Vehicular Demand Management in Traffic Networks Comprised by Macroscopically Homogeneous Regions under Inter-boundary Constraints

Vehicular demand management strategies emerge as a potential solution to traffic congestion. These strategies aim to regulate the inflow of vehicles across specific regions of a vehicular transportation network. This work investigates the stability and optimality of vehicular demand management schemes in regional traffic networks, considering inter-boundary flow constraints and triangular macroscopic fundamental diagram relationships between densities and flows. We first formulate an optimization problem that aims to maximize the total vehicular throughput at steady-state. Due to the triangular macroscopic fundamental diagram relationships and inter-boundary flow constraints, the optimization problem is nonconvex. We tackle this challenge by reformulating the problem as a Mixed Integer Linear Program that can be solved with standard mathematical programming solvers to determine the optimal operating set-points. Nonetheless, it has been demonstrated that operating at maximum throughput set-points, particularly near local critical density points, may lead to instability and gridlock. To address this issue, we propose a decentralized proportional vehicular demand management controller, accompanied by proper local design conditions, such that stability is guaranteed. The effectiveness and practicality of the proposed approach are demonstrated through numerical simulations in a six-region traffic network system, that showcase the impact, in terms of maximum throughput, of incorporating inter-boundary constraints.