Normal law of distribution of wind velocity vector in polar coordinates

Authors: Shkolnyy E. P.

Year: 2017

Issue: 20

Pages: 5-17

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 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,a) r – where is a module of wind speed and a – is a polar angle appears to be more preferable. The article shows a transform from a normal law of distribution of probabilities with density ф(u,v) to a normal law distribution with density ф(r,a) completed by means of functional transformation with elliptic dispersion in place. Based on a normal law of distribution ф(r,a) and through integration with respect to corresponding variables (r,a) individual distributions of probabilities ф(r) and ф(a) as well as conditional distributions of probabilities ф(r/a)and ф(a/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.

Tags: dispersion; Fisher criterion; individual and conditional distribution; probability density; wind speed; дисперсія; критерій Фішера; частковий та умовний розподіли; швидкість вітру; щільність ймовірностей

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