Indonesian J
ournal of Ele
c
trical Engin
eering and
Computer Sci
e
nce
Vol. 1, No. 3,
March 20
16, pp. 419 ~ 4
3
0
DOI: 10.115
9
1
/ijeecs.v1.i3.pp41
9-4
3
0
419
Re
cei
v
ed
De
cem
ber 2, 20
15; Re
vised Janua
ry 1
9
, 20
16; Accepted
February 6, 2
016
Pareto Optimal Reconfiguration of Power Distribution
Systems with Load Uncertainty and Recloser
Placement Simultaneousely U
s
ing a Genetic Algorithm
Based on NSGA-II
Sina Khajeh
Ahmad
Attar
i
*, Mahmoud Reza Shaka
rami, Farhad
Namdari
Dep
a
rtement o
f
Electrical Eng
i
ne
erin
g, Lores
tan Univ
ersit
y
,
Dan
e
shg
ah Str
eet, 712
34-9
8
6
53, Khorram
a
b
ad, Loresta
n, Iran
e-mail: sinaattari@gmail.com
A
b
st
r
a
ct
Reconfi
gurati
o
n, by excha
n
g
in
g the funct
i
on
al li
nks bet
w
een the el
e
m
e
n
ts of the
system,
repres
ents o
n
e
of th
e
most
i
m
porta
nt
me
asures w
h
ich
can i
m
prov
e t
he
oper
atio
nal
perfor
m
a
n
ce
of a
distrib
u
tion sy
stem. Besi
des
, reclosers
u
s
e to el
i
m
i
n
a
t
e transie
nt faults, faults i
s
olati
on, n
e
tw
ork
ma
na
ge
me
nt
and
en
ha
nce r
e
lia
bi
lity to re
d
u
ce cust
omer
outag
es. F
o
r l
oad
unc
ertaint
y
a n
e
w
met
h
od
base
d
on
prob
abil
i
stic interv
a
l
arith
m
etic a
p
p
roac
h is use
d
to incorp
orat
e
uncertai
n
ty in
loa
d
de
mand t
hat
can forec
a
st reasonably acc
u
rate
operational condit
ions
of radial system
distribution
(RDS) with better
computati
o
n
a
l
efficiency.In thi
s
paper, the o
p
timi
z
a
ti
on pro
c
ess is perfor
m
e
d
by cons
id
erin
g pow
er lo
ss
reducti
on al
on
g w
i
th reliab
ilit
y index as o
b
j
e
ctive functio
n
s
. Simul
a
tio
n
results on ra
di
al 33 b
u
ses tes
t
system
indicat
e
that s
i
multaneous
optim
i
z
a
tion of thes
e two issues
has signific
ant im
pact on syst
em
perfor
m
a
n
ce.
Ke
y
w
ords
: p
o
w
e
r distrib
u
ti
on syste
m
s, reconfi
gurati
on,
loa
d
unc
ertai
n
ty, genetic
a
l
gorit
hms,
mu
l
t
i-
obj
ective o
p
timi
z
a
t
i
o
n
, pareto
opti
m
a
lity, recl
oser pl
ace
m
e
n
t
1. Introduc
tion
The m
o
st
imp
o
rtant m
e
a
s
u
r
es
whi
c
h
ca
n imp
r
ove
the
pe
rform
a
n
c
e
in th
e o
p
e
r
at
ion of
a
distrib
u
tion
system a
r
e:
(i) reconfigu
r
ation of
th
e
syste
m
, ex
cha
ngin
g
the
functio
nal li
nks
betwe
en it
s e
l
ements (syst
e
m/netwo
rk/feede
r
re
confi
guratio
n p
r
o
b
l
em);
(ii) va
ri
ation a
nd
co
n
t
rol
of the rea
c
tive power flo
w
throug
h the system (
optim
al rea
c
tive po
wer
dispatch probl
em), u
s
i
n
g
ban
k ca
pa
citors, p
o
wer g
enerators, etc.; (iii)
variati
on and
co
ntrol of the voltage by u
s
ing
on-
load tap-ch
a
ngers for po
wer tra
n
sfo
r
mers (by us
i
ng automati
c
voltage reg
u
lators); and
(iv)
cha
ngin
g
the
ope
rating
scheme
of the
parall
e
l
c
o
nne
c
te
d p
o
w
e
r
tr
a
n
s
for
m
ers
,
e
t
c
.
T
h
is
pa
pe
r
focu
se
s on o
p
timization th
roug
h the re
confi
guration o
f
power di
stri
bution sy
ste
m
s.
The
re
config
uration
p
r
obl
em is on
e of
the mu
lti
-
criteria
optimi
z
at
ion type
s, where
the
solutio
n
is
chosen after t
he evaluatio
n of som
e
in
dice
s (e.g., active power l
o
sse
s
, relia
bi
lity
indices, b
r
a
n
c
h l
oad li
mits, voltage d
r
o
p
limits, et
c.),
whi
c
h
re
pre
s
ent multiple
p
u
rpo
s
e
s
. T
h
e
s
e
crite
r
ia can b
e
gro
uped i
n
two differe
nt categ
o
rie
s
:
(i
) obje
c
tive function
s: criteria that must be
minimized; a
nd (ii)
con
s
traints (re
s
tri
c
tions): cr
ite
r
ia
that must be
inclu
ded
withi
n
som
e
bou
n
d
s.
On the other
hand, the criteria a
r
e in
co
mpatible fr
om
the point of view of mea
s
u
r
eme
n
t units
and
are often con
f
licting.
