A Car-Following Model for Mixed Traffic Flows in Intelligent Connected Vehicle Environment Considering Driver Response Characteristics

Abstract

Autonomous driving technology and vehicle-to-vehicle communication technology make the hybrid driving of connected and automated vehicles (CAVs) and regular vehicles (RVs) a long-existing phenomenon in the coming future. Among the existing studies, IDM models are mostly used to study the performance of homogeneous traffic flow. To explore the stability of mixed traffic flow, an extended intelligent driver model (IDM) based car-following model was proposed for mixed traffic flow (MTF) with both CAVs and RVs, considering the headway, the speed and acceleration of multiple front vehicles, as well as the response characteristics of RV drivers. Through the linear stability analysis, the criterion for the stability of MTFs was derived, and the relationship among the penetration rate of CAVs, equilibrium velocity and traffic stability in MTF are discussed. Based on the above theoretical model, a numerical simulation was conducted in two typical scenarios of starting and braking. The results showed that, at the microscopic scale, the vehicle in the Cooperative Adaptive Cruise Control (CACC) mode could significantly decelerate in response to the interference from other vehicles in the same traffic environment. At the macroscopic scale, as the penetration rate of CAVs increased, the overall acceleration fluctuation of the traffic flow decreased. At the same penetration rate of CAVs, the higher density of CAVs coincided with the higher stability of the MTF. When the penetration rate of CAVs was 50%, the degree of distribution had the greatest impact on the MTF. When the penetration rate of CAVs exceeded 70%, the degree of distribution had little impact on the MTF. This research can provide basic theoretical support for the management and control of MTF in the future.

1. Introduction

With the continuous advancement in information and network technology, connected and automated vehicles (CAVs) have emerged at the request of the times. With the support of advanced communication technology, traffic information can be transmitted in real time between vehicles, roads, and drivers. CAVs can offer great help in improving the efficiency of traffic flow and optimizing users’ riding experience. The traffic flow will be mixed up with the gradually popularized CAVs and regular vehicles (RVs, i.e., human-driven vehicles) in the future. Therefore, it is significant to study the MTF composed of intelligent CAVs and RVs.

Car-following behavior is the most common behavior in the driving process that describes the interactive mechanism between vehicles traveling along the same lane from a microscopic perspective. Car-following theorization is conducive to higher efficiency and stability of traffic flows. On a theoretical basis, car-following modeling further deals with the characteristics of car-following behavior. In the late 1950s, Herman et al. [1] put forward a famous multi-vehicle following hypothesis. Later, Newell et al. [2] proposed a car-following model based on the optimized speed function to characterize the effect of headway on the following vehicles. Based on Newell’s model, Bando et al. [3] analyzed the effect of the speed difference and established the optimal vehicle model (OVM). Subsequently, numerous scholars [4,5] completely extended the OVM model according to the impact of multi-vehicle information on car-following behavior. Taking a different approach, Treiber et al. [6] established an intelligent driver model (IDM) with physical significance which was easy to calibrate. The idea of this model is to obtain the following rules of vehicles according to the relationship between the actual speed of the considered vehicle and the speed and position of the front vehicle. The development of the IDM involved two main aspects. First, such factors as time delay were analyzed from the perspective of artificial driving characteristics. For example, focusing on the time delay, measurement error, and time prediction, Treiber et al. [7] proposed the human driver model (HDM). Saifuzzaman et al. [8] established the task difficulty car-following framework on intelligent driver model (TDIDM) by introducing the difficulty of driving tasks as the dynamic interaction between driving task requirements and the driver’s capability. Second, the influences of multi-vehicle speed, acceleration, and headway on car-following behavior were explored based on the cooperative vehicle infrastructure system. With consideration of the acceleration information about the front vehicle, Deng H et al. [9] improved the extended IDM proposed by Li et al. [10] with consideration of the expected acceleration of the front vehicle and verified its stability through theoretical analysis.

With the advancement of Internet of Vehicles technology, CAV is closer to people’s real life, research on the MTF composed of CAV and RV has become more abundant. Qin et al. [11] deduced a basic graph model for MTF at distinct penetration rates of CAVs and analyzed and optimized the stability of MTF to improve the safety of traffic flow. Yang et al. [12] considered the four types of car-truck following combinations and explored the effect of different combinations on the stability by the original Bando’s optimal velocity (OV) model to heterogeneous form. According to the systematic framework for the analysis of mixed-traffic problems, Li et al. [13] analyzed the stability of MTF and the energy consumption of vehicles in three different modes of intelligent networked vehicle control (ACC, CACC, and CCC), using the infinite norm theory of transfer function. Zong et al. [14] established a car-following model (MFRHVAD) suitable for mixed traffic flow based on the traditional optimal velocity model (OV) and verified that CAV can make the fleet more stable through microscopic numerical simulation. Most of the existing studies are focused on the stability of MTF and the penetration rate of CAVs while overlooking the impact of the distribution degree of CAVs on MTF.

The instability of mixed traffic flow is mostly caused by the operation of human drivers. Qiu et al. [15] divides the driving style based on the cluster analysis method and constructs a car following model considering the driving style of the individual driver. Through experiments, Hei et al. [16] determined the main parameters of drivers with different styles of car-following behavior and established a quantitative model of vehicle-to-vehicle interaction behavior considering driving styles. However, few studies have discussed the influence of driver style on the stability of MTF. Therefore, according to the analyses above, based on the IDM, besides considering the effect of the intelligent features of CAVs on traffic flow, this paper analyzes the influence of the response characteristics of human drivers on the stability of MTF and explores the influence of penetration rate and distribution degree of CAVs on MTF through numerical simulation.

The rest of this paper is organized as follows: Section 2 presents an extended IDM applicable for RVs and CAVs, which lays a foundation for the study of MTF composed of CAVs and RVs. Section 3 firstly derives the criterion for the stability of the extended IDM, then analyses the traffic flow stability of different driver type, compares RV, ACC and CCAC traffic flow stability, and finally studies the relationship among the penetration rate of CAVs, equilibrium velocity and traffic stability in MTF. Section 4 compares the acceleration fluctuation of RV, ACC and CACC under two typical disturbances of braking and starting by numerical simulation at the microscopic scale, analyzes the influence of the penetration rate and distribution degree of CAVs and the driver style of RVs on MTF by numerical simulation at the macroscopic scale. Finally, Section 5 concludes this paper.

2. The Extended IDM

For RVs, the IDM is compatible with the real data within a reasonable error range [17]. For CAVs with the function of acquiring the driving state information about the front vehicle, the IDM can reflect its “intelligent” features to a great extent [11]. Based on the classical IDM, combined with the improvement in later studies, this paper proposes a car-following model with consideration of the headway, speed difference, and acceleration of multiple front vehicles. The model can reflect the car-following characteristics of human-driven vehicles and intelligent connected vehicles by modifying parameters. Acceleration and desired safe headway can be calculated by the following Equations (1) and (2):

$$\dot{a_n\left(t\right)=a_0[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4-{\left(\frac{{\tau }_n{s}_n^{*}\left(t\right)}{{{\displaystyle \sum }}_{q=1}^Q{\phi }_q{s}_{n+q-1}\left(t\right)}\right)}^2}]+\mu {\displaystyle \sum }_{q=1}^Q{\beta }_q{a}_{n+q-1}\left(t\right)$$

(1)

$$s_n^{*}\left(t\right)=s_0+T{v}_n\left(t\right)+\frac{v_n\left(t\right)\left({\sum }_{q=1}^Q{\alpha }_q\Delta v_{n+q-1}\left(t\right)\right)}{2\sqrt{a_{0}b}}$$

(2)

where $a_0$ and $b$ is are the maximum acceleration and desired deceleration of vehicle n, respectively; $v_0$ is the desired velocity in free flow; $s_0$ is the minimum space interval for completely quiescent traffic; $T$ is the constant desired time interval; $\mu$ is the sensitivity coefficient of the vehicle to the acceleration of front vehicles; $Q$ is the number of the front vehicles whose state information is accessible to the considered vehicle; ${\tau }_n$ is the expected coefficient of car-following distance of RVs, which is applicable only to the car-following model for RVs; $v_n\left(t\right)$ and $s_n^{*}\left(t\right)$ are the velocity and desired safe headway of vehicle $n$ at time $t$ respectively; $s_{n+i-1}\left(t\right)$ and $\Delta v_{n+i-1}\left(t\right)$ are the headway and speed difference between vehicle $n+i-1$ and the front vehicle at time $t$; ${\alpha }_q$, ${\beta }_q$ and ${\phi }_q$ are the weights of speed difference, acceleration, and headway of vehicle $n+q$.

2.1. Car-Following Model for RVs

The MTF studied in this paper is composed of RVs and CAVs. In the driving process, the acceleration and deceleration of RVs depend completely on the driver’s judgment and responsiveness to the surrounding situation. Drivers with different response characteristics tend to have different reactions and decisions under the same traffic conditions. The characteristics of driving response, as a relatively stable habitual driving style, vary among individuals or groups [18]. Qiu et al. [15] classified drivers into four types by their car-following behavior and response characteristics that affected the velocity, acceleration, headway, and other indicators of their vehicles, as shown in

Table 1. The tpes and characteristics of drivers.

Driver Type	Characteristics of Car-Following Behavior	Response Characteristics
Type I	Low speed but a long headway	Less perceptive and highly responsive
Type II	Highest speed and short headway	More perceptive and steadily responsive
Type III	A medium level of speed and headway	Standard
Type IV	Lowest speed and longest headway	Less perceptive and less responsive

.

RVs acquire no information about the velocity and acceleration of front vehicles, so Formula (1) can be modified as:

$$\dot{a_n\left(t\right)=a_0\left[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4-{\left(\frac{{\tau }_n{s}_n^{*}\left(t\right)}{s_n\left(t\right)}\right)}^2\right]}$$

(3)

$$s_n^{*}\left(t\right)=s_0+T{v}_n\left(t\right)+\frac{v_n\left(t\right)\Delta v_n\left(t\right)}{2\sqrt{a_{0}b}}$$

(4)

According to the research [19], the parameters that can more accurately describe driver’s response characteristics in car-following behavior include $a_0=1 \mathrm{m}/{\mathrm{s}}^2$, $\mathrm{b}=2.8 \mathrm{m}/s^2$, $s_0=2 \mathrm{m}$, and $T=1.5 \mathrm{s}$.

When a large-size vehicle is in front, the desired headway of the considered RV will be subject to such objective factors as vision and slow braking [9]. Similarly, drivers with different response characteristics also tend to choose different parameters such as the desired safe headway, maximum velocity, and acceleration.

Table 2. Parameter settings for different driver types.

Driver Type	${\mathit{\tau }}_{\mathit{n}}$	${\mathit{v}}_0$
Type I	1.1	$11 \mathrm{m}/\mathrm{s}$
Type II	0.9	$13 \mathrm{m}/\mathrm{s}$
Type III	1	$12 \mathrm{m}/\mathrm{s}$
Type IV	1.2	$10 \mathrm{m}/\mathrm{s}$

below shows the parameter settings of the expected coefficient of car-following distance and the desired velocity according to the driver types and characteristics.

2.2. Car-Following Model for CAVs

In an intelligent network environment, information can be accurately transmitted between CAVs in real time via vehicle-to-vehicle (V2V) communication technology. If the considered vehicle and the front vehicle happens to be CAVs, the former runs in the CACC mode. If the vehicle considered is a CAV while the front vehicle is an RV, that is, the V2V communication conditions are not met, then the former runs in the Adaptive Cruise Control (ACC) mode. ACC mainly relies on onboard sensors to acquire information about front vehicles, so that the vehicle considered can be controlled by the model algorithm. Based on the function of ACC, CACC also has a connected driver assistant system, which can not only identify the driving state of front vehicles but also acquire the state information about other CAVs in the traffic flow through information technology. The above two scenarios are further discussed below.

