Indonesian J
ournal of Ele
c
trical Engin
eering and
Computer Sci
e
nce
Vol. 2, No. 3,
Jun
e
201
6, pp. 566 ~ 58
2
DOI: 10.115
9
1
/ijeecs.v2.i2.pp56
6-5
8
2
566
Re
cei
v
ed Ma
rch 2, 2
016;
Re
vised
Ma
y 11, 2016; Accepted Ma
y 24
, 2016
Optimal Capacitors in Radial Distribution System for
Loss Reduction and Voltage Enhancement
S Bhongad
e
*, Sachin Ar
y
a
S.G.S Institute
of T
e
chnol
ogy
& Science, Indore/R.G.P.V B
hopal (M.P)-India
Department of Electrical E
ngineer
i
ng, S.G.S Institute of
T
e
chnology
& Science, Indore
-452003, India
F
a
x No. +
9
1
7
3
124
32
540
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: bhon
ga
desa
nde
ep@
gmai
l.com
A
b
st
r
a
ct
T
he w
o
rk pr
e
s
ented
in
this
pa
per
is c
a
rri
ed
out
w
i
th th
e o
b
jectiv
e
of id
entifyin
g
th
e o
p
ti
ma
l
locati
on
an
d si
z
e
(Kvar r
a
tin
g
s) of s
hunt c
apac
itors to
b
e
pl
ace
d
i
n
ra
dial
d
i
stributi
o
n syste
m
, to
h
a
v
e
overa
ll eco
n
o
m
y cons
id
erin
g
the savin
g
du
e to ener
gy los
s
min
i
mi
z
a
ti
on.
T
o
achieve th
i
s
objectiv
e
, a tw
o
stage
metho
d
o
l
ogy
is
ad
opte
d
i
n
th
is p
a
p
e
r. In th
e firs
t sta
ge, th
e b
a
se
c
a
se
loa
d
fl
ow
of unc
o
m
p
ens
ate
d
distrib
u
tion sy
stem is c
a
rri
e
d
out. On th
e bas
is of b
a
s
e case
lo
ad
flow
solutio
n
,
no
mi
nal v
o
lt
age
ma
gn
itudes
an
d loss se
nsitiv
ity fact
ors are
calcul
ated
an
d the w
eak b
u
ses are s
e
le
cted for cap
a
c
i
tor
plac
e
m
ent. In the sec
ond sta
ge, partic
l
e sw
arm
opti
m
i
z
at
io
n (PSO) algor
ithm
is use
d
to i
dentify the s
i
z
e
o
f
the capac
itors to be plac
ed at
the selected b
u
ses fo
r mi
ni
mi
z
i
ng th
e pow
e
r
loss. T
he develo
ped a
l
g
o
rith
m
is tested for 10-bus, 34-bus and 85-
bus
radial distribution system
s. The
results show that
there has been
an en
ha
nce
m
e
n
t in voltag
e pr
ofile a
nd re
duc
tion in p
o
w
e
r lo
ss thus resultin
g in much a
n
n
ual sav
i
ng.
Ke
y
w
ords
: C
apac
itor pl
ace
m
e
n
t, Loss se
nsitivity factors
,
Particle Sw
a
r
m Opti
mi
z
a
t
i
o
n
(PSO), Radi
a
l
Distributi
on Sy
stem (RDS)
Copy
right
©
2016 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Electri
c
po
wer indu
stry i
s
advan
cin
g
rapi
dly and
the environ
ment is mov
i
ng to
a
comp
etitive powe
r
supply
market. One
of the most
importa
nt concern
s
in t
oday’s life i
s
to
minimize the
po
wer lo
ss
and i
n
crea
se
the ove
r
al
l
efficien
cy
of the
sy
stem. Furthe
rmo
r
e,
the
voltage p
r
ofil
e of the
system is to b
e
ke
pt withi
n
a p
r
e
s
cribe
d
limit. The
e
l
ectri
c
al
ene
rg
y
prod
uced at
the gene
rat
i
ng station
i
s
d
e
livere
d
to the
con
s
umer throug
h a
network of
transmissio
n
and di
stri
buti
on sy
stem.
Distri
bution
system is
one
of the mai
n
three
part
s
o
f
a
power
syste
m
, re
spo
n
si
bl
e for tran
sfe
r
ring el
ectri
c
al
energy to th
e
end
u
s
ers. T
he a
nalysi
s
o
f
a
distrib
u
tion
system i
s
a
n
i
m
porta
nt are
a
of a
c
tivi
ty, as
distri
butio
n sy
stem
s p
r
ovide the
vital link
betwe
en the
bulk p
o
wer
system and th
e co
nsu
m
ers.
A
distributio
n
circuit no
rma
lly uses
prim
ary
or m
a
in fee
d
e
rs an
d late
ral di
stributo
r
s. Many
dist
ribution
syste
m
s u
s
e
d
in
pra
c
tice
hav
e a
singl
e circuit
main feede
r and a
r
e d
e
fined a
s
ra
dial dist
ributi
on syst
e
m
s (RDS).
Radi
al
Distri
bution S
y
stems a
r
e p
opula
r
be
cau
s
e of
their
si
mple de
sign
and ge
nerally low co
st.
The
distri
buti
on n
e
two
r
ks
have a
typical feat
u
r
e th
at the voltag
es
at b
u
se
s
(nod
es)
redu
ce
if mo
ved away fro
m
su
bstatio
n
. This
de
cr
ease in voltage is mai
n
ly due to ins
u
ffic
i
ent
amount of reactive po
wer. It is also
well kn
own
that losse
s in a distrib
u
tion syste
m
are
signifi
cantly h
i
gher comp
ared to th
at in
a
tran
smi
ssi
on
syste
m
. Mo
st of t
he l
oad
s
are
indu
ctive
in
nature
an
d
re
quire
rea
c
tive po
wer, if
re
a
c
tive po
we
r i
s
fed
to the
m
locally than
the lin
e
curre
n
t
can be red
u
ced.
