Modeling car-following behavior in heterogeneous traffic mixing human-driven, automated and connected vehicles: considering multitype vehicle interactions

In this study, a car-following model is put forth to explain the car-following behavior of HDVs (human-driven vehicles), AVs (autonomous vehicles) and CAVs (connected and automated vehicles) in mixed traffic. Based on the IDM (intelligent-driver model) and molecular dynamic theory, the model includes the velocity of surrounding vehicles along with the difference of velocity and the headway between each pair of vehicles. The drivers’ sensitivity to the deceleration of the nearest front vehicle was considered in HDV car-following modeling and the influences of various kinds of nearest front and rear vehicles were distinguished in the CAV model. Based on the information obtained from the real road test mixed with HDVs, AVs and CAVs, the optimum value of the model parameters was obtained and the accuracy of model was verified with simulation. The results show that the simulation for HDVs’, AVs’ and CAVs’ car-following behavior under the proposed model is more accurate than that for the IDM, ACC (adaptive cruise control) model and CACC (cooperative adaptive cruise control) model. The model can be applied to car-following simulations of HDVs, AVs and CAVs in mixed traffic, in the model the tools and concepts of nonlinear dynamics especially molecular dynamic theory is applied which can benefit road traffic vehicle interaction analysis. In addition, this study provides valuable suggestions and guidance for effectively guiding AVs and CAVs to follow vehicles and improving the stability of car-following behavior.


Introduction
Car-following (CF) is an essential element of microscopic traffic flow characteristics in which two vehicles closely follow each other in a single lane [1][2][3][4]. With the development and improvement of CAVs, information can be exchanged between vehicles and vehicles (V2V) and between vehicles and infrastructure (V2I) [5]. In the long term, there will be a mixed driving scenario of multiple types of vehicles, which include mainly HDVs, AVs and CAVs [6,7], CF modeling for mixed traffic flow has always been a hot topic [8,9].
The majority of recent research uses various CF models to define the CF behaviors of HDVs, AVs and CAVs. For example, Orosz et al. [10] used the full velocity difference (FVD) model and CACC model to describe the CF behavior of HDVs and CAVs, respectively. They found that the stability of the mixed traffic flow gradually increased as the ratio of CAVs in the traffic flow increased, which indicates that increasing the ratio of CAVs could effectively improve the traffic capacity. However, they did not consider the effects of communication delay. Kuang et al. [11] considered the effects of driver's memory and mean expected velocity field, and proposed an improved CF model in intelligent transportation system. The simulation results indicated that the coupling effect of driver's memory and the mean expected velocity field is helpful to improve the stability of traffic flow. Sun et al. [12] used the optimal velocity model (OVM) for HDVs, a multianticipative CF model for AVs and multi-cooperative CF models for CAVs. The simulation results indicated that string stability was greatly impacted by communication delay as well as connectivity. However, they did not consider the impact of surrounding vehicles on CF behavior. Kuang et al. [13] considered multianticipative average velocity effect in a new CF model. They found that the multi-anticipative average velocity effect can effectively improve the stability of traffic system. Zhang et al. [14] proposed an improved CAV car-following model taking into account data from multiple front vehicles. The results of stability analysis showed that anticipating the future motion of multiple front vehicles was helpful for CAVs to achieve stability. However, front vehicles at different positions had different effects on the host vehicle [15]. Zong et al. [16] took into account the data from multitype vehicle and proposed an improved IDM. They discovered that as the distance between the vehicles increased, the influence of the front vehicles on the movement state of the host vehicle steadily decreased. Kuang et al. [17] proposed an extended CF model by considering average headway of preceding vehicles group in intelligent transportation systems environment. They found that the traffic jam can be suppressed efficiently with taking into account the average headway effect. Based on V2V communication technology, Xiao et al. [18] considered the information and lateral gaps effect from the front and rear vehicles to describe car-following behavior. The results of stability analysis indicated that the new proposed model has the larger stability region, which means that in the case of disturbances, it could recover stability faster. However, they did not the impact of CAV ratio or the velocity on traffic flow stability.
