Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 3480368, 9 pages

http://dx.doi.org/10.1155/2016/3480368

## Asymptotic Stability Analysis of Binary Heterogeneous Traffic Based on Car-Following Model

^{1}Jiangsu Key Laboratory of Urban ITS, Southeast University, Nanjing 210096, China^{2}Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies, Nanjing 210096, China^{3}Hualan Design & Consulting Group, Nanning 530011, China

Received 26 November 2015; Revised 11 March 2016; Accepted 16 March 2016

Academic Editor: Juan R. Torregrosa

Copyright © 2016 Hao Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the asymptotic stability of Chandler Model for a heterogeneous traffic by using numerical simulations. A simple binary platoon is considered which consists of two types of vehicles. Platoon stabilities under various kinds of combinations of parameters are investigated. It is found that the stability of the binary platoon cannot be determined by the mean values of individual vehicle’s parameters. Some combinations of parameters that benefit to the platoon stability are found. Several interesting properties of binary platoon’s stability are summarized. The analytic stability criterion of heterogeneous traffic reported in the historical literature is studied. The result indicates the analytic criterion is not rigorous, which is apt to overestimate the stability of heterogeneous platoon.

#### 1. Introduction

The stability of traffic flow is a very interesting problem in the research of traffic flow theory. It depicts how perturbation evolves with time and along the platoon. The pioneer work related to the stability analysis of traffic flow was started by Chandler et al. [1] and Herman et al. [2] in the late 1950s. In their work, the following simple linear car-following model which is also called Chandler Model was examined:where denotes the position of th vehicle at time , is the response delay of the following driver, and is the sensitivity to stimulus. Based on Chandler Model, two types of stability were studied: local stability and asymptotic stability. Local stability is concerned with the time variations of the response of the following vehicle to a change in the motion of its direct leading vehicle. Asymptotic stability, unlikely, focuses on the manner in which perturbation on the leading vehicle is propagated down a line of traffic. The stability was examined analytically by Fourier analysis in Chandler et al.’s work [1] and Laplace transforms in Herman et al.’s study [2]. Perhaps the most important result of the stability analysis was that Chandler Model is asymptotically stable when the condition holds.

In the following five decades, Chandler Model was further developed into some more general forms [3–6] by substituting a nonlinear sensitivity for the original constant one. Meanwhile, many other kinds of car-following models were also proposed, such as Newell Model [7], Optimal Velocity Model [8], Full Velocity Model [9], Intelligent Driver Model [10], and elaborate models [11, 12]. A lot of efforts were devoted into the analytic stability studies of these car-following models [13–17]. More recently, Treiber and Kesting introduced a new classification between convective and absolute stability [18] and described a mathematical framework for linear stability analyses of all sorts of microscopic models [19]. Most of the abovementioned works are concerned with the traffic consisting of identical drivers. In other words, all driver-vehicles are governed by uniform car-following model with uniform parameters in stability analyses. However, such an ideal homogeneous traffic does not exist in the real world. Field observations reveal that there are considerable heterogeneities between the driving behaviors of individual drivers as well as the performances of different types of vehicles [20–22]. In view of this fact, Ossen et al. investigated the asymptotic stability of Chandler Model in heterogeneous traffic via numerical simulation. They found that it cannot be simply stated that a heterogeneous platoon becomes asymptotically instable when the mean values of the model parameters fall outside the stable region for homogeneous platoons [21].

Compared with homogeneous traffic, only a few of attempts [23–25] were undertaken to examine the analytic stability criterion of heterogeneous platoon in last two decades, due to the complexity of this problem. Zhang and Jarrett [23] extended the traditional stability analysis method to consider different parameter values for different drivers in the platoon, while they only gave the sufficient condition for the asymptotic stability. Holland proposed a generalized method to analyze the stability of platoon, which was also extended to nonidentical drivers [24]. However, the explicit stability criterion for heterogeneous traffic was not provided. Though some works were carried out for analyzing the asymptotic stability of heterogeneous traffic, there are still several issues needed to be deeply studied as follows:(a)Is that possible to describe the stability of heterogeneous traffic with the mean of the model parameters?(b)How does the heterogeneity influence the stability of the platoon?(c)Does the sequence of individual vehicles influence the stability of platoon?

In order to answer the above questions, this paper starts effects from a simple scenario, namely, a binary heterogeneous traffic which only consists of two types of vehicles (or drivers) under the car-following rule of Chandler Model. A simulation based method is used to investigate the asymptotic stability of the binary platoon with various combinations of car-following model parameters. The rest of the paper is organized as follows. In next section, the simulation based methodology of our work is introduced. Then, several simulation experiments are carried out and the properties of the asymptotic stability of a binary platoon are presented in Section 3. In Section 4, some possible explanations are presented for the results obtained from simulations. In Section 5, some discussions are given based on the simulation of Holland’s criterion. Finally, conclusions are summarized in Section 6.

#### 2. Methodology

Numerical simulations were performed to investigate the asymptotic stability of platoon. The platoon consists of two types of vehicles (or drivers), which are defined as type A and type B. Both of them were governed by Chandler Model. There are numerous possible combinations for type A vehicles and type B vehicles to constitute such a platoon. It hardly makes any sense to say whether a platoon is stable or not, if type A vehicles and type B vehicles distribute randomly in the platoon. In view of this fact, we mainly focus on the stability of binary platoon with periodic structures in our work, such as “ABAB…” and “AABBAABB….” Without loss of generality, we start the studies from the case “ABABAB…,” within which the proportions of both type A vehicles and type B vehicles are equal to 50%. The parameters of type A and type B are denoted as , and , , respectively.

The platoon used in the simulation consists of 20 type A vehicles and 20 type B vehicles, as shown in Figure 1. We give the first vehicle in the platoon the ID number “1” and the th vehicle number “.” Then all the vehicles with odd-numbered ID are type A vehicles, and the even-numbered vehicles are type B vehicles. At the beginning of the simulation, all vehicles move at the same velocity ; the headway of each vehicle obeys the implicit velocity-headway function of Chandler Model as follows [24]:where denotes the headway at jam state. Then we add small perturbation on the first vehicle and see how it propagates along the platoon.