Real-time wind power forecast

10/04/2016 ∙ by Aurélie Fischer, et al. ∙ 0

We focus on short-term wind power forecast using machine learning techniques. We show on real data provided by the wind energy company Maia Eolis, that parametric models, even following closely the physical equation relating wind production to wind speed are out-performed by intelligent learning algorithms. In particular, the CART-Bagging algorithm gives very stable and promising results. Besides, we show on this application that the default methodology to select a subset of predictors provided in the standard random forest package can be refined, especially when there exists among the predictors one variable which has a major impact.

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1. Introduction

In France, wind energy represents today 3.9% of the national electricity production. The United Nations Conference on Climate Change COP21 has set a goal of 30% renewable energy in the overall energy supply in the country by 2020, and more precisely, the French wind production should double by 2020 [19].

Since electricity can hardly be stored, forecasting tools are essential to appropriately balance the production of the different renewable energies. Today, in France, wind energy is produced by more than 1400 wind farms scattered all over the country. The production of each wind farm is highly dependent of the meteorological conditions and especially of the wind. It is well known that the behavior of the wind is very different from one region to another, and this seems especially significant in France, where several quite different climates are present despite the relatively small area of the country [19]. So, to be accurate, the global wind electricity forecast should rely on local models, dedicated to each wind farm. Consequently, an important first step is to quantify the modeling performances of wind production in the different French regions, using real operational data.

Two kinds of framework are usually investigated today for wind power prediction. On the one side, physical models rely on the modeling of each wind turbine based on equations [5]

. On another side, a trend of new mathematical tools tends to model the power production by learning the phenomenon directly on the data, without integrating any knowledge on the physical behavior of the wind turbines. Such techniques using statistical models and data mining methods have been investigated in many complex situations, for instance considering short term prediction. Among others, parametric regression models, Support Vector Machines for regression, regression trees, random forests, neural networks have been considered. For instance, the use of neural networks has been investigated in

[18, 12] and in [17]. A special network, called extreme learning machine, has been used in [21] for probabilistic interval forecasting. The -nearest neighbor method has been studied for wind power modeling by [13]. In [15], the -nearest neighbor algorithm is used for probabilistic forecasts in the frame of the Global Energy Forecasting Competition 2014.

Support vector machines for regression have been proposed in this context in [11], whereas [13] provides a comparison between several data-mining approaches. Besides, time series-based models have also contributed to the field of wind power forecast (see, e.g., [16, 22]). For an overview of different modeling and forecasting methods for wind power, the reader may further refer to the surveys [5, 8, 9, 10].

In the present paper, adopting the second point of view, we investigate and compare different techniques for modeling the electrical power for several wind farms in France. For each farm, we first model the electrical power of each wind turbine of the farm using local inputs coming from sensors directly installed on each wind turbine. The predictive power of the farm is then given by the sum of the predictive powers computed for each wind turbine. In a second step, we quantify the modeling performances by using more global inputs as may be provided by a meteorologist forecaster as for example, Météo France. This approach helps to quantify the performance of the different models running in an operational environment, using only average input information at a farm scale.

The CART-Bagging algorithm appears to perform the best on our data and gives very satisfactory predictions.

The paper is organized as follows. In Section 2, we thoroughly describe the data set at hand. Section 3 introduces the different methods investigated in our study. Section 4 presents and discusses the modeling performances obtained using the local information on each turbine. The results found when replacing this information by the more global one, relying on averages, are given in Section 5.

2. Data set

The data set has been provided by the wind energy company Maïa Eolis. In a farm, each wind turbine provides 10 minute measurements of electrical power, wind speed, wind direction, temperature, as well as an indicator of the working state of the turbine. The electrical power output of the whole farm is also provided on a 10 minute basis. All measures are recorded simultaneously. Data is available for 3 different farms made up of 4 to 6 turbines, in the North and East of France, from 2011 to 2014.

