A Comparison of the Greenshields, Pipes, and Van Aerde Car-Following and Traffic Stream Models

Transcription

1 A Comparison o the Greenshields, Pipes, and Van Aerde Car-Following and Trai Stream Models by: Hesham Raha and Brent Crowther ABSTRACT The paper ompares three ar-ollowing models. These models inlde the Greenshields single-regime model, the Pipes two-regime model, and a or-parameter single-regime model that amalgamates both the Greenshields and Pipes models. The or-parameter model that was proposed by Van Aerde (995) and Van Aerde and Raha (995), is less nown bt is rrently implemented in the INTEGRATION.3 sotware. The Greenshields and Pipes models are onsidered bease they represent state-o-the-pratie models or a nmber o mirosopi and marosopi sotware. First, the Greenshields model is widely sed in marosopi transportation planning models. In addition, the Pipes model is implemented in a nmber o mirosopi trai simlation sotware inlding the CORSIM and VISSIM sotware. The paper also relates steady-state ar-ollowing behavior to marosopi trai stream models in order to develop alibration proedres that an be ahieved sing marosopi loop detetor data. The stdy onldes that the additional degree-o-reedom that reslts rom inlding a orth parameter (Van Aerde model) overomes the shortomings o the rrent state-o-pratie trai stream models by aptring both marosopi and mirosopi steady-state trai behavior or a wide range o roadway ailities and trai onditions. The paper also provides a proedre or alibrating the Pipes ar-ollowing model sing marosopi ield measrements that an be obtained rom loop detetors. This alibration proedre, while it does not overome the inherent shortomings o the Pipes model, does provide an opportnity to alibrate the CORSIM and VISSIM ar-ollowing behavior to existing roadway onditions more eiiently and withot the need to ollet mirosopi trai data. Key words: Trai stream models, ar-ollowing models, alibration o mirosopi simlation tools, CORSIM, VISSIM, and INTEGRATION.. INTRODUCTION Mirosopi simlation models se ar-ollowing behavior to model the interation o a vehile and the preeding vehile while traveling in the same lane. The proess o ar-ollowing is modeled as an eqation o motion nder steady-state onditions pls a nmber o onstraints that govern the behavior o vehiles while moving rom one steady-state to another (deelerating and aelerating). Typially p to two onstraints are onsidered. The irst onstraint governs the vehile aeleration behavior, whih is typially a ntion o the vehile dynamis. The seond and inal onstraint ensres that vehiles have a sae position relative to the lead vehile in order to deelerate to a omplete stop withot olliding with the preeding vehile in the event that the preeding vehile deelerates to a omplete stop. Conseqently, the alibration o the ar-ollowing behavior within a mirosopi simlation model an be viewed as a two-step proess. In the irst step the steady-state behavior is alibrated ollowed by a alibration o the non-steady state behavior. The alibration o the steady-state behavior is ritial bease it ditates the maximm roadway throghpt Assistant Proessor, Charles Via Department o Civil and Environmental Engineering, Virginia Teh. Virginia Teh Transportation Institte, 35 Transportation Researh Plaza (536), Blasbrg, VA hraha@vt.ed Trai Engineer, Kimley-Horn & Assoiates, Site 5, 76 North 5 th Street, Phoenix, AZ Brent.rowther@imley-horn.om.

