IEEE Systems, Man and Cybernetics Magazine - January 2023 - 52

where N represents the total number of agents, l
denotes the dimension of the search space, and
dm ,,l
m6f=
i
1
1
dk
2
ii
1
2
== and dki
3
= [refer to (24)]. Thus, in the
3
positive constant. The mass of the agent is updated in
the following way:
denotes the decision variable represented
through agent i. In [2] and [11], the controller gains
were represented as positions of the agents such that
dk ,,
optimization process, the dimension of the search
space was l = 3.
In the proposed method, by modulating the values of
d the parameters 2d and 3d can be obtained as per
1 ,
(22) and (23), respectively. Consequently, the optimization
approach has only one decision variable
given in
1 ,
Dd iN1
i== ,
[]6f, .
i
l
1 ,
d as
(25)
The search space is initially populated by the values of
the parameter d which are drawn randomly by satisfying
the following equation:
(, )
i
lb
m
ub
m
where dlb
m
and dub
m
dd dm li N11
,, and
,,
m ! 6f f== (26)
the decision variable, respectively. Each ()D 0i
are the lower and upper boundaries of
needs to
satisfy (8) to populate the search space, which imposes
additional computation in [2], [11], and [23]. On the contrary,
in the proposed method, the lower boundary of the
agents at (26) is defined as
dlb
=+e
a
a
11
10
considering e as a small positive number. All of the
agents can be initialized from the stable operating zone,
and the search space remains bounded by the stable
operating zone. Thus, the proposed method can influence
the controller gains (k1, k2, and k3) and the system performances
by only modulating the parameter
d The force
1 .
acting on the ith agent by the jth agent at iteration n is
defined in
Fn Gn () ()# Rn
=
ij
l
Mn Mn() ()
ij() + v
ij
#
# dd^ ^^
j
l
i
l
nn .hhh
-
(27)
G(n) represents the gravitational constant; () andMni
Mnj () are the masses of agents i and j, respectively; v is
a small positive constant; and the Euclidean distance
between agents i and j is represented by ()
Rn as
ij
Rn Dn Dn
ij ij
=
() (),( ).
G(n) is defined in the following way:
()
GGn =-0 # b`
nm
n j.
(28)
M n
i ()
N
k=1
The objective function value as obtained by the agent i
is denoted by (),Oni
values are denoted by () and (),Onw
and the best and the worst objective
Onb
respectively. In
the proposed method, besides stability, a process to
improve the transient and the steady state performances is
also incorporated. Accordingly, the following objective
functions are considered in this study:
OF1
OF MP=+3_
OK3
2
= min 3
min
= min
OF MKP
4=+3min_
+
i .
i
(31)
(32)
(33)
(34)
The minimization problems of the error and the norms
of the feedback gain vector are represented through O1 in
(31) and O3 in (33) [24], respectively. In the LFC problem,
one key aspect is to maintain the maximum variation of
frequency within the tolerance limit. Thus, an objective
function including a combination of the steady state and
the transient performances is considered as in (32) and
(34). Here,
K () () ()=+ +
kkk [24], and Mp
12 22
32
defines the maximum peak of the output response. The
parameters ()
Onw
and () are defined as
On max ()
Onb
On min On
{, ,}
w ()=kN
b ()= k! f1 N
1
! f{, ,}Onk
k().
(35)
(36)
The force exerted on agent i by other agents is given by
() .
Fn =() |rand Fn
= !1,
i
l
kk i
N
k
ik
l
The acceleration of agent i is given by
;
anh 00i^
otherwise
i
l
^
==
=
if Mnh
Mn
Fn
i
l
i
^
^
h
h
;.
(38)
The velocity and the position of an agent are updated as
()
vn 1
i
l
dn dn vn
+= ++
i
l
i
l
i 11
l
+= # () ()
() () ()
rand vn an
i
i
l
+
i
l
== -
-
| ()
mn
mn
i ^
k
h
wheremnh
^
()
i bw
i
On On
On On
() ()
w
()
.
(30)
(37)
(39)
(40)
(29)
The initial value of G(n) is represented by G0, the
maximum number of iterations is nm, and b is a
52 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE January 2023
where randk and randi are random numbers drawn from a
uniform distribution from [0,1]. The index at which the
global best agent is available is given by
(,)()@E
b
qrOnq kr n.!! (41)
,
=
indexand
,,
,,
;
kN
nnm
!
!
"
"
1
1
f
f
min 6
,
IEEE Systems, Man and Cybernetics Magazine - January 2023

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