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IEEE: Transactions on Power D elivery, Vol. 12, No. 1, January 1997 157
Frequency Estimation by Demodulation of Two Complex Signals
Magnus Akke
Sydkraft ABS-205 09 Malmo
Sweden
Abstract
This paper presents a m ethod for frequen cy estimation in
power system by demodulation of two complex signals.
In power system analysis, the @transfo rm is used toconvert three phase quantities to a complex quantity
where the real part is the in-phase component and theimaginary part is the quadrature component. This
complex signal is demodulated with a known complex
phasor rotating in opposite direction to the input. The
advantage of this method is that the demodulation does
not introduce a double frequency component. For signals
with high signal to noise ratio, the filtering demand for
the double frequency com ponent can often limit the speed
of Ihe frequency estimator. Hence, the method can
improve fast frequency estimation of signals with goodnoise properties. The method looses its benefits for noisy
signals, where the filter design is governed by the demand
to filter harmon ics and white noise. The method has been
previously published, but not explored to its potential.
The paper presents four examples to illustrate the
strengths and weaknesses of the method.
1. Introduction
Fast and accurate frequency estimation in presence ofnoise is a challenging problem that has attracted a lot ofattention. Many solutions have been suggested, both in
signal processing and in power system publications.
Che 3per com putational power has boosted the use of
mor12 refined signal processing methods. A new research
area, known as time-frequency signal analysis, has
emerged and is discussed in [ 7 ] , [8]. This area deals with
instantaneous frequency estimation and is, to some extent,also applicable to power system frequency estimation.
96 Shrl379-8 PWRD A paper recommended and approved by the IEEEPower System Relaying Committee of the IEEE Power Engineering
Society for presentation at the 1996 IEEWPES Summer Meeting, July 28- August I 996, in Denver , Colorado. Manuscript submitted December
28, 1 395; made available for printing May 21, 1996.
The typical use of frequency estimation in power systems
is for protection scheme against loss of synchronism [lo],
under-frequency relaying and for power system
stabilisation [5]. Frequency estimation in power system
has evolved along several paths. Some are
Chang e of angle for phasor measurem ents [11
Kalman filters [ 2 ]
Zero crossing and m odification thereof [31
Demo dulation with fixed frequency [ 3 ] , [4]
Demo dulation with varying frequency. A feedback loop
controls the frequency, i.e., a phase locked loop (PLL).This has been used in [5].
Estimation using identification theory, such as recursive
least squares, least mean squares, see [6]
Numerical optimisation. A Newton type method has
been used in [9].
The applications can be categorised based on their time
demand , that is,
critical real time applications, such as relay protection;
on-line data monitoring in control room;
off-line data analysis of co mputer recordings.
The classification is useful since the different time
demands put restrictions on what type of frequency
estimator and filter technique that can be u sed. In off-line
data analysis we have access to the full time series and the
estimation and filtering can be improved by using non-
causal forward-backward filtering. The term causal is
explained in [111, but basically it mean s that on ly sam ples
at and before time k can be used to calculate the output at
time k. For example, the relation y(k)=u(k)-u(k- 1) is
causal, whereas y(k)=u(k +l)-u(k-1) is non-causal.
To compare different methods we need a test criterionthat reflects relevant demands. Three such demands are:
speed of convergen ce; accuracy; and no ise rejection. Thekey problem is to find a method that improves all thesedemands and not just compromise one demand foranother.
0885-8977/97/$10.00 996 IEEE
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To m ake the analysis straightforward we first assume that
the input voltages VI , v2, v3 do not have any negative
sequence voltage nor any noise. We then have
4)(k) =A[cos(w,t, + I+ jsin(w,t, + I]
A ej(wltk+@)
where A is the phase to phase RMS-value.
The demodulation is done with a complex signal Z , thatrotates in the opposite direction, i.e., negative sequence,
compared to the input signal V.
Z(k) = e joOtk
V(k) = A&(mltk+ ) (k) = Aej[ ol-wO)tk+~l
Figure 2. New demodulation of two complex signals.
