An Iterative Group-based Signal Optimization Scheme for Traffic Equilibrium Networks

Transcription

1 Journal of Advanced Transportation, Vol. 33, No. 2, pp An Iterative Group-based Signal Optimization Scheme for Traffic Equilibrium Networks S.C. WONG Chao YANG This paper presents an iterative scheme for a combined signal optimization and assignment problem, using a traffic model from the well-known procedure TRANSYT. The signal settings are optimized by means of a group-based technique, in which the signal timings are specified by the common cycle time, the start time and duration of the period of right of way for each signal group in the network. The optimization problem was formulated as an integer program and solved by a set of heuristics. Given the optimized signal settings determined from the group-based technique, a path-based assignment algorithm is employed to obtain the equilibrium traffic pattern using the sensitivity information for TRANSYT model and a Frank-Wolf method. Based on the equilibrium flow pattern, the group-based optimization algorithm is then used to determine a better set of signal timings. The procedure is repeated until certain convergence criteria are satisfied. A numerical example is employed to demonstrate the benefits obtained from this iterative scheme. Encouraging results are obtained. Introduction Traffic signals have been used world-wide to resolve conflicting traffic movements, both vehicular and pedestrian, especially at intersections. In urban streets where the spacing between adjacent intersections is comparatively short, better operational performance of the signal-controlled intersections can often be obtained by taking into S.C. Wong is at the Department of Civil Engineering, The University of Hong Kong, Hong Kong, P.R. China. Chao Yang is at the Department of Road and Traffic Engineering, Tongji University, Shanghai, P.R. China. Received December 1998; Accepted March 1999.

2 202 S.C. Wong and Chao Yang account the interaction between adjacent intersections in the determination of signal settings. Such co-ordination among intersections over an area is called area traffic control. Early experience with area traffic control (Holroyd and Hillier, 1971) showed that substantial benefits can be achieved by coordinating adjacent signals using fixedtime plans. TRANSYT (Vincent et al, 1980) has been found to be one of the most effective analysis tools for calculating settings for coordinated signals for an urban road network. The model has also been used in many applications, such as the implementation of dynamic advisory speed signs (Van Leersum J., 1985), and others. The TRANSYT program consists of two main modules, the traffic model and the signal optimizer. In the traffic model, TRANSYT simulates the movement of traffic through a network and takes into account of the effects of platoon dispersion. The model provides the means of calculating a performance index which is a weighted linear combination of estimated vehicular delays and stops on all the streets and is used to measure the overall cost of traffic congestion. The signal optimizer adjusts the signal timings and checks, using the traffic model, whether the adjustment reduces the performance index. The signal timings are successively improved by adopting only those adjustments which reduce the performance index. This method, which is characterized by its trial and error basis, is called a hill-climbing technique. The recent development of microprocessor controller technology for signal-controlled junctions provides a higher degree of flexibility for the specification of traffic signal settings. A recent method which takes full advantages of this flexibility by controlling each of the signal groups of a signal-controlled junction independently is called group-based technique, in which the sets of traffic streams and pedestrian movements controlled as single units are referred to as signal groups. The capability of groupbased techniques to include the structure of interstage periods and some aspects of sequence selection in the optimization of timings at individual junctions is now well established (Heydecker and Dudgeon, 1987; Gallivan and Heydecker, 1988; Allsop, 1992; Heydecker, 1992). Recently, this group-based technique has been extended to the optimization of fixed-time signal timings of a signal-controlled network, using the traffic model from TRANSYT to evaluate the performance index (Wong, 1995, 1996, 1997). The problem was formulated as a set of non-linear mathematical programs, in which the performance index of a network is minimized subject to certain constraints on the group-based

3 An Iterative Group-based Signal Optimization Scheme control variables, such as the common cycle time, the start time and duration of the period of right of way for each signal group. Since the performance index of a network is a non-convex function of the groupbased control variables, optimization heuristics were formulated to optimize the fixed-time signal timings for the network. In the above group-based method, it is assumed that the traffic flows in the network are given and fixed during the course of signal optimization procedure. However, it has been observed that the equilibrium flow pattern of a network is strongly related to the traffic signal settings. By changing the control strategies, the traffic will arrange itself in an user optimal manner (Wardrop, 1952), for which users traveling from any origin to any destination react to the new control strategy by choosing paths such that their individual cost are minimized. This redistribution effect on equilibrium traffic flow pattern will affect the performance of the network. Several authors have pointed out the importance of considering signal control in conjunction with traffic equilibrium and studied the general characteristics of this non-convex optimization problem. Among them, Allsop (1974) was one of the first who suggested the signal control can be explored to affect the distribution and assignment of traffic on an equilibrium network, and provided a rigorous mathematical framework for the problem. This problem has been known as equilibrium traffic signal settings in the literature (Wong, et al., 1998a). The difficulties associated with detailed junction modeling in road traffic assignment have been discussed in Heydecker (1983), in which it was shown that for most signal control policies the conditions for good convergence behavior in traffic assignment are violated. Despite the fact that it is assumed in this paper that the signal settings are fixed, the problem is still difficult to solve as the cost functions, in contrast to the conventional network assignment problems, are generally asymmetric and non-convex. This problem was also studied in Hall et a1 (1980), in which a simulation-assignment model, SATURN, for the evaluation of traffic management schemes was developed. The simulation module was based on the TRANSYT traffic model and the assignment module was basically a conventional assignment procedure (LeBlanc et al, 1975). The problem was then solved by a diagonalization algorithm in which at each iteration the set of separable cost functions (i.e. the travel time on a link is unaffected by the flows on all the other links in the network), which were approximated by polynomial curves whose parameters were estimated from the last simulation results, was generated for use in the

4 204 S.C. Wong and Chao Yang assignment module. It was however mentioned in Hall et a1 (1980) that (even though the assignment module guaranteed to converge in itself) the simulatiotdassignment loop does not necessarily converge to the equilibrium solution where neither flows nor delays change significantly. Oscillation of results have been found in certain networks, especially when the interaction of traffic at adjacent signal-controlled intersections is significant. To overcome the shortcoming associated with the diagonalization algorithm, a path-based algorithm to solve this strongly asymmetric traffic assignment problem using the TRANSYT traffic model was recently developed (Wong et al., 1998b). Based on the group-based optimization heuristics and this path-based assignment algorithm, an iterative scheme is employed in this paper to optimize the signal settings taking into account the full advantages of the group-based techniques, modeling capability of TRANSYT traffic models and consideration of re-routing characteristics of road users in response to the signal control. The iterative scheme consists of two main modules: group-based signal optimization module and path-based assignment module. In the signal optimization module, the flow pattern is assumed fixed and the groupbased optimization heuristics (Wong, 1996) are employed to optimize the signal timings. In the assignment module, given the optimized signal settings, the equilibrium traffic pattern is determined by means of the path-based assignment algorithm in response to the signal settings obtained thus far (Wong et al., 1998b). These two modules iterate until certain convergence criteria are satisfied. In Section 2, the TRANSYT traffic model is briefly described. The signal optimization module and assignment module are then discussed in Sections 3 and 4 respectively. The iterative scheme is formulated in Section 5, and computational results for a small trial network are shown in Section 6 to demonstrate the benefits of the scheme. The Transyt Traffic Model The TRANSYT traffic model (Robertson, 1969; Vincent et al, 1980) simulates the movement of traffic through a network and takes into account of the effect of platoon dispersion. It is a widely used procedure to determine the queues and delays in a signal-controlled network with explicit consideration of the signal coordination effects. However, the traffic model does not consider the re-routing of traffic in the network in

5 An Iterative Group-based Signal Optimization Scheme response to the traffic conditions. In this paper, the TRANSYT traffic model is employed for the evaluation of delays in the network, which forms the basic module of the problem in the paper. In our model, the cruise time on a link is fixed, but the delays at the end of the link depends on the flows of many other links in the network. The travel time on a link, therefore, consists of two components: the cruise time and the delay at the end of the link. This travel time function is generally asymmetric with respect to the link flows, and therefore the problem falls into the class of asymmetric network equilibrium problems. In the TRANSYT traffic model, the delay is divided into two components: the uniform and random-and-oversaturation delays. The uniform delay represents the delay incurred with an identical pattern of traffic arrives during every cycle. To determine this uniform delay, the following traffic patterns as functions of time during a cycle are defined to describe how the vehicles arrive at and depart from a link: (i) IN pattern: the pattern of traffic that would arrive at the stop line at the end of the link if the traffic were not impeded by the signals at the stop line; (ii) OUT pattern: the pattern of traffic leaving a link; (iii)go pattern: the pattern of traffic that would leave the stop line if there was enough traffic to saturate the green. These definitions were employed in Vincent et a1 (1980). The uniform component of delay was obtained through simulation of two cycles of the IN, OUT and GO patterns to obtain the queue formation patterns of all links, which were then used to calculate the uniform delay. The GO pattern depends on the link characteristics such as saturation flow and the signal settings. From the simulation process, the OUT pattern from the link is determined from the IN and GO patterns. in the network, the traffic entering into a link is affected by all the OUT patterns from the upstream links. These traffic will travel along the link in accordance with a linear recursive platoon dispersion function to determine the IN pattern at the stop line of the link. For a network with a tree structure, the traffic model only needs to iterate once. However, for a general network structure, the traffic model has to iterate until the traffic patterns stabilize. Then the uniform delay is calculated based on the stabilized (or converged) traffic patterns. The random-and-oversaturation delay takes into account respectively the variations in traffic arrivals from cycle to cycle and the steady increase in queues on oversaturation links. To determine the randomand-oversaturation delay, approximate delay formulae, adopting the

6 206 S.C. Wong and Chao Yang coordinate transformation method (Kimber & Hollis, 1979), were employed to estimate their values. In this paper, the performance index for a network is defined as the sum of the cruise time, uniform delay and random-and-oversaturation delay, incurred by all users. Group-Based Signal Optimization Module Groupbased variables and constraints The group-based control variables for a signal-controlled network are given as follows. The junctions are operated at a common cycle time. The period during which a particular signal group at a junction has right of way is specified by two control variables: the start and duration of green for the signal group. The start of green is measured from a master clock for all junctions. All the control variables are expressed in time units. The offsets of junctions are very common variables in most linked signal calculations, but it is one of the advantages of group-based method that the offset variables have been implicitly included in the group-based control variables. These group-based variables are usually subject to the following constraints: For the case of unspecified cycle time, the common cycle time is also considered as a control variable and is confined to a certain practicable range. A minimum acceptable duration of green indication is usually specified for a vehicular or pedestrian stream. For a pedestrian stream, this minimum duration depends on the width of crossing and the walking speed of pedestrians. For any two incompatible signal groups, a clearance time is required between the end of green of a signal group and the start of green of another group so that all the vehicles from the former group have left the conflict points before the vehicles from the latter group arrive. When a traffic stream operates near to its capacity, delay to vehicles is substantial. Sometimes, one may need to specify the maximum acceptable degree of saturation for a traffic stream so as to make sure the stream is always operating below a certain acceptable congestion level.

7 An Iterative Group-based Signal Optimization Scheme For reasons of practicability, there may be some constraints to be imposed on the relative timing of starts and ends of green for different signal groups. ODtimization heuristics The optimization problem can be formulated as the following integer program, Minimize P(q*,s) S subject to where P(q*,s) is the value of the performance index of the network * when the equilibrium traffic pattern is q and the signal settings are s. All control variables are expressed in integer seconds, and the coefficient matrix and constant vector in equation (1) take integer forms. The integer program specified in equation (1) is solved by means of the heuristics developed in Wong (1996), which consists of a mixture of network-wide and junction-based steps. In network-wide steps, the control variables at all junctions are changed simultaneously, whereas in junction-based steps, the control variables at all junctions are changed in turn as happens in TRANSYT. For more detailed description of the heuristics, the reader can refer to the earlier paper (Wong, 1996). Path-Based Assignment Module Under user equilibrium assumption, there exists a such that the equilibrium flow pattern can be determined for a given set of group-based signal parameters s, i.e.