Re
du
ced curre
n
t
free
s up
capa
city
; the same
circuit can
se
rve
more l
oad
s a
n
d
also
signifi
ca
ntly lowers th
e
line losse
s
. Hen
c
e, in order to imp
r
ov
e the voltage profile an
d to
minimize
the
losse
s
prop
er rea
c
tive
p
o
we
r com
p
e
n
satio
n
i
s
re
quire
d.
O
ne su
ch exampl
e
of
rea
c
tive power co
mpe
n
sation in distri
bu
tion system i
s
shunt capa
ci
tors.
The shunt
ca
pacito
r
s
su
pp
ly part of the
rea
c
tive po
wer de
man
d
, there
b
y re
du
cing the
curre
n
t and
p
o
we
r flow in l
i
nes. In
stallati
on of s
hunt capa
citors
on
distrib
u
tion
n
e
twork will
he
lp
in re
du
cing
e
nergy l
o
sse
s
,
pea
k
dema
n
d
lo
sses an
d
also
hel
ps in
improvin
g the
syste
m
volta
ge
profile,
system stability and power
fact
or. However, to achieve thes
e
objectives,
keeping in m
i
nd
the overall e
c
onomy, an
op
timal si
ze a
n
d
location
of
cap
a
cito
rs ne
ed to b
e
d
e
ci
ded. Th
e
syst
em
benefits attai
ned du
e to the appli
c
ati
on
of shunt capa
citors incl
ude:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
2, No. 3, Jun
e
2016 : 566
– 582
567
Reac
tive power s
u
pport.
Voltage profile improvem
e
n
ts.
Line lo
ss red
u
ction
s
.
Rele
ase of powe
r
syste
m
cap
a
city.
Savings du
e to decrea
s
e
d
energy loss.
Due to the n
eed of impro
v
ing the overall e
fficiency
of the system
, the loss min
i
mization
in distrib
u
tion
systems i
s
cons
i
dered hig
h
ly significa
nt since a
bout
13% of total
power ge
nera
t
ed
is wa
sted in
the form of losses at t
he dist
ributio
n level [1].
Since, the o
p
timal ca
pacitor
placement
is a
com
p
licated
com
b
inat
orial
optim
i
z
a
t
ion p
r
oble
m
, many
different optimi
z
ati
on
techni
que
s a
nd alg
o
rithm
s
have b
een
p
r
opo
se
d in th
e pa
st. Ne
agl
e and
Sam
s
o
n
[4] develo
p
ed
a ca
pa
citor
placement a
ppro
a
ch for
uniformly
di
st
ributed li
ne
s
and
sho
w
ed
that the opti
m
al
cap
a
cito
r lo
cation is the
p
o
int on the
ci
rcuit
whe
r
e t
he re
active p
o
we
r flow
eq
uals
half of the
cap
a
cito
r var
rating. From this, they develope
d t
he 2/3 rule for
sel
e
cting a
nd pl
acin
g ca
pa
citors.
For a u
n
iform
l
y distribute
d
load, the opti
m
al si
ze
cap
a
citor i
s
2/3 o
f
the var requ
ireme
n
ts of the
circuit. The o
p
timal pla
c
e
m
ent of this capa
citor i
s
2/3 of the dista
n
ce from the
sub
s
tation to
the
end of th
e lin
e. For thi
s
op
timal pla
c
em
ent for
a
unif
o
rmly di
strib
u
t
ed load, th
e
sub
s
tation
so
urce
provide
s
va
rs for the
first
1
/
3 of the
circu
i
t, and t
he
ca
pacito
r
p
r
ovid
es va
rs for th
e la
st 2/3 of t
h
e
circuit. Grai
n
ger an
d Lee
[5] provided anothe
r si
mple and o
p
timal metho
d
for cap
a
ci
tor
placement. T
h
is m
e
thod
i
s
u
s
eful
for
circuits
with
any loa
d
p
r
o
f
ile, not ju
st for
uniforml
y
distrib
u
ted lo
ad profile. Here al
so the
main pri
n
ci
pl
e is to pla
c
e
the cap
a
cito
r at the point
of
circuit whe
r
e
the rea
c
tive p
o
we
r eq
ual
s
one half of
ca
pacito
r
rating.
With this 1/2
-
k
VAR rule,
the
cap
a
cito
r su
p
p
lies h
a
lf of its VARs d
o
wn
strea
m
and h
a
lf are se
nt upstre
a
m.
Baran
et al
[6] propo
se
d a
mixed i
n
teger p
r
og
ramming
tech
nique
for ca
pacito
r
placement
problem, in
whi
c
h th
e p
r
o
b
le
m is de
com
p
ose
d
into
two
levels.
The
p
r
oble
m
at th
e
top
level is
call
e
d
the m
a
ste
r
pro
b
lem
whi
c
h i
s
a
n
inte
ger
pro
g
ram
m
ing p
r
obl
e
m
and
is
used to
place the ca
pacito
r
(i.e. to determi
ne
the num
be
r
and the lo
ca
tion of the capa
citors). T
h
e
probl
em
at th
e bottom l
e
vel is called th
e sl
ave p
r
obl
em an
d i
s
u
s
ed by th
e ma
ster
problem
to
determi
ne th
e types and
the setting
s of capa
citors
place
d
. Here the co
st of the capa
cito
r is
taken a
s
a dif
f
erentiabl
e fu
nction of its
size. Bara
n et
al [7] also de
veloped a
sol
u
tion algo
rith
m
for the
ca
pa
ci
tor ba
se
d o
n
a fea
s
ible
sol
u
tion ap
proa
ch. Also
a ne
w po
wer flow
e
quation
s
a
nd
a
solutio
n
meth
od call
ed ‘Di
s
c flow’ is p
r
op
ose
d
.
