Transportation Research Part B

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1 Transportation Research Part B 45 (211) Contents lists available at ScienceDirect Transportation Research Part B ournal homepage: A generic class of first order node models for dynamic macroscopic simulation of traffic flows Chris M.J. Tampère *, Ruben Corthout, Dirk Cattrysse, Lambertus H. Immers Department of Mechanical Engineering, CIB/Traffic & Infrastructure, Katholieke Universiteit Leuven, Celestinenlaan 3A, PO Box 2422, 31 Leuven, Belgium article info abstract Article history: Received 1 November 28 Received in revised form 16 June 21 Accepted 16 June 21 Keywords: Macroscopic node model Dynamic network loading Dynamic traffic assignment Node models for macroscopic simulation have attracted relatively little attention in the literature. Nevertheless, in dynamic network loading (DNL) models for congested road networks, node models are as important as the extensively studied link models. This paper provides an overview of macroscopic node models found in the literature, explaining both their contributions and shortcomings. A formulation defining a generic class of first order macroscopic node models is presented, satisfying a list of requirements necessary to produce node models with realistic, consistent results. Defining a specific node model instance of this class requires the specification of a supply constraint interaction rule and (optionally) node supply constraints. Following this theoretical discussion, specific macroscopic node model instances for unsignalized and signalized intersections are proposed. These models apply an oriented capacity proportional distribution of the available supply over the incoming links of a node. A computationally efficient algorithm to solve the node models exactly is included. Ó 21 Elsevier Ltd. All rights reserved. 1. Introduction Simulation-based dynamic traffic assignment (DTA) models can be applied for various researches on road traffic (e.g. reliability studies, traffic management, road pricing and network planning). In DTA models, traffic is propagated through the network by a dynamic network loading (DNL) model. DNL models separately treat traffic flows on links and through nodes in a link model and a node model, respectively. Flow propagation on links has been extensively studied and various adequate link models exist. Though being an equally important component of DNL models (especially in congested networks), node models have attracted much less attention in the literature. As a result, the effect of the node model on shockwave propagation and congestion spillback is often not satisfactorily incorporated in DNL models. Node models have two functions in DNL models. The first is to impose constraints on the outflow of each incoming link (limited supply of the node itself or node supply constraints); the second to seek consistency between the demand and supply constraints imposed by the incoming and outgoing links (and the node itself), respectively. Constraints imposed by the node inherent to the presence of traffic lights, conflicts between crossing flows, etc. have been studied for several decades. Chapters 8 and 9 in Gartner et al. (2) summarize earlier work. These formulations assume an uncongested traffic state downstream, ignoring the possibility of congestion spillback over the node. This renders them not directly compatible with macroscopic node models for DNL applications. Therefore, node supply constraints are generally not considered in state-of-the-art macroscopic node models. At best, the process responsible for the capacity restrictions (e.g. signal phases) is explicitly modeled. * Corresponding author. Tel.: ; fax: address: chris.tampere@cib.kuleuven.be (C.M.J. Tampère) /$ - see front matter Ó 21 Elsevier Ltd. All rights reserved. doi:1.116/.trb

2 29 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) In order to guarantee consistency between demand and supply constraints, a distribution of the available downstream supply over the incoming links has to be determined in case of congestion. Hereby the various constraints interact with each other and with the flows transferred over the node, which is captured by what is referred to in this paper as a supply constraint interaction rule (SCIR). This rule should represent the aggregate driver behavior at a congested intersection. This paper presents a critical review of state-of-the-art node models, highlighting both their contributions and shortcomings. Based on that, a set of requirements for macroscopic node models is compiled, herewith defining a generic class of first order macroscopic node models. 1 This node model class is not only applicable to intersections of vehicular traffic flow. In principle, it is transferrable to any kind of multi-commodity flow 2 as long as the commodities have the propensity to move whenever possible. Node model instances for specific intersections are obtained by introducing a SCIR and node supply constraints. By doing so, this paper gives the theoretical onset for realistic and consistent macroscopic node model formulations that combine internal node constraints and up- and downstream link boundary constraints within a single generic class. This class encompasses all types of nodes, ranging from simple merges and diverges to roundabouts, priority and signal controlled intersections. From this generic class, two specific node model instances (including an efficient solution algorithm) are derived: one for priority unctions and one for signalized intersections. Despite some simplifications, both models are superior to all state-ofthe-art node models. The paper is structured as follows. Section 2 provides an overview of the state-of-the-art on macroscopic node models. Section 3 discusses a set of requirements, formulating a generic class of macroscopic node models. In order to define specific node model instances, the SCIR (and node supply constraints) needs to be specified (Section 4). In Section 5 specific node models for unsignalized and signalized intersections are presented, accompanied by an efficient solution algorithm. Conclusions and future research directions are formulated in Section State-of-the-art on macroscopic node models Many macroscopic node models are generalizations of simple merge and diverge models. The latter have been described by, e.g. Daganzo (1995), Lebacque (1996) and Jin and Zhang (23). As an introductorily step, Daganzo s merge and diverge models are outlined, as well as the merge distribution schemes of Jin and Zhang and of Ni and Leonard (25). Section 2.3 provides a numerical example, illustrating some of the less obvious shortcomings of state-of-the-art