The description of the wind by histograms and distribution functions is suitable for the yield calculation, but the information on the time history is lost.
Vibrations and the dynamic loading on blades, drive train and tower occur in the frequency range of 0.1 to approx. 30.0 Hz. The structure is very sensitive at frequencies close to the individual natural frequencies of the components and re- acts very “nervous”. The spectral representation of the wind is very suitable to in- vestigate this frequency range which is found to the right in Fig. 4-23. The Fast Fourier Transform is used to transfer the time domain data (the measured time se- ries) into the frequency domain to obtain the power spectral density S. It has the dimension (m/s)2/Hz = W/kg and describes the share of the individual frequency f (in Hz) to the variance v2, equation (4.8) (resp. to the standard deviation
v2
Vv )
f³
0 2 S(f)df
v , (4.14)
The spectral representation is also very suitable for the generation of “synthetic winds” for the digital simulation, cf. section 4.3.
Two spectral models of the wind are commonly used. The Kaimal spectrum model was determined empirically from wind measurements
Kaimal spectrum model: 5 3
v 1 v 1 v 2
) / 6 1 (
/ ) 4
( v /
v L f
v f L
S V . (4.15)
The given turbulence length scale parameters L1v and L2v (Fig. 4-13) vary slightly depending on the chosen guide lines.
The Kármán spectrum model is the second model and describes very well the tur- bulence in wind tunnels and pipes, but is also often applied to describe the wind. It allows quite easy to formulate the correlation between neighbouring points (e.g.
hub centre – middle of blade).
von-Kármán spectrum model: 2 5/6
v 2 v 2 v2
v (1 70,8( / ) )
/ ) 4
( f L v
v f L
S V (4.16)
In general the power spectral density, Fig.s 4-23 and 4-24, is displayed versus the frequency, but in Fig. 4-23 the time scale is added to relate the spectrum to Fig. 4-13.
Fig. 4-23 Power spectrum of wind speed based on a continuous measurement in flat and homo- geneous terrain [11]
The power spectrum in Fig. 4-23 is based on a one-year measurement with a sam- pling rate of 8 Hz. The fluctuations in the range of seconds and minutes, to the right in the spectrum, are caused by atmospheric turbulence. The diurnal cycle of the wind at the site creates the peak at the frequency (1/86400 s = 1.16*10-5 Hz) corresponding to the period of one day. The maxima in the spectrum in the range of several days reflect large-scale weather events, e.g. the passage of an Atlantic low-pressure system.
Fig. 4-24 shows schematically, for the case of flat, even terrain, the spectra for the three different states of atmospheric stability, cf. section 4.2.1. The area below the curve is proportional to the variance. Under neutrally stable conditions (L = infinite) the spectrum is dominated by a broad maximum. At higher frequen- cies f, it declines with the exponent f –5/3. Low frequencies are characterized by a high variation and a high uncertainty. As explained in section 4.2.2 the tempera- ture profile has a large influence on the vertical mass transfer and therefore on the turbulence, cf. Fig. 4-10. The Fig. 4-24 shows a strong increase of the turbulence
4.2 Atmospheric boundary layer 140
in an unstable atmosphere (Monin-Obukhov stability length L = -30 m), whereas stable stratification suppresses turbulence.
Up to now, only the longitudinal turbulence was considered since it dominates in flat terrain. But in reality, turbulence is a three-dimensional phenomenon and in flat terrain the ratio of the kinetic energies is approx. 1.0 : 0.8 : 0.5 for the longitu- dinal, lateral and vertical direction. In complex terrain the proportions change, and the lateral component becomes as strong as the longitudinal component (1.0 : 1.0 : 0.8), [19].
ffSV
Fig. 4-24 Model spectra of the longitudinal wind speed component 50 m above ground in flat terrain for neutrally stable (L = infinite), stable (L = 30 m) and unstable (L = - 30 m) conditions;
L stands for the Monin-Obukhov stability length [1]
Cross spectra and coherence functions
The turbulence spectra discussed above describe the temporal wind speed fluctua- tions of the turbulent components at a single point in the rotor swept area of the wind turbine. But since the turbine blades are moving through the turbulent wind field it is not enough to consider the spectra at one single point. The spatial change in lateral and vertical direction is also of importance because the blade is “collect- ing” all the spatial changes, cf. section 8.1.
In order to reflect these effects, the spectral models of the turbulence have to be expanded by the cross correlation of the turbulent fluctuations at two points of dif- ferent lateral and vertical positions. The correlation is obviously decreasing with increasing radial difference 'r between the two points. Moreover, the correlation is for the high-frequency changes smaller than for low-frequency changes. The coherence Coh('r, f) is measure for the relation between the turbulent fluctuations at the two individual points “1“ and “2“ in the rotor plane. The coherence is described depending on the frequency spectrum and the distance. It is defined as follows
) ( ) (
) ) (
, (
22 11
12
f S f S
f f S
r
Coh' (4.17)
where S12 (f) is the cross spectrum of the two points with the distance 'r, and S11 (f) und S22 (f) are the auto spectra of the individual points. More details on the spectral and coherence analysis is found in the literature [35, 37].