Two-Lane Lattice Hydrodynamic Modeling at Sag Sections With the Empirical Lane-Changing Rate

Sag sections are freeway sections where the gradient varies significantly from top to bottom. Due to their special geometric characteristics, sag sections are often bottlenecks in motorway networks. However, we are still unclear about the underlying mechanisms of traffic congestion in the two-lane sag sections. Based on the actual lane change experience, a two-lane lattice traffic flow model at the sag sections considering the empirical lane change rate is proposed. The model is developed to reveal the evolutionary characteristics of traffic congestion at the sag section, and then to alleviate the traffic congestion problem at the sag section. In this paper, the empirical lane change rate is described as a function that depends on the road traffic density. First, the condition of traffic flow stability was obtained by analyzing the linear stability at sag sections when lane changes were allowed. The stability condition of traffic system show that lane change parameter and slope have important effects on the stability of traffic system. Then, the mKdV equation describing the phase transition of the traffic flow and its solution are obtained using nonlinear analysis. Finally, numerical simulations were performed. The results show that the increase of slope increases the traffic flow stability in uphill sections and the opposite in downhill sections. Moreover, the lane-change rate parameter can effectively increase the stability region and suppress traffic congestion.

SECTION I. Introduction

Road traffic is the primary condition of urban development, and traffic congestion is an inevitable problem faced during the accelerated development of urbanization. In recent years, the growing number of cars has brought pressure to the urban road-driving environment. At present, the problem of traffic congestion has been widely concerned by many scholars. In order to alleviate traffic congestion more effectively, a scientific theoretical approach is needed. It is necessary to draw on fundamental knowledge from the fields of applied mathematics, fluid dynamics, and traffic engineering for cross-sectional studies, which can help scholars understand the formation of traffic congestion based on realistic conditions. Therefore, many scholars have proposed various traffic flow models to study and explore the essential characteristics of traffic flow. The aforementioned models include two main categories: microscopic models [1], [2], [3], [4], [5], [6] and macroscopic models [7], [8], [9], [10], [11], [12].

In 1998, Nagatani [13] transformed the higher-order continuum model and proposed the lattice hydrodynamic model for the first time. The model is a combination of the ideas of macroscopic and microscopic models, which is more useful to study the characteristics of traffic flow. At the same time, with the traffic environment changes and the passage of time, there are complications in actual traffic: for example, the complexity of roads and the unpredictability of the traffic environment. Based on this, the researchers developed a variety of extended lattice models. For example, driver expectation effect [14], [15], [16], memory effect [17], [18], density difference effect [19], [20], [21], [22], and gradient road [23], [24], [25], [26], etc. The continuous improvement and enrichment of the traffic flow model not only enriches the theoretical knowledge of traffic flow, but also alleviate traffic congestion, improve traffic efficiency and stability of traffic flow.

However, many of the above models describe only single-lane traffic flows. In real traffic situations, most roads are two-lane or multi-lane, so the lane changing behavior is not rare. When lane changing is prohibited, the two-lane traffic flow model will collapse into the classical single-lane traffic flow model. In order to be more consistent with the actual traffic scene, it is necessary to study the operation mechanism of vehicles on two-lane roads. Thus, Nagatani [27] introduced the lane change effect to the lattice model in 1999. The analysis concluded that the lane change rate of vehicles could positively effect on traffic flow stability. Since then, many scholars have considered other influencing factors in the above two-lane model to study complex traffic problems. With several examples, based on previous studies, Peng et al. [28] developed a model for traffic congestion mitigation by considering the flow differential memory integration effect. Subsequently, Zhai et al. [29] established a two-lane model by accounting the influence of lane change and memory effects. In real road traffic flows, the driver’s behavior in the next moment is influenced by the vehicle in front. Therefore, Zhang et al. [30] constructed a two-lane model considering the expectation effect. Other related studies can be found in references [31], [32], [33], [34]. However, many scholars believe that previously used constant lane change rate does not fully match the real traffic conditions. Since the traffic flow evolves towards more complexity, in 2019 Zhu et al. [35] considered the actual road environment and therefore replaced the constant lane change rate with the empirical lane change rate (ELCR). He found it more convincing to introduce the ELCR into the two-lane lattice model for analytical studies.

In the actual traffic flow, geographic factors affect drivers’ driving behavior, such as gradient. Sag section traffic capacity is obvious lower than the flat section, traffic congestion often occurs, so sag section often become the bottleneck of the highway. Due to the special geometric characteristics of sag section, the traffic dynamic behavior of vehicles will change, the speed will drop, and the disturbance will make the traffic flow unstable, resulting in traffic jams. According to the previous research analysis, the congestion evolution pattern of two-lane road at sag section has not been studied, and it is necessary to model this type of bottleneck for reality. For the present, there are few research achievements on two-lane lattice model, and lane change rate is a constant in most existing literatures. In other words, if the lattice density of the current lane exceeds that of the adjacent lane, a lane change will occur. However, in real traffic, the assumption of constant change rate is too strong. Therefore, based on previous studies, we apply the ELCR at sag sections. In this case, the ELCR is described as a function of the traffic state uniquely determined by the road density.

