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Energy Conversion and Management 49 (2008) 1309–1320

A joint probability density function of wind speed and directionfor wind energy analysis

Jose A. Carta a,*, Penelope Ramırez b, Celia Bueno b

a Department of Mechanical Engineering, University of Las Palmas de Gran Canaria, Campus de Tafira s/n,

35017 Las Palmas de Gran Canaria, Canary Islands, Spainb Department of Renewable Energies, Technological Institute of the Canary Islands, Pozo Izquierdo Beach s/n, 35119 Santa Lucia, Gran Canaria,

Canary Islands, Spain

Received 21 May 2007; accepted 6 January 2008Available online 4 March 2008

Abstract

A very flexible joint probability density function of wind speed and direction is presented in this paper for use in wind energy analysis.A method that enables angular–linear distributions to be obtained with specified marginal distributions has been used for this purpose.For the marginal distribution of wind speed we use a singly truncated from below Normal–Weibull mixture distribution. The marginaldistribution of wind direction comprises a finite mixture of von Mises distributions. The proposed model is applied in this paper to winddirection and wind speed hourly data recorded at several weather stations located in the Canary Islands (Spain). The suitability of thedistributions is judged from the coefficient of determination R2.

The conclusions reached are that the joint distribution proposed in this paper: (a) can represent unimodal, bimodal and bitangentialwind speed frequency distributions, (b) takes into account the frequency of null winds, (c) represents the wind direction regimes in zoneswith several modes or prevailing wind directions, (d) takes into account the correlation between wind speeds and its directions. It cantherefore be used in several tasks involved in the evaluation process of the wind resources available at a potential site. We also concludethat, in the case of the Canary Islands, the proposed model provides better fits in all the cases analysed than those obtained with themodels used in the specialised literature on wind energy.� 2008 Elsevier Ltd. All rights reserved.

Keywords: Circular–linear distributions; Wind direction probability distributions; Wind speed probability distributions; Wind rose; Wind power density;Wind turbine energy output

1. Introduction

As pointed out by Koeppl [1], certain wind characteris-tics are important for the evaluation of wind resources andthe design and performance of wind turbines. Theseinclude wind speed and direction probability distributionfunctions. The use of continuous wind speed probabilitydensity functions (pdf) is common [1–5]. However, lesscommon is the use of the proposed model of continuouswind direction probability density functions [6–12].

0196-8904/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2008.01.010

* Corresponding author. Tel.: +34 928 45 14 83; fax: +34 928 45 14 84.E-mail address: jcarta@dim.ulpgc.es (J.A. Carta).

In the evaluation of wind resources available at a givensite of complex terrain or with several prevailing winddirections, it is interesting to have use of a joint probabilitydensity function. That is, to have use of a continuousmodel of the wind rose1 that enables analysis of the vari-ability of the energy characteristics of the wind in termsof speed and direction. Knowledge of these characteristicsenables the wind turbines to be positioned in such a way asto maximise the capturable energy. It should be borne inmind that it could be that the most intense winds are not

1 The wind rose usually provides simultaneous information about thewind direction (normally for eight direction sectors) and intensity in graphor table form.

2 h0 represents the angles that the wind speeds form with the prevailingwind direction, y0 (Fig. 1).

3 The term in square brackets represents the Rayleigh probabilitydensity function, referred to in numerous wind energy papers [20].

1310 J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320

those which blow most hours over the year from a partic-ular direction. This is an important aspect to be taken intoaccount when deciding on the orientation of the windturbines.

The use of a joint probability density function of windspeed and direction is also useful in the modelling of themeasure-correlate-predict method (MPC) [13]. However,the proposed use of continuous [9–12] and discrete [14]joint probability functions of wind speed and direction isscarce in the specialised literature on wind energy and otherrenewable energy sources.

The isotropic Gaussian model of McWilliams et al.[9,10] and Weber [11] was derived from the assumptionsthat the wind speed component along the prevailing winddirection (longitudinal component of the wind vector) isnormally distributed with non-zero mean and a given var-iance, while the wind speed component along a directionorthogonal to that (lateral component of wind vector) isindependent and normally distributed with zero meanand the same variance. The anisotropic Gaussian modelof Weber [12] is a generalization of the model of McWil-liams et al. [9]. In Weber [12], no restrictions are imposedon the standard deviations of the longitudinal and lateralfluctuations.

