Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 115

It can be noted that for the smallest possible delay value (k = 1) Eqs. 8.7a , b
represents signal value and its derivative. Using them for magnitude measurement
(e.g. by substituting to ( 8.50 )) results in one of the simplest and fastest estimation
algorithms.

Orthogonalization by Double Time Delay

The method can be derived easily taking the signal samples at given instant and
delayed by 2k sampling periods:

x ð n 2k Þ¼ x ð n k k Þ
¼ X cos ½ð n k Þ X 1 þ u cos ð kX 1 Þþ X sin ½ð n k Þ X 1 þ u sin ð kX 1 Þ

x ð n Þ¼ x ð n k þ k Þ
¼ X cos ½ð n k Þ X 1 þ u cos ð kX 1 Þ X sin ½ð n k Þ X 1 þ u sin ð kX 1 Þ

It is seen at once that orthogonal components are given by equations:

x c ð n Þ¼ x ð n k Þ ð 8 : 10a Þ

x s ¼ x ð n 2k Þ x ð n Þ
2 sin ð kX 1 Þ

ð 8 : 10b Þ

''Pure'' signal delay, required to get the first of orthogonal components, is not a
digital filter, however, it introduces certain phase shift depending on frequency.
The second of orthogonal components is obtained using a transformation as certain
digital filter. Its frequency response calculated in the same way as before is given
by:

G 2s ð jX Þ¼ exp ð j2kX Þ 1
2 sin ð kX 1 Þ

¼ cos ð 2kX Þ 1 j sin ð 2kX Þ
2 sin ð kX 1 Þ

ð 8 : 11 Þ

The magnitude of frequency response has especially simple form:

j G 2s ð jX Þj ¼ sin ð kX Þ
sin ð kX 1 Þ

ð 8 : 12 Þ

Figure 8.4 shows the curves of the magnitude ( 8.12 ) versus frequency
depending on the parameter k (number of samples of delay) for the range from 1 to
5, assuming sampling frequency of 1 kHz. It is seen that the frequency response of
the method is better than in the case of orthogonalization by single delay, espe-
cially in the range of high frequencies. However, one must remember that for the
same value of a parameter k, the orthogonalization time is two times longer in the
case of double delay.