traffic control is modified such that congestion on a certain route dis- .... H. J. van Zuylen, Delft University of Technology, P.O. Box 5048, 2600 GA ... traffic and traffic control conditions will cause the system to slide ... are inclined to change their behavior sooner. ... determine the choices made by travelers, is too simplistic.
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Anticipatory Optimization of Traffic Control Henk J. van Zuylen and Henk Taale Traffic control and travelers’ behavior are two mutually influential processes with different objectives. Decisions made in traffic control influence travelers’ possibilities in choosing their preferred mode, route, and time of departure; and the choices made by travelers influence the optimization possibilities for traffic control. This research presents the results of simulation studies and a mathematical analysis of this bilevel optimization problem. Under certain conditions, multiple stable situations are possible, but some of these situations are sensitive to small disturbances by which the system moves away from the original equilibrium state. There appears to be a nonlinear relationship between system parameters and the character and location of the equilibrium situations. The details of the travel time model appear to have a large influence. If road authorities want to optimize traffic control, they have to anticipate the reaction of travelers. This makes the optimization process much more complicated. Iterative optimization, where traffic control is adjusted as soon as traffic conditions change, generally does not lead to a system optimum. Methods are therefore necessary that allow for the optimization of traffic control while taking into account that traffic flows will change as a result of traffic control.
There is a tradition in traffic control to adapt control structure and parameters to actual traffic flows and conditions such that total delay, average queue length, or some other collective objective function is optimized. For fixed-time and traffic-actuated controllers, methods and guidelines have been developed (such as Webster’s formula for cycle time and green splits, the rule to minimize delays by choosing green splits that minimize the maximum degree of saturation, the rules for maximum green times and maximum gap times for vehicleactuated controllers, etc.). Apart from the optimization of the total network, often special treatment is given to selected groups of road users. In urban areas pedestrians may get preferred treatment, at bus routes the bus may get priority, and at bicycle path crossings a cyclist may get a green phase more frequently. Such priority control is the consequence of a policy to assign the use of traffic space to certain preferred groups of road users because they play an important role in the local situation. In the case of priority for public transport, principal reasons might be that it minimizes the total waiting time of all road users and also improves the operating speed for busses and trams. Another reason would include reducing the operating costs of a transport mode heavily subsidized by public funding. The authorities may also have a certain expectation that preferential treatment of certain traffic classes—and in most cases one has public transport and pedestrians in mind—will reduce the growth of H. J. van Zuylen, Delft University of Technology, P.O. Box 5048, 2600 GA Delft, Netherlands. H. Taale, Transport Research Center AVV, Rijkswaterstaat, P.O. Box 1031, 3000 BA Rotterdam, Netherlands.
auto traffic and influence the modal choice in favor of collective public transport or walking. Because traffic control has an influence on travel behavior, a change in traffic control may result in changes in traffic volumes. If traffic control is modified such that congestion on a certain route disappears and delays at intersections decrease, traffic might be attracted from other areas where congestion still exists or that are part of a longer route. The queues that originally had disappeared, then, could return, and delays may come back to their original levels. If we assume that a modification of traffic control results in a change in travel behavior, then it is necessary to anticipate this change. For example, if we want to optimize delays, it should be for the traffic volumes that will be present after the introduction of the optimized traffic control and not for the traffic volumes that exist before the implementation. It is of course possible to follow an iterative approach, where after each shift in traffic volumes the control scheme is adjusted until equilibrium has been reached. Or one may use self-adjusting traffic control. It has been shown for certain examples, however, that the process of traffic control adjustment followed by a shift in traffic volumes does not necessarily lead to a system optimum. It is even possible that the system oscillates between two or more states. After the first papers by Allsop (1) and Charlesworth (2), several studies have reported on traffic control optimization taking into account driver reaction in terms of route choice. Issues concerning changes of departure time as a response to congestion and traffic management measures have also been studied [e.g., by Chang and Mahmassani (3)]. The problems associated with conducting bilevel optimization are rather complex, and in order to keep the computation feasible, simplifying assumptions have been made by most researchers (4) about travel behavior and the travel time function in controlled networks. Some authors (4, 5) have tried to use trafficresponsive traffic control, but they assume that traffic responsive signals are still fixed-time signals adapted to the prevailing traffic conditions. The objective of this study is to show how the characteristics of the traffic control (fixed-time or vehicle-actuated) and the ways in which control schemes are adapted to changing traffic flows may have a large influence on system equilibrium and its stability. This study begins with a discussion of the relation between traffic management and travel behavior. In a following section, a simple road network with traffic control on one of the links is studied, and the occurrence of stable, consistent equilibrium situations is investigated. A simulation study shows that one stable condition exists where flows are consistent with travel times and traffic control is optimized with respect to total delay. This stable situation depends on the kind of traffic control that is applied. A further analysis of a slightly simplified situation shows the possibility of multiple stable and metastable states in route choice for a simple road network. The
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structure of the solution space depends on details of the control scheme, such as minimum green time and lost time. In a later section the combination of route choice and traffic control optimization is studied in more detail. It is shown that in the two-level optimization, optimum solutions exist that can be interpreted as metastable saddlepoints. This means there is equilibrium between traffic control and route choice but that small disturbances in the traffic and traffic control conditions will cause the system to slide to another state.
