Modeling and Analysis of Car-Following for Intelligent Connected Vehicles Considering Expected Speed in Helical Ramps

Abstract

In this paper, to explore the influence of expected speed on traffic flow in helical ramps, a new car-following model for intelligent connected vehicles (ICVs) was established for helical ramps, mainly considering the expected speed provided in the vehicle-to-everything (V2X) environment. On this basis, sufficient conditions to ensure the stability of the traffic stream were met and the congestion propagation mechanism was discussed by using a linear stability analysis and nonlinear stability analysis. The results showed that the ICVs can effectively increase the stability of the traffic flow by considering the expected speed of the helical ramps. When the feedback coefficients of the expected speed of the helical ramps were 0.3 and 0.5, the stability of the traffic flow changed significantly, especially in the uphill section; the feedback coefficient was 0.5 when the traffic flow was completely restored to the initial steady state even under the action of small disturbances. In a difficult field-driving test, this paper showed through a numerical simulation that broadcasting an expected speed to the ICVs in the helical ramps can effectively improve the stability of traffic flow, which provides a theoretical basis for future landing applications of ICVs in complex road scenarios.

1. Introduction

As a typical ramp in an interchange, the helical ramp is an important traffic node in mountainous city roads, plays an important role in the conversion of traffic flow and the connection between roads, and is also a concentrated area for vehicle diversion, as well as merging and weaving operations. Its traffic efficiency will directly affect the overall performance of a road network [1,2,3,4]. Due to the combined effect of centripetal force, gravity and friction in helical ramps and their unique characteristics of long-distance lifting and large-angle whirling, helical ramps are prone to low traffic efficiency and safety problems [5,6,7,8]. In helical ramps, the driving processes of vehicles will be jointly affected by many factors, such as the geometric characteristics of the road (such as the radius of the curvature, gradient, etc.) and the road conditions (road friction), which will lead to greater differences in the driving behavior characteristics of vehicles in this area compared with those of vehicles on general plane roads, as well as greater difficulty in following between vehicles [9,10,11]. In fact, the occurrence of such problems is not only related to the characteristics of the helical ramp itself, but also to the limited range of information obtained by vehicles in this area, the randomness of drivers and other factors [12,13]. Therefore, it is of great significance to study the traffic congestion mechanism of traffic flow on helical ramps to improve traffic safety and traffic efficiency in helical ramps.

With the rapid development of information technology and automatic driving technology, vehicle-to-everything (V2X) technology is being developed. It enables vehicles to communicate with other vehicles, infrastructure and base stations, so as to obtain a series of traffic information, such as real-time road conditions, road information, pedestrian information, etc., thereby improving driving safety and traffic efficiency and reducing congestion [14,15,16]. In a V2X environment, ICVs have become the focus of attention domestically and internationally in recent years [17,18,19,20]. With the emergence of ICVs that are similar to traditional human-driven vehicles (HVs), scholars have also conducted extensive research on their basic micro driving behaviors. These results can be applied to traffic planning, traffic management and traffic control, thereby improving existing traffic facilities and the environment, as well as solving many practical problems in existing road traffic.

The research on the car-following model of ICVs has mainly focused on the improvement of the traditional car-following model by considering the network communication and driving characteristics of ICVs. For example, Tang et al. [21] improved the classic full-speed difference model by considering multi-vehicle information at an early stage, thus proposing a new car-following model for ICVs. Wang et al. [22] proposed an intelligent connected car-following model considering vehicle-to-vehicle communication for the scenario of unsignalized intersections. Peng et al. [23] proposed an extended car-following model based on the cooperative information transmission delay effect of time headway and speed in a V2X environment. Sun et al. [24] improved the traditional optimization speed model based on the feedforward information obtained from communication and then obtained a new ICV model. Milan é s et al. [25] proposed a car-following model that uses adaptive cruise control and cooperative adaptive cruise control technology, considering parameters such as the error between the headway and the expected headway, the headway error weight, the speed difference weight and the differential weight of the headway. For non-planar road scenarios, Sun et al. [26] proposed a suitable car-following model suitable that considers the driver’s uncertainty characteristics. Li et al. [27] explored the effect of a two-sided lateral gap with uncertain velocity on the stability of the traffic flow on a curved road. The results showed that driving between two lanes and inaccurate speed estimates both had a negative effect on traffic flow stability, and the stability also decreased with the increase in the radius of the curved road. Wang et al. [28] explored how curved roads and lane-changing rates affect the stability of traffic flow and an extended two-lane lattice hydrodynamic model on a curved road accounting for the empirical lane-changing rate was presented. The results showed that the empirical lane-changing rate on a curved road was an important factor that can alleviate traffic congestion. The traffic flow phenomenon for a curved road with a slope was analyzed by using the lattice hydrodynamic approach, considering the driver’s anticipation effect on the two-lane system. It was concluded that the slope of a curved road played a significant role in influencing the stability of traffic flow in a two-lane traffic system [29].

