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In order to deeply analyze and describe the characteristics of car-following behaviour of turning vehicles at intersections, the features and application conditions of classic car-following models were analyzed firstly. And then, through analysing the relationship between the maximum velocity of car-following vehicles and the turning radius of intersection, the differences in key variables between turning and straight car-following behaviour were identified. On the basis of Optimal Velocity (OV) model, a Turning Optimal Velocity (TOV) car-following model with consideration of turning radius and sideway force coefficient at intersections was developed. PreScan simulation was employed to build the scene of turning car-following process at an intersection. Based on linear stability analysis, the stability conditions of the TOV model were derived. And it was found that (1) the turning radius has a significant effect on the car-following behaviour of turning vehicles at intersections; (2) with the increase of the distance between vehicles, the driver’s response sensitivity coefficient increases and then decreases and reaches the maximum value when the distance reaches the minimum safe distance; (3) with the increase of turning radius, the stability of the car-following fleet tends to decrease, and it is more likely to become a stop-and-go traffic flow. In addition, the numerical simulation results indicate that the TOV model can describe the car-following behaviour of turning vehicles more accurately with consideration of turning radius. The findings of this study can be used in the development of microscopic traffic simulation software and for improving traffic safety at intersections.

With the rapid development of economy and the acceleration of urbanization, traffic pollution, traffic congestion, and other related problems have become social hotspots [

An appropriate space gap between vehicles in the car-following state is difficult to determine. Leaving sufficient space with the vehicle in front can ensure safety, but this gap increases the probability of cut-ins by other vehicles. Dou et al. [

Car-following model has been studied from multiple perspectives for nearly seven decades. It can be classified from two perspectives: traffic engineering aspect and statistical physics aspect. Car-following models from the traffic engineering aspect include Stimulus-Response models, Safety Distance models, Psycho-Physical models, and Artificial Intelligence models. The statistical physics aspect includes Optimal Velocity (OV) models, Intelligent Driver models, and Cellular Automata models. Among them, the OV model was proposed by Bando et al. [

The core idea of the OV model is to optimize the optimal velocity of the following car according to headway spacing. It is not sufficient to only consider the effect of the single factor on the car-following behaviour. Hence, traffic scholars have added some variables to the OV model such as headspace, velocity difference, and the acceleration characteristics of the leading vehicle [

A new car-following model was established to address the effect of the optimal velocity changes with memory. Both linear stability and nonlinear analyses were performed. And it was found that the stability of the traffic flow could be enhanced by considering the influence of memory on the optimal velocity of car-following behaviour [

It was also found that the following vehicle is influenced by the leading vehicle and itself, which can be expressed by the space headway [

In recent years, driver information navigation system, intelligent transportation system, and intelligent cruise control system have been developed rapidly. Car-following model is playing a more important role in developing advanced driver-assistance systems by accurately modeling driver’s car-following behaviour from a microscopic perspective [

However, most of the existing models are suitable for the straight traffic, which are difficult to accurately describe the car-following behaviour of turning vehicles at intersections. The car-following behaviour of turning vehicles is significantly different with that of through vehicles, due to the influence of turning radius, sideway force coefficient, and other related factors. Hence, it is necessary to improve the classical model according to the differentiated influencing factors. In this paper, the turning radius of intersection and sideway lateral force coefficient will be introduced into the OV model, and a Turning Optimal Velocity (TOV) car-following model at intersections will be proposed. The linear stability analysis and numerical simulation experiments will be carried out to verify the validity of the proposed model.

Taking left-turn vehicles at intersections as an example, the basic scenario of car-following behaviour is shown in Figure

The basic mode of turning car-following behaviour at an intersection.

In the initial acceleration stage, the following cars, which are influenced by the vehicle operation rules, are more likely to travel at the similar speed of the leading car. To minimize the speed difference among vehicles, drivers might pay less attention to their current speeds. During car-following process, they might pay more attention on the headway space from the vehicle ahead to ensure driving safety. In the dissipation process, the headway increases gradually, and the velocities of following cars begin to show some discrete characteristics.

The OV model, which uses acceleration to describe the car-following behaviour, is easy to understand and can accurately present the running state of vehicles in the through lanes. Hence, it conforms pretty well to the traffic flow in reality and becomes one of the most important models to analyze the car-following behaviour. The OV model can be expressed by the changes in acceleration, which is given as follows:

In 1995, Bando first used the optimal velocity function of space headway to determine the dynamic change process of the optimal velocity. And the OV model was established, which can be expressed as follows:

The OV model was established mainly for through lanes, which did not consider turning radius, road friction coefficient, and other turning-related factors. Therefore, it could not accurately describe the car-following behaviour of turning at intersections. And thus, Turning Optimal Velocity (TOV) car-following model was established for turning vehicles at an intersection:

As to left turn lanes at intersections, the super elevation can be ignored; thus,

Substituting equation (

In order to accurately examine the car-following characteristics of turning vehicles at an intersection, an actual scene was selected for analysis. On April 18, 2018, the traffic data of left turning vehicles were collected using a digital video camera, at the intersection of Xincheng Avenue and Boxue Road, Changchun City, including a total of 67 cycles and 804 vehicles. An individual photo was extracted every five frames (at 0.2 s interval) of a video clip. Left-turn vehicles were numbered sequentially (1, 2, ...,

Professor Ibuyama of the Kyoto University of Japan found that, under complex road conditions, significant changes can be made in driver’s body function because driver’s nerve center and sympathetic nerve are being stimulated by overcomplicated traffic information. The results showed that the pulse would increase, the blood flow rate would accelerate, and the blood pressure would rise in challenging driving conditions.

