# Modelling the Wind Speed Using Exponentiated Weibull Distribution: Case Study of Poprad-Tatry, Slovakia

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}), and the information criteria—Akaike’s information criterion (AIC) and Bayesian information criterion (BIC)—were used to assess the suitability of the considered probability distributions and to compare their performances.

## 2. Description of Studied Area

## 3. Wind Speed Data Characteristics and Analysis

**Monthly wind speed analysis**(Table 1)

**:**It was observed that during the studied period, the lowest monthly mean wind speed with value of 2.75 m/s was in August, while in March there was the highest mean wind speed with value of 3.85 m/s. The standard deviation was used to assess the variability in the wind speed. Here, the standard deviation varied from 1.86 m/s in August to 2.87 m/s in January. In general, the winter and spring months have a higher mean wind speed and variability in the wind speed than the summer and autumn months. The coefficient of variation (CV) is useful for identifying months with a higher variability in the wind speed. According to [56], the value of the CV > 40% is classified as a very high variability and CV > 70% indicates an extremely high variability in the wind speed. The coefficient of variation ranged from 66.10% in June to 86.49% in January. Based on this, the results imply that the wind speed in all months could be classified as having very high variability. During the months of September–March, there was an observed extremely high variability in the wind speed in this location. Skewness and kurtosis measure the asymmetry and the peakness of the wind speed distribution, respectively. The coefficients of skewness ranged from 0.95 in March to 1.31 in November, indicating that all distributions were right skewed. Because skewness for all months was greater than 1, the wind speed data could be regarded as highly right skewed, except for March, when the skewness of 0.95 corresponded to a moderately right skewed distribution. The coefficient of kurtosis ranged from 3.31 in March to 4.57 in August. This indicated a highly leptokurtic distribution when compared to the normal distribution.

**Seasonal wind speed analysis**(Table 2): The results show that the highest seasonal mean wind speed was observed in spring with value of 3.56 m/s, whereas in autumn, there was observed the lowest mean wind speed with value of 2.97 m/s. CV ranged from 67.81% to 85.27% what indicates a very high variability of wind speed in this location. The coefficient of skewness for all seasons ranged from 1.10 to 1.32. That implies that distributions are highly right skewed. The coefficient of kurtosis ranged from 3.91 to 4.44, therefore the distributions can be regarded as highly leptokurtic distributions.

**Analysis of wind speed data as a whole**(Table 3): The mean wind speed of 3.19 m/s was observed with a standard deviation of 2.41 m/s. The CV of 75.56%, skewness of 1.25, and kurtosis of 4.32 revealed that the wind speed data had extremely high variability in terms of the wind speed, were highly right skewed, and highly leptokurtic.

## 4. Wind Direction Analysis

## 5. Methods

#### 5.1. Probability Distributions

#### 5.2. Parameter Estimation

^{2}test, they demonstrated better performance of the MLM as compared to the others. Many authors have used the MLM as the method for parameter estimation when modelling the wind speed using various probability distributions, for example, in [63,64]. Therefore, the MLM was utilised for estimating the parameters in this study.

#### 5.3. Goodness-of-Fit and Model Selection Criteria

^{2}) were used. Employed GOF tests and model selection criteria are briefly described below.

^{2}is greater than the critical value of the AD test. Again, the smaller value of the test statistic A

^{2}indicates a better fit.

^{2}) and the root mean square error (RMSE) are considered to decide on the best fitting model. The RMSE determines the accuracy of the model by calculating the average of the square difference between the observed and the predicted probabilities of the theoretical distribution. The R

^{2}is used to measure the linear relationship between the observed and the predicted probabilities of the theoretical distribution. The RMSE and R

^{2}are calculated by:

^{2}indicate better fit of the theoretical distribution to the wind speed data as compared to the others.

## 6. Results and Discussion

- Respective months:

^{2}, and RMSE. In this case, when some GOF tests and model selection criteria favoured the EW distribution, whereas the others the W3 distribution, we made a conclusion according to the value of the AD test since the AD test is considered as a more powerful GOF test. Furthermore, according to conclusions drawn in [69], R

^{2}appears to be more informative than the other indicators in such cases. Therefore, when choosing the most suitable distribution for a given month, the probability distribution with the smallest value of the AD test and the highest R

^{2}value was selected as the best fitting distribution for the wind speed data in that month. According to this, the EW distribution was more suitable for January and December. Such assumptions agree with the visualization in Figure 7, where the theoretical distributions fitted to the observed data in the months January and December show that the EW distribution was closer to the empirical distribution than the W3 distribution.

- The entire period and the seasons:

^{2}and the lowest values of all the other criteria, except for the winter. In the winter, the W3 distribution obtained better results for the information criteria. However, the EW distribution had the highest value of the coefficient of determination and the lowest values of the AD test and the RMSE among the discussed distributions. This indicated that the EW distribution demonstrated a better fit than the other two distributions. Thus, the EW distribution was more suitable for modelling the wind speed in this location for the entire studied period and for all the seasons. According to the criteria, W3 performed as the second best.

- The datasets in both other papers had high values of kurtosis—in the dataset from [49], the kurtosis was 2.502; in the datasets from [50], the values of kurtosis ranged from 3.877 to 8.806. The datasets here, with the EW distribution as the most suitable one, also possessed high values of the coefficient of kurtosis, ranging from 3.31 to 4.57.
- The datasets in both other papers had positive values of skewness—in the dataset from [49], the skewness was 0.633; in the datasets from [50], the values of skewness ranged from 0.888 to 2.014. The datasets, modelled here, also had values of the coefficient of skewness ranging from 0.95 to 1.31. All of these datasets can be regarded as moderate to highly right skewed.

