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IEEE Transactions on Power D elivery, Vol. 13, No. 4, October 1998
ult
Location Using Wavelets
Fernando
H .
Magnago and Ali Abur
Department of Electrical Engineering
Texas
A& M University
College Station, TX
77843,
U.S.A.
Abstract
This paper describes the use of wavelet transform for
analyzing powe r system fault transients in order to determine the
fault location. Traveling wave theory is utilized in capturing the
travel time of the transients along the monitored lines between the
fault point and the relay. Time resolution for the high frequen cy
components of the fault transients, is provided by the wavelet
transform. Thi s information is related to the travel time of the
signals which are already decomposed into their modal compo-
nents. A erial mode is used for all fault types, whereas the ground
mode is used to resolve problems associated with certain special
cases. Wavelet transform is found to be an excellent discriminant
for identifying the traveling wave reflections from the fault irre-
spective of the fault type and impedance. EM TP simu lations are
used to test and validate the proposed fault location approach f or
typical power system faults.
Keywords: Power System Faults, Electromagnetic Transients,
Wavelet Transform, Fault Location, Traveling Waves.
1
Introduction
Transmission line fault location has lo ng been on e of the primary
concerns of the power industry. Methods of locating power sys-
tem faults introduced so far, can be broadly classified under two
categories: one based on the power frequency components, and
the other utilizing the higher frequency contents of the transient
fault signals. The latter is also referred to
as
traveling wave or
ultra high speed fault location method, due to its use of traveling
wave theory and shorter sampling windows.
The use of traveling wave theory for fault detection was
initially proposed by Dommel and Michels in [I] , where a dis-
criminant was defined based on the transient voltage an d current
waveforms in order to detect
a
transmission line fault. McLaren
et al. have later developed
a
correlation based technique where
the cross correlation between stored sections of the forward and
backward traveling waves were used to estimate the travel times
of transient signals from the relays to the fault point [2,3,4]. An
overview of traveling wave based fault location methods can be
found in [ 5 , 6 ] .
PE-303-PWRD-0-12-1997
A
paper recommended and approved by
the IEEE Transmission and Distribution Committee of the IEEE Power
Engineering Society for publication in the IE EE Transactions on Power
Delivery. Manuscript submitted July 28, 1997; made available for
printing Decem ber 12,
1997.
1475
Among the limitations of the traveling
wave
methods, the
requirement of high sam pling rate is frequently stated. Other
stated problems include the uncertainty in the choice of sampling
window and problems of distinguishing between traveling waves
reflected from the fault and from the remote end of the line.
Recent developments in optical current transducers technology
enabled high sampling rate recording of transient signals during
faults [7]. Availability of such broad bandwidth sampling capa-
bility facilitates better and more efficient use of traveling wave
based methods for fault analysis.
The correlation based fault location method introduced in
[ 2 ] s very effective as long
as
the width of the time window
to save the forward moving wave is properly selected. Since
this selection depends on the fault location which is unknown,
the window width selection remains an unresolved issue for the
practical implem entation of this method . Com bined use of a
short and
a
long window has been proposed as one solution for
this problem in [4].
In this paper,
a
different approach, based on the wavelet
transform of the fault transients, is presented. Wavelet transform
possesses some unique fea tures that make it very suitable for this
particular application. It maps a given function from the time do-
main into time-scaling dom ain. The wavelet, the basis function
used in the wavelet transform, has band pass characteristics wh ich
makes this mapping similar to a mapping to the time-frequency
plane. Unlike the basis func tions used
in
Fourier analysis, the
wavelets are not only localized in frequency but
also
in time.
This localization allows the detection of the time of occurence
of abrupt disturbances, such
as
fault transients. Fault generated
traveling waves appear
a:;
such disturbances superposed on the
powe r frequency signals recorded by th e relays. Processing these
signals using the wavelet transform reveals their travel times
between the fault and the relay locations.
The potential benefits of a pply ing wavelet transform for anal-
ysis of transient signals in pow er systems have been recognized
in the recent years. Robertso n et al. present a comparative
overview of Fourier, short time Fourier and wavelet transforms,
give examples of applying wavelet transform to analyze power
system transients and extraction of their particular features in [8].
