Page 193 - Decision Making Applications in Modern Power Systems

156 Decision Making Applications in Modern Power Systems

where
z k 5 a 1 sin kω 1 T s 1 φ

1 ð6:2Þ
where ω 1 is the fundamental of angular frequency, φ is the fundamental of
1
phase angle, and a 1 is the fundamental amplitude of the signal.
The observation noise, v k , is a Gaussian white noise with zero mean and
2

T
variance, σ , and the covariance of measured errors is R k 5 Ev k v . The
v k
sinusoid can be represented by using three complex state variables as
x kð1Þ 5 e jω 1 T s ð6:3Þ
x kð2Þ 5 a 1 e jðkω 1 T s 1φ 1 Þ ð6:4Þ

x kð3Þ 5 a 1 e 2jðkω 1 T s 1φ 1 Þ ð6:5Þ
The state-space model can be formulated by using state and measurement
equations as given in the following equations:

State equation x k11 5 fð x k Þ 1 Gw k ð6:6Þ
Measurement equation y k 5 Hx k 1 v k ð6:7Þ
where
T

x k 5 x kð1Þ x kð2Þ x kð3Þ ð6:8Þ
The state transition matrix can be obtained from state equation using
Taylor series expansion as
1 0 0
2 3
F k 5 4 x kð2Þ x kð1Þ 0 5 ð6:9Þ
2x kð3Þ =x 2 0 1=x kð1Þ
kð1Þ
The measurement matrix is given by

H k 5 0 20:5i 0:5i ð6:10Þ
^
Frequency, f , and amplitude, ^ aðkÞ, can be estimated from state variables
ðkÞ
as shown in the following equations:
f ^ 5 1 Imðlnð^ x kð1Þ ÞÞ ð6:11Þ
ðkÞ
2πΔT
^ aðkÞ 5 j^ x kð1Þ j ð6:12Þ

6.2.1.2 Signal model for harmonic estimation
Similar complex state-space model can also be used to estimate harmonic
parameters and decaying DC components [20]. If the power signal is