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Delegates are invited to meet and discuss with the poster presenters in this topic directly after the session 'How does the wind blow behind wind turbines and in wind farms?' taking place on Tuesday, 11 March 2014 at 16:30-18:00. The meet-the-authors will take place in the poster area.

Emilio Gomez-Lazaro Renewable Energy Research Institute, Spain
Sergio Martin-Martinez (1) F P Andrés Honrubia-Escribano (1) Miguel Cañas-Carretón (1) Emilio Goméz-Lázaro (1) Angel Molina-García (2)
(1) Renewable Energy Research Institute, Albacete, Spain (2) Universidad Politecnica de Cartagena, Cartagena, Spain

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Wind power forecast error probability distribution function using Pearson system for different timescales


The problem of accurately forecasting wind energy has represented a great deal of attention in recent years. There are always some errors associated with any forecasting methodology.

It is necessary for the Transmission System Operators (TSOs) and the market participants to understand these errors. Reserves addressed to system balance are calculated as a function of Wind Power Forecasting Errors (WPFE). As a result of this, WPFE have an important impact in market costs [1].

This paper addresses the problem of modelling the Probability Distribution Function (PDF) of WPFE using Pearson System to fit WPFE PDF for different timescales and production ranges.


Originally, wind power forecast error was been assumed to have a near Gaussian distribution. A simple analysis demonstrates that this is not valid. In the literature we can find several works about wind power forecast error distributions. The errors are often assumed to follow a normal distribution [2-5], though Weibull [6], Beta [7] and Cauchy [10] distributions have also been utilized. In [8], the distribution of wind power forecast errors is studied for different timescales between 6 and 48 hours ahead with a particular focus on the additional errors created from converting wind speed forecasts created by numerical weather prediction to wind power output. The study demonstrates that while the NWP errors are well represented by a Gaussian curve, the power forecast error distributions exhibit both skewness and excess kurtosis. In [9] the smoothing of forecast errors for multiple wind farms is showed at timescales from 6 to 48 hours ahead. In [7], variable kurtosis values between different time scales (with a minimum of ten-minute averaged output data) were measured, and model the error distributions using a beta function. This function was then applied to the sizing of an energy storage system that will act to smooth wind power output. In [10], The Cauchy distribution was proposed as a means of representing the forecast error distributions and was compared with some of the other distributions.

The Pearson system is a parametric family of distributions used to model a broad scale of distributions with various third and fourth moments. This method of moments is a statistical technique to estimate probability distributions by equating their theoretical moments with the moments of empirical distributions. This system has also been used in wind energy potential distribution analysis [11].

Main body of abstract

The main goal of the present work is to obtain PDF to fit WPFE distributions for different timescales and production ranges. Having established that the normal distribution is a poor fit for the forecast error distributions at the timescales under study, the next step in generating better forecasting intervals is to find a model that can more accurately represent the observed distributions. To accomplish this goal we have chosen Pearson system distributions that resemble those observed in the data.

Data used for the validation of the method selected consist on hourly WPFE measured for Spanish power system from 2007 to 2012 for four different timescales, 1, 6 ,12 and 24 hours. Additionally, hourly wind power generation is used to characterize WPFE in production ranges.

The presented method is developed as follows. First, WPFE data is organized in different bins according to the production range. These bins are defined from 0 to 1 pu referred to installed capacity every 0.05 pu, resulting 20 bins. Then, statistical parameters -- mean, standard deviation, kurtosis and skewness -- are calculated for each bin. Results for mean and standard deviation with a timescale of 12 hours during last years are shown in figures and , respectively. Bins with less than 50 data are not shown. The mean error trend is marked by a continuous decrease as production range increase. For standard deviation, dispersion is increased from 0 to 0.2 pu, then values stabilize for medium power bins and finally standard deviation decreases at high power bins for most of the years.

In figures and kurtosis and skewness values are represented. The kurtosis values have a strong decreasing trend with increasing bin. In addition to the strength of the peak, the symmetry of the distributions is another very important aspect of the forecast error distribution characterization. Low production bins are characterized by a pronounced asymmetry. For medium and high production bins, different results are obtained depending on year. Negative values are shown for high production bins representing left side asymmetry.

Using mean, standard deviation, kurtosis and skewness, a Pearson distribution is obtained. Pearson distributions are defined by a separable first order differential equation. According to the equation parameters, a K value is calculated. Depending on the value of K parameter, different types of Pearson curves can be obtained such:
1. If K < 0, roots are real and of opposite signs. This corresponds to beta distribution or type-I distribution in the Pearson system.
2. If K > 1, roots are real and have the same sign. This corresponds to beta distribution of the second kind, or type-VI distribution.
3. If 0 < K < 1, roots are complex. This corresponds to type IV-distribution.
Pearson proposed further class distinctions by taking into account certain distributions and boundaries between classes and classified the solutions into types numbered 1 to 12.

Finally, K parameter is obtained for every year and bin, and a distribution type is proposed in every case. Figure shows the evolution of K parameter and the ranges of the different Pearson distributions. Extreme bins are defined by the three kinds of distributions depending on the year, while medium bins are represented by Pearson IV distribution.


In this paper we have examined the shape of wind power forecast error distributions through a statistical parameters analysis. The wind power forecast models are becoming a key tool used to facilitate the integration of wind power in power systems with a great amount of wind power capacity.

The main statistical parameters of WPFE probability distributions have been analysed for every bin from 0 to 1 pu with a period of 0.05 pu. Evolution of PDF mean and standard deviation has been stablish as a function of wind power generation. Moreover, kurtosis and skewness values have been calculated to evaluate the shape of the real probability distributions.

Moreover, the distributions were found to differ greatly from the commonly assumed normal distribution. The kurtosis and the skewness of the distribution were found to vary with timescale and production bins. The Pearson system has been proposed as a means of representing the forecast error distributions and was compared with some of the other distributions used in the literature. Finally, the effect that the new forecast error distribution model has on the calculation of power system reserves confidence intervals was illustrated.

The most used distribution for data analysed is the Pearson type IV distribution. Nevertheless, in extreme bins, other distribution types, as beta and beta2 distribution, obtain a better fit of real data.

The use of the Pearson curve in approximating the probability distribution of WPFE has the advantages in terms of capability to take on variety shapes of distribution which makes it particularly applicable to the analysis of error determination characterized by random variability.

Learning objectives
This work, based on real measurements, contributes to a better knowledge of the probabilistic characteristics of the WPFE, which is the main parameter related to the reserves dispatched in power systems with great amounts of wind power capacity installed.

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