To detect freeze, wind speed is measured on each turbine both by a classical anemometer and a heated one. Since more measures are available from the heated anemometer, the study has been conducted with this data. Wind direction is provided by a weather vane and has been recoded into two variables corresponding to the cosine and the sine of the angle. The state of the turbine may correspond to start, stop or full working of the turbine, depending on the wind speed and maintenance operations. For the sake of simplicity, this study focuses on fully operating times. Besides, the data has been averaged over 30 minutes in order to slightly smooth the signals. However, it should be stressed that most often the results obtained on a 10 minute basis are quite similar to those presented in the sequel.

Taking advantage of the 30 minutes averages, two new variables have been introduced: the variance of the wind speed, and the variance of the wind direction over 30 minutes. The second variable (complex-valued), has been decomposed into its real part and its imaginary part, leading to a total of 7 explanatory variables.

3. Predictive methods

In this section, all the measures are assumed to be observed in real time. Based on this data, our aim is to model the farm power. More precisely, the variables are observed at time and the sum of the power of each turbine of the farm at time

is predicted. We recall that the studied model is applied at each turbine, providing an estimate of its power. Then the estimated farm power is computed by summing the estimated turbine powers. The error is calculated at the farm scale.

Our intention is to compare parametric statistical methods closely reflecting the related physical equation, to more elaborated techniques inspired from machine learning. These methods are especially designed to learn a phenomenon in a completely agnostic way and may be suitable for high dimensional data or complex data. In particular, they can easily accommodate non-linear modeling as well as dependence between variables, which is the case here.

3.1. Theoretical equation

According to theoretical studies on wind turbines (see, e.g., [14]), the delivered power obeys the following equation :

(1)

where is the wind speed, the air density, the rotor surface, which is the area swept by the blades, and the power coefficient, corresponding to the fraction of wind energy that the wind turbine is able to extract. Thus, as expected, the power significantly depends on the wind speed and a good approximation of the power curve could lead to good predictions using wind speed measurements. Figure 1 shows the raw observations and the fit to the theoretical curves for a wind turbine. Figure (a)a plots power versus wind-speed, whereas Figure (b)b plots power versus the cube of the wind speed. The two plotted theoretical curves correspond to two different values of : the maximal theoretical value (, red curve), and a more realistic value given in Table 8 of [4] (blue curve). The third curve (in green) is provided by the turbine builder, based on his experiments.

Notice that the cloud of observations is quite dispersed and we can already anticipate difficulties for prediction.

In particular, it should be underlined that the parameters of the physical equation (1) are in practice difficult to guess, so that the theoretical curves may not fit very well. Furthermore, to better reflect the observations, the theoretical formula is often used only for a range of wind speeds, outside which the power is assumed to be constant. However, the knowledge of this range requires the estimation of both endpoints of the interval.

Although these curves correspond to some trend, there is obviously room for improvement to produce a better prediction.

(a)
(b)
Figure 1. Empirical observations for a wind turbine and theoretical power curves for different power coefficient values, compared to the curve provided by the turbine builder.

3.2. Parametric methods

Several methods have been tested to approximate the power curve and model the production. In this section, we present the parametric statistical methods, directly inspired from the physical equation.

Parametric modeling according to the wind speed only

We first investigated the simplest parametric models, namely linear regression and logistic regression, with the wind speed as unique explanatory variable. If the predicted power at time

is denoted by , the linear model is given by

where denotes the wind speed at time , and and

are computed using ordinary least squares (OLS). The logistic regression model may be written

where the parameters , are obtained using maximum likelihood.

Introducing a third degree polynomial of the wind speed in the logistic regression has also been considered to mimic more closely Equation (1). More precisely, the model is then defined by:

where , and are estimated parameters. This model is denominated in the sequel as polynomial logistic regression.

3.2.1. Parametric modeling using more variables

Linear regression, logistic regression and polynomial logistic regression have also been studied with more variables, using not only wind speed as a predictor, but also wind direction, (coded by its cosine and sine : and ), temperature and the variances of the wind speed and direction, and . Denoting for each variable by its value at time , the corresponding equations describing the estimated power may be written as follows.

(2)
(3)
(4)

In the last equation, corresponding to polynomial logistic regression, only the wind speed occurs in the expression as a polynomial of order 3, to be consistent with the theoretical equation (1). All the parameters of the different methods are estimated as in the previous paragraph.

