Peilin Zhao 赵沛霖

Peilin Zhao 赵沛霖

PhD Candidate @ NTU CEE

Modeling and analysis of the platoon size of Connected Autonomous Vehicles in a mixed traffic environment

Published in Transportation Research Part E: Logistics and Transportation Review, 2025

Abstract

In a mixed traffic environment that consists of both Connected Autonomous Vehicles (CAVs) and Human-driven Vehicles (HVs), the platoon sizes of CAVs play a significant role in traffic flow analysis. However, the statistical properties of these platoon sizes have not been thoroughly addressed in existing research. This study aims to fill this critical gap by modeling CAV platoon sizes as a random variable, analyzing scenarios both with and without a Maximum Platoon Size (MPS) constraint. Specifically, the frequencies and corresponding probability distributions of CAV platoon sizes under these conditions are derived. Furthermore, the distribution derivations are extended by incorporating platooning willingness. Through numerical analysis, the results reveal that the proposed probability distributions align closely with numerical observations, demonstrating the consistency and reliability of the model. The study also explores the characteristics of these distributions, as well as the effects of the MPS constraint and platooning willingness. By examining the platooning behaviors in mixed traffic and providing analytical derivations for CAV platoon size probability distributions, this research lays a robust mathematical foundation for further analysis of mixed traffic dynamics, enhancing traffic management and efficiency in increasingly automated traffic environments.

Highlights

  • This paper unveils the statistical properties of CAV platoon sizes in mixed traffic.

  • Accounting for scenarios with or without Maximum Platoon Size (MPS).

  • The frequencies and distributions of various CAV platoon sizes are formulated.

  • Distribution derivations are extended to incorporate platooning willingness.

  • The corresponding probability distributions are derived and verified by numerical studies.

Assumptions

  • Only two classes of vehicles are considered, i.e., CAVs and HVs, which are randomly distributed on a single-lane roadway.

  • Lane changes and other lateral movements are excluded; the analysis focuses solely on longitudinal platoon formation.

  • In Foundational model and MPS constraint model, consecutive CAVs are assumed to automatically form a platoon.

  • Platoon size is defined as the number of consecutive CAVs within a platoon (which are typically interspersed by HVs), on a single-lane road. HVs are treated as size-0 platoons.

  • In MPS constraint model, when a platoon exceeds the MPS, the next CAV in line will serve as the leader of a new platoon.

  • In platooning willingness model, each CAV is further assumed to have a probability w of forming the platoon with its leading vehicle.

Models

  • Foundational model: The probability of size-m CAV platoon follows: \(P_m = \begin{cases} \displaystyle \frac{1}{1 + P_{\mathrm{CAV}}}, & m = 0,\\[1em] \displaystyle \frac{P_{\mathrm{CAV}}^m \,(1 - P_{\mathrm{CAV}})}{1 + P_{\mathrm{CAV}}}, & m \ge 1. \end{cases}\)

  • MPS constraint model: When MPS is set as \(L\): \(P'_{m'} = \begin{cases} \displaystyle \frac{1 - P_{\mathrm{CAV}}^L}{\,1 - P_{\mathrm{CAV}}^L + P_{\mathrm{CAV}}\,}, & m' = 0, \\[1em] \displaystyle \frac{(1 - P_{\mathrm{CAV}})\,P_{\mathrm{CAV}}^{m'}}{\,1 - P_{\mathrm{CAV}}^L + P_{\mathrm{CAV}}\,}, & m' \in \{1,2,\dots,L-1\}, \\[1em] \displaystyle \frac{P_{\mathrm{CAV}}^L}{\,1 - P_{\mathrm{CAV}}^L + P_{\mathrm{CAV}}\,}, & m' = L. \end{cases}\)

  • Platooning willingness model: \(P_m^{(w)} \;= \begin{cases} \displaystyle \frac{1 - P_{\mathrm{CAV}}}{1 - w\,P_{\mathrm{CAV}}^2}, & m = 0,\\[1em] \displaystyle \frac{(1 - w\,P_{\mathrm{CAV}})^2\,w^{\,m-1}\,P_{\mathrm{CAV}}^m}{1 - w\,P_{\mathrm{CAV}}^2}, & m \in \{1,2,\dots\}\,. \end{cases}\)

  • MPS constraint & platooning willingness model: \(P'^{(w)}_{m'} = \begin{cases} \displaystyle \frac{(1 - P_{\mathrm{CAV}})\,(1 - w^{L}P_{\mathrm{CAV}}^{L})} {1 - w^{L}P_{\mathrm{CAV}}^{L} + w^{L}P_{\mathrm{CAV}}^{L+1} - w\,P_{\mathrm{CAV}}^{2}}, & m' = 0, \\[1em] \displaystyle \frac{(1 - w\,P_{\mathrm{CAV}})^{2}\,w^{\,m'-1}\,P_{\mathrm{CAV}}^{\,m'}} {1 - w^{L}P_{\mathrm{CAV}}^{L} + w^{L}P_{\mathrm{CAV}}^{L+1} - w\,P_{\mathrm{CAV}}^{2}}, & m' \in \{1,2,\dots,L-1\}, \\[1em] \displaystyle \frac{(1 - w\,P_{\mathrm{CAV}})\,w^{\,L-1}\,P_{\mathrm{CAV}}^{\,L}} {1 - w^{L}P_{\mathrm{CAV}}^{L} + w^{L}P_{\mathrm{CAV}}^{L+1} - w\,P_{\mathrm{CAV}}^{2}}, & m' = L. \end{cases}\)

Visualization

  • Foundation model & MPS constraint model:

    1

  • Platooning willingness model:

    2

  • MPS constraint & platooning willingness model (MPS = 5):

    3

Milestone

  • Apr 2023: Began research in modeling the platoon size of CAV.

  • Jul 2023: Formulated the foundation model for CAV platoon size.

  • Oct 2023: Incorporated MPS constraint into foundation model.

  • Sep 2024: Presented the foundation model and MPS constraint model at 4th Annual Next-Generation Transport Systems Conference (NGTS-4) (Slides)

  • Sep 2024: Incorporated platooning willingness into both foundation model and MPS constraint model.

  • Apr 2025: Published the article in Transportation Research Part E: Logistics and Transportation Review.

DOI: 10.1016/j.tre.2025.104130
License: © 2025. This manuscript version is made available under the CC BY-NC-ND 4.0 license.

Recommended citation: Zhao, P., Wong, Y. D., & Zhu, F. (2025). Modeling and analysis of the platoon size of Connected Autonomous Vehicles in a mixed traffic environment. Transportation Research Part E: Logistics and Transportation Review, 199, 104130.
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