Train delays have a great impact on the schedule of high-speed railways including overall efficiency and the quality of the travel service of the passengers. Therefore, the development of an approach to recover a schedule via timely and rapid operation adjustments must be investigated. In this paper, a complete final real-time adjustment scheme is proposed for the train dispatcher of a railway enterprise for delayed trains. A train operation process model based on the Max-Plus algebra method is proposed. Six operation adjustment strategies are analyzed including section acceleration, operation based on the original timetable, dwell time reduction, increase in overtaking, reduction in overtaking, and train postponement. An approximate model is then built based on the minimum number of delayed trains considering the constraints of the adjustment strategies and the feasible adjusted schemes can be quickly obtained without any record specific time data and constraints. The set of feasible solutions of the approximate model is then regarded as the importation of the second model. In addition, the second model is an optimization model for operation adjustment with the least average delay time of each train at each station by updating the state matrixes of the train operation process model. The algorithms are designed for these models, and this approach can reduce the computation time. Finally, the timetable of the Beijing-Shanghai high-speed railway is considered as the actual case for analysis. Thus, the method was proven feasible for operation adjustment of delayed trains.
A high-speed railway is characterized by several parameters such as large density and high speed. Moreover, the operation of high-speed trains is sensitive to traffic delay. As such, efficient and on-time operation of high-speed trains is very important given the emphasis of passengers on on-time service. Therefore, real-time operation adjustment for delayed trains should be investigated. Theoretical results can thus be provided for studies on train schedules, timetable evaluation, and passenger service level.
The train delay propagation methods have been studied for a long time. The simulation model and the analysis model of train delay were initially proposed by Pertersen [
The operation adjustment method for delayed trains has been investigated in numerous studies. The method for operation adjustment of delayed trains is usually studied using an optimization, simulation, or intelligent algorithm method. It is necessary to obtain the optimal adjustment scheme based on a specific target for a new timetable. For this reason, two main aspects of these studies have been addressed: (1) adjustment strategies are proposed and (2) the optimal targets of the adjustment method are different.
Regarding the first aspect, Qian et al. [
For the second aspect, the three main optimal targets in these studies are as follows.
(1)
(2)
(3) Some relative contents of the train delay such as the railway capacity and the buffer time of the timetable were studied. In [
When the trains are delayed for a long time, different trains are adjusted at different stations, which cause a continuous change of the timetable. A large-scale solving scheme for the adjustment will therefore be produced. The algorithm is complex and has a low efficiency, which cannot meet the needs of real-time adjustment. In this investigation, the method of ordinal optimization for the operation adjustment of the delayed train is used. Xie et al. [
Apparently, “order” is much easier to be compared than “value.” A simple rough model is not sufficient to accurately determine the difference in performance among different solutions, but it can clearly show the difference between the solutions. An approximate model is usually sufficient because it has a significant advantage in terms of computation time compared to the exact model. In addition, the set of solutions of the approximate model is the input parameters of the exact model.
This report analyzes the delay time of each train at each station to decrease the impact on delays, and it is structured as follows. Section
As shown in Figure
The methodology structure of the paper.
After adding the train delays, the first station affected by the delay is selected. Then the delayed trains are selected in order and adjusted using the alternative adjustment strategies. Six alternative adjustment strategies are utilized including the acceleration in section, operation based on the timetable, dwell time reduction, increase in overtaking, and reduction in overtaking, including train postponement with the minimum headway in Section
Based on the method of ordinal optimization, an optimal method is proposed in Sections
The good-enough solutions are the feasible adjustment schemes based on the target value (the minimal number of delayed trains) and the constraints of the adjustment strategies. The solution of the approximate model is not just one solution scheme, but a set of adjustment schemes. These good-enough solutions are calculated using the second model. The second model is the accurate model that has the comprehensive constraints and target values of the total minimum delayed time for all trains. In the second model, only the delayed time of the trains needs to be calculated. It is not necessary to choose the adjustment strategy again.
An optimal solution based on the accurate model is the final operation adjustment scheme of the delayed trains.
