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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 163565, 5 pages
Wind Farm Power Forecasting
1Laboratoire de Mathématiques (UMR CNRS 6620), Université Blaise Pascal (Clermont-Ferrand II), 63177 Aubière Cedex, France
2Computational Mathematics Laboratory, Monastir, Tunisia
Received 19 April 2013; Accepted 4 July 2013
Academic Editor: Wei-Jen Lee
Copyright © 2013 Nabiha Haouas and Pierre R. Bertrand. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Forecasting annual wind power production is useful for the energy industry. Until recently, attention has only been paid to the mean annual wind power energy and statistical uncertainties on this forecasting. Recently, Bensoussan et al. (2012) have pointed that the annual wind power produced by one wind turbine is a Gaussian random variable under a reasonable set of assumptions. Moreover, they can derive both mean and quantiles of annual wind power produced by one wind turbine. The novelty of this work is the obtainment of similar results for estimating the annual wind farm power production. Eventually, we study the relationship between the power production for each turbine of the farm in order to avoid interaction between them.
The energy industry is one of the most important types of modern industries. In recent years, wind power has become increasingly popular as a renewable energy source that can both develop the economy and protect the environment. Thereby, the development of suitable mathematical models becomes necessary. A detailed review on the existing tools used in wind power prediction is provided by , which proposes a perspective of future developments. A more recent update can be found in  where they used two alternative numerical prediction models: an empirical one and a computational one, in order to forecast the power output of two Greek wind farms before their installation. Different models for monitoring and forecasting the turbine output are considered such as those in the studies by the authors of [3–7] or  and recently in . But, to our knowledge, the performance of a wind power farm has not been adequately studied. In this work, we suggest a statistical analysis based on central limit theorem as in . Firstly, by using the wind speed, as input variables, we can forecast the annual energy production and its quantiles. Secondly, we study the relationship between the power outputs for each turbine in the farm to avoid the effect of interaction between them.
The rest of this paper is organized as follows. In Section 2, we present the dataset. In Section 3, we first recall the state on the art on the wind farm power forecasting, then we give the theoretical results for the forecasting of the annual wind power production, and we apply these results to real datasets. Moreover, we study the relationship between the power outputs for each turbine in the farm. Finally, discussions are available in Section 4.
2. Data Presentation
In this case study, we have processed ten-minute wind speed and ten-minute wind power production corresponding to a wind farm with four turbines. The duration of observation is 29 months leading to large series. The wind farm is located in a flat area close to the sea.
The wind farm power production depends on wind speed. Thus we begin with data representation. The wind speed series is intermittent; that is, it presents very irregular variations, as shown in Figure 1. This intermittency induces forecasting difficulties. A different approach has been considered, but also with a different horizon of time; see, for example, [10, 11] or . Let us point out that the previous methods take only into account the absolute value of wind speed, leading to 1D time series. Other modern methods include the 2D or 3D behavior of wind; see, for example,  or .
By a simple calculation, the average wind speed over the period is equal to 7.25 m/s, which is enough to guarantee a good profitability of the project but does not allow a detailed forecast.
Regarding the energy output, we can note the large amount generated by this wind farm which reaches 2870 kW on average every ten minutes. This perfectly fits the building of a wind farm on a flat area close to a windy sea. We can also plot the ten-minute wind speed versus the ten-minute power; see Figure 3. We then get a cloud of points around the nominal power law. Let us recall that the nominal power law is provided by the manufacturer and indicates the power produced for a given wind speed , which corresponds to the map (see Figure 2).
Moreover, the turbine is cut for wind speed outside the interval . The large dispersion in Figure 3 around the nominal power law is due to outliers and error of measurement. We see that the power (i)is null if the wind speed is less than the starting speed ( m/s) and beyond the cut out speed m/s, (ii)is proportional to the wind speed rise between and the rated speed (about 13 m/s), (iii)is constant between the rated speed and the cut out speed. Let us remark that the energy output is not the same for each of the four turbines. Indeed, there is a turbine that produces on average less than the others although the four turbines have the same power law, being of the same type. A possible explanation is the wake effect. In this frame, modeling by 2D wind series could enhance wind power forecasting (see Table 1).
3. Wind Farm Power Forecasting
3.1. State of the Art
The traditional method is based on modeling ten-minute wind speed probability density function (pdf) and then calculating the average ten-minute wind power production as , where denotes the probability of zero wind; , denote the wind speed for cutting in and cutting off the turbine; denotes the nominal power law; denotes the wind speed probability density function (pdf). Most often the ten-minute wind speed is assumed to follow a Weibull distribution, or a hybrid Weibull law; see, for example, [15, 16], or . This model gives a good estimation of both mean ten-minute wind speed and mean ten-minute wind power production and after that provides us with a good forecasting of the mean annual wind power production. However, this traditional method discards the time structure of wind speed and wind power and thus it does not allow to forecast the variance, nor the quantiles of annual wind power production.
Recently, the authors of  have proposed a new method which takes into account the dynamical structure of the annual wind power production. This method provides forecasting not only of the mean annual wind power production, but also of its variance and the quantiles of annual wind power production. The techniques rely on the central limit theorem (CLT) which asserts that the pdf of annual wind power production is almost Gaussian under natural assumptions. In the next section, we slightly adapt this method to the case of a wind farm.