Sundha
raja
n
and Pah
w
a
[8] formulate
d
a de
sign
me
thodolo
g
y for determi
ning
the si
ze,
locatio
n
, type and n
u
mbe
r
of cap
a
cito
rs to be
pl
a
c
e
d
on radial
di
stribut
io
n sy
stem. Sensitivity
analysi
s
i
s
u
s
ed
to sele
ct the candi
dat
e location
s f
o
r pl
aci
ng th
e capa
citor in the di
stri
bu
tion
system. Ying
-Tun
g Hsia
o
et al [10] con
s
ide
r
ed
thre
e
obje
c
tive fun
c
tion
s an
d a
non-differe
ntial
optimizatio
n probl
em for
minimizi
ng th
e total cost
f
o
r en
ergy lo
ss. A combin
a
t
ion of fuzzy
and
geneti
c
algo
ri
thm was u
s
e
d
to resolve t
he ca
pa
ci
tor
placement p
r
oblem. The o
b
jective fun
c
tion
wa
s formul
ated in fuzzy sets to asse
ss their im
preci
s
e natu
r
e. Das [11]
prese
n
ted the prob
lem
by using F
u
zzy-GA meth
o
d
, in that sen
s
itivity
analysis ha
s bee
n use
d
to identi
f
y the candid
a
te
buses fo
r
sh
unt ca
pa
citor placement.
Only th
re
e l
oad level
s
were
co
nsid
ered an
d sy
stem
voltage impro
v
ement analy
s
is
wa
s not carri
ed out.
Injeti et al [12] implement
ed two bi
o-in
sp
ired al
gorit
hms
(Bat Algorithm a
nd
Cu
ckoo
Search Algo
ri
thm) to solve
optimal cap
a
citor pl
acem
ent probl
em i
n
two way
s
that is, Variab
le
Location
s
Fix
ed Capa
cito
r
ban
ks
(VLF
Q
)
an
d Vari
abl
e Lo
cation
s
Variabl
e Sizi
ng of
Cap
a
cit
o
rs
(VLVQ) for
real po
we
r lo
ss minimi
zati
on a
nd n
e
twork saving
s
maximizatio
n
. Wu
et al [1
3]
prop
osed the
disp
atch
of cap
a
cito
rs i
n
distri
b
u
tion
systems fo
r d
a
ily operation
,
base
d
on l
o
op-
analyzi
ng me
thods. Here
switchi
ng of ca
pac
ito
r
s fo
r varying loa
d
is optimized.
Rao
et al [1
4] develop
ed
a two
stage
methodol
og
y for ca
pa
citor pla
c
e
m
ent
in radi
al
distrib
u
tion systems. In pa
rt one, they calcul
ated
loss sensitivity factors to sele
ct the candida
te
locatio
n
s fo
r
the ca
pa
citor placeme
n
t a
nd in
part t
w
o they empl
o
y
ed Plant g
r
owth Simul
a
tion
Algorithm (P
GSA) to esti
mate the optimal size of
capa
citors at the optimal bu
se
s determin
ed in
part on
e. Elsheikh et al [26] pre
s
ente
d
the pr
o
b
lem
as di
screte o
p
timization
problem of fixed
shu
n
t capa
cit
o
r pl
aceme
n
t and
si
zing
a
nd impl
ement
ed
Clu
s
terin
g
Base
d O
p
timization
(CB
O
)
for minimi
zin
g
the po
we
r l
o
ss an
d capa
citor
co
sts,
consi
deri
ng ov
er-com
pen
sat
i
on an
d volta
g
e
con
s
trai
nts.
In this pap
er an attempt is mad
e
to redu
ce the lo
sses a
nd to
improve the
system
voltage p
r
ofil
e by pla
c
e
m
ent of
cap
a
ci
tors
at t
he ca
ndidate
bu
se
s sele
cted
by Lo
ss Sen
s
itivity
Facto
r
a
n
d
the
sizi
ng
of
optimal
ca
pa
citor is
do
ne
by Pa
rticle
Swarm
O
p
timization.
Th
e lo
ss
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Optim
a
l Capa
citors in Ra
di
al Distri
bution
System
for Loss Re
du
ctio
n and …
(S Bhong
ade
)
568
sen
s
itivity factor is a very
important to
ol for
predi
ction of bu
se
s whe
r
e pl
acin
g cap
a
cito
r
will
prod
uce be
st result
s i.e. maximum loss red
u
ctio
n. Therefore, th
ese
se
n
s
itive
buse
s
can serve
as candi
date
location
s for the capa
cito
r pla
c
eme
n
t. To estimate
the requi
red
level of shu
n
t
cap
a
citive co
mpen
sation t
o
minimize the losse
s
and
to improve th
e voltage prof
ile of the system
PSO is u
s
e
d
. The
ca
pa
cito
r pla
c
em
ent p
r
oble
m
in
dist
ribution
sy
ste
m
nee
ds re
p
eated lo
ad flo
w
solutio
n
s. MATPOWE
R
[17] version 3.
2 packag
e
is used for Ne
wton-Ra
ph
so
n (NR) loa
d
flow
analysi
s
in this pape
r. The
propo
se
d method is test
ed on 10-bu
s, 34-bu
s and
85-bu
s Ra
di
al
Distri
bution S
y
stems a
nd result
s sh
ow t
he effectiven
ess of the pro
posed metho
d
.