CHOICE PROCESSES OF TRAVELERS Traffic control has an influence on travel times, and it is believed that travel time is the most important decision criterion for route choice. This section reviews literature that shows that although travel times may be important, in reality there are also several other issues that influence route choice. Therefore, the model used in the following sections, which is based on rational route choice determined by travel times, is only a rough approximation of reality and, for the moment, must be seen as a theoretical exercise. Traffic must be considered as the result of a process in which travelers, shippers, carriers, and managers of transport companies make many choices. It starts with the choice to move someone or something. Destination and time of departure or arrival are subsequent choices. The transport mode vehicle and route also have to be chosen, and during the transport process several other choices have to be made (e.g., lanes and speed). The usual assumption is that choices are made such that the resulting experienced benefits, expressed in some monetary unit, after subtraction of the generalized costs, are at a maximum. For the choice of transport mode, the benefit of a car and public transport does not differ much as far the trip is concerned. In that case the choice would have been made based on the costs. In reality, this assumption appears to be much too simple and unrealistic because modes are not chosen based on arguments that concern only the trip itself, but they are also based on arguments concerning other activities during the day before or after the trip, as well as from chaining of trips. For example, in a survey performed in Washington, D.C., Pendyala et al. (6) found that automobile drivers combine several activities within one trip and that they revise their choices less when circumstances change than other drivers do. When traffic conditions change, choices made in the past may also change. Jou and Mahmassani (7) found in Dallas and Austin that route switching and departure time depend on traffic conditions and on personal characteristics of the traveler (e.g., sex, age, and job conditions). Mokhtarian and Raney (8) show that travelers adapt their behavior by changing their existing choices with minimum costs and effort (e.g., by changing the moment of departure, choosing other routes, or by spending the time during the trip in a more comfortable way such as by the purchase of a better audio system or a cellular phone). If traffic conditions become worse, possible choices for different travel behavior include working more at home, working on fewer weekdays, or choosing a different mode of transport. Other, larger-impact changes of behavior in this circumstance include opting for a part-time job, moving to another residence, and taking a different full-time job. It also appears that the social and economic status of the traveler has an influence on willingness to change behavior. For instance, people with a relatively lower status are inclined to change their behavior sooner. The resulting conclusion is that congestion and traffic management are experienced in different ways by different groups of travelers.
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The usual approach, in which costs and benefits are assumed to determine the choices made by travelers, is too simplistic. A more realistic approach, such as the triad model proposed by Poiesz (9), takes other conditions into account. Poiesz assumes that behavior is determined by three factors: 1. Motivation (does the traveler want to use the alternative?); 2. Potential (is the traveler able to use the alternative?); and 3. Opportunity (is the alternative available?). Only if all conditions are satisfied will a new alternative be chosen. Costs and benefits determine the motivation, but if no suitable alternative mode of transport is available (opportunity), the motivation will not lead to different behavior. If a traveler has various kinds of activity during the trip, it is likely that one of these activities will render it impossible to use an alternative transport mode. Even if people are willing to use an alternative route, departure time, or transport mode, the second factor, potential, may prohibit them from making a different choice. Dynamic traffic management has a large influence on travel times and the distribution of travel times on different routes. The reaction of travelers toward such measures remains complex and rather uncertain. Even if possible to model the (aggregated) reaction, it would still be necessary to separate short-term and long-term impacts. The influence of individual factors is large, and realistic models have to take this into account. The analysis in subsequent sections is still based on rational travel behavior, where travelers choose the fastest routes. The purpose of this study (which, it must be emphasized, is of a purely academic character) is to investigate the system dynamics of the traffic control and route choice mechanisms. It looks into the question of whether such systems carry stable solutions. In the future, the research will be enhanced, and a more realistic model of travelers’ behavior will be included.