Some studies have employed the same CF model to capture the CF behaviors of HDVs, AVs and CAVs. Ngoduy [19] used the IDM to describe the behavior of CAVs and HDVs. Herman et al. [20] investigated the car-following behavior (without passing) of the vehicles on a highway and the propagation of disturbances to the platoon. They derived local stability and asymptotic stability criteria for vehicle chains. Barrachina and Conejero [21] described the car-following dynamical behavior of linear operators by adding the concept of distributional chaos to the study of linear dynamics of operators and C 0 -semigroups of operators. They provided a good reference for researchers interested in the study of linear dynamics in heterogeneous traffic flow. Based on the IDM [22], Talebpour and Mahmassani [23] proposed an overall acceleration framework that distinguishes connectivity and automation while simulating a novel connected environment. The analysis revealed that CAVs can make strings more stable. However, the parameters in these models cannot accurately describe the motion of AVs or CAVs in mixed traffic. Barrachina et al. [24] considered the linear Forward and Backward Control traffic model for vehicles based on a nonlinear model. They also proved the existence of chaotic behavior solutions using conclusions about linear dynamics models and analyzed solutions for the initial stabilization configuration. Yu et al. [25] analyzed the relationship between relative velocity difference and car-following behavior, and considered ACC strategy to simulate AV-following behavior, thereby exploring how relative velocity difference affects velocity fluctuation. However, ACC strategy only considers the movement state of the target vehicle, and ignores overall influence of rear vehicles on velocity fluctuation.
In addition, a few studies found that the CF behavior of CAVs in mixed traffic may change with the type of nearest front and rear vehicles and their positions. Wang et al. [26] presented a framework for analytical studies on the stability of the heterogeneous flow mixed with CAVs and HDVs. They found that if CAVs followed right behind HDVs, the CACC model would degrade to the ACC model. Yao et al. [27] analyzed the stability of mixed traffic by considering the degradation of CAVs into AVs due to communication failure. The results indicated that when CAVs degenerated into AVs, the safety risk of mixed traffic increased significantly. Hence, the type of front vehicle nearest to a CAV in mixed traffic should be considered. However, they did not consider the influence of a CAV's degradation on its CF behavior or the impact of the surrounding CAVs on it by incorporating their motion information [28]. The car-following tests mixed with CAV, AV and HDV Thus, we can summarize previous research  on mixed-vehicle CF modeling as follows. First, previous research only discusses the microscopic behavior such as the position of the front vehicle relative to the host vehicle, nor in-depth propose and study the influence of micro-information such as target vehicle on host vehicle. Second, the influence of CAVs' degradation on the CF behaviors of CAVs has not been fully considered and discussed, the motion information for other CAVs in different positions also has not been considered.
In this paper, the molecular dynamics theory was used to quantitatively describe the interaction between surrounding vehicles and host vehicle. To express this influence, a model that considers the velocity of surrounding vehicles, as well as the headway, will be employed. To construct mixed-vehicle CF modeling, the impact of CAV degradation on the CF behaviors of the CAV model and the different driving characteristics of HDVs, AVs and CAVs was considered.
This paper is organized as follows: In Sect. 2, the behaviors of CF were analyzed and a CF model considering multitype vehicle interactions was established. In Sect. 3, the model calibration was analyzed. In Sect. 4, the stability analysis of the CF model under different CAV penetrations was analyzed and its performance was validated by comparing it with the IDM. In Sect. 5, the numerical simulation was analyzed. Some conclusions and suggestions for future research are given in Sect. 6.

Data
To investigate the different CF characteristics of HDVs, AVs and CAVs in mixed traffic, we employ three sets of data in this study. As shown in Fig. 1, the car-following tests mixed with CAV, AV and HDV are conducted in test field. The first set comprises data on CAVs following AVs in a mixed traffic flow with AVs and CAVs. The data from 32 real-world CF behaviors in the situation of an HDV following an AV are introduced in the second part. The third set includes the data that we collected from AVs' CF behavior in the scenario of an AV following an HDV. All CF scenarios include acceleration, deceleration and constant-speed driving. The items of the survey data include velocity and acceleration of each vehicle and headway.

