Energy Conversion and Management 52 (2011) 15–26

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Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

Probability distributions for offshore wind speeds

Eugene C. Morgan a,*, Matthew Lackner b, Richard M. Vogel a, Laurie G. Baise a

a Dept. of Civil and Environmental Engineering, Tufts University, 200 College Ave., Medford, MA 02155, United Statesb Wind Energy Center, University of Massachusetts, Amherst, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 September 2009Accepted 2 June 2010Available online 14 July 2010

Keywords:Wind speedProbability distributionMixture distributionWind turbine energy outputWeibull distributionExtreme wind

0196-8904/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.enconman.2010.06.015

* Corresponding author. Tel.: +1 617 627 3098.E-mail address: eugene.morgan@tufts.edu (E.C. Mo

In planning offshore wind farms, short-term wind speeds play a central role in estimating various engi-neering parameters, such as power output, extreme wind load, and fatigue load. Lacking wind speed timeseries of sufficient length, the probability distribution of wind speed serves as the primary substitute fordata when estimating design parameters. It is common practice to model short-term wind speeds withthe Weibull distribution. Using 10-min wind speed time series at 178 ocean buoy stations ranging from1 month to 20 years in duration, we show that the widely-accepted Weibull distribution provides a poorfit to the distribution of wind speeds when compared with more complicated models. We compare dis-tributions in terms of three different metrics: probability plot R2, estimates of average turbine power out-put, and estimates of extreme wind speed. While the Weibull model generally gives larger R2 than anyother 2-parameter distribution, the bimodal Weibull, Kappa, and Wakeby models all show R2 values sig-nificantly closer to 1 than the other distributions considered (including the Weibull), with the bimodalWeibull giving the best fits. The Kappa and Wakeby distributions fit the upper tail (higher wind speeds)of a sample better than the bimodal Weibull, but may drastically over-estimate the frequency of lowerwind speeds. Because the average turbine power is controlled by high wind speeds, the Kappa and Wake-by estimate average turbine power output very well, with the Kappa giving the least bias and meansquare error out of all the distributions. The 2-parameter Lognormal distribution performs best for esti-mating extreme wind speeds, but still gives estimates with significant error. The fact that different dis-tributions excel under different applications motivates further research on model selection based uponthe engineering parameter of interest.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The offshore environment shows greater wind energy potentialthan most terrestrial locations [1]. Accurate characterization of off-shore winds will be essential in tapping this tremendous powersource. In wind turbine design and site planning, the probabilitydistribution of short-term wind speed becomes critically impor-tant in estimating energy production. Here short-term wind speedU refers to mean wind speed averaged on the time scale of 10 min,which is much longer than the time scale associated with turbu-lence and gusts. In engineering practice, the average wind turbinepower Pw associated with the probability density function (pdf) ofwind speeds U is obtained from

bPw ¼Z 1

0PwðUÞf ðUÞdU; ð1Þ

where f(U) is the pdf of U and where the turbines power curve Pw(U)describes power output versus wind speed [2]. The largest uncer-

ll rights reserved.

rgan).

tainty in the estimation of Pw lies in the choice of the wind speedpdf f(U), since the turbine manufacturer knows Pw(U) fairly accu-rately. Minimizing uncertainty in wind resource estimates greatlyimproves results in the site assessment phase of planning [3], and thiscan be accomplished by utilizing a more accurate distribution (pdf).

By far the most widely-used distribution for characterization of10-min average wind speeds is the 2-parameter Weibull distribu-tion (W2) [2,4–9]. Sometimes the simple 1-parameter Rayleigh(RAY) distribution offers a more concise fit to a sample, since it isa special case of the W2 [2,10], but ultimately having only a singlemodel parameter makes the RAY much less flexible. Despite thewidespread acceptance of the W2 and RAY distributions, Cartaet al. [11] and others have noted that under different wind regimesother distributions may fit wind samples better. Other distribu-tions used to characterize wind speed include the 3-parameterGeneralized Gamma (GG), 2-parameter Gamma (G2), inverseGaussian, 2-parameter Lognormal (LN2), 3-parameter Beta, singlytruncated from below Normal, distributions derived from the Max-imum Entropy Principle, and bimodal (two component mixture)Weibull model (BIW). Kiss and Janosi [12] recommend the GG forthe ERA-40 dataset (6-hourly mean) after testing the RAY, W2

16 E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26

and LN2 distributions. Simiu et al. [13] use probability plot corre-lation coefficient (PPCC) goodness-of-fit tests to document thatthe RAY distribution provides a poor approximation to hourly windspeeds, but the authors do not advocate an alternative.

Recent studies show that Weibull mixture models (includingBIW) out-perform the W2 distribution [14–16]. However, thesestudies only use a limited number of samples from confined geo-graphic regions, and often explain the better fits of the mixturemodels as unique exceptions to the rule. For example, Akpinarand Akpinar [15] show that mixture models out-perform the Wei-bull distribution and other distributions derived from the Maxi-mum Entropy Principle, but only use 4 samples from a singleregion in Turkey. Carta and Ramirez [16] arrive at similar findingsfor 16 samples in the Canary Islands, as do Jaramillo and Borja [14]for one sample from La Ventosa, Mexico.

One goal of this study is to exploit a much larger and geograph-ically diverse dataset than previous studies in order to evaluatealternative wind speed distributional hypotheses, such as theBIW model. We use data spread out over many different wind re-gimes, albeit all from ocean environments, in order to assess thecommunicability of such models. While the offshore setting forall samples in this study may limit the utility of some of our find-ings for onshore data, we find the use of this data essential inremoving topographic effects, which would complicate a similaranalysis performed on terrestrial wind speed samples.

Another primary goal of this study is to illustrate that finding anoptimal wind speed distributional model is contingent upon theapplication of interest. Measures of goodness-of-fit commonlyused in wind literature (i.e. R2) [17,16,18,12,10,5,11] do not neces-sarily indicate how well a model predicts wind energy parameters.Following Chang and Tu [19], a model validation approach basedon parameter estimation instead of R2 may find theoretical modelsthat are best suited for specific wind energy applications. In prac-tice, one may resort to a variety of models to give optimal predic-tions of in situ conditions when lacking observations.

