An Improved Lattice Hydrodynamic Model by considering the Effect of “Backward-Looking” and Anticipation Behavior

In order to prevent the occurrence of traffic accidents, drivers always focus on the running conditions of the preceding and rear vehicles to change their driving behavior. By taking into the “backward-looking” effect and the driver’s anticipation effect of flux difference consideration at the same time, a novel two-lane lattice hydrodynamic model is proposed to reveal driving characteristics. +e corresponding stability conditions are derived through a linear stability analysis. +en, the nonlinear theory is also applied to derive the mKdV equation describing traffic congestion near the critical point. Linear and nonlinear analyses of the proposed model show that how the “backward-looking” effect and the driver’s anticipation behavior comprehensively affect the traffic flow stability. +e results show that the positive constant c, the driver’s anticipation time τ, and the sensitivity coefficient p play significant roles in the improvement of traffic flow stability and the alleviation of the traffic congestion. Furthermore, the effectiveness of linear stability analysis and nonlinear analysis results is demonstrated by numerical simulations.

By incorporating the ideas of car-following as well as continuum models, the first lattice hydrodynamic model [26] with a conservation equation (1) and a flow equation (2) was proposed in 1998.
In addition to the information from the preceding vehicle, drivers always focus on the running conditions of the rear vehicles from the rear-view mirror to change their driving behavior. By considering the "backward-looking" effect, Ge and Cheng [39] presented the BL-LV model as follows: where H(·) and ρ c represent the Heaviside function and the safety density, respectively. ρ denotes a parameter close to 1. V F (·) is the optimal velocity function for "forward-looking." While V B (·) is the optimal velocity function for "backwardlooking," which works if and only if ρ c ≤ ρ j−1 (t) ≤ ρ holds. Functions V F (·) and V B (·) have exactly the opposite effects and are defined as where c is a positive constant that represents the relative role of the "backward-looking." For the sake of safe and comfortable driving, drivers always predict the velocity of the preceding vehicles so that they can adjust their driving velocity at the next moment. is driving behavior is known as the anticipation effect [42] or predictive effect [43,44]. Wang et al. [43] added the driver's anticipation effect into the optimal velocity (OV) function in a single-lane lattice model. Later, Kaur and Sharma [44] extended the singlelane model [43] to the two-lane case along with the optimal current difference (OCD) effect. By considering the anticipation effects in the OCD term as well as the OV function, Sharma [45] constructed a new lattice hydrodynamic model with an anticipation coefficient α, where the positive α represents the anticipation effect while the negative α represents the delay effect. en, Li et al. [32] adopted this anticipation effect to the multiple optimal current differences' anticipation version. Meanwhile, numerical results indicated that only the information of three preceding lattices [46] was enough for the improvement of traffic stability. In view of the anticipation effect of flux difference, Chang and Cheng [47] proposed a new lattice model as follows: where τ represents the driver's anticipation time and k 1 and k 2 denote the deviation degree of speed v j and the feedback gain, respectively. Some previous studies have considered the "backward-looking" effect and the anticipation effect of flux difference separately. In a real traffic system, the velocity of the current vehicles is not only affected by the preceding vehicles but also affected by the rear vehicles. Generally, the information of the rear vehicles could be received by the rear-view mirror. According to the information, drivers could change their driving behavior to avoid the traffic accidents. Meanwhile, the velocity of the preceding vehicles could be predicted by drivers on the basis of the vehicle distance information, so that they can adjust their driving velocity in a period of anticipation time. However, the above two factors have never been considered simultaneously in the existing lattice models. is motivates us to develop a new lattice model by incorporating the "backward-looking" effect with the anticipation effect of flux difference. e outline of this paper is organized as follows. In Section 2, we introduce a new lattice hydrodynamic model considering the flux anticipation difference effect and the "backward-looking" effect and use the linear stability theory to derive the stability condition of the traffic flow. In Section 3, the propagation behavior of traffic density waves is presented by means of the nonlinear analysis method. In Section 4, numerical simulations are carried out to validate the results of linear and nonlinear stability analyses. At last, the conclusions are drawn in Section 5.

The Extended Lattice Hydrodynamic Model and Linear Stability Analysis
In light of the aforementioned facts, a novel lattice model is presented by taking the flux anticipation difference effect and the "backward-looking" effect (FADBE) into account as follows: where p is the sensitivity coefficient of the anticipation flux difference. e greater p is, the stronger the driver's anticipation effect is. τ means the anticipation time, and ρ j+1 (t + τ)v j+1 (t + τ) − ρ j v j represents the anticipation flux difference between the preceding lattice j + 1 and the current lattice j. e optimal velocity functions and the function H(·) are the same as those in Ge and Cheng's model [39].
After eliminating the velocity term v j in equations (1) and (6), the following density equation can be derived: 2 Complexity When p � 0, the model reduces to Ge and Cheng's model [39]. In addition, the lattice model coincides with Nagatani's model [26] when p � 0 and τ � 0. e influence of the "backward-looking" effect and the anticipation effect has been studied by linear stability analysis. e uniform traffic stream with the fixed density ρ 0 and the optimal velocity V(ρ 0 ) is defined as the stable state. One can obtain the solution of the steady-state vehicular system as follows: en, y j is supposed as a small perturbation added into the steady-state solution at lattice j, and the perturbed solution is Equation (9) is substituted into equation (7), and the derived equation is linearized. e equation is obtained as Let the small perturbation y j (t) � exp(ikj + zt), and inserting it into equation (10), we gain Assuming z � z 1 (ik) + z 2 (ik) 2 + · · · and putting it into equation (11), we reserve the first-order and second-order terms of ik by neglecting higher-order terms. en, one can get e uniform steady-state flow remains stable as z 2 is positive. On the contrary, the uniform flow tends to be unstable if the value of z 2 is negative. When z 2 is equal to zero, the formula of the neutral stability condition is acquired as where e stability condition for the uniform traffic flow is obtained as It is obvious that the stable condition (14) is reduced to that of Ge and Cheng's model [39] when p � 0. Equation (14) shows that the parameters p, τ, and c play important roles in the performance of the traffic system. e phase diagram of (ρ, a) is shown in Figure 1, where ρ is density and a denotes sensitivity. Note that c denotes a positive constant considering the "backward-looking" effect [40,41].
ree solid curves in each pattern of Figure 1 represent the neutral stability curves under c � 0.1. e phase diagram is divided into two regions: the stable and unstable regions are above and below the solid neutral stability curve, respectively. In the stable region, the traffic flow will remain stable with a perturbation. In the unstable region, the congested traffic will happen even with a small perturbation. Figure 1 depicts the neutral stability curves of the FADBE model for different values of driver's anticipation time τ and the sensitivity coefficient p of the anticipation flux difference when c � 0.1. As shown in Figure 1, the corresponding stable region expands gradually with the increasing values of τ and p, which means that the driver's anticipation effect can contribute to the stabilization of traffic flow obviously.

Nonlinear Stability Analysis
To investigate the influence of the "backward-looking" effect and the anticipation effect of flux difference, we carried out the reduction perturbation method to obtain the mKdV equation. e slow variables X and T are defined as follows: where ε (0 < ε ≪ 1) is a small positive scaling parameter.
Substituting equation (15) into equation (7), each term of equation (7) is expanded up to the fifth-order of ε with the Taylor expansion technique as follows: Complexity where the value of a c is defined as a c � a(1 + ε 2 ). e squared and cubic terms of ε are eliminated in equation (16) with the One can get the simplified equation as follows:  Complexity To derive the standard mKdV equation, the following transformations are applied: us, equation (17) is converted into the standard mKdV equation as follows: When the term O(ε) is ignored, the kink-antikink soliton solution of the mKdV equation can be rewritten as where c represents the propagation velocity. e specific value of c can be provided only when the following solvability condition is satisfied: where . By solving equation (22), the general solution of propagation velocity c can be calculated as Hence, the corresponding kink-antikink solution is derived as where ε 2 � ((a c /a) − 1), and the amplitude of the density wave A is defined as ��������� � (g 1 ε 2 C/g 2 ). According to the kinkantikink solution equation (24), one can get the coexisting phases which include the freely moving phase with low density (ρ j � ρ c − A) and the congested phase with high density (ρ j � ρ c + A).
Based on the linear and nonlinear theory analyses, numerical simulations are carried out with the comprehensive consideration of the anticipation effect of flux difference and the "backward-looking" effect.

