Page 276 - Wind Energy Handbook

250 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES
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n
S jk (n) @ 2 0:12 2 A
C jk (n) ¼ ¼ exp 8:8˜s jk þ (5:53)
S u (n) U L
(Note that coherence is sometimes termed coherency, and that some authors define
coherence as the square of the normalized cross spectrum.) See Section 2.6.6 for
details of the coherence corresponding to the Von Karman spectrum.
Three distinct approaches have been developed for generating simulation time
histories:

(1) the transformational method, based on filtering Gaussian white noise signals;

(2) the correlation method, in which the velocity of a small body of air at the end of
a time step is calculated as the sum of a velocity correlated with the velocity at
the start of the time step and a random, uncorrelated increment;

(3) the harmonic series method, involving the summation of a series of cosine
waves at different frequencies with amplitudes weighted in accordance with
the power spectrum.

This last method is probably now the one in widest use, and is described in more
detail below. The description is based on that given in Veers (1988).

Wind simulation by the harmonic series method

The spectral properties of the wind-speed fluctuations at N points can be described
by a spectral matrix, S, in which the diagonal terms are the double-sided single point
power spectral densities at each point, S kk (n), and the off-diagonal terms are the
cross-spectral densities, S jk (n), also double sided. This matrix is equated to the
T
product of a triangular transformation matrix, H, and its transpose, H :
2 3 2 32 3
... ...
S 11 S 21 S 31 H 11 H 11 H 21 H 31
6 ... 7 6 76 ... 7
6 S 21 S 22 S 32 7 ¼ 6 H 21 H 22 76 H 22 H 32 7
4 5 4 54 5
S 31 S 32 S 33 ... H 31 H 32 H 33 H 33 ...
... ... ... S NN ... ... ... H NN H NN
resulting in a set of N(N þ 1)=2 equations linking the elements of the S matrix to
the elements of the H matrix:

2
S 11 ¼ H 2 S 21 ¼ H 21 :H 11 S 22 ¼ H þ H 2 S 31 ¼ H 31 :H 11
11 21 22
2
2
S 32 ¼ H 31 :H 21 þ H 32 :H 22 S 33 ¼ H þ H þ H 2
31 32 33
(5:54)
X X
k
k
S jk ¼ H jl :H kl S kk ¼ H kl
l¼1 l¼1