Joint optimization of high-speed train timetables, speed levels and stop plans for increasing capacity based on a compressed multilayer space-time network

With the steady increase in passenger volume of high-speed railways in China, some high-speed railway sections have faced a difficult situation. To provide more transport services, it is necessary to add as many trains as possible in a section to increase capacity. To solve this problem, a compressed multilayer space-time network model is constructed with the maximum number of trains that can be scheduled in the train timetable as the objective. The combination of the train stop plan and speed level is represented by the layer of network where the train is located, and constraints such as train selection, train safety, train overtake and cross-line trains are considered. A method based on timing-cycle iterative optimization is designed to decompose the original problem into multiple subproblems, and the solving order of the subproblems is determined by a heuristic greedy rule. Taking the Beijing-Jinan section of the Beijing-Shanghai high-speed railway as an example, the maximum number of trains was increased by 12.5% compared with the timetable before optimization. The saturated timetables provide detailed schedules, which helps decision-makers better adjust the timetable to run more trains.


Introduction
With the rapid development of high-speed railways worldwide, this good travel experience has attracted an increasing passenger flow, causing the capacity of some busy high-speed railways to approach saturation. To provide more services, operation strategy improvements with existing resources (e.g., modifying timetables) can be applied to increase the capacity (the maximum number of trains that can be scheduled) of the railway system in order to satisfy the higher transport demand.
A typical passenger train scheduling process consists of the following sequential steps: passenger demand surveying, train service plan development, and train timetabling. Under normal circumstances, these three steps are carried out sequentially. However, it should be noted that the speed level and the stop plan of trains are two factors that affect capacity that are determined in the second step (train service plan development). This will result in an inability to adjust the train's speed level and stop plan in the timetabling process, which finally determines the number of trains. Optimizing each stage of transportation planning individually and in order may be able to obtain the optimal solution of this stage, but for the whole system, it can only achieve a suboptimal solution. Therefore, the purpose of this paper is to find a way to optimize the train timetable, speed level and stop plan at the same time. A space-time network is a kind of modeling method that is often used to solve railway timetabling problems. First, an abstract train operation network is constructed according to the physical railway network. Then, according to the optimization objectives and constraints, the train selects the arcs in the space-time network to determine the process of train operation in time and space, that is, the train timetable. However, similar to the literal meaning of a spacetime network, the typical space-time network contains only two kinds of information: time and space. To achieve the purpose of the joint optimization of the timetable, speed level and stop plan, one important challenge is to express speed and stop information in the network.
To solve this problem, the multilayer space-time (MLST) network is proposed in this paper. For each combination of stop plans and speed levels, a space-time network will be constructed. Therefore, the MLST network is composed of multiple space-time networks. For each train, there will be multiple alternative space-time networks. The train's choice of a space-time network entails the choice of a speed level and stop plan. Just as a MLST network is composed of multiple space-time networks, the number of arcs in MLST network is several times of that in a space-time network. To reduce the number of arcs in the MLST network to improve the computational efficiency, a network compression technology is proposed according to the characteristics of the timetabling problem to generate the compressed multilayer space-time (CMLST) network. For large-scale real-world problems, the timing-cycle iterative method is used to decompose a large-scale problem into multiple subproblems that can be solved in a short time. Finally, the proposed model and method are verified through the example of the Beijing-Shanghai high-speed railway.
The remainder of this paper is organized as follows: A literature review is presented in Section 2. In Section 3, we first describe the passenger flow section and four aspects that need to be satisfied in the timetabling process. Then, the construction process of the MLST network and the CMLST network is described in detail. Section 4 develops a linear integer programming model with six kinds of constraints. Section 5 describes the proposed timing-cycle iterative method with a greedy rule that is integrated with the Gurobi solver. Section 6 shows the results of a real-world case to verify the validity of the proposed model and method. Finally, conclusions are presented in Section 7.

