Applied Energy 88 (2011) 3869–3881

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Applied Energy

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Comparison between ANNs and linear MCP algorithms in the long-termestimation of the cost per kW h produced by a wind turbine at a candidate site:A case study in the Canary Islands

Sergio Velázquez a, José A. Carta b,⇑, J.M. Matías c

a Department of Electronics and Automatics Engineering, Universidad de Las Palmas de Gran Canaria, Campus de Tafira s/n, 35017 Las Palmas de Gran Canaria, Canary Islands, Spainb Department of Mechanical Engineering, Universidad de Las Palmas de Gran Canaria, Campus de Tafira s/n, 35017 Las Palmas de Gran Canaria, Canary Islands, Spainc Department of Statistics, University of Vigo, Lagoas Marcosende, 36200 Vigo, Spain

a r t i c l e i n f o

Article history:Received 8 November 2010Received in revised form 31 March 2011Accepted 4 May 2011Available online 25 May 2011

Keywords:Artificial Neural NetworkLong-term estimationMeasure correlate predictVariance Ratio MethodWind energy cost

0306-2619/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.apenergy.2011.05.007

⇑ Corresponding author. Tel.: +34 928 45 96 71; faxE-mail address: jcarta@dim.ulpgc.es (J.A. Carta).

a b s t r a c t

In the work presented in this paper Artificial Neural Networks (ANNs) were used to estimate the long-term wind speeds at a candidate site. The specific costs of the wind energy were subsequently deter-mined on the basis of the knowledge of these wind speeds. The results were compared with thoseobtained with a linear Measure–Correlate–Predict (MCP) method. The mean hourly wind speeds anddirections recorded over a 10 year period at six weather stations located on different islands in the CanaryArchipelago (Spain) were used as a case study. The power-wind speed curves for five wind turbines ofdifferent rated power were also used. The mean absolute percentage error (MAPE), Pearson’s correlationcoefficient and the Index of Agreement (IoA) between measured and estimated data were used to eval-uate the errors made with the different metrics analysed.

Amongst the conclusions that can be drawn from the study it can be stated that, in all the cases ana-lysed, the MAPE of the specific cost of the energy obtained using ANNs was lower than that obtained withthe linear MCP method that was employed. In some cases reductions in the error of up to 26.5% wereachieved with the ANNs with respect to the method used for purposes of comparison. However, whenthe correlation coefficients between the short-term data series of the candidate and reference sites wererelatively low, the MAPEs generated by all the analysed metrics were higher when the estimation modelswere used than when the data for the short-term period at the candidate site was taken as being repre-sentative of the long-term wind performance at that site.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The economic feasibility of a wind farm is fundamentallydependant on the specific cost entailed in its production of electri-cal energy (cost per kilowatt hour). The specific cost of the electri-cal energy production of a wind turbine depends on its annualpower output as well as on the investment, financing and operat-ing and maintenance costs [1–6]. The annual power output of awind turbine has a strong influence on the specific cost. Giventhe cubic relationship between wind speed and wind power den-sity [7], the economic feasibility of a project that aims to exploitwind energy at a potential wind energy conversion site dependsto a large extent on the conditions and characteristics of the winddriving the turbine during the useful life of the installation [2,5,8].Since wind characteristics vary over time (diurnal, seasonal, inter-annual) [9–15], historical series of wind data measurements for the

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: +34 928 45 14 84.

candidate site are required to enable an evaluation of that variabil-ity. In this way, the element of uncertainty can be reduced when itcomes to estimating the economic feasibility of installing a windturbine or farm at a particular candidate site. Some authors [11]maintain that at least 5 years worth of wind data for a candidatesite are required, though ten would be preferable. According toHiester and Pennell [12], accurate estimation of the mean valuesof the wind performance at a candidate site is difficult with lessthan 10 years of data. Indeed, some authors [13–15] argue that20 or even 30 years of data are indispensable for the long-termcharacterisation of a wind resource. However, on many occasionsthere are no historical wind measurement data series availablefor a candidate site. One option to resolve this problem is to carryout sufficiently long measurement campaigns at the candidate site,but this involves an often unacceptably lengthy postponement ofthe final decision as to whether to install the wind farm or not.In the particular case of the Canary Islands, the installation of windfarms is regulated by wind power tenders promoted by the Auton-omous Canary Government through legislative decrees [16]. Theperiod of time between publication of the legislative documents

Nomenclature

a slope of the regression line, Eq. (1)AEMET Spanish initials of the State Meteorological Agency of

the Ministry of the Environment and Rural and MarineEnvirons of the Spanish Government

ANN Artificial Neural Networkb y-intercept of the regression line, Eq. (1)BN Bayesian networkCI initial cost of a wind installation, Eqs. (6) and (7)cI specific initial cost of a wind installation, Eqs. (6) and (7)COE specific cost of the energy generated by a wind turbine,

Eq. (6)CO&M annual operating and maintenance cost, Eqs. (6) and (7)cO&M specific operating and maintenance cost, Eqs. (6) and (7)corr(�, �) linear correlation coefficient between two variablesD number of nodes of the power curve of a wind turbine,

Eq. (6)�E mean value of the estimated values of a variable, Eq. (9)Ea annual electrical energy produced by a wind system,

