## 4 Numerical Simulation
### Results and Discussion
#### 4.2.2 The Scenario with Multiple Traffic Waves  
Based on experimental parameters, multiple traffic waves induced by speed were simulated. The results of a scenario with multiple traffic waves are summarized in Figure 4, where the initial trajectory is represented by a black line. In Figure 4a,b, these trajectories without control indicated that the vehicle experienced frequent acceleration and deceleration rates. Similarly, the optimized vehicle and velocity trajectories are shown in Figure 4c,d. The results indicated that, compared to the initial trajectory, the vehicle trajectories tended to become stable after optimizing for smooth speed. According to the constraint conditions, the vehicle velocity was maintained within 30 m/s and exhibited improvement. In Figure 4e, the optimized acceleration trajectory illustrated that the vehicle initially decelerated at a rate of \(-0.71\) m/s\({}^{2}\), gradually increased to 0.65 m/s\({}^{2}\), and finally tended to be stable and remained at zero after about 570 s. The optimized acceleration became smoother, indicating that the JAD optimal control improved the comfort of the human driver. The simulation results demonstrated that JAD optimal control within the MPC framework can effectively eliminate traffic oscillations. In addition, the elevation of vehicle speed and the provision of smooth acceleration contribute to an enhancement in fuel economy, traffic efficiency, and the comfort of human drivers. Most importantly, the cost function takes into account the speed difference between the last vehicle and the maximum allowed speed, ensuring the safety of the vehicles. To demonstrate the feasibility of the proposed optimization control, the change in traffic throughput and traffic safety for the JAD strategy and the optimal control method were quantified in Table 2. It is evident that compared to the JAD strategy, the safety performance indicators of the optimized method showed significant improvement, with a reduction rate of 8.14% in TET and 4.22% in TIT. Simultaneously, there is a notable increase in traffic throughput by 4.25%. The results indicated that the optimal method has better performance in terms of traffic throughput and safety. Considering the comprehensive analysis above, it can be readily inferred that the proposed optimization control is capable of efficiently mitigating traffic oscillations, enhancing both traffic efficiency and safety, thereby alleviating traffic congestion. It can be found that the performance improvement of the optimized method in the scenario of multiple traffic waves is slightly better than that in the scenario of a single wave. This reveals the proposed method is robust and makes complex traffic flow more stable on a single-lane freeway. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{4}{c}{**The Change in Performance Metrics (\%)**} \\  & **Throughput** & **TET** & **TIT** \\ \hline No control & 0 & 0 & 0 \\ JAD & 58.46 & \(-65.96\) & \(-73.65\) \\ Optimal control & 61.11 & \(-72.01\) & \(-75.31\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of performance metrics for different methods in the scenario of a single stop-and-go wave. Figure 4: The results of a scenario with multiple stop-and-go waves: (**a**) The vehicle trajectories without control; (**b**) The velocity trajectories without control; (**c**) The optimized vehicle trajectories; (**d**) The optimized velocity trajectories; (**e**) The acceleration profiles. (The black line represents the initial trajectory without control for (**a–d**). For (**e**), the blue and red lines correspond to the initial and optimized acceleration profiles, respectively).