8 208 S.C. Wong and Chao Yang The major difficulty associated with the problem in equation (2) is that using the TRANSYT traffic model for the evaluation of junction delays the assignment problem is asymmetric and non-convex. Owning to the constantly increasing power of computers which helps in relaxing the computational constraints, the path-based assignment algorithms have recently received much attention (Jayakrishnan et al., 1994; Cascetta et al., 1997; Bell et al., 1997; Lo, 1998). In a recent paper (Wong et al., 1998b), the asymmetric assignment problem using TRANSYT traffic model mentioned above was formulated as a minimization problem using path-based notation. A post-simulation sensitivity analysis was developed to evaluate the derivatives information of the TRANSYT traffic model. This enables the descent direction to be determined for a line search algorithm. The Frank-Wolfe solution algorithm is then employed to solve the resulting minimization problem to obtain the equilibrium traffic pattern. The Iterative Scheme For general iterative procedures, the problem is divided into two subproblems: optimize the signal settings subject to fixed flow pattern and determine user-equilibrium flow pattern subject to fixed signal control parameters. These two sub-problems are solved alternately until certain convergence criteria are satisfied. The iterative scheme used in this paper is summarized below: Optimization of signal settings with fixed equilibrium flow Dattern For a given equilibrium flow pattern, the group-based signal optimization module discussed in Section 3 is employed to optimize the signal settings: Minimize P(q*,s) S (34 subject to As2b

9 An Iterative Group-based Signal Optimization Scheme where qt is the equilibrium traffic pattern obtained from the assignment module based on the optimized signal timings in last iteration in Section 4. The solution of the problem provides the optimized signal settings denoted by S*. These optimized settings are then fed into the assignment module. Determination of equilibrium flow pattern with fixed signal settings With the optimized signal settings, the equilibrium flow pattern is determined by the path-based assignment algorithm given in Section 4 as: 4* = m*) (4) where s* is the optimized signal settings obtained from the signal optimization module thus so. The equilibrium flow pattern obtained from equation (4) is then fed into the signal optimization module. These two modules are solved alternately until certain convergence * * criteria are satisfied. In such case, the solution obtained (q,s ) is a mutually consistent point satisfying equations (3, 4). COMPUTATIONAL RESULTS Consider a network shown in Figure 1 consisting of 9 signalcontrolled intersections, 60 links and 4 origirddestination nodes. The length and speed of the links are 500 meters and 50 kmhr respectively. All clearance times between incompatible traffic streams and minimum acceptable greens of signal groups are 5 seconds. The cycle time is 120 seconds. The saturation flows for straight ahead and right turning links are 1,600 vehhr and 1,200 vehhr respectively, and those for all other links are 1,400 vehhr. The performance index of the network is considered as the total travel time in veh-hrhr. The OD matrix is given as r o

10 "" tr 30 [ it c Origiddestination node Signal-coii(rollcd junction TRANSYT link Figure 1. The example network.

11 An Iterative Group-bused Signal Optimization Scheme > v 220\ Figure 2. Typical convergence characteristics of the iterative scheme. The convergence characteristics of the example problem is shown in Figure 2. Note that each point in the figure refers to the performance index of the network at equilibrium of traffic in response to the optimized signal timings obtained from the last group-based optimization module. Good convergence of the algorithm is observed. However, it is also found that occasional rebounds of the performance index occurs at certain iterations. This is a common characteristics of iterative schemes. When the result stabilizes, a mutually consistence flow pattern between signal settings and equilibrium assignment is found. Further optimization of signal timings keeping traffic flows fixed would not improve the performance index further. However, this may not necessarily a locally minimum to the problem concerned. Therefore, the optimized signal settings are chosen as the one with best performance index out of all the iterations during the solution process. For the case as shown in Figure 2, the best performance index occurs at the eighth iteration. The corresponding initial and final optimized timing plans for the central junction 5 are plotted in Figure 3 for comparison. It can be

