Environmental Engineering Reference
In-Depth Information
2
4
0000010000
0000001000
0000000100
0000000010
0000000001
3
5
C
¼
ð
14
:
19
Þ
where:
h
r
h
g
h
l
T
State variables : x
¼½
y
t
c
ð
14
:
20
Þ
ch
r
h
g
h
l
y
t
T
g
T
Inputs : u
¼½
F
T
T
r
ð
14
:
21
Þ
h
r
h
g
h
l
T
Outputs : y
¼½
y
t
ð
14
:
22
Þ
c
According to the rotor aerodynamics and the characteristics of C
p
and C
T
(see
[
2
] ), the inputs of Eqs. (
14.16
) and (
14.21
), F
T
and T
r
, depend on v
1
, b and X
r
in a
nonlinear way. If the aerodynamic part of these equations is linearized around a
working point (v
10
, b
0
, X
r0
), and the bias components are ignored, the inputs F
T
and T
r
can be described by a transfer matrix whose elements are just gains, so that,
2
3
¼
X
r
ðÞ
v
1
ðÞ
b
ðÞ
F
T
ðÞ
T
r
ðÞ
K
FX
K
FV
K
Fb
4
5
ð
14
:
23
Þ
K
TX
K
TV
K
Tb
where the gains are calculated by using the C
T
and C
p
curves. Now, the transfer
matrix description G(s) of the wind turbine is calculated by using the transfor-
mation G
ð
s
Þ¼
C sI
A
Þ
1
B for y
ð
s
Þ¼
G
ð
s
Þ
u
ð
s
Þ
.
ð
2
3
y
t
ðÞ
c
ðÞ
X
r
ðÞ
X
g
ðÞ
X
l
ðÞ
4
5
þ
D
ð
s
Þ
v
1
ðÞ
b
d
ðÞ
T
gd
ðÞ
¼
P
ð
s
Þ
ð
14
:
24
Þ
where the plant matrix and the disturbance matrix are,
2
4
3
5
ð
Þ
þ
K
Fb
l
11
ðÞ
l
32
ðÞ
K
FX
K
Tb
K
Fb
K
TX
l
11
ðÞ
l
33
ðÞ
K
FX
A
b
ð
s
Þ
1
l
32
ðÞ
K
TX
A
T
ð
s
Þ
1
l
32
ðÞ
K
TX
ð
Þ
þ
K
Fb
l
21
ðÞ
l
32
ðÞ
K
FX
K
Tb
K
Fb
K
TX
l
21
ðÞ
l
33
ðÞ
K
FX
A
b
ð
s
Þ
1
l
32
ðÞ
K
TX
A
T
ð
s
Þ
1
l
32
ðÞ
K
TX
P
ð
s
Þ¼
K
Tb
1
l
32
ðÞ
K
TX
A
b
ð
s
Þ
l
32
ðÞ
l
33
ðÞ
1
l
32
ðÞ
K
TX
A
T
ð
s
Þ
1
K
Tb
1
l
32
ðÞ
K
TX
A
b
ð
s
Þ
l
42
ðÞ
l
33
ðÞ
K
TX
þ
l
43
ðÞ
l
43
ðÞ
l
32
ðÞ
K
TX
1
l
32
ðÞ
K
TX
l
42
ðÞ
A
T
ð
s
Þ
K
Tb
1
l
32
ðÞ
K
TX
A
b
ð
s
Þ
l
52
ðÞ
l
33
ðÞ
K
TX
þ
l
53
ðÞ
l
53
ðÞ
l
32
ðÞ
K
TX
1
l
32
ðÞ
K
TX
l
52
ðÞ
A
T
ð
s
Þ
ð
14
:
25
Þ