A similar overview along with application of wavelet transform
to detect and classify pow er quality d isturbances, are given in
[9].
Advantages of using w avelet transform for analyzing transients
and solution of linear time-invariant differential equations using
wavelet transform is demonstrated in
[101.
In this paper, another
useful application of the wavelet transform in solving the prob-
lem of fault location, will be presented. A brief introduction to
wavelet transform will be given before formulating the problem
and presenting the proposed solution method.
0885-8977/98/ 10.00 0 1997 IEEE
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Wavelet transform has been introduced rather recently in math-
ematics, even though the essential ideas that lead to this devel-
opment have been around for
a
longer period
C P ~
ime. It is a
linear transformation much like the Fourier transform, howev-
er with one important difference: it allows time localization of
different frequency components of a given signal. Windowed
Fourier transform also partially achieves this sam e goal, but with
a
limitation of using
a
fixed width windowing function.
As a
result, both frequency and time resolution of the resulting trans-
form will be apriori fixed. In the case of the wavelet transform,
the analyzing functions, which are called wavelets, will adjust
their time-widths to their frequency in such a way that, higher
frequency wavelets will be very narrow and lower frequency
ones will be broader. This property of multi resolution is partic-
ularly useful for analyzing fault transients which contain local-
ized high frequency components superposed on power frequency
signals. Thus, wavelet transform is better suited for analysis of
signals containing short lived high frequen cy disturbances super-
posed on lower frequency continuo us wav eforms by virtue of this
zoom-in capability.
Given a function f ( t ) , its continuous wavelet transform
(WT) will be calculated as follows:
where,
a
and
b
are the scaling (dilation) and translation (tim e
shift) constants respectively, and IQ is the wavelet function which
may not b e real as assumed in the above equation for simplicity.
The choice of the wavelet function (mother wavelet) is flexible
provided that it satisfies the
so
called admissibility conditions
Wavelet transform of sampled wav eforms can b e obtained by
implementing the discrete wavelet transform which is given by:
P11.
where, the parameters a and
b
in
Eq. 1)
are replaced by a; and
ka?, k and m being integer variables. In a standard discrete
wavelet transform, the coefficients are sampled from the contin-
uous WT on a dyadic grid, a
=
2 and
bo = 1
yielding a:
= 1,
U;
= i , tc.
b
= k
x
2- , being an integ er variable.
Actual implementation of the discrete wavelet transform, in-
volves successive pairs of high-pass and low-pass filters at each
scaling stage of the wavelet transform. This can be thought of
as successive approximations of the same function, each approx-
imation providing the incremental information related to a par-
ticular scale (frequency range), the first scale covering a broad
frequency range at the high frequency end of the spectrum and the
higher scales covering the lower end of the frequency spectrum
however with progressively shorter bandwidths. Conversely, the
first scale will have the highest time resolution, higher scales w ill
cover increasingly longer time intervals. While, in principle any
admissible wavelet can be used in the wavelet analysis, we have
chosen to use the Daubechies4 [9],[12] wavelet s the mother
wavelet in all the transformations.
lem
Consider a single phase lossless transmission line of length
e
connected between buses A and B , with a characteristic
impedance
2
and traveling wave velocity of U . If a fault occurs
at a distance x from bus
A,
this will appear as an abrupt injection
at the fault point. T his injection will travei like
a
surge along the
line in both directions and will continue to bounce back and forth
between the fault point, and the two terminal buses
until
the post
fault steady state is reached. Hence, th e recorded fault transients
at the terminals of the line will contain abrup t changes at intervals
commensurate with the travel times of signals between the fault
to the terminals. Using the knowled ge of the velocity oftrav eling
waves along the given line, the distance to the fault point can
be deduced easily. Thi s is the essential idea behind traveling
wave methods. Unlike the correlation based methods where the
forward and backward traveling wave components are computed
and used for the cross correlation, in the wavelet based appro ach,
the composite signal (voltage or current) at the relay location is
directly analyzed.
