Mathematical Problems in Engineering

Volume 2016, Article ID 6820394, 9 pages

http://dx.doi.org/10.1155/2016/6820394

## The Analysis and Calculation Method of Urban Rail Transit Carrying Capacity Based on Express-Slow Mode

^{1}College of Urban Rail Transportation, Shanghai University of Engineering Science, Shanghai 201620, China^{2}College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China^{3}College of Transportation Engineering, Tongji University, Shanghai 201804, China^{4}College of Transportation, Beijing Jiaotong University, Beijing 100001, China

Received 26 April 2016; Revised 27 July 2016; Accepted 3 August 2016

Academic Editor: Chunlin Chen

Copyright © 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 properly cited.

#### 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 trains and the slow trains, the station where overtaking takes place, and the number of overtaking maneuvers. 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 modes are applied to improve the carrying capacity of the suburban railway.

#### 1. Introduction

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 as “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 window.

Several researchers have investigated the methodologies and applications of the operation of heterogeneous trains in railway transport. Li and Mao [1] 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 et al. [2] 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. Chen et al. [3] 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. Wang et al. [4] 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 literatures have been dedicated to developing a systematic methodology to measure the carrying capacity of suburb lines when applying an express-slow mode [5, 6]. 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 situation.

The remainder of this paper is organized as follows. Section 2 explores 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 maneuvers, 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

The carrying capacity of an urban rail refers to the maximum frequency of a train [7] 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.

We 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 [8, 9]:where is 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”) and is the minimum time interval of two departure trains (expressed in “s”).

is generally calculated as follows:where is the tracking duration of a train (expressed in “s”), is 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 “s”).

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 [10]. 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 [11–14]. 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 include 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 station [15].

Based on the specific characteristics of the urban rail under the express-slow mode and factors influencing its operation, we remodify (1) and propose a methodology to measure the maximum carrying capacity of a suburb line applying the express-slow mode.where is the maximum carrying capacity of a line under an express-slow mode (train); is the cycle time of a combined express-slow train (“s”); is the total number of express and slow trains within a cycle time (train).

As shown in (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 Capacity

In this section, we explore a combined impact of the ratio of the number of express and slow trains and the total number of overtaking maneuvers on the line carrying capacity under the express-slow mode. We assume that 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 [16–19].

As the situation that merely express trains are operated on a line does not allow the operated express trains to stop at any intermediate stations, which is unrealistic and would lead to large demand loss [20], this paper does not consider this situation. Given the ratio of the number of express and slow trains, the total number of overtaking maneuvers 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.

*Scenario 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 (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.

*Scenario 2 ( k :m = 1 : 1 and no overtaking). *In this scenario, as no overtaking takes place, one express train is operated between two slow trains as shown in Figure 1.

Measuring the cycle time should consider the following two situations due to the higher speed of express trains [8, 21, 22]. 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:where

*n*is the stopping frequency of a slow train; is the sum of the time for acceleration and deceleration as well as the waiting time of a train at one station.

According to (3), the carrying capacity in Scenario 2, therefore, is calculated as follows:As can be seen from (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 scheduled.