AIAA-2001-0047

1

PARAMETRIC MODELS FOR ESTIMATING WIND TURBINEFATIGUE LOADS FOR DESIGN

Lance Manuel1

Paul S. Veers2

Steven R. Winterstein3

1Department of Civil Engineering, University of Texas at Austin, Austin, TX 787122Sandia National Laboratories, Wind Energy Technology Department, 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-counted range data for eachten-minute segment are characterized by parametersrelated to their first three statistical moments (mean,coefficient of variation, and skewness). QuadraticWeibull distribution functions 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. First,we 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 distributionis represented by 95% confidence bounds on predictedloads.

This paper is declared a work of the U.S. Government and is notsubject to copyright protection in the United States. Sandia is amultiprogram laboratory operated by Sandia Corporation, a LockheedMartin company, for the U.S. Department of Energy under contractDE-AC04-94AL85000.

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 Winterstein1. Inboth cases, the loads must be determined as functions ofwind speed (or other climatic conditions).

Parametric models define the response, statistically,with respect to input conditions. Such models fitanalytical distribution functions to the measured orsimulated data. The parameters of these distributionfunctions can be useful in defining the response/loadsas a function of the input conditions. The end result,then, 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 aeroelastic 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 turbineat those particular conditions. The full 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 data, a similar

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approach has been described by McCoy et al.2 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 almost impossible to characterize.

With 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 frequent loadlevels, (2) map the response to the input conditions, and(3) calculate load uncertainty. For example, Ronold etal.3 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 loading distribution to a quadratic(transformed by a squaring operation) Weibulldistribution.

Veers and Winterstein4 described a parametricapproach, quite similar to that employed by Ronold etal.3, that can be used with either simulations ormeasurements, and have shown how it may be used inan 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 standards5. 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

limited duration of the sample, whether based onsimulation 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 Specification6. No consensuscould be obtained on how to use measured loads toeither create or substantiate a fatigue load spectrum atthe conditions specified in the Safety Standard5. Allthat is offered are several examples of differingapproaches in an annex of the specifications6.

Here, we present a methodology for using measured orsimulated loads to produce a long-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 Weibull distributionfunction based on three statistical moments of theranges (mean, coefficient 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-term” distribution of ranges may be predicted for anycombination of average wind speed and turbulenceintensity. The “long-term” distribution of ranges is,then, 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 95% 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 Model5. The standard provides an equationfor the standard deviation of the ten-minute wind speed,σ 1, that depends on the hub-height wind speed and twoparameters, I15 and a.

)1()m/s15( /151 ++= aaVI hubσ (1)

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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 interest,”8 and does notrepresent any single site. σ 1 is intended to represent acharacteristic value of wind-speed standard deviation.Certification guidelines are provided for high (A) andmoderate (B) turbulence sites. I15 defines thecharacteristic value of the turbulence intensity at anaverage wind speed of 15 m/s, and a is a slopeparameter when σ 1 is plotted versus hub-height windspeed. The values of these parameters for eachcategory are shown in Table 1.

CATEGORYA

(HIGH)B

(MODERATE)I15 0.18 0.16a 2 3

Table 1: Parameters for IEC turbulence categories.

The Category B moderate turbulence specification isintended to roughly envelope (i.e., be higher than) themean plus one sigma level of turbulence for all thecollected data above 15 m/s. Similarly, Category Aenvelopes all collected values of turbulence intensity(with the exception of one southern California site) formean wind speeds above 15 m/s and is above theoverall mean plus two sigma level in high winds8.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 untilsufficient 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 program7 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 further analysis; thus, only the rainflow-counted ranges were employed. The number of cyclecounts was tallied in 50 bins ranging from 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 foredge moment. Plots showing exceedance counts forspecified flap and edge loads are shown in Figures 2band 2d, respectively.

0.08

0.12

0.16

0.20

0.24

15 16 17 18 19Wind speed (m/s)

Tur

bule

nce

Inte

nsit

y

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

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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 distribution 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, conditional on windspeed and turbulence level.

