# Exploring the Stability and Capacity Characteristics of Mixed Traffic Flow with Autonomous and Human-Driven Vehicles considering Aggressive Driving

Authors: Yun Li; Shengrui Zhang; Yingjiu Pan; Bei Zhou; Yanan Peng
Journal: Journal of Advanced Transportation
 (2023)

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## Abstract

With the popularization of autonomous vehicle (AV) technology, mixed traffic flows that consist of AVs and human-driven vehicles (HDVs) will appear in the real world. Although many studies of the features of mixed traffic flow have been carefully evaluated, few studies have focused on the effect of aggressive driving performance on mixed traffic flow. This study aims to develop an approach to evaluate the effects of aggressive driving on the stability and capacity performance under the conditions of AV and HDV mixed traffic flow. First, since a car-following model can describe the relationship between vehicles, we calibrate a car-following model for aggressive driving and nonaggressive driving behaviors based on real traffic data and previous research results. Then, in a mixed traffic flow environment, a basic linear stability formula and capacity calculation expression are developed that consider the effects of vehicle order on the capacity. Finally, because the proportion of aggressive driving and aggressive driving parameters may change, nine combinations of three aggressive driving proportions and three driving parameter cases are used for the sensitivity analysis. The results indicate that the effect of aggressive driving on mixed traffic flow is complex. When the proportion of aggressive driving is less than 35%, the increase in the proportion of aggressive driving increases the traffic capacity and reduces the unstable part. However, when the proportion of aggressive driving is greater than 35%, the increase in the proportion of aggressive driving increases the unstable part. When the penetration rate of AVs exceeds 0.490, mixed traffic flow remains stable at all aggressive driving proportions. In addition, the capacity of a mixed traffic flow may be improved as the penetration rate of AVs increases. To a certain extent, these conclusions provide a theoretical basis for formulating different management modes of AVs and HDVs.


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## Body

## 1. Introduction

Autonomous vehicle (AV) technologies have attracted increasing attention due to their significant economic and time-saving advantages [1]. Some scholars also use connected and autonomous vehicles (CAVs) to represent autonomous vehicles [2–4]. With the development of automatic driving and communication technology, the difference between connected autonomous vehicles (CAVs) and autonomous vehicles (AVs) is decreasing.AVs are considered to significantly enhance transportation systems through communicability and predictability [5]. In particular, an increasing number of technologies achieve coordinated driving or platooning of AVs, which has great potential for increasing road capacity and enhancing flow stability [6].The effect of driving behavior on traffic flow has been studied in detail by many scholars [7–9], who generally postulate that the effect of aggressive driving on traffic flow is very complex, especially in specific scenarios with different characteristics.With the advent of AVs, mixed traffic flows that consist of AVs and human-driven vehicles (HDVs) will appear in the real world [10]. The effect of aggressive driving on mixed traffic flow has attracted the attention of some scholars. Relevant studies have shown that when a HDV follows an AV, the HDV driver tends to become more aggressive since AV systems take on the operational characteristics of ensuring safety at all costs [11]. The aggressive behavior of drivers may affect many traffic flow parameters such as the headway between platoons, the headway within a platoon, and the platoon size [12]. Research on the headway between platoons and within platoons has been highly valued for traffic modeling and platoon control. However, the effect of HDV driver behavior on transportation systems has not been well studied.A driver’s character may have a considerable influence on mixed traffic flow. Aggressive drivers may seek smaller headways to obtain more benefits that likely improve the capacity of mixed flow [13], which will increase the risk of traffic collisions [14–16]. According to the trade-off between stability and capacity [15], aggressive diving is suggested to be a non-negligible factor for mixed traffic flow. Nevertheless, no quantitative analysis of the effect of the proportion of HDV aggressive driving has been provided in previous studies. Furthermore, although few studies have examined how aggressive driving affects the relevant parameters of a platoon, some of these studies considered the effect of aggressive driving on traffic flow and conducted sensitivity analysis on aggressive driving parameters but did not consider that driving parameters may change at the same time [13].On the other hand, although research on the stability and capacity of mixed traffic flow has achieved substantial progress, much work must be completed. When many scholars describe mixed traffic flow, they usually only consider the situation of a platoon composed of AVs and HDVs [14–17]. This type of platoon often requires all AVs to be at the forefront, maintaining both vehicle-to-vehicle (V2V) communication between the AV and a relatively small headway. However, this research assumes that all AVs are at the forefront of a platoon and thus does not fully describe the scenario of mixed traffic flow. An HDV following the AVs is extremely unlikely to stop and instead passes them, thereby complicating the issue; for aggressive driving, the possibility of this situation may even be higher. Accordingly, the order of AVs and HDVs in a platoon composed of mixed traffic flow must be considered. Many scholars typically use the parameters commonly used in car-following models when analyzing the stability and capacity of mixed traffic flow. The parameters of the car-following model for HDVs usually change in different traffic scenarios [18], and thus the parameters of the car-following model for HDVs must be calibrated using real traffic data.This study focuses on the effect of the HDV driver’s perception of a mixed flow on the capacity and flow stability to address the limitations of existing research. Thus, we separate HDVs into those operating in a normal mode and those operating in an aggressive mode, and we fully consider the interaction between AVs and HDVs. Specifically, when calibrating the car-following model, different parameters are assigned for different models of driving behavior based on real traffic data. Then, we combine the calibrated car-following model parameters with a linear stability formula to analyze the stability of the mixed traffic flow and combine the results with the fundamental diagram model (FDM) and the order between vehicles to study the capacity of a mixed traffic flow. Finally, considering that the aggressive driving proportion and its related parameters may change with the public’s acceptance of AVs, we conduct a sensitivity analysis of different scenarios to study the stability and capacity in an environment with changing parameters.The major contributions of this research are outlined as follows:(1)
This study represents an initial attempt to consider the typical value of aggressive driving as a non-negligible factor in the analysis of stability and capacity in a mixed traffic flow environment. Typical values of the proportion of aggressive driving are based on actual survey data, and the car-following model parameters of typical values are calibrated using real traffic data.(2)
According to an FDM, we propose a capacity calculation model considering scenarios with different vehicle orders. The average headway has been suggested to exert a substantial effect on the capacity of a mixed traffic flow that includes a changing vehicle order [12]. Therefore, this study utilizes the typical values for aggressive driving and the varying market penetration rate of AVs to characterize the changes in the average headway such that accurate capacity values can be obtained.(3)
We evaluate the effects of aggressive driving and changes in its relevant parameters on stability and capacity by performing a sensitivity analysis. Unlike other studies, we do not perform a sensitivity analysis of a single parameter because the parameter may change at the same time during aggressive driving. Therefore, different scenarios of aggressive driving are provided to analyze the stability and capacity.The remaining sections of this article are arranged as described below. Section2 establishes a mixed traffic flow model considering the HDV driving mode. The stability and capacity models of mixed traffic flow are developed in Section 3, and the capacity model is verified with simulation of urban mobility (SUMO). In Section 4, a sensitivity analysis is conducted to explore the effects of changes in HDV aggressive driving and AV market penetration rates on the stability and capacity of mixed traffic flow. Finally, a discussion and conclusions are presented in Section 5.


## 2. Car-Following Model of Mixed Traffic Flow

The behavior of drivers can be microscopically analyzed using car-following models [19]; thus, car-following models must be studied before analyzing stability and capacity, and a mixed flow scenario must be identified first. In Section 2.1, the scenario described in this study is presented. Moreover, the car-following model of mixed traffic flow is discussed in Sections 2.2 and 2.3 for HDVs and AVs, respectively.


### 2.1. Description of the Mixed-Flow Scenario

The major objective of this study was to evaluate the capacity and stability of a mixed traffic scenario composed of both HDVs and AVs. Generally, scholars use cooperative adaptive cruise control (CACC) or adaptive cruise control (ACC) vehicles to represent AVs [6–9].At present, vehicles with ACC technology are generally defined by scholars as low-level AVs [20]. Although the vehicle with ACC represents the AVs to a certain extent, the key parameter of the car-following model does not change significantly compared with the HDV. If ACC is used as the model of AVs, the heterogeneity of traffic flow may not be reflected in the low market penetration rate.Thus, in AVs, only CACC is allowed. The situation in which a CACC vehicle degrades to an ACC vehicle as a result of long platoon and V2V communication failures is not considered in this study [20].With advances in intelligent connected vehicle (ICV) technology in the future, we believe that CACC vehicles will be able to obtain some information about the HDVs in front of them to judge the state of those vehicles in advance, or they will have a shorter reaction time, which could ensure a small headway.Assume that a mixed traffic flow hasN vehicles, the market penetration rate of CACC vehicles is p [21, 22], and the HDV market penetration rate is 1−p. The proportion of aggressive driving in HDVs is pA, and the proportion of other driving styles (normal driving) is 1−pA. In this article, HDV-A represents an HDV with aggressive driving, and HDV-N represents an HDV with normal driving. Thus, the numbers of three types of vehicles are as follows:(1)nC=pN,nN=1−p1−pAN,nA=1−ppAN,where nC, nN, and nA are the numbers of CACC, HDV-N, and HDV-A, respectively.


### 2.2. HDV Car-Following Model

#### 2.2.1. Basic Car-Following Model

In an ICV environment, many researchers use an intelligent driver model (IDM) [23–25] as the HDV car-following model, which properly reflects the car-following interrelation. The IDM is defined using equation (2) as follows:(2)v˙it=a1−vitv04−s0+vitT+vitΔvit/2abhit−l2,where vit represents the speed of vehicle i at time t; v˙it represents the derivative of vit; v0 denotes the free flow velocity; a is the maximum acceleration; T denotes the safe headway of the Q type of vehicle; s0 is the minimum distance between adjoining vehicles when stationary; b represents the desired deceleration; Δvit represents the speed difference between adjoining vehicles; hit represents the spatial distance between vehicle i and the vehicle behind at time t; and l is the vehicle length.Kesting et al. [26, 27] calibrated IDMs and classified HDV drivers into three types (normal, timid, and aggressive) based on their driving behaviors. The values for the parameters included in the IDM are shown in Table 1 [27].Table 1
Values of typical parameters for different types of HDV driving in the IDM.
ParameterTimidNormalAggressivev0m/s33.333.333.3Ts1.81.51.0s0m4.02.01.0lm5.05.05.0am/s21.01.42.0bm/s2−1.0−2.0−3.0


#### 2.2.2. Calibration of the Car-Following Model According to Driver Behavior

We are not sure whether the values presented in Table1 have been tested using real traffic data; therefore, we are uncertain whether the values reflect the real situation of aggressive driving.We plan to use Chinese data to calibrate the IDM and classify HDV drivers into three types in accordance with Table1 to better reflect the situation of mixed traffic flow in China. Therefore, the most important function of Table 1 is to provide us with a classified reference. We selected the local road traffic trajectory database in China as the research data for this article. The commonly used road traffic trajectory databases in China include Ubiquitous Traffic Eyes by Southeast University (PRC), the Tongji Road Trajectory Sharing (TJRD TS) platform by Tongji University (PRC), Mirror-Traffic by Tsinghua University (PRC), and the Traflow platform by Fudan University (PRC). Since the research scenario of this article is the traffic flow under the free state of an expressway, we choose the Traflow platform as the validation database to determine whether the data in Table 1 match the real situation. The process of model calibration is illustrated in Figure 1.Figure 1
The process used to calibrate and divide driving behaviors.Step 1.
Data processing.
(1) Data Filtering. The Traflow platform provides trajectory data for the G2, G15w, G45, and G60 expressways. A comparison shows that the flow of G2 is relatively moderate; therefore, the data for G2 are selected for analysis in this article. The basic information of the Traflow dataset (G2) is shown in Table 2.
Using the data from the G2 expressway in Jiangqiao, Shanghai, China, the data collection time in the study was 14:45–16:12 pm on April 3, 2012, which lasted for approximately an hour and half. We prefer that these data only consider cars and no other types of vehicles to remove interference from other factors; thus, we filter the car data from the database.
(2) Processing Rules. (a)
The car-following behavior data are extracted in the form of two vehicles continuously driving in the same lane as a vehicle group.(b)
Neither the following car nor the leading car will change lanes for a period of time.(c)
Every following car has a front car for a period of time, and thus the first car in the observation range is not used as the research object.(d)
The car-following behavior of each pair of car groups lasts for more than 30 s to ensure that the vehicle has a relatively stable car-following state.(e)
The distance between the following vehicle and the leading vehicle is 2–50 meters. A distance that is too large indicates that the interaction between the two vehicles is weak.(f)
We further screened the data and selected the vehicles that completed lane changing within 16 s. According to references [16, 17], all normally running vehicles can successfully complete a single lane change within 16 s.
(3) Data Processing Method. Many sources of error and noise are present in the Traflow data obtained based on processing videos. If the original Traflow data are directly used, a greater deviation in the results of the analysis will occur, which will affect the calibration of the micro traffic flow model and reduce the accuracy of subsequent analysis. Therefore, this article used the symmetric exponential moving average (SEMA) proposed by Thiemann et al. [28] to smooth the data.Table 2
Basic information of the Traflow dataset (G2).
Road IDG2Number of lanes4Date/Periods2012.04.03/14:45–16.12WeatherSunnyNumber of vehicles4284Step 2.
Driving behavior extraction.
(1) Driving Behavior Parameters. There is no uniform standard for the selection of driving behavior evaluation indicators. Different scholars choose different evaluation indicators. The more the evaluation indicators selected, the more comprehensively the driver’s driving behavior can be represented. This article selects 21 indicators as driving behavior parameters based on existing research results [29], as shown in Table 3. X1–X3, X7–X9, and X12–X17 are directly obtained from the data set. Other parameters are calculated as follows:
ForX4–X6, the lateral speed vyt is required for calculation.(3)vyt=yt−yt−1Δt,where yt is the lateral displacement at time t, Δt is the time interval from (t−1) to t, and Δt = 0.1 s in this study.
ForX10 and X11, the average value of the overall vehicle (No) speed vm is required for calculation.(4)vm=1No∑i=1No116si,where vm is the average speed of No vehicles in 16 s and si is the driving distance in 16 s.
ForX18, we need to determine when to start and end the lane change. So we make the following definitions:(5)tcl=t2−t1,where tcl is the lane change time and t1 and t2 are critical time, when the lateral speed of the vehicle is less than 0.2 m.s⁻1, and the distance from the target lane line is greater than half of the vehicle width. t1 searches forward from the lane change time, and t2 searches backward from the lane change time.
ForX19, X20, and X21, we need to determine the acceleration change rate. The specific calculation formula is as follows:(6)Jt=at−at−1Δt,where Jt is the acceleration change rate and at is the acceleration at time t.
(2) Principal Component Analysis. The 21 preliminarily selected evaluation indicators can comprehensively reflect the driving behavior in the 16 s driving trajectory including the lane-changing process, but the importance of each indicator is different, and there is information redundancy between indicators.
We also performed principal component analysis (PCA) which is a statistical method of dimension reduction. Several studies use PCA to identify key factors affecting traffic flow [30, 31], and the main calculation formula is as follows:(7)CPV=∑i=1Aλi∑i=1Mλi,where CPV is the cumulative contribution percentage of variance, λi represents the top A larger eigenvalues of the covariance matrix, and M represents the number of original variables. Usually, when the CPV is greater than 70%, the former A principal component is assumed to explain most of the variance in the data.
We perform PCA and factor analysis on the data using Python. The rotation method is the maximum variance method, and the eigenvalue and the variance contribution rate are obtained. We use the obtained factor loading matrix to analyze the correlation between principal components and indicators.Table 3
Driving behavior parameters.
ParametersDenotationMaximum speedX1Average speedX2Standard deviation (SD) of speedX3Maximum lateral speedX4Average lateral speedX5SD of lateral speedX6Maximum absolute value of accelerationX7Average value of absolute accelerationX8SD of absolute accelerationX9Time proportion above average total speedX10Driving distance within 16 sX11Minimum following distanceX12Average following distanceX13SD of following distanceX14Minimum headwayX15Average headwayX16SD of headwayX17Lane change timeX18Maximum absolute value of the acceleration change rateX19Average absolute value of the acceleration change rateX20SD of absolute value of the acceleration change rateX21Step 3.
Behavior analysis.
K-means is one of the most popular clustering algorithms based on partitioning and unsupervised learning. Therefore, this article uses the k-means algorithm to classify driving behaviors. The essence of this algorithm is to continuously determine the new cluster center through iterative operations, and the calculation will converge when the cluster center does not change:(8)DC,X=∑ik∑xjϵCixj−mi2,where C is the label of each cluster, k = 3; X is the sample car-following set, with each group of data in the car-following dataset used as a sample; xj=Tj,QTj where Tj is the clustering variable and represents the four parameters shown in Table 3 in this study; QTj is the coefficient of variation of each clustering variable; mi is the clustering center of class i; and xj−mi2 is the Euclidean distance.
We use k-means to cluster principal components. According to the clustering results obtained from the K-means analysis, the driving style of each cluster is judged, and the driving behavior label of each cluster (timid, normal, and aggressive) is determined.Step 4.
Model calibration.
(1) Calibration Method. In the previous step, we have classified the sample data according to different driving behaviors, so we calibrated the models for different driving behaviors.
A genetic algorithm is a relatively mature algorithm. This article intends to directly adopt this type of algorithm. Through the analysis of the IDM model, because the research data in this article are free flow, parameterv0 is generally considered to be the maximum speed limit (33.3 m/s), and the length of the car is generally considered to be 5 m. Combined with the research in the literature, the value ranges of other parameters to be calibrated are given in Table 4.
(2) Validation Indicators. Using the aforementioned genetic algorithm to calibrate the car-following model, the error between the simulated driving trajectory of each type of car-following model and the actual vehicle driving trajectory is obtained. The mean error (ME), mean absolute error (MAE), mean absolute relative error (MARE), and root mean square error (RMSE) are calculated as indicators to evaluate and verify the calibration of the parameters in the vehicle model results and to evaluate the results of trajectory fitting in this Section [30]. The calculation formulas are as follows:(9)ME=∑m=1Mhmreal−hmsimM,MARE=∑m=1Mhmreal−hmsim/hmrealM,MAE=∑m=1Mhmreal−hmsimM,RASE=∑m=1Mhmreal−hmsim2M.
Among them,hmreal represents the actual data that correspond to the speed, acceleration, position, and other parameters of the mth vehicle; hmsim represents the analog calibration data that correspond to the speed, acceleration, position, and other parameters of the mth vehicle [30].Table 4
The range of values for parameters included in the IDM.
ParameterRangeTs[0, 5]s0m[0.1, 10]am/s2[0.1, 5]bm/s2[−5, −0.1]