2 Raha and Crowther (apaity), the speed at whih vehiles travel at dierent levels o ongestion (trai stream behavior), and the spatial extent o qees when lly stopped (am density). Alternatively, the alibration o the non-steady state behavior, whih in most ases is less ritial, inlenes how vehiles move rom one steady-state to another and ths aptres the apaity redtion that reslts rom trai brea down and the loss o apaity dring the irst ople o seonds as vehiles depart rom a trai signal (ommonly nown as the start loss). Under ertain irmstanes, the non-steady state behavior an inlene steady-state behavior. For example, vehile dynamis may prevent a vehile rom attaining state-state onditions. A typial example o sh a ase is the motion o a tr along a signiiant pgrade setion. In this ase the atal speed o the tr is less than the desired steady-state speed bease the vehile dynamis does not permit the vehile rom attaining its desired speed. The analysis o sh instanes is beyond the sope o this paper.. Paper Obetives The obetives o this paper are two-old. First, the paper presents and ompares the Greenshields, Pipes, and Van Aerde ar-ollowing models or the modeling o steady-state onditions that or when the lead and ollower vehiles rise at eqal speeds, or the dierene in speeds are relatively small that they annot be pereived by the driver, or the distane headway between the two vehiles is signiiantly long that the behavior o the lead vehile has no inlene on the behavior o the ollowing vehile. An analysis o non-steady state onditions, whih inlde moving between steady states that reslt rom dierenes in vehile speeds is beyond the sope o this paper and will be presented in a orthoming paper. Seond, the paper develops an analytial ormlation that relates the roadway apaity to the Pipes model driver sensitivity ator. This important relationship between mirosopi and marosopi trai behavior provides a niqe opportnity to alibrate mirosopi ar-ollowing models sing marosopi loop detetor data. It shold be noted, however, that the mehanis o the alibration tool that is reqired to ondt sh a alibration eort is beyond the sope o this paper bt will be presented in a orthoming eort. Instead, the paper ompares the varios alibrated models to demonstrate dierenes in ar-ollowing behavior withot desribing how these models were alibrated to ield data.. Signiiane o Researh The signiiane o this researh eort lies in the at that it not only demonstrates the similarities and dierenes in the three ar-ollowing models, bt also explains why dierenes in model reslts may or. The seond ontribtion o this paper is that it provides a proedre or alibrating the Pipes model to marosopi ield data that an be obtained rom loop detetors. This alibration proedre wold be extremely beneiial or alibrating CORSIM and VISSIM model inpt parameters to ield data. Crrently, the alibration o CORSIM is ahieved in a rather adho manner by hanging a driver sensitivity ator ntil a desired roadway apaity is observed. Speiially, May et al. () pointed ot that diret relationships between desired apaity estimates and CORSIM inpt parameters were not lear. Researh is needed to determine the relationships between the appropriate CORSIM model inpt parameters and obtaining the desired segment apaity otpt. Alternatively, the literatre proposes alibrating the VISSIM arollowing behavior by driving a lly-instrmented probe vehile within the trai stream to ollet ar-ollowing data. This approah is extremely expensive and ineasible in most ases..3 State-o-the-Art Researh A ew stdies have attempted to establish the relationship between the Pipes sensitivity ator that is embedded in the CORSIM model and desired apaity estimates. Speiially, Payne et al. (997) perormed a series o simlation rns in whih the FRESIM model (reeway model within CORSIM) parameters were systematially varied, and the resltant apaity measred. A series o harts were developed rom whih the ar-ollowing sensitivity ator old be obtained by reerening the desired apaity. However, these harts did not aont or variations in am density and ree-speed. Frthermore, the relationships were empirially derived, and thereore senario-speii.

3 Raha and Crowther 3 In addition, Halati et al. (997) ondted a series o simlation rns to determine the relationship between the distribtion o ar-ollowing sensitivity ators and roadway apaity. The stdy onlded that the mean driver sensitivity ator is a main determinant o roadway apaity. Speiially, the stdy onlded that the mean vale o the deile distribtion o sensitivity ators was important, and that the variane appeared to have a lesser impat on the roadway apaity. Halati et al. inding is ritial to the proedres that are developed in this paper bease the proposed analytial proedres estimate the average driver sensitivity ator that is to be inpt to the simlation model in order to obtain a desired roadway apaity..4 Paper Layot The paper irst provides a brie bagrond o ar-ollowing models that desribe the behavior o vehiles as they ollow a lead vehile. Having desribed the behavior o a pair o vehiles, the behavior o a trai stream o vehiles is desribed. Speiially, a nmber o standard marosopi ntional relationships are desribed inlding single- and mltiple-regime models. The relationship between ar-ollowing and marosopi trai low behavior is then introded. Sbseqently, three steady-state ar-ollowing models are presented. The paper then develops analytial relationships that allow the ser to alibrate these three ar-ollowing models sing readily available loop detetor data. Finally, the onlsions o the paper are presented.. BACKGROUND This setion desribes a nmber o standard single- and mlti-regime marosopi trai stream models that have been desribed in the literatre. These marosopi models identiy the relationship between the three trai low parameters, namely low, speed, and density. While marosopi trai low models desribe the behavior o a stream o vehiles along a roadway streth, mirosopi ar-ollowing models desribe the behavior o a pair o vehiles within a trai stream. The importane o relating steady-state ar-ollowing behavior to marosopi trai low parameters stems rom the at that, nlie mirosopi behavior, marosopi trai low parameters an be measred airly easily in the ield sing standard loop detetors or trai onters. Conseqently, the alibration o the ar-ollowing models is more easily ahieved sing marosopi parameters as will be desribed later in the paper.. Mirosopi Steady-State Car-Following Models Initially the ar-ollowing behavior o a single vehile is explained prior to desribing how the motion o a single vehile aets the low o an entire trai stream. It shold be noted at this point that this setion only desribes ar-ollowing behavior nder steady-state onditions, when the lead vehile is traveling at a onstant speed and both the lead and ollower vehiles have an idential ar-ollowing behavior. Steady-state ar-ollowing is extremely ritial given that it inlenes the overall behavior o the trai stream. Speiially, steady-state behavior determines the desirable speed o vehiles at dierent levels o ongestion, the roadway apaity, and the spatial extents o qees. Clearly, these ators have signiiant impats on the ormation o qees and extent to whih qees spillba onto pstream roadways... Under steady-state onditions given that the speed dierential between the lead and ollowing vehile approahes zero, rrent state-o-pratie ar-ollowing behavior resolves to a relationship between the vehile speed and its distane headway rom the lead vehile. The relationship tends to an asymptoti maximm speed (termed the ree-speed) as the headway approahes ininity, as illstrated in Figre. In other words, at ree-speed the vehile s desired speed is not governed by the srronding trai; instead it is governed by roadway rition, reglatory, and/or roadway geometri onditions. As the distane headway between the sbet and lead vehile dereases the desired vehile speed dereases ntil the vehile omes to a omplete stop. It shold be noted rom Figre that ield data tend to demonstrate a airly linear inrease in speed as a ntion o the distane headway or reeway ailities where the vehile is less onstrained by the roadway geometry and/or rition bt is more onstrained by reglatory onditions

