# Stability Analysis of a Car-Following Model on Two Lanes

Authors: Yuhan Jia; Yiman Du; Jianping Wu
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

Considering lateral influence from adjacent lane, an improved car-following model is developed in this paper. Then linear and nonlinear stability analyses are carried out. The modified Korteweg-de Vries (MKdV) equation is derived with the kink-antikink soliton solution. Numerical simulations are implemented and the result shows good consistency with theoretical study.


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## Body

## 1. Introduction

Traffic flow is a system of consecutive vehicles with interaction [1]. Recently various models have been developed, including general models, safety distance models, action point models, optimal velocity models (OVM), cellular automaton models, and fuzzy logic models [2–4]. Among those models, OVM developed by Bando et al. [5, 6] is well known for its accuracy and rationality. Afterwards Helbing and Tilch [7] calibrated the OV model by experimental data and developed a generalized force model (GFM) to overcome the deficiencies. But both OVM and GFM cannot describe the phenomenon that the following vehicle may not decelerate when the leading vehicle is much faster even if the headway distance is smaller than safety distance. Then the inconsistencies of previous models were overcome by a continuous microscopic single-lane model, the intelligent driver model, developed by Treiber et al. [8] by the analysis of German freeways data. After this, a full velocity difference model (FVDM) was developed by Jiang et al. [9, 10] to solve the disadvantage. However, there are still some problems in previous models, which are discussed in detail and improved by Treiber and Kesting [11]. Also the gas-kinetic-based model was investigated by observed data and simulation experiments, which showed good agreement with phenomena in reality [12]. In this century, many new models have been established by considering decentralized delayed-feedback control [13], delay time due to driver’s reaction [14], extended OV function for acceleration difference [15], multiple velocity difference [16], and optimal velocity difference [17].To study traffic jam waves in OVM, Komatsu and Sasa [18] firstly derived the modified Korteweg-de Vries (MKdV) equation to describe kink waves. Then Muramatsu and Nagatani [19] derived Korteweg-de Vries (KdV) equation from OVM to describe sliton waves in traffic jam, and Nagatani also found triangular shock wave solved Burgers equation [20]. From then many models have been analyzed by nonlinear stability theory aforementioned. Nagatani [21] derived MKdV equation near critical point in two continuum models: partial differential and discrete lattice model. Yu [22] presented a simplified OVM considering relative velocity and derived KdV and MKdV equations. Ge et al. developed several intelligent transportation system (ITS) based models with KdV and MKdV analysis [23] and also did similar research in three OVM based models [24]. Yu et al. [25] recently build a two-delay model with MKdV investigation and implemented numerical simulations. More studies show that the triangular wave, soliton wave, and kink wave occur in stable region, metastable region, and unstable region, respectively [20, 26, 27].However, only a few researches focused on car-following with lateral impact, in which case the lateral influence from adjacent lane should be considered. Nagatani [28] presented two lattice models to simulate traffic flow wave on a two-lane highway with lane changing. Jin et al. [29] considered the lane-width influence and developed a non-lane-based FVDM with simulation experiments. Ge et al. [30] studied the influence from neighbor vehicle or nonmotor vehicle by considering two more OV functions and analyzed the stability condition by control theory method. Based on previous work, this paper investigates a new car-following model considering lateral influence by introducing the combination of two OV functions. In Section 2 the new model is developed and linear stability analysis is carried out in Section 3. In Section 4 the MKdV equation is derived to obtain kink-antikink soliton solution. Then numerical simulation experiments are performed to verify the theoretical study in Section 5. The summary is given in Section 6.


## 2. Improved OVM

The typical OV model is presented as [5, 6]
(1)d2xn(t)dt2=αVopΔxnt-vnt,
where xn(t) and vn(t) are the position and velocity of the nth vehicle, Δxn(t) is the headway distance between the nth and its leading vehicle, α is the sensitivity parameter of the driver, and Vop(·) is the optimal velocity function described as [5]
(2)VopΔxnt=vmax⁡2tanhΔxnt-hc+tanhhc,
where vmax⁡ is the maximum velocity on a particular roadway and hc means the safety headway distance.However, as noticed in the study on roadway, a driver usually focuses not only the leading vehicle on the present lane, but also the vehicle on adjacent lane, especially when the neighbor vehicle decelerates. This phenomenon occurs because of the potential action of lane changing or the avoidance of collision when the lane width is small [29]. Hence the lateral influence should be considered in car-following model even if lane changing does not occur.It is assumed that the driver makes his decision upon the combination impact of leading vehicle and neighbor vehicle by introducing a second OV function, which can be defined as(3)V-op(ln(t))=Vop(Δxl,n+1(t)),lv≤Δxl,n+1t<d,0,others,
where Δxl,n+1(t) is the headway distance between the nth vehicle and its leading vehicle on the adjacent lane, lv is the length of a normal vehicle, and d is a preset constant.Referring to previous study,Δvn(t) and Δvl,n(t) are introduced [7], where Δvn(t)=vn+1(t)-vn(t) and Δvl,n(t) can be given as
(4)Δvl,n(t)=vl,n(t)-vn(t),lv≤Δxl,n+1t<d,0,others,
in which vl,n(t) is the velocity of the leading vehicle on adjacent lane.The new model can be expressed as(5)d2xn(t)dt2=αpVopΔxnt+qV-opΔxl,nt-vnt+λ1Δvn(t)+λ2Δvl,n(t),
where p and q are the weights of the two OV functions and λ1 and λ2 are the weights of velocity difference.


## 3. Linear Stability Analysis

According to linear stability analysis method [6], stable condition of the uniform traffic flow is given by
(6)xn0t=nhc+Vopht,xl,n0t=nhc+V-opht.Letyn(t) and yl,n(t) be small deviations from xn(0)(t) and xl,n(0)(t) as xn(t)=x(0)(t)+yn(t) and xl,n(t)=xl(0)(t)+yl,n(t). The linearized equation can be obtained:
(7)d2yntdt2=αqV-′(Δxl0)Δyl(t)-vn(t)pV′Δx0Δythhhh+qV-′(Δxl0)Δyl(t)-vn(t)+λ1Δvn(t)+λ2Δvln(t),
where V′ and V-′ are the derivatives of OV functions Vop(Δxn(t)) and V-op(ln(t)). By expanding yn(t)∝exp⁡(ikn+zt) and yl,n(t)∝exp⁡(ikn+zt), (7) can be rewritten as
(8)z2+[α-(λ1+λ2)(eik-1)]z-α(pV′+qV-′)(eik-1)=0.Then expandz by the order of ik at the point of ik≈0 as z=z1ik+z2(ik)2+⋯ and insert it into (8). The following terms can be obtained:
(9)z1=pV′+qV-′,z2=12+λ1+λ2αpV′+qV-′-pV′+qV-′2α.According to previous study, the vehicle system is stable whenz2>0, which is
(10)α>2pV′+qV-′-2λ1+λ2,
and the neutral stability condition has the following form:
(11)αc=2pV′+qV-′-2λ1+λ2.The stability surface is described in Figure1. Parameters are set as vmax⁡=4, hc=7, p=1, q=0, λ1=0.2, and λ2=0 in Figure 1(a), while p=0.8, q=0.2, λ1=0.16, and λ2=0.04 in Figure 1(c). As can be seen, the unstable region is smaller considering influence from adjacent lane [25].Headway-sensitivity space for (a) and (b):p=1, q=0, λ1=0.2, and λ2=0 and (c) and (d): p=0.8, q=0.2, λ1=0.16, and λ2=0.04.
(a)
(b)
(c)
(d)


## 4. Nonlinear Stability Analysis

For the convenience of nonlinear analysis, (5) is rewritten as
(12)d2Δxntdt2=αdΔxntdtpVΔxn+1t-pVΔxnthhhh+qV-Δxl,n+1thhhh-qV-(Δxl,n(t))-dΔxn(t)dt+λ1dΔxn+1tdt-λ1dΔxntdt+λ2dΔxl,n+1tdt-λ2dΔxl,ntdt.MKdV equation is obtained in unstable region around the critical point(hc,αc), where V′′=0. By the analysis method in [18], the long wave expansion is applied in this section. Two slow scales for space variable n and time variable t are introduced. We define slow variables X and T as
(13)X=εn+bt,T=ε3t,
where b is a constant determined later and ε=αc/α-1.Headways for two lanes are set as(14)Δxnt=hc+εRX,T,Δxl,nt=hc+εRX,T.Expanding (12) to the fifth order of ε then gives
(15)ε2αb-αpV′+qV-′∂XR+ε3b2-αpV′+qV-′2-λ1+λ2b∂X2R+ε4α∂TR-α(pV′′′+qV-′′′)6∂XR3hhhhhhh-α(pV′+qV-′)+3b(λ1+λ2)6∂X3R+ε5α(pV′′′+qV′′′-)12(2b-λ1-λ2)∂X∂TRhhhhhhh-α(pV′+qV-′)+4b(λ1+λ2)24∂X4Rhhhhhhh-α(pV′′′+qV-′′′)12∂X2R3=0.It is noticed that the∂X∂TR in the sixth order term of (15) can be eliminated by taking the derivative of X in the fifth order term. Then insert b=pV′+qV-′ and αc/α-1=ε2 into (15); that is,
(16)[∂TR-m1∂X3R+m2∂XR3]+ε[m3∂X2R+m4∂X4R+m5∂X2R3]=0,
where
(17)m1=(α+3λ1+3λ2)(pV′+qV-′)6α,m2=-pV′′′+qV-′′′6,m3=(pV′+qV-′)2,m4=42pV′+2qV-′-λ1-λ2α+3λ1+3λ224α2hhh-α+4λ1+λ224α42pV′+2qV′--λ1-λ2α+3λ1+3λ224α2pV′+qV-′,m5=22pV′+2qV-′-λ1-λ2-α12αpV′′′+qV-′′′.In order to have standard MKdV equation, the following transformations are made in (16):
(18)T=1m1Tm,R=m1m2Rm.Then (16) can be rewritten as
(19)[∂TmRm-∂X3Rm+∂XRm3]+εm1[m3∂X2Rm+m4∂X4Rm+m1m5m2∂X2Rm3]=0.Ignoring theO(ε) term, we have MKdV equation with a kink-antikink soliton solution expressed as
(20)Rm0X,Tm=BtanhB2X-BTm.To determine the value of amplitudeB, the solvable condition is considered:
(21)(Rm0,M[Rm0])≡∫-∞∞Rm0MRm0dX=0,
where M[Rm0] means the O(ε) term in (19). By performing the integration of (21), the value of amplitude B can be obtained:
(22)B=5m2m32m2m4-3m1m5.The kink-antikink soliton solution of headway can be written as follows:(23)Δxn=hc+(αcα-1)(5m1m32m2m4-3m1m5)×tanh(αcα-1)5m2m34m2m4-6m1m5hhhhhhhh×5m2m3t2m2m4-3m1m5n+(pV′+qV-′)thhhhhhhhhh-(αcα-1)5m2m3t2m2m4-3m1m5(αcα-1)5m2m34m2m4-6m1m5.With the amplitude of (23), we have the coexisting surface in Figures 1(b) and 1(d) based on Figures 1(a) and 1(c), respectively. The space is divided into three regions: stable region above the coexisting surface, metastable region between coexisting surface and stability surface, and unstable region below stability surface. However, the limitation should be noted that this nonlinear analysis is only valid at the critical point when h=hc.


