Download (f-fo0) sin(2rrf0t) + (2n+l)!

IEEE Transactions on Power Apparatus and Systems, Vol. PAS-104, No. 2, February 1985
437
A LEAST ERR)R SQUARES THNIQUE FOR
y
DETERMINING POWER SYSTEM
M.M. Giray
M.S. Sachdev, FIEE
Power Systems Research Group
University of Saskatchewan
Saskatocn, Saskatchewan
Canada S7N OWO
Abstract
An algorithm for measuring frequency at a power
system bus is presented in this paper. The algorithm
is based am the least error squares curve fitting
technique and uses digitized samples of voltage at a
relay locaticn.
Mathematical developnent of the
algorithm is presented and the effects of key
parameters, that affect the performance of the
algorithm, are discussed. The algorithm was tested
using simulated data and data recorded fran a dynamic
frequency source. Results of sample tests are also
presented in this paper.
ODUCTIO2N
Frequency is an important operating parameter of a
power system. Generation-load mismatches cause the
system frequency to deviate fran its naninal value.
Frequencies lower than the naninal value indicate that
the system is overloaded.. Underfrequency relays are
used to detect these conditions and disconnect load
blocks to restore the frequency to its normal value.
These relays provide outputs when the frequency
decreases to prespecified thresholds.
Frequencies
higher than the naninal value indicate that the system
has more generation than load. These conditions are
detected by overfrequency relays provided at generator
terminals.
Overfrequency relays are also used to
protect generators fram overspeeding during start-ups.
Frequency relays available at present are of the
electranagnetic and electronic types. The accuracy of
electramagnetic relays ranges fran + 0.1 to + 0.2 Hz
of the set frequency (1). These relays are being
gradually replaced by the solid state types which
measure
time
durations
between
successive
zero-crossings of the input voltage and determine
frequency fram these measurements. Developments of
digital frequency and frequency trend relays have been
reported (2, 3, 4). These relays also measure time
duraticns between successive zero-crossings of the
system voltage for canputing frequencies and frequency
trends. The performace of solid state and digital
relays is adversely affected by the presence of
distorticn and noise which shift the zero-crossings or
create multiple zero-crossings.
Lately, canputer based frequency relays using
discrete fourier transform technique have been
developed (5, 6).
System voltage at the relay
location is sampled and digitized. These values are
then used to estimate frequency and rate of change of
protection of transformers (8, 9). In this paper, the
technique is extended to simultaneously measure
frequency and amplitude of the system voltages fran
sampled values. Simultaneous measurement of voltage
and frequency is useful in detecting overexcitation of
This is also suitable for
transformers (10).
protection of generators during start-up and shut-dow
operations and while operating off-line for spinning
reserve.
DEVELEPMENr OF TE ALGORITHM
This sedticn presents the algorithm which measures
the frequency of a voltage signal. It assumes that
the system frequency does not change during a data
window used for measurement.
The algorithm is
developed using the least error squares approach and
uses digitized values of the voltage sampled at the
relay location.
The Algorithm
The frequency at a power system bus is usually not
required to be measured during a fault transient. The
system voltage sampled for measuring frequency may,
therefore, be expressed as:
where; V is
wm is
t is
0 is
the peak value of the voltage.
the radian frequency and is equal to 2Trf.
time in seconds.
an
arbitrary phase angle.
Using the well known trigonanetric identity,
sin(27rft+G) = cos6 sin(2 Trft) + sine cos(27rft),
Equation 1 can be expanded as follows:
v(t) = Vm (cose) sin(2'rft) + Vm (sine) cos(2Trft) (2)
Using the Taylor series sin(2vrft) and cos(2-rrft) can be
expanded in the neighborhood of an expected value f
These series ccntain infinite terms and are given n
Equations 3.1 and 3.2.
sin(2Trft)= E
frequency.
This paper presents a new algorithm which is based
on the least error squares curve fitting technique
which has previously been used for impedance
protecticn of transmission lines (7) and differential
(1)
v(t) = V sin(wt+6)
Z
(-1)
sin(2rrf0t)
+
n+
~~2n+l
(1l)n (2Trt) | (f-f )
cos(2Trf t) (3.1)
(
(2n+l)!
Z()m, (2vt
2m
m=--0
84 SM 632-6
A paper recommended and approved
by the IEEE Power System Relaying Committee of the
IEEE Power Engineering Society for presentation at
the IEEE/PES 1984 Summer Meeting, Seattle,
Washington, July 15 - 20, 1984. Manuscript submitted February 1, 1984; made available for printing May 15, 1984.
(f-fo0)
co
n=
cos(27Tfft)=
(2t!
co
n=0
(f-f )
oo
2n+l
)n+l(2Tt)
I
2+l
sin (2TTf t)
0
o
cS (2vTf 0t) +
2n+l
(3.2)
Using the first three terms of these series, sin(2Trft)
and cos(27rft) can be approximated by
0018-9510/85/0002-0437$01.00© 1985 IEEE
438
sin (27rft)
-=
sin (27rf0t) + 271t (f-f0) cos (27rf0t)
(f-f0)
2
sin(27Tf 0t)
-
(4. 1)
cos (27rf0t) - 2irt (f-f0) sin (2Trf0t)
cos (2Trft)
(2Trt)2
(f-f0)
2
2
cos
-
(t2
[sin(2Trfot)
00
(f_f 0)
Vm(sine) [cos(27Tf t)
+
2.rt(f-fo) 00tsin(2Tff (2ft2 t) (f-fo)
o
-
cos (27Tfot)]
The following 2quation :Fan now ye obtained by
replacing (f-f ) with (f -2ff + f ) in Equation 5
and rearrangincg the resulting eguati&?.
o + [217tos (2rf0t) ] (f-fY)V00ose
v (t) = [sin (2rT ft) ] Vms
+[cos(2Ef t)]VmsinO
tn 2(-
+[t si(27Trft)
+
[-2Ttsin(2TrfOt)1 (f-fo)Vmsine
22
2
+
(27T)22
2
2
(271) ff
a16x6
a21 x1 + a22 x2
+ a25x5 + a26 X6
Vmcose + [t 2cos(27vf o t)l(
2 +
fT)~
(27T) 2 ff o
2
+2 2
~fO2
f02 )Vmsin0
+
(6)
If the voltage is sampled at a pre-selected rate, its
samples would be obtained at equal time intervals,
say, At seconds. A set of m samples may be designated
as v(t ), v(t +At), v(t +2At), ... v(t +mAt) where t
is an &rbitra4y time rekerence. The bltage sampleA
at t=t can now be expressed by substituting t for t
in EquAticn 6. Making the following substitutions in
the resulting equation and rearranging, Equation 7 is
obtained.
1
X2= (f-f )V%cose
x4= (f-f )Vmsine
m
x3= Vmsine
(271f2+(21)2ff
x5
x
=
(2_
6
2
2+ (2 7T12ff+_
2
a13 =cos(2
a
=
(2Tr)
2
all= sin(271Tf0t)
1fotl)
t2sin(271f0t1)
2o)V
27)
2
2 w
mx6 6xl
~0
Vms
a12= 21Tt10os(27fot1)
a14
=
2vt1sin(2Trf0t1)
2
a16 = tc1os (2rrf0t1)
+
a24
x4
(8)
=
[VI
(9)
mxl
The elements of the matrix [A] depend on the time
reference and sampling rate used and can be
pre-determined in an off-line mode. To determine the
six unknowns, at least six equations must be
established. In other words, at least six samples of
the voltage would be required.
As a general case, assume that m sanples are
available and m is greater than six. The matrix [Al
is now a mx6 rectangular matrix. Pre-multiplying both
sides of Equation 9 with the left pseudo-inverse of
[A], values of the mknown can be determined as
follows:
[V]
(10)
where [A]
[[A]T [A I [AlT. It can be shown that
this approach provides the least error squares
estimate of the mknowns (7).
Out of all the elements of the vector [XI, x J x2
x3* and X are of interest for calculating the Aystem
frequency. One possible approach is to use x and x2
for calculating frequency deviations using the
following equation.
x2 -=
x1
(f-f0)
vmoose
= f-f 0
Vmcoso
(11)
Another possible approach is to estimate frequency
deviations using the variables x3 and x4 as follows.
x4
x3
(f-f0)V sino
Vmsine
=_
_
=
f_f 0
(12)
Frequency of the sanpled signal is, thereEore, given
by
s
s
a23 x3
+
Notice that all x's are unknowns and are functions of
V , 0, f and f . Also t1 is the time reference and
tle voltage is Samrpled at a pre-selected rate (At is
The values of the "la" coefficients of
known).
Equations 7 and 8 can, therefore, be evaluated. The
values of v (t ) .2nd v (t2) are also known; these are
digitized sampLes of the voltage. Proceeding in this
manner, m digitized values of the voltage sampled from
the system can be expressed as m equations in six
unknowns.
These equations can be written in the
matrix form as follows:
[XI = [Al
2
(7)
=
[A] [XI
(5)
+axa
x
133 + ax
144N +a 155
Similarly, the next voltage sample, taken at time
t2=t1+At, can be expressed by the equation.
(4.2)
(2-77f0t)
+ 2Trt(f-f ) cos(2Trf t) 0
sin(2lTf t)I
+
v(t2)
Substituting Equations 4.1 and 4.2 in Equation 2, the
following equation is obtained.
v(t) = Vm (cosO)
x
aj 122
v (t+)
t = 1a
f
=
ff+0 X2
1
or
f = fo+
x-X43
When the value of V oose is small, frequency
estimated by Equation 11 innot sufficiently accurate.
Similarly when the value of V sinO is small, the
frequency estimated by Equation 1 is not sufficiently
accurate. However, when the value of Vm cosO is small,
the value V sirte is close to 1.0 p.u. and vise versa.
The estimaTe using one of these equations is,
439
therefore, expected to be within acceptable accuracy
bounds.
Yet another approach is to use all the variables
>,tx2, x3 and x4 to estimate f-fo. It can be shown
(ff0o)
2
X2f+ X4
2 +2
xl+
X3
(13)
this approach obviates the necessity of deciding
whether Equaticn 11 or 12 should be used.
