Page 57 - Wind Energy Handbook

EXTREME WIND SPEEDS 31

hourly means, since the high-wind tail of the distribution cannot be considered to
be reliably known. However, Fisher and Tippett (1928) demonstrated that for any
cumulative probability distribution function which converges towards unity at least
exponentially (as is usually the case for wind speed distributions, including the
Weibull distribution), the cumulative probability distribution function for extreme
^
U
values U will always tend towards an asymptotic limit
^
^
F(U) ¼ exp( exp( a(U U9)))
U
U
as the observation period increases. U9 is the most likely extreme value, or the mode
of the distribution, while 1=a represents the width or spread of the distribution and
is termed the dispersion.
This makes it possible to estimate the distribution of extreme values based on a
fairly limited set of measured peak values, for example a set of measurements of
^
U
the highest hourly mean wind speeds U recorded during each of N storms. The N
measured extremes are ranked in ascending order, and an estimate of the cumula-
tive probability distribution function is obtained as
^
U
m(U)
~ ^
U
F F(U) ffi
N þ 1
^
^
U
where m(U) is the rank, or position in the sequence, of the observation U. Then a
U
~ ^
^
F
plot of ln( ln(F(U))) against U is used to estimate the mode U9 and dispersion
U
U
1=a by fitting a straight line to the datapoints. This is the method due to Gumbel.
Lieblein (1974) has developed a numerical technique which gives a less biased
estimate of U9 and 1=a than a simple least squares fit to a Gumbel plot.
Having made an estimate of the cumulative probability distribution of extremes
^
U
F(U), the M year extreme hourly mean wind speed can be estimated as the value of
^
U U corresponding to the probability of exceedance F ¼ 1 1=M.
According to Cook (1985), a better estimate of the probability of extreme winds is
obtained by fitting a Gumbel distribution to extreme values of wind speed squared.
This is because the cumulative probability distribution function of wind speed
squared is closer to exponential than the distribution of wind speed itself, and it
converges much more rapidly to the Gumbel distribution. Therefore by using this
method to predict extreme values of wind speed squared, more reliable estimates
can be obtained from a given number of observations.
2.8.1 Extreme winds in standards
The design of wind turbines must allow them to withstand extremes of wind speed,
as well as responding well to the more ‘typical’ conditions described above. There-
fore the various standards also specify the extremes of wind speed which must be
designed for. This includes extreme mean wind speeds as well as various types of
severe gust.
Extreme conditions may be experienced with the machine operating, parked or
idling with or without various types of fault or grid loss, or during a particular