Improved Car-Following Model for Connected Vehicles on Curved Multi-Lane Road

Abstract

Under the development of intelligent network technology, drivers can obtain the surrounding traffic situation in real time, which is conducive to improving the stability of traffic flow. Therefore, this paper proposes a new curve-car-following model considering multi-vehicle information of adjacent lanes in connected environment, and conducts linear and nonlinear stability analyses of the model to demonstrate the effectiveness of the proposed model and its ability to improve the stability of traffic system; in addition, numerical simulation experiments of traffic flow convoys are designed to analyze the effects of different parameters in the proposed model on the stability of the traffic flow and test the proposed model’s ability to maintain the following behavior in a convoy. Furthermore, numerical simulation experiments are designed to analyze the effects of different parameters in the proposed model on the stability of traffic flow, and to test the ability of the proposed model to maintain the following behavior in the convoy. The model can provide theoretical guidance to alleviate traffic congestion and improve safety, and extend the application of the following model in curved multi-lane road scenarios.

1. Introduction

Car-following (CF) depicts the interaction between two neighboring vehicles traveling in a platoon in a single lane and is a fundamental microscopic driving behavior of traffic flow. The car-following model employs a dynamic approach to investigate the corresponding behavior of the following vehicle (FV) brought on by a change in the leading vehicle’s motion condition (LV). The macro-traffic flow can be used to examine the micro-driver behavior using the single-lane traffic flow characteristics. The car-following model is useful for a variety of applications, including self-cruise control, traffic capacity analysis, micro-traffic simulation, and traffic safety evaluation.

At present, the car-following model driven by theory is widely studied. Through observation and research on car-following behavior, different scenarios and factors are considered, and the corresponding car-following model is proposed based on practical assumptions. Among them, Pipes [1] first performed a dynamic analysis of the platooning traffic flow from a kinematics perspective and assumed that the driver expects to follow the vehicle at a prescribed safe distance. Newell [2] developed a car-following model taking the optimal speed into account, and Bando et al. [3], according to the basic assumption that the speed is adjusted by the distance between two adjacent vehicles, proposed the optimal velocity (OV) model. The OV model has the advantages of a relatively simple model structure and easy application and can be used to describe more traffic phenomenon formation mechanisms. However, Helbing et al. [4] calibrated the optimal speed model and pointed out its existing problems; as a result, they constructed the Generalized Force (GF) model. Later, Jiang et al. [5] pointed out that in the process of modeling the car-following model, one should consider the impact of the velocity difference between the leading and following vehicles on the driving behavior and safety of the two consecutive vehicles in the GF model. On this basis, a Full Velocity Difference (FVD) model is proposed. Scholars have given the FVD model a great deal of attention and research because it is better able to capture the peculiarities of microscopic traffic flow in the real environment. A number of expanded models have also been created as a result of this research.

On the one hand, in the non-connected environment, the above more classical follow-through model plays an important role, and many scholars have extended the modeling based on it to meet the follow-through modeling in different traffic scenarios. Sun et al. [6] improved a car-following model regarding the Backward-Looking and Velocity Difference (BLVD), considered the information of the following vehicle in the FVD model, and proved that the information of the rear-vehicle was used in the research of the car-following process significantly. In the connected environment, Xiao et al. [7] developed a car-following model considering the forward and backward effect on unstructured roads. Tang et al. [8] considered the influence of the whistle effect on the process of car-following and constructed a new car-following model. Peng et al. [9] constructed an extended model considering the change of optimal speed with memory. Ma et al. [10] proposed an extended vehicle-following model considering the driver’s response delay by improving the FVD model and discussed the relationship between traffic flow stability and driver response delay through theoretical analysis. In addition, there are many car-following models considering various factors, such as driver characteristic [11], driver memory [12], honk environment [13], energy sustainability [14,15], mixed traffic flow [16], and so on. In SUMO, the following model is used to simulate the moving behavior of a vehicle on the road, especially how the vehicle follows the vehicle in front of it. SUMO implements a variety of following models, but the Krauss model [17] is the default following model in SUMO, which calculates the maximum safe speed of the vehicle based on the concept of safe distance. The IDM [18] is a more complex following model, which takes into account the current speed of the vehicle, the distance to the vehicle in front, and the difference in speed between the two vehicles to dynamically adjust the speed of the vehicle.

