ZHENG Xun(郑勋) ,LI Hai-ying(李海鹰) ,MENG Ling-yun(孟令云) ,XU Xin-yue(许心越) ,CHEN Xu(陈旭)

1.State Key Laboratory of Rail Traffic Control and Safety (Beijing Jiaotong University),Beijing 100044,China;2.School of Traffic and Transportation,Beijing Jiaotong University,Beijing 100044,China

1 Introduction

With continuously growing urban population and increasing mass events,pedestrian simulation has attracted more and more people’s attention.Models of pedestrian behaviors have been proposed in the late 1950s.In recent decades,simulation models have provided important information in analyzing and designing railway or subway stations,airports,stadiums,cinemas,and other public facilities and places.Meanwhile,computer simulation is the most economical and feasible method to research the emergency situations,such as fire YANG et al [1].There are three types of simulation models:macroscopic models,medium models and microscopic models.Macroscopic models focus on the overall situations of the simulated scenes and base on mean values and their relationship,such as relationships between speed,density and flow.Medium models mainly focus on group characteristics of pedestrian flow.Microscopic models mainly research the individual behaviors of pedestrians,such as field of view,obstacle or pedestrian avoidance,path selection.Since the microscopic models can describe single pedestrian behaviors and the interaction among pedestrians,they are widely used today.Common models in pedestrian microscopic simulation are social force model,cellular automata model,multi-agent model and hybrid model.

Social force model is one of the famous microscopic pedestrian simulation models (e.g.HELBING and MOLNAR [2]and HELBING et al [3]).The model assumes a mixture of psychological and physical forces influencing the behaviors of pedestrians in a crowd.The concept of social force doesn’t refer to real existent force in physics,but in a virtual manner represents interaction among pedestrian with other pedestrians and environment.Researchers have modified the model so that the movement of pedestrians can be more realistic.LAKOBA et al [4]explored a range of numerical values of parameters of the social force model and proposed some modifications to produce reasonable behaviors of an isolated pedestrian or a small number of pedestrians while maintaining the realism for simulating larger crowds.PARISI et al [5]modified the model by introducing a“self-stopping”mechanism to prevent pedestrian from pushing over other pedestrians in simulation process,and the simulation results showed that it could match the empirical data in normal conditions well.ZANLUNGO et al [6]introduced a modified social force model in which pedestrians can explicitly predict the time and place of the next collision in order to collision avoidance.MOUSSAID et al [7]analyzed the organization of pedestrian social behavior by collecting empirical data of the motion of pedestrian group,and developed an individual-based model which can describe how a pedestrian interacts with other members in the same group and with the other group members.MA and WANG [8]researched the relationship between the view radius and the evacuation time.

Exit selection is one of the most important issues of simulation models.YUAN and TAN [9]took spatial distance and occupant density into account in order to simulate evacuation from a room with multiple exits.GUO and HUANG [10]proposed a logit-based exit selection model of evacuation in rooms with internal obstacles and multiple exits,and HUANG and GAO [11]proposed a modified floor field model that could calculate the static floor field for every lattice site and put forward a logit-based discrete choice principle to formulate the exit selection behavior.LO et al [12]proposed a game theory based on exit selection model for pedestrian evacuation.LOVREGLIO et al [13]improved the exit selection model based on the random utility models (RUMs) approach during an emergency situation.HELIÖVAARA et al [14]took an experimental study about pedestrian behavior and exit selection in evacuation of a corridor,and the results showed that the egress time would be shorter when the pedestrians behave selfishly instead of cooperatively.

In recent years,many people are migrating into cities.Subways have playing an important role in the urban transportation systems.Some researchers have developed some pedestrian simulation models in subway station.LENG et al [15]presented an improved floor field (FF) model based on local view,and in order to understand the improved model,a simulation scenario was presented on a virtual MTR station.A simulation of pedestrians’ evacuation from a huge transit terminal subway station presented in Ref.[16].LI et al [17]introduced a pedestrian estimation method based on gray scale edge histogram.A modification method based on the social force was introduced in Ref.[18],and it was applied for modeling the guided pedestrian group.