models. The time dimension (t) is omitted in the equations for notational convenience. All equations hold for one time step in discrete models or for one point in time for continuum models Merge and diverge models For the coherence with the remainder of the paper, abstraction is made of the division of links into cells in Daganzo s cell transmission model and his models are explained in terms of links Merge model The merge model connects two incoming links i (i = 1, 2) to one outgoing link, maximizing the total flow q: Max q ¼ q 1 þ q 2 Let us first define demand constraints S i formally as the maximum flow that incoming link i could possibly send if the node and outgoing link would impose no constraint whatsoever on the outflow of link i (i.e. as if link i was directly connected to a reservoir with infinite capacity instead of being connected by the node to the outgoing link). Obviously, the demand constraint is determined purely by traffic conditions inside link i. Analogously, the supply constraint R is the maximum inflow that outgoing link could receive if the node and link(s) upstream would impose no inflow constraint whatsoever (i.e. as if it had been connected to a reservoir capable of sending an infinite flow into link ). Also the supply constraint is determined purely by traffic conditions inside link and is therefore a constraint external to the node model. The demand and supply constraints of a merge model express that the total inflow q into the outgoing link is constrained by supply R, and the outflows q 1 and q 2 by the demands S 1 and S 2, respectively. q 1 6 S 1 ; q 2 6 S 2 ; q 1 þ q 2 6 R ð2þ ð1þ 1 The node models discussed in this paper are limited for use with first order link models, which consider only one independent variable (e.g. flow). Higher order models, like Metanet (Messmer and Papageorgiou, 199), need more complex node models that, in addition to transferring flow, also transfer speeds or momentum. The restriction to first order models is ustified because in interrupted flow the influence of capacity restrictions at intersections is dominant, rendering second order traffic flow phenomena (instability, capacity drop, stop and go traffic) not relevant. For uninterrupted flow, second order effects can be relevant, but then node models connecting three links (merges and diverges) suffice. 2 Multiple commodities are not indifferent with respect to their destination or route. Examples are vehicular flow, mail networks, container transportation and telecommunication networks.

3 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) It is straightforward to see that (1) and (2) imply q = min{s 1 + S 2,R}. In case the demand exceeds the supply, a queue will form on at least one of the incoming links. Daganzo introduced distribution fractions d i (d 1 + d 2 = 1) to reflect priorities. A share of the supply d i R is assigned to each link i. IfS i < d i R for one i, the remaining or reduced supply e R ¼ R S i is appointed to the other link i, so that q = R. Summarizing, three regimes can be identified corresponding to the following solutions: ð1þ Free flow state on both incoming links : q 1 ¼ S 1 ; q 2 ¼ S 2 ; q ¼ S 1 þ S 2 ð2þ Free flow state on i; congestion on i : q i ¼ S i ; q i ¼ R S i ; q ¼ R ð3þ Congestion on both incoming links : q 1 ¼ d 1 R; q 2 ¼ d 2 R; q ¼ R ð3þ Jin and Zhang (23) incorporated a fairness requirement into Daganzo s merge model, with distribution fractions proportional to the demands as in (4) (demand proportional distribution). Many node models are based on this assumption (see Section 2.2). Demand proportional models can only yield solutions of the first and third regime d 1 ¼ S 1 ) q S 1 þ S 1 ¼ minðs 1 ; d 1 RÞ 2 d 2 ¼ S 2 ) q S 1 þ S 2 ¼ minðs 2 ; d 2 RÞ 2 ð4þ Ni and Leonard (25) introduced distribution fractions proportional to the capacities C i of the incoming links (capacity proportional distribution) into Daganzo s merge model, also making the extension to a general merge with any number of incoming links i. For a traditional merge model as discussed in this section (with two incoming links) the distribution fractions of Ni and Leonard s model are represented by (5). Although the distribution fractions are defined differently, the solution space of Ni and Leonard s model is equivalent to that of Daganzo, encompassing all three regimes in (3). Ni and Leonard supported the validity of their model with empirical tests at a freeway merge d 1 ¼ C 1 C 1 þ C 2 ; d 2 ¼ C 2 C 1 þ C 2 ð5þ To represent the flows as a function of upstream variables, a level of reduction a is introduced. As an example, this level of reduction for the merge model of Jin and Zhang is given by (6). This notation is used in the remainder of the paper; it is straightforward to see that it is equivalent to (4) a ¼ R ) q S 1 þ S 1 ¼ minð1; aþs 1 ; q 2 ¼ minð1; aþs 2 ð6þ Diverge model A diverge connects one incoming link to two outgoing links ( = 1, 2). Again the total flow q is maximized (1); q now being the outflow of the incoming link, which is divided over the outgoing links according to turning fractions f (f 1 + f 2 = 1). These can be fixed or variable to the route choice. It is assumed that vehicles exit in a FIFO sequence, delaying successive vehicles regardless of their destination. This assumption usually suffices for the level of accuracy required for large scale macroscopic modeling. 3 Consequently, if the inflow into one of the links is constrained, the inflow into the other is constrained accordingly (as in Newell (1993)). The demand and supply constraints can be formulated as: q 1 ¼ f 1 q 6 R 1 ; q 2 ¼ f 2 q 6 R 2 ; q ¼ q 1 þ q 2 6 S ð7þ From (1) and (7) the following solution results: q ¼ minfs; R 1 =f 1 ; R 2 =f 2 g Analogously to (6), demand proportional levels of reduction can be defined for both outgoing links. Then, solution (8) can be written as: a 1 ¼ R 1 f 1 S ; a 2 ¼ R 2 f 2 S ) q 1 ¼ minð1; a 1 ; a 2 Þf 1 S; q 2 ¼ minð1; a 1 ; a 2 Þf 2 S ð9þ ð8þ Overview of macroscopic node models The node models in the following represent general intersections with n incoming links i (i = 1,..., n) and m outgoing links ( = 1,..., m), for which flow distribution is significantly more complicated than for merges and diverges, yielding flows from every i to every, denoted as q i (q i ¼ P q i and q ¼ P i q iþ. The definitions of partial demands S i and turning fractions f i are 3 The FIFO assumption neglects the separation of traffic in different turning lanes. Thus, congestion on one turn immediately affects the other turns. If this assumption is unacceptable for some application, one should consider the beginning of the turning lanes as the diverge point and model an extra node there.

