Fábio SİLVEİRA, Frank GOMES-SİLVA, Cícero BRİTO, Jader JALE, Felipe GUSMÃO, Silvio XAVİER-JÚNİOR, João ROCHA

Modelling wind speed with a univariate probability distribution depending on two baseline functions

Characterizing the wind speed distribution properly is essential for the satisfactory production of potential energy in wind farms, being the mixture models usually employed in the description of such data. However, some mixture models commonly have the undesirable property of non-identifiability. In this work, we present an alternative distribution which is able to fit the wind speed data decently. The new model, called Normal-Weibull-Weibull, is identifiable and its cumulative distribution function is written as a composition of two baseline functions. We discuss structural properties of the class that generates the proposed model, such as the linear representation of the probability density function, moments and moment generating function. We perform a Monte Carlo simulation study to investigate the behavior of the maximum likelihood estimates of the parameters. Finally, we present applications of the new distribution for modelling wind speed data measured in five different cities of the Northeastern Region of Brazil.

Keywords:

Goodness-of-fit, identifiability, L-BFGS-B algorithm, maximum likelihood, Monte Carlo simulation,

PDF

___

[1] S. Akdag, H. Bagiorgas and G. Mihalakakou, Use of two-component Weibull mixtures in the analysis of wind speed in Eastern Mediterranean, Appl. Energy 87, 2566-2573, 2010.
[2] A. Alzaatreh, C. Lee and F. Famoye, A new method for generating families of continuous distributions, Metron 71, 63-79, 2013.
[3] J.C.H. Araújo, W.F. Souza, A.J.A. Meireles and C. Brannstrom, Sustainability challenges of wind power deployment in coastal Ceará state, Brazil, Sustainability 12 (14), 5562, 2020.
[4] A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Stat. 12 (2), 171-178, 1985.
[5] T.G. Bali and P. Theodossiou, Risk measurement performance of alternative distribution functions, J. Risk Insur. 75 (2), 411-437, 2008.
[6] C.R. Brito, L.C. Rêgo, W.R. Oliveira and F. Gomes-Silva, Method for generating distributions and classes of probability distributions: the univariate case, Hacet. J. Math. Stat. 48 (3), 897-930, 2019.
[7] R.H. Byrd, P. Lu, J. Nocedal and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Comput. 16 (5), 1190-1208, 1995.
[8] J. Carta and P. Ramírez, Analysis of two-component mixture Weibull statistics for estimation of wind speed distributions, Renew. Energy 32, 518-531, 2007.
[9] T. Chang, Estimation of wind energy potential using different probability density functions, Appl. Energy 88, 1848-1856, 2011.
[10] G. Chen and N. Balakrishnan, A general purpose approximate goodness-of-fit test, J. Qual. Technol. 27 (2), 154-161, 1995.
[11] A. Eltamaly, Design and implementation of wind energy system in Saudi Arabia, Renew. Energy 60, 42-52, 2013.
[12] Q. Han, S. Ma, T. Wang and F. Chu, Kernel density estimation model for wind speed probability distribution with applicability to wind energy assessment in China, Renew. Sustain. Energy Rev. 115, 109387, 2019.
[13] B.E. Hansen, Autoregressive conditional density estimation, Int. Econ. Rev. 35 (3), 705-730, 1994.
[14] B. Hu, Y. Li, H. Yang and H. Wang, Wind speed model based on kernel density estimation and its application in reliability assessment of generating systems, J. Mod. Power Syst. Clean Energy 5 (2), 220-227, 2017.
[15] U. Ilhan and Y.M. Kantar, Analysis of some flexible families of distribution for estimation of wind speed distributions, Appl. Energy 89, 355-367, 2012.
[16] INMET, National Institute of Meteorology of Brazil. Official website, URL https://portal.inmet.gov.br/ accessed in 11/01/2021.
[17] O. Jaramillo and M. Borja, Wind speed analysis in La Ventosa, Mexico: A bimodal probability distribution case, Renew. Energy 29, 1613-1630, 2004.
[18] R. Kollu, S. Rayapudi, S. Narasimham and K. Pakkurthi, Mixture probability distribution functions to model wind speed distributions, Int. J. Energy Environ. Eng. 3 (27), 2012.
[19] G. McLachlan and D. Peel, Finite Mixture Models, Wiley Interscience, 2000.
[20] K. Mohammadi, O. Alavi and J. McGowan, Use of Birnbaum-Saunders distribution for estimating wind speed and wind power probability distributions: A review, Energy Convers. Manag. 143, 109-122, 2017.
[21] E.C. Morgan, M. Lackner, R.M. Vogel and L.G. Baise, Probability distributions for offshore wind speeds, Energy Convers. Manag. 52 (1), 15-26, 2011.
[22] G.S. Mudholkar and D.K. Srivastava, Exponentiated weibull family for analyzing bathtub failure-rate data, IEEE Trans. Reliab. 42 (2), 299-302, 1993.
[23] J. Von Neumann, Various techniques used in connection with random digits, Applied Mathematics Series 12, National Bureau of Standards, Washington, DC, USA, 1951.
[24] H.D. Nguyen, D. Wang, G.J. McLachlan, Randomized mixture models for probability density approximation and estimation, Inf. Sci. 467, 135-148, 2018.
[25] S. Perkin, D. Garrett and P. Jensson, Optimal wind turbine selection methodology: A case-study for Búrfell, Iceland, Renew. Energy 75, 165-172, 2015.
[26] S. Pishgar-Komleh, A. Keyhani and P. Sefeedpari, Wind speed and power density analysis based on Weibull and Rayleigh distributions (a case study: Firouzkooh county of Iran), Renew. Sustain. Energy Rev. 42, 313-322, 2015.
[27] Z. Qin, W. Li and X. Xiong, Estimating wind speed probability distribution using kernel density method, Electr. Power Syst. Res. 81 (12), 2139-2146, 2011.
[28] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2018.
[29] B. Safari, Modeling wind speed and wind power distributions in Rwanda, Renew. Sustain. Energy Rev. 15, 925-935, 2011.
[30] D. Weisser, A wind energy analysis of Grenada: an estimation using the Weibull density function, Renew. Energy 28 (11), 1803-1812, 2003.