Mo
re
over,
some criteria can
be
(or it is i
m
po
rtant for the
m
to be) mod
e
l
ed,
at the same t
i
me as obj
ect
i
ves and
con
s
traint
s. For i
n
stan
ce, the
active po
wer
losse
s
must
be
minimized bu
t we can
sim
u
ltaneo
usly impose a max
i
mal accepta
b
le value (co
n
strai
n
t). Thu
s
, in
orde
r to solv
e the proble
m
, first of all, a proper m
odel ha
s to be cho
s
e
n
. The pro
b
lem
of
optimizatio
n throu
gh the re
config
uratio
n of a powe
r
di
stributio
n system, in terms of its definition,
is a hi
sto
r
ical
singl
e obj
ect
i
ve probl
em
with co
n
s
trai
nts. Since 19
75, wh
en M
e
rlin an
d Ba
ck [1]
introdu
ce
d th
e idea of di
st
ribution
syste
m
re
config
u
r
ation for a
c
ti
ve powe
r
lo
ss re
du
ction,
until
nowaday
s, a lot of rese
arche
r
s
have p
r
opo
se
d di
ve
rse
method
s
and alg
o
rith
ms to solve the
reconfigu
r
atio
n pro
b
lem
a
s
a
singl
e o
b
jective
p
r
obl
em. The m
o
st freq
uently
use
d
on
e is
the
main crite
r
io
n
method (
ε
-constraint) wh
ere the probl
em is def
ined in the following conditions:
a
ma
in
c
r
iter
ion
is
c
h
os
en
,
c
o
nc
om
itant
ly indicatin
g
accepta
b
le
v
a
lue
s
for th
e
other
criteri
a
.
Usually, active power lo
sses a
r
e ad
opte
d
as the
m
a
in
criteri
on [1–7
]. This app
roa
c
h ha
s a maj
o
r
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
1, No. 3, March 20
16 : 419 – 430
420
wea
k
n
e
ss
be
cau
s
e
there
is m
o
re
than
one
ind
e
x that mu
st b
e
take
n into
accou
n
t in t
he
optimizatio
n pro
c
e
ss a
nd,
without any p
r
ior info
rm
ati
on abo
ut the different crite
r
ia, ch
oosi
ng
the
accepta
b
le v
a
lue
ca
n b
e
p
r
oble
m
atic. A
dditionally,
thi
s
a
p
p
r
oa
ch
consi
ders load
un
ce
rtainty. On
the othe
r ha
n
d
, som
e
auth
o
rs have
stud
ied this
probl
em u
s
ing
agg
regatio
n fun
c
t
i
ons,
conve
r
ti
ng
the multi-obje
c
tive problem
into a
sin
g
le
obje
c
ti
ve one
that assu
me
s a
(weig
h
ted
or
not)
su
m
of
the sele
cted
obje
c
tive functions [8–1
0]. The majo
r di
ffic
u
lty in this
kind of problem c
o
ns
is
t
s
in the
incom
patibilit
y of different criteria. To create a gl
obal
function, all criter
ia must be converted to
the sam
e
me
asu
r
em
ent u
n
it; a freque
n
t
ly used met
hod i
s
to co
n
v
ert them int
o
co
sts,
whi
c
h is
usu
a
lly a tri
c
ky an
d often
inaccu
rate
op
eration.
In
ad
dition, subje
c
tivity appears, cau
s
e
d
by t
he
introdu
ction
o
f
weightin
g fa
ctors for diffe
rent
crit
e
r
ia.
Thus, th
e exi
s
ten
c
e of a
model that
could
take into
con
s
ide
r
ation
m
o
re
obje
c
tive
function
s
an
d co
nst
r
aint
s at the
same
time is
of great
intere
st. To
el
iminate the
subje
c
tivity and rigi
di
ty of th
e cl
assi
c met
hod
s, the
aut
hors p
r
o
p
o
s
e
an
origin
al ap
proach to form
ulate this pro
b
lem u
s
in
g the Pareto o
p
t
imality con
c
ept that defin
es a
dominate rela
tion
among solution
s.
Re
clo
s
er
use
to eliminate
transi
ent fa
ults
, faults i
s
olation, network
man
age
ment and
enha
nce relia
bility indice
s.
A recl
oser i
s
a dev
ice
with
the ability to
detect
pha
se
and
pha
se
-to-
earth ove
r
cu
rre
nt con
d
itions, to inte
rru
pt t
he circuit if the overcurrent
persist
s after a
pred
etermi
ne
d time, and t
hen to auto
m
atically re
cl
o
s
e to re-ene
rgi
z
ed th
e line.
Optimum
swit
ch
placement has carried out with QEA-based algori
t
h
m to improve customer
servi
c
e reliability
[11]. A new comp
osite
obje
c
tive fun
c
tion of
inve
stment
co
st and
reliabili
ty on optim
um
placement of
line swit
ch is
p
r
e
s
ente
d
[12].
Simula
ted ann
ealin
g optimi
z
ing
algo
rithm h
a
ve
discu
s
sed [13
]
.
In this pape
r
an origi
nal m
e
thod, aiming
the
optimizat
ion throu
gh the re
config
uration of
distrib
u
tion systems an
d reclo
s
e
r
pla
c
e
m
ent
simulta
neou
sly are
prop
osed whi
c
h ca
n impro
v
e
reliability ind
e
x
and redu
ce
power l
o
sse
s
of the
dist
ri
butionn
etwo
rk. The
novelt
y
of the meth
od
c
o
ns
is
ts
in:
1)
The criteria fo
r optimization
are eval
uate
d
on active p
o
we
r dist
ribut
ion system.
2)
The o
r
igin
al
formulatio
n o
f
the optimizati
on proble
m
, as a Pa
reto optimal
one, with t
w
o
obje
c
tive functions (
a
c
tive power lo
sses
and
system
average inte
rruption freq
ue
ncy inde
x
).
3)
The pro
babili
stic dist
ributi
on-b
a
sed int
e
rval
arithm
e
t
ic approa
ch
is used to inco
rpo
r
ate
uncertainty in
load dema
n
d
.
4)
An ori
g
inal
g
enetic algo
rit
h
m (ba
s
ed
o
n
NSGA
-II) to
solve
the p
r
o
b
lem
(a
s a P
a
reto
optimal
one) in a n
o
n
-
prohibitive e
x
ecution time
.
All the sim
u
l
a
tions
are carri
ed o
u
t in
MA
TLAB software.
The
re
st of the
pape
r i
s
orga
nized a
s
follo
ws:
section
2 hi
ghlight
s pro
b
lem form
ul
ation with t
he criteri
a
for
optimizatio
n.Section 3,
4
rep
r
e
s
ent Pa
reto optim
ality proble
m
formulation a
n
d
solving
meth
od.
Geneti
c
ope
rators a
r
e introdu
ced in section 5.
In se
ction 6 be
st comp
ri
se
solutio
n
will be
discu
s
sed. T
he
simulatio
n
re
sult
s a
r
e ill
ustrate
d
in
se
ction
7 a
nd fi
nally con
c
ludi
ng rema
rks a
r
e
dra
w
n in secti
on 8.
2. Problem Formulation
2.1. Load Un
certaint
y
Mo
deling
Assu
ming tha
t
the real and
rea
c
tive power
load va
ry as per
Gau
s
sian distri
butio
n
2
1
2
2
()
1
()
2
i
i
x
fx
e
(1)
Whe
r
e
is
cons
tant,
i
s
the sta
nda
rd
deviation, a
n
d
is them
e
an valu
e. x
i
is the
no
rmali
z
ed
real o
r
rea
c
tive load at the ithbus. Fo
r re
al and re
activ
e
load
()
()
()
()
ii
rr
PL
i
Q
L
i
xa
n
d
x
PL
i
Q
L
i
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Pareto Optim
a
l Re
config
uration of Powe
r Di
st
ributio
n System
s with
Load … (Sin
a Khajeh AA)
421
Whe
r
e PL
r
(i) and
Q
L
r
(i
)
are
rated (n
omi
nal) real and rea
c
ti
ve load
at ith bus. L
e
t
the deg
ree
of
belon
gingn
ess (memb
e
rshi
p) fo
r the
real
and
re
acti
ve load
at all
of
t
he
bu
se
s be repre
s
e
n
ted b
y
PL
(k) a
nd
QL
(k), respe
c
tively, where
k is the
num
ber
of d
egre
e
of belon
gin
gne
ss. F
r
om
the
cha
r
a
c
teri
stic cu
rve
sh
own
in Fi
gu
re
1, t
he m
ean
valu
e of th
e n
o
rm
alize
d
real
an
d rea
c
tive loa
d
is 1.0 for the
degree of bel
ongin
gne
ss
1
.
0.
()
1.
0
(
)
1
.
0
()
PL
r
PL
i
fo
r
k
PL
i
()
1.
0
(
)
1
.
0
()
QL
r
QL
i
an
d
f
or
k
QL
i
Equation (1)
can b
e
writte
n as
2
2
[(
(
)
/
(
))
]
2
()
1
()
(
)
()
2
r
PL
i
P
L
i
PL
r
PL
i
kf
e
PL
i
(2)
For
=1.0 an
d
PL
(k)=1.0. From (2), we get
=0.398. Usi
ng
=0.39
8
and
=1.0, (2)
res
u
lts in
1/
2
ln
(
(
)
)
()
()
11
.
0
.
()
(
)
PL
rr
k
PL
i
P
L
i
fo
r
PL
i
P
L
i
(3)
A simila
r eq
u
a
tion can b
e
derived f
o
r th
e re
active lo
ad. In the a
b
o
ve expressi
on,
PL
(k)
is t
h
e
degree of be
longin
gne
ss
and it can take any value
between
PL
(k
)
PLma
x
(k)
/
N
L
to
PL
(k)
=
PLma
x
(k) for t
he spe
c
ified
numbe
r of i
n
tervals N
L
wh
ere N
L
i
s
the
numbe
r of
po
int of linea
rization
of the
curve
(Figure 1
)
a
n
d
PLma
x
(k) i
s
t
he maxim
u
m
possibl
e de
gree of
belo
ngi
ngne
ss. Let t
he
right-han
d sid
e
(RHS) of (3
) be
sym
boli
c
ally represent
ed as [-l
n
PL
(k
)
/
]
1/2
=
k
, k = 1, 2, 3, …
,
N
L
. Thus, (3
)
can
be
re
written a
s
PL(i
) /
PL
r
(i) =
1
k
, whe
r
e the
sign
re
sult
s i
n
t
he f
o
llo
win
g
lowe
r and u
p
per limit of the real po
we
r l
oad at ith bus:
(4)
Similar analy
s
is
results in t
he followi
ng lowe
r and u
p
p
e
r limit of rea
c
tive load at ith bus:
()
()
[
1
]
(
)
(
)
[
1
]
lr
k
u
r
k
QL
i
Q
L
i
and
Q
L
i
QL
i
(5)
For the p
r
e
s
e
n
tation of the
results, linea
rizatio
n
is
co
ndu
cted atthree point
s tha
t
result in three
distin
ct load i
n
tervals
(re
gi
ons), whi
c
h a
r
e as follo
ws:
1
22
2
33
3
()
,
(
)
,
.
.
,
1
()
[
1
]
,
()
[
1
]
,
.
.
,
2
()
[
1
]
,
()
[
1
]
,
.
.
