2023風(fēng)能控制技術(shù)建模

Windplantpoweroptimizationthroughyawcontrolusingaparametricmodelforwakeeffects—aCFDsimulationstudy

ABSTRACT

Thisarticlepresentsawindplantcontrolstrategythatoptimizestheyawsettingsofwindturbinesforimprovedenergyproductionofthewholewindplantbytakingintoaccountwakeeffects.TheoptimizationcontrollerisbasedonanovelinternalparametricmodelforwakeeffectscalledtheFLOwRedirectionandInductioninSteady-state(FLORIS)model.TheFLORISmodelpredictsthesteady-statewakelocationsandtheeffectiveflowvelocitiesateachturbine,andtheresult-ingturbineelectricalenergyproductionlevels,asafunctionoftheaxialinductionandtheyawangleofthedifferentrotors.TheFLORISmodelhasalimitednumberofparametersthatareestimatedbasedonturbineelectricalpowerproductiondata.Inhigh-fidelitycomputationalfluiddynamicssimulationsofasmallwindplant,wedemonstratethattheoptimizationcontrolbasedontheFLORISmodelincreasestheenergyproductionofthewindplant,withareductionofloadsontheturbinesasanadditionaleffect.Copyright?2014JohnWiley&Sons,Ltd.

KEYWORDS

windplantcontrol;windturbineyawcontrol;windturbinewakes;optimization

Correspondence

J.W.vanWingerden,Mekelweg2,2628CDDelft,TheNetherlands.E-mail:

J.W.vanWingerden@TUDelft.nl

Received21February2014;Revised6October2014;Accepted28October2014

INTRODUCTION

Eachwindturbineinaclusterofwindturbines(awindpowerplant)caninfluencetheperformanceofotherturbinesthroughthewakethatformsdownstreamofitsrotor.Thewakeisaflowstructurethatischaracterizedbyareducedwindspeedbecausetheturbinerotorextractskineticenergyfromtheincomingflowandanincreasedturbulencebecausetheturbineobstructstheflow.Ifanotherturbineisstandinginthepathofawakeatalocationwheretheflowhasnotyetfullyrecoveredtofreestreamconditions,thereducedwindspeedresultsinalowerelectricalenergyproductionofthatturbine.Inaddition,theincreasedturbulenceandshearinthewakemayinduceanincreaseindynamicloadsonthedownstreamturbine.Thesewakeinteractioneffectshavebeenstudiedextensively;seethestudiesofVermeeretal.,Crespoetal.andSanderseforreviewsoftheliterature.1–3Thetopologyandamountofthewakeinteractiondependontime-varyingatmosphericconditions(e.g.,winddirection,windspeed,turbulence,andatmosphericstability)andontheoperatingpointofeachturbinethatcanbeadjustedbychangingtheircontrolsettings(generatortorque,pitchanglesoftheblades,4oryawangle).5–7

Incurrentindustrialpractice,windturbinesinwindplantsarestillcontrolledtomaximizetheirownindividualperfor-mance,ignoringtheeffectthattheturbineshaveonotherturbinesthroughtheirwakes.8Recently,thewakeinteractioneffectshavebecomeamoresignificantfieldofstudyintheresearchonwindturbinecontrolalgorithmsbecausewindplantshavegrowninsize,andmoreknowledgehasbecomeavailableonthelossofefficiencybecauseofthewakeinteractioneffect.ThestudyofBarthelmieetal.,9forexample,reportsanaverageenergyproductionlossof12%inanoffshorewindplantcausedbythewakeeffects(thispercentageisaveragedoverthewinddirections).

Copyright?2014JohnWiley&Sons,Ltd.

Windplantoptimizationbyyawcontrolusingaparametricwakemodel

P.M.O.Gebraadetal.

P.M.O.Gebraadetal.

