MODELLING A TRANSIT STATION WITHIN A TRANSIT ASSIGNMENT MODEL: A SYSTEMIC APPROACH

Transcription

1 MODELLING A TRANSIT STATION WITHIN A TRANSIT ASSIGNMENT MODEL: A SYSTEMIC APPROACH EKTORAS CHANDAKAS, UNIVERSITÉ PARIS-EST, LVMT, EKTORAS.CHANDAKAS@ENPC.FR FABIEN LEURENT, UNIVERSITÉ PARIS-EST, LVMT, FABIEN.LEURENT@ENPC.FR This is n bridged version of the pper presented t the conference. The full version is being submitted elsewhere. Detils on the full pper cn be obtined from the uthor.

2 MODELLING A TRANSIT STATION WITHIN A TRANSIT ASSIGNMENT MODEL: A SYSTEMIC APPROACH Ektors Chndks, Université Pris-Est, LVMT, ektors.chndks@enpc.fr Fbien Leurent, Université Pris-Est, LVMT, fbien.leurent@enpc.fr ABSTRACT A model is provided to cpture the cpcity phenomen occurring in sttion for the pssenger trffic ssignment to trnsit network. These pertin to the circultion cpcity for pssengers of the horizontl nd verticl elements of the trnsit sttion. If the circultion within the sttion is isolted nd the sttion pltforms nd sttion entrnces ct s the pssengers entry nd exit points of the sttion system, verge circultion conditions my be clculted for ech origin destintion trip inside the sttion. These conditions, long with the resulting trvel disutility my ffect the generl trvel conditions of trip nd the ssocited trvel disutility. At the network level, tht plys significnt role on the pth choice of the users within the trnsit system. A more relistic trnsit ssignment model coupled with the sttion sub-model is of vlue when estimting the trvel demnd. Along with the models deling with cpcity effects within trnsit line, the sttion model is developed to evlute the qulity of service offered nd to ssess the effect of the cpcity constrints to the generl trffic conditions. Keywords: Trffic Equilibrium, Bi-lyer Assignment, Sttion Cpcity, Trnsit Cpcity, Sttion Model INTRODUCTION In the trnsit network of lrge urbn res, it frequently occurs tht the trnsit system is submitted to hevy congestion t the pek hours on working dys, especilly so t the morning pek in the centrl prt of the urbn re. Along with the congestion on the trnsit services, relted to the vehicles or lines chrcteristics, congestion ppers t the trnsit sttions. Tht influences the in-sttion wlking time for the ccess-egress trips nd the trnsfers of the pssengers, hence the qulity of service of the trnsporttion system. The principl design constrint of trnsit sttion is the security evcution, especilly for underground fcilities in the cse of fire. Vrious sfety guidelines designte the evcution flows nd determine the pssenger behviour ssocited with the sfety evcution of trnsit sttion, such s the NFPA130 (NFPA, 2010) in the United Sttes nd IN1724 (SNCF- GI, 2002) in Frnce. The respect of sfety regultions is one of the resons of the development of micro nd mcro simultions models, such s Legion (Still, 2000), PEDROUTE (Buckmn nd Lether, 1994), NOMAD (Hoogendoorn, 2001) nd SimPed 1

3 (Dmen, 2002). They re often simultion-bsed models where the circultion or the storge of pssengers in the trnsit sttion is modelled into very high level of detil. The effect of the cpcity constrints in trnsit sttion is further discussed in the Trnsit Cpcity nd Qulity of Service Mnul (TRB, 2003). It is focused in the nominl opertion of the trnsit system nd cts s guideline for sttion design. It provides some opertionl chrcteristics of the sttion elements tht ct s reference vlues for sttion design nd suggests simple frmework for estimting the effect of pssenger flow. However, fundmentl issues such s the trnsfer for the trnsit system nd the circultion inside trnsit sttion re not given thorough ttention in trnsit plnning, in prt due to the bsence of efficient trnsit ssignment models. The scientific literture bosts vrious trnsit sttion models for the ssessment of the sttion design (Hoogendoorn et l, 2004) nd other opertionl purposes (Hrris, 1991). Other reserch proposes methods for the ssessment of the cost of the trnsfer in prticulr sttions (Guo nd Wilson, 2011). Nevertheless, these pproches tret trnsit sttion seprtely, independently from the trnsit system. The insttion pth flows re ssumed of fixed origin nd destintion nd the network effect is not tken into considertion. A pssenger flow ssignment on trnsit network focuses on the pth choice behviour in reltion to the qulity of the trnsit service. Recent models re cpble of efficiently cpturing the cpcity effects of the trnsit services on the pth choice (such s Ceped et l., 2006; Kuruchi et l., 2003; nd Leurent et l., 2011). Nevertheless, they omit the effect of the insttion pssenger flow on the qulity of trnsfers nd ccess-egress trips nd the flow ssignment on the network. This pper is purported to bridge the gp between the microscopic pedestrin simultion nd the mcroscopic trnsit ssignment model. A simple model for the trnsit sttion is developed, where the effect of the in-sttion pssenger flows on the in-sttion pth trvel time nd the qulity of circultion is tken into ccount. Tht model cn be integrted in trnsit ssignment model, such s CpTA, described in Leurent et l (2012). The rest of the pper is orgnised into four sections. First systemic nlysis of trnsit sttion is conducted in Section 2, where the sttion elements re described. Then, section 3 provides the definition of the network topology. The model ssumptions nd the formultion of the cost flow reltionships is outlined in section 4. Section 5 trets the rticultion of the trnsit sttion model to the network model. Section 6 dels with n ppliction instnce, dpted from the Ntion sttion in the centre of the Pris trnsit network. Lstly, Section 6 concludes bout the model contents nd some points of potentil developments. SYSTEMIC ANALYSIS OF A STATION The systemic nlysis is essentil s led-in to the modelling process. The boundries of the system re defined, long with the reltions to the externl elements nd the other systems. Furthermore, the elements of the system re described, long with their reltions. We perform systemic nlysis of the trnsit sttion, focusing on the prt of the sttion used by the pssengers nd the pssengers ctivities inside. First, n overview llows the reder to plce the trnsit sttion within the trnsit system nd focus on the functionlities involved nd the reltions mong the elements. Then the sttion elements re defined, first the 2

