Measurement Algorithms for Digital Protection - Digital Signal Processing in Power System Protection and Control - page 146

Both equations can be written in matrix form:

" # R

L

¼ :

i k þ 1 i k 1
2T S

i k

u k
u k þ 1

ð 8 : 113 Þ

i k þ 2 i k
2T S

i k þ 1

Having voltage and current samples, we can solve the equation to find unknown
R, L:

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

; ð 8 : 114 Þ

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

L ¼ 2T S ð 8 : 115 Þ

A very interesting feature of the solution is the result independence of the
possible presence of decaying DC component. This is important advantage of the
method since in many algorithms it is a source of comparatively high errors that
are usually difficult to remove. The disadvantage of this method is its sensitivity to
high frequency noise. It can be eliminated by the method described below.

• calculation of R, L using discrete model of integral of differential equation of
fault loop.
The fundamental method of suppressing oscillatory noise in the above method
relies on calculating an integral of the differential equation of fault loop ( 8.110 ).
Doing that for two short time periods yields two equations:

Z

t k þ 1

u ð t Þ dt ¼ R Z

t k þ 1

i ð t Þ dt þ L ½ i ð t k þ 1 Þ i ð t k Þ; ð 8 : 116a Þ

t k t k

Z

t k þ 2

u ð t Þ dt ¼ R Z

t k þ 2

i ð t Þ dt þ L ½ i ð t k þ 2 Þ i ð t k þ 1 Þ: ð 8 : 116b Þ

t k þ 1 t k þ 1

Calculating these integrals assuming trapezoidal integration rule one gets:

Z

t k þ 1

u ð t Þ dt ¼ T S
2

½ u ð t k Þþ u ð t k þ 1 Þ ð 8 : 117a Þ

t k

and

Z

t k þ 1

i ð t Þ dt ¼ T S
2

½ i ð t k Þþ i ð t k þ 1 Þ: ð 8 : 117b Þ

t k