# Use of an Improved Car-Following Model to Explain the Influence of Traffic Composition on Saturation Headway at Signalized Intersections

Authors: Yi Wang; Jian Rong; Wei Luo; Yacong Gao
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

Previous studies mainly used statistical methods to analyze the impact of traffic composition on saturation flow rate from the mesolevel, and there is insufficient research on how traffic composition affects driving behavior. Thus, the purpose of this paper is to establish a more accurate car-following model, establish the relationship between microbehavior and mesostatistical regularity, and explain the influence of vehicle composition on saturation headway. In this paper, an improved full velocity difference (FVD) model is proposed, which abstracts the driver characteristics of a heterogeneous flow into four scenarios: car-car, car-bus, bus-car, and bus-bus. The measured data are used to calibrate and verify the basic FVD and the improved FVD models. The performance of the improved model is significantly improved. The RMSE and RMSPE are reduced by 15.29% and 22.32%, respectively. Finally, through numerical simulation experiments, the variation of saturation headway with different proportions of buses is analyzed. The saturation headway increases with the increase of the proportion of heavy vehicles. Moreover, another important finding is that the saturation headway is not significantly influenced by the position of the buses but only by the proportion of the buses. The research results could provide theoretical support for the control and management of fleets composed of different vehicles at intersections.


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## Body

## 1. Introduction

In recent years, with the increasing number of vehicles on the roads, traffic delays and exhaust pollution have become serious problems faced by many cities around the world [1]. An intersection is a place where a variety of mixed traffic flows gather. The traffic capacity of intersections is less than 50% of the road section, and the delays at intersections account for 80% of the road network [2]. The intersections are extremely prone to congestion. Frequent acceleration, deceleration, and stopping at intersections are important factors causing delay and exhaust pollution [3]. How to improve traffic capacity and alleviate traffic congestion has become an important and difficult problem. The main cause of traffic congestion is the imbalance between traffic supply and demand [4]. There are two common methods to solve congestion. The first method is to improve the traffic capacity of the entrance road through fine design [5]. The second method is to develop a traffic signal optimization model to reduce delay. All methods are inseparable from the guidance of capacity theory. The Highway Capacity Manual (HCM) points out that capacity theory can help engineers make intelligent decisions on operation, design, and planning for different purposes [6]. Therefore, it is very important to accurately grasp the traffic capacity and influencing factors of the signalized intersection.Saturation headway is an important parameter for capacity estimation of signalized intersections, which is affected by a variety of factors. America Transportation Research Board listed the main influencing factors in the Highway Capacity Manual, including the vehicle composition, lane width, approach grade, existence of a parking lane and parking activity adjacent to lane group, blocking effect of local buses that stop within intersection area, area type, lane utilization, left‐turn vehicle presence in a lane group, right‐turn vehicle presence in a lane group, pedestrian for left‐turn groups, and pedestrian for right‐turn groups [7]. Among them, vehicle composition is the main influencing factor [8]. Therefore, many scholars have studied the relationship between vehicle composition and saturation headway and constructed the proportion adjustment coefficient of large vehicles [9, 10]. For example, Padmakumar et al. [9] found that HCM has a limited ability to solve heterogeneity. A new saturation flow model was developed based on a dynamic passenger car unit (PCU) using the regression method. Biswas et al. [10] studied the influence of multiple factors on saturation flow rate under mixed traffic conditions.In summary, scholars have obtained a series of meaningful research results on the relationship betweenf vehicle composition and saturation headway. However, in previous studies, the adjustment coefficient model neglected two factors. The first is whether the vehicle position will affect the headway, and the second is how the heavy vehicles in the queue affect the saturation headway. In other words, capacity can only be adjusted by changing the proportion of heavy vehicles. When the proportion of heavy vehicles cannot be changed, whether the capacity can be improved by adjusting the position of large vehicles in the queue or changing the driving characteristics needs further analysis and verification.By constructing a microscopic traffic flow model, the actual movement of each vehicle at the intersection can be well described, and then how the vehicle composition affects the saturation headway can be analyzed [11]. During the green light, it is typical following behavior for the vehicles in the queue to pass the stop line in turn. Because the entrance lanes are separated by solid lines, vehicles cannot overtake other vehicles in the process of dissipation, and vehicles must follow the preceding vehicles through the stop line. The speed and acceleration of the rear vehicle will be affected by the front vehicle. Therefore, the micro car-following model can describe the process of vehicle following and explain the relationship between influencing factors and saturation headways. The car-following model is an important method to study micro traffic flow, which describes the longitudinal interaction between front and rear vehicles [12]. The car-following model can be used to understand the characteristics of traffic flow and reveal the internal mechanisms of traffic phenomena. Vehicle state is usually described by a group of variables xn,vn,a, which refer to the spatial position, velocity, and acceleration of the vehicle, n, traveling on the road. The following state of the vehicle is shown in Figure 1. Based on the above considerations, the following model is used to further explore the relationship between vehicle composition and saturation headway in this paper. At present, the following models are mainly used in a continuous flow environment. These models mainly describe the following behavior of drivers, ignoring the influence of the external environment on drivers. Some studies have shown that drivers tend to drive more carefully when surrounded by heavy vehicles [13, 14]. How to consider the influence of vehicle composition on driving behavior to improve the existing following model is crucial for analyzing the influence of vehicle composition on saturation headway.Figure 1
The car-following model and its variables.Therefore, the purpose of this paper is to establish a more accurate car-following model, establish the relationship between microbehavior and mesostatistical regularity, and explain the influence of vehicle composition on saturation headway. The research results can further improve applicable scenarios of the car-following model, clarify the essence of the influence of vehicle composition on headway, and provide theoretical support for the control and management of traffic composed of different vehicles at intersections.


## 2. Data Collection

The data were collected from field videos. On July 24 and 25, 2018, during the afternoon rush hour (17:00–19:00), the authors took videos at the northbound entrance lane at the intersection of Andingmenwai Street and Waiguan Diagonal, the northbound entrance lane at the intersection of Anli Road and Huizhong Road, and the northbound entrance lane at the intersection of Anli Road and Datun Road.The survey objects were the following behavior of cars and normal bus in the straight lane of the entrance lane during green light periods. To facilitate the extraction of data, the camera was set up on a high-rise building near the intersection and the angle of the camera was adjusted to include part of the solid line of the entrance lane and clearly show the stop line. Meanwhile, a wheel rangefinder was used to measure the width of each lane in the filming area and the length of the solid line of the entrance lane, so that a coordinated system could be established in the process of data extraction and track data could be easily obtained. Figure2 shows the data acquisition diagram of the entrance lane at the signalized intersection.Figure 2
Setting of field video collections.The vehicle data used in this paper were extracted from real intersections in Beijing. In reality, multistrip bus lines pass through the intersection. More than 80% of these buses are ordinary buses, and few articulated buses appear. In other words, among the vehicles extracted in the green light period, there are almost no car-following scenarios of car-articulated bus, articulated bus-car, and articulated bus-articulated bus. In addition, during rush hour, heavy trucks are not allowed to pass inside Beijing’s Fourth Ring Road. So heavy trucks were hardly seen during extracting intersection data. There were only two green light periods in which heavy trucks existed, which are sanitation vehicles. Thus, during data processing, these green light periods’ data were deleted. The ordinary buses are mostly 12 m long and 5 m wide. Before extracting the trajectory data, the video segments of different bus models in the queue are removed, and only the video segments of cars followed by passenger cars and ordinary buses are retained. There are two considerations. (1) The 1st to 5 h vehicles in the queue generally have a start-up delay, which is not included in the formula when calculating the saturation headway. Therefore, when counting the trajectory data, the trajectory data of the 1st to 5 h vehicles in the queue in each period are not extracted and deleted. (2) In order to eliminate the interference of other types of heavy vehicles (sanitation trucks, hinged buses, etc.), only the periods of cars and normal buses in the queue are extracted. Other periodic videos are deleted. The integrity of the period is preserved. Then, the tracking fragments are imported into the image processing software to extract the trajectory data and thereby obtain the complete vehicle trajectory data (xn, vn, and an).When the green light comes on, the queueing vehicles in the straight lane of the entrance lane change from stationary to start and then quickly pass the stop line. This process is repeated in the behavior of the following vehicles. Ultimately, 168 segments of data of 3 signalized intersections were extracted. According to the types of leading vehicles and following vehicles, the segment data could be divided into the four following kinds: passenger car following passenger car (88 segments of data), passenger car following bus (25 segments of data), bus following passenger car (28 segments of data), and bus following bus (27 segments of data). From these data, the typical car-following process with measurement data is shown in Table1, where v1 is the speed of the leading vehicle, v2 is the speed of the following vehicle, s is the vehicle distance, and a is the acceleration of the following vehicle. The total length of time of the following process is 16.125 s, with 8 groups of data per second, and there are a total of 129 groups of data. Examples of acceleration and relative speed curves of the following vehicle in the following process are shown in Figure 3.Table 1
A typical car-following process with measurement data.
Time (s)v1 (m/s)v2 (m/s)s (m)a (m/s2)0.1250.200.024.210.460.2500.210.044.330.60.3750.230.074.470.730.5000.250.094.620.870.6250.270.124.800.990.7500.290.154.991.120.8750.310.185.211.231.0000.330.215.441.34…………………………7.6253.082.2622.240.22Figure 3
Example of acceleration and relative speed curves of following vehicle in the following process.


## 3. Methodology

### 3.1. Analysis of Basic Car-Following Model

#### 3.1.1. Selection of Basic Car-Following Model

In the past 70 years, the integration of scholars’ ideas in various fields has made great progress in research related to the car-following model. Car-following models can be roughly divided into theory-driven and data-driven types. The theory-driven models can be divided into two categories: those based on traffic engineering and those based on statistical physics. The car-following model based on traffic engineering describes driving behavior through mathematical formulas, which include the psychophysical GM (General Motor) model [15], the safe distance Gipps’ model [16], and the physiological-psychological Wiedemann 74 model [17]. The car-following model based on statistical physics describes the changes of vehicle velocity and displacement through differential equations, including the OV (Optimized Velocity) model [18], the IDM (Intelligent Driver Model) model [19], and the cellular automaton NaSch model [20]. The modeling idea of data-driven car-following models is not limited to theoretical assumptions and does not pursue mathematical derivation. The data-driven car-following model uses nonparametric methods to mine following vehicle trajectory data and establish car-following models with high prediction accuracy. The classical car-following models include the fuzzy logic model [21], the artificial neural network model [22], the case learning model [23], the support vector regression model [24], and the deep learning model [25].Since this paper focuses on the influence of vehicle composition on following behavior in order to further explain how vehicle composition affects the saturation flow rate on a mechanism level, the kind of model which is most suitable is one in which driving behavior is characterized by differential equations to show complex macroscopic traffic characteristics. This kind of model, which is based on statistical physics, can accurately describe the complex phenomena of actual traffic flow by introducing the variable factors that can describe the basic characteristics of actual traffic flow. Further analyzing these models, it is found that, by introducing the basic characteristics of actual traffic flow, the car-following model aimed at velocity optimization can accurately describe the actual traffic flow. There are three car-following models based on optimal velocity which are the OV (Optimized Velocity) model [18], the GF (Generalized Force) model [26], and the FVD (Full Velocity Difference) model [27]. Through comprehensive analysis, it is found that the OV and GF models are inadequate to a certain extent, especially in the intersection area, as some features cannot be accurately described. The IDM and CA (Cellular Automata) models have complex structures and many parameters and are difficult to calibrate. The FVD model not only has the advantages of simple structure, easy calibration, and high precision but also can describe the common traffic flow characteristics at the intersection. Moreover, by adding the influencing factors or improving the model structure, the traffic flow characteristics under the influence of different factors can be described. In this way, the influencing factors can be correlated with traffic flow characteristics, and the influence principle of the influencing factors on the traffic flow characteristics can be further analyzed. Therefore, the FVD model is selected as the basic model for improvement research.


#### 3.1.2. Parameter Calibration of Basic Car-Following Model

The core of parameter calibration of the car-following model is the least squares method. The specific steps are as follows:Step 1: Determine the equation structure of the car-following model, put in the track data of the leading vehicle, and use the initial value of the required calibration parameters in the car-following model. Then the acceleration data of the following vehicle from the following model can be simulated and, through derivation, the speed and position data can be simulated.Step 2: Compare the simulated acceleration data of the following vehicle with the measured data, and the error sum of squares index can be obtained, which is related to the selection of the model parameters.Step 3: Reselect the parameter value to be calibrated and repeat Step 1 and Step 2. A selected set of calibration parameters will be corresponded to an error sum of squares index.Step 4: The first three steps are classified as solving the optimization problem of evaluation indexes, and intelligent algorithms are used to solve them.The genetic algorithm has the advantages of strong robustness and unlikelihood of falling into a local optimum. Thus, the genetic algorithm is used to calibrate the relevant parameters in the full-speed difference model in this paper.(1) Determination of Error Index. MSE (Mean Square Errors), RMSE (Root Mean Square Errors), and RMSPE (Root Mean Square Percentage Errors) [28] are selected as the decision indexes. The specific calculation processes are shown in the three following formulas:(1)MSE=1N∑i=1Nobservedi−predictedi2,(2)RMSE=1N∑i=1Nobservedi−predictedi2,(3)RMSPE=∑i=1Nobservedi−predictedi2∑i=1Nobservedi2,where “observed” is the measured trajectory value, such as the measured position, velocity, and acceleration, “predicted” is trajectory value predicted by simulation, and N is observed sample size.(2) Genetic Algorithm Flow. The genetic algorithm (GA) is an evolutionary computing algorithm [29, 30], which is a part of artificial intelligence technology. John Holland [30] first invented and improved GA. In the process of species evolution, in order to better adapt to the environment and repopulate the Earth, good genes will be inherited, and the bad genes will be eliminated. After many generations of genetic changes, species retain the best genes in the present time. GA is an algorithm constructed by imitating this phenomenon in the biological world. This algorithm has been widely used in optimization and search to find the optimal solution.The GA flow chart is as shown in Figure4, which can be divided into 6 steps:Step 1: Set the genetic algorithm parameters. The population size is set in the range of 20–100. From 20 to 100, one parameter is taken every 10, and 9 parameters are taken in total. Similarly, the maximal iteration algebra is set as 5 parameters, the crossover probability is set as 6 parameters, the mutation probability is set as 5 parameters, and the terminative evolutionary algebra is set as 2 parameters. Each combination scheme is traversed through programming.Step 2: Randomly select a leading vehicle trajectory data, set all the parameters in the FVD model to 1, and generate the simulation data of the car-following from the model.Step 3: Based on the simulation data, the parameters of the car-following model are calibrated by the genetic algorithm. Continuously change the parameter combination scheme in the genetic algorithm and select the combination whose parameters of the calibrated car-following model are closest to 1 as the best combination. Then, the parameters of the genetic algorithm are selected.Step 4: Obtain all trajectory data to be calibrated, and set the initial parameter value to 1 (a = 1, λ = 1, and hc = 1). Then the trajectory data of the leading vehicle is taken into the FVD model. After numerical simulation, velocity, acceleration, and position of the subject vehicle are obtained.Step 5: Compare the simulated velocity, acceleration, and position of the subject vehicle with the measured data, and the MSE (Mean Square Error) is taken as the evaluation index to determine whether to select the initial parameter. If the error satisfies the termination condition in the genetic algorithm, the initial parameter is used. If not, proceed to step 6.Step 6: Reselect the parameter values to be calibrated, and cycle Step 4 and Step 5 to obtain multiple groups of parameters and corresponding evaluation indexes. The optimal set of parameters is taken as the final result of parameter calibration of the car-following model. So far, the process of calibrating parameters by the genetic algorithm is over.Figure 4
Genetic algorithm flow chart.The genetic algorithm toolbox in MATLAB is used to calculate the optimal solution. The toolbox can only find the minimum value of fitness function as the optimization goal. Thus, when inputting the fitness function, the mean absolute error of the whole sample is taken as the final fitness result of the function. The related parameters are as follows: fitness function is set as mean absolute error of simulated and measured accelerations; population size is set as 60; maximal iteration algebra is set as 500; crossover probability is set as 0.9; mutation probability is set as 0.2; convergence tolerance error is set as 1 × 10−6; terminative evolutionary algebra is set as 100.To verify the effectiveness of the proposed method in this paper, simulated data are used for testing. A leading vehicle trajectory is randomly selected, and all the parameters in the FVD model are set to 1. From the model, the simulation data of the following vehicle are generated. Based on the simulation data, the car-following model parameter calibration method based on the genetic algorithm is used for calibration. If the parameter results are close to 1, it is shown that the calibration method in this paper can find the optimal solution. In the end, the results show thata = 0.898, λ = 1.029, and hc = 1.050; therefore the calibration error is quite small and it can be considered that the calibration method in this paper can find the optimal solution of the model. Then, the same method is used to calibrate the measured data, and the final FVD model calibration results are as follows: α is 0.010; λ is 0.328; hc is 3.334.


#### 3.1.3. Validation of Basic Car-Following Model

With acceleration, velocity, and spacing as performance indexes, the corresponding judgment indexes are obtained. The model is verified from multidimensional dimensions. The specific results are shown in Figure5. It can be seen from Figure 5 that there is a large deviation between the measured acceleration value and the simulation value. On the one hand, RMSPE is sensitive when the acceleration is near 0. On the other hand, the driving characteristics of different vehicles vary, and the model ignores the differences of vehicle types, resulting in some errors.Figure 5
Performance of FVD model.Therefore, one following process is randomly selected to further analyze the acceleration, velocity, and distance with respect to the measured values and the simulation value. In Figure6, the curve represents the state change of following vehicles. Through a comparison between the simulation and the measured values, it is found that the acceleration and velocity change trend of the model simulation is still similar to the measured value, but the error is large and the difference of the following distance is obvious. It is indicated that the performance of the model is poor, and improvement is needed in subsequent studies.Figure 6
The curves of subject vehicle’s acceleration, speed, and distance during car-following.