Windplantoptimizationbyyawcontrolusingaparametricwakemodel

Previousworkonwindplantcontrolhasmainlyfocusedonreducingwakeinteractionbyadjustingtheaxialinductionofturbinestoimprovetheoverallwindplantperformance,whichcanbeachievedbyadjustingpitchandtorque.Thisconceptwasfirstproposedinthelate1980sbySteinbuchetal.10Staticmodel-basedoptimizationstrategies,basedonsimplifiedparametricwakemodels,aretestedbyJohnsonandThomas8andBitarandSeiler11(basedontheJensenwakemodel)12,13andbyGonzalezetal.14(basedontheFrandsenwakemodel).15IntheworkofSchepersandvanderPijl,16asimilarstaticoptimizationisperformedbasedonacomputationalfluiddynamics(CFD)model.Axial-induction-basedwindplantcontrolstrategiesthatadjustthecontrolsettingstochanginginflowconditionsweredevelopedbyMardenetal.andGebraadandvanWingerden(model-freedata-drivenapproaches),17,18Heeretal.(amodel-basedcontrolapproachusingtheJensenmodel),19andSoleimanzadehetal.(model-basedcontrolapproachesusingsimplifiedCFDmodels).20–22Thegoaloftheworkpresentedhereistooptimizetheyawanglesofthewindturbinesforincreasedtotalelectricalpowerproductionofthewindplant.Bychangingtheyawangleofaturbine,theflowdirectionofthewake,aswellastheaxialinductionoftherotor,ischanged.Bycontrollingthedeflectionofthewakethroughyawing,thewakecanbedirectedawayfromdownstreamturbines.ThisapproachwasshowntohavegreatpotentialinCFDsimulationsinthestudiesof

Jiménezetal.andFlemingetal.7,23,24Theconceptwasalsotestedinwindtunnelexperimentswithscaledturbines5andonasmallwindplant.6Thesetestsconfirmedthatthewakecanberedirectedusingyaw,butsinceonlyalimitedamountofdatacouldbegathered,noquantitativeanalysiscouldbemade.AninterestingworkinthiscontextisthatofKragh

andHansen25inwhichitisshownthatmisaligningtherotoryawofaturbinewiththewinddirectioncanalsobeusedtoreducetheloadsonthemisalignedturbine.Theresponseofthecompletewindplantsystemwithrespecttocontrolsettingchangesisslowbecauseoflargedelaysassociatedwiththeflowinthewaketravelingfromoneturbinetothenext.26,27Thisisadisadvantageformodel-freeglobaloptimizationapproachessuchastheoneproposedbyMardenetal.17Thelongtimeitwouldtakeaglobaloptimizationtoiterativelytestcontrolsettingsontherealsystemandconvergeisproblematicifthecontrollerhastoadapttotime-varyingconditionssuchaswinddirectionandinflowvelocity.InthestudyofGebraad

andvanWingerden,18theproblemofhavinglargedelaysinthesystemisaddressed,andanalternativegradient-based‘localized’approachformodel-freeoptimizationisproposedtoimprovethetimeefficiencyofthewindplantcontrol.Thislocalizedoptimizationalgorithmonlytakesintoaccounttheeffectofcontrolsettingchangesonthenearestdownstreamneighboringwindturbines.ThisapproachwasusedbyGebraadandvanWingerdenforpitch-basedandtorque-basedwindplantcontrol,18butitislesssuitedforyawcontrolifthegoalistonotonlydeflectthewakeawayfromthenearestdownstreamturbinebutalsotoavoidthewakehittingturbinesfartherdownstream.Therefore,inthiswork,weproposeamodel-basedcontrolscheme,inwhichanoptimizationalgorithmcantestalargenumberofpossiblecontrolsettingsonthemodel,inordertoiterativelyfindtheoptimalsettingsbasedonthemodelpredictionsbeforeapplyingthemtotherealsystem.