4 292 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) analogous. These partial demands S i can either be derived from the total demand S i (delivered by the link model) and fixed turning fractions at the node level, or be directly provided by the link model (if this computes the demand towards each separately). In the latter case, the turning fractions are variable to the demand and result from the ratios of each S i and S i.in both cases, the relationship between f i, S i and S i can be written as f i ¼ S i S i. One of the more simplified models include that of Holden and Risebro (1995). They defined an entropy condition that maximizes the total flow, subect to the following constraints: 6 q i 6 S i ; 6 q 6 R 8i; ; X q i ¼ X i q Turning fractions are not considered, assuming drivers choose their destination link solely based on the principle of least resistance. We find this to be an oversimplification that does not properly represent vehicular traffic. Herty and Klar (23) adopted this assumption and formulated a multilane model. Coclite and Piccoli (25) applied a similar approach as Holden and Risebro, maximizing the total flow subect to the constraints in (1) and considering fixed turning fractions so that q i = f i q i. However, they need to impose the artificial constraint f i f i (for each, with i i) to guarantee a unique solution. Lebacque and Khoshyaran (25) introduced the invariance principle to avoid discontinuous changes in the flows. This principle states that under constant demand and supply constraints, flows should be invariant during an infinitesimal time step. An example of violation of the invariance principle is provided in Section 2.3. Following this principle, Lebacque and Khoshyaran assumed that demand and supply cannot be linked directly in order to find the flows. Their model thus incorporates node demands and supplies, derived from a global zone fundamental diagram (Buisson et al., 1995, 1996a,b; Lebacque, 23). The node supply is distributed over the incoming links proportional to the number of lanes. However, deriving a global zone fundamental diagram for each node does not seem practicable since this would require detailed information about every intersection. This cumbersome procedure can be omitted, since the models presented in this paper do link demand and supply directly without violating the invariance principle (so do some of the models proposed in Adamo et al. (1999) and the model by Gentile et al. (27)). Jin (21) incorporated interior states into merge models so that asymptotically the invariance principle is always satisfied, which unnecessarily renders the problem more complex since satisfaction of the invariance principle can also be assured without these interior states, as is shown in the remainder of this paper. Several researchers have suggested node models that generalize the demand proportional distribution scheme of Jin and Zhang (23), all of which do not satisfy the invariance principle. In Jin and Zhang (24) the merge and diverge models are merely weld together, with the outflow of an incoming link being the demand proportional part of the total flow. A similar model is presented in Nie et al. (28). Both of these models do not obey FIFO conservation of turning fractions (CTF), which requires that all flows of link i are coupled, with turning fractions equal to those in the demand (see Section 3.1): ð1þ f i ¼ S i S i ¼ q i q i ð11þ Bliemer (27) formulated a node model in which the total flow is maximized subect to the constraints defined in (1) and (11). The turning fractions f i are variable to the partial demands S i, provided by the link model. Bliemer stated that solving this maximization problem results in a demand proportional distribution, in which each flow q i is limited according to either demand or its most restricted (demand proportionally distributed) supply. In the numerical example of Section 2.3 however, it is demonstrated that this node model does not always maximize the flow with regards to the available supply. A very similar node model was suggested earlier by Rubio-Ardanaz et al. (21). To ensure that the invariance principle is satisfied, the distribution of supply must be independent of the ratio of the demands S i. Therefore, node models that generalize the distribution scheme of Daganzo (1995) based on priority constants or that of Ni and Leonard (25) based on capacities are valid in that sense. Node models that fall into these categories include that of Gentile et al. (27) and some of the models of Adamo et al. (1999). Adamo et al. present, e.g. two models with fixed turning fractions, one of which applies CTF yet does not maximize flows, while the other maximizes flows without ensuring CTF. Gentile et al. formulated a SCIR similar to the one presented in Section 5.1, applying a distribution based on capacities. Analogously to the numerical example in Section 2.3 it can be shown that their model does not always maximize flow within all constraints Numerical example The invariance principle and the difficulty of flow maximization are illustrated by means of Bliemer s model applied to a numerical example. This model defines for each outgoing link a level of reduction a equal to the ratio of R and the total demand P i S i towards : a ¼ R P i S ; 8 ð12þ i

5 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) If several links impose a reduction on a link i, the smallest a determines all q i from i (CTF). For each i, there is thus one that imposes the strongest supply constraint on i (denoted as * (i)). The corresponding a determinant for iis denoted as a ðiþ. Hereby, the assumption is made that drivers who are unable to enter their destination link do not hinder flows from other incoming links. This means that each a is only imposed on links ithat wish to send flow towards (expressed by "S i >in (13)). a ðiþ ¼ min a ; 8i ð13þ 8S i > Note that in Bliemer s model, i is only demand constrained if a P 1 "S i >. The flows q i are then determined as follows: q i ¼ minðs i ; a ðiþ S i Þ; 8i; ð14þ For this example a standard general intersection is considered, with four incoming and four outgoing links (Fig. 1). The demand is specifically set up in order to demonstrate the violation of the invariance principle and flow maximization. The partial demands are exhibited in Table 1, in vehicles per hour (veh/h). Assuming no congestion is spilling back from the outgoing links, R = C. From Table 1, it can easily be derived that for outgoing links 7 and 8: P i S i > R and a 7 = a 8 =.8. All demands are thus reduced with a factor.8, yielding the following solution according to (14) (see Table 2). Congestion thus starts building up on all incoming links. In the congested regime, S i = C i (discharge rate at the head of the queue at link capacity). Conserving the turning fractions, the partial demands of Table 3 apply after an infinitesimal time increment. From Table 3 results that a 7 =.649 and a 8 =.678. From (13) follows that a 7 holds for links 1, 2 and 4, while a 8 holds for link 3. Again applying (14). The invariance principle is now violated by incoming link 1. Table 4 shows that q 1 = veh/h. Since this is higher than the demand at the tail of the queue, which is still 5 veh/h (assuming unchanged conditions upstream), the queue on link 1 Fig. 1. Standard general intersection. Table 1 Partial demands S i. Table 2 Flows q i (initial solution).