### 3.2. Model Improvement

#### 3.2.1. Analysis of Traffic Flow Characteristics at Intersections

The verification of the FVD model shows that the model cannot completely and accurately describe the car-following characteristics of heterogeneous vehicles and that the accuracy of the FVD model needs to be improved. Therefore, it is necessary to further analyze the characteristics of departure flow at the intersection entrance lane to provide ideas for the improvement of the FVD model.The common characteristics of departure flow are shown in Figure7. Figure 7(a) shows the characteristics of departure flow. When the green light turns on, the vehicles behind the stop line start and pass the stop line in turn. The flow rate soon increases to a stable value, which is the saturated flow rate. As shown in Figure 7(b), through the statistics of the headway of each position of queueing vehicles passing through the stop line, it is found that the headway decreases gradually with the increase of position and maintains a relatively stable state. At this time, the corresponding headway is the saturation headway. This parameter is the key to signalized intersection control and capacity analysis. However, queueing vehicles are often composed of different models in reality, which makes the headway of these vehicles different. Figure 7(b) shows the headway distribution when the 7th vehicle in line in one period is a bus. After averaging the saturation headway, it can be considered that it increases in relation to the increased proportion of buses. Figure 7(b) shows a common following scenario in mixed traffic, which can be divided into 4 categories, namely, car-car, car-bus, bus-car, and bus-bus [31].Figure 7
The features of departure vehicles passing the stop line during car-following. (a) Diagram of departure flow at the intersection entrance lane. (b) A common following scenario in mixed traffic.
(a)(b)To further verify the difference of headway, the videos of the three measured intersections are further extracted. Only the headway under saturation condition is extracted, from the 4th to 6th vehicle to the end of the queue. The extracted headway is classified for statistics, and the probability distribution of headway is shown in Figure8. The rank-sum test is used to test four distributions. The data are substituted into SPSS statistical analysis software and the Kolmogorov-Smirnov test method is used for the statistics of the data. The test results are as follows: car-car and car-bus (Z = 5.582, Sig. = 0.000); car-car and bus-car (Z = 1.451, Sig. = 0.030); car-car and bus-bus (Z = 6.691, Sig. = 0.000); car-bus and bus-car (Z = 3.777, Sig. = 0.000); car-bus and bus-bus (Z = 1.460, Sig. = 0.028); and bus-car and bus-bus (Kolmogorov-Smirnov test statistics, Z = 4.614, Sig. = 0.000). The results show that when the confidence interval is 99%, the distributions of car-car and bus-car have a certain similarity; the distributions of car-bus and bus-bus also have a certain similarity; the other distributions are significantly different. This is mainly caused by two factors: one is the difference of vehicle length; the other is the difference in vehicle performance and driving behavior. Therefore, car-following models should be built separately for different scenarios.Figure 8
The distribution of headway in four car-following scenarios [30].


#### 3.2.2. Improvement of Basic Model

Vehicle performance and driving behavior are mainly represented by three parameters in the car-following model: the desired velocityVmax in the optimized velocity function; the driver’s sensitivity coefficient to vehicle distance, α; and the driver’s sensitivity coefficient to velocity difference λ. Differences in vehicle performance will lead to differences in the desired velocity. The measured speed distribution of cars and buses in a nonfollowing state is shown in Figure 9. There is a significant difference between the desired velocities of the two vehicle types (Kolmogorov-Smirnov test, Z = 3.590, Sig. = 0.00). The average desired velocity of cars is 11.28 m/s and that of buses is 9.67 m/s. The main reason for the difference in driving behavior is that drivers have different sensitivities to the velocity difference and distance difference of the vehicle in front of them in different scenarios. The original FVD model can be described in segments according to different scenarios.Figure 9
The distribution of car and bus desired velocity [30].The improved FVD model is shown in the following formula:(4)dvntdt=αc−cVΔxnt−vnt+λc−cΔvntifcar−carαc−bVΔxnt−vnt+λc−bΔvntifcar−busαb−cVΔxnt−vnt+λb−cΔvntifbus−carαb−bVΔxnt−vnt+λb−bΔvntifbus−bus,and, in formula (3), αc-c,λc-c is the sensitivity coefficient in a scenario of car following car, αc-b¸λc-b is the sensitivity coefficient in a scenario of car following bus, αb-c, λb-c is the sensitivity coefficient in a scenario of bus following car, αb-b, λb-b is the sensitivity coefficient in a scenario of bus following bus, V⋅ is velocity optimization function, Δxnt is the distance between the following vehicle and the leading vehicle, Δvnt is the relative velocity difference between the following vehicle and the leading vehicle, and vnt is the velocity of the following vehicle at time t.The optimized velocity function is shown in the following formula:(5)VΔxnt=Vcarmax2tanhΔxnt−hc+tanhhc,Vbusmax2tanhΔxnt−hc+tanhhc,whereVcarmax is the desired velocity of a car, Vbusmax is the desired velocity of a bus, and hc is the safe distance between vehicles.To further test the improved model, a group of following data are randomly selected in each scenario. The changed curves of the measured values, FVD model, and simulation values of the improved model are drawn, as shown in Figures10 to 13. In Figure 10(in the car-car scenario), the variation trends of acceleration, velocity, and distance between the FVD model and the improved model are basically consistent with the measured values; that is, there is little difference between them.Figure 10
The curves of a subject vehicle’s acceleration, speed, and distance in a car-car scenario.Figure 11
The curves of a subject vehicle’s acceleration, speed, and distance in a car-bus scenario.Figure 12
The curves of a subject vehicle’s acceleration, speed, and distance in a bus-car scenario.Figure 13
The curves of a subject vehicle’s acceleration, speed, and distance in a bus-bus scenario.In Figure11 (in the car-bus scenario), the simulation results of the FVD model have a large deviation from the measured values. In particular, the vehicle distance not only has great differences in values but also has inconsistent trends. The acceleration, speed, and vehicle distance curves of the improved model are basically consistent with the measured curves.In Figure12 (in the bus-car scenario), the simulation results of the FVD model are consistent with the measured values, but there is a certain deviation between the values. However, the simulation curve of the improved model is closer to the measured curve.In Figure13 (in the bus-bus scenario), the simulation results of the FVD model are different from the measured results. However, the simulated curve of the improved model is closer to the measured curve.To analyze the error of the whole sample, the RMSE and RMSPE of the improved model are calculated. It can be seen from Figure14 that the acceleration error decreases by 1.89% and 1.92%, respectively. This is because when the acceleration value approaches 0, a slight difference will lead to a sharp increase in statistical errors. In addition, the velocity error decreases by 4.24% and 15.29%, respectively. Moreover, the distance error decreases by 22.31% and 22.32%, respectively. To sum up, in a heterogeneous flow, the improved model gives a better performance and can better describe the following behavior of departure vehicles at the intersection entrance lane.Figure 14
Performance of proposed car-following model. Note: IM is improved model.


#### 3.2.3. Parameter Calibration and Validation of Improved Model

A genetic algorithm is still used to calibrate the improved model, and the calibration results are shown in Figure15. A bus driver is more sensitive to the distance between vehicles than a car driver, indicating that a bus driver will always change his velocity and maintain a certain safe distance from the leading vehicle. A bus driver is less sensitive to velocity difference than a car driver, indicating that a bus driver is more interested in maintaining his own velocity than a car driver. The safety distance between a car and a bus is the largest, indicating that the car driver is affected by the leading vehicle being a bus. To avoid frequent acceleration and deceleration, a large safety distance is maintained.Figure 15
Calibrated parameters of proposed car-following model in different scenarios.


