- Email: [email protected]
A Congestion Eliminating Control Method for Large-scale Urban Traffic Networks*
13th IFAC Symposium on Large Scale Complex Systems: Theory and Applications July 7-10, 2013. Shanghai, China
A Congestion Eliminating Control Method for Large-scale Urban Traffic Networks Zhao Zhou ∗ Shu Lin ∗ Dewei Li ∗ Yugeng Xi ∗ ∗
Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai, 200240, China (e-mail: zzhou553, [email protected] dwli, [email protected]).
Abstract: This paper investigates an efficiency of signal control methodology, which mainly focuses on dealing with the traffic congestion problem in those key congested links and is applicable to be implemented in a hierarchical control structure in large-scale heterogeneous urban traffic networks. In this methodology, an algorithm for finding the most congested path is presented firstly, and the urban traffic flow is modeled by using a simplified macroscopic modeling framework. Then the problem of network-wide signal control is formulated as a linear programming problem that aims at minimizing the number of vehicles(or densities) in congested links so as to improve the mobility of the network and mitigate the traffic congestion. For the application of this method in real time, the multi-variables optimization problem including constraints is embedded in a model-based dynamic control procedure. Finally, different traffic demand scenarios are designed and four evaluation criteria are applied to measure the performance of the proposed method in a hypothetical road network. Compared with the fixedtime control strategy, the simulation results show that it is an effective and feasible way to regulate the traffic flow and mitigate the congestion in large-scale urban networks. Keywords: Urban traffic networks, large-scale systems, congested path, model-based control, hierarchical structure. 1. INTRODUCTION With the fast increasing number of vehicles and the rapid urbanization, traffic congestion became a big problem for metropolis, both in developed countries and developing countries. For example, in Shanghai, China, the citizens suffer from the problem of traffic congestion for a long period, especially during every morning and evening peak. Although this problem could be solved by constructing new transportation infrastructures and extending road networks, it is both costly and time-consuming. Therefore, a potential solution is to adopt network-wide traffic signal control strategies on the basis of the existing transportation infrastructures. For large-scale urban networks, the traffic flow system is a complex dynamic process. This network is composed of numerous intersections and roads, which means the dimension of traffic system is very high. The unpredictable activities of vehicles will lead to the uncertainty of traffic state on each road. There also exists complex interactions between each roads. Therefore, the urban traffic system is usually a dynamic large-scale system with multi-inputs and multi-outputs, the scale of optimization problem increases with the scale of the networks, and it will bring This work was supported in part by the National Science Foundation of China (Grant No. 60934007, 61203169, 61104160), China Postdoctoral Science Foundation (Grand No. 2011M500776), Shanghai Education Council Innovation Research Project (Grant No. 12ZZ024) and International Cooperation Project of National Science Committee (Grant No. 71361130012).
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high computation to design real-time efficient centralized traffic control strategies. Since the early 1980s, some hierarchical control systems like SCOOT and SCATS respectively proposed by Hunt et al. (1982) and Lowrie (1982) have been used for real-time adaptive traffic control. Gartner (1983) put forward a three-layer control architecture called OPAC. The OPAC control strategy is the adaptive algorithm implemented as the local controller of the hierarchical framework. Mirchandani and Head (2001) developed a multi-level hierarchical structure system—RHODES for real-time traffic signal control. Lin et al. (2011, 2012) presented an efficient network-wide model-based predictive control for urban traffic networks, which can be used as the local controller for subnetworks in a hierarchical control structure. Despite these systems play an important role in mitigation of urban traffic congestion, it does not consider the characteristic of heterogeneity, i.e. the densities of vehicle is uneven distributed in road networks. It means that if the traffic congestion in those serious oversaturated links could not be eliminated as soon as possible, it will lead to the phenomenon of queue spillback and even the gridlock in whole network. In this paper, we first develop an algorithm to find the most congested path, in which a great deal of vehicles is accumulated. Then, a simplified urban traffic model based on the dynamic of traffic flow is introduced and lays the foundation for future control. Finally, a model-based control strategy is proposed and used to alleviate the traffic congestion for large-scale urban traffic networks in real time.
10.3182/20130708-3-CN-2036.00032
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
The rest of this paper is organized as follows: Section 2 puts forward an algorithm to find the most oversaturated links. Section 3 introduces an urban traffic model and proposes an efficient model-based control method to eliminate the congestion in the congested path. Section 4 demonstrates the network-wide signal control method by examples. Section 5 concludes this paper.
