Car-Following Model Optimization and Simulation Based on Cooperative Adaptive Cruise Control

Abstract

This study aims to improve the desired distance adaptability of the cooperative adaptive cruise control (CACC) during car-following. In this study, the characteristics of the desired distance in different traffic flow states were analyzed. The general functional form of the desired distance in the car-following process was formulated. Then, a car-following platoon was constructed to compare the car-following effect of the platoon under different conditions, using the following speed and the lead vehicle disturbance, as the observed variable and the simulation condition, respectively. The car-following effect of the platoon under different parameters was also compared, based on the improved CACC model. The results show that the improved CACC model exhibited more advantages in car-following efficiency, it can better describe the state of the car-following queue under different traffic flow parameters and car-following behavior conditions, it has a strong anti-interference ability for the fluctuation of the car-following queue and is conducive to further improving the intelligent operation of car-following queue.

1. Introduction

With increasingly severe traffic congestion, traffic safety and pollution problems are becoming more prominent and causing more harm to human beings. Thus, many traffic practitioners and researchers have been attempting to develop solutions to address these traffic problems from various aspects [1,2]. With the rapid development of modern mathematics and computer science, new technologies have been used to guide and manage current relatively chaotic traffic states, thus facilitating more orderly and efficient traffic. This has increased some hot traffic topics in traffic engineering [3]. (1) Intelligent transportation system. An intelligent transportation system is a large-scale and all-round system established by effectively integrating advanced information technology, data communication transmission technology, electronic sensing technology, control technology and computer technology, into the entire ground traffic management system; (2) Real-time, accurate and efficient integrated transportation management system. It can effectively utilize the existing traffic facilities, reduce environmental pollution, ensure traffic safety and improve transportation efficiency. Thus, this type of system has attracted increasing attention, worldwide.

In order to seize the technological development opportunities brought to China by intelligent transportation technology, China’s autonomous vehicles and intelligent networked transportation systems have developed to a national strategic level. The connected and automated vehicle (CAV) industry is listed as one of the key areas in “Made in China 2025” (a national strategic plan and industrial policy of China) and the “14th Five-Year Plan”. In China, the top-level design of autonomous driving has been strengthened. Documents, such as the “Action Plan for the Development of the Intelligent Connected Vehicle Industry”, “Intelligent Vehicle Innovation and Development Strategy” and “Guiding Opinions on Promoting the Development and Application of Road Traffic Autonomous Driving Technology”, were issued to promote the development and application of autonomous driving technology. In July 2021, the “Management Specification for Road Test and Demonstration Application of Intelligent Connected Vehicles (Trial)” and the “Opinions on Strengthening the Administration of Access of Intelligent and Connected Vehicle Producers and Products”, were released. In August 2022, a notice on the pilot application of high-precision maps for intelligent connected vehicles was released to regulate the testing and application of autonomous vehicles. In December 2021, the State Council of China issued the “14th Five-Year Plan for the Development of a Modern Comprehensive Transportation System”. The document outlined the following aspects: “adhere to innovation-driven development, promote the deep integration of new technologies (such as the Internet, big data, artificial intelligence and blockchain) with the transportation industry, promote the application of advanced technology and equipment and build a ubiquitously interconnected, flexible and collaborative intelligent transportation system with global competitiveness”.

Other countries have also formulated and issued relevant documents and policies for CAV development planning. Since 2016, the U.S. Department of Transportation has successively issued policy documents, such as The Federal Autonomous Vehicle Policy, Automated Driving Systems 2.0: A Vision for Safety and Preparing for the Future of Transportation: Automated Vehicles 3.0. Automated Vehicles 3.0 further relaxed the autonomous driving policies, encouraged the establishment of a unified regulatory framework and operating environment nationwide, improved public participation, supported the development of voluntary technical standards and updated the relevant laws [4]. The European Union believes that the travel mode characterized by collaboration, networking and automation will be the future trend. In May 2018, the European Future Travel Strategy was released. This is an important document to guide the development of CAVs in Europe, which defines the strategic plan for developing autonomous driving and intelligent travel. The development goal of CAVs was also set: by 2022, all vehicles will have communication functions; By 2030, fully autonomous driving will be popularized [5]. In Japan, the automobile information-sharing organization was established. The Approaches for Vehicle Information Security was released in August 2013. The approaches put forward the automobile information security model “IPACar” and formulated the security strategies and measures for each stage of the automobile life cycle, for the information security of CAVs [6].

