A New Family of Measurement Technique for

Tracking Voltage Phasor, Local System

Frequency, Harmonics and DC offset
Jun-Zhe Yang Chih-Wen Liu

Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan

Abstract - A series of precise digital algorithms based on of DFT is very small.

Discrete Fourier Transforms @FT) to calculate the frequency 2. Fast computation: recursive and simple computation
and phasor in real-time are proposed. These algorithms, we process make DFT faster than other methods.
called the Smart Discrete Fourier Transforms (SDFT) family, 3. Strong immunity: DFT immunes to harmonics that are
succeeded in overcoming several problems of DFT, which
include frequency deviation, harmonics and DC offset. Moreover, multiple of 60Hz.
if using smoothing windows to filter noise, SDFT family is not A series of precise digital algorithms that we called
affected by phase shift and amplitude decay. Factional cycle SDFT family offer utilities a flexible solution for
computing is allowed. Also, the advantages of DFT are still be estimation. SDFT family can take integral, non-integral
reserved in SDFT family. These make the SDFT family more harmonics and DC offset into consideration by very
accurate than conventional DFT. Besides, SDFT family is very simple rules. Besides flexibility, SDFT family keeps
easy to implement, so it is very suitable for use in power systems. advantages of DFT and adds these new advantages:
We provide the general form of SDFT family and simulation 1. No Leakage error: the range of frequency, which
results compared with conventional DFT method to verify the SDFT family can estimate, is 0 to half of sampling
claimed benefits of SDFT family.
fiequency.
Keywords: Discrete Fourier Transform (DFT), Frequency 2. Arbitrary window size: there are only two types of
estimation, phasor measurement DFT using sampled data: full-cycle DFT and
half-cycle DFT (HDFT). But in SDFT family, except
I. Introduction zero, how many sampled data use to compute is
Frequency and phasor are the most important quantities arbitrarily.
in power system operation because they can reflect the 3. Complete immunity: both integral and non-integral
whole power system situation. Frequency reflects the harmonics can be estimated while system frequency is
dynamic energy balance between load and generating 60Hzor not.
power, while operators use phasor to constitute the state of Therefore, SDFT family is a flexible and reliable
system and, moreover, phasor based line relays are solution for utilities to fit need in practice.
currently used in most power systems. So frequency and This paper is organized into four sections, the first of
phasor are regarded as indices for the operating power which is introduction. In section 11, we present the rules to
systems in practice. produce every member of SDFT family. Furthermore, we
However, utilities have difficulty in calculating those prove that SDFT family can avoid the phase shift and
quantities precisely. There are many devices, such as amplitude decay caused by smoothing windows. Fifteen
power electronic equipment and arc &maces, etc. members of SDFT family are tested by fifteen examples in
generating lots of integral, non-integral harmonics, flicker section Ill. Finally, we give a conclusion in section IV.
and noise in modem power systems. Moreover, when a
fault occurs, DC offset component is presented in fault 11. The Proposed Digital Algorithm
current waveforms, Therefore, it is essential for utilities to In this section, we presented how we found the rules of
seek and develop a flexible and reliable method that can SDFT family to take harmonics and DC offset into
measure fiequency and phasor in presence of harmonics, consideration at same time. First, we only consider the
DC offset and noise. fundamental component of waveform.
There have been many digital algorithms applied to
estimating frequency or phasor during recent years, for x(t) = X ,COS( WI + +, )
example Modified Zero Crossing Technique [I], Level
Crossing Technique [2], Least Squares Error Technique where X,: the amplitude,
[;I, Newton method [4], Kalman Filter [ 5 ] , Prony Method q51 : the phase angle.
[6], and Discrete Fourier Transform (DFT) [7], etc.
Among the aforementioned methods, DFT is the most
used method in phasor and frequency estimation [SI. The signal x ( t ) is conventionally represented by a phasor
When fundamental frequency is 60Hz, DFT has following (a complex number) XI
advantages:
1. Highly precise: under the assumption of system h X,cos+, + jx,sin+,
2, = ~ , e J =
frequency is very close 60Hz in normal, leakage error

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 where X I , X, : the amplitude,
Then x(t) can be expressed as
4, , : the phase angle.

