Characteristics of Measurement of Criterion Values and Adaptive - Digital Signal Processing in Power System Protection and Control - page 181

however, the frequency of the power system may change, with the deviations as
high as a few percent noted during power imbalance cases and close generator
faults. Crucial for proper operation of digital relays is susceptibility of employed
algorithms to frequency deviations. Secondly, sometimes it is needed to apply the
protection algorithms in wide range of frequencies, e.g., for reversible machines in
power plants. Thirdly, the input current and voltage signals may contain compo-
nents of various levels and frequencies. All these factors should be taken into
account during design and evaluation of the protection algorithms [ 5 ].
The frequency spectra of measurement algorithms are directly associated with
applied filters and the shape of measurement equations themselves. The filters,
usually used as pairs of orthogonal ones, are linear units of well-known frequency
responses. Unfortunately, the measurement algorithms mostly employ nonlinear
operations like multiplications, squaring, divisions or calculating of absolute
values; therefore one should be very careful by drawing suitable conclusions or
making generalizations.
In order to enable analysis of frequency features of measurement algorithms
three types of characteristics will be determined: sensibility characteristics for
small frequency deviations, algorithms' spectra in full range of considered fre-
quencies (up to half of sampling frequency) and error characteristics for additive
distortions. The considerations and analyses will have both theoretical and simu-
lative character. One should, however, remember that all of the analyzed algo-
rithms are nonlinear and thus concluding with respect to their advantages and
drawbacks are to be conducted with high cautiousness.
The algorithms of the averaging family ( Sect. 8.1.2 ) apply either averaging of
absolute or squared values of signals samples for magnitude estimation and
averaging of current and voltage samples products for measurement of active and
reactive powers. For the sake of preserving the clarity of conducted analyses the
considerations for algorithms with absolute values averaging will be omitted. It is
justified also by the fact that their accuracy is also dependent on sampling fre-
quency, which would make analysis doubly difficult.
Let the current and voltage signals for further considerations be given in the
form:

u ð n Þ¼ cos ð nX þ u U Þþ g k cos ð nX k þ u Uk Þ;

ð 9 : 20a Þ

i ð n Þ¼ cos ð nX þ u I Þ; ð 9 : 20b Þ

where

g k ¼ U kM = U 1m :

The analysis will be performed for two exemplary protection criterion signals—
magnitude and active power. Application of averaging principle requires calcu-
lation of the components of addition that in considered cases for the signals
( 9.20a , b ) are: