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1、Applicationsof Statisticsand Probabilityin Civil EngineeringKanda, Takada & Furuta (eds)? 2007 Taylor & Francis Group, London,ISBN 978-0-415-45211-3Fatiguereliability and effective turbulence models in wind farmsJ.D. S?rensenAalborg University, Aalborg, DenmarkRis? National Laboratory, Roskilde, Den
2、markS.Frandsen & N.J.Tarp-JohansenRis? National Laboratory, Roskilde, DenmarkABSTRACT:Offshore wind farms with 100 or more wind turbines areexpectedto be installed many placesduring the next years.Behind a wind turbine a wake is formed where the mean wind speeddecreasesslightlyand theturbulence inte
3、nsity increasessignificantly. This increasein turbulence intensity in wakes behind windturbines can imply asignificant reduction in the fatigue lifetime of wind turbines placed in wakes.In this paperthe design code model in the wind turbine code IEC 61400-1 (2005) is evaluated from a probabilistic p
4、oint ofview, including the importance of modeling the SN-curve by linear or bi-linear models. Further, the influenceon the fatigue reliability is investigated from modeling the fatigue responseby a stochastic part related to theambient turbulence andthe eigenfrequenciesof the structure anda determin
5、istic, sinusoidal part with frequencyof revolution of the rotor.1INTRODUCTIONWind turbines for electricity production haveincreasedsignificantly the last years both in production capa-bilityand in size. This development is expected tocontinue also in the coming years. Offshore windturbines with an i
6、nstalled capacity of 510 MW areplanned.Typically, theselargewind turbinesareplacedin offshore wind farms with 50100 wind turbines.Behind a wind turbine a wake is formed where themean wind speed decreasesslightly and the turbu-lence intensity increasessignificantly. The change isdependenton the dista
7、ncebetween the wind turbines.In this paperfatigue reliability of main componentsin wind turbines in clustersis considered.The increasein turbulence intensity in wakesbehind wind turbinescan imply a significant reduction in the fatigue life-time of wind turbine components.In thewind turbinecodeIEC 61
8、400-1 (IEC 2005) and in (Frandsen2005)are given a model to determine an effective turbu-lenceintensity in wind farms. This model is basedonfatigue strengthsmodeledby linear SN-curveswithoutendurancelimit.The reliability level for fatigue failure for wind tur-bines in a wind farm is evaluated using t
9、he effectiveturbulenceintensity modelin IEC 61400-1 (IEC 2005)for deterministic design, and the uncertainty modelsin (Frandsen 2005) are used in the reliabilityanaly-sis. The fatigue strength is assumed to be modeledusingtheSN-approachandMiners rule.The stochasticmodels recommended inthe JCSS PMC(JC
10、SS2006) are used. Linear and bi-linear SN-curves areconsidered.Further, the influence on the fatigue reliabilityinwind farms with wakesisconsideredin thecasewherethe fatigue load spectrum is modeled not only byastochasticpart related to theambient turbulenceandtheeigenfrequenciesof thestructure, but
11、alsoaddition-ally a deterministic, sinusoidal part with frequency ofrevolution of therotor.This deterministic part is impor-tantfor many fatigue sensitive details in wind turbinestructures.Devising the model for effective turbulence, theunderlying assumption is a simple power law SNcurve. However, w
12、hen applying the effective turbu-lencethatassumptionshall notbepropagated,andthusthe load spectrashould be derived from the responseseries by e.g. rainflow counting and the fatigue lifeshould be derived by integrating with the actual SNcurve. In this paper, the possible error/uncertaintyintroduced b
13、y the simple, one-slope power law SNcurve, is investigated.2WAKES IN WIND FARMSIn figure 1 from (Frandsen, 2005) the changein tur-bulence intensity I (standard deviation of turbulencedivided by mean 10-minutes wind speed) and mean10-minutes wind speedU is illustrated.1名师资料总结 - - -精品资料欢迎下载 - - - - -
14、- - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 1 页，共 8 页 - - - - - - - - - Figure 1.Illustrative ratios of wind velocity U athub (rotor)height and turbulence uinside (Uhand wf) and outside (Uand 0) a wind farm, as function of wind velocity. The fulllines are the model predictions from (Frandsen, 2
15、005).Figure 2.Layout of the Vindeby offshore wind farm from(Frandsen, 2005).In figures 34 from (Frandsen, 2005) data areshown, obtained from the Vindeby wind park (southof Denmark). It is seenthat the standarddeviation oftheturbulence andthe wind load(flapwise bladebend-ing moment) increasesignifica
16、ntly behind otherwindturbines. Forwind turbine W4 the directions 23?,77?,106?, 140?, 320?and 354?, and for wind turbine E5the directions 140?, 203?,257?, 286?, 298?and 320?.3EQUIVALENT FATIGUE LOADThe fatigue load spectrum for fatigue critical detailsin wind turbines is modeled not only by a stochas
17、-tic part related to the ambient turbulence and theeigenfrequencies of the structure, but also addition-ally a deterministic, sinusoidal part with frequency ofrevolution of the rotor, seefigure 5.Figure 3.Equivalent load (flap-wise bending) for two windturbines, 4W and5E in figure 2,asfunction of wi
18、nd direction,in the Vindeby wind farm, 8 U 9 m/s from (Frandsen,2005).Figure 4.Vindeby data: Standard deviation of wind velocitymeasured at hub height and flapwise blade bending moment(normalized with free flow conditions in 260300?) of windturbine 4W from (Frandsen, 2005). The bending momentis scal
19、ed to fitat free-flowconditionsat wind directions260300?. Scale of ordinate is arbitrary.Figure 5.Typical response time series for fatigue analysis(from Frandsen 2005).In case the response is Gaussian and narrow-banded/cyclic, e.g. if the responseis dominated by aneigenfrequency, the stressranges be
20、comes Rayleighdistributed and the number of fatigue load cyclestypically = 107/year.2名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 2 页，共 8 页 - - - - - - - - - A linear SN relation is assumed:whereSis the fatigue load stressrange,N is the num-berof stresscyclestofail
21、ure with constantstressrangeS andK , m arematerial parameters.Assuming a linear SN curve, it has been demon-stratedby (Rice 1944), (Crandall and Mark 1963) andadopted by (Frandsen 2005) that an equivalent loadmeasureof fatigue loading for a combined Gaussian,narrow-bandedstochastic anddeterministic
22、load maybewritten aswhereeis the double-amplitude of thesinusoidal withthe process up-crossing frequency, that causes thesamedamageasthe combined process.A is theampli-tude of the deterministic part, and xis the standarddeviation of thezero mean-valuestochastic part.(*)is the gamma function, and M (
23、*;*;*)is the confluenthypergeometric function. The function e has the fol-lowing asymptotic values for vanishing random andsinusoidal component, respectively:It is noted that for zero amplitude of the determinis-tic component andconditioned onm the characteristicamplitude eissimply proportional toth
24、e standarddevi-ation, x, of the responseof the stochasticcomponentalone.For wake and non-wake conditions, the model foreffective turbulence implies that a design turbulencestandard deviation may be calculated as, see alsosection 4.1.2where 0is turbulence standard deviation under freeflow conditions,
25、 wis themaximum wake turbulence,pw(= 0.06)is the probability of wakecondition andNwis the number of wakesto which the consideredwindturbine is exposed to. We assumethat the standarddeviation of responseis proportionnal to the standarddeviation of turbulence (a constant between the twoquantities have
26、 no consequencefor the argumentstofollow).Is the model for effective turbulence sensitive towhether there is a super imposed deterministic loadFigure6.Equivalentloadas functionofdeterministicamplitude. Circles are with SN-curve exponent m = 4 andsquaresm= 10. The full lines are with the application
27、of themodel for effective ponent?This is testedby calculating directly theequivalent loads for the two levels of turbulence andsubsequentlyweighting thesesimilarly to (4):On theotherhand,according tothemodel for effectiveturbulence,thesamequantity maybeapproximatedbyee,m= e(e,m, A, m).ForthecasesNW=
28、 4,pW= 0.06,0= 1,w= 2andm= 4and10,thetwo quantities areplotted in Figure 6.The model for effectiveturbulence is slightly conserva-tive (with up to 34%), mostly sowhen the stochasticanddeterministic component have approximately thesameweight.4PROBABILISTICMODELS FOR FATIGUEFAILUREDesignandanalysisfor
29、 fatigue areassumedto beper-formedusing linear andbilinear SN-curves, eventuallywith lower cut-off limit ase.g. used in (Eurocode 3,2003).It is assumedthat Miner s rule with linear damageaccumulation can be used.Further, fatigue contribu-tions from eventsasstart andstop of the wind turbinearenot inc
30、luded.In the following first a linear SN-curve describedby equation (1) is considered, and next a bi-linearSN-curve.3名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 3 页，共 8 页 - - - - - - - - - 4.1Linear SN-curve4.1.1Freewind flowFor a wind turbine in free wind flowthe
31、 designequation in deterministic design is written:whereis the expectedvalue ofmgiven standarddeviationandmeanwind speedU , andtotal number of fatigue load cycles per year (deter-mined by e.g. Rainflow counting)TLdesign life timeFDFFatigue Design Factor(equal to (fm)mwherefandmarepartial safety fact
32、orsfor fatigue loadandfatigue strength)KCcharacteristic value of K (obtained from log KCequal to meanof log K minus two standarddevia-tions of log K )Uincut-in wind speed(typically 5 m/s)Uoutcut-out wind speed(typically 25m/s)f(s| (U) density functionfor stress rangesgiven standarddeviation of (U )
33、at meanwindspeedU .This distribution function canbeobtainedby e.g.Rainflow counting of response,andcane.g.beassumedto beWeibull distributed. It is assumedthat the standarddeviation (U ) can bewritten:with(U ) influence coefficient for stress rangesgivenmean wind speedUu(U ) standard deviation of tur