4 Raha and Crowther 4 (e.g. speed limit). Alternatively, data that were obtained rom the Holland Tnnel in New Yor demonstrate a non-linear ar-ollowing behavior. The Holland Tnnel is a two-lane diretional roadway nderneath the Hdson River that onnets New Jersey to New Yor. The design eatres o the tnnel are somewhat restritive, with 3.3-meter lanes (- oot lanes), no sholders, and a typial tnnel alignment onsisting o a downgrade ollowed by an pgrade. These data that were reorded by the Port o New Yor Athority as part o their srveillane system were aggregated into 5-minte averages (May, 99). The ar-ollowing model that is alibrated to the data demonstrates a less aggressive driving behavior ompared to reeway driving as demonstrated by the milder slope in the speed-headway relationship. Finally, the arterial data that were obtained rom a Newastle University researh team in the United Kingdom who were stdying inident detetion tehnologies (May, 99), demonstrate an even less aggressive driving behavior with a signiiant non-linear ar-ollowing model. Unlie the reeway and tnnel-driving environment, the arterial roadway onstittes driving in an environment that is lassiied as nontrolled with vehiles enontering signiiant roadway rition. It shold be noted that the ield data that are presented in Figre were extrated rom a nmber o data sores, as was desribed earlier. These data represent driving onditions in a nmber o ontrolled and nontrolled environments. The ar-ollowing models that are alibrated to the ield data, whih will be desribed later in the paper, represent the steady-state ar-ollowing behavior o a vehile. As was mentioned earlier, vehiles will typially move rom one steady state to another along this ar-ollowing relationship. The vehile s ability to move between steady states is onstrained by its aeleration and deeleration apabilities, by the driver aggressiveness, and by weather and pavement onditions... State-o-the-Art Car-Following Models Theories desribing how vehiles ollow one another were developed in the 95s and 96s (May, 99) and ontine to be developed and ompared (Ayin and Beneohal, 999). As was mentioned earlier, the maority o ar-ollowing models onsider two independent variables in the driver s speed seletion deision. These variables inlde the distane headway and the speed dierential between the lead and ollower vehile. The models mae a simpliying assmption that drivers only respond to the vehile ahead o them withot observing other vehiles rther downstream. In some instanes the models onsider a driver reation time with its trai stability reperssions. In order to lill the obetives o the paper, three ar-ollowing models are presented. These models inlde the Pipes model bease it serves as the basis or a nmber o mirosopi simlation models inlding the CORSIM and VISSIM steady-state ar-ollowing logi, the ar-ollowing model that evolves rom the Greenshields marosopi trai stream model, and a or-parameter model (Van Aerde model) that ombines both the Pipes and Greenshields models. The Van Aerde model onstittes the steady-state ar-ollowing logi within the INTEGRATION.3 sotware. Pipes haraterized the motion o vehiles in a trai stream as ollowing a linear relationship, as illstrated in Figre. The ar-ollowing behavior o a vehile is onstrained by a maximm speed, whih is ommonly nown as the reespeed. Free-low speed or ree-speed is the desired travel speed o a vehile when the distane headway tends to ininity (i.e. no other vehiles are on the roadway). The ree-speed is typially slightly higher than the speed limit o the roadway and depends on the level o highway srveillane. The slope o the Pipes model has been alibrated to relet the data or the varios ailities that are presented in the igre. Figre rther illstrates the ar-ollowing behavior that reslts rom the Greenshields marosopi trai stream model that is desribed in the orthoming setion and the Van Aerde model that is also illstrated in Figre. Figre learly illstrates that the Pipes model strtre (linear ar-ollowing model) does ails to aptre ield observed arollowing behavior over a wide range o aility types and ar-ollowing domain.. Marosopi Trai Stream Models Having haraterized the motion o a pair o vehiles, this setion desribes the behavior o an entire trai stream that evolves rom the ar-ollowing logi o the individal vehiles that onstitte this trai stream. Trai stream models

