The method presented in this paper employs recently introduced S-transform which is an important development of STFT with improved properties.  Proposed methods allow tracking changes in amplitude and frequency with better precision than STFT and Wigner-Ville transform.  Possible applications in diagnosis and power quality problems are targeted. The S-transform outperforms the STFT in that it has a better resolution in phase space giving a fundamentally more sound time frequency representation. Investigations of the representation error show that optimally adjusted S-transform can also outperform the Wigner-Ville transform when dealing with time-frequency representations of the signal. The S- transform is also tested on nonstationary electric signals where it shows excellent tracking capability. These properties show that S-transform can be effectively used for analysis of electric signals, especially when dealing with multi-component time-varying waveforms

Leonowicz, Z

Lobos, T

Wozniak, K

English

Repozytorium Eny Politechnika Wrocławska

 
 
ZBIGNIEW   LEONOWICZ,   TADEUSZ   ŁOBOS,   KRZYSZTOF   WOŹNIAK 
WROCLAW UNIVERSITY OF TECHNOLOGY, POLAND 
 
ANALYSIS OF POWER QUALITY DISTURBANCES USINGTHE S-TRANSFORM 
 
Keywords: S-Transform, Power Quality, Disturbances, 
Time-Frequency Analysis 
 
Introduction 
 
Contemporary power systems, which contain a 
considerable number of non-linear loads, require 
advanced methods of spectral analysis for their 
investigation and control. Fourier-based spectral methods 
are useful in stationary signal analysis but insufficient in 
the case of numerous real-life problems when signal 
contents changes in time. Nonstationary signals are 
usually analyzed with STFT (Short-Time Fourier 
Transform). The method presented in this paper employs 
recently introduced S-transform which is an important 
development of STFT with improved properties. 
 Proposed methods allow tracking changes in 
amplitude and frequency with better precision than STFT 
and Wigner-Ville transform.  Possible applications in 
diagnosis and power quality problems are targeted. 
 
 
1. THE S TRANSFORM 
The S Transform, introduced by Stockwell [4] is 
defined by the general equation: 
 
( ) 2( , ) ( , ) j ftS f h t g t f e dπτ τ∞ −
−∞
= −∫ t       (1)  
 
where ),( fg τ is a window function. Window 
function is a modulated Gaussian function, 
expressed by: 
 
)2/( 22
2
),( σπτ
te
f
fg −=                   (2) 
 
where σ is defined as: 
 
1
f
σ =                                 (3) 
 
Finally, the general formula in (1) takes the form: 
  
( ) 2 2(( ) / 2) 2( , )
2
t f j ftfS f h t e e dτ πτ π
∞
− − −
−∞
= ∫ t
tτ
   (4) 
 
where f represents frequency and t , τ represent time. 
      The above derivation of the definition of S Transform 
originates from the STFT idea. However, the S Transform 
is related to the Wavelet Transform, as well.  
Continuous Wavelet Transform W(τ ,d) of the function 
h(t) is defined by: 
( ) dtdtwthdW ∫∞
∞−
−= ),(),( ττ                (5) 
As previously stated, the S – Transform can be also 
presented as continuous wavelets transform using a 
specific mother wavelet multiplied by a phase coefficient: 
 
2( , ) ( , )j fS f e W d dπ ττ =                (6) 
 
Where the mother wavelet ψ(t,d) is defined as: 
 
2 2( / 2) 2( , )
2
t f j ftft f e e πψ π
− −=                (7) 
 
where the coefficient d represents the inverse of the 
frequency.  
Taking into account the above considerations, we can 
derive the general formula of the S-transform as in (1). 
The equation (1) assumes that the width of the window 
function is inversely proportional to the local frequency. 
We can also adjust the resolution of the S-transform by 
introducing a modified coefficient σ: 
 
    
k
f
σ =                                      (8) 
 
Then, the equation (4) takes the form: 
 
( ) 2 2 2(( ) / 2 ) 2( , )
2
t f k j ftfS f h t e e d
k
τ πτ π
∞
− − −
−∞
= ∫ t   (9) 
 
where the coefficient k controls the resolution in such way 
that if k>1 the resolution in frequency increases and when 
k<1 the resolution in time increases.  
The S-transform is a linear transform of the signal h(t). 
The additive noise can be modeled as follows:       
hnoisy(t)= h(t) + η(t). The S-transform of a noisy signal 
represents a sum of transformed signal and 
transformed noise, as follows: 
 
{ } { } { )()()( tSthSthS noisy }η+=             (10) 
 
The S-transform output is a complex matrix, where the 
rows correspond to frequencies and the columns to time. 
Each column thus represents a “local spectrum” for that 
point in time. Since ( , )S fτ  is complex valued, in 
practice, usually the module ( , )S fτ  is plotted and this 
gives time-frequency S spectrum. The S-transform 
outperforms the STFT in that it has a better resolution in 
phase space (i.e. a more narrow time window for higher 
frequencies), giving a fundamentally more sound time 
frequency representation [4]. 
 
 
2. COMPUTATION OF THE ERROR OF 
REPRESENTATION (S-TRANSFORM, STFT, 
WIGNER-VILLE TRANSFORM) 
 
Selected power quality disturbances are investigated 
using the S-transform (ST) and compared to previously 
investigated Short-Time Fourier Transform (STFT) and 
Wigner-Ville transform (WV) [?].  
Signal parameters are as follows: 
- sampling frequency 1,6 kHz 
- signal length equal to 8 periods of main harmonic 
50 Hz 
- duration of the disturbance equal to 4 periods 
Simulated power quality disturbances include: 
- voltage drop 
- over-voltage (5-50% of the nominal voltage) 
- voltage sag (10-100% of the nominal voltage) 
 
Investigations were carried out as follows, 
independently for each kind of disturbance: 
1.  Analysis using the S-transform for different values 
of the k coefficient, modifying its resolution in time and 
in frequency. The range of k varied from 0.1 to 1.0 
which assures better resolution in time domain of the 
S-transform. After the transformation a time-frequency 
of the signal was obtained. An exemplary time-
frequency plot is shown in Figure 1. for a 50% voltage 
sag and k=1.0. 
2. Analysis using the STFT for analysis window length 
equal to  ¼ of the signal length. 
3. Analysis using the WV transform with smoothing.  
 
From each time-frequency representation the main 
harmonic 50 Hz was filtered out and its amplitude was 
compared to the reference waveform. Additionally, for the 
S-transform, the investigations included the evaluation of 
the influence of the k coefficient on the error of 
representation.  
Reference signal composed of one main harmonic of 
50 Hz was generated independently for each analysis 
method with randomly variable phase. 
 
 
Fig.1. Time-frequency representation of a 50% voltage 
sag and k=1.0 using the S-transform. 
 
Figure 2 shows the representation of the reference 
signal using the S-transform, the Short-Time Fourier 
Transform and the Wigner-Ville Transform for a 50% 
voltage sag. For each group of waveforms (as in Figure 
2), the error of representation was calculated, according 
to the formula: 
 
    ( )256 250 Repr 50 Ref
1
1 ( ) ( )
256 Hz Hzn
MSE A n A n
=
= −∑ (11)  
 
where n- number of samples , 50 Rep ( )HzA n - 
representation of the amplitude of the 50 Hz main 
harmonic for the time instant corresponding to the sample 
number n, - reference amplitude of the 50 Hz 
main harmonic for the time instant corresponding to the 
sample number n . 
50 Ref ( )HzA n
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
time [s]
am
pl
itu
de
 [p
.u
.]
50% zapada -skladowa 50Hz
 
