Abstract

Stability of Wardrop equilibria is analyzed for dynamical transportation networks in which the drivers' route choices are influenced by information at multiple temporal and spatial scales. The considered model involves a continuum of indistinguishable drivers commuting between a common origin/destination pair in an acyclic transportation network. The drivers' route choices are affected by their, relatively infrequent, perturbed best responses to global information about the current network congestion levels, as well as their instantaneous local observation of the immediate surroundings as they transit through the network. A novel model is proposed for the drivers' route choice behavior, exhibiting local consistency with their preference toward globally less congested paths as well as myopic decisions in favor of locally less congested paths. The simultaneous evolution of the traffic congestion on the network and of the aggregate path preference is modeled by a system of coupled ordinary differential equations. The main result shows that, if the frequency of updates of path preferences is sufficiently small as compared to the frequency of the traffic flow dynamics, then the state of the transportation network ultimately approaches a neighborhood of the Wardrop equilibrium. The presented results may be read as a further evidence in support of Wardrop's postulate of equilibrium, showing robustness of it with respect to non-persistent perturbations. The proposed analysis combines techniques from singular perturbation theory, evolutionary game theory, and cooperative dynamical systems. Comment: 21 pages, 2 figures, submitted

Abstract

This paper introduces an optimization problem (P) and a solution strategy to design variable-speed-limit controls for a highway that is subject to traffic congestion and uncertain vehicle arrival and departure. By employing a finite data-set of samples of the uncertain variables, we aim to find a data-driven solution that has a guaranteed out-of-sample performance. In principle, such formulation leads to an intractable problem (P) as the distribution of the uncertainty variable is unknown. By adopting a distributionally robust optimization approach, this work presents a tractable reformulation of (P) and an efficient algorithm that provides a suboptimal solution that retains the out-of-sample performance guarantee. A simulation illustrates the effectiveness of this method. Comment: 10 pages, 2 figures, submitted to ECC 2019

Abstract

A well-known optimal velocity (OV) model describes vehicle motion along a single lane road, which reduces to a perturbed modified Korteweg–de Vries (mKdV) equation within the unstable regime. Steady travelling wave solutions to this equation are then derived with a multi-scale perturbation technique, where the travelling wave propagation coordinate depends upon slow and fast variables. The leading order solution in the hierarchy is then written in terms of these multi-scaled variables. At the following order, a system of differential equations is highlighted that govern the slowly evolving properties of the leading solution. Next, it is shown that the critical points of this system signify travelling waves without slow variation. As a result, a family of steady waves with constant amplitude and period are identified. When periodic boundary conditions are satisfied, these solutions’ parameters, including the wave speed, are associated with the driver’s sensitivity, a ˆ , which appears in the OV model. For some given a ˆ , solutions of both an upward and downward form exist, with the downward type corresponding to traffic congestion. Numerical simulations are used to validate the asymptotic analysis and also to examine the long-time behaviour of our solutions. Document type: Article