Characteristics of Measurement of Criterion Values and Adaptive - Digital Signal Processing in Power System Protection and Control - page 177

outputs during transients, the second one can use unique transient response of
reactive power and its application in development of corrected algorithms for other
criterion values. Both methods require zero initial conditions and precise esti-
mation of fault instant. With the viewpoint of protection systems that instant
should be estimated as soon as possible.
The basis for the first method is Eqs. 9.7a , b which describes output signals of a
pair of orthogonal filters (during transients) as a function of variable coefficients
and orthogonal components of the signal. The pair of non-stationary equations
allows to calculate orthogonal signal components:

x C ð n Þ¼ d C ð n Þ
d ð n Þ
x S ð n Þ¼ d S ð n Þ

ð 9 : 11 Þ

d ð n Þ ;

where

d ð n Þ¼ d CC ð n Þ

d CS ð n Þ

¼ d CC ð n Þ d SS ð n Þ d CS ð n Þ d SC ð n Þ;

d SC ð n Þ d SS ð n Þ

d C ð n Þ¼ y C ð n Þ

d CS ð n Þ

¼ d SS ð n Þ y C ð n Þ d CS ð n Þ y S ð n Þ;

y S ð n Þ d SS ð n Þ

d S ð n Þ¼ d CC ð n Þ

y C ð n Þ

¼ d CC ð n Þ y S ð n Þ d SC ð n Þ y C ð n Þ:

d SC ð n Þ y S ð n Þ

These orthogonal components known during transients allow to calculate
magnitude of the signal using standard algorithm:

X 1m ¼ x C ð n Þþ x S ð n Þ:

ð 9 : 12 Þ

The trajectory of magnitude measurement applying this method of dynamical
correction is shown in Fig. 9.4 . Substantial speed-up of reaching the steady state
result can be observed. It should be added here that both calculations and simu-
lations were made for pure sinusoidal signals without noise. When noise is present
in the signal then the result changes oscillatory around steady state and variance of
error decreases until filter window length is reached.
It is evident that in the same way dynamical correction of the other criterion
values can be arranged.
The basis of the second method of dynamical correction is the algorithm of
measurement of reactive power. It was mentioned before that any algorithm of
reactive power has unique, phase independent transient response. It can be easily
proved for standard algorithm:

1
2F 1C F 1S

Q 1 ¼ ½ u F1S ð n Þ i F1C ð n Þ u F1C ð n Þ i F1S ð n Þ; ð 9 : 13 Þ