# Turnback Capacity Assessment and Delay Management at a Rail Transit Terminal with Two-Tail Tracks

Authors: Zhibin Jiang; Yuyan Tan; Feng Wang; Lei Bu
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

Terminal capacity and performance have become a major concern for rail transit agencies in China due to the ever increasing passenger demand. This paper develops a mixed integer programming (MIP) optimization model to estimate the turnback capacity and performance of a rail transit terminal with two-tail tracks. The capacity evaluation and delay propagation are described and assessed as anN-track integrated model with minimal time span and train delay. Operations and design parameters such as tail track allocation strategies, maximum layover time, headway pattern, buffer time distribution scheme, and primary delay are also considered in this model. The effectiveness of the model is tested by a case study with computation results drawn from one terminal station in Shanghai, China. The case study results show that unfixed platform time and flexible tail track allocation strategies can improve the capacity of turnback operation, and the strategy of allowing swapping of the tail tracks has a significantly positive impact on delay absorption.


---

## Body

## 1. Introduction

A rail transit terminal is a key point for train movements, where trains reverse direction (turn back) to provide continuing service in the opposite direction. Careful planning and operations of a terminal are essential for transit agencies to provide quality service to passengers effectively and efficiently [1]. The line capacity of a rail transit is often governed by the frequency with which trains can be reversed at terminal stations and intermediate turnback sites [2]. Terminal capacity and performance have become a major concern for rail transit agencies in China due to the ever increasing passenger demand, which has constantly driven the need for more frequent service up to the capacity that the system was originally designed for. Take Shanghai as an example; as of March 2014, there are 14 rail transit lines operating in Shanghai, with a total operating route length of 538 kilometers and 329 stations. On each weekday, over 8 million passengers ride on the lines of Shanghai rail transit network, and five of the 14 rail lines in Shanghai have become nearly saturated due to the limiting capacities of the terminals. Similar issues are happening in other cities in the world as well. From a survey of the Transit Cooperative Research Program (TCRP), nine out of 58 responding transit systems cited turnbacks as a constraint [3].However, there is a lack of well-established concepts and tools in the existing rail transit literature to optimize the tail track allocation strategies, maximum layover time, headway pattern, recovery time, and buffer time distributions at a heavily utilized terminal. This paper deals with the problem of estimating the capacity of a rail transit terminal with two-tail tracks, which is a popular configuration for rail transit terminals with high demands in China. The main objectives of this paper are to construct a new MIP optimization model for the turnback capacity and disruption management problems of a rail transit terminal and to develop a robust numerical tool that can help transit agencies to better assess the capacity and performance of their terminals to serve the ever increasing demands, while considering the major factors that affect the capacity of a transit terminal. The application of this model may also be applied to the planning, design, and safety assessment of a rail transit terminal.


## 2. Background

Both in literature and in practice, quite a lot of effort has been devoted to the assessments of the capacity and delays of high-intensity terminals in a rail transit network. A simple model was presented in TCRP Report 13 [3], and the capacity of a 2-track stub-end terminal based on infrastructure layout and train movement characteristics was estimated. More recently, in TCRP Report 165, Kittelson et al. [4] presented a formula to extend the approach to estimating terminal layover time and found that the layout and configuration design of a terminal played a key role in providing delay recovery and service performance.Lee [1] stated that terminal capacity and performance have become a major concern for public transport authorities, and the research results confirmed the effects of terminal configuration on service quality, but the results were limited to only one type of terminals (with no tail track). Based on simulation studies, van Oort and van Nes [5] presented their research of the effects of various terminal configurations on reliability of service and calculated the average delay per vehicle for three main types of terminals.Obviously, terminal capacity is closely related to timetable scheduling. Yet, the train scheduling problem in a rail transit system is quite complex [6]. Numerous proposed scheduling models and algorithms have addressed different but related optimization objectives, including capacity, route service, minimal delay, passenger demand, operating cost, and transit unit (TU) circulation [7–12]. However, although the parameters of minimum headway and layover time at terminals were considered in these studies, other important factors, such as the tail track allocation strategy and delay recovery time distribution, were not included in these studies.Actually, the major factors that affect the capacity and reliability of a high-intensity terminal should include track allocation strategies, train operation process, and recovery time and buffer time distributions. Since terminal congestion happens most often when high frequencies are scheduled [13, 14], it would be necessary to understand the distributions of the recovery time and buffer time to study the terminal capacity. Furthermore, maximum capacity has a significant effect on terminal reliability and delay management, which has often been identified as the main performance criterion for customer satisfaction on rail transit system [15].Due to the high passenger volumes and high utilization of rail transit lines, delay management has also received much attention from the perspectives of passengers or rail transit companies. Schmöcker et al. [15] stated that the terminal recovery times can often be used to reduce both the cumulative delay and the number of affected trains. Carrel et al. [13] also proposed that the capacity of terminals and reversing points has a direct influence on the propagation of delays throughout the line. Puong and Wilson [16] stated that delay recovery is preferably performed at terminal stations to minimize the negative impacts of the disruption on the reverse direction. The result of Wang et al. [17] showed that sections with switches and turnbacks at terminal stations are both capacity bottlenecks for a moving block railway line. Furthermore, Dollevoet et al. [18] developed a model for the delay management problem that considered the capacities of the stations and allowed the rescheduling of the platform track assignment under a real-time operating scenario.There are quite some published works on the real-time delay management in rail transit terminals. Cacchiani et al. [19] presented an overview of recovery models and algorithms for real-time railway disturbance and disruption management. Flamini and Pacciarelli [20] addressed a scheduling problem arising in the real-time management of a metro rail terminus in a rail transit system. The scheduling problem is modeled as a bicriteria job shop scheduling problem with additional constraints. Mannino and Mascis [21] developed a real-time automated traffic control system to operate trains in metro stations. For each candidate routing an instance of the blocking, no-wait job-shop scheduling problem with convex costs is solved to optimality by branch and bound.However, in these studies, a lot of attention has been paid to passenger demand, train unit circulation, or line capacity, but few of the studies looked at service regularity, tail track allocation strategies, maximum layover time, buffer time distribution scheme, and the delay management at terminals. In practice, in China terminal capacity is calculated by transit agencies who always allocate trains to one fixed tail in the process of timetable scheduling, and, in actual operation, the trains would maintain the turnback track as planned in the original timetable. Obviously, this operating strategy is not an optimal method of using the resources of tail tracks and platforms at a terminal.Therefore, this study introduces a number of new features to capture the influence of major elements that have not been comprehensively considered in past studies on the problem of estimating terminal capacity and delay management. Firstly, we will focus on developing and refining capacity concepts relevant to rail transit terminals with tail tracks. Secondly, anN-track based MIP model, which considers important operating and design parameters, such as tail track allocation strategies, maximum layover time, and buffer time distribution scheme, will be developed. Finally, a case study in Shanghai will be used to illustrate the practical value of the developed model.


## 3. Problem Description

### 3.1. Rail Transit Terminal Types and Characteristics

Rail transit terminals may be designed in many forms: hundreds of types exist over the world. In China, a stub-end terminal with one of the four main types is commonly chosen in rail transit lines, as shown in Figure1.Four main types of stub-end terminal.
(a)
Terminal with crossovers located in advance of station
(b)
Terminal with far-side crossovers and two-tail tracks beyond the station
(c)
Terminal with far-side crossovers and four-tail tracks beyond the station
(d)
Terminal with far-side crossovers and three-tail tracks beyond the stationFigure1(a) is a typical stub-end terminal with crossovers located in advance of the station. Trains arrive and depart from the same platform and all the passenger and crew activities happen during the time the train stays on the platform. Figure 1(b) is a typical terminal with far-side crossovers and tail tracks beyond the station; it can also be called a relay terminal [1]. Trains use separate tracks/platforms for arrival and departure. The arriving train pulls into one platform, then pulls into one of the tail tracks, changes direction, and then returns to pick up passengers from the other platform.Figures1(c) and 1(d) are terminals with four and three-tail tracks, respectively. These terminals are often designed as a function of operation by short turning at the middle position of a line and allow a storage track or tracks for spare or disabled trains—a useful, if not essential, failure management facility. Normally, if all the trains need to turn back at the terminal, the terminal types of Figures 1(b) and 1(c) have the same procedure for train turnbacking, because only two-tail tracks (tail tracks 1 and 2) of Figure 1(c) can be used for turnback operation. Because of its popularity in use for high demands, we will take the terminal with two-tail tracks (Figure 1(b)) as the terminal type for our study.


### 3.2. Turnback Operation at Terminal with Two-tail Tracks

Under low frequency operations, only 1 tail track (the preferred) may be used for train turnbacks. In high frequency operations, both tail tracks would be needed for train processing with three operation modes: only at tail track 1 (T1), only at tail track 2 (T2), and both at tail tracks 1 and 2 (T1 + T2). Figure2 shows the two turnback operation procedures of a terminal with two-tail tracks.Two turnback operation procedures of a terminal with two-tail tracks.
(a)
Turnback operation with one tail track (T1)
(b)
Turnback operation with two trail track (T1 + T2)It is evident that tail track allocation is an integral component at both the strategic planning level and the operational level. In determining the capacity of a rail transit line at the strategic planning level, one must allocate the associated turnback track over a certain time horizon in such a way that the maximum number of trains can be scheduled. Furthermore, on an operational level, when a disruption to the planned timetable occurs, one must reallocate the tracks in such a way that the negative impact of the disruption is minimized.


### 3.3. Layover Time

Layover time is the time between the scheduled arrival and departure of a vehicle at a transit terminal station. The layover time at a terminal with tail tracks consists of several time elements, which are illustrated in Figure3, and the detail explanation of these elements was introduced by van Oort and van Nes [5]. Adequate buffer time at a terminal is always needed to absorb delays due to variability of train arrivals and terminal processes. This helps provide reliability of train departure times. However, too much buffer time reduces productivity, constrains terminal capacity, and may result in terminal congestion.Figure 3
Time compositions of layover time at terminal with tail track.


### 3.4. Theoretical Capacity and Delay Management of Terminal

Theoretical capacity of a terminal is defined as [1]: maximum number of incoming trains a terminal can process per unit time based on the existing infrastructure and type of operations. When a system operates at close to its capacity, any significant delay may use up a train’s terminal layover time, resulting in potential delay to the trip in the reversed direction [4]. The capacity of terminals and reversing points has a direct influence on the propagation of delays throughout the line, because the layover time scheduled at a terminal helps put trains back into sequence or on schedule. A “delay recovery strategy” can be defined as the series of decisions made by the operations management to maximize the service performance following the delays that occur after an incident. Since terminal capacity is closely related to delay management, this study followed the result of Törnquist and Persson [22], which considered the rescheduling of trains under disturbance on a so called N-tracked network, where N refers to the number of tracks in a segment. We extended their model by adding crossovers or block sections to form multiple parallel tracks in a segment of high service demand.


## 3.1. Rail Transit Terminal Types and Characteristics

Rail transit terminals may be designed in many forms: hundreds of types exist over the world. In China, a stub-end terminal with one of the four main types is commonly chosen in rail transit lines, as shown in Figure1.Four main types of stub-end terminal.
(a)
Terminal with crossovers located in advance of station
(b)
Terminal with far-side crossovers and two-tail tracks beyond the station
(c)
Terminal with far-side crossovers and four-tail tracks beyond the station
(d)
Terminal with far-side crossovers and three-tail tracks beyond the stationFigure1(a) is a typical stub-end terminal with crossovers located in advance of the station. Trains arrive and depart from the same platform and all the passenger and crew activities happen during the time the train stays on the platform. Figure 1(b) is a typical terminal with far-side crossovers and tail tracks beyond the station; it can also be called a relay terminal [1]. Trains use separate tracks/platforms for arrival and departure. The arriving train pulls into one platform, then pulls into one of the tail tracks, changes direction, and then returns to pick up passengers from the other platform.Figures1(c) and 1(d) are terminals with four and three-tail tracks, respectively. These terminals are often designed as a function of operation by short turning at the middle position of a line and allow a storage track or tracks for spare or disabled trains—a useful, if not essential, failure management facility. Normally, if all the trains need to turn back at the terminal, the terminal types of Figures 1(b) and 1(c) have the same procedure for train turnbacking, because only two-tail tracks (tail tracks 1 and 2) of Figure 1(c) can be used for turnback operation. Because of its popularity in use for high demands, we will take the terminal with two-tail tracks (Figure 1(b)) as the terminal type for our study.


## 3.2. Turnback Operation at Terminal with Two-tail Tracks

Under low frequency operations, only 1 tail track (the preferred) may be used for train turnbacks. In high frequency operations, both tail tracks would be needed for train processing with three operation modes: only at tail track 1 (T1), only at tail track 2 (T2), and both at tail tracks 1 and 2 (T1 + T2). Figure2 shows the two turnback operation procedures of a terminal with two-tail tracks.Two turnback operation procedures of a terminal with two-tail tracks.
(a)
Turnback operation with one tail track (T1)
(b)
Turnback operation with two trail track (T1 + T2)It is evident that tail track allocation is an integral component at both the strategic planning level and the operational level. In determining the capacity of a rail transit line at the strategic planning level, one must allocate the associated turnback track over a certain time horizon in such a way that the maximum number of trains can be scheduled. Furthermore, on an operational level, when a disruption to the planned timetable occurs, one must reallocate the tracks in such a way that the negative impact of the disruption is minimized.


## 3.3. Layover Time

Layover time is the time between the scheduled arrival and departure of a vehicle at a transit terminal station. The layover time at a terminal with tail tracks consists of several time elements, which are illustrated in Figure3, and the detail explanation of these elements was introduced by van Oort and van Nes [5]. Adequate buffer time at a terminal is always needed to absorb delays due to variability of train arrivals and terminal processes. This helps provide reliability of train departure times. However, too much buffer time reduces productivity, constrains terminal capacity, and may result in terminal congestion.Figure 3
Time compositions of layover time at terminal with tail track.


## 3.4. Theoretical Capacity and Delay Management of Terminal

Theoretical capacity of a terminal is defined as [1]: maximum number of incoming trains a terminal can process per unit time based on the existing infrastructure and type of operations. When a system operates at close to its capacity, any significant delay may use up a train’s terminal layover time, resulting in potential delay to the trip in the reversed direction [4]. The capacity of terminals and reversing points has a direct influence on the propagation of delays throughout the line, because the layover time scheduled at a terminal helps put trains back into sequence or on schedule. A “delay recovery strategy” can be defined as the series of decisions made by the operations management to maximize the service performance following the delays that occur after an incident. Since terminal capacity is closely related to delay management, this study followed the result of Törnquist and Persson [22], which considered the rescheduling of trains under disturbance on a so called N-tracked network, where N refers to the number of tracks in a segment. We extended their model by adding crossovers or block sections to form multiple parallel tracks in a segment of high service demand.


## 4. Optimization Model

### 4.1. Problem Definition

In this paper we denote a segment as a collection of one or multiple tracks between two points where a point merely is the connection between two segments. A track can only be occupied by one train at a time and the track can be either unidirectional (i.e., only permitting one-way traffic) or bidirectional. In the model of Törnquist and Persson [22], the switches between the tracks were not defined explicitly. That means, if two trains use the same track within a segment and thereby are separated by time, their paths are not considered to be in conflict, but in practice they could be. The proposed model in this paper has handled this issue.Figure4 shows the illustration of the terminal with two-tail tracks, and the terminal is divided into three segments. Segment 1 represents the platform tracks (A and B), segment 2 represents double crossovers (D) and two main tracks (C and E), and segment 3 represents tail tracks (G and F). Segments 1 and 3 are double-tracked and segment 2 is triple-tracked. It is also important to know in which direction(s) the track can be operated. A bidirectional track can be used in any direction (but still only by one train at a time) while a unidirectional only in one specific direction. Therefore, according to the operations practice of turning back, the directional rule of each track at all segments is shown in Figure 4.Figure 4
TheN-tracked model of terminal with two-tail tracks.The temporary resource request by a train to occupy a segment is hereafter referred to as an event. An event has initial start and end times for a track within the segment but then needs to be updated with new start and end times (and possibly track name) during operations. Figures5(a) and 5(b) illustrate the platform and tail track occupation diagram and event allocation diagram. The event allocation diagram shows the resources for segments 1–3 and how the segments are allocated to the events. Each box represents a scheduled event and is numbered with respect to the train the event is associated with.Track occupation diagram and events allocation diagram ofN-tracked model.
(a)
Platform and tail track occupation diagram
(b)
Events allocation diagramFigure6 shows the possible sequence of events of each train trip. For inbound trains, the events must start at track A in segment 1, and then select one track (track C or track D) in segment 2. If one train selects track C in segment 2, it must select track F in segment 3; otherwise it must select track D in segment 2 and track G in segment 3. Then the first event of the connecting trip, which is linked to the same TU, must select the same track in segment 3. If an outbound train selects track F, it must select track D and track B subsequently, or if it selects track G, then it must select track E and track B. The time taken to traverse segments 1 and 2 must equal the minimum required traversing time, which means that none of the trains can add extra stop time in segment 1 and segment 2.Figure 6
Possible sequence of events of each train.Every transit unit (TU) has six events and must follow a fixed sequence of segments. In modeling, we define two main train trips: inbound train trips and outbound train trips and all trips are grouped by pairs, with each pair of train trips denoting a same TU. In Figure5 there are 8 train trips and therefore 4 TUs, for example, trip 101 and trip 102 are connected by TU01, and trip 102 must share the same track with trip 101 at segment 3.Different events of train movements may cause conflicts due to the share use of the same equipment (track and/or switch). The conflicts are detected whenever different events try to occupy the same track at the same time, and these events require a gap time to separate them (i.e., a gap time between a train leaving the track and another train entering the track). The length required for the gap time is different depending on the moving direction of the conflicting events (meeting at opposite directions or following at a same direction). Figure7(a) shows an example of events conflicting at track F, where train 2 arriving at track F must ensure that train 1 has left track F. Figure 7(b) shows an example of events conflicting at track D (crossover), where train 4 arriving at track F (through track D) must ensure that train 3 has left track D.Events conflict detection at different track.
(a)
Conflict of events at track F
(b)
Conflict of events at track D


### 4.2. Model Formulation

In this section, a Mixed integer programming (MIP) optimization model is developed for the problem of terminal capacity assessment with delay management. It is noted that the time units of following parameters and decision variables are all in seconds.


#### 4.2.1. Sets

The sets below are used for the mathematical model:U: the set of TUs,
S: the set of segments, in the model of Figure 4, S
=
1,2
,
3,
S
T: the set of segments with tail tracks, in the model of Figure 4, S
T
=
3,
S
P: the set of segments with platform tracks, in the model of Figure 4, S
P
=
1,
S
C: the set of segments with crossover tracks, in the model of Figure 4, S
C
=
2,
S
W: the set of segments where trains can wait extra time at tracks,
K: the ordered set of events where an event is a resource request by a certain train for a specific segment, K
=
[
(
i
,
j
,
in
/
out
)
,
i
∈
U
,
j
∈
S
], which means the event of TU i at segment j in inbound or outbound direction,
K
i
,
j
in: the ordered set of events for TU i at segments j in inbound direction, i
∈
U, K
i
,
j
in
∈
K, j
∈
S,
K
i
,
j
out: the ordered set of events for TU i at segments j in outbound direction, i
∈
U, K
i
,
j
out
∈
K, j
∈
S,
K
j
S: the ordered set of events of segment j, j
∈
S,
P
j: the set of parallel tracks of segment j, j
∈
S,
f
i: the ordered set of sequence numbers for first events in K for TU i, i
∈
U,
l
i: the ordered set of sequence numbers for last events in K for TU i, i
∈
U.Accordingly, the ordered set of eventsK for TU01 in Figure 5 can be expressed as K
=
[
(
1,1
,
in
), (
1,2
,
in
), (
1,3
,
in
)
,
(
1,3
,
out
)
,
(
1,2
,
out
)
,
(
1,1
,
out
)
].


#### 4.2.2. Parameters

The model uses the following parameters, which are assumed to be in integer values:n
TU: the number of initial TUs,
n: the number of events of each TU, in the model of Figure 5, n
=
6,
fe: the sequence number of the first event in K, in the model of Figure 4, fe
=
1,
le: the sequence number of the last event in K, in the model of Figure 4, le
=
n
×
n
TU,
d
j
,
t
in: the minimal occupation time for track t at segment j in the inbound direction, it specifies the minimum nonstop running time for the segment, or it represents the minimal dwell time (including the running time of the train entering or leaving the track) at platform or tail track, j
∈
S, t
∈
P
j, j
∈
S,
d
j
,
t
out: the minimal occupation time for track t at segment j in the outbound direction, j
∈
S, t
∈
P
j, j
∈
S,
h
j
,
t
M: the required gap time for track t at segment j if the trains meet, j
∈
S, t
∈
P
j, j
∈
S,
h
j
,
t
F: the required gap time for track t at segment j if one train follows the other, j
∈
S, t
∈
P
j, j
∈
S,
t
b: the start time of the first event fe,
t
delay: the primary delay time,
π
k
begin: the start time of event k in the initial timetable, k
∈
K,
π
k
end: the end time of event k in the initial timetable, k
∈
K,
o
k: the run direction of event k, k
∈
K,
M: a sufficiently large positive constant, here given the value 3600 * 24 = 86,400,
θ
k
,
t: = {1, if k uses track t (as in the initial timetable), k
∈
K
j
S, t
∈
P
j, j
∈
S
T; 0, otherwise},
b
k
,
j: = {1, if event k runs inbound direction at segment j, k
∈
K
j
S, j
∈
S; 0, otherwise},
N
fix: the sequence number of fixed tail track.


#### 4.2.3. Decision Variables

The following decision variables are defined in the model:x
k
begin: the start time of event k, k
∈
K,
x
k
end: the end time of event k, k
∈
K,
q
k
,
t: = {1, if event k uses track t, k
∈
K
j
S
,
t
∈
P
j
,
j
∈
S; 0, otherwise},
γ
k
k
^: = {1, if event k occurs before event k
^, k
,
k
^
∈
K
j
S
,
j
∈
S
,
k
<
k
^; 0, otherwise}.


#### 4.2.4. Objective Function

We consider objectives in the view of the following two aspects.(a)
The maximal number of TUs can be operated in a given time period, which can be represented by minimizing the time span of all trains:(1)
Min
⁡
⁡
x
le
end
-
x
fe
begin
.
(b)
Least delay of all train operations, which can be presented by minimizing the total arrival and departure delays of trains on platform tracks:(2)
Min
⁡
⁡
∑
i
∈
U
x
f
i
begin
-
π
f
i
begin
.


#### 4.2.5. Constraints

In this section, we will focus on the constraints of the model, which are listed as follows.(1) Event Time Constraints. Consider
(3)
x
fe
begin
=
t
b
,
(4)
x
k
+
1
begin
=
x
k
end
,
k
∉
K
i
,
j
out
,
i
∈
U
,
j
∈
S
,
(5)
x
f
i
+
1
begin
≥
x
f
i
begin
,
i
∈
U
,
i
<
n
TU
,
(6)
x
l
i
+
1
end
≥
x
l
i
end
,
i
∈
U
,
i
<
n
TU
,
(7)
x
k
end
≥
x
k
begin
+
b
k
,
j
·
d
j
,
t
in
+
1
-
b
k
,
j
·
d
j
,
t
out
,
hhhhhhhhhhhh
k
∈
K
,
j
∈
S
W
,
t
∈
P
j
,
(8)
x
k
end
=
x
k
begin
+
b
k
,
j
·
d
j
,
t
in
+
1
-
b
k
,
j
·
d
j
,
t
out
,
hhhhhhhhhhh
k
∈
K
,
j
∉
S
W
,
t
∈
P
j
,
(9)
x
f
i
+
2
begin
-
x
f
i
+
1
begin
=
x
f
i
+
1
begin
-
x
f
i
begin
,
hhhhhhhh
i
∈
U
,
i
<
n
TU
-
1
,
(10)
x
l
i
+
2
end
-
x
l
i
+
1
end
=
x
l
i
+
1
end
-
x
l
i
end
,
hihhh
i
∈
U
,
i
<
n
TU
-
1
.Constraint (3) defines the start time of first event to be t
b. Constraint (4) specifies that each train event is directly succeeded by the next one in the ordered set of events for the train trip. Constraint (5) defines the entering sequence of all arrival trains. Constraint (6) ensures that the first entering train must depart first; namely, all the trains must obey the first in first out rule. At the segments that train can wait a long time; actual occupation time should be no less than the minimum occupation time; otherwise, actual occupation time must be equal to the minimum occupation time. These facts are depicted in constraints (7) and (8). Constraints (9) and (10) specify identical gaps between all arrival or departure trains at platform tracks.(2) Constraints at Segments and Tracks. Consider
(11)
∑
t
∈
P
j
q
k
,
t
=
1
,
k
∈
K
j
S
,
j
∈
S
,
(12)
q
k
,
1
=
1
,
k
∈
K
i
,
j
in
,
i
∈
U
,
j
∈
S
P
,
(13)
q
k
,
2
=
1
,
k
∈
K
i
,
j
out
,
i
∈
U
,
j
∈
S
P
,
(14)
q
k
,
t
=
q
k
-
1
,
t
,
k
∈
K
i
,
j
in
∪
K
i
,
j
out
,
hihhhhh
t
∈
P
j
,
i
∈
U
,
j
∈
S
T
,
(15)
q
k
,
3
=
q
k
-
1,2
,
k
∈
K
i
,
j
out
,
i
∈
U
,
j
∈
S
C
,
(16)
q
k
,
2
=
q
k
-
1,1
,
k
∈
K
i
,
j
out
,
i
∈
U
,
j
∈
S
C
,
(17)
x
k
^
begin
-
x
k
end
≥
h
j
,
t
M
·
γ
k
k
^
-
M
·
1
-
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
k
<
k
^
-
1
,
o
k
^
≠
o
k
,
hhhhhhhhhhhhhh
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
,
(18)
x
k
^
begin
-
x
k
end
≥
h
j
,
t
F
·
γ
k
k
^
-
M
·
1
-
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
hihhhhhhhh
k
<
k
^
-
1
,
o
k
^
=
o
k
,
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
,
(19)
x
k
begin
-
x
k
^
end
≥
h
j
,
t
M
·
1
-
γ
k
k
^
-
M
·
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
hihhhhhhhh
k
<
k
^
-
1
,
o
k
^
≠
o
k
,
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
,
(20)
x
k
begin
-
x
k
^
end
≥
h
j
,
t
F
·
1
-
γ
k
k
^
-
M
·
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
hhihhhhhhh
k
<
k
^
-
1
,
o
k
^
=
o
k
,
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
.Constraint (11) enforces that every event must use exactly one track per relevant segment. Constraints (12) and (13) ensure that the entering and departing trains at segment 1 must select track 1 and track 2, respectively. Constraint (14) ensures that if a train enters segment 3, it must select the same track ID of the foregoing event. Constraints (15) and (16) ensure that if a train departs from track G, it must select track E, and if a train departs from track F, it must select track D. Constraints (17) through (20) specify that one event at a segment must end with the elapse of a required gap time before a next event may start on the same segment, if the events are using the same track of the segment. The length of the gap time depends on whether the conflicting trains meet (h
j
,
t
M) or follow (h
j
,
t
F) on each other.


### 4.3. Evaluation Strategy

Four strategies, strategies 1 through 4, are considered in this study regarding the rule of turnbacking and delay management. Strategies 1 and 2 are applied to quantify the maximal turnback capacity of the terminals with different operating methods, and strategies 3 and 4 are mainly applied to assess the delay propagation by different ways of tail track allocation.


#### 4.3.1. Strategy 1: Select Fixed Tail Track

This strategy allows trains to select one fixed tail track. The strategy adopts a formulation including objective function (1) and constraints (3) though (20) in addition to the following constraint:
(21)
q
k
,
t
=
1
,
t
=
N
fix
,
k
∈
K
j
S
,
j
∈
S
T
.


#### 4.3.2. Strategy 2: Select from the Two-tail Tracks and Every Two TUs Are Grouped and Operated with a Cyclic Pattern

This strategy allows trains to select any one of the tail tracks. In order to assure that all trains selecting the same turnback track keep the same occupation process, every two TUs are grouped and run in the cyclic pattern; that is, the third TU has the same occupation time and track as the first TU and the same for the fifth. The strategy adopts a formulation including objective function (1) and constraints (3) through (20) in addition to the following constraints:
(22)
x
k
+
3
n
begin
-
x
k
+
n
begin
=
x
k
+
2
n
begin
-
x
k
begin
,
k
∈
K
,
k
≤
n
×
n
TU
-
3
,
x
k
+
3
n
end
-
x
l
k
+
n
end
=
x
l
k
+
2
n
end
-
x
k
end
,
k
∈
K
,
k
≤
n
×
n
TU
-
3
,
q
k
+
2
n
,
t
=
q
k
,
t
,
k
∈
K
,
t
∈
P
j
,
j
∈
S
,
k
≤
n
×
n
TU
-
2
,
x
k
+
2
n
end
-
x
k
+
2
n
begin
=
x
k
end
-
x
k
begin
,
k
∈
K
,
k
≤
n
×
n
TU
-
2
.Constraints (22) are used to group and operate TUs with a cyclic pattern.


#### 4.3.3. Strategy 3: Delay Recovery, Allowing Swaps of Tail Tracks

This strategy allows trains to swap the turnback tracks. The strategy adopts a formulation including objective function (2) and constraints (3) through (20) in addition to the following constraints:
(23)
t
b
=
π
fe
begin
+
t
delay
,
(24)
x
f
i
begin
≥
π
f
i
begin
,
i
∈
U
,
(25)
x
l
i
end
≥
π
l
i
end
,
i
∈
U
.Constraint (23) sets the first arrival train with the delay time of t
delay and Constraints (24) and (25) ensure that the actual arrival or start time at the platform must be no earlier than as planned.


#### 4.3.4. Strategy 4: Delay Recovery, Maintaining the Tail Track in Original Timetable

This strategy requires that the trains to maintain the turnback track as in the original timetable, which means that the trains maintain the same turnback track at segmentS
T. The strategy adopts a formulation including objective function (2), constraints (3) through (20), and (23) through (25) in addition to the following constraint:
(26)
q
k
,
t
=
θ
k
,
t
,
k
∈
K
j
S
,
t
∈
P
j
,
j
∈
S
T
.


## 4.1. Problem Definition

In this paper we denote a segment as a collection of one or multiple tracks between two points where a point merely is the connection between two segments. A track can only be occupied by one train at a time and the track can be either unidirectional (i.e., only permitting one-way traffic) or bidirectional. In the model of Törnquist and Persson [22], the switches between the tracks were not defined explicitly. That means, if two trains use the same track within a segment and thereby are separated by time, their paths are not considered to be in conflict, but in practice they could be. The proposed model in this paper has handled this issue.Figure4 shows the illustration of the terminal with two-tail tracks, and the terminal is divided into three segments. Segment 1 represents the platform tracks (A and B), segment 2 represents double crossovers (D) and two main tracks (C and E), and segment 3 represents tail tracks (G and F). Segments 1 and 3 are double-tracked and segment 2 is triple-tracked. It is also important to know in which direction(s) the track can be operated. A bidirectional track can be used in any direction (but still only by one train at a time) while a unidirectional only in one specific direction. Therefore, according to the operations practice of turning back, the directional rule of each track at all segments is shown in Figure 4.Figure 4
TheN-tracked model of terminal with two-tail tracks.The temporary resource request by a train to occupy a segment is hereafter referred to as an event. An event has initial start and end times for a track within the segment but then needs to be updated with new start and end times (and possibly track name) during operations. Figures5(a) and 5(b) illustrate the platform and tail track occupation diagram and event allocation diagram. The event allocation diagram shows the resources for segments 1–3 and how the segments are allocated to the events. Each box represents a scheduled event and is numbered with respect to the train the event is associated with.Track occupation diagram and events allocation diagram ofN-tracked model.
(a)
Platform and tail track occupation diagram
(b)
Events allocation diagramFigure6 shows the possible sequence of events of each train trip. For inbound trains, the events must start at track A in segment 1, and then select one track (track C or track D) in segment 2. If one train selects track C in segment 2, it must select track F in segment 3; otherwise it must select track D in segment 2 and track G in segment 3. Then the first event of the connecting trip, which is linked to the same TU, must select the same track in segment 3. If an outbound train selects track F, it must select track D and track B subsequently, or if it selects track G, then it must select track E and track B. The time taken to traverse segments 1 and 2 must equal the minimum required traversing time, which means that none of the trains can add extra stop time in segment 1 and segment 2.Figure 6
Possible sequence of events of each train.Every transit unit (TU) has six events and must follow a fixed sequence of segments. In modeling, we define two main train trips: inbound train trips and outbound train trips and all trips are grouped by pairs, with each pair of train trips denoting a same TU. In Figure5 there are 8 train trips and therefore 4 TUs, for example, trip 101 and trip 102 are connected by TU01, and trip 102 must share the same track with trip 101 at segment 3.Different events of train movements may cause conflicts due to the share use of the same equipment (track and/or switch). The conflicts are detected whenever different events try to occupy the same track at the same time, and these events require a gap time to separate them (i.e., a gap time between a train leaving the track and another train entering the track). The length required for the gap time is different depending on the moving direction of the conflicting events (meeting at opposite directions or following at a same direction). Figure7(a) shows an example of events conflicting at track F, where train 2 arriving at track F must ensure that train 1 has left track F. Figure 7(b) shows an example of events conflicting at track D (crossover), where train 4 arriving at track F (through track D) must ensure that train 3 has left track D.Events conflict detection at different track.
(a)
Conflict of events at track F
(b)
Conflict of events at track D


## 4.2. Model Formulation

In this section, a Mixed integer programming (MIP) optimization model is developed for the problem of terminal capacity assessment with delay management. It is noted that the time units of following parameters and decision variables are all in seconds.


### 4.2.1. Sets

The sets below are used for the mathematical model:U: the set of TUs,
S: the set of segments, in the model of Figure 4, S
=
1,2
,
3,
S
T: the set of segments with tail tracks, in the model of Figure 4, S
T
=
3,
S
P: the set of segments with platform tracks, in the model of Figure 4, S
P
=
1,
S
C: the set of segments with crossover tracks, in the model of Figure 4, S
C
=
2,
S
W: the set of segments where trains can wait extra time at tracks,
K: the ordered set of events where an event is a resource request by a certain train for a specific segment, K
=
[
(
i
,
j
,
in
/
out
)
,
i
∈
U
,
j
∈
S
], which means the event of TU i at segment j in inbound or outbound direction,
K
i
,
j
in: the ordered set of events for TU i at segments j in inbound direction, i
∈
U, K
i
,
j
in
∈
K, j
∈
S,
K
i
,
j
out: the ordered set of events for TU i at segments j in outbound direction, i
∈
U, K
i
,
j
out
∈
K, j
∈
S,
K
j
S: the ordered set of events of segment j, j
∈
S,
P
j: the set of parallel tracks of segment j, j
∈
S,
f
i: the ordered set of sequence numbers for first events in K for TU i, i
∈
U,
l
i: the ordered set of sequence numbers for last events in K for TU i, i
∈
U.Accordingly, the ordered set of eventsK for TU01 in Figure 5 can be expressed as K
=
[
(
1,1
,
in
), (
1,2
,
in
), (
1,3
,
in
)
,
(
1,3
,
out
)
,
(
1,2
,
out
)
,
(
1,1
,
out
)
].


