Evaluating Line Capacity with an Analytical UIC Code 406 Compression Method and Blocking Time Stairway

energies

Article Evaluating Line Capacity with an Analytical UIC Code 406 Compression Method and Blocking Time Stairway

Ruxin Wang, Lei Nie and Yuyan Tan *

School of Traﬃc and Transportation, Beijing Jiaotong University, Beijing 100044, China; [email protected] (R.W.); [email protected] (L.N.) * Correspondence: [email protected]

Received: 17 January 2020; Accepted: 6 April 2020; Published: 10 April 2020

Abstract: Railways around the world are experiencing growth in traffic flow, but the problem concerning how to optimize the utilization of capacity is still demands significant research. To accommodate the increasing traffic demand, the high-speed railway operator in China is interested in understanding the potential benefit of adopting reasonable headway to balance the safety and efficiency of train operations. In this study, a compress timetable scheduling model based on the UIC Code 406 method is presented to evaluate the line capacity. In this model, train headway is not pre-fixed as in the existing research, but considers the actual operating conditions and is calculated using actual running data. The results of the case study show that refined headway calculations generally have positive capacity effects.

Keywords: high-speed railway capacity; train headway; blocking time theory; UIC Code 406

1. Introduction The China high-speed railway has made remarkable achievements after operations. However, with the rapid development of economy and society, the China high-speed railway is facing a big problem of capacity shortage. The railway operator in China, the National Railway Administration of People’s Republic of China (CR), has found good ways to increase its system capacity. Since the first high-speed railway in China was open to traffic in 2008, the CR upgraded its infrastructure, such as building new rail lines and stations and purchasing rolling stocks, to provide high-quantity and high-quality services. In recent years, improving the signaling systems has also been regarded as a potential solution to increase the railway system capacity [1]. However, the above projects need a great amount of capital investment. Therefore, exploring other different kinds of methods to increase the China high-speed railway capacity seems especially important. Train timetables play an important role in daily actual railway transportation operation. There are many elements in making railway timetables, such as the train journey time, train dwell time and the train headway, which can greatly affect the utilization of the railway system capacity. Based on timetables, the UIC Code 406 method compresses the gap between two adjacent trains under the condition of meeting the standard interval time required for safe railway operation. In this manner, we can get a closely scheduled timetable and the capacity consumption, which we can use to represent the line capacity utilization. In this paper, we want to use the UIC Code 406 method to calculate the consumption time, in order to measure line capacity. The structure of the manuscript is as follows: the next section of this paper describes the literature review. Then, we present a linear programming model for railway capacity evaluation and some formulas of blocking time in Section3 and then a solution based on Python is

Energies 2020, 13, 1853; doi:10.3390/en13071853 www.mdpi.com/journal/energies Energies 2020, 13, 1853 2 of 16 proposed in Section4. In Section5, we analyze the Beijing–Tianjin Intercity Railway as a research case. In Section6, the outcomes and the signiﬁcance of the paper are summarized.

2. Literature Review In general, a railway line capacity is defined as the maximum number of trains that would be able to operate on a given railway infrastructure and operation conditions during a specific time interval. Moreover, Heydar [2] and Petering [3] proposed a different definition for railway capacity in a cyclic train timetable, i.e., the capacity is the minimum cycle length that can feasibly accommodate a given number of trains over a given section of track in each cycle. Inspired by them, we define capacity in this paper as the minimum occupation time of a railway line under the conditions of a given infrastructure and the number of trains within one day, i.e., a makespan of the timetable. Numerous approaches and tools have been developed to evaluate railway capacity; Abril et al.[4] categorized the capacity computation methods in three types: analytical methods, optimization methods and simulation methods. Among these three types of methods, although analytical methods and simulation methods present a good estimate, they cannot give an exact amount. Optimization methods are based on obtaining optimally saturated timetables; therefore, one can acquire a more accurate amount of capacity. The International Union of Railways (UIC) put forward a method of capacity estimation, namely the UIC Code 406, which is based on a timetable compression method and calculates the consumption time and capacity consumption [5]. Table1 presents some studies on the UIC Code 406 method and other evaluation methods based on timetables and compares them with our research within the key dimensions of the model structure: objective, headway and problem size. From Table1, we can find that there have been many researchers using capacity consumption, cycle time or train travel time as a measure to evaluate railway line capacity. Therefore, in this paper, we apply the UIC Code 406 compression method to calculate the consumption time, which is also the minimum makespan of a timetable. In addition, some researchers have studied the calculation method of station capacity, such as Corriere [14], Armstrong [15] and Chen [16]. However, the evaluation approach of station capacity is very different from the estimation of line capacity. Lindner [17] analyzed the applicability of the UIC Code 406 method for evaluating the line and station capacity. He showed an example where the node capacity determines the maximum capacity of the whole infrastructure. Using this example, he explained that the UIC Code 406 method cannot be applied for node capacity research. In our study, we regard the station as two block sections and the train operations in the station are analyzed in detail, which can be found in Section3. In this manner, the station capacity is transformed into line capacity which we can use our method to evaluate. Moreover, we can find in Table1 that many researchers set “headway” to a certain value. Headway is the time interval between two following trains and the minimum line headway is the minimum time interval in which two successive trains of either direction may enter a section of line without any overlapping of the blocking time stairways [18]. Additionally, the headway is not a certain or fixed value, but related to the actual condition of the railway line, train speed, signaling system and the operation schedule [19] and it is one of the most important elements of the timetable because it can ensure running efficiency and safety. If the headway is too small, knock-on delays would occur due to occasional disruptions on the lines. However, if the headway is too long, it will cause a problem of capacity shortage. Therefore, finding a good method to calculate reasonable train headway in a more refined way is a very important issue related to railway capacity. In China, the calculation method of train headway I is taking the maximum of four types of headways based on the train control system, which are section headway I , departure headway I , arrival headway n o s d Ia, passing headway Ip, that is to say, I = max Is, Id, Ia, Ip . In addition, there are four formulas of four types of headways in China, which give full consideration to the issue of the safety. Figure1 is a diagram of the train headways in China and Table2 gives definitions of the parameters in Figure1. Energies 2020, 13, 1853 3 of 16

Table 1. Studies about the railway capacity evaluation and timetabling.

Publication Model Structure Objective Headway Problem Size Minimum required UIC Code 406 6 stations, Minimize headway applied A. Jamili [6] compression 9 trains, consumption time by the signaling method 7 sections system The minimum UIC Code 406 headway time 6 stations, Rob M.P. Goverde Minimize cycle compression depends on the 10 trains in one [7] time method critical block cycle section UIC Code 406 The minimum Higher occupancy Friederike Chu [8] compression headway time is 20 trains per hour rate in one cycle method 2.5 minutes The minimum Mixed-integer Minimize the cycle headway time is a linear lengthand the total 10–70 stations, Mojtaba Heydar [2] certain value programming dwell time of trains 3–64 trains varying from 1 to 5 model at all stations minutes Mixed-integer Minimize the cycle Headways are 5–35 intermediate Matthew E. H. linear length and the total different certain stations, Petering [3] programming journey time of all values in different 3–11 train types model trains examples Integer 7 types of 23 stations, Minimize the cycle Xin Zhang [9] programming headways with 18 trains in one time model certain values cycle Maximize the number of Mixed-integer Consist of 4 types 5 stations, train-pairs and Feng Li et al. [10] programming of headways with 4 train-pairs, Minimize the total model certain values 4 segments travel times of all trains The adopted headway depending upon A methodology for the adopted A. Dicembre et al. Maximize the capacity signaling system, 10 stations [11] number of trains quantification such as minimum headway of 3 min during rush hours in Milan Headway is a certain value, such A comprehensive Maximize capacity as 2.5 min on Yung-Cheng Lai et evaluation utilization and 07:00–09:00 24 stations al. [12] framework of minimize expected weekday and 5 min capacity recovery time on 06:00–23:00 weekend Minimize travel Mixed-integer time, The minimum 14 stations, linear Fei Yan [13] empty-seat-hour headway time is 3 25 trains in one programming and the number of minutes cycle model lines Based on a large number of real train operations Linear Minimize 5 stations, Ruxin Wang et al. data, the minimum programming makespan of 131 trains, (this paper) headway is model timetable 56 sections calculated by using blocking time stairway Energies 2020, 13, x FOR PEER REVIEW 4 of 17

In China, the calculation method of train headway I is taking the maximum of four types of headways based on the train control system, which are section headway Is , departure headway Id , arrival headway Ia , passing headway I p , that is to say, ImaxIIII= {}sdap,,, . In addition, there are four formulas of four types of headways in China, which give full consideration to the issue of the safety. Figure 1 is a diagram of the train headways in China and Table 2 gives definitions of the parameters in Figure 1. Energies 2020, 13, 1853 4 of 16

Station A G103 G101

Is Station B Lad Lbr Lpr Lbl Ltr

G103

Id G101 Station A Lbl Lta Ltr

Lwa Ld

G101 G103

Station A Lta Lbr Lpr Lth Ltr Ia La

G103 G101

Station A Lta Lbr Lpr Lbl Lbl Ltr Ip Lp

Station A

Sections

Station B I I = max{Is,Id,Ia,Ip} Figure 1. Diagram of the train headways in China. Figure 1. Diagram of the train headways in China. Table 2. Deﬁnitions of the parameters in Figure1. Table 2. Definitions of the parameters in Figure 1. Notation Description Notation Description Lad Distance traveled by a train during additional time of interval tracking operation Lbr Distance traveled by a trainBraking during distance additional time of interval Lad Lpr trackingSafety protection operation distance Lbl Length of the block section Lbr Braking distance Ltr Length of the train Lpr Ls Train intervalSafety trackingprotection distance distance between sections Lwa Lbl DistanceL betweenength of train the stopblock sign section and departure signal Lta Distance traveled by a train during station operation time Ltr Length of the train Ld Train interval tracking distance when departing from a station Ls Train interval tracking distance between sections Lth Length of throat area of station La Lwa DistanceTrain between interval tracking train stop distance sign whenand departure arriving in signal a station Lp Train interval tracking distance when passing a station Lta Distance traveled by a train during station operation time

Ld Train interval tracking distance when departing from a station Based on the Chinese Train Control System (CTCS-2/CTCS-3) and the technical characteristics of Lth Length of throat area of station Centralized Traﬃc Control System (CTC) in high-speed railway, Tian [20] proposed the calculation La Train interval tracking distance when arriving in a station method of train headway with the automatic block sections of a high-speed railway, discussed the Lp Train interval tracking distance when passing a station choices of the values of every parameter in formulas and then calculated the realizable train headway on automatic block sections based on the characteristics and operation status of the China high-speed railway. In his research, the train headway of the China high-speed railway was mainly limited by Id, which can basically be 4.5–5 minutes. Wang [21] optimized the arrival headway, calculated the arrival headway and the section headway by using CRH EMU traction calculation system (Version 1.0, National Railway Train Diagram Research and Training Center, Chengdu, China) and then used cellular automata to simulate it. Wu Gao and Mu [22] analyzed some types of train headways and Energies 2020, 13, x FOR PEER REVIEW 5 of 17

Based on the Chinese Train Control System (CTCS-2/CTCS-3) and the technical characteristics of Centralized Traffic Control System (CTC) in high-speed railway, Tian [20] proposed the calculation method of train headway with the automatic block sections of a high-speed railway, discussed the choices of the values of every parameter in formulas and then calculated the realizable train headway on automatic block sections based on the characteristics and operation status of the China high-speed railway. In his research, the train headway of the China high-speed railway was mainly limited by

