Dynamic traffic assignment approximating the kinematic wave model: System optimum, marginal costs, externalities and tolls

Carey, Malachy; Watling, David · 2012 · OpenAlex-citations

DOI: 10.1016/j.trb.2012.01.008

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Summary

This paper addresses the derivation of system marginal costs, externalities, and optimal congestion tolls for dynamic traffic networks. While traditional dynamic system optimizing (DSO) models often rely on "whole link" approximations, this study develops a DSO formulation that more closely aligns with traffic flow theory, specifically the Lighthill-Whitham-Richards (LWR) kinematic wave model. The motivation is to provide a tractable, theoretically consistent framework for calculating optimal tolls that internalize congestion externalities, thereby inducing a user equilibrium that coincides with the system optimum. The methodology employs a finite difference approximation (FDA) of the LWR model, extending the Cell Transmission Model (CTM) to accommodate general nonlinear flow-density functions rather than the standard triangular or trapezoidal forms. This extension allows for differentiability, which is crucial for deriving marginal costs. The network is discretized into cells and time intervals, and the DSO problem is formulated as a convex nonlinear program that minimizes total network travel costs. The model incorporates constraints on cell occupancy, sending and receiving capacities, and flow conservation at nodes. Crucially, the formulation endogenously determines merge and diverge proportions at junctions and allows for "holding back" of traffic flows to prevent congestion-induced throughput reductions. The paper also extends the model to include cost-elastic travel demand. The main findings establish the relationship between the DSO solution and dynamic user equilibrium (DUE). By analyzing the Kuhn-Tucker optimality conditions, the author identifies dual variables as system marginal costs. The paper demonstrates that if tolls are imposed on users equal to the difference between the system marginal cost and the experienced travel time, the resulting user equilibrium will match the system optimum. Specifically, Proposition 2 proves that tolls defined as the system marginal cost minus the path travel time for utilized paths (and higher for unutilized paths) ensure that users internalize the externality. Furthermore, the study shows that these path tolls can be decomposed into link-level tolls based on differences in dual variables between successive nodes. The paper also notes that a constant flat toll can be added to ensure non-negative tolls while maintaining the DSO-DUE equivalence. The significance of this work lies in providing a rigorous mathematical foundation for dynamic congestion pricing based on realistic traffic flow dynamics. By bridging the gap between the LWR model and traffic assignment, the paper offers a method to compute optimal tolls that account for spillbacks and moving queues, phenomena often ignored in simpler models. This approach supports the design of intelligent transport systems where variable tolls or flow controls can be used to optimize network performance, ensuring that individual user decisions align with overall system efficiency.

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• mental model of traffic