A very flexible joint probability density function of windspeed and direction is presented in this paper for windenergy analysis. A method proposed by Johnson and Weh-rly [15] to obtain angular–linear distributions with specifiedmarginal distributions has been used for this purpose. Forthe marginal distribution of wind speed we use a singlytruncated from below Normal–Weibull mixture distribu-tion, TNW-pdf, [4]. The marginal distribution of winddirection comprises a finite mixture of von Mises distribu-tions [16]. The parameters of the model are estimated usingthe Least Squares method [17], which is resolved in thispaper using the Levenberg–Marquardt algorithm [18].The suitability of the distributions is judged from the coef-ficient of determination R2 [18]. A comparison is madebetween the models presented by McWilliams et al. [9]and Weber [12] and the proposed joint distribution. Thiscomparison is based on an analysis of the level of fit tothe cumulative frequencies of hourly wind speeds and winddirections recorded at four weather stations located in theCanary Islands (Spain).

2. Models

2.1. Isotropic Gaussian model

The isotropic Gaussian model used by McWilliams et al.[9,10] and Weber [11] takes as its starting point the follow-ing hypotheses: (a) the existence of a prevailing wind direc-tion; (b) the wind speed components for the prevailingwind direction (longitudinal, vy0) and that which is perpen-dicular to it (lateral, vx0) are random variables which aredescribed by a Gaussian distribution; (c) the longitudinaland lateral components are statistically independent of

each other; (d) the variances of the longitudinal, r2y0 , and

lateral, r2x0 , components are the same; (e) the mean of the

longitudinal component, ly0 , is other than zero, and themean of the lateral component, lx0 , is null.

According to this model, the longitudinal componentand the lateral component are described by the probabilitydensity functions (pdfs) given by Eqs. (1) and (2)

fy0 ðvy0 Þ ¼1ffiffiffiffiffiffi

2pp

ry0exp �

ðvy0 � ly0 Þ2

2r2y0

" #ð1Þ

fx0 ðvx0 Þ ¼1ffiffiffiffiffiffi

2pp

ry0exp �ðvx0 Þ2

2r2y0

" #ð2Þ

In accordance with hypothesis (c), the joint pdf is given bythe product of the two functions defined in Eqs. (1) and (2).

Performing a polar coordinate transformation, Eq. (3),2

the joint pdf is obtained as a function of the speed andangle, Eq. (4)

vx0 ¼ v sin h0; vy0 ¼ v cos h0 ð3Þ

fV ;Hðv; h0Þ ¼v

2pr2y0

exp �l2

y0

2r2y0

" #exp �

v2 � 2ly0v cos h0

2r2y0

" #ð4Þ

The marginal pdf of the angle h0, Eq. (5), is obtained afterintegration over v in Eq. (4)

fHðh0Þ ¼1

2pexp � k2

2

� �1�

ffiffiffipp

n expðn2Þ/ðnÞ� �

;

0 < h0 < 2p ð5Þ

where

k ¼ly0

ry0; n ¼ � k cos h0ffiffiffi

2p ; /ðnÞ ¼ 2ffiffiffi

pp

Z 1

nexpð�t2Þdt

ð6ÞThe marginal pdf of the wind speed v, Eq. (7),3 is obtainedafter integration over h0 in Eq. (4)

fV ðvÞ ¼vr2

y0exp � v2

2r2y0

!" #exp �

l2y0

2r2y0

!I0 v

ly0

r2y0

!;

0 < v <1 ð7Þ

where I0(u) is the modified Bessel function of the first kindand order zero [19].

2.2. Anisotropic Gaussian model

The anisotropic Gaussian model proposed by Weber[12], uses as its starting point the same hypotheses as theisotropic model, with the exception of hypothesis (d). Inthe anisotropic model the two variances do not have tobe the same.

J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320 1311

According to this model, the longitudinal and lateralcomponents are described by the probability density func-tions given by Eqs. (1) and (8)

fx0 ðvx0 Þ ¼1ffiffiffiffiffiffi

2pp

rx0exp �ðvx0 Þ2

2r2x0

" #ð8Þ

In accordance with hypothesis (c), the joint pdf is given bythe product of the two functions defined in Eqs. (1) and (8).

