Short
-
time Fourier and wavelet transforms for fault
detection in power transformers
during impulse
tests
L.Satish
Indexing terms: Fault detection, Short
-
time
Fourier
trunqorm,
Wuvelet
trunsform,
Impulse tests,
Power
trunsfimners,
Time-Jequency unulysi.s
Abstract:
A
novel approach using the short
-
time
Fourier transform and wavelet transform
(time-
frequency analysis tools) for fault detection
during impulse testing of power transformers is
described. The neutral
and/or capacitively
transferred currents which are recorded during an
impulse test can be directly analysed with this
approach. These currents are considered to be
evolving in time,
i.e. as nonstationary signals,
especially when there is a fault. Results from
simulation studies are presented wherein the fault
condition is modelled as a fast decaying transient
superposed on the neutral current.
A comparison
of the two transforms is made to assess their
ability to detect as small a fault as possible and
other implemenational issues. Advantages of
these methods over the conventional
transfer-
function method are demonstrated, and it appears
that the wavelet transform is better suited for this
I
task
1
Introduction
Impulse testing of power transformers is a routine
to
ascertain the integrity of insulation. This involves
application of a specific number of predefined levels of
impulse voltage and wave shapes. The resulting neutral
and/or capacitively transferred currents are recorded.
Standards stipulate comparison of these currents for
detecting any visible variation in their shapes and, if
none are present, the apparatus
is
adjudged as having
passed the test. It is quite clear that this procedure is
simple to adopt when the apparatus has a major fault
resulting in large changes in shape of the pertinent cur
-
rents, but is rather difficult when only a minor fault,
say a sparkover between adjacent
coilsiturns lasting for
a few microseconds occurs
[l].
In such cases, the judg
-
ment
is
based on the expertise of the inspector and
often becomes controversial. However, detection of
With
the availability
of
fast digital recorders and per
-
sonal computers these waveforms are now being
acquired and stored digitally, thus enabling their
indepth analysis than mere visual examination of oscil-
lographic traces. Initially, differences in waveforms
were amplified and compared. Subsequently, develop
-
ment of the transfer function
(TF)
approach for fault
diagnosis
[2-41
was a milestone. The computed transfer
functions at different voltage levels were compared and
any deviation amongst them was considered to be due
to a fault in the transformer. The main philosophy of
this approach is that it is independent (at least theoret
-
ically speaking) of the shape of the applied voltage,
chopping times, amplitude, bushing flashover and
impulse
-
generator component faults, if any, which
would not show up in the transfer function. But, con
-
trary to this belief, practical experience in transformer
testing has indicated some problems with regard to it
being independent of the input excitation
and chopping
times
[5].
The possible sources leading to errors and
ambiguity in the transfer function comparison can be
due to one or more of the following reasons:
(a) noise inherent in the acquired data, in spite
of
good
shielding
(h)
errors due to sampling, quantisation, A/D errors,
finite record length
(cj
different signal processing methods being adopted:
windowing, filtering etc.
It
is,
perhaps that due to these many unanswered ques
-
tions, this approach has not yet found its way into the
relevant standards
[6].
This paper presents simulation results of an entirely
new approach, based on the time
-
frequency analysis of
signals, for detecting faults from the neutral current
waveforms. Of course, detection of major faults has
never been an issue. Hence, the focus is on being able
to detect accurately the smallest or minor type of
faults, which are hard to detect using the
TF
approach.
2
The underlying principle
frequency varies with time) can be analysed with this
tcchnique.
The
primary
rcasoii
for
most
real
-
world
sig-
rials
being nonstationary is that the production of Par
-
ticular frequencies depends on the physical parameters
and conditions
of
the system, which may change in
time due to many reasons. It is important to be able to
detect these changes accurately. There are a number of
ways in which the input signal can be subjected to
time
-
frequency analysis and, among them, the short-
time Fourier transform and wavelet transform are pop
-
ular and, hence, have been considered here (Wigner
distributions,
Gabor transforms, bilinear transforms
etc. have also been used for this purpose).
