Validation of an Urban Traffic Network Model using ... - Semantic Scholar

Validation of an Urban Traffic Network Model using Colored Timed Petri Nets Mariagrazia Dotoli Dipartimento di Elettrotecnica ed Elettronica Politecnico di Bari Via Re David 200 70125 Bari, Italy

Maria Pia Fanti Dipartimento di Elettrotecnica ed Elettronica Politecnico di Bari Via Re David 200 70125 Bari, Italy

Giorgio Iacobellis Dipartimento di Elettrotecnica ed Elettronica Politecnico di Bari Via Re David 200 70125 Bari, Italy

[email protected]

[email protected]

[email protected]

Abstract - This paper validates a Colored Timed Petri Net (CTPN) model proposed to describe urban traffic networks. In particular, a CTPN models the dynamics of signalized traffic networks and timed Petri nets describe the traffic lights controlling the area. To this aim, the modeling framework is applied to a real intersection located in Bari, Italy. Discrete event simulations of the controlled intersection test the signal timing plan under different traffic scenarios and give a confirmation of the model capability to correctly predict the traffic performance. Keywords: modeling, urban systems, colored timed Petri nets, validation, performance measures.

1

Introduction

Traffic signal control is the most common technique to regulate and manage urban traffic areas and surface street networks. The related literature offers a wide variety of traffic signal control strategies [8], each sharing the need for appropriate modeling of the Traffic Network (TN), either for simulation purposes or in order to determine on line some states of the transportation network that are not available due to detector absence or failures. Urban TNs exhibit high degree of concurrency and are characterized by resource sharing and conflicts. Thanks to the well known ability of Petri Nets (PNs) to capture concurrency and asynchrony, PN based models may be suitably derived for urban traffic systems, both for describing traffic signals controlling urban signalized areas, as well as for modeling concurrent activities that are typical of TNs. In particular, Di Febbraro, et al. [4] present a traffic model in a timed PN framework, where tokens are vehicles and places are parts of lanes and intersections. Fluid stochastic PNs are proposed in [1] and hybrid PNs in [3] to model urban networks of signalized intersections. Colored PNs are introduced in [2] to distinguish among different vehicles and their associated routes. Moreover, Dotoli and Fanti [5] have proposed a Colored Timed Petri Net (CTPN) model to describe the flow of vehicles in an urban TN. The token color represents the path that a vehicle has to follow and places are cells accommodating

one vehicle at a time. Moreover, the TN traffic lights is modeled by ordinary Timed Petri Nets (TPNs). The model presents three main advantages. First, the parameters required by the model dynamics are the number of vehicles entering each road lane and their average rate. These parameters can be directly measured and, respectively, evaluated at each input branch of every intersection. Second, the model is modular and easy to build by the TN topology. Third, the code to simulate real intersections can be directly generated in order to test both fixed signal timing plans or actuated control strategies. The aim of the paper is to validate the model proposed in [5] by simulating a case study that describes a real intersection located in the city of Bari, Italy: the traffic dynamics is studied under different scenarios. The flexibility of the model is enlightened by the description of a junction branch where buses only are admitted. Moreover, simulation results give a confirmation of the model capability to correctly predict traffic performance measures and to test different signal timing plans. The paper is organized as follows. Section 2 recalls the CTPN model of signalized areas. Moreover, Section 3 describes the case study and Section 4 reports the results of some simulations performed under different traffic scenarios. Finally, Section 5 summarizes the conclusions.

