DOI : 10.17577/IJERTV3IS071210
- Open Access
- Total Downloads : 258
- Authors : Chayanika Baruah, Dr. Dipankar Chanda
- Paper ID : IJERTV3IS071210
- Volume & Issue : Volume 03, Issue 07 (July 2014)
- Published (First Online): 28-07-2014
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
A Comparative Study of Wavelet Transform Technique & FFT in the Estimation of Power System Harmonics and Interharmonics
Chayanika Baruap, Dr. Dipankar Chanda2
1Post-Graduate scholar, Electrical Engineering Department, Assam Engineering College, India
2Associate professor, Electrical Engineering Department, Assam Engineering College, India
AbstractThis 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
-
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 FFTs 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.
-
FAST FOURIER TRANSFORM THEORY
Let x0,x1….xN-1 be a vector of complex numbers. The DFT is defined by the equation-
1 1
2
Xk = xn
=0
, k=0,1.N-1 (1)
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
order of harmonics=
IV. Experiments with Simulated Waveform The first signal considered is given by
x(t)= 100cos (240t) + 50cos (2217t) + 40cos (2760t)
(6)
signal, i.e. Periodogram=
|F(signal )|2 (2)
N
+ 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
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.
-
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-
waveform described by (7). Sampling frequency is taken as
2000 Hz for both the methods.
-
FFT
FFT estimates the frequencies present in (7) as shown in the following figures.
200
150
100
1
50
CWT(a,b)= | |
1
( ) (3)
0
Where | | is the normalization factor, (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.
-
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]:
(t)=exp(jowt- 0.5t2) (4)
Where ow= 2fow ; fow is frequency of Morlet wavelet.The relationship between scale and frequency in CWT is given by:
Amplitude
-50
-100
-150
-200
100
90
80
70
amplitude
60
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
time in sec
Fig .1. First signal
f =
(5)
a 50
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 rom the following expression as [3]:
40
30
20
10
0
0 10 20 30 40 50 60
Frequency (Hz)
Fig .2. Fundamental frequency estimated by FFT
100
90
80
70
amplitude
60
50
40
30
20
1600
1400
1200
frequency
1000
800
600
400
200
20 40 60 80 100 120
scale
10
0
215 215.5 216 216.5 217 217.5 218 218.5 219 219.5 220
Frequency (Hz)
Fig .5. Scale Vs frequency curve for Morlet wavelet with sampling frequency of 2000 Hz
Analyzed Signal
100
90
80
70
amplitude
60
50
40
30
20
10
Fig .3. Interharmonic estimated by FFT as 216.7 Hz
200
0
-200
62
59
56
53
50
Scales a
47
44
41
38
35
32
29
26
23
20
200 400 600 800 1000 1200 1400 1600 1800 2000
Scalogram
Percentage of energy for each wavelet coefficient
200 400 600 800 1000 1200 1400 1600 1800 2000
Time (or Space) b
-3
x 10
8
7
6
5
4
3
2
1
0
750 755 760 765 770 775 780
Frequency (Hz)
Fig .6.Fundamental frequency estimated as 40.625 Hz scale 40 by WT
Analyzed Signal
Fig .4. Estimation of 19th Harmonic as 760 Hz by FFT
200
0
B. Wavelet Transform
The CWT is applied to the signal with Morlet as the mother wavelet and with the sampling frequency of 2000 Hz.
-200
9
Scales a
8.5
8
7.5
7
200 400 600 800 1000 1200 1400 1600 1800 2000
Scalogram
Percentage of energy for each wavelet coefficient
200 400 600 800 1000 1200 1400 1600 1800 2000
Time (or Space) b
Fig .7. Interharmonic estimated as 216.67 Hz at scale 7.5
0.025
0.02
0.015
0.01
0.005
200
0
-200
2.9
2.75
Scales a
2.6
2.45
2.3
2.15
Analyzed Signal
500 1000 1500 2000 2500 3000 3500 4000
Scalogram
Percentage of energy for each wavelet coefficient
Selected signal: 60 cycles. FFT window (in red): 2 cycles
400
200
0
-200
-400
-3
x 10
12
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Time (s)
1
10
Fundamental (60Hz) = 327.2 , THD= 63.02%
120
100
80
60
40
20
8
Mag (% of Fundamental)
6
4
2 2
500 1000 1500 2000 2500 3000 3500 4000
Time (or Space) b
Fig .8.19th Harmonic estimated as 755.81 Hz at scale 2.15
-
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.
-
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.
0
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency (Hz)
Fig .9.Simulated voltage signal and its FFT estimation
-
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.
5000
4500
4000
3500
frequency
3000
2500
2000
1500
1000
500
10 20 30 40 50 60 70 80 90 100
scale
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
0
-500
99
96
93
90
87
Scales a
84
81
78
75
72
69
66
63
60
Analyzed Signal
50 100 150 200 250 300 350 400 450 500
Scalogram
Percentage of energy for each wavelet coefficient
50 100 150 200 250 300 350 400 450 500
Time (or Space) b
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
500
0
-500
2.96
Scales a
2.64
2.32
2
Analyzed Signal
50 100 150 200 250 300 350 400 450 500
Scalogram
Percentage of energy for each wavelet coefficient
50 100 150 200 250 300 350 400 450 500
Time (or Space) b
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
Fig .11.The fundamental frequency estimated as 59.77 Hz Hz at scale 87
Fig .14. Frequency of 2241.4 Hz estimated at scale 2.32
500
0
-500
5.8
5.7
5.6
5.5
Scales a
5.4
5.3
5.2
5.1
5
4.9
4.8
Analyzed Signal
50 100 150 200 250 300 350 400 450 500
Scalogram
Percentage of energy for each wavelet coefficient
0.05
0.04
0.03
0.02
0.01
500
0
-500
1.9
1.89
1.88
1.87
1.86
1.85
Scales a
1.84
1.83
1.82
1.81
1.8
1.79
1.78
1.77
1.76
1.75
Analyzed Signal
50 100 150 200 250 300 350 400 450 500
Scalogram
Percentage of energy for each wavelet coefficient
50 100 150 200 250 300 350 400 450 500
Time (or Space) b
0.05
0.04
0.03
0.02
0.01
50 100 150 200 250 300 350 400 450 500
Time (or Space) b
Fig .15. Frequency of 2872.9 Hz estimated at scale 1.81
500
0
-500
2.5
2.48
2.46
2.44
Scales a
2.42
2.4
2.38
2.36
2.34
2.32
2.3
Fig .12. Frequency of 1040 Hz estimated at scale 5
Analyzed Signal
50 100 150 200 250 300 350 400 450 500
Scalogram
Percentage of energy for each wavelet coefficient
50 100 150 200 250 300 350 400 450 500
Time (or Space) b
0.07
0.06
0.05
0.04
0.03
0.02
0.01
500
0
-500
1.75
1.74
1.73
1.72
1.71
1.7
Scales a
1.69
1.68
1.67
1.66
1.65
1.64
1.63
1.62
1.61
1.6
Analyzed Signal
50 100 150 200 250 300 350 400 450 500
Scalogram
Percentage of energy for each wavelet coefficient
50 100 150 200 250 300 350 400 450 500
Time (or Space) b
0.07
0.06
0.05
0.04
0.03
0.02
0.01
ig .13. Frequency of 2139.9 Hz estimated at scale 2.44
Fig 16.Frequency of 3076.9 Hz estimated at scale 1.69
-
-
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
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.
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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
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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
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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
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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.