CRITICAL PARAMETERS OF THE ALGORITHM
An algorithm which estimates local frequency fran
sampled bus voltages has been developed in the last
secticn. This algorithm is affected by many factors,
such as, the size of data window, sampling frequency,
time reference and the level of truncation of the
Taylor series expansions of sine and cosine terms. As
described in the last secticn, each voltage sample
obtained fran the system yields one equation in six
Lnkno1ns. Considerable freedan in selecting the size
of data window exists when the least error squares
aproach is used. Many cinbinations of the number of
equaticns and the sampling rate can be used for
ahieving a pre-specified size of data window. In
this secticn, implications of using data windows of
different sizes, different sampling rates and the
chice of the time reference are examined. Also
investigated are the effects of the level of
truncation of the Taylor series expansions of sine and
oosine terms.
Data Window Size
The effect of varying the size of the data window
The elemets of the coefficient
was investigated.
matrices and their left pseudo-inverses were
calculated for many cases. Sane of the combinaticns
of parameters examined are listed in Table I. In
these cases, time was considered to be zero at the
middle of the selected- data windows.
Table II lists the elements of the first four rows
of the left pseudo-inverse of the coefficient matrix
of case I.l. All elements of the 1st and 3rd rows are
numerically less than 1.0 whereas most elements of 2nd
and 4th rows are larger than 1.0.
The magnitudes of coefficients of a filter
determine if the filter would suppress or amplify
noise. Sun of the squares of the filter coefficients
is the measure of noise amplification (11). If this
sum is less than 1.0 the noise present in the
digitized input is suppressed. On the other hand, if
the su'm is greater than 1.0 the noise is mplified.
The sums of the squares of the filter coefficients
are also listed in Table II.
The elements and the sum of the squares of the
elements of the first four rows of other cases listed
in Table I were also calculated. The sum of the
squares of the elements as a function of data window
are shown in Figure 1. It shows that the sum of the
quares of the row elemnts reduce as the size of the
data window is increased. A study of Figure 1 reveals
that the effect of noise cn frequency measurement
would be reduced if larger data windows are used. One
disadvantage of increasing the data window is that the
speed of frequency measurement decreases. Also using
more samples per data window increases the computaticn
requirements.
Table I A Partial List of the Conbinations of
Parmeters Used in the Algorithm.
# OF
SAMPLES
CASE
(Hz)
36
48
60
I.3
I1.4
I.5
.6
I.7
(msec)
16.67
33.33
50.00
66 .67
83.33
100.00
116.67
720
720
720
720
720
720
720
12
24
I.1
1.2
DATA
WINDOW
SAMPLING
FRE QUENCY
72
84
Table II Numerical Values of the Elements of the 1st,
2nd, 3rd and 4th Rows of At and Sum of the
Squares of the Cdefficients of Each Row for
Case I.1
Numerical values of the row elements
1st Row:
-0.3509
-0.4873
0.3313
-0.3061
-0.2256
0.1029
-0. 1029
0.2256
0.3061
-0.3313
0.4873
0.3509
0.8084
-2.6393 -4.8659
4.7814 -15.9566
2nd Row:
15.9566
4.8659
-0 .8084
-4 .7814
-5 .8514
3rd Row:
0.0605
0.2169
0.0486
-0.0721
0.3339
-0.0732
-0.0732
0.3339
-0.0721
0.0486
0.2169
1th Row: -2.9083
2.4637
5.1311
-0.4939
-0.4939
8.0614
6.3383
2.4637
2.6393
5.8514
0.0605
Sum of the squares of the row elements
1st Row:
2nd Row:
1.2625
686.0659
c,b
x -
+ -
-i
0.3503
292.5212
3rd Row:
4th Row:
FIRST ROt(Xi I
SECOND ROW (X2)
0 ,
WV
+
IL
4
4
cr
.0.
U)
+
UL.
C)
Uz,
:
a
&
U
'nVI
a+
0
1.20
2.40
3.5
4.
6.W
U
7.0
WINOOW SIZE (NO OF SAMPLES) *10'
8.4
Figure 1. Sum of the Squares of the Row Elements of
Each Row of the Pseudo-Inverse Matrix as a
Function of Data Window Size; Sampling Rate
= 720 Hz.
440
Sampling Rate
To examine the effects of sampling frequency,
pseudo-inverse matrices for different sampling rates,
540, 600, 660, 720, 780, 840, 900, 960, 1020 and 1080
Hz, were calculated and the sum of the squares of
these rows were canputed. The sun of the squares of
the elements of row 2 are shown in Figure 2. It
reveals that increasing the sampling rate improves the
noise imnmity characteristic. A suitable conbination
cf data window size and sampling rate is required to
be selected.
C)-
3-
1--
z
LJ5
W.S
.S W
I
S I ZE
ZOOW
-
Table III Sum of the Squares of the Row Elements for
Case I.5
o
0z
_
a:
CO
Number of
unknowns
.
-
L
Ur)
Table I. The algorithm was also developed using 4 and
5 terms of the Taylor series expansicns. The nunber
of uknowns increased to 8 and 10 respectively. The
nature of the first four unknowns remained the same as
that of the cases of Table I. Pseudo-inverse matrices
were calculated for the algorithms also using 4 and 5
terms for the case I.5. The sum of the squares of the
first four rows for each case are listed in Table III.