Thesupervisorywindplantcontrolschemeproposedinthispaperincreasesthetotalelectricalenergyproductionofthewindplantbymodel-basedoptimizationoftheyawcontrolsettings.Figure

1

givesanoverviewoftheproposedcon-trolscheme.Animportantpartoftheworkpresentedinthispaperisthedevelopmentofthe‘internalmodel’forthewindplantcontroller.Theinternalmodelpredictsthewakeeffectsinthewindplant.ApreliminaryversionofthisworkappearedinthestudyofGebraadetal.28High-fidelityCFD-basedmodelsthatarebasedonacouplingofdetailedtur-binedynamicsmodelswithaccuratewindflowmodels,suchastheonespresentedbyYangandSotiropoulos,Larsenet

al.,Churchfieldetal.andSchepersandvanderPijl,29–32haveanimportantroleinwindplantcontrolsdevelopment,as

theyallowthealgorithmstobetestedinacontrolledenvironment.Becauseoftheircomputationalcomplexity,accurate

Figure1.Overviewofthedata-drivenmodel-basedwindplantcontroloptimizationapproach(windplantphotocourtesy:Vattenfall,

C.Steiness).

CFD-basedmodelsarelesssuitedforuseasinternalmodelsforreal-timecontrollers.ThesimplifiedparametricJensenandFrandsenmodelsthatwereusedinaxial-induction-basedwindplantcontrolstrategiesmentionedearlier8,11,14,19donotincludetheabilitytopredicttheeffectofyawcontrolonwakeredirection.Therefore,wehavedevelopedanovel

control-orientedmodelthatisabletopredictthesteady-stateeffectsofyawcontrolonthewakesaswellastheresult-ingeffectsontheturbineelectricalpowerproductions.Themodelhasparametersthatcanbeidentifiedbyfittingthepredictionsofthemodeltoturbinepowermeasurements,anapproachreferredtoas‘gray-box’systemidentification.Inaddition,themodelusesmeasurementsfromthewindplanttoestimaterelevantpropertiesoftheinflowintothewindplant.Thecombinationofthemodelidentificationandthemodel-basedoptimizationstepsinthecontrolschemeisillustratedintheoverviewinFigure

1

.Thefactthatweusemeasurementstoidentifythemodelparametersandtheinflowconditionsisthereasonwhywerefertothecontrolschemeas‘data-driven’.Furthermore,themodelhasarelativelysimplestructurethatallowsforquickcomputation,meaningthatitissuitedforreal-timecontrolbasedonmodel-basedoptimizationofthecontrolsettings.

Becausewedonothaveaccesstoareal-worldwindplantonwhichtoperformyawcontrolexperiments,inthiswork,weuseahigh-fidelityCFDwindplantmodeltogeneratethedataneededtodevelopthesimplifiedparametricmodelandidentifythemodelparameters.Next,weimplementthemodelinawindplantcontrolschemethatperformsmodel-basedoptimizationoftheyawsettingsofeachturbineusingagame-theoretic(GT)approach.Finally,wetestthismodel-basedoptimizationcontrolstrategyinthehigh-fidelitywindplantsimulationinwhichtheeffectsonpowerproductionandloadsarecalculated.Inthisway,thehigh-fidelitysimulationisusedtoprovideaproofofconceptforthedata-drivenoptimizationcontrolschemebasedonthesimplifiedparametricmodel.Previousworkonyawoptimizationforwindplants33didnotincludevalidationoftheoptimizedsettingsusinghigh-fidelitynumericalsimulations.

Theremainderofthispaperisorganizedasfollows.InSection

2

,wedescribethesimulationexperimentsperformedinthehigh-fidelityCFDsimulatortoobtainidentificationdatafortheparametricmodel.Thesimplifiedparametricmodel,calledFLOwRedirectionandInductioninSteady-state(FLORIS),ispresentedinSection

3

.InSection

4

,weexplaintheGTapproachtocalculateoptimalyawcontrolsettingsbasedonthesimplifiedmodel.InSection

5

,wepresentsimulationstudiestovalidatethedata-drivenmodel-basedoptimizationapproachinthehigh-fidelityCFDwindplantsimulation.Finally,wediscussourconclusionsinSection

6

.

CHARACTERIZINGWAKEEFFECTSTHROUGHSIMULATIONSINSOWFA,AHIGH-FIDELITYCFDWINDPLANTSIMULATOR

Inthissection,wedescribethesimulationsweperformedinaCFDsimulatortoobtainidentificationdataforthepara-metricmodel.WeusetheSimulatorforOnshore/OffshoreWindFarmApplications(SOWFA),whichisalarge-eddysimulation(LES)ofthethree-dimensionalwindflowaroundoneormoreturbinerotorsintheatmosphericboundarylayer.Therotatingrotorbladesaremodeledthroughanactuatorlineapproach.3TheactuatorlinesarecoupledwiththeFatigue,Aerodynamics,StructuresandTurbulence(FAST)turbineaeroelasticssimulatortool34thatcalculatestheloads,powerandrotorspeedofeachturbine,inadditiontotheforcesthateachturbinebladeexertsontheflow.EachturbineinthesimulationcanbecontrolledusinganindividualcontrolalgorithmimplementedinFASTandalsothroughasuper-visoryordistributedplant-widecontroller.MoredetailsontheCFDcalculationsinSOWFAcanbefoundinthestudyofChurchfieldetal.,31andFlemingetal.35,36givemoreexplanationoncontrolsimplementationinSOWFA.Inaddition,Churchfieldetal.37presentavalidationofSOWFAwithtime-averagedturbinepowersmeasuredattheLillgrundwindplantin?resund,Sweden.

SOWFAsimulationresultswerepresentedbyFlemingetal.23,24thatshowthefollowing:

●

Theeffectivenessoftheyawtechniquesinredirectingthewake.

●

Theeffectofyawwakeredirectiontechniquesontheelectricalenergyproductionandloadsofdownstreamturbinesthatarestandinginthewakeoftheyawingturbine.

●

Theeffectonelectricalenergyproductionandloadsonaturbineofrepositioningthatturbinesuchthattheoverlapwithawakeofanupstreamturbineisreduced.

D

1

Moreinparticular,Flemingetal.24presenttheresultsofSOWFAsimulationsofasetupoftwoNationalRenewableEnergyLaboratory(NREL)5-megawatt(MW)baselineturbines.38TheseturbineshavearotordiameterD126.4m.Inthissetup,theturbinesarealignedinthewinddirectionwithadownwindspacingof7rotordiameters.7D/.Theturbinesareplacedinadomainthatis3km(horizontallength)by3km(horizontalwidth)by1km(height).Theturbulentinflowintothedomainhasameanhub-heightfree-streamwindspeedUof8ms—1andaturbulenceintensityof6%.Thisturbulentinflowisgeneratedbyaprecursorsimulationoftheneutralboundarylayerinthesamedomain,withanaerodynamicsurfaceroughnessthathasalowvalueof0.001m,whichistypicalforoffshoreconditions.

Figure2.SetupandresultsfortheSOWFASimulationSeries1and2asdescribedinSection

2

.Thepowerdatawereusedtofindtheparametersoftheparametricmodel(Section

3

).(a)Experimentalsetups.(b)Time-averagedpowerdata.

Inthispaper,weusedthedatafromthefollowingtwoseriesofsimulationsperformedbyFlemingetal.24(Figure

2

(a)):

InSOWFASimulationSeries1,theupstreamturbine(turbine1)isyawedtoredirectitswakeawayfromthedownwind

turbine(turbine2),resultinginanelectricalpowerproductiondecreaseonturbine1causedbyalossofrotorefficiencyandanelectricalpowerproductionincreaseonturbine2causedbyanincreaseofthevelocityoftheinflowintoturbine2.

●

InSOWFASimulationSeries2,turbine2ismovedinthecrosswinddirectiontoreducetheoverlapofitsrotorwiththewakeofturbine1,alsocausinganincreaseintheelectricalpowerproductionofturbine2.

Foreachyawsettingandposition,a600-ssimulationwasrun.Thewakeswereallowedtodevelopduringthefirst200softhesimulationandthen400sofsimulateddatawerecollected.Byaveragingthepowersignalsoftheturbinesoverthese400s,theresultspresentedinFigure

2

(b)weregenerated.Ineachcase,theturbinesusethebaselinepitchandtorquecontrollersdefinedbyJonkmanetal.38Forthesimulatedflowconditions,boththeupstreamandthedownstreamturbineoperateinabelow-ratedoperatingregion(region2)andthususeconstantpitch,variabletorquecontroltomaximizepowerproduction.38,39Formostcases,thedownwindturbineproduceslesselectricalenergythantheupwindonebecauseitissubjectedtothelow-speedwakeoftheupwindturbine.Figure

2

(b)alsoincludesthepredictionsofthesimplifiedparametricmodel(FLORIS)thatispresentedinSection

3

.

SOWFAhigh-fidelityCFDsimulationsaretypicallyrunforafewdaysonaclusterwithafewhundredprocessors.23,24

BecauseofthecomplexityandcomputationalcostsoftheSOWFAmodel,itisnotsuitableasaninternalmodelforawindplantcontroller.ThedatageneratedbySOWFA,however,canbeusedtodevelopsimplifiedmodelsthatcanbedirectlyusedbythecontroller.InSection

3

,wedescribehowthepowerdatafromSOWFASimulationSeries1and2areusedtodevelopandidentifyparametersofsuchasimplifiedcontrol-orientedmodel(FLORIS).InSection

5

,weuseSOWFAtoevaluatethecontroltechniquesbasedonthesimplifiedinternalmodelinhigh-fidelitysimulations.

FLORIS,ADATA-DRIVENPARAMETRICWINDPLANTMODEL

Inthissection,weexplainthestructureofaparametricmodelpredictingthesteady-stateeffectsofyawmisalignmentofdifferentturbinesontheelectricalenergyproductionsofwindturbinesinawindplant.Itcapturestheeffectsoftheyawcontrolonboththeredirectionofthewakebehindtheturbineandonthevelocityinthewake.Thisisimportantforpredictingtheelectricalenergyproductionsondownstreamturbines,asisalsopointedoutbyChoiandShan.27Sinceitincludesbotheffects,fortheremainderofthepaper,werefertothemodelastheFLORISmodel.

TheFLORISmodelisacombinationoftheJensenmodel12,13andamodelforwakedeflectionthroughyawfirst

presentedbyJiménezetal.7Further,augmentationsweremadetotheJensenmodelinordertobettermodelsituationswithpartialwakeoverlapandtothewakedeflectionmodelinordertoincludewakepositionoffsetscausedbyrotorrotationaleffects.TheseaugmentationsalsoallowbetterfitsofthemodelwiththepowermeasurementsobtainedinSOWFASimulationSeries1and2.

Figure

3

givesanoverviewofthedifferentpartsofthemodelandofhowitinteractswiththeyawoptimizationalgorithmofthewindplantcontroller.Italsoshowsthatmeasuredpowerandyawsettingofturbines,aswellaswinddirectionmea-surementsateachturbine,arefedintothemodel.Themeasurementsareusedtoestimatecertainatmosphericconditions,beingthecurrentdirectionandfree-streamvelocityoftheinflowintothewindplant.Theseyawmeasurementsshouldbedistinguishedfromtheprelimarytestyawsettingsthattheoptimizationalgorithmfeedsintothemodel,andthecorre-spondingpredictedturbinepoweroutputsthatthemodelgeneratesonthebasisofthosetestyawsettingsandtheestimatedinflowproperties,andfeedsbacktotheoptimizationalgorithm.

Inthissection,wepresentthedifferentpartsoftheFLORISmodel.First,inSection

3.1,

weexplainhowtheelectricalpowerproductionlevelsoftheturbinesarecalculated(theturbinepowermodelinFigure

3

).Tocalculatethesepowerlevels,estimatesoftheeffectiveinflowspeedsareused.Theseinflowspeedestimatesfollowfromthewakemodel.Inthewakemodel,weuseaspecificdownwind/crosswindcoordinateframe.Figure

3

showsthattheturbinecoordinatesaretransformedtothesecoordinatesusingmeasuredwinddirectionsattheturbines.ThisstepisfurtherexplainedinSection

3.2.

Submodelsfordifferentwakepropertiesarethewakedecay,deflectionandexpansionmodels,whicharealsoshowninFigure

3

.ThesesubmodelsareexplainedinSections

3.3–3.5.

