Page 197 - Decision Making Applications in Modern Power Systems

160 Decision Making Applications in Modern Power Systems

values. The Kalman filter is basically used for the estimation of a state vec-
tor in a linear model of a dynamical system. But if the model is nonlinear,
Kalman filtering can be extended through a linearization procedure. The
resulting filter is referred to as the EKF in which estimation process is
divided into state prediction and state update.
1. State prediction

~ x k11jk 5 fð^ x kjk Þ ð6:30Þ

P k11jk 5 F k P kjk F k 1 Q k ð6:31Þ
The symbols B and ^ stand for predicted and estimated values,
respectively.
2. State update
The state update equation is formulated by using predicted state variable
and innovation vector.
y k 2 H k ~ x k21jk ð6:32Þ

^ x k11jk 5 ~ x k21jk 1 K k ðy k 2 H k ~ x k21jk Þ ð6:33Þ
Along with the state vector, the Kalman gain is also updated, which plays
a significant role in the improvement of the tracking behavior of the
algorithm.
T T 21
K k 5 P kjk21 H k ½H k P kjk21 H k 1R k Š ð6:34Þ
By using the update value of the Kalman gain, estimation error covari-
ance can also be updated as per the following equation:
P kjk 5 P kjk21 2 K k H k P kjk21 ð6:35Þ

6.2.3 Sparse model based adaptive filters
Sparse modeling of adaptive filters is the current research focus due to
reduction in computational complexity, which will help to design low com-
plex PQ estimation models for real-time applications. In this section, norm-
based sparsity is introduced in standard EKF algorithm.
The inherent sparsity of the filter is exploited by incorporating an ‘ 1
norm penalty into the quadratic cost function. Inclusion of ‘ 1 relaxation to
the cost function will help one to obtain the original sparse solution as com-
pared to ‘ 0 and ‘ 2 norms. The modified cost function with ‘ 1 norm penalty
can be expressed as

1
2
J 1 ðnÞ 5 e ðnÞ 1 δ:wðnÞ: ð6:36Þ
2 1