5 Raha and Crowther 5 provide the ndamental relationships between three marosopi trai stream parameters or steady-state onditions, as illstrated in Figre 3. The trai stream parameters inlde low, speed, and density. A niqe low parameter is the maximm low or apaity (q ), whih orresponds to the x-oordinate at the nose o the speed-low relationship. Two niqe parameters are identiied on the speed-low relationship, whih inlde the ree-speed ( ) and speed-at-apaity ( ). The speed-at-apaity is the speed that exists at maximm low onditions and orresponds to the y-oordinate o the nose o the speed-low relationship, as illstrated in Figre 3. The two niqe density parameters inlde am density ( ) and density-at-apaity ( ), oten reerred to as the optimm density. Jam density is the density o trai when both low and speed approah zero. The basi trai stream model establishes low as the prodt o density and speed, as demonstrated in Eqation. q = [].. In order to demonstrate how trai stream models vary or dierent aility types, the three dierent data sets that were presented earlier are presented again in the speed-low-density domain. Speiially, Figre 3 illstrates the I-4 reeway data rom Orlando, Florida, while Figre 4 illstrates the data rom the Holland Tnnel in New Yor, and Figre 5 illstrates the loop detetor data that were olleted on an arterial roadway in the United Kingdom. Sperimposed on the igres are alibrated speed-low-density relationships that were generated sing a alibration based on the Van Aerde model strtre with a alibration tool that will be desribed in a orthoming pbliation. The reeway speed-low relationship demonstrates a airly onstant speed in the nongested regime o the relationship (speeds in exess o speed-at-apaity), as illstrated in Figre 3. This trend is ased by the at that the vehiles traveling nder nongested onditions are not onstrained by vehile-to-vehile interation; instead vehiles are onstrained by the speed limit o the aility (9 m/h). The igre also demonstrates that the speed-density relationship is highly non-linear exhibiting an S-shaped relationship. The low-density relationship exhibits a low level o variability abot the alibrated model in the nongested regime (densities less than the density-at-apaity) with more variability in the ongested regime. The tnnel data exhibit a more paraboli type o speed-low relationship with a more linear trend in the speed-density domain, as illstrated in Figre 4. Comparing Figre 3 to Figre 4 it is evident that the vehiles are more onstrained with the vehile interations in the ase o the tnnel than in the ase o the reeway aility. The igres also demonstrate a higher lane apaity in the range o veh/h on the reeway verss a 3 veh/h apaity along the tnnel aility. The apaity o the arterial roadway is rther reded to 6 veh/h as a reslt o varios orms o rition along the aility and the impat o trai signal timings on the aility apaity... State-o-the-Art Trai Stream Models Over the years a nmber o trai stream models have been proposed and desribed in the literatre. The earlier models assmed a single regime or the ongested and nongested onditions. Later models attempted to improve on the single-regime models by introding separate models or the nongested and ongested regimes. The most amos o the single regime models is Greenshields model that was developed in 934 based on observations o speed-density measrements obtained rom an aerial photographi stdy (May, 99). Using these data Greenshields onlded that speed was a linear ntion o density, as demonstrated in Eqation. Using the linear speed-density relationship together with the basi trai stream model that was presented in Eqation, the speed-low relationship is represented as a paraboli relationship, as demonstrated in Eqation 3. The speed-at-apaity an be ompted by taing the derivative o the low with respet to speed, as demonstrated in Eqation 4. This relationship reslts in a speed-at-apaity that is eqal to hal the ree-speed. The trai stream relationships that orrespond to the Greenshields model are illstrated in Figre 6. It is lear by omparing Figre 6 to the ield data that are presented in Figre 3, Figre 4, and Figre 5 that the paraboli speed-low relationship is reletive o ield data nder limited onditions. Finally, it shold be noted that the alibration o the Greenshields model reqires the estimation o two parameters, namely the ree-speed and either the am density or the apaity.

6 Raha and Crowther 6 = [] q q = ( ) = = = [3] [4].3 Relationship Between Marosopi Trai Stream and Steady-State Car-Following Behavior As was mentioned earlier given that steady-state mirosopi ar-ollowing models haraterize the relationship between the vehile s desired speed and the distane headway between the lead and ollower vehiles, this model an be related to the marosopi speed-density relationship assming all vehiles in the trai stream maintain the same headway distane or have an average headway distane that is onsistent with the ar-ollowing model. For example, Drew (968) indiates, the similarities in the marosopi and mirosopi approahes have been emphasized. The ormer solves a dierential eqation o stream motion and a dierential eqation o ontinity, both expressed in terms o speed and density to obtain an eqation o state (the eqation o the ndamental q-- trai srae). The latter ombines the dierential eqation o motion or an individal vehile together with the appropriate bondary onditions to obtain an eqation o state. 3. GREENSHIELDS CAR-FOLLOWING MODEL Bease the or-parameter or Van Aerde model that is desribed later provides more degrees o reedom to aptre the range o behavior aross dierent regimes and aility types; as illstrated in Figre 3, Figre 4, and Figre 5; omparisons are made against the more general Van Aerde model. The ar-ollowing model that orresponds to the Greenshields trai stream model is derived rom Eqation, as demonstrated in Eqations 5, 6, and 7. The orresponding ar-ollowing behavior that evolves rom the Greenshields model demonstrates a airly aggressive ar-ollowing behavior at short distane headways with a less aggressive arollowing behavior at longer headways, as demonstrated in Figre. ( ) = ( ) = [5] ( ) [6] = h h = [7] ( ) A omparison o the Greenshields and Van Aerde models or the I-4 reeway data, as illstrated in Figre 6, demonstrates that the Greenshields paraboli speed-low relationship and its orresponding linear speed-density relationship are inonsistent with the ield data. Speiially, the data indiate that vehiles travel at a speed that is higher than hal the ree-speed when traveling at apaity. It shold be noted at this point that the alibrated am density or the Greenshields model is dierent rom the alibrated am density or the Pipes and Van Aerde models bease the Greenshields model reqires the alibration o two

