PARAMETRIC MODELS FOR ESTIMATING WIND TURBINEFATIGUE LOADS FOR DESIGN

LanceManuel’Paul S. Veers2

Steven R. Winterstein3

lDepartment of Civil Engineering, University of Texas at Austim Austin, TX 787122Sandia National Laboratories, Wind Energy Technology DepartrnenL Albuquerque, NM 87185-07083Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305-4020

ABSTRACT

International standards for wind turbine certificationdepend on finding long-term fatigue load distributionsthat are conservative with respect to the state ofknowledge for a given system. Statistical models ofloads for fatigue application are described anddemonstrated using flap and edge blade-bending datafrom a commercial turbine in complex terrain.Distributions of rainflow-cotmted range data for eachten-minute segment are characterized by parametersrelated to their first three statistical moments (mean,coefficient of variation, and skewness). QuadraticWeibull distribution fi,mctions based on these threemoments are shown to match the measured loaddistributions if the non-damaging low-amplitude rangesare first eliminated. The moments are mapped to thewind conditions with a two-dimensional regression overten-minute average wind speed and turbulenceintensity. With this mapping, the short-termdistribution of ranges is known for any combination ofaverage wind speed and turbulence intensity. The long-term distribution of ranges is determined by integratingover the annual distribution of input conditions. FirsGwe study long-term loads derived by integration overwind speed distribution alone, using standard-specifiedturbulence levels. Next we perform this integrationover both wind speed and turbulence distribution for theexample site. Results are compared between standard-driven and site-driven load estimates. Finally, usingstatistics based on the regression of the statisticalmoments over the input conditions, the uncertainty (dueto the limited data set) in the long-term load dktributionis represented by 95°A confidence bounds on predictedloads.

This paper is decked a work of the U.S. Government and is notsubject to eopyrigtrt protection in the United States. Sandia is amultiprograrnlaboratoryoperated by Sandia CqoradorL a Lo&heedMartin contpany, for the U.S. Departmentof Energy under contractDE-AC04-94ALS5000.

INTRODUCTION

Design constraints for wind turbine structures fall intoeither extreme load or fatigue categories. In the case ofextreme load design drivers, the load estimationproblem is limited to finding a single maximum loadlevel against which to assess the structural strength.For design against fatigue, however, loads must bedefined over all input conditions and then summed overthe distribution of input conditions weighted by therelative frequency of occurrence. While this mightseem to be a more daunting task, it is in many waysquite similar to the extreme load problem, as can beseen by comparing with Fitzwater and Wmtersteinl. Inboth cases, the loads must be determined as functions ofwind speed (or other cliitic conditions).

Parametric models define the response, statistically,with respect to input conditions. Such models titanalytical distribution fimctions to the measured orsimulated data. The parameters of these distributionfimctions can be useful in defining the responsefloadsas a function of the input conditions. The end resultjthen, is a full statistical definition of the loads over allinput conditions.

In the most prevalent alternative to parametricmodeling, an empirical distribution of loads (i.e., ahistogram describing frequency of occurrence of themodeled response quantity) is used to define the turbineresponse at the conditions of the measurement orsimulation. When using simulations, a ten-minute timeseries is generated at specified environmentalconditions using an aeroehstic analysis code. The timeseries is rainflow-counted and the number of ranges inspecified intervals is summarized in histograms. Thehistograms serve as empirical distributions that aretaken to be representative of the response of the timeat those particular conditions. The fill lifetimedistribution is then obtained by summing thedistributions after weighting by the frequency ofoccurrence of the wind speed associated with eachsimulated data segment included in a histograminterval. In the case of measured &@ a. similar

1

DISCLAIMER

This report was prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessari~y constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.

DISCLAIMER

Portions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.

approach has been described by McCoy et al.z but withan innovative weighting scheme to account for theirregular input conditions of field measurements.

The empirical approach uses only the measured orsimulated data without any fitting of distributions orextrapolation to higher values that would be seen ifmore data were obtained. One of the disadvantages ofusing a purely empirical approach is, therefore, that theloading distribution may not be representative. Perhapsa subtler shortcoming is that the uncertainty in the loadsis ahnost impossible to characterize.

Whh regard to uncertainties in loads and how theymight be dealt with in design, one might expect thatthese uncertainties could be covered by the use ofstandard specifications of characteristic loads (derivedfrom a specified high turbulence level) and safetyfactors. However, current standard load definitions usesafety factors that do not depend on the relativeuncertainty in the load estimates. Either the marginsare larger than they need to be when load estimates arereasonably well established (i.e., exhibit lowuncertainty), or they fail to cover the uncertainty whenload estimates are based on limited data (i.e., largeuncertainty cases).

Parametric load distribution models offer significantadvantages over empirical models; they provide ameans to (1) extrapolate to higher, less fkequent loadlevels, (2) map the response to the input conditions, and(3) calculate load uncertainty. For example, Ronold etaL3 have published a complete analysis of theuncertainty in a wind turbine blade fatigue lifecalculation. They used a parametric definition of thefatigue loads, matching the first three moments of thewind turbine cyclic load~ng dkibution to a quadratic(transformed by a squaring operation) Weibulldistribution.

