Energy and Power Engineering, 2013, 5, 603-607
doi:10.4236/epe.2013.54B116 Published Online July 2013 (
Application of Atomic Sparse Decomposition to
Feature Extraction of the Fault Signal in Small
Current Grounding System*
Nanhua Yu1, Rui Li1, Jun Yang2, Bei Dong2
1Electric Power Research Institute of Guangdong Power Grid, Guangzhou, China
2School of Electrical Engineering Wuhan University, Wuhan, China
Received January, 2013
Applying the atomic sparse decomposition in the distribution network with harmonics and small current grounding to
decompose the transient zero sequence current that appears after the single phase to ground fault is occurr ed. Based on
dictionary of Gabor atoms and matching pursuit algorithm, the method extracts the atomic components iteratively from
the feature signals and translated them to damped sinusoidal components. Then we can obtain the parametrical and
analytical representation of atomic components. The termination condition of decomposing iteration is determined by
the threshold of the initial residual energy with the purpose of extract the features more effectively. Accordingly, the
proposed method can extract the starting and ending moment of disturbances precisely as well as their magnitudes, fre-
quencie s and othe r f ea tures. The nu merical examples demonstrate its effectiveness.
Keywords: Small Current Grounding System; Fault Line Selection; Atomic Sparse Decomposition; Matching Pursuit;
Damped Sinusoids
1. Introduction
The small current grounding system include ungrounded
neutral system, arc suppression coil compensated neutral
system and high resistance-grounded neutral system. In
our country, scientists and engineers have done a lot of
researches on fault line selection in small current grou nd-
ing system, a variety of line selection methods were put
forward and some successes were achieved. But the
problem of fault line detection hasn’t been well settled
because of that the fault current is small, the fault fea-
tures are unobvious, the fault circumstances are complex
and the mode of the system runs various. This will hinder
the smooth development of power distribution automa-
tion system, damage the security and stability of the
power system.
Fourier transforms and wavelet transforms are exam-
ples of time-frequency signal decomposition that have
been used for many years. The Fourier basis provided a
poor representation of functions well localized in time,
and wavelet basis are not well adapted to represent func-
tions whose Fourier transforms have a narrow high fre-
quency support. In both cases, it is difficult to detect and
identify the signal patterns from their expansion coeffi-
cients for the information is diluted across the whole ba-
sis. To extract information from complex signals, it is
often necessary to adapt the time-frequency decomposi-
tion to the particular signal structures. [1]
Domestic and foreign scholars have proposed a variety
of methods about fault line selection. Existing line selec-
tion method of stable state quantity [2-4] is difficult to
meet the site operational requirements. There is still fault
transient detection method [5-12]. The paper [5] pro-
poses a distribution network fault line selection method
about fusion of multiple sampling points’ poll results
based on S-transform through studying S-transform to
extract the amplitude and frequency characteristic and
phase-frequency characteristic of signal. This method
needs to collect to the correct the feeder phase angle and
frequency information. The paper [6,7] extract fault line
travelling wave through using wavelet transform and
structure criterion in order to determine fault line. Al-
though wavelet transform has good enough time domain
- frequency domain localization features to provide cha-
racteristics of the signal in different scales, it has ineffec-
tive application as it is vulnerable to the effects of noise.
The paper [8] realizes the fusion line selection method
through introducing the concept of fault measures and
using Dempster-Shafer theory. The paper [11] captures
feature band by comparing transient zero-sequence cur-
*This work is supported by Key Technology Projects of China South-
ern Power Grid
Copyright © 2013 SciRes. EPE
rent amplitude and gets characteristic band signal
through filtering. References[9,10] use wavelet packet to
extract the transient signal, determined band which the
transient capacitive cu rrent most concentrate by the prin-
ciple of maximum energy and elected fault feeder. How-
ever, the feature band that can be used for line selection
is influenced by the system parameters, failure modes
and so on. Various feeders transient capacitive current
concentration of energy bands are not the same, if the
band is selected improperly, it is easy to make the fail
selection of the fault line.
The paper [12] uses S-transform to process zero se-
quence current of each feeder, and determines the domi-
nant frequency of the capacitive current by comparing
transient energy of different frequency points, and selects
the fault line according to the size of the energy.
