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

Authors: Jin Wan; Xin Huang; Wenzhi Qin; Xiuge Gu; Min Zhao
Journal: Complexity
 (2021)

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## Abstract

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. The corresponding stability conditions are derived through a linear stability analysis. Then, the nonlinear theory is also applied to derive themKdV 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. The results show that the positive constant γ, 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.


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## Body

## 1. Introduction

With the rapid development of wireless technologies and intelligent vehicles in the past years, the vehicle-to-vehicle (V2V) communication [1, 2] can not only alleviate the congested traffic flow but also provide pleasurable driving experiences. To reveal the realistic traffic phenomena, there have emerged various traffic models such as car-following models [3–10], cellular automation models [11, 12], macrotraffic models [13–15], lattice hydrodynamic models [16–21], continuum models [22–24], and gas kinetic models [25].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.(1)∂tρj+ρ0ρjvj−ρj−1vj−1=0,(2)∂tρjvj=aρ0Vρj+1−ρjvj,where ρ0Vρj+1 and ρjvj denote the optimal flow and the actual flow, respectively. ρ0 is the average density, and a refers to the driver’s sensitivity. Then, the stability condition and mKdV equation can be, respectively, obtained with the linear analysis and nonlinear analysis methods. Subsequently, many extended works have been developed by taking different factors into accounts, such as driver’s memory [27–31], driver’s anticipation effect [32–34], density difference [35], traffic interruption probability [36–38], and “backward-looking” effect [39–41].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:(3)ρjt+τvjt+τ=ρ0VFρj+1t+Hρ¯−ρj−1t⋅Hρj−1t−ρcρ0VBρj−1t,where H· and ρc represent the Heaviside function and the safety density, respectively. ρ¯ denotes a parameter close to 1. VF· is the optimal velocity function for “forward-looking.” While VB· is the optimal velocity function for “backward-looking,” which works if and only if ρc≤ρj−1t≤ρ¯ holds. Functions VF· and VB· have exactly the opposite effects and are defined as(4)VFρj+1=tanh2ρ0−ρj+1ρ02−1ρc+tanh1ρc,VBρj−1=γ−tanh2ρ0−ρj−1ρ02−1ρc+tanh1ρc,where γ 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. This 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 single-lane 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. Then, 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:(5)∂tρjvj=aρ0Vρj+1−a1+k1ρjvj+k2ρj+1t+τvj+1t+τ−ρjvj,where τ represents the driver’s anticipation time and k1 and k2 denote the deviation degree of speed vj 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. This motivates us to develop a new lattice model by incorporating the “backward-looking” effect with the anticipation effect of flux difference.The outline of this paper is organized as follows. In Section2, 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.


## 2. 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:(6)∂tρjvj=aρ0VFρj+1t+Hρ¯−ρj−1t·Hρj−1t−ρcρ0VBρj−1t−aρjvj+apρj+1t+τvj+1t+τ−ρjvj,where p is the sensitivity coefficient of the anticipation flux difference. The greater p is, the stronger the driver’s anticipation effect is. τ means the anticipation time, and ρj+1t+τvj+1t+τ−ρjvj represents the anticipation flux difference between the preceding lattice j+1 and the current lattice j. The optimal velocity functions and the function H· are the same as those in Ge and Cheng’s model [39].After eliminating the velocity termvj in equations (1) and (6), the following density equation can be derived:(7)∂t2ρjt+aρ02Vρj+1t−VFρjt+VBρj−1t−VBρj−2t+a∂tρj+ap∂tρj−∂tρj+1t+τ=0.Whenp=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.The influence of the “backward-looking” effect and the anticipation effect has been studied by linear stability analysis. The uniform traffic stream with the fixed densityρ0 and the optimal velocity Vρ0 is defined as the stable state. One can obtain the solution of the steady-state vehicular system as follows:(8)ρjt=ρ0,vj=VFρ+VBρ.Then,yj is supposed as a small perturbation added into the steady-state solution at lattice j, and the perturbed solution is(9)ρjt=ρ0+yit.Equation (9) is substituted into equation (7), and the derived equation is linearized. The equation is obtained as(10)∂t2yjt+aρ02VF′ρ0yj+1t−yjt+aρ02VB′ρ0yj−1t−yj−2t−ap∂tyj+1t+τ+ap+a∂tyj=0.Let the small perturbationyjt=expikj+zt, and inserting it into equation (10), we gain(11)z2+aρ02VF′ρ0eik−1+VB′ρ0eik−e−2ik−apzezτ+ik+ap+az=0.Assumingz=z1ik+z2ik2+⋯ and putting it into equation (11), we reserve the first-order and second-order terms of ik by neglecting higher-order terms. Then, one can get(12)z1=−ρ02VF′ρ0+VB′ρ0,z2=2apz1−2z12+2apz12τ−aVF′ρ0ρ02+3aVB′ρ0ρ022a.The uniform steady-state flow remains stable asz2 is positive. On the contrary, the uniform flow tends to be unstable if the value of z2 is negative. When z2 is equal to zero, the formula of the neutral stability condition is acquired as(13)a=2VF′ρ0+VB′ρ02ρ02M,where M=−VF′ρ0+3VB′ρ0−2VF′ρ0+VB′ρ0p+2VF′ρ0+VB′ρ02pτρ02. The stability condition for the uniform traffic flow is obtained as(14)a>2VF′ρ0+VB′ρ02ρ02M.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 γ play important roles in the performance of the traffic system.The phase diagram ofρ,a is shown in Figure 1, where ρ is density and a denotes sensitivity. Note that γ denotes a positive constant considering the “backward-looking” effect [40, 41]. Three solid curves in each pattern of Figure 1 represent the neutral stability curves under γ=0.1. The 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 γ=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.Figure 1
The neutral stability curves forγ=0.1 for different values of p and τ: (a) p=0.1; (b) p=0.2; (c) p=0.3; (d) p=0.4.
(a)(b)(c)(d)Figure2 demonstrates the neutral stability curves of the FADBE model for different values of τ and γ when p=0.1. As shown in Figure 2, the corresponding neutral stability curves and critical points decrease gradually with the increasing values of τ and γ. When the value of γ is fixed in each pattern of Figure 2, the stability of the traffic flow can be improved efficiently with the increasing value of anticipation time τ.Figure 2
The neutral stability curves forp=0.1 for different values of τ and γ: (a) τ=0.5; (b) τ=1.5; (c) τ=2.5; (d) τ=3.5.
(a)(b)(c)(d)Figure3 exhibits that the phase diagram of ρ,a of the FADBE model for different values of the positive constant γγ=0.05,0.1,0.15,0.2 and the sensitivity coefficient pp=0.1,0.2,0.3 when τ=1.5. With the increasing value of the positive constant γ in each pattern of Figure 3, the amplitudes of neutral stability curves all fall down gradually when the value of the parameter p is fixed. For different values of γ, it can be seen from each pattern of Figure 3 that the corresponding peak value of solid curves lowers down with the increasing value of p. These phenomena demonstrate that when the value of γ or p increases, the area of the stable region becomes larger. Thus, it can be concluded that the traffic flow stability can be improved by incorporating the positive constant γ and the sensitivity coefficient p.Figure 3
The neutral stability curves forτ=1.5 for different values of γ and p: (a) γ=0.05; (b) γ=0.1; (c) γ=0.15; (d) γ=0.2.
(a)(b)(c)(d)


## 3. 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. The slow variablesX and T are defined as follows:(15)X=εj+bt,T=ε3t,ρj=ρc+εRX,T,where ε0<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:(16)ε2b+ρc2VF′+VB′∂XR+ε3b2a−b2pτ−32ρc2VB′+12ρc2VF′∂X2R+ε4∂TR−12b3pτ2−76ρc2VB′−16ρc2VF′∂X3R+16ρc2VB′″+16ρc2Vf#x2032;″∂XR3+ε52ba−2bpτ∂X∂TR+−16b4pτ3−58ρc2VB′+124ρc2VF′∂X4R+112ρc2Vf′″−14ρc2VB′″∂X2R3=0,where VF′=dVρ/dρ|ρ=ρcVF′″=d3Vρ/dρ3|ρ=ρc and VB′=dVρ/dρ|ρ=ρcVB′″=d3Vρ/dρ3|ρ=ρc. Near the critical point ρc,ac, the value of ac is defined as ac=a1+ε2. The squared and cubic terms of ε are eliminated in equation (16) with the consideration of b=−ρc2VF′+VB′. One can get the simplified equation as follows:(17)ε4∂TR−g1∂X3R+g2∂XR3+ε5g3∂X2R+g4∂X4R+g5∂X2R3=0,where(18)g1=−16ρc27VB′+VF′+3pρc4τ2VB′+VF′3,g2=ρc2VF′″+VB′″,g3=12ρc23VB′−VF′−2pVB′+VF′+2pρc2τVB′+VF′2,g4=−124ρc215VB′−VF′+4pρc6τ3VB′+VF′4−13aρc4apτ−1VB′+VF′3pρc4τ2VB′3+VF′−13aρc4apτ−1VB′+VF′3pρc4τ2VF′3+VB′7+9pρc4τ2VF′2,g5=−112ρc23VB′′′−VF′′′−13aρc4apτ−1VB′+VF′VB′′′+VF′′′.To derive the standardmKdV equation, the following transformations are applied:(19)T=1g1T′,R=g1g2R′.Thus, equation (17) is converted into the standard mKdV equation as follows:(20)∂T′R′=∂X3R′−∂XR′3+εg3g1∂X2R′+g4g1∂X4R′+g5g2∂X2R′3.When the termOε is ignored, the kink-antikink soliton solution of the mKdV equation can be rewritten as(21)R0′X,T′=ctanhc2X−cT′,where c represents the propagation velocity. The specific value of c can be provided only when the following solvability condition is satisfied:(22)R0′,MR0′≡∫−∞∞dXR0′MR0′=0,where MR0′=1/g1g3∂X2R′+g4∂X4R′+g1g5/g2∂X2R′3. By solving equation (22), the general solution of propagation velocity c can be calculated as(23)c=5g2g32g2g4−3g1g5.Hence, the corresponding kink-antikink solution is derived as(24)ρj=ρc+εg1cg2tanhc2X−cg1T,where ε2=ac/a−1, and the amplitude of the density wave A is defined as g1ε2C/g2. According to the kink-antikink solution equation (24), one can get the coexisting phases which include the freely moving phase with low density ρj=ρc−A and the congested phase with high density ρj=ρc+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.


## 4. Numerical Simulations

As a positive constantγ 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:(25)ρj0=ρ0=0.25,j∈1,N,ρj1=ρ0;j≠N2,N2−1,ρ0+Δρ;j=N2−1,ρ0−Δρ;j=N2,where Δρ=0.01 is the initial perturbation and other parameters are set as N=200, a=1.2, and ρ0=ρc=0.25.


### 4.1. The Effect of Anticipation Timeτ

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 timet=10000−10200s is depicted in Figure 4 for different anticipation time τ with fixed parameters p=0.1 and γ=0.05. Since the given parameters in Figures 4(a)–4(c) do not satisfy the linear stable condition equation (14), kink-antikink 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 subfigure shown in Figure 4 at t = 10200. Moreover, Figures 5(a)–5(c) reveal that the amplitudes of density waves corresponding to Figures 4(a)–4(c) weaken as the anticipation time τ increases. When τ=2.5, i.e., the parameters in 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). This phenomenon indicates that driver’s anticipation effect of flux difference could significantly promote the stabilization of traffic flow.Figure 4
The evolution of the traffic densities for different values of parameterτ: (a) τ=1.0; (b) τ=1.5; (c) τ=2.0; (d) τ=2.5.
(a)(b)(c)(d)Figure 5
The density profile att=10200 with different values of τ: (a) τ=1.0; (b) τ=1.5; (c) τ=2.0; (d) τ=2.5.
(a)(b)(c)(d)


