Wind Speed Stochastic Models:A Case Study for the Mediterranean AreaGiovanni Bonanno, Riccardo Burlon, Davide Gurrera, Claudio Leone

Dipartimento di Fisica e Tecnologie RelativeUniversit degli Studi di Palermo

EMBED MS_ClipArt_Gallery

IntroductionNew power capacity installed in 2008 (source: EWEA)

IntroductionTop 10 installed wind power capacity at the end of 2008 (source: EWEA)

IntroductionSecond to second or minute to minute variations in wind energy production are rarely a problem for installing wind power in a grid.Wind energy production may, however, vary from hour to hour, just as demand from electricity customers will vary from hour to hour. In both cases, other generators on the grid have to provide power at short notice to balance supply and demand.The cost of providing this balancing service depends both on the type of other available generating equipment and on the predictability of the variation in net electricity demand.

IntroductionOn the spot market, demand and supply bids have to be submitted typically 12-48 hours in advance and by equalising demand and supply the spot prices are found for a 24-hour period.If forecast production and actual demand are not in balance, the balancing market has to be activated. This is especially important for wind-based power producers.Experience from different countries (Germany, Spain and Ireland) has shown that the accuracy of the forecast can be improved in several ways, ranging from improvements in meteorological data supply to the use of a combination of different methods in the prediction process.

Modelling...The idea of using a mathematical model to predict the behaviour of a physical phenomenon is well established.Sometimes it is possible to derive a deterministic model, and sometimes it is not...In many problems, we have to consider a time-dependent phenomenon in which there are many unknown factors or large experimental resources needed to produce a deterministic model are not available. In such cases, it may be possible to derive a stochastic model.

...and ForecastingAssuming a stochastic data generating process, an observed time series is regarded as a sample realization from an infinite population of such time series that could have been generated by the stochastic process.Although it may be possible to increase the sample size by varying the length of the observed time series, there will only be a single outcome of the investigated stochastic process and time series analysis is essentially concerned with evaluating the properties of this underlying data generating process from the observed time series.Once a hypothetical probability model to represent the data has been set up, it may be used to draw useful inferences from the time series.

SARIMA ProcessesMany time series contain a seasonal periodic component which repeats every s observations. We expect relationships to occur between adjacent observations and between observations separated by s units of time. A SARIMA(p,d,q) (P,D,Q)s process {Zt} is defined by the relation:

where B is the backward shift operator;F(BS), Q(BS), f(B) and q(B) are polynomials;{Wt} is an i.i.d. Gaussian noise.

Our ContributionOur work has aimed at providing a general class of stochastic models for hourly average wind speed time series, taking into account all the main features of wind speed data, namely autocorrelation, non-Gaussian distribution, seasonal and diurnal nonstationarity.

The proposed approach for wind speed modelling and forecasting has been applied to the HAWS time series recorded, from 2003 to 2006, in 29 sites of Sicily (Italy), at 10 metres above ground by the Servizio Informativo Agrometeorologico Siciliano.

The Spectral AnalysisCammarataMazzarrone

The Frequency DistributionCammarataMazzarrone

Data PretreatmentSeasonal nonstationarity is removed by a monthly stratification. After such a stratification, the only residual periodicity is expected to be the daily cycle typical of wind speed on a land area.The stochastic models employed in this study were rigorously developed for random variables possessing a Gaussian distribution. A Box-Cox transformation has been applied to the time series utilized in the present work, but in no case the transformed series have passed the Shapiro-Wilk test for normality. Hence, a new approach is presented.The transformed series have the same diurnal cycle of the original data. The proposed approach attains stationarity by the use of the differencing technique and employs seasonal models to allow for randomness in the daily cycle and to take into account the day to day correlation.

A Different TransformationThe new time series is obtained by solving the following equation for zt:

{vt} being the empirical non-Gaussian series, {P(vt)} the series of the estimated empirical cumulative probabilities, the mean and the standard deviation of {vt}.

The Proposed Class of ModelsThe proposed class

SARIMA (p,1,q)x(P,1,Q)24p, q, P, Q 3

has proved adequate for 93% of the analysed series.

Since it comes out that P 1, it results that it is possible to model the investigated stochastic process (however after a nonlinear transformation) by a suitable linear model with at most 10 parameters (in some cases just 3 parameters are sufficient).

ResultsThe developed automatic approach has provided satisfactory forecasts. Below we show the results for Cammarata (a) and for Mazzarrone (b). Poor predictions, more common during winter, are a consequence of sudden disturbances.

ResultsThe developed automatic approach has provided satisfactory forecasts. Below we show the results for Cammarata (a) and for Mazzarrone (b). Poor predictions, more common during winter, are a consequence of sudden disturbances.

Results

Results

Thank You