Macroscopic traffic state estimation using relative flows from stationary and moving
observers
van Erp, Paul B.C.; Knoop, Victor L.; Hoogendoorn, Serge P.
DOI
10.1016/j.trb.2018.06.005
Publication date
2018
Document Version
Final published version
Published in
Transportation Research Part B: Methodological
Citation (APA)
van Erp, P. B. C., Knoop, V. L., & Hoogendoorn, S. P. (2018). Macroscopic traffic state estimation using
relative flows from stationary and moving observers. Transportation Research Part B: Methodological, 114,
281-299. https://doi.org/10.1016/j.trb.2018.06.005
Important note
To cite this publication, please use the final published version (if applicable).
Please check the document version above.
Copyright
Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy
Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.
This work is downloaded from Delft University of Technology.
Paul
B.C.
van
Erp
∗,
Victor
L.
Knoop,
Serge
P.
Hoogendoorn
Department of Transport & Planning, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1 Delft,
2628 CN, The Netherlands
a
r
t
i
c
l
e
i
n
f
o
Article history: Received 16 March 2018 Revised 7 June 2018 Accepted 8 June 2018 Keywords:Traffic state estimation Cumulative vehicle number Relative flow data
a
b
s
t
r
a
c
t
Thisarticlepresentsaproceduretoestimatethemacroscopictrafficstateinapre-defined space-timemeshusingrelative flowdatacollectedbystationary and movingobservers. Theprocedureconsistoftwoconsecutiveandindependentprocesses:(1) estimatepoint observationsofthecumulativevehiclenumberinspace-time,i.e.,N(x,t),basedonrelative flowdatafromthe observersand (2)estimateflow and density inapre-define space-timemeshbasedonthepointobservationsofN.Inthispaper,theprinciplesbehindthe firstprocessareexplainedandamethodology(thePoint-ObservationsN(PON)estimation methodology)isintroduced forthesecondprocess.Thismethodologydoesnot incorpo-rate informationinthe formof atrafficflowmodel orhistorical data.To evaluatethis performanceand improveourunderstandingofthemethodology,amicroscopic simula-tionstudyisconducted.Theestimationperformanceiseffectedbythehomogeneityand stationarityoftrafficinestimationareaandinthesamplearea.Incaseoflarge changes intrafficconditions,e.g.,fromfree-flowtocongestionorstop-and-gowaves,abetter sam-plingresolutionwillimprove localizingthesechangesinspaceand timeandhence im-provetheestimationperformance.Inthesimulationstudy,theproposedmethodologyis alsocomparedwithestimatesbasedonloop-detectordata.Thisindicatesthatthe combi-nationoftheproposedmethodologyanddatayieldsanalternativeforexisting combina-tionsofmethodologyand data.Especially,intermsofdensityestimationtheintroduced methodologyshowspromisingresults.
© 2018TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBYlicense. (http://creativecommons.org/licenses/by/4.0/)
1. Introduction
Thispaperaddressesmacroscopic traffic stateestimation. Estimatesof themacroscopic trafficflow variables, i.e.,flow
q, density k and speed u, can be used as input for control decisions within dynamic traffic management applications (Papageorgiouetal.,1991;Smaragdisetal.,2004).
Theestimation procedureintroduced in thispaperallows ustoestimate the macroscopictraffic flow variableswithin apre-definedspace-timemeshusingstationaryandmovingobservers.Thisprocedure consistsoftwo main(independent andconsecutive)processes.Theseare:(1)estimatethecumulativevehiclenumberNforpointsalongtheobservedpathsin
∗ Corresponding author.
E-mail addresses: p.b.c.vanerp@tudelft.nl (P.B.C. van Erp), v.l.knoop@tudelft.nl (V.L. Knoop), s.p.hoogendoorn@tudelft.nl (S.P. Hoogendoorn).
https://doi.org/10.1016/j.trb.2018.06.005
0191-2615/© 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )
space-time(wewillcallthesepoint-observations)usingtraffic sensingdatafromstationaryandmovingobserversand(2) estimateqandkinapre-definedmeshbasedonpoint-observationsofN.
This paperhastwo important contributions.The first andmore generic contributionis the full estimationprocedure. We propose touse equippedand/or automated vehicles that observethe relativeflow withrespect to their trajectoryin combinationwithstationaryobservers.Thetrafficsensingdatafromtheseobserverscanbefusedonthecumulativevehicle numberlevel(firstprocess)andcanbeusedtoestimatethemacroscopictrafficconditions(secondprocess).Bothprocesses areexplainedinthispaper.Thesecondandmorespecificcontributionisthemethodologydesignedforthesecondprocess. ThismethodologyiscalledthePoint-ObservationN(PON)estimationmethodology.Thetwoprocessesrequireindependent methodologiesandshouldinourviewnotbediscussedindetailinasinglepaper.Therefore,inthispaper,weexplainthe secondprocessindetail,whilewesolelyexplaintheprinciplesbehindthefirstprocess.
Thispaperisorganizedasfollows.Section2providestheexistingmethodologicalbasisforourworkandpositions this workwithin theresearchfield. Next,we explainthetwo processesinSections3and4.ThePONestimationmethodology (explained inSection 4)isevaluated usingasimulationstudyinSections5and6.Weconcludewiththeconclusionsand discussioninSection7.
2. Backgroundonmacroscopictrafficstateestimation
In thissection, we discussthe topicof macroscopictraffic state estimation andexplain how the proposed methodol-ogydiffersfromexistingwork. First,Edie’sgeneralizeddefinitions ofthe macroscopictraffic flowvariables andthe three-dimensionalrepresentationoftrafficflowareprovided.Second,thecategorizationdiscussedbySeoetal.(2017)isusedto categorizetheproposedestimationapproach.Third,weelaborateondifferenttypesoftrafficsensingdatausedfor macro-scopic trafficstate estimationandwhichdataare usedintheproposed estimationprocedure.And finally,wediscussour estimationoutputandhowthisrelatestoexistingwork.
Thegeneralizeddefinitionsofflowq,densitykandspeedu,foranareaD inspace-timeareprovidedbyEdie(1965): qD= idi AD (1) kD= iri AD (2) uD=qD kD (3)
wherediandrirespectivelydenotethedistancetraveledandtimespentbyvehicleiwithintheareaD andAD denotesthe
surfaceofD.
idiand
irirespectivelydenotetheTotalTravelDistance(TTD)andTotalTimeSpent(TTS)inD.
Makigamiet al.(1971)proposed the three-dimensional representationoftraffic flow. The three dimensionsare space, time and the cumulative vehicle number, where N(x, t) denotes the cumulative vehicle number at location x and time instantt.Asvehiclesarediscrete,Ncanberepresentedasadiscretevariable.Here,N(x,t)increasesinstantlybyonevehicle atthetimeinstanttwhenavehiclepasseslocationx.
We want to describe traffic flow on a macroscopic level. For this purpose, the discrete N can be smoothed (Makigamietal., 1971). Forthe smoothedandcontinuouslydifferentiable N,themacroscopictraffic flowvariables canbe describedbasedonthethreedimensions. Themacroscopicvariablesforapoint inspace-time,i.e.,(x,t), aregivenbythe timeandspacederivativesofN(x,t):
q
(
x,t)
=∂
N∂
(
x,t)
t (4) k(
x,t)
=−∂
N(
x,t)
∂
x (5) u(
x,t)
= q(
x,t)
k(
x,t)
(6)Seoetal.(2017)categorizestheestimationapproachintothreecategories(i.e.,model-driven,data-drivenand streaming-data-driven)basedoninformationinput.Followingthiscategorization,themethodologypresentedinthisstudycanbe cat-egorizedasastreaming-data-driventrafficstateestimationmethodology.Examplesofotherstreaming-data-driven method-ologies are WardropandCharlesworth (1954), Seo andKusakabe (2015) and Florin andOlariu (2017). A streaming-data-drivenmethodologydoesnotdependoninformationintheformofatrafficflowmodel,fundamentaldiagramorhistorical data,butsolelyreliesonreal-timedataand‘weak’assumptionssuchastheconservation-of-vehicles.Therefore,‘itisrobust against uncertainphenomena andunpredictable incidents’ (Seo etal., 2017). At the sametime, Seo etal. (2017)denotes twolimitationsofstreaming-data-driven methodologies:(1)additionalinformation(e.g.,atrafficflowmodel)isneededto
HerreraandBayen,2010), space-headway(SeoandKusakabe,2015).Thesedata haveapotential bias-problem (Seo etal., 2017)asequippedvehiclescanhavealargelyconsistentdifferentdrivingbehaviorthantheaverageroaduser.Forinstance, wemaycollectspeeddataforafleetoftrucks,whichwillalowerspeedthanaverage.Furthermore,wemaycollect head-waydatafromvehiclesequippedwithheadwaysensorsandautomated drivingtechnologies,whichmayactdifferentthan manualvehicles.