4 horizontl nd verticl elements nd then the sttion res. Finlly, the effect of the trnsit informtion on the qulity of service nd the pth choice is clrified. Overview of the trnsit sttion The systemic nlysis of the trnsit system performed in Leurent (2011) identifies trnsit sttion s seprte subsystem of the trnsporttion network. The sttion fulfils two primry functions with respect to the trnsit users. First, it cts s n interfce between the urbn re (city) nd the trnsit network (trnsit service) providing ccess to nd egress from the trnsit system. Second, when multiple trnsit lines serve trnsit sttion, connections between them re vilble vi the trnsit sttion for trnsfer, hence incresing the connectivity nd performnce of the system. Such functionlities imply strightforwrd nlogies with the role of the irport terminl to the commercil vition system. The boundries of the trnsit sttion subsystem re defined in rticultion with the multimodl trnsporttion system. Two input nd output sides re distinguished. On the city side, trnsit user cn ccess to (or egress from) the trnsit sttion by privte modes: wlking, using the pedestrin network, by privte cr, using the prk & ride (s driver) or the kiss & ride (s cr pssenger) fcilities vilble or by cycling, using bike shelter. On the trnsit service side, user bords on (or lights from) trnsit vehicle: bus vehicle t stop or ril vehicle t pltform. These correspond to the origins nd destintions of the trips within trnsit sttion. The min functionlities of the intermedite components of trnsit sttion re the ccommodtion, consolidtion nd dispersion of the pssenger flows, from the origin nd destintion points. Secondry functionlities of such sttion components re relted with the production of the trnsit service (eg. fre control brriers nd ticket mchines) nd the nontrnsit services to pssengers (eg. shops) s wy to improve the qulity of the service. Figure 1 Flowchrt of trnsit sttion 3

5 The flowchrt in Figure 1 illustrtes the trnsit sttion nd its components. The rounded corner rectngles correspond to the externl points of the trnsit sttion, or the origin nd the destintion of the pedestrin trips within the sttion, both city side nd trnsit service side. The rectngles correspond to the res of the trnsit sttion. Their min functionlity is the consolidtion nd dispersion of the pssenger flows nd the storge of pssengers stocks. These res my include vrious menities. Finlly, the links represent the liner circultion elements, both horizontl nd verticl. They hve n nlogue function nd the use of horizontl or verticl element depends on the sttion design. The trjectory of pssenger who ccesses trnsit lines origintes t the pedestrin network (wlking) nd the intermodl fcilities: prk & ride (s cr driver), kiss & ride (s cr pssenger) nd bike shelter (s cycler). Then, he enters the sttion nd circultes through the sttion concourse (free nd pid re) to rech the pltform. Some witing is needed before vehicle becomes vilble for bording. An egress trip is defined in n inverse wy. Similrly, trnsferring pssenger lighting t the stop or pltform circultes t the sttion concourse nd the horizontl nd verticl circultion elements to rech the preferred pltform. Some witing my be needed to bord the preferred vehicle. In the following prts we describe the components of the trnsit sttion. We first ddress the liner circultion elements horizontl nd verticl nd then the sttion res. The description of the liner circultion elements is relted to their functionlities, their geometric chrcteristics, the pssenger flow cpcity, the trvel time nd their prticulrities. The description of the sttion res involves their functionlities, their geometric forms nd their position within the opertion of the trnsit system. If not stted otherwise, the vlues used in the following section stem from the Trnsport Cpcity nd Qulity of Service Mnul (TRB, 2003). Horizontl Circultion Elements Wlkwy A wlkwy is liner circultion element connecting two entry/exit points, where usully the length of the elements is fr superior of its width. A smll rmp my exist, however the verticl displcement is lrgely inferior to the horizontl movement of the pssengers. The wlkwy my be used for single or double direction pssenger flow, ccording to the sttion design. The cpcity of wlkwy depends on the pedestrin wlking speed, their density nd chrcteristics nd the effective width of the wlkwy t its nrrowest point. According to TCQSM, 2003 its cpcity (LOS E) is locted t 82 p/m/min or equivlent to 4920 p/m/h. Its cpcity my be reduced for bi-directionl pssenger flow on the wlkwy ccording to the reltive rtio of these flows (Pushkrev nd Zupn, 1975). At free flow conditions pssenger my rech the mximum wlking speed on tht element, equivlent to 81 m/min or 4,87 km/h. A reduction up to 46 m/min (2,76 km/h) t cpcity cn be observed with the increse of pssenger flows. When the density of the wlking pssengers is greter thn the cpcity, the speed, s well s the flow, is reduced until reching hlt. 4

6 Moving Wlkwy A moving wlkwy resembles the functionlity of wlkwy for the circultion of pssenger flow. It offers mechnicl support for the pssenger horizontl movement, usully operting t lower thn the wlking speed, t 1,8-3 km/h. The pssenger speed depends on the pssenger flow, s suggested for the wlkwy, incresed by the operting speed of the moving wlkwy. The cpcity depends on the entry width, similrly to the escltor. A stndrd double width moving wlkwy cn trnsfer up to 5400 p/h. A moving wlkwy is operted t single direction. Nevertheless, for long corridors, vrious lyouts exist from single moving wlkwy operting t the direction of the dominnt flow or one or more for ech direction. Fre Control Brriers Fre control is n importnt function, linked to the performnce of the trnsit opertion. At trnsit sttion, the fre control divides the sttion to the controlled res (or pid res), linked to the trnsit services, nd the uncontrolled res (or free res) which provide the link to the city. In cse of different trnsit fres mong systems, dditionl fre control brriers my be plced between pid res inside the trnsit sttion. An elementry fre control brrier my hve predetermined usge for entry or exit of pssengers, or lternting the opertion direction ccording to the principl pssenger flow. The ltter cse mens tht the cpcity of group of fre control brriers my be dynmiclly llocted ccording to the chrcteristics of the flow. The operting mode of the fre control brriers resembles tht of toll plzs on motorwys. A service time my be defined for the elementry trnsction (vlidte nd cross). It depends on the pssenger chrcteristics nd the type of fre control mechnism nd vries from 0,5 to 4 seconds per elementry trnsction (TRB, 2003). The cpcity of group of fre control brriers depends on the service time nd the number of elementry fre control gtes used for ech direction. Sttion Entrnce The sttion entrnce is defined s the limit between the sttion exterior nd the interior of the sttion nd its min functionlity is to offer the sttion concourse protected environment from externl conditions. While for typicl ground trnsit sttion, such s rilwy sttion, the sttion entrnce cn be esily noticed, corresponding to the entrnce of the building, the concept is hrder to define for underground trnsit sttions. In tht cse, the verticl circultion elements ply the role of the sttion entrnce. In the sme wy to the fre control brriers, it constitutes n obstcle to the pssengers. Tht hs result to slow down the pssengers. Nevertheless, setting the entrnce elements to operte t predetermined directions cn enhnce fluidity by seprting conflicting pssenger flows. The cpcity of the elements depends on the type of mechnism used for seprtion of externl nd internl re simple, revolving or sliding door nd the pssge chrcteristics. 5