(1)

RV followed by CAV

When a CAV is following behind an RV, the CAV is considered to run in the ACC mode. In other words, the CAV can acquire the speed and acceleration information about the front RV via the onboard sensors. From Formula (1), therefore, the acceleration of the CAV can be expressed as:

$$\dot{a_n\left(t\right)=a_0[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4-{\left(\frac{s_n^{*}\left(t\right)}{s_n\left(t\right)}\right)}^2}]+\mu a_{n-1}\left(t\right)$$

(5)

$$s_n^{*}\left(t\right)=s_0+T{v}_n\left(t\right)+\frac{v_n\left(t\right)\Delta v_n\left(t\right)}{2\sqrt{a_{0}b}}$$

(6)

Combined with the parameter settings of RV and CACC, the characteristic parameters of the car-following behavior in the ACC mode are set as follows:$a_0=2 \mathrm{m}/{\mathrm{s}}^2$, $\mathrm{b}=2 \mathrm{m}/{\mathrm{s}}^2$, $v_0=10 \mathrm{m}/\mathrm{s}$, $s_0=2 \mathrm{m}$, $T=2 \mathrm{s}$, and $\mu =0.16$ [9].

(2)

CAV followed by CAV

When a CAV is following behind another CAV, the follower is considered to run in the CACC mode, and the state information about the front CAV can be acquired via the follower’s onboard sensors. In addition, the vehicle considered can get access to data of multiple front vehicles in the fleet via the network [20]. From Formula (1), therefore, the acceleration of the follower CAV can be expressed as:

$$\dot{a_n\left(t\right)=a_0[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4-{\left(\frac{{\tau }_n{s}_n^{*}\left(t\right)}{{{\displaystyle \sum }}_{q=1}^Q{\phi }_q{s}_{n+q-1}\left(t\right)}\right)}^2}]+\mu {\displaystyle \sum }_{q=1}^Q{\beta }_q{a}_{n+q-1}\left(t\right)$$

(7)

$$s_n^{*}\left(t\right)=s_0+T{v}_n\left(t\right)+\frac{v_n\left(t\right)\left({{\displaystyle \sum }}_{q=1}^Q{\alpha }_q\Delta v_{n+q-1}\left(t\right)\right)}{2\sqrt{a_{0}b}}$$

(8)

According to existing research [21], the parameters that can more accurately describe the car-following characteristics in the CACC mode are set as follows:$a_0=2 \mathrm{m}/{\mathrm{s}}^2$, $\mathrm{b}=2 \mathrm{m}/{\mathrm{s}}^2$, $v_0=10 \mathrm{m}/{\mathrm{s}}^2$, $s_0=2 \mathrm{m}$, $T=2 \mathrm{s}$, and ${\tau }_n=1$. ${\phi }_q$, ${\alpha }_q$, and ${\beta }_q$ are the weight factors in the model, with ${\phi }_q\ge 0$, ${\alpha }_q\ge 0$, ${\beta }_q\ge 0$, and ${\displaystyle \sum }_{q=1}^Q{\phi }_q=1$, ${\displaystyle \sum }_{q=1}^Q{\alpha }_q=1$, ${\displaystyle \sum }_{q=1}^Q{\beta }_q=1$. The weights ${\phi }_q$, ${\alpha }_q$, and ${\beta }_q$ are assigned by the following formula [22]:

$${\phi }_q,{\alpha }_q,{\beta }_q=\{\begin{array}{c}\frac{Q-1}{Q^q},q\ne Q\\ \frac{1}{Q^{q-1}},q=Q\end{array}$$

(9)

3. Linear Stability Analysis

3.1. Stable Analysis of the Model

When all vehicles drive on the road at the same headway and speed, the traffic system is in an equilibrium state. Once the driving state of a vehicle in the fleet changes, an “interference signal” is created, leading to a speed fluctuation of the rear vehicles. If this fluctuation weakens over time while spreading until the traffic system regains equilibrium, then the traffic system is considered stable; else, it is considered unstable. Therefore, it is very important to analyze the stability of the car-following model.

In previous studies, the Laplace transforms, and the difference equation are usually used to analyze the stability of the car-following model. Due to the complexity of the Laplace transform process, it is more suitable for the stability study of homogeneous traffic flow. Therefore, this paper chooses to use the difference equation to linearly analyze the stability of the extended IDM model. Proceed as follows:

Step 1. Initial state settings

Simplifying Formulae (1) and (2) gives:

$$a_n\left(t\right)=f\left(v_n\left(t\right),s_n\left(t\right),\dots s_{n+i-1}\left(t\right),\Delta v_n,\dots ,\Delta v_{n+i-1},a_n\left(t\right),\dots ,a_{n+i-1}\left(t\right)\right)$$

(10)

Assuming that the traffic system is in equilibrium in the initial state and all vehicles are traveling at the same headway spacing and velocity, the position of any vehicle in the traffic flow at time t can be expressed as:

$$x_n^0\left(t\right)=\left(N-n\right)\overline{s}+\overline{v}t n=1,2,\dots N$$

(11)

where $N$ is the total number of vehicles in the fleet.

The fluctuation in the steady traffic flow can be expressed as:

$$y_n\left(t\right)=c{e}^{i{\alpha }_{k}n+zt}=x_n\left(t\right)-x_n^0\left(t\right), y_n\left(t\right)\to 0$$

(12)

The position and headway spacing of the vehicle in the traffic flow can be expressed as:

$$x_n\left(t\right)=y_n\left(t\right)+x_n^0\left(t\right)$$

(13)

$$\Delta x_n\left(t\right)=\Delta y_n\left(t\right)+x_n^0\left(t\right)$$

(14)

Taking the first and second derivatives of Formula (13) gives:

$$v_n\left(t\right)=\dot{y_n\left(t\right)}+\overline{x_n}\left(t\right)$$

(15)

$$a_n\left(t\right)=\ddot{y_n }\left(t\right)$$

(16)

Substituting Formula (16) into Formula (10) gives:

$$y_n^{″}\left(t+t_d\right)=f\left(v_n\left(t\right),s_n\left(t\right),\dots s_{n+i-1}\left(t\right),\Delta v_n,\dots ,\Delta v_{n+i-1},a_n\left(t\right),\dots ,a_{n+i-1}\left(t\right)\right)$$

(17)

