# Stochastic Extreme Wind Speed Modeling and Bayes Estimation under the Inverse Rayleigh Distribution
## 1 Introduction  
Inverse Rayleigh distribution (IRD) has raised growing interest in modeling positive random variables and especially for its applications in reliability and survival times studies [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. Being also a particular case of the inverse Weibull distribution, which has been adopted as a valid model for the characterization and estimation of extreme values of wind speed (WS) [12; 13; 14; 15; 16; 17]--or briefly, \"extreme wind speed\" (EWS)--the IRD model is studied in this paper in relation to such kinds of applications. Before introducing the above model, its generalization, and their application to EWS analyses, it is recalled that studies on wind speed statistical distribution are more and more largely widespread, since wind energy production strongly depends on such distribution. For instance, the annual electricity production (AEP) indicator is frequently used to quantify the electricity output expected in a particular system. The following expression can be considered for the evaluation of the AEP indicator [15]:  
\[AEP=T\int_{u_{act,in}}^{u_{act,eff}}f(u)P(u)du \tag{1}\]  
where* AEP is in MWh,
* T is the average number of hours per year (8760),
* f(u) is the wind speed probability density function,
* _P(u)_ is the power curve of a turbine as function of wind speed, and
* ucut-in and ucut-out are the cut-in and cutoff speed values of a turbine. Hence, accurate wind speed distribution modeling is the first step to achieving accurate wind energy production estimation. Moreover, as discussed in [16; 17; 18; 19; 20], the randomness of WS also has a great impact on the mechanical reliability of wind power systems, since extreme values of wind speed may damage sensible components of the structures, such as towers and wind blades, so that EWS characterization also constitutes a basic tool for an officiate wind turbine design. Furthermore, as also pointed out in [19; 20], values of WS that are greater than the \"cutoff\" value of the wind generator are generally undesirable, since the electric generator has to be disconnected from the wind turbine to not compromise the electrical section of the wind power system; consequently, the \"cutoff\" value of the generator must be chosen according to the characterization of EWS in the particular location, since it has a great impact on aggregate power production [21; 22; 23]. Therefore, EWS is also indirectly related to the AEP formulation of Equation (1). However, in this paper, primary reference to the issue of mechanical safety will be dealt with, with emphasis on the most adequate methods for the correct choice or identification of the various probability models involved, which is a delicate step in all applied statistical analysis [24]. Leaving more details on the IRD model to Section 2 and the Appendices, for the subsequent discussion, it is sufficient to report the following definition of the model: given a positive random variable (RV) X (the EWS in this study), the IRD cumulative distribution functions (CDF), \(F(x)\), with positive scale parameter \(\alpha\) is expressed in the paper by the following equation:  
\[F(x|\alpha)=e^{-(\frac{\alpha}{x^{2}})},\ x>0 \tag{2}\]  
to which the following probability density function (PDF) corresponds:  
\[f(x|\alpha)=2\alpha x^{-3}e^{-(\frac{\alpha}{x^{2}})},\ x>0 \tag{3}\]  
In particular, in this paper, a practical method for the Bayes estimation [7; 8; 9; 10; 11; 12] of the IRD model for EWS is proposed, in the framework of the assessment of a dynamical \"risk index\", to be outlined in the following. The method is also extended to a generalization of the IRD obtained by means of a continuous mixture of its parameter. The mixture of the scale parameter has been recently adopted also for other models [19; 20] in order to allow the randomness on environmental conditions of the windfarm under study.