Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 139

u F1C ð n Þ¼ X

11

u 1 ð n k Þ cos ½ 0 : 1p ð 5 : 5 k Þ:

k ¼ 0

For the half-cycle data window (N ¼ 10) the filter gains are identical and
equal to 5. Thus the algorithm equations take the form:

(b)

P 1 ¼ 0 : 02 ½ u F1C ð n Þ i F1C ð n Þþ u F1S ð n Þ i F1S ð n Þ;

Q 1 ¼ 0 : 02 ½ u F1S ð n Þ i F1C ð n Þ u F1C ð n Þ i F1S ð n Þ;

where orthogonal components of voltage (similar for the current) are

obtained from u F1C ð n Þ¼ P

9

u 1 ð n k Þ cos ½ 0 : 1p ð 4 : 5 k Þ:

k ¼ 0

(c) In this case the filter gains are the same as in (b). Additionally one calculates:

sin ð kX 1 Þ¼ sin ð 2 0 : 1p Þ¼ 0 : 588 and thus

P 1 ¼ 0 : 034 ½ u F1S ð n Þ i F1C ð n 2 Þ u F1S ð n 2 Þ i F1C ð n Þ;

Q 1 ¼ 0 : 034 ½ u F1 ð n 2 Þ i F1 ð n Þ u F1 ð n Þ i F1C ð n 2 Þ:

(d) Here the components of power are calculated as in (b), with different way of

reaching the orthogonal components of the signals. If the single-delay
method is applied (with k = 2 samples) the specific equations become
(here—for voltage):

u F1S ¼ u 1 ð n 2 Þ u 1 ð n Þ cos ð 2 0 : 1p Þ
sin ð 2 0 : 1p Þ

¼ u 1 ð n 2 Þ 0 : 809u 1 ð n Þ
0 : 588

¼ 1 : 7u 1 ð n 2 Þ 1 : 376u 1 ð n Þ:

Similar equation can be used for orthogonalization of current signal.

8.2.3 Measurement of Impedance and Its Components

8.2.3.1 Application of Orthogonal Components

The algorithms allowing to measure impedance and its components as well as
conductance and its components result from complex relations similar to those
used before. One can write the following equations:

Z ¼j Z j exp ð ju Þ¼j Z j cos ð u Þþ j j Z j sin ð u Þ¼ R þ jX : ð 8 : 90 Þ

These quantities can easily be calculated from power and magnitude obtained
before: