A rigorous multi-population multi-lane hybrid traffic model and its mean-field limit for dissipation of waves via autonomous vehicles

In this paper, a multi-lane multi-population microscopic model, which presents stop and go waves, is proposed to simulate traffic on a ring-road. Vehicles are divided between human-driven and autonomous vehicles (AV). Control strategies are designed with the ultimate goal of using a small number of AVs (less than 5\% penetration rate) to represent Lagrangian control actuators that can smooth the multilane traffic flow and dissipate the stop-and-go waves. This in turn may reduce fuel consumption and emissions. The lane-changing mechanism is based on three components that we treat as parameters in the model: safety, incentive and cool-down time. The choice of these parameters in the lane-change mechanism is critical to modeling traffic accurately, because different parameter values can lead to drastically different traffic behaviors. In particular, the number of lane-changes and the speed variance are highly affected by the choice of parameters. Despite this modeling issue, when using sufficiently simple and robust controllers for AVs, the stabilization of uniform flow steady-state is effective for any realistic value of the parameters, and ultimately bypasses the observed modeling issue. Our approach is based on accurate and rigorous mathematical models, which allows a limit procedure that is termed, in gas dynamic terminology, mean-field. In simple words, from increasing the human-driven population to infinity, a system of coupled ordinary and partial differential equations are obtained. Moreover, control problems also pass to the limit, allowing the design to be tackled at different scales.