2011-a new probabilistic method to estimate the long-term wind speed characteristics at a potential...

7/30/2019 2011-A New Probabilistic Method to Estimate the Long-term Wind Speed Characteristics at a Potential Wind Energ

1/15

A new probabilistic method to estimate the long-term wind speed characteristicsat a potential wind energy conversion site

Jos A. Carta a,*, Sergio Velzquez 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 Electronics and Automatics Engineering, University of Las Palmas de Gran Canaria, Campus de Tafira s/n, 35017 Las Palmas de Gran Canaria, Canary Islands, Spain

a r t i c l e i n f o

Article history:

Received 14 October 2010Received in revised form3 February 2011Accepted 5 February 2011Available online 11 March 2011

Keywords:

Conditional distributionsMeasureecorrelateepredict methodWind speedStratified cross-validationRoot relative squared errorCoefficient of determination

a b s t r a c t

This paper proposes the use of a new MeasureeCorrelateePredict (MCP) method to estimate the long-term wind speed characteristics at a potential wind energy conversion site. The proposed method usesthe probability density function of the wind speed at a candidate site conditioned to the wind speed ata reference site. Contingency-type bivariate distributions with specified marginal distributions are usedfor this purpose. The proposed model was applied in this paper to wind speeds recorded at six weatherstations located in the Canary Islands (Spain). The conclusion reached is that the method presented inthis paper, in the majority of cases, provides better results than those obtained with other MCP methodsused for purposes of comparison. The metrics employed in the analysis were the coefficient of deter-mination (R2) and the root relative squared error (RRSE). The characteristics that were analysed were thecapacity of the model to estimate the long-term wind speed probability distribution function, the long-term wind power density probability distribution function and the long-term wind turbine power outputprobability distribution function at the candidate site.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

As a consequence of the interannual variability of wind, the firstconcern about a site where the installation of a wind power stationis being considered lies with itslong-term (many years) wind speedcharacteristics [1e4]. However, for many places often only short-term wind data are available. To overcome this drawback variousmethods have been proposed in the scientific literature for esti-mation of the long-term wind speed characteristics at such sites[2e15].

These methods use simultaneous measurements of the windspeed at the site in question and at one [2e10] or several [11e15]nearby reference sites with a long history of wind data measure-

ments. The methods that have been proposed use different types offunctions to relate the long-term set of wind speeds with the short-term setof data. The type of relationshipis often linear [7,8], thoughprobabilistic relationships have also been proposed [9,10,14].

This paper proposes the use of a new MeasureeCorrelateePredict(MCP) method to estimate the long-term wind speed characteristicsat a potential windenergy conversion site. Theproposedmethod usesthe probability density function of the wind speedat a candidate site

conditioned to the wind speed at a reference site. Contingency-typebivariate distributions with specified marginal distributions are usedfor this purpose [16,17]. The proposed model was applied in thispaper to wind speeds recorded at six weather stations located in theCanary Islands (Spain).

Three types of analysis were undertaken for the purpose ofevaluating the capacity of the model to accurately represent thelong-term wind characteristics: (a) an analysis of the degreeoffit ofthe cumulative distribution functions (cdf) for wind speed to thecumulative relative frequency histograms of hourly mean windspeeds recorded at six weather stations located in the CanarianArchipelago; (b) an analysis of the degree offit of the cdfs for windpower density to the cumulative relative frequency histograms of

the cube of hourly mean wind speedsrecordedat the same weatherstations; (c) an analysis of the degree offit of the cdfs for windturbine power output to the cumulative relative frequency histo-grams of the wind turbine power output. A commercial pitch-regulated 900 kW rated power wind turbine was used for this thirdanalysis [18].

Two metrics were used in the three analyses, namely the coef-ficient of determination R2 and the root relative squared error(RRSE). As part of the analysis of the model proposed in this paper,a comparison of these metrics with those obtained from otherprobabilistic and linear models referenced in the scientific litera-ture, which also use a single reference station [8e10], was also

* Corresponding author. Tel.: 34 928 45 96 71; fax: 34 928 45 14 84.E-mail address: [email protected] (J.A. Carta).

