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3RDINTERNATIONAL CONFERENCE ON MODERN POWERSYSTEMS MPS2010,18-21MAY 2010,CLUJ-NAPOCA,ROMANIA

1AbstractHarmonic and interharmonic analysis in power

systems are usually based on the nominal frequency of 50Hz or

60Hz, which is an approximation of the fundamental frequency.

Many spectrum analyzers consider this frequency constant and

therefore, their measurement is not very accurate. This paper

presents a new fundamental frequency estimation method based

on an improved zero crossing algorithm. The electric signal is

first filtered using the wavelet transform and the frequency isthen computed using a method that counts the number of zero

crossings in time using an adaptive window that detects the false

zero crossings. The paper continues with a virtual instrument

that computes the frequency from a sampled signal. Several

computer simulations test results are presented in the paper to

highlight the usefulness of this approach in estimating near

nominal power system frequencies.

Index TermsFundamental Frequency, Harmonics and

Interharmonics, Virtual Instrument, Wavelet Denoising, Zero

Crossings

I. INTRODUCTION

PECTRUM estimation of discretely sampled processes isusually based on procedures employing the Fast Fourier

Transform (FFT). This approach is computationally efficient

and produces reasonable results for a large class of signal

processes. However, there are several performance limitations

of the FFT.

The most prominent limitation is that of frequency resolution,

i.e. the ability to distinguish the spectral responses of two or

more signals. A second limitation is caused by data

windowing, which manifests as leakage in the spectral

domain. These performance limitations are particularly

troublesome when analyzing short data records, which occur

frequently in practice, because many measured processes arebrief.

Modern frequency converters generate a wide spectrum of

harmonic components. Large converter systems and arc

furnaces can also generate non-characteristic harmonics and

All authors are with the Electrical Power Systems Department, Technical

University of Cluj-Napoca, 15 C. Daicoviciu St., RO 400020, Cluj Napoca.

D. Gheorghe and R.B. Vasiliu are PhD students (e-mails:

[email protected], [email protected]), A.

Cziker is an associate professor (e-mail:[email protected]) and

M. Chindri is a full professor (e-mail: [email protected]).

The firs author D. Gheorghe received the License Degree in Electrical

Engineering from the Technical University of Cluj-Napoca, Cluj-Napoca,

Romania in 2009. Since 2009 he is working towards his Ph.D. at the Electrical

Power Systems Department of Technical University of Cluj-Napoca. His

research project is concentrated on studying the power quality parameters.

interharmonics, which strongly deteriorate the quality of the

power supply voltage. Periodicity intervals in the presence of

interharmonics can be very long. Parameter estimation of the

components is very important for control and protection tasks.

The design of harmonics filters relies on the measurement of

distortions in both current and voltage waveforms [1].

Digital control and protection of power systems require the

estimation of supply frequency and its variation in real-time.Variations in system frequency from its normal value indicate

the occurrence of a corrective action for its restoration. A

large number of numerical methods is available for frequency

estimation from the digitized samples of the system voltage.

Discrete Fourier transforms , Least error squares technique,

Kalman filtering , Recursive Newton-type algorithm , adaptive

notch filters etc. are known signal processing techniques used

for frequency measurements of power system signals [2].

The real-time performance of a fundamental frequency

estimation algorithm depends not only on its computational

efficiency but also on its ability to obtain accurate estimates

from short signal segments.

The algorithm proposed in this paper is based on a DiscreteWavelet Transform filter that attenuates the high frequency

harmonics which would create false zero crossings near the

real one, an adaptive window of search and an algorithm that

tracks the fundamental frequency and approximates it each

period.

II. WAVELET FILTERDESIGN

A. Discrete Wavelet Transform

The Wavelet Series is just a sampled version of Continuous

Wavelet Transform and its computation may consume

significant amount of time and resources, depending on the

resolution required. The Discrete Wavelet Transform (DWT),which is based on sub-band coding is found to yield a fast

computation of the Wavelet Transform. It is easy to

implement and reduces the computation time and resources

required.

Filters are one of the most widely used signal processing

functions. Wavelets can be realized by iteration of filters with

rescaling. The resolution of the signal, which is a measure of

the amount of detail information in the signal, is determined

by the filtering operations, and the scale is determined by

upsampling and downsampling (subsampling) operations[3].

