Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 136

This fundamental equation can be simplified when filter gains are identical: i.e.
F 1C ¼ F 1S ¼ F 1 (it is true for full-cycle sine, cosine or Walsh FIR filters). And then:

1
2F 1

P 1 ¼ ½ u F1C ð n Þ i F1C ð n Þþ u F1S ð n Þ i F1S ð n Þ; ð 8 : 81 Þ

which presents the rescaled version of fundamental Eq. 8.60 .
Substituting components ( 8.79a , b , c , d ) to fundamental Eq. 8.61 one can get
equation of reactive power:

1
2F 1C F 1S

Q 1 ¼ ½ u F1S ð n Þ i F1C ð n Þ u F1C ð n Þ i F1S ð n Þ: ð 8 : 82 Þ

This is a rescaled fundamental Eq. 8.61 , similar to ( 8.81 ).
Among the fundamental equations of power measurements ( 8.64 - 8.67 )
employing delayed orthogonal components, the most important are the last two.
Substituting there orthogonal components from Eqs. 8.79a , b , c , d one gets:

1
2F 1C F 1S sin ð kX 1 Þ ½ u F1S ð n Þ i F1C ð n k Þ u F1S ð n Þ i F1C ð n Þ;

P 1 ¼ ð 8 : 83 Þ

1
2F 1 sin ð kX 1 Þ ½ u F1 ð n k Þ i F1 ð n Þ u F1 ð n Þ i F1 ð n k Þ:

Q 1 ¼ ð 8 : 84 Þ

It is seen in Eq. 8.84 that in this case one can use any arbitrary selected filter for
voltage and current (the same for both signals) or avoid using them at all. Both
algorithms are rescaled version of fundamental Eqs. 8.64 and 8.67 . If the delay is
equal to unity (k ¼ 1), the fastest algorithm is obtained, while for the number of
samples of delay being equal to a quarter of number of samples per cycle of
fundamental frequency component (k ¼ N 1 = 4), the simplest one is reached. In this
case there is no need for orthogonal filters at all (similar to ( 8.74 )):

1
2F 1

P 1 ¼ ½ u F1 ð n Þ i F1 ð n Þþ u F1 ð n N 1 = 4 Þ i F1 ð n N 1 = 4 Þ; ð 8 : 85 Þ

1
2F 1

Q 1 ¼ ½ u F1 ð n N 1 = 4 Þ i F1 ð n Þ u F1 ð n Þ i F1 ð n N 1 = 4 Þ: ð 8 : 86 Þ

When orthogonal components are obtained from standing phasor (correlation
method, DFT and others), they can be described by the formulae:

u 1C ð n Þ¼ U 1C ¼ U 1m cos ð u 1U Þ; ð 8 : 87a Þ

u 1S ð n Þ¼ U 1S ¼ U 1m sin ð u 1U Þ; ð 8 : 87b Þ

i 1C ð n Þ¼ I 1C ¼ I 1m cos ð u 1I Þ; ð 8 : 87c Þ

i 1S ð n Þ¼ I 1S ¼ I 1m sin ð u 1I Þ: ð 8 : 87d Þ