,
3
rr
rr
rr
DP
L
i
P
L
i
i
e
f
o
r
k
DP
L
i
P
L
i
i
e
f
o
r
k
DP
L
i
P
L
i
i
e
f
o
r
k
(6)
Equation
(6
)
reflect
s
th
at
D
1
, D
2
, an
d
D
3
a
r
e
ce
rtai
nly in
boun
dform
and,
he
nce,
an
interval
arithmeti
c
op
eration h
a
s t
o
be pe
rform
ed to inco
rp
o
r
ate the
s
e va
riation
s
in co
nventional lo
ad
flow.
()
()
[
1
]
(
)
(
)
[
1
]
lr
k
u
r
k
PL
i
P
L
i
a
n
d
P
L
i
PL
i
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IJEECS
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16 : 419 – 430
422
Figure 1. Gau
ssi
an di
stribut
ion of load de
mand
2.2. Interv
al
Arithmetic
Let M an
d
N
be two
interv
al num
bers
with a supp
orti
ng inte
rval [m
1
, m
2
] and
[n
1
, n
2
],
respec
tively. If, in partic
u
lar, m
1
=
m
2
=
m, the inte
rval nu
mbe
r
M
redu
ce
s to th
e real
numb
e
r
m
= [m , m],
whi
c
h is
call
ed a point i
n
terval or
si
ngleton. Rul
e
s for ad
dition, subtracti
on,
multiplicatio
n, and divisio
n
are defin
ed a
s
:
11
2
2
[,
]
MN
m
n
m
n
(7)
12
2
1
[,
]
MN
m
n
m
n
(8)
11
1
2
2
1
2
2
11
1
2
2
1
2
2
[
m
i
n
(
,,,
)
,
ma
x
(
,
,
,
)
]
M
N
mn
mn
m
n
m
n
mn
m
n
m
n
m
n
(9)
1
M
MN
N
(10
)
Whe
r
e N
-1
=
[1/n
2
, 1/n
1
] with
0
[n
1
,
n
2
]. Howev
e
r, for the
pu
rpo
s
e of
po
wer flo
w
an
alysis,
cal
c
ulatio
ns i
n
volving com
p
lex num
ber,
rathe
r
tha
n
real
numb
e
rs are
nee
ded.
Hen
c
e,
ba
sic
interval num
b
e
rs in [14] a
r
e
used.
2.3. Interv
al
Po
w
e
r Flo
w
Analy
s
is
The ba
si
c power flo
w
analysi
s
m
e
thod u
s
ed
in this wo
rk i
s
esse
n
t
ially the
backward/forward
sweep
po
wer flow algo
rithm
a
nd given
in
[15]. The
uncertaintie
s
are
con
s
id
ere
d
varying a
s
[1
6]. The de
gre
e
of belon
ging
n
e
ss was
obtai
ned a
s
[17] f
o
r the
sp
ecifi
ed
numbe
r of int
e
rvals
N
L
. Fo
r differe
nt de
gree
of belo
n
g
ingn
ess (
-cuts), is
esti
mated for
all
k.
Usi
ng the val
ue of
k
, the i
n
tervals are
obtaine
d over whi
c
h
real
a
nd rea
c
tive lo
ads
are varyi
n
g
usin
g (4
) and
(5), respectiv
e
ly, in the form of lower a
n
d uppe
r bou
n
d
.
2.4. Activ
e
Po
w
e
r Los
ses
(
∆
P
)
Active po
wer losse
s
rep
r
e
s
ent th
e mo
st im
porta
nt criterio
n an
d
cannot b
e
ig
n
o
red
in
reconfigu
r
atio
n probl
em
s [1–10]. In orde
r to evaluate
this criteri
on it is necessa
ry to perform th
e
load flow
calcul
us. As mentione
d
before th
e most recommen
ded
approa
che
s
are
backward/forward swee
p based algo
rit
h
ms an
d use
d
in this articl
e. Due to loa
d
unce
r
tainty and
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a
l Re
config
uration of Powe
r Di
st
ributio
n System
s with
Load … (Sin
a Khajeh AA)
423
three
a
c
tive p
o
we
r lo
ss i
n
te
rvals gen
erated by
differe
n
t
PL
(k
),
weighted
s
u
m for total power los
s
cal
c
ulatio
n were u
s
ed. Th
e basi
c
con
c
e
p
ts are:
W
1
=1,
W
2
=0.
2
,
W
3
=0.
6
;
W
total
=W
1
+W
2
+W
3
;
P
Los
s_
kW
=
(
W
1
/W
total
)*Plo
s
s
(
PL
(k
)
=
1)
+
(
W
2
/W
total
)*m
ean(Pl
oss(
PL
(
k
)=
0.2)
)+(
W
3
/W
total
)*
mean
(Ploss(
PL
(k)
=
0.6
)).
2.5. Reliability
of the Distribution S
y
st
em
The
esse
ntial attribute
s
of interruptio
ns in
the
po
we
r su
pply of th
e custo
m
ers
are
th
e
freque
ncy an
d duratio
n. While d
u
ratio
n
is pre
dom
i
nantly influen
ced by the di
stributio
n system
stru
cture (rad
ial, meshed,
wea
k
m
e
she
d
) a
nd th
e
ex
isting a
u
toma
tions, the fre
quen
cy is ma
inly
influen
ced
by the ado
pted
operational
configuration; i
t
can b
e
mi
ni
mized by
the suitabl
e
choi
ce
of the effective configu
r
a
t
ion and o
p
timal re
clo
s
e
r
place
m
ent.
In other word
s, thro
ug
h
reconfigu
r
atio
n and optima
l
reclo
s
e
r
pla
c
eme
n
t,
we can improve those relia
bility indice
s whi
c
h
refer to the in
terru
ption fre
quen
cy [18]. Otherwise
, the reliability of a distrib
u
tion
system can
be
con
s
id
ere
d
from two different angle
s
:
1)
Reliability of
a parti
cular customer: e.g.
