Theoretical and experimental study of wind turbine drivetrain fault diagnosis by using torsional vibrations and modal estimation

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Journal of Sound and Vibration

journalhomepage:www.elsevier.com/locate/jsv

Special Issue: Recent Advances in Acoustic Black Hole Research

Theoretical and experimental study of wind turbine drivetrain fault diagnosis by using torsional vibrations and modal

estimation

Farid K. Moghadam

^∗

, Amir R. Nejad

Department of Marine Technology, Norwegian University of Science and Technology, Trondheim, Norway

a rt i c l e i nf o

Article history:

Received 29 August 2020 Revised 9 April 2021 Accepted 19 May 2021 Available online 21 May 2021 Keywords:

Drivetrain system Torsional measurements Modal analysis Fault diagnosis Sensitivity analysis Floating wind turbines

a b s t ra c t

Thispaperprovidesananalyticalproofandthetheoreticaldevelopmentoftheideaofus- ingthetorsionalvibrationmeasurementsforasystem-levelconditionmonitoringofthe drivetrainsystem.Themethodreliesonmodalparameterestimationofthedrivetrainsys- tembyusingthetorsionalmeasurementsandsubsequentmonitoringofthevariationsin thesystemeigenfrequenciesandnormalmodes.Angularvelocityerrorfunctionextracted from encoderoutputsatbothinput andoutput ofdrivetrainisused toestimatemodal parametersincludingnaturalfrequenciesand dampingcoefficients.Intheproposedcon- dition monitoringapproach,itisshownthatanyabnormaldeviation fromthereference valuesofthedrivetrainsystemdynamicpropertiescanbetranslatedintotheprogression ofaspecificfaultinthesystem.Inordertoextracttheconditionmonitoringfeatures,lo- calsensitivityanalysisisengagedtoestablisharelationshipbetweendifferentcategories ofdrivetrainfaultswiththesystemdynamicpropertiesandtheamplitudeoftorsionalre- sponse,whichhelpswithbothtoidentifythestateoftheprogressivefaultsandtolocalize them.Localsensitiveanalysisshowsthatabnormaldeviationsinstiffnessandmomentof inertiaduetothepresenceoffaultsresultinconsiderablechangesinnaturalfrequencies andmodalresponseswhichcanbemeasuredandusedasfaultdetectingfeaturesbyus- ingtheproposedanalyticalapproach.Sensitivityanalysisisalsoemployedalongwiththe estimated modalfrequency forestimationofmodal dampingfromthe amplitudeofre- sponse atthe naturalfrequencies andtheirsubsequentuseforestimationofundamped naturalfrequencieswhicharelaterusedintheproposedconditionmonitoringapproach.

Theproposedapproachiscomputationallyinexpensiveandcanbeimplementedwithout additionalinstrumentation. Twotestcases, using10MWsimulatedand1.75MWopera- tionaldrivetrainshavebeendemonstrated.

© 2021TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Bothpredictiveandcondition-basedmaintenancesare proposedintheliterature aspotential gamechangersandmea- sureswhichcould betakentoflattenthegapbetweenOPEXinoffshoreandland-basedwind turbinesaimedatrealizing

∗Corresponding author.

E-mail address: [email protected] (F.K. Moghadam).

https://doi.org/10.1016/j.jsv.2021.116223

0022-460X/© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

the EU 2050plan by reduction of downtimeandsubsequently levelizedcost of energy(LCOE) of offshorewind [1].The motivation ofthisresearch isreducing the costlyoperation andmaintenance of offshoreturbines -more specificallythe drivetrainsystemoffloatingoffshorewindturbines-andimprovingtheriskofinvestmentbyusingcondition-basedmain- tenance and a subsequentreduction in downtimeasone of themostinfluential consequences ofdrivetrain failures.The latterisinvestigatedbasedon developingthemethodswhichcanuseonlythe existingsensors,database, communication networkandcanbeimplementedforbothfaultdiagnosis(asakeycomponentofturbineoperationsmanagementautoma- tionsystem)andofflineconditionmonitoring(CM)purposes.TheCMsystemisinadditiontotheperformancemonitoring, andtheconceptbehindismonitoringoftheconditionsoftheturbinesystemswiththehighestriskoflossofturbineavail- ability considering both likelihood and consequenceof failures,because monitoring the conditionof all systems maybe economicallyandtechnologically infeasible.AccordingtothestudybyPfaffeletal. [2]whichprovides acautious compar- ison onreliabilitycharacteristicsofboth onshore andoffshore windturbines,drivetrainsystemwhichingeneralincludes all rotatingcomponentsinpowerconversionsystemi.e.hub,rotor,mainbearings,gearbox,generatorandpowerconverter accounts for57% of turbinetotal failures and65% ofturbine total downtime. These numbers are expected to be higher in floating offshore turbines.The latteris dueto morecostly marine operations speciallyin deepwaters, thelarger and moreexpensivecomponents,andawiderrangeofexcitationsourcesduetothesynergisticimpactsofwaves,currentsand wind turbulenceswhichcall forinnovativeapproachesto achievea betterunderstanding aboutthesystemdynamicsand excitation sources. Thefocus ofthisresearch isproposing a system-levelCM solution bythe drivetrainmodal estimation anda subsequentmonitoring ofabnormalvariationsofsystemmodes. This goalisperformedby developinga numerical model ofthedrivetrainasa dynamicsystembasedonits measuredtorsionalresponse andthesubsequentestimation of the drivetraintorsional modes.In contrastwiththe other systems(e.g.bridges andbuildings), the dynamicpropertiesof the drivetrain donot experience a significantchange over normaloperations. The lattercan be usedto monitoranyab- normality caused byfaults. Therefore, variationsinthedrivetrain can be monitoredby trackingthe changes inthethree modal parameters(modal frequency,modal dampingandmodeshape (amplitudeandphase)) ofthedominant modesof thissystem[3].Estimationofmechanicalsystemsdynamicalcharacteristicsismainlybasedonoperationalmodalanalysis (OMA) which ischallengingfordrivetrainasa complexdynamical system.The latterismainly basedon translationalvi- brationmeasurements[4],andthereportedresultsintheliteratureshowthehighpossibilityofharmonicstobemistaken withtheeigenfrequencies[3,5].Drivetrainisacomplexrotationalsystemwithdifferentsources ofexternal excitationand componentsdefectfrequencies.TheuncertaintiesintheestimatedmodeshavemadeavailableOMAtechniquesless-efficient forcondition-basedmaintenance.

ThecurrentCMapproachesofthewindturbinedrivetrainarebasedononeoracombinationoffivecategoriesoftech- niques, namely vibrationanalysis[6],electrical signature(current andpower signals)[7],acoustic emissionsanalysis[8], thermography[9]andtemperatureanalysis[10],andanalysisofoilparticles[11].Today,vibrationanalysisismainlybased on systemtranslationalresponses obtainedby accelerometers(e.g.see [12])withaminorattentionto torsionalmeasure- ments.TheonlycommerciallyavailabledrivetrainCMbasedontorsionalmeasurements,isassociatedwiththemeasurement oftorqueasthesystemappliedload[13].Thelatterisnotwidelyusedduetothematterofcost,technologicallimitations related to operating speed and torque ranges and shafts dimensions, intrusive nature of the torque measurement tech- niques, and alsoa lack ofa standardized approach and theimmature andinsufficient knowledge toanalyze andextract featuresfromthetorsionalmeasurements.

