Characteristics of Measurement of Criterion Values and Adaptive - Digital Signal Processing in Power System Protection and Control - page 173

where

d CC ð n Þ¼ X

n

a c ð k Þ b c ð k Þ;

k ¼ 0

d CS ð n Þ¼ X

n

a c ð k Þ b s ð k Þ;

k ¼ 0

d SC ð n Þ¼ X

n

a s ð k Þ b c ð k Þ;

k ¼ 0

d SS ð n Þ¼ X

n

a s ð k Þ b s ð k Þ:

k ¼ 0

For given orthogonal filters, i.e., for their impulse responses being known, it is
possible to calculate the above coefficients as a function of actual window length n.
If for instance full-cycle sine, cosine orthogonal filters are used, their impulse
responses are given by:

a c ð k Þ¼ cos ½ð k þ 0 : 5 Þ X 1 ¼ b c ð k Þ; ð 9 : 8a Þ

a s ð k Þ¼ sin ½ð k þ 0 : 5 Þ X 1 ¼ b s ð k Þ; ð 9 : 8b Þ

the coefficients of equations ( 9.7a , b ) are following:
d CC ð n Þ¼ X

n

cos 2 ½ð k þ 0 : 5 Þ X 1 ¼ n þ 1
2

þ sin ð nX 1 Þ

2 sin ð X 1 Þ cos ð nX 1 Þ;

ð 9 : 9a Þ

k ¼ 0

d SS ð n Þ¼ X

n

sin 2 ½ð k þ 0 : 5 Þ X 1 ¼ n þ 1
2

sin ð nX 1 Þ

2 sin ð X 1 Þ cos ð nX 1 Þ;

ð 9 : 9b Þ

k ¼ 0

d CS ð n Þ¼ d SC ð n Þ¼ X

n

cos ½ð k þ 0 : 5 Þ X 1 sin ½ð k þ 0 : 5 Þ X 1 ¼ sin ð nX 1 Þ

2 sin ð X 1 Þ sin ð nX 1 Þ:
ð 9 : 9c Þ

k ¼ 0

Substituting the above coefficients into output signals of orthogonal filters ( 9.7a , b )
and applying them in standard algorithm of magnitude measurement (sum of squared
orthogonal components) after simple rearrangements one obtains:

( n þ 1
2

sin ð nX 1 Þ
sin ð X 1 Þ

! 2

2

4
N 1

X 1m ð n Þ¼ X 1m

þ

2 sin ð X 1 Þ cos ½ð n þ 1 Þ X 1 þ 2u 1 )

þð n þ 1 Þ sin ð nX 1 Þ : ð 9 : 10 Þ

According to expectations the measured magnitude depends on initial phase
shift of the signal. Transients and steady state of magnitude measurement is shown