Car-following models have been studied not only on horizontal single-lane roads, but also on curvy roads and two-lane roads by some scholars. In considering a two-lane car-following model, Ge et al. [19] accounted for the frictional interference of neighboring lanes and analyzed stability using feedback signals. According to Ponnu et al.’s [20] hypothesis, the behavior of the automobile trailing also depends on the relative speed to the next lane. Jiang et al. [21] developed an extended-view car-following model that considers the visual angles of the same lane and adjacent lane and their rate of change. Gao et al. [22] proposed a left- and right-lane speed difference car-following model considering the relative speed of the following vehicles in the left and right adjacent lanes, and through the stability analysis, it was concluded that the model can more effectively stabilize the middle lane traffic flow. With the aid of their innovative car-following model and inter-vehicle communication, Ou et al. [23] investigated how each vehicle moves in a two-lane traffic system when an event takes place in a lane. Regarding the curved road following model, Li et al. [24] developed an extended following model that considered the effect of the lateral gap on the curvy road with uncertain speed. Sun et al. [25] established an extended following model taking the driver’s willingness to drive the vehicle smoothly into consideration when driving on a curved road. An expanded car-following model that takes into account the impact of roadways with both curved and straight sections was suggested by Zheng et al. [26].

On the other hand, in a connected environment, technology for communication and information is a rapidly evolving subject. Using vehicle-to-vehicle (V2V) technology, researchers have built many car-following models for connected cars and have been able to gather motion information of adjacent vehicles or a larger range of vehicles by employing V2V communication technology. Jiao et al. [27] investigated the effects of drivers’ traits on car-following behavior in a V2V communication environment and put forth an expanded car-following model that can enhance vehicle mobility, safety, fuel efficiency, and emissions in various traffic situations. Chen et al. [28] suggested an enhanced car-following model for connected cars that took into consideration the differences in electronic throttle angle and average headway. In order to construct extended models that take into account the effect of average speed and mean expected velocity on the car-following behavior, Sun et al. [29] and Zhu and Zhang [30], respectively, used V2V communication technology to acquire the speed and position information of numerous cars. The model of following the ground in a connected environment is not limited to ground vehicle traffic; Wiseman, Y. [31], in “Autonomous Vehicles”, suggests that autonomous flying vehicles will also have the ability to connect and move in platoons. Similarly, Venzao [32] suggests that the aerial vehicle could follow the ground one, maintaining it in its field of view.

In conclusion, a number of achievements have been made by researchers who have studied micro-traffic flow models in the context of vehicular ad hoc networks (VANETs) and under varied road conditions. Nonetheless, the car-following models for micro-traffic flow that are examined in these works have potential for enhancement. For the foreseeable future, drivers will still play a major part in driving, even while the evolving driving environment made possible by VANETs gives cars the ability to gather driving information outside the line of sight. The combined deployment of vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V) technologies in the VANET environment can give drivers information about road conditions and other cars within communication range. The entire influence of drivers obtaining information about surrounding cars connected to the curved road characteristics on the driving process and traffic flow evolution mechanisms has still to be researched in particular car-following situations on curved multi-lane highways.

The paper’s primary organization is as follows: A curved multi-lane road car-following model is proposed in Section 2, and its stability under various parameter conditions is described in Section 3. Section 4 summarizes the outcomes of the numerical simulation, examines the impact of improved car-following model, and focuses on the traffic flow stability. The overall framework of the thesis is shown Figure 1.

2. The Extended Car-Following Model

At present, most micro-traffic flow car-following models focus on different road traffic scenarios. Obviously, compared to straight road dual-lane scenarios, vehicles following a leading vehicle on a curved road are more susceptible to the influence of multiple lanes, because of its specific road type and vehicle travel characteristics. This implies that drivers of following vehicles in the self lane will pay some attention to vehicles in the adjacent lane while following vehicles on the same lane, which leads to driving behavior that is different from that on a straight road, particularly interfering with the vehicle following the vehicle in front in the self lane.

