Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
Vol.
5, No. 6, Decem
ber
2015, pp. 1234~
1
244
I
S
SN
: 208
8-8
7
0
8
1
234
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Non-Convex E
c
onomic Dispat
ch
with P
r
ohibited Operati
n
g
Zones through Gravitatio
nal Search Algorithm
P.K. Ho
ta
*,
N.C
.
Sa
hu**
* Department of
Electrical Eng
i
n
eering
,
Veer Su
r
e
ndra Sai Univ
er
sity
of
Technolo
g
y
, Burla, India
** Departmen
t
o
f
Electr
i
cal
Engineering
,
I
TER
, S
OA University
,
Bhubaneswar, In
dia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Dec 28, 2014
Rev
i
sed
Jun
29,
201
5
Accepte
d
J
u
l 18, 2015
This pap
e
r presents a n
e
w ap
proach
to th
e
solution of op
timal power
generation for
economic lo
ad
dispatch
(
ELD)
using grav
itational sear
ch
algorithm
(GSA) when a
ll
the
genera
tors inc
l
u
d
e va
lve po
int
effec
t
s an
d
some/all
of the generators
have prohib
ited
oper
a
ting zon
e
s. In
this paper
a
gravitation
a
l search algorithm
is sugge
sted th
at deals with equality
and
inequa
lit
y
constr
aints in E
L
D problem
s. A constra
i
nt tre
a
tm
ent m
e
chanism
is
als
o
dis
c
us
s
e
d to acce
lera
te
the optim
izat
i
on proces
s
.
To verif
y
th
e
robustness and superiority
of th
e propos
ed GSA
based approach
, a practical
s
i
zed 40-g
e
ner
a
t
o
rs
cas
e wi
th v
a
lve
po
int
effects and prohib
ited operating
zones
is
cons
idered.
The s
i
m
u
lation res
u
l
t
s
rev
eal th
at th
e pro
pos
ed GS
A
approach
ensures convergen
ce within
an acceptab
le ex
ecution time and
provides high
ly
optimal solution
as compar
ed to
the results ob
tained from
well establ
ished heuristi
c
optimization
approaches
.
Keyword:
Econom
ic load dis
p
atch
Ev
ol
ut
i
o
nary
p
r
o
g
ram
m
i
ng
Grav
itatio
n
a
l search algo
rithm
Pro
h
i
b
i
t
e
d ope
rat
i
n
g
zo
nes
Valv
e po
in
t
effects
Copyright ©
201
5 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
P.
K.
Ho
ta,
Depa
rt
m
e
nt
of
El
ect
ri
cal
Engi
neeri
n
g
,
Veer Su
ren
d
ra
Sai Un
iv
ersity o
f
Techno
log
y
,
B
u
rl
a, PI
N:
76
80
1
8
, O
d
i
s
ha
, In
di
a.
Em
a
il: p
_
h
o
t
a@red
i
ffm
ail.co
m
1.
INTRODUCTION
Econ
o
m
ic lo
ad d
i
sp
atch
is an
i
m
p
o
r
tan
t
po
wer system
o
p
timizatio
n
task
an
d on
e
o
f
th
e
fu
nd
am
en
tal
i
ssues o
f
p
o
w
e
r sy
st
em
operat
i
on f
o
r sc
he
dul
i
ng ge
ne
rat
i
on
am
ong t
h
e co
m
m
i
t
t
e
d gener
a
t
o
rs w
h
i
l
e
sat
i
s
fy
i
n
g
sy
st
em
const
r
ai
nt
s an
d m
i
nim
i
zi
ng t
h
e cos
t
of e
n
er
gy
re
q
u
i
r
em
ent
s
. Fo
r
sol
v
i
ng
ELD
pr
o
b
l
e
m
s
, prev
i
ousl
y
classical
m
e
thods [1] have
been
s
u
ccess
f
ully em
ployed with s
o
m
e
approxim
atio
ns du
e to
nonline
a
r
charact
e
r
i
s
t
i
c
s of p
r
act
i
cal
sy
st
em
s [2]
.
Ho
weve
r, s
u
ch a
p
pr
o
x
i
m
at
i
ons m
a
y
cause t
o
huge r
e
ve
n
u
e l
o
ss ove
r
the passa
ge
of tim
e
. The c
l
assical
m
a
the
m
atical
prog
ra
m
m
i
ng such
as l
i
n
ear
pr
og
ram
m
i
ng, q
u
a
d
rat
i
c
pr
o
g
ram
m
i
ng and i
n
t
e
ri
or
po
i
n
t
al
gori
t
h
m
,
et
c., pr
od
uce p
r
om
i
s
i
ng econ
o
m
i
c generat
i
o
n sche
dul
i
ng r
e
sul
t
s
whe
n
t
h
e
fuel
cost
c
u
r
v
e i
s
co
nsi
d
e
r
ed
as
m
onot
o
n
i
cal
l
y
i
n
creasi
n
g
o
n
e.
H
o
we
ve
r,
whe
n
t
h
e
pr
o
b
l
em
i
s
hi
g
h
l
y
n
o
n
l
i
n
e
a
r
or
has
n
o
n
-
s
m
oot
h cost
f
unct
i
o
ns
, s
o
m
e
o
f
t
h
e
s
e t
ech
ni
q
u
es m
a
y
no
t
be a
b
l
e
t
o
pr
od
uc
e
g
ood
so
lu
tion
s
.
In
past two decades, stoc
ha
stic search algorith
m
s
like genetic algorithm (GA)
[3], e
vol
utiona
ry
pr
o
g
ram
m
i
ng (
E
P)
[4]
a
nd si
m
u
l
a
t
e
d an
nea
l
i
ng [
5
]
m
a
y
p
r
o
v
e t
o
be
ver
y
effi
ci
ent
i
n
s
o
l
v
i
n
g c
o
m
p
l
e
x E
L
D
pr
o
b
l
e
m
s
but
i
t
s co
nt
rol
para
m
e
t
e
rs t
uni
n
g
i
s
di
f
f
i
c
ul
t
task
. Tabu
sear
ch
[6
],
p
a
r
ticle swarm
o
p
t
i
m
iza
t
i
o
n
[7
]-
[8] and
neural
network approaches
[9]-[11] have
been a
ppl
ied success
f
ul
l
y
but, these m
e
thods do not always
gua
ra
nt
ee t
o
h
a
ve t
h
e
gl
o
b
al
l
y
opt
im
al
sol
u
t
i
o
n
.
T
h
e
recen
t
research
has id
en
tified
few drawb
a
ck
s o
f
th
e
st
ochast
i
c
m
e
tho
d
s l
i
ke p
r
em
at
ure c
o
nve
rge
n
ce
of
G
A
ca
u
s
i
ng
de
gra
d
at
i
o
n
i
n
per
f
o
rm
ance a
n
d
re
d
u
ct
i
on i
t
s
search
cap
a
b
ility an
d
un
su
it
ab
le wh
en
app
lied
to
h
i
gh
ly ep
istatic o
b
j
ectiv
e fun
c
tion
s
(i.e.,
wh
ere th
e
param
e
t
e
rs bei
ng
o
p
t
i
m
i
zed are hi
ghl
y
c
o
r
r
el
at
ed).
M
a
ny
researc
h
e
r
s
h
a
ve s
o
l
v
e
d
t
h
e
ELD
pr
o
b
l
e
m
wi
t
h
val
v
e p
o
i
n
t
ef
f
ect
s of ge
nerat
o
rs e
ffi
ci
ent
l
y
by
usi
n
g t
h
e a
b
o
v
e m
e
nt
i
one
d he
uri
s
t
i
c
opt
i
m
i
zat
i
on t
echn
i
ques
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Non-Convex Economic Dis
patch with Prohibited
Op
erating Zones t
h
rough
Gr
avit
ational… (P.K. Hota)
1
235
Bu
t, in
all th
ese m
e
th
o
d
s, t
h
e wh
o
l
e
o
f
th
e un
it op
era
ting r
a
ng
e is av
ail
a
b
l
e fo
r op
erat
ion. In
practic
e, the
ope
rat
i
n
g ran
g
e
i
s
broke
n i
n
t
o
seve
ral
di
sj
o
i
nt
sub
-re
gi
o
n
s
whe
n
pr
o
h
i
b
i
t
e
d zo
nes are p
r
esent
.