Step 2. Linearize and derive the difference equation

Linearize Formula (17) to obtain:

$$y_n^{″}\left(t+t_d\right)=f_n^s{\displaystyle \sum }_{q=1}^Q{\phi }_q\left(t\right)\left(y_{n-q}\left(t\right)-y_{n-q+1}\left(t\right)\right)+f_n^v{y}_n^{\prime }\left(t\right)+f_n^{\Delta v}\rho {\displaystyle \sum }_{q=1}^Q{\alpha }_q\left(y_{n-q+1}^{\prime }\left(t\right)-y_{n-q}^{\prime }\left(t\right)\right)+\mu f_n^a{\displaystyle \sum }_{q=1}^Q{\beta }_q{y}_{n-q}^{″}\left(t\right)$$

(18)

where: $f_n^s=\frac{\partial f_n}{\partial s}\ge 0$, $f_n^v=\frac{\partial f_n}{\partial v}\ge 0$, $f_{\Delta v}^s=\frac{\partial f_n}{\partial \Delta v}\le 0$, $f_n^a=\frac{\partial f_n}{\partial a}\ge 0$.

Rewrite Formula (18) into the following difference equation:

$$\begin{array}{l}{y}_n^{\text{'}}\left(t+2t_d\right)-y_n^{\text{'}}\left(t+t_d\right)\\ =t_d{f}_n^s{\displaystyle \sum }_{q=1}^Q{\phi }_q\left(y_{n-q}\left(t\right)-y_{n-q+1}\left(t\right)\right)+f_n^v\left(y_n\left(t+t_d\right)-y_n\left(t\right)\right)\\ +\mu f_n^a{\displaystyle \sum }_{q=1}^Q{\beta }_q\left(y_{n-q}^{\text{'}}\left(t+t_d\right)-y_{n-q}^{\text{'}}\left(t\right)\right)\\ +\rho f_n^{\Delta v}{\displaystyle \sum }_{q=1}^Q{\alpha }_q\left(y_{n-q+1}\left(t+t_d\right)-y_{n-q+1}\left(t\right)-y_{n-q}\left(t+t_d\right)+y_{n-q}\left(t\right)\right)\end{array}$$

(19)

Substituting $y_n\left(t\right)=c{e}^{i{\alpha }_{k}n+zt}$ and $y_n‘\left(t\right)=zc{e}^{i{\alpha }_{k}n+zt}$ into Formula (19) gives the simplified formula below:

$$\left(e^{z{t}_d}-1\right)[z{e}^{z{t}_d}-f_n^v+\rho f_n^{\Delta v}{\displaystyle \sum }_{q=1}^Q{\alpha }_q\left(e^{-i{\alpha }_{k}q}-e^{-i{\alpha }_k\left(q-1\right)}\right) - \mu f_n^a{\displaystyle \sum }_{q=1}^Q{\beta }_{q}z{e}^{-i{\alpha }_{k}q}]=t_d{f}_n^s{\displaystyle \sum }_{q=1}^Q{\phi }_q\left(e^{-iq{\alpha }_k}-e^{-i{\alpha }_k\left(q-1\right)}\right)$$

(20)

Step 3. Expand and organize the formula

Further substituting $e^{z{t}_d}=1+t_{d}z+\frac{t_d^2{z}^2}{2}+\dots$ and $e^{-i{\alpha }_k}=1-i{\alpha }_k+\frac{{(i{\alpha }_k)}^2}{2}$ into Formula (20) gives:

$$\begin{array}{l}\left(t_{d}Z+\frac{t_d^2+Z^2}{2}\right)[Z\left(1+t_{d}Z+\frac{t_d^2{Z}^2}{2}\right)-f_n^v\\ +\rho f_n^{\mathsf{\Delta }v}{\displaystyle {\displaystyle \sum }_{q=1}^Q}{\alpha }_q\left({\left[1-i{\alpha }_k+\frac{{\left(i{\alpha }_k\right)}^2}{2}\right]}^q-{\left[1-i{\alpha }_k+\frac{{\left(i{\alpha }_k\right)}^2}{2}\right]}^{(q-1)}\right)\\ -\mu f_n^a{\displaystyle {\displaystyle \sum }_{q=1}^Q}{\beta }_{q}Z{\left[1-i{\alpha }_k+\frac{{\left(i{\alpha }_k\right)}^2}{2}\right]}^q]\\ =t_d{f}_n^s{\displaystyle {\displaystyle \sum }_{q=1}^Q}{\phi }_q\left({\left[1-i{\alpha }_k+\frac{{\left(i{\alpha }_k\right)}^2}{2}\right]}^q-{\left[1-i{\alpha }_k+\frac{{\left(i{\alpha }_k\right)}^2}{2}\right]}^{(q-1)}\right)\propto \end{array}$$

(21)

Expand $\mathrm{z}=z_1\left(i{\alpha }_k\right)+z_2{\left(i{\alpha }_k\right)}^2+\dots$, insert it into Formula (21), and ignore high-order infinitesimals:

$$\begin{array}{l}\left(t_d\left(Z_1\left(i{\alpha }_k\right)+Z_2{\left(i{\alpha }_k\right)}^2\right)+\frac{t_d^2{\left(Z_1\left(i{\alpha }_k\right)+Z_2{\left(i{\alpha }_k\right)}^2\right)}^2}{2}\right)\\ *{\displaystyle [}\left(Z_1\left(i{\alpha }_k\right)+Z_2{\left(i{\alpha }_k\right)}^2\right)\left(1+t_d\left(Z_1\left(i{\alpha }_k\right)+Z_2{\left(i{\alpha }_k\right)}^2\right)+\frac{t_d^2{\left(Z_1\left(i{\alpha }_k\right)+Z_2{\left(i{\alpha }_k\right)}^2\right)}^2}{2}\right)-f_n^v\\ +\rho f_n^{\Delta v}{\displaystyle {\displaystyle \sum }_{q=1}^Q}{\alpha }_q\left({\left[1-i{\alpha }_k+\frac{{\left(i{\alpha }_k\right)}^2}{2}\right]}^q-{\left[1-i{\alpha }_k+\frac{{\left(i{\alpha }_k\right)}^2}{2}\right]}^{\left(q-1\right)}\right)\\ - \mu f_n^a{\displaystyle {\displaystyle \sum }_{q=1}^Q}{\beta }_q\left(Z_1\left(i{\alpha }_k\right)+Z_2{\left(i{\alpha }_k\right)}^2\right){\left[1-i{\alpha }_k+\frac{{\left(i{\alpha }_k\right)}^2}{2}\right]}^q{\displaystyle ]}\\ =t_d{f}_n^s{\displaystyle {\displaystyle \sum }_{q=1}^Q}{\phi }_q\left({\left[1-i{\alpha }_k+\frac{{\left(i{\alpha }_k\right)}^2}{2}\right]}^q-{\left[1-i{\alpha }_k+\frac{{\left(i{\alpha }_k\right)}^2}{2}\right]}^{\left(q-1\right)}\right)\end{array}$$

