# Optimization of the distribution of wind speeds using convexly combined Weibull densities

- Jonas Gräbner
^{1}and - Johannes Jahn
^{1}Email authorView ORCID ID profile

**4**:7

https://doi.org/10.1186/s40807-017-0045-9

© The Author(s) 2017

**Received: **4 May 2017

**Accepted: **4 December 2017

**Published: **16 December 2017

## Abstract

This paper presents a new approach for the determination of the wind speed distribution based on wind speed data. This approach is based on the fact that, in general, wind speed distributions restricted to seasons of year or months are different. Therefore, instead of one Weibull density function, a convex combination of Weibull density functions is considered for a calendar year. This model improves the maximum likelihood of the estimated wind speed distribution. Numerical results including a Kolmogorov–Smirnov test are given for a site at Jamaica. Numerical comparisons are carried out for different sites and various known methods for the estimation of the wind speed distribution.

## Keywords

## Mathematics Subject Classification

## Introduction

For the forecast of the annual revenue of wind power stations, one needs a good estimate of the probability distribution of wind speeds (compare also Wang et al. 2016b; Zhao et al. 2016; Sohoni et al. 2016a). By default, one generally works with a Weibull probability density function (PDF) for wind power potential calculations (e.g. see Hennessey 1977; Bowden et al. 1983; Genc et al. 2005; Sohoni et al. 2016b). Quite often, such an estimated PDF leads to an incorrect prediction of the produced energy so that additional costs may occur (e.g. see Tye et al. 2014). The use of only one Weibull PDF seems to be problematic, and at special sites, e.g. the wind farm Chungtun located at a small island in Taiwan Trait (see Liu et al. 2014 for details), a bimodal mixture Weibull PDF has shown to be more useful (see also Jaramillo and Borja 2004). Other approaches such as the truncated normal-Weibull PDF, the mixture Gamma–Weibull PDF and the mixture truncated normal PDF are known from the special literature (e.g. see Chang 2011; Akpinar and Akpinar 2009; Carta and Mentado 2007; Wang et al. 2016a; Tian Pau 2011; Kollu et al. 2012). Better PDF estimates can be expected, as proposed by Bischoff and Jahn (2016), using convex combinations of different Weibull PDFs. The present paper extends these investigations in such a way that monthly distributions are taken into account. This leads to an improvement of the estimate, which is achieved by a high numerical effort for the solution of a constrained optimization problem with a highly nonlinear objective function.

The goal of this paper is to present this new approach. This method is based on a highly nonlinear optimization problem, which can be solved by standard algorithms of numerical smooth optimization. Since this approach uses much more parameters than the known methods, one gets an improved resulting PDF of wind speeds.

This paper is organized as follows: the next section describes preliminaries, and then, convex combinations of Weibull PDFs are investigated. The algorithmic approach is presented in the fourth section followed by numerical results and a Kolmogorov–Smirnov test. In the last section, numerical comparisons are carried out for known estimation methods applied to different sites.

## Preliminaries

*V*describe the wind speed (in m/s) at an arbitrary site of a wind farm. The PDF of

*V*is very often assumed to be a Weibull density function \(f_{k,c}\) given as

*k*and

*c*(e.g. see Gupta et al. 1998). It is outlined by Akdağ and Dinler (2009) that there are different methods for the computation of these parameters. We restrict ourselves to the maximum likelihood estimation, which estimates the parameters

*k*and

*c*in such a way that the data are generated by the corresponding distribution with maximal probability. From a mathematical point of view, one solves the nonlinear optimization problem

The standard Weibull PDF is certainly not appropriate for site 1. This already shows the known fact that a Weibull PDF is not always the best choice. Wind power potential calculations require a better approximation of the PDF.

- 1.
The data of wind speeds \(v_{1},\ldots ,v_{8760}\) are ordered in time. This ordering is not considered in problem (1). Therefore, the structure of the wind profiles is not completely used.

- 2.
If there are wind speeds of the form \(v_{i}=0\) for some \(i\in \{ 1,\ldots ,8760\}\), then this information is unused in problem (1). This leads to an incorrect estimate of the PDF.

