# An Extended Car-Following Model considering the Driver’s Desire for Smooth Driving and Self-Stabilizing Control with Velocity Uncertainty

Authors: Shihao Li; Ting Wang; Rongjun Cheng; Hongxia Ge
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, an extended car-following model with consideration of the driver’s desire for smooth driving and the self-stabilizing control in historical velocity data is constructed. Moreover, for better reflecting the reality, we also integrate the velocity uncertainty into the new model to analyze the internal characteristics of traffic flow in situation where the historical velocity data are uncertain. Then, the model’s linear stability condition is inferred by utilizing linear stability analysis, and the modified Korteweg-de Vries (mKdV) equation is also obtained to depict the evolution properties of traffic congestion. According to the theoretical analysis, we observe that the degree of traffic congestion is alleviated when the control signal is considered, and the historical time gap and the velocity uncertainty also play a role in affecting the stability of traffic flow. Finally, some numerical simulation experiments are implemented and the experiments’ results demonstrate that the control signals including the self-stabilizing control, the driver’s desire for smooth driving, the historical time gap, and the velocity uncertainty are of avail to improve the traffic jam, which are consistent with the theoretical analytical results.


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## Body

## 1. Introduction

Along with the improvement of modern transportation system, the complexity of transportation system is becoming larger and larger. Thus, there are more concerns about the fact that the transportation system’s growing complexity will undoubtedly increase the burden of people’s travel. A large number of researchers have been involved in the research of revealing the dynamic properties of traffic flow over the years [1–5]. In the past few decades, lots of models based on the actual traffic conditions have been built to study traffic flow. Normally, traffic flow models can be categorized into microscopic models [1, 6–12], lattice models [3, 4, 13–22], and macroscopic hydrodynamic models [23–32].The car-following model theory aims at analyzing the motion properties of a single vehicle and establishes the relationship between current vehicle and preceding vehicle on the road without the case of overtaking. Besides, the car-following model can describe many evolution processes of traffic flow and many traffic factors can be integrated into the car-following model to simulate real traffic situations [33–36]. It has now been almost seventy years since the first paper about car-following model was published, which was constructed by Pipes [37] in 1953. However, the acceleration process of single vehicle cannot be described by the model correctly. Later in 1961, Newell [38] suggested a novel car-following model, which established the expression of velocity instead of acceleration. But the acceleration of vehicle will become infinite when the traffic light changes to green from red in intersection. For solving the defects of previous models, Bando et al. [39] proposed the classical optimal velocity model (OVM). In OVM, the delay time of vehicle is not considered. Moreover, the detailed vehicle characters and realistic driver behavior are included in OVM, many traffic phenomena observed in real traffic can be simulated in terms of OVM. Since then, on account of the OVM, a legion of researchers proposed new models integrating different traffic factors to study traffic flow. In 1998, Helbing and Tilch [40] identified the OVM by utilizing the measured information and the simulation results showed that the OVM can generate the unrealistic deceleration and acceleration rates. Hence, Helbing et al. suggested the general force model (GFM). In 2001, based on the GFM, Jiang et al. [41] constructed the full velocity difference model (FVDM) with consideration of the effect of positive velocity difference on vehicle dynamics.In real traffic environment, on the one hand, drivers wish to keep vehicle’s velocity consistent with traffic flow’s equilibrium velocity for reducing the frequency of adjusting speed. With that in mind, Wang et al. [42] proposed the extended car-following model considering the driver’s desire for smooth driving in the difference between current and historical velocity. In their studies, some simulation experiments were implemented to verify the impact of the driver’s desire for smooth driving, and the results corroborated that the factor plays a positive role in stabilizing traffic flow. On the other hand, the desire for driving smoothly is just one of drivers’ psychological characteristics, because the drivers may also wish to keep vehicle’s speed consistent with the velocity of previous moment for reducing energy consumption. Based on this, Li et al. [43] constructed the novel car-following model taking the self-stabilizing control effect in the difference of historical velocity data into account. From the investigation by using analytical and numerical methods, researchers found that the traffic flow will be more stable by means of the self-stabilizing control, just like the cooperatively driving control. According to the analysis above, we can conclude that the driver’s desire for smooth driving and the self-stabilizing effect represent the drivers’ two psychological characteristics; different drivers will choose different driving strategies. Meanwhile, we can realize that the historical velocity data can be introduced into traffic flow models to study traffic flow’s properties.Recently, many researchers believe that the operation of intelligent transportation system will be affected by numerous uncertain factories, such as network fluctuation, fault of equipment, driver’s personality, and traffic interruption. Hence, the historical traffic data including historical velocity is not usually accurate when uncertain factories happen. With that in mind, Zhang et al. [44] put forward an enhanced FVDM incorporating the velocity uncertainty in cooperation driving system and found that velocity uncertainty of preceding vehicle has a significant influence on traffic flow. Nevertheless, relatively little attention has been drawn toward velocity uncertainty of current vehicle. In reality, the driver will not always pay attention to the velocity of current vehicle; there will be a difference between the velocity perceived by driver and the actual velocity of the current vehicle. The velocity uncertainty of this study indicates the difference between real velocity and velocity perceived by drivers in historical time.In conclusion, for better mirroring the reality and revealing the evolution characteristics of traffic flow when these traffic conditions are taken into account, this paper constructed an enhanced car-following model incorporating the driver’s desire for smooth driving and the self-stabilizing control with velocity uncertainty in Section2. The remainder of the paper is arranged as follows. In Section 3, linear stability analysis of enhanced model is derived. The mKdV equation is inferred to display the evolution properties of traffic congestion in Section 4. After that, some simulation experiments are conducted to verify the correctness of numerical analysis and to discuss the influences of proposed factors on traffic flow in Section 5. The final section gives some conclusions.