Currently,there are many simulation softwares[19-20],such as Anylogic,Simwalk,SimPed,NOMAD,Legion,STEPS.These softwares have different charateristics,and their own specific application range.Pedestrians in subway station have some special characteristics such as explicit destinations,different familiarities with subway station.So,an improved social force model based on exit selection is proposed to simulate pedestrians’ microscopic behaviors in subway station.Three contributions to simulation of the microscopic behaviors of passengers in subway station are offered:1) Except for spatial distance,exit efficiency which combines the occupant density and exit width is proposed in the improved social force model.2) The problem of pedestrians selecting exit frequently is solved as follows:not changing to other exits in the affected area of one exit,using the probability of remaining preceding exit (SHI et al [21]) and invoking function of exit selection after several simulation steps.3) On the basis of appropriately simplified pedestrian behaviors and station facilities,some assumptions and approaches are proposed,and then passenger flow simulation in Beijing Zoo Subway Station is presented with the C#programming language.The effectiveness of simulation results is verified by comparing and analyzing the actual data and simulation data.

2 Modeling

In this section,a brief review of the original social force model is introduced.Then,the improved social force model based on exit selection is presented,and the problem of pedestrian selecting exit frequently is solved by some methods.What’s more,a simulation experiment to verify the validity of the improved model is designed.

2.1 Original social force model

First of all,a brief review of the original social force model was provided [22].The social force model for pedestrian flow was proposed by HELBING and MOLNAR [2]first.Another social force model that simulated panic situations was put forward in HELBING et al [3].Pedestrians are normally driven by three forces:desire forceFiDfrom pedestrian himself or herself,“interaction forces”fijandfiWfrom other pedestriansjand wallsW,respectively.The corresponding equations are given by

wheremiis the mass of the pedestriani;is the desired speed;is the unit vector pointing to the desired target;viis the actual velocity;τis a certain characteristic time related to the relaxation time of the pedestrian to achieve

The forceFijexerted on pedestrianiby pedestrianjhas the following form as

where,the first term of the equation’s right hand side describes social force between pedestrianiand pedestrianj;AiandBiare constants which represent the strength and range of social interaction,respectively;rij=(ri+rj) is the sum of their radiiriandrj;dij=||ri-rj||denotes the distance between their centers of mass;is the normalized vector pointing from pedestrianjtoi.The second term of the equation’s right hand side describes the physical force between pedestrianiand pedestrianj.If pedestrian collision occurs,two additional forces will appear,namely a“body force”,k(rij-dij)nij,counteracting body compression and a“sliding friction force”,κ(ri j-dij)tij,impeding relative tangential motion.tij=)represents the tangential direction and=(v j-v i)·tijrepresents the tangential velocity difference,whilekandκare large constants.At last,the functiong(x) is zero if pedestrians do not touch each other;otherwise,it is equal to the argumentx.

The interaction to wallWis treated analogously,which is given by

2.2 Modifications to social force model based on exit selection

An improved method on the social force is used as follows:People in front of the current pedestrian often have larger influence than those behind,so the vision field factor is considered by HELBING et al [23].

2.2.1 Method of exit selection considering three factors

A considerable number of existing methods solving the problem of exit selection are used for cellular automata models.Some new improvements on the basis of the modified model about exit selection by YUAN and TAN [9]are presented,which is used for social force model.

Exit width plays a significant role when pedestrian selects exits,so apart from spatial distance and occupant density,exit width is also taken into account.Improved model is shown as follows.

As shown in Fig.1,limeans the distance from the pedestrian to exiti(i=1,2,…,N).The large semicircle(the light blue one) surrounding an exit is called exit area.We introduce a concept of exit efficiencyρi,which is equal to the number of pedestrians within the exit area divided by exit width.

