Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
Vol
.
5
,
No
. 5, Oct
o
ber
2
0
1
5
,
pp
. 88
7~
89
5
I
S
SN
: 208
8-8
7
0
8
8
87
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
Optimal Design of Switched
Reluctance Motor Using PSO
Based FEM-EMC Modeling
Mouellef Sihe
m, Ben
t
ounsi
Amar, Ben
a
lla H
o
cine
LGEC,
Labor
ato
r
y
of Electrotech
ni
cs Dept., Mentouri University
of Constantine 1
,
Alger
i
a
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Mar 25, 2015
Rev
i
sed
Jun
3
,
2
015
Accepted
Jun 25, 2015
This paper
aim
s
to optim
iz
e th
e
desi
gn of a
pr
ototy
p
e of
a 6/4 Switched
Reluct
anc
e
Mot
o
r (SRM) using the Part
icl
e
Swarm
Optim
ization (PSO)
algorithm. The
geometrical p
a
r
a
meters to op
timize ar
e th
e w
i
dths of th
e
stator and rotor
teeth due to th
eir significan
t eff
e
cts on the prototy
p
e d
e
sign
and th
e p
e
rfor
m
ances
in
term
s
of incr
eas
ed
averag
e
torque
and redu
ced
torque ripp
le. Th
e studied
3kW SRM is
modeled
using a numerical-an
a
ly
tical
approach b
a
sed
on a coupled
Finite
Element Method with
Equivalent
Magnetic Circu
it (FEM-EMC). The
simulatio
ns are perfor
m
ed under
MATLAB environment with u
s
er-friendl
y
sof
t
ware.
The op
timal results
found are discussed, compared agai
nst those obtained b
y
the Genetic
Algorithms (GA) and showed
a significa
nt improvement in
av
erage torqu
e
.
Keyword:
Eq
ui
val
e
nt
M
a
gnet
i
c
C
i
rc
ui
t
Fin
ite Elem
en
t
Meth
od
Gen
e
tic Algo
ri
th
m
s
Particle Swarm Op
ti
m
i
zatio
n
Switche
d Reluctance Mot
o
r
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
:
M
ouel
l
e
f Si
he
m
,
LGEC
,
Lab
o
r
a
t
ory
of El
ect
rot
echni
cs
De
pt
.,
M
e
nt
o
u
ri
Uni
v
ersi
t
y
of
C
o
nst
a
nt
i
n
e 1,
47
R
u
e
E-
A-
K,
K
h
r
o
u
b
,
C
o
nst
a
nt
i
n
e,
Al
ge
ri
a
.
Em
a
il: m
o
u
e
lle
f_
si
h
e
m
@
yah
o
o
.
fr
1.
INTRODUCTION
To m
eet
chal
l
e
ngi
ng
re
qui
re
m
e
nt
s, ne
w de
si
gn a
n
d m
o
re efficient struct
ures
of electrical
m
achines
are investigate
d
by m
a
nufacture
r
s a
n
d rese
arche
r
s.
In
th
i
s
con
t
ex
t,
Perman
en
t Magn
et Syn
c
hr
ono
us Mo
tor
(PM
S
M
)
, B
r
us
hl
ess dc m
o
t
o
r (B
L
D
C
)
,
Li
near
Swi
t
c
he
d Reluctance
Motor
(LSRM
)
and rotary Switched
Relu
ctan
ce Mo
tors (SRMs)
h
a
v
e
b
e
en
exp
l
ored
in
t
h
e
literatu
re as th
ey are an
att
r
activ
e altern
ativ
e to
in
du
ctio
n
an
d
syn
c
hrono
u
s
mach
in
es [1
-4
]. Du
e to
th
eir rob
u
s
t
n
ess,
reliab
ility, h
i
g
h
p
e
rform
a
n
ce an
d
red
u
ced
cost
, t
h
e SR
M
fou
n
d
n
u
m
e
rous ap
pl
i
cat
i
ons
at
hi
gh spee
d dri
v
e or l
o
w s
p
eed
gene
ra
tor (electrical vehicles,
air-conditione
rs, extractors,
centrifuge
s,
fl
ywheel ene
r
gy
storage
,
ship
buildi
n
g, aeronautics, a
nd gearless
w
i
nd
g
e
n
e
r
a
tor)
[5-
7
].
M
e
t
a
heuri
s
t
i
c
m
e
t
hods a
r
e g
e
neral
opt
i
m
i
z
ing al
go
ri
t
h
m
s
appl
i
cabl
e
t
o
a
wi
de v
a
ri
et
y
of
pr
o
b
l
e
m
s
.
Th
ey app
eared
in
th
e 19
80s, with
a commo
n
a
m
b
iti
o
n
:
to
so
lv
e
efficien
tly th
e d
i
fficu
lt op
timizatio
n
pr
o
b
l
e
m
s
, for whi
c
h t
h
ere i
s
no
kn
o
w
n m
o
st
effect
i
v
e cl
assi
cal
m
e
t
hod
[
8
]
,
[9]
.