## 2. The Improved Model

Bando et al. established the well-known OVM, and the dynamic formula of OVM can be shown as(1)d2xntdt2=αVopΔxnt−vnt,where α represents the sensitivity coefficient of drivers; Δxnt=xn+1t−xnt stands for distance between vehicle n and vehicle n+1; vnt represents velocity of vehicle n; the desired optimal velocity of vehicle n is as follows:(2)VopΔxnt=vmax2tanhΔxnt−hc+tanhhc,where vmax and hc represent the maximum speed and the safety space headway distance.Since the historical velocity information of current vehicle is uncertain, we introduce the effects of the driver’s desire for smooth driving and self-stabilizing control with velocity uncertainty on the OVM, and the extended model can be expressed as(3)d2xntdt2=αVopynt−vnt+λ1Vh−1+pvnt−τ+λ2vnt−1+pvnt−τ,where ynt=Δxnt; h represents the average space headway distance; τ is the time gap between the historical time and the current time; Vh−vnt−τ denotes the driver’s desire for smooth driving; vnt−vnt−τ denotes the self-stabilizing control effect in the difference between the current and history velocity; λ1 and λ2 mean the reaction coefficients of two introduced factors, respectively; p denotes the reaction coefficient reflecting the uncertainty of vehicle’s velocity; −1<p<0 stands for the vehicle’s real velocity which is larger than the historical velocity perceived by drivers, impelling that the drivers overestimate the driving condition and choose to increase the acceleration of vehicle, which will likely cause the traffic accident and traffic jam; p=0 means that the historical velocity perceived by drivers equals the vehicle’s real velocity; 0<p<1 represents the historical velocity perceived by drivers which is larger than the vehicle’s real velocity, spurring that the drivers underestimate the driving condition and make a deceleration decision, which will probably cause the inessential delay and traffic congestion.From (3), we can get the space headway equation of novel model which is written as(4)d2Δxntdt2=αVΔxn+1t−VΔxnt+λ2−αdΔxntdt−λ1+λ21+pdΔxnt−τdt.


