###### Abstract  
The cross-line operation mode, based on interoperability technology, is becoming increasingly common in urban rail transits (URTs). Compared to trains running on a single line, cross-line trains can greatly facilitate transfer passengers. Taking the scenario of emergent large transfer passenger flow as an example, this paper explores the train rescheduling problem for serving transfer passengers by adding a cross-line backup train. To maximize the number of transfer passengers served by the cross-line backup train, a nonlinear optimization model is constructed by taking into account the operation parameters of planned trains on relevant lines, the deviation degree of the planned timetable, the utilization of the cross-line backup train, and the passenger flow calculation as constraints, and some linearization lemmas are proposed to transform it into a mixed integer programming (MIP) model with quadratic terms. A case study is conducted to discuss the impact of parameter changes on the objective function value and the applicability of different solution approaches. The results suggest that the operation trajectory of the cross-line backup train has an effect on the objective function value, which is related to the demand, the deviation tolerance of the planned timetable, and the running efficiency tolerance of the cross-line backup train. The corresponding methods help guide the organization of the cross-line backup train for large transfer passenger flow scenarios. url  
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facility and equipment failure, operation error, etc. These factors are uncertain and have varying degrees of impact on operations. For example, if there are too many passengers on the platform, it is easy to cause passengers to be hindered from getting on and off, and there is a risk of a stampede accident. When the onboard signal equipment malfunctions, the train may stop in a section for maintenance, thereby affecting the operation of other trains. Due to the different characteristics of different scenarios, the corresponding train rescheduling methods are also different. In previous studies, [PERSON] and [PERSON] [6] aimed to minimize the negative impact of train delay on passengers and explored the problem of train rescheduling by combining delay recovery and passenger selection behavior. [PERSON] et al. [7] constructed a multi-objective optimization model based on the strategy of \"adding backup train to replace failure train\" to improve the balance of train departure and reduce total delay. [PERSON] et al. [8] suggested that trains skip to some stations to accelerate train turnover, constructed an optimization model considering train operation efficiency and the number of passengers served, and designed an iterative algorithm in conjunction with the Gurobi solver. For train operation disruptive events, [PERSON] et al. [9] constructed a multi-objective mixed integer linear programming (MILP) model considering the deviation of the train timetable, the number of train cancellations and the deviation of train operation interval, and designed a two-stage algorithm. Meanwhile, [PERSON] et al. [10] considered the randomness of disruptive events and constructed a two-stage optimization model that takes into account the allocation of backup train positions and the online rescheduling of train diagrams. For normal and abnormal large passenger flows, to ensure operation safety and service quality, management can be implemented from two aspects: passenger flow organization and train organization. Passenger flow organizations use passenger flow control methods to reduce platform pressure, including gate control [11; 12], escalator control [12], and passenger flow restriction at station entrance [13; 14]. Some studies consider flow control problems together with train organization, including timetable optimization [15; 16] and train rescheduling [17; 18].