The main parameters are defined as shown in Table
Variables.
Variables | Corresponding meaning |
---|---|
| The station set, |
| |
| The train set, |
| |
| The arrival time of train |
| |
| The arrival time of train |
| |
| The departure time of train |
| |
| The departure time of train |
| |
| The minimum operation time of train |
| |
| The delay time of train |
| |
| The minimum arrival headway at station |
| |
| The minimum departure headway at station |
| |
| The minimum stop time of train |
| |
| The maximum stop time of train |
| |
| The maximum acceleration time of train |
| |
| The adjustment time of train |
| |
| The adjustment time of train |
| |
| The state matrix records the operation time of the trains with different adjustment schemes. |
| |
| The state matrix records the departure sequence of the trains at each station. |
| |
| The number of sections. |
| |
| The number of sections. |
| |
| The number of trains in the section |
| |
| The Max-Plus algebra model of train |
| |
| Binary variable, |
| |
| The set of good-enough solutions that represent the input parameter for Model II. |
| |
| Binary variable. |
| |
| The minimum average delay time of each train |
| |
| The beginning time and the ending time of the effective time period in section |
The Max-Plus algebra method (Max-Plus) can be defined as follows, assuming
Based on the Max-Plus algebra method, the train operation process model of a high-speed railway can be built based on Eq. (
The elements in matrixes
The Max-Plus algebra model
When a train is delayed, the timetable changes and the matrices in the model are updated. The state matrices after
Six adjustment strategies are designed for the adjustment method of high-speed delayed trains. For each delayed train at each station, at least one adjustment strategy can be selected. The constraints of these adjustment strategies are analyzed as follows.
In addition, the time can be reduced for train
The condition of the strategy (2).
The constraint on this strategy is that the departure time of train
The condition of the strategy (4).
This adjustment strategy is constrained by Eq. (
The condition of the strategy (5).
The constraint conditions of this strategy include the following: (a) the station
Based on different adjustment strategy, the adjustment time of trains at each station and in each section is different and these values can be calculated using Eqs. (
Therefore, the target function of the approximate model is the minimum number of delayed trains after adjustment as described by Eq. (
Based on the parameters of the original timetable and Max-Plus algebra method, the algorithm of the train operation process is as shown in Algorithm
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
Based on the original delay, the algorithm for the rough model is as shown in Algorithm
( the solution set ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
Based on the set
( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
In this case, the timetable of the Beijing-Shanghai High-Speed railway in May 2017 is analyzed. The operation parameters are described as follows:
The speed of the trains is 300 km/h; the additional time of departure is 2 min and additional time of arrival is 3 min; the minimum departure headway of the original station is 5 min; the minimum arrival headway and departure headway at other stations are 4 min; the minimum stop time at the station is 2 min and the minimum operation time of each section is 12 min, 12 min, 18 min, 21 min, 12 min, 12 min, 14 min, 11 min, 7 min, and 13 min.
The original timetable is shown as in Figure
Original timetable.
A 20-min delay time is added to train G103 including 10 min from the Langfang Station to Tianjin South Station and 10 min from Tianjin South Station to Cangzhou West Station. The conflicts caused by the delay of train G103 and the other trains are shown as in Figure
Timetable after delay.
The optimal adjustment scheme is shown in Figure
The original timetable and adjusted timetable.