Contents lists available at ScienceDirect

Energy

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n e r g y

0360-5442/$ e see front matter 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.energy.2011.02.008

Energy 36 (2011) 2671e2685
mailto:[email protected]://www.sciencedirect.com/science/journal/03605442http://www.elsevier.com/locate/energyhttp://dx.doi.org/10.1016/j.energy.2011.02.008http://dx.doi.org/10.1016/j.energy.2011.02.008http://dx.doi.org/10.1016/j.energy.2011.02.008http://dx.doi.org/10.1016/j.energy.2011.02.008http://dx.doi.org/10.1016/j.energy.2011.02.008http://dx.doi.org/10.1016/j.energy.2011.02.008http://www.elsevier.com/locate/energyhttp://www.sciencedirect.com/science/journal/03605442mailto:[email protected]


2/15

carried out. The variance ratio method (VRM) was chosen from the

linear models as, in accordance with the conclusions of reference[8], of the four MCPs that were compared in that study only theVariance Ratio method appeared to give consistently reliablepredictionsof allof the metrics. From the fewprobabilistic methodsthat have been published in the scientific literature related torenewable energies one discrete type was chosen [9], which takesinto account wind direction, and one continuous type [10].

2. Methods used for comparison

The Joint Probabilistic Approach [9], the Weibull Scale method[10] and the Variance Ratio method [8] were chosen to comparetheir results with those obtained using the alternative methodproposed by the authors.

2.1. The joint probabilistic approach (JPA)

The procedure used by Garca-Rojo [9] is based on calculation ofthe joint probability mass functionpSTcervc;dc; vr;dr ofthe speed (v)and direction (d) of the wind over the short-term (ST) at thecandidate site (c) and the reference site (r).

From the joint probability mass function of the short-term windspeed and direction and the probability mass function of the long-term wind speed and direction for the reference site, pLTr v;d,estimation was made of the probability mass function of the long-term (LT) wind speed and direction for the candidate site, pLTc v;dEq. (1).

pLTc vi;dj 1NF XNwk 1

XNdz 1

pSTcervi;dj; vk;dzpLTr vk;dz (1)where NF is a normalisation factor1, Eq. (2)

NF XNwi 1

XNdj 1

XNwk 1

XNdz 1

pSTcervi;dj; vk;dz

pLTr vk;dz (2)

Nw and Nd are the wind speed and wind direction bins of the jointprobability mass function. In this study twelve 30 direction binswere considered for the purpose of defining the wind direction atthe candidate and reference sites. The widths of the wind speed

bins were 1 m/s. The Mathcad Software 2001i programme of

MathSoft Engineering & Education, Inc [19] was used for applica-tion of this method. One of the features provided by this softwarewas used in the programming of the short-term joint probabilitymass function to create nested matrices. That is, matrices whoseelements are also matrices.

The marginal wind probability mass function for the Nw binscan be obtained from Eq. (3),

pLTc vi1

NF

XNdj1

XNwk1

XNdz1

pSTcervi;dj;vk;dz

pLTr vk;dz; i 1.Nw (3)

2.2. The Weibull scale method (WSM)

This empirical method is based on the hypothesis that thereference site and candidate site wind speeds follow a twoparameter Weibull distribution, whose probability density function(pdf), W-pdf[20], is given by Eq. (4)

fv;a;b ab

v

b

a1exp

v

b

a!(4)

In addition, the existence is considered of a linear relationshipbetween the scale parameters (b) and the shape parameters (a) ofthe probability density functions of the candidate site and thereference site, Eq. (5)

lSTc

l

ST

r

lLTc

l

LT

r

0lLTc

lSTc

l

ST

r

lLTr ; (5)

In Eq. (5), l represents the scale or shape parameter which is beingconsidered.

The parameters of the Weibull distribution were estimatedusing the least-squares method [21]. The Mathcad Software 2001i

Fig. 1. Location of the weather station used.

Table 1

Distances between the weather stations (in km).

WS-1 WS-2 WS-3 WS-4 WS-5 WS-6

WS-1 0 61.1 208.6 307.7 438.5 407.8WS-2 61.1 0 160.4 269.3 401.0 381.5WS-3 208.6 160.4 0 116.9 246.2 244.7WS-4 307.7 269.3 116.9 0 131.8 132.3WS-5 438.5 401.0 246.2 131.8 0 89.3WS-6 407.8 381.5 244.7 132.3 89.3 0

1

Probably due to an erratum, this factor is not included in reference [9].

J.A. Carta, S. Velzquez / Energy 36 (2011) 2671e26852672


3/15

Fig. 2. Relative frequency histograms of the wind speed of the year 2008 for the six stations used. Each histogram also shows the two-component mixture Weibull-pdf and fitted

Weibull-pdf.

Table 2

Means, standard deviations (S.D.), kurtosis and skew of the wind speeds recorded at the different anemometer weather stations.

