Environmental Engineering Reference
In-Depth Information
Fig. 4.1 Power coefficient
C
P
for a 5 MW variable-
speed variable-pitch wind
turbine [
6
]
0.5
0.4
0.3
0.2
0.1
0
−10
0
15
10
10
20
5
30
0
λ
β
Fig. 4.2 Two-mass model
describing the first drive-train
mode
T
sh
T
g
T
r
K
s
Ω
r
B
s
Ω
g
J
g
J
t
the purpose of this work that the torque reference of the power converter coincides
with the electrical torque T
g
imposed to the wind rotor. That is, it can be assumed
that T
g
is a control input.
The pitch actuator is a highly nonlinear mechanic and hydraulic system [
5
]. For
control-oriented purposes, it is usually modeled as a first-order low-pass filter with
saturation in the amplitude b and rate of change b. The pitch system is showed in
Fig.
4.3
. In the linear zone, the pitch actuator dynamics can be described by
b
¼
1
s
b
þ
1
s
b
r
;
ð
4
:
4
Þ
where s is the time constant and b
r
the pitch angle command.
The drive-train dynamics (Eq.
4.3
) is highly nonlinear. This nonlinearity comes
mainly from the aerodynamic torque (Eq.
4.2
). For optimal control design, a linear
representation of the system dynamics is necessary. With this aim, the aerody-
namic torque is linearized around the operating point:
T
r
ð
V
;
b
;
X
r
Þ¼
B
r
ð
V
;
b
;
X
r
Þ
X
r
þ
k
V
ð
V
;
b
;
X
r
Þ
V
þ
k
b
ð
V
;
b
;
X
r
Þ
b
;
ð
4
:
5
Þ