The DWT is computed by successive lowpass and highpass

filtering of the discrete time-domain signal as shown in figure

1. This is called the Mallat algorithm or Mallat-treedecomposition. Its significance is in the manner it connects

Fundamental Frequency Estimation Using

Wavelet Denoising TechniquesD. Gheorghe, M. Chindri, A. Cziker and R. B. Vasiliu

S

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the continuous-time mutiresolution to discrete-time filters. In

the figure, the signal is denoted by the sequence x[n], where n

is an integer. The low pass filter is denoted by G0

while the

high pass filter is denoted by H0. At each level, the high pass

filter produces detail information, d[n], while the low pass

filter associated with scaling function produces coarseapproximations, a[n].

Fig. 1. Three-level wavelet decomposition tree.

At each decomposition level, the half band filters producesignals spanning only half the frequency band. This doubles

the frequency resolution as the uncertainty in frequency is

reduced by half. In accordance with Nyquists rule if the

original signal has a highest frequency of, which requires a

sampling frequency of2 radians, then it now has a highest

frequency of/2 radians. It can now be sampled at a

frequency of radians thus discarding half the samples with

no loss of information. This decimation by 2 halves the time

resolution as the entire signal is now represented by only half

the number of samples. Thus, while the half band low pass

filtering removes half of the frequencies and thus halves the

resolution, the decimation by 2 doubles the scale.

With this approach, the time resolution becomes arbitrarilygood at high frequencies, while the frequency resolution

becomes arbitrarily good at low frequencies.

The filtering and decimation process is continued until the

desired level is reached. The maximum number of levels

depends on the length of the signal. The DWT of the original

signal is then obtained by concatenating all the coefficients,

a[n] and d[n], starting from the last level of decomposition.

Fig. 2. Three-level wavelet reconstruction tree.

Figure 2 shows the reconstruction of the original signal

from the wavelet coefficients. Basically, the reconstruction is

the reverse process of decomposition. The approximation and

detail coefficients at every level are upsampled by two, passed

through the low pass and high pass synthesis filters and then

added. This process is continued through the same number of

levels as in the decomposition process to obtain the original

signal.

B. Analytical Approach

Wavelets are similar to continuous wavelets, but the scale

a and the location parameterb are measured in discrete

intervals. The Wavelet used must be Orthogonal translation on

itself and in dilation, the discrete wavelet is defined as:

=

ma

manbt

ma

tnm

0

00

0

1)(

, (1)

Discrete Wavelets involve a scaling function or father

wavelets this function must be orthogonal on translation and

orthogonal to other wavelets in its family. The scaling

function is defined as:

= ntm

m

tnm

222)(,

(2)

Using the inner product, the projection of a function onto

the wavelets is found. The same rule applies for the scaling

function. This is generated using the following formulas:

= dttmn

txnm

T )(,

)(,

(3)

The discrete wavelet transform.

= dttmn

txnm

S )(,

)(,

(4)

Discrete transform of scaling functions.

The wavelets and scaling functions are orthogonal and form

a suitable basis. The function can now be reconstructed as a

linear combination of wavelet and scaling function and theircorresponding coefficients:

= =

=

+=n

m

m n

mnnmmnnm tTtStx0

)()()( ,,,, (5)

The n indices represent the translation through time will the

m indices represent the Dilation. The actual wavelets behave

like a band pass filter and the scaling behaves like a low pass

filter.

= )()( ,, tTtmd mnnm (6)

= )()( ,, tStma mnnm (7)

Intuitively as m increases the wavelet is compressed and

fluctuates more rapidly, like a sine wave with increasing

frequency. The signal can be reconstructed in time with

different levels of detail, hence the reconstructed function

called the m detail. The higher values of m dictate higher

levels of detail and much of the high frequency components.

C. Case Study

The following signal contains a fundamental component of

50hzand two harmonics, a low frequency harmonic (rank 3)

and a high frequency one (rank 25).

)25sin(17

2

3sin23)sin(230)( ttttf

+

++= (8)

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This signal will be sampled at 512 points and it will be

stored in vector F:

)(2

02.012...09 iini

n tfFitin ==== (9)

0 0.005 0.01 0.015 0.02

200

0

200 Voltage Waveform

Time [s]

Magnitude[V]

Fig. 3. Voltage Waveform

1 2 5 8 11 14 17 20 23 26 29401020

50

80

110

140

170

200

230260

Rank

Magnitude[V]

Fig. 4. Magnitude spectrum of the original voltage waveform

The discrete wavelet transform of function f(t):

+

=

n

n

x

x

dtnxtWtfxnfDWT2

2

),()(),,( (10)

The mother wavelet chosen is Daubechies 14:

Fig. 5. Daubechies 14 wavelet and scaling function

Where S is the scaling function and W is the wavelet

function.