Wang et al. [30] proposed a traffic flow model considering the dynamic effect of electronic throttle, and based on this, analyzed the complex dynamic impact of road geometric characteristics on traffic flow stability. Zhu et al. [31] proposed a car-following model based on the optimized speed model by considering the relevant influencing factors of helical ramps. Wang et al. [32] studied the relationship between the speed of a vehicle on a particular curve and the recommended speed. Shallama et al. [33] used the goodness of fit parameter to analyze the relationship between different road conditions (such as radius, deflection angle and transition curve) and the percentile speed, and used SPSS to fit the percentile speed and establish the corresponding relationship model. Zirkel et al. [34] discussed the influence of driving speed discreteness and sight distance on accident rates. Based on the measured data of a radar speed-measuring gun, Perco et al. [35] established a vehicle running-speed model for the entrance and exit of an annular interchange. Field-driving tests on four helical ramps were conducted and the trajectory, speed and acceleration of vehicles under normal driving conditions were collected. The speed characteristics and speed patterns on helical ramps, as well as their affecting factors, were obtained and analyzed [36]. Meng et al. [37] presented a car-following model based on speed difference information on a curved road with a slope, and the linear stability of the model was discussed based on control theory. In addition, many scholars have also proposed many traffic flow models to study complex road scenes [38,39,40].

By analyzing the above research, it can be found that the existing research either focuses on establishing new car-following models considering the driving characteristics of ICVs in a V2X environment, or proposes new car-following models specifically for complex scenes similar to helical ramps, but the former ignores the geometric characteristics of the road, while the latter does not fully consider the macro traffic information obtained by ICVs. Therefore, this paper took helical ramps as the scenario of interest, considered roadside equipment that could broadcast the expected speed of the current road to ICVs driving on helical ramps under a V2X environment, proposed a new helical ramp ICV car-following model and obtained sufficient conditions to ensure the stability of the traffic stream through a local stability analysis. At the same time, the congestion propagation mechanism under the proposed model was discussed by using a linear stability analysis and nonlinear stability analysis of the periodic boundary conditions. Finally, the validity of the model was verified by a numerical simulation.

This paper is organized as follows. Following the introduction, Section 2 presents the vehicle-following model considering the expected speed on helical ramps. Section 3 describes the model stability analysis. Section 4 presents the nonlinear analysis. Section 5 presents the numerical simulation. Section 6 summarizes the main findings of this study.

2. Vehicle-Following Model Considering Expected Speed on Helical Ramps

First of all, we consider ICVs that run on helical ramps, as shown in Figure 1. In this area, ICVs are affected by a variety of forces, in which mg represents the gravity of the vehicle, mgsinθ represents the component force of mg whose direction is tangent to the pavement of the helical ramp, mgcosθ is the component force of mg whose direction is perpendicular to the pavement of the helical ramp, m is the total mass of the vehicle, g is the acceleration of gravity, and θ is the slope angle of the helical ramps. In addition, the ICVs are also affected by centripetal force, namely F = μmgcosθ, in which μ is the lateral friction coefficient of the helical ramps. At the same time, if $\varphi$ is the arc length from point P to point Q and r represents the radius of the circle, then the radius of the curvature of the helical ramps is $\rho =r/{\cos }^2\theta$ (see Appendix A) and the total length of the roads in the helical ramps is $L=\rho \varphi$.

In a V2X environment, roadside equipment can broadcast the expected speed of the current road section to ICVs driving on helical ramps, and the ICVs can further adjust their own motion state according to this information. Based on this, we propose an ICV following model that takes into account the expected speed of the helical ramps, and its kinematic expression is:

$$\rho {\ddot{\varphi }}_i(t)=\alpha \left[G\left(\rho \Delta {\varphi }_i(t)\right)-\rho {\dot{\varphi }}_i(t)\right]+\lambda \rho \Delta {\dot{\varphi }}_i(t)+\gamma \left(v_g-\rho {\dot{\varphi }}_i(t)\right)$$

(1)

where $\rho {\varphi }_i(t)$ represents the position of the i-th vehicle at time t, $\alpha =1/\tau$ represents the sensitivity coefficient, and $\tau$ represents the delay time. $v_g$ represents the expected speed of the helical ramps, $\lambda$ is the speed difference feedback coefficient, and $\rho \Delta {\varphi }_i(t)=\rho {\varphi }_{i+1}(t)-\rho {\varphi }_i(t)$ represents the headway between the i-th car and the (i + 1)-th car. $\rho {\dot{\varphi }}_i(t)$ represents the speed of the i-th vehicle at time t, and $\rho \Delta {\dot{\varphi }}_i(t)=\rho {\dot{\varphi }}_{i+1}(t)-\rho {\dot{\varphi }}_i(t)$ represents the speed difference between the i-th vehicle and the (i + 1)-h vehicle. $\rho {\ddot{\varphi }}_i(t)$ represents the acceleration of the i-th vehicle at time t, and $\gamma$ represents the reaction coefficient. $G\left(\rho \Delta {\varphi }_i(t)\right)$ represents the optimal speed function of the i-th vehicle at time t, and the specific expression is as follows [41]:

$$G\left(\rho \Delta {\varphi }_i(t)\right)=\frac{\rho {\omega }_{\max }\mp v_{g,\max }}{2}\left[\tanh (\rho \Delta {\varphi }_i(t)-g_s(\theta ))+\tanh (g_s(\theta ))\right]$$

(2)

where ${\omega }_{\max }$ represents the maximum angular velocity of the vehicle, + represents the downhill situation, and − represents the uphill situation. According to the literature [42], the safe distance of vehicles $g_s(\theta )$ on the helical ramps can be defined as follows:

$$g_s(\theta )=g_s(1\mp \beta \sin \theta )$$

(3)

where $g_s$ represents the safe distance without an angle, and $\beta$ is a constant, usually $\beta =1$. $v_{g,\max }$ is the maximum increase or decrease speed of the helical ramps, and its expression is:

$$v_{g,\max }=\frac{mg\sin \theta }{\chi }$$

(4)

$\chi$ represents the longitudinal friction coefficient; for simplicity, we use $v_{g,\max }=\sin \theta$.

