The Analysis and Calculation Method of Urban Rail Transit Carrying Capacity Based on Express-Slow Mode
Xiaobing Ding1, Shengrun
Zhang2, Zhigang Liu1, Hua Hu1, Xingfang Xu3,
(1. College of Urban Rail
Transportation, Shanghai University of Engineering Science, Shanghai, 201620,
College of Civil Aviation, Nanjing University of Aeronautics and Astronautics,
211106, Nanjing, China)
College of Transportation Engineering, Tongji University, Shanghai, 201804,
College of Transportation, Beijing Jiaotong, University, Beijing,100001, China)
should be addressed to Xiaobing Ding; firstname.lastname@example.org
Received 26 April 2016; Revised 27 July
2016; Accepted 3 August 2016
Academic Editor: Chunlin Chen
2016 Xiaobing Ding et al. This is an open access article distributed under the
Creative Commons Attribution License,which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is
Abstract: Urban railway transport that connects suburbs and city areas is
characterized by uneven temporal and spatial distribution in terms of passenger
flow and underutilized carrying capacity. This paper aims to develop
methodologies to measure the carrying capacity of the urban railway by
introducing a concept of the express-slow mode. We first explore factors
influencing the carrying capacity under the express-slow mode and the
interactive relationships among these factors. Then we establish seven
different scenarios to measure the carrying capacity by considering the ratio
of the number of the express train and the slow train, the station where
overtaking takes place, as well as the
number of overtaking. Taking Shanghai metro line 16 as an empirical study, the
proposed methods to measure the carrying capacity under different express-slow mode
are proved to be valid. This paper contributes to the literature by remodifying
the traditional methods to measure the carrying capacity when different
express-slow mode are applied to improve the carrying capacity of the suburban
words: Urban railway transportation；carrying capacity；express-slow mode；schedule optimization； suburban-urban line
Since the beginning of the 21st century, urban rail transit system in China has been rapidly developed, which
accelerates the urbanization process of China. However, the traffic jam in
megacities has become more and more severe alongside the urbanization. One of the solutions is to develop strong economy of suburban areas
to attract people from city areas. The coverage of the urban railway transport
therefore, needs to be enlarged in order to facilitate the commuting between
suburbs and city areas. As the suburb passenger flow increases rapidly, the
regular rail transit system reveals its limitation in accommodating the
increased flow and providing efficient services in terms of travel time. Comparing to the regular rail
transit line, the length of suburb rail transit line (hereinafter referred to “suburb
line”) is longer and the passenger flow shows uneven spatial and temporal
distribution. Drawing upon these characteristics, a concept of the express-slow
mode has been introduced to provide flexible and efficient services on suburb
lines. The express-slow mode is a scheme in which express train(s) and slow
train(s) are operated in an alternative and coordinate way within a time
Several researchers have
investigated the methodologies and applications of the operation of
heterogeneous trains in railway transport. Mao Baohua et al. have investigated the differences
of the tracking time interval between a heterogeneous train and a homogeneous
train and found that the carrying capacity of the former was largely dampened.
They also analyzed the dynamic relationships between the ratio of the number of
the heterogeneous trains and their tracking time intervals. Bai Yun developed an optimization model in order to find an equilibrium between the
delay time and the number of departures and arrivals of trains. As the passenger
flows accumulated on the delayed trains would cause severe safety issues, they
also analyzed the carrying capacity in the context of a mixture of regular
trains and trains delayed due to external factors. Ni Shaoquan established
an optimization model in order to maximize the carrying capacity at peak hour
of a high-speed railway passenger station. His model can not only explore
factors influencing the carrying capacity of a train line, but also provide an
efficient algorithm to find a better solution for the model. Qin Yong et al. have applied the fuzzy
Markov chain theory to measure the carrying capacity on an intersectional line
by considering the random factors influencing the carrying capacity.
However, to date few literature
has been dedicated to develop a systematic methodology to measure the carrying
capacity of suburb lines when applying an express-slow mode [26, 27].
The objective of this paper, therefore, is to examine factors influencing the
carrying capacity of suburb lines under the express-slow mode and develop a
systematic methodology to calculate the carrying capacity in the same
The remainder of this paper is
organized as follows. Section 2 explore factors influencing the carrying
capacity of suburb lines under the express-slow mode. In particular, we focus
on the impacts of (i) the ratio of the number of express and slow trains, (ii)
the number of overtaking, and (iii) the
overtaking location of express trains based on the demand of a certain urban
rail station. In addition, the total en-route travel time is also considered as
a significant factor. Applying the express-slow mode may reduce the travel time
of express trains, but prolonging that of slow trains. Hence, the calculation
of this indicator should consider the balance of travel time of different
trains, which may have impacts on the service level of the urban railway
system. Given these factors, this section also develops a systematic
methodology to calculate the carrying capacity under the express-slow mode. Section
3 presents an empirical study to illustrate the application of the methodology
established in section 2. In section 4, we summarize the main implications of
our analysis and outline some avenues for further research.
