Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 132

In fundamental algorithm ( 8.50 ) the signal components obtained from any
known method of orthogonalization can be used. For instance, in the case of
orthogonalization by single delay one receives:

s

x 1 ð n Þþ x 1 ð n k Þ x 1 ð n Þ cos ð kX 1 Þ

2

X 1m ¼ : ð 8 : 73 Þ

sin ð kX 1 Þ

Depending on specific value of delay we can get either very fast (k = 1) or very
simple algorithm (k ¼ N 1 = 4), i.e. for the latter case:

q

x 1 ð n Þþ x 1 ð n N 1 = 4 Þ X 1m ¼ : ð 8 : 74 Þ

It should be noticed that applying orthogonalization by delay one must be
careful, since for very small delays the algorithm frequency response is bad, even
with possible noise amplification.
The second group of methods of magnitude measurement is related to Eq. 8.54 ,
which uses delayed orthogonal components. Substituting the components ( 8.70a , b )
to ( 8.54 ) one can get:

p

y 1S ð n Þ y 1C ð n k Þ y 1S ð n k Þ y 1C ð n Þ

1
F 1C F 1S sin ð kX 1 Þ

X 1m ¼ p : ð 8 : 75 Þ

This is an important and interesting result. There is one common gain factor,
which changes according to filter gains, delay value and sampling frequency,
however, the fundamental equation remains the same. This simple result is
obtained even when filter gains are different and, what more, the algorithm is less
sensitive

to frequency deviation, especially for filters with sine and cosine

windows.
When one uses correlation, methods of MSE or DFT (see Chap. 4 ) to get
required orthogonal components, then there is much less choice of possibilities.
Since orthogonal components are then of DC type instead of AC, it is impossible
to use methods and algorithms with time delay. We can simply use fundamental
Eq. 8.50 only. Assuming orthogonal components in the form:

x 1C ð n Þ¼ X 1C ¼ X 1m cos ð u 1 Þ ð 8 : 76a Þ

x 1S ð n Þ¼ X 1S ¼ X 1m sin ð u 1 Þ ð 8 : 76b Þ

the following equation of magnitude measurement is obtained:

q

X 1C þ X 1S X 1m ¼ ð 8 : 77 Þ

Example 8.6 Provide exemplary equations of the magnitude measurement algo-
rithms with application of separate orthogonalization procedures. Assume sam-
pling rate f S = 1000 Hz.