J. Cent. South Univ. (2017) 24: 236-244

DOI: 10.1007/s11771-017-3424-x

Metro passenger flow control with station-to-station cooperation based on stop-skipping and boarding limiting

JIANG Man(姜曼)^{1, 2}, LI Hai-ying(李海鹰)¹, XU Xin-yue(许心越)¹, XU Shi-peng(徐仕鹏)², MIAO Jian-rui(苗建瑞)¹

1. State Key Laboratory of Rail Traffic Control & Safety, Beijing Jiaotong University, Beijing 100044, China;

2. School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China

 Central South University Press and Springer-Verlag Berlin Heidelberg 2017

Abstract: Metro passenger flow control problem is studied under given total inbound demand in this work, which considers passenger demand control and train capacity supply. Relevant connotations are analyzed and a mathematical model is developed. The decision variables are boarding limiting and stop-skipping strategies and the objective is the maximal passenger profit. And a passenger original station choice model based on utility theory is built to modify the inbound passenger distribution among stations. Algorithm of metro passenger flow control scheme is designed, where two key technologies of stopping-station choice and headway adjustment are given and boarding limiting and train stopping-station scheme are optimized. Finally, a real case of Beijing metro is taken for example to verify validity. The results show that in the three scenarios with different ratios of normal trains to stop-skipping trains, the total limited passenger volume is the smallest and the systematic profit is the largest in scenario 3.

Key words: metro; passenger flow control; stop-skipping; boarding limiting; passenger original station choice

1 Introduction

Along with rapid construction and development of urban rail transit in both home and abroad, these exist crowded passenger flow problems in morning and evening peak hours and holidays in metropolises. Take Beijing metro for example. It is with the characteristics of large and increasing passenger volume in morning peak hours, while headway of trains is the minimum and the line will not supply more capacity for passengers during this period. It is in need to limit certain passengers outside the station in a more scientific way because there is larger space outside the entrances, passenger falling tracks and treading on others will not happen and passenger safety will be well guaranteed. Section loading rates are in high levels, some of which are even beyond 120% and the overall distribution of section loading rates is unbalanced, which makes it possible to apply stop-skipping strategy to dynamically assign train capacity among stations. For example, ratio of line mileage where section loading rate is beyond 100% to total line mileage in Line 13 of Beijing metro is 22%, which is more than one fifth of the whole length. In fact, faced with serious passenger crowding problem in stations in morning peak hours, many metro stations take the measure of boarding limiting and the proportion of stations with boarding limiting reaches to 18% in total station number. But the measures in these stations are usually aimed at a single station without consideration of other adjacent stations, where time and volume of limited passenger flow are according to experience. It is meaningful theoretically and practically to focus on metro passenger flow control method of a single line based on boarding limiting and stop-skipping to balance limited passenger volume among stations.

Relevant researches include boarding limiting and train stopping-station scheme adjustment. In the field of boarding limiting, HE [1] researched coordination boarding limiting problem in two stations and proposed a parameter calculation method based on priority. STUBENSCHROTT et al [2] presented a dynamic pedestrian route choice model based on continuous observations of perceived time estimations and validated the model with real world passenger flow data. XU [3] analyzed influencing factors of metro boarding limiting, developed a metro boarding limiting model and realized total passenger service time minimum and capacity matching problem remission in significant stations. HAYWOOD et al [4] used original survey data from Paris to assess the distribution of comfort costs of congestion in public transport and estimate willingness to pay for less crowded trips at different levels of in-vehicle passenger density. In the field of train stopping-station scheme adjustment and capacity matching, SUN et al [5] discussed the probability of stop-skipping strategy application in real-time operation control. JIANG et al [6] studied crowding during the morning peak hours at a platform staircase in a metro station in China. The evacuation process was simulated and predicted results on the total evacuation time and maximum flow capacity of the selected staircase were compared with the field observations. FELIPE et al [7] focused on the problem of bus bunching and proposed a mathematical programming model to control passenger flow to realize minimal delay time in road corridor. They suggested mixing bus holding and boarding limiting strategies to improve systematic operation efficiency and successfully applied the model to St. Paul metro in Brazil. CORTES et al [8, 9] proposed mixed strategies of bus holding and stop-skipping to control bus lines and considered bi-objective of minimum waiting time and uniform consistency of time interval in stochastic demand. Few of present researches refer to influence of boarding limiting in a single station on adjacent stations and consider train stopping-station adjustment at the same time.