2. Factors influencing the line carrying capacity
2.1 Line carrying capacity
capacity of an urban rail refers to the maximum frequency of a train  that
passes through an urban rail line within a unit time (generally the peak hour)
based on different types of trains, signal facilities and traffic organization.
first introduce a method to calculate the line carrying capacity of the regular
rail traffic which is characterized by the parallel and periodic operation
diagram. In addition, the intersections between urban rail traffic lines and
stations along the lines are regarded as an integrated system. Therefore, the
carrying capacity of the regular rail can be calculated as follows (Equ. 1) [6,
QUOTE QUOTE （1）
Where: QUOTE QUOTE —— the maximum frequency of a train passing a line in one direction
within a unit time (i.e., 1 hour in this paper) (expressed in “train/h”)
—— the minimum time interval of two departure trains(expressed
is generally calculated as follows
QUOTE QUOTE （2）
Where: QUOTE QUOTE —— the tracking duration of a train (expressed in “s”)
QUOTE QUOTE —— the minimum time interval between the arrival time of a train at the
destination and the departure time of the same train starting from another direction of a line (expressed
In formula 1,
the maximum time interval of train tracking and the maximum time interval of
train turning back are considered.
In a combined
mode with the coexistence of express and slow trains, overtaking should be
taken in account when measuring the carrying capacity of an urban rail .
Overtaking occurs when a slow train is operated ahead of an express train and
the interval of departure time between these two trains is less than the
minimum time interval[9-12]. In other words, overtaking that allows
an express train passing a station without stopping or departing ahead of a
slow train in the same direction can maximize the carrying capacity under the
express-slow mode. In addition, the stop frequency and waiting time of a slow
train also influence the line carrying capacity. As the stop frequency and
waiting time of the slow train increase, the interval of departure time between
the express train and the slow one may decrease. Other factors influencing the
line carrying capacity includes: the distance between two neighboring
overtaking stations, the interval of a train’s tracking time, the ratio of the
number of express trains and slow trains and the location of an overtaking
Based on the specific
characteristics of the urban rail under the express-slow mode and factors
influencing its operation, we remodify Equation 1 and propose a methodology to
measure the maximum carrying capacity of a suburb line applying the
express-slow mode (Equ. 3). （3）
—— The maximum carrying capacity
of a line under an express-slow mode (train);
QUOTE QUOTE —— The cycle time of a combined
express-slow train (“s”);
QUOTE QUOTE —— The total number of express and slow trains within a cycle time QUOTE QUOTE (train).
As shown in Equation 3, the carrying capacity of a suburb line under
the express-slow mode increases as the cycle time is reduced.
2.2 The impacts of the express-slow train mode on the line carrying
section, we explore a combined impact of the ratio of the number of express and
slow trains and the total number of overtaking on the line carrying capacity
under the express-slow mode. Assuming the time intervals of two departing
trains are the same and the ratio of the number of express and slow trains is
equal to k:m within a cycle[28-31].
As the situation that merely express trains
are operated on a line does not allow the operated express trains stop at any
intermediate stations, which is unrealistic and would lead to large demand loss , this paper does not consider this situation. Given the ratio of the
number of express and slow trains, the total number of overtaking is also
considered. To simplify the calculation, we only consider three types of
situation when overtaking occurs once, twice or no overtaking occurs. In this
way, seven different scenarios are designed to develop different methods to
measure the line carrying capacity.
1: only slow trains
In this scenario, as merely slow trains are
operated and no overtaking occurs, the carrying capacity is measured based on
the minimum time intervals between two slow trains. The equation is as Equ.1:
This scenario is designed as the baseline of
other proposed scenarios in order to figure out which scenario serves the best
combination to maximize the line carrying capacity.
（2）Scenario 2: k:m=1:1 & no overtaking
In this scenario, as no overtaking takes place, one express train is
operated between two slow trains as shown in Fig. 1.
Fig.1 Scenario 2: k:m=1:1 and no overtaking
Measuring the cycle time should
consider the following two situations due to the higher speed of express trains[14-16].
