Train Occupancy Time of a Railway Section and the Combined Occupancy Method for Capacity Assessment

Featured Application

This work can be applied to the rapid and concise evaluation of high-speed railway section capacity.

Abstract

Accurate and convenient assessment of a section’s capacity is an essential problem in the operation of high-speed railway companies. The concept of occupancy time is widely used in the field, but there is still a lack of quantitative analysis to support follow-up studies and application. In this research, the calculation method for one train, two trains and multiple trains is discussed. The occupancy time of the two trains was divided into three cases according to the relationship between them, and the calculation formulas are given. For the calculation of multiple trains, the circumstances were too intricate to be classified. To solve this problem, the concept of a continuous train group was proposed. The three properties of the continuous train group were analyzed and proved by mathematical derivation to provide theoretical support to transform a multi-train problem into many two-train problems. Finally, a combined occupancy method for capacity assessment was formed and applied to the Beijing–Shanghai high-speed railway. A series of numerical cases were designed to evaluate the performance compared with the UIC406 method. The results demonstrate that the combined occupancy method can better assess the capacity and is more convenient to implement.

1. Introduction

The sections of a high-speed railway are the basic space units of train movement. A reasonable and accurate assessment of the carrying capacity of railway sections is an important basis for train timetabling and dispatching. As the passenger volume of high-speed railways increases year by year [1], the capacity of some lines tends to become saturated. As a result, railway companies pay increasing attention to the fine management of capacity and put forward higher requirements for the accuracy of capacity assessment results.

A railway section in this paper refers to the railway line between two main stations, from which a train can depart and arrive. A railway section consists of two terminal stations and many intermediate stations. For the capacity assessment of high-speed railway sections, numerous approaches have been developed. The most relevant methods can be classified into three levels: analytical methods, optimization methods, and simulation methods [2]. Analytical methods are designed to model the railway environment by means of mathematical formulae or algebraic expressions. They usually obtain theoretical capacities and determine practical capacities either as a percentage of the theoretical capacity or by including regularity margins when they calculate the theoretical capacity.

In Russia and China, some scholars have discussed the application of the deduction coefficient method in high-speed railways in the early stage [3,4,5,6]. However, with the gradual in-depth study of the deduction coefficient method, due to the excessive factors affecting capacity, the situations that need to be considered and the form of the formulas are becoming increasingly complex, which makes subsequent research difficult and is not conducive to practical applications [7]. The deduction coefficient method is a calculation method based on statistical analysis to obtain the average deduction coefficient of various types of trains by simulating a train timetable under various situations to evaluate the occupancy of the trains to capacity. Zhang and Tian [8] analyzed the essence of the deduction coefficient method and believed that the traditional deduction coefficient method is no longer suitable for the calculation of the capacity of high-speed railways. It is necessary to start with the overlap relationship between the trains and the time occupancy of the train timetable to carry out follow-up quantitative research more directly and effectively.

In Germany and some other European countries, capacity assessment methods based on the average minimum headway and queuing theory are widely applied [9]. Burdett and Kozan [10] have developed several methods to calculate the absolute capacity of railway lines and networks. Weik, Niebel and Niessen [11] discussed a stochastic model for the capacity analysis of railway lines with applications in long-term planning, and a generalization of an approach widely used in Germany was derived that is valid for a wide range of timetable requirements. The extensions of these approach are given in [12,13,14,15].

However, none of these methods consider a timetable with its scheduled arrival and departure times as an input for the capacity assessment. The International Union of Railways proposed the UIC method [16]. It calculates the capacity of line sections to identify bottlenecks and takes into account the order of trains, and a buffer time is inserted to achieve an acceptable quality of service. This method was officially dropped some years ago and is no longer recognized as a standard. It has been superseded by more general recommendations [17] that establish a link between railway capacity and railway quality. In 2004, the UIC406 leaflet [18] was published and proposed the compression method for capacity assessment. In 2013, the second edition [19] was published, a methodology was presented enabling the calculation of nodes’ capacity based on the same principles. Based on the existing train timetable of the railway section, the capacity utilization of the section was evaluated through two steps: compression and encryption.