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2. CONGESTED PATH In the previous studies, it mainly focuses on optimizing the signals to improve the mobility of whole traffic network or minimizing and balancing the number of vehicles in all links so as to reduce the risk of queue spillback under oversaturated traffic conditions. In reality, however, most large-scale traffic networks are usually heterogeneous networks with uneven distribution of congestion, especially in peak hours. In other words, due to the different traffic supply and demand, some links or subnetworks experience more congestion scenarios than the others. If the vehicles in these oversaturated links could not cross the intersections as soon as possible, it will produce queuing to block the inflow from upstream links, and then lead to the congestion of whole network. Therefore, it is necessary to regulate the traffic flow in these congested links. Recently, it has been verified by Geroliminis and Daganzo (2007, 2008) that there existing a well-defined macroscopic fundamental diagram(MFD) with unimodal and low scatter relationship between network vehicle density and space-mean flow in large-scale urban traffic networks. The maximum value of traffic flow appears at the critical point of MFD. With the increase of vehicles in network, the flow decreases and the traffic conditions become more congested. So the critical point of MFD indicates the optimal condition of traffic flow. Based on this, we first propose a congestion index to reflect the congestion degree of whole network. Consider that an urban traffic network can be expressed as a directed weighted network N = (V, E), where V = v1 , v2 , · · · , vn is the set of intersections and E = e(vi , vj )vi ,vj ∈V is the set of links between two adjacent intersections. The weight wij of edge e(vi , vj ) is the vehicle density between intersection i and j, which represents the traffic condition. wij = ni→j /li,j
]
(1)
where ni→j is the number of vehicles from intersection vi to vj , li,j is the link length. By collecting a great number of traffic data from different scenarios and periods in practice, we can derive a well-defined MFD and then obtain the vehicle density and flow corresponding to the critical point. The average value of traffic condition of each link at critical point is calculated and taken as the reference state. This work can be done off-line. According to the real-time traffic information directly collected using loop detectors and the physical characteristics of road network, the weighted value of each link can be obtained by (1). However, the biggest problem is how to find the most congested path in the heterogeneous network, for the reason that our aim is to guide and regulate the traffic flow in this key path. Here, we develop a method to realize the purpose. Due to that the weighted value 497
Fig. 1. An urban link denotes the traffic condition of each link, we take its reciprocal and assign it to the corresponding link as the new weighted value. After the preprocess, the lower weighted value means the more congested traffic condition. Thus, the problem mentioned above will be converted to the problem of searching the shortest path in whole network, which can be solved by using the popular algorithms in graph theory. In this paper, a most classical shortest path algorithm proposed by Dijkstra (1959) is applied to find the congested path, whose computational complexity is O(|E| + |V |log(|V |)), where |V | is the number of vertices and |E| is the number of links. The developed method can not only be suitable for searching the most congested path in large-scale urban traffic network, but also avoid the endless loop. Through this method, we can obtain a number of congested paths between all original-destination(OD)of the network, and then calculate the sum of weighted values of all links in each paths. There is only one maximum value, which indicating the most congested path in the network. Therefore, we can define the congestion index as follows: δ = Lmax /loptimum
(2)
where Lmax is the current weighted value of most congested path, loptimum is the weighted value of this path corresponding to the critical state. If 0 < δ < 1, it means that the traffic condition has not reached the critical state; otherwise, the traffic condition would become more congestion and it need to take measures to control the traffic signals in this most congested path. 3. MODELING AND CONTROL APPROACH 3.1 Urban Traffic Modeling In this subsection, the store-and-forward model of urban traffic networks proposed by Gazis and Potts (1963), and improved by Aboudolas et al. (2007, 2009, 2010), is briefly introduced, meanwhile, some simplifications are employed to make the traffic model more appropriate for control. An urban traffic network is represented as a directed graph with links z ∈ Z and intersections j ∈ J. Each signalized junction j has its own sets of incoming Ij and outgoing Oj links. Consider a typical urban road (link z) between two adjacent intersections (M, N ) as illustrated in Fig. 1 such that z ∈ OM and z ∈ IN . The dynamic of link z is described by the conservation equation:
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
xz (k + 1) = xz (k) + T [iz (k) − oz (k)]
(3)
where xz (k) is the number of vehicles within link z at time kT ; iz (k) and oz (k) are the inflow and outflow, respectively, of link z in the sample period [kT, (k + 1)T ]; T is the discrete time step and k = 0, 1, ... is the discrete time index. For simplicity we assume that there is no endogenous traffic flow and no vehicles terminating their trips within the link. Moreover, if the turning rates tw,z from the links that enter junction M towards link z with w ∈ IM also are assumed to be known, then the inflow to the link z is derived by iz (k) = Σw∈IM tw,z ow (k)
(4)
where ow (k) is the outflow from upstream links of z. To further improve the urban traffic model for the implementation of control, the discrete time step T is set to be equal to C, where C is the cycle time for each intersection j ∈ J. Supposed that there are enough vehicles in link z and the space is available in the downstream links, the outflow oz of link z is obtained by oz (k) = uz (k)Sz /C
(5)
where Sz is the saturation flow and always considered constant in practice, uz is the green time(control input) of link z during the period [kT, (k + 1)T ]. If the link z has right of way, the outflow is equal to the saturation flow, and zero otherwise. By combining (3), (4) and (5), it will lead to a linear scalar equation for the traffic movements of a given link. For a large-scale urban traffic network with arbitrary topology and characteristics, it can construct a linear state space representation to describe the dynamics of traffic system. This representation is composed of state vectors, control input vectors and the related coefficient system matrices. The general form of linear state equation is given by following: x(k + 1) = x(k) + Bu(k) + d(k)
(6)