Fig.1 Social force model of a room with multiple exits

In order to select one exit to move towards,the pedestrian has to consider these factors,namely spatial distance,exit efficiency (occupant density and exit width).

If a pedestrian only considers spatial distance,the probability of exiti(i=1,2,…,N) to be selected is given byPi-l:

whereliis the distance between the pedestrian and the exitiandclis a constant that is used to adjust the spatial distance effect.Equation (5) is easy to show that:

Then,pedestrian considers occupant density and exit width together.ρODmeans the number of occupants within the exit area,andaexitis the exit width.So,ρiis assumed to denote exit efficiency of exiti,which is equal toρODdivided byaexit.The probability of exiti(i=1,2,…,N) to be selected is given byPi-l:

wherecρis a constant that is used to adjust the sensitivity of exit efficiency.Equation (7) is also can be proved.

At last,the interaction betweenPi-l(spatial) andPi-ρ(exit efficiency) is considered.In the social force model,the probability of selecting exiti(i=1,2,…,N) is given as follows:

wherecαandcβare scalar constants that represent the relative importance of spatial distance and exit efficiency,respectively.

2.2.2 Further optimization of exit selection method

The improved method about exit selection through a combination of three factors has been proposed in the above section.However,there is a common problem when pedestrian selecting an exit,that is,pedestrians may select exit frequently,which is not consistent with reality.To solve the problem,three feasible methods are proposed:

1) Keeping the exit instead of changing to other exits in the affected area of one exit.As shown in Fig.1,the small semicircle (the dark blue one) surrounding an exit is called area of prohibiting replacing the exit,which means that pedestrians can not change the exit in the area.

2) The probability to maintain the preceding exit[21].Though pedestrians meet the requirement of replacing exits,they have the probability to maintain the previous exit,which can prevent pedestrians from changing the target exit frequently.

3) In simulation process,the function of exit selection is invoked after several simulation steps.

2.2.3 Model validation

A simulation experiment to verify the validity of the improved model is designed.There are three exits in the room in Fig.2.The length and width of the room are 15 m and 20 m,respectively,and the widths of exits from top to bottom are 1.5,1 and 2 m,respectively.Pedestrians are uniformly generated on the left of the room,and they’ll leave the room from one of the three exits.Finally,the number of pedestrians in front rectangular area (6 m×6 m) of each exit is calculated.

Figure 2 shows the results of original exit selection model and improved exit selection model.In original model (see Fig.2(a)),lots of pedestrians gather at the middle narrowest exit,which is unreasonable in reality.In improved model (see Fig.2(b)),pedestrians in the middle of the room are more willing to select the top or bottom exit rather than the middle exit,even though the occupant density of the middle exit is relatively small.It is because the width of the middle exit is small and the exit efficiency is relatively small.

To further illustrate this problem,the numbers of pedestrians in the rectangular area at two different simulation steps are compared,and the result is shown in Table 1.These exits are considered three parallel service desks.If service desk has larger passing capability,more people will select this service desk to receive service.At simulation step=1432,there is 11 pedestrians in gray rectangular area in original model and the number of pedestrians is 7 in improved model.Similarly,at simulation step=3351,the number of pedestrians is 17 in original model and 7 in improved model.It shows that the number of pedestrians in the rectangular area of the middle exit in the original model is more than the number of pedestrians in the improved model.In improved model,pedestrians select the exits more reasonably.The result shows that the improved social force model based on exit selection is feasible.

Fig.2 Simulation comparison on exit selection Based on original model at simulation step 26 (a1),1432 (a2) and 3351 (a3),and based on improved model at simulation step 24 (b1),1432 (b2) and 3351 (b3) (The gray line denotes the rectangular area in front of each exit)

Table 1 Number of pedestrians in front rectangular area of each exit at different simulation steps

3 Pedestrian microscopic simulation in subway station

In the previous section,the improved social force based on exit selection was proposed and the validity of the model was verified.There are many pedestrians’ exit selection behaviors in passenger transport terminal and station,so Beijing Zoo Subway Station is taken as an example to realize the pedestrian simulation in complex environment.

