Characteristics of Measurement of Criterion Values and Adaptive - Digital Signal Processing in Power System Protection and Control - page 193

R ðÞ¼ 2P ðÞ
I m ðÞ ¼ 2CP

¼ R ; ð 9 : 38 Þ

CI m

and

R ¼ u FS ðÞ i FC n k

ð Þ u FS n k ð Þ i FC ðÞ

Þ i FC ðÞ :

ð 9 : 39 Þ

i FS ðÞ i FC n k ð Þ i FS n k ð

Since the same factor C appears in numerator and denominator it is evident that
the resistance measurement is completely independent of frequency (except for
C equal to zero).
Similarly, writing Eq. 9.37 for measurement of voltage magnitude and dividing
voltage by current, after canceling the same constant factors in numerator and
denominator one gets an accurate frequency independent impedance algorithm:

Z 2 ðÞ¼ U m ðÞ
I m ðÞ ¼ CU m

¼ Z 2 ð 9 : 40 Þ

CI m

Unfortunately, there is no similar result for reactance algorithms. However,
possible solution is calculation of reactance using measured impedance and
resistance:

X ð X Þ¼

p

Z 2 ð X Þ R 2 ð X Þ ð 9 : 41 Þ

Since both quantities of the right-hand side of the equation are frequency
independent then the same is valid for the left side, i.e. reactance measurement is
also frequency independent.
Equations 9.36 - 9.41 describe the algorithms that are less sensitive or insensi-
tive to frequency changes. The algorithms of impedance components ( 9.39 - 9.41 )
are completely frequency independent. All of the algorithms can be applied with
good result and small errors to measurements of criterion values during small
frequency deviations in the range ±2.5 Hz around nominal frequency.
When frequency of the signals changes in wider range (protections of reversible
generation units) then even those algorithms less sensitive to frequency deviations
may give too big errors. Measured active power and magnitude differ more from
their accurate value since coefficient C differs much more from unity. Even in
measured impedance some errors may appear due to another reason not taken into
account till now. This is noise which can cause errors not only during transients
but also during steady state. Simply during substantial frequency changes one must
take into consideration reshaping of the filters frequency responses. The situation
is shown in Fig. 9.15 presenting a part of frequency response of full period sine,
cosine orthogonal filters. If the filters were designed for nominal frequency 50 Hz
but actual frequency is equal, say, 25 Hz (1h 0 in Fig. 9.15 ) then normalized cosine
filter gain is equal to 0.4 for this frequency and the 2nd harmonic is gained 2.5
times, approximately (2h 0 ). In general, quite good band pass filter becomes sub-
stantially worse when frequency changes in wider range.