# Robustness-Oriented Optimization of Train Platforming Problem in Busy Complex Railway Station

Authors: Qiongfang Zeng; Yinggui Zhang; Huanyin Su; Ruihua Hu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we present a track-circuits-based model for robust train platforming problem (RTPP) at busy complex stations. First, we explicitly explained the operation process of different train types and fixed track utilization rule. Second, we propose a multicriterion scheduling model for RTPP with objectives of minimizing the weighted number of delayed trains, minimizing trains that break rules, and maximizing robustness based on sectional-release interlocking system. Third, we design a hybrid heuristic algorithm to solve the above model based on dispatching rules and present different new solution generation strategies to further optimize rule-based solutions. Performance of the new solution generation strategies and algorithm is evaluated based on a case study for Guangzhou East Station. To validate the effectiveness of robustness indicator, we compared its performance on delay propagation counteraction with a TPP model without robustness objective under the same perturbation scenario.


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## Body

## 1. Introduction

The market share of railway transportation has been increasing in many countries all over the world. Among them, China has the largest high-speed and regular passenger railway network which will be increased to 38000 km by 2025 according to an issued Chinese government plan. The large network scale and large passenger traffic volume require high efficiency of train operations in busy complex stations. Infrastructure utilization within busy complex stations is at a saturation level. It is a challenging and tedious work for station dispatchers to make conflict-free train operation plans due to complex station layout and dense traffic in busy complex stations. Train platforming problem coordinates several different subproblems at strategical, tactical, and operational levels of a railway system as described in the work of Zhang et al. [1]. Capacity management at the strategic level is crucial for the safe reception of a greater number of trains in busy complex stations. At the tactical level of the railway planning process, an assignment plan of routes through station areas is subsequent to train timetabling stage. Train routing problem (TRP) is focused on making a macroscopic feasible routes plan, and this problem was first introduced by Zwaneveld et al. [2] to increase the work efficiency of dispatchers. Train platforming problem (TPP) was introduced when some researchers solved the TRP, while they put more emphasis on the allocation of platforms and tracks. Extensive overviews related to TRP and TPP are given in the works of Sels et al. [3], Cacchiniani et al. [4], and Zhang et al. [5]. In recent years, more and more scholars have paid attention to the robustness of railway planning. We refer the readers to the work of Lusby et al. [6] for excellent surveys on robustness in different railway planning phases.It is widely acknowledged that light perturbations are light disturbances produced by delayed trains in a neighboring dispatching area and disruptions are much heavier disturbances (e.g., block sections breakdowns) of the scheduled times and routes. Train delays in busy complex stations are unavoidable and recurrent due to internal and external stochastic disturbances. In a saturated level of train operations, those initial or primary delays may propagate to other trains and cause knock-on delays spreading over the network. Therefore, routes and track assignments need to incorporate some proactive measures to obtain higher robustness. In this paper, we consider an extension of the original TPP with robustness. Given a microscopic station topology and interlocking information, train dynamic features, alternative route or platform sets, fixed track utilization rule, a macroscopic timetable, and allowable time window for train operation times, robust train platforming problem (RTPP) focuses on obtaining a preventive track assignment plan to reduce the knock-on train delays caused by perturbations in busy complex station area.Publications considering robustness and stability while solving the train platforming problems at busy complex station areas can be divided into the two following streams:(1)
Robustness indicators (e.g., buffer times). In most of extant literature, buffer times are introduced to absorb small recurrent perturbations. Corman et al. [7] proposed an Alternative Graph formulation for rescheduling and rerouting using an incompatibility graph and blocking times. A more efficient aggregated route formulation is furthermore introduced which computes slightly conservative microscopically conflict-free solutions with small extra buffer times. According to Miao et al. [8], stability could be achieved by maximizing the time interval between two adjacent occupations of the same track. In the work of Caprara et al. [9], a recoverable robust method is proposed and applied to the TPP that considers assigning each train in a specific timetable a stopping pattern, which includes platform, an arrival route, and a departure route. Burggraeve et al. [10] propose an iterative approach to construct a passenger robust railway routing plan and timetable from scratch which also takes supplement and buffer time allocation into account on the signaling level. Dewilde et al. [11] point out that increasing a buffer time span between two trains that occupied the same routes is beneficial for the robustness of a route assignment plan. However, they assume a rigid platform or track allocation. In most of extant station-level robustness-related literature, interlocking constraints of track occupations deduce that buffer times between two adjacent track occupations are a posteriori robustness parameters in station scheduling [12, 13]. However, the location and size of two trains’ buffer time may be deduced from interlocking constraints of adjacent track-circuits occupations. Moreover, train weights should also be considered for buffer time distribution to reduce the total knock-on delay propagation in the whole network. In this paper, a robustness indicator will be formulated and bottleneck station resources under perturbations will be identified considering different train types and sectional-release interlocking system. Feasibility of solutions in practice is more guaranteed with explicit modeling of sectional-release interlocking system and train operations process of different train types.(2)
Robustness evaluation or delay propagation based on simulation. Some researchers put their interest in robustness evaluation based on a posteriori simulation of disturbed route plans within the station area. For example, Bešinović et al. [13] pointed out that stability is achieved by minimizing and evenly distributing total infrastructure capacity occupation because the occupation of resources determines how frequent maintenance should be undertaken. They also proposed a delay propagation algorithm in the station area using the max-plus algebra method.Existing papers rarely considered train platforming with operation requirements of different train types (e.g., rolling stock connections between turnaround passenger trains). Fixed track utilization rule and breaking rule were rarely considered in the studies of the previous TPP. Station dispatchers usually formed a fixed track utilization rule that groups the applicable tracks depending on train-, track-, or usage-related multiple factors such as trains length, trainload, train speed, train operations, and trains directions. They may break the fixed rules to increase the track assignment plan’s feasibility under perturbations. In this paper, we incorporated the above measures in our algorithm and developed several new solution generation strategies analogous to the manual methods currently used in practice.The comparison of this paper with other classical papers on the TTP is summarized in Table1.Table 1
Comparison of our research with other literature related to RTPP.
ReferencesTrack circuitsTrack utilization ruleCarriage turnover planBuffer timesDelay propagationZhang et al. [1]√√√××Corman et al. [7]――×√×Miao et al. [8]×―×√×Burggraeve et al. [10]×―×√×Dewilde et al. [12]×××√√Bešinović et al. [13]√―×√√This paper√√√√√The contribution of this paper is described as follows.Most of RTPP with robustness only considered aggregated routes due to the size of problem instances and thus cannot guarantee the real insights into the impacts of perturbations. We propose a microscopic robustness assessment approach based on time spans between train pairs that occupy the same station resource (e.g., track circuits).We formulate the RTPP by considering the fixed track utilization rule and breaking rule which are traffic control measures taken by dispatchers to increase the probability of achieving a feasible track assignment plan under perturbations.We formulated a delay propagation process based on a sectional-release interlocking system and identified bottleneck station resources in station area. By simulating small perturbations with stochastic perturbation distributions, we verified the effectiveness of RTPP on reducing the weighted delay propagation within busy complex stations.The remainder of this paper is organized as follows. In <u>Section2</u>, the definitions, assumptions, and notations are given. Section 3 presents a job shop scheduling model for the RTPP based on route-locking and sectional-release interlocking system. Section 4 designs different optimizing strategies and introduces a hybrid heuristic algorithm to solve the above model. In Section 5, a case study is conducted on Guangzhou South Railway Station in China to illustrate the optimized method and verify its effectiveness in reducing delays. In <u>Section 6</u>, the paper concludes with a summary and discussion of further researches.