Train number | Train station | original arrival time | original departure time | Delayed arrival time | Delayed departure time | Adjustment scheme | Delay Time |
---|---|---|---|---|---|---|---|
G103 | Beijing South | 0 | 7:05 | 0 | 7:05 | 0 | |
G103 | Langfang | 7:23 | 7:23 | 7:23 | 7:23 | 0 | |
G103 | Tianjin South | 7:36 | 7:36 | 7:46 | 7:46 | 10 | |
G103 | Cangzhou West | 7:56 | 7:58 | 8:16 | 8:18 | 20 | |
G103 | Dezhou East | 8:23 | 8:23 | 8:41 | 8:41 | section acceleration 2min | 18 |
G103 | Jinan West | 8:44 | 8:46 | 8:56 | 8:58 | section acceleration 6min | 12 |
G103 | Taian | 9:03 | 9:04 | 9:15 | 9:16 | train postponement | 12 |
G103 | Qufu East | 9:21 | 9:21 | 9:33 | 9:33 | train postponement | 12 |
G103 | Tengzhou East | 9:35 | 9:37 | 9:47 | 9:49 | train postponement | 12 |
G103 | Zaozhuang | 9:46 | 9:46 | 9:58 | 9:58 | overtaking increasing | 12 |
G103 | Xuzhou East | 10:02 | 0 | 10:14 | 0 | train postponement | 12 |
G11 | Beijing South | 0 | 8:00 | 0 | 8:00 | 0 | |
G11 | Langfang | 8:18 | 8:18 | 8:18 | 8:18 | 0 | |
G11 | Tianjin South | 8:28 | 8:28 | 8:28 | 8:28 | 0 | |
G11 | Cangzhou West | 8:48 | 8:48 | 8:48 | 8:48 | 0 | |
G11 | Dezhou East | 9:10 | 9:10 | 9:10 | 9:10 | 0 | |
G11 | Jinan West | 9:32 | 9:34 | 9:32 | 9:34 | 0 | |
G11 | Taian | 9:48 | 9:48 | 9:48 | 9:48 | 0 | |
G11 | Qufu East | 10:06 | 10:08 | 10:06 | 10:08 | section acceleration 1min | 0 |
G11 | Tengzhou East | 10:21 | 10:21 | 10:21 | 10:21 | 0 | |
G11 | Zaozhuang | 10:28 | 10:28 | 10:28 | 10:28 | 0 | |
G11 | Xuzhou East | 10:41 | 0 | 10:44 | 0 | train postponement | 3 |
G133 | Jinan West | 0 | 8:54 | 0 | 8:54 | Operation based on the original timetable | 0 |
G133 | Taian | 9:08 | 9:08 | 9:08 | 9:08 | 0 | |
G133 | Qufu East | 9:29 | 9:30 | 9:29 | 9:30 | 0 | |
G133 | Tengzhou East | 9:43 | 9:43 | 9:43 | 9:43 | 0 | |
G133 | Zaozhuang | 9:54 | 9:59 | 9:54 | 10:00 | 0 | |
G133 | Xuzhou East | 10:18 | 0 | 10:18 | 0 | section acceleration 1min | 0 |
G177 | Beijing South | 0 | 7:25 | 0 | 7:25 | 0 | |
G177 | Langfang | 7:43 | 7:43 | 7:43 | 7:43 | 0 | |
G177 | Tianjin South | 7:59 | 8:03 | 8:02 | 8:06 | train postponement | 3 |
G177 | Cangzhou West | 8:21 | 8:21 | 8:32 | 8:32 | train postponement | 11 |
G177 | Dezhou East | 8:50 | 8:52 | 8:59 | 9:01 | section acceleration 2min | 9 |
G177 | Jinan West | 9:16 | 9:16 | 9:20 | 9:20 | section acceleration 5min | 4 |
G177 | Taian | 9:32 | 9:32 | 9:34 | 9:34 | section acceleration 2min | 2 |
G177 | Qufu East | 9:46 | 9:46 | 9:48 | 9:48 | train postponement | 2 |
G177 | Tengzhou East | 10:00 | 10:06 | 10:06 | 10:08 | reducing the dwell time 4min | 6 |
G177 | Zaozhuang | 10:18 | 10:20 | 10:20 | 10:22 | 2 | |
G177 | Xuzhou East | 10:38 | 0 | 10:40 | 0 | 2 | |
G234 | Jinan West | 0 | 9:25 | 0 | 9:25 | 0 | |
G234 | Taian | 9:43 | 9:43 | 9:43 | 9:43 | 0 | |