## 3. Linear Stability Analysis

The method of linear stability analysis will be utilized to obtain the stability condition of enhanced model in this section. First of all, letN represent the number of all vehicles and L denote the road’s length. Traffic flow’s steady state will be defined as that all vehicles possess the same headway and velocity. Then the uniform steady state of extended model is(5)Δxn∗t=h,vn∗t=Vh,where h equals L/N.Supposing thatεnt represents a small perturbation, we impose it into the steady state to gain the linear stability condition. Then the new state of traffic flow can be represented as(6)Δxnt=h+εnt.Substituting (6) into (4), we can get the new equation which can be expressed as(7)z2=αV′hεn+1t−εnt−λ1+λ21+pdεnt−τdt+λ2−αdεntdt.Transformingεnt into Fourier form of εnt=Aexpikn+zt, then (7) will be translated to(8)z2=αV′heik−1+λ2−αz−λ1+λ21+pze−zτ.Further expandingz and eik with the form of z≈z1ik+z2ik2 and eik≈1+ik+ik2/2 in (8), we gain the following formula omitting the terms of ik of which the order number exceeds two:(9)αV′h+λ2−αz1−λ1+λ21+pz1ik+12αV′h+λ2−αz2−z12+λ1+λ21+pz12τ−λ1+λ21+pz2ik2=0.Then, there exists(10)αV′h+λ2−αz1−λ1+λ21+pz1=0,12αV′h+λ2−αz2−z12+λ1+λ21+pz12τ−λ1+λ21+pz2=0.Following that, we get the value ofz1 and z2 by calculating (10) as(11)z1=αV′hλ1+λ1+λ2p+α,z2=λ1+λ21+pτ−1α2V′h2+1/2αλ1+λ1+λ2p+α2λ1+λ1+λ2p+α3.The traffic congestion will be avoidable whenz2 is positive. Thus, we can get the linear stability criterion as(12)λ1+λ21+pτ−1αV′h+12λ1+λ1+λ2p+α2>0.From (12), we can obtain the neutral stability curve which can be shown as(13)αc=−A+V′hB+2V′hAB+V′h2B2−3A2,where A=λ1+λ1+λ2p and B=λ1+λ21+pτ−1.From (13), when λ1=λ2=0, the novel model reduces to the OVM, which is in accordance with the results of OVM.In order to discuss the influences of the driver’s desire for smooth driving, the self-stabilizing control effect, the velocity uncertainty, and the historical time gap on traffic flow, the neutral stability lines for different parameters are drawn in Figures1–4.Figure 1
The neutral stability lines for differentλ1 with λ2=0,p=0,andτ=1.Figure 2
The neutral stability lines for differentλ2 with λ1=0.01,p=0,andτ=1.Figure 3
The neutral stability lines for differentτ with λ1=0.01,λ2=0.02,andp=0.Figure 4
The neutral stability lines for differentp with λ1=0.01,λ2=0.02,andτ=1.For given neutral stability curves, the region below the line is the unstable region where the traffic jam will show up with a small derivation. On the contrary, the region above the line is the stable region where the traffic congestion will never occur when the outside perturbation is imposed on traffic flow. Based on this, we conduct the following analyses.In Figure1, the solid line is the neutral stability line of OVM with λ1=0 and λ2=0. When the value of λ1 increases with fixed coefficients τ=1,λ2=0,andp=0, the height of lines will decrease which means the stable region becomes bigger. In other words, if the effect of the driver’s desire for smooth driving is considered, the traffic flow becomes more stable by increasing the value of λ1. Then in Figure 2, the fixed parameters are τ=1,λ1=0.01,andp=0; factors including the driver’s desire for smooth driving and the self-stabilizing control are both considered. With increasing λ2 from 0 to 0.08, the stable region will increase little by little. Namely, the traffic flow becomes more stable when two effects are considered. From Figure 3, one can note that the stable region will also become bigger when the historical time gap τ increases. Finally, in Figure 4, we begin to discuss the influence of velocity uncertainty on traffic flow and the fixed parameters are λ1=0.01,λ2=0.02,andτ=1. When p=−0.25, the driver overestimates the driving condition and increases the acceleration of vehicle, which may cause the appearance of traffic accident, so the unstable region is larger. When p=0, the historical velocity perceived by driver equals the real velocity. When p=0.25 and p=0.45, the driver underestimates the driving condition and makes a deceleration decision. One can notice that the stable region is larger with increasing the value of p, whereas the stability in this situation is “fictitious” at the price of inferior traffic efficiency, as can be confirmed as follows.To sum up, in view of the figures of neutral stability curves and corresponding analyses, it is clear that the factors considered in the new model play a significant role in influencing the traffic flow’s stability.


## 4. Nonlinear Analysis

The mKdV equation is inferred to describe the varying behaviors of traffic flow for critical pointhc,αc in unstable region [45]. At first, define the slow variables T and X as(14)T=ε3t,X=εn+bt,where 0<ε≪1 and b is an unknown constant which can be calculated.The space headway distanceΔxnt with a disturbance will be expressed ast(15)Δxnt=hc+εRX,T.Inserting (14) and (15) into (4) and extending to the fifth order of ε, we will obtain(16)αV′+λ2−αb−Ωbε2∂XR+12αV′+b2+Ωb2τε3∂X2R+16αV′+12Ωb3τ2∂X3R+16αV‴∂XR3−λ2−α−Ω∂TRε4+124αV′+16Ωb4τ3∂X4R+112αV‴∂X2R3+2Ωbτ−2b∂XTRε5=0,where ∂XkR=∂kR/∂Xk, ∂XTR=∂2R/∂X∂T, ∂XR=∂R/∂X, V′=∂Vh/∂hh=hc, R=RX,T, V‴=∂3Vh/∂h3h=hc, and Ω=λ1+λ21+p.Takingb=αV′/λ1+λ1+λ2p+α and α=αc1−ε2 into (16), then the coefficients of ε2 and ε3 can be eliminated; (16) will be translated into(17)16αV′+12Ωb3τ2∂X3R+16αV‴∂XR3−λ2−α−Ω∂TRε4+124αV′+16Ωb4τ3∂X4R+112αV‴∂XR3+2Ωbτ−2b∂XTR+λ1+λ1+λ2p+α241−Ωτ∂X2Rε5=0.Equation (17) can be simplified as follows:(18)ε4∂TR−ξ1∂X3R+ξ2∂XR3+ε5ξ3∂X2R+ξ4∂X4R+ξ5∂X2R3=0,where(19)ξ1=−16αV′+12λ1+λ21+pb3τ2,(20)ξ2=16αV‴,(21)ξ3=λ1+λ1+λ2p+α241−λ1+λ21+pτ,(22)ξ4=124αV′+16λ1+λ21+pb4τ3,(23)ξ5=112αV‴.Conducting the conversion ofT′=ξ1T and R′=ξ2/ξ1R, we will obtain the regularized mKdV equation with a correction term οε:(24)∂T′R′−∂X3R′+∂XR′3+εξ3ξ1∂X2R′+ξ4ξ1∂X4R′+ξ5ξ2∂X2R′3=0.Ignoring the termοε, we can get a soliton solution of (24) which can be expressed as(25)R0′X,T′=ctanhc2X−cT′.SupposingR′X,T′=R0′X,T′+εR1′X,T′, only when solvability condition is fulfilled, we can gain the value of c; the solvability condition is(26)R0′,MR0′≡∫−∞+∞dXR0′MR0′=0,where MR0′=ξ3/ξ1∂X2R′+ξ4/ξ1∂X4R′+ξ5/ξ2∂X2R′3.Based on the solvability condition above, we can get the propagation speedc:(27)c=5ξ2ξ32ξ2ξ4−3ξ1ξ5.Thus, the kink-antikink soliton solution of the space headway distance is given by(28)Δxnt=hc+ξ1cξ2ααc−1×tanhc2ααc−1×n+1−cξ1ααc−1t.Here, the amplitude of traffic jam wave is(29)A=cξ1ξ21−ααc.One can note that the density wave can represent the property of traffic congestion, which is decided by the mKdV equation. From (19), (20), and (29), the parameters of factors considered in the new model have a momentous influence on influencing the amplitude of traffic jam wave.