## 2. Definitions and Notations

The station topology gives information about mapping relationships between directions, routes, track circuits, tracks, and platforms. The interlocking table shows an electronic interlocking relation between related switches, track circuits, routes, and signals. A macroscopic timetable consists of train direction, train number, and arrival or departure time. In this paper, we identify routes and their related information already existing in the interlocking table, making our solution more practical.Train properties give information about train locomotive properties, train length, maximum allowable speed in the station area, mean braking deceleration, mean acceleration, train marshaling, set-up and release times, and safe distance for driver’s sight and reaction time.A train type is defined according to different train services and technical operational processes. Zeng et al. [16] classified trains into five train types: originating passenger trains (ORPTs), terminating passenger trains (TEPTs), turnaround passenger trains (TUPTs), nonstop through passenger trains (NSTHPTs), and stop through passenger trains (STHPTs).Fixed track utilization rule groups several alternative tracks that support specific train type, directions, and properties into an alternative track set according to station safety regulations and rules. Limitation denotes the total number of alternative tracks that are allowed to be claimed by the train according to the fixed track utilization rule. When the preferred alternative tracks are not feasible, station dispatchers will consider the other tracks. Therefore, we set a special utilization rule called the breaking rule that includes alternative tracks for all passenger train types to increase feasibility under perturbations. For instance, trains in up direction may be allowed to claim track that is usually used to accommodate trainsT1 and T2 in the down direction to deal with perturbations or disruptions as shown in Figure 1.Figure 1
Instance of breaking fixed track utilization rule.An earliness and lateness time window (ELTW) refers to a time range within which station dispatchers can make modifications of train’s arrival and departure times in macroscopic timetables. Usually, dispatchers do not proactively adjust the arrival and departure times of NSTHPTs and STHPTs for passenger service operation stability and flexibly adjust the arrival of TEPTs or departure time of ORPTs. If standard service time for ORPTs ranging from 30 min to 50 min and standard service time for TEPTs ranging from 25 min to 40 min are acceptable, other carriages-related operations will be ensured and penalty cost of time adjustment can be ignored. Table2 presents notations and variables used in this paper.Table 2
Notations for parameters and variables.
SymbolDescriptionxiArrival time of traini in macroscopic timetableyiDeparture time of traini in macroscopic timetableJi∈JSet of passenger trains,i=1,2,…,n. n denotes the total number of trains within given fixed periodMk∈MSet of all tracks indexed byk. Km is the total number of tracksZp∈PSet of all platforms indexed byp. Kp is the total number of platformsRl∈RSet of all routes indexed byl. Kr is the total number of routesSq∈SetSet of all track circuits indexed byq. Kq is the total number of track circuitsωiWeight of trainiJ−MMapping relationships between trains and alternative tracksM−PMapping relationships between tracks and platformsM−RMapping relationships between tracks and alternative inbound (outbound) routesO−WindowOperations time windows for shunting locomotives and station infrastructure maintenanceJ-J′Mapping relationships between passenger train numbers marshaled by the same carriages according to carriage turnover planR−SMapping relationships between routes and track circuitsθiLimitationαpMaximum allowable trains number of platformZpβkMaximum allowable trains number of trackMkJ′Processing sequencing list,J′=J1′,…,Ji′,Jj′…,Jn′. For any different trains Ji′ and Jj′ (i≠j), Jj′ is prior to Ji′Si,kVariable for storing the claim (start) time instant of trackkk∈Qik by train i when mapping relationship k⊕i=1 is ensured.Fi,kVariable for storing the release (finish) time instant of trackk by train i when mapping relationship k⊕i=1 is ensuredSi,qVariable for storing the claim time instant of track circuitq in route l by train i when mapping relationships k⊕i=1,k⊕l≠0,l⊕q=1 are ensured.Fi,qVariable for storing the release time instant of track circuitq in route l by train i when mapping relationships k⊕i=1,k⊕l≠0,l⊕q=1 are ensuredoik0–1 binary variable. If trackk is claimed by train i, then oik=1; otherwise oik=0πi0–1 binary variable. If train breaks the fixed track utilization rule and is assigned with a trackk in alternative track set of breaking rule, then πi=1; otherwise πi=0oilRoute variable. If mapping relationshipsk⊕i=1,k⊕l=1 are ensured, route l is claimed by train i as an inbound route ending at track k, and then oil=1; if mapping relationships k⊕i=1,k⊕l=2 are ensured, route l is claimed by train i as an outbound route starting from track k, and then oil=2; otherwise oil=0oiq0–1 binary track-circuit variable. If mapping relationshipsk⊕i=1,k⊕l≠0,l⊕q=1 are ensured, track circuit q is claimed by train i as part of route l that is connected with track k, and then oiq=1; otherwise oiq=0oip0–1 binary platform variable. If mapping relationshipi⊕p=1 is ensured, platform p is occupied by train i, and then oip=1; otherwise oip=0Di0–1 binary variable of delayed train. If trainJi is assigned with a track in alternative track set according to fixed track utilization rule, then Di=1; otherwise Di=0


## 3. RTPP Model Formulation

Train’s operation in tracks can be viewed as the processing of jobs (trains) on paralleled machines (tracks) being subjected to process constraint and machine constraint. A job (train)i is characterized by its operation processing time pi, and the processing time of train i can be denoted by pi=yi∗−xi∗. Each job has attributes such as ready time ri and due date ci, which are related to the arrival and departure time of train i as described in the work of Zeng et al. [16]. The attached resources such as track circuits, signals, and switches were released or recovered after processing. TPP satisfies the basic assumptions of classic job shop scheduling problem: (1) We assume that each track can accommodate one train at any time. (2) Arrival time and departure time of each train are given in a macroscopic timetable. However, TPP violates part of those assumptions such as single resources, certainty, single objective, and regularity. TPP in busy complex passenger stations can be formulated as a modern job shop scheduling problem with restrained resources and processing features such as weight and limitation.If we set minimizing the total number of weighted delayed trains as the first objective and minimizing the weighted number of trains that break the fixed track utilization rule as the second objective, TPP can be formulated as a job shop scheduling problem with 3-field problem classificationα|β|γ denotation as shown in (1).TPP is as follows:(1)Pm|ri,pi,ci,ωi,ϑi,J−M,J-J′,M−P,M−R,R−S,ELTW|Ff1,f2.Furthermore, we proposed pertinent objective criteria including minimizing weighted delayed trains, minimizing weighted number of trains that break the fixed track utilization rule, and maximization of the robustness. Hence, the RTPP is formulated as (2).RTPP is as follows:(2)Pm|ri,pi,ci,ωi,ϑi,J−M,J-J′,M−P,M−R,R−S,ELTW|Ff1,f2,f3.


### 3.1. Objectives

#### 3.1.1. Weighted Number of Delayed Trains

LetDi denote whether train Ji is not assigned with track and delayed. Di=1 when there is no alternative track k in set Qi and the alternative track set of breaking rules. Let ξ1 denote the penalty cost of unassigned trains. Weighted number of delayed trains can be expressed as follows:(3)f1=∑i=1nξ1ωiDi.


#### 3.1.2. Weighted Number of Trains Breaking Rule

Letπi denote whether train Ji is assigned with track in alternative track set of breaking rules. Let ξ2 denote the penalty cost of trains breaking fixed track utilization rule. The second subobjective is minimizing weighted number of trains breaking rule as shown in the following equation:(4)f2=∑i=1nξ2ωiπi.


#### 3.1.3. Robustness

Buffer times in platforming process can impede the formation of train conflicts and counteract the knock-on delays due to primary delays or perturbations. Dewilde et al. (2014) proposed that the total buffer time between any two trains that go through station areas is a good indicator of robustness in station areas. To formulate the robustness indicator more precisely, we build an evaluation objective function considering the buffer time between any two trains that occupy station resources based on route-locking and sectional-release interlocking system. Buffer times between pairs of consecutive train movements on the same track circuit are computed as shown in equation (5). Buffer times between pairs of consecutive train movements on the same track are computed as shown in equation (6).(5)Bi,q,j,q=Fi,q−Sj,q∀Ji,Ji∈J,i≠j,oiq=ojq=1,(6)Bi,k,j,k=Fi,k−Sj,k∀Ji,Ji∈J,i≠j,oik=ojk=1.If any two trains in the timetable occupy different tracks with partially overlapped inbound or outbound routes, the smallest buffer time of those trains may be the buffer time between track circuits’ occupation in overlapped train routes (e.g., trainsT3 and T4 in Figure 1). If any two trains with same directions and inbound routes occupy the same track, the buffer time of those trains may be the buffer time between track circuits’ occupation due to sectional-release interlocking relationship (e.g., trains T1 and T2 in Figure 1). Unit penalty cost of the buffer times to absorb perturbations can be expressed as shown in equations (7)-(8), similar to the objective function from Dewilde et al. [7].(7)Ci,q,j,q=∞ifBi,q,j,q=0,ξ3ωiωjBi,q,j,qifBi,q,j,q<Bmax,0otherwise,(8)Ci,k,j,k=∞ifBi,k,j,k=0,ξ4ωiωjBi,k,j,kifBi,k,j,k<Bmax,0otherwise.In the two above equations,ξ3 denotes a big parameter and ξ4 denotes adjusted parameters for unit penalty cost. Bmax denotes the average safe buffer time needed to deal with daily perturbations. Delay-damping effect decreases with the increase of buffer magnitude when its buffer time is bigger than Bmax. When Bij is smaller than Bmax, the penalty cost is nonlinear and proportional to the train weights ωi and ωj and inversely proportional to its buffer time because the possibility of conflicts increases with the decrease of buffer times under perturbations. Therefore, robustness indicator for a track assignment plan can be calculated as shown in the following equation:(9)minf3=∑∀Sq∈Set∑∀Ji,Ji∈J,i≠jCi,q,j,q+∑∀Mk∈M∑∀Ji,Ji∈J,i≠jCi,k,j,k.


### 3.2. Constraints

A pair of conflict routes have the same track circuits in their track-circuits sequences or adjoining track circuits that have double crossovers or insulation joints as shown in interlocking table. The route-lock sectional-release interlocking system means the track circuits within or related to a route are claimed all at once and released one by one in sequence. All the track circuits within or related to one route are listed in the track-circuits column of the interlocking table. Only if all the track circuits are vacant during their occupation time is the given route during its occupation time interval safe and feasible. Thus, to check the safety and feasibility of a given route, we should check all the track circuits in the track-circuits column of the interlocking table after knowing their start and end time instants. The occupation time interval of station resources between adjacent trains should be nonoverlapping. This constraint is expressed by the two following equations:(10)Si,q,Fi,q∩Sj,q,Fj,q=∅,i≠j,∀oiq=ojq=1,∀Sq∈Set,(11)Si,k,Fi,k∩Sj,k,Fj,k=∅,i≠j,∀oik=ojk=1,∀Mk∈M.The constraint that each train can claim one platform track and platform at most is expressed in the two following equations:(12)∑k=1Kmoik≤1∀Ji∈J,(13)∑p=1Kpoip≤1∀Ji∈J.The constraint that each track can be claimed by train number that is no more than the maximum allowable train numberβk is identified as follows:(14)∑i=1nk⊕ioikSi,k,Fi,k≤βk,∀Mk∈M.Besides, the maximum allowable train numberαp for track and platform is generally 1 or 2. The constraint that each trackside platform p should alight or board passenger of train number no larger than the maximum allowable number αp to avoid overcrowdedness on platform is expressed as follows:(15)∑i=1n∑k=1Kmk⊕pk⊕ioikzipSi,k,Fi,k≤αp∀Zp∈P.


## 3.1. Objectives

### 3.1.1. Weighted Number of Delayed Trains

LetDi denote whether train Ji is not assigned with track and delayed. Di=1 when there is no alternative track k in set Qi and the alternative track set of breaking rules. Let ξ1 denote the penalty cost of unassigned trains. Weighted number of delayed trains can be expressed as follows:(3)f1=∑i=1nξ1ωiDi.


### 3.1.2. Weighted Number of Trains Breaking Rule

Letπi denote whether train Ji is assigned with track in alternative track set of breaking rules. Let ξ2 denote the penalty cost of trains breaking fixed track utilization rule. The second subobjective is minimizing weighted number of trains breaking rule as shown in the following equation:(4)f2=∑i=1nξ2ωiπi.