Theestimationprocedureproposedinthisstudysolelyusesobservationsof
N,i.e.,flowdatafromstationaryobservers andrelativeflowdatafrommovingobservers.Stationarysensing-equipementthatobserveflow(e.g.,detectorsorcameras) arewidelyappliedinpractice(e.g.,PeMS).AsproposedbyRedmilletal.(2011)andFlorinandOlariu(2017),equipped(e.g., automated)vehiclescanserveasmoving observersthat recordtheflowrelativetotheirpositionovertime.Thesevehicles observe
Nalongtheirtrajectory.Earlierstudies(e.g.,ClaudelandBayen,2010a)haveusedtrajectorydataofvehiclesthat cannotobservetherelativeflowinthesamemanner.Inthiscase,theyoftenassumethat
N=0,i.e.,noovertaking,along thevehicletrajectory.Althoughthisassumptionisexpectedtoholdforsingle-lanetraffic,itislikelytoviolatedinmulti-lane traffic.Therefore,thesemethodologiescouldalsobenefitfromusingmovingobserversthatobserve
N.
Interms ofestimationoutput, we candifferentiate various typesofestimations. Thesecan differboth inthe variable typesthatareestimatedasthespatial-temporal characteristicsrelatedtotheestimates.Theestimationmethodology pro-posedin thisstudy estimatesthe flow anddensity ina pre-defined space-time mesh.A potential space-time mesh is a discritisationofspace inroad-segments (cells) andtime inperiods, e.g.,Nanthawichit etal.(2003), Wangand Papageor-giou(2005)andHerreraandBayen (2010).Incontrasttothesemethodologies,ourmethodology isfreetowork withany other pre-definedmesh.In theremainder ofthis paper,ourfocus lieson thetwo directoutput ofthe methodology, i.e., flowanddensity.However,aswe canobtainthespeed fromtheflowanddensity,all threemacroscopicvariablescanbe estimated. Other methodologies existthat only estimate the speed using datafrom individual vehicles (e.g.,Work etal., 2010;DelArcoetal.,2011).Newell’sthree-detectormethod(Newell,1993a;1993b;1993c;Lavaletal., 2012)andClaudel’s approach (Claudel and Bayen, 2010a; 2010b) estimate the cumulative vehicle number for different pointsin space-time. Althoughthisisrelatedtoour estimationoutput,comparing theoutputs wouldrequirean additionalstepforone ofthe estimationmethodologies.
3. Point-observationsofthecumulativevehiclenumber
Wecombinetraffic sensingdatafromstationaryandmovingobserversto estimateNforpointsinspace-time,i.e.,N(x, t).Inthissection,weexplaintheprinciplesbehindthisprocess.
We define observationspaths aspaths in space-timeover which we observe
N. Fig.1 showsa combination offive observationpaths.Forstationaryobservers(dashedlines),theobservationpathisahorizontallineinspace-time,i.e.,afixed locationover timeis observed.Formovingobservers (solidlines),theobservationpath isthetrajectory oftheconnected automated vehicle.The relative flow observed betweenpoints along theseboundariesis the changein N betweenthese points.IndividualobserversthusprovidetherelativeNalong asingleboundary. However,weareinterestedinhavingthe relativeNforallcombinationsofpoint-observationsinspace-time.
Combining sensing data from stationary andmoving observers allows us to relate the data from different observers and to deal with potential observation errors. As stationary and moving observers move with different speeds through space-time,the observationpaths intersect.Observation paths that havenot yet beeninitialized (e.g.,a moving observer entering thelink) can be initializedbased on their firstinteraction (e.g.,witha stationaryobserveratthe upstream link boundary).Anintersectionoftwoalready-initializedobservationpathscanbeusedforerrorcorrection.Observation errors canleadtoadiscrepancybetweenNattheintersectionpointonthetwoobservationpaths.Asbothpathsshouldhavethe sameN-valueattheintersectionpoint,wecancorrectforobservationerrorsbasedonthedifferenceinN.Thisprincipleis simple;however,indesigninganerrorcorrectionmethodology,weneedtodefinehowthedifferenceinNtranslatesintoa correctionin
Novertheobservationpaths.Asexplainedintheintroduction,designingthismethodologyisnotthefocus ofthispaper.However,webelieveitisvaluabletoshowthatthefull estimationprocedureisrobusttoobservationerrors.
Fig. 1. Visualization of observation boundaries and point-observations along these boundaries.
For thispurpose,we design a simple methodology andevaluate the estimationperformance under differentobservation errorsinAppendixA.
ToapplythePON-estimationmethodology(whichwillbeexplainedinthenextsection),allpoint-observationsofNhave tobedefinedinasingleframework.Therefore,itshouldbepossibletotravelfromanobservationpathtoanyother obser-vationpath.ThisholdsforthecombinationofobservationpathspresentedinFig.1.Eachmovingobserverisinitializedby intersectingwiththestationaryobserverattheupstreamboundaryandthemovingobserverscrossallstationaryobservers. Therefore,wehavea
Nforeachcombinationoftwopointsthatlieonanobservationpath.
4. ThePONestimationmethodology
Threesteps aretakentoestimateflow qanddensitykinapre-definedmeshusingpoint-observationsofN.Theseare (1)subdivide thespace-timedomain intriangularareas,wherethethreecorners arepoint-observations ofN, i.e.,N(x,t), (2)estimate qandkforeachtriangulararea T,i.e., qT andkT,basedonthe threepoint-observationsand(3)estimate q
andkforeachareaD,i.e.,qD andkD,inapre-definedmeshbasedonqT andkT.Intheformerstep,weuseanexisting methodology (Delaunaytriangulation). Thelattertwo steps arederived based onthe three-dimensionalrepresentationof trafficflow(Makigamietal.,1971)andEdie’sgeneralizeddefinitionsofthemacroscopictrafficflowvariables(Edie,1965).
Thissectioncontainsfoursub-sections.Sections4.1and4.4respectivelyexplainthefirstandthirdsteps.Sections4.2and
4.3bothrelatetothesecondstep(estimatingqT andkT).InSection4.2,wederivetheequationsusedforthisstepbasedon thethree-dimensionalrepresentationoftrafficflow (Makigamietal.,1971). However,thereisasubtledifference between theinterpretationoftheestimatesobtainedusingthederivedequationsandestimatesfora triangulararea inspace-time. InSection4.3,weuseEdie’sgeneralizeddefinitions(Edie,1965)toshowthatwecanusetheequationstoestimateqT and
kT arequantifytherelatedestimationerrors.
4.1. Subdividespace-timeintotriangularareas
Givena setofpoint-observations,we wantto subdividethe space-timedomaininto triangularareas,wherethethree cornersarepoint-observations.WeuseDelaunaytriangulation,whichavoidssliver,i.e.,narrow,triangularareas,tosubdivide space-timebasedoncoordinatesofthecornerpoints.
