Characteristics of Measurement of Criterion Values and Adaptive - Digital Signal Processing in Power System Protection and Control - page 187

value are observed for active power, moderate—for voltage magnitude, and the
lowest and unique (phase independent) for reactive power. Such results are a
derivative of the normalized filters' characteristics (see Fig. 9.7 b). Since one of the
filter gain coefficients is slightly higher and the other one slightly lower than unity,
their product is close to unity and this is why we have almost constant level of
reactive power (Fig. 9.10 c). On the other hand, squared values of gains differ
much more from the gains themselves (in considered range of frequency), there-
fore deviations for active power are higher than that for magnitude. Considering
the spectra in wider range one can conclude that the results for higher frequencies
are different than before. The values of active and reactive power are lower than
for magnitude, since for low values of filter gains their squared values or products
are much lower than the gains themselves.
Simulation studies have also been performed for the algorithms with time delay
( 8.41 ), ( 8.49 ) and ( 8.50 ), with application of the sine/cosine filters for signal
orthogonalization and sine filter only in case of the algorithm for reactive power
estimation. The results are presented in Fig. 9.11 . Characteristic here is phase
independent shape of spectra (unique curve) for all considered algorithms, which
may allow for eventual adaptation of estimators to varying signals frequency. One
can see that the signal magnitude U estimator spectrum has the side bands at the
level between the values resulting from the sine/cosine filters frequency responses.
The spectra of both P and Q estimators exhibit only the main band, whereas the
signal frequencies above 100 Hz are almost fully suppressed, which is a significant
advantage over the algorithms without time delay. One should remember, how-
ever, that this superiority is achieved at the cost of elongated response in time
domain, due to introduction of additional time delay.
Analytical evaluation of the accuracy of standard measurement algorithms of
magnitude, active and reactive power for additive distortion components can be
performed when one substitutes the voltage and current terms according to ( 9.24a , b )
and ( 9.25a , b ) to the equations of particular measurement equations, assuming
constant value of fundamental frequency (X = X 1 ). Measured quantities are now
described as follows:

U m ¼ 1 þ g Ck cos 2 ð b Þþ 2g Ck cos ð a Þ cos ð b Þþ g Sk sin 2 ð b Þþ 2g Sk sin ð a Þ sin ð b Þ;
ð 9 : 29 Þ

P ¼ 0 : 5 ½ cos ð u U u I Þþ g Ck cos ð c Þ cos ð b Þþ g Sk sin ð c Þ sin ð b Þ; ð 9 : 30 Þ

Q ¼ 0 : 5 ½ sin ð u U u I Þþ g Sk cos ð c Þ sin ð b Þ g Ck sin ð c Þ cos ð b Þ; ð 9 : 31 Þ

where

a ¼ nX 1 þ u U ;

b ¼ nX k þ u Uk ;

c ¼ nX 1 þ u I :