Car-following behavior analysis
For HDVs, the driver makes a car-following decision by observing the movement of the target vehicle [29]. Therefore, in the HDV model, we consider the data from the target vehicle only, such as velocity and headway.
AVs do not have networked communication equipment and rely instead on sensors (camera, radar, OBU, etc.) to obtain real-time information on the nearest front  Fig. 2 The mixed-vehicle platoon consisting of HDVs, AVs and CAVs vehicle and rear vehicle [30,31]. Thus, we consider information from the nearest front and rear vehicle only. In addition, HDVs and AVs rely only on the driver and sensors to obtain the driving behavior of the front and rear vehicles, respectively, they do not need information feedback from the front and rear vehicles. CAVs take advantage of lidars, cameras, OBUs, etc., to receive information from the nearest front vehicle and rear vehicle. CAVs can also capture motion information about other CAVs in different positions of the platoon through connected equipment [32]. Thus, we assume that the CF behavior of CAVs is affected by not only the movement state of its target vehicle and the nearest rear vehicle but also other CAVs in different positions of the platoon.
An example of a CF fleet consisting of HDVs, AVs and CAVs is shown in Fig. 2. Based on lidar, OBUs and RSUs, CAVs can obtain the motion state not only of the nearest front and rear vehicles (e.g., in Fig. 2, CAV2 can obtain information about HDV1 and AV1 and the headway between them) but also of all CAVs in the platoon by V2V (e.g., CAV1, CAV2 and CAV3 in Fig. 2 can mutually obtain information from the other two CAVs). In addition, the types of nearest front and rear vehicles will affect the CF behavior of CAVs [26].
In this paper, an IDM-based CF model taking into account the data of the surrounding vehicles and including the types of vehicles to describe the CF behaviors of AVs was proposed, HDVs and CAVs in mixed traffic.

Modeling
According to the IDM [22], the host vehicle's acceleration is divided into two components. The acceleration strategy, which is the first component, is as follows: where a 0 n represents the maximum acceleration, v 0 n represents the velocity of the nth vehicle, v n (t) represents the velocity of the nth vehicle at time t.
The deceleration strategy, which is the second component, is as follows: where α I is the sensitivity of the braking deceleration. I correspond to the vehicle types. I = 1, 2, 3 represent an HDV, AV and CAV, respectively. τ I F and τ I R represent the influence weight, in which F and R (F, R ∈ N ) are the numbers of front and rear vehicles in consideration, [22] and 0 ≤ τ I F + τ I R ≤ 1 [16]. In the platoon, f is the f th front vehicle and r is the rth rear vehicle, f = 1, 2, 3..., F and r = 1, 2, 3..., R. Then, D F is the deceleration of F front vehicles. D R is the deceleration of R rear vehicles. D F and D R can be expressed as follows: where a 0 n+ f −1 is the maximum acceleration of the n + f − 1th vehicle. a 0 n−r represents the maximum acceleration of the n − rth vehicle. Δx n+ f −1 and Δx n−r are the headways between each pair of vehicles at time t. θ j (θ j ≥0, F j=1 θ j =1, R j=1 θ j =1) is the influence factor of the jth vehicle on the host vehicle.
Studies [33,34] have shown that according to the velocity of each molecule and distance between them, molecular dynamics theory can describe the attraction and repulsion forces [35,36]. Accordingly, we regard each vehicle as a molecule and calculate the attraction and repulsion between each vehicle. [37,38]. The influence factor of the jth vehicle on the host vehicle is calculated as follows: where M j (·) is the total energy of attraction and repulsion for the jth target vehicle, which is a function of velocity and position. E 1 is the total energy of q vehicles. E 1 and E 2 are constants that represent the parameters of field strength. V j (·) is the velocity function, a, b and c are the parameters of the quadratic polynomial [39]. According to Ref [40], Zong et al. selected the intersection of Dongling South Street and South Huancheng Road in Changchun City as a typical scene for aerial camera survey, and extracted the spacetime location information of vehicles, and obtained 127 valid samples. According to the measured track data of vehicles, they calibrated the normal field gain coefficients of left turn, right turn and straight travel at the intersection. By using the velocities of measured vehicles at the intersection, they calibrated the terminal gain coefficient and calculated the values of E 1 and E 2 , i.e., In Eqs. (3) and (4), Δx * n+ f −1 and Δx * n−r are the desired gap. They have equilibrium terms Thus, their expressions are shown in Eq. (6): where Δx 0 is the minimum space gap for completely stopped traffic.