, t
he average number of inte
rruptions to the power
sup
p
ly. This index can
rep
r
esent a
po
ssible
obj
e
c
tive and/o
r
con
s
traint i
n
the
optimizati
o
n
probl
em (be
c
ause so
me
cu
stome
r
s
can impo
se
maximal/mini
mal limits in
their supply
cont
r
a
ct
s).
2)
Reliability of
the ent
ire supply
sy
stem: e.g.,
the num
ber of interrup
ted
customers per year
[18], system
averag
e inte
rruption f
r
eq
u
ency in
dex (SAIFI) [19] (d
efined a
s
: total numb
e
r
of
cu
stome
r
inte
rru
ption
s
long
er than 3 min
u
tes pe
r total numbe
r of cu
stome
r
s
se
rved).
Knowin
g the
failure
rate
s at the level
of ea
ch
su
pplied
nod
e
(load
poi
nt),
we
can
estimate the
SAIFI using the relatio
n
shi
p
(in a simil
a
r form to the one given in [2
0]):
1
.
n
ii
i
N
SAI
F
I
N
(11
)
w
h
er
e
N
re
prese
n
ts th
e to
tal num
ber of
cu
stom
ers
served;
N
i
is th
e total
numb
e
r
of custo
m
ers
sup
p
lied from
node
i
;
n
i
s
the num
ber
of load no
de
s
of the system
;
λ
i
is
the total failure rate
of
the equivale
nt element
correspon
ding
to the re
lia
bility block di
agra
m
at the
level of nod
e i
[year
−
1
].
With a
sim
p
le
example, th
e
method
of
calcul
at
ion of t
he indi
cato
rs
descri
bed.
Figure
2
sh
ows a
sampl
e
radi
al
distributio
n system of four lines A,
B, C
and D with four loa
d
point
s 1, 2, 3, and 4.
Hypotheti
c
al
recl
osers R1, R2, R3 h
a
ve
been pl
ace
d
in approp
riate locatio
n
s. If X
i
=0, it means
the ab
sen
c
e
and X
i
=1, me
ans
presen
ce
of re
clo
s
e
r
s.
Unavail
ability and failu
re
rate of loa
d
p
o
int
can b
e
cal
c
ul
ated in the followin
g
app
ro
ach.
D
i
s
t
r
i
but
i
on
s
ubs
t
a
t
i
o
n
1
2
3
A
B
C
D
R1
R2
R3
4
Figure 2. Sample ra
dial di
stributio
n syst
em
First, unavail
ability to each
of lines
A
,
B
,
C,
and
D
can
be cal
c
ulate
d
.
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16 : 419 – 430
424
Acco
rdi
ng to
the netwo
rk stru
cture
sh
own
in Fi
gure 2. Load po
int 1, in the followin
g
scena
rio
s
are
without ele
c
tricity:
AA
A
B
B
B
CC
C
D
D
D
Ur
U
r
Ur
U
r
(12
)
1)
A fault on the line A.
2)
A fault on the line B in the
absen
ce of
R
1
(X
1
'
=1 or X
1
=0).
3)
A fault on the line C in the absen
ce of
R
1
,
R
2
(X
1
'
. X
2
'
=1).
4)
A fault on the line D in the absen
ce of
R
1
,
R
3
(X
1
'
. X
3
'
=1).
Therefore load point 1 unavailability can be formulated in (13).
''
'
'
'
11
1
2
1
3
AB
C
D
uU
U
x
U
x
x
U
x
x
(13
)
In this equati
on
X
'
is co
mpl
i
mentary to
X
(X
'
= 1–X
).
Similarly, una
vailability of l
oad poi
nts 2, 3,
and 4 ca
n be formul
ated
as equ
ation
s
(14-16).
'
23
AB
C
D
uU
U
U
U
x
(14
)
''
32
3
AB
C
D
uU
U
U
x
U
x
(15
)
'
42
A
BC
D
uU
U
U
x
U
(16
)
Failure rate for load poi
nt 1 to 4 can be acce
ss
ed
by replacin
g
with line unavailability. Thus,
reliability indi
ces can be calcul
ated.
2.6. Other
Cr
iteria
1)
Nod
e
Voltag
es
(Vi): Ba
si
cally, ea
ch v
o
lt
age
r.m.s.
value of th
e
network n
o
des mu
st be
framed
within
the allowa
ble
limits.
2)
Bran
ch Lo
ad
Limits thro
ug
h Line
s (I
ij
): a typical co
nstraint on
the re
config
uratio
n probl
em.
3)
Safegua
rd of
power
sup
p
li
es for
all customer
s: The
attache
d
gra
ph of the ele
c
tri
c
syst
em
sho
u
ld be
co
nne
cted (a tree or a forest
).
Config
uratio
n
of the Distri
but
ion Syste
m
: Generally, electri
c
al di
stribution sy
st
ems a
r
e
operated in radial configu
r
ation. This
co
ndition can b
e
expre
s
sed
as follo
ws:
ij
ij
E
np
(17
)
Whe
r
e
α
ij
is a
binary varia
b
le, represen
ting the st
atus of a tie line (0–op
en, 1–
clo
s
ed
);
n
is the
numbe
r of el
ectri
c
sy
stem
node
s;
E
is the set of po
wer
system li
nes
(bran
c
he
s) a
nd
p
is
th
e
numbe
r of
co
nne
cted com
pone
nts
. In graph the
o
ry term
s, for a system with o
ne so
urce (
p =
1)
we are talkin
g about an
optimal tree
and for a syst
em with more
than one fee
der (
p >
1
)
we
are
talking
ab
out
an
optimal fores
t
with
a
nu
mber of tree
s (con
ne
cted
compon
ents)
equal
to that
of
sou
r
ce nod
es.