2. Data

The National Data Buoy Center (NDBC) [20] provides 10-minmean wind speed observations recorded at 178 buoys aroundNorth America (Fig. 1). We analyze the 10-min mean wind speeddata because the International Electrotechnical Commission re-quires the evaluation of this form of wind measurement in turbine

Fig. 1. Locations of buoys providing wind speed observations for this study. Datasource: http://www.ndbc.noaa.gov.

site assessment [21]. Depending on the type of buoy, the anemom-eters on the buoys measure wind speed at either 5 or 10 m abovesea level. Sample sizes (n) range from 4,314 to 889,699, corre-sponding to time spans of approximately 30 days to 20 years. Thisdataset comprises the largest collection of first-order wind speedobservations that we could find readily available online and thatprovides a relatively reliable and consistent measurement ofshort-term wind speeds.

3. Wind speed distributions

We consider some of the most common wind speed distribu-tions used in previous studies. We also consider numerous otherclosely-related distributions, such as the Generalized Rayleigh(GR), Kappa (KAP), and Wakeby (WAK) distributions, that havenot previously been investigated as wind speed models, but aresimple extensions or generalizations of commonly used models.The 2-parameter GR distribution is a generalization of the 1-parameter RAY distribution, and the KAP and WAK distributions(with four and five parameters, respectively) are generalizationsof numerous other distributions.

Since the record lengths of the time series of wind speeds arerather large, differences among parameter estimation methods willnot be nearly as important as differences among distributions. Wepreferentially use maximum likelihood estimators (MLE) becausethey usually yield lower mean square errors (MSE) associated withmodel parameter estimates than method of moments (MOM) esti-mators for the large samples considered here. However, for somedistributions we use MOM, least squares (LS), or a parameter esti-mation method involving L-moments when MLEs are either toocumbersome or unavailable, as discussed in each case. When usingMOM, we calculate the sample mean ðUÞ, standard deviation (S),and skewness (G) as

U ¼ 1n

Xn

i¼1

Ui; ð2Þ

S ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n� 1

Xn

i¼1

ðUi � UÞ2vuut ; ð3Þ

G ¼ 1n

Pni¼1ðUi � UÞ3

S3 ; ð4Þ

where n is the number of observations in U, our random variable forwind speed. The following describes each distribution in our studyand the parameter estimation methods used.

3.1. Rayleigh

The Rayleigh (RAY) distribution is the simplest distributioncommonly used to describe 10-min average wind speeds [2,4,17]because it only has a single model parameter b. Its pdf f(U), andcumulative distribution function (cdf) F(U), are

f ðU; bÞ ¼ U

b2 exp �12

U2

b2

!; ð5Þ

and

FðU; bÞ ¼ 1� exp �12

U2

b2

!; ð6Þ

for U P 0 and b P 0. We estimate the single scale parameter b usingthe MLE:

b ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2n

Xn

i¼1

U2i

vuut : ð7Þ

E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26 17

Often databases report wind speeds as two orthogonal compo-nents (u and v). If u and v are independent and identically distrib-uted Gaussian random variables with means of zero and standarddeviations of b2, U ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2p

follows a Rayleigh distribution [22].However, NDBC does not report wind speeds in separate u, vcomponents.

3.2. Weibull

The Weibull (W2) distribution constitutes the most widely ac-cepted distribution for wind speed [2,4–9]. It is a generalizationof the RAY distribution, and has been shown to fit wind samplesbetter than RAY due to its more flexible form (with an additionalshape parameter) [17,12]. The Weibull pdf and cdf are

f ðU; b;aÞ ¼ bUb�1

abexp � U

a

� �b" #

; ð8Þ

and

FðU; b;aÞ ¼ 1� exp � Ua

� �b" #

; ð9Þ

with U P 0. The RAY is a special case of the W2 with b = 2 [23].MLEs are obtained by solvingPn

i¼1 Ubi ln Ui

� �Pni¼1

Ubi

� 1b� 1

n

Xn

i¼1

ln Ui ¼ 0 ð10Þ

iteratively for the shape parameter estimate b, and then using

a ¼ 1n

Xn

i¼1

Ubi

!1b

ð11Þ

for the scale parameter estimate a.

3.3. Generalized Rayleigh

Another generalization of the RAY distribution is the general-ized Rayleigh (GR) model, which is sometimes referred to as the2-parameter Burr Type X distribution [24]. While the GR has notpreviously been applied to wind speed samples, it is a naturalextension of the RAY distribution (like the W2 distribution). Ithas a pdf

f ðU;a; kÞ ¼ 2ak2U exp½�ðkUÞ2�ð1� exp½�ðkUÞ2�Þa�1; ð12Þ

and cdf

FðU;a; kÞ ¼ ð1� exp½�ðkUÞ2�Þa; ð13Þ

for U > 0. We estimate the shape a and scale k parameters using themodified moment estimators introduced by Kundu and Raqab [24]:

W ¼ 1n

Xn

i¼1

U2i ; ð14Þ

V ¼ 1n

Xn

i¼1

U2i �W2; ð15Þ

V

W2 ¼w0ð1Þ � w0ðaþ 1Þðwðaþ 1Þ � wð1ÞÞ2

; ð16Þ

which requires solving iteratively for a, and

k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwðaþ 1Þ � wð1Þ

W

r; ð17Þ

where w() is the digamma function and w0() is the polygamma func-

tion of order one.

3.4. 3-parameter Weibull

The 3-parameter Weibull (W3) is a generalization of the W2distribution, where the location parameter s establishes a lowerbound (which was assumed to be zero for the W2 distribution).Stewart and Essenwanger [25] found that the W3 fits wind speeddata better than the W2 model. The W3 pdf and cdf are

f ðU; b;a; sÞ ¼ bUb�1

abexp � U � s

a

� �b" #

; ð18Þ

and

FðU; b;a; sÞ ¼ 1� exp � U � sa

� �b" #

; ð19Þ

respectively, for U P s. We estimate the parameters using the mod-ified MLEs recommended by Cohen [26]:Pn

i¼1ðUi � sÞb lnðUi � sÞPni¼1ðUi � sÞb

� 1b� 1

n

Xn

i¼1

lnðUi � sÞ ¼ 0; ð20Þ

a ¼ 1n

Xn

i¼1

ðUi � sÞb !1

b

; ð21Þ

sþ a

n1b

C 1þ 1b

!¼ U1; ð22Þ

where n is the number of observations in sample U, and U1 indicatesthe minimum value of U. First, s is found by iteratively solving Eqs.(20)–(22), and then b is found by iteratively solving Eq. (20). Finally,a is given by Eq. (21) using the estimates of the other two parame-ters [23,26].