Year 2008 Years 1999e2008

WS-1 WS-2 WS-3 WS-4 WS-5 WS-6 WS-1 WS-2 WS-3 WS-4 WS-5 WS-6

Mean (m s1) 6.12 6.23 7.98 6.01 6.66 5.47 5.83 5.83 7.14 5.64 5.97 4.82S.D (m s1) 2.91 2.46 3.79 3.08 2.50 2.74 3.02 2.53 3.73 3.17 2.89 2.32Kurt (e) 0.37 0.40 0.92 0.45 0.21 0.96 0.13 0.39 0.86 0.15 0.46 1.02Skew (e) 0.39 0.01 0.02 0.42 0.29 0.80 0.38 0.12 0.06 0.57 0.21 0.69

J.A. Carta, S. Velzquez / Energy 36 (2011) 2671e2685 2673


4/15

programme of MathSoft Engineering & Education, Inc [19] wasusedfor application of this method.

2.3. The variance ratio method (VRM)

The variance ratio method, proposed by Rogers et al. [8],predicts the long-term wind speed at the candidate site, vLTc , by

means of the linear relationship given by Eq. (6)

vLTc

v

STc

sSTcsSTr

v

STr

!

sSTcsSTr

v

LTr (6)

where vSTr ,vSTc , s

STr and s

STc are the short-term means and standard

deviations of the reference and candidate site wind speeds. Theseparameters were calculated for the twelve 30 direction bins whichwere defined for the wind direction at the reference site. vLTrrepresents the observed long-term wind speeds at the referencesite. If Eq. (6) provides negative wind speed values, then a null valueis assigned to these speeds. The Mathcad Software 2001i pro-gramme of MathSoft Engineering & Education, Inc [19] was used forapplication of this method.

3. Proposed method

Let VSTc and VSTr be two random variables which represent the

short-term wind speed at the candidate and reference sites,respectively. If the joint probability density function fSTcr vc; vrexists and is known, then the conditional probability densityfunction of the random variable VSTc , represented by f

STc vcjvr, for

a value vr ofVSTr , is defined by Eq. (7) [22].

fSTc vcjvr fSTcrvc; vr=fSTr vr (7)

where fSTr vr is the probability density function of VSTr such that

f

ST

r v

r > 0.

In the method proposed in this paper, from the short-term windspeed conditional probability density function, Eq. (7), and thelong-term wind speed probability density function for the refer-ence site, fLTr v, estimation is made of the long-term wind speedprobability density function for the candidate site, fLTc v, Eq. (8).

fLTc vc

1

CNFZN

0f

STc vcjvrf

LTr vrdvr (8)

where CNF is a normalisation factor, Eq. (9)

CNF ZN0

ZN0

fSTc vcjvrfLTr vrdvrdvc (9)

Thereare variousmethodsof constructingbivariate distributionsfrominformation regarding the form of the marginal distribution. This

Table 3

Linear correlation coefficients between the wind speeds of the different anemometer weather stations.

Year 2008 Years 1999e2008

WS-1 WS-2 WS-3 WS-4 WS-5 WS-6 WS-1 WS-2 WS-3 WS-4 WS-5 WS-6

WS-1 1.000 0.660 0.736 0.497 0.410 0.509 1.000 0.649 0.663 0.489 0.430 0.485WS-2 0.660 1.000 0.671 0.491 0.488 0.501 0.649 1.000 0.674 0.514 0.456 0.521WS-3 0.736 0.671 1.000 0.502 0.472 0.552 0.663 0.674 1.000 0.492 0.469 0.566WS-4 0.497 0.491 0.502 1.000 0.248 0.319 0.489 0.514 0.492 1.000 0.259 0.377

WS-5 0.410 0.488 0.472 0.248 1.000 0.449 0.430 0.456 0.469 0.259 1.000 0.480WS-6 0.509 0.501 0.552 0.319 0.449 1.000 0.485 0.521 0.566 0.377 0.480 1.000

Fig. 3. Schematic representation of the methodology used to train and test the proposed model.

Fig. 4. Powere

wind speed characteristic curve of the wind turbine used in the study.



5/15

paper used the family of distributions introduced by Plackett [16,17],which are called contingency-type or simple c-type distributions.

So, Eq. (7) can be expressed as shown in Eq. (10)

where S1 is given by Eq. (11) and S2 is given by Eq. (12)

S1 1 h

FSTc vc FSTr vrij 1 (11)

S2 S12 4jj 1FSTc vcFSTr vr (12)In the above equations FSTc vc and FSTr vr are the cumulativedistribution functions (cdf) of the wind speed at the candidate andreference site, respectively. fSTc vc is the wind speed probabilitydensity function (pdf) of the candidate site. j is the coefficient of

association which has to be estimated.2 In this paper, to estimate jthe values of the coefficient of correlation of the sample (namelythe short-term observed wind speeds at the candidate and refer-ence site) and the model were equated [16,17].