### 3.1.3. Robustness

Buffer times in platforming process can impede the formation of train conflicts and counteract the knock-on delays due to primary delays or perturbations. Dewilde et al. (2014) proposed that the total buffer time between any two trains that go through station areas is a good indicator of robustness in station areas. To formulate the robustness indicator more precisely, we build an evaluation objective function considering the buffer time between any two trains that occupy station resources based on route-locking and sectional-release interlocking system. Buffer times between pairs of consecutive train movements on the same track circuit are computed as shown in equation (5). Buffer times between pairs of consecutive train movements on the same track are computed as shown in equation (6).(5)Bi,q,j,q=Fi,q−Sj,q∀Ji,Ji∈J,i≠j,oiq=ojq=1,(6)Bi,k,j,k=Fi,k−Sj,k∀Ji,Ji∈J,i≠j,oik=ojk=1.If any two trains in the timetable occupy different tracks with partially overlapped inbound or outbound routes, the smallest buffer time of those trains may be the buffer time between track circuits’ occupation in overlapped train routes (e.g., trainsT3 and T4 in Figure 1). If any two trains with same directions and inbound routes occupy the same track, the buffer time of those trains may be the buffer time between track circuits’ occupation due to sectional-release interlocking relationship (e.g., trains T1 and T2 in Figure 1). Unit penalty cost of the buffer times to absorb perturbations can be expressed as shown in equations (7)-(8), similar to the objective function from Dewilde et al. [7].(7)Ci,q,j,q=∞ifBi,q,j,q=0,ξ3ωiωjBi,q,j,qifBi,q,j,q<Bmax,0otherwise,(8)Ci,k,j,k=∞ifBi,k,j,k=0,ξ4ωiωjBi,k,j,kifBi,k,j,k<Bmax,0otherwise.In the two above equations,ξ3 denotes a big parameter and ξ4 denotes adjusted parameters for unit penalty cost. Bmax denotes the average safe buffer time needed to deal with daily perturbations. Delay-damping effect decreases with the increase of buffer magnitude when its buffer time is bigger than Bmax. When Bij is smaller than Bmax, the penalty cost is nonlinear and proportional to the train weights ωi and ωj and inversely proportional to its buffer time because the possibility of conflicts increases with the decrease of buffer times under perturbations. Therefore, robustness indicator for a track assignment plan can be calculated as shown in the following equation:(9)minf3=∑∀Sq∈Set∑∀Ji,Ji∈J,i≠jCi,q,j,q+∑∀Mk∈M∑∀Ji,Ji∈J,i≠jCi,k,j,k.


## 3.1.1. Weighted Number of Delayed Trains

LetDi denote whether train Ji is not assigned with track and delayed. Di=1 when there is no alternative track k in set Qi and the alternative track set of breaking rules. Let ξ1 denote the penalty cost of unassigned trains. Weighted number of delayed trains can be expressed as follows:(3)f1=∑i=1nξ1ωiDi.


## 3.1.2. Weighted Number of Trains Breaking Rule

Letπi denote whether train Ji is assigned with track in alternative track set of breaking rules. Let ξ2 denote the penalty cost of trains breaking fixed track utilization rule. The second subobjective is minimizing weighted number of trains breaking rule as shown in the following equation:(4)f2=∑i=1nξ2ωiπi.


## 3.1.3. Robustness

Buffer times in platforming process can impede the formation of train conflicts and counteract the knock-on delays due to primary delays or perturbations. Dewilde et al. (2014) proposed that the total buffer time between any two trains that go through station areas is a good indicator of robustness in station areas. To formulate the robustness indicator more precisely, we build an evaluation objective function considering the buffer time between any two trains that occupy station resources based on route-locking and sectional-release interlocking system. Buffer times between pairs of consecutive train movements on the same track circuit are computed as shown in equation (5). Buffer times between pairs of consecutive train movements on the same track are computed as shown in equation (6).(5)Bi,q,j,q=Fi,q−Sj,q∀Ji,Ji∈J,i≠j,oiq=ojq=1,(6)Bi,k,j,k=Fi,k−Sj,k∀Ji,Ji∈J,i≠j,oik=ojk=1.If any two trains in the timetable occupy different tracks with partially overlapped inbound or outbound routes, the smallest buffer time of those trains may be the buffer time between track circuits’ occupation in overlapped train routes (e.g., trainsT3 and T4 in Figure 1). If any two trains with same directions and inbound routes occupy the same track, the buffer time of those trains may be the buffer time between track circuits’ occupation due to sectional-release interlocking relationship (e.g., trains T1 and T2 in Figure 1). Unit penalty cost of the buffer times to absorb perturbations can be expressed as shown in equations (7)-(8), similar to the objective function from Dewilde et al. [7].(7)Ci,q,j,q=∞ifBi,q,j,q=0,ξ3ωiωjBi,q,j,qifBi,q,j,q<Bmax,0otherwise,(8)Ci,k,j,k=∞ifBi,k,j,k=0,ξ4ωiωjBi,k,j,kifBi,k,j,k<Bmax,0otherwise.In the two above equations,ξ3 denotes a big parameter and ξ4 denotes adjusted parameters for unit penalty cost. Bmax denotes the average safe buffer time needed to deal with daily perturbations. Delay-damping effect decreases with the increase of buffer magnitude when its buffer time is bigger than Bmax. When Bij is smaller than Bmax, the penalty cost is nonlinear and proportional to the train weights ωi and ωj and inversely proportional to its buffer time because the possibility of conflicts increases with the decrease of buffer times under perturbations. Therefore, robustness indicator for a track assignment plan can be calculated as shown in the following equation:(9)minf3=∑∀Sq∈Set∑∀Ji,Ji∈J,i≠jCi,q,j,q+∑∀Mk∈M∑∀Ji,Ji∈J,i≠jCi,k,j,k.


