WT-ERA user’s manual

Program for Wind Turbine - Extreme ResponseAnalysis

J.M. Peeringa

ECN-E–10-055

Acknowledgement

The publication of the WT-ERA user’s manual is funded by the SenterNovem project 2020-04-11-10-003 "Wind turbine design and optimalsiation tool FOCUS-6", ECN project 7.9413.

AbstractSince the IEC61400-1 (2005) edition 3 standard has been issued the statistical extrapolationof responses for the ultimate strength analysis is part of the design of wind turbines. At ECNa software tool WT-ERA is developed to facilitate statistical extrapolation for the wind turbineindustry. This report is a user’s manual for the WT-ERA software.

2 ECN-E–10-055

Contents

Notations 3

1 Introduction 7

2 Statistical extrapolation of wind turbine responses 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Data selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Short-term probability distributions . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Long-term distribution and 50-year response . . . . . . . . . . . . . . . . . . . . 13

3 Using the program 15

3.1 Starting the program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Input file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

References 19

A L-moments 21

B L-moments for Weibull distribution 23

ECN-E–10-055 3

4 ECN-E–10-055

Notations

E[] ExpectationFlong(L) Long-term extreme response distributionFshort(L|v10) Conditional short-term extreme response distributionF10−min(L|v10) 10-minute equivalent conditional short-term extreme response distributionfv(v10) Marginal Weibull probability density function of mean wind speedj Order (statics) extremesk Shape parameter of the probability distributionL Random variable of extreme responseL50 50-year extreme responseMp,r,s Probability weighted moments (PWM)N Total number of extremesn Sample sizep, r, s Real values in probability weighted moment Mp,r,s

Qlong(L) Long-term extreme response exceedance distributionT Duration of time series (10-minutes)v10 10-minute mean speed at hub heightvi Wind speed bin centrexj:n jth order statistics

α Scale parameter of probability distributionαr Probability weighted moment M1,0,r

βr Probability weighted moment M1,r,0

∆vi Wind speed bin widthΦ Standard normal distributionλr L-momentξ Location parameter probability distributionτr L-moment ratioτ3 L-skewnessτ4 L-kurtosis

ECN-E–10-055 5

6 ECN-E–10-055

1 Introduction

Since the IEC61400-1 (2005) edition 3 standard has been issued the statistical extrapolation ofresponses is part of the design of wind turbines. At ECN a software tool WT-ERA is developedto facilitate statistical extrapolation for the wind turbine industry.

In this user’s manual the statistical extrapolation process for wind turbines is discussed brieflyfirst. Next the use of the WT-ERA tool is described.

ECN-E–10-055 7

8 ECN-E–10-055

2 Statistical extrapolation of wind turbine responses

2.1 Introduction

During its life time a wind turbine is subjected to a variety of wind conditions and wave conditionsin case of an offshore wind turbine. Both wind and waves are of stochastic nature and should beaccounted for during the wind turbine design. For the wind turbine responses driven by theenvironmental conditions like the blade and tower loads and displacements an extreme responseanalysis is required. In the case of ultimate wind turbine responses extreme value statistics shouldbe applied to the wind turbine responses. A general introduction in extreme value statistics canbe found in for instance Gumbel (2004), Castillo (1988) or Coles (2001). More informationabout the extreme value analysis for wind energy application can be found in for example Cheng(2002), Moriarty et al. (2003) and Peeringa (2009).

In load case DLC 1.1 of the third edition of the IEC61400-1 (2005) standard the ultimate re-sponses during power production are estimated using extreme value theory. Aim of this statisticalextrapolation is to determine the extreme load with a return period of 50 year. This means that themean period between two fifty year responses is 50 year. In order to calculate the 50-year extremeresponse L50 the long-term extreme response distribution has to be estimated. This long-term ex-treme response distribution is found by integrating the long-term environmental distribution withthe short-term extreme response distribution. The weighted average (convolution) of the condi-tional short-term probability distribution per mean wind speed of the response L is given by:

Flong(L) =

∫v10

Fshort(L|v10)fv(v10) dv10 (1)

In IEC61400-1 (2005) the long-term environmental distribution is the well known Weibull dis-tribution of the 10-minute mean wind speed fv(v10).

