Statistical Tools - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 164

As the relation between the mean wind speed ( A.20 ) and the wind energy
( A.23 ) is non-linear, different combinations of A and k can lead to the same mean
wind energy (see Fig. A.4 upper right for an example). Likewise, for a given mean
wind speed, win d energy from ( A.23 ) increases with an increasing variation of the
wind speed r 3 =½ u (see Fig. A.4 bottom). Thus, for a correct estimation of the wind
energy, the parameters A and k have to be known, not just the mean wind speed.
For a practical determination of the two Weibull parameters A and k from a
time series of wind speed values, we take the double logarithm of the relation
( A.18 ) following Justus et al. ( 1976 ):

¼ ln

k k

1 1 exp u
A

exp u
A

y ¼ ln ln ln

¼ k ln A k ln u ¼ a þ b ln u ð A : 24 Þ

From ( A.24 ) A and k can be determined by fitting a straight line into a plot of
y against ln u. We get the scale factor A from the intersection a of the fitted line
with the y-axis:

a
k

A ¼ exp ð A : 25 Þ

and the form factor k from the negative slope b of this line:

k ¼ b ð A : 26 Þ

Inversion of ( A.20 ) and an exponential fit to ( A.21 ) gives alternative ly (Justus
et al. 1978 ) a useful relation between A and k and the mean wind speed ½ u and the
standard deviation r 3 :

½ u
C ð 1 þ k Þ

A ¼ ð A : 27 Þ

and

1 : 086

r 3
½ u

k ¼

ð A : 28 Þ

Relation ( A.28 ) is plotted in the lower frame of Fig. A.4 for a constant value of
½ u . Sensitivity calculations show that the wind energy estimate from ( A.23 )is
much more sensitive to the correct value of A than to the value of k. An uncertainty
in A of 10 % leads to a deviation of 30 % in the estimated wind energy. An
uncertainty in k of 10 % on the other hand only leads to a deviation of 9 % in the
estimated wind energy. An overestimation of k yields an underestimation of the
wind energy and vice v er sa.
Please note that r 3 =½ u in ( A.28 ) is different from the turbulence intensity I u
defined in ( A.6 ). Usually r 3 =½ u is considerably larger than I u , because it represents
the much larger diurnal, synoptic and seasonal fluctuations of the 10 min-mean
wind speeds, while I u describes the smaller short-term fluctuations during a 10 min