## 7. Conclusions

^{2}. The EW distribution performed the best (in comparison to values of the indicators for the W2 and W3 distributions, respectively) and provided a better fit to the seasonal and monthly wind speed data, except for February and March, when the W3 performed better, but the EW was the second best. Therefore, the EW distribution can be considered as a suitable wind speed distribution and can be applied to forecast and estimate the wind speed at the meteorological station Poprad. Based on the character of the studied data in terms of their skewness, we can also recommend the EW distribution as a good model for highly right skewed data. In addition, the EW distribution is flexible enough in terms of modelling the peakness of the data. To sum it up, the EW distribution proved to be a good alternative to the W2 and W3 distributions due to its flexibility in modelling the data of the wind speed with high positive skewness and kurtosis.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Safari, B.; Gasore, J. A statistical investigation of wind characteristics and wind energy potential based on the Weibull and Rayleigh models in Rwanda. Renew. Energy
**2010**, 35, 2874–2880. [Google Scholar] [CrossRef] - Chen, H.; Birkelund, Y.; Anfinsen, S.N.; Staupe-Delgado, R.; Yuan, F. Assessing probabilistic modelling for wind speed from numerical weather prediction and observation in the Artic. Sci. Rep.
**2021**, 11, 7613. [Google Scholar] [CrossRef] [PubMed] - Lun, I.Y.F.; Lam, J.C. A study of Weibull parameters using long-term wind observations. Renew. Energy
**2000**, 20, 145–153. [Google Scholar] [CrossRef] - Davenport, A.G. The relationship of wind structure to wind loading. In Proceedings of the Symposium No. 16—Wind Effects on Buildings and Structures, Teddington, UK, 26–28 June 1963; pp. 53–111. [Google Scholar]
- Hemanth Kumar, M.B.; Saravanan, B.; Sanjeevikumar, P.; Holm-Nielsen, J.B. Wind Energy Potential Assessment by Weibull Parameter Estimation Using Multiverse Optimization Method: A Case Study of Tirumala Region in India. Energies
**2019**, 12, 2158. [Google Scholar] [CrossRef] [Green Version] - Abbas, G.; Gu, J.; Asad, M.U.; Balas, V.E.; Farooq, U.; Khan, I.A. Estimation of Weibull Distribution Parameters by Analytical Methods for the Wind Speed of Jhimpir, Pakistan—A Comparative Assessment. In Proceedings of the 2022 International Conference on Emerging Trends in Electrical, Control, and Telecommunication Engineering (ETECTE), Lahore, Pakistan, 2–4 December 2022. [Google Scholar]
- Hussain, I.; Haider, A.; Ullah, Z.; Russo, M.; Casolino, G.M.; Azeem, B. Comparative Analysis of Eight Numerical Methods Using Weibull Distribution to Estimate Wind Power Density for Coastal Areas in Pakistan. Energies
**2023**, 16, 1515. [Google Scholar] [CrossRef] - Kang, S.; Khanjari, A.; You, S.; Lee, J.-H. Comparison of different statistical methods used to estimate Weibull parameters for wind speed contribution in nearby an offshore site, Republic of Korea. Energy Rep.
**2021**, 7, 7358–7373. [Google Scholar] [CrossRef] - Salami, A.A.; Ouedraogo, S.; Kodjo, K.M.; Ajavon, A.S.A. Influence of the random data ampling in estimation of wind speed resource: Case study. Int. J. Renew. Energy Dev.
**2022**, 11, 133–143. [Google Scholar] [CrossRef] - Dhakal, R.; Sedai, A.; Pol, S.; Parameswaran, S.; Nejat, A.; Moussa, H. A Novel Hybrid Method for Short-Term Wind Speed Prediction Based on Wind Probability Distribution Function and Machine Learning Models. Appl. Sci.
**2022**, 12, 9038. [Google Scholar] [CrossRef] - Singh, K.A.; Khan, M.G.M.; Ahmed, M.R. Wind Energy Resource Assessment for Cook Islands with Accurate Estimation of Weibull Parameters Using Frequentist and Bayesian Methods. IEEE Access
**2022**, 10, 25935–25953. [Google Scholar] [CrossRef] - Alsamamra, H.R.; Salah, S.; Shoqeir, J.A.H.; Manasra, A.J. A comparative study of five numerical methods for the estimation of Weibull parameters for wind energy evaluation at Eastern Jerusalem, Palestine. Energy Rep.
**2022**, 8, 4801–4810. [Google Scholar] [CrossRef] - Al-Hussieni, A.J.M. A prognosis of wind energy potential as a power generation source in Basra city, Iraq state. Eur. Sci. J.
**2014**, 10, 163–176. [Google Scholar] - Okakwu, I.; Akinyele, D.; Olabode, O.; Ajewole, T.; Oluwasogo, E.; Oyedeji, A. Comparative Assessment of Numerical Techniques for Weibull Parameters’ Estimation and the Performance of Wind Energy Conversion Systems in Nigeria. IIUM Eng. J.
**2023**, 24, 138–157. [Google Scholar] [CrossRef] - Shu, Z.R.; Jesson, M. Estimation of Weibull parameters for wind energy analysis across the UK. J. Renew. Sustain. Energy
**2021**, 13, 023303. [Google Scholar] [CrossRef] - Truhetz, H.; Krenn, A.; Winkelmeier, H.; Müller, S.; Cattin, R.; Eder, T.; Biberacher, M. Austrian Wind Potential Analysis (AuWiPot). In Proceedings of the 12. Symposium Energieinnovation, Graz, Austria, 15–17 February 2012. [Google Scholar]
- Pobočíková, I.; Sedliačková, Z.; Šimon, J.; Jurášová, D. Statistical analysis of the wind speed at mountain site Chopok, Slovakia, using Weibull distribution. IOP Conf. Ser. Mater. Sci. End.