 
Wigner-Ville'a
STFT
ST
wzorzec
Fig. 2. Representation of the reference signal using 
the S-transform (ST) for K=1, the Short-Time Fourier 
Transform (STFT) – window length = 0.04 s and the 
Wigner-Ville Transform with smoothing window length 
of 0.04s – of the test waveform:  50% voltage sag. 
50% voltage sag, 50 Hz harmonic 
refe ence 
 
 
 2.1. Comparison of representation error. 
 
The representation error computed as in (11) for 
exemplary power disturbances is shown in Figures 3 and 
4. In Figure 3 voltage sag of variable amplitude from 10% 
to 100% is investigated and in Figure 4 a case of over-
voltage with variable amplitude from 5% to 50% is shown. 
The S-transform shows overall smallest representation 
error, especially for the case when k=0.1 (small value of k 
means higher resolution in time domain).  
Best performance of the S-transform can be explained 
by the multi-resolution capabilities of this representation –
see Figure 5. The S-transform has better time resolution 
for higher harmonics (narrow analysis window) and better 
frequency resolution for low harmonics (wide analysis 
window). 
 
10 20 30 40 50 60 70 80
10-3
10-2
10-1
100
M
S
E
 
 
STFT zapada
WV a z
ST za
ST za
 
Fig.  3. Representation error for varying amp
voltage sag. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig.  4. Representation error for varying amp
over-voltage. 
 
 
As seen from Figures 2-4, the S-tra
comparison to smoothed Wigner-Ville Tran
Short-Time Fourier Transform allows more
representation of dynamic changes of sign
components.  The S-transform, providing t
setting of the parameter k, has the smallest rep
error.  
 
 
t
f
 
Fig.  5. Frequency-dependent resolution of the S-
transform. 
 
2.2. Analysis of switching of capacitor banks 
a) Transformer
DY 110/15 kV
25MVA
0.2km 1km
Line Line
  
 
5 10 15 20 25 30 35 40
10-3
10-2
10-1
100
 
STFT 
M
S
E
prze
WV przep
ST epi
przepiecie [%]
prz
ST przepi
 
 
amplitude of over-voltage [%] 90 100
pada
da k=0.1pa
pada k=1
0.03s 0.09s
Lo
adglebokosc zapady[%]amplitud  f  volt ge sag [%]  
litude of 
litude of  
nsform in 
sform and 
 accurate 
al spectral 
he optimal 
resentation 
source
50Hz
900kVAr 1200kVAr
0 0.04 0.08 0.12 0.16 0.2
-8
-6
-4
-2
0
2
4
6
8
10
time (s)
cu
rre
nt
 (A
)
x10 3
switching on
1200kVar
switching on
900kVar
b)
 
45 50
 
Fig.  6. a) One-phase diagram of the simulated distribution 
system. b) Voltage waveform at the beginning of the 
feeder. 
 
The investigation results in a distribution system as in 
Figure 6a are shown in this section. Two capacitor banks 
(CB) were installed along the feeder. Several cases were 
simulated and both currents and voltages were recorded. 
Figure 6b shows the voltage waveform at the beginning of 
the feeder for the case that the first CB 900 (kVAr) was 
switched on at 0.03 s and the second CB 1200 (kVAr) at 
0.09 s. Figure 7 shows three-dimensional time frequency 
representations obtained using a) S-transform b) STFT - 
of the complex waveform. 
piecie
ieiec
ecie k=0.1
ecie k=1
 
a) |S-transform|
600
500
400
200
100
0
0 0.05 0.10 0.15
time (s)
300
700
fre
qu
en
cy
 (H
z)
0.20
270Hz(a-t)
475Hz(a-t)
50Hz(a-t)
(a-t): auto-terms
(c-t): cross-terms
b)
0.20
600
500
400
200
100
0
0 0.05 0.10 0.15
time (s)
300
700
fre
qu
en
cy
 (H
z)
270Hz(a-t)
475Hz(a-t)
50Hz(a-t)
( )h τ|Short-Time Fourier Transform|- Spectrogram                   - Hamming, width - 0.04s
(a-t): auto-terms
(c-t): cross-terms
 