6 294 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) Table 3 Partial demands S i after infinitesimal time increment. Table 4 Flows q i after infinitesimal time increment. will dissolve. This node model produces a discontinuous solution, with the queue alternatively growing and dissolving under constant link boundary conditions. Furthermore, inconsistency can be observed for outgoing link 8, which is underutilized with an inflow of only 1936 veh/h (<R 8 ). Yet, q 3 is restricted assuming an insufficient supply R 8. Since the other competing flows for R 8 (q 18 and q 28 ) are restricted according to a 7 (<a 8 ), a bigger share of R 8 should have been attributed to q 38. In general: if flow q i of link i is constrained in direction (= * (i)), flows q i in other directions are constrained accordingly (CTF). Due to this restriction on q i, the other flows towards experience less competition for R and the share of R that is not claimed by i (due to its external constraint in ) should be distributed among the other flows towards. We conclude that in order to maximize flow consistent with the prevailing distribution scheme, it is necessary to consider the interaction of the various supply constraints when determining the flows. Therefore, a mere distribution scheme is insufficient, but rather a supply constraint interaction rule (SCIR) needs to be formulated (see Section 4.2). 3. Requirements defining a generic class of first order macroscopic node models The models described in Section 2.2 either produce results that violate certain conditions or that are illogical in some cases (contradicting the assumptions made in the model or traffic observations) apart from the model of Lebacque and Khoshyaran (25), which does not seem feasible in large scale macroscopic DNL due to the necessity to determine node demands and supplies. From the literature study, a list of requirements that need to be satisfied by any first order macroscopic node model in order to yield a realistic, consistent solution is composed Requirements 1. General applicability, irrespective of the number of incoming and outgoing links Node models for general intersections should be applicable to any combination of number of incoming links (n) and outgoing links (m). Note that this requirement is by definition not met by merge or diverge models. However, adequate merge and diverge models (that comply to all other requirements listed in this chapter) exist in the literature (e.g. Daganzo, 1995; Ni and Leonard, 25). 2. Maximizing flows The maximization of flows can be seen as an extension at the node level of the maximization of entropy presented by Ansorge (199). It follows from the fact that drivers will always try to move whenever possible. This means that the maximization is to be understood as each flow should be actively restricted by one of the constraints, otherwise it would increase until it hits some constraint. These constraints are partly generic in nature (listed in this section) and partly specific for the type of node that is considered (discussed in Sections 4.1 and 4.2). Note that the additional SCIR constraints that dictate the distribution of supply (see Section 4.2) should reflect the prevailing aggregate driving behavior. As a consequence, as drivers are selfish and thus seek for a user optimum, maximization under this non-cooperative, selfish distribution scheme will not yield the absolute maximum (or system optimum) total flow that could possibly be transferred over the node.

7 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) Although most state-of-the-art node models assume maximization of flows, the solution algorithms that were proposed often violate this requirement, as was illustrated in Section 2.3. In order to fulfill this requirement, the supply distribution scheme needs to be imbedded in a SCIR (see Section 4.2). 3. Non-negativity Traffic never flows backwards and therefore all flows need to be non-negative. 4. Conservation of vehicles Since flow neither disappears nor is created atan intersection, the outflow of the incoming links must equal the inflow of the outgoing links, both in terms of total flow P i q i ¼ P q as in terms of partial flows, i.e. q i coming from i and heading towards equals q i going into and originating from i. In this paper, conservation of vehicles is implicitly guaranteed by defining only one flow q i between i and and defining the outflow of i as q i ¼ P q i and the inflow into as q ¼ P i q i. 5. Satisfying demand and supply constraints The demand constraints state that the outflow q i from an incoming link i can never exceed the demand S i at the downstream boundary of i. Demand constraints are imposed only on the flows from i. The supply constraints state that the inflow q into an outgoing link can never exceed the supply R that is available at the upstream boundary of. Such constraints are typically imposed on all flows competing for R (i.e. is i > ). 6. Obeying conservation of turning fractions (CTF) Drivers have the intention of traveling towards a specific destination, following a certain route. As a result, it is necessary to consider this route choice instead of merely maximizing flows irrespective of the destination link after the intersection. The latter would be acceptable for fluids or any other commodity having no preference for a route, but not for vehicular traffic with autonomous route choice. In order to ensure first-in-first-out (FIFO) on an incoming link, it is required that traffic flows out of this link and into different outgoing links in the same order they reached the end of the incoming link. Vehicles that are unable to exit into their preferred outgoing link prevent all those behind, regardless of destination, to continue. In other words, if either one of the outgoing links is unable to accommodate its allocation of flow, all outflow is restricted accordingly (Daganzo, 1995). It is trivial to see that this FIFO requirement is equivalent to requiring conservation of turning fractions (CTF). At the end of the link i, the total demand S i can be split into partial demands S i to each. In order to ensure that a constraint on