**2020**, 776, 012114. [Google Scholar] [CrossRef] - Wais, P. A review of Weibull functions in wind sector. Renew. Sust. Energ. Rev.
**2017**, 70, 1099–1107. [Google Scholar] [CrossRef] - Carta, J.A.; Ramírez, P.; Velázquez, S. A review of wind speed probability distributions used in wind energy analysis: Case studies in the Canary Islands. Renew. Sust. Energ. Rev.
**2009**, 13, 933–955. [Google Scholar] [CrossRef] - Akgül, F.G.; Şenoğlu, B.; Arslan, T. An alternative distribution to Weibull for modeling the wind speed data: Inverse Weibull distribution. Energy Convers. Manag.
**2016**, 114, 234–240. [Google Scholar] [CrossRef] - Sukkiramathi, K.; Seshaiah, C. Analysis of wind power potential by the three-parameter Weibull distribution to install a wind turbine. Energy Explor. Exploit.
**2020**, 38, 158–174. [Google Scholar] [CrossRef] [Green Version] - Pobočíková, I.; Sedliačková, Z.; Michalková, M. Application of four probability distributions for wind speed modeling. Procedia Eng.
**2017**, 192, 713–718. [Google Scholar] [CrossRef] - Sarkar, A.; Singh, S.; Mitra, D. Wind climate modeling using Weibull and extreme value distribution. Int. J. Eng. Sci. Technol.
**2011**, 3, 100–106. [Google Scholar] [CrossRef] - Morgan, E.C.; Lackner, M.; Vogel, R.M.; Baise, L.G. Probability distributions for offshore wind speeds. Energy Convers. Manag.
**2011**, 52, 15–26. [Google Scholar] [CrossRef] - Kang, D.; Ko, K.; Huh, J. Determination of extreme wind values using the Gumbel distribution. Energy
**2015**, 86, 51–58. [Google Scholar] [CrossRef] - Alavi, O.; Mohammadi, K.; Mostafaeipour, A. Evaluating the suitability of wind speed probability distribution models: A case study of east and southeast parts of Iran. Energy Convers. Manag.
**2016**, 119, 101–108. [Google Scholar] [CrossRef] - Mohammadi, K.; Alavi, O.; McGowan, J.G. Use of Birnbaum-Saunders distribution for estimating wind speed and wind power probability distributions: A review. Energy Convers. Manag.
**2017**, 143, 109–122. [Google Scholar] [CrossRef] - Jung, C.; Schindler, D. Global comparison of the goodness-of-fit of wind speed distributions. Energy Convers. Manag.
**2017**, 133, 216–234. [Google Scholar] [CrossRef] - Kantar, Y.M.; Usta, I.; Arik, I.; Yenilmez, I. Wind speed analysis using the Extended Generalized Lindley Distribution. Renew. Energy
**2018**, 118, 1024–1030. [Google Scholar] [CrossRef] - Ahmad, Z.; Mahmoudi, E.; Roozegarz, R.; Hamedani, G.G.; Butt, N.S. Contributions towards new families of distributions: An investigation, further developments, characterizations and comparative study. Pak. J. Stat. Oper. Res.
**2022**, 18, 99–120. [Google Scholar] [CrossRef] - Marshall, A.W.; Olkin, I. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika
**1997**, 84, 641–652. [Google Scholar] [CrossRef] - ul Haq, M.A.; Rao, G.S.; Albassam, M.; Aslam, M. Marshall-Olkin Power Lomax distribution for modeling of wind speed data. Energy Rep.
**2020**, 6, 1118–1123. [Google Scholar] [CrossRef] - Carta, J.A.; Ramírez, P. Analysis of two-component mixture Weibull statistics for estimation of wind speed distributions. Renew. Energy
**2007**, 32, 518–531. [Google Scholar] [CrossRef] - Akpinar, S.; Akpinar, E.K. Estimation of wind energy potential using finite mixture distribution models. Energy Convers. Manag.
**2009**, 50, 877–884. [Google Scholar] [CrossRef] - Kollu, R.; Rayapudi, S.R.; Narasimham, S.V.L.; Pakkurthi, K.M. Mixture probability distribution functions to model wind speed distributions. Int. J. Energy Environ. Eng.
**2012**, 3, 27. [Google Scholar] [CrossRef] [Green Version] - Ouarda, T.B.M.J.; Charron, C. On the mixture of wind speed distribution in a Nordic region. Energy Convers. Manag.
**2018**, 174, 33–44. [Google Scholar] [CrossRef] - Mahbudi, S.; Jamalizadeh, A.; Farnoosh, R. Use of finite mixture models skew-t-normal Birnbaum-Saunders components in the analysis of wind speed: Case studies in Ontario, Canada. Renew. Energy
**2020**, 162, 196–211. [Google Scholar] [CrossRef] - Gupta, R.C.; Gupta, P.L.; Gupta, R.D. Modeling failure time data by Lehman alternatives. Commun. Stat.-Theory Methods
**1998**, 27, 887–904. [Google Scholar] [CrossRef] - Mudholkar, G.S.; Srivastava, D.K. Exponentiated Weibull family for analysing bathtub failure-rate data. IEEE Trans. Reliab.
**1993**, 42, 299–302. [Google Scholar] [CrossRef] - Mudholkar, G.S.; Srivastava, D.K.; Freimer, M. The exponentiated Weibull family: A reanalysis of the bus-motor-failure data. Technometrics
**1995**, 37, 436–445. [Google Scholar] [CrossRef] - Mudholkar, G.S.; Hutson, A.D. The exponentiated Weibull family: Some properties and a flood data application. Commun. Stat.-Theory Methods
**1996**, 25, 3059–3083. [Google Scholar] [CrossRef] - Nassar, M.M.; Eissa, F.H. On the exponentiated Weibull distribution. Commun. Stat.-Theory Methods
**2003**, 32, 1317–1336. [Google Scholar] [CrossRef] - Nassar, M.M.; Eissa, F.H. Bayesian estimation for the exponentiated Weibull model. Commun. Stat.