2 Description of metro passenger flow control problem

Multi stations and lines among them form physical conditions of station-to-station transportation system in a single line (see Fig. 1). Passenger OD distribution determines passenger demand and specific train stopping-station scheme determines train capacity supply. Metro passenger flow control problem will be analyzed under given total inbound passenger demand. When the system begins to run and passenger flow propagates in lines, a dynamic transportation system is formed.

Different section loading rates are unbalanced in a line and passenger demand distribution among sections and stations are various. Passenger volumes in some sections and stations are huge and far beyond section capacity and station capacity. It is effective to limit passenger volume in stations which locate ahead of the target station to preserve line capacity for passenger demand in the target station. Stopping-station adjustment can dynamically control and assign train capacity and boarding limiting scheme relieves the passenger crowding problem of key stations by controlling related station demand in advance. Finally, the metro passenger flow control will be realized by the mixture of these two strategies.

Take a zone of station A to station C for example where two scenarios are set. In scenario 1, there are two normal trains, which refer to trains that will stop at each station on the line, running. And in scenario 2, there is one similar train with scenario 1 and another stop-skipping train, which is a train that will skip some stations and only stop at the rest stations. These two kinds of trains are illustrated as the following Fig. 2. Normal train will stop at stations A, B and C, while stop-skipping train will only stop at stations A and C. In another word, station B is skipped by the stop-skipping train. An assumption is made that the passengers detained on the platform should be limited not to enter the station to ensure their safety, which means they are treated as limited passengers. The scheme results are as following Table 1. Compared with scenario 1, total limited passenger volume is the same as in scenario 2. While in scenario 1, only some passengers in station C are controlled and the volume is 6; in scenario 2, station B shares some limited passengers and limited passenger volume in station C decreases to 2. The combination of stop-skipping and boarding limiting strategies relieves the pressure of passenger crowding in station C and helps optimizing the overall condition of the zone, which are exactly the goals of metro passenger flow control.

In the above-mentioned case, the definition of metro passenger flow control is confirmed. Metro passenger flow control is the mixture of train stopping-station scheme, which dynamically assign train capacity to station to station, and boarding limiting, which includes control of inbound speed and inbound passenger volume to relieve passenger crowding pressure of key stations by station staff with the help of auxiliary facility like inbound passages in metro stations, where the large passenger volume is beyond station capacity in morning peak hours of workdays. The train stopping-station and boarding limiting strategies are in the views of capacity supply adjustment and passenger demand control, respectively, to realize metro passenger flow control.

Fig. 1 Illustration of station-to-station transportation system in a single line

Fig. 2 Comparison of boarding limiting schemes before and after stop-skipping strategy

Table 1 Limited passenger volume in stations B and C

3 Metro passenger flow control model

This chapter builds a metro passenger flow control model. First, a passenger original station choice model is proposed to determine the distribution of inbound passenger volume. Then, metro passenger flow control is realized by boarding limiting and stop-skipping strategies. Furthermore, feedback of these two strategies to passenger original station choice is considered. Finally, the optimal metro passenger flow control scheme is made.

3.1 Assumptions

1) In passenger flow control scheme, only inbound passengers and transfer passengers who transfer from other lines to the objective line are considered. And for simplification, inbound passengers and these transfer passengers are treated equally.

2) Train stopping-station adjustment refers to stop- skipping trains added to normal trains at certain rates.

3.2 Constraints

Influencing factors of passenger original station choice include waiting time, travel time, ticket fare, distance from home to original station and so on [10-16]. Probabilities of passengers choosing each original station are got by adding these factors with weights, shown as

             (1)

In Eq. (1), n is the number of stations, n=1, 2, …, N; P_n is the probability of passengers choosing station n as an original station; A_n is the choice set of station n; θ=[θ₁, θ₂, …, θ_J] is the vector of unknown parameters; θ′ is the transposition of θ; X_i=[x_i1, x_ij, …, x_iJ] is the feature vector of station n.

The above probability will decide passenger volume waiting outside stations [17, 18], as shown in Eq. (2):

                           (2)

In Eq. (2), k is the number of trains, k=1, 2, …, K;  is the passenger volume waiting outside station n during the period of train k-1 leaves and train k arrives; B^w is the total inbound passenger demand during headway time, which is the given total inbound passenger volume.

