Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 157

Dy ¼ y ð n Þ y ð n 1 Þ¼ 2CU sin ð 0 : 5X Þ sin ½ð n 0 : 5 Þ X þ u þ b : ð 8 : 150 Þ

If the ratio of that difference and input magnitude is denoted by D (Dy = U ¼ D),
then:

DX ¼ 2

N 1 arcsin ð D = 2 Þ:

ð 8 : 151 Þ

This is the equation of measurement of frequency deviation which does not
require assumption of approximate value of frequency. When value of D is small
the sine function can be substituted by its argument and then:

DX ¼ D = N 1 ð 8 : 152 Þ

where DX ¼ 2p Df
f S

:

Example 8.16 Calculate relative error of frequency measurement (with respect to
the nominal value) realized with use of the simplified methods with Walsh zero-
order function. The measured frequency is equal to 49 Hz and the input signal is
sampled at 1000 Hz.

Solution According to Eq. 6.12 the ratio of output and input signals magnitudes
for given signal and sampling frequencies is equal: to

¼ sin ð 20p 49 = 1000 Þ
sin ð p 49 = 1000 Þ

Y m
U m

¼ C ¼ sin ð N 1 X = 2 Þ
sin ð X = 2 Þ

¼ sin ð N 1 pf = f s Þ
sin ð pf = f s Þ

¼ 0 : 413 :

This ratio should be measured by applying selected algorithm for magnitude
measurement, the same for the Walsh filter input and output signals, which creates
possibility of cancelling coefficients in numerator and denominator. On the other
hand, one should remember that the separated methods of orthogonalization or
orthogonalization with delay by a quarter of period cannot be used, simply because
the real period of the signal is not known (it is to be determined). Taking that into
account one can propose application of the following algorithms for calculation of
the sought magnitudes ratio, with use of the described higher algorithms for
magnitude measurement ( 8.55 ), ( 8.74 ) and ( 8.75 ):
- for any delay values (assuring non-zero values of the cancelable coefficients)

s

y ð n k Þ y ð n m Þ y ð n Þ y ð n m k Þ
u ð n k Þ u ð n m Þ u ð n Þ u ð n m k Þ

Y m
U m

¼ C ¼ ;

• for the smallest possible delay values

s

y 2 ð n 1 Þ y ð n Þ y ð n 2 Þ
u 2 ð n 1 Þ u ð n Þ u ð n 2 Þ

Y m
U m

¼ C ¼ ;