(3) As the same steps as above, we define

where * denotes complex conjugate. 2 m

We take frequency deviation ( w = 2a(60 +AY) ) into z, =Real(am)=cos(-(Af+60))
60N
consideration, and suppose that x(t) is sampled with a
sampling rate (60*N) Hz to produce the sample set And the result is
{ x ( k ) }. Then, we define A, , B, , and a as
- 4 X x(r +2)
+2 x (x(r+l)+x(r+3)) x
(4)
-1 x (x(r)+2x(r+2)+x(r+4))

Once again, we add non-integral harmonic into the signal.

x(t)= XIc o s ( r ~ + ~ ~ ) +cXo s, ( m o t + h )

(19)
Then, the sample set { x ( k ) } can be expressed as
+x,cos(w,t+(b,)
where X,,X, ,X, : the amplitude,
X(f) = A,+ B,
4, ,4, ,4,, : the phase angle.
x(r + 1) = Ala+ Bp-'
x ( r + 2 ) = A , a 2 +B,o-2 Also, wedefine a, and z , as

We multiply a on both sides of (8) and (9), respectively,

then, subtract (7) from (8) and subtract (8) from (9). We
canerase B, andobtain 2a
z, = Real(a, ) := cos(- (60 + 4,
)) (21)
60N
x ( r ) - mc(r + 1) = A, (a2 -1)
And the result i:
-
x(r + 1) mc(r + 2) = ~,a(a'- 1)
E x x(r + 3) x :,:&,
Dividing (1 1) by (lo), we get -4 x (x(r + 2 ) + x(r + 4)) x (q:" +:,zn + Z " S " ) (22)
+2 x (x(r+l)t Z J ( r + 3 ) + x ( r + S ) ) x ( z , + :",+ :*)
x(r -
+ 1) ar(f + 2 ) -I x ( N r ) +3+(r + 2 ) + 3r(r +4) + x(r + 6)) -
- 0
=U
-
x ( r ) me(r + 1)
Finally, we add DC offset into the signal.
There is only one unknown variable in (12), and after
some algebraic manipulations we obtain: x(t) =x,COS(UX+q5,)~+X, cos(mwt+(b,)
(23)
+X,cos(4Bnt+(b,)+Xde-m
a2x(r)-a(x(r)+x(r + 2 ) ) +x(r +1 ) = 0 (13)
where X , ,X , , X , : the amplitude,
Andwedefine i, as 4, ,4, ,4,, :the phase angle.

2n -I= r : the time constant of the signal

z, = Real(a) = cos(-(Af + 60)) (14) a
60N
Xde-m : DC offset
Then (1 3) can be expressed as:
Also,wedefine ad and zd as
2 x x(r+l) x iI
(15) -
-a
- 1 x (x(r)+.r(r+2)) = 0 --,60N
d -
Now, we add one integral harmonic into the signal. -
-a -
a
e m +,60N
~
zd = cosh(ad) :=
x ( t ) = XI cos(ut+4,)+X, cos(mut++,) (16) 2

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 And the result is
-16 x x(r +4) x ZlZ,Z,Z, +k=4
+8 x (x(r+ 3) + x(r + 5)) x (zlzmz,+ZIZ,Zd +Z1ZdZ, +ZJZ,Z,) +k=3
-4 x (x(r+2)+2~(~+4)+x(r+6)) x (zI Z
, + z~z,,+ ZI zd + z,z,, + Z,Z~ + z,z~) t- k = 2
+2 x (x(r + 1)+ 3x(r + 3) +3x(r + 5)+x(r +7)) x (ZI+z, +zn +z,) +k=I (26)
- 1 x ( ~ ( r+)4x(r + 2) + 6x(r + 4) + 4x(r + 6) + X ( P+ 8)) - 0 +k=O
t t t
Pm1 Pad Part3

Now, it is very easy to find the rules from (15), (18), (22) Af
and (26). There are three parts of these equations. The first
4 Z
where &,=-,and e,, =--(2+-),
E
60 N N 60
part of rules is (-1)k+'2k , k=O ... i . Where i is
dependent on the number of components considered in .e,, = -(m
IC - 1 + -)m 4 ,emb= --(m
IC + 1 + -)mAf
N 60 N 60
signal. The second part is similar to Pascal's triangle, and
ena=- nafn e =--(2+--) Afn
third part is a type of C i . Using these rules, we can easily 60N ' nb N 60
express any type of waveform containing harmonics and
DC offset. Then, we combine these rules with DFT. As
(23), we consider integral harmonic, non-integra1 Actually, the development from (27) to (29) is the same
harmonic and DC offset at the same time. as the conventional DFT method. So the SDFT family can
keep all advantages of DFT such as recursive computing
manner, but then we use the steps of SDFT family to
x ( f ) = x, cos(wr+(b,)+X, cos(mwr+(b,,)
(27) investigate (29). We define A , , B , , C,, D, , E r , F, ,
+ X , cos(w,r + +, ) + Xde-m andG, as