In three phase transmission lines, the traveling waves are
mutually coupled and therefore
a
single traveling wave velocity
does not exist. In order to implement the traveling wave method
in three phase systems, the phase domain signals are first decom-
posed in to their modal compon ents by means of the modal trans-
formation matrices. In this study, all transmission line models
are assumed to be fully transposed, and therefore the well known
Clarke's constant and real transformation matrix given by:
is
used. The phase signals are transformed in to their modal com-
ponents by using this transformation matrix as follows:
S i n o d e
=
T S p h a s e 4)
where, Smo nd Spha.ve are themodal and phase signals (voltages
or currents) vectors respectively.
Clarke's transformation is real and can be used with any
transposed line. If the studied line is untransposed, then an eigen-
vector based transformation matrix, which is frequency depen-
dent, will have
to
be used. This matrix should be computed at
a frequency equal or close to the frequency of the initial fault
transients.
Recorded phase signals are first transformed into their modal
components. The first mode (mode l ) , s usually referred to as the
ground
mode, and its magnitude is significant only during faults
having
a
path to ground. Hence, this component can not be used
for all types of faults. The second mode (mode 2 ,also known as
the
aerial
mode, how ever is present fo r any kind of fault. Acco rd-
ingly, the fault location problem is formulated based essentially
on the aerial mode, m aking occasional use of the ground m ode
signal for purposes of distinguishing b etween certain peculiar sit-
uations, which will be discussed in the next section. Depending
on the existing communication scheme between the two ends of
the line, fault location problem can be solved in two different
ways described below.
Two
ended sync ronized recording
Fault signals are recorded simultaneously at both ends of the
line by two separate channels both of which are using the same
time reference synchronized using Global Positioning Satellite
(GPS ) receivers. The recorded waveforms w ill be transformed
into modal signals, after which the modal signals will be analyzed
using their wavelet transforms. Let
t~
and t g correspond to the
times at which the modal signal wavelet coefficients in scale
1,
show their initial peaks for the signals recorded at bus
A
and
B
respectively. Assum ing that the recorded sign als at the two
ends of the
l ine
are fully synchronized, the delay between the
fault detection times at the two ends, i.e. t d
=
t g
-
t A can be
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1477
Insignificant coefficients will imply that the fault is in the remote
half of the line, and vice versa.
If the fault is determined to be in the near half
o f the
line, then
td in Eq.(6) will simply be the time interval between the first two
peaks of the scale 1 WTC’s for the aerial mode.
If the fault is suspected to be in the second half of the line,
then td in
Eq. 6)
will be replaced by:
7)
d
= t x
where:
t s the travel time for the entire line length, and e is the time
interval between the first two peaks of aerial mode WTC’s in
scale 1.
Figure 1 shows th e flowchart for the proposed fault location
algorithm based on the wavelet transform coefficients. Next
section contains results of simulations used to test this proposed
algorithm for various fault types and line configurations.
determined [13]. The distance between the fault point to bus A
will then be given by:
e - t d
2
=
where,
e is the length of the line,
x
is the distance to fault from bus A,
and U, is the speed of the traveling waves for mode
m .
3.2
Single ended recording
A more robust configuration that does not require remote end
synchronization is when the fault location is determined based
solely on th e recorded signals at one end of the line. However, in
such a case, du e to the lack of any other time reference, all time
measurements will be with respect to the instant when the fault
is first detected. Therefore, fau lt location calculations will be
based on the reflection tim es of the traveling waves fr om the fault
point. Unfortunately, for faults involving a ground connection,
not only those reflections from the fault point, but also from the
remote end bus will be observed at the sending end of the line.
Proper algorithms should therefore be devised in order to dis-
tinguish between close-in and remote faults which may produce
similar reflection patterns f or the grounded faults. T he following
sections describe our proposed approach to accomplish this task.
3.2.1
Approach
I:
Ungrounded faults
It has long been observed that ungrounded faults such as line-to-
line or ungrounded three-phase, do not cause significant reflec-
tions from the remote e nd bus during the fault transients. Thus,
by measuring the time delay between the two consecutive peaks
in the wavelet transform coefficients of the recorded fault signal
at scale
1,
and taking the product of the wav e velocity and half of
this time delay, the distance to the fault can easily be calculated
for these kinds of faults. The fault distance will be given by the
equation:
where,
x is the distance to the fault, v is the wave velocity (for the mode
used), and td is the time difference between two consecutive
peaks of the wavelet transform coefficients.
3.2.2
Approach
11:
Grounded faults
When the fault involves
a
connection to ground, then sending end
signals may contain significant reflections from the remote end
bus in addition to the ones from the fault point. Also, depending
on the location of the fault, the reflections from the remote end
may arrive before or after those reflected from the fault point.
It can be easily verified by using the Lattice diagram method,
that the remote end reflections will arrive later than the fault
reflections if the fault occurs within half the length of the line,
close to the relay location. The opposite will be true if the fault
is situated in the second half of the line. It is observed that, in
the former case the
ground mode
wavelet transform coefficient
(WTC) for scale 1, show s significant peaks, w hile the latter case
ground mode
WTC for scale 1 remains insignificant below the
chosen detection threshold.
Therefore first a decision is made on whether
or
not the
fault
is
grounded, based on scale 2 WTC’s of the ground mode
signals. If these coefficients are found significant, then th e fault
will be assumed to be a ground fault. Next decision will be made
on which half of the line the fault is actually located. Thi s is
done by observing scale 1 WTC’s of the ground mode signals.
Transducer
output
Transformation
Wavelet
Transformation
L-r’
Ungrounded Fault
YE§
calculate the fault loc.
as
in
Section
3.2.1
’
r
Remote half of the line.
Based on Scale
1
Mode
2
calculate td s in Eq. 7)
then calculate the fault
Grounded Faull
Grounded
Fault
Near half of the line.
Based OD Scale 1Mode
2
calculate the fault
loc.
as
in
Section 3.2.1
igure 1: Flowchart of the proposed fa ult location algorithm
4 Simulation results
The ATF’EMTP program
[
141 is used to calculate the transient
signals in the power system. Figure 2 show s the system config-
uration used in the simulations. The frequency dependent model
and
B
for the do uble ended configuration and at busbar A for the
single ended configuration.
For this tower configuration, mode 2 (aerial mode) has
a
propagation velocity of 1.8 182
x
lo5miles/sec. A sampling time
o f l o p s is used. The system is simulated using double and single
is
used
t
model the line [151 The
relays are
ocated at hushar
A
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1478
1=
200 miles
c
X
I 345 Kv
- x
D c h
345
KV
Figure
2:
Schematic diagram of the simulated system
ended configurations under various kind of faults. Different type
of balanced and unbalanced faults at different locations along
the line and at different inception angles are simulated. Results
of single phase to ground, phase to phase and three phase to
ground faults for an inception angle of
108
degrees are reported
to illustrate the method. The modal signals are decomposed using
the Daubechies4 wavelet where number 4 represents the number
ofw ave let coefficients. O nly the first two scales, scale
1
and scale
2
of the WTC are used in the proposed fault location method.
In order to minimize the noise effect, we squared the wavelet
coefficients at each scale as also done in [9].
A lattice diagram illustrating the reflection and refraction of
traveling waves initiated by the fault transients, is shown in Fig-
ure
3.
On th e left side of the figure, a line connecting buses A and
B
is
drawn vertically. The line
is
200 miles long.
A
single phase
to ground fault is assumed to occur at point F,120miles from bus
A. The horizontal axis starting from point F, epresents the time.
A set of arrows are shown below the lattice diagram, indicating
the arrival times of various w aves reflected from th e fault as well
as bus
B.
Mode
2
(aerial mode) is considered only. The travel
times from the fault
to
bus A, and from the fault
to
bus
B
are
designated by
TI
and
T2
respectively. G iven the traveling wave
velocity v2 or mode 2,
TI
and
T2
will be given by 12 0 mi I
v
and
80 mi
1 U:
respectively.
Figure
4
shows the WTC at scale
1
calculated fo r the example
of Figure 3. Comparing the WTC peak times with the arrival
times of waveform reflections at bus A, it can be observed that
there is a on e to one correlation between them. Simulation results
for both the tw o ended and single ended fault location approaches
will now be given.
T2 3T2 2Tl T2 5T2
R
,
I I .**‘
‘.
.’