T
h
e f
u
e
l
cost
cu
rv
e
o
f
a
un
it with
p
r
oh
ib
ited
o
p
e
rating
zo
n
e
s is a d
i
sco
n
tinuo
us fun
c
tio
n
.
Thu
s
, th
e trad
ition
a
l m
e
th
ods
cann
o
t
be di
re
ct
l
y
em
pl
oy
ed t
o
s
o
l
v
e
t
h
i
s
di
spat
ch
p
r
obl
e
m
. Howe
ver, t
h
e
he
uristic se
arch technique
s
suc
h
as
GA, SA, PSO, etc., a
r
e ca
pa
ble
of ta
k
i
ng
i
n
to accoun
t the un
it’s
proh
ibited
zon
e
s, since th
ey
d
o
n
o
t
req
u
i
re
th
e fun
c
tio
n to
b
e
co
n
tinuo
us.
Ore
r
o
,
et al
. [1
2]
have a
p
p
l
i
e
d genet
i
c
al
go
ri
t
h
m
appr
o
ach t
o
sol
v
e t
h
e eco
n
o
m
i
c
di
spat
c
h
o
f
g
e
n
e
rators with
p
r
oh
ib
ited
op
erating
zon
e
s.
In
t
h
is p
a
pe
r,
t
h
ey
have
u
s
e
d
t
h
e
penal
t
y
f
unct
i
o
n
ap
pr
oa
ch t
o
han
d
l
e
t
h
e pr
o
h
i
b
i
t
e
d o
p
e
r
at
i
ng z
one co
nst
r
ai
nt
. C
h
en,
et al
. [1
3]
have a
l
so sol
v
e
d
t
h
e sam
e
pro
b
l
e
m
usi
n
g
g
e
n
e
tic algo
rith
m
wh
ere, ram
p
-rate li
mits
are also
co
n
s
i
d
ere
d
apa
r
t
fr
om
prohi
bi
t
e
d o
p
erat
i
n
g z
one
s.
Ev
ol
ut
i
o
nary
p
r
o
g
ram
m
i
ng b
a
sed ec
o
nom
i
c
di
s
p
at
ch
of
ge
nerat
o
r wi
t
h
p
r
ohi
bi
t
e
d ope
rat
i
ng
z
o
nes has bee
n
pr
o
pose
d
by
J
a
y
a
barat
h
i
,
et al.
[1
4]
. In
a
not
her
pa
pe
r, Perei
r
a
-
Net
o
,
et al
. [1
5]
ha
ve use
d
a
n
e
f
fi
ci
ent
ev
o
l
u
tio
n
a
ry strateg
y
op
ti
m
i
zatio
n
pro
cedu
r
e to
so
l
v
e th
e no
n-co
nv
ex
ELD prob
l
e
m
with
p
r
o
h
ib
ited
ope
rat
i
n
g z
one
co
nst
r
ai
nt
.
I
n
t
h
e a
b
o
v
e m
e
nt
i
one
d t
ech
ni
q
u
e
s, o
n
l
y
sm
al
l
si
ze ELD
p
r
o
b
l
em
s wi
t
h
p
r
o
h
i
bi
t
e
d
ope
rat
i
n
g
zo
ne
s
ha
ve been s
o
l
v
e
d
. H
o
we
v
e
r,
C
h
at
ur
ve
di
,
et al
. [16] ha
ve s
o
lve
d
a la
rge s
cale non-conve
x
ELD
pr
obl
em
wi
t
h
p
r
o
h
i
b
i
t
e
d o
p
erat
i
n
g z
o
nes usi
n
g a sel
f
-
o
r
g
a
n
i
z
i
ng
hi
erarc
h
i
cal
PSO
t
echni
q
u
e. Si
m
i
l
a
rl
y
,
Selvakum
ar,
et al
. [
17]
ha
ve
pr
o
pose
d
a ne
w pa
rt
i
c
l
e
swa
r
m
opt
im
i
z
at
i
on (
N
PS
O) s
o
l
u
t
i
on p
r
oce
d
ure
t
o
n
o
n
-
con
v
e
x
EL
D
p
r
o
b
l
e
m
wi
t
h
p
r
ohi
bi
t
e
d
op
erat
i
ng z
o
ne c
o
nst
r
ai
nt
. C
o
el
h
o
,
et al
. [18] have
com
b
ined
chaotic
di
ffe
re
nt
i
a
l
evol
ut
i
o
n an
d q
u
a
drat
i
c
p
r
o
g
ra
m
m
i
ng t
echni
q
u
e f
o
r ec
on
om
i
c
di
spat
ch
opt
im
i
zati
on wi
t
h
val
v
e
poi
nt effect.
Recently, a new he
uristic s
earch algor
ithm
,
na
m
e
ly gravitationa
l sea
r
ch al
gorithm (GSA)
m
o
ti
vat
e
d by
g
r
avi
t
a
t
i
onal
l
a
w an
d l
a
w
o
f
m
o
ti
on h
a
s bee
n
p
r
op
ose
d
by
R
a
shedi
,
et al
.
[19]
. T
h
ey ha
ve been
applied s
u
cces
sfully in solvi
n
g va
ri
ous non l
i
near functi
ons
. Recently, GS
A has been s
u
ccessfully applied to
ELD a
n
d
hy
d
r
ot
he
rm
al
schedul
i
n
g
p
r
o
b
l
e
m
s
[20]
-[
2
2
]
.
The
o
b
t
a
i
n
ed
r
e
sul
t
s
co
n
f
i
r
m
t
h
e
hi
g
h
per
f
o
rm
ance
and ef
fi
ci
ent
con
v
e
r
ge
nt
ch
aract
eri
s
t
i
c
s of
t
h
e pro
p
o
sed
m
e
t
hod. Fu
rt
her
,
GS
A has
a fl
exi
b
l
e
and wel
l
-
b
a
lan
c
ed
m
ech
an
ism
to
en
h
a
nce ex
p
l
oration
ab
ility. Main
o
b
j
ective o
f
th
is p
a
p
e
r is to
p
r
esen
t th
e u
s
e of GSA
o
p
tim
izat
io
n
tech
n
i
q
u
e
in
ob
t
a
in
in
g th
e
ELD resu
lts.
Hen
c
e, an
attem
p
t h
a
s b
e
en
mad
e
in
t
h
is pap
e
r to
exp
l
ore th
e
p
o
ssib
ility of ap
p
l
ying
recen
t heuristic
o
p
tim
izat
io
n
tech
n
i
q
u
e
n
a
m
e
ly g
r
av
itational sear
ch algor
ith
m
in
so
lv
i
n
g th
e larg
e scale n
on-
convex
ELD
p
r
ob
lem
with
p
r
oh
ib
ited
op
eratin
g
zon
e
s. A 4
0
-u
n
it n
on-co
nv
ex
ELD pr
ob
lem
w
ith
all practical c
o
nstraints
suc
h
as
ram
p
-rat
e
co
nst
r
ai
nt
,
pr
o
h
i
b
i
t
e
d
o
p
e
rat
i
n
g
zo
ne c
onst
r
ai
nt
, et
c.
,
has
bee
n
s
o
l
v
ed ef
fect
i
v
el
y
usi
n
g
g
r
av
itatio
n
search
algo
rith
m in
th
is
p
a
p
e
r. To inv
e
st
i
g
a
t
e t
h
e p
o
t
e
nt
i
a
l
of t
h
e
pr
o
pos
ed a
p
pr
oac
h
, t
h
e
si
m
u
latio
n
resu
lts are co
m
p
ared
t
o
th
at of recen
t
app
r
oach
es repo
rted in
th
e literatu
re. Th
e
p
r
opo
sed
m
e
t
hod
ol
o
g
y
gi
ves t
h
e c
h
ea
pest
ge
nerat
i
o
n sche
d
u
l
e
an
d o
u
t
p
e
r
f
o
rm
s pre
v
i
o
usl
y
repo
rt
ed
ot
her
m
e
t
hods
p
a
rticu
l
arly when
ap
p
lied to larg
e-scale ELD
p
r
ob
lem
s
.