The UIC method (compressed method) is simple and effective, and many scholars have subsequently extended and expanded it based on the UIC406 method, forming a mainstream capacity assessment system in Europe. Landex [20] extended the UIC406 method to assess the capacity of single-track railways. Lindner and Pachl [21] compared and combined the UIC406 method and a method independent of a timetable. Liu, Han and Li [22] proposed a compression optimization model and encryption method for train paths to evaluate the utilization rate and available capacity of the station. Yaghini et al. [23] generated a compressed timetable based on the UIC406 method and applied it to Iranian railways. Beinovi, Goverde and Quaglietta [24] studied the application of the compression method in station capacity evaluation and realized the mathematical calculation of the compression method through maximum addition algebra. Wang, Tian and Zhang [25] calculated the passing capacity of the railway section based on the buffer time of the compressed timetable and the encryption of trains in the idle time.

Each section of the railway line, due to the differences in their status and function in the railway network, has a different internal structure and faces a different external environment. Therefore, the person responsible for timetable planning will adopt different strategies for different sections, which will have a great impact on the section carrying capacity. After a long period of operation and iterative optimization, high-speed railways in various countries have formed relatively stable train timetable characteristics, which will not be easily changed, and this information is contained in the train timetable structure [26]. The timetable-dependent method can use information on existing timetables to make the result more pertinent, which is more accurate than the analytical method based on the principle of statistical averaging in specific scenarios. Although the compression method can maintain the train timetable structure, the method changes the train arrival and departure time in the compression process without considering the relationship between the target section and the adjacent sections. In addition, the buffer time is deducted in this process, which will lead to larger capacity calculation results.

Summarizing the above research in

Table 1. Summary of section capacity calculation methods.

Area	Method	Type	Disadvantage
Russia and China	Deduction coefficient method	Timetable independent	Lack of accuracy
Germany	Average minimum headway	Timetable independent	Lack of accuracy
France	UIC method	Timetable dependent	Timetable structure is changed

, it can be found that the core of the capacity assessment is the time occupied by trains of railway facilities and equipment. The purpose of this paper was to take the occupancy time of trains on timetables as the research focus directly to form a conceptual system and propose a derived method for the railway section carrying capacity assessment based on train timetables to make the capacity calculation process faster and the results more accurate. The contributions of this paper are: (1) a conceptual system centered on train occupancy time is formed, including standard trains, virtual trains, combined trains, and continuous train groups, to provide theoretical support for capacity assessment based on occupancy time; (2) a combined occupancy method of capacity assessment is formed, and the process and formulas are given.

The remainder of this paper is structured as follows. Section 2 explains the concept of occupancy time and conducts a graphical analysis of the calculation of occupancy time of one train and two trains. A method combining two or more trains together is proposed to solve repeated occupancy problem in Section 3, and a derived capacity assessment is formed in Section 4. In Section 5, a case study is conducted for the Beijing–Shanghai high-speed railway. In the end, Section 6 summarizes the conclusions.

2. Train Occupancy Time and Graphic Analysis

The train occupancy time on the train timetable refers to the occupancy of the train’s path line to the available time horizon of the train timetable. Its essence is the reflection of the occupancy of section facilities by the train. Train occupancy time is related to train type, speed level, operation, train control system, blocking mode of passenger flow section, section length and other factors. The concept of train occupancy time on train timetables has formed a certain consensus in previous studies [27,28,29], yet there has been no detailed description and quantitative analysis of this term.

2.1. The Standard Train and Occupancy Time

For the nonparallel train timetable containing multiple types of trains, the headway time between different types of trains is different because the train speed level and dwelling will affect the minimum headway time between trains. To unify the benchmark of comparison, the concept of the standard train is introduced. In research on high-speed railway capacity, the standard train is usually selected as the nonstop train with the highest speed. Since this kind of train occupies the least time on the timetable compared with other types of trains, this setting is also adopted in this paper. For the parallel timetable including only high-speed nonstop trains (standard trains), as shown in Figure 1, the time distance between the latter train and the former train is the tracking headway, $I_s$. We can see that the occupancy of the timetable by each standard train is the tracking headway $I_s$ between standard trains.

2.2. Graphic Analysis of One Train

With the concept of a standard train, this section analyzes the occupancy time of one single train on the train timetable. For one single train i, the occupancy time of train i on the timetable starts at the departure time at the departure station. However, it is more important to determine the end time when train i occupies the train timetable. To solve this problem, the concept of virtual (additional) trains in UIC406 was introduced. As shown in Figure 2, an auxiliary virtual standard train is added after train i with the minimum tracking headway time I_is.Then, the departure time of the virtual standard train at the departure station is the time point when the occupancy of train i on the timetable ends.