Veers and Wmterstein4 described a parametric

ap roach, quite similar to that employed by Ronold et!?

al. , that can be used with either simulations or

measurements, and have shown how it may be used in

an uncertainty evaluation. Although Reference 4describes how to use the statistical model to estimatethe complete load spectrum, it does not indicate howthese models compare with the design standards. It iscritical that the load distributions generated by anystatistical methodology be adaptable for use in existingdesign standards. Moreover, it is arguably even moreimportant that the load model provide insight into howthe design standards might be improved in futurerevisions. The standards should require an accuratereflection of the load distribution with sufficientconservatism to cover the uncertainties caused by the

. r-, pJ T Qyi,f*,f&~,

. i:: 4J f%if’%w &M

~~~ 2!: 2W0

$) fiseTo\limited duration of the sample, whesimulation or field measurements. Only then candesign margins be trimmed to the point of least costwhile still maintaining sufficient margins to keepreliability levels high.

The approach to load modeling is not uniform acrossthe wind community by any measure. This lack ofcommonality in approach was reflected in the workinggroup that produced IEC’S Mechanical LoadMeasurement Technical Specifications. No consensuscould be obtained on how to use measured loads toeither create or substantiate a fatigue load spectrum atthe conditions specified in the Safety Standards. Allthat is offered are several examples of differingapproaches in an annex of the specifications.

Here, we present a methodology for using measured orsimulated loads to produce a Iong-term fatigue-loadspectrum at specified environmental conditions and atdesired confidence levels. To illustrate, examplemeasurements of the two blade-root moments (flap andedge) from a commercial turbine in complex terrain areused. The ten-minute distributions of rainflow rangesare modeled by a quadratic We]%ull distributionfunction based on three statistical moments of theranges (mean, coefilcient of variation, and skewness).The moments are mapped to the wind conditions by atwo-dimensional regression over ten-minute averagewind speed and turbulence intensity. Thus, the “short-terrn” distribution of ranges may be predicted for anycombination of average wind speed and turbulenceintensity. The “long-term” distribution of ranges is,them easily obtained by integrating over the annualdistribution of input conditions. Results are comparedbetween standard-driven and site-driven load estimates.Finally, using statistics based on the regression of thestatistical moments over the input conditions, theuncertainty (due to the limited data set) in the long-termload distribution is represented by 95V0 confidencebounds on predicted loads.

IEC LOAD CASES

The loads specified by IEC 61400-1 Wind TurbineGenerator Safety Requirements for design must bedefined for a specified combination of mean wind speedand turbulence intensity known as the NormalTurbulence Mode15. The standard provides an equationfor the standard deviation of the ten-minute wind&c,, that depends on the hub-height wind speed and twoparameters, 115and a.

c1 = 115(15m/s+ aVti)/(c7 + 1) (1)

2

Equation 1 is based on wind speed standard deviationdata gathered from around the world and aggregatedinto a common data set. The equation was created to be“broadly representative of sites with reasonableinternational marketing interes~”s and does notrepresent any single site. 01 is intended to represent acharacteristic value of wind-speed standard ‘deviation.Certification guidelines are provided for high (A) andmoderate (B) turbulence sites. 11~ defines thecharacteristic value of the turbulence intensity at anaverage wind speed of 15 m/s, and a is a slopeparameter when cr~ is plotted versus hub-height windspeed. The values of these parameters for eachcategory are shown in Table 1.

CATEGORYA B

(HIGH) (MODERATE)11< 0.18 0.16

II . .4

Table 1: Parameters for IEC turbulence categories.

The Category B moderate turbulence specification isintended to roughly envelope (i.e., be hQher than) themean plus one sigma level of turbulence for all thecollected data above 15 mh. Similarly, Category Aenvelopes all collected values of turbulence intensity(with the exception of one southern California site) formean wind speeds above 15 MIS and is above theoverall mean plus two sigma level in high winds*.Clearly, the IEC Normal Turbulence Model is intendedto be conservative for all but the most turbulent sites.

It is a relatively straightforward matter to create aloading distribution that meets the standard criteriawhen using an aeroelastic simulation code. Input windscan be generated for any combination of wind speedand turbulence intensity. Representative loadings can,in theory, be generated by simulating repeatedly untilsutllcient data are produced to drive the uncertainty toan arbitrarily small level. Practically, however, itwould be beneficial to generate a loading distributionwith small, or at least known, uncertainty from asmaller data set. This is where the parametric approachprovides significant value. By means of regression ofload statistics (e.g., moments) over the entire range ofwind speeds and turbulence levels, the uncertainty inthe values of the parameters defining the short-termdistributions at any specified turbulence condition canbe estimated.

