Estimation of regional trip length distributions for the calibration of the aggregated network traffic models

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Estimation of regional trip length distributions for the

calibration of the aggregated network traffic models

Sergio Batista, Ludovic Leclercq, Nikolas Geroliminis

To cite this version:

Sergio Batista, Ludovic Leclercq, Nikolas Geroliminis. Estimation of regional trip length

distribu-tions for the calibration of the aggregated network traffic models. Transportation Research Part B:

Methodological, Elsevier, 2019, 122, pp192-217. �10.1016/j.trb.2019.02.009�. �hal-02052638�

ContentslistsavailableatScienceDirect

Transportation

Research

Part

B

journalhomepage:www.elsevier.com/locate/trb

Estimation

of

regional

trip

length

distributions

for

the

calibration

of

the

aggregated

network

traffic

models

S.F.A.

Batista

a,b

_,

_Ludovic

_Leclercq

a,∗

_,

_Nikolas

_Geroliminis

b

a University of Lyon, ENTPE, IFSTTAR, LICIT, Lyon F-69518, France

b School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne (EPFL), Switzerland

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 3 December 2018 Revised 15 February 2019 Accepted 19 February 2019 Keywords: Trip lengths Regional network Regional path

Macroscopic fundamental diagram Trip-based macroscopic fundamental diagram model

a

b

s

t

r

a

c

t

Thecalibrationoftriplengthsisanimportantchallengeformulti-regionalMFD-based ap-plications, as itcan influence the dynamicsofregional densities (and speeds). Existing researchhasnotpaidsignificantattentioninthetopic,givingopportunitiesforanswering somefundamentalquestions.Inthispaper,weproposeanoriginalmethodologyto explic-itlydeterminetriplengthdistributionsbasedonasubsetoftripsinthecitynetworkand itspartitioning.Sincethefull descriptionofallrealizedtripsisdifficulttoestimatefora largecitynetwork,weproposetodefineasetofvirtualtripscorrespondingtoafull cov-erageofpotentiallocalorigin-destinationpairsandshortest-pathsindistance.We inves-tigatehowdifferentlevelsofinformationcaninfluencetheaccuracyofthemulti-regional dynamicMFD-based models,throughtheestimated triplengthdistributions.This infor-mation rangesfromregional triplengthswithoutanyinformation oforigin,destination, previousornextregionsuptothespecificregionalpathassociatedtoeachtrip.

Wetestthismethodologyonasimulatedenvironmentwithasignificantamountofreal informationforanetworkwith757linksthatcorrespondstothe6thdistrictofLyon.We firstprovideguidanceonhowtoproperlydefinethesetofvirtualtrips.Weshowthata singleaveragetriplengthisnotrepresentativeofallpossibletriplengthsestimatedbythe mostdetailedlevelsofinformationforoneregion.Wealsohighlightthatthefirstlevelof informationassignssimilartriplengthsforregionalpathsthatarecomposedbythesame regions,butinareverseorderedsequence.Nevertheless,thisisnotthecaseforthemost detailedlevelsofinformationsincetheycapturethedirectedlinkscitynetworktopology. Weproposeaproceduretoupdatethetriplengthswhenanupdateoftheregional Origin-Destination matrixchanges, withoutaneed forafull recalculation.We alsoinvestigate the influenceofthetriplengthtuningonthe trafficdynamicsoftheregional network. Thetrafficstatesaremodeledbyatrip-basedMFDmodel.Weshowthatthesettingofthe triplengthsinfluencesthetrafficdynamicsintheregions.Moreover,thetrafficconditions predictedbythetwomostdetailedlevelsofinformationareclose.

© 2019TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

∗ _{Corresponding author.}

E-mail address: ludovic.leclercq@entpe.fr (L. Leclercq).

https://doi.org/10.1016/j.trb.2019.02.009

0191-2615/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

1. Introduction

Anetwork level aggregated traffic model wasearly introduced by Godfrey(1969) and later revisitedby Hermanand Prigogine (1979) and Mahmassani et al. (1984). But only after the works of Daganzo (2007) and Geroliminis and Da-ganzo(2008)didthiskindoftrafficmodelsattractmoretheattentionfromthescientific community.Intheseaggregated trafficmodels,thecity network(Fig.1(a)) ispartitionedintoregions (Fig.1(b)),wherethetraffic conditionsare approx-imatelyhomogeneous(JiandGeroliminis,2012).Thecitynetworkencompassesalltheconnectedlinksthat correspondto theexistingroads.LetAbethesetofdirected linksandN bethesetofnodesthatdefinethecitynetwork.Topartitiona networkinanumberofregions, itispossibleto applytheclusteringtechniquesdiscussedintheliterature (Saeedmanesh andGeroliminis,2016;2017;Lopezetal.,2017;Casadeietal.,2018).LetXbethesetoftheregionsthatdefinetheregional network(Fig.1 (c)).The systemdynamicsinside each regionis governed bya conservationequation, where theoutflow is given by a well-defined relation between the mean flow and the accumulation, called the Macroscopic Fundamental Diagram(MFD)(Daganzo,2007).UsingtrafficdatafromthecityofYokohama(Japan),GeroliminisandDaganzo(2008) pro-videdgroundtruthevidenceoftheexistenceoftheMFD.Theexistence andpropertiesofthisrelationshiphavealsobeen confirmedbyotherauthors(GeroliminisandSun,2011;AmbühlandMenendez,2016;Loderetal.,2017).

ThemathematicalformulationoftheMFDwasintroducedbyDaganzo(2007)forasingleregionr.Thetrafficdynamics isgoverned by a state equation that associatesthe accumulationof vehicles (nr(t)) withthebalance betweenthe inflow (Qin,r(t))andtheoutflow(Qout,r(t)):

dnr

dt =Qin,r

(

t

)

− Qout,r

(

t

)

,t>0 (1)

Dependingonthe assumptionmade onQout,r(t), two modelscan be distinguishedinthe literature:the accumulation-based model(Daganzo, 2007; Geroliminis andDaganzo, 2008);and the trip-based model(Arnott, 2013; Fosgerau, 2015; LamotteandGeroliminis,2016;Mariotteetal., 2017; Leclercq etal., 2017;MariotteandLeclercq, 2018). Akey ingredient forimplementing thesemodels is the definitionof the triplengths L,i.e., the travel distance ofvehicles in each region. The simplest assumption is to consider a constant average trip length forall vehicles traveling inthe same region (see e.g., Daganzo,2007). Up to today, theapplications ofthe MFD-basedmodels havemostly beenfortesting different con-trolalgorithmsordesign perimetercontrolstrategies.The firstperimeterflow controlstrategies weredesignedforsingle regioncities (see e.g., Daganzo, 2007; Keyvan-Ekbatanietal., 2012; Ekbatani etal., 2013), wherea constant average trip lengthisassumed.Severalother authorsconsideredtheoutflowMFDapplicationintheoreticalstudies ona2-region net-work(seee.g.,Geroliminisetal.,2014;SirmatelandGeroliminis,2018;Haddad,2017;Zhongetal.,2017;Yangetal.,2018), wherethetravel distances wereequal forall thevehicles inthe sameregion.Other perimetercontrol studies havebeen applied to real city networks, in detailed simulation environments. For example, Aboudolas and Geroliminis (2013) de-signeda perimeter control strategywhere theSan Francisco (USA) downtownnetworkwas partitionedinto threezones. The MFD model was implemented considering a similar trip length value for all vehicles travelingin the same region. Kouvelasetal.(2017)implementedanoutflowMFD-basedmodelforaperimetercontrolapplicationinpartoftheBarcelona (Spain)citynetwork,dividedintofourregions,bycontrollingonlyinter-transfersamongregionsandnottheinflowofthe outer regions ofthe network (as happened in Aboudolas andGeroliminis (2013)). As with the previous studies, a simi-lartriplength per region associatedwiththe outflowMFDassumption, isconsidered. Comparedtothe previous studies, Ramezanietal.(2015) andSirmatel andGeroliminis(2018) considered variablemeantrip lengthsinthe implementation ofanMFDmodelforperimetercontrolpurposes.Theaveragetrip lengthswere estimateddynamicallyforatoy network, basedontheexchangeflowsbetweenadjacentregionsandtheaccumulationofvehicles.Moreover,thetriplengthsarekept constantperOrigin-Destinationregionalpairineachregion.Routeguidancecoupledwithan MFD-basedmodelshasbeen recentlyaddressedinYildirimogluetal.(2015)andYildirimogluetal.(2018),withsimilarapproachestodeterminethetrip lengths.

