## 6 Discussion  
In this section, we further discuss the validity of the proposed methods and the three hypotheses in Section 2. For the methods in this paper, the results verify the effectiveness of the models and solution approaches, which can provide theoretical references for large transfer passenger flow and emergency scenarios of cross-line operation. The optimization model focuses on evacuating passengers at the source line of large transfer passenger flow, considering the relevant conditions for train cross-line operation. For the scenarios involved, compared to the planned timetable, adopting the \"adding a cross-line backup train\" strategy can better ensure the operational safety of stations, including interchange stations, and improve service quality. The result accords with the expected effect of adding a backup train [12; 19; 20; 21]. For solution approaches, the first and second can obtain effective solutions within seconds. For the first solution approach, it fixes the deviation tolerance and running efficiency tolerance, resulting in a relatively fixed solution. For the second solution approach, multiple sets of approximate optimal solutions can be obtained by dynamically adjusting the two parameters. In addition, the third solution approach can obtain a train rescheduling scheme without considering passenger flow data. It should be noted that when adding a cross-line backup train, corresponding operating conditions are required, which involve cross-line operating conditions between lines, \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Timetable**} & \multirow{3}{*}{**Transfer Passengers**} & \multirow{3}{*}{**Remaining Capacity/Person 1**} & \multicolumn{6}{c}{**Transfer Passengers on Planned**} & \multicolumn{6}{c}{**Maximum Stranded**} \\  & & & & & & **Trains/Person** & & & & **Passengers/Person** \\ \cline{3-14}  & & & **Ta2** & **Ta3** & **Ta4** & **Ta5** & **Ta6** & **Ta7** & **Ta8** & **A1** & **A2** & **A3** & **A4** \\ \hline Rescheduled timetable & 836 & 484 & 556 & 476 & 605 & 438 & 483 & 440 & 428 & 0 & 0 & 0 & 170 \\ Original timetable & \(-\) & \(-\) & 556 & 669 & 558 & 596 & 556 & 550 & 428 & 0 & 0 & 0 & 1001 \\ \hline \hline \end{tabular}  
* The remaining capacity refers to the remaining capacity when the cross-line backup train arrives at station B3.  
\end{table} 
Table 2: Results comparison for original timetable and rescheduled timetable. large operation intervals in the original timetable, and may have a certain impact on operating costs. In this regard, this paper assumes a suitable optimization premise, and in practice, emergency strategies can be adopted based on the characteristics of the scenarios. For example, when the cross-line operation condition is not available, strategies such as implementing passenger flow control on the source line of large transfer passenger flow or adding backup trains on the target line of large transfer passenger flow can also be adopted. For the first of the three hypotheses discussed, the operation trajectories of the cross-line backup train in Figures 9 and 10 are different, and the corresponding objective function values are also different. The result indicates that the arrival time and departure time of the cross-line backup train at each station have an impact on the number of transfer passengers served by the backup train. Therefore, when adding a backup train, it is necessary to determine its appropriate online time based on passenger flow data [21]. For the second hypothesis, Figure 11 obtained according to the second solution approach shows when the tolerance parameters of timetable deviation or backup train running efficiency are small, the objective function value is also small, and when the two parameters increase, the objective function value tends to increase, but this trend will disappear when the two parameters are large enough.