## 5. Numerical Simulation

Consider a two-lane system with 100 vehicles running on each lane under a periodic boundary condition without overtaking or lane changing. The initial values arevmax⁡=4, hc=7, and d=10 on both lanes. In this section, only the situation of p+q=1 is simulated, because we suppose that the sum of OV functions’ weights should remain as 1, which means the driver has no extra attention when considering lateral influence.First add a disturbance on one lane as follows:(24)forlane1,Δxn(0)=7–0.345<n<50Δxn(0)=7others;forlane2,Δxn(0)=70<n≤100.Setα=2.85, p=0.8, q=0.2, λ1=0.16, and λ2=0.04 for the first simulation test. Figure 2 shows the space-time evolution of headway for lane1 in (a) and lane2 in (b) from 900 s to 1000 s. Then set α=2.85, p=0.6, q=0.4, λ1=0.12, and λ2=0.08 for the next simulation. The results are shown in Figures 2(c) and 2(d). It is noticed that when the proportion of lateral influence increases from 0.2 to 0.4, the density wave on lane2 becomes heavier, while on lane1 the perturbation decays.Space-time evolution of headway for lane1 in (a) and lane2 in (b) whenp=0.8, q=0.2, λ1=0.16, and λ2=0.04 and for lane1 in (c) and lane2 in (d) when p=0.6, q=0.4, λ1=0.12, and λ2=0.08.
(a)
(b)
(c)
(d)Second, we consider both lanes have small disturbances, which are defined as follows:(25)forlane1,Δxn(0)=7–0.145<n<50Δxn(0)=7others;forlane2,Δxn(0)=7–0.345<n<50Δxn(0)=7others.Letα=2.85, p=1, q=0, λ1=0.2, and λ2=0, which cannot satisfy the stable condition in (10). Figure 3 shows the space-time evolution of headway for lane1 in (a) and lane2 in (b) from 900 s to 1000 s. The headway profiles of traffic wave at t=950 s are in (c) for lane1 and (d) for lane2. It is observed that the small perturbation propagates into traffic jam on both lanes. Furthermore, traffic jam is more serious in lane2 due to the larger initial headway perturbation.Space-time evolution of headway for lane1 in (a) and lane2 in (b) and headway profile of traffic wave att=950 s for lane1 in (c) and lane2 in (d). (α=2.85, p=1, q=0, λ1=0.2, and λ2=0).
(a)
(b)
(c)
(d)Then supposeα=2.85, p=0.8, q=0.2, λ1=0.16, and λ2=0.04. Figure 4 describes the space-time evolution of headway and the headway profiles of traffic wave at t=950 s corresponding to Figure 3. The initial perturbation decays after sufficient time on both lanes. Thus the consideration of lateral impact from adjacent lane can suppress traffic jam. The amplitude of traffic wave in lane2 is larger just like in Figure 3.Space-time evolution of headway for lane1 in (a) and lane2 in (b) and headway profile of traffic wave att=950 s for lane1 in (c) and lane2 in (d). (α=2.85, p=0.8, q=0.2, λ1=0.16, and λ2=0.04).
(a)
(b)
(c)
(d)Finally whenα=2.2, p=0.8, q=0.2, λ1=0.16, and λ2=0.04, the system is more unstable and serious kink-antikink waves are observed in Figure 5. Unlike Figures 3 or 4, there is no significant difference between the two lanes, because all the vehicles get influenced heavily by lateral impact.Space-time evolution of headway for lane1 in (a) and lane2 in (b) and headway profile of traffic wave att=950 s for lane1 in (c) and lane2 in (d). (α=2.2, p=0.8, q=0.2, λ1=0.16, and λ2=0.04).
(a)
(b)
(c)
(d)Then we exchange the disturbances of the two lanes, and simulations show similar results. In other words, the situation on lane1 after exchange is like lane2 before exchange, the same for lane 2. By simulation, the theoretical analysis of MKdV solution can be described.


## 6. Conclusions

In this paper, a new car-following model is proposed considering lateral influence from adjacent lane. Linear and nonlinear stability analyses are carried out, from which MKdV equation is obtained. Numerical simulations show that new model has good consistency with theoretical study and can suppress traffic jam wave. However, whenα satisfied the unstable condition, both lanes will have serious traffic jam of same level despite different initial headway perturbation. In conclusion, even though lane changing does not happen, considering lateral impact has influence on car-following behavior indeed and can keep the vehicle system more stable.



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*Source: 727560-2014-12-17.xml*