With the commercialization of 5G communication technology, CAVs can drive in a coordinated adaptive cruise control mode through vehicle-to-vehicle (V2V) communication technology, i.e., cooperative adaptive cruise control (CACC) vehicles. In recent years, in the United States, Japan, the United Kingdom and other countries, a large amount of money and manpower has been invested in the research and application of intelligent transportation. Intelligent transportation uses advanced technology to improve the scientific nature of traffic management decisions, guide rational traffic behavior and fully utilize existing traffic facilities. The real-world vehicle tests in actual road environments show that [7] the gradual popularization of CAVs is conducive to alleviating traffic congestion and improving traffic safety. Building a modern intelligent transportation system has become a common goal for all countries.

Since more traffic information can be obtained during the operation of CACC traffic flow, there are many models describing the car-following behavior of CACC traffic flow at present. The information, including the speed, location, acceleration and deceleration of the front and rear vehicles, allows the models to be more intelligent. However, the adaptability of the models to the car-following headway is limited. Therefore, The main objectives of this study are: (1) to analyze the characteristics of the classic CACC car-following model and then to propose a CACC car-following model that can adapt to various traffic operating conditions in combination with the traffic operation characteristics; (2) to verify the advantages of the improved CACC car-following model, in terms of adaptability, by building a simulation analysis environment and comparing with the traditional CACC car-following model; (3) to analyze the impact of the different model parameters on traffic flow.

This paper is organized as follows. Following the introduction, Section 2 presents the literature review on the relevant studies of the car-following model establishment and stability analysis. Section 3 describes the car-following models and improvements. Section 4 presents the simulation and comparison between the original car-following model and the improved model. Section 5 analyzes the characterization of the improved CACC model. Section 6 summarizes the main findings of this study.

2. Literature Review

The car-following model has always been a research hotspot in traffic flow theory. Modeling car-following behavior can quantify the longitudinal interaction between the following vehicles so as to understand the operating characteristics of traffic flow and reveal the internal mechanism of traffic congestion and other traffic phenomena. With the development of new technologies, such as the Internet of Vehicles and big data, new changes have occurred in the content and methods of car-following modeling. The research on car-following models has made significant progress in content and methodology. In recent years, more scholars have combined a variety of factors affecting micro-driving behavior into the research of car-following models and also applied new model theories to study car-following behavior. They have also applied new machine learning methods to the car-following behavior modeling. These innovations in content and methods have further improved the accuracy of the model. Therefore, this section will introduce the following two parts: (1) car-following study on theoretical derivation and (2) car-following study on data-driven methodology.

2.1. Car-Following Study on the Theoretical Derivation

In the early research of car-following models, scholars were devoted to using different modeling methods to describe the car-following behavior of drivers. However, the influencing factors in the driver’s car-following process have been simplified to varying degrees. In recent years, more scholars have combined a variety of factors that affect micro-driving behavior into the research of car-following models. Some have applied the new theory to the study of car-following behavior. New machine learning methods have also been applied to the modeling of car-following behavior. These innovations in content and methods have further improved the accuracy of the model. A series of representative achievements have been made. Ci et al. [8,9] established a car-following model at signalized intersections, based on V2I and revealed that it can improve the traffic efficiency of the vehicles at the signalized intersection. Wang et al. [10] proposed the car-following model, based on drivers’ psychological characteristics, used the genetic algorithm and NGSIM (new generation simulation) data to calibrate the parameters. Peng [11] proposed an improved multiple desired speed difference model, considering the influence of multi-vehicle information on the following vehicles. Zhang [12] presented a new optimal velocity function incorporating the desired safety distance, to describe the dynamic performance of car-following behavior. Tang [13] improved the full speed difference model by considering the influence of dual lead vehicle information, and thus enhanced the stability of the following vehicle. Zhang [14] conducted a stability analysis of the optimal full speed difference model using the state space method and the system stability criterion, and obtained the stability conditions of traffic flow on a curved road. He [15] established a car-following model, considering the lateral separation and overtaking expectations by introducing lateral separation parameters. For the car-following models, in the context of connected and autonomous vehicles (CAV), the stability analysis method of conventional car-following models is commonly used. For example, Sun et al. [16], Gu et al. [17] and Qin et al. [18] reviewed the stability analysis methods of conventional car-following models and CAV car-following models, in their recent research and these methods were compared and analyzed for these types of models. Qin [19] established a CACC following model, based on a nonlinear dynamic space headway strategy and derived the basic graphical model of a mixed traffic flow at different CACC ratios. Milanes [20] applied the intelligent driving model to mass-produced vehicles and tested the controller response under different traffic conditions.