Frequencyresponsefunction(FRF)isacommontoolwhichisusedformodalestimationby theestimationofasystem transfer function.However, thecomplexityofthedrivetrain systemandinadequacyof modelsinconsidering theinternal dynamicsandinteractionsbetweensystems,nonlinearandsynergisticimpactsofdifferentexcitationsources,uncertainties inestimationofloads aresome reasonswhichcauseinexplicableharmonicsandlimit theapplicationofFRFfortheesti- mation ofdrivetraindynamicproperties.Inthiswork, theoperationalmodalanalysisandfaultdiagnosisarebasedonthe systemtorsionalresponses.Ananterior estimationofthedrivetrain loadscanprovidemoreoptionstotheproposed algo- rithm. Thepossibilityofobservingdrivetraintorsionalnaturalfrequenciesinthetorsionalresponseisreportedin[14]for differentapplicationssuchasjetenginehigh-pressuredisk,ahydrostationturbineandacoal-firedpowerplant.Thepossi- bilityofobservingbladenaturalfrequenciesindrivetrainshafttorsionalresponseandthepotentialsforCMofthebladesis alsoreportedin[14,15].Suominenetal.[16]hasreportedthevisibilityofshippropulsionsystemnaturalfrequenciesinthe torquemeasurementsofthepropulsionshaftduetothepropellerbladecontactwithice.However,thesereportsarebased onobservationsonexperimentalstudiesandarenotreliantonananalyticaltorsionalmodelofthedrivetrainsystems.

The drivetrainsystemtorsionalresponseandthenaturalfrequencies areproposed intheliterature fordetecting faults initiatedbytorsionalsources.Patel etal.[17]proposestheuseofangulardisplacementtosupportthelateralresponseto recognise therubbingfaults inthedrivetrain,sothattheexcitedtorsionalfrequencyandtheamplitudeofresponseinthe naturalfrequency andtheside bandsare utilized tocharacterize thefault. Fenget al.[18] proposestheuseof themea- surements oftorqueinstead oftransversevibration signalstodiagnose planetarygearboxlocal/distributed faults,because theyarefreefromtheamplitudemodulationeffectcausedbytimevariantvibrationtransferpaths,thustheyhavesimpler spectralcontentthantransversesignals.Leboldetal.[19]suggestsmonitoringthecharacteristicchangesintorsionalnatural frequencies,andclaimsthatthosechangesareassociatedwiththeshaftcrackpropagation.Kiaetal.[20]proposestheesti- matedelectromagnetictorqueoftheelectricalmachineasanoninvasivetorsionalmeasurementinthedrivetraintomonitor the torsional stress on the components includingshaft, bearings, and gearbox, andthe method is used to detect a gear failure. Theelectromagnetictorqueestimationiscommonlyusedinelectrical drivestocontroltheelectrical machine,and

implementationofthemethoddoesnotneedanyadditionalsensor.Notonlythedrivetrainfaults,butalsorotorandtower excited modesmay resultin frequencycomponents in thedrivetrain torsional response [21].The amplitude ofresponse atbladeedgewiseandtowerbendingnaturalfrequencies canprovideinsightsaboutresonancesinthesecomponents.The monitoringofthevariationsofthesecomponentsfrequenciesisalsousefulforsome otherpurposessuch asicedetection in blades,andhealth monitoringof blades (detectrootcrackswithin turbineblades) andtower. Theidea ofusingangu- larvelocitymeasurements forthewindturbinedrivetrainfaultdetectionwasoriginallyproposed byNejadetal.[22].The input dataisprovided bytheencodersinstalled onthedrivetrain forthe turbinecontrolpurposes.The latterisnormally accessibleinbothturbineandfarmlevels,whichhelpstorealizeCMbymeansofsupervisorycontrolanddataacquisition (SCADA)systemavailable measurements.Therefore,anyalgorithmbasedonthosemeasurementscanbeintegratedintoei- therturbineorfarmcontroltosupporttheonlineCMofthedrivetrain.Moghadametal.[15]hasexperimentallyevaluated thepossibilityofdetectingsomecategoriesoffaultsinearlystagesbyadirectutilizationoftorsionalresponse,andthere- sultsofthestudyarecomparedtoaconventionalmethodbasedontranslationalvibrationsobtainedbyaccelerometers.The authorsdemonstratedhowtorsionalmeasurements cancomplementtheconventionalapproachesbyprovidinginsightson theexcitationsourceswhicharesignificantlyinfluencingonthedrivetrainlifetimewhichisusefulasbothadesignfeedback andearlierstagefaultdetection.

Eventhoughthetorsionalresponsecannotdirectlybeusedformonitoringofsomesortoffaults,itcontainsthedrive- trainsystem-leveldynamicpropertieswhichcanprovideanearreal-timemodalestimationofsystem.Fromthisperspective, torsionalmeasurements are indirectlyusedforthedrivetrainsystemCM purposes.Forthispurpose,thesemeasurements arefirstusedtoestimatethedynamicpropertiesofthedrivetrainasarotationalsystem.Thesepropertiesareonlyrelatedto thesystemphysicalparametersandnottheloadingorspecificoperationalcondition,sothattheycanbeusedinthesecond steptomonitorthevariationsinthedrivetrain,whichcanbetranslatedintoafaultincaseofpassingaprespecifiedlevel.

Moghadam etal. [23]hasstartedananalyticalapproachtoturntorsionalmeasurementsintomeaningfulfeatures forfault detectionpurposesbyspecificationoftheanalyticalrelationshipbetweenthesystemnaturalfrequencyvariationsandfaults, andasubsequentpotentialfordetectionofsystemfaults.Thecurrentworkisdedicatedtothetheoreticaldevelopmentand simulation/experimentalvalidationoftheideaoriginallyproposedby[23]forthemodalestimationofthedrivetrainbyus- ingtorsionalmeasurements,andasubsequentuseofthisideatodevelopamethodforthedrivetrainsystem-levelCM.The influence ofshaftcrack propagationon thetorsionalnaturalfrequencywasdiscoveredby [14,24].However, thosestudies lackananalyticalmodelwhichdescribesthevariationsinordertoestablishameaningfulfeatureformonitoringthecondi- tionofcrack.Forthisfeaturetobeusedasacriterion asashaftcrackingmonitoringtechnique, asufficientmodelshould beprovidedtobeabletorelatethevariationstothestateofthefault.Inaddition,thereareothercategoriesofsystem-level faultswhichcanalsoinfluencedrivetraintorsionalmodeswhicharenotconsideredinearlierstudies.

TheCMofthedrivetrainatsystemlevelbyusingtheestimatedtorsionalnaturalfrequencies,thenormalmodes,theam- plitudeofresponseinthenaturalfrequenciesandthedampingofthesystematnaturalfrequenciesindifferentoperations isdiscussedinthispaper.Forthispurpose,onlineoperationalmeasurementsofthedrivetraindifferenttorsionalresponses includingtheangularvelocityresidualfunctionandfilteredangularvelocityareemployed.Drivetrainfaultsatsystemlevel can influence the drivetrain model parameters,so that they can be categorizedinto the faults that change the torsional stiffnessmost(e.g.crackintheshaftsandbearingwear speciallyingearbox),andfaults thatchangemostlythe inertiaof the drivetrain(changesinmassbalance/distrinutionwhichcan beduetoe.g. lossofmass,wear andunbalance;andalso changeintheaxisofrotationwhichcanbeduetoe.g.misalignmentandlooseness).Regardingtherelationofinertiawith thesquareofthecenterofmassdistancefromthecenterofrotation,thefaultswhichvariatethecenterofrotationdemon- strate moresignificantinfluence ontheinertia andthus aremoreinfluentialonthe torsionaldynamicproperties.Among the faults that influencethe inertia ofthebodies indrivetrain equivalentmodel,there are some typeswhichhave more considerableimpactontheboundaryconditionsbetweenrotatingandstationaryelementsandthusinfluencemoredrive- train lateralresponsesthan thetorsionalresponse(e.g.looseness(pedestal,shaft andbearings,coupling),[25]).The latter influencessignificantlylateralstiffnessparametersandthelateralresponsesofthedrivetrain,sothatthedetectionofthose faultsbyusingthelateralresponseandmonitoringthevariationsofsystemlateralpropertiescouldbeapracticalapproach.