The Lasso method, which simultaneously performs variable selection and regularization through the least squares criterion penalized by the norm of the regression coefficients has been investigated as well (see for instance [20]). For this model, the predicted power at time is a linear combination of all the previous variables as in equation (2), the coefficients being estimated with OLS, under the usual constraint for some constant .

3.3. Non-parametric methods and machine learning algorithms

It is well-known that non-parametric and non-linear methods are very useful to model complex phenomena. The following algorithms do not generally lead to closed formulas as in the previous section. We will describe them briefly and refer to the literature for more details.

3.3.1. Knn

The -nearest-neighbor procedure consists in computing the average power corresponding to the nearest neighbors in the feature space (see for instance [7]). More precisely, the rule is given by

where corresponds to the wind power of the -th nearest neighbor of the observation at time , according to the Euclidean distance of the variables at time . The features are standardize to have mean 0 and variance 1 since they are measured in different units.

As for the previous method, the number of neighbors is optimized on a grid.

3.3.2. CART, Bagging and RF

Tree-based methods like CART [3] and Random Forests [2] are also applied. CART grows a binary tree by choosing at each step the cut minimizing the intra-node variance, over all explanatory variables already introduced in equation (2) (here denoted by , ) and all possible thresholds (denoted by hereafter). More specifically, the intra-node variance, usually called deviance, is defined by

where (respectively ) denotes the average of the wind power in the area (respectively ). Then, the selected variable and associated threshold is given by . To avoid over-fitting, the tree is usually pruned by cross-validation. The prediction is provided by the value associated to the leaf in which the observation falls.

Another way to reduce variance and avoid over-fitting is to use Bagging [1]. Bagging consists in generating bootstrap samples, fitting a method on every sample (here growing a full tree by CART) and averaging the predictions. Hence, for bootstrap samples, the predicted power is given by

(5)

where denotes the power predicted by CART for the -th bootstrap sample. To produce more diversity in the trees to be averaged, an additional random step may be introduced in the previous procedure, leading to Random Forests. In the Random Forests procedure, each tree is grown following the same principle as in CART (with no pruning), but, here, the best cut is chosen among a smaller subset of randomly chosen variables. The predicted value is the mean of the predictions of the trees, as in (5).

3.3.3. SVM for regression

The SVM method for regression maps the inputs into a non-linear feature space, using a kernel representation, for example a Gaussian kernel (see for instance [6]). A non-linear regression function is computed by minimizing the sum of the losses on the points giving rise to an error exceeding some threshold. The threshold parameter is here calibrated using a grid.


In the next section, all the experiments have been conducted using the R software. The previous procedures are implemented respectively in the packages lars, kernlab, FNN, rpart and randomForest. For the Random Forests, the default parameters, advocated by Breiman, were used: 500 trees were grown in each forest and the size of the subset of randomly chosen variables, commonly denoted by mtry, is the floor of the third of the number of variables. Note that the CART-Bagging algorithm is a particular case of Random Forests where mtry equals the total number of variables.

3.4. The naive method

Finally, the so-called “persistence method” uses the last observation as prediction: if denotes the electric production at time , the predicted production at time is given by . It is interesting to introduce this very naive method as a benchmark in comparison to more sophisticated methods to precisely quantify their gain.

3.5. From turbine to farm modeling

As mentioned, the evaluation of the performances is made at the farm scale. Therefore, each turbine is modeled using the evaluated method, then the estimation of each wind turbine power is provided on test points. Finally, the estimated power of the farm is computed by summing theses estimations. More precisely, if the farm comprises six turbines and the linear regression is considered, six linear regression models are adjusted, then predictions for the test set are computed on each turbine: , and finally the estimation of the farm power is given by .

4. Modeling performance results

As we are interested in evaluating the predictive power of each method, the data set is split as usual into a training and a test set. In order to quantify the variability of the predictive ability, several test sets are used. An average performance, as well as a standard deviation, are then computed.