A
T1 T1 2TZ
3T1 T1 4T2
Scale
1
mode 2 (aerial mode)
t
I
t
I I
20
205 21 21 22
5
23
time
(ms)
Figure 4: Single phase to ground fault at 12 0 miles from A Peaks
correspond to the predicted ones in Fig. 3.
5(a) and (b) show the WTC for scale 1, of the voltage transients
recorded at bus A and
B
respectively.
In
this example, the first
WTC
peak at bus A occurs at
tA
=
20.15 m s , and at bus B at
t B =
21ms, yielding
f d
= 0.85ms and
using Eq. 5):
200-
1.81 x
io5
x
0.85
x
1 0 - ~
x = 22.99 miles.
2
B us A: Scale 1
mode
2
21
7
B us
B: Scale
1 mode
2
3 5 3 j
time (ms)
Figure 5: Three phase fault at 20 miles from A.
The arrival time of the first transient peak depends on the
velocity of the line and the f ault distance, it is independent of
the type of fault, hence the method applies to all type of faults
provided the tw o terminal recordings are synchronized in time.
Ilj
igure
3 :
Lattice d iagram for a single phase to ground fau lt at
120
miles from
A.
4.2 Single ended recording
4.2-1 Ungrounded
faults:
Figure 6 shows the WTC’s for an example of a phase to phase
fault at 30 miles from A. It can be seen fro m the figure that mode
1 (ground mode) signals are zero, therefore mod e 1 WTC’s can
be used
to
identify this as an ungrounded fault.
4. ronized recording
Assuming synchronized recording of fau lt transients at both ends
of the line, a three phase fau lt is simulated at
20
miles away from
bus A. Mode 2 (aerial mode) voltage signals ar e used only. Figu re
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d)Scale
1 ,
mode
2
c)Scale
1,
mode
1
101 1
t ime ms) time ms)
b)Scale
2.
mode 2
d)Scale 2, mode
1
151 I 4 ,
1
I
J
I
Figure 8: P hase to ground fault at 1 70 miles from A.
345 Kv
45
KV
I \ ‘
345
KV
345 Kv
I I I
100 miles 100
miles
Figure 9: Circuit diagram of the simulated system with mutually
coupled lines.
5
This paper presents
a
new, wavelet transform based fault loca-
tion method. Using the traveling wave theory of transmission
lines, the transient signals are first decoupled into their modal
components. Modal signals are then transformed from the time
domain into the time-frequency domain by applying the wavelet
transform. The w avelet transform coefficients at the two lowest
scales are then used to determine the fault location for various
types of faults and line configurations. The proposed fault loca-
tion method is independent of the fault impedance and is shown
to be suitable for mutually coupled tower geom etries as well as
series capacitor compensated lines. The method can be used both
with single ended and synchronized two ended recording of fault
transients. The fault location estimation error is related to the
sampling time used in recording the fault transient. Furthermo re,
for grounded faults near
the
middle of
the line, mode
1 signals
from the fault and from the far end become comparable increas-
ing the error of the fault location algorithm. Simulation resu lts
are given to demonstrate the performance of the method.
References
[2] S. Wajendra and P.G. McLaren, “Traveling-W ave Tech-
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P.
G. McLaren, “Traveling Wave Tech-
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Fernando
N.
Magnago
obtained the B.S. degree from UNRC,
Arge ntinain 1990 and his M.S. degree from Texas A&M Univer-
sity, College Station, TX in 1997. H e is currently a Ph.D. student
at Texas A&M University.
Ali Abur (SM ’90) received the B.S. degree f rom M E W , Turkey
in 1979, the M.S. and Ph.D. degrees from The O hio State Uni-
versity, Columbus, OH, in 198
1
and 198 5 respectively. Since
late 19 85, he has been with the Dept. of Elect. Eng. at Texas
A&M University, College Station, TX, where he
is
currently an
H.
W.
Dommel, and J. M. Michels, “High Speed Relaying
using Traveling Wave Transient Analysis”, IEEE Publica-
tions NO. 78CH1295 -5 PWR, paper no. A78 21 4-9, IEEE
PES Winter Power Meeting, New York, January 1978,
pp.1-7. Assoc iate Professo r.