New t
e
chni
que
s i
n
spi
r
ed by
art
i
f
i
c
i
a
l
i
n
t
e
l
l
i
gence
ha
ve em
erge
d an
d
dev
e
l
ope
d t
o
of
fer as p
o
t
en
tial al
tern
ativ
e techniq
u
e
s to
im
p
r
ov
e the
q
u
a
lity of th
e so
lu
tion
,
n
a
m
e
l
y
Gen
e
tic
Algorith
m
s
(G
A),
Particle Swarm
Op
tim
izat
io
n
(PSO), and
so
on
.
Th
e PSO is a
still relativ
ely
u
nkn
own
and
relativ
ely yo
un
g
tech
n
i
qu
e in
th
e
field
o
f
d
e
sign
[10
]
,
[11
]
. It is an
alo
gou
s to
GA in
th
e sense th
at th
e syste
m
is
in
itialized
with
a ran
d
o
m
p
o
p
u
l
atio
n
o
f
so
lu
t
i
o
n
s
; it
is com
p
ared t
o
all its nei
g
hbors
by m
a
intaining each time the best
re
sult [12].
Unlike the
GA a
n
d
othe
r
metah
e
u
r
istic alg
o
rith
m
s
, PSO h
a
s t
h
e flex
ib
ility to
co
n
t
rol th
e b
a
lan
ce
between
g
l
ob
al an
d
l
o
cal exp
l
o
r
ati
on
of the
searc
h
space
[13], [14]. T
h
e
PSO
has ac
hieve
d
rapid de
velopmen
t
following adva
ntage
s
[15-17]:
sim
p
l
e
concept
,
easy
im
pl
em
ent
a
t
i
on,
ro
b
u
st
ness an
d com
putational efficiency. In [1
8]
, t
h
e t
o
r
q
ue pr
o
d
u
ct
i
o
n
i
s
im
prove
d u
s
i
ng P
S
O al
g
o
r
i
t
h
m
t
o
opt
i
m
i
z
e t
h
e st
at
or an
d r
o
t
o
r an
gl
es of a
8/
6 SR
M
.
In [
1
9]
, t
h
e P
S
O i
s
appl
i
e
d
t
o
t
h
e
r
o
t
o
r
pol
e a
r
c
of
a 4/
2 SR
M
t
o
m
i
nim
i
ze t
h
e t
o
r
q
ue ri
ppl
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
887
–
8
95
88
8
One as
pect of the contri
but
ion of this work lie
s with
th
e ap
p
lication
of th
e Parti
c
le Swarm
Opt
i
m
i
zati
on m
e
t
hod (
P
S
O
)
fo
r o
p
t
i
m
i
zi
ng t
h
e av
era
g
e
t
o
r
que
of a
6/
4 SR
M
t
h
r
o
ug
h va
ri
o
u
s ge
o
m
et
ri
c
param
e
ters. The othe
r as
pect is the com
p
arat
ive study
of the perform
a
nce of
PSO
and
GA algo
rith
m
s
ap
p
lied
t
o
t
h
e m
achi
n
e GA
[
20]
. T
h
e
resul
t
s
s
h
o
w
t
h
at
t
h
e PS
O-
bas
e
d ap
p
r
oac
h
gi
ves t
h
e
best
pe
rf
orm
a
nce i
n
t
e
rm
s
of solution
quality, accuracy and
conver
ge
nce tim
e. The
m
a
in contri
bution
of
this
work is related to t
h
e
num
eri
cal
-anal
y
t
i
cal
appr
oac
h
use
d
t
o
m
odel
t
h
e
st
u
d
i
e
d SR
M
usi
n
g
a use
r
-
fri
e
ndl
y
p
r
o
g
ram
carri
ed
out
un
de
r M
A
TLA
B
.
The
pa
per i
s
o
r
ga
ni
zed
as
fo
l
l
o
ws:
Sect
i
o
n
2
desc
ri
bes
t
h
e F
E
M
-
EM
C
ap
pr
oac
h
m
odel
i
n
g
o
f
t
h
e
stu
d
i
ed
SRM.
In
Sectio
n 3
t
h
e PSO algo
rithm
is p
r
esen
ted with
th
e fo
rm
u
l
atio
n
o
f
th
e
p
r
ob
lem
.
Th
e resu
lts
obt
ai
ne
d a
r
e
di
scusse
d i
n
Sect
i
on
4.
T
h
e
pap
e
r c
oncl
udes
i
n
Sect
i
o
n
5
.
2.