The signal Z with a known frequency is
Z(k) = cos(-o, tk )+ s in(-motk) =e- Joo tk. 5 )
The resulting signal, Y, after the m ultiplication becomes
Y(k) = V(k) . Z(k) = A @I tk+@)e-jwOk
A ej[(ol-wO)tk+@]
= A(cos[(w, -w,>t ,+$]+ jsin[(w, -w ,) t ,+ ~] ). 6 )
Note that the demodulation does not create the double
frequency component. Hence, the d emodulation does not
add demands to filter away the double frequencycomponent. However, there still might be a need to filter
due to noise. The frequency estimation is done as in [3].
To find the phase difference, we define the complex
variable U as
U(k) = Y (k ) .Y( k 7 7)
where * stands for conjugate. We separate Y in real and
imaginary part and find that
U(k ) = Re[Y (k)] Re[Y (k )]+ m[ Y (k)] Im[ Y(k )]
+ { Im[Y( k)] Re[Y ( k )] Re[Y (k)] Im[Y( k )]}. (8)
The phase difference y between two consecutive samples
is calculated from the real and imaginary part of U.
(9)
The dev iation in a ngular frequency is estimated from1
AtAm( k .5) = -[y( k) Y(k )] = f, [y(k) (k )] (10)
The time index k-0.5 is used to point out that the estimateis best in the middle of the time interval [k , k] . For real-
time application we are restricted to causal relations and
get
1t [ y ( k ) ( k111= fs .[y k) ( k 11 (1 1)
The unknown frequency for the signal V is estimated as
where fo is the nom inal, and f, is the sampling frequency.
Typical use of this demodulation is for frequency
estimation by demodulation with a fixed frequency or by
a PLL where the demodulation frequency is c ontrolled byfeedback.
3. Demodulation Examples
The purpose of this section is to give examples to
illustrate the strengths and weakness of the proposed
method. The used examples are:
1. Step in frequency under ideal noise conditions; no
noise, no negative sequence, no additional filters are used.
2. Test signal from [3] with low noise; no negative
sequence. No additional filters.
3. Test signal from [3], with medium white noise,
3:rd harmonics, 5:th harmonics and negative sequence.
Additional filters that a re c ausal.
4. Same test signal as 3) but filters that are non-causal.
The program M atlab has been used for calculations. The
code for Example 3 and 4 are given in Appendix A.
Example 1: Step in frequency under ideal conditions
The test signals are three noise-free symmetrical phase
voltages. There is a step change in frequency from 50 to
51 Hz at t=100 ms. T his signal is unrealistic since power
frequency can not chan ge instantaneously. The test signal
is only chosen to illustrate that, with a perfect symmetry
and without any noise, the demodulation gives an exact
frequency estimate within one sam pling interval.
Figures 3-5 illustrate the new demodulation. Note thedifferent time scales.
Real p n af vmaglnsry pa Dl v 2,
I I I01 0.15 2 25
Time ( 1$ 1 015 02 0 2 5
Time E)
R e a l pa,, O l Zpa 01z
Figure 3. Real and imaginary part of the complex input signal V
and the demodulationsignal Z.
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Io z 0 4 5 8 1
T mB a)“a 2 01 6 8 1
Time Is1
Figure 4. Real and imagina ry part of the demodulated signal Y as
well as amplitude and phase of the same sign al.
Estimated Frequency
Freq
Figure 5. Frequency estimate from the demodulated signal Y. No
noise present, nor any nega tive sequence voltage.
From this exam ple we see that under ideal conditions, we
can nearly make an arbitrary fast frequency estimator.
The only limitation is the analogue anti-aliasing filters.
Example 2: Test signal [3] and very low noise
In this example we use the test signal from reference [3 ]
The three phase voltages a re
v, (k) = &A rms sin($, (k)) + N, (m,6 i = 1,2,3
where the angles are calculated from
@ i ( k ) = + , ( k - l ) + o ( k ) A t ; for k 2 l
with the initial values
The frequency is time varying,
o ( k ) = 2 . ~ [ 5 0 sin(2 . .n.l. k )+ 0 .5 , s in (2 .7~6 . k)].
The notation N(m,o) is used for normally distributed
white noise. In this example the standard deviation is
0=0.0001 and the mean m=O. The signal used has anRM S-value of 1p.u. giving a signal to no ise ratio of
SNR = 20 log(- ) = 8 0 d B .0.0001
True freq.= Solid; Estimate=DashDot; SNR=80 dB51.5
495’ 0:1 0’2 0’3 0:4 0’5
Time s)
Figure 6. True and estimated frequency for SN RS O dB and nofiltering of estimate.
This example illustrates that the algorithm works well for
SNR above 80 dB. For lower signal to noise ratios the
frequency estimate needs to be filtered. The two
following examples show filtering in two alternative
situations. Example 3 shows filtering for real-time
applications, such as relay protection. Example 4 shows a
filtering method that can be used for off line calculations,
for exam ple filtering of fault recordings.
Example 3: Test signal [3] with medium noise; causal
filter.
We consider the same type of test signal as in Example 2,
but now distorted with
-Negative sequence of 1%;
-white noise with SN R 40 dB;
-3:rd harmonic, 5 %, mainly z ero sequence;
-5:th harmonic, 2 %, mainly negative sequence.
Matlab’s Signal Processing Toolbox was used to test
various filters. The final c hoice was a 3:rd order low pass
Butterworth filter with a cross over frequency of 20 Hz.
The code is given in Appendix A. Figure 7 shows the true
signal and the frequency estimate be fore filtering.
True freq.=Solid Unfiltered estimate=DashDot S N R=40 dB60
400 0.1 0.2 0 3 0.4 0 5
Time (s)
Figure 7. True and estimated frequency for SNR=40 dJ3 beforefiltering.
Figure 6 shows the simulation result, Note that no
filtering has been used, neither before, nor after thefrequency estimation.
Figure 8 shows the effect of the Butterworth filter.
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The proposed demodulation can be made very fastfor signals with high signal to noise ratio
(SNR>80dB).
The frequency e stimate needs filtering when the S N R
value falls below 8 0 dB.
Causal filtering introduce a time delay. This is true
also for filters such as Bessel with a maximal flat
phase, that implies constant group delay. A constant(non-zero) group delay results in a fixe d time delay.
For off-line calculation we can use non-causal filters
to reduce the time delay. This gives significant
improvements.
True and filtered estimate; SNR=40 dB51.5
49.5' I0.1 0.2 0.3 0.4 0.5
Time s)
Figure 8. True and estim ated frequency filtered in 3:rd order low-pass Butterworth filter with crossover frequency of 20 Hz.
SNR=40 dB.
We see that the filter has introduced a lag, so the estimate
lags the true frequency by around 20 ms. A cross-over
frequency of 20 Hz, still introduces some phase shift at
lower frequencies. This can be seen in the Bode plot of
the filter.
For off-line applications where the raw data have been
sampled and stored, it is possible to reduc e the lag effect.
This is shown by the next example.
Example 4: Same test signal as Example 3, but non-
causal filter.
In this example the full time series is used. The phase
shift is reduced by applying forw ard filtering followed bybackward filtering. The filtering is performed in the
Matlab package by the command Jil@lt and is described
in reference [12]. We use the same filters as for Example
3 and filters the estimate twice, first forward and then
backward. The resulting sequence has precisely zero-phase distortion and double the filter order. We get Figure
9 that shows significant improvements, except at the
beginning and e nd, w here initial transients from the filter
show up.
True freq. = Solid; Filtered=DashDot; SNR=40 dB51.5
0.1 0.2 0.3 0.4 0.5Time (s)
Figure 9. True and estimated frequency forward-backwardfiltered in 3:rd order low-pass Butterworth filter with crossoverfrequency of 20 Hz. SNR=4O dB.
