Optimal velocity difference model for a car-following theory

In this Letter, we present a new optimal velocity difference model for a car-following theory based on the full velocity difference model. The linear stability condition of the new model is obtained by using the linear stability theory. The unrealistically high deceleration does not appear in OVDM. Numerical simulation of traffic dynamics shows that the new model can avoid the disadvantage of negative velocity occurred at small sensitivity coefficient λ in full velocity difference model by adjusting the coefficient of the optimal velocity difference, which shows that collision can disappear in the improved model.     

多速度差模型及稳定性分析

在全速度差(Full Velocity Difference, FVD)模型的基础上，提出了一个扩展模型——多速度差(Multiple Velocity Difference, MVD)模型. 在MVD模型中，尝试利用多辆前车信息以提高交通流的稳定性，除了考虑前车与本车的速度差外，进一步利用了多辆前车间的速度差信息. 通过线性稳定性分析，对两个模型进行比较，发现在MVD模型中，自由流稳定的敏感系数临界值变小，稳定区域有明显增加. 数值仿真结果表明，MVD模型能有效地抑制交通流堵塞. 关键词： 交通流 稳定性分析 速度差 节能     

An asymmetric full velocity difference car-following model

This paper presents a car-following model that considers the asymmetric characteristic of the velocity differences of the vehicles in a traffic stream. The problems of the prevalent general force (GF) model and the full velocity difference (FVD) model were solved. Furthermore, the optimal velocity (OV) model, the GF model, and the FVD model are proved to be the special cases of the proposed asymmetric full velocity difference (AFVD) model. The mathematical model is presented, followed by simulation analysis which demonstrates the properties of the AFVD model.     

A lattice hydrodynamical model considering turning capability

A new two-dimensional lattice hydrodynamic model considering the turning capability of cars is proposed. Based on this model, the stability condition for this new model is obtained by using linear stability analysis. Near the critical point, the modified KdV equation is deduced by using the nonlinear theory. The results of numerical simulation indicate that the critical point a c increases with the increase of the fraction p of northbound cars which continue to move along the positive y direction for c = 0.3, but decreases with the increase of p for c = 0.7. The results also indicate that the cars moving along only one direction (eastbound or northbound) are most stable.     

Traffic bottleneck characteristics caused by the reduction of lanes in an optimal velocity model

In this paper, the lane reduction bottleneck is investigated using the optimal velocity model, in which two kinds of vehicles (fast and slow) are introduced. The asymmetric lane changing rules in the slowdown section and the lane squeezing behaviors at the bottleneck are taken into account. Under the periodic boundary condition, the numerical simulations are performed. The traffic states change with increasing density. And an interesting phenomenon of ratio inversion appears. When the current saturates, the headway and velocity discontinuously vary with the position. In addition, traffic patterns and the phase transition points depend greatly on the speed limit and the length of the slowdown section.     

The KdV-Burgers equation in speed gradient viscous continuum model

Based on the microscopic two velocity difference model, a macroscopic model called speed viscous continuum model is developed to describe traffic flow. The relative velocities are added to the motion equation, which leads to viscous effects in continuum model. The viscous continuum model overcomes the backward travel problem, which exists in many higher-order continuum models. Nonlinear analysis shows that the density fluctuation in traffic flow leads to density waves. Near the onset of instability, a small disturbance could lead to solitons described by the Korteweg-de Vries-Burgers (KdV-Burgers) equation, which is seldom found in other traffic flow models, and the soliton solution is derived.     

The stabilization effect of the density difference in the modified lattice hydrodynamic model of traffic flow

A modified lattice hydrodynamic model of traffic flow is proposed by introducing the density difference between the leading and the following lattice. The stability condition of the modified model is obtained through the linear stability analysis. The results show that considering the density difference leads to the stabilization of the system. The Burgers equation and mKdV equation are derived to describe the density waves in the stable and unstable regions respectively. Numerical simulations show that considering the density difference not only could stabilize traffic flow but also makes the lattice hydrodynamic model more realistic.     

多前车速度差的车辆跟驰模型的稳定性与孤波

通过线性稳定性分析,得到了多前车速度差模型的稳定性条件, 并发现通过调节多前车信息,使交通流的稳定区域明显扩大. 通过约化摄动方法 研究了该模型的非线性动力学特性:在稳定流区域,得到了描述密度波的Burgers方程;在交 通流的不稳定区域内,在临界点附近获得了描述车头间距的修正的Korteweg-de Vries (modified Korteweg-de Vries, mKdV)方程; 在亚稳态区域内,在中性稳定曲线附近获得了描述车头间距 的KdV方程. Burgers的孤波解、mKdV方程的扭结-反扭结波解及KdV方程的 孤波解描述了交通流堵塞现象.     

