Efficient particle-based estimation of marginal costs in a first-order macroscopic traffic flow model

Marginal costs in traffic networks are the extra costs incurred to the system as the result of extra traffic. Marginal costs are required frequently e.g. when considering system optimal traffic assignment or tolling problems. When explicitly considering spillback in a traffic flow model, one can use a numerical derivative or resort to heuristics to calculate the marginal costs. Numerical derivatives are computationally demanding, restricting its use to simple networks. Heuristic approaches in most cases approximate the marginal costs by only considering the extra costs on the links which are traveled by the extra traffic, excluding the possibly external costs incurred on other links due to spillback. This paper proposes a novel way to estimate the true marginal costs of traffic in a dynamic discrete LWR model which correctly deals with congestion onset, spillback and dissolution. The proposed methodology tracks virtual changes in density through the network by means of particles which travel along with the characteristics of traffic. By using density based cost functions, the virtual changes in density can be directly related to the marginal costs. The computational efficiency of the methodology stems from the fact that only local conditions are considered when propagating the virtual change in density. The paper discusses the methodology and necessary model extensions, provides a numerical validation experiment illustrating the exact detail of the solution by comparison to a numerical derivative and discusses some generalizations