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RETRACTED ARTICLE: Application of extreme learning machine for estimation of wind speed distribution

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This article was retracted on 29 November 2018

An Erratum to this article was published on 16 July 2015

This article has been updated

Abstract

The knowledge of the probabilistic wind speed distribution is of particular significance in reliable evaluation of the wind energy potential and effective adoption of site specific wind turbines. Among all proposed probability density functions, the two-parameter Weibull function has been extensively endorsed and utilized to model wind speeds and express wind speed distribution in various locations. In this research work, extreme learning machine (ELM) is employed to compute the shape (k) and scale (c) factors of Weibull distribution function. The developed ELM model is trained and tested based upon two widely successful methods used to estimate k and c parameters. The efficiency and accuracy of ELM is compared against support vector machine, artificial neural network and genetic programming for estimating the same Weibull parameters. The survey results reveal that applying ELM approach is eventuated in attaining further precision for estimation of both Weibull parameters compared to other methods evaluated. Mean absolute percentage error, mean absolute bias error and root mean square error for k are 8.4600 %, 0.1783 and 0.2371, while for c are 0.2143 %, 0.0118 and 0.0192 m/s, respectively. In conclusion, it is conclusively found that application of ELM is particularly promising as an alternative method to estimate Weibull k and c factors.

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Change history

  • 29 November 2018

    Editors have retracted this article because validity of the content of this article cannot be verified.

  • 16 July 2015

    An erratum to this article has been published.

  • 29 November 2018

    Editors have retracted this article because validity of the content of this article cannot be verified.

  • 29 November 2018

    Editors have retracted this article because validity of the content of this article cannot be verified.

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Acknowledgments

The authors would like to thank the University of Malaya for the research grants allocated (UMRG RP015C-13AET and High Impact Research Grant, HIR-D000006-16001). Special appreciation is also credited to the Malaysian Ministry of Education, MOE for the Fundamental Research Grant Scheme (FP053-2013B). The authors would like to thank the Bright Spark Unit of University of Malaya for the financial support.

Author information

Authors and Affiliations

  1. Department of Computer System and Technology, Faculty of Computer Science and Information Technology, University of Malaya, 50603, Kuala lumpur, Malaysia

    Shahaboddin Shamshirband

  2. Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran

    Kasra Mohammadi

  3. Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50603, Kuala Lumpur, Malaysia

    Chong Wen Tong & Dalibor Petković

  4. Technical University Federico Santa Maria, Valparaíso, Chile

    Emilio Porcu

  5. Industrial Engineering Department, Yazd University, Yazd, Iran

    Ali Mostafaeipour

  6. Department of Civil and Environmental Engineering, ITM University, Gurugaon, 122017, India

    Sudheer Ch

  7. Department of Mechanical Engineering, School of Engineering, Australian College of Kuwait, P.O. Box 1411, 13015, Safat, Kuwait

    Ahmad Sedaghat

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  1. Shahaboddin Shamshirband
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  2. Kasra Mohammadi
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  7. Sudheer Ch
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Correspondence to Shahaboddin Shamshirband.

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The Editors have retracted this article because validity of the content of this article cannot be verified. This article showed evidence of peer review manipulation. None of the authors agree to this retraction.

Appendix

Appendix

1.1 Coefficient of determination (R2)

The R2 provides a measure of level of the linear relationship between the estimated and the actual values. The R2 is obtained by:

$$ R^{2} = \frac{{\sum\nolimits_{i = 1}^{n} {\left( {X_{i,act} - X_{act,avg} } \right)}^{2} - \sum\nolimits_{i = 1}^{n} {\left( {X_{i,est} - X_{i,act} } \right)}^{2} }}{{\sum\nolimits_{i = 1}^{n} {\left( {X_{i,act} - X_{act,avg} } \right)}^{2} }} $$

1.2 Mean absolute percentage error (MAPE)

The MAPE shows the mean absolute percentage difference between the estimated and actual data. The MAPE is calculated by following equation:

$$ MAPE = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left| {\frac{{X_{i,est} - X_{i,act} }}{{X_{i,act} }}} \right| \times 100} $$

1.3 Mean absolute bias error (MABE)

The MABE represents the average quantity of total absolute bias errors between estimated and actual values and is defined by:

$$ MABE = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left| {X_{i,est} - X_{i,act} } \right|} $$

1.4 Root mean square error (RMSE)

The RMSE determines the precision of the model by comparing the deviation between the estimated and actual data. The RMSE has always a positive value and is calculated by:

$$ RMSE = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {X_{i,est} - X_{i,act} } \right)}^{2} } $$

where X i,est, X i,act and X act,avg represent the ith estimated value based upon the applied approaches, ith actual value and average of actual values, respectively. Also, n is the number of data.

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Shamshirband, S., Mohammadi, K., Tong, C.W. et al. RETRACTED ARTICLE: Application of extreme learning machine for estimation of wind speed distribution. Clim Dyn 46, 1893–1907 (2016). https://doi.org/10.1007/s00382-015-2682-2

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