For purposes of the present discussion, the first threemoments µi, (i=1,2,3) of the rainflow-range amplitudes,r, are defined here as:

rrE == ][1µ , (2)

[ ]222 )( ; rrE

r rr −== σσµ , (3)

[ ]3

3

3)(

r

rrE

σµ −= , (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 coefficient 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 first threestatistical moments10.

To illustrate the fit of the quadratic Weibull distributionto a ten-minutes sample, one of the 101 samples shownin Figure 2 is studied. This data sample is taken fromthe middle of the measured wind conditions; V = 17 m/sand I = 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 transformed from the

0.1

1

10

100

1000

0 10 20 30 40 50 60 70 80

Edge-wise bending moment range (kN-m)

No.

of

occu

rren

ces

Figure 2c Histogram of edge-wise bending momentranges for 101 10-minute data sets.

0.1

1

10

100

1000

10000

0 10 20 30 40 50 60 70 80

Edge-wise bending moment range, r (kN-m)

No.

of

exce

edan

ces

of r

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

0.1

1

10

100

1000

0 10 20 30 40 50 60 70 80

Flap-wise bending moment range (kN-m)

No.

of

occu

rren

ces

Figure 2a Histogram of flap-wise bending momentranges for 101 10-minute data sets.

0.1

1

10

100

1000

10000

0 10 20 30 40 50 60 70 80

Flap-wise bending moment range, r (kN-m)

No.

of

exce

edan

ces

of r

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

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Cumulative Distribution Function (CDF) as (-ln(1-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

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

0.1

1.0

10.0

1 10 100

Edge bending moment range (kN-m)

Tra

nsfo

rmed

CD

F

(a) Weibull scale plot.

0.1

1.0

10.0

1 10 100

Edge bending moment range - 32 (kN-m)

Tra

nsfo

rmed

CD

F

(b) Weibull scale plot (truncation at 32.0 kN-m)

0.1

1

10

100

1000

0 10 20 30 40 50 60 70 80

Edge bending moment ranges (kN-m)

Num

ber

of e

xcee

danc

es

Data set with V = 17 m/s, I = 0.18

Quadratic Weibull Fit (with shift = 32 kN-m)

(c) Exceedance plot format (truncation at 32.0 kN-m).

Figure 4 Quadratic Weibull model fits to data on edge-bending moment ranges (V = 17.0 m/s, I =0.18).

0.1

1.0

10.0

1 10 100

Flap bending moment range (kN-m)

Tra

nsfo

rmed

CD

F

(a) Weibull scale plot.

0.1

1.0

10.0

1 10 100

Flap bending moment range - 11.5 (kN-m)

Tra

nsfo

rmed

CD

F

(b) Weibull scale plot (truncation at 11.5 kN-m).

0.1

1

10

100

1000

10000

0 10 20 30 40 50 60 70 80

Flap bending moment range (kN-m)

Num

ber

of e

xcee

danc

es

Data Set with V = 17 m/s, I = 0.18

Quadratic Weibull Fit (with shift = 11.5 kN-m)

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

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

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The proliferation of small amplitude cycles seen inFigure 2 produces a distribution difficult 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 kN-m. The flap loads have a lessdistinctive kink at around 10-13 kN-m (11.5 kN-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.Clearly, the fits of the quadratic Weibull are improveddramatically. Thus, the short-term data are wellmodeled by a quadratic Weibull distribution thatpreserves the first 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 to

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 frequent, higher amplitude loads.

The low amplitude cycles (that make distribution fitsdifficult as described in the preceding) can only bediscarded 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 5a, and due to atruncation at 32 kN-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 b=9, more characteristic of fiberglasscomposites. In no case does the truncation removemore that 12% of the damage, and that only when b=3.In almost 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 damage is not unexpectedsince it has long been known that eliminating most ofthe small amplitude cycles has a negligible effect ondamage11.

