WT-ERA user’s manual Program for Wind Turbine - Extreme Response Analysis J.M. Peeringa

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Acknowledgement The publication of the WT-ERA user’s manual is funded by the SenterNovem project 2020-0411-10-003 "Wind turbine design and optimalsiation tool FOCUS-6", ECN project 7.9413.

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

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Contents Notations

3

1

Introduction

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2

Statistical extrapolation of wind turbine responses

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2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2

Data selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3

Short-term probability distributions . . . . . . . . . . . . . . . . . . . . . . . .

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2.4

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

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3

Using the program

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3.1

Starting the program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2

Input file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3

Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References

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A L-moments

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B L-moments for Weibull distribution

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Notations E[] Flong (L) Fshort (L|v10 ) F10−min (L|v10 ) fv (v10 ) j k L L50 Mp,r,s N n p, r, s Qlong (L) T v10 vi xj:n

Expectation Long-term extreme response distribution Conditional short-term extreme response distribution 10-minute equivalent conditional short-term extreme response distribution Marginal Weibull probability density function of mean wind speed Order (statics) extremes Shape parameter of the probability distribution Random variable of extreme response 50-year extreme response Probability weighted moments (PWM) Total number of extremes Sample size Real values in probability weighted moment Mp,r,s Long-term extreme response exceedance distribution Duration of time series (10-minutes) 10-minute mean speed at hub height Wind speed bin centre j th order statistics

α αr βr ∆vi Φ λr ξ τr τ3 τ4

Scale parameter of probability distribution Probability weighted moment M1,0,r Probability weighted moment M1,r,0 Wind speed bin width Standard normal distribution L-moment Location parameter probability distribution L-moment ratio L-skewness L-kurtosis

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1

Introduction

Since the IEC61400-1 (2005) edition 3 standard has been issued the statistical extrapolation of responses is part of the design of wind turbines. At ECN a software tool WT-ERA is developed to facilitate statistical extrapolation for the wind turbine industry. In this user’s manual the statistical extrapolation process for wind turbines is discussed briefly first. Next the use of the WT-ERA tool is described.

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2 2.1

Statistical extrapolation of wind turbine responses Introduction

During its life time a wind turbine is subjected to a variety of wind conditions and wave conditions in case of an offshore wind turbine. Both wind and waves are of stochastic nature and should be accounted for during the wind turbine design. For the wind turbine responses driven by the environmental conditions like the blade and tower loads and displacements an extreme response analysis is required. In the case of ultimate wind turbine responses extreme value statistics should be applied to the wind turbine responses. A general introduction in extreme value statistics can be found in for instance Gumbel (2004), Castillo (1988) or Coles (2001). More information about 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 responses during power production are estimated using extreme value theory. Aim of this statistical extrapolation is to determine the extreme load with a return period of 50 year. This means that the mean period between two fifty year responses is 50 year. In order to calculate the 50-year extreme response L50 the long-term extreme response distribution has to be estimated. This long-term extreme response distribution is found by integrating the long-term environmental distribution with the short-term extreme response distribution. The weighted average (convolution) of the conditional short-term probability distribution per mean wind speed of the response L is given by: Z

Flong (L) =

Fshort (L|v10 )fv (v10 ) dv10

(1)

v10

In IEC61400-1 (2005) the long-term environmental distribution is the well known Weibull distribution 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 simulations are performed with a sophisticated aeroelastic model. This way all the important features of the wind turbine are captured. The design loads are calculated using a turbulence wind model to account for the stochastic nature of the wind. For every wind bin a number of simulations are performed using different turbulent wind fields having the same mean wind speed and turbulence. Because of the different turbulent wind fields every simulation will result in a different maximum load. Using these maxima of the simulations a short-term extreme response distribution Fshort (L|v10 ) conditional on the wind speed is estimated for every wind speed bin. By using wind speed bins the integration of equation 1 is changed in a summation P r{l ≤ L} = Flong (L) =

X

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

(2)

i

P r{l ≥ L} = Qlong (L) = 1 − Flong (L) =

X

(1 − Fshort (L|vi ))fv (vi ) 4vi

(3)

i

Instead of the long-term extreme response distribution Flong the probability of exceedance distribution Qlong is used to determine the 50-year response L50 . In wind energy the wind loads and responses are simulated over 10-minute periods. This means that the probability of exceedance level associated with a 50-year return period should be based on the number of 10-minute periods during 50 year Qlong (L50 ) =

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10 = 3.8 × 10−7 50 × 365 × 24 × 60

(4) 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 turbine responses is the number of simulations needed to estimate a 50-year response. A check of convergence 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 affects the 50-year response. The extreme value analysis is performed based on the selected maxima in the 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 Peak Over 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 Figure 1. The Block maximum method en the Peak Over Threshold (POT) method use more peaks 10

