heuristic and the allocation problem. We conclude confirming that this heuristic is an effective locational algorithm if we want to solve problems of realistic size in a reasonable amount of time. 2 Iberian Gas Network Model The Iberian high pressure (60 bar) natural gas network is defined in terms of sources (GSUs), demand points and junction nodes, that are connected by pipelines. The Iberian pipeline transport system has an extension of approximately 7500 km and links all the demand points connected in these physical pipes. The 65 demand points are aggregation loads, including electrical producers, distribution gas utilities, industrial, commercial and domestic consumers. These loads are geographically organized in 18 Portuguese districts and 47 Spanish provinces as shown in figure 1. We can see in figure 1 some demand points which are not connected in the physical pipelines. These are five Portuguese aggregation loads - Vila Real, Bragança, Évora, Beja and Faro, and four Spanish provinces Ávila, Almaria, Teruel and Albacete. These demand points are supplied by road trucks with gas in liquefied form - the called virtual pipelines. These virtual pipelines are, in fact, every connection between two nodes without a physical pipeline. To avoid design congestion they are not drawn in figure 1. The presented network is the real transport gas system, so, it has already some GSUs installed to supply the actual NG demand. These are drawn in figure 1 by nine circle points, they are the actual existing gas sources. Seven of these points are marine gas terminals and the other two - La Rioja and Cadiz - are supply injection gas points, respectively from France and Algeria. Because of economic reasons, these existing GSUs cannot be moved. So, our location model is asked to find the location of the new aditional GSUs to supply the growth forecasted natural gas demand for Gas Demand Forecast Analising the current 65 nodes gas demand and according to recent gas forecast studies [6], we can evaluate the natural gas demand values for This forecast is presented in the fourth column of table 1 (demand gas units are expressed in million cubic meter Mm 3 ). As referred, these values are aggregated loads for each geographic area. We should not be surprised if we find a higher gas demand in Porto than in Lisbon. The reason is that in Porto geographic node demand, an important electricity generator is included, which raises demand value. Near Lisbon, there is an important electricity generator but its demand is included in the Santarem geographic area. 2.2 Geographic data The fifth and sixth columns from table 1 present the 65 load points geographic coordinates, respectively,

3 latitude and longitude. With this information we can calculate the distance matrix [d ij ], which is an important parameter of the location model. Observing figure 1, we identify that there are 60 pipelines that interconnect the nodes. For the 65x65 distance matrix values, the 2 x 60 values are filled with the physical pipelines lengths. The others, are the Euclidian distances between each pair of nodes. In our program, this data can be replaced by the real road distance. So, the principal diagonal line numbers from distance matrix are zero and all the others are positive values. 