## 3.1. Analysis of Basic Car-Following Model

### 3.1.1. Selection of Basic Car-Following Model

In the past 70 years, the integration of scholars’ ideas in various fields has made great progress in research related to the car-following model. Car-following models can be roughly divided into theory-driven and data-driven types. The theory-driven models can be divided into two categories: those based on traffic engineering and those based on statistical physics. The car-following model based on traffic engineering describes driving behavior through mathematical formulas, which include the psychophysical GM (General Motor) model [15], the safe distance Gipps’ model [16], and the physiological-psychological Wiedemann 74 model [17]. The car-following model based on statistical physics describes the changes of vehicle velocity and displacement through differential equations, including the OV (Optimized Velocity) model [18], the IDM (Intelligent Driver Model) model [19], and the cellular automaton NaSch model [20]. The modeling idea of data-driven car-following models is not limited to theoretical assumptions and does not pursue mathematical derivation. The data-driven car-following model uses nonparametric methods to mine following vehicle trajectory data and establish car-following models with high prediction accuracy. The classical car-following models include the fuzzy logic model [21], the artificial neural network model [22], the case learning model [23], the support vector regression model [24], and the deep learning model [25].Since this paper focuses on the influence of vehicle composition on following behavior in order to further explain how vehicle composition affects the saturation flow rate on a mechanism level, the kind of model which is most suitable is one in which driving behavior is characterized by differential equations to show complex macroscopic traffic characteristics. This kind of model, which is based on statistical physics, can accurately describe the complex phenomena of actual traffic flow by introducing the variable factors that can describe the basic characteristics of actual traffic flow. Further analyzing these models, it is found that, by introducing the basic characteristics of actual traffic flow, the car-following model aimed at velocity optimization can accurately describe the actual traffic flow. There are three car-following models based on optimal velocity which are the OV (Optimized Velocity) model [18], the GF (Generalized Force) model [26], and the FVD (Full Velocity Difference) model [27]. Through comprehensive analysis, it is found that the OV and GF models are inadequate to a certain extent, especially in the intersection area, as some features cannot be accurately described. The IDM and CA (Cellular Automata) models have complex structures and many parameters and are difficult to calibrate. The FVD model not only has the advantages of simple structure, easy calibration, and high precision but also can describe the common traffic flow characteristics at the intersection. Moreover, by adding the influencing factors or improving the model structure, the traffic flow characteristics under the influence of different factors can be described. In this way, the influencing factors can be correlated with traffic flow characteristics, and the influence principle of the influencing factors on the traffic flow characteristics can be further analyzed. Therefore, the FVD model is selected as the basic model for improvement research.


### 3.1.2. Parameter Calibration of Basic Car-Following Model

The core of parameter calibration of the car-following model is the least squares method. The specific steps are as follows:Step 1: Determine the equation structure of the car-following model, put in the track data of the leading vehicle, and use the initial value of the required calibration parameters in the car-following model. Then the acceleration data of the following vehicle from the following model can be simulated and, through derivation, the speed and position data can be simulated.Step 2: Compare the simulated acceleration data of the following vehicle with the measured data, and the error sum of squares index can be obtained, which is related to the selection of the model parameters.Step 3: Reselect the parameter value to be calibrated and repeat Step 1 and Step 2. A selected set of calibration parameters will be corresponded to an error sum of squares index.Step 4: The first three steps are classified as solving the optimization problem of evaluation indexes, and intelligent algorithms are used to solve them.The genetic algorithm has the advantages of strong robustness and unlikelihood of falling into a local optimum. Thus, the genetic algorithm is used to calibrate the relevant parameters in the full-speed difference model in this paper.(1) Determination of Error Index. MSE (Mean Square Errors), RMSE (Root Mean Square Errors), and RMSPE (Root Mean Square Percentage Errors) [28] are selected as the decision indexes. The specific calculation processes are shown in the three following formulas:(1)MSE=1N∑i=1Nobservedi−predictedi2,(2)RMSE=1N∑i=1Nobservedi−predictedi2,(3)RMSPE=∑i=1Nobservedi−predictedi2∑i=1Nobservedi2,where “observed” is the measured trajectory value, such as the measured position, velocity, and acceleration, “predicted” is trajectory value predicted by simulation, and N is observed sample size.(2) Genetic Algorithm Flow. The genetic algorithm (GA) is an evolutionary computing algorithm [29, 30], which is a part of artificial intelligence technology. John Holland [30] first invented and improved GA. In the process of species evolution, in order to better adapt to the environment and repopulate the Earth, good genes will be inherited, and the bad genes will be eliminated. After many generations of genetic changes, species retain the best genes in the present time. GA is an algorithm constructed by imitating this phenomenon in the biological world. This algorithm has been widely used in optimization and search to find the optimal solution.The GA flow chart is as shown in Figure4, which can be divided into 6 steps:Step 1: Set the genetic algorithm parameters. The population size is set in the range of 20–100. From 20 to 100, one parameter is taken every 10, and 9 parameters are taken in total. Similarly, the maximal iteration algebra is set as 5 parameters, the crossover probability is set as 6 parameters, the mutation probability is set as 5 parameters, and the terminative evolutionary algebra is set as 2 parameters. Each combination scheme is traversed through programming.Step 2: Randomly select a leading vehicle trajectory data, set all the parameters in the FVD model to 1, and generate the simulation data of the car-following from the model.Step 3: Based on the simulation data, the parameters of the car-following model are calibrated by the genetic algorithm. Continuously change the parameter combination scheme in the genetic algorithm and select the combination whose parameters of the calibrated car-following model are closest to 1 as the best combination. Then, the parameters of the genetic algorithm are selected.Step 4: Obtain all trajectory data to be calibrated, and set the initial parameter value to 1 (a = 1, λ = 1, and hc = 1). Then the trajectory data of the leading vehicle is taken into the FVD model. After numerical simulation, velocity, acceleration, and position of the subject vehicle are obtained.Step 5: Compare the simulated velocity, acceleration, and position of the subject vehicle with the measured data, and the MSE (Mean Square Error) is taken as the evaluation index to determine whether to select the initial parameter. If the error satisfies the termination condition in the genetic algorithm, the initial parameter is used. If not, proceed to step 6.Step 6: Reselect the parameter values to be calibrated, and cycle Step 4 and Step 5 to obtain multiple groups of parameters and corresponding evaluation indexes. The optimal set of parameters is taken as the final result of parameter calibration of the car-following model. So far, the process of calibrating parameters by the genetic algorithm is over.Figure 4
Genetic algorithm flow chart.The genetic algorithm toolbox in MATLAB is used to calculate the optimal solution. The toolbox can only find the minimum value of fitness function as the optimization goal. Thus, when inputting the fitness function, the mean absolute error of the whole sample is taken as the final fitness result of the function. The related parameters are as follows: fitness function is set as mean absolute error of simulated and measured accelerations; population size is set as 60; maximal iteration algebra is set as 500; crossover probability is set as 0.9; mutation probability is set as 0.2; convergence tolerance error is set as 1 × 10−6; terminative evolutionary algebra is set as 100.To verify the effectiveness of the proposed method in this paper, simulated data are used for testing. A leading vehicle trajectory is randomly selected, and all the parameters in the FVD model are set to 1. From the model, the simulation data of the following vehicle are generated. Based on the simulation data, the car-following model parameter calibration method based on the genetic algorithm is used for calibration. If the parameter results are close to 1, it is shown that the calibration method in this paper can find the optimal solution. In the end, the results show thata = 0.898, λ = 1.029, and hc = 1.050; therefore the calibration error is quite small and it can be considered that the calibration method in this paper can find the optimal solution of the model. Then, the same method is used to calibrate the measured data, and the final FVD model calibration results are as follows: α is 0.010; λ is 0.328; hc is 3.334.


### 3.1.3. Validation of Basic Car-Following Model

With acceleration, velocity, and spacing as performance indexes, the corresponding judgment indexes are obtained. The model is verified from multidimensional dimensions. The specific results are shown in Figure5. It can be seen from Figure 5 that there is a large deviation between the measured acceleration value and the simulation value. On the one hand, RMSPE is sensitive when the acceleration is near 0. On the other hand, the driving characteristics of different vehicles vary, and the model ignores the differences of vehicle types, resulting in some errors.Figure 5
Performance of FVD model.Therefore, one following process is randomly selected to further analyze the acceleration, velocity, and distance with respect to the measured values and the simulation value. In Figure6, the curve represents the state change of following vehicles. Through a comparison between the simulation and the measured values, it is found that the acceleration and velocity change trend of the model simulation is still similar to the measured value, but the error is large and the difference of the following distance is obvious. It is indicated that the performance of the model is poor, and improvement is needed in subsequent studies.Figure 6
The curves of subject vehicle’s acceleration, speed, and distance during car-following.


## 3.1.1. Selection of Basic Car-Following Model

In the past 70 years, the integration of scholars’ ideas in various fields has made great progress in research related to the car-following model. Car-following models can be roughly divided into theory-driven and data-driven types. The theory-driven models can be divided into two categories: those based on traffic engineering and those based on statistical physics. The car-following model based on traffic engineering describes driving behavior through mathematical formulas, which include the psychophysical GM (General Motor) model [15], the safe distance Gipps’ model [16], and the physiological-psychological Wiedemann 74 model [17]. The car-following model based on statistical physics describes the changes of vehicle velocity and displacement through differential equations, including the OV (Optimized Velocity) model [18], the IDM (Intelligent Driver Model) model [19], and the cellular automaton NaSch model [20]. The modeling idea of data-driven car-following models is not limited to theoretical assumptions and does not pursue mathematical derivation. The data-driven car-following model uses nonparametric methods to mine following vehicle trajectory data and establish car-following models with high prediction accuracy. The classical car-following models include the fuzzy logic model [21], the artificial neural network model [22], the case learning model [23], the support vector regression model [24], and the deep learning model [25].Since this paper focuses on the influence of vehicle composition on following behavior in order to further explain how vehicle composition affects the saturation flow rate on a mechanism level, the kind of model which is most suitable is one in which driving behavior is characterized by differential equations to show complex macroscopic traffic characteristics. This kind of model, which is based on statistical physics, can accurately describe the complex phenomena of actual traffic flow by introducing the variable factors that can describe the basic characteristics of actual traffic flow. Further analyzing these models, it is found that, by introducing the basic characteristics of actual traffic flow, the car-following model aimed at velocity optimization can accurately describe the actual traffic flow. There are three car-following models based on optimal velocity which are the OV (Optimized Velocity) model [18], the GF (Generalized Force) model [26], and the FVD (Full Velocity Difference) model [27]. Through comprehensive analysis, it is found that the OV and GF models are inadequate to a certain extent, especially in the intersection area, as some features cannot be accurately described. The IDM and CA (Cellular Automata) models have complex structures and many parameters and are difficult to calibrate. The FVD model not only has the advantages of simple structure, easy calibration, and high precision but also can describe the common traffic flow characteristics at the intersection. Moreover, by adding the influencing factors or improving the model structure, the traffic flow characteristics under the influence of different factors can be described. In this way, the influencing factors can be correlated with traffic flow characteristics, and the influence principle of the influencing factors on the traffic flow characteristics can be further analyzed. Therefore, the FVD model is selected as the basic model for improvement research.