7 Raha and Crowther 7 parameters; the ree-speed and apaity. One these two parameters are alibrated the am density an be ompted as twie the density at apaity, whih is based on the roadway apaity and ree-speed (4q / ). The Holland Tnnel data that were illstrated in Figre 4 indiate a more paraboli speed-low relationship, as illstrated in Figre 7. However, the speed-density domain learly indiates that a linear speed-density relationship does not relet the data. The Holland Tnnel data demonstrate a very important onept, namely that a good it in one domain (e.g. speed-low) does not neessarily imply a good it in another domain (e.g. speed-density). Similarly, data olleted along an arterial street, as illstrated in Figre 5 indiate a paraboli behavior in the speed-low domain with a non-linear speed-density relationship, as illstrated in Figre 8. The three aility types that are illstrated in Figre 6, Figre 7, and Figre 8 learly demonstrate the deiieny o the Greenshields model in aptring the steady-state behavior o a wide variety o aility types or the ll range o trai low data domains. 4. PIPES CAR-FOLLOWING MODEL As was mentioned earlier the Pipes model onstittes the steady-state ar-ollowing model in a nmber o mirosopi simlation sotware, inlding CORSIM and VISSIM. This setion briely desribes the ar-ollowing models that are inorporated in the CORSIM and VISSIM sotware and demonstrates that nder steady-state onditions these models revert to the Pipes ar-ollowing model. This setion provides a very brie overview o the ar-ollowing logi within the CORSIM and VISSIM models. More detailed inormation an be obtained rom the literatre. The trai stream models that evolve rom the Pipes ar-ollowing model are mlti-regime in the sense that a dierent model is tilized or the ongested verss nongested regimes, as illstrated in Figre 6. Speiially, the Pipes model assmes that the trai stream speed is insensitive to the trai density in the nongested regime. This assmption best its roadways that are designed nder high geometri standards when vehile speeds are onstrained by reglatory onstraints rather than the srronding trai (e.g. speed limit), however, the assmption is less valid or roadways with sb-standard geometri designs or with signiiant roadway rition, as demonstrated in Figre 6, Figre 7, and Figre CORSIM Car-Following Behavior CORridor SIMlation (CORSIM) is the miro-simlation omponent o the TRAF amily o models that were developed by the US Federal Highway Administration (FHWA) or simlation o trai behavior on integrated rban transportation networs o reeway and srae streets. CORSIM ombines the NETwor SIMlation (NETSIM) and FREeway SIMlation (FRESIM) models into an integrated paage. Both NETSIM and FRESIM simlate trai behavior at a mirosopi level with detailed representation o individal vehiles and their interation with their physial environment and other vehiles. 4.. FRESIM Car-Following Behavior The FRESIM model tilizes the Pitt ar-ollowing behavior that was developed by the University o Pittsbrgh (Halati et al., 996). The basi model inorporates the distane headway and speed dierential between the lead and ollower vehile as two independent variables, as demonstrated in Eqation 8. As was mentioned earlier, given that steady-state onditions are haraterized by travel at an eqal onstant speed by both the lead and ollower vehiles, the third term o the ar-ollowing model tends to zero nder steady state driving. Conseqently, the steady-state ar-ollowing model that is inorporated within FRESIM an be haraterized by Eqation 9 in the ongested regime and a onstraint o maximm speed on the roadway, as demonstrated in Eqation (nongested regime). This ar-ollowing behavior is idential to the Pipes model that was desribed earlier. h = h + + [8] 3 b3 = h + [9] h 3

