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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:
Daniel.GHEORGHE@eps.utcluj.ro, Razvan.VASILIU@eps.utcluj.ro), A.
Cziker is an associate professor (e-mail:Andrei.CZIKER@eps.utcluj.ro) and
M. Chindri is a full professor (e-mail: Mircea.CHINDRIS@eps.utcluj.ro).
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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