In addition, since the angular velocity of the moving vehicle can be obtained from the derivative of radians, Equation (1) can be further expressed as follows:

$$\rho {\dot{\omega }}_i(t)=\alpha \left[G\left(\rho \Delta {\varphi }_i(t)\right)-\rho {\omega }_i(t)\right]+\lambda \rho \Delta {\omega }_i(t)+\gamma \left(\rho {\omega }_v-\rho {\omega }_i(t)\right)$$

(5)

where ${\omega }_i(t)={\dot{\varphi }}_i(t)$, ${\dot{\omega }}_i(t)={\ddot{\varphi }}_i(t)$, $v_g=\rho {\omega }_v$.

According to the centripetal force formula, the maximum angular velocity of the vehicle is related to the lateral friction coefficient, namely:

$$m{\omega }_{\max }^2\rho =\mu mg\cos \theta$$

(6)

Then the maximum angular velocity of the vehicle can be obtained as:

$${\omega }_{\max }=\sqrt{\frac{\mu g\cos \theta }{\rho }}$$

(7)

Generally speaking, in actual traffic situations, the maximum angular velocity of the vehicle will be lower than the theoretical calculation value in Equation (7). Therefore, a constant coefficient $\xi (0<\xi \le 1)$ can be introduced into Equation (7), and the following optimal speed function can be obtained:

$$G\left(\rho \Delta {\varphi }_i(t)\right)=\frac{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta }{2}{G}_0\left(\rho \Delta {\varphi }_i(t)\right)$$

(8)

where $G_0\left(\rho \Delta {\varphi }_i(t)\right)=\tanh (\rho \Delta {\varphi }_i(t)-g_s(\theta ))+\tanh (g_s(\theta ))$, and the other parameters are the same as before.

3. Model Stability Analysis

3.1. Local Stability Analysis of Open Boundary Conditions

The current vehicle response to disturbance is particularly important during vehicle driving. This section will study the disturbance immunity of the model of ICVs proposed in this paper through the local stability analysis of open boundary conditions. In general, local stability means that when the disturbance propagates upstream, the speed fluctuation amplitude of the following vehicle is not greater than that of the adjacent vehicle in front. The specific definition is as follows:

$${‖\frac{W_i(s)}{W_{i+1}(s)}‖}_{\infty }=\underset{w\in \left[0,\infty \right)}{\sup }\left|\frac{W_i(s)}{W_{i+1}(s)}\right|=\left|\frac{W_i(jw)}{W_{i+1}(jw)}\right|\le 1 \forall w\in [0,\infty )$$

(9)

where $W_i(s)$ and $W_{i+1}(s)$ represent the Laplace transform of vehicle i and vehicle i + 1 speed fluctuation, respectively, and $w$ represents the signal angular frequency. When all the vehicles running in the helical ramps meet the local stability conditions, the traffic flow in this section has string stability. Therefore, we will give the local stability conditions of the model in the form of a theorem.

Theorem 1. 

When the following conditions are met:

$$\Lambda <\frac{2\lambda \left(\alpha +\lambda \right)+{\left(\alpha +\gamma \right)}^2}{2\alpha }$$

(10)

the model proposed in this paper can ensure the stability of the traffic flow in the helical ramps.

Proof of Theorem 1. 

It is assumed that the ICV travels in the same lane of the helical ramps in a car-following manner, and all vehicles travel at the same speed as the preceding vehicle with the same spacing. At this time, a small disturbance is applied to the leading vehicle of the current platoon, and all subsequent following vehicles will change with the motion state of the preceding vehicle. For the convenience of our analysis, it can be assumed that the expected speed of the lead vehicle after a small disturbance is $\rho {\omega }_v$, then all vehicles on the road will have the following steady state:

$${\left[\rho {\omega }^{\ast }(t)\hspace{1em}\hspace{1em}\rho \Delta {\varphi }^{\ast }(t)\right]}^T={\left[\rho {\omega }_v\hspace{1em}\hspace{1em}{G}^{-1}\left(\rho {\omega }_v\right)\right]}^T$$

(11)

where $G^{-1}$ is the inverse function of the optimized speed function $G$. □

The corresponding linear error dynamic equation can be obtained by linearizing Equation (5) in a steady state as follows:

$$\left\{\begin{array}{l}\frac{d\sigma \rho {\omega }_i(t)}{dt}=\alpha \left[\Lambda \sigma y_i(t)-\sigma \rho {\omega }_i(t)\right]+\lambda \left(\sigma \rho {\omega }_{i+1}(t)-\sigma \rho {\omega }_i(t)\right)-\gamma \left(\sigma \rho {\omega }_i(t)\right)\\ \frac{d\sigma y_i(t)}{dt}=\sigma \rho {\omega }_{i+1}(t)-\sigma \rho {\omega }_i(t)\end{array}\right.$$

(12)

where $y_i(t)=\rho \Delta {\varphi }_i(t)$, $\sigma \rho {\omega }_i(t)=\rho {\omega }_i(t)-\rho {\omega }_v$, $\sigma \rho {\omega }_{i+1}(t)=\rho {\omega }_{i+1}(t)-\rho {\omega }_v$, $\sigma y_i(t)=y_i(t)-\frac{G^{-1}\left(\rho {\omega }_v\right)}{\rho }$, $\Lambda ={\left.\frac{G\left(\rho \Delta {\varphi }_i(t)\right)}{\Delta {\varphi }_i(t)}\right|}_{\Delta {\varphi }_i(t)=\frac{G^{-1}\left(\rho {\omega }_v\right)}{\rho }}$.