Eqs. (6) and (7)Ei estimated value of a variable, Eqs. (8) and (9)FCR fixed charge rate per year, Eqs. (6) and (7)IoA Index of Agreement, Eq. (10)M number of neurons of the hidden layer of an ANNMAPE mean absolute percentage error, Eq. (8)MCP Measure–Correlate–PredictMLP multilayer perceptronsN number of reference stationsn number of wind speed or direction data of a time series,

Eqs. (8) and (9)�O mean value of the values observed of a variable, Eq. (9)Oi observed value of a variable, Eqs. (8) and (9)Pwtj power of the node j of the power curve of a wind tur-

bine, Eq. (6)PWT(vi) power curve of a wind turbine, Eq. (6)

R Pearson’s linear correlation coefficient, Eq. (9)rc,s linear correlation coefficient between the cosine and

sine of the wind direction h, Eq. (4)rws,c linear correlation coefficient between the wind speed

and the cosine of the wind direction h, Eq. (4)rws,s linear correlation coefficient between the wind speed

and the sine of the wind direction h, Eq. (4)Rws,h correlation coefficient between a linear variable and an

angular variable, Eq. (3)sST

c standard deviation of the short-term (ST) wind data ofthe candidate station (c), Eq. (2)

sSTr standard deviation of the short-term (ST) wind data of

the reference station (r), Eq. (2)VRM Variance Ratio Method, [20]Vcut-in cut-in wind speed of a wind turbine, Eq. (6)Vcut-out cut-out wind speed of a wind turbine, Eq. (6)vi element i of a time series of n wind speed data, Eqs. (5)

and (6)Vj wind speed of the node j of the power curve of a wind

turbine, Eq. (6)vc wind speed of the candidate station (Fig. 1)vrj wind speed of the reference station j (Fig. 1)vLT

c long-term (LT) wind speed of the candidate site (c)vLT

r long-term (LT) wind speed of the reference site (r), Eq.(2)

vSTc short-term (ST) wind speed of the candidate site (c), Eq.

(2)vST

r short-term (ST) wind speed of the reference site (r), Eq.(2)

WS weather stationws wind speedh wind direction angle, Eq. (4)hrj wind direction angle of the reference station j (Fig. 1)

3870 S. Velázquez et al. / Applied Energy 88 (2011) 3869–3881

and the deadline for presentation of proposals by potential inves-tors is limited. Consequently there is little time available to carryout such wind data measurement campaigns. Investors rarely findthey are given more than 1 year to analyse the wind resource at acandidate site. While it may be possible in such a short period oftime to acquire information about the seasonal wind variationsat a candidate site, knowledge of the interannual variability ofthe wind characteristics is unattainable.

The use of statistical methods, commonly known as Measure–Correlate–Predict (MCP) techniques, is often proposed in the scien-tific literature as a way of getting round this lack of long-term data.These methods make use of historical wind data series which areavailable for nearby climatological weather stations used as refer-ence stations. An additional requirement with these methods isthat a short-term measuring period for the wind data of the refer-ence station must coincide in length and date with a measuringperiod for the candidate site. The concept is based on the assump-tion of the existence of a certain relationship between the windperformance at the candidate and reference sites.

The MCP methods that were first used [9,10,17–19] only esti-mated the long-term annual mean wind speed. Later, the mostcommonly used MCP methods made use of linear regression algo-rithms [20,21] to establish the relationship between hourly (or oflower intervals of time) wind characteristics of the candidate siteand of the weather station chosen as the reference site. Modelshave also been proposed that establish non-linear type relation-ships [22–24]. Likewise, automatic learning techniques have alsobeen proposed [25–31], of which particular mention can be given

to the techniques which employ statistical learning algorithms,such as Bayesian networks (BNs) [25] and techniques such as Arti-ficial Neural Networks (ANNs) based on biology-inspired algo-rithms [26–31].

In this work, ANNs are used to carry out long-term estimation ofthe wind speeds and, based on these speeds, of the specific costs ofwind energy. [32,33]. The mean absolute percentage error (MAPE)and Pearson’s correlation coefficient between measured and esti-mated data were used to evaluate the errors made with the differ-ent metrics analysed. These have been used previously by otherauthors [27,29]. The Index of Agreement (IoA) has also been usedin this paper [34]. The errors were compared with those obtainedusing a linear MCP method. A linear MCP method was chosen asit is the standard method used [35]. The specific linear methodused is the method proposed in Ref. [20]. The choice of the so-called Variance Ratio Method (VRM) was based on the conclusionsreached by the authors in the aforementioned paper, where it wasstated that of the four MCP methods which were compared in theirstudy only the VRM appeared to give consistently reliable predic-tions of all the metrics analysed.

The originality of this paper lies in the fact that this is the firsttime, following an exhaustive search and consultation by theauthors of the information available, that a comparison has beencarried out between ANNs and linear MCP algorithms in thelong-term estimation of the cost per kW h produced by a windturbine at a candidate site. Likewise, this is the first time that theerrors made by the models used to undertake the long-term esti-mation of various parameters have been compared with the errors

S. Velázquez et al. / Applied Energy 88 (2011) 3869–3881 3871

generated when the short-term wind data of the candidate stationare used to represent the long-term wind performance.

