Wind speed parameter estimation from one-month sample via Bayesian approach

Research Article

Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/qre.1184

Wind Speed Parameter Estimation fromOne-month Sample Via Bayesian ApproachPasquale Erto, Antonio Lanzotti and Antonio Lepore∗†

Site-specific wind potential assessment shows difficulties mainly because it needs very long on-site anemometricmonitoring. This paper proposes to reduce the long-term monitoring by exploiting other initial information aboutparameters to be estimated via MCMC (Markov chain Monte Carlo). The proposed Bayesian approach allows theintegration of prior information (e.g. obtained from atlases, databases and/or fluid-dynamic assessment) with samplingdata, and furnishes effective and timely posterior information about Weibull parameters of the wind speed distribution.Real sampling data, collected from a southern Italian site, are analysed in order to illustrate the main features of themethodology. Moreover, the effectiveness of both the filtering strategy adopted to deal with the high correlation thatusually characterizes the anemometric data, and the seasonal adjustment proposed to obtain a sample unbiased byseasonal effect is highlighted. The results of the application show that the proposed methodology fits the applicativeneeds very well. A bootstrap simulation remarks that the attained precision of the Bayesian estimates carried out froma one-month sample is comparable to the maximum likelihood estimates obtained from an actual one-year sample.Copyright © 2010 John Wiley & Sons, Ltd.

Keywords: Markov chain Monte Carlo; Bayesian estimators; wind speed distribution; Weibull distribution

1. Introduction

In the last decade, wind energy has proven to be one of the most competitive and fastest growing sources of renewableenergy. Such growth is driven by the wider context of energy supply and demand, the rising profile of environmental issuesand the impressive improvements in the technology itself. These factors have combined in many regions of the world to

encourage political support for the industry’s development. Renewable energy is also supported in Europe, by a Kyoto-led target,for 22% of electricity supply to come from renewables by 2010. This has now been extended into a new target for 20% of thefinal energy consumption to be renewable by 2020, which will be binding on all the 27 member states1.

However, the growth of new wind projects continues to be hampered by the lack of wind resource data. Such data are neededto enable governments and/or private developers to make timely decisions about areas potentially suitable for development. Forall these reasons, rapid evaluation of site-specific wind potential is crucial.

In such a context, wind speed is one of the most important factors to consider, since its probability distribution significantlyinfluences the wind energy system performance. The two-parameter Weibull distribution is the most widely used and acceptedto model wind speeds and to assess wind energy potential; see e.g. Van der Auwera et al.2, Tuller and Brett3, Carta et al.4. Thisis mainly due to its flexibility5, the dependence on only two parameters and the form of the probability density function (pdf)

expressible in closed form. Accordingly, Weibull distribution is adopted in most commercial wind software/packages (e.g. WAsP� ,

WindPro�) and is used in this paper.For the sake of exhaustiveness, it has to be mentioned that the two-parameter Weibull model has not been able to represent

all the wind regimes encountered in nature, and new wind speed probability models have continued to be proposed in thespecialized literature. Takle and Brown6 advise the use of a hybrid pdf, employed in the following studies; see e.g. Tuller andBrett3, Takle and Brown6, Nfaoui et al.7, Persaud et al.8, Merzouk9, Castino et al.10. However, Tuller and Brett3 indicate thatsuch pdf in many cases produces a worse fit than the Weibull distribution, after having analysed the data collected in sevenwind sites on the British Columbia coast. Van der Auwera et al.2 propose the three-parameter Weibull distribution in the windenergy context. It obviously better fits the data than the two-parameter Weibull distribution, but the added location parameterintroduces complications in the estimation procedure and in some cases produces unrealistic conditions11 or is not significantlyneeded12.

Department of Aerospace Engineering, University of Naples Federico II, P. le V. Tecchio n. 80, 80125 Napoli, Italy∗Correspondence to: Antonio Lepore, Department of Aerospace Engineering, University of Naples Federico II, P. le V. Tecchio n. 80, 80125 Napoli, Italy.†E-mail: [email protected]

Copyright © 2010 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2010, 26 853--862

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P. ERTO, A. LANZOTTI AND A. LEPORE

The task of a more effective determination of the distribution parameters is often more crucial than the selection of abetter model. In most studies presented in the literature on renewable energy sources, graphical methods (e.g. Tar13, Jowder14,Akpinar and Akpinar15), the Maximum Likelihood (ML) method (e.g. Auwera et al.1, Shata and Hanitsch16, Carta and Ramirez17,Carta et al.18) and moment methods (e.g. Chadee and Sharma11, Auwera et al.1, Gökçek et al.19, Weisser20) are commonly usedto estimate Weibull distribution parameters. The ML method has a high reputation since it provides consistent and efficientestimators that are asymptotically unbiased21. Therefore, it has been used in this paper as the reference estimation method.

In the renewable energy technical literature, papers that introduce alternative estimation methods or Bayesian approachesare few. Akdag and Dinler22 propose a new method, namely Power Density (PD) method, which has a simpler formulation thanthe ML method to estimate Weibull parameters and it is more suitable especially when estimating wind power pdf for energyapplications. A Bayesian approach to wind speed is found in Pang et al.12 and in Li and Shi23. The former paper highlights thesuitability of a Bayesian estimation of Weibull parameters instead of the ML method. The latter presents a method based onBayesian Model Averaging (BMA) on more than one model indicated as suitable for wind speeds. Such an approach is known tobe able to model not only the usual ‘within-model’ variance but also the so-called ‘between-model’ one. Both methods utilizeMarkov chain Monte Carlo techniques in order to overcome computational problems. However, different from the approachproposed in this paper, none of the aforementioned papers uses informative prior distributions and short-run data.

In fact, in order to hasten wind potential assessment, this paper proposes to integrate the generally available prior knowledge(on model parameters) with sampling data via Bayesian approach. Such methodology, combining prior information (obtainedfrom atlases, databases and/or fluid-dynamic assessment) with sampling data, furnishes robust and timely posterior information.Furthermore, filtering and seasonal adjustment strategies are proposed to eliminate autocorrelation and seasonality from the data(before performing the estimation procedure).

In Section 2, the explicit definition of the problem is given. In Section 3 a Bayesian solution is discussed facing the mathematicalaspects of the prior distributions choice and elicitation. In Section 4, a case study from a southern Italian wind-site is presentedto highlight the potentiality and the practical steps of the proposed method. Section 5 points out the conclusions and findingsof the research, highlighting the broader perspective of the results.

2. Definition of the problem

In order to model wind speed measurements collected during a specific time window, the Weibull random variable W with twoparameters � (scale) and � (shape) is used. This variable has the following cumulative distribution function (cdf)

FW (w;�,�)=1−exp[−(w / �)�], w>0;�,�>0 (1)

pdf

fW (w;�,�)=��−�w�−1 exp[−(w / �)�], w>0;�,�>0 (2)

and expected value

v =E(W)=��(1+1 / �) (3)

where �(·) and E(·) are the Euler Gamma and the expected value functions, respectively.

Likelihood function

Wind speed measurements are usually affected by left censoring because of the presence of an offset value wc in the dataacquisition system. Thus, the corrected likelihood function L(w|�,�) can be formulated as follows:

L(�,�|w)={1−exp[−(wc / �)�]}nc (� / �)n−nc exp

[−

n∑i=nc+1

(w(i) / �)�]

n∏i=nc+1

(w(i) / �)�−1 (4)

where w(i) is the ith ordered observation from the original sample w= (w1,. . . , wn) and nc is the number of censored observations.The function L(�,�|w) (or more efficiently its natural logarithm) can be numerically maximized with respect to � and � in orderto get the ML estimates �ML and �ML. Then, from Equation (3) the ML estimate vML of v is obtained as �ML�(1+1 / �ML).

Data filtering

In the application, rapid parameter estimation is desirable in order to reduce the time of analysis for potential investments onwind energy sources. In such conditions data collection has to be as short as possible.

Unfortunately, hourly mean wind speed measurements are intrinsically highly autocorrelated. Although the use of suchmeasurements has no significant effect on �ML and �ML

21, the use of an effective number of non-autocorrelated observationsis crucial when determining the efficiency of such estimators as well as in the Bayesian estimation procedure being explainedbelow. In order to suitably select the sample of non-autocorrelated observations, the filtering algorithm idea found in Ramirezand Carta21 is utilized.

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At each iteration h(1�h�n)

Step 1. The subsample w= (w1,. . . , wi ,. . . , wnh ) is derived from the original sample w by deleting h−1 observations every h,where wi =w1+(i−1)h and nh is the smallest integer greater than or equal to n / h.

Step 2. The Box–Cox24 transformation

xi =w�

i −1

�(5)

is performed to achieve normality, obtaining x= (x1,. . . , xi,. . . , xnh ). � is a suitable constant, calculated by utilizing the ML methodas explained in Box and Cox24.

In order to determine if the subsample x can be considered non-autocorrelated, the Ljung–Box25 statistic

Q=nh(nh +2)L∑

l=1

r2l

nh −k(6)

(approximately distributed as a chi-square random variable with L degrees of freedom) is utilized to test whether theautocorrelation coefficients at lag l (l =1, ..., L) can be considered zero, where

rl =nh−l∑i=1

[(xi − x)(xi+l − x)] /nh∑

i=1(xi − x)2 (7)

is the sample autocorrelation at lag l, x is the mean value of x and L is assumed equal to the nearest integer to nh / 4 (accordingto Anderson26).

The algorithm stops when the estimate of Q, based on the subsample x, is less than the quantile q1−� of order 1−�=0.95 ofthe chi-square distribution with L degrees of freedom. Thus, the effective sample size nh can be determined and the correspondingsubsample w can be somehow considered as the effective experimental information.

Seasonal scale transformation

If the temporal monitoring window is less than one year, data are highly affected also by the period when the anemometercampaign is performed (seasonal effect). In order to obtain a seasonally adjusted (i.e. unbiased by seasonal effect) sample wannual,the transformation

wannual =kw (8)

of the data is found suitable. An estimate of the coefficient k can be obtained if at least one-year monitoring data wsannual is

available from neighbour survey stations. For instance, if the monitoring days (when collecting w) are in winter, the correspondingestimate is deducted as follows:

k = wsannual

wswinter

(9)

where wswinter (resp. ws

annual) is the mean value of the measurements taken during the winter months (resp. the whole year)by the neighbour survey station. For simplicity, without loss of generality, the year is roughly split into winter (from January toMarch), spring (from April to June), summer (from July to September) and autumn (from October to December).

In the case where the monitoring period includes days from more than one season, a weighted mean of the seasonal k’scorresponding to such seasons can be plausibly used in Equation (8), with the actual number of monitoring days in each season asweights. In the case that more than one k are available (since more than one year monitoring data are available from neighboursurvey stations) one could better exploit the available information by utilizing the mean of the annual seasonal coefficient k’s. Inaddition, commercial software, which is based on numeric fluid-dynamic algorithms, could be eventually used to make neighbourdata coherent (e.g. when the data are collected at different heights). Thus, in order to target the one-year monitoring data onthe current station, the filtered and seasonally adjusted sample

wannual = kw (10)

is obtained from the filtered sample w, by using the appropriate point estimate k. Currently, we are not able to take into accountthe variability of k since the formulation of its exact distribution is not trivial and constitutes an open research issue.

It is important to emphasize that the transformation (8) does not conflict with the Weibull model assumption. In fact, thanks toits scaling property, if k is a positive real constant, then kW is still a Weibull random variable with scale parameter k� and shapeparameter �. The implicit assumption of common � between seasons is generally plausible, as it is supported by the seasonalanalysis of Akpinar and Akpinar15 and several case studies (e.g. Lun and Lam27).


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3. Bayesian approach

As anticipated, the problem of evaluating site-specific wind potential is faced by using on-site anemometric long-term monitoring,neglecting any other initial information about parameters. Although wind atlases/databases represent the usual starting point fora preliminary site analysis, they fail to give site-specific information about the wind speeds. On the contrary, the on-site directanemometric monitoring gives reliable and complete information, but it is costly and lengthy because wide temporal windows(at least one year) are required to accurately characterize wind potential. Moreover, the standard commercial software used bypractitioners is still a long way from giving them the possibility to integrate their experience and technical knowledge withsampling data.

