Environmental Engineering Reference
In-Depth Information
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