Optimal Control of Motorway Tidal Flow

Abstract

When inbound and outbound traffic on a bi-directional motorway is unbalanced throughout the day a lane management strategy called tidal (reversible) flow lane control is usually applied. In this control case, the direction of one or more contraflow buffer lanes is reversed according to the needs of each direction. This paper introduces a basic dynamical model for tidal traffic flow and considers the minimum traveltime, minimum-time, and maximum throughput optimal control problems for efficient motorway tidal flow lane control. Lane management is effectuated by a control variable, indicating the number of lanes opened or closed in each direction of traffic. To derive the analytical form of optimal control, the Pontryagin's maximum principle is employed. The obtained optimal control is intuitively natural of bang-bang type, as also shown in a previous work by the authors [1]. It takes only the values ±1 and switches between these values at most once. In other words, the optimal control strategy consists of switching between opening and closing in each direction of traffic one contraflow buffer lane. Of course it is an open-loop control, and thus the switch time (if applicable) depends on the initial conditions. In the case of the maximum throughput optimal control problem, semi-state feedback control is obtained and singular arcs might exist. Finally, cumulative arrival rate and output curves for both directions of traffic are used to provide a graphical interpretation of the minimum travel-time optimal control problem and obtained bang-bang control.