### 4.2. The Effect of Sensitivity Coefficientp

To expose the sensitivity coefficient’s influence on the stability of the traffic system, the temporal and spatial evolution of density waves between timet=10000−10200s is exhibited in Figure 6 for different values of p with fixed γ=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). Therefore, 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(d), the congested state entirely disappears when p=0.15. The relationship between Figures 6 and 7 is similar to the relationship between Figures 4 and 5. 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. This indicates that the sensitivity coefficient p contributes to suppressing the traffic congestion.Figure 6
The evolution of the traffic densities for different values of parameterp: (a) p=0; (b) p=0.05; (c) p=0.1; (d) p=0.15.
(a)(b)(c)(d)Figure 7
The density profile att=10200 with different values of p: (a) p=0; (b) p=0.05; (c) p=0.1; (d) p=0.15.
(a)(b)(c)(d)Figures8(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 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.Figure 8
Hysteresis loops of traffic flux and density for (a)τ=1,1.5,2,and2.5 with fixed γ=0.05 and (b) p=0,0.05,0.1,and0.15 with fixed γ=0.05.


## 4.1. The Effect of Anticipation Timeτ

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 timet=10000−10200s is depicted in Figure 4 for different anticipation time τ with fixed parameters p=0.1 and γ=0.05. Since the given parameters in Figures 4(a)–4(c) do not satisfy the linear stable condition equation (14), kink-antikink 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 subfigure shown in Figure 4 at t = 10200. Moreover, Figures 5(a)–5(c) reveal that the amplitudes of density waves corresponding to Figures 4(a)–4(c) weaken as the anticipation time τ increases. When τ=2.5, i.e., the parameters in 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). This phenomenon indicates that driver’s anticipation effect of flux difference could significantly promote the stabilization of traffic flow.Figure 4
The evolution of the traffic densities for different values of parameterτ: (a) τ=1.0; (b) τ=1.5; (c) τ=2.0; (d) τ=2.5.
(a)(b)(c)(d)Figure 5
The density profile att=10200 with different values of τ: (a) τ=1.0; (b) τ=1.5; (c) τ=2.0; (d) τ=2.5.
(a)(b)(c)(d)


## 4.2. The Effect of Sensitivity Coefficientp

To expose the sensitivity coefficient’s influence on the stability of the traffic system, the temporal and spatial evolution of density waves between timet=10000−10200s is exhibited in Figure 6 for different values of p with fixed γ=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). Therefore, 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(d), the congested state entirely disappears when p=0.15. The relationship between Figures 6 and 7 is similar to the relationship between Figures 4 and 5. 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. This indicates that the sensitivity coefficient p contributes to suppressing the traffic congestion.Figure 6
The evolution of the traffic densities for different values of parameterp: (a) p=0; (b) p=0.05; (c) p=0.1; (d) p=0.15.
(a)(b)(c)(d)Figure 7
The density profile att=10200 with different values of p: (a) p=0; (b) p=0.05; (c) p=0.1; (d) p=0.15.
(a)(b)(c)(d)Figures8(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 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.Figure 8
Hysteresis loops of traffic flux and density for (a)τ=1,1.5,2,and2.5 with fixed γ=0.05 and (b) p=0,0.05,0.1,and0.15 with fixed γ=0.05.


## 5. Conclusion

In this paper, a novel lattice hydrodynamic model with the anticipation effect of flux difference and the “backward-looking” effect is constructed to explore the traffic driving behavior for a single-lane traffic system. The stability conditions and the solution of themKdV equation are deduced, respectively, based on the linear stability analysis and nonlinear theory. The 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.



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*Source: 4642202-2021-11-05.xml*