REGRESSION ANALYSIS

Because a good match can be obtained to the short-termdistribution of rainflow ranges given the first threemoments and a fixed data truncation, it is sufficient 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, µ2 andµ3 describe the COV and skewness, respectively, of theshifted range r’ =r-r t. More precisely, by eliminating allranges below the truncation level rt, we obtain theshifted values r’ =r-rt of the remaining ranges andconsider models based on statistics of rt. This is doneto conform with the quadratic Weibull model, whichgenerally assigns probability to all outcomes r’ > 0.For example, the second moment is

2t

r

rr −=

σµ . (5)

-12

-9

-6

-3

0

15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0Wind speed (m/s)

Per

cent

dif

fere

nce

in d

amag

e du

e to

shi

ft o

f fl

ap d

ata

b = 3

b = 6

b = 9

(a) Flap-wise bending

-12

-9

-6

-3

0

15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0Wind speed (m/s)

Per

cent

dif

fere

nce

in d

amag

e du

e to

shi

ft o

f ed

ge d

ata

b = 3

b = 6

b = 9

(b) Edge-wise bending

Figure 5 Effect on damage estimation of shift inblade bending moment range data.

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Because these definitions of µ2 and µ3 aredimensionless,, they are insensitive to the choice oftruncation level. For example, if the loads areexponentially distributed, µ2=1 and µ3=2 regardless ofthe truncation level. The mean value, µ1, presented isthe same as r , the mean with respect to zero of theranges retained after eliminating the small-amplitudecycles.

As in Reference 4, the first three moments (µi, i=1-3)are fitted to a power law function of wind speed, V, andturbulence intensity, I.

i

ref

i

refii

c

I

Ib

V

Va

=µ (6)

The reference wind speed, Vref, and referenceturbulence intensity, Iref, are determined from thegeometric mean values of the data4. For the Lavrio dataset, Vref=17.1m/s and Iref=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. In 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 line 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

of ci, the less the importance of I in the estimate of theith moment.

Table 3 summarizes the regression uncertainties interms of the widely used R2 and t statistics. A high R2

value, approaching unity, indicates that a largepercentage of the data variation is explained by theregression. In contrast, a low R2 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 coefficient is statisticallysignificant. A t value less than about two wouldindicate that the coefficient is not significantly differentfrom zero at about the 95% confidence level. Since theleading coefficients, ai, are estimates of the moments atthe reference conditions, they are always significantlydifferent from zero, and t is not reported for them.However, the t values of bi and ci indicate whether anyfunctional variation with respect to wind speed orturbulence intensity, 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 indicate that thedistribution shapes are relatively constant over all input

COEFFICIENT FLAP EDGE

a1 21.49 40.02b1 0.808 0.359

Mea

n

c1 0.202 0.039a2 0.722 0.635b2 0.031 -0.573

CO

V

c2 0.080 0.063a3 0.963 0.980b3 -1.260 -0.468

Ske

w-

Nes

s

c3 0.033 0.132Reference values: Vref =17.1m/s; Iref =0.145

Table 2: Coefficients from regression analysis.

PARAMETER FLAP EDGE

R2 – Mean, µ1 0.51 0.76a1 (σ) 0.108 0.050b1 (σ, t statistic) (0.081, 9.9) (0.020, 17.5)c1 (σ, t statistic) (0.032, 6.4) (0.008, 4.9)

R2 – COV, µ2 0.05 0.43a2 (σ) 0.003 0.003b2 (σ, t statistic) (0.076, 0.3) (0.080, 7.1)c2 (σ, t statistic) (0.030, 2.7) (0.031, 2.0)

R2 – Skewness, µ3 0.17 0.05a3 (σ) 0.018 0.016b3 (σ, t statistic) (0.301, 4.2) (0.266, 1.8)c3 (σ, t statistic) (0.117, 2.8) (0.104, 1.3)Avg. Cycle Rate 1.75 Hz 1.38 Hz

Table 3: Regression parameter summary.

8

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 I.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.

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 divisions over

15

17

19

21

23

25

15 16 17 18 19Wind speed (m/s)

Mea

n F

lap

BM

Ran

ge (

kN-m

)

Measured 10-minute data sets

Regression prediction at measured V and I

Regression at I-ref

(a) Mean

0.5

0.6

0.7

0.8

0.9

15 16 17 18 19Wind speed (m/s)

CO

V o

f F

lap

BM

Ran

ge

Measured 10-minute data sets

Regression prediction at measured V and I

Regression at I-ref

(b) Coefficient of variation (COV)

0.0

0.4

0.8

1.2

1.6

15 16 17 18 19

Wind speed (m/s)

Skew

ness

of

Fla

p B

M R

ange

Measured 10-minute data sets

Regression prediction at measured V and I

Regression at I-ref

(c) Coefficient of skewness

Figure 6 Regression results for flap-wise bendingmoment range.

