# Stochastic Extreme Wind Speed Modeling and Bayes Estimation under the Inverse Rayleigh Distribution
## 8 Conclusions  
The correct assessment of the probabilistic distribution for wind and extreme wind speeds is very important for a technical and economical point of view and gives the possibility to optimize the electricity production of a wind farm. In particular, the occurrence of very high values of wind speed is an undesirable event, since wind power systems provide zero power output for values greater than their cutoff thresholds. Also, the mechanical safety of the installations can be seriously compromised by extreme values of wind speed. The IRD is proposed in the paper as a valid model for extreme wind speeds; such a model is proposed for the first time in the literature on the object, and its validity is illustrated with reference to real wind-speed data. Then, this paper investigates the problem of a proper and efficient Bayes estimation of a risk index described in terms of a stochastic process of extreme values of wind speed based upon a practical parameter estimation of the IRD using the gamma and/or the prior beta distribution according to which kind of prior information is available. The methodology proposed is based upon the use of real data of EWS and using different estimation procedures, also comparing such estimation procedures to the ML procedure. The proposed model is verified by means of many numerical simulations. The obtained results show the efficiency and precision of the analyzed model. The considered Bayes estimator appears also to be robust with respect to incorrect prior assumptions. It is highlighted that, beyond the theoretical aspects which have been developed in the paper, its basic purpose is to address a real engineering problem, since it deals both with wind power production evaluation and the mechanical safety of the installations. This is a clearly basic topic to be addressed both in the planning and the operation of a wind farm and is afforded in the manuscript by a novel risk index which the authors deem to be an important tool in helping a decision making procedure for deciding if a wind farm is worth being installed and whether its operation will be safe or reliable enough in time. Further extension of the proposed approach is worth being developed, extending the hypotheses assumed for the Poisson process of exceedances, allowing for non-homogeneous Poisson processes [39; 40; 41; 42; 43; 44]. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameters** & **n = 3** & **n = 10** & **n = 30** & **n = 50** \\ \hline _AV(R)_ & 0.8112 & 0.8110 & 0.8100 & 0.8125 \\ _AV(S)_ & 0.1696 & 0.1696 & 0.1696 & 0.1699 \\ _AV(\(\phi\))_ & 9.9955 & 10.0037 & 9.9497 & 10.012 \\ _AV(RL)_ & 0.1198 & 0.2649 & 0.5637 & 0.7316 \\ _AV(RB)_ & 0.7875 & 0.7912 & 0.8018 & 0.8110 \\ _RMSEL_ & 0.7153 & 0.5784 & 0.3023 & 0.1652 \\ _RMSEB_ & 0.0503 & 0.0485 & 0.0419 & 0.0395 \\ _RARE_ & 17.0349 & 14.1524 & 7.5331 & 3.8165 \\ _RMRE_ & 3.4950 & 5.1857 & 3.5330 & 4.8343 \\ _REFF_ & 14.2283 & 11.9186 & 7.2107 & 4.1840 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Simulation results for cases C’22. **Author Contributions:** Conceptualization, E.C. and L.P.D.N. ; methodology E.C. ; software, E.C. ; validation, E.C. and L.P.D.N. ; writing--original draft preparation, E.C. and L.P.D.N. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding. **Acknowledgments:** The authors express their most sincere thanks to the reviewers for their insightful and very helpful comments, which considerably improved the manuscript. **Conflicts of Interest:** The authors declare no conflict of interest. **Nomenclature**