Characteristics of Measurement of Criterion Values and Adaptive - Digital Signal Processing in Power System Protection and Control - page 182

u 2 ð n Þ¼ 0 : 5 ½ 1 þ cos ð 2nX þ 2u U Þ þ 0 : 5g k ½ 1 þ cos ð 2nX k þ 2u Uk Þ
þ g k cos ½ n ð X k þ X Þþ u uk þ u U þ cos ½ n ð X k X Þþ u Uk u U

f g;

ð 9 : 21 Þ

u ð n Þ i ð n Þ¼ 0 : 5 ½ cos ð u U u I Þþ cos ð 2nX þ u U þ u I Þ
þ 0 : 5g k cos ½ n ð X k þ X Þþ u Uk þ u I þ cos ½ n ð X k X Þþ u Uk u I f g:

ð 9 : 22 Þ

The above components are summed up for a number of samples being a
multiple of signal half-cycle. One can see that the estimation error may result from
either non-zero sub-sums of AC components or the existing constant parts of
( 9.21 ), ( 9.22 ). It is worth to add that during analyses of the algorithm spectra,
either for small or bigger frequency deviations, the additive distortions should be
assumed zero (g k = 0). Determination of the characteristic for distortion influence
is done under assumption of signal frequency being equal to nominal value
(X ¼ X 1 ) and the magnitude of distortion components equal to unity (g k ¼ 1).
Changing the frequency of distortion component X m from zero to half of the
sampling frequency one gets measurement oscillations around accurate value, with
the magnitude dependent on distortion frequency.
In the following considerations calculation of the sums of cosine components,
being typical components of Eqs. 9.21 and 9.22 , could be quite useful. Simple
transformations, facilitated with application of exponential complex functions,
yield:

X

N 1

cos ð nX þ u Þ¼ cos ½ 0 : 5 ð N 1 Þ X þ u sin ð 0 : 5NX Þ
sin ð 0 : 5X Þ

ð 9 : 23 Þ

n ¼ 0

The above expression can be utilized for determination of algorithm's spectra,
as well as specific conditions for zeroing of error components. One can conclude
that for the frequency characteristics the source of error of magnitude and power
measurement is possible non-zero sum of the second harmonic. This sum is always
zero when the signal frequency is a full multiple of the fundamental frequency,
i.e., for X ¼ kX 1 : From ( 9.23 ) results that the algorithms susceptibility for small
frequency deviations can be quite significant.
The above observations are confirmed by the simulative investigations, illus-
trated in Figs. 9.7 and 9.8 .
Considering the errors due to oscillating distorting components one should
assume the values X ¼ X 1 and g k ¼ 1 : The sums of second harmonic are here
equal to zero and do not cause errors. On the other hand, deviations from accurate
measurement values result from non-zero sums of components of frequencies
X k ; X k þ X 1 ; X k X 1 and additionally non-zero sum of constant component
in voltage signal (0 : 5g k ). For certain frequencies at least the sums resulting from
AC components are zero, it is so for instance for the third harmonic (X k ¼ 3X 1 )
and then the measurement error is minimal, though also quite significant.