More precisely, the procedures are trained on around instant-points and 10 data sets of 724 points are used to evaluate the performances. The error criterion is the Root Mean Squared Error (RMSE), defined between a vector of predictions and a vector of observed wind power productions by

A quantity which is also of interest for industries is the error in term of percentage of the installed power (% of IP in the results tables), defined by the average RMSE divided by the theoretical power of the farm. For example, if the farm is composed of 6 turbines of theoretical power GW (specified by the turbines builder), the error in term of percentage of the installed power is

This quantity sometimes appears under the denomination Normalized RMSE.

Method Mean of RMSE Sd of RMSE % of IP
Persistence 855.52 141.14 6.96
using wind speed only Linear Regression 373.61 86.91 3.04
Logistic Regression 404.86 76.74 3.29
Polynomial Log. Reg. 290.36 73.87 2.36
CART 314.46 57.74 2.56
CART-Bagging (=RF) 250.52 46.52 2.04
SVM for regression 269.94 64.21 2.19
using all variables Linear Regression 364.21 102.39 2.96
Logistic Regression 362.76 107.58 2.95
Polynomial Log. Reg. 292.57 100.53 2.38
LASSO 364.21 102.39 2.96
CART 314.46 57.74 2.56
CART-Bagging 203.50 39.72 1.65
RF 425.78 161.53 3.46
SVM for regression 382.16 134.34 3.11
kNN (k=2) 355.29 109.96 2.89
Table 1. Modeling performances using local measures for one farm (IP= Installed Power).
Figure 2. Boxplots of the RMSE for the different procedures using local measures for one farm.

Let us comment the main conclusions drawn thanks to Table 1 and Figure 2.

4.0.1. General observations

As can be observed in Table 1, the learning algorithms have been investigated either using the wind speed variable only, in which case the emphasis is on the non-linear added value of the method, or using all variables, insisting then on both the non-linear and regularization aspects.
We first observe that all the methods investigated show a much better performance than the naive persistence method, substantially reducing the mean error, with a standard deviation almost always better.

4.0.2. Wind speed only

Concerning the methods using only the wind speed as predictor, their performances are pretty good, more than twice better than persistence.

The polynomial logistic regression shows a very good performance, which was expected since this model is directly inspired from the physical equations as illustrated in Figure 1. However, the variability of the prediction is a bit high.

The SVM and RF methods for regression show the best results with the best stability. Note that in that particular case, RF and the CART-Bagging procedure coincide.

4.0.3. All variables

Regarding the parametric methods, the results show that adding more variables, namely the wind direction, the variances of the wind speed and direction, and the temperature, do not lead to any substantial improvement. Among these procedures, polynomial logistic regression shows the best performances.

The LASSO procedure is not very promising. This is probably due to multiple factors : the method uses the predictors in a linear way – compared to SVM or CART, which are bringing different kinds of non-linearity – and the predictors are highly correlated. We observe also that the results are the same for LASSO and the classical linear regression due to the fact that no selection has been in fact performed by the method : all the variables are kept.

It appears that the CART algorithm does not take advantage of the additional variables and seem to choose its cuts only according to the wind speed. This may be explained by the prevailing importance of the wind speed over other measures.

Among the agnostic machine learning algorithms, the SVM shows one of the poorest performances. It should be noted that several tested kernels were not able to compete with the polynomial logistic regression for example.

The kNN method has a performance similar to the SVM procedure.

The CART-Bagging algorithm outperforms all the investigated statistical models. The case of Random Forests is quite interesting and has to be discussed separately. Looking at the Table 1 and Figure 2, we can observe that RF surprisingly seem less efficient than other methods and especially CART when dealing with all variables. However this poor result has to be refined.

As explained above, the RF algorithm, instead of considering all the variables to grow a tree (as CART does), operates a random selection among these variables. The default choice for this random selection is the uniform distribution to choose a subset of the original variables, of size

mtry, the floor of the third of the number of predictors. In the CART-Bagging procedure, all the variables are selected. In our data set, obviously, the importance of the wind speed prevails over all other variables: for instance, CART performs nearly all its cuts according to the wind speed. Therefore, if the wind speed variable is often not selected in a random sample, the resulting cut is often not appropriate. Choosing more variables increases the probability to select a specific variable, namely, here, the wind speed. Very different performances are then observed between RF with the default parameter for mtry and the CART-Bagging method, corresponding to RF with mtry equal to the number of predictors.