7 Verticl Circultion Elements Stirwy The stirwy is the min element for verticl circultion of the pssenger flow nd functionlly resembles wlkwy. It is used to connect res with height difference, such s different levels of trnsit sttion. Due to the higher effort needed thn wlking, the pssengers scend nd descend stirwy in slower speeds thn wlking. In ddition, the scending speed vries ccording to pssenger flow from 33m/min (or 1,98 km/h) to 20 m/min (or 1,2 km/h) nd it is usully lower thn the descending speed. According to the TCQSM (TRB, 2003), the cpcity of stirwy per unit of width, corresponding to the upper limit of the LOS E is 56 p/m/min, or 3360 p/m/h. If not considered otherwise, cendign nd descending pssengers my coincide t given stirwy. The produced friction cuses reduction of the stirwy s totl cpcity. Escltor The incresed effort needed for scending nd descending stirwy mkes the escltor n essentil element for the verticl circultion. It provides mechnicl ssistnce nd in tht wy increses the qulity of the circultion within the sttion nd the pssenger flow for the sme width. By equivlence to the moving wlkwy, it opertes in single direction, nd functionlly llows the seprtion of pssenger flows. The operting speed of n escltor vries from 27,4 m/min (1,6 km/h) to 36,6 m/min (2,2 km/h). Therefore, the speed of the pssenger flow depends on the operting speed of the escltor nd the individul pssenger on the escltor. Its cpcity depends on the entry width nd vries from 2700 p/h to 5400 p/h for single nd double width escltor respectively. The sttion design usully plces the escltors djcent to the stirwy. When instnt cpcity is not sufficient, queue is creted t the bse of the escltor. Assuming the pssengers wish to reduce their trvel time, pssenger will ccept mximum witing time. This is relted to the trvel time difference between the escltor nd the djcent stirwy. The pssengers t the end of the queue who perceive their witing time s greter thn tht trvel time difference will use the djcent stirwy. Elevtor An elevtor cn be used in two wys: either s primry verticl circultion element to ccess levels with n importnt height difference (such s in deep underground sttions) or s secondry element to fcilitte prt of the pssenger demnd which is excluded from other verticl circultion elements (such s persons with reduced mobility). The circultion speed nd cpcity is independent of the pssenger flow nd it is linked with the elevtor s chrcteristics. In fct, opertionlly it cn be distinguished from other circultion elements. By its opertionl chrcteristics it resembles trnsit service with verticl movement. It is chrcterized by discrete vilbility, nd witing time until it 6

8 becomes vilble, trvel time between stops nd person s cpcity of the vehicle (or cge). The elevtor used in trnsit service cn be ctegorized s fixed-route, demndresponsive trnsit service. Sttion Ares The Sttion Concourse The sttion concourse is defined s n re within the trnsit sttion with multiple locl origins nd destintions of wlking trjectories, cting s junction of different pths. The primry functionlities re the consolidtion nd dispersion of the pssenger flows nd the storge of pssengers stocks. The secondry functionlities re derived by the menities locted t the sttion concourses, which offer vrious services linked to the trnsit service (ex. ticket sle, informtion point, witing re) or not (newspper stnd, etc.). Fre control brriers re usully instlled t the sttion concourse, dding further distinction between uncontrolled (free) nd controlled (pid) re. In ddition, it gives ccess to non-public technicl res. The sttion menities induce high impct on the pssenger flow. On the one hnd, these menities need minimum spce for their opertion, limiting the re vilble for pssenger circultion nd slowing down the pssenger trffic. On the other, their positioning t the sttion concourse my led to more efficient pssenger trffic orgniztion with reduced conflicts nd hence improve the performnce of the fcility. The wlking speed is influenced by the pssenger density nd the friction with conflicting or djcent pssenger flows nd cn be relted to the number of the pssengers present per surfce re. The pssenger flow is usully orgnized in virtul directed corridors within the sttion re, whose width fluctutes ccording to the composition of the flows in the re. The Sttion Pltform The pltform is prticulr element of the trnsit system, cting s pivot point between the trnsit services nd the sttion. A sttion pltform s primry functionlities ly on the pssenger exchnge with the trnsit vehicles (for bording nd lighting), by wy of consolidting the pssenger flows nd bording, s well s lighting nd dispersing t the sttion. Furthermore, it serves s storge of pssengers tht wit for the trnsit service to rrive. Secondry functionlities re relted with improving the qulity of the witing with vrious menities. The form of the pltform resembles rectngle where the longer sides usully correspond to its input nd output points. The one side long the trcks in the cse of ril trffic is used for vehicle berthing, while the other offers connection by the circultion elements with the trnsit sttion vi entry nd exit points, s illustrted in Figure 2. When vehicle is present the pssengers bord nd light the vehicles. The pedestrins in the sttions nd the pssengers on the vehicles re chrcterized by different flow forms: continuous flow is ssocited with the in-sttion flow, while the on-bord pssengers re orgnised in pckets, due to the discrete form of the trnsit service. 7

