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Hybrid multi-population traffic flow model: Optimal control for a mean-field limit

Maria Teresa Chiri, Christopher Denaro, Xiaoqian Gong, Benedetto Piccoli

TL;DR

The paper develops a rigorous hybrid-multipopulation framework for multi-lane traffic with cars ($P$), trucks ($S$), and autonomous vehicles ($Q$), combining convolution-based car-following and lane-changing rules. It derives a mean-field limit with a density PDE for human-driven vehicles and controlled ODEs for AVs, and proves existence of optimal controls in both finite- and infinite-dimensional settings via $\Gamma$-convergence. A microscopic simulator demonstrates how truck penetration affects velocity variation and overall traffic efficiency, highlighting safety and efficiency implications of heavy vehicles. The work provides a solid theoretical and computational foundation for optimizing AV control in mixed traffic and informs policy and infrastructure considerations regarding heterogeneous vehicle mixes.

Abstract

Modeling heterogeneous and multi-lane traffic flow is essential for understanding and controlling complex transportation systems. In this work, we consider three vehicle populations: two classes of human-driven vehicles (cars and trucks) and autonomous vehicles, the latter characterized by controlled acceleration. Compared to single-population models, multi-population modeling poses greater challenges, primarily due to the increased number of parameters required to describe lane-changing behavior and the added complexity in passing to the mean-field limit. We model multi-lane traffic as a hybrid dynamical system, combining continuous dynamics within each lane and discrete events corresponding to lane-changing maneuvers. We then formulate and analyze the optimal control problem associated with such hybrid systems from both microscopic and mesoscopic perspectives. Using techniques from $Γ$-convergence, we prove the existence of solutions to the optimal control problem in the mean-field limit of a finite-dimensional hybrid system. Finally, we present numerical simulations illustrating the impact of trucks on overall traffic efficiency.

Hybrid multi-population traffic flow model: Optimal control for a mean-field limit

TL;DR

The paper develops a rigorous hybrid-multipopulation framework for multi-lane traffic with cars (), trucks (), and autonomous vehicles (), combining convolution-based car-following and lane-changing rules. It derives a mean-field limit with a density PDE for human-driven vehicles and controlled ODEs for AVs, and proves existence of optimal controls in both finite- and infinite-dimensional settings via -convergence. A microscopic simulator demonstrates how truck penetration affects velocity variation and overall traffic efficiency, highlighting safety and efficiency implications of heavy vehicles. The work provides a solid theoretical and computational foundation for optimizing AV control in mixed traffic and informs policy and infrastructure considerations regarding heterogeneous vehicle mixes.

Abstract

Modeling heterogeneous and multi-lane traffic flow is essential for understanding and controlling complex transportation systems. In this work, we consider three vehicle populations: two classes of human-driven vehicles (cars and trucks) and autonomous vehicles, the latter characterized by controlled acceleration. Compared to single-population models, multi-population modeling poses greater challenges, primarily due to the increased number of parameters required to describe lane-changing behavior and the added complexity in passing to the mean-field limit. We model multi-lane traffic as a hybrid dynamical system, combining continuous dynamics within each lane and discrete events corresponding to lane-changing maneuvers. We then formulate and analyze the optimal control problem associated with such hybrid systems from both microscopic and mesoscopic perspectives. Using techniques from -convergence, we prove the existence of solutions to the optimal control problem in the mean-field limit of a finite-dimensional hybrid system. Finally, we present numerical simulations illustrating the impact of trucks on overall traffic efficiency.
Paper Structure (14 sections, 3 theorems, 48 equations, 3 figures, 2 algorithms)

This paper contains 14 sections, 3 theorems, 48 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Car-following models for heterogeneous traffic
  4. Lane-changing rules for multi-lane traffic
  5. Cool-down time
  6. Generalization to multi-population models
  7. The optimal control problem on a finite-dimensional hybrid system
  8. The optimal control problem on an infinite-dimensional hybrid systems
  9. Microscopic Hybrid Multi-Population Simulations
  10. Model Equations
  11. Simulation Setting
  12. Algorithmic Flow
  13. Model Parameters and Experimental Results
  14. Conclusion and future work

Key Result

Theorem 3.1

The finite horizon optimal control problem eqn_FPSk associated to the finite dimensional hybrid system has solutions.

Figures (3)

  • Figure 1: A schematic view of the setting: we consider autonomous vehicles (red), cars (black), and trucks (black) traveling on $L$ lanes. Vehicles may change lanes according to a probability given by \ref{['eqn: probability_function']}, which depends on the Incentive and Safety criteria specified in \ref{['lane-changing_acc']}--\ref{['lane-changing_safe']}.
  • Figure 2: Branched violin plots showing the difference between maximum velocity and minimum velocity for each vehicle across 100 trials of each scenario. Violin plots correspond to 10%, 20%, and 30% truck penetration rates from left to right. Each violin plot shows the max velocity difference for cars (blue, left side) and trucks (orange, right side).
  • Figure 3: Branched violin plots showing the total variation of velocity for each vehicle across 100 trials of each scenario. Violin plots correspond to 10%, 20%, and 30% truck penetration rates from left to right. Each violin plot shows the total variation of velocity for cars (blue, left side) and trucks (orange, right side).

Theorems & Definitions (13)

  • Definition 1: The controlled finite-dimensional hybrid system
  • Definition 2: Optimal control problem associated with a finite-dimensional hybrid system
  • Theorem 3.1
  • proof
  • Definition 3: The controlled infinite-dimensional hybrid system
  • Definition 4: Optimal control problem associated with an infinite-dimensional hybrid system
  • Definition 5
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • ...and 3 more