2 2.1

Review of the Model The CTPN modeling the signalized area

In a TN the following fundamental components are considered: signalized intersections, links, vehicles and traffic lights. More precisely, each link represents the space available between two adjacent intersections and may include one or several lanes. Hence, a generic signalized urban area comprises a number of junctions controlled by traffic lights pertaining to a common signal timing plan, including a set L={Li| i=1,…I} of I links. A generic link Li of length li with i=1,…I has a finite capacity Ci>0 denoting the maximum number of Passenger Car Units (PCUs) that

the link can simultaneously accommodate. Hence, each link is usually divided in Ci cells with unit capacity. Moreover, it is necessary to take into account the physical space that a vehicle crossing the intersection occupies. Such a physical space, named intersection cell, can be occupied by only one vehicle and may coincide with the whole intersection area in a very simple intersection or with a part of the physical space in a multi-lane intersection. According to the model defined in [5], TNs may be modeled by two nets: a Colored Timed Petri Net (CTPN) [6] describes roads and junctions of the TN, while a Timed Petri Net (TPN) [7] describes the traffic light pertaining to a common signal timing plan. Obviously, if some intersections have two or more independent signal timing plans, an independent TPN have to model the traffic lights of each timing plan. Moreover, the TPNs modeling the traffic lights may be considered as a subnet of the CTPN modeling the signalized traffic network. More precisely, the resulting CTPN modeling the TN is CTPN=(P, T, Co, H, Pre, Post, FT), where: P is a set of places; T is a set of transitions; Co is a color function that maps each place pi∈P to the set of possible token colors Co(pi); H is a weight function defined for an inhibitor arc that connects a transition to a place; Pre and Post are the pre-incidence and the post-incidence matrices; FT denotes a timing vector and the firing time of each transition tj is the positive number FTj (i.e., the j-th element of FT). For each place pi∈P, we define the marking mi of pi as a non-negative multiset over Co(pi). The mapping mi: Co(pi)→Ν associates with each possible token colour in pi a non-negative integer representing the number of tokens of that colour that is contained in pi. In the following we denote by mi the vector of non-negative integers, whose h-th component mi(h) is equal to the number of tokens of colour ah∈Co(pi) that are contained in pi. The marking M of a CPN is a column vector of multisets, i.e., M=[m1…m|P||]T. Finally, a coloured Petri net system is a CPN with initial marking M0. A place pi∈P denotes a resource, i.e., a cell in a lane or in a crossing section that a vehicle can occupy. More precisely, two types of places are considered: places representing link cells and places corresponding to intersection cells. In addition, transitions from T are timed transitions and model the flow of vehicles into the system between consecutive cells. We consider five types of transitions: 1) 2) 3) 4)

input transitions t0,i∈TI, modeling the arrival of a vehicle in a link from the outside; output transitions ti,0∈TO, modeling the departure of a vehicle from a link to the outside; flow transitions tj∈TF, modeling the flow of vehicles between two consecutive link cells; intersection transitions tj∈TC, modeling vehicles

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entering or leaving a crossing section. The intersection transitions that model a vehicle entering the crossing sections are regulated by a multi-phase traffic light; lane changing transitions tj∈TL, modeling vehicles that change lane in road.

The firing times of t0,j∈TI are equal to the interval times in which vehicles can enter the system network: such values may change during the observation time in the same traffic scenario and may be determined by detectors appropriately placed in the links. In addition, the value of FTj assigned to each tj∈TO∪TF∪TC∪TL is equal to the average time interval in which a vehicle passes from a cell to the subsequent one or occupies an intersection cell and depends on the average vehicle speed in the considered traffic scenario. However, a longer time is assigned to transitions tj∈TL modeling lane changing. A colored token in a place represents a vehicle. The color of each token is the route assigned to each vehicle indicating the different paths that a vehicle can follow starting from a particular position. Finally, inhibitor arcs are introduced to model the finite capacity of each cell. More precisely, for each transition t∈TI∪TF∪TC∪TL and for each pi such that Pre(pi,t)>0 there exists an inhibitor arc between pi and t, i.e., H(pi,t)=1.