Under a connected environment, drivers have real-time access to multi-lane multi-vehicle information, so when their own vehicle is following the behavior of the vehicle in front of this lane, the driver is easily affected by the influence of the vehicles from both sides of the lane (the vehicle driving on a multi-lane straight road is shown in Figure 2a; the curved multi-lane traffic scenario is displayed in Figure 2b). This influence will be amplified due to the driver on the curved road driving more carefully and due to the curved road characteristics.

First of all, in the existing car-following model, the full speed difference model (FVD) improved the optimal speed model (OV) considering the influence of speed difference, and many scholars have also extended and improved it on this basis. Its specific formula is as follows:

$$\frac{d{V}_n\left(t\right)}{dt}=\alpha \left[V\left(\Delta x_n\left(t\right)\right)-v_n\left(t\right)\right]+\lambda \Delta v_n\left(t\right),$$

(1)

where $\alpha =\frac{1}{\tau }$ represents the driver’s sensitivity coefficient; τ is the delay time for the nth vehicle to reach the ideal speed; $\lambda$ is the responding factor to the velocity difference; $v_n\left(t\right)$, $\Delta x_n\left(t\right)$ represent the velocity and head distance between the nth vehicle and the front vehicles; the head distance of $\Delta x_{n+1}\left(t\right)-\Delta x_n\left(t\right)$; $V\left(\Delta x_n\left(t\right)\right)$ is the optimal speed of the vehicle under the head distance $\Delta x_n\left(t\right)$; the optimized speed function $V\left(\Delta x_n\left(t\right)\right)$ is described as follows:

$$V\left(\Delta x_n\left(t\right)\right)=\frac{v_{max}}{2}\left[\tanh \left(\Delta x_n\left(t\right)-h_c\right)+\tanh \left(h_c\right)\right]$$

(2)

Figure 3 shows basic scenarios for modeling the improved car-following model. The green car indicates the ego vehicle following the leading vehicle. In a connected environment, connected vehicles are able to obtain information about the traveling status of multiple vehicles around them in real time, which is conducive to improving traffic flow stability, especially when traveling in curves. Therefore, this paper proposes a new microscopic traffic flow following model for curved roads, which takes into account the driving information of both sides of the lane.

This study made some assumptions about communication technology connecting the vehicles during the modeling process:

(1)

The vehicle itself is able to sense the driving information of multiple vehicles around it and interact with the information;

(2)

The information interaction process is secure and in real time.

Modeling the driver’s following behavior in a curved multi-lane traffic scenario is divided into three steps: firstly, we define parameter p to represent the sensitivity coefficient of the driver’s following behavior in his own lane, and $(1-p)$ represents the sensitivity coefficient on the traffic dynamics of the two neighboring lanes; secondly, we use different optimized speed functions to represent the influence of the two lanes on the following behavior of his own vehicle; finally, we combine the driving dynamics characteristics of vehicles on curved roads. The specific modeling steps are established below.

$$\frac{d^2{s}_n\left(t\right)}{d{t}^2}=\alpha \left[p{V}_{self}\left(\Delta S_n\left(t\right)\right)+{\displaystyle (1-p)}{V}_{adj}\left(\Delta S_n\left(t\right)\right)-\frac{d{s}_n\left(t\right)}{dt}\right]+\gamma \frac{d\Delta s_n\left(t\right)}{dt},$$

(3)

When driving on a curved road, the relationship between the radian and radius is expressed as follows:

$$\begin{gathered} s_n\left(t\right)=r{\theta }_n\left(t\right), \\ {\Delta s}_n\left(t\right)=r\Delta {\theta }_n\left(t\right), \end{gathered}$$

(4)

The improved multi-lane curve following model is obtained by bringing Equation (4) into Equation (3):

$$\frac{d^2{\theta }_n\left(t\right)}{d{t}^2}=\frac{\alpha }{r}\left[p{V}_{self}\left(r\Delta {\theta }_{n,n+1}\left(t\right)\right)+\left(1-p\right)V_{adj}\left(r\Delta {\theta }_{n,m}\left(t\right)\right)-r\frac{d{\theta }_n\left(t\right)}{dt}\right]+\gamma \frac{d\Delta {\theta }_{n,n+1}\left(t\right)}{dt}$$