2. MCP method used for comparison

The Variance Ratio Method, proposed by Rogers et al. [20], isbased on the assumption of the existence of a linear relationshipbetween the long-term wind speeds at the candidate site vLT

c andthe reference site vLT

r ,

vLTc ¼ avLT

r þ b ð1Þ

The values of the slope a and y-intercept b are deduced when forc-ing the mean and variance of the wind speeds estimated during theperiod of concurrence (short-term) of data between the candidateand reference sites to coincide with the mean and variance of thewind speeds observed over that period. With this criterion, Eq. (1)is transformed into following equation

vLTc ¼ �vST

c �sST

c

sSTr

� ��vST

r

� �þ sST

c

sSTr

� �vLT

r ð2Þ

MCP methods generally bin or group together the wind speeds inaccordance with the wind direction at the reference site. The dataare divided for this purpose into a specific number of direction sec-tors, normally 8 of 45� or 12 of 30�. Then the parameters a and b ofthe model are estimated, Eq. (1), for each direction sector. That is,Eq. (2) is applied to each direction sector. So, the short-term means�vST

r , �vSTc and the standard deviations sST

r , sSTc of the wind data of the

reference and candidate station, respectively, must be calculatedfor each of the direction sectors under consideration.

3. Architecture of the ANNS used

The ANNs used in this paper are comprised of multilayer per-ceptron topologies (MLPs) [36,37]. The architectures were de-signed with a single layer of hidden neurons (Fig. 1). This designhas demonstrated its ability to satisfactorily approximate any con-tinuous transformation [36,37] and its use has been proposed byvarious authors [26,28,30]. In addition, restricting the number ofhidden layers to one reduces training times. Both the mean hourlywind speeds and directions of the reference station are used as theinput signals. So, the number of neurons in the input layer is twice

Fig. 1. Schematic diagram of an ANN with 2 N wind speed and angular winddirection.

the number N of reference stations employed. In the hypothesesconsidered in this paper the output layer has only one neuron.The output signal provided by that neuron is the estimated meanhourly wind speed of the candidate station.

The backpropagation algorithm with sigmoid activation func-tion was used as a training algorithm [36–38]. The Levemberg–Marquardt algorithm was used as a minimisation algorithm ofthe mean-square error made during the training [36].

4. Meteorological data and wind turbines used

Despite the considerable number of weather stations that havebeen installed in the Canary Islands in the past [39], for variousreasons those used in this work are the only ones which theauthors of this paper are aware of that have sufficiently long andreliable data records. The meteorological data used in this paper(mean hourly wind speeds and directions) were recorded overthe period 1999–2008 at six weather stations installed on six is-lands of the Canarian Archipelago (Spain) (Fig. 2). This informationwas provided by the State Meteorological Agency (Spanish initials:AEMET) of the Ministry of the Environment and Rural and MarineEnvirons of the Spanish Government.

Fig. 2 shows the names of the islands (Lanzarote, Fuerteventura,Gran Canaria, Tenerife, La Palma and El Hierro), their surface areas(in km2), the distances between the weather stations (in km s) andthe reference codes assigned to each station (WS-1, WS-2, WS-3,WS-4, WS-5 and WS-6). The names of the stations are given in Ta-ble 1 (Lanzarote Airport, Fuerteventura Airport, Gran Canaria Air-port, Reina Sofía Airport, La Palma Airport and Los CangrejosAirport), together with the codes assigned to them and their geo-graphic coordinates. The height above ground level of these sta-tions is 10 m. Fig. 3 shows the interannual mean wind speeds foreach station for the long-term period (1999–2008). Also shownare the intervals between these mean wind speeds and one andthree standard deviations as well as the annual mean wind speedscorresponding to the final year (2008) of the data series availablefor all the stations analysed. As explained in the methodology sec-tion, in all the candidate stations the short-term period for whichwind data series are available is considered to be the period whichcorresponds to the year 2008.

Table 2 shows the linear correlation coefficients between themean hourly wind speeds of the various stations. These coefficientswere obtained for the wind speed data of 2008 and for the set of allthe wind speed data (1999–2008).

Fig. 4 shows the wind roses for each of the stations. The left-hand column represents the wind roses for 2008, while the right-hand column represents the wind roses constructed with winddirections of all the available years (1999–2008).

Table 3 contains the correlation coefficients Rwshi,j between thews wind speeds of the candidate station i and the wind directions hof the reference station j. To calculate the correlation between alinear variable (wind speed) and an angular variable (wind direc-tion) the expression proposed by Mardia and Jupp [40] was used,

R2ws;h ¼

r2ws;c þ r2

ws;s � 2rws;crws;src;s

1� r2c;s

ð3Þ

where rws,c, rws,s and rc,s are given by following equation.

rws;c ¼ corrðws; cos hÞ; rws;s ¼ corrðws; sin hÞ; rc;s

¼ corrðcos h; sin hÞ ð4Þ

The wind turbine models used in this work (E44 and E70) [32],(G52, G58 and G80) [33] were chosen for their different ratedpower ranges. These ranges include the rated powers of wind tur-bines which are currently in operation in the Canary Archipelago

Fig. 2. Location of the weather stations used.

Table 1General data of the weather stations (WS) used in the study.

Station name Code Geographical coordinates

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and of turbines for which there are installation plans in the nearfuture. The power-wind speed curves of these turbines can be seenin Fig. 5.

Latitude(north)

Longitude(west)

Altitude (m)

Lanzarote Airport WS-1 28�570700 13�360 10Fuerteventura Airport WS-2 28�2701000 13�5105400 24Gran Canaria Airport WS-3 27�5504400 15�2302000 16Reina Sofía Airport WS-4 28�0203500 16�3401600 51Los Cangrejos Airport WS-5 27�4805000 17�5301000 30La Palma Airport WS-6 28�3604700 17�4503600 85

5. Methodology

In the methodology developed in this study and as steps prior tothe long-term estimation of the specific cost of the electrical en-ergy (c€/kW h) at a candidate site, the wind speeds and power out-put at the site are estimated.

