Stability Analysis of a Car-Following Model on Two Lanes

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.


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][3][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 twolane 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.

Improved OVM
The typical OV model is presented as [5,6] where ( ) and V ( ) are the position and velocity of the th vehicle, Δ ( ) is the headway distance between the th and its leading vehicle, is the sensitivity parameter of the driver, and op (⋅) is the optimal velocity function described as [5] where V max is the maximum velocity on a particular roadway and ℎ 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 carfollowing 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 where Δ , +1 ( ) is the headway distance between the th vehicle and its leading vehicle on the adjacent lane, V is the length of a normal vehicle, and is a preset constant. Referring to previous study, ΔV ( ) and ΔV , ( ) are introduced [7], where ΔV ( ) = V +1 ( ) − V ( ) and ΔV , ( ) can be given as in which V , ( ) is the velocity of the leading vehicle on adjacent lane.
The new model can be expressed as where and are the weights of the two OV functions and 1 and 2 are the weights of velocity difference.

Linear Stability Analysis
According to linear stability analysis method [6], stable condition of the uniform traffic flow is given by Let ( ) and , ( ) be small deviations from (0) ( ) and where and are the derivatives of OV functions op (Δ ( )) and op ( ( )). By expanding ( ) ∝ exp( + ) and , ( ) ∝ exp( + ), (7) can be rewritten as Then expand by the order of at the point of ≈ 0 as = 1 + 2 ( ) 2 + ⋅ ⋅ ⋅ and insert it into (8). The following terms can be obtained: According to previous study, the vehicle system is stable when 2 > 0, which is and the neutral stability condition has the following form: The stability surface is described in Figure 1. Parameters are set as V max = 4, ℎ = 7, = 1, = 0, 1 = 0.2, and

Nonlinear Stability Analysis
For the convenience of nonlinear analysis, (5) is rewritten as MKdV equation is obtained in unstable region around the critical point (ℎ , ), where = 0. By the analysis method in [18], the long wave expansion is applied in this section.
Two slow scales for space variable and time variable are introduced. We define slow variables and as where is a constant determined later and = √ / − 1.
Headways for two lanes are set as Expanding (12) to the fifth order of then gives Mathematical Problems in Engineering It is noticed that the in the sixth order term of (15) can be eliminated by taking the derivative of in the fifth order term. Then insert = + and / − 1 = 2 into (15); that is, In order to have standard MKdV equation, the following transformations are made in (16): Then (16) can be rewritten as Ignoring the ( ) term, we have MKdV equation with a kink-antikink soliton solution expressed as To determine the value of amplitude , the solvable condition is considered: where [ 0 ] means the ( ) term in (19). By performing the integration of (21), the value of amplitude can be obtained: The kink-antikink soliton solution of headway can be written as follows: 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 = ℎ .   The amplitude of traffic wave in lane2 is larger just like in Figure 3.

Numerical Simulation
Finally when = 2.2, = 0.8, = 0.2, 1 = 0.16, and 2 = 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.
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.

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, when 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.