Literature review
Optimizing the train timetables is an important transportation organization strategy to increase the capacity. In recent years, many remarkable studies have been devoted to train timetabling problems. Goverde [1] presented a model and an algorithm to compute the propagation of initial delays over a periodic railway timetable. The model proposed by Meng L and Zhou X [2] periodically optimizes schedules for a relatively long rolling horizon while selecting and disseminating a robust meet-pass plan for every roll period. Four fundamental timetable formulations suitable for optimization were defined by Harrod [3]. Corman et al. [4] considered the biobjective problem of minimizing train delays and missed connections to provide a set of feasible nondominated schedules to support this decisional process. Cacchiani and Toth [5] surveyed the main studies dealing with the train timetabling problem in its nominal and robust versions. Niu and Zhou [6] focused on optimizing a passenger train timetable in a heavily congested urban rail corridor. Sun et al. [7] proposed a multiobjective optimization model for train routing on a high-speed railway network, which can provide an important reference for train plans in providing better service. Arenas et al. [8] proposed an alternative mathematical model to address the train timetabling problem (TTP) and presented a genetic algorithm implementing the model to rapidly obtain near-optimal train timetables. Wang et al. [9] proposed an event-driven model that involves three types of events, i.e., departure events, arrival events, and passenger arrival rate change events. Samà et al. [10] and Fu and Dessouky [11] examined various important topics in train scheduling, including timetabling under complex network or N-track conditions and real-time scheduling to mitigate delay propagation under various degrees of disturbances. Hu X et al. [12] found that the high circuity of train paths can be decreased with the construction of a high-speed railway line, which indicates that the structure of the high-speed rail (HSR) network in China should become more complete and that the HSR network can make the Chinese railway network more efficient. Liu S et al. [13] studied the performance and mutual influence of a syncretic railway network (SRN) that comprises high-speed railways, regional railways, and urban rail transit under the condition of traffic overload during peak hours.
As mentioned above, there have been many studies and results on the train timetabling problem. However, there are many stages of train transport planning, such as line planning, timetabling and vehicle scheduling, which depend greatly on one another. One of the weaknesses of the timetabling planning solutions is that it does not optimize the plans of each stage comprehensively, so it can only reach the local optimization. It is therefore beneficial to solve them in an integrated way rather than sequentially. The trend of follow-up research has been to combine the train timetabling problem with other problems. In recent years, many integrated approaches for railway operation optimization have been proposed.
Lusby et al. [14] analyzed the structure of the resulting integer program with respect to decomposition approaches and showed that decompositions that are superior to the canonical decomposition exist in planning stages, line planning, timetabling, and vehicle scheduling. Veelenturf et al. [15] modeled and solved the crew rescheduling problem with retiming. Kaspi M and T Raviv [16] formulated an integrated line planning and timetabling model with the objective of minimizing both user inconvenience and operational costs. Corman and Meng [17] offered a systematic review of real-time train dispatching, rescheduling, and disposition under stochastic and dynamic conditions. Schmidt and Schöbel [18] studied the integrated problem of determining a timetable and passengers' routes simultaneously. Zhang and Nie [19] proposed a minimum cycle time calculation (MCTC) model based on the periodic event scheduling problem (PESP) for a given train line plan for integrating capacity analysis with timetabling. Burggraeve S [20] proposed a heuristic algorithm to build a railway line plan from scratch that minimizes passenger travel time and operator cost and for which a feasible and robust timetable exists. Schöbel [21] developed and discussed a generic biobjective model for integrating line planning, timetabling, and vehicle scheduling. Shi J et al. [22] proposed an effective method for collaboratively optimizing the train timetable and accurate passenger flow control strategies on an oversaturated metro line.
These studies provide strong support for the joint optimization of train timetables and other issues. The recent publications associated with the integrated optimization of timetabling are listed and compared in Table 1.
Among the research performed on train timetabling, models based on a space-time network have demonstrated great power. The space-time network representation of the timetabling problem is well-developed since Caprara et al. [27] propose a integer programming based on a directed graph for solving the classic train timetabling problem. Harrod [28] proposed a directed hypergraph formula based on time discrete spatiotemporal network to deal with the conflict between train paths in space-time network, and applied the model to the train timetabling problem at the micro level of busy railways in America.
When the train diagram problem is combined with other problems, the space-time network needs to be extended to consider more variables. Some research have added the "state" dimension to form the space-time-state network modeling method. Zhou et al. [29] proposed a space-time speed network modeling method to achieve the collaborative optimization of train diagram problem and train speed with the goal of minimizing the total travel cost. Xu et al. [30] considered the "state" dimension of space-time state network as the state of locomotive traction train, and built a model to realize the collaborative optimization of train timetable and locomotive turnover. Meng et al. [31] optimized passenger service and train timetable by comprehensively considering passenger demand, infrastructure and vehicle capacity.
Space-time network is also used in the research of public transport and rail transit [32][33][34]. However, the goal of schedule optimization of public transport and rail transit is to optimize the departure interval at initial station in different periods according to the passenger flow demand. The similarity between these studies and high-speed railway is that they all need to build space-time network. These problems are essentially a network flow problem, and the solution process is to select the arcs in the network. In the application of MLST, the most important difference of high-speed railway vs other transportation is that the relevant constraints of time headway are required to avoid the conflict of trains in time and space. These constraints are the most difficult constraints in railway optimization problems, which will greatly improve the difficulty of finding a feasible solution. Therefore, for the improvement of railway capacity, there is little reference experience in public transport and rail transit.
According to the analysis of literatures, the method based on space-time network shows strong adaptability in railway transportation organization optimization. However, for the increasing of railway capacity, it is necessary to consider the train speed level and stop plan to construct a dedicated space-time network. There are few studies in the current timetabling research and joint optimization research on the train timetable, speed level and stop plan is lacking. By solving these problems, this paper makes the following improvements to the literature:  3. A timing-cycle iterative optimizing method approach is combined with a greedy rule for solving the problem. We use the timing-cycle iterative optimizing method to decompose the original problem into subproblems. To improve the accuracy of the results, the subproblems are solved by the branch-and-bound algorithm using the Gurobi solver. The whole algorithm process implemented by Python.

