Methods for solving the carrying capacity problem for High-Speed Railways (HSRs) have received increasing attention in the literature in the last few years. As important nodes in the High-Speed Railway (HSR) network, large stations are usually the carrying capacity bottlenecks of the entire network due to the presence of multiple connections in different directions and the complexity of train operations at these stations. This paper focuses on solving the station carrying capacity problem and considers train set utilization constraints, which are important influencing factors that have rarely been studied by previous researchers. An integer linear programming model is built, and the CPLEX v12.2 software is used to solve the model. The proposed approach is tested on a real-world case study of the Beijing South Railway Station (BS), which is one of the busiest and most complex stations in China. Studies of the impacts of different train set utilization constraints on the practical station carrying capacity are carried out, and some suggestions are then presented for enhancing the practical carrying capacity. Contrast tests indicate that both the efficiency of the solving process and the quality of the solution show huge breakthroughs compared with the heuristic approach.

As important nodes in the High-Speed Railway (HSR) network, large stations are usually the carrying capacity bottlenecks of the entire network due to the presence of multiple connections in different directions and the complexity of train operations at these stations. The China Railway Corporation (CRC), which owns the entire network and is also responsible for railway operation in China, has paid considerable attention to solving the carrying capacity problem of large stations to run more trains and satisfy the greatly increasing travel demands. However, the corporation currently lacks an accurate and effective approach for assessing the station carrying capacity of HSRs. From 2013 to 2014, we developed a software system named the “Intelligent Aided Train Timetable Programing Network Collaboration System” (IATPS), which was financially supported by the CRC [

Many studies have been carried out worldwide to explore the calculation of the station carrying capacity problem. In Europe, this problem is also defined as the railway infrastructure saturation problem [

Some examples in the literature focus on estimating the infrastructure capacity under disturbed conditions. Goverde et al. [

Other researchers studied some other factors that affect the railway capacity. Lai and Wang [

From the above literature review, the following conclusions were observed:

Previous researchers have proposed many meaningful approaches, considering various factors or under different situations. However, they rarely considered the train set utilization constraints. These are important factors that greatly influence the station carrying capacity.

The majority of the studies solved the problem using stochastic event simulation methods or heuristic algorithms, and the accuracy and efficiency of these methods cannot meet the current demands of HSR operations.

This paper offers the following contributions to the growing body of research work on the station carrying capacity problem:

The obtained practical station carrying capacity refers to a carrying capacity value that can be put into practice; it is not merely a theoretical number. Additionally, we study the influence of the train set utilization constraints on the station carrying capacity.

We propose an integer linear programming approach that is more accurate and efficient than the genetic algorithm approach applied in the IATPS. The performance of the IATPS station carrying capacity calculation function is improved.

A real-world case study is carried out to verify the feasibility of our approach.

The paper is organized as follows. In Section

The goal of solving the station carrying capacity problem is to calculate the maximum number of trains that the station can possibly serve within 24 hours [

The minimum connecting time affects the station carrying capacity.

Lack of allocated train sets affects the station carrying capacity.

The saturated timetable is the foundation of this approach because it limits the solution space within a reasonable range. If we generate an oversized timetable, the solving time of the model will increase; if the scale of the timetable is too small, the potentially realizable carrying capacity of the station will not be achieved even though the solving time may be short.

The generation principles of the saturated timetable are as follows:

Set a reasonable interval time for trains intensively arriving and departing the station, according to the minimum headway of the railway signaling system.

Set a reasonable station service time for trains arriving and departing considering the infrastructure maintenance during the late night period.

To satisfy the passenger travel demand as much as possible, the train companies pay attention to designing a good passenger train service plan and to scheduling a good train timetable. To ensure that the eventual capacity meets the passenger demand, reasonable proportions of different types of trains should be made available according to the current timetables or according to the passenger demand forecasts for newly built railway lines.

All the possible operation chains for every train in the saturated timetable should be generated, and they will make up the operation chain set. A complete operation chain for a train contains the following: get-in, stop or nonstop, and get-out. The inbound route and the outbound route must connect to the same track that the train stops at or runs through. The operation chain set contains all the possible operation chains of every train at the station. Generating a proper operation chain set is the premise of route conflict checking.

The generation principles of the operation chain set are as follows:

Generate all the possible operation chains for every train in the saturated timetable, referred to as the “train-track,” according to the track utilization rules at the station and the train running directions.

In some cases, multiple routes exist that connect one point to one track; all the feasible choices should be generated and referred to as “train-track-

Some trains are scheduled to add water or discharge sewerage at the station, while only a few tracks run near the relevant facilities. In such circumstances, we only generate the corresponding choices in the operation chain set.

Only generate the feasible operation chains for certain trains due to some other special operational rules. In this way, the operational constraints in the model are effectively reduced; because the optional operation chains that do not meet the rules are not generated, the corresponding constraints can be ignored.