one of these partial flows also restricts the others accordingly, the turning fractions f i ¼ S i in the demand S i should be conserved in the resulting flows, i.e. q i = f i q i. Because of this requirement, the q i are mutually coupled. Note that from the perspective of the node model, it is not relevant how the turning fractions are determined at the link end. One could consider fixed turning rates, fixed route flows (requiring the link to distinguish between separate route subflows), fixed destinations with some local route choice logic like dynamic user optimal routing (Ran et al., 1993) requiring the link to distinguish between separate destination subflows turning rates imposed onto the drivers by some controller (e.g. by police officers in the case of evacuations), etc. If fixed turning fractions are chosen (e.g. based on number of turning lanes), CTF is obviously guaranteed. However, whether driver s route choice is adequately modeled using fixed turning fractions is questionable. In Adamo et al. (1999) CTF was introduced as an optional requirement. However, we feel this requirement to be strictly necessary, since otherwise the node model would unrealistically favor some flows over others in order to maximize flow. 7. Compatibility with link traffic flow dynamics: satisfaction of the invariance principle If in the solution of the node model q i < S i, i enters a congested regime. As a consequence of traffic flow dynamics, the demand of i (at the downstream link end) increases after some infinitesimally small time increment to the queue discharge rate (or link capacity) C i. Any node model that would predict a different value for q i because of this change from S i to C i contradicts its own initial solution and is said to violate the invariance principle (Lebacque and Khoshyaran, 25). In order to be compatible with link traffic flow dynamics, a node model should therefore yield solutions that are invariant to replacing S i by C i if q i is supply constrained (q i < S i ). Lebacque and Khoshyaran analogously defined the invariance principle for supply. This condition is automatically satisfied in this paper (and also in other node models found in the literature) because the solution is derived by distributing the supply and not the demand. Therefore, the invariance principle for supply is not relevant here and any further reference to the invariance principle aims at demand. An example of a violation of the invariance principle was shown in Section 2.3. It must be stressed that any node model applying a demand proportional supply distribution can never satisfy the invariance principle. Any macroscopic node model for vehicular flows should comply with requirements 2 7, irrespective of the type of intersection (highway merge, highway diverge, (un)signalized intersection, roundabout) and irrespective of the driver behavior that might vary for instance due to visibility, signposting, legislation, interpersonal or cultural differences (e.g. urge to enter the intersection if the outflow is blocked) Mathematical formulation The following formulation defines a generic class of first order node models for macroscopic simulation of vehicular traffic flow over intersections:

8 296 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) max s:t: X i! X q i non negativity : q i P ; 8i; demand constraints : q i ¼ X q i 6 S i ; 8i supply constraints : q ¼ X i q i 6 R ; 8 ð15þ CTF constraints : f i ¼ S i ¼ q i ; 8i; S i q i invariance principle : if 9iq i < S i ; q i in variant to S i! C i SCIR constraints ðoptionalþ node supply constraints Formulation (15) encompasses requirements 2 7 of Section 3.1. Note that conservation of vehicles is implicitly guaranteed by defining only one variable q i that represents the flow from i to. In order to define a specific instance of this generic class, the supply constraint interaction rule (SCIR) and the corresponding constraints it imposes and (optionally) node supply constraints need to be specified: Without specification of the SCIR in (15), cooperative, non-selfish driver behavior would be assumed as the distribution of supply would be solely determined by the maximization of the total flow. Since this assumption is indefensible, the supply constraint interaction rule (determining supply distribution and the interaction of constraints) is mandatory for any specific node model instance. Even with a SCIR defined, (15) does not account for constraints that may be imposed by limited supply of the node itself. Such constraints are usually needed in order to obtain a realistic representation of traffic flows. Since there is not an irrefutable necessity for the inclusion of node supply constraints for every type of intersection (e.g. merges and diverges) or every traffic state (e.g. heavy downstream congestion) node supply constraints may be considered optional to the specification of a node model. 4. Defining node model class instances for specific intersections A rule describing how limiting supply is distributed over the competing flows and how supply constraints interact with each other and with the flows over the node which constraint is actually limiting each flow and how is the supply distributed over the competing flows? is an obligatory addition to any node model. This rule is denoted as the supply constraint interaction rule (SCIR). Moreover, the node itself typically imposes some node supply constraints. These might be caused traffic lights or some internal capacity (e.g. shared space on conflict points on priority unctions or on the arcs of a roundabout). So far, macroscopic node models found in the literature have not considered node supply constraints. In contrast to the requirements listed in Section 3.1, these additional specifications are specific for different types of intersections (priority, traffic light controlled, roundabout). Therefore, these specifications are the only means available to the modeler to distinguish between different types of intersections, or between different types of behavior. Another important difference with the generic requirements is that it is not postulated in this paper that there exist one