Eventhoughthesefaultscancauseasmallvariationofthetorsionalparameters,theimpactmaynotbesignificantenough for fault detectionpurposes. For example,a pedestal looseness maycause increasedrubbing which leadsto a nonlinear smallincreaseoftorsionalstiffness.

By specification of the parameter in the equivalent reduced ordermodel which will be significantly influenced by a fault,itispossibletolookfortheexpectedconsequencesoftheassociatedvariationsoftheparameterasaresultoffault onthe systemdynamicproperties,astheCMindicators.Thefirstcategoryoffaults,whichare detectablebytheproposed CM approach,influencethetorsionalstiffness. Cracksinthedrivetrainshaftsareone ofthe mostinfluentialfaults inthis category. Theinitialcracksoccurduetomaterialimperfections andtemperaturevariationsinthepartsofmainshaftwith severestressconcentration,whichcangrowworseunderlargealternatingloadsduetowindturbulence.Todetecttheshaft cracksofdifferentrelativedepths,anapproachbasedonnonlinearoutputFRFisproposedby[26]andexperimentallytested onasimpledouble-diskrotorsystem,wherethelineardisplacementsandthebendingmomentsaretheunderconsideration responses butthe torsionalvibrations areneglected. In shaftcrack faults,variation instiffnessis influencedby the crack depthandtheshapeofthecrackfront.Thelattermakesthedetectionofdifferenttypesofcracksquitechallengingsothat a detectionmethodsuitable foronetypeofcrackcannot begeneralizedtotheothertypes.Thisfaultdoesnottake place asfrequentlyasshaft unbalanceormisalignmentbuttheconsequences arevery high,so thatthe detectioninveryearly

stages is ofa highimportance. If shaft cracks are not detected inearly stages, thelater stages ofthis category offaults may causesevere damageof theshaft andthe occurrenceof considerablesecondary faults withhighrisk of injuriesfor the plantpersonnel.Asdiscussed byChattertonetal. [27],crackdetectionbyusingtranslational/axialvibrations obtained fromaccelerometers ischallengingdueto theinfluenceofthe dynamiceffects causedby differentcomponents andtheir consequentinducedvibrations.Theinterpretationofthedatainthesemethodsisalsodependentonadeepunderstanding of the type of crack, its physical properties and the specific operational conditions so that the realization of an online monitoringmaynotbe possible.The frequencyandtimedomainsanalysisofaccelerometersistheconventionalapproach to detect increasedvibrations in the component-level in higherstages of a progressive fault (e.g. gear tooth and rolling element bearingfatigue issues).Thesecondcategoryoffaultsdetectableby theproposedapproachinfluencetheinertia of thecomponentsintheequivalentmodel.Inthisgroup,misalignmentandunbalancearesignificantlymorecommonthanthe other faults.Unbalanceintherotorbladesisoneofthemostinfluentialandprevalentfaultswhichcanbeduetodifferent reasons e.g. excessiveweight followinga bladerepair, icing, waterpenetration throughcracks andloosematerial moving insidetheblades.Thelattercauseslossinthepowerproduction.Thereasonforplacinganemphasisontherotorunbalance isthatthehighestunbalanceinthedrivetrainarisesfromthecomponentwiththehighestmomentofinertiawhichisthe rotor assembly in the wind turbine drivetrain. The mass unbalance also causes additional loads on the entire structure andspeciallythedrivetraincomponentssothatitresultsinaperiodictorsional(inearlierstages)andtransversal(inlater stages)oscillationsinthewindturbine’sdrivetrain.Itdirectlyincreasesthewearofthebladeonthedrivetrainbearingsand gearsbygeneratingasymmetricalloads.Therotormassunbalancecanbemeasuredbymonitoringitsconsequentvariations in thedrivetrain dynamicproperties.As aprognostic measure, theunbalancemass canbe estimatedandifthe detected unbalanceexceedsalimit,therotorbladesshouldbebalancedwithabalancingdevice.

CMismostlydesignedincomponentlevel,whichishelpingwhenthefaultispropagatedtotheindividual components and causesphysical changes inthe component level.However, the rootcauseof a wide rangeof faults are system-level issuessuch asunbalance, misalignment,loosenessandshaftcracks. Intheproposed drivetrainCMapproachofthispaper, itisassumedthatsystem-leveldrivetrainfaults(e.g.shaftcracksandunbalance)canrevealthemselvesbyvariationsinthe systemstiffnessandmoment ofinertia.Therefore,by monitoringthe consequencesof variationsof drivetrainparameters (i.e.stiffnessandmomentofinertiamatrices)inchangeofthedrivetraindynamicproperties(i.e.naturalfrequencies,mode shapesanddampingcoefficients),itispossibleto monitortheprogressoffaults.Forthispurpose,ananalytical modelof thedrivetrainwhichrepresentstherelationshipbetweenthesystemparametersanddynamicpropertiesandasubsequent sensitivityanalysishelpstorealizewhatarethemostinfluentialsystemparameters/faultswhichcanvariatethedrivetrain dynamicproperties.Inthefeatureselectionalgorithm,thetorsionaldynamicalmodelofthedrivetrainandthelocalsensitivity analysisareengaged.Thealgorithmisdesignedtoanextentthatcouldofferrobust,fastandaccurateonlinemonitoring.

The mainfocusofthisworkareon geareddrivetrainsystemsusedforwindturbines.Based onthetheoreticalstudies in thisresearch work,a 3-DOFequivalenttorsionalmodelofthegeared drivetrainissufficient fordetectionofthe drive- train faults ata system-level, becausesystem-levelfaults representthemselvesmainly by changingthe torsionalstiffness andthe momentofinertia parameters ofthe3-DOF equivalentmodel.Inthe firststepof thework, theproposed modal estimation approach byusing the torsionalmeasurements ispresented, whichis proved inthe generalcasefora n-DOF torsionalmodelofthedrivetrain,followedbyadetailedparametricproofbasedon3-DOFmodel.Asthesecondstepofthis research, theanalyticalrelationshipbetweenthe3-DOFequivalentmodelparametersanddrivetraindynamicpropertiesis established, whichhelpstoidentifythedrivetrain systemcondition/state-of-operation bymonitoringthe variationsinthe drivetraindynamicproperties(undampednaturalfrequenciesandnormalmodes)whichcanbeestimatedfromtheopera- tionalmeasurementsbyusingtheproposedmodalestimationapproachortheotherapproachesproposedbytheliterature.

The other reasonfor sticking to 3-DOF model,is that the closed-form parametric expressions of the drivetrain dynamic propertiesasa functionofequivalentmodelparameters canbe obtainedforthissimplified model.Thoseexpressions are therequiredinputsfortheproposedfaultdetectionapproachbasedonmonitoringthevariationsofthedrivetraindynamic properties.Thoseexpressionsprovidequantifiablefaultdetectionfeatures,whichareimplementableinmicrocontrollersand canbe integratedwithturbinefullyautomatedcontrolandmonitoringsystems.Bytheincreaseoftheorderofequivalent model,moredynamicproperties(highernaturalfrequencieswhicharenotseenby3-DOFmodel)canbe employed,which cansupportamoredetailedfaultdetectioninthedrivetrain.However,itisalittlechallengingforcurrentlyavailablemodal estimationapproachestoobservehighermodeswhichappearwithalowamplitudeinthefrequency-domainresponse.In other words,therealconditionsforthemodalestimationproblemisrestrictive,sothatthehighereigenfrequenciesofthe drivetrainsystem,whichmaybeexcitedbytheinputtorquewithalowerenergy,maynotbeobservable.