Suppose that x ( t ) is sampled with a sampling rate (60*N)

Hz to produce the sample set { ~ ( k }.) Moreover, the
fundamental frequency (60Hz) component of DFT of
{ ~ ( k1)is given by

21rk
2 M-i -I-
X' =- C x(k + r ) e
M k=O

M is the window size used in DFT. Taking frequency

deviation ( w = 2 ~ ( 6 +
0 Af) ) into consideration, at last,
we obtain:

Then (29) can be expressed as

ir= A r + B r + C r + D r + E , + F r + G ,

And from (29), we can find the following relations

?r+l = A,a + B+-' +Cram+ Dra-" + Eran+ F+G' +Grad (38)

-- And the result is

ar
e 60N - 1 --60N
+-2Xd e
IM -_-
a .2z
e 60N 'c-1

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 (39)
+2 x +32,+3 +3j2,+5 +i,+,) x (2,+z, +zn + Z d )
-1 x ( i r+ 4 i , + , +6gr+4+4ir+, +kr+8) =O
From (39), we can get the solutions of frequency and time Consider a sampled set { x ( k ) } becomes a filtered set
constant of DC offset.
{ z(k)} by a smoothing window { SW(n)I s I , s 2 , ~ - ~},s n,
60N is window size.
f = 6 0 + A f =cos-'(real(zl))- (40)
2n
60N
z(k) = slx(k) t ~ s z x ( k + l ) + . + .
+s,x(k + n - 1) (50)
f, =60+Af, =co~-~(real(z,))--- (41)
2n
Moreover, the :DFT of { z(k) } is given by
1
5=
60N log a d .2nk
2 M-1 -J-
z, = - z(k+r)e
Moreover, by some algebraic operation we can get the M k=O
value of A, , B, , C, , D, , E , , F, , and G, after getting
exact frequency and time constant of DC offset. Then
phasors can be obtained by the following equations:

M sin (e,,)
X , =abs(A,) (43)
sin (Me,,)

M sin(O,,)
X, =abs(C,) (45) From the (37), we can obtain:
sin(M8 ma )

(47)

(48)

(49)

Conventional DFT methods incur leakage error in

estimating fiequency and phasor when frequency deviates The relations of (37) and (38) are still kept in (52).
from nominal frequency (60Hz). However, in SDFT Therefore, we can estimate frequency without modifying
family we get exact solutions of frequency and phasor. equations, but we have to do some change in (43-49) when
For off-line analysis, we can take all of the harmonics we estimate phsaor. For example:
into consideration, but for on-line applications, we need
smoothing windows to decay noise and high order
harmonics. Since the more harmonics taken into
consideration will take more time in computing. The
advantages of smoothing windows are no voltage drop, no
temperature drift, no noise addition, and don't care any
filter element feature. Besides these, smoothing windows
can be easily implemented in microprocessor-based The phasor getling from (53) and (54) will allay the phase
equipment. These make us choose the smoothing windows shiA and amplitude decay caused by smoothing windows.
to filter noise and high order harmonics for on-line
applications. There are many smoothing windows that we 111. Simulation Results
can choose e.g., Hanning, Hamming, and Blackman Simulation results presented in this section are all
window. simulated from. Matlab. Sampling frequency is 1920Hz
Now, we take smoothing windows into consideration. (N=32).