undisputed definition for the SCIR or the node supply constraints. Until extensive empirical research is conducted, several definitions are conceivable, even for the same intersection type. Moreover, empirical research can probably not take away the need for different definitions, e.g. due to differences in driver behavior culture. Therefore, multiple plausible node models for a specific intersection can fit into the generic class defined by (15) Node supply constraints Usually the node itself imposes constraints, as different flows use some shared part of the internal node infrastructure with limited capacity. Examples are conflict points of crossing flows and the arcs of a roundabout. At signalized intersections, traffic control manages the use of most conflict points by alternatively blocking the flows. The supply is limited by the signal controller and has to be shared among different incoming flows belonging to the same green phase. Traffic lights can thus be seen as special cases of ointly used internal node infrastructure. Formally, let us define a set of node supply constraints N p. These constraints set upper bounds to a selection U p of all flows that make use of the internal infrastructure p (e.g. green phase, conflict point, arc on a roundabout). internal node supply : X U p f ðq i Þ 6 N p ; 8p ð16þ

9 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) Eq. (16) is the general formulation of the node supply constraints where f(q i ) is a function of the flows q i. This function f(q i ) and the exact definition of the supply N p are dependent on the type of the internal node infrastructure the constraint represents. For a conflict point of crossing or merging flows, a constraint of the following form (17) is provisionally suggested (based on Brilon and Wu, 21). A thorough analysis of node supply constraints for conflict points is a topic of future research. X U p q i t i 6 1; 8p ð17þ In (17), t i is the time (in hour) during which 1 vehicle of the flow from i to occupies the conflict point. The total occupancy (N p )by P Up q i (veh/h) can of course not exceed 1 h. If t i can be considered as a constant for each movement i (independent of q i ) then it is direct input to the node model. Otherwise, the function defining how t i depends on q i would be input to the node model. If and how t i has to be considered dependent of q i is a question that will need to be answered in future research. A formulation for the node supply constraints induced by traffic lights is presented in Section Theoretically all constraints of the node model (both the node supply constraints and the constraints imposed by the outgoing links) need to be considered simultaneously, since they may interact (as discussed in the next section) in yielding the resulting flows Supply constraint interaction rule The supply constraints limit a sum of flows q i (i.e. P i q i 6 R Þ that are as opposed to the q i in the demand constraints not mutually coupled by CTF. To overcome this difficulty, the supply constraints are to be translated to individual constraints on each q i (or, due to CTF, to each q i ) by the SCIR. In case of active supply constraint(s), the SCIR needs to answer two questions: For each flow q i : by which of the constraints (demand, internal supply, supply) is it limited? For each supply constraint (R, N p ): how is the supply distributed to each one of the competing flows towards or through p (i.e. that make a non-zero claim for the supply)? In general, these two questions cannot be detached: whether or not the flow of an incoming link is limited by some supply constraint depends on the share of this supply that would be distributed to this incoming link, and this share depends on which flows are limited by which constraint. More specifically, a SCIR seeks for a consistent solution of the following interdependent steps: Composition of sets U: dependent on how each supply constraint is distributed over competing flows, exactly one (demand or supply) constraint can be identified for each link i as the most restrictive one. A link i therefore belongs to exactly one set U i, U or U p, which consists of incoming links i that are constrained by S i, R or N p, respectively: i 2 U i () q i ¼ S i i 2 U () ¼ ðiþ and q i < S i ð18þ i 2 U p () p ¼ p ðiþ and q i < S i In (18), * (i)(p * (i)) denotes the (or p) that imposes the most restrictive (internal) supply constraint on i. Of course, a set U i can only consist of i itself, or be empty. A (possibly empty) set can be composed for each i, (and p). Supply distribution scheme: dependent on the composition of the above sets, each active supply (i.e. with a non-empty set) can be unambiguously distributed via a distribution scheme that describes the aggregate driver behavior (e.g. based on capacities, number of turning lanes, etc.). Hereby it needs to be considered that if i is constrained by S i, R * (i) or N p * (i) then (due to CTF) it uses less than its rightful share of other supplies R (and N p ) it was competing for (i.e. for which S i > ). Thus, only the part of R and N p that is not actually used by such links (the reduced supply e R or e N P Þ should be divided among the links of sets U (and U p ). Thus, the most restrictive constraint determines the reduced claim a link i makes for other, less restrictive supply constraints. The SCIR describes which constraint is actually limiting each flow (composing sets U for each constraint) and defines the distribution of supply. However, because of the interdependency between memberships of sets U and distribution of supply, the SCIR may be difficult to define. Ideally, it should take into account all characteristics of driver behavior, turning fractions, intersection geometry, priority rules and traffic control (if applicable). In this paper, this complex problem is identified but not entirely solved. In the next section, simplified models that properly combine the requirements in Section 3.1 with a SCIR are proposed for unsignalized and signalized intersections. Establishing improved models that avoid the current simplifications, as well as roundabout models are open issues for future research.