Thetriangularareasaredefinedontwodifferentdimensions,i.e.,spaceandtime.Inordertodefinethreeinteriorangles oftriangularareas,we needtodefine therelationbetweendistancesinspaceandtime.Thisyieldsthesoleparameter of our traffic state estimator,i.e., the space-timeratio
υ
used inDelaunay triangulation.Bydefiningυ
we define a desired dimensions ofthe triangular areas. Forinstance, reducingυ
yields triangular areas witha smaller space-time ratio, i.e., widerintermsoftimeandmoresliverintermsofspace.Therefore,υ
influenceswhichinformationisusedtoestimatethe trafficconditionswithintheindividualareasofthepre-definedmesh.4.2. Three-pointtrafficstateestimation
Weproposeequationstoestimatethemacroscopictrafficconditionsbasedonthreepoint-observationsofthecumulative vehiclenumberinspace-time,i.e.,N(x,t).Letusconsidertwopointsinthespace-timedomain,i.e.,(x1,t1)and(x2,t2),for
whichthedifferenceinN,i.e.,
N12 =N
(
x2 ,t2)
− N(
x1 ,t1)
isknown.Fig. 2a shows two pointsin space-time and the considered path between these points. Using (4) and (5) a relation between q and k based on
N12 ,
x12 and
t12 is derived. To derive the relation, homogeneous and stationary traffic
Fig. 2. Visualizations related to three-point traffic state estimation.
conditionsare assumed, i.e., q
(
x,t)
=q andk(
x,t)
=k forany(x, t) inthe space-timedomain. Therefore,overspace and timethechangesinNarerespectivelyequalto−kx12 andq
t12.Thisyields:
N12 =q
t12 − k
x12 (7)
Notethat anypath betweenthe two points will yield the sameequation for
N12 as we assume homogeneous and stationarytrafficconditions.Re-arranging(7)allowustowriteqasafunctionofkandviceversa:
q=
N12 t12 +
x12
t12 k (8) k=−
N12
x12 +
t12
x12 q (9)
Anextra(third)pointisaddedtoestimateqandkasafunctionof
x,
tand
N.Incombinationwiththetwoinitial points,thethirdpointprovides
N13 ,
x13 ,
t13 and
N23 ,
x23 ,
t23 .Toincorporatetheinformationofallthreepoints, itsufficestoconsidertwocombinationofpoints.Inthiscase,weconsidercombinations12and23.Forbothcombinations, therelations(8)and(9)areobtained.Below,weshowthestepstakentoderivetheequationforq,i.e.,(14):
q=
N12 t12 +
x12
t12 −
N23 x23 +
t23
x23 q (10) q 1−
xt12 12
t23
x23 =
Nt12 12 −
N23
x23
x12
t12 (11) q
t12
t12
x23
x23 −
x12
t12
t23
x23 =
N12 t12
x23
x23 −
N23
x23
x12
t12 (12) q
(
t12
x23 −
t23
x12
)
=N12
x23 −
N23
x12 (13) q=
1 t12
x23 −
t23
x12
(
N12
x23 −
N23
x12
)
(14)Similarstepscanbetakentoderivetheequationfork,whichresultsinthefollowingequation:
k=
1
t12
x23 −
t23
x12
(
N12
t23 −
N23
t12
)
(15)Theresulting q andk canbe interpreted asthe homogeneous andstationarytraffic conditionsthat satisfy the cumu-lative vehiclenumber forthreepoints inthe space-timedomain. Fig.2b provides a visual interpretationof theestimate homogeneousandstationaryconditions.Here,thebluelinesdenotetrajectoriessatisfyingqandk.
Ifthe conditions are indeed homogeneous andstationary, any three observations of N are sufficient to estimate the trafficconditionsforthefull space-timedomain.However,one conditionshastohold.Thethree pointsshould notlie on astraight lineinspace-time, i.e.,
x12/
t12=
x23/
t23. If
x12 /
t12 =
x23 /
t23 thedenominatorwill bezero.In this
casethethirdpointdoesnotprovideanyadditionalinformation.Itisfurthermoreimportanttonote that(14)and(15)are invariantto thenumberingofthe threepoints,i.e., thesameresults areobtainedifthesamethreepointsare numbered differently.
4.3. Space-timeareatrafficstateestimationbasedonthemeanboundaryconditions
WeconsiderEdie’sgeneralizeddefinitions(Edie,1965)torelatetheequationsderivedabovetoatriangularareain space-timeandexplaintherelatedestimationerrors.Edie’sgeneralizeddefinitions(1)and(2)canberewrittenonanaggregated level:
q= Id¯
A (16)
k=I¯r
A (17)
where d¯and ¯rare respectively the meandistance traveledand time spent by the Inumber ofvehicles. d¯and ¯rcan be describedasfunctionofthemeanentry
(
¯xin ,¯tin)
andexit(
¯xout ,¯tout)
points,i.e.,d¯= ¯xout − ¯xin and ¯r=¯tout − ¯tin .Letusconsiderareasinthespace-timedomainwithstraight boundaries,e.g.,a triangleorpentagon. Eachboundaryb
hasanetflow
Nb.Hereoutflowandinflowarerespectivelydenotedbypositiveandnegativevaluesof
Nb.Furthermore,
each b hasa meanintersectionpoint
(
¯xb,¯tb)
,which reflects themeanpoint inspace-time wherevehicles intersect withthe boundary.As we consider straight boundaries,this pointlies onthe boundary. Inthe casethat traffic conditionsare homogeneousandstationaryandNisconsideredcontinuous,
(
¯xb,¯tb)
lieatthemiddlepointofboundaryb.For thespecific case ofhaving an area witha finite number ofstraight boundaries, (16)and (17)can rewritten asa functionoftheboundaryconditions:
q= b
Nb¯xb A (18) k= b
Nb¯tb A (19)
Letusconsidera triangulararea forwhichall three
Nb areknown.ThisresemblesthecasepresentedinSection 4.2,
inwhichweknow
Nforthreecombinationsoftwopointsinspace-time.Drawingastraightline(boundary)betweenthe combinationsofpointsinspace-timeresultsinatriangle.
Theorem 4.1. For triangular areas in which the mean intersection of each boundary coincides with the middle pointof the boundary,(18)and(19)areequaltorespectively(14)and(15).
Proof. Weprove that(18)and(19)areequalto respectively(14)and(15)inthe casea triangularareaisconsidered and themeanintersectionofeachboundarycoincideswiththemiddlepointoftheboundary,i.e.,:
b
Nb¯xb A = 1
t12
x23 −
t23
x12
(
N12
x23 −
N23
x12
)
(20) bNb¯tb A = 1
t12
x23 −
t23
x12
(
N12
t23 −
N23
t12
)
(21)For triangular areas we have three corner points, i.e., 1, 2 and 3,and three boundaries, i.e., 12, 23 and 31. For this situation,theleftpartoftheequationcanberewrittenas
b
Nb¯xb
A =
1
A
(
N12 ¯x12 +
N23 ¯x23 +
N31 ¯x31
)
(22)where the conservations of vehicles conditiondepicts that
N12 +
N23 +
N31 =0, thus
N31 =−
(
N12 +
N23
)
. Fur-thermore,forcaseinwhichthemeanintersectionofeach boundarycoincides withthemiddlepointofthe boundary, ¯xb canbedescribedasafunctionofthecornerpoints,e.g., ¯x12 =(
x1 +x2)
/2.Thisallowsustorewrite(22):b
Nb¯xb A = 1 A
N12 x1 +x2 2 +
N23 x2 +x3 2 −
(
N12 +
N23
)
x1 +x3 2 (23) =1 AN12 x2 − x3 2 +
N23 x2 − x1 2 (24) wherex2 − x3 =−
x23 andx2 − x1 =
x12 ,thus:
b
Nb¯xb
A =−
1
2A
(
N12
x23 −
N23
x12
)
(25)Next, we consider the surface ofthe triangular area, i.e., A. Tocalculate the surface of a triangular area we can use
A=1
2 wh,wherewisthewidthandhistheheight.Fig.3aprovidesthetriangularareaanddefinitionsofwandhwhich
Fig. 3. Important dimensions to find an equation for A .
boundariesare alsoconsidered forthe rightsidesofEqs.(20)and(21).Furthermore,note thatthere aremultiplecorrect approachestofindthedesiredfunctionforA.