If $\sigma \rho {\omega }_i(t)$ and $\sigma y_i(t)$ are selected as state variables, the state space expression of system (12) can be expressed as:

$$\left\{\begin{array}{l}\left[\begin{array}{c}\frac{d\sigma \rho {\omega }_i(t)}{dt}\\ \frac{d\sigma y_i(t)}{dt}\end{array}\right]=\left[\begin{array}{cc}-\alpha -\lambda -\gamma & \alpha \Lambda \\ -1& 0\end{array}\right]\left[\begin{array}{c}\sigma \rho {\omega }_i(t)\\ \sigma y_i(t)\end{array}\right]+\left[\begin{array}{c}\lambda \\ 1\end{array}\right]\sigma \rho {\omega }_{i+1}(t)\\ \sigma \rho {\omega }_i(t)=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}\sigma \rho {\omega }_i(t)\\ \sigma y_i(t)\end{array}\right]\end{array}\right.$$

(13)

Set $A=\left[\begin{array}{cc}-\alpha -\lambda -\gamma & \alpha \Lambda \\ -1& 0\end{array}\right], B=\left[\begin{array}{c}\lambda \\ 1\end{array}\right], C=\left[\begin{array}{cc}1& 0\end{array}\right],$$W_i(s)=L\left(\sigma \rho {\omega }_i(t)\right)$,$W_{i+1}(s)=L\left(\sigma \rho {\omega }_{i+1}(t)\right)$ where $L\left(•\right)$ represents the Laplace transform of the corresponding variable. From the frequency domain perspective, the motion state relationship between the following vehicle and its preceding vehicle can be expressed in the form of transfer function, that is:

$$W_i(s)=\Gamma (s)W_{i+1}(s)$$

(14)

where $\Gamma (s)$ is the error transfer function. According to the transfer function theory:

$$\begin{array}{l}\Gamma (s)=C{\left[sI-A\right]}^{-1}B\\ =\left[\begin{array}{cc}1& 0\end{array}\right]{\left[\begin{array}{cc}s+\alpha +\lambda +\gamma & -\alpha \Lambda \\ 1& s\end{array}\right]}^{-1}\left[\begin{array}{c}\lambda \\ 1\end{array}\right]\\ =\frac{\lambda s+\alpha \Lambda }{s^2+\left(\alpha +\lambda +\gamma \right)s+\alpha \Lambda }\end{array}$$

(15)

where $d(s)=s^2+\left(\alpha +\lambda +\gamma \right)s+\alpha \Lambda$ represents the characteristic equation of the system. From the definition of local stability:

$${‖\Gamma (s)‖}_{\infty }=\underset{w\in \left[0,\infty \right)}{\sup }\left|\Gamma (s)\right|=\sqrt{\frac{{\left(\alpha \Lambda \right)}^2+{\left(w\lambda \right)}^2}{{\left(\alpha +\lambda +\gamma \right)}^2{w}^2+{\left(\alpha \Lambda -w^2\right)}^2}}\le 1$$

(16)

Then the following inequality can be obtained:

$$w^2+2\lambda \left(\alpha +\lambda \right)+{\left(\alpha +\gamma \right)}^2-2\alpha \Lambda \ge 0$$

(17)

Therefore, we can obtain a sufficient condition for Equation (15) to hold:

$$\Lambda <\frac{2\lambda \left(\alpha +\lambda \right)+{\left(\alpha +\gamma \right)}^2}{2\alpha }$$

(18)

The proof is completed.

3.2. Linear Stability Analysis of Periodic Boundary Conditions

Section 3.1 focuses on the local stability of the model. In this section, the linear stability of the proposed ICVs model under periodic conditions is analyzed by the perturbation method, and then the influences of the expected speed, slope of helical ramps and friction coefficient on the traffic flow are explored. Suppose that all ICVs on helical ramps drive on the road with the same spacing h and the same speed $G\left(h\right)$, the steady state of traffic flow can be expressed as:

$$\rho {\varphi }_m^0(t)=hm+G(h),h=Y/N=\rho \varphi /N$$

(19)

where L represents the length of the road, and N is the total number of vehicles. For the convenience of our analysis, through the forward difference and letting $\gamma =\alpha \eta \tau$, Equation (1) can be changed to the following difference equation:

$$\begin{array}{ll}\rho \Delta {\varphi }_i(t+2\tau )& =\rho \Delta {\varphi }_i(t+\tau )+\tau \left[G\left(\rho \Delta {\varphi }_{i+1}(t)\right)-G\left(\rho \Delta {\varphi }_i(t)\right)\right]\\ & +\lambda \tau \left(\rho \Delta {\varphi }_{i+1}(t+\tau )-\rho \Delta {\varphi }_{i+1}(t)-\rho \Delta {\varphi }_i(t+\tau )+\rho \Delta {\varphi }_i(t)\right)\\ & -\eta \left(\rho \Delta {\varphi }_i(t+\tau )-\rho \Delta {\varphi }_i(t)\right)\end{array}$$

(20)

We add a small disturbance ${\delta }_m(t)$ to the steady-state position of the vehicle, namely:

$$\rho {\varphi }_m(t)=\rho {\varphi }_m^0(t)+{\delta }_m(t)$$

(21)

