Speed ​​trajectory optimization scheme for connected autonomous vehicles passing through signalized intersections

Article Summary

Taking the Connected Autonomous Vehicle (CAV) as the research object, the speed trajectory optimization control strategy of the CAV fleet passing through the urban signalized intersection is studied. Based on the optimal control theory, the CAV autonomous driving model is used to describe the vehicle-to-vehicle interaction, and the total fuel consumption of all CAV vehicles during driving is taken as the optimization target. The model constraints are established according to the timing information of the signal lights. By optimizing the speed trajectory of the CAV lead vehicle, the entire CAV fleet is guaranteed to pass the intersection quickly under the green light phase and achieve the minimum fuel consumption.

为了对该优化控制进行高效求解，采用离散Pontryagin极小值原理建立最优解的必要条件，利用基于神经网络训练的弹性反向传播（Resilient backpropagation，RPROP）算法设计了数值求解算法。多个典型场景的仿真结果显示：整个CAV车队均能在不停车的情形下通过信号交叉口，避免因在红灯时间窗到达停车线造成的停车、启动等过程，总油耗量最高可减少69.74%。该控制方法利用网联自动驾驶技术的优势，显著改善了城市交通通行效率和燃油经济性。

Proposed method

1. CAV speed trajectory optimization model

This paper adopts the automatic following algorithm proposed by C. Letter to describe the driving behavior of CAV, as shown in the following formula.

Where: i = 1, 2, …, n; a**i ( t ) is the instantaneous acceleration of the i-th vehicle at time t, m/s ^2^; s**i ( t ) is the instantaneous displacement of the i-th vehicle at time t, m; v**i ( t ) is the instantaneous velocity of the i-th vehicle, m/s; h**t is the expected headway, s; d0 is the minimum safe vehicle spacing, m; k1 and k2 are model parameters.

Taking the displacement and velocity of each vehicle in the CAV convoy as the state variables (dimension is 2n), x =[s1v1s2v2 … snv**n ] ^T^, and the acceleration of the CAV lead vehicle u, m/s ^2^, as the optimized control variable, we can establish the state equation model of CAV convoy driving: x ˙( t )= *f * [ *x* ( *t* ), *u* , *t* ], as shown in the following formula.

In order to calculate the fuel consumption of the vehicle during driving, a fuel consumption model needs to be established. There are many vehicle fuel consumption models. Considering that the calculation amount of the model will increase with the increase of the number of vehicles in the CAV fleet, in order to improve the calculation efficiency, the author uses the COPERT fuel consumption model based on average rate developed by the European Environment Agency (EEA), as shown in the following formula.

Where: ρ1, ρ2, ρ3 are COPERT model parameters.

The total fuel consumption of all CAV vehicles is taken as the objective function of optimization control, as shown in the following formula.

Where: t0 and tf are the initial and terminal time of control, respectively, s.

Using V2I communication, CAV obtains the signal timing of the intersection, assuming the green light phase period [tg1, tg2] closest to the current time, and establishes the system constraints as shown in the following formula.

The above formula constrains the displacement of the head vehicle of the CAV convoy at time tg1 and the displacement of the tail vehicle at time tg2, ensuring that all vehicles in the CAV convoy can pass through the intersection during the green light phase period [tg1, tg2].

Considering the capacity limitations of the car, the driving speed needs to meet the constraints, see the following formula.

Where: vmin is the minimum vehicle speed, m/s, vmax is the maximum vehicle speed, m/s.

In summary, equations (2) and (4) to (7) are CAV speed trajectory optimization models based on optimal control. By optimizing the speed trajectory of the CAV lead vehicle, it can be ensured that all vehicles in the CAV fleet can efficiently pass through the intersection under the green light phase, while minimizing the total fuel consumption of all vehicles and improving fuel economy.

2 Solution method based on RPROP

2.1 Necessary conditions for the optimal solution

In order to efficiently solve the above optimal control problem on a computer, it is discretized, and then the inequality constraints are processed by the penalty function method. By introducing Lagrange multipliers, it is converted into an unconstrained optimal control problem. Finally, the Pontryagin minimum principle of discrete systems is used to establish the necessary conditions for the optimal solution.

