Research on high-speed railway operation adjustment model based on priority

Purpose – This paper aim at providing decision support for operational adjustment. In order to effectively reducetheexternal interference impactontrain operation. Design/methodology/approach – According to the reality that there are two kinds of high-speed trains running in China, considering the reality that there are two speed level train running in Chinese high-speed railway, this paper proposed the high-speed railway operation adjustment model based on priority, and the objective functionis to minimizeinterference impact ontrain operation. Findings – Adjustment strategy based on priority than based on sequence can effectively reduce the interference in ﬂ uence on train operation, which makes the train resume the planned operation back on schedulemore quickly. Research limitations/implications – The solution of large-scale cases is too slow, and the practical application is limited. Originality/value – The model was veri ﬁ ed by a case, and the case results proved that adjustment strategy based on priority than based on sequence can effectively reduce the interference in ﬂ uence on train operation, whichmakesthetrain resumetheplanned operationback onschedule more quickly.


Introduction
In the train running process, all kinds of interference are inevitable, such as the equipment failure and bad weather, these external interference usually cause the section passing capacity to fail, when these section passing capacity is restored, under the premise of ensuring operation safety, how to reasonably arrange the train departure sequence and departure time so that the deviation between the actual train operation and the planned operation diagram after interference is as small as possible is the key to train operation adjustment.
In essence, train operation adjustment is an optimization problem of train resource redistribution. There are two main ideas for establishing mathematical models: one is to abstract train operation as the occupation of train resources from the micro level (Hirai et al., 2009;Meng et al.,2012;Wang et al., 2019;Liao et al., 2021); the other is to abstract train operation as event activity network for modeling from the macro level (Shimizu et al., 2008;Nakamura et al.,2011;Zhan et al., 2015;Peng et al., 2018). Hirai et al. (2009) proposed his train operation adjustment model that focused on the parking scheme of trains under severe interference, and he designed the Petri net algorithm to solve the problem. Meng et al. (2012) proposed a two-stage stochastic expectation model with compensation and an incomplete continuous multistage decision model designed a branch and bound algorithm to solve the problem. Both models could improve the train punctuality level to a certain extent. However, this study was based on the modeling of single-track railways, which was not practical for the actual situation in China. Wang et al. (2019) further considered the train running path and established the train operation adjustment model to minimize the total train delay time. However, when the scale of the problem was large, the model could not quickly obtain a feasible solution. Liao et al. (2021) established an optimization model of train operation diagram based on cumulative flow variables. In view of the difficulty of solving large-scale problems with traditional algorithms, the author designed a Lagrange algorithm to solve the problem. Shimizu et al. (2008) established a constraint model to manage the train stop scheme and the train departure sequence under severe delay. This constraint programming technology can effectively reduce the train delay in earthquake scenarios. The model established by Nakamura et al. (2011) can adjust train operation and optimize rolling stocks at the same time. Zhan et al. (2015) abstracted the high-speed railway network into an event activity network established a train adjustment model with the objective function of minimizing the impact of interference on train operation. The author designed a two-stage algorithm to solve the problem. However, all the test examples are single lines, so the applicability of the model needs to be further verified. Peng et al. (2018) further considered the problem of train section and station parking in case of section failure, and his model described the process of train operation adjustment more detailed.
The research results of train operation adjustment are rich, but there are few researches for the real situation of Chinese high-speed railways, and their applicability is insufficient. Therefore, this paper establishes a priority-based train operation adjustment model based on Chinese high-speed railway real situation and researched train operation adjustment under the failure of the passing capacity of the high-speed railway section of the line.