Id , which can basically be 4.5–5 minutes. Wang [21] optimized the arrival headway, calculated the arrival headway and the section headway by using CRH EMU traction calculation system (Version 1.0,Energies National2020 ,Railway13, 1853 Train Diagram Research and Training Center, Chengdu, China) and then used5 of 16 cellular automata to simulate it. Wu Gao and Mu [22] analyzed some types of train headways and optimized the high-speed railway interval tracking and arrival tracking models under the signaling systemoptimized by the theresearch high-speed of the railwaytrain tracking interval model. tracking Chen and [23 arrival] studied tracking the calculation models under method the of signaling train headwayssystem under by the the research CTCS- of3 Train the train Operation tracking Control model. System. Chen By [23 presetting] studied the the condition calculation of the method train of trackingtrain headwaysinterval, he under analyzed the CTCS-3 the influence Train Operationof the rear Controltrain on System. the front By train presetting in different the condition tracking of circumstances.the train tracking interval, he analyzed the inﬂuence of the rear train on the front train in diﬀerent trackingThe scheduled circumstances. headway between two trains must consist of the minimum line headway plus the requiredThe scheduled buffer time headway to compensate between small two trainsdelays. must Many consist scholars of the in minimumother countries line headway considered plus to the compressrequired train buffer headways time to or compensate buffer times small in order delays. to improve Many scholars high-speed in other railway countries capacity. considered The computationto compress of minimum train headways headway or on buffer a line timesis based in on order the blocking to improve time high-speed stairway [16 railway]. This mean capacity.s thatThe not computation only one block of minimum section but headway the whole on a blocking line is based time on stairways the blocking of a time line stairwayshould be [16 considered]. This means to calculatethat not only the line one headway. block section However, but the the whole blocking blocking times time depend stairways not ofonly a line on shouldthe signal be consideredspacing andto train calculate length, the but line also headway. from the However,actual train the speed, blocking deceleration times depend rate and not the only conditions on the signalof the block spacing sections,and train which length, we present but also in from the thenext actual section. train For speed, a given deceleration timetable, rate we andcan push the conditions the blocking of the time block stairwayssections, together which weas close presently as in possible the next without section. any For buffer a given times timetable, left, which we can is shown push the in Figure blocking 2. time stairways together as closely as possible without any buffer times left, which is shown in Figure2.

Distance

Time

th,12

th,23

Station A

Blocking Sections

Headway Station B Blocking Time Figure 2. Diagram of the train headway proposed by Pachl. Figure 2. Diagram of the train headway proposed by Pachl.

Sangphong et al. [24] presented a primary model to maximize rail line capacity by minimizing the train headway and the influence of the length of the block sections and the train headway on capacity is analyzed. Medeossi et al. [25] presented a method for introducing stochastic blocking times to support timetable planning. Liu and Han [19] analyzed the influencing factors of the train headway, established an integer linear programming model to solve the problem of a high-speed railway train schedule under large capacity and considered the utilization of capacity under different train headways. Lai, Liu and Lin [26] developed headway-based models and computational processes for the standard rail capacity unit, which would be instrumental in the assessment of the impact of heterogeneous trains. Xu [27] proposed an optimization method to minimize the headway of the mainline in the moving Energies 2020, 13, 1853 6 of 16 block and the case study showed that this method could effectively reduce the train headway and there would be a negligible increase in train running time. In addition, there were some Chinese scholars researching this aspect. Based on the blocking time under the moving block system, Liu and Miao [28] designed a new headway calculation method and made a quantitative analysis of different influencing factors on the law of time headways. Lu [29] calculated the minimum train headway with reference to the UIC Code 406 capacity analysis method and analyzed the influence of the maximum running speed and approach speed of trains on the minimum train headway. From the different calculation methods of train headway, it can be found that the calculation of train headways in China is mostly based on the characteristics of the train operation control system, which can only take the maximum values or the published empirical values and these values are mostly based on the premise of the full consideration of safety and therefore, there will be more safety separations and buffer times in these headways, which cannot truly reflect the train headways under different infrastructure and transportation conditions, but instead would reduce the capacity of the whole lines to some extent. However, the calculation method of train headway proposed by Pachl is mostly based on the blocking time theory, which is based on the train timetable and can greatly reduce buffer times between two train lines, in order to get the maximum utilization of railway capacity. This method is also a vacancy in the field of train headways and capacity calculation in China. The main innovation of this paper is that a large number of actual train operation control data was analyzed to calculate the headway. These data came from the Beijing–Tianjin Intercity Railway, containing the real operation of different types of trains on this line. Giorgio Medeossi et al. [25] collected the GPS data of trains and proposed a method to introduce stochastic blocking times to improve timetable planning. Inspired by them, we processed these data and obtained the minimum headway between two adjacent trains based on the blocking time stairway. The data processing and our innovation is shown in Section4. Therefore, the blocking times and headways we used are based on real data and could reflect all kinds of situations encountered in the process of train operations. This paper can be divided into three parts. Firstly, we combined the calculation method of the minimum headway proposed by Pachl and established a compress timetable scheduling model based on the UIC Code 406 method. The input of this model is the actual timetable and blocking time parameters. Secondly, we presented the solution process. Finally, capacity was evaluated based on this method and real data. In this part, we also focused on the impact of the distance interval between trains on train speed and we reached useful conclusions.

3. Mathematical Formulas

3.1. Model The notations involved in our model are listed in Tables3 and4. As we mentioned in Section2, railway line capacity can be deﬁned as the minimum occupation time of a railway line under the condition of a given infrastructure and the number of trains within one day, i.e., a makespan of the timetable. Therefore, we proposed a model with the objective function of minimizing the total time of trains occupying high-speed railway sections under the given number of trains and train sequence. This is formally stated in Equation (1):

min Z = max xd , i M min xa , j M (1) { in ∈ } − { j1 ∈ } Energies 2020, 13, 1853 7 of 16

Table 3. Deﬁnition of the sets, indices and the parameters.

Notation Description m Number of trains n Number of block sections M Set of trains, M Ms Set of stopped trains, Ms M ∈ N Set of block sections, N Ns Set of block sections of stations, Ns N ∈ i, j Train index, i, j M ∈ k Block section index, k N ∈ tik Journey time for train i in block section k ts ik Dwell time for train i in block section k The time when train i passes the blocking signal of block section k (the arrival time of the t s,ik train at the block section is set to 0) The beginning time when block section k begins to lock for the occupation of train i (the t bb,ik arrival time of the train at the block section is set to 0) The ending time when block section k ends to lock for the occupation of train i (the arrival t be,ik time of the train at the block section is set to 0) Ts The beginning time of maintenance Te The ending time of maintenance

Table 4. List of variables.

Variable Description xa ik The arrival time for train i arriving at block section k xd ik The departure time for train i leaving block section k

The objective of the model is to minimize the occupation time of the timetable, by using the time when the last train leaves the last block section minus the time when the ﬁrst train enters the ﬁrst block section. However, there are some constrains that must be considered, such as the train headway, journey time, commercial stop and maintenance, which ensure the safety of trains and compress the train diagram as much as possible: xd xa = t , i M, k N (2) ik − ik ik ∈ ∈ d a x = x , i M, k 1, 2, ... , n 1 and k < Ns (3) ik i(k+1) ∈ ∈ { − } a a x x t t , i, j M, i > j, k N and k < Ns (4) ik − jk ≥ be,jk − bb,ik ∈ ∈ d a s x x t , i Ms, k Ns (5) ik − i(k+1) ≥ ik ∈ ∈ d a x , x , i M, k 1, 2, ... , n 1 and k Ns (6) ik i(k+1) ∈ ∈ { − } ∈ a x Te, i M, k N (7) ik ≥ ∈ ∈ d x Ts, i M, k N (8) ik ≤ ∈ ∈ We now introduce each of these in turn. Constraints (2) and (3) are journey time constraints and the arrival time or departure time of each train in each block section should satisfy these two constraints. Constraint (4) is a headway constraint in each block section, which refers to that if a train is running on a line, the blocking time block of each block section formed by this train cannot be overlapped with the other blocking time blocks of a certain block section formed by other trains. Constraints (5) and (6) are commercial stop constraints and the dwell time of a stopped train at a station must satisfy this constraint. Constraints (7) and (8) are maintenance constraints, where Ts and Te are the beginning and end of two maintenance periods surrounding one operation period and the arrival and departure times of trains shall be within this operation period. Table5 shows a graphical representation of each constraint. Energies 2020,, 13,, xx FORFOR PEERPEER REVIEWREVIEW 8 of 17

a a xTi k e  , i M k N, (7) xTi k e  , i M k N, (7)

d d xTi k s  , (8) xTi k s  , (8)

We now introduce each of these in turn. Constraints (2) and (3) are journey time constraints and thethe arrival arrival time time or or departure departure time time of of each each train train in in each each block block section section should should satisfy satisfy these these two two constraints. Constraint (4) is a headway constraint in each block section, which refers to that if a train isis runnrunninging onon aa line,line, thethe blockingblocking timetime blockblock ofof eacheach blockblock sectionsection formedformed byby thisthis traintrain cannotcannot bebe overlapped with thethe other blocking time blocks of a certain block section formed by other trains. Constraints (5) and (6) are commercial stop constraints and the dwell time of a stopped traintrain atat aa station must satisfy this constraint. Constraints (7) and (8) are maintenance constraints, where Ts and station must satisfy this constraint. Constraints (7) and (8) are maintenance constraints, where Ts and

Te are the beginning and end of two maintenance periods surrounding one operation period and the Te are the beginning and end of two maintenance periods surrounding one operation period and the Energies 2020, 13, 1853 8 of 16 arrival and departure times of trains shall be within this operation period. Table 5 shows a graphical representation of each constraint. Table 5. Constraints of the capacity evaluation model. Table 5. Constraints of thethe capacitycapacity evaluationevaluation model.. Constraint Graphical Representation Constraint Graphical Representation

Nk Nk +1 Nk Nk +1

Journey time Journey time a Journey time constraint xa constraint xik constraint ik t tik a d ik xa xikd xik +1 xik ik +1

Nk Nk Nk Nk

tbbjk, Train j tbbjk, Train j tbb, ik Train j tbb, ik Train j a a a xaik and xajk xajk xik and x jk x jk Train ii tbe, ik tbe, ik

tbe, jk tgap tbe, jk tgap t 'bb, ik a t 'bb, ik HeadwayHeadway constraint xaik Headway xik Train i constraint Train i t 'be, ik t 'be, ik ' aa ' aa where tbb,,,,, ik= t bb ik+(a x ik − xa jk) = t bb ik +( t be jk − t bb ik) + t gap wheret t=bb,,,,, ikt= t bb ik++x x ik −x x jk= = t bb ik ++ t bet jk − t bbt ik + t+ gapt where bb0 ,ik bb,ik (ik jk ) bb,ik ( be,jk bb),ik gap ' aa− − ' aa ttxxttttbe,,,,, ikbe= ikikjkbea+−=+−+ ikbe( jkbba ikgap ) ( ) t = tttxxttttbe,,,,, ikbe=+ ikikjkbex +−=+−+ ikbe( x jkbb ikgap= )t + t ( t )+ tgap be0 ,ik be,ik ik − jk be,ik be,jk − bb,ik t istgap the is time the intervaltime interval between between the two the blocking two blocking time blocks time after blocks the delay after of the gap tgap is the time interval between the two blocking time blocks after the EnergiesEnergies 20 202020, ,1 133, ,x x FOR FOR PEER PEER REVIEW REVIEWthe latter train line and the value of tgap should be greater than or equal to 0 99 ofof 1717 delay of the latter train line and the value of should be greater than or equal to 0