36

38

40

42

44

15 16 17 18 19Wind speed (m/s)

Mea

n E

dge

BM

Ran

ge (

kN-m

)

Measured 10-minute data sets

Regression

Regression at I-ref

(a) Mean

0.4

0.5

0.6

0.7

0.8

15 16 17 18 19Wind speed (m/s)

CO

V o

f E

dge

BM

Ran

geMeasured 10-minute data sets

Regression

Regression at I-ref

(b) Coefficient of variation (COV)

0.0

0.4

0.8

1.2

1.6

15 16 17 18 19

Wind speed (m/s)

Skew

ness

of

Edg

e B

M R

ange

Measured 10-minute data sets

Regression

Regression at I-ref

(c) Coefficient of skewness

Figure 7 Regression results of edge-wise bendingmoment range.

9

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 distribution of fatigue loads is obtainedby integrating the short-term distributions (for loadsconditional on wind conditions) over the specifieddistribution of wind conditions. Current IEC standardsspecify a Rayleigh distribution of wind speed with theannual average depending on class. Class I sites have a10 m/s 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

designer. The turbulence intensity is a deterministicfunction 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

∫= dVVfIVrFrF )(),|()( , (7)

where F(r) is the long-term distribution of stress ranges,r, and F(r|V,I) is the short-term distribution of stressranges conditional on the ten-minute average windspeed, V, and the specified turbulence intensity, I. f(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 densityfunction for wind speed, f(V).

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 I and by determining the short-term distributions, F(r|V,I), from the moments. Asexamples we will calculate the long-term distributionsfor two standard-driven and two site-drivenenvironments.

Figures 9a and 9b show the long-term distributions offlap and edge loads respectively for the IEC Class Iwind speed distribution (Rayleigh 10m/s) 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:

∫∫= dIdVIVfIVrFrF ),(),|()( (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, f(V,I) in Eq. 8 is simply theproduct of the individual marginal PDFs, f(V) and f(I).)

1.E-8

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IEC Class A Intensity

IEC Class B Intensity

V = 17 m/s

(a) Flap-wise bending

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IEC Class A Intensity

IEC Class B Intensity

V = 17 m/s

(b) Edge-wise bending

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

10

The wind speed is a 10 m/s-average Rayleighdistribution as prescribed for IEC Class I sites. Theturbulence is assumed to be normally distributed withmean, I , defined by VI /5.2≅ , 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.

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%(Class A). This similarity in turbulence levels isevident 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 development, 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.

In 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 reliabilityacross a range of cases. In contrast, current standardsmay lead to potential over-design of machines that areparticularly sensitive to turbulence, and under-design inturbulence-insensitive cases.

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I assumed normal from data

Average I at each wind speed

(a) Flap-wise bending

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IEC-B Turbulence Intensity

I assumed normal random variable

Average I at each wind speed

(b) Edge-wise bending

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

0.08

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0.24

15 16 17 18 19Wind speed (m/s)

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DataI = 2.5/V (regression)regression mean + 1 std devregression mean - 1 std devIEC-AIEC-B

Figure 10 Lavrio site turbulence intensity as a functionof wind speed, regression fits, and IECCategory A and B definitions.

11

ESTIMATING UNCERTAINTY IN LONG-TERMLOADS

To review, the parametric load modeling proposed hereproceeds by (1) modeling loads by their statisticalmoments µi (i=1,2,3) and (2) modeling each moment µi

as a parametric function of V and I (Eq. 6). Themoment-based model in step (1) is in principleindependent of the turbine characteristics (although theoptimal choice among such models may be somewhatcase-dependent). Hence, in this parametric approach,the turbine characteristics are reflected solely throughthe moment relations in Eq. 6; specifically, the 9coefficients ai, bi, ci (i=1,2,3). For clarity, we organizethese here into a vector,

},,,,,,,,{ 333222111 cbacbacba= .

