Wind Energy

19 The Statistical Distribution of Turbulence Driven Velocity Extremes 113
As the introduced transformation, g(∗), is strictly monotone, every local
extreme in the “real” process, u(z,t), is transformed into a local extreme
in the “fictitious” transformed process, v(z,t). Thus, the number of local
extremes (and their position on the time-axis) is unaltered by the performed
transformation.
In the v-domain, the process is Gaussian, and the analysis performed by
Cartwright – Longuet-Higgens applies. The derived asymptotic probability
density function for the largest maxima, during a time span T , is thus given
in terms of excursions normalized with σu , ηm, as [1], [5]
fmax η (ηm) =|ηm| exp
�
1 −
−e 2 η2
m +ln(Tυ)�
1 −
e 2 η2
m +ln(Tυ) , (19.6)
where υ is the zero-up-crossing frequency multiplied by 2π.
The final step is to transform this asymptotic result from the v-domain
to the u-domain. Denoting ζ as the velocities in the u-domain normalized
by σu, we finally obtain the requested asymptotic distribution for the largest
maxima, ζm, as
fmax ζ (ζm) =
1
2C (z) exp
�
1 −
−e 2C(z) |ζm|+ln(Tυ)� 1 −
e 2C(z) |ζm|+ln(Tυ) . (19.7)
The asymptotic probability density function expressed in (19.7) is seen to
be of the EV1 type [8], which is consistent with the finding from several experimental
studies [5–7]. It can be shown that, contrary to the asymptotic
PDF described by Cartwright – Longuet-Higgens, the root mean square associated
with (19.7) is independent of the time span considered, and the present
asymptotic extreme value distribution thus does not narrow with increasing
time span.
What remains now is to determine the transformation constant, C(z),
such that the best possible agreement between the upper tails of measured
PDF’s and the proposed asymptotic fit is obtained. This calibration has been
performed based extensive full scale measuring campaigns, representing three
different terrain types – offshore/coastal, flat homogeneous terrain and hilly
scrub terrain.
A common feature of the C-values resulting from the performed data fit,
is a moderate, but significant, dependence with height, which may be interpret
as the effect of the gradually increasing size of the biggest turbulence
eddies contributing to extreme events, as the distance to the blocking ground
increases. The following approximation result for the transformation constant
has been obtained
C (z) =az + b, (19.8)
where the values of a and b for the three investigated terrain categories, are
given in Table 19.1.