2.2

The TPN modeling the traffic lights

The traffic lights of a generic TN are defined according to a signal timing plan, including green, red and amber signals, that in most American and European cities correspond respectively to clear way, stop and caution signal after green and before red. In addition, we take into account the lost (or intergreen) times, i.e., short duration phases in which all traffic lights in one intersection are red, in order to let vehicles, previously allowed to occupy the crossing area and late due to congestion, clear the junction. Among the main decision variables in a timing plan, we recall cycle time and green splits [8]. Cycle time is defined as the duration of time from the centre of the red phase to the centre of the next red phase. Green split for a signal in a given direction of movement is defined as the fraction of cycle time when the light is green in that direction. Moreover, a phase is the time interval during which a given combination of traffic signals in the area is unchanged. Finally, during each phase different streams may be allowed to proceed, where a stream of vehicles in a junction is a portion of traffic formed by all vehicles that cross the intersection from the same departure link and are directed to the same arrival link in the considered phase. We model the traffic light controller pertaining to a common signal timing plan by the definition of the TPN=(P, T, Pre, Post, FT) [4]. In particular, places from P

represent phases and transitions from T model the succession of red, yellow and green phases.

3

The Case Study

In this section we describe the case study selected to validate the model. The real traffic intersection shown in Fig. 1 is composed by 6 links Li (i=1,…,6) with length l1= 40m, l2=l3=45m and l4=l5=l6=60m, respectively. The capacities of the links are C1=16 PCUs, C2=C3=9 PCUs C4=C6=24 PCUs and C5=12 PCUs, derived by assuming that one PCU is l0=5m long and taking into account the number of lanes in each link.

and L6 by 24 places. In addition, links L3 and L4 are dedicated only to buses and, considering each bus of 3PCUs, links L3 and L5 are modeled by 3 and 4 places, respectively. With reference to Fig. 2, transitions t0,1, t0,2, t0,3, t0,4 and t0,5 are input transitions and transitions t1,0, t2,0, t3,0 and t4,0 are output transitions. Transitions tj with j=12,…,21 model lane changing in the two-lanes intersection links (i.e., links L1, L4 and L6), transitions t1, t’1, t3, t6, t’6 model vehicles entering a crossing section and are controlled by the traffic light TPN. Vehicles traveling in the intersection are tokens of 5 colors. More precisely, colors a1 and a2 are associated with vehicles following the routings (L1, L4) and (L1, L2), respectively. Moreover, colors a3, a4 refer to vehicles following routings (L6, L2) and (L6, L4), respectively. Finally, color a5 represents vehicles following the routing (L3, L5). Having defined the CTPN structure, the matrices Post and Pre may be easily defined [5, 6].

3.2

Figure 1. Layout of a real traffic intersection located in the urban area of Bari, Italy. a3 t 0,3

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Figure 2. The CTPN modeling the case study.

3.1

The CTPN model of the intersection

Figure 2 shows the CTPN modeling the intersection. Places pi, p’i, with i=1,…,9 are a subset of places modeling cell links and places pi, p’i with i=10,11,12 model the crossing area composed of six cells. For the sake of simplicity, Fig. 2 shows just two places for each lane in an input link and one place for each lane in an output link. Moreover, considering the number of lanes and capacity of each link, L1 is modeled by 16 places, L2 by 9 places, L4

The TPN model of the signal timing plan

In this section we describe the TPN representing the traffic light of the case study. For this intersection, the streams allowed to proceed during the phases of the signal timing plan are depicted in Fig. 3 and labeled with numbers from 1 to 5. Moreover, amber and intergreen times are taken into account, so that the considered fixed timing plan comprises 8 phases, including 3 amber phases (i.e., phases 2, 4 and 7) and 2 lost times (i.e., phases 5 and 8). Figure 4 shows the phases of the signal timing plan and the duration of the phases in seconds is indicated by τi with i=1,…,8, while the cycle time is CT. For the sake of simplicity, in Fig. 3 phases 5 and 8 are omitted, being lost times (see Fig. 4): indeed, no stream is allowed to proceed during such phases. Note that each stream is modeled by a token color of the CTPN, ruled by the described signal timing plan. The signal timing plan, described by Fig. 3 and 4, is realized by three traffic lights, each one with three phases modeled by three places representing the red, yellow and green phases, respectively. Hence, to model the phases of the cycle, nine places are necessary. In particular, a neutral color token in place p13 enables transitions t1 and t1’ to rule the vehicles in L1. Moreover, places p14, p15 and p16 describe the state of green, yellow and red, respectively. Hence, in order for the TPN in Fig. 5 to describe the signal timing plan in Fig. 4, we select FT22=τ6 (the duration of the green signal controlling L1), FT23=τ7 (the length of the amber phase ruling L1) and FT24=τ8 (the duration of the subsequent lost time). In a similar way, we describe the traffic light ruling L6 and modeled by places p17, p18, p19 and p20 with FT25=τ1 (the duration of the green signal for L6) and FT26=τ2 (the duration of the amber phase for L6). Finally, the traffic light controlling L3 is modeled by places p21, p22, p23 and p24 with FT27=τ1+τ2+τ3 (the green signal duration for L3), FT28=τ4 (the amber phase duration for L3).