2.3 Fixed Location Costs Most of gas sources need storage facilities infrastructures and their construction is not suitable or feasible in certain geographic locations. In our model we will consider an explicit cost of locating a GSU at each candidate location node, which is the necessary infrastructure investment to build a GSU. Because the most appropriate sites to locate the GSUs are seaside points, we consider lower prices for these points than for the interior points. For the results presented in this work we consider that, the necessary infrastructure investment, is 9000 thousands euro for seaside nodes and for interior nodes. For the nine circle nodes presented in figure 1, we consider a fixed cost of zero. This is a manipulation in order to maintain these location points in the model solution. Imposing zero fixed costs, the model will find them as an optimal location and they are the first nodes to locate. This is appropriate for our system because these GSUs cannot be relocated. Figure 1: Iberian Natural Gas System

4 Table 1: Demand nodes data Nodes Name Simbol Demand Latitude Longitude (Mm3) ( º ) ( º ) 1 Viana Castelo VCT 375,5 41,42-8,50 2 Braga BRA 1247,3 41,33-8,26 3 Porto PRT 7642,8 41,09-8,37 4 Vila Real VLR 335,7 41,18-7,45 5 Bragança BRG 223,3 41,19-6,45 6 Aveiro AVE 1070,6 40,38-8,39 7 Guarda GRD 260,6 40,14-7,16 8 Viseu VIS 592,5 40,78-7,87 9 Castelo Branco CTB 312,2 39,49-7,29 10 Coimbra CMB 662,0 40,12-8,25 11 Leiria LEI 689,3 39,75-8,48 12 Portoalegre PRL 190,6 39,03-7,43 13 Santarém STR 5682,6 39,14-8,41 14 Lisboa LSB 3204,7 38,73-9,15 15 Setúbal STB 1183,0 38,32-8,54 16 Évora EVR 260,2 38,50-7,90 17 Beja BEJ 241,9 38,01-7,52 18 Faro FAR 592,9 37,01-7,56 19 Corunha COR 3441,8 43,40-8,40 20 Lugo LUG 1268,6 43,00-7,30 21 Pontevedra PTV 2909,4 42,43-8,63 22 Orense ORS 1217,5 42,40-7,86 23 Astúrias AST 4074,3 43,20-6,00 24 Cantábria CAN 2621,0 43,20-4,00 25 Bilbao BLB 5446,6 43,15-2,55 26 San Sebastian SSB 4182,2 43,10-2,10 27 Vitoria VTR 3083,7 42,50-2,45 28 Leon LEO 1723,8 42,60-5,50 29 Palencia PLC 812,7 42,01-4,32 30 Burgos BUR 1342,7 42,35-3,07 31 Zamora ZMR 882,1 41,45-6,00 32 Valladolid VLD 1776,8 41,65-4,73 33 Soria SOR 584,7 41,46-2,28 34 Segóvia SGV 762,0 41,10-4,00 35 Ávila AVI 472,0 40,39-4,42 36 Salamanca SLM 1318,3 40,95-5,65 37 Córdoba CRB 3395,9 37,88-4,77 38 Huelva HLV 2546,5 37,25-7,00 39 Sevilha SEV 6305,1 37,40-6,00 40 Jaén JAE 3045,2 37,77-3,78 41 Cádiz CDZ 4516,1 36,50-6,28 42 Málaga MLG 5286,4 36,72-4,42 43 Granada GRN 3612,1 37,18-3,60 44 Almeria ALM 1730,2 36,83-2,47 45 Huesca HSC 1384,0 42,10-1,00 46 Zaragoza ZRG 3351,3 41,63-0,88 47 Teruel TER 398,7 40,21-1,06 48 Lérida LRD 3837,8 41,62 0,62 49 Tarragona TRG 4700,9 41,12 1,25 50 Barcelona BAR 17477,3 41,38 2,17 51 Gerona GRN 4586,8 42,10 2,40 52 Guadalajara GUA 1349,7 40,50-2,30 53 Cuenca CUE 1361,7 40,04-2,08 54 Albacete ALB 1086,9 39,00-2,00 55 Cidade Real CDR 2187,1 39,00-4,00 56 Toledo TLD 2464,5 39,52-4,01 57 Cáceres CAC 1165,8 39,29-6,22 58 Badajoz BDJ 1896,9 39,00-6,59 59 La Rioja RIO 1882,8 42,17-2,24 60 Madrid MAD 21497,1 40,24-3,41 61 Múrcia MUR 6870,7 38,00-1,10 62 Navarra NAV 2709,0 42,45-1,40 63 Castellon CTL 3427,6 40,00-0,02 64 Valencia VAL 8720,8 39,20-0,23 65 Alicante ALC 6787,3 38,35-0,50 TOTAL ,8 3 Location Algorithm 3.1 Applied P-median Problem To exemplify the application of lagrangean relaxation to the location p-median problem, we will consider the following binary integer programming problem, with m GSUs potencial sites and n demand nodes: m n ij ij (1) i=1 j=1 Z = Min α.d.x subject to: m X=1 ij j = 1,,n (2) i=1 m X= ii P (3) i=1 X ij ij X i = 1,..,m; j = 1,..,n (4) ii { } X 0,1 i = 1,..,m; j = 1,..,n (5) Where: α.