2.
PROBLEM FORMUL
ATION
ELD
pr
o
b
l
e
m
is ab
out
m
i
nim
i
zi
ng t
h
e f
u
el
c
o
st
o
f
gene
rating
unit real power
outputs for a s
p
ecified
peri
od
of
operation so a
s
to
accom
p
lish optim
a
l di
spatch
am
ong t
h
e com
m
i
tted units,
while satisfying t
h
e
syste
m
co
n
s
train
t
s. Th
e
g
e
n
e
rato
rs wit
h
m
u
l
tip
le v
a
lv
e steam tu
rb
in
es
p
o
ssess a wid
e
v
a
riatio
n
in
th
e i
n
pu
t-
out
put
c
h
aract
eri
s
t
i
c
s. The
val
v
e
poi
nt
effect introduce
s ripples in t
h
e
heat
rate curves a
nd ca
nnot
be
represen
ted
b
y
th
e po
lyno
m
i
a
l
fun
c
tio
n.
Hence, th
e actual co
st curv
e is a co
m
b
in
atio
n
o
f
sin
u
s
o
i
d
a
l
fun
c
tion
and
q
u
a
d
rat
i
c
f
unct
i
o
n
re
prese
n
t
e
d
by
t
h
e
f
o
l
l
o
wi
ng
eq
uat
i
o
n.
s
i
n
,
(1)
Whe
r
e,
a
i
,
b
i
a
nd
c
i
are the
fuel-cost coe
ffici
ents of t
h
e
i
th
uni
t
an
d
e
i
,
f
i
are the constants of the
i
th
u
n
i
t with
v
a
lv
e po
in
t
effects.
Th
e
p
r
im
e o
b
jectiv
e o
f
th
e ELD
p
r
ob
lem
is to
d
e
term
in
e t
h
e m
o
st eco
nomic lo
ad
in
gs of g
e
n
e
rato
rs
to m
i
nimize the ge
neration c
o
st s
u
ch that t
h
e load dem
a
nds
P
D
in
t
h
e sch
e
du
ling
ho
r
i
zo
n can
b
e
m
e
t and
si
m
u
ltan
e
o
u
s
ly
th
e o
p
e
ration co
n
s
trai
n
t
s are satisfied
. Here, th
is con
s
train
e
d
op
timiza
tio
n
p
r
ob
lem c
a
n
b
e
written
as:
Minim
i
ze
∑
∈
Ω
(2)
Whe
r
e,
Ω
is t
h
e set of all commi
tted
u
n
its.
Th
is m
i
n
i
m
i
za
t
i
o
n
pr
ob
lem
is su
bj
ected
to a
v
a
r
i
ety of
con
s
tr
ain
t
s
depe
n
d
i
n
g u
p
o
n
assum
p
t
i
ons
and p
r
act
i
cal
im
pl
i
cati
ons l
i
k
e po
we
r bal
a
nce co
nst
r
ai
n
t
s, gene
rat
o
r o
u
t
p
ut
li
mits, ram
p
rate li
m
its an
d
pro
h
i
b
ited
o
p
erat
in
g
zon
e
s.
The
s
e constrai
nts a
n
d lim
its are di
scusse
d as
follows:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJECE
Vol. 5, No. 6, D
ecem
ber
2015 :
1234 –
1244
1
236
1)
Pow
e
r
bal
anc
e con
s
t
r
ai
nt
o
r
dem
a
n
d
co
n
s
t
r
ai
nt
: Th
e t
o
tal g
e
n
e
ration
sh
ou
l
d
b
e
equal to
th
e
to
tal syste
m
d
e
man
d
D
P
plus
the t
r
ansm
ission loss
P
LOSS
. Th
at is rep
r
esen
ted as:
∑
∈
Ω
(3)
Th
e tran
sm
issi
o
n
loss m
u
st b
e
tak
e
n
in
to
acco
un
t in
ord
e
r to
ach
iev
e
tru
e
econo
m
i
c
d
i
sp
atch
. To
calcu
late th
e tran
sm
issio
n
loss,
B
c
o
e
ffi
ci
ent
s
m
e
t
hod i
s
use
d
i
n
ge
ne
ral
.
T
h
e l
o
s
s
i
s
re
prese
n
t
e
d
by
B
coefficients.
∑∑
∑
∈
Ω
∈
Ω
∈
Ω
(4)
2)
Th
e g
e
n
e
ra
to
r
limits
: The generation output of each unit
shoul
d
be bet
w
een its
m
i
nim
u
m and
maxi
m
u
m
li
mits. That is, the
followi
ng ine
q
uality
const
r
aint for each ge
nerator should be satisfied.
,
,
∀
∈
Ω
(5)
Whe
r
e,
θ
is the set o
f
all commi
tted
u
n
its h
a
v
i
n
g
proh
ib
i
t
ed
zon
e
s,
(
Ω
-
θ
) is th
e set of all co
mmitted
u
n
its
whi
c
h are
n
o
t
havi
ng
p
r
o
h
i
b
i
t
ed zo
nes,
P
i
is th
e pow
er
ou
tpu
t
of
i
th
ge
nerat
o
r a
n
d
P
i,
min
and
P
i,max
ar
e
th
e
m
i
nim
u
m
and
m
a
xim
u
m
real
po
we
r
out
put
o
f
i
th
gene
rat
o
r.
3)
Ramp
ra
te
limits
:
In EL
D
pr
obl
em
s, t
h
e ge
nerat
o
r
o
u
t
p
ut
i
s
us
ual
l
y
assum
e
d t
o
be a
d
ju
st
e
d
sm
oot
hl
y
and i
n
st
ant
a
neo
u
sl
y
.
H
o
weve
r,
u
n
d
er
p
r
act
i
cal
circum
stances ra
m
p
rate limit restricts the ope
r
ating
rang
e
of all the on
lin
e
un
its
for adj
u
sting th
e
g
e
n
e
ration
o
p
e
ration
b
e
tween
t
w
o op
eratin
g p
e
riod
s. In o
t
h
e
r
w
o
r
d
s, t
h
is constr
ain
t
r
e
st
r
i
cts th
e
o
p
e
r
a
tin
g ran
g
e
of
th
e physical lo
w
e
r
and
u
p
p
e
r li
m
it t
o
th
e eff
ectiv
e
lo
w
e
r
li
mit
,
an
d upp
er
limit
,
, res
p
ectively.
These lim
its [15] are
de
fine
d
as:
,
= m
a
x [
P
i, min
,
P
i
0
-
DR
i
]
;
(6)
,
= min [
P
i, max
,
P
i
0
+
UR
i
]
;
(
7
)
Whe
r
e,
P
i
and
P
i
0
are the current and
pre
v
ious powe
r output of
i
th
g
e
n
e
rato
r, resp
ectiv
ely;
DR
i
and
UR
i
are the
d
o
wn
ram
p
an
d up
ram
p
limits o
f
th
e
i
th
ge
nerat
o
r as gene
ration dec
r
eases
a
n
d
inc
r
eases,
res
p
ect
ively.
Accord
ing
l
y, it is ob
tain
ed
as:
,
,
.
(8)
4)
Proh
ib
ited
opera
ting
zo
n
e
s
:
Th
e i
n
pu
t-ou
tp
u
t
ch
aracteristics o
f
m
o
d
e
rn
u
n
its are inh
e
ren
tly
n
o
n
lin
ear
b
ecau
s
e
o
f
th
e stea
m
v
a
lv
e
po
int lo
ad
i
n
g
s
.
T
h
e operating zones
due to
valve
poi
nt loa
d
i
n
g or
vi
b
r
at
i
on
due
t
o
shaft
bea
r
i
n
g i
s
gene
ral
l
y
avoi
de
d i
n
o
r
de
r t
o
achi
e
v
e
best
econ
o
m
y
, cal
l
e
d pro
h
i
bi
t
e
d
ope
rat
i
n
g zo
ne
s of a u
n
i
t
,
w
h
i
c
h
m
a
ke t
h
e cost
cur
v
e
di
sco
n
t
i
n
u
o
u
s i
n
nat
u
re
. The
pr
ohi
bi
t
e
d o
p
erat
i
n
g
zone
constraints a
r
e
descri
bed as:
Fo
r all
i
∈
,
,
(9)
,
,
2,
…
,
(10)
,
,
(11)
Whe
r
e,
,
and
,
are the lower a
n
d upper lim
its
of
k
th
pr
ohi
bi
t
e
d zo
ne f
o
r
i
th
un
it an
d
is th
e nu
m
b
er o
f
p
r
oh
ib
ited
zones of
u
n
it
i
.