(5)

Based on vehicle dynamics on curved roads, the optimized speed function in the model is rewritten as follows:

$$V_{self}\left(·\right)=\omega \frac{\sqrt{\mu gr}}{2}\left[\tanh \left(r\Delta {\theta }_n\left(t\right)-s_c\right)+\tanh \left(s_c\right)\right],$$

(6)

$$\begin{gathered} V_{adj}\left(·\right)=\omega \frac{\sqrt{\mu g(r-l)}}{2k}{\sum }_{i=1}^k\left[\tanh \left((r-l)\Delta {\theta }_n\left(t\right)-s_c\right)+\tanh \left(s_c\right)\right]+\omega \frac{\sqrt{\mu g(r+l)}}{2k}{\sum }_{i=1}^k\left[\tanh \left((r+ \\ l)\Delta {\theta }_n\left(t\right)-s_c\right)+\tanh \left(s_c\right)\right], \end{gathered}$$

(7)

In the formula, $\omega$ is the driving safety factor under the condition of curved road; $\mu$ is the friction coefficient of the road; $r$ is the radius of the curve; $s_c$ is the safe following head distance under the radius of the curve $r$; furthermore, safe headway distances are usually affected by various factors, such as traffic flow, weather conditions, etc., so this parameter in the model should be considered comprehensively based on the relevant influences when the model is applied; $l$ is the width of lanes.

3. Stability Analysis

Examining the stability of traffic flow involves determining how road disturbances affect the flow’s overall condition. Small disturbances will spread upstream along the vehicle flow if the system is unstable, and the normally constant and smooth flow of traffic will become congested. If the system is stable, the small disturbance will finally be regulated in a relatively limited range as it gradually shrinks and vanishes during the propagation phase.

3.1. Linear Stability Analysis

Therefore, in this section, we will examine the stability of this model using the linear stability approach. First, it is presumed that the flow of traffic is balanced, each vehicle maintains the same head-to-head distance b, and each vehicle moves at the same ideal speed $V(s_1,s_2)$. Based on this condition, the location of each vehicle at time t of the traffic flow is as follows:

$$\left\{\begin{array}{c}{x}_n^0\left(t\right)=bn+V\left(s_1,s_2\right)t\\ x_m^0\left(t\right)=bm+V\left(s_1,s_2\right)t\end{array}\right.,$$

(8)

Apply disturbances $y_n\left(t\right)$, $y_m\left(t\right)$ to the n and mth vehicles in the main lane and adjacent lane, respectively, and the position information after adding the disturbance is represented as follows:

$$\left\{\begin{array}{c}{x}_n\left(t\right)=x_n^0\left(t\right)+y_n\left(t\right)\\ x_m\left(t\right)=x_m^0\left(t\right)+y_m\left(t\right)\end{array}\right.,$$

(9)

$$\left\{\begin{array}{c}{y}_n\left(t\right)=Aexp\left(ikn+zt\right)\\ y_m\left(t\right)=\delta Aexp\left(ikn+zt\right)\end{array}\right.,$$

(10)

According to Formulas (6) and (7), we can get

$$\left\{\begin{array}{c}\Delta y_{n,n+1}\left(t\right)=Aexp(ikn+zt){(e}^{ik}-1)\\ \Delta y_{n,m}\left(t\right)=(\delta -1)Aexp(ikn+zt)\end{array}\right.,$$

(11)

$$\left\{\begin{array}{c}{\dot{y}}_n\left(t\right)=zAexp\left(ikn+zt\right)\\ {\ddot{y}}_n\left(t\right)=z^{2}Aexp\left(ikn+zt\right)\end{array}\right.,$$

(12)

Linearize Formula (3), perform Taylor expansion, and ignore higher-order terms:

$$\frac{d^2{y}_n\left(t\right)}{d{t}^2}=\frac{\alpha }{r}\left[p{V}_{self}^{\prime }\left(s_c\right)\Delta y_{n,n+1}\left(t\right)+{\displaystyle (1-p)}{V}_{adj}^{\prime }\left(s_c\right)\Delta y_{n,m}\left(t\right)-r\cdot \frac{d{y}_n\left(t\right)}{dt}\right]+\gamma p\frac{d\Delta y_{n,n+1}\left(t\right)}{dt}$$

