Monte Carlo Comparison of the Methods for Estimating the Weibull Distribution Parameters - Wind Speed Application

Wind energy represents a form of solar energy that appears due to uneven heating of the Earth. Approximately 12% of energy emitted by the Sun is transformed to wind energy. This kind of energy has been used for centuries as a drive for wind mills, water pumps or for ships. The interest in the wind energy has been regained recently because of its high potential in electrical energy production. It is motivated by the efforts to lessen the pollution and its negative impacts on the global environment since one of the major greenhouse gases producers are electric power plants through use of fossil fuels. On the other hand, wind provides a renewable, sustainable and environmentally friendly energy source. The most important parameter for modelling wind energy is the wind speed. It is a random variable changing noticeably in time and space and is influenced by local climatic conditions and terrain [1 2]. Thus, for wind energy applications it is necessary to describe variation of wind speed using statistical methods. In the literature several probability distributions (e.g. lognormal, Rayleigh, Weibull, gamma, normal) have been studied to describe the wind speed [3 6] For applications in wind energy studies the two parameter Weibull distribution is frequently used in modelling the wind speed distributions [4, 7 8] It is due to its easy implementation and flexibility. There are several methods for estimating the Weibull distribution parameters [5, 7, 9 12]. In recent years these methods have been compared several times and in different ways [13 17]. Since the Weibull distribution parameters play an important role in the wind speed applications, it is essential to compare a variety of methods for estimation of the Weibull distribution parameters in order to find the best fitting one. The suitability of the method may vary with the sample size, sample data distribution, sample data format and goodness of fit test [12]. In this paper we study six methods, namely, maximum likelihood method (MLM), the method of moments (MOM), the empirical method (EM), the empirical method of Lysen (EML), the power density method (PDM), the least squares method (LSM) and the weighted least squares method (WLSM). The performances of these methods are compared and discussed through the Monte Carlo simulation. The values of the parameters that are frequently occurring in the wind speed applications are chosen in simulation. The methods are compared using bias and the root mean square error (RMSE). All computations for simulation and parameter estimation are done using Matlab. MONTE CARLO COMPARISON OF THE METHODS FOR ESTIMATING THE WEIBULL DISTRIBUTION PARAMETERS – WIND SPEED APPLICATION Ivana Pobocikova Zuzana Sedliackova Maria Michalkova Florence George*


Introduction
Wind energy represents a form of solar energy that appears due to uneven heating of the Earth. Approximately 1-2% of energy emitted by the Sun is transformed to wind energy. This kind of energy has been used for centuries as a drive for wind mills, water pumps or for ships. The interest in the wind energy has been regained recently because of its high potential in electrical energy production. It is motivated by the efforts to lessen the pollution and its negative impacts on the global environment since one of the major greenhouse gases producers are electric power plants through use of fossil fuels. On the other hand, wind provides a renewable, sustainable and environmentally friendly energy source.
The most important parameter for modelling wind energy is the wind speed. It is a random variable changing noticeably in time and space and is influenced by local climatic conditions and terrain [1 -2]. Thus, for wind energy applications it is necessary to describe variation of wind speed using statistical methods.
In the literature several probability distributions (e.g. lognormal, Rayleigh, Weibull, gamma, normal) have been studied to describe the wind speed [3 -6] For applications in wind energy studies the two parameter Weibull distribution is frequently used in modelling the wind speed distributions [4, 7 -8] It is due to its easy implementation and flexibility.
There are several methods for estimating the Weibull distribution parameters [5, 7, 9 -12]. In recent years these methods have been compared several times and in different ways [13 -17]. Since the Weibull distribution parameters play an important role in the wind speed applications, it is essential to compare a variety of methods for estimation of the Weibull distribution parameters in order to find the best fitting one. The suitability of the method may vary with the sample size, sample data distribution, sample data format and goodness of fit test [12].
In this paper we study six methods, namely, maximum likelihood method (MLM), the method of moments (MOM), the empirical method (EM), the empirical method of Lysen (EML), the power density method (PDM), the least squares method (LSM) and the weighted least squares method (WLSM). The performances of these methods are compared and discussed through the Monte Carlo simulation. The values of the parameters that are frequently occurring in the wind speed applications are chosen in simulation. The methods are compared using bias and the root mean square error (RMSE). All computations for simulation and parameter estimation are done using Matlab.

