Wind Distribution Functions

From Wind wiki

If we were to plot a graph of the frequency of wind speeds that occur at various wind speeds, we would find that this graph will take the form of a probability distribution function (which essentially assigns a probability to a specified wind speed).

The Weibull Distribution

The Weibull distribution is the most commonly used statistical distribution for describing wind speed data. It is described by the equation:

F(v)=e^(-(v/λ)^k )

,where F(v) is the fraction of time for which the hourly mean wind speed exceeds v. It has two adjustable parameters (λ and k) that enable it to fit a wide range of probability density functions. λ is a scale parameter related to the mean wind speed while k controls the shape of the Weibull distribution and describes the variability about the mean. λ is related to the annual mean wind speed x ̅ by the relationship

v=cΓ(1+1⁄k)

where Γ is the complete gamma function. This can be obtained by consideration of the probability density function

f(v)={ (〖k/λ (v/λ)〗^(k-1) e^(-(v⁄λ)^k ),&x≥0@0,&x<0)┤

because the mean wind speed is given by

v ̅=∫_0^∞▒vf(v)dv

A special Weibull Distribution with k=2 is called the Rayleigh Distribution. This is the typical value for wind distributions at many sites. In this case, the factor Γ(1+1⁄k) has the value( √π)⁄2=0.8862. The larger the value of k, the less variability around the annual mean wind speed the hourly wind speed will be. The smaller the k value, the larger the variability will be.