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

In the actual traffic environment, the driver’s aggressive driving behaviors are closely related to the traffic conditions at the nextnearest grid point at next time step. )e 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. )e linear stability method is used to analyze the current stability of the new model, and the neutral stability curve is obtained. )e 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. )e 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.


Introduction
e 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. e 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 shortterm results. If the city is compared to the human body, then traffic is equivalent to the flow of human blood. e 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][2][3] have broadened the field of traffic flow research from other perspectives in recent years.
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. erefore, 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 v max � 1. In recent years, two-lane traffic flow models [54][55][56][57][58][59][60][61][62][63][64][65][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.
is paper consists of the following parts. Firstly, the development history of traffic flow related models is introduced in Section 1. 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.

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. is 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. Figure 1 shows the diagrammatic sketch of the lane change in a two-lane expressway traffic flow model, in which continuous twolane traffic flow is discretized into two rows of uniform grid points. e direction of the arrow shown in Figure 1 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 In the absence of lane diversion, the conservation equation of the site j on the first lane is given as and the conservation equation of the site j on the second lane is given as where ρ 1,j and ρ 2,j denote the density of the lattice j on the first lane and the second lane, respectively. ρ 1,j v 1,j − ρ 1,j−1 v 1,j−1 represents the relative flux difference between the lattices j and j − 1 of the first lane. ρ 2,j v 2,j − ρ 2,j−1 v 2,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 twolane continuity equation is obtained as follows:

Mathematical Problems in Engineering
where ρ 0 is the initial density, ρ j � ((ρ 1,j + ρ 2,j )/2) and 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: where driver's sensitivity coefficient is defined as a, and a � (1/τ). V(ρ j+1 ) in equation (4) represents the optimal velocity function, and V(ρ j+1 ) � ((V(ρ 1,j+1 ) + V(ρ 2,j+1 ))/2). e optimal velocity function is adopted as follows: where v max � 2 is the maximal velocity and ρ c � 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: represents the integral of the flow difference between t − t 0 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]. e dynamic equation of traffic density can be obtained by eliminating the velocity v in equations (6) and (7). e equation is expressed as follows:

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 density ρ 0 and constant velocity V(ρ), so the steady-state solution of the traffic flow for this new model is given as follows: y j is assumed to be a small disturbance of steady-state flow on lattice j, and let Inserting equation (10) into equation (8), one obtains e following equation about z is derived by expanding y j � exp(ikj + zt): where V ′ (ρ 0 ) � (dV(ρ)/dρ)|ρ � ρ 0 , and the sum L l�1 e −lzτ can be expressed as L l�1 e −lzτ � L. Let z � z 1 (ik) + z 2 (ik) 2 + · · ·, in the equation, omit the higher order terms of ik larger than the second power, we obtain When the value of z 2 is less than 0, the uniform steadystate flow will be unstable for long-wavelength modes. In contrast, when the value of z 2 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: e stable region of two-lane traffic flow system can be obtained by satisfying the following condition: Figure 2 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.
In Figure 2, 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, c � 0 and β values are changed to 0, 0.05, and0.1, respectively. e three solid lines represent the neutral stability curves when P � 0.1, c � 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 c � 0.1, changing the value of β, the value of neutral stability curve decreases gradually, and the area of unstable region decreases accordingly. e two lines with the same color are the neutral stability curves when the values of P and β are fixed and the values of c are 0 and 0.1, respectively. By comparing each solid line with the dashed line, we find that when c � 0.1, the neutral stability curve is below the neutral stability curve when c � 0. It can be explained that increasing the lane change rate c has a stabilizing effect on traffic flow. Figure 2(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. ree color dotted lines are neutral stability curves with c � 0, and three color solid lines are neutral stability curves with c � 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 c � 0 and c � 0.1, respectively, the two curves with the same c value can be seen that when c takes 0.1, the stability region is larger. We can conclude that increasing lane change rate c is beneficial to traffic flow stability in a certain range.
rough 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.

The mKdV Equation
In this section, the reduced perturbation method is used to analyze the above model near the critical point (ρ c , a c ). In the unstable area of traffic flow, the slow variables X and T for a small direct scaling parameter ε(0 < ε ≪ 1) are supposed as follows: where b alleges a constant to be determined. e density is defined as follows: By introducing equations (17) and (18) into equation (8), the following nonlinear partial differential expression is obtained: where the coefficients values h i (i � 1, 2, . . . , 7) contained in equation (19) are given by where (19), the simplified equation is obtained as follows: where the parameter values g i (i � 1, 2, . . . , 5) in equation (21) are given by In order to derive the regularization equation, the following transformations should be made: It is drawn out that the mKdV equation with an O(ε) correction term is given as follows: where M[R 0 ′ ] � (g 3 /g 1 )z 2 X R ′ + (g 4 /g 1 )z 4 X R ′ + (g 1 g 5 /g 2 ) z 2 X R ′ 3 . Subsequently, when ignoring the O(ε), the kinkantikink soliton solutions of the mKdV equation are proved as follows: where c means the determined velocity of the kink-antikink solution. By solving the following integral equation, the value of c will be obtained: With the method Mathematical Problems in Engineering described in Ref. [48], the general velocity c is obtained as follows: erefore, the kink-antikink solution of the mKdV equation near the critical point can be rewritten as follows:

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][65][66]. In order to facilitate numerical simulation, equation (8) is simplified by using difference method as follows: where the time step Δt is 0.05. e initial conditions for this extended traffic lattice model are chosen as follows: where the value of N which defined the sum number of the sites is opted as 100, a � 1.25. ρ j (0) � ρ 0 � 0.25, ρ j (1) � ρ 0 � 0.25, for j ≠ 55, 56, ρ j (1) � 0.25 − 0.1, for j � 50, ρ j (1) � 0.25 + 0.1, for j � 51, and the value of σ which defined the initial disturbance is opted as 0.05.  Figure 3(a) reveals when β and P is equal to 0.05 and 0.15, let c 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 c � 0.1 and c � 0.2, respectively. When the value of c 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). rough the continuous debugging of numerical simulation, with the gradual increase of c 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 twodimensional phase diagrams with density wave corresponding to  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  Mathematical Problems in Engineering 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.      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. rough 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. e numerical simulation results also validate the correctness of our theoretical results in practical operation.

Conclusion
According to the actual traffic environment, a novel twolane lattice hydrodynamic model is proposed, which considers the effects of the driver's aggressive effect and the relative flow difference integral. e 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. e 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.

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.