Traffic assignment and signal control in saturated road networks

Transpn. Res.-A. Vol. 29A, No. 2, pp. 125-139, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

Pergnmoo

0%5-8561195$9.50 + .oo

TRAFFIC ASSIGNMENT AND SIGNAL CONTROL IN SATURATED ROAD NETWORKS HAI YANG Department of Civil and Structural Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong and SAM YAGAR Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada, N2L 3Gl (Received 7 July 1993; in revisedform 25 January 1994) Abstract-This article presents a model and a procedure for determining traffic assignment and optimizing signal timings in saturated road networks. Both queuing and congestion are explicitly taken into account in predicting equilibrium flows and setting signal split parameters for a fixed pattern of origin-to-destination trip demand. The model is formulated as a bilevel programming problem. The lower-level problem represents a network equilibrium model involving queuing explicitly on saturated links, which predicts how drivers will react to any given signal control pattern. The upper-level problem is to determine signal splits to optimize a system objective function, taking account of drivers’ route choice behavior in response to signal split changes. Sensitivity analysis is implemented for the queuing network equilibrium problem to obtain the derivatives of equilibrium link flows and equilibrium queuing delays with respect to signal splits. The derivative information is then used to develop a gradient descent algorithm to solve the proposed bilevel traffic signal control problem. A numerical example is included to demonstrate the potential application of the assignment model and signal optimization procedure. 1. INTRODUCTION

Conventional methods for setting traffic signals assume given flow patterns, whereas demands are assigned to networks assuming fixed signal settings. This method is not fully satisfactory in the normal case in which traffic flow and signal settings are mutually interdependent. In an attempt to eliminate the inconsistency between signal control and traffic assignment, Allsop (1974) suggested that the effects of signal settings on the traffic flow pattern should be taken into account explicitly by combining traffic control and route choice. This problem has been known as equilibrium traffic signal settings and has been examined extensively in two different ways: the global optimization models and the iterative optimization and assignment procedure (Cantarella et al., 1991). The global optimization model seeks signal control patterns in which a system performance such as the total travel time is minimized, whereas the drivers’ route choice behavior is described with an equilibrium model. Gartner et al. (1980) and Fisk (1984) described the global optimal signal setting problem as a Stackelberg or leader-follower game between network users and a traffic agency. Marcotte (1983), Sheffi and Powell (1983), Heydecker and Khoo (1990) and others proposed heuristic algorithrhs to solve a small network problem. The main difficulty with the global optimization models is that as yet there are no efficient solution algorithms for calculating optimal settings in general road networks while anticipating driver responses in terms of route choice (Smith & Van Vuren, 1993; Tan et al., 1979). The iterative optimization assignment procedure is to update alternatively the signal setting for fixed flows and solve the traffic equilibrium problem for fixed signal settings until the solutions of the two problems are considered to be mutually consistent (Cantarella et al., 1991; Gartner et al., 1980; Smith & Van Vuren, 1993; Smith et al., 1987; Van Vuren & Van Vliet, 1992). This approach has the advantages that the traditional traffic assignment and signal setting techniques can be employed to solve the problem and can 125

126

H. YANG and S. YA~AR

be applied to large networks. However, it is demonstrated theoretically and empirically that the approach does not necessarily minimize total travel time and might lead to a decline in network performance rather than an improvement (Dickson, 1981). When we seek to combine signal settings and traffic assignment using either of the aforementioned approaches, the cost or delay formula is very important. Smith and Van Vuren (1993) showed that the same control policy considered under different cost assumptions may possess completely different theoretical properties and thus may be expected to give rise to completely different practical performance results. Delay functions commonly used in signal settings have been chosen either according to their ability to predict queue effects or because of their convenient functional form for use in traffic assignment. However, their behavior in signalized networks will be completely different, especially in congested situations. For example, Webster’s delay formula, which was originally derived through theoretical queuing analysis for isolated intersections, predicts infinite values of delay when flows approach capacity, whereas in realistic situations, queues will not grow infinitely even when we consider a steady-state saturated network. Drivers will change to alternative routes to avoid large queues. The resultant queued steady state lasts for less than one peak period and not an infinite length of time, as assumed in Webster’s formula. On the other hand, when urban traffic operations are near or above saturation, a control scheme is needed for queue management. It should at least avoid queues growing to block upstream intersections. This may require a traffic model for predicting queues in saturated conditions and hence allowing explicit constraints for queue lengths when optimizing signal settings. Recently, the authors have developed a bilevel programming model and an efficient algorithm for combined traffic assignment and traffic control in general freeway/arterial corridor systems (Yang & Yagar, 1993). This article describes an application of this approach to the global optimal signal setting problem, particularly under saturated conditions in which queue management is critical. In the next section, we give a simple example to illustrate the traffic assignment and signal control problem. In Section 3, we present a bilevel programming formulation of the problem. The intersection between drivers’ route choices and signal controls under conditions of congestion and queuing is explicitly considered. Section 4 describes application of the sensitivity analysis procedure developed for the bilevel traffic control problem. Section 5 contains a numerical example to demonstrate the bilevel control model and the solution algorithm. Conclusions are summarized in Section 6. 2. BASICCONSIDERATIONS A static queued system such as one described herein can develop due to either stochastic variation or a temporary oversaturation period preceding our queued static equilibrium. In the latter, one can conceive of a situation in which demands in excess of capacity meter themselves at acceptable levels of queuing. The objective of this static model is to identify that equilibrium state rather than to describe how queues would develop, which would involve dynamics. Throughout this study, we assume that the origin-destination demands are given and fixed, and drivers have sufficient and perfect knowledge of queuing delays and travel times via all routes and make routing decisions in a user-optimal manner for any given signal control pattern. Furthermore, we assume that the cycle time and link offsets are known, and only red-green splits are variables in the signal settings. This is consistent with congested conditions, in which a minimum cycle length is used and it is not possible to obtain signal progression. First we present a simple example to illustrate our basic ideas for the equilibrium signal settings with queuing. 2.1. A simple example Consider the simple network shown in Fig. 1, with a traffic signal at the intersection of links 1 and 3. We assume that link travel cost is a sum of flow-dependent running time and signal delay. Signal delay is assumed to be a function of flow, v, , and the fraction of