Theproposedmethodinthispapercandetectstiffnessorinertiarelatedfaultsbymonitoringtheconsequencesoffaults on onlineestimatedmodesandamplitudeofresponse.Themethodiscomputationally inexpensive sinceitreliesononly fewdatasamplesandamoderatedataresolutionandsamplingfrequency.Onthisbasis,themaincontributionsandnovelty ofthisworkare:

1. Analyticalproofofadrivetrainmodalanalysisapproachbyusingtorsionalmeasurements,

2. Introducingan analyticalapproachforestimationofdampingcoefficientsofthesystemmodesbyanalyzingtheampli- tudeoftorsionalresponseerrorfunctionatthenaturalfrequencies,

3. Theoreticaldevelopmentandsimulation/experimentalvalidationofadrivetrainsystem-levelhealthmonitoringapproach basedonestimatedmodalparameters,andcomparisonwithothermethodsintheliterature.

The paper is organized as follows: Modal estimation by using torsional measurements is analytically elaborated in Section2.Ananalyticalapproachfordrivetrainconditionmonitoringbyusingtorsionalresponseandtheestimatedmodesis proposedanddiscussedindetailsinSection3.Theproposeddrivetrainmodalestimationandconditionmonitoringapproach arevalidatedandcomparedwiththeapproachesintheliteraturethroughsimulation/experimentalstudiesinSection4.The paperisconcludedinSection6.

2. Operationalmodalanalysisbyusingtorsionalmeasurements 2.1. Torsionalnaturalfrequencyestimationtheory

Drivetrainisoftenmodelledasone-degree-of-freedom(1-DOF)rotationalsysteminglobaldynamicresponsetools.The forcedtorsionalvibrationresponseoftheequivalent1-DOFdampedrotationalmodelofdrivetraininfluencedbytherandom excitation

τ

(^t)^{infrequencydomainand}non-dimensionalformcanbeexpressedby

| θ () |

=

|τ ()| kt

(

¹−

(

n

)

²

)

²+

(

²

ζ

t^ω

(

n

))

^{2, (1)}

where

θ

() ^and

τ

() ^{are the Fourier transforms of angularposition and the excitation torque,} respectively. ^{n is the} undampedtorsionalnaturalfrequencyofthesystem,ktisthetorsionalstiffnessoftheshaft,and

ζ

^{t isthetorsionaldamping}

coefficient ofthemoden attheoperatingspeed

ω

^{.Asitcan beseen,an amplifiedfrequencyinthedrivetraintorsional}

responsecanbeduetoasignificantexcitationamplitudeorthevicinityofexcitationfrequencywithnaturalfrequencies.

Naturalfrequenciesappearintorsionalresponsese.g.angularvelocitymeasurementsduetoimpulsivebehaviorofwind whichexcitesthosefrequencies.AninitialvelocityappliedonasystemasdescribedbyThomsonetal.[28]canplayarole asanimpactwhichisabletoexcitethesystemtorsionalfrequencies.Inthewindturbine,theceaselessvariationsofwind resultsincontinualvariationsinangularvelocitywhichisphysicallysimilartoaninitialvelocityappliedtothesystem.Even though thesevariationsinspeedandsubsequentlytorquehappen inverylow frequency,they canintroduce considerable energyindifferentfrequenciesincludingthecharacteristicfrequenciesofthesystem.Duetotheexistenceofdampingina realsystem,themeasurednaturalfrequenciesfromthetorsionalresponsearethedampedfrequencies.Byfilteringtheshafts revolutionfrequencies,componentsdefectfrequenciesandexcitations(verylowfrequencyduetowind,lowfrequencydue towave towershadoweffect,andhighfrequencyduetogenerator),thedrivetraintorsionalnaturalfrequencies,andsome torsionalinducedmotionsduetoexcitededgewiserotorbladeandtowerbendingmodesareacquired.Basedonaprimary knowledgeonthetorsionalfrequenciesforeachpowerrange,itispossibletoseparatetheobservednaturalfrequenciesfor drivetrain,bladesandtower.Thevariationsinthenaturalfrequenciesandnormalmodescanbeusedascriteriatoidentify somesortsoffaultsinthedrivetrain.Toestimatethedampednaturalfrequencies,angularvelocityresidual/errorfunctionis proposed.Theinputofthismethodisprovidedbytwoencoderslocatedatthehigh-andlow-speedshaftsofdrivetrain,and subsequentlytheresidualfunctionisconstructedbasedonthesubtractionofthesetwosignals.Somedrivetrainsystemsare onlyequippedwithoneangularvelocitymeasurementontheshaft,sothattheimplementationofthemethodmayrequire an additionalmoderateresolutionencodertoprovidethesufficientinputs.Theangularvelocityresidualfunctione^ω_tot from thehigh-speedsideisexpressedby

e^ω_tot=

ω

^HS−a1a2a3

ω

^{LS, (2)}

where

ω

^{HS and}

ω

^{LS are the rotationalspeed in rad}s obtained fromthehigh- and low-speed encoders, respectively.a1,a2

anda₃ aretheinverseofgearratiosofthegearboxstages.Theerrorfunction mainfeatureiscancellationoftheimpacts oftheexcitationswhicharetransferred tothedrivetrainfromthehousing,fromtheresultanttorsionalresponse.Angular displacementandaccelerationaretheothertorsionalresponsesofthedrivetrainsystemwhichcouldtheoreticallybeused similar toangularvelocitytoobtain thesystemtorsionalparameters.Forthispurpose,similartoe^ω_tot,theangularposition errorfunctione^θ_tot andtheangularaccelerationerrorfunctione_tot^α aredefinedby

e^θ_tot=

θ

HS−a₁a₂a₃

θ

LS, e_tot^α =

α

HS−a₁a₂a₃

α

LS. (3) In particular,angular accelerationisthe torsionalresponse whichhasa directrelation withtheapplied, andcontains usefulinformationonhowtheappliedtorqueinteractswiththesystem.Ifaccelerationordisplacementisusedforevalua- tion,theassessmentcriteriatendstovarywithfrequency,becausetherelationbetweenthemandvelocityisproportional tofrequency.TheFourierseriesofe_tot^ω,e^α_tot ande^θ_tot aredefinedbye^ω_tot()=_∞

n=−∞Cne^ikn, e^α_tot()=_∞

n=−∞Cn(^ikn)^eikn, e^θ_tot()=_∞

n=−∞Cn(^ikn)^−1eikn.Differentiationandintegrationarelinearoperationsthat aredistributiveoveraddition.As itcanbeseen,ine^α_tot comparedtoe_tot^ω,theamplitudeofthefrequencycomponentshigherthan1Hzismagnifiedwiththe gain kn,andthefrequencieslower than1Hzare weakenedwiththesameproportion.Ine^θ_tot comparedtoe_tot^ω,theampli- tudeofthefrequencycomponentslowerthan1Hzismagnifiedwiththegaink⁻_n¹,andthefrequencieshigherthan1Hzare weakenedwiththesameproportion.The1stnaturalfrequencyofthedrivetrainsystemsofthesametechnologydecreases as the rated power increases. However, even for 10 MWwind turbine which is the biggest commercially available and evenforthehigh-speedtechnologieswhichhavelowerfirstnaturalfrequencies,the1st torsionalfrequencyishigherthan 1 Hz[32].Therefore,theangularaccelerationerrorfunctionis theoreticallyslightlybetterthanthe othertwo approaches

inhighlightingthetorsionalfrequencies.Theother benefitisweakeningthefrequencies lowerthan1Hzwhichappearin thedrivetraintorsionalresponsemostlyduetowindturbulenceandstructuralmotions,butdonotcontainanyinformation onthedrivetrainnaturalfrequencies.However,anadditionalderivationoperationisrequiredtoattainaccelerationfromthe velocitymeasurementswhichincreasesthecomputationalcostofthismethod.