The ANN architectures and the VRM as described above wereused in the estimation process of the long-term mean hourly windspeeds at the candidate stations. The study is based on the assump-tion of the availability of mean hourly wind speed and directiondata over a matching short-term period at both the candidateand reference stations. In this work, the final year of available data(2008) was chosen as the short-term period of time of simulta-neous availability of data for both candidate and reference stations.This choice was made on the basis of the points made in Section 1

Fig. 3. Mean interannual wind speeds of each weath

of this paper. That is, a wind data measurement campaign for acandidate site is normally undertaken during the year prior tothe proposed installation. It is likewise assumed that historical ser-ies (long-term) of wind speed and direction data for the referencestations are available.

Using the 2008 wind data, a model was generated characteris-ing the relationship between the candidate and reference station.

er station for the long-term period (1999–2008).

Table 2Linear correlation coefficients between the hourly wind speeds of the different anemometer weather stations.

Year 2008 Years 1999–2008

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 0.377WS-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

S. Velázquez et al. / Applied Energy 88 (2011) 3869–3881 3873

This stage is represented in Fig. 6 by a ‘1’ enclosed in a circle. Byfeeding the model with the known long-term mean hourly windspeeds and directions for the reference station, estimation is madeof the long-term mean hourly wind speeds for the candidate sta-tion. These stages are represented in Fig. 6 by a ‘2’ and ‘3’ enclosedin circles.

In the methodology that has been developed in this study andusing the available weather stations, different cases were consid-ered in which one (or various in the case of the ANNs) of these sta-tions was taken as the reference station and another as thecandidate station.

Two scenarios were analysed for the purpose of comparing theresults obtained with the VRM and the proposed ANNs:

(A) Like the VRM, the ANNs use a single reference station.(B) The ANNs use two reference stations, while the VRM uses

only one.

5.1. Scenario A

The nineteen cases which can be seen in Table 4 were analysedin Scenario A. Each of the nineteen models of ANNs was built andused following the methodology outlined in Fig. 6.

The architectures of the ANNs used in this scenario can be rep-resented through Fig. 1, in the particular case where N is equal toone. The idea was for the number of neurons of the hidden layer tobe the same for all the cases analysed. With this purpose in mind,the number of neurons was increased in twos (one, three, five, etc.)as long as the validation results showed improvement. After fifteenneurons the validation results ceased to improve and, therefore,the number of neurons in the single existing hidden layer wasset at fifteen. For the purpose of evaluating the quality of the mod-els designed with the available short-term wind data series (2008)for the reference and candidate stations, these data series were di-vided into three random subsets of different data: training, valida-tion and test data [38,41]. The percentage of data chosen for eachsubset was, respectively, 60%, 20% and 20%.

The training data subset was used to estimate the parameters ofthe ANNs. The validation data subset was used to check the pro-gress of the ANN training, optimising their parameters [38,41].That is, they were used to measure the degree of generalisationof the ANNs. The test data subset, unused in either the trainingor validation stages, was used to calculate the error percentageof the final optimised network. In other words, it was used as anindependent measure of how well the network was performingafter training. However, the models used to perform long-termestimation were built using 100% of the available short-term data.Of these data 80% was used for training and the remaining 20% forvalidation.

The various tests were performed using the algorithm known asTrainlm, available in the MATLAB toolbox of neural networks [41].

With regard to the parameters of the models developed in theVRM, it should be mentioned that in this paper they were calcu-lated for 30� direction bins. If Eq. (2) gave negative wind speed val-

ues then null value was assigned to these speeds. To evaluate thequality of the models a simple validation was performed [38]. Thatis, 20% of the data was kept in reserve as a test subset, and this 20%was not used in the construction of the model. The remaining 80%was used for training. The division of data into these two groupswas done randomly. The VRM models used to carry out long-termestimation were constructed using 100% of the available short-term period data. The various tests were programmed using theMATLAB software.

5.2. Scenario B

In this scenario the ANN models used can be representedthrough Fig. 1, in the particular case where N is equal to two.

The nineteen cases analysed in this scenario are shown in Table4. It can be seen that these cases were obtained by adding to thereference stations of Scenario A (Table 4) an additional referencestation. The extra station was chosen for its having, after the refer-ence station used in Scenario A, the highest correlation with thecandidate station (Table 2). Each of the nineteen models of ANNswas built and used following the methodology outlined in Fig. 6.

It should be mentioned that cases 13 and 14 of this scenariocoincide with the corresponding cases of Scenario A, as there isno second station with a correlation coefficient lower than thatheld by station WS-4 with WS-5 and WS-6 (Table 2).

6. Parameters analysed

The parameters analysed in this paper were the wind speeds,the wind turbine power outputs and the estimated long-term spe-cific costs of the electrical energy (c€/kW h) at the candidate sites.

6.1. Wind speeds

When using both the ANN and the VRM models analysis wasmade of the errors produced when estimating the n mean hourlywind speeds which they generated. That is, for the ANNs an anal-ysis was carried out of the errors produced between the windspeeds estimated with the architectures represented in Fig. 1(cases N = 1 and N = 2) and by the VRM models represented byEq. (2).

