A linear Lagrangian model predictive controller of macro- and micro- variable speed limits to eliminate freeway jam waves

Variable speed limits (VSLs) are a common traffic control measure to resolve freeway jam waves. State-of-the-art model predictive control (MPC) approaches of VSLs are developed based on Eulerian Lighthill-Whitham and Richards (LWR) models, where the decision variables are flows between road segments. It is difficult to implement constraints on speeds that are necessary in typical real-world speed limit systems, because converting flow to speed results in nonlinear and non-convex optimization formulations. In this paper, we develop a new MPC of VSLs based on a discrete Lagrangian LWR model, in which the decision variables are average speeds of vehicle groups. This allows formulating speed constraints as control constraints rather than state constraints in the MPC problem. The optimization of vehicle groups speeds is formulated as a linear programming problem which can be solved efficiently. We further integrate the presented MPC to a hierarchical VSL control framework leveraging connected vehicles. The presented MPC decides the optimal target speed of each vehicle group led by a connected automated vehicle (CAV) at the upper macroscopic level with a prediction horizon of 20 min. At the lower microscopic level, CAVs randomly distributed in mixed traffic are regarded as actuators of the upper layer. Microscopic CAV accelerations are optimized in a short horizon of the order 5-10 s so that the human-driven vehicles following them reach the target speed from the upper layer in an efficient and smooth manner. The presented MPC and the hierarchical control approach are tested in microscopic simulation environments. Simulation results show that (i) the presented MPC resolves freeway jam waves efficiently with reasonable safety constraints implemented, and (ii) the presented hierarchical control approach can effectively resolve jam waves in a single-lane freeway, even though the penetration rate of CAVs is as low as 5%.