T v n+ f −1 (t) and T v n−r (t) are the velocity-dependent distances, and v n+ f −1 (t) is the velocity of the n + f −1th vehicle. v n−r (t) is the velocity of the n−rth vehicle. Δv n+ f −1 (t) and Δv n−r (t) are the velocity differences between each pair of vehicles, respectively. d n+ f −1 and d n−r are the optimal decelerations of the n + f − 1th vehicle and n − rth vehicle, respectively. Successively, we introduce two correction items to control the deceleration strategy of the host vehicle, i.e., the correction item of the front vehicle ψ Δx * v n+ f −1 (t), Δv n+ f −1 (t) and the correction item of the rear vehicle ψ Δx * (v n−r (t), Δv n−r (t)) , which are defined as follows: Hence, the CF model can be expressed as follows: where a n (t + t d ) is the acceleration m/s 2 and t d (t d ∈ R) and is the time delay (s).
Successively, according to the different CF behaviors of HDVs, AVs and CAVs, we define the models for each of them in detail.
(1) Host vehicle of HDV (I = 1) For HDV, the driver solely takes into account the data from the closest front vehicle. Hence, the HDV model can be expressed as follows: In real-world traffic, the deceleration strategy α 1 of HDV may change with the type of front vehicle. For example, when the front vehicle is an AV, some drivers think that the automatic driving technology is not sufficiently mature. Therefore, the driver may increase the distance to ensure driving safety. In contrast, some drivers believe in automatic driving technology, which will lead them to follow the front vehicle to accelerate or decelerate.
(2) Host vehicle of AV (I = 2) For AV, based on on-board devices, the information between the nearest vehicles and the host vehicle can be accurately obtained. In addition, without V2V communication, the AV will not obtain information from vehicles at positions beyond its perceived range. Hence, the AV model can be described as follows: (3) Host vehicle of CAV (I = 3) For CAV, it can sense the surrounding traffic environment and capture more accurate infrastructure information through cameras, radars and V2V communication. Therefore, CAVs can obtain the movement state not only of the nearest front and rear vehicle but also of other CAVs at different positions in the platoon. Since the CF behavior of CAVs is affected by the vehicle type, we consider the following four scenarios: (a) As shown in Fig. 3, when only the nearest front vehicle is an HDV or AV and the other vehicles are all CAVs, the deceleration sensitivity of the nearest front vehicle in the CAV model is the same as that of the AV, and information on the other CAVs in the platoon can be obtained: (b) As shown in Fig. 4, when the target vehicle of the host vehicle is an HDV or AV, the deceleration sensitivity of the target vehicle in the CAV model is the same as that of the AV. Meanwhile, the CAV can also obtain information from other CAVs in the platoon by V2V communication. Thus, the CAV model in this case is as follows: (c) As shown in Fig. 5, when both the nearest front and rear vehicles of the host vehicle are HDVs or AVs, due to the degradation of CAVs, the deceleration sensitivities of the nearest front and rear vehicles in the CAV model are the same as those of AVs. However, compared with the AV model, the CAV model can also obtain information from other CAVs in the pla- (d) As shown in Fig. 6, when all vehicles in the platoon are CAVs, the CAV model is expressed as follows:

Calibration for HDV model
To analyze the CF behavior of mixed flow driving and calibrate the car-following model, we carried out a real vehicle test. In the test, we used a Level-4 AV as the leading vehicle and HDVs as the following vehicles.
The HDV data follow an AV to calibrate the parameter α 1 in the HDV model. Based on Ref [22], the range of α 1 is set to 0.5−1.5. Figure 7 shows the velocity distribution (solid line clusters) of the HDV model under different α 1 values. We use one-way analysis of variance (ANOVA) to obtain the deviation between the actual speed and the simulation speed. As shown in Table 1, when α 1 = 0.94, the mean maximum error (MME) is 1.15 m/s, the mean error (ME) is −0.44 m/s and Rsquared is 87.15%. When α 1 = 0.94, the fitting accuracy of the HDV model is the highest, and the results are shown in Fig. 8.
Thus, the HDV model can be expressed as follows:

Model calibration for AV
Then, we calibrate τ 2 F=1 , τ 2 R=1 and α 2 in the AV model. The experimental data comes from the real vehicle data, which collected by an AV following an HDV in the test field. During t = 0-12 s, the speed of the leading HDV is 0. Then, the leading HDV accelerates at a constant of 1m/s 2 for 20 s until the speed is 20 m/s. Then, the leading HDV accelerates at a constant acceleration of 0.125m/s 2 for 24 s until the speed reaches 23 m/s. and τ 2 R=1 is 0.9−0.99, and that of α 2 is 0.8−1.5. In order to obtain the optimal values of them, we calculated the deviation between the simulation speed and the actual speed, and carried out ANOVA. As shown in Fig. 9, when τ 2 F=1 = 0.95, τ 2 R=1 = 0.05, α 2 = 1.04, the MME, ME and R-Squared are 1.22 m/s, −0.86 m/s and 87.68%, respectively. In this case, the error of the model is the minimum. Therefore, the AV model can be expressed as:

Model calibration for CAV
Then, we calibrate τ 3 F , τ 3 R and α 3 in the CAV model. Similarly, the experimental data comes from the real vehicle data, which collected by a CAV following an AV in the test field. During t = 0-10 s, the speed of the leading AV is 0. Then, the leading AV accelerates at a constant of 1m/s 2 [16], the range of τ 3 F , and τ 3 R is 0.9−0.99, and that of α 3 is 0.8−1.5. In order to obtain the optimal values of them, we calculated the deviation between the simulation speed and the actual speed, and carried out ANOVA. As shown in Fig. 10

Stability analysis
According to Eq. (8), the proposed model in this paper is a function about the velocity of the host vehicle, the headway and velocity difference between each front and rear vehicle. Therefore, it can be expressed as: wherein, f n (·) is the function of the acceleration. θΔx (t + t d ) represents the headway between each vehicle. θΔv (t + t d ) is velocity difference between each vehicle. Due to considering the information of multiple front and rear vehicles.
θΔx (t + t d ) and θΔv (t + t d ) are expressed as follows: Therefore, the proposed model in this paper can be rewritten as: By linearizing v n (t + t d ), θΔx (t + t d ) and θ Δv (t + t d ), we obtain the following equations: where f v n is the partial differential of velocity. f Δx n and f Δv n represent the partial differential of the headway and velocity difference, respectively. Δx is the average headway, andv represents the average velocity of each vehicle in the platoon.
Then, we can get the linearization result of Eq. (21): The disturbances to velocity and headway are expressed as follows: where ε n (t + t d ) represents the disturbance to velocity. z n (t + t d ) and z n−1 (t + t d ) represent the disturbance to the headway of each surrounding vehicle. By substituting Eq. (26) into Eq. (22) and Eq. (24), we obtain the disturbances to velocity and velocity difference in the proposed model: By substituting Eq. (27) and Eq. (28) into Eq. (23), we obtain the disturbances to headway in the proposed model: Therefore, the disturbances to the proposed model are expressed as follows: .