S-transform is the development of continuous wavelet
transform and the short time Fourier transform, and has
good time-frequency characteristic, but amount of infor-
mation is so much after decomposition. As well as there
is some line selection methods which combine steady
state with transient state, but neural network algorithm
exists local optimum problem, poor convergence, longer
training time and limited reliability.
To compensate for these shortcomings, this paper pro-
vides an algorithm that can decomposes signal into a
linear expansion of waveforms that belong to a redundant
dictionary of functions, these waveforms are selected in
order to best match the signal structures. For a particular
signal, according to the characteristics of the signal, the
algorithm can choose the best spread functions adap-
tively. By using less function more accurate represent the
signal, it can decompose a signal into coherent compo-
nents. The new feature extraction of the fault signal me-
thod in the neutral indirectly grounded system which is
based on the atomic sparse decomposition.
2. Time-Frequency Atom Dictionaries
Time-frequency signal decomposition such as Fourier
transforms and wavelet transforms has been used for
many years. It is often essential to adapt the time-fre-
quency decomposition to the particular signal structures
to extract information from complex signals. The algo-
rithm provided in this paper can decomposes signal into a
linear expansion of waveforms that belong to a redundant
dictionary of functions, the selected waveforms are the
best match of the signal structures. For a particular signal,
the algorithm can choose the best spread functions adap-
tively on the basis of the characteristics of the signal. The
functions more accurate to represent the signal are called
atom, the possibly redundant dictionary of functions
called atoms.
The matching pursuit algorithm is often used in the
atomic decomposition process, it is a greedy adaptive
decomposition to decompose signal into a linear expan-
sion of waveforms that belong to atoms. The selected
atoms have good time-frequency characteristic and can
represent the inherent characteristics and critical infor-
mation of the signal.
2.1. The Gabor Dictonary
Decomposing a signal over a redundant dictionary im-
proves the compression efficiency, especially at low bit
rate where most of the signal energy is captured by only
few elements. We often use Gabor dictionary in atomic
sparse decomposition. Define
()( )
The real Gabor atoms:
()( )cos()
gt gt
() 2t
is Gaussian window function, (,, , )s
is the index of ()
, s is Scaling parameters,
Displacement parameters,
is Frequency parameters,
is Phase parameters. Such atomic space is boundless,
and in practice, we can’t search a boundless space, so the
atomic database should be dispersed. (,
apa ,
while 2,a
 2
0l ogjN ,
02pN ,
 1
 Then we can get a Gabor
()(2)cos( 2)
rd j
gn gnpnk
 (3)
() 0
()(2)[1, )
0,1,..., 1
nKgn jL
2.2. Matching Pursuits Algorithm
In contrast to orthogonal transforms, over complete ex-
pansions of signals are not unique. The number of feasi-
ble decompositions is infinite, and finding the best solu-
tion under a given criteria is a NP-complete problem. In
compression, one is interested in representing the signal
to be coded with the smallest number of elements, which
is in finding the solution with most of the energy on only
a few coefficients. Matching Pursuit is one of the sub-
optimal approaches that greedily approximate the solu-
tion to this NP-complete problem [13].
Matching Pursuits algorithm, which is a greedy adap-
tive decomposition that has the potential of decomposing
a signal into coherent components.
Copyright © 2013 SciRes. EPE
N. H. YU ET AL. 605
()1 ()
arg max,
mm mmm
xx x
f is the original signal. After m iterations, we decompose
f into the concatenated sum
1(1) ()()()
If we stop the algorithm at this stage, the summation of
(6) recovers an approximation of f, with an error equal to
2.3. Building a Dictionary for Power System
In a very simplistic way, power systems can be consid-
ered to be built from transmission lines, sources, and
loads. Besides, we should add to this model the discon-
tinuities due to circuit switching caused by operative
maneuvers and by the protection system. The employed
model is given by [14-16].
() cos(2)
sq eq
ftAft e
tt utt
 
where each component is a damped sinusoid represented
by a six-tuple (,,,,,
qq qqsqeq
), where q
is its
amplitude, q
is the frequency, q
is the damping
factor, q
is the phase, and are the component
starting and ending times. Note that the model in (7) by
no means compactly represents “all” possible phenomena
in power systems signals.
