Digital Signal Processing Reference
In-Depth Information
the hypotheses advanced could represent normal and faulty/abnormal operating
conditions. When distinguishing between two hypotheses is required, one can use
the Sequential Probability Ratio Test (SPRT) [
12
]. The approach can also be
adapted for multiple hypotheses testing; then so-called Multi-hypotheses
Sequential Probability Ratio Test (MSPRT) can be used [
1
]. The SPRT methods
use conditional PDFs of the decision vector (for given classes of events) to gen-
erate the decision.
The algorithm of SPRT to be used in power system relaying was first proposed
by Sakaguchi in 1980. His work [
12
] may be treated as first attempt to solve a
decision-making problem with probabilistic method. The author applied the
method to decide whether a phase is faulted or not, by using differential instan-
taneous signal as criterion value (d'Alembert relay). It must be said that the
criterion signal was chosen arbitrary and the required PDFs were estimated very
roughly (with 32 fault cases only). In testing phase their further approximation
with normal distribution was used. The paper showed only possibilities of the
SPRT application in digital relaying without any optimization suggestions.
SPRT belongs to decision methods in which number of samples of the decision
vector necessary to issue the decision is not pre-defined. The algorithm of the
SPRT can be written in the form:
H
k
¼
Y
k
f
l
ð
X
l
j
H
1
Þ
f
l
ð
X
l
j
H
0
Þ
;
ð
10
:
4
Þ
l
¼
1
8
<
9
=
8
<
9
=
H
k
A
A [ H
k
[ B
H
k
B
stop & accept H
1
continue test
stop & accept H
0
If
then
;
ð
10
:
5
Þ
:
;
:
;
A
¼
1
e
1
e
0
e
1
1
e
0
;
B
¼
ð
10
:
6
Þ
;
where:
f
l
ð
X
l
j
H
i
Þ
PDFs of the decision vector X at the instant l after test starting for
hypothesis H
i
;
H
k
test index at the instant k,
A, B
probabilistic thresholds,
e
0
;
e
1
assumed values of the first and second order error probabilities.
The above procedure minimizes values of the first and the second type error
probabilities e
0
;
e
1
[
3
]. The first type error comprises situations of recognizing
hypothesis H
1
instead of H
0
(underfunction), while the second type error corre-
sponds to the decision H
0
instead of H
1
(overfunction). Flow chart of the SPRT
algorithm is shown in Fig.
10.7
.
Application of the SPRT method is possible when the following conditions are
satisfied:
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