The existing traffic flow models mainly study the stability of road traffic flow under ideal road conditions, without studying the traffic flow congestion evolution process under special traffic conditions. The sag road section as a special case of road conditions, due to the slope causes much lower capacity than conventional road sections, and the traffic behavior at sag section is also significantly different from the behavior on ideal roads. In order to fill this research gap in the lattice hydrodynamic model, this paper takes the sag section as the research background, considers the actual lane change experience, and constructs a two-lane lattice model for the ELCR of the sag section. Then, the theoretical analysis and numerical simulation are carried out to deeply understand the evolution mechanism of traffic flow at sag sections.

The main work of this paper is: In Section II, the classical lattice hydrodynamic model is reviewed and a new lattice model is proposed. In Section III, the condition for model stability is determined. In Section IV, the mKdV equation and its solution is derived by nonlinear analytical analysis. In Section V, numerical simulation experiments are performed. In Section VI, the experimental results of this paper are discussed.

SECTION II. Model

In 1998, Nagatani [13] proposed a classical lattice hydrodynamic model with the help of the idea of microscopic following model. After discretizing the equation of motion and the continuity equation, the following form can be obtained:

$$\begin{align*}\partial _{t} \rho _{j} +\rho _{0} \left ({{\rho _{j} v_{j} -\rho _{j-1} v_{j-1}} }\right)&=0 \tag{1}\\ \partial _{t} \left ({{\rho _{j} v_{j}} }\right)&=a\rho _{0} V\left ({{\rho _{j+1}} }\right)-a\rho _{j} v_{j} \tag{2}\end{align*}$$ where $a$ is the sensitivity coefficient, $\rho _{0}$ is the initial density. $\rho _{j}$ and $v_{j}$ denote the density and velocity at the $jth$ lattice, respectively. $V\left ({\cdot }\right)$ denotes the optimal velocity function and its expression is: $$\begin{align*} V\left ({{\rho _{j}} }\right)=\frac {v_{\max }}{2}\left [{ {\tanh \left ({{\frac {2}{\rho _{0} }-\frac {\rho _{j}}{\rho _{0}^{2}}-\frac {1}{\rho _{c}}} }\right)+\tanh \left ({{\frac {1}{\rho _{c}}} }\right)} }\right] \\{} \tag{3}\end{align*}$$ where $v_{\max } =2$ denotes the maximum speed, $\rho _{c}$ denotes the vehicle safety density. when $\rho _{0} =\rho _{c}$ , the function has an inflection point at $\rho =\rho _{c}$ .

Considering the real traffic conditions of the road, Nagatani [27] proposed a two-lane lattice hydrodynamic traffic flow model by adding the lane-changing effect in the continuity equation. Figure 1 contains two lanes divided into lattice points, and vehicles may change from one lane to another.

FIGURE 1.

Vehicle lane change behavior in a two-lane scenario.

The lane change rule is as follows: First, when the instantaneous density of vehicles at lane L2 lattice point $j-1$ is higher than the instantaneous density at lane L1 lattice point $j$ , some vehicles will be shifted from lane L2 to lane L1 lattice point $j$ with a certain density difference ratio, and the corresponding lane change flow is $\gamma \left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\left ({{\rho _{2,j-1} -\rho _{1,j}} }\right)$ . The constant coefficient of proportionality $\gamma$ and the constant $\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |$ are chosen to make it dimensionless. Second, if the instantaneous density at lane L1 lattice point $j$ is higher than the instantaneous density at lane L2 lattice point $j+1$ , some vehicles will change from the L1 lane to the L2 lane and the lane change flow is $\gamma \left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\left ({{\rho _{1,j} -\rho _{2,j+1}} }\right)$ , then the conservation equations of lane L1 and lane L2 are:

$$\begin{align*} &\hspace {-1.2pc} \partial _{t} \rho _{1,j} +\rho _{0} \left ({{\rho _{1,j} v_{1,j} -\rho _{1,j-1} v_{1,j-1}} }\right) \\ & =\gamma \left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\left ({{\rho _{2,j+1} -2\rho _{1,j} +\rho _{2,j-1}} }\right) \tag{4}\\ &\hspace {-1.2pc} \partial _{t} \rho _{2,j} +\rho _{0} \left ({{\rho _{2,j} v_{2,j} -\rho _{2,j-1} v_{2,j-1}} }\right) \\ & =\gamma \left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\left ({{\rho _{1,j+1} -2\rho _{2,j} +\rho _{1,j-1}} }\right) \tag{5}\end{align*}$$ where $\rho _{1,j}$ and $v_{1,j}$ denote the density and speed at lattice point $j$ in lane L1, respectively. $\rho _{2,j-1}$ and $v_{2,j-1}$ denote the density and speed at lattice point $j-1$ in lane L2, respectively.

The conservation (6) is obtained by (4) and (5), so the equation for the two-lane model is:

$$\begin{align*}& \partial _{t} \rho _{j} +\rho _{0} \left ({{\rho _{j} v_{j} -\rho _{j-1} v_{j-1}} }\right) \\ & =\gamma \left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\left ({{\rho _{j+1} -2\rho _{j} +\rho _{j-1}} }\right) \tag{6}\\ &\partial _{t} \left ({{\rho _{j} v_{j}} }\right) \\ &=a\left [{ {\rho _{0} V\left ({{\rho _{j+1}} }\right)-\rho _{j} v_{j}} }\right] \tag{7}\end{align*}$$ where $\rho _{j} =\frac {\rho _{1,j} +\rho _{2,j} }{2}$ , $\rho _{j} v_{j} =\frac {\rho _{1,j} v_{1,j} +\rho _{2,j} v_{2,j}}{2}$ .