Characteristics of three sites

Site 1 | Site 2 | Site 3 | |
---|---|---|---|

Geographical coordinates | |||

Latitude | 18.504 | 21.42028 | 26.35561 |

Longitude | − 77.9125 | − 77.8475 | 127.76763 |

Country | Jamaica | Cuba | Japan |

Characteristics of data sets | |||

Time period | 2011/9/1 | 2011/9/1 | 2011/9/1 |

Until | Until | Until | |

2016/9/1 | 2016/9/1 | 2016/9/1 | |

Mean (m/s) | 3.643 | 3.985 | 5.061 |

Variance (m | 4.159 | 2.106 | 4.390 |

Standard deviation (m/s) | 2.039 | 1.451 | 2.095 |

For sites in the Caribbean, it is well known (compare Wang 2007) that mean wind speeds have two local maxima in summer and winter and two local minima in fall and spring. Figure 2 illustrates monthly mean wind speeds for sites 1 and 2 given in Table 1. Based on these observations, it certainly makes sense to incorporate monthly distributions into an approach with convexly combined Weibull PDFs. This leads to an significant improvement of the PDF for difficult sites.

## Convex combinations of Weibull PDFs

- 1.
The original objective function appears in a logarithmic form.

- 2.
All observed wind speeds are taken into account including wind speeds with 0 m/s.

- 3.
The positivity of the parameters \(k_j\) (\(j\in \{ 1,\ldots , 12\}\)) and \(c_j\) (\(j\in \{ 0,\ldots ,12\}\)) is ensured by the lower bounds \(\varepsilon \) and \(\delta \).

- 4.
The last constraint ensures the right PDF value at 0 m/s.

## Procedure

Instead of the SQP method, one can also choose any numerical method of smooth constrained optimization. Since the objective function in problem (4) is highly nonlinear, one cannot expect that an SQP method finds the global solution of this problem. It is known that the computed solution strongly depends on the choice of the starting point. Therefore, the SQP method, which is not a method of global optimization, is repeatedly applied to different starting points. Among all computed points, one then selects this one with largest objective function value. This leads to more realistic numerical results.

## Numerical results

The algorithm in the previous section is now applied to the wind speeds at site 1. At this site, we have \(h_0=0\), i.e. there are no wind speeds with 0 m/s.

Starting vector and solution vector for \(\ell _{\text {max}}=1\)

\(\tilde{\lambda }_{\text {start}}\) | \(\tilde{\lambda }_{\text {opt}}\) | \(\bar{k}_{\text {start}}\) | \(\bar{k}_{\text {opt}}\) | \(\tilde{c}_{\text {start}}\) | \(\tilde{c}_{\text {opt}}\) | |
---|---|---|---|---|---|---|

January | 0.085 | 0.000 | 1.366 | 3.400 | 4.400 | 10.644 |

February | 0.077 | 0.311 | 1.338 | 3.229 | 4.253 | 3.044 |

March | 0.085 | 0.018 | 2.068 | 3.523 | 4.375 | 2.733 |

April | 0.082 | 0.160 | 1.510 | 5.669 | 3.894 | 6.523 |

May | 0.085 | 0.010 | 1.365 | 3.525 | 3.790 | 5.607 |

June | 0.082 | 0.002 | 1.276 | 1.289 | 4.193 | 50.000 |

July | 0.085 | 0.001 | 1.691 | 2.405 | 4.342 | 11.741 |

August | 0.085 | 0.186 | 1.763 | 5.608 | 3.945 | 5.772 |

September | 0.082 | 0.274 | 1.268 | 3.214 | 3.395 | 1.806 |

October | 0.085 | 0.019 | 1.710 | 2.827 | 3.330 | 3.512 |

November | 0.082 | 0.000 | 2.011 | 4.720 | 4.491 | 6.344 |

December | 0.085 | 0.020 | 1.818 | 26.295 | 4.332 | 7.201 |

Solution vector for \(\ell _{\text {max}}=45,000\)

Component | \(\tilde{\lambda }_{\text {opt}}\) | \(\bar{k}_{\text {opt}}\) | \(\tilde{c}_{\text {opt}}\) |
---|---|---|---|

1 | 0.012 | 50.000 | 1.027 |

2 | 0.053 | 18.575 | 7.081 |

3 | 0.022 | 8.270 | 1.468 |

4 | 0.100 | 2.596 | 4.774 |

5 | 0.041 | 10.272 | 7.480 |

6 | 0.112 | 5.866 | 3.290 |

7 | 0.000 | 2.909 | 4.330 |

8 | 0.388 | 3.052 | 2.085 |

9 | 0.003 | 1.000 | 29.918 |

10 | 0.011 | 50.000 | 1.936 |

11 | 0.211 | 7.252 | 5.792 |

12 | 0.047 | 8.700 | 4.199 |

The constrained optimization problem (4) is solved by the SQP method of the optimization toolbox of MATLAB. The components of the obtained solution vector can be found in the columns \(\tilde{\lambda }_{\text {opt}}\), \(\bar{k}_{\text {opt}}\) and \(\tilde{c}_{\text {opt}}\) of Table 2. The parameters of the exponential PDF are unchanged. It is evident from the data in Table 2 that the components of the starting vector are quite different from the components of the solution vector. This optimization leads to an improvement of the value of the objective function by 7.08% in comparison with the objective function value at the starting vector.