The Korteweg-de Vires equation for the bidirectional pedestrian flow model considering the next-nearest-neighbor effect

This paper focuses on a two-dimensional bidirectional pedestrian flow model which involves the next-nearest-neighbor effect. The stability condition and the Korteweg-de Vries （KdV） equation are derived to describe the density wave of pedestrian congestion by linear stability and nonlinear analysis. Through theoretical analysis, the soliton solution is obtained.     

A new car-following model considering velocity anticipation

The full velocity difference model proposed by Jiang et al. [2001 Phys. Rev. E 64 017101] has been improved by introducing velocity anticipation. Velocity anticipation means the follower estimates the future velocity of the leader. The stability condition of the new model is obtained by using the linear stability theory. Theoretical results show that the stability region increases when we increase the anticipation time interval. The mKdV equation is derived to describe the kink--antikink soliton wave and obtain the coexisting stability line. The delay time of car motion and kinematic wave speed at jam density are obtained in this model. Numerical simulations exhibit that when we increase the anticipation time interval enough, the new model could avoid accidents under urgent braking cases. Also, the traffic jam could be suppressed by considering the anticipation velocity. All results demonstrate that this model is an improvement on the full velocity difference model.     

The density wave in a new anisotropic continuum model

In this paper the new continuum traffic flow model proposed by Jiang et al is developed based on an improved car-following model, in which the speed gradient term replaces the density gradient term in the equation of motion. It overcomes the wrong-way travel which exists in many high-order continuum models. Based on the continuum version of car-following model, the condition for stable traffic flow is derived. Nonlinear analysis shows that the density fluctuation in traffic flow induces a variety of density waves. Near the onset of instability, a small disturbance could lead to solitons determined by the Korteweg-de-Vries （KdV） equation, and the soliton solution is derived.     

The density wave in a new anisotropic continuum model

In this paper the new continuum traffic flow model proposed by Jiang {\it et al is developed based on an improved car-following model, in which the speed gradient term replaces the density gradient term in the equation of motion. It overcomes the wrong-way travel which exists in many high-order continuum models. Based on the continuum version of car-following model, the condition for stable traffic flow is derived. Nonlinear analysis shows that the density fluctuation in traffic flow induces a variety of density waves. Near the onset of instability, a small disturbance could lead to solitons determined by the Korteweg--de-Vries (KdV) equation, and the soliton solution is derived.     

The theoretical analysis of the lattice hydrodynamic models for traffic flow theory

The lattice hydrodynamic model is not only a simplified version of the macroscopic hydrodynamic model, but also connected with the microscopic car following model closely. The modified Korteweg-de Vries (mKdV) equation related to the density wave in a congested traffic region has been derived near the critical point since Nagatani first proposed it. But the Korteweg-de Vries (KdV) equation near the neutral stability line has not been studied, which has been investigated in detail for the car following model. We devote ourselves to obtaining the KdV equation from the original lattice hydrodynamic models and the KdV soliton solution to describe the traffic jam. Especially, we obtain the general soliton solution of the KdV equation and the mKdV equation. We review several lattice hydrodynamic models, which were proposed recently. We compare the modified models and carry out some analysis. Numerical simulations are conducted to demonstrate the nonlinear analysis results.     

Nonlinear analysis of traffic jams in an anisotropic continuum model

This paper presents our study of the nonlinear stability of a new anisotropic continuum traffic flow model in which the dimensionless parameter or anisotropic factor controls the non-isotropic character and diffusive influence.In order to establish traffic flow stability criterion or to know the critical parameters that lead,on one hand,to a stable response to perturbations or disturbances or,on the other hand,to an unstable response and therefore to a possible congestion,a nonlinear stability criterion is derived by using a wavefront expansion technique.The stability criterion is illustrated by numerical results using the finite difference method for two different values of anisotropic parameter.It is also been observed that the newly derived stability results are consistent with previously reported results obtained using approximate linearisation methods.Moreover,the stability criterion derived in this paper can provide more refined information from the perspective of the capability to reproduce nonlinear traffic flow behaviors observed in real traffic than previously established methodologies.     

A new car-following model with the consideration of anticipation optimal velocity

In this paper, a new anticipation optimal velocity model (AOVM) is proposed by considering anticipation effect on the basis of the full velocity difference model (FVDM) for car-following theory on single lane. The linear stability condition is derived from linear stability analysis. Starting and braking process is investigated for the car motion under a traffic signal, which shows that the results accord with empirical traffic values. Especially AOVM can avoid the disadvantage of the unrealistically high deceleration appearing in FVDM. Furthermore, numerical simulation shows that AOVM might avoid the disadvantage of negative velocity and headway that occur at small sensitivity coefficients in the FVDM since the anticipation effect is taken into account in AOVM, which means that collision disappears with the consideration of an appropriate anticipation parameter.     

