Wind Energy

16 W. Bierbooms and D. Veldkamp
u(t) =
K�
ak cos ωkt + bk sin ωkt (3.1)
k=1
The vector u(t) represents the longitudinal velocity fluctuations in N points
in the rotor plane (K frequencies). The Fourier coefficients ak and bk (row
vectors) are independent for different frequencies and for the same frequency
the following relation holds:
E[ak a T k ]=E[bk b T k ]= 1
T Sk
(3.2)
with T the total time of the sample and Sk the matrix of the cross-power
spectral densities of the wind speed fluctuations in the different space points.
They are found from the auto power spectral densities and the (root) coherence
function γ.
In order to generate a spatial wind field we have to factorize the Sk matrix
for each frequency component. In case the wind is measured at some location,
it is straightforward to determine the set of Fourier coefficients for that space
point. We can now improve the simulation of the wind speeds at the other
space points on basis of the knowledge of the wind speed at that point. The
accompanying equations can be found in the appendix of [1]. The wind fields
which are simulated on basis of measured wind fields are called constrained
stochastic simulations (they are generated under the constraint that at one or
more points the measured wind speeds have to be reproduced). The generated
wind fields may be considered to be wind fields which could have occurred
during the measuring period; the actual wind field can never be reproduced
in case of just a few measurement locations, since turbulence is a stochastic
process.
3.3 Stochastic Wind Fields which Encompass Measured
Wind Speed Series
With the method given in Sect. 3.2 several sets of wind time series are
generated. As turbulence model the Kaimal spectrum and coherence function
of IEC 61400-1 are chosen. Wind fields are created with mean wind speeds
below, around and above rated wind speed of the reference turbines. Turbulence
intensity is taken according to the IEC standard. For each situation at
least 20 wind fields have been generated of 10 min length and a time step of
0.04 s. Nowadays it is common to generate all three velocity components; since
the measurements (see Sect. 3.5) involved the longitudinal component only,
the constrained simulations are restricted to that component.
The constrained simulations have been performed with two in-house packages.
One uses a polar grid (30 azimuth by 5 radial, plus a point at the rotor