(22)

The first and second-order expressions of parameter Z can be derived by sorting as:

$$Z_1={\displaystyle \sum }_{q=1}^Q{\phi }_q\frac{f_n^s}{f_n^v}$$

(23)

$$Z_2=\frac{\left[\left(1-\frac{t_d}{2}{f}_n^v\right)Z_1^2-f_n^{\Delta v}{{\displaystyle \sum }}_{i=1}^q{\alpha }_{Fi}{Z}_1-Z_1^2\mu f_n^a{{\displaystyle \sum }}_{i=1}^q{\beta }_{Fi}-f_n^s{{\displaystyle \sum }}_{i=1}^p{\phi }_{Fi}\left(p-\frac{1}{2}\right)\right]}{f_n^v}>0$$

(24)

Step 4. Derive the condition of traffic flow stability

If $Z_2>0$, the disturbed traffic flow will gradually tend to stability; else, a traffic disorder will result from the disturbance to the traffic flow. Therefore, the condition for traffic flow stability is:

$$\left(1-\frac{t_d}{2}{f}_n^v\right)Z_1^2-f_n^{\Delta v}{\displaystyle \sum }_{q=1}^Q{\alpha }_{Fi}{Z}_1-Z_1^2\mu f_n^a{\displaystyle \sum }_{q=1}^Q{\beta }_q-f_n^s{\displaystyle \sum }_{q=1}^Q{\phi }_q\left(q-\frac{1}{2}\right)>0$$

(25)

Substituting $Z_1={\displaystyle \sum }_{q=1}^Q{\phi }_q\frac{f_n^s}{f_n^v}$ into Formula (25) gives the simplified formula below:

$$\frac{t_d}{2}<\frac{1-\mu f_n^a{{\displaystyle \sum }}_{q=1}^Q{\beta }_q}{f_n^v}-\frac{f_n^{\Delta v}{{\displaystyle \sum }}_{q=1}^Q{\alpha }_q+f_n^v{{\displaystyle \sum }}_{q=1}^Q{\phi }_q\left(q-\frac{1}{2}\right)}{f_n^s}$$

(26)

Set μ = 0, Q = 1, t_d = 0, then Formula (26) will degenerate to the stability condition of the classical IDM model:

$$f_n^v{f}_n^{\Delta v}+\frac{1}{2}{\left(f_n^v\right)}^2-f_n^s>0$$

(27)

From Formulas (1) and (2), the expressions of $f_n^s$, $f_n^v$, $f_n^{\Delta v}$ and $f_n^a$ are given as follows:

$$f_n^s=\frac{2a_0\left[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4\right]\sqrt{1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4}}{{\tau }_n\left(s_0+v_n\left(t\right)T\right)}$$

(28)

$$f_n^v=-\frac{4a_0{v}_n^3\left(t\right)}{v_0^4}-\frac{2a_{0}T\left[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4\right]}{s_0+v_n\left(t\right)T}$$

(29)

$$f_n^{\Delta v}=\frac{\left[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4\right]v_n\left(t\right){{\displaystyle \sum }}_{q=1}^Q{\alpha }_q\sqrt{\raisebox{1ex}{$a_0$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}}{{\tau }_n\left(s_0+v_n\left(t\right)T\right)}$$

(30)

$$f_n^a=\mu {\displaystyle \sum }_{q=1}^Q{\beta }_q$$

(31)

Substituting Formulas (28)–(31) into Formula (26) gives:

$$\frac{t_d}{2}<\frac{1-{\left(\mu {{\displaystyle \sum }}_{q=1}^Q{\beta }_q\right)}^2}{-\frac{4a_0{v}_n^3\left(t\right)}{v_0^4}-\frac{2a_{0}T\left[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4\right]}{s_0+v_n\left(t\right)T}}-\frac{\frac{\left[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4\right]v_n\left(t\right){\left({{\displaystyle \sum }}_{q=1}^Q{\alpha }_q\right)}^2\sqrt{\raisebox{1ex}{$a_0$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}}{{\tau }_n\left(s_0+v_n\left(t\right)T\right)}+\left\{-\frac{4a_0{v}_n^3\left(t\right)}{v_0^4}-\frac{2a_{0}T\left[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4\right]}{s_0+v_n\left(t\right)T}\right\}{{\displaystyle \sum }}_{q=1}^Q{\phi }_q\left(q-\frac{1}{2}\right)}{\frac{2a_0\left[1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4\right]\sqrt{1-{\left(\frac{v_n\left(t\right)}{v_0}\right)}^4}}{{\tau }_n\left(s_0+v_n\left(t\right)T\right)}}$$

(32)

3.2. Stability Analysis for Different Driver Types

According to Formula (32), the critical stability curves for homogeneous traffic flows corresponding to four driver types are plotted in Figure 1, from which one can summarize the influence of the driver’s response type on the stability of traffic flow. Through comparison, the stability regions of four driver types, from the largest to the smallest, are the Type IV, Type I, Type III (standard type), and Type II drivers. Due to the distinct values of parameters ${\tau }_n$ and $v_0$ corresponding to different driver types, the zero point and peak value of the critical stability curves are different. Combined with the different driver types and car-following response characteristics in Table 1, it can be concluded that the area of stability region increases with the increase of desired headway and the decrease of desired speed. This is because there is enough time and space for the driver to take countermeasures against the disturbance to the fleet while keeping a great following distance and low traveling speed.