8 Raha and Crowther 8 h h = min, [] 3 Where: h = distane headway between ront bmper o lead vehile and ront bmper o ollower vehile (m), h = distane headway when vehiles are ompletely stopped in a qee (m), 3 = driver sensitivity ator (h), b = alibration onstant that eqals. i the speed o the ollower vehile exeeds the speed o the lead vehile, otherwise it is set to zero, = dierene in speed between lead and ollower vehile, and = roadway ree-speed. 4.. NETSIM Car-Following Behavior The basi ar-ollowing model within NETSIM is deined in Eqation. Using the onditions o steady-state travel Eqation an be simpliied to Eqation. This simpliiation reslts rom the at that the stopping distane or both the lead and ollower vehiles are idential given their eqal speeds. In addition, no driver reation is reqired bease the ollower and lead vehiles are traveling at onstant speeds (steady-state low). Frthermore, given that the distane traveled dring a time inrement t is the prodt o the vehile speed and the time dration ( t) and bease NETSIM tilizes a time step o seond, the steady-state ar-ollowing model reverts to the Pipes model with a driver sensitivity ator o. (assming speed is in t/s and distane is in t), as demonstrated in Eqation 3. h = h + s + r + S S [] F L h = h + t [] h = h + [3] Where: s = distane traveled by ollower vehile over time interval t (m) r = distane traveled by ollower vehile dring its reation time (m) S F = distane reqired by ollower vehile to ome to a omplete stop (m) S L = distane reqired by lead vehile to ome to a omplete stop (m) 4. VISSIM Car-Following Model The VISSIM model ses a psyho-physial ar-ollowing model (Fellendor and Vortish, ). The model inorporates the Pipes ar-ollowing logi at its ore, as deined in Eqation. The model also onsiders other ators that inlde; randomness in the am density headway, the speed dierene between lead and ollower vehiles, the distane headway that the vehile reats to a speed dierene, and the driver s pereption o the speed dierene. Conseqently, as is the ase or the CORSIM model, nder steady-state onditions the ar-ollowing logi within VISSIM reverts to the Pipes model. 4.3 Proposed Proedres or Calibrating the Pipes Car-Following Model The Pipes model represents a linear inrease in the travel speed as the distane headway inreases. The driver sensitivity ator ( 3) deines the slope o the speed-headway relationship, while the interept with the x-axis deines the am density headway (h ) o trai, as illstrated in Figre. A third parameter that is reqired in haraterizing the speed-headway relationship is the ree-speed, or the maximm speed o travel when a vehile is not onstrained by the srronding trai. Conseqently, the Pipes ar-ollowing model reqires the alibration o three parameters, namely; the ree-speed, the am density headway, and the driver sensitivity ator. While ree-speed is relatively easy to estimate

9 Raha and Crowther 9 in the ield and generally lies between the speed limit and the design speed o the roadway, am density headway is more diilt to alibrate bt typially ranges between to 5 vehiles/m/lane. The driver sensitivity ator is extremely diilt to alibrate bease it is not measred sing standard srveillane tehnologies (e.g. loop detetors). Two ommon proedres have been tilized to alibrate the Pipes model within the CORSIM and VISSIM sotware. The irst proedre involves a trial-and-error approah that entails hanging the driver sensitivity ator ntil a desired roadway apaity is observed. This approah, however, is networ speii and time onsming. The alternative proedre, whih has been applied to VISSIM, involves driving an instrmented probe vehile along the roadway to ollet data. This approah, however sers rom a nmber o shortomings. First, it is extremely expensive to instrment a vehile or prposes o data olletion deeming it nrealisti. Seond, the approah is driver dependent., whih wold reqire extensive data olletion in order to inlde dierent driver behaviors. Third, it is diilt to ollet data along the entire range o the speed-headway relationship. Forth, dierenes in the speed-headway behavior will or depending on the aility type, as was learly demonstrated earlier in the paper. Conseqently, this paper proposes a alibration proedre that an be applied sing readily available loop detetor data. The derivation o the proposed ntional orm involves two steps. The irst step is to demonstrate that the ntional orm o the Pipes model is a strit monotoni ntion. Seond, having demonstrated that the ntion is stritly monotoni the driver sensitivity ator an be related to the roadway apaity, whih an be estimated rom loop detetor data. The Pipes ar-ollowing relationship in the ongested regime is haraterized by Eqation 4. Sine trai stream density is the inverse o the spae headway, Eqation 5 desribes the basi speed-density relationship that evolves rom the Pipes ar-ollowing model. = h + [4] h 3 = [5] h + 3 Using the basi trai stream relationship that is deined in Eqation in onntion with Eqation 5 the speed-low relationship an be derived, as demonstrated in Eqation 6. q = [6] h + 3 The slope o the speed low relationship is ompted as the derivative o low with respet to speed as demonstrated in Eqation 7. The inal orm o the slope that as ompted by Eqation 8 is a strit monotoni ntion given that the am density headway o vehiles is non-negative and non-zero. Conseqently, the maximm low will or at the extreme point (i.e. at the maximm speed whih is the ree-speed). q = q = h ( h + ) h = h Where : h > ( h ) Using Eqation 6 and reognizing the at that the speed-low relationship is stritly monotoni, the maximm low (q ) ors at the bondary o the relationship (i.e. at ree-speed). Conseqently, the maximm low an be derived rom Eqation 6 by sbstitting the low or the apaity o the roadway (q ) and the speed or the ree-speed ( ). Using Eqation 9, the driver sensitivity oeiient an be ompted, as deined in Eqation. Eqation reqires three parameters that an be obtained rom standard loop detetor data. These parameters inlde the roadway apaity (maximm low rate), the spaing o vehiles at am density, and the roadway ree-speed. [7] [8]