#### 2.2.3. Results for the Calibration of the IDM Based on Driver Behavior

(1) Data Processing. After data processing and screening, 1434 groups of car-following records were obtained from 6721 trajectories; among them, the data were obtained from 906 cars.(2) Driving Behavior Extraction. We perform PCA and factor analysis on the data using Python. The final number of principal components is 4. The rotation method is the maximum variance method, and the eigenvalue and the variance contribution rate are obtained.We rotate the factors to obtain the factor loading matrix, as shown in Table5.Table 5
The factor loading matrix.
F1F2F3F4X10.81570.32060.26000.0623X20.91680.17990.1562−0.0793X30.23600.32740.36360.3094X40.09020.11170.8670−0.0393X50.0381−0.00880.8620−0.0137X60.04820.11030.8247−0.0553X70.05220.79000.02520.0070X80.26150.72480.31310.0499X90.07910.78200.21250.0829X100.88300.15790.0889−0.0898X110.93630.17890.1567−0.0805X120.5852−0.0001−0.08970.0642X130.58180.0099−0.05820.6980X140.30740.02440.00100.7774X15−0.2518−0.1966−0.29080.2187X16−0.3334−0.0943−0.16690.8660X17−0.2632−0.0077−0.05770.8048X18−0.05220.0220−0.72240.0857X190.03350.8043−0.1266−0.0761X200.28610.76300.1299−0.0530X210.09820.8793−0.0752−0.0759The values in bold show a strong correlation.As shown in Table5, there is a strong correlation between principal components and some indexes, and the detailed analysis is described below.(a)
The correlation coefficient betweenF1 and X1∼X2, X10∼X11 is large, which mainly reflects the longitudinal motion characteristics of vehicles, and is named as the “longitudinal factor.”(b)
The correlation coefficient betweenF2 and X7∼X9, X19∼X21 is large, which mainly reflects the acceleration characteristics of the vehicle, and is named as the “acceleration factor.”(c)
The correlation coefficient betweenF3 and X4∼X6 is large, which mainly reflects the lateral driving characteristics of the vehicles, and is named as the “lateral factor.”(d)
The correlation coefficient betweenF4 and X13∼X14, X16∼X17 is large, which mainly reflects the car-following characteristics of vehicles, and is named as the “car-following factor.”Using Tables5 and 6, we obtain the component score coefficient matrix, as shown in Table 7.Table 6
Relevant data of factor analysis.
Principal componentsEigenvalueVariance contribution rateF14.58610.2213F24.13690.2016F33.24260.1567F42.79850.1325Table 7
The component score coefficient matrix.
F1F2F3F4X10.17790.07750.08020.0223X20.19990.04350.0482−0.0283X30.05150.07910.11210.1106X40.01970.02700.2674−0.0140X50.0083−0.00210.2658−0.0049X60.01050.02670.2543−0.0198X70.01140.19100.00780.0025X80.05700.17520.09660.0178X90.01720.18900.06550.0296X100.19250.03820.0274−0.0321X110.20420.04320.0483−0.0288X120.12760.0000−0.02770.0229X130.12690.0024−0.01800.2494X140.06700.00590.00030.2778X15−0.0549−0.0475−0.08970.0781X16−0.0727−0.0228−0.05150.3094X17−0.0574−0.0019−0.01780.2876X18−0.01140.0053−0.22280.0306X190.00730.1944−0.0390−0.0272X200.06240.18440.0401−0.0189X210.02140.2126−0.0232−0.0271According to Table7, we can get the score expression of the principal component, as shown in equations (15)–(18).(10)F1=0.1779X1+0.1999X2+⋯0.0214X21,(11)F2=0.0775X1+0.0435X2+⋯0.2126X21,(12)F3=0.0802X1+0.0482+⋯−0.0232X21,(13)F4=0.0223X1−0.0283X2+⋯−0.0271X21.(3) Behavior Analysis. We use k-means to cluster F1, F2, F3, and F4. The cluster center factor values and the number of clusters are shown in Table 8.(a)
By comparingF1 with the largest difference in cluster centers, it can be seen that Clusters 1 have the smallest longitudinal factor, tend to drive at a lower speed, and are more cautious in driving behavior; Clusters 3 had the largest longitudinal factor, preferred to drive at a higher speed, and had a more aggressive driving behavior.(b)
By comparingF2, it can be seen that the absolute value of the acceleration factor of Clusters 1 is the smallest, which indicates that their driving behavior is the most cautious in the performance of acceleration; Clusters 3 drivers have the largest absolute acceleration factor, which indicates that their driving behavior is the most aggressive in terms of acceleration performance.(c)
By comparingF3, it can be seen that the absolute value of the lateral factor of Clusters 1 is the smallest, which indicates that their driving behavior is the most cautious in the performance of lateral movement; Clusters 3 drivers have the largest absolute lateral factor, which indicates that their driving behavior is the most aggressive in terms of lateral movement.(d)
F4 does not show the same characteristics as F1, F2, and F3, mainly because its low contribution to variance leads to its low contribution to driving behavior clustering.Table 8
The central value and the number of clusters.
ClustersF1F2F3F4Number (proportion)124.4115−0.3330−3.27634.516498/11%239.8161−2.4688−3.99843.5426587/65%354.5711−4.3355−4.80444.6925221/24%Thus, the label of Clusters 1, 2, and 3 are defined as timid, normal, and aggressive, respectively.(4) Model Calibration. Since the genetic algorithm is a mature classical algorithm, its specific calculation process will not be described in this article. The genetic algorithm is set as follows: selection method: roulette selection method; crossover rate: 0.85; crossing method: uniform crossing method; variation rate: 0.01; and allowable error of convergence: 1.0E − 10.According to the process diagram shown in Figure1, we first determined the error of parameter calibration. First, this article uses the commonly used genetic algorithm for calibration. Table 9 shows that the indicators of speed, acceleration, and position are relatively small. Thus, these parameters are successfully calibrated, and the next step can be performed.Table 9
Error in the calibration of the parameters in the IDM.
DataMEMAEMARERMSESpeed−0.0511.8750.3271.457Acceleration0.0611.7880.2551.304Position−0.7034.2810.0241.316Finally, we used the genetic algorithm to calculate the calibration results of three different driving behaviors, and the results are shown in Table10.Table 10
Calibration results of the IDM of different driving behaviors.
ClustersModel parameterMeanAggressivea0m/s21.8b0m/s2−3.1T (s)1.3s0 (m)1.5Normala0m/s21.45b0m/s2−2.0T (s)1.5s0 (m)2.3Timida0m/s21.14b0m/s2−1.23T (s)2.3s0 (m)4.3The comparison for timid driving is relatively small; thus, this article does not consider timid driving in the research model to explore the effect of aggressive drivers on a mixed traffic flow; only aggressive driving and normal driving are considered.


### 2.3. AV Car-Following Model

Section2.1 explains that AVs only include CACC vehicles in this research; therefore, this study uses the CACC vehicle model proposed by the path laboratory to reflect the characteristics of AVs [32]. The CACC vehicle model is shown in equation (19).(14)vit+Δt=vit+kpet+kde˙t,et=hit−s0−l−tcvit,where vit, hit, l, and s0 are identical to the parameters in equation (2); vit+Δt represents the target speed at time t+Δt; et denotes the error between the actual and the expected gap distance at time t; e˙t denotes the derivative of et; tc represents the expected headway; and kp and kd are the controlling parameters.The derivative of velocityvit in the CACC vehicle model is as follows:(15)v˙it=kphit−l−s0−kptcvit+kdΔvitΔt+kdtc.The typical parameters of the CACC vehicle model are verified through real experiments, and the results are shown in Table11 [32].Table 11
Typical values for the parameters of the CACC vehicle model.
ParameterValuekp0.45kd0.25tc (s)0.6l (m)5.0s0 (m)2.0Δt (s)0.01


## 2.1. Description of the Mixed-Flow Scenario

The major objective of this study was to evaluate the capacity and stability of a mixed traffic scenario composed of both HDVs and AVs. Generally, scholars use cooperative adaptive cruise control (CACC) or adaptive cruise control (ACC) vehicles to represent AVs [6–9].At present, vehicles with ACC technology are generally defined by scholars as low-level AVs [20]. Although the vehicle with ACC represents the AVs to a certain extent, the key parameter of the car-following model does not change significantly compared with the HDV. If ACC is used as the model of AVs, the heterogeneity of traffic flow may not be reflected in the low market penetration rate.Thus, in AVs, only CACC is allowed. The situation in which a CACC vehicle degrades to an ACC vehicle as a result of long platoon and V2V communication failures is not considered in this study [20].With advances in intelligent connected vehicle (ICV) technology in the future, we believe that CACC vehicles will be able to obtain some information about the HDVs in front of them to judge the state of those vehicles in advance, or they will have a shorter reaction time, which could ensure a small headway.Assume that a mixed traffic flow hasN vehicles, the market penetration rate of CACC vehicles is p [21, 22], and the HDV market penetration rate is 1−p. The proportion of aggressive driving in HDVs is pA, and the proportion of other driving styles (normal driving) is 1−pA. In this article, HDV-A represents an HDV with aggressive driving, and HDV-N represents an HDV with normal driving. Thus, the numbers of three types of vehicles are as follows:(1)nC=pN,nN=1−p1−pAN,nA=1−ppAN,where nC, nN, and nA are the numbers of CACC, HDV-N, and HDV-A, respectively.


## 2.2. HDV Car-Following Model

### 2.2.1. Basic Car-Following Model

In an ICV environment, many researchers use an intelligent driver model (IDM) [23–25] as the HDV car-following model, which properly reflects the car-following interrelation. The IDM is defined using equation (2) as follows:(2)v˙it=a1−vitv04−s0+vitT+vitΔvit/2abhit−l2,where vit represents the speed of vehicle i at time t; v˙it represents the derivative of vit; v0 denotes the free flow velocity; a is the maximum acceleration; T denotes the safe headway of the Q type of vehicle; s0 is the minimum distance between adjoining vehicles when stationary; b represents the desired deceleration; Δvit represents the speed difference between adjoining vehicles; hit represents the spatial distance between vehicle i and the vehicle behind at time t; and l is the vehicle length.Kesting et al. [26, 27] calibrated IDMs and classified HDV drivers into three types (normal, timid, and aggressive) based on their driving behaviors. The values for the parameters included in the IDM are shown in Table 1 [27].Table 1
Values of typical parameters for different types of HDV driving in the IDM.
ParameterTimidNormalAggressivev0m/s33.333.333.3Ts1.81.51.0s0m4.02.01.0lm5.05.05.0am/s21.01.42.0bm/s2−1.0−2.0−3.0


### 2.2.2. Calibration of the Car-Following Model According to Driver Behavior

We are not sure whether the values presented in Table1 have been tested using real traffic data; therefore, we are uncertain whether the values reflect the real situation of aggressive driving.We plan to use Chinese data to calibrate the IDM and classify HDV drivers into three types in accordance with Table1 to better reflect the situation of mixed traffic flow in China. Therefore, the most important function of Table 1 is to provide us with a classified reference. We selected the local road traffic trajectory database in China as the research data for this article. The commonly used road traffic trajectory databases in China include Ubiquitous Traffic Eyes by Southeast University (PRC), the Tongji Road Trajectory Sharing (TJRD TS) platform by Tongji University (PRC), Mirror-Traffic by Tsinghua University (PRC), and the Traflow platform by Fudan University (PRC). Since the research scenario of this article is the traffic flow under the free state of an expressway, we choose the Traflow platform as the validation database to determine whether the data in Table 1 match the real situation. The process of model calibration is illustrated in Figure 1.Figure 1
The process used to calibrate and divide driving behaviors.Step 1.
Data processing.
(1) Data Filtering. The Traflow platform provides trajectory data for the G2, G15w, G45, and G60 expressways. A comparison shows that the flow of G2 is relatively moderate; therefore, the data for G2 are selected for analysis in this article. The basic information of the Traflow dataset (G2) is shown in Table 2.
Using the data from the G2 expressway in Jiangqiao, Shanghai, China, the data collection time in the study was 14:45–16:12 pm on April 3, 2012, which lasted for approximately an hour and half. We prefer that these data only consider cars and no other types of vehicles to remove interference from other factors; thus, we filter the car data from the database.
(2) Processing Rules. (a)
The car-following behavior data are extracted in the form of two vehicles continuously driving in the same lane as a vehicle group.(b)
Neither the following car nor the leading car will change lanes for a period of time.(c)
Every following car has a front car for a period of time, and thus the first car in the observation range is not used as the research object.(d)
The car-following behavior of each pair of car groups lasts for more than 30 s to ensure that the vehicle has a relatively stable car-following state.(e)
The distance between the following vehicle and the leading vehicle is 2–50 meters. A distance that is too large indicates that the interaction between the two vehicles is weak.(f)
We further screened the data and selected the vehicles that completed lane changing within 16 s. According to references [16, 17], all normally running vehicles can successfully complete a single lane change within 16 s.
(3) Data Processing Method. Many sources of error and noise are present in the Traflow data obtained based on processing videos. If the original Traflow data are directly used, a greater deviation in the results of the analysis will occur, which will affect the calibration of the micro traffic flow model and reduce the accuracy of subsequent analysis. Therefore, this article used the symmetric exponential moving average (SEMA) proposed by Thiemann et al. [28] to smooth the data.Table 2
Basic information of the Traflow dataset (G2).
Road IDG2Number of lanes4Date/Periods2012.04.03/14:45–16.12WeatherSunnyNumber of vehicles4284Step 2.
Driving behavior extraction.
(1) Driving Behavior Parameters. There is no uniform standard for the selection of driving behavior evaluation indicators. Different scholars choose different evaluation indicators. The more the evaluation indicators selected, the more comprehensively the driver’s driving behavior can be represented. This article selects 21 indicators as driving behavior parameters based on existing research results [29], as shown in Table 3. X1–X3, X7–X9, and X12–X17 are directly obtained from the data set. Other parameters are calculated as follows:
ForX4–X6, the lateral speed vyt is required for calculation.(3)vyt=yt−yt−1Δt,where yt is the lateral displacement at time t, Δt is the time interval from (t−1) to t, and Δt = 0.1 s in this study.
ForX10 and X11, the average value of the overall vehicle (No) speed vm is required for calculation.(4)vm=1No∑i=1No116si,where vm is the average speed of No vehicles in 16 s and si is the driving distance in 16 s.
ForX18, we need to determine when to start and end the lane change. So we make the following definitions:(5)tcl=t2−t1,where tcl is the lane change time and t1 and t2 are critical time, when the lateral speed of the vehicle is less than 0.2 m.s⁻1, and the distance from the target lane line is greater than half of the vehicle width. t1 searches forward from the lane change time, and t2 searches backward from the lane change time.
ForX19, X20, and X21, we need to determine the acceleration change rate. The specific calculation formula is as follows:(6)Jt=at−at−1Δt,where Jt is the acceleration change rate and at is the acceleration at time t.
(2) Principal Component Analysis. The 21 preliminarily selected evaluation indicators can comprehensively reflect the driving behavior in the 16 s driving trajectory including the lane-changing process, but the importance of each indicator is different, and there is information redundancy between indicators.
We also performed principal component analysis (PCA) which is a statistical method of dimension reduction. Several studies use PCA to identify key factors affecting traffic flow [30, 31], and the main calculation formula is as follows:(7)CPV=∑i=1Aλi∑i=1Mλi,where CPV is the cumulative contribution percentage of variance, λi represents the top A larger eigenvalues of the covariance matrix, and M represents the number of original variables. Usually, when the CPV is greater than 70%, the former A principal component is assumed to explain most of the variance in the data.
We perform PCA and factor analysis on the data using Python. The rotation method is the maximum variance method, and the eigenvalue and the variance contribution rate are obtained. We use the obtained factor loading matrix to analyze the correlation between principal components and indicators.Table 3
Driving behavior parameters.
ParametersDenotationMaximum speedX1Average speedX2Standard deviation (SD) of speedX3Maximum lateral speedX4Average lateral speedX5SD of lateral speedX6Maximum absolute value of accelerationX7Average value of absolute accelerationX8SD of absolute accelerationX9Time proportion above average total speedX10Driving distance within 16 sX11Minimum following distanceX12Average following distanceX13SD of following distanceX14Minimum headwayX15Average headwayX16SD of headwayX17Lane change timeX18Maximum absolute value of the acceleration change rateX19Average absolute value of the acceleration change rateX20SD of absolute value of the acceleration change rateX21Step 3.
Behavior analysis.
K-means is one of the most popular clustering algorithms based on partitioning and unsupervised learning. Therefore, this article uses the k-means algorithm to classify driving behaviors. The essence of this algorithm is to continuously determine the new cluster center through iterative operations, and the calculation will converge when the cluster center does not change:(8)DC,X=∑ik∑xjϵCixj−mi2,where C is the label of each cluster, k = 3; X is the sample car-following set, with each group of data in the car-following dataset used as a sample; xj=Tj,QTj where Tj is the clustering variable and represents the four parameters shown in Table 3 in this study; QTj is the coefficient of variation of each clustering variable; mi is the clustering center of class i; and xj−mi2 is the Euclidean distance.
We use k-means to cluster principal components. According to the clustering results obtained from the K-means analysis, the driving style of each cluster is judged, and the driving behavior label of each cluster (timid, normal, and aggressive) is determined.Step 4.
Model calibration.
(1) Calibration Method. In the previous step, we have classified the sample data according to different driving behaviors, so we calibrated the models for different driving behaviors.
A genetic algorithm is a relatively mature algorithm. This article intends to directly adopt this type of algorithm. Through the analysis of the IDM model, because the research data in this article are free flow, parameterv0 is generally considered to be the maximum speed limit (33.3 m/s), and the length of the car is generally considered to be 5 m. Combined with the research in the literature, the value ranges of other parameters to be calibrated are given in Table 4.
(2) Validation Indicators. Using the aforementioned genetic algorithm to calibrate the car-following model, the error between the simulated driving trajectory of each type of car-following model and the actual vehicle driving trajectory is obtained. The mean error (ME), mean absolute error (MAE), mean absolute relative error (MARE), and root mean square error (RMSE) are calculated as indicators to evaluate and verify the calibration of the parameters in the vehicle model results and to evaluate the results of trajectory fitting in this Section [30]. The calculation formulas are as follows:(9)ME=∑m=1Mhmreal−hmsimM,MARE=∑m=1Mhmreal−hmsim/hmrealM,MAE=∑m=1Mhmreal−hmsimM,RASE=∑m=1Mhmreal−hmsim2M.
Among them,hmreal represents the actual data that correspond to the speed, acceleration, position, and other parameters of the mth vehicle; hmsim represents the analog calibration data that correspond to the speed, acceleration, position, and other parameters of the mth vehicle [30].Table 4
The range of values for parameters included in the IDM.
ParameterRangeTs[0, 5]s0m[0.1, 10]am/s2[0.1, 5]bm/s2[−5, −0.1]


