A New Numeric Technique of Accurate Frequency and Harmonics

Estimation for Power System Protection and Power Quality Applications

L.ASNIN

V.BACKMUTSKY

Holon Academic Institute of Technology

52, Golomb St.,58 102 Holon, Israel

Abstract.

This paper presents a new method of a

measuring a magnitude I frequency spectrum of

periodical signals, based on Discrete Fourier

Transform (DFT). The main purpose of this method

is to reduce the leakage errors under condition of

desynchronization between the signal and the

generator of samples. Digital modeling of the

suggested method shows that accuracy of the estimation of magnitudes of the signal frequency

components increases by factor 10-100 if one

compares this method with well known others based

on the interpolation of samples. This method can be

used especially in power system dynamics

investigation (power quality, relay protection, UPS

tuning, etc).

1. Introduction. In this paper we consider the case of digital

spectrum analysis of the periodic multifrequency

signal with main frequency f, and sampling

frequency f, . The spectrum of the signal consists

of k harmonics with frequenciesfoA = kf, (h=

0,l. ..., M) that are placed within [0, f , ] , where f, - Nyquist frequency.

It is well known that a desynchronization

between signal and generator of samples is an

important cause of errors in signal's spectrum

analysis by means DFT. Those errors appear due to

the frequencies of harmonics are not equal to zeros

of frequency response of the FIR filter with

rectangular window, which is used in algorithm

DFT. There are two effective methods for reducing

these errors: quasi - synchronous interpolation [l], [2], [4] and an optimization of the frequency

response of the window [;I. The first method provides an ideal synchronization, but errors of the

spectrum measurement are caused by the

interpolation and grow if the ratio f, I fs increases. The optimization in the second method

results from connection 4 FIR filters with

rectangular windows. This connection provides low

value of its frequency response within frequency

intervals near f O k , but this method needs to use

number of samples greater by factor 4 as compared

with usual DFT. Therefore. improvement of this

approach, achieved by the suzgested technique. is

very desired.

2. Principles of the suggested algorithm.

The algorithm of measuring ma-miNdes of

harmonic components of periodical signal by means

of DFT uses following operations (Fig. 1):

- -- - - - -,.. - - __ - - - - h'rrr. :

1

.: i: ' 'i I . .

w'tk,~. I : L -2

. . L - - - - _ - - - - - - _ --.(

Fig.1 Block scheme for DFT algorithm realization. O;

A, =Js: +c, 2 ;

(4)

where u(n)- samples of the signal, k- harmonic's

number, N- number of samples in the period T, of

signal, i.e. integer part of ratio N , = f, i fo . It is obvious that the spectrum of signals (1) and (2)

consists of frequency components

O,f , , ,2f , , ,..., f,. Values S, , cI are two samples of output signals of FIR filters F with rectangular

windows (the width of the window is N samples),

which aim is to suppress frequency components

f n .2 f ",..., f N of the signals sI andc, .

In the case of synchronization, N,=N ,

frequency response of F, K(n = 0 for

f = f,,2 f,, ..., f N and thus only the frequency components w i t h p 0 pass through the filters. In the

case of desynchronization the mentioned above

undesirable components pass through the F. This is

the main cause of measurement errors in the case of

desynchronization.

The main idea of the suggested method is to use

FIR filters with K m =0, forf = f,,2f ,..., f,, in spite of desynchronization. Such filter consists of

N simple FIR filters of order 2 ( N is an integer part of the ratio N/2). It is well known, that

frequency response of FIR filter with window [l,b,

. I ] has zero at frequency f, , if b, COS(^& / f,) .

-

The desired FIR filter constitutes a connection of

h' FIR filters of order 2 with different -

b, = 2 ~ 0 ~ ( 2 n k f , / f , ) , k 1 , 2 ,..., N . Samplesof

the window of this filter can be easily obtained by

preliminary calculation of the coefficients of the

polynomial

2 N

P ( z ) = Cc,z- ' = i=n

N

= n , z - > + 2 c o s ( 2 ~ " i f , ) z - ' + l ) . ( 5 ) ,=I

Since K@)=l it is necessary to multiply cj by the

coefficient k , = (Cc, )" . Thus samples of the window of Fare

& = k,c , . (6) Examples of the window and the tiequency

response of F. if N = 13, N, =27.3 are shown in

Fig 2,3.

,=n

The suggested algorithm consists of following

steps:

- measurement of f, ;

- calculation of the samples of the window of F,

wi according to (5),(6); - calculation of usx (n ) and uck (n) according to

(1)Q); - calculation of the values :

W ) 0

- calculation of the magnitudes A, according to

(4).

0.05 I

s o 2 0.025

0 5 10 15 20 25

3. Simulation results.

A simulation program in MATLAB has been

used to estimate properties of the suggested

algorithm. The program generates multifrequency

periodical signal with fn=50 Hz. M=8 harmonics.

