The modern traffic system is one of the most important symbols of social modernization. Nevertheless, traffic problems such as the pollution of fog and haze, traffic congestion and fuel consumption have influenced human daily life. To address these problems, there has been a lot of studies on capturing the complex mechanisms behind the phenomena of vehicular traffic flow from the microscopic or macroscopic viewpoints. Consequently, various driving assistant systems, such as adaptive cruise control (ACC) and cooperative adaptive cruise control (CACC), have been introduced over recent decades to improve driving efficiency and safety. Studies have been done on investigating the mechanism of traffic flow as much as possible, and some interesting results have been acquired.^[1–5]

To date, some regional traffic signal control systems have been used to enhance driving efficiency since the 1960s. The early Urban Traffic Control System (UTCS) project was completed in the US. Afterwards, the Split, Cycle and Offset Optimization Technique (SCOOT) was developed by the Transport and Road Research Laboratory (TRRL) in the UK.^[6] Now, this system has been introduced in several cities of China due to its good performance in application. Sydney Coordinated Adaptive Traffic System (SCATS) was developed by New South Wales Roads and Transport Authority (NSWRTA) in Australia.^[7] These systems have been successfully introduced to improve traffic conditions.

Many car-following models have been proposed to investigate traffic behaviors on a single lane without overtaking.^[8–10] The early linear car-following model is presented by Chandler et al.,^[11] and a nonlinear model is put forward by Pipes.^[12] Newell^[13] extended a car-following model with a differential equation and gave a graphic description of the optimal velocity (OV) function. Konishi et al.^[14] proposed the famous chaotic car-following model by setting the time delay feedback control signals, and provided further results through theoretical analysis. Subsequently, Bando et al.^[15] presented the classical microscopic traffic model called optimal velocity model (OVM). Later, much work has been done based on the optimal velocity model by introducing multiple comprehensive information into relative velocity or headway in various ways.^[16–20]

As a famous traffic control system, the adaptive cruise control(ACC) system allows drivers to keep a suitable headway or velocity with respect to the preceding vehicle based on the sensor devices.^[21] Studies have shown that traffic jams can be suppressed in the mixed traffic of human-driven cars with the adaptive cruise control cars constituting at least 20% of the traffic flow.^[22] To further improve highway throughput and safety, cooperative adaptive cruise control system was introduced in 1997 to integrate the adaptive cruise control system and wireless communication. Later, many studies have shown the advantages of using signals received from multiple cars farther ahead.^[23–25]

The cooperative adaptive cruise control system, which can receive more comprehensive information from multiple vehicles ahead via vehicle to vehicle communication or radar sets, has gained more and more interest of scholars. So far, many studies have been done in this research field. Tang et al.^[26] proposed an improved car-following model considering inter-vehicle communication and extended the OV model by taking account of the effect of the driver’s memory. Based on the full velocity difference (FVD) model, Peng and Sun^[27] presented a car-following model considering multiple preceding cars’ information in 2010. Li et al.^[28] investigated the intelligent driver model and carried out its numerical simulation under the open boundary condition. All in all, much literature has shown the great potential and broad prospects of the cooperative adaptive cruise control system to suppress traffic congestion.

Up to the present time, much work has been done on the traffic with ACC/CACC vehicles.^[29,30] Nevertheless, there is little study of the CACC system from the viewpoint of control methods. Moreover, the safety distance should be related to the velocity of the considered car. In this study, some vital comprehensive information, such as multiple preceding cars’ speed differences and headway, variable safety distance (VSD) and time-delay effect are investigated. Further details are given in Section 2 and Section 3.

The organization of this paper is as follows: In Section 2, a comprehensive control system which can handle three traffic conditions to guarantee driving efficiency and safety is designed. In Section 3, the stability criterion for VC model and GC model are obtained by using linear stability theory. In Section 4, numerical simulations are carried out to validate the theoretical results. Some conclusions are given in Section 5.

The existing car-following models can be formulated as follows

To better study the car-following mechanisms in the cooperative adaptive cruise control (CACC) system, an improved control system considering more comprehensive information is presented, which consists of three models: (i) velocity control (VC) model, (ii) gap control (GC) model, and (iii) safety control (SC) model. The longitudinal control system is typically designed as a hierarchical two level controller. The upper level controller is determined by the desired acceleration while the lower level controller is determined by the throttle or brake commands required to track the desired acceleration.^[31] In this paper, we also adopt a hierarchical two-layer controller for CACC as shown in Fig. 1.

Fig. 1.

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	Fig. 1. The controller and the diagram of interaction between the three control models.

The VC control model can capture multiple preceding cars’ headway and velocity difference based on sensor devices. So, vehicles can maintain the desired velocity . In the GC mode which captures following behavior, vehicles follow its preceding vehicle and maintain an appropriate space with constant time gap T_g. In the SC mode which captures driving in emergency conditions, maximum deceleration is applied to avoid vehicle collisions. The desired acceleration for these three modes are a_v, a_g, and a_s respectively. When g ≥ g_safe (g is space gap and g_safe is safety space gap), there is no risk of collision and vehicles operate in VC or GC mode. The intended acceleration is determined by the minimum of a_v and a_g, thus a vehicle can operate without exceeding the desired velocity or getting too close to its preceding vehicle. For safety concern, vehicles will switch into SC model when g < g_safe. Then, a comprehensive control system which can handle different traffic conditions to ensure driving efficiency and safety is presented based upon these three models.