### 2.2.3. Results for the Calibration of the IDM Based on Driver Behavior

(1) Data Processing. After data processing and screening, 1434 groups of car-following records were obtained from 6721 trajectories; among them, the data were obtained from 906 cars.(2) Driving Behavior Extraction. We perform PCA and factor analysis on the data using Python. The final number of principal components is 4. The rotation method is the maximum variance method, and the eigenvalue and the variance contribution rate are obtained.We rotate the factors to obtain the factor loading matrix, as shown in Table5.Table 5
The factor loading matrix.
F1F2F3F4X10.81570.32060.26000.0623X20.91680.17990.1562−0.0793X30.23600.32740.36360.3094X40.09020.11170.8670−0.0393X50.0381−0.00880.8620−0.0137X60.04820.11030.8247−0.0553X70.05220.79000.02520.0070X80.26150.72480.31310.0499X90.07910.78200.21250.0829X100.88300.15790.0889−0.0898X110.93630.17890.1567−0.0805X120.5852−0.0001−0.08970.0642X130.58180.0099−0.05820.6980X140.30740.02440.00100.7774X15−0.2518−0.1966−0.29080.2187X16−0.3334−0.0943−0.16690.8660X17−0.2632−0.0077−0.05770.8048X18−0.05220.0220−0.72240.0857X190.03350.8043−0.1266−0.0761X200.28610.76300.1299−0.0530X210.09820.8793−0.0752−0.0759The values in bold show a strong correlation.As shown in Table5, there is a strong correlation between principal components and some indexes, and the detailed analysis is described below.(a)
The correlation coefficient betweenF1 and X1∼X2, X10∼X11 is large, which mainly reflects the longitudinal motion characteristics of vehicles, and is named as the “longitudinal factor.”(b)
The correlation coefficient betweenF2 and X7∼X9, X19∼X21 is large, which mainly reflects the acceleration characteristics of the vehicle, and is named as the “acceleration factor.”(c)
The correlation coefficient betweenF3 and X4∼X6 is large, which mainly reflects the lateral driving characteristics of the vehicles, and is named as the “lateral factor.”(d)
The correlation coefficient betweenF4 and X13∼X14, X16∼X17 is large, which mainly reflects the car-following characteristics of vehicles, and is named as the “car-following factor.”Using Tables5 and 6, we obtain the component score coefficient matrix, as shown in Table 7.Table 6
Relevant data of factor analysis.
Principal componentsEigenvalueVariance contribution rateF14.58610.2213F24.13690.2016F33.24260.1567F42.79850.1325Table 7
The component score coefficient matrix.
F1F2F3F4X10.17790.07750.08020.0223X20.19990.04350.0482−0.0283X30.05150.07910.11210.1106X40.01970.02700.2674−0.0140X50.0083−0.00210.2658−0.0049X60.01050.02670.2543−0.0198X70.01140.19100.00780.0025X80.05700.17520.09660.0178X90.01720.18900.06550.0296X100.19250.03820.0274−0.0321X110.20420.04320.0483−0.0288X120.12760.0000−0.02770.0229X130.12690.0024−0.01800.2494X140.06700.00590.00030.2778X15−0.0549−0.0475−0.08970.0781X16−0.0727−0.0228−0.05150.3094X17−0.0574−0.0019−0.01780.2876X18−0.01140.0053−0.22280.0306X190.00730.1944−0.0390−0.0272X200.06240.18440.0401−0.0189X210.02140.2126−0.0232−0.0271According to Table7, we can get the score expression of the principal component, as shown in equations (15)–(18).(10)F1=0.1779X1+0.1999X2+⋯0.0214X21,(11)F2=0.0775X1+0.0435X2+⋯0.2126X21,(12)F3=0.0802X1+0.0482+⋯−0.0232X21,(13)F4=0.0223X1−0.0283X2+⋯−0.0271X21.(3) Behavior Analysis. We use k-means to cluster F1, F2, F3, and F4. The cluster center factor values and the number of clusters are shown in Table 8.(a)
By comparingF1 with the largest difference in cluster centers, it can be seen that Clusters 1 have the smallest longitudinal factor, tend to drive at a lower speed, and are more cautious in driving behavior; Clusters 3 had the largest longitudinal factor, preferred to drive at a higher speed, and had a more aggressive driving behavior.(b)
By comparingF2, it can be seen that the absolute value of the acceleration factor of Clusters 1 is the smallest, which indicates that their driving behavior is the most cautious in the performance of acceleration; Clusters 3 drivers have the largest absolute acceleration factor, which indicates that their driving behavior is the most aggressive in terms of acceleration performance.(c)
By comparingF3, it can be seen that the absolute value of the lateral factor of Clusters 1 is the smallest, which indicates that their driving behavior is the most cautious in the performance of lateral movement; Clusters 3 drivers have the largest absolute lateral factor, which indicates that their driving behavior is the most aggressive in terms of lateral movement.(d)
F4 does not show the same characteristics as F1, F2, and F3, mainly because its low contribution to variance leads to its low contribution to driving behavior clustering.Table 8
The central value and the number of clusters.
ClustersF1F2F3F4Number (proportion)124.4115−0.3330−3.27634.516498/11%239.8161−2.4688−3.99843.5426587/65%354.5711−4.3355−4.80444.6925221/24%Thus, the label of Clusters 1, 2, and 3 are defined as timid, normal, and aggressive, respectively.(4) Model Calibration. Since the genetic algorithm is a mature classical algorithm, its specific calculation process will not be described in this article. The genetic algorithm is set as follows: selection method: roulette selection method; crossover rate: 0.85; crossing method: uniform crossing method; variation rate: 0.01; and allowable error of convergence: 1.0E − 10.According to the process diagram shown in Figure1, we first determined the error of parameter calibration. First, this article uses the commonly used genetic algorithm for calibration. Table 9 shows that the indicators of speed, acceleration, and position are relatively small. Thus, these parameters are successfully calibrated, and the next step can be performed.Table 9
Error in the calibration of the parameters in the IDM.
DataMEMAEMARERMSESpeed−0.0511.8750.3271.457Acceleration0.0611.7880.2551.304Position−0.7034.2810.0241.316Finally, we used the genetic algorithm to calculate the calibration results of three different driving behaviors, and the results are shown in Table10.Table 10
Calibration results of the IDM of different driving behaviors.
ClustersModel parameterMeanAggressivea0m/s21.8b0m/s2−3.1T (s)1.3s0 (m)1.5Normala0m/s21.45b0m/s2−2.0T (s)1.5s0 (m)2.3Timida0m/s21.14b0m/s2−1.23T (s)2.3s0 (m)4.3The comparison for timid driving is relatively small; thus, this article does not consider timid driving in the research model to explore the effect of aggressive drivers on a mixed traffic flow; only aggressive driving and normal driving are considered.


## 2.2.1. Basic Car-Following Model

In an ICV environment, many researchers use an intelligent driver model (IDM) [23–25] as the HDV car-following model, which properly reflects the car-following interrelation. The IDM is defined using equation (2) as follows:(2)v˙it=a1−vitv04−s0+vitT+vitΔvit/2abhit−l2,where vit represents the speed of vehicle i at time t; v˙it represents the derivative of vit; v0 denotes the free flow velocity; a is the maximum acceleration; T denotes the safe headway of the Q type of vehicle; s0 is the minimum distance between adjoining vehicles when stationary; b represents the desired deceleration; Δvit represents the speed difference between adjoining vehicles; hit represents the spatial distance between vehicle i and the vehicle behind at time t; and l is the vehicle length.Kesting et al. [26, 27] calibrated IDMs and classified HDV drivers into three types (normal, timid, and aggressive) based on their driving behaviors. The values for the parameters included in the IDM are shown in Table 1 [27].Table 1
Values of typical parameters for different types of HDV driving in the IDM.
ParameterTimidNormalAggressivev0m/s33.333.333.3Ts1.81.51.0s0m4.02.01.0lm5.05.05.0am/s21.01.42.0bm/s2−1.0−2.0−3.0


## 2.2.2. Calibration of the Car-Following Model According to Driver Behavior

We are not sure whether the values presented in Table1 have been tested using real traffic data; therefore, we are uncertain whether the values reflect the real situation of aggressive driving.We plan to use Chinese data to calibrate the IDM and classify HDV drivers into three types in accordance with Table1 to better reflect the situation of mixed traffic flow in China. Therefore, the most important function of Table 1 is to provide us with a classified reference. We selected the local road traffic trajectory database in China as the research data for this article. The commonly used road traffic trajectory databases in China include Ubiquitous Traffic Eyes by Southeast University (PRC), the Tongji Road Trajectory Sharing (TJRD TS) platform by Tongji University (PRC), Mirror-Traffic by Tsinghua University (PRC), and the Traflow platform by Fudan University (PRC). Since the research scenario of this article is the traffic flow under the free state of an expressway, we choose the Traflow platform as the validation database to determine whether the data in Table 1 match the real situation. The process of model calibration is illustrated in Figure 1.Figure 1
The process used to calibrate and divide driving behaviors.Step 1.
Data processing.
(1) Data Filtering. The Traflow platform provides trajectory data for the G2, G15w, G45, and G60 expressways. A comparison shows that the flow of G2 is relatively moderate; therefore, the data for G2 are selected for analysis in this article. The basic information of the Traflow dataset (G2) is shown in Table 2.
Using the data from the G2 expressway in Jiangqiao, Shanghai, China, the data collection time in the study was 14:45–16:12 pm on April 3, 2012, which lasted for approximately an hour and half. We prefer that these data only consider cars and no other types of vehicles to remove interference from other factors; thus, we filter the car data from the database.
(2) Processing Rules. (a)
The car-following behavior data are extracted in the form of two vehicles continuously driving in the same lane as a vehicle group.(b)
Neither the following car nor the leading car will change lanes for a period of time.(c)
Every following car has a front car for a period of time, and thus the first car in the observation range is not used as the research object.(d)
The car-following behavior of each pair of car groups lasts for more than 30 s to ensure that the vehicle has a relatively stable car-following state.(e)
The distance between the following vehicle and the leading vehicle is 2–50 meters. A distance that is too large indicates that the interaction between the two vehicles is weak.(f)
We further screened the data and selected the vehicles that completed lane changing within 16 s. According to references [16, 17], all normally running vehicles can successfully complete a single lane change within 16 s.
(3) Data Processing Method. Many sources of error and noise are present in the Traflow data obtained based on processing videos. If the original Traflow data are directly used, a greater deviation in the results of the analysis will occur, which will affect the calibration of the micro traffic flow model and reduce the accuracy of subsequent analysis. Therefore, this article used the symmetric exponential moving average (SEMA) proposed by Thiemann et al. [28] to smooth the data.Table 2
Basic information of the Traflow dataset (G2).
Road IDG2Number of lanes4Date/Periods2012.04.03/14:45–16.12WeatherSunnyNumber of vehicles4284Step 2.
Driving behavior extraction.
(1) Driving Behavior Parameters. There is no uniform standard for the selection of driving behavior evaluation indicators. Different scholars choose different evaluation indicators. The more the evaluation indicators selected, the more comprehensively the driver’s driving behavior can be represented. This article selects 21 indicators as driving behavior parameters based on existing research results [29], as shown in Table 3. X1–X3, X7–X9, and X12–X17 are directly obtained from the data set. Other parameters are calculated as follows:
ForX4–X6, the lateral speed vyt is required for calculation.(3)vyt=yt−yt−1Δt,where yt is the lateral displacement at time t, Δt is the time interval from (t−1) to t, and Δt = 0.1 s in this study.
ForX10 and X11, the average value of the overall vehicle (No) speed vm is required for calculation.(4)vm=1No∑i=1No116si,where vm is the average speed of No vehicles in 16 s and si is the driving distance in 16 s.
ForX18, we need to determine when to start and end the lane change. So we make the following definitions:(5)tcl=t2−t1,where tcl is the lane change time and t1 and t2 are critical time, when the lateral speed of the vehicle is less than 0.2 m.s⁻1, and the distance from the target lane line is greater than half of the vehicle width. t1 searches forward from the lane change time, and t2 searches backward from the lane change time.
ForX19, X20, and X21, we need to determine the acceleration change rate. The specific calculation formula is as follows:(6)Jt=at−at−1Δt,where Jt is the acceleration change rate and at is the acceleration at time t.
(2) Principal Component Analysis. The 21 preliminarily selected evaluation indicators can comprehensively reflect the driving behavior in the 16 s driving trajectory including the lane-changing process, but the importance of each indicator is different, and there is information redundancy between indicators.
We also performed principal component analysis (PCA) which is a statistical method of dimension reduction. Several studies use PCA to identify key factors affecting traffic flow [30, 31], and the main calculation formula is as follows:(7)CPV=∑i=1Aλi∑i=1Mλi,where CPV is the cumulative contribution percentage of variance, λi represents the top A larger eigenvalues of the covariance matrix, and M represents the number of original variables. Usually, when the CPV is greater than 70%, the former A principal component is assumed to explain most of the variance in the data.
We perform PCA and factor analysis on the data using Python. The rotation method is the maximum variance method, and the eigenvalue and the variance contribution rate are obtained. We use the obtained factor loading matrix to analyze the correlation between principal components and indicators.Table 3
Driving behavior parameters.
ParametersDenotationMaximum speedX1Average speedX2Standard deviation (SD) of speedX3Maximum lateral speedX4Average lateral speedX5SD of lateral speedX6Maximum absolute value of accelerationX7Average value of absolute accelerationX8SD of absolute accelerationX9Time proportion above average total speedX10Driving distance within 16 sX11Minimum following distanceX12Average following distanceX13SD of following distanceX14Minimum headwayX15Average headwayX16SD of headwayX17Lane change timeX18Maximum absolute value of the acceleration change rateX19Average absolute value of the acceleration change rateX20SD of absolute value of the acceleration change rateX21Step 3.
Behavior analysis.
K-means is one of the most popular clustering algorithms based on partitioning and unsupervised learning. Therefore, this article uses the k-means algorithm to classify driving behaviors. The essence of this algorithm is to continuously determine the new cluster center through iterative operations, and the calculation will converge when the cluster center does not change:(8)DC,X=∑ik∑xjϵCixj−mi2,where C is the label of each cluster, k = 3; X is the sample car-following set, with each group of data in the car-following dataset used as a sample; xj=Tj,QTj where Tj is the clustering variable and represents the four parameters shown in Table 3 in this study; QTj is the coefficient of variation of each clustering variable; mi is the clustering center of class i; and xj−mi2 is the Euclidean distance.
We use k-means to cluster principal components. According to the clustering results obtained from the K-means analysis, the driving style of each cluster is judged, and the driving behavior label of each cluster (timid, normal, and aggressive) is determined.Step 4.
Model calibration.
(1) Calibration Method. In the previous step, we have classified the sample data according to different driving behaviors, so we calibrated the models for different driving behaviors.
A genetic algorithm is a relatively mature algorithm. This article intends to directly adopt this type of algorithm. Through the analysis of the IDM model, because the research data in this article are free flow, parameterv0 is generally considered to be the maximum speed limit (33.3 m/s), and the length of the car is generally considered to be 5 m. Combined with the research in the literature, the value ranges of other parameters to be calibrated are given in Table 4.
(2) Validation Indicators. Using the aforementioned genetic algorithm to calibrate the car-following model, the error between the simulated driving trajectory of each type of car-following model and the actual vehicle driving trajectory is obtained. The mean error (ME), mean absolute error (MAE), mean absolute relative error (MARE), and root mean square error (RMSE) are calculated as indicators to evaluate and verify the calibration of the parameters in the vehicle model results and to evaluate the results of trajectory fitting in this Section [30]. The calculation formulas are as follows:(9)ME=∑m=1Mhmreal−hmsimM,MARE=∑m=1Mhmreal−hmsim/hmrealM,MAE=∑m=1Mhmreal−hmsimM,RASE=∑m=1Mhmreal−hmsim2M.
Among them,hmreal represents the actual data that correspond to the speed, acceleration, position, and other parameters of the mth vehicle; hmsim represents the analog calibration data that correspond to the speed, acceleration, position, and other parameters of the mth vehicle [30].Table 4
The range of values for parameters included in the IDM.
ParameterRangeTs[0, 5]s0m[0.1, 10]am/s2[0.1, 5]bm/s2[−5, −0.1]