Finally,Figure

3

showsthewakecombinationsubmodel,whichdefineshowthewakeeffectsofthedifferentturbinesarecombinedtofindtheeffectiveinflowspeeds

?Σ ?j2Σ

Figure3.OverviewoftheFLORISdata-drivenparametricmodelasitisimplementedinthewindplantcontroller.Figure

1

showsthesamebasiccontrolscheme,butinthisfigure,thedifferentsubmodelsoftheFLORISsimplifiedwindplantmodelareshown,andtheidentificationblockisomitted.TheFLORISmodelusessomemeasurementsfromthewindplant(shownbelowthemodelscheme)toestimateintheinflowproperties(speedanddirection).TheGToptimizationalgorithm(topright)usestheFLORISmodeltotestyawsettingsfortheseparticularinflowconditionsandfinallysendsoptimizedyawsettingsasreferencesignalstothewindplant.Intheschemeshownhere,theshorthandnotation8iisusedfortheset8iiF,where8iisacertainpropertyofaturbineiinthe

windplant.

ateachturbine.WeexplainthissubmodelinSection

3.6.

Inexplainingthemodel,weintroducedifferentcoefficientsthatserveasmodelparametersthataretobetunedtomeasurementsfromawindplant.Inthiswork,weusethepowermeasurementsfromSOWFAtofindtheFLORISmodelparameters,asdiscussedinSection

3.7.

Turbinepower

2F

FDf ··· g

Let 1,2,,Ndenoteasetofindicesthatnumberthewindturbinesinawindplant,withNdenotingthetotalnumberofturbinesintheplant.Thesteady-stateelectricalpowerofaturbinei,denotedasPi,iscalculatedasfollows:40

2

i

PiD1pAiCP.ai,μi/U38i2F (1)

D—

wherepistheairdensity,Aiistherotorsweptarea,CPisthepowercoefficientoftheturbine,andUiistheeffectivewindspeedattheturbine.Innonyawedidealizedconditions,thepowercoefficientisrelatedtotheaxialinductionfactorofeachturbine,definedasai1Ui,D=Ui,withUi,Dbeingthewindspeedattherotor,andUithefree-streamwindspeedinfront

D —

ofturbinei,asCP.ai/4ai?1ai]2.40Inthemodelpresentedhere,weapplyacorrectiononthisrelationshiptoaccount

fortheeffectoftheyawmisalignmentangleμiontherotorpowercoefficient,followingtheexampleoftheexperimentalstudiesbyMedici.41Inaddition,weuseaconstantscalingoftheCPvalue,щ,toaccountforotherlosses.Thisresultsin

CP.ai,μi/D4ai?1—ai]2щcos.μi/pP (2)

D

D

D

D

TomatchthemaximumCP0.482and94.4%generatorefficiencyreportedbyJonkmanetal.38fortheNREL5-MWturbinethatisusedinSOWFASimulationSeries1and2,weusealossfactorщ0.768.InthestudyofMedici,41aparametervaluepP2wasfoundtofitdatafromwindtunneltestswithyawingturbines,buttheparametersettingslistedinTable

I

arefoundtofittheyaw-powerrelationshipoftheupstreamturbine(turbine1)inSOWFASimulationSeries1(Figure

2

(b)).Tofindtheseparameters,weassumeanidealizedaxialinductionofai1=3forbelow-ratedconditions.Themodelcanbeextendedtoabove-ratedoperationbymakecorrectionsonaiasafunctionofinflowspeedUi(basedonamaximumratedpower),orbymakingaiafunctionofpitchandtip-speedratio,usingknowlegdeoftheCP-curve.

Intheremainderofthissection,wedescribehowtheeffectiveinflowspeedsUiateachturbineareestimatedbythemodelbypredictingthesteady-statewakecharacteristicsasafunctionoftheyawangles.

Inflowdirectionandthedownwind–crosswindcoordinateframe

82F

Todescribethespatialpropertiesofthewakesbehindtheturbines,weadoptaCartesiancoordinateframework.x,y/inwhichthex-axisispointingdownwindalonganestimatedmeaninflowdirectioninthewindplant,andthey-axisispointingorthogonaltothex-axisinthehorizontaldirection(i.e.,alongthecrosswinddirection,asillustratedinFigure

4

).Thez-axisthenrepresentsthealtitude.Inthiswork,weassumethateachturbinehasthesamehub-height,andtheturbinelocationsinthisdownwind–crosswindcoordinateframearedenotedas.Xi,Yi/i.