In (6), the elements of state vector x(k) is the number of vehicles at step k of each controlled link in road network. The second term of state equation is the product of constant matric B and control input u. Vector u contains the green times of all stages of all intersections in the road network and should be optimized at each cycle. Matric B is related to the parameters saturation flow Sz , cycle time C and sample interval T . The last term d is the inflow of each link, which is determined by the outflow o and turning rates t from upstream links. It have been discussed above that the discrete time step T is equal to cycle time C, and S and t are constant, thus the elements of d(k) can be calculated by (4). In fact, it can be considered that the inflow of each link is a fixed value in the store-and-forward model. As the state space equation describes the dynamics and states of the urban traffic networks, there are several constraints which have to be taken into account. The first and most constraint of this model is the capacity of each link, which expresses the maximum number of vehicles the link contains. It is obvious that the element is determined by the topology and physical structure of road network. 498
This factor is defined by the product of the length and number of lanes of link between two adjacent intersections. Thus, the states are subject to the constraints: (7) 0 ≤ xz (k) ≤ xz,max The control input is the next variable restricted by some constraints. In order to guarantee that there is no all red stage appearing at intersection j ∈ J, it should allocate a minimum permissible green time for the stage. Moreover, an upper bound uz,max for the green time is also designed. Thus, the control input constraints are given by uz,min ≤ uz (k) ≤ uz,max (8) Another constraint the control input should satisfy is represented by the linear combination of green times at junction j ∈ J Nj
uz,i (k) = C
(9)
i=1
where Nj is the number of stages at junction j ∈ J. 3.2 Model-based Urban Traffic Control The aim of our research is to develop a control method to generate a set of optimal traffic signal timing dynamically according to the present traffic condition. The corresponding algorithm should be embedded in a rollinghorizon framework so that the optimal control problem can be solved on-line before every cycle. To this end, model predictive control technique is chosen since it is able to implement and repeatedly apply optimal control in a rolling-horizon way and take all the constraints into consideration in the process. Since our aim is to guide and regulate the traffic flow in the most congested path and further improve the mobility of whole network, the control objective is to minimize the risk of oversaturated and spill back of number of vehicles in this path. To this end, we attempt to minimize and balance the links’ occupancies xz /xz,max via the following finite-horizon linear criterion: k0 +K−1 xz (k) J= (10) xz,max k=k0
z∈Zcongestion
where K is the finite control horizon, Zcongestion is the most congested path searched in each step. On the basis of the linear model (6), the linear constraints (7), (8), (9) and linear objective function (10), the optimization problem of model-based control can be formulated as follows: min J = min
k0 +K−1 k=k0
s.t.
z=Zcongestion
xz (k) xz,max
xz (k + 1) = xz (k) + Buz (k) + dz (k) 0 ≤ xz (k) ≤ xz,max
uz,min ≤ uz (k) ≤ uz,max Nj
uj,i (k) = C
i=1
for k = k0 , . . . , k0 + K − 1, all z ∈ Z
(11)
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
with dash line in the figure. Then, we implement the proposed method to control the network. The fixed-time control strategy are used to compare with our method. For each control approach, four evaluation criteria are calculated for comparison via the simulation model. MFD is used to indicate the relationship between network vehicle density and average space-mean traffic flow. The number of congested links(NCL) denotes the number of oversaturation links, in this paper, we assume that if xz ≥ 0.7xz,max for all z ∈ Z, the link z is considered to be overloaded. Total Time Spent(TTS) is the accumulation amount of time spent by all the vehicles inside the road network since the beginning of the simulation, including both the vehicles freely running on a link and the vehicles slowing down or waiting in queues. TTS can be calculated as follows:
TTS =
K
T xz (k)
(12)
k=0 z∈Z
where K is the time horizon. Total Vehicle Departure(TVD) is the accumulation number of vehicles terminating their trips and leaving the network, and it reflects the mobility of the network.
Fig. 2. A hypothetical urban traffic network 4. SIMULATION To evaluate the effectiveness of the proposed control method in urban traffic management, we constructed a hypothetical test urban traffic network. The test network is shown in Fig. 2. There are 60 nodes including 16 source nodes and 44 intersections, and 90 two-way links with length varying from 213 to 913 meters in it. We carried out the simulation by using CORSIM, C++ and MATLAB. CORSIM is a microscopic traffic simulation software for analyzing traffic operations. It is able to simulate urban traffic network consisting of several intersections and allow the use of external control algorithm in the control processes. MATLAB is used to solve the rollinghorizon optimization problem. C++ provides the interface between CORSIM and MATLAB. We design two traffic scenarios with different peripheral inflow to investigate the efficiency of our method. The fixed cycle time in the network is C = 60s, and T = C is taken as a control interval for all signalized intersections. The turning rate tw,z and the saturation flow Sz are assumed to be constant for simplification. The results are shown below. 4.1 Case 1 In this case, we simulate the traffic scenarios in peak hour. The simulation time corresponds to 5 hours. The peak hour is divided into 10 half-hour demand periods. The demand variation for each source node is illustrated in Fig. 3, around 133,600 vehicles will be generated during this time. First of all, the optimal traffic state corresponding to the critical point in MFD should be obtained to compute the value of loptimum in (2). So we run 4 times with different supply flow rates for the network to derive a well-defined MFD as shown in Fig. 4, and the critical point is denoted 499
The simulation results are shown in Fig. 5. As illustrated in Fig. 5(a), it shows a comparison of the stationary MFDs for the two control strategy. In the low density region, the performance of each approach is quite similar, for the reason that the congestion index δ is less than 1. With the increase of supply inflow, the network become more congested, leading to the decrease of average space-mean flow. However, the proposed method obviously allows the network to reach a higher capacity keeping a higher traffic flow. The number of overloaded links at each control step is illustrated in Fig. 5(b). It can be noted that the proposed method makes the number of congested links maintaining a low value. Compared with fixed-time control, it represents a 10.3% decrease by using our method in the maximum number of congested links. Fig. 5(c) gives the comparison of TVD for the two traffic signal control strategy. Because this evaluation criteria denotes the throughput of the network, the developed method allows more vehicles to get through the network, and further improves the mobility of traffic. As a final evaluation observed from Fig. 5(d), we note that the proposed method performs slightly better than the fixed-time strategy. This can be further improved by implementing our method in a hierarchical control structure, which coordinates the interacted traffic flow between each subnetworks. The detailed analysis and comparison for each control strategy are shown in table 1. Table 1. Comparison of evaluation criteria Criterion TTS TVD NCL
