Environmental Engineering Reference
In-Depth Information
as a candidate residual vector. It is indeed easily seen from Eq.
10.23
that the
probability law for r
ð
k
Þ
is
L
ð
r
ð
k
ÞÞ ¼
N
ð
0
;
N
C
RN
C
Þ
when f
ð
k
Þ¼
0
L
ð
r
ð
k
ÞÞ ¼
N
ð
N
C
f
ð
k
Þ;
N
C
RN
C
Þ
when f
ð
k
Þ 6¼
0
r
ð
k
Þ
can thus be used as a fault indicator.
A hypothesis on the fault type is needed to be able to apply the decision system
described in Sect.
10.3
in order to process the residual vector defined in Eq.
10.24
.
Bias-like faults will be considered from now on. In this case, the fault vector f
ð
k
Þ
can be written f
ð
k
Þ¼
b
j
e
j
1
f
k
k
0
g
where e
j
¼
0 ... 010...
½
T
is
the jth standard basis vector, b
j
is the fault magnitude and 1
f
k
k
0
g
is the indicator
function of event
f
k
k
0
g
. Notice that both positive and negative bias should be
detected and isolated; hence, the ± symbol in the expression of f
ð
k
Þ
. In these
conditions, the change in the mean due to the fault can be detected and isolated
using the algorithm presented in Sect.
10.3.2
in which the number of faults is
n
f
¼
2p. The required log-likelihood ratio for each residual sample takes the form
s
'
0
ð
k
Þ¼
e
j
N
C
b
j
ð
N
C
RN
C
Þ
1
ð
r
ð
k
Þ
1
2
N
C
b
j
e
j
Þ
' ¼
1...
;
p
s
'
0
ð
k
Þ¼
e
j
N
C
b
j
ð
N
C
RN
C
Þ
1
ð
r
ð
k
Þþ
1
2
N
C
b
j
e
j
Þ
' ¼
p
þ
1...
;
2p
By insertion of these expressions into Eq.
10.11
, Eqs.
10.11
-
10.13
yield the
required decision system.
10.5 Fault Detection and Isolation Based on Analytical
Redundancy
In this section, the relationships between different physical quantities is exploited
through a mathematical model in order to achieve detection and isolation of
incipient faults on the measurements of these quantities. This is the principle of
analytical redundancy. The approach is illustrated for the monitoring of stator
current and voltage measurements in a wind-driven DFIG. A specific property of
these signals is exploited, namely the fact that they consist of balanced three-phase
signals. This feature is very helpful for detecting and locating small additive faults.
Notice that the monitoring of the rotor current sensors has to be performed with a
more involved approach. Indeed, due to closed-loop control, a sensor fault on a
rotor current is attenuated and propagates to the other measurement channels,
while this situation is not so pronounced with stator current and voltage mea-
surements. Therefore, the monitoring of stator voltages and currents sensors can be
performed without resorting to a model of the DFIG. A simple signal model is
