## 4 Simulation Results
### Results
In contrast, the mean speed of vehicles under NC or FSC is kept at a smaller value, reflecting the inefficiency of the traffic. Figure 10 shows the number of stops for case 3 in 15 min under the different strategies. It is easy to observe that almost no stops occur in the scenario of 100% PRA with the proposed scheme, and a small number of stops exist under 70% PRA. When NC or FSC is adopted, many vehicles stop near the intersection, leading to queuing. Figures 9 and 10 indicate that the proposed scheme in this paper can significantly improve mobility and smooth traffic flows compared to the NC operation and FSC strategy. Figure 8: The acceleration of vehicles for case 3 in 5 min under the proposed strategy. Figure 7: The speeds of vehicles for case 3 in 5 min under the different strategies. Since few of the current state-of-practice strategies consider mixed-automated traffic at signal-free intersections, we have compared the results of the proposed scheme with a recently developed strategy that jointly optimized the signal timing plans with the trajectory of vehicles in [PERSON] et al. [24]. The paper developed a centralized mixed-integer non-linear program to coordinate the trajectories of AVs and signal timing plans under mixed-automated traffic. We adopt the same traffic parameters as [24] for comparative simulations under the two-levels of traffic demands, i.e., demand 1 and demand 2. Demand 1 denotes low traffic demand, with 300 veh/h/lane through demand and 24 veh/h/lane left-turn demand, and demand 2 represents high traffic demand, with 600 veh/h/lane through demand and 48 veh/h/lane left-turn demand. Table 6 shows the total delay between the scheme proposed in this paper and in [24] for the two demands under different PRAs in 5 min. The results indicate that the total delay increases with PRA reduction in both schemes. In addition, our proposed scheme performs better than [24] under high PRAs and low traffic demand. However, when the PRA drops below 70%, especially under high demand, the total delay in our scheme is higher than in [24]. This suggests that the role of signal timing optimization in [24] is more significant under high demand and a low PRA. Table 7 shows the computation performance between the proposed scheme in this paper and [24] for demand 2 under different PRAs. The parameters # Var. and # Con. denote the mean number of variables and constraints processed per second, and # Con. represents the mean number of communications required per second. The results show that the mean number of variables, constraints, and communications mostly decrease as PRA decreases in both schemes. In addition, the proposed scheme in this paper has fewer  
Figure 10: Comparison of the number of stops for Case 3 in 15 min under the different strategies: (a) proposed scheme under 100% penetration rate of AVs; (b) proposed scheme under 70% penetration rate of AVs. Figure 9: Comparison of the mean speed for case 3 in 15 min under the different strategies: (a) proposed scheme under 100% penetration rate of AVs; (b) proposed scheme under 70% penetration rate of AVs. computation parameters to process per second than [24], which means that our scheme has an advantage in computational complexity.