Tofindhwefirst considerFig.3b.Inthisfigure,weobservetwotriangles whichhavea commonangle
α
.Thisallows ustorelatetheratioofdifferentknownandunknowntrianglesides:sin
(
α
)
= ha=wt12 (26)
Thushisgivenby:
h=a
t12
w (27)
ThisallowsustosimplifytheequationforA:
A=1
2a
t12 (28)
As
t12 isknownweonlyneedtofinda.Fig.3cshowstherelevantdimensionsusedtofinda.Thisfigureprovidestwo basicrelations,whichweusetofinda,namely:
a=b−
x23 (29)
b
t32 =
x21
t12 (30)
where
t32 =−
t23 and
x21 =−
x12 ,thus: b=
t
23
x12
a=
t
23 t
x12
12 −
x23 (32)
ThisallowsustofindAgiven(28):
A=1 2
t23
x12
t12 −
x23
t12 (33) =1 2
(
t23
x12 −
t12
x23
)
(34)Substitutingthisequationinto(25)yields: b
Nb¯xb A =− 1 2A
(
N12
x23 −
N23
x12
)
(35) =− 1 2·1 2(
t23
x12 −
x23
t12
)
(
N12
x23 −
N23
x12
)
(36) =1 x23
t12 −
t23
x12
(
N12
x23 −
N23
x12
)
(37)Similarstepscanbetakentoproveequalityfork,i.e.,(21).
ThecomparisonmadeinTheorem4.1isimportantfortworeasons.Firstly,itshowsthat(14)and(15)provideaproxyfor theTTS andTTDwithin thetriangularareaenclosed bythethreepoints. Secondly,itallowsustodescribetheestimation error in(14) and(15) based on the meanlevel of inhomogeneity andnon-stationarity of thetraffic conditionsover the threeboundaries.Themeanlevelofinhomogeneityandnon-stationarityofthetrafficconditionsoverbaffects
(
¯xb,¯tb)
.Thedifference betweenthispointandthemiddlepoint isdenoted as
¯xb and
¯tb. Asboth themeanintersectionpoint and
middlepoint lie on theboundary, the inhomogeneousand non-stationarytraffic conditionscan be describedby a single variable,i.e.,thefractional differencebetweenthetwopoints
μ
¯b,where¯xb=
μ
¯bxband¯tb=
μ
¯btb.Thisallows ustoquantifytheerrorin(14)and(15),i.e.,
ε
qandε
k:ε
q= bNb
μ
¯bxb A (38)ε
k= bNb
μ
¯btb A (39)Theseequationsprovideinsightintheestimationerrordependentontheareadimensions,i.e., A,xbandtb,andtraffic
conditions,i.e., Nb and
μ
¯b.Theyshow that theestimatorestimates thetraffic conditionsperfectly, i.e.,ε
q=0 andε
k=0, inhomogeneous andstationaryconditions,i.e.,
μ
¯b=0 forall b. Furthermore,the equationsprovideinsight intotheestimationerrorsoftwotriangularareaswhichshareaboundary,i.e., adjacenttriangularareas.Inhomogeneousand non-stationary traffic conditions over the shared boundary result in an estimation error in both triangular areas. There are, however,twodifferences,i.e.,(1)theflowisopposite(inflowvs.outflow)and(2)theerrorisscaledbytherelatedtriangular area surface. Therefore, theerrorinduced by a shared boundary isinverselyproportional to the surfacesof theadjacent triangularareas.
4.4. Trafficstateestimationinapre-definedmesh
Therelationsintroducedinprevioussectionsallowustoestimateqandkfortriangularareasinspace-time,seeFig.4a. However,dependingontheapplication, wemaywanttoselectthedimensionsoftheestimationarea,i.e., definethe esti-mation mesh.Asan example,inthisresearch,we considera meshwhichsubdividesspace-timeinrectangularareas,see
Fig.4b.Theestimatesfortherectangularareascanbeinterpretedasthemeantrafficconditionsforaroadsegment(cell) duringatimeperiod.Theselectedmesh,i.e.,discretizingspaceandtime, allowsformodel-basedpredictionwiththeCell TransmissionModel(CTM)(Daganzo,1994;1995).Therefore,ourestimatescanbeusedasinputfortrafficstateestimation usingtheCTM.Priorproposedestimators(WangandPapageorgiou,2005;HerreraandBayen,2010)andcontrolalgorithms (Smaragdisetal.,2004)alsoconsideradiscretespaceandtimemesh.
Letusconsiderthecaseinwhichthecompletespace-timedomainissubdividedintriangularareas.Foreachtriangular area T we knowtheflowqT anddensitykT.Toestimate thetrafficconditionsforadefinedarea inspace-time,the con-ditionswithin T areassumedtobe homogeneousandstationary.Thismeansthat theTTSandTTDwithin asubareaofT
Fig. 4. Visualisation of three-point traffic state estimation and estimation in a defined mesh.
isproportional totherelative sizeofthe subarea.Basedon Edie(1965),weknown thatTTDT =qTAT andTTST =kTAT. Combinedwiththeassumptionstatedabove,theTTDandTTSforasubareaS ofthetriangularareaT aregivenby:
TTDS=TTDTAAS
T =qTAS (40)
TTSS=TTSTAAS
T =kTAS (41)
LetusconsiderarectangularareaD withasurfaceAD.ThesurfaceoftherelevantsubareaAS isthesurfaceoftriangle
T,i.e.,AT,inthedesiredareaD,i.e.,AT ∈AD.Basedonthetrafficconditionsinallrelevantareas,i.e.,qT andkT,andtheir contribution,i.e.,
(
AT ∈AD)
/AD,thetrafficconditionsintherectangularareacanbeestimated:qD= T
(
AT ∈AD)
qT AD (42) kD= T(
AT ∈AD)
kT AD (43)Fig.4bprovidesavisualizationoftherelevanttrianglesubareasforasinglerectangulararea.
Theaccuracyofestimatesobtainedwith(42)and(43)depend ontwofactors.Firstly, ifqT andkT are estimates,they maycontainestimationerrors.Iftheseestimatesareobtainedusingthree-pointestimation,theestimationerrorsofadjacent areasinducedbythesharedboundaryarenegativelycorrelated.Inthiscasetheerrorsarepartiallycorrectedbycombining adjacentareas.Secondly,weassumedthatconditionswithinT arehomogeneousandstationary.IfT partlyfallsoutsidethe desiredareaD,inhomogeneityandnon-stationaritywithinT caninduceestimationerrorforD.
5. Simulationstudy
Asexplainedbefore,theestimationprocedureconsistsoftwoprocesses:(1)estimatingN(x,t)basedonmeasurements fromstationaryandmovingobserversand(2)estimatingqandkinapre-definedspace-timemeshbasedonN(x,t).These processeswere consecutivelydiscussed inSections3 and4.Inthe simulationstudywe focus onthe second process,for whichthePON estimationmethodologyisdesigned. Thesimulation studyis explainedinthissectionandtheresults are discussedinSection6.
Toinvestigateandexplaintheworkingofthemethodologyinmoredetail,itisassumedthatthefirstprocessisperfect, sowecorrectlyknowN(x,t)forthepoint-observationsalongtheobservationspathsoftheavailablestationaryandmoving observers.Thisrequiresthattheobservationpathsarecorrectlyinitializedandthereareno(orfullycorrected)observation errors.Inpractice,therewillbecount-errorsinstationary(andmoving)observers.Therefore,errorcorrectionisrequiredfor methodologiesthat onlyrequirecumulative counts,e.g.,Van Lintetal.(2014).Inourstudy,asexplainedinSection3,the cumulativedriftproblemislessofanissue.Wewillthereforefurtherelaborateonthesecondprocess,assumingnoerrors, inthisandthenextsection.Additionally,AppendixAshowshowobservationserrorsmightaffecttheperformance.