Problem statement
This paper considers a high-speed railway passenger flow section, as illustrated in Fig

Passenger demands
A typical passenger train scheduling process consists of the following sequential steps: passenger demand surveying, train service plan development, and train timetabling. This paper focuses on the third step. Indirectly, passenger demands are considered in this timetabling process as the number of intermediate stops.

Number of different types of trains
There are many types of trains operating in the passenger flow section, which is mainly reflected by the speed level. Due to the limitation of the number of electrical multiple units (EMUs), the number of trains of a certain speed level is limited.

Safety headway constraints
To ensure the safety of train operation, it is necessary to preserve a minimum time interval (headway) between trains to guarantee that there is no conflict in the train schedule.

PLOS ONE
Joint optimization of high-speed train timetables, speed levels and stop plans for increasing capacity The stop plan and the speed level of the train will affect the running time of the train in the passenger flow section, thereby affecting the maximum number of trains in the passenger flow section. To maximize the number of trains, the problem studied in this paper is to determine the train stop plan, speed level and train timetable at the same time to get a global optimal solution.

Network construction
Before describing the network, the related sets and parameters are defined in Table 2.
Typical space-time network. The network proposed in this paper is based on the typical space-time network. A typical space-time network is illustrated in

Starting arc set A o
For the starting station o l of train l and the start time window, there is a starting arc from the dummy source � o to the point (o l , t), which means that the train starts to run from the starting station at time t.

Segment arc set A e
For segment e and time t, there is a segment arc from the beginning (s, t) to the end of segment (s',t'), which means that the train departs from station s at time t and arrives at station s' at time t'. It should be noted that s and s' must be two successive stations, and the difference between t and t' is the running time of train l in section e. When solving a train timetabling problem, the first step is to generate the corresponding space-time network according to the parameters of the train, that is, to determine which arcs exist in the network. Then, the process of solving the timetabling problem is the process of selecting arcs in the space-time network, and all the selected arcs form the final train timetable. In Fig 2, the starting arcs are indicated by green arrows, the segment arcs by blue arrows, the stop arcs by yellow arrows and the end arcs by red arrows. All arcs can be selected. The selected arcs are indicated by highlighted black arrows and finally form the path of the train.

Dwell arc set
Multilayer space-time (MLST) network construction. In contrast to the traditional train timetabling problem, the difficulty of this paper is to determine the train stop plan and speed level at the same time. However, the stop plan and speed level will affect the arcs existing in the space-time network.
For each combination of stop plans and speed levels, a different space-time network will be formed, so for each train, there will be multiple alternative space-time networks. The train must select the space-time network first and then select the arcs in the selected network. This

PLOS ONE
Joint optimization of high-speed train timetables, speed levels and stop plans for increasing capacity kind of model is composed of multiple two-dimensional space-time networks, which is called an MLST network in this paper.
An MLST network is illustrated in Fig 3. The MLST network G = (V, A) also consists of vertices and arcs. The vertex set V of the MLST network G is V ¼ f� o; � dg [ fðp; k; s; tÞjp 2 P; k 2 K; s 2 S; t 2 Tg, where p and k represent the space-time network where the vertex is located. Arc set A of the MLST network also contains four types of arcs, but now 6 subscripts are needed to represent them, which have the form (p,k,s,s',t,t').

PLOS ONE
Although the form of the CMLST network is the same as that of the typical space-time network, it is different from the typical space-time network in that not all arcs in the network can be selected by trains. A certain type of train can only select some of the arcs. The CMLST network has a form between those of a space-time network and a space-time-state network. Compared with the space-time network, it can express more information, while compared with the space-time-state network, it can reduce the dimensionality of the network and effectively reduce the computational complexity.