3.5. Lognormal

The 2-parameter Lognormal (LN2) pdf and cdf are

f ðU; l;rÞ ¼ 1rU

ffiffiffiffiffiffiffi2pp exp

�ðlnðUÞ � lÞ2

2r2

" #; ð23Þ

and

FðU;l;rÞ ¼ 12þ 1

2erf

lnðUÞ � lrffiffiffi2p

� �; ð24Þ

where erf() is the error function from the Normal distribution, andthe parameters l and r are the mean and standard deviation of thenatural logarithm of U [5,12,17,27]. Since some observations may bezero, we employ MOM estimators [28] instead of MLEs as follows:

l ¼ lnUffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ S2

U2

q0B@

1CA; ð25Þ

r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln 1þ S2

U2

!vuut : ð26Þ

For both the 2- and 3-parameter Lognormal distributions, wechose MOM over MLE because it showed considerably better fitsto the data and because it enables us to handle samples with zeros.The results of Carta et al. [17] corroborate this statement withshort-term wind speed observations from the Canary Islands,showing that MOM for LN2 fits the data better than MLE for 5 of8 samples.

18 E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26

3.6. 3-parameter Lognormal

The 3-parameter Lognormal (LN3) model is a shifted version ofthe LN2, with pdf

f ðU; l;r; fÞ ¼ 1rU

ffiffiffiffiffiffiffi2pp exp

� lnðU � fÞ � lð Þ2

2r2

" #; ð27Þ

and cdf

FðU;l;rÞ ¼ 12þ 1

2erf

lnðU � fÞ � lrffiffiffi2p

� �; ð28Þ

where the location parameter f establishes a lower bound on thedistribution. This distribution has yet to be fit to wind speeds, butwe expect it to perform as well or better than the LN2 model.MOM estimates the three parameters via the following equations:

exp½r2� ¼ 1þ 12

G2 þ Gð4þ G2Þ12

h i� �13

þ 1þ 12

G2 � Gð4þ G2Þ12

h i� �13

� 1; ð29Þ

l ¼ 12

lnS2

exp r2ð Þ exp r2ð Þ � 1½ �

!; ð30Þ

f ¼ U � exp lþ 12r2

� �; ð31Þ

which allow us to keep null values (U = 0) [28].

3.7. Generalized Normal

The Generalized Normal (GNO) model encompasses Lognormaldistributions with both positive and negative skewness, as well asthe Normal distribution [29]. We consider this pdf due to its moregeneral form than the LN2 pdf, which has previously been recom-mended for use with wind speed data [5,12,17,27]. The pdf and cdfare

y ¼ �k�1 lnð1� kðU � nÞ=aÞ; k–0ðU � nÞ=a; k ¼ 0

(; ð32Þ

f ðU; a; k; nÞ ¼ expðky� y2=2Þaffiffiffiffiffiffiffi2pp ; ð33Þ

and

FðU;a; k; nÞ ¼ UðyÞ; ð34Þ

where U() is the standard Normal cdf. The location n, scale a, andshape k parameters relate to standard parameters for the LN2 andLN3 distributions above by

k ¼ �r; ð35Þa ¼ r expðlÞ; ð36Þ

and

n ¼ fþ expðlÞ; ð37Þ

and we estimate these parameters using sample L-moments k1 andk2 and the sample L-moment ratio s3:

k � �s32:0466� 3:6544s2

3 þ 1:8397s43 � 0:2036s6

3

1� 2:0182s23 þ 1:242s4

3 � 0:21742s63

; ð38Þ

a ¼ k2k expð�k2=2Þ1� 2U �k=

ffiffiffi2p� � ; ð39Þ

n ¼ k1 �akð1� expðk2=2ÞÞ: ð40Þ

Hosking and Wallis [29] present this distribution, as well as anexplanation of the sample L-moments and their ratios.

3.8. Gamma

The 2-parameter Gamma (G2) distribution has been applied towind speed data by Sherlock [30], Kaminsky [27], and Auwera et al.[31], for example. This distribution is often referred to as a PearsonType III, however in this paper we make a distinction between thetwo distributions in order to evaluate the effect of the added loca-tion parameter which we include in the Pearson Type III (P3). TheG2 pdf and cdf are

f ðU; b;aÞ ¼ Ua�1exp � U

b

� �baCðaÞ ; ð41Þ

and

FðU; b;aÞ ¼ c a;Ub

� �CðaÞ; ð42Þ

where c() is the lower incomplete gamma function [22]. In order tokeep null wind speeds in the sample, we estimate the shape a andscale b parameters via MOM:

b ¼ S2

U; ð43Þ

a ¼ U2

S2 ð44Þ

as given by Carta et al. [17]. Again, we do not use the MLEs for thisdistribution because they require removing all Ui = 0.

3.9. Pearson type III

The Pearson type III (P3) distribution is a G2 distribution with athird parameter for location, whose use for wind speed was ini-tially advocated by Putnam [32]. Its pdf and cdf are

f ðU; s; b;aÞ ¼ ðU � sÞa�1exp � U�s

b

� �baCðaÞ ; ð45Þ

and

FðU; s;b;aÞ ¼ c a;U � s

b

� �CðaÞ: ð46Þ

Again, to enable us to handle null wind speeds, we estimate thelocation s, scale b, and shape a parameters using MOM [33]:

s ¼ U � 2SG

� �; ð47Þ

b ¼ SG2; ð48Þ

a ¼ 4

G2 : ð49Þ

The MLEs for this distribution require the removal of null windspeeds [34].

3.10. Log Pearson type III

The log Pearson type III (LP3) distribution describes a randomvariable whose logarithms follow a P3 distribution [33]. Thus, wefit the LP3 distribution to the wind speed sample U by transform-ing U by the natural logarithm and fitting the P3 distribution. TheLP3 distribution is one of the most flexible 3-parameter distribu-tions that exists. Thus, we sought to evaluate its performance, eventhough we could not find any previous studies which evaluate itsuse for wind speeds.

E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26 19

3.11. Generalized Gamma

The 3-parameter generalized Gamma (GG) distribution includesthe W2, P3, and other distributions as special cases, and has beenproposed as a wind speed model by several authors [12,17,31].Its pdf and cdf are

f ðU; b;a;pÞ ¼ jpjUpa�1exp � U

b

� �p

bpaCðaÞ ; ð50Þ

and

FðU; b;a;pÞ ¼ c a;Ub

� �p� �: ð51Þ

In order to avoid removing null wind speeds from the samplesbefore fitting the model (as required by the MLEs), we estimatethe scale b and shape parameters a, p by MOM. First, we solvethe following equation iteratively for a:

�jGj ¼ w00ðaÞ=ðw0ðaÞÞ3=2; ð52Þ

and then find the remaining parameters with the followingequations:

p ¼ ðw0ðaÞÞ1=2=S; ð53Þ

b ¼ expðU � wðaÞ=pÞ; ð54Þ

where w() is the digamma function, and w0() and w

00() are the polyg-

amma functions of order one and two [35].