9 Figure 2 Schemtic representtion of sttion pltform Witing occurs before bording, s result of to the trnsition between the pssenger flow forms. When the number of witing pssengers increses, the pssengers choose witing positions t the edge of the pltform. A wlking corridor is liberted t the rer fcilitting the ccess nd egress circultion on the sttion pltform. The witing t the pltform is ssocited with the storge function of the elements nd two forms my be distinguished. On the one hnd, pssengers witing t the trck-side of the pltform, ensure priority from lter rrivls to enter the vehicle nd occupy set if vilble. On the other, pssengers t witing re benefit from the sets vilble but my fce difficulties into ensuring better comfort stte on-bord. Therefore, witing trde-off ppers, between the witing comfort stte nd the in-vehicle comfort stte. The witing choice of n individul pssenger depends on the reltive time of the witing time nd the in-vehicle trvel time s well s the in-vehicle comfort conditions nd the probbility to ccess n in-vehicle comfort stte. To summrize the pltform functionlities, the bording pssengers use the circultion elements to ccess the pltform, suffer some witing nd when the pproprite vehicle rrives they bord. The witing pssengers re not uniformly distributed long the sttion pltform, but certin elements my ffect the distribution, such s obstcles, the position of the pltform entries nd the knowledge of the exct position of the doors when vehicle hs stopped either by prior knowledge due to homogenous rolling stock, n informtion by the opertor or pltform screen doors. The qulity nd the performnce of the pssenger bording nd lighting linked to the vehicle s dwell time depends on the reltive horizontl nd verticl gp between the vehicle nd the pltform, the number nd width of the doors nd ese of circulting, due to the density of both the on-bord pssengers nd witing pssengers t pltform. When pssengers light vehicle, they re in front of the witing pssengers nd circulte towrds n exit, ccording to their in-sttion destintion. The exit my be verticl or horizontl circultion element, depending on the sttion design. However, the discrete lighting process mens tht pltform exit fces recurrent pcket of pssengers rriving. Tht infers to the need of dditionl witing for evcuting the pltform. Even when the hourly (verge) cpcity of the pltform exits is sufficient, bottleneck my be creted t the entrnce of the circultion element due to the discrete chrcter of the trnsit services nd the pssenger rrivl rte ssocited. 8

10 Trnsit Informtion in the sttion nd effect on route choice Intelligent trnsporttion systems re being deployed worldwide s wy to improve the performnce of the trnsit system nd the qulity of the service offered. Tht informtion produced by informtion nd communiction technologies - is highly vlued by both existing nd potentil users (Hickmn nd Wilson, 1995). Its presence t the vrious decision points influences the route choices of the pssengers. By pssenger informtion we group ll the informtion, sttic (timetbles nd service mps) nd dynmic (rel-time informtion) which hve n effect the pssenger s perception of the trnsit service. According to its loction it cn be divided into three ctegories: 1. Informtion prior to the journey 2. Informtion in the sttion 3. Informtion on-bord vehicle Vrious medi of informtion exist, such s pper plns nd signs, screens nd more recently the use of cellphones nd smrtphones. An dditionl medium of dynmic informtion is the visul informtion, bsed on rel-time observtion. The informtion distributed usully concerns the next vehicle rrivl or the witing time, but the recent brekthrough in informtion technology my llow shring more composite informtion such s the dynmic comfort stte of the next rrivls or the stte of the trnsit network. The dynmic informtion t sttion usully involving disply screens with the next vehicle rrivls or the witing time hs n importnt impct on the pth choice. Without informtion on the imminent rrivls t the pltforms, pssenger chooses the pltform on the bsis of minimizing the expected trvel time to the destintion. Therefore, given the loction of the decision point, without ny informtion, the fesible pth set is reduced to single pth. On the contrry, when informtion is vilble t tht decision point, the pth choice depends on the reltive rrivls of the vehicles on different pltforms. Furthermore, the presence of congestion on the pltform lters the chrcteristics of the witing time nd hence the pth probbilities. The witing time cn be split in vehicle witing time (due to the rndom rrivl distribution of the vehicles) nd queuing (due to congestion pssengers re unble to bord the first vehicle). However, queuing cts s buffer for witing of the next vilble vehicle. Thus, the combined witing time when congestion is present cn be no longer considered s rndom vrible. In tht cse the combined witing time hs deterministic vlue nd the line option with the witing time is reduced to pedestrin option evluted t its verge generlized time. NETWORK TOPOLOGY We define the network topology of the trnsit sttion used within pssenger flow ssignment model for trnsit networks. As shown previously, the trnsit sttion is composed of vrious elements with different functionlities, geometricl nd opertionl chrcteristics. The topology of the trnsit infrstructure network is described by directed grph G = ( N, A), with N the set of nodes n of the trnsit system nd A N N the set of rcs with endpoints in N. The demnd representtion follows some stndrd principles. The re is divided into zones; ech zone includes set of loctions with trip endpoints. Let o nd s denote trip origin 9