Figure 3. The streams of the signal timing plan controlling the intersection in Fig. 1. Phases Links 1 2 3 4 5 6 7 8 1, 2 1 Streams 5 3 3, 4 6 Phase duration [s] τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8 Cycle duration [s] CT Figure 4. The signal timing plan of the case study ( = green, = amber, = red). t 24 P16

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Figure 5. The TPN modeling the case study traffic light. In addition, transition t29 models the duration of the intergreen signal subsequent to the allowed movement of the stream originating by L3 and hence it holds FT29=τ5. The start of the signal timing plan cycle is with a token in p14 and in p13 (green for L1). When a token is in p16, the red phase begins for L1. Hence, after the lost time FT24 transition t24 fires and the green phases start for links L6 and L3 (tokens in p17, p20, p21 and p22). If there is a token in p20, p21, p24 and p19, then t29 can fire when it is ready (that is after the lost time FT29) and the green phase starts for L1, so that the cycle begins again.

4 4.1

The Model Validation The system specification

To validate the CTPN model of the case study, on the basis of real traffic data, we present some simulation results referring to the TN behavior of the described case study. Moreover, to determine the deterministic firing

times of the input transitions t0,j∈TI, the time instants of the vehicle arrivals are obtained for two scenarios from the direct measures at the input links registered at the considered time of the day, during the red and green-amber signals for 20 cycles. Hence, we approximate the interarrival time of the vehicles during green and amber phases and during the red phase of each cycle as shown in Table 1 for scenario 1 (weekday around 3 p.m.) and scenario 2 (weekday around 6 p.m.). Being the cycle time CT=70 s, each registration has been performed for 1400 s. In particular, Table 1 shows that the second scenario features a stronger congestion than the first case with regard to links L1 and L6 (with increased arrivals per cycle on average), while link L3 is always under-saturated, corresponding to the fact that such a link is dedicated to buses, that run with fixed and non-overlapping schedules. On the basis of experimental evidence, v=40 km h-1 is assumed to be the vehicle average speed. Hence, the firing time of the transitions tj∈TO∪TF∪TC is equal to FTj=l0/v=0.45 s in each scenario. However, if transitions tj∈TO∪TF belong to links L3 and L5, then we assume FTj=1.35 s to model the bus movements. For the sake of simplicity, lane changing is not allowed in the present case study. Traffic in the considered intersection is currently ruled by a fixed time control strategy with a given signal timing plan. The fixed durations of the traffic light phases applied to the real intersection and independent from the traffic scenarios are listed in Table 2. In particular, all amber phases last 4s and lost times are 2s long. Moreover, as described in section 3.2, the firing times FTi with i=22,…,29 referring to the corresponding transitions of the TPN modeling the traffic light are obtained from the phases of the signal timing plan.