Comparing CART and CART-Bagging highlights the advantages of bootstrapping and averaging. This step allows to reduce the error by a third, when dealing with all the predictors.

Note that, according to the renewable energy union [19], French industries obtain a root mean squared error of 2.4 % of the installed power of farm productions, which illustrates the benefits of using CART-Bagging (1.65 %).

4.0.4. Comparison of different farms

The results given in the previous paragraphs concern a farm in the East of France. Data from two different sites in the North of France were also available. For every farm, the hierarchy between procedures is quite similar, the procedure ranking first most often is CART-Bagging.

To make a fair comparison between the farms, a new experiment has been conducted. A common test set, with observed variables available at the same time for each farm, with at least one turbine fully operational, has been drawn. The test set has been divided into ten subsets of 1440 instant-points, each covering a period of around thirty days, to quantify the average performance and its variability. The training set consists in around 7200 instant-points, satisfying a ratio of of the data dedicated to learning and used for test.

Only the best procedure, CART-Bagging, has been applied. We also compare the results with the turbine builder’s power curve, used on each turbine to model the farm and represented by the green curve in Figure 1. Figure 3 highlights the good results of CART-Bagging on the first and the third farms. It performs reasonably well on the second farm, but is not as good as the power curve’s builder. It may be explained by the difference between the wind speed in the training sample and in the test set. Few high wind speed levels are observed in the training sample on the second farm compared to the test sample, so the CART-Bagging prediction may not be accurate.

Figure 3. Comparison of the RMSE for the turbine builder’s power curve and the CART-Bagging procedure on several farms using local measures.

5. Towards forecast : a stability investigation

Forecasting electrical power requires two steps: one is to provide forecasts of the explanatory variables, and the other one, which is our aim here, consists in constructing an accurate model to plug the previous forecasts in. Indeed, if we have at hand an efficient model, then the performance of a forecasting procedure of electrical wind power will directly depend on wind forecasts. Up to now, the best wind forecasts are directly inspired from physics: these meteorological forecasts are obtained thanks to ensemble methods based on numerical computations, mostly based on the Navier-Stokes equations, like the climate reanalyzes performed for instance by the European Centre for Medium-Range Weather Forecasts (ECMWF), or, for France, the French Weather Agency Météo France.

At a farm scale, on a daily use, many observations are recorded in real time on each wind turbine. For example, as already mentioned, each turbine has its own anemometer and vane, which provide very localized information about wind speed and wind direction. Thanks to the analysis conducted in the previous section, we have been able to identify an accurate model, built with this kind of observations. It should be noted that, in a wind farm, in general, two wind turbines are at a distance of about 300 m from each other. However, concerning numerical models, for instance, the finest grid resolution for forecast of wind and temperature provided by the French Weather Agency Météo France is brought by the AROME model, which proposes a resolution of about km (5 times larger). Consequently, an interesting question is also to quantify the predictive power not using very local information, but information on a much broader scale.

To mimic the scale of meteorological wind forecasts, which are in the frame of this project not available, we decided to introduce virtual sensors: for each variable, a global information is computed by averaging all the localized variables coming from the set of turbines installed on the wind farm. Studying, thanks to this kind of data, which is less precise and less spatially localized than the true wind information, the stability of the different procedures, helps to quantify the loss of accuracy due to the replacement of all the localized data with a unique global information and is a first step towards forecast.

The same methods have been used and the results are available in Table 2. The deterioration of the prediction can easily be seen in Figure 4. We observe that polynomial logistic regression is remarkably robust, performing similarly to the context with local measures, contrary to SVM and kNN. When only wind speed is considered, polynomial logistic regression competes with CART-Bagging, whereas the latter outperforms all the considered procedures when dealing with all the variables. Therefore, we argue that CART-Bagging should be a good choice for providing electrical forecasts in combination with appropriate meteorological wind forecast inputs.

