Adjoint-Based Optimal Coordinated Ramp Metering Using a Differentiable Second-Order Macroscopic Traffic Flow Model

Ramp metering regulates the rate at which vehicles enter a freeway and can reduce congestion when several entrance ramps compete for limited mainline capacity. Coordinating multiple ramps requires a traffic model that captures how present metering decisions influence later traffic conditions across the corridor. This study develops a coordinated ramp-metering framework based on METANET (Modèle d’Écoulement de Trafic sur Autoroute NETworks), a second-order macroscopic model in which traffic density and average speed evolve dynamically. Minimum and bounding operations are replaced with smooth differentiable approximations while preserving vehicle conservation, nonnegative flows and queues, and physically admissible traffic states. A discrete adjoint formulation computes sensitivities to the full sequence of metering decisions, and the control problem is solved using bound-constrained optimization with penalty continuation. The framework is evaluated on a 12-mile westbound section of Interstate 210 in Pasadena, California, with six metered ramps. The optimized policy reduces total delay by 5.86% relative to uncontrolled operation, compared with 0.27% for a jointly tuned ALINEA benchmark, while maintaining identical vehicle-miles traveled and remaining within the prescribed ramp-storage tolerance. The improvement is driven mainly by temporal redistribution of ramp arrivals and is limited by ramp storage and metering capability.