Performing a polar coordinate transformation, Eq. (3),the joint pdf is obtained as a function of the speed andangle that the wind speed forms with the axis of the prevail-ing wind direction, y0, Eq. (9)

fV ;Hðv; h0Þ ¼v

2pry0rx0

� exp �ðv cos h0 � ly0 Þ

2

2r2y0

� v2 sin2 h0

2r2x0

" #ð9Þ

The marginal pdf of the angle h0, Eq. (10), is obtained afterintegration over v in Eq. (9)

fHðh0Þ ¼1

2pexp � k2

2

� �c

c2 sin2 h0 þ cos2 h0

� �� 1�

ffiffiffipp

g expðg2Þ/ðgÞ� �

; 0 < h0 < 2p ð10Þ

where

c ¼ ry0

rx0; g ¼ � k cos h0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 cos2 h0 þ 2c2 sin2 h0q ; ð11Þ

The marginal pdf of the wind speed v, Eq. (12), is obtainedafter integration over h0 in Eq. (9)

fV ðvÞ ¼v

ry0rx0exp � k2

2

� �I0 v

ly0

r2y0

!

�Z 2p

0

exp � v2

2r2y0ðcos2 h0 þ c2 sin2 h0Þ

" #dh0;

0 < v <1 ð12Þ

5 TNW-pdf can provide many good fits for unimodal, bimodal andbitangential wind speed frequency distributions. Bitangentiality occurs if

2.3. Proposed model

There are various methods of constructing joint distri-butions from information about the shape of the marginaldistributions [21]. However, very little has been written inthe scientific literature about angular–linear distributionmodels. In this paper, we have used the method proposedby Johnson and Wehrly [15] to obtain angular–lineardistributions.

Johnson and Wehrly [15] define the probability densityfor an angular–linear distribution through Eq. (13)4

fV ;Hðv; hÞ ¼ 2pgðfÞfV ðvÞfHðhÞ; 0 6 h < 2p; �1 6 v <1ð13Þ

4 h represents the angles that the wind speeds form with the y axisdirection.

where g(�) is the pdf of the circular variable f, given by Eq.(14)

f ¼ 2p½F V ðvÞ � F HðhÞ� ð14ÞIn the model proposed in this paper we use for wind speedpdf, fV(v), a Singly Truncated from below Normal Weibullmixture distribution, TNW-pdf. [4], Eq. (15). According toCarta and Ramırez [4], the mixture distribution proposedhere provides very flexible models5 for wind speed studiesand can be applied in a widespread manner to representthe wind regimes in many regions. In addition, the TNW-pdf takes into account the frequency of null winds and,therefore, can represent wind regimes with high percent-ages of null wind speeds

fV ðvÞ ¼x0

Ið/1;/2ÞZðv;/1;/2Þ þ ð1� x0Þ

ab

vb

� �a�1

� exp �ðvbÞa

� �; 0 6 v <1 ð15Þ

where Z(v,/1,/2) and I(/1,/2) are given by Eq. (16)

Zðv;/1;/2Þ ¼1ffiffiffiffiffiffi2pp exp �ðv� /1Þ

2

2/22

" #;

Ið/1;/2Þ ¼1

/2

Z 1

0

Zðv;/1;/2Þdv ð16Þ

where b is a scale parameter with the same units as the ran-dom variable and a is a shape parameter; /1 and /2 areparameters with the same units as the random variable;x0 is the weight in the mixture of the Singly Truncatedfrom below Normal distribution (0 6 x0 6 1).

For the wind direction pdf, fH(h), we use a mixture ofvon Mises distributions [16], Eq. (17). This mixture pro-vides a very flexible model for wind direction studies andcan be applied in a widespread manner to represent thewind direction regimes in zones with several modes or pre-vailing wind directions [16]

fHðhÞ ¼XN

j¼1

xj

2pI0ðjjÞexp½jj cosðh� ljÞ�; 0 6 h < 2p

ð17ÞIn Eq. (17), N is the number of components of the mixture;jj P 0 and 0 6 lj < 2p are parameters; xj are nonnegativequantities that sum to one; that is, Eq. (18)

0 6 xj 6 1; ðj ¼ 1; . . . ;NÞ andXN

j¼1

xj ¼ 1 ð18Þ

The parameter lj is the mean direction and the parameterjj is known as the concentration parameter. Here, I0(jj)

there are two distinct points, v1, v2, at which there is a common tangent tothe density curve. Thus, bitangentiality is implied by, but does not implybimodality. Informally, bimodality implies an extra hump, but bitangen-tiality merely an extra bump.

