Page 279 - Wind Energy Handbook

BLADE LOADS DURING OPERATION 253

section will be restricted to the consideration of extreme loads in the absence of
dynamic effects.
As described in Section 5.4, it is customary for wind turbine design codes to
specify extreme operating load cases in terms of deterministic gusts. The extreme
blade loadings are then evaluated at intervals over the duration of the gust, using
blade element and momentum theory as described in Section 5.7.2.
Although the discrete gust models prescribed by the codes have the advantage of
clarity of definition, they are essentially arbitrary in nature. The alternative
approach of adopting a stochastic representation of the wind provides a much more
realistic description of the wind itself, but is dependant on assumptions of linearity
as far as the calculation of loads is concerned.
Normally the loading under investigation, for example the blade root bending
moment, will contain both periodic and random components. Although it is
straightforward to predict the extreme values of each component independently,
the prediction of the extreme value of the combined signal is quite involved.
Madsen et al. (1984) have proposed the following simple, approximate approach,
and have demonstrated that it is reasonably accurate.
The periodic component, z(t), is considered as an equivalent 3 level square wave,
in which the variable takes the maximum, mean (ì z ), and minimum values of the
original waveform, for proportions å 1 , å 2 and å 3 of the wave period respectively. It
is easy to show that:
ó 2 ó 2
å 1 ¼ z å 3 ¼ z (5:58)
(z max ì z )(z max z min ) (ì z z min )(z max z min )
Extreme values of the combined signal are only assumed to occur during the
proportion of the time, å 1 , for which the square wave representation of the periodic
component is at the maximum value, z max .
Davenport (1964) gives the following formula for the extreme value of a random
variable over a time interval T:
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ª
x max
¼ 2ln(íT) þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5:59)
ó x 2ln(íT)
where í is the zero up-crossing frequency (i.e., the number of times per second the
variable changes from negative to positive) given by Equation (A5.46) and
ª ¼ 0:5772 (Euler’s constant). Thus the extreme value of the combined periodic and
random components is taken to be
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ª
z max þ x max ¼ z max þ ó x 2ln (íå 1 T) þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ z max þ g 1 :ó x (5:60)
2ln (íå 1 T)
where g 1 is termed the peak factor.
The variation of x max =ó x with exposure time, T, is shown Table 5.3 for a zero up-
crossing frequency of 1 Hz. The periodic component is assumed to be a simple
sinusoid, giving å 1 ¼ 0:25.
The method for determining the extreme load described above has to be applied
with caution when the wind fluctuations exceed the rated wind speed. In the case