Wind Energy

27 Phenomenological Response Theory to Predict Power Output 157
L(T ) − L(0)
=< LPC(U(t)) > − , (27.6)
Tr0
which implies that the left hand side of the equation tends to zero in the limit
of large T .
In reality, a constant relaxation is not observed, see e.g., [3]. In order to
implement a frequency-dependent response to wind fluctuations, we made the
following linear response ansatz [1]:
r(t) =r0( Ū)+r1( ˙ U); r1( ˙ U(t)) =
� t
−∞
dt ′ g(t − t ′ ) ˙ U(t ′ ). (27.7)
Here, r0 describes relaxation at constant wind speed with ˙ U(t) = 0, compare
Fig. 27.1a. The dynamic part r1(t) of the relaxation function takes into
account the delayed response to wind speed fluctuations ˙ U(t). The function
g(t), which simulates the response properties of the turbine, principally may
include the control strategies of the turbine at various mean (10-min) wind
speeds Ū; thus one will generally set g(t) =gŪ (t), see also [3]. In view of the
convolution integral in (27.8), one has factorization in frequency space with
ˆr(f) =ˆg(f)[2πI Û(f)]; I = √ −1. (27.8)
If the response function g(t) is known together with the power curve, then the
average power output can be predicted within this model after an elementary
numerical integration of system (27.4) with a given wind field as input. Formally,
the dynamic effect can be defined as a correction term, Ddyn, tothe
usual estimate by means of the power curve:
=< LPC(U(t)) > (1 + Ddyn). (27.9)
For low turbulence intensities the dynamic correction factor Ddyn canbeobtained
analytically within our model [1]. It has the same structure as the
dynamic correction introduced in an ad hoc way in [3]. The comparison with
their results [3] allowed us, to deduce the response function g(t) in a unique
way, for details see [1]. The response function is shown in Fig. 27.4.
We remark that the ad hoc ansatz made in [3] is limited to small turbulence
intensities, whereas our response model can deal, in principle, with arbitrary
wind fields.
27.4 Discussion and Conclusion
We have started to derive the response function g(t) from the measurement
data [4] by solving (27.4) for r(t) and determining r(ti) from L(ti) and
LPC(U(ti)). This has to be done conditioned to different wind speed bins. It
turned out that the data base used so far is much too small. There is also the