Characteristics of Measurement of Criterion Values and Adaptive - Digital Signal Processing in Power System Protection and Control - page 185

F C ¼ H C ð jX 1 Þ j j;

F S ¼ H S ð jX 1 Þ j j:

Similarly as before for analysis of algorithms' spectra it is assumed that
g Ck ¼ g Sk ¼ 0 ; while h C and h S are functions of X. For analysis of additive dis-
tortion influence one should assume X ¼ X 1 ; which induces unity values of the
coefficients h C ¼ h S ¼ 1 :
One has to mention that for investigations discussed here additive distortions in
one of the signals only (voltage) were introduced. Moreover, since g Ck and g Sk are
very small, the errors resulting from products of various distortion components can
also be neglected. Thus, assuming free change of distortion components phase
shifts one can conclude that the errors due to a number of components are roughly
the same as the sum of errors for particular components separately. With this in
mind the conclusions from consideration for single distortion components can be
extended accordingly.
It is essential that the orthogonal FIR filters, having linear phase, deliver at their
outputs orthogonal components of input signal of any frequency, i.e., this holds
both for fundamental frequency component and all distortion components. This
also explains the form of Eqs. 9.24a , b .
Basing on the above models of current and voltage signals one can determine
sought spectra of the measurement algorithms, for assumed zero level of additive
distortions (g Ck ¼ g Sk ¼ 0). Considering standard algorithms for signal magnitude,
active and reactive power ( 8.37 ), ( 8.47a , b ), ( 8.48 ), after simple transformations
one gets:

U m ¼ 0 : 5 ð h C þ h S Þþ 0 : 5 ð h C h S Þ cos ð 2nX þ 2u U Þ;

ð 9 : 26 Þ

P ¼ 0 : 25 ð h C þ h S Þ cos ð u U u I Þþ 0 : 25 ð h C h S Þ cos ð 2nX þ u U þ u I Þ;

ð 9 : 27 Þ

Q ¼ 0 : 5h C h S sin ð u U u I Þ:

ð 9 : 28 Þ

It is seen that measurement result of both magnitude and active power contains,
apart from constant value proportional to measured variable, also the second
harmonic term, being a source of error. The latter one appears always when the
normalized filter gains h C and h S differ one from another (for given frequency).
Depending on the initial phase of the signals the maximum error values of mag-
nitude measurement may vary between the values of h C and h S . When the filter
gains are identical, which is a case for frequencies being integer multiple of the
base frequency of applied orthogonal filters, this error disappears.
The result of reactive power contains only a constant term proportional to the
measured value, multiplied by the product of filter gains. The measured value
differs from accurate one when this product differs from unity.
The calculated measurement algorithms' spectra (signal magnitude, active and
reactive power) in wide range of frequency changes are presented in Fig. 9.10 .In
the vicinity of the fundamental frequency the highest deviations of the measured