## 3.1.2. Parameter Calibration of Basic Car-Following Model

The core of parameter calibration of the car-following model is the least squares method. The specific steps are as follows:Step 1: Determine the equation structure of the car-following model, put in the track data of the leading vehicle, and use the initial value of the required calibration parameters in the car-following model. Then the acceleration data of the following vehicle from the following model can be simulated and, through derivation, the speed and position data can be simulated.Step 2: Compare the simulated acceleration data of the following vehicle with the measured data, and the error sum of squares index can be obtained, which is related to the selection of the model parameters.Step 3: Reselect the parameter value to be calibrated and repeat Step 1 and Step 2. A selected set of calibration parameters will be corresponded to an error sum of squares index.Step 4: The first three steps are classified as solving the optimization problem of evaluation indexes, and intelligent algorithms are used to solve them.The genetic algorithm has the advantages of strong robustness and unlikelihood of falling into a local optimum. Thus, the genetic algorithm is used to calibrate the relevant parameters in the full-speed difference model in this paper.(1) Determination of Error Index. MSE (Mean Square Errors), RMSE (Root Mean Square Errors), and RMSPE (Root Mean Square Percentage Errors) [28] are selected as the decision indexes. The specific calculation processes are shown in the three following formulas:(1)MSE=1N∑i=1Nobservedi−predictedi2,(2)RMSE=1N∑i=1Nobservedi−predictedi2,(3)RMSPE=∑i=1Nobservedi−predictedi2∑i=1Nobservedi2,where “observed” is the measured trajectory value, such as the measured position, velocity, and acceleration, “predicted” is trajectory value predicted by simulation, and N is observed sample size.(2) Genetic Algorithm Flow. The genetic algorithm (GA) is an evolutionary computing algorithm [29, 30], which is a part of artificial intelligence technology. John Holland [30] first invented and improved GA. In the process of species evolution, in order to better adapt to the environment and repopulate the Earth, good genes will be inherited, and the bad genes will be eliminated. After many generations of genetic changes, species retain the best genes in the present time. GA is an algorithm constructed by imitating this phenomenon in the biological world. This algorithm has been widely used in optimization and search to find the optimal solution.The GA flow chart is as shown in Figure4, which can be divided into 6 steps:Step 1: Set the genetic algorithm parameters. The population size is set in the range of 20–100. From 20 to 100, one parameter is taken every 10, and 9 parameters are taken in total. Similarly, the maximal iteration algebra is set as 5 parameters, the crossover probability is set as 6 parameters, the mutation probability is set as 5 parameters, and the terminative evolutionary algebra is set as 2 parameters. Each combination scheme is traversed through programming.Step 2: Randomly select a leading vehicle trajectory data, set all the parameters in the FVD model to 1, and generate the simulation data of the car-following from the model.Step 3: Based on the simulation data, the parameters of the car-following model are calibrated by the genetic algorithm. Continuously change the parameter combination scheme in the genetic algorithm and select the combination whose parameters of the calibrated car-following model are closest to 1 as the best combination. Then, the parameters of the genetic algorithm are selected.Step 4: Obtain all trajectory data to be calibrated, and set the initial parameter value to 1 (a = 1, λ = 1, and hc = 1). Then the trajectory data of the leading vehicle is taken into the FVD model. After numerical simulation, velocity, acceleration, and position of the subject vehicle are obtained.Step 5: Compare the simulated velocity, acceleration, and position of the subject vehicle with the measured data, and the MSE (Mean Square Error) is taken as the evaluation index to determine whether to select the initial parameter. If the error satisfies the termination condition in the genetic algorithm, the initial parameter is used. If not, proceed to step 6.Step 6: Reselect the parameter values to be calibrated, and cycle Step 4 and Step 5 to obtain multiple groups of parameters and corresponding evaluation indexes. The optimal set of parameters is taken as the final result of parameter calibration of the car-following model. So far, the process of calibrating parameters by the genetic algorithm is over.Figure 4
Genetic algorithm flow chart.The genetic algorithm toolbox in MATLAB is used to calculate the optimal solution. The toolbox can only find the minimum value of fitness function as the optimization goal. Thus, when inputting the fitness function, the mean absolute error of the whole sample is taken as the final fitness result of the function. The related parameters are as follows: fitness function is set as mean absolute error of simulated and measured accelerations; population size is set as 60; maximal iteration algebra is set as 500; crossover probability is set as 0.9; mutation probability is set as 0.2; convergence tolerance error is set as 1 × 10−6; terminative evolutionary algebra is set as 100.To verify the effectiveness of the proposed method in this paper, simulated data are used for testing. A leading vehicle trajectory is randomly selected, and all the parameters in the FVD model are set to 1. From the model, the simulation data of the following vehicle are generated. Based on the simulation data, the car-following model parameter calibration method based on the genetic algorithm is used for calibration. If the parameter results are close to 1, it is shown that the calibration method in this paper can find the optimal solution. In the end, the results show thata = 0.898, λ = 1.029, and hc = 1.050; therefore the calibration error is quite small and it can be considered that the calibration method in this paper can find the optimal solution of the model. Then, the same method is used to calibrate the measured data, and the final FVD model calibration results are as follows: α is 0.010; λ is 0.328; hc is 3.334.


## 3.1.3. Validation of Basic Car-Following Model

With acceleration, velocity, and spacing as performance indexes, the corresponding judgment indexes are obtained. The model is verified from multidimensional dimensions. The specific results are shown in Figure5. It can be seen from Figure 5 that there is a large deviation between the measured acceleration value and the simulation value. On the one hand, RMSPE is sensitive when the acceleration is near 0. On the other hand, the driving characteristics of different vehicles vary, and the model ignores the differences of vehicle types, resulting in some errors.Figure 5
Performance of FVD model.Therefore, one following process is randomly selected to further analyze the acceleration, velocity, and distance with respect to the measured values and the simulation value. In Figure6, the curve represents the state change of following vehicles. Through a comparison between the simulation and the measured values, it is found that the acceleration and velocity change trend of the model simulation is still similar to the measured value, but the error is large and the difference of the following distance is obvious. It is indicated that the performance of the model is poor, and improvement is needed in subsequent studies.Figure 6
The curves of subject vehicle’s acceleration, speed, and distance during car-following.


## 3.2. Model Improvement

### 3.2.1. Analysis of Traffic Flow Characteristics at Intersections

The verification of the FVD model shows that the model cannot completely and accurately describe the car-following characteristics of heterogeneous vehicles and that the accuracy of the FVD model needs to be improved. Therefore, it is necessary to further analyze the characteristics of departure flow at the intersection entrance lane to provide ideas for the improvement of the FVD model.The common characteristics of departure flow are shown in Figure7. Figure 7(a) shows the characteristics of departure flow. When the green light turns on, the vehicles behind the stop line start and pass the stop line in turn. The flow rate soon increases to a stable value, which is the saturated flow rate. As shown in Figure 7(b), through the statistics of the headway of each position of queueing vehicles passing through the stop line, it is found that the headway decreases gradually with the increase of position and maintains a relatively stable state. At this time, the corresponding headway is the saturation headway. This parameter is the key to signalized intersection control and capacity analysis. However, queueing vehicles are often composed of different models in reality, which makes the headway of these vehicles different. Figure 7(b) shows the headway distribution when the 7th vehicle in line in one period is a bus. After averaging the saturation headway, it can be considered that it increases in relation to the increased proportion of buses. Figure 7(b) shows a common following scenario in mixed traffic, which can be divided into 4 categories, namely, car-car, car-bus, bus-car, and bus-bus [31].Figure 7
The features of departure vehicles passing the stop line during car-following. (a) Diagram of departure flow at the intersection entrance lane. (b) A common following scenario in mixed traffic.
(a)(b)To further verify the difference of headway, the videos of the three measured intersections are further extracted. Only the headway under saturation condition is extracted, from the 4th to 6th vehicle to the end of the queue. The extracted headway is classified for statistics, and the probability distribution of headway is shown in Figure8. The rank-sum test is used to test four distributions. The data are substituted into SPSS statistical analysis software and the Kolmogorov-Smirnov test method is used for the statistics of the data. The test results are as follows: car-car and car-bus (Z = 5.582, Sig. = 0.000); car-car and bus-car (Z = 1.451, Sig. = 0.030); car-car and bus-bus (Z = 6.691, Sig. = 0.000); car-bus and bus-car (Z = 3.777, Sig. = 0.000); car-bus and bus-bus (Z = 1.460, Sig. = 0.028); and bus-car and bus-bus (Kolmogorov-Smirnov test statistics, Z = 4.614, Sig. = 0.000). The results show that when the confidence interval is 99%, the distributions of car-car and bus-car have a certain similarity; the distributions of car-bus and bus-bus also have a certain similarity; the other distributions are significantly different. This is mainly caused by two factors: one is the difference of vehicle length; the other is the difference in vehicle performance and driving behavior. Therefore, car-following models should be built separately for different scenarios.Figure 8
The distribution of headway in four car-following scenarios [30].


