ZBIGNIEW LEONOWICZ , TADEUSZ ŁOBOS , KRZYSZTOF WO ŹNIAK 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-bad spectral methods are uful in stationary signal analysis but insufficient in the ca of numerous real-life problems when signal contents changes in time. Nonstationary signals are usually analyzed with STFT (Short-Time Fourier Transform). The method prented in this paper employs recently introduced S-transform which is an important development of STFT with improved properties.
Propod 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 ft S f h t g t f e d πττ∞
−−∞
=
−∫
t (1)
where ),(f g τis a window function. Window function is a modulated Gaussian function, expresd by:
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2/(222),(σπ
τt e
f f
cyril
g −=
(2)
where σis defined as:
1f
σ=
(3)
老鹰乐队好听的歌Finally, the general formula in (1) takes the form:
(
2
2
(()
/2)2(,)t f j ft
S f h t e d τπτ∞
−−−−∞
=
∫t t τ (4)
where f reprents frequency and t , τ reprent 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:
()dt d t w t h d W ∫∞
∞
−−=
),(),(ττ (5)
As previously stated, the S – Transform can be also prented as continuous wavelets transform using a specific mother wavelet multiplied by a pha coefficient:
2(,)(,)j f S f e W d d πττ= (6)
Where the mother wavelet ψ(t,d) is defined as:
22(/2)2(,)t f j ft
t f e
πψ−−=
(7)
where the coefficient d reprents the inver 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 inverly proportional to the local frequency. We can also adjust the resolution of the S-transform by introducing a modified coefficient σ:
k f
σ
=
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(8)
Then, the equation (4) takes the form:
(
2
22(()
/2)2(,)t f k j ft生活大爆炸第八季
S f h t e d τπτ∞
−−−−∞
=
∫
t (9)
where the coefficient k controls the resolution in such way that if k>1 the resolution in frequency increas and when k<1 the resolution in time increas.
The S-transform is a linear transform of the signal h (t). The additive noi can be modeled as follows:
h noisy (t)= h(t) + η(t). The S-transform of a noisy signal reprents a sum of transformed signal and transformed noi, as follows:
{}{}{)()()(t S t h S t h S 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 reprents a “local spectrum” for that point in time. Since (,)S f τ is complex valued, in practice, usually the module
(,)
S f τ is plotted and this
gives time-frequency S spectrum. The S-transform outperforms the STFT in that it has a better resolution in pha space (i.e. a more narrow time window for higher frequencies), giving a fundamentally more sound time frequency reprentation [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
fue- 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 fr
equency. 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 reprentation 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 reprentation.
Reference signal compod of one main harmonic of 50 Hz was generated independently for each analysis method with randomly variable pha.
Fig.1. Time-frequency reprentation of a 50% voltage sag and k=1.0 using the S-transform.
Figure 2 shows the reprentation of the reference signal using the S-transform, the Short-Time Fou
rier Transform and the Wigner-Ville Transform for a 50% voltage sag. For each group of waveforms (as in Figure 2), the error of reprentation was calculated, according to the formula:
256
1
1256n MSE ==(11)
where n- number of samples ,
50Rep ()Hz A n -
reprentation 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 .
50Ref ()Hz A n
2.1. Comparison of reprentation error.
The reprentation 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 ca of over-voltage with variable amplitude from 5% to 50% is shown. The S-transform shows overall smallest reprentation error, especially for the ca 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 rep
rentation –e 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).
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Fig. 3. Reprentation error for varying amplitude of
voltage sag.
Fig. 4. Reprentation error for varying amplitude of
over-voltage.
As en from Figures 2-4, the S-transform in comparison to smoothed Wigner-Ville Transform and Short-Time Fourier Transform allows more accurate reprentation of dynamic changes of signal spectral components. The S-transform, providing the optimal tting of the parameter k , has the smallest reprentation error.
t
f
Fig. 5. Frequency-dependent resolution of the S-transform.
2.2. Analysis of switching of capacitor banks
amplitude of voltage sag [%]
Fig. 6. a) One-pha 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 ction. Two capacitor banks (CB) were installed along the feeder. Several cas were simulated and both currents and voltages were recorded. Figure 6b shows the voltage waveform at the beginning of the feeder for the ca that the first CB 900 (kVAr) was switched on at 0.03 s and the cond CB 1200 (kVAr) at 0.09 s. Figure 7 shows three-dimensional time frequency reprentations obtained using a) S-transform b) STFT - of the complex waveform.
b)
|Short-Time Fourier Transform|- Spectrogram
Fig. 7. Comparison of S-transform reprentation and
放声大笑STFT spectrum of the waveform.
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 compod of multiple spectral components and noi (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 reprentation 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 reprentation 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 ud 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
reprentations - 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”,
牛的英文44(4), pp.998--1001, 1996.
6. SHORT ABSTRACT
The S-transform outperforms the STFT in that it has a better resolution in pha space giving a fundamentally more sound time frequency reprentation. Investigations of the reprentation error show that optimally adjusted S-transform can also outperform the Wigner-Ville transform when dealing with time-frequency reprentations of the signal. The S- transform is also tested on nonstationary electric signals where it shows excellent tracking capability.
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The properties show that S-transform can be effectively ud 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
krzysztof.w5.wozniak@pwr.wroc.pl