FEM-EMC MODELING OF
THE SRM
Modeling of e
l
ectric
m
achines can be cla
ssified in
to t
h
ree categories:
analytical
m
o
dels, fi
nite
el
em
ent
anal
ysi
s
(FEA
) an
d
equi
val
e
nt
m
a
gnet
i
c
ci
rcui
t
s
(EM
C
), w
h
i
c
h can be c
o
nsi
d
e
r
ed as a
sem
i
-
anal
y
t
i
cal
m
e
tho
d
[
2
1]
. M
o
d
e
l
i
ng usi
ng E
M
C
has been
chosen for furt
her investigati
on
beca
use it see
m
ed a
good technique with great
speed a
n
d acce
ptable accurac
y
. The m
odel
produce
d
will be use
d
later in an
opt
i
m
i
zati
on
pr
ocess t
h
at
ai
m
s
t
o
fi
n
d
t
h
e
bes
t
sy
st
em
param
e
t
e
rs.
The m
achi
n
e t
o
p
o
l
o
gy
st
udi
e
d
i
s
a
d
o
ubl
e
s
a
l
i
e
ncy
three
-
phase
6/4 SRM
with
Ns=
6
stat
or teeth a
n
d
Nr=
4
r
o
t
o
r t
eet
h as rep
r
esent
e
d "Fi
g
ure 1"
. It
s ope
rat
i
ng
pri
n
ci
pl
e, si
m
i
l
a
r
t
o
t
h
e st
eppe
r m
o
t
o
r, has l
o
n
g
be
e
n
k
nown
:
b
y
excitin
g
su
ccessiv
e
ly th
e th
ree stato
r
p
h
a
ses, th
e ro
tor teeth
are
po
sitio
ned
to
m
a
x
i
m
i
ze th
e
inductance
of t
h
e power
ph
as
e, unde
r the
rul
e
of
'ma
x
imum flu
x'
(align
e
d
p
o
s
ition
)
; b
y
tu
rn
ing
o
f
f
th
e p
o
wer,
the m
o
tor will
continue
its m
ove
m
e
nt unt
il it reaches
a
position c
o
rre
sponding to the mini
m
u
m
value of
in
du
ctan
ce
o
r
flu
x
(un
a
lig
n
e
d
p
o
s
ition
)
. On
th
e lin
k
e
d
fl
u
x
(
)-c
ur
rent (
i
) characte
r
istics, the area betwe
e
n t
h
e
p
r
ev
iou
s
two
ex
trem
e p
o
s
ition
s
represen
ts t
h
e electrical
en
erg
y
co
nv
erted
i
n
to
m
ech
an
i
cal en
erg
y
p
e
r
cycle,
W=
W
a
-W
u
, as
sh
own
in "Figur
e
2
"
.
As d
e
scrib
e
d
in
[22
]
, to
d
e
termin
e an
alytica
lly
th
e relatio
ns flu
x
-At fro
m
o
n
l
y sev
e
n
characteristics
equal
-flux line
s
traced by the
fin
ite elem
ent
m
e
thod (FEM
) and c
o
rres
po
ndi
ng to se
ve
n
m
a
gnetic equi
valent
circu
its (EMC), we im
p
l
e
m
en
ted
a prog
ram
i
n
MATLAB
package
softwa
re for the iterative calculation
of t
h
e
satu
r
a
ted aligned
an
d un
aligned
indu
ctan
ces, r
e
sp
ectiv
ely
L
a
and
L
u
, an
d
th
e cor
r
e
spon
din
g
en
er
g
i
es,
W
a
and
W
u
, f
r
om
whi
c
h
one
can
de
d
u
ce t
h
e a
v
era
g
e
t
o
r
que
as
depi
c
t
ed i
n
"
F
i
g
ure
3":
2
u
a
r
av
W
W
qN
T
(1
)
i
W
n
a
*
2
1
.........
2
1
(2
)
p
u
u
I
W
2
1
(3
)
n
I
p
i
(4
)
Th
e i
n
teg
r
atio
n step
i
is t
h
e ratio
of th
e p
e
ak
v
a
lu
e
of cu
rren
t
I
p
on
th
e num
b
e
r
o
f
i
n
tervals
n
.
Fi
gu
re 1.
C
r
oss
-
sect
i
o
n
o
f
t
h
e st
udi
e
d
6/
4 SR
M
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
Opt
i
m
al
Desi
g
n
of
Sw
i
t
c
he
d
Rel
u
ct
a
n
ce M
o
t
o
r U
s
i
n
g P
S
O
Base
d FE
M-E
M
C
Mo
del
i
n
g
(Mou
ellef S
i
h
e
m)
88
9
Fig
u
re
2
.
Ex
t
r
emes
m
a
g
n
e
tic ch
aracteristics flux
v
s
. ex
citatio
n
mm
f
Fi
gu
re
3.