In practice the wind speed is divided in a number of wind speed bins and for every bin simula-tions are performed with a sophisticated aeroelastic model. This way all the important featuresof the wind turbine are captured. The design loads are calculated using a turbulence wind modelto account for the stochastic nature of the wind. For every wind bin a number of simulationsare performed using different turbulent wind fields having the same mean wind speed and tur-bulence. Because of the different turbulent wind fields every simulation will result in a differentmaximum load. Using these maxima of the simulations a short-term extreme response distribu-tion Fshort(L|v10) conditional on the wind speed is estimated for every wind speed bin. By usingwind speed bins the integration of equation 1 is changed in a summation

Pr{l ≤ L} = Flong(L) =∑i

Fshort(L|vi)fv(vi)4vi (2)

Pr{l ≥ L} = Qlong(L) = 1− Flong(L) =∑i

(1− Fshort(L|vi))fv(vi)4vi (3)

Instead of the long-term extreme response distribution Flong the probability of exceedance distri-bution Qlong is used to determine the 50-year response L50. In wind energy the wind loads andresponses are simulated over 10-minute periods. This means that the probability of exceedancelevel associated with a 50-year return period should be based on the number of 10-minute periodsduring 50 year

Qlong(L50) =10

50× 365× 24× 60= 3.8× 10−7 (4)

ECN-E–10-055 9

Important issues in the statistical extrapolation process are:

- The number of time series (simulations) needed per wind speed bin

- Selection of maximum or minimum responses from time series for every wind bin

- Estimation of conditional short-term probability distributions

- Estimation of 50-year response

These issues will be described briefly in the next paragraphs.

2.2 Data selection

The first question to be answered in the case of a statistical extrapolation analysis of wind turbineresponses is the number of simulations needed to estimate a 50-year response. A check of con-vergence is required by GL. According to the Best Engineering Practice (Argyriadis et al., 2008)an increase in the number of data points ≥ 10% must not lead to a change in the results of ≥ 5%.

Not only the number of time series is of importance also the way the maxima are selected affectsthe 50-year response. The extreme value analysis is performed based on the selected maxima inthe time series. Three data selection methods are available in WT-ERA:

1 Global maxima

2 Block maxima

3 Peak Over Threshold (POT) maxima

The main difference between the Global maxima method, the Block maxima method and the PeakOver Threshold method is the amount of data used from every 10-minute time series.

Figure 1 Example of a global maximum

The Global maximum selection uses only one maximum of the entire time series. See Figure1. The Block maximum method en the Peak Over Threshold (POT) method use more peaks

10 ECN-E–10-055

(maxima) of the time series. In the Block maximum method the time series is divided in ahorizontal sense by cutting the time series in a number of blocks and taking the maximum in eachblock. See Figure 2. The number of blocks should be not to large, because in the statisticalextrapolation analysis it is assumed that the maxima are independent.

Figure 2 Example of selection of block maxima

Another popular method is the POT method. In this method the time series is divided in a verticalsense by a threshold value. The maximum between two up-crossing of the threshold is selected.See Figure 3. The determination of the threshold value is not straight forward.

Figure 3 Example of POT maxima

2.3 Short-term probability distributions

In the statistical extrapolation analysis it is not known beforehand which probability distributiongives the best fit to the extracted maxima. Therefore different theoretical probability distribution

ECN-E–10-055 11

should be used. In the WT-ERA software five probability distributions are available to model theshort term extreme responses. Depending on the data selection method used different short-termextreme response distributions Fshort(L|v10) can be applied.

1 Gumbel distribution

2 Generalized Extreme Value (GEV) distribution

3 Lognormal distribution

4 3-parameter Weibull distribution

5 Generalized Pareto Distribution (GPD)

An overview of the relation between the data selection method and the probability distributionis given in Table 1. The parameters of the probability distributions are calculated using the so-called L-moment method (Hosking, 1990; Hosking and Wallis, 1997). A short description of theL-moment method is given in Appendix A.

