![](bg5.png)
Overlapping decentralized state estimator L
S
can be
implemented in the pair subsystems S
ij
by L
S
=
M
TLU
directly, or by L
S
S =
M
LU
, VL
S
=
M
TL
indirectly, based on the Theorem 3 and Theorem 4.
4.2 Fully Decentralized Estimation
Although the estimators described above have been
designed in a decentralized way, they are, essentially,
centralized. The desired features for an efficient
decentralized AGC require that each decentralized
dynamic controller and/or estimator should be
applied to its subsystem, using the measurements
only accessible to its own area (Calovic, 1972 and
1984). In order to comply with these requirements, a
modification of the overlapping decentralized
methodology has been done, leading to a fully
decentralized estimator.
The tie-line power variations depend, essentially,
on the states in both the i-th and the j-th areas.
According to (3) and (8), fully decentralized
estimators can be designed, starting from the
estimator of tie-line power variations defined by
)
ˆ
(
ˆ
eeme
PPLP −=
&
(15)
where, L
m
is an properly chosen constant, adapted to
both dynamics of the tie-line power variations and
the measurement noise. It is obvious that this
estimator is completely autonomous, independent of
the remaining parts of the state vector, having in
mind that the estimators for the remaining parts of
the local state vectors become completely decoupled.
The estimator gain matrix is now modified from (14)
to
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
=
××
××
××
jj
jj
m
m
ii
M
LL
LLL
L
LL
L
4919
1111
1949
00
00
000
00
~
. (16)
As far as the types of expansion are concerned, fully
decentralized estimator schemes are designed in
parallel with the overlapping decentralized ones.
5 EXPERIMENTAL RESULTS
The efficiency of the described estimation schemes
applied to AGC has been tested by simulation. All
the experiments have been done in the case that the
estimators have been implemented together with the
corresponding gain matrices mapping the state
estimates to the control signals. These gain matrices
have been obtained by using the methodology
(Stankovic et al., 1999), based on expansion,
decomposition to subsystems and the local
application of the LQG optimal design. In order to
get a better practical feeling about the quality of
different estimators, responses to a step load
disturbance in area i have been analyzed.
For the pair of subsystems S
ij
, without losing
generality, assume i = 1, j = 2, let the parameters of
the system matrices in (7) correspond to the
references (Chen, 1994; Calovic, 1984; Stankovic et
al., 1999), and have expanding matrices be (10).
Consider the non-balance case of area 1 and area 2,
that is a steady load normalization factor α
12
= P
10
/
P
20
=10. Therefore, choose β = 0.1 and step
disturbance is 0.01 with 5% white noises in ξ
1
.
When y
i
= [P
T
, P
H
, f, v, P
e
]
i
T
, i = 1, 2, the estimators
are designed for full measurement sets; while y
i
= [f,
v, P
e
]
i
T
, i=1,2, the estimators for reduced
measurement sets. The following notation has been
adopted for estimator designs: (1) Overlapping
decentralized (OD) scheme, full measurement sets
(FMS); (2) OD scheme, reduced measurement sets
(RMS); (3) Fully decentralized (FD) scheme, FMS;
(4) FD scheme, RMS. In the case of OD scheme,
L
m
=0; and L
m
=120 for FD.
In Figure 1 (a), differences between the globally
optimal estimation errors (obtained by implementing
the globally optimal LQG regulator for the entire
model (7)) and the estimation errors obtained by the
proposed estimators are depicted for f
1
, P
e
and f
2
, all
the noise terms are set to zero, in order to provide a
better insight into the corresponding dynamics.
Smooth overlapped curves are for the cases of 1 and
2, whereas fluctuant overlapped ones for 3 and 4.
Obviously, FD schemes are only slightly inferior to
OD schemes; the number of measurements does not
influence the estimation accuracy significantly.
Figure 1 (b) corresponding to the general situation,
when the stochastic effects are present. It is
interesting to observe that the estimator
decentralization does not degrade the noise
immunity significantly; however, the reduction of
the number of measurements leads in both OD and
FD cases to a visible increase of the estimation error.
This estimator parameter L
m
plays an important role
in achieving the desired overall system performance.
Figure 1 (c) shows the estimation error differences
when L
m
= 20 for FD schemes, corresponding to
Figure 1 (b). Obviously, the estimation quality is
deteriorated.
In Figure 2 (a) and Figure 2 (b), the true states
are represented, together with their estimates, for
OD scheme / FMS case and FD scheme / RMS case.
The estimation accuracy is obvious; the bias,
especially pronounced in f
1
, represents a
DECENTRALIZED ESTIMATION FOR AGC OF POWER SYSTEMS
261