COMBINED TRAFFIC ASSIGNMENT AND TRAFFIC CONTROL OPTIMIZATION PROBLEM Already by the 1970s, Allsop (1) and Charlesworth (2) had shown the relevance of the interdependence of traffic control and route choice. The initial challenge was the search for a traffic control scheme that optimized total delay for traffic volumes consistent with the travel times influenced by the control scheme (i.e., a traffic condition where no traveler could improve his or her travel time by choosing another route). In this context, two autonomous actors try to achieve their own goals, each with their own objective function and domain of possible choices. One of the actors, the infrastructure manager, tries to optimize the road system to maximize utilization while minimizing total delays and stops. One of the tools available for this task is the setting of traffic signals, but there are also others. The second group of actors comprises the drivers, who choose their routes such that they minimize their travel time. The travel times are partly determined by the traffic signal settings, while the traffic signal settings are optimized for certain traffic flows that, in turn, are the consequence of the behavior of the drivers. Fisk (10) showed that this situation could be viewed as an example of a noncooperative game in which two players have their own objectives and strategies. The strategy is known and the choices are predictable, however, so that it is possible to choose an optimal strategy that takes into account the predictable reaction of the other party.
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Road users can choose their route under the assumption that traffic control will be optimized for the total delays. The infrastructure manager can optimize the traffic control knowing that the road users will shift their roads after the modification of the control scheme.
EXAMPLE A first attempt to analyze this problem was by simulation. The objective was to analyze the distribution of flows between Origin A and Destination B on two alternative routes. One route had a controlled intersection; the other route was a bypass (Figure 1). For this junction a two-lane bypass for the north-south movement was created. This bypass was situated on the west side of the junction and was 2.5 km longer than the route across the junction. The free speed for the bypass was 100 km/h; for the route with the junction it was 50 km/h. In this example, the morning peak (AM) flows have been used. A number of control strategies were simulated for this situation to determine how they would affect route choice. For the calculation of travel times, the microscopic simulation model FLEXSYT-IIwas used (11–13). The travel times had been calculated as the travel time at cruising speed plus the average delay at the intersection. Because FLEXSYT-II- has no assignment other than that specified by the user, route choice in this study was investigated by changing the splitting rate. First, it was assumed that only 1 percent of the traffic took the bypass (to measure travel time), then the splitting rate
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was increased to 10 percent, 20 percent, and so on up to 99 percent. In this way it was possible to find the equilibrium. This same method was performed for four control types. First, the existing fixed-time control plan was simulated for all splitting rates, then the optimized fixed-time control plan was simulated in like manner. (The optimized control plan was derived using Webster’s formula for cycle time and green times for the current flows.) Third, the existing vehicle actuated control plan was simulated, and, finally, an optimized vehicle-actuated control plan was simulated. (Optimized vehicleactuated control was accomplished using Webster’s formula and by incorporating the green times as maximum green times in the control plan.) In Figure 2, travel times on both routes (controlled and bypass) are shown for all control types. The travel time for the bypass is the same for all types. For normal fixed time, control equilibrium is never reached. For optimized fixed-time control, the equilibrium is around 50 percent. For both vehicle-actuated control types, equilibrium is reached around 60 percent. The total delay appears to be monotonically decreasing over increasing flow on the bypass. The system optimum is realized for all control strategies at the extreme assignment of all traffic to the bypass. So, clearly, the user optimum is not the same as system optimum. So far, the green times have been optimized only for the situation in which only 1 percent of the traffic uses the bypass. When the optimization is performed for both equilibrium situations, the result is as indicated in Table 1.
FIGURE 1 Simple road network and layout of central intersection, with morning-peak flows to illustrate coupling between route choice and traffic control.
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FIGURE 2
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Travel times for different distributions of flow on the controlled route and bypass.
Table 1 shows that for fixed-time control, the travel time for the two separate routes does not change and the total delay decreases 18 percent. For vehicle-actuated control, the situation does not change at all: the travel times on the two routes and the total delay stay the same.
MULTIPLE SOLUTIONS The simulation study showed the existence of one single solution for the combined assignment and traffic control problem. To investigate the possibilities for equilibrium solutions and to look for multiple solutions, the network had been changed slightly without changing the degrees of freedom. In Figure 3, the second network layout is given.
