Normal law of distribution of wind velocity vector in polar coordinates

Abstract

It is impossible to organize wind energy systems without studying of wind speed regime at the surface layer of the atmosphere within a specific area and at climatic scales. Such studies are often accompanied by approximations of probabilities of wind speed performed in the form of normal law of a system of random values presented by a zonal u and a meridional u which are components of a wind speed vector. It is suggested that, for the purposes of wind energy, display of a wind speed vector in polar coordinates (r, α) where r is a module of wind speed and α is a polar angle appears to be more preferable. The article shows a transform from a normal law of distribution of probabilities with density f(u,u) to a normal law distribution with density f(r,α) completed by means of functional transformation with elliptic dispersion in place. Based on a normal law of distribution f(r,α) and through integration with respect to corresponding variables individual distributions of probabilities f(r) and f(α)  as well as conditional distributions of probabilities f(r/α)  and f(α/r)  were obtained in the areas of their existence. The article shows individual distributions in case of circular and elliptic dispersion of a wind speed vector. It shows that an individual distribution of a wind speed probability in case of circular dispersion and in the absence of correlated dependence turns into the Rayleigh's distribution and a conditional distribution of a polar angle degenerates in an even distribution. The cases of distributions with dispersions of a wind speed module having elliptic properties subject to availability of correlated connection between wind speed components were also studied. Calculation of probabilities of a polar angle being within different sections of the area 0≤α≤2π with set values of a wind speed module also took place. Numerical experiments proved the advantage of such modeling of distributions of wind speed vector.