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 I, 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 θ.

dVVfIVrFrF )(),,|()|( ∫= (9)

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

A further advantage of the parametric model lies in itsusefulness in estimating the effects of statisticaluncertainty. To clarify, it is useful to distinguishbetween the various terms in Eq. 9. The quantities Vand I are “random variables;” that is, their futureoutcomes will show an intrinsic randomness that cannotbe reduced by additional study of past wind conditions.In contrast, the 9 coefficients in θ are in principle fixed(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 additional sampling. Theconsequence of having only limited data can bereflected through 95% confidence levels, for example,on the exceedance probability 1-F(r). These areconceptually straightforward to establish by simulation.Assuming the entries of θ are each normally distributed,for example, one may (1) simulate multiple outcomesof θ; (2) estimate F(r) for each θ as in Eq. 9; and (3)sort the resulting F(r) values (at each fixed r value) toestablish confidence bands; e.g., in which 95% of thevalues lie.

Figures 11a and 11b show the 95% confidence level onthe exceedance probability, 1-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 fromFigs. 9a-b are repeated in Figs. 11a-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% confidence, is found tobe relatively modest. This reflects the benefit of havingas many as 101 10-minute samples. If the same mean

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(b) Edge-wise Bending

Figure 11 95% Confidence levels on the exceedanceprobability of fatigue loads for the Lavriosite with turbulence set to the average valuefor each wind speed.

12

trends had resulted from fewer samples, the resulting95% confidence results would be correspondinglyhigher than the mean results. Note also that, 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 intensitydistributions) by using parametric models to fit theshort term load spectrum to the first 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 turbulenceCategory. 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-length 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 FragiskosMouzakis who found time in a busy schedule to supplythe portion of the MOUNTURB data set used in theexamples. The authors also gratefully acknowledge theefforts of LeRoy Fitzwater, of Stanford University, 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

Energy Symp., at the AIAA Aerospace SciencesMtg., Reno, Nevada, AIAA-2001-0046, January2001.

2. McCoy, T. J., Malcolm, D. J., and Griffin, D. A.,“An Approach to the Development of TurbineLoads in Accordance with IEC 1400-1 and ISO2394,” A collection of the 1999 ASME WindEnergy Symp., at the AIAA Aerospace SciencesMtg., Reno, Nevada, AIAA-99-0020, January1999, pp. 1-9.

3. 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.

4. Veers, P. S. and Winterstein, S. R., “Application ofMeasured Loads to Wind Turbine Fatigue andReliability Analysis,” Journal of Solar EnergyEngineering, Trans. of the ASME, Vol. 120, No. 4,November 1998.

5. IEC/TC88, 61400-1 Wind Turbine GeneratorSystems – Part 1: Safety Requirements,International Electrotechnical Commission (IEC),Geneva, Switzerland, 1998.

6. IEC/TC88, Draft IEC 61400-13 TS, Ed. 1: Windturbine generator systems – Part 13: Measurementof mechanical loads, 88/120/CDV, InternationalElectrotechnical Commission (IEC), Geneva,Switzerland, 1999.

7. MOUNTURB, Load and Power MeasurementProgram on Wind Turbines Operating in ComplexMountainous Regions, Volumes. I - III, Editor P.Chaviaropoulos, Coord. A. N. Fragoulis, CRES,RISO, ECN, NTUA-FS, published by CRES,Pikermi, Greece, Nov. 1996.

8. Butterfield, S., Holley, B., Madsen, P. H. andStork, C., Report on 88/69/CD – Wind TurbineGenerator Systems Part 1: Safety Requirements, 2.Edition, National Renewable Energy Laboratory,Golden, CO, Oct. 1997.

9. Manuel, L., Kashef, T. and Winterstein, S. R.,Moment-Based Probability Modelling and ExtremeResponse Estimation The Fits Routine, Version 1.2,SAND99-2985, Sandia National Laboratories,Albuquerque, NM, Nov. 1999.

10. Lange, C. H., Probabilistic Fatigue Methodologyand Wind Turbine Reliability, SAND96-1246,Sandia National Laboratories, Albuquerque, NM,May 1996.

11. 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.