A Comparative Study of Wavelet Transform Technique & FFT ...
Transcript of A Comparative Study of Wavelet Transform Technique & FFT ...
A Comparative Study of Wavelet Transform
Technique & FFT in the Estimation of Power
System Harmonics and Interharmonics
Chayanika Baruah1, Dr. Dipankar Chanda
2 1Post-Graduate scholar, Electrical Engineering Department, Assam Engineering College, India 2Associate professor, Electrical Engineering Department, Assam Engineering College, India
Abstract—This paper presents a comparison of Wavelet
Transform technique and Fast Fourier Transform technique in the
estimation of power system harmonics and interharmonics.
Fourier analysis converts a signal in time domain to frequency
domain. Fast Fourier Transformation performs the same
conversion but with a faster rate. Now-a-days, Wavelet
Transformation is one of the most popular candidates of time-
frequency transformation. Because Wavelet Transformation can
provide time & frequency information simultaneously and it is
suitable for the analysis of non-stationary signal. To investigate
these methods, a number of studies have been performed using
simulated signals. The analysis of the voltage waveform of a 2 level
PWM converter sypplying an Induction motor has been
investigated employing these two methods with the same sampling
period.
Index Terms— CWT, DFT, FFT, Harmonics , Interharmonics
I. INTRODUCTION
An ideal power system is defined as the system where a perfect
sinusoidal voltage signal is seen at load-ends. In reality,
however, such idealism is hard to maintain [2]. It is because the
widespread applications of electronically controlled loads have
increased the harmonic distortion in power system voltage and
current waveforms. As power semiconductors are switched on
and off at different points on the voltage waveform, damped
high-frequency transients are generated. If the switching occurs
at the same points on each cycle, the transient becomes periodic
[1]. Harmonic frequencies in the power grid are a frequent cause
of power quality problems. Harmonics in power systems result
in increased heating in the equipment and conductors, misfiring
in variable speed drives, and torque pulsations in motors. So,
.
estimation and reduction of harmonics is very important. Many
algorithms have been proposed for the evaluation of harmonics.
The design of harmonic filters relies on the measurement of
harmonic distortion [1]. Harmonics state estimation (HSE)
techniques have been used since 1989 for harmonics analysis in
power systems. Many mathematical methods have been
developed over the years. It is proved that
by using only partial or selected measurement data, the
harmonic distortion of the actual power system can be obtained
effectively. In this paper, the performances of Fast Fourier
Transform (FFT) and Wavelet Transform (WT) technique have
been compared in estimating power system harmonics.
FFT is an algorithm for calculation of the Discrete Fourier
Transform (DFT). Usually, a Power spectra indicates the
frequencies containing the power of the signal. The frequencies
can be estimated by distributing the value of the power as a
function of frequency, where power is considered as the average
of the square of the signal. In the frequency domain, this is
equivalent to the square of FFT’s magnitude. It is suitable for
stationary signals only.
In WT method, Continuous Wavelet Transform(CWT) is
applied to the signal. The Morlet wavelet is applied as the
mother wavelet to estimate the frequencies of the signal. It is
suitable for the analysis of non-stationary signal.
The principles of these two methods are explained in
section II and III, the experimental results are given in section
IV and V and conclusion is given in the final section.
II. FAST FOURIER TRANSFORM THEORY
Let x0,x1……....xN-1 be a vector of complex numbers. The DFT is
defined by the equation-
Xk =1
𝑁 xn𝑒
−𝑗2𝑘𝜋𝑛
𝑁
𝑁−1
𝑛=0 , k=0,1……….N-1 (1)
International Journal of Engineering Research & Technology (IJERT)
IJERT
IJERT
ISSN: 2278-0181
www.ijert.org
Vol. 3 Issue 7, July - 2014
IJERTV3IS071210
(This work is licensed under a Creative Commons Attribution 4.0 International License.)
1377
Evaluation by this definition, directly requires O(N2) operations
as there are N outputs of Xk, and each output requires a sum of
N no. of terms. An FFT is a method to compute the same results
in O(N logN) operations. To estimate the frequencies, the
periodogram is obtained. The periodogram computes the power
spectra of the entire input
signal, i.e.
Periodogram= |F(signal )|2
N (2)
Where F(signal) is the fourier transform of the signal and N is
the normalization factor. The spectrum power is maximum at
the frequencies present in the signal.
III. WAVELET TRANSFORM THEORY
In this approach the signal is subjected Continuous Wavelet
Transform to estimate the harmonics and interharmonics.
A. Continuous Wavelet Transform
The CWT of a continuous, square-integrable function x(t) at
scale a >0 and translational value b € R is expressed by the
following integral-
CWT(a,b)= 1
|𝑎 | 𝑥 𝑡 𝛹∗∞
−∞(𝑡−𝑏
𝑎)𝑑𝑡 (3)
Where 1
| 𝑎 | is the normalization factor, ψ(t) is called mother
wavelet which is a continuous function both in time domain and
frequency domain. The main purpose of the mother wavelet is
to provide a source function to generate the daughter wavelets
which are simply the translated and scaled version of mother
wavelet.
B. Harmonics and Interharmonics Estimation
To estimate the harmonics and interharmonics, CWT is applied
to the signal. The Morlet wavelet is selected to be the mother
wavelet. It is defined in time domain as follows [3]:
ψ(t)=exp(j𝜔owt- 0.5t2) (4)
Where 𝜔ow= 2πfow ; fow is frequency of Morlet wavelet.The relationship between scale and frequency in CWT is given by:
fa=𝑓𝑜𝑤
𝑎 ∆ (5)
where a= scale, fa =frequency corresponding to the scale a, ∆= sampling period. The table showing the scales and their corresponding frequencies is first determined and then the scalograms are obtained for the signal at different scales for the estimation. The maximum energy points represent the scales corresponding to the frequencies present in the signals. Order of harmonics and interharmonics can be found from the following expression as [3]:
order of harmonics= 𝐻𝑎𝑟𝑚𝑜𝑛𝑖𝑐 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑠𝑦𝑠𝑡𝑒𝑚 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 (6)
IV. Experiments with Simulated Waveform
The first signal considered is given by
x(t)= 100cos (2π40t) + 50cos (2π217t) + 40cos (2π760t)
+ kse(t) (7)
where e(t) is a white Gaussian noise of zero mean and variance
equal to 1. The signal to noise ratio (SNR) is 10. To investigate
the methods, several experiments have been performed with the
waveform described by (7). Sampling frequency is taken as
2000 Hz for both the methods.
A. FFT
FFT estimates the frequencies present in (7) as shown in the
following figures.
Fig .1. First signal
Fig .2. Fundamental frequency estimated by FFT
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-200
-150
-100
-50
0
50
100
150
200
time in sec
Am
plitu
de
0 10 20 30 40 50 600
10
20
30
40
50
60
70
80
90
100
Frequency (Hz)
am
plitu
de
International Journal of Engineering Research & Technology (IJERT)
IJERT
IJERT
ISSN: 2278-0181
www.ijert.org
Vol. 3 Issue 7, July - 2014
IJERTV3IS071210
(This work is licensed under a Creative Commons Attribution 4.0 International License.)
1378
Fig .3. Interharmonic estimated by FFT as 216.7 Hz
Fig .4. Estimation of 19th Harmonic as 760 Hz by FFT
B. Wavelet Transform
The CWT is applied to the signal with Morlet as the mother
wavelet and with the sampling frequency of 2000 Hz.
Fig .5. Scale Vs frequency curve for Morlet wavelet with sampling frequency
of 2000 Hz
Fig .6.Fundamental frequency estimated as 40.625 Hz scale 40 by WT
Fig .7. Interharmonic estimated as 216.67 Hz at scale 7.5
215 215.5 216 216.5 217 217.5 218 218.5 219 219.5 2200
10
20
30
40
50
60
70
80
90
100
Frequency (Hz)
am
plitu
de
750 755 760 765 770 775 7800
10
20
30
40
50
60
70
80
90
100
Frequency (Hz)
am
plitu
de
20 40 60 80 100 120
200
400
600
800
1000
1200
1400
1600
scale
frequency
200 400 600 800 1000 1200 1400 1600 1800 2000-200
0
200Analyzed Signal
Scalogram
Percentage of energy for each wavelet coefficient
Time (or Space) b
Scale
s a
200 400 600 800 1000 1200 1400 1600 1800 2000202326293235384144475053565962
1
2
3
4
5
6
7
8x 10
-3
200 400 600 800 1000 1200 1400 1600 1800 2000-200
0
200Analyzed Signal
Scalogram
Percentage of energy for each wavelet coefficient
Time (or Space) b
Scale
s a
200 400 600 800 1000 1200 1400 1600 1800 2000 7
7.5
8
8.5
9
0.005
0.01
0.015
0.02
0.025
International Journal of Engineering Research & Technology (IJERT)
IJERT
IJERT
ISSN: 2278-0181
www.ijert.org
Vol. 3 Issue 7, July - 2014
IJERTV3IS071210
(This work is licensed under a Creative Commons Attribution 4.0 International License.)