The PSO me
thod is b
e
co
ming very po
pular
be
ca
u
s
e of its simpli
city of implementation
as
well as ability to swiftly converge t
o
a good
sol
u
tion.
As compared with other
optimization
method
s, it is faster,
che
a
p
e
r a
nd mo
re
efficient. In a
ddition, the
r
e
are fe
w p
a
ra
meters to a
d
just
in PSO.
That’s
why PSO is an idea
l optimization
probl
em solver in optimi
z
a
t
ion probl
em
s.
The
remai
n
in
g pa
rt of the
pape
r i
s
stru
cture
d
a
s
foll
ows: Sectio
n
2 give
s th
e
probl
em
formulatio
n; Section
3 de
scrib
e
s sen
s
itivity analys
is
and lo
ss
sen
s
itivity factors to dete
r
mine
th
e
optimal lo
cati
on of shunt
capa
citors an
d
Section
s
4
gives b
r
ief de
scriptio
n of the
parti
cle swa
r
m
optimizatio
n
and al
so th
e
algorith
m
for
cap
a
cito
r pla
c
eme
n
t u
s
ing
PSO. In se
ction 5
re
sults
on
the 10
-bu
s
,
34-b
u
s an
d
85-b
u
s Radi
al Di
strib
u
ti
o
n
System
s radial di
stri
bu
tion sy
stem
are
pre
s
ente
d
an
d finally the concl
u
si
on is g
i
ven in Sectio
n 6.
2. Problem Formulation
Shunt capa
ci
tors
pla
c
ed i
n
dist
ribution
sy
stem
ca
n
provide
re
a
c
tive po
wer
and al
so
redu
ce
s th
e
voltage d
r
op
in the
ra
dial
distri
but
ion
system.
Opti
mal capa
cito
r pla
c
eme
n
t i
s
a
compl
e
x optimization p
r
o
b
l
em in whi
c
h
we try to
“opt
imally” set th
e values of control vari
abl
es
i.e. reactive p
o
we
r output o
f
s
hunt com
p
ensators (cap
acitors) to
minimize the tot
a
l active power
losse
s
whil
e satisfying a gi
ven set of co
nstrai
nts.
2.1. Assump
tions
There are m
a
ny variable
s
whi
c
h are to be co
nsi
dere
d
for cap
a
cito
r placeme
n
t probl
em
inclu
d
ing
size of the cap
a
citors, locations
wh
e
r
e capa
citors are
to be place
d
, cost of the
cap
a
cit
o
r.
Fo
r sim
p
licit
y
o
n
ly
t
he f
i
x
e
d
t
y
pe ca
pa
ci
tors
are
taken into
con
s
ideratio
n whi
l
e
followin
g
assumption
s are made:
The sy
stem is bala
n
ced
All loads a
r
e time invaria
n
t
2.2. Objectiv
e Functio
n
The obje
c
tive
function of the optimal ca
pacito
r
pla
c
e
m
ent is to minimize the tot
a
l active
power lo
sses and thus min
i
mizing the to
tal annual co
st due to ene
rgy loss whil
e
con
s
ide
r
ing t
h
e
co
st of capa
ci
tor placeme
n
t.
2.3. Cons
trai
nts
While
doi
ng
a
ca
pa
citor
pla
c
eme
n
t p
r
obl
em the
r
e
are
a num
be
r of
con
s
trai
nts
which
are
to be taken in
to account
Voltage at the buse
s
mu
st
remain withi
n
the
permi
ssible limits bef
ore an
d after the cap
a
cito
r
placement.
Rea
c
tive p
o
wer
com
pen
sat
i
on by
cap
a
ci
tor pla
c
e
m
ent
at a
bu
s is li
mited an
d i
s
available i
n
disc
r
e
te s
i
zes.
For p
r
a
c
tical
use th
ere
exist a finite nu
mber
of stan
dard
discrete
size capa
citors
an
d
there cost i
s
not linearly p
r
oportio
nal to size of the ca
pacito
r
ban
k.
2.4. Mathem
atical Re
pre
sentation
Mathemati
c
al
ly, the objective function of
the proble
m
is de
scribe
d a
s
:
B
B
N
k
ij
j
i
j
i
ij
N
k
kloss
V
V
V
V
G
P
1
2
2
1
cos
2
min
(1)
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2016 : 566
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569
W
h
er
e
k
The bran
ch b
e
twee
n bu
s
and
B
N
Total no. of bran
che
s
B
N
k
kloss
P
1
Total power l
o
ss in radi
al distrib
u
tion sy
stem
ij
G
C
o
nd
uc
ta
nc
e o
f
b
r
a
n
c
h
k
(p.u.)
j
i
V
V
,
Magnitud
e
(p.