In the case of measured loads, it may be simplyimpossible to gather data at the specified turbulence

conditions because of the limitations of the test site. Inthat case, the parametric approach provides a method tointerpolate to a specified turbulence level using all ofthe data collected (thus adding to the confidence of theinterpolation), or to extrapolate beyond the limits of themeasurements. In either case, the parametric approachsimplifies the generation of fatigue loads to Standardspecifications.

EXAMPLE DATA SET

An example data set taken from the copiousmeasurements of the MOUNTURB program’ is used toillustrate the parametric modeling process. The data arecomprised of 101 ten-minute samples of rainflow-counted flap-wise and edge-wise bending-momentranges at the blade root. The test turbine is a WINCON110XT, a 110kW stall-regulated machine operated byCRES (the Centre for Renewable Energy Systems,Pikermi, Greece) at their Lavrio test site. The terrain ischaracterized as complex.

The original time series of the loads and winds were notavailable for Iin-ther analysis; thus, only the rainflow-counted ranges were employed. The number of cyclecounts was tallied in 50 bins ranging ti-om zero to themaximum range in each sample. A single ten-minutesample is categorized by the mean wind speed and theraw turbulence intensity at hub height. The averagewind speeds are limited to the range between 15 and 19m/s and thus reflect response in high wind operation.Turbulence intensities cover a wide range of operatingconditions as can be seen in Figure 1. The measuredloads are summarized by frequency of occurrence inFigure 2a for flap moment ranges and in Figure 2C for ‘edge moment. Plots showing exceedance counts forspecified flap and edge loads are shown in Figures 2band 2d, respectively.

0.24 0

0.08I I I

15 16 17 18 19Wind SPA (IS/$)

Figure 1 Wmd speed and turbulence intensity valuesfor the 101 10-minute data samples.

3

0 102030405060 7080

Flap-wisebendingmoment range (kN-m)

Figure 2a Histogram of flap-wise bending momentranges for 101 1O-minute data sets.

100QO,

0 102030405060 7080

Flap-wite bending moment rang%r (ICN-m)

Figure 2b Cumulative counts of flap-wise bendingmoment ranges for 101 10-minute &ta sets.

SHORT-TERM ANALYSIS

The FITS9 software for fitting probability distributionmodels to empirical data was used to analyze each ten-minute sample. FITS calculates the central moments ofthe data and estimates the best fit distribution model tomatch a user-specified set of moments (e.g., the usercan request a dh-ibution model fit based on twomoments or one based on three moments). FITS is,therefore, a tool for examining the fit of a probabilitymodel to the short-term response, condhional on windspeed and turbulence level.

For purposes of the present discussion, the frst threemoments M, (*1,2,3) of the rainflow-range amplitudes,r, are defined here as:

p, =.E[r]=7, (2)

P2 =+; 0r2 =E[(r-i)2], (3)

4

0 102030405060 7080

Edge-wisebending moment range (kN-m)

Figure 2C Histogram of edge-wise bending momentranges for 101 1O-minute data sets.

0.1

0 1020304050d0 70S0

Edgewise bending moment rmngq r (W&m)

Figure 2d Cumulative counts of flap-wise bendingmoment ranges for 101 10-minute data sets.

,3=ZMI9

0,3(4)

where E[.] is the expectation (or averaging) operator.The first moment is the mean or average range, ameasure of central tendency. The second moment is theCoefficient of Variation (COV), which is the standarddeviation divided by the mean, a measure of thedispersion or spread in the distribution. The thirdmoment is the coefllcient of skewness, which providesinformation on the tail behavior of the distribution.Load range data are often well fit by a Weibulldistribution, a slight distortion of the Weibulldistribution is used to exactly match the fu-st threestatistical moments lO.

To illustrate the fit of the quadratic Weibull distributionto a ten-minutes sample, one of the 101 samples shownin Figure 2 is studied. TMs &ta sample is taken fromthe middle of the measured wind conditions; V= 17 m/sand 1 = 0.18. The data are plotted on a Weibull scalefor the flap loads in Figure 3 and for the edge loads inFigure 4. The vertical scale is @ansformec,fiom the

,!!

10 100

Flapbanding mommt range (kN-m)

(a) Weibull scale plot.

..

I 10 lCO

Flap bending moment range -11.5 (kN-m)

(b) Weibull scale plot (truncation at 11.5 lcN-m).1000O

-Data Setwitb V= 17m/s, l= 0.18

0.1

-l ‘@wlrntic WeibullFit(withsMI= 11.5kN-@

o 1020304050150 70

Flapbending moment range (kN-m)

(c) Exceedance plot format (truncation at 11.5 lcN-m).

Figure 3 Quadratic Weibull model fits to data on flap-bending moment ranges (1’ = 17.0 m/s, 1 =0.18).

Cumulative Distribution Function (CDF) as (-ln(l -CDF)) so that a Weibull distribution will be a straightline on a log-log plot. (Recall that the CDF is thecomplement of the traditional exceedance diagram;exceedance = 1-CDF.)

Figures 3a and 4a show attempts to fit the entire flapand edge data with quadratic Weibull models. As seen

10.0

1.0

,/

. ...

..

.

0.1

10 100

Edge bending moment rmtge (kN-m)

(a) Weibull scale plot.

1 10 lCII

Edgebending moment range -32 (kN-m)

(b) Weibull scale plot (truncation at 32.0 IcN-m)1000 –

