Using Prony Method

Spectral Estimation of Distorted Signals
Using Prony Method
Tadeusz Lobos, Jacek Rezmer
Ah.rfracf -- Mudern Frequency power converters
generate a widc spectrum of harmonic components.
L a r p converter systems can also generale noncharacteristic harmonics and jnterharrnonics. Standard
tools oC harmonic analysis based on the Fourier
transform assume that o n l j harmonics are present and
the periodicity intervals are fixed, while periodicity
intervals in the presence of interharmonics are variable
and very long. I n the case of frequency converters the
periods of the main component are unknown. The Prony
method as applied for signal analysis in power frequency
converter was tested in the paper. The method does not
show the disadvantages o f the traditional tools and algow
cxact estimation the frequencies of all or dominant
companents, cvcn when the periodicity intervals are
unknown. To investigate the appmprjatencss of the
method several experiments were performed. For
comparison, similar experiments were repeated using the
FlT. The comparison proved the superiority of the
Prony method.
The quality of voItage waveforms is nowadays an
issue- of the ubnost importance for power utilities,
electric energy consumers and also for the
manufactures of electric and eiectronic equipment.
The liberalization of European energy market will
strengthen the competition and 1s expected to drive
down the energy prices. This is reason for the
requirements concerning the power quality. The
voltage waveform is expected to be a pure sinusoldal
with a given frequency and amplitude. Modem
frequency power converters generate a wide spectmn
of harmonic components witch deteriorate the quality
of the delivered energy, increase the energy losses as
well as decrease the reliab~lityof a power system. In
some cases, large converters systems generate not only
characteristic harmonics typical for the ideal converter
operatjon, but also considerable amount of noncharactenstic harmonics and interharmonics which
may strongly deteriorate the quality of the power
supply voltage [I]. Interharmon~csare defined as noninteger harmonics of the main fundamental under
consideratjon. The estlmation of the components is
very important for control and protection tasks. The
design of harmonics filters relies on the measurement
of distorlions in both current and voltage waveforms.
There are many different approaches for measuring
harmonics, like FFT, application of adaptive filters,
artificial neural networks, SVD, higher-order spectra,
Authors arc wlth the Institute of Rlcctr~cal Engineering
Fundamentals, Wrnclaw Un~verrityof Tcchnolog, 50-370 Wmclaw
Poland, c-mall: [email protected] wrm.pI
etc [2,3,4,5]. Most of them operate adequately only in
the narrow range of frcquencres and a t moderate noise
levels. The linear methods of system spectrum
estlmation (Rlachan-Tukey), based on the Fourier
transform, suffer from the major problem of
resolution. Because of some invalid assumptions (zero
d a ~or repetitive data outside the durat~on of
observation) made in these methods, the est~rnated
spectrum can be a smeared version of the true
spectrum [6, 71.
These methods usually assume that only harmonics
are present and the periodicity intervals are fixed,
while periodicity intervals in the presence of
lnterhmonics are variable and very long [I]. It is
very important to devclop better tools of parameter
estimation of signal frequency camponen$.
In the case of power frequency converter the
periodicity intervals are unknown. Identification of
some power converter faults can be a difficult task,
especially in under-load- conditions. Different faults
cause specific additional distortions of voltage and
current waveforms. Detection o f add~tionalfrequency
components can be used for fault identification.
In this paper the frequencies of slgnal components
are estimated usmg the Prony model. Prony method is
a technique for modelling sampled data as a linear
combmation of exponentials. Al~hough i t is not a
spectral estimation technique, Prony method has a
close relationship to the least squares linear prediction
algorithms used for AR and ARMPL parameter
estimation. Prony method seeks to fit a deterministic
exponential model to the data in contrast to AR and
ARMA methods that seek to fit a random modcl 20 the
second-order data statistics.
TI. PRONY METHOD
Prony method is a technique for extracting sinusold
or exponential signals from time serles data, by
solving a set of linear equations.
Assuming the N complex data samples w[ll ..., x [ ~ ]
the investigated function can be approximated
by M exponential functions:
k=l
n = 1,2, ..., )V
Tp- sampling period,
where
AL - amplitude
an - damping factor,
ok- angular velocity
vn - ~nitialphase.
