Environmental Engineering Reference
In-Depth Information
BK
j
P
1
E
ð
p
Þ
P
11
¼
X
11
; P
12
¼
P
1
0
P
1
R
0
P
22
¼
lower right block
to do variable change, the following Lemma is required:
Lemma 1 (Congruence) Consider two matrices P and Q, if P is positive definite
and if Q is a full column rank matrix, then the matrix
Q
P
Q
T
is positive
definite.
2
4
3
5
X
1
000
0 I 00
00I 0
000I
;
P
1
1
0
Let Q
¼
and X
¼
0
X
Then Q
P
ij
Q
T
\0 is also true and can be written as:
\0
P
1
P
11
P
1
P
1
P
12
X
1
ð
7
:
36
Þ
XP
22
X
Inequality [
36
] implies that P
22
\0 so that the following inequality holds true
[
18
,
24
]:
T
P
22
X
þ
lP
1
0
,
XP
22
X
2lX
l
2
P
1
X
þ
lP
1
22
ð
7
:
37
Þ
22
22
where l is a scalar.
By substituting [
37
] into [
36
] and using the Schur complement Theorem, then
[
36
] holds if the following inequality holds:
2
4
3
5
\0
P
1
1
P
11
P
1
P
1
1
P
12
X
0
1
XP
12
P
1
ð
7
:
38
Þ
2lX
lI
1
0
lI
P
22
After substitution for P
11
;
P
12
;
P
12
;
P
22
from [
35
] and by simple manipulation,
the LMI in [
29
] is obtained. This completes the proof
7.6.1 Simulation Results
The rotor and generator sensor faults are represented by two-scale errors. The scale
factors of 1.1 and 0.9 are multiplied by the simulated real generator and rotor
rotational speeds. The expected fault effects represent a deviation of the wind
turbine from the optimal operation. Figure
7.2
shows how the wind turbine
operation is affected by the two fault scenarios and helps to illustrate the success of