## Kolmogorov–Smirnov test

In the previous sections, we have concentrated ourselves to a good type of approximation of the CDF of wind speeds at a specific site. But now we test the hypothesis that the wind speed as random variable has the optimized CDF obtained by the presented algorithm. One accepts this hypothesis, if the optimized CDF and the empirical CDF are in a certain sense close together. The well-known Kolmogorov–Smirnov (KS) test (e.g. see D’Agostino and Stephens 1986) can be used for the test of this hypothesis.

For the Kolmogorov–Smirnov test, the wind speeds (8760 numbers) at site 1 are randomly splitted into two data sets with 4380 numbers. The first data set is used for the application of the algorithm of this paper. This leads to an optimized CDF, which is then compared with the empirical CDF of the second data set. Then, the Kolmogorov–Smirnov test is applied to these two CDFs. We get the result that with a level of significance of 5% the hypothesis is accepted that the optimized CDF is the true CDF of the second sample. In fact, the calculated test statistic value 0.017 is less than the critical value 0.021 of the Kolmogorov–Smirnov test. This shows that the approach of this paper is suitable for a good determination of the CDF of wind speeds.

If one works with the whole data set of 8760 wind speeds per year, the critical value in the Kolmogorov–Smirnov test at a level of significance of 5% is given by \(1.358/\sqrt{8,760}\approx 0.0145\), i.e. for the supremum of deviations below this value, the hypothesis is accepted that the calculated CDF is the true CDF of the wind speeds as random variable. Assuming the correctness of the hypothetical CDF, there is a maximum probability of 5% observing test statistic values above the critical value, thus rejecting the hypothesis falsely.

*F*and the empirical CDF \(\hat{F}\) have the derivatives \(F'(0)=0\) and \(\hat{F}'(0)=h_{0}\) (given in the algorithm). So, the expression \(\sup_{v\ge 0}| F(v)-\hat{F}(v)|\) may be greater than the critical value 0.0145 so that the tested hypothesis is rejected. The convex combination presented in this paper tries to avoid this disadvantage.

