Statistical Tools - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 165

interval after any longer trends and variations have been subtracted from the data
in this interval. Eq ua tion ( A.28 ) can be inverted in order to estimate the order of
magnitude of r 3 =½ u : This ratio is of the order of 1/k, i.e., 0.4-0.5 while the
turbulence intensity over land is in the order of 0.2 and the offshore turbulence
intensity is usually below 0.1.

A.3 Extreme Mean Wind Speeds and the Gumbel Distribution

Extreme mean wind speeds are important for load estimations for wind turbines.
Usually, they have to be specified for a certain return period which is related to the
time period for which the turbine is expected to operate. The probability of
occurrence of extreme values can be described by a Gumbel distribution (Gumbel
1958 ). This distribution is a special case of a generalized extreme value
distribution or Fisher-Tippett distribution as is the Weibull distribution (Cook
1982 ; Palutikof et al. 1999 ). It is named after the German mathematician Emil
Julius Gumbel (1891-1966).
The probability density function for the occurrence of a largest value x reads:

f ð x Þ¼ e x e e x ð A : 29 Þ

Due to its form this distribution is often call double exponential distribution.
The related cumulative frequency distribution reads:

F ð x Þ¼ e e x ð A : 30 Þ

The inverse of ( A.30 ) is the following percent point function:

G ð p Þ¼ ln ð ln ð p ÞÞ ð A : 31 Þ

The 98th percentile (p = 0.98) of this percent point function has the value 3.9,
the 99th percentile the value 4.6, and the 99.9th percentile the value 6.9.
The practical calculation from a given time series may be done as follows: In a
first step, independent maxima of a wind speed time series (e.g., annual extreme
values) are identified. Then, these maxima are sorted in ascending order forming a
new series of maxima with N elements. The cumulated probability p that a value of
this new series is smaller than the mth value of this series is p(m) = m/(N + 1).
Finally, the sorted values are plotted against the double negative logarithm of their
cumulative probability, i.e., they are plotted against -ln(-ln(p)). Data which follow
a Gumbel distribution organize along a straight line in such a graph. Once the graph
is plotted, estimations of extreme values for a given return period are easy. For
example, from a statistics of annual extreme values, u max the extreme value which is
expected to appear once in 50 years is found where the extrapolated straight line

u max ¼ a ð ln ð ln ð p ÞÞÞ þ b ð A : 32 Þ

crosses the value 3.9 (p = 1-1/50 = 0.98 and -ln(-ln(0.98)) = 3.9).