[d ij ], is the symmetric cost [distance] matrix, with d ii = 0, i ; α is the kilometric gas transported cost; [X ij ] is the allocation matrix, with X ij = 1 if a node i is allocated to node j, and X ij = 0, otherwise; X ii = 1 if node i has a GSU and X ii = 0, otherwise; P is the number of GSUs (medians) to be located. The objective function (1) minimises the distance of each pair of nodes in network, weighted by α. Constraints (2) ensure that each node j is allocated to a source node i. Constraint (3) determines the exact number of GSUs to be located (P). Constraint (4) sets that a demand node j is allocated to a node i, if there is a UFG at node i. Constraint (5) states the integer conditions. In the Iberian gas system we assume that all demand points are also potential GSUs sites, resulting in the same dimension for indexes, i. e., m = n. The parameter value α is the kilometric cost of natural gas transported by physical pipelines, with natural gas in its gas form. The value β, is the same cost for gas transported in virtual pipelines. The β cost is usually greater than α and both values will influence the locating solution of GSUs on natural gas network. These different transport costs are implicitly assumed in the algorithm. To simplify mathematical formulation, these values α and β are included in distance matrix, obtaining [D ij ]: n n ij ij (6) i=1 j=1 Z = Min D.X

5 subject to (2), (3), (4) and (5) To solve this p-median problem (6) applied to Iberian gas system, we propose a lagrangean heuristic, based upon lagrangean relaxation and subgradient optimization. 3.2 Problem Relaxation We can relax the original problem Z by eliminating constraint (2) and including it in the objective function, weighted by the lagrangean multipliers λ i. The resulting relax problem is: n n n n Z Rel = Min D ij.xij λi X-1 ij i=1 j=1 i=1 j=1 or, written in a different way: n n n Z Rel = Min ( Dij λ i ).Xij + λi (7) subject to (3), (4) and (5). i=1 j=1 i=1 The resulting problem Z Rel is the relax or lagrangean problem and λ i are the parameters of the multiplier lagrangean vector, a non-negative vector. For a given λ > 0, problem (7) is solved considering implicitly constraint (3). The problem is solved for different values of P, until reaching the right number of GSUs that should be located to minimise costs. It can be easily demonstrated [7] that ν(z Rel ) ν(z). So, we can find that ν(z Rel ) is the inferior bound of the optimal solution to the original problem (6). The next step is to find the upper bound. It can be founded by solving a lagrangean dual, the problem that maximises the relax problem: W = Max Z Rel (λ) (8) Due to its nature, problem (8) has some properties that make it difficult to find a dual solution gap duality. We propose a lagrangean heuristic, combining the subgradient optimization with the feasibility of the solution founded in each iteration. In this case, we obtain a feasible solution which can be improved step by step. These upper and lower bounds found for the p- median objective function will reduce the required computational cost of the original problem Z. 3.3 The Lagrangean Heuristic The following algorithm is used as the base to the proposed lagrangean heuristic. Consider the next terminology for the problem: C set of nodes already fixed: C = {i X ii =1} UB upper bound for problem (6) LB low bound for problem (6) X λ - relax problem solution X f relax problem solution after feasibility ν f relax problem value after feasibility ν(z Rel ) relax problem value for a given λ Lagrangean Heuristic Given λ > 0; Set LB = +, UB = +, C = 0; Repeat Solve Z Rel, obtaining X λ and ν(z Rel ); Obtain a feasible solution, X f and its value ν f (SolutionFeasibility); Update LB = max [LB, ν(z Rel )]; Update UB = min [UB, ν f ]; Fix X ii = 1 if ν(z Rel X ii = 0) UB, i N-C Update C: n Set g λ λ j = 1 - Xij, j N ; i=1 Update the step size θ (Updateθ): Set λ j = max {0, λ j + θ.g λ j }, j N Until (StoppingCriterion) The three auxiliary procedures bold outlined in the main algorithm, are briefly described. The SolutionFeasibility identifies the median nodes that can be used to produce feasible solutions to original problem Z. As we know x λ is a solution to Z Rel but not necessarily feasible to Z. In this procedure, the median nodes are reallocated to their nearest medians producing x f. The step sizes used in Updateθ procedure are: π ( LS - LI ) θ = λ 2 g The control of parameter π is made using the classic control [8]. The StopCriterion procedure used is: a) π 0.005; b) LS LI < 1; c) λ 2 g 0 = ; d) Every GSUs were fixed