6.2. Electrical power

The power-wind speed curves of the wind turbines chosen forthis study were used to estimate their power output (Fig. 5)[32,33]. These deterministic models of the power output of thewind turbines were used for the purpose of simplifying the analy-sis. This has no effect on the objective of this paper, namely thecomparison of the ANNs and linear MCP algorithms. Manufacturersusually provide the power curves of their wind turbines in a dis-crete form with D nodes (Pwtj, Vj). The electrical power obtainablebetween two nodes ‘‘j’’ and ‘‘j + 1’’ of the power curve can be calcu-

Fig. 4. Wind roses for each of the stations. The left-hand column represents thewind roses for 2008, and the right-hand column represents the wind rosesconstructed with wind directions of all the available years (1999–2008).

3874 S. Velázquez et al. / Applied Energy 88 (2011) 3869–3881

lated approximately [42]. One of the possible means of approxima-tion is to assume that the variation between two nodes of thepower-wind speed curve is linear, Eq. (5)

PWTðv iÞ ¼Pwtjþ1�Pwtj

Vjþ1�Vjðv i � VjÞ þ Pwtj

0; Vcut-out < v i < Vcut-in

(1 6 j 6 D� 1 ð5Þ

where Vcut-in and Vcut-out are, respectively, the cut-in and cut-outspeeds of the wind turbine. It should be mentioned that the windspeeds used in Eq. (5) are those estimated for the candidate stationat the height of the stations used, namely 10 m above ground level.This has been done so as not to introduce the errors that are gener-ated with the models of extrapolation of wind speed with height.

6.3. Energy cost

Various economic analysis models have been published in thescientific literature related to renewable energies [1–6]. Since theaim of this paper is to compare the ANNs and linear MCP algo-rithms, a simplified calculation model of the specific cost of energy(COE) was chosen. This method, similar to the EPRI TAG method[1–3] estimates the cost as the ratio between the total costs andthe annual electrical energy produced by the wind-based system,Ea. In general, two terms can be distinguished in the total cost pro-duced by a wind turbine: the initial cost of installation CI and thetotal annual operating and maintenance costs CO&M, Eq. (6)

COE ¼ CI � FCRþ CO&M

Ea¼ cI þ cO&M ð6Þ

where

cI ¼CI � FCR

Ea; cO&M ¼

CO&M

Eað7Þ

In Eqs. (6) and (7) FCR is the fixed charge rate per year and its valuein this paper is 0.05.

To calculate CI a specific investment is assumed of 1200€ perkW installed. This value was chosen by the authors from theirknowledge and experience of this type of installation in the CanaryIslands. CO&M covers preventive and corrective operating and main-tenance costs, land leasing costs, administration costs, etc. Theauthors of this paper consider from experience that the specific an-nual operating and maintenance costs, cO&M, can be estimated at1.6c€/kW h.

Eq. (6) assumes that the debt period is equal to the working lifeof the wind turbine. In this study the working life period was takenas 20 years, a length of time which is normally covered by themanufacturer’s guarantee.

7. Evaluating numeric estimation

Of the various alternatives that have been proposed [34,38,43]to numerically evaluate the estimations made by the models, threehave been chosen in this paper to analyse the parameters de-scribed in Section 6. These are the so-called mean absolute per-centage error (MAPE), the correlation coefficient R and the Indexof Agreement (IoA). MAPE has been used by several authors[27,29] and is generally expressed by Eq. (8)

MAPE ¼ 100n

Xn

i¼1

jOi � EijOi

ð8Þ

R, Eq. (9), measures the existing statistical correlation between themeasured or observed data and the estimated data. The correlationcoefficient varies between 1 and �1. It takes the value 1 for

Table 3Correlation coefficients between the mean hourly wind speeds of the candidate stations and the mean hourly wind directions of the reference stations.

Candidate Year 2008 Years 1999–2008Reference weather stations Reference weather stations

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 0.332 0.335 0.342 0.247 0.361 0.486 0.254 0.311 0.282 0.265 0.292 0.454WS-2 0.270 0.325 0.341 0.220 0.343 0.446 0.280 0.337 0.292 0.268 0.345 0.453WS-4 0.211 0.443 0.280 0.402 0.198 0.360 0.217 0.389 0.257 0.384 0.203 0.377WS-5 0.281 0.301 0.272 0.162 0.399 0.314 0.273 0.273 0.238 0.186 0.338 0.324WS-6 0.163 0.134 0.124 0.179 0.241 0.415 0.199 0.189 0.120 0.124 0.224 0.367

Fig. 5. Power-wind speed characteristic curves of the wind turbines used in thestudy.

Fig. 6. Stages of the process of estimation of the long-term wind speeds.

S. Velázquez et al. / Applied Energy 88 (2011) 3869–3881 3875

perfectly correlated results and 0 when there is no relation. Nega-tive values indicate the existence of a negative correlation [38].

R ¼Pn

i¼1ðEi � EÞðOi � OÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½Pn

i¼1ðEi � �EÞ2�½Pn

i¼1ðOi � OÞ2�q ð9Þ

The Index of Agreement IoA [34], is given by Eq. (10).

IoA ¼ 1�Pn

i¼1ðEi � OiÞ2Pni¼1ðjEi � Oj þ jOi � OjÞ2

; 0 6 IoA 6 1 ð10Þ

In Eqs. (8)–(10), n is the number of analysed data, Oi are the mea-sured or observed data, Ei are the estimated values, O and E arethe means of the values Oi and Ei, respectively.