Fig.  7.  Comparison of S-transform representation and 
STFT spectrum of the waveform. 
0 0.02 0.04 0.06 0.08 0.1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
5
time (s)
|e
ne
rg
y|
 (J
)
Dynamism of tracking the energy changes -
475Hz component
Short-Time Fourier
Transform with Hamming
window, width 0.04s
S-Transform
t=0.03s
switching on
900kVar
 
Fig.  8. Comparison of tracking capabilities of dynamic 
amplitude changes of the 475 Hz components. 
The S-transform shows again better energy 
concentration and ,therefore, lower error when estimating 
the amplitude of time-varying amplitude of one spectral 
component in a signal composed of multiple spectral 
components and noise (Figure 8).  
 
 
3. CONCLUSIONS 
 
In all performed experiments (in the paper only limited 
number of them is reported) the S-transform showed the 
lowest error when comparing the error of representation to 
STFT and Wigner-Ville transform, especially when 
analyzing dynamically changing spectral components. 
Sometimes it was necessary to adjust the k coefficient to 
obtain the optimal representation for a given problem and 
adjust in this way the multi-resolution property of the S-
transform.  The same advantageous tracking capability 
showed the S-transform when analyzing the time-varying 
multi-component signal. 
Performed experiments allow concluding that the S-
transform is a reliable and accurate tool for the analysis 
non-stationary waveforms in power systems and its 
properties can be used for diagnostic and power quality 
applications.  
 
 
5. REFERENCES 
 
1. Leonowicz, T. Lobos and J. Rezmer, Advanced 
Spectrum Estimation Methods for Signal Analysis in 
Power Electronics, “IEEE Trans. on Industrial 
Electronics”, pp. 514-519, June 2003. 
2. Mertins A., Signal Analysis: Wavelets, Filter Banks, 
Time-Frequency Transforms and Applications, John 
Wiley & Sons, 1999. 
3. Sikorski T., Wozniak K.: Joint time-frequency 
representations - comprehensive method for analysis 
of nonstationary phenomena in electrical engineering. 
Proceedings of the 15th International Conference on 
Power System Protection. PSP 2006. Bled, Slovenia, 
September 6-8, 2006, pp. 1-10. 
4. Stockwell R.G., Mansinha L and Lowe R.P.. 
Localization of the complex spectrum: The S-
Transform. “IEEE Transactions on Signal Processing”, 
44(4), pp.998--1001, 1996. 
 
 
6. SHORT ABSTRACT  
The S-transform outperforms the STFT in that it 
has a better resolution in phase space giving a 
fundamentally more sound time frequency 
representation. Investigations of the representation 
error show that optimally adjusted S-transform can 
also outperform the Wigner-Ville transform when 
dealing with time-frequency representations of the 
signal.  The S- transform is also tested on 
nonstationary electric signals where it shows 
excellent tracking capability.  
These properties show that S-transform can be 
effectively used for analysis of electric signals, 
especially when dealing with multi-component time-
varying waveforms. 
 
 
 
 
 
 
 
 
prof. dr hab. inż. Tadeusz ŁOBOS 
dr inż. Zbigniew LEONOWICZ  
mgr inż. Krzysztof WOŹNIAK 
 
INSTYTUT PODSTAW ELEKTROTECHNIKI 
I ELEKTROTECHNOLOGII  
POLITECHNIKA WROCŁAWSKA 
pl. Grunwaldzki 13, 50-370 Wrocław 
tadeusz.lobos@pwr.wroc.pl 
leonowicz@ieee.org
krzysztof.w5.wozniak@pwr.wroc.pl 
 


Analysis of power quality disturbances using the S-Transform

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Analysis of power quality disturbances using the S-Transform

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