## 3.2. Constraints

A pair of conflict routes have the same track circuits in their track-circuits sequences or adjoining track circuits that have double crossovers or insulation joints as shown in interlocking table. The route-lock sectional-release interlocking system means the track circuits within or related to a route are claimed all at once and released one by one in sequence. All the track circuits within or related to one route are listed in the track-circuits column of the interlocking table. Only if all the track circuits are vacant during their occupation time is the given route during its occupation time interval safe and feasible. Thus, to check the safety and feasibility of a given route, we should check all the track circuits in the track-circuits column of the interlocking table after knowing their start and end time instants. The occupation time interval of station resources between adjacent trains should be nonoverlapping. This constraint is expressed by the two following equations:(10)Si,q,Fi,q∩Sj,q,Fj,q=∅,i≠j,∀oiq=ojq=1,∀Sq∈Set,(11)Si,k,Fi,k∩Sj,k,Fj,k=∅,i≠j,∀oik=ojk=1,∀Mk∈M.The constraint that each train can claim one platform track and platform at most is expressed in the two following equations:(12)∑k=1Kmoik≤1∀Ji∈J,(13)∑p=1Kpoip≤1∀Ji∈J.The constraint that each track can be claimed by train number that is no more than the maximum allowable train numberβk is identified as follows:(14)∑i=1nk⊕ioikSi,k,Fi,k≤βk,∀Mk∈M.Besides, the maximum allowable train numberαp for track and platform is generally 1 or 2. The constraint that each trackside platform p should alight or board passenger of train number no larger than the maximum allowable number αp to avoid overcrowdedness on platform is expressed as follows:(15)∑i=1n∑k=1Kmk⊕pk⊕ioikzipSi,k,Fi,k≤αp∀Zp∈P.


## 4. Algorithms

In this section, we will present a hybrid heuristic algorithm for achieving the best-so-far robust track assignment plans and a delay propagation algorithm to validate the effectiveness of the previous a posteriori robustness evaluation parameters and heuristic algorithm.


### 4.1. Hybrid Heuristic Algorithm

Scheduling jobs on parallel machines with fixed start and end time are a typical NP-hard problem. RTPP problem complexity can be come down to jobs scheduling problem complexity on parallel machines with fixed start and end time. Therefore, we design an effective hybrid heuristic that combines simple or composite dispatching rules to generate a good quality solution within acceptable computation time. The main steps of hybrid heuristic algorithm are constructed by simulating the manual working process as shown below.Step1Initialization and precalculation of parametersStep2Achieving initial processing sequence listSelect simple, composite, orθ composite dispatching rules described in the work of Zeng et al. [16] and achieve the initial processing sequence list J′ for all jobs.Step3Achieving initial solution (solution based on dispatching rule)According to train order in listJ′, we call conflicts check algorithm in Section 5.2 of Zeng et al. [16] to check feasibility of each track in alternative track set according to their sequence in fixed track utilization rule. If all the above tracks are not feasible, we assign tracks in alternative track set of breaking rules and check feasibility with Algorithm 2. Otherwise, the train is added to unassigned train set. Compute objective values of rule solutions (RS).Step4Generating new current solutionGenerate new solutions by choosing different optimal solution strategies. If the new solution is better than the best-so-far one, then replace the best-so-far solution with the new solution. Otherwise, the deteriorated solution is accepted with certain possibility.If any two trains have same train types, train attributes, and train direction and their arrival or departure times are close to each other, the trains are more likely to conflict with each other. We can define that trains are in the same group if they have same alternative track and route sets. We proposed the seven following strategies analogous to manual methods to generate new solutions: (1) random-track mutation, (2) single-track mutation in group, (3) multiple tracks’ mutation in group, (4) single train pair crossover in group, (5) multiple train pairs’ crossover in group, (6) single-time mutation in group (e.g., we can adjust arrival or departure time of single ORPT or TEPT within ELTW and check feasibility), and (7) multiple times’ mutation in group.Step5: Judgment of algorithm termination conditionsIf algorithm termination conditions are satisfied, the algorithm ends with the best-so-far solution. Output the best-so-far track assignment plans and objective values. Otherwise, go to Step 4.


### 4.2. Delay Propagation Algorithm

The total weighted knock-on delays of a track assignment plan based on a particular distribution of primary delays provide a direct and concrete means of judging the plan’s robustness performance. It is also useful in the effectiveness analysis of the previous a posteriori robustness evaluation parameters. In real operation scenario, each train has a planned departure time and cannot depart from the station before the macroscopic departure time. Knock-on delays occur when a primary delayed train impedes other trains by still occupying the assigned route or tracks. In an intuitive way, the following affected trains will right shift their arrival or departure time forward along timeline to avoid train conflicts. For small primary delays, train order, track, and route assignment in original operation plan usually remain the same for stability within the time horizon and no traffic control or rescheduling measures are considered to avoid conflicts. Based on the above analysis and time-space conflict model, we simulate the delay propagation process and identify the effect of all primary delays within a time horizon in station area. The prediction information of small primary delay distribution from 18 : 00 to 18 : 00 the next day should be given to capture the propagation over the time horizon.Delay propagation process can be partitioned into stages. In thetth stage, delay value set is denoted as Ωt. Let δitδit∈Ωt denote delay of train Ji at the tth stage. Let Ψt denote waiting checked delay train set at the tth stage. Let φi=0 denote whether train Ji in set Ψt is primary delay train in set Ψ0. In the tth stage, the set of trains affected by δit is denoted as Αt, and Λτ denotes the total number of affected trains in set At=J1′,…,Ja′,…JΛit′. Bijt denotes the minimum buffer time between two adjacent trains that have overlapped route patterns at the tth stage.Step 1.
Initialization. Setφi=0 and t=0.Step 2.
Choose the earliest trainJ0′ in set Ψt and renew delay set Ψt=Ψt−J0′. ηt=0.Step 3.
Achieve affected train operation set.
Compare all the buffer times between trainJa′=Ji and its adjacent train Jj with primary delay δit. Renew buffer time of train Jj according to the equations (16)-(17), and renew track and track-circuits occupation end time instants of train Ji.(16)Bi,q,j,qt=0ifδit≥Bijt,oiq=ojq=1,Bi,q,j,qt−δitifδit<Bijt,oiq=ojq=1,(17)Bi,k,j,kt=0ifδit≥Bijt,oik=ojk=1,Bi,k,j,kt−δitifδit<Bijt,oik=ojk=1.
IfJjJj∈J satisfied conditions oiq=ojq=1, δit<Bijt, and knock-on flag φi=1, ηt=ηt+δitωi. If JjJj∈J satisfied conditions oiq=ojq=1, δit<Bijt, and knock-on flag φi=0, ηt=ηt. If JjJj∈J satisfied conditions oiq=ojq=1 and δit≥Bijt, train Jj is added to affected trains set Αt.Step 4.
Achieve knock-on delay produced at thetth stage.Step 4.1.
a=0.(18)δjt=δit−Bij−δjtifδit−Bij≥δjt,oik=ojk=1,δjt−δit−Bijifδjt>δit−Bij,oik=ojk=1,(19)φi=1ifδit−Bij≥δjt,oik=ojk=1,0ifδjt>δit−Bij,oik=ojk=1.Step 4.2.
IfJa′=Jj already exists in set Ψt, renew the delay and knock-on flag φi with equations (18) and (19), respectively. Otherwise, train Ja′=JjJj∉Ψt is added to delay set Ψt+1=Ψt∪Ja′ with delay δjt=δit−Bij and knock-on flag φi=1. Renew knock-on delay ηt=ηt+Bijωj. If a≤Λt, a=a+1 and go to Step 4.2.Step 5.
Achieve knock-on delayft=ft+ηt. If Ψt≠∅, t=t+1. Go to Step 2. If Ψt=∅, output current total knock-on delay ft as total knock-on delay.