100-

10-

1--Data aetwith V=17 m/5, 1=0.18

—Quadmtic WeibuU Fit (with shift= 32 kN-m)

o 1020304050d0 7080

Edge bending naxaent raogea (W+.)

(c) Exceedance plot for!nat (truncation at 32.0 IcN-m).

Figure 4 Quadratic Weibull model fits to data on edge-bending moment ranges (V = 17.0 rds, Z =O.18).

in Figure 3a and especially Figure 4A the data have akinked appearance which the smooth probabilitydistribution, in spite of the quadratic distortio~ can notmatch. Closer examination of the data reveals that thekink is due to a very large number of cycles atrelatively low amplitudes.

5

The proliferation of small amplitude cycles seen inFigure 2 produces a distribution difilcult to duplicatewith a simple analytical form, but these small cyclesproduce relatively little damage. By truncating theloads at a threshold, the kink in the data can beeliminated without significantly reducing the damage.In the edge case, there are obviously a great number ofcycles of smaller amplitude than the dominant gravityload at about 32 ld$m. The flap loads have a lessdktinctive kink at around 10-13 kN-m (1 1.5 k%m wasused as the filtering threshold). Figures 3b and 4b aresimilar to Figures 3a and 4a but include only a subset ofthe data and can be thought of as applying a “shift” toall loads that effectively discards the smallest cycles.ClearIy, the fits of the quadratic Weibull are improveddramatically. Thus, the short-term data are wellmodeled by a quadratic Weibull distribution thatpreserves the fwst three central moments of thetruncated rainflow ranges.

Figures 3C and 4C show the same data as do Figures 3band 4b but with the axes in the more commonexceedance plot format. These plots are included

o

$g:.3 o 0 .

9 Cg;4

o : 000 c

c: o 0 Oco $@<’060

j: or o 8=00

0 0 00 0-6- ;:=

g? 8 Oo 05- : %0 o 0 0~~yw .9 .bo o

ob=3 —go

0 ❑ b=6

Ab=9

-12. I ,

Figure

15,0 15.5 16.0 16.5 17,0 17.5 18.0 18.5 19.0Wind sped (IO/s)

(a) Flap-wise bending

to

o

-3

0 04-

C~o

<>

0-9 .’

0-o 0° 0

3=o 0

0b=3 -% 0 0

❑ b=6

o Ab=9

-12 + I

15,0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0Wind Se (ink)

(b) Edge-wise bending

5 Effect on darnage estimation of shifl inblade bending moment range data.

reorient the reader back to the original summaries of thedata shown in Figure 2. They also serve to illustratehow the analytical distribution functions may be used toextrapolate to less fiequen~ higher amplitude loads.

The low amplitude cycles (that make distribution fitsdii%cult as described in the preceding) can only bedkcarded if they produce an insignificant amount ofdamage. The damage unaccounted for due to thetruncation of rainflow range data at 11.5 kN-m for theflap loads is represented in Figure 5% and due to atruncation at 32 kl+m for the edge loads in Figure 5b.All 101 ten-minute data segments are represented inFigures 5a and 5b. Lost damage is plotted for threefatigue exponents, b, representing typical values ofwind turbine materials ranging from b=3 in weldedsteel up to &9, more characteristic of fiberglasscomposites. In no case does the truncation removemore that 12V0 of the damage, and that only when b=3.In ahnost all cases for the higher b values, the lostdamage is less than 3%. In both flap and edge bendingcases, over 80’% of the rainflow-counted ranges areremoved by truncating the data sets. Our findings thatdiscarding so much of the data does not lead to grosslyunconservative estimates of darnage is not unexpectedsince it has long been known that eliminating most ofthe small amplitude cycles has a negligible effect ondamagell.

REGRESSION ANALYSIS

Because a good match can be obtained to the short-termdkribution of rainflow ranges given the fust threemoments and a fixed data truncatio% it “issufficient toknow the moments over all the operating conditions inorder to fully define the turbine fatigue loading.Regression of the moments over ten-minute averagewind speed and turbulence intensity can achieve thedesired result and assist in understanding the loadingdependence on and sensitivity to both turbulence andwind speed. Results from a regression analysis can alsoprovide information on the uncertainty of the loads.

The moments presented in the following figures, M andP3 describe the COV and skewness, respectively, of theshifted range r ‘=r-r,. More precisely, by eliminating allranges below the truncation level r,, we obtain theshifted values r ‘=r-r, of the remaining ranges andconsider models based on statistics of rl. This is doneto conform with the quadratic Weibull model, whichgenerally assigns probability to all outcomes r’ >0.For example, the second moment is

p2=& . (5)r

6

.

I COEFFICIENT I FLAP I EDGE IIG al 21.49 40.02

3bl 0.808 0.359c, 0.2(-)2 (-).039a2 0.722 0.635

~ bl 0.031 -0.573u

c] 0.080 0.063

$5a3 0.963 0.980

$Z b3 -1.260 -0.468C3 0.033 0.132

Reference values: Vnf=17.1m/s; Id =0.145 II

Table 2: Coetllcients fkom regression analysis.

Because these definitions of P2 and P3 aredimensionless,, they are insensitive to the choice oftruncation level. For example, if the loads areexponentially distributed M= 1 and W=2 regardless ofthe truncation level. The mean value, PI, presented isthe same as F, the mean with respect to zero of theranges retained afier eliminating the small-amplitudecycles.