## 2.2.3. Results for the Calibration of the IDM Based on Driver Behavior

(1) Data Processing. After data processing and screening, 1434 groups of car-following records were obtained from 6721 trajectories; among them, the data were obtained from 906 cars.(2) Driving Behavior Extraction. We perform PCA and factor analysis on the data using Python. The final number of principal components is 4. The rotation method is the maximum variance method, and the eigenvalue and the variance contribution rate are obtained.We rotate the factors to obtain the factor loading matrix, as shown in Table5.Table 5
The factor loading matrix.
F1F2F3F4X10.81570.32060.26000.0623X20.91680.17990.1562−0.0793X30.23600.32740.36360.3094X40.09020.11170.8670−0.0393X50.0381−0.00880.8620−0.0137X60.04820.11030.8247−0.0553X70.05220.79000.02520.0070X80.26150.72480.31310.0499X90.07910.78200.21250.0829X100.88300.15790.0889−0.0898X110.93630.17890.1567−0.0805X120.5852−0.0001−0.08970.0642X130.58180.0099−0.05820.6980X140.30740.02440.00100.7774X15−0.2518−0.1966−0.29080.2187X16−0.3334−0.0943−0.16690.8660X17−0.2632−0.0077−0.05770.8048X18−0.05220.0220−0.72240.0857X190.03350.8043−0.1266−0.0761X200.28610.76300.1299−0.0530X210.09820.8793−0.0752−0.0759The values in bold show a strong correlation.As shown in Table5, there is a strong correlation between principal components and some indexes, and the detailed analysis is described below.(a)
The correlation coefficient betweenF1 and X1∼X2, X10∼X11 is large, which mainly reflects the longitudinal motion characteristics of vehicles, and is named as the “longitudinal factor.”(b)
The correlation coefficient betweenF2 and X7∼X9, X19∼X21 is large, which mainly reflects the acceleration characteristics of the vehicle, and is named as the “acceleration factor.”(c)
The correlation coefficient betweenF3 and X4∼X6 is large, which mainly reflects the lateral driving characteristics of the vehicles, and is named as the “lateral factor.”(d)
The correlation coefficient betweenF4 and X13∼X14, X16∼X17 is large, which mainly reflects the car-following characteristics of vehicles, and is named as the “car-following factor.”Using Tables5 and 6, we obtain the component score coefficient matrix, as shown in Table 7.Table 6
Relevant data of factor analysis.
Principal componentsEigenvalueVariance contribution rateF14.58610.2213F24.13690.2016F33.24260.1567F42.79850.1325Table 7
The component score coefficient matrix.
F1F2F3F4X10.17790.07750.08020.0223X20.19990.04350.0482−0.0283X30.05150.07910.11210.1106X40.01970.02700.2674−0.0140X50.0083−0.00210.2658−0.0049X60.01050.02670.2543−0.0198X70.01140.19100.00780.0025X80.05700.17520.09660.0178X90.01720.18900.06550.0296X100.19250.03820.0274−0.0321X110.20420.04320.0483−0.0288X120.12760.0000−0.02770.0229X130.12690.0024−0.01800.2494X140.06700.00590.00030.2778X15−0.0549−0.0475−0.08970.0781X16−0.0727−0.0228−0.05150.3094X17−0.0574−0.0019−0.01780.2876X18−0.01140.0053−0.22280.0306X190.00730.1944−0.0390−0.0272X200.06240.18440.0401−0.0189X210.02140.2126−0.0232−0.0271According to Table7, we can get the score expression of the principal component, as shown in equations (15)–(18).(10)F1=0.1779X1+0.1999X2+⋯0.0214X21,(11)F2=0.0775X1+0.0435X2+⋯0.2126X21,(12)F3=0.0802X1+0.0482+⋯−0.0232X21,(13)F4=0.0223X1−0.0283X2+⋯−0.0271X21.(3) Behavior Analysis. We use k-means to cluster F1, F2, F3, and F4. The cluster center factor values and the number of clusters are shown in Table 8.(a)
By comparingF1 with the largest difference in cluster centers, it can be seen that Clusters 1 have the smallest longitudinal factor, tend to drive at a lower speed, and are more cautious in driving behavior; Clusters 3 had the largest longitudinal factor, preferred to drive at a higher speed, and had a more aggressive driving behavior.(b)
By comparingF2, it can be seen that the absolute value of the acceleration factor of Clusters 1 is the smallest, which indicates that their driving behavior is the most cautious in the performance of acceleration; Clusters 3 drivers have the largest absolute acceleration factor, which indicates that their driving behavior is the most aggressive in terms of acceleration performance.(c)
By comparingF3, it can be seen that the absolute value of the lateral factor of Clusters 1 is the smallest, which indicates that their driving behavior is the most cautious in the performance of lateral movement; Clusters 3 drivers have the largest absolute lateral factor, which indicates that their driving behavior is the most aggressive in terms of lateral movement.(d)
F4 does not show the same characteristics as F1, F2, and F3, mainly because its low contribution to variance leads to its low contribution to driving behavior clustering.Table 8
The central value and the number of clusters.
ClustersF1F2F3F4Number (proportion)124.4115−0.3330−3.27634.516498/11%239.8161−2.4688−3.99843.5426587/65%354.5711−4.3355−4.80444.6925221/24%Thus, the label of Clusters 1, 2, and 3 are defined as timid, normal, and aggressive, respectively.(4) Model Calibration. Since the genetic algorithm is a mature classical algorithm, its specific calculation process will not be described in this article. The genetic algorithm is set as follows: selection method: roulette selection method; crossover rate: 0.85; crossing method: uniform crossing method; variation rate: 0.01; and allowable error of convergence: 1.0E − 10.According to the process diagram shown in Figure1, we first determined the error of parameter calibration. First, this article uses the commonly used genetic algorithm for calibration. Table 9 shows that the indicators of speed, acceleration, and position are relatively small. Thus, these parameters are successfully calibrated, and the next step can be performed.Table 9
Error in the calibration of the parameters in the IDM.
DataMEMAEMARERMSESpeed−0.0511.8750.3271.457Acceleration0.0611.7880.2551.304Position−0.7034.2810.0241.316Finally, we used the genetic algorithm to calculate the calibration results of three different driving behaviors, and the results are shown in Table10.Table 10
Calibration results of the IDM of different driving behaviors.
ClustersModel parameterMeanAggressivea0m/s21.8b0m/s2−3.1T (s)1.3s0 (m)1.5Normala0m/s21.45b0m/s2−2.0T (s)1.5s0 (m)2.3Timida0m/s21.14b0m/s2−1.23T (s)2.3s0 (m)4.3The comparison for timid driving is relatively small; thus, this article does not consider timid driving in the research model to explore the effect of aggressive drivers on a mixed traffic flow; only aggressive driving and normal driving are considered.


## 2.3. AV Car-Following Model

Section2.1 explains that AVs only include CACC vehicles in this research; therefore, this study uses the CACC vehicle model proposed by the path laboratory to reflect the characteristics of AVs [32]. The CACC vehicle model is shown in equation (19).(14)vit+Δt=vit+kpet+kde˙t,et=hit−s0−l−tcvit,where vit, hit, l, and s0 are identical to the parameters in equation (2); vit+Δt represents the target speed at time t+Δt; et denotes the error between the actual and the expected gap distance at time t; e˙t denotes the derivative of et; tc represents the expected headway; and kp and kd are the controlling parameters.The derivative of velocityvit in the CACC vehicle model is as follows:(15)v˙it=kphit−l−s0−kptcvit+kdΔvitΔt+kdtc.The typical parameters of the CACC vehicle model are verified through real experiments, and the results are shown in Table11 [32].Table 11
Typical values for the parameters of the CACC vehicle model.
ParameterValuekp0.45kd0.25tc (s)0.6l (m)5.0s0 (m)2.0Δt (s)0.01


## 3. Assessing Stability and Capacity considering Aggressive Driving for the HDV

According to the results of previous studies using real traffic data, aggressive drivers account for approximately 25% of the total number of drivers [33]. Therefore, in this section, pA=0.25 is selected as a typical value for calculating the mixed traffic model.


### 3.1. Stability

Stability is an intrinsic characteristic of traffic flow. Decreasing stability may increase the interference amplitude, propagation range, and interference duration, resulting in traffic congestion. The main characteristic of stability is closely related to safety. In this section, beginning with the inherent characteristics of the car-following model, we establish a structure for analyzing the stability of mixed traffic flow.


#### 3.1.1. Homogeneous Traffic Flow

The stability of homogeneous traffic flow has been evaluated using the linear stability condition presented by Ward [34], which is as follows:(16)ffQ=12fQv2−fQΔvfQv−fQh<0,where ffQ is the value of the discriminant, ffQ<0 shows that the traffic flow is in an unstable state; Q represents vehicle type identification, i.e., Q=A refers to aggressive driving vehicles, and Q=N refers to normal driving vehicles; fQv, fQ△v, and fQh represent partial differentiation for vehicle speed, speed difference, and space headway, respectively, of the Q type vehicle.The conditions for evaluating the stability of HDV traffic flow are obtained by calculating equation (2), and the result is as follows:(17)fQv=−4aQvit3v04−2aQTQ1−vit/v0sQ+vitTQ,fQΔv=aQbQvit1−vit/v04sQ+vitTQ,fQh=2aQ1−vit/v041−vit/v04sQ+vitTQ.Similarly, the stability conditions of homogeneous traffic flow with CACC vehicles are calculated using equation (14), and the result is as follows:(18)fCν=kptcΔt+kdtc,fCΔν=kdΔt+kdtc,fCh=kpΔt+kdtc.Figure2illustrates the stability discrimination function ffQ of a traffic flow that consists of HDV-A and HDV-N vehicles at different equilibrium speeds after substituting equation (17) into equation (21). Table 12 shows the statistics of this unstable speed range. The unstable speed range of HDV-A vehicles is only slightly smaller than that of HDV-N vehicles. However, the minimum ffQ value of HDV-A vehicles is also lower than that of HDV-N vehicles, indicating that the instability caused by HDV-A vehicles is worse than that caused by HDV-N vehicles in an unstable state.Figure 2
Analysis of HDV stability.Table 12
Unstable speed range of different types of traffic flow.
TypeSpeed range in the unstable state (m/s)Length of the speed range (m/s)CACC——HDV-N(2.24, 24.98)22.74HDV-A(2.71, 25.16)22.45Using equations (14) and (15) and the values of the parameters provided in Table 11, the value of the CACC vehicle traffic flow stability discriminant function ffQ is obtained. The results show that ffQ=1.2480>0, implying that the CACC vehicle traffic flow is unconditionally stable.


#### 3.1.2. Mixed Traffic Flow

Based on the form of the traditional car-following model, Ward [24] computed the following stability conditions in a mixed traffic flow:(19)∑nN12fnv2−fnΔvfnv−fnh∏m≠nfmv2<0,where n represents the n th vehicle in a heterogeneous traffic flow; m represents the m th vehicle in a heterogeneous traffic flow; and the number N denotes the total number of vehicles in the heterogeneous traffic flow.The stability discriminant of mixed traffic flow does not depend on the relative spatial position of vehicles but only on the number of vehicles of each type, according to equation (24). Therefore, the three vehicle types (HDV-N, HDV-A, and CACC) discussed in Section 2.1 are considered in the stability analysis.Substituting equations (1) and (3) into equation (19), we extend the stability discriminant to the investigation of the stability of mixed traffic flow, and the result is as follows:(20)nN12fNv2−fNΔvfNv−fNhfNhnN−1fAhnAfChnC2+nA12fAv2−fAΔvfAv−fAhfAhnA−1fNhnNfChnC2+nC12fCv2−fCΔvfCv−fChfChnC−1fNhnNfAhnA2<0.Equation (20) is reduced to the following formula:(21)F=1−p1−pAFN+1−ppAFA+pFC<0,FN=1/2fNv2−fNΔvfNv−fNhfNh2,FA=1/2fAv2−fAΔvfAv−fAhfAh2,FC=1/2fCv2−fCΔvfCv−fChfCh2,where FN, FA, and FC are the stability factors of the HDV-N, HDV-A, and CACC vehicles, respectively, and F is the value of the stability discriminant.An investigation of equation (11) shows that the stability of the mixed traffic flow is related to the proportion of aggressive driving, CACC vehicle market penetration rate, and equilibrium speed. Figure 3 shows the F function graph of a mixed traffic flow in the equilibrium state when pA=0.25 and under the conditions of different CACC vehicle market penetration rates and speeds.Figure 3
Stability analysis at differentp and equilibrium speed ve (when pA=0.25). (a) ve∈0,2.24. (b) ve∈2.24,2.71. (c) ve∈2.71,25.16. (d) ve∈25.16,33.3.
(a)(b)(c)(d)By analyzing the homogeneous traffic flow stability, the equilibrium speedve is divided into five parts for a separate analysis of the heterogeneous traffic flow, which are (0, 2.24), (2.24, 2.71), (2.71, 24.98), (24.98, 25.16), and (25.16, 33.3). However, in the actual stability analysis, we established four ranges to more intuitively observe the changes in stability, as shown in Figure 3.The horizontal axis in Figure3 depicts the equilibrium speed, and the vertical axis depicts the CACC vehicle market penetration rate. Furthermore, F is the value of the stability evaluation. The black curve marked as 0.00 indicates F=0, which is the critical line for mixed traffic flow stability. Table 13 summarizes the results from the final analysis of the data presented in Figure 3.Table 13
Unstable speed range in mixed traffic flow.
CACC market penetration rateSpeed range in the unstable state (m/s)Length of the speed range (m/s)(0, 0.445)(2.56, 25.16)22.60(0.445, 1)None0Figure3 and Table 13 show that when the CACC vehicle market penetration rate p exceeds 0.445, the mixed traffic flow is unconditionally stable (for pA=0.25). We define p=0.445 as the lowest limit of the CACC vehicle market penetration rate for ensuring absolute stability. When p is 0–0.445, an unstable situation occurs. Specifically, when p∈0,0.445 and v∈2.56,25.16, the mixed traffic flow will be in an unstable state, and the length of the unstable speed range is 22.60 m/s, which is the mean of HDV-N and HDV-A vehicles in the homogeneous traffic flow. Moreover, from the analysis of Figure 3(c), when the value of v is close to 13.8 m/s and p∈0.3,0.485, the value of F does not vary continuously from 0.634 to 1.27, which does not affect the analysis of the stability results.By analyzing the stability results of mixed traffic flow whenpA=0.25, the length of the unstable speed range influenced by HDV-A and HDV-N vehicles is the same when the CACC market penetration rate p is low. When p increases to 0.445, the unstable part will disappear; thus, increasing p will increase the stability of the mixed traffic flow. The effect of pA on stability is mentioned in Section 4.1.


