Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 960630, 6 pages

http://dx.doi.org/10.1155/2015/960630

## A New Macro Model Considering the Average Speed of Preceding Vehicles Group in CPS Environment

^{1}Key Laboratory of Dependable Service Computing in Cyber Physical Society (Chongqing University), Ministry of Education, Chongqing 400030, China^{2}College of Automation, Chongqing University, Chongqing 400030, China^{3}School of Computer Science, Guangxi University of Science and Technology, Liuzhou, Guangxi 545006, China

Received 14 December 2014; Accepted 6 February 2015

Academic Editor: Maria Gandarias

Copyright © 2015 Yi-rong Kang 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

A new car following model considering the effect of average speed information of preceding vehicles group in real traffic is presented. Based on the new car following model, a new macro model for traffic flow is proposed employing the relationship between the micro and macro variables. The linear stability condition of the macro model is obtained by using the linear stability theory. The numerical tests show that the new model can not only simulate the dynamic process of shock, rarefaction wave, and small perturbation, but also can further stabilize the traffic flow.

#### 1. Introduction

Traffic jams has attracted much attention of physicists and engineers. Various traffic models [1–6], including microscopic models and macroscopic models from different scales, have been proposed to describe the complex traffic phenomena. The physical mechanisms of traffic phenomena such as the stop-and-go traffic, the ghost jams, local clusters, and others are revealed. However, these models are unsuited to describe the complex phenomena resulting from the effects of average speed of preceding vehicles group (called local average speed, LAS), since they do not involve the factors. In reality, as we know, along with the remarkable development of the wireless communication technology, the transportation cyber physical systems (T-CPS) are now becoming widely available. T-CPS enable exchanging the velocity information among vehicles and provide average speed about preceding cars on road. Depending on this on-line traffic data, drivers can adjust the speed to the optimal state in advance.

In fact, LAS is described as the key parameter to reflect the front traffic situation, as well as the multivehicle interactions effect in multiple vehicles following situation. So the LAS will lead to some complex phenomena. However, until now few scholars studied the effect of LAS on car following process.

In view of the reasons given above, we in this paper present a new car following model with considering the average speed of preceding cars. And a macro model for traffic flow is constructed by the relevancy relation of micro and macro variables. The analytical analysis and numerical simulation results indicate that the new consideration can stabilize the traffic flow.

#### 2. The Car Following Model

The car following model focus on describing the motion of vehicles following each other on a single lane. In general, the motion equation is reduced as [3] where and are, respectively, the velocity and position of the th car. denotes the relative velocity. represents the space headway between two successive vehicles. The stimulus function depends on the factors velocity , relative velocity , and headway . In this general framework, many classical car following models [4–6] have been developed and several important achievements have been attained.

Recently, in order to improve the stability of traffic system, scholars developed some extended car following models by incorporating multivehicle consideration in real traffic. Some were extended by introducing multiple information of headway [7, 8] or relative velocity [9, 10], whereas others considered both factors at the same time [11–13]. In 2005, considering the multiple headway information, Ge et al. [7] proposed the cooperative driving (CD) model withIn 2006, Wang et al. [9] developed a multiple velocity difference (MVD) model; that is,In 2010, Peng and Sun [11] presented the multiple car following (MCF) model with simultaneously taking the multiple velocity difference and multiple headway information into account; that is,Stability analysis in the aforementioned models all turn out that the stability of traffic flow is greatly improved by incorporating multivehicle consideration.

The aforementioned models can describe some complex traffic phenomena (e.g., congestion, instability, and stop-and-go waves in traffic flow). However, these models are unsuited to study the average speed effect of preceding vehicles group because they do not consider this factor at all. In fact, the average speed of preceding vehicles group reflects the whole front traffic situation on segment, that is, whether the traffic flow of preceding cars on segment will cluster, dissipate, or simply maintain a constant velocity. By applying T-CPS, the follower can sense the front traffic situation through the indicator of average speed of preceding vehicles group; then the following one can make a decision and adjust its vehicle in advance to adapt the state change of preceding cars based on sensing information. In view of the above reason, we developed a new car following model that considers the average speed information of preceding vehicles group. The model’s dynamics equation is as follows:where the term is the average speed of preceding vehicles group, which reflect the front traffic situation consisting of car and its leading cars at time . denotes the number of the vehicles ahead considered. is the responding factor to the difference between the velocity and average speed term . The idea of the extended model is that the acceleration of the th car is determined not only by the velocity and the headway , but also by the velocity difference between velocity and front local average speed at time . When , the new model is reduced to the FVD model [6].