LittleattentionhasbeenpaidintheliteraturetothepropersettingoftriplengthsforMFD-basedapplications.Ingeneral, a similar distance for all vehicles travelinginside the same region is considered. This is a rough approximation, clearly

Fig. 2. (a) Partition of the city network with three trips represented. (b) Regional paths that correspond to the blue and green trips. (c) Zoom in the gray region, where the trip lengths distributions L 1 and L 2 of the green and blue regional paths are represented, respectively. A scheme of an MFD diagram is

also represented. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

influencingthemodeledtrafficstatesevenwhenthetriplengthsaredistinguishedconsideringdifferentregionalpaths,e.g., per regional OD pairinside the regions, thetrip lengths areassumed to be giventhe MFDmodel. Inreality, theproper settingandcalibrationoftriplengthsfortheapplicationofMFD-basedmodelsisacomplextask.Tobetterhighlightthis, considerasanexamplethepartitionedcitynetwork inFig.2(a),wherethreetripsare represented.Thesetrips represent vehicletrajectoriesinthecitynetwork.Ascanbeobserved,thesetripswillcrossadifferentsequenceofregionsfromtheir origintotheir destinationnodes,definingdifferentregionalpaths (YildirimogluandGeroliminis,2014).Aregionalpathis theordered sequenceofregions crossed by thetrips fromtheirOrigin (O) totheir Destination(D) regions.Generically, a regionalpathpcanbedefinedas:

p=

(

p1,...,pm,...,pR

)

,

∀

m=1,...,R∧m∈X (2) where R is thenumber of regions that define p; p1 and pR are the Origin (O) andDestination (D) regions, respectively. BatistaandLeclercq(2018)distinguishesbetweenregionalpathsthatcrossdifferentregionsandinternalpathsdefinedby tripsthatonlytravelinsidethesameregion.

The associatedregionalpaths totheblue andgreen tripsare representedinFig.2 (b).Letusnow focusonthegreen trips andthe region highlighted in gray in Fig. 2 (b).As can be seen in Fig. 2 (a),the green trips have differenttravel distancesinsidethisgrayregion.Thelatterarealsodifferentfromthetraveldistanceofthebluetripinsidethissameregion. The greenandblue regionalpaths are thereforecharacterized by differenttriplength distributions L1 andL2 (Fig.2 (c)),

respectively.Asfarasweknow,YildirimogluandGeroliminis(2014)isoneofthefirsteffortsthatproposesamethodology to set up different travel distances inside the same region, for MFD-based model applications. The authors discussed a methodology where the time-varyingaverage trip lengths are implicitly estimated. Nevertheless, thisapproach does not allowtakingintoaccountexplicitlyestimateddistributionsoftriplengths.Asfarasweknow,nostudyhasbeenperformed asyet to describe a methodological framework based ontrip patterns andcity network partitioning, aimed atexplicitly definingdistributionsoftriplengthsforthecalibrationofdynamicMFDmodels.Thepurposeofthispaperistofillthisgap. Weproposeamethodologytoexplicitlyestimatedistributionsoftriplengthsbasedonasetoftripsinthecitynetworkand itspartitioning,inordertocalibrateanMFD-basedmodel.Thetruetrippatternsinacitynetworkareunknownandchange overtime.ThequestionnowishowtocalibratetheMFDmodelwhenthetruesetoftripsisnotavailable.Tothisend,we constructasetofvirtual tripsbyrandomlysamplingorigin-destination(od)nodesinthecitynetworkandcalculatingthe shortest-pathindistanceforallodpairs.ThisprocessisperformedonlyonetimebeforerunningtheMFDmodel.Let



be thesetofvirtual tripsandNod bethetotal numberofodpairsthat areuniformlysampledinthecitynetwork.Oncethe setofvirtualtrips hasbeenobtained, thequestionregardingthecalculationoftriplengthdistributions ishowshouldwe differentiatethe tripsinsideeach region?Totacklethisquestion,we proposea methodologyinthispaper,that considers differentlevelsofinformationfromtheregionalnetwork.Thesedifferentlevelsinclude:

• Level1:noinformationontheOriginandDestinationregionsofthetrips. • Level2:thenextregiontobetraveledbythetrips.

• Level3:thepreviousandnextregionstraveledbythetrips. • Level4:thespecificregionalpathassociatedwitheachtrip.

The mathematical formulation of the proposed methodology in this paper is introduced in Section 2. We test this methodology onthe 6thdistrict ofthe Lyonnetwork (France),which is dividedinto eightregions. In Section 3,we first investigatehowtheset



should beproperlycalibratedintermsofN_od.Wethenanalyzethedistributions oftriplengths foreach region, considering thefourlevels of information.Changesin theorigins anddestinationsofthe trips canhave a significant influence onthe trip lengths(Leclercq etal., 2015). Therefore,we introduce a procedure to update thetrip lengths, whentheOD flowsare modifiedinsidetheregionalOD matrix,withouttheneedto re-samplethe setofvirtual tripsinthecitynetwork.TheupdatedODmatrixhasthesamezoningfororiginsanddestinationsbutadifferentnumber oftripsbetweenODpairs.InSection3,weconsiderthestaticcasewhentheODmatrixisconstantovertime.InSection4, weinvestigatehowthetriplengthsestimatedonthebasisofthefourlevelsofinformationinfluencethetrafficdynamics

intheregions.Todothis, weconsiderthe trip-basedMFDmodelasdescribedinMariotteetal.(2017).We considertwo scenarios.Inthefirstscenario,weassumethatwehavethefullinformationonthetrippatternsinthecitynetwork.So,we cannotonlydeterminethetime-dependentevolutionoftheODmatrixbutalsothetripdistributionsforallregionalpaths. Inthesecond scenario,westill knowthetime-dependentODmatrixbutwe computethetrip distributionsbased onthe DeterministicUserEquilibrium.InSection5,weoutlinetheconclusionsofthispaper.

2. Methodologicalframework:triplengthdistributionsforthecalibrationofMFD-basedmodels

OneofthecorevariablesforthecalibrationofanMFD-basedmodelisthesettingofthetriplengths,i.e.thedistancethat vehicleshavetotravelinaregion.InSection2.1,wedescribetheroleoftriplengthsintheformulationofthe accumulation-based(Daganzo,2007;GeroliminisandDaganzo,2008)andtrip-based(Arnott,2013;Fosgerau,2015;Lamotteand Gerolim-inis,2016;Mariotteetal.,2017;Leclercqetal.,2017;MariotteandLeclercq,2018)MFDmodels.InSection2.2,weintroduce themathematicalbasisofthemethodologicalframeworkproposedinthispapertoestimatethedistributionsoftriplengths. InTable1wesummarizethenotationsofallthesymbolsandvariablesusedinthispaper.

2.1. TheroleoftriplengthsintheformulationofMFD-basedmodels

Thetraffic statesinsidearegion raremodeled throughthetemporal evolutionofastate variablecalledaccumulation

nr(t)(seeEq.(1)).Intheliterature,twoMFD-basedmodelscanbedistinguished:theaccumulation-basedmodel(Daganzo, 2007;GeroliminisandDaganzo,2008);andthetrip-basedmodel (Arnott, 2013;Fosgerau, 2015;LamotteandGeroliminis, 2016;Mariotteetal.,2017;Leclercqetal.,2017;MariotteandLeclercq,2018).

In the accumulation-based model,the outflowfunction Qout,r(t) (Daganzo, 2007) is defined asthe ratio between the production-MFDPr(nr(t))andtheaveragetriplengthLr:

Qout,r≈Pr

(

nr

(

t

))

Lr

(3)

whereLristhedistancethatvehicleshavetotravelinregionrtocompletetheirtrips.

Thisoutflow function (see Eq. (3)) is then embeddedin the dynamic equation that describes the evolution oftraffic statesinthesystem. Forasingleregion,the stateistheaccumulationnr(t) oftheregion.Formulti-regionsthestate rep-resentationismorecomplicatedasOrigin-Destinationflowsmatter.Thestaterepresentationformulti-regionalsystemscan havedifferentdegreesofcomplexity.Themostparsimoniousmodelsutilizestatesoftypenij,whereiisthecurrentregion andj thefinalone.Then thesametriplengthcan beassumedforall vehiclescrossinga regioniindependentlyofwhere theyare comingfromorwhat istheregional pathtowards their destination(Geroliminisetal., 2014). Morecomplicated applications(YildirimogluandGeroliminis,2014;Geroliminis,2015;MariotteandLeclercq,2018)considerdifferentregional pathscrossing thesame regionrare assigneddifferenttrip lengths. LetLr p bethe averagetrip lengthofregional pathp insideregionr.TheoutflowfunctionQout,r(t)isrewrittenas:

Qout,rp≈n_nr p

(

t

)

r

(

t

)

Pr

(

nr

(

t

))

Lr p

(4)

wherenrp(t)istheaccumulationofvehiclestravelingonregionalpathpandinsideregionr;andnr(t)isthetotal accumu-lationofvehiclesinregionr.Forthesimplestmodels,Eq.(4)simplifiestoQout,i j≈

n_{i j}(t) n_i(t)Pr(nLr_i(t)).