u
.) of bus
and
res
p
ec
tively
ij
Load a
ngle di
fference between bu
s
and
(rad
)
With co
nst
r
ai
nts:
max
min
i
i
i
V
V
V
(2)
C
C
i
Q
Q
max
(3)
Whe
r
e,
i
V
The voltage
magnitud
e
of bus
min
i
V
Minimum voltage limit
max
i
V
Maximum voltage limit
C
i
Q
The re
active
power comp
e
n
satio
n
at bu
s
C
Q
max
The maximu
m amount of rea
c
tive power co
mpe
n
sation at any bu
s
3. Sensitivit
y Anal
y
s
is an
d Loss Sensiti
v
it
y
Factors
Con
s
id
er a distrib
u
tion
li
ne with
an
i
m
peda
nce
jX
R
and a
loa
d
of
eff
eff
Q
P
con
n
e
c
ted be
tween ‘
p
’ an
d ‘
q
’ buse
s
a
s
given belo
w
:
Figure 1. Electri
c
al equival
ent of one branch of RDS
Ac
tive power
loss
in the
th
k
lin
e is given by,
∗
which ca
n be
expresse
d a
s
,
2
2
2
q
V
k
R
q
Q
q
P
q
P
eff
eff
lineloss
(4)
Similarly the reactive po
we
r loss in
th
k
line is given by
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IJEECS
ISSN:
2502-4
752
Optim
a
l Capa
citors in Ra
di
al Distri
bution
System
for Loss Re
du
ctio
n and …
(S Bhong
ade
)
570
2
2
2
q
V
k
X
q
Q
q
P
q
Q
eff
eff
lineloss
(5)
Whe
r
e,
q
P
eff
Total effective active power su
pplie
d b
e
yond the no
de ‘
q
’
q
Q
eff
Total effective rea
c
tive po
wer
sup
p
lied
beyond
the node ‘
q
’.
No
w, both the Loss Sen
s
itivity Factor
s can be obtai
ne
d as sho
w
n b
e
low:
2
2
q
V
k
R
q
Q
Q
P
eff
eff
lineloss
(6)
2
2
q
V
k
X
q
Q
Q
Q
eff
eff
lineloss
(7)
For
cal
c
ulatin
g the Lo
ss Sensitivity Fact
ors
eff
lineloss
Q
P
the ba
se
ca
se lo
ad flo
w
s
are
con
s
id
ere
d
and i
s
cal
c
ulated fo
r a
ll the line
s
of the
given
system,
these
value
s
of
eff
lineloss
Q
P
are a
rra
nge
d in decrea
s
in
g
orde
r.
The sequ
en
ce in whi
c
h b
u
s
e
s
are arran
ged de
cid
e
s t
he se
que
nce
in whi
c
h the
buse
s
are to b
e
con
s
ide
r
ed fo
r ca
pacito
r
pla
c
e
m
ent. The L
o
ss Se
nsitivity Facto
r
s i
s
sol
e
ly respon
sib
l
e
for the se
que
nce in
whi
c
h
buses a
r
e to be co
nsi
der
e
d
therefo
r
e it is very
powerful and useful
in
capacitor pl
acement.It is i
m
portant
to fi
nd the candi
date buses
wh
ere capacitors
can be placed
in ord
e
r to re
duce the po
wer l
o
sse
s
in
radial
di
stri
b
u
tion syste
m
. Since capa
citors
can
not be
placed
at a
b
u
s
with
healt
h
y voltage,
some m
e
thod
is
req
u
ire
d
to
find th
e
wea
k
b
u
ses in
th
e
radial dist
ribu
tion
system. Normali
z
ed
v
o
ltage
ma
gnit
ude i
s
one
such m
e
thod t
o
find the
we
ak
buses in the radial di
stributi
on network.
Normali
z
ed v
o
ltage ma
gnit
ude,
i
norm
for bus
i
can b
e
calcu
l
ated by co
nsiderin
g the
base ca
se vol
t
age magnitu
des a
nd is giv
en by:
95
.
0
i
V
i
norm
(8)
The
b
u
ses
with
i
norm
value
greate
r
tha
n
1.01 a
r
e
h
ealthy bu
se
s and
are
not
con
s
id
ere
d
for Capa
citor
Placem
ent. The we
ak
bu
ses in the
se
quen
ce
01
.
1
i
norm
nee
ds
comp
en
satio
n
and are co
n
s
ide
r
ed fo
r the cap
a
cito
r pl
acem
ent.
4. Particle Sw
arm Optimi
z
a
tion
In this p
ape
r,
Particl
e
Swa
r
m O
p
timizati
on is u
s
ed
to
identify the
size
s of the
ca
pacitor
for minimi
zin
g
the co
st of
energy loss. The PSO
al
gorithm m
o
tivated by so
cial activities
of
individual
s
such
a
s
sch
o
o
ling of fi
sh
es
and
bird
flocking
wa
s p
r
op
osed
by Eberhart
and
Kennedy i
n
1995
an
d
si
nce
then, it
ha
s b
een
su
ccessfully utilized
in di
fferent
p
r
a
c
tical
optimizatio
n. Suppo
se a group of bi
rd
s is ran
domly searchin
g for
food in an are
a
. There is o
n
ly
one pi
ece of
food in the
area bei
ng
sea
r
ch
ed. All
the
bird
s do
not
kno
w
whe
r
e t
he food i
s
, b
u
t
they kno
w
ho
w far the foo
d
is in ea
ch it
eration.
So th
e best
strate
gy to find the food is to foll
ow
the bird
whi
c
h is ne
are
s
t
to the food. The flo
cks si
multaneo
usly
achieve th
ei
r be
st co
nditi
on
throug
h com
m
unication a
m
ong mem
b
ers
who al
re
ady have a better co
nditi
on. This ha
p
pen
s
repe
atedly un
til the piece o
f
food is disco
v
ered.
4.1. Mathem
atical Model
of PSO
In PSO, each
singl
e solution is
a “bi
r
d
”
in the search
spa
c
e
and it
is called
a ‘p
article’.
Each p
a
rti
c
le
has it
s fitness value
and t
he velo
city.
The velo
city direct
s the flyin
g
of the pa
rticle
and influe
nce
s
its po
sition.