These four exam ples show that:
4. Discussion
In frequency estimation by de mod ulation there is a need
to filter signals for various reasons. If we exclude anti-
aliasing filters, the m ost imp ortant reasons are reduction
of white noise, harmonics and the double frequencycom ponen t caused by the d emo dulation. The proposed
method does not introduce the double frequency
component. As a consequence, we do not need to filter
for this specific reason. Therefore the proposed method
will show its advantage for applications where the main
concern of the filtering has been the double frequency
compon ent from the demo dulation. In contrast, the new
method will only give minor improvements for signals
with a large content of white noise and harmonics that
need a lot of filtering for these reasons.
Unsymmetric phase voltages
The proposed method w orks excellent when the negative
sequence component is small. If the input contains
negative sequence, the demodulation introduce a double
frequency com ponen t that is proportiona l to the negative
sequence amplitude. This gives the same type of doubIe
frequency component as the traditional demodulation
method. Even though, for most cases our situation is
better, because of the proportionality to the negative
sequence, the amplitude of the double frequency
compon ent is small. How ever, at unsymmetrical fau lts the
negative sequence component can be large. In these
situation the proposed method will give a double
frequency component with a large amplitude and will
work similar to the old demodulation m ethod.
Discrete Hilbert transform filter
The ab-transform is used to get the complex phasor
representation of the three phase inputs. The real and
imaginary parts hav e a phase difference of 90 degrees. An
alternative way to ac hieve this is to use a discrete Hilberttransform filter, as suggested in [lo]. This filter can be
designed to have 90 degrees phase shift and unity gain for
the frequency band of interest. The drawback with a
discrete Hilbert transform filter is that the filter introduces
a time delay of half the filter length. An ad vantag e is that
we can use the three phasors inde pend ently and use the
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162
mean value as a filtered estimate. In signal processing, th e
Hilbert transform plays a key role for frequencyestimation, see [7], [8], [ I l l . It might be possible that
improvements can be made by using a discrete Hilbert
transform filter instead of the a@-trans form .
Yet another alternative is to use sine and cosine filters.
Reference [131has used this to divide a scalar input signal
into two orthogonal components. The drawback with thismethod is that-in addit ion to the filter tim e delay-the
two filter gains are equal only at the nominal frequency.
Phased Locked Loop PLL)
The proposed demodulation method can also be applied
to PLL as already done in [5]. In the design in [SI, four
FIR filters of a total order of 130 were used inside the
control loop. Filters inside the co ntrol loop puts an upper
limit to PLL performance. From a control point of view it
seems better to filter away harmonic and noise before the
signal enters the PLL. Without any filters inside the PLL
loop, the PLL can be made much faster without stability
problems.In our work we have not found any improvements by
using PLL demodulation instead of demodulation with a
fixed frequency.
5. Conclusion
Demodulation is a promising method for power system
frequency estimation, but one drawback is that the
demodulation itself, introduces a double frequency
component that needs to be filtered. This paper has
demonstrated a demodulation method that solve this
problem. The method uses three phases as inputs and the
ap-transform to convert these inputs to a complex vector
with two orthogonal components. This vector is
demodulated using a complex vector with known
frequency, rotating in the opposite direction. The
resulting signal does not contain the double frequency
component. Hence, a filter for this specific purpose is not
needed.
Advantages with the proposed dem odulation:
* No need to filter the double frequency;
Can be m ade extremely fast for low noise signals
Disadvantages with the proposed demodulation
e The advantages are much reduced if the input signal
contains a large negative sequence component, that might
appear under fault conditions.
* All three phases are used for one calculation. Othermethods that use all the three phases independently, canuse the mean value of the them as a filtered estimate.
Acknowledgement
This work has been supported by a research grant from
Sydkraft. I am also grateful for help from my supervisors
S. Lindahl, Professor G. Olsson and L . Messing.
References
[1] A. G. Phadke, J. S. Thorp, M. G. Adamiak, A New
Measurement Technique for Tracing Voltage Phasors,
Local System Frequency, and Rate of Change ofFrequency , IEEE Trans. on Power Apparatus and
Systems, Vol. PAS-102, No. 5, 1983, pp. 1025-1038.
