Publishers version - DTU Orbit

Figure 138: Plots of the magnitude frequency response of the closed-loop transfer functions
from the u, v, and w components of wind speed (m/s) to (from upper to lower plots)
generator speed error (rpm) and the collective, vertical, and horizontal components of blade
root bending moment (kN·m) for the NREL 5-MW turbine, with all 16 degrees of freedom,
linearized at 13 m/s.
domain model of the wind field. The wind field used here is based on the IEC von Karman
isotropicmodelforneutral atmosphericstability(Jonkman, 2009).An above-rated mean wind
speed U = 13 m/s is used, along with a turbulence intensity of 10%. Wind speeds are spatially
correlated in the transverse y and vertical z directions using the IEC coherence formula
(Jonkman 2009; IEC 2005). Whereas the IEC standard only defines spatial coherence for the
u component of wind speed, we apply the coherence formula to the spatial correlation of the
v and w components as well.
Lidar measurements and blade effective wind speeds are calculated as weighted sums of
wind speeds along the lidar beam and blade span. Conveniently, the spectra of weighted
sums of wind speeds are easy to calculate. For example, the CPSD between signals a(t) =
M
N
αmam(t) (A(f) in the frequency domain) and b(t) = βnbn(t) (B(f) in the fre-
m=0
quency domain), which are linear combinations of other signals, can be calculated as
n=0
Sab(f) = A(f)B ∗ (f)
M
N
= αmAm(f) β ∗ nB ∗
n(f)
=
m=0
M
N
m=0n=0
n=0
αmβ ∗ n Sambn (f),
(233)
as long as the CPSDs Sambn (f) are known for each m and n. The symbol {} ∗ represents the
complexconjugate operator. The wind field model used here defines the CPSDs between wind
speeds at any two locations, allowing the spectra of lidar measurements and blade effective
wind speeds to be calculated.
202 DTU Wind Energy-E-Report-0029(EN)