5.1. MicroscopicsimulationinFOSIM
ThesimulationstudyisconductedwiththemicroscopicsimulationprogramFOSIM(DijkerandKnoppers,2006).FOSIM iscalibratedandvalidatedforDutchfreeways(MinderhoudandKirwan,2001;Henkensetal.,2017).However,thesimulated
Fig. 5. Considered road lay-out in the microscopic simulation program FOSIM.
trafficconditionsmaystilldeviatefromrealtrafficconditionsforsimilarroadinfrastructurepropertiesandtrafficdemand. In estimation we wantto reconstruct theconditions based on a limiteddata-set.In our casestudy, thismeans that we want toreconstruct the simulatedtraffic conditions usingthe proposed methodologiesand data.It thereforeis essential that thefullgroundtruth, whichshould bereconstructed,isknow,whichisthecaseforsimulation.Therefore,wecollect twosynthesizeddata-sets;(1)adata-setofallvehicletrajectoriestoobtainthetruetrafficconditions,i.e.,thegroundtruth, and(2)alimiteddata-setoftrafficsensingdatatoreconstructthetruetrafficconditions.
Weconsidera10kmfreewaylinkwithabottleneckatx=7km.Thenumberoflanesdropsfromthreetotwoatthis locationandgoesbacktothreeatx=8.5km.Thisroadlay-out,asvisualizedinFOSIM,isshowninFig.5.
The trafficconditionsare describedbythe macroscopictrafficflow variablesin arectangularmesh,i.e., discretespace andtime. Here, thefreeway issubdivided in500m long roadsegments and15 s time-periods. Giventhe speed limit of 120km/h thiscombinationofroadsegment lengthandtime-periodduration satisfiestheCourant–Friedrichs–Lewy condi-tions(Courantetal.,1928).Althoughthisconditionisnotaprerequisitefortheproposedmethod,itisimportantfor numer-icalstabilityinmodel-basedestimationandprediction.Astheestimatesmaybeusedasinitialconditionsformodel-based trafficstateprediction,wedecidedtoevaluatethemethodologyinthismesh.
Trafficissimulatedfortwoone-hourperiods,onewithsolelyfree-flowconditionsandonewithfree-flowandcongested conditions.Intheremainderofthispaperthesetwocasesarerespectivelyreferredtoasthefree-flow andcongestedcase.
Fig.6aandbshowthetrafficdemandthatisusedasinputintothemicroscopicsimulationprogramisshown.Inbothcases trafficiscomposedof90%passengercarsand10%trucks.Moredetailsaboutthemicroscopicmodelsandparametersused inFOSIMcan befound intheusersmanual (Dijker andKnoppers,2006).Thetrue qD,kD anduD,are obtainedfromthe trajectoryinformationforallvehiclesusing(1),(2)and(3)(Edie,1965).ThisgroundtruthisshowninFig.6.
Fig.6d,fandh,i.e.,thecongestedcase,showpatternsinthecongestedarea.Tounderstandthesepatterns,wedetermine thegroundtruth forafinermesh,i.e., smallerroadsegments andshortertimeperiods. Fig.7showsthegroundtruth in terms ofdensity andspeed forroad segments of 100m and5 s time-periods. Thisfigure shows that thereare multiple waves which move upstream withan approximate speed of 25km/h. The speed within these wavesis close to 0km/h, whilethespeedbetweenthewavesreaches40km/h.Thesepatternsarethusstop-and-gowavesincongestion.Asexplained above,ourobjectiveistoreconstructthetruesimulatedtrafficconditions.Therefore,wewillnotaddresstherealismofthe observedpatterns.However,weareinterestedintheabilityofourestimatortoreconstructthesepatterns.
Inadditiontothefree-flowandthecongestedcase,wesimulatedandevaluatedtheestimationperformanceforan inci-dentsituation.Astheresultsshowlargesimilaritieswiththecongestedcase,weshortlydiscusstheestimationperformance, butdonotprovidedetailedinformationinthisandthenextsection.
5.2. Thereferencetrafficstateestimator
Existingtypesoftrafficsensingdataandestimationmethodologiescanbeusedtoestimatethemacroscopictrafficflow variables. To evaluate theadded value of our work, we compare theestimation performance witha referenceestimator which uses loop-detector data. It is assumed that a loop-detector is installed in the middle of each cell, i.e., the loop-detectorspacingisequalto 500m(therebyfollowingtheDutchstate-of-the-art).The choiceofconsideringloop-detectors installedinthemiddleofeachcellisbeneficialforitsestimationperformancecomparedtootherlocations, e.g.,upstream ordownstreamboundaries.
Theloop-detectordatacharacteristicsarebasedontheloop-detectorsinstalledontheDutchfreeways,i.e.,lane-specific one-minuteaggregatedspeeduT
l andflowql.Theflowforalllanesqisobtainedbysummingthelane-specificflows ql.To
approximatethemeanspeed,thefollowingequationsareconsidered(KnoopandHoogendoorn,2012):
q=λ l=1 ql (44) u= λ l=1 ql λ l=1 uqTl l (45) k= q u (46)
Fig. 6. Demand and ground truth for free-flow and congested cases.
where
λ
denotesthenumberoflanes.The resulting estimates relate to the one-minute loop-detector data period, i.e., four consecutive 15 s periods in the consideredestimationmesh.Theestimatesareassignedtoeach15speriodwiththeone-minuteperiod.Astheconditions obtainedusingEdie(1965)differwithin thisone-minuteperiod,theone-minutedataaggregationperiodisacauseofthe estimationerrors.
Fig. 7. Ground truth of the congested case for a finer mesh.
Loop-detector datacharacteristics differacross the world.A downsideofthe Dutchstate-of-the-art isthat time-mean speeds are observed. These data overestimate the speed (and thus underestimate the density) is congested conditions (Knoopetal.,2009).Thisbias-problemcanbeaddressedbycollectingharmonicmeanspeedsinsteadoftime-meanspeed. AssomesystemsoutsidetheNetherlandscancollectharmonicmeanspeeds,wewillalsodiscussthistypeofmeanspeeds intheresultssection.
5.3. ThePONtrafficstateestimator
The dataused bythe proposed trafficstate estimatoris retrievedfromsensingequipment installedon fixedlocations (stationary observers) andvehicle-based sensing equipment (moving observers). In our study,all observers have a fixed samplingperiod of15 s.Every 15s, i.e.,t=0s, 15s, 30s, ...,the observersshare their position,which yields aset of point-observations alongthe observationpath.Foreach periodbetweenthese points
Nis observed,i.e., thenumberof vehiclesthatpassedtheobserverminusthenumberofvehiclesthatarepassedbytheobserverwithinthisperiod.
We fix three observationpaths, which are thus used for all evaluated PON estimators. These are: the upstream, i.e.,
x=0km,anddownstream,i.e.,x=10km,linkboundaries,andthefirstvehicleenteringthelink.Additionaltothisfixed information,a fraction of thevehicles serve as moving observers. Inthis set-up all observationpaths can be connected, therebyyieldinga
Nforeachcombinationoftwopoint-observations.Furthermore,asweobservetheupstreamand down-streamboundaries,allcornerpointsofspace-timedomainuntilthecurrenttimeTareobserved,i.e.,(0,0),(10,0),(0,T)and (10,T).Observingthecornerpointsisimportantasthismeansthatthecompletespace-timedomainwillbesubdividedinto triangularareas.
Initialevaluationshaveshownthatthespace-timeratiousedinDelaunaytriangulationcanaffecttheestimation perfor-mance. However,theseevaluationsalsoindicatedthat therelationsare notstraightforward.Therefore,tokeep thisarticle conciseandto-the-point,wehavedecidedtoleaveadditionalanalysisrelatedtosubdivisionofspace-timeintriangular ar-easforfutureresearch.Thisalsomeansthattherestillisroomforimprovementintheresultsshowninthenextsection.In thisstudy,thespace-timeratiousedinDelaunaytriangulationisselectedtobeequaltothespace-timeratioofthedefined estimationmesh,i.e.,
υ
=120km/h.Whenthetriangularareadimensionsareinlinewithestimationmeshdimensions,the PONestimatorisexpectedtousetheobservationsnearest(inspace-time)tothedesiredestimationarea.ThePON estimatorisused atthesametime intervalasthereferenceestimator,i.e.,everyminute.Forbothestimators itis assumedthat thereisno dataavailabilitylatency. Therefore,atthe endofeach minutebothestimators estimatethe trafficconditionsforthefour15speriodswithinthisminute.Incontrasttothereferenceestimator,thePON estimatoris abletoutilizeinformationpriortothisminute.