Traffic assignment and signal control

127

@ I 3’

Fig. 1. A simple example.

the cycle that is green time for that link. Let XI be the proportion of green time given to link 1. For simplicity, we assume no lost time, and hence (1 - X,) is the proportion of green time given to link 3. The cost functions used here are t, (v,X) = 2 + VI + 2 1

t2 = 4 + 2v,

t3 (v,h) = 4 + vj +

v3 5(1 - A,)

Let & = 10 and Q., = 15 be the demands from 1 to 2 and 3 to 4, respectively. In this simple example, only the drivers from 1 to 2 have a route choice, and we have one independent flow parameter v, and one signal parameter X, . Assuming that drivers choose their minimum cost route, it is straightforward to obtain 2 + v, + 2

= 4 + 2v, I

Vl

+

10,

v2 =

v3

=

15

giving rise to v, =

22

,vg = 15

3+& 1

This result is true when no queue is formed. Namely, it holds when demand is small and a well-balanced proportion of green times are assigned between links 1 and 3 so that flow on each link is below its exit capacity. Assuming that the saturation flows of links 1 and 3 are S, = 10 and S, = 50, respectively, the undersaturated conditions can be expressed as v1 < s,X, or

22

< 10x,

3+& 1

and v3 < s,(l That is,

- X,) or 15 < 50(1 - X,)

H.

128

and S. YAGAR

YANG

Here we assume that the capacity of link 2 is large enough to accommodate rerouting of traffic. Note that v3 < s,(l - A,) must be satisfied in the equilibrium signal setting of the current network structure to have a feasible solution. The condition, v, < s, A,, is not necessary and may not hold when demand D12 is large and/or A, is small. For given demand Di2 = 10, demand on link 1 will become greater than its exit capacity at the intersection when 0 I A, I 2/3, and hence a queue behind the intersection forms. As the queue grows, drivers will shift from route 1 to alternative route 2 until a temporary steady-state queue is reached. In this state, v, equals sI A,, and the physical length of queue is determined so that travel times on routes 1 and 2 are equal. The queuing network equilibrium conditions can be written as 2 + v, + c

+ d, = 4 + 2vz I

v,

+

v2

=

10, v3

=

15

VI = s*A,( = 10X1) where d, is the queuing delay on link 1. The last equation implies that queuing occurs only on saturated links. Solving the equations leads to VI = 10x, v2

=

lO(1 - A,)

v3

=

15

d, = 20 - 30, In a saturated road network, one of the major objectives in signal control is to restrict the queues developing at the approaches of the intersection so as not to block upstream intersections. Because delay equals queue size divided by service rate, the number Q, of queued vehicles on link 1 equals the product of queuing delay, d, , and its exit flow, v, . The maximum storage capacity Qy” (assumed here to be 25 vehicles) on link 1 can thus be enforced by a queue length constraint d, v1 I Qy” or (20 - 30hl) * lOAl I 25 It is solved by

The total travel time spent on the network is the sum of flow-dependent running time, delay due to the signal and queuing delay due to limited exit capacity. In the present example, it can be obtained easily as

F(v(A),A)

=

The central problem considered here becomes to seek a feasible A which minimizes F(v(A),A) subject to the queue length constraint and given lower and upper bounds. In the preceding example, a global optimal solution can be obtained analytically as