[2] A. A. Girgis, W . L. Peterson, Adap tive Estimation of
Power System Frequency Deviation and its Rate of
Change for Calculating Sudden Power System
Overloads , IEEE Trans. on Power D elivery, Vol. 5 ,
No. 2, April, 19 90, pp. 585-594.
[3] M . M. Begovic, P. M. Djuric, S. Dunlap, A. G.Phadke, Frequency Tracking in Power Networks in
the Presence of Harmonics , IEEE Trans. on Power
Delivery, Vol. 8, No. 2, April, 1 993, pp. 480-486.
[4] A. G. Phadke et al, Synchronized Sampling and
Phasor Meas urements for Relaying and C ontrol ,
IEEE Trans. on Power Delivery, Vol. 9, No. 1,January, 1 994, pp. 442-452.
[ 5 ]V. Eckhardt, P. Hippe, G. Hosemann, Dynamic
Measuring of Frequency and Frequency Oscillations
in Multiphase Power Systems , IEEE T rans. on Power
Delivery, Vol. 4, No. 1, January, 19 89, pp. 95-102.
[6] I. Kamwa, R. Grond in, Fast Adap tive Schem es for
Tracking Voltage Phasor and Local Frequency in
Powe r Transmission and Distribution Systems , IEE E
Trans. on Power Delivery, Vol. 7, No. 2, April, 1992,
pp. 789-795.
[7] B. Boas hash, Estimating and Interpreting T he
Instantaneous Frequency of a Signal - Part 1:
Fundamentals , Proc. of IEEE, Vol. 80, No. 4, April,
[SI B. Boashash , Estimating and Interpreting Th eInstantaneous Frequency of a Signal - Part 2:
Algorithms and Application , Proc. of IEEE, Vol. 80,No. 4, A pril, 1992 , pp. 540-568.
1992, pp. 520-538.
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163
Epsilonl =zeros(size(t));
Epsilon2=zeros(size( )) ;
Epsilon3=zeros size t));
Epsilonl 1)=0;
Epsilon2(1)=-125 @/I80;
Epsilon3(1)=1 5 * p i 480;
for k=2:length(t);
Epsilonl k)=rem Epsilonl k-l)-5*wl k) dt,2*pi):
Epsilon2 k)=rem Epsilon2 k-l)-5*wl k) d t ,2*pi);
Epsilon3 k)=rem Epsilon3 k-I)-5*wl k) dt,2*pi);endqo.......................................................
Hurml =zeros(size(t));Hurm2=zeros(size(t)); Hurm 3=zeros(si~e(t));
Hurml =O.O5*VI *sin(G ummu l)+0.02*VI *sin(Epsilonl);
Hurm2=0.05 V2'Sin(Gummu2)+0.02 V2 *sin(Epsilon2);
Hurm3 =O. 05*V3 *sin(Gumma3)+0.02 *V3*.~in(Epsil0?~3);
% Phusors
V I =VI *sin(TetuI +Sigmu*rundn(size( ))+Hurml;
v2=V2 *sin Tetu2)+Sigmu*rundn size t))+HurmZ;
v3= V3*sin(Tetu3)+Sigma*randn(size(t))+Hurm2;
% Alpha Betu Components
Alpha=sqrt(2/3) ( V I -0.5*v2 -0.5 *v3);
Betu =sqrt(lQ)*(v2 - v3);
% Complex input signulV=Alpha+j*Betu;
% Modulutiqn signulZ=cos(-2*piTf *t)+j*sin(-2*pi?jO*t);
% Demoduluted signul
Y= v.*z;Im-Y=imug(Y); Re_Y=reul(Y);
Amp-Y=sqrt(Re-Y. *Re-Y+lm-Y. *Im-Y);
Pha_Y=utan2(lm_Y,Re_Y);
% Create the signul U
NN= ength( Y) ;
Re-U=[O Re-Y(2:NN). *Re-Y l:NN-l)+Im-Y 2:NN).Im-Y(I: NN-I)] ;
lm-U=[0 Im-YI2:NN). *Re-Y I:NN-I)-Re-Y 2:NN).Im-Y(I:NN-I)l;
70
Arg_U=utun2(lm_U,Re_LI);
.flhut= fo+fi Arg-V./(2 *pi);7 ___----
N=3; % jilter order
fc=20; % ( Hz ) Cut o f frequency
fn= f YQ;
filter estimate ________--
% spec cution in normulize djrequency
Wn= d f n ;
% design LP Butterworth,fi'lter
[B,Al=butter(N,W n ) ;%jilte r the estimate
,fLhutfilt-ex3 = ilter(B,A,,fLhut;fO)+fO;
, f ~ h a t f i l t ~ e x 4 = ~ l ~ i l t ( B , A , . f ~ h u t ~ f ~ ) i f ~ ;
[9] V. V. Terzija et al, Voltage Phasor and Local System
Frequency Estimation Using Newton Type
Algorithm , Paper 94 WM 016-6 PWRD, IEE EPE S
1994 Winter Meeting, New York, January 30 -February 3, New Y ork, 1 994. Later published in IEEE
Trans. on Power Delivery (T-PWR D), July, 1994.