Therefore, the occupancy time of train i on train timetable $O{T}_i$ is the period between the departure time of train i at the departure station and the departure time of the subsequent virtual standard train at the departure station. According to Figure 2, the occupancy time of single train i can be calculated by Equation (1).

$$O{T}_i=T{T}_i+I_{is}-T{T}_s$$

(1)

2.3. Graphic Analysis of Two Trains

For two trains i and j, their total occupancy time on the timetable cannot be obtained by simply adding up their own occupancy time because the two trains may occupy the same time period repeatedly. In this article, this phenomenon is called the section repeated-occupancy problem. If the section repeated occupancy of trains on a timetable is not considered, only a simple sum will lead to larger calculation results.

For ease of explanation, there are assumed to be two trains i and j, and the departure time of train i is earlier than the departure of train j from the departure station. In the train timetable, there may be three situations between two trains: no repeated occupancy, partial repeated occupancy and full repeated occupancy.

As shown in Figure 3, the two trains have no repeated occupancy. In this situation, the total occupancy time $O{T}_{ij}$ can be obtained by adding the occupancy time of train i and train j.

$$O{T}_{ij}=O{T}_i+O{T}_j$$

(2)

Figure 4 shows a situation of the partial repeated occupancy of two trains. In this situation, the travel time of train j is longer than the travel time of train i due to the differences in speed or stop plans. Therefore, train j can start from the departure station before the occupancy of train i is over, resulting in overlapping part $R{T}_{ij}$ between the occupancy time $O{T}_i$ and $O{T}_j$. In this case, the overlapping part shall be removed from the total occupancy time of the two trains.

$$O{T}_{ij}=O{T}_i+O{T}_j-R{T}_{ij}$$

(3)

Figure 5 shows the situation of full repeated occupancy of two trains. In this situation, because train i is overtaken by train j when it stops at the intermediate station, the occupancy time of train j is completely covered by the occupancy time of train i. In this case, the overlapping part between the two trains is equal to the occupancy time of train j. Therefore, the total occupancy time of the two trains on the train timetable is the occupancy time of train i.

To summarize the above three situations, the repeated occupancy time between the two trains $R{T}_{ij}$ can be calculated by Equation (4), and the total occupancy time of the two trains $O{T}_{ij}$ can be calculated by Equation (5):

$$R{T}_{ij}=\{\begin{array}{cc}0& R{T}_{ij}<0\\ O{T}_i-(d_j-d_i)& 0\le R{T}_{ij}\le O{T}_j\\ O{T}_j& R{T}_{ij}>O{T}_j\end{array}$$

(4)

$$O{T}_{ij}=O{T}_i+O{T}_j-R{T}_{ij}$$

(5)

3. Combined Method for Occupancy Time Calculation

In Section 2.3, we introduced the calculation method for the occupancy time of two trains. Through further research, we found a combined method in which the two trains can be seen as a combined train for calculation. This method is the basis for the calculation of multiple trains.

First, the concept of the repeated occupancy relationship is given. For two trains i and j, if there is an overlap in their occupancy time in the timetable, that is, $R{T}_{ij}>0$, it is considered that there is a repeated occupation relationship between the two trains, corresponding to the two trains with partial or full repeated occupancy in Section 2.3. For the two trains i and j with a repeated occupation relationship, we can use the combined method for calculation. The specific proof is as follows.

3.1. Occupancy Time of Two Trains

For two trains with partial repeated occupancy, substituting Equation (4) into Equation (5), we can obtain:

$$O{T}_{ij}=O{T}_i+O{T}_j-(O{T}_i-(d_j-d_i))=O{T}_j+d_j-d_i$$

(6)

According to Equation (1), OT_j = TT_j + I_js − TT_s. Substituting into Equation (6), we can obtain:

$$O{T}_{ij}=T{T}_j+I_{js}-T{T}_s+d_j-d_i$$

(7)

Substituting TT_j = a_j − d_j  into Equation (6), we can obtain:

$$O{T}_{ij}=(a_j-d_i)+I_{js}-T{T}_s$$

(8)

By observing the form of Equation (8), if train i and train j are regarded as a combined train $\overline{ij}$, as shown in Figure 6, taking $d_i$ as the departure time and $a_j$ as the arrival time, the content in the parentheses of Equation (8) is the section travel time $T{T}_{\overline{ij}}$ of train $\overline{ij}$.

$$O{T}_{ij}=O{T}_{\overline{ij}}=T{T}_{\overline{ij}}+I_{js}-T{T}_s$$

(9)

For two trains with full repeated occupancy, substituting $R{T}_{ij}=O{T}_j$ into Equation (5), we can obtain:

$$O{T}_{ij}=O{T}_i=T{T}_i+I_{is}-T{T}_s$$

(10)

Therefore, we can regard train i and train j as a combined train $\overline{ij}$, as shown in Figure 7, taking $d_i$ as the departure time and $a_i$ as the arrival time.