### 3.2. Capacity

As is well known, AVs can improve the status of traffic flow by significantly enhancing traffic capacity. The effect of AVs on traffic capacity has received increasing attention. In traffic capacity research, the FDM is widely used as a classical model. However, with the development of vehicle platoon technology, AVs have smaller headways than HDVs, and thus the order of vehicles must be considered in the traffic capacity analysis. In this section, the fundamental diagram model (FDM) will be introduced first. Second, the theoretical capacity considering different vehicle orders in a mixed platoon based on a FDM will be discussed.


#### 3.2.1. FDM

The FDM can be used to calculate the capacity based on the assumption that when the traffic is in equilibrium, the flow volume is numerically equal to the capacity.When the traffic flow is in equilibrium, all of the vehicle speeds are equal to the equilibrium speedve. The difference in speed (between vehicles) is 0 m/s, and the acceleration for all vehicles is 0 m/s2 [35]. In other words, equation (22) is satisfied:(22)v=ve,Δv=0ms,v˙=0ms2.By replacing equation (22) with equation (18), we obtain the average space headway of HDV-N and HDV-A vehicle equilibrium traffic flows using the following equation:(23)hA,N=sA,N+veTA,N1−ve/v04+l.For CACC vehicles, by replacing equation (22) with equation (14), we obtain the average CACC vehicle space headway in the equilibrium state using the following equation:(24)hC=vetc+l+s0.Based on the meaning of traffic flow density, the density in mixed traffic flow is calculated using the following equation:(25)k=11−p1−pAhN+1−ppAhA+phC.Subsequently, the FDM of the mixed traffic flow can be obtained by substituting equations (23) and (24) into equation (25) and combining the volume-density-speed relationship of the basic traffic flow, as presented in equation (31).(26)k=11−p1−pAsN+TNve/1−ve/v04+l+1−ppAsA+TAve/1−ve/v04+l+ptcve+s0+lCACC,q=kve.The parameters in Tables10 and 11 are input in equation (26), and the capacity of mixed traffic flow at various p values is calculated, as illustrated in Figure 4.Figure 4
Capacity analysis of mixed traffic flow at differentp values (when pA=0.25).By analyzing the data presented in Figure4, the equilibrium speed ve at different p values can be obtained, as shown in Table 14.Table 14
The FDM of the theoretical capacity.
CACC market penetration rateCapacity (veh/h)ve (m/s)0197418.90.1208119.30.2219319.80.3231120.30.4245520.80.5260821.40.6277422.20.7301823.20.8319224.60.9366326.71.0444333.3Figure4 and Table 14 show that as the CACC vehicle market penetration rate p increases, the capacity of mixed traffic also increases, consistent with previous studies [1, 22]. Moreover, with the increase in the CACC vehicle market penetration rate p, the equilibrium speed also increases. Further analysis of the data presented in Table 14 shows that the increasing proportion of capacity increases when p increases from 0 to 1, the mixed traffic flow becomes the CACC vehicle homogeneous traffic flow, and the capacity is approximately 2.23 times higher than that of HDVs alone.


#### 3.2.2. Theoretical Capacity Based on the FDM considering Different Vehicle Orders

For a mixed traffic flow of AVs and HDVs, the headway will be directly affected by the orders of vehicles [36], while the FDM does not consider the interaction between vehicles to calculate the headway, assigning each type of vehicle the same value. Therefore, this research examines the effect of headway changes on traffic capacity from the perspective of studying the relationship between vehicles.The density of traffic flow is calculated using a basic definition, which is calculated using the following equation:(27)k=NL,where N represents the number of vehicles on the road and L represents the total length of the road occupied by a vehicle. Equation (27) is deformed to equation (28) to accurately study the density of the mixed traffic flow:(28)k=nL0,⟶k=nn.l+n−1h¯ve˜p+ve˜pT1ε,L0=n.l+n−1h¯ve˜p+ve˜pT2ε,where n represents the maximum number of the vehicles, and n=16 in this article. The number is identified according to previous studies [12]; L0 represents the road length occupied by n vehicles; l has the same meaning as described above; h¯ represents the average headway; ve˜p is the equilibrium speed at different CACC vehicle market penetration rates p, whose values are shown in Table 14 (when pA  = 0.25); T1 is the interplatoon time gap (s), and T1=2.0 s in this article. ε is the number of the platoon.The first task is to study the average headwayh¯ in different vehicle orders. When AVs and HDVs are randomly mixed in the stream, this is not an insignificant task because the average headway is affected by the HDV-N, HDV-A, and CACC vehicle headways. Ramezani et al. [37, 38] proposed a method for calculating the average headway between different types of vehicles, where hnv−nv is the headway between NVs; hav−av is the headway between AVs; hav−nv is the headway between an NV and an AV when an NV follows an AV; hhv−nv is the headway between an AV and an NV when an AV follows an NV; and hnv−nv, hav−nv, hhv−nv, and hav−av satisfy equations (29) and (30).(29)hav−nv=hnv−nv,(30)hhv−av=hav−av+hnv−nv−hav−av3.In this case, thehnv−nv and hav−av depend on the safe headway. Since three types of vehicles are studied in this article, nine combinations of headways between AVs and HDVs exist.Based on the aforementioned analysis, we extend the calculation formula originally applicable to two types of vehicles in equations (29) and (30) to three types of vehicles. Specifically, the method for calculating the nine combinations is as follows:(31)hC−N=hN−N,(32)hC−A=hA−A,(33)hA−N=hN−N,(34)hN−C=hC−C+hN−N−hC−C3,(35)hA−C=hC−C+hA−A−hC−C3,(36)hN−A=hA−A+hN−N−hA−A3,where the hN−N, hA−A, and hC−C depend on the safe headway.The results calculated using equations (31)–(36) are shown in Table 15.Table 15
Nine combinations of headways in a mixed traffic flow.
No.NomenclatureLeaderFollowerValue (s)1hC−CCACCCACC0.62hC−NCACCHDV-N1.53hC−ACACCHDV-A1.04hN−NHDV-NHDV-N1.55hN−CHDV-NCACC0.96hN−AHDV-NHDV-A1.47hA−AHDV-AHDV-A1.38hA−CHDV-ACACC0.89hA−NHDV-AHDV-N1.5The next task is to determine how to use the nine combinations of headways to calculateh¯. Considering all possible vehicle order combinations in a mixed traffic flow, some arrangements are considered on one-lane roads. Three scenarios will include all possible vehicle orders to explore the characteristics of the equilibrium flow [39]. (i) The first is a general scenario where HDV-N, HDV-A, and CACC vehicles are mixed randomly. (ii) The second is the best scenario where the three types of vehicles have their own platoons without interacting with other types of vehicles, thereby ensuring the lowest average headway. (iii) The third is the worst scenario considering the interaction between vehicles shown in Table 15, where a vehicle with a large headway follows a vehicle with a small headway, leading to the largest average headway. The complexity of this scenario is that the CACC vehicle market penetration p will directly affect the vehicle order.The average headway of the general scenario is between the best and the worst cases, and its calculation is very difficult; thus, this scenario is not considered in this study. Figure5 shows the two situations. Considering the importance of p to the headway analysis, the approaches described below will be used to investigate the effect of p on the headway in the two scenarios.Figure 5
Best and worst scenarios for average headway at differentp values.(1)p=0. In this case, when ε=2 in equation (28), the average headway of the best scenario is calculated using the following equation:(37)h¯=nA−1hA−A+nN−1hN−Nn−1.Since no AV is present, we arrange the two types of HDV cars sequentially to ensure a minimum headway. Similarly, using an HDV-A following an HDV-N as an example, the maximum headway is obtained, which will be the worst scenario. The average headway is calculated using the following equation:(38)h¯=nAhN−A+nA−1hA−N+nN−nA−1hN−Nn−1.(2)0<p<1. The best scenario occurs when the three types of vehicles have their own platoons, and the average headway is calculated using equation (39); in this case, ε=3 in equation (33).(39)h¯=nC−1hC−C+nA−1hA−A+nN−1hN−Nn−1.The worst scenario is relatively complex compared with the best scenario, and the vehicle order depends on the numbers of the three types of vehicles. According to previous research results [33], the proportion of aggressive driving will not exceed 50%; therefore, this article only considers pA less than 50%. This presumption plays a primary role in modeling the average headway in the worst scenario.Case 1.
0≤p<0.5−pA/1−pA
When the value ofp is small, the CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon one. Because the number of CACC vehicles is low, the number of remaining HDV-N vehicles is greater than the number of HDV-A vehicles, and we can arrange the HDV-A and HDV-N vehicles in turn where each HDV-N is a follower. This combination forms platoon two; the other HDV-Ns become platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because three platoons are present.Table 16
Calculation ofh¯ when p∈0,1 in the worst scenario.
Case IDThe value ofpεCalculation formula ofh¯10≤p<0.5−pA/1−pA3nChC−N+nC−1hN−C+nAhA−N+nA−1hN−A+nN−nC−nA−1hN−N/n−120.5−pA/1−pA≤p<1−pA/2−pA3nChC−N+nC−1hN−C+nN−nChA−N+nN−nC−1hN−A+nA−nN−nC−1hA−A/n−131−pA/2−pA≤p<0.53nNhC−N+nN−1hN−C+nC−nNhC−A+nC−nN−1hA−C+nA−nC−nN−1hA−A/n−14p=0.52nNhC−N+nN−1hN−C+nAhC−A+nA−1hA−C/n−150.5<p<13nNhC−N+nN−1hN−C+nAhC−A+nA−1hA−C+nC−nN−nA−1hC−C/n−1Case 2.
0.5−pA/1−pA≤p<1−pA/2−pA
Although the value ofp is greater than that of Case 1, the number of HDV-Ns is larger than the number of CACC vehicles, the CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon one. Because the number of CACC vehicles is larger than that in Case 1, the number of remaining HDV-A vehicles is greater than the number of HDV-N vehicles, which is the main difference from Case 1. The HDV-A and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon two; the remaining HDV-A becomes platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because there are three platoons.Case 3.
1−pA/2−pA≤p<0.5
Although the value ofp is less than 50%, the number of CACC vehicles exceeds the number of HDV-N vehicles. The CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower, forming platoon one. Due to the number of CACC vehicles, the remaining vehicles are also CACC vehicles, but the number of remaining vehicles is less than the number of HDV-A vehicles. CACC vehicles and HDV-A vehicles are arranged in turn, and each HDV-A is a follower, forming platoon two. The remaining HDV-A becomes platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because three platoons are present.Case 4.
p=0.5
When the number of AVs is consistent with the number of HDVs, CACC vehicles are combined with HDV-N and HDV-A, ensuring maximum headway. The CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon one. The number of remaining CACC vehicles is equal to that of HDV-A, and thus CACC vehicles and HDV-A are arranged in turn, and each HDV-A is a follower. This combination forms platoon two; in this case,ε=2 because two platoons are present.Case 5.
20.5<p<1
When the value ofp is greater than 50%, all HDVs are combined with CACC vehicles; HDV-N and HDV-A are followers, and this combination forms platoons one and two. The remaining CACC vehicles form platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because three platoons are present.
(3)p=1. When p=1, indicating homogeneous CACC vehicle traffic flow, h¯ has the same value as hC−C, and only one platoon is present; therefore, ε=1. In this case, the best scenario and the worst scenario are one scenario.


#### 3.2.3. Comparison of the Models and Verification of the Simulation

The capacity for different vehicle orders is calculated using the method proposed in Section3.2.2, and the results are shown in Table 17.Table 17
Capacity with different vehicle orders.
CACC vehicle market penetration rateCapacity (veh/h)Mean−FDM/Mean×100 (%)Best headwayWorst headwayMean01965192619461.460.12107207920930.570.22211215221820.530.32324223122781.470.42440218523136.160.52591232124566.190.62753255226534.580.72930276128466.060.83141301330773.740.93423332833768.521.04203420342035.71Whenp<0.4, the calculated capacity value of the FDM is less than that of the best scenario and slightly greater than that of the worst scenario. When p<0.4, the calculated capacity value is greater than the best headway. The mean capacity of the best scenario and the worst scenario are compared with the FDM, showing that the difference between the two models is not large and that the largest Mean−FDM/Mean is only 8.52%.In addition to the numerical comparison, the simulation is also verified. In this study, SUMO is selected as the simulation platform, which deeply describes the interaction between AVs and HDVs.This study uses a typical expressway case as the simulation scenario to ensure that the simulation results are close to the actual value. A detector is installed 5 km downstream of the ramp to detect the number of passing vehicles, with an upstream ramp speed limit of 120 km/h, a speed limit section of 80 km/h, and a weaving section of 50 km/h, as illustrated in Figure6. The maximum flow is detected by randomly generating vehicles without limiting the upstream traffic demand.Figure 6
Scenario of the capacity SUMO.The SUMO result shows that whenp<0.4, the simulation model is evenly distributed near the three curves, and when p<0.4, it is basically between the best scenario and the worst scenario, as shown in Figure 7.Figure 7
Comparison of capacity calculation models for different values ofp.As shown in Figure7, the differences in the results between the theoretical analysis and SUMO cannot be ignored, and the detailed analysis is described below.(a)
The SUMO platform is an open source simulation platform that deeply describes the interaction between AVs and HDVs. We can set or modify most models and corresponding model parameters involved in vehicle operation, but some built-in models cannot be accessed and adjusted, which may lead to differences between the simulation and the theory.(b)
When we conduct the simulation, we use some common random algorithm to generate traffic flow. Because a consideration of all random situations is impossible, our simulation does not completely reproduce the actual traffic flow situation. On the other hand, considering the randomness of traffic flow in theoretical calculations is insufficient. Therefore, this limitation may also explain the difference between the theoretical results and the simulation results.(c)
For a specific traffic capacity calculation model, the FDM is too idealistic and does not consider the relationship between vehicles, and thus the value for capacity is relatively large. Based on the FDM, the calculation method proposed in this study further considers the relationship between vehicles, and it is close to the simulation result.By analyzing the calculation formulas for the FDM and the proposed model,T1 (interplatoon time gap) shows the largest difference between the two models. T1 has little effect on the calculation results at a low CACC vehicle market penetration p, but with increasing p, the capacity results calculated with the FDM without T1 are lower than the best case. Meanwhile, the value of T1 in this research is 2 s, which covers most scenarios, but it may be slightly higher for scenarios with a high CACC vehicle market penetration p (the value of T1 will be studied in the future). Thus, the capacity calculation result is higher than the best scenario value after p<0.4. However, the FDM is too ideal to truly describe the relationship between vehicles. Therefore, after p exceeds 0.4, the SUMO values are near the blue and purple curves. Using this analysis of a mixed traffic flow, we verified that the proposed model accurately calculates the capacity.


## 3.1. Stability

Stability is an intrinsic characteristic of traffic flow. Decreasing stability may increase the interference amplitude, propagation range, and interference duration, resulting in traffic congestion. The main characteristic of stability is closely related to safety. In this section, beginning with the inherent characteristics of the car-following model, we establish a structure for analyzing the stability of mixed traffic flow.


### 3.1.1. Homogeneous Traffic Flow

The stability of homogeneous traffic flow has been evaluated using the linear stability condition presented by Ward [34], which is as follows:(16)ffQ=12fQv2−fQΔvfQv−fQh<0,where ffQ is the value of the discriminant, ffQ<0 shows that the traffic flow is in an unstable state; Q represents vehicle type identification, i.e., Q=A refers to aggressive driving vehicles, and Q=N refers to normal driving vehicles; fQv, fQ△v, and fQh represent partial differentiation for vehicle speed, speed difference, and space headway, respectively, of the Q type vehicle.The conditions for evaluating the stability of HDV traffic flow are obtained by calculating equation (2), and the result is as follows:(17)fQv=−4aQvit3v04−2aQTQ1−vit/v0sQ+vitTQ,fQΔv=aQbQvit1−vit/v04sQ+vitTQ,fQh=2aQ1−vit/v041−vit/v04sQ+vitTQ.Similarly, the stability conditions of homogeneous traffic flow with CACC vehicles are calculated using equation (14), and the result is as follows:(18)fCν=kptcΔt+kdtc,fCΔν=kdΔt+kdtc,fCh=kpΔt+kdtc.Figure2illustrates the stability discrimination function ffQ of a traffic flow that consists of HDV-A and HDV-N vehicles at different equilibrium speeds after substituting equation (17) into equation (21). Table 12 shows the statistics of this unstable speed range. The unstable speed range of HDV-A vehicles is only slightly smaller than that of HDV-N vehicles. However, the minimum ffQ value of HDV-A vehicles is also lower than that of HDV-N vehicles, indicating that the instability caused by HDV-A vehicles is worse than that caused by HDV-N vehicles in an unstable state.Figure 2
Analysis of HDV stability.Table 12
Unstable speed range of different types of traffic flow.
TypeSpeed range in the unstable state (m/s)Length of the speed range (m/s)CACC——HDV-N(2.24, 24.98)22.74HDV-A(2.71, 25.16)22.45Using equations (14) and (15) and the values of the parameters provided in Table 11, the value of the CACC vehicle traffic flow stability discriminant function ffQ is obtained. The results show that ffQ=1.2480>0, implying that the CACC vehicle traffic flow is unconditionally stable.


### 3.1.2. Mixed Traffic Flow

Based on the form of the traditional car-following model, Ward [24] computed the following stability conditions in a mixed traffic flow:(19)∑nN12fnv2−fnΔvfnv−fnh∏m≠nfmv2<0,where n represents the n th vehicle in a heterogeneous traffic flow; m represents the m th vehicle in a heterogeneous traffic flow; and the number N denotes the total number of vehicles in the heterogeneous traffic flow.The stability discriminant of mixed traffic flow does not depend on the relative spatial position of vehicles but only on the number of vehicles of each type, according to equation (24). Therefore, the three vehicle types (HDV-N, HDV-A, and CACC) discussed in Section 2.1 are considered in the stability analysis.Substituting equations (1) and (3) into equation (19), we extend the stability discriminant to the investigation of the stability of mixed traffic flow, and the result is as follows:(20)nN12fNv2−fNΔvfNv−fNhfNhnN−1fAhnAfChnC2+nA12fAv2−fAΔvfAv−fAhfAhnA−1fNhnNfChnC2+nC12fCv2−fCΔvfCv−fChfChnC−1fNhnNfAhnA2<0.Equation (20) is reduced to the following formula:(21)F=1−p1−pAFN+1−ppAFA+pFC<0,FN=1/2fNv2−fNΔvfNv−fNhfNh2,FA=1/2fAv2−fAΔvfAv−fAhfAh2,FC=1/2fCv2−fCΔvfCv−fChfCh2,where FN, FA, and FC are the stability factors of the HDV-N, HDV-A, and CACC vehicles, respectively, and F is the value of the stability discriminant.An investigation of equation (11) shows that the stability of the mixed traffic flow is related to the proportion of aggressive driving, CACC vehicle market penetration rate, and equilibrium speed. Figure 3 shows the F function graph of a mixed traffic flow in the equilibrium state when pA=0.25 and under the conditions of different CACC vehicle market penetration rates and speeds.Figure 3
Stability analysis at differentp and equilibrium speed ve (when pA=0.25). (a) ve∈0,2.24. (b) ve∈2.24,2.71. (c) ve∈2.71,25.16. (d) ve∈25.16,33.3.
(a)(b)(c)(d)By analyzing the homogeneous traffic flow stability, the equilibrium speedve is divided into five parts for a separate analysis of the heterogeneous traffic flow, which are (0, 2.24), (2.24, 2.71), (2.71, 24.98), (24.98, 25.16), and (25.16, 33.3). However, in the actual stability analysis, we established four ranges to more intuitively observe the changes in stability, as shown in Figure 3.The horizontal axis in Figure3 depicts the equilibrium speed, and the vertical axis depicts the CACC vehicle market penetration rate. Furthermore, F is the value of the stability evaluation. The black curve marked as 0.00 indicates F=0, which is the critical line for mixed traffic flow stability. Table 13 summarizes the results from the final analysis of the data presented in Figure 3.Table 13
Unstable speed range in mixed traffic flow.
CACC market penetration rateSpeed range in the unstable state (m/s)Length of the speed range (m/s)(0, 0.445)(2.56, 25.16)22.60(0.445, 1)None0Figure3 and Table 13 show that when the CACC vehicle market penetration rate p exceeds 0.445, the mixed traffic flow is unconditionally stable (for pA=0.25). We define p=0.445 as the lowest limit of the CACC vehicle market penetration rate for ensuring absolute stability. When p is 0–0.445, an unstable situation occurs. Specifically, when p∈0,0.445 and v∈2.56,25.16, the mixed traffic flow will be in an unstable state, and the length of the unstable speed range is 22.60 m/s, which is the mean of HDV-N and HDV-A vehicles in the homogeneous traffic flow. Moreover, from the analysis of Figure 3(c), when the value of v is close to 13.8 m/s and p∈0.3,0.485, the value of F does not vary continuously from 0.634 to 1.27, which does not affect the analysis of the stability results.By analyzing the stability results of mixed traffic flow whenpA=0.25, the length of the unstable speed range influenced by HDV-A and HDV-N vehicles is the same when the CACC market penetration rate p is low. When p increases to 0.445, the unstable part will disappear; thus, increasing p will increase the stability of the mixed traffic flow. The effect of pA on stability is mentioned in Section 4.1.


## 3.1.1. Homogeneous Traffic Flow

The stability of homogeneous traffic flow has been evaluated using the linear stability condition presented by Ward [34], which is as follows:(16)ffQ=12fQv2−fQΔvfQv−fQh<0,where ffQ is the value of the discriminant, ffQ<0 shows that the traffic flow is in an unstable state; Q represents vehicle type identification, i.e., Q=A refers to aggressive driving vehicles, and Q=N refers to normal driving vehicles; fQv, fQ△v, and fQh represent partial differentiation for vehicle speed, speed difference, and space headway, respectively, of the Q type vehicle.The conditions for evaluating the stability of HDV traffic flow are obtained by calculating equation (2), and the result is as follows:(17)fQv=−4aQvit3v04−2aQTQ1−vit/v0sQ+vitTQ,fQΔv=aQbQvit1−vit/v04sQ+vitTQ,fQh=2aQ1−vit/v041−vit/v04sQ+vitTQ.Similarly, the stability conditions of homogeneous traffic flow with CACC vehicles are calculated using equation (14), and the result is as follows:(18)fCν=kptcΔt+kdtc,fCΔν=kdΔt+kdtc,fCh=kpΔt+kdtc.Figure2illustrates the stability discrimination function ffQ of a traffic flow that consists of HDV-A and HDV-N vehicles at different equilibrium speeds after substituting equation (17) into equation (21). Table 12 shows the statistics of this unstable speed range. The unstable speed range of HDV-A vehicles is only slightly smaller than that of HDV-N vehicles. However, the minimum ffQ value of HDV-A vehicles is also lower than that of HDV-N vehicles, indicating that the instability caused by HDV-A vehicles is worse than that caused by HDV-N vehicles in an unstable state.Figure 2
Analysis of HDV stability.Table 12
Unstable speed range of different types of traffic flow.
TypeSpeed range in the unstable state (m/s)Length of the speed range (m/s)CACC——HDV-N(2.24, 24.98)22.74HDV-A(2.71, 25.16)22.45Using equations (14) and (15) and the values of the parameters provided in Table 11, the value of the CACC vehicle traffic flow stability discriminant function ffQ is obtained. The results show that ffQ=1.2480>0, implying that the CACC vehicle traffic flow is unconditionally stable.


## 3.1.2. Mixed Traffic Flow

Based on the form of the traditional car-following model, Ward [24] computed the following stability conditions in a mixed traffic flow:(19)∑nN12fnv2−fnΔvfnv−fnh∏m≠nfmv2<0,where n represents the n th vehicle in a heterogeneous traffic flow; m represents the m th vehicle in a heterogeneous traffic flow; and the number N denotes the total number of vehicles in the heterogeneous traffic flow.The stability discriminant of mixed traffic flow does not depend on the relative spatial position of vehicles but only on the number of vehicles of each type, according to equation (24). Therefore, the three vehicle types (HDV-N, HDV-A, and CACC) discussed in Section 2.1 are considered in the stability analysis.Substituting equations (1) and (3) into equation (19), we extend the stability discriminant to the investigation of the stability of mixed traffic flow, and the result is as follows:(20)nN12fNv2−fNΔvfNv−fNhfNhnN−1fAhnAfChnC2+nA12fAv2−fAΔvfAv−fAhfAhnA−1fNhnNfChnC2+nC12fCv2−fCΔvfCv−fChfChnC−1fNhnNfAhnA2<0.Equation (20) is reduced to the following formula:(21)F=1−p1−pAFN+1−ppAFA+pFC<0,FN=1/2fNv2−fNΔvfNv−fNhfNh2,FA=1/2fAv2−fAΔvfAv−fAhfAh2,FC=1/2fCv2−fCΔvfCv−fChfCh2,where FN, FA, and FC are the stability factors of the HDV-N, HDV-A, and CACC vehicles, respectively, and F is the value of the stability discriminant.An investigation of equation (11) shows that the stability of the mixed traffic flow is related to the proportion of aggressive driving, CACC vehicle market penetration rate, and equilibrium speed. Figure 3 shows the F function graph of a mixed traffic flow in the equilibrium state when pA=0.25 and under the conditions of different CACC vehicle market penetration rates and speeds.Figure 3
Stability analysis at differentp and equilibrium speed ve (when pA=0.25). (a) ve∈0,2.24. (b) ve∈2.24,2.71. (c) ve∈2.71,25.16. (d) ve∈25.16,33.3.
(a)(b)(c)(d)By analyzing the homogeneous traffic flow stability, the equilibrium speedve is divided into five parts for a separate analysis of the heterogeneous traffic flow, which are (0, 2.24), (2.24, 2.71), (2.71, 24.98), (24.98, 25.16), and (25.16, 33.3). However, in the actual stability analysis, we established four ranges to more intuitively observe the changes in stability, as shown in Figure 3.The horizontal axis in Figure3 depicts the equilibrium speed, and the vertical axis depicts the CACC vehicle market penetration rate. Furthermore, F is the value of the stability evaluation. The black curve marked as 0.00 indicates F=0, which is the critical line for mixed traffic flow stability. Table 13 summarizes the results from the final analysis of the data presented in Figure 3.Table 13
Unstable speed range in mixed traffic flow.
CACC market penetration rateSpeed range in the unstable state (m/s)Length of the speed range (m/s)(0, 0.445)(2.56, 25.16)22.60(0.445, 1)None0Figure3 and Table 13 show that when the CACC vehicle market penetration rate p exceeds 0.445, the mixed traffic flow is unconditionally stable (for pA=0.25). We define p=0.445 as the lowest limit of the CACC vehicle market penetration rate for ensuring absolute stability. When p is 0–0.445, an unstable situation occurs. Specifically, when p∈0,0.445 and v∈2.56,25.16, the mixed traffic flow will be in an unstable state, and the length of the unstable speed range is 22.60 m/s, which is the mean of HDV-N and HDV-A vehicles in the homogeneous traffic flow. Moreover, from the analysis of Figure 3(c), when the value of v is close to 13.8 m/s and p∈0.3,0.485, the value of F does not vary continuously from 0.634 to 1.27, which does not affect the analysis of the stability results.By analyzing the stability results of mixed traffic flow whenpA=0.25, the length of the unstable speed range influenced by HDV-A and HDV-N vehicles is the same when the CACC market penetration rate p is low. When p increases to 0.445, the unstable part will disappear; thus, increasing p will increase the stability of the mixed traffic flow. The effect of pA on stability is mentioned in Section 4.1.


## 3.2. Capacity

As is well known, AVs can improve the status of traffic flow by significantly enhancing traffic capacity. The effect of AVs on traffic capacity has received increasing attention. In traffic capacity research, the FDM is widely used as a classical model. However, with the development of vehicle platoon technology, AVs have smaller headways than HDVs, and thus the order of vehicles must be considered in the traffic capacity analysis. In this section, the fundamental diagram model (FDM) will be introduced first. Second, the theoretical capacity considering different vehicle orders in a mixed platoon based on a FDM will be discussed.


### 3.2.1. FDM

The FDM can be used to calculate the capacity based on the assumption that when the traffic is in equilibrium, the flow volume is numerically equal to the capacity.When the traffic flow is in equilibrium, all of the vehicle speeds are equal to the equilibrium speedve. The difference in speed (between vehicles) is 0 m/s, and the acceleration for all vehicles is 0 m/s2 [35]. In other words, equation (22) is satisfied:(22)v=ve,Δv=0ms,v˙=0ms2.By replacing equation (22) with equation (18), we obtain the average space headway of HDV-N and HDV-A vehicle equilibrium traffic flows using the following equation:(23)hA,N=sA,N+veTA,N1−ve/v04+l.For CACC vehicles, by replacing equation (22) with equation (14), we obtain the average CACC vehicle space headway in the equilibrium state using the following equation:(24)hC=vetc+l+s0.Based on the meaning of traffic flow density, the density in mixed traffic flow is calculated using the following equation:(25)k=11−p1−pAhN+1−ppAhA+phC.Subsequently, the FDM of the mixed traffic flow can be obtained by substituting equations (23) and (24) into equation (25) and combining the volume-density-speed relationship of the basic traffic flow, as presented in equation (31).(26)k=11−p1−pAsN+TNve/1−ve/v04+l+1−ppAsA+TAve/1−ve/v04+l+ptcve+s0+lCACC,q=kve.The parameters in Tables10 and 11 are input in equation (26), and the capacity of mixed traffic flow at various p values is calculated, as illustrated in Figure 4.Figure 4
Capacity analysis of mixed traffic flow at differentp values (when pA=0.25).By analyzing the data presented in Figure4, the equilibrium speed ve at different p values can be obtained, as shown in Table 14.Table 14
The FDM of the theoretical capacity.
CACC market penetration rateCapacity (veh/h)ve (m/s)0197418.90.1208119.30.2219319.80.3231120.30.4245520.80.5260821.40.6277422.20.7301823.20.8319224.60.9366326.71.0444333.3Figure4 and Table 14 show that as the CACC vehicle market penetration rate p increases, the capacity of mixed traffic also increases, consistent with previous studies [1, 22]. Moreover, with the increase in the CACC vehicle market penetration rate p, the equilibrium speed also increases. Further analysis of the data presented in Table 14 shows that the increasing proportion of capacity increases when p increases from 0 to 1, the mixed traffic flow becomes the CACC vehicle homogeneous traffic flow, and the capacity is approximately 2.23 times higher than that of HDVs alone.