Inthe trip-based model (Arnott, 2013; Fosgerau, 2015; Lamotteand Geroliminis, 2016;Mariotte etal., 2017; Leclercq etal.,2017;MariotteandLeclercq,2018),theMFDdynamicsiscenteredonthevehicletriplengthLir:

Lir=

texit

tentry

v

r

(

n

(

s

))

ds (5)

wheretentryandtexitaretheentryandexittimesofthevehicleiinregionr,respectively;andvr(n(s))isthespeed-MFD.The traveltimeofvehicleiinregionristtravel=texit− tentry.Weassumethatallvehiclestravelingonthesameregionalpathp andinside regionrhavethesametravel distance.Moreover, weassume thatLir istime-invariantandthat bothttravel and

nr(t)arecontinuousfunctionsanddifferentiableatt,basedonEq.(5)andunderfluidconditions,itispossibletogatherthe followingrelationfortheoutflowQout,r(t)(Arnott,2013;Mariotteetal.,2017;MariotteandLeclercq,2018):

Qout,r

(

t

)

=Qin,r

(

t− ttravel

)

v

r

(

nr

(

t

))

v

r

(

nr

(

t− ttravel

))

(6)

ThesettingoftriplengthsforthecalibrationofanMFDmodelcannotbeneglected.Inan MFDframework,thevehicles speedisassumedtobehomogeneous.Thus,thetriplengthsdictatethetimethatvehicleshavetospendtocompletetheir tripsinside aregion r.Longertriplengthsmeanthat vehiclesspend moretime tocomplete theirtrips inside theregion, increasingthe accumulationnrandthusthe modeledtraffic states.Inthenext section,we introducethemethodology to estimatethe distancetobe traveledby thevehicles insideeach region,that willdependon thelevelofinformationthat weconsiderfromtheregionalnetworkasstatedintheintroduction.Theinfluenceofthetriplengthstuningonthetraffic dynamicsmodeledbyatrip-basedMFDmodel,willbeinvestigatedinmoredetailinSection4.

Table 1

Nomenclature used in this paper.

City and regional networks:

o Origin node.

d Destination node.

O Origin region.

D Destination region.

W Set of regional OD pairs.

p Regional path.

R Number of regions that define regional path p . X Set of regions that define the regional network.

r Region.

k Virtual trip in the city network. N Set of nodes of the city network. A Set of directed links of the city network.

Nnodes Total number of nodes in the city network.

Nlinks Total number of links in the city network.

OD _{Regional choice set of regional OD pair.}

MFD models:

t Time instant.

T Simulation period.

nr ( t ) Accumulation of vehicles in region r .

Qin,r ( t ) Inflow function of region r .

Qout,r ( t ) Outflow function of region r .

Pr ( n r ( t )) Production MFD.

vr ( n ( s )) Speed-MFD.

nij Accumulation of vehicles that travels on region i and goes to j .

Lr Average trip length inside region r .

i Vehicle.

tentry Entry time of vehicle i in region r .

texit Exit time of vehicle i of region r .

ttravel Travel time of vehicle i in region r .

Lr p Average trip length of regional path p inside region r .

Set of virtual trips:

 Set of virtual trips.

Nod Number of virtual trips listed in .

Nnodes

used(Nod) Total number of nodes visited in the city network.

Nlinks

used(Nod) Total number of links visited in the city network.

 Set of regional paths defined by the virtual trips listed in .

LMx

p Average trip length of regional path p considering the level x ( M x ) of information.

Z and Z ∗ _{Regional OD matrices.}

Level 1:

Lr Distribution of trip lengths of region r .

Lr Average trip length of region r .

lrk Length of trip k in region r .

δrk Binary variable that equals 1 if trip k travels on region r ,or 0 otherwise.

Level 2:

j Next adjacent region to r .

Lrj Distribution of trip lengths to go from region r to region j .

Lr j Average trip length to go from region r to j .

δrjk Binary variable that equals 1 if trip k travels in region r and goes to region j , or 0 otherwise.

 Set of adjacent regions to r .

γ Binary variable that equals 1 if p is composed of 2 or more regions, or 0 otherwise.

Level 3: internal trips

Lr Distribution of trip lengths of region r .

Lr Average trip length of region r .

ηrk Binary variable that equals 1 if k is an internal trip of region r ,or 0 otherwise.

Level 3: crossing trips

h Previous origin region.

Lhrj Distribution of trip lengths of region r ,for trips coming from region h and going to region j .

Lhr j Average trip length to go from region h to j and crossing region r .

δhrjk Binary variable that equals 1 if trip k comes from region h and goes to j by crossing region r , or 0 otherwise.

γ Binary variable that equals 1 if p is composed 3 or more regions, or 0 otherwise.

Table 1 ( continued )

Level 4:

Lp_i Distribution of trip lengths for p on region r .

Lpr Average trip length of p on region r .

δp

rk Binary variable that equals 1 if trip k travels in region r and both define regional path p , or 0 otherwise.

Other variables:

ρ Relative differences of the average trip lengths calculated for the i −th level of information, between the sets of N od = 400 and 5000 and the reference set of N od = 10 , 0 0 0 .

δlr Binary variable that equals 1 if link l lies inside region r .

δnr Binary variable that equals 1 if node n lies inside region r .

βip Relative differences between the average trip lengths calculated by the j−th level of information ( j = 1 , 2 , 3 ) and the reference level of information M 4 .

αOD

r Scaling factor.

ˆ

L Estimated trip length based on the updated OD matrix.

θ Relative differences between ˆ L and L .

ζs Descent step of the Method of Successive Averages.

s Descent step iteration of the Method of Successive Averages.

Nmax Total number of descent step iterations.

Gap Relative gap between the regional paths travel time and the travel time at the network equilibrium.

tol Tolerance for the Gap criterion.

2.2.Methodologicalframeworkfortheestimationoftriplengthdistributions

Themethodologyproposedinthispapertoestimatethetriplengthdistributions,itutilizesarepresentativesetoftripsin thecitynetworkaswellasthedefinitionofitspartitioning.Topartitionthecitynetwork,wecanapplyoneofthedifferent methodsdescribedintheliterature(SaeedmaneshandGeroliminis,2016;2017;Lopezetal.,2017;Casadeietal., 2018).To gatherthesetoftrips,theidealscenariowouldbetouserealtripsinthecitynetwork.However,thefullsetofthelatteris usuallyunknownanddifficulttoestimate.Tobypassthisobstacle,weconstructasetofvirtualtripsinthecitynetwork,by uniformlysamplingN_ododpairsandcalculatingapaththatconnectseachofthem.Inthispaper,weconsiderthe shortest-pathsindistanceestimatedbytheDijkstraalgorithm.Wefurthernote thatother approachescan beusedtoestimate the virtual trips.To nameafew examples, we canconsider thelink elimination(Azevedoet al.,1993) orlink penalty (de la Barraetal., 1993) approaches, thebranch-and-boundalgorithm(PratoandBekhor,2006),the doublystochastic approach (Nielsen,2000)ortheimplementationoftheMetropolis–Hastingalgorithm,asproposedby FlötterödandBierlaire (2013). Exploringtheinfluenceofthe localtripscalculation onthegeneration ofthevirtual tripssetis beyondthescopeofthis study.

Oncethesetofvirtualtrips



isdefined, toestimate thedistributionsoftrip lengths,we differentiatethetrips inside eachregionraccordingtofourlevelsofinformationfromtheregionalnetwork.Theselevelsare:

• Level 1(M1): no information on the regional Originand Destination of the trips. This level followsthe original idea of

Daganzo(2007)forMFD-basedmodelsapplicationsandconsistsofasingleaveragetriplength(Lr)forallvehicles trav-eling inregion r. Alltrips that crossregionr areconsidered, independentlyofthe previous andthenext regions that thesetripstravel,toestimateanaveragetriplengthLras:

Lr=  klrk  k

δ

rk ,

∀

k∈



(7)

wherelrkisthelengthofthepartofthetripkthatoccursinregionr;and

δ

rkisabinaryvariablethatequals1iftripk travelsonregionr,or0otherwise.

