Solving the Integrated Corridor Control Problem Using Simultaneous Perturbation Stochastic Approximation

Integrating various control measures within a transportation corridor is believed to improve the overall operational performance of the entire corridor. In this study, we formulate a corridor control problem that considers two control actions: traffic signal timing and ramp metering, and propose a solution method for the formulated problem. In our formulation, traffic dynamics within a general corridor is modeled on a coherent platform based on the kinematic wave traffic flow model, and the traffic control actions of urban street signals and ramp meters are embedded in the platform. One solution algorithm based on the simultaneous perturbation stochastic approximation (SPSA) scheme is developed to optimize the integrated control. Numerical experiments show that the SPSA algorithm is computationally much more efficient than genetic algorithms (GA) and the hill-climbing algorithm, while its solutions are better than or comparable to the solutions obtained from the two other methods. Ma, Nie and Zhang 3 – INTRODUCTION A transportation corridor is operationally (rather than geographically or organizationally) defined as “a combination of discrete parallel surface transportation networks (e.g., freeway, arterial, transit networks) that link the same major origins and destinations” [1]. A corridor usually includes various types of facilities (e.g., freeway sections, ramps and urban streets), which are typically managed by different agencies and jurisdictions. In current practices, most corridors are operated separately with little consideration to the coordination of individual facilities [2-3], although it has long been recognized that integrating the control measures can improve the operational performance of the entire corridor (e.g., [4]). Two components are fundamental to modeling an integrated corridor control system (e.g., [5]). The first is the traffic flow model that realistically represents traffic evolution, and the other is the optimization method to generate optimal control plans. Three major categories of traffic flow models have been developed and applied in traffic control studies: the point-queue (P-Q) or vertical queue model, the spatial queue (S-Q) or horizontal queue model and the LighthillWhitham-Richards (LWR) model. Most studies, including the classical ones such as Webster's [6] and later HCM methods, used the P-Q model. In this model the vehicles are assumed to travel at the design speed uniformly along the road section and arrive at the stop line at a constant rate. The vehicles behind the stop line take no physical space and will be discharged at the saturation flow rate during the effective green time. The platoon dispersion model in TRANSYT [7] uses an empirical formula to depict the cyclic flow profiles (CFP) on road sections and thus relaxes the constant arrival assumption, but vehicles are still queued at the stop line. TRANSYT version 8 [8] advanced to allow vehicles to join at the end of the stopped queue. Link storage capacity constraint is enforced and no traffic can enter a link if it is occupied by stopped vehicles. This SQ model is seeing more applications recently in other traffic control studies [9-11]. Both the P-Q and S-Q models can provide good estimates of the queue size, i.e., the number of stopped vehicles, under low to medium traffic loads in particular. But when the traffic load is high and the intersection is near or over saturated, the traffic densities behind the stop line will be in frequent transitions because of the varying arrival rates and intermittent signal services [12]. Shock and acceleration waves, interfaces between two differing traffic states, will be generated in such a complicated way that neither the P-Q nor the S-Q model could capture the spatial extent of queue formations and dissipations. Consequently, queue lengths cannot be estimated accurately. In this study, queue length is stated as “the length of the roadway section behind the stop line where traffic conditions are in the congested region of the flow-density curve, i.e., they range from capacity to jammed.” [12]. While Michalopoulos and Stephanopoulos [13] further argued that under these circumstances the control action would be dictated by minimizing queue length instead of delay, Stephanopoulous et al [12] incorporated the more elaborate LWR model to analyze the complicated queuing phenomena at the signal intersections. In their analysis the linear speed-density relation [14] and the resulting parabola fundamental diagram (flow-density curve) was used to compute the maximum queue length analytically. Using the triangular fundamental diagram, Helbing [15] recently derived the formula for queue dynamics and travel time variations with respect to the arrival and departure flow rates. A self-organized control method was later developed based on these results [16]. Ma, Nie and Zhang 4 – Recently, researchers make use of a finite difference solution scheme to the LWR model, the socalled cell transmission model (CTM) [17-18], to perform traffic control studies. A linear transformation of the CTM model has been carefully designed to study the global optimal ramp metering strategies [19-20]. In an earlier work, Lo [21-22] also modified the original CTM to formulate the signal control problem into a mixed integer program. The program only considered the intersections without turns; and generally it is tedious to solve. He later applied a genetic algorithm (GA) based solution algorithm to optimize control plans for more general intersection layouts [23]. Optimization methods used to solve a traffic control problem are highly tied to the underlying traffic flow models. For instance, in [9-10], the researchers used the store-and-forward approach to depict the flow dynamics of urban streets, ramps and freeway mainline. This approach is essentially similar to the S-Q model, and the formulated integrated corridor control problem is a linear one with a sparse constraint set, for which highly efficient algorithms exist. But typically the store-and-forward approach requires the control updating time period to be no less than a signal’s cycle length; this feature rules out the possibility of synchronizing the control actions and thus make the model only suitable as a strategic queue-management tool [10]. Later they adopted the high-order flow model in METANET [24] and studied integrated ramp metering and Variable Message Sign (VMS) controls. Conjugate gradient algorithms were deployed to solve the integrated control problem [25]. The same algorithm was applied in [5], where a forward time centered space method was used to model traffic evolution. The resulting system state equations are also twice-differentiable. However, both studies can only guarantee local optima, which can be sensitive to the initial guess of the solution [5]. To summarize, mathematical programming methods [e.g., 5, 9, 10, and 25] usually require the traffic flow models to be simplified so that the gradient information can be computed. Such simplification often compromises the underlying traffic flow models. On the other hand, heuristic optimization methods such as the genetic algorithm can search for a near-global optimal control plan while allowing more realistic representation of traffic flow (e.g., [23]). However, heuristic methods usually need a large number of evaluations of system performance and usually lead to high computational costs. In this paper, we explore a stochastic approximation technique that can be viewed as a compromise of the above two types of approaches. The proposed simultaneous perturbation stochastic approximation (SPSA) has been used in other fields [26] and shown satisfying performances. In this study, an SPSA-based algorithm is developed to compute the time-of-day optimal corridor control plan, while the corridor operational performances under various control plans are evaluated on a CTM-based platform. The platform embeds signal control and ramp metering in a generic way and can thus model any general corridor network. Numerical examples are used to investigate the effectiveness of the method as compared to other heuristics methods. Practical guidelines of applying the SPSA method are also discussed. Ma, Nie and Zhang 5 – MODELING DYNAMIC NETWORK FLOW In this section, a cell transmission model (CTM) based network flow model is built, in which the control actions from traffic signals and ramp meters are modelled in a coherent way. Flow Dynamics on a General Corridor Roadway Section The well accepted LWR model states the following: ) , , ( 0 t x f q t x q ρ ρ = = ∂ ∂ + ∂ ∂ (1) where q is the flow rate on a road section; ρ is the density; x and t are the space and time variables, respectively. Daganzo [17] developed a stable numerical approximation scheme that approximates the LWR model. He shows that, if the relationship between traffic flow q and density ρ is in the form )} ( , , { min max ρ ρ ρ − = j w q v q (2) where v is the free flow speed, max q is the maximum flow rate, w is the backward shockwave speed and j ρ is the jam density, then LWR model can be approximated by a set of difference equations. The model discretizes the entire time horizon T (assignment period) into small intervals t , the loading interval. Conforming to the loading interval, the model divides every road section of the network into homogeneous segments called cells, in a way that the cell length equals the distance traversed by one typical vehicle at free flow speed in one loading interval. The flows are updated by the following difference equations: ))} ( ( , ), ( { min ) ( max , 1 t n N q t n t y i i i i i − = − δ (3) and ) ( ) ( ) ( ) 1 ( 1 t y t y t n t n i i i i + − + = + (4) where ) (t yi , ) ( 1 t yi+ are the fluxes that entering cell i and 1 + i at time t , respectively, ) ( ), ( ), ( 1 1 t n t n t n i i i + − are the numbers of vehicles in the cell ) 1 ( − i , i and 1 + i at time t , respectively, max , i q is the capacity flow into i at t , i i n N − is the space available in i , v w / = δ . Essentially equation (4) tells that the number of vehicles staying in cell i at loading interval 1 + t is the number of vehicles from interval t plus the incoming vehicles and minus the outgoing vehicles. Daganzo [18] extended the model to a general network by carefully dividing various roadway junctions into basic merges and diverges. Since control actions take places at junctions, we will introduce the flow updating rules at general junctions including signalized intersections and metered ramps. Ma, Nie and Zhang 6 – Flow Updating at Signalized Urban Intersections In [21], Lo showed that CTM can be deployed to model the flow updates at urban intersections with a few changes. If the flow capacity max q in equation (2) is replaced by one that depends on the signal timing variable ) (t gi , ⎩ ⎨ ⎧ ∈ = otherwise 0 green ) ( max max t q t q (5) where it switches between max q (green) and zero (red), the end cell of an intersection approach will serve as a functioning signal, and the flow dynamics still approximates the LWR model. At a typical intersection, traffic is grouped into movements or streams. A generalized four-leg intersection with all vehicular movements can be illustrated in Figure 1.