The p
r
ohi
bi
t
e
d
ope
rat
i
ng z
o
n
e
const
r
ai
nt
s (
9
-1
1) a
v
oi
d t
h
e
ope
rat
i
o
n o
f
u
n
i
t
s
i
n
t
h
e pr
ohi
bi
t
e
d zo
nes.
Th
e p
r
oh
ib
ited
zon
e
s
o
f
t
h
e
di
s
p
at
cha
b
l
e
uni
t
s
di
vi
de t
h
e o
p
erat
i
n
g
re
gi
o
n
bet
w
ee
n
t
h
e m
i
nim
u
m
and
max
i
m
u
m
g
e
neratio
n lim
i
t
s in
to
+1
di
sj
oi
nt
o
p
e
r
at
i
n
g
s
u
b
-re
gi
o
n
s.
T
h
e
pr
o
pose
d
m
e
t
hod
fo
r
m
i
nim
i
zi
ng
Eq
uat
i
o
n
(
2
)
w
i
t
h
co
nst
r
ai
nt
s
defi
ned
by
E
q
u
a
t
i
on
(3
),
(5
-1
1) is
p
r
esen
ted in
th
e fo
llowing sectio
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Non-Convex Economic Dis
patch with Prohibited
Op
erating Zones t
h
rough
Gr
avit
ational… (P.K. Hota)
1
237
3.
GSA
BASE
D ECON
O
M
IC
DISP
ATC
H
In
t
h
i
s
se
ct
i
o
n
,
a
bri
e
f
descri
pt
i
o
n
of
GS
A,
t
h
e
p
r
oce
d
ure
f
o
r
G
S
A
ba
se
d ec
o
nom
i
c
l
o
ad
di
spat
c
h
,
con
s
t
r
ai
nt
sat
i
s
fact
i
o
n
t
ech
ni
q
u
e a
n
d
t
h
e
o
v
er
al
l
com
put
at
i
onal
p
r
oced
ure
have
bee
n
e
x
pl
ai
ned.
3.
1.
Grav
ita
t
iona
l Sea
rch
Al
g
o
r
i
t
hm
Th
e grav
itation
a
l search
al
g
o
rith
m
(GSA), is o
n
e
o
f
the re
cent heuristic search algorit
h
m develope
d
b
y
Rash
ed
i et
al. in
20
09
[
1
7
]
.
G
S
A is
b
a
sed
on
th
e physical law
of
g
r
av
ity and
the law
o
f
m
o
ti
o
n
. The
g
r
av
itatio
n
a
l
fo
rce b
e
t
w
een two p
a
rticle
s is
d
i
rectly propo
rtio
n
a
l to th
e
p
r
o
d
u
c
t
o
f
th
eir
masses and
inversely
pr
o
p
o
r
t
i
onal
t
o
t
h
e s
qua
re
of
t
h
e
di
st
ance
bet
w
een
t
h
em
.
In
the propose
d
a
l
gorithm
,
agents are c
onsi
d
ered as
ob
ject
s a
n
d t
h
e
i
r pe
rf
orm
a
nce
i
s
m
easured
b
y
t
h
ei
r m
a
sses:
m
i
p
p
p
P
n
i
d
i
i
i
,
2
,
1
,
,
,
,
(1
2)
Whe
r
e,
P
i
d
is t
h
e po
sitio
n
of th
e
i
th
mass in
th
e
d
th
di
m
e
nsi
on a
nd
n
is the
dim
e
nsion of the search s
p
a
ce. At
specific tim
e ‘
t
’
a g
r
av
itatio
n
a
l
force form
mass
j
act on m
a
ss
i
an
d is
d
e
fined
as fo
llo
ws:
ε
(1
3)
Whe
r
e,
M
pi
is t
h
e
p
a
ssi
v
e
g
r
av
itatio
n
a
l m
a
ss related to
ag
en
t
i
,
M
aj
is t
h
e activ
e
g
r
av
itatio
n
a
l m
a
ss related
to
agent
j
,
G
(t
)
is th
e g
r
av
itation
a
l co
nstan
t
at ti
me
t
,
R
ij
(
t
) i
s
th
e Eu
clid
ian d
i
stan
ce between the two
objects
i
and
j
, a
n
d
is a
sm
a
ll co
n
s
tan
t
.
,
(
1
4
)
The t
o
tal force
acting
o
n
th
e ag
e
n
t
i
i
n
t
h
e
di
m
e
nsi
o
n
d
is calcu
l
ated
as
fo
llo
ws.
∑
,
(
1
5
)
Whe
r
e,
ran
d
j
is a random
num
b
er in the interval [0,
1]. A
ccording to the
law of m
o
tion, the acceleration
of
the age
n
t
i
, at ti
m
e
t
, in
th
e
d
th
di
m
e
nsi
on,
α
i
d
(
t
) is
g
i
v
e
n
as fo
llo
ws:
(16)
The
next
veloc
ity of an a
g
ent
is a function
of its
c
u
rrent
velocity adde
d to its curre
nt acceleration.
Th
erefo
r
e, th
e
n
e
x
t
po
sitio
n an
d n
e
x
t
v
e
lo
cit
y
o
f
an ag
en
t can
b
e
calcu
lated
as fo
llo
ws:
1
(
1
7
)
1
1
(
1
8)
Whe
r
e,
ran
d
i
is a u
n
i
fo
rm
rand
o
m
v
a
riab
le in
th
e in
terv
al [0
, 1
]
. Th
e gravitatio
n
a
l co
n
s
tan
t
,
G
, is in
itialized
at
the be
ginning
and
will be decreased
with
the ti
m
e
to control the
searc
h
accuracy. In other
words,
G
is
fun
c
tion
o
f
t
h
e
in
itial v
a
lu
e (
G
0
) and tim
e (
t
):
,
(
1
9
)
/
(20)
The m
a
sses of the agents are calculated using f
itnes
s evaluation.
A he
avier m
a
ss
me
ans a
m
o
re
efficient age
n
t. This
m
eans that be
t
t
e
r agent
s
have hi
ghe
r at
t
r
act
i
ons an
d
m
oves
m
o
re sl
owl
y
. Su
p
posi
ng t
h
e
eq
u
a
lity o
f
th
e
g
r
av
itatio
n
a
l an
d in
ertia m
a
ss, th
e
v
a
lu
es
of
masses is calcu
lated
u
s
ing
the m
a
p
o
f
fitn
ess. Th
e
g
r
av
itatio
n
a
l an
d in
ertial
m
a
sses are upd
ated b
y
th
e fo
llowin
g
equ
a
tion
s
.
(
2
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJECE
Vol. 5, No. 6, D
ecem
ber
2015 :
1234 –
1244
1
238
∑
(
2
2
)
Whe
r
e,
represen
ts th
e
fitn
ess v
a
lu
e
of the ag
en
t
i
at time
t
, and the
bes
t
(
t
) an
d
wo
rst
(
t
) in
th
e
population
res
p
ectively, indi
cate the st
ron
g
est an
d
th
e
weak
n
e
ss ag
en
t acco
rd
ing
to
their fitn
ess
v
a
lu
e. For a
m
i
nim
i
zat
i
on p
r
o
b
l
e
m
:
∈
1,
…
,
(
2
3
)
∈
1,
…
,
(24)
3.
2.