-Theory Methods
**2004**, 33, 2343–2362. [Google Scholar] [CrossRef] - Alizadeh, M.; Bagheri, S.F.; Baloui Jamkhaneh, E.; Nadarajah, S. Estimates of the PDF and the CDF of the exponentiated Weibull distribution. Braz. J. Probab. Stat.
**2015**, 29, 695–716. [Google Scholar] [CrossRef] - Nadarajah, S.; Kotz, S. On the moments of the exponentiated Weibull distribution. Commun. Stat.-Theory Methods
**2005**, 34, 253–256. [Google Scholar] [CrossRef] - Salem, A.M.; Abo-Kasem, O.E. Estimation for the parameters of the exponentiated Weibull distribution based on progressive hybrid censored samples. Int. J. Contemp. Math. Sci.
**2011**, 6, 1713–1724. [Google Scholar] - Pal, M.; Ali, M.M.; Woo, J. Exponentiated Weibull distribution. Statistica
**2006**, 66, 136–147. [Google Scholar] [CrossRef] - Barrios, R.; Dios, F. Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves. Opt. Express
**2012**, 20, 13055–13064. [Google Scholar] [CrossRef] [Green Version] - Shittu, O.I.; Adepoju, K.A. On the exponentiated Weibull distribution for modeling wind speed in South Western Nigeria. J. Mod. Appl. Stat. Methods
**2014**, 13, 431–445. [Google Scholar] [CrossRef] [Green Version] - Akgül, F.G.; Şenoğlu, B. Comparison of wind speed distributions: A case study for Aegean coast of Turkey. Energy Sources A Recovery Util. Environ. Eff.
**2019**, 45, 2453–2470. [Google Scholar] [CrossRef] - Unesco. Tatra Transboundary Biosphere Reserve, Poland/Slovakia. Available online: https://www.unesco.org/en/man-and-biosphere/tatra-transboundary-biosphere-reserve-poland/slovakia-0 (accessed on 11 March 2023).
- Ilčin, J. Problematics of Flight in Mountainous Terrain. Bachelor’s Thesis, Brno University of Technology, Brno, Czech Republic, 2017. [Google Scholar]
- Bochníček, O.; Hrušková, K.; Zvara, I. Climate Atlas of Slovakia, 1st ed.; Slovak Hydrometeorological Institute: Bratislava, Slovakia, 2015; p. 131. [Google Scholar]
- Google Earth. Available online: https://earth.google.com/web/search/Poprad,+Slovakia/@49.05877989,20.29744794,672.117089a,24378.01462403d,35y,0h,0t,0r/data=CnsaURJLCiUweDQ3M2UzYTk0YmNmZGExOTE6MHg5Mzk3NGEwYWVmODZiOTIwGSrNQSuLhkhAIdHHfECgSzRAKhBQb3ByYWQsIFNsb3Zha2lhGAIgASImCiQJI8yHUE9QNUARIcyHUE9QNcAZKLUxbvu7QkAhhILfAjV3UMA (accessed on 8 March 2023).
- Manwell, J.F.; McGowan, J.G.; Rogers, A.L. Wind Energy Explained: Theory, Design and Application, 2nd ed.; Wiley: Chichester, UK, 2010; p. 704. [Google Scholar]
- Hare, W. Assessment of Knowledge on Impacts of Climate Change—Contribution to the Specification of Art. 2 of the UNFCCC: Impacts on Ecosystems, Food Production, Water and Socio-Economic Systems; External Expertise Report for German Advisory Council on Global Change: Berlin, Germany, 2003; p. 104. [Google Scholar]
- Al Mac. Wind_Rose (Wind_Direction, Wind_Speed). MATLAB Central File Exchange. Available online: https://www.mathworks.com/matlabcentral/fileexchange/65174-wind_rose-wind_direction-wind_speed (accessed on 13 February 2023).
- Akpinar, E.K.; Akpinar, S. Determination of the wind energy potential for Maden-Elazig, Turkey. Energy Convers. Manag.
**2004**, 45, 2901–2914. [Google Scholar] [CrossRef] - Akdag, S.A.; Dinler, A. A new method to estimate Weibull parameters for wind energy applications. Energy Convers. Manag.
**2009**, 50, 1761–1766. [Google Scholar] [CrossRef] - Chang, T.P. Performance comparison of six numerical methods in estimating Weibull parameters for wind energy application. Appl. Energy
**2011**, 88, 272–282. [Google Scholar] [CrossRef] - Azad, A.K.; Rasul, M.G.; Yusaf, T. Statistical Diagnosis of the Best Weibull Methods for Wind Power Assessment for Agricultural Applications. Energies
**2014**, 7, 3056–3085. [Google Scholar] [CrossRef] [Green Version] - Shoaib, M.; Siddiqui, I.; Rehman, S.; Khan, S.; Alhems, L.M. Assessment of wind energy potential using wind energy conversion system. J. Clean. Prod.
**2019**, 216, 346–360. [Google Scholar] [CrossRef] - Al-Mhairat, B.; Al-Quraaan, A. Assessment of Wind Energy Resources in Jordan Using Different Optimization Techniques. Processes
**2022**, 10, 105. [Google Scholar] [CrossRef] - Chang, T.P. Estimation of wind energy potential using different probability density functions. Appl. Energy
**2011**, 88, 1848–1856. [Google Scholar] [CrossRef] - Evans, J.W.; Johnson, R.A.; Green, D.W. Two- and Three-Parameter Weibull Goodness-of-Fit Tests; Res. Pap. FPL-RP-493; U.S. Department of Agricultural, Forest Service, Forest Products Laboratory: Madison, WI, USA, 1989; p. 27.
- Krit, M.; Gaudoin, O.; Remy, E. Goodness-of-fit tests for the Weibull and extreme value distributions: A review and comparative study. Commun. Stat.-Simul. Comput.
**2021**, 50, 1888–1911. [Google Scholar] [CrossRef] - Akaike, H. A new look at the statistical model identification. IEEE Trans. Automat. Contr.
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Schwarz, G. Estimating the Dimension of a Model. Ann. Stat.
**1978**, 6, 461–464. [Google Scholar] [CrossRef] - Chicco, D.; Warrens, M.J.; Jurman, G. The coefficient of determination R-squared is more informative than SMAPE, MAE, MAPE, MSE and RMSE in regression analysis evaluation. PeerJ Comput. Sci.
**2021**, 7, e623. [Google Scholar] [CrossRef]