11 nd destintion respectively, where o O nd s S (not to confuse with the sttion symbols), let W the set of zone pirs from origin to destintion. An origin-destintion pir is defined s ( o, s) O S. Pssenger demnd is modelled s set of trip mkers: ech trip mker chooses network pth so s to minimize his own cost of trvel between his origin nd destintion points. For given trip-mker, the in-sttion trjectory corresponds to pth, yielding generlized cost nd trvel time on the bsis of locl chrcteristics defined by network element, node or link of the sttion. The Lower nd Upper Network Lyers S [ S The representtion of the trnsit sttion within trnsit ssignment model extends the bilyer network representtion of the trnsit services proposed in Leurent et l (2011) to the trnsit sttion. By tht pproch, we define the flow chrcteristics of ech sttion element in detil t the lower lyer, while keeping the upper lyer sufficiently consistent to the network flow ssignment. On the lower lyer of the network ech trnsit sttion is modelled s specific sub-network corresponding to the infrstructure network. In other words, ech trnsit sttion is hndled seprtely nd the physicl elements of the sttion (simulting to the supply side) re defined long with their opertionl chrcteristics nd the ssocited cost flow functions. On the upper lyer trip-mker is fced to service network where ech sttion leg corresponds to prticulr origin destintion trjectory in the sttion. Such leg rc represents one or severl sttion physicl elements (wlkwy, stirwy, escltor etc.) long the trnsit sttion in the infrstructure network. Therefore the upper lyer cts s representtion of the demnd side, where for n origin destintion trip is mde by line leg rcs (corresponding to the ccess egress sttion couples long line, introduced in Leurent et l., 2011) for the inter-sttion journey, sttion leg rcs for the intr-sttion nd the ccess-egress rcs for the respective trips. There is twofold reltionship between the two lyers by sttion s : top-down, vector of pssenger flows x = x : A ] by leg long the sttion is ssigned to the sttion submodel, yielding the locl pssenger flows; bottom-up, the sttion sub-model yields the vector g of the verge generlized cost. In other words, the sttion model mounts to n S elborte cost-flow reltionship in vector form on the upper lyer of the network. The Lower lyer representtion The lower lyer corresponds to the infrstructure network. A trnsit sttion is represented s ( s) ( s) ( s) (s) specific directed grph G = ( N, A ) mde up of set N of sttion nodes n, (s) together with set A of rcs with endpoints in N. We define three subsets of the set (s) N of sttion nodes: N E the set of sttion entry nd exit nodes, directly connected with the origin nd destintion nodes o O nd s S of the trnsit network, the set N P of pltform edges, cting s n intermedite between the sttion nd the trnsit service nd the set N (s) S 10

12 of sttion nodes. The first two subsets ct s the origin nd destintion points for the insttion pths of the sttion subsystem. (s) An rc represents stte trnsition for trip-mker nd the set A is split between the following subsets on the infrstructure network: the subsets A V of verticl circultion elements, A H of horizontl circultion elements nd A P of pltform rcs. Furthermore the subset of ccess nd egress rcs is denoted by A. The infrstructure network rcs correspond to the physicl elements used for the pssenger circultion nd storge within trnsit sttion. Ech sttion element (horizontl nd verticl circultion nd sttion re) is chrcterised by prticulr flow-dependent trvel time t ( x ) nd generlized time g x ), described in the following section. ( R The Upper lyer representtion The upper lyer corresponds to the service network, s representtion of the demnd side for the network model. As the trnsit sttion representtion is concerned, the service network is mde up of sttion legs. A sttion leg corresponds to the verge conditions of pssenger circulting from n origin to his destintion in the trnsit sttion subsystem. A sttion leg is composed by consecutive physicl circultion elements, s defined on the lower lyer nd its generlized cost nd trvel time is built upon these elements. The sttion leg rcs on the upper lyer re denoted by the subset A A A A. Overll, the service network is mde up of the set S L R S A A A A nd the ssocited node sets. A L denotes the subset of line legs, representing the verge conditions of the trnsit service of n ccess egress sttion couple on trnsit line nd A the subset of legs for ccessing to nd egressing from the trnsit network. The trnsit sttion is defined s subsystem of the trnsit network offering connection between the city nd the trnsit lines nd mong the trnsit lines, where the pssengers perform ccess egress trip nd trnsfers respectively. In the service network, n ccess itinerry is composed of n ccess rc nd series of sttion physicl rcs from n origin node o O connected to sttion entry nd exit node, N E, to pltform edge node N P, tht form sttion leg. The inverse is pplied for the egress trip. Respectively, trnsfer is represented by sttion leg A, connecting two pltform edges within the sttion or bus stop nd pltform edge. THE STATION MODEL FORMULATION S R H V P In the first prt of this section we present the form of pssenger flow nd their prticulrities when deling with cpcity constrints. Subsequently, we develop locl models ddressing the locl cpcity effects within the sttion: pltform entry nd exit flowing, the pedestrin trvel time by horizontl nd verticl circultion element nd the modelling of sttion re. Then we sketch the outline of sttion model deling with these effects. 11

13 The pssenger flow form nd their influence on cpcity The pssenger flow t the trnsit systems tkes two distinct forms: the continuous pssenger flow, relted to pedestrin wlking for trnsferring nd ccessing to (or egress from) the trnsit service nd the discrete pssenger flow, relted to the trnsit services, where pssengers re orgnized into repeting on-bord pckets. A phse trnsition occurs between these forms t the trnsit sttion. On the one hnd, the trnsition from continuous pssenger flow to discrete, upon bording vehicle, is ssocited with witing time due to the discrete vilbility of the trnsit services. On the other hnd, trnsition of the discrete pssenger flows to continuous ones tkes plce t the sttion upon pssenger lighting trnsit vehicle. The ltter trnsition impcts the sttion opertion. The circultion elements fce recurring pckets of rriving pssengers nd my be subject to congestion. At this point, we define two types of congestion: Instnt congestion: due to the trnsition between pssenger flow forms, the instnt dischrge rte of the circultion element is inferior to the instnt rrivl rte of pssengers, resulting in the formtion of congestion. However, the flow is dissipted before the rrivl of the next pcket of pssengers, mening tht recurring instnt congestion is susceptible with ech pssenger pcket rrivl. The cpcity during the simultion period is sufficient for the pssenger demnd; Continuous congestion: either by trnsition phse or the presence of bottleneck, the congested flow is not dissipted during the simultion period. Pssenger congestion is present throughout the entire simultion period nd the totl cpcity during the simultion period is not sufficient for the pssenger demnd. The continuous congestion of the sttion elements hs considerble impct on the qulity of the trnsit service. Nevertheless, the instnt congestion my punctully influence the pth choices t the sttion nd t the network level nd hs to be ddressed thoroughly. Pltform entry nd exit flowing A sttion pltform is n re of the trnsit sttion where vehicles of the trnsit services re mde vilble for bording or lighting. The edge of the pltform hs sufficient length for the vehicle berthing nd the vehicle s doors ct s n interfce between the sttion nd the service. Opposite, the pltform s entries or exits re locted; verticl or horizontl circultion elements distributed on the length of the pltform cting s connection between the pltform nd the rest of the sttion. A pltform entry flowing consists of the trjectory from the pltform s entry to the vehicle s door where the pssenger ccesses the trnsit service. Tht is ssocited with some witing, clculted by the line model, vi the rrivl distribution of the pssengers nd vehicles nd the vehicle bording cpcity effect. However, the presence of lrge pssenger stock on the pltform witing for the next vehicle deteriortes the circultion conditions. The pssenger trvel time on the pltform is ssocited with the reltion between pssenger stock present nd the storge cpcity of the pltform. The pltform exit flowing is relted to the rrivl of vehicle nd the trjectory of the lighting pssengers from the doors to the sttion s exit. In ddition to the difficulty of circultion due to the pssenger stock, instnt congestion on the entry of the element my be observed. In 12