3.12. Kappa

The 4-parameter Kappa (KAP) distribution is a generalization ofmany other distributions and includes as special cases the General-ized Logistic, Generalized Extreme Value, and Generalized Paretodistributions [29]. Its pdf and cdf are

f ðU; n;a; k;hÞ ¼ a�1ð1� kðU � nÞ=aÞ1=k�1ðFðUÞÞ1�h; ð55Þ

and

FðU; n;a; k;hÞ ¼ ð1� hð1� kðU � nÞ=aÞ1=kÞ1=h; ð56Þ

respectively. We estimate the location n, scale a, and two shapeparameters k, h by solving the following relationships betweenthese parameters and the sample L-moments k1, k2 and L-momentratios s3, s4:

k1 ¼ nþ að1� g1Þ=k; ð57Þk2 ¼ aðg1 � g2Þ=k; ð58Þs3 ¼ ð�g1 þ 3g2 � 2g3Þ=ðg1 � g2Þ; ð59Þs4 ¼ ð�g1 þ 6g2 � 10g3 þ 5g4Þ=ðg1 � g2Þ; ð60Þ

where

gr ¼rCð1þkÞCðr=hÞ

h1þkCð1þkþr=hÞ; h > 0

rCð1þkÞCð�k�r=hÞð�hÞ1þkCð1�r=hÞ

; h < 0

8><>: : ð61Þ

Given s3 and s4, we solve Eqs. (57) and (58) iteratively for k andh [29].

3.13. Wakeby

The 5-parameter Wakeby (WAK) distribution is a generalizationof many existing pdfs (including LN2 and P3), but the WAK pdf canexhibit shapes that other distributions cannot mimic [36]. It hasbeen considered as a super population or parent distribution inmany previous studies in the earth sciences [37,38] and was orig-

inally introduced by Harold Thomas [39,40] who lived on Wakebypond on Cape Cod. The pdf and cdf are not explicitly defined, andthe WAK distribution is most easily defined by its quantilefunction:

F�1ðFðUÞ; n;a; b; c; dÞ ¼ nþ abð1� ð1� FðUÞÞbÞ

� cdð1� ð1� FðUÞÞ�dÞ: ð62Þ

Parameter estimation follows an algorithm outlined in Hoskingand Wallis [29] and Hosking [41] which relies on sample L-mo-ments. We implement this distribution and its parameter estima-tion using the package lmomco [42] for the statistical software R[43].

3.14. Bimodal Weibull mixture

The bimodal Weibull mixture distribution (BIW) is a mixture oftwo W2 distributions. The BIW model exhibits two different modesand has been recommended for use with wind speed data by Cartaet al. [17], Akpinar and Akpinar [15], Carta and Ramirez [18], andJaramillo and Borja [14]. The pdf and cdf are

f ðU; b1;a1; b2;a2;xÞ ¼ xb1Ub1�1

ab11

exp � Ua1

� �b1" #

þ ð1�xÞb2Ub2�1

ab22

exp � Ua2

� �b2" #

; ð63Þ

and

FðU; b1;a1;b2;a2;xÞ ¼ x 1� exp � Ua1

� �b1" # !

þ ð1�xÞ 1� exp � Ua2

� �b2" # !

; ð64Þ

where the shape b and scale a parameters have subscripts corre-sponding to the two different modes, and the mixing parameter xdescribes the proportion of observations belonging to the first com-ponent. We first estimate b1 and a1 using the MLE for the W2, andassign b2 = 1, a2 = 1, and x = 0.5. Then, using these estimations asour initial parameters, we optimize the five parameters to givethe LS fit of the model cdf to the empirical (wind sample) cdf. Theleast squares method is usually applied to the cdf [17].

4. Results and discussion

We evaluate the goodness-of-fit of the various fitted distribu-tions to the buoy wind speed samples using the coefficients ofdetermination associated with the resulting P–P probability plots,an example of which is presented in Fig. 2. R2 is used widely forgoodness-of-fit comparisons and hypothesis testing because itquantifies the correlation between the observed probabilities andthe predicted probabilities from a distribution. Here we employP–P probability plots instead of Q–Q probability plots (which plotpredicted quantiles versus observed quantiles) because P–P plotsdo not require specially designed quantile unbiased plotting posi-tions for each pdf considered. Instead, we use the Weibull plottingposition F(U) = i/(n + 1), where i=1,. . ., n, in all cases because it al-ways gives an unbiased estimate of the observed cumulative prob-abilities regardless of the underlying distribution considered, anddoes not estimate the highest observed wind speed as the maxi-mum possible wind speed (i.e. F(Un) – 1).

A larger value of R2 indicates a better fit of the model cumula-tive probabilities bF to the empirical cumulative probabilities F.We determine R2 as:

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Observed Probability

W2

Prob

abili

tyR^2 = 0.9882

Fig. 2. Example P–P probability plot from buoy 41040. Red line denotes a 1:1relationship between the model-predicted probabilities and the observed proba-bilities (i.e. R2 = 1). (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

20 E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26

R2 ¼Pn

i¼1ðbF i � FÞ2Pni¼1ðbF i � FÞ2 þ

Pni¼1ðFi � bF iÞ2

; ð65Þ

where F ¼ 1n

Pni¼1bF i. The estimated cumulative probabilities bF are

obtained from the various models’ cdfs described previously. Wechose R2 as the goodness-of-fit measure for ease of comparison be-tween distributions, and also because of its widespread use in sim-ilar studies [5,17,44].

Fig. 3 ranks the distributions (from left to right) in terms of theirgoodness-of-fit as measured by their median R2 values. Althoughthe W2, the most widely accepted distribution for modeling windspeeds, performs the best out of all 2-parameter distributions andeven better than some three-parameter distributions, it is not themodel with the best goodness-of-fit in terms of R2. The BIW, KAP,and WAK all show significant improvement in fit over the other pdfs,with the BIW exhibiting the highest R2 values and the highest med-ian R2. In comparison to the WAK, the BIW model exhibits a widervariability in R2 values and lower R2 values for the poorly fit sites.

4.1. The probability distribution of offshore wind speeds

The poor performance of the W2 distribution for these samplescompared to some of the more complex models can be further ex-plained using moment diagrams as shown in Fig. 4. The theoreticalrelationships for coefficient of variation ðCv ¼ S=UÞ and skewness

G2 LN2 RAY LP3 GR LN3 G

0.95

0.96

0.97

0.98

0.99

1.00

R2

Fig. 3. Boxplots of R2 values for the distribution fits to the buoy wind speed samples. Disline in each box).

