Page 194 - Decision Making Applications in Modern Power Systems

Adaptive estimation and tracking of power quality disturbances Chapter | 6 157

considered with fundamental, third, and fifth harmonics and decaying DC
component, state-space model can be formulated using nine complex state
variables.
x k ð1Þ 5 e
8 jω 1 T s 9
> >
> >
> >
x k ð2Þ 5 a 1 e
> jðkω 1 T s 1φ 1 Þ >
> >
> >
> >
> >
x k ð3Þ 5 a 1 e
> 2jðkω 1 T s 1φ 1 Þ >
> >
> >
> >
> >
> jðk3ω 1 T s 1φ 3 Þ >
> x k ð4Þ 5 a 3 e >
> >
< =
x k ð5Þ 5 a 3 e 2jðk3ω 1 T s 1φ 3 Þ ð6:13Þ
> >
> >
> x k ð6Þ 5 a 5 e jðk5ω 1 T s 1φ 5 Þ >
> >
> >
> >
> >
x k ð7Þ 5 a 5 e
> 2jðk5ω 1 T s 1φ 5 Þ >
> >
> >
> >
x k ð8Þ 5 a DC e
> >
> 2αkT s >
> >
> >
> >
: ;
x k ð9Þ 5 e 2αkT s
The corresponding state vector can be expressed as
T
x k 5 x k ð1Þ x k ð2Þ x k ð3Þ x k ð4Þ x k ð5Þ x k ð6Þ x k ð7Þ x k ð8Þ x k ð9Þ
ð6:14Þ
The state transition matrix, F k , and measurement matrix, H k , can be gen-
erated by Taylor series expansion neglecting higher order derivative terms.
1 0 0 0 0 0 0 0 0
0 1
B C
x k ð2Þ x k ð1Þ 0 0 0 0 0 0 0
B C
B C
2 x k ð3Þ 1
B C
0 0 0 0 0 0 0
2
B C
B x k ð1Þ x k ð1Þ C
B C
B C
B x k ð4Þ 0 0 x k ð1Þ 0 0 0 0 0 C
B C
B C
2 x k ð5Þ 1
B C
F k 5 B 2 0 0 0 0 0 0 0 C
x k ð1Þ x k ð1Þ
B C
B C
B C
x k ð6Þ 0 0 0 0 x k ð1Þ 0 0 0
B C
B C
B 2 x k ð7Þ 1 C
B 0 0 0 0 0 0 0 C
2
B C
x k ð1Þ x k ð1Þ
B C
B C
0 0 0 0 0 0 0
B C
@ x k ð9Þ x k ð8Þ A
0 0 0 0 0 0 0 0 e 2αT s
ð6:15Þ

H k 5 0 20:5i 0:5i 2 05i 0:5i 2 0:5i 0:5i 11 ð6:16Þ
The amplitudes and phases of the harmonics can be estimated as shown
from Eqs. (6.17) to (6.20).