Time
-
frequency analysis of nonstationary signals
indicates the time instants at which different frequency
components of
the signal come into reckoning. One
direct consequence of such
a
treatment will be the pos
-
sibility to accurately locate in time all abrupt changes
in the signal and estimate their frequency components
as well. It is in this very application that we are inter
-
ested in here, i.e. to locate low
-
magnitude abrupt
changes in the neutral current waveforms. In this con
-
text, we can visualise the neutral current as a nonsta-
tionary signal whose properties change or evolve in
time, when there is a fault. This is particularly
so
when
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frequency,
MHz
a
there is a momentary short
-
circuit between adjacent
turns due to high stresses. This short
-
circuit lasts for
a
short interval only, say
1-2p,
and can be considered
to
manifest itself as
a
fast decaying oscillation superim
-
posed on the neutral current. During such momentary
short circuits, the circuit conditions are definitely
altered, but due to the low magnitude and transient
nature, the influence of such a short
-
circuit on the
transfer function is almost insignificant, as will be
shown later in the simulations.
3
transform (STFT)
[71
Brief introduction to short
-
time Fourier
The primary objective of time
-
frequency analysis is to
be able to define a function that will describe the
energy density of
a
signal simultaneously in time and
frequency, and is commonly used in applications
to
speech, sonar and acoustic signals. Among the few
4
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I
0
1
2
3
4 0
frequency,
MHz
C
Fig.
1
a
Case
1,
no
fault
b
Case
1,
with
fault
c
Case
2,
no
fault
d
Case
2,
with
fault
ESfeect
of
superposing
faults
on
neutral
cuwents
on
the transJer
function
magnitude
I
2
3
4
frequency,
MHz
b
.
1
2
3
4
frequency.
MHz
d
tools, STFT happens to be very common and popular,
because the concept behind it is simple yet powerful.
The basic idea
of
STFT is to slice up the signal into
suitable overlapping time segments (using windowing
methods) and then to Fourier analyse each slice to
ascertain the frequencies contained in it. The accumula
-
tion
of
such spectra indicates how the spectrum is vary
-
ing in time and is called the spectrogram.
It
is assumed
that frequency information is associated with the time
index in the middle
of
each slice
of
windowed data.
STFT
of
a continuous
-
time signal
x(t)
is defined as:
STFT(f,
T
)
=
J’
z(t)w(t
-
T
)
exp(-j27~ft)dt
where
w(t)
is the window function whose position is
translated in time by
z.
There are some limitations
associated with STFT, the first being the window
length. It is obvious that a wide window yields a good
--oo<t<oo
(1)
50
100
150
200
250
time
(7
resolution in the frequency domain, but poor resolu
-
tion in the time domain, and vice versa.
So,
in practical
situations, a compromise between the two resolutions
has to be made. Secondly, raw STFT is
computation-
ally expensive, but ways
of
accelerating it by avoiding
redundant calculations are available in the literature.
These drawbacks notwithstanding, STFT is an ideal
tool in many respects, the most important being its
excellent spectrogram structure, which is consistent
with our intuition regarding frequency spectra, thereby
qualifying as a good visualisation tool.
4
[8,91
Signal processing using wavelet theory has emerged as
a powerful tool over the past ten years and has led to
significant developments in data analysis, data com
-
pression, image and speech processing, multiresolution
Brief introduction to wavelet transforms
(WT)
50
100
150
200
250
time
b
5
10
%
0
7
U
u
15
2
-
20
25
3c
50
100
150
200
250
time
C
Fig.2
,
STFTnzugnitude
undphuse
for
use
I
with window
length
=
64
u Magnitude,
no fault
h
Phase,
no
fault
c
Magnitudc,
with
fault
d
Phase.
with
fault
50
100
150
200 250
time
d
IEE
Pvoc.-Sci.
Meas.
Technol.,
Vol.
145, No.