The influence of turning lane design factors on vehicle operation at intersections is mainly reflected in vehicle velocity and acceleration. In order to make a safe turn, drivers must drive carefully so as not to take emergency avoidance or braking measures in case of sudden accidents. For safety reasons, critical values of maximum speed at different turning radii were suggested, as shown in Figure

The relationship between velocity and radius: (a) variation of maximum velocity under different turning radii and (b) the frequency distribution of velocity, when radius is 40 m.

In Figure

In order to verify the effectiveness of the model, PreScan simulation software was used in this study. By using the GUI and the traffic environment model (including grassland and trees), infrastructure model (including some highway sections, signs, and buildings), traffic actors’ model (including various vehicle models and pedestrian models), and sensor models (laser/lidar, camera, millimeter wave radar, infrared, etc.), the turning traffic scene at intersection was built. Millimeter wave radar, laser radar, and vehicle communication sensor were installed on the vehicle to obtain the surrounding driving environment parameters. Building the control system model correctly is the core to demonstrate the normal driving of vehicles properly. The simulation model of the control system was established in Simulink window, and then the model and control algorithm were transformed into Simulink logic diagram. According to the designed Simulink logic diagram and software co-simulation, the driving process and effect diagram of the vehicle can be observed in the 3D viewer interface of PreScan, as shown in Figure

Simulation of the car-following process of turning vehicles at an intersection.

According to the car-following characteristics, the linear stability of the car-following queue is directly related to traffic efficiency and safety. It is necessary to determine whether the following vehicles in a platoon are stable or not, when driver’s operating characteristics or the geometric design elements of intersection have changed. On the one hand, the driving states of relative vehicles are needed to be determined. For example, if the fluctuation of the distance between two vehicles is large, the car-following fleet will be unstable, and it will be stable otherwise (called local stability). On the other hand, the speed change of the leading vehicle affects the vehicles behind, which has an influence on the stability of the traffic flow. If the velocity fluctuation becomes larger, the car-following queue will be unstable. Otherwise, the car-following fleet will be stable (called asymptotic stability).

Assuming that the initial state of the current traffic flow is steady, all the vehicles move at the same distance

Suppose that

Substitute equation (

Following the expansion of

Based on the long wave expansion, the stability condition of the model is derived by substituting into equation (

Suppose

The real and imaginary parts of equation (

Substituting equation (

At the same time, let

The stability conditions of the TOV model are expressed by

When the stability conditions (equation (

The derivation of

Substituting equation (

According to the linear stability condition (equation (^{−1}) of driver’s response is a dependent variable, which reflects the change of driver response caused by the change of independent variable (

Medium linear stability conditions of the TOV car-following model.

In Figure

When the turning radius of intersection is fixed, the sensitivity coefficient increases rapidly and then decreases rapidly with the increase of distance between vehicles. This is because when the distance between the front and rear vehicles is lower than the safety distance. It means that the front and rear vehicles are very close, and the two vehicles are running at low speed or queuing state. When the vehicle stops completely, although the distance between the front and rear vehicles is very small, the driver’s response sensitivity is very low. When the distance between the front and rear vehicles increases gradually and approaches the safe distance gradually, the driver’s sensitivity coefficient gradually increases. Drivers always pay attention to the dynamical changes of the front vehicle and make corresponding response, accelerating with the acceleration of the front vehicle or decelerating with the deceleration of the front vehicle. When the distance between two vehicles exceeds the safety distance, the driver response sensitivity coefficient is also at a relatively high value. However, with the increase of distance, the influence of the running state of the front vehicle on the running state of the rear vehicle gradually decreases, and the sensitivity coefficient gradually decreases until it tends to a small value.

In order to further illustrate the stability of the traffic flow in the TOV car-following model and verify the correctness of the theoretical analysis, the corresponding numerical simulation experiments were carried out. Under the critical conditions, the length of turn lane

Given driver response sensitivity coefficient

The fluctuation of velocities in different car-following models: (a) the fluctuation of velocity in OV model and (b) the fluctuation of velocity in TOV model.

As shown in Figure

Figure

Obviously, compared to Figure

The simulation results show that the turning radius has a strong influence on the stability of turning traffic stream at intersections. Under the influence of disturbance, the TOV model with consideration of turning radius reaches a more stable state than the OV model. It indicates that the model can describe the car-following behaviour more accurately with consideration of turning radius.

This study analyzed the characteristics of car-following behaviour in turning process at intersections and proposed a Turning Optimal Velocity (TOV) car-following model with consideration of turning radius. The main findings are summarized as follows:

With consideration of the turning radius, the sideway force coefficient, and other turning-related factors at intersections, the TOV car-following model can more accurately describe the car-following behaviour of vehicles in a turning process.

Linear stability analysis of the TOV car-following model shows that the stable area of traffic stream increases with the decrease of the turning radius of intersection. With the increase of turning radius, the following vehicle is more sensitive to the stimulation of the car in front of it. The turning radius of intersection has a significant effect on the stability of the traffic flow. Numerical simulation experiments further verify the influence of turning radius on the stability of turning traffic flow at intersections. This paper determined the performance and description of car-following behaviour under the turning condition at intersections. It effectively shows the influence of turning radius and other key factors on car-following behaviour. These findings can be used in the development of traffic simulation software and for improving traffic safety at intersections. In future work, we will be interested in analysing the effect on right-turn traffic flow at intersections. We will also be interested in improving the TOV car-following model by considering more factors, such as weather condition and super-high transverse gradient of highway.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation (Grant nos. 71901134 and 51942806), National Science Fund for Distinguished Young Scholars (Grant no. 51925081), and Jiangsu Postdoctoral Research Funding Scheme (Grant no. 2018K118C).