Numerical results for different sites and various approaches

Site 1 | Site 2 | Site 3 | |
---|---|---|---|

Weibull PDF | |||

| 1.459 | 2.947 | 2.436 |

| 4.044 | 4.474 | 5.688 |

KS test statistic value | 0.084 | 0.089 | 0.091 |

Bimodal Weibull PDF | |||

\(\lambda _1\) | 0.002 | 0.397 | 0.070 |

\(k_1\) | 1.294 | 5.451 | 2.310 |

\(c_1\) | 50.000 | 3.114 | 10.095 |

\(\lambda _2\) | 0.998 | 0.603 | 0.930 |

\(k_2\) | 1.869 | 3.789 | 3.521 |

\(c_2\) | 4.023 | 5.222 | 5.299 |

KS test statistic value | 0.051 | 0.035 | 0.019 |

Gamma–Weibull PDF | |||

\(\lambda _1\) | 0.765 | 0.514 | 0.979 |

\(\alpha \) | 3.353 | 15.843 | 8.344 |

\(\beta \) | 1.268 | 0.194 | 0.586 |

\(\lambda _2\) | 0.235 | 0.486 | 0.022 |

| 3.351 | 4.076 | 3.746 |

| 1.852 | 5.462 | 14.424 |

KS test statistic value | 0.079 | 0.025 | 0.020 |

Convexly combined Weibull PDF | |||

\(\lambda _1\) | 0.000 | 0.039 | 0.065 |

\(k_1\) | 3.400 | 9.134 | 11.811 |

\(c_1\) | 10.644 | 6.812 | 5.666 |

\(\lambda _2\) | 0.311 | 0.021 | 0.100 |

\(k_2\) | 3.229 | 6.800 | 4.251 |

\(c_2\) | 3.044 | 3.409 | 7.414 |

\(\lambda _3\) | 0.018 | 0.000 | 0.087 |

\(k_3\) | 3.523 | 1.000 | 5.551 |

\(c_3\) | 2.733 | 5.773 | 3.165 |

\(\lambda _4\) | 0.160 | 0.011 | 0.052 |

\(k_4\) | 5.669 | 12.425 | 4.891 |

\(c_4\) | 6.523 | 7.931 | 2.612 |

\(\lambda _5\) | 0.010 | 0.032 | 0.045 |

\(k_5\) | 3.525 | 18.863 | 18.392 |

\(c_5\) | 5.607 | 4.731 | 4.921 |

\(\lambda _6\) | 0.002 | 0.148 | 0.000 |

\(k_6\) | 1.289 | 9.826 | 1.000 |

\(c_6\) | 50.000 | 3.949 | 3.081 |

\(\lambda _7\) | 0.001 | 0.047 | 0.121 |

\(k_7\) | 2.405 | 5.713 | 4.676 |

\(c_7\) | 11.741 | 5.077 | 5.114 |

\(\lambda _8\) | 0.186 | 0.000 | 0.005 |

\(k_8\) | 5.608 | 2.828 | 19.063 |

\(c_8\) | 5.772 | 4.997 | 11.806 |

\(\lambda _9\) | 0.274 | 0.302 | 0.358 |

\(k_9\) | 3.214 | 6.298 | 5.062 |

\(c_9\) | 1.806 | 2.736 | 5.980 |

\(\lambda _{10}\) | 0.019 | 0.082 | 0.025 |

\(k_{10}\) | 2.827 | 14.432 | 3.242 |

\(c_{10}\) | 3.512 | 3.365 | 13.675 |

\(\lambda _{11}\) | 0.000 | 0.000 | 0.139 |

\(k_{11}\) | 4.720 | 5.270 | 9.266 |

\(c_{11}\) | 6.344 | 4.508 | 4.048 |

\(\lambda _{12}\) | 0.020 | 0.319 | 0.004 |

\(k_{12}\) | 26.295 | 5.217 | 4.399 |

\(c_{12}\) | 7.201 | 5.553 | 5.338 |

KS test statistic value | 0.017 | 0.024 | 0.012 |

## Numerical comparisons

For these sites, the PDF of wind speeds is calculated for various standard approaches. First of all, the (standard) Weibull PDF is determined for the three sites. Moreover, the bimodal Weibull PDF also known as Weibull–Weibull PDF and the mixture Gamma–Weibull PDF are calculated with the wind speed data. Figures 8, 9 and 10 illustrate the histograms of the measured wind speeds together with the PDFs obtained with the standard Weibull approach, the bimodal Weibull method, the Gamma–Weibull approach and the new method of this paper with \(\ell _{\text {max}}=1\). All numerical results are listed in Table 4.

Figures 8, 9 and 10 and Table 4 show that there are significant differences between the computed PDFs. It is obvious that the standard Weibull approach is not suitable for difficult international sites.

Furthermore, the new method of this paper seems to be superior in contrast to the other methods. These discrepancies between the PDFs of the considered approaches are certainly smaller, if one investigates wind sites with a more uniform PDF.

The Kolmogorov–Smirnov test is carried out for all four approaches and all three sites. For every site, the KS test statistic value of the convexly combined Weibull PDF is the smallest among all used methods, which means that the new method determines the best approximation of the CDF. But this better performance of the new approach is reached by a higher numerical effort.

## Conclusion

## Declarations

### Authors' contributions

JG carried out the statistical studies including the numerics and drafted the greater part of the manuscript in German. JJ provided the underlying model together with the algorithm and translated the German manuscript into English. Figures 1, 2, 3, 4, 5, 6, 8, 9 and 10 were produced by JG, whereas Fig. 7 was drawn by JJ. Both authors read and approved the final manuscript.

### Acknowledgements

The authors thank Dr. Martin Bischoff (Siemens AG, München, Germany), Dr. Karl Gutbrod (meteoblue.com, Basel, Switzerland) and Prof. Dr. Christoph Richard (Department of Mathematics, Friedrich-Alexander University of Erlangen-Nürnberg, Erlangen, Germany) for valuable suggestions concerning this paper.

### Competing interests

The authors declare that they have no competing interests.

### Ethics approval and consent to participate

Not applicable.