### 3.2.2. Theoretical Capacity Based on the FDM considering Different Vehicle Orders

For a mixed traffic flow of AVs and HDVs, the headway will be directly affected by the orders of vehicles [36], while the FDM does not consider the interaction between vehicles to calculate the headway, assigning each type of vehicle the same value. Therefore, this research examines the effect of headway changes on traffic capacity from the perspective of studying the relationship between vehicles.The density of traffic flow is calculated using a basic definition, which is calculated using the following equation:(27)k=NL,where N represents the number of vehicles on the road and L represents the total length of the road occupied by a vehicle. Equation (27) is deformed to equation (28) to accurately study the density of the mixed traffic flow:(28)k=nL0,⟶k=nn.l+n−1h¯ve˜p+ve˜pT1ε,L0=n.l+n−1h¯ve˜p+ve˜pT2ε,where n represents the maximum number of the vehicles, and n=16 in this article. The number is identified according to previous studies [12]; L0 represents the road length occupied by n vehicles; l has the same meaning as described above; h¯ represents the average headway; ve˜p is the equilibrium speed at different CACC vehicle market penetration rates p, whose values are shown in Table 14 (when pA  = 0.25); T1 is the interplatoon time gap (s), and T1=2.0 s in this article. ε is the number of the platoon.The first task is to study the average headwayh¯ in different vehicle orders. When AVs and HDVs are randomly mixed in the stream, this is not an insignificant task because the average headway is affected by the HDV-N, HDV-A, and CACC vehicle headways. Ramezani et al. [37, 38] proposed a method for calculating the average headway between different types of vehicles, where hnv−nv is the headway between NVs; hav−av is the headway between AVs; hav−nv is the headway between an NV and an AV when an NV follows an AV; hhv−nv is the headway between an AV and an NV when an AV follows an NV; and hnv−nv, hav−nv, hhv−nv, and hav−av satisfy equations (29) and (30).(29)hav−nv=hnv−nv,(30)hhv−av=hav−av+hnv−nv−hav−av3.In this case, thehnv−nv and hav−av depend on the safe headway. Since three types of vehicles are studied in this article, nine combinations of headways between AVs and HDVs exist.Based on the aforementioned analysis, we extend the calculation formula originally applicable to two types of vehicles in equations (29) and (30) to three types of vehicles. Specifically, the method for calculating the nine combinations is as follows:(31)hC−N=hN−N,(32)hC−A=hA−A,(33)hA−N=hN−N,(34)hN−C=hC−C+hN−N−hC−C3,(35)hA−C=hC−C+hA−A−hC−C3,(36)hN−A=hA−A+hN−N−hA−A3,where the hN−N, hA−A, and hC−C depend on the safe headway.The results calculated using equations (31)–(36) are shown in Table 15.Table 15
Nine combinations of headways in a mixed traffic flow.
No.NomenclatureLeaderFollowerValue (s)1hC−CCACCCACC0.62hC−NCACCHDV-N1.53hC−ACACCHDV-A1.04hN−NHDV-NHDV-N1.55hN−CHDV-NCACC0.96hN−AHDV-NHDV-A1.47hA−AHDV-AHDV-A1.38hA−CHDV-ACACC0.89hA−NHDV-AHDV-N1.5The next task is to determine how to use the nine combinations of headways to calculateh¯. Considering all possible vehicle order combinations in a mixed traffic flow, some arrangements are considered on one-lane roads. Three scenarios will include all possible vehicle orders to explore the characteristics of the equilibrium flow [39]. (i) The first is a general scenario where HDV-N, HDV-A, and CACC vehicles are mixed randomly. (ii) The second is the best scenario where the three types of vehicles have their own platoons without interacting with other types of vehicles, thereby ensuring the lowest average headway. (iii) The third is the worst scenario considering the interaction between vehicles shown in Table 15, where a vehicle with a large headway follows a vehicle with a small headway, leading to the largest average headway. The complexity of this scenario is that the CACC vehicle market penetration p will directly affect the vehicle order.The average headway of the general scenario is between the best and the worst cases, and its calculation is very difficult; thus, this scenario is not considered in this study. Figure5 shows the two situations. Considering the importance of p to the headway analysis, the approaches described below will be used to investigate the effect of p on the headway in the two scenarios.Figure 5
Best and worst scenarios for average headway at differentp values.(1)p=0. In this case, when ε=2 in equation (28), the average headway of the best scenario is calculated using the following equation:(37)h¯=nA−1hA−A+nN−1hN−Nn−1.Since no AV is present, we arrange the two types of HDV cars sequentially to ensure a minimum headway. Similarly, using an HDV-A following an HDV-N as an example, the maximum headway is obtained, which will be the worst scenario. The average headway is calculated using the following equation:(38)h¯=nAhN−A+nA−1hA−N+nN−nA−1hN−Nn−1.(2)0<p<1. The best scenario occurs when the three types of vehicles have their own platoons, and the average headway is calculated using equation (39); in this case, ε=3 in equation (33).(39)h¯=nC−1hC−C+nA−1hA−A+nN−1hN−Nn−1.The worst scenario is relatively complex compared with the best scenario, and the vehicle order depends on the numbers of the three types of vehicles. According to previous research results [33], the proportion of aggressive driving will not exceed 50%; therefore, this article only considers pA less than 50%. This presumption plays a primary role in modeling the average headway in the worst scenario.Case 1.
0≤p<0.5−pA/1−pA
When the value ofp is small, the CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon one. Because the number of CACC vehicles is low, the number of remaining HDV-N vehicles is greater than the number of HDV-A vehicles, and we can arrange the HDV-A and HDV-N vehicles in turn where each HDV-N is a follower. This combination forms platoon two; the other HDV-Ns become platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because three platoons are present.Table 16
Calculation ofh¯ when p∈0,1 in the worst scenario.
Case IDThe value ofpεCalculation formula ofh¯10≤p<0.5−pA/1−pA3nChC−N+nC−1hN−C+nAhA−N+nA−1hN−A+nN−nC−nA−1hN−N/n−120.5−pA/1−pA≤p<1−pA/2−pA3nChC−N+nC−1hN−C+nN−nChA−N+nN−nC−1hN−A+nA−nN−nC−1hA−A/n−131−pA/2−pA≤p<0.53nNhC−N+nN−1hN−C+nC−nNhC−A+nC−nN−1hA−C+nA−nC−nN−1hA−A/n−14p=0.52nNhC−N+nN−1hN−C+nAhC−A+nA−1hA−C/n−150.5<p<13nNhC−N+nN−1hN−C+nAhC−A+nA−1hA−C+nC−nN−nA−1hC−C/n−1Case 2.
0.5−pA/1−pA≤p<1−pA/2−pA
Although the value ofp is greater than that of Case 1, the number of HDV-Ns is larger than the number of CACC vehicles, the CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon one. Because the number of CACC vehicles is larger than that in Case 1, the number of remaining HDV-A vehicles is greater than the number of HDV-N vehicles, which is the main difference from Case 1. The HDV-A and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon two; the remaining HDV-A becomes platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because there are three platoons.Case 3.
1−pA/2−pA≤p<0.5
Although the value ofp is less than 50%, the number of CACC vehicles exceeds the number of HDV-N vehicles. The CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower, forming platoon one. Due to the number of CACC vehicles, the remaining vehicles are also CACC vehicles, but the number of remaining vehicles is less than the number of HDV-A vehicles. CACC vehicles and HDV-A vehicles are arranged in turn, and each HDV-A is a follower, forming platoon two. The remaining HDV-A becomes platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because three platoons are present.Case 4.
p=0.5
When the number of AVs is consistent with the number of HDVs, CACC vehicles are combined with HDV-N and HDV-A, ensuring maximum headway. The CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon one. The number of remaining CACC vehicles is equal to that of HDV-A, and thus CACC vehicles and HDV-A are arranged in turn, and each HDV-A is a follower. This combination forms platoon two; in this case,ε=2 because two platoons are present.Case 5.
20.5<p<1
When the value ofp is greater than 50%, all HDVs are combined with CACC vehicles; HDV-N and HDV-A are followers, and this combination forms platoons one and two. The remaining CACC vehicles form platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because three platoons are present.
(3)p=1. When p=1, indicating homogeneous CACC vehicle traffic flow, h¯ has the same value as hC−C, and only one platoon is present; therefore, ε=1. In this case, the best scenario and the worst scenario are one scenario.


### 3.2.3. Comparison of the Models and Verification of the Simulation

The capacity for different vehicle orders is calculated using the method proposed in Section3.2.2, and the results are shown in Table 17.Table 17
Capacity with different vehicle orders.
CACC vehicle market penetration rateCapacity (veh/h)Mean−FDM/Mean×100 (%)Best headwayWorst headwayMean01965192619461.460.12107207920930.570.22211215221820.530.32324223122781.470.42440218523136.160.52591232124566.190.62753255226534.580.72930276128466.060.83141301330773.740.93423332833768.521.04203420342035.71Whenp<0.4, the calculated capacity value of the FDM is less than that of the best scenario and slightly greater than that of the worst scenario. When p<0.4, the calculated capacity value is greater than the best headway. The mean capacity of the best scenario and the worst scenario are compared with the FDM, showing that the difference between the two models is not large and that the largest Mean−FDM/Mean is only 8.52%.In addition to the numerical comparison, the simulation is also verified. In this study, SUMO is selected as the simulation platform, which deeply describes the interaction between AVs and HDVs.This study uses a typical expressway case as the simulation scenario to ensure that the simulation results are close to the actual value. A detector is installed 5 km downstream of the ramp to detect the number of passing vehicles, with an upstream ramp speed limit of 120 km/h, a speed limit section of 80 km/h, and a weaving section of 50 km/h, as illustrated in Figure6. The maximum flow is detected by randomly generating vehicles without limiting the upstream traffic demand.Figure 6
Scenario of the capacity SUMO.The SUMO result shows that whenp<0.4, the simulation model is evenly distributed near the three curves, and when p<0.4, it is basically between the best scenario and the worst scenario, as shown in Figure 7.Figure 7
Comparison of capacity calculation models for different values ofp.As shown in Figure7, the differences in the results between the theoretical analysis and SUMO cannot be ignored, and the detailed analysis is described below.(a)
The SUMO platform is an open source simulation platform that deeply describes the interaction between AVs and HDVs. We can set or modify most models and corresponding model parameters involved in vehicle operation, but some built-in models cannot be accessed and adjusted, which may lead to differences between the simulation and the theory.(b)
When we conduct the simulation, we use some common random algorithm to generate traffic flow. Because a consideration of all random situations is impossible, our simulation does not completely reproduce the actual traffic flow situation. On the other hand, considering the randomness of traffic flow in theoretical calculations is insufficient. Therefore, this limitation may also explain the difference between the theoretical results and the simulation results.(c)
For a specific traffic capacity calculation model, the FDM is too idealistic and does not consider the relationship between vehicles, and thus the value for capacity is relatively large. Based on the FDM, the calculation method proposed in this study further considers the relationship between vehicles, and it is close to the simulation result.By analyzing the calculation formulas for the FDM and the proposed model,T1 (interplatoon time gap) shows the largest difference between the two models. T1 has little effect on the calculation results at a low CACC vehicle market penetration p, but with increasing p, the capacity results calculated with the FDM without T1 are lower than the best case. Meanwhile, the value of T1 in this research is 2 s, which covers most scenarios, but it may be slightly higher for scenarios with a high CACC vehicle market penetration p (the value of T1 will be studied in the future). Thus, the capacity calculation result is higher than the best scenario value after p<0.4. However, the FDM is too ideal to truly describe the relationship between vehicles. Therefore, after p exceeds 0.4, the SUMO values are near the blue and purple curves. Using this analysis of a mixed traffic flow, we verified that the proposed model accurately calculates the capacity.


## 3.2.1. FDM

The FDM can be used to calculate the capacity based on the assumption that when the traffic is in equilibrium, the flow volume is numerically equal to the capacity.When the traffic flow is in equilibrium, all of the vehicle speeds are equal to the equilibrium speedve. The difference in speed (between vehicles) is 0 m/s, and the acceleration for all vehicles is 0 m/s2 [35]. In other words, equation (22) is satisfied:(22)v=ve,Δv=0ms,v˙=0ms2.By replacing equation (22) with equation (18), we obtain the average space headway of HDV-N and HDV-A vehicle equilibrium traffic flows using the following equation:(23)hA,N=sA,N+veTA,N1−ve/v04+l.For CACC vehicles, by replacing equation (22) with equation (14), we obtain the average CACC vehicle space headway in the equilibrium state using the following equation:(24)hC=vetc+l+s0.Based on the meaning of traffic flow density, the density in mixed traffic flow is calculated using the following equation:(25)k=11−p1−pAhN+1−ppAhA+phC.Subsequently, the FDM of the mixed traffic flow can be obtained by substituting equations (23) and (24) into equation (25) and combining the volume-density-speed relationship of the basic traffic flow, as presented in equation (31).(26)k=11−p1−pAsN+TNve/1−ve/v04+l+1−ppAsA+TAve/1−ve/v04+l+ptcve+s0+lCACC,q=kve.The parameters in Tables10 and 11 are input in equation (26), and the capacity of mixed traffic flow at various p values is calculated, as illustrated in Figure 4.Figure 4
Capacity analysis of mixed traffic flow at differentp values (when pA=0.25).By analyzing the data presented in Figure4, the equilibrium speed ve at different p values can be obtained, as shown in Table 14.Table 14
The FDM of the theoretical capacity.
CACC market penetration rateCapacity (veh/h)ve (m/s)0197418.90.1208119.30.2219319.80.3231120.30.4245520.80.5260821.40.6277422.20.7301823.20.8319224.60.9366326.71.0444333.3Figure4 and Table 14 show that as the CACC vehicle market penetration rate p increases, the capacity of mixed traffic also increases, consistent with previous studies [1, 22]. Moreover, with the increase in the CACC vehicle market penetration rate p, the equilibrium speed also increases. Further analysis of the data presented in Table 14 shows that the increasing proportion of capacity increases when p increases from 0 to 1, the mixed traffic flow becomes the CACC vehicle homogeneous traffic flow, and the capacity is approximately 2.23 times higher than that of HDVs alone.


## 3.2.2. Theoretical Capacity Based on the FDM considering Different Vehicle Orders

For a mixed traffic flow of AVs and HDVs, the headway will be directly affected by the orders of vehicles [36], while the FDM does not consider the interaction between vehicles to calculate the headway, assigning each type of vehicle the same value. Therefore, this research examines the effect of headway changes on traffic capacity from the perspective of studying the relationship between vehicles.The density of traffic flow is calculated using a basic definition, which is calculated using the following equation:(27)k=NL,where N represents the number of vehicles on the road and L represents the total length of the road occupied by a vehicle. Equation (27) is deformed to equation (28) to accurately study the density of the mixed traffic flow:(28)k=nL0,⟶k=nn.l+n−1h¯ve˜p+ve˜pT1ε,L0=n.l+n−1h¯ve˜p+ve˜pT2ε,where n represents the maximum number of the vehicles, and n=16 in this article. The number is identified according to previous studies [12]; L0 represents the road length occupied by n vehicles; l has the same meaning as described above; h¯ represents the average headway; ve˜p is the equilibrium speed at different CACC vehicle market penetration rates p, whose values are shown in Table 14 (when pA  = 0.25); T1 is the interplatoon time gap (s), and T1=2.0 s in this article. ε is the number of the platoon.The first task is to study the average headwayh¯ in different vehicle orders. When AVs and HDVs are randomly mixed in the stream, this is not an insignificant task because the average headway is affected by the HDV-N, HDV-A, and CACC vehicle headways. Ramezani et al. [37, 38] proposed a method for calculating the average headway between different types of vehicles, where hnv−nv is the headway between NVs; hav−av is the headway between AVs; hav−nv is the headway between an NV and an AV when an NV follows an AV; hhv−nv is the headway between an AV and an NV when an AV follows an NV; and hnv−nv, hav−nv, hhv−nv, and hav−av satisfy equations (29) and (30).(29)hav−nv=hnv−nv,(30)hhv−av=hav−av+hnv−nv−hav−av3.In this case, thehnv−nv and hav−av depend on the safe headway. Since three types of vehicles are studied in this article, nine combinations of headways between AVs and HDVs exist.Based on the aforementioned analysis, we extend the calculation formula originally applicable to two types of vehicles in equations (29) and (30) to three types of vehicles. Specifically, the method for calculating the nine combinations is as follows:(31)hC−N=hN−N,(32)hC−A=hA−A,(33)hA−N=hN−N,(34)hN−C=hC−C+hN−N−hC−C3,(35)hA−C=hC−C+hA−A−hC−C3,(36)hN−A=hA−A+hN−N−hA−A3,where the hN−N, hA−A, and hC−C depend on the safe headway.The results calculated using equations (31)–(36) are shown in Table 15.Table 15
Nine combinations of headways in a mixed traffic flow.
No.NomenclatureLeaderFollowerValue (s)1hC−CCACCCACC0.62hC−NCACCHDV-N1.53hC−ACACCHDV-A1.04hN−NHDV-NHDV-N1.55hN−CHDV-NCACC0.96hN−AHDV-NHDV-A1.47hA−AHDV-AHDV-A1.38hA−CHDV-ACACC0.89hA−NHDV-AHDV-N1.5The next task is to determine how to use the nine combinations of headways to calculateh¯. Considering all possible vehicle order combinations in a mixed traffic flow, some arrangements are considered on one-lane roads. Three scenarios will include all possible vehicle orders to explore the characteristics of the equilibrium flow [39]. (i) The first is a general scenario where HDV-N, HDV-A, and CACC vehicles are mixed randomly. (ii) The second is the best scenario where the three types of vehicles have their own platoons without interacting with other types of vehicles, thereby ensuring the lowest average headway. (iii) The third is the worst scenario considering the interaction between vehicles shown in Table 15, where a vehicle with a large headway follows a vehicle with a small headway, leading to the largest average headway. The complexity of this scenario is that the CACC vehicle market penetration p will directly affect the vehicle order.The average headway of the general scenario is between the best and the worst cases, and its calculation is very difficult; thus, this scenario is not considered in this study. Figure5 shows the two situations. Considering the importance of p to the headway analysis, the approaches described below will be used to investigate the effect of p on the headway in the two scenarios.Figure 5
Best and worst scenarios for average headway at differentp values.(1)p=0. In this case, when ε=2 in equation (28), the average headway of the best scenario is calculated using the following equation:(37)h¯=nA−1hA−A+nN−1hN−Nn−1.Since no AV is present, we arrange the two types of HDV cars sequentially to ensure a minimum headway. Similarly, using an HDV-A following an HDV-N as an example, the maximum headway is obtained, which will be the worst scenario. The average headway is calculated using the following equation:(38)h¯=nAhN−A+nA−1hA−N+nN−nA−1hN−Nn−1.(2)0<p<1. The best scenario occurs when the three types of vehicles have their own platoons, and the average headway is calculated using equation (39); in this case, ε=3 in equation (33).(39)h¯=nC−1hC−C+nA−1hA−A+nN−1hN−Nn−1.The worst scenario is relatively complex compared with the best scenario, and the vehicle order depends on the numbers of the three types of vehicles. According to previous research results [33], the proportion of aggressive driving will not exceed 50%; therefore, this article only considers pA less than 50%. This presumption plays a primary role in modeling the average headway in the worst scenario.Case 1.
0≤p<0.5−pA/1−pA
When the value ofp is small, the CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon one. Because the number of CACC vehicles is low, the number of remaining HDV-N vehicles is greater than the number of HDV-A vehicles, and we can arrange the HDV-A and HDV-N vehicles in turn where each HDV-N is a follower. This combination forms platoon two; the other HDV-Ns become platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because three platoons are present.Table 16
Calculation ofh¯ when p∈0,1 in the worst scenario.
Case IDThe value ofpεCalculation formula ofh¯10≤p<0.5−pA/1−pA3nChC−N+nC−1hN−C+nAhA−N+nA−1hN−A+nN−nC−nA−1hN−N/n−120.5−pA/1−pA≤p<1−pA/2−pA3nChC−N+nC−1hN−C+nN−nChA−N+nN−nC−1hN−A+nA−nN−nC−1hA−A/n−131−pA/2−pA≤p<0.53nNhC−N+nN−1hN−C+nC−nNhC−A+nC−nN−1hA−C+nA−nC−nN−1hA−A/n−14p=0.52nNhC−N+nN−1hN−C+nAhC−A+nA−1hA−C/n−150.5<p<13nNhC−N+nN−1hN−C+nAhC−A+nA−1hA−C+nC−nN−nA−1hC−C/n−1Case 2.
0.5−pA/1−pA≤p<1−pA/2−pA
Although the value ofp is greater than that of Case 1, the number of HDV-Ns is larger than the number of CACC vehicles, the CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon one. Because the number of CACC vehicles is larger than that in Case 1, the number of remaining HDV-A vehicles is greater than the number of HDV-N vehicles, which is the main difference from Case 1. The HDV-A and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon two; the remaining HDV-A becomes platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because there are three platoons.Case 3.
1−pA/2−pA≤p<0.5
Although the value ofp is less than 50%, the number of CACC vehicles exceeds the number of HDV-N vehicles. The CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower, forming platoon one. Due to the number of CACC vehicles, the remaining vehicles are also CACC vehicles, but the number of remaining vehicles is less than the number of HDV-A vehicles. CACC vehicles and HDV-A vehicles are arranged in turn, and each HDV-A is a follower, forming platoon two. The remaining HDV-A becomes platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because three platoons are present.Case 4.
p=0.5
When the number of AVs is consistent with the number of HDVs, CACC vehicles are combined with HDV-N and HDV-A, ensuring maximum headway. The CACC vehicle and HDV-N are arranged in turn, and each HDV-N is a follower. This combination forms platoon one. The number of remaining CACC vehicles is equal to that of HDV-A, and thus CACC vehicles and HDV-A are arranged in turn, and each HDV-A is a follower. This combination forms platoon two; in this case,ε=2 because two platoons are present.Case 5.
20.5<p<1
When the value ofp is greater than 50%, all HDVs are combined with CACC vehicles; HDV-N and HDV-A are followers, and this combination forms platoons one and two. The remaining CACC vehicles form platoon three. The formula used to calculate h¯ is shown in Table 16; in this case, ε=3 because three platoons are present.
(3)p=1. When p=1, indicating homogeneous CACC vehicle traffic flow, h¯ has the same value as hC−C, and only one platoon is present; therefore, ε=1. In this case, the best scenario and the worst scenario are one scenario.