Maximum likelihood method
The likelihood function of the Weibull distribution is given by By differentiating the logarithm of the function (2) with respect to k and c, respectively, and equating them to zero, we obtain The estimate k t of the parameter k can be obtained by solving (3) with respect to k. The estimate c t of the parameter c can be obtained using equation (4). The Newton method was used for the numerical computations.

Method of moments
The estimates of the parameters k and c can be obtained by equating the moments of the Weibull distribution with the corresponding sample moments. The shape parameter k and the scale parameter c can be estimated by following equations [3] c k

The Weibull distribution
The Weibull distribution is named after Waloddi Weibull who described it in detail in 1951. The probability density function f(x) and the cumulative distribution function F(x) of the twoparameter Weibull distribution are given by respectively, for x >0, k >0 and c >0. Here x is the wind speed, k is the dimensionless shape parameter and c is the scale parameter in units of the wind speed.
The mean E(X) and the variance D(X) of the Weibull distribution are given by a C^h is the gamma function defined by , a x e dx a 0 a x 1 It is known that the Weibull shape parameter k generally ranges from 1.5 to 3 for most wind speed conditions in the world [1]. The Weibull distribution is right skewed, reflecting the fact that the strong winds are rare while the moderate and fresh winds are more common. The higher value of the shape parameter k indicates more stability in the wind speed while the higher value of the scale parameter c indicates that the wind speed is higher.

Methods for estimating the Weibull distribution parameters
In this section we give a brief description of the methods that will be used for estimation of the Weibull distribution parameters. Let , , , X X Xn  be ordered observations. The estimates of the parameters k and c we denote k t and c t , respectively.

/ /
Based on our previous study [19] to estimate the values of the cumulative distribution function F(x) the mean rank is used

Weighted least squares method
The estimates of the regression parameters a and b minimize the function In this paper we use the weight factor proposed in [20] , The shape parameter k and the scale parameter

Empirical method
The empirical method can be used as a special case of the method of moments. The shape parameter k can be estimated by following equation [6,12] k s x .

Empirical method of Lysen
In the empirical method suggested by Lysen [18] the shape parameter k can be estimated using (8) and the scale parameter c can be estimated using . .

Power density method
The energy pattern factor is defined as [12] E x

Least squares method
The cumulative distribution function (1) can be linearized as follows Let ln ln Then the equation (9) can be rewritten as

Results and discussion
The results of the Monte Carlo simulation are presented in Tables 1 -3. From the simulation results, we observe the following with regards to bias and RMSE of the two parameters: •

Simulation study
A simulation study is conducted to compare the performance of the aforementioned methods for estimating the Weibull distribution parameters.
In simulation, we choose parameter values which appear frequently in the wind speed applications. We consider k=1.5, 2, 2.5, c=1 and sample sizes n=30, 100, 500, 1000 in order to cover small, medium and large sample sizes. For each combination k, c and n, we generate by the Monte Carlo simulation, N 5000 = random samples from the Weibull distribution using the inverting the cumulative distribution function (1) ln where U has uniform distribution over the interval , 0 1 h.
For each method discussed, we obtain 5000 estimates of the parameter k and 5000 estimates of the parameter c. Then we compute the sample means and the sample variances    performs better than the other methods discussed in the paper. MOM and PDM also perform well and can be considered as alternative methods for estimating the Weibull distribution parameters.

Acknowledgement
This paper was supported by the Slovak Grant Agency VEGA through the project No. 1/0812/17.

Conclusions
In this paper, we compared the performance of the maximum likelihood method (MLM), the method of moments (MOM), the empirical method (EM), the empirical method of Lysen (EML), the power density method (PDM), the least squares method (LSM) and the weighted least squares method (WLSM) for estimation of the Weibull distribution parameters. The comparisons were conducted in terms of the bias and root mean square error (RMSE) using the Monte Carlo simulation study. Based on the simulation results we have concluded that MLM Simulation results, k = 2.5, c = 1