10 298 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) Macroscopic models for unsignalized and signalized intersections In this section two specific node models are derived from the generic class defined by (15). The first includes a SCIR specific for unsignalized intersections. This model is simplified in the sense that no internal node supply constraints are considered yet. The second node model is specific for signalized intersections. It builds on the node model formulation for the unsignalized intersection, complemented by a set of node supply constraints modeling the green times. This is also a simplified model, in the sense that it does not consider node supply constraints due to conflicts between crossing flows of the same green phase and that it does not yet consider the grouping of flows in green phases in the SCIR. The proposed models are applicable to merges and diverges, but not roundabouts. However, despite these simplifications both models are nevertheless superior to all state-of-the-art node models, since they satisfy all the requirements in Section Node model for unsignalized intersections Supply constraint interaction rule: oriented capacity proportional distribution In this section, a supply constraint interaction rule (SCIR) is defined. It implicitly defines the flows q i based on definitions of the sets U containing all incoming links i whose flows are constrained by if any; some sets U may be empty. Furthermore, the SCIR prescribes how R is distributed over the constrained links i 2 U. By doing so, the SCIR translates the supply constraints to individual (SCIR) constraints for each incoming link i of set U (see Section 4.2). Later in this section, it appears that no explicit definition of the sets U and the flows can be written, but in Section an algorithm is presented that iteratively solves the flows q i from the set of implicit equations given in this section. Rather than merely being a mathematical construction to solve the node model, the SCIR is actually an aggregate representation of driver behavior on intersections with at least one active supply constraint. However, there are limitations to how a stable and consistent SCIR may be formulated: as was already discussed in Section 3.1, a distribution based on demand is not applicable, since this does not satisfy the invariance principle. A distribution scheme based on constant ratios could be applied, based on, e.g. priority rules, number of (turning) lanes or capacities. However, it is not advisable to base the distribution solely on priority rules, since observation shows that these are obeyed less strictly under congested conditions (Troutbeck and Kako, 1999). In this paper, the claim that traffic from an incoming link i makes in the distribution of each R is determined by the capacity C i and the turning fractions f i. The supply distribution is thus based on oriented capacities C i, applying CTF C i ¼ f i C i ¼ S i S i C i ; 8i; ð19þ The behavioral interpretation of choosing oriented capacities as a basis for the distribution scheme is as follows. In case of an active constraint R, different links i are competing for a limited number of entering opportunities into. The number of opportunities taken by a movement i is proportional to: Turning fractions: e.g. a link i that sends all of its traffic to exploits twice as many opportunities than one with the same capacity that sends only half of its traffic to and the other half to other. Capacity: a link i having more lanes, better visibility, or a smaller turning angle into exploits more opportunities than one with less lanes, worse visibility, or a sharper turn into. Note that the suggested SCIR based on oriented capacity proportional distribution reflects aggregate selfish, non-cooperative behavior of the drivers: it is assumed that drivers competing for limited available space in the outgoing link do not give way to other drivers to help realize higher flows at a system level. Rather, the share of supply taken is only proportional to the number of opportunities. As a consequence, the maximization of (15) does not produce a system optimal flow, but rather a maximization under selfish non-cooperative behavior (comparable to user optimum). Also note that alternative definitions of the SCIR than the one presented in this section are conceivable if empirical evidence would indicate that an alternative SCIR would describe reality better. As long as the SCIR is compatible with all requirements in Section 3.1, it would yield an alternative node model instance of the generic class discussed in this paper with different SCIR constraints. Obviously such alternative node model instances may yield different maximized flows than the model instance presented here, even if all demand and supply constraints are the same. Maximization of flows in (15) is to be understood as a constrained maximum, with one of the constraints being the prevailing aggregate driver behavior(expressed by the SCIR constraints). Alternative driver behavior may thus yield higher or lower constrained maxima Definition of flows dependent on set membership. Maximization of the flows q i in (15) implies that each q i has to be restricted by some constraint; otherwise this flow could be trivially (and selfishly) increased by raising q i until it hits some constraint. Since no internal node supply constraints are considered, there are only two possible solutions to this maximization problem for each q i :

11 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) either q i is constrained by demand, which means that link i belongs to set U i and therefore not to any set U : i 2 U i () i R U ; 8 ð2þ In this case, the flow is equal to the demand: q i ¼ S i ; 8iU i ; ð21þ or there is some supply constraint * (i) that is most stringent on all flows coming from link i (because of CTF coupling), implying that: i 2 U ðiþ () i R U ; 8 ðiþ and i R U i ð22þ In this case, q i is actively constrained by the SCIR constraint, which is the translation of the supply constraint R into individual constraints for each i (see (23)). Eq. (23) constitutes the specification for the SCIR constraints in the generic formulation (15) q i 6 a ðiþc i ; 8i; ð23þ Thus, if supply constraint * (i) is most stringent on the flows from i, the q i are equal to the oriented capacity C i times the reduction level a ðiþ imposed by this most stringent supply constraint on all members of its set: q i ¼ a ðiþc i ; 8i 2 U ðiþ ð24þ The rationale behind solution (21) for demand constrained flows is trivial. For supply constrained flows, the rationale of solution (24) is the following. Since the outflow from i is constrained, a queue will build up in link i. Therefore, it will make the maximal claim for supply, being the fraction of C i that is oriented towards : C i. Of course, this maximal claim cannot be granted by lack of supply R, so only a fraction a of this maximal claim can actually flow out of i towards. This reduction level a, and how it comprises the supply constraint R and the oriented capacity proportional distribution thereof, is defined hereafter Definition of the reduction levels a. If outgoing link is active, i.e. if U is non-empty, each turning movement i for which there is demand (S i > ) makes a claim