Analytical Formulation of Autonomous Vehicle Freeway Merging Control with State-Dependent Discharge Rates

The core of the freeway merging control problem lies in dynamic queue propagation and dissipation linked to merging vehicle behavior. Traditionally, queuing is modeled through demand-supply interactions with time varying demand and fixed capacity. However, field observations show flow rates decrease during congestion at freeway merges due to the impact of intersecting traffic, a factor overlooked in fundamental diagrams. This manuscript introduces an analytical approach to characterize and control the dynamic multi-stage merging of autonomous vehicles, prioritizing traffic efficiency and safety. For the first time, the effective discharge rate at the merging point, reduced by the multi-stage dynamic merging process, is analytically derived using a closed form formulation. Leveraging this expression, performance metrics such as queue length and traffic delay are derived as the first objective. Additionally, a crash risk function is established to quantitatively assess potential collisions during the merging process, serving as the second objective. Finally, the problem is formulated as a dynamic programming model to jointly minimize delay and crash risk, with the merging location and speed as decision variables. Given the terminal state, the ramp vehicle merging task is formulated as a recursive optimization problem, employing backward induction to find the minimum cost solution. Numerical experiments using the NGSIM dataset validate the derived effective discharge rate. The results indicate that the proposed model outperforms two benchmark algorithms, leading to a more efficient and safer merging process.