**Figure 2.**The topography of the Poprad basin and its surroundings. Source [54].

**Figure 5.**Modelled probability distributions fitted to the histogram of the wind speed data–the entire considered period.

**Figure 6.**Modelled probability distributions fitted to the histogram of the wind speed data—the seasons.

**Figure 7.**Modelled probability distributions fitted to the histogram of the wind speed data—the respective months.

Period | Mean | Standard Deviation | Min | Max | Lower Quartile | Median | Upper Quartile | Coefficient of Variation (%) | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|---|---|

January | 3.31 | 2.87 | 0.1 | 16.7 | 1.1 | 2.2 | 5.0 | 86.49 | 1.21 | 3.78 |

February | 3.23 | 2.70 | 0.1 | 16.6 | 1.2 | 2.3 | 4.7 | 83.69 | 1.23 | 4.13 |

March | 3.85 | 2.73 | 0.1 | 18.5 | 1.7 | 3.1 | 5.5 | 71.04 | 0.95 | 3.31 |

April | 3.49 | 2.33 | 0.1 | 16.7 | 1.7 | 2.9 | 4.8 | 66.75 | 1.12 | 4.15 |

May | 3.33 | 2.26 | 0.1 | 15.3 | 1.6 | 2.7 | 4.5 | 67.77 | 1.17 | 4.21 |

June | 3.12 | 2.06 | 0.1 | 12.4 | 1.6 | 2.5 | 4.3 | 66.10 | 1.14 | 4.05 |

July | 3.16 | 2.17 | 0.1 | 13.6 | 1.6 | 2.5 | 4.3 | 68.46 | 1.18 | 3.99 |

August | 2.75 | 1.86 | 0.1 | 12.2 | 1.4 | 2.2 | 3.7 | 67.62 | 1.28 | 4.57 |

September | 2.92 | 2.11 | 0.1 | 14.1 | 1.4 | 2.3 | 3.9 | 72.41 | 1.29 | 4.47 |

October | 2.96 | 2.25 | 0.1 | 14.7 | 1.3 | 2.2 | 4.1 | 75.97 | 1.28 | 4.38 |

November | 3.03 | 2.52 | 0.1 | 14.3 | 1.1 | 2.1 | 4.2 | 83.15 | 1.31 | 4.23 |

December | 3.15 | 2.69 | 0.1 | 14.9 | 1.1 | 2.2 | 4.6 | 82.27 | 1.28 | 4.17 |

Period | Mean | Standard Deviation | Min | Max | Lower Quartile | Median | Upper Quartile | Coefficient of Variation (%) | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|---|---|

Spring | 3.56 | 2.46 | 0.1 | 18.5 | 1.7 | 2.9 | 4.9 | 69.14 | 1.10 | 3.91 |

Summer | 3.01 | 2.04 | 0.1 | 13.6 | 1.5 | 2.4 | 4.1 | 67.81 | 1.21 | 4.24 |

Autumn | 2.97 | 2.30 | 0.1 | 14.7 | 1.3 | 2.2 | 4.1 | 77.44 | 1.32 | 4.44 |

Winter | 3.23 | 2.76 | 0.1 | 16.7 | 1.1 | 2.2 | 4.8 | 85.27 | 1.24 | 4.02 |

Period | Mean | Standard Deviation | Min | Max | Lower Quartile | Median | Upper Quartile | Coefficient of Variation (%) | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|---|---|