Substituting Equations (19) and (21) into Equation (20) and linearizing them, the following linear discrete error equation can be obtained:

$$\begin{array}{ll}\Delta {\delta }_m(t+2\tau )& =\Delta {\delta }_m(t+\tau )+\tau \frac{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta }{2}{G}_0^{\prime }\left[\Delta {\delta }_{m+1}(t)-\Delta {\delta }_m(t)\right]\\ & +\lambda \tau \left(\Delta {\delta }_{m+1}(t+\tau )-\Delta {\delta }_{m+1}(t)-\Delta {\delta }_m(t+\tau )+\Delta {\delta }_m(t)\right)\\ & -\eta \left(\Delta {\delta }_m(t+\tau )-\Delta {\delta }_m(t)\right)\end{array}$$

(22)

where $\Delta {\delta }_m(t)={\delta }_{m+1}(t)-{\delta }_m(t)$, $G_0^{\prime }={\left.\frac{d{G}_0\left(\rho \Delta {\varphi }_m(t)\right)}{d\rho \Delta {\varphi }_m(t)}\right|}_{\rho \Delta {\varphi }_m(t)=h}$.

If we let $\Delta {\delta }_m(t)=e^{ikm+zt}$, Equation (22) can be further rewritten as:

$$e^{2z\tau }-e^{z\tau }-\tau \frac{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta }{2}{G}_0^{\prime }\left(e^{ik}-1\right)-\lambda \tau \left(e^{ik}-1\right)\left(e^{z\tau }-1\right)+\eta \left(e^{z\tau }-1\right)=0$$

(23)

Expanding z to $z=z_1(ik)+z_2(ik)+\cdots$ and substituting it into Equation (15), the coefficients of the first-order and second-order terms of ik are:

$$z_1=\frac{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta }{2\left(1+\eta \right)}{G}_0^{\prime }$$

(24)

$$\begin{array}{l}{z}_2=\frac{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta }{4\left(1+\eta \right)}{G}_0^{\prime }+\frac{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta }{2{\left(1+\eta \right)}^2}{G}_0^{\prime }\lambda \tau \\ -3\frac{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta }{4{\left(1+\eta \right)}^3}{G}_0^{\prime }\tau -\frac{\eta \tau {\left(\frac{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta }{2}{G}_0^{\prime }\right)}^2}{2{\left(1+\eta \right)}^3}\end{array}$$

(25)

For the small disturbance of the long-wave mode, if $z_2<0$, the system will evolve from an initial equilibrium state to an unstable state. The size of the initial disturbance will increase with time, and finally the traffic flow will evolve into a traffic congestion phenomenon with stop-and-go waves. On the contrary, a small disturbance will dissipate over time, and the traffic flow will gradually return to the initial stable state. The critical stability curve of traffic flow can be obtained as follows:

$$\tau =\frac{2{\left(1+\eta \right)}^2}{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta G_0^{\prime }\left(3+\eta \right)-4\lambda \left(1+\eta \right)}$$

(26)

Correspondingly, the stability condition of the model is:

$$\tau <\frac{2{\left(1+\eta \right)}^2}{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta G_0^{\prime }\left(3+\eta \right)-4\lambda \left(1+\eta \right)}$$

(27)

It can be seen from Equation (27) that the parameter $\eta$, the angle of the helical ramps and the radius of curvature will affect the stability of the traffic flow.

Figure 2, Figure 3 and Figure 4 show the two-dimensional phase diagram of the headway sensitivity of the proposed model with different $\eta$ values when the parameters $k=0.1$, $r=60$, $\lambda =0.2$, $h=4$, and $g=0.98$. In Figure 2, for example, the headway sensitivity phase plane is divided by the critical stability curve of the model into an upper stable region and a lower unstable region, and each curve has a vertex, which we call the critical point $(y_c,{\alpha }_c)$. In the stable area, the small disturbance of the traffic flow will be effectively suppressed, and will eventually evolve into a stable traffic flow operation state. In the unstable area, a small disturbance in the traffic flow will be amplified by the increase in time, which will eventually lead to traffic congestion and make the traffic flow stop and go.

In Figure 2, Figure 3 and Figure 4, the three critical stability curves correspond to the uphill, downhill and no-inclination angles of the helical ramps. It can be seen from Figure 3 that the stable region is the largest for uphill, the smallest for downhill, and the ramp without any inclination is in the middle. Therefore, when the vehicle is driving on helical ramps, driving uphill will be more beneficial to the stability of the traffic flow. In addition, compared with Figure 2, Figure 3 and Figure 4, it can be found that the larger the feedback coefficient of the expected speed of the helical ramps, all the critical stability curves will become ‘shorter’; that is, the stability region of the model will also increase. This shows that in helical ramps, ICVs can effectively improve the stability of traffic flow by considering the expected speed broadcasted by roadside equipment.

4. Nonlinear Analysis

This section will further explore the traffic characteristics of the model proposed in this paper near the critical point $(y_c,{\alpha }_c)$ through the reductive perturbation analysis method, and then reveal the influence of the expected speed of helical ramps on the spatiotemporal evolution characteristics of the traffic flow. In the vicinity of the critical point $(y_c,{\alpha }_c)$, given a positive small quantity $\epsilon (0<\epsilon <1)$, the slow-scale spatial variable m and the time variable t are extracted, and the auxiliary slow variables T and $\rho \mathsf{\Phi }$ are defined as follows:

$$\rho \mathsf{\Phi }=\epsilon \left(m+yt\right), T={\epsilon }^{3}t$$

(28)

where $y$ is a constant to be determined.