Fortunately, as is known, Bayesian estimators can incorporate the above two sources of knowledge by fusing both a priori andsample information in a timely and iterative way, i.e. as soon as the information is collected or updated.

The proposed Bayesian approach requires the specification of only those prior parameters in which the expert knowledge canbe really converted28. Under the assumption of Weibull distribution of the wind speed, stated before, the following paragraphshows the practical way to convert expert information (from both technical knowledge and past experience) into the priordistributions of the Weibull scale and shape parameters.

Prior and posterior distributions

In expert opinion, it is generally known that the shape parameter � does not vary considerably. A usually narrow interval (�1,�2)of possible values is easy to define, depending upon the experience of the considered geographic area. Therefore, a prior uniformpdf in such an interval is assumed as

f�(�)={

1 / (�2 −�1) 0<�1<�<�2

0 elsewhere.(11)

In order to define the prior distribution for the scale parameter �, the information about the mean annual wind speed watl anduncertainty, generally reported in wind atlases/databases, can be utilized as follows.

Conditional on �, let us consider a large number of wind speed observations (supposed to be Weibull distributed) and thecorresponding sample mean estimator W , which is approximately Gaussian (by the Central Limit Theorem) with mean ��(1+1 / �)and known variance �2

atl (deduced from wind atlases/databases). Let us furthermore assume a diffuse (i.e. locally uniform, by

using the Jeffreys rule) prior for � and consider watl the only observation of W , then the posterior pdf for � is given by O’Hagan29

f�(�|�)= �(1+1 / �)√2��atl

exp

{−1

2

(��(1+1 / �)−watl

�atl

)2}

(12)

becoming the prior before new data are collected. Thus, in the proposed procedure, the pdf (12) is assumed as prior distributionof � conditional on �. Strictly speaking, f�(�|�) is nonzero also for �<0. Fortunately, in the majority of applications (which usuallyinvolve windy areas) watl and �atl are such that the probability of negative values for � is negligible. The joint prior pdf f��(�,�)can be then obtained as the product of the pdf (11) and (12).

Combining the joint prior pdf with the likelihood (4) of the censored sample, the joint posterior pdf is given by

�(�,�|w)= L(�,�|w)f��(�,�)∫ �2�1

∫ +∞0 L(�,�|w)f��(�,�)d�d�

= L(�,�|w)f�(�|�)f�(�)∫ �2�1

∫ +∞0 L(�,�|w)f�(�|�)f�(�)d�d�

. (13)

As is known, the larger the sample, the more reliable the posterior estimates. In order not to disprove the method in the caseof an autocorrelated sample, the size must be reduced by using the aforementioned filtering algorithm thus avoiding the useof an incorrect likelihood. The Bayesian estimators of � and � are the expected values E(�|w) and E(�|w) based on the marginalpdfs �(�|w) and �(�|w) of �(�,�|w) in Equation (13). The Bayesian estimator of v is the expected value E(v|w) as follows:

E(v|w)=∫ �2

�1

∫ +∞

0��(1+1 / �)�(�,�|w)d�d�. (14)

However, since the expectations of the posterior marginal pdfs involve complex integral calculations, we must have recourse to anumerical technique. The outline of the proposed estimation procedure is given in Figure 1 and can be iteratively implementedusing �(�,�|w) as the joint prior pdf for the following step.

4. A case study using MCMC: A southern Italian site

In the last few years, Markov chain Monte Carlo (MCMC) technique30 has become popular in complex statistical models12. Byusing this method, samples from a required distribution (the posterior distribution, in our case) can be drawn by running asuitably constructed Markov chain for a sufficiently long time. The estimation procedure outlined in Figure 1 can be numerically

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Figure 1. The outline of the proposed estimation procedure

Table I. Data monitoring windows

A 1 March 2008 to 30 March 2008B 1 May 2008 to 30 May 2008C 1 September 2008 to 30 September 2008D 1 October 2008 to 30 October 2008

Table II. Effective number of non-autocorrelated observations (nh)

nh n

A 56 720B 52 720C 66 720D 48 720

Table III. �, Q and q0.95 values achieved by the filtering algorithm

� Q q0.95

A 0.168 16.549 22.362B 0.176 19.590 21.026C 0.400 23.478 24.996D 0.115 14.415 19.675

implemented by using WinBUGS31, a package designed expressly to carry out MCMC computations for a wide variety of Bayesianapplicative scenarios.