Process of passenger entering a station is related to dynamic change of passenger volume inside the station. When much too many passengers gather inside and the volume is beyond station capacity, passengers outside cannot keep entering the station. And when train available capacity can satisfy the passengers waiting on the platform, all of them succeed in boarding the train. Otherwise, some have to be detained and wait for another train [19-21]. These constrains of limited passengers are as following Eq. (3)-(5):

                             (3)

             (4)

         (5)

where  is the passenger volume who arrives at the station entrance and enter station n before train k arrives; w_k,n is the passenger volume who is limited not to enter station n before train k arrives; b_k,n is the passenger volume who boards train k after waiting for it on the platform in station n;  is the passenger volume who is detained on the platform after train k leaves station n; s_k,n is train available capacity when train k arrives at station n.

Process of passengers waiting on the platform and boarding a train is described by passenger location (on the platform or inside a train) and state (waiting, boarding, alighting or in a train). When a train arrives at a station, alighting passenger volume is at the ratio of alighting passenger volume to passenger volume inside the train. These constrains are as follows:

                             (6)

                    (7)

                                   (8)

                        (9)

where a_k,n is the alighting passenger volume in train k when it arrives at station n; is the passenger volume inside train k before it arrives at station n;  is the alighting rate of train k at station n; η^max is the maximal loading rate; TC is the train capacity;  is the passenger volume inside train k when it departs from station n.

Process of train operation and dwelling is determined by departure time, arrival time and dwell time of each train at each station and running time in sections. Trains depart from original station at the same headway. And adjacent running trains satisfy the requirement of minimum headway. These constraints are as follows:

                              (10)

                           (11)

                              (12)

where  is the departure time of train k at station n;  is the arrival time of train k at station n;  is the dwell time of train k at station n;  is the section running time of train k from station n-1 to station n, including pure running time in the section and added time because of starting and stopping processes; I is the minimal headway of departure trains.

These constrains are all linear. They cover passenger original station choice behavior, passenger entering a station, waiting for and boarding a train and train operation processes.

3.3 Objective function

Utilization rates of stations in a line in morning peak hours vary greatly and passenger crowding differs in each station. Importance of stations is distinguished by giving their passengers different weights in the objective function. Also ticket fare is added. The profit of passengers who board the first train arriving at the station is

              (13)

where j is any station number of following stations when train k arrives at station n;  is the ticket fare of OD (n, j) according to its mileage;  is the ratio of passengers whose OD is (n, j) to passengers boarding train k at station n; γ_n is the importance weight of station n.

At the same time, due to boarding limiting strategy, some passengers are limited and forbidden to enter stations for now. And thus, their waiting time outside stations is increased and efficiency of boarding a train decreased [22-24]. So, punishment for their waste time is added to the objective function by punishment parameter as

           (14)

where φ is the punishment coefficient of limited passengers.

The final objective function is comprehensive consideration of weighted profit of boarding passengers and limited passengers as

                        (15)

The objective is linear and considers comprehensive profit of both boarding passengers and limited passengers.

4 Algorithm

Combined with the model features, a metro passenger flow control solving algorithm based on stopping-station choice and train headway adjustment is designed. In this algorithm, how to choose the stopping stations in the stop-skipping strategy and how to adjust train headway are considered and then influence of metro passenger flow control on passenger original station choice is added to the passenger flow distribution model. The relevant technologies and algorithms are as follows.

4.1 Two key technologies of stop-skipping strategy

Overtaking operations in stop-skipping strategy is not incorporated in our consideration and a train will skip some stations and keep safe headway with both trains before and after it. There are two crucial technologies in the algorithm, which are stopping-station choice algorithm and headway adjustment algorithm. The steps for these two vital technologies will be described in detail as follows.

4.1.1 Stopping-station choice algorithm

Stop-skipping strategy dynamically assigns train capacity supply by adding stop-skipping trains to normal trains at certain rate and determining which stations should be stopped at. The steps of stopping-station choice algorithm are as following:

First, compare inbound and outbound passenger volumes in each station on the line and sort them. And then, put weights to the two orders and analyze them with passenger flow features in each station. Certain rate (like 60%) of the final order is chosen to determine the stopping stations, while other stations will be skipped. There is urgent need for train dwelling and passenger boarding and alighting trains in the stations with large inbound and outbound passenger volume. If the need cannot be satisfied in time, there will be serious problem in the station.