Thedistributionoftriplengthsofregionris:

Lr=

{

δ

rklrk

}

,

∀

k∈



(8)

InFig.3(a)we showtheapplicationofthisaggregationlevelforthegreenregion.Thisregionishighlightedandfour trips are shown:their originnodesare representedby blackcircles andthedestination nodesby arrows.Eachtrip is identifiedbyanidentificationnumber.Threeofthesetrips(i.e.,trips1,2and4)simplycrossthegreenregion,whileone hasitsoriginnodeinsidethegreenregion(i.e.,trip3).Toestimatethedistributionoftriplengthsforthegreenregion, we aggregatethelengthofthepartofthesetripsthat occurinsidethegreenregion.Theselengthsare representedby thesolidblacklinesinFig.3(a).Notethat,althoughtheinternaltrips(i.e.thathaveoriginanddestinationnodesinside thegreenregion)arenotshown,theyarenonethelessconsidered.

• Level2(M2):nextregiontobetraveledbythetrips.Theideaistoconsiderallthetripsthatcrossregionrandthatgoto

thesamenextadjacentregion,independentlyoftheirpreviousadjacentregion.Wefilterthetripson



thatcrossregion

Fig. 3. Example of application of the methodology to estimate the distributions of trip lengths, considering the four levels of information. Levels 1 to 4 are represented in the sub figures (a) to (d), respectively.

thatare utilizedindifferenteffortsforperimetercontrolofmulti-regionalnetworks(see forexampleRamezanietal., 2015). Fortheseworksitis notimportant toknowfromwherevehicles startedtheir trips,butonly whatis thenext regiontowardsthedestination.

TheaveragetriplengthLr jtogofromregionrtojisestimatedas:

Lr j=  k

δ

r jklrk  k

δ

r jk ,

∀

k∈



∧

(

∀

j∈



∨j=r

)

(9)

where

δ

rjkisabinaryvariablethatequals1iftripktravelsinregionrandgoestoregionj;and



isthesetofadjacent regionstor,or0otherwise.Notethat j₌r representsthecaseofadestinationregionofthetrips.

Thedistributionoftriplengthsofregionrthatgotojis:

Lr j=

{

δ

r jklrk

}

,

∀

k∈



(10)

InFig.3 (b)weshow theapplicationofthisaggregation level.Fiveexamplesoftrips are shownforthegreenregion. Allofthesetripshavethesameadjacentdestinationregion(thepinkone).Trip2hasadestinationnodeontheborder betweenthe greenandthe pinkregions. The length ofthesetrips that occur inside the greenregion are aggregated. Thisisrepresentedbythesolidblacklines.NowletusconsiderthepinkregionshowninFig.3(b),whichisacommon destinationregionofthetripsrepresentedinit.Trip6representsanexampleofaninternaltrip.Inthiscase,thelengths oftrips1to6thatoccurinsidethepinkregionareaggregatedtodefinethedistributionoftriplengthsforthisregion. • Level3(M3):thepreviousandthenextregionstraveledbythetrips.Theideaistoconsiderall thetrips thatcrossregion

randthat comefrom thesameprevious adjacent regionh andgoto the samenext adjacentregion j. Thereasoning behindthiscalculation isthat whenpartitioned, regions haveasymmetricanduneven shapesand vehiclesthat come fromdifferentprevious regionsmighthavesignificantlydifferenttriplengthsinthecurrentregionofconsideration.In thiscase,wemustconsidertwodistincttypesoftrips:

– Thefirsttype correspondstotheinternaltripsofregionr,i.e.internalpaths.Inthiscase,theaveragetriplengthLr is: Lr=  k

η

rklrk  k

η

rk ,

∀

k∈



(11)

wherelrkisthelengthoftripk;and

η

rk isabinaryvariablethatequals1iftripkisaninternaltripofregionr,or0 otherwise.

Thedistributionoftriplengthsofregionris:

Lr=

{

η

rklrk

}

,

∀

k∈



(12)

– Thesecondtypecorrespondstotripsthatcrossregionr.Inthiscase,alltripstravelinginregionr,comingfromthe sameprevious adjacent regionh andgoing tothe nextadjacent regionj are aggregatedto define theaverage trip lengthLhr j,togofromregionhtojandcrossingregionr:

Lhr j=  k

δ

hr jklrk  k

δ

hr jk ,

∀

k∈



∧

∀

(

h,j

)

∈



∧h=j (13)

where

δ

hrjk isabinaryvariablethatequals1iftripkcomesfromthepreviousregionhandgoestothenextregion

j,bycrossingregionr,or0otherwise.

Thedistributionofthetriplengthsofregionr,fortripscomingfromregionhandgoingtoregionjis:

Lhi j=

{

δ

hi jklik

}

,

∀

k∈



(14)

InFig.3(c), weshow an exampleofanapplicationofthiscaseofLevel3forintermediate regions.Todothis, we consider thetwo trips that comefromthe yellowregion (the equivalentto region h), crossthe green region(the equivalenttothe regionr),andgo tothepinkregion (the equivalentto regionj).The lengthsofthepartofthese two trips inside the green region are aggregatedtogether to estimate the distribution of trip lengthsLhrj. This is representedbythesolidblacklines.

• Level4(M4):therelatedregionalpathpdefinedbyeachtrip.Thisisthemostgeneralcaserequiringasignificantnumber

ofinformation.Thatis,weonlyconsiderthetripsthatcrossregionr,followingthesamespecificsequencehijofregions andthatdefinethesameregionalpathp.Thus,allthetripsthatcrossregionranddefinethesameregionalpathp,are consideredtoestimatetheaveragetriplengthofpinregionr(L_rp):

Lrp=  k

δ

p rklrk  k

δ

p rk ,

∀

k∈



(15)

where

δ

_rkp isabinaryvariablethatequals1iftrip ktravelsinregionrandisassociatedwiththeregionalpathp,or0 otherwise.

Thedistributionofthetriplengthsoftheregionalpathpinsideregionris:

Lrp=

{

δ

rkplrk

}

,

∀

k∈



(16)

InFig.3(d),weshowtheapplicationofthisaggregationmethod.Twoexamplesoftrips thatdefinethesameregional path,definedbythesequence oftheblue-yellow-green-pinkregions,are shown.TodetermineLrpforthegreenregion, weaggregatethelengthofthepartofthesetripsinsidethisregion.Thisisrepresentedbythesolidblacklines. Level1isthemoregenericone,whereallthetripsthatcrossthesameregionareconsideredfortheaggregation.Level 4isthemostrefinedintermsofthelevelofinformationconsidered.Itallowsdefiningaspecificdistributionoftriplengths foraregionalpaththatcrossesagivenregion,basedonthespecificsub-setoftripsthatdefinethisregionalpath.Thus,we considerLevel4asthereferencethroughoutthispaper,evenifitmightbethemostchallengingtocomputewithrealdata. WereinforcetheideathatourgoalistoassesstheinfluenceofthetriplengthstuningforMFD-basedmodelapplications.In thelattercase,itisM1 whichisusuallyconsideredintheabsenceofmoreinformationonapropercalibrationframework.

Thepurposeofthispaperistosetupthisframeworkandtoinvestigatetheinfluenceonthetrafficdynamicsintheregions, whenthetriplengthsareestimatedbetweenM1,M2 andM3 andthereferenceM4.

Theaveragetriplengthofregionalpathpcanbedirectlyestimatedconsideringanyofthepreviouslydiscussedmethods asfollows: 1.Level1(M1): LM1 p = R  m=1 Lpm (17) 2.Level2(M2): LM2 p =

γ

R−1  m=1 R  l=m+1 Lpmpl+LpRpR (18)

where

γ

isabinaryvariablethatequals1ifR≥ 2,or0otherwise. 3.Level3(M3): LM3 p =



LpR ifR=1 Lp1p2+

γ

R−1 m=2Lpm−1pmpm+1+LpR−1pR ifR≥ 2 (19)

where

γ

isabinaryvariablethatequals1ifR≥ 3,or0otherwise.Notethat:Lp_R isestimatedaccordingtoEq.(11);Lp₁p₂ andLpR−1pR areestimatedaccordingtoEq.(9);andLpm−1pmpm+1 isestimatedaccordingtoEq.(13).