4.2

The simulation results

The CTPN model is implemented and simulated in the Matlab environment. Considering the two described traffic scenarios, two simulations are performed for 20 timing plan cycles, for a total run time T=1400s. Starting from the initial state describing the CTPN in scenario 1 (scenario 2), with 4 (8) vehicles present in link L1, 0 (0) vehicles in link L3 and 0 (4) vehicles in link L6, the discrete event simulation program executes the CTPN in the fixed

Table 2. The signal timing plan phases [s].

run time. The simulation model generates as an output the number of vehicles ni at the beginning of each cycle in the input links Li for i=1,6. Such performance indices are

τ1 31

determined as follows: ni = ∑ m j ( h ) for each ah∈Co(pj) j ,h

and for each pj in the input links Li (i=1,6), where the marking is determined at the beginning of each cycle. Note that we do not consider the number of vehicles in input link L3. Indeed, since such a link is dedicated to buses, it is extremely under-saturated and most buses entering the junction during a cycle are able to cross it in the same cycle. Moreover, the simulation model determines the average time Tsi spent in the input link Li with i=1,3,6. More precisely, Tsi denotes the average time interval that elapses from a vehicle entrance through Li to its exit from an output link of the intersection. Hence, each performance index Tsi with i=1,3,6 is computed by determining the average time interval between the firing of the input transition of Li enabled by a colored token and the firing of the output transition that is enabled by the same colored token. In addition, the model evaluates the average time spent in the network, indicated by Ts, i.e. the average time interval that elapses from a vehicle entrance in the junction to its exit from the intersection. Table 1. Deterministic firing times of the input transitions for Scenario 1 (S1) and Scenario 2 (S2). Green Red and amber phase phases FT0,1= FT0,1= =FT0,2 =FT0,2

Green Green and Red and amber phase amber phases phases FT0,3= FT0,3= FT0,5 =FT0,4 =FT0,4

Red phase

τ2 4

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The values of n1 and n6 for links L1 and L6 are respectively reported in Figs. 6 and 7 for scenario 1 and in Figs. 8 and 9 for scenario 2. In particular, the figures show that the simulation results are consistent with real traffic data. The slight differences are due to accidental events that are not modeled in the proposed CTPN model, such as parking and temporary stops at the sides of the roads (that are more frequent in scenario 2 because of the increased traffic). Moreover, comparing Figs. 6 and 8 shows that under both scenarios link L1 is always under-saturated (its capacity being C1=16 PCUs), but it is more congested in the second scenario, with longer queues at the beginning of the cycles in its lanes. Similarly, the comparison of Figs. 7 and 9 shows that also link L6 is always under-saturated (since C6=24 PCUs), but it is more congested in the second scenario, with longer queues at the beginning of the cycles in its lanes. Finally, Table 3 reports performance indices Ts,i with i=1,3,6 and Ts for both scenarios. The increased values of such indices for scenario 2 with respect to scenario 1 confirm that the junction is more congested under scenario 2. Table 3. The delay time for each input link [s].

FT0,5

Cycle S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 1 8.8 14.7 88.0 11.0 20.0 20.0 15.0 30.0 20.7 15.5 23.3 23.3 2 8.8 8.8 17.6 17.6 13.3 40.0 30.0 30.0 20.7 10.3 70.0 14.0 3 7.3 8.8 14.7 8.0 13.3 20.0 30.0 15.0 15.5 12.4 70.0 23.3 4 6.3 7.3 17.6 17.6 20.0 20.0 30.0 30.0 20.7 7.8 35.0 17.5 5 11.0 6.3 17.6 9.8 10.0 20.0 30.0 30.0 12.4 12.4 35.0 23.3 6 6.3 44.0 17.6 14.7 40.0 40.0 30.0 30.0 12.4 12.4 35.0 8.8 7 3.7 11.0 12.6 11.0 40.0 40.0 30.0 30.0 15.5 31.0 35.0 10.0 8 5.5 14.7 22.0 12.6 20.0 40.0 30.0 30.0 15.5 31.0 35.0 17.5 9 3.7 7.3 14.7 11.0 40.0 40.0 30.0 30.0 12.4 8.9 14.0 17.5 10 4.4 14.7 44.0 14.7 40.0 40.0 30.0 30.0 12.4 10.3 35.0 14.0 11 4.9 11.0 12.6 11.0 20.0 40.0 30.0 30.0 8.9 10.3 17.5 14.0 12 8.8 8.8 29.3 14.7 40.0 20.0 30.0 30.0 15.5 10.3 23.3 70.0 13 3.1 8.8 29.3 29.3 40.0 20.0 30.0 30.0 15.5 6.8 35.0 14.0 14 4.4 5.5 11.0 11.0 13.3 13.3 30.0 30.0 12.4 31.0 23.3 70.0 15 7.3 14.7 12.6 17.6 10.0 40.0 30.0 30.0 31.0 7.8 17.5 14.0 16 4.4 5.5 14.7 17.6 40.0 13.3 30.0 30.0 7.8 31.0 35.0 70.0 17 6.3 4.9 17.6 22.0 40.0 20.0 30.0 30.0 15.5 20.7 35.0 14.0 18 7.3 6.3 9.8 14.7 40.0 20.0 30.0 30.0 20.7 6.9 23.3 14.0 19 8.8 14.7 8.8 17.6 13.3 20.0 30.0 30.0 31.0 89 23.3 23.3 20 8.8 6.3 1.0 1.3 40.0 40.0 30.0 30.0 20.7 15.5 23.3 35.0