## 5. Numerical Simulation

Some simulation experiments are conducted for the extended model to verify the influence of factors considered on traffic flow in this part. At first, during the experiment,N=100 cars are running on the roadway and the relative fixed coefficients are set as follows:(30)vmax=2m/s,α=1.7s−1,hc=5m.In addition, the initial conditions are set as follows:(31)ΔxnΔt=Δxn0,Δxn0=h,n≠50,51,h−0.5,n=50,h+0.5,n=51.Figures5–7 are the headway profiles, the velocity profiles, and the hysteresis loop for different value of λ1 with λ2=0,p=0,andτ=1, respectively. The hysteresis loop is the main reason that induces the traffic flow to be unstable state and causes the traffic phenomenon of stop and go. When λ1=0, the novel model reduces to the OVM; Figures 5(a), 6(a), and 7(a) describe the traffic jam of OVM. With increasing the value of λ1, namely, the driver’s desire for smooth driving is incorporated, we can observe that traffic jam is mitigated.Figure 5
The profiles of headway for differentλ1 under λ2=0,p=0,andτ=1. (a) λ1=0andλ2=0, (b) λ1=0.01andλ2=0, (c) λ1=0.02andλ2=0, and (d) λ1=0.03andλ2=0.
(a)(b)(c)(d)Figure 6
The profiles of velocity for differentλ1 under λ2=0,p=0,andτ=1. (a) λ1=0andλ2=0, (b) λ1=0.01andλ2=0, (c) λ1=0.02andλ2=0, and (d) λ1=0.03andλ2=0.
(a)(b)(c)(d)Figure 7
The hysteresis loop for differentλ1 under λ2=0,p=0,andτ=1. (a) λ1=0andλ2=0, (b) λ1=0.01andλ2=0, (c) λ1=0.02andλ2=0, and (d) λ1=0.03andλ2=0.
(a)(b)(c)(d)Then, in Figures8–10, λ1=0.01,p=0,andτ=1 are fixed, and λ2 is a variable. In this case, the driver’s desire for smooth driving and the self-stabilizing control effect are considered simultaneously. Three figures show the headway profiles, the velocity profiles, and the hysteresis loop, respectively. One can note that when increasing the value of λ2 from 0 to 0.1, the amplitude of traffic jam wave decreases, vehicles’ velocity tends to be unified, and the hysteresis loop reduces to be a red dot. That is to say, the two factors considered can effectively diminish the traffic jam.Figure 8
The profiles of headway for differentλ2 under λ1=0.01,p=0,andτ=1. (a) λ1=0.01andλ2=0, (b) λ1=0.01andλ2=0.05, (c) λ1=0.01andλ2=0.07, and (d) λ1=0.01andλ2=0.1.
(a)(b)(c)(d)Figure 9
The profiles of velocity for differentλ2 under λ1=0.01,p=0,andτ=1. (a) λ1=0.01andλ2=0, (b) λ1=0.01andλ2=0.05, (c) λ1=0.01andλ2=0.07, and (d) λ1=0.01andλ2=0.1.
(a)(b)(c)(d)Figure 10
The hysteresis loop for differentλ2 under λ1=0.01,p=0,andτ=1. (a) λ1=0.01andλ2=0, (b) λ1=0.01andλ2=0.05, (c) λ1=0.01andλ2=0.07, and (d) λ1=0.01andλ2=0.1.
(a)(b)(c)(d)For investigating the effect of velocity uncertaintyp on traffic flow, fixed parameters are λ1=0.01,λ2=0.05,andτ=1. Figures 11–13 represent the headway profiles, the velocity profiles, and the hysteresis loop, respectively. When p<0, the historical velocity perceived by driver is smaller than the real velocity, the driver overestimates the traffic condition and makes an aggressive response to increase the acceleration of vehicle, and the traffic accident may appear, so the traffic flow will be more unstable, while when p>0, the opposite is true, whereas the time delay will show up. Therefore, from Figures 11–13, what we can see is that the amplitude and the hysteresis loop of traffic congestion decrease by increasing the value of p from negative to positive; however, the improvement of traffic congestion is at the price of unnecessary time delay, which conforms with the analysis above.Figure 11
The profiles of headway for differentp under λ1=0.01,λ2=0.05,andτ=1. (a) p=−0.25, (b) p=0, (c) p=0.25, and (d) p=0.45.
(a)(b)(c)(d)Figure 12
The profiles of velocity for differentp under λ1=0.01,λ2=0.05,andτ=1. (a) p=−0.25, (b) p=0, (c) p=0.25, and (d) p=0.45.
(a)(b)(c)(d)Figure 13
The hysteresis loop for differentp under λ1=0.01,λ2=0.05,andτ=1. (a) p=−0.25, (b) p=0, (c) p=0.25, and (d) p=0.45.
(a)(b)(c)(d)Finally, for analyzing the influence of time gapτ, the coefficients of λ1, λ2, and p are fixed to be λ1=0.01,λ2=0.05,andp=0. From Figures 14–16, one can note that when increasing the value of τ, the headway and the velocity of traffic flow in the first two figures tend to be stable; the hysteresis loop in the last figure shrinks gradually. Namely, the stability of traffic flow becomes higher.Figure 14
The profiles of headway for differentτ under λ1=0.01,λ2=0.05,andp=0. (a) τ=0, (b) τ=0.75, (c) τ=1, and (d) τ=1.5.
(a)(b)(c)(d)Figure 15
The profiles of velocity for differentτ under λ1=0.01,λ2=0.05,andp=0. (a) τ=0, (b) τ=0.75, (c) τ=1, and (d) τ=1.5.
(a)(b)(c)(d)Figure 16
The hysteresis loop for differentτ under λ1=0.01,λ2=0.05,andp=0. (a) τ=0, (b) τ=0.75, (c) τ=1, and (d) τ=1.5.
(a)(b)(c)(d)


## 6. Conclusions

This paper constructs an enhanced car-following model integrating the driver’s desire for smooth driving and the self-stabilizing control effect. Both of the factors considered are based on the historical velocity data of current vehicle, while the current vehicle’s historical velocity perceived by driver does not always conform with its real velocity. Therefore, the velocity uncertainty of current vehicle in the difference between actual velocity and historical velocity perceived is included in the model. For investigating the influences of these factors on traffic flow, the novel model is analyzed by linear and nonlinear stability analysis methods. At last, a series of simulation experiments are performed to verify the theoretical analytical results. The experimental results show that factors including the driver’s desire for smooth driving and the self-stabilizing control can enhance the stability of traffic flow clearly. Then the larger historical time gap can improve the traffic congestion effectively. In addition, the value of velocity uncertainty can result in the degraded traffic efficiencyp>0 and the traffic accident p<0.



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