1379
Fig .8.19th Harmonic estimated as 755.81 Hz at scale 2.15
V. SIMULATION OF A FREQUENCY
CONVERTER
A PWM converter with modulation frequency of 1080 Hz
supplying a 4 pole, 3 hp asynchronous motor(U=220 V) is
simulated in simulink. The simulated converter has a
modulation index of 0.92. The output voltage waveform of the
converter corrupted with noise having zero mean value & unity
variance is taken for analysis. Fig.9 shows the noise corrupted
voltage waveform at the converter output for the frequency 60
Hz and estimation of harmonics for this signal using FFT.
A. FFT
The simulated voltage signal and its FFT estimation are shown
in Fig.9. The signal is sampled with frequency 6400 Hz. The
first two cycles (in red) of the waveform are considered for this
estimation. FFT estimates the major frequencies as 60, 960,
1200, 1770, 2100, 2220, 2370, 2910, 3000, 3030 and 3120 Hz.
Fig .9.Simulated voltage signal and its FFT estimation
B. Wavelet Transformation
Continuous Wavelet Transform is applied to the voltage
signal with sampling frequency 6400 Hz. Again, Morlet wavelet
is considered as mother wavelet. The first 512 samples are taken
for this analysis.
Fig .10. Scale Vs Frequency curve for Morlet wavelet at sampling frequency
6400 Hz
Some of the major frequencies estimated by CWT are shown
below.
500 1000 1500 2000 2500 3000 3500 4000-200
0
200Analyzed Signal
Scalogram
Percentage of energy for each wavelet coefficient
Time (or Space) b
Scale
s a
500 1000 1500 2000 2500 3000 3500 4000 2
2.15
2.3
2.45
2.6
2.75
2.9
2
4
6
8
10
12x 10
-3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-400
-200
0
200
400Selected signal: 60 cycles. FFT window (in red): 2 cycles
Time (s)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000
20
40
60
80
100
120
Frequency (Hz)
Fundamental (60Hz) = 327.2 , THD= 63.02%
Mag (%
of F
undam
ental)
10 20 30 40 50 60 70 80 90 100
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
scale
frequency
International Journal of Engineering Research & Technology (IJERT)
IJERT
IJERT
ISSN: 2278-0181
www.ijert.org
Vol. 3 Issue 7, July - 2014
IJERTV3IS071210
(This work is licensed under a Creative Commons Attribution 4.0 International License.)