The discrete-time function
expressed in the form
may be
concisely
t=l
m=O
pp
where hk = A,e''& , z, = e
The estimation problem bases on the rn~nimizationof
the squared error over the data values:
-Pi
M
where
E [n] = x[n] - v[n]= x
'Theright-hand sulnmation in (10) may be recognize
as polynorn~aldefined by (81, evaluated a1 each of ils
mots zk yielding the zero result-
h ]-
hi z;-'
(4)
k-l
This turns out to be a difficult nonlinear problem. It
can be solved using the Prony method that utilizes
linear equation solutions.
If as many data samples are used as there are
exponential parameters, then an exact exponential fit
to the data may be made.
Consider the M-exponent discrete-time function:
The equatron can be solved for the polynvm~al
coeficients. In the second step the roots of the
polynomial defined by (8) can be calculated. The
damping factors and sinusoidal frequenc~esmay hc
determined from the roots z k
For pract~calsituations, the number of data points N
usually exceeds the minimum number needed to fit a
model of exponentials, i.e. N > 2M . In the
overdetermined data case, the linear equatlon ( 1 I )
must be modified to:
m=O
The estimation problem bases on the minimization of
the total squared error:
n=M+l
The M equations of ( 5 ) may be expressed in matrix
from as:
The matrix equation represents a set of linear
equations that can be solved for the unknown vector of
amplitudes.
Prony proposed to define the polynomial that has
the zt exponents as its roots:
When estimating the parameters of the srgnaI
components, in the case of heavy distorted waveforms,
we identify apart horn some sinusoidal components,
also some components with relativ great damping
factors. In reality they do not exist, but are caused by
noise and computation errors. To eliminate the
components we divide the estimated amplltudcs by
damping factors and choose the components 1~1ththe
biggest results of the division.
The investigations have been carried out using also
the modified Prony algorithm [&I, which maximises
the likelihood of the fitting problem. For different
numbers of the estimated exponential components M
(5...AJZ, changed by step of 5) the square error of the
approximation was cdculated, As optrmal was chosen
the number whrch assures the minimal error. From the
calculated components, the components with the
snlallest relation AJa has been neglected.
The polynomial may be represented as the sum:
III. INDUSTRIAL FREQUENCY CONVERTER
(8)
Shifiing the index on ( 5 ) from n to n-m a"d
muEtiplying by the parameter ~ [ myield:
]
(9)
The (9) can be modified into:
(ID)
The investigated dnve represents a typical
configuration of industrial drives, consisting
o f a three-phase asynchronous motor and a power
converter,
composed of a single-phase halfcontrolled bridge rectifier and a voltage source
converter [Fig. 1) ( 1 01.
Thecurrents waveforms at the converter input
(I-input), at the converter ot~tput(I-output)as well as
in the intermediate circuit (1-internled), during normal
conditions and under fault conditions (capacitor or
switch failure) were investigated usmg the Prony and
FFT methods. The main frequency of the wavrform at
thc converter input was 50 Hz, and at the converter
oulput 40 Hz.
Figs 2, 3 and 4 show the current waveforms In the
in~ermediatecircuit (I-interned). Under fault free nor-
Fig. 1 Industrial convener drive
0
-1
0.015
Time is]
0.01
0.005
0.02
0.025
L -
0
b
-
0.01
i - _ - L - -
0.02
0.03
Time Is3
0.04
0.05
lr
....
1.5
:
- - - - -
-
-
.
.
i
N - -
0
1000
2000
3000
4000
Frequency [Hz]
C
t
.
0
500
1000
Frequency [Hz]
1500
1.5
I
Frequency [Hz]
Frequency [Hz]
Fig. 2. Cumnt In thc ~ntcrmed~atc
clrcu~t(I-~nterrned),undcr
normal conditions (a), investiption results: Prony N=250,
M=12O @); FFT N=250 (c), f,=10 kHz
Fig. 3. Current ~nthc intcrmcdiatc circuit (I-~ntcrmcd),after
capacltar failure (a). investigation results Frony N=250,
M=105 (b), FFT N=250 (c), fp=5kHz
ma1 conditions (Fig. 2), apart from dc component,
the following frequencies have been detected, when
applying the sampling window equal to 25 rns.,: ca.
240, 720, 970, 1460, 1700, 2400, 3100 Hz.
Capacitor or switch failure changes significantly the
current wavefornl in the intermediate circuit (Figs.
3, 4, 5). Figs. 3 and 5 show the investigation results
for [he sampling window equaI to 50 ms. For
comparison, Fig. 4 shows the results for the
sampling window equal to 30 ms. In the first case
the results arc more exact. Additional frequency
compunent havc been detccled. In thc case of
capacitor failure (Fig. 3) 80 and 640 Hz and in the
case of switch failure (Fig. 5) 40 Hz. Detection of
the component indicates a fault.