Stop-skipping trains mainly realize dynamic distribution of train capacity in each section. But it also decreases total conveyed passenger volume at certain degree. So for operability, rate of normal trains to stop- skipping trains is set m:1, whereand The scheme number of stop-skipping strategy is

4.1.2 Train headway adjustment algorithm

When headway of passing time of stop-skipping train at non-stopping-station and departure time or arrival time of its former or latter train is not satisfied, arrival time or departure time of the objective train or the following trains needs to be adjusted. Detailed steps of train headway adjustment algorithm are as following:

If station n-1 is to be skipped by stopping-skipping train k, its dwell time is cancelled, which is

Step 1: Judge if the headway of arrival time of train n at station k and departure time of train k-1 at station n satisfies the minimum headway between departure and arrival time (I^fd). If not, i.e.  turn to Step 2; otherwise, the adjustment ends.

Step 2: Update departure time of train k at station n-1 and n-2. Set  then  , and turn to Step 3.

Step 3: Take a look at if station n-2 is the original station. If so, turn to Step 4; otherwise, turn to Step 6.

Step 4: Make a decision if headway of train k and train k+1 satisfies the minimal departure time headway (I^f). If not, i.e.  turn to Step 5; otherwise, the adjustment ends (see Fig. 3).

Step 5: From train k+1 on, postpone the following trains at timeand the adjustment ends.

Step 6: Decide if the headway of departure time of train k at station n-2 and arrival time of train k+1 at station n-2 satisfies the minimal headway between departure time and arrival time. If not, i.e.  turn to Step 7; otherwise, the adjustment ends.

Fig. 3 An illustration of train headway adjustment

Step 7: Renew the departure time of train k+1 at station n-3 and set  then  Make n=n-1, turn to Step 3.

4.2 Solving algorithm of metro passenger flow control model

Combined with the above technologies, the chart flow of the solving algorithm of metro passenger flow control model is as in Fig. 4. Detailed steps are as follows:

Step 1: In the module of original station choice, total inbound passenger volume is given. Passenger original stations are determined by original station probability. Turn to Step 2.

Step 2: Perform the stopping-station choice algorithm and set ratio of normal trains to stop-skipping trains, which gives train capacity supply and total number of stop-skipping schemes (N). From n=1 on, judge if headway requirements of each train with adjacent trains are satisfied. If not, perform train headway adjacent algorithm and turn to Step 3; otherwise, turn to Step 3 immediately.

Step 3: Use the metro passenger flow control model to realize processes of train running, passenger entering and leaving stations and boarding and alighting from trains. Then passenger flow control schemes and relevant weighted profit Z_n are got at each stop-skipping strategy scenario. Then, decide if Z_n needs to be updated and make n=n+1. Repeat these processes until all stop-skipping strategy scenarios are calculated. And the maximal weighted profit Z_n is got and turn to Step 4.

Step 4: Stop-skipping and boarding limiting schemes will change influencing factors of passenger original station choice, such as waiting time and crowding degree. So probability of passenger choosing each station as an original station will change and passenger flow distribution differs under given total inbound passenger volume. And thus passenger original station choice behavior gets modified after initial metro passenger flow control scheme is made. Then, repeat Step 1-Step 3. In Step 3, when the maximal weighted profit Z_max is got, the algorithm ends.

5 Case study

A real case study in a morning peak hour is chosen to verify the validity of the model and algorithm. In this case, a west zone of Line 13 in Beijing metro in the up direction is analyzed, which incorporates five stations:

Huoying (HY), Huilongguan (HLG), Longze (LZ), Xierqi (XEQ) and Shangdi (SD). The case study will be described in detail as the structure of parameter setting, scenario design, results and analysis in this section.

Fig. 4 Flow chart of metro passenger flow control model

5.1 Parameters

Of the five stations, HY and XEQ are transfer stations, where most passengers are transfer passengers who transfer from another line to this line in the up direction. Here, these transfer passengers are treated equally to inbound passengers. Probability of passenger choosing each station as an original station and passenger alighting rates at each station are as following

Table 2, where choice probability is determined by adding the five weighted influencing factors and then inbound passenger volume at each station can be calculated.

Furthermore, parameters related to trains are set as following. Train capacity is 1464 persons. The maximal loading rate is 130%, which means there can be 1904 persons at most inside a train. Headway in the peak hour is 2 min 40 s.