Alimitationwithaforedescribedapproachisthedependencyontwoencoders,becauseinseveralturbinesthereisonly oneencoderavailablelocatedinthelow-speedshaft.Asdiscussedearlier,oneofthesignificantfeaturesoftheerrorfunction iscancellationoftheinfluencesofstructuralmotionsfromthetorsionalresponse.Thelatterresultsinacleansignalwhich isabletohighlightthesystemcharacteristicfrequencies.Thosemotionsaremainlyinfluencedbywind,waveandstructural resonances,andnaturalfrequenciesofstructuralmotionsandlowfrequencyinteractionsbetweenrotor, towerandsupport structureandhavealowfrequencycontent.Therefore,thefilteredangularvelocityoflow-speedshafthassome potentials inhighlightingthedrivetraintorsionalfrequencies.ThefilteredsignalX(HP)^{isextractedby}

X

()

=a1a2a3

ω

^LS

()

, X

(

HP

)

=X

()

^H

(

HP

)

, (4)

whereH(HP)^{isthetransferfunctionofthehigh-passfilterappliedtothelow-speedshaftencodersignaltoattenuatethe} low frequencynoisesresulted wind inducedlow frequencymotions. The performanceof filteredangularvelocity oflow- speed shaftinhighlightingthetorsionalfrequenciescompared tothedifferenttorsionalresponseerrorfunctionsistested withbothsimulationandoperationalmeasurementsasreportedinSection4.

Inorder tocapturebetterthedrivetraindynamicsatsystemlevelforthesubsequentusefordrivetrainfault diagnosis at systemlevel while minimizing the computational complexityof the model,3-DOF torsional modelis offered andthe performance of the model is evaluated throughout the paper. Forthis purpose, to evaluate the observability of natural frequencies onthe torsionalresponse errorfunctionsandthesubsequent applicationfordrivetrain CM,a3-DOF torsional modelofthedampeddrivetrainisengaged.The1st and2ndundamped naturalfrequencies(nonrigid modes)basedon3- DOFlumped-mass-springmodelofageareddrivetraincanbecalculatedbytheequationsreportedinA.1.Theeigenvectors ofthedampeddrivetrainmodeltakecomplexvalues.Byassumingdampingequaltozero,thenormalmodestakerealvalues whichshowtherelativeangularmotionofthedifferentinertiasinthemodel.Theclosedformoftwonormalmodesrelated to thetwonon-rigid modesofthe underconsideration drivetrainmodelwhichare scaledto unitylengthare reportedin A.1.

Bothundamped frequenciesandnormalmodesareuniqueforthesystemso thatanydeviationoftheparameterscan indicatevariationsinthedrivetrainsystemwhichcanbeusedforfaultdetection.Thecontinueddiscussionisdedicatedon ananalyticalproofoftheideaofobservingnaturalfrequenciesfromthetorsionalresponse.Thetheoryisfirstpresentedfor thegeneralformofresponseobtainedfromthegeneraln-DOFtorsionaldrivetrainmodel.Thenthepossibilityofobserving torsional modesinamplitudeofangularvelocity errorfunction basedon a3-DOF modelismathematically proven tobe usedintheproposedmodel-basedfaultdetectionapproach.

Theorem1. Torsionalnaturalfrequenciesbelongtothesetofextremepointsofthetorsionalresponseinthefrequencydomain.

Proof. The general form ofthe discrete multi-DOF torsional model of drivetrain withn degreesof freedom in the time domainisdefinedby

J

θ

¨+C

θ

˙+K

θ

=T

(

^t

)

, (5)

whereJ,CandK arethemomentofinertia,dampingandstiffnessmatriceswiththesizen×n.

θ

^{andT aretheresponse}

andloadvectorswiththesizen×1,whereeachelementofthesetwovectorsrepresentatimeseriesdata.Therepresenta- tioninfrequencydomainbyusingthefrequencyvariableSandLaplacetransformwillbe

[JS²+CS+K]n×n[

(

^S

)

^]n×1=T

(

^S

)

n×1. (6)

ByreplacingthecharacteristicequationJS²+CS+KwithM,thefrequencydomainresponse(^S)^willbe

(

^S

)

=adj

(

^M

)

det

(

^M

)

^T

(

^S

)

, (7)

whereadj(^M)^{istheadjugateofMwhichisapolynomial functionwiththematrixvariableM.det}(^M)^isthedeterminant ofthesystemcharacteristicequation.Asitcan beseen,therootsofthedet(^M)^{are theextremepointsofresponse}(^S)^. However,therootsofthedeterminantofcharacteristicequationofasystemarethesystem’seigenfrequencies.Therefore,the torsionalnaturalfrequencies ofthesystembelong totheset ofextremepointsoftheresponse. Intheundamped system (C=0), the rootswill be pure imaginarywhich representthe undamped naturalfrequencies ⁱn.In thegeneral damped system,therootsarethedampednaturalfrequenciesⁱdwiththefollowingrelationwiththeundampedfrequencies

ⁱd=

ζ

ⁱ

ⁱn+j

ⁱn

1−

( ζ

ⁱ

)

^{2 i}∈1,..,n. (8)

where

ζ

i is thedamping coefficient relatedto themode i.The torsional naturalfrequencies inboth cases ofdamped or undampedsystembasedontheprovidedproofwhichreferstothegeneralformofdampedsystemaretheextremepoints oftheresponsefrequencydomainfunction.

Thus,wecompletetheproofofTheorem1.

The other extreme pointsof (^S) ^{are dueto the loaddynamics, the system unmodelled internal dynamics andthe} interactions betweenthesetwo. Asdiscussed earlier,inordertopick outthe naturalfrequencies,other harmonicswhich also demonstrate themselves asother extremepoints inresponse must be filtered.For thispurpose, the response error function isengaged whichis ableto filtertheinfluences ofthe loadstransferred tothe drivetrainthrough the structure, whichisveryusefulspeciallyinoffshorewindturbinesequippedwithfloatingsupportstructureswhichcaninduceawider rangeofharmonicsinthedrivetrainresponse.

The typical signal for frequency domain fault detectionstudy is the single-sided amplitude spectrum ofresponse. In A.2,thepossibilityofextendingtheresultsofTheorem1totheamplitudeoftorsionalresponseandmorespecificallythe amplitude of angular velocity error function based on an equivalent 3-DOF model is investigated. For this purpose,the general 3-DOF damped torsionalmodel ofthe geared drivetrain systemis selected, andthe detailed analytical proof for observingthenaturalfrequenciesintheamplitudespectrumofangularvelocityerrorfunctionispresented.