for the limited supply R. Consider the situation where all incoming links i have as the most stringent constraint the outgoing link that we are currently considering, i.e. i 2 U " i. In that case, all incoming links will make their maximal claim, being their oriented capacity C i, for the outgoing supply R. The rightful share of R for each incoming link i is then proportional to C i, with the sum of all rightful shares exactly consuming R. Consider now the more general situation where not all incoming links i are members of the set U (i.e. not all i are constrained by the outgoing link currently under consideration). This means that links i R U are constrained either by demand or by another outgoing link * (i). As a consequence, the flows q i from these links into use less than their rightful (i.e. oriented capacity proportional) share of the supply R. Therefore, before distributing the available supply R among the members of U, first the less-than-rightful claims of the non-competing i R U are subtracted from R after which the remaining, reduced supply R e is distributed among the members of U, proportional to their maximal claims C i. The reduced supply R e available for distribution among the members of U is defined as: er ¼ R X q i ¼ R X S i X X a C i ð25þ iru iu i ; i2u The reduction factors a follow from the fact that the members of U fully consume the available supply R e : P i2u q i ¼ P i2u a C i ¼ R e, or: a ¼ e R P a ¼ 1; i2u C i ; if U ; otherwise ð26þ In case of an empty set U, P i q i 6 R, so there are no incoming links that are constrained by. In this case, a will not determine any of the resulting flows and is by definition set equal to the default value 1. With these definitions for the reduction factor, the most stringent outgoing constraint * (i) (used in definitions (22) (24)) for links i that are not demand constrained is defined by: ðiþ ¼arg min a S i > ð27þ Eq. (27) actually defines the sets U based on the definition of a that in turn uses the sets. Therefore, all definitions given in this section are implicit. Also the flows resulting from the problem constituted by (15) (complemented by (23)) are implicitly defined, since they contain the mutually dependent set and level of reduction definitions (26) and (27). Summarizing, the resulting flows can be written as in (28), which combines the two previously defined possible solutions (21) and (24)

12 3 C.M.J. Tampère et al. / Transportation Research Part B 45 (211) q i ¼ minðs i ; a ðiþc i Þ; 8i; ð28þ It is straightforward to see that in this solution (28), each q i is actively constrained and thus cannot be increased. Furthermore, in Appendix A it is shown that (28) (with the underlying definitions (19) (27)) satisfy all the requirements in Section 3.1. In order to solve the system of implicit definitions (19), (2), (22) (27) to find solution (28), iterative solution algorithms can be used, as is shown in the next section Solution algorithm From (27) follows immediately that for i 2 U in (25) holds that either a > a or S i ¼ () f i ¼ () C i ¼. This means that when calculating a according to (25) and (26), only the a < a need to be considered. Thus, the smallest level of reduction over all (denoted as a^ of ^) is given by: a^ ¼ R P iu i ; S i Pi2U C i ; if U ; ð29þ All terms on the right side of (29) are input to the node model. Therefore, if the solution algorithm can correctly pick out ^, the smallest level of reduction a^ can be easily calculated from (29). Given this a^, the second smallest a can be calculated and so on, leading towards the exact solution of Section for all a, the corresponding sets U, and thus q i. It is shown in the following that the proposed solution algorithm indeed finds this exact solution (see also Appendix B.1). The iterative solution algorithm (k iterations, k starting at ) for the presented node model can be subdivided into five steps. Note that the first two steps are initialization steps that are only executed in the first run. In the algorithm, the sets U i for the demand constraints are not explicitly considered, since this is not necessary to find the solution. The algorithm is first explained briefly in words and subsequently in commented code lines. 1. S i and R (to which e R ðþ is initialized) are determined from a link model. The sets U ðþ are initialized as all is i > and a set J () is initialized as all outgoing links under consideration disregarding P i S i ¼. This set J is merely an aid to the algorithm and has no physical meaning in the solution. 2. The oriented capacities C i are determined from (19). 3. In iteration k, all a ðkþ for all still under consideration ( 2 J (k) ) are calculated from (26). The smallest is saved as a ðkþ ; the ^, only the interaction from smaller a (definitively fixed in pre- corresponding as ^ ðkþ. Recall that in the calculation of a ðkþ vious iterations) is to be considered. This means that for the first a to be determined the smallest over all no interaction with other supply constraints must be considered (see (29)). Furthermore, when k =, R eðþ constraints are considered yet either. 4. a ðkþ ^ determined in step 3 is imposed on all i 2 U ðkþ ^. ¼ R ; 8 and no demand (a) If link(s) i are found to be demand constrained, they are removed from all U ðkþ. For i results q i = S i. According to (25),a share of each R eðkþ equal to S i is assigned to i. All U = ; are removed from consideration ( removed from J (k) ). (b) In case no links are demand constrained, each i 2 U ðkþ ^ i 2 U ðkþ ^ each i 2 U ðkþ ^ is constrained according to a ðkþ ^ is removed from all other sets U ^ ðkþ. According to (25), a share of each R eðkþ. All U = ; and ^ ðkþ are removed from consideration (removed from J (k) )., resulting in q i ¼ a ðkþ ^ equal to a ðkþ ^ C i. Each C i is assigned to 5. Stop criterion: if in the next iteration there would be no left to consider (J (k+1) = ;), the algorithm stops. Otherwise k? k+1 and the algorithm returns to step 3. In both of the substeps 4(a) and 4(b), the available supply of the still under consideration (to be distributed in the next iteration) is reduced and thus the corresponding a need to be recalculated in step 3 of the next iteration. Since the share that the treated links i (no longer 2 U ðkþ Þ consume of these supplies is less than their rightful share either due to a more restrictive demand constraint (a) or a more restrictive supply constraint in ^ ðkþ (b) a bigger share of supply will be available for remaining i (still 2 U ðkþ ). Consequently, the corresponding a can only increase (or be set to 1) as they are recomputed. This makes it possible to pick out ^ ðkþ in each iteration and definitively fix the corresponding a ðkþ (i.e. the smallest over all still ^ under consideration) and iteratively find the exact solution for all a and U (and thereby all q i ). In Appendix B.1 the above statement is proven. In code, the algorithm can be written as follows. 1. Retrieve link constraints and initialize supplies and sets For all i, : Determine S i, R and C i via a link model. nninitialisation: er ðþ ¼ R nnsupply constraint determined by the link model U ðþ ¼fiS i > g nn add all i competing for R to initial set U ðþ J () ={S >} nn add all towards which non-zero demand is nn directed to initial set J