Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 147

Substituting the integrals ( 8.117a ) and similar sum to ( 8.113 ) the matrix
equation is reached:

R
L

¼ :

0 : 5T S ð i k þ 1 þ i k Þ i k þ 1 i k 0 : 5T S ð u k þ 1 þ u k Þ

0 : 5T S ð u k þ 2 þ u k þ 1 Þ

ð 8 : 118 Þ

0 : 5T S ð i k þ 2 þ i k þ 1 Þ i k þ 2 i k þ 1

Solving it we get sought values R and L:

R ¼ ð u k þ 1 þ u k Þð i k þ 2 i k þ 1 Þð u k þ 2 þ u k þ 1 Þð i k þ 1 i k Þ
2 ð i k i k þ 2 i k þ 1 Þ

; ð 8 : 119 Þ

ð u k þ 1 þ u k Þð i k þ 2 i k þ 1 Þð u k þ 2 þ u k þ 1 Þð i k þ 1 i k Þ
2 ð i k i k þ 2 i k þ 1 Þ

L ¼ T S : ð 8 : 120 Þ

The method is immune to high frequency noise, however, certain problem is
oscillation of denominator and its small value for substantial decaying DC in
current signal.

8.2.3.3 Measurement of Impedance Components Using Averaging

At the end of presentation the impedance measurement algorithms with application
of averaging approach are described. They result directly from general Eqs. 8.91 -
8.93 and calculation of power and magnitude using averaging ( 8.39 ), ( 8.43 ), ( 8.45 )
and ( 8.46a , b ). Assuming identical coefficients very simple equation can be reached:
Z 1 ¼ P mN 1 = 2

j u 1 ð n k Þj

k ¼ 0

ð 8 : 121 Þ

P mN 1 = 2
k ¼ 0

j i 1 ð n k Þj

or using ( 8.9a ):

Z 1 ¼ P mN 1 = 2

u 1 ð n k Þ

k ¼ 0

i 1 ð n k Þ :

ð 8 : 122 Þ

P mN 1 = 2
k ¼ 0

Constant coefficients in numerator and denominator are cancelled thanks to the
same data windows and identical algorithms for voltage and current processing.
In case of resistance and reactance the choice is smaller and the best solution is
got using ( 8.43 ):

R 1 ¼ P N 1

k ¼ 0 u 1 ð n k Þ i 1 ð n k Þ
P N 1
k ¼ 0

ð 8 : 123 Þ

i 1 ð n k Þ

and applying Eqs. 8.43 and 8.46a , b the reactance becomes:
X 1 ¼ P N 1

u 1 ð n k N 1 = 4 Þ i 1 ð n k Þ
P N 1
k ¼ 0

k ¼ 0

: ð 8 : 124 Þ

i 1 ð n k Þ