Determining Weibull Parameters in Wind Speed Distribution

The precise estimation of Weibull parameters in wind speed distribution is crucial for analyzing and developing wind energy projects. In this article, we provide a practical step-by-step example that demonstrates how to accurately determine these parameters. Through statistical analysis and the use of mathematical methods, you will learn how to estimate the scale parameter (c) and shape parameter (k) of the Weibull distribution. A deep understanding of these parameters is essential for planning and optimizing wind energy production. However, it’s important to note that the effectiveness of the methods may vary depending on the wind speed conditions, i.e., the specific study location.

Example statement

We have available the data for the average magnitude of wind speeds, measured every hour at a height of 120 meters, for an entire year. This provides us with a total of 8,760 data points of wind speeds.

Wind data used in the application: Download

Figure 1 displays the magnitudes of wind speeds over the course of a year from the data obtained from the previous link. Analyzing wind speeds in this manner is complex and not suitable for analysis.

Figure 1: Magnitude of wind speed during one year, recorded hourly.

On the other hand, representing wind speeds in the form of a histogram with wind speed intervals of 1 m/s is much simpler and easier to understand.

Figure 2: Histogram of wind speeds during one year.

Below, we obtain the k and c parameters of the Weibull distribution obtained using different numerical methods.

a) By the least squares estimation method

The 8760 data points are divided into equal wind speed ranges to calculate the wind speed distribution and the distribution of relative frequencies. Table 1 shows the established ranges and the calculations performed to determine the “k” and “c” parameters of the Weibull distribution.

Table 1: Table of relative frequencies, cumulative frequencies, and auxiliary calculations.

Then, using the equations to calculate “a”, “b”, and “c”, the following values are obtained:

a = k = 3.334 ; b = -6.808 ; c = 7.707

Figure 3 shows the plot of the least squares linear fit.

Figure 3: Adjustment by least squares method.

Therefore, the Weibull distribution:

b) By the energy pattern factor method

The value of the Energy Pattern Factor is calculated using the wind speed data from the previous link.

The value of the shape parameter is calculated:

The value of the scale parameter is calculated:

Therefore, the Weibull distribution:

c) By the empirical method

The value of the shape parameter is calculated:

The value of the scale parameter is calculated:

Therefore, the Weibull distribution:

d) By the maximum likelihood method

For the maximum likelihood method, we use the MATLAB function wblfit(v), which is used to fit a Weibull distribution to a dataset and outputs a matrix containing the estimated values of the Weibull parameters. In another publication, the algorithm will be explained in more detail step by step. Below is the code to calculate the parameters:

Wind=xlsread('source of wind speed information');

% We calculate the parameters of the Weibull distribution
[parametros, intervalo] = wblfit(Wind);

% We display the results
disp(['The parameters of the Weibull distribution are:']);
disp(['c = ', num2str(parameters(1))]);
disp(['k = ', num2str(parameters(2))]);
disp(['The confidence interval for the parameters is:']);
disp(['[', num2str(interval(1)), ', ', num2str(interval(2)), ']']);

Therefore, the obtained Weibull parameters are:

c = 8.1721 ; k = 4.9566

Therefore, the Weibull distribution:

Method comparison

The following table shows a summary of the values obtained for the parameters c and k using the different methods mentioned earlier.

Method	Scale parameter (c)	Shape parameter (k)
Least squares fitting	7.707	3.334
Energy pattern factor	8.319	3.741
Empirical	8.235	4.469
Maximum likelihood	8.1721	4.9566

In Figure 4, the comparison of the Weibull probability curve is shown with the different parameters k and c obtained using the various numerical methods.

Figure 4: Comparison of numerical methods for calculating Weibull parameters.

Conclusion

There are several methods to calculate the parameters of the Weibull distribution in wind speed analysis, such as least squares fitting, energy pattern factor, empirical methods, and maximum likelihood estimation. Generally, the maximum likelihood method tends to be the most accurate. Each method has its advantages and limitations, and the choice depends on the conditions and available data.