8. Analysis of results

Fig. 7 shows the MAPEs obtained when estimating the long-term mean hourly wind speeds in the nineteen cases analysed,both in Scenario A and Scenario B (Table 4). In other words,Fig. 7 shows the MAPEs generated when use is made of the VRMand the ANN architectures. The errors are shown of the ANN archi-tectures which use as input signals the mean hourly wind speedsand directions from both a single reference station (Scenario A),and from two reference stations (Scenario B). Fig. 7 also showsthe MAPEs generated when the short-term wind data period(2008) was considered to be representative of the long-term peri-od. It can be observed in Fig. 7 that the models developed using theVRM are the ones which generated the higher MAPEs. With theexception of cases 12 and 15, where the results are similar, thereare cases where substantial reduction of the MAPE obtained with

the ANNs of Scenario A was achieved with respect to the MAPEproduced by the VRM. To give a specific example, the MAPE of case4 was reduced by 28.7%. Expressed in another way, a MAPE of49.61% (VRM) was reduced to a MAPE of 35.35% (ANN).

The results for the correlation coefficient R are shown in Fig. 8,where a comparison can be made of the results obtained with bothmethods. Also shown in this figure is the correlation coefficient cal-culated between the mean hourly wind speeds of the candidatestation and the reference station for the short-term period (Table2). It can be observed that the Rs obtained with the ANNs betweenthe long-term estimated and measured values of the candidate sta-tion are higher than the Rs measured during the short-term periodbetween the reference and candidate stations (2008).

Fig. 9 shows the results for the Index of Agreement. It can be ob-served that, in general, when the existing coefficients of correlationbetween the candidate and reference stations (Table 2) are verysmall, the models developed using the VRM generated higher IoAvalues than the ANN models (Scenario A). In Scenario A, the biggest

Table 4Cases studied in Scenarios A and B.

Casenumber

Scenario A Scenario B

Referencewind station

Candidatewind station

Referencewind station

Candidatewind station

1 WS-3 WS-1 WS-3, WS-2 WS-12 WS-3 WS-2 WS-3, WS-1 WS-23 WS-3 WS-4 WS-3, WS-1 WS-44 WS-3 WS-6 WS-3, WS-2 WS-65 WS-3 WS-5 WS-3, WS-6 WS-56 WS-1 WS-2 WS-1, WS-6 WS-27 WS-1 WS-4 WS-1, WS-2 WS-48 WS-1 WS-6 WS-1, WS-5 WS-69 WS-1 WS-5 WS-1, WS-4 WS-5

10 WS-2 WS-4 WS-2, WS-6 WS-411 WS-2 WS-6 WS-2, WS-1 WS-612 WS-2 WS-5 WS-2, WS-3 WS-513 WS-4 WS-6 WS-4 WS-614 WS-4 WS-5 WS-4 WS-515 WS-6 WS-5 WS-6, WS-4 WS-516 WS-1 WS-3 WS-1, WS-2 WS-317 WS-2 WS-3 WS-2, WS-6 WS-318 WS-4 WS-3 WS-4, WS-5 WS-319 WS-6 WS-3 WS-6, WS-4 WS-3

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differences between the IoA values of the VRM based models andthe ANN based models occurred in cases 13 and 14. That is, inthe cases where the lowest correlation coefficients (0.319 and0.248, respectively) between the reference and candidate stationwere recorded.

It can also be seen in Figs. 7 and 8 that the ANNs of Scenario Bproduced, in all the cases analysed, lower MAPEs and higher Rsthan the ANNs of Scenario A. In other words, the use of a secondreference station improved, in all the cases analysed, estimationof the long-term mean hourly wind speeds. It can also be seen fromFig. 9 that the ANNs of Scenario B produced, in 82% of the casesanalysed, higher IoA values than the ANNs of Scenario A. In Sce-nario B, when the correlation coefficients between the referenceand candidate stations were higher than 0.502, the IoA values ofthe VRM based models were lower than those obtained with theANN based models.

Fig. 7. MAPEs obtained when estimating the long-term

Fig. 10 shows the ratios between the IoA values of the ANNmodels (Scenario B) and the VRM models, when considering eachof the years of the data set used (1999–2008) as the short-termperiod. It can be observed in this figure that the year used as theshort-term period does not qualitatively affect the results obtainedwhen the year 2008 is taken as being representative of this period.

The bubble graph of Fig. 11 shows the differences between theMAPEs generated by the ANNs of Scenario A and the MAPEs ob-tained when considering the data of the short-term period(2008) to be representative of the long-term wind performance.

It can be observed that for the correlation coefficients windspeed-wind speed used in this work (0.248–0.736), the MAPEs ob-tained when considering the data of the short-term period (2008)to be representative of the long-term wind performance can belower than those obtained with all the estimation methods used.

Fig. 12 has a diagram similar to that of Fig. 11, but with theMAPEs of the ANNs of Scenario A replaced with the correspondingMAPEs of Scenario B. A comparison of Figs. 11 and 12 reveals twodifferences. Firstly, the ANNs of Scenario B increase the number ofcases in which the MAPEs generated by the models are lower thanthose obtained when considering the short-term data (2008) to berepresentative of the long-term wind performance. Secondly, it canbe seen that the differences between the MAPEs generated withthe models and the MAPEs obtained with the 2008 data are higherwhen Scenario B is used than when Scenario A is used. In caseswhere the correlation coefficients are lower than 0.7, the methodsused in this paper are not an appropriate solution when it comes toestimating the long-term mean hourly wind speeds. In these cir-cumstances, better estimations can be obtained by investigatorsif they rely on the data measured over 1 year at the candidate siterather than using the results obtained from the methods analysedin this paper. However, this question has not been fully analysedwhen using data from stations with a limited correlation to designestimation models of the long-term wind resources [20,29].