3.2. Forecasting Wind Farm Power
The annual wind power production of the farm is defined by where denotes the number of ten-minute periods during one year. The farm production is the sum of the individual production of the turbines, that is, . The most simple but quite large model is to assume that each series is wide sense stationary. Wide sense stationary means that on the one hand both mean and variance are not depending on time and that on the other hand the dynamical structure read on the covariance does not depend on the time . As a corollary, the farm production series is also wide sense stationary with mean and variance as follows: and correlation coefficient which does not depend on the time . Moreover, the family of random variables is weakly dependent, and it admits a finite second-order moment (i.e., for each ). In addition, the annual wind production is also random with mean and variance as follows: where with which was introduced to variance analysis for characterizing wind energy conversion as in .
Let us stress that the variance of annual production of the farm depends both on the variance of ten-minute wind power and its correlogram , which corresponds to the time structure of the series. We will also need the two following assumptions. (A1) The second-order moment of is finite, that is, size ; (A2) The family of random variables is weakly dependent. After having made precise assumptions and notation, by using the same tricks as in Proposition 3.1 in , we can deduce the following CLT.
Theorem 1 (CLT for wind farm annual production). (i) If the family of r.v. is wide sense stationary, and assumptions (A1) and (A2) are fulfilled, then one has where and is a zero mean r.v. with variance 1 which converges towards a standard Gaussian law; that is, when .(ii) Moreover, one can deduce the quantiles of annual production as follows:
Remark 2. Theorem 1 is warrant under the assumptions (A1) and (A2), which correspond to the stationarity of wind series and consequently of power series . However, this result can be enhanced by taking into account the wind direction as in [13, 14] or by replacing the stationarity assumption by seasonality.
To sum up, the quantiles and the mean annual wind power production depend on three parameters, that is the mean of ten-minute wind power of the farm, the variance of ten-minute wind power, and which depends on the correlation structure of the wind power time series. Before starting the estimation procedures, we check the correlation coefficient as described previously, which proves that wind speed and the average wind farm power of four turbines are strongly correlated at a time scale smaller than ten hours and become uncorrelated at scale 48 hours (confirmed by Ljung-Box test; see e.g., ) as shown in the following Figure 4.
In order to carry out the overestimation of the mean wind farm power production, first, we take again the nominal power law proposed in  based on the use of a numerical sample of the ten-minute wind power derived from the sample of wind speed. Secondly, we use the traditional method which is modeled by a Weibull distribution law described in Section 3.1. We compare the overestimation rate for each turbine and for all the farm as illustrated in Table 2.
We should note that the overestimation of the mean power farm production based on a Weibull model is 10.43% and with the numerical wind power series it becomes 1.8%. The sources of this overestimation are out of the scope of this article. However, starting from the estimation of the autocorrelation coefficient, defined in Section 3.2, we proceed to calculate the mean and the quantiles of the annual power production for the whole farm. It is based on a sample of ten-minute wind power as a postproduction approach. Then, we use the new method, that is, the preproduction approach which has already been used in the computed precedents.
3.3.1. The Postproduction Approach
For the measurements of the existing turbines, we calculate the quantity provided by the entire farm and for each turbine, that is, , and we get the values stored in Table 3.
Then, we can deduce the mean and the quantiles of the annual power production (see Table 4).
3.3.2. The Preproduction Approach
First, we calculate , , , and based on the wind speed measurements provided by each turbine through power law. Thereafter, we follow the same procedure as mentioned previously. We get Table 5.
We can also forecast the twenty-year wind farm power production (20yPP), which corresponds to the lifespan of some turbines (see Table 7).
We find that the uncertainties decrease with the length of the forecast period, rising from 13.8% for one year to 3% for twenty years on total wind farm power, whereas it is decreased from 14% for one year to 3% for twenty years at each turbine.
3.4. The Relationship between the Measurements of Each Turbine in the Farm
To achieve a good characterization of the relationship between the power output and the wind speed for the four turbines, we calculate the correlation coefficients of wind power production (denoted by , , , and ) and the correlation coefficients of wind speed (denoted by , , , and ) of the four turbines. Let us recall that the correlation coefficient for two series and is calculated as . The correlations for the existing turbines can be summarized in Table 8.
Let us point out that the coefficient correlation is always superior to 0.9 for the wind power production of the four turbines. This assessment is of importance to appreciate the quality of the location of the turbines. We should take into account the effect of interactions between neighbour turbines which can eventually present loss in wind power production. The results obtained by the data generated through the power law are very close to this one. This once again underlines the contribution of the proposed approach to study the relationship that may exist between the energy outputs of wind turbines on a farm before their installation.
Similarly, regarding the correlation between the four wind speeds (denoted by , , , and ), we can distinguish the strong correlation between all the turbines which is nearly equal to 1 (see Tables 9 and 10).
The grouping of several wind turbines on the same site reduces the investment costs. However, it is important to make an optimal configuration of the turbines locations. For this wind farm site, there will be certain directions for which other turbines affect the production of single wind turbine. A more detailed analysis of this dependence will be made in further work.
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