3.3. Stability Analysis of Homogeneous Traffic Flows of RV, ACC, and CACC Vehicles

To compare the stability of homogeneous traffic flows of RVs (with standard response characteristics), ACC, and CACC vehicles, the critical stability curves for the three types of traffic flows are plotted in Figure 2. Evidently, the critical stability curves for the three types show similar trends. By comparison, the stability regions of these three types of traffic flows are in descending order: CACC vehicles > ACC vehicles > RVs. Among them, the instability region of CACC vehicles is significantly reduced compared with those of ACC vehicles and RVs. It can be concluded that considering the headway, the speed difference, and the acceleration of multiple front vehicles can significantly improve the stability of traffic flows. This means higher vehicle intelligence coincides with the higher stability of the operating state of the fleet.

3.4. Stability Analysis of MTF with CAVs and RVs

Based on the traditional common car-following model, Ward [23] studied the analytical method for the stability of heterogeneous traffic flows, deriving the following criterion on the stability of heterogeneous traffic flows:

$${\displaystyle \sum _n^N\left(\left[1/2{\left(f_n^v\right)}^2+f_n^{\Delta v}{f}_n^v-f_n^s\right]{\left[{\displaystyle \prod _{m\ne n}{f}_m^s}\right]}^2\right)}<0$$

(33)

Later, Qin et al. [11] extended the above formula to the car-following model for MTF, as follows:

$$\begin{gathered} N\left(1-p\right)\left[\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\left(f_R^v\right)}^2-f_R^{\Delta v}{f}_R^v-f_R^s\right]{\left[{\left(f_R^s\right)}^{N\left(1-p\right)-1}{\left(f_A^s\right)}^{N\left(1-p\right)p}{\left(f_C^s\right)}^{N{p}^2}\right]}^2+ \\ N\left(1-p\right)p\left[\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\left(f_A^v\right)}^2-f_A^{\Delta v}{f}_A^v-f_A^s\right]{\left[{\left(f_R^s\right)}^{N\left(1-p\right)}{\left(f_A^s\right)}^{N\left(1-p\right)p-1}{\left(f_C^s\right)}^{N{p}^2}\right]}^2+ \\ N{p}^2\left[\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\left(f_C^v\right)}^2-f_C^{\Delta v}{f}_C^v-f_C^s\right]{\left[{\left(f_R^s\right)}^{N\left(1-p\right)}{\left(f_A^s\right)}^{N\left(1-p\right)p}{\left(f_C^s\right)}^{N{p}^2-1}\right]}^2<0 \end{gathered}$$

(34)

Simplifying Formula (34) yields the following results:

$$\{\begin{array}{l}\left(1-p\right)F_R+\left(1-p\right)p{F}_A+p^2{F}_C<0\\ F_R=\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\left(f_R^v\right)}^2-f_R^{\Delta v}{f}_R^v-f_R^s}{{\left(f_R^s\right)}^2}=\frac{f{f}_R}{{\left(f_R^s\right)}^2}\\ F_A=\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\left(f_A^v\right)}^2-f_A^{\Delta v}{f}_A^v-f_A^s}{{\left(f_A^s\right)}^2}=\frac{f{f}_A}{{\left(f_A^s\right)}^2}\\ F_C=\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\left(f_C^v\right)}^2-f_C^{\Delta v}{f}_C^v-f_C^s}{{\left(f_C^s\right)}^2}=\frac{f{f}_C}{{\left(f_C^s\right)}^2}\end{array}$$

(35)

where $p$ is the penetration rate of CACC vehicles; $\left(1-p\right),\left(1-p\right)p$ and $p^2$ are the mathematical expectations for number of RV, ACC, and CACC vehicles, respectively; $f{f}_R$, $f{f}_A$ and $f{f}_C$ are the criteria on the stability of RV ACC, and CACC vehicles, respectively; $F_R$, $F_A$, and $F_C$ are stability factors of RV, ACC, and CACC vehicles, respectively. For the convenience of description, the overall stability factor of MTF is denoted as $F$, as follows:

$$\mathrm{F}=\left(1-p\right)F_R+\left(1-p\right)p{F}_A+p^2{F}_C$$

(36)

According to Formulas (28) and (30), the partial differentials of RV, ACC, and CACC vehicles with respect to speed $v$, speed difference $\Delta v$, and headway $s$ are calculated and substituted into Formula (35) to determine the value of $F$ of MTF at ditinct values of equilibrium speed $v$ and penetration rate of CAVs, so as to derive the stability region of MTF, as shown in Figure 3. From the figure, when $p=0$ the curved surface intersects the abscissa axis (velocity) at points 0.72 and 5.424, which are the critical speeds at which the RV reaches stability. When $p>0.104$, that is, the penetration rate of CAVs exceeds $10.4\%$, the MTF can reach stability at any equilibrium speed. When the speed is less than $0.72 \mathrm{m}/\mathrm{s}$ or greater than $5.424 \mathrm{m}/\mathrm{s}$, the MTF is stable regardless of the penetration rate of CAVs.

4. Numerical Simulation Analysis and Results

4.1. Microscopic Numerical Simulation

To study the feasibility of the car-following model for different vehicle types, the traffic environment was set as follows: Car 1 is the leading vehicle with a preset value of acceleration; Car 2, Car 3, and Car 4 are followers, corresponding to an RV, an ACC vehicle, and a CACC vehicle, respectively; Car 5 is the test vehicle designated as an RV (with each of four driver types) or an ACC or CACC vehicle, as shown in Figure 4. Through numerical simulation, the acceleration fluctuations of different vehicles in the starting and braking processes were analyzed.