By the Laplace transformation, we obtain the transfer function: where G(s) represents the transfer function for the disturbance. s represents the Laplace domain. In order to analyze the stability of traffic flow mixed with CAVs, AVs and HDVs, we need to obtain the transfer function about disturbance in traffic flow by Laplace transformation. G 1 (s), G 2 (s) and G 3 (s) are the Laplace transformation functions of HDVs, AVs and CAVs, respectively. When the host vehicle is HDV, the driver considers the information of the nearest front vehicle only. Thus, G 1 (s) is the transfer function about the velocity of the host vehicle, the headway and velocity difference between the host vehicle and the nearest front vehicle, which can be expressed as follows: Δx+2 f Δxn n 1 (34) When the host vehicle is AV, the transfer function G 2 (s) is not only related to the velocity of the host vehicle, but also related to motion information of the nearest front and rear vehicle. Therefore, it can be described as follows: is the transfer function of CAVs, which is related to the headway, velocity difference and acceleration difference between the host vehicle and surrounding vehicles, and illustrated below: By converting Eqs. (34), (35), and (36) to the frequency domain, we rewrite them as follows: where j is the imaginary number and ω is the frequency. q j=1 θ j = 1. According to Ref. [41], we assume that there are n vehicles that form mixed traffic. The proportion of CAVs and HDVs is p : q, where p + q = 1. The proportion of AVs degraded by CAVs is pq, and the proportion of degraded CAVs is p − pq = p 2 . Based on the stability criteria [42], the stability condition of the proposed model can be expressed as: By substituting Eqs. (37), (38), and (39) into the Eq. (40), we obtain: a 0 Substituting Eqs. (51), (52), and (53) into Eq. (41), the open-loop amplitude-phase characteristics of the mixed-vehicle CF model (shown in Fig. 11) can be obtained. According to Ref. [28], a 0 n = 3m/s 2 , Δx 0 = 2 m, T = 2 s, d n = 2m/s 2 and v 0 = 33 m/s. As shown in Fig. 11, the transfer function of the mixed-vehicle CF model is a closed-loop system, where the direction of the open-loop amplitude-phase characteristic curve (OACC) with different speeds is the direction of low relative stability of the system. The point (−1, j 0 ) is the Nyquist criterion point [43], which is the intersection of the real axis −1 and the imaginary axis j 0 , representing the dividing point between the stable and unstable regions. When the OACC has no poles in the right half-plane and does not pass through the point (−1, j 0 ), the system is stable based on the Nyquist stability criterion [43]. In contrast, if the OACC passes through the point (−1, j 0 ), the system is unstable. In addition, the OACC gradually moves away from point (−1, j 0 ), which means that the stability of the system gradually increases. Based on the stability criterion, the proposed model is stable. Moreover, we find that regardless of the penetration of CAVs, with increasing velocity, the stability of traffic flow gradually increases. For example, as shown in Fig. 11a (p = 0), when v n = 0, the cutoff frequency (blue dot) ω c = 1rad/s, and the phase margin γ (ω c ) = 90 • ; when v n = 20m/s (green line), the cutoff frequency ω c = 0.17rad/s, and the phase margin γ (ω c ) = 160 • . Similarly, as shown in Fig. 11b (p = 20%), compared with v n = 0 m/s, when v n = 20 m/s (green line), ω c decreases by 0.49rad/s and γ (ω c ) increases by 28.7%).
In addition, we compare the stability of the mixed traffic under different p. The data show that when p = 0 and v n = 0, the cutoff frequency (blue dot) ω c = 1 rad/s, the phase margin γ (ω c ) = 90 • and the corresponding overshoot δ% = 0.08%. When p = 20%, compared with p = 0, the cutoff frequency ω c corresponding to v n = 0 decreases by 0.084rad/s, the phase margin increases to 114 • , and the overshoot δ% is less than 0.03%. When p = 40% and v n = 0, the cutoff frequency ω c (blue dot) of G ( jω) (the blue line in Fig. 11c) is further reduced to 0.71rad/s, and the overshoot (δ%) decreases by 1.3% compared with that of p = 20%. The decrease in the overshoot (δ%) show that the stability of mixed traffic gradually increases. When p is greater than 55.1%, there is no cutoff frequency ω c in the G ( jω) plane, hence, the traffic flow is the most stable. The results indicate that with the increase in the CAV ratio, the stability under the mixed-vehicle CF behavior increases due to CAVs under the proposed model can receive information about not only the nearest front and rear vehicles but also the motion state of other CAVs in the platoon.

