Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 158

• with application of the orthogonal components and any delay values (also under
condition of non-zero filter gain values, to be cancelled in numerator and
denominator)

s

y S ð n Þ y C ð n k Þ y C ð n Þ y S ð n k Þ
u S ð n Þ u C ð n k Þ u C ð n Þ u S ð n k Þ

Y m
U m

¼ C ¼ :

Having measured the value of magnitudes ratio C the frequency deviation and
the sought frequency itself can now be determined Having in mind that this value
resulting from the Walsh filter characteristic is equal to 0.413 for the measured
frequency of 49 Hz and substituting this value to the simpler, yet less accurate,
Eq. 8.149 , one obtains:

DX
X 1

¼ Df
f 1

C
N 1 C ¼

0 : 413
20 0 : 413 ¼ 0 : 021 :

¼

Therefore Df ¼ f 1 0 : 021 ¼ 50 0 : 021 ¼ 1 : 05 Hz and the relative error of
measurement, defined as the relative difference of real and measured values of
frequencies with respect to the nominal frequency, is equal:

df ¼j Df real Df j= f 1 ¼j 1 0 : 5 j= 50 ¼ 0 : 001 :
Applying the more accurate formula ( 8.114 ) one obtains:

DX
X 1

¼ Df
f 1

¼ C sin ð p = N 1 Þ
p

¼ 0 : 413 sin ð 0 : 05p Þ
p

¼ 0 : 0205 ;

which yields Df ¼ f 1 0 : 0205 ¼ 50 0 : 0205 ¼ 1 : 025 Hz and the relative error value
equal

df ¼j Df real Df j= f 1 ¼j 1 1 : 025 j= 50 ¼ 0 : 0005 :

One can see that applying more accurate Eq. 8.148 allows getting twice lower
error values, at the cost of slightly higher computational complexity (necessity of
calculation of the sine value). Generally, it is seen that both methods deliver quite
satisfactory results, with relative errors of 0.1 and 0.05%, respectively.

8.2.5.3 Measurement of Frequency and Its Deviation Using Orthogonal
Signal Components

The last from described methods adopts in a way the algorithm of magnitude
measurements with application of delayed orthogonal components. Fundamental
role plays the following function of orthogonal and delayed orthogonal

components: