# Analysis of a Novel Two-Lane Hydrodynamic Lattice Model Accounting for Driver’s Aggressive Effect and Flow Difference Integral

Authors: Xinyue Qi; Hongxia Ge; Rongjun Cheng
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In the actual traffic environment, the driver’s aggressive driving behaviors are closely related to the traffic conditions at the next-nearest grid point at next time step. The driver adjusts the acceleration of the driving vehicle by predicting the density of the front grid points. Considering the driver’s aggressive effect and the relative flow difference integral, a novel two-lane lattice hydrodynamic model is presented in this paper. The linear stability method is used to analyze the current stability of the new model, and the neutral stability curve is obtained. The nonlinear analysis of the new model is carried out by using the theory of perturbations, and the mKdV equation describing the density of the blocked area is derived. The theoretical analysis results are verified by numerical simulation. From the analysis results, it can be seen that the driver’s aggressive effect and the relative flow difference integral can improve the stability of traffic flow comprehensively.


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## Body

## 1. Introduction

The rapid improvement of urbanization level and the explosive growth of household car ownership have made urban traffic contradiction more and more prominent and even become an important factor restricting urban economic development. Especially since 1990, it is the fastest growing period of motor vehicles. The traffic flow is more concentrated, unable to move and stop, and the contradiction is extremely sharp. In order to change the traffic situation, many big cities began to construct ring roads, large interchanges, elevated roads, and metro. Because of the hasty decision and improper decision-making, they often only pay attention to local improvement and can only achieve short-term results. If the city is compared to the human body, then traffic is equivalent to the flow of human blood. The flow of blood is not smooth, accidents occur frequently, the light cause is short-term paralysis, and the heavy cause is long-term poor flow and the formation of “dead city,” which will bring immeasurable losses to the country and people. Traffic flow theory comes into being under such circumstances. Furthermore, some researchers [1–3] have broadened the field of traffic flow research from other perspectives in recent years.In the face of many hazards brought by traffic congestion, the construction and development of urban roads urgently need the support and guidance of theoretical knowledge, and the theoretical model of traffic flow is applied. In order to explore the intrinsic mechanism of traffic congestion, scholars at home and abroad have carried out a series of studies on the physical phenomena of traffic flow [4–28]. Based on the characteristics of traffic flow, various traffic models have been developed through research, such as car-following models [26, 29–35], cellular automata models [36–41], macro-traffic models [42–47], and lattice hydrodynamic model [30, 48–52].Lattice hydrodynamic modeling and nonlinear analysis are important tools for macro-traffic flow simulation and traffic pressure relief. Lattice hydrodynamics model has been widely used to simulate the phase transition phenomena in real traffic because it can describe the macroscopic characteristics of traffic flow. Nagatani [12, 13] discretized the hydrodynamic model to obtain a simple lattice hydrodynamic model, which incorporated the optimized velocity function of the micromodel into the macromodel. Therefore, the lattice hydrodynamic model has both the characteristics of macromodel and micromodel, and it is convenient for simulation calculation and prediction. Based on the lattice hydrodynamic model of Nagatani [13], many extended models are proposed to study the nonlinear phenomena in traffic flow under intelligent transportation environment. Xue [14] established a one-dimensional traffic flow lattice hydrodynamic model considering the interaction between the nearest neighbor and the next-nearest neighbor lattices. Using the linear stability theory, they deduced the neutral stability conditions of traffic flow and the mKdV equation describing the phase transition of traffic congestion by using the nonlinear analysis method. Considering the influence of any number of lattice information in front, Ge et al. [15] proposed an extended lattice hydrodynamic model. Ge and Cheng [16] further improved the lattice model by combining “backward looking” effect and deduced the mKdV equation near the neutral stable line. Considering the optimal traffic flow, Zhu et al. [17, 18] extended the lattice model and found that this model can significantly alleviate traffic congestion. On the basis of previous studies, Peng et al. [19, 20] combined various road information to further promote the study of lattice hydrodynamics model. Tian et al. [21, 22] considered different traffic information to further improve and apply the lattice hydrodynamic model of Nagatani.Using lane changing rules, Nagatani [53] constructed a new extended two-lane traffic model with vmax=1. In recent years, two-lane traffic flow models [54–66] have been developed gradually due to the restrictions of single lane being unable to change lanes and overtaking. In these two-lane traffic flow models, more attention is paid to the relative displacement, speed between two adjacent vehicles. However, in the real traffic flow, the driver’s aggressive driving behaviors also have a significant impact on the traffic situation. In real traffic life, the aggressive driving behaviors are closely related to the starting, braking, overtaking, and lane changing of driving vehicle. At the same time, the integral form of flow difference can describe traffic flow more accurately and objectively. In the two-lane lattice hydrodynamic model, a new model is proposed with consideration of driver’s aggressive effect and flow difference integral.This paper consists of the following parts. Firstly, the development history of traffic flow related models is introduced in Section1. Based on the existing models, an extended two-lane lattice hydrodynamic model is proposed in Section 2. In Section 3, the linear stability analysis of the new model is presented. In Section 4, the new model is analyzed by nonlinear analysis method and the mKdV equation is solved. Numerical simulations are carried out in Section 5. Finally, the conclusions are provided in Section 6.


## 2. The Novel Two-Lane Lattice Hydrodynamic Model

With the rapid development of the modern urban process, the situation of single lane in the real road environment is gradually reduced, which makes the study of single lane traffic model cannot meet the needs of the actual road traffic. This contradiction has gradually attracted the attention of traffic flow researchers, and the two-lane traffic model came into being. Compared with single lane, the two-lane traffic model can reflect the complexity of the actual traffic, and more and more academic papers on the two-lane model gradually increase. But at present, the research of the two-lane model has a lot of room to expand, which is the significance of this paper. Figure1 shows the diagrammatic sketch of the lane change in a two-lane expressway traffic flow model, in which continuous two-lane traffic flow is discretized into two rows of uniform grid points.Figure 1
The schematic model of traffic flow on a two-lane highway.The direction of the arrow shown in Figure1 represents the forward flow direction of traffic flow in actual traffic operation, and each circle represents a grid point. When the traffic flow gap between two rows at a grid point is too large, the traffic flow will be adjusted according to the actual situation: part of the traffic flow will be transferred from the large part to the small part of the traffic flow. As shown in Figure 1, when the traffic flow in site j−1 of the second lane is larger than that in site j of the first lane, a large part of the traffic flow flows from point j−1 of the second lane to point j of the first lane. If the traffic flow in point j of the first lane is larger than that in point j+1 of the second lane, some traffic flow will be transferred from point j to point j+1. In the course of lane changing of two-lane traffic flow, we define lane changing rate as γρ02V′ρ0, with dimensionless constant coefficient represented by γ.In the absence of lane diversion, the conservation equation of the sitej on the first lane is given as(1)∂tρ1,j+ρ0ρ1,jv1,j−ρ1,j−1v1,j−1=γρ02V′ρ0ρ2,j−1−2ρ1,j+ρ2,j+1,and the conservation equation of the site j on the second lane is given as(2)∂tρ2,j+ρ0ρ2,jv2,j−ρ2,j−1v2,j−1=γρ02V′ρ0ρ1,j−1−2ρ2,j+ρ1,j+1,where ρ1,j and ρ2,j denote the density of the lattice j on the first lane and the second lane, respectively. ρ1,jv1,j−ρ1,j−1v1,j−1 represents the relative flux difference between the lattices j and j−1 of the first lane. ρ2,jv2,j−ρ2,j−1v2,j−1 represents the relative flux difference between the lattices j and j−1 of the second lane.By substituting equation (1) into equation (2), the two-lane continuity equation is obtained as follows:(3)∂tρj+ρ0ρjvj−ρj−1vj−1=γρ02V′ρ0ρj−1−2ρj+ρj+1,where ρ0 is the initial density, ρj=ρ1,j+ρ2,j/2 and ρjvj=ρ1,jv1,j+ρ2,jv1,j/2.Assuming that the evolution equation of traffic flow on each lane is not affected by the lane change proposed by Nagatani, the evolution equation of two-lane traffic is as follows:(4)∂tρjvj=aρ0Vρj+1−aρjvj,where driver’s sensitivity coefficient is defined as a, and a=1/τ. Vρj+1 in equation (4) represents the optimal velocity function, and Vρj+1=Vρ1,j+1+Vρ2,j+1/2.The optimal velocity function is adopted as follows:(5)Vρ=vmax2tanh2ρ0−ρρ02−1ρc+tanh1ρc,where vmax=2 is the maximal velocity and ρc=4 is the safety density.A large number of studies have shown that driver’s habits and personality will affect traffic flow in varying degrees. However, at this stage, no scholars have taken the driver’s aggressive effect into account in two-lane traffic. Based on this, we take the driver’s aggressive driving behaviors into two-lane to explore what the specific impact of driver’s aggressive effect will have on two-lane traffic operation. We use the integral form of flow difference to give the specific density of the current road. To sum up, a new extended two-lane model of traffic flow is constructed, and its continuity equation and the motion equation are listed as follows:(6)∂tρjvj=aρ01−PVρj+1t+PVρj+2t+t0−aρjvj+aβ∫t−t0tρj+1svj+1s−ρjsvjsds,(7)∂tρj+ρ0ρjvj−ρj−1vj−1=γρ02V′ρ0ρj−1−2ρj+ρj+1,where ∫t−t0tρj+1svj+1s−ρjsvjs represents the integral of the flow difference between t−t0 and t, besides, gives the integral a coefficient β. In actual traffic, compared with ordinary drivers, some drivers who are confident of their driving skills will drive vehicles close to their front because they can quickly predict the driving information of the front car, adjust, and accelerate in time. P is the weighting value, 0≤P≤0.5, which means the intensity of the driver’s aggressive effect. As P=0, this new expanded traffic model returns to Nagatani’s model [13].The dynamic equation of traffic density can be obtained by eliminating the velocityv in equations (6) and (7). The equation is expressed as follows:(8)aρ021−PVρj+1t−Vρjt+PVρj+2t+t0−Vρj+1t+t0+∂ρt2jt−γρ02V′ρ0∂tρj−1−2∂tρj+∂tρj+1+a∂tρj−aγρ02V′ρ0ρj−1s−2ρjs+ρj+1s−aβρj+1t−ρj+1t−t0−ρjt+ρjt−t0+aβτγρ02V′ρ0∑l=1L3ρjt−lτ−3ρj+1t−lτ+ρj+2t−lτ−ρj−1t−lτ=0.


## 3. Linear Stability Analysis

In this section, linear analysis is used to discuss the driver’s aggressive effect, the relative flow difference integral, and the lane change rate on traffic operation. Obviously, the uniform traffic flow means constant densityρ0 and constant velocity Vρ, so the steady-state solution of the traffic flow for this new model is given as follows:(9)ρjt=ρ0,vjt=Vρ,yj is assumed to be a small disturbance of steady-state flow on lattice j, and let(10)ρjt=ρ0+yjt.Inserting equation (10) into equation (8), one obtains(11)∂t2yj+a∂tyj−aβyj+1−yj+1t−t0−yj+yjt−t0+aρ02PV′ρ0yj+2−yj+1+aρ02PV′ρ0t0∂tyj+2−∂tyj+1+aγρ02V′ρ0yj−1−2yj+yj+1+aγρ02V′ρ0yj−1−2yj+yj+1+aρ021−PV′ρ0yj+1−yj+γρ02V′ρ0∂tyj−1−2∂tyj+∂tyj+1−aβγρ02V′ρ0τ∑l=1L3yjt−lτ−3yj+1t−lτ+yj+2t−lτ−yj−1t−lτ=0.The following equation aboutz is derived by expanding yj=expikj+zt:(12)aρ021−PV′ρ0eik−1+PV′ρ0e2ik−eik+Pt0V′ρ0ze2ik−eik+aγρ02V′ρ0e−ik−2+eik−aβeik−eik−zt0−1+e−zt0+z2+az+γρ02V′ρ0ze−ik−2+eik−aβτγρ02V′ρ0∑l=1Le−lzτ3−3eik+e2ik−e−ik=0,where V′ρ0=dVρ/dρρ=ρ0, and the sum ∑l=1Le−lzτ can be expressed as ∑l=1Le−lzτ=L. Let z=z1ik+z2ik2+⋯, in the equation, omit the higher order terms of ik larger than the second power, we obtain(13)z1=−ρ02V′ρ0,(14)z2=−ρ02V′ρ02a−ρ02V′ρ012−P−Pt0z1−γ+βt0z1.When the value ofz2 is less than 0, the uniform steady-state flow will be unstable for long-wavelength modes. In contrast, when the value of z2 is greater than 0, the uniform steady-state flow will remain stable. After introducing equation (13) into equation (14), we get the neutral stability condition is as follows:(15)a=−2ρ02V′ρ01−P−Pt0z1−γ+βt0.The stable region of two-lane traffic flow system can be obtained by satisfying the following condition:(16)a>−2ρ02V′ρ01−P−Pt0z1−γ+βt0.Figure2 reveals the phase diagram in the ρ,a-plane. With the change of the influence coefficient of different parameters, the correlation between the variation and parameters of the stable and unstable regions can be shown in Figure 2.Figure 2
Phase diagram in parameter spaceρ,a for equation (16). (a) P=0.1. (b) β=0.05.
(a)(b)In Figure2, the solid and dotted lines of color represent the neutral stability curves presented under different constraints. Figure 2(a) shows the neutral stability curves of three color dotted lines when P=0.1,γ=0 and β values are changed to 0, 0.05, and0.1, respectively. The three solid lines represent the neutral stability curves when P=0.1,γ=0 and β is changed to 0, 0.05, and 0.1. Among them, the unstable region is shown below the curve and the stable region is shown above the curve. By comparing dotted lines separately, we can see that the value of β increases gradually, the area of unstable region decreases gradually, and the surface value of stable region increases gradually. Similarly, by comparing the three solid lines in the graph, we deduce that when P=0.1 and γ=0.1, changing the value of β, the value of neutral stability curve decreases gradually, and the area of unstable region decreases accordingly. The two lines with the same color are the neutral stability curves when the values of P and β are fixed and the values of γ are 0 and 0.1, respectively. By comparing each solid line with the dashed line, we find that when γ=0.1, the neutral stability curve is below the neutral stability curve when γ=0. It can be explained that increasing the lane change rate γ has a stabilizing effect on traffic flow.Figure2(b) shows the neutral stability curve when the density difference integral β is 0.05 and P is 0, 0.1, and 0.2, respectively. Three color dotted lines are neutral stability curves with γ=0, and three color solid lines are neutral stability curves with γ=0.1. It is obvious that increasing the value of P can gradually expand the stability region by changing the curve in the graph. Similarly, when comparing γ=0 and γ=0.1, respectively, the two curves with the same γ value can be seen that when γ takes 0.1, the stability region is larger. We can conclude that increasing lane change rate γ is beneficial to traffic flow stability in a certain range. Through the current stability analysis, we can get the neutral stability curve, and the simulation results can be verified. Finally, it is concluded that increasing the values of the driver’s aggressive effect, the relative flow difference integral, and the lane change rate is helpful to relieve traffic pressure.


## 4. The mKdV Equation

In this section, the reduced perturbation method is used to analyze the above model near the critical pointρc,ac. In the unstable area of traffic flow, the slow variables X and T for a small direct scaling parameter ε0<ε≪1 are supposed as follows:(17)X=εj+bt,T=ε3t,where b alleges a constant to be determined.The density is defined as follows:(18)ρjt=ρc+εRX,T.By introducing equations (17) and (18) into equation (8), the following nonlinear partial differential expression is obtained:(19)ε2h1∂XR+ε3h2∂X2R+ε4∂TR+h3∂X3R+h4∂XR3+ε5h5∂X∂TR+h6∂X4R+h7∂X2R3=0,where the coefficients values hii=1,2,…,7 contained in equation (19) are given by(20)h1=b+ρc2V′ρ0,h2=b2a+ρc2V′ρ02+Pb+γρc2V′ρ0+βτL,h3=γbρc2V′ρ0a+ρc2V′ρ06−βbτ2∑l=1Ll+Pb2−Pγρc2V′ρ0h4=ρc2V″′ρ06,h5=2ba−P,h6=ρc2V′ρ024+γρc2V′ρ012+βb2τ32∑l=1Ll2+βτL12−Pb6−Pγρc2V′ρ02,h7=ρc2V″′ρ012,where V′=dVρ/dρρ=ρc and V″′=d3Vρ/dρ3ρ=ρc.By substitutingb=−ρc2V′ρc, τ=1+ε2τc into equation (19), the simplified equation is obtained as follows:(21)ε4∂TR−g1∂X3R+g2∂XR3+ε5g3∂X2R+g4∂X4R+g5∂X2R3=0,where the parameter values gii=1,2,…,5 in equation (21) are given by(22)g1=−bγρc2V′ρ0ac−ρc2V′ρ06+βbτ2∑l=1Ll+Pb2+Pγρc2V′ρ0,g2=ρc2V″′ρ06,g3=−ρc2V′ρ02−Pρc2V′ρ0−γρc2V′ρ0+βτL,g4=γρc2V′ρ012+ρc2V′ρ024+12βb2τ3∑l=1Ll2+βτL12−Pb6−Pγρc2V′ρ02−2bac−P×γbρc2V′ρ0ac+ρc2V′ρ06−βbτ2∑l=1Ll+Pb2−Pγρc2V′ρ0,g5=ρc2V″′12−ρc2V″′6×2bac−P.In order to derive the regularization equation, the following transformations should be made:(23)T=1g1T′,R=g1g2R′.It is drawn out that the mKdV equation with anOε correction term is given as follows:(24)∂TR′=∂X3R′−∂XR′3+εMR0′,where MR0′=g3/g1∂X2R′+g4/g1∂X4R′+g1g5/g2∂X2R′3. Subsequently, when ignoring the Oε, the kink-antikink soliton solutions of the mKdV equation are proved as follows:(25)R0′X,T′=ctanhc2X−cT′,where c means the determined velocity of the kink-antikink solution. By solving the following integral equation, the value of c will be obtained:(26)∫−∞+∞cg1g2g2g3∂X2R′+g2g4∂X4R′+g1g5∂X2R′3tanhc2X−cT′dX=0.SupposeR′X,T′=R0′X,T′+εR1′X,T′ takes account of Oε correction. Among them, R0′,MR0′≡∫−∞+∞dX′R0′MR0′. With the method described in Ref. [48], the general velocity c is obtained as follows:(27)c=5g2g32g2g4−3g1g5.Therefore, the kink-antikink solution of the mKdV equation near the critical point can be rewritten as follows:(28)ρjt=hc±g1cg2ττc−1×tanhc2ττc−1×j+1−cg1ττc−1t.