Commercial Nk N Commercial Nk N k +1 k +1 stop constraint

a a xik +1 xik +1 Commercial stop s s constraint tik tik

d d xik xik

Maintenance Maintenance MaintenanceMaintenance StationStation A A BlockingBlocking SectionsSections

MaintenanceMaintenance Maintenance constrain constrainconstrain

StationStation B B T T T e T s e s

3.2.3.2. BlockingBlocking TTimeime PParametersarameters InIn orderorder toto obtainobtain thethe minimumminimum highhigh--speedspeed railwayrailway capacity,capacity, wewe shouldshould determinedetermine thethe traintrain headways.headways. As As mentioned mentioned before, before, the the minimum minimum headway headway on on a a line line with with a a fixed fixed block block system system depends depends onon thethe blockingblocking time,time, soso wewe shouldshould analyzeanalyze thethe compositioncomposition ofof thethe blockingblocking timetime andand calculatecalculate threethree parameters of the blocking time, ts i, k , tb b i, k and tb e i, k , which are: the time when the train i passes parameters of the blocking time, ts i, k , tb b i, k and tb e i, k , which are: the time when the train i passes thethe blockingblocking signalsignal ofof thethe blockblock sectionsection kk,, thethe beginningbeginning timetime whenwhen thethe blockblock sectionsection beginsbegins toto lolockck andand thethe endingending timetime whenwhen thethe blockblock section section endsends toto lock,lock, respectively.respectively. TTherehere are are many many running running states states for for a a train train running running on on a a high high--speedspeed railway railway line line,, such such as as running, running, arrivarrivinging andand departdepartinging andand thethe compositioncomposition ofof thethe blockblock timetime isis differentdifferent underunder differentdifferent conditions.conditions. WeWe analyzeanalyzedd thethe blockingblocking timetime andand ggaaveve somesome formulas formulas toto calculatecalculate thethe blockingblocking timetime variables.variables. PachlPachl proposedproposed thatthat thethe blockingblocking timetime ofof aa blockblock sectionsection isis usuallyusually muchmuch longerlonger thanthan thethe timetime forfor whichwhich thethe train train occupiesoccupies thatthat blockblock sectiosectionn and and thereforetherefore,, for for aa train train withoutwithout a a scheduled scheduled stop,stop, hehe divideddivided thethe blockingblocking timetime of of aa blockblock sectionsection into into six six parts, parts, whichwhich areare:: thethe timetime for for clearingclearing thethe signal, signal, thethe signal signal watching watching time, time, approach approach time, time, the the time time between between bl blockingocking signals, signals, clearing clearing time time and and release release timetime [1[188],], asas shownshown inin Figure Figure 33..

Energies 2020, 13, 1853 9 of 16

3.2. Blocking Time Parameters In order to obtain the minimum high-speed railway capacity, we should determine the train headways. As mentioned before, the minimum headway on a line with a fixed block system depends on the blocking time, so we should analyze the composition of the blocking time and calculate three parameters of the blocking time, ts,ik, tbb,ik and tbe,ik, which are: the time when the train i passes the blocking signal of the block section k, the beginning time when the block section begins to lock and the ending time when the block section ends to lock, respectively. There are many running states for a train running on a high-speed railway line, such as running, arriving and departing and the composition of the block time is different under different conditions. We analyzed the blocking time and gave some formulas to calculate the blocking time variables. Pachl proposed that the blocking time of a block section is usually much longer than the time for which the train occupies that block section and therefore, for a train without a scheduled stop, he divided the blocking time of a block section into six parts, which are: the time for clearing the signal, the signal watching time, approach time, the time between blocking signals, clearing time and release time [18], as shown in Figure3. Energies 2020, 13, x FOR PEER REVIEW 10 of 17

The Minimum Distance Between Two Trains

Clearing Point

A Block Section Length of Train tbb, ik Tt Time for clearing the signal

Ts Signal watching time

Ta Approach time Blocking tsik, Time

T Time between o blocking signals

Tc Clearing time

Tr Release time tbe, ik

FigureFigure 3. 3. DiagramDiagram of of the the composition composition of of the the blocking blocking time time for for a a train train without without a a scheduled scheduled stop stop..

WhenWhen a a train train has has a a scheduled scheduled stop stop at at a a station, station, the the bloc blockingking time time of of a a block block section section at at the the station station would be changed. We We use usedd the the clearing clearing point point to to divide divide the the block block section section at at the the station station into into two two parts parts:: oneone is the throat points and the other is the arrival–departurearrival–departure tracks. After After a a train train with with a a stop stop at at a a statistationon pass passeses through thethe ﬁrstfirst partpart ofof thethe block block section, section, the the blocking blocking time time of of this this part part goes goes directly directly to tothe the clearing clearing time. time. Figure Figure4 shows 4 shows this this situation. situation. After a commercial stop, a train should leave that station. At this time, the approach time does not apply. Moreover, we used a clearing point to divide the block section behind the station into two parts. Figure5 shows this situation.

tbb ik, Tt Time for clearing the signal

Ts Signal watching time

Ta Approach time

ts, ik T Time between o blocking signals

tbeik,

Tc Clearing time

Tr Release time

Figure 4. Diagram of the composition of the blocking time when a train has a scheduled stop.

After a commercial stop, a train should leave that station. At this time, the approach time does not apply. Moreover, we used a clearing point to divide the block section behind the station into two parts. Figure 5 shows this situation.

Energies 2020, 13, x FOR PEER REVIEW 10 of 17

The Minimum Distance Between Two Trains

Clearing Point

A Block Section Length of Train tbb, ik Tt Time for clearing the signal

Ts Signal watching time

Ta Approach time Blocking tsik, Time

T Time between o blocking signals

Tc Clearing time

Tr Release time tbe, ik

Figure 3. Diagram of the composition of the blocking time for a train without a scheduled stop.

When a train has a scheduled stop at a station, the blocking time of a block section at the station would be changed. We used the clearing point to divide the block section at the station into two parts: one is the throat points and the other is the arrival–departure tracks. After a train with a stop at a statiEnergieson 2020pass, 13es, 1853through the first part of the block section, the blocking time of this part goes directly10 of 16 to the clearing time. Figure 4 shows this situation.

tbb ik, Tt Time for clearing the signal

Ts Signal watching time

Ta Approach time

ts, ik T Time between o blocking signals

tbeik,

Tc Clearing time

Tr Release time

Energies 20FigureFigure20, 13, x4.4. FOR DiagramDiagram PEER REVIEW of of the the composition composition of of the the blocking blocking time time when when a a train train has has a a scheduled scheduled stop stop.. 11 of 17

tbbik, Tt Time for clearing the signal

Ts Signal watching time ts, ik

T Time between o blocking signals

Tc Clearing time tbe, ik T Release time r FigureFigure 5. DiagramDiagram of of the the composition composition of of the the blocking blocking time time when when a train leave leavess a station station..

Based on these three situations, the calculation formulas of tbb,ik and tbe,ik are as follows: Based on these three situations, the calculation formulas of tb b i, k and tb e i, k are as follows:

tbb,ik = ts,ik (Tt,ik + Ts,ik + Ta,ik ) (9) ttTTTbb,,,,, iks=−++ ikt iks ika− ik() (9)

tbe,ik = ts,ik + (To,ik + Tc,ik + Tr,ik) (10) ttTTTbe,,,,, iks=+++ iko ikc ikr ik() (10) In principle, the six parts of the blocking time should be determined based on the actual conditions of theIn trains principle, and the the fundamental six parts of infrastructures the blocking and time the should values be of determined each parts are based given on in the this actual paper, conditionsas shown in of Table the trains6. and the fundamental infrastructures and the values of each parts are given in this paper, as shown in Table 6.

Table 6. Values of the six parts of the blocking time.

Each Part of the Blocking Time Values (in Seconds)

Time for clearing the signal Tt ik, 5

Signal watching time Tt ik, 3 Approach time T Lvaa/ a i, k ikik T Lv/ Time between blocking signals o i, k kik Clearing time T Lvc / c i, k kik (1)+ Release time Tr i, k 5

a a In Table 6, Lnk is the distance from the location of the train i to the block section k , vik is the

average speed of the train when the train is located in the approach intervals, Lk is the length of c the block section , vik is the average speed of the train in block section , Lk is the distance from the length of a train after the clearing point to the block section .

4. Solution Based on the JetBrains Pycharm Community Edition 2018.1.2, we used the Python 3.6 to process the data and program the model and the programming language was Python. The solution flow chart is shown in Figure 6.

Energies 2020, 13, 1853 11 of 16

Table 6. Values of the six parts of the blocking time.

Each Part of the Blocking Time Values (in Seconds)

Time for clearing the signal Tt,ik 5 Signal watching time Tt,ik 3 La va Approach time Ta,ik ik/ ik Time between blocking signals To,ik Lk/vik Lc v Clearing time Tc,ik k/ i(k+1) Release time Tr,ik 5

a a In Table6, Lnk is the distance from the location of the train i to the block section k, vik is the average speed of the train i when the train is located in the approach intervals, Lk is the length of the block c section k, vik is the average speed of the train i in block section k, Lk is the distance from the length of a train after the clearing point to the block section k.

Energiesis shown 2020 in, 1 Figure3, x FOR6 PEER. REVIEW 12 of 17

Onboard Train event GPS data collected recorder data tracking

Stochastic blocking times

Balance efficiency and delay

Potential conflict probability on Blocking time parameters corresponding blocking time

Data processing

Constraints

Headway constraint Objective Timetable makespan Compress Journal time constraint timetable scheduling model Evaluate capacity Commercial stop constraint consumption

Maintenance constraint Model

Compressed timetable Capacity consumption Results

Figure 6. The flow ﬂow chart of the solution solution..

5. Case Study We obtained a large number of data of actual train operations on the Beijing–Tianjin Intercity Railway and therefore, the Beijing–Tianjin Intercity Railway was selected as the research case and the railway capacity evaluation model proposed above was used to make a quantitative analysis of the case.

5.1. Background The Beijing–Tianjin Intercity Railway is an intercity railway connecting Beijing and Tianjin, it was the first high-standard and high-speed railway and it has a 350 km/h design speed in mainland China. The Beijing–Tianjin Intercity Railway has been in operation since August 1, 2008 and the total length of whole line is 120 kilometers. There are five stations in total, namely Beijing South Railway Station, Yizhuang Railway Station, Yongle Railway Station, Wuqing Railway Station and Tianjin Railway Station, as shown in Figure 7, among which Yizhuang Railway Station and Yongle Railway Station are not handling passenger transport business temporarily. In addition, there is a Nancang Block Post on the Beijing–Tianjin Intercity Railway line. On January 5, 2019, the national railways adjusted the train timetables and the stop stations of 26 pairs of Beijing–Tianjin intercity trains changed from Tianjin Railway Station to Tianjin West Railway Station.

Energies 2020, 13, 1853 12 of 16

5.1. Background The Beijing–Tianjin Intercity Railway is an intercity railway connecting Beijing and Tianjin, it was the ﬁrst high-standard and high-speed railway and it has a 350 km/h design speed in mainland China. The Beijing–Tianjin Intercity Railway has been in operation since August 1, 2008 and the total length of whole line is 120 kilometers. There are ﬁve stations in total, namely Beijing South Railway Station, Yizhuang Railway Station, Yongle Railway Station, Wuqing Railway Station and Tianjin Railway Station, as shown in Figure7, among which Yizhuang Railway Station and Yongle Railway Station are not handling passenger transport business temporarily. In addition, there is a Nancang Block Post on the Beijing–Tianjin Intercity Railway line. On January 5, 2019, the national railways adjusted the train timetables and the stop stations of 26 pairs of Beijing–Tianjin intercity trains changed from Tianjin

EnergiesRailway 2020 Station, 13, x FOR to TianjinPEER REVIEW West Railway Station. 13 of 17

Yizhuang Beijing South Railway Railway Station Station

Yongle Railway Station

Wuqing Railway Station

Nancang Block Tianjin West Post Railway Station Tianjin Railway Station FigureFigure 7. 7. DiagramDiagram of of the the Beijing Beijing–Tianjin–Tianjin Intercity Intercity Railway Railway line line..

BasedBased on on the blocking time theory, wewe onlyonly needed needed to to calculate calculate the the minimum minimum occupation occupation time time of ofthe the trains trains in in the the critical critical sections sections with with the the largest largest number number of trains.of trains. According According to theto the model model above, above, we wechose chose the downthe down direction direction of the of Beijing–Tianjin the Beijing–Tianjin Intercity Intercity Railway Railway from Beijing from SouthBeijing Railway South StationRailway to StationNancang to Block Nancang Post, Block which Post, contains which 56 block contains sections, 56 block as the sections, critical sectionsas the critical for our sections research. for our research.Using the actual data of the Beijing–Tianjin Intercity Railway to produce the train paths in one day,Using only considering the actual data the downof the directionBeijing–Tianjin and sorting Intercity out theRailway data ofto allproduce the trains the (131train trains pathsin in total) one dayrunning, only on considering the critical the sections, down wedirection obtained and some sorting relevant out the data data of of the all original the trains timetable, (131 trains as shown in total) in runningTable7 and on drewthe critical the original sections, timetable we obtained as shown some in relevant Figure8. data of the original timetable, as shown in Table 7 and drew the original timetable as shown in Figure 8.

Table 7. Statistical table of some of the relevant data of the original timetable.

Data Name Value The earliest departure time 6:02:00 The latest arrival time 0:29:15 Occupation time on sections 1107.25 min The average train headway 8.34 min The average journey time 25.46 min

Figure 8. An original timetable of the Beijing–Tianjin Intercity Railway.

Energies 2020, 13, x FOR PEER REVIEW 13 of 17

Yizhuang Beijing South Railway Railway Station Station

Yongle Railway Station

Wuqing Railway Station

Nancang Block Tianjin West Post Railway Station Tianjin Railway Station Figure 7. Diagram of the Beijing–Tianjin Intercity Railway line.

Based on the blocking time theory, we only needed to calculate the minimum occupation time of the trains in the critical sections with the largest number of trains. According to the model above, we chose the down direction of the Beijing–Tianjin Intercity Railway from Beijing South Railway Station to Nancang Block Post, which contains 56 block sections, as the critical sections for our research. Using the actual data of the Beijing–Tianjin Intercity Railway to produce the train paths in one day, only considering the down direction and sorting out the data of all the trains (131 trains in total) running on the critical sections, we obtained some relevant data of the original timetable, as shown Energiesin Table2020 7 ,and13, 1853 drew the original timetable as shown in Figure 8. 13 of 16

Table 7.7. Statistical table of some of the relevant datadata ofof thethe originaloriginal timetable.timetable.

DataData Name Name Value Value The earliest departure time 6:02:00 The earliest departure time 6:02:00 TheThe latest latest arrival arrival time time 0:29:15 0:29:15 OccupationOccupation time ontime sections on sections 1107.25 1107.25 min min TheThe average average train headwaytrain headway 8.34 8.34 min min TheThe average average journey journey time time 25.46 25.46 min min

Energies 2020, 13, x FOR FigurePEER REVIEW 8. An original timetable of the Beijing Beijing–Tianjin–Tianjin Intercity Railway.Railway. 14 of 17

5.2.5.2. Results Results WeWe perform performeded several several computational computational cases cases using using the the train train operation operationss data. data. WithWith the the allowable allowable time window ofof thethe journeyjourney from from 6:00 6:00 a.m. a.m to. to 0:00 0:00 a.m., a.m a., totala total of of 18 18 hours, hours, we weobtained obtained the the result result which which showed show thated that the the minimum minimum occupation occupation time time of the of trainsthe trains on the on high-speed the high- speedrailway railway line is line 777.33 is 777.33 minutes minutes and the and corresponding the corresponding timetable timetable is drawn is drawn as shown as shown in Figure in Figure9. 9.

FigureFigure 9. 9. TimetableTimetable of of the the Beijing Beijing–Tianjin–Tianjin Intercity Intercity Railway Railway during during th thee minimum minimum occupation occupation time time..

