Estimacion Espectral Con Prony

8/15/2019 Estimacion Espectral Con Prony

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Spectral Estimation of

Distorted Signals

Using

Prony M ethod

Tadeusz

Lobos

Jacek Rezmer

Ah rfracf

M u d e r n

Frequency

power

converters

generate

a widc

spectrum

o f

harmonic

components.

L a r p

converter systems can

also

generale

non-

characteristic harmonics and

jnterharrnonics

Standard

tools

oC

harmonic analysis based on the

Fourier

transform

assume

that

o n l j harmonics

are

present

a nd

the periodicity intervals

are fixed,

while

periodicity

intervals in the

presence o f interharmonics are

variable

and very long. n the case

of

frequency

converters

the

periods

o f the

main

component

ar e 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 tradit ional tools and algow

cxact estimation the

frequencies

of

all or

dominant

companents,

cvcn

w he n

the periodicity intervals a re

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 i s

nowadays

a n

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 rel iab~li ty

of 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

non-

charactenstic harmonics and interharmonics

which

may

strongly deteriorate the quality of the power

supply voltage

[I].

In terharmon~cs

re

defined

as

non-

integer

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 m a n y

different approaches

for

measuring

harmonics,

l ike FFT,

application

of

adaptive filters,

artificial

neural networks, SVD,

higher-order

spectra

uthors arc wlth

the Institute of

Rlcctr~ca l Engineering

Fundamentals Wrnclaw

U n ~ v e r r i t y f Tcchnolog

50 370 mclaw

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

( R l a c h a n - T u k e y ) ,

based

on the

Fourier

transform, suffer from the major problem of

resolution.

Because of some invalid assum ptions (zero

d a ~

r repetitive data

outside the

d u r a t ~ o n o f

observation)

made

in these methods,

the est~rnated

spectrum

can be

a smeared v rsion

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

l n t e r h m o n i c s

are variable and very long [ I ]

It

is

very important to devclop better tools

of parameter

estimation of signal frequency campon en .

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~ t ional requency

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 t is not a

spectral

estimation

technique, Prony

method

has a

close

relationship to

the least squares linear

prediction

algorithms

used for

AR and

RMPL

parameter

estimation. Prony method

seeks to fit a deterministic

exponential model to

the data

in contrast

to

AR and

ARMA methods that seek

to

f i t a random

modcl

2 t he

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 l l

...

x [ ~ ]

the

investigated

function

can be

approximated

by

M

exponential

functions:

k l

where

=

1 2

... V

Tp

ampling

period

L- amplitude

n

damping

factor,

ok ngular velocity

vn

~ni t ia l hase .



The discrete-time function

may be

concisely

Theright-hand sulnmation

in (10)

may

be

recognize

expressed in the fo rm as polynorn~aldefined by

81,

evaluated a1 e a c h

of

ils

mots

zk yielding the zero result-

t=l

where hk

= A,e''& , z = e

P i

pp

The estimation problem bases on the rn~nimizationo f

the squared

error

over

the

data

values:

where E [n]= x[n] n]=

x 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

s

there

re

exponential parameters

then

an exact

exponential fit

to the data may bemade.

Consider the M-exponent discrete-time function:

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:

The

polynomial m y be represented as the

sum:

Shifiing

the index o n 5 ) from

n

t

n m

muEtiplying

by

the

parameter ~ [ m ]ield:

The 9) can be modified into:

m=O

The equatron can be solved for

the polynvm~al

coef ic ients .

In the second

step

the roots

of

the

polynomial

defined

b y (8) can be calculated. The

damping factors and sinusoidal frequenc~esmay hc

determined

from the

roots z k

For pract~cal ituations, the number of data points

N

usually

exceeds

the

minimum

number

needed

to fit a

model

of

exponentials,

i.e.

> 2 M . In the

overdetermined data

case

the linear equatlon (

I )

must be

modified

to:

m=O

The estimation problem

bases

o n

the

minimization of

the

total

squared

error:

n= M l

When

estimating

the parameters

of

the srgnaI

components,

in the

case

of heavy distorted waveforms,

we identify apart horn some sinusoidal components,

a lso

some

components with

relativ

great damping

factors.

