The Modified Optimal Velocity Model: Stability Analyses and Design Guidelines
For researchers and engineers in traffic flow theory, this work provides rigorous stability conditions and bifurcation analysis for the MOVM, which is an incremental theoretical contribution.
This paper analyzes the local stability of the Modified Optimal Velocity Model (MOVM) under no delay, small delay, and arbitrary delay, deriving necessary and sufficient conditions for stability and non-oscillatory convergence, and characterizing Hopf bifurcations. The results provide design guidelines to ensure smooth traffic flow and ride quality.
Reaction delays are important in determining the qualitative dynamical properties of a platoon of vehicles traveling on a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics of the Modified Optimal Velocity Model (MOVM). Specifically, we analyze the MOVM in three regimes -- no delay, small delay and arbitrary delay. In the absence of reaction delays, we show that the MOVM is locally stable. For small delays, we then derive a sufficient condition for the MOVM to be locally stable. Next, for an arbitrary delay, we derive the necessary and sufficient condition for the local stability of the MOVM. We show that the traffic flow transits from the locally stable to the locally unstable regime via a Hopf bifurcation. We also derive the necessary and sufficient condition for non-oscillatory convergence and characterize the rate of convergence of the MOVM. These conditions help ensure smooth traffic flow, good ride quality and quick equilibration to the uniform flow. Further, since a Hopf bifurcation results in the emergence of limit cycles, we provide an analytical framework to characterize the type of the Hopf bifurcation and the asymptotic orbital stability of the resulting non-linear oscillations. Finally, we corroborate our analyses using stability charts, bifurcation diagrams, numerical computations and simulations conducted using MATLAB.