Method Mean of RMSE Sd of RMSE % of IP
Persistence 855.52 141.14 6.96
using wind speed only Linear Regression 393.09 77.25 3.20
Logistic Regression 541.37 103.15 4.40
Polynomial Log. Reg. 288.28 75.23 2.34
CART 349.17 53.20 2.84
CART+Bagging (=RF) 293.26 48.96 2.38
using all variables Linear Regression 387.71 89.73 3.15
Logistic Regression 524.30 92.58 4.26
Polynomial Log. Reg. 297.16 92.79 2.42
LASSO 387.44 89.86 3.15
CART 349.17 53.20 2.84
CART + Bagging 228.75 43.35 1.86
RF 447.77 161.84 3.64
SVM 424.15 143.02 3.45
kNN 428.05 125.84 3.48
Table 2. Modeling performances using deteriorated wind measures (average).
Figure 4. Boxplots of the RMSE for the different procedures using the mean of wind speed measures.

5.0.1. Comparison of different farms

Just as in the previous framework, CART-Bagging and the turbine builder’s power curve prediction have been tested on several farms. Figure 5 stresses the good results of CART-Bagging, which seems robust to the difference between the mean wind speed and the local wind speed on each turbine, contrary to the use of the power curve, suffering from the aggregation of sensor data.

Figure 5. Comparison of the RMSE for the turbine builder’s power curve and the CART-Bagging procedure on several farms using local measures.

6. Conclusion and perspectives

A first interesting conclusion that may be drawn from our study is that, depending on the wind farm, a method which is the best with true local wind information inputs may not perform well any more when using averaged data designed to mimic meteorological wind forecasts. So, despite the good performance of the constructor power curve for one farm, CART-bagging shows to be more robust when turning to aggregated data.

More generally, this observation raised the following question: in the frame of this work, the data, provided by the company Maïa Eolis, comes from 3 wind farms, all located in the North and East of France, the turbines installed by the company so far being essentially located in these regions, but it could be of prime interest to have access to wind data from farms in other regions of France, in order to check if there is some universality in the good behavior of the CART-bagging procedure. In other words, does it remain everywhere the best method for aggregated inputs? Note that, in the comparison of data-mining approaches conducted by [13] for US wind farm data, the kNN method, which does not perform particularly well in our study, appears to outperform the other methods, which reinforces the usefulness of continuing our modeling task for other French wind farms.

Here, we calibrated our models using stationary data, that is, data corresponding to full functioning of the wind turbines. A complementary work may be to enrich our models including the time slots where wind turbines are working in a non stationary regime (corresponding essentially to start-up regime). This would allow to compute predictions over a (complete) long time of use, taking into account transitory phenomena of a turbine.

Another direction for future research regards the final step of effective forecasting. In fact, if we get access to meteorological wind forecasts provided by Météo France or the ECMWF as mentioned above, an intermediate step has to be accomplished before simply plugging this information in our models. Indeed, these previsions suffer inevitably from a noticeable bias due to several causes, which has to be corrected in order to build accurate forecasting at the end. For example, the wind speed prevision is provided by the meteorologists for a given height, but the wind speed at the height of the wind turbine may be different. This shows that it is necessary to elaborate a so-called downscaling method, in other words to find the best possible relationship between real wind at a wind turbine and the meteorological wind forecasts at hand.

Acknowledgements

We are extremely grateful to Nicolas Girard and Sophie Guignard from the Maïa Eolis Company, for providing the data and for helpful discussions.
This research has been supported by a public grant overseen by the French National Research Agency (ANR) as part of the project FOREWER (reference: ANR-14-CE05-0028).

