A new car-following model with consideration of the prevision driving behavior

doi:10.1016/j.cnsns.2014.03.012

Communications in Nonlinear Science and Numerical Simulation

Volume 19, Issue 10, October 2014, Pages 3820-3826

https://doi.org/10.1016/j.cnsns.2014.03.012 Get rights and content

Highlights

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The new car-following model considering the prevision driving behavior is proposed.
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The stable condition of the new model is obtained.
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The mKdV equation is deduced to describe the jamming transition.

Abstract

In the paper, a new car-following model is presented with the consideration of the prevision driving behavior on a single-lane road. The model’s linear stability condition is obtained by applying the linear stability theory. And through nonlinear analysis, a modified Korteweg–de Vries (mKdV) equation is derived to describe the propagating behavior of traffic density wave near the critical point. Numerical simulation shows that the new model can improve the stability of traffic flow by adjusting the driver’s prevision intensity parameter, which is consistent with the theoretical analysis.

Introduction

Traffic jam has been a serious problem in modern city traffic and many traffic models [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], such as car-following models, cellular automaton models, gas kinetic models, and hydrodynamic models, have been developed to investigate the properties of traffic jams. The optimal velocity (OV) model proposed by Bando et al. [11], one of the favorable car-following traffic models on studying traffic flow, has successfully described the formation of traffic jams and revealed the transition mechanism in a simple way. Subsequently, inspired by the OV model, some new car -following models were successively put forward to more realistically describe the traffic nature. Some of them were extended by introducing multiple headway or relative velocity information of car [11], [12], [13], [14], and others considered the two factors at the same time [15], [16], [17], [18], [19], [20].

These car-following models mentioned above can reproduce many complex actual traffic phenomena, but they cannot be employed to study the influence of the prevision driving behavior since they did not consider the factor. In the transportation cyber physical systems (T-CPS) [21], a leading car can send the future speed control command of the leading car to the follower before the leading car’s the current speed changes, and thus the following one can obtain in advance the leading car’s future running state information to control his current acceleration for achieving the optimal state. However, few scholars have studied the prevision driving behavior in car-following model [22].

In this paper, a new car following model is proposed by taking the prevision driving behavior on a single-lane road into account. In the following section, the new car following model is introduced. In Section 3, linear stability analysis is conducted. In Section 4, nonlinear analysis is done. In Section 5, numerical simulation is carried out to validate the analytic results. Conclusions are given in Section 6.

Section snippets

Models

In 1995, Bando et al. [11] proposed the OV model to describe the car-following behavior on a single lane highway. The motion equation is given as follows: $\frac{{dv}_{j} (t)}{dt} = a [V (Δ x_{j} (t)) - v_{j} (t)],$ where $x_{j} (t)$ is the position of car $j$ at time t, $Δ x_{j} (t) = x_{j + 1} - x_{j}$ represents the headway of two successive vehicles, $a$ is the sensitivity of a driver, and $V (Δ x_{j} (t))$ is the optimal velocity function. The comparison with empirical data shows that the OV model appears too high acceleration and unrealistic deceleration.

Linear stability analysis

In order to investigate the impact of the prevision driving behavior on the traffic flow, the linear stability analysis can be conducted for PD-CF model. The vehicles move with the uniform headway $b$ and the optimal velocity $V (b) .$ Therefore, the steady-state solution is given as $x_{j}^{0} (t) = bj + V (b) t, b = L / N$ where $L$ is the road length and $N$ is the car number. Suppose $y_{j} (t)$ is a small deviation from the steady state: $x_{j} (t) = x_{j}^{0} (t) + y_{j} (t) .$

Substituting Eq. (8) into Eq. (6) and linearizing the resulting

Nonlinear analysis and mKdV equation

In order to investigate the prevision driving effect of the value of k on the traffic flow, the nonlinear analysis is carried out to study the slowly varying behavior near the critical point $(h_{c}, a_{c})$ . For extracting slow scales with the space variable j and the time variable t, accordingly, we define slow variable $X$ and $T$ as follows: $X = ε (j + bt) and T = ε^{3} t, 0 < ε ⩽ 1,$ where b is a constant determined later. Letting $Δ x_{j} (t) = h_{c} + ε R (X, T) .$

Substituting Eqs. (16), (17) into Eq. (6) and making the Taylor expansion to

Numerical simulation

Computer simulation is carried out to check the validity of our theoretical results above. Under the periodic boundary condition, the following initial conditions are chosen as follows: $\begin{matrix} Δ x_{j} (0) = Δ x_{j} (1) = 4.0, for j \neq 50, 51, Δ x_{j} (1) = 4.0 + 0.1, for j = 50, \\ Δ x_{j} (1) = 4.0 - 0.1, for j = 51 \end{matrix}$

The total number of cars is N = 100 and the sensitivity a = 1.4.

Fig. 2 shows the space–time evolution of the headway after $t = 10^{4}$ time steps under the different parameter k. The patterns (a)–(d) in Fig. 2 exhibit the time evolution of the

Summary

Based on the property of the prevision driving behavior on a single-lane road, a new car following model is developed to suppress traffic jams. The extended model has been analyzed by using the linear stability theory and the nonlinear analysis. The stability condition of the extended model is obtained and the results show that the stability of traffic flow is improved by taking driver’s prevision effect into account. Moreover, The kink–antikink soliton solution of mKdV equation near the

Acknowledgements

This work is supported by the Chinese Academy of Engineering Major Consulting Project (No. 2012-ZX-22), the Natural Science Foundation of CSTC of China (No. 2012jjB40002), the Engineering Center Research Program of CSTC of China (No. 2011pt-gc30005) and the Key Technology R&D Project of CSTC of China (No. 2012gg-yyjsB30001).

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A new car-following model with consideration of the prevision driving behavior

Highlights

Abstract

Introduction

Section snippets

Models

Linear stability analysis

Nonlinear analysis and mKdV equation

Numerical simulation

Summary

Acknowledgements

Commun Nonlinear Sci Numer Simul

Commun Nonlinear Sci Numer Simul

Physica A

Physica A

Commun Nonlinear Sci Numer Simul

Commun Nonlinear Sci Numer Simul

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Commun Nonlinear Sci Numer Simul

Commun Nonlinear Sci Numer Simul

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Phys Lett A

Physica A