Statistical Tools - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 161

where s is the time lag between the correlated time series. See Fig. A.6 for an
example. The autocorrelation function R(s) is via a Fourier transformation related
to the spectral density of the time series (Morales et al. 2010 ) and the power
spectrum S(f).

Z

1

S ð f Þ¼ 1
2p

R ð s Þ e if s ds ð A : 15 Þ

1

More general two-point statistics can be made by analysing the distributions of
wind speed increments du:

du ð t ; s Þ¼ u ð t þ s Þ u ð t Þ ð A : 16 Þ

The moments of these increments are the structure functions Sf:

Sf n ð s Þ¼ du ð t ; s Þ n ð A : 17 Þ

Increment probability density functions of wind speed time series are always
non-Gaussian (Morales et al. 2010 ).

A.2 Mean Wind Speed Spectrum and the Weibull Distribution

The wind speed spectrum shows a minimum in the range of about 1 h or
*0.0003 Hz (van der Hoven 1957 ; Gomes and Vickery 1977 ; Wieringa 1989 ).
Higher frequencies are usually termed as turbulence. In wind energy this high-
frequency turbulence is usually characterized by one variable, the turbulence
intensity [see Eq. ( A.6 ) above]. It will be neglected when now looking at
frequency distributions for 10 min me an wind speeds, i.e., we will now
concentrate on time series of the values u ð t Þ which appear as the first term on
the right-hand side of the decomposition ( A.1 ). These 10 min mean wind speeds
show temporal variations as well. The power spectrum of these low-frequency
variations show secondary maxima around 1 day (this is the diurnal variation of
the wind), 5-7 days (this is the variation due to the moving weather systems such
as cyclones and anticyclones), and around 1 year (the annual variation). The
diurnal variation exhibits a phase change with height (Wieringa 1989 ). The
reversal height is roughly at 80 m above ground but values for the reversal height
between 40 and 177 m are cited in Wieringa ( 1989 ). The phenomenon of the
reversal height is closely related to the occurrence of the nocturnal low-level jet in
Sect. 3.4 . The other long-term variations do not show this phase change.
The frequency distribution of the wind speed for the low-frequency end of the
spectrum (i.e., frequencies less than 0.01-0.001 Hz) is usually described by the
Weibull distribution. This distribution, which is named after the Swedish engineer,
scientist, and mathematician Ernst Hjalmar Waloddi Weibull (1887-1979), is
governed

by two parameters: a scale factor A (given in m/s, principally