Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis
Introduction
The Weibull distribution is a two-parameter function commonly used to fit the wind speed frequency distribution. This family of curves has been shown to give a good fit to measured wind speed data [1]. The Weibull function provides a convenient representation of the wind speed data for wind energy calculation purposes. It is important to note that the analysis presented here does not consider extreme wind speed analysis, for this see [2].
Before sufficiently powerful computers were widely available, the preferred method of calculating the Weibull parameters was a graphical technique which entailed generating the cumulative wind speed distribution, plotting it on special Weibull graph paper, and drawing a line of best fit. This procedure is now commonly implemented by performing a linear regression on a computer. A more accurate and robust approach, however, is given by the maximum likelihood method. The purpose of this paper is to demonstrate that the maximum likelihood method is a more suitable computer-based method for estimating the Weibull parameters. Both methods are presented and demonstrated in this paper, as are a variation of the maximum likelihood method and a qualitative method for estimating the parameters using the average wind speed.
Section snippets
The Weibull distribution
This family of curves is widely used in statistical analysis. In wind energy analysis it is used to represent the wind speed probability density function, commonly referred to as the wind speed distribution. The Weibull distribution function is given bywhere is the Weibull scale parameter, with units equal to the wind speed units, is the unitless Weibull shape parameter, is wind speed, is a particular wind speed, is an incremental wind speed,
Wind speed data
Measured wind speed data are commonly available in time-series format, in which each data point represents either an instantaneous sample wind speed or an average wind speed over some time period. An example of such data (giving hourly averages over a 24 h period) is given in Table 1. In some instances, wind speed data may instead be available in frequency distribution format. In this format, the frequency with which the wind speed falls within various ranges (bins) is given. An example of such
Determination of Weibull parameters
Three methods of estimating the parameters of the Weibull wind speed distribution are presented: two variations of the maximum likelihood method as well as the popular graphical method.
Demonstration of the methods
In this section, each method is applied to the same sample data set. The sample time-series data set is shown in Table 3. Table 4 gives the frequency distribution and the cumulative frequency distribution of this data set. For clarity, the sample data set consists of only three days of hourly wind speed data. It is important to note that a true evaluation would require many months or years of measured wind speed data.
Accuracy of the Methods
Two tests were employed to determine the accuracy of the three methods given in this article. In the first test, each method was applied to sample sets of wind speed data drawn from known Weibull distributions, and the estimated Weibull parameters were compared to the known values. In the second test, a wind turbine power curve was used to translate wind speeds into energy outputs. The “reference energy output” was calculated using the sample time-series data set shown in Table 3. This value
Conclusions
When wind speed data is available in time-series format, the maximum likelihood method is the recommended method for estimating the parameters of the Weibull distribution for wind energy analysis. The graphical method is not only less accurate than the maximum likelihood method, it is also less robust since its accuracy is affected by external variables such as the bin size in the cumulative frequency distribution. When wind speed data is available in frequency distribution format, the modified
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Ph. D. Candidate and ASME, CRES, ASES Member.