4.1.1. Microscopic Numerical Analysis in the Vehicle Starting Scenario

Initial state: all vehicles started with an initial speed of 0, and the positions of Car 1 to Car 5 were 37.5 m, 30 m, 22.5 m, 15 m, and 7.5 m, respectively. Starting process: at time t = 0, the leading vehicle (Car 1) started to speed up with a initial acceleration $3 \mathrm{m}/{\mathrm{s}}^2$ that decreasesd in magnitude to 0 until traveling at a constant speed. Following the start of Car 1, Car 2 to Car 5 started one after another by the extended IDM model proposed in this paper. And record and plot the acceleration changes of the five vehicles at 0.1 s intervals. Figure 5a–f shows the acceleration variations of the five vehicles over time with Car 5 being an RV (with each of four driver types) or an ACC or a CACCin the starting scenario.

In the starting process, the maximum acceleration of Car 5 as an RV with each of four driver types was $0.8974 \mathrm{m}/{\mathrm{s}}^2$, $0.9053 \mathrm{m}/{\mathrm{s}}^2$, $0.9021 \mathrm{m}/{\mathrm{s}}^2$, and $0.8906 \mathrm{m}/{\mathrm{s}}^2$, respectively. From the simulation results, the maximum acceleration of Car 5 as an RV with a Type IV driver is the smallest, $1.2748\%$ lower than that with a standard driver; the maximum acceleration of Car 5 as an RV with a type II driver is the largest, $0.3547\%$ higher than that with a standard driver. The reason is that the Type IV driver maintains a low driving speed, while the Type II driver pursues a higher speed. The maximum acceleration of Car 5 as an ACC vehicle is $0.8984 \mathrm{m}/{\mathrm{s}}^2$, slightly lower than that of Car 5 as an RV with the standard driver; the maximum acceleration of Car 5 as a CACC vehicle is $0.8703 \mathrm{m}/{\mathrm{s}}^2$, $3.5251\%$ lower than that of Car 5 as an RV with the standard driver and $3.1281\%$ lower than that of Car 5 s an ACC vehicle. Therefore, Car 5 started more smoothly and steadily in the case of being a CACC vehicle other than an RV or an ACC vehicle. The results of the comparisons among four driver types and among the RV, ACC, and CACC vehicles are consistent with the results of the stability analysis in Section 3.

4.1.2. Microscopic Numerical Analysis in the Vehicle Braking Scenario

Initial state: all vehicles traveled at a constant initial speed, and the positions of Car 1 to Car 5 were 150 m, 120 m, 90 m, 60 m, and 30 m, respectively. Braking process: at time t = 0, the leading vehicle (Car 1) started to speed down with an intial deceleration $-3 \mathrm{m}/{\mathrm{s}}^2$ that decreased in magnitude to 0 until coming to a complete stop. Following the brake of Car 1, Car 2 to Car 5 decelerated one after another by the extended IDM model proposed in this paper. And record and plot the acceleration changes of the five vehicles at 0.1 s intervals. Figure 6a–f shows the acceleration variations of the five vehicles over time in each type of Car 5 in the braking scenario.

In the braking process, the difference between the maximum and minimum deceleration of Car 5 being an RV with each of four driver types is $0.6558 \mathrm{m}/{\mathrm{s}}^2$,$0.5181 \mathrm{m}/{\mathrm{s}}^2$, $0.5804 \mathrm{m}/{\mathrm{s}}^2$, and $0.7390 \mathrm{m}/{\mathrm{s}}^2$, respectively. It can be concluded from Figure 6 that the difference is smallest for the Type II driver and largest for the Type IV driver. The reason is that the Type II driver tends to be more skilled in driving and more steadily responsive to the braking of the front vehicle compared with the extraordinarily cautious Type IV driver. The difference between the maximum and minimum deceleration of Car 5 as an ACC vehicle is $0.5737 \mathrm{m}/{\mathrm{s}}^2$, slightly lower than that with a standard driver; the difference is $0.4540 \mathrm{m}/{\mathrm{s}}^2$ for Car 5 as a CACC vehicle, $21.78\%$ lower than that for Car 5 as an RV with a standard driver, and $20.86\%$ lower than that for Car 5 as an ACC vehicle. This means that when the fleet is disturbed by braking, in the case of Car 5 as a CACC vehicle other than an RV or an ACC vehicle, the fluctuation of speed can be reduced to a great extent.

4.2. Macroscopic Numerical Analysis

To examine the influence of penetration rate and distribution degree of CAVs onMTF at a macro level, the traffic environment was set as follows: Car 1 is the leading vehicle with a preset value of acceleration; the next 20 vehicles include RVs and ACC and CACC vehicles that make up an MTF. The numerical simulation in starting and braking scenarios was divided into four groups according to the response characteristics of RV drivers, namely Type I, Type II, Type III, and Type IV drivers. During the numerical simulation, the interval at which to continuously increase the penetration rate of CAVs was set as $10\%$, so as to analyze the impact of penetration rate on the fleet. At the same penetration rate, the arrangement of CAVs was divided into two cases: decentralized and centralized. Under the distributed arrangement, the CAVs are dispersed into the fleet at the maximum distance; under the centralized arrangement, the CAVs are gathered in the fleet. The effect of the distribution degree of CAV arrangement on the fleet was analyzed. In the specific situation, the CAVs in the fleet were driven in the separate modes of ACC and CACC.

4.2.1. Macroscopic Numerical Analysis in the Vehicle Starting Scenario

In the initial state, 21 vehicles were lined up with a headway of 7.5 m and an initial speed of 0. At time $t=0$, the leading vehicle (Car 1) started to speed up with a initial acceleration 3m/s^2 that decreasesd in magnitude to 0 until traveling at a constant speed. The remaining 20 test vehicles accelerated in response to the speed change of Car 1 by the extended IDM model proposed in this paper. Finally, all vehicles reached a uniform speed of $12 \mathrm{m}/ \mathrm{s}$. Calculate and record the average acceleration of 20 test vehicles at 0.1 s intervals, as shown in Figure 7.

