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Mixed platoon control of automated and human-driven vehicles at a signalized intersection: dynamical analysis and optimal control

Chaoyi Chen, Jiawei Wang, Qing Xu, Jianqiang Wang, Keqiang Li

arXiv:2010.16105v4math.OCeess.SY

TL;DR

The paper addresses the limited evidence on controlling CAVs in signalized intersections where HDVs and CAVs coexist. It proposes a “1+n” mixed-platoon model with direct CAV control, theoretical dynamics analysis, optimal control, and event-triggered coordination. Simulations across traffic volumes and MPRs report greater traffic-efficiency and fuel-consumption benefits than single-CAV trajectory optimization.

  • Problem

    Existing research on CAV control at signalized intersections has mostly considered fully autonomous traffic, while mixed-traffic intersection control has not been fully discussed.

  • Method

    The paper models one leading CAV and n following HDVs, analyzes linearized stability and controllability, and formulates constrained optimal control with hierarchical event-triggered coordination.

  • Results

    Traffic simulations across multiple traffic volumes and MPRs found the mixed-platoon control method surpassed traditional single-CAV intersection control in traffic efficiency and fuel consumption.

  • Takeaways & Limitations

    Directly controlling a leading CAV can be used to improve the performance of an entire mixed traffic intersection through the proposed mixed-platoon structure.

  • Takeaways & Limitations

    Under high-density traffic, queuing is inevitable and the mixed-platoon algorithm has little optimization space, limiting improvements in efficiency and fuel consumption.

Abstract

from arXiv · show

The emergence of Connected and Automated Vehicles (CAVs) promises better traffic mobility for future transportation systems. Existing research mostly focused on fully-autonomous scenarios, while the potential of CAV control at a mixed traffic intersection where human-driven vehicles (HDVs) also exist has been less explored. This paper proposes a notion of "1+n" mixed platoon, consisting of one leading CAV and n following HDVs, and formulates a platoon-based optimal control framework for CAV control at a signalized intersection. Based on the linearized dynamics model of the "1+n" mixed platoon, fundamental properties including stability and controllability are under rigorous theoretical analysis. Then, a constrained optimal control framework is established, aiming at improving the global traffic efficiency and fuel consumption at the intersection via direct control of the CAV. A hierarchical event-triggered algorithm is also designed for practical implementation of the optimal control method between adjacent mixed platoons when approaching the intersection. Extensive numerical simulations at multiple traffic volumes and market penetration rates validate the greater benefits of the mixed platoon based method, compared with traditional trajectory optimization methods for one single CAV.

1. Introduction

The paper addresses the limited study of directly controlling CAV trajectories in mixed traffic intersections, where HDVs and CAVs coexist. It proposes a “1+n” mixed platoon framework, analyzes its dynamics, and reports improved intersection efficiency and fuel consumption in simulations.

  • Existing intersection research largely studied fully autonomous traffic with a 100% CAV market penetration rate.
  • CAV control in mixed traffic intersections has not been fully discussed, with prior methods often treating HDVs as disturbances or focusing on collision avoidance.
  • The proposed “1+n” mixed platoon consists of one leading CAV and n following HDVs, allowing the CAV to influence the entire mixed traffic intersection.
  • The framework analyzes open-loop stability and controllability of the linearized mixed-platoon dynamics, finding controllability under a mild condition regardless of platoon size n.
  • The optimal control formulation considers velocity deviations and fuel consumption for all platoon vehicles and optimizes terminal velocity to improve throughput.
  • Large-scale simulations across traffic volumes and MPRs found the mixed-platoon method surpassed single-CAV intersection control in traffic efficiency and fuel consumption.

2. Problem Statement

The problem formulation models a signalized intersection with coexisting CAVs and HDVs, organized into “1+n” mixed platoons. It combines connected-vehicle information, linearized car-following dynamics, and direct CAV control.

  • 2.1. Scenario Setup: The scenario contains a central traffic light, connected CAVs and HDVs, and a cloud coordinator that computes CAV velocity trajectories.
  • 2.1. Scenario Setup: The intersection is divided into an Observation Zone for lane changes, a Control Zone for direct CAV control, and a Merging Zone for potential lateral collisions.
  • 2.1. Scenario Setup: The model assumes ideal communication, fully autonomous CAVs following assigned trajectories in the Control Zone, human-controlled HDVs, and no lane changes in that zone.
  • 2.2. Dynamical Modeling of Mixed Platoon Systems: Each “1+n” mixed platoon has one leading CAV and n following HDVs, with the CAV intended to lead the platoon through the intersection.
  • 2.2. Dynamical Modeling of Mixed Platoon Systems: HDV acceleration depends on headway distance, relative velocity, and own velocity, and the formulation uses linearization around an equilibrium headway and velocity.
  • 2.2. Dynamical Modeling of Mixed Platoon Systems: The leading CAV’s acceleration is the only external control input, while the state-space model includes the CAV and all following HDVs.

