A New Family of Measurement Technique for
Tracking Voltage Phasor, Local System
Frequency, Harmonics and
DC
offset
Jun-Zhe
Yang
Chih-Wen LiuDepartment of Electrical Engineering, National Taiwan University, Taipei, Taiwan Abstract
-
A series of precise digital algorithms based onDiscrete Fourier Transforms @FT) to calculate the frequency and phasor in real-time are proposed. These algorithms, we called the Smart Discrete Fourier Transforms (SDFT) family, succeeded in overcoming several problems of DFT, which include frequency deviation, harmonics and DC offset. Moreover, if using smoothing windows to filter noise, SDFT family is not affected by phase shift and amplitude decay. Factional cycle computing is allowed. Also, the advantages of DFT are still be reserved in SDFT family. These make the SDFT family more accurate than conventional DFT. Besides, SDFT family is very easy to implement, so it is very suitable for use in power systems. We provide the general form of SDFT family and simulation results compared with conventional DFT method to verify the claimed benefits of SDFT family.
Keywords: Discrete Fourier Transform (DFT), Frequency estimation, phasor measurement
I. Introduction
Frequency and phasor are the most important quantities in power system operation because they can reflect the whole power system situation. Frequency reflects the dynamic energy balance between load and generating power, while operators use phasor to constitute the state of system and, moreover, phasor based line relays are currently used in most power systems. So frequency and phasor are regarded as indices for the operating power systems in practice.
However, utilities have difficulty in calculating those quantities precisely. There are many devices, such as power electronic equipment and arc &maces, etc. generating lots of integral, non-integral harmonics, flicker and noise in modem power systems. Moreover, when a fault occurs, DC offset component is presented in fault current waveforms, Therefore, it is essential for utilities to seek and develop a flexible and reliable method that can measure fiequency and phasor in presence of harmonics, DC offset and noise.
There have been many digital algorithms applied to estimating frequency or phasor during recent years, for example Modified Zero Crossing Technique [I], Level Crossing Technique [2], Least Squares Error Technique [;I, Newton method [4], Kalman Filter [ 5 ] , Prony Method
[6], and Discrete Fourier
Transform
(DFT) [7], etc.Among the aforementioned methods, DFT is the most used method in phasor and frequency estimation [SI. When fundamental frequency is 60Hz, DFT has following advantages:
1. Highly precise: under the assumption of system frequency is very close 60Hz in normal, leakage error
of DFT is very small.
2. Fast computation: recursive and simple computation process make DFT faster than other methods.
3. Strong immunity: DFT immunes to harmonics that are multiple of 60Hz.
A series of precise digital algorithms that we called SDFT family offer utilities a flexible solution for estimation. SDFT family can take integral, non-integral harmonics and DC offset into consideration by very simple rules. Besides flexibility, SDFT family keeps advantages of DFT and adds these new advantages:
1. No Leakage error: the range of frequency, which SDFT family can estimate, is 0 to half of sampling fiequency.
2. Arbitrary window size: there are only two types of DFT using sampled data: full-cycle DFT and half-cycle DFT (HDFT). But in SDFT family, except zero, how many sampled data use to compute is arbitrarily.
3. Complete immunity: both integral and non-integral harmonics can be estimated while system frequency is 60Hz or not.
Therefore, SDFT family is a flexible and reliable solution for utilities to fit need in practice.
This paper is organized into four sections, the first of which is introduction. In section 11, we present the rules to produce every member of
SDFT
family. Furthermore, we prove that SDFT family can avoid the phase shift and amplitude decay caused by smoothing windows. Fifteen members of SDFT family are tested by fifteen examples in section Ill. Finally, we give a conclusion in section IV.11. The Proposed Digital Algorithm
In this section, we presented how we found the rules of SDFT family to take harmonics and DC offset into consideration at same time. First, we only consider the fundamental component of waveform.
x ( t ) = X , COS( W I
+
+,
) whereX,
: the amplitude,q51 : the phase angle.
The signal x ( t ) is conventionally represented by a phasor (a complex number) XI
where X I , X, : the amplitude,
4, ,
: the phase angle. Then x ( t ) can be expressed as(3) As the same steps as above, we define
where
*
denotes complex conjugate.We take frequency deviation ( w = 2a(60 +AY) ) into consideration, and suppose that x ( t ) is sampled with a sampling rate (60*N)
Hz
to produce the sample set{ x ( k ) }. Then, we define A, , B, , and a as
2 m
60 N z, =Real(am) =cos(-(Af+60))
And the result is
- 4 X x(r + 2 )
+ 2 x ( x ( r + l ) + x ( r + 3 ) ) x -1 x ( x ( r ) + 2 x ( r + 2 ) + x ( r + 4 ) )
(4)
Once again, we add non-integral harmonic into the signal.
x ( t ) = XI c o s ( r ~ + ~ ~ ) + X , c o s ( m o t + h )
(19)
+x,
cos(w,t+(b,) Then, the sample set { x ( k ) } can be expressed aswhere X, , X, , X, : the amplitude,
4, ,4, ,4,,
: the phase angle. X ( f ) = A, + B,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 (21) 2 a 60N z, = Real(a, ) := cos(- (60
+
4,
)) x ( r )-
mc(r+
1) = A, (a2-
1 )x(r + 1)
-
mc(r+
2) = ~,a(a'-
1) And the result i::E x x(r + 3) x :,:&,
+ 2 x (x(r+l)t Z J ( r + 3 ) + x ( r + S ) ) x ( z , + :", + :*)
- 4 x (x(r + 2 ) + x(r + 4)) x (q:" +:,zn + Z " S " ) (22)
-
I x ( N r ) +3+(r + 2 ) + 3r(r +4) + x(r + 6)) --
0 Dividing (1 1) by (lo), we getx(r
+
1)-
ar(f + 2 )x ( r )
-
me(r+
1) = UFinally, 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 ,4,
,
4,
,4,,
- = r I a : the amplitude, : the phase angle.: the time constant of the signal Andwedefine i, as (14) 2n 60 N z, = Real(a) = cos(-(Af
+
60)) Xde-m : DC offset Then (1 3) can be expressed as:Also,wedefine ad and zd as (15) 2 x x ( r + l ) x iI - 1 x ( x ( r ) + . r ( r + 2 ) ) = 0
-
-a --,60N d -Now, we add one integral harmonic into the signal.
-
-a-
ae m ~ +,60N
zd = cosh(ad) :=
2 x ( t ) = XI cos(ut+4,)+X, cos(mut++,) (16)
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 xt
t
Pm1
P a d (x(r+
1)+
3x(r+
3) +3x(r + 5)+x(r +7))-
1 x ( ~ ( r )+
4x(r+
2)+
6x(r+
4)+
4x(r+
6) + X ( P+
8))Now, it is very easy to find the rules from (15), (18), (22) and (26). There are three parts of these equations. The first part of rules is (-1)k+'2k , k=O
... i
.
Where i is dependent on the number of components considered in signal. The second part is similar to Pascal's triangle, and third part is a type of C i . Using these rules, we can easily 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 harmonic and DC offset at the same time.
x ( f ) =
x,
cos(wr+(b,)+X, cos(mwr+(b,,)(27)
+
X , cos(w,r+
+,,
)+
Xde-mSuppose 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 { ~ ( k )
1
is given by21rk
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: 2Xd+-
IM ar--
e 60N - 1--
e 60N a .2z e-_-
60N'c-1
x (ZI +z, +zn +z,) + k = I (26)-
-
0 + k = Ot
Part3 4 Z E Afwhere &,=-,and
e,,
=--(2+-),60 N N 60
.e,,
= -(m IC-
1+
-) m 4,emb
= --(m IC+
1+
-) mAfN 60 N 60
n a f n
e
=--(2+--) Afnena
=-60N
'
nb N 60Actually, the development from (27) to (29) is the same 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 investigate (29). We define A , , B , , C,, D, , E r , F, , andG, as
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 (3 8) And the result is+ 2 x +32,+3 +3j2,+5
+i,+,)
-1 x( i r
+ 4 i , + , +6gr+4 +4ir+, +kr+8) From (39), we can get the solutions of frequency and time constant of DC offset. (40) (41) 60N f = 6 0 + A f =cos-'(real(zl))- 2n 60N f, = 6 0 + A f , =co~-~(real(z,))--- 2n 1 60N log a d 5 =Moreover, by some algebraic operation we can get the 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,,)
sin (Me,,) X , =abs(A,) (43) M sin(O,,) sin(M8 ma ) X, =abs(C,) (45) (47) (48) (49)Conventional DFT methods incur leakage error in estimating fiequency and phasor when frequency deviates from nominal frequency (60Hz). However, in SDFT family we get exact solutions of frequency and phasor.