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(maxima) of the time series. In the Block maximum method the time series is divided in a horizontal sense by cutting the time series in a number of blocks and taking the maximum in each block. See Figure 2. The number of blocks should be not to large, because in the statistical extrapolation 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 vertical sense 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

2.3

Example of POT maxima

Short-term probability distributions

In the statistical extrapolation analysis it is not known beforehand which probability distribution gives the best fit to the extracted maxima. Therefore different theoretical probability distribution ECN-E–10-055

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should be used. In the WT-ERA software five probability distributions are available to model the short term extreme responses. Depending on the data selection method used different short-term extreme 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 distribution is given in Table 1. The parameters of the probability distributions are calculated using the socalled L-moment method (Hosking, 1990; Hosking and Wallis, 1997). A short description of the L-moment method is given in Appendix A. Table 1

Overview of data selection methods and probability distributions used

Distribution Gumbel GEV Lognormal Weibull GPD

Global maximum x x x x

Block maximum x x x x

POT maximum

x x

The distributions are defined for a response L with α as the scale parameter, ξ as the location parameter 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), the Frechet distribution (k < 0) and the reverse Weibull distribution (k > 0). See equation 5

FGEV (L|v10 ) =

 k(L−ξ) 1    exp(−(1 − α ) k )

k 6= 0

(5)

     exp(− exp − (L−ξ) ) k=0 α

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

FGumbel (L|v10 ) = exp(− exp −

(L − ξ) ) α



(6)

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

FLognormal (L|v10 ) =

   log(1− k(L−ξ) )  α  k 6= 0  k  Φ −       Φ (L−ξ) α

(7)

k=0

with Φ the standard Normal distribution. 12

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

L−ξ FW eibull (L|v10 ) = 1 − exp α 

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.

FGP D (L|v10 ) =

 k(L−ξ) 1    1 − (1 − α ) 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 are modeled using different probability distributions. To see how well the probability distributions fit with the extracted maxima the probability distributions are plotted together with the so called empirical distribution. This empirical distribution is determined as follows. First the maxima of the time series are ordered in ascending order. The smallest maxima will have a value i = 1. The largest maximum will be i = N . Now every maximum is given a probability according to the plotting position in equation 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 response of the wind turbine. For this 50-year response to be estimated a long-term distribution of the extreme response is needed. The procedure for the extreme response analysis is given in the algorithm below. Select an extreme analysis for minimum values or maximum values Select for the global maxima, block maxima or peak over threshold maxima to be used for every BIN do for every time series do Get maximua or minima from the 10-minute time series end for if Minimum then Minimum value is multiplied by minus one end if Sort the selected maxima in ascending order Link probability to every maximum according to the Gumbel plotting position Estimate the L-moments Estimate the parameters of the short-term distributions end for Compute the long-term exceedance probability Estimate numerically Qlong (L) = 3.8 · 10−7 using Regula Falsi Check Qlong (L) = 3.8 · 10−7 graphically ECN-E–10-055

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In case the statistical extrapolation is applied for minimum values like for example the tower clearance the minimum values are multiplied by minus 1. This way the extrapolation of a minimum value is transformed to an extrapolation of a maximum value and the same routines in WT-ERA can be used. For every wind speed bin the short-term conditional distributions are estimated for an observation period T of 10 minutes. The short-term distribution Fshort (L|v10 ) in equation 1 is used when only one maximum is extracted from each time series. This is the so called Global maxima method. For the Block maxima method and the POT method the short-term distribution has to be transformed to the 10-minute equivalent short-term distribution F10−min (L|v10 ). When only the global 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 a block. For a distribution of the maxima in a period T (10-minute) the short-term distribution of block 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 local maximum occurring during a period T. The peak distribution is transformed as in equation 13 with np the average number of peaks per time period T . F10−min (L|v10 ) = Fshort (L|v10 )np T

(13)

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

X

(1 − F10−min (L|vi ))fv (vi ) 4vi

(14)

i

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3 3.1

Using the program 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 the time 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 input file should start with a " # ! < % *" or a space for an empty line. An example of the input file looks 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

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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 present in the input file. sessionId Identification of the extreme response session for instance by giving the name of the wind turbine. inputDirectory This is the directory were the input files with the time series are saved. outputDirectory This is the directory were the output files with the results of extreme response analysis are saved. loadLabel This is the name of the response analysed in this statistical extrapolation session. For instance flap bending moment. extremeType This keyword can have two values, MINIMUM or MAXIMUM. Both MINIMUM and MAXIMUM values can be extremes. For loads the maximum value is searched for. In case of a tower clearance the minimum value is of importance. extremeMethod Three extreme data methods can be selected, GLOBAL, BLOCK and POT. Each method selects the maximum and minimum values of the time series in a different way. numberOfBlocks This keyword is used when extremeMethod "BLOCK" is selected. It gives the number of blocks the time series should be divided in. POTthreshold This keyword is only used when extremeMethod "POT" is selected. POTthreshold is a factor with which the standard deviation of the time series is multiplied. The POTthreshold value should be positive. The threshold for the extremeType "MAXIMUM" is the mean + POTthreshold*standard deviation. In case extremeType "MINIMUM" the threshold is the mean - POTthreshold*standard deviation.