Gra
v
i
t
ati
o
nal
Search
Al
gori
t
hm B
a
sed
E
c
on
omi
c
L
o
ad
Di
spa
t
ch
In
o
r
d
e
r to
h
a
nd
le th
e co
nstrai
n
t
s conv
en
ien
tly, th
e stru
cture o
f
so
l
u
tio
ns for ELD
p
r
o
b
l
em so
lv
ed
b
y
t
h
e p
r
o
p
o
sed
m
e
t
hod i
s
c
o
m
pos
ed
of a set
of
real
p
o
we
r o
u
t
p
ut
deci
si
o
n
vari
a
b
l
e
s f
o
r e
ach u
n
i
t
i
n
al
l
ove
r t
h
e
sche
dul
i
n
g
per
i
ods
. T
h
e sect
i
on
p
r
o
v
i
d
e
s
t
h
e sol
u
t
i
o
n m
e
tho
d
o
l
o
gy
t
o
t
h
e ab
ove
-m
ent
i
one
d ec
o
nom
ic l
o
a
d
d
i
sp
atch
p
r
o
b
l
em
s th
ro
ug
h grav
itatio
n
a
l search
algorith
m
.
3.
2.
1.
Initia
liza
t
io
n
Th
e in
itial p
opu
latio
n
is carefu
lly g
e
n
e
rated as it d
ecid
e
s
for reach
i
ng
the o
p
tim
u
m
so
lu
tio
n. It is
com
posed of
m
m
a
sses. The ele
m
ents of
each m
a
ss
are the
n
-d
im
en
sio
n
a
l po
sition
s
o
f
search
sp
ace. The
ele
m
en
ts o
f
a
mass ar
e r
a
ndomly cr
eated
p
e
r
m
u
t
atio
n
o
f
po
w
e
r
ou
tpu
t
s of
th
e g
e
n
e
r
a
ting
un
its. Each
ele
m
en
t
is u
n
i
fo
rm
ly
d
i
strib
u
t
ed
within
its feasib
le ran
g
e
. Th
e i
n
itializatio
n
mu
st satisfy all
co
n
s
trai
n
t
s g
i
v
e
n
in
sect
i
on-
2
an
d a
ccor
d
i
n
gl
y
i
s
g
e
nerat
e
d as
des
c
ri
be
d
bel
o
w.
P
i
i
s
ha
vi
n
g
u
n
i
form
l
y
di
st
ri
b
u
t
e
d
ge
nerat
i
o
n l
e
v
e
l
ran
g
i
n
g ove
r [
P
i,min
,
P
i,max
] for un
its wit
h
who
l
e of its op
eratin
g
rang
e av
ailab
l
e fo
r
op
erat
io
n
.
Bu
t,
fo
r all
un
its
with
p
r
o
h
i
b
ited
o
p
e
rating
zon
e
s, i
n
itially a rando
m
in
teg
e
r nu
m
b
er
u
r
bet
w
een 1 and
n
i
+ 1
bo
th
i
n
clusiv
e is
gene
rat
e
d
.
Thi
s
num
ber i
s
the o
p
erat
i
n
g s
u
b
-re
gi
o
n
o
f
uni
t
i
, in
wh
i
c
h
its g
e
n
e
rati
o
n
lev
e
l shou
ld
fall.
Gene
rat
i
o
n
P
i
m
u
st satisfy co
n
s
train
t
Eq.
(9) if
u
r
=1
.
P
i
m
u
st
sat
i
s
fy
c
o
n
s
t
r
ai
nt
E
quat
i
o
n
(1
1)
whe
n
u
r
=
n
i
+
1
wh
ereas, all
in
term
ed
iate n
u
m
b
er g
e
n
e
rated
b
y
u
r
resul
t
i
n
generat
i
o
n
l
e
vel
s
const
r
ai
ned
by
Eq. (
1
0
)
. Th
e
ab
ov
e m
e
n
tio
n
e
d
in
itializatio
n
p
r
o
c
edu
r
e lead
s to
p
e
rm
u
t
atio
n
of g
e
n
e
ratio
n
ou
tpu
t
s co
nfin
ed
to
operati
ng
su
b-reg
i
o
n
s
al
o
n
e
. Ho
wev
e
r, th
e in
itialized
so
lu
tion
s
, i.e.,
n
-dim
ensional m
a
sses are
P
i
= [
P
1,
P
2,
…,
P
n
],
i
=
1, 2, …,
m
a
n
d
n
-t
he n
u
m
b
er of ge
ne
rat
i
n
g
uni
t
s
. In
or
der
t
o
sat
i
s
fy
t
h
e
exact
po
we
r ba
l
a
nce con
s
t
r
ai
n
t
(Eq.
3
)
,
u
s
u
a
lly th
e
larg
est
g
e
n
e
rat
o
r withou
t proh
ib
ited op
erat
i
n
g zon
e
s is arbitrarily sel
ected as
a
de
pende
n
t unit.
Accord
ing
l
y, its ou
tpu
t
is calcu
lated
as:
∑
(
2
5
)
The powe
r
loss
P
LOSS
, is ob
tain
ed using
th
e
B
-m
at
ri
x l
o
ss f
o
rm
ul
a as de
sc
ri
be
d
by
Eq
uat
i
on
(
4
).
3.
2.
2.
Fitness E
v
alu
a
ti
on
(Ob
j
ecti
v
e F
uncti
on
)
The fitne
ss evaluation in ea
ch ag
en
t in
the p
opu
latio
n
set is ev
alu
a
ted
u
s
i
n
g
t
h
e Eq
u
a
tion
(2).
Iteratio
n co
un
t
fro
m
th
is step
,
t
=1. Update
G
(
t
), best (
t
),
w
o
rst (
t
) and
M
i
(
t
) f
o
r
i
=1, 2...
m
3.
2.
3.
Age
n
t Force
Calcul
ati
o
n
Th
e t
o
tal fo
rce
actin
g
o
n
th
e ag
en
t
i
i
n
t
h
e
di
m
e
nsi
o
n
d
is calcu
l
ated
in Equ
a
tio
n (15
)
.
3.
2.
4.
Evaluati
on of Acceleration
of an Age
n
t
The acceleration
of an a
g
ent
in
d
th
di
m
e
nsion ove
r
T
di
sp
at
ch peri
od ha
s eval
uat
e
d u
s
i
ng E
quat
i
o
n
(1
6)
.
3.
2.
5.
Upd
a
te
t
h
e A
g
en
ts’ Posi
tio
n
The
next
velocity of an a
g
e
n
t is calculated by
a
ddi
ng t
h
e acceleration of an a
g
e
n
t to the curre
nt
v
e
lo
city and
al
so
p
o
sitio
n
of an
ag
en
t
will upd
ated
.
3.
2.
6.
Stoppin
g
Criterion
There
are
va
ri
ous
cri
t
e
ri
a a
v
ai
l
a
bl
e t
o
st
o
p
a st
oc
hast
i
c
opt
i
m
i
zati
on a
l
go
ri
t
h
m
.
In t
h
i
s
pape
r, t
o
co
m
p
are with th
e
p
r
ev
iou
s
resu
lts, m
a
x
i
m
u
m
n
u
m
b
er
o
f
i
t
erations is
chose
n
as t
h
e stoppi
ng criterion. If the
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Non-Convex Economic Dis
patch with Prohibited
Op
erating Zones t
h
rough
Gr
avit
ational… (P.K. Hota)
1
239
sto
p
p
i
ng
criterio
n
is no
t satisfied
, th
e ab
ov
e p
r
ocedu
r
e
is repeated from
fitness
ev
alu
a
ti
o
n
with
in
cremen
ted
iteratio
n
.
3.
3.
Constr
aints S
a
tis
f
ac
tion
Te
chnique
Th
e elem
en
ts
o
f
in
itial
m
a
sses co
n
t
ain
g
e
nerated
powers o
n
l
y with
in
t
h
e op
erati
n
g
su
b-reg
i
on
s.
Ho
we
ver
,
a
f
t
e
r
u
p
d
a
t
i
n
g
p
r
oc
ess o
f
GS
A
al
go
ri
t
h
m
,
t
h
ey
m
a
y
vi
ol
at
e t
h
e co
nst
r
ai
nt
s
gi
ven
by
E
quat
i
o
ns,
(
5
-
11
).
The
p
r
oce
d
u
r
e
fo
r c
o
nst
r
ai
nt
s sat
i
s
f
actio
n is
d
ealt with as fo
llows
[12].