## 4.1. Hybrid Heuristic Algorithm

Scheduling jobs on parallel machines with fixed start and end time are a typical NP-hard problem. RTPP problem complexity can be come down to jobs scheduling problem complexity on parallel machines with fixed start and end time. Therefore, we design an effective hybrid heuristic that combines simple or composite dispatching rules to generate a good quality solution within acceptable computation time. The main steps of hybrid heuristic algorithm are constructed by simulating the manual working process as shown below.Step1Initialization and precalculation of parametersStep2Achieving initial processing sequence listSelect simple, composite, orθ composite dispatching rules described in the work of Zeng et al. [16] and achieve the initial processing sequence list J′ for all jobs.Step3Achieving initial solution (solution based on dispatching rule)According to train order in listJ′, we call conflicts check algorithm in Section 5.2 of Zeng et al. [16] to check feasibility of each track in alternative track set according to their sequence in fixed track utilization rule. If all the above tracks are not feasible, we assign tracks in alternative track set of breaking rules and check feasibility with Algorithm 2. Otherwise, the train is added to unassigned train set. Compute objective values of rule solutions (RS).Step4Generating new current solutionGenerate new solutions by choosing different optimal solution strategies. If the new solution is better than the best-so-far one, then replace the best-so-far solution with the new solution. Otherwise, the deteriorated solution is accepted with certain possibility.If any two trains have same train types, train attributes, and train direction and their arrival or departure times are close to each other, the trains are more likely to conflict with each other. We can define that trains are in the same group if they have same alternative track and route sets. We proposed the seven following strategies analogous to manual methods to generate new solutions: (1) random-track mutation, (2) single-track mutation in group, (3) multiple tracks’ mutation in group, (4) single train pair crossover in group, (5) multiple train pairs’ crossover in group, (6) single-time mutation in group (e.g., we can adjust arrival or departure time of single ORPT or TEPT within ELTW and check feasibility), and (7) multiple times’ mutation in group.Step5: Judgment of algorithm termination conditionsIf algorithm termination conditions are satisfied, the algorithm ends with the best-so-far solution. Output the best-so-far track assignment plans and objective values. Otherwise, go to Step 4.


## 4.2. Delay Propagation Algorithm

The total weighted knock-on delays of a track assignment plan based on a particular distribution of primary delays provide a direct and concrete means of judging the plan’s robustness performance. It is also useful in the effectiveness analysis of the previous a posteriori robustness evaluation parameters. In real operation scenario, each train has a planned departure time and cannot depart from the station before the macroscopic departure time. Knock-on delays occur when a primary delayed train impedes other trains by still occupying the assigned route or tracks. In an intuitive way, the following affected trains will right shift their arrival or departure time forward along timeline to avoid train conflicts. For small primary delays, train order, track, and route assignment in original operation plan usually remain the same for stability within the time horizon and no traffic control or rescheduling measures are considered to avoid conflicts. Based on the above analysis and time-space conflict model, we simulate the delay propagation process and identify the effect of all primary delays within a time horizon in station area. The prediction information of small primary delay distribution from 18 : 00 to 18 : 00 the next day should be given to capture the propagation over the time horizon.Delay propagation process can be partitioned into stages. In thetth stage, delay value set is denoted as Ωt. Let δitδit∈Ωt denote delay of train Ji at the tth stage. Let Ψt denote waiting checked delay train set at the tth stage. Let φi=0 denote whether train Ji in set Ψt is primary delay train in set Ψ0. In the tth stage, the set of trains affected by δit is denoted as Αt, and Λτ denotes the total number of affected trains in set At=J1′,…,Ja′,…JΛit′. Bijt denotes the minimum buffer time between two adjacent trains that have overlapped route patterns at the tth stage.Step 1.
Initialization. Setφi=0 and t=0.Step 2.
Choose the earliest trainJ0′ in set Ψt and renew delay set Ψt=Ψt−J0′. ηt=0.Step 3.
Achieve affected train operation set.
Compare all the buffer times between trainJa′=Ji and its adjacent train Jj with primary delay δit. Renew buffer time of train Jj according to the equations (16)-(17), and renew track and track-circuits occupation end time instants of train Ji.(16)Bi,q,j,qt=0ifδit≥Bijt,oiq=ojq=1,Bi,q,j,qt−δitifδit<Bijt,oiq=ojq=1,(17)Bi,k,j,kt=0ifδit≥Bijt,oik=ojk=1,Bi,k,j,kt−δitifδit<Bijt,oik=ojk=1.
IfJjJj∈J satisfied conditions oiq=ojq=1, δit<Bijt, and knock-on flag φi=1, ηt=ηt+δitωi. If JjJj∈J satisfied conditions oiq=ojq=1, δit<Bijt, and knock-on flag φi=0, ηt=ηt. If JjJj∈J satisfied conditions oiq=ojq=1 and δit≥Bijt, train Jj is added to affected trains set Αt.Step 4.
Achieve knock-on delay produced at thetth stage.Step 4.1.
a=0.(18)δjt=δit−Bij−δjtifδit−Bij≥δjt,oik=ojk=1,δjt−δit−Bijifδjt>δit−Bij,oik=ojk=1,(19)φi=1ifδit−Bij≥δjt,oik=ojk=1,0ifδjt>δit−Bij,oik=ojk=1.Step 4.2.
IfJa′=Jj already exists in set Ψt, renew the delay and knock-on flag φi with equations (18) and (19), respectively. Otherwise, train Ja′=JjJj∉Ψt is added to delay set Ψt+1=Ψt∪Ja′ with delay δjt=δit−Bij and knock-on flag φi=1. Renew knock-on delay ηt=ηt+Bijωj. If a≤Λt, a=a+1 and go to Step 4.2.Step 5.
Achieve knock-on delayft=ft+ηt. If Ψt≠∅, t=t+1. Go to Step 2. If Ψt=∅, output current total knock-on delay ft as total knock-on delay.


## 5. Experimental Result and Discussion

The performance of our algorithm is tested on the complex and highly used station area in China. 213 passenger trains are served in Guangzhou East Station from 18 : 00 to 18 : 00 the very next day, including 68 STHPTs, 7 of NSHPTs, and 58 pairs of ATUPTs and DTUPTs, as well as 16 pairs of ORPTs and TEPTs. There are 315 routes in interlocking table and 108 track circuits in station layout. Maintenance time window for track 7G is from 18 : 30 to 20 : 00. The algorithm is performed on a ThinkPad T570 laptop with 2.90 GHz CPU and 64-bit operating system. In this section, we will analyze objective values of those new solution generation strategies for RTPP with the same algorithm framework and explore best parameters for composite new solution generation strategies. Second, we also compare initial solution and good quality solution in acceptable computation time based on dispatching rules and new solution generation strategy for RTPP. Finally, we will compare the differences of their total knock-on delays and validate the effectiveness of robustness indicators using the same delay propagation simulation.


### 5.1. Results of Different New Solution Generation Methods

If we set 500 optimal solution iteration times (around 180 s) as terminating condition, good quality solutions based on dispatching rule FIFO and different new solution generation strategies are illustrated in Figure2.Figure 2
Effectiveness of different new solution generation strategies.Objective values of new solution generation strategies 3, 5, and 7 are much better than those of strategies 1, 2, 4, and 6 in RTPP with the same algorithm framework. In composite new solution generation strategy, we set probabilities of single-track mutation, multiple tracks’ mutation, single train pair crossover, multiple train pairs’ crossover, single-time mutation, multiple times’ mutation, and random-track mutation as 0.05, 0.3, 0.05, 0.4, 0.05, 0.05, and 0.1, respectively.


### 5.2. Solution Quality and Algorithm Stability

In this paper, we set penalty costξ1=6000s of unassigned trains higher than penalty cost ξ2=4000s of trains assigned to tracks in alternative track set of breaking fixed track utilization rule. Let Bmax=10min. The average computation time for initial or rule solution is about 1500 ms. Weight factors of those two subobjectives are adopted to 2 and 2, respectively. The third subobjective is expressed in penalty cost unit and weight factor takes 1. The rule-based solution (RS) and good-quality solution (GQS) of the RTPP after 150 iterations (around 15 s) are shown in Figures 3 and 4.Figure 3
Average subobjective values of rule-based solution.Figure 4
Rule-based solution and good-quality solution of RTPP.Based on composite new solution generation strategy and 150 iterations, we conclude that solutions of dispatching rules 1, 10, 12, and 13 are feasible solutions without breaking fixed track utilization rules. Solutions of dispatching rules 3, 4, 6, 7, and 9 are feasible solutions which break fixed track utilization rules. Solutions of dispatching rules 2, 5, 8, and 11 are not feasible. Based on composite new solution generation strategy and 150 iterations, optimized feasible solutions are shown in Figure4. Therefore, we would better consider breaking track utilization rule and choose appropriate dispatching rules to access a feasible and relatively good quality solution within short computation time in real-time level.


### 5.3. Identification of the Bottleneck Station Resources and Key Operations under Perturbations

A feasible and robust track assignment plan is achieved by metaheuristic algorithm after 150 iterations with dispatching rule BW-SL-EED-FIFO and composite solution improvement strategy. If we find out the buffer time between two adjacent resource occupation times, we can identify the bottleneck station resources under perturbations. Assume that arrival and departure primary delay ranges from 0 s to 30 s. If two adjacent resource occupation buffer times are less than those primary perturbations, related trains and their operations need to be conducted carefully to avoid primary or knock-on delays in station operation. The bottleneck station resources are shown in Table3.Table 3
Bottleneck track circuits and related train operations.
Track circuitsBuffer timeTrain 1Occupation end time 1Train 2Occupation start time 2Route of train 1Route of train 2128–232DG4.98T01817 : 59 : 35T11517 : 59 : 30R0120R0213128–232DG7.22T0567 : 56 : 41T0587 : 56 : 48R0211R012021–65DG11.18T1307 : 43 : 19T1397 : 43 : 30R0001R0086244DG15.37T0567 : 56 : 33T0587 : 56 : 48R0211R0120128–232DG18.07T0618 : 25 : 48T0578 : 25 : 30R0123R0209128–232DG19.88T04422 : 28 : 41T04522 : 29 : 01R0213R0123IBG27.09T02218 : 54 : 27T13318 : 54 : 54R0223R0001128–232DG27.56T02118 : 40 : 41T02318 : 41 : 08R0211R0121174DG27.92T04422 : 28 : 33T04522 : 29 : 01R0213R0123164DG29.19T07410 : 57 : 59T12110 : 57 : 30R0209R0299


### 5.4. The Total Knock-On Delays of TPP and RTPP

We mainly consider two kinds of primary delays for station operation: (1) arrival primary delays (e.g., sectional running time deviations caused by weather or train control issues) and (2) departure primary delays (e.g., additional dwell time caused by passenger over late passengers, crowdedness, reaction time variation of platform staff, or door-lock failure). Corman et al. [17] proposed that both arrival primary delay and departure primary delay fit Weibull distribution with certain time offset. Other researches such as those of Dewilde et al. [12], Goverde et al. [13], and Jensen et al. [18] regard departure primary delay as constant value and think that arrival primary delays fit exponential distribution. For example, En/6,Pn/830 represents a specific primary delay distribution scenario. An/6 denotes that half of the macroscopic arrival times have primary delays and fit an exponential distribution, and Pn/830 represents that 1/8 of the macroscopic departure time has a 30 s additional dwell delay. We continuously monitor and log arrival and departure train delays in the associated traffic management system. The best fitted primary delay distribution function can be acquired through mathematical statistical analysis based on field research data. The effectiveness of robustness improvement based on RTPP is compared by total knock-on delays within time horizon (between 18 : 00 and 18 : 00 the next day).If we set 500 iteration times as terminating condition and choose dispatching rule FIFO and composite new solution generation strategy, the objective values of TPP and RTPP are shown in Table4. The total weighted knock-on delays for the same primary delay distribution are shown in the last column.Table 4
The amount of total weighted knock-on delays in scenarioEn/6,Pn/830.
Model formulationWeighted number of delayed trainsWeighted number of trains that break ruleRobustnessTotal weighted knock-on delaysTPP model003.776.88RTPP model003.045.25The total knock-on delay of track assignment plan corresponding to RTPP solution is 23% better than the total knock-on delay of track assignment plan corresponding to TPP solution. Therefore, a proactive approach that considers a better distribution of buffer times in planning phases of track assignment plan is effective for impeding train conflicts and counteracting knock-on delays under real perturbations.


## 5.1. Results of Different New Solution Generation Methods

If we set 500 optimal solution iteration times (around 180 s) as terminating condition, good quality solutions based on dispatching rule FIFO and different new solution generation strategies are illustrated in Figure2.Figure 2
Effectiveness of different new solution generation strategies.Objective values of new solution generation strategies 3, 5, and 7 are much better than those of strategies 1, 2, 4, and 6 in RTPP with the same algorithm framework. In composite new solution generation strategy, we set probabilities of single-track mutation, multiple tracks’ mutation, single train pair crossover, multiple train pairs’ crossover, single-time mutation, multiple times’ mutation, and random-track mutation as 0.05, 0.3, 0.05, 0.4, 0.05, 0.05, and 0.1, respectively.