## 5. Numerical Simulation

In this section, the influence of the driver’s aggressive driving behaviors and the relative flow difference integral of traffic flow are verified by numerical simulation [64–66]. In order to facilitate numerical simulation, equation (8) is simplified by using difference method as follows:(29)aβτ2LL+12Δtρj+1t+Δt−ρj+1t−2ρjt+Δt+2ρjt+ρj−1t+Δt−ρj−1t−γρ02V′ρ0Δtρj−1t+Δt−ρj−1t−2ρjt+Δt+2ρjt+ρj+1t+Δt−ρj+1t−ρjt+2Δt+2ρjt+Δt−ρjt−aβτLΔt2ρj+1t−2ρjt+ρj−1t−aγρ02V′ρ0Δt2ρj−1t−2ρjt+ρj+1t+aPΔtρj+1t+Δt−ρj+1t−ρjt+Δt+ρjt−aρ02Vρj+1t−Vρjt−aΔtρjt+Δt−ρjt+aPγρ02V′ρ0Δt23ρjt−3ρj+1t+ρj+2t−ρj−1t=0,where the time step Δt is 0.05.The initial conditions for this extended traffic lattice model are chosen as follows:(30)ρj1=ρj0=ρ0,j≠N2,N2+1,ρ0−σ,j=N2,ρ0+σ,j=N2+1,where the value of N which defined the sum number of the sites is opted as 100, a=1.25. ρj0=ρ0=0.25, ρj1=ρ0=0.25, for j≠55,56, ρj1=0.25−0.1, for j=50, ρj1=0.25+0.1, for j=51, and the value of σ which defined the initial disturbance is opted as 0.05.Figures3(a)–3(d) express the three-phase diagram of the new expanded two-lane model under the condition of changing the lane change rate step by step. It shows that when the values of P and β are fixed and unchanged, the vibration amplitude of density wave decreases gradually and the vibration frequency decreases as the value of the lane change rate γ increases from 0 to 0.3. Figure 3(a) reveals when β and P is equal to 0.05 and 0.15, let γ be 0. As can be seen from Figure 3(a) intuitively and effectively, the density wave oscillates, ranging from 0.2 to 0.9, and the vibration frequency is the highest than the other three figures. Figures 3(b) and 3(c) show the phase diagrams at γ=0.1 and γ=0.2, respectively. When the value of γ is gradually increased to 0.3, the vibration amplitude of density wave is about 0.3 to 0.7 which is shown in Figure 3(d). Through the continuous debugging of numerical simulation, with the gradual increase of γ value, the density wave curve becomes more and more stable. For the actual traffic flow, this means that the traffic operation capacity can be enhanced and the traffic efficiency is greatly improved. Figure 4 is the two-dimensional phase diagrams with density wave corresponding to Figure 3. They finally verify that considering lane change rate is of positive value to traffic flow. Increasing the value of γ in a certain range can improve the ability of traffic flow to run smoothly.Figure 3
The evolution of the traffic densities with differentγ values. (a) γ=0. (b) γ=0.1. (c) γ=0.2. (d) γ=0.3.
(a)(b)(c)(d)Figure 4
The density profile att=10300 with different γ values. (a) γ=0. (b) γ=0.1. (c) γ=0.2. (d) γ=0.3.