G234 | Qufu East | 9:59 | 10:01 | 9:59 | 10:01 | 0 | |
G234 | Tengzhou East | 10:14 | 10:14 | 10:14 | 10:14 | 0 | |
G234 | Zaozhuang | 10:25 | 10:34 | 10:25 | 10:34 | 0 | |
G234 | Xuzhou East | 10:52 | 0 | 10:52 | 0 | 0 | |
G261 | Beijing South | 0 | 7:15 | 0 | 7:15 | 0 | |
G261 | Langfang | 7:33 | 7:33 | 7:33 | 7:33 | 0 | |
G261 | Tianjin South | 7:49 | 7:51 | 7:54 | 7:56 | train postponement | 5 |
G261 | Cangzhou West | 8:10 | 8:10 | 8:24 | 8:24 | train postponement | 14 |
G261 | Dezhou East | 8:35 | 8:37 | 8:49 | 8:51 | train postponement | 14 |
G261 | Jinan West | 9:01 | 9:03 | 9:10 | 9:12 | train postponement | 9 |
G261 | Taian | 9:21 | 9:21 | 9:26 | 9:26 | section acceleration 4min | 5 |
G261 | Qufu East | 9:38 | 9:38 | 9:40 | 9:40 | section acceleration 3min | 2 |
G261 | Tengzhou East | 9:49 | 9:49 | 9:54 | 9:54 | 5 | |
G261 | Zaozhuang | 9:56 | 9:56 | 10:05 | 10:05 | overtaking reduction | 9 |
G261 | Xuzhou East | 10:11 | 0 | 10:21 | 0 | 10 | |
G265 | Beijing South | 0 | 7:48 | 0 | 7:48 | 0 | |
G265 | Langfang | 8:05 | 8:05 | 8:05 | 8:05 | 0 | |
G265 | Tianjin South | 8:22 | 8:24 | 8:22 | 8:24 | 0 | |
G265 | Cangzhou West | 8:43 | 8:43 | 8:43 | 8:43 | 0 | |
G265 | Dezhou East | 9:05 | 9:05 | 9:05 | 9:05 | 0 | |
G265 | Jinan West | 9:27 | 9:39 | 9:27 | 9:39 | 0 | |
G265 | Taian | 9:55 | 9:57 | 9:55 | 9:57 | 0 | |
G265 | Qufu East | 10:14 | 10:14 | 10:14 | 10:14 | 0 | |
G265 | Tengzhou East | 10:25 | 10:25 | 10:25 | 10:25 | 0 | |
G265 | Zaozhuang | 10:32 | 10:32 | 10:32 | 10:32 | 0 | |
G265 | Xuzhou East | 10:48 | 0 | 10:48 | 0 | 0 | |
G471 | Beijing South | 0 | 7:10 | 0 | 7:10 | 0 | |
G471 | Langfang | 7:28 | 7:28 | 7:28 | 7:28 | 0 | |
G471 | Tianjin South | 7:44 | 7:45 | 7:50 | 7:51 | train postponement | 6 |
G471 | Cangzhou West | 8:05 | 8:05 | 8:20 | 8:20 | train postponement | 15 |
G471 | Dezhou East | 8:30 | 8:32 | 8:47 | 8:49 | train postponement | 17 |
G471 | Jinan West | 8:56 | 0 | 9:06 | 0 | section acceleration 7min | 12 |
G57 | Beijing South | 0 | 7:20 | 0 | 7:20 | 0 | |
G57 | Langfang | 7:38 | 7:38 | 7:38 | 7:38 | 0 | |
G57 | Tianjin South | 7:54 | 7:58 | 7:58 | 8:02 | train postponement | 4 |
G57 | Cangzhou West | 8:18 | 8:18 | 8:28 | 8:28 | train postponement | 10 |
G57 | Dezhou East | 8:45 | 8:48 | 8:55 | 8:57 | reducing the dwell time 1min | 10 |
G57 | Jinan West | 9:11 | 9:11 | 9:16 | 9:16 | section acceleration 4min | 5 |
G57 | Taian | 9:26 | 9:26 | 9:30 | 9:30 | section acceleration 1min | 4 |
G57 | Qufu East | 9:42 | 9:42 | 9:44 | 9:44 | section acceleration 2min | 2 |
G57 | Tengzhou East | 9:53 | 9:53 | 9:58 | 9:58 | train postponement | 5 |
G57 | Zaozhuang | 10:03 | 10:04 | 10:09 | 10:10 | train postponement | 6 |
G57 | Xuzhou East | 10:22 | 0 | 10:28 | 0 | 6 |
The timetable after adjustment.
