Characteristics of Measurement of Criterion Values and Adaptive - Digital Signal Processing in Power System Protection and Control - page 192

u FS ð n Þ i FC ð n k Þ u FS ð n k Þ i FC ð n Þ
¼ U m I m cos ð u U u I Þ H C ð jX Þ j j H S ð jX Þ j j sin ð kX Þ: ð 9 : 33 Þ

If filter gains for nominal frequency X 1 are:

j H C jX ðÞ j ¼ F C

H S jX ðÞ

ð 9 : 34 Þ

j j ¼ F S

then one can calculate active power using algorithms with delayed orthogonal
components. To do that Eq. 9.33 should be divided by constant coefficient

ð 2F C F S sin kX 1 Þ obtaining:

P ðÞ¼ u FS ðÞ i FC n k

ð Þ

ð Þ u FS n k ð Þ i FC ðÞ sin k ðÞ

sin kX 1

¼ h C h S

Þ P ¼ CP

ð 9 : 35 Þ

2F C F S sin kX 1 ð Þ ð

where h C ; h S are normalized filter gains (ratios of gain at given frequency and
nominal frequency).
The conclusion is that during frequency variation measured active power is
equal to the accurate value multiplied by a constant factor. When the frequency is
equal to nominal value the coefficient C is equal to unity, otherwise it is different
giving certain measurement error.
In the same way using Eq. 9.33 , substituting current instead of voltage samples,
one obtains an algorithm of current magnitude measurement depending on fre-
quency variations:

ðÞ¼ i FS ðÞ i FC n k

ð Þ i FS n k ð Þ i FC ðÞ sin k ðÞ

sin kX 1

I m

Þ I 2 ¼ CI 2

¼ h C h S ð 9 : 36 Þ

2F C F S sin kX 1 ð Þ ð

with coefficient C depending on frequency being the same as in ( 9.35 ).
For standard algorithm of reactive power we get in turn:

Q ðÞ¼ h C h S Q ð 9 : 37 Þ

However, in this case one obtains different factor depending on frequency
(there is no sine function).
Time responses of standard algorithms and those using delayed orthogonal
components are shown in Fig. 9.14 . According to the above considerations in case
of in standard algorithms the second harmonic component appears additionally
when frequency changes, however, for algorithms with delayed orthogonal com-
ponents a constant deviation of measured values is observed. In case of resistance
measurement it is even better since after transient period disappears the algorithm
is again accurate, irrespective of signal frequency. This is very important and
advantageous result, which can be proved easily. The resistance is calculated as a
ratio of active power ( 9.35 ) and current magnitude squared ( 9.36 ):