Fixed-time 2.6 × 106 56339 68
Proposed methd 2.5 × 106 61215 61
Variation ↓ 3.9% ↑ 8.7% ↓ 10.3%
4.2 Case 2 In this subsection, another 5 hours traffic demand is chosen, including a demand increase as well as a decrease, to simulated the congestion onset and resolution. The alternative demand variation for each source node is illustrated
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
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As illustrated in Fig. 7(a), it shows a comparison of the relationship between network accumulation and spacemean traffic flow for each control strategy. When the 0.2 network passes the critical state, the traffic flow presents a trend of decrease. However, the proposed method also 0.15 clearly allows the network to reach a higher capacity keeping a higher traffic flow and prevents the network from 0.1 more accumulated vehicles. The number of overloaded links at each control step is illustrated in Fig. 7(b). It can 0.05 be noted that the proposed method makes the number of congested links maintaining a low value. From the results of TVD given in Fig. 7(c), it is not obviously to evaluate 0 0 5000 Critical point 10000 15000 the efficiency of two control strategies for the reason that Accumulation(veh) the traffic congestion has been resolved at the end of the simulation, most vehicles could through the network Fig. 4. MFD of the network and arrive their destinations outside. Therefore, we design a parameter, named as difference of TVD(D-TVD), to measure the performance of our method. The result is shown in Fig. 8. We note that the maximum difference of TVD of two methods is 895, which means that our method improves the throughput of the network, especially under the congested traffic states. The final comparison shown in Fig. 7(d) gives the results of TTS. It can be seen that the proposed method clearly decreases the TTS of the whole network. The detailed analysis and information are (b) Number of congested links illustrated in table 2, where max D-TVD represents the (a) MFDs value of TVD corresponding to the maximum difference of TVD, and mean NCL is the average number of congested links in whole simulation time. Traffic flow(veh/s)
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IFAC LSS 2013 July 7-10, 2013. Shanghai, China
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REFERENCES
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Aboudolas, K., Papageorgiou, M., and Kosmatopoulos, E. (2007). Control and optimization methods for traffic (b) Number of congested links (a) MFDs signal control in large-scale congested urban road networks. In American Control Conference, 2007. ACC’07, 3132–3138. IEEE. Aboudolas, K., Papageorgiou, M., and Kosmatopoulos, E. (2009). Store-and-forward based methods for the signal control problem in large-scale congested urban road networks. Transportation Research Part C: Emerging Technologies, 17(2), 163–174. Aboudolas, K., Papageorgiou, M., Kouvelas, A., and Kos(c) Total Vehicle Departure (d) Total Time Spent matopoulos, E. (2010). A rolling-horizon quadraticFig. 7. Simulation results of case 2 for each signal control programming approach to the signal control problem in strategy large-scale congested urban road networks. Transportation Research Part C: Emerging Technologies, 18(5), 680–694. 900 Dijkstra, E. (1959). A note on two problems in connexion with graphs. Numerische mathematik, 1(1), 269–271. Gartner, N. (1983). Opac: A demand-responsive strategy for traffic signal control. Transportation Research 600 Record, (906), 75–81. Gazis, D. and Potts, R. (1963). The oversaturated intersection. In Proceedings of the Second International 300 Symposium on Traffic Theory, London, UK, 221–237. Geroliminis, N. and Daganzo, C. (2007). Macroscopic modeling of traffic in cities. In Transportation Research Board 86th Annual Meeting, 07-0413. 0 Geroliminis, N. and Daganzo, C. (2008). Existence of −100 urban-scale macroscopic fundamental diagrams: Some 0 100 200 300 Time step experimental findings. Transportation Research Part B: Methodological, 42(9), 759–770. Hunt, P., Robertson, D., Bretherton, R., and Royle, M. Fig. 8. Difference of TVD at each control step in case 2 (1982). The scoot on-line traffic signal optimisation 5. CONCLUSION technique. Traffic engineering and control, 23(4), 190– 192. In this paper, we present an efficiency of signal control Lin, S., De Schutter, B., Xi, Y., and Hellendoorn, H. (2011). Fast model predictive control for urban road methodology to mitigate the traffic congestion, which can networks via milp. Intelligent Transportation Systems, be implemented as a local controller in a hierarchical IEEE Transactions on, 12(3), 846–856. control structure in large-scale heterogeneous urban traffic networks. Firstly, consider the uneven distribution of Lin, S., De Schutter, B., Xi, Y., and Hellendoorn, H. (2012). Efficient network-wide model-based predictive congestion degree in road network, we design a congestion control for urban traffic networks. Transportation Reindex by using the theory of MFD to indicate the traffic search Part C: Emerging Technologies, 24, 122–140. conditions and develop an algorithm to find the most congested path. Then, a simplified macroscopic urban traffic Lowrie, P. (1982). The sydney coordinated adaptive traffic system-principles, methodology, algorithms. In model is introduced to describe the dynamic process of International Conference on Road Traffic Signalling, traffic flow. Thirdly, the problem of network-wide signal 1982, London, United Kingdom, 207, 67–70. control is formulated as a linear programming problem that aims at minimizing the number of vehicles in congest- Mirchandani, P. and Head, L. (2001). A real-time traffic signal control system: architecture, algorithms, and ed links so as to improve the mobility of the network and analysis. Transportation Research Part C: Emerging mitigate the traffic congestion. For the application of this Technologies, 9(6), 415–432. method in real time, model-based control strategy is used to solve the traffic congestion problem in real time. Finally, we measure the performance of the proposed method under two different traffic demand scenarios by using four evaluation criteria in a hypothetical road network. Compared with the fixed-time control strategy, the simulation results show that it is an effective and feasible way to regulate the traffic flow and mitigate the congestion in 0
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A Congestion Eliminating Control Method for Large-scale Urban Traffic Networks Zhao Zhou ∗ Shu Lin ∗ Dewei Li ∗ Yugeng Xi ∗ ∗
Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai, 200240, China (e-mail: zzhou553, [email protected] dwli, [email protected]).