Table 1 Overview of data selection methods and probability distributions used

Distribution Global maximum Block maximum POT maximumGumbel x xGEV x xLognormal x xWeibull x x xGPD x

The distributions are defined for a response L with α as the scale parameter, ξ as the locationparameter and in case of a three parameter distribution with k as the shape parameter.

Theoretically the extreme responses can be modelled using the Generalized Extreme Value (GEV)distribution only. The GEV distribution contains the classical Gumbel distribution (k = 0), theFrechet distribution (k < 0) and the reverse Weibull distribution (k > 0). See equation 5

FGEV (L|v10) =

exp(−(1− k(L−ξ)

α )1

k ) k 6= 0

exp(− exp(− (L−ξ)

α ))

k = 0

(5)

Since the GEV distribution only asymptotically reaches k = 0, the Gumbel distribution is definedseparately.

FGumbel(L|v10) = exp(− exp

(−(L− ξ)

α)

)(6)

In practice the Lognormal and three parameter Weibull distribution show good results and aretherefore included for the statistical extrapolation analysis in WT-ERA (Freudenreich and Ar-gyriadis, 2007; Cheng, 2002; Ragan and Manuel, 2007).

FLognormal(L|v10) =

Φ

(− log(1− k(L−ξ)

α )k

)k 6= 0

Φ((L−ξ)α

)k = 0

(7)

with Φ the standard Normal distribution.

12 ECN-E–10-055

Although the Weibull distribution was developed to model minimum values in practise the Weibulldistribution is also used to model maxima.

FWeibull(L|v10) = 1−[exp

(L− ξα

)k]L− ξ, α, k > 0 (8)

In case threshold models like the POT method are selected the Generalized Pareto Distribution(GPD) is used to model the data.

FGPD(L|v10) =

1− (1− k(L−ξ)

α )1

k k 6= 0

1− exp(− (L−ξ)

α

)k = 0

(9)

For the statistical extrapolation method the maxima extracted from the response time series aremodeled using different probability distributions. To see how well the probability distributionsfit with the extracted maxima the probability distributions are plotted together with the so calledempirical distribution.

This empirical distribution is determined as follows. First the maxima of the time series areordered in ascending order. The smallest maxima will have a value i = 1. The largest maximumwill be i = N . Now every maximum is given a probability according to the plotting position inequation 10.

F (L|v10) =i

N + 1(10)

In literature equation 10 is called the Gumbel plotting position (Gumbel, 2004).

2.4 Long-term distribution and 50-year response

Aim of the statistical extrapolation of the wind turbine responses is to determine the 50-year re-sponse of the wind turbine. For this 50-year response to be estimated a long-term distribution ofthe extreme response is needed. The procedure for the extreme response analysis is given in thealgorithm below.

Select an extreme analysis for minimum values or maximum valuesSelect for the global maxima, block maxima or peak over threshold maxima to be usedfor every BIN do

for every time series doGet maximua or minima from the 10-minute time series

end forif Minimum then

Minimum value is multiplied by minus oneend ifSort the selected maxima in ascending orderLink probability to every maximum according to the Gumbel plotting positionEstimate the L-momentsEstimate the parameters of the short-term distributions

end forCompute the long-term exceedance probabilityEstimate numerically Qlong(L) = 3.8 · 10−7 using Regula FalsiCheck Qlong(L) = 3.8 · 10−7 graphically

ECN-E–10-055 13

In case the statistical extrapolation is applied for minimum values like for example the towerclearance the minimum values are multiplied by minus 1. This way the extrapolation of a min-imum value is transformed to an extrapolation of a maximum value and the same routines inWT-ERA can be used.