To show how the outlined procedure works, real sampling data are analysed. Data are collected from January 2008 in aSouthern Italian site, located on the border between Puglia and Campania. The current wind station (rotating-cup anemometer)is placed at a height of 30 m above ground level. The hourly mean wind speed is calculated from wind data collected every 10 s.The offset value wc is equal to 0.4 m/s.

In order to test the procedure of parameter estimation from a one-month sample, it is separately applied to the four (arbitrarychosen) monitoring windows reported in Table I, ignoring any available data outside such windows.

Data filtering

In each monitoring window (Table I) the effective numbers of non-autocorrelated observations nh, which is obtained by applyingthe filtering algorithm explained in Section 2, is considered (Table II) instead of the original number n=720 (total monthly hours).�, Q and q0.95 values achieved by the filtering algorithm are reported in Table III, for each monitoring window as well.


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Table IV. Seasonal coefficient estimates (second station)

k 2005 2006 2007 Mean

Winter 0.915 0.797 0.846 0.853Spring 1.005 1.084 1.238 1.109Summer 1.020 1.020 0.952 0.997Autumn 1.058 1.212 1.022 1.098

Figure 2. Italian wind atlas32

Seasonal coefficient estimates

Wind speed measurements are also available during the period 2005–2007 from a second survey station situated 2 km away fromthe current one at a height of 30 m.

Such measurements are utilized to estimate the seasonal coefficient k by using Equation (9) for each season of the threeavailable years (Table IV) and their mean values is considered, as indicated in Section 2. Then the data from the current stationare transformed by Equation (8), by using the corresponding mean value of the aforementioned seasonal estimates for eachmonitoring window of Table I.

Prior distribution parameters

As outlined in Figure 1, in order to start the procedure the prior distribution parameters must be anticipated. Consulting theItalian wind atlas32 (Figure 2) around the considered wind site, the values watl =6.5 m / s and �atl =1.25 m / s can be plausiblyassumed in Equation (12). Furthermore, on the basis of both the expert opinion on the second station and the fluid dynamic

tool of the widely used WindPro� software, the values �1 =1 and �2 =2 are set in Equation (11). According to practitioners andwind experts, the parameter � is related to the specific wind regime and does not vary greatly once the area of interest hasbeen defined.

Calculation and analysis of the results

A WinBUGS model is implemented to numerically calculate Equation (13) based on the filtered and seasonally adjusted observationswannual. In order to obtain the posterior expectations, sufficiently large samples from �(�,�|w), �(�|wannual) and �(�|wannual) mustbe drawn. For each data monitoring window (Table I) two chains are run for 5000 nodes (discarding the appropriate burn-invalues for each chain). Both convergence to stationary conditions and mixing of every two chains are visually tested by usingWinBUGS diagnostic tools. The mean value of �(�|wannual) and �(�|wannual), namely �B and �B, are the Bayesian estimates of the

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Figure 3. Posterior distributions and the correlogram with 95% confidence bounds (obtained before and after the filtering algorithm) based on the observationsin the monitoring window D (Table I)

Table V. Bayesian and ML estimates for each monitoring window (Table I)

�B �B vB �ML �ML vML

A 7.423 1.318 6.864 7.399 1.304 6.829B 5.756 1.429 5.250 5.375 1.407 4.896C 6.004 1.728 5.360 5.810 1.728 5.178D 6.501 1.220 6.115 6.129 1.178 5.793

annual parameters � and �, respectively. On the basis of the sample from �(�,�|w), we are able to draw a large sample v of thewind speed expected value v as well. The sample mean of v is the Bayesian estimate vB.