Abstract: This paper investigates an efficiency of signal control methodology, which mainly focuses on dealing with the traffic congestion problem in those key congested links and is applicable to be implemented in a hierarchical control structure in large-scale heterogeneous urban traffic networks. In this methodology, an algorithm for finding the most congested path is presented firstly, and the urban traffic flow is modeled by using a simplified macroscopic modeling framework. Then the problem of network-wide signal control is formulated as a linear programming problem that aims at minimizing the number of vehicles(or densities) in congested links so as to improve the mobility of the network and mitigate the traffic congestion. For the application of this method in real time, the multi-variables optimization problem including constraints is embedded in a model-based dynamic control procedure. Finally, different traffic demand scenarios are designed and four evaluation criteria are applied to measure the performance of the proposed method in a hypothetical road network. Compared with the fixedtime control strategy, the simulation results show that it is an effective and feasible way to regulate the traffic flow and mitigate the congestion in large-scale urban networks. Keywords: Urban traffic networks, large-scale systems, congested path, model-based control, hierarchical structure. 1. INTRODUCTION With the fast increasing number of vehicles and the rapid urbanization, traffic congestion became a big problem for metropolis, both in developed countries and developing countries. For example, in Shanghai, China, the citizens suffer from the problem of traffic congestion for a long period, especially during every morning and evening peak. Although this problem could be solved by constructing new transportation infrastructures and extending road networks, it is both costly and time-consuming. Therefore, a potential solution is to adopt network-wide traffic signal control strategies on the basis of the existing transportation infrastructures. For large-scale urban networks, the traffic flow system is a complex dynamic process. This network is composed of numerous intersections and roads, which means the dimension of traffic system is very high. The unpredictable activities of vehicles will lead to the uncertainty of traffic state on each road. There also exists complex interactions between each roads. Therefore, the urban traffic system is usually a dynamic large-scale system with multi-inputs and multi-outputs, the scale of optimization problem increases with the scale of the networks, and it will bring This work was supported in part by the National Science Foundation of China (Grant No. 60934007, 61203169, 61104160), China Postdoctoral Science Foundation (Grand No. 2011M500776), Shanghai Education Council Innovation Research Project (Grant No. 12ZZ024) and International Cooperation Project of National Science Committee (Grant No. 71361130012).
978-3-902823-39-7/2013 © IFAC
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high computation to design real-time efficient centralized traffic control strategies. Since the early 1980s, some hierarchical control systems like SCOOT and SCATS respectively proposed by Hunt et al. (1982) and Lowrie (1982) have been used for real-time adaptive traffic control. Gartner (1983) put forward a three-layer control architecture called OPAC. The OPAC control strategy is the adaptive algorithm implemented as the local controller of the hierarchical framework. Mirchandani and Head (2001) developed a multi-level hierarchical structure system—RHODES for real-time traffic signal control. Lin et al. (2011, 2012) presented an efficient network-wide model-based predictive control for urban traffic networks, which can be used as the local controller for subnetworks in a hierarchical control structure. Despite these systems play an important role in mitigation of urban traffic congestion, it does not consider the characteristic of heterogeneity, i.e. the densities of vehicle is uneven distributed in road networks. It means that if the traffic congestion in those serious oversaturated links could not be eliminated as soon as possible, it will lead to the phenomenon of queue spillback and even the gridlock in whole network. In this paper, we first develop an algorithm to find the most congested path, in which a great deal of vehicles is accumulated. Then, a simplified urban traffic model based on the dynamic of traffic flow is introduced and lays the foundation for future control. Finally, a model-based control strategy is proposed and used to alleviate the traffic congestion for large-scale urban traffic networks in real time.
10.3182/20130708-3-CN-2036.00032
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
The rest of this paper is organized as follows: Section 2 puts forward an algorithm to find the most oversaturated links. Section 3 introduces an urban traffic model and proposes an efficient model-based control method to eliminate the congestion in the congested path. Section 4 demonstrates the network-wide signal control method by examples. Section 5 concludes this paper.
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2. CONGESTED PATH In the previous studies, it mainly focuses on optimizing the signals to improve the mobility of whole traffic network or minimizing and balancing the number of vehicles in all links so as to reduce the risk of queue spillback under oversaturated traffic conditions. In reality, however, most large-scale traffic networks are usually heterogeneous networks with uneven distribution of congestion, especially in peak hours. In other words, due to the different traffic supply and demand, some links or subnetworks experience more congestion scenarios than the others. If the vehicles in these oversaturated links could not cross the intersections as soon as possible, it will produce queuing to block the inflow from upstream links, and then lead to the congestion of whole network. Therefore, it is necessary to regulate the traffic flow in these congested links. Recently, it has been verified by Geroliminis and Daganzo (2007, 2008) that there existing a well-defined macroscopic fundamental diagram(MFD) with unimodal and low scatter relationship between network vehicle density and space-mean flow in large-scale urban traffic networks. The maximum value of traffic flow appears at the critical point of MFD. With the increase of vehicles in network, the flow decreases and the traffic conditions become more congested. So the critical point of MFD indicates the optimal condition of traffic flow. Based on this, we first propose a congestion index to reflect the congestion degree of whole network. Consider that an urban traffic network can be expressed as a directed weighted network N = (V, E), where V = v1 , v2 , · · · , vn is the set of intersections and E = e(vi , vj )vi ,vj ∈V is the set of links between two adjacent intersections. The weight wij of edge e(vi , vj ) is the vehicle density between intersection i and j, which represents the traffic condition. wij = ni→j /li,j
]
(1)
where ni→j is the number of vehicles from intersection vi to vj , li,j is the link length. By collecting a great number of traffic data from different scenarios and periods in practice, we can derive a well-defined MFD and then obtain the vehicle density and flow corresponding to the critical point. The average value of traffic condition of each link at critical point is calculated and taken as the reference state. This work can be done off-line. According to the real-time traffic information directly collected using loop detectors and the physical characteristics of road network, the weighted value of each link can be obtained by (1). However, the biggest problem is how to find the most congested path in the heterogeneous network, for the reason that our aim is to guide and regulate the traffic flow in this key path. Here, we develop a method to realize the purpose. Due to that the weighted value 497
Fig. 1. An urban link denotes the traffic condition of each link, we take its reciprocal and assign it to the corresponding link as the new weighted value. After the preprocess, the lower weighted value means the more congested traffic condition. Thus, the problem mentioned above will be converted to the problem of searching the shortest path in whole network, which can be solved by using the popular algorithms in graph theory. In this paper, a most classical shortest path algorithm proposed by Dijkstra (1959) is applied to find the congested path, whose computational complexity is O(|E| + |V |log(|V |)), where |V | is the number of vertices and |E| is the number of links. The developed method can not only be suitable for searching the most congested path in large-scale urban traffic network, but also avoid the endless loop. Through this method, we can obtain a number of congested paths between all original-destination(OD)of the network, and then calculate the sum of weighted values of all links in each paths. There is only one maximum value, which indicating the most congested path in the network. Therefore, we can define the congestion index as follows: δ = Lmax /loptimum
(2)
where Lmax is the current weighted value of most congested path, loptimum is the weighted value of this path corresponding to the critical state. If 0 < δ < 1, it means that the traffic condition has not reached the critical state; otherwise, the traffic condition would become more congestion and it need to take measures to control the traffic signals in this most congested path. 3. MODELING AND CONTROL APPROACH 3.1 Urban Traffic Modeling In this subsection, the store-and-forward model of urban traffic networks proposed by Gazis and Potts (1963), and improved by Aboudolas et al. (2007, 2009, 2010), is briefly introduced, meanwhile, some simplifications are employed to make the traffic model more appropriate for control. An urban traffic network is represented as a directed graph with links z ∈ Z and intersections j ∈ J. Each signalized junction j has its own sets of incoming Ij and outgoing Oj links. Consider a typical urban road (link z) between two adjacent intersections (M, N ) as illustrated in Fig. 1 such that z ∈ OM and z ∈ IN . The dynamic of link z is described by the conservation equation:
IFAC LSS 2013 July 7-10, 2013. Shanghai, China
xz (k + 1) = xz (k) + T [iz (k) − oz (k)]
(3)
where xz (k) is the number of vehicles within link z at time kT ; iz (k) and oz (k) are the inflow and outflow, respectively, of link z in the sample period [kT, (k + 1)T ]; T is the discrete time step and k = 0, 1, ... is the discrete time index. For simplicity we assume that there is no endogenous traffic flow and no vehicles terminating their trips within the link. Moreover, if the turning rates tw,z from the links that enter junction M towards link z with w ∈ IM also are assumed to be known, then the inflow to the link z is derived by iz (k) = Σw∈IM tw,z ow (k)