Model formulation
The optimization model aims to incorporate the train speed level and stop plan into the timetable design process considering realistic operating constraints to maximize the number of trains in the passenger flow section. Table 3 lists the notation used in the optimization model.

Model notations
Objective function and constraints. Objective function. Within the CMLST network framework, the objective function for maximizing the total number of trains in the passenger flow section is stated in Eq (1).
Objective function (1) aims to maximize the number of trains for which stop plan k is not 0 (i.e., the trains are scheduled in the timetable).

PLOS ONE
from all the ending arcs of train l from (d l , t) to � d. Otherwise, if y p;k l ¼ 0 for all p 2 P and k 2 � K , then no arc should be selected from the ending arcs of train l. Constraint (4) means that for all vertices (s, t) 2 V, the number of arcs from (s, t) to (s', t') should equal the number of arcs from (s', t') to (s, t).
These constraints ensure that when a train is scheduled, it will start from the dummy origin and end at the dummy sink after a series of network vertices, and the selected arcs will finally form a complete path.
Train selection constraint. The selection of trains in set L is not arbitrary. It must not only meet the service frequency demand of the stations but also consider the limitation of the number of consecutive trains and the resources of the EMU. A set of train selection constraints is constructed as follows: Constraint (5) means that for all trains in L, a combination of a speed level p and stop plan k must be selected, and if k = 0 in the selected combination, it means that the train will not be scheduled in the timetable. Constraint (6) means that for intermediate station m, the number of trains stopping at station m (r k m ¼ 1) should be greater than or equal to the demand of station m. Constraint (7) means that the number of trains with speed level p should be less than the maximum number of EMU resources.
Consecutive trains with the same speed level can improve the utilization rate of the capacity, but it will lead to a strong homogeneity of trains, which will cause concentrated train stops and long waiting times of overtaken low-speed trains at the stations. Constraint (8)  The purpose of constraint (12) is to ensure that the difference between t and t' of the selected arc (s,s',t,t') is equal to the standard travel time plus the additional starting and stopping time.
Train safety (time headway) constraint. To guarantee the safety of high-speed railway operation, there is a minimum time headway between the departure, arrival and passing through of two consecutive trains at the same station. In this paper, seven kinds of train safety constraints

PLOS ONE
Constraint (18) means that for segment e = (s, s'), time t' and two trains, if the first train arrives at station s' at time t', then the second train, which passes through station s' at time t"', cannot select arcs in the time range ½t 0 ; t 0 þ t s dt �. Otherwise, at most one arc can be selected by the latter train.
Constraint (19) means that for segment e = (s, s'), time t' and two trains, if the former train passes through station s' at time t', then the latter train, which arrives at station s' at time t"', cannot select arcs in time range ½t 0 ; t 0 þ t s td �. Otherwise, at most one arc can be selected by the latter train.
Train overtaking constraint. Train overtaking can only occur in a station, not in a segment; otherwise, there will be conflict. Cross-line train constraint. For high-speed railway passenger flow sections with long mileage, there are usually some trains from other lines called cross-line trains, which usually have a higher priority than the trains in the research section. In this model, cross-line train l with speed level � p and stop plan � k can be scheduled in advance by setting the relevant variable to 1, as shown in Eq (21). The arcs occupied by the cross-line trains in the network cannot be selected by other trains again.
Binary variable constraint. There are also binary definitional constraints for variables x p;k;a l and y p;k l . Timing cycle iterative optimizing method The CMLST network can reduce the arcs of the network and the calculation time significantly. However, due to the complexity of railway systems, the scale of real-world problems is often beyond control. For the space-time network model, the variable size is equal to the number of arcs in the network. When the time dimension and space dimension of the case increase linearly, the number of available arcs in the network will increase exponentially, resulting in very complex problem solving. Therefore, we need to find an effective method to address this situation.
The timing-cycle iterative optimizing method is often used in a large-scale railway timetabling problem. The idea of the method is as follows: divide the time horizon into several subdomains, and schedule the trains in each subdomain in turn. When a subdomain is saturated, it is no longer possible to add new trains in this subdomain. Therefore, when a subdomain is scheduled, only the arcs of the current subdomain need to be constructed, reducing the scale of the problem. In the calculation of a subdomain, it is necessary to consider the influence of trains in other subdomains and to take the trains that may conflict as the known input to ensure that there is no conflict. In this way, the original problem is divided into several subproblems, and each subproblem can find the optimal solution in a limited time. After all the subproblems are solved, the solutions of all the subproblems are combined into a complete solution set, and the solution of the original problem is obtained. The flow diagram for the timing-cycle iterative optimizing method is presented in Fig 6. For the determination of the order of the subproblems, this paper introduces the idea of greedy rules. It is known that trains scheduled in one subdomain will affect the adjacent subdomains. Since our goal is to maximize the number of trains in the timetable, we should solve the most intense time domain first. In terms of specific quantitative indicators, if there are more stop demands in a time domain, the order of this time domain should be higher.