This table indicates that for a constant data window,
the sums of the squares of the elements of each row
increase as the number of terms of the Taylor series
expansion are increased. To reduce the susceptibility
to noise, the size of the measurement window may have
to be increased if additicnal terms of the Taylor
series expansions are incorporated in the design
procedure.
-
_
-
LL
cr.i:
'
7~
S3.a
rsecs
-
i'SCS.
6.00
7.Zo
8
10
1st
row
0.0704
0.0765
0.1222
2nd
row
1.4809
8.9195
9.1641
3rd
row
0.0745
0.0751
0.1160
4th
row
1.4968
9.7922
10.0011
.
8.40
SAIMPL1NG RATE (HZ)
Figure 2.
6
9.60
*ICo"
REUEN2Y RESPCNSE OF THE AIlORITHM
10.60
Sum of the Squares of the 2nd Row of the
Pseudo-Inverse as a Function of Sampling
Rate.
Time Reference
If the time reference (t=0) is selected to be at
the middle of the data window, the numerical values of
the elements of each row becane symnetrical. This is
obvious fran the results shown in Table II where time
reference was chosen to be at the middle of the data
window. The effect of shifting time reference fran
the middle of the data window was also examined.
Pseudo-inverse matrices canputed showed that the
symmetry is upset if the time reference is moved fran
the middle of the window.
Symnetry reduces the
ocroputation burden and, therefore, is an important
factor that should be considered while designing an
algorithm.
To predict the behaviour of the algorithm in the
presence of distorted waveforms, frequency response of
the algorithms were examined. Sane.of the results are
presented in this secticn.
Figure 3 presents the frequency response of the
algorithm which uses a data window of 12 samples taken
at of 720 Hz. The plot indicates that the. filter is
sensitive to high frequencies especially the
components of the 2nd, 5th and 6th harmcnics and 150
Hz and 210 Hz. Figure 4 shows the frequency response
of the algorithm designed for a data window of 24
This figure shows that the
samples at 720 Hz.
sensitivity to higher frequencies has been
significantly reduced. Further increasing the window
size further improves the respcnse at higher
frequencies.
21
Taylor Series Expansicns
While developing this algorithm in the last
section, first three terms of the Taylor series
expansicns of sine and cosine functions were used.
When the measurement window is small, a few terms
these functions with reasonable
would represent
accuracy.
If the measurement window is large, more
terms are needed for accurate representation of the
sine and cosine functions, failing which truncations
wuld adversely affect the accuracy of measurements.
In power system applicaticns, frequencies may have to
It is, therefore,
be measured over a wide range.
essential to establish the accuracy with which the
The
sine and cosine terms must be represented.
effects of truncating the Taylor series was,
therefore, examined and is reported in this section.
As menticned earlier, three terms of the Taylor
series expansion were used in the designs listed in
0.0
60
120
20I1
FREQUENCY
Figure 3.
240
( Hz
300
)
Frequency Response of the Algorithm for
Case I.1
360
441
Table V Maximnm Frequency Errors Observed for Case
I.10 when 3 and 4 Terms of the Taylor Series
Expansion are used.
C)
NUMBER OF
UNKNOW-NS
H
SIGNAL
FREQ.
0-
(Hz)
60
59
11
0.0
60
120
240
182
FREQUENCY
(
300
360
OWF-LIE TESTING
In the preceding secticn, effects of sampling
rate, data window size and truncation of Taylor Series
expansicn on the behaviour of the algorithm have been
The effects of these parameters on
discussed.
frequency measurements are investigated in this
For this purpose, frequency measurements
section.
The program
were simulated in a software program.
generates a voltage which is sampled at pre-selected
rates. It then calculates the frequency of the signal
using the algorithm and the digitized values of the
voltage.
The effects of data window size were, first,
examined. The algor-ithm based cn equations with six
unknowns, 720 Hz sampling frequency and 12- sample
The
data window (case I.1, Table I) were used.
frequency of the signal was varied fran 50 Hz to 64 Hz
in steps of 2 Hz. The components, X1, X2, X3 and X
were caomuted and the frequency of the signal was
Errors in the measured frequency were
calculated.
Similar
ncoputed and are given on Table IV.
measurements were made by increasing the data window
The errors
size to 24, 36, 48 and 60 samples.
observed in these cases are also given in Table IV.
The results given in this table show that, for each
data window size, errors in the measured frequency
increase as the deviation of the signal frequency fran
the nominal value increases. Also, the measurement
errors increase as the window size is increased. This
hapens because the inaccuracies of the truncated sine
and cosine terms increase as the window size is
increased.
Maximum Frequency Errors Observed; Sampling
Rate = 720 Hz; Nunber of Unknowns = 6.
MEASUREMENT ERROR (Hz)
SIGNAL
DATA WINDOW
FREQ.
