# Nonlinear Analysis of the Car-Following Model considering Delay under the V2X Environment

Authors: Zhipeng Xu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper proposes a model that analyzes the delay effect in V2X communication. V2X technology in the intelligent transportation system can realize the exchange of information between vehicles, people, and the environment. Still, more V2X links occupy many channel resources, and the occupation of channel resources leads to longer communication delays. Therefore, based on the optimal velocity of the car-following model, we model the channel delay in the V2X environment. The stability analysis results show that delay increases lead to system instability and traffic congestion. However, if more links are established, that is, the number of vehicles in the vehicle group ahead increases, traffic jams are less likely to occur. The numerical simulation results show that the longer the delay, the more serious the traffic congestion. In addition, we find an apparent phase transition region in the numerical simulation results, which is consistent with the theoretical analysis results. Therefore, in the intelligent transportation system, the delay caused by the complex electromagnetic environment is a factor that cannot be ignored, and thus the allocation of spectrum resources needs to be reasonably arranged. Otherwise, the V2X technology leads to traffic congestion.


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## Body

## 1. Introduction

With the development of communication technology, more research combining communication networks and traffic flow has been conducted in recent years. The intelligent transportation system technology represented by V2X is the best option to alleviate the increasing traffic congestion and accidents. In particular, the low-latency characteristics of advanced communication technologies in the technical part of V2X [1]—for example, 5G technology—provide the connection between each vehicle. Additionally, V2X technology can provide the link between environments, such as traffic signal lights, real-time traffic information, and other factors. Therefore, combined with autonomous driving technology, V2X can effectively improve the efficiency of road traffic.Many scholars [2–8] have investigated various factors and behaviors in intelligent transportation systems, such as delayed feedback [9], two-lane information interaction [10], and other behaviors [11–15]. For example, Ge et al. [11] analyzed the influence of the rear-view mirror effect on driving behavior. The result shows that the cooperative driving behavior of drivers observing vehicles following their vehicle could improve traffic flow stability. Kuang et al. [13] proposed the model considering the influence of the local density in the front view. Analytical and simulation results show that the driving strategy considering multiple vehicles ahead could significantly reduce traffic congestion on the road. In addition, Zhu and Zhang [14] found that referring to the average speed of the vehicle group ahead also had a positive effect on the traffic system. Then, Kuang et al. [15] extended the conclusion by combining the average density and speed. Ngoduy [16] and Luo et al. [17] investigated the effect of the mixed flow with the impact of the penetration of connected and automated vehicles (CAVs) on the traffic flow stability. Jia et al. [18] developed the model for a heterogeneous platoon considering the communication between manual manipulation vehicles and autonomous vehicles. Also, theoretical and numerical analysis showed that the multiclass microscopic model [18] for heterogeneous platoon is linearly stable. Kesting and Treiber [19] analyzed three characteristic time impacts that may affect traffic flow stability at the micro-level: reaction time, update time, and adaptation time. The reaction time and the update time would affect the traffic system via “short-wavelength mechanisms” as well as transient responses. In contrast, the velocity adaptation time introduces instabilities via collective “long-wavelength mechanisms” like a steady-state response. Yao et al. [20] found that delay time significantly impacts intelligent vehicles because vehicles in a platoon run with such a small gap that even a tiny delay can induce platoon instability. Additionally, Han et al. [21] conducted corresponding research from macro and micro-perspectives. Hence, theoretical analysis has shown that introducing additional information for the vehicle group ahead can significantly improve traffic flow stability.Many theoretical studies have shown that V2X technology can improve the traffic efficiency of the transportation system, but note that more V2X links may have adverse effects. In particular, many V2X links occupy many electromagnetic spectrum resources, while cell networks have used many frequency bands. Limited wireless channel capacity and resource conflicts also introduce many unpredictable delays into V2X communication. Therefore, in the V2X network, the delay time is a factor that cannot be ignored. Hence, we conduct corresponding research on the impact of delay on traffic flow stability in V2X technology.