The practical carrying capacity requires every train counted towards the capacity to be assigned to a conflict-free operation chain, so that the station operation plan can be put into practice. In the process of generating an operation chain set, there are usually multiple operation chains that are suitable for one train. The interaction between the routes in the operation chains is so high that many pairs of operation chain assignments lead to conflicts. Whenever two trains request the same infrastructure equipment (track, switch, intersection, etc.) at the same time, the two trains are defined as conflicting according to interlocking rules, and at least one of the trains should be rerouted or cancelled for safety reasons. We classify the conflicts into two groups: one contains the conflicts that occur on the tracks that trains stop on and the other contains the conflicts that occur in routes.

Let

When trains stop at platforms for passenger boarding and alighting, it is easy to check whether they are conflicting with other trains present on the track. Let both train

Let

This is different from the above method for checking whether the routes in the operation chains are conflicting. Before arriving at or departing from a station, a train usually requests the use of several track sections before occupying them. While passing through the route, the train will release the appropriate track sections along the route successively after the tail of the train has left these sections. When two routes request the same switches or intersections, there may be a conflict, as shown in Figure

A conflict between trains in operation chains.

Let train

Therefore, the time train

If the route is inbound, the request time is

Therefore, if

Let

To build an integer linear programming mathematical model, we make the following assumptions:

To ensure that we will always obtain a feasible solution, we propose a virtual operation chain that can be assigned to any train. Trains running though the virtual operation chain will not conflict with any train, but they will not be counted in the station carrying capacity.

Different from the real-time railway traffic management model, in which the trains may be rescheduled, we treat the arrival and the departure times of the trains in the saturated timetable as unalterable. This means the trains in the saturated timetable will be either rerouted or assigned to the virtual operation chain due to route conflicts, but the arrival and the departure times of the trains cannot be changed.

The sets

The notations for the proposed formulation are introduced in Definitions of Symbols in the Model, including their descriptions.

A 0-1 integer linear programming model was built as follows.

In Group I, constraints (

In Group II, constraint (

Because the model we built is a 0-1 integer linear programming model, after generating all the related data in the station module of the IATPS, we can directly solve the model using the integer programming solver CPLEX.

To deal with a series of problems in train timetable programing, we developed the software system named the “Intelligent Aided Train Timetable Programing Network Collaboration System” (IATPS), which was financially supported by the CRC. The core functions of the station module are as follows: station carrying capacity calculation; automatic scheduling of the station operation plan, given the station timetable; operation plan conflict detection and robustness optimization; and operation plan simulation based on station interlocking. The main interface of the station module is shown in Figure

The main interface of the station module of the IATPS.

The genetic algorithm is widely applied in the field of railway station operation optimization [

Flowchart of the genetic algorithm in the IATPS.

In general, the trains in the saturated station timetable are ordered chronologically. Each chromosome is encoded to a one-dimensional string; that is, every gene gets an operation chain index, which means that the corresponding train

Because of the heavy mutual influences among train routes, the algorithm is so time-consuming that the software performance needs improvements. Comparative experiments between the genetic algorithm approach applied in the IATPS and the integer linear programming approach proposed by this paper were carried out, and the comparison results are shown in Section

To verify the feasibility of our approach, the Beijing South Railway Station (BS) on the Beijing-Shanghai HSR was used as an example. The BS also serves the Beijing-Tianjin intercity railway, and the track map for the BS Beijing-Shanghai HSR yard is shown in Figure

The track map of the BS station yard for the Beijing-Shanghai HSR.

The BS is a dead-end station; both arriving and departing trains run through the left side. There are 4 borders in the yard named B1 to B4. B2 and B3 lead to Shanghai and are used by trains arriving or departing, respectively. B1 and B4 connect to the same depot, and the connecting lines are isolated from each other and from other railway lines. Train sets scheduled to be shunted to or from the depot can run in either of the two directions through B1 or B4 without influencing other trains. The 12 tracks in the yard are named track 8 to track 19. Because the BS is a dead-end station, the track utilization rule is simple: regardless of whether they are arriving or departing, trains can stop on any track in this yard.

The station infrastructure maintenance time interval is 0:00–7:00. Trains can arrive between 9:00 and 24:00 and depart between 7:00 and 22:00. The interval time for arriving and departing is 3 min.