If
ge
nerat
o
r l
i
m
i
t
s
const
r
ai
nt
(
5
) i
s
vi
ol
at
ed
t
h
en,
P
i
=
P
i, min
if
P
i
<
P
i, min
a
n
d
P
i
=
P
i, max
if
P
i
>
P
i, max
(26)
If p
r
ohi
bi
t
e
d o
p
erat
i
n
g zo
nes
const
r
ai
nt
(9
-
1
1
)
are
vi
ol
at
ed, t
h
e
n
t
h
e m
i
d-
p
o
i
n
t
s
o
f
t
h
e
pr
ohi
bi
t
e
d
ope
rat
i
n
g z
o
n
e
s f
o
r
eac
h
gen
e
rat
o
r
are
c
o
m
put
e
d
.
The
m
i
d-
po
in
ts
o
f
th
e proh
ib
ited zon
e
co
rresp
ond
in
g to a
gene
rat
i
o
n l
e
ve
l
P
i
l
y
i
ng
bet
w
een
,
and
,
i
s
gi
v
e
n as:
,
,
,
fo
r
n
= 1, 2,
…,
n
i
and
,
if
,
and
,
if
,
(
2
7
)
If ram
p
-rate l
i
m
i
ts constrai
nts a
r
e
violat
ed, the
n
the l
i
m
i
ts
max
,
min
,
,
i
i
P
P
i
n
E
q
ua
t
i
on
(2
6
)
a
r
e
repl
ace
d by
,
and
,
to
satisfy t
h
ese con
s
trai
n
t
s.
3.
4.
C
o
mp
ut
at
io
na
l P
r
o
ced
u
r
e
The purposed
GSA approach for ec
o
n
o
m
ic
lo
ad
d
i
sp
atch p
r
ob
lem
with
v
a
lv
e- po
in
t
effect and
pr
o
h
i
b
i
t
e
d
o
p
er
at
i
ng z
one
s ca
n
be s
u
m
m
ari
z
ed as
f
o
l
l
o
w
s
.
Step 1. Searc
h
space i
d
entific
ation
Step
2
.
Gen
e
rate
in
itial
p
o
p
u
l
atio
n
b
e
tween
min
i
m
u
m
an
d
max
i
m
u
m
v
a
lues.
Step 3. Fitness evaluation of
a
g
ents
.
Step
4
.
Upd
a
te g
r
av
itatio
n
a
l co
nstan
t
G
(
t
), b
e
st (
t
) an
d w
o
rst (
t
) in
th
e
p
opu
latio
n
and u
p
d
a
te th
e
mass of the
object
M
i
(
t
).
St
ep 5. Fo
rce
c
a
l
c
ul
at
i
on
i
n
di
ffe
rent
di
rect
i
o
n.
Step 6.
Calc
ulation of
accelerati
on a
n
d vel
o
c
ity of a
n
a
g
ent.
Step
7
.
Up
d
a
ti
n
g
th
e
po
sitio
n of an
ag
en
t.
Step
8
.
Rep
eat
step
3
to step
7 un
til th
e stop
criteria is satisfied
Step 9. Stop.
4.
SYSTE
M
AN
D RESULTS
Th
e
p
r
esen
t
work h
a
s b
e
en
im
p
l
e
m
en
te
d
in
co
mm
an
d
lin
e i
n
Matlab
-
7
.
0
fo
r the so
lu
tion
of
econ
o
m
i
c l
o
ad di
spat
ch
wi
t
h
no
n
-
sm
oot
h co
st
funct
i
ons
. T
h
e p
r
o
g
ram
was ru
n o
n
a 3.
06
G
H
z, Pe
nt
i
u
m
-IV,
with
256
MB RAM PC. After
sev
e
ral trials, t
h
e setup
fo
r t
h
e p
r
o
p
o
s
ed
algo
rith
m
is ex
ecu
ted
with
fo
llowing
param
e
ters:
m
= 100 (m
asses),
G
i
s
set
us
i
ng E
q
. (
2
0),
w
h
ere
G
0
i
s
set
to 1
00 a
nd
α
is set to
8
,
and
T
is th
e to
tal
num
ber
of
i
t
e
r
a
t
i
ons
wi
t
h
a
m
a
xi
m
u
m
val
u
e
of
1
0
0
0
.
To
dem
onst
r
at
e t
h
e r
o
bust
n
es
s o
f
t
h
e
propos
ed a
p
proac
h
, a
practical si
zed test syste
m
co
n
s
istin
g of
40
ge
nerat
o
rs
wi
t
h
val
v
e p
o
i
n
t
l
o
adi
ng e
ffe
ct
s, ram
p
ra
t
e
lim
it
s const
r
ai
n
t
s and
pr
ohi
bi
t
e
d o
p
erat
i
ng z
one
s i
s
con
s
i
d
ere
d
.
A
l
o
ad d
e
m
a
nd o
f
1
0
,
5
0
0
M
W
i
s
consi
d
ere
d
i
n
t
h
i
s
case. The
i
nput
dat
a
of
40
uni
t
s
i
s
sh
o
w
n i
n
Tabl
e 1 an
d
2.
The o
p
t
i
m
a
l
r
e
sul
t
s
by
t
h
e p
r
o
p
o
sed
GS
A algorithm
are
com
p
ared w
ith th
o
s
e ob
tain
ed
fro
m
sev
e
n
o
t
h
e
r
meth
od
s [1
4
]-[16
]
an
d
sh
own in
Tab
l
e 3
.
Th
e o
t
h
e
r
well estab
lish
e
d
heu
r
istic m
e
th
od
s ar
e
im
pro
v
ed fast
evol
ut
i
ona
ry
p
r
o
g
ram
m
i
ng
(I
FEP) [
15]
,
m
odi
fi
ed part
i
c
l
e
swarm
opt
i
m
i
zat
i
on
(M
P
S
O
)
[1
5]
,
p
a
rticle swarm op
ti
m
i
zatio
n
-
lo
cal ra
n
dom
search
(
P
S
O
-LR
S
)
[1
5]
,
new
p
a
rt
i
c
l
e
swarm
opt
i
m
i
zati
on (
N
P
S
O
)
[1
5]
, ne
w pa
rt
i
c
l
e
swarm
opt
im
i
zati
on-l
o
cal
ran
dom
search (
N
PS
O-
LR
S
)
[1
5]
, sel
f
-
o
rg
ani
z
i
ng
hi
era
r
c
h
i
cal
part
i
c
l
e
swa
r
m
opt
i
m
i
z
at
i
on (
S
O
H
PS
O)
[1
4]
and c
h
a
o
t
i
c
di
ffe
rent
i
a
l
ev
ol
ut
i
on
(
D
E)
[1
6
]
The m
i
nim
u
m
cost
obt
ai
ne
d by
p
r
op
ose
d
GS
A i
s
12
1,
44
7.
5
47
$
/
h whe
r
eas, t
h
e
m
i
nim
u
m
cost
s obt
ai
ne
d by
t
h
ese seve
n m
e
tho
d
s
are m
o
re t
h
an
t
h
e pr
o
p
o
s
ed
GS
A m
e
t
hod
. The
r
ef
ore
,
t
h
e G
S
A m
e
t
hod
gi
ves t
h
e
cheape
s
t
ge
ne
rat
i
o
n
sche
dul
e,
w
h
i
c
h m
a
y
be co
n
s
i
d
ere
d
as
gl
o
b
al
o
n
e.
T
h
e
gene
rat
i
o
n
out
put
s
an
d c
o
rre
spo
n
d
i
n
g c
o
st
of
t
h
e
opt
i
m
al
sol
u
t
i
o
n
by
p
r
o
p
o
se
d
GS
A m
e
t
h
o
d
a
r
e
pr
o
v
i
d
e
d
i
n
Ta
bl
e
4.
Hence
,
Ta
bl
e
3 a
n
d
4
val
i
d
at
e t
h
e
sup
e
ri
o
r
i
t
y
of
t
h
e
GS
A m
e
t
hod.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
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:
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08
IJECE
Vol. 5, No. 6, D
ecem
ber
2015 :
1234 –
1244
1
240
Fi
gu
re
1.
C
o
nv
erge
nce c
h
a
r
ac
t
e
ri
st
i
c
s of t
h
e
pr
o
pose
d
GS
A
m
e
t
hod
Tab
l
e
1
.