## 2. Model

Bando et al. [22] proposed an ordinary differential equation to study traffic congestion behavior based on the assumption that the driver makes a corresponding speed decision based on the distance from the vehicle ahead. In other words, under the maximum speed limit, the driver accelerates when the distance from the car ahead is enough; otherwise, they decrease the speed conversely. The behavior of the car following can be written as(1)dvntdt=aVΔxnt−vnt,where Δxnt represents the distance at time t between the nth vehicle and the n−1th vehicle and vn represents the nth car’s speed correspondingly. a means the sensitivity, reflecting the driver’s reaction time. The expected speed VΔxnt related to the gap between two cars can be written as(2)VΔxnt=vmax2tanhΔxnt−hc+tanhhc,where hc and vmax correspond to the maximum safe distance and speed. The velocity VΔxnt rises with the increase of space Δxnt and approaches the limited speed vmax, also called the optimal velocity.The optimal velocity car-following model can reflect the nonlinear behavior in the traffic flow system and describe the evolution of traffic congestion. Therefore, based on the optimal velocity model, many scholars have proposed that the model reflects the driving behavior in the intelligent transportation system. Kuang et al. [13] proposed a car-following model considering the density of the vehicle group ahead to discuss the influence of the average density on the system, and the model can be written as(3)dvntdt=aVΔxnt,Δxnt¯−vnt+λΔvnt.Compared with the original Bando’s model (equation (2)), the average distance of the vehicle ahead Δxnt¯ is introduced here as(4)Δxnt¯=1m∑l=1mΔxn+lt.Correspondingly, the optimal velocity can be written as(5)VΔxnt,Δxnt¯=V1−pΔxnt+pΔxnt¯,where m represents the number of vehicles in the vehicle group ahead and the coefficient p represents the intensity factor of the average distance of the vehicle group ahead. Generally, the coefficient p should not be too large; otherwise, the vehicle collides with the vehicle ahead. Kuang et al. [13] took p∈0,0.3 since the influence of current vehicle’s headway is more important than the average headway of preceding vehicles group. λ is the responding factor of the velocity difference to avoid unrealistically high acceleration [23].In the network of vehicles, the allocation of communication channel resources leads to delay, so we extend the model of Kuang by introducing the delay time and propose the following model:(6)dvntdt=aVΔxnt,Δxnt−τ¯−vnt+λΔvnt.The definition of variables is consistent with Kuang et al.’s [13] model (see equation (3)), and the time delay term τ is introduced into the average distance between the preceding vehicle groups, reflecting the acquisition and processing delay of the preceding vehicle group. In addition, the ahead vehicle’s information is almost instantaneous, so we can ignore the delay. The gap and optimal velocity functions are(7)Δxnt−τ¯=1m∑l=1mΔxn+lt−τ,where Δn+kt−τ=xn+k+1t−τ−xn+kt−τ. Here, Δxnt−τ¯ represents the average distance between preceding vehicles n+1,n+2,…,n+m at time t−τ. Next, the optimal velocity VΔxnt,Δxnt−τ¯ as well as expected velocity can be obtained as(8)VΔxnt,Δxnt−τ¯=vmax2tanh1−pΔxnt+pΔxnt−τ¯−hc+tanhhc.Also,hc and vmax represent the maximum safe distance and speed. Similarly, we also take p∈0,0.3 to ensure that the expected velocity considers the average density of the preceding vehicle group and ignores the essential vehicle ahead distance.