14 tht cse n verge witing time is dded t the pssenger s pltform trvel time. Figure 3 illustrted the bottleneck produced on single exit element when rriving flow (in blue) is confronted to the dischrge cpcity (in red). Figure 3 The Bottleneck creted when exiting sttion pltform A Let t the wlking time on the pltform of the most distnt pssenger to the exit, V c the e wlking speed nd the ctchment re of tht exit element. It is strightforwrd tht A e Q t = V. Let t be the queuing time of the lst pssenger. The totl evcution time of c E the pltform, t, is clculted through the time needed for the lst pssenger to evcute E A Q e nd t = t + t. If N denotes the number of lighting pssengers heding to n e E e elementry exit nd k its nominl cpcity, the evcution time is t = N k. By mking the pproprite lgebric trnsformtions, the verge witing time, 13 e e w, t the exit element due to the instnt bottleneck, is given by the following formul: e e 1 N e w = ( ) e 2 k Vc If the distribution of the pssengers lighting e zi from vehicle of the service z t sttion i is homogenous nd the sttion exits re eqully distributed, the number of lighting pssenger t ech sttion mounts to N e E, where E is the number of exits vilble. e = Also, the ctchment re of n exit element will be length. The previous formul tkes the form: P e 1 ezi w = ( ) e 2 E k 2 E V c zi e P = 0, 5 E, where P is the pltform The unbounded bottleneck presented previously, ssumes tht pssenger who presented himself t the bse of the escltor remins indefinitely, until evcuting. However, usully the sttion design includes n escltor djcent to stirwy, the ltter hving sufficient cpcity to bsorb the excess demnd. In such cses, when the witing time of the Q pssenger exceeds certin vlue, t mx, the pssengers t the bck of the queue prefer using the stirwy. The mximum witing time t the escltor is subject to the estimted

15 difference of the trvel time between the escltor nd the stirwy. An equivlent decision cse would be the choice between discontinuous bus service to the next stop or wlking. Q The pssengers tht give up queuing suffer only the dditionl time of t mx. The verge witing time for the pssengers t the exit is given by: e w = e zi t Q mx t Q 2 mx E k 2 e P k e t 2 V Figure 4 illustrtes the verge witing time of the pssengers on RER pltform with 4 exits nd n escltor in ech exit, for the bounded nd unbounded bottleneck. We ssume Q tht the pssengers who would wit longer thn t min use the djcent stirwy. c Q mx mx =1 Although t tht cse we do not ssume ny cpcity constrints for the stirwy, n dditionl bottleneck could be creted for n importnt pssenger flow. Figure 4 Averge Additionl Time of n lighting pssenger to exit sttion pltform Three pltform exit phses re designted t Figure 4. For smll pssenger flow, the rrivl rte t the exit is inferior to the dischrge rte, mking the exit immeditely vilble, without ny dditionl time due to witing. The second trffic stte is limited to the rrivl rte being higher thn the dischrge rte. Then bottleneck is creted t the bse of the exit. If the lighting pssengers re distributed homogenously, the dditionl time due to witing t the bottleneck increses linerly. However, in the cse of the bounded model n dditionl trffic stte cn be distinguished. When the queuing time of the lst rriving pssengers exceeds their mximum dmissible queuing time then, the lst rriving pssengers use the djcent stirwy, setting n upper bound to the witing time. Pedestrin trvel time by circultion element In liner elements, such s wlkwys nd stirwys, the reltion between pssenger flows, pssenger density nd wlking speed is strightforwrd. TCQSM (TRB, 2003) indictes the speed-flow reltions on the elementry circultion elements of trnsit sttion: the wlkwy nd the stirwy (scending). These suggest reduction of the wlking speed with the increse of the pssenger flows. Higher pssenger densities result in lower speed nd flows; such effects re not tken into ccount in trnsit ssignment model. In this cse, the wlking 14

16 speed of the pssengers in ny circultion element is mde flow dependent. The model proposed trnsforms the reduction in the wlking speed to n dditionl trvel time of the elements (Figure 5); the ltter cn be expressed s function of the flow-to-cpcity rtio of the element. Figure 5 Trvel Time digrm in function of the flow-to-cpcity rtio The sttion s circultion elements re rrely subject to instnt congestion, since the scrce cpcity t the pltform exit leds to the downstrem homogenistion of the pssenger flows. However, the elements my be subject to continuous congestion if the vilble cpcity during the simultion period is not sufficient. A simple modelling pproch is used, bsed on queuing theory for clculting the dditionl time due to continuous congestion. During the simultion period H, if y is the in-sttion pssenger flow rriving t the element s entry, demnding the pssge. Let N d = y H is the number of pssenger k the dischrge rte of the circultion element nd H the time needed for cooling the demnd. It exists, Nd = y H = k H nd therefore n dditionl time w is clculted for the pssengers, where: w y k = k 0 if y > k otherwise Pedestrin Trvel Time by Are Element A sttion re, such s the sttion concourse, is n element with vrious converging nd diverging pssenger flows. Nevertheless, the pressure of the pssenger flow results in the formtion of virtul wlkwys. Therefore, neglecting the idle pssengers using the menities, the pssenger flows in sttion re re modelled by directed rcs, where virtul cpcity my be defined. A sttion re is reduced in set of horizontl circultion rcs with 15

