Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 148

Example 8.11 For the sampling frequency 1000 Hz propose: (a) possibly simplest
algorithms for resistance and reactance measurement for undistorted voltage and
current signals or when only harmonic distortions are expected; (b) possibly fastest
algorithms insensitive to frequency deviations.

Solution

(a) the simplest algorithms for impedance components measurement result from

Eqs. 8.101 and 8.102 , for k = 5:

R 1 ¼ u 1 ð n Þ i 1 ð n Þþ u 1 ð n 5 Þ i 1 ð n 5 Þ
i 1 ð n Þ i 1 ð n Þþ i 1 ð n 5 Þ i 1 ð n 5 Þ

X 1 ¼ u 1 ð n 5 Þ i 1 ð n Þþ u 1 ð n Þ i 1 ð n 5 Þ
i 1 ð n Þ i 1 ð n Þþ i 1 ð n 5 Þ i 1 ð n 5 Þ

or from averaging algorithms ( 8.123 ), ( 8.124 ):
R 1 ¼ P k ¼ 0 u 1 ð n k Þ i 1 ð n k Þ

;

P k ¼ 0 i 1 ð n k Þ

X 1 ¼ P k ¼ 0 u 1 ð n k 5 Þ i 1 ð n k Þ P k ¼ 0 i 1 ð n k Þ :

In case when harmonic distortions may appear one can apply the same algo-
rithms with appropriate prefiltering of the current and voltage. In case of
averaging one can also apply longer averaging time (a multiple of half-cycle).

(b) the fastest possible algorithms insensitive to frequency deviations

From Eqs. 8.101 - 8.103 for the shortest possible delay value (1 sample) one
obtains:

R 1 ¼ u F1S ð n Þ i F1C ð n 1 Þ u F1S ð n 1 Þ i F1C ð n Þ
i F1S ð n Þ i F1C ð n 1 Þ i F1S ð n 1 Þ i F1C ð n Þ

;

Z 1 ¼ u F1S ð n Þ u F1C ð n 1 Þ u F1S ð n 1 Þ u F1C ð n Þ
i F1S ð n Þ i F1C ð n 1 Þ i F1S ð n 1 Þ i F1C ð n Þ

;

q

Z 1 R 1 X 1 ¼ :

The resulting speed of the algorithms is dependent on the type and window
length of applied orthogonal filters.

As it is seen, to measure impedance components and similarly power and
magnitudes one can use either orthogonal components or averaging. Features of
the algorithms will first of all depend on introductory signal processing (mainly
digital filters) and in some way on the algorithms themselves. Application of
identical filters leads to substantial simplification of the algorithms of impedance