(c) of the W2 distribution are obtained solely in terms of the shapeparameter b from the following relations [23]:

Cv ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC 1þ 2

b

� �� C 1þ 1

b

� �� �2r

C 1þ 1b

� � ; ð66Þ

and

c ¼C 1þ 3

b

� �� C 1þ 1

b

� �3

C 1þ 2b

� �� C 1þ 1

b

� �2� �3=2

�3C 1þ 1

b

� �C 1þ 2

b

� �� C 1þ 1

b

� �2� �

C 1þ 2b

� �� C 1þ 1

b

� �2� �3=2 : ð67Þ

Here moment diagrams are constructed for both synthetic andempirical samples. Fig. 4 illustrates that the coefficient of variationand skewness for synthetic W2 samples with the same nj, aj, and bj

as the buoy samples closely mimic the W2 theoretical relationship,whereas the moments of the actual buoy samples do not. Fig. 4provides a clear documentation of how poorly the tail behaviorof the empirical samples is mimicked by the W2 distribution.The scatter about the theoretical relationships in Fig. 4b is shownto arise from real distributional departures alone, and not samplingvariability.

While we cannot compare the buoy sample moments and BIWsynthetic sample moments to a theoretical BIW moment relation-ship (because theoretical moment relationships are incalculable forthis model), we can compare BIW R2 values for the buoy samples(from Fig. 3) to BIW R2 values for synthetic BIW samples. Fig. 5 pro-vides a comparison of the goodness-of-fit of the W2 and BIW dis-tributions to both the real buoy samples and the synthetic buoysamples. For each real buoy sample, we randomly generate 2 syn-thetic samples: one from the W2 model and one from the BIWmodel. We use the same parameters and sample sizes as the actualsamples, and then fit the two distributions to their respective syn-thetic samples using the same parameter estimation methods de-scribed previously. The W2 and BIW models fit the synthetic W2and BIW samples almost exactly, as expected, with any departuresof R2 values away from 1 reflecting sampling error alone. If thebuoy samples truly came from a W2 or BIW parent distribution,they would have R2 values comparable to those of the W2 orBIW synthetic samples.

G PE3 W2 GNO W3 WAK KAP BIW

tributions are ordered from left to right by ascending median R2 value (heavy black

−0.5 0.0 0.5 1.0 1.5

0.3

0.4

0.5

0.6

0.7

Synthetic Weibull Samples

Skewness

Cv

Syn. WeibullTheoretical Weibull

−0.5 0.0 0.5 1.0 1.5

0.3

0.4

0.5

0.6

0.7

Real Buoy Samples

Skewness

Cv

Buoy Wind SpeedsTheoretical Weibull

Fig. 4. Comparison of moments between synthetic W2 samples (left) with the real buoy samples (right). The red line represents the theoretical relationship betweenmoments for the W2 distribution. Note the wide dispersion of points around the red line for the buoy samples versus the proximity to the red line that the synthetic sampleshave. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0.975

0.980

0.985

0.990

0.995

1.000

R2

a

W2 Real W2 Synthetic BIW Real BIW Synthetic

0.975

0.980

0.985

0.990

0.995

1.000

b

Fig. 5. Boxplots of R2 values for (a) W2 fits to the real buoy samples (left) and to synthetic samples of same size and parameters (right), and (b) BIW fits to the real buoysamples (left) and to synthetic samples of same size and parameters (right). The greater similarity between the real and synthetic BIW R2 values (versus those for W2)illustrates that the buoy wind speed samples are better modeled with a BIW distribution.

BIW mixing parameter value

Freq

uenc

y

0.5 0.6 0.7 0.8 0.9 1.0

0

20

40

60

80

10068 values (out of 178) = 1

Fig. 6. Distribution of BIW mixing parameters x from fits to buoy samples. Valuesof 1 indicate W2 distributed samples.

E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26 21

The departures of the R2 values from unity for the W2 real sam-ples are much too great to be explained by random sampling erroralone (Fig. 5a). The discrepancy between the two boxplots in Fig. 5aresults from the inadequacy of the W2 distribution to accuratelymodel the buoy wind speed samples, and cannot be explained byrandom sampling error. We observe that the BIW model does amuch better job at fitting the buoy wind speed samples (Fig. 5b)than the W2 model.

4.2. BIW mixing parameter

Further comparison of the W2 and BIW fits can be found in themixing parameter x of the BIW distribution, which tells us aboutthe significance of the second mode in the buoy samples. x = 0.5indicates that the two Weibull modes are comparable in size,whereas a value of 1 indicates that no second mode exists. We con-strain x between 0.5 and 1 because we interpret it as the weight ofthe ‘‘strongest” mode in the model. That is, a BIW model withx < 0.5 can be considered as an alternative model with xalt = 1 �xand with the shape and scale parameters switched between thetwo modes (i.e. b1, a1, b2, and a2 for the initial model respectivelybecome b2,alt, a2,alt, b1,alt, and a1,alt for the alternative model).

Fig. 6 illustrates the distribution of mixing parameters for theBIW fits to the buoy samples. 68 of the 178 samples have x = 1,which implies that they follow a W2 distribution. For these 68

samples, any improvement in R2 from W2 to BIW (e.g. Fig. 7, right-side histogram) only highlights the benefit of using least squaresparameter estimation instead of MLE for W2. The remaining samples

22 E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26

display a wide range of bimodality, with x having approximatelyuniform frequency from 0.5 to 0.95 (Fig. 6). Forty-eight samples haveclose to 1 (that is 0.95 6x < 1), indicating diminutive second modes.Thus, a little more than 1/3 of the samples (62 samples) show varyingdegrees of bimodality (0.5 6x < 0.95).