(a)(b)(c)(d)Figures5(a)–5(d) express the three-dimensional phase diagram of density wave when the values of β and γ are fixed to 0.05 and 0.1, and the values of the coefficient P of the driver’s aggressive effect are gradually changed. Figure 5(a) reveals that the curve fluctuation range of density wave ranges from 0.1 to 0.9 without considering the driver’s aggressive effect. In Figures 5(b) and 5(c) with the increase of P value to 0.1 and 0.2, the fluctuation range of density wave decreases obviously, and the change of vibration frequency is not significant. When the value of P is increased to 0.3, the density wave presents a straight line, which means that the traffic flow will reach an ideal state without blockage. Figure 6 is the two-dimensional headway profile phase diagram of density wave matching Figure 5. In the two-lane lattice hydrodynamics model, considering the driver’s aggressive effect, traffic congestion can be effectively alleviated.Figure 5
The evolution of the traffic densities with differentP values. (a) P=0. (b) P=0.1. (c) P=0.2. (d) P=0.3.
(a)(b)(c)(d)Figure 6
The density profile att=10300 with different P values. (a) P=0. (b) P=0.1. (c) P=0.2. (d) P=0.3.
(a)(b)(c)(d)Figures7 and 8 show the three-dimensional phase diagram and corresponding two-dimensional phase diagram when the value of β is gradually increased from 0 to 0.15 when the values of P and γ are fixed as 0.1 and 0.1. From Figures 7(b)–7(d), with the increase of the influence parameter value β of the relative flow difference integral, the density wave tends to be stable. When the value of β is 0.15, the density wave approximates a straight line. Considering the influence of the relative flow difference integral has a positive effect on traffic flow.Figure 7
The evolution of the traffic densities with differentβ values. (a) β=0. (b) β=0.05. (c) β=0.1. (d) β=0.15.
(a)(b)(c)(d)Figure 8
The density profile att=10300 with different β values. (a) β=0. (b) β=0.05. (c) β=0.1. (d) β=0.15.
(a)(b)(c)(d)Through numerical simulation, it is proved that considering the factors of the driver’s aggressive effect, the relative flow difference integral, and the lane change rate in two-lane traffic flow model is beneficial to improve traffic operation. The numerical simulation results also validate the correctness of our theoretical results in practical operation.


## 6. Conclusion

According to the actual traffic environment, a novel two-lane lattice hydrodynamic model is proposed, which considers the effects of the driver’s aggressive effect and the relative flow difference integral. The analytical stability conditions are obtained by linear stability analysis method, and the spatial phase diagram of sensitivity coefficient density is given. Based on the change of the neutral stability curve in the phase diagram, this paper analyzes the three influencing factors: driver’s radical effect, relative flow difference integral, and lane change rate, which play an important role in improving the stability of traffic flow. Nonlinear method is used to analyze the model in advance and the corresponding numerical validation is made. The numerical simulation results are consistent with the theoretical analysis. In conclusion, it is reasonable to consider the driver’s aggressive effect and the relative flow difference integral comprehensively in a two-lane lattice hydrodynamic model.



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