## 3. Linear Stability Analysis

To discuss the system’s stability, that is, the critical condition when uniform flow turns into congestion, we introduce a perturbation into the system to analyze its system robustness. For a system withN vehicles and a road of length L, there must be a stationary solution (the uniform flow) to the equation. Under this condition, all of the vehicles move with the same space bb=L/N and velocity Vb,b. Hence, each vehicle’s position xn0t for steady-state solution at time t can be expressed in the following form:(9)xn0t=bn+Vb,bt.If we add a perturbation,ynt, that is, xnt=xn0t+ynt, and substitute it into equation (6), then we can obtain(10)d2yndt2=a1−pΔynt+pm∑l=1mΔyn+lt−τV′b−dyntdt+λdΔyntdt.Here, the definitions ofV′b and Δynt can be written as V′b=dVΔxn/dΔxn|Δxn=b and Δynt=yn+1t−ynt.Next, letyn=expikn+zt; that is, express the perturbation in the form of dispersion and substitute this equation into formula (10). Then, we can get(11)z2−zλeik−1−a−a1−peik−1V′b+pmeikm+1−zτ−eik−zτV′b=0.The parameterz can be expanded as(12)z=ikz1+ikz2+⋯.By substitutingz into equation (11), we can obtain first and second-order expansions of ik as follows:(13)z1=V′bz2=121+p1+m−2z1τV′b+1az1λ−z12.According to the analysis of stability theory, whenz2<0, the disturbance decays exponentially and tends to be stable. Corresponding to the traffic flow, the speed fluctuation of individual vehicles does not cause traffic congestion. Conversely, when z2>0, the velocity fluctuations of cars can easily cause congestion in the entire system. Therefore, letting z2=0, we can get the critical condition of system stability, that is, the neutral stability condition:(14)a=2V′b−λ1+p+mp−2pτV′b.To keep the system stable, the sensitivity coefficient needs to satisfy the following condition:(15)a>2V′b−λ1+p+mp−2pτV′b.Whenτ=0, the stability results are consistent with Kuang et al.’s [13] results; also, in the condition of p=0,τ=0, this model reduces into Jiang et al.’s [23] full-velocity difference model.Figure1 shows the parameter space phase diagram of vehicle distance Δx and the sensitivity coefficient a, where τ=0.3,p=0.3, and λ=0.1. The region in the figure can be divided into three areas: the stable area, the metastable area, and the unstable area. The system has strong robustness when the parameter space Δx,a is located in the stable region. At higher sensitivities or lower vehicle densities, random disturbances between individual vehicle speeds and vehicle spacing eventually dissipate, and the traffic flow can remain stable. Compared with Kuang’s results, we can find that the introduction of delay τ increases the area of the unstable region, which is consistent with the fact that delay leads to system instability. Importantly, in V2X systems, especially when the electromagnetic environment becomes more complex, the delay is a nonnegligible factor. In Ge et al.’s [24] results, the number of vehicles in the current vehicle group m=3 is the optimal state for cooperative driving, so we only analyze it with m=3 in Figure 1.Figure 1
The phase space of headway density.Figure2 shows the stability curves under different m values, and we notice that as the number of vehicles in the vehicle group ahead m increases, the instability area shrinks. This result shows that the stability of the traffic flow system can be promoted if more vehicles ahead are considered. However, under the same m, as the delay time τ increases, the unstable region also increases, but the gap becomes smaller as m increases. The analysis also reminds us that when τ increases in the intelligent transportation system, we can introduce more reference vehicle information in exchange for improving system stability.Figure 2
The phase space of headway density with differentm and τ values.