(13)

Expanding $y_n\left(t\right)$ with Taylor series, the equation for $z$ can be obtained:

$$z^2=\left[V^{\prime }\left(s_c\right)p\left(e^{ik}-1\right)+V^{\prime }\left(s_c\right)(1-p){\delta (e}^{ik}-1)-z\right]+\gamma \left(e^{ik}-1\right)z$$

Substitute the formula $z=z_1\left(ik\right)+z_2(ik{)}^2+\cdots \mathrm{and} e^{ik}=1+ik+\frac{1}{2}(ik{)}^2+\cdots$ into Formula (11):

$$\left\{\begin{array}{c}{z}_1=V_{self}^{\prime }\left(s_c\right)p+V_{adj}^{\prime }\delta (1-p)\\ z_2=\frac{1}{2}\left[V_{self}^{\prime }\left(s_c\right)p+V_{adj}^{\prime }(1-p)\delta \right]+\frac{1}{\alpha }\left[\gamma z_1-{z_1}^2\right]\end{array}\right.,$$

(14)

when $z_2<0$, the traffic flow under interference is in an unstable state, and when $z_2>0$, the traffic flow under interference is in a stable state, so we take $z_2=0$ to get the traffic flow stability under the interference condition. We can derive the following:

$$\alpha >2\left[V_{self}^{\prime }\left(s_c\right)p+V_{adj}^{\prime }(1-p)\delta \right]-2\gamma ,$$

(15)

3.2. Nonlinear Stability Analysis

In order to further analyze the impact of multi-lane perimeter vehicle information on traffic flow on curved roads, the proposed model is analyzed nonlinearly using the perturbation method [33]. The main analysis method is to introduce a small perturbation parameter $\epsilon$ near the critical point $\left(a_c{,h}_c\right)$ to study the nonlinear behavior of the system near the critical value $a_c$. And since it is used to study the slow variable and slow behavior in time and space near the critical point, the two slow variables in time and space are defined separately as follows:

$$\begin{gathered} X=\epsilon (n+bt), \\ T={\epsilon }^{3}t, \end{gathered}$$

(16)

where $b$ is a parameter to be determined and the headspace is set to the following:

$$\Delta x_n\left(t\right)=s_c+\epsilon R(X,T),$$

(17)

In order to facilitate the nonlinear analysis of the dynamical equations of the vehicle system using perturbation methods, the equations first need to be written as follows:

$$\begin{array}{c}\frac{d^2\Delta x_n\left(t\right)}{d{t}^2}=a\left[\begin{array}{c}p\left(V_{self}\left(\Delta x_{n+1}\left(t\right)\right)-V_{self}\left(\Delta x_n\left(t\right)\right)\right)\\ \left.+(1-p)\left(V_{adj}\left(\Delta x_n\left(t\right)\right)-V_{adj}\left(\Delta x_{n-1}\left(t\right)\right)\right)-\frac{d\Delta x_n\left(t\right)}{dt}\right]\end{array}\right]\\ +\lambda \left[\frac{d\Delta x_{n+1}\left(t\right)}{dt}-\frac{d\Delta x_n\left(t\right)}{dt}\right],\end{array}$$

(18)

Equation (19) is the intermediate formula in the nonlinear derivation process, which is a Taylor expansion to ${\epsilon }^5$ after bringing Equation (17) into Equation (18).

$$\begin{gathered} {\epsilon }^2\alpha \left[p{V}_{self}^{\prime }+(1-p)V_{adj}^{\prime }-b\right]{\partial }_{S}R+{\epsilon }^3\left[\left({-b}^2+\lambda b+\frac{\alpha [p{V}_{self}^{\prime }+\left(1-p\right)V_{adj}^{\prime }]}{2}\right){\partial }_S^{2}R\right]+{\epsilon }^4\left[\left(\frac{\alpha [p{V}_{self}^{\prime }+\left(1-p\right)V_{ad}^{\prime }]}{6}+ \\ \frac{\lambda b}{2}\right){\partial }_S^{3}R+\frac{\alpha [p{V}_{self}^{\prime \prime \prime }+\left(1-p\right)V_{adj}^{\prime \prime \prime }]}{2}{\partial }_S{R}^3-{\alpha \partial }_{T}R\right]+\left\{(-2b+\lambda ){\partial }_{ST}R+\frac{1}{24}\left(\alpha [p{V}_{self}^{\prime }+\left(1-p\right)V_{adj}^{\prime }]+4b\lambda \right){\partial }_S^{4}R+ \\ \frac{\alpha [p{V}_{self}^{\prime \prime \prime }+\left(1-p\right)V_{adj}^{\prime \prime \prime }]}{4}{\partial }_S^2{R}^3\right\}=0, \end{gathered}$$

(19)

where

$$\left\{\begin{array}{c}{V}_{self}^{\prime }=V_{self}^{\prime }\left(s_c\right)={\left.\frac{d{V}_{self}\left(\Delta x_n\right)}{d\Delta x_n}\right|}_{\Delta x_n=s_c}\\ V_{self}^{\prime \prime \prime }=V_{self}^{\prime \prime \prime }\left(s_c\right)={\left.\frac{d^3{V}_{self}\left(\Delta x_n\right)}{d^3\Delta x_n}\right|}_{\Delta x_n=s_c}\\ V_{adj}^{\prime }=V_{adj}^{\prime }\left(s_c\right)={\left.\frac{d{V}_{adj}\left(\Delta x_{n-1}\right)}{d\Delta x_{n-1}}\right|}_{\Delta x_{n-1}=s_c}\\ V_{adj}^{\prime \prime \prime }=V_{adj}^{\prime }\left(s_c\right)={\left.\frac{d^3{V}_{adj}\left(\Delta x_{n-1}\right)}{d^3\Delta x_{n-1}}\right|}_{\Delta x_{n-1}=s_c}\end{array}\right.,$$

(20)

In the vicinity of the critical point $({\alpha }_c,h_c)$, take $b=p{V}_{self}^{\prime }+(1-p)V_{adj}^{\prime }$ and $\alpha =(1-{\epsilon }^2){\alpha }_c$, eliminating the second- and third-order terms of the parameter $\epsilon$, and simplifying and rewriting the above equation as follows:

$${\epsilon }^4\left[{\partial }_{T}R-g_1{\partial }_S^{3}R+g_2{\partial }_S{R}^3\right]+{\epsilon }^5\left[g_3{\partial }_S^{2}R+g_4{\partial }_S^{2}R+g_5{\partial }_S^2{R}^3\right]=0,$$

(21)

where the coefficients $g_i(i=\mathrm{1,2},\dots ,5)$ are as follows:

$$\begin{gathered} g_1=-\left[\frac{{\alpha }_c\left[p{V}_{self}^{\prime }+\left(1-p\right)V_{adj}^{\prime }\right]}{6}+\frac{\lambda {\left(\left[p{V}_{self}^{\prime }+\left(1-p\right)V_{adj}^{\prime }\right]\right)}^3}{2}\right], \\ {g}_2=\frac{{\alpha }_c\left[p{V}_{self}^{\prime \prime \prime }+\left(1-p\right)V_{adj}^{\prime \prime \prime }\right]}{2}, \\ {g}_3=\frac{\alpha \left[p{V}_{self}^{\prime }+\left(1-p\right)V_{adj}^{\prime }\right]\left[{\alpha }_c^2+\left(\left[p{V}_{self}^{\prime }+\left(1-p\right)V_{adj}^{\prime }\right]-\lambda \right){\alpha }_c\right]}{2} \\ {g}_4=\frac{1}{24}\left[{\alpha }_c^3+4{\alpha }_c^2\lambda \right]\left[p{V}_{self}^{\prime }+\left(1-p\right)V_{adj}^{\prime }\right] \\ {g}_5=\frac{{\alpha }_c^3\left[p{V}_{self}^{\prime \prime \prime }+\left(1-p\right)V_{adj}^{\prime \prime \prime }\right]}{4} \end{gathered}$$

(22)