Fl
o
w
chart
of
m
a
i
n
p
r
o
g
ram
of
si
m
u
l
a
t
i
on
u
n
d
e
r
M
A
TLAB
0
100
0
20
00
3
000
4000
5
000
6000
7000
800
0
9000
0
2
4
6
8
10
12
S
t
a
t
o
r
E
x
c
i
t
a
tio
n
(
A
t)
M
agn
et
i
c
Fl
ux
(
m
W
b
)
A
l
i
g
n
ed P
o
si
t
i
on
U
nal
i
gne
d P
o
s
i
t
i
on
C
o
n
ver
t
e
d
E
n
er
g
y
W
=
(W
a
-
W
u
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
887
–
8
95
89
0
3.
PARTICLE SWARM OPTIMIZ
A
TION
ALGORITHM
3.
1.
B
a
sic
Concepts
of P
S
O
The Pa
rt
i
c
l
e
Swarm
Opt
i
m
i
zat
i
on m
e
t
hod (
P
S
O
) i
s
a
rel
a
t
i
v
el
y
rec
e
nt
he
uri
s
t
i
c
pr
o
pose
d
by
Eber
ha
rt
an
d
Ken
n
e
d
y
f
o
r t
h
e fi
rst
t
i
m
e
in t
h
e ea
rl
y
9
0
s
[
23]
,
base
d
on
a st
oc
hast
i
c
p
o
p
u
l
a
t
i
o
n
ca
ndi
dat
e
s
sol
u
t
i
o
ns t
o
de
vel
o
p an
opt
i
m
al
sol
u
t
i
on t
o
t
h
e p
r
o
b
l
e
m
prese
n
t
e
d
.
Thi
s
m
e
t
hod i
s
pa
rt
i
c
ul
arl
y
sui
t
a
bl
e fo
r
n
on-lin
ear syste
m
s; it d
o
e
s
no
t requ
ire t
h
e
calcu
latio
n
o
f
t
h
e
first an
d seco
nd
d
e
riv
a
tive, un
lik
e th
e
grad
ien
t
t
y
pe
m
e
t
hods.
It
s basi
c i
d
ea i
s
i
n
spi
r
e
d
f
r
om
t
h
e act
i
ons o
f
ani
m
al
gro
u
p
s
(swa
rm
s) i
n
t
h
ei
r search
fo
r t
h
e bes
t
sub
s
i
s
t
e
nce a
r
e
a
s. T
h
us, e
ach
i
ndi
vi
dual
i
n
t
h
e
p
o
p
u
l
a
t
i
o
n
has t
h
e m
e
m
o
ry
of
i
t
s
p
r
e
v
i
o
us e
x
peri
e
n
ce
and
t
h
e
i
n
f
o
rm
at
i
on p
r
o
v
i
d
e
d
by
t
h
e gr
ou
p o
n
t
h
e m
o
st
pro
m
i
s
i
ng regi
on
s. Thi
s
co
nt
ri
but
i
o
n t
o
t
h
e
ove
ral
l
expe
rience
, in
addition t
o
personal experie
n
ce is one
of
the features
of PSO
which
e
n
sure it success i
n
global
searche
s
.
A swarm
o
f
particles, wh
ich are
po
ten
tial so
lu
tion
s
t
o
th
e op
ti
m
i
zatio
n
p
r
ob
lem
,
"flies
"
th
e search
space, i
n
the s
earch
for the
global
op
tim
u
m
. The m
ovem
e
nt of a pa
rticle is influe
nce
d
by three c
o
m
pone
nts
[2
4]
:
1.
A c
o
m
pone
nt
of
i
n
ert
i
a
:
t
h
e
p
a
rt
i
c
l
e
t
e
nds
t
o
fol
l
o
w i
t
s
c
u
r
r
e
n
t
t
r
a
v
el
di
rect
i
on;
2.
A c
o
gnitive c
o
m
ponent: the
particle te
nds
to
rely on its own experie
n
ce, a
n
d thus
to m
ove
towa
rds the
b
e
st site in
wh
ich
it h
a
s already p
a
ssed
;
3.
A s
o
ci
al
com
pone
nt
:
t
h
e
pa
rt
i
c
l
e
t
e
nds
t
o
rel
y
on
t
h
e e
x
peri
ence
of
i
t
s
co
n
g
ene
r
s,
an
d t
h
u
s
t
o
m
ove
towa
rds
the
be
st sites already
r
eache
d
c
o
llectively by the
swarm
.
3.
2.
B
a
sic Pri
n
ciple of PSO
In a sea
r
ch s
p
ace of
dim
e
nsion
D
, th
e algo
rit
h
m
starts with a rand
o
m
in
itializatio
n
o
f
the p
a
rticle
swarm
.
Particle i o
f
th
e swarm
i
s
m
odel
e
d by
i
t
s
posi
t
i
on vect
or
iD
i
i
i
x
x
x
x
,...,
,
2
1
an
d
th
e
v
e
lo
city
vector
iD
i
i
i
v
v
v
v
,...,
,
2
1
.