2.2. Car-Following Study on the Data-Driven Methodology

In recent years, the development of science and technology has facilitated the acquisition of large-scale and high-precision trajectory data [21]. Data-driven car-following models, based on non-parametric methods are gradually developed, such as artificial intelligence [22], machine learning [23] and deep learning [24]. Data-driven car-following models do not adhere to various theoretical assumptions and pursue a strict mathematical derivation. However, these models use non-parametric methods to obtain the internal information of the track data and establish car-following models with a high prediction accuracy [25]. At present, data-driven car-following models are divided into fuzzy logic, artificial neural network and case learning (including the local weighted regression algorithm and the K-nearest neighbor algorithm), support vector regression (SVR) class and deep learning algorithm class. Based on the powerful data learning ability of data-driven models, driving characteristics are extracted from microscopic vehicle trajectory data. Theoretically, as long as the data samples used for training the model are large enough, a data-driven car-following model with a high prediction accuracy can be obtained.

The artificial neural network method establishes a general description, by learning sample data and has a high prediction accuracy. Therefore, using the artificial neural network to model the car-following behavior, has commonly been a research hotspot. The BP neural network [26], gray neural network [27], fuzzy neural network [28] and other methods, have been continuously applied to the car-following behavior modeling. With the progress of neural network technology, researchers pay more attention to deep neural networks with more hidden layers.

Some scholars have used the local weighted regression, K-nearest neighbor [29,30] and other machine learning methods, to model car-following behavior and take the car-following models as case learning models. In the initial training, the model simply stores the training samples. When predicting new instances, the model does not need to estimate the objective function once in the entire instance space, but makes local estimates for the new instances to be solved.

SVR introduces a regression algorithm into the idea of a support vector machine (SVM), which can be used for the regression fitting of vehicle trajectory data. This method follows the principle of structural risk minimization and theoretically has a stronger data learning ability and generalization ability, than the artificial neural network [31].

Deep learning is a branch of machine learning algorithms. Its core comes from artificial neural networks. Deep learning can improve the accuracy of the classification or prediction, by constructing a multi-layer hidden layer model and analyzing the training data. Deep reinforcement learning combines reinforcement learning algorithms and deep learning algorithms to create an agent that can act intelligently in complex situations [32].

All of the above studies have improved the car-following theory and its application from different perspectives. However, the response state of the rear vehicle is still unclear when the driving state of the front vehicle changes. The expected distance in the classic CACC car-following model is fixed and the model adaptability to the operation state of traffic flow is limited. Therefore, in this paper, the CACC model was improved, considering the following behavior of the front and rear vehicles. The car-following effectiveness of the improved model was compared. The influence of the different parameters on the following platoon was analyzed to obtain the time-varying law of the following platoon. This will provide a theoretical basis for developing and improving traffic flow theory.