Total | 3.19 | 2.41 | 0.1 | 18.5 | 1.4 | 2.4 | 4.5 | 75.56 | 1.25 | 4.32 |

Distribution | MLM Estimate | |
---|---|---|

W2 | Log-likelihood function | $\mathrm{ln}L=n\mathrm{ln}\frac{a}{{b}^{a}}-\frac{1}{{b}^{a}}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{a}+\left(a-1\right){\displaystyle {\displaystyle \sum}_{i=1}^{n}}\mathrm{ln}{x}_{i}$ |

Likelihood equations | $\frac{1}{a}-\frac{{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}{}^{a}\mathrm{ln}{x}_{i}}{{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}{}^{a}}+\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\mathrm{ln}{x}_{i}=0$ | |

$b={\left[\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}{}^{a}\right]}^{1/a}$ | ||

W3 | Log-likelihood function | $\mathrm{ln}L=n\mathrm{ln}\frac{a}{{b}^{a}}-\frac{1}{{b}^{a}}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left({x}_{i}-c\right)}^{a}+\left(a-1\right){\displaystyle {\displaystyle \sum}_{i=1}^{n}}\mathrm{ln}\left({x}_{i}-c\right)$ |

Likelihood equations | $\frac{1}{a}-\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-c\right)}^{a}\mathrm{ln}\left({x}_{i}-c\right)}{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-c\right)}^{a}}+\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\mathrm{ln}\left({x}_{i}-c\right)=0$ | |

$b={\left[\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left({x}_{i}-c\right)}^{a}\right]}^{1/a}$ | ||

$\frac{a}{1-a}-\frac{1}{n}\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-c\right)}^{a}}{{{\displaystyle \sum}}_{i=1}^{n}{\left({x}_{i}-c\right)}^{a-1}}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\frac{1}{{x}_{i}-c}=0$ | ||

EW | Log-likelihood function | $\mathrm{ln}L=n\mathrm{ln}\left(\frac{\gamma a}{{b}^{a}}\right)-\frac{1}{{b}^{a}}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{a}+\left(a-1\right){\displaystyle {\displaystyle \sum}_{i=1}^{n}}\mathrm{ln}{x}_{i}+\left(\gamma -1\right){\displaystyle {\displaystyle \sum}_{i=1}^{n}}\mathrm{ln}\left[1-\mathrm{exp}\left(-{\left(\frac{{x}_{i}}{b}\right)}^{a}\right)\right]$ |

Likelihood equations | $\frac{n}{a}-{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left(\frac{{x}_{i}}{b}\right)}^{a}\mathrm{ln}\left(\frac{{x}_{i}}{b}\right)+{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\mathrm{ln}\left(\frac{{x}_{i}}{b}\right)+\left(\gamma -1\right){\displaystyle {\displaystyle \sum}_{i=1}^{n}}\frac{\mathrm{exp}\left(-{\left(\frac{{x}_{i}}{b}\right)}^{a}\right){\left(\frac{{x}_{i}}{b}\right)}^{a}\mathrm{ln}\left(\frac{{x}_{i}}{b}\right)}{\left[1-\mathrm{exp}\left(-{\left(\frac{{x}_{i}}{b}\right)}^{a}\right)\right]}=0$ | |

$-\frac{na}{b}+\frac{a}{b}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left(\frac{{x}_{i}}{b}\right)}^{a}-\frac{a}{b}\left(\gamma -1\right){\displaystyle {\displaystyle \sum}_{i=1}^{n}}\frac{\mathrm{exp}\left(-{\left(\frac{{x}_{i}}{b}\right)}^{a}\right){\left(\frac{{x}_{i}}{b}\right)}^{a}}{\left[1-\mathrm{exp}\left(-{\left(\frac{{x}_{i}}{b}\right)}^{a}\right)\right]}=0$ | ||

$\gamma =-\frac{n}{{{\displaystyle \sum}}_{i=1}^{n}\left[1-\mathrm{exp}\left(-{\left(\frac{{x}_{i}}{b}\right)}^{a}\right)\right]}$ |

**Table 5.**Maximum likelihood estimates of parameters of the modelled distributions—the entire considered period and the seasons.

Distribution | Estimated Parameters | Entire Period | Spring | Summer | Autumn | Winter |
---|---|---|---|---|---|---|

W2 | a | 1.3893 | 1.5146 | 1.5680 | 1.3650 | 1.2059 |

b | 3.5133 | 3.9596 | 3.3696 | 3.2592 | 3.4510 | |

W3 | a | 1.3263 | 1.4591 | 1.5046 | 1.2978 | 1.1305 |

b | 3.3802 | 3.8383 | 3.2479 | 3.1216 | 3.2817 | |

c | 0.0894 | 0.0875 | 0.0914 | 0.0919 | 0.0945 | |

EW | a | 0.8726 | 1.0340 | 0.8823 | 0.7791 | 0.7642 |

b | 1.6847 | 2.3591 | 1.3894 | 1.2294 | 1.5294 | |

γ | 2.5451 | 2.1045 | 3.3732 | 3.2179 | 2.4516 |

**Table 6.**The GOF tests and model selection criteria for the entire considered period and for the seasons.

Period | Distribution | KS | AD | ln L | AIC | BIC | R^{2} | RMSE ^{a} |
---|---|---|---|---|---|---|---|---|