### 3.2.2. Improvement of Basic Model

Vehicle performance and driving behavior are mainly represented by three parameters in the car-following model: the desired velocityVmax in the optimized velocity function; the driver’s sensitivity coefficient to vehicle distance, α; and the driver’s sensitivity coefficient to velocity difference λ. Differences in vehicle performance will lead to differences in the desired velocity. The measured speed distribution of cars and buses in a nonfollowing state is shown in Figure 9. There is a significant difference between the desired velocities of the two vehicle types (Kolmogorov-Smirnov test, Z = 3.590, Sig. = 0.00). The average desired velocity of cars is 11.28 m/s and that of buses is 9.67 m/s. The main reason for the difference in driving behavior is that drivers have different sensitivities to the velocity difference and distance difference of the vehicle in front of them in different scenarios. The original FVD model can be described in segments according to different scenarios.Figure 9
The distribution of car and bus desired velocity [30].The improved FVD model is shown in the following formula:(4)dvntdt=αc−cVΔxnt−vnt+λc−cΔvntifcar−carαc−bVΔxnt−vnt+λc−bΔvntifcar−busαb−cVΔxnt−vnt+λb−cΔvntifbus−carαb−bVΔxnt−vnt+λb−bΔvntifbus−bus,and, in formula (3), αc-c,λc-c is the sensitivity coefficient in a scenario of car following car, αc-b¸λc-b is the sensitivity coefficient in a scenario of car following bus, αb-c, λb-c is the sensitivity coefficient in a scenario of bus following car, αb-b, λb-b is the sensitivity coefficient in a scenario of bus following bus, V⋅ is velocity optimization function, Δxnt is the distance between the following vehicle and the leading vehicle, Δvnt is the relative velocity difference between the following vehicle and the leading vehicle, and vnt is the velocity of the following vehicle at time t.The optimized velocity function is shown in the following formula:(5)VΔxnt=Vcarmax2tanhΔxnt−hc+tanhhc,Vbusmax2tanhΔxnt−hc+tanhhc,whereVcarmax is the desired velocity of a car, Vbusmax is the desired velocity of a bus, and hc is the safe distance between vehicles.To further test the improved model, a group of following data are randomly selected in each scenario. The changed curves of the measured values, FVD model, and simulation values of the improved model are drawn, as shown in Figures10 to 13. In Figure 10(in the car-car scenario), the variation trends of acceleration, velocity, and distance between the FVD model and the improved model are basically consistent with the measured values; that is, there is little difference between them.Figure 10
The curves of a subject vehicle’s acceleration, speed, and distance in a car-car scenario.Figure 11
The curves of a subject vehicle’s acceleration, speed, and distance in a car-bus scenario.Figure 12
The curves of a subject vehicle’s acceleration, speed, and distance in a bus-car scenario.Figure 13
The curves of a subject vehicle’s acceleration, speed, and distance in a bus-bus scenario.In Figure11 (in the car-bus scenario), the simulation results of the FVD model have a large deviation from the measured values. In particular, the vehicle distance not only has great differences in values but also has inconsistent trends. The acceleration, speed, and vehicle distance curves of the improved model are basically consistent with the measured curves.In Figure12 (in the bus-car scenario), the simulation results of the FVD model are consistent with the measured values, but there is a certain deviation between the values. However, the simulation curve of the improved model is closer to the measured curve.In Figure13 (in the bus-bus scenario), the simulation results of the FVD model are different from the measured results. However, the simulated curve of the improved model is closer to the measured curve.To analyze the error of the whole sample, the RMSE and RMSPE of the improved model are calculated. It can be seen from Figure14 that the acceleration error decreases by 1.89% and 1.92%, respectively. This is because when the acceleration value approaches 0, a slight difference will lead to a sharp increase in statistical errors. In addition, the velocity error decreases by 4.24% and 15.29%, respectively. Moreover, the distance error decreases by 22.31% and 22.32%, respectively. To sum up, in a heterogeneous flow, the improved model gives a better performance and can better describe the following behavior of departure vehicles at the intersection entrance lane.Figure 14
Performance of proposed car-following model. Note: IM is improved model.


### 3.2.3. Parameter Calibration and Validation of Improved Model

A genetic algorithm is still used to calibrate the improved model, and the calibration results are shown in Figure15. A bus driver is more sensitive to the distance between vehicles than a car driver, indicating that a bus driver will always change his velocity and maintain a certain safe distance from the leading vehicle. A bus driver is less sensitive to velocity difference than a car driver, indicating that a bus driver is more interested in maintaining his own velocity than a car driver. The safety distance between a car and a bus is the largest, indicating that the car driver is affected by the leading vehicle being a bus. To avoid frequent acceleration and deceleration, a large safety distance is maintained.Figure 15
Calibrated parameters of proposed car-following model in different scenarios.


## 3.2.1. Analysis of Traffic Flow Characteristics at Intersections

The verification of the FVD model shows that the model cannot completely and accurately describe the car-following characteristics of heterogeneous vehicles and that the accuracy of the FVD model needs to be improved. Therefore, it is necessary to further analyze the characteristics of departure flow at the intersection entrance lane to provide ideas for the improvement of the FVD model.The common characteristics of departure flow are shown in Figure7. Figure 7(a) shows the characteristics of departure flow. When the green light turns on, the vehicles behind the stop line start and pass the stop line in turn. The flow rate soon increases to a stable value, which is the saturated flow rate. As shown in Figure 7(b), through the statistics of the headway of each position of queueing vehicles passing through the stop line, it is found that the headway decreases gradually with the increase of position and maintains a relatively stable state. At this time, the corresponding headway is the saturation headway. This parameter is the key to signalized intersection control and capacity analysis. However, queueing vehicles are often composed of different models in reality, which makes the headway of these vehicles different. Figure 7(b) shows the headway distribution when the 7th vehicle in line in one period is a bus. After averaging the saturation headway, it can be considered that it increases in relation to the increased proportion of buses. Figure 7(b) shows a common following scenario in mixed traffic, which can be divided into 4 categories, namely, car-car, car-bus, bus-car, and bus-bus [31].Figure 7
The features of departure vehicles passing the stop line during car-following. (a) Diagram of departure flow at the intersection entrance lane. (b) A common following scenario in mixed traffic.
(a)(b)To further verify the difference of headway, the videos of the three measured intersections are further extracted. Only the headway under saturation condition is extracted, from the 4th to 6th vehicle to the end of the queue. The extracted headway is classified for statistics, and the probability distribution of headway is shown in Figure8. The rank-sum test is used to test four distributions. The data are substituted into SPSS statistical analysis software and the Kolmogorov-Smirnov test method is used for the statistics of the data. The test results are as follows: car-car and car-bus (Z = 5.582, Sig. = 0.000); car-car and bus-car (Z = 1.451, Sig. = 0.030); car-car and bus-bus (Z = 6.691, Sig. = 0.000); car-bus and bus-car (Z = 3.777, Sig. = 0.000); car-bus and bus-bus (Z = 1.460, Sig. = 0.028); and bus-car and bus-bus (Kolmogorov-Smirnov test statistics, Z = 4.614, Sig. = 0.000). The results show that when the confidence interval is 99%, the distributions of car-car and bus-car have a certain similarity; the distributions of car-bus and bus-bus also have a certain similarity; the other distributions are significantly different. This is mainly caused by two factors: one is the difference of vehicle length; the other is the difference in vehicle performance and driving behavior. Therefore, car-following models should be built separately for different scenarios.Figure 8
The distribution of headway in four car-following scenarios [30].