Fig. 1. Reference axes for the measurement of wind speed direction.

1312 J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320

is the modified Bessel function of the first kind and orderzero [19], and is given by Eq. (19)

I0ðjjÞ ¼1ffiffiffiffiffiffi2pp

Z 2p

0

exp½jj cos h�dh ¼X1k¼0

1

ðk!Þ2jj

2

2kð19Þ

The cumulative distribution functions for the speed, FV(v),and direction, FH(h), of the wind are given by Eqs. (20) and(21), respectively

F V ðvÞ ¼Z v

0

fV ðvÞdv ð20Þ

F HðhÞ ¼Z h

0

fHðhÞdh ð21Þ

For the pdf of the circular variable f, in this paper we haveused a mixture of von Mises distributions, Eq. (17).

3. Estimation of the parameters of the several models

3.1. Isotropic and anisotropic Gaussian models

Normally, in meteorology wind direction is measured ina clockwise direction6 using the North as starting point (they-axis in Fig. 1). However, in the models proposed byMcWilliams et al. [9] and Weber [12], the angle measure-ment starting point is with respect to the prevailing winddirection (the y0-axis in Fig. 1). hp defines the position ofthe prevailing wind direction y0 (and perpendicular x0) interms of the xy Cartesian system.

To estimate the parameters on which the isotropic andanisotropic models depend we will use the method ofmoments. In other words, the sample moments are equatedto the corresponding moments in the population,7 Eqs. (22)and (23)

6 In this paper the angle corresponding to the northerly direction istaken as angle 0�.

7 This is the method used by McWilliams et al. [9] in an initial paper.McWilliams and Sprevak [10], in a second paper, used a method which hassince been criticised [12,16].

lx0 ¼ �vx0 ¼1

n

Xn

i¼1

ðvx0 Þi ¼1

n

Xn

i¼1

vi cos h0i;

ly0 ¼ �vy0 ¼1

n

Xn

i¼1

ðvy0 Þi ¼1

n

Xn

i¼1

vi sin h0i ð22Þ

rx0 ¼ sx0 ¼1

n� 1

Xn

i¼1

½vi cos h0i � lx0 �2

( )12

;

ry0 ¼ sy0 ¼1

n� 1

Xn

i¼1

½vi sin h0i � lx0 �2

( )12

ð23Þ

where n is the number of data of the sample and h0 is thewind direction with respect to the prevailing axis y0. If his the wind direction with respect to north of the xy Carte-sian system, then we have, Eq. (24)

h0 ¼h� hp if h P hp

360� hp þ h if h < hp

(ð24Þ

McWilliams and Sprevak [10] assume that the prevailingwind direction is given by the centre of the sector withthe largest marginal frequency of occurrence. In this paper,hp is determined through Eq. (25) [16]

hp ¼

arctan�vy

�vx

; �vy P 0; �vx > 0

p2; �vy > 0; �vx ¼ 0

pþ arctan�vy

�vx

h i; �vx < 0

8>>>><>>>>:

hp ¼

p; �vy ¼ 0; �vx ¼ �1

2pþ arctan�vy

�vx

h i; �vy < 0; �vx > 0

3p2

; �vy < 0; �vx ¼ 0

8>>><>>>:

ð25Þ

where �vx and �vy are the sample mean components with re-spect to the xy-axes.

3.2. Proposed model

As indicated in Section 2.3, the proposed model is builtfrom marginal distributions of wind speed and wind direc-tion. The parameters of these models are estimated in thispaper using the Least Square (LS) method,8 as described inRefs. [4,16].

From n sample wind speed and direction data, n valuesare calculated of the variable f, defined in Eq. (14), throughthe use of Eqs. (20), (21) and (26)

fi ¼ 2p½F V ðviÞ � F HðhiÞ�; i ¼ 1; . . . ; n ð26Þ

8 The Levenberg–Marquardt algorithm (LMA) [18] is used. TheMathcad? Software 2001i programme of MathSoft Engineering &Education, Inc., [22] is used to find the values of the parameters.