3. Methods and Methodology

3.1. CACC Model Description

The classical following model was used to describe the influence of different driving environments on the driving behavior of drivers. This model was derived from the driving dynamics model and expressed as the acceleration of the following vehicle. The acceleration is proportional to the speed difference between the front and rear vehicles, and inversely proportional to the space headway between the two vehicles. The acceleration is directly related to vehicle speed. The classical car-following model can completely describe the car-following process. The model is expressed as

$$a_{n+1}(t+T)=\alpha v_{n+1}^m(t+T)\frac{\Delta v(t)}{\Delta x^l(t)}$$

(1)

where $a_{n+1}(t+T)$ indicates the acceleration of the (n + 1)th vehicle at moment (t + T); $\Delta v(t)$ and $\Delta x(t)$ indicate the speed difference and distance between the following and lead vehicles at moment t, respectively; $\alpha$, $m$ and $l$ indicate the constant to be calibrated.

Based on the previous research on the CACC model, a CACC car-following model with a constant headway was proposed at the University of California, Berkeley [9]. In their model, it is considered that the acceleration of the following vehicle depends on three parts: the acceleration of the lead vehicle, the speed differences between the lead and following vehicles and the errors between the actual and desired headways. The constant headway is regarded as the key factor affecting the headway errors. The model is expressed as

$$\begin{gathered} {\ddot{X}}_{n+1}(t+T)=\alpha {\ddot{X}}_n(t)+\beta (X_n(t)-X_{n+1}(t) \\ -t_g\dot{X}(t)-L-S_0)+\gamma ({\dot{X}}_n(t)-{\dot{X}}_{n+1}(t)) \end{gathered}$$

(2)

where ${\dot{X}}_n(t)$ and ${\dot{X}}_{n+1}(t)$ are the velocities of the nth and the (n + 1)th vehicle at the moment $t$, respectively; ${\ddot{X}}_{n+1}(t)$ and ${\ddot{X}}_n(t)$ are the accelerations of the nth vehicle and the (n + 1)th vehicle at the moment $t$, respectively; $X_n(t)$ and $X_{n+1}(t)$ are the positions of the nth vehicle and (n + 1)th vehicle at the moment $t$, respectively; $t_g$ is the desired headway; $L$ is the vehicle length; $S_0$ is the safe stopping distance; $T$ is the reaction time; $\alpha$, $\beta$ and $\gamma$ are the coefficients to be determined.

The PATH laboratory at the University of California, Berkeley, integrated the CACC system into four real-world vehicles. The actual trajectory data of the CACC vehicles were obtained and the parameters of the CACC car-following model were calibrated. The real-world vehicle tests show that the calibrated model exhibited consistent car-following characteristics with the actual vehicles. The parameter values [33] were also recommended: $\alpha$ = 1.0, $\beta$ = 0.2 and $\gamma$ = 3.0. This model has a simple structure and explicit meaning and is the most commonly used car-following model, based on a constant headway.

3.2. CACC Model Improvement

Existing research results have indicated that there is a specific correlation between the headway and the speed of the lead and following vehicles [34], i.e., the desired distance is different under different speed conditions:

(1)

In a free flow state, the expected headway tends to be unconstrained, with a significantly large value.

(1)

In a congested flow state, the expected headway is approximately the sum of the vehicle length and safety distance, i.e., $L$ + $S_0$.

(3)

In a normal traffic flow state, the expected headway should be an increasing function of the vehicle speed.

Based on the above characteristics, the expected headway ($H_d$) can be expressed as

$$H_d=(L+S_0)/(1-{(v(t)/v_0)}^k)$$

(3)

The improved model in this paper fully retains the original CACC model structure and reflects the dynamic change characteristics of the space headway at different vehicle speeds. Substituting Equation (3) into Equation (2),

$${\ddot{X}}_{n+1}(t+T)=\alpha {\ddot{X}}_n(t)+\beta (X_n(t)-X_{n+1}(t)-\frac{L+S_0}{1-{({\dot{X}}_n(t)/v_0)}^k})+\gamma ({\dot{X}}_n(t)-{\dot{X}}_{n+1}(t))$$

(4)