Themeaninflowdirection,denotedby?,canbeestimatedinseveralways.Inthemodel,asusedintheCFDsimulation

examplesinSection

5

,itisfoundusingwinddirectionmeasurementsatthemostupwindturbine.Todeterminewhichisthefrontturbine,weneedsomeinitialguessofthefree-streamwinddirection,soweusetheiterativeproceduredescribedinthenumberedlistthatfollows.Thestepsofthisprocedurearealsoillustratedintheleft-mostblockintheFLORISmodelschemeinFigure

3

.Thestepsareasfollows:

i

2F

Weaveragetheflowdirectionmeasurementsatthehubofeachturbinei,denotedas?measured,toprovideafirstestimateoftheinflowdirection.IntheCFDsimulationexamples,thewinddirectionattheturbinehubsisestimatedbysamplingthehorizontalvelocitycomponents.uNi,vNi/atthehublocationofeachturbinei2Ffromtheflowfield

TableI.Parametricmodelparameters.

Turbine

power

Deflection

Expansion

Velocity

y

0.768

kd 0.15

ke 0.065

MU,1 0.5

pP

1.88

ad —4.5

me,1 —0.5

MU,2 1

MU,3 5.5

me,3 1

aU 5

bU 1.66

Wake

bd —0.01 me,2 0.22

Figure4.Thethreedifferentwakezonesoftheparametricmodel.Thefree-streamwindvectorsareindicatedasarrowswithlength

Ui(thefree-streamvelocity).Insidethewakezones,thewindvectorshaveareducedvelocity(Section

3.5).

Theareasoverlapping

i,j,q

withadownstreamrotorj,Aolareusedtocalculatetheeffectivewindspeedatturbinej(equations

(20)

and(

22)).

(a)Topview.(b)Cut-throughatdownstreamturbine.

calculatedbytheCFDsimulatorandbycalculatingthedirectioninthehorizontalplaneas?iDtan—1.vNi=uNi/:

X

1N

i

?DN

iD1

?measured (3)

? Σ NN

Theturbinepositionsindownwind/crosswindcoordinatesarecalculatedaccordingtotheestimatedwinddirection.IfXNi,YNiaretheturbinecoordinatesrelativetothesameCartesiancoordinates.x,y/towhichthewinddirection?ismeasured(Figure

4

),thedownwind–crosswindturbinecoordinatesare

ΣXiΣDΣcos.—?/—sin.—?/ΣΣXNiΣ (4)

Yi sin.—?/cos.—?/

YNi

Itisestablishedwhichturbineisthefront(mostupwind)turbine,anditisassumedthatthemeaninflowdirectionisequaltothewinddirectionmeasuredatthatturbine(i.e.,weassumeauniformdirectionofthefree-streaminflowtothewindplant):

D

f argminXi (5)

i2F

f

?D?measured

Werepeatsteps2and3untilconvergence(i.e.,nochangeinestimatedwinddirection?).

(6)

Thewinddirectionestimationiterativeprocedurewillgenerallyconvergetoawinddirectionmeasuredatacertainturbinewithintwoorthreeiterationsinoursimulationexamples.NotethatinourimplementationofthemodelasillustratedinFigure

3

,thewinddirectionmeasurementsatthehuboftheturbines,definedrelativetothemeshcoordinates,arelow-passfilteredtofilteroutsmall-scaleturbulenceeffects.

Wakedeflection

. Σ

i

40

Yawingaturbinerotorcausesthethrustforcethattherotorexertsontheflow,FD,torotateinsuchawaythatacrosswindcomponentisinduced,7whichcausesthewindflowtodeflectinthedirectionoppositetotheyawrotation(Figure

4

(a)).Becausethewakedeflectionisinducedbythethrustforce,theamountofdeflectionisafunctionofthethrustcoefficientoftheturbineCTD2FD=pAiU2.Whentheyawisnotmisalignedwithrespecttothewinddirection(i.e.,μiD0),the

thrustcoefficientisrelatedtotheaxialinductionfactoraioftherotorofaturbine