First, if a slow train is operated before an express train, the time interval
of the two trains arriving at the destination should not be larger than the
minimum time interval of departures I. Second, if the situation is in the
opposite, then the departure time difference of the two trains should be
calculated. The cycle time of scenario 2 is calculated as follows (Equ. 4):
n—— the stopping frequency of a slow
—— the sum of the time for acceleration and deceleration as well as the
waiting time of a train at one station.
According to Equation 3, the carrying
capacity in scenario 2, therefore, is calculated as follows (Equ. 5):
As can be seen from Equation 5, the
carrying capacity of scenario 2 is determined by the stopping frequency n and the
departure time interval I, which is constant when the operation scheme has been
（3）Scenario 3: k:m=1:1 & overtaking occurring once
In scenario 2 when overtaking is ignored, the longer time interval
between two departures can guarantee safety, but may deteriorate the carrying
capacity. Scenario 3 thus considers the overtaking occurring once.
Fig. 2 Scenario 3: k:m=1:1 & overtaking
As shown in Fig. 2, A-G represent
terminal stations; station D is available for overtaking. In order to guarantee
the following express train to safely overtake station D, the departure time
interval between the slow train and the express train should be extended to I1. The time interval between the slow train and the express train at station D should
not be less than I. Meanwhile, I1 should be larger than I. The cycle
time, therefore, is measured as follows (Equ. 6):
According to the Formula (3), Then
the carrying capacity can be calculated by embedding Equation 6 into Equation
3. Comparing to scenario 2, the carrying capacity of scenario 3 increases due
to the reduced number of stopping and the waiting time at the intermediate
4: k:m=1:2 & overtaking occurring once
Scenario 4 not only considers the
condition of overtaking, but also doubles the number of slow trains. In this
situation, an additional slow train is supplemented between every two
Fig. 3 Scenario
4: k:m=1:2 & overtaking occurring once
The operation period is detailed as follows
Although the number of slow trains increases,
whether the carrying capacity also increases depends on the operation in
practice. The carrying capacity of scenario 4 is calculated as follows(Equ. 8):
5: k:m=1:2 & overtaking occurring twice
5 increases the number of overtaking from 1 to 2, comparing to scenario 4.
Fig. 4 Scenario
5: k:m=1:2 & overtaking occurring twice
in Fig. 4, an express train departs lagging behind two slow trains and
overtakes the two slow trains at station E and C, respectively. The cycle time
of this scenario is measured as follows(Equ. 9):
—— the waiting time of the first slow train until the overtaking
express train departs the same station.
In such a condition, the carrying capacity is
calculated as follows(Equ. 10):
is calculated separately given
the entire section from station C to G. As an additional slow train is operated
between a slow train and an express train at station E within a cycle, the time
interval between the first departing slow train and the departing express train
at station E depends on: (i) the minimal time interval between the second departing
train and the departing express train at station E, and (ii) the departure time
interval between the first slow train and the second slow train.
6: k:m=2:1 & overtaking occurring once
considers the situation when the overtaking occurs once and the number of
express trains are doubled. The relationships between SAB and SCD should be taken into account.
Fig. 5 Scenario 6: k:m=2:1 & overtaking occurring once
in Fig. 5, overtaking can occur at either station E or station C. In the later condition, in order to ensure the
safe operation conditions, it is required to select the longer operation time
in the C~A section and G~E section as the integral part of the period; where
the period expression is detailed as follows (Equ. 11):
calculation of the carrying capacity under the train service percentage and
service condition is similar to that of the Scheme (5); however since the
station stop time is shortened, so the carrying capacity can be calculated
through the Equation 3.
7: k:m=2:3 & overtaking occurring twice
In scenario 7, the two preceding slow trains
are overtaken by the following express train and a third slow train can be
operated between two express trains in a cycle. Scenario 5 will be a simplified
version of this scenario if the two express trains in a cycle is combined.
Fig. 6 Scenario
7: k:m=2:3 & overtaking occurring twice
in Fig. 6, on the basis of the Scheme (5), one and
two are added,
where is the longest operation time between all
sections, where the period expression is detailed as follows (Equ. 12):
Where:is the longest operation time of all sections (expressed in “s”).
In this scenario, applying the
express-slow mode can increase the carrying capacity of a suburb line, which
further improves the serve quality for passengers. However, if the distance
between two overtaking stations is too long, the longer waiting time for slow
trains can also deteriorate the service level for passengers. Therefore, we
suggest that the number of overtaking should be reduced if a station cannot be
restructured as an overtaking station in the short run.