## 3.2.2. Improvement of Basic Model

Vehicle performance and driving behavior are mainly represented by three parameters in the car-following model: the desired velocityVmax in the optimized velocity function; the driver’s sensitivity coefficient to vehicle distance, α; and the driver’s sensitivity coefficient to velocity difference λ. Differences in vehicle performance will lead to differences in the desired velocity. The measured speed distribution of cars and buses in a nonfollowing state is shown in Figure 9. There is a significant difference between the desired velocities of the two vehicle types (Kolmogorov-Smirnov test, Z = 3.590, Sig. = 0.00). The average desired velocity of cars is 11.28 m/s and that of buses is 9.67 m/s. The main reason for the difference in driving behavior is that drivers have different sensitivities to the velocity difference and distance difference of the vehicle in front of them in different scenarios. The original FVD model can be described in segments according to different scenarios.Figure 9
The distribution of car and bus desired velocity [30].The improved FVD model is shown in the following formula:(4)dvntdt=αc−cVΔxnt−vnt+λc−cΔvntifcar−carαc−bVΔxnt−vnt+λc−bΔvntifcar−busαb−cVΔxnt−vnt+λb−cΔvntifbus−carαb−bVΔxnt−vnt+λb−bΔvntifbus−bus,and, in formula (3), αc-c,λc-c is the sensitivity coefficient in a scenario of car following car, αc-b¸λc-b is the sensitivity coefficient in a scenario of car following bus, αb-c, λb-c is the sensitivity coefficient in a scenario of bus following car, αb-b, λb-b is the sensitivity coefficient in a scenario of bus following bus, V⋅ is velocity optimization function, Δxnt is the distance between the following vehicle and the leading vehicle, Δvnt is the relative velocity difference between the following vehicle and the leading vehicle, and vnt is the velocity of the following vehicle at time t.The optimized velocity function is shown in the following formula:(5)VΔxnt=Vcarmax2tanhΔxnt−hc+tanhhc,Vbusmax2tanhΔxnt−hc+tanhhc,whereVcarmax is the desired velocity of a car, Vbusmax is the desired velocity of a bus, and hc is the safe distance between vehicles.To further test the improved model, a group of following data are randomly selected in each scenario. The changed curves of the measured values, FVD model, and simulation values of the improved model are drawn, as shown in Figures10 to 13. In Figure 10(in the car-car scenario), the variation trends of acceleration, velocity, and distance between the FVD model and the improved model are basically consistent with the measured values; that is, there is little difference between them.Figure 10
The curves of a subject vehicle’s acceleration, speed, and distance in a car-car scenario.Figure 11
The curves of a subject vehicle’s acceleration, speed, and distance in a car-bus scenario.Figure 12
The curves of a subject vehicle’s acceleration, speed, and distance in a bus-car scenario.Figure 13
The curves of a subject vehicle’s acceleration, speed, and distance in a bus-bus scenario.In Figure11 (in the car-bus scenario), the simulation results of the FVD model have a large deviation from the measured values. In particular, the vehicle distance not only has great differences in values but also has inconsistent trends. The acceleration, speed, and vehicle distance curves of the improved model are basically consistent with the measured curves.In Figure12 (in the bus-car scenario), the simulation results of the FVD model are consistent with the measured values, but there is a certain deviation between the values. However, the simulation curve of the improved model is closer to the measured curve.In Figure13 (in the bus-bus scenario), the simulation results of the FVD model are different from the measured results. However, the simulated curve of the improved model is closer to the measured curve.To analyze the error of the whole sample, the RMSE and RMSPE of the improved model are calculated. It can be seen from Figure14 that the acceleration error decreases by 1.89% and 1.92%, respectively. This is because when the acceleration value approaches 0, a slight difference will lead to a sharp increase in statistical errors. In addition, the velocity error decreases by 4.24% and 15.29%, respectively. Moreover, the distance error decreases by 22.31% and 22.32%, respectively. To sum up, in a heterogeneous flow, the improved model gives a better performance and can better describe the following behavior of departure vehicles at the intersection entrance lane.Figure 14
Performance of proposed car-following model. Note: IM is improved model.


## 3.2.3. Parameter Calibration and Validation of Improved Model

A genetic algorithm is still used to calibrate the improved model, and the calibration results are shown in Figure15. A bus driver is more sensitive to the distance between vehicles than a car driver, indicating that a bus driver will always change his velocity and maintain a certain safe distance from the leading vehicle. A bus driver is less sensitive to velocity difference than a car driver, indicating that a bus driver is more interested in maintaining his own velocity than a car driver. The safety distance between a car and a bus is the largest, indicating that the car driver is affected by the leading vehicle being a bus. To avoid frequent acceleration and deceleration, a large safety distance is maintained.Figure 15
Calibrated parameters of proposed car-following model in different scenarios.


## 4. Simulation Experiment

The essence of numerical simulation is to update the vehicle trajectory information through the established micro driving behavior model. Then, the macro traffic flow parameters are obtained, such as the saturation flow rate. The performance of the microscopic model is further discussed by comparing the flow rate observed in practice.Trajectory information mainly includes displacement, velocity, and acceleration. According to the kinematics theorem, velocity is the first derivative of displacement and acceleration is the second derivative of displacement. The improved car-following model in this paper is the differential equation of the theoretical model of acceleration. Therefore, the process of simulation is the process of solving differential equations. Then, the numerical simulation can be transformed into the problem of solving differential equations.


### 4.1. Simulation Process Method

In mathematics and computer science, Euler’s method is a first-order numerical method for solving ordinary differential equations (or initial value problems) with given initial values. It is a basic explicit method for solving numerical ordinary differential equations [32]. The Euler method is simple and intuitive and easy to program. The improved following model in this paper is a typical first-order differential equation, as is the relationship between displacement, velocity, and acceleration. Therefore, the Euler method is chosen to realize the simulation process.


### 4.2. Experimental Design

To study the traffic composition impact on saturation headway at signalized intersections, six groups of simulation experiments with the proportion of buses at 0%, 0–10%, 10–20%, 20–30%, 30–40%, and 40–50% were constructed, respectively. The simulation experiment parameters are shown in Table2, the lane width is 3.5 m, and the lane type is straight lane.Table 2
The simulation parameters in 0% heavy vehicles scenario.
Simulation parametersCar length: 5 mSimulation area: the signalized intersection entrance lane, with a length of 80 m.Number of simulated vehicles: 12 vehiclesSimulation time: 50 sSimulation precision: 0.1 sPosition of stop line: 0 m (a negative value is recorded after the vehicle passes the stop line)Simulation period: 100个Meanwhile, the measured saturation headway data with the same environmental scene and bus ratio were selected as the comparison item. The saturation time headway of each simulation experiment was calculated and compared with the measured values in the same scene. This eliminates the interference of other factors. Table3 shows the distribution of measured headways in different scenarios of vehicle composition.Table 3
The distribution of measured headways in different scenarios of vehicle composition.
Proportion of buses0%0–10%10–20%20–30%30–40%40–50%Average saturation headway (s)2.202.472.572.782.923.22


## 4.1. Simulation Process Method

In mathematics and computer science, Euler’s method is a first-order numerical method for solving ordinary differential equations (or initial value problems) with given initial values. It is a basic explicit method for solving numerical ordinary differential equations [32]. The Euler method is simple and intuitive and easy to program. The improved following model in this paper is a typical first-order differential equation, as is the relationship between displacement, velocity, and acceleration. Therefore, the Euler method is chosen to realize the simulation process.


## 4.2. Experimental Design

To study the traffic composition impact on saturation headway at signalized intersections, six groups of simulation experiments with the proportion of buses at 0%, 0–10%, 10–20%, 20–30%, 30–40%, and 40–50% were constructed, respectively. The simulation experiment parameters are shown in Table2, the lane width is 3.5 m, and the lane type is straight lane.Table 2
The simulation parameters in 0% heavy vehicles scenario.
Simulation parametersCar length: 5 mSimulation area: the signalized intersection entrance lane, with a length of 80 m.Number of simulated vehicles: 12 vehiclesSimulation time: 50 sSimulation precision: 0.1 sPosition of stop line: 0 m (a negative value is recorded after the vehicle passes the stop line)Simulation period: 100个Meanwhile, the measured saturation headway data with the same environmental scene and bus ratio were selected as the comparison item. The saturation time headway of each simulation experiment was calculated and compared with the measured values in the same scene. This eliminates the interference of other factors. Table3 shows the distribution of measured headways in different scenarios of vehicle composition.Table 3
The distribution of measured headways in different scenarios of vehicle composition.
Proportion of buses0%0–10%10–20%20–30%30–40%40–50%Average saturation headway (s)2.202.472.572.782.923.22


