Stability analysis and variable speed limit control of a traffic flow model

Abstract

The cell transmission traffic flow model (CTM) has attracted considerable interest in the field of transportation due to its simplicity as well as the ability to capture most of the macroscopic traffic flow characteristics. The stability properties of the CTM under different demand and capacity constraints are not always obvious. In addition, the impact of microscopic phenomena such as forced lane changes at bottlenecks leading to capacity drop is not captured by the CTM. In this paper, we start with a single section and modify the CTM to account for capacity drop. We analyze the stability properties of the CTM under all possible demand and capacity constraints as well as all possible initial density conditions. The analysis is used to motivate the design of variable speed limit (VSL) control to overcome capacity drop and achieve the maximum possible flow under all feasible traffic situations. The results are extended to multiple sections, where the stability properties of the open-loop system are analyzed and a VSL control scheme is designed and shown to achieve the objective of maximizing the traffic flow under different demand and capacity constraints. Unlike the open loop system where an infinite number of equilibrium points exist under certain demand levels, the proposed nonlinear VSL scheme guarantees exponential convergence to a unique equilibrium point that corresponds to maximum possible flow and speed under all possible demand levels and capacity constraints.