It can also be seen in Fig. 9 that the highest IoA values are gen-erated in cases 1, 2, 6, 16 and 17 (Table 4). In other words, in thecases where the correlation coefficients (wind speed–wind speedand wind speed–wind direction) calculated for the short-term per-iod between the candidate and reference stations are highest. Ananalysis of the results represented in Fig. 7 also reveals high MAPEs

mean hourly wind speeds in the 19 cases analysed.

Fig. 8. Correlation coefficients obtained when estimating the long-term mean hourly wind speeds in the 19 cases analysed.

Fig. 9. IoA values obtained when estimating the long-term mean hourly wind speeds in the nineteen cases analysed.

S. Velázquez et al. / Applied Energy 88 (2011) 3869–3881 3877

obtained with the estimation methods that were used. This is aconsequence of the relatively low correlation coefficients (0.248–0.736) between the wind data series of the reference and candidatestations (Table 2).

Fig. 13 shows in a similar way to that of Fig. 7 the mean and±three times the standard deviation of the MAPEs obtained whenestimating the mean hourly power output of the wind turbinesconsidered in this paper (Fig. 5).

It can be observed that in all the cases analysed the ANNs ofScenario A gave lower errors than those obtained when using theVRM. However, the lower errors generated by the ANNs of ScenarioA against those produced by the ANNs of Scenario B in the estima-tion of the mean hourly wind speeds (Fig. 7) were not observed inall the cases analysed when the mean hourly power outputs of thewind turbines used were estimated. Various factors, acting simul-taneously, are involved in this question. One of these is the smalldifference between the errors which in some cases existed be-tween Scenario B and A in the estimation of the mean hourly windspeeds. Another factor is the existence of high MAPEs and low Rs

between the estimated long-term mean hourly wind speeds andthe observed mean hourly wind speeds. A third factor is a conse-quence of the low frequencies [24] which were recorded for windspeeds higher than the rated wind speed of the wind turbines, at10 m above ground level, at the stations analysed. Such windspeeds mean that the wind turbines operate with greater fre-quency in the range between the cut-in wind speed and the ratedwind speed. This can be deduced from the magnitudes which theannual capacity factors of the wind turbines (CFs) reach in the can-didate stations (Fig. 14). It can be observed that the CFs, or ratiosbetween the mean annual power output of the wind turbinesand their rated power [1,3,5], are reasonably low at 10 m aboveground level. So, the higher the MAPE of the mean hourly windspeeds in the partial load operating range of the wind turbine,the more significant the effect of the electrical power oscillationswhich are generated in that range on the MAPE obtained whenestimating the mean hourly power output of the turbines. Giventhat the wind turbine power output depends on the shape of thepower curve in its partial load speed range, the MAPEs obtained

Fig. 10. Ratio between the IoA values obtained with the ANN models (Scenario B) and the IoA values obtained with the VRM models.

Fig. 11. Differences between the MAPEs generated by the ANNs of Scenario A and the MAPEs obtained when considering the data of the short-term period (2008) to berepresentative of the long-term wind performance.

3878 S. Velázquez et al. / Applied Energy 88 (2011) 3869–3881

when estimating the mean hourly power output also depend onthe characteristics of that curve. For a given MAPE of the meanhourly wind speeds, if wind turbines of the same cut-in wind speedand the same rated wind speed are used, then those turbines witha higher power curve slope will give rise to higher MAPEs of themean hourly power output than those with a lower slope.

The combined effect of these three factors also causes the num-ber of cases (cases 1–4, 7–8, 10 and 13) (Fig. 13) to increase to eightin which the ANNs give rise to lower MAPEs than those producedwhen considering the short-term wind data series (2008) to berepresentative of the long-term wind performance. Given this,the small number of stations analysed and the limited range ofavailable correlation coefficients, it is difficult to provide a guideas to how low the magnitude of the correlation coefficient has tobe before use of the reference station cannot be recommended inthe long-term estimation of the power output of a given windturbine.

Fig. 15 shows in a similar way to that of Figs. 7 and 13 the meanMAPEs obtained when estimating the specific costs of the electrical

energy in terms of the investment and set-up costs (cI) of the windturbines used in this paper. It can be observed that in all the casesanalysed the errors obtained when using the models based on theVRM are higher than those obtained when the models based onANNs were used. In Scenario A the difference in the errors madebetween the two methods ranged between 2.66% in case 1 and26.57% in case 13.

In the estimation of this parameter it was observed that, similarto the estimation of the mean hourly power outputs, there werealso eight cases (cases 1–4, 7–8, 10 and 13) (Fig. 15) which gaverise to lower MAPEs than those produced when considering theshort-term wind data period (2008) to be representative of thelong-term wind performance.

There is a tendency for the MAPEs produced by the VRM andANN models analysed in this paper to decrease when the correla-tion coefficients increase. Given the limited number of stationsanalysed, it has not been possible to deduce a rule of thumb basedon the magnitude of the correlation coefficient of the short-termperiod which would help to decide whether it would be more

Fig. 12. Differences between the MAPEs generated by the ANNs of Scenario B and the MAPEs obtained when considering the data of the short-term period (2008) to berepresentative of the long-term wind performance.