Th
e
q
u
a
lity of its p
o
s
ition
is d
e
term
in
ed
by th
e v
a
l
u
e of th
e obj
ectiv
e
fun
c
tion
at th
at p
o
i
n
t
. Th
is
p
a
rticle rem
e
m
b
ers th
e
b
e
st po
sitio
n in
wh
ich it h
a
s al
read
y p
a
ssed, wh
ich
is n
o
t
ed
iD
i
i
i
pbest
pbest
pbest
best
P
,...,
,
2
1
. The
best
p
o
si
t
i
on achi
e
ve
d
by
i
t
s
nei
g
hb
o
r
i
n
g pa
rt
i
c
l
e
s i
s
not
e
d
D
gbest
gbest
gbest
best
G
,...,
,
2
1
.
Ind
e
ed
, at iteratio
n
t +1
, t
h
e velo
city v
ecto
r
an
d
t
h
e po
sition
v
ect
o
r
are calcu
l
ated
fro
m
t
h
e equ
a
tion
(5
) a
n
d
eq
uatio
n
(6
),
res
p
ectiv
ely
.
t
j
i
t
j
t
j
t
j
i
t
j
i
t
j
t
j
i
t
j
i
x
gbest
r
c
x
pbest
r
c
wv
v
,
,
2
2
,
,
,
1
1
,
1
,
(5
)
D
j
v
x
x
t
j
i
t
j
i
t
j
i
,....,
2
,
1
,
1
,
,
1
,
(6
)
whe
r
e :
t
j
i
t
j
i
v
v
,
1
,
,
: Are t
h
e sp
eed
o
f
the p
a
rticle to
t an
d
t +1
iteratio
n
s
.
Pbest
: Is th
e b
e
st p
o
s
ition
o
f
th
e
p
a
rticle.
Gbest
: Is th
e b
e
st po
sitio
n
o
f
a
n
e
ig
hbo
r at iteratio
n
t.
t
j
i
x
,
: Is th
e po
sitio
n
o
f
the particle at
iteratio
n
t.
w
:
Is ge
neral
l
y
cal
l
e
d a
co
nst
a
nt
fact
or
of
i
n
ert
i
a, i
t
kee
p
s
a
b
a
l
a
nce
bet
w
ee
n e
x
pl
orat
i
o
n
and
expl
oi
t
a
t
i
on.
1
c
and
2
c
: are two c
onsta
nts called acceleration c
o
effi
cients, the
y
keep t
h
e bal
a
nce
betwee
n
i
ndi
vi
dual
an
d
soci
al
beha
vi
or
o
f
t
h
e
part
i
c
l
e
whe
n
t
h
e
y
are e
qual
[
25]
.
1
r
and
2
r
:
are t
w
o
ra
n
dom
ly
gene
rated
num
b
ers with a
range
of
[0,1],
for each iteration a
n
d for each
dim
e
nsion
j.
3.
3.
Pr
obl
em Formul
a
ti
on
Th
e ob
jectiv
e fu
n
c
tion
u
x
f
,
used t
o
fo
rm
ul
at
e t
h
e SR
M
pro
b
l
e
m
represe
n
t
s
a
m
a
xim
i
zi
ng
avera
g
e torque
. In the case
of an
op
ti
m
i
zatio
n
prob
lem
wh
ere th
e obj
ective is to
b
e
m
a
x
i
mized
, th
e functio
n
is co
n
s
id
ered
with
th
e op
po
site sig
n
u
x
f
,
; th
e eq
u
a
lity co
n
s
train
t
s ex
pressed
b
y
th
e fun
c
tion
u
x
g
,
are re
p
r
ese
n
t
e
d
by
t
h
e e
quat
i
o
ns
of
t
h
e
m
a
xim
u
m
and m
i
nim
u
m
i
nduct
a
n
ces
L
a
and
L
u
as well as
W
a
an
d
W
u
en
erg
i
es of th
e two
ex
trem
e p
o
s
itio
n
s
o
f
wh
i
c
h
th
e av
er
ag
e
to
rq
u
e
an
d
in
eq
u
a
lity co
n
s
train
t
s will b
e
d
e
d
u
c
ed
whi
c
h
refl
ect
t
h
e l
o
we
r a
n
d
u
ppe
r
de
nt
al
o
p
e
ni
n
g
s
w
h
i
c
h a
r
e
gi
ve
n
by
eq
uat
i
o
n
s
l
i
m
i
t
s
[26]
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Opt
i
m
al
Desi
g
n
of
Sw
i
t
c
he
d
Rel
u
ct
a
n
ce M
o
t
o
r U
s
i
n
g P
S
O
Base
d FE
M-E
M
C
Mo
del
i
n
g
(Mou
ellef S
i
h
e
m)
89
1
r
s
r
sm
N
qN
45
30
2
(9
)
sm
r
r
sm
rm
60
30
(1
0)
90
2
r
r
r
s
N
a
(1
1)
There
f
ore, t
h
e
pr
op
ose
d
s
o
l
u
t
i
o
n
s
m
u
st
t
a
ke t
h
e c
onst
r
a
i
nt
s of c
o
nst
r
u
c
t
i
on i
n
t
o
acc
ou
nt
. T
h
ese
constraints a
r
e taken i
n
to
account by
penalizing
pr
oportionally the objective
function for
c
o
nstraint
v
i
o
l
atio
ns.