Numerical simulation analysis
We design a numerical simulation to analyze the dynamic performance. We assume that there are 10 The results show that for the whole platoon control, under the CAV model, all vehicles in the platoon complete the acceleration at t = 8.1 s, which is 2.3 s earlier than the AV model and 4.3 s earlier than the HDV model. The reason is that based on the motion data of surrounding vehicles, the CAV model can effectively control the CF behavior of drivers and reduce the impact of disturbance to improve traffic efficiency and stability of the fleet.
In addition, in order to compare the motion states of each vehicle under proposed model, as shown in Fig. 12d, we obtained the time when the velocity of each vehicle reach 20 m/s in the deceleration process. We find that compared with HDV model, the velocity of each vehicle under AV model and CAV model can reach stable state earlier. The reason is that based on the movement information of multiple front and rear vehicles, the CAV model can be effectively utilized to control the car-following behavior of drivers and reduce the effect of disturbance during the car-following process to improve the operation efficiency of the fleet and stability of the traffic flow.

Model verification for HDV
In this study, to confirm the reliability of the developed HDV model, we conduct a simulation of HDVs following AVs by using the HDV model and the IDM. The test data of an HDV following an AV that we present above are used to calculate model accuracy and compare the two models. Every experiment is performed in MAT-LAB with 0.01 time step, and all data are reported with two decimal places. The results are displayed in Fig. 13. A one-way ANOVA is used to obtain the error between the simulation speed and the actual speed. The results show that for the HDV model, the MME is 0.92 m/s. The ME is −0.58 m/s, which is less than that under the IDM (MME = 1.04 m/s, ME = −0.71 m/s). In addition, compared with the IDM, the R-Squared of the HDV model is 87.15%, which is an improvement of 2.23%. The results indicated that the HDV model in this paper fits better than the IDM.

Verification for AV model
To confirm the fitting accuracy of the proposed AV model and ACC model in PATH laboratory, we use the experimental data which collected by an AV following an HDV in the test field. In the initial state, the leading vehicle runs at a speed of 12 m/s. When t = 20 s, it starts to accelerate, and the speed reaches 17 m/s after 2 s. Then, within 1 s, the leading vehicle suddenly decel-erates to 16 m/s, and then accelerates to 20 m/s with constant acceleration. During t = 23-28 s, the leading vehicle decelerates to 10 m/s at a constant deceleration of 2m/s 2 . When t = 30 s, it accelerates to 14 m/s and moves at a constant speed for 2 s. During t = 32-34 s, the vehicle decelerates at a constant deceleration until the speed is 9 m/s. When t = 34 s, it suddenly accelerates again for 1 s, then decelerates until the speed is 0. We can obtainthe error between the simulation speed and the actual speed form the fitting results of the two models (Fig. 14). The results show that both models can well fit the car-following behavior of AV. Among them, the ME and MME of the proposed model are 0.64 m/s and −0.38 m/s, respectively, which is 0.58 m/s and 0.22 m/s less than those of ACC model. We find that the proposed AV model is better than ACC model. Especially, when the host vehicle suddenly accelerates or decelerates, the car-following decisions of the two models will change. Among them, the AV model can change the CF behavior of the host vehicle in time according to the movement information of the target vehicle, so it has high fitting accuracy with the following speed of the real vehicle. The car-following decision of ACC model ignores the drastic changes of the moving state of the front car and chooses a more stable operation.

