## 7 Conclusions  
We conducted a study on how to better utilize a cross-line backup train to cope with a large transfer passenger flow. The train rescheduling model and solution approaches of adding the cross-line backup train and planned trains are studied to guide the emergency response in such scenarios. The experiments show that the mixed integer programming model is effective, and the solution efficiency is also high. The operation trajectory of the cross-line backup train has an impact on the objective function value, which is limited by the demand, the deviation tolerance of the planned timetable, and the running efficiency tolerance of the cross-line backup train. When the requirements for timetable deviation and running efficiency are reduced, the objective function value tends to increase. For example, when both are set at 30%, the number of transfer passengers served by the cross-line backup train is 836, which is an increase of 18.1% compared to when both are set at 0. After tripling the demand, the number of transfer passengers served remained stable at 846 and is not affected by changes in the two parameters. Choosing an appropriate tolerance coefficient for the demand situation is meaningful. When the demand does not exceed the service capacity, dynamically adjusting the two parameters to obtain multiple sets of rescheduled timetables is a suitable way to deal with such scenarios. When the demand exceeds the service capacity, we can only focus on the emergency efficiency of adding the cross-line backup train. For the cross-line service of transfer passengers, this study can provide a reference for train rescheduling in the context of networked operation and cross-line operation.  
Regarding the possible directions for further research on the emergency scenario, we further suggest considering the details of the problem more finely. For example, operating costs can be considered as one of the optimization goals, as organizing a backup train requires a large amount of electricity consumption. At the same time, when the passenger flow is large, it can also be managed in combination with passenger flow control, especially as to whether it can achieve good operational control effects at a lower cost, including passenger flow organization and train organization. Furthermore, the operation of many trains usually follows a certain target speed profile, which determines the interval operation time and energy consumption. If the optimization of train organization can be carried out in conjunction with the target speed profile, more excellent results may be obtained.  
Conceptualization, J.Y. and P.C.; methodology, J.Y. and P.C.; software, P.C.; validation, J.Y. and P.C.; formal analysis, P.C.; investigation, J.Y. and X.Z.; resources, J.Y. and P.C.; data curation, J.Y. and P.C.; writing--original draft preparation, J.Y. and P.C.; writing--review and editing, J.Y. and P.C.; visualization, P.C.; supervision, X.Z.; project administration, X.Z.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript.  
The work is sponsored by the Shanghai Collaborative Innovation Research Centre for Mul-ti-network & Multi-modal Rail Transit, and the Key Technologies of Operation Control and Environment & Disaster Monitoring of 600 km/h High Speed Maglev Train under grant 2022 YFG0085.  
Not applicable.  
The data used to support the findings of this study are available from the corresponding author upon request.  
The authors declare no conflict of interest.