3. Example Analysis
3.1. Harmonics
The original analog signal model:
()cos(100 )()0.4cos(700)
() 0
q( )0
utt ptt
qt t
st s
pt otherwise
st s
 
 
The total time is 0.5 s, Adding seven times harmonic
at 0.04 - 0.25 s; 13th harmonic at 0.1 - 0.5 s, sampling at
4 KHz.
Figure 1(a) shows the Waveforms of decomposed
atoms, and the Table 1 shows the atom parameters of
each iteration. From the Figure 1(a) we can see that the
atomic decomposition has iterated only three times to
separate the fundamental component and harmonic
component effectively, and the feature extraction of the
Correlative Component is obtained. Figure 1(b) shows
that the decomposed signal is closed to the original signal,
and the error of this method reached 10-3, continuing
decomposition can make the approximate degree higher.
Besides, the data in Table 1 shows that the frequency,
phase angle, and the starting and termination time got
from the atomic decomposition are basically identical to
the original signal.
3.2. Earth Fault Line Selection
1) Put up a model of 10 kV distribution network with
PSCAD. As Figure 2 shows, there are four lines named
m, n, w, p. The single-phase- to-ground fault happened at
0.5 s. Collecting and analyzing the zero sequence current
in the four lines.
200 400600800 1000 1200 1400 1600 1800 2000
11st 原子
200 400600800 1000 1200 1400 1600 1800 2000
-0. 4
-0. 2
0.4 2nd 原子
200 400600800 1000 1200 1400 1600 1800 2000
-0. 05
0.05 3rd 原子
20040060080010001200 14001600 18002000
t /ms
20040060080010001200 14001600 18002000
t /ms
Origi nal Sig nal (blue)
20040060080010001200 14001600 18002000
Const ructed S i ngal (red)
Figure 1. (a) Waveforms of decomposed atoms; (b) the re-
construction for harmonics.
Table 1. Decomposed atom parameters of signal.
Atom Amplitude
(V) Frequency
(Hz) Phase Angle
(rad) Start Time
(s) Terminal
1ed atom0.999 50.00000 0 0.5
2rd atom0.388 350.00000.524 0.04 0.25
3rd atom0.200 650.0001.075 0.1 0.5
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Copyright © 2013 SciRes. EPE
Figure 2. The model of 10 kV distribution network.
00.05 0.10.15 0.20.25 0.30.350.40.45
-0. 4
-0. 2
00.05 0.10.15 0.20.25 0.30.350.40.45
-0. 1
0.2 2nd
00.05 0.10.15 0.20.25 0.30.350.40.45
-0. 05
00.05 0.1 0.150.2 0.40.45
-0. 1
t /s
00.05 0.1 0.150.2 0.40.45
-0. 5
t /s
Origi nal Si gnal(blue)
00.05 0.1 0.150.2 0.40.45
-0. 5
Constructed S ingal (red)
Figure 3. (a) Waveforms of decomposed atoms; (b) the reconstruction for zero sequence transient current of m line.
Table 2. Decomposed atom parameters of signal.
Line Atom
(V) Frequency
(Hz) Phase
2ed atom 0.2048 4 88.08 -5.6284
m line 3rd atom 0.0812 547.03 26.8752
2ed atom 0.1320 487.87 179.0498
n line 3rd atom 0.0603 549.71 -209.0478
2) It is clearly observed from Figure 3(a) that the
atomic decomposition can separate the fundamental
component and harmonic component effectively with
only three iterations. Figure 3(b) shows that the decom-
posed signal is similar to the original signal. From Table
2 we can see that the phase angles of the second atom is
opposite to the third atom. Through which we can iden-
tify Single-Phase-to-Ground Fault line.
4. Summary
By the above analysis of the actual case, we apply a
compression algorithm for signals measured during
power system disturbances that obtains good effect while
preserving important features for signal analysis. The
damped sinusoidal dictionary by no means compactly
represents “all” possible phenomena in power systems
signals. Adaptive, analyticity and the sparse of Atomic
decomposition make the algorithm has obvious advan-
tages in power system fault line selection. Therefore, the
method is feasible and practical, and can be applied
broadly in power system.
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