Initialization
of PSO
is do
ne by grou
p
of rando
m pa
rticle
s and th
e
n
by means of
updating the
i
r gene
ration
they find the
optimal solu
tion. In every iteration, ea
ch
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IJEECS
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e
2016 : 566
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571
particl
e upd
ate their fitness -
local be
st
(
pbe
st
)
i.e. the best solutio
n
it has achie
v
ed so far an
d
the bes
t fitnes
s
-
global b
e
s
t (gbe
st)
a
c
h
i
eved so far b
y
any particle
in the popula
t
ion.
The pa
rticle u
pdate
s
its velocity
)
(
t
v
i
as:
)
1
(
)
(
(
1
1
2
2
1
1
t
x
t
gbest
r
c
t
x
t
pbest
r
c
t
wv
t
v
i
i
i
i
(9)
Whe
r
e,
1
c
and
2
c
are accel
e
ratio
n
coefficie
n
ts and
and
are rand
om vectors and
is ada
ptive inertia wei
ght a
nd is given by
:
iter
iter
w
w
w
w
max
min
max
max
)
(
(10
)
Whe
r
e,
w
var
i
es
fr
om
max
w
to
min
w
,
max
itr
is total numbe
r of iteration
s
and
it
r
is the
numbe
r of cu
rre
nt iteration
.
Let
denote the positio
n of particle
in th
e sea
r
ch spa
c
e at time step. The chan
g
e
in the particl
e
’
s po
sition is
done by ad
di
ng a velocity,
to the curren
t position:
1
1
t
v
t
x
t
x
i
i
i
(11
)
Whe
r
e,
is
the c
o
ns
tric
tion fac
t
or.
4.2. Parameters of PSO
There a
r
e
so
me pa
ram
e
te
rs i
n
PSO al
g
o
rithm
that m
a
y affect its
p
e
rform
a
n
c
e.
For a
n
y
given optimi
z
ation proble
m
, some of t
hese pa
ra
me
ter’s valu
es
a
nd choices
h
a
ve larg
e im
pact
on the efficie
n
cy of the PSO method, an
d other
pa
ra
meters have
small o
r
no ef
fect [19].
Population Si
ze: Pop
u
latio
n
si
ze o
r
swa
r
m si
ze i
s
the
numbe
r of p
a
rticle
s
n
in th
e swarm. A
big swa
r
m
g
enerates larg
er p
a
rt
s of th
e search
spa
c
e to
be
cov
e
red
pe
r ite
r
ation. A larg
e
numbe
r
of pa
rticle
s m
a
y re
duce the
num
ber
of it
eratio
ns
nee
d to o
b
t
ain a
good
o
p
timization
result. In con
t
rast, hug
e a
m
ounts
of p
a
rticle
s in
cre
a
se th
e com
putational
co
mplexity per
iteration, and
more time co
nsumi
ng. Fro
m
a
numbe
r of empiri
cal studies, it has
been sho
w
n
that most of the PSO implementation
s
use a
n
interv
al of
]
60
,
20
[
n
f
o
r t
he swar
m si
ze.
Iteration Nu
mber: The n
u
mbe
r
of iteration
s
to obtain a goo
d result is a
l
so problem
-
depe
ndent. A
too low n
u
m
ber of ite
r
atio
ns may
stop t
he search
proce
s
s premat
urely, whil
e
too large iterations ha
s th
e con
s
eq
uen
ce of
unne
ce
ssary add
ed comp
utationa
l complexity
and mo
re time need
ed [20]
.
Accel
e
ration Coeffici
ents: The
a
c
celeration
co
efficient
s
1
c
and
2
c
, together
with the random
values
1
r
and
2
r
, maintain the
stoch
a
sti
c
in
fluence of the cognitive an
d so
cial com
pone
nts
of the pa
rticl
e
’s velo
city resp
ectively. T
he con
s
tant
expre
s
se
s how mu
ch confiden
ce a
particl
e ha
s i
n
itself, whil
e
expre
s
ses
h
o
w mu
ch
co
n
f
idence a p
a
rticle ha
s in it
s nei
ghb
ours
[20]. Normally,
1
c
and
2
c
a
r
e
static,
with th
eir o
p
timized
value
s
b
e
in
g foun
d e
m
pi
rically.
Wrong initiali
zation of
1
c
and
2
c
may result in diverge
n
t or cy
clic b
e
h
a
viour [20]. F
r
om the
different emp
i
rical
re
sea
r
ches, it ha
s b
een p
r
op
ose
d
that the two accele
ratio
n
con
s
tant
s
sho
u
ld be
2
2
1
c
c
.
Inertia weight
: The inertia
weig
ht plays
a ve
ry import
ant role in th
e conve
r
g
e
n
c
e behavio
ur
of the PSO algorithm. The
inertia wei
g
h
t
is em
ployed
to control th
e impact of the previo
us
history
of vel
o
citie
s
o
n
th
e current
on
e. Usually th
e be
st
choi
ce of the
ine
r
tia wei
ght i
s
arou
nd 1.2, a
nd as the al
g
o
rithm prog
re
sses
thi
s
valu
e is gra
dually
decrea
s
ed to
0.
Con
s
tri
c
tion
f
a
ctor: The co
nstri
c
tion co
e
fficient
was d
e
velope
d
by Clerc.
T
h
is coefficient
i
s
extremely important to contro
l the exploration and exploi
tation trade-off, to ens
u
re
conve
r
ge
nce
behavio
ur.
The
con
s
triction
coe
fficient gua
rant
ees
co
nverg
ence of the
particl
es ove
r
time and also prevent
s co
llapse [21].
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Optim
a
l Capa
citors in Ra
di
al Distri
bution
System
for Loss Re
du
ctio
n and …
(S Bhong
ade
)
572
Maximum Vel
o
city: Maxim
u
m velocity
max
v
controls the g
r
anul
arity of t
he sea
r
ch sp
ace
by
clampi
ng vel
o
citie
s
a
nd
create
s
a b
e
tter
balan
ce
b
e
twee
n gl
ob
al explo
r
atio
n an
d lo
cal
exploitation. If a particle’
s
velocity goe
s beyond it
s
sp
ecified maxi
mum velocity
, this velocity
is set to the
value
max
v
. If the
maximum veloc
i
ty
max
v
is too large, the
n
the parti
cle
s
may
move errati
ca
lly and jum
p
over the
opt
i
m
al
solution. On
the other hand,
if
max
v
is too sm
all,
the particl
e’s
movement is
limited and th
e swar
m may
not explore sufficie
n
tly or the swa
r
m
may beco
m
e
trappe
d in a local o
p
timum
.
The flowcha
r
t
for the basi
c
PSO algorith
m
is sh
own in
figure 2.
4.3. Algorith
m
of Capa
citor Placemen
t Using PSO
The
Com
put
ational
step
s involved i
n
finding th
e o
p
timal lo
catio
n
s
and
si
ze
s of the
cap
a
cito
rs to
minimize the losse
s
in a ra
dial distri
butio
n system a
r
e
summ
ari
z
ed i
n
followin
g
:
1.
Input load
an
d line d
a
ta of
the test
ca
se and
ru
n th
e NR loa
d
flo
w
p
r
og
ram. E
v
aluate real
and re
active
power flows i
n
lines a
nd a
s
well a
s
lo
sses.
2.
Cal
c
ulate
Lo
ss Sen
s
itivity Facto
r
s a
nd Normalized
Voltag
e Magnitud
e
. And
find
th
e
can
d
idate b
u
s
e
s
for capa
citor placeme
n
t
in the radial distrib
u
tion sy
stem.
3.
Define vari
ab
les (cap
acito
r
s to be place
d
at
the can
d
idate bu
se
s) within their permi
ssible
rang
e, defin
e po
pulation
si
ze, n
o
. o
f
iteration
a
nd a
s
sume
suitabl
e valu
es
of PSO
para
m
eters.
4. Take
ite
r
=0
5.
Ran
domly ge
nerate the p
o
pulation
of pa
rticle
s and th
eir velocitie
s
6.
For ea
ch p
a
rt
icle ru
n NR lo
ad flow to find out losse
s
.
7.
Cal
c
ulate the
fitness fun
c
tio
n
of each p
a
rt
icle u
s
ing e
q
u
.
(1)
8.
Find out “p
erson
a
l be
st (p
best)
” of all p
a
rticle
s an
d “global be
st (gbe
st)” p
a
rticle from their
fitnes
s
.
9.
Incre
m
ent iteration co
unt
10.
Cal
c
ulate the
velocity of each pa
rticle u
s
i
ng equ
ation (9) and a
d
ju
st it if
its limit gets violated
11.
Cal
c
ulate the
new p
o
sitio
n
of each p
a
rticle usin
g equ
a
t
ion (11
)
12.
For ea
ch p
a
rt
icle ru
n NR lo
ad flow to find out losse
s
.
13.
Cal
c
ulate the
fitness fun
c
tio
n
of each p
a
rt
icle u
s
ing e
q
u
a
tion (1
)
14.
For ea
ch p
a
rt
icle if curre
n
t fitness(P) i
s
b
e
tter than pb
est then pb
est = p
15.
Set best of pbest a
s
gbe
st
16.
Go to step no
. 9, until maximum numb
e
r of iterations i
s
co
mpleted.
17.
Coo
r
din
a
te of gbest parti
cl
e gives optim
ized valu
e
s
o
f
control varia
b
les an
d its fitness gives
minimized val
ue of losse
s
.
Figure 2. Flow ch
art depi
cting the PSO Algorithm
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02-4
752
IJEECS
Vol.
2, No. 3, Jun
e
2016 : 566
– 582
573
4.4. Algorith
m
of Capa
citor Placemen
t Using PSO
The
Com
put
ational
step
s involved i
n
finding th
e o
p
timal lo
catio
n
s
and
si
ze
s of the
cap
a
cito
rs to
minimize the losse
s
in a ra
dial distri
butio
n system a
r
e
summ
ari
z
ed i
n
followin
g
:
1.
Input load
an
d line d
a
ta of
the test
ca
se and
ru
n th
e NR loa
d
flo
w
p
r
og
ram. E
v
aluate real
and re
active
power flows i
n
lines a
nd a
s
well a
s
lo
sses.
2.
Cal
c
ulate
Lo
ss Sen
s
itivity Facto
r
s a
nd Normalized
Voltag
e Magnitud
e
. And
find
th
e
can
d
idate b
u
s
e
s
for capa
citor placeme
n
t
in the radial distrib
u
tion sy
stem.
3.
Define vari
ab
les (cap
acito
r
s to be place
d
at
the can
d
idate bu
se
s) within their permi
ssible
rang
e, defin
e po
pulation
si
ze, n
o
. o
f
iteration
a
nd a
s
sume
suitabl
e valu
es
of PSO
para
m
eters.
4. Take
ite
r
=0
5.
Ran
domly ge
nerate the p
o
pulation
of pa
rticle
s and th
eir velocitie
s
6.
For ea
ch p
a
rt
icle ru
n NR lo
ad flow to find out losse
s
.
7.
Cal
c
ulate the
fitness fun
c
tio
n
of each p
a
rt
icle u
s
ing e
q
u
.
(1)
8.
Find out “p
erson
a
l be
st (p
best)
” of all p
a
rticle
s an
d “global be
st (gbe
st)” p
a
rticle from their
fitnes
s
.