In
t
h
e con
t
ex
t o
f
tak
i
ng
in
t
o
acco
u
n
t
th
e con
s
train
t
s, it is to
deg
r
ad
e th
e
p
e
rfo
r
m
a
n
ce of in
feasib
le
indivi
duals i
n
function of the
i
r prox
imity
to the feasible a
r
ea of the sea
r
ch space.
For
each elem
ent
of t
h
e
search s
p
ace, i
t
s proxim
ity
to the feasible
regi
on ca
n
be
measured through the le
vel of vi
olation
of each
co
nstrain
t
.
Usi
n
g
t
h
is m
easu
r
e o
f
i
n
feasi
b
ility o
f
th
e ind
i
v
i
d
u
a
l
x
fro
m
ea
ch
co
n
s
t
r
ain
t
, t
h
e pen
a
lty fu
nctio
n
in
t
h
e ge
ne
ral
f
o
r
m
can be i
n
t
r
o
duce
d
:
n
i
x
x
x
F
x
f
F
Min
u
i
i
i
penalty
obj
,......,
1
,
1
(1
2)
4.
SIM
U
LATI
O
N
RESULTS
AN
D DIS
C
US
SION
A co
m
p
arativ
e stu
d
y
with
Gen
e
tic Algo
rithm
s
(GA)
has been m
a
de to verify the performance of the
pr
o
pose
d
al
go
r
i
t
h
m
.
The PS
O an
d G
A
pa
ram
e
t
e
rs use
d
f
o
r sim
u
l
a
tion are summ
ariz
ed in Table
1. For the
im
ple
m
entatio
n of PSO, seve
ral param
e
ters
m
u
st
be specified, suc
h
as acceleration fact
ors (
1
c
and
2
c
), the
in
ertia fact
o
r
(
w
), t
h
e size
of t
h
e s
w
arm
s
and the st
op criteri
on.
T
h
e P
S
O
al
go
ri
t
h
m
has
been
a
ppl
i
e
d
t
o
t
h
e
ob
ject
i
v
e
f
unct
i
on acc
o
r
di
ng
t
o
t
h
e
fl
owc
h
art
i
n
Fi
gu
re
4.
Fi
gu
re
4.
The
f
l
owc
h
art
of
ad
apt
i
v
e P
S
O
f
o
r
SR
M
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 5
,
O
c
tob
e
r
20
15
:
887
–
8
95
89
2
Tabl
e 1. Si
m
u
lat
i
on param
e
t
e
rs
GA
PSO
Population size:20
Nu
m
b
er
of Par
t
icles:20
Gener
a
tions:100
I
t
er
ations:100
Cr
ossover
r
a
te:0.
7
1
c
=
2
c
=2
Mutation rate:0.09
w
= -0.1
To c
o
nfi
r
m
t
h
e pe
rf
orm
a
nce o
f
t
h
i
s
m
e
t
hod, a
c
o
m
p
ari
s
o
n
of
i
t
s
res
u
l
t
s
wi
t
h
t
h
e
res
u
l
t
s
of
g
e
net
i
c
algorithm
s
has
been m
a
de. The co
m
p
ar
ison
i
s
show
n in
Table 2
.
The co
nve
r
g
en
ce charact
eri
s
t
i
c
s of t
h
e t
w
o
m
e
t
hods:
PS
O
and G
A
fo
r a vari
abl
e
rel
u
c
t
ance
m
o
t
o
r
are s
h
ow
n i
n
F
i
gu
res
5,
6
,
a
n
d
7.
Acc
o
r
d
i
n
g
t
o
t
h
e
res
u
l
t
s
,
i
t
can
be
n
o
t
i
ced t
h
at
t
h
e
PS
O e
xpl
ore
s
a
s
o
l
u
t
i
o
n
sup
e
ri
o
r
t
o
t
h
e
genet
i
c
al
go
ri
t
h
m
for t
h
e sam
e
n
u
m
b
er o
f
po
pul
at
i
o
n a
n
d
g
e
nerat
i
o
n.
Tab
l
e 2
.
C
o
m
p
ar
ison
r
e
su
lts of
PSO
and
G
A
f
o
r
Fa=
169
1At
par
a
m
e
ter
s
Pr
ototy
p
e
Opti
m
i
zation
GA
PSO
Stator
pole ar
c (
d
eg.