According to the above analysis, two trains with a repeated occupation relationship can be regarded as a combined train whose departure time is the earliest departure time of two trains and whose arrival time is the latest arrival time of two trains. Then, the occupation time can be calculated according to Equation (1). This method does not need to calculate the repeated occupancy time between two trains, nor does it need to calculate the occupancy time of the two trains, which greatly simplifies the calculation process.

3.2. Continuous Train Group and Occupancy Time of Multiple Trains

For multiple trains, as the number of trains increases, the relationship between trains may be very complex, and it is impossible to analyze all situations. To solve this problem, we can combine multiple trains into one train for calculation to avoid dealing with the complex relationship between trains.

First, the concept of the continuous train group is proposed and defined as follows: a continuous train group refers to a group of trains where the trains all have a repeated occupancy relationship with at least one other train in the group. That is, for a continuous train group L and a train i, if there is a repeated occupancy relationship between train i and any train j in the L, then train i belongs to L. The trains in the continuous train group L have the following three properties:

Property 1.

For train $i\in L$, if there is a repeated occupancy relationship between train i and train j, and there is a repeated occupancy relationship between train j and train k, then train $k\in L$.

Proof of Property 1.

Text of the proof. Note that train $i\in L$, and there is a repeated occupancy relationship between train i and train j, according to the definition of the continuous train group, train $j\in L$. There is also a repeated occupancy relationship between train j and train k; therefore, by definition, train $k\in L$. This property reflects the continuity of the train group. Through the repeated occupancy relationship, trains i, j and k form a train group. □

Property 2.

Any train on the timetable belongs to and belongs only to one continuous train group.

Proof of Property 2.

The proof by the contradiction method was adopted to prove this property. Assuming that a certain train I belongs to two train groups, $L_1$ and $L_2$, then there must be a train $j\in L_1$ that has a repeated occupation relationship with train i, and there must be a train $k\in L_2$ that has a repeated occupation relationship with train i. According to Property 1, train j and train k belong to the same train group, which contradicts the preconditions; thus, train i cannot belong to two train groups. □

Property 3.

A continuous train group L composed of m trains can be regarded as one combined train whose departure time is the earliest departure time of the trains, and the arrival time is the latest arrival time of trains.

Proof of Property 3.

In Section 3.1, we have proven that two trains with a repeated occupation relationship can be regarded as one combined train whose departure time is the earliest departure time of two trains and whose arrival time is the latest arrival time of two trains. Therefore, we can combine the train in pairs to form one combined train. □

As shown in Figure 8, the trains in train group L were sorted according to the departure time from the departure station. Based on the earliest train $l_1$ from the departure station, train $l_1$ and train $l_2$ can be combined into train $l_{12}$. Furthermore, train $l_{12}$ and train $l_3$ can be combined into train $l_{123}$. Each combination is a pairwise combination based on the previous combination. Finally, after m − 1 combinations, the combined train $l_L$ can represent this train group.

The occupancy time of the continuous train group L is:

$$O{T}_L=T{T}_L+I_{Ls}-T{T}_s$$

(11)

In this way, the calculation of multiple train occupancy time can be transformed into the calculation of one combined train, which greatly simplifies the calculation process. This method is also applicable to one single train or two trains.

4. Combined Occupancy Method for Capacity Assessment

The above analysis and proof provide the combined method of occupancy time of one single train, two trains and multiple trains on a train timetable. On this basis, this paper formed the combined occupancy method for the capacity assessment of railway sections. As shown in Figure 9, a method flow chart is described below with an example.

Step 1: Sorting

According to the departure time from the departure station of the section, all trains in the operation diagram are sorted and named $l_1$, $l_2$… in sequence. The occupancy time of each train is calculated according to Equation (1).

Step 2: Grouping

Starting from the first train $l_1$, the relationship between the trains is judged according to Equation (4), and the trains are divided into continuous train groups.

Step 3: Combine

Combine all train groups into combined trains according to the earliest departure time and the latest arrival time of the trains in the train group.