Finally, parameters in the objective function are as following:

γ₁=1, γ₂=1, γ₃=0.5, γ₄=1, φ=0.5

5.2 Scenario design

In scenario design, the ratio of normal trains to stop-skipping trains need to be determined. Referring to real train operation data and considering train capacity decrease caused by stop-skipping trains in peak hours, the ratio is not proper to be set too high. So, two ratios are chosen as 2:1 and 3:1. And specially, 2:1 means two normal trains in a row and then a stop-skipping train, and 3:1 means three normal trains in a row and then a stop-skipping train.

As to determine the stations that need to be skipped, inbound, transfer and outbound passenger volume should be analyzed together and calculated as the following Table 3, where the data is accumulated in one hour.

First, the original station HY and the terminal station SD of the line are chosen to be stopping stations. Then sum of inbound and transfer passenger volume, and outbound volume of other stations are sorted and the top two-third of them are chosen as stopping stations. In both two orders, XEQ is the first and thus chosen as a stopping station. In the first order, HLG is the second and 4272 larger than the third, which is LZ. While in the second order, LZ is the second and 481 larger than the third, which is HLG. Analyzing the two orders together, LZ is chosen as the last stopping station.

There are three scenarios in the stopping-station strategy. Scenario 1 is that there are only normal trains running. Scenario 2 is that the ratio of normal trains to stop-skipping trains is 2:1. And scenario 3 is that the ratio of normal trains to stop-skipping trains is 3:1. And the stopping stations are HY, HLG, XEQ and SD.

5.3 Results and analysis

Initial data of passenger original station choice and relevant parameters are put into the solving program. Then the metro passenger flow control scheme and weighted profit are calculated as the following Fig. 5, where the data is got in one hour.

In scenario 3, the total limited passenger volume in stations HLG, LZ and XEQ is 30726 and its systematic weighted profit is the maximal 112701, which is the optimal metro passenger flow control scheme. So 3:1 is the best ratio of normal trains to stop-skipping trains.

Furthermore, due to the influence of stop-skipping and boarding limiting schemes, passenger original station choice probability is adjusted by the utility function. New distribution of original station choice probability and inbound passenger volume is calculated as Table 4.

The inbound passenger flow input is renewed and the weighted profit is calculated once more. The results are as the following Fig. 6.

In Fig. 6, after adjusting inbound passenger volume distribution, the final metro passenger flow control scheme is changed. Total limited passenger volume increases from 30726 to 30814 and weighted profit decreases from 112701 to 109995. Thus, it is necessary to modify passenger original station choice behavior.

This case study is an application of the metro passenger flow control model. The results show combination of boarding limiting and stop-skipping strategy relieves passenger crowding problems in key stations of a single line and optimizes the systematic profit. This is better than with the measures of passenger flow control in a single station, and thus it is effective to guide the work of passenger flow control in stations.

Table 2 Original station choice probability of each station

Table 3 Stopping-station confirmation according to inbound, transfer (from another line to objective line) and outbound passenger volume

Fig. 5 Solving results of three scenarios

Table 4 Original station choice probability and inbound passenger volume after being adjusted

Fig. 6 Solving results after updating inbound passenger volume in scenario 3

6 Concluding remarks

1) A dynamic metro passenger flow control model is proposed in the view of passenger demand and train capacity supply.

2) The innovations of the model are that boarding limiting and train operation organization are optimized at the same time and an original station choice model is used to modify the inbound passenger distribution.

3) In the research, dynamic propagation features of passenger flow on the line are considered and metro passenger flow control theory is built.

4) A real case of the model applied to Beijing metro verifies the validity.

5) Only metro passenger flow control problem in a single line is considered in this research, so the future direction will add multi lines and transfer passengers.

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(Edited by DENG Lü-xiang)

Cite this article as: JIANG Man, LI Hai-ying, XU Xin-yue, XU Shi-peng, MIAO Jian-rui. Metro passenger flow control with station-to-station cooperation based on stop-skipping and boarding limiting [J]. Journal of Central South University, 2017, 24(1): 236-244. DOI: 10.1007/s11771-017-3424-x.

Foundation item: Projects(RCS2015ZZ002, RCS2014ZT25) supported by State Key Laboratory of Rail Traffic Control & Safety, China; Project(2015RC058) supported by Beijing Jiaotong University, China

Received date: 2015-07-22; Accepted date: 2015-12-16

Corresponding author: LI Hai-ying, Professor, PhD; Tel: +86-10-51688063; E-mail: 13120855@bjtu.edu.cn