Magnitude of the first harmonic A , =1,

magnitudes A, of high harmonics are randomly

distributed ,but

J A i + . . . + A i , THD = = const

AI

for each realization of the signal . Some realizations of this signal are shown in Fig 4.

1 SIGNAL 1 1

"SIGNAL 2

Fig.4 Realizations of multifrequency periodical

signal, THD=20%.

The generalized results of the measurement are

shown in Table 1. An accuracy of the measurement

is estimated as a maximum of values

A, = [ A , - A , l / A , % , where A i is an

estimation value of A , . Two methods with similar

length of the windows are compared: the suggested

algorithm (upper line in Table I ) and the

interpolation method (lower line). For the estimation

of f, in both cases IZC method [4] was used. A frequency deviation was chosen from 49.5 Hz to

50.5 Hz ( Af = f0.5 Hz).

-

An accuracy of the measurement may be

increased by factor IO, if we use connection of 2

FIR filters with K(t) =0 at frequencies f, = yo or FIR filter F with quasi-triangular window, but

length of this window is equal to 2 N .

It is necessary to mark, that counting samples q. of filter's window in accordance with (5) in

MATLAB leads to inadmissible enom if N >20. Considerably hener result can he obtained by using (5).

if variable k changes randomly in interval ( 1, N ).

An accuracy of the frequency measurement by

IZC with its further improvements was considered in

our last publication [SI. One can show, that under

quite acceptable conditions this accuracy is not less

than 0.005 Hz. It is enough for all accurate harmonics estimation by QSl (Quasi-synchronous Sample

Interpolation) or resampling and for suggested

method, but as it is shown above, the harmonic

estimation by suggested method is more accurate, at

least for the static case, i.e. slowly changed

frequency.

Table.1

AJ =o.l

1 f s ( H z ) I 2000 I 3000 1 4000 1 5000 I 0.0131 0.0013 0.0036 0.0018 0.4900 0.2280 0.1651 0.0912

I I I I I I I q- =o.2 I ;:;O;; I 0.0103 I 0.0019 I 0.0018 I 0.3350 0.2707 0.4040

I 4f =o,3 1 0.0124 1 0.0160 I 0.0044 1 0.0013 I 1.5231 0.7015 0.2070 0.2440

I I I 1 I I I ~f =o,4 1 0.0128 0.0291 0.0032 0.0043 1.0013 I 1.2609 1 0.1964 1 0.0765 I

I 1 0.0138 I 0.OlOl 1 0.0031 I 0.0021 1 f =05 1.2270 0.9953 0.0975 0,1006

4. Conclusion.

Obtained results show that above mentioned

technique is much more accurate than regular

sample interpolation (resampling). Influence of

dynamic errors of frequency change prediction will

be investigated separately because of its separate

role in power system dynamics investigation.

REFERENCES

[ I ] T.Grandke. Interpolation algorithms for

Discrete Fourier Transforms of weighted signals.

IEEE Trans. on Instrum. and Meas.,vol.36, pp. 350- 355, June 1983.

[2] Jiangtao Xi, Joe F. Chicharo, A new algorithm

for improving the accuracy of periodic signal

analysis.IEEE Trans. on Instrum. and Meas., vo1.45,

N.4, August 1996.

[3] X.Day. R. Ciretsch, Quasi-synchronous sampling

algorithm and its applications. IEEE Trans. on

Instrum. and Meas., vo1.43, N.2, April, 1994.

[4] Backmutsky V.,Zmudikov V. A DSP and data

acquisition method for application in power systems

with variable frequency. Proc.of Boston Conf. on

Signal Processing Application and Technoloe. USA.

1994,pp.415-419.

[ 5 ] L.Asnin, V.Backmutsky, M.Gankin, J.Blashka and

M.Sedlachek.DSP methods for dynamic estimation

of frequency and magnitude parameters in power

systems transients,2001 IEEE Porto Power Tech

Conference, September, Porto, Portugal.

Biography Dr L.Asnin .MSc. from University of Electrical Communication , Tashkent, USSR,1960, PhD from Polytechnical 1nstiNte in Samara, USSR, 1972. From1968 to 1987 - senior lecturer at University of Elecmcal Communication in Samara, USSR, from 1998 up to now - research worker in laboratory of microprocessors at Holon Academic Institute of Technology, Electricity and Electronic Dpt., Israel. The main professional interests and publications are in the field of DSP (Digital Signal Processing), especially for Power System Applications, and in the field of Digital Communication Systems.

Dr. V.Backmutsky, IEEE member #01003474, MSc from Lvov University, USSR,I957, PhD from Lvov Polyiechnical InstiNte,USSR,1968. From 1988 up to now

~ senior IecNrer at Holon Academic Institute of Technology, Electricity and Electronic Dpt.. Israel. The main professional interests and publications are in the field of DSP (Digital Signal Processing), especially for Power System Applications.