## 4 Case Study
### Virtual Coupling Train Diagram Analysis  
Utilize the genetic-simulated annealing algorithm to solve the case and obtain the theoretical optimal train diagram under virtual coupling. Relevant parameters are detailed in Table 6. The algorithm starts converging after 90 iterations, with the convergence of the fitness function value depicted in Figure 9. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Symbol** & **Description** & **Value** \\ \hline \(pop\_size\) & Population size & 100 \\ \(p_{c}\) & Crossover probability & 0.9 \\ \(p_{m}\) & Mutation probability & 0.1 \\ \(max\_gen\) & The maximum number of iterations & 150 \\ \(t_{max}\) & The initial temperature & 100 \\ \(\varepsilon\) & Cooling coefficient & 0.9 \\ \(t_{min}\) & Termination temperature & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameter Values Used in GA-SA. Figure 8: (a) Section Passenger Flow(7:00 to 8:00); (b) Section Passenger Flow(8:00 to 9:00). Figure 7: Passenger Arrival Rate of Station 1. We assign a value of 0 or 1 to the gene segment representing the route type in the initial chromosome. Then, we set the departure interval between adjacent trains at the starting station to \(h_{1}\in[h_{min},h_{max}]\), thereby generating a traditional single-route train diagram. We add the short-turn route train to the gene fragment, initialize \(h_{1}\in[h_{min},h_{max}]\), and obtain the traditional full-length and short-turn route train diagram. The theoretical optimal train diagram under virtual coupling is shown in Figure 10. The cost values for the three operation plans are shown in Table 7.  
Figure 10: Optimized Train Diagram under Virtual Coupling Technology. Figure 9: Convergence Curve of GA-SA. Table 7 illustrates that introducing short-turn route trains increases passenger waiting costs but significantly reduces enterprise operating costs. Implementing virtual coupling technology further decreases various cost values. In the traditional single-route mode, 50 full-length route trains operate in both directions. Station 1's depot uses 26 sets of rolling stock, and station 23's depot uses 24 sets. In the full-length and short-turn routes mode, 35 full-length route trains and 18 short-turn route trains operate in the up direction, while 32 full-length route trains and 18 short-turn route trains operate in the down direction. The inclusion of short-turn route trains expedites rolling stock circulation, utilizing 47 sets of rolling stock to complete 103 trains. Regarding rolling stock circulation, eight upbound short-turn route trains depart from depot 1, reverse at station 15, and continue as downbound short-turn route trains. After completing the downbound task, they continue to reverse. Thus, the same rolling stock continuously serves three operating lines. After adopting virtual coupling technology, 44 groups of rolling stock left the depot and completed 106 train tasks. This included 30 full-length route trains and 25 short-turn route trains traveling upbound, as well as 26 full-length route trains and 25 short-turn route trains traveling downbound. The virtual coupling technology resulted in a reduction of 10.29% in the objective function, 9.13% in passenger waiting cost, 12% in train procurement cost, and 11.86% in train mileage cost compared to the traditional single-route mode. Moreover, when compared to the traditional full-length and short-turn routes mode, the reductions were 7.39%, 10.22%, 6.38%, and 2.22%, respectively. Based on the virtual coupling time of trains, operations primarily occur between 7:45 and 8:30, demonstrating improved synchronization between dynamic passenger flow and train formation. In the up direction, 14 full-length and short-turn routes trains form large eight-group formations at the starting station to accommodate heavy passenger flow, uncoupling at the short-turn turnaround station. Short-turn trains then reverse to service down trains, while full-length trains continue to the terminal.