## 2 Problem Description  
In the context of networked operation, the coupling between lines makes passenger flow management at interchange stations crucial [28; 29; 30]. The cross-line operation mode not only improves the travel experience of transfer passengers through cross-line trains, but also helps to serve emergency scenarios. For example, when a large number of passengers transferring out (i.e., a large passenger flow from a line to other lines via the interchange station) is suddenly detected on a line, a cross-line backup train can be added to alleviate the passenger flow. At the same time, when rescue trains are needed for adjacent lines, trains on the line can be used as cross-line trains. As shown in Figure 1, when an emergency occurs on Line B, trains (backup or mainline trains) of Line A can be used as rescue trains. At the same time, if a large transfer passenger flow from Line A to Line B suddenly occurs, it can also improve the service quality and alleviate the risk of passenger detail in the transfer station (i.e., station B3) by adding cross-line backup trains.  
This paper focuses on train rescheduling strategies for large transfer passenger flow scenarios, involving two or more lines. As shown in Figure 1, to implement a cross-line service, Line A adds a cross-line train, and Line B also adds a temporary train service. The use of backup trains involves processes such as startup, arrival, departure, and return, thus it is necessary to optimize the schedule of the backup train and the planned trains by combining the routing of the backup train and the change of large passenger flow. This is because the train operation has strict safety constraints, each train needs to maintain a safe interval with its front and rear trains [31], and there is a significant coupling relationship between passenger flow and train flow, as shown in Figures 2 and 3.  
Figure 1: Schematic of train cross-line operation.  
Figure 2: Train operation interval and its coupling relationship with passenger flow.  
For the impact of adding backup trains on the train timetable, when the train operation interval is large enough, backup trains can be added without affecting other trains. When the train operation interval is small, the addition of backup trains will affect the operation of planned trains, as shown in Figure 3. At the same time, in addition to considering the impact of adding cross-line backup train on the train timetable, it is also necessary to consider the online timing, the train routing, and the passenger flow evacuation effect for providing a complete scheme and quantitative decision-making basis for the cross-line train in emergency scenarios. Specifically, the problem considered in this paper is how to coordinate and optimize the operation plans of trains on the origin line (i.e., the origin line of the large transfer passenger flow), trains on the adjacent line, and the cross-line backup train in the early stages of large transfer passenger flow scenarios in order to better serve the transfer passengers and reduce the risk of passenger detain.  
We use optimization theory to investigate the cooperative rescheduling problem of planned trains and the cross-line backup train. At the same time, while solving the problem, we discuss three hypotheses: (1) the operation trajectory of the cross-line backup train has an impact on the number of transfer passengers served by the backup train; (2) the number of transfer passengers served by the backup train will be limited when the deviation of the planned timetable or the running efficiency of the backup train is limited; (3) when the passenger flow is large enough, whether passenger data is taken into account has a limited impact on the number of transfer passengers served by the backup train. It should be noted that the analysis of these hypotheses is based on the proposed optimization models and the experiment data.