## 3.2.3. Comparison of the Models and Verification of the Simulation

The capacity for different vehicle orders is calculated using the method proposed in Section3.2.2, and the results are shown in Table 17.Table 17
Capacity with different vehicle orders.
CACC vehicle market penetration rateCapacity (veh/h)Mean−FDM/Mean×100 (%)Best headwayWorst headwayMean01965192619461.460.12107207920930.570.22211215221820.530.32324223122781.470.42440218523136.160.52591232124566.190.62753255226534.580.72930276128466.060.83141301330773.740.93423332833768.521.04203420342035.71Whenp<0.4, the calculated capacity value of the FDM is less than that of the best scenario and slightly greater than that of the worst scenario. When p<0.4, the calculated capacity value is greater than the best headway. The mean capacity of the best scenario and the worst scenario are compared with the FDM, showing that the difference between the two models is not large and that the largest Mean−FDM/Mean is only 8.52%.In addition to the numerical comparison, the simulation is also verified. In this study, SUMO is selected as the simulation platform, which deeply describes the interaction between AVs and HDVs.This study uses a typical expressway case as the simulation scenario to ensure that the simulation results are close to the actual value. A detector is installed 5 km downstream of the ramp to detect the number of passing vehicles, with an upstream ramp speed limit of 120 km/h, a speed limit section of 80 km/h, and a weaving section of 50 km/h, as illustrated in Figure6. The maximum flow is detected by randomly generating vehicles without limiting the upstream traffic demand.Figure 6
Scenario of the capacity SUMO.The SUMO result shows that whenp<0.4, the simulation model is evenly distributed near the three curves, and when p<0.4, it is basically between the best scenario and the worst scenario, as shown in Figure 7.Figure 7
Comparison of capacity calculation models for different values ofp.As shown in Figure7, the differences in the results between the theoretical analysis and SUMO cannot be ignored, and the detailed analysis is described below.(a)
The SUMO platform is an open source simulation platform that deeply describes the interaction between AVs and HDVs. We can set or modify most models and corresponding model parameters involved in vehicle operation, but some built-in models cannot be accessed and adjusted, which may lead to differences between the simulation and the theory.(b)
When we conduct the simulation, we use some common random algorithm to generate traffic flow. Because a consideration of all random situations is impossible, our simulation does not completely reproduce the actual traffic flow situation. On the other hand, considering the randomness of traffic flow in theoretical calculations is insufficient. Therefore, this limitation may also explain the difference between the theoretical results and the simulation results.(c)
For a specific traffic capacity calculation model, the FDM is too idealistic and does not consider the relationship between vehicles, and thus the value for capacity is relatively large. Based on the FDM, the calculation method proposed in this study further considers the relationship between vehicles, and it is close to the simulation result.By analyzing the calculation formulas for the FDM and the proposed model,T1 (interplatoon time gap) shows the largest difference between the two models. T1 has little effect on the calculation results at a low CACC vehicle market penetration p, but with increasing p, the capacity results calculated with the FDM without T1 are lower than the best case. Meanwhile, the value of T1 in this research is 2 s, which covers most scenarios, but it may be slightly higher for scenarios with a high CACC vehicle market penetration p (the value of T1 will be studied in the future). Thus, the capacity calculation result is higher than the best scenario value after p<0.4. However, the FDM is too ideal to truly describe the relationship between vehicles. Therefore, after p exceeds 0.4, the SUMO values are near the blue and purple curves. Using this analysis of a mixed traffic flow, we verified that the proposed model accurately calculates the capacity.


## 4. Sensitivity Analysis

With the further development of AV technology, the public’s acceptance of it will improve, and aggressive drivers will tend to moderate their driving behavior [40]. Therefore, the corresponding parameters of HDV-A will also change and become increasingly similar to those of HDV-N.Based on the analysis described above, we established three cases (Cases1, 2, and 3) with different HDV-A parameters, and Cases 1–3 gradually approach the HDV-N parameters, as shown in Table 18. As the proportion of aggressive driving may also change, three values of pA, 0.25, 0.35, and 0.45, are set in the sensitivity analysis, and the three proportional values are combined with Cases 1–3 to form nine combinations.Table 18
HDV-A parameters in the different cases.
CaseTA (s)aA (m/s2)bA (m/s2)hA−AshA−CshC−AshN−As11.31.8−3.11.30.81.31.421.41.7−2.71.40.851.41.531.51.6−2.21.50.91.51.53


### 4.1. Sensitivity Analysis of Stability

Using the parameters listed in Table18, different values of pA, and the previous formulas, the stability range of mixed traffic flow with different combinations is calculated. In particular, the combination of pA  = 0.25 and Case 1 has been analyzed earlier, and thus it will not be repeated here. The result of the sensitivity analysis for stability is shown in Figure 8.Figure 8
Stability comparison of mixed traffic flow under different conditions. (a)pA=0.25. (b) pA=0.35. (c) pA=0.45.
(a)(b)(c)Figure8 illustrates that when pA=0.25, the unstable speed range gradually decreases from 2.28 and 25.16 in Case 1 to 2.92 and 25.03 in Case 3, and the lowest limit of the CACC vehicle market penetration rate for ensuring absolute stability gradually decreases from 0.445 in Case 1 to 0.408 in Case 3. When pA=0.35, the unstable speed range gradually decreases from 2.58 and 25.16 in Case 1 to 3.02 and 24.99 in Case 3, and the lowest limit of the CACC vehicle market penetration rate for ensuring absolute stability gradually decreases from 0.440 in Case 1 to 0.392 in Case 3. When pA=0.45, the unstable speed range gradually decreases from 2.58 and 25.16 in Case 1 to 3.23 and 25.16 in Case 3, and the lowest limit of the CACC vehicle market penetration rate for ensuring absolute stability is 0.490, which does not change as the case changes. From the perspective of pA, the unstable part decreases for any pA from Case 1 to Case 3. From the perspective of the case, for any case, the unstable part of pA=0.35 is less than that of pA=0.45 and greater than that of pA=0.25.By analyzing the data presented in Figure8, the following conclusions are obtained:(a)
Using the same aggressive driving parameters, the effect of the increase in the aggressive driving proportion on the instability part decreases first and then increases, and the critical point is near 1/3 (35%).(b)
Using the same proportion of aggressive driving, the stability gradually improves as the aggressive driving parameters become closer to the HDV-N parameters.(c)
When the CACC vehicle market penetration rate exceeds 0.490, mixed traffic flow remains stable for allpA and cases.


### 4.2. Sensitivity Analysis of Capacity

Using the different cases listed in Table18 and different values of pA, we calculate the capacity of mixed traffic flow for different combinations. The results for the analysis of the capacity in different combinations are shown in Table 19.Table 19
Mean capacity for different HDV-A parameters (considering vehicle order).
ppA0.250.350.45Case1Case2Case3Case1Case2Case3Case1Case2Case301946191318721986193218801997196718830.12093203019692102204119782171214220880.22182211520582188212520642257222921800.32278220621542290221921622359233222850.42313230322502392232122592460244423900.52456240923542497242923582565253324990.62653260625572688262725602757271326180.72846281027672882282827712949289427960.83077305830253112307230273175315231250.93376336333453397337233473444340833771420342034203420342034203420342034203As shown in Table19, when pA=0.25, the capacity gradually decreases from Case 1 to Case 3; similarly, when pA=0.35 and 0.45, the same trend is present, as shown in Figures 9(a)–9(c). For Case 1, the capacity gradually increases from pA=0.25 to pA=0.45; similarly, for Cases 2 and 3, a similar trend is observed, as shown in Figures 9(d)–9(f). By comparing the capacity values for the same case, the growth ratio from Case 3 to Case 2 is greater than that from Case 2 to Case 1. Additionally, for a fixed pA or case, an increase in the CACC vehicle market penetration rate will directly lead to an increase in traffic capacity.Figure 9
Analysis of the capacity of mixed traffic flow for differentpA values and cases. (a) pA=0.25. (b) pA=0.35. (c) pA=0.45. (d) Case 1. (e) Case 2. (f) Case 3.
(a)(b)(c)(d)(e)(f)By analyzing the data presented in Table19 and Figure 9, the following conclusions can be obtained:(a)
Using the same aggressive driving parameters, increasing the proportion of aggressive driving will increase the capacity in a mixed traffic flow.(b)
Using the same proportion of aggressive driving, when the aggressive driving parameters approach the HDV-N parameters, the capacity gradually decreases in a mixed traffic flow.


## 4.1. Sensitivity Analysis of Stability

Using the parameters listed in Table18, different values of pA, and the previous formulas, the stability range of mixed traffic flow with different combinations is calculated. In particular, the combination of pA  = 0.25 and Case 1 has been analyzed earlier, and thus it will not be repeated here. The result of the sensitivity analysis for stability is shown in Figure 8.Figure 8
Stability comparison of mixed traffic flow under different conditions. (a)pA=0.25. (b) pA=0.35. (c) pA=0.45.
(a)(b)(c)Figure8 illustrates that when pA=0.25, the unstable speed range gradually decreases from 2.28 and 25.16 in Case 1 to 2.92 and 25.03 in Case 3, and the lowest limit of the CACC vehicle market penetration rate for ensuring absolute stability gradually decreases from 0.445 in Case 1 to 0.408 in Case 3. When pA=0.35, the unstable speed range gradually decreases from 2.58 and 25.16 in Case 1 to 3.02 and 24.99 in Case 3, and the lowest limit of the CACC vehicle market penetration rate for ensuring absolute stability gradually decreases from 0.440 in Case 1 to 0.392 in Case 3. When pA=0.45, the unstable speed range gradually decreases from 2.58 and 25.16 in Case 1 to 3.23 and 25.16 in Case 3, and the lowest limit of the CACC vehicle market penetration rate for ensuring absolute stability is 0.490, which does not change as the case changes. From the perspective of pA, the unstable part decreases for any pA from Case 1 to Case 3. From the perspective of the case, for any case, the unstable part of pA=0.35 is less than that of pA=0.45 and greater than that of pA=0.25.By analyzing the data presented in Figure8, the following conclusions are obtained:(a)
Using the same aggressive driving parameters, the effect of the increase in the aggressive driving proportion on the instability part decreases first and then increases, and the critical point is near 1/3 (35%).(b)
Using the same proportion of aggressive driving, the stability gradually improves as the aggressive driving parameters become closer to the HDV-N parameters.(c)
When the CACC vehicle market penetration rate exceeds 0.490, mixed traffic flow remains stable for allpA and cases.


## 4.2. Sensitivity Analysis of Capacity

Using the different cases listed in Table18 and different values of pA, we calculate the capacity of mixed traffic flow for different combinations. The results for the analysis of the capacity in different combinations are shown in Table 19.Table 19
Mean capacity for different HDV-A parameters (considering vehicle order).
ppA0.250.350.45Case1Case2Case3Case1Case2Case3Case1Case2Case301946191318721986193218801997196718830.12093203019692102204119782171214220880.22182211520582188212520642257222921800.32278220621542290221921622359233222850.42313230322502392232122592460244423900.52456240923542497242923582565253324990.62653260625572688262725602757271326180.72846281027672882282827712949289427960.83077305830253112307230273175315231250.93376336333453397337233473444340833771420342034203420342034203420342034203As shown in Table19, when pA=0.25, the capacity gradually decreases from Case 1 to Case 3; similarly, when pA=0.35 and 0.45, the same trend is present, as shown in Figures 9(a)–9(c). For Case 1, the capacity gradually increases from pA=0.25 to pA=0.45; similarly, for Cases 2 and 3, a similar trend is observed, as shown in Figures 9(d)–9(f). By comparing the capacity values for the same case, the growth ratio from Case 3 to Case 2 is greater than that from Case 2 to Case 1. Additionally, for a fixed pA or case, an increase in the CACC vehicle market penetration rate will directly lead to an increase in traffic capacity.Figure 9
Analysis of the capacity of mixed traffic flow for differentpA values and cases. (a) pA=0.25. (b) pA=0.35. (c) pA=0.45. (d) Case 1. (e) Case 2. (f) Case 3.
(a)(b)(c)(d)(e)(f)By analyzing the data presented in Table19 and Figure 9, the following conclusions can be obtained:(a)
Using the same aggressive driving parameters, increasing the proportion of aggressive driving will increase the capacity in a mixed traffic flow.(b)
Using the same proportion of aggressive driving, when the aggressive driving parameters approach the HDV-N parameters, the capacity gradually decreases in a mixed traffic flow.


## 5. Discussion and Conclusions

This study considers HDV aggressive driving, with HDVs divided into aggressive driving and normal driving vehicles, to assess the stability and capacity of a mixed traffic flow with AVs and HDVs. The focus of this research is to explore the effects of aggressive drivers on the stability and capacity of a mixed flow. According to numerical and sensitivity analyses, the following conclusions are drawn:(a)
A method considering vehicle order based on an FDM is proposed here to calculate the capacity of mixed traffic flow through simulation verification. The maximum capacity is obtained by accurately modeling the average headway for different vehicle orders.(b)
The capacity of mixed traffic increases with the increase in the proportion of aggressive driving, but whenpA is less than 1/3 (or 35%), the increase in the proportion of aggressive driving reduces the unstable part; however, when pA is greater than 1/3 (or 35%), it will increase the unstable part.(c)
When aggressive driving parameters are increasingly similar to normal driving, the capacity will be relatively reduced and the stability will be relatively improved.(d)
Consistent with the research of other scholars, the increase in the CACC vehicle market penetration rate will increase the capacity of mixed traffic flow and reduce the unstable part [41–44].The results of this study indicate that the effect of aggressive driving on mixed traffic flow cannot be ignored. Specifically, with an increasing CACC vehicle market penetration rate, the aggressive potential of HDVs will be increased; thus, the proportion of aggressive driving will be further increased. However, considering the popularity of AVs, the public’s acceptance of AVs will be further improved; thus, aggressive driving parameters may not be as problematic at that point as they were in the early stages of popularization. These situations are considered in this article. Notably, the proportion of aggressive driving in HDVs is approximately 1/3, which is a critical point. Once the proportion of aggressive driving is below the critical value, the traffic capacity will be temporarily increased and the unstable part will be reduced. However, when it exceeds the critical value, the unstable part will increase rapidly. Aggressive driving can increase the capacity of mixed traffic flow, but AVs are the main factor contributing to increased traffic; thus, in the long term, reducing the proportion of aggressive driving or changing the level of aggressive driving is very important to improving stability.This article represents an original attempt from the perspective of assessing the effect of HDV aggressive driving on stability and traffic capacity. First, traffic flow stability and safety are directly related; thus, the effect of aggressive driving on safety is the focus of our next study. Second, some scholars have verified that the traffic capacity can be improved through right-of-way reallocation [45–49]. Thus, improving the traffic capacity through right-of-way reallocation without increasing unstable parts while considering aggressive driving will be an important focus of future research. Finally, when the proportion of aggressive driving in mixed traffic flow increases, improving the entire traffic flow state through the control of AVs will be the focus of continued future research.



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*Source: 2578690-2023-02-10.xml*