## 4. Nonlinear Analysis

The theoretical analysis in most literature shows that the propagation process of density waves represents the traffic congestion behavior so that these equations can be expanded into the corresponding Burgers, KdV, TDGL, and mKdV equations under specific conditions [25–31]. Here, we only discuss the case of the mKdV equation. We rewrite equation (6) in the form of d2xn+1t/dt2−d2xnt/dt2 to get the following equation:(16)d2Δxntdt2=aV1−pΔxn+1t+pm∑l=1mΔxn+l+1t−τ−V1−pΔxnt+pm∑l=1mΔxn+lt−τ−dΔxntdt+λdΔxn+1tdt−dΔxntdt.Then, we introduce the slow variablesX and T as(17)X=εj+bt,T=ε3t,0<ε≤1.Among them,b is the parameter related to the critical point. Next, we introduce the function RX,T and redefine the vehicle distance Δxnt as follows:(18)Δxnt=hc+εRX,T.Substituting equations (17) and (18) into equation (16) and then performing the reductive perturbation method [11, 13, 15, 24, 25], we get(19)dΔxntdt=bε2∂XR+ε4∂TR,(20)d2Δxntdt2=b2ε3∂X2R+2bε5∂X∂TR,(21)dΔxn+1tdt=bε2∂XR+bε3∂X2R+ε412b∂X3R+∂TR+ε516b∂X4R+∂X∂TR,(22)Δxnt−τ=hc+εR−bε2τ∂XR+12b2ε3τ2∂X2R+ε4−τ∂TR−16b3τ3∂X3R+ε5bτ2∂X∂TR+124b4τ4∂X4R,(23)1m∑l=1mΔxn+lt−τ=hc+εR+ε2m−bmτ+∑l=1ml∂XR+ε32mbmτbτ−1−m+∑l=1ml2∂X2R+ε46m−12bmτ2b2τ2−3bm+1τ+2m2+3m+1+∑l=1ml3∂X3R−ε4τ∂TR−ε512τ−2bτ+m+1∂X∂TR+ε524m−bmτ−bτ+m+1−bmτ+bτbτ−1+m2+m+∑l=1ml4∂X4R,(24)1m∑l=1mΔxn+l+1t−τ=hc+εR+ε2m−bmτ+∑l=1ml+1∂XR+ε32mbmτbτ−3−m+∑l=1ml+12∂X2R+ε46m−12bmτm9−3bτ+bτ2bτ−9+2m2+13+∑l=1ml+13∂X3R−ε4τ∂TR−ε512τ−2bτ+m+3∂X∂TR+ε524m−bmτ−bτ+m+3m3−bτ+bτbτ−3+m2+4+∑l=1ml+14∂X4R,(25)V1−pΔxnt+pm∑l=1mΔxn+lt−τ=Vhc+1−pΔxnt+pm∑l=1mΔxn+lt−τ−hcV′+161−pΔxnt+pm∑l=1mΔxn+lt−τ−hc3V‴,(26)V1−pΔxn+1t+pm∑l=1mΔxn+l+1t−τ=Vhc+1−pΔxn+1+pm∑l=1mΔxn+l+1t−τ−hcV′+161−pΔxn+1+pm∑l=1mΔxn+l+1t−τ−hc3V‴.Here, the definition ofV′ and V‴ can be written as(27)V′=V′hc=dV1−pΔxn+pΔxn¯dΔxnΔxn=hc,Δxn¯=hc,V‴=V‴hc=d3V1−pΔxn+pΔxn¯d3Δxn|Δxn=hc,Δxn¯=hc.Next, we substitute equations (20)–(27) into equation (16) and get(28)ε2b−V′∂XR+ε3bb−λa−mp+p+1−2bpτV′2∂X2R+ε4∂TR−m2p+3mp+2p−3bpm2+mτ+1V′6+bλ2a∂X3R−V‴6∂XR3+ε52b−λa+2pτV′∂X∂TR−mp+p+1V‴12∂X2R3+124a−4bλ−a1+pm24−4bτ+6mbτ−12+m3+3−4bτbτbτ−3+3∂X4R=0.Letac=a1+ε2 and b=V′; then, the terms in equation (28) of ε2 and ε3 can be eliminated, and we finally get the equation for the fourth-order and fifth-order terms of ε as(29)ε4∂TR−g1∂X3R+g2∂XR3+ε5g3∂X2R+g4∂X4R+g5∂X2R3=0.Here, each coefficient can be written as(30)g1=16+p1+m2+m6+λ2ac+pτV′τV′−2−m2V′,g2=−16V‴,g3=1+p+mp−2pτV′2V′,g4=−16a2λ−apτ+2V′3apτV′τV′−m−2+am+1m+2p+a+3λ−124aa+4λ+apm+1mm+3+3+2τV′−2mm+3−6+τV′−2τV′+3m+6,g5=−acp+mp+1−41+apτV′+2λ12acV‴.To obtain the standard mKdV equation, including the high-orderε, we can introduce the transformation as follows:(31)T=T′g1,RX,T=g1g2R′X,T′.The obtained mKdV equation includingOε can be written as(32)∂T′R′−∂X3R′+∂XR′3+εMR′=0.The term ofMR′ is(33)MR′=1g1g3∂X2R′+g4∂X4R′+g1g5g2∂X2R′3.If we neglectOε, then we can get the kink-antikink solution of the standard mKdV equation as(34)R0′X,T′=ctanhc2X−cT′.To determine the speed velocityc in equation (34), we can introduce the soluble condition as(35)R0′,MR0′≡∫−∞+∞dXR0′MR0′=0.In equation (35), we have MR0′=MR0. By integrating equation (35), we can calculate c for the kink-antikink wave as(36)c=5g2g32g2g4−3g1g5.Finally, we get the solution for the mKdV equation as(37)RX,T=g1cg2tanhc2X−cg1T.By substituting equation (37) into equation (18), we get the kink-antikink solution of the headway as(38)Δxn=hc+g1cg2aca−1tanhc2aca−1n+1−cg1aca−1t.The amplitude for the kink-antikink wave can be written as(39)A=g1cg2aca−1.The coexisting curves can expressed asΔxn=hc+A, as shown in Figures 1 and 2.