Byreplacingn fromeq.(A.1)insteadof

|

^S

|

^{ineq.(A.5),theamplitudeofresponseatthetwonatural}frequenciesinthe generalcaseofadampedsystemhastherelationshipwiththesystemparametersandloadsas

|

^eωtot

(

^tor1

) |

=

|

^Tg

( ω

¹

) |

²

FA+H+J_r²J_gr²A²√

A+

|

^Tr

( ω

¹

) |

²

EA+G+J²_grJ_gn²A²√ A

4

A²I²+

(

^Jr

√E+cLJgrJgn

)

^2A3+D²

(

^Jr+Jgr+Jgn

)

^2A+k²_Lk²_H

(

^Jr+Jgr+Jgn

)

²+J_r²J_gr²J²_gnA⁴

, (9a)

|

^eωtot

(

^tor2

) |

⁼

|

^Tg

( ω

²

) |

²

FB+H+J_r²J²_grB²√

B+

|

^Tr

( ω

²

) |

²

EB+G+J²_grJ_gn²B²√ B

4

B²I²+B³

(

^Jr√E+c_LJgrJgn

)

²+k²_Lk²_H

(

^Jr+Jgr+Jgn

)

²+D²

(

^Jr+Jgr+Jgn

)

^2B+J_r²J_gr²J²_gnB⁴

, (9b)

whereT_randT_garetherotorandgeneratortorques,respectively.

The frequency domainangular velocity errorfunction ofa theoretically undamped systemunder excitationat natural frequencies is unbounded.Therefore, performing a sensitivity analysis to findthe relation betweenthe variations ofthe amplitudeofresponseatnaturalfrequenciesandthevariationsofsystemparameterswhichcanrepresentthesystemfaults isnotpossible.However,aphysicalsysteminpracticehasdamping.Theresponseofadampedsystematnaturalfrequencies isboundedduetotheinfluenceofdampinginthesystem.Therefore,monitoringthevariationsoftheamplitudeofresponse in the natural frequencies can be relatedto variations of the system parameters andfaults. In the continued part, the possibilityofusingthe amplitudeofresponse ofa dampedsystematnaturalfrequencies formonitoringthevariationsin thesystemisdiscussed.Theresultsofthisstudycanalsobeusedforestimationofdampinginthesystem.

As it canbe seen fromeq. (9),indifference withthe equationsforthe systemnaturalfrequencies andmode shapes, theamplitudeofresponseatthefirstandsecondnaturalfrequenciesisproportionaltonotonlythesystemparametersbut alsotheloads.Thelatterlimitstheapplicationofamplitudeofresponseasafaultprecursor.However,itcanbeusedasa criteriontoevaluatetheresultsobtainedbytheproposedfaultdetectionalgorithm,sothatitprovidesinputsfordrivetrain CM based onmonitoring the variationsofamplitudeofresponse atthe naturalfrequencies interms ofvariationsin the systemparametersbysensitivityanalysiswhichiselaboratedinSection 3.Theresultsofanalysisofamplitudeofresponse alsoprovidesnecessaryinputsfortheestimationofdampinginthesystem.

Byusingthesimplifiedmodelineq.(1),thepeakfrequencyoftheamplitudeofresponsehastherelationasdescribed by eq.(10)withtheassociatedundamped naturalfrequency.Thisresultcanbeextendedtothehigherordersystemsand higherordernaturalfrequencies.Ouranalyticalstudyontheextremepointsoftheamplitudeofresponseinhigherorder models shows that these points are highlynonlinear andcomplicated functions of system parameters which make the utilization ofthese equationscomputationally expensive. However, thesepointscan be relatedto the undamped natural frequenciesbyusingthedampingcoefficientsas[29]

ⁱpeak=

ⁱn

1−2

ζ

i². (10)

Thetwofollowingequationscanbeusedtoestimatethedampingcoefficientsofdifferentmodesatdifferentoperating speeds,byusingthepeakfrequenciesandtheamplitudeofresponseatthosefrequencies,asfollows:

ⁱ_peak^,t1

^i,t_peak^{2 =}

1−2

ζ

i²,t1

1−2

ζ

_i,t²₂^{, (11a)}

|

^etot^ω

(

ⁱ_peak^,t1

) |

|

^etot^ω

(

^i,t_peak²

) |

^=f

⁽

^{Tr,Tg,cL,cH,kL,kH,Jr,Jgr,Jgn}

⁾

^{, (11b)}

where

ζ

i,t isthedampingcoefficientrelatedtothemodeiandoperationt.

|

^eωtot(peak)

|

^{istheamplitudeofresponseatthe}

peak frequency. Boththe peak frequencyandamplitudeofresponse atpeakfrequency areestimatedfromthe frequency domainresponsebasedonthetheoryelaboratedearlierinthisSection.Theeq.(11b)islongandnonlinearwithdependency toallthesystemparametersandloadssothatrelatingthevariationsintheamplitudeofresponsetovariationsindamping

coefficientseemstobeachallengingtask.Thetheoryrelatedtotheemploymentofsensitivityanalysisforrelatingtheratio ofamplitudeofresponse tothe ratioofdampingcoefficientsfromeq.(11) andtheimplementationofthisapproach will be discussedinthecontinuedSection.From thepeakfrequencyandtheapproximateddampingcoefficient,theundamped frequencycanbeestimatedbyusingeq.(10).

3. Drivetrainconditionmonitoringbyusingtorsionalmeasurements

The estimationofsystemmodesby usingtheangularvelocityerrorfunctionwaselaboratedinSection2.Asdiscussed earlier,theestimatedmodesandtheamplitudeofresponse atthosefrequenciescan berelatedtothesystemparameters andfaults.Inordertoestablishthisrelationshiptobeusedintheproposedfaultdetectionapproach,sensitivityanalysisis employed.

3.1. Sensitivityanalysis

Thispartisaimedtoobtaintheclosedformmathematicalrelationshipsbetweenthedrivetraindynamicpropertiesand amplitudeofresponsewiththedrivetrainreduced-ordermodelparametersthrougha sensitivityanalysisforasubsequent useintheproposedfaultdiagnosisalgorithm.SimilartoinSection2,thegeneral3-DOFdampedtorsionalmodelofdrive- trainisselectedfortheanalyticalstudiesinthisSection.

Asdiscussedearlier,faultslikecrackinshaftsandrotor,couplingdamage,ordamageingearboxareexamplesofpoten- tialfaultswhichcanchangethedrivetrainstiffness.Forexample,ashaftcrackresultsinreductionofthetorsionalstiffness of theshaft [30].A changeinthestiffnessinfluences thedrivetrain systemfrequencymodes. Therefore,by obtaining the mathematical relationship betweenthe stiffnessofdifferentshafts andthesystemmodes, it is possibleto monitortheir conditionsby monitoringthevariationsinthesystemnaturalfrequencies andnormalmodes. Theother parameterwhich can influencethedrivetraintorsionalmodesisthemomentofinertiaofthedrivetraincomponents.Variationsinthemo- ment ofinertia matrixrepresentsthe othercategoryoffaults inthe drivelinewiththeunbalanceandlossofmassasthe foremost. Forexample,unbalancefaultsare characterizedby theincreaseofmomentofinertia duetoan additionalforce that isgeneratedduringthose conditionsbasedonthe parallelaxistheorem.The mathematicalrelationship betweenthe drivetraintorsionalnaturalfrequenciesandthemomentofinertiaofcomponentscanhelptoidentifythesefaults.Thevari- ationsin stiffnessandinertia can resultin similarnaturalfrequencyvariation patterns.Therefore,todistinguish between variationsinthe naturalfrequencies becauseofvariationsinthe shafts’stiffnesswiththosedueto variationsin moment of inertiamatrix (source offault), determiningthecorrelation betweenthe systemparametersandthe normalizedmode shapescanprovideausefuldirectiontofindthesourceoffault.Thecorrelationbetweentheamplitudeofresponseatthe systemnaturalfrequenciescanalsobeusefulintwoways:first,toestimatethedampingcoefficientsandsubsequentlythe undamped naturalfrequencies fromthe estimated naturalfrequencies;second, to confirm orrepeal the results obtained aboutthesystemfaultstakenbasedontheanalysisofnaturalfrequenciesandnormalmodes.