For off-line analysis, we can take all of the harmonics 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 equipment. These make us choose the smoothing windows to filter noise and high order harmonics for on-line applications. There are many smoothing windows that we can choose e.g., Hanning, Hamming, and Blackman
window.
Now, we take smoothing windows into consideration.
x (2, +z, + z n + Z d )
= O
(39)
Consider a sampled set { x ( k ) } becomes a filtered set { z(k)} by a smoothing window { SW(n)I s I , s 2 , ~ - ~ s , }, n is window size.
z(k) = slx(k) t ~ s z x ( k + l ) + . + . +s,x(k
+
n - 1) (50) Moreover, the :DFT of { z(k) } is given by.2nk
2 M-1 -J-
M k=O z, = - z(k+r)e
From the (37), we can obtain:
The relations of (37) and (38) are still kept in (52). Therefore, we can estimate frequency without modifying equations, but we have to do some change in (43-49) when we estimate phsaor. For example:
The phasor getling from (53) and (54) will allay the phase shiA and amplitude decay caused by smoothing windows.
111. Simulation Results
Simulation results presented in this section are all simulated from. Matlab. Sampling frequency is 1920Hz (N=32).
We choose fifteen members from SDFT family and use three different window sizes 16, 32 and 48 to evaluate performance. All of them are tested by three different conditions of fundamental frequency. Meanwhile, each of them includes 5 cases. Table 1 lists frequency errors
(Hz)
of every method under every condition and shows the average CPU time of Pentium 133. For comparison, we also listed the performances of Half-Cycle DFT (HDFT) and DFT in Table 1.To distinguish every member of SDFT family easily, SDFT means calculating frequency for only fundamental component. We add suffix to the others, for example SDFTISn means takes 3d and 5‘ and one non-integral harmonics into consideration. SDFTd means calculating fundamental frequency and DC offset. SDFT,, which combined smoothing window in SDFT family, uses Blackman window (n-32) for filtering in simulations.
From Table 1, many topics can be discussed. Generally speaking, the accuracy of SDFT family is as well as DFT at 60Hz and no leakage error when frequency deviated from 60Hz. Moreover, SDFT family tracks frequency variation well and some members of SDFT family deal with non-integral harmonic and DC offset certainly. It is
worthy to note that the performance of SDFT, iS equal to SDFTd,
because they have to solve the same equation. SDFT,
- 4 X i *+* z1Zn
+ 2 x (;r+l + i , + 3 ) x (21 + z n ) -1 x ( j i , t 2 i , + , t k r + 4 ) = 0
The only deference between SDFTn and SDFTd, is the definitionof z, and zd.
The last column of Table 1 is CPU Time of each method. Since this is the average computing time of each method under every condition, it should reflect real relation in practice. We can find there are 3 methods to increase accuracy of SDFT family in Table 1. First, increasing window size of SDFT family. Since all SDFT family use recursive computing, increasing window size doesn’t increase CPU time, but reduce CPU time instead. Because increasing window size is helpful to reduce the 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 about 4.8sec per 1920 data. But as the same of increasing window size, adding smoothing window is also helpful to reduce the times of iteration. It is very clear at SDFT,, series. Thirdly, take more harmonics into consideration, but the more harmonics taken into consideration will take more time in computing, especially taking non-integral harmonics into consideration.
IV. Conclusion
In this paper we present the SDFT family and demonstrate their performance in Table 1. SDFT family
provides flexible and reliable solutions for estimating. Not only keeps the advantages of DFT but adds new advantages in taking harmonics and DC offset into consideration, combining smoothing windows, and fractianal cycle computing. These aspects make SDFT family more efficient and suitable for power systems under real-time demands.
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Biographies
Chih-Wen Liu was born in Taiwan in 1964. He received the
B.S.
degree in Electrical Engineering from National Taiwan University in 1987, Ph.D. degree in electrical engineering from Comell University in ,1994. Since 1994, he has been with National Taiwan University, where he is associate professor of electrical engineering. His research interests include application of numerical methods to power system, motor control and GPS time transfer.Jun-Zhe Yang was born at Tainan, Taiwan, in 1971. He received his B.S. degree in electrical engineering from Tatung Institute of Technology in I994 and M.S. degree from National Taiwan University in 1997. He is presently a graduate student in the electrical engineering department, National Taiwan University, Taipei, Taiwan.