34、bulence given meanwind speedUzdesignparameter(e.g.proportional to crosssectionalarea)The characteristic value of the standard deviationof turbulence, ? u(U ) given averagewind speedU ismodeled by, see(IEC 2005):where Irefis the referenceturbulence intensity (equalto 0.14 for medium turbulence charac
35、teristics) and ? uis denotedthe ambient turbulence.The corresponding limit stateequation is writtenwhereis a stochastic variable modeling the model uncer-tainty related to the Miner rule for linear damageaccumulationt time in yearsXWmodel uncertainty related to wind load effects(exposure,assessmento
36、f lift anddragcoefficients,dynamic responsecalculations)XSCFmodel uncertainty relatedto local stressanalysisu(U ) standarddeviation of turbulence given averagewind speedU u(U) is modeled asLogNormaldistributed with characteristicvalue ? u(U )definedas the 90% quantile and standarddeviation equalto I
37、ref 1.4 m/s:The design parameter z is determined from thedesign equation (6) and next used in the limitstateequation (10) to estimatethe reliability index or prob-ability of failure with thereferencetime interval 0; t.4.1.2Wind turbines in clustersFora wind turbine in afarm the design equationbasedo
38、n IEC 61400-1 (IEC 2005) canbewritten:whereNWnumber of neighboring wind turbinespWprobabilityof wake from a neighboring windturbine (equal to 0.06)? ustandarddeviation of turbulence given by (9)? u,jstandarddeviation of turbulence from neighboringwind turbine no jwhere djis the distancenormalized by
39、 rotor diameterto neighboring wind turbine no j and c constantequalto 1 m/s.Alternatively,an effective/equivalentturbulencemodel can be used, where the same model as usedfor a single wind turbine is used,but with aneffective4名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - -
40、 - 第 4 页，共 8 页 - - - - - - - - - turbulencestandarddeviation. In (Frandsen2005) andimplementedin IEC 61400-1(IEC 2005) themodel foreffective turbulence is presented.It is conditioned onwind speedandSN-curve slopem,i.e. fatigue loadcal-culations should becarriedout separatelyfor a numberof wind speed
41、bins, and for each wind speedbin theeffective turbulence intensity should be established.Further, it is in the equivalent model assumedthatthe standarddeviation of turbulent wind speedfluc-tuations, ? u( ,U ), is a deterministic function of winddirection with a superimposed,wind-direction inde-pende
42、nt random component. The model is basedontheassumptionthat theSN curveis linear: N = KS- m.Disregarding other flow variables than standarddeviation of wind speedfluctuations, the equivalentload (stressranges)is assumedto be written aswherewhere (U ) is the influence coefficient for windspeedfluctuat
43、ions fu(u|U , ) is the density functionu, conditioned of mean wind speedand direction.u,eff(U , ) is the fixed standard deviation of windspeedthat causesthesamefatigue asthe varying quan-tity. Denominating the distribution of wind directionconditioned on wind speedfwd( |U ), the integratedequivalent
44、 load at wind speedU becomeswhereis the effective turbulence intensity for mean windspeed U . Througha set of assumptions on theshapeof the wake turbulence profile and by assum-ing the density function of wind direction uniform,fwd=1360 deg- 1, and the ambient turbulence inten-sity is independentof
45、wind direction, (16) reducesto,see(Frandsen2005)where u,jis maximum turbulence standarddeviationfor wake numberj andNwis thenumberof neighboringwind turbines takeninto accountand pw0.06.Figure 7.Bilinear SN-curve.The resulting design equation is written:Note that for linear SN-curves equations (11)
46、and(18) areidentical.The limit stateequation corresponding to eithertheof the abovedesignequations is written:whereXUmodel uncertainty related to wake generatedtur-bulencemodel.The design parameter z is determined from thedesign equation (10) or (17) and next used in thelimitstate equation (18) to e
47、stimate the reliabilityindex or probability of failure with the referencetimeinterval 0; t.4.2Bilinear SN-curveNext, it is assumedthat the SN-curve is bilinear, seefigure 7 (thickness effect not included) with slopechangeat ND= 5 106:whereK1,m1material parametersfor SDK2,m2material parametersfor SD5
48、名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 5 页，共 8 页 - - - - - - - - - Figure 8.Number of load cycles in 10 minutes period forflap moment and mudline bending moment. Mean wind speedequal to 14m/s.In casetheSN-curve is bilinear DL(m; ) in designequations and limit
49、 state equations in section 4.1 isexchangedwith(24) caneasily bemodified to include a lower thresh-oldth.5EXAMPLESWind turbines areconsidered with a design life timeTL= 20 years and fatigue life time TF= 60 years,corresponding to FDF = 60/ 20= 3. The mean windspeedis assumedto beWeibull distributed:
50、with A = 10.0m/s and k = 2.3. It is assumedthat thereferenceturbulence intensity is Iref= 0.14, andthat5wind turbines areclose to thewind turbine consideredwith di= 4.5.1Modeling of stressrangesThe stress ranges are assumed to be Weibull dis-tributed. In figure 8 is shown typical distributions ofFig