With Δt as the discrete step length, [0, tg2] as the optimization control cycle, there are K discrete moments in total, kg1 corresponds to the start moment tg1 of the green light time window, and the end moment tg2 of the green light time window is the control end moment K. The CAV fleet dynamic model is discretized using the difference method to obtain the following form.

According to the model constraints, the following penalty function is constructed:

The penalty factor is introduced to construct the augmented objective function of discrete optimal control:

Where: is the penalty factor.

Construct the Hamiltonian function as shown below.

The necessary conditions for establishing the optimal solution are established using the Pontryagin minimum principle.

2.2 Solution algorithm based on RPROP

The Resilient Back Propagation (RPROP) algorithm based on neural network training is used to construct the gradient direction and design the solution algorithm. The basic idea is: starting from a certain initial value, the gradient direction is obtained according to the Pontryagin minimum principle (see Section 2.1), and the search step size is dynamically updated according to the historical gradient information to speed up the solution. The solution algorithm based on RPROP is designed.

The conditions for the minimum of the discretized system are:

In the process of searching for solutions, the RPROP method determines the search direction according to the gradient sign and dynamically adjusts the search step size according to the results of the search process, which can ensure the rapidity of the solution. Although the RPROP method cannot guarantee convergence to the global minimum, it can often obtain a relatively satisfactory optimal solution in practical applications.

The gradient value h (k) can be expressed as follows based on the Hamiltonian function.

The iterative formula for establishing the control vector is shown in equation (17).

Simulation experiment

Through simulation experiments, the CAV speed trajectory optimization strategy designed in this paper is verified. In order to facilitate comparative analysis, the results before and after optimization control are compared and analyzed. In the simulation process, it is assumed that there is a CAV fleet of 5 vehicles 550 m upstream of the intersection stop line, with an initial speed of 10 m/s, an initial acceleration of 0 m/s^2^, and a maximum communication distance of V2I of 350 m (point A). In other words, once the CAV enters the communication range of the ICU, it can communicate with the roadside infrastructure.

At the initial moment, the signal light is in the green light phase, and the green light time window is [0, 30 s]. The simulation results are shown in Figure 1.

As shown in Figure 1, in this scenario, the convoy initially travels at a speed of 10 m/s. Without optimization control, the CAV vehicle continues to travel at a constant speed, misses the first green light phase (time window is [0 s, 30 s]), and reaches the stop line at the red light phase time, and has no choice but to stop and wait for the next green light phase. After adopting the speed trajectory control strategy designed in this paper, once the CAV enters the communication distance of the ICU, it can receive the signal timing information broadcast by the ICU and start trajectory optimization. During the optimization process, the CAV leader not only considers the vehicle itself, but also the traffic efficiency of the entire convoy. Therefore, the CAV leader begins to accelerate, so that all CAV vehicles in the convoy can pass through the intersection at the first green light phase without stopping and waiting.

由于优化过程中考虑了燃油经济性的优化，故加速过程中并未出现较大幅度的变速（整个运动过程中，0 m/s ^2^ ≤ a ≤2.93 m/s ^2^ ），经过COPERT模型的计算，总油耗量减少了69.74%。可见，通过本文设计的轨迹优化控制策略，CAV头车及时地根据信号配时信息进行轨迹优化，保证车队所有车的行驶效率，避免了因在红灯时间窗到达停车线造成的减速、停车、加速行为，显著减少了燃油消耗量。

Figure 1 Comparison of simulation results for scenario 1

Reading experience

This paper constructs a speed trajectory optimization control model for CAV vehicles passing through signalized intersections based on optimal control, and uses the Pongryagin minimum principle of discrete systems to establish the necessary conditions for the optimal solution. The solution method is designed using the RPROP algorithm method, which improves the solution speed while ensuring the solution quality.

仿真结果显示，CAV根据基于V2I通信获得实时信号配时信息，提前对自身速度轨迹进行调整，保证所有CAV车辆在绿灯相位时间窗无停车通过信号交叉口，避免因在红灯时间窗到达停车线造成的减速、停车、启动加速等过程，显著减少了所有车辆的总油耗，提高了通行效率。由于本文只考虑了CAV在单车道行驶的情况，在未来的研究中，将进一步研究CAV在多车道的行驶情况，考虑CAV的换道行为，对CAV的速度轨迹进行优化。