Step 4: Calculate occupancy time

Calculate the occupancy time of all combined trains on the train timetable and add them to obtain the total occupancy time $O{T}_{total}$.

Step 5: Calculate available time

Due to the vertical maintenance window, the available time of the train timetable needs to subtract the time occupied by maintenance, $T_{maintenance}$, and the invalid time caused by the triangle area, $T_{triangle}$, which can be calculated by Equation (12).

$$T_{available}=1440-T_{maintenance}-T_{triangle}=1440-T_{maintenance}-60\cdot \theta /v$$

(12)

Step 6: Assess carrying capacity

Divide the total occupancy time of all trains on the train timetable by the available time to obtain the capacity utilization rate $\mu$, which indicates the degree of capacity to be used. In other words, for the train timetable with $N_{total}$ trains, $\mu$ of the available time is used. Therefore, we can deduce that if all available time is used, the maximum number of trains in the train timetable is the carrying capacity $N_{capacity}$.

$$O{T}_{total}/T_{available}=\mu =N_{total}/N_{capacity}$$

(13)

5. Case Study

Taking the Beijing–Shanghai high-speed railway as the background to carry out a case study, the busiest section, Xuzhou–Bengbu, was taken as the research section to calculate the capacity utilization and carrying capacity. Then, compared with the compression method of UIC406, the algorithm characteristics were analyzed. Finally, the carrying capacity of each section of the whole line of the Beijing–Shanghai high-speed railway was calculated to verify the universality of the algorithm.

5.1. Data of Beijing–Shanghai High-Speed Railway

There are 24 stations along the whole line of the Beijing–Shanghai high-speed railway. According to the arrival and departure of trains, the whole line is divided into six sections as shown in

Table 2. Sections of the Beijing–Shanghai high-speed railway.

Sections	Departure Station	Arrival Station	Number of Stations	Length (km)	TTs
Beijing–Tianjin	Beijing South	Tianjin South	3	131	0:26:40
Tianjin–Jinan	Tianjin South	Jinan West	4	288	0:51:10
Jinan–Xuzhou	Jinan West	Xuzhou East	6	269	0:50:45
Xuzhou–Bengbu	Xuzhou East	Bengbu South	3	156	0:27:20
Bengbu–Nanjing	Bengbu South	Nanjing South	4	174	0:34:00
Nanjing–Shanghai	Nanjing South	Shanghai Hongqiao	8	284	0:55:05

:

At present, the maintenance window adopted by the Beijing–Shanghai high-speed railway is set as a rectangular window, and the maintenance time range $T_{maintenance}$ is from 0:00 to 6:00, a total of 360 minutes.

5.2. Capacity Assessment of Xuzhou–Bengbu Section

The Xuzhou–Bengbu Section is generally considered to be the busiest and most tightly utilized section of the Beijing-Shanghai high-speed railway, including three stations of Xuzhou East, Suzhou East, and Bengbu South.

According to the statistics of the existing timetable, the number of trains in the Xuzhou–Bengbu section is 152. As shown in Figure 10, these trains are divided into four continuous train group groups after sorting and grouping steps.

The occupancy time of each continuous train group is calculated in turn and summed to obtain a total occupancy time $O{T}_{total}$ of 911.4 min. The total available time in this section is: $T_{available}=1052.67(\min )$. The capacity utilization rate of timetable is: $\mu =911.4/1052.67=86.57\%$. The maximum number of trains in the timetable is: $N_{capacity}=152/86.57\%=175$.

Based on the existing train timetable, the algorithm characteristics are compared with the compression method by randomly generating train timetables with different train numbers for thousands of times. The capacity, utilization and computing time under different train number of both methods are recorded. A part of the results is shown in

Table 3. Calculation results under different train numbers using the two methods.