Train operation adjustment
The train will inevitably be disturbed by external interference during operation. When the interference occurs, it will usually lead to the loss of the passing capacity of the section, resulting in the two-way trains of the section being unable to pass, and subsequent trains must wait at the previous station or section until the passing capacity is restored; Figure 1 shows this train adjustment process.
The train adjustment measures include changing the departure sequence and time as well as the stop scheme. On the Chinese high-speed railway network, there are two speed levels of train running, 250 km/h and 300 km/h. If two trains were stop at the same station due to a fault, when the passing capacity is restored, the higher-level 300 km/h train is arranged to depart first. This departure adjustment strategy that prioritizes the train speed level is a priority-based adjustment strategy, as shown in Figure 1(a). If the train departure is carried out in the order of arrival at the station, it is a common order-based adjustment strategy, as shown in Figure 1

Model establishment
Aiming at the reality of mixed running of high-speed trains of 250 km/h and 300 km/h in Chinese high-speed railway network, to describe the process of train operation adjustment more scientifically, this paper establishes a priority-based train operation adjustment model.

Model assumptions
The time when the pass capacity is lost and recovered is known. The arrival and departure lines of the station are available for the trains going up and down without restriction. The trains run as planned before the section passing capacity fails.

Symbols of parameters and variables
The process of a train traveling from one station to the next station is called a train running activity, and those parameters and variables related to train operation activity are shown in Table 1.

Objective function
The main purpose of train operation adjustment is to restore operation as graph þ as soon as possible and reduce the deviation between actual operation and planned operation. This model takes the minimum deviation between the actual operation of all trains and the planned operation as the objective function, and the degree of deviation is weighted and added by the difference between the actual departure time and the planned departure time of each train operation activity, and the difference between the actual arrival time and the planned arrival time.
Tdr a À Tdp a þ f a X a2A Tar a À Tap a (1)

Constraints
In this model, it is assumed that the start time of all running activities is greater than or equal to the planned start times.
Tdr a ! Tdp a ; a 2 A Section running time constraint: When the train is running in the section, the running time must be greater than or equal to the minimum running time of section.
Tdr N a ð Þ À Tar a ! Tr a ; a 2 A : Sa a ð Þ 6 ¼ s a^S a a ð Þ 6 ¼ s e Station stop time constraint: When the train stops at each station, the stop time must meet the minimum stop time requirement.
Tar a À Tdr a ! Ts a ; a 2 A (4) Departure interval constraint: When two trains leave continuously at the same station, the departure time interval must be greater than or equal to the minimum departure interval.
Tdr a À Tdr a 0 þ M*u a;a 0 ! Id; a; a 0 2 A : a 6 ¼ a 0^S d a ð Þ ¼ Sd a 0 ð Þ Running activity a actual arrival time fa Running activity a priority parameter N(a) Running activity a next running activity Sd(a) Running activity a departure station Sa(a) Running activity a arrival station Tra Running activity a section minimum running time Tsa Running activity a station minimum stop time s a Line started station s e Line terminal station Id Minimum departure interval Ia Minimum arrival interval Iad Minimum departure and arrival interval Te1 Start time of section passing capacity failure Te 2 Recover time of section passing capacity Se 1 Station 1 corresponding to the section where the passing capacity is invalid Se 2 Station 2 corresponding to the section where the passing capacity is invalid u a,a' Running activity a and a' whether overtaking 0-1variable v a,a' Running activity a and a' whether meet the arrival and departure interval 0-1variable x a ,y a ,z a 0-1 variable Railway operation adjustment model u a;a 0 þ u a 0 ;a ¼ 1; a; a 0 2 A: a 6 ¼ a 0^S d a ð Þ ¼ Sd a 0 ð Þ Arrival interval constraint: When two trains continuously arrive at the same station, the arrival time interval must be greater than or equal to the minimum arrival interval.
Tar a À Tar a 0 þ M*u a;a 0 ! Ia; a; a 0 2 A: a 6 ¼ a 0^S a a ð Þ ¼ Sa a 0 ð Þ (7) u a;a 0 þ u a 0 ;a ¼ 1; a; a 0 2 A: a 6 ¼ a 0^S a a ð Þ ¼ Sa a 0 ð Þ Departure and arrival interval constraint: When two trains arrive and depart at the same station, the time interval between them must meet a certain interval.
Tar a À Tdr a 0 þ M*v a;a 0 ! Iad; a; a 0 2 A: a 6 ¼ a 0^S a a ð Þ ¼ Sd a 0 ð Þ Tdr a À Tar a 0 þ M* 1Àv a;a 0 ð Þ! Iad; a; a 0 2 A: a 6 ¼ a 0^S a a ð Þ ¼ Sd a 0 ð Þ This model assumes that the train runs as planned before the failure occurs.
Tar a ¼ Tap a ; a 2 A : Tap a < Te 1 Whether depart or arrive at the end of failure.
Te 2 þ1 ð Þ1 À y a ð Þ# Tar a # Te 2 þ M 1 À y a ð Þ; a 2 A Te 1 Á z a # Tar a # Te 1 À 1 þ M 1 À z a ð Þ; a 2 A: Sd a ð Þ ¼ Se 1 jjSd a ð Þ ¼ Se 2 (14) x a þ z a # 1; a 2 A: Sd a ð Þ ¼ Se 1 jjSd a ð Þ ¼ Se 2 Station capacity constraint: At the same time, the number of trains at the station must be smaller than or equal to its number of arrival and departure lines.  Tables 2 and 3.

