Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 130

1
2 sin ð kX 1 Þ ½ u 1 ð n k Þ i 1 ð n Þ u 1 ð n Þ i 1 ð n k Þ:

Q 1 ¼ ð 8 : 67 Þ

The first two out of four equations are more general versions of the fundamental
relations ( 8.60 ) and ( 8.61 ). They are equal when the value of delay is equal to zero. The
two remaining equations have interesting features: the second ( 8.67 ) does not require
orthogonal components at all, the first, ( 8.66 ), removes or decreases low-frequency
noise present in the signals including decaying DC components.
Among many other possibilities very simple algorithms of power measurements
can be a result of orthogonalization by delay of a quarter of period of fundamental
component:

u 1C ð n N 1 = 4 Þ¼ u 1S ð n Þ
u 1S ð n N 1 = 4 Þ¼ u 1C ð n Þ:

ð 8 : 68 Þ

All of these fundamental equations can be used to measure criterion values
applying different methods for obtaining of orthogonal components.
Example 8.5 Assuming the same as before sampling rate (1000 Hz) specify
examples of power measurement algorithms that do not require usage of orthog-
onal filters.

Solution Among the algorithms for power estimation only the one for reactive
power, described by Eq. 8.67 , does not require orthogonal components. Depending
on the selected delay value one may obtain:
for k = 1:

1
2 sin ð X 1 Þ ¼ 1 : 618

Q 1 ¼ k 4 ½ u ð n 1 Þ i ð n Þ u ð n Þ i ð n 1 Þ; with k 4 ¼

and for the delay equal to a quarter of cycle:

Q 1 ¼ 0 : 5 ½ u ð n 5 Þ i ð n Þ u ð n Þ i ð n 5 Þ

that in fact realizes signal orthogonalization by time delay. Now one can also write
similar algorithm for active power measurement:

P 1 ¼ 0 : 5 ½ u ð n Þ i ð n Þþ u ð n 5 Þ i ð n 5 Þ:

8.2 Measurement of Protection Criterion Values

Above the fundamental versions of measurement algorithms of electrical quanti-
ties on basis of either averaging or orthogonal components have been described.
Some particular solutions resulting from different methods of orthogonalization
are described and analyzed below [ 4 , 6 , 8 , 10 ].