3.1 Assumption

Pedestrians in subway station have some characteristics:Pedestrians have explicit destinations;Passenger flow is uneven.There are generally early peak and late peak passenger flow.Pedestrians have different familiarities with subway station.

Station environment and pedestrian behaviors are much too complex in reality,so some simplifications and necessary instructions are proposed as follows.

1) The selection of automatic fare collection (AFC)is a special exit selection.There are two characteristics of these special exits:one is that these AFC(s) are too close;another is that only one person can go through the AFC every time.Two methods are proposed to solve the problem of special exit selection as follows.

(1) If the AFC is occupied by one pedestrian,the extra occupants (e.g.10 occupants) add toρOD,which denotes the number of occupants within the exit area.By this means,it can prevent other pedestrians from selecting the exit as much as possible.

(2) Other pedestrians may be pushed by the crowd into the occupied AFC,so the resistance coming from the AFC that preventing pedestrians from entering the occupied AFC should be increased.However,due to the increasing resistance and relatively small driving force coming from pedestrian himself or herself,pedestrian can not enter the free AFC.Attractive force coming from the free AFC is introduced.

2) When pedestrians generate,they have clear macro paths.There are two types of macro paths in intermediate station:some pedestrians coming from the entrances go through AFC,station hall and station platform in turn,and they get on the train when the train comes.Other pedestrians coming from the train go through station platform,station hall and ticket gates in sequence,and they leave the station finally.There is a principle that pedestrians get off the train first and then others get on the train when the train arrives at the station.

3) Simplify station facilities.There is no safety check,ticket lobby and other complicated facilities.

4) Pedestrians have different radii,masses and desired speeds.These parameters are randomly generated according to the Gaussian distribution.In addition,the desired speeds are different when the pedestrians go upstairs or go downstairs,and the pedestrians have uniform motion on the escalator.

Some parameters of the model are specified in Table 2.

Table 2 Parameters of improved social force model

3.2 Pedestrian simulation in subway station and validation

Beijing Zoo Subway Station is one of the intermediate stations in Beijing subway network system.Sketch map of Beijing Zoo Subway Station is shown in Fig.3.There are five exits,one station hall and one station platform.The train that leaves for the direction of Anheqiao North is assumed to up train,and similarly,the train leaving for the direction of Tiangongyuan is down train.

Fig.3 Sketch map of Beijing Zoo Subway Station

The C# programming language was used to realize the simulation system.The simulation environment of station facilities has the same proportion as real subway station.Pedestrian simulation in Beijing Zoo Subway Station is shown Fig.4.The simulation results show that the improved model can depict pedestrians’ microscopic behaviors in subway station.

An experiment is designed to verify the simulation result.The selection of AFC is an important node selection in the subway station simulation.The entrance AFC for entering the station as an example for analyzing the problem is researched.A real investigation in evening peak is conducted to obtain the actual data,because the actual traffic data can be used in the simulation system.And then,the actual data and simulation data are compared and analyzed to verify the effectiveness of the simulation results.

In Fig.3,there are 12 AFC(s) at the bottom of the station hall for pedestrians entering the station.On the day when we surveyed the pedestrian data,only eight AFC(s) on the left side were open.From left to right,the number of every AFC was from 1 to 8.The number of pedestrians passing every AFC in 20 min and the percentages of pedestrians selecting every AFC are listed in Table 3.

The number of pedestrians to reach AFC in unit time is considered consistent with the actual survey data.Then simulation experiment about the percentage of pedestrians selecting every AFC is conducted in ten groups,and the numbers of simulation pedestrians are selected from 100 to 1000.In order to guarantee the accuracy of the data,every group is measured three times.And then,compared with realistic data,mean absolute percent error (MAPE) of every AFC in every group data is calculated.ui(i=1,2,…,8) is the MAPE of pedestrians selecting thei-th AFC in every group and the equation is given as follows:

wheren1is the sample size of simulation in every group,son1=3 in this simulation experiment;(j=1,2,…,n1)is percentage of pedestrians selecting thei-th AFC in thej-th simulation sample and(i=1,2,…,8) is percentage of pedestrians selectingi-th AFC in survey data.

