# An Optimal Scheduling Model for Connected Automated Vehicles at an Unsignalized Intersection
## 4 Numerical Experiment
### Result Analysis  
Based on the model calculation results, the vehicle delays and model computation times for the four scenarios are presented in Tables 4 and 5, respectively. Table 4 shows the average delay per vehicle and the total delay for the three control methods, while Table 5 displays the computation times for the three control methods.  
Table 4 shows that under the three control methods, the average vehicle delay gradually increases with the rise in traffic demand. Compared to the FCFS control method, the proposed algorithm and the solver significantly reduce the average vehicle delay. In traffic demand scenario 1, the proposed algorithm and the solver can reduce the average vehicle delay by 45.87% and 46.62%, respectively, while in traffic demand scenario 4, the proposed algorithm can reduce the average vehicle delay by up to 76.22%. This indicates that the proposed algorithm can obtain better solutions across different traffic demand scenarios  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Definition** & **Value** \\ \hline \(l\) & control area circular radius & 300 m \\ \(T\) & total optimization duration & 100 s \\ \(\Delta T\) & optimized time window length & 20 s \\ \(v_{i}\) & free-flow speed of road segments & 15 m/s \\ \(\tau\) & safe headway distance for vehicles traveling in the same direction & 0.5 s \\ \(\omega\) & minimum safe time headway for vehicles entering the conflict zone from conflicting directions & 1 s \\ \hline \hline \end{tabular}
\end{table}
Table 2: Model parameters in numerical analysis.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Indices**} & \multicolumn{4}{c}{**Traffic Volume (veh/h)**} \\ \cline{2-5}  & **Direction 1** & **Direction 2** & **Direction 3** & **Direction 4** \\ \hline
1 & 900 & 900 & 1200 & 1200 \\
2 & 1200 & 1200 & 1200 & 1200 \\
3 & 1200 & 1200 & 1800 & 1800 \\
4 & 1800 & 1800 & 1800 & 1800 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Experimental scene setting.  
than the solver. Moreover, the vehicle delay varies depending on different traffic demands. When traffic demand is saturated, vehicles will queue outside the control zone and wait to enter the control zone. This is not the focus of this paper, but readers can refer to [13; 14].  
Table 5 shows that compared to the solver Gurobi and the proposed algorithm, the computation time of FCFS is extremely short, with an average time of less than 1 millisecond, and it is not affected by data randomness. This can be attributed to the sequential calculation adopted by FCFS, which does not involve optimization, resulting in very short computation times. The computation time of the solver Gurobi increases exponentially with traffic demand, and from the standard deviation, it is evident that it is significantly influenced by data randomness. This is because as traffic demand increases, both the decision variables and constraints of the model increase exponentially, leading to an exponential increase in solution time. Compared to the solver Gurobi, the solution time of the proposed algorithm only increases linearly with traffic demand, and the solution time is shorter. Therefore, based on Tables 4 and 5, it can be concluded that the proposed model and algorithm can maximize the optimization effect of the model while reducing the solution time. However, from Table 5, it is also evident that for scenarios 3 and 4, the solution time of the proposed algorithm is significantly longer than 100 s, which does not meet the practical requirements for solution time. Therefore, further steps could be taken to reduce the optimized time window length or design a more efficient algorithm.