129

Traffic assignment and signal control A: =

0.5256, v: = 5.5256 and F(v(X*),h*) = 514.74

We note that this optimum occurs at a network equilibrium state with queuing. The queue is formed at link 1 with a queuing delay, d: = 4.232, and queue size, Q, = v: x d: = 23.384 < QI”“, which is within the allowable storage limit. 2.2. Comparison with other equilibrium signal setting methods From the preceding simple example, it would seem sufficient to observe some characteristics of our approach which can be compared and contrasted with the previous equilibrium signal setting methods. Assumption on travel delay. When we seek to develop a combined traffic signal control and traffic assignment model, the cost or delay function plays an important role. For example, in TRANSYT after each iteration in the green time optimization, the resulting delays must be evaluated in the traffic model (Robertson, 1969). The delay function employed has evidently great effects not only on resulting green times and network performance results, but also on the theoretical properties of the control policy adopted (Smith & Van Vuren, 1993). Traditionally, delay to traffic is estimated as the sum of uniform and random delay and considered for isolated intersections. For example, Webster’s two-term delay formula for signalized intersections (Webster, 1958) is often used in determination of signal splits (Smith & Van Vuren, 1993; Van Vuren & Van Vliet, 1992). That is, d=

C(l

-Al2 +

2( 1 - v/s)

x2 2v(l

- x)

where x = v/c is the degree of saturation and C the cycle time. The first term is the uniform delay for a traffic stream with a uniform distribution of arrivals. The second is the random delay due to fluctuations in arrivals and can be predicted from theoretical queuing analysis. The random delay component is small compared with the uniform delay when the degree of saturation is small, but it increases rapidly at the higher v/c ratios and goes asymptotically to infinity as demands approach capacity (Fig. 2). This property is obviously unrealistic (Hurdle, 1985; Yagar, 1977). As shown by Van As (1991), the estimation of delay by Webster’s isolated intersection formula is not appropriate for a network operating under heavy traffic conditions. Because of the unrealistic property of Webster’s delay formula, a delay formula with a polynomial form (see Fig. 2, the transformed curve) has been developed and used in traffic assignment. The polynomial cost function can also not be ensured to represent

A

0.0

1.0

v/c

Fig. 2. Steady-state delay, deterministic queuing delay, and transformed delay curves.

130

H. YANG and% YAGAR

delays properly at signalized intersections and is thus inappropriate for use in signal optimization (Van Vuren & Van Vliet, 1992, Chapter 5). In addition, when a polynomial cost function is used in iterative traffic assignment and signal control, the signalcontrolled intersection may become oversaturated, especially under heavy traffic demands. This would be incompatible with the steady-state assumption used in the static network signal settings, in which link flow should be less than or equal to exit capacity. When we consider a network-level signal control for a temporary steady state, drivers will switch to alternative routes when queuing delay through a link reaches a certain value, and hence queue length will not grow indefinitely (we assume only flows on a subset of links reach their exit capacities, but the network as a whole has sufficient capacities to carry out rerouting of traffic). The formula developed for isolated intersections thus becomes inapplicable to signalized networks, especially under saturated traffic conditions. As shown in the preceding example, for realistic cost calculation, travel delay in signalized networks should be divided explicitly into two kinds, signal delay and queuing delay, and treated separately (see Fig. 3). The former is due to interruption of traffic by the traffic signal and could be determined by a formula developed from local traffic conditions, whereas the latter is due to limited capacity and should be determined from network equilibrium conditions. Although in reality it cannot be expected that a permanent steady state will be reached, one can imagine that a temporary saturated steady state (Fig. 4) exists for a short duration of a peak period, where flow equals exit capacity and temporary steady-state queuing holds. Our point of view, as already expressed in the example, is that for a more realistic traffic model under saturated conditions, explicit capacity constraints should be imposed so that the resulting link flow for any given signal settings will never exceed capacity. Furthermore, a steady-state queue is formed only on a saturated approach, and the queue lengths will never approach infinity. The queuing delay should be determined endogenously rather than predicted approximately by an analytical delay formula. This would be a close representation of the drivers’ rerouting behavior under steady-state congested conditions. Queue length constraints. When urban traffic operations are near saturation, a control scheme is needed for the management of the inevitable queues. A signal-control approach should be oriented toward controlling queue lengths and not blocking upstream intersections. Previous analytical equilibrium network signal setting methods are difficult to use for queue management because they use an approximate delay formula which cannot predict queue length properly. In contrast, in our example, queue length is treated as a variable and determined from queuing network equilibrium conditions. This would allow explicit constraints on queue lengths in setting traffic signals.

Delay Determined endogenously from queuing network equilibrium conditions Predicted by a formula from local traffic ,,

0.0

Queuing delay due to limited exit capacity

/

1.0 Fig. 3. Delay curve in queuing network equilibrium assignment.

v/c

131

Traffic assignment and signal control

A Vehicles

Arrivals

Oversaturated

I Saturated

1 Undersaturated

Time

Fig. 4. Arrival and departure curves for a saturated link.