Since single lane cannot describe the lane changing behavior that exists in real traffic phenomena, Nagataini extended the classical single lane lattice model to two lanes. However, the models are all only applicable to describe the traffic flow in ideal road conditions and do not take into account the effect of special traffic environment on traffic flow stability (e.g., slope). Therefore, this paper proposes a two-lane model that can describe the traffic evolution at sag sections for the special road conditions at sag sections.

However, the lane-changing behavior of the model is deterministic. That is, when the real-time density of one lane exceeds that of the adjacent lane at any lattice point, the lane-changing behavior is bound to occur. Such an assumption is too strong to real traffic phenomena. To solve this problem, an empirical lane change rate function based on real-time road density information to replace the constant coefficient $\gamma$ . As a result, the continuity equations of L1 lane and L2 lane at lattice point $j$ are modified as:

$$\begin{align*} &\hspace {-1pc} \partial _{t} \rho _{1,j} +\rho _{0} \left ({{\rho _{1,j} v_{1,j} -\rho _{1,j-1} v_{1,j-1}} }\right) \\ & =\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\left [{ {\begin{array}{l} \gamma \left ({{\rho _{j}} }\right)\left ({{\rho _{2,j-1} -\rho _{1,j}} }\right) \\ -\gamma \left ({{\rho _{j+1}} }\right)\left ({{\rho _{1,j} -\rho _{2,j+1}} }\right) \\ \end{array}} }\right] \tag{8}\\ &\hspace {-1pc} \partial _{t} \rho _{2,j} +\rho _{0} \left ({{\rho _{2,j} v_{2,j} -\rho _{2,j-1} v_{2,j-1}} }\right) \\ & =\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\left [{ {\begin{array}{l} \gamma \left ({{\rho _{j}} }\right)\left ({{\rho _{1,j-1} -\rho _{2,j}} }\right) \\ -\gamma \left ({{\rho _{j+1}} }\right)\left ({{\rho _{2,j} -\rho _{1,j+1}} }\right) \\ \end{array}} }\right] \tag{9}\end{align*}$$

By (8) and (9), we can be obtain:

$$\begin{align*}& \partial _{t} \rho _{j} +\rho _{0} \left ({{\rho _{j} v_{j} -\rho _{j-1} v_{j-1}} }\right) \\ & =\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\left [{ {\gamma \left ({{\rho _{j}} }\right)\left ({{\rho _{j-1} -\rho _{j}} }\right)-\gamma \left ({{\rho _{j+1}} }\right)\left ({{\rho _{j} -\rho _{j+1}} }\right)} }\right] \\{} \tag{10}\end{align*}$$ where $\gamma \left ({\rho }\right)$ is the lane change function, which is expression as: $$\begin{equation*}\gamma \left ({\rho }\right)=\gamma _{\max } \frac {1-\rho \mathord {\Big /{ {\vphantom {\rho {\rho _{m}}}} } } {\rho _{m} }}{1+E\left ({{\rho \mathord {\Big /{ {\vphantom {\rho {\rho _{m}}}} } } {\rho _{m}}} }\right)^{4}} \tag{11}\end{equation*}$$ where $\gamma _{\max }$ denotes maximum channel change rate, $\rho _{m}$ indicates critical density, and $E$ means a constant.

Considering that the urban road environment will not always ideal, the traffic behavior at sag Section is different from that of ideal sections, as shown in Figure 2. For this purpose, a lattice model describing the operation of vehicles at sag sections is proposed.

FIGURE 2.

Force analysis diagram of vehicle at sag section.

The continuity and motion equations are (7) and (10), and the expression of dynamic optimal speed function equation is:

$$\begin{align*}& V\left ({\rho }\right)=\frac {v_{f,\max } -H\left ({\sigma }\right)\nu _{g,\max }}{2} \\ &\quad \qquad \qquad \times \left [{ {\tanh \left ({{\frac {1}{\rho }-\frac {1}{\rho _{c,\theta }}} }\right)+\tanh \left ({{\frac {1}{\rho _{c,\theta }}} }\right)} }\right] \tag{12}\end{align*}$$ where $H\left ({\cdot }\right)$ denotes the Heaviside function, $v_{f,\max }$ denotes the maximum frictional speed of the vehicle on the ideal road, $v_{g,\max }$ denotes the maximum speed on a slope road. The specific manifestations of $H\left ({\cdot }\right)$ is: $$\begin{align*} H\left ({\sigma }\right)=\begin{cases} 1& \sigma >0 \\ 0& \sigma =0 \\ -1& \sigma < 0 \\ \end{cases} \tag{13}\end{align*}$$ where $\sigma$ is the gradient. $\sigma >0$ denotes uphill, $\sigma =0$ denotes no gradient, and $\sigma < 0$ is downhill. Thus, the optimal velocity functions are expressed as: $$\begin{align*} V\left ({\rho }\right)&=\frac {v_{f,\max } -v_{g,u,\max }}{2} \\ &\quad \times \left [{ {\tanh \left ({{\frac {1}{\rho }-\frac {1}{\rho _{c,u,\theta } }} }\right)+\tanh \left ({{\frac {1}{\rho _{c,u,\theta }}} }\right)} }\right] \tag{14}\\ V\left ({\rho }\right)&=\frac {v_{f,\max } +v_{g,d,\max }}{2} \\ &\quad \times \left [{ {\tanh \left ({{\frac {1}{\rho }-\frac {1}{\rho _{c,d,\theta } }} }\right)+\tanh \left ({{\frac {1}{\rho _{c,d,\theta }}} }\right)} }\right] \tag{15}\end{align*}$$ where $\rho _{c,u,\theta } =1/h_{c,u,\theta }$ and $\rho _{c,d,\theta } =1/h_{c,d,\theta }$ are expressed as the safety density of uphill and downhill.