4. Level4(M4): LMp4= pR  m=p1 Lmp (20)

Althoughthe fourlevelsofinformation mightgivesimilar estimations ofLp (Eqs.(17)–(20)) forthemean valuewhen the shape of the regions is symmetrical and the distribution of demand and congestion is homogeneous, this might not be the case with the variance. For accumulation-based MFD models only the average trip lengths matter. How-ever, forthe trip-basedMFD models, thevariance ofthe trip length distributions plays a significant role in theoutflow. Lamotteetal.(2018)showsthattriplengthdistributionscaneasilybeimplementedinthisframework.

Thetriplengthsdependonfactors suchasthedefinitionofN_od,theregionalODmatrix(Leclercq etal., 2015) andthe trafficconditions(i.e.congestionpatterns).Inthenextsections,weinvestigatetheinfluenceofthefirsttwofactorsonthe triplengths.Theinfluenceofcongestiontogetherwithamethodologytoestimatetime-varyingtriplengthdistributionswill beinvestigatedinfutureresearch.

3. Analysisofthetriplengthdistributionsobtainedbythefourlevelsofinformation

In thissection, we analyze the differencesbetweenthe trip lengthsestimated through thefour levels ofinformation (M1toM4).Thetestnetwork isthe6th _{districtoftheLyonnetwork(France),dividedintoeightregions.Thetestnetwork} and respective MFDfunctions are introduced in Section 3.1. In Section 3.2, we investigate the importance of calibrating

Nodwell toobtain thesetofvirtualtrips andestimatethe triplengthdistributions. Basedon thisanalysis, we choosean appropriate setof virtualtrips andin Section3.3we analyzethe regionaltrip lengthdistributions estimatedby thefour levelsofinformation.InSection3.4,weinvestigatetheinfluenceoftheregionalODmatrixonthetriplengths.Wepropose aprocedurethatallowsupdatingtheregionaltriplengthswhenthenumberoftripsbetweenthedifferentODpairschange, withouttheneedtore-samplethesetofvirtualtrips



.ThisisveryusefultorundifferentMFD-basedmodelapplications fordifferentdemandscenarios.

3.1. Networkdefinition

ThecitynetworkconsideredfortestingthemethodologyproposedinSection2.2isthe6thdistrictoftheLyonnetwork (France)showninFig.4.Itiscomposedof757 linksand431nodes.Thecitynetworkispartitionedintoeightregions,for whichwefittedtheMFDfunctions(Fig.4)consideringthedimensionsofeachone.Weassumeabi-parabolicshapeforthe MFD(oneparabolafortheincreasingandoneforthedecreasingpartoftheMFD,withafirst derivativeequaltozerofor thecriticalaccumulationthatmaximizesproduction).

3.2. DefinitionofN_odforthecalculationoftriplengths

Inthissection,weanalyzetheimportanceofproperlysettingthevalueofNodtocalibratethesetofvirtualtrips



.We alsoinvestigateitsinfluenceonthecalculationofthetriplengths,consideringthefourlevelsofinformation(i.e.M1toM4).

The propersettingofNod isessential toensurethat thefull city networkhasbeencovered bytheset



.Alow value ofNoddoesnotguarantee thatall thelinksandnodesofthecitynetwork havebeenvisited.Tobetter elucidatethis, we execute100trialsof



fordifferentvaluesofNod:200;400;600;800;1000;5000;and10,000.Theodpairsareuniformly sampledinthecitynetworkforallNodtrials.Forthese100 trialsandforeachNodvalue,weestimatethefractionoflinks andnodesthathavebeenvisitedbythesetsofvirtualtrips.Wedefinetwocriteriathatestimatethepercentageofthecity networknodes(Nnodes

cov

(

Nod

)

)andlinks(Ncolinksv

(

Nod

)

)visited:

Nnodes cov

(

Nod

)

= Nnodes i=1 Nnodes used

(

Nod

)

Nnodes (21) Nlinkscov

(

Nod

)

= Nlinks i=1 Nlinks used

(

Nod

)

Nlinks (22) where Nnodes

used

(

Nod

)

andNusedlinks

(

Nod

)

are the totalnumber ofnodesandlinks coveredin thecity network forone trialofa fixedNod,respectively;andNnodesandNlinksrepresentthetotalnumberofnodesandlinksofthecitynetwork,respectively. Inthiscase,Nnodes=431andNlinks=757.

ForeachvalueofNodpreviouslystatedandbasedonthe100trials,weestimatetheaverageandstandarddeviationsfor

Nnodes

used

(

Nod

)

(Eq.(21))andNusedlinks

(

Nod

)

(Eq.(22)). Theseresultsare showninFig.5.WhenNod=200,we coveraround90% ofthenodesand78%ofthelinksofthecitynetwork.AsN_odincreases,bothNnodes

used

(

Nod

)

→1andNlinksused

(

Nod

)

→1withlow standarddeviations,showingthatthefractionofnodesandlinkstraveledby thevirtualtripslistedin



are verycloseto thetotalnumberofnodesandlinksinthecitynetwork.Moreover,thestandarddeviationiscloseto0forNod≥ 5000.This highlightsthat thefractionofnodesandlinksvisitedbythe100 trialsfora fixedNod valueare equal.Nevertheless,what

Fig. 4. Top: Lyon 6th district network divided into 8 regions. Bottom: MFD function of each region.

istheinfluenceofgoodcitynetworkcoverage forthecalculationofthetriplengthdistributions?Toanswerthisquestion, we analyze therelative differences

ρ

between average trip lengthsestimated through all fourlevels of information (M1

to M4). We choose three sets of Nod=400,5000 and 10,000. The last value of Nod is considered as the referencesince all nodes (Nnodes

used

(

Nod

)

=1) have been visited by the virtual set of trips (see Fig. 5). We also consider Nod=400 where

Nnodes

used

(

Nod

)

∼ 0.95andNod=5000whereNusednodes

(

Nod

)

→1.Thesevaluesguaranteethatwecompareacasewithasituation similartothereference(i.e.Nod=5000)andanotherthatisnot(i.e.Nod=400).Thisanalysisprovidesbetterunderstandof thelinkbetweentheresultsshowninFig.5andthecalculationofthetriplengths.Weestimatetherelativedifferences

ρ

, betweentheaveragetriplengthsestimatedfromthesetsofN_od=400and5000andthereferencesetofN_od=10_,000_,as:

ρ

= L x Mi− L re f Mi Lre fMi ,

∀

i=1,...,4∧x=400,5000 (23)

whereM_ireferstoithlevelofinformation(i.e.M1toM4)andxreferstotheNod=400,5000.

Thedistributionsoftherelativedifferences

ρ

forallfourlevelsofinformationaresummarizedintheboxplotsshownin Fig.6(a)forNod=400andinFig.6(b)forNod=5000.Thehorizontalredlinesrepresenttheaverageofthe

ρ

distributions whilethe red crossesrepresenttheoutliers. Levels3 (M3) and4(M4) aresplit into thefourpossible cases discussedin

Section2.2,i.e.forthe calculationofthe triplength distributions fortheOrigin (O), Intermediate(I)andDestination(D) regionsoftheregionalpathsaswellasfortheinternal paths(referredtoasint.inFig.6).Weobservethatthevariances ofthe

ρ

distributionsarelowerforNod=5000comparedtoNod=400.Theaverageofthe

ρ

distributionsarealsobasically

Fig. 5. Average and standard deviation for N nodes

used(Nod) (a) and N linksused(Nod) (b) estimated based on the 100 trials and for each N od value considered. The blue dots represent the average values and the vertical bars represent the standard deviations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Distributions of the relative differences ρfor the four levels of information ( M 1 to M 4 ) and both N od = 40 0 , 50 0 0 compared to the reference

Nod = 10 , 0 0 0 . Levels M 3 and M 4 have been divided into the cases for Origin (O), intermediate (I) and destination (D) of the regional paths as well as the

internal paths (int.).

0forNod=5000.Thismeansthattheaveragetriplengthsareverysimilarforsetsofvirtual tripsassociatedwithdistinct valuesofNod,whenNusednodes

(

Nod

)

=1.Thismeansthatanodehasbeenvisitedbythevirtualtripsatleastonetime.Forthis specificcase,thisoccursforN_{od≥ 50}00.WenowanalyzetheresultsshowninFig.6inmoredetail.Level1(M1)isthemost

robustregardingthedefinitionofNod,i.e.wheretherelativedifferencesoftheaveragetriplengthsforbothNod=400,5000 andthe referencevalue Nod=10,000are thelowest. Indeed,forM1,we considerall the virtualtrips that travel ineach

region toestimate thetrip length distributions.It isthus moreprobableto obtaina statisticalnumberofvirtual trips to estimate thedistributions oftriplengthsforM1 thanwiththeother threelevels ofinformationM2 toM4.ForNod=400 comparedtothereferenceNod=10,000,thelargestrelativedifferencesare ∼10mforregions4and5,meaningthat

ρ

∼ 4%. Thenumbersofvirtualtripsusedtodefinethedistributionsoftriplengthsforregions4and5are95and162,respectively.