3. MODEL FORMULATION

As described in Gartner et al. (1980) and Fisk (1984), the global signal setting problem can be viewed as a leader-follower or a Stackelberg game in which the traffic agency is the leader and the network users are the followers. It is assumed that the traffic agency can influence, but cannot control, the users’ route choice behavior by changing signal settings. In light of this control decision, the network users make their route choice decisions by choosing individual minimum cost routes. This interaction game between the traffic agency and the network users can be formulated as a bilevel programming problem. 3.1. The queuing network equilibrium model In the previous section, we have shown that an efficient signal control pattern should consider finite capacities for link queues, especially in saturated road networks. This requires the development of a model for predicting drivers’ route choices under conditions of both queuing and congestion. The previous standard network equilibrium models (Sheffi, 1985) are inapplicable in this case. Thompson and Payne (1975) dealt with traffic assignment in transportation networks with capacity constraints and queuing. Under the simplifying assumption that the running times on links are constant, they showed that at network equilibrium, queuing delay at the exit of a link corresponds to the Lagrange multiplier associated with the link capacity constraint in simple linear programming. Smith (1987) extended the result of Thompson and Payne to a signal-controlled network and gave the equilibrium conditions of the interactions between flows, queues and green times when a control policy developed by himself is used. Inouye (1986b) proposed a traffic equilibrium model involving both queuing and congestion. By assuming that the travel time on a link is equal to the sum of a flow-dependent running time and queuing delay at the link exit, both link flows and equilibrium queuing delays can be obtained from a convex programming problem. This convex programming problem is the standard network equilibrium problem with link exit capacity constraints included explicitly. The Lagrange multipliers associated with the capacity constraints imply the queuing delays. We shall now give a simple description of the queuing network equilibrium model of Inouye (1986b) which can be adopted to describe drivers’ route choice under traffic signal control in saturated networks. Consistent with the assumption that each driver traveling from an origin to a destination will have perfect knowledge of the travel costs and queues via all routes and will

H. YANGand S. YAGAR

132

choose the route in a user-optimized manner, the following equilibrium relationships are satisfied for every origin-destination (O-D) pair w E W and every path r E R,:

where A W RV

is the set of links in the network is the set of origin-destination pairs is the set of routes between origin-destination pair w E W is the queuing delay at link a E A $ is the flow on route r E R, to;v.,ho) is the travel time on link a E A described as a function of link flow v, and green split A, 6”lV is 1 if route r between O-D pair w uses link a, and 0 otherwise is the minimal travel time between O-D pair w E W. PM’ Note that t,(v,,A,) should be considered as a sum of flow-dependent running time, tl(v,), and signal delay, ti(v,, X,). We do not specify their detailed forms, but each should be regarded as a continuous increasing function of flow v,. On the other hand, queues may form only on saturated links: d, = 0 if v, < h,s, a ul do 1 0 if v, = h,s, 1 where S, is the saturation flow of link a E A. For a given vector of signal splits, A, the steady-state traffic assignment in a saturated road network with queuing involves determination of the vector of link flows, v, and a set of queues, d, satisfying the flow conservative conditions and the aforementioned queuing equilibrium conditions. This problem is equivalent to the following nonlinear mathematical optimization program (Inouye, 1986b; Yang & Yagar, 1993): minimize c ”

oul

1:

t,, (x,X,)dx

(la)

subject to

c c f*;‘Sl’i= v,,acA

v, I

X,s,,aEA

f^: 1 O,reR,,we

(lb)

(14 W

(le)

where D, is the demand for O-D pair w E W, R is the set of routes in the network, eqns lb and lc are flow conservation constraints and eqn le is a nonnegativity constraint. This problem is distinguished from the standard network equilibrium problem by explicit inclusion of exit capacity constraint (eqn Id). The queue delay can be shown to correspond exactly to the Lagrange multiplier associated with the capacity constraint. Note that because the constraint set is convex and the objective function is strictly convex with respect to link flow variables, the equilibrium link flow pattern will be unique.

Traffic assignment and signal control

133

3.2. The bilevel traffic control model Given the drivers’ route choice behavior described by the aforementioned model, we shall now formulate the model to determine optimal signal splits such that a particular system performance criterion or objective function is optimized for a given origindestination demand. A meaningful objective is to minimize the total network travel time expressed as the sum of running time, signal delay and queuing delay spent in the network by all vehicles over a given time period:

Furthermore, an acceptable control should consider queue length constraints for saturated links. This means that the queues developing on the approaches to an intersection should be restricted so that upstream intersections are not blocked. The number of queuing vehicles is equal to the product of queuing delay, d,,, and exit flow, v,, a E A. The maximum storage capacity Qf” (vehicles) on link a E A can thus be enforced by the constraint

dclv, I Q:“,aEA Let Z be the set of signalized intersections in the network. The proportions of green times for links approaching a given signalized intersection i E Z should satisfy some linear constraints, the detailed form of which depends on the specific phase structure. One of the possible constraint relations may be written as

c

Aa = 1.0, ieZ fJ.54; where Ai denotes the set of approaching links for intersection i E I. Here we have assumed that lost times are zero for simplicity. Finally, the signal split parameter should be fitted into given lower and upper bounds:

are lower and upper bounds of proportion

where ,ri” and h:”

of green time for link

aEA. In summary, the global optimal signal setting problem under queuing network equilibrium conditions can be formulated as the following bilevel programming problems: minimize F(X,v(X),d(X))

= c

x

v, *

(t,(v,,h,,)

+ d,)

(W

ClEA

subject to

d,(X) * v,(X) I Qf”, acA

(W

c A, = l.O,iEZ

(2c)

(IEA,

h rax,aeA

(W

where v(X) and d(h) are obtained by solving minimize 2

V

l;” t,(x,X,)dx

(W

subject to c weW

cf: rsR,

S,W,= v,,aEA

(W

134

H.

YANG

and S. YAGAR

f; = D,,wE

c

W

=Rw

v, I

f,”

1

W)

X,s,,aEA

O,~ER,,WE

(33)

W

(29

4. SENSITIVITY ANALYSIS ALGORITHM

Recently, the authors have developed an efficient heuristic algorithm for solving a bilevel optimal ramp metering problem in general freeway/arterial corridor systems based on sensitivity analysis results for the queuing network equilibrium problem. This algorithm can be easily adapted to solve the current bilevel optimization problem (eqns 2a-2i) with only slight modifications (the two problems are essentially very similar to each other in model structure). For a detailed description of the development of the sensitivity analysis algorithm, the reader may refer to Yang and Yagar (1993). Here we present some discussions about theoretical difficulties in solving the bilevel network optimization problem, some interesting past developments and their major characteristics. 4.1. Previous algorithmic developments The global optimal signal setting problem (eqns 2a-2i) may be regarded as one kind of network design problem (NDP), which has long been considered an important yet complicated and challenging problem in transportation research. This NDP is to determine optimum network improvements in which some measure of total network cost is minimized while taking into account drivers’ route choice in response to these improvements. The main difficulty that arises in solving the signal setting problem as an NDP lies in the fact that the equilibrium flow, v(X), and queuing delay, d(X), do not vary linearly with the control parameter, X, and their functional forms are implicitly defined by the lower-level equilibrium problem. Exact evaluations of the objective function, determination of its descent direction and treatment of the constraints in the upper-level problem are thus difficult. Furthermore, because the implicit reaction functions, v(X) and d(X), are nonlinear, the problem is a nonconvex problem. Nonconvexity portends existence of local minima, and hence a global optimum is difficult to find even with the most computationally efficient procedures (Friesz et al., 1990; Yang et al., 1993). A number of researchers have proposed various approaches to find solutions to the equilibrium network design problem (Cantarella et al., 1991; Fisk, 1984; Marcotte, 1983; Marcotte & Marquis, 1992; Tan et al., 1979 are but a few). Van Vuren and Van Vliet (1992) presented a good overview and concluded that as yet no efficient solution algorithms to the NDP exist. The existing methods suffer from various shortcomings in terms of either computational complications or simplifying assumptions. Here we should specifically mention two methods by Sheffi and Powell (1983) and Heydecker and Khoo (1990) for solving the signal setting problem with which the sensitivity analysis heuristic algorithm may be contrasted. Sheffi and Powell (1983) proposed a feasible descent procedure. To determine a descent direction, a partial derivative of flow for every link with respect to all green split parameters in the network must be determined. This is carried out numerically but would require in each iteration a number of equilibrium assignments equal to the overall number of split parameters in the network. On the other hand, in Heydecker and Khoo (1990), the nonlinear, implicit reaction function, v(h), is approximated by linear relationships, which have to be fitted numerically to a number of userequilibrium flow patterns. 4.2. Major characteristics of the sensitivity analysis algorithm In contrast to the aforementioned methods, the algorithm developed by the authors formulates local linear approximations of both the objective function and constraints in the upper-level problem using the derivatives of the reaction functions with respect to

Traffic assignment and signal control

135

split parameters. The derivative information is obtained by implementing sensitivity analysis for a given solution of the queuing network equilibrium problem. The sensitivity analysis algorithm is summarized as follows: Step 0. Determine a set of initial signal splits X(O).Set n = 0. Step 1. Solve the queuing network equilibrium assignment problem (eqn la-le) for given Xcn)using the inner penalty function method, and hence get (v(“), d’“‘). Step 2. Calculate the derivatives, &/ax and ad/& using the sensitivity analysis method. Step 3. Formulate local linear approximations of the upper-level objective function and constraints using the derivative information and solve the resulting linear programming problem to obtain an auxiliary solution y. Step 4. Compute Xcn+‘) = X(“) + (~(“)(y - X’“‘). Step 5. If a convergence criterion is satisfied, then stop. Otherwise let n: = n + 1 and go to Step 1. At Step 1, the lower-level queuing network equilibrium traffic assignment specifies the routes and distribution of demand among these routes for each origin-destination pair, and the queues at saturated links for the given signal control pattern. This traffic assignment is distinguished from conventional equilibrium assignment algorithms such as the Frank-Wolfe method (LeBlanc et al., 1975) by the explicit capacity constraints. The algorithm adopted by the authors is an inner penalty function method which was originally developed by Inouye (1986a) and used later by Yang and Yagar (1993). After proper transformation of the augmented objective function, the method gives rise to a similar subproblem structure to the Frank-Wolfe convex combination algorithm and hence can be implemented easily even for large networks. In the final output, the algorithm generates information necessary for implementing sensitivity analysis, including a complete set of link flow pattern, a subset of saturated links and their corresponding queuing delay and a subset of minimum time routes used by the drivers between each O-D pair. At Step 2, sensitivity analysis is implemented to compute the derivatives of equilibrium link flows and queuing delays with respect to signal splits for a given solution of the queuing network equilibrium problem. These derivatives can be used to predict changes in the link flow pattern and queuing lengths and hence to evaluate the total travel cost through the network in response to any small changes in signal splits. The sensitivity analysis method employed by the authors is essentially an extension of the results of Tobin and Friesz (1988) to the network equilibrium problem with capacity constraints. At Step 3, the derivative information obtained from sensitivity analysis is used to formulate a local linear approximation for both the objective function and constraints in the upper-level problem. This idea has been applied successfully in finding optimal ramp metering rates for urban freeway networks and freeway-arterial corridors with useroptimal flows (Yang & Yagar, 1993; Yang et al., 1993). This approximation will result in a linear programming problem, to which the solution indicates the farthest point in the descent direction of the upper-level objective function. At Step 4, the step size, CX”‘),is either determined by any standard one-dimensional search method or determined a priori and decreases monotonically with the number of iterations. If a one-dimensional search is adopted, it will require repeated evaluation of the upper-level objection function along the descent direction. Each of these evaluations involves an equilibrium assignment to determine link flow and queuing delay. Thus the computational requirements of the one-dimensional search may be prohibitively expensive. On the other hand, if a predetermined step length sequence is employed, it cannot always be guaranteed that the objective will be improved at each iteration, and little can be said theoretically about the convergence of the algorithm (Friesz et al., 1990). 5. A NUMERICAL EXAMPLE

In this section, a numerical example is presented to illustrate the application of the bilevel programming model and the sensitivity analysis algorithm for setting traffic signals in saturated networks. The example network, shown in Fig. 5, includes one signalized

H. YANGand S. YAGAR

136

Fig. 5. Network for numerical example.

intersection. The link cost function (quadratic form), free-flow travel time and saturation flows are shown in Table 1. It is assumed that there are two origin-destination pairs from node 1 to node 5 and 2 to 4, with fixed demands (veh/min); D15 = 70, & = 80, respectively. The lower and upper bounds for signal splits of links 1 and 2 are 0.05 I X, I 0.95,O.OO I AZ I 0.95, and the maximum storage capacities are Qy = Qy = 100 (veh). Note that each of the two O-D pairs has one or more alternative routes available that do not pass the signal-controlled intersection. We first examine the sensitivity analysis results. Table 2 presents the derivatives of link flows and queuing delays with respect to signal splits at X, = hz = 0.5. These results represent the variations of link flows and the variations of queuing delays on saturated links when the signal splits at each link in the network are changed by one unit. The derivative information has many important implications in the design and operation of signalized networks. The importance lies in that it not only serves for the implementation of the algorithm to find the optimal solution but also indicates the signal splits to which the queuing equilibrium flow pattern is the most sensitive and, therefore, the signal control which deserves the most consideration to improve the current operation of the saturated system. For example, the derivative information can be incorporated efficiently into queue and access control in congested urban networks. When the queue length on a link approaches its critical value, simply increasing its split rate may not significantly reduce the queue on this link because it may attract more traffic. Thus there may be a need to move queues to locations which can better accommodate longer queues or allow only a limited number of vehicles to enter the critical intersection. This would involve adjusting signal splits at neighboring controlled intersections to balance queues from a network-level viewpoint. The derivative information also allows one to estimate nearby solutions for any combination of signal split changes once an equilibrium solution has been calculated. Table 3 shows the unperturbed solutions (link flow and queuing delay on saturated link) of the queuing network equilibrium problem at h, = XZ = 0.5 and the perturbed solutions when the signal splits are varied by 6X, = -0.05 and 6X, = 0.05. The estimation is made using a local linear approximation based on the derivatives in Table 2. We next examine the results for the application of the sensitivity analysis algorithm to the optimal setting of traffic signals. Figure 6 shows the changes in signal splits as represented by A, (AZ = 1.0 - A,) with iterations, starting with different initial values. The corresponding changes in the upper-level objective function (total cost) are shown in

Table 1. Input data to the test network Link number Free-flow time t: Saturation flows, Delay formula

1 4.0 60.0

2 5.0 50.0

3 2.0 50.0

4 8.0 50.0

5 8.0 80.0

6 4.0 60.0

1 4.0 50.0

8 2.0 60.0

9 3.0 50.0

Traffic assignment and signal control

137

Table 2. Derivatives obtained from sensitivity analysis at X1 = A, = 0.50

a(.yax,

Solution Variable

a( .)/ax2 0.0000 50.0000

6o.oooo 0.0000

“I “1 “3 “4 “5 “6 “7 “6 “¶

4 dz

- 36.5174 - 23.4826 - 36.5174 0.0000 60.0000 0.0000 -23.4826

17.9382 - 17.9382 -32.0618 5o.OOOo 0.0000 - 50.0000 - 17.9382

- 16.1502 -5.8564

-4.8803 - 15.9792

Table 3. Estimated and exact solutions for perturbed signal splits at h, = 12 = 0.50 Perturbed with 6X, = -0.05,&Q = +0.05 Solution Variable

0

10

Unperturbed Solution

Exact

Estimated

29.9930 24.9947 9.1113 30.9161 64.1488 24.9947 29.9930 55.0263 30.9161

26.9941 27.4931 11.8072 3 1.2058 64.3258 27.4931 26.9941 52.5148 31.2058

26.9930 27.4947 11.8341 31.1933 64.3715 27.4947 26.9930 52.5263 31.1933

1.7439 2.3188

2.0827 1.7798

2.3074 1.8127

2415.22

2411.67

2403.14

20

30

40

ITERATION Fig. 6. Convergence of signal split A, with different initial values.

50

H. YANG andS. YAGAR

m. L

2.48

&

2.46

s 2

2.44

E

2.42 2.40 2.38

0

10

20

30

40

50

ITERATION Fig. 7. Changes of total travel cost with iterations.

Fig. 7. From this simple example, it can be observed that the proposed algorithm converges quickly to the optimum X: = 0.2361 (F* = 2400), which has been obtained by direct search. Note that the total travel cost does not monotonically decrease with each iteration because we have used a predetermined step length sequence, CY(“)= (n + l), in the linear combination. In addition, the intermediate objective values in the iterations are occasionally below its minimum F* = 2400. This is due to incomplete convergence of the inner penalty function method for assignment, which may give rise to underestimated queuing delay and hence underestimated total travel cost. The variation of total travel costs associated with signal splits as the optimum is approached is not significant. Thus a strict optimal solution may not necessarily be required and a nearby optimum may be a good solution in practical applications. Although this simple network example demonstrates a better performance of the sensitivity analysis algorithm, more extensive numerical tests need to be performed to validate the algorithm. Bilevel optimization models are generally difficult to solve, especially in the current case in which both the objective function and the constraints in the upper-level problem are nonlinear and implicitly defined. The solution, even if it is convergent, cannot be confirmed as globally optimal. Local optimal points may exist due to nonconvexity. Furthermore, the problem may have no feasible solution. The nonexistence of a feasible solution arises when either the demand is too great to be accommodated by the capacity constrained network or the queue storage capacities are so small that queue length constraints at two or more links cannot be satisfied simultaneously. 6. CONCLUSIONS

Some developments in model formulation and solution procedure for the static queuing equilibrium network signal setting problem have been described. In the newly formulated traffic assignment and signal control model, delay to traffic at signalized intersections is explicitly divided into signal delay and queuing delay. The latter is due to limited exit capacity and is determined endogenously from network equilibrium conditions rather than predicted by an analytical formula. This is preferable to previous methods in two ways. First, the model avoids the difficulties encountered by link cost functions such as Webster’s delay formula in the combined signal control assignment problem. Second, the model predicts link queues explicitly and hence makes it easier to implement queue length control in saturated networks.

Traffic assignment and signal control

139

On the other hand, sensitivity analysis is implemented to obtain the derivatives of link flows and queue delay with respect to signal splits and hence produce the directions in which the queuing network equilibrium pattern can move if the signal settings are changed. This information has been used efficiently to determine optimal signal settings such that total network travel time is minimized. Although the algorithm cannot be guaranteed to converge to a global optimum, it could be expected to obtain a good suboptimal solution. Further work should be concentrated on extensive numerical tests for larger networks to assess the efficiency of the algorithm. Acknowledgements-The

authors wish to express their thanks to the Natural Science and Engineering Research Council of Canada for an international postdoctoral research fellowship and a research grant which made this work possible. They also wish to acknowledge an associate editor and two anonymous referees for their helpful suggestions and valuable comments. REFERENCES

Allsop R. E. (1974). Some possibilities for using traffic control to influence trip destinations and route choice. In D. J. Buckley (Ed.), Proceedings of the Sixth International Symposium on Transportation and Traffic Theory, Sydney, Australia, pp. 345-374. Elsevier, Amsterdam. Cantarella G. E., Improta G. and Sforza A. (1991). Iterative procedure for equilibrium network traffic signal setting. Transpn. Res., 24A, 241-249. Dickson T. J. (1981). A note on traffic assignment and signal timings in signal controlled road network. Transpn. Res., ISB, 267-271.

Fisk C. S. (1984). Game theory and transportation systems modeling. Transpn. Res., 18B, 301-313. Friesz T. L., Tobin R. L., Cho H. J. and Mehta N. J. (1990). Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints. Mathemat. Program., 48,265-284. Gartner N. H., Gershwin S. B., Little J. D. C. and Ross P. (1980). Pilot study of computer-based urban traffic management. Transpn. Res., 148,203-217. Heydecker B. G. and Khoo T. K. (1990). The equilibrium network design problem. Proceedings of AIR0 PO Conference on Models and Methodsfor Decision Support, Sorrento, pp. 587-602. Hurdle V. F. (1985). Signalized intersection delay model-a primer for the uninitiated. Transpn. Res. Rec., 971, 96-105.

Inouye H. (1986a). Equilibrium traffic assignment on a network with congested flows. Infrastructure Planning Rev., 3, 177-183 (in Japanese). Inouye H. (1986b). Traffic equilibrium in congested networks and their numerical solution. Proceedings of the Japan Society of Civil Engineers, No. 365/IV-4, 125-133 (in Japanese). LeBlanc L. J., Morlok E. K. and Pierskalla W. (1975). An efficient approach to solving the road network equilibrium traffic assignment problem. Transpn. Res., 9, 309-318. Marcott P. (1983). Network optimization with continuous control parameters. Transpn. Sci., 17, 181-197. Marcotte P. and Marquis C. (1992). Efficient implementation of heuristics for the continuous network design problem. Annal. Oper. Res., 34, 163-176. Robertson D. I. (1969). Transyt: A traffic network study tool. Road Research Laboratory Report 253. Sheffi Y. (1985). Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice Hall, Englewood Cliffs, NJ. Sheffi Y. and Powell W. B. (1983). Optimal signal setting over transportation networks. Transpn. Eng., 109(6), 824-839.

Smith M. J. (1987). Traffic control and traffic assignment in a signal-controlled network with queuing. In N. H. Garnter and N. H. M. Wilson (Eds.), Proceedings of the 10th International Symposium on Transportation and Traffic Theory, July 1987, pp. 61-77. MIT Press, Cambridge, MA. Smith M. J., Van Vuren T., Heydecker B.J. and Van Vliet D. (1987). The interaction between signal control policies and route choices. In N. H. Garnter and N. H. M. Wilson (Eds.), Proceedings of the 10th International Symposium on Transportation and Traffic Theory, July 1987, pp. 319-338. MIT Press, Cambridge, MA. Smith M. J. and Van Vuren T. (1993). Traffic equilibrium with responsive traffic control. Transpn. Sci., 27, 118-132.

Tan N. H., Gershwin S. B. and Athans M. (1979). Hybrid optimization in urban transport networks. Laboratory for Information and Decision Systems Technical Report DOT-TSC-RSPA-97-7, published by MIT, Cambridge, MA. Thompson W. A. and Payne H. J. (1975). Traffic assignment on transportation networks with capacity constraints and queuing. Paper presented at the 47th National ORSA/TIMS North America Meeting. Tobin R. L. and Friesz T. L. (1988). Sensitivity analysis for equilibrium network flows. Transpn. Sci., 22, 242250.

Van Vuren T. and Van Vliet D. (1992). Route Choice and Signal Control. Athenaeum Press Ltd., Newcastle upon Tyne. Van As S. C. (1991). Overflow delay in signalized networks. Transpn. Res., 2M, l-7. Webster F. V. (1958). Traffic signal settings. Road Research Laboratory Technical Paper No. 39. Yagar S. (1977) Minimizing delays for transient demands with application to signalized road junctions. Transpn. Res., II, 53-62.

Yang H. and Yagar S. (1993). Traffic assignment and traffic control in general freeway-arterial corridor systems. Transpn. Res., 28B, in press. Yang H., Yagar S., Iida Y. and Asakura Y. (1993). An algorithm for the inflow control problem on urban freeway networks with user-optimal flows. Transpn. Res., 28B.