6 We have implemented this heuristic approach to the Iberian gas system, with the data presented in table 1, which is a realist forecast instance of The heuristic result will inform us about the right number P to locate the GSUs and their optimal location on the network. After these medians are located, it is necessary to allocate the demand points to the gas sources. With the location problem already done, the inherent problem of the allocation will have is computational dimension reduced. To solve the allocation problem we use the capacitated fixed charge GSU location problem [3], because it was considered that the GSUs production capacity cannot be exceeded. 3.4 Location Model Results In table 2 we can see the behavior of the implemented location-allocation problem, for α = and β = /m 3 /km. The lagrangean heuristic solution reports 16 gas source locations, including the first 9 existing GSU s in the Iberian gas system. As we imposed zero fixed costs, the algorithm found it attractive to install GSUs at these points. The additional 7 GSUs installed complete the optimal location solution for the stated problem. For each GSU located we can verify the optimal demand allocation and the adequate transport mode, respectively in the third and fifth columns. For demand points included in the physical system, the gas is injected by high pressure pipelines. The other demand points are supplied by appropriate gas trucks. The sixth column summarizes the total gas demand allocated to each GSU in order to completely satisfy the load point (compare last rows from tables 1 and 2). The last column reports the total cost for locating each GSU site, which includes the fixed cost and the associated NG transport cost (which depends on the allocation result). This can be better understood by observing figure 2. There we can visualize the gas load allocation to the GSUs. The 16 candidate nodes are represented by a triangular shape; the other 49 load nodes by bold points. The total 65 nodes are identified by their symbols (presented in the third column from table 1). For each GSU located we can visualize the demand allocation result. In figure 2, the dash lines represent the gas transport made by truck and the solid ones, the pipeline transport mode. As expected, to minimize transport cost, the loads are allocated to the nearest gas sources. We can find GSUs with a great number of allocated gas loads, like GSU 11 Madrid and GSU 15 Santarém, and GSUs with fewer loads. The GSU 10 Cadiz and GSU 14 Malaga have no gas load allocated. In these cases, the associated transport costs are zero. The GSU 10 is an existing gas source, so, we consider that it hasn t any fixed cost, resulting in a total cost of zero for this location. In table 2, all the demand points in shadow rows are self supplied nodes, i. e., they are simultaneously source and demand nodes. To minimize expenses each source point should supply its own demand. With this arrangement, the total demand is completely satisfied with minimum expenses. GSU Nº Location Node Table 2: Location-allocation results Demand Allocation (nodes) Simbols Transport Mode Demand Allocation (Mm3) Total Cost (thousand euros) 1 Setúbal Setúbal STB , ,40 Lisboa LSB pipe Beja BEJ truck Évora EVR truck 2 Corunha