In ordertoachieve the abovedescribedpurposes, twodifferent setsofsensitivity analysesare performedinthisSec- tion.First,asensitivityanalysisontorsionalfrequencies andnormalmodesoftheequivalentundampedsystemtoextract drivetrain system-levelCM features.Second,a sensitivityanalysisonthe amplitudeofresponseatthenaturalfrequencies primarily toestimate thedampingcoefficient andsubsequentlytheundamped naturalfrequencies whichare requiredfor thefirstanalysis,andthentosupporttheCMfeaturesobtainedinthefirstsensitivityanalysis.

Inordertocheckhowthevariationsinstiffnessandmomentofinertiainfluencethesystemtorsionalnaturalfrequencies and modeshapes, a sensitivityanalysis isperformed. There are twoclasses ofsensitivityanalysis methods,namely local andglobalsensitivityanalysis.Morioetal.[31]hasreportedthesamekindofresultsbyusingthesetwomethodforsimple models. Local sensitivitydetermines howa smallperturbation nearan input parametervalue influences thevalue ofthe output.InthisSection,inordertofindtheparameterswiththegreatestimpactonthedrivetraindynamiccharacteristics, localsensitivityanalysisisemployedduetotwomainreasons.First,themotivationofthisworkisdetectingfaultsinearly stages for predictive maintenance purposes so that variationsin the drivetrain system parameters happen witha slight change around theset point values.Second, local sensitivityanalysisderivesa closed formexpression forthe sensitivity valuewhichmakestheresultmorereliableandeasiertoimplement.Localsensitivityisdefinedasthepartialderivativeof theoutputfunctionwithrespecttotheinputparameters[33]as

S^Loc_k,l =

δ

^yk

δ

^xl

, y_k∈

{

^y1,...,y_p

}

^andxl∈

{

^x1,...,xq

}

^{, (12)}

wherey_kisthek^thoutputandx_l isthel^thinput.Toneutralizetheimpactoflarge/smallinputsandsmall/largeoutputs,the localsensitivitycanbenormalizedbythenominalvaluesofinputsandoutputsby

S^Norm_k,l = x^{re f}_l y^{re f}_k

δ

^yk

δ

^xl

, (13)

withx^{re f}_l andy^{re f}_k asthenominalvaluesofx_l andy_k.Forthe3-DOFtorsionalmodeldescribedinSection2.1,theinputand outputvectorsforsensitivityanalysisare

x=

{

^{kL, kH, Jr, Jgr, Jgn, cL, cH,Tr,Tg}

}

^{, (14a)}

y=

{

^tor1 ,

^tor2 ,

rot¹,

rot²,

gear¹,

gear²,

gen¹,

gen²,

|

^eωtot

(

^tor1

) |

^,

|

^etot^ω

(

^tor2

) |}

^{. (14b)}

where

|

^etot^ω(^tor₁ )

|

^and

|

^eωtot(^tor₂ )

|

^{are theamplitudeofresponse atthe first andsecond natural}frequencies, respectively.

Theclosedformofequationsafterapplyingnormalizedlocalsensitivitytheoryoneqs.(A.1)and(A.2)areshowninB.1and B.2. Apositive value inthisanalysisstandsfora direct relationshipbetweenthe inputparameter andoutput, whereasa negativevaluerepresentsthat theparameterandoutputareinverselycorrelated.Thenormalizedlocalsensitivityanalysis cantakedifferentvalues.Iftheabsolutevalueisequalto1,itmeansthattherelativevariationininputparameterisequally transmitted totheoutput,whereas theabsolutevalue higherthan1showsthatthe relativevariation ismagnifiedin the output.However,theabsolutevaluelowerthan1representsthattherelativevariationsoftheinputisshrunkintheoutput.

Inthe secondstudy,inorderto findtheparameters/variableswhichhavethehighestcontributioninvariationsofthe amplitude of response at the response peak frequencies, a local sensitivity analysis is performed. For thispurpose, two methodsareproposed.First,thepeakfrequenciesareapproximatedwiththeassociatednaturalfrequencyandsubsequently the eq. (9)is used.The closed formequationswhich specifythe correlation betweenthe angularvelocity errorfunction amplitude at the naturalfrequencies with the system parameters and loads are derived by performing local sensitivity analysisasshowninB.3.Thisapproximationcanbeimprovedbyusingtheapproximateddampingcoefficientsandupdating the eq.(9) byusing theeq. 10.Anotherapproach whichisbased onnumericalcalculationsand isalsoused laterin the simulationstudiesforcomparisonpurposesistonumericallyfindthepeakfrequenciesoftheresponseintheeq.(A.5)and findingthesensitivityoftheresponseequationstotheparametersafterreplacingthenumericallycalculatedfrequenciesin the responsefunction.The precision inestimationofthedamping coefficientby thetwo proposedmethods comparedto theapproximationproposedin[23]ispresentedinsimulationstudies.Inordertoattaintheaccuracyofthesemethods,the resultsarecomparedtotheexactvaluesofthecoefficientsbasedonthemodelparameters.

Thefollowingproceduresummarizesthemodalestimationapproach:

1. The torsionalresponseerrorfunction (orinterchangeablythe low-passfilteredsignal ofa singletorsionalresponse)is generated.Theresponsecanbeangularvelocity/acceleration.

2. The resultant signal ispreprocessed sothat thedefect frequencies ofthecomponents andstructuralmotions-induced harmonicsarefiltered.Theresultwillgivethedampedtorsionalnaturalfrequenciesofthedrivetrainsystem.

3. The measured natural frequencyis validated by the analysisof variationsof amplitudeof response inthe suspicious frequencies at different operating speeds. In simple words, the variation of the amplitudeof response in the system naturalfrequency(dampednaturalfrequencies)duetothevariationofdampingcoefficientismoresignificantcompared tothevariationoftheamplitudeofresponseintheharmonics.Thevariationofdampingcoefficientisduetothefrequent variationsintheoperatingspeedinwindturbinedrivetrains.

4. Dampingatthenaturalfrequencydependsonthe operatingspeed.Thedampingcoefficientattwoensuing operations intwodifferentspeedscanbeestimatedbyapplyingthetheorydevelopedinthisSectionandmodeledbyeq.11,based onmonitoringthevariationsofthenaturalfrequencyandamplitudeofresponsebetweentwosequentialoperations.

5. Byusingtheestimateddampednaturalfrequenciesfromtorsionalresponseandtheestimateddampingcoefficientfrom the analysis of amplitudeof response, the undamped natural frequencies are obtained, which provide inputs forthe drivetrainsystemhealthmonitoringapproachbasedonmonitoringthevariationsofsystemdynamicproperties.

ThealgorithmwhichsummarizestheproposeddrivetrainmodalestimationandtheensuingCMapproachisillustrated bytheflowchartinFig.1.m¹andm² arethenormalmodesrelatedtothe1stand2ndnaturalfrequencies,respectively.

m variesfrom1 up tothe degree ofthemodel, whichaccountsforthe differentbodies intheequivalent reducedorder model.

τ

^{tor and}

τ

m are thelow-limit thresholdnatural frequencyand normal mode related tonormal operations.Close modesaremoredifficulttoidentifythanwell-separatedmodesandtheir identificationoftenhasan uncertainty.Forwind turbinedrivetrain,lower naturalfrequenciesareingeneralwell separatedandclosemodesmighthappen onlyforhigher modes[34].