[ l o ] P. Denys, C. Counan, L. Hossenlopp, C. Holweck,
Measurement of Voltage Phase for the French FutureDefence Plan Against Losses of Synchronism , IEEE
Trans. on Power Delivery, Vol. 7, No. 1, Jan, 1992,
pp. 62-69.
[111 A . V. Oppen heim, and R. W. Schafer, Discrete-Time
Signal Processing, Prentice-Hall, Englewood Cliffs,
New Jersey, USA, 1989.
Natick, Mass.
[13] P. J. Moore, R. D. Carranza, A. T. Johns, " A NewNumeric Technique for High-speed Evaluation of
Power System Frequency , IEE Proc.-Gener. Transm.
Distrib., Vol. 141, No. 5 Sept, 1 994, pp. 529-536.
[121 Matlab-Reference Guide, The Mathworks, Inc.,
Appendix
A. Matlab code for new demodulation
% % test qfcomplex demodulution
% Ex 3 und 4 in paper
,ji=I000; % ( H z ) sumplingfrequency
Sigmu=le-Z; % stundurd deviutionfiv noise
j0=50; % Hz;
V-rm.i=l.O;
V I =,ryrt(2) V-rms; Phil =0.3;
V2=1 015 V I ; Phi2 =Ph il -2 *pi/3;
V3=1.015*VI; P hi3=Phil+2*pi/3;;
SNR=2O*logI 0( V-rms/Sigmu);
t=[0:dt:0.5];
. f l=(to) ones(size(t))+ *sin(2*pi31*t)+0. *si42*pi*6*t);
wl=2*pi;y ' l ;%-------Angles,fi,r,fundumantul phusor quuntities----------
Tetul =zeros(size(t));
Tetu2 =zeros(size(t));
Tetu3=zeros(size(t));
Tetul(I)=Phil; Tetu2(1)=PhiZ; Tetu3(1)=Phi3;
.for k=2:length (t);
dt=14r;
Tetul(k)=rem(Tetul k- l)+w l(k)*dt ,2 pi);
Tet~Z k)=rem Tetu2 k-l)+wl k)*dt,2~~pi);
Tetu3 k)=rem Tetu3 k-l)+wl k)*dt,2 pi);
end
%------- Angles,for 3:rd harmonic ----------Gumm ul =zeros(size.e(tJ);Gummu2=zeros(size(t));
Gumma3 =zeror(size(t));
Gummul I ) =O ;
Gummu2(1)= lO;Kpi/ l0;
GammuJ 1 = I O f p i / 1 8 0 ;
,for k=2: ength(t);
Gummul (k)=rem(Gummul k-I)+3*w l k) d t ,2*pi);
Gummu2(k)=rem(Gummu2(k-l)+3*wl(k)*dt,2*pi);
Gummu3 k)=rem Gummu3 k-1)+3wl k)*dt,2 pi);
end% ...............................................
%-------Anglesfrir5:th hurmonic ..........