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 We choose fifteen members from SDFT family and use provides flexible and reliable solutions for estimating. Not
three different window sizes 16, 32 and 48 to evaluate only keeps the advantages of DFT but adds new
performance. All of them are tested by three different advantages in taking harmonics and DC offset into
conditions of fundamental frequency. Meanwhile, each of consideration, combining smoothing windows, and
them includes 5 cases. Table 1 lists frequency errors (Hz) fractianal cycle computing. These aspects make SDFT
of every method under every condition and shows the family more efficient and suitable for power systems
average CPU time of Pentium 133. For comparison, we under real-time demands.
also listed the performances of Half-Cycle DFT (HDFT)
and DFT in Table 1. Reference
To distinguish every member of SDFT family easily, G. Missout and P. Girard, “Measurement of Bus Voltage Angle
SDFT means calculating frequency for only fundamental Between Montresl and Sept-Ilcs”, IEEE Transactions on Power
Apparatus and5jvrems. Vol. PAS-99, No. 2, MarcWApril 1980, pp,
component. We add suffix to the others, for example 536-539.
SDFTISnmeans takes 3d and 5‘ and one non-integral C. T. Nguyen and K. Srinivasan, “A New Technique for Rapid
harmonics into consideration. SDFTd means calculating Tracking of Frequency Deviations Based on Level Crossings”,
fundamental frequency and DC offset. SDFT,, which IEEE Transactions on Power Apparatus and $stems, Vol.
PAS-103, No.8, August 1984, pp.2230-2236.
combined smoothing window in SDFT family, uses M. S.Sachdev and M. M. Giray. “A Least Error Squares Technique
Blackman window (n-32) for filtering in simulations. For Determining Power System Frequency”, lEEE Transactions on
From Table 1, many topics can be discussed. Generally Power ApparatuJ and Qslems. Vol. PAS-104, No. 2, Februw
speaking, the accuracy of SDFT family is as well as DFT 1985, pp. 437-443.
V. V. Terzija, M.B. Djuric, and B. D. Kovacevic. “Voltage Phasor
at 60Hz and no leakage error when frequency deviated and Local System Frequency Estimation Using Newton Type
from 60Hz. Moreover, SDFT family tracks frequency Algorithm’’; IEEE Transactions on Power Dellveiy, Vol. 9, No. 3,
variation well and some members of SDFT family deal July 1994, pp. 1368-1374
with non-integral harmonic and DC offset certainly. It is M. S. Sachdev, H. C. Wood, and N. G. Johnson, “Kalman Filtering
Applied to Power System Measurements for Relaying”, IEEE
worthy to note that the performance of SDFT, iS equal to SDFTd, Transactionson Power Apparahu and Systems, Vol. PAS-I 04,No.
because they have to solve the same equation. 12, December 1985, pp. 3565-3573.
T. Lobos and J. Rezmer, “Real-Time Determination of Power
SDFT, System Frequency”, IEEE Transactions on lnstrumentation and
measurement, Vol. 46, No. 4, August 1997, pp. 877-881.
-4X i *+* z1Zn A. 0. Phadke, J. S. Thorp, and M. G. Adamiak, “A New
+2 x (;r+l + i , + 3 ) x (21 + z n ) Measurement Technique for Tracking Voltage Phason, Local
System Frequency, and Rate of Change of Frequency”, K E E
-1 x ( j i , t 2 i , + , t k r + 4 ) = 0 Transactions on Power Apparams and Systems, Vol.102, No. 5 ,
May 1983, pp, 1025-1038.
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Biographies
Chih-Wen Liu was born in Taiwan in
The only deference between SDFTn and SDFTd, is the 1964. He received the B.S. degree in
definitionof z, and zd. Electrical Engineering from National
The last column of Table 1 is CPU Time of each Taiwan University in 1987, Ph.D.
method. Since this is the average computing time of each degree in electrical engineering from
method under every condition, it should reflect real Comell University in ,1994. Since
relation in practice. We can find there are 3 methods to 1994, he has been with National
increase accuracy of SDFT family in Table 1. First, Taiwan University, where he is
increasing window size of SDFT family. Since all SDFT associate professor of electrical
family use recursive computing, increasing window size engineering. His research interests
doesn’t increase CPU time, but reduce CPU time instead. include application of numerical methods to power system,
Because increasing window size is helpful to reduce the motor control and GPS time transfer.
times of iteration. Secondly, adding smoothing window to
SDFT family is useful method and CPU time addition is
constant. The computing time of smoothing window is Jun-Zhe Yang was born at Tainan,
about 4.8sec per 1920 data. But as the same of increasing Taiwan, in 1971. He received his B.S.
window size, adding smoothing window is also helpful to degree in electrical engineering from
reduce the times of iteration. It is very clear at SDFT,, Tatung Institute of Technology in
series. Thirdly, take more harmonics into consideration, I994 and M.S.degree from National
but the more harmonics taken into consideration will take Taiwan University in 1997. He is
more time in computing, especially taking non-integral presently a graduate student in the
harmonics into consideration. electrical engineering department,
National Taiwan University, Taipei,
IV. Conclusion Taiwan.
In this paper we present the SDFT family and
demonstrate their performance in Table 1. SDFT family

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