9.
Incre
m
ent iteration co
unt
10.
Cal
c
ulate the
velocity of each pa
rticle u
s
i
ng equ
ation (9) and a
d
ju
st it if
its limit gets violated
11.
Cal
c
ulate the
new p
o
sitio
n
of each p
a
rticle usin
g equ
a
t
ion (11
)
12.
For ea
ch p
a
rt
icle ru
n NR lo
ad flow to find out losse
s
.
13.
Cal
c
ulate the
fitness fun
c
tio
n
of each p
a
rt
icle u
s
ing e
q
u
a
tion (1
)
14.
For ea
ch p
a
rt
icle if curre
n
t fitness(P) i
s
b
e
tter than pb
est then pb
est = p
15.
Set best of pbest a
s
gbe
st
16.
Go to step no
. 9, until maximum numb
e
r of iterations i
s
co
mpleted.
17.
Coo
r
din
a
te of gbest pa
rticl
e
gives optimi
z
ed va
lu
es of
control va
ria
b
les a
nd its fitness give
s
minimized val
ue of losse
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Optim
a
l Capa
citors in Ra
di
al Distri
bution
System
for Loss Re
du
ctio
n and …
(S Bhong
ade
)
574
Figure 3. Flow Ch
art of O
p
timal Cap
a
ci
tor Placem
en
t through PS
O
5. Test Resu
lts
The Algo
rith
m given in S
e
ction IV ha
s been p
r
og
ra
mmed u
s
ing
MATLAB and
run o
n
an
Intel Co
re i3,
2.20-G
H
z p
e
rson
al comp
uter
with
2.00
GB RAM a
n
d
is te
sted o
n
10-b
u
s,
34-b
u
s
and 8
5
-b
us radial di
strib
u
tion sy
stem
s
and the
obtai
ned
re
sults
a
r
e expl
ained
i
n
this
se
ction
to
demon
strate
the effectiven
ess of this
m
e
thod. Fo
r ca
lculatio
n of th
e co
st, the $
rate ha
s b
een
con
s
id
ere
d
in
ord
e
r to
me
et the interna
t
ional st
a
nda
rds. Th
e cost
value can
be
conve
r
ted i
n
to
any curren
cy
value
s
with
the u
s
e
of resp
ective
multiplication factor. In
this diss
ertation, the
equivalent
₹
f
o
r $, multiplication factor h
a
s be
en a
s
su
med as
₹
62 /$.
Comm
ercially
availabl
e ca
pacito
r
s size
s with
real
co
sts/KVar are
use
d
in
the
a
nalysi
s
.
Table I
sho
w
s the exa
m
pl
e of su
ch
dat
a. It was
d
e
ci
ded that the
maximum pe
rmissi
ble
cap
a
c
itor
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ISSN: 25
02-4
752
IJEECS
Vol.
2, No. 3, Jun
e
2016 : 566
– 582
575
siz
e
max
c
Q
placed
a
t
any bu
s
sh
o
u
ld n
o
t excee
d
12
00
KVa
r. The
equival
e
nt ann
ual
co
st per unit
of po
wer lo
ss in
$
/ (kW-y
ear) i
s
sele
ct
ed to
be
168
$/ (k
W-yea
r) [23]. The
fixed
co
st of th
e
cap
a
cito
rs is
taken
a
s
$
1
0
00 [9] a
nd th
e data
given
in Tabl
e II is
use
d
to
cal
c
u
l
ate the
annu
al
inst
allat
i
o
n
co
st
of the cap
a
c
itor.
Table 1. Available Th
ree P
hase Ca
pa
citor Size
s an
d Co
sts
Size
(KVAr)
150 300 450 600 900 1200
Cost
($)
750
975
1140
1320
1650
2040
Table 2. Possible Siz
e
s
of Capac
i
tors
and Siz
e
s
In $/KVAR
j
1 2 3 4
5 6 7
c
j
Q
150 300 450 600
750 900 1050
$/KVAR
0.50
0
0.35
0
0.25
3
0.22
0
0.27
6
0.18
3
0.28
8
j
8
9
10 11
12 13 14
c
j
Q
1200
1350
1500
1650
1800
1950
2100
$/KVAR
0.17
0
0.20
7
0.20
1
0.19
3
0.18
7
0.21
1
0.17
6
j
15 16 17 18
19 20 21
c
j
Q
2250
2400
2550
2700
2850
3000
3150
$/KVAR
0.19
7
0.17
0
0.18
9
0.18
7
0.18
3
0.18
0
0.19
5
j
22 23 24 25
26 27 --
c
j
Q
3300
3450
3600
3750
3900
4050
--
$/KVAR
0.17
4
0.18
8
0.17
0
0.18
3
0.18
2
0.17
9
--
The sel
e
cte
d
para
m
eters o
f
PSO are sh
own in Ta
ble
3.
Table 3. Sele
cted Paramet
e
rs of PSO
Population Size
50
Acceleration Constants (
,
2.1 and 2.0
Inertia w
e
ights
(
,
1 and 0.2
Max. and
min. velocity
of pa
rticles
0.003 and
-0.003
Constriction Factor
0.729
A. 10-Bus
RDS
The test ca
se
I is a 10-bus,
9-line, singl
e
f
eeder ra
dial
distributio
n system [23] sh
own in
figure
4
with
the rate
d lin
e
voltage
of 2
3
kV, a
c
ti
ve and rea
c
tive power
l
oad
o
n
the system
are
12368KW
and 4186 KVAR respectively.
The line and load data
of the
10-bus radi
al di
stribution
system i
s
given in Table I
V
.
Figure 4. 10-Bus Ra
dial Di
stributio
n System
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