2.3 Discussions of seven
impacts of different ratios of the number of express and slow trains on the carrying
subsection attempts to generalize the method to calculate the carrying capacity
in a situation when overtaking merely occurs once, while the ratios of the
number of express and slow trains can be random . Taking scenario 3
and 4 as a comparison, the travel time of the latter increases by as the number of slow
trains doubled. Assuming that only one additional slow train is introduced
within one cycle time, then we can generalize the method to the situation when
the ratio of the number of express and slow trains is 1:m (m≥2). The average travel time of each train takes can be measured as follows
formula (13), and are fixed; therefore
infinitely, will approach
to; therefore when m increases, the line carrying capacity will
be improved. The similar conclusions can be deduced to the n:m express-slow train service percentage.
In Equation 13, and are fixed. An extra is added to the equation in
order to correspond to one addition of the slow train. Whenincreases infinitely,will approach to. In other words, the line carrying capacity also increases,
which may imply potential improvement.
impacts of overtaking frequencies on the carrying capacity
Recalling scenario 4 and 5, we
summarize that overtaking twice can increase the carrying capacity given the
ratio of the number of express and slow trains is equal to 1:2. As the number
of overtaking increase, the reduced waiting time of the slow train means the
decrease of the cycle time (Fig. 7).
Fig. 7 A comparison of different scenarios based on different
However, when the number of
overtaking increases to three or more, the passenger service level in practice
can be dampened due to the increased number of stops of slow trains
(References) [18,19]. This paper, therefore, do not consider the
impact of overtaking number over three on the carrying capacity.
2.3.3 Procedure to calculate the carrying capacity based
on the express-slow mode
This section introduces a procedure
to calculate the carrying capacity based on the aforementioned seven scenarios
under different ratios of the number of express and slow trains and the number
of overtaking. Three steps should be taken as follows:
Step 1: Given a predefined ratio of the
number of express and slow trains, the cycle time and the time interval between
two departing trains are calculated. Then the theoretical carrying capacity is measured based on Equation 5.
Step 2: This step determines
whether an overtaking should be considered based on the following criterion,
i.e., . If this
criterion is satisfied, then the procedure goes to step 3; otherwise, the line
carrying capacity will be equal to measured in step 1.
Step 3: This step determines the number of
overtaking and the location of overtaking stations based on the ratio of the
number of express and slow trains and the characteristics of passenger flow on
a suburb line. The carrying capacity are then measured based on equation 6 to
It is important to systematically examine the
changes of the line carrying capacity drawing upon different ratios of the
number of express and slow trains and different number of overtaking. In this
way, operators can figure out the combination scenario with the largest
carrying capacity and establish a denser schedule upon which the time interval
between two departure trains in the neighbor stations can be at its minimum.
3. Empirical study
3.1 Line introduction
The empirical study of this paper is Shanghai
Metro Line 16 that applies the express-slow mode since 2013. In order to
guarantee the safe overtaking of express trains, five stations of Line 16 are
reconstructed. First, Luoshan Station has constructed one island and four
tracks. Second, two islands and four tracks have been built at Hangtou East
Station, Wild Zoo Station, and Huinan East Station, respectively. In addition,
junction lines are set at every 1 or 2 railway section(s) . The
construction structure of Line 16 is shown in Figure 8.
Fig. 8 Schematic diagram of
Shanghai Metro Line 16
Shanghai Metro Line 16 adopts the CBTC mode
upon which the maximum speed can reach 110 km/h. In a recurrent situation, if a
train stops at every station, then the total travel time is 51.4 mins; if a
train stops merely at large stations (give specific station names), then the
total travel time is 35.4 mins[20,21]. In general, Type A trains
with three carriages are operated on Line 16 and can carry in total 648
passengers with seats. The passenger flow is strictly regulated in peak hours
due to the severe crowding problem on Line 16 and the theoretical time interval
of two departing trains in peak hours is 8 mins. Based on the
aforementioned parameters, the maximum number of passengers that Line 16 can
accommodate can be estimated by the following equation (Equ. 14): QUOTE QUOTE
3.2 Changes of the carrying capacity of Line 16
Following the calculation procedure
introduced in section 2.3.3, this section examines the changing carrying
capacity of Line 16 based on the scenarios proposed in section 2.3.1. We first
measure the carrying capacity of Line 16 given the three simplest scenarios,
i.e., k:m=1:1, k:m=1:2 and k:m=2:1, respectively.
（1）When and 2 express-slow trains run during the
QUOTE QUOTE trains； QUOTE QUOTE QUOTE QUOTE
（2）When and 3 express-slow trains run
during the combined period:
trains; QUOTE QUOTE QUOTE QUOTE
（3）When and 3 express-slow trains run
during the combined period:
In practice, the tracking duration of every express/slow
train varies due to the differences of distance between stations, which will
further influence the impacts of the ratio of the number of express and slow
trains on the carrying capacity. Therefore, we also systematically examine the
variation of the carrying capacity given different ratios of the number of
express and slow trains (Table 1 & Fig. 9), when the waiting time and the
tracking duration of trains are predefined.