5.4. Evaluationoftheestimators
Toevaluatetheoverallperformance ofthedifferentestimatorsweconsiderthebias(whereapositivebiasindicatesan underestimation)andRootMeanSquaredError(RMSE).Thesestatisticsareusedtoevaluatetheperformanceintermsofthe flowanddensity,andforthetwocases,i.e.,thefree-flowandthecongestedcase.Althoughwefocusonflowanddensity (which arethedirect outputsofthe PONestimationmethodology), wewill brieflydiscusstheestimation performancein termsofspeed.Weuseawarmupperiodof15min,whichensuresallobservationpathsareconnected.
AtfirstwewillfocussolelyonthePONestimator.AsexplainedinSection4,thehomogeneityandstationarityofthe traf-ficconditionsisexpectedtoinfluencetheestimationaccuracy.ThetwocasesshowninFig.6,i.e.,free-flowandcongested, allowustoevaluatetheestimationperformancefordifferentlevelsofinhomogeneityandnon-stationarity.Furthermore,as theconstructedtriangularareasdependonthedata-availabilityitisinteresting tovarythisfactoranddiscusstherelated
6.1. EstimationperformanceofthePONestimator
Theestimationperformanceforthedifferentpenetrationratesdependsonthehomogeneityandstationarityofthetraffic conditions.Thelowerthepenetrationrate,thelargertheconstructedtriangularareas.Therefore,forlowerpenetrationrates thesetriangularareasareexpectedtoindividuallybepartofmorerectangularareasinthedefinedestimationmesh.Ifthe trafficconditionsare inhomogeneousandnon-stationarywithinthetriangulararea, themacroscopictrafficconditionsare notrepresentativeforallrelatedrectangularareasyieldingestimationerrors.Thisexplainsthecleardifferencebetweenthe RMSEsin thefree-flow andthe congested caseobserved inFig. 8.At low penetration rates,theRMSEs are larger inthe congestedcase;however,atapenetrationrateof10.0%theperformanceintermsofRMSEissimilarforbothcases.Interms offlowestimates,thebiasimproveswiththepenetrationrate(similartotheRMSE).However,intermsofdensityestimates, weobserveanear-zerobiasforallpenetrationrates.
Toillustratetheimportanceoflocalizingchangesintrafficconditionsoverspace-time,welookatFig.9.Thisfigureshows that the estimatesforPON estimator atthree different penetration rates,i.e., p= 0.10,1.0and 10.0%,and thereference estimator.These estimatescan be compared to the groundtruth givenin Fig. 6d.The patternat p=0.10% iscaused by threeindividual vehiclesofwhichthetrajectoriesareshownusingtheblacklines.Thesevehiclesdonotprovidesufficient informationto localize the congestion(delay). Althoughthe congested patternis more inline with theground truth at
p=1.0%,the identification ofthis patternclearlyimproves when we move to p=10.0%.When estimates for p=10.0%,
i.e.,Fig.9c,withthegroundtruth,i.e.,Fig.6d,weseethattheestimatorisabletoapproximateboththecorrectvaluesof densityandthepatterncausedbythestop-and-gowavesincongestion.
Inadditiontothefree-flowandcongestedcase,thePONestimatorwasappliedtoanincidentcase.Thisrepresentsaform ofnon-recurrentcongestionandasituationin whichthetraffic flowbehavior temporarilychanges duetoa laneclosure. Theresultingestimationperformancewassimilartothecongestedcaseandthereforewedonotpresentthedetailedresults inthispaper.SimilarresultstothecongestedcasewereexpectedasthePONestimatordoesnotuseanyinformationinthe formofatrafficflowmodelorhistoricalinformation.Therefore,theperformanceofthePONestimatorshouldanddidnot differbetweenrecurrentandnon-recurrentcongestion,orbeeffectedbyusinganinaccuratedescriptionofthetrafficflow behavior.
6.2.ComparisonbetweenthePONandreferenceestimator
Thesimulationstudyallowsustocomparetheestimationperformance ofthePONestimatorwiththereference (loop-detectordata)estimator. Wediscusstwo elementsoftherelative performance,i.e., (1)therelationto theinhomogeneity andnon-stationarityintrafficconditionsand(2)thedifferencesbetweenthemacroscopicvariables.
Therelativeperformanceofthetwoestimatordependsonthelevelofinhomogeneityandnon-stationarityofthetraffic conditions.As discussedabove, thePON estimatorrequiresa higherdata-availability fortraffic phaseidentification ifthe inhomogeneityandnon-stationarityoftrafficconditionsincreases.Thereferenceestimatorusesobservationsfromthe mid-dleofeachcell,i.e.,sensingequipmentisinstalledevery500m.Asaresult, thereferenceestimatoraccuratelylocatesthe differenttrafficphasesinspace-timeforthecongestedcase,seeFig.9.ThisexplainsthelargerdifferencebetweenthePON andreferenceestimatorsinthecongestedwithrespecttothefree-flowcaseatlowpenetrationrates.
With respect to the referenceestimator, we observe a better relative performance of the PON estimator for density estimatesthanforflowestimates.Intermsofflow,therelativeestimationperformanceisbetterinthefree-flowcasethan thecongestedcase,i.e.,thePONestimatoroutperformsthereferenceestimatorintermsofRMSErespectivelyatpenetration rateshigherthanapproximately2.5%and5.0%.ThebiasofthePONestimatoratpenetrationrateshigherthan5%andthe referenceestimator are both (near) zero. In termsof density, the PON estimatoroutperforms the referenceestimator in termsofRMSEstartingfroma penetration between1.0%and 2.5%forboth cases.As showninFig. 8fandby comparing
Figs.6dand9d,thereferenceestimatorhasalargebiasincongestedconditions.Thisbiasisreducediftheloop-detectors wouldobserve harmonic mean speedsinstead of lane-specific time-mean speeds(ceteris paribus).In this case the PON
Fig. 8. Estimation performance (bias and RMSE) of the PON and reference (loop-detector) estimator for the free-flow and congested case.
estimatoroutperformsthereferenceestimatorintermsofdensityRMSEstartingfromapenetrationratebetween2.5%and 5.0%forthecongestedcase.
Incontrasttodensityandflowestimation,wedonotrecommendusingthePONestimatorforspeedestimation.Existing methodologiesthat usetrajectorydataorprobespeed data(e.g.,Worketal., 2010;DelArco etal.,2011) areexpectedto outperform speedestimateswiththePON estimator.However,ourapproach tocombinestationaryandmoving observers isvaluablefor(thespeed-related)dynamiclinktravel-timeestimation.Thisapproachallowsustoestimatethecumulative curvesattheupstreamanddownstream linkboundaries,whichinturncan beusedto estimatethelink travel-time,e.g.,
Fig. 9. Density estimation in space-time for the PON estimator at p = 0.10, 1.0 and 10.0%, and the reference estimator. The moving observers trajectories (black lines) are shown for p = 0 . 10% and 1.0%.
7. Conclusionsanddiscussion
Inthisstudy,weproposeanestimationproceduretoestimatethemacroscopictrafficconditionsinapre-defined space-timemeshusingtrafficsensingdatacollectedbystationaryandmovingobservers.Thisprocedureconsistsoftwoprocesses: (1)obtainpoint-observations ofthecumulative vehiclenumberN usingstationaryandmovingobserversand(2)estimate themacroscopictraffic conditionsbasedon thepoint-observationsofN.Forthesecond, wedesigneda full methodology, whichwe denoteasthePoint-ObservationsN(PON)methodology.Thismethodologydoesnotuseanyinformationrelated totrafficbehavior(e.g.,afundamentaldiagram)orhistoricaldataandisthusastreaming-data-drivenmethodology.