3. Pareto Optimalit
y
Problem Formulation
The criteri
a
pre
s
ente
d
ab
ove are n
o
t uniqu
e,
but we con
s
ide
r
them to be the mo
st
importa
nt on
e
s
. Ta
king
into
acco
unt the
s
e criteri
a
, we
can
be
gin to
perceive th
e
real dim
e
n
s
io
ns
of the proble
m
. These crit
eria are inco
mpatible
from
the point of view of mea
s
u
r
eme
n
t units and
can
be
group
ed in
two
different
cate
go
ries: o
b
je
ct
ive functio
n
s an
d con
s
traint
s
(re
stri
ction
s
).
In
Pareto
optimi
z
ation, th
e
central
con
c
ep
t is
name
d
n
on-d
o
min
a
ted
sol
u
tion. T
h
i
s
solutio
n
m
u
st
satisfy the followin
g
two condition
s: (i)
there is n
o
other solution t
hat is su
peri
o
r at least in o
ne
obje
c
tive function; (ii
)
it is eq
ual o
r
superi
o
r
with
respe
c
t to other o
b
je
ctive function val
ues.
Usually, the solution is not
uniqu
e and consi
s
ts of
a set of acce
pta
b
le optimal solution
s (Pareto
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IJEECS
ISSN:
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752
Pareto Optim
a
l Re
config
uration of Powe
r Di
st
ributio
n System
s with
Load … (Sin
a Khajeh AA)
425
optimal). T
h
e
set of Pa
reto
solutio
n
s fo
rms the Pa
ret
o
front a
s
so
ci
ated with
a p
r
oblem. Fig
u
re 3
pre
s
ent
s
a p
o
ssible
Pareto fro
n
t for th
e optimi
z
atio
n p
r
oble
m
b
a
s
ed
on
two
o
b
jective
s
(
∆
P
and
SAIFI). The
Pareto f
r
ont
allows
an i
n
forme
d
d
e
ci
si
on to
be
ma
de by
visu
alizing
an
exte
nsiv
e
rang
e of optio
ns si
nce it co
ntains the
sol
u
ti
ons that are optimal fro
m
an overall
stand
point.
A
c
t
i
ve
P
o
w
e
r L
o
s
s
e
s
,
P(kW)
S
y
s
t
e
m
A
v
e
r
a
g
e
Int
e
rrupt
i
o
n
F
r
e
q
ue
nc
y Inde
x, S
A
IF
I
Figure 3. A Pareto fro
n
t for a bi-obje
c
tive reconfigu
r
at
ion pro
b
lem
As a Pareto o
p
timal multi-o
b
jective probl
em, we propo
se the follo
wi
ng form:
Obje
ctive function
Con
s
trai
nts:
mi
n
[
,
]
PS
A
I
F
I
mi
n
m
ax
ii
i
VV
V
ma
x
ij
ij
II
ij
ij
E
np
(18
)
(19
)
4. Problem Solv
ing
4.1. Genetic
Encoding
The i
n
itial p
opulatio
nis g
enerated
u
s
i
ng the
b
r
an
ch-exch
ang
e
heu
ri
stic
al
gorithm
pre
s
ente
d
in
[3]. In this i
m
plementatio
n, the
rep
r
e
s
en
tation u
s
ing
the
br
an
c
h
lists
wa
s c
h
os
en
becau
se a p
o
w
er
syste
m
n
ode is
only linke
d with a
small part of t
he othe
r nod
es (it results i
n
a
rare gra
ph,
i.e.
, the asso
ci
ated matrix contain
s
many
ze
ro ele
m
en
ts). Con
s
e
q
u
ently, the graph
asso
ciated
wi
th the
ele
c
tri
c
n
e
two
r
k
ca
n be
d
e
scri
b
ed by
a
matri
x
with
2 lin
es and
m
c
olu
m
ns
(wh
e
re m is the numb
e
r of
the bran
che
s
), each
co
lum
n
indicatin
g
the two end
s
of a bran
ch. This
matrix doe
s n
o
t contain
ze
ro eleme
n
ts. Therefore,
u
s
ing the re
pre
s
entatio
n via the bra
n
ch lists,
a bina
ry codi
fication of th
e pro
b
lem (b
inary chro
mo
some
with fixed length
)
ca
n be obtai
ne
d.
Binary value
s
of the chromosome
wil
l
indicate
th
e
status
of every ele
c
tri
c
line: 0–op
en,
1–
clo
s
ed. Fi
gure4a exe
m
plifies the
graph
(which i
ndi
cates the
net
work to
polog
y) attach
ed t
o
a
distrib
u
tion
system, re
pre
s
e
n
ted by bran
ch lists
(
α
a
nd
β
), and the
bi
nary attached
chromo
som
e
g (sy
s
tem/gri
d
encodin
g
).
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16 : 419 – 430
426
1
2
5
6
3
4
:
:
1
2
1 5 6 1
4
2
3
5 6 3 4
3
g:
1
2
5
6
3
4
a:
1
0
1
1 0
1
1
:
1
2
1
5 6
1
4
:
2
3
5
6
3
4
3
Figure 4. A powe
r
dist
ribut
ion system: (a) t
he bran
ch
lists of the attache
d
gra
ph
(
α
an
d
β
)
and the attached chro
mo
some (g
); (b
) Bran
ch list
s
(
α
and
β
) obta
i
ned by de
co
ding
the chromo
so
me a.