In order to show a typical output of the procedure, the posterior pdfs �(�|wannual) and �(�|wannual) as well as the correlogram(obtained before and after the filtering algorithm) corresponding to the monitoring window D are reported in Figure 3 (noessential differences are found with respect to the other windows). The graphical analysis of the correlograms with 95% confidencebounds (±1.96 /

√n) shows the absence of autocorrelation, which has been already tested at Step 3 of the data filtering algorithm.

In Table V the Bayesian estimates (�B, �B, vB) are compared with the ML (reference) ones (�ML, �ML, vML), carried out on the basisof the same (filtered and seasonally adjusted) sample.

The rather large bias sometimes observed in the estimates �B and �B as well as �ML and �ML (e.g. case A and B of Table V)generally has a low effect on vB and vML, respectively.

As data from the whole year 2008 monitoring window is also available in this case, the one-year ML estimates of the scaleand the shape parameters

�ML,year =6.32 m / s �ML,year =1.19 (15)

and of the wind speed expected value (by using Equation (3))

vML,year = �ML,year�(1+1 / �ML,year)=5.93 m / s (16)

are calculated too, on the actual (neither filtered nor seasonally adjusted) observations, following the common practice (e.g. Cartaand Ramirez4, Erdem and Shi33). These estimates—being those commonly required for business plans—represent the actualvalues to be targeted by the proposed Bayesian estimates (�B, �B, vB) and the ML ones (�ML, �ML, vML) from a one-month sample.The differences between one-month and one-year estimates are reported in Table VI.

The performances of the Bayesian and ML estimators are studied through a simulation based on the bootstrap. For eachmonitoring window of Table I, R=1000 bootstrap samples are repeatedly drawn from the filtered and seasonally adjusted(censored) data and both Bayesian (�B, �B, vB) and ML (�ML, �ML, vML) estimates, which are below indistinctly referred to as �j , are

calculated for each bootstrap sample j. In Table VII the mean, variance and covariance of the estimates (�1,. . . , �R) are reported.


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Table VI. Differences between one-month Bayesian estimates (resp. ML estimates) and the one-yearML estimates (13) for each monitoring window (Table I)

�B − �ML,year �B − �ML,year vB − vML,year �ML − �ML,year �ML − �ML,year vML − vML,year

A 1.105 0.127 0.927 1.081 0.113 0.892B −0.562 0.238 −0.687 −0.943 0.216 −1.041C −0.314 0.537 −0.577 −0.508 0.537 −0.759D 0.183 0.029 0.178 −0.189 −0.013 −0.144

Table VII. Bootstrap mean, variance and MSE (from the one-year ML estimates) of the Bayesian andML estimates for each monitoring window (Table I)

�B �B vB �ML �ML vML

A Mean 7.080 1.339 6.526 7.361 1.318 6.802Variance 0.427 0.018 0.324 0.628 0.016 0.477

Covariance 0.026 — 0.043 —MSE 1.008 0.040 0.671 1.797 0.032 1.226

B Mean 5.503 1.457 5.016 5.353 1.432 4.877Variance 0.277 0.024 0.217 0.330 0.021 0.262

Covariance 0.009 — 0.020 —MSE 0.942 0.094 1.065 1.261 0.079 1.385

C Mean 5.831 1.778 5.207 5.792 1.758 5.166Variance 0.174 0.029 0.133 0.197 0.028 0.153

Covariance 0.011 — 0.015 —MSE 0.410 0.373 0.667 0.474 0.350 0.747

D Mean 6.120 1.273 5.734 6.103 1.230 5.782Variance 0.416 0.016 0.336 0.599 0.032 0.579

Covariance −0.010 — −0.007 —MSE 0.455 0.022 0.377 0.645 0.034 0.603

The Mean Square Errors (MSE) from the actual one-year ML estimates in Equation (15) are furthermore calculated as follows:

MSE(�)= 1

R

R∑i=1

(�i − �ML,year)2. (17)

For monitoring windows A, B and C the ML estimator of the shape parameter � is slightly better than the corresponding Bayesianestimates. This is probably due to the support of the prior for � that is not well centred on the target value �ML,year. However, itis evident that the estimators of the wind speed expected value v and scale parameter �, carried out by the proposed Bayesianapproach, achieve smaller values for the MSE than the ML ones.

