Wind Energy

112 G.C. Larsen and K.S. Hansen
19.2 Model
The basic idea behind the model is, in analogy with Cartwright – Longuet-
Higgens, to derive an asymptotic expression for the distribution of the largest
excursion from the mean level during an arbitrary recurrence period, however
based on a “mother” distribution (different from the Gaussian distribution)
that reflects the observed exponential-like behaviour for large wind speed excursions.
This is achieved on the expense of an acceptable distribution fit in
the data population regime of small to medium excursions which, however,
for an extreme investigation is considered unimportant. More specifically, we
postulate the following conjecture: the tails of a total population of velocity
fluctuations can be approximated by a Gamma distribution with shape
parameter 1/2.
We introduce a (stationary) stochastic wind speed process U(z,t) as
U (z,t) =U (z)+u (z,t) , (19.1)
where an upper bar denotes the mean value operator, u(z,t) are turbulence
excursions, z denotes the altitude above terrain, and t is the time co-ordinate.
The PDF of the extreme segment of the excursions, ue(z,t), is thus expressed
as:
fue (ue)
1
=
2 � �
2πC (z) σu |ue| exp
�
− |ue|
�
, (19.2)
2C (z) σu
where σu is the standard deviation of the total data population, and C(z) is
a dimensionless, but site- and height-dependant, positive constant.
We further introduce the monotone and memory-less transformation, g(∗),
defined by
�
σu
v = g (ue) =
C (z) sign (ue) � |ue| ; C (z) > 0 , (19.3)
with the inverse transformation
ue = g −1 C (z)
(v) = sign (v) v 2 ; C (z) > 0 . (19.4)
σu
Formulated in terms of the Gamma PDF, fue , the PDF of the transformed
variable, fv, is expressed as
�
C (z) � � �
fv (v) =2 |g−1 −1
(v)|fue g (v) =
σu
which is recognized as a Gaussian distribution.
σu
�
1
√ exp −
2π v2
2σ2 �
u
, (19.5)