Scenarios design
Three interference scenarios are designed for numerical experiments. Scenario 1, the section between stations C and D, breaks down at 13:40; Scenario 2, the section between stations E and F, breaks down at 11:00; Scene 3, the section between stations H and I, breaks down at 11:00. Assuming that the break down causes the passing capacity of the section to be lost, the duration takes 20, 40, 60, 80, 100 and 120 min, respectively, to be calculated. The prioritybased model calculation results of the are shown in Table 4 below (Tables 5 and 6). For the above three scenarios, most of the train adjustment schemes can be solved in a short time (20 s), indicating that the model can solve the operation adjustment scheme in a short time for the majority of failure scenarios, but the in third scenario when the duration is 120 min, the calculation time reaches 178.54 s. It is presumed that there is a high frequency  2  2  3  D  7  5  4  E  2  2  3  F  3  2  3  G  2  2  3  H  3  2  3  I  5  3  4  J  2  2  2  K --4 Note: The slow train is 250 km/h train, and the fast train is 300 km/h train Railway operation adjustment model of departures during this period, so more trains need to be adjusted, and the calculation amount is relatively large (Figure 2).

Comparison with sequence-based adjustment strategy
For the six failure durations in Scenario 2, sequence-based adjustment strategy was applied respectively to adjust the train operation after the interval was recovered through capacity. The deviation degree between the actual train and the planned operation was used to measure the influence degree of interference on the train operation. The calculation results are as shown in Table 7 ( Figure 3). Compared with the order-based adjustment strategy, the priority-based adjustment model solution established in this paper has a certain degree of reduction in train operation deviation (ranging from 20.84% to 9.26%), indicating that priority-based model can effectively reduce the impact of interference on train operation compared with the sequencebased adjustment strategy.
Sensitivity analysis: To explore the influence of section passing capability failure time on the deviation degree caused by the train, sensitivity analysis was conducted, and the start time of the failure was set to be the same. The last time of section passing capability was set to be 20, 40, 60, 80, 100 and 120 min, respectively, for experiments. The calculation of the three failure scenarios was carried out, and the following figure was obtained ( Figure 4). As can be seen from the above figure, in the three scenarios, when the failure starts at the same time, the longer the failure time of interval passing ability is, the greater the deviation between the train and the planned operation is. This conclusion is consistent with the   To restore the planned operation as soon as possible, the fault must be solved as soon as possible, and the passing ability must be restored.

Conclusion
Based on the practical characteristics of Chinese high-speed railways, this paper establishes a double track operation adjustment model based on train priority under the losing of section passing capacity. Taking a double track high-speed railway as a case background, three failure scenarios under six durations time are designed to carry out to verify the model; the solution results show that the model can solve the train operation adjustment plan in a short time (20 s). Compared with the sequence-based adjustment strategy, the model can effectively reduce the deviation of the actual operation between the planned graph further reduce the impacts of interference on train operation, which can provide decision support for dispatchers to adjust train operation in the event of a sudden failure in the section.