For the optimal adjustment scheme, there are five trains affected by the original delayed train G103. Train G133 operates according to original timetable because of strategy (2) and trains G234 and G265 are also not affected by the delay. The average delay time of each train at each station is 355 min. Using a standard PC with an Intel (R) Core (TM) i5-6500 3.2 GHz and 4 Gb of RAM, the calculation time is within 20 s using MATLAB 2016a and has good solution efficiency.
The average delay time of the trains is shown in Figure
The delay time of each train.
The delay time of each train at each station.
In this case, the delay time can be absorbed within the time period of five trains by optimal adjustment. The buffer time includes two parts; one is the headway buffer time between the two adjacent trains (the difference of the actual headway and the minimum headway between two adjacent trains) and the other is the operation buffer time of the train (the difference between the actual operation time and the minimum operation time in the same section). In this case, most delay of the trains can be reduced by the strategies of the section acceleration and train postponement. This indicates that the original timetable has a certain buffer time.
The buffer time of each train operation in each section of the original timetable and the adjusted timetable is shown in Figures
The operation buffer time of each train in each section.
Original timetable
Adjusted timetable
The buffer time of each train at each station by arrival headway.
Original timetable
Adjusted timetable
The difference of the buffer time between the original timetable and adjusted timetable.
As shown in Figures
Three schemes with different delay distributions are studied.
Scheme #1 is the scheme in Section
Scheme #2: 10 min delay time is added to trains G103 and G471, respectively, from Langfang Station to Tianjin South Station.
Scheme #3: 10 min delay time is added to trains G103 and G261, respectively, from Langfang Station to Tianjin South Station.
For Scheme #2, the adjusted timetable of the optimal adjustment scheme is shown in Figure
The optimal adjustment solution of the delay scheme #2.
Figure
The delay time of each train in different delay schemes.
In this report, a train operation process model is established for high-speed railway based on the Max-Plus algebra method. For operation adjustment of the delayed trains, six alternative adjustment strategies are proposed including the section acceleration, operation in advance, stop time reduction, increase of overtaking, reduction of overtaking, and train postponement. The constraints are analyzed for each adjustment strategy. Then, the train operation adjustment method is established with the two models and three algorithms. Finally, the portion of the delayed timetable of the Beijing-Shanghai High-Speed Railway is analyzed. The conclusions can be summarized as follows. (1) After the high-speed trains are delayed, the optimal adjustment scheme can be obtained by the operation adjustment method proposed in this report. The optimal adjustment scheme is a new timetable with the minimum number of delayed trains and the minimum average delay time of each train at each station. The adjusted new timetable can be directly applied in the real-time scheduling for the delay condition of the high-speed railway. (2) The buffer time, which includes the headway buffer time and the operation buffer time, can effectively absorb train delays and rapidly recover the schedule. However, the buffer time will simultaneously reduce the carrying capacity of the railway line. (3) This method can be applied in real-time railway rescheduling and it proves that the proposed method is valid and feasible for the operation adjustment of delay.
The train operation data and timetable data used to support the findings of this study are included within the article.
We certify that we have participated sufficiently in the work to take public responsibility for the appropriateness of the experimental design and method and the collection, analysis, and interpretation of the data.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the Natural Science Foundation of Inner Mongolia [grant numbers: 2017BS0501]; National Natural Science Foundation of China [grant number: 51668048]; and Inner Mongolia Autonomous Region University Scientific Research Project [grant number: NJZY18012].