3.3. Numerical Simulation Environment Construction

The simulation environment of the car-following model was constructed, using MATLAB to simulate the operation of the vehicle platoon. It is assumed that: the vehicle platoon consists of five standard cars moving at a certain uniform speed v (m/s); the standard car length is $L$; the initial space headway is $d_0$; at a certain time t, the vehicle platoon is disturbed by traffic. In actual operation, the lead vehicle in the vehicle platoon can be disturbed and cause the corresponding acceleration or deceleration. In this paper, the acceleration and deceleration were regarded as sinusoidal fluctuations. Thus, the sinusoidal fluctuations were applied to the lead car in the simulated vehicle platoon, which determines the acceleration and deceleration status of the following vehicles. The sinusoidal fluctuation function is expressed as

$$v=A\sin (\omega t)$$

(5)

where $A$ indicates the acceleration amplitude of the first vehicle, due to the disturbance, m/s²; $\omega$ is the angular frequency of the disturbance signal, rad/s.

The simulation parameters were arranged as follows: the driving speed of the vehicle platoon, $v_0$ = 10 m/s; the headway, $L$ = 30 m; by considering the driving comfort, the maximum acceleration and deceleration were set as 0.6 m/s² (i.e., the amplitude of the applied sinusoidal disturbance $A$ = 0.6 m/s² and the angular frequency, $\omega$ = 1.0 rad/s).

4. Model Analysis and Simulation

In order to study the effects of the different parameters and simulations on the vehicle platoon in different states, a large number of simulations were conducted in this paper. Only one of the parameters was changed in each simulation to observe the changes in the operating parameters and space headway in the vehicle platoon, and the remaining simulation conditions were unchanged. The parameters were set as follows: simulation duration, 20; car length,$L$ = 5 m; safety distance, $S_0$ = 5 m.

4.1. Speed Variation of the Vehicle Platoon Using the Original CACC Model

The recommended parameters of the following model by Arem [10], were adopted as the input of the simulated vehicle platoon. In order to simplify the simulation process, it is assumed that $k$ = 1 in Equation (4). Based on the vehicle platoon and the arranged disturbance conditions, the speed variation of each vehicle in the platoon is shown in Figure 1.

From Figure 1, the vehicle speed of the platoon showed the same changing trend as that of the lead vehicle. The speed fluctuations gradually decreased, which is consistent with the law of traffic wave transmission. This also indicates that the running state of the rear vehicle depended on the front vehicle. With the increase of the following time, the vehicle speed in the platoon tended to converge to a constant, thus maintaining a stable distance between the vehicles.

4.2. Speed Variation of the Vehicle Platoon, Using the Improved CACC Model

Based on the improved CACC model, the speed variation of each vehicle in the platoon is shown in Figure 2.

From Figure 2, the vehicle platoon showed the same speed-changing trend. The maximum speed difference occurred in the first wave and the subsequent speed difference was gradually reduced. The speed fluctuation of the vehicle platoon gradually decreased until a uniform speed was reached, in order to keep the platoon stable.

4.3. Speed Variation Comparison, before and after the Model Improvement

The second vehicle in the vehicle platoon was selected in order to compare the simulation effectiveness of the two models, as shown in Figure 3. The speed state in the original and improved CACC models varied synchronously with the speed of the lead vehicle. The speed difference between the following vehicle and the lead vehicle decreased as the platoon proceeded, which is related to the initial following distance. The car-following distance of the platoon gradually approached a stable state. It is also found that the speed difference between the vehicles in the platoon from the improved CACC model was smaller at the same time point. The vehicle distance in the improved CACC platoon can converge to a stable state more rapidly with a smaller distance.

The simulation comparison shows that the improved CACC model can better guide the following behavior and reduce the space headway between the following vehicles and provide better vehicle control under the same conditions.

5. Characterization of the Improved CACC Model

5.1. Analysis of the Following Characteristics at Different $k$

For the original CACC model, $k$ = 1 was assumed and the other parameters were unchanged. Then, $k$ = 1, 0.5 and 2 were used as the input of the improved model, respectively, to observe the speed variation of the second vehicle in the following platoon, as shown in Figure 4.