)
30
39.
940
9
37.
832
4
Rotor
pole ar
c (
d
eg.
)
30
49.
933
3
52.
167
6
aver
age tor
que (
N
m
)
11.
139
9
15.
543
15.
617
8
It
can be see
n
fr
om
fi
gure 5
,
fi
rst
l
y
, t
h
at
t
h
e PSO al
g
o
r
i
t
h
m
conve
rge
s
t
o
wa
r
d
t
h
e gl
o
b
al
o
p
t
i
m
u
m
from
the thirty sixth iteration (36),
while the converge
nce
of the
GA al
go
rithm
is reached
at iteration (26)
with a
n
optimal value l
o
we
r
com
p
ared
t
o
t
h
e PSO algo
r
ith
m
.
Th
is pr
oves th
at th
e pow
er
of
co
nv
erg
e
n
c
e t
o
th
e
g
l
ob
al op
tim
u
m
in
th
e PSO m
e
th
o
d
ex
ceed
s t
h
at
o
f
th
e meth
o
d
of g
e
netic
alg
o
r
it
h
m
s
(GA), th
is
will h
a
ve
a di
rect
im
pact
on t
h
e t
i
m
e
requi
red
fo
r co
n
v
er
ge
nce o
f
t
h
e t
w
o m
e
t
hods
. Fu
rt
he
rm
ore, t
h
e ro
b
u
st
nes
s
of t
h
e
PSO al
go
rithm
is
m
o
re rem
a
rkable.
The
dif
f
e
rence i
n
ave
r
age torque
between t
h
e two
o
p
tim
izat
io
n
meth
od
s
(15
.
61
78
Nm
c
o
m
p
ared
with
1
5
.543
Nm
) is
v
i
rtu
a
lly in
sign
ifican
t or n
e
glig
ib
le (a slig
ht d
i
fferen
ce of ab
ou
t
0
.
4
8
%). Th
is
will co
nfirm
ou
r find
ing
s
in
t
e
rm
s o
f
rob
u
stn
e
ss
o
f
th
e PSO co
nv
erg
e
n
c
e.
The res
u
l
t
s
p
r
e
s
ent
e
d i
n
Fi
g
u
r
e
s 6 an
d 7 s
h
o
w
va
ri
at
i
ons
of
s
and
r
arou
nd
th
eir
op
tim
u
m
v
a
lu
es.
The sa
fety constraints are
als
o
c
h
ecke
d
for t
h
ese
t
w
o angle
s
. T
h
ese a
r
e
qualified in t
h
eir
ranges.
Fi
gu
re
5.
O
b
je
ct
i
v
e f
unct
i
o
n
10
20
30
40
50
60
70
80
90
100
15
15.
1
15.
2
15.
3
15.
4
15.
5
15.
6
15.
7
Ge
ner
at
i
o
n
s
F
i
tn
e
ss V
a
l
u
e
PSO
GA
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Opt
i
m
al
Desi
g
n
of
Sw
i
t
c
he
d
Rel
u
ct
a
n
ce M
o
t
o
r U
s
i
n
g P
S
O
Base
d FE
M-E
M
C
Mo
del
i
n
g
(Mou
ellef S
i
h
e
m)
89
3
Fi
gu
re
6.
C
h
a
nge
o
f
s
arou
nd
th
e op
tim
al v
a
lu
e
Fi
gu
re
7.
C
h
a
nge
o
f
r
arou
nd
th
e op
tim
al v
a
lu
e
5.
CO
NCL
USI
O
N
The
pa
per
p
r
o
p
o
ses a
perm
eance net
w
o
r
k
m
odel
i
n
g
a
n
d
p
r
es
ent
s
a
di
rect
c
o
upl
i
n
g
bet
w
ee
n t
h
e
fi
ni
t
e
el
em
ent
m
e
t
hod (
F
EM
)
an
d t
h
e e
qui
val
e
nt
m
a
gnet
i
c
ci
rcu
i
t
m
e
t
hod
(EM
C
) t
o
m
odel
t
h
e swi
t
c
he
d
rel
u
ct
ance
m
achi
n
e. The
m
odel
i
ng t
ool
prese
n
t
e
d i
s
de
si
gne
d t
o
be i
n
t
e
grat
ed i
n
t
o
an o
p
t
i
m
i
zati
on pr
ocess t
h
at
m
odi
fi
es
t
h
e ge
om
et
ry
of t
h
e
en
gi
ne
.
