Publishers version - DTU Orbit

T
C
FB
F
FF
t
Hpre
gen
wm
Lidar
worig
Evo.
Figure 134: Block diagram of the feedforward control scenario. P is the linearized turbine
modelandC isthefeedbackcontroller,controllingbladepitchandgeneratortorquedeviations
from the operating point using a measurement of generatorspeed error yFB. T represents the
turbine with feedback control, as enclosed in the dashed box. y is the output error variable of
interest, which is intended to be regulated to 0. F represents the ideal feedforward controller
(assuming perfect measurements) and Hpre is the prefilter used to estimate a preview of the
wind disturbance wt based on the lidar measurement wm. worig is the original upstream wind
speed which becomes wm after lidar measurement and wt after experiencing wind evolution
and arriving at the turbine.
where Tywt is the transfer function from the wind disturbance wt to y. With the introduction
of feedforward control without a prefilter (Hpre = 1), y becomes a function of both wt and
wm:
y = Tywtwt +TyβFFFwm
(215)
where TyβFF is the transfer function from the feedforward blade pitch command βFF to
y. When measurements are perfect (wm = wt) and there is perfect turbine modeling, the
feedforward controller that completely cancels the effect of the disturbance wt on the turbine
is given by
F = −T −1
yβFF Tywt, (216)
which yields the output y = 0 in Eq. (215). In reality, non-minimum phase (unstable) zeros
in TyβFF can cause the ideal F to be unstable (Dunne et al., 2011b). In this case, a stable
approximation to the ideal F is used (Dunne et al., 2011b). For the analysis in this chapter,
however,itissimplyassumedthatF = −T −1
yβFFTywt. Whenthemeasurementsarenotperfect
(wm = wt), the output is
y = Tywt (wt −wm). (217)
Due to the distortingeffects ofthe lidarand wind evolutionalongwith the spatial averaging
of the wind caused by the area of the turbine rotor, in general wm = wt. In this case, the
variance, or mean square magnitude, of y in the frequency domain is given by
|Y(f)| 2 = |Tywt(f)| 2 |Wt(f)−Wm(f)| 2
P
wt
yFB
= |Tywt(f)| 2 (Swtwt (f)+Swmwm (f)−2ℜ{Swtwm (f)})
y
(218)
whereSwtwt (f)representsthepowerspectraldensity(PSD)ofthewinddisturbance,Swmwm (f)
is the PSD of the lidar measurement, Swtwm (f) is the cross-power spectral density (CPSD)
between the wind disturbance and the lidar measurement, and ℜ{} indicates the real part of
a complex value. The variance of y in the time domain is calculated by integrating Eq. (218)
over all frequencies:
Var(y) =
∞
−∞
|Tywt(f)| 2 (Swtwt (f)+Swmwm (f)−2ℜ{Swtwm (f)})df. (219)
196 DTU Wind Energy-E-Report-0029(EN)