A Comparison of Static and Dynamic Traffic Assignment Under Tolls: A Study of the Dallas-Fort Worth Network

There is great interest in developing pricing models for congestion relief. However, most of the work in the literature uses static transportation models for analysis. The benefits of accounting for traffic dynamics under congestion pricing are unclear. This research performs a systematic comparison of static traffic assignment with the VISTA model, a simulation-based dynamic traffic assignment approach, and with an approximation to DTA using an add-in for TransCAD software. A novel demand profiling algorithm based on piecewise linear curves is developed, and a method to enable reasonable comparisons of static traffic assignment and the TransCAD add-in is provided. The results indicate that traditional static models have the potential to significantly underestimate network congestion levels in traffic networks, and the ability of DTA models to account for variable demand and traffic dynamics under a policy of congestion pricing can be critical. INTRODUCTION AND BACKGROUND As the number of drivers increases in urban areas around the world, the search for policies to counteract congestion continues in earnest, as does the search for models to reliably predict the impacts of these policies. One such policy is congestion pricing, which assesses users a fee for traveling certain links at certain times, in an effort to efficiently allocate space on the network. While the idea of congestion pricing has existed for some time (see, for instance, (1), (2), and (3)), it has gained considerable acceptance in practice as technological advancements address various implementation issues. Techniques for finding firstand second-best pricing schemes under a variety of scenarios have been developed, as in (4, 5, 6, 7, and 8), and different versions of congestion pricing has been tested or implemented in a number of locations around the world (9). Techniques for predicting the impact of such policies also have improved in recent years. Dynamic traffic assignment models have attracted recent attention, due to their ability to account for time-varying properties of traffic flow (10). However, these formulations generally lead to extremely complicated solution procedures. Nevertheless, progress has been made using techniques such as simulation for solving large networks (11 and 12). In particular, many state agencies are currently evaluating pricing as a potential congestion relief policy, and the Texas Department of Transportation (TxDOT) is exploring the value of using DTA models to predict the impact of such policies. This paper investigates the difference in results obtained from using static and dynamic traffic assignment (STA and DTA) to evaluate congestion pricing policies on a real traffic network, and its contributions of this paper are twofold. First, an algorithm is proposed that quickly and reliably generates a timevarying demand profile from aggregate demand data (static OD trip tables). Such profiles are a required input to perform the DTA analysis. Second, this investigation compares the results from a DTA approximation, a simulation-based DTA approach, and traditional static assignment when applied to the Dallas-Ft. Worth (DFW) network, where TxDOT is considering implementation of congestion pricing on selected links. This paper is organized as follows. First, the TransCAD add-in and the VISTA model, which were used to perform the DTA analysis, are described. Following this is a description of key issues that arise when attempting to compare static and dynamic assignment, and a method to facilitate comparison between the approximator and static assignment. Next, an algorithm is presented that creates time-varying demand data from aggregate data, followed by the DFW network results. Finally, the contributions are summarized and the principal findings reiterated. DTA APPROXIMATION USING TRANSCAD'S ADD-IN The add-in to TransCAD, developed by Caliper Corporation, approximates DTA by utilizing an algorithm developed by Janson and Robles (13) which converges to a dynamic user equilibrium solution. This algorithm approximates the variability in travel demand and link flows by dividing the analysis period into smaller, discrete time intervals, over which demand is assumed to be uniform. The procedure employs the notion of a node-time interval , , d t r i α , a binary variable which is set to unity if the last unit of flow leaving origin r during a particular time interval d passes through node i during time interval t, and set to zero otherwise. Intuitively, the node time intervals trace the paths of the last vehicles leaving each origin at each time interval. While the mixed integer program developed by Janson and Robles is non-convex over all possible node time intervals, for a fixed set of node-time intervals, the problem is convex. Thus, the algorithm employs an iterative procedure whereby a set of node time intervals is assumed and then the traffic assignment problem is solved. A new set of node time intervals is calculated from the traffic assignment results, which are used to generate another traffic assignment. This iterative process converges to a dynamic user-equilibrium solution within a given tolerance, defined as sufficiently few node-time intervals changing from one iteration to the next. Constraints in the mathematical program guarantee that FIFO conditions are satisfied, and the algorithm also includes procedures to resolve problems with spillback queues that may occur due to incidents or other changes in link capacity. However, since the approximator in its current state only uses Bureau of Public Roads-type (BPR) link performance functions in TransCAD to calculate travel times, this feature cannot be used. SIMULATION-BASED DTA USING VISTA VISTA (Visual Interactive System for Transport Algorithms) is a network-enabled software that integrates spatio-temporal data and models for a wide range of transport applications: planning, engineering, and operational (14). VISTA can be accessed via a cross-platform JAVA client or a web page. In particular, VISTA can perform dynamic traffic assignment using a cell transmission model (CTM). The cell transmission model was developed by Daganzo (15) as a discrete version of the hydrodynamic traffic flow model. The CTM can be thought of as a simulation-based model which divides network links into shorter "cells," which then tracks the number of vehicles in each cell through a series of discrete time steps on the order of five seconds. Limits on the maximum number of vehicles in each cell and the maximum number of vehicles that can move from one cell to the next between iterations correspond to maximum densities and capacity for links in the network. A key feature of the CTM is that flows are explicitly prohibited from exceeding capacity. This is in contrast to static assignment methods, or the DTA approximator, where it is possible to have link volumes exceed capacity. In the CTM, if demand for a cell exceeds the available capacity, queuing forms to maintain flows less than capacity. This ability to model queues in a somewhat realistic manner is one of the prime attractions of the CTM. The simulator used in VISTA is an extension of the basic CTM. The main enhancements over the basic cell transmission model are the concept of adjustable size cells that improves the flexibility, accuracy and computational requirements of the model, and a modeling approach to represent signalized intersections. The basic cell transmission model along with the enhancements yields a model that can simulate integrated freeway/surface street networks with varying degree of detail. ISSUES IN COMPARING STA AND DTA RESULTS The question of how to compare static assignment to DTA is a nontrivial one. Typical measures of comparison, such as volumes on individual links or total system travel time (TSTT), cannot be applied in a naïve fashion due to fundamental differences between the modeling approaches. Moreover, the behavioral assumptions are so different that parameter assumptions are not really comparable. With regard to the DTA approximator, the need for a more sophisticated way to compare results arose from preliminary investigations into a smaller test network, where, contrary to intuition, the DTA approximator predicted a lower total system travel time than a static assignment. The cause of this phenomenon was determined to be the presence of clearance intervals in DTA after the assignment periods, which continue to model vehicles in the network until they reach their destinations, even though no additional trips are loaded. Since this results in some links having volume for a longer period of time than in static assignment (which has no need for clearance intervals), the result is an effective increase in link capacities. Fundamentally, this occurs because STA has no concept of arrival or departure times; thus, the issue of trips departing late in the peak period is not relevant. For this reason, a procedure was sought to make the results of static and dynamic assignment commensurable. The procedure used here first assigns vehicles with DTA; then, for each link, the number of clearance intervals needed before the last vehicle leaves is noted. Finally static assignment is performed, with the capacity of each link altered based on the number of clearance intervals needed in DTA. For instance, for a link with a two-hour capacity of 2000 veh/hr, if the last vehicle departed this link 15 minutes after the end of the assignment period, STA link capacity was increased by (2000)(15/120) = 250 veh/hr (for a total of 2250 veh/hr). Essentially, this procedure provides the same additional capacity that clearance intervals provide in DTA, on a link-by-link basis. Capacities were only increased using this method (that is, if the last vehicle on a particular link left before the end of the assignment periods, no capacity reduction was made). This method was chosen for several reasons. First, it is a fairly straightforward way to accommodate the effective increase in capacity provided by clearance intervals in DTA, and is specific to each link in the network. Additionally, it avoids manipulation of DTA results, thereby preserving arrival times, departure times, and all the other time-varying properties of the network which make DTA attractive in the first place. Furthermore, the additional computational burden imposed by this method is negligible. The issues in comparing DTA with STA are further compounded when attempting to compare STA with VISTA results. While clearance intervals are a major issue in comparing results from the DTA approximator, the approximator still uses the same BPR link performance functions that static assignment used. VISTA, on the other hand, uses no link performance functions at all, but instead uses the simulation-based cell transmission model to propagate traffic. For this reason, global measures of comparison were chosen to compare the two assignment procedures. Individual link flows are not directly comparable because of the vast differences between the assignment procedures, and measures such as v/c ratios have different meanings (flow-to-capacity in VISTA, demand-to-capacity in static assignment). For each of five functional classes of roadways (freeways, arterials, etc.), the total travel time was compared, as was the total system travel time for the entire network.