Numerical Simulations
As a positive constant c considering the "backward-looking" effect has been studied fully in some previous studies [36,40,41], this paper will focus on the effects of parameters p and τ in this part. Periodic boundary conditions are adopted by a step function as follows: where Δρ � 0.01 is the initial perturbation and other parameters are set as N � 200, a � 1.2, and ρ 0 � ρ c � 0.25.

e Effect of Anticipation Time τ.
In order to analyze the anticipation time's influence on the stability of the traffic system, the temporal and spatial evolution of density waves between time t � 10000 − 10200 s is depicted in Figure 4 for different anticipation time τ with fixed parameters p � 0.1 and c � 0.05. Since the given parameters in Figures 4(a)-4(c) do not satisfy the linear stable condition equation (14), kinkantikink density waves appear and the initial stable traffic flow evolves into nonuniform traffic flow with the added perturbation. To reveal the amplitude of the density wave more clearly, Figure 5 describes the density profile of each   Figure 4(d) satisfy the linear stability condition, the added perturbation will disappear, and the traffic flow will eventually return to a uniform state. Figure 5(d) can get results similar to Figure 4(d).
is phenomenon indicates that driver's anticipation effect of flux difference could significantly promote the stabilization of traffic flow.

e Effect of Sensitivity Coefficient p.
To expose the sensitivity coefficient's influence on the stability of the traffic system, the temporal and spatial evolution of density waves between time t � 10000 − 10200 s is exhibited in Figure 6 for different values of p with fixed c � 0.05 and τ � 2. When p � 0, the FADBE model degrades into Ge and Cheng's model [39]. Note that the parameters in Figures 6(a)-6(c) do not satisfy the linear stable condition equation (14). erefore, the stop-go traffic density waves occur with the small added perturbation in Figures 6(a)-6(c). Meanwhile, the amplitude of the density wave decreases gradually as the value of p increases. Especially in Figure 6 Figure 7 describes the density profile of each subfigure shown in Figure 6 at t � 10200. As parameter p increases, the traffic flow becomes more gentle and stable. is indicates that the sensitivity coefficient p contributes to suppressing the traffic congestion. Figures 8(a) and 8(b) describe the flux against the density with different parameters corresponding to the traffic flow in Figures 4 and 6, respectively. In Figure 8, the amplitude of the hysteresis loops reduced with the increasing value of τ or p. When the corresponding parameter τ ≥ 2.5 or p ≥ 0.15, the hysteresis loop turns into a stable point, which means that the traffic flow eventually becomes stable even with the 10 Complexity added small perturbation. To sum up, as the anticipation time τ or the sensitivity coefficient p increases, the stability of the system can be enhanced by incorporating the driver's anticipation effect and the "backward-looking" effect.

Conclusion
In this paper, a novel lattice hydrodynamic model with the anticipation effect of flux difference and the "backwardlooking" effect is constructed to explore the traffic driving behavior for a single-lane traffic system. e stability conditions and the solution of the mKdV equation are deduced, respectively, based on the linear stability analysis and nonlinear theory. e results of the theoretical analysis display that the increasing values of both the driver's anticipation time and the sensitivity coefficient of the anticipation flux difference can contribute to enlarging the stable region. Numerical simulations reveal that the FADBE model is effective in stabilizing traffic flow even with perturbations. By considering the anticipation effect of flux difference and the "backward-looking" effect simultaneously, the traffic congestion can be alleviated effectively, which is in accordance with the conclusions of the theoretical analysis. Future work will focus on solving the traffic congestion on multilane road problem, the automation for longitudinal driving, and the vehicular emergency warning system.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.