TABLE 1
The Figure 3 example is symmetric. Both routes between A and B cross the route between C and D at a controlled intersection. The traffic control at this type of intersection is fixed time optimized with Webster’s method. Depending on the magnitude of the flow and the internal lost time, it appears that at least two different solutions exist: 1. A symmetrical solution with a 50/50 percent distribution between Routes 1 and 2; and 2. An asymmetrical solution where more drivers choose one route over the other. Of course, for every asymmetrical solution a mirror solution exists: if a stable solution is obtained with x percent on Route 1 and 100 – x percent on Route 2, another solution is the distribution 100 – x percent on Route 1 and x percent on Route 2.
Results Before and After Optimization at Equilibrium
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phenomenon of the multiple asymmetric solution has not yet been found for vehicle-actuated controllers. Apparently the system of route choice and traffic control is, under certain circumstances, critically dependent on system parameters. In the following section this behavior is investigated in more detail for the original network indicated in Figure 1. FURTHER ANALYSIS FIGURE 3 Example of a network with symmetric choice possibilities.
The assignment problem in the case of deterministic route choice based on individual shortest routes can be formulated mathematically as min ( V ) Z (Vi )
In Figure 4 the difference in travel time between Routes 1 and 2 is given for different percentage values of traffic using Route 1. The travel time difference is the same as the difference in delay, which was calculated using the two-term delay formula derived by Webster (14). The stable situations occur if the difference in travel time is zero. For high volumes, Figure 4 gives a symmetrical situation where the equilibrium and optimum is obtained for 50 percent distribution. The high-volume situation was calculated for 1,000 vehicles/h flow from A to B, 500 vehicles/h flow from C to D, saturation flows of 1,800 vehicles/h and internal lost times of 9 s, with a minimum green time of 6 s. If the flow from A to B is changed to 600 vehicles/h, the picture changes dramatically (Figure 4 low-volume curve): two asymmetric stable equilibrium states exist (20 and 80 percent) in addition to the symmetrical (50/50 percent) distribution equilibrium state. If the system is in the 50/50 percent state and small changes occur in this distribution, the change is enhanced by the subsequent adaptation of the traffic control scheme—the control scheme for the route with the largest flow gives the shorter average delays. The total travel time in both cases (for the high and low volumes) is minimum for the 50/50 percent distribution. Changing the parameters of the control scheme (lost time or minimum times) or the flow or saturation flows significantly changes the appearance of the time-difference curves so that a small change of the parameters can result in the equilibrium states moving over large distances and the asymmetric solution disappearing suddenly. The
(1)
where Vi + is the volume on Link i. The function Z is defined as Vi
Z = Σ i ∫ Ti ( z, C, tg )dz
(2)
where Ti is the travel time on Link i including delays for Volume z, Cycle Time C, and Green Time tg. The minimization of delays on controlled intersections can be represented by the following formal expression: min {t } Σ i Di (Vi , t j )
(3)
where tj = time parameters of the traffic control, Di = delay for Link i, and Vi = volumes to be calculated from the solution of Equation 1. To gain better insight into the characteristics of the problem, traffic control is reduced to a single dimension. The combined assignment and optimization problem can be visualized in a two-dimensional space, making further analysis easier. Beginning with the assumption that the cycle time remains fixed, the only parameter left is the green split. The Delay d for a single controlled flow is given by (14) −1 d ( V, C, tg ) ≈ 0.9[ 1 2 (C − tg ) (1 − V s )−1 C + 2
FIGURE 4 Travel time difference between Routes 1 and 2, giving single symmetric equilibrium for high volumes and three equilibrium states for lower volumes.
1
x V (1 − x )] 2
2
( 4)
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where V = volume, x = (V/s) (C/tg), s = saturation flow, C = cycle time, and tg = green time. For both approaches of the intersection the delay is given by
[
D ≈ 0.9 V1 {1 2 (C − tg1 ) (1 − V1 s1 )−1 C −1 +
1
2
2
(V1 C tg1s1 )2
V1 (1 − V1 C tg1s1 )}
+ V0 {1 2 (C − tg 0 ) (1 − V0 s0 )−1 C −1 2
+
1
2
(V0 C
]
tg 0 s0 ) V0 (1 − V0 C tg 0 s0 )} 2
(5)
where 0 ≤ V1 ≤ V, tg1 + tg0 = C − tl, and tl = internal lost time of the control scheme. In Figure 5 the function D is represented as two-dimensional isocurves. The states with optimum green time for a fixed volume V1 are connected in Figure 5 by a thick line (i.e., the thick line gives the green time that minimizes the total delay for a given V1). The route choice problem can be formulated in the following way: min {V } Z {V1 , V2 } with V1 + V2 = V
(6)
where Z can be elaborated as
[
Z = V1 L1 v + V2 L2 v + 0.45 (C − tg1 ) C ∫ (1 − z s) dz + ( C s t g1 )
2
2
∫ z(1 − z C s t ) dz] g1
[
= V2 L2 v + V1 L1 v + 0.45 s(C − tg1 ) C ln(1 − V1 s)−1 2
]
− V1C s tg1 − ln(1 − V1C s tg1 )
FIGURE 5 plane.
