IEEE Systems, Man and Cybernetics Magazine - January 2023 - 50

for each considered number of agents in each iteration.
Furthermore, the components of the controller gains are
generally considered as the decision variables in these
algorithms. This motivated the proposed work, which formulates
the optimization algorithm with a reduced number
of decision variables. The key contributions of this work
are highlighted as follows:
◆ A generalized range of the system stability of an LFC
system is defined. It is dependent on parameters of the
considered system.
◆ In the existing literature, a study to define the feasible
variations of a system parameter for a particular
controller gain is missing [4], [5], [6], [7], [9], [11].
Since a power system model has mixed uncertainties
of different parameters, this issue becomes more challenging.
The proposed method is useful to address
such issues.
◆ The proposed method ensures a stable range of controller
operations and reduces the number of decision
variables of the optimization algorithm. Here, GSA is
used as an optimization algorithm.
◆ Four objective functions are explored to influence
both the steady state and the transient performances
of the system.
The article is arranged in the following way. The proposed
methodology is presented in the " Methodology " section,
the results and the comparative performances are
presented in the " Results " section, and the final section
provides the conclusion of this study.
Methodology
In this section, the considered system and the proposed
optimized controller are presented.
Design of an LFC
An LFC system [11], [13] with an SFC as shown in Figure
1 is considered. The details of the system without the
controller are available in [11] and [13]. The considered
−
+
−
+
k3
+
k2
kt
x3
1 + sTt
Turbine
x2
ksg
1 + sTsg
Speed
Governor
+
∆PC(s)
−
1
R
Speed
Regulation
Figure 1. An LFC system with an SFC. (Source: [11]; used with
permission.)
50 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE January 2023
k1
+
∆PD(s)
Generator and Load Model
kps
1 + sTps
system has a generator and load model, a turbine, a
speed governor, and a speed regulation system. The
parameters kps, kt, and ksg characterize the gains, and Tps,
Tt, and Tsg characterize the time constants of the generator
and load model, the turbine, and the speed governor,
respectively. The parameters R and PD
D characterize
the speed regulation of the governor and the load disturbances.
The controller is represented by
K [, ,],kkk
12 3
=
where the components k1, k2, and k3 modulate the system
output considering the outputs of the generator and load
model, the speed governor, and the turbine, respectively.
The state equations of the system in Figure 1 considering
a free governor operation ()P 0C
D = are as follows [14]:
xo =1
-
c
c
xo
=-c
kk +1
Tps
1
ps
kk 1
Tps
ps ()
-
3
21 2
sg
x T
k
RT
k
x T
k
32 3
t
sg
o =-a
t
t
k
x T x
1
a
-a
k
m
3 -c
k
sg
.
The characteristic equation of the system is given by
sA sA sA 0
3++ +=
2
1
where
A =
1
A =
2
A =
3
TT TT TT kk 1
t
ps
sg ps
RT Tk kRTk kk T
t
()() ps
tt
++ +- sg ps
RTTT
sg
Rk kk kk kk k
RT TTt
() ()+- +ps
12
3
sg ps
1
sg ps
tt
.
ps
1
12
ps sg
t ++ +1^ht sg ps
TT Tps sg
(5)
(6)
(7)
The system stability can be ensured using the Routh-
Hurwitz criterion by satisfying the necessary and sufficient
conditions as
AA A
A
A =
4
x1
∆F(s)
12 and
-
22 2
2
,, ,
.
3
12 3
1
AA A
00 0
(8)
Next, modulation of the values of k1, k2,
and k3 while satisfying (8) will ensure system
stability and improve system performance.
The next section deals with the
problem pertaining to the stable ranges of
controller operations.
Stable Ranges of Operation
With a Reduced Number
of Decision Variables
To define the stability range, (8) with the
help of (5) can be written as
kP1
2- 1
(9)
23
x T x
1
mx1 -c
m
kk
Tps
ps
2
ps
ps
m
m
x2
TPD
(1)
(2)
(3)
(4)

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