In addition, during the operation of the vehicle, the speed will change in the case of uphill (downhill). The safe distance for uphill and downhill can be expressed as:

$$\begin{align*} \begin{cases} h_{c,u,\theta } =h_{c} \left ({{1-\kappa \sin \theta } }\right) \\ h_{c,d,\theta } =h_{c} \left ({{1+\delta \sin \theta } }\right) \\ \end{cases} \tag{16}\end{align*}$$ where $h_{c}$ is the safe distance. $\kappa$ and $\delta$ are control coefficients and take $\kappa =\delta =1$ . $v_{g,u,\max }$ is the value of the reduced velocity for uphill, and $v_{g,d,\max }$ is the value of the increased velocity when going downhill. The expression is given as follows: $$\begin{equation*} v_{g,u,\max } =v_{g,d,\max } =\frac {mg\sin \theta }{\mu } \tag{17}\end{equation*}$$ where $\mu$ is the friction coefficient, for simplicity, the relevant parameters are taken as $v_{f,\max } \textrm {=2}$ , ${mg} \mathord {\Big /{ {\vphantom {mg \mu }} } } \mu =1$ , $-9^{\circ }\le \theta \le 9^{\circ }$ , so the optimal velocity function obtained from the above equation is: $$\begin{align*} V\left ({\rho }\right) &=\frac {2\pm \sin \theta }{2}V_{0} \left ({\rho }\right) \tag{18}\\ V_{0} \left ({\rho }\right)&=\tanh \left ({{\frac {2}{\rho _{0}}-\frac {\rho }{\rho _{0}^{2}}-\frac {\left ({{1-\sin \theta } }\right)}{\rho _{c}}} }\right) \\ &\quad +\tanh \left ({{\frac {\left ({{1-\sin \theta } }\right)}{\rho _{c}}} }\right) \tag{19}\end{align*}$$ By relating (7) and (10) and eliminating the velocity $v$ , the evolution equation of the density wave is obtained: $$\begin{align*}& \partial _{t}^{2} \rho _{j} +a\rho _{0}^{2} \left [{ {V\left ({{\rho _{j+1}} }\right)-V\left ({{\rho _{j}} }\right)} }\right]+a\partial _{t} \rho _{j} \\ &\quad -a\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right | \\ &\quad \times \left [{ {\gamma \left ({{\rho _{j}} }\right)\left ({{\rho _{j-1} -\rho _{j}} }\right)-\gamma \left ({{\rho _{j+1}} }\right)\left ({{\rho _{j} -\rho _{j+1}} }\right)} }\right] \\ &\quad -\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right | \\ &\quad \times \partial _{t} \left [{ {\gamma \left ({{\rho _{j}} }\right)\left ({{\rho _{j-1} -\rho _{j}} }\right)-\gamma \left ({{\rho _{j+1}} }\right)\left ({{\rho _{j} -\rho _{j+1}} }\right)} }\right]=0 \\{} \tag{20}\end{align*}$$

SECTION III. Linear Stability Analysis

In order to study the effect of ELCR on the stability of traffic flow at the sag section, we apply linear stability theory to analyze the stability condition of the model. Assuming in the steady state, $\rho _{0}$ and $V\left ({{\rho _{0}} }\right)$ are the density and velocity in the ideal state, respectively. The expressions are respectively:

$$\begin{equation*} \rho _{j} \left ({t }\right)=\rho _{0}, v_{j} \left ({t }\right)=V\left ({{\rho _{0}} }\right) \tag{21}\end{equation*}$$

Adding small perturbation $y_{j} \left ({t }\right)$ to the initial stable traffic flow, at which point the density is $\rho _{j} \left ({t }\right)=\rho _{0} +y_{j} \left ({t }\right)$ . Taking this and (21) into the density equation, it can be obtained:

$$\begin{align*}& \partial _{t}^{2} y_{j} \left ({t }\right)+a\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)\left ({{y_{j+1} -y_{j}} }\right)+a\partial _{t} y_{j} \left ({t }\right) \\ &\quad -a\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\left [{ {\gamma \left ({{\rho _{0}} }\right)\left ({{y_{j-1} -2y_{j} +y_{j+1}} }\right)} }\right] \\ &\quad -\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\gamma \left ({{\rho _{0}} }\right)\left ({{\partial _{t} y_{j+1} -2\partial _{t} y_{j} +\partial _{t} y_{j-1}} }\right)=0 \\{} \tag{22}\end{align*}$$ where ${V}'\left ({{\rho _{0}} }\right)=\frac {dV\left ({{\rho _{j}} }\right)}{d\rho _{j}}\Big |{ {_{p_{j} =\rho _{0}}} }$ .