The
opt
i
m
i
z
at
ion m
e
t
hod c
h
ose
n
i
s
t
h
e pa
rt
i
c
l
e
swarm
opt
im
i
zat
i
on (P
SO)
,
t
h
e st
och
a
st
i
c
nat
u
re
,
metah
e
u
r
istics, allo
ws th
e app
licatio
n
to
d
i
fficu
lt an
d
n
o
n
-l
i
n
ea
r
pr
obl
e
m
s. The pri
n
ci
ple of the m
e
thod is
expl
ai
ne
d as w
e
l
l
as t
h
e di
ffer
e
nt
coef
fi
ci
ent
s
of t
h
e al
g
o
ri
t
h
m
and t
h
e i
n
fl
u
e
nce t
h
ey
ha
ve
on t
h
e e
vol
ut
i
on
of
t
h
e al
go
ri
t
h
m
.
The
opt
i
m
i
z
at
ion
pr
oce
d
ure
of t
h
e desi
gn
of SR
M
usi
n
g
PSO i
s
prese
n
t
e
d
wi
t
h
t
h
e
aim
of
m
a
xim
i
zi
ng t
h
e avera
g
e t
o
r
q
ue f
o
r an e
ffi
ci
ent
sol
u
t
i
o
n al
l
by
act
i
ng on t
h
e t
oot
h ge
om
et
ry
whi
c
h
has
a great
influe
nce o
n
m
o
to
r per
f
o
rm
an
ce. The di
ffe
re
nce in the ave
r
age torque
estimated by the two algorithm
s
, PSO
an
d GA, is
n
e
glig
ib
le sugg
estin
g th
e
sim
u
lta
n
e
ou
s conv
ergen
ce to th
e same q
u
a
si-op
timal so
lu
tio
n.
Fro
m
th
e sim
u
latio
n
resu
lts, i
t
can
b
e
fou
nd th
at th
e PSO
can
lead to
op
ti
m
a
l feasib
le so
lu
tion
,
and
th
at is th
e rel
a
tiv
e ease o
f
i
m
p
l
e
m
en
tatio
n
and
ab
ility
to
prov
id
e
reaso
n
ab
ly goo
d
so
lu
tion
s
for
man
y
co
m
b
in
ato
r
ial p
r
ob
lem
s
.
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BIOGRAP
HI
ES OF
AUTH
ORS
S.
M
o
ue
lle
f
r
e
ceived th
e M.S.
degrees from Mentouri University
of Constantine 1. She is
current
l
y
As
s
i
s
t
ant r
e
s
ear
cher
in
the D
e
partm
e
nt
of El
ec
tric
al
E
ngineer
ing a
t
th
e Univers
i
t
y
of
Cons
tantine
1, A
l
geri
a. S
h
e
is
cur
r
entl
y pr
epar
ing
his PhD degree
in Constantin
e U
n
iversity
, His
research
activities focus on the optimization s
y
stem using methaeuristique methods. She has
worked toward
t
h
e dev
e
lopm
ent
of the
s
w
itch
e
d r
e
luc
t
anc
e
m
ach
i
n
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
Opt
i
m
al
Desi
g
n
of
Sw
i
t
c
he
d
Rel
u
ct
a
n
ce M
o
t
o
r U
s
i
n
g P
S
O
Base
d FE
M-E
M
C
Mo
del
i
n
g
(Mou
ellef S
i
h
e
m)
89
5
A.
Be
ntounsi
:
(1953) After receiving its ''
docto
rate-
e
ngine
er
''
in
Paris 6, France, in 1980, he
joined th
e University
of Constantine 1, Alg
e
ria,
in 1984, as an A
ssociate Profess
o
r. Since 1995
,
he is working on his Ph.D. dissertation in co
llabor
ation with
th
e Cegely
Lab
.
of
Ecole C
e
ntr
a
le,
Ly
on
, Fran
ce.
Professor Bentounsi is the d
i
r
ect
or o
f
th
e Labo. d
e
Génie Electr
ique d
e
Cons
tantine (
L
G
E
C), Alger
i
a
.
Hi
s
current res
e
arc
h
inter
e
s
t
s
are C
AD of the ele
c
tr
ica
l
m
achines
and ren
e
wable
energ
y
conv
ersio
n
.
H. Ben
a
l
l
a
: (1
957) He receiv
ed the B.S., M.S., a
nd Do
ctorate Eng
i
neer deg
r
ees in power
electronics, fro
m the National
Poly
technic Institute of Tou
l
ouse,
France, respectiv
ely
in 1981,
1984. In 1995, he receiv
e
d th
e Ph.D. degree
in Electr
i
c
a
l E
ngineer
ing from
Universit
y
of
Jussieu-Paris VI, France. Since 1996, h
e
is w
ith th
e dep
a
rtment of
Electrotechnics,
at
Cons
tantine Uni
v
ers
i
t
y
Algeri
a,
as
a P
r
ofes
s
o
r. His
current res
e
arch fi
eld inc
l
ud
es
Active P
o
wer
Filters,
PW
M Inverters,
E
l
ec
tri
c
Machines,
and
AC Drives.
Evaluation Warning : The document was created with Spire.PDF for Python.