Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 128

Example 8.4 Specify the algorithms for current or voltage magnitude measure-
ment (other than averaging) that do not employ orthogonal components. Sampling
rate is assumed at 1000 Hz.

Solution Similar to the previous example, the number of samples per fundamental
frequency cycle N 1 equals to 20 and the relative angular frequency X 1 ¼ 2p = N 1 ¼
p = 10 : There is only one algorithm fulfilling the requirements, given by Eq. 8.55 .
Its parameters to be selected are the values of delay, which allows getting either
very fast algorithms or the ones with very simple equation. Examples:

(a) both delays equal one sample

X 1m ¼ k 1

p

x 2 ð n 1 Þ x ð n Þ x ð n 2 Þ ;

1
sin ð X 1 Þ ¼

1
sin ð p = 10 Þ ¼ 3 : 236 ;

where k 1 ¼

(b) both delays equal to quarter of cycle, i.e. five samples

X 1m ¼

p

x 2 ð n 5 Þ x ð n Þ x ð n 10 Þ

;

(c) m = 1, k = 5 samples

p

x ð n 1 Þ x ð n 5 Þ x ð n Þ x ð n 6 Þ

X 1m ¼ k 3 ;

where

1
sin ð X 1 Þ

k 2 ¼ p ¼ 1 : 799 :

8.1.3.2 Measurement of Power

Digital algorithms of power measurement can be found in the same way as the
ones of magnitude. To calculate them it is assumed that orthogonal components of
current and voltage are on disposal:

u 1 ð n Þ¼ u 1C ð n Þ¼ U 1m cos ð nX 1 þ u 1U Þ;

ð 8 : 56a Þ

i 1 ð n Þ¼ i 1C ð n Þ¼ I 1m cos ð nX 1 þ u 1I Þ; ð 8 : 56b Þ

u 1S ð n Þ¼ U 1m sin ð nX 1 þ u 1U Þ; ð 8 : 56c Þ

i 1S ð n Þ¼ I 1m sin ð nX 1 þ u 1I Þ; ð 8 : 56d Þ

The components can be combined to give phasors:

u 1 ð n Þ¼ u 1C ð n Þþ ju 1S ð n Þ¼ U 1m exp ½ j ð nX 1 þ u 1U Þ; ð 8 : 57a Þ