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where Li is the Kalman gain that can be determined by aMATLAB function called “KALMAN.”
The full dynamical model of the ith subsystem including
the dynamics of the actual system in (1) and the Kalman es-timator in (26) is expressed in (28).
Dii
ii
ii
i
i
i
i
iii
i
i
i P
F
F z
Gu
B
B
x
x
AC L
A
x
x∆
+
+
+
=
0ˆˆ
0&̂&
(28)
In Fig. 3, the input of the subsystem using the estimates of state variables and interface is defined as
i Diiii z K x K u ˆˆ +−= (29)
IV. PERFORMANCE ANALYSIS
The stability of the entire interconnected system, called
here the composite or actual system, when the proposed con-trollers are implemented, is of a major concern rather thanthat of individual subsystems. The composite system has the
following state-space model and control input:
D P F Bu Ax x ∆++=& (30)
Kxu −= (31)
whereT
T N
T T T N
T T x x x x x x x
=
&L
&&&L&& ˆˆˆ
2121
[ ]T N uuuu L21= , [ ]T
DN D D D P P P P ∆∆∆=∆ L21
=
N N N
N N N
N
N
AC L
AC L
AC L
AGG
G AG
GG A
A
ˆ0000
0ˆ000
00ˆ00
00
00
000
222
111
21
2221
1121
LL
MOMMOM
LL
LL
OMMOM
M
LL
T
T N
T N
T T
T T
B B
B B
B B
B
=
LLOMOM
MM
LL
00
00
0000
22
11
jijiij S T GG = , [ ]4 4 34 4 21
L
j xof rowsof number
jS 0001=
=
N
NxN
K
K
K
K
L
OM
M
L
0
0
00
0 2
1
, [ ] Diii K K K −=
The system matrix ( A) can be written as
G A A~~
+= (32)
where
A~
: Ideal system matrix without interface consideration, i.e.,
A A =~
where any 0=ijG
G
~
: Interface matrix
The closed loop system is:
Dcl D P F x A P F x BK A x ∆+=∆+−= )(& (33)
( ) G AG BK A A cl cl ~~~~
+=+−= (34)
The performance of the composite closed-loop system ( Acl )
is analyzed based on its eigenvalues. Let µ be any eigenvalue
of the composite closed-loop system, i.e. an eigenvalue of Acl ,
and λ i be the ith eigenvalue of cl A~
. The objective of this sec-
tion is to estimate the maximum departure of µ from the ei-
genvalues of cl A~ . For this purpose let:
{ }ncl diag DT AT ~211
,,,~
λ λ λ L==− (35)
then
Γ+=+= −−− DT GT T AT T AT cl cl ~~ 111
(36)
Gershgorin Circle Theorem [6] states:
∑ Γ≤−∈=ℑ=
n
jijii d C d
~
1
: λ (37)
iℑ : Gershgorin disk’s region
C : Complex domain
ijΓ : (i-j) entity of the matrixΓ From Gershgorin’s theorem, a Gershgorin disk’s region
gives an estimate of the maximum departure of the composite
closed-loop system eigenvalues (µ) from the eigenvalues (λi)
of the closed-loop control area model. Since the strength of the interconnection is preset, one can guarantee closed-loop
stability of the composite system by properly designing the
area controllers.
V. CASE STUDY
A power system, which consists of three control areas in-
terconnected through a number of tie-lines as shown in Fig. 4
This test system is more realistic than the previous one. Now area 1 has an additional generating unit identical to the
one in area 2. It is used to demonstrate the ability of the pro-
posed controllers to allocate generating units’s outputs ac-cording to a given load following contract described as
pu
P
P
P
P
P
P
P
dc
dc
dc
G
G
G
G
−=
∆
∆
∆
=
∆
∆
∆∆
6.0
0325.0
0475.0
05.0
8.000
1.08.00
1.004.0
02.06.0
3
2
1
4
3
2
1
4 4 34 4 21α
Later the identical changes in demands in the previous test
system are applied. Then the total change in demand for
which each area has to be responsible can be obtained as
pu
P
P
P
P
P
P
P
G
G
G
G
D
D
D
=
∆
∆
∆
∆
=
∆
∆
∆
0.06
0.0325-
0975.0
1000
0100
0011
4
3
2
1
3
2
1
After the simulation is completed, the changes in turbine
power (∆PT) of each unit are shown in Fig. 9.
Fig. 9. Changes in turbine power for test system #2.
The area control error and frequency deviation of each area
are shown as Fig. 10 and 11 respectively.
Fig. 10. Area control error for test system #2
Fig. 11. Frequency deviation for test system #2.
VII. CONCLUSION
This paper proposes a decentralized controller for the load
frequency control operated as a load following service. Decen-
tralization is achieved by developing a model for the interfacevariables, which is a combination of frequencies of other sub-
systems. To account for the modeling uncertainties, a local
Kalman filter is designed to estimate each subsystem’s ownand interface variables. The controller uses these estimates,
optimizes a given performance index, and allocates generat-ing units’s outputs according to a deregulation scenario. The
performance of the proposed controllers is assessed through
eigenanalysis and simulation of two test systems, each con-
sisting of a three-area power system. The first system has onegenerator in each area, and the second has one area with two
units operating under a deregulation scenario. It is shown that
the proposed technique gives good results.
VIII. REFERENCES
[1] C. E. Fosha, Jr. and O. I. Elgerd, “The Megawatt-Frequency ControlProblem: A New Approach Via Optimal Control Theory,” IEEE Trans-actions on Power Systems, vol. 89, no. 4, pp. 563-577, April 1970.
[2] M. L. Kothari, N. Sinha and M. Rafi, “Automatic Generation Control of an Interconnected Power System Under Deregulated Environment,”
Power Quality’ 98, pp. 95-102, 1998.
[3] R. D. Christie and A. Bose, “Load Frequency Control Issues in Power System Operations after Deregulation”, IEEE Transaction on Power Systems, Vol. 11, No. 3, pp. 1191-1200, August 1996.
[4] R. D. Christie and A. Bose, “Load Frequency Control In Hybrid Elec-
tric Power Markets,” Proceedings of the 1996 IEEE International Con- ference on Control Applications, pp. 432-436, September, 1996.
[5] J. B. Burl, Linear Optimal Control , Addison Wesley Longman, Inc.,1999.
[6] G. H. Golub and A.F. Van Loan, “Matrix Computations,” 2nd edition,
Baltimore: The John Hopkins university Press, 1989.
IV. BIOGRAPHIES
Dulpichet Rerkpreedapong received his MSEE from the CSEE Depart-
ment, West Virginia University in 1999. He is currently working towards a
Ph.D. in Electrical Engineering at WVU. His research interests are in power systems control and operation, and power systems restructuring.
Ali Feliachi received the MS and PhD degree in electrical engineering
from Georgia Tech in 1979 and 1983 respectively. He joined the faculty of Electrical and Computer Engineering at West Virginia University in January
1984 where he is now a Full Professor and the holder of the endowed Electric
Power Systems Chair position. His research interests are in modeling,
simulation, control and estimation of large-scale systems with emphasis onelectric power systems.