Average time spent Ts1 [s] Ts3 [s] Ts6 [s] Ts [s]

5

Scenario 1 18.43 11.00 15.11 17.09

Scenario 2 22.43 11.26 16.01 19.35

Conclusions

This paper validates a modeling technique proposed by two of the authors to describe the behavior of urban traffic network systems. Colored timed Petri nets model links, intersections and vehicles of the urban area and timed Petri nets synthesize the traffic lights. The obtained model is able to provide a sufficiently accurate and valid representation of the traffic network system using data that are collected by detectors, positioned at the input links of the traffic area. Moreover, the model can be easily translated in a simulation software that is not costly to develop. In order to show the model efficacy, the paper presents a case study describing a real intersection located in the city of Bari, Italy. The flexibility of the model is enlightened by the description of a link where buses only

are admitted. In addition, simulation results give a confirmation of the model capability to correctly predict traffic performance measures and to test different traffic scenarios.

References [1] M. Bouyekhf, A. Abbas-Turki, O. Grunder and A. El Moudni, “Fluid stochastic Petri net for control of an isolated two-phase intersection, Proc. IEEE Multiconf. on Computational Engineering in Systems Applications, Lille, France, July 2003. [2] F. Di Cesare, P.T. Kulp, M. Gile and G. List, “The application of Petri nets to the modelling, analysis and control of intelligent urban traffic networks”, Application and Theory of Petri Nets, Lecture Notes in Computer Science, Vol. 815, Springer, Berlin, 1994. [3] A. Di Febbraro and N. Sacco, “On modelling urban transportation networks via hybrid Petri nets”, Control Engineering Practice, Vol. 12, No. 10, pp. 1225-1239, October 2004.

[5] M. Dotoli and M.P. Fanti, “An urban traffic network model via coloured timed Petri nets”, Proc. WODES04 – the 7th IFAC Workshop on Discrete Event Systems, Reims, France, September 2004. [6] K. Jensen, Colored Petri nets: basic concepts, analysis methods and practical use, Vol. 1, Springer, New York, 1992. [7] T. Murata, “Petri Nets: Properties, Analysis and Applications”, Proceedings of the IEEE, Vol. 77, No. 4, pp. 541-580, April 1989. [8] M. Papageorgiou, C. Diakaki, V. Dinopoulou, A. Kotsialos and Y. Wang, “Review of road traffic control strategies”, Proceedings of the IEEE, Vol. 91, No. 12, pp. 2043-2067, December 2003. Simulated results for link 1

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[4] A. Di Febbraro, D. Giglio and N. Sacco, “On applying Petri nets to determine optimal offsets for coordinated traffic light timings”, Proc. 5th IEEE Int. Conf. on Intelligent Transportation Systems, Singapore, pp. 687706, September 2002.

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