The transform coefficients are computed :

+

+==

12

12

,2...0

pn

ipDWT

ipAnp (11)

0 200 400100

0

100

200

300

Coefficient 2 + 250V

Coefficient 4 +100V

Coefficient 5

Time [s]

Magnitude[V]

Fig. 6. Discrete Wavelet Transform coefficients (2, 4 and 5)

The signal is now decomposed in 10 levels of coefficients,each representing a frequency band. The first two elements, 0

and 1, are called approximation coefficients. The remaining

elements are the detailcoefficients. The transform DWT

contains 8 levels of detail. The last 256 entries represent

information at the smallest scale, the preceding 128 entries

represent a scale twice as large, and so on.

The parameterscalein the wavelet analysis is similar to the

scale used in maps. As in the case of maps, high scales

correspond to a non-detailed global view (of the signal), and

low scales correspond to a detailed view. Similarly, in terms

of frequency, low frequencies (high scales) correspond to a

global information of a signal (that usually spans the entire

signal), whereas high frequencies (low scales) correspond to adetailed information of a hidden pattern in the signal (that

usually lasts a relatively short time).

The large scale coefficients will contain the fundamental

frequency and the low frequency harmonics and the high

frequency harmonics will be stored in the low scale

coefficients. Therefore, if the low scale coefficients are

truncated, the remaining coefficients will contain only low

frequencies. The signal is reconstructed by taking the Inverse

Discrete Wavelet Transform of the remaining coefficients.

0 0.005 0.01 0.015 0.02

200

0

200

Original Waveform

Filtered Waveform

Time [s]

Magnitude[V]

Fig. 7. Original voltage waveform and filtered voltage waveform

Magnitude spectrum of the reconstructed signal taken with

the Discrete Fourier Transform:

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1 2 5 8 11 14 17 20 23 26 29401020

5080

110

140

170

200

230

260

Rank

Mag

nitude[V]

Fig. 8. Magnitude spectrum of the filtered waveformIII. FREQUENCY ESTIMATION METHOD

The zero crossing detection algorithm is based on the sign

difference between two consecutive samples. When a group of

ti and ti+1 samples are found, a linear interpolation is

performed in order to find an approximate coordinate in timewhen the signal value is 0.

Zero crossing condition:

( ) ( )( ) 02 21 =+ +ii tsigntsign (12)Linear interpolation method :

Fig. 9. Linear interpolation

( ) ( )( ) ( )ii

iiiii

tftf

tttftt

=

+

+

1

10 (13)

After the first zero crossing, the axes are translated into that

location in time so the first non integer cycle is truncated. The

frequency estimation is determined by dividing the number of

zero crosses counted and the total duration of the integer

cycles.

timetotal

crossesofnumberf= (14)

Fig. 10. First zero crossing and the adaptive search window

The rectangular search window is dimensioned after the

first zero crossing when the width and the translation are

defined. This window will be then shifted along the whole

signal.

Multiple zero crossing can occur on a single period duration

leading to major measuring errors. Usually these problems arecaused by harmonics and the most difficult ones to detect are

the ones very close to the fundamental zero crossing point.

Luckily, if the false zero crossing are very close to the

real crossing points, the harmonic frequency that produces

them is also very high. Since the harmonic frequency is high,

it will not pass through the low pass wavelet filter. If the false

zero crossings are further than the real ones, they are produced

by low rank harmonics with high amplitudes. These effects

are solved by adjusting a rectangular window of search around

the expected fundamental frequency value. Furthermore, the

window of search will considerably reduce the number of

iterations per period since the samples outside the window are

not being read.

IV. VIRTUAL INSTRUMENT

A virtual instrument has been implemented with the help of

LabView. The current or voltage waveform can be either

inputted as an array of samples or as an analytical expression.

The instrument requires the width of the search window, the

total analysis time (if the waveform is an analytical

expression) and the sampling frequency.

The instrument returns a figure of the frequency

approximation in time, the frequency being approximated

every cycle, the frequency deviation from the real frequency

(analytical case), the total integer cycles time, the number of

detected false zero crossings and the total analysis duration.The input waveform and the wavelet filtered waveform are

also displayed.