Mapping x indicates some general spatial structure in the de-gree of bimodality of the wind speed distributions (Fig. 7; seehttp://engineering.tufts.edu/cee/geohazards/peopleMorgan.asp fora Google Earth file showing plots for all buoys). We can see thatthe California coast has a majority of low x values, where theBIW substantially improves the fit over W2, but higher values ofx also exist in that region. The east coast of North America shows

130˚W

130˚W

125˚W

125˚W

120˚W

120˚W 115˚W

35˚N

40˚N

45˚NBuoy 46063

0 5 10 15

0.00

0.04

0.08

0.12

Wind Speed (m/s)

Den

sity

W2 , R^2 = 0.9862BIW , R^2 = 1.0000

Mixing Para

0.5 ≤ ω 0.7 ≤ ω 0.9 ≤ ω

Fig. 7. Map of BIW mixing parameters. Histograms with fitted W2 and BIW distributionsmuch better fit given by the BIW model (x = 0.53). Right: Massachusetts buoy data wit

GNO PE3 G2 LN2 RAY W3 W2

−10

−5

0

5

10

% E

rror

in P

w (a)

GNO LN2 PE3 G2 BIW LP3 W3

−10

−5

0

5

10

% E

rror

in P

w (b)

Fig. 8. Boxplots of percent error between time series (empirical) and model-predicted amean square error from left to right, and (b) ranks by descending bias.

predominantly high values of x (with little difference between theBIW and W2 fits), with lower values interspersed. Thus, no trulyhomogenous region exists, and one cannot say for which regionsto use which distribution. Therefore, if one is concerned with over-all goodness-of-fit, the conservative practice would be to use theBIW as the default wind speed distribution.

4.3. Model performance in terms of average power output

Here we evaluate how well each fitted pdf performs in terms ofits ability to agree with the empirical average wind turbine poweroutput computed using

meter

< 0.7

< 0.9

≤ 1.0

85˚W

85˚W

80˚W

80˚W

75˚W

75˚W 70˚W

25˚N

30˚N

35˚N35˚N

40˚N

45˚N

Buoy 44028

0 5 10 15 20

0.00

0.04

0.08

0.12

Wind Speed (m/s)

Den

sity

W2 , R^2 = 0.9985BIW , R^2 = 0.9993

exemplify two mixing parameter end-member cases. Left: California sample with ah little difference between W2 and BIW (x = 1).

LP3 GG GR BIW LN3 WAK KAP

LN3 RAY W2 GR GG WAK KAP

verage wind machine power output for each distribution. (a) Ranks by descending

E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26 23

Pw ¼1n

Xn

i¼1

PwðUiÞ ð68Þ

for each buoy sample with size n. The model estimates of the poweroutput bPw are obtained from Eq. (1), and we use the regressionequation for the Vestas V47-666 KW turbine power curve Pw(U)provided by Chang and Tu [19]. Because turbines do not exist at

Buoy 46086

Den

sity

0 5 10 15

0.00

0.05

0.10

0.15

010

020

030

040

050

060

0

W2 , P = 59.6BIW , P = 53.3KAP , P = 59WAK , P = 58.8P(U) , P = 58.7

a

Buoy 51001

Den

sity

0 5 10 15 20

0.00

0.05

0.10

0.15

010

020

030

040

050

060

0

W2 , P = 196.6BIW , P = 198.7KAP , P = 203.8WAK , P = 204.4P(U) , P = 203.7

b

Buoy cdrf1

Wind Speed (m/s)

Den

sity

0 5 10 15 20

0.00

0.05

0.10

0.15

0.20

010

020

030

040

050

060

0

W2 , P = 48.2BIW , P = 41.2KAP , P = 47.7WAK , P = 47.8P(U) , P = 47.5

c

Fig. 9. Histograms with fitted W2, BIW, and WAK pdfs. Power curves P(U) are inserted ooutput.

any of our buoy locations, and because we generally have largesamples in this study, we assume the empirical time series estimateas our ‘‘true” capacity factor. Chang and Tu [19] find that the timeseries approach generally estimates capacity factors closer to theactual measured (from the turbines) capacity factors than the distri-butional approach.

Buoy 42057

0 5 10 15 20 25 30

0.00

0.05

0.10

0.15

010

020

030

040

050

060

0

Pow

er (

KW

)

W2 , P = 187.4BIW , P = 181.9KAP , P = 187.1WAK , P = 187.6P(U) , P = 185.9

d

Buoy 46014

0 5 10 15 20

0.00

0.02

0.04

0.06

0.08

0.10

0.12

010

020

030

040

050

060

0

Pow

er (

KW

)

W2 , P = 156.3BIW , P = 162.8KAP , P = 164.9WAK , P = 165P(U) , P = 165.1

e

Buoy 46081

Wind Speed (m/s)

0 5 10 15 20

0.00

0.05

0.10

0.15

0.20

0.25

0.30

020

040

060

0

Pow

er (

KW

)

W2 , P = 130BIW , P = 141.1KAP , P = 142.3WAK , P = 139.6P(U) , P = 143.1

f

nly to illustrate that higher wind speeds matter most for estimating average power

24 E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26

Fig. 8 presents the percent error associated with the averagepower estimates for all the pdfs considered previously. Here thepercent error is defined as

Pw � bPw

Pw

� 100%: ð69Þ

Because the power curve weights the influence that probabili-ties of higher speeds have on bPw, we see a different performanceof the distributions in Fig. 8 than we did in Fig. 3, which ranksbased on an unweighted goodness-of-fit to all wind speeds in asample. In Fig. 8a, we rank the distributions byMSE ¼ 1

N

PNj¼1ð

bPw;j � Pw;jÞ2, where N = 178 is the number of buoysamples, and in Fig. 8b we rank by bias ¼ 1

N

PNj¼1ð

bPw;j � Pw;jÞ.Fig. 8 illustrates that the BIW model produces average power esti-mates with lower MSE than the W2 model (16.8 versus 34.8), butthese estimates have more bias (�3.6 versus �1.8). It should benoted here that BIW under-predicts the mean power at every buoy.Also note here that the W2 model does not perform the best out ofthe 2-parameter models, like it did when comparing R2. Instead,the GR distribution gives slightly better estimates of average poweroutput than the W2 distribution.

Fig. 8 shows that the KAP distribution yields the least MSE andbias in estimation of mean power output (0.3 and 0.1, respec-tively), followed closely by the WAK model (0.4 and 0.2). Inspect-ing the fits of these two distributions to the wind speed histogramsin Fig. 9, we begin to understand why the KAP and WAK performworse than BIW in terms of R2 and better than BIW in terms ofpower output. While the BIW provides a good fit to the entire his-togram, the KAP and WAK models fit better at the upper tail (i.e.,

LP3 GG KAP WAK BIW RAY

LP3 GG KAP WAK BIW RAY

LP3 GG KAP WAK BIW W2

1000

500010000

50000100000

5000001000000

a

100

1000

10000

100000

1000000

|E[x

] −

Tot

al E

xcee

danc

es (

x)|

b

10

1000

100000

c

Fig. 10. Total number of observations from all 178 buoys exceeding the (100 � p)th perc1.9 � 10�7, respectively. Distributions are ranked from left to right by increasing agreem

the wind speeds that matter most in bPw: 4–25 m/s for this powercurve), but then diverge from the samples at low wind speeds.The WAK distribution grossly over-predicts the frequency of nullwind speeds in every case and sometimes over-predicts the high-est density (e.g., Fig. 9a–d). The KAP either over-predicts (Fig. 9a,c and d) or under-predicts (Fig. 9b and e) the peak probability den-sity, and sometimes gives wildly inaccurate null value densities(Fig. 9f), which accounts for some of the lower KAP R2 values. Thus,the agreement between the models’ average power estimates bPw

and the empirical average power estimates Pw depends on howwell the models fit the wind speeds between the cut-in and cut-out values (4 m/s and 25 m/s, respectively) of the power curve,and this predictive capability does not necessarily correspond tothe models’ R2 values.