The mKdv equation with a correction term $o\left(\epsilon \right)$ can be obtained by passing the equation through $T=\frac{1}{g_1}{T}^{\prime }$ and $R=\sqrt{\frac{g_1}{g_2}}{R}^{\prime }$:

$${\partial }_{T^{\prime }}{R}^{\prime }={\partial }_S^3{R}^{\prime }-{\partial }_S{R}^{\prime 3}-\epsilon \left[\frac{g_3}{g_1}{\partial }_S^2{R}^{\prime }+\frac{g_4}{g_1}{\partial }_S^4{R}^{\prime }+\frac{g_5}{g_2}{\partial }_S^2{R}^{\prime 3}\right],$$

(23)

The standard mKdv equation is obtained by neglecting the correction term in Equation (23). The model kink and anti-kink wave solution is obtained as follows:

$$R_o^{\prime }\left(S,T^{\prime }\right)=\sqrt{c}\tanh \sqrt{\frac{c}{2}}\left(S-c{T}^{\prime }\right),$$

(24)

According to the equation, the solution for the propagation velocity $c$ of the kink wave can be found when the condition is satisfied by the propagation velocity $c$:

$$\left(R_o^{\prime },M\left[R_o^{\prime }\right]\equiv {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }} d{S}^{\prime }{R}_o^{\prime }M\left[R_o^{\prime }\right]\right),$$

(25)

In Equation (25), $M\left[R_o^{\prime }\right]=M\left[R^{\prime }\right]$, and integrating Equation (25) yields a chosen propagation velocity $c$:

$$c=\frac{5g_2{g}_3}{3g_1{g}_5-2g_2{g}_4},$$

(26)

From the mKdV equation, the kink and anti-kink soliton solution of the headway is given by

$$\Delta S_n(t)=s_c\pm \sqrt{\frac{g_{1}c}{g_2}\left(\frac{\alpha }{{\alpha }_c}-1\right)}\times \tanh \sqrt{\frac{c}{2}\left(\frac{\alpha }{{\alpha }_c}-1\right)}\times \left[n+\left(1-c{g}_1\right)\left(\frac{\alpha }{{\alpha }_c}-1\right)t\right]$$

(27)

4. Numerical Simulation and Result Analysis

In this section, numerical simulation experiments are conducted to investigate the effects of traffic flow in a connected environment where drivers are able to perceive information about the surrounding vehicles in a curved multi-lane scenario. The main purpose of the interference propagation is to test the adaptability of the proposed vehicle-following model to changes in traffic flow. Therefore, the vehicle-following rule in the simulation scenario is the extended model proposed in this paper.

We assume that 100 cars have the same parameters running on a curvy road. The total simulation step size is $\mathrm{15,000} s$; some parameters are set and the virtual loop simulation approach is used in the simulation, and the car-following rules are provided by Equation (28):

$$\begin{gathered} s_c=5m,\alpha =2,T=0.1,\omega =0.1,\mu =0.2, \\ \left\{\begin{array}{c}{v}_n(t+\Delta t)=v_n\left(t\right)+\Delta t\cdot a_n\left(t\right)\\ x_n(t+\Delta t)=x_n\left(t\right)+\Delta t\cdot v_n\left(t\right)+0.5\cdot a_n\left(t\right)\cdot \Delta t\cdot \Delta t\end{array}\right. \end{gathered}$$

(28)

Figure 4 and Figure 5 show the evolution diagram of head distance after 14,500 time steps and the headway of the vehicle distribution diagram at 15,000 time steps under the parameters $p$ = 1, $p$ = 0.9, $p$ = 0.8, and $p$ = 0.7 with a fixed $r$ = 20, respectively. It can be seen in Figure 6a–d that no matter what value of $p$ is taken, the evolution of the headspace gradually decreases with the increase in the simulation time, which shows that the proposed following model is able to adjust the following state over time and gradually reduce the influence of the following fleet by the interference. Furthermore, as you can see from the diagram, when parameter p decreases, the following vehicles increasingly consider the influence of the neighboring lanes, and the range of headway changes between vehicles gradually decreases in the same simulation timeframe. That is, when driving on a curved road, the traffic flow is more stable when the driver no longer only pays attention to the current lane, but also pays more attention to the surrounding vehicles in adjacent lanes, and with the increasing attention to adjacent lanes, traffic congestion can be well relieved.