ThePONestimationmethodologyassumeshomogeneous andstationarytraffic conditions.Ifthisassumptionholds the methodologyperfectlyestimatesthemacroscopictrafficconditionsinspace-time.Ifthisassumptionisviolated,errorsare induced.Here,wemake adifferencebetweenerrorsinducedwhenestimatingtheconditionsforthetriangularareasthat formthebasicestimationunitandtheerrorsinduced whengoing fromthetriangularareastothedesiredareas.The re-lation betweenthese assumptions and the estimation errors explain that a positive relation exists between the level of inhomogeneityandnon-stationarityofthetrafficconditionsandtherequireddata-availabilitytoreachasimilarestimation performance.Ifthetrafficconditionschangehighlyoverspaceandtime, e.g.,fromfree-flowtocongestionorstop-and-go waves,having sufficient point-observations is importantto localize the differenttraffic phases. Nevertheless,in the con-ductedcase-studywe stillonly neededtoobserve1.0–2.5%ofthevehicles incombinationwiththeupstream and down-stream10kmlinkboundarytoreachthesamedensityestimationperformance intermsofRMSEashavingloop-detectors installedevery 500m.Inthiscase,the flowestimationperformance issimilarfora penetrationrateofapproximately5%. Theestimationperformance forthesamelinkwasalsoevaluatedforsolelyfree-flowconditions.Here,thePONestimation methodologyoutperformsloop-detectordata-basedestimatedintermsofbothflowanddensityestimatesatapenetration rateof1.0–2.5%.Atlowerpenetrationratestherelativeestimationperformanceisclearlybetterthaninthecongestedcase. ThePONestimationmethodologydoesnothaveadvantageswhenestimatingspeedinadiscretespace-timemesh.However, theapproachtocombineobservationsofstationaryandmovingobserverscanbeusedtoaccuratelyestimatethecumulative
vehiclenumberattheupstreamanddownstreamlinkboundaries,whichinturncanbeusedtoaccuratelyestimatethelink travel-times.
The estimationperformance seemsto belargely determinedby theabilityto localizethechanges intrafficconditions inspaceandtime.Especially,miss-localizinglarge changesintraffic conditions,i.e.,fromfree-flow tocongestionor stop-and-gowaves,resultsinlargeerrors.The abilitytolocalizethechangesintraffic conditionsdependsonthesampling(in spaceandtime)of thepoint-observationsof N.In thesimulationstudy,the influenceofthe penetrationrate(defined as the fractionofvehicles that is observed) on theestimation performance wasdiscussed. However, thepenetration rateis not the only factoraffecting the samplingofthe moving observers. The numberof moving observers inspace andtime respectively depend on the penetration ratein combination with the densityand flow. Furthermore, a fixed numberof availableobserverscanbespreaddifferentlyinspaceandtime,whichcan(dependingonthechangesintrafficconditions) affecttheestimationperformance.
InadditiontothePONestimationmethodology,themoregeneralestimationprocedureisanimportantcontributionof thispaper.Equippedand/orautomatedvehiclescanbeusedtocollecttrafficsensingdataontherelativeflowwithrespect totheirtrajectories.Thisdescribesthechangeinthecumulativevehiclenumber
Noverapathinspace-time.Thesedata (togetherwithdatafromstationaryobservers)canbefusedontheN-levelusingtheprinciplesdiscussedinthispaper.This is highlyvaluable informationas N isthe coremacroscopic traffic flow variable.Knowing N over spaceand time allows forderivingallthreemacroscopictrafficflowvariables.ThePONestimationmethodologycanbeusedtoestimatetheflow anddensityina pre-definedspace-time mesh;however, different(existing)methodologiesmay alsobe usedto estimate macroscopictrafficconditions,e.g.,Natnon-observedpointsinspace-timeorthetravel-time.
Acknowledgments
We thank theNetherlands Organisation for Scientific Research,i.e., NWO inDutch, forproviding the fundingused to performthisresearch.Thegrant-numberassignedtothisprojectis022.005.030.
AppendixA. Performanceunderobservationerrors
This paperproposesandevaluates new methodology, i.e., thePON estimation methodology. Up tillnow, we assumed that the data allow us to obtain error-free point-observations of N. However, in reality, it is likely that the data is not perfect.Toinvestigatetheeffectsofobservationerrors,werelaxthepriorassumptions,i.e.,observationerrorsinNoverthe observationpathsareintroduced.AsexplainedinSection3thedata,i.e.,acombinationofsensingdatafromstationaryand movingobservers,allowsustodealwithobservationerrors;however,weneedtodesignanerrorcorrectionmethodology todoso.
TheobjectiveofthisappendixistoshowthatthecombinationofsensingdataandthePONestimatoralsoyieldsagood estimation output whenhaving to deal withobservationerrors. Forthispurpose, a simpleerror-correction methodology isdesigned. We optto show thateven a simplemethodologysuffices todeal withobservationerrors.Similar tothe TSE methodologydiscussed inSection 4,wedonot wanttorelyonparameters thathavetobe calibrated,asthisisawayof incorporatingadditionalinformationintheformofhistoricaldata.
A1. Methodology
Section 3explainsthebasic conceptbehindtheerrorcorrection methodology:An intersectionoftwo already-initialized observation pathscanbe usedforerror correction.Observation errorscanlead toa discrepancy between Nattheintersection pointon thetwo observation paths.As bothpathsshouldhavethe sameN-value atthe intersectionpoint, we cancorrect for observation errorsbasedonthedifferencein N.Thisprincipleis simple;however,in designinganerror correctionmethodology, weneedtodefinehowthedifferenceinNtranslatesintoacorrectionin
Novertheobservationpaths.
Letusconsider(aswedointhesimulationstudy)alinkwithstationaryobserversattheupstreamanddownstreamlink boundaries,andmorethanonemovingobserver.ToinitializeNovertheobservationpaths,theupstreamlinkboundary(x= 0m)istakenasthereferencepointandN
(
0,0)
=0.Uponenteringthelink,thuswhenthemovingobserverinteractswith theupstreamboundary,eachmoving observerisassignedaninitialN.Thedownstreamlinkboundary(x=L)isinitialized bythefirstmovingobserverexitingthelink.Initializationofthedownstreamboundarydoesnotoccuratt=0s.Therefore, inthesimulationstudy,awarmupperiodof15minisused.Anyother intersectionofobservationpaths isanintersectionoftwoalready-initializedobservation pathsandcanthusbe usedtoforerrorcorrection(seeabove).Inoursimplemethodology,foreachoftheseintersections,wedefine aleadingand followingpath.Theleadingpathhasthemostrecentinformationfromtheupstreamlinkboundary. Incaseofinteraction betweenthedownstream linkboundaryanda movingobserver,the movingobserveris leading.Incaseofan interaction betweentwomovingobservers,theovertaking(i.e.,fastest)movingobserverisleading.
AdifferenceinNintheintersectionpointbetweenthetwoobserversisaccountedforbythefollowingobserverinthe period betweenits last andcurrentinteraction. Inthis wayevery observationis atmostcorrected one time. Toaccount forthedifference,whichistheerror,the
Narealtered. thedifferenceisspreadoutevenlyovertheabsolutenumberof overtakingsregistered bythe observer.Forinstance,ifthere isan errorof+4 vehandthefollowing vehicleregistered16
Fig. 10. Estimation performance (bias and RMSE) of the PON estimator for the free-flow and congested case under observation errors.
overtakings,weadd−0.25vehtoeachofthe16registeredovertakings.Hereweassumethattheconnectedvehicleprovides informationoneachovertakingitobserves.Forinstance,iftheconnectedvehicleregistersthatitovertakesonevehicleand isovertakenbyanothervehicleinonedata-period,wewanttoknowthattwoovertakingswereregisteredandthat
N=0 vehwithinthisperiod.Ifthe followingobserverdidnot registeranyovertakings,thedifferenceisspreadout evenlyover thetimeperiods.
This simple error correction methodology doesnot require anyparameters. Furthermore, foreach observed point in space-timetherearemaximallytwo valuesofN,thatis,beforeandaftercorrection.Iftheintersecting ofthetworelevant observationspaths(whichareusedforcorrection)fallswithintheavailabledataperiod,weusethecorrectedvalue.
A2. Implementationinthesimulationstudy
The performanceunderobservationerrorsis evaluatedforthecases thatwere discussedinSections5and6.The dif-ference betweenthesimulationstudyinthesesectionsandthisappendix liesintheobservationsandimplementationof an error correction methodology (see above).Instead of assuming that we know N forspecific points inspace-time, we considerpotentiallyerroneousobservationsof
NandestimateNbasedontheseobservations.
Stationaryandmovingobserversshouldobservevehiclesthattheypassorthatpassthem.However,theseobserverscan miss orincorrectlydefine vehicle-passings.Inthisstudy,we considerthefollowingerrors: eachpassing caneitherbe(1) correctlyobserved,(2)missedor(3)double-counted.Furthermore,weassignequalprobabilitiestoevents(2)and(3).
Intheory,itispossiblethatwithonedata-period(5s)avehicleovertakesanothervehicleandisdirectlyovertakenagain bythesamevehicle.Theseovertakingsaremissedinourstudy,asweexposethepassingsbasedonthedifferencebetween the vehiclepositions in consecutivetime-instances, which are 5s apart. However, asvehicle speedsare not expectedto showlargeshort-termfluctuations,webelievethatweonlymissaverysmallfractionoftheovertakings.
Inline withthemain simulationstudyofthis article,weevaluate theRMSEof density,flow andspeedestimates for differentpenetrationrates.However,inthissimulationstudy,wealsoconsiderdifferentprobabilitiesformissingor double-countingovertakings,i.e.,0,5,10,15,20,25%.Amiss/double-countprobabilityof10%meansthattheprobabilitiesofcorrect, missedanddouble-countedobservationsarerespectively90%,5%and5%.
A3. Results
Fig.10showstheestimationperformancefordifferentmiss/double-countprobabilities.As expected,observationerrors leadtoalowerestimationperformance.However, evenwithlargeobservationerrors(e.g.,25.0%),weobserveagood esti-mationperformance.
Theestimationperformancestatistics,i.e.,biasandRMSE,underobservationerrors(Fig.10)haveasimilarshapetothose withoutobservationerrors.TheRMSEerrorreduceswithincreasingpenetrationrateinbothcases(free-flowandcongested) andforbothvariables(flowanddensity).However,theinfluenceoftheobservationerrors,i.e.,thedifferencebetweenthe lines with differentmiss/double-count probabilities, is smallest forlow penetration rates.At theselow penetration rate, theinfluenceofincorrect
Nvaluesbetweenpointsinspace-timemaybeoflower significancecomparedtotheexisting estimationerrors (whichare discussedin Section6). Furthermore,theerrorscausedby incorrect observationsof
N can becomemorelocal,e.g.,anunderestimationandoverestimationofthedensityrespectivelyupstreamanddownstreamofa movingobservercancomehand-in-handandbothcontributetoalargerRMSE.
Foreachobservationerrorprobabilitythebiasmovestowardsalevelthatdependsontheerrorsintheupstream station-arydetector.However,dependingontheseerrorsthislevelliesatacertainnon-zerovalue.Thismakes sense,asourerror correction methodologydoesnot trytocorrectallerrors(e.g.,nocorrectionsareperformedontheupstream link bound-ary),butoptstolimittheerrorsbetweenpointobservationsinnearproximityinspaceandtime.Thismaybeareasonto designabettererrorcorrectionmethodology;however,astatedbefore,thisisoutofthescopeofthisappendix.Incontrast tothe RMSE,thebias ofoneobservationerrorprobability canbe better(i.e.,closertozero)than foralower observation errorprobability.Again,thismakessense, aswe considerazero-meanerrordistribution.Therefore,additionalerrorsmay actuallycompensateearliererrors.
Wediscussedthattheintersectionbetweenobservationpathsisimportantforinitializationanderrorcorrection. How-ever,theobserversconsideredinthisstudy,donotdirectlyobservetheseintersectionpoint.Instead,intheerrorcorrection methodologyweneededtoestimate theintersectionpointsbasedonthetrajectoryobservationpointsthathavean inter-valof15s. Therefore,itwouldbebeneficial ifthestationaryandmoving observersobservetheexactintersectionpoints. However, asthisputsanextra strainontherequireddatacharacteristics,inthisstudy,we wanttokeepthisasa recom-mendationforpractice.
Supplementarymaterial
Supplementarymaterialassociatedwiththisarticlecanbefound,intheonlineversion,atdoi:10.1016/j.trb.2018.06.005.
References
Claudel, C.G. , Bayen, A.M. , 2010. Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: theory. IEEE Trans. Automat. Contr. 55 (5), 1142–1157 .
Claudel, C.G. , Bayen, A.M. , 2010. Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: computational meth- ods. IEEE Trans. Automat. Contr. 55 (5), 1158–1174 .
Courant, R. , Friedrichs, K. , Lewy, H. , 1928. Uber die partiellen Differenzengleiehungen der mathematischen Physik. Math. Ann. 100 (1), 32–74 .
Daganzo, C.F. , 1994. The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory. Transp. Res. Part B 28B (4), 269–287 .
Daganzo, C.F. , 1995. The cell transmission model, Part II: network traffic. Transp. Res. Part B 29B (2), 79–93 .
Del Arco, E. , Morgado, E. , Ramiro-Barguene, J. , More-Jimenez, I. , Caamano, A. , 2011. Vehicular sensor networks in congested traffic: Linking STV field recon- struction and communications channel. In: 14th International IEEE Conference on Intelligent Transportation Systems. Washington, DC, USA, pp. 606–613 . Dijker, T. , Knoppers, P. , 2006. FOSIM 5.1 Gebruikershandleiding (Users Manual). Technical Report. Technische Universiteit Delft .
Redmill, K.A. , Coifman, B. , Mccord, M. , Mishalani, R.G. , 2011. Using transit or municipal vehicles as moving observer platforms for large scale collection of traffic and transportation system information. In: 14th International IEEE Conference on Intelligent Transportation Systems .
Seo, T. , Bayen, A.M. , Kusakabe, T. , Asakura, Y. , 2017. Annual reviews in control traffic state estimation on highway : a comprehensive survey. Annu. Rev. Control 43, 128–151 .
Seo, T. , Kusakabe, T. , 2015. Probe vehicle-based traffic state estimation method with spacing information and conservation law. Transp. Res. Part C 59, 391–403 .
Smaragdis, E. , Papageorgiou, M. , Kosmatopoulos, E. , 2004. A flow-maximizing adaptive local ramp metering strategy. Transp. Res. Part B 38 (3), 251–270 . Treiber, M. , Helbing, D. , 2002. Reconstructing the spatio-temporal traffic dynamics from stationary detector data. Cooper@tive Tr@nsport@tion Dyn@mics 1,
3.1–3.24 .
Van Lint, H. , Bertini, R.L. , Hoogendoorn, S.P. , 2014. Data fusion solutions to compute performance measures for urban arterials. In: Celebrating 50 Years of Traffic Flow Theory, pp. 1–5 . Portland, Oregon
Wang, Y. , Papageorgiou, M. , 2005. Real-time freeway traffic state estimation based on extended Kalman filter: a general approach. Transp. Res. Part B 39 (2), 141–167 .
Wardrop, J. , Charlesworth, G. , 1954. A method of estimating speed and flow of traffic from a moving vehicle. Proc. Inst. Civ. Eng. 3 (1), 158–171 . Work, D.B. , Blandin, S. , Tossavainen, O.-P. , Piccoli, B. , Bayen, A.M. , 2010. A traffic model for velocity data assimilation. Appl. Math. Res. eXpress 1, 1–35 .
