## 6 Simulation Solution  
### Free Lane Change under Barrier-Free Traffic Vehicle Scenarios  
When intelligent electric vehicles are driving on highways, they generate the intention to change lanes, and if there are no other vehicles around, they can freely change lanes. The vehicle used in the simulation experiment has a length of 4.2 m and a lane width of 3.75 m. The speed of the lane is 25 m/s, and the speed at the end of the lane change is 30 m/s. According to the aforementioned weight design section, the allocation of weight coefficients for each indicator is provided. When solving multi-objective functions, the obtained comfort demand weight vector \(W_{11}\), efficiency demand weight vector \(W_{12}\), and economic demand weight vector \(W_{13}\) can be replaced with the objective function weight \((\beta_{1},\beta_{2},\beta_{3})\) to calculate the optimal lane change time in each case, the value obtained from the AHP algorithm used in Section 5 is Equation (21). In order to obtain the accurate lane-changing time, here, we select the particle swarm optimization algorithm (PSO), the basic idea of which is to find the optimal solution of the function through the individual extreme value and the global optimal value in the group, and this function is the fitness function. Equations (24) and (25) below are the solution formulas of particle swarm optimization (PSO). \[V_{i}^{t+1}=\omega V_{i}^{t}+c_{1}r_{1}\big{(}\mathit{Pbest}_{i}^{t}-X_{i}^{t} \big{)}+c_{2}r_{2}\big{(}\mathit{Gbest}_{g}^{t}-X_{i}^{t}\big{)} \tag{24}\]  
\[X_{i}^{t+1}=X_{i}^{t}+V_{i}^{t+1} \tag{25}\]  
where \(\omega\) is the inertia weight; \(\mathit{Pbest}_{i}^{t}\) is the historical optimal solution searched from the \(i\)th particle to the \(t\)th generation; \(\mathit{Gbest}_{g}^{t}\) is the optimal solution currently searched by the entire particle swarm; \(X_{i}^{t}\), and \(V_{i}^{t}\) are the current position and flight speed of the \(i\)th particle, respectively; \(c_{1}\) and \(c_{2}\) are acceleration factors; \(r_{1}\) and \(r_{2}\) are random numbers between [0, 1]. We set the population size to 10 particles; the maximum velocity is 0.5, \(c_{1}\) and \(c_{2}\) both are 2. The fitness function is a multi-objective function. As shown in Figures 4 and 5, the solution method achieved stability after six iterations. The objective function J curve in Figure 5 shows that the PSO algorithm has successfully converged to the optimal value. The PSO algorithm provides the optimal solution and value of the objective function in the form of particle position and fitness function value. The optimal solutions given are the optimal lane changing time under comfort demand is 5.2 s, the optimal lane changing time under efficiency demand is 2.8 s, and the optimal lane-changing time under economic demand is 2.9 s. From Figure 5, it can be seen that the objective function value at this time has indeed reached its lowest point under the three demands. In the obstacle-free traffic scenario, based on the optimal lane change time and Equation (11), the optimal lane change trajectory and lane change speed of the lane change vehicle under various demands are obtained by simulating the theoretical curve using MATLAB, as shown in Figure 6a,b. The yaw rate and lateral acceleration of lane-changing vehicles are shown in Figure 6c,d. Figure 4: Particle swarm optimization (PSO) iteration. The simulation results in Figure 5 indicate that the planned trajectory and speed are smooth and consistent with the actual lane-changing situation. The efficiency demand and economic demand result in shorter lane-changing times and more rapid lane changing. The efficiency demand aims to shorten the lane-changing time and achieve driving needs through faster acceleration. The economic demand is also achieved by shortening the lane change time to reduce energy consumption during the lane change period. The trajectory obtained under comfort demand has a longer lane changing time and smoother  
Figure 5: Lane change time solution in barrier-free scenarios.