1Y.CONCLUSIONS
It has been shown that a high-resolution spectrum
est~mationmethod, such as Prony method could be
effectively used for estimation of the frequencies of
signal components. The accuracy of the estimation
depends on the slgnal distortion, the sampling
window and on number of samples taken into the
esti~nationprocess.
ol0
b
-
I
0.02
0.01
7-:
0.03
Time [s]
.
--. -!
r-.-.-..----
I
Frequency [Hz]
Frequency [Hz]
21- ---
500
Time [s]
-..-,-----
1000
1500
Frequency [Hz]
----
2000
Frequency [Hz]
F i g 4. C w e n t in the lntcrmediate circuit (I-intcrmed), aftcr
capacitor fa~lurc(a), ~nvcstigationrcsults: Prony N=150,
M=?5 @); FFT N= 150 (c), f,=5 kHz
Fig 5 . Current In thc intermediate circuit [I-intcmcd), aftcr
switch failure [a)$ investigation rcsuIts: Prony N=5OO,
M=250 (b); F R N=500.(c), f,=10 kHz
The proposed method was investigated under
different conditions and found to be a variable and
efficient b o l for detection of all higher harmonlc
existing in a slgnal. It also makes it possible the
estimation of frequencies of interhmonics.
Detection of some non-characteristic signal
components can be used for identification of
frequency converter f a d t s.
[4] f.Loboq, T. Korina and H.J. Koglin, "Powcr System
Harrnonlcs Estrmation Using Linear Least Squares Mclhod
and SVD", IEE Pmc. on Gem Transrn, and Drstrib., vol.
V. REFERENCES
[ I ] R Carbone, D. Menniti, N. Sorrcntino, and A. Tcsta,
"Itcmtrvc H m o n l c s and Intcrhanwn~c Analysts in
Mult~convcncrIndustrial Systems", qhIn!. Conjerence on
Harmontcs and Qwalrp of Power, Afhens (Greece), t998,
pp. 4 3 2 4 3 8 .
and
K. Eichhom, "Rccurslvc Real-T!mc
Calculailon o f Basic Waveforms of Signals", IEb Proc
Pf.C,vol. 138, no 6, Novembcr 1991, pp. 469-470.
[31 Z.h n o w ~ c z ,T. Lobos, P. Ruczcwskl and 1. Siymanda
"Application o f Highcr.Odcr Spectra for S~gnal Proccs i n g In Elcckical Power Enginccting", Int Journal for
Computation and Mathmatics In Elcctronlc Englnccring
COMPEL., vol. 17,na 516,pp. 602-61 i , 1998.
[2] T.L o h s
1 48. NO.6, pp. 567-572Nov.2001
(51 M Meunier and F Brouage, "Fourier Transform, Wavelcts,
Prony Analysis: Too1 for Harmonics and Quality of
Power'' ", @ Ini Conference on Hurmonics and Qualrry l
u
j
Power, Athcns (Greocc), 1998, pp. 71-76,
[ 6 ] S L.Marplc, DigitolSpc!ruIAno!ys~~,
Pr~nbcc-Hall, Englcwood Cliffs. 19Rf [7] S M. K a y , Modern Spectral
EsrimaIion: T h w y and Applicarron. Englcwood Cliffs:
Prcnticc-Hall, 1988, pp 224-225
[ I ] S. M. Kay, Modern Specrral Esrimarion: Theory and
Rpplrcotrrm.. Englcwood Cliffs: Prcntrce-Hall, 1988, pp
224-22s.
[ 8 ) M R Osborne and C K Srnyth "A m d ~ t i c dPmny a l p
rrthm for cxponcntia1 function f tting. SIAMJ. 5ci. Stu~isr.
Compur, vol 16, pp 1 19-1 3R.
191 T.Lobs and J . R c m r , "Rcal Timc Dctcrmination o f
Pawcr System Frcqucncy", IEEE T r u n ~on In~trurnenfation
andMensuramenr, August 1997, vol. 46, no 4, pp. 877-88 1.
[lo] C Gcntrlc, N. Rotondalc, M . Franccschini and C. Tassonr,
"Fault opctation of ~nvcner-fcdtnduct~onmotor", In Pmc.
31" Wnrversrtres Power Engin~ermngConf Vol. 2, pp. 353360.