How to Estimate Weibull Parameters for a Wind Speed Distribution

The Weibull distribution is widely used in wind energy to mathematically model the distribution of wind speeds at a study site. This distribution is characterized by two parameters, the scale factor (c) and the shape factor (k). Accurately estimating these parameters is essential for predicting the energy generated by a wind farm and making informed decisions. In this article, we will explore how to estimate Weibull parameters from a wind speed distribution using different methods such as the maximum likelihood method, the energy pattern factor method method, moment method and others.

The scale parameters and can be estimated using the following methods:

1. Maximum likelihood method

The application of the maximum likelihood method involves maximizing the probability that the observed data fits the Weibull distribution with the estimated parameters. This method provides accurate and reliable estimates of the Weibull parameters, which are essential for predicting the energy generated by a wind farm. On the other hand, it is difficult to solve, since numerical iterations are needed to determine the parameters k and c. To estimate the Weibull parameters, the following equations are used:

Where is the number of observations and is the average wind speed recorded in time interval i.

2. Energy pattern factor method

This method is particularly useful when data availability is limited. The energy pattern factor is calculated from wind power and used to fit a wind speed distribution to the energy distribution. This method is related to averaged wind speed data and is defined by the following equations:

Where E_pf is the energy pattern factor and is the gamma function defined by:

3. Moment method

This method involves equating the moments of the Weibull distribution with the moments of the observed wind speed data. The estimation of the parameters is done by solving a system of non-linear equations. In this method, information from the moments of the wind speed data is used to obtain accurate estimates of the Weibull parameters. The moment method can be used as an alternative to the maximum likelihood method. In this case, the parameters k and c are determined by the following equations:

Where is the mean wind speed and is the standard deviation of the data of the wind speed.

4. Empirical method

The empirical method is considered a special case of the moment method. It is a quick and simple method where the Weibull parameters k and c can be calculated by the following equations:

5. Method of least squares

This method is based on the Weibull cumulative distribution function. The wind speed values are interpolated by a straight line using the concept of least squares.

: Wind speed (m/s).
: Weibull cumulative distribution function.
: Scaling factor (m/s), value close to the annual mean velocity.
: Shape factor characterizing the asymmetry or skewness of the F(v) function.

The above equation can be transformed linearly by taking the Neperian logarithm twice. This gives it:

We proceed to least squares fitting. By substituting the standard form of the linear regression equation can be obtained:

Where “a” and “b” are calculated using linear regression of the cumulative distribution function.

The slope (k) can be calculated:

The value of “b” can be calculated:

The scale parameter (c) is equal to:

Comparison of Methods for Estimating Weibull Parameters: Which One is the Most Accurate?

To determine which method is the most efficient for estimating the parameters and , the following tests are used: chi-square , root mean square error , and squared multiple correlation coefficient . These tests are used as criteria to determine which method fits the actual wind speed data the best. These tests are defined by:

Where is the number of observations, is the frequency of observations, is the frequency of Weibull, is the mean wind speed, and is the number of constants used.

The coefficient of determination, , is commonly used as a measure of goodness of fit, as it provides information on the amount of variability in the data that can be explained by the model. Therefore, higher values of indicate a better fit of the model to the data. Generally, a value of close to 1 indicates a good fit of the data model, while a value close to 0 indicates that the model does not explain the variability in the data well. However, in some research fields, it is common for models not to explain all of the variability in the data due to the complexity of the factors influencing the studied phenomenon. In these cases, an value of 0.2 or 0.3 may be considered a good fit value.

On the other hand, is a measure of the discrepancy between the observed values and the values expected according to the model. Therefore, lower values of indicate that the model fits the observed data better.

Finally, the (Root Mean Squared Error) is a measure of the accuracy of the model in predicting the data. Lower values of RMSE indicate greater accuracy in prediction.

Conclusion

In conclusion, estimating the Weibull parameters for wind speed distribution is a crucial task for wind energy applications as it helps to understand the potential of wind resource at a specific site. The process involves using various statistical techniques to obtain the shape and scale parameters of the Weibull distribution. The accuracy of the estimated parameters can be evaluated using measures such as , and , which indicate the goodness of fit and the accuracy of the model in predicting the data. Overall, obtaining accurate estimates of the Weibull parameters is essential for optimizing the design and operation of wind energy systems.

Reference

[1] Costa Rocha, P. A., de Sousa, R. C., de Andrade, C. F., & da Silva, M. E. V. (2012). Comparison of seven numerical methods for determining Weibull parameters for wind energy generation in the northeast region of Brazil. Applied Energy, 89(1), 395–400. https://doi.org/10.1016/j.apenergy.2011.08.003.

[2] Wang, W., Chen, K., Bai, Y., Chen, Y., & Wang, J. (2022). New estimation method of wind power density with three‐parameter Weibull distribution: A case on Central Inner Mongolia suburbs. Wind Energy, 25(2), 368–386. https://doi.org/10.1002/we.2677.

[3] Villarrubia López M. (2012). Ingeniería de la Energía Eólica. Facultad de Física, Universidad de Barcelona. Alfaomega Grupo Editor, S.A. de C.V.