12 212 S.C. Wong and Chao Yang R I - v) W 1 I 0 co &.- v) W 0 W 1 ln N 1 0 LJ ;I (3 N Ic Ic W 0 a, 3 : cd c M N

13 An Iterative Group-based Signal Optimization Scheme seen that the advantages of group-based specification of signal timings have been fully utilized, where new phases (stages) were generated automatically during optimization. Although the results obtained from the iteration scheme may not necessarily be the global minimum, nor even a local minimum, it is observed from the computational results that a substantial reduction in performance index is achieved. This provides a very useful tool for setting signal timings in a network taking into account the rerouting behavior of road users. Ignoring this assignment characteristics, the benefits obtained from a signal optimization schemecould be misleading, as travelers may react to the derived signal plan in a way to cancel some of the expected benefits. On the other hand, for practicality considerations, when a truly optimal solution may not be strictly required, the iterative scheme can produce a near optimal solution, or at least a solution lies in a region likely containing the optimal settings, with which remarkable improvements in the performance of the network can be achieved. Table 1. The results from the random offset calculation. Note: (A) Statistics for the set of initial values of performance index (B) Statistics for the set of final optimized values of performance index (C) Statistics for the set of percent improvement values calculated as the difference between the initial and final optimized performance index divided by the initial value for a particular optimization

14 214 S.C. Wong and Chao Yang To test the reliability of the iterative scheme in producing better mutually consistence flow pattern in the network, a random offset method (Wong, 1996) is employed in which the iterative scheme is applied to a number of starting points each with a distinct set of initial offsets at junctions. For the example network, 40 sets of initial offsets are generated randomly and each of this starting points is input to the iterative scheme to compute the best performance index. The results are summarized in Table 1. It can be seen that the iterative scheme is always able to produce good results even with different starting points. The improvements are also consistent and reliable. Over 15% of improvement on the performance index is obtained for most cases in this example network. The distribution of optimized performance index for the 40 sets of results is also plotted in Figure 4. It is interesting to note that there are two clusters of local minimums obtained from the random offset method. While almost half of the results fall into the region of the right-hand cluster (with larger performance index), there are about 20% of results hitting the left-hand cluster with better performance index D 2 P, " I I I I85 I90 Performance index (veh-hrhr) Figure 4. The distribution of optimized performance index.

15 An Iterative Group-based Signal Optimization Scheme Conclusions In a recent research work, the group-based optimization heuristics employing the TRANSYT traffic model for the evaluation of network performance were developed to optimize the signal settings in a network (Wong, 1996). However, no rerouting was considered in the optimization problem and therefore the optimal signal plan obtained from the method may not fully reflect the true benefits of signal coordination, as road users might switch to other routes leading to a different traffic pattern on which the signal optimization is based. On the other hand, the determination of user equilibrium pattern with TRANSYT is a difficult task due to the asymmetric and non-convex nature of the assignment problem. In a more recent paper (Wong et al., 1998b), the difficulties associated with the solution of user equilibrium pattern using TRANSYT traffic model are resolved by employing a path-based solution algorithm. This opens the way for the formulation of a more realistic signal optimization algorithm taking into account the rerouting characteristics of road users in response to the signal plan. In this paper, an iterative scheme has been developed to solve the combined signal optimization and assignment problem mentioned above. Encouraging results were obtained. For the example network shown in Figure 1, over 15% of improvement in the performance index of the network has been obtained. From the random offsets calculations, it has also been shown that the iterative scheme is reliable in producing a near optimal signal timing plan with mutually consistent flow pattern between signal settings and equilibrium assignment. Acknowledgements This research was supported by a research grant HKU 7015/97E from the Hong Kong Research Grant Council. Transport Research Laboratory is acknowledged for supplying the source code of the TRANSYT program for this research.

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