# Train Service Design for Rail Transit Cross-Line Operation Applying Virtual Coupling
## 5 Case Study
### In terms of additional VC time cost  
In CO, the uncertainty of the train arrival time can result in an average additional VC time cost for trains on different lines during the coupling process. In terms of the additional VC time cost, SG-No-CO, SG-CO, and MG-CO schemes incur no costs, while only the VC-CO scheme requires CNY 2230 (assuming an average VC time is 30 s). To analyze the variation in objective values under different average VC times, we examined scenarios with average VC times ranging from 10 s to 150 s. Figure 15 illustrates the changes in objective values as average VC times change. \begin{table}
\begin{tabular}{c c c c c} \hline
**Objective (CNY)** & **SG-No-CO** & **SG-CO** & **MG-CO** & **VC-CO** \\ \hline (1) Passenger travel cost & 184,752 & 186,987 & 178,017 & 171,679 \\ residence time cost & 59,127 & 61,362 & 52,392 & 43,824 \\ in-vehicle cost & 125,625 & 125,625 & 125,625 & 125,625 \\ additional VC time cost & 0 & 0 & 0 & 2230 \\ (2) Train capacity equity & 45,230 & 52,680 & 9934 & 8778 \\ (3) Operation cost & 116,671 & 134,600 & 57,005 & 62,074 \\ usage cost & 6720 & 6840 & 2452 & 2794 \\ operating cost & 109,951 & 127,760 & 54,553 & 59,280 \\ Objective value & 112,895 & 109,594 & 81,652 & 80,843 \\ \hline \end{tabular}
\end{table} 
Table 7: The table of passenger classification. Figure 14: The resistance time cost of different case passengers. With the increase in average VC times, objective values also increase gradually. This is because average VC times affect the coupling between different lines. As the average VC times increase, the coupling between different lines will result in higher train travel costs and affect the objective value. The objective value in MG-CO is CNY 81,652. As can be seen in Figure 15, when the average VC time is equal to 80 s, the objective value is 82,074, which is higher than that in MG-CO. Thus, when the average VC time is less than 80 s, VC can play an effective role and improve operational efficiency. **(3)**: In terms of train capacity equity  
Compared with SG-No-CO, train capacity equity under SG-CO has increased by 16.5%, as shown in Figure 16. Regarding CO, using MG-CO and VC-CO can further balance the problem of capacity inequity between different lines by changing the train compositions. Among them, train capacity equity is reduced by 78.0% and 80.1% under MG-CO and VC-CO, respectively. MG-CO can determine the corresponding number of train compositions according to passenger demand between different lines, but the optimization effect needs to be improved due to the limitation of train departure frequency. VC-CO can flexibly couple/decouple trains of different lines, which can more flexibly adapt to passenger flow, thus making train capacity match more with passenger demand and providing a more equitable train service. **(4)**: In terms of operation cost  
Figure 17 shows the train departure frequency on different lines, the number of train compositions, and turn-back stations under MG-CO and VC-CO. In addition, the minimum  
Figure 16: The train capacity equity in different zones. Figure 15: The average VC time. headway is set as 3 min, so the maximum number of trains on one line under MG is 20. However, since VC technology can allow trains to be virtually reconnected into a virtual coupled formation, up to 40 trains can be run on one line. In MG-CO, train 1 and train 2 are operated with two and six carriages, respectively, as shown in Figure 17. Train 1 and train 2 are operated with 16 pairs and 12 pairs in line 1 and line 2, respectively. Due to the constraints of traditional operating, cross-line trains operate in groups of two carriages per hour with four pairs. As shown in Figure 17, in VC-CO, train 1 operates in 2 e in the S1-S10 interval and S15-S16 interval, train 1 coupled with train 3 operates with 2 + 2 carriages in the S10-S15 interval, train 2 operates with 4 carriages in the S19-S27 interval, and train 2 coupled with train 3 operates with 2 + 4 carriages in the S10-S20 interval.