Setting $y_{j} \left ({t }\right)=e^{ikj+zt}$ and bringing into (22), the following equation is obtained:

$$\begin{align*}& z^{2}+a\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)\left ({{e^{ik}-1} }\right)+az \\ &\quad -a\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\gamma \left ({{\rho _{0}} }\right)\left ({{e^{-ik}-2+e^{ik}} }\right) \\ &\quad -\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\gamma \left ({{\rho _{0}} }\right)\left ({{ze^{-ik}-2z+ze^{ik}} }\right)=0 \tag{23}\end{align*}$$

Letting $z=z_{1} (ik)+z_{2} (ik)^{2}+\ldots$ be substituted into (23), we obtain the coefficients of $ik$ and $(ik)^{2}$ respectively as:

$$\begin{align*} z_{1}& =-\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right) \tag{24}\\ z_{2}& =-\frac {\rho _{0}^{4} {V}'\left ({{\rho _{0}} }\right)^{2}}{a}-\frac {1}{2}\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right) \\ &\quad +\left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right |\gamma \left ({{\rho _{0}} }\right) \tag{25}\end{align*}$$

Therefore, from the theory derivation, the stability of the system is related to $z_{2}$ . When $z_{2} < 0$ , the traffic flow will gradually become unstable; on the contrary, when $z_{2} >0$ , the uniform flow is met stable. Therefore, the neutral stability condition that can be obtained by solving for $z_{2} =0$ is specified as:

$$\begin{equation*} a=\frac {-2\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)}{1+2\gamma \left ({{\rho _{0}} }\right)} \tag{26}\end{equation*}$$

At this point, the stability condition for uniform traffic flow is:

$$\begin{equation*} a>\frac {-2\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)}{1+2\gamma \left ({{\rho _{0}} }\right)} \tag{27}\end{equation*}$$

SECTION IV. Nonlinear Stability Analysis

In order to investigate the influence of ELCR at the sag section on the traffic flow density wave evolution process, we further discussed the model in the unstable region of traffic flow and near the critical point $\left ({{\rho _{c},a_{c}} }\right)$ . In the following, the theoretical derivation is carried out using nonlinear analysis, mainly to find the mKdV equation of the model and the solution of the equation. First, a small quantity $\varepsilon$ ($0 < \varepsilon \le 1$ ) is introduced, assuming that the slow variables are defined as $S$ and $T$ :

$$\begin{equation*} S=\varepsilon \left ({{j+bt} }\right)\textrm {, T=}\varepsilon ^{3}t \tag{28}\end{equation*}$$ where $b$ is a deterministic constant, let the density $\rho _{j} \left ({t }\right)$ be: $$\begin{equation*} \rho _{j} \left ({t }\right)=\rho _{c} +\varepsilon R\left ({{S,T} }\right) \tag{29}\end{equation*}$$

Bringing (28) and (29) into (20), and as a Taylor expansion to the fifth-order term of $\varepsilon$ :

$$\begin{align*}& \varepsilon ^{2}\left ({{b+\rho _{c}^{2} {V}'} }\right)\partial _{S} R+\varepsilon ^{3}\left ({{\frac {b^{2}}{a}+\frac {1}{2}\rho _{c}^{2} {V}'\left ({{1+2\gamma \left ({{\rho _{c}} }\right)} }\right)} }\right)\partial _{S}^{2} R \\ &\quad +\varepsilon ^{4}\left ({{\partial _{T} R+\frac {\rho _{c}^{2} {V}'}{6}\left ({{1+\frac {6b\gamma \left ({{\rho _{c}} }\right)}{a}} }\right)\partial _{S}^{3} R+\frac {\rho _{c}^{2} {V}'''}{6}\partial _{S} R^{3}} }\right) \\ &\quad +\varepsilon ^{5}\left ({{\begin{array}{l} \frac {2b}{a}\partial _{T} \partial _{S} R+\frac {\rho _{c}^{2} {V}'}{24}\partial _{S}^{4} R \\ +\frac {\gamma \left ({{\rho _{c}} }\right)\rho _{c}^{2} {V}'}{12}\partial _{S}^{4} R+\frac {\rho _{c}^{2} {V}'''}{12}\partial _{S}^{2} R^{3} \\ \end{array}} }\right)=0 \tag{30}\end{align*}$$

At the critical point $\left ({{\rho _{c},a_{c}} }\right)$ , $a_{c} =a\left ({{\varepsilon ^{2}+1} }\right)$ , substituting $b=-\rho _{c}^{2} {V}'$ to eliminate the second and third order terms of $\varepsilon$ in (30), the final equation is as follows:

$$\begin{align*}& \varepsilon ^{4}\left [{ {\partial _{T} R-g_{1} \partial _{S}^{3} R+g_{2} \partial _{S} R^{3}} }\right] \\ &\qquad \qquad \qquad +\varepsilon ^{5}\left [{ {g_{3} \partial _{S}^{2} R+g_{4} \partial _{S}^{4} R+g_{5} \partial _{S}^{2} R^{3}} }\right]=0 \tag{31}\end{align*}$$ where $g_{1} =-\frac {\rho _{c}^{2} {V}'}{6}-\frac {\gamma \left ({{\rho _{c}} }\right)b\rho _{c}^{2} {V}'}{a_{c}}$ , $g_{2} =\frac {\rho _{c}^{2} {V}'''}{6}$ , $g_{3} =-\frac {\rho _{c}^{2} {V}'}{2}\left ({{1+2\gamma \left ({{\rho _{c}} }\right)} }\right)$ , $g_{4} =\frac {\rho _{c}^{2} {V}'}{24}+\frac {\gamma \left ({{\rho _{c}} }\right)\rho _{c}^{2} {V}'}{12}+\frac {2b}{a_{c}}g_{1}$ , $g_{5} =\frac {\rho _{c}^{2} {V}'''}{12}-\frac {2b}{a_{c}}g_{2}$ .