5. Conclusion

The proposed Bayesian approach allows the integration of prior information (obtained from atlases, databases and/or fluid-dynamicassessment) with sampling data and furnishes effective posterior information. Such an approach requires practitioners to specifyonly those parameters about which prior knowledge generally exists. Such knowledge can also incorporate predictions obtained

by commercial wind software (e.g. WAsP� , WindPro�).A filtering strategy is utilized in order to consider only the effective non-autocorrelated experimental information taken from

the intrinsically autocorrelated hourly wind speed measurements. Moreover, in order to exploit one-month data, a seasonaladjustment is proposed by means of a data transformation that does not affect the Weibull model assumption. This originaltechnique gives effective results on the considered case study and opens new research issues.

The MCMC technique, utilized to implement the Bayesian estimation procedure, is extremely flexible and it can be extendedto any alternative prior distribution and wind speed model (e.g. in case of unsuitability of Weibull distribution to represent the

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specific wind regime). The applicative example and the simulation study confirm that the proposed Bayesian estimates, based onboth one-month data and a priori information, can successfully compete with the usual one-year estimates.

Acknowledgements

The authors are extremely grateful to the anonymous referees and to the editor for their suggestions and help in significantlyimproving the manuscript. They also deeply thank Fortore Energia S.p.A. (www.fortorenergia.comH) for having transferred anemo-metric data and Engineer Massimo Lepore (Renewable Energy Sources Systems) for his experienced discussion useful in definingthe case study and in eliciting the prior information.

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Authors’ biographies

Pasquale Erto is a Senior Professor of ‘Statistics and Calculus of Probability’ at the University of Naples Federico II, Italy. He isa member of the ‘International Statistical Institute,’ the ‘IEEE Reliability Society’ and the ‘American Society for Quality.’ He is afellow of both the ‘Safety and Reliability Society’ and the ‘Royal Statistical Society’. Having graduated in engineering, he worked


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as a researcher at the National Council of Research and as a university professor, teaching in Faculties of Engineering, Medicineand Political Science. In 1974 and 1975, at the University of Calabria, he gave the first ‘Reliability Theory’ courses included inEngineering curricula in Italy. He was Associate Professor of ‘Reliability and Quality Control’ at the University of Naples until 1990.

He has worked with major industries as a statistical consultant on engineering and scientific applications involving reliabilityanalysis, maintenance and quality. He has been an invited lecturer on quality topics in Japan, Russia, Australia, Korea, Turkey. Hehas published two books and more than 100 papers in journals that include IEEE Transaction on Reliability, Journal of AppliedStatistics, Quality and Reliability Engineering International, Reliability Engineering & System Safety, Total Quality Management,Quality Technology and Quantitative Management, Quality Engineering, International Statistical Review.

Antonio Lanzotti is a Senior Professor in Design and Methods of Industrial Engineering at the University of Naples Federico II.Having graduated (cum laude) in Mechanical Engineering (1985), he received the PhD in Materials Technology and IndustrialEngineering (1990). He teaches ’Fundamentals and Methods of Industrial Eng. and Eng. Drawing; he taught Statistics for Innovationat the Faculty of Engineering and Robust Design at the Faculty of Architecture. He is Regional Editor of Int. Journal of InteractiveDesign and Manufacturing (Springer) and referee of international journals. He has published more than 90 papers in internationaljournals and conferences and has been an organizer and member of the Scientific Committee of International Conference. From2008, he is the President of the Italian Society for Quality Culture of South Italy. From 2004, he is the President of Italian Associationof Product Design.

Antonio Lepore is currently grant holder in the Department of Aerospace Engineering, University of Naples Federico II, Italy.He received his PhD degree in Total Quality Management and the MS degree (cum laude) in Industrial Engineering from theUniversity of Naples Federico II. He is a member of the Bernoulli Society for Mathematical Statistics and Probability and the ItalianSociety for Quality Culture in South Italy.

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Wind speed parameter estimation from one-month sample via Bayesian approach

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