## 5. Results Analysis and Discussion

In the first group of simulation experiments, the proportion of buses is 0%; that is to say, the vehicles on the lane are all cars. When the signal light changes from red to green, the displacement and velocity of all vehicles are studied.Figure16 shows the displacement and velocity changes of 12 vehicles in a simulation period. From the displacement simulation diagram, it can be intuitively seen that when the green light is on, the vehicles pass the stop line in turn. When the simulation time is at about 30 s, all 12 vehicles have passed the stop line, and when the simulation time is at about 40 s, the vehicles enter a stable state. At the stop line, the time interval between vehicles, namely, the headway, is different. The headway of vehicles at the front is longer, and that of vehicles at the back is shorter, which is the same as the observation result. It can be seen from the velocity simulation diagram that the vehicle reaches a stable state and tends to adopt a uniform speed at about 40 s. In addition, the velocity of the leading vehicle is greater than that of the following vehicle at the same simulation moment, which is mainly due to the safety distance between the following vehicle and the leading vehicle, and this is consistent with the actual situation.Figure 16
The numerical simulation diagram in 0% heavy vehicles scenario. (a) Velocity simulation diagram. (b) Displacement simulation diagram.
(a)(b)A detector is set in the simulation program to detect the time when the vehicle passes through the stop line in order to calculate the time difference and then the headway. The headway of the first vehicle is the sum of the time when the first vehicle passes through the stop line, the start-up time, and reaction time. A total of 100 periods are simulated to calculate the average headway of vehicles at each position, which can eliminate random errors. The average headway distribution of vehicles at each position is shown in Figure17. Generally, the 4th–6th vehicles in the queue begin to enter the saturation flow state. The 6th to 12th vehicles are selected as the saturation flow state, and the saturation headway is calculated. The result is 2.19 s.Figure 17
The simulation value of headways in 0% heavy vehicles scenario.In the second group of simulation experiments, the proportion of buses is 0–10%; that is, the proportion of buses in the lane is between 0 and 10% of the queue and the other vehicles are cars. Since there are 12 simulated vehicles, one bus is needed to make the proportion of buses 0–10% of the total number. To further analyze whether the saturation headway is related to the position of the bus, the bus is placed between the 6th and 12th vehicles for simulation, and the saturation headways are recorded. Figure18 shows the displacement simulation diagram of all vehicles at different positions. As shown in Figure 18, the distance between buses and leading vehicles is significantly larger than that of cars in other positions, which is consistent with the actual situation.Figure 18
The numerical simulation diagram in 0–10% heavy vehicles scenario. (a) The bus at the 6th position. (b) The bus at the 7th position. (c) The bus at the 8th position. (d) The bus at the 9th position. (e) The bus at the 10th position. (f) The bus at the 11th position. (g) The bus at the 12th position.
(a)(b)(c)(d)(e)(f)(g)Each position is simulated for 20 periods, and the saturation headway corresponding to each position (the sixth to the 12th vehicle) is recorded. The saturation headway distribution corresponding to each position is shown in Table4. The saturation headway is not significantly influenced by the position of the buses but only by the proportion of the buses. The average saturated time headway is 2.37 s.Table 4
The distribution of saturation headways with different positions of one bus in queue.
Bus position6789101112Saturated headway (s)2.422.402.372.362.362.362.34In the third group of simulation experiment, the proportion of buses is 10–20%. Two buses are required in the queue to make the proportion of buses 10–20% of the total. Similarly, how to allocate the position of two buses needs further discussion and analysis of whether different position combinations affect the saturation headway. To simplify the analysis, the comparison is made between the simulation values of saturation headway when the buses are located at the 6th and 7th, 6th and 8th, 6th and 9th, and 6th and 10th positions, respectively. Figure19 shows the simulation diagram of vehicle displacement with different position combinations. It shows that the distance between the bus and the leading vehicle is significantly larger than that of the car in other positions.Figure 19
The numerical simulation diagram in 10–20% heavy vehicles scenario. (a) The bus at the 6th and 7th positions. (b) The bus at the 6th and 8th positions. (c) The bus at the 6th and 9th positions. (d) The bus at the 6th and 10th positions.
(a)(b)(c)(d)The four position combinations are simulated for 50 periods, and the saturation headway corresponding to each position is recorded. The saturation time headway distribution is shown in Table5. There is no significant relationship between saturation headway and position combination of the buses. The average saturation time headway is 2.39 s.Table 5
The distribution of saturation headways with different positions of two buses in queue.
Combination of bus positions6, 76, 86, 96, 10Saturated headway (s)2.422.402.372.36In the fourth group of simulation experiment, the proportion of buses is 20–30%. Three buses are needed in the queue to make the proportion of buses 20–30% of the total. According to the previous analysis, the saturation headway is only related to the proportion of buses. Therefore, in this experiment, only three buses need to be placed in positions 6, 7, and 8 to simulate 100 periods. Figure20 shows the simulation diagram of vehicle displacement in one period. The average saturation time headway is 2.69 s.Figure 20
The numerical simulation diagram in 20–30% heavy vehicles scenario.In the fifth group of simulation experiment, the proportion of buses is 30–40%. Four buses are needed in the queue to make the proportion of buses 30–40% of the total. In this experiment, four buses are placed in positions 6, 7, 8, and 9 to simulate 100 periods. Figure21 shows the simulation diagram of vehicle displacement in one period. The average saturation time headway is 2.83 s.Figure 21
The numerical simulation diagram in 30–40% heavy vehicles scenario.In the sixth group of simulation experiment, the proportion of buses is 40–50%. Five or six buses are needed in the queue to make the proportion of buses 40–50% of the total. In this experiment, five buses are placed in positions 6, 7, 8, 9, and 10 to simulate 50 periods. Figure22 shows the simulation diagram of vehicle displacement in one period. The average saturation time headway is 2.97 s.Figure 22
The numerical simulation diagram in 40–50% heavy vehicles scenario. (a) The 5th bus in the queue. (b) The 6th bus in the queue.
(a)(b)The measured saturation headway is compared with the simulated saturation headway with different bus ratios. The distribution diagram is shown in Figure23. There is no significant difference between the measured value and the simulated value (Kolmogorov-Smirnov test, Z = 0.289, Sig. = 1.00). This also fully shows that the improved car-following model can well describe the phenomenon that the saturation headway is affected by the proportion of buses.Figure 23
Comparison of simulated and measured values of saturation headway in different percentages of heavy vehicles scenarios.


## 6. Conclusion

In this paper, the measured data are used to calibrate and verify the basic full velocity difference model. However, the model cannot characterize accurately the operation of traffic flow in a heterogeneous flow environment, and the simulation values of vehicle speed, acceleration, distance, and other parameters are quite different from the measured values. This is because the basic FVD model does not consider the mutual interference between vehicles in a heterogeneous flow environment, that is to say, the impact of traffic composition. Therefore, an improved FVD model is proposed, which abstracts the driver characteristics of a heterogeneous flow into four scenarios: car-car, car-bus, bus-car, and bus-bus. The performance differences between vehicles and driver response characteristics under different scenarios are considered. The performance of the proposed model is significantly improved. The RMSE and RMSPE are reduced by 15.29% and 22.32%, respectively, which can reflect well the operation of traffic flow in a heterogeneous flow environment. Finally, through numerical simulation experiments, the variation of saturation headway with different proportions of buses is analyzed. There is no significant difference between the simulation results of the improved model and the measured results, which shows that the improved model explains the impact of vehicle composition on drivers’ driving behavior on the micro level, resulting in the change of traffic flow spacing and the change of saturation headway. It is identified from this research that when the following vehicle is a bus, the acceleration and deceleration performance of the vehicle is significantly different from that of a car. When the leading vehicle is a bus, it will affect the reaction time of the driver of the following vehicle. In addition, the saturation headway increases with the increase of the proportion of large vehicles. Moreover, another important finding is that the saturation headway is not significantly influenced by the position of the buses but only by the proportion of the buses.The research results could provide theoretical support for the control and management of fleets composed of different vehicles at intersections. The model in this paper can be applied to two fields. (1) Balance the proportion of heavy vehicles in the lane group and improve the traffic capacity of the lane group. When the proportion of heavy vehicles exceeds 30%, the saturation headway increases significantly, and the capacity decreases significantly. This is an example of an application scenario. It is assumed that one lane group has three straight lanes, in which the proportion of heavy vehicles in two lanes is 30% and the other is 0%, and the saturation headway is 2.52 s. If heavy vehicles are evenly distributed in three lanes, which means the proportion of heavy vehicles in each lane is 20%, the saturation headway would be down to 2.39 s. The improvement effect is obvious. In addition, bus priority lane and the rule of buses driving on the right can be considered. (2) Optimize driver reaction coefficient, reduce the safe distance, and improve overall capacity. In the future, through the vehicular networking technology, this model can be applied to improve the driver’s response characteristics, reduce the response time, and reduce the safe distance, so as to improve the traffic capacity and relieve the status of traffic jams. However, this paper only considers the following behavior of ordinary buses and cars and does not consider the following behavior of other vehicle combinations. On the other hand, this analysis method is consistent, so it can be analyzed comprehensively in the future. Meanwhile, the performance of the improved model needs to be further improved. In the future, a better model can be proposed by combining with other car-following models. Besides, more factors will be considered for the model in the future, such as bus priority lane and the rule of buses driving on the right.



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*Source: 5025393-2022-06-08.xml*