The current actual timetable shows that there are only two train direction types at the BS: “terminally arrived” and “originally departed.” Therefore, the generated saturated timetable for the BS contains 300 terminal trains and 300 original trains:

After one train has terminally arrived at the station, the train set will be shunted to the depot or depart without shunting according to the “train set circulation scheme.” Because there is no train set circulation scheme corresponding to the generated saturated timetable, we propose a series of

We generate all the relevant data corresponding to different

The generation of relevant data.

| 20 min | 30 min | 40 min | 50 min | 60 min |
---|---|---|---|---|---|

Number of trains after connecting | 347 | 351 | 354 | 357 | 361 |

| 4,511 | 4,563 | 4,602 | 4,641 | 4,693 |

| 60,000 | 85,104 | 103,680 | 122,040 | 146,184 |

| 13.30 | 18.65 | 22.53 | 26.30 | 31.15 |

| 64,796 | 67,802 | 70,057 | 72,314 | 75,321 |

| 14.36 | 14.86 | 15.22 | 15.58 | 16.05 |

The

From Table

The CPLEX v12.2 was used for problem solving, and the solving time limit was set to 300 seconds. After running in the PC (Core E3, Frequency 3.30 GHz, Memory 8 GB), the integer solutions and the upper bounds of the practical carrying capacity corresponding to different

Practical carrying capacity corresponding to different

The computational results above show that the practical carrying capacity decreased by 24% when the

We can explain this phenomenon using Table

The maintenance capacity of the depot is limited, so the number of train sets of terminally arriving trains shunted to the depot cannot exceed

We propose a series of

Practical carrying capacity corresponding to different

Practical carrying capacity corresponding to different

The computational results above show that the practical carrying capacity has a roughly linear relationship to

Figures

The above results show the effectiveness of two measures for enhancing the carrying capacity of the only studied station. Considering the train set circulation on the rail network, they could provide more train services from the network point of view, but this is beyond the scope of this article.

To verify the high efficiency of this approach, two groups of comparative experiments were carried out in the CPLEX and the IATPS software based on the same data, using

Convergence processes of CPLEX and IATPS.

The convergence process of CPLEX was faster than IATPS. It required 54 seconds, while IATPS was much more time-consuming and required approximately 180 seconds. Moreover, the capacity obtained by our approach was 435 (optimal), which is much better than the value obtained by IATPS (385).

Comparing the two curves in Figure

Because calculating the practical station carrying capacity problem is a large-scale combinatorial optimization problem and due to the heavy mutual influence between train routes, the solving efficiency of the genetic algorithm is low.

Part of the station operation plan corresponding to a capacity of 435 is shown in Table

Part of the corresponding station operation plan.

Train name | Operation chain |
---|---|

X32 | B4-Track 9-B3 |

X31 | B4-Track 11-B3 |

X33 | B1-Track 15-B3 |

X35 | B1-Track 14-B3 |

X34 | B1-Track 16-B3 |

X30 | B4-Track 10-B3 |

X26 | B1-Track 16-B3 |

X25 | B1-Track 19-B3 |

X27 | B1-Track 17-B3 |

X29 | B4-Track 8-B3 |

X36 | B1-Track 17-B3 |

⋯ | ⋯ |

Y1-X48 | B2-Track 16-B3 |

Y2-X49 | B2-Track 15-B3 |

Y3-X50 | B2-Track 14-B3 |

Y4-X51 | B2-Track 12-B3 |

Y5-X52 | B2-Track 8-B3 |

Y6-X53 | B2-Track 11-B3 |

Y7-X54 | B2-Track 9-B3 |

Y8-X55 | B2-Track 10-B3 |

Y11-X58 | B2-Track 15-B3 |

Y12-X59 | B2-Track 17-B3 |

Y13-X60 | B2-Track 19-B3 |

⋯ | ⋯ |

In this paper, we proposed an integer linear programming model for calculating the practical carrying capacity of railway stations. A real-world case study showed that, compared with the IATPS, the solution time decreased by 126 seconds, and the solution value increased by 22%; both the efficiency of the solving process and the quality of the solution were significantly improved.

Moreover, we studied the impacts of different train set utilization constraints on the practical carrying capacity of the station. The results show that to enhance the practical carrying capacity of the station increasing the efficiency of the train set utilization (decreasing the minimum connecting time) is a good approach. Enhancing the maintenance capacity of the depots and allocating more train sets to the station are also helpful in certain ranges, but when

Our future research will focus on two major areas. First, we will study the impact of different types of saturated timetables (such as cyclic timetable and acyclic timetable) on station practical carrying capacity. Second, we will extend our approach from a station to a railway network, the solving scale will be larger, and it will be more challenging.

The generated saturated timetable

The operation chain set, which contains all the possible operation chains for every train in the saturated timetable

The virtual operation chain

The set of all the operation chains that are suitable for train

The set of assignment pairs that are in conflict on the tracks where the trains stop—for the generation of

The set of assignment pairs that are in conflict on the routes—for the generation of

The maintenance capacity of the depot—when the depot serves multiple railway lines,

The number of train sets that are allocated to this station

The condition of whether the train set of train

The condition of whether the train set of train

The arrival time of terminal train

The departure time of original train

The minimum connecting time for train sets of terminally arriving trains departing as original trains without shunting.

The grants do not lead to any conflict of interests regarding the publication of this manuscript. The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is financially supported by two Science and Technology Research Programs of China Railway Corporation (Grants 2014X001-B, 2016X005-D) and two Fundamental Research Funds of the Central Universities (Grants 2015JBM057, 2014JBZ008).