In

reality

they do not

exist

but a re caused by

noise and computation errors. T o

eliminate

the

components we

divide the estimated amplltudcs by

damping factors and choose the components 1~1th he

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 o 5)

the

square error of the

approximation was cdculated,

s

optrmal was chosen

the

number whrch ssures the minimal error. From

the

calculated

components the components

with the

snlallest relation AJa has been neglected.

III.

INDUSTRIAL

FREQUENCY CONVERTER

8)

The

investigated

dnve

represents a typical

configuration of industrial drives,

consisting

ofa

three-phase synchronous

motor and

a power

converter,

composed

of

a

single-phase

half-

a d

controlled bridge

rectifier

and a

voltage

source

converter

[Fig. 1) 1

01.

9)

Thecurrents waveforms

at the

converter input

(I-input), at the

converter

ot~tput I-output)as well as

in

the

interm ediate circuit (1-internled), during normal

conditions

and

under

faul t conditions (capacitor or

switch failure)

were

investigated

usmg

the

Prony and

FFT methods. The main frequency of the wavrfo rm at

I D )

t h c

converter in p u t

was

50 Hz and at the

converter

oulput 40

Hz.

Figs

2, 3 and 4 show t he current

waveforms

n the

in~e rm edia te ircuit (I-interned). Under fault

free

nor-



Fig.

1 Industrial convenerdrive

0 0.005 0.01 0.015

0.02

0.025

Time

is

-1 i _ L

0.01 0 02

0.03

0.04 0.05

Time Is3

b

lr

1.5

. . . .

:

- - - - - -

.

N

t

0 500

1000 1500

Frequency

[Hz]

1000 2000 3 4000

Frequency [Hz]

1.5

Frequency [Hz] Frequency [Hz]

C

Fig. 2. Cumnt n

thc

~ntcrmed~atc

lrcu~tI-~nterrned),

undcr

Fig 3. Current

~

hc intcrmcdiatc circuit I-~ntcrmcd), fter

normal

conditions

a),

investiption results: Prony

N=250,

capacltar

failure a). investigation r sults

Frony N=250,

M=12O @);

FFT

N=250 c),

f,=10 kHz

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 t 25 rns.,:

ca

240

720

970

1460

1700

2400 31

Hz

Capacitor

or

swi t ch fai lure 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

a r c more exact. Additional

f requency

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 O N CL U SIO N S

It has been shown

that a

high-resolution spectrum

e s t~ m a t ionmethod, 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

es t i~nat ion

rocess.



ol-

0.01 0.02 0.03

Time [s]

b

7

. .

--

I

Frequency [Hz]

500

1000 1500 2000

Frequency

[Hz]

F i g 4 . C w e n t in

the

lntcrmediate circuit (I-intcrmed),

ftcr

capacitor fa~lurc

a),

~nvcstigation

csults:Prony N=150

M=?5

@);

FFT N =

150 (c) ,

f,=5

kHz

The

proposed method

was

investigated under

different

conditions nd 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 i n t e r h m o n i c s .

Detection

of

some

non-characteristic signal

components

can be

used

for identification

of

frequency converter f a d

s.

V.

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Carbone, D.

Menniti,

N.

Sorrcntino, and A .

Tcsta,

"Itcmtrvc Hm o nl cs and Intcrhanwn~c Analysts in

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ndustrial

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Conjerence

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Harmontcs

and Qwalrp of

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pp. 4 3 2 4 3 8 .

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Z.

n o w ~ c z ,T. Lobos, P.Ruczcwskl and 1.

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Application of Highcr.Odcr Spectra for

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516 pp.

602-61 i 1998.

Time

s]

r . . .. -..- -----

Frequency

[Hz]

21

---

----

Frequency [Hz]

Fig 5 .

Current

In thc intermediate circuit

[I-intcmcd),

aftcr

switch failure

[a)$

investigation rcsuIts: Prony

N=5OO,

M=250

(b);

F R

=500.

c),

f =10 kHz

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00

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Marplc,

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En-

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odern

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1

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Estimacion Espectral Con Prony