Model verification for CAV
To confirm the fitting accuracy of the proposed CAV model and CACC model in PATH laboratory, we use the experimental data which collected by a CAV following an AV in the test field. In the initial state, the leading vehicle runs at a speed of 50 km/h. Then, it starts to decelerate and the speed reaches 48 km/h after 10 s. During t = 10-20 s the leading vehicle accelerates to 55 km/h at a constant acceleration. Successively, during t = 20-30 s, the leading vehicle decelerates to 50 km/h at a constant deceleration. Then, it starts to accelerate until the speed reaches 60 km/h after 10 s, and moves at a constant speed for 6 s. When t = 46 s, it suddenly decelerates until the speed is 28 km/h. We can obtain the error between the simulation speed and the actual speed form the fitting results of the two models (Fig. 15). The results show that both models can well fit the CF behavior of CAV. Among them, the ME and MME of the proposed model are 0.71 m/s and −0.45 m/s, respectively, which is 0.25 m/s and 0.33 m/s less than those of ACC

Conclusion
A CF models to describe the CF behaviors of HDVs, AVs and CAVs in mixed traffic was proposed. Based on the IDM, the model calculates the host vehicle's acceleration using data from surrounding vehicles. It takes into account the sensitivity of deceleration as well as the velocity of each vehicle, the velocity difference, and the headway between each pair of vehicles. The key conclusions are summarized as follows: (1) For the CAV model, we consider the influence of multitype of nearest surrounding vehicles on the CF behavior of CAVs and obtain CAV models for four cases: a the nearest front vehicle is an HDV or AV, and the other vehicles are CAVs, b the nearest rear vehicle is an HDV or AV, and the other vehicles are CAVs, c both nearest front and rear vehicles are an HDV or AV, and the other vehicles are CAVs, and (d) the vehicles in the platoon are all CAVs. In addition, based on molecular dynamics theory, we add θ j for each surrounding vehicle to quantitatively describe its degree of influence. By employing data from the actual test field, we obtain the optimum values of the CAV model, which are τ 3 F = 0.93, τ 3 R = 0.07 and α 3 = 1.06. The analysis revealed that the MME and ME of the CACC model are reduced by 26.04% and 42.31%, respectively, compared to the CACC model. R-Squared is 90.61%, which is an improvement of 1.24% over the CACC model. Especially when the vehicle begins to accelerate or decelerate, compared to the CACC model, the proposed model offers a more quicker and smoother acceleration and deceleration process to the target velocity.
(2) For the AV model, the information of the nearest surrounding vehicles was considered, and the sensitivity of the braking deceleration α 2 was added.
The optimum values of the AV model are τ 2 F=1 = 0.95, τ 2 R=1 = 0.05 and α 2 = 1.04. The results show that compared with the ACC model, the MME and ME value of the AV model are reduced by 3.79% and 14.28%, respectively. The reason is that the AV model reduces the coefficient of the deceleration strategy, resulting in a reduce in peak acceleration, thus avoiding large disturbances caused by excessive changes in velocity, as well as efficiently suppressing the generation of frequent stops and starts to reduce traffic congestion and accidents. The analysis revealed that compared with the ACC model, the acceleration curve of the AV model is smoother.
(3) For the HDV model, the driver's sensitivity to deceleration α 1 was considered. Based on data of an HDV following an AV, the optimum parameters of the HDV model are obtained. The analysis revealed that compared with the IDM model, both MME and ME were reduced by 11.53% and 18.31% in the HDV model. In addition, a one-way ANOVA shows that the P value of the HDV model is improved by 6.25% than the IDM, indicating a superior accuracy for the HDV model. (4) The results of the stability analysis show that regardless of the penetration of CAVs, with the increase in CAV velocity, the stability of traffic flow under the proposed model has been greatly improved. In addition, with the increase in the CAV ratio, the stability of the traffic flow increases gradually. The reason is that CAVs under the proposed model can receive information on not only the nearest front and rear vehicles but also the motion state of other CAVs in the platoon. The proposed CF model can be applied to car-following behavior control for platoons composed of vehicles of the same type or mixed types concerning HDVs, AVs and/or CAVs. Notably, further research can focus on the impacts of some factors (e.g., driving types of HDVs, the stochasticity of vehicle type in the platoon) on the stability of mixed traffic flow. Additionally, the impact of the efficiency of the platoon mixed with HDVs, AVs and CAVs on fuel saving must be further discussed.
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