Page 195 - Decision Making Applications in Modern Power Systems

158 Decision Making Applications in Modern Power Systems

^ a 1 ðkÞ 5 j^ x kð2Þ j
^ a 3 ðkÞ 5 j^ x kð4Þ j ð6:17Þ
^ a 5 ðkÞ 5 j^ x kð6Þ j
" !#
Λ x kð2Þ
1
φ 5 imag log ð6:18Þ
1 k
a 1 x
kð1Þ
" !#
Λ
1
φ 5 imag log x kð4Þ ð6:19Þ
3
a 3 x 3k
kð1Þ
" !#
Λ
1
φ 5 imag log x kð6Þ ð6:20Þ
5
a 5 x 5k
kð1Þ
6.2.2 Adaptive filtering algorithms for power quality estimation
The state-space model discussed in the previous section cannot track and
estimate the time-varying PQ disturbances if the state variables are not
updated recursively. The updated state variables will provide the estimated
values of amplitude, frequency, and phase parameters of distorted power sig-
nals. The mathematical formulation and weight update equations have been
described in this section. The adaptive filtering algorithm starts from a prede-
termined set of initial conditions, which represents some statistical behavior
of the environment. In the case of stationary environment, the algorithm con-
verges to the optimum Wiener solution in some statistical sense after succes-
sive iterations. In a nonstationary environment such as PQ disturbances, the
algorithm can track time variations in the statistics of the input data.

6.2.2.1 Least mean square algorithm
LMS algorithm is simple to implement and is a class of stochastic gradient
algorithm. According to LMS algorithm, recursive relation for updating the
tap weight vector can be expressed as

^ wðn 1 1Þ 5 ^ wðnÞ 1 μuðnÞe ðnÞ ð6:21Þ
In the weight updating expression, the filter output is given by
H
yðnÞ 5 ^ wðnÞu ðnÞ ð6:22Þ
and estimation error is given by

e ðnÞ 5 d ðnÞ 2 yðnÞ ð6:23Þ
The step size parameter, μ, plays a vital role for the convergence of the
algorithm.