## 4 Construction of Multi-Objective Function of Lane-Changing Trajectory
### Economy  
Here, the so-called economy refers to the energy consumption of the entire car. For electric vehicles, electrical energy is mainly consumed in terms of efficiency of power devices, loss of transmission mechanisms, overcoming driving resistance, and on-board electrical equipment. Due to the fact that losses and other factors are only related to the efficiency coefficient, and the power consumption of on-board appliances and other devices is not closely related to lane changing, research will not be considered in lane changing. The driving resistance closely related to vehicle movement is related to the process of lane-changing behavior. Therefore, this article focuses on optimizing the energy consumption of lane-changing resistance. We will use the energy consumed to complete the entire lane-changing task as an economic indicator. Driving resistance energy consumption is mainly used to overcome driving resistance, including rolling resistance, air resistance, slope resistance, and acceleration resistance. Among them, rolling resistance and slope resistance are mainly related to road conditions and generally do not change during lane change, which can be regarded as fixed values. The generation of acceleration resistance is mainly used to increase vehicle speed. Due to the high-speed driving conditions, the driving resistance is mainly air resistance. So, we only consider the energy consumed by air resistance here.  
\[E\approx F\cdot S=\frac{C_{D}Av^{3}t_{f}}{21.15} \tag{18}\]  
In the above Equation (18), \(E\) is the energy consumed in the process of changing lanes, \(F\) is the air resistance, \(S\) is the displacement of the vehicle in the process of changing lanes; \(C_{D}\) is the air resistance coefficient, which is generally between 0.30 and 0.41 for cars [32]; \(A\) is the windward area of the car, which is generally 1.7-2.1 m\({}^{2}\)[32]; \(v\) is the speed of the car, and \(t_{f}\) is the lane change time.  
In summary, we establish the final multi-objective function as follows, see Equation (19).  
\[\begin{array}{l}\min J=\beta_{1}\frac{\sqrt{\ddot{x}(t)^{2}+\ddot{y}(t)^{2}} }{a_{max}}+\beta_{2}\frac{t_{f}}{t_{max}}+\beta_{3}\frac{C_{D}A\left(\sqrt{ \ddot{x}(t)^{2}+\ddot{y}(t)^{2}}\right)^{3}dt}{21.15E_{max}}\\ \text{s.t.}\quad 0<y(t)<w\\ \quad 0<\dot{x}(t)<v_{x,\max}\\ \quad 0<\dot{y}(t)<v_{y,\max}\\ \quad\left|\ddot{x}\right|<a_{x,\max}\\ \quad\left|\ddot{y}\right|<a_{y,\max}\\ \quad MSS(i,M)+d_{0}<d(i,M)\\ \quad\beta_{1}+\beta_{2}+\beta_{3}=1\end{array} \tag{19}\]  
The three items in Equation (19) represent the acceleration, the lane-changing time, and the energy consumption during the lane-changing, respectively, where \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\) are the weight coefficients of each objective in the multi-objective function with the values all between 0 and 1, and the sum of them is 1. As mentioned above, \(t_{f}\) is the lane change time, \(\dot{x}(t)\) and \(\dot{y}(t)\) are the speeds in the longitudinal and lateral directions of the lane change, respectively; \(\ddot{x}(t)\) and \(\ddot{y}(t)\) are the longitudinal and lateral accelerations of lane changing, respectively, \(a_{max}\), \(t_{max}\), and \(E_{max}\) are the maximum acceleration, the maximum lane-changing time, and the maximum energy consumption constraint value, respectively, which will be used to normalize the three items on the right side of the objective function. \(a_{max}\) can be obtained from the lateral and longitudinal acceleration thresholds, and \(t_{max}\) is the maximum time for changing lanes, which is generally taken as 6 s; \(E_{max}\) can be obtained from the maximum lane change time and the end speed of the lane change. In different lane-changing scenarios and conditions, we can adjust the three weight coefficients to find the optimal lane-changing time and complete the lane-changing.