We substitute ${T}'=g_{1} T$ , $\textrm {R=}\sqrt {{g_{1}} \mathord {\Big /{ {\vphantom {{g_{1}} {g_{2}}}} } } {g_{2}}} {R}'$ into (31) to transform them and derive the standard mKdV equation as:

$$\begin{equation*} \partial _{T'} {R}'-\partial _{S}^{3} {R}'+\partial _{S} {R}'^{3}+\varepsilon M\left [{ {R'} }\right]=0 \tag{32}\end{equation*}$$ where $M\left [{ {R'} }\right]=\frac {1}{g_{1}}\left ({{g_{3} \partial _{S}^{2} {R}'+\frac {g_{1} g_{5}}{g_{2}}\partial _{S}^{2} {R}'^{3}+g_{4} \partial _{S}^{4} {R}'} }\right)$ .

Under the condition that the term $O(\varepsilon)$ of (32) is neglected, then the solution of the kink-antikink is:

$$\begin{equation*} {R}'_{0} \left ({{S,{T}'} }\right)=\sqrt {c} \tanh \left [{ {\sqrt {\frac {c}{2}} \left ({{S-c{T}'} }\right)} }\right] \tag{33}\end{equation*}$$

To determine the propagation speed in (33), the condition to be satisfied is:

$$\begin{equation*} \left ({{R'_{0},M\left [{ {R'_{0}} }\right]} }\right)\equiv \int _{-\infty }^{+\infty } {dS{R}'_{0}} M\left [{ {R'_{0}} }\right]=0 \tag{34}\end{equation*}$$ where $M\left [{ {R'_{0}} }\right]=M\left [{ {R'} }\right]$ . Through the above (34), the propagation velocity $c$ of the kink-antikink density wave is obtained as: $$\begin{equation*} c=\frac {5g_{2} g_{3}}{2g_{2} g_{4} -3g_{1} g_{5} } \tag{35}\end{equation*}$$

Thus, the obtained kink-antikink solution is:

$$\begin{equation*} \rho _{j} =\rho _{c} +\varepsilon \sqrt {\frac {g_{1} c}{g_{2}}} \tanh \left ({{\sqrt {\frac {c}{2}} \left ({{S-cg_{1} T} }\right)} }\right) \tag{36}\end{equation*}$$ where $\varepsilon ^{2}=\frac {a_{c} }{a}-1$ , the amplitude $Q$ is: $$\begin{equation*} Q=\sqrt {\frac {g_{1}}{g_{2}}\varepsilon ^{2}c} \tag{37}\end{equation*}$$ The kink-antikink solution can represent the coexistence phase, including the free-moving phase and jamming phase. This suggests that the coexistence curves can be expressed as $\rho =\rho _{c} \pm Q$ . The free-flowing phase density is expressed as $\rho =\rho _{c} -Q$ , and the jamming phase density is denoted as $\rho =\rho _{c} +Q$ . The coexistence curves as shown by the dashed lines in Fig. 3 and Fig. 4.

FIGURE 3.

Phase diagram of the corresponding neutral stability curves (blue) and coexistence curves (red) for different values of $\gamma _{\max }$ when $\theta =3^{\circ }$ , $E=10$ .

FIGURE 4.

Phase diagrams of the corresponding neutral stability curves (blue) and coexistence curves (red) for different values of $\theta$ when $\gamma _{\max } =0.2$ , $E=10$ . (a) denotes uphill, (b) indicates downhill.

Figures 3 and 4 plot the neutral stability curve and coexistence curve of the model. The vertices of each curve are critical points. As can be seen in Figures 3 and 4, when the system parameters are in the steady state region, any small disturbances do not induce traffic congestion and the system is uniform flow. In the metastable region, congestion occurs in the traffic flow only when the disturbance present in the system is large enough, otherwise the congestion gradually disappears. When the system parameters are located in unstable area, regardless of the size of the disturbance, the system will be destabilized after a period of evolution, forming a stop-and-go wave and finally evolving into a jamming flow. Figure 3 shows the effect of different values of $\gamma _{\max }$ on the traffic flow stabilization area in the case of vehicle lane change. It is observed from the figure that with the parameter $\gamma _{\max }$ increases, the critical point $\left ({{\rho _{c},a_{c}} }\right)$ decreases and the stability region expands. This indicates that the larger the parameter $\gamma _{\max }$ is, the more conducive the traffic flow system is to reach the steady state.