## 5.2. Solution Quality and Algorithm Stability

In this paper, we set penalty costξ1=6000s of unassigned trains higher than penalty cost ξ2=4000s of trains assigned to tracks in alternative track set of breaking fixed track utilization rule. Let Bmax=10min. The average computation time for initial or rule solution is about 1500 ms. Weight factors of those two subobjectives are adopted to 2 and 2, respectively. The third subobjective is expressed in penalty cost unit and weight factor takes 1. The rule-based solution (RS) and good-quality solution (GQS) of the RTPP after 150 iterations (around 15 s) are shown in Figures 3 and 4.Figure 3
Average subobjective values of rule-based solution.Figure 4
Rule-based solution and good-quality solution of RTPP.Based on composite new solution generation strategy and 150 iterations, we conclude that solutions of dispatching rules 1, 10, 12, and 13 are feasible solutions without breaking fixed track utilization rules. Solutions of dispatching rules 3, 4, 6, 7, and 9 are feasible solutions which break fixed track utilization rules. Solutions of dispatching rules 2, 5, 8, and 11 are not feasible. Based on composite new solution generation strategy and 150 iterations, optimized feasible solutions are shown in Figure4. Therefore, we would better consider breaking track utilization rule and choose appropriate dispatching rules to access a feasible and relatively good quality solution within short computation time in real-time level.


## 5.3. Identification of the Bottleneck Station Resources and Key Operations under Perturbations

A feasible and robust track assignment plan is achieved by metaheuristic algorithm after 150 iterations with dispatching rule BW-SL-EED-FIFO and composite solution improvement strategy. If we find out the buffer time between two adjacent resource occupation times, we can identify the bottleneck station resources under perturbations. Assume that arrival and departure primary delay ranges from 0 s to 30 s. If two adjacent resource occupation buffer times are less than those primary perturbations, related trains and their operations need to be conducted carefully to avoid primary or knock-on delays in station operation. The bottleneck station resources are shown in Table3.Table 3
Bottleneck track circuits and related train operations.
Track circuitsBuffer timeTrain 1Occupation end time 1Train 2Occupation start time 2Route of train 1Route of train 2128–232DG4.98T01817 : 59 : 35T11517 : 59 : 30R0120R0213128–232DG7.22T0567 : 56 : 41T0587 : 56 : 48R0211R012021–65DG11.18T1307 : 43 : 19T1397 : 43 : 30R0001R0086244DG15.37T0567 : 56 : 33T0587 : 56 : 48R0211R0120128–232DG18.07T0618 : 25 : 48T0578 : 25 : 30R0123R0209128–232DG19.88T04422 : 28 : 41T04522 : 29 : 01R0213R0123IBG27.09T02218 : 54 : 27T13318 : 54 : 54R0223R0001128–232DG27.56T02118 : 40 : 41T02318 : 41 : 08R0211R0121174DG27.92T04422 : 28 : 33T04522 : 29 : 01R0213R0123164DG29.19T07410 : 57 : 59T12110 : 57 : 30R0209R0299


## 5.4. The Total Knock-On Delays of TPP and RTPP

We mainly consider two kinds of primary delays for station operation: (1) arrival primary delays (e.g., sectional running time deviations caused by weather or train control issues) and (2) departure primary delays (e.g., additional dwell time caused by passenger over late passengers, crowdedness, reaction time variation of platform staff, or door-lock failure). Corman et al. [17] proposed that both arrival primary delay and departure primary delay fit Weibull distribution with certain time offset. Other researches such as those of Dewilde et al. [12], Goverde et al. [13], and Jensen et al. [18] regard departure primary delay as constant value and think that arrival primary delays fit exponential distribution. For example, En/6,Pn/830 represents a specific primary delay distribution scenario. An/6 denotes that half of the macroscopic arrival times have primary delays and fit an exponential distribution, and Pn/830 represents that 1/8 of the macroscopic departure time has a 30 s additional dwell delay. We continuously monitor and log arrival and departure train delays in the associated traffic management system. The best fitted primary delay distribution function can be acquired through mathematical statistical analysis based on field research data. The effectiveness of robustness improvement based on RTPP is compared by total knock-on delays within time horizon (between 18 : 00 and 18 : 00 the next day).If we set 500 iteration times as terminating condition and choose dispatching rule FIFO and composite new solution generation strategy, the objective values of TPP and RTPP are shown in Table4. The total weighted knock-on delays for the same primary delay distribution are shown in the last column.Table 4
The amount of total weighted knock-on delays in scenarioEn/6,Pn/830.
Model formulationWeighted number of delayed trainsWeighted number of trains that break ruleRobustnessTotal weighted knock-on delaysTPP model003.776.88RTPP model003.045.25The total knock-on delay of track assignment plan corresponding to RTPP solution is 23% better than the total knock-on delay of track assignment plan corresponding to TPP solution. Therefore, a proactive approach that considers a better distribution of buffer times in planning phases of track assignment plan is effective for impeding train conflicts and counteracting knock-on delays under real perturbations.


## 6. Conclusions

In this paper, we presented track-circuits-based formulations for RTPP in busy complex passenger stations and proposed a proactive approach to make a preventative track assignment plan. First, we propose a microscopic formulation of train operations of different train types and explain station dispatcher’s fixed track utilization rule and breaking rule under perturbations to achieve a feasible plan. Second, we propose a multicriterion scheduling model for RTPP with objectives of minimizing weighted number of delayed trains, minimizing trains that break rules, and maximizing robustness based on sectional-release interlocking system. Third, we propose delay propagation modeling based on microscopic station resources under specific perturbation scenario. Fourth, we design hybrid heuristic algorithm to solve the above model based on dispatching rules and present different new solution generation strategies to further optimize rule-based solutions. A case study of a typical busy complex station area in China shows that composite new solution generation strategies have the biggest potential in improving solution quality. To validate the effectiveness of robustness indicator, the RTPP is tested for a perturbation scenario in busy complex station area. The computational results indicate an improvement in robustness of 19% together with a decrease in knock-on delay propagation in station area about 23%.Subsequent research should explore a model that integrates placing-in and taking-out operations of ORPTs and TEPTs with the arrival and departure operations and simulates the delay propagation process considering interference of placing-in and taking-out technical operations in throat area. More attention should be focused on ensuring platforming plan’s feasibility under other disruptions such as track maintenance [14, 15].



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*Source: 5922653-2022-03-12.xml*