3. Methodology

The methodology analyzes mixed-platoon dynamics, formulates optimal CAV control, and coordinates adjacent platoons through event triggering.

  • The paper analyzes open-loop stability and controllability of the proposed “1+n” mixed platoon systems.
  • It establishes an optimal control framework for optimizing the CAV’s driving strategy in a single mixed platoon.
  • It designs an event-triggered algorithm to address collisions between different mixed platoons.

3.1. Open-Loop Stability Analysis

The linearized “1+n” mixed platoon is open-loop stable under the same coefficient condition as a single HDV, independently of platoon size. Its stability is Lyapunov rather than asymptotic because of two zero eigenvalues.

  • The analysis evaluates open-loop stability with the leading CAV’s external input set to u(t) = 0.
  • For the linearized car-following model, stability is characterized by α1 > 0 and α2 > 0.
  • Two zero eigenvalues make the full mixed-platoon system Lyapunov stable but not asymptotically stable, while the following-HDV subsystem is strictly asymptotically stable.
  • Under the OVM model, the corresponding linearized stability condition is κ > 0 and V2C1 > 0.

3.2. Controllability Analysis

The paper analyzes controllability of the linearized “1 + n” mixed platoon, establishing when direct control of the leading CAV can steer the platoon dynamics.

  • Controllability motivation: Controllability determines whether the mixed platoon can reach a prescribed equilibrium velocity under control of the leading CAV.The paper frames controllability as feasibility of moving the platoon to any desired state through the CAV input.
  • Controllability criterion: The analysis uses the Popov-Belevitch-Hautus criterion to characterize controllability of the linear time-invariant platoon model.The criterion tests controllability through eigenvalues and the rank condition on (λI − A, B).
  • Theoretical result: The “1 + n” mixed platoon is controllable when the theorem’s parameter condition holds, independently of platoon size n.The proof proceeds by assuming uncontrollability and deriving a contradiction from the resulting eigenvector equations.
  • Interpretation: Under the stated condition, directly controlling the leading CAV provides complete control over the following n HDVs without changing their natural driving behaviors.This property supports designing the CAV input to improve the performance of the entire mixed platoon.

3.3. Optimal Control Framework

The paper formulates constrained optimal control for a “1 + n” mixed platoon approaching a signalized intersection. It selects a platoon-level terminal velocity and cost function to coordinate intersection passage, velocity stabilization, and fuel consumption.

  • Cost function and objectives: The optimal-control framework directs the leading CAV to the stopping line at green while stabilizing following HDVs at equilibrium velocity v∗ and minimizing platoon fuel consumption.The Bolza cost combines terminal-state deviation with transient fuel consumption across the CAV and HDVs.
  • Cost function and objectives: Fuel consumption is modeled for both the leading CAV and following HDVs, extending individual-CAV eco-approaching objectives to the entire mixed platoon.The framework uses Akcelik’s fuel-consumption model and includes vehicle power and inertial-drag terms.
  • Terminal velocity: The target equilibrium velocity v∗ maximizes the number of following HDVs passing during a constant green phase, subject to the equilibrium car-following condition.The equilibrium headway depends on v∗, and the velocity is obtained from the stated optimization problem.
  • Terminal velocity: Figure 3 shows that HDV equilibrium headway generally increases with equilibrium velocity, while passing number first rises and then falls.For constant traffic SPAT, the red point identifies a velocity maximizing the passing number of HDVs.
  • Terminal velocity: The optimal velocity v∗ depends on the car-following model and equilibrium equation, whereas the maximum passing number also depends on TGreen.Thus, increasing terminal velocity is not universally beneficial for mixed-platoon throughput.
  • Numerical formulation: The nonlinear optimal-control problem is converted into nonlinear programming with a pseudo-spectral method, using OVM for computational tractability.The authors note limitations for more complex models such as IDM and identify more efficient numerical methods as future work.

3.4. Algorithm Design

The paper compares a single-CAV predictive cruise-control benchmark with a hierarchical event-triggered algorithm for mixed platoons at signalized intersections. The proposed design plans CAV trajectories using traffic-signal information and mixed-platoon interactions, while recomputing when safety or prediction concerns arise.