In Figure 4(a), it can be clearly seen that when $\theta =0^{\circ }$ , there is no slope, and the curve is at the top. With the value of $\theta$ increases, the area of the stable region becomes progressively larger. Therefore, the larger the value of $\theta$ is, the more stable the system is. This indicates that when going uphill, considering $\theta$ is more beneficial to the stability of the system than when there is no slope. That is, considering the effect of $\theta$ is beneficial to the stability of traffic flow. In Figure 4(b), when $\theta =0^{\circ }$ , the curve is at the bottom. As the slope continues to increase, the position of the vertical coordinate of the curve becomes larger, the critical point $\left ({{\rho _{c},a_{c}} }\right)$ gradually increases, while the area of the stabilized zone decreases. Therefore, the increase of $\theta$ value in downhill will make the system less stable.

SECTION V. Numerical Simulation

To facilitate numerical calculation, the difference form of the model can be obtained from (20):

$$\begin{align*} &\hspace {-0.5pc} \rho _{j} \left ({{t+2\tau } }\right)-\rho _{j} \left ({{t+\tau } }\right) \\ &=\tau \left |{ {\rho _{0}^{2} {V}'\left ({{\rho _{0}} }\right)} }\right | \\ &\quad \times \left [{ {\begin{array}{l} \gamma \left ({{\rho _{j} \left ({{t+\tau } }\right)} }\right)\left ({{\rho _{j-1} \left ({{t+\tau } }\right)-\rho _{j} \left ({{t+\tau } }\right)} }\right) \\ -\gamma \left ({{\rho _{j+1} \left ({{t+\tau } }\right)} }\right)\left ({{\rho _{j} \left ({{t+\tau } }\right)-\rho _{j+1} \left ({{t+\tau } }\right)} }\right) \\ \end{array}} }\right] \\ &\quad -\tau \rho _{0}^{2} \left [{ {V\left ({{\rho _{j+1}} }\right)-V\left ({{\rho _{j}} }\right)} }\right] \tag{38}\end{align*}$$ where $\tau$ is the time difference step.

A series of numerical simulation studies were performed in this section, and conclusions are drawn by simulating the variation of density waves using the control variables approach. The simulation scenario is in a periodic boundary environment. The initial conditions are taken as:

$$\begin{align*}\rho _{j} \left ({0 }\right)&=\rho _{0} =0.25,\quad j\in \left [{ {1,N} }\right] \tag{39}\\ \rho _{j} \left ({1 }\right)&=\begin{cases} \rho _{0}: j\in \left [{ {1,N} }\right] and j\ne N \mathord {\Big /{ {\vphantom {N {2,{N} \mathord {\Big /{ {\vphantom {N {2-1}}} } } {2-1}}}} } } {2,{N} \mathord {\Big /{ {\vphantom {N {2-1}}} } } {2-1}} \\ \rho _{0} -\xi:j=N \mathord {\Big /{ {\vphantom {N 2}} } } 2 \\ \rho _{0} +\xi:j=N \mathord {\Big /{ {\vphantom {N {2-1}}} } } {2-1} \\ \end{cases} \tag{40}\end{align*}$$ where $\xi$ is the initial perturbation, $N$ represents the total number of lattices. The related parameters can be: $\xi =0.05$ , $N=100$ .

Firstly, the influence of parameter $E$ is discussed by numerical simulation. Then, the influence of important parameters ($\theta$ and $\gamma _{\max }$ ) in this model on the stability at sag section traffic flow is further studied.

Figure 5 shows the spatio-temporal evolution of density waves when $E=10,50,100,150$ , $\theta =3^{\circ }$ , $\gamma _{\max } =0.2$ . It can be clearly seen from Figure 5 that with the increase of parameter $E$ , density waves appear and their amplitude of oscillation gradually increases, which is not conducive to the stability of traffic flow. When $E=10$ , it means that the disturbance added to the system disappears with the evolution of traffic flow and the traffic flow is in a stable state. Figure 6 corresponds to the density curve in two-dimensional form shown in Figure 5.

FIGURE 5.

Spatial and temporal evolution of the density waves of $E=10,50,100,150$ , when $\theta =3^{\circ }$ , $\gamma _{\max } =0.2$ , after $t=10^{4}s$ .

FIGURE 6.

Density waves distribution of $E=10,50,100,150$ , when $\theta =3^{\circ }$ , $\gamma _{\max } =0.2$ , after $t=10^{4}s$ .

Figure 7 and Figure 8 show the spatio-temporal evolution diagram and density distribution diagram of the density waves obtained for $\gamma _{\max } =0.2,0.4,0.6,0.8$ when setting $a=0.8$ , $E=10$ , and $\theta =3^{\circ }$ . The corresponding critical sensitivity coefficients in Fig. 7 are $a_{c} =1.5,1.3\textrm {,1.1,}0.9$ , respectively. It can be seen from the figure, when parameter $\gamma _{\max }$ increases, the density waves with decreasing amplitude. This indicating that appropriate lane change behavior can accelerate the elimination of disturbance, relieve traffic pressure and improve traffic efficiency. Figure 8 shows the distribution of density waves corresponding to Figure 7. We can clearly see the relationship between traffic flow stability and the parameter $\gamma _{\max }$ . As the parameter $\gamma _{\max }$ increases, the fluctuation is gradually gentle, and the stability of the system gradually enhanced. This further illustrates the necessity of considering the parameter $\gamma _{\max }$ .

FIGURE 7.

Spatial and temporal evolution of the density waves of $\gamma _{\max } =0.2,0.4,0.6,0.8$ , when $\theta =3^{\circ }$ , $E=10$ , after $t=10^{4}s$ .