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## Authors’ Affiliations

## References

- Akdağ, S. A., & Dinler, A. (2009). A new method to estimate Weibull parameters for wind energy applications.
*Energy Conversion and Management*,*50*, 1761–1766.View ArticleGoogle Scholar - Akpinar, S., & Akpinar, E. K. (2009). Estimation of wind energy potential using finite mixture distribution models.
*Energy Conversion Management*,*50*, 877–884.View ArticleGoogle Scholar - Bischoff, M., & Jahn, J. (2016). Economic objectives, uncertainties and decision making in the energy sector.
*Journal of Business Economics*,*86*, 85–102.View ArticleGoogle Scholar - Bowden, G. J., Barker, P. R., Shestopal, V. O., & Twidell, J. W. (1983). The Weibull distribution function and wind power statistics.
*Wind Engineering*,*7*, 85–98.Google Scholar - Carta, J. A., & Mentado, D. (2007). A continuous bivariate model for wind power density and wind turbine energy output estimations.
*Energy Conversion and Management*,*48*, 420–432.View ArticleGoogle Scholar - Chang, T. P. (2011). Estimation of wind energy potential using different probability density functions.
*Applied Energy*,*88*, 1848–1856.View ArticleGoogle Scholar - D’Agostino, R., & Stephens, M. (1986).
*Goodness-of-fit techniques*. New York: Marcel Dekker.MATHGoogle Scholar - Genc, A., Erisoglu, M., Pekgor, A., Oturanc, G., Hepbasli, A., & Ulgen, K. (2005). Estimation of wind power potential using Weibull distribution.
*Energy Sources*,*27*, 809–822.View ArticleGoogle Scholar - Gupta, L. P., Gupta, R. C., & Lvin, S. J. (1998). Numerical methods for the maximum likelihood estimation of weibull parameters.
*Journal of Statistical Computation and Simulation*,*62*, 1–7.MathSciNetView ArticleMATHGoogle Scholar - Hennessey, J. P. (1977). Some aspects on wind power statistics.
*Journal of Applied Meteorology*,*16*, 119–128.View ArticleGoogle Scholar - Jaramillo, O. A., & Borja, M. A. (2004). Wind speed analysis in La Ventosa, Mexico: A bimodal probability distribution case.
*Renewable Energy*,*29*, 1613–1630.View ArticleGoogle Scholar - Kollu, R., Rayapudi, S. R., Narasimham, S. V. L., & Pakkurthi, K. M. (2012). Mixture probability distribution functions to model wind speed distributions.
*International Journal of Energy and Environmental Engineering*,*3*, 1–10.View ArticleGoogle Scholar - Liu, F.-J., Ko, H.-H., Kuo, S.-S., Liang, Y.-H., & Chang, T.-P. (2014). Study on wind characteristics using bimodal mixture Weibull distribution for three wind sites in Taiwan.
*Journal of Applied Science and Engineering*,*17*, 283–292.Google Scholar - Rinne, H. (2008).
*The Weibull distribution: A handbook*. Boca Raton: CRC Press.View ArticleMATHGoogle Scholar - Sohoni, V., Gupta, S., & Nema, R. (2016). A critical review on wind turbine power curve modelling techniques and their applications in wind based energy systems.
*Journal of Energy*,*2016*, 1–18.View ArticleGoogle Scholar - Sohoni, V., Gupta, S., & Nema, R. (2016). A comparitive analysis of wind speed probability distribution functions for wind power assessment of four sites.
*Turkish Journal of Electrical Engineering & Computer Sciences*,*24*, 4724–4735.View ArticleGoogle Scholar - Tian Pau, C. (2011). Estimation of wind energy potential using different probability density functions.
*Applied Energy*,*88*, 1848–1856.View ArticleGoogle Scholar - Tye, M. R., Stephenson, D. B., Holland, G. J., & Katz, R. W. (2014). A Weibull approach for improving climate model projections of tropical cyclone wind-speed distributions.
*Journal of Climate*,*27*, 6119–6133.View ArticleGoogle Scholar - Wang, C. (2007). Variability of the Caribbean low-level jet and its relations to climate.
*Climate Dynamics*,*29*, 411–422.View ArticleGoogle Scholar - Wang, J., Hu, J., & Ma, K. (2016a). Wind speed probability distribution estimation and wind energy assessment.
*Renewable and Sustainable Energy Reviews*,*60*, 881–899.View ArticleGoogle Scholar - Wang, J., Song, Y., Liu, F., & Hou, R. (2016b). Analysis and application of forecasting models in wind power integration: A review of multi-step-ahead wind speed forecasting models.
*Renewable and Sustainable Energy Reviews*,*60*, 960–981.View ArticleGoogle Scholar - Zhao, J., Guo, Z.-H., Su, Z.-Y., Zhao, Z.-Y., Xiao, X., & Liu, F. (2016). An improved multi-step forecasting model based on WRF ensembles and creative fuzzy systems for wind speed.
*Applied Energy*,*162*, 808–826.View ArticleGoogle Scholar