Fig. 2. Horizontal components, vx and vy, of the wind speed and estimated prevailing wind directions.

J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320 1313

Those values of f obtained through Eq. (26) which are low-er than zero are recalculated with Eq. (27), in such a waythat 0 6 fi < 2p for all i = 1, . . . ,n

fi ¼ 2pþ fi if fi < 0; i ¼ 1; . . . ; n ð27Þ

In this paper, a mixture of von Mises distributions is fittedto this data sample of f and, following the procedure setout in Ref. [16], the parameters on which this mixture de-pends are determined.

9 This value is usually 1.225 kg m�3, corresponding to standard condi-tions (sea level, 15 �C). Carta and Mentado [24] recommend the use of themean air density of the area when the variability of the air density and itscorrelation with the wind speed are not taken into account.10 The power curve of a wind turbine PWT(v), can be approximated by a

piecewise linear function with a few nodes [25], or through cubic splineinterpolation [24].

4. Wind power density distribution and energy output of awind turbine

An indicator of the size of the local wind energyresource is its annual mean wind power density [1,23,24].Carta and Mentado [24] have proposed a model for windpower density and wind turbine energy output estimations.This model takes into account the time variability of airdensity q and wind speed, as well as the correlation existingbetween both variables.

However, in the literature related to wind energy, it isnormally assumed that air density and wind speed arenot correlated and is assumed that the air density is con-stant.9 On this assumption, if we multiply the wind powerdensity with the probability of each wind speed and winddirection, we have calculated the probability of wind powerdensity at different wind speed and direction

P ðv; hÞ ¼ 1

2qfV ;Hðv; hÞv3 ð28Þ

The output of a wind turbine, E(v,h), with a power curve,PWT(v), in a wind regime with a joint probability densityfunction, fV,H(v,h), in a period t, can be expressed as afunction of the wind speed and direction using10

1314 J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320

Eðv; hÞ ¼ tZ h2

h1

Z v2

v1

PWT ðvÞfV ;Hðv; hÞdv� �

dh ð29Þ

5. Meteorological data used

In order to determine the flexibility of the proposedmodel to represent different wind regimes a study was car-ried out of samples recorded at different anemometer sta-tions located in the Canarian Archipelago [4,26]. Fourstations were selected which are representative of the mostcomplex wind speed and direction distributions in the Can-ary Islands [4]. For the station called Amagro [4], mean

Table 1Mean speeds, standard deviations and linear correlation coefficients of the wi

W. station Cartesian axes xy

�vx (m s�1) �vy (m s�1) sx (m s�1) sy (m s�1) rxy (

Amagro 5.56 1.01 5.46 3.47 0.41R. Prieto 4.13 1.35 4.60 2.27 0.15P. Vargas 3.51 3.48 3.43 3.83 0.40Granadilla 4.53 2.75 4.97 3.25 0.58

Fig. 3. Amagro station: (a) marginal distributions of wind direction, (b) wind sanalysed. (c) Probability density function of the variable f.

hourly wind direction and wind speed data have been usedfrom 7 years (1997–1999, 2001–2003, 2005). For the stationcalled R. Prieto [4], data have been used from 4 years(1997–1998, 2001, 2005). For the station called P. Vargas[4], data have been used from 3 years (2002–2004). Forthe station called Granadilla [4], data have been used from5 years (1998–2000, 2002, 2004). All the wind direction andwind speed records data were taken at a height of 10 mabove ground level.

In Fig. 2 we can see the horizontal components, vx andvy, of the wind speed at the four stations under study, aswell as the prevailing wind directions estimated using Eq.(25).

nd speed for axes xy and x0y0

Axes x0y0

–) �vx0 (m s�1) �vy0 (m s�1) sx0 (m s�1) sy0 (m s�1) rx0y0 (–)

9 0 5.65 3.02 5.65 �0.2299 0 4.69 2.24 4.35 0.3123 0 4.94 2.82 4.30 �0.123

0 5.30 2.46 5.40 0.145

peed and (d) wind power probability density function, for the three models

Fig. 4. R. Prieto station: (a) marginal distributions of wind direction, (b)wind speed and (d) wind power probability density function, for the three modelsanalysed. (c) Probability density function of the variable f.