17 predefined cpcities. A penlty my be imputed for tking into considertion the trvel time losses due to trjectory conflicts. A Trnsit Sttion Model Algorithm The conditions externl to the sttion pertin to the pssenger lod vector by sttion leg, x S = [ quv : u, v VS ], where VS N E N P denotes the set of origin nd destintions nodes of the in-sttion pths. By pltform element, we denote ϕ the frequency of the services tht serve the sttion, inherited from the line model, P i E i, the number of pltform exits nd pltform length. We ssocite by circultion element, free flow trvel time P i the 0 t nd n element cpcity k. A network equilibrium model with fixed demnd, stisfying Wrdrop s user optimlity cn be (s) defined for ech trnsit sttion subsystem. The nodes i N represent the origin, (s) destintion nd intersections of the physicl rcs, A. The origin nd destintion nodes o, s coincide with the endpoints of the sttion legs nd we define n origin destintion pir 2 (, v) VS u. These crete rc flows y nd the cost of trvelling on n rc is flow-depended, (s) g ( y ), A, s described in the previous sections. The pssenger flow in the trnsit sttion is described by the in-sttion flow vector y = [ y : A ]. The network equilibrium model for the trnsit sttion cn be formulted s fixed point problem. Ech sttion s S is treted seprtely by n itertive lgorithm, bsed on the Method of Successive Averges (MSA) cn be used to solve the trffic equilibrium problem of sttion ssignment, using decresing sequence of positive numbers ( λ µ ) µ 0 with λ 0 = 1, s follows: Initilistion: Set y s = 0 nd µ = 0; Cost Formtion: by sttion element bsed on y s. We tret ech physicl element on the trnsit sttion s. Auxiliry Stte: Shortest pth serch for ll the origin-destintion pirs nd lod the OD flows long them, yielding n uxiliry trffic stte ŷ. By destintion, shortest pth is built on the sttion network from ech node recursively, in clssicl wy (Dijkstr, 1959). Convex Combintion: Let y s = ( 1 λ µ ) y s + λµ yˆ s Evlute convergence criterion between y s nd y s. If it is sufficiently smll then stop with solution y s, evlute the cost of the sttion legs A S. Else increment µ, replce y s by y s nd go to Cost Formtion. A convergence criterion cn be bsed on the flow gp of the rcs between two consecutive trffic sttes, y y. s s s (s) s 16

18 ARTICULATION WITH THE NETWORK MODEL The trnsit sttion model is used s n elborte cost-flow reltionship within network ssignment model, in order to evlute the generlized cost of the sttion legs for ech trnsit sttion. Route Choice on the upper (network) lyer On the upper lyer, sttion legs hve generlized cost witing time w nd revised frequency g, but they re chrcterized y ϕˆ, relted to tht of the line leg t the hed of the sttion leg. Let us define the revised frequency of sttion leg, s function of the conditions inherited by the line leg t the sttion leg s endpoint nd the presence of in-sttion informtion. In CpTA the tretment of the revised frequency is stndrd when =. If w α / ϕ w α / ϕ > due to crowding congestion on the line pltform, the pssenger my not be ble to bord in the first incoming vehicle. In this cse the leg option is kin to pedestrin option evluted t its verge cost only. To integrte continuous (pedestrin) nd discrete (uncongested trnsit) vilbility, let us define discontinuity ttenution function denoted ψ tht is continuous nd decreses from 1 t 0 to ψ ( x) = 0 whtever x ε smll positive prmeter. The verge witing w nd effective operting frequency ϕ delivered by the line model yield revised frequency of ˆ ϕ = ϕ / β if β ψ ( w α / ϕ ) > 0, or ϕˆ = if β = 0 In the presence of informtion, line combintion will occur t the sttion with the formtion of line bundle, s considered in the bsic model of line combintion (Chriqui nd Robillrd, 1975), due to the rndom cost of witing for line leg AL. In tht cse the verge witing cost of line bundled is evluted by the combined revised frequency of the line legs of the bundle. Then, line bundling on the upper lyer is bsed on the revised frequencies nd proceeds in the clssicl wy, yielding route-bsed hyperpths s in (De Ce nd Fernndez, 1993). The discontinuity ttenution function is in fct development of the line model, s it involves only results of tht model. It is further coordinted with the sttion model, bsed on the leg representtion of the network lyer. APPLICATION INSTANCE A trnsit sttion of the Greter Pris trnsit network is chosen s n ppliction instnce for the trnsit sttion model. The Ntion sttion (Figure 6) is n intermodl trnsit terminl, locted t the dense zone of the trnsit system. Aprt from the RER A, - for which is the first sttion in Pris from the Est suburbs - it is served by 4 metro lines (2 of them s terminl sttion), nd 4 bus lines. 17

19 Figure 6 A schemtic mp of the trnsit system t Ntion sttion (source: RATP) The input dt re extrcted by network ssignment on the Greter Pris network for the morning hyper-pek hour with n increse by 30% of the verge pek flow. While there is significnt number of ccess egress trips (8250 pssengers, whose one endpoint is Ntion), it is lrgely inferior to the number of trnsfer pssengers, who mke the 83% of the in-sttion pssenger flow. The upper lyer network of the Ntion sttion is composed of 14 sttion leg endpoints, corresponding to the origin nd destintion points inside the sttion: 10 line pltform edges, 2 bus stops nd 2 entries/exits, s n rticultion with the ccess/egress trips. The pths within the sttion re represented by 134 sttion legs, connecting ech leg endpoint (except the entries/exits with ech other). The exit flow on the pltform induces bottleneck t the entries of the circultion elements. Tble I illustrtes some selected pltforms, the A/R indicte the directions of the trnsit lines: Tble 1 Chrcteristics of sttion pltforms nd the dditionl witing time of the pltform exit flow Pltform Exit Flow (pss/h) Nb of Exits Frequency (veh/h) Pltform Length (m) Combined Exit Cpcity (pss/min) RERA A ,45 RERA R ,61 M1 R ,56 M2 R ,40 M6 R ,79 M9 A ,57 Additionl Witing Time (in min) The exogenous pssenger flow induces n increse in the trvel time of the circultion elements from 8% to 41%, depending on their cpcity, type nd pssenger flow. The trvel time of the sttion legs, connecting the sttion input/output nodes, fces n increse of 15% to 79%. However lrge prt of tht increse is ttributed to the bottleneck creted during the pltform exit flowing. Here some results for the trnsfer between lines: 18