(Hz)
50
52
54
56
58
60
62
64
12
SAMPLES
0.397
0.218
0.094
0.029
0.004
0.000
0.004
0.027
60
48
24
36
SAMPLES SAMPLES SAMPLES SAMPLES
1.069
0.567
0.238
0.071
0.009
0.000
0.009
0.069
2.203
1.284
0.518
0.158
0.020
0.000
0.020
0.151
57
56
Hz
Figure 4. Frequency Response of the Algorithm for
Case I.2
Table IV.
58
3.489
1.945
0.879
0.273
0.036
0.000
0.035
0.263
4.794
2.771
1.298
0.416
0.055
0.000
0.054
0.403
55
54
53
52
51
50
6
8
6
8
MEASUREMENT ERROR (Hz) USING
X
X1 AND
0.00000
0.00440
0.03605
0.11717
0.27340
0.52261
0.87898
1.35267
1.94505
2.65870
3.48939
0.00000
0.00002
0.00026
0.00114
0.00307
0.00618
0.01008
0.01363
0.01473
0.01005
0.00512
D
0.00000
0.00427
0.03282
0.10848
0.25048
0.47435
0.79157
1.20088
1.73374
2.36523
3.11486
X4
0.00000
0.00003
0.00048
0.00272
0.00940
0.02475
0.05470
0.10705
0.19151
0.31984
0.50610
The effect of sampling rate on frequency
Algorithms designed
measurements was then studied.
for 540 Hz and 720 Hz sampling frequency were used to
The
measure frequencies of signals of 50-64 Hz.
errors observed in these cases indicated that, for a
omstant window size, the frequency measurement
doesn't change significantly if the sampling
rate is varied.
The effects of the level of truncation of the
Taylor series were also examined. The algorithm of
case I.4 (Table I) was used for measuring frequencies
The errors
of signals in the range of 50-60 Hz.
observed in these cases are given in Table V. A study
of these results indicates that the error is zero when
the signal frequency is 60 Hz. In the Taylor series
expansion of sine and cosine terms which were
expressed by Equaticns 4.1 and 4.2, 60 Hz replaced f
Therefore, if the frequency of the input signal is .0
Hz, the model used in developing the algorithm
represent this signal- correctly. This model, however,
does not measure frequencies of signals of less than
60 Hz frequency accurately.
It is reasonable to expect that if more terms of
the Taylor series expansion are used to approximate
the sine and oosine terms, measurements at off-naninal
frequencies would be more accurate. This is confirmed
by the errors observed during off-line testing and
recorded in Table V.
accuracy
EXPERIMAL RESULTS
Sane results of off-line tests have been presented
in the last section. The proposed method was also
tested using voltage samples taken from a dynamic
Results of these tests are
frequency source.
presented in this secticn.
Test data was obtained frcan a dynamic frequency
source manufactured by Doble Engineering Company. The
output voltage of this source can be adjusted fran 0
to 140 V rms in one V steps. The frequency can be
adjusted fran 40 Hz to 79.99 Hz in 0.01 Hz steps. The
output voltage and frequency are accurate to +1% and
+0.001% of the setting respectively. The voltage of
the source was adjusted to 110 volts and the frequency
was set at various levels between 40 and 65 Hz. The
level of the output was reduced to 3 volts peak value
by an isolating transformer and voltage divider
The reduced level output was then
combination.
awplied to a +5 volts 12-bit A/D cnvertor which
sampled the output at 2160 Hz and stored them on a
442
magnetic tape via a PDP-11/60 minicomputer. Data of
required sampling rate was extracted. It was then
used in conjunction with the algorithms to ompute the
frequency and peak values.
Maximum errors observed when measuring frequencies
of signals of 56-60 Hz using the algorithm I.3 (Table
I) are listed in Table VI. Maximum errors observed
when the algorithms I.4, I.5 and I.6 were used are
also listed. The frequency has been estimated with
aceptable accuracy in all cases and for signals
The accuracy of the
ranging fron 58 to 60 Hz.
measurements deteriorates with increasing deviation of
the signal frequency fram the nominal value. The two
main sources of errors are the noise in the sampled
voltage and the truncation of the Taylor series
expansions of sine and cosine functions.
When
mesuring the frequency of a 60 Hz signal, the errors
are exceptionally small and are caused mainly by the
noise in the voltage samples.
A study of Table VI reveals that maximum errors in
estimating frequency of 60 Hz signal decrease as the
data window size is increased.; This is because the
noise associated with the signal is suppressed more
effectively as the window size is increased. Errors
in estimating off naninal frequencies increase as the
size of the data window is increased.
This
deterioraticn is caused by the trmcaticn of sine and
cosine terms while designing the algorithm.
Table VI Maximum Frequency Errors Observed for Cases
I.2, I.4, I.5 and I.6; Number of Unknowns
=6
DATA WINDOW OF
(Hz)
36
48
SAMPLES
SAMPLES
SAMPLES
60
72
SAMPLES
0.018
0.027
0.032
0.076
0.167
0.013
0.015
0.009
0.007
0.017
0.083
60
59
58
57
56
0.042
0.126
0.284
0.016
0.059
0.187
0.425
0.262
0.587
Table VII Maximum Frequency Errors Observed for Cases
I.2, I.4J I.5 and I.6; Number of Unknowns
8.