The headway between m and m + 1 vehicles is:

$$\rho \Delta {\varphi }_m(t)=y_c+\epsilon R(\rho \mathsf{\Phi },T)$$

(29)

According to Equations (28) and (29), Equation (22) can be expanded to the fifth-order term of $\epsilon$ by using Taylor series to obtain the corresponding partial differential equation:

$$\begin{array}{l}{\epsilon }^2\left(y-\frac{\mathsf{\Omega }{G}_0^{\prime }}{1+\eta }\right){\partial }_{\rho \mathsf{\Phi }}R+{\epsilon }^3\frac{y\tau \left[\left(3+\eta \right)y-2\lambda \right]-\mathsf{\Omega }{G}_0^{\prime }}{2\left(1+\eta \right)}{\partial }_{{}_{\rho \mathsf{\Phi }}}^{2}R\\ +{\epsilon }^4\left(\frac{\left(\eta +7\right)y^3{\tau }^2-\mathsf{\Omega }{G}_0^{\prime }-3\lambda \left(y\tau +1\right)y\tau }{6\left(1+\eta \right)}{\partial }_{\rho \mathsf{\Phi }}^{3}R-\frac{\mathsf{\Omega }{G}_0^{‴}}{6\left(1+\eta \right)}{\partial }_{\rho \mathsf{\Phi }}{R}^3+{\partial }_{T}R\right)\\ +{\epsilon }^5\left(\begin{array}{l}\frac{\left(\eta +15\right)y^4{\tau }^3-\mathsf{\Omega }{G}_0^{\prime }-2\lambda \left(2y^2{\tau }^2+3y\tau +2\right)y\tau }{24\left(1+\eta \right)}{\partial }_{{}_{\rho \mathsf{\Phi }}}^{4}R\\ -\frac{\mathsf{\Omega }{G}_0^{‴}}{12\left(1+\eta \right)}{\partial }_{{}_{\rho \mathsf{\Phi }}}^2{R}^3+\frac{\left[-\lambda +\left(3+\eta \right)y\right]}{\left(1+\eta \right)}{\partial }_T{\partial }_{\rho \mathsf{\Phi }}R\end{array}\right)=0\end{array}$$

(30)

where $\mathsf{\Omega }=\frac{\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta }{2}$, ${\partial }_{\rho \varphi }=\frac{\partial }{\partial \rho \varphi }$, ${\partial }_T=\frac{\partial }{\partial T}$, ${\partial }_T{\partial }_{\rho \varphi }=\frac{{\partial }^2}{\partial T\partial \rho \varphi }$, $G_0^{\prime }={\left.\frac{d{G}_0\left(\rho \Delta \varphi \right)}{d\rho \Delta \varphi }\right|}_{\rho \Delta \varphi =y_c}$, $G_0^{‴}={\left.\frac{d^3{G}_0\left(\rho \Delta \varphi \right)}{d{\left(\rho \Delta \varphi \right)}^3}\right|}_{\rho \Delta \varphi =y_c}$.

Near the critical point $(y_c,{\alpha }_c)$, let $y=\frac{\mathsf{\Omega }{G}_0^{\prime }}{1+\eta }$, $\tau =\left(1+{\epsilon }^2\right){\tau }_c$, ${\tau }_c=\frac{{\left(1+\eta \right)}^2}{\left(3+\eta \right)\mathsf{\Omega }{G}_0^{\prime }-2\lambda \left(1+\eta \right)}$, by eliminating the second-order and third-order terms of $\epsilon$ in Equation (30), a simplified equation can be further obtained:

$${\partial }_{T}R-{\Xi }_1{\partial }_{{}_{\rho \mathsf{\Phi }}}^{3}R+{\Xi }_2{\partial }_{\rho \mathsf{\Phi }}{R}^3+\epsilon \left({\Xi }_3{\partial }_{{}_{\rho \mathsf{\Phi }}}^{2}R+{\Xi }_4{\partial }_{{}_{\rho \mathsf{\Phi }}}^{4}R+{\Xi }_5{\partial }_{{}_{\rho \mathsf{\Phi }}}^2{R}^3\right)=0$$

(31)

where

$${\Xi }_1=-\frac{(\eta +7)y^3{\tau }_c^2-\mathsf{\Omega }{G}_0{}^{\prime }-3\lambda (y{\tau }_c+1)y{\tau }_c}{6(1+\eta )}$$

$${\Xi }_2=-\frac{\mathsf{\Omega }{G}_0^{‴}}{6(1+\eta )}$$

$${\Xi }_3=\frac{\mathsf{\Omega }{G}_0^{\prime }}{2(1+\eta )}$$

$$\begin{array}{l}{\Xi }_4=\frac{(\eta +15)y^4{\tau }_c^3-\mathsf{\Omega }{G}_0^{\prime }-2\lambda (2y^2{\tau }_c^2+3y{\tau }_c+2)y{\tau }_c}{24(1+\eta )}\\ -\frac{{\tau }_c\left[(3+\eta )y-2\lambda \right]\left[(\eta +7)y^3{\tau }_c^2-\mathsf{\Omega }{G}_0^{\prime }-3\lambda (y{\tau }_c+1)y{\tau }_c\right]}{6{(1+\eta )}^2}\end{array}$$

$${\Xi }_5=\frac{\left[2{\tau }_c(3y+y\eta -\lambda )-\eta -1\right]\mathsf{\Omega }{G}_0^{‴}}{12{(1+\eta )}^2}$$

The following transformation is made for Equation (31):

$$T^{\prime }={\Xi }_{1}T, R=\sqrt{\frac{{\Xi }_1}{{\Xi }_2}}{R}^{\prime }$$

(32)