Train Number	Combined Method			Compressed Method
	Capacity (Train)	Utilization (%)	Time (Millisecond)	Capacity (Train)	Utilization (%)	Time (Millisecond)
12	68.04201	17.63616	1	139.503	8.601963	8
17	75.61408	22.48258	1	142.0265	11.9696	13
22	75.71078	29.05795	1	146.5738	15.0095	20
27	76.13036	35.46548	1	138.6552	19.47277	29
32	83.87782	38.15073	1	142.6136	22.43825	43
37	89.77105	41.21596	1	141.5286	26.14313	50
42	92.55827	45.37682	1	145.3067	28.90437	69
47	93.7921	50.11083	1	142.048	33.0874	83
52	98.09211	53.0114	1	145.08	35.84231	111
57	102.5325	55.59215	1	147.0517	38.76187	132
62	105.4452	58.79829	2	154.1944	40.20899	154
67	105.8671	63.28689	2	154.1891	43.45313	188
72	111.7794	64.4126	5	157.7193	45.65073	232
77	115.2828	66.79227	2	158.1053	48.70171	273
82	116.1107	70.62223	2	161.1425	50.88664	310
87	119.7986	72.62191	2	165.4698	52.57758	358
92	124.4505	73.92495	3	169.0342	54.42685	402
97	122.9509	78.89329	3	175.3691	55.31191	520
102	123.3122	82.71691	3	174.5319	58.44205	552
107	119.1761	89.78309	4	173.2807	61.74953	625
112	124.1583	90.20741	6	176.3674	63.5038	679
117	133.7191	87.49683	4	179.0275	65.35307	822
122	138.571	88.04148	5	175.3167	69.58835	877
127	142.9265	88.85687	5	176.444	71.97752	982
132	154.1513	85.63015	5	177.4271	74.39677	1133
137	162.3803	84.36985	5	180.5965	75.85972	1216
142	166.4326	85.31982	5	186.487	76.14471	1312
147	170.0711	86.43445	5	190.44	77.18968	1441
152	175.5503	86.58486	9	193.1537	78.69379	2416

.

The results show that, for both of the two methods, as the number of trains in the timetable increased, the section capacity also increased. This phenomenon occurs because when the number of trains is small, the trains are laid sparsely, and the average occupancy time is longer. When the number of trains in the timetable increases, to make full use of the time, the new trains will overlap with the original trains so that the average occupancy time decreases, and the calculated carrying capacity becomes larger. In addition, we can see that although the calculation time of the two methods is both increasing, the combined method is obviously much faster than the compression method because the relationship between trains is simplified.

Figure 11 shows the calculation results for the capacity of the two methods of the combined method and compression method when the number of trains in the timetable is different. The calculation result of the compression method is higher than the calculation result of the combined method in all cases because the compression method does not consider the buffer time between trains, which will cause the calculation result to be relatively large. In addition, when the number of trains in the timetable is small, the calculation results differ greatly because when the number of trains is small, the trains are often spread evenly, and the compression process for the compression method changes the trains into centralized, which changes the characteristics of the train timetable. The calculation process of the combined occupancy method will not change the structure of the timetable and can better reflect the characteristics of the train timetable.

5.3. Capacity Assessment of the Beijing–Shanghai High-Speed Railway

Apply the combined occupancy method to calculate the passing capacity of the entire Beijing-Shanghai railway and obtain the passing capacity, capacity utilization rate, operation chart occupancy time and traffic density of each section. The results are shown in Figure 12:

According to Figure 12, under the existing conditions, among the sections of the Beijing–Shanghai high-speed railway, the Bengbu–Xuzhou section (red bar) has the largest capacity of 179 trains. The capacity utilization rate of the whole line of the Beijing Shanghai high-speed railway is more than 80%, except for the Beijing-Tianjin section.

6. Conclusions

This paper took train occupancy time on the timetable as the research focus and proposes a combined occupancy method, which can assessment the carrying capacity and capacity utilization of each section based on the existing train timetable.

The main work of this paper is as follows:

• A conceptual system centered on occupancy time was formed. The main concepts introduced or proposed included a standard train, virtual (additional) train, combined train, repeated occupancy relationship and continuous train group. This system can provide support for subsequent research on occupancy time;
• The meaning of the occupancy time in the train timetable was demonstrated by the method of graphical analysis, and the calculation method of the occupancy time of one single train and two trains was discussed;
• A combined method for calculating the occupancy time of two trains was proposed and the three properties of a continuous train group are proven. On this basis, a combined method for calculating the occupancy time of multiple trains was proposed, which can greatly reduce the computational complexity;
• Based on the concepts and results of the above research, the combined occupancy method of assessment the section carrying capacity was formed. Through thousands of calculations, compared with compressed method, the calculation time is greatly reduced, and the capacity is more reasonable because the structure of the existing train timetable is maintained. The results show that the combined occupancy method was more accurate and convenient to implement. The capacity of all sections of the Beijing–Shanghai high-speed railway is calculated to verify the generality of the method.

The method proposed in this paper can reflect the characteristics of train timetables and capacity utilization of sections without changing the structure of train timetables. The calculation results can provide a reference for timetabling and provide support for more refined capacity management.