Case study
We take the real-world case of the Beijing South to Jinan West passenger flow section of the Beijing-Shanghai high-speed railway as an example, which is one of the most important highspeed rail lines in China.

Data input
In terms of infrastructure, there are 6 stations, 5 segments and 4 intermediate stations in the Beijing South-Jinan West passenger flow section, and the total mileage is 419 kilometers. This case considers a full day of operation, from 6:00 to 24:00. The number of available trains is set to 216 (12 trains per hour). The details are shown in Table 4.
According to the data obtained from the survey of railway company. There are two speed levels of trains in the studied section, namely, 300 km/h and 350 km/h. The travel time of trains at the two speed levels in each segment and the additional starting and stopping time of each station are shown in Table 5.
The minimum headway time and stop time at each station are the same, as shown in Table 6.
We used Gurobi 9.1 and the timing-cycle iterative optimizing method (implemented in Python 3.7) to solve the CMLST model. In the process of problem solving, Python's function is to implement iterative optimization method proposed in section 5, including processing input data, dividing the problem into subproblems and integrating the results of  subproblems. The function of Gurobi solver is to solve subproblems in turn. The specific method used is the branch pricing method. For the Gurobi solver, we can set the parameters according to the characteristics of the problem to improve the calculation speed, as shown in Table 7. The MIPGap refers to the max relative difference between the solution and the theoretical optimal solution. The value of MIPGap affects the calculation time and the quality of solution. For the large-scale optimization problem in this paper, the theoretical optimal solution usually cannot be found in an acceptable time. Therefore, it is necessary to find an appropriate MIPGap value to obtain a satisfactory solution in an acceptable time.
Through a large number of numerical experiments, we find that it is appropriate to set the value to 0.1, that is, to ensure that the relative difference between the obtained solution and the optimal solution is within 10%.

Computational results
Given the real-world schedule of the Beijing-Shanghai high-speed rail line shown in Fig 7, we generate an optimized real-world train timetable in Fig 8. The overall total train number is increased from 152 trains to 171 trains (with an equivalent capacity enhancement of approximately 12.5%). Cross-line trains are indicated by red lines, and some adjustments can be made for times when the capacity is too tight. There is no conflict between trains, and all the kinds of constraints set by the model are satisfied, which proves that the model is effective. The calculation results for each hour are shown in Table 8. As the busiest high-speed rail line in China, the capacity utilization rate of the case section is 100% in 11 hours of the day. The available trains are all in periods of less passenger flow, which indicates that the utilization of capacity in the current transportation plan is close to saturation, so it is necessary to take further measures to strengthen the capacity or optimize the transportation organization. In terms of the calculation process, the gap value of the solution in each period can be controlled to below 10%, which indicates that the quality of the solution is good.

Conclusion
This paper studies a railway capacity increasing problem that integrates train timetabling with the train speed level and stop plan. The capacity increasing problem is solved as a joint optimization model timetable saturation problem. The saturated timetable for capacity increasing purposes has the maximum number of trains given the infrastructure-and passenger demand- PLOS ONE related aspects. We use a compressed multilayer space-time network to model this optimization problem and propose integer programming that maximizes the number of trains in the railway sections. A timing-cycle iterative optimizing method is proposed to solve this problem efficiently. According to the case test, the integration problem is managed well by the iterative approach proposed. Academically, this paper extends the current space-time network-based modeling method to a multilayer space-time network, and compression technology is proposed to reduce the number of arcs in the network. This is a beneficial proposal. In practical applications, the train timetable generated by the CMLST model can intuitively show the carrying capacity and its usage in each hour, indicating at which periods more trains can be added and providing a reference schedule for the railway company.
In future research, we will consider designing a multiobjective model based on the approach proposed in this paper to find a Pareto solution that can balance the total train number, total travel time, total dwell time and so on. For larger-scale real-world problems, it may be necessary to design heuristic methods to reduce the number of combinations of speed levels and stop plans according to the specific passenger demand.