4.4. Model performance in terms of extreme wind speeds

In order to evaluate how well the models estimate extremewind speeds, we compare the number of observations that eachmodel predicts will exceed various percentiles (total exceedancesx) to the expected number of exceedances E[x], an analysis basedon simple probability theory and previously applied to flood flows[45]. For various low probabilities p = 1.9 � 10�5, 1.9 � 10�6,1.9 � 10�7, we generate model-predicted extreme events fromthe quantile functions F�1(1 � p), where the p is the probabilitythat the event will be exceeded by any observation Ui. We chosethese particularly low p values because with such large samplesizes, we want to examine sufficiently rare events. Predicting ex-treme wind speeds plays an important role in turbine design,

W2 W3 GR GNO PE3 LN3 LN2

W2 W3 GR GNO PE3 LN3 LN2

RAY W3 GR GNO PE3 LN3 LN2

p = 1.9e−05E[x] = 1375

p = 1.9e−06E[x] = 137

p = 1.9e−07E[x] = 13

entile of each fitted distribution. a, b, and c correspond to p = 1.9 � 10�5, 1.9 � 10�6,ent with E[x] (i.e. decreasing jE[x] � Total exceedances (x)j).

E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26 25

where one normally considers the 50-year return gust value. Tradi-tionally determined via annual maximum data, this extreme gustvalue can be computed as 1.4 times the 50-year return event deter-mined from the distribution of 10-min mean values [4].

For choosing a suitable model for estimation of extreme designevents, such as the wind speed with an average return period of 50years, we recommend fitting extreme value distributions, such asthe Generalized Extreme Value distribution to series of annualmaximum wind speeds. We have not considered the predictionof such events here. Instead, in this section, we simply evaluatethe ability of various models to capture the extreme tails of 10-min wind speeds. Such analyses can be related to annual frequencyanalyses and design return period estimation, but we have not ad-dressed those issues here.

We count the number of observations Ui in each sample exceed-ing the extreme quantiles generated by each model. These quan-tiles are generated for each sample and each distribution usingthe fitted parameters from the previous sections. For each distribu-tion, we sum these exceedances over all samples to give the totalnumber of exceedances x used in Fig. 10. The expected numberof exceedances is E[x] = mp for each p, where m is the total numberof 10-min observations (�72 million, or �1376 site-years). Thus,we expect higher wind speed events (lower p) to be exceeded few-er times and correspond to a decrease in E[x].

Fig. 10 compares the absolute difference between the distribu-tions total number of exceedances x and E[x] because the bestmodel for estimating an extreme wind speed from 10-min averagedata should be in best agreement with E[x]. We see that for all p, alldistributions perform roughly the same relative to each other, withthe LN2 distribution performing the best. However, the errors(jE[x] � xj) associated with the LN2 model are still significant(1148, 78, and 15 exceedances, for Fig. 10a, b, and c, respectively;which come out to 83%, 57%, and 115% of their respective E[x] val-ues). Unfortunately, constructing confidence intervals for the esti-mated number of exceedances as Vogel et al. [45] have done ischallenging in our case because our observations cannot be consid-ered independent.

5. Conclusion

In terms of characterizing entire wind speed samples, the BIW,KAP, and WAK distributions offer significantly better fits than othermodels, with the BIW giving the highest R2 values. These modelsare all quite complex, with each having 4–5 parameters. Consider-ing only the simpler models (2–3 parameters), the traditional W2distribution generally gives larger R2 values than all the 2-param-eter models, and some of the 3-parameter models. Using manylarge samples spread over a wide geographic region, we show thatthe superior fit of the BIW to the data (versus the W2 distribution)cannot be explained by sampling error. BIW synthetic samplesmore closely resemble the real data than W2 synthetic samples.The BIW model led to better goodness-of-fit than the W2 at alllocations, to varying degrees. Mapping the BIW mixing parametershows the degree to which the BIW differs from the W2 at all loca-tions, and we could not discern any regions showing homogeneityof the mixing parameter.

When predicting power output, the BIW no longer performs aswell as other distributions. Because the BIW model does not neces-sarily model the wind speeds between the cut-in and cut-out val-ues of the power curve as well as the KAP or WAK, it led toconsiderably greater errors in average wind turbine power outputestimates. Among all the models considered, the KAP distributionpredicted average power output with the lowest MSE and bias.Interestingly, the LN2 distribution yielded the best estimate of ex-treme wind speeds, but still exhibited large errors.

The fact that different distributions perform better for differentmetrics suggests that one’s selection of model depends on the windenergy application. While the KAP model best estimates averagepower output, there may be yet another distribution that is bestwhen estimating fatigue loads, for example. If one’s goal is to ob-tain that distribution which performs best under a particular appli-cation, further analyses are needed of the type performed here thatvalidate models based on average power output and extreme windspeed.

Acknowledgements

We thank the Wittich Foundation for supporting this researchthrough the Peter and Denise Wittich Family Fund for AlternativeEnergy Research at Tufts University. Additional funding was pro-vided by NSF Grant OISE-0530151.

References

[1] US Dept. of Energy, Wind powering America; September 2009. <http://www.windpoweringamerica.gov/>.

[2] Manwell JF, McGowan JG, Rogers AL. Wind energy explained: theory, designand application; 2002.

[3] Lackner MA, Rogers AL, Manwell JF. Uncertainty analysis in MCP-based windresource assessment and energy production estimation. In: Journal of solarenergy engineering – transactions of the ASME: AIAA 45th aerospace sciencesmeeting and exhibit, vol. 130; 2008.

[4] Burton T, Sharpe D, Jenkins N, Bossanyi E. Wind energy handbook. Wiley; 2001.[5] Garcia A, Torres JL, Prieto E, Francisco AD. Fitting wind speed distributions: a

case study. Solar Energy 1998;62(2):139–44.[6] Harris RI. Generalised pareto methods for wind extremes. useful tool or

mathematical mirage? J Wind Eng Ind Aerodyn 2005;93(5):341–60.doi:10.1016/j.jweia.2005.02.004.