For every wind speed bin the short-term conditional distributions are estimated for an observationperiod T of 10 minutes. The short-term distribution Fshort(L|v10) in equation 1 is used whenonly one maximum is extracted from each time series. This is the so called Global maximamethod. For the Block maxima method and the POT method the short-term distribution has to betransformed to the 10-minute equivalent short-term distribution F10−min(L|v10). When only theglobal maximum of each time series is used the short-term distribution is

F10−min(L|v10) = Fshort(L|v10) (11)

The short-term distribution for the Block maxima refers to a distribution for a maxima in the in ablock. For a distribution of the maxima in a period T (10-minute) the short-term distribution ofblock maxima is transformed with the number of blocks nb in a time period T as in equation 12.

F10−min(L|v10) = Fshort(L|v10)nb (12)

In case of the POT method the short-term distribution corresponds with a distribution for localmaximum occurring during a period T. The peak distribution is transformed as in equation 13with np the average number of peaks per time period T .

F10−min(L|v10) = Fshort(L|v10)npT (13)

To determine the 50-year response the long-term exceedance probability distribution Qlong inequation 14 is used. The 50 year response is estimated numerically using the Regula Falsimethod. In the Regula Falsi method we are looking for the response L corresponding with thefunction g(L) = Qlong(L)− 3.8 · 10−7 = 0.0. When there is no solution after 100 iterations thesearch is stopped. To check the calculated value of the 50-year response a visually inspection ofthe long-term distribution graph is strongly recommended. See Figure 5

Qlong(L) =∑i

(1− F10−min(L|vi))fv(vi)4vi (14)

14 ECN-E–10-055

3 Using the program

3.1 Starting the program

WT-ERA is a console application that is executed by typing:

WT-ERA.exe example.inp

The input file example.inp contains the keyword values for the extreme response analysis of thetime series. The content is explained in the next section.

3.2 Input file

In the input file example.inp the keywords and their values are given. Comment lines in the inputfile should start with a " # ! < % *" or a space for an empty line. An example of the input filelooks like:

sessionId Calculated

inputDirectory P: \phatasdata4 \

outputDirectory P: \ExtremenFocus6 \testLongterm \

loadLabel ’flap bending moment’

extremeType maximum

#extremeMethod global

extremeMethod block

#extremeMethod pot

numberOfBlocks 10

# POTthreshold give factor to be multiplied with the standard deviation

POTthreshold 1.4

columnNumber 7

numberOfBins 3

# [binOccurence numberOfSeries]

binTable

0.0880 3

0.0892 3

0.0866 3

binTimeseries

# binNumber FileName

ECN-E–10-055 15

1 0600.tim

1 0601.tim

1 0602.tim

2 0700.tim

2 0701.tim

2 0702.tim

3 0800.tim

3 0801.tim

3 0802.tim

Below the keywords are discussed briefly. The program checks whether all keywords are presentin the input file.

sessionIdIdentification of the extreme response session for instance by giving the name of the wind turbine.

inputDirectoryThis is the directory were the input files with the time series are saved.

outputDirectoryThis is the directory were the output files with the results of extreme response analysis are saved.

loadLabelThis is the name of the response analysed in this statistical extrapolation session. For instanceflap bending moment.

extremeTypeThis keyword can have two values, MINIMUM or MAXIMUM. Both MINIMUM and MAXI-MUM values can be extremes. For loads the maximum value is searched for. In case of a towerclearance the minimum value is of importance.

extremeMethodThree extreme data methods can be selected, GLOBAL, BLOCK and POT. Each method selectsthe maximum and minimum values of the time series in a different way.

numberOfBlocksThis keyword is used when extremeMethod "BLOCK" is selected. It gives the number of blocksthe time series should be divided in.

POTthresholdThis keyword is only used when extremeMethod "POT" is selected. POTthreshold is a factor withwhich the standard deviation of the time series is multiplied. The POTthreshold value should bepositive. The threshold for the extremeType "MAXIMUM" is the mean + POTthreshold*standarddeviation. In case extremeType "MINIMUM" the threshold is the mean - POTthreshold*standarddeviation.