References

  • [1] Leo Breiman, Bagging predictors, Machine Learning, 1996, pp. 123–140.
  • [2] Leo Breiman, Random forests, Machine Learning 45 (2001), no. 1, 5–32.
  • [3] Leo Breiman, Jerome H. Friedman, Richard A. Olshen, and Charles J. Stone, Classification and regression trees, Wadsworth Statistics/Probability Series, Wadsworth Advanced Books and Software, Belmont, CA, 1984. MR 726392
  • [4] C. Carrillo, A.F. Obando Montaño, J. Cidrás, and E. Díaz-Dorado, Review of power curve modelling for wind turbines, Renewable and Sustainable Energy Reviews 21 (2013), 572 – 581.
  • [5] Alexandre Costa, Antonio Crespo, Jorge Navarro, Gil Lizcano, Henrik Madsen, and Everaldo Feitosa, A review on the young history of the wind power short-term prediction, Renewable and Sustainable Energy Reviews 12 (2008), no. 6, 1725–1744.
  • [6] Harris Drucker, Christopher J. C. Burges, Linda Kaufman, Alex J. Smola, and Vladimir Vapnik, Support vector regression machines, Advances in Neural Information Processing Systems 9 (M. C. Mozer, M. I. Jordan, and T. Petsche, eds.), MIT Press, 1997, pp. 155–161.
  • [7] E. Fix and J. L. Hodges, Discriminatory analysis, nonparametric discrimination: Consistency properties, US Air Force School of Aviation Medicine Technical Report 4 (1951), no. 3, 477+.
  • [8] Aoife M Foley, Paul G Leahy, Antonino Marvuglia, and Eamon J McKeogh, Current methods and advances in forecasting of wind power generation, Renewable Energy 37 (2012), no. 1, 1–8.
  • [9] Gregor Giebel, Richard Brownsword, George Kariniotakis, Michael Denhard, and Caroline Draxl, The state-of-the-art in short-term prediction of wind power: A literature overview, Tech. report, ANEMOS. plus, 2011.
  • [10] Jaesung Jung and Robert P Broadwater, Current status and future advances for wind speed and power forecasting, Renewable and Sustainable Energy Reviews 31 (2014), 762–777.
  • [11] Oliver Kramer and Fabian Gieseke, Short-term wind energy forecasting using support vector regression, Soft Computing Models in Industrial and Environmental Applications, 6th International Conference SOCO 2011, Springer, 2011, pp. 271–280.
  • [12] Oliver Kramer, Fabian Gieseke, and Benjamin Satzger, Wind energy prediction and monitoring with neural computation, Neurocomputing 109 (2013), 84–93.
  • [13] Andrew Kusiak, Haiyang Zheng, and Zhe Song, Wind farm power prediction: a data-mining approach, Wind Energy 12 (2009), no. 3, 275–293.
  • [14] M. Lydia, S. Suresh Kumar, A. Immanuel Selvakumar, and G. Edwin Prem Kumar, A comprehensive review on wind turbine power curve modeling techniques, Renewable and Sustainable Energy Reviews 30 (2014), 452 – 460.
  • [15] Ekaterina Mangalova and Olesya Shesterneva, K-nearest neighbors for {GEFCom2014} probabilistic wind power forecasting, International Journal of Forecasting (2016), –.
  • [16] Michael Milligan, Marc Schwartz, and Yih-Huei Wan, Statistical wind power forecasting for us wind farms, the 17th Conference on Probability and Statistics in the Atmospheric Sciences/2004 American Meteorological Society Annual Meeting Seattle, Washington, January11-15, 2004.
  • [17] Hao Quan, Dipti Srinivasan, and Abbas Khosravi, Short-term load and wind power forecasting using neural network-based prediction intervals, IEEE transactions on neural networks and learning systems 25 (2014), no. 2, 303–315.
  • [18] George Sideratos and Nikos D Hatziargyriou,

    Probabilistic wind power forecasting using radial basis function neural networks

    , IEEE Transactions on Power Systems 27 (2012), no. 4, 1788–1796.
  • [19] Syndicat des Énergies Renouvelables, Panorama de l’électricité renouvelable en 2015, Tech. report, Syndicat des Énergies Renouvelables, 2015.
  • [20] Robert Tibshirani, Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society. Series B (Methodological) 58 (1996), no. 1, 267–288.
  • [21] Can Wan, Zhao Xu, Pierre Pinson, Zhao Yang Dong, and Kit Po Wong, Probabilistic forecasting of wind power generation using extreme learning machine, IEEE Transactions on Power Systems 29 (2014), no. 3, 1033–1044.
  • [22] Bingheng Wu, Mengxuan Song, Kai Chen, Zhongyang He, and Xing Zhang, Wind power prediction system for wind farm based on auto regressive statistical model and physical model, Journal of Renewable and Sustainable Energy 6 (2014), no. 1, 013101.