J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320 1315

Table 1 shows the mean speeds, standard deviations andlinear correlation coefficients, rxy and rx0y0 , of the compo-nents of the wind speed for axes xy and x0y0, respectively.

6. Analysis of results

Fig. 3 shows, for the case of Amagro, the marginal dis-tributions of the wind direction (Fig. 3a), wind speed(Fig. 3b), and wind power probability density functions(Fig. 3d) for the three models analysed. Also shown is theprobability density function of the variable f (Fig. 3c),involved in the proposed model, Eqs. (13) and (14). Thesame graphs are represented in Figs. 4–6, but for the sta-tions of R. Prieto, P. Vargas and Granadilla, respectively.

The parameters of the proposed marginal distributionsof wind speed for the four stations under study are shownin Table 2. The proposed marginal distributions of windspeed have been built from a mixture of six von Mises dis-tributions11 and their parameters are shown in Table 3.

11 The influence of the number of components of the von Mises mixtureon the degree of fit is discussed in Ref. [16].

It can be seen in these figures that the proposed marginaldistributions have a better degree of fit to the sample histo-grams than the marginal distributions obtained from theisotropic (Eqs. (5) and (7)) [9] and anisotropic models(Eqs. (10) and (12)) [12].

The probability density function g(f), Eq. (13), in whichwe see the existing relation between wind speed and direc-tion has been represented by a mixture of two von Misesdistributions (Table 4). In the cases analysed, the use ofmixture distributions of more than two components hasnot provided an increase in the coefficient of determination,R2, of the joint probability density functions. We shouldpoint out that when we have used the Maximum Likeli-hood method instead of the Least Squares method to esti-mate the parameters of the distribution, g(f), we haveobtained a uniform distribution,12 g(f) = 1/2p. This indi-cates, as can be seen in Eq. (13), that according to the Max-imum Likelihood method wind speed and wind directionare independent. However, we should point out that, inthe cases analysed, the use of mixture distributions of

12 When j = 0, von Mises distribution is the uniform distribution [27].

Fig. 5. P. Vargas station: (a) marginal distributions of wind direction, (b) wind speed and (d) wind power probability density function, for the threemodels analysed. (c) Probability density function of the variable f.

13 The value of R2 varies between 0 and 1. The higher R2 is, the greaterthe fit.14 This might be due to the fact that the hypotheses on which the

analysed Gaussian models are based are not fully met when applied to thedata observed at the weather stations considered.15 To represent each curve in the Figure, the considered angle has been

put in Eq. (28) and the wind speed vhas been varied between 0 and /. Wehave taken q = 1.225 kg m�3.

1316 J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320

two components has provided coefficients of determina-tion, R2, of the joint probability density functions slightlyhigher than those obtained when using the uniformdistribution.

In order to determine the existing correlation betweenthe hourly sample wind speeds (linear variable) and winddirections (circular variable), we have used the correlationcoefficient proposed by Mardia [28]

r2 ¼ r2vc þ r2

vs � 2rvcrvs

1� r2cs

ð30Þ

where

rvc ¼ corrðv; cos hÞ; rvs ¼ corrðv; sin hÞ;rcs ¼ corrðcos h; sin hÞ ð31Þ

As can be seen in Figs. 7a, 8a, 9a and 10a, the values of thecoefficients of correlation r2, at the stations under study,are low.

Fig. 7 shows, for the case of Amagro, the joint probabil-ity density function obtained with the proposed model(Fig. 7a) and a graph which represents the theoreticalcumulative distribution against the sample cumulative dis-

tribution (Fig. 7b), and in which we can see the degree of fitto the sample data of the three bivariable distributions ana-lysed. In Figs. 8–10 the same graphs are shown, but for thestations R. Prieto, P. Vargas and Granadilla, respectively.

It can be seen that, for all the stations analysed, the pro-posed joint probability density function has a higher degreeof fit13 to the sample data than the bivariable distributionsproposed in the specialised literature on wind energy.14

The proposed joint probability function can be used toestimate the wind power density per rotor swept areaand wind speed. As an example, we can see in Fig. 11a,for Amagro, the wind power density for five wind direc-tions (60�, 79.7�, 90�, 100� and 260�).15 For the samestation, we can see in Fig. 11b the wind power density

Fig. 6. Granadilla station: (a) marginal distributions of wind direction, (b) wind speed and (d) wind power probability density function, for the threemodels analysed. (c) Probability density function of the variable f.