4
0
-un
it syste
m
with
v
a
lv
e po
in
t
load
ing
effect
Unit P
i
,
m
in
P
i
,
m
ax
a
b
c e
f
1 36
114
0.
0069
0
6.
73
94.
705
100
0.
084
2 36
114
0.
0069
0
6.
73
94.
705
100
0.
084
3 60
120
0.
0202
8
7.
07
309.
54
100
0.
084
4 80
190
0.
0094
2
8.
18
369.
03
150
0.
063
5 47
97
0.
0114
5.
35
148.
89
120
0.
077
6 68
140
0.
0114
2
8.
05
222.
33
100
0.
084
7 110
300
0.
0035
7
8.
05
287.
71
200
0.
042
8 135
300
0.
0049
2
6.
99
391.
98
200
0.
042
9 135
300
0.
0057
3
6.
60
455.
76
200
0.
042
10
130
300
0.
0060
5
12.
9
722.
82
200
0.
042
11
94
375
0.
0051
5
12.
9
635.
20
200
0.
042
12
94
375
0.
0056
9
12.
8
654.
69
200
0.
042
13
125
500
0.
0042
1
12.
5
913.
40
300
0.
035
14
125
500
0.
0075
2
8.
84
1760.
4
300
0.
035
15
125
500
0.
0070
8
9.
15
1728.
3
300
0.
035
16
125
500
0.
0070
8
9.
15
1728.
3
300
0.
035
17
220
500
0.
0031
3
7.
97
647.
85
300
0.
035
18
220
500
0.
0031
3
7.
95
649.
69
300
0.
035
19
242
550
0.
0031
3
7.
97
647.
83
300
0.
035
20
242
550
0.
0031
3
7.
97
647.
81
300
0.
035
21
254
550
0.
0029
8
6.
63
785.
96
300
0.
035
22
254
550
0.
0029
8
6.
63
785.
96
300
0.
035
23
254
550
0.
0028
4
6.
66
794.
53
300
0.
035
24
254
550
0.
0028
4
6.
66
794.
53
300
0.
035
25
254
550
0.
0027
7
7.
10
801.
32
300
0.
035
26
254
550
0.
0027
7
7.
10
801.
32
300
0.
035
27
10
150
0.
5212
4
3.
33
1055.
1
120
0.
077
28
10
150
0.
5212
4
3.
33
1055.
1
120
0.
077
29
10
150
0.
5212
4
3.
33
1055.
1
120
0.
077
30
47
97
0.
0114
0
5.
35
148.
89
120
0.
077
31
60
190
0.
0016
0
6.
43
222.
92
150
0.
063
32
60
190
0.
0016
0
6.
43
222.
92
150
0.
063
33
60
190
0.
0016
0
6.
43
222.
92
150
0.
063
34
90
200
0.
0001
8.
95
107.
87
200
0.
042
35
90
200
0.
0001
8.
62
116.
58
200
0.
042
36
90
200
0.
0001
8.
62
116.
58
200
0.
042
37
25
110
0.
0161
5.
88
307.
45
80
0.
098
38
25
110
0.
0161
5.
88
307.
45
80
0.
098
39
25
110
0.
0161
5.
88
307.
45
80
0.
098
40
242
550
0.
0031
3
7.
97
647.
83
300
0.
035
The convergence characterist
i
c of
propos
ed GSA m
e
thod
is illustrate
d in Figure 1. To assess the
robustness a
n
d effective
n
ess
of the
propose
d
GSA m
e
thod in com
p
aris
on to
othe
r m
e
thods [14]-[16] in a
statistical
m
a
nner, the relative fre
que
ncy of conve
r
ge
nce
i
s
provide
d
for
each ra
nge of
cost am
ong
100 trials
i
n
Ta
bl
e 5
.
On
e can
see t
h
e
d
o
m
i
nat
i
ng nat
u
re
of
GS
A m
e
tho
d
o
v
er
ot
her
exi
s
t
i
n
g
m
e
t
hods.
The
pe
rf
o
r
m
a
nce
of
GS
A i
s
com
p
are
d
wi
t
h
t
h
o
s
e o
f
ot
he
r
heu
r
i
s
t
i
c
m
e
t
hods
.
It
i
s
cl
ear
t
h
at
t
h
e
GS
A m
e
t
hod
o
u
t
p
e
r
f
o
rm
s an
d
pr
o
v
i
d
es t
h
e
c
h
eape
s
t
ge
nera
t
i
on sc
hed
u
l
e
f
o
r
w
h
i
c
h
hu
ge
reve
n
u
e i
s
sa
ved
o
v
er a l
o
n
g
pe
ri
o
d
, say
y
earl
y
.
The
heu
r
i
s
t
i
c
m
e
t
hods a
r
e st
ocha
st
i
c
m
e
t
hods w
h
e
r
e t
h
e s
o
l
u
t
i
o
ns
obt
ai
n
e
d m
a
y
not
be
sam
e
at
every
ru
n
.
Whe
n
t
h
e p
r
og
ram
i
s
run
10
0
t
i
m
e
s, t
h
e ran
g
es
of t
h
e c
o
st of the system
obtaine
d a
r
e cl
assified int
o
10 s
u
b-
0
200
40
0
600
800
1000
1.
2
1
5
1.
22
1.
2
2
5
1.
23
1.
2
3
5
1.
24
x 1
0
5
C
onve
r
g
e
n
c
e
c
h
a
r
a
c
t
e
r
i
s
t
i
c
s
of
40
u
n
i
t
s
I
t
er
at
i
o
n
s
Ge
n
e
r
a
t
i
o
n
C
o
st
(
$
/
h
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Non-Convex Economic Dis
patch with Prohibited
Op
erating Zones t
h
rough
Gr
avit
ational… (P.K. Hota)
1
241
ran
g
es as sh
ow
n i
n
Tabl
e 5. T
h
e cha
o
t
i
c
DE m
e
t
hod p
r
o
v
i
d
es al
l
cost
s
m
o
st
l
y
i
n
l
a
st
t
h
re
e range
s suc
h
as 8
th
ran
g
e (4
tim
es), 9
th
r
a
ng
e
(
31 ti
m
e
s)
an
d 10
th
r
a
ng
e
(6
5 ti
mes)
.
Th
e co
st
ob
ta
ined by t
h
e
chaotic
DE m
e
thod
l
i
e
s i
n
bet
w
ee
n
12
0,
0
0
0
$/
h t
o
1
2
2
,
5
0
0
$/
h
wi
t
h
65 t
i
m
es in t
h
e
1
0
th
r
a
n
g
e. Th
e pr
opo
sed
GSA,
N
P
SO-
L
R
S
and
SO
HP
SO
m
e
t
hods
pr
o
v
i
d
e t
h
e al
l
cost
s
obt
ai
ne
d i
n
1
0
0
t
r
i
a
l
s
i
n
t
h
e
9
th
and
1
0
th
ranges on
ly. Bu
t, the co
st
obt
ai
ne
d by
pr
op
ose
d
GS
A m
e
t
hod
l
i
e
s
i
n
bet
w
ee
n 1
2
0
,
00
0 $/
h
t
o
1
2
2
,
5
0
0
$/
h wi
t
h
9
2
t
i
m
es
i
n
t
h
e
1
0
th
rang
e as sho
w
n
in
Tab
l
e 5
.
Hen
c
e, propo
sed
GSA can
prov
id
e m
o
re reliab
l
e an
d
quality
so
lu
tio
n
s
th
an
M
B
FA m
e
t
hod
.
Tabl
e
2.