4.2. Genetic
Deco
ding
The ope
ratio
n
scheme of the system
wil
l
be
obtained
by making th
e pre
s
ervatio
n
of the
corre
s
p
ondin
g
bran
ch valu
e equal 1 (i
n operation
)
. For insta
n
ce, by decodi
ng th
e chromo
som
e
a,
the radi
al operation sch
e
me w
ill be o
b
tained
(with
co
rrespon
di
ng
α
an
d
β
li
sts) (Figure 4b).
Usi
ng thi
s
co
dification,
we
have a
pop
u
l
ation that
co
nsi
s
ts
of a
se
t of ch
romo
somes of type
a.
By decoding each chrom
o
some, a
parti
c
ular configuration will
be
obtained and its perform
ance
can b
e
tested
.
5. Genetic O
p
erators
5.1. Selectio
n
The goal of the sel
e
ction
operator i
s
to assu
re m
o
re
chan
ce
s to replicate for the be
st
chromo
som
e
s of a popul
a
t
ion. The sel
e
ction is p
e
rfo
r
med taki
ng in
to account th
e fitness of the
chromo
som
e
s. The
mo
st
use
d
sele
ctio
n metho
d
s a
r
e M
onte
Ca
rlo a
nd to
urn
a
ment. Fo
r t
h
is
multi-obj
ectiv
e
optimizatio
n probl
em, the author
h
a
s
use
d
the ecol
ogical niche
method [21].
5.2. Cross
o
v
e
r
Cho
o
si
ng th
e numb
e
r a
nd po
sition
of cro
s
sover points for t
he cro
s
sover operator
depe
nd
s on t
he sy
stem to
pology. If these p
o
ints
are
sele
cted i
n
an ina
deq
uat
e mod
e
we
will
obtain “ba
d
” chromo
som
e
s:
(i
)
u
n
-co
n
necte
d syst
e
m
s with
i
s
ol
ated
n
ode
s; or (ii)
conne
cted
system
s with
loops (m
esh
ed). In orde
r
to redu
ce
the
numbe
r of these ca
se
s, we propo
se t
hat
the numbe
r o
f
cut points b
e
equal to CN
−
1. CN re
pre
s
ent
s the
cyclom
atic nu
mber
(the nu
mbe
r
of fundam
ent
al ci
rcuits/loo
ps) corre
s
po
n
d
ing to the
attach
ed g
r
ap
h: CN =
m
−
n +
p (whe
re
m
is
the nu
mbe
r
o
f
bra
n
che
s
, n
is the
num
b
e
r
of no
de
s/vertice
s
an
d p
is the
num
b
e
r
of conn
ect
e
d
comp
one
nts).
5.3. Mutation
One
of the
two
co
ndition
s in
o
r
de
r to
have a
tre
e
or
a forest
is to a
s
sure n
-
p cl
osed
bran
ch
es
(in
operation
)
, a
s
in
(17
)
. A radial
c
onfig
uration cann
ot be conv
ert
e
d
to anothe
r
ra
dial
one by
simpl
y
altering th
e
value of a
chosen g
ene.
Therefore,
we use thi
s
op
erato
r
only i
n
the
ca
se when, b
y
performi
ng
the cr
ossove
r operator, no
n-radial
conf
i
guratio
ns a
r
e
obtained. Th
us,
if in a
ch
rom
o
som
e
the
r
e
are
more o
r
f
e
we
r tha
n
n
-
p ge
ne
s e
q
u
a
l to 1, th
e
mutation o
p
e
r
ator
rand
omly re
p
l
ace
s
the ex
ce
ss/in
suffici
ency of
ge
ne
s eq
ual to 1
(in orde
r to have n-p ge
nes
equal to 1).
5.4. In
v
e
rsion
The
se
co
nd
con
d
ition i
n
o
r
de
r to
have
a tre
e
o
r
a
fo
rest
is to h
a
ve a
co
nne
cte
d
g
r
ap
h
(for a tree) o
r
a graph
with con
n
e
c
ted
comp
one
nt
s
(for a fo
re
st). Thus, thi
s
o
perato
r
ma
ke
s
some
b
r
an
ch
-exch
ang
es (each inve
rsi
o
n bet
wee
n
two ge
ne
s of
a
ch
rom
o
som
e
be
have
s
a
s
a
bran
ch
-ex
c
ha
nge), repai
ri
ng existing
non conn
ect
ed
graph ch
romo
som
e
s (whi
ch are not
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Pareto Optim
a
l Re
config
uration of Powe
r Di
st
ributio
n System
s with
Load … (Sin
a Khajeh AA)
427
con
n
e
c
ted
bu
t whi
c
h
have
n-p
ge
ne
s e
q
ual to
1) an
d
increa
se
s t
h
e
diversity of a
pop
ulation. I
n
our alg
o
rithm,
this is an inte
nsively used
oper
ator after performing crossove
r an
d mutation
6. Best
Com
p
rise Solutio
n
using Fuzzy
Set Theor
y
A multiobjecti
ve optimizati
on algo
rithm
gener
ates t
he non
-domi
nated set of solutio
n
s
kno
w
n a
s
th
e Pareto
-opti
m
al sol
u
tion
s. The de
ci
sio
n
maker
who
is in this
co
ntext the power
system o
p
e
r
ator may ha
ve impre
c
ise
or fuzzy go
als for
ea
ch
obje
c
tive function. To ai
d
the
operator in selectin
g an o
peratin
g point
from the
obtained set of Pareto-optim
al solution
s, fuzzy
set the
o
ry i
s
applie
d to e
a
c
h
obje
c
tive functi
ons to obtain a fuzz
y members
h
ip func
tion
μ
fi
as
follows
[22]:
mi
n
max
mi
n
m
a
x
max
m
i
n
ma
x
1
0
ii
k
k
ii
ii
i
i
ii
ii
ff
ff
ff
f
ff
ff
(20
)
The b
e
st
no
n-do
minate
d
solutio
n
can
be fou
nd
wh
en Equ
a
tion
(21
)
i
s
a
ma
ximum
whe
r
e the no
rmalize
d
su
m of membershi
p
function val
ues for
all obj
ectives i
s
hig
hest:
1
11
i
i
N
k
f
k
i
MN
k
f
ki
(21
)
Whe
r
e Mi
s the numbe
r of non-domi
nate
d
solutio
n
s
a
nd Ni
s the nu
mber of obj
ective functions.