Figure 6 and Figure 7 represent the effect of considering different numbers of vehicles in adjacent lanes on traffic flow at curve $r$ = 20, $p$ = 0.8. With the increase in k value, the traffic flow fluctuation decreases, which indicates that the more information about the surrounding vehicles’ movements is considered by the driver of the following car, the more beneficial it is for the traffic flow to evolve to a stable state after being disturbed.

In conclusion, we focus on the effect of model parameters on fleet evolution when the fleet is disturbed by speed changes. The main finding is that the fleet evolution tends to stabilize as the attention to the information of the surrounding vehicles increases when following a curve. Furthermore, the increase in attention to the neighboring lanes also affects the fleet evolution trend.

5. Conclusions

This paper proposes a novel car-following model by considering drivers’ different levels of attention to cars in adjacent lanes on a curved road. This paper analyzes the linear and nonlinear stability of the new model and verifies the influence of the model on the traffic flow fleet-following under different parameters through numerical simulation. The main findings are concluded as follows:

(1)

An improved car-following model applied to a curved multi-lane road is proposed, and the model validity and stability are verified;

(2)

In the connected environment, when the following vehicles increase their ability to perceive the information of the surrounding adjacent lanes, the traffic flow based on the proposed model is more stable, which helps to alleviate traffic congestion;

(3)

With the increase in the number of connected vehicles and the number of vehicles that can acquire information, the following model is more stable, and the trend of traffic flow evolution is smoother.

By accurately simulating the driving behavior and traffic flow of vehicles at curved multi-lane locations, it not only helps to improve the efficiency and safety of transportation systems in curved multi-lane traffic scenarios, but also provides an important theoretical basis and practical guidance for the integration and development of automated driving technologies to adapt to more complex and dynamic traffic environments. For traffic management, the model can be used to develop dynamic traffic management systems, such as variable speed limits, dynamic lane management, and intelligent traffic signal systems. This enables real-time monitoring of traffic flow data to adjust traffic control measures to reduce congestion, increase road capacity, and improve traffic flow.

Furthermore, in a connected environment, connected vehicles are able to provide richer and more complex datasets, such as information on nearby multi-vehicle driving, road conditions, etc., allowing for deeper analysis and more accurate modeling and prediction of driving behavior. This is difficult to achieve in a non-connected environment. In other words, connected vehicles that take into account nearby multi-vehicle information are better able to analyze and model the surrounding traffic flow dynamics and accurately understand driving behavior in dynamic traffic.

Compared with the non-connected vehicle-following model, the connected vehicle has a stronger ability to perceive information, so more information can be considered in the following model building process to make the model more accurate. The main improvement of the model proposed in this paper is that it can be applied in curved multi-lane traffic scenarios compared to other connected vehicle-following models. However, this paper simplifies the influence of surrounding vehicle information on drivers in the modeling process, and the specific patterns of influence on drivers in a connected environment need to be further investigated. In addition to this, the calibration of the model will be the focus of future work due to the lack of a suitable dataset.

For a better presentation, this paper summarizes the main parameters in

Table 1. Explanation of parameters in the formulas of this paper.

Parameter	Unit	Definition
$p$	$\backslash$	The driver’s sensitivity coefficient to self lane car-following behavior
$\alpha$	${\mathrm{s}}^{-1}$	The driver’s sensitivity coefficient to current speed and optimal speed
r	$\mathrm{m}$	The curve radius
$\gamma$	${\mathrm{s}}^{-1}$	Response coefficient to speed difference
$\omega$	\	The driving safety factor under the condition of a curved road
$\mu$	${\mathrm{s}}^{-1}$	The friction coefficient of the road
$s_c$	$\mathrm{m}$	The safe distance between the car and the car in front under the condition of a curved road
$\Delta S_n\left(t\right)$	$\mathrm{m}$	Distance between this vehicle and the vehicle ahead under the condition of a curved road
$V_{self}\left(·\right)$	$\mathrm{m}/\mathrm{s}$	Optimal velocity function on curved road for self lane
$V_{adj}\left(·\right)$	$\mathrm{m}/\mathrm{s}$	Optimal velocity function on curved road for adjacent lane

.