From Figure 4, $k$ did not affect the speed-changing trend of the second vehicle in the platoon, which was the same as that of the lead vehicle. At a k above 1, the following vehicle had a larger speed than the lead vehicle, indicating that the following vehicle tended to further reduce the space headway. At a k less than 1, the following vehicle had a smaller speed than the lead vehicle, indicating that the following vehicle tended to increase the space headway. When an acceptable headway was reached, the speed of the platoon gradually converged.

5.2. Analysis of the Following Characteristics at Different $\alpha$

For the improved CACC model, $\alpha$ was set as 1.0, 0.2 and 2.0, and the other parameters were kept constant. The speed variation of the second vehicle in the following platoon is shown in Figure 5.

From Figure 5, when $\alpha$ > 1.0, the speed fluctuation amplitude and frequency of the following vehicle increased. This indicated that the state of the following vehicle was unstable and the further accumulation of the following vehicle’s state may cause potential accidents. When $\alpha$ < 1.0, the state of the following vehicle slightly changed, and the fluctuation amplitude and frequency of the speed were slightly reduced, The following platoon was stable and had a certain self-healing ability.

5.3. Analysis of the Following Characteristics at Different $\beta$

For the original CACC model, $\beta$ was set as 0.2 and the other parameters were unchanged. For the improved CACC model, $\beta$ were set as 0.2, 0.1 and 0.5, respectively, to observe the speed variation of the second vehicle in the following platoon, as shown in Figure 6.

From Figure 6, $\beta$ did not affect the speed of the following platoon. As $\beta$ increased, the speed fluctuation of the following vehicles in the platoon decreased, indicating that the space headway showed a reciprocal variation. When the lead vehicle was in an acceleration state, the space headway between the vehicles increased gradually. When the speed of the lead vehicle decreased to the same speed as the following car, the headway reached the maximum value and then gradually decreased until reaching the minimum headway. Then, the headway showed a reciprocal variation within a certain fixed range.

5.4. Analysis of the Following Characteristics at Different $\gamma$

For the original CACC model, $\gamma$ was set as 3.0 and the other parameters were kept unchanged. For the improved CACC model, $\gamma$ were set as 3.0, 1.0 and 5.0, respectively, to observe the speed variation of the second vehicle in the following platoon, as shown in Figure 7.

From Figure 7, as $\gamma$ increased, the speed difference between the following and lead vehicles decreased. When the speed reached a stable state, the fluctuation frequency of speed remained stable. At a larger $\gamma$, the time for the following and lead vehicles to reach a stable state was shorter, and the space headway in the stable following state was smaller.

6. Conclusions

From the above discussion, the primary conclusions can be reached as following.

(1)

Based on the dynamic time headway, the classical CACC car-following model was improved. The simulation analysis verified that the improved following model showed a greater car-following effectiveness.

(2)

In the improved CACC model, the disturbance condition of the first car is proposed, and the operation state in the car-following queue is observed. The simulation results show that the space headway in a stable state is negatively related to the speed difference between the front and rear vehicles. The space headway of the queue is positively correlated with the extreme speed; The anti-interference capability of the car-following queue is positively related to the information coordination ability.

The optimization model proposed in this paper may require more experimental verification, especially in the different communication technology scenarios and the high-speed running condition. The dynamic analysis of the expected distance in this study can provide ideas for the future continuous optimization and improvement of the car-following model. Thus, the car-following characteristics of the traffic flow with CAVs can be more accurately analyzed so as to better solve the scientific management and control problem of the CAV traffic flow. This reflects the consistency between the traffic flow car-following modeling and macro traffic flow evolution. It is also shown that the car-following model in this study meets the current research needs of the CAV car-following behavior, which reflects the practical value of this study.

This study is focused on the pure CACC traffic flow, which is also the future trend of traffic flow development. The mixed traffic flow with CAVs and manually driven vehicles will be the traffic flow state for a long time. The car-following characteristics of multiple mixed vehicles need further research. Then, the car-following model is calibrated and verified with the observed data of the mixed traffic flow at different proportions of CAVs. Thus, the car-following theory of traffic flow can be improved to further study the intelligent management of traffic flow in the context of future autonomous driving.