7. Simulation Resul
t
s
The Pa
reto
front allo
ws a
n
inform
ed
de
cisi
on to
be
made
by visualizi
ng
an
e
x
tensive
rang
e of
opti
ons si
nce it
contain
s
the
solution
s
that
are
optimal
from a
n
ove
r
all
stan
dpoi
nt. The
impleme
n
tation was
ada
pted in o
r
de
r to
work
with two obje
c
tives
(
∆
P and
SAIFI). Thu
s
, the u
s
er
can ch
oo
se (in
a
flexible mode
)
∆
P
as the obj
ective
function
an
d
som
e
criteri
a
as con
s
trai
nts
(voltage
devi
a
tion, load
s li
mits through
lines,
et
c.)
or a vecto
r
o
b
j
e
ctive fun
c
tio
n
with
∆
P and
SAIFI as va
ri
able
s
(at the
same
time
) a
nd oth
e
r cr
ite
r
ia as
con
s
tra
i
nts.
Th
e stop
ping criterio
n of
the algorith
m
is an impo
sed maximum
numbe
r of
generation
s
. Result
s we
re o
b
tained for 1
00
run
s
a
nd 50 popul
ation.
T
he stoppi
ng criterio
n
of
th
e
algo
rithm i
s
an imp
o
sed
maximum n
u
m
ber
of generation
s
. Table 1, p
r
esents
singl
e-obj
ective
(active po
wer losses). The
evolution of the
active po
wer l
o
sse
s
alon
g the se
archin
g pro
c
e
ss i
s
prese
n
ted in Fi
gure 5.
Performi
ng
reco
nfiguratio
n for the te
st
syst
em [3] a
s
a Pa
reto
problem, the
o
peratin
g
config
uratio
n
s
are o
b
taine
d
by optimizin
g two criteria:
∆
P and SAIFI (Table 2
)
.
The p
r
opo
se
d algo
rithm h
a
s obtai
ned
a
Pareto fro
n
t with ten soluti
ons
(Figu
r
e 6
)
. In this
ca
se, the first non
-d
omin
ated solution
wa
s o
b
tain
ed from
initi
a
l pop
ulation
.
After the first
gene
ration, the algo
rithm found the se
con
d
non
-d
o
m
inated solut
i
on (the Pare
to front conta
i
ns
two solution
s). The
sea
r
ching p
r
o
c
e
s
s contin
ued a
nd the third
non-domi
nate
d
sol
u
tion was
found in g
ene
ration 2
(at the end of ge
n
e
ration
2, the Pareto front
contai
ns th
re
e solutio
n
s). The
sea
r
ching
proce
s
s contin
ued u
n
til ge
neratio
n 8,
whe
r
e the
ei
ghth an
d final non
-domi
n
ated
solutio
n
wa
s found. In the e
nd, the Paret
o
front co
ntains six no
n-d
o
m
inated solut
i
ons.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
1, No. 3, March 20
16 : 419 – 430
428
Figure 5. The
evolution of the active po
wer losse
s
alo
n
g the se
arch
ing pro
c
e
ss
Table 1. Re
sults for Single
-
Obje
ctive Re
config
uratio
n
Config
ur
ation
Op
en b
r
a
n
ches
(t
ie lines)
Active po
wer l
o
sses (kW)
Base case
8-2
1
, 9
-
15
, 12
-2
2
,
18
-33,
25
-29
202.7
MOReco
7–8, 9–10, 1
4–1
5, 28–29, 32
–33
139.55
Table 2. Re
sults for Pareto Re
c
onfig
uration with two
Objective
s
Co
nfi
g
ura
t
io
n
Op
en
br
an
che
s
(ti
e li
ne
s)
Ac
t
i
ve
po
w
e
r
loss
es
S
A
IFI
Recl
os
er
plac
em
en
t
Initial po
pulatio
n
7-8,
9-
10,
12
-13
,
27-
2
8,
18-
33
149.5
8
1.42
3-4,
17
-18
gene
ratio
n
2
7–8,
9–1
0, 1
2–1
3, 28
–29
,
18–3
3
146.3
3
1.5
3-4,
29
-30
gene
ratio
n
4
7–8,
9–1
0, 1
2–1
3, 28
–29
,
32–3
3
1
4
5
.
73
1.51
3-4,
30
-31
gene
ratio
n
5
7–8,
9–1
0, 1
4–1
5, 27
–28
,
18–3
3
145.0
1
1.63
3-4,
10
-11
,
3
-
23
gene
ratio
n
6
7–8,
9–1
0, 1
4–1
5, 28
–29
,
18–3
3
141.7
6
1.71
3-4,
10
-11
,
13
-1
4
,
20-
21
gene
ratio
n
8
7–8,
9–1
0, 1
4–1
5, 28
–29
,
32–3
3
140.1
1.73
3-4,
10
-11
,
3
-
23
,
12-22
Figure 6. The
Pareto front
0
10
20
30
40
50
60
70
80
90
100
13
0
14
0
15
0
16
0
17
0
18
0
19
0
20
0
21
0
I
t
er
at
i
o
n
A
c
t
i
ve
p
o
w
e
r
l
o
ss
e
s
(
k
W
)
Evaluation Warning : The document was created with Spire.PDF for Python.