Statistical Tools - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 162

Fig. A.3 Weibull probability density distribution ( A.19 ) for A = 10 and k = 2.5 as function of
wind speed u

proportional to the mean wind speed of the whole time series) and a form factor k
(also called shape parameter, dimensionless, describing the shape of the
distribution). The probability F(u) of the occurrence of a wind speed smaller or
equal to a given speed u is expressed in terms of the Weibull distribution by:

k

F ð u Þ¼ 1 exp u
A

ð A : 18 Þ

The respective probability density function f(u) (see Fig. A.3 ) is found by
taking the derivative of F(u) with respect to u:

¼ k exp u

A

k 1 k k

u k 1
A k

f ð u Þ¼ dF ð u Þ
du

¼ k
A

u
A

exp u
A

ð A : 19 Þ

The mean of the Weibull distribution (the first central moment) and thus the
m ean wind speed of the whole time series described by the Weibull distribution,
½ u is given by:

½ u ¼ AC ð 1 þ 1

k Þ

ð A : 20 Þ

where the square brackets denote the long-term average of the 10 min-mean wind
speeds and C is the Gamma function. The variance (the second central moment)
of this distribution and thus the variance of the 10 min-mean horizontal wind
speeds is:

h i ¼ A 2

C 1 þ 2
k

1 þ 1
k

Þ 2

r 3 ¼ u ½ u C 2

ð ð A : 21 Þ

r 2 is equal to the second term on the right-hand side of ( A.5 ), u 000 2 if the angle
brackets defined for that equation denote an average over a day or much longer,
and thus become identical with the square brackets. For k = 1, the Weibull
distribution is equal to an exponential distribution. For k = 2, it is equal to the
Rayleigh distribution and for about k = 3.4 it is very similar to the Gaussian