20 From RERA to M9, the trvel time psses from 2,90 to 3,94 min (of which 0,45 min for pltform exit flowing), n increse by 36% From M1 to M6, n increse by 66% to 1,99 min, of which 0,56 min of pltform exit flowing From M2 to RERA, the trnsfer time increses from 4,00 to 5,12 min (+28%, where 0,40 min of pltform exit flowing). These results indicte significnt increse of the trvel times within the sttion. It certinly does not hve the sme order of mgnitude s the witing time nd the in-vehicle trvel time. However, its impct on the pth choices of the pssengers nd the pth flows on the trnsit network remin to be evluted. In the cse of the Greter Pris trnsit network, the trnsfer nd ccess/egress trips on the network mount pproximtely to 30% of the verge generlized time of trip. CONCLUSION The trnsit sttion is complex system, composed by vriety of elements, ech one with distinct geometricl chrcteristics, functionlities nd opertionl ttributes. In the sme time, it constitutes n importnt subsystem of the trnsit network, cting s n intermediry for the trnsfers between trnsit lines nd for the ccess/egress trips to the trnsit services. The sttion model constitutes frmework to represent the fetures nd phenomen, of physicl nd microeconomic nture, tht tke plce in the sttion nd impct the trnsit opertions nd pssenger pth choices. With the bi-lyer representtion the model hndles the physicl interction of the pssengers in the sttion on the lower lyer nd the economic pprisl of the in-sttion trips on the upper lyer. The prcticl implictions of the trnsit sttion model cn be significnt in pssenger flow ssignment in lrge-scle trnsit network. Although the influence of the in-sttion trvel time on the totl trvel time is inferior to tht of the in-vehicle trvel time, more relistic representtion of the cpcity effects within sttion is expected to hve n impct on the pssenger pth choices, minly when they involve trnsfers through congested trnsit sttions. In tht cse, pssengers my choose lterntive pths due to deteriorted circultion conditions of the congested sttions. The ppliction t Ntion intermodl sttion shows the effect of the pssenger flows on the in-sttion pedestrin trips. The bi-lyer representtion leds to n increse, in the first plce, of the upper lyer sttion rcs, but the degree of detil chosen for the lower lyer depends on the sttion prticulrities. The results in Ntion indicte tht most of the pltform-to-pltform pths corresponding to sttion legs re subject to trvel time increses from 8% to 41%. Further step would be to include the sttion model to network ssignment model nd evlute the trvel time increse due to sttion congestion nd its interction with the cpcity constrints on the trnsit lines. Certin cpcity effects re included in the current model such s the recurring bottleneck t the pltform exits when vehicle rrives, or the impct of the pssenger flow to the trvel time. Nonetheless, the frmework is essentilly systemic nd modulr: some prts my be replced by more pproprite sub-models nd other effects my be dded. Mny developments cn be thought of: dditionl cpcity effects my be included, such s the 19

21 impct of the pssenger stock in the trvel time in the sttion res, of the congestion on the witing conditions or the opertion of the elevtor s principl circultion element. ACKNOWLEDGEMENT This reserch is supported by the Reserch nd Eduction Chir on The socio-economics nd modeling of urbn public trnsport, operted by Ecole des Ponts PrisTech in prtnership with the Trnsport Orgnizing Authority in the Pris Region (STIF), which we would like to thnk for its support. REFERENCES Ceped M., Cominetti R., Florin M. (2006). A frequency-bsed ssignment model for congested trnsit networks with strict cpcity constrints: chrcteriztion nd computtion of equilibri, Trnsporttion Reserch Prt B, Chriqui, C. nd Robillrd P. (1975). Common bus lines, Trnsporttion Science, Vol 9, Dmen W (2002). SimPed : Pedestrin Simultion Tool for Lrge Pedestrin Ares, conference proceedings EuroSIW, June 2002 Dijkstr E.W. (1959). A note on two problems in connexion with grphs, Numerische Mthemtik, Vol 1, Dly P.N., McGrth F, Annesley T.J. (1991). Pedestrin speed/flow reltionships for underground sttions, Trffic Engineering nd Control, Vol. 32, Issue 2, De Ce, J. nd Fernández J.E. (1993). Trnsit Assignment for Congested Public Trnsport Systems: An Equilibrium Model. Tptn. Sc., 27(2), Fire Protection Assocition, NFPA 130: Stndrd for Fixed Guidewy Trnsit nd Pssenger Ril Systems, 2010 Edition Guo Z, Wilson N.H.M. (2011). Assessing the cost of trnsfer inconvenience in public trnsport systems: A cse study of the London Underground, Trnsporttion Reserch Prt A, Vol 45, Hrris N.G (1991). Modelling wlk link congestion nd the prioritistion of congestion relief, Trffic Engineering nd Control, Vol 32, Issue 2, Hickmn M.D. nd Wilson N.H.M. (1995). Pssenger Trvel Time nd Pth Choice Implictions of Rel-Time Trnsit Informtion, Trnsporttion Reserch Prt C, Vol 3, No. 4, Hoogendoorn, S.P. (2001). Normtive Pedestrin Flow Behvior: Theory nd Applictions. Reserch Report Vk Trnsporttion nd Trffic Engineering Section, Delft University of Technology Kuruchi F., Bell MGH, Schmocker J-D. (2003). Cpcity Constrined Trnsit Assignment with Common Lines, Journl of Mthemticl Modelling nd Algorithms, Vol. 2, Leurent F., Chndks E. nd Poulhes A. (2011). User nd service equilibrium in structurl model of trffic ssignment to trnsit network, in Zk J. (ed) The Stte of the Art in the Europen Quntittive Oriented Trnsporttion nd Logistics Reserch 14 th 20