Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 116

Fig. 8.4 Frequency response
of orthogonalization by
double time delay

G ( j Ω )

2 S

G ( j Ω )

2 S 1

k = 1

2.5

k = 2

2

k = 5

1.5

1

0.5

0

0 2 4 6 8 Ω Ω

1

To calculate phase difference resulting from ( 8.10a , b ) one can use frequency
response ( 8.11 ) and the frequency response of the transformation required to get
direct signal component x c ð n Þ given by:

G 2c ð jX Þ¼ exp ð jkX Þ: ð 8 : 13 Þ

The tangent of this phase difference is equal to:

tg ð w 2c w 2s Þ¼ tg ð w 2c Þ tg ð w 2s Þ

tg ð kX Þ ctg ð kX Þ
1 þ½ tg ð kX Þ ctg ð kX Þ ¼1; 1 þ tg ð w 2c Þ tg ð w 2s Þ ¼

ð 8 : 14 Þ

where w 2c ¼ arg ð G 2c Þ and w 2s ¼ arg ð G 2s Þ; which proves that the phase difference
is equal p = 2 for all frequencies.

8.1.1.3 Least Squares Estimation Technique

The FIR orthogonal filters presented in Chap. 6 are aimed at extracting particular
(mostly—fundamental) frequency component and at producing the signal
orthogonal components useful for further processing in the measurement unit. The
transient period of FIR filters lasts as long as their window length and cannot be
shortened without significant loss of quality in frequency domain. In order to
minimize the transient state after sudden change of signal parameters one can
apply the LSE techniques.
The LSE method is a commonly known approach to signal parameters esti-
mation with a minimum of least square error. The technique is applied in many
areas of engineering,also including electrical power engineering [ 2 , 5 , 8 ]. In many
problems of power system protection and control parameters of input signal
(phasor magnitude and phase) are often to be determined whereas the signal model
structure is usually clearly defined. For a single component discrete fundamental
frequency signal such a model consists of two terms plus noise, according to: