Leonid Engelson

Congestion Pricing of Road Networks with Users Having Different Time Values

We study congestion pricing of road networks with users differing only in their time values. In particular, we analyze the marginal social cost (MSC) pricing, a tolling scheme that charges each user a penalty corresponding to the value of the delays inflicted on other users, as well as its implementation through fixed tolls. We show that the variational inequalities characterizing the corresponding equilibria can be stated in symmetric or nonsymmetric forms. The symmetric forms correspond to optimization problems, convex in the fixed-toll case and nonconvex in the MSC case, which hence may have multiple equilibria. The objective of the latter problem is the total value of travel time, which thus is minimized at the global optima of that problem. Implementing close-to-optimal MSC tolls as fixed tolls leads to equilibria with possibly non-unique class specific flows, but with identical close-to-optimal values of the total value of travel time. Finally we give an adaptation, to the MSC setting, of the Frank-Wolfe algorithm, which is further applied to some test cases, including Stockholm.

On dynamics of traffic queues in a road network with route choice based on real time traffic information

Introducing real time traffic information into transportation network makes it necessary to consider development of queues and traffic flows as a dynamic process. This paper initiates a theoretical study of conditions under which this process is stable. A model is presented that describes within-one-day development of queues when drivers affected by real-time traffic information choose their paths en route. The model is reduced to a system of differential equations with delay. Equilibrium points of the system correspond to constant queue lengths. Stability of the system is investigated using characteristic values of the linearised minimal face flow. A traffic network example illustrating the method is provided.

Properties of Expected Travel Cost Function with Uncertain Travel Time

The paper presents a theoretical analysis of travelers’ scheduling preferences and the resulting form of the expected utility that includes travel time reliability measures. A series of research papers and reports used the mean–standard deviation approach to evaluate policies that improve travel time reliability. Recently, this approach was theoretically substantiated under the conventional assumptions of constant marginal utility of time (MUT) at the origin, two discrete MUT values at the destination, and constant standardized travel time distribution. In this paper, properties of the minimal expected travel cost are investigated with smooth MUTs at the origin and destination of the trip. The influence of small variations in travel time on travel cost is well approximated by a term proportional to the travel time variance and independent of the distribution form of travel time. Two examples of MUT functions are provided: the minimal expected travel cost can be analytically expressed through moments or through a moment generating function of travel time, and conditions are stated guaranteeing that the expected travel cost is exactly additive by independent parts of the trip. These results provide justification in particular for the mean–variance approach to modeling drivers’ decisions under uncertain travel times. This formulation is convenient especially for scheme evaluation in large road networks because it allows the use of conventional network assignment routines by just modifying the volume delay functions to include the travel time variability term.

Optimal toll locations and toll levels in congestion pricing schemes: a case study of Stockholm

As congestion pricing has moved from theoretical ideas in the literature to real-world implementation, the need for decision support when designing pricing schemes has become evident. This paper deals with the problem of finding optimal toll levels and locations in a road traffic network and presents a case study of Stockholm. The optimisation problem of finding optimal toll levels, given a predetermined cordon, and the problem of finding both optimal toll locations and levels are presented, and previously developed heuristics are used for solving these problems. For the Stockholm case study, the possible welfare gains of optimising toll levels in the current cordon and optimising both toll locations and their corresponding toll levels are evaluated. It is shown that by tuning the toll levels in the current congestion pricing cordon used in Stockholm, the welfare gain can be increased significantly, and furthermore improved by allowing a toll on a major bypass highway. It is also shown that, by optimising both toll locations and levels, a congestion pricing scheme with welfare gain close to what can be achieved by marginal social cost pricing can be designed with tolls being located on only a quarter of the tollable links.

Multi-Class User Equilibria under Social Marginal Cost Pricing

In the congested cities of today, congestion pricing is a tempting alternative. With a single user class, already Beckmann et al. showed that “system optimal” traffic flows can be achieved by social marginal cost (SMC) pricing. However, different user classes can have wildly differing time values. Hence, when introducing tolls, one should consider multi-class user quilibria, where the classes have different time values. With SMC pricing, Netter claims that multi-class equilibrium problems cannot be stated as an optimization problems. We show that, depending on the formulation, the multi-class SMC-pricing equilibrium problem (with different time values) can be stated either as an asymmetric or as a symmetric equilibrium problem. In the latter case, the corresponding optimization problem is in general non-convex. For this non-convex problem, we devise descent methods of Frank-Wolfe type. We apply the methods and study a synthetic case based on Sioux Falls.

The role of volume-delay functions in forecasting and evaluating congestion charging schemes: the Stockholm case

This paper uses observations from before and during the Stockkholm congestion charging trial in order to validate and improve a transportation model for Stockholm. The model overestimates the impact of the charges on traffic volumes while at the same time it substantially underestimates the impact on travel times. These forecast errors lead to considerable underestimation of economic benefits which are dominated by travel time savings. The source of error lies in the static assignment that is used in the model. Making the volume-delay functions (VDFs) steeper only marginally improves the quality of forecast but strongly impacts the result of benefit calculations. We therefore conclude that the dynamic assignment is crucial for an informed decision on introducing measures aimed at relieving congestion. However, in the absence of such a calibrated dynamic model for a city, we recommend that at least a sensitivity analysis with respect to the slope of VDFs is performed.

Convexification of the Traffic Equilibrium Problem with Social Marginal Cost Tolls

In an earlier paper, we have demonstrated that traffic equilibria under social marginal cost tolls can be computed as local optima of a nonconvex optimization problem. The nonconvexity of this problem implies in particular that linearizations, e.g. in the Frank-Wolfe method, do not give underestimates of the optimal value. In this paper we derive the convex hull of nonconvex arc cost functions of BPR type. These convexifications can be used to get underestimates of the optimal value, or to get better search directions in the initial phase of the Frank-Wolfe method. Computational results for the Sioux Falls and Stockholm networks are reported

Tolled multi-class traffic equilibria and toll sensitivities

We review properties of tolled equilibria in road networks, with users differing in their time values, and study corresponding sensitivities of equilibrium link flows w.r.t. tolls. Possible applications include modeling of individual travellers that have different trip purposes (e.g. work, business, leisure, etc.) and therefore perceive the relation between travel time and monetary cost in dissimilar ways. The typical objective is to reduce the total value of travel time (TVT) over all users. For first best congestion pricing, where all links in the network can be tolled, the solution can be internalized through marginal social cost (MSC) pricing. The MSC equilibrium typically has to be implemented through fixed tolls. The MSC as well as the fixed-toll equilibrium problems can be stated as optimization problems, which in general are convex in the fixed-toll case and non-convex in the MSC case. Thus, there may be several MSC equilibria. Second-best congestion pricing, where one only tolls a subset of the links, is much more complex, and equilibrium flows, times and TVT are not in general differentiable w.r.t. tolls in sub-routes used by several classes. For generic tolls, where the sets of shortest paths are stable, we show how to compute Jacobians (w.r.t positive tolls) of link flows and times as well as of the TVT. This can be used in descent schemes to find tolls that minimize the TVT at least locally. We further show that a condition of independent equilibrium cycles, together with a natural extension of the single class regularity condition of strict complementarity, leads to genericity, and hence existence of said Jacobians.