[7] Harris RI. Errors in gev analysis of wind epoch maxima from Weibull parents.Wind Struct 2006;9(3):179–91.

[8] Ramirez P, Carta JA. Influence of the data sampling interval in the estimation ofthe parameters of the Weibull wind speed probability density distribution: acase study. Energy Convers Manage 2005;46(15–16):2419–38.

[9] Hennessey JP. Some aspects of wind power statistics. J Appl Meteorol1977;16(2):119–28.

[10] Celik AN. A statistical analysis of wind power density based on the Weibull andRayleigh models at the southern region of Turkey. Renew Energy2004;29(4):593–604.

[11] Carta JA, Ramirez P, Velazquez S. Influence of the level of fit of a densityprobability function to wind-speed data on the WECS mean power outputestimation. Energy Convers Manage 2008;49(10):2647–55.

[12] Kiss P, Janosi IM. Comprehensive empirical analysis of ERA-40 surface windspeed distribution over Europe. Energy Convers Manage 2008;49(8):2142–51.doi:10.1016/j.enconman.2008.02.003.

[13] Simiu E, Heckert N, Filliben J, Johnson S. Extreme wind load estimates based onthe Gumbel distribution of dynamic pressures: an assessment. Struct Safety2001;23(3):221–9.

[14] Jaramillo OA, Borja MA. Wind speed analysis in La Ventosa, Mexico: a bimodalprobability distribution case. Renew Energy 2004;29(10):1613–30.

[15] Akpinar S, Akpinar EK. Estimation of wind energy potential using finitemixture distribution models. Energy Convers Manage 2009;50(4):877–84.

[16] Carta JA, Ramirez P. Use of finite mixture distribution models in the analysis ofwind energy in the Canarian Archipelago. Energy Convers Manage2007;48(1):281–91.

[17] Carta JA, Ramirez P, Velazquez S. A review of wind speed probabilitydistributions used in wind energy analysis Case studies in the CanaryIslands. Renew Sustain Energy Rev 2009;13(5):933–55.

[18] Carta JA, Ramirez P. Analysis of two-component mixture Weibull statistics forestimation of wind speed distributions. Renew Energy 2007;32(3):518–31.

[19] Chang T-J, Tu Y-L. Evaluation of monthly capacity factor of WECS usingchronological and probabilistic wind speed data: a case study of Taiwan.Renew Energy 2007;32(12):1999–2010.

[20] NDBC. Noaa’s national data buoy center; 2009. <http://www.ndbc.noaa.gov/>[07.01.09].

[21] IEC. Wind turbines – part 1: design requirements. Tech. rep. 61400-1 Ed.3,International Electrotechnical Commission; 2005.

[22] Krishnamoorthy K. Handbook of statistical distributions withapplications. New York: Chapman & Hall; 2006.

[23] Murthy DNP, Xie M, Jiang R. Weibull models 2004.[24] Kundu D, Raqab M. Generalized Rayleigh distribution: different methods of

estimations. Comput Stat Data Anal 2005;49(1):187–200. doi:10.1016/j.csda.2004.05.008.

[25] Stewart DA, Essenwanger OM. Frequency-distribution of wind speed near-surface. J Appl Meteorol 1978;17(11):1633–42.

26 E.C. Morgan et al. / Energy Conversion and Management 52 (2011) 15–26

[26] Cohen AC. Multi-censored sampling in 3 parameter Weibull distribution.Technometrics 1975;17(3):347–51.

[27] Kaminsky FC. Four probability densities (log-normal, gamma, Weibull, andRayleigh) and their application to modelling average hourly wind speed. In:Proceedings of the international solar energy society; 1977. p. 19.6–19.10.

[28] Stedinger JR. Fitting log normal-distributions to hydrologic data. Water ResourRes 1980;16(3):481–90.

[29] Hosking J, Wallis J. Regional frequency analysis: an approach based on L-moments. Cambridge University Press; 1997.

[30] Sherlock RH. Analyzing winds for frequency and duration. Meteorol Monogr1951;1:42–9.

[31] Auwera L, Meyer F, Malet L. The use of the Weibull three-parameter model forestimating mean wind power densities. J Appl Meteorol 1980;19:819–25.

[32] Putnam PC. Power from the wind. New York: D. Van Nostrand Company, Inc.;1948.

[33] Griffis VW, Stedinger JR. Log-Pearson type 3 distribution and its application inflood frequency analysis. II: parameter estimation methods. J Hydrol Eng2007;12(5):492–500.

[34] Hirose H. Maximum likelihood parameter estimation in the three-parametergamma distribution. Comput Stat Data Anal 1995;20(4):343–54.

[35] Stacy EW, Mihram GA. Parameter estimation for a generalized Gammadistribution. Technometrics 1965;7(3):349–58.

[36] Houghton JC. Birth of a parent – Wakeby distribution for modeling flood flows.Water Resour Res 1978;14(6):1105–9.

[37] Vogel RM, Mcmahon TA, Chiew FHS. Floodflow frequency model selection inAustralia. J Hydrol 1993;146(1–4):421–49.

[38] Guttman NB, Hosking JRM, Wallis JR. Regional precipitation quantile values forthe continental United States computed from L-moments. J Climate1993;6(12):2326–40.

[39] Landwehr JM, Matalas NC, Wallis JR. Estimation of parameters and quantiles ofWakeby distributions. 1. Known lower bounds. Water Resour Res1979;15(6):1361–72.

[40] Landwehr JM, Matalas NC, Wallis JR. Estimation of parameters and quantiles ofWakeby distributions. 2. Unknown lower bounds. Water Resour Res1979;15(6):1373–9.

[41] Hosking JRM. Fortran routines for use with the method of L-moments, version3. Research report RC 20525, IBM Research Division; 1996.

[42] Asquith WH. Lmomco: L-moments, trimmed L-moments, L-comoments, andmany distributions, R package version 0.96.3; 2009.

[43] R Development Core Team. R: a language and environment for statisticalcomputing, R Foundation for Statistical Computing, Vienna, Austria; 2009.<http://www.R-project.org>.

[44] Ramirez P, Carta JA. The use of wind probability distributions derived from themaximum entropy principle in the analysis of wind energy: a case study.Energy Convers Manage 2006;47(15–16):2564–77.

[45] Vogel RM, Thomas WO, Mcmahon TA. Flood-flow frequency model selection inSouthwestern United States. J Water Resour Plan Manage – ASCE1993;119(3):353–66.