Environmental Engineering Reference
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2
4
3
5 ¼
2
4
3
5
2
4
3
5 þ
2
4
3
5
x r
x g
h D
a 11
a 12
a 13
x r
x g
h D
b 11 0
0 b 22
00
T a
T g
a 21
a 22
a 23
ð 7 : 4 Þ
a 31
a 32
a 33
where:
a 11 ¼ B dt þ B r
ð
Þ
a 12 ¼ B dt
n g J r
a 21 ¼ B dt
n g J g
a 13 ¼ K dt
J r
J r
a 22 ¼ B dt þ n g B g
a 23 ¼ K dt
n g J g
a 32 ¼ 1
n g
a 31 ¼ 1
n g J g
b 11 ¼ 1
J r
b 22 ¼ 1
J g
a 33 ¼ 0
where J r is the rotor inertia, B r is the rotor external damping, J g is the generator
inertia, x g and T g are the generator speed and torque, B g is the generator external
damping, n g is the gearbox ratio, K dt is the torsion stiffness, B dt is the torsion
damping coefficient, and h D is the torsion angle.
The hydraulic pitch system is modelled as a second-order transfer function
between the pitch angle b and its reference b r as follows:
x n
s 2 þ 2fx n s þ x n
b ¼
b r
ð 7 : 5 Þ
where f is the damping factor and x n is the natural frequency. A transfer function
is associated with each of the three pitch systems.
Finally, the generator subsystem is given by the following linear relation:
T g ¼ 1
s g T g þ 1
s g T gr
ð 7 : 6 Þ
Where T gr is the generator torque reference signal and s g is the time constant.
For controller design purposes, the state space model of wind turbine is
required. The non-linear model of a wind turbine is established by combining the
individual systems given above. However, it is clear that the main source of non-
linearity is the aerodynamic subsystem which is usually linearised in order to
predict its effects on all model states. Hence, the state space model of wind turbine
is given as:
x ¼ Ax þ Bu þ Ev
y ¼ Cx
ð 7 : 7 Þ
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