The analytical test waveform contains a 50hz fundamental

frequency, a low frequency interharmonic (rank 3.3) and a

high frequency interharmonic (rank 25.7), each component

with a phase shift of 0.36 rad, 0.7 rad and 1.6 rad.

Fig. 11. Virtual instrument front panel. The wavelet filter is activated.

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The signal has been sampled over 10 seconds, which

corresponds to 500 cycles of 20ms each, with a sampling

frequency of 6400 samples/second.

The total numbers of samples was 64000 but the program

performed only 5163 iterations. This reduced calculation

complexity is given by the rectangular search window. Thewindow slides along the sampled signal and the virtual

instrument reads only the samples that are inside the window.

TABLE I

FREQUENCY ESTIMATION RESULTS

Input

Frequency

[Hz]

Estimated

Frequency

[Hz]

Error

[mHz]

Input

Frequency

[Hz]

Estimated

Frequency

[Hz]

Error

[mHz]

42.5 42.497 2.731 50.5 50.496 3.245

43 42.997 2.763 51 50.996 3.277

43.5 43.497 2.795 51.5 51.496 3.309

44 43.997 2.827 52 51.996 3.342

44.5 44.497 2.860 52.5 52.496 3.374

45 44.997 2.892 53 52.996 3.40645.5 45.497 2.924 53.5 53.496 3.438

46 45.497 2.956 54 53.996 3.470

46.5 46.497 2.988 54.5 54.496 3.502

47 46.996 3.020 55 54.996 3.534

47.5 47.496 3.052 55.5 55.496 3.566

48 47.996 3.084 56 55.996 3.599

48.5 48.496 3.117 56.5 56.496 3.631

49 48.996 3.149 57 56.996 3.663

49.5 49.496 3.181 57.5 57.496 3.695

50 49.996 3.213 58 57.996 3.727

Fig. 12. Measurement error on the 42.5Hz 57.5Hz domain

The figure above presents a set of measurements on the

interval 42.5Hz to 58Hz, comparing the analytically inputfrequency and the estimated frequency. The frequency

deviation is calculated as the difference between the estimated

frequency and the real frequency. As seen from the table, the

maximum error for the analyzed domain is 3.72mHz.

According to the 61000-4-30 standard, the instrument

complies with class A equipment where the maximum

allowable deviation for a 10 seconds analysis is 10mHz [5].

The total analysis duration for a 10 seconds waveform is

about 5 seconds so the instrument is able to perform a real

time frequency measurement.

Fig. 13. Virtual instrument front panel. Wavelet filter disabled.

The second test has been performed with the same input

dates but without the wavelet filter in order to point out the

difference in error.

The measurement error without the wavelet filter for a

50Hz input signal is -95.34mHz

V. CONCLUSIONS

In the last decades power quality estimation has become a

very important issue in power systems so the necessity of

accurate parameters measurement has grown in importance

too.

The algorithm presented and the results obtained aresatisfactory. The new signal processing techniques, like the

discrete wavelet transform offered important accuracy

improvements and solved most of the false zero crossings

instances.

The paper describes briefly the wavelet denoising process

which can be useful in many other applications in electrical

engineering.

The virtual instrument developed performed well in high

harmonic polluted conditions. The total analysis duration is

less than the total waveform duration so the instrument can be

used online.

VI. REFERENCES

[1]C. Mattavelli, Analysis of Interharmonics in DC Arc FurnaceInstallations, 8th Int. Conf. on Harmonics and Quality of Power,

Athens, Greece, vol.2, pp.1092-1099, 1998.

[2]P. K. Dash, "Frequency Estimation of Distorted Power SystemSignals Using Extended Complex Kalaman Filter", IEEE Transactions

on Power Delivery, Vol. 14, No. 3, July 1999

[3]C. Taswell, "The What, How, and Why of Wavelet ShrinkageDenoising," Computing In Science And Engineering, vol. 2, no. 3,

May/June 2000, p. 12-19.

[4]Golovanov, Carmen .a. Metode moderne de msurare nelectroenergetic. Bucharest: Editura Tehnic, 2001.

[5]IEC 61000-4-30 Ed.2: Electromagnetic compatibility (EMC) Part4-30: Testing and measurement techniques Power quality measurement

methods , April 2007

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