16 ECN-E–10-055

columnNumberThis is the column number in the input file of the time series to be analysed.

numberOfBinsThis is the number of wind bins (environmental conditions) that are used in the extreme responseanalysis.

binTableThe keyword gives for every wind speed bin the occurrence of that wind speed bin and the num-ber of time series to be used for that bin.

binTimeseriesThis keyword gives for every time series first the bin number it is associated with and next the filename of the file that contains the time series.

3.3 Output

The results of the extreme response analysis with WT-ERA are written in ASCII format to outputfiles.

For every bin two output files are created. One output file, extremeMethodBin001Empirical.txt,contains the short term empirical distribution based on the extracted extremes of the time series.The second output file, extremeMethodBin001Model.txt, shows the results of the estimation ofthe theoretical short term distributions.

The theoretical short-term extreme response distributions are used to estimate the long term ex-ceedance probability distribution Qlong (equation 14). The results of the long term responsedistribution are saved in the output file extremeMethodlongtermResponse.txt.

For example the output files for extremeMethod = "BLOCK" are called:

• BLOCKBin001Empirical.txt

• BLOCKBin001Model.txt

• BLOCKlongtermResponse.txt

One way to check the results of the extreme value analysis is to plot for every wind bin the shortterm distributions. For every bin the empirical and theoretical short term distributions are plottedtogether. This way one can see which theoretical distributions fits the empirical distribution best.Also outliers can be identified and studied. See Figure 4.

The long term exceedance distribution Qlong is presented in a graph like Figure 5. The plotof Qlong can be used to check the numerically estimated 50-year response given in the ex-tremeMethodlongtermResponse.txt file.

The messages, warnings and errors produced (encountered) during the extreme response analysisare stored in a log file logsessionID.dat. This log file is saved in the directory of the WT-ERAexecutable.

ECN-E–10-055 17

Figure 4 Example of a short term distribution graph

Figure 5 Example of a long term distribution graph

18 ECN-E–10-055

References

Argyriadis, K., K. Freudenreich and N. Hille (2008): Suggestions for "Best Engineering Practise"for application of IEC 61400-1:2005, Revision 5. Memo.

Castillo, E. (1988): Extreme Value Theory in Engineering. Academic Press Inc., San Diego,USA.

Cheng, P.W. (2002): A reliability based design methodology for extreme responses of offshorewind turbines. Ph.D. thesis, Delft University of Technology, Delft.

Coles, S. (2001): An introduction to statistical modeling of extreme values. Springer.

Freudenreich, K. and K. Argyriadis (2007): The Load Level of Modern Wind Turbines accordingto IEC 61400-1. The Science of Making Torque from Wind. Technical University of Denmark,Journal of Physics, volume 75.

Greenwood, J.A., J.M. Landwehr, N.C. Matalas and J.R. Wallis (1979): Probability weightedmoments: definition and relation to parameters of several distributions expressable in inverseform. Water Resources Research, 15(5):1049 –1054.

Gumbel, E.J. (2004): Statistics of extremes. Dover Publications.

Hosking, J.R.M. (1990): L-moments analysis and estimation of distributions using linear com-bination of order statistics. Journal of the Royal Statistical Society, Series B, 52(1):105 –124.

Hosking, J.R.M. and J.R. Wallis (1997): Regional Frequency Analysis: An approach based onL-moments. Cambridge Univerity Press.

IEC61400-1 (2005): Wind turbine generator systems - Part 1: Safety requirements. InternationalElectrotechnical Commission (IEC), IEC 61400-1, third edition.

Landwehr, J.M. and N.C. Matalas (1979): Probability weighted moments compared with sometraditional techniques in estimating Gumbel parameters and quantiles. Water Resources Re-search, 15(5):1055–1064.

Moriarty, P.J., W.E. Holley and S.P. Butterfield (2003): Extrapolation of extreme and fatigueloads using probabilistic methods. Report NREL/TP-500-34421, National Renewable EnergyLaboratory NREL, Golden, Colorado, USA.