Entire period | W2 | 0.059 | 662.6 | −308,150.5 | 616,305.0 | 616,324.8 | 0.9909 | 0.028 |

W3 | 0.050 | 470.5 | −306,925.1 | 613,856.2 | 613,885.9 | 0.9936 | 0.023 | |

EW | 0.034 | 243.3 | −306,288.0 | 612,582.0 | 612,611.8 | 0.9970 | 0.016 | |

Spring | W2 | 0.053 | 124.4 | −80,074.0 | 160,152.1 | 160,169.1 | 0.9933 | 0.024 |

W3 | 0.047 | 92.7 | −79,880.8 | 159,767.6 | 159,793.1 | 0.9950 | 0.021 | |

EW | 0.035 | 54.6 | −79,757.7 | 159,521.4 | 159,547.0 | 0.9972 | 0.016 | |

Summer | W2 | 0.067 | 227.4 | −73,395.1 | 146,794.3 | 146,811.4 | 0.9873 | 0.032 |

W3 | 0.061 | 179.7 | −73,121.5 | 146,248.9 | 146,274.5 | 0.9899 | 0.029 | |

EW | 0.037 | 54.3 | −72,601.7 | 145,209.5 | 145,235.0 | 0.9972 | 0.015 | |

Autumn | W2 | 0.066 | 218.7 | −74,507.1 | 149,018.2 | 149,035.2 | 0.9880 | 0.032 |

W3 | 0.057 | 156.8 | −74,116.8 | 148,239.7 | 148,265.3 | 0.9915 | 0.027 | |

EW | 0.033 | 64.5 | −73,827.2 | 147,660.4 | 147,685.9 | 0.9970 | 0.016 | |

Winter | W2 | 0.063 | 215.2 | −78,083.0 | 156,170.0 | 156,187.0 | 0.9884 | 0.032 |

W3 | 0.049 | 143.7 | −77,557.3 | 155,120.5 | 155,146.0 | 0.9927 | 0.025 | |

EW | 0.046 | 130.6 | −77,737.6 | 155,481.1 | 155,506.6 | 0.9936 | 0.024 |

^{a}The best results of the GOF tests and of the model selection criteria are highlighted in bold.

**Table 7.**Maximum likelihood estimates of parameters of the modelled distributions—the respective months.

Distribution | Estimated Parameters | Jan. | Feb. | Mar. | Apr. | May | June |
---|---|---|---|---|---|---|---|

W2 | a | 1.1862 | 1.2171 | 1.4476 | 1.5725 | 1.5599 | 1.6024 |

b | 3.5240 | 3.4546 | 4.2539 | 3.9058 | 3.7221 | 3.4978 | |

W3 | a | 1.1133 | 1.1360 | 1.3928 | 1.5199 | 1.5009 | 1.5403 |

b | 3.3507 | 3.2860 | 4.1266 | 3.7968 | 3.5984 | 3.3781 | |

c | 0.0961 | 0.0903 | 0.0875 | 0.0803 | 0.0925 | 0.0910 | |

EW | a | 0.7044 | 0.9292 | 1.1745 | 1.0970 | 0.9501 | 0.9537 |

b | 1.3078 | 2.2820 | 3.2900 | 2.4519 | 1.8160 | 1.6601 | |

γ | 2.8359 | 1.6514 | 1.4615 | 2.0194 | 2.7514 | 2.9133 | |

Distribution | EstimatedParameters | July | Aug. | Sept. | Oct. | Nov. | Dec. |

W2 | a | 1.5523 | 1.5765 | 1.4656 | 1.3873 | 1.2650 | 1.2178 |

b | 3.5363 | 3.0818 | 3.2430 | 3.2543 | 3.2742 | 3.3750 | |

W3 | a | 1.4918 | 1.5082 | 1.4001 | 1.3198 | 1.1964 | 1.1473 |

b | 3.4131 | 2.9617 | 3.1175 | 3.1203 | 3.1214 | 3.2118 | |

c | 0.0922 | 0.0905 | 0.0888 | 0.0902 | 0.0953 | 0.0963 | |

EW | a | 0.8530 | 0.8841 | 0.8586 | 0.8306 | 0.6598 | 0.6786 |

b | 1.3725 | 1.2609 | 1.3794 | 1.3990 | 0.8914 | 1.0755 | |

γ | 3.5642 | 3.4373 | 3.0535 | 2.8575 | 3.9939 | 3.3284 |

Period | Distribution | KS | AD | ln L | AIC | BIC | R^{2} | RMSE ^{a} |
---|---|---|---|---|---|---|---|---|

Jan. | W2 | 0.069 | 93.9 | −27,316.8 | 54,637.6 | 54,652.4 | 0.9855 | 0.036 |

W3 | 0.056 | 65.3 | −27,117.3 | 54,240.5 | 54,262.8 | 0.9903 | 0.029 | |

EW | 0.051 | 57.6 | −27,174.5 | 54,355.0 | 54,377.3 | 0.9920 | 0.027 | |

Feb. | W2 | 0.053 | 42.2 | −24,131.0 | 48,266.0 | 48,280.6 | 0.9924 | 0.026 |

W3 | 0.038 | 28.8 | −24,000.8 | 48,007.5 | 48,029.5 | 0.9958 | 0.019 | |

EW | 0.041 | 30.0 | −24,089.7 | 48,185.5 | 48,207.5 | 0.9950 | 0.021 | |

Mar. | W2 | 0.052 | 39.2 | −27,971.6 | 55,947.1 | 55,962.0 | 0.9938 | 0.023 |

W3 | 0.045 | 30.0 | −27,911.0 | 55,828.0 | 55,850.3 | 0.9953 | 0.020 | |

EW | 0.042 | 31.8 | −27,946.2 | 55,898.5 | 55,920.8 | 0.9951 | 0.021 | |

Apr. | W2 | 0.053 | 35.7 | −25,790.8 | 51,585.7 | 51,600.5 | 0.9939 | 0.023 |

W3 | 0.048 | 27.3 | −25,743.8 | 51,493.6 | 51,515.8 | 0.9954 | 0.020 | |

EW | 0.035 | 13.5 | −25,687.6 | 51,381.3 | 51,403.5 | 0.9977 | 0.014 | |

May | W2 | 0.056 | 53.3 | −26,080.9 | 52,165.7 | 52,180.6 | 0.9916 | 0.026 |

W3 | 0.050 | 40.5 | −25,999.1 | 52,004.2 | 52,026.5 | 0.9936 | 0.023 | |

EW | 0.034 | 15.4 | −25,890.7 | 51,787.4 | 51,809.7 | 0.9977 | 0.014 | |

June | W2 | 0.065 | 59.4 | −24,247.7 | 48,499.4 | 48,514.2 | 0.9900 | 0.029 |

W3 | 0.059 | 46.4 | −24,170.1 | 48,346.3 | 48,368.5 | 0.9922 | 0.025 | |

EW | 0.037 | 17.7 | −24,045.0 | 48,095.9 | 48,118.1 | 0.9971 | 0.016 | |

July | W2 | 0.071 | 89.1 | −25,472.5 | 50,949.1 | 50,964.0 | 0.9853 | 0.035 |

W3 | 0.065 | 72.4 | −25,380.1 | 50,766.1 | 50,788.5 | 0.9880 | 0.032 | |

EW | 0.041 | 25.4 | −25,198.1 | 50,402.1 | 50,424.5 | 0.9962 | 0.018 | |

Aug. | W2 | 0.068 | 77.2 | −23,489.9 | 46,983.8 | 46,998.7 | 0.9871 | 0.032 |

W3 | 0.062 | 60.0 | −23,391.6 | 46,789.2 | 46,811.5 | 0.9899 | 0.028 | |

EW | 0.035 | 14.3 | −23,197.7 | 46,401.4 | 46,423.7 | 0.9978 | 0.014 | |

Sept. | W2 | 0.070 | 77.2 | −23,997.5 | 47,999.1 | 48,013.9 | 0.9864 | 0.033 |

W3 | 0.063 | 59.7 | −23,899.7 | 47,805.4 | 47,827.6 | 0.9895 | 0.029 | |

EW | 0.040 | 20.8 | −23,766.2 | 47,538.4 | 47,560.6 | 0.9967 | 0.017 | |

Oct. | W2 | 0.061 | 66.0 | −25,246.6 | 50,497.2 | 50,512.1 | 0.9894 | 0.030 |

W3 | 0.052 | 47.0 | −25,127.5 | 50,261.0 | 50,283.3 | 0.9926 | 0.025 | |

EW | 0.033 | 21.8 | −25,051.8 | 50,109.5 | 50,131.9 | 0.9970 | 0.016 | |

Nov. | W2 | 0.068 | 89.8 | −25,124.7 | 50,253.4 | 50,268.3 | 0.9858 | 0.035 |

W3 | 0.057 | 63.5 | −24,947.6 | 49,901.2 | 49,923.4 | 0.9902 | 0.029 | |

EW | 0.040 | 33.8 | −24,869.7 | 49,745.4 | 49,767.6 | 0.9955 | 0.020 | |

Dec. | W2 | 0.066 | 85.5 | −26,617.7 | 53,239.3 | 53,254.2 | 0.9871 | 0.034 |

W3 | 0.055 | 58.5 | −26,418.7 | 52,843.4 | 52,865.8 | 0.9913 | 0.027 | |

EW | 0.048 | 46.7 | −26,426.6 | 52,859.3 | 52,881.6 | 0.9935 | 0.024 |

^{a}The best results of the GOF tests and of the model selection criteria are highlighted in bold.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pobočíková, I.; Michalková, M.; Sedliačková, Z.; Jurášová, D.
Modelling the Wind Speed Using Exponentiated Weibull Distribution: Case Study of Poprad-Tatry, Slovakia. *Appl. Sci.* **2023**, *13*, 4031.
https://doi.org/10.3390/app13064031

**AMA Style**

Pobočíková I, Michalková M, Sedliačková Z, Jurášová D.
Modelling the Wind Speed Using Exponentiated Weibull Distribution: Case Study of Poprad-Tatry, Slovakia. *Applied Sciences*. 2023; 13(6):4031.
https://doi.org/10.3390/app13064031

**Chicago/Turabian Style**

Pobočíková, Ivana, Mária Michalková, Zuzana Sedliačková, and Daniela Jurášová.
2023. "Modelling the Wind Speed Using Exponentiated Weibull Distribution: Case Study of Poprad-Tatry, Slovakia" *Applied Sciences* 13, no. 6: 4031.
https://doi.org/10.3390/app13064031