Finally,yis the weighted-average of MAPE in every group,which is given by

wheren2is the number of AFC(s) in this simulation experiment,son2=8;ωi(i=1,2,…,n2) is the weight of thei-th AFC,and it is equal to ˆ.ixThe MAPE and weighted-average of MAPE are shown in Table 4.

Fig.4 Pedestrian simulation in Beijing Zoo Subway Station (Blue circles mean pedestrians who have got in station already and want to get on subway,and red circles denote pedestrians who have got off subway already and want to get out station):(a) Layer of station hall;(b) Layer of station platform

Table 3 Survey data of pedestrian selecting every AFC

Table 4 MAPE and weighted-average of MAPE in simulation

Fig.5 Change rule of weighted-average of MAPE with changing number of pedestrians

In Fig.5,it shows the change rule of weightedaverage of MAPE with increasing the number of the pedestrians.In peak hours,the weighted-average of MAPE is around 8%,which means that the effectiveness of simulation is good.Meanwhile,the possible source of weighted-average of MAPE is analyzed.On one hand,there are finiteness and error with the actual data;on the other hand,the model has certain defects and the pedestrian characteristics are more complicated in reality.

4 Conclusions

1) To simulate pedestrians’ microscopic behaviors in subway station,an improved social force model based on exit selection is proposed.Except for spatial distance,exit efficiency which combines the occupant density and exit width is proposed in the improved social force model.In order to solve the problem of pedestrians selecting exits frequently,three methods are introduced,including keeping the exit instead of changing to other exits in the affected area of one exit,using the probability of remaining preceding exit and invoking function of exit selection after several simulation steps.Finally,a simulation experiment is designed to verify the validity of the improved model.The result shows that the improved model based on exit selection is feasible.

2) Station environment and pedestrian behaviors are too complex in reality,so some assumptions and approaches are proposed on the basis of appropriately simplified station facilities and pedestrian behaviors.Beijing Zoo Subway Station is taken as an example.Finally,the effectiveness of pedestrian simulation in subway station is verified by comparing and analyzing the actual data and simulation data.

3) The improved model can simulate some complex environment,such as subway station,but the model also has some deficiencies.There are three prospects for future work.Firstly,the example in Section 2 has verified the validity of the improved model,such as the factor of exit width.These improvements are considered applicable in pedestrian simulation in subway station.So in future work,this problem needs further research and validation.Secondly,the exit area and the area of prohibiting replacing the exit are influenced by some factors,such as the exit width,pedestrian’s density in current condition,pedestrian’s preference about exit selection.So,the calibration of these parameters will continue to research in future work.Thirdly,pedestrian properties have significant impact on the simulation model,such as familiarities with subway station.More station facilities,pedestrian behaviors and properties will be added to the simulation system.

[1]YANG Gao-shang,PENG Li-min,ZHANG Jing-hua,AN Yong-lin.Simulation of people’s evacuation in tunnel fire [J].Journal of Central South University of Technology,2006,13(3):307-312.

[2]HELBING D,MOLNAR P.Social force model for pedestrian dynamics [J].Physical Review E,1995,51(5):4282-4286.

[3]HELBING D,FARKAS I,VICSEK T.Simulating dynamical features of escape panic [J].Nature,2000,407(6803):487-490.

[4]LAKOBA T I,KAUP D J,FINKELSTEIN N M.Modifications of the Helbing-Molnar-Farkas-Vicsek social force model for pedestrian evolution [J].Simulation,2005,81(5):339-352.

[5]PARISI D R,GILMAN M,MOLDOVAN H.A modification of the social force model can reproduce experimental data of pedestrian flows in normal conditions [J].Physica A:Statistical Mechanics and its Applications,2009,388(17):3600-3608.