Peeringa, J.M. (2009): Comparison of extreme load extrapolations using measured and calcu-lated loads of a MW wind turbine. European Wind Energy Conference. EWEA, Marseille,France.

Ragan, P. and L. Manuel (2007): Statistical extrapolation methods for estimating wind turbine ex-treme loads. A collection of the 2007 ASME Wind Energy Symposium, at the AIAA AerospaceMeeting. American Institute of Aeronautics and Astronautics, Reno, Nevada, USA. AIAA-2007-1221.

ECN-E–10-055 19

20 ECN-E–10-055

A L-moments

For the selected distributions the scale parameter α, the location parameter ξ and the shapeparameter k are estimated using the L-moment method (Hosking, 1990; Hosking and Wallis,1997). L-moments are linear combination of probability weighted moments PWM. Greenwoodet al. (1979) defined the probability weighted moment PWM for a distribution function F (x) =P (X ≤ x) as:

Mp,r,s = E[XpF (X)r(1− F (X))s] (15)

where E[] is the expectation.

Useful probability weighted moments are αr = M1,0,r and βr = M1,r,0, where αr is related tominima and βr to maxima. For a distribution that has a quantile function x(u) αr and βr aregiven by:

αr =

1∫0

x(u)(1− u)r du βr =

1∫0

x(u)ur du (16)

Unbiased estimators of the PWM αr and βr are defined by Landwehr and Matalas (1979):

ar =1

n

(n− 1

r

)−1 n∑j=r+1

(n− jr

)xj:n (17)

br =1

n

(n− 1

r

)−1 n∑j=r+1

(j − 1

r

)xj:n (18)

where xj:n is the jth order statistics in a sample of size n

Hosking and Wallis (1997) give the L-moments λr as:

λ1 = α0 = β0λ2 = α0 − 2α1 = 2β1 − β0λ3 = α0 − 6α1 + 6α2 = 6β2 − 6β1 + β0λ4 = α0 − 12α1 + 30α2 − 20α3 = 20β3 − 30β2 + 12β1 − β0

(19)

Beside L-moments λr, L-moment ratios are defined:

τr =λrλ2

r > 2 (20)

where τ3 is called the L-skewness and τ4 the L-kurtosis.

The expression of the probability distribution parameters in L-moments λ1, λ2, τ3 and τ4 is givenby Hosking and Wallis (1997) for the GEV, the Gumbel, the Lognormal and the GPD distribution.For the Weibull distribution the definitions in appendix B are used.

ECN-E–10-055 21

22 ECN-E–10-055

B L-moments for Weibull distribution

The Weibull distribution is defined in equation 21 with α as the scale parameter, ξ as the locationparameter and k as the shape parameter.

FWeibull(X) = 1−[exp

(X − ξα

)k]X − ξ, α, k > 0 (21)

Greenwood et al. (1979) defines the probability weighted moments (PWM) for the Weibull dis-tribution as follows:

αr = M1,0,r =ξ

1 + r+αΓ(1 + 1

k )

(1 + r)1+1

k

(22)

Expressing the Weibull distribution in L-moments using equation 19 gives the following expres-sions for the L-moments:

λ1 = ξ + αΓ(1 + 1k )

λ2 = (1− 2−1

k )αΓ(1 + 1k )

τ3 = 3− 2(1−3− 1k )

(1−2− 1k )

τ4 = 5(1−4− 1k )−10(1−3− 1

k )+6(1−2− 1k )

(1−2− 1k )

(23)

Greenwood et al. (1979) gives the definition of the parameters of the Weibull distributions inPWM. The shape parameter k is defined as:

k =log(2)

log(

α0−2α1

2(α1−2α3)

) =log(2)

log(

5τ4−5τ3+4

) (24)

The scale parameter α is:

α =λ2

(1− 2−1

k )Γ(1 + 1k )

(25)

The location parameter ξ is:

ξ = λ1 − αΓ(1 +1

k) (26)

ECN-E–10-055 23