As in Reference 4, the fmt three moments @i, i=l-3)are fitted to a power law fimction of wind speed, V,andturbulence intensity, Z.

(6)

The reference wind speed, l’~. and referenceturbulence intensity, l=fi are determined from thegeometric mean values of the data4. For the Lavrio datase~ V~~17. lm/s and lw~0.145. The calculatedregression coefficients are shown in Table 2.

The regression results for the flap bending moments areshown in Figure 6 and for the edge bending moments inFigure 7. Mean, COV, and skewness are plotted in theparts (a), (b), and (c), respectively, of the figures. h allcases, the regression line uses the reference value forturbulence intensity. The circles correspond to themeasured data from the 101 samples. The solidsymbols (squares) show the regression prediction usingthe measured wind speed and turbulence intensity foreach ten-minute data set. A large spread in solidsymbols about the regression lime indicates a sizeabledependence on turbulence level (e.g., Figure 6a’s meanflap range) while a small variation in the solid symbolsindicates that the turbulence has little effect on thatparticular moment (e.g., Figure 6C’S flap skewness).This sensitivity can also be inferred from the magnitudeof the ci coefficients in Table 2. The smaller the value

PARAMETER FLAP

@-Mean. u, 0.51

al (a) 0.108 0.050 v

b, (q t statistic) (0.081, 9.9) (0.020, 17.5)Cf(CZt statistic) (0.032, 6.4) (0.008, 4.9)

1? - Cov.U> 0.05 0.43a2 (0) 0.003 0.003b2(c, f statistic) (0.076, 0.3) (0.080, 7.1)C2(0, tstatistic) (0.030, 2.7) (0.03 1, 2,0)

@– Skewness, Pj 0.17 0.05a3(a) 0.018 0.016bj (q t statistic) (0.301, 4.2) (0.266, 1.8)C3(0, t statistic) (0.1 17, 2.8) (0.104, 1.3)

Avg. Cycle Rate 1.75 Hz 1.38 Hz

Table 3: Regression parameter summary.

of Ci,the lessthe importance of Z in the estimate of the2Amoment.

Table 3 summarizes the regression uncertainties interms of the widely used Z# and tstatistics. A high ~value, approaching unity, indicates that a largepercentage of the data variation is explained by theregression. In eontras4 a low Z? value suggests thepresence of other influences, not included in theregression model, that induce the scatter in the data.Note that the goal here is not to predict the momentstatistics in a single 10-minute history, but rather thelong-run average of such 10-minute samples over theentire turbine lifetime.

The t statistic, which is the estimated coefficientdivided by the standard deviation of the estimate,indicates whether a particular coefllcient is statisticallysignificant. A t value less than about two wouldindicate that the coefficient is not significantly differenttlom zero at about the 95% cotildence level. Since theleading coefficients, ajj are estimates of the moments atthe reference conditions, they are always significantlydifferent from zero, and t is not reported for them.However, the tvalues of bi and Ciindicate whether anyfunctional variation with respect to wind speed orturbulence intensiq, respectively, is supported by thedata.

Examination of Tables 2 and 3 suggests that the meanload range is strongly related to both wind speed andturbulence, although the relation to turbulence has smallexponents (0.202 and 0.039). The only higher momentrelationships that have high t statistics are the edgeCOV relation to wind speed and the flap skewnessrelation to wind speed. The overall low exponents and tstatistics for the higher moments indkate that thedistribution shapes are relatively constant over all input

7

.’.

2s 7

s&23-*

% 0

❑ 21J a

=k >

0

;17-o Measuredlo-minutedets*ts

z ■ Regression prediction st meswred V snd I

— Regression st kef157

15 16 17 18 19

Wind speed (m/s)

(a) Mean

m1

= 0.7- 0 0E >b

00 00 0 0

$0.6- 0 Measured IO-minutedam sets 1

■ Regression prediction at messured V and I

— Regression at I-ref0.5.

15 16 17 18 19

Wmd speed (m/s)

(b) Coefficient of variation (COV)

t 0.4 I ( Io Measured 10-minutedata sets

#m Regression prediction at mes.wrsd V snd I

— Regession st I-ref0.0

15 16 17 18 19

Wind speed (M/s)

(c) Coefficient of skewness

Figure 6 Regression results for flap-wise bendingmoment range.

conditions. The variation seen in Figures 6 and 7beyond that indicated by the solid symbols is sample-to-sample variation not indicative of a systematicrelationship with the independent variables, V and 1.Part of this remaining variation will be irreducible, anatural outcome of random processes, but some couldpossibly be reduced with regression over betterturbulence descriptors than the simple turbulenceintensity.

a I I I I I

— Regression st Lref36

15 16 17 18 19

Wmd sped (m/s)

(a) Mean

15 16 17 18 19

Wmd speed (m/s)

(b) Coefficient of variation (COV)

0& 00

~o 00 0 0 a&zm% 0

~ 0.8 - . 7 0 0 00 ~

% 0 00“ ~

g0

j 0.40 Messured 10-minutedats sets

mm Regression

— Regression st I-r&0.03

15 16 17 18 19

Wmd speed (m/s)

(c) Coefficient of skewness

Figure 7 Regression results of edge-wise bendingmoment range.

The Lavrio data set used in this example is limited to arange of wind speeds from 15 to 19 m/s. The long-termanalysis in the next section will, for example purposesonly, assume the regression trends found in high windsapply to all wind speeds. In an actual application thedata from a particular turbine will need to be examinedover the entire range of damaging wind speeds. Itmight be amenable to regression fits that run all the wayfrom cut-in to cut-out. More likely, the wind speedrange will have to be partitioned into divMons over

8

which the response moments are well behaved enoughto be fit with simple regression. For example, it islikely that the response will have differentcharacteristics above and below rated wind speed. Inthat case, the analysis presented here would have to berepeated for each wind-speed division beforeproceeding with the long-term analysis in the nextsection. It may also be the case that the response in lowwinds has an insignificant contribution to the fatiguedamage and the analysis can safely deal with only highwind response. The individual application willdetermine the constraints.

LONG-TERM ANALYSIS

The long-term dktribution of fatigue loads is obtainedby integrating the short-term distributions (for loadsconditional on wind conditions) over the specifieddistribution of wind conditions. Current IEC standardsspeci~ a Rayleigh distribution of wind speed with theannual average depending on class. Class I sites have a10 rnh average and Class II sites have a 8.5 m/saverage. Wind-speed classes defined as “Special” arealso allowed with conditions that may be defined by the

l,E+O

m&

%~ 1.E-2

Q.

1x* I.E-t -%k’~+.= I.E-d -— — Reference Intensityf

--- IEC Class A Intensity

-- IEC Class B Intensity

1.E-S . I

o 20 40 60 80Flap bending moment ran% r

(a) Flap-wise bending

1.E+O \

1—Refer- hnensity\ .

-‘- lEC Class A Intensity

-- IEC Claw B Intensity 1

1.E-8 I

20 40 64 80Edge bending moment rangei r

(b) Edge-wise bending

Figure 8 Distribution of bending moment rangesconditional on wind speed and turbulenceintensity.

designer. The turbulence intensity is a deterministicfimction of wind speed given by Eq. 1. A lifetime loaddistribution must sum all the short-term distributions ateach wind speed and associated turbulence intensityweighting them by the annual wind speed distribution.This can be written as

F(r) = p(r \v, l)f(v)dv , (7)

where F(r) is the long-term distribution of stress ranges,r, and F(rl V,f) is the short-term distribution of stressranges conditional on the ten-minute average windspeed, V, and the specified turbulence intensity, 1. j(v)is the wind speed probability density function (PDF).The integration is carried out over all damaging windspeeds. The distribution functions of r can be either theCDF or the exceedence (1-CDF). However, theintegration must be over the probability densityfhnction for wind speed, flV).

Any environmental conditions can be used with Eq. 7once the response moments have been defined withrespect to the turbulence levels and wind speeds. Thishas been accomplished by the regression of themoments over V and 1 and by determining the short-term distributions, F@!VJ), from the moments. Asexamples we will calculate the long-term distributionsfor two standard-driven and two sitedrivenenvironments.

Figures 9a and 9b show the long-term distributions offlap and edge loads respectively for the IEC Class Iwind speed distribution (Rayleigh 10tis) and for bothturbulence Categories A and B. Both of these standardenvironments define the turbulence level as a functionof wind speed by Eq. 1.

The specification of a fixed turbulence intensityfunctionally related to the wind speed is somewhatartificial; measurements indicate that the turbulenceintensity varies over a range of values for each ten-minute sample (see Figure 1). A more realisticrepresentation than Eq. 7 for the long-term distributionmight be to include turbulence intensity as a randomvariable by integrating over both wind speed andturbulence intensity as follows:

F(r) = Jp(r [V,l)j-(v,z)dldv (8)

Figures 9a and 9b show the long-term flap and edgedistributions derived by integrating over both windspeed and turbulence assuming wind speed andturbulence variations are independent. (Independenceimplies that the joint PDF, flVJ) in Eq. 8 is simply theproduct of the individual marginal PDF$fl~ andj(l).),..,

,.

9

. ..

lE+O

: IE-2

!5% lE-4b~a -A- [EC-A Turbulence Inteosity~

~ lE.6 -- -A- IEC.B Turbulence Intensity

+IsmmncdmxnU16mndats I

0.24 . 3%.5(V(mgm.3im)--- W’=W- +1**-= K”--’”ti

●

... .

* -. .--:-----s- . ● .0

*. . *.--------

e . ● -:.“. ..

\ +Avemgc lstescbwindspeed

IE-S 0.08 ~

o 20 40 60 80 15 16 17 18 19Flap bending moment rsnge (kN-m) wind speed (Ids)

(a) Flap-wise bending Figure 10 Lavrio site turbulence intensity as a fimctionlE+O , ! 1

; 1J32 .

i5* Icd A

g

II-A- IEC-A Turbulemx Intemily

z~-A- IEC-B Turbulence intensity

& IE.6-o- I ss.smoednommi mudom vsriable

+ Average I at each wind speedJI

IE-S I I 1

0 20 40 60 80Edgebendingmomentr8nge(kN-m)

(b) Edge-wise bending

Figure 9 Long-term dktibution of edge-wise bendingmoment ranges (Rayleigh distributed windspeed with mean = 10 m/s).

The wind speed is a 10 m/s-average Rayleighdistribution as prescribed for IEC Class I sites. Theturbulence is assumed to be normally distributed with

mean, ~, defined by ~s 2.5/V, and standarddeviation equal to 0.025, based on a best fit to the dataof turbulence vs. wind speed shown in Figure 10,