The coefficient of correlation of the short-term model, zST, isgiven by Eq. (13).

xST Evc; vr EvcEvrscsr

(13)

where the expected values, E, are given by Eq. (14) and the vari-ances s2c and s

2r by Eq. (15)

Evcvr ZN0

ZN0

vcvrfST

cr vc; vrdvcdvr; Evc

ZN0

vcfSTc vcdvc; Evr

ZN0

vrfSTr vrdvr (14)

s2c ZN0

vc Evc2fSTc vcdvc; s2r ZN0

vr Evv2fSTr vrdvr 15

The coefficient of correlation of the sample

br can be calculated

using the n sample valuesv

c;i;v

r;i, through Eq. (16)

In this paper Eq. (15) is equated with Eq. (16) and the Secant/Mueller method [23] is used to find j.3

3.1. Marginal distributions used and their estimation

Various pdfs have been proposed in the scientific literature torepresent the wind regimes at a site [21]. The most appropriate pdfscan be selected from this wide range of possibilities to representfSTc v and fSTr v. However, an analysis of the wind data recorded inthe Canary Islands showed that the distribution which best fits the

fSTc

vc

jvr

1

2 ffiffiffiffiffiffiS2p fST

c

vc

fj 1

hS1 2jFSTc vc

ihS1 2jFSTr vr

iS2

j

1g (10)

br nPn

i 1 vc;ivr;i Pn

i 1 vc;i

Pni 1 vr;i

"

nPn

i 1 v2c;i Pn

i 1 vc;i

2#12"

nPn

i 1 v2r;i Pn

i 1 vr;i

2#12

(16)

Table 4

Parameters of the two-component mixture Weibull probability density functions of the different weather stations.

WS Year 2008 Years 1999e2008

a1 (e) b1 (m/s) a2 (e) b2 (m/s) u (e) a1 (e) b1 (m/s) a2 (e) b2 (m/s) u (e)

WS-1 2.729005 3.745895 2.950677 7.947805 0.292472 1.827694 5.816552 4.929111 8.922942 0.823541WS-2 3.034438 3.519389 3.828031 7.60603 0.209921 2.517273 3.256737 3.365667 7.263881 0.227528WS-3 1.810713 5.7846 4.254421 11.267293 0.468408 1.588097 4.849404 3.841308 10.334497 0.465121WS-4 2.597023 3.752469 3.358039 8.591042 0.417332 2.268574 3.43214 2.936784 8.265112 0.429719WS-5 1.735932 5.765048 4.670338 7.957536 0.347225 1.41332 3.998037 3.137729 7.591305 0.304605WS-6 9.822167 3.660911 2.092843 6.141108 0.0602 6.273016 3.154643 2.289994 5.452468 0.066275

2

This bivariate distribution has only one parameter, in addition to the marginals.

3 First the secant method is applied. If this secant method fails to find a root j,then the Mueller method is used. The Mueller method is a modified version of thesecant method. This algorithm is implemented in the Mathcad Software 2001iprogramme of MathSoft Engineering & Education, Inc [19], which was used in this

paper.



6/15

data was a two-component mixture Weibull probability densityfunction (WW-pdf) [21,24,25].

The WW-pdf, which depends on 5 parametersa1;b1;a2; b2 > 0; 0 u 1, is given by Eq. (17)

fv;a1;b1;a2;b2;u

u&

a1

b1v

b1a11

expv

b1a1

!'1u

&a2b2

v

b2

a21exp

v

b2

a2!'(17)

Fig. 5. RRSE metrics of the models used in estimation of the long-term wind speed distributions.

Fig. 6. R

2

metrics of the models used in estimation of the long-term wind speed distributions.

Table 5

Coefficient of association j of the bivariate distributions. Year 2008.

WS-1 WS-2 WS-3 WS-4 WS-5 WS-6

WS-1 e 11.0275 16.892751 5.265359 3.912902 5.69398WS-2 11.0275 e 11.246719 5.151877 5.186045 5.743455WS-3 16.892751 11.246719 e 5.346644 4.845632 6.82522WS-4 5.265359 5.265359 5.346644 e 2.214407 2.805872WS-5 3.912902 5.186045 4.845632 2.214407 e 4.680542

WS-6 5.69398 5.743455 6.82522 2.805872 4.680542e



7/15

The cumulative distribution function (WW-cdf) is given by Eq. (18).

Fv;a1; b1;a2;b2;u u&

1 exp

v

b1

a1!' 1 u

&1

exp

v

b2

a2!'(18)

b1 and b2 are scale parameters with the same units as the randomvariable and a1 and a2 are shape parameters.

The least-squares method was used to estimate the five param-eters on which the WW-pdf depends. The LevenbergeMarquardtalgorithm [26] was used to solve the problem.4

4. Meteorological data used

The meteorological data used in this paper (mean hourly windspeeds and directions) were recorded over the period 1999e2008at six weather stations installed on six islands in the Canarian

Archipelago (Spain) (Fig. 1). These data were provided by the StateMeteorological Agency (Spanish initials: AEMET) of the Ministry ofthe Environment and Rural and Marine Environs of the SpanishGovernment.

These weather stations are located at the public airports in theCanary Archipelago. The Spanish Meteorological Agency (Spanishinitials: AEMET) issues certifications related to past atmosphericweather data kept in the official records. The existing distancesbetween the various stations are shown in Table 1.

The height above ground level of each of the stations is 10 m.Fig. 2 shows, for 2008, the relative frequency histograms of the

wind speed for each of the stations with bin widths of 1 m s1. Eachhistogram also shows the WW-pdf and the W-pdf.

Table 2 shows the mean, standard deviation, coefficients ofvariation, kurtosis and skewness of thewind speed in the year 2008and over the period 1999e2008 for all six analysed stations.

Table 3 shows the linear correlation coefficients between thewind speeds of the different stations. These coefficients wereobtained for the wind speed data of 2008 and for the period1999e2008. If the data from Table 1 are compared with the cor-responding data from Table 3, a fact can be observed that haspreviously been demonstrated in a number of references [4,27e29].That is, it can be observed that the correlations between windspeeds recorded at different geographical sites decrease with thedistance between them. However, it should also be added that thiscoefficient is influenced by the orography of the areas where thesestations are installed.

With regard to the length of the period of time for which dataareavailable in this study, it should be mentioned that, according toHiester and Pennell [1], accurate estimation of the mean values ofthe wind performance at the candidate site is possible using 10years of data. However, other authors [30,31] assume that twentyor thirty years of data are required to undertake long-term char-

acterisation of a wind resource.

5. Methodology

Fig. 3 shows a schematic diagram of the procedure that iscommonly employed. It can be seen that there is a candidate sitefor which a short period of mean hourly wind data is available(in our case, for the year 2008) and a reference station for whichmean hourly wind data are available for a long period of time (inour case, 10 years: 1999e2008). Using the model describedabove in Section 3, the objective was to estimate the wind speedcharacteristics at the candidate station for the unknown long-term period. For this purpose, models were trained and tested

from the set of known data recorded for the year 2008 (this

Fig. 7. RRSE metrics of the models used in estimation of the long-term wind power density distributions.

4 The Mathcad Software 2001i programme of MathSoft Engineering & Educa-

tion, Inc [19] is used tofi

nd the values of thefi

ve parameters.



8/15

stage is indicated by the circled number 1 in Fig. 3). The strat-ified k-fold cross-validation method [32,33] was used to trainand test the model. This method is based on randomly parti-tioning the set of data available into similar-sized disjointsubsets. The model is trained with the data set comprising theunion of k 1 subsets, while the remaining subset is used fortesting purposes. This procedure is repeated k times, each timeusing a different subset to test and determine a partial error di.The overall error d is calculated as the arithmetic mean of the kerrors, Eq. (19) [14].

d

1

kXk

i 1di (19)

The standard deviation Sd

of the errors is given by Eq. (20)

Sd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1k 1

Xki 1

di d2vuut (20)

If the hypothesis is considered that the errors are independent anddistributed according to a normal law, then the error confidenceinterval can be estimated through Students t test. That is, byapplying the statistic indicated in Eq. (21)

d mSd=ffiffiffikpwtk1 (21)

Fig. 9. RRSE metrics of the models used in estimation of the long-term distributions of the electrical power generated by a wind turbine.

Fig. 8. R2 metrics of the models used in estimation of the long-term wind power density distributions.



9/15

This statistic has a tdistribution with k 1 degrees of freedom. So,the confidence level for the mean error m will be given by Eq. (22).Where a is the level of significance.

P

8

2011-a new probabilistic method to estimate the long-term wind speed characteristics at a potential...

Documents