Equation (31) can be transformed into an mKdV normalized equation with a correction term:

$${\partial }_T{R}^{\prime }-{\partial }_{{}_{\rho \mathsf{\Phi }}}^3{R}^{\prime }+{\partial }_{\rho \mathsf{\Phi }}{R^{\prime }}^3+\epsilon M\left[R^{\prime }\right]=0$$

(33)

where $M\left[R^{\prime }\right]=\sqrt{\frac{1}{{\Xi }_1}}\left({\Xi }_3{\partial }_{{}_{\rho \mathsf{\Phi }}}^2{R}^{\prime }+{\Xi }_4{\partial }_{{}_{\rho \mathsf{\Phi }}}^4{R}^{\prime }+\frac{{\Xi }_1{\Xi }_5}{{\Xi }_2}{\partial }_{{}_{\rho \mathsf{\Phi }}}^2{R^{\prime }}^3\right)$.

Ignoring the correction term in Equation (33), we can further obtain the kink–antikink density wave solution of the mKdV equation:

$$R_0^{\prime }(\rho \mathsf{\Phi },T^{\prime })=\sqrt{c}\tanh \left(\sqrt{\frac{c}{2}}(\rho \mathsf{\Phi },c{T}^{\prime })\right)$$

(34)

The parameter c represents the propagation velocity of the kink–antikink wave, which can be obtained by the integral equation as follows:

$${\displaystyle {\int }_{-\infty }^{+\infty }\frac{\sqrt{c}}{{\Xi }_1{\Xi }_2}}\left({\Xi }_2{\Xi }_3{\partial }_{{}_{\rho \mathsf{\Phi }}}^2{R}^{\prime }+{\Xi }_2{\Xi }_4{\partial }_{{}_{\rho \mathsf{\Phi }}}^4{R}^{\prime }+{\Xi }_1{\Xi }_5{\partial }_{{}_{\rho \mathsf{\Phi }}}^2{R^{\prime }}^3\right)\tanh \left(\sqrt{\frac{c}{2}}(\rho \mathsf{\Phi },c{T}^{\prime })\right)d\rho \mathsf{\Phi }=0$$

(35)

Solving the above equations, we can obtain:

$$c=\frac{5{\Xi }_2{\Xi }_3}{2{\Xi }_2{\Xi }_4-3{\Xi }_1{\Xi }_5}$$

(36)

Therefore, near the critical point $(y_c,{\alpha }_c)$, the spacing kink–anti-kink density wave solution of the proposed model is:

$$\rho \Delta {\mathsf{\Phi }}_m(t)=y_c+\sqrt{\frac{{\Xi }_{1}c}{{\Xi }_2}\left(\frac{\tau }{{\tau }_c}-1\right)}\tanh \sqrt{\frac{c}{2}\left(\frac{\tau }{{\tau }_c}-1\right)}\left[m+\left(\frac{1}{1+\eta }-c{\Xi }_1\left(\frac{\tau }{{\tau }_c}-1\right)\right)t\right]$$

(37)

The corresponding amplitude A is:

$$A=\sqrt{\frac{2c\left[\left(\eta +7\right)y^3{\tau }_c{}^2-\mathsf{\Omega }{G}_0^{\prime }-3\lambda \left(y{\tau }_c+1\right)y{\tau }_c\right]}{\left(\xi \sqrt{\mu g\rho \cos \theta }\mp \sin \theta \right)G_0^{‴}}\left(\frac{\tau }{{\tau }_c}-1\right)}$$

(38)

The above analysis shows that in the unstable region of the critical stable point $(y_c,{\alpha }_c)$, the small disturbance encountered by the traffic flow will form a congestion wave with the evolution of time, and the specific value is determined by Equation (36). At the same time, the road traffic will appear in the high-density congestion phase and low-density free flow phase, in which the corresponding headway is $\rho \Delta {\varphi }_m(t)=y_c+A$ and $\rho \Delta {\varphi }_m(t)=y_c-A$. In addition, it can be seen from Equations (36) and (38) that the expected speed of the helical ramps has a direct impact on the headway amplitude A and the propagation speed c. Subsequently, the theoretical analysis results are verified by simulation experiments, and the influence law is analyzed.

5. Numerical Simulation

In order to verify the theoretical analysis results, this section will numerically simulate the proposed model under periodic boundary conditions to reveal the evolution characteristics of the traffic flow density wave in its unstable region. The simulation parameters are selected as:

$$N=100, a=1, \mu =0.5, k=0.1, r=60, \lambda =0.2, y_c=4, g=0.98$$

The initial condition is set as: $\rho \Delta {\varphi }_m(0)=4(m\ne 50,51)$, $\rho \Delta {\varphi }_m(0)=4-0.1(m=50)$, $\rho \Delta {\varphi }_m(0)=4+0.1(m=51)$.

The slope angle $\theta =4^{\circ }$ is selected. Figure 5 and Figure 6 show the evolution of traffic flow corresponding to the uphill and downhill conditions of the helical ramps under the value of $\eta$ at 0, 0.3 and 0.5, respectively.

It can be seen from Figure 5a that when driving uphill, because $\eta =0$, that is, the expected speed of the helical ramps is not considered at this time, the headway of the whole traffic flow fluctuates strongly, and the fluctuation range is large. Figure 5b,c shows that the expected speed of the helical ramps is considered. In Figure 5b, because the parameter values do not meet the stability conditions (Equation (27)), the traffic flow will still become unstable under the action of small disturbances, and the traffic congestion will propagate upstream in the form of kink–anti-kink waves. However, compared with Figure 5a, the headway fluctuation is obviously slowed down, and the fluctuation range is smaller. In Figure 5c, the whole traffic flow is completely stable, and the fluctuation of the headway also disappears. The main reason is that the value of the parameter $\eta$ satisfies the stability condition, and the disturbed traffic flow will return to the initial stable state. In addition, we can also find that as the value of parameter $\eta$ increases, the severity of traffic congestion decreases.