10 Raha and Crowther q = [9] h + 3 h [] 3 q = 5. VAN AERDE CAR-FOLLOWING MODEL The INTEGRATION model (M. Van Aerde and Assoiates, a and b) ses a steady-state ar-ollowing model that was proposed by Van Aerde (995) and Van Aerde and Raha (995), whih ombines the Pipes and Greenshields models into a single-regime model. The model, whih reqires or inpt parameters, an be alibrated sing ield loop detetor data. The details o the alibrating proedre are beyond the sope o this paper bt will be desribed in detail in a orthoming pbliation. The ntional orm o the Van Aerde model amalgamates the Greenshields and Pipes ar-ollowing models, as demonstrated in Eqation. This ombination provides the ntional orm with an additional degree o reedom by allowing the speed-at-apaity to be dierent rom the ree-speed. Speiially, the irst two parameters provide the linear inrease in the vehile speed as a ntion o the distane headway, while the third parameter introdes rvatre to the model and ensres that the vehile speed does not exeed the ree-speed. The addition o the third term allows the model to operate with a speed-at-apaity that does not neessarily eqal the ree-speed, as is the ase with the Pipes model. In smmary, the Van Aerde single-regime model ombines the Greenshields and Pipes models in order to address the main laws o these models. Speiially, the model overomes the shortoming o the Pipes model in whih it assmes that vehile speeds are insensitive to trai density in the nongested regime, whih has been demonstrated to be inonsistent with a variety o ield data rom dierent aility types, as illstrated in Figre 3, Figre 4, and Figre 5. Alternatively, the model overomes the main shortoming o the Greenshields model, whih assmes that the speedlow relationship is paraboli, whih again is inonsistent with ield data rom a variety o aility types, as demonstrated in Figre 3, Figre 4, and Figre 5. It is siient to note at this time that the alibration o the ar-ollowing model reqires estimating or parameters, namely,, 3 and tilizing Eqations a, b,, and d. These or parameters are a ntion o the roadway ree-speed, the speed-at-apaity, apaity, and am density. The derivation o these or eqations is presented in Appendix C sing a nmber o bondary onditions inlding the maximm low and am density bondary onditions. h = ( ) m = [a] = + m = m [] q = [d] 3 + [] [b]

11 Raha and Crowther Where: = ixed distane headway onstant (m), = irst variable distane headway onstant (m /h), 3 = seond variable distane headway onstant (h), = ree-speed (m/h), = speed at apaity (m/h), q = low at apaity (veh/h), = am density (veh/m), and m = is a onstant sed to solve or the three headway onstants (h/m). 6. FINDINGS AND CONCLUSIONS Three marosopi trai stream models and their orresponding mirosopi steady-state ar-ollowing models were presented in this paper. The simplest o these models, namely the Greenshields single-regime model reqires two alibrated parameters: ree-speed and either apaity or am density. Alternatively, the Pipes two-regime ar-ollowing model reqires three alibrated parameters: ree-speed, am density, and a driver sensitivity ator. Finally, the orparameter single-regime model that was proposed by Van Aerde (995) and Van Aerde and Raha (995), while reqiring or parameters or alibration, provides more degrees o reedom to relet dierent trai behavior aross dierent roadway ailities. Speiially, Figre 3, Figre 4, and Figre 5 demonstrate the ability o the Van Aerde model to relet trai stream behavior rom a nmber o varying aility types inlding a reeway, an arterial, and a tnnel roadway. These its demonstrate a good representation o the steady-state marosopi behavior o a trai stream in the low, speed, and density domains. Frthermore, Figre illstrates a good representation o steady-state arollowing behavior or the three aility types that were onsidered. The additional degree-o-reedom that is provided by inlding or parameters in the mirosopi ar-ollowing and marosopi trai stream models overomes the shortomings o the rrent state-o-pratie models by aptring both marosopi and mirosopi steady-state trai behavior or a wide range o roadway ailities. Speiially, Figre 6, Figre 7, and Figre 8 learly demonstrate the shortomings o both the Greenshields and Pipes models in aptring the ll range o trai stream behavior or a single aility in addition to aptring dierenes in trai behavior over dierent ailities. Finally, the proposed modiiation to the alibration proedres o the Pipes model oers an avene to alibrate mirosopi ar-ollowing behavior sing marosopi ield measrements that are readily available rom loop detetors. This alibration proedre, while it does not overome the inherent shortomings o the Pipes model, does provide an opportnity to better alibrate the CORSIM and VISSIM ar-ollowing behavior to existing roadway onditions. REFERENCES Ayin M.F. and Beneohal R.F. (999), Comparison o Car-Following Models or Simlation, Transportation Researh Reord, No. 678, pp Drew D. (968), Trai Flow Theory and Control, MGraw Hill. Halati A., Lie H., and Waler S. (997) CORSIM- Corridor Trai Simlation Model, Proeedings ASCE Conerene on Trai Congestion and Saety in the st Centry, Chiago. May A.D. (99), Trai Flow Fndamentals, Prentie Hall, Englewood Clis, New Jersey. May A.D., Rophail N., Bloomberg L., Hall, F. and Urbani T. () Freeway Systems Researh Beyond the HCM Presented at the 8 th Annal Meeting o the Transportation Researh Board, Washington D.C. Paper No