Corunha COR , ,30 Lugo LUG pipe Pontevedra PTV pipe Orense ORS pipe 3 Bilbao Bilbao BLB , ,00 Vitoria VTR pipe San Sebastian SSB pipe 4 La Rioja La Rioja RIO , ,50 Navarra NAV pipe Zaragoza ZRG pipe Huesca HSC pipe Burgos BUR pipe Palencia PLC pipe 5 Barcelona Barcelona BAR , ,50 Lérida LRD pipe Tarragona TRG pipe Gerona GRN pipe 6 Valencia Valencia VAL , ,90 Castellon CTL pipe 7 Cuenca Cuenca CUE , ,50 Teruel TER truck Albacete ALB truck 8 Murcia Murcia MUR , ,00 Alicante ALC pipe 9 Huelva Huelva HLV , ,70 Faro FAR truck Sevilla SEV pipe 10 Cadiz Cadiz CDZ ,20 0,00 11 Madrid Madrid MAD , ,90 Valladolid VLD pipe Soria SOR pipe Segovia SGV pipe Avila AVI truck Guadalajara GUA pipe Toledo TLD pipe Cidade Real CDR pipe 12 Asturias Asturias AST , ,80 Cantabria CAN pipe Leon LEO pipe Zamora ZMR pipe Salamanca SLM pipe 13 Porto Porto PRT , ,70 Braga BRA pipe Viana do Castelo VCT pipe Vila Real VLR truck Bragança BRG truck Aveiro AVE pipe 14 Malaga Malaga MLG , ,00 15 Santarem Santarem STR , ,10 Guarda GRD pipe Viseu VIS pipe Castelo Branco CTB pipe Coimbra CMB pipe Leiria LEI pipe Portoalegre PRL pipe Caceres CAC pipe Badajoz BDJ pipe 16 Jaen Jaen JAE , ,40 Granada GRN pipe Cordoba CRB pipe Almeria ALM pipe TOTAL , ,70

7 Figure 2: GSUs allocation results 4 Conclusion Our work aims at studying the gas source unit location-allocation problem applied to the Iberian natural gas transportation system. Because of the forecasted growth in natural gas demand in Portugal and Spain, more GSUs need to be installed in the Iberian market to completely supply all customers. For the location resolution we present a lagrangean heuristic to solve the p-median problem. With its gratifying results, the allocation problem finds its computational size decreased and therefore we can use an exact approach to assign the demand nodes to the located GSU. The location model proposed shows a good performance in a reduced computational time of few seconds. Apart from that, the developed algorithm is able to capture the richness of the real Iberian network with a proved accuracy of maintaining the existing facilities and installing additional ones. This heuristic location model can be applied to larger gas networks with the same good performance. Acknowledgements The authors would like to acknowledge FCT, FEDER, POCTI, POSI, POCI and POSC for their support to R&D Projects and GECAD Unit. References: [1] Nogueira, T., Vale, Z. and Cordeiro M., Trends in Energy Markets and Systems: The Case of Natural Gas and Facility Location Techniques, 9CHLIE, Spain, R-220, 2005, pp [2] Daskin, M. Network and Discrete Location: Models, Algorithms and Applications, John Wiley & Sons, 1995 [3] Nogueira, T., Vale, Z. and Cordeiro, M., Advanced Techniques for Facility Location Problems in Natural Gas Networks, ICKEDS 06, Lisbon, 2006, pp [4] Beasley, J. E. Lagrangean Heuristics for Location Problems, European Journal of Operational Research, 1993, vol.65, pp [5] Lorena, L. and Narciso, M., Relaxation Heuristics for a Generalized assignment Problem, European Journal of Operational Research, 1996, vol 91, pp [6] Ministério de Industria, Turismo Y Comercio, Futuro del Sector de Gas Natural, 2006, [7] Fisher, M. L., The Lagrangian Relaxation Method for Solving Integer Programming Problems, Management Science, 1981, 27(1): 1-18 [8] Senne, E. and Lorena, L., "Lagrangean/Surrogate Heuristics for p-median Problems", in Computing Tools for Modeling, Optimization and Simulation: Interfaces in Computer Science and Operations Research, Kluwer Academic Publishers, 2000, pp [9] Lorena, L. and Narciso, M., "Relaxation heuristics for a generalized assignment problem", European Journal of Operational Research, 1996, 91(1),