Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 125

and similarly, when the current is delayed:

X

N 1

Q 1 ð n Þ¼ 1
N

u 1 ð n k Þ i 1 ð n k N 1 = 4 Þ; ð 8 : 46b Þ

k ¼ 0

where N ¼ mN 1 = 2 :
It can be noticed that it is possible to get very simple algorithms using aver-
aging, which results in Eqs. 8.39 , 8.43 , 8.45 and 8.46a , b . In the first case only the
measurement error depends on sampling frequency. On the other hand, the algo-
rithms are sensitive to noise of either high or low frequency. The resulting errors
can be removed by prefiltering.

Example 8.3 With the averaging approach provide the equations of half- and full-
cycle algorithms for signal magnitude and power components measurement for the
sampling rate equal to 1000 Hz. Assess influence of sampling frequency on the
measurement errors. Specify sampling frequency and/or measurement period to
get errors less than 0.1%.

Solution For the assumed sampling frequency the number of samples per full
cycle of fundamental frequency component equals N 1 ¼ f S = f 1 ¼ 1000 = 50 ¼ 20.
Thus, for half-cycle algorithms one obtains N ¼ N 1 = 2 ¼ 10 and from Eqs. 8.39 ,
8.43 , 8.45 and 8.46a , b one gets:
X 1m ð n Þ¼ 0 : 157 X

9

j x 1 ð n k Þj;

k ¼ 0

t

X

9

x 1 ð n k Þ X 1m ð N Þ¼ 0 : 447 ;

k ¼ 0

P 1 ¼ 0 : 1 X

9

u 1 ð n k Þ i 1 ð n k Þ;

k ¼ 0

Q 1 ¼ 0 : 1 X

9

u 1 ð n k Þ i 1 ð n k 5 Þ:

k ¼ 0

Full-cycle algorithms are obtained by taking twice longer averaging window, i.e.
for N ¼ N 1 ¼ 20 : Comparing to the above the scaling coefficients and summing
limits are changed, yielding:
X 1m ð n Þ¼ 7 : 85 10 3 X

19

j x 1 ð n k Þj;

k ¼ 0

t

X

19

x 1 ð n k Þ

X 1m ð n Þ¼ 0 : 316 ;

k ¼ 0