Also plotted in Fig. 9 is the result of assuming theturbulence at the Lavrio site is defined by the averagevalue at each wind speed (the line with the black circlesin Figure 10). This simpler assumption allows the useof Eq. 7. The comparison indicates that the integrationover all turbulence levels, which is the most realisticreflection of the measurements, produces a much lowerload spectrum. The simplified alternative, i.e., fixingthe turbulence at the mean level, leads to a moreconservative result although the conservatismdiminishes at very high load levels for the flap bendingmoment ranges. However, these results may besensitive to the choice of a normal distribution forturbulence intensity.

of wind spee~ regression ‘fits, and IECCategory A and B definitions.

Discussion

The Lavrio site’s mean-plus-one-sigma turbulenceintensity at 15 m/s wind speed is quite similar to theIEC standard specification of 16% (Class B) to 18?Z0

(Class A). This similarity in turbulence levels is

evident throughout the high wind range as shown inFigure 10. The differences between the distributions inFigures 9a and 9b therefore provide an indication of theconservatism built into the IEC load cases relative to afairly turbulent site.

Within the context of standards developmen~ it may bereasonable to argue for lower turbulence specificationsif differences as seen above can be shown to besignificant and consistent. However, because thestandards are based on past experience and industryconsensus rather than objective risk-based analysis, itmay be dangerous to remove conservatism from onearea without also checking elsewhere to insure that thisconservatism isn’t covering for an unknown lack ofconservatism elsewhere in the design process.

k general, the current standards give a load calculation“recipe” that is meant to meet some specific reliabilitycriteria. If these current reliability levels are deemedadequate on average (over various cases), one cannotreduce conservatism in turbulence specification withoutadjusting the recipe to compensate elsewhere; e.g.,through use of a higher load factor. Note that thisalternative procedure - unbiased turbulence with higherload factor - may result in more uniform relitillityacross a range of cases. In contrast current standardsmay lead to potential overdesign of machines that areparticularly sensitive to turbulence, and underdesign inturbulence-insensitive cases.

10

.,.,

ESTIMATING UNCERTAINTY IN LONG-TERMLOADS

To review, the parametric load modeling proposed hereproceeds by (1) modeling loads by their statisticalmoments A (P 1,2,3) and (2) modeling each moment Mas a parametric function of V and 1 (Eq. 6). Themoment-based model in step (1) is in principleindependent of the turbine characteristics (although theoptimal choice among such models may be somewhatcasedependent). Hence, in this parametric approach,the tubine characteristics are reflected solely throughthe moment relations in Eq. 6; specifically, the 9coefficients aj, bfi Ci(i= 12,3). For clarity, we organizethese here into a vector,O={fzl, bl,cl,a2, h2, c2, a3, b3,c3).

Simpler 2-moment models would require only 6coefficients.

The preceding section has shown one benefit of thisparametric model. Because it permits load statistics tobe estimated for arbitrary V and Z, the results can beweighted to form the long-term loads distribution as inEqs. 7-8 (and Figs. 9a-b). Symbolically, we rewrite Eq.7 here, noting explicitly its dependence on the vector 9.

w Ie) = JF(r Iv, I,e)j(v)w (9)

@q. 8 can be rewritten analogously.) The foregoingresults (Figs 9a-b) have used our best estimates for theentries of 0; i.e., the mean values of each entry in 0.These are the values of ai, bi, and ci cited in Table 2.

A fiirther advantage of the parametric model lies in itsusefulness in estimating the effects of statisticaluncertainty. To clari~, it is usefi,d to distinguishbetween the various terms in Eq. 9. The quantities Vand 1 are “random variables: that is, their fitureoutcomes will show an intrinsic randomness that cannotbe reduced by additional study of past wind conditions.In contrast the 9 coefllcients in 0 are in principle fwed(under the model’s assumptions). We may, however,be uncertain as to their values due to limited responsedata. This “uncertainty” (as opposed to “randomness”)can be reduced through addhional sampling. Theconsequence of having only limited data can bereflected through 95°A confidence levels, for example,on the exceedance probability 1-F(r). These areconceptually straightforward to establish by simulation.Assuming the entries of Oare each normally distributed,for example, one may (1) simulate multiple outcomesof 0; (2) estimate F’(r) for each 0 as in Eq. 9; and (3)sort the resulting F(r) values (at each freed r value) toestablish confidence bands; e.g., in which 95’% of thevalues lie.

IE+O

-(

+ Oeteminktic E3time4e)

Y ,+ 95% ConfidenceLevel

,,-8 ~o 20 40 60 80

F18pbendiogmomentrange(kN-m)

(a) Flap-wise banding

+ Deterministic EstimateI

b

+-95% ConadenceLevel

tI I I

o

Figure 11

20 40 60 80

Edge bending momeot range (kN-m)

(b) Edge-wise Bending

95?40Confidence levels on the exceedanceprobability of fatigue loads for the Lavriosite with turbulence set to the average valuefor each wind speed.

Figures 1la and 1lb show the 95% confidence level onthe exceedance probability, l-F(r), which result fromthe simulation procedure described above. Each of the9 coefficients in Eq. 9 were generated as statisticallyindependent, normally distributed random variables,with means and standard deviations given by Tables 2and 3, respectively. (Correlation among these variablescan also be included; however, this was not done here.)All of these results adopt the site-specific meanturbulence model; i.e., the results labeled “Average I ateach wind speed” in Figs. 9a-b. These results hornFigs. 9a-b are repeated in Figs. 1la-b, and referred tothere as “deterministic” results. Also shown are 95%confidence results; i.e., probability levels below which95% of the simulations fall.

The increase in probability, over the deterministicresults in order to achieve 95’%0confidence, is found tobe relatively modest. This reflects the benefit of havingas many as 101 10-xninute samples. If the same mean

11

.

trends had resulted from fewer samples, the resulting95% confidence results would be correspondinglyhigher than the mean results. Note also thaq at least forflap-wise loads, the conservatism induced by the IECturbulence models exceeds that required to cover ourstatistical loads uncertainty, based on the data at hand.Of course, as noted earlier, this IEC conservatism maybe desirable to cover other sources of uncertainty.Finally, we caution again that these long-term loadresults are intended for example purposes only;accurate numerical values would require data across abroader range of wind speeds.

SUMMARY

Fatigue load spectra are generated for arbitrary siteconditions (wind speed and turbulence intensi~distributions) by using parametric models to fit theshort term load spectrum to the fmt three moments ofthe truncated rainflow range distributions andregressing the moments over wind speed and turbulenceintensity. The spectra are generated to specified IECconditions for wind speed Class and turbulenceCatego~. The spectra are also generated for as-measured scatter in the turbulence levels across all windspeeds. The comparison of the two approaches revealsthe level of conservatism that results from assumedhigh turbulence levels written into the currentstandards. The selected confidence level can becalculated using the statistics from regression analysis.Since the confidence interval depends on theuncertainty in the load characterization, it could providea better margin of safety on the loads than can beaccomplished with an inflated turbulence level. Theparametric approach presented here illustrates howstatistically based standards may be able to reflect theuncertainty in the loading definition caused by finite-Iength data records.

ACKNOWLEDGMENTS

The authors would like to extend a special debt ofgratitude to the Centre for Renewable Energy Sources(CRES), Pikermi, Greece and especially to FragiskosMouzakk who found time in a busy schedule to supplythe pofiion of the MOUNTURB data set used in theexamples. The authors also gratefidly acknowledge theefforts of LeRoy Fitzwater, of Stanford Universi~, whoprovided the uncertainty analysis described in the finalsection of this paper.

REFERENCES

1. Fitzwater, L. M. and Winterstein, S. R, “PredictingDesign Wind Turbine Loads from Limited Data:Comparing Random Process and Random PeakModels: A collection of the 2001 ASME Wind

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Energy Symp., at the AMA Aerospace SciencesMfg., Reno, Nevad4 AIAA-2001-O046, January2001.McCoy, T. J., Malcolm, D. J., and Griffin, D. A.,“An Approach to the Development of TurbineLoads in Accordance with IEC 1400-1 and 1S02394/ A collection of the 1999 ASME WindEnergy Symp., at the AMA Aerospace SciencesMtg., Reno, NevadA AIAA-99-0020, January1999, pp. 1-9.Ronold, K. O., Wedel-Heinen, J. and Christensen,C. J., “Reliability-based fatigue design of wind-turbine rotor blades; Elsevier, Engineering.Structures 21, 1999, pp. 1101-1114.Veers, P. S. and Winterstein, S. R., “Application ofMeasured Loads to Wmd Turbine Fatigue andReliability Analysis: Journal of Solar EnergyEngineering, Trans. of the ASME, Vol. 120, No. 4,November 1998.IEC/TC88, 61400-1 Wind Turbine GeneratorSystems – Part 1: Safety Requirements,International Electrotechnical Commission (IEC),Geneva, Switzerland, 1998.IEC/TC88, Draji IEC 61400-13 7S, Ed. 1: Windturbine generator systems – Part 13: Measurementof mechanical Ioadr, 88/1201CDV, InternationalElectrotechnical Commission (IEC), Genev~Switzerland, 1999.MOUNTURB, Load and Power MeasurementProgram on Wind Turbines Operating in ComplexMountainous Regions, Volumes. I - III, Editor P.Chaviaropoulos, Coord. A. N. Fragoulis, CRES,IWO, ECN, NTUA-FS, published by CRES,Pikermi, Greece, Nov. 1996.Butterfield, S., Honey, B., Madse% P. H. andStorlq C., Report on 88/69/CD – Wind TurbineGenerator Systems Part 1: Safety Requirements, 2.Edition, National Renewable Energy Laboratory,Golden, CO, Oct. 1997.Manuel, L., Kashef, T. and Wmterste@ S. R,Moment-Based Probability Modelling and ExtremeResponse Estimation The Fits Routine, Version 1.2,SAND99-2985, Sandia National Laboratories,Albuquerque, NM, Nov. 1999.Lange, C. FL, Probabilistic Fatigue Methodologyand Wind Turbine Reliability, SAND96- 1246,Sandia National Laboratories, Albuquerque, NM,May 1996.Nelson, D. V., and Fuchs, H. O., “Predictions ofCumulative Fatigue Damage using CondensedLoad Histories,” Fatigue under Complex Loading:Analysis and Experiments, Advances inEngineering, Vol. 6, Ed. R M. Wetzel, SAE,Warrendale, PA, 1977, pp. 163-187.

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