Table 2

Average trip lengths [in meters] for the outliers of the ρdistributions shown in Fig. 6 , for M 2 and the three N od = 40 0 , 50 0 0 , 10 , 0 0 0 values. The number of trips that define the associated trip length distributions are also listed.

Region Nod = 400 Nod = 50 0 0 Nod = 10 , 0 0 0 r j Lr j k∈δrjk Lr j k∈δrjk Lr j k∈δrjk 3 7 259 8 ∼ ∼ 341 233 5 4 19 1 232 14 161 19 7 6 220 36 ∼ ∼ 180 1010 8 6 171 1 443 1 327 8 Table 3

Average trip lengths [in meters] for the outliers of the ρdistribu- tions shown in Fig. 6 , for M 3 . The number of trips that define the

associated trip length distributions are also listed. Region Nod = 400 Nod = 10 , 0 0 0 ρ

h r j Lhr j k∈δhrjk Lhr j k∈δhrjk

1 2 3 103 1 339 95 −0.70 4 5 6 144 1 321 50 −0.55 4 6 5 185 1 344 42 −0.46

Thesenumbersarehigherthanthenumberoflinksandnodesofthecitynetworkthatlieinsidethesetworegions.Let

δ

kr beabinaryvariableequalto1ifvirtualtripktravelsonregionr,or0otherwise;and

δ

lr and

δ

nrbebinaryvariablesthat equal1iflinklornodeklie insideregionr,respectively,or0otherwise.Thus k∈δkr

n∈Nδnr ∼ 1.2and  k∈δkr  l∈Aδlr ∼ 1.7forregion r=4;and  k∈δkr  n∈Nδnr ∼ 2.5and  k∈δkr  l∈Aδlr ∼ 3.9forregionr=5.

TheotherthreelevelsofinformationM2 toM4requirealargervirtualtripsetsize,Nodforagoodestimationofthetrip lengthsduetotheirnature.ForlevelM2,weobservethepresenceofoutliersinFig.6forbothNod=400,5000values.Let usfirstanalyzethecaseofNod=400tounderstandthepresenceofthesefouroutliers.Theseoutliersoccurfortheaverage triplengthsL37(i.e.fortheaveragetriplengthtogofromregionr=3totheadjacentregionj=7-seeEq.(9)),L54,L86and

L76.TheseaveragetriplengthsarelistedinTable2forthethreeNodvalues,aswellasthenumberofvirtualtripsassociated withthesetriplengthdistributions.Notethat,asdefinedinSection2.2forM2,



k∈

δ

rjkdefinesthetotalnumberoftrips thattravelinregionrandgothesameadjacentregionj.Ascanbeseen,todefine thetriplengthdistributionwhengoing fromregionr₌3to j₌7 thereare only8virtual trips forN_od=400_, whilethereare 233virtual tripsforthe reference

Nod=10,000.The averagetrip length L76 representsthelowest outlierin Fig.6,where

ρ

∼ 0.2.In thiscase, thenumber

ofassociated virtual trips is36, whichis statisticallymore significant.For theother two trip length distributions (i.e. to go from regions r₌5 andr₌8 to j₌4 and j₌6_, respectively) the number of associated virtual trips is low, despite itsincrease verified forthethree Nod values.Indeed,the entryandexitlinksofthe nodeslocatedon theborders ofthe city network partitioning might not allow traveling inboth directionsbetween two adjacent regions. This is because of thepresenceof one-waylinks,which occursinmostofthe6thdistrict oftheLyon network.Fig.4 showsthatthere are 5nodeslocated on theborderbetween regions 4and5,while 4nodes locatedon theborder betweenregions 8and6. Nevertheless,onlytwo nodesallowtravelingfromregions 5to 4andonlyone node allowstravelingfromregion8 to6. WiththeincreaseofN_od,weincreasetheprobabilityofhavingalargernumberoflongervirtualtripsthatcrosstheborders ofthe citynetworkpartitioningatthesenodes.Thus,forveryspecifictriplength distributions,thenumberofconnection nodesbetweenadjacentregionsalsoplaysarole.Thisbecomesmoresignificantasweincreasethelevelofinformationfor

M3 andM4.Thismightfurtherreducethetotalnumberofvirtualtripsusedtodefinethetriplengthdistributions.

Wenow focusouranalysison M3 andM4.Ascan be seenin theboxplots showninFig.6,the distributions of

ρ

are

narrower forNod=5000 compared to Nod=400. The increase of Nod, increases the probability that a larger number of virtualtripsareconsideredtoestimatethedistributionsoftriplengths.However,severaloutlierscanbeseen,especiallyfor theintermediateanddestinationregions.Letusfirstanalyzetheoutliersofthe

ρ

distributionsforM3 andNod=400.The averagetriplengthsLhr j(seeEq.(13))forthesethreeoutliersarelistedinTable3.NotethatasdefinedinSection2.2forM3,



k∈

δ

hrjkdefinesthetotalnumberofvirtualtrips kassociatedwiththetriplengthdistributionLhrj.AsshowninTable3, theseoutliersoccurbecauseforN_od=400thereisonlyonevirtual triptodefinethetriplengthdistribution.However,the numberofvirtualtripsbecomemuchmoresignificantforthereferenceN_od=10,000.Wealsoanalyzethelargestoutlierof the

ρ

distributionfortheintermediateregions andforNod=5000.ThisoutlieroccursforthetriplengthdistributionL645,

forvirtual trips travelingon region r₌4_,coming fromregion h₌6andgoing toregion j₌5.This sequenceof regions is not covered by the virtual trip for N_od=400. The average trip lengthsL645 are 179m for Nod=5000 and 105m for

Nod=10,000.Thesevaluesareestimatedfromasetof3virtual tripsforNod=5000and2virtual tripsforNod=10,000. ThelowernumberofvirtualtripsassociatedwiththesesetsoftriplengthsforlargerNodvaluesisrelatedwiththetopology

Table 4

Number of virtual trips between each regional OD pair, for the 6th Lyon district network.

Destination region 1 2 3 4 5 6 7 8 Origin re g io n 1 165 258 135 153 96 133 125 171 2 215 298 164 176 147 202 184 282 3 160 188 114 135 97 103 128 185 4 156 247 105 104 96 106 155 175 5 108 139 69 93 43 99 86 138 6 140 159 102 67 85 65 89 162 7 147 227 115 81 75 79 135 152 8 199 272 141 132 103 149 151 175

of the city network aswell asthe definition of its partitioning.These five virtual trips have an Origin in region 8 and Destinationinregion5,definingtheregionalpath p=

{

87645

}

.InthesetofNod=5000thereareatotalof42virtualtrips connectingthisregional ODpair,while intheset ofN_od=10_,000there are94virtual trips. Outof thetotalnumbers of virtualtrips mentionedpreviously,thereare ∼ 90%thatdefine theregionalpath p=

{

875

}

.Travelisthereforemorelikely ontheregionalpathp=

{

875

}

ratherthanon p=

{

87645

}

,fortheregionalODpair8-5(seealsoFig.4).

Fig.6 showsthat themostrefined levelof informationM4 isthat mostsensitive to thecalibration ofthevirtual trip

sets



.Italsoshowsseveraloutliersinthe

ρ

distributionsthatcomefromthecalculationoftriplengthsbasedonregional paths that are notstatisticallysignificant. When theregional pathsare not statisticallysignificant, thebias induced from thesamplingoforiginanddestination nodesplays animportantroleinthecalculationofthetriplengthdistributions for theOriginandDestinationregions.Nevertheless,thisbiasisconsiderablyreducedfordistributionsoftriplengthsestimated onthe basisofstatisticallysignificantregional paths.Tobetterelucidate thelatter,we analyzeseveralexamples.Wefirst consider theregional path p=

{

357

}

which is not statisticallysignificant. Thisregional pathis definedby 3 virtual trips forN_od=5000 andby 4virtual trips forN_od=10_,000.We estimate therelative differences

ρ

forthe three regions.The

ρ

valuesare −0.41 fortheOrigin region,0.73 forthe intermediateregion and−0.35forthe Destinationregion.Wenow analyze theregional path p=

{

578

}

which isstatistically significant.This regional pathis definedby 37 virtual trips for

N_od=5000andby66virtualtripsforN_od=10_,000.The

ρ

valuesare0.02fortheOriginregion,_−0.03fortheintermediate regionand−0.05fortheDestinationregion.Weobservethat fortheregionalpath p=

{

578

}

,theaveragetriplengthsare verycloseforthethreeregions.However,thisdoesnotoccurfortheregionalpathp=

{

357

}

.