FIGURE 8.

Density waves distribution of $\gamma _{\max } =0.2,0.4,0.6,0.8$ , when $\theta =3^{\circ }$ , $E=10$ , after $t=10^{4}s$ .

Figure 9 shows that the other parameters are set to $\gamma _{\max } =0.2$ , $a=1.85$ , and $E=10$ for different $\theta$ values at uphill. In Figure 9(a)-(c), the evolution process of kink-antikink waves is shown. Figure 9(a) indicates no slope, which can be compared with the traffic flow state after different slopes. The effect of slope on the system is studied by varying the value of $\theta$ with other parameters held constant. According to the stability conditions obtained, the critical sensitivity coefficients $a_{c} =2.1,\textrm {2.0,1.9}$ in (a), (b), and (c) of Fig. 9, so the modes of traffic flow are all unstable. Unlike the first three traffic flow patterns, the critical sensitivity coefficient $a_{c} =1.8$ in Fig. 9(d), the system is in a steady state at this time. The disturbance is eventually absorbed with the evolution of time, and after a long enough time, the traffic flow is uniform. This indicates that considering the effect of slope when going uphill can suppress traffic flow congestion and enhance system stability. Figure 10 is the two-dimensional of the density map corresponding to Figure 9.

FIGURE 9.

Spatial and temporal evolution of the density waves for uphill of $\theta =0^{\circ },3^{\circ },6^{\circ },9^{\circ }$ , when $\gamma _{\max } =0.2$ , $E=10$ , after $t=10^{4}s$ .

FIGURE 10.

Density waves distribution for uphill of $\theta =0^{\circ },3^{\circ },6^{\circ },9^{\circ }$ , when $\gamma _{\max } =0.2$ , $E=10$ , after $t=10^{4}s$ .

In Figure 10, (a), (b), (c), and (d) correspond to $\theta =0^{\circ },3^{\circ },6^{\circ },9^{\circ }$ , respectively. From the graph, it can be seen that as $\theta$ increases, the fluctuation range of the density curve gradually decreases. As time goes by, the added disturbance gradually disappears, and finally the system reaches a stable state.

Figure 11 analyzes the density profiles in three-dimensional form for different values of $\theta$ at downhill, when the other parameters are set to $\gamma _{\max } =0.2$ , $a=1.85$ and $E=10$ . $\theta =0^{\circ },-3^{\circ },-6^{\circ },-9^{\circ }$ correspond to (a), (b), (c), and (d) of Figure 11, respectively. The corresponding critical sensitivity coefficients in Fig. 11(a)-(d) are $a_{c} =2.1,\textrm {2.2, 2.3, 2.4}$ . And for a given value of $a=1.85$ , the traffic flow patterns of all four maps are in an unstable state. From the figure, it can be seen that as $\theta$ increases, the fluctuation degree of density curve gradually increases, and the traffic flow is congested flow. This indicates that the increase of value is not conducive to relieving traffic congestion. From Figure 11(a), it can be seen that no slope is more beneficial to system stability than increasing the slope when going downhill compared to the other three plots. Figure 12 is the density distribution corresponding to Figure 11. When $\theta =0^{\circ }$ gradually increases to −9°, the fluctuation of the density wave becomes more and more violent. Therefore, we can get the same conclusion as Figure 11.

FIGURE 11.

Spatial and temporal evolution of the density waves for downhill of $\theta =0^{\circ },-3^{\circ },-6^{\circ },-9^{\circ }$ , when $\gamma _{\max } =0.2$ , $E=10$ , after $t=10^{4}s$ .

FIGURE 12.

Density waves distribution for downhill of $\theta =0^{\circ },-3^{\circ },-6^{\circ },-9^{\circ }$ , when $\gamma _{\max } =0.2$ , $E=10$ , after $t=10^{4}s$ .

SECTION VI. Discussions

In this paper, a two-lane lattice model at the sag section considering ELCR is constructed to overcome the deficiency of the lane change rate as a constant in the traditional lattice model. Through linear and nonlinear analytical analysis, the stability condition of the model is obtained, the mKdV equation and the corresponding analytical solution is derived. The results show the parameters $\gamma _{\max }$ and $\theta$ are closely related to the stability of the traffic system, and the contribution of each parameter is different. As the parameter $\gamma _{\max }$ increases, the traffic congestion is suppressed. However, the increase of $\theta$ stabilizes the traffic at uphill and the opposite at downhill. The theoretical analysis results are verified by computer simulation experiments. To the best of our knowledge, this is the first time that a macroscopic traffic flow has quantified the impact of the ELCR on traffic congestion at sag road sections. Compared with the existing literature, this paper is modeled and analyzed for its special traffic conditions with the sag section as the research background. This not only provides insight into the relationship between lane change behavior and traffic congestion at sag sections, but also provides theoretical support for the development and design of subsequent traffic control strategies at sag sections.

Although the research in this paper has alleviated traffic congestion to a certain extent. However, the actual traffic flow system is a dynamically changing and is a complex system consisting of various controllable and uncontrollable factors. The behavioral characteristics of drivers are also important and can have a certain impact on the traffic road passage as well as safety. Therefore, in future research, we need to consider the influence of driver behavior factors in the model to better solve the problems in actual traffic.