2,
Mwdi
I998
analysis etc. The
WT
like the Fourier transform
decomposes
a
given signal into its frequency compo
-
nents, but differs in providing a nonuniform division of
the frequency domain. In addition, unlike the Fourier
transform which gives
a
global representation
of
the
signal, WT provides
a
local representation in both time
and frequency. This results from the fact that the
ana-
lysing basis functions in the case of the Fourier trans
-
form (namely sines and cosines) extend over infinite
time, whereas they are compactly supported functions
in the case of WT, thus giving them the localisation
property. This property greatly facilitates analysis of
nonstationary signals, transient detection etc.
A
mathe
-
matical definition of
WT
follows:
Let
x(t)
denote
a
continuous
-
time finite energy
signal, then
WT
of
x(t)
is defined as:
z(i)g(,,b)(t)dt
-
00
<
t
<
00
(2)
where
g(a,b)(t)
=
lal(-1’2)g((t
-
b)/a)
(3)
is called the base function
or
mother wavelet.
a,
b
(real,
a
#
0)
are the dilation and translation parameters,
respectively.
A
restriction
on
the choice of
g(t)
is
that it
must have
a
zero average value and be of short dura
-
tion, which, mathematically, is called the admissibility
condition
on
g(t).
Daubechies’ wavelet, Morlet wavelet,
Harr wavelet are some examples of popularly used
functions for
g(t)
[8,
91.
In general, STFT and WT may be interpreted
as
inner products
of
the signal
and
a
set
of
analysing
functions located throughout in the time
-
frequency
and time
-
scale planes, respectively. The function
WT(a,
b)
gives an idea
of
the contributions to the signal
around time
b,
at
a
scale
u
and hence leads to a time-
scale decomposition when compared to the time-fre-
50
100
150
200
250
50
100
150
ZOO
250
time time
CI
b
50
100
150
200
250 50
100
150
200
250
time time
C
d
Fig.3
n
Magnitude, no fault
0
Phase,
no
fault
c
Magnitude, with fault
d
Phase, with fault
STFT
mugnitude
undphusefor
case
2
with
window
length
=
64
80
IEE
Proc.-Sci.
Meas
Technol..
Val.
14S,
No.
2,
March
1998
quency decomposition obtained from STFT. The quan
-
tities frequency and scale are inversely related, i.e. a
small value of
a
implies
a
high frequency and vice
versa. Time
-
scale decomposition obtained from the WT
is referred to as a scalogram. Among the many wavelet
functions, the Morlet wavelet
g(t)
=
exp(-jwot)
exp(-t2/2)
(4)
has been shown to yield the best time
-
frequency locali-
sation [lo], and, hence, has been chosen here. Briefly,
the steps involved in computing scalograms are given
by the following pseudocode:
Input: sampling rate, minimum, maximum and step
-
size of scale
-
factor
a,
input signal
Do
Loop scale
-
factor
=
minimum, maximum, step
-
size:
for each scale
-
factor
compute length, lw,
of
Morlet wavelet required
compute Morlet wavelet sequence with
b
=
0
convolve this with the input signal
accumulate convolution result after discarding ini
-
tial and final lw/2 points
End
Loop
Plot magnitude and phase
of
accumulated results to
yield scalograms
(Note: For computing the Morlet wavelet, a value
of
-
6
to
+6
time units is used because its values are negli
-
gibly small for
Abs(t)
>
6
time units. Based on this and
the sampling rate,
Iw
is obtained, using which a linear
time
-
space is generated for computing the Morlet
wavelet sequence. Anonymous FTP sites for download
-
ing useful wavelet transform and related programs are:
playfair.stanford.edu in
/pub/wavelab, ftp.tsc.uvigo.es
in IpubNvi-Waveimatlabl)
5
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15
20
25
t ime,
11s
a
0
5
10
time,
~5
b
0.4
z1
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0.8
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5
10
15
20
25
time,,us
C
Fig.4
n
Magnitude,
no fault
b
Phase,
no
fault
c
Magnitude,
with
fault
d
Phase, with
fault
Sculogrum
magnitude
und
phuse
jor
ruse
I
0.
h
C
U
%
0.
2
+-
0.
1.
0
5
10
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