#### 3. The Macro Model

In order to construct the corresponding macro model, we should transform the micro variables of individual vehicles in (5) into the macro ones with using the method by Liu et al. [14]; that is,where the , are the speed and density at the point , respectively; denotes the distance between the following and leading vehicles; is the equilibrium speed; is the relaxation time and is the time needed for the backward propagated disturbance to travel a distance of . Thus, (5) can be rewritten as follows:Expanding (7), neglecting the nonlinear terms, and combining with the conservation equation of traffic flow, we can obtain a new macro model with considering the local average speed information of preceding vehicles group:where denotes the propagating speed of the small perturbation. When , the new model reduced to the speed gradient (SG) model [15]. Comparing our model with SG model, we can see that our model differs from the SG model in the motion equation, in which the number of preceding cars is involved.

In order to analyze the characteristic speed problem for the new model, we rewrite (8) in a vector forms as follows:where , , and .

The eigenvalues, , of the matrix are found by settingwhere is identity matrix. From (10), one can obtainSince , the characteristic speeds in (11) are no greater than the macroscopic flow velocity . This result ensures that our model conforms to the anisotropy property which is the fundamental principle in single lane traffic flow [15, 16].

#### 4. Linear Stability Analysis

Assuming that and are the steady-state solutions for (8) and introducing the small perturbations and into the steady-state solutions, one obtains the perturbed solution:Substituting the perturbed solutions (12) into (8) and then making the Taylor expansion at and neglecting higher-order terms of and , we obtain the following expression:where , , , , and . Eliminating from (13) results in the equationwhere , and . According to the traditional way of linear stability analysis, substituting into (14), one can obtainFor to be the nontrivial solution of (15), we must haveIt is obvious that the solution is stable if and only if the imaginary part of both of the roots is nonpositive. According to the method using in [15], the requirement for this is When condition (17) cannot be met, unstable traffic will appear and lead to many complex traffic phenomena like stop-go waves or cluster jams.

#### 5. Numerical Simulation

For numerical computations, we adopt the finite difference method to discretize (10), and the corresponding difference equations are as follows:If the traffic is heavy, ,Otherwisewhere , , , and are the space index, the time index, the spatial step, and the time step, respectively.

First, we use numerical tests to investigate whether our model can capture the shock and rarefaction wave of the real traffic flow. The initial conditions are as follows [15, 17]:where and are the upstream and downstream densities, respectively, in the case of shock wave; and are the upstream and downstream densities, respectively, in the case of rarefaction wave. The initial speeds are set below:Comparing with the results of [15], we adopt the following equilibrium speed here [18]:where is the free speed and the kinematic wave speed at jam density . Here, the free boundary conditions are adopted, that is, and on both sides. And 20 km test road section is divided into 100 meshes equally. Other inputting parameters are as follows: , , , , , and .

The computational results for the new model () under the initial conditions of (20a) and (20b) are shown in Figures 1(a) and 1(b), respectively. Equation (20a) simulates the free-flow traffic meets a queue of nearly stopping vehicles where shock waves will appear while (20b) describes the rarefaction wave corresponding to that in the process of dissolution. Figures 1(a) and 1(b) show the density evolution in shock and rarefaction waves, respectively; the results are similar to those reported in [15, 17], so our model can provide correct predictions in the formation and propagation of the waves. In addition, owing to the local average speed of preceding cars considered in our model, the shock front between the congested and the free-flow traffic is smoother than that in [15], which means that the new consideration affects the smoothness of shock front positively.