Comparing Figure 7a−d, the averages of acceleration differences of the fleet in the four cases follow similar trends. Take the standard driver (Figure 7c) as an example. When the penetration rate of CAVs is below $50\%$, compared with the centralized arrangement of CAVs, the average acceleration differences of the fleet decrease more slowly under the decentralized arrangement of CAVs. This indicates that the distribution degree of CAVs has a great impact on the stability of the fleet, because some CAVs running in the ACC mode cannot give full play to their intelligence level. When the penetration rate of CAVs is $50\%$, the average acceleration of CAVs under the centralized arrangement is $7.68\%$ lower than that under the decentralized arrangement. This coincides with the greatest influence of the distribution degree of CAVs on the stability of traffic flow. When the penetration rate of CAVs is above $50\%$, the average of acceleration differences of the fleet decreases faster with the increase of penetration rate under the decentralized arrangement of CAVs. When the penetration rate of CAVs increases to $70\%$, the average of acceleration differences of the fleet shows a similar trend under both arrangements, suggesting that the distribution degree of CAVs has little impact on the stability of the fleet at this time.

It can be summarized from Figure 7a–d that the average of acceleration differences of the fleet increases slightly with the increase of penetration rate for RV drivers with Type I and Type IV response characteristics and under the decentralized arrangement of CAVs. The curve reaches the inflection point where the penetration rate is $50\%$, namely, where the acceleration difference begins to decrease significantly. This means that mixing a small number of CAVs with the RVs driven by these two types of drivers is of no help in improving the stability of the traffic flow. Only when the proportion of CAVs is more than half can some of them give full play to their intelligent features, hence alleviating the interference to a great extent. From the perspective of the whole fleet, the average of acceleration differences of the fleet is slightly lower for RV drivers with Type I and Type IV response characteristics than for the standard RV drivers. The reason is that Type I and Type IV drivers tend to be insensitive and conservative and maintain a low speed, so the overall acceleration fluctuation of the fleet is small during the starting process. For RV drivers with Type II response characteristics, the curve of the average of acceleration differences of the fleet with respect to the penetration rate of CAVs shows a similar trend to that for RV drivers with the standard type of response characteristics. In addition, the range of acceleration difference variation is wide, mainly because the Type II drivers tend to be more perceptive and pursue higher speeds. To sum up, for RV drivers with Type IV response characteristics, the whole fleet started more smoothly, which is consistent with the results of the stability analysis in Section 2.

4.2.2. Macroscopic Numerical Analysis in the Vehicle Braking Scenarios

Initial state: 21 vehicles were driven at the initial speed of $12 \mathrm{m}/\mathrm{s}$ with a headway of 30 m. At time $t=0$, the leading vehicle (Car 1) started to speed down with an initial deceleration −3 m/s² that decreased in magnitude to 0 until coming to a complete stop. The remaining 20 test vehicles decelerated in response to the speed change of the leading vehicle by the extended IDM model. Finally, all vehicles came to a complete stop. Calculate and record the averages of the acceleration difference of the fleet at 0.1 s intervals, as shown in Figure 8.

Comparing Figure 8a–d, the averages of deceleration difference of the fleet show similar trends in the four types of RV drivers. Under the most centralized arrangement of CAVs, the average of deceleration differences decreases more evenly with the increase of the penetration rate of CAVs. When the penetration rate of CAVs is below $50\%$, the average of deceleration differences of the fleet decreases more pronounced under the centralized arrangement of CAVs than under the decentralized arrangement of CAVs. When the penetration rate increases to $50\%$, the curve reaches the turning point where the positive effect of the centralized arrangement of CAVs on the fleet is most significant; the average of deceleration differences of the fleet in the four types of RV drivers is 15.08%, 15.69%, 15.86%, and 13.93% lower, respectively, than that of the fleet with decentralized CAVs. When the penetration rate of CAVs is above $70\%$, the averages of deceleration differences are similar under both arrangements of CAVs, which is consistent with the results in the starting scenario. Unlike the starting scenario, however, when the penetration rate of CAVs is 70%, the deceleration difference of the fleet with decentralized CAVs is slightly lower than that of the fleet with centralized CAVs, indicating that the decentralized arrangement of CAVs can better alleviate the disturbance caused to the fleet by braking.

5. Conclusions

Based on the IDM model, this paper has proposed a car-following model suitable for MTF composed of RVs and CAVs. With consideration of the influences of headway, speed difference, and acceleration difference, the car-following model has focused on the effect of RV drivers with distinct response characteristics on the stability of MTF.

Through analysis of the stability of homogeneous traffic flows, the criteria for the stability of CACC and ACC vehicles and RVs have been derived. It has been found that, in the case of a homogeneous traffic flow, CACC vehicles can improve the stability of the traffic flow to a great extent. The traffic flow composed of RVs with Type IV drivers is more stable. In other words, a longer headway and a lower speed are more conducive to drivers responding to sudden disturbances. It should be noted that this paper has given mere consideration to the stability of a traffic flow, without considering whether the traffic flow is reduced.

The stability analysis of the MTF has demonstrated the relationship among the penetration rate, the equilibrium speed, and the stability of CAVs. When the equilibrium speed is below $0.72 \mathrm{m}/\mathrm{s}$ or above $5.424 \mathrm{m}/\mathrm{s}$, the MTF is stable regardless of the penetration rate of CAVs; when the penetration rate of CAVs exceeds $10.4\%$, the MTF can reach stability at any equilibrium speed.

The numerical simulation results have shown that under the two typical disturbances of starting and braking, the acceleration fluctuation of CACCs is steadier than that of ACC vehicles and RVs. With the increase in the penetration rate of CAVs, the average of acceleration differences of the whole fleet can be effectively reduced. When the penetration rate of CAVs is below 70%, the centralized arrangement of CAVs can alleviate the fluctuation of traffic flow acceleration to a greater extent than the decentralized arrangement of CAVs. When the penetration rate of CAVs is above 70%, the positive impact of CAVs on the fleet is the same regardless of the distribution degree of CAVs. For RVs, higher stability coincides with the Type VI response characteristics of drivers in the starting scenario, and deceleration fluctuation is less significant for the Type II response characteristics of drivers in the braking scenario.

In the future management and control of MTF, the penetration rate of CAVs can be increased by promoting CAV to improve the stability of traffic flow. Besides, adjusting the distribution degree of CAVs according to the penetration rate is also a way to improve the stability of MTF. The influence of RV drivers also cannot be neglected, so related departments can improve the comprehensive quality of drivers through publicity and training.