(Hz)
60
59
58
56
54
52
50
36
SAWLES
48
60
SAMPLES
0.033
0.045
0.047
0.034
0.046
0.104
02141
0.021
0.029
0.019
0.023
0.025
0.111
0.191
0.520
0.017
0.012
0.024
0.113
0.362
0.991
The method presented in this paper measures
frequencies in the neighborhood of the nominal value
quite accurately. It can, however, be modified to
accurately measure off-noninal frequencies.
Using
combinations of algorithms designed for different
nominal frequencies, frequencies over a large range,
say 10-70 Hz can be measured with reasonable accuracy.
In underfrequency relaying measurement of rate of
change of frequency is also important. The proposed
algorithm can be used to calculate the rate of change
of frequency of the input signal using consecutive
frequency measurements and curve fitting techniques.
This paper describes the least error squares
The magnitude of
the input voltage can also be determined using this
Data window size, sampling. rate, time
approach.
reference and truncaticon of the Taylor series
expansion are critical parameters which affect the
performance of the algorithm. Off line tests and
experimental results indicate that the proposed
technique is suitable for measuring frequency and peak
values of a signal.
The technique presented here is general enough to
be applied for different relaying purposes, such as
frequency relaying (umderfrequency and overfrequency)
and over-excitation protection.
approach for measuring frequency.
CES
1. Westinghouse Electric Corporation,
"Applied
Protective Relaying", a book, Newark, N.J., 1976.
DATA WINDOW OF
SAMPLES
EU1jRHER CONSIDERATI(NS
EEF
MEASUREMENT ERROR (Hz)
SIGNAL
FREQUENCY
The results given on
Table VII demonstrate this improvement when cinpared
with the results given on Table VI. Using more terms
of the Taylor series expansions affects the noise
imnunity characteristics adversely.
The maximum
errors observed when measuring frequencies close of
the nominal value, therefore, increase slightly.
The results given in this section reveal that
parameters used in designing the algorithm affect
frequency measurements significantly. Larger windows
are required to suppress noise effectively. On the
other hand, large windows may adversely affect the
results by increasing errors caused by truncating the
Taylor series expansion of sine and cosine functions.
Suitable compromises need to be made while selecting
the data window size, sampling rate and truncation of
Taylor series expansion.
CCtCLUSION
MEASUREMENT ERROR (Hz)
SIGNAL
FREQUENCY
capared to the previous case.
72
SAMPLES
0.013
0.011
0.009
0.031
0.179
0.632
1.732
Frequency measurements were reported using the
algorithms developed by approximating the sine and
oosine functicns by the first four terms of their
Taylor series expansicns. The maximum errors observed
whe algorithms I.3, I.4, I.5 ad I.6 were used are
given on Table VII. Since more terms of the Taylor
sries expansion are used in developing these
algorithms, the measurement errors due to the
truncation of the sine and cosine terms are reduced
2. Widrevitz, B.C., Armington, R.E., "A Digital Rate
of Change Underfrequency Protective Relay for
Power Systems", Trans . IEEE, Power Apparatus and
Systems, Vol. PAS-96, Number 5, 1977, pp.
1707-1713.
3. Leibold, H., Overdorfer, E.W., "Microprocessor
Based Frequency Relay", Siemens Power Engineering,
Vol. IV, December 1982, pp. 302-303.
4. Sachdev, M.S., Giray, M.M., "A Digital Frequenc'y
aid Rate of Change of Frequency Relay",, Trans.
CEA, Engineering and Operation Division, Vol. 17,
Part 3, 1978, No. 78-Sp-145.
5. Phadke, A.G., Thorp, J.S., Adamiak, M.G., "A.New
Measurement Technique for Tracking Voltage
Phasors, Local System Frequency, and Rate of
Change of Frequency", Trans. TIEE, Poer Apparatus
and Systems, Vol. PAS-102, No. 5, May 1983, pp.
1025-1034.
443
6.
Girgis, A.A., Ham, F.M., "A New FFT-based Digital
Frequency Relay for Load Shedding". Proceedings
of PICA, 1981, Philadelphia.
7.
Sachdev, M.S., Baribeau, M.A., "A New Algorithm
for Digital Impedance Relay", Trans. IEEE, Power
Apparatus and Systems, Vol. PAS-98, No. 6,
Nov/Dec. 1979, pp 2232-2240.
8.
"Transformer
D.V.,
Shah,
M.S.,
Sachdev,
Differential and Restricted Earth Fault Protecticn
CEA,
Trans.
a
Digital Processor",
Using
Ehgineering and Operatian Divisicn, Vol. 20, Part
4, 1981, No. 81-SP-155.
"Numerical
N.,
Abu-Nasser,
R.,
9. Yacamini,
Calculatian of Inrush Current in Single Phase
Proc, Vol. 128, Part B, No. 6,
Transformers",
Nov. 1981, pp. 327-334.
10. Goff, L.E., "Volt-per-Hertz Relays Protects
Transformers Against Overexcitatian", Electrical
World, No. 168, Sept. 1967, pp. 47-48.
U.
HamnTng, R.W., "Digital Filters",
Prentice-Hall, Inc., New Jersey, 1983.
book,
a
Mahmut M. Giray (S'83) was
born in Ankara, Turkey an
December, 1949. He received
the B.Sc. degree in Electrical
f rm
Istanbul
Engineering
University,
Technical
Istanbul, Turkey in 1973. He
then joined Turkish Power
Authority where he worked
until 1975.
In September, 1975, he joined
the
University of Saskatchewan
for
worked
Graduate
in
the
Studies
area
of
and
digital
loomputer
completed all the requirements
relaying.
He
for M.Sc. degree in November 1979. Presently, he is
working for a Ph.D. degree at the University of
Saskatchewan in the area of Power System Protection.
Mohindar
Sachdev
S.
(Mt67,
lS'73, F'83) was born in
Amritsar, India, in 1928. He
received the B.Sc. degree in
.~Electrical
mhgineering
and
fran
Mechanical
the
Benares
Hindu University, India, the
M.Sc.
degrees
in
Electrical
!ngineering fram the Panjab
University, Chandigarh, India
Of
the
and
University
Saskatoon
Saskatchewan,
fran
Ph.D.
the
Degree
and
the
University of Saskatchewan,
Saskatoon, Canada.
He joined the Punjab
P.W.D. Electricity Branch in
1951 when he was posted in
Amritsar for management of subtransmissicn and
Later, in 1956, he was
distributicn systems.
appointed Power Controller and Senior Substaticn
Engineer. He was Assistant Resident Engineer of the
Uhl River Hydroelectric power plant at Joginder Nagar,
India, during 1959. Fran 1960 to 1961, he worked at
the Bhakra Power Plant I on the design of control
circuits and ooordination of the plant equipment. His
services were then loaned to the Punjab Engineering
College, Chandigarh, India, where he was an Associate
Professor until 1965.
In 1968 he joined the Faculty of Engineering at
the University of Saskatchewan where he is presently a
Professor of Electrical Engineering, a menber of the
Power Systems Research Group and Chairman of the
Research and Graduate Studies Cammittee of the
Department of Electrical Engineering. His areas of
interest are Power System Analysis, High Voltage DC
Transmissicn and Power System Protecticn.
Sachdev is Chairman of the Canadian
Dr.
Subcarmittee of the Intematicnal Electrotechnical
Cacnissicn Cammittee #IC41 Electrical Relays. He is
a Fellow of the Instituticn of Engineers (India), a
Fellow of the Institutian of Electrical Engineers,
Iondon (England) and a Fellow of the Institute of
Electrical and Electronics Engineers, New York. Dr.
Sachdev is a Registered Professianal Engineer in the
Province of Saskatchewan.
-
444
Discussion
R. W. Beckwith (Beckwith Elec. Co., Largo, FL): The authors present
a new and rather complex algorithm for determining frequency from
amplitude samples of a wave. The algorithm could be useful where these
samples are available and zero crossing information is not. The paper
implies that the method will be superior to these using zero crossing information, but no proof is given of the superiority. While the algorithm
presented extracts the frequency information from noise, so does the
measurement of periods from zero crossings if digital or analog filtering is used. Tests are needed using both methods on various noisy signals.
A problem will arise in the comparison of results, however, as to which
is correct. What is needed is a method much better than either so that
those making the test will know the "correct" answer. Of course, if we
knew how to get such a "correct" answer, we would throw both other
methods out and use the latter.
Another method of comparison is to select a problem to be solved
by frequency detection. The early remote detection of the tripping of
a large generator would be such a problem. The two methods could be
compared as to the ratio of correct detections to false alarms. Here the
question of which frequency is correct is replaced by which method best
detects the target event.
It is hoped that such a critical evaluation can be made so that the
usefulness of this excellent mathematical work can be known and the
method applied where shown to be superior.
Manuscript received August 9, 1984.
Adly A. Girgis (North Carolina State University, Raleigh, NC): The
authors have shown with clarity the effect of the different parameters
on measuring the frequency deviation of the fundamental frequency of
power systems, using the determininstic least-square estimation technique.
The discrete measurements of the continuous voltage waveform is a
nonlinear function of the frequency deviation and voltage magnitude
as indicated by Eq. (2). One of the successful methods to handle the
nonlinear estimation problem is to use Taylor series to expand the function about a known vector that is close to the exact vector. This goal
is usually achieved by two methods. One is to expand the function about
a nominal average vector. The second is to expand the function about
the current estimate of the state vector. The second method led to the
extended Kalman filter model which advantageously uses the statistical
properties of the signal and the noise in a recursive estimation algorithm
[Al.
The Taylor expansion used by the authors assumes that the voltage
magnitude and frequency are constants and the quantity 2ir(f-fo)t is small.
However, disturbances that cause the frequency to deviate create changes
in the voltage. In this case, what is the dynamic range of voltage and
frequency variations as functions of window size? The same disturbances
induce transients in the voltage waveforms. These transients are less severe
than fault-induced transients but have longer duration. Have the authors
tested the time response of the algorithm in the presence of non-60 Hz
components? The frequency response methods are valid techniques in
characterizing linear systems. However, the problem is essentially
nonlinear. Therefore, should not the frequency responses shown in Fig.
3 and 4 need to be modified to include the effect of the approximation
in the Taylor series expansion?
The calculations of xl, x2, x3 and x4, needed for Eq. (13), require
twelve multiplications and twelve additions for each component. That
is more than fifty multiplications and fifty additions to compute the frequency deviation. That is considerably more computation than those required for the extended Kalman filter model which optimally updates
the trajectory of the state vector on-line.
Finally, Eq. (13) has two answers for (f-fo). Which one is to be selected?
It appears to me that the last positive sign in the expression of x6 is in
error.
REFERENCE
[A] Adly A. Girgis and T. D. Hwang, "Optimal Estimation of Voltage
Phasors and Frequency Deviation Using Linear and Nonlinear
Kalman Filtering Theory and Limitations" IEEE PES paper #84
WM 104-6.
Manuscript received August 9, 1984.
M. S. Sachdev and M. M. Giray: We thank Mr. R. W. Beckwith and
Dr. A. A. Girgis for their interest in our paper. The discussers have raised
issues which are relevant to the methods that measure frequency from
sampled data and the comments are, therefore, of particular interest to
us. We offer the following in response to the discussions.
It is true that the proposed algorithm is useful where sampled values
of system voltages are available instead of the information on zero crossings. Such would be the situation in digital relays for generator,
transformer and transmission line protection. These relays are normally inactive except for performing system monitoring functions periodically. The proposed algorithm is, therefore, suitable for use in these relays
to monitor local frequency as a background task.
The proposed technique estimates simultaneously the amplitude and
frequency of a signal. In several situations, this feature can be of substantial advantage. For example, if a generator is being started, is being shut
down, or has been isolated on load rejection, the generator frequency
would deviate from the nominal value. Relays of several designs do not
operate properly at off nominal frequencies. The proposed technique
is suitable for designing digital relays that would provide reasonable
estimates of voltages and currents at off nominal frequencies and thus
provide adequate protection during those operating conditions.
We did not intend to leave the impression that the proposed method
is superior to the method of measuring time between zero crossings and
computing frequency from those measurements. The proposed method
is, however, immune to the noise that produces multiple zero crossings.
Our understanding is that several utilities have, in the past, experienced
multiple zero crossing related frequency measurement problems resulting
in maloperation of control functions. Of course, a combination of properly designed filters and zero crossing detection logic helps in alleviating
the multiple zero crossing problems.
Mr. Beckwith's proposal of comparing methods by investigating their
effectiveness to detect target events is interesting and very appropriate.
We would be pleased to participate in such a project.
Dr. Girgis' interpretation that the use of the Taylor series expansion
and truncation are justified on the assumption that the terms 2
is small. This is valid for the technique and the manner in which it has
been applied. Recollect that the proposed procedure is a nonrecursive
filter and a two-cycle data window has been used. In the selected wins and, for 27r(f-fo)t ranges from
dow, time t ranges from -0.016 to
-0.60 to 0.60. These deviations are less than one percent of 60 Hz, the
nominal power system frequency in North America. The largest term
that is neglected due to the truncation of the Taylor series is 0.036 for
the selected example. The sum of all terms that are truncated is even
smaller.
Figs. 3 and 4 include the effects of the nonlinearities due to the approximations introduced by truncating the Taylor series expansions. Like
all other techniques, the method described in the paper is not completely immune to the presence of nonfundamental frequency components
in the input signals. However, it is important to appreciate that, in a
power system application, analog filters would be used to suppress high
frequency components. Also, the effects of the nonfundamental frequencies that are usually experienced in power systems would be minimal if
the proposed method is used for measuring frequency. Figs. 3 and 4 clearly demonstrate this characteristic of the proposed technique.
The symmetrical nature of the filter coefficients determined by the proposed technique should be exploited to reduce the number of multiplications while implementing the algorithm. For implementing a two-cycle
window, 36 multiplications and 60 additions are required if both Eqs
(11) and (12) are implemented.
Our understanding of the Kalman filter approach (suggested in Ref.
A) is that the results obtained by its application in the presence of harmonic components in the signal, are substantially inaccurate. Also the
Kalman gains have to be computed in the real-time mode. On the other
hand, the computations required by the proposed technique are not too
many for application on modern microprocessors. Moreover, Figs. 2,
6 and 10 of Ref. A clearly indicate that the two-and three-state Kalman
filters provide frequency estimates that are substantially less accurate
than the estimates provided by the method proposed in our paper. Also,
the Kalman filters take 0.13 s or more time to start reasonable tracking
of a one Hz frequency change. This is much more time than required
by the least error squares technique described in our paper.
It is true that Eq.
has two answers, one positive and the other
negative. The correct answer can be selected by comparing the signs of
and (or and x4). The frequency deviation,
is positive if
both xI and x2 are positive (or are both negative). The frequency deviation is negative if the computed values of xl and x2 are of opposite signs.
We are grateful to Dr. Girgis for pointing out the typographical error
in the expression for x6.
Once again, we thank Mr. Beckwith and Dr. Girgis for their interest
in our paper a'nd for providing thought-provoking discussions.
ir(f-fo)t
0.016
(13)
xl
x2 x3
Manuscript received September 17, 1984.
(f-fo)