1380
Fig .11.The fundamental frequency estimated as 59.77 Hz Hz at scale 87
Fig .12. Frequency of 1040 Hz estimated at scale 5
ig .13. Frequency of 2139.9 Hz estimated at scale 2.44
Fig .14. Frequency of 2241.4 Hz estimated at scale 2.32
Fig .15. Frequency of 2872.9 Hz estimated at scale 1.81
Fig 16.Frequency of 3076.9 Hz estimated at scale 1.69
VI. CONCLUSION
The estimation of Harmonics and Interharmonics present
in a power system has been investigated using FFT and
Continuous Wavelet Transform for different test signals with
same sampling period. It is observed that Continuous Wavelet
50 100 150 200 250 300 350 400 450 500-500
0
500Analyzed Signal
Scalogram
Percentage of energy for each wavelet coefficient
Time (or Space) b
Scale
s a
50 100 150 200 250 300 350 400 450 50060
63
66
69
72
75
78
81
84
8790
93
96
99
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
50 100 150 200 250 300 350 400 450 500-500
0
500Analyzed Signal
Scalogram
Percentage of energy for each wavelet coefficient
Time (or Space) b
Scale
s a
50 100 150 200 250 300 350 400 450 5004.8
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
0.01
0.02
0.03
0.04
0.05
50 100 150 200 250 300 350 400 450 500-500
0
500Analyzed Signal
Scalogram
Percentage of energy for each wavelet coefficient
Time (or Space) b
Scale
s a
50 100 150 200 250 300 350 400 450 500 2.3
2.32
2.34
2.36
2.38
2.4
2.42
2.44
2.46
2.48
2.5
0.01
0.02
0.03
0.04
0.05
0.06
0.07
50 100 150 200 250 300 350 400 450 500-500
0
500Analyzed Signal
Scalogram
Percentage of energy for each wavelet coefficient
Time (or Space) b
Scale
s a
50 100 150 200 250 300 350 400 450 500 2
2.32
2.64
2.96
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
50 100 150 200 250 300 350 400 450 500-500
0
500Analyzed Signal
Scalogram
Percentage of energy for each wavelet coefficient
Time (or Space) b
Scale
s a
50 100 150 200 250 300 350 400 450 5001.751.761.771.781.79 1.81.811.821.831.841.851.861.871.881.89 1.9
0.01
0.02
0.03
0.04
0.05
50 100 150 200 250 300 350 400 450 500-500
0
500Analyzed Signal
Scalogram
Percentage of energy for each wavelet coefficient
Time (or Space) b
Scale
s a
50 100 150 200 250 300 350 400 450 500 1.61.611.621.631.641.651.661.671.681.69 1.71.711.721.731.741.75
0.01
0.02
0.03
0.04
0.05
0.06
0.07
International Journal of Engineering Research & Technology (IJERT)
IJERT
IJERT
ISSN: 2278-0181
www.ijert.org
Vol. 3 Issue 7, July - 2014
IJERTV3IS071210
(This work is licensed under a Creative Commons Attribution 4.0 International License.)
1381
transform is not as accurate as FFT in estimating frequencies in
case of stationary signal.
Distinct Estimation
of higher order
frequencies is difficult with Continuous Wavelet Transform
because the frequency decreases exponentially with scale in
case of CWT as shown in the figures (6) and (11). In general,
the techniques best suited for estimation of frequency of
stationary signal is based on FFT. However, wavelets, though
not specifically dedicated to this type of analysis, can recover
some of the spectral information. In case of non-stationary
signal, Wavelet analysis can estimate
the frequencies as well as
instants
of occurrence of the frequencies.
REFERENCES [1]` Adley A. Girgis , W. Bin Chang, Elham, B. Macram(1991, july 3). “A
DIGITAL RECURSIVE MEASUREMENT SCHEME FOR ON- LINE
TRACKING OF POWER HARMONICS”. IEEE Transactions on power delivery. Vol.6 No.3. pp-1153-1160. Available-www.ieeexplore.ieee.org
[2] U. Arumugam, N.M.Nor and M.F.Abdullah( 2011, November). “A Brief
Review on Advances of Harmonic State Estimation Techniqures in Power System.” International Journal of Information and Electronics
Engineering.Vol.1 No.3. page-217-222. Available-
www.ijiee.org/papers/34-1040.pdf [3] T. Keaochantranond and C. Boonseng(2002, October), “Harmonics and
Interharmonics Estimation Using Wavelet Transform”, IEEE.proc.-Trans.
Distrib., page-775-779. Available-www.ieeexplore.ieee.org [4] IEEE working Group on Power System Harmonics(1983, August).
“Power System Harmonics: An Overview”. IEEE Transactions on Power
Apparatus and System”. Vol.PAS-102, No.8. page.2455-2460. Available www.ieeexplore.ieee.org
[5] Fernando H. Magnago and Ali Abur(1998, October). “Fault Location
UsingWavelets”. IEEE Trans. on Power Delivery. Vol. 13, No. 4. page 1475-1480. Available- www.ieeexplore.ieee.org.
Wavelet Transform estimates the major frequencies as- 59.77, 1040,
2031.3, 2071.7, 2131.5 , 2241.4, 2872.9, 3076.9, 3322.7 Hz.
International Journal of Engineering Research & Technology (IJERT)
IJERT
IJERT
ISSN: 2278-0181
www.ijert.org
Vol. 3 Issue 7, July - 2014
IJERTV3IS071210
(This work is licensed under a Creative Commons Attribution 4.0 International License.)
1382