Nonlinear analysis of traffic continuum jams in an anisotropic model

This paper presents our study of the nonlinear stability of a new anisotropic continuum traffic flow model in which the dimensionless parameter or anisotropic factor controls the non-isotropic character and diffusive influence. In order to establish traff~c flow stability criterion or to know the critical parameters that lead, on one hand, to a stable response to perturbations or disturbances or, on the other hand, to an unstable response and therefore to a possible congestion, a nonlinear stability criterion is derived by using a wavefront expansion technique. The stability criterion is illustrated by numerical results using the finite difference method for two different values of anisotropic parameter. It is also been observed that the newly derived stability results are consistent with previously reported results obtained using approximate linearisation methods. Moreover, the stability criterion derived in this paper can provide more refined information from the perspective of the capability to reproduce nonlinear traffic flow behaviors observed in real traffic than previously established methodologies.     

The time-dependent Ginzburg-Landau equation for the two-velocity difference model

A thermodynamic theory is formulated to describe the phase transition and critical phenomenon in traffic flow.Based on the two-velocity difference model,the time-dependent Ginzburg-Landau (TDGL) equation under certain condition is derived to describe the traffic flow near the critical point through the nonlinear analytical method.The corresponding two solutions,the uniform and the kink solutions,are given.The coexisting curve,spinodal line and critical point are obtained by the first and second derivatives of the thermodynamic potential.The modified Korteweg de Vries (mKdV) equation around the critical point is derived by using the reductive perturbation method and its kink-antikink solution is also obtained.The relation between the TDGL equation and the mKdV equation is shown.The simulation result is consistent with the nonlinear analytical result.     

Time-dependent Ginzburg-Landau equation in a car-following model considering the driver’s physical delay

In this paper, an extended car-following model considering the delay of the driver’s response in sensing headway is proposed to describe the traffic jam. It is shown that the stability region decreases when the driver’s physical delay in sensing headway increases. The phase transition among the freely moving phase, the coexisting phase, and the uniformly congested phase occurs below the critical point. By applying the reductive perturbation method, we get the time-dependent Ginzburg-Landau (TDGL) equation from the car-following model to describe the transition and critical phenomenon in traffic flow. We show the connection between the TDGL equation and the mKdV equation describing the traffic jam.     

随机计及相对速度的交通流跟驰模型

从研究微观个体车辆行为出发，考虑车辆加速过程的不确定性，提出了随机计及相对速度的 交通流跟驰模型(SR-OV模型).对随机相对速度的跟驰模型的动力学方程进行稳定性分析，得 到与Bando跟驰模型不同的稳定性判据，其稳定性优于Bando模型.运用摄动理论分析交通过 程中密度波的变化，结果表明，在发生交通阻塞相变时，交通密度波以mKdV方程描述的扭结 -反扭结波演化.对随机相对速度跟驰模型进行数值模拟和分析，结果发现车流速度的变化小 于Bando模型的速度变化，而且与随机概率有关，当随机考虑相对速度的概率增大时，初始 的小扰动不会放大对车流产生影响，甚至长时间就消失，这与Bando模型完全不同.数值模拟 所得到的相图与解析解相符合,而且交通流稳定区域大于Bando模型.从车间距-速度演化图上 ，随着随机概率的增大，SR-OV模型在初始时存在的滞后现象，随着时间的增长，趋于稳定 状态后，滞后曲线收敛于一小区域，滞后效应被削弱.这完全不同于Bando模型，在Bando模 型中，滞后曲线由一点向外扩散，滞后曲线区域越来越大，车流趋于不稳定状态. 关键词： 交通流 跟驰模型 稳定性判据 相对速度     

Intermittent unstable structures induced by incessant constant disturbances in the full velocity difference car-following model

In this paper, we study traffic flow patterns induced by incessant constant disturbances in the full velocity difference car-following model. It is found that intermittent unstable structures may occur in the convectively unstable traffic flow under certain situations. A phenomenological explanation of the phenomenon is given: the incessant constant disturbances mainly function to maintain the stationary oscillating structure while the stationary oscillating structure is not always stable, the intermittent instability of it leads to the intermittent unstable structures. The similarity of the stationary oscillating structure to the transition layer in the local cluster effect is pointed out. The dependence of the phenomenon on the headway of the initially uniform traffic, the safety distance x_c, the sensitivity parameters κ and λ, and the noise term is also investigated.