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)

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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


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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


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


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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


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


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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


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


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


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


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


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


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


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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.


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A Comparative Study of Wavelet Transform Technique & FFT ...

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