Table 3Parameters of the marginal distributions of wind direction

Number of components j Amagro R. Prieto P. Vargas Granadilla

lj (rad) jj (–) xj (–) lj (rad) jj (–) xj (–) lj (rad) jj (–) xj (–) lj (rad) jj (–) xj (–)

1 0.678 32.501 0.034 0.725 14.792 0.081 0.544 11.412 0.463 0.218 11.149 0.1182 1.348 19.360 0.621 1.323 15.512 0.519 0.994 4.869 0.297 1.056 18.754 0.6013 1.339 1.920 0.154 0.403 0.001 0.038 1.776 3.776 0.09 1.683 6.689 0.0724 3.593 1.029 0.130 0.000 1.140 0.189 3.806 3.105 0.081 3.326 1.031 0.0835 4.404 19.395 0.044 1.876 13.260 0.125 5.182 5.43 0.041 4.454 2.945 0.0856 6.283 21.494 0.018 4.401 31.906 0.048 5.905 28.484 0.026 5.473 18.540 0.042

Table 2Parameters of the marginal distributions of wind speed

W. station Normal truncated Weibull

/1 (m s�1) /2 (m s�1) x0 (–) a (–) b (m s�1) 1 � x0 (–)

Amagro 4.272 3.783 0.462 3.471 10.845 0.538R. Prieto �0.36 5.504 0.364 3.145 7.816 0.636P. Vargas 2.791 2.28 0.396 4.008 9.237 0.604Granadilla 9.193 3.017 0.635 2.272 3.54 0.365

J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320 1317

Table 4Parameters of the probability density function of the variable f

Number of components j Amagro R. Prieto P. Vargas Granadilla

lj (rad) jj (–) xj (–) lj (rad) jj (–) xj (–) lj (rad) jj (–) xj (–) lj (rad) jj (–) xj (–)

1 3.589 0.8 0.593 4.390 0.620 0.947 2.594 0.687 0.366 2.718 0.968 0.2492 5.446 1.264 0.407 5.531 4.435 0.053 5.016 1.052 0.634 4.746 0.879 0.751

Fig. 7. Amagro station: joint probability density function (a), probability graph (b).

Fig. 8. R. Prieto station: (a) joint probability density function, (b) probability graph.

1318 J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320

per rotor swept area and wind direction.16 In a similar way,for a given wind turbine the obtainable energy can be

16 Each curve was obtained by integrating the wind speed in an interval(0–10 m/s, 10–25 m/s and 0–1m/s) in Eq. (28), and representing the windpower density as a function of the variations in wind direction between 0�and 360�. We have taken q = 1.225 kg m�3.

calculated as a function of the wind direction using Eq.(29).17

17 Integrating the wind speed between 0 and1 in Eq. (29), and the anglesin the desired intervals.

Fig. 9. P. Vargas station: (a) joint probability density function, (b) probability graph.

Fig. 10. Granadilla station: (a) joint probability density function, (b) probability graph.

Fig. 11. Amagro station: (a) wind power density per rotor swept area and wind speed, (b) wind power density per rotor swept area and wind direction.

J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320 1319

1320 J.A. Carta et al. / Energy Conversion and Management 49 (2008) 1309–1320

7. Conclusions

The conclusions reached are that the joint probabilitydensity function proposed in this paper is very flexibleand complete: (a) can represent unimodal, bimodal andbitangential wind speed frequency distributions, (b) takesinto account the frequency of null winds, (c) representsthe wind direction regimes in zones with several modes orprevailing wind directions, (d) takes into account the corre-lation between wind speed and direction. It can thereforebe used in several tasks involved in the evaluation processof the wind resources available at a potential site. We alsoconclude that, in the case of the Canary Islands, the pro-posed model provides better fits in all the cases analysedthan those obtained with the models used in the specialisedliterature on wind energy. It should be pointed out that theanalysed stations have been installed in areas suitable forthe development of the large scale exploitation of windenergy [26], and have thus avoided terrain of complextopography. We therefore consider that the results of theproposed model are even better when set against those ofother models for sites with more complex wind directionhistograms.

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