## 4 Simulation Results and Case Study  
### Simulation Results and Visualization  
To verify the model's validity, we first established the ramp merging scenario in SUMO and simulated the proposed model. In our simulation, the total length of the mainline was 1000 m, of which the length of the lane-changing zone was 350 m, the length of the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Description** \\ \hline \(v_{l,n,t}\) & m/s & The speed of the \(n\)-th vehicle on lane \(l\) at time step \(t\) \\ \(u_{l,n,t}\) & m/s\({}^{2}\) & The acceleration of the \(n\)-th vehicle on lane \(l\) at time step \(t\) \\ \(s_{l,n,t}\) & m & The position of the \(n\)-th vehicle on lane \(l\) at time step \(t\) \\ \(v_{\max}\) & m/s & Maximum speed allowed \\ \(u_{\min}\) & m/s\({}^{2}\) & Minimum acceleration allowed \\ \(u_{\max}\) & m/s\({}^{2}\) & Maximum acceleration allowed \\ \(\omega_{1,t\geq 2}\) & Weight coefficient \\ \(T_{\min}^{TO}\) & s & The minimum safety time gap \\ \(\Delta t\) & s & Time interval of trajectory optimization zone \\ \(t\) & s & Timesetp \\ \(l\) & & Lane identifier (Outer Lane is 1 and Ramp Lane is 2) \\ \(n\) & & The index of each vehicle in the corresponding lane \\ \(N_{l}\) & & Total number of vehicles in the lane \(l\) \\ \(T^{TO}\) & s & Total simulation step for trajectory optimization \\ \(i\) & & Index of the vehicle in the Outer Lane \\ \(j\) & & Index of the vehicle in the Ramp Lane \\ \(T\) & s & Total simulation time \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter and symbol description of the trajectory optimization model. trajectory optimization zone was 300 m, and the length of the adjustment zone was 250 m; refer to [10; 26]. The simulation parameters are shown in Table 2. In the simulation, we limited the maximum traffic flow of the main line to 2000 vehicles per hour, and the maximum traffic flow of the ramp to 1000 vehicles per hour; refer to [10]. At the beginning of the simulation, the vehicle enters the simulation scenario with random speed and acceleration that meets the parameters shown in Table 2. When vehicles enter the lane-changing zone, the traffic controller controls the vehicle to change lanes. Vehicles then enter the trajectory optimization zone, and the traffic controller specifies the speed and acceleration for the vehicle before it passes the merging point. In the trajectory optimization zone, optimization was performed while the vehicle filled the trajectory optimization zone. The first optimization results are shown in Figure 8, where veh1-1 represents the first vehicle in the Outer Lane, veh2-1 represents the first vehicle in the Ramp Lane, and so on. The optimization results showed that the vehicles traveled at different speeds before optimization. The vehicles maintained a safe distance from the front and rear vehicles \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline \(v_{\text{max}}\) & m/s & 25 \\ \(u_{\text{min}}\),\(u_{\text{max}}\) & m/s\({}^{2}\) & \(-5\), 5 \\ \(\omega_{1}\),\(\omega_{2}\) & - & 1 \\ \(H_{\text{m}}\) & s & 1.5 \\ \(T_{\text{min}}^{\text{TO}}\) & s & 1.0 \\ \(T^{\text{TO}}\) & s & 10 \\ \(T\) & s & 100 \\ \(L\) & \(m\) & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 8: The first optimization in the trajectory optimization zone. through acceleration and deceleration during the optimization process. At the last optimization time step, all vehicles drove at the maximum speed, and the distance-time curve showed that a safe driving gap could be maintained between the same and different lanes after optimization. We took the first optimization results as an example to make a brief analysis. In 32 s, the traffic controller started to optimize the vehicles in the trajectory optimization zone. Veh1-1 was located on the main road, and veh2-1, veh2-2, veh2-3, and veh2-4 were located on the ramp. In 32-33 s, veh2-2 and veh2-4 slowed down to maintain the distance with the vehicle in front.