Fig. 13. Mean MAPEs obtained when estimating the mean hourly power outputs of the wind turbines considered in this paper.

S. Velázquez et al. / Applied Energy 88 (2011) 3869–3881 3879

realistic to use the ANNs to estimate the long-term mean hourlypower output or whether it would be better to rely on the observedshort-term wind data at the candidate station. However, in theanalysis of the data used in this paper it was noted that for the lin-ear MCP method to generate lower MAPEs than the 2008 wind data(considering this to be representative of the long-term wind per-formance), the degree of correlation between the reference andcandidate station must be higher than that required by the ANNs.

The degree of correlation is generally related to the distance be-tween the candidate and reference station [44,45]. In this paper, inaddition to the distances between the stations used, the orographyof the islands on which the stations are located also affects thedegree of correlation between the wind data recorded at thosestations. However, on each island there are areas where the corre-lation coefficients between the wind speeds recorded at the sta-tions in those areas and the wind speeds of the stations used inthis paper are relatively high. In many of these areas correlationcoefficients in the range of 0.88–0.95 were obtained [31].

Given the shortness of the wind speed time series which havebeen recorded to date at these stations, it has only been possibleto carry out analyses corresponding to the stage represented inFig. 6 by a ‘1’ enclosed in a circle [31]. However, it has not beenpossible to check the effect of these levels of correlation on the

long-term estimation of the parameters analysed in this paper.According to Ref. [46], as a general rule, where the coefficient ofdetermination (square of Pearson’s correlation coefficient) of a cor-relation of the monthly wind speed in all directions is lower than0.8, there exists substantial uncertainty when using the long-termdata of the reference station to deduce the long-term wind condi-tions at the candidate site. If this rule is accepted, with the afore-mentioned range of correlation coefficients it is predicted thatthe proposed ANN models will provide lower errors than those thatwould be generated when using the recorded short-term data asbeing representative of the long-term wind regime performance.Likewise, in accordance with this rule and with the results ob-tained in this paper, the uncertainty of the prediction will be lessthan that generated by the VRM and will not be substantial.

Figs. 16 and 17 were drawn for the purpose of analysing the de-gree of generalisation of the models used. Fig. 16 shows the MAPEsobtained between the estimated and observed long-term meanhourly wind speeds as a function of the MAPEs generated in thetest stage of the models (the stage represented by a ‘1’ enclosedin a circle in Fig. 6). Fig. 17 is similar to Fig. 16, but with the MAPEsreplaced by the correlation coefficients R.

It can be seen in Fig. 16 that the long-term MAPEs are some-what higher than those obtained in the test stage. Likewise, it

Fig. 14. Annual capacity factors of the wind turbines at the candidate stations.

Fig. 15. Mean MAPEs obtained when estimating the specific costs of the electrical energy generated by the investment and set-up costs of the wind turbines considered inthis paper.

Fig. 16. MAPEs obtained between the long-term estimated and observed meanhourly wind speeds as a function of the MAPEs generated in the test stage of themodels.

Fig. 17. Correlation coefficients obtained between the between the long-termestimated and observed mean hourly wind speeds as a function of the correlationcoefficients generated in the test stage of the models.

3880 S. Velázquez et al. / Applied Energy 88 (2011) 3869–3881

S. Velázquez et al. / Applied Energy 88 (2011) 3869–3881 3881

can be seen in Fig. 17 that the calculated long-term Rs are lowerthan the Rs obtained in the test stage (stage 1 in Fig. 6). Of impor-tance in this question is the degree of representation that theshort-term period data have of the long-term data [47] and the le-vel of correlation between the reference and candidate stations.

9. Conclusions

A series of conclusions were drawn after analysing the resultsobtained in this paper:

(a) The models based on ANNs whose input signals are the windspeeds and directions of a reference station (Scenario A)gave lower MAPEs in 87% of the cases than the models basedon the VRM, when estimating the long-term mean hourlywind speeds at the candidate stations.

(b) The inclusion of a second reference station in the ANNs (Sce-nario B) lowered in all the cases analysed the MAPEsobtained when estimating, with the ANNs of Scenario A,the long-term wind speeds.

(c) In 100% of the cases analysed the errors obtained in the esti-mation of the long-term specific costs of the wind energythrough the use of models based on the VRM were higherthan those obtained when the models based on ANNs wereused.

(d) It was observed that with correlation coefficients of 0.736(wind speed–wind speed) and 0.342 (wind speed–winddirection), measured between the short-term mean hourlywind data of the candidate and reference stations, thelong-term mean hourly wind speeds analysed generatedlower MAPEs when the ANNs (Scenario B) were used thanwhen the wind data for 2008 were used as being represen-tative of the long-term wind performance. However, it wasnot possible to establish a rule of thumb based on themagnitude of these coefficients due to the low number ofstations analysed and the limited range of available correla-tion coefficients.

(e) There are stations located in areas of each island with shortrecords of wind data for which the coefficients of correlationwith the data series of the stations considered in this paperare high [31]. If the rule as stated in Ref. [46] is followed, theuncertainty of the estimation will not be substantial. In addi-tion, in accordance with the results obtained in this paper, itis considered that the uncertainty will be less than that pro-duced by the VRM. Likewise, it is believed that the proposedANNs will generate lower errors than would be produced bythe recorded short-term data considered as being represen-tative of the long-term wind regimes.

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