In the downhill situation in Figure 6, the evolution of traffic flow also appears similar to that in Figure 5. However, it is worth noting that in Figure 6c, although the parameter $\eta$ also takes the same value as Figure 5c, the traffic flow in Figure 6c is not completely stable. It can be seen from Section 3.2 that in the same case, the corresponding stability range downhill is lower than that uphill, so the traffic flow in Figure 6c is not completely stable in the same situation.

In helical ramps, ICVs driving on different slopes will also show different following behaviors. Therefore, in order to further describe the change in headway at different angles, this paper selects the parameter $\eta =0.3$ to analyze the evolution of the headway. Figure 7 and Figure 8, respectively, give the distribution map of the downhill and uphill headways at t = 12,000 steps under different angles. It can be seen from Figure 7 that when driving downhill, the smaller the inclination angle of the road, the lower the fluctuation amplitude of the headway. As shown in Figure 7a, when the angle $\theta =0^{\circ }$, the maximum headway is about 5 m, and as the angle increases, the maximum headway also increases from 5 m. When the angle $\theta =6^{\circ }$, the maximum headway exceeds 5.5 m. This shows that a large downhill angle is not conducive to the stability of traffic.

When driving uphill, it can be seen from Figure 8a–d that the maximum headway will gradually decrease as the road inclination angle increases. As shown in Figure 8b, when the angle $\theta =2^{\circ }$, the maximum headway of the traffic flow exceeds 4.5 m and is close to 5 m. When the angle increases to $\theta =6^{\circ }$, the maximum headway of the traffic flow is less than 4.5 m. This shows that a large uphill angle is conducive to the stability of traffic. At the same time, comparing Figure 7 and Figure 8, it can be seen that when there is an inclination angle, the maximum headway of the traffic flow uphill is lower than that downhill, which is also consistent with the analysis results in Section 3.2.

In order to highlight the contribution of this paper, we compare our model with a model proposed in the literature [37] (called Meng’s model). The slope angle $\theta =5^{\circ }$ is selected, the parameter $\eta =0.3$ and the other parameters are consistent with the previous. Figure 9 and Figure 10 represent the space–time evolution diagram of the headway and the headway profiles at t = 10,000 steps corresponding to our model and Meng’s model when driving uphill, respectively. Comparing Figure 9a and Figure 10a, it can be clearly seen that under the same conditions, the traffic flow corresponding to our model is more stable and the traffic congestion is lower. In addition, it can be seen from Figure 9b and Figure 10b that when the time t = 10,000 steps, the headway in Figure 10b fluctuates more. Since Meng’s model does not consider the expected speed of the helical ramp, the speed of the vehicle cannot be adjusted in advance according to the macro information when following the vehicles in front, resulting in violent fluctuations in traffic flow. Similarly, Figure 11 and Figure 12 present the space–time evolution diagram of the headway and the headway profiles at t = 10,000 steps corresponding to our model and Meng’s model when driving downhill, respectively. The comparison shows that the stability of traffic flow and the fluctuation range of the headway are consistent with when driving uphill. This further shows that it is important to consider the expected speed information of the helical ramp for the regulation of traffic flow and the alleviation of congestion in the region.

6. Conclusions

This study aimed to explore the impact of expected speed in a V2X environment on traffic flow in helical ramps, and a microscopic car-following model considering the expected speed of the helical ramps was proposed for ICVs. For the proposed model, a local stability analysis of the open boundary and a linear and nonlinear stability analysis of the periodic boundary conditions were carried out. The theoretical results and numerical simulation results showed that the expected speed can effectively improve the traffic flow stability of the helical ramp and alleviate traffic congestion.

The findings of this study revealed that the stability of traffic flow in helical ramps was related to the slope, curvature radius, friction coefficient and the expected speed. The results found that when the vehicle is driving uphill, it will be more beneficial to the stability of the traffic flow. Further, it can be seen that the large angle downhill is not conducive to the stability of the traffic flow, and the large angle uphill can enhance the stability of the traffic flow. In addition, the results showed that the expected speed received by ICVs can effectively enhance the stability of the overall traffic flow and avoid the occurrence of stop-and-go waves, whether on the uphill or downhill of the helical ramp. What is more, because of the driving speed $v=\rho \omega$ and $\rho =r/{\cos }^2\theta$, when the θ was fixed, it was easy to know that the driving speed was proportional to the radius of the helical ramp; that is, the greater the ramp radius, the greater the driving speed. This result was consistent with the conclusion obtained through the field-driving data analysis in the literature [36].

The research results of this paper lay a foundation for better ICV applications in mountainous cities. However, this study has a few limitations that must be acknowledged. First, due to the limited conditions and the high test risk of ICVs in the helical ramps of mountainous cities, the effect of the proposed model was mainly verified by numerical simulation, which inevitably ignored some factors. In order to make the model have better applicability, the model should be verified by a field-driving test in the future. Second, this paper only considered ICVs, and did not involve other types of vehicles. In recent years, mixed traffic composed of human-driven vehicles and ICVs has received much attention. Therefore, future research can consider a variety of types of vehicles and target the problem of mixed traffic in helical ramps.