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columnNumber This is the column number in the input file of the time series to be analysed. numberOfBins This is the number of wind bins (environmental conditions) that are used in the extreme response analysis. binTable The keyword gives for every wind speed bin the occurrence of that wind speed bin and the number of time series to be used for that bin. binTimeseries This keyword gives for every time series first the bin number it is associated with and next the file name 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 output files. 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 of the theoretical short term distributions. The theoretical short-term extreme response distributions are used to estimate the long term exceedance probability distribution Qlong (equation 14). The results of the long term response distribution 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 short term distributions. For every bin the empirical and theoretical short term distributions are plotted together. 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 plot of Qlong can be used to check the numerically estimated 50-year response given in the extremeMethodlongtermResponse.txt file. The messages, warnings and errors produced (encountered) during the extreme response analysis are stored in a log file logsessionID.dat. This log file is saved in the directory of the WT-ERA executable.

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Figure 4 Example of a short term distribution graph

Figure 5 Example of a long term distribution graph

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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 offshore wind 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 according to 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 weighted moments: definition and relation to parameters of several distributions expressable in inverse form. 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 combination 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 on L-moments. Cambridge Univerity Press. IEC61400-1 (2005): Wind turbine generator systems - Part 1: Safety requirements. International Electrotechnical Commission (IEC), IEC 61400-1, third edition. Landwehr, J.M. and N.C. Matalas (1979): Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resources Research, 15(5):1055–1064. Moriarty, P.J., W.E. Holley and S.P. Butterfield (2003): Extrapolation of extreme and fatigue loads using probabilistic methods. Report NREL/TP-500-34421, National Renewable Energy Laboratory NREL, Golden, Colorado, USA. Peeringa, J.M. (2009): Comparison of extreme load extrapolations using measured and calculated 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 extreme loads. A collection of the 2007 ASME Wind Energy Symposium, at the AIAA Aerospace Meeting. American Institute of Aeronautics and Astronautics, Reno, Nevada, USA. AIAA2007-1221.

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A

L-moments

For the selected distributions the scale parameter α, the location parameter ξ and the shape parameter k are estimated using the L-moment method (Hosking, 1990; Hosking and Wallis, 1997). L-moments are linear combination of probability weighted moments PWM. Greenwood et al. (1979) defined the probability weighted moment PWM for a distribution function F (x) = P (X ≤ x) as: Mp,r,s = E[X p F (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 to minima and βr to maxima. For a distribution that has a quantile function x(u) αr and βr are given by: Z1

αr =

Z1

r

x(u)(1 − u) du

βr =

0

x(u)ur du

(16)

0

Unbiased estimators of the PWM αr and βr are defined by Landwehr and Matalas (1979): 1 n−1 ar = n r

!−1

n X j=r+1

!−1

1 n−1 br = n r

n X j=r+1

!

n−j xj:n r

(17)

!

j−1 xj:n r

(18)

where xj:n is the j th order statistics in a sample of size n Hosking and Wallis (1997) give the L-moments λr as: λ1 λ2 λ3 λ4

= = = =

α0 α0 − 2α1 α0 − 6α1 + 6α2 α0 − 12α1 + 30α2 − 20α3

= β0 = 2β1 − β0 = 6β2 − 6β1 + β0 = 20β3 − 30β2 + 12β1 − β0

(19)

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

λr λ2

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 given by 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.

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B

L-moments for Weibull distribution

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

X −ξ FW eibull (X) = 1 − exp α 

k #

X − ξ, α, k > 0

(21)

Greenwood et al. (1979) defines the probability weighted moments (PWM) for the Weibull distribution as follows: αr = M1,0,r =

αΓ(1 + k1 ) ξ + 1 1 + r (1 + r)1+ k

(22)

Expressing the Weibull distribution in L-moments using equation 19 gives the following expressions for the L-moments: λ1 = ξ + αΓ(1 + k1 ) 1 λ2 = (1 − 2− k )αΓ(1 + k1 ) 1

3−

τ3 =

2(1−3− k )

(23)

−1 k

(1−2 ) 1 1 1 5(1−4− k )−10(1−3− k )+6(1−2− k )

τ4 =

1

(1−2− k )

Greenwood et al. (1979) gives the definition of the parameters of the Weibull distributions in PWM. The shape parameter k is defined as: log(2)

k= log



α0 −2α1 2(α1 −2α3 )

 =

log(2) log



5 τ4 −5τ3 +4



(24)

The scale parameter α is: α=

λ2 (1 − 2

− k1

)Γ(1 + k1 )

(25)

The location parameter ξ is: 1 ξ = λ1 − αΓ(1 + ) k

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(26)

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