In thissection,we show that the criterion Nnodes

cov

(

Nod

)

provides goodguidancefor ensuringa proper definitionofthe virtualtripsets



forthecalculationofthetriplengthdistributions.Wealsocarriedoutadetailedanalysisoftheinfluence oftheNod settingonthecalculationofthetriplengthsandidentifiedthatthepresenceofoutliersinthe

ρ

distributions, especiallyforthemoredetailedlevelsofinformation(i.e.M3andM4),originatefromtriplengthsetsthatarenotstatistically

significant. As mentionedpreviously,these outliersoriginate fromthe scalingof thevirtual trips to theaggregatedlevel, takingintoaccountthedefinitionofthecitynetworkpartitioning.Onesolutiontoavoidthisproblemistofirstanalyzethe setofregionalpathsdefinedbythevirtualtrips set



.Let



bethissetofregionalpaths.First,itispossibletofilterthe non-statisticallysignificantregionalpathsofthisset



,i.e.filterthevirtualtripsfromtheset



thatdefinetheseregional paths.Then,basedonthesevirtualtrips,thedistributionsoftriplengthscanbereestimatedaccordingtothefourlevelsof informationM1toM4.Byapplyingthisprocedure,wereducethenumberofoutliersonthe

ρ

distributionsto2forMI3and

MI

4andto3forMD4.

Basedontheanalysisdoneinthissection,weselecttheN_od=10,000setofvirtualtrips



asthereferenceforthenext two Sections3.3 and3.4. The latteryields Nod

N_links∼ 13.2. The virtual trips are uniformlysampledin thecity network and define theregionalODmatrixlisted inTable4.LetZbe thisregionalODmatrix.Thesevirtual tripsdefine atotal of205 regionalpaths.

3.3. Analysisofthetriplengthdistributions

Inthissection,westartby analyzingthedifferencesbetweentheaveragesofthetriplengthdistributionsestimatedby the differentlevels of information M1 to M4. We consider M4 asthe reference distributionas statedin Section 2.2.The

averageofthetriplengthdistributionsoftheotherthreelevelsofinformation,M1toM3,areanalyzedagainstthereference

levelM4.Wethen analyzethedifferencesofthetriplengthdistributions estimatedby thefourlevelsofinformationfora

representativeregionalpath p=

{

124

}

.

Weinvestigatethedifferencesbetweentheaveragetrip lengthsestimatedbythelevels ofinformationM1 toM3

com-paredto the referencelevel M4.We do thisanalysis atthe region level.First, we focus ouranalysis atthe region level.

Fig.7depictstheaverage triplengthsestimatedbythe fourlevelsofaggregationfortheeightregions ofthe6thdistrict Lyonnetwork (seeFig.4). Thebluedots representtheaveragetriplengthsestimatedforeachregion byM1 (see Eq.(7)).

Fig. 7. Average regional trip lengths L [in meters] estimated through the four levels of information M 1 to M 4 , for the eight regions. The blue dots refer to

M1 (see Eq. (7) ). The green triangles refer to M 2 (see Eq. (9) ). The black crosses refer to M 3 (see Eqs. (11) and 13 ). The red dots refer to M 4 (see Eq. (15) ).

(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

internaltrips ofeach specificregion. Theblackcrossesrepresenttheaverage triplengthsestimatedforM3.Thisincludes

theaveragetriplengthsfortheOrigin,intermediateandDestination(see Eq.(11))regionsoftheregionalpaths aswellas theinternalpaths(seeEq.(13)).ThereddotsrepresenttheaveragetriplengthsestimatedforM4 (seeEq.(15)).Eachofthe

reddotsrepresentedinFig.7foreachregioncorrespondstoadifferentregionalpaththatcrossesthisspecificregion.The resultsshowninFig.7showthatconsideringonlyoneaveragetriplengthforeachregion(i.e.M1)isnotrepresentativeof

allthepossibleaveragetriplengthsoftheregionalpathsthatcrosseachregion(i.e.M4).Thelevelofinformationincreases

thediversityofthepossibleaveragetriplengthsineachregion.

Wediscussthesedifferencesinmoredetailforregion5,whichhas4adjacentregions



=

{

2,4,6,7

}

.Ascanbeseenin Fig.4,therearefiveregionsadjacenttoregion5.However,thecitynetworknodethatliesontheborderbetweenregions3 and5onlyallowstravelingfromregion5to3.Notethatsimilarconclusionscanbemadefortheotherregionsaswell.The firstlevelofinformationM1providesanaveragetriplengthL=273meters(seeEq.(7)).ForM2,wehavefiveaveragetrip

lengthsthatcorrespondtotheadjacentregionstoregion5andtotheinternalpathp=

{

5

}

.Theestimatedtriplengthsrange fromL54=203meterstoL57=321meters.AscanbeseenM2alreadyprovidesaheterogeneityofaveragetriplengthsthat

Fig. 8. Distribution of the relative differences β, between method l = 1 , 2 , 3 , 4 (from the left to the right) and the reference method 4.

M1isunabletodeliver.InthecaseofM3,weobserveatotalof20blackcrosses.Theycorrespondtotheorigin,intermediate

anddestinationregions oftheregionalpaths aswell asthe internalpath p=

{

5

}

. Thesetof20trip lengthsgivenby M3

hasan averageof296m andstandarddeviation111m. Asshown, by increasingthe levelofinformation,we increasethe rangeofpossibleaveragetriplengthscomparedtoM2 and,inparticular,toM1.Thisisbecauseweconsiderinmoredetail

allthepossibletripsthatmightoccurinsidearegion.WhenweconsiderthelevelofinformationM4,therangeofpossible

averagetriplengthsincreasesevenmore.Atotalof124regionalpathsoutof205crossregion5.Theaveragetriplengthsof these124regionalpaths arerepresentedby thereddotsinFig.7.Theaverageoftheseregionalpathsinsideregion5can rangefromafewmetersupto ∼ 600 m.Forexample,theregionalpaths p=

{

5687

}

andp=

{

645

}

areamongtheshortest. Whilst, the regionalpaths p₌

{

45768

}

and p₌

{

245768

}

are longestonesthat cross region 5.As can be seen,the more detailedlevelofinformationM4 allowsdistinguishingbetweenregionalpathsthatjustcrossaregionbyafewmetersfrom

othersthathavemuchlongertriplengths.

ThesefourlevelsofinformationM1toM4provideregionaltriplengthsforeachregionalpathcrossingeachregionofthe

regionalnetwork. Therefore,itisrelevantto estimatethe relativedifferencesbetweentheregionaltrip lengthsestimated through thelevels of informationM1 to M3 andthe referencelevel M4, whenassigningregional trip lengthstoregional

paths(p)insidetheregionsi.Thisenhancesamoreaggregatedcomparisonbetweenthedifferentlevelsofinformationthan inFig.7.Therelativedifferences

β

ipareestimatedas:

β

ip= LMj i − L M4 i LM4 i

δ

p i,

∀

j=1,2,3∧

∀

p∈



(24)

where

δ

_ipisabinaryvariablethatequals1ifregionalpathpcrossesregioni,or0otherwise.LM4

i istheaverageregionaltrip lengthestimatedthroughEq.(15)(i.e.levelofinformationM4).ForlevelofinformationM1,L

M_j

i isestimatedthroughEq.(7). ForM2,L

M_j

i isestimatedthroughEq.(9),wherei=pmisthecurrentregionand j=pm+1 isthenextregionadjacenttoi onthesequenceofregionalpathp.ForM3,L

M_j

i isestimatedthroughEq.(13),wherei=pmisthecurrentregion,h=pm−1

isthepreviousregionand j₌pm+1isthenextregionadjacenttoionthesequenceofregionalpathp.

The estimated relative differences

β

ip are illustrated in Fig. 8. As shown, the relative differences

β

ip decrease as we increase the level of information, i.e. from M1 to M3. This clearly shows how far the average trip lengths provided by

thelevelofinformationM1 are fromall thepossibletrip lengthsoftheregional pathscrossing thedifferentregions.The

settingofthetriplengthsiscrucialfortheMFD-basedmodels.Indeed,asshownbyLeclercqetal.(2015),thehypothesisof consideringan averagetriplengthforallthevehiclestravelinginsidethesameregionforMFD-basedmodelsapplications isastronglimitation.Disaggregatingthedefinitionoftriplengthsbythedifferentregionalpathsthatcrosseachregionis clearlyabetterapproach.Thisisbecause itallowsgatheringshorttriplengthsforcertainregionalpaths thatjustcrossa givenregion.However, onecanalsoassignlongertriplengthsforotherregionalpaths.Thishasanon-negligibleinfluence onthemodeledtrafficstates,apointthatwillbediscussedinSection4.

WenowanalyzethedifferencesbetweenthetriplengthdistributionsestimatedbythefourlevelsofinformationM1to

M4fortheregionalpathp=

{

124

}

.Thedistributionsoftriplengthsforeachregionofthisregionalpathandforeachofthe

fourlevelsofinformationarepresentedinFig.9.Theblackhorizontaldashedlinesrepresenttheaverageofeachtriplength distributioninside eachregion. Ascan beobserved,thevariance ofthetrip lengthdistributions decreasesaswe increase thelevelofinformationconsidered.Thediversityofvirtualtrips,i.e.thenumberofvirtualtripsthatisconsideredtogather thedistributionsoftriplengthsdecreasesfromM1 toM4.Inparticular,thisclearlyinfluencestheshapeofthetriplength

Fig. 9. Regional trip length distributions estimated through the four levels of information M 1 to M 4 for each region of the regional path p = { 124 } . The

total number of trips considered for each distribution is identified at the top of each subplot. Each row of the subplots represents the results for each level of information, while each column of subplots represents each region of the regional path. The horizontal dashed lines represent the average of the trip length distributions.

thetriplengthdistributionsforintermediateregions givesprioritytothevirtual tripsthatspecificallytravelthis interme-diatesequence.Thisexplainsthepeakobservedforthetriplengthdistributions fortheintermediateregion2gatheredby

M3 andM4.Indeed,thefractionofthecity networkthatliesinside region2issimilartoa gridnetwork(see Fig.4).The

linksofthecity networkthatlie insideregion2aremostlyonedirectional. Thisincreasestheprobability thatthevirtual tripstravelingonthisspecificsequenceofintermediateregions,useaspecificsequence oflinksofthedirectional linksof thecitynetworkmoreoften.Whereasthetriplength distributionsforregion2gatheredfromM3 andM4 aresimilar,this

mightnot be the caseforall intermediate regions. Note that, forM3 we consider all virtual trips that cross the specific

sequenceofregions 4-2-1.ForM4,weonlyconsiderthevirtualtripsthat definethesameregionalpathp.Thetriplength

distributionsfortheintermediateregionsareestimatedbasedonthelengthofeachtripinsidetheseregions.Asshownin Fig.9,thereare122virtualtripsthatdefinetheregionalpath p=

{

124

}

andthatarethenusedtoestimatethetriplength distributionforregion2.ForM3 thereare12additionalvirtualtripsthatdefineotherregionalpathsthatareconsideredto

definethetriplengthdistributionforregion2.

Wealsoelucidatethatsequencep=

{

124

}

definesaregionalpathdifferentfromthesequence p=

{

421

}

.InTable5,we listtheaveragetriplengthsforeach regionofthesetwo regionalpaths,estimatedthroughthefourlevelsofinformation.

M1 attributessimilaraveragetriplengthsforthethreeregions ofthesetworegionalpaths.Duetoitsnature,weconsider

allthevirtualtrips thatcrosseachoftheseregionstodefinethetriplengthdistributions.Nevertheless,asweincreasethe levelof informationM2 to M4, we assign differentaverage trip lengthsfor thethree regions of each regionalpath. This

Table 5

Average trip lengths for the regional paths p = { 124 } (left) and p = { 421 } (right) estimated by the four levels of information for each region.

Region Level of information Region Level of information M1 M2 M3 M4 M1 M2 M3 M4 1 312 277 253 314 4 311 362 419 439 2 364 353 390 393 2 365 396 355 355 4 311 268 243 296 1 312 345 247 285 Table 6 Regional OD matrix Z ∗_. Destination region 1 2 3 4 5 6 7 8 Origin re g io n 1 135 231 81 933 173 881 142 838 2 153 198 140 208 119 285 137 377 3 137 153 75 95 65 91 90 133 4 102 156 83 84 77 72 94 106 5 76 119 64 58 38 61 65 99 6 96 112 68 46 42 37 77 104 7 105 137 78 72 53 43 80 102 8 147 157 107 81 73 106 114 142

hasa clearinfluence on thetraffic statesmodeled by an MFD-basedmodel. Alonger averagetrip length insidea region meansthatavehiclehastospendmoretimetocompleteitstripinsidetheregion.Considertwovehiclesdepartingatthe sametime,buttravelingondifferentregionalpaths p=

{

124

}

andp=

{

421

}

.Furthermore,assumethatwehavefluidtraffic conditionsinthenetwork.AccordingtotheaveragetriplengthsassignedbyM1,thesevehicleswillcompletetheir tripsat

thesametime.M2 toM4 assigndifferentaveragetriplengthsforthesevehiclestotravelinsidethesameregion,leadingto

differenttraveltimesforvehiclestravelinginthesameregionandalongdifferentregionalpaths.

Inthissection,we emphasizetheimportanceofconsideringhigherlevels ofinformationfromtheregionalnetworkto definethetriplengthdistributions.First,wehighlightthatconsideringanaveragetriplengthforallvehiclestravelinginside thesameregionandondifferentregionalpaths (i.e.M1)isnotrepresentativeofallthepossibleaveragetriplengthsofall

the regionalpaths that crossthisregion (i.e.M4).Second, we show that thelevel ofinformationconsidered alsoplays a

roleintheshapeofthetriplengthdistributions.Moreover,weshowanotherlimitationofM1comparedtotheotherthree

levelsofinformation.Weshowthatfortworegionalpathsthatare definedbythesameregions,butinreverseorder,the assigned averagetrip lengths foreach region is similar. However, the other three levels of informationM2 to M4 assign

different averagetrip lengths. Becauseof their nature, they are able to captureimportant informationfromthe possible directionsofthecitynetworklinks.InSection4,wediscusstheimportanceoftriplengthssettingforMFD-basedmodels.

3.4. InfluenceoftheODmatrixonthetriplengths

ThetriplengthdistributionsareclearlyinfluencedbytheregionalODmatrix(Leclercqetal.,2015).Infact,theregional ODmatrixdefinesthedistributionofodpairsinthecitynetwork,whichmayleadtoverydifferentsetsofvirtualtrips.This happensmostlywhenconsideringtherelative quantitiesthatgofromoneregiontoanother.Inthissection,weintroduce a methodologytoupdate thetriplengthsforwhenthenumberoftrips betweenODpairsis modified.Thismethodology isonlyapplicableforchangesoftheODmatrixcausedbydifferencesinthetrippatternsandnotforupdatingtriplength distributionsinfluencedby thespatialdistributionofthecitynetworkodpairs.Asdiscussedpreviously,thisisthecaseof thetriplengthdistributionsfortheOrigin andDestinationregions estimatedbythelevelofinformationM3.Nevertheless,

weintroduceanestimationprocedureforthesetwocases,tocoverallthreepossibilitiesforM3dissectedinSection2.2.We

consideranupdatedregionalODmatrixZ∗,alsocomposedofNod=10,000virtualtrips.Outofthese10,000virtualtrips,a totalof1000vehiclestravelspecificallyoneachofthefollowingregionalODpairs:1-4;1-6;and1-8.Theremaining7000 virtual tripsareuniformlydistributedoverthecitynetwork.FortheupdatedOD matrix,we re-estimatethesetofvirtual trips



andthedistributions oftriplengthsaccordingtothefourlevelsofinformationM1 toM4.TheupdatedODmatrix

Z∗islistedinTable6.

Themethodologyappliedtoupdate theregionaltriplengthsisbasedontheregionalODmatrix(Z) definedbytheset of virtual trips



aswell asthe trip lengthsthat are estimated andthe updated regionalOD matrix Z∗.One solutionis to estimate anew setofvirtual trips based onZ∗ andupdate thetrip length distributions accordingly.Nevertheless,the computationalcosttodosolargelyincreasesforlargecitynetworks.Weproposeanalternativemethodologytocircumvent thecomputationalpowerrequiredtoupdatethetriplengthdistributions.Thegeneralideaistobalancetheweightofvirtual trips toadjusttheirfrequencytotheupdated regionalOD matrix.Thisprocedureisvery usefulasitupdatestheregional triplengthsconsideringthecurrentsetoftrips



whentheregionalODmatriceschangeovertime,orwhenanother