  • 3.4.1. Benchmark Algorithm: PCC selects a feasible green-phase velocity window from signal-phase timing and CAV distance, then sets the target velocity to v_high.The window [v_low, v_high] represents velocities that allow the CAV to pass without idling.
  • 3.4.1. Benchmark Algorithm: PCC+ adjusts the CAV’s stopping-line distance using upstream traffic-flow shock-wave information to account for preceding queues.The paper retains PCC+ as a one-CAV benchmark because its core optimization remains centered on a single CAV.
  • 3.4.2. Algorithm Design for Mixed Platoon: The algorithm uses event-triggered replanning because trajectory interference is uncommon below saturated traffic density and continuous receding-horizon planning imposes a large computation burden.The design seeks to maintain safety with minimum additional calculation.
  • 3.4.2. Algorithm Design for Mixed Platoon: The mixed-platoon algorithm uses four CAV states—uncontrolled, computed, controlled, and re-computed—to coordinate planning and safety updates near the intersection.Optimization begins at the OZ–CZ boundary, and the controlled state executes the planned velocity trajectory.
  • 3.4.2. Algorithm Design for Mixed Platoon: When a mixed platoon has fewer HDVs or additional CAVs than its nominal 1+n composition, other CAVs use the HDV car-following model so the leading-CAV optimization remains applicable.The paper identifies cooperative control of multiple CAVs in one mixed platoon as future work.
  • 3.4.2. Algorithm Design for Mixed Platoon: During controlled operation, the leading CAV checks its safety distance from the preceding vehicle at every step and enters re-computed state when the constraint is violated.If sufficient distance remains, it replans; otherwise, it follows the HDVs’ car-following model through the remaining intersection approach.

4. Simulation Results and Discussion

Large-scale simulations evaluate the mixed platoon framework across traffic volumes and CAV market penetration rates. The MP algorithm generally improves traffic efficiency and fuel consumption, especially in intermediate-to-high traffic volumes where HDV–CAV interactions matter.

  • Simulation setup: Simulations used SUMO with 100 vehicles, evaluated average travel time delay and fuel consumption, and focused on the last 50 vehicles for steady-state assessment.A 50% MPR case and comparisons across traffic volumes were conducted against PCC+.
  • Case study at 50% MPR: At 750 veh/(hour · lane), MP constrained queue accumulation within the 300 m control zone, unlike uncontrolled traffic and PCC+ scenarios.This limited queue propagation toward the upstream intersection.
  • Case study at 50% MPR: MP reduced idling and smoothed vehicle velocity trajectories, improving fuel economy relative to no control and PCC+.PCC+ optimized individual CAV trajectories but did not account for the following HDVs in the same way.
  • Different traffic volumes: MP produced obvious improvements in traffic efficiency and fuel consumption between 600−1200 veh/(hour · lane) at 50% MPR.Below 600 veh/(hour · lane), no algorithm created queues; above 1200 veh/(hour · lane), CAV optimization space was limited.
  • Different traffic volumes and MPRs: The highest reported improvements were 20% in ATTD and 60% in fuel consumption, occurring near 35−40% MPR at 1100 veh/(hour · lane) and 80−85% MPR at 1000 veh/(hour · lane).Across the tested range, MP improved both measures in general between 600−1200 veh/(hour · lane).
  • Different traffic volumes and MPRs: MP benefits increased when HDVs were incorporated into mixed platoons, but the improvement percentage dropped slightly at extremely high MPRs above 90%.The authors identify around 70% MPR as a condition where almost all HDVs can be incorporated.

5. Conclusions

The paper concludes that the “1+n” mixed platoon provides a theoretically tractable and practically implementable basis for controlling CAVs in mixed-traffic signalized intersections. Simulations verify the effectiveness of the resulting optimal control method, while several extensions remain open.

  • Conclusions: The proposed “1+n” mixed platoon places one leading CAV ahead of n following HDVs to improve global traffic mobility.The framework targets mixed traffic intersections rather than fully autonomous traffic only.
  • Conclusions: The mixed platoon is open-loop stable and controllable under a mild condition independent of platoon size n.The control framework also considers whole-platoon velocity deviation, fuel consumption, and terminal velocity.
  • Conclusions: A hierarchical event-triggered algorithm addresses collision prevention between adjacent mixed platoons and is applicable across mixed-traffic MPRs.Traffic simulations verified the effectiveness of the proposed optimal control method.
  • Conclusions: Future work includes heterogeneous HDV dynamics, model uncertainty, cooperative control of multiple CAVs, lane changing, and field validation.The current study focuses on longitudinal CAV control and forbids lane changing in the control zone.

Appendix A. Proof of Theorem 1

Appendix A proves stability of the linearized “1+n” mixed platoon by induction from the one-HDV case. The resulting criterion depends on α1 and α2, not on platoon size.

  • Base case: For n = 1, the characteristic equation contains two zero eigenvalues, so the system is not asymptotically stable but can be critically stable.The stability criterion is α1 > 0 and α2 > 0.
  • Inductive step: The quadratic λ2 + α2λ + α1 = 0 is stable if and only if α1 > 0 and α2 > 0.This completes the induction proving the theorem for arbitrary platoon size.

Appendix B. Performance Indexes of Simulations under Different Traffic Volumes and MPRs

Appendix B reports average travel time delay and fuel consumption under MP control for different traffic volumes and MPRs. These indexes support comparisons between 0% and 100% MPR conditions.

  • Performance indexes: Average travel time delay is reported for 0% and 100% MPR across traffic volumes from 600 to 1200 veh/(hour · lane).The metric is presented in seconds.
  • Performance indexes: Fuel consumption is reported for 0% and 100% MPR across traffic volumes from 600 to 1200 veh/(hour · lane).The metric is presented in L/100km.
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