### 4.2.2. Parameters

The model uses the following parameters, which are assumed to be in integer values:n
TU: the number of initial TUs,
n: the number of events of each TU, in the model of Figure 5, n
=
6,
fe: the sequence number of the first event in K, in the model of Figure 4, fe
=
1,
le: the sequence number of the last event in K, in the model of Figure 4, le
=
n
×
n
TU,
d
j
,
t
in: the minimal occupation time for track t at segment j in the inbound direction, it specifies the minimum nonstop running time for the segment, or it represents the minimal dwell time (including the running time of the train entering or leaving the track) at platform or tail track, j
∈
S, t
∈
P
j, j
∈
S,
d
j
,
t
out: the minimal occupation time for track t at segment j in the outbound direction, j
∈
S, t
∈
P
j, j
∈
S,
h
j
,
t
M: the required gap time for track t at segment j if the trains meet, j
∈
S, t
∈
P
j, j
∈
S,
h
j
,
t
F: the required gap time for track t at segment j if one train follows the other, j
∈
S, t
∈
P
j, j
∈
S,
t
b: the start time of the first event fe,
t
delay: the primary delay time,
π
k
begin: the start time of event k in the initial timetable, k
∈
K,
π
k
end: the end time of event k in the initial timetable, k
∈
K,
o
k: the run direction of event k, k
∈
K,
M: a sufficiently large positive constant, here given the value 3600 * 24 = 86,400,
θ
k
,
t: = {1, if k uses track t (as in the initial timetable), k
∈
K
j
S, t
∈
P
j, j
∈
S
T; 0, otherwise},
b
k
,
j: = {1, if event k runs inbound direction at segment j, k
∈
K
j
S, j
∈
S; 0, otherwise},
N
fix: the sequence number of fixed tail track.


### 4.2.3. Decision Variables

The following decision variables are defined in the model:x
k
begin: the start time of event k, k
∈
K,
x
k
end: the end time of event k, k
∈
K,
q
k
,
t: = {1, if event k uses track t, k
∈
K
j
S
,
t
∈
P
j
,
j
∈
S; 0, otherwise},
γ
k
k
^: = {1, if event k occurs before event k
^, k
,
k
^
∈
K
j
S
,
j
∈
S
,
k
<
k
^; 0, otherwise}.


### 4.2.4. Objective Function

We consider objectives in the view of the following two aspects.(a)
The maximal number of TUs can be operated in a given time period, which can be represented by minimizing the time span of all trains:(1)
Min
⁡
⁡
x
le
end
-
x
fe
begin
.
(b)
Least delay of all train operations, which can be presented by minimizing the total arrival and departure delays of trains on platform tracks:(2)
Min
⁡
⁡
∑
i
∈
U
x
f
i
begin
-
π
f
i
begin
.


### 4.2.5. Constraints

In this section, we will focus on the constraints of the model, which are listed as follows.(1) Event Time Constraints. Consider
(3)
x
fe
begin
=
t
b
,
(4)
x
k
+
1
begin
=
x
k
end
,
k
∉
K
i
,
j
out
,
i
∈
U
,
j
∈
S
,
(5)
x
f
i
+
1
begin
≥
x
f
i
begin
,
i
∈
U
,
i
<
n
TU
,
(6)
x
l
i
+
1
end
≥
x
l
i
end
,
i
∈
U
,
i
<
n
TU
,
(7)
x
k
end
≥
x
k
begin
+
b
k
,
j
·
d
j
,
t
in
+
1
-
b
k
,
j
·
d
j
,
t
out
,
hhhhhhhhhhhh
k
∈
K
,
j
∈
S
W
,
t
∈
P
j
,
(8)
x
k
end
=
x
k
begin
+
b
k
,
j
·
d
j
,
t
in
+
1
-
b
k
,
j
·
d
j
,
t
out
,
hhhhhhhhhhh
k
∈
K
,
j
∉
S
W
,
t
∈
P
j
,
(9)
x
f
i
+
2
begin
-
x
f
i
+
1
begin
=
x
f
i
+
1
begin
-
x
f
i
begin
,
hhhhhhhh
i
∈
U
,
i
<
n
TU
-
1
,
(10)
x
l
i
+
2
end
-
x
l
i
+
1
end
=
x
l
i
+
1
end
-
x
l
i
end
,
hihhh
i
∈
U
,
i
<
n
TU
-
1
.Constraint (3) defines the start time of first event to be t
b. Constraint (4) specifies that each train event is directly succeeded by the next one in the ordered set of events for the train trip. Constraint (5) defines the entering sequence of all arrival trains. Constraint (6) ensures that the first entering train must depart first; namely, all the trains must obey the first in first out rule. At the segments that train can wait a long time; actual occupation time should be no less than the minimum occupation time; otherwise, actual occupation time must be equal to the minimum occupation time. These facts are depicted in constraints (7) and (8). Constraints (9) and (10) specify identical gaps between all arrival or departure trains at platform tracks.(2) Constraints at Segments and Tracks. Consider
(11)
∑
t
∈
P
j
q
k
,
t
=
1
,
k
∈
K
j
S
,
j
∈
S
,
(12)
q
k
,
1
=
1
,
k
∈
K
i
,
j
in
,
i
∈
U
,
j
∈
S
P
,
(13)
q
k
,
2
=
1
,
k
∈
K
i
,
j
out
,
i
∈
U
,
j
∈
S
P
,
(14)
q
k
,
t
=
q
k
-
1
,
t
,
k
∈
K
i
,
j
in
∪
K
i
,
j
out
,
hihhhhh
t
∈
P
j
,
i
∈
U
,
j
∈
S
T
,
(15)
q
k
,
3
=
q
k
-
1,2
,
k
∈
K
i
,
j
out
,
i
∈
U
,
j
∈
S
C
,
(16)
q
k
,
2
=
q
k
-
1,1
,
k
∈
K
i
,
j
out
,
i
∈
U
,
j
∈
S
C
,
(17)
x
k
^
begin
-
x
k
end
≥
h
j
,
t
M
·
γ
k
k
^
-
M
·
1
-
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
k
<
k
^
-
1
,
o
k
^
≠
o
k
,
hhhhhhhhhhhhhh
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
,
(18)
x
k
^
begin
-
x
k
end
≥
h
j
,
t
F
·
γ
k
k
^
-
M
·
1
-
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
hihhhhhhhh
k
<
k
^
-
1
,
o
k
^
=
o
k
,
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
,
(19)
x
k
begin
-
x
k
^
end
≥
h
j
,
t
M
·
1
-
γ
k
k
^
-
M
·
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
hihhhhhhhh
k
<
k
^
-
1
,
o
k
^
≠
o
k
,
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
,
(20)
x
k
begin
-
x
k
^
end
≥
h
j
,
t
F
·
1
-
γ
k
k
^
-
M
·
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
hhihhhhhhh
k
<
k
^
-
1
,
o
k
^
=
o
k
,
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
.Constraint (11) enforces that every event must use exactly one track per relevant segment. Constraints (12) and (13) ensure that the entering and departing trains at segment 1 must select track 1 and track 2, respectively. Constraint (14) ensures that if a train enters segment 3, it must select the same track ID of the foregoing event. Constraints (15) and (16) ensure that if a train departs from track G, it must select track E, and if a train departs from track F, it must select track D. Constraints (17) through (20) specify that one event at a segment must end with the elapse of a required gap time before a next event may start on the same segment, if the events are using the same track of the segment. The length of the gap time depends on whether the conflicting trains meet (h
j
,
t
M) or follow (h
j
,
t
F) on each other.


## 4.2.1. Sets

The sets below are used for the mathematical model:U: the set of TUs,
S: the set of segments, in the model of Figure 4, S
=
1,2
,
3,
S
T: the set of segments with tail tracks, in the model of Figure 4, S
T
=
3,
S
P: the set of segments with platform tracks, in the model of Figure 4, S
P
=
1,
S
C: the set of segments with crossover tracks, in the model of Figure 4, S
C
=
2,
S
W: the set of segments where trains can wait extra time at tracks,
K: the ordered set of events where an event is a resource request by a certain train for a specific segment, K
=
[
(
i
,
j
,
in
/
out
)
,
i
∈
U
,
j
∈
S
], which means the event of TU i at segment j in inbound or outbound direction,
K
i
,
j
in: the ordered set of events for TU i at segments j in inbound direction, i
∈
U, K
i
,
j
in
∈
K, j
∈
S,
K
i
,
j
out: the ordered set of events for TU i at segments j in outbound direction, i
∈
U, K
i
,
j
out
∈
K, j
∈
S,
K
j
S: the ordered set of events of segment j, j
∈
S,
P
j: the set of parallel tracks of segment j, j
∈
S,
f
i: the ordered set of sequence numbers for first events in K for TU i, i
∈
U,
l
i: the ordered set of sequence numbers for last events in K for TU i, i
∈
U.Accordingly, the ordered set of eventsK for TU01 in Figure 5 can be expressed as K
=
[
(
1,1
,
in
), (
1,2
,
in
), (
1,3
,
in
)
,
(
1,3
,
out
)
,
(
1,2
,
out
)
,
(
1,1
,
out
)
].


## 4.2.2. Parameters

The model uses the following parameters, which are assumed to be in integer values:n
TU: the number of initial TUs,
n: the number of events of each TU, in the model of Figure 5, n
=
6,
fe: the sequence number of the first event in K, in the model of Figure 4, fe
=
1,
le: the sequence number of the last event in K, in the model of Figure 4, le
=
n
×
n
TU,
d
j
,
t
in: the minimal occupation time for track t at segment j in the inbound direction, it specifies the minimum nonstop running time for the segment, or it represents the minimal dwell time (including the running time of the train entering or leaving the track) at platform or tail track, j
∈
S, t
∈
P
j, j
∈
S,
d
j
,
t
out: the minimal occupation time for track t at segment j in the outbound direction, j
∈
S, t
∈
P
j, j
∈
S,
h
j
,
t
M: the required gap time for track t at segment j if the trains meet, j
∈
S, t
∈
P
j, j
∈
S,
h
j
,
t
F: the required gap time for track t at segment j if one train follows the other, j
∈
S, t
∈
P
j, j
∈
S,
t
b: the start time of the first event fe,
t
delay: the primary delay time,
π
k
begin: the start time of event k in the initial timetable, k
∈
K,
π
k
end: the end time of event k in the initial timetable, k
∈
K,
o
k: the run direction of event k, k
∈
K,
M: a sufficiently large positive constant, here given the value 3600 * 24 = 86,400,
θ
k
,
t: = {1, if k uses track t (as in the initial timetable), k
∈
K
j
S, t
∈
P
j, j
∈
S
T; 0, otherwise},
b
k
,
j: = {1, if event k runs inbound direction at segment j, k
∈
K
j
S, j
∈
S; 0, otherwise},
N
fix: the sequence number of fixed tail track.


## 4.2.3. Decision Variables

The following decision variables are defined in the model:x
k
begin: the start time of event k, k
∈
K,
x
k
end: the end time of event k, k
∈
K,
q
k
,
t: = {1, if event k uses track t, k
∈
K
j
S
,
t
∈
P
j
,
j
∈
S; 0, otherwise},
γ
k
k
^: = {1, if event k occurs before event k
^, k
,
k
^
∈
K
j
S
,
j
∈
S
,
k
<
k
^; 0, otherwise}.


## 4.2.4. Objective Function

We consider objectives in the view of the following two aspects.(a)
The maximal number of TUs can be operated in a given time period, which can be represented by minimizing the time span of all trains:(1)
Min
⁡
⁡
x
le
end
-
x
fe
begin
.
(b)
Least delay of all train operations, which can be presented by minimizing the total arrival and departure delays of trains on platform tracks:(2)
Min
⁡
⁡
∑
i
∈
U
x
f
i
begin
-
π
f
i
begin
.


## 4.2.5. Constraints

In this section, we will focus on the constraints of the model, which are listed as follows.(1) Event Time Constraints. Consider
(3)
x
fe
begin
=
t
b
,
(4)
x
k
+
1
begin
=
x
k
end
,
k
∉
K
i
,
j
out
,
i
∈
U
,
j
∈
S
,
(5)
x
f
i
+
1
begin
≥
x
f
i
begin
,
i
∈
U
,
i
<
n
TU
,
(6)
x
l
i
+
1
end
≥
x
l
i
end
,
i
∈
U
,
i
<
n
TU
,
(7)
x
k
end
≥
x
k
begin
+
b
k
,
j
·
d
j
,
t
in
+
1
-
b
k
,
j
·
d
j
,
t
out
,
hhhhhhhhhhhh
k
∈
K
,
j
∈
S
W
,
t
∈
P
j
,
(8)
x
k
end
=
x
k
begin
+
b
k
,
j
·
d
j
,
t
in
+
1
-
b
k
,
j
·
d
j
,
t
out
,
hhhhhhhhhhh
k
∈
K
,
j
∉
S
W
,
t
∈
P
j
,
(9)
x
f
i
+
2
begin
-
x
f
i
+
1
begin
=
x
f
i
+
1
begin
-
x
f
i
begin
,
hhhhhhhh
i
∈
U
,
i
<
n
TU
-
1
,
(10)
x
l
i
+
2
end
-
x
l
i
+
1
end
=
x
l
i
+
1
end
-
x
l
i
end
,
hihhh
i
∈
U
,
i
<
n
TU
-
1
.Constraint (3) defines the start time of first event to be t
b. Constraint (4) specifies that each train event is directly succeeded by the next one in the ordered set of events for the train trip. Constraint (5) defines the entering sequence of all arrival trains. Constraint (6) ensures that the first entering train must depart first; namely, all the trains must obey the first in first out rule. At the segments that train can wait a long time; actual occupation time should be no less than the minimum occupation time; otherwise, actual occupation time must be equal to the minimum occupation time. These facts are depicted in constraints (7) and (8). Constraints (9) and (10) specify identical gaps between all arrival or departure trains at platform tracks.(2) Constraints at Segments and Tracks. Consider
(11)
∑
t
∈
P
j
q
k
,
t
=
1
,
k
∈
K
j
S
,
j
∈
S
,
(12)
q
k
,
1
=
1
,
k
∈
K
i
,
j
in
,
i
∈
U
,
j
∈
S
P
,
(13)
q
k
,
2
=
1
,
k
∈
K
i
,
j
out
,
i
∈
U
,
j
∈
S
P
,
(14)
q
k
,
t
=
q
k
-
1
,
t
,
k
∈
K
i
,
j
in
∪
K
i
,
j
out
,
hihhhhh
t
∈
P
j
,
i
∈
U
,
j
∈
S
T
,
(15)
q
k
,
3
=
q
k
-
1,2
,
k
∈
K
i
,
j
out
,
i
∈
U
,
j
∈
S
C
,
(16)
q
k
,
2
=
q
k
-
1,1
,
k
∈
K
i
,
j
out
,
i
∈
U
,
j
∈
S
C
,
(17)
x
k
^
begin
-
x
k
end
≥
h
j
,
t
M
·
γ
k
k
^
-
M
·
1
-
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
k
<
k
^
-
1
,
o
k
^
≠
o
k
,
hhhhhhhhhhhhhh
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
,
(18)
x
k
^
begin
-
x
k
end
≥
h
j
,
t
F
·
γ
k
k
^
-
M
·
1
-
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
hihhhhhhhh
k
<
k
^
-
1
,
o
k
^
=
o
k
,
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
,
(19)
x
k
begin
-
x
k
^
end
≥
h
j
,
t
M
·
1
-
γ
k
k
^
-
M
·
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
hihhhhhhhh
k
<
k
^
-
1
,
o
k
^
≠
o
k
,
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
,
(20)
x
k
begin
-
x
k
^
end
≥
h
j
,
t
F
·
1
-
γ
k
k
^
-
M
·
γ
k
k
^
,
k
,
k
^
∈
K
j
S
,
j
∈
S
,
hhihhhhhhh
k
<
k
^
-
1
,
o
k
^
=
o
k
,
q
k
,
t
+
q
k
^
,
t
=
2
,
t
∈
P
j
.Constraint (11) enforces that every event must use exactly one track per relevant segment. Constraints (12) and (13) ensure that the entering and departing trains at segment 1 must select track 1 and track 2, respectively. Constraint (14) ensures that if a train enters segment 3, it must select the same track ID of the foregoing event. Constraints (15) and (16) ensure that if a train departs from track G, it must select track E, and if a train departs from track F, it must select track D. Constraints (17) through (20) specify that one event at a segment must end with the elapse of a required gap time before a next event may start on the same segment, if the events are using the same track of the segment. The length of the gap time depends on whether the conflicting trains meet (h
j
,
t
M) or follow (h
j
,
t
F) on each other.


## 4.3. Evaluation Strategy

Four strategies, strategies 1 through 4, are considered in this study regarding the rule of turnbacking and delay management. Strategies 1 and 2 are applied to quantify the maximal turnback capacity of the terminals with different operating methods, and strategies 3 and 4 are mainly applied to assess the delay propagation by different ways of tail track allocation.


### 4.3.1. Strategy 1: Select Fixed Tail Track

This strategy allows trains to select one fixed tail track. The strategy adopts a formulation including objective function (1) and constraints (3) though (20) in addition to the following constraint:
(21)
q
k
,
t
=
1
,
t
=
N
fix
,
k
∈
K
j
S
,
j
∈
S
T
.