4
0
-
u
n
i
t
sy
st
em
wi
t
h
ram
p
rat
e
s an
d
pr
o
h
i
b
i
t
e
d
o
p
er
at
i
ng z
one
s
Unit Pi,min
Pi,max
P
i0
UR
i
DR
i
P
r
ohibited Zones (
MW)
1 36
114
100
114
114
-
2 36
114
100
114
114
-
3 60
120
90
120
120
-
4 80
190
150
100
150
-
5 47
97
80
97
97
-
6 68
140
120
80
125
-
7 110
300
280
165
200
-
8 135
300
200
165
200
-
9 135
300
230
165
200
-
10
130
300
240
155
190
[13
0
-
150]
[2
00 23
0]
[270-
29
9]
11
94
375
210
150
185
[10
0
-
140]
[2
30-
28
0]
[300-
35
0]
12
94
375
210
150
185
[10
0
-
140]
[2
30-
28
0]
[300-
35
0]
13
125
500
230
206
235
[15
0
-
200]
[2
50-
30
0]
[400-
45
0]
14
125
500
355
260
290
[20
0
-
250]
[3
00-
35
0]
[450-
49
0]
15
125
500
350
186
215
-
16
125
500
350
186
215
-
17
220
500
460
240
270
-
18
220
500
470
240
268
-
19
242
550
500
290
315
-
20
242
550
500
290
315
-
21
254
550
510
335
360
-
22
254
550
520
335
360
-
23
254
550
520
335
362
-
24
254
550
450
350
378
-
25
254
550
400
350
380
-
26
254
550
520
350
380
-
27
10
150
20
95
145
-
28
10
150
20
95
145
-
29
10
150
25
98
145
-
30
47
97
90
97
97
-
31
60
190
170
90
145
-
32
60
190
150
90
145
-
33
60
190
190
90
145
-
34
90
200
190
105
150
-
35
90
200
150
105
150
-
36
90
200
180
105
150
-
37
25
110
60
110
110
-
38
25
110
40
110
110
-
39
25
110
50
110
110
-
40
242
550
512
290
315
-
Tabl
e
3.
C
o
m
p
ari
s
on
o
f
Si
m
u
l
a
t
i
on
res
u
l
t
s
bet
w
ee
n
GS
A
and
ot
her
m
e
t
h
ods
f
o
r
4
0
-
u
n
i
t
sy
st
em
M
e
thod
IFEP
[15]
MPSO
[15]
PSO-
L
R
S[15]
NPSO
[15]
NPSO-
L
R
S[15]
SOHPSO
[14]
Chaotic-
DE
[16]
GSA
Min
i
m
u
m C
o
s
t
($
/h
)
122,
62
4.
3
5
122,
25
2.
2
6
122,
03
5.
7
9
121,
70
4.
7
4
121,
66
4.
4
3
121,
50
1.
1
4
121,
74
1.
9
8
121,
44
7.
5
5
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJECE
Vol. 5, No. 6, D
ecem
ber
2015 :
1234 –
1244
1
242
Table
4.
Ge
ne
ration outputs
of each ge
nerator
and t
h
e c
o
rresponding c
o
st
in
40-unit Syst
e
m
Unit
Pmin Pmax
Gen. (MW)
Cost ($/h)
1
36
114
114.0000
978.156
2
36
114
114.0000
978.156
3
60
120
97.3995
1190.547
4
80
190
179.7330
2143.550
5
47
97
87.7999
706.500
6
68
140
139.9996
1596.463
7
110
300
259.5997
2612.885
8
135
300
284.5996
2779.837
9
135
300
284.5996
2798.230
10
130
300
130.0000
2502.065
11
94
375
167.2422
2949.744
12
94
375
167.2553
2967.697
13
125
500
214.7590
3792.067
14
125
500
394.2754
6414.843
15
125
500
304.5195
5171.198
16
125
500
394.2711
6436.551
17
220
500
489.2793
5296.711
18
220
500
489.2793
5288.765
19
242
550
511.2793
5540.929
20
242
550
511.2794
5540.910
21
254
550
523.2793
5071.290
22
254
550
523.2790
5071.290
23
254
550
523.2794
5057.224
24
254
550
523.2793
5057.223
25
254
550
523.2794
5275.089
26
254
550
523.2793
5275.089
27
10
150
10.0000
1140.524
28
10
150
10.0000
1140.524
29
10
150
10.0000
1140.524
30
47
97
89.4748
734
.279
31
60
190
190.0000
1643.991
32
60
190
190.0000
1643.991
33
60
190
190.0000
1643.991
34
90
200
164.7998
1585.544
35
90
200
164.7997
1539.870
36
90
200
164.7998
1539.870
37
25
110
110.0000
1220.166
38
25
110
110.0000
1220.166
39
25
110
110.0000
1220.166
40 242
550
511.2793
5540.929
Total Gen. and
Total Cost
10,500.000
1,21,447
.547
Tabl
e
5.
C
o
m
p
ari
s
on
o
f
di
f
f
e
rent
m
e
t
hod
s
on
rel
a
t
i
v
e
fre
q
u
ency
o
f
c
o
n
v
e
r
ge
nce i
n
t
h
e
r
a
nge
s
of c
o
st
(
k$/
h)
fo
r 40
-
unit
Sy
s
t
em
M
e
thods
126.5---127.0
126.0---126.5
125.5---126.0
125.0---125.5
124.5---125.0
124.0---124.5
123.5---124.0
123.0---123.5
122.5---123.0
120.0---122.5
I
F
E
P
[15]
10
4 -
16
22
42
4
2 -
-
MPSO [
15]
6 -
4
2
10
20
26
24
6 -
PSO-LR
S
[
15]
- -
- -
-
1
4
2
6
5
0
1
0
-
NPSO [
15]
-
-
2 -
4
4
18
50
22
-
NPSO-
L
RS [
15]
- -
- -
- -
- -
5
3
4
7
SOHPSO
[
14]
- -
- -
- -
- -
1
8
8
2
Chaotic DE [16]
- -
- -
- -
-
0
4
3
1
6
5
GSA
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I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Non-Convex Economic Dis
patch with Prohibited
Op
erating Zones t
h
rough
Gr
avit
ational… (P.K. Hota)
1
243
5.
CO
NCL
USI
O
N
Thi
s
pa
per
pre
s
ent
s
a new st
ocha
st
i
c
search t
echni
que
na
m
e
d GSA t
o
s
o
l
v
e t
h
e n
o
n
-s
m
oot
h ELD
p
r
ob
lem
with
v
a
lv
e
po
in
t effect, ram
p
rate
li
mits c
onst
r
ai
nt
s a
n
d
p
r
ohi
b
i
t
e
d o
p
erat
i
n
g
zone
s c
onst
r
ai
nt
s.
A
practical sized ELD test syste
m
ha
s be
en c
onsi
d
ere
d
. T
h
e
sim
u
lation re
s
u
lts dem
onstra
t
e the effectiveness
and
ro
b
u
st
ness
of t
h
e p
r
o
p
o
se
d GS
A m
e
t
hod
t
o
sol
v
e EL
D pr
o
b
l
e
m
i
n
m
o
der
n
p
o
w
er sy
s
t
em
s. The obt
a
i
ned
resul
t
s
o
f
t
h
e p
r
o
p
o
sed
GS
A m
e
t
hod h
a
ve
b
een com
p
ared
wi
t
h
t
h
e res
u
l
t
s
obt
ai
ne
d f
r
o
m
publ
i
s
he
d m
e
t
h
o
d
s
in
th
e literatu
re. Th
e co
m
p
arison
con
f
i
r
ms th
e effec
tiven
ess,
h
i
gh
qu
ality so
lu
tio
n, stab
le con
v
e
rg
ence
charact
e
r
i
s
t
i
c
, go
o
d
com
put
at
i
on e
ffi
ci
ency
.
Hence
,
t
h
e
su
peri
ori
t
y
of
t
h
e pr
o
pose
d
GS
A m
e
t
hod
ove
r ot
he
r
heuristic techniques in term
s of sol
u
tion
quality is
validated. T
h
e proposed m
e
t
hodol
ogy can be appl
ied to
large-scale EL
D
problem
s as
well optim
a
l dispatch pr
ob
lems un
der
d
e
r
e
gulated
environm
ent efficiently.
ACKNOWLE
DGE
M
ENTS
The s
u
pp
ort
f
r
om
t
h
e El
ect
ri
cal
En
gi
nee
r
i
ng
De
part
m
e
nt
o
f
Veer
S
u
re
n
d
ra
Sai
U
n
i
v
e
r
si
t
y
of
Tech
nol
ogy
, B
u
rl
a,
In
di
a, e
x
t
e
nde
d t
o
t
h
e s
econ
d
a
u
t
h
or
f
o
r t
h
e w
o
rk
re
po
rt
ed i
n
t
h
i
s
pape
r i
s
grat
ef
ul
l
y
ack
now
ledg
ed.
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