Page 186 - Modern Control of DC-Based Power Systems

150 Modern Control of DC-Based Power Systems

source is considered as an unknown dynamic disturbance, the local
dynamic states are augmented by x di .

x i;ext 5 x i (5.106)
x di

Applying (5.106) on (5.119), (5.120) enables to write the local state-
space model:

_ x i;ext 5 A i;ext x i;ext 1 B i;ext u i (5.107)
(5.108)
_ y 5 C i;ext x i;ext 1 H i;ext x di
i
On this local state-space model the Kalman filter is designed, where
the matrices A i;ext ;B i;ext ;C i;ext ;H i;ext include the local states and the virtual
disturbance source [32]. Applying the augmented Kalman filter will lead
to (5.109) where K i;ext is augmented Kalman gain computed by solving
the algebraic Riccati equation, and ^x i;ext is the estimation of x i;ext .
_
^ x i;ext 5 A i;ext ^x i;ext 1 B i;ext u i 1 K i;ext ðx i 2 C i;ext ^x i;ext Þ (5.109)
As the observer estimates an additional state, the 2DOF of the LQR is
augmented with L ext , a term expressing the online deviation of the system
set-point due to the virtual disturbance.

5.4.1.4 Shaping Filter
The virtual disturbance was presented in the previous section. To include
the virtual disturbance inside the state estimation of the Kalman filter an
appropriate model is needed. In control and estimation theory unknown
inputs are assumed to be white noise. However, this model is not suffi-
cient to describe the influences of the rest of the network since external
currents can have a nonzero mean value and correlation time. Therefore,
a non-Gaussian signal (say, UðsÞ) can be treated by synthesizing a filter
transfer function (GðsÞ) such that:

b d
GsðÞ 5 5 (5.110)
WðsÞ
s 1 a d
Us ðÞ
Where WðsÞ is a white noise signal. Thus, by adding a new element
G to the process model, the requirement that w shall be a Gaussian signal
of zero mean may be met [47]. The element G used in this way is some-
times referred to as a coloring or shaping filter.