### 4.3.2. Strategy 2: Select from the Two-tail Tracks and Every Two TUs Are Grouped and Operated with a Cyclic Pattern

This strategy allows trains to select any one of the tail tracks. In order to assure that all trains selecting the same turnback track keep the same occupation process, every two TUs are grouped and run in the cyclic pattern; that is, the third TU has the same occupation time and track as the first TU and the same for the fifth. The strategy adopts a formulation including objective function (1) and constraints (3) through (20) in addition to the following constraints:
(22)
x
k
+
3
n
begin
-
x
k
+
n
begin
=
x
k
+
2
n
begin
-
x
k
begin
,
k
∈
K
,
k
≤
n
×
n
TU
-
3
,
x
k
+
3
n
end
-
x
l
k
+
n
end
=
x
l
k
+
2
n
end
-
x
k
end
,
k
∈
K
,
k
≤
n
×
n
TU
-
3
,
q
k
+
2
n
,
t
=
q
k
,
t
,
k
∈
K
,
t
∈
P
j
,
j
∈
S
,
k
≤
n
×
n
TU
-
2
,
x
k
+
2
n
end
-
x
k
+
2
n
begin
=
x
k
end
-
x
k
begin
,
k
∈
K
,
k
≤
n
×
n
TU
-
2
.Constraints (22) are used to group and operate TUs with a cyclic pattern.


### 4.3.3. Strategy 3: Delay Recovery, Allowing Swaps of Tail Tracks

This strategy allows trains to swap the turnback tracks. The strategy adopts a formulation including objective function (2) and constraints (3) through (20) in addition to the following constraints:
(23)
t
b
=
π
fe
begin
+
t
delay
,
(24)
x
f
i
begin
≥
π
f
i
begin
,
i
∈
U
,
(25)
x
l
i
end
≥
π
l
i
end
,
i
∈
U
.Constraint (23) sets the first arrival train with the delay time of t
delay and Constraints (24) and (25) ensure that the actual arrival or start time at the platform must be no earlier than as planned.


### 4.3.4. Strategy 4: Delay Recovery, Maintaining the Tail Track in Original Timetable

This strategy requires that the trains to maintain the turnback track as in the original timetable, which means that the trains maintain the same turnback track at segmentS
T. The strategy adopts a formulation including objective function (2), constraints (3) through (20), and (23) through (25) in addition to the following constraint:
(26)
q
k
,
t
=
θ
k
,
t
,
k
∈
K
j
S
,
t
∈
P
j
,
j
∈
S
T
.


## 4.3.1. Strategy 1: Select Fixed Tail Track

This strategy allows trains to select one fixed tail track. The strategy adopts a formulation including objective function (1) and constraints (3) though (20) in addition to the following constraint:
(21)
q
k
,
t
=
1
,
t
=
N
fix
,
k
∈
K
j
S
,
j
∈
S
T
.


## 4.3.2. Strategy 2: Select from the Two-tail Tracks and Every Two TUs Are Grouped and Operated with a Cyclic Pattern

This strategy allows trains to select any one of the tail tracks. In order to assure that all trains selecting the same turnback track keep the same occupation process, every two TUs are grouped and run in the cyclic pattern; that is, the third TU has the same occupation time and track as the first TU and the same for the fifth. The strategy adopts a formulation including objective function (1) and constraints (3) through (20) in addition to the following constraints:
(22)
x
k
+
3
n
begin
-
x
k
+
n
begin
=
x
k
+
2
n
begin
-
x
k
begin
,
k
∈
K
,
k
≤
n
×
n
TU
-
3
,
x
k
+
3
n
end
-
x
l
k
+
n
end
=
x
l
k
+
2
n
end
-
x
k
end
,
k
∈
K
,
k
≤
n
×
n
TU
-
3
,
q
k
+
2
n
,
t
=
q
k
,
t
,
k
∈
K
,
t
∈
P
j
,
j
∈
S
,
k
≤
n
×
n
TU
-
2
,
x
k
+
2
n
end
-
x
k
+
2
n
begin
=
x
k
end
-
x
k
begin
,
k
∈
K
,
k
≤
n
×
n
TU
-
2
.Constraints (22) are used to group and operate TUs with a cyclic pattern.


## 4.3.3. Strategy 3: Delay Recovery, Allowing Swaps of Tail Tracks

This strategy allows trains to swap the turnback tracks. The strategy adopts a formulation including objective function (2) and constraints (3) through (20) in addition to the following constraints:
(23)
t
b
=
π
fe
begin
+
t
delay
,
(24)
x
f
i
begin
≥
π
f
i
begin
,
i
∈
U
,
(25)
x
l
i
end
≥
π
l
i
end
,
i
∈
U
.Constraint (23) sets the first arrival train with the delay time of t
delay and Constraints (24) and (25) ensure that the actual arrival or start time at the platform must be no earlier than as planned.


## 4.3.4. Strategy 4: Delay Recovery, Maintaining the Tail Track in Original Timetable

This strategy requires that the trains to maintain the turnback track as in the original timetable, which means that the trains maintain the same turnback track at segmentS
T. The strategy adopts a formulation including objective function (2), constraints (3) through (20), and (23) through (25) in addition to the following constraint:
(26)
q
k
,
t
=
θ
k
,
t
,
k
∈
K
j
S
,
t
∈
P
j
,
j
∈
S
T
.


## 5. Application on Shanghai Rail Transit Line 1

The applicability of the proposed model for estimating the terminal capacity and delay was tested by a case study of the Xinzhuang terminal of line 1 in Shanghai, which is one of the most congested terminals of the Shanghai rail transit network. The length of this line is approximately 36.39 km, with 28 stations (including 8 transfer stations). This line uses TUs each composed of 8 high capacity vehicles (310 persons/vehicle). The minimal headway of this line is 2.75 min in peak hours and the line is operated extremely busy with a daily ridership exceeding 1 million passengers. A schematic track layout of the terminal area of this station is shown in Figure8.Figure 8
Track map of Xinzhuang (XZ) Station of line 1 in Shanghai, China.The proposed models were implemented in Visual Studio 2013 using IBM ILOG CPLEX 12.5 as a black-box MIP solver running on a personal computer with an Intel (R) Core (TM) i7-3520M CPU at 2.90 GHz and 4 GB of RAM under Windows 8 64-bit environment. The original timetable was obtained from the TPM (Train Plan Maker) software [9], which has been applied to designing operational timetables for all rail transit lines in Shanghai. The general input parameters of this terminal, as shown in Table 1, were adopted from the rail transit operation company of Shanghai.Table 1
Terminal train operations parameters (unit: second).
Segment track
Inbound occupation time (d
j
,
t
in
)
Outbound occupation time (d
j
,
t
out
)
Separation time of meeting events  (h
j
,
t
M)
Separation time of following events (h
j
,
t
F)
A
30
—
—
60
B
—
30
—
60
C
45
—
—
60
D
50
50
20
60
E
—
45
—
60
F
20
20
115
60
G
20
20
115
60


### 5.1. Theoretical Capacity Analysis

As summarized in Table2, four scenarios were tested to evaluate the proposed model under different operational conditions: these scenarios differed in three ways: (1) allocation strategy of tail tracks, (2) fixed or unfixed platform time, and (3) layover time. Other operating parameters are n
TU
=
8, n
=
6, fe
=
1, le
=
48, and t
b
=
0.Table 2
Computational scenarios.
Scenario ID
Operational scheme of tail tracks
Strategy
Platform time
Layover time (sec)
1
T2
1
Unfixed (S
W
=
1
,
3)
200,210,220
,
…
,
480
2
T2
1
Fixed (S
W
=
3)
200,210,220
,
…
,
480
3
T1 + T2
2
Unfixed (S
W
=
1
,
3)
200,210,220
,
…
,
480
4
T1 + T2
2
Fixed (S
W
=
3)
200,210,220
,
…
,
480For every combination of scenario and layover time, 112 computation runs were carried out. The result from the experiments is shown in Figure9.Figure 9
Minimum headway versus layover time and track occupation diagrams.These diagrams in Figure9 show that both the operational schemes of tail tracks and the platform time have very significant impacts on mininum headway of arrival or departure trains. Figure 9 shows that scenario 2 and scenario 4 result in a high minimum headway when layover time is less than 310 s. However, the minimum headway of scenario 4 is much lower than scenario 2 when layover time is more than 310 s. Figure 9 also indicates that the mininum headway reaches the lowest 116 s when layover time is 300 s at scenario 3. It is also interesting to notice that the mininum headway is not a linear relationship with layover time when layover time is less than 300 s at scenario 3, and the results do not change when layover time is less than 320 s at scenario 1.The comparison of strategy 1 versus strategy 2 shows that unfixed platform time and flexible tail track allocation strategies can impove the capacity of turnback operation. The figure also suggests that the mininum headway could vary quite significantly even at a same layover time. Figures9(a)–9(d) show the track occupation diagrams for the four different scenarios when the layover time is 320 s. Apparently, the fixed platform time produces longer wait times at the tail track, which makes the tail track constraining to the terminal capacity. Figure 9 also indicates that the reason for the headway to drop between 290 s and 300 s in scenario 3 is due to the identical gaps between all arrival or departure trains at platform tracks.