12 Raha and Crowther Payne H. et al. (997) Calibration o FRESIM to Ahieve Desired Capaities, Transportation Researh Reord, n. 59. p Van Aerde M. (995), Single Regime Speed-Flow-Density Relationship or Congested and Unongested Highways, Presented at the 74 th TRB Annal Conerene, Washington D.C., Paper No Van Aerde M. and Raha H. (995) Mltivariate Calibration o Single Regime Speed-Flow-Density Relationships, Proeedings o the Vehile Navigation and Inormation Systems (VNIS) onerene, Seattle, Washington, Agst. Vellendor M. and Vortish P. (), Integrated modeling o Transport Demand, Rote Choie, Trai Flow and Trai Emissions, Presented at the 79 th TRB Annal Conerene, Washington D.C., Paper No APPENDIX A The or-parameter trai stream model that was proposed by Van Aerde (995) and Van Aerde and Raha (995) is deined in Eqation (A-). The alibration o the model reqires estimating the parameters,, and 3. The allation o these parameters reqires estimating or parameters, as will be demonstrated. These parameters inlde the reespeed ( ), the speed-at-apaity ( ), the apaity (q ), and the am density ( ). = [A-] Using the basi trai stream relationship that is deined in Eqation (A-) in onntion with Eqation (A-), the relationship between the trai stream low rate and speed an be derived, as demonstrated in Eqation (A-3). q = [A-] q = [A-3] Two bondary onditions exist. The irst bondary ondition is that the maximm low rate (apaity) ors at the speed-at-apaity when the derivative o the low rate (q) with respet to speed () is eqal to zero. Conseqently, by dierentiating Eqation (A-3) as demonstrated in Eqation (A-4), Eqation (A-7) is inally derived throgh a nmber o variable maniplations, as smmarized in Eqations (A-5) and (A-6). q = = [A-4] ( ) = [A-5] + = [A-6] ( )

13 Raha and Crowther 3 = m Where : m = [A-7] ( ) The seond bondary ondition is that at am density the trai stream speed is zero. Conseqently, Eqation (A-8) an be derived rom Eqation (A-) by setting the speed to zero at am density. Sbseqently, Eqation (A-9) an be derived rom Eqation (A-8) throgh variable maniplations. = = [A-8] + + = [A-9] By inorporating Eqation (A-9) in Eqation (A-7), the headway oeiient ( ) an be ompted sing Eqation (A-) ater ompting the variable m. Sbseqently, the headway oeiient ( ) an be ompted sing Eqation (A-7). = + m [A-] By solving Eqation (A-3) at apaity where low eqals the roadway apaity (q ) and the speed o the trai stream is the speed-at-apaity ( ), the headway oeiient ( 3) an be estimated sing Eqation (A-). + q = [A-] 3 LIST OF FIGURES Figre : Sample Steady-State Calibrated Car-Following Model Figre : Comparison o Car-Following Models Figre 3: Sample Trai Stream Models (I-4, Orlando) Figre 4: Sample Trai Stream Models (Holland Tnnel, New Yor City) Figre 5: Sample Trai Stream Models (Arterial Road) Figre 6: Comparison o Trai Stream Models (I-4, Orlando) Figre 7: Comparison o Trai Stream Models (Holland Tnnel, New Yor) Figre 8: Comparison o Trai Stream Models (Arterial, UK)

14 Raha and Crowther a. Freeway, Orlando Headway (m) b. Holland Tnnel, New Yor Headway (m) Arterial Road, United Kingdom Headway (m) Figre : Sample Steady-State Calibrated Car-Following Model

15 Raha and Crowther a. Freeway, Orlando Distane Headway (m) b. Holland Tnnel, New Yor Distane Headway (m) Arterial, United Kingdom Distane Headway (m) Figre : Comparison o Car-Following Models

16 Raha and Crowther Flow (vph/lane) Flow (veh/h) Figre 3: Sample Trai Stream Models (I-4, Orlando)

17 Raha and Crowther Flow (vph/lane) Flow (veh/h) Figre 4: Sample Trai Stream Models (Holland Tnnel, New Yor City)

18 Raha and Crowther Flow (vph/lane) Flow (veh/h) Figre 5: Sample Trai Stream Models (Arterial Road)

19 Raha and Crowther Flow (veh/h) Flow (veh/h) Figre 6: Comparison o Trai Stream Models (I-4, Orlando)

20 Raha and Crowther Flow (veh/h) Flow (veh/h) Figre 7: Comparison o Trai Stream Models (Holland Tnnel, New Yor)

21 Raha and Crowther Flow (veh/h) Flow (veh/h) Figre 8: Comparison o Trai Stream Models (Arterial, UK)