Table 1. Carrying capacity of
Line 16 under different ratios of the number of express and slow trains
train service percentage
Fig. 9 Diagram for relationship between
express-slow train service ratio and carrying capacity
As can be seen from Fig. 9, the changes of
the carrying capacity presents a V-shape curve. Only if the number of express
and slow trains within a cycle are not equal, the carrying capacity can be
increased. Specifically, when the number of express trains are more than that
of slow ones, the carrying capacity will increase as the ratio of the number of
express and slow trains increase, and vice versa [22, 23, 25]. When
the number of express trains is equal to that of slow ones, the carrying capacity
reaches its minimum. However, it should be noted that the determination of
ratios is also affected by the actual demand at a certain time period of a day.
3.3 Better rail transit management and discussion
This section discusses how the
proposed methodology can be applied in practice and provide suggestions for
policy makers and metro operators. This is achieved by measuring the actual
number of trains required to carry passengers at different period of a day.
Then the ratio of the number of express and slow trains can be determined by
comparing these numbers with the maximum carrying capacity under the
express-slow mode as shown in Table 1.
We divide the
time periods of a day into peak and off-peak hours due to their distinguished
characteristics of demand. Two time periods, i.e., 7:00～9:00 and 17:00～19:00 are considered as peak
hours, while the left is defined as off-peak hours. Table 2
presents the average number of passengers per hour during the two time periods.
Table 2. Passenger flow of Metro Line 16 in
the peak and off-peak period
Huaxia Zhonglu Station
Wild Animal Zoo Station
The number of trains corresponding to the
actual demand is measured as follows (Equ.15)
Where, represents the number
of trains required to carry passengers, represents the total
number of passengers in a specific period, represents the designed
capacity of a carriage,represents the overloading coefficient, and represents the number of
According to table 2, the total number of passengers
during the peak period is 16,259, so
this number with the theoretical carrying capacity of Line 16 under the
express-slow mode as shown in table 1, the suggested ratio of the number of
express and slow trains is 1:4.
The same calculation procedure also applies
for the situation in the off-peak period, and the required number of trains is
6.10. The suggested ratio of the number of express and slow trains, therefor,
In their daily
operation, the operator does not take into account the different
characteristics of passenger flow during morning and evening peak hours on Shanghai
Metro Line 16. As illustrated by this example, the proposed framework can not
only determine the number of express and slow trains but also establish an
equilibrium between demand and carrying capacity [35, 36]. This
paper, therefore, provides guidance for both metro operators and policy makers.
paper attempts to establish a systematic framework to measure the carrying
capacity of a suburb line applying the express-slow mode by considering both
the ratio of the number of express and slow trains and the number of
overtaking. We establish seven individual scenarios to develop different
methods to measure the carrying capacity under this novel technique. We find
that the proposed methods can be generalized into the situation when overtaking
merely occurs once and the ratio of the number of express and slow trains is
random. In addition, the number of overtaking should be more than twice due to
the deteriorated service level for passengers.
Shanghai Metro Line 16 as a case study, we find that only if the number of
express and slow trains within a cycle are not equal, the carrying capacity can
be increased. Specifically, when the number of express trains are more than
that of slow ones, the carrying capacity will increase as the ratio of the
number of express and slow trains increase, and vice versa. When the number of
express trains is equal to that of slow ones, the carrying capacity reaches its
minimum. We further discuss how the proposed framework can help policy makers
and metro operators make decisions by considering the balance between demand
and carrying capacity.
future, the framework can be improved by establishing a comprehensive model to
consider the ratio of the number of express and slow trains as a continuous
variable instead of a discrete variable in this paper.
Additional, it is also a challenging task to measure and optimize the carrying
capacity under the express-slow mode from a network perspective, upon which
several different metro lines are crossed with each other.
authors declare that they have no competing interests.
work was supported in part by the National Natural Science Foundation under
Grant 61272029, the key technologies of integrated test and authentication
platform for Urban Rail Transit under Grant 2015BAG19B02-28.
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About the corresponding author:
Ding Xiao-bing (1982~), male, Jingsu,China.
Doctor’s degree of Transportation Engineering, Tongji University, majored in
rail transit operation organization optimization, E-mail：email@example.com;
correspondence address is: Room 304, No. 202, Lane 3688, Wenxiang Road,
Songjiang District, Shanghai. Postal code: 201620; contact telephone: 18017349689;
fixed-line telephone: 021-67791165.