### 5.2. Delay Analysis

The initial timetable of line 1 was an actual Friday operation timetable in April 2014. This timetable was named 148-2. The average headway of arriving or departing trains at the Xinzhuang terminal was planned at 164 s during the morning peak. In our study, we selected 6:58 am to 8:06 am as the computational time horizon, which included 46 trains and 23 TUs. The minimum alighting and boarding times were 45 s and the minimal tail track time was 40 s, assuming that the platform time was not fixed and the primary delay train was the first arrival train.Table3 presents the results yielded by strategy 3 and strategy 4; in this table, the values for CPU time, total delay, delay trains, and delay recovery time are reported with respect to different primary delays. Figure 10 shows more details for the arrival and departure delays. The figure and table show that strategy 3 (which allows swapping of the tail tracks) has a significantly positive impact on absorbing of the delay. When the primary delay is less than 180 seconds, the trains do not suffer from knock-on delay at all, whereas the total delay increased rapidly when the primary delay increases to 240 seconds. Furthermore, the total delay of the trains is affected directly by the allocation strategy of the tail tracks. Figures 10(a) and 10(b) show the new tail track occupation schemes obtained by the optimization results with a primary delay of 300 seconds and constraint modeling strategy 4 and strategy 3, respectively.Table 3
Results of delay estimation.
Primary delay (s)
Strategy 3
Strategy 4
CPU Time
Total delay
Delay trains
Delay recovery time
CPU Time
Total delay
Delay trains
Delay recovery time
(s)
(s)
(s)
60
24
60
1
60
5
60
1
60
120
23
120
1
120
5
120
1
120
180
24
219
3
395
5
291
8
870
240
24
435
7
715
6
895
12
1,240
300
294
876
9
1,035
5
1,935
26
2,385
360
95
1,468
11
1,095
5
3,589
31
3,065
420
277
2,191
13
1,420
5
5,863
44
4,055
480
335
2,981
15
1,570
5
8,608
46
4,115
540
795
3,986
17
1,770
5
11,368
46
4,175
600
410
5,152
21
2,090
5
14,128
46
4,235
660
869
6,450
24
2,395
5
16,888
46
4,295
720
277
7,960
26
2,440
5
19,648
46
4,355
780
1,175
9,579
28
2,760
5
22,408
46
4,415
840
848
11,359
32
3,065
5
25,168
46
4,475
900
754
13,316
33
3,113
5
27,928
46
4,535Figure 10
Delay index and track occupation diagram by the different primary delay.The choice of strategy had a noticeable effect on computation time. Strategy 3 performed longer CPU time than strategy 4 at a same primary delay. The CPU time was related to the primary delay under strategy 3, which needed very long computation times when the primary delay was more than 240 seconds. However, the CPU time was quite short and did not vary when strategy 4 was applied.Scenario 2 in Table2 is the most popular and practical operation in China, because it is quite simple and easy to be applied in daily operations. Actually, Scenario 3 is the best operational strategy in terminal with maximal capacity and more flexibility in operation according to the computational results.


## 5.1. Theoretical Capacity Analysis

As summarized in Table2, four scenarios were tested to evaluate the proposed model under different operational conditions: these scenarios differed in three ways: (1) allocation strategy of tail tracks, (2) fixed or unfixed platform time, and (3) layover time. Other operating parameters are n
TU
=
8, n
=
6, fe
=
1, le
=
48, and t
b
=
0.Table 2
Computational scenarios.
Scenario ID
Operational scheme of tail tracks
Strategy
Platform time
Layover time (sec)
1
T2
1
Unfixed (S
W
=
1
,
3)
200,210,220
,
…
,
480
2
T2
1
Fixed (S
W
=
3)
200,210,220
,
…
,
480
3
T1 + T2
2
Unfixed (S
W
=
1
,
3)
200,210,220
,
…
,
480
4
T1 + T2
2
Fixed (S
W
=
3)
200,210,220
,
…
,
480For every combination of scenario and layover time, 112 computation runs were carried out. The result from the experiments is shown in Figure9.Figure 9
Minimum headway versus layover time and track occupation diagrams.These diagrams in Figure9 show that both the operational schemes of tail tracks and the platform time have very significant impacts on mininum headway of arrival or departure trains. Figure 9 shows that scenario 2 and scenario 4 result in a high minimum headway when layover time is less than 310 s. However, the minimum headway of scenario 4 is much lower than scenario 2 when layover time is more than 310 s. Figure 9 also indicates that the mininum headway reaches the lowest 116 s when layover time is 300 s at scenario 3. It is also interesting to notice that the mininum headway is not a linear relationship with layover time when layover time is less than 300 s at scenario 3, and the results do not change when layover time is less than 320 s at scenario 1.The comparison of strategy 1 versus strategy 2 shows that unfixed platform time and flexible tail track allocation strategies can impove the capacity of turnback operation. The figure also suggests that the mininum headway could vary quite significantly even at a same layover time. Figures9(a)–9(d) show the track occupation diagrams for the four different scenarios when the layover time is 320 s. Apparently, the fixed platform time produces longer wait times at the tail track, which makes the tail track constraining to the terminal capacity. Figure 9 also indicates that the reason for the headway to drop between 290 s and 300 s in scenario 3 is due to the identical gaps between all arrival or departure trains at platform tracks.


## 5.2. Delay Analysis

The initial timetable of line 1 was an actual Friday operation timetable in April 2014. This timetable was named 148-2. The average headway of arriving or departing trains at the Xinzhuang terminal was planned at 164 s during the morning peak. In our study, we selected 6:58 am to 8:06 am as the computational time horizon, which included 46 trains and 23 TUs. The minimum alighting and boarding times were 45 s and the minimal tail track time was 40 s, assuming that the platform time was not fixed and the primary delay train was the first arrival train.Table3 presents the results yielded by strategy 3 and strategy 4; in this table, the values for CPU time, total delay, delay trains, and delay recovery time are reported with respect to different primary delays. Figure 10 shows more details for the arrival and departure delays. The figure and table show that strategy 3 (which allows swapping of the tail tracks) has a significantly positive impact on absorbing of the delay. When the primary delay is less than 180 seconds, the trains do not suffer from knock-on delay at all, whereas the total delay increased rapidly when the primary delay increases to 240 seconds. Furthermore, the total delay of the trains is affected directly by the allocation strategy of the tail tracks. Figures 10(a) and 10(b) show the new tail track occupation schemes obtained by the optimization results with a primary delay of 300 seconds and constraint modeling strategy 4 and strategy 3, respectively.Table 3
Results of delay estimation.
Primary delay (s)
Strategy 3
Strategy 4
CPU Time
Total delay
Delay trains
Delay recovery time
CPU Time
Total delay
Delay trains
Delay recovery time
(s)
(s)
(s)
60
24
60
1
60
5
60
1
60
120
23
120
1
120
5
120
1
120
180
24
219
3
395
5
291
8
870
240
24
435
7
715
6
895
12
1,240
300
294
876
9
1,035
5
1,935
26
2,385
360
95
1,468
11
1,095
5
3,589
31
3,065
420
277
2,191
13
1,420
5
5,863
44
4,055
480
335
2,981
15
1,570
5
8,608
46
4,115
540
795
3,986
17
1,770
5
11,368
46
4,175
600
410
5,152
21
2,090
5
14,128
46
4,235
660
869
6,450
24
2,395
5
16,888
46
4,295
720
277
7,960
26
2,440
5
19,648
46
4,355
780
1,175
9,579
28
2,760
5
22,408
46
4,415
840
848
11,359
32
3,065
5
25,168
46
4,475
900
754
13,316
33
3,113
5
27,928
46
4,535Figure 10
Delay index and track occupation diagram by the different primary delay.The choice of strategy had a noticeable effect on computation time. Strategy 3 performed longer CPU time than strategy 4 at a same primary delay. The CPU time was related to the primary delay under strategy 3, which needed very long computation times when the primary delay was more than 240 seconds. However, the CPU time was quite short and did not vary when strategy 4 was applied.Scenario 2 in Table2 is the most popular and practical operation in China, because it is quite simple and easy to be applied in daily operations. Actually, Scenario 3 is the best operational strategy in terminal with maximal capacity and more flexibility in operation according to the computational results.


## 6. Conclusions

In this paper, we have proposed a model and problem formulation for estimating turnback capacity and delay at a rail transit terminal with two-tail tracks and considered the capacity evaluation and delay propagation as an integrated optimization model with two objectives of minimizing time span and minimizing train delay. Operations parameters such as tail track allocation strategies, maximum layover time, headway pattern, buffer time distribution scheme, and primary delay were also considered in the model. Since the model built in this paper is a generic model, the method can be easily changed to adapt to the changes in track occupation time or initial timetable and can also be fit for other types of terminals (crossovers in advance of terminal with two or more platform tracks) with small changes in constraints. A case study in Shanghai rail transit line 1 illustrated that unfixed platform time and flexible tail track allocation strategies can impove the capacity of turnback operation, and the strategy of allowing swapping of the tail tracks has a significantly positive impact on absorbing of the delay. The example also illustrated that rail transit agencies could use this proposed optimization model to compute for a reasonable capacity value with different managerial goals and operational strategies and for a delay recovery plan with high reliability.



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*Source: 369767-2015-09-08.xml*