IEEE Systems, Man and Cybernetics Magazine - July 2021 - 32
where (),( ), and ()GM Maa a are the gravitational concji
stant,
mass of agent i, and mass of agent j, respectively, at
iteration ;a e is a small constant; and ()Rij
Euclidean distance between agents i and j at iteration a
and defined as
RK Kaa a
ij ij
=
G ()c
a is updated as
() ()
c
where ()G 0c
GG exp0
cab#
() (),( ).
=- a`` jj,
am
is the initial value of ();Gc
M ()i a = N
=
i()
j
aa and m
(9)
(10)
a are
the current and maximum iteration numbers, respectively;
and b is a constant. The mass is defined as
m a
,
| ()m a
j 1
where
m a =
i ()
OO
OO
-
-
i() ()
()
aa
aa
bw
w
()
iteration a is denoted by (),Ow
,
(12)
where the objective value of agent i at iteration a is denoted
by (),Oi
among all of the agents at iteration a is denoted by ().Ob
As discussed earlier, one of the objectives of this work is
a
to minimize the error in frequency variations and net change
in tie-line power flow for stable power system operation. To
achieve this, a minimization problem is considered. The
objective function considered in the proposed method is
nt
Of ,Pdt
1
ik tie
k
()
=+
==
k
1
a TT# ; hhE||
n
t
^^ 2
2
,k
where nt is the total number of tie lines, and P ,tie k
(13)
T represents
the net change in power flow in the tie lines between
the connected areas in the power systems. The worst value
of the objective function ()Ow
objective function ()Ob
a and the best value of the
OO{, ,}1
w
OO{, ,}1
b
() iN
() iN
! f
aa= ! f
min
a are given as
aa= max ()
i().
i
(14)
(15)
The total force acting on agent i imposed by all of the
agents in the search space along the dimension m is
FF(),
1,
m ()
i
aa=
= !
jj i
N
where rand [, ].01j
!
| randj
ij
m
(16)
Now, mimicking the Newtonian laws
of motion, the acceleration of agent i in dimension m at
iteration a is
ai
m ()
a
=
=
00M if
otherwise.
i a =
()
M
F
m
i
i
()
()
a
a
It is noteworthy that, in each iteration, the mass of the
agent with the worst objective value will be zero.
32 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE July 2021
a the worst objective value among all of the agents at
a and the best objective value
Klh
a is called the
Thus, in (17), acceleration is considered as zero for the
worst agent in each iteration. The velocity denoted by
vi
m and position ki
updated by
vv() ()aaai#
m
m () randi
i +=
1
aa a
+= ++
i
m
i
+
kk v11i
m
where rand [, ].01i
!
hl
m
i a
() () (),
m
(18)
(19)
The index of the global best agent
can be defined as
(,)()
= aa
f
f
inde Oh il .!! (20)
1
xand
{, ,}
{, ,}
IEEE Systems, Man and Cybernetics Magazine - July 2021
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