4. Simulationstudies 4.1. Simulationtestcase

Forthesimulationstudies,DTU10MWreferencewindturbineisselected.Inordertoevaluateiftheinputtorqueisable toexcite thedrivetrainnaturalfrequencies andsubsequentlytostudythepossibilityofobservingthosefrequencies inthe differentdrivetraintorsionalresponses,aneffectiveapproachisinvolvingdecoupledsimulationtechnique.Forthispurpose, therotortorqueT_rofadetailedmodelof10MWturbinewithasparfloatingplatformobtainedfromSIMAglobalsimulation software[35] isused,andtheimpacts onthedrivetrainisstudied byusingadecoupled analysis.Thegenerator torqueTr

is also decided toset the speed onthe shaft undervariable input torque,but theinternal dynamics ofthe generator in trackingthedefinedsetpointisneglected.

The decoupledsimulationapproach consistsoftwo separatedphases.Inthefirst phase,globalsimulation analysesare performedunderdifferentenvironmentalconditions.Intheglobalsimulation,theblades andhubassembly, thestructural moduleincludingtheflexiblemulti-bodysystemsfortowerandplatformandthenacellearemodelled.Thismodeliscoping

Fig. 1. Proposed algorithm for driveline condition monitoring by using torsional measurements and estimated modes.

withcombined aerodynamicandhydrodynamicloadingby using numericalandprobabilistic models ofwind,waves and currentintheglobalsimulationsoftwaretocapturetheintegratedeffectoftheloadsandthewindturbinecontrolsystem on the turbinemodel. The resultsof the globalanalysisin this studyare the loadstransmitted to the drivetrain by the rotorspecifiedbythetimeseriesoftheresultantmomentontherotor.Thesecondphaseofdecoupledanalysisisthatthe offline globalsimulationresultswillthen beapplied asinputson thedrivetrainmodelinSimpack multi-bodysimulation

Table 1

10 MW medium-speed drivetrain 3-DOF model specification.

Parameter Value

Equivalent rotor moment of inertia J r(kg.m ²) 800,000,000 Equivalent gearbox moment of inertia J gr(kg.m ²) 1,239,300 Equivalent generator moment of inertia J gn(kg.m ²) 15,716,775 Equivalent low-speed shaft torsional stiffness k L(N.m/rad) 2 , 452 , 936 , 425 Equivalent high-speed shaft torsional stiffness k H(N.m/rad) 245 , 293 , 642 , 500

Table 2

Sensitivity of natural frequencies and normal modes to variations of model parameter (stiffness and inertia).

Sensitivity / Variable k L k H J r J gr J gn

^tor1 0.50 0.00 −0 . 01 –0.03 –0.45

^tor2 0.00 0.50 0.00 -0.45 –0.04

φ1^rot 0.00 0.00 –1.00 0.07 0.93

φ1^gear 0.00 0.00 0.00 0.00 0.00

φ1^gen 0.00 0.00 0.00 0.00 0.00

φ2^rot 0.99 –0.99 –1.00 0.89 0.08

φ2^gear 0.00 0.00 0.00 −0 . 01 0.01

φ2^gen −0 . 01 0.01 0.00 0.96 –1.00

(MBS)software[36] tocalculateandanalysethedrivetraincomponentslocaldynamicresponsesformodalestimationand faultdetectionpurposes.Thedrivetrainmodelinthesecondphaseofdecoupledsimulationutilizesthecomponentsreduced ordermodels.Asacomplementarystep,thepostprocessingoflocalresponsesprovidesusefulinformationforthedrivetrain secondarystudies.Withoutlossofgenerality,ageareddrivetraintechnologyisselectedforthesimulationstudies.However, the 3-DOFreferencemodel canalso beused fordirect-drivetechnology faultdetectionbased onthe proposed approach, where regarding the considerable mass of main shaft, it should be modelled as a separate mass to improve the model accuracy,andthenasimilarapproachcanbeengaged.

TheoperatingconditionfortheglobalsimulationisclosetotheratedoperationwithanaveragewindspeedUw=11m/s, significantwaveheightH_s=3.5mandpeakperiodT_p=7.5s.Intheunderconsideration3-DOFtorsionalmodelofthegeared drivetrain,rotor,gearboxandgeneratoraremodelledwithequivalentmomentofinertia,andthelow-andhigh-speedshafts areeachmodelledwithconstanttorsionalstiffnesses.Thegeneratorandgearboxspecificationsareusedfromtheoptimized 10 MWmedium-speed drivetrain system proposed in [32]. The parameters of this model are listed inthe Table 1.The undamped natural frequencies of this model calculated by eq. (A.1), and validatedby Simpack are 1.9 Hz and73.9 Hz.

Theactualdampingofthelow-andhigh-speedshaftsarealsoassumedtobe5%and10%ofthelow-speedshaftstiffness, respectively.ThetorsionalresponsesofrotorandgeneratorshaftsareobtainedfromtheMBSmodeltoinvestigatepossibility ofobservingthenaturalfrequenciesfromtheangularvelocity,accelerationanddisplacementerrorfunctions.Theproposed methods for estimation ofdamping coefficients indifferent operating speedsare tested on thedamped model of under consideration 10 MWgeared drivetrain. The possibility ofdetecting different stagesof system-level inertia and stiffness related faults fromthe torsionalresponse obtainedfrom the10 MWMBS modelare investigated by usingthe proposed algorithm.

Inordertocapturethesystemdynamicpropertiesintheproposedapproachandtogetstatisticallycomparableresults, thetime intervalofeach blockofdatashouldbe largeenoughtocapturethelowestnaturalfrequency.Thesamplingfre- quencyshould behighenoughto capturethehigherfrequencymodeswhichare ofsignificance,andontheother side is limitedtotheSCADAsamplingfrequencyincaseofimplementationinthefarmlevel.Sincetherealizationofthemethodis basedonthe1stand2ndnonrigidmodes,forobservingthesetwomodes,therequiredlengthofdatablockisonlyafraction ofonesecondandtherequiredsamplingfrequencyisaround400Hzfor10MWmedium-speeddrivetraintechnology.

4.2. Sensitivityanalysisresults

Theresultsofthenormalizedlocalsensitivityanalyseswithnaturalfrequencies(^tor1 ,^tor2 )^{andnormalmodes}(

φ

1,

φ

2) astheoutputs andshaftstiffnesses(^kL,k_H)^{astheinputswithvariationofonlyonemodelparameteratatimeareshown} inthe Table2.Theinterpretationofthe localsensitivityanalysisvaluesis disclosedinSection3.1.The reportednumbers show thenormalizedsensitivityvalueswhichare calculatedbasedon10 MWdrivetrainmodelparameters.The valuesof thetableinbolddesignatetheabsolutesensitivityvalueshigherthan0.01,whichisusedasthecriterionthattheassociated parameterandoutputare correlated.Thevalueswhicharenothighlighted designatetheabsolutesensitivityvalueslower than 0.01representinga negligiblesensitivity,sothat theassociatedparameter andoutputareuncorrelated. Asit canbe seen,there isadirectrelationshipbetweenthe1st frequencyandk_L,andthe2ndfrequencyandk_H.Therefore,variations in thenaturalfrequencies can be translatedintothe variationsinthe shaftstiffness andsubsequentlythe defectsin the drivetrainshafts.Theinfluenceoftheshaftsdefect(stiffnessvariation)onthenormalmodeofthe1stnaturalfrequencyis