Wind Energy

76 François G. Schmitt
10 4
100
1
0.01
0.0001
E(f)
E(f)
slope −5/3
10
0.001 0.01 0.1
−6
1 10
f (Hz)
Fig. 13.2. The power spectrum of atmospheric turbulent data, in log–log plot, with
a straight dotted line of slope −5/3 for comparison
IV(t+3)−V(t)I
3.5
3
2.5
2
1.5
1
0.5
0
0
10 20 30 40 50 60 70 80
t (s)
Fig. 13.3. A portion of the 3−s increment time series; large fluctuations correspond
to 3 − s gusts. Gusts associated to a velocity > 1ms −1 are shown, together with the
construction of their recurrent times
process is Brownian motion: using dimensional arguments, Feller [10] showed
that the pdf of their first-passage times is of the form:
p(T ) � T −µ
(13.3)
with µ =3/2. For fractional Brownian motion characterized by an exponent
H = ζ(1), it has been shown that the form of (13.3) is still valid, with µ =
2 − H [11, 12]. For multifractal processes, the choice of a given threshold δ
selects a fractal black-and-white process: the “gust” state occurs on a set
whose support has a fractal dimension ds; the larger the threshold δ, the
smaller the dimension ds characterizing the occurrence of gust events. In such
case, the tail behavior of recurrence times has been shown to obey (13.3), with