(4)
where ow (k) is the outflow from upstream links of z. To further improve the urban traffic model for the implementation of control, the discrete time step T is set to be equal to C, where C is the cycle time for each intersection j ∈ J. Supposed that there are enough vehicles in link z and the space is available in the downstream links, the outflow oz of link z is obtained by oz (k) = uz (k)Sz /C
(5)
where Sz is the saturation flow and always considered constant in practice, uz is the green time(control input) of link z during the period [kT, (k + 1)T ]. If the link z has right of way, the outflow is equal to the saturation flow, and zero otherwise. By combining (3), (4) and (5), it will lead to a linear scalar equation for the traffic movements of a given link. For a large-scale urban traffic network with arbitrary topology and characteristics, it can construct a linear state space representation to describe the dynamics of traffic system. This representation is composed of state vectors, control input vectors and the related coefficient system matrices. The general form of linear state equation is given by following: x(k + 1) = x(k) + Bu(k) + d(k)
(6)
In (6), the elements of state vector x(k) is the number of vehicles at step k of each controlled link in road network. The second term of state equation is the product of constant matric B and control input u. Vector u contains the green times of all stages of all intersections in the road network and should be optimized at each cycle. Matric B is related to the parameters saturation flow Sz , cycle time C and sample interval T . The last term d is the inflow of each link, which is determined by the outflow o and turning rates t from upstream links. It have been discussed above that the discrete time step T is equal to cycle time C, and S and t are constant, thus the elements of d(k) can be calculated by (4). In fact, it can be considered that the inflow of each link is a fixed value in the store-and-forward model. As the state space equation describes the dynamics and states of the urban traffic networks, there are several constraints which have to be taken into account. The first and most constraint of this model is the capacity of each link, which expresses the maximum number of vehicles the link contains. It is obvious that the element is determined by the topology and physical structure of road network. 498
This factor is defined by the product of the length and number of lanes of link between two adjacent intersections. Thus, the states are subject to the constraints: (7) 0 ≤ xz (k) ≤ xz,max The control input is the next variable restricted by some constraints. In order to guarantee that there is no all red stage appearing at intersection j ∈ J, it should allocate a minimum permissible green time for the stage. Moreover, an upper bound uz,max for the green time is also designed. Thus, the control input constraints are given by uz,min ≤ uz (k) ≤ uz,max (8) Another constraint the control input should satisfy is represented by the linear combination of green times at junction j ∈ J Nj
uz,i (k) = C
(9)
i=1
where Nj is the number of stages at junction j ∈ J. 3.2 Model-based Urban Traffic Control The aim of our research is to develop a control method to generate a set of optimal traffic signal timing dynamically according to the present traffic condition. The corresponding algorithm should be embedded in a rollinghorizon framework so that the optimal control problem can be solved on-line before every cycle. To this end, model predictive control technique is chosen since it is able to implement and repeatedly apply optimal control in a rolling-horizon way and take all the constraints into consideration in the process. Since our aim is to guide and regulate the traffic flow in the most congested path and further improve the mobility of whole network, the control objective is to minimize the risk of oversaturated and spill back of number of vehicles in this path. To this end, we attempt to minimize and balance the links’ occupancies xz /xz,max via the following finite-horizon linear criterion: k0 +K−1 xz (k) J= (10) xz,max k=k0
z∈Zcongestion
where K is the finite control horizon, Zcongestion is the most congested path searched in each step. On the basis of the linear model (6), the linear constraints (7), (8), (9) and linear objective function (10), the optimization problem of model-based control can be formulated as follows: min J = min
k0 +K−1 k=k0
s.t.
z=Zcongestion
xz (k) xz,max
xz (k + 1) = xz (k) + Buz (k) + dz (k) 0 ≤ xz (k) ≤ xz,max
uz,min ≤ uz (k) ≤ uz,max Nj
uj,i (k) = C
i=1
for k = k0 , . . . , k0 + K − 1, all z ∈ Z
(11)
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with dash line in the figure. Then, we implement the proposed method to control the network. The fixed-time control strategy are used to compare with our method. For each control approach, four evaluation criteria are calculated for comparison via the simulation model. MFD is used to indicate the relationship between network vehicle density and average space-mean traffic flow. The number of congested links(NCL) denotes the number of oversaturation links, in this paper, we assume that if xz ≥ 0.7xz,max for all z ∈ Z, the link z is considered to be overloaded. Total Time Spent(TTS) is the accumulation amount of time spent by all the vehicles inside the road network since the beginning of the simulation, including both the vehicles freely running on a link and the vehicles slowing down or waiting in queues. TTS can be calculated as follows:
TTS =
K
T xz (k)
(12)
k=0 z∈Z
where K is the time horizon. Total Vehicle Departure(TVD) is the accumulation number of vehicles terminating their trips and leaving the network, and it reflects the mobility of the network.