[6]ZANLUNGO F,IKEDA T,KANDA T.Social force model with explicit collision prediction [J].EPL (Europhysics Letters),2011,93(6):68005.

[7]MOUSSAID M,PEROZO N,GARNIER S,HELBING D,THERAULAZ G.The walking behaviour of pedestrian social groups and its impact on crowd dynamics [J].PLOS ONE,2010,5(4):e10047.

[8]MA Pei-jie,WANG Bing-hong.The escape of pedestrians with view radius [J].Physica A:Statistical Mechanics and its Applications,2013,392(1):215-220.

[9]YUAN W,TAN K H.An evacuation model using cellular automata[J].Physica A:Statistical Mechanics and its Applications,2007,384(2):549-566.

[10]GUO Ren-yong,HUANG Hai-jun.Logit-based exit choice model of evacuation in rooms with internal obstacles and multiple exits[J].Chinese Physics B,2010,19(3):030501.

[11]HUANG Hai-jun,GUO Ren-yong.Static floor field and exit choice for pedestrian evacuation in rooms with internal obstacles and multiple exits [J].Physical Review E,2008,78(2):021131.

[12]LO S M,HUANG H C,WANG P,YUEN K K.A game theory based exit selection model for evacuation [J].Fire Safety Journal,2006,41(5):364-369.

[13]LOVREGLIO R,BORRI D,DELL’OLIO L,IBEAS A.A discrete choice model based on random utilities for exit choice in emergency evacuations [J].Safety Science,2014,62:418-426.

[14]HELIÖVAARA S,KUUSINEN J M,RINNE T,KORHONEN T,EHTAMO H.Pedestrian behavior and exit selection in evacuation of a corridor—An experimental study [J].Safety Science,2012,50(2):221-227.

[15]LENG Biao,WANG Jian-yuan,ZHAO Xiao-xi,FANG Jun,XIONG Zhang.Out flow in a virtual MTR station using a local view floor field model [J].International Journal of Modern Physics C,2013,24(6):1350037.

[16]LEI Wen-jun,LI An-gui,GAO Ran,HAO Xin-peng,DENG Bao-shun.Simulation of pedestrian crowds’ evacuation in a huge transit terminal subway station [J].Physica A:Statistical Mechanics and its Applications,2012,391(22):5355-5365.

[17]LI De-wei,ZHANG Lin,SUN Yu-xing.Crowd estimation using image processing based on gray scale edge histogram in subway station of Beijing [C]// Measuring Technology and Mechatronics Automation (ICMTMA),2013 Fifth International Conference on Mechatronics Automation.Hong Kong.IEEE,2013:415-419.

[18]YANG Xiao-xia,DONG Hai-rong,WANG Qian-ling,CHEN Yao,HU Xiao-ming.Guided crowd dynamics via modified social force model [J].Physica A:Statistical Mechanics and its Applications,2014,411:63-73.

[19]HOOGENDOORN S P,BOVY P H L.Pedestrian route-choice and activity scheduling theory and models [J].Transportation Research Part B,2004,38(2):169-190.

[20]YANG Ye-di,LI Jin,ZHAO Qun-xin.Study on passenger flow simulation in urban subway station based on anylogic [J].Journal of Software,2014,9(1):140-146.

[21]SHI Zheng-wei,CHEN Zhi-ya,ZHOU Le,LING Jing-wen.A cellular automata model of pedestrian evacuation under multi-exit conditions [J].Systems Engineering,2010,28(9):51-56.(in Chinese)

[22]YUEN J K K,LEE E W M.The effect of overtaking behavior on unidirectional pedestrian flow [J].Safety Science,2012,50(8):1704-1714.

[23]HELBING D,FARKAS I J,MOLNAR P,VICSEK T.Simulation of pedestrian crowds in normal and evacuation situations [J].Pedestrian and Evacuation Dynamics,2002,21(2):21-58.