( 7)
where Li is the length of Route i and v is the speed. With the boundary condition V1 + V2 = V, the function Z also becomes a function of two variables (V1 and tg). Figure 6 gives a graphical representation of Z(V1, tg). The equilibrium solutions are on the line where δZ/δV1 = 0. The unfeasible domain contains the states where one of the approaches of the intersection is oversaturated. In fact Webster’s formula is not applicable for that domain nor for the states close to that domain; Webster’s formula is not realistic and has to be replaced [ e.g., by Akcelic’s formula (15)]. If the lines that give the optimum green split (Figure 5) and the equilibrium assignment (Figure 6) are combined, the result is as given in Figure 7. In this example, three situations exist in which traffic control is optimized with respect to traffic volumes, and traffic volumes are consistent with travel times. If it is assumed that the process of adjustment of traffic control and route choice is iterative, the patterns given by the arrows in Figure 7 are found. That is, an adjustment in traffic control will give a change in travel time, resulting in some drivers choosing another route. The changed traffic volumes, among other things, make it necessary to adjust the control scheme. The process stops if a situation has been reached where the unbroken and dotted curves intersect (i.e., Points 1, 2, and 3). If at the equilibrium situation of Point 2, the route choice would change slightly and the traffic control were adapted to the changed flows, a positive feedback mechanism would exist. That is, a small variation in route choice would be reinforced by the mechanism in which more traffic leads to more green time, which, among other things, reduces delay and attracts more traffic. Only at the extremes, where all traffic chooses the same routes or where congestion prevents further growth, will the positive feedback disappear. Thus, in the example in Figure 7 two stable equilibrium situations exist: at Points 1 and 3. The total travel time is minimum (for this calculation) for the situation at Point 3. Also, in this case the form of the two curves in Figure 7 is critically dependent on control parameters and (saturation) flows. The optimal green split depends on minimum green time and the internal lost time. This means that the shape of the curves in Figure 7 is determined a great deal by the boundaries of the space of feasible solutions. Changes in the boundaries of the feasible domain will change the shape of the curves, which can result in the curves intersecting at one or more points and the intersection point moving irregularly after small changes in the system parameters or boundary conditions.
Iso-curves with equal total delay in the t g − V 1 FIGURE 6
Iso-lines with equal values of objective function Z.
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travel, and choice of destination has been quantified by a several researchers (3, 6–8, 17) but more needs to be done to apply these results to a method which optimizes traffic control taking into account the expected behavioral response. Apart from the lack of practical tools that would make it possible to optimize traffic control, predict the impact on travel behavior, and anticipate future changes in behavior, there is also the need for an analytical framework to study the existence of equilibrium conditions in a system of traffic control and individual travelers.
REFERENCES
FIGURE 7 Optimum green time (dotted line) and equilibrium assignment (unbroken line), with three equilibrium situations.
CONCLUSIONS With simulation studies and a further mathematical and graphical analysis of the problem, it was shown that in rather simple traffic situations, very complex processes can arise if the system is allowed to move to equilibrium. The equilibrium situation is not always uniquely determined, and it is even possible that small variations in traffic conditions can lead to a large shift of the equilibrium. The equilibrium situation achieved after an iterative adjustment of traffic control to changing route choice is not always a system optimum. This leads to the conclusion that the traffic-dependent optimization of traffic control may result in a suboptimal situation and that it might therefore be more effective to use traffic control as a management tool for steering traffic flows rather than as a means to accommodate traffic volumes. The necessary analytical tools for such a strategic approach are still limited: the combination of traffic control (especially vehicleactuated traffic control) and traffic assignment that support the search for a system optimum is still to be developed. Furthermore, knowledge of the occurrence of instabilities is still very limited. However, empirical data on this subject exist; there are even realtime systems that estimate travel behavior from real-time traffic data (16). There are also several studies that address the problem by simultaneously optimizing delays and route choice (4). However, as far as is known at present, there is no analysis of the existing traffic data that looks for the existence of multiple stable equilibria and their dependence on traffic control characteristics. The problem becomes even more complex when realizing that in the real world the degrees of freedom for travelers are much larger than just route choice. The influence of traffic control on such elements as time of departure, modal choice, frequency of
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