A Smoothing Method for Ramp Metering

Ramp metering offers great potential to mitigate traffic congestion and improve freeway management efficiency under traffic congestion conditions. This paper proposes an optimization program for freeway dynamic ramp metering based on Cell Transmission Model (CTM). This problem has been formulated as a discrete time optimal control problem with smooth state equations and constraints to meter traffic inflow from on-ramps. In the proposed model, the `min' operators in the primal CTM are non-differentiable and thus, the corresponding optimal control problem cannot be solved directly using conventional gradient based methods. In this paper, we introduce a smooth approximation to approximate the `min' operators and then a unified computational approach is developed to solve the problem. Theoretical analysis is carried out, showing that the optimal solution obtained from the approximated problem converges to the optimal solution of the primal CTM. Compared to the classical inequality relaxation method, our method can resolve the flow holding-back problem and reduce under fundamental diagram phenomenon. Compared with the Big-M method, our method has better efficiency. To achieve the desired traffic response control in real application, a series of online optimal control problems are solved using Model Predictive Control (MPC). Simulation studies show that our method can significantly improve freeway traffic management efficiency.