Fig. 2. A hypothetical urban traffic network 4. SIMULATION To evaluate the effectiveness of the proposed control method in urban traffic management, we constructed a hypothetical test urban traffic network. The test network is shown in Fig. 2. There are 60 nodes including 16 source nodes and 44 intersections, and 90 two-way links with length varying from 213 to 913 meters in it. We carried out the simulation by using CORSIM, C++ and MATLAB. CORSIM is a microscopic traffic simulation software for analyzing traffic operations. It is able to simulate urban traffic network consisting of several intersections and allow the use of external control algorithm in the control processes. MATLAB is used to solve the rollinghorizon optimization problem. C++ provides the interface between CORSIM and MATLAB. We design two traffic scenarios with different peripheral inflow to investigate the efficiency of our method. The fixed cycle time in the network is C = 60s, and T = C is taken as a control interval for all signalized intersections. The turning rate tw,z and the saturation flow Sz are assumed to be constant for simplification. The results are shown below. 4.1 Case 1 In this case, we simulate the traffic scenarios in peak hour. The simulation time corresponds to 5 hours. The peak hour is divided into 10 half-hour demand periods. The demand variation for each source node is illustrated in Fig. 3, around 133,600 vehicles will be generated during this time. First of all, the optimal traffic state corresponding to the critical point in MFD should be obtained to compute the value of loptimum in (2). So we run 4 times with different supply flow rates for the network to derive a well-defined MFD as shown in Fig. 4, and the critical point is denoted 499
The simulation results are shown in Fig. 5. As illustrated in Fig. 5(a), it shows a comparison of the stationary MFDs for the two control strategy. In the low density region, the performance of each approach is quite similar, for the reason that the congestion index δ is less than 1. With the increase of supply inflow, the network become more congested, leading to the decrease of average space-mean flow. However, the proposed method obviously allows the network to reach a higher capacity keeping a higher traffic flow. The number of overloaded links at each control step is illustrated in Fig. 5(b). It can be noted that the proposed method makes the number of congested links maintaining a low value. Compared with fixed-time control, it represents a 10.3% decrease by using our method in the maximum number of congested links. Fig. 5(c) gives the comparison of TVD for the two traffic signal control strategy. Because this evaluation criteria denotes the throughput of the network, the developed method allows more vehicles to get through the network, and further improves the mobility of traffic. As a final evaluation observed from Fig. 5(d), we note that the proposed method performs slightly better than the fixed-time strategy. This can be further improved by implementing our method in a hierarchical control structure, which coordinates the interacted traffic flow between each subnetworks. The detailed analysis and comparison for each control strategy are shown in table 1. Table 1. Comparison of evaluation criteria Criterion TTS TVD NCL
Fixed-time 2.6 × 106 56339 68
Proposed methd 2.5 × 106 61215 61
Variation ↓ 3.9% ↑ 8.7% ↓ 10.3%
4.2 Case 2 In this subsection, another 5 hours traffic demand is chosen, including a demand increase as well as a decrease, to simulated the congestion onset and resolution. The alternative demand variation for each source node is illustrated
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REFERENCES
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Aboudolas, K., Papageorgiou, M., and Kosmatopoulos, E. (2007). Control and optimization methods for traffic (b) Number of congested links (a) MFDs signal control in large-scale congested urban road networks. In American Control Conference, 2007. ACC’07, 3132–3138. IEEE. Aboudolas, K., Papageorgiou, M., and Kosmatopoulos, E. (2009). Store-and-forward based methods for the signal control problem in large-scale congested urban road networks. Transportation Research Part C: Emerging Technologies, 17(2), 163–174. Aboudolas, K., Papageorgiou, M., Kouvelas, A., and Kos(c) Total Vehicle Departure (d) Total Time Spent matopoulos, E. (2010). A rolling-horizon quadraticFig. 7. Simulation results of case 2 for each signal control programming approach to the signal control problem in strategy large-scale congested urban road networks. Transportation Research Part C: Emerging Technologies, 18(5), 680–694. 900 Dijkstra, E. (1959). A note on two problems in connexion with graphs. Numerische mathematik, 1(1), 269–271. Gartner, N. (1983). Opac: A demand-responsive strategy for traffic signal control. Transportation Research 600 Record, (906), 75–81. Gazis, D. and Potts, R. (1963). The oversaturated intersection. In Proceedings of the Second International 300 Symposium on Traffic Theory, London, UK, 221–237. Geroliminis, N. and Daganzo, C. (2007). Macroscopic modeling of traffic in cities. In Transportation Research Board 86th Annual Meeting, 07-0413. 0 Geroliminis, N. and Daganzo, C. (2008). Existence of −100 urban-scale macroscopic fundamental diagrams: Some 0 100 200 300 Time step experimental findings. Transportation Research Part B: Methodological, 42(9), 759–770. Hunt, P., Robertson, D., Bretherton, R., and Royle, M. Fig. 8. Difference of TVD at each control step in case 2 (1982). The scoot on-line traffic signal optimisation 5. CONCLUSION technique. Traffic engineering and control, 23(4), 190– 192. In this paper, we present an efficiency of signal control Lin, S., De Schutter, B., Xi, Y., and Hellendoorn, H. (2011). Fast model predictive control for urban road methodology to mitigate the traffic congestion, which can networks via milp. Intelligent Transportation Systems, be implemented as a local controller in a hierarchical IEEE Transactions on, 12(3), 846–856. control structure in large-scale heterogeneous urban traffic networks. Firstly, consider the uneven distribution of Lin, S., De Schutter, B., Xi, Y., and Hellendoorn, H. (2012). Efficient network-wide model-based predictive congestion degree in road network, we design a congestion control for urban traffic networks. Transportation Reindex by using the theory of MFD to indicate the traffic search Part C: Emerging Technologies, 24, 122–140. conditions and develop an algorithm to find the most congested path. Then, a simplified macroscopic urban traffic Lowrie, P. (1982). The sydney coordinated adaptive traffic system-principles, methodology, algorithms. In model is introduced to describe the dynamic process of International Conference on Road Traffic Signalling, traffic flow. Thirdly, the problem of network-wide signal 1982, London, United Kingdom, 207, 67–70. control is formulated as a linear programming problem that aims at minimizing the number of vehicles in congest- Mirchandani, P. and Head, L. (2001). A real-time traffic signal control system: architecture, algorithms, and ed links so as to improve the mobility of the network and analysis. Transportation Research Part C: Emerging mitigate the traffic congestion. For the application of this Technologies, 9(6), 415–432. method in real time, model-based control strategy is used to solve the traffic congestion problem in real time. Finally, we measure the performance of the proposed method under two different traffic demand scenarios by using four evaluation criteria in a hypothetical road network. Compared with the fixed-time control strategy, the simulation results show that it is an effective and feasible way to regulate the traffic flow and mitigate the congestion in 0
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