A PDE-ODE Model for a Junction with Ramp Buffer

Monache, Maria Laura Delle; Reilly, Jack; Samaranayake, Samitha; Krichene, Walid; Goatin, Paola; Bayen, A.M. · 2014 · OpenAlex-citations

DOI: 10.1137/130908993

archive: archived pipeline: cataloged verified

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Summary

This paper addresses the modeling of macroscopic traffic flow at a junction comprising a mainline, an onramp, and an offramp. The authors aim to improve upon existing network models, which often fail to accurately represent ramp metering dynamics or result in unrealistic flux distributions (such as blocking onramp flow). To resolve this, they propose a coupled Partial Differential Equation-Ordinary Differential Equation (PDE-ODE) system. The mainline traffic evolution is governed by the scalar Lighthill-Whitham-Richards (LWR) conservation law, while the onramp is modeled as an infinite-capacity buffer whose queue length evolves according to an ODE based on the difference between incoming and outgoing fluxes. This buffer approach prevents information loss associated with backward-moving shock waves at the ramp boundary. The core of the methodology involves defining the solution to the Riemann problem at the junction through a Linear Programming optimization framework. Unlike previous models that maximize total flux through the junction, this model maximizes the flux on the outgoing mainline while incorporating a right-of-way parameter to balance priority between the mainline and the onramp. This ensures that the onramp demand is respected proportionally to available lanes, avoiding the zero-flux scenarios common in other models. The authors prove the existence and uniqueness of admissible solutions for this system. For numerical approximation, they adapt the classical Godunov scheme to handle the discontinuous solutions and the specific boundary conditions imposed by the junction and the ODE buffer. The scheme is modified to account for the creation of additional shocks when the buffer empties. The study presents theoretical proofs regarding the consistency of the Riemann solver and demonstrates the effectiveness of the numerical scheme through simulations. The results confirm that the modified Godunov scheme provides accurate approximations of the traffic density and queue length. The numerical tests verify the convergence of the method and illustrate how the model handles various traffic scenarios, including demand-limited and supply-limited cases. The inclusion of the buffer allows the model to capture the dynamic evolution of queue lengths and the resulting impact on mainline traffic, such as the formation of new waves when the buffer transitions from non-empty to empty. The significance of this work lies in providing a more realistic and mathematically rigorous framework for traffic management at junctions with ramp buffers. By integrating the ODE buffer with the PDE mainline model and optimizing flux distribution via a priority parameter, the model better reflects real-world traffic behavior compared to discrete engineering models or continuous models that ignore ramp dynamics. This approach facilitates more accurate simulations for traffic control strategies, particularly ramp metering, by ensuring that flux allocation respects physical constraints and priority rules without artificially suppressing onramp flow.

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• traffic density