WeWe also also calculate calculatedd the the corresponding corresponding occupation occupation time time of of trains trains based based on on the the train train control control system. system. AccordingAccording to to the the block block system system of of the the high high-speed-speed railway railway and and the the characteristics characteristics of of the the train train operation operation control system in China, we took I s = 2.5 min, I p = 3 min, Id = 3 min and Ia = 5 min to calculate the time. The occupation time we obtained for the journey was 842.75 min. Compared with the occupation time calculated based on the characteristics of the train control system, there are 65.42 minutes less when using blocking time and train headway theories and obviously, the average train headway also decreases, as shown in Table 8. It can be seen that the calculation method of the high-speed railway capacity based on the blocking time and train headway theories is better than the calculation method based on the characteristics of the train control system, in terms of capacity improvement.

Table 8. Comparison of the relevant indicators.

Indicator Result of the Refined Model Result of the Existing Method The earliest departure time 6:00:00 6:00:00 The latest arrival time 18:57:20 20:02:45 Occupation time on sections 777.33 min 842.75 min The average train headway 5.78 min 6.29 min

In order to show to what degree the different factors have an effect on high-speed railway capacity, we considered two more cases with different factors, one with less interval distance and the other with higher speed. In the actual operation of the China high-speed railway at present, the minimum train headway is 4 minutes and the interval distance between two adjacent trains is about seven block sections. In the three-aspect automatic block system, when the signal shows green, it means that there are at least three block sections free in front of it, that is, when there are at least three block sections free in front of the block section where a train is running at normal speed. Then, we simulated the case that the interval distance is three block sections and compared it with seven block sections. The relevant indicators are arranged as shown in Table 9.

Energies 2020, 13, 1853 14 of 16

control system in China, we took Is = 2.5 min, Ip = 3 min, Id = 3 min and Ia = 5 min to calculate the time. The occupation time we obtained for the journey was 842.75 min. Compared with the occupation time calculated based on the characteristics of the train control system, there are 65.42 minutes less when using blocking time and train headway theories and obviously, the average train headway also decreases, as shown in Table8. It can be seen that the calculation method of the high-speed railway capacity based on the blocking time and train headway theories is better than the calculation method based on the characteristics of the train control system, in terms of capacity improvement.

Table 8. Comparison of the relevant indicators.

Indicator Result of the Reﬁned Model Result of the Existing Method The earliest departure time 6:00:00 6:00:00 The latest arrival time 18:57:20 20:02:45 Occupation time on sections 777.33 min 842.75 min The average train headway 5.78 min 6.29 min

In order to show to what degree the different factors have an effect on high-speed railway capacity, we considered two more cases with different factors, one with less interval distance and the other with higher speed. In the actual operation of the China high-speed railway at present, the minimum train headway is 4 minutes and the interval distance between two adjacent trains is about seven block sections. In the three-aspect automatic block system, when the signal shows green, it means that there are at least three block sections free in front of it, that is, when there are at least three block sections free in front of the block section where a train is running at normal speed. Then, we simulated the case that the interval distance is three block sections and compared it with seven block sections. The relevant indicators are arranged as shown in Table9.

Table 9. Comparison of relevant indicators when diﬀerent interval distance.

Indicator Result of Seven Block Sections Result of Three Block Sections The earliest departure time 6:00:00 6:00:00 The latest arrival time 18:57:20 15:12:51 Occupation time on sections 777.33 min 552.85 min The average train headway 5.78 min 4.06 min

From the data above, we can see that with the decrease of the interval distance between two adjacent trains, the occupation time and average train headways also decrease, which shows the positive correlation. Meanwhile, the maximum capacity of the Beijing–Tianjin Intercity Railway can be significantly improved. Speed determines the journey time of a train in line sections and plays a decisive role in capacity. Due to the different trains and line plans, the maximum speed of high-speed railway trains in China is different, such as 350 km/h or 300 km/h. We simulated two cases in which the maximum speed of trains was 350 km/h and 300 km/h. The relevant indicators are arranged as shown in Table 10.

Table 10. Comparison of relevant indicators when diﬀerent train speed.

Indicator Result of 350 km/h Result of 300 km/h The earliest departure time 6:00:00 6:00:00 The latest arrival time 18:57:25 20:54:50 Occupation time on sections 775.41 min 894.83 min The average train headway 5.77 min 6.66 min Energies 2020, 13, 1853 15 of 16

From the data above, it can be seen that with the decrease of the maximum speed of trains, the occupation time and average train headway will increase, which shows a negative correlation. Meanwhile, the maximum capacity of Beijing–Tianjin Intercity Railway will decrease.

6. Conclusions The evaluation method of high-speed railway capacity was discussed in this paper. In our research, a compress timetable scheduling model was established to evaluate line capacity and the blocking time was not pre-ﬁxed but considered the state of actual operations. A case study on the Beijing–Tianjin Intercity Railway line showed that this method can improve the capacity to a certain extent and that if the interval distance between two adjacent trains can be decreased or the speed of trains can be increased, the capacity can be improved eﬀectively. Moreover, we can also get the corresponding highly accurate train timetable with the minimum occupation time.

Author Contributions: Conceptualization and methodology, L.N. and Y.T.; validation and analysis, R.W., L.N. and Y.T.; resources, L.N.; writing—original draft preparation and visualization, R.W.; writing—review and editing, R.W. and Y.T.; project administration and funding acquisition, L.N. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Key Research and Development Plan of China, grant number 2016YFE0201700; the 111 Project of China, grant number No.B18004; National Natural Science Foundation of China, grant number Grant Nos.U1734204,61703030; and China Railway Corporation Project: Research on the construction of intelligent Beijing Zhangjiakou technical regulation system, grant number T19d00210. Acknowledgments: The authors would like to thank referees and the editor for very helpful and constructive comments throughout the reviewing process. This work was supported by the National Key Research and Development Plan of China under grant (2016YFE0201700), the 111 Project of China (No.B18004), National Natural Science Foundation of China (Grant Nos.U1734204,61703030) and China Railway Corporation Project: Research on the construction of intelligent Beijing Zhangjiakou technical regulation system (T19d00210). Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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