2.1. Velocity Control Model

The velocity control model considering multiple preceding cars’ headway and speed differences is constructed. The intended acceleration for this model can be expressed as follows

where a_v is intended acceleration and K_sc is the the speed control gain; is the extended optimal velocity (EOV) function and h is the typical safety distance; T_v is the time step unit and d is the reaction coefficient for v_n(t). The physical meaning of h_v is the acceptable safety distance of a driver. ȳ_n(t) is named as the weighted headway and λ_i, γ_j are weighting factor of Δx_n+i−1(t) and Δv_n+i−1(t), respectively; λ = 2,3,4,…, γ = 2,3,4,… being a constant. In real traffic flow, the farther a car is away from the current car, the weaker the influence. So we can get λ_i+1/λ_i < 1, i = 1,2,…,m, and γ_j+1/γ_j < 1, j = 1,2,…,m, λ_i, γ_j are the monotone decreasing function of i and j, respectively.

2.3. Safety control model

When the space gap is smaller than the safe gap (g < g_safe), the vehicle will switch to SC model. The function of SC model is to avoid vehicle collisions by applying the maximum deceleration. The safe gap g_safe is determined by velocity and maximum deceleration ability of the current and the preceding vehicle.

where τ is delay time and a_max, are maximum deceleration of the considered vehicle and the preceding vehicle.

In this simulation, several numerical simulations are carried out to verify the model by simulating the real traffic flow. We just consider some simple situations for the presented models. The parameters for the vehicles are set as K_sc = 2 s⁻¹, d = 0.5, v* = 15 m/s, y* = 5.5 m, v_max = 20 m/s, and T = 0.1 s. It is assumed that all vehicles have the same parameters in the traffic system. Particularly, in Fig. 2, the parameters are set to be λ = γ = 3, v_max = 20 m/s and in Fig. 3, the parameters are set to be λ = γ = 3, τ = 0.25. The initial condition is the steady state for the model, and the initial positions and speeds are set as , n = 1,2,3,…,N, and N = 120 is the total number of vehicles. It is consider a case where the leading vehicle stops suddenly for t = nT = 100–103.

Fig. 2.

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	Fig. 2. Numerical simulations for the VC model with (a) τ = 0.25, (b) τ = 0.5.

Fig. 3.

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	Fig. 3. Numerical simulations for the VC model with (a) vmax = 21.5 m/s, (b) vmax = 20.5 m/s.

Figure 2 illustrates the speed–time patterns of the 1st, the 25th, and the 50th vehicles with different parameter values of τ in VC model. It can be seen from Figs. 2(a) and 2(b) that, as the coefficient τ decreases from 0.5 to 0.25, the stability of the traffic system is strengthened. The vehicles can reach a steady running state in a relatively short time as τ decreases.

Figure 3 illustrates the space–time plot of the traffic system for the VC model. It can be observed that the area of a traffic jam will be aggravated and the arrival time of the steady state will be delayed as the speed of the vehicle increases.

The simulation results in Fig. 4 show the advantage of the VC control model. As reaction coefficient γ = 3, λ = 3, the area of the traffic jam is weakened and the arrival of the steady state will be in advance, when compared with the case γ = 2, λ = 2,. That is to say, it is very useful for guaranteeing driving efficiency and safety if vehicles capture multiple preceding cars’ information.

Fig. 4.

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	Fig. 4. Numerical simulations for the VC model with (a) τ = 0.25, γ = λ = 2, (b) τ = 0.25, γ = λ = 3.

Now we consider the system with GC control model. The simulation results in Fig. 6 show that as we consider the preceding car’s acceleration, the area of traffic jam is weakened and the arrival of the steady state is quicker when compared with Fig. 5. In addition, the amplitude of the speed fluctuation for the 25th vehicle decreases and the 50th vehicle runs more smoothly in Fig. 6. In other words, the preceding car’s acceleration is an important factor in the vehicle control system.

Fig. 5.

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	Fig. 5. (a) Space-time plot of the traffic system. (b) Temporal velocity behavior of the 1st, 25th, and 50th vehicles. (Kv = 0.85, Kg = 0.8, Ka = 0.35, τ = 0.25 s).

Fig. 6.

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	Fig. 6. (a) Space–time plot of the traffic system. (b) Temporal velocity behavior of the 1st, 25th, and 50th vehicles. (Kv = 0.85, Kg = 0.8, Ka = 0.75, τ = 0.25 s).

Based upon the above simulation results, it can see that the extended control system considering more comprehensive information from multiple preceding vehicles ahead can suppress the increasing traffic jams and enhance traffic safety.

In this study, a comprehensive control system which can handle different traffic conditions to guarantee driving efficiency and safety is designed based on the classical cooperative adaptive cruise control theory. In this control system, some vital comprehensive information (such as multiple preceding cars’ speed differences and headway, variable safety distance (VSD) and time-delay effect) have been investigated. The stability criterion for the VC model and GC model are derived via linear stability theory. Several simulations are carried out to verify the model by simulating the real traffic flow, and the results prove that the extended control system can suppress traffic jams and enhance traffic safety.