## 5. Numerical Simulation

We simulate the equation to verify the theoretical analysis results. Here, we use periodic boundary conditions, and the initial conditions are as follows:(40)Δxn0=Δxn1=Δx0,forn≠0.5N,0.5N+1,Δxn0=Δxn1=Δx0+0.1,forn=0.5N,(41)Δxn0=Δxn1=Δx0−0.1,forn=0.5N+1,where Δx0 represents the vehicle distance of the uniform solution and N is the total number of vehicles. We introduce a small disturbance with the length of 0.1 m, as shown in equation (41). In addition, other parameters of the system are N=100,Δx0=4.0,p=0.2,λ=0.2, and a=0.88.Figure3 gives the phase diagram results of the space–time evolution when 19000<t<19200, where Figures 3(a)–3(d) represent τ=0,0.1,0.2,0.3. When τ is relatively small, the disturbance introduced in equation (6) will disappear. The disturbance is amplified when τ is rather large and the stability condition is not met. Finally, the propagation of the density wave is formed, showing the stop-and-go phenomenon. The phenomenon is well described by expanding the model to a kink-antikink wave solution of the mKdV equation at the critical point. In addition, with the increase of τ, the amplitude of the density wave also increases, which also means that the rise of the delay in the complex environment leads to traffic congestion. However, the accumulation of vehicles caused by traffic congestion further worsens the transmission delay, and the positive feedback deteriorates the traffic congestion. Therefore, the intelligent transportation system needs delay control and a good electromagnetic environment.Figure 3
Space-time evolution of the headway for different values ofα when FΔx=−2Δx−1. (a)τ=0. (b)τ=0.1. (c)τ=0.2. (d)τ=0.3.
(a)(b)(c)(d)Figure4 shows the distribution of density waves when t=19200. When τ=0.3, the amplitude becomes more significant in that the distance between vehicles becomes larger. When τ decreases, the amplitude gradually decreases until a steady state of uniform flow is maintained. The steady-state system with low delay time can suppress the disturbance, so the minor communication delay helps improve the traffic efficiency of the intelligent transportation system.Figure 4
The distribution of density waves whent=19200.Figure5 shows the hysteresis loop in phase space Δx,v at different τ values, where the x-axis represents the distance between vehicles, and the y-axis represents the speed. It can be observed from the phase diagram that after a long-term evolution, the speed and the distance between cars exhibit oscillation, and the amplitude of the oscillation shrinks as τ decreases until τ=0 converges to the H point. Therefore, the size characteristics of the hysteresis loop are consistent with the theoretical analysis results.Figure 5
The hysteresis loop in phase spaceΔx,v for τ=0,0.1,0.2,0.3.Finally, to determine the accuracy of the stability analysis, we define the difference between the maximum and minimum headways in the system whent=20000 steps, as follows:(42)δx=maxΔxn−minΔxn.Similarly, we can define the maximum and minimum speed difference as(43)δv=maxvn−minvn.We are curious whether the numerical results on the region but not specified points are consistent with the theoretical analysis results. Hence, simulations on the parameter space withΔx∈2,6×a∈0.02,1.5 are performed. From the phase diagram of the distance difference between vehicles, shown in Figure 6, it can be found that there is a prominent phase transition area, and the black dotted line corresponds to its neutral stability curve. It can be seen that the results of the numerical simulation throughout the parameter space are consistent with the results of the theoretical analysis. The blue area represents the uniform free flow, while the area below the curve represents the occurrence of traffic congestion. As the sensitivity coefficient decreases, the gap between the maximum and minimum distance between vehicles becomes larger. Also, in Figure 7, we observe the phase transition, but the maximum and minimum speed difference caused by the speed limit is only about 2.Figure 6
Heatmap ofmaxΔxn−minΔxn for t=20000 in the phase space of Δx,α.Figure 7
Heatmap ofmaxΔvn−minΔvn for t=20000 in the phase space of Δx,α.In addition, we observe that the phase diagrams shown in Figures6 and 7 show symmetry with Δx=4 as the symmetry axis. Under the same sensitivity coefficient a, traffic congestion still occurs even if the vehicle density is low. This also shows that there is no absolute relationship between road congestion and the level of vehicle density. Even so, this significantly relates to the driver’s reaction speed and driving behavior. In the case of manual driving, it is difficult to improve the sensitivity coefficient a, while automatic driving can effectively overcome the problems of decreased sensitivity caused by long-term driving and inattention. Therefore, it can be found that autonomous driving in the V2X environment can effectively improve road traffic efficiency and reduce congestion.


## 6. Conclusions

With the rapid development of artificial intelligence, self-driving vehicles have become a reality. 5G technology has been shown to effectively improve the speed of mobile communication networks and reduce communication delays, laying the foundation for the realization of automotive networks. Therefore, this paper introduces the communication delay based on the average density of the vehicle group ahead in the intelligent transportation system and analyzes the influence of the communication delay on traffic flow composed of intelligent vehicles. The theoretical analysis results show that excessive communication delay causes instability in the traffic system. Even so, if the number of cars in the vehicle group is increased, we can effectively reduce the possibility of traffic jams caused by the communication delay.Numerical simulation results verify our conclusions; at the same time, we conduct large-scale numerical simulations for the results of stability analysis. The numerical results are consistent with the theoretical analysis results and exhibit an obvious phase transition phenomenon, and the phase diagram shows a high degree of symmetry. This indicates that road congestion is mainly caused by driving behavior and to a lesser degree by too many vehicles. Therefore, fast response and consistent driving behavior in an intelligent transportation system can effectively alleviate traffic congestion.Autonomous driving is developing rapidly, which inspired us to analyze possible factors in the independent driving system through theoretical analysis and conduct simulation accordingly.



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