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n
m
a
d
e
m
o
re
c
o
m
p
re
h
e
n
siv
e
l
y
.
P
e
rf
o
rm
a
n
c
e
o
f
th
e
p
ro
p
o
se
d
p
a
rti
ti
o
n
o
f
sp
a
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e
s
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h
m
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v
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ted
in
sta
n
d
a
rd
IEE
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b
u
s
sy
ste
m
s
a
n
d
sim
u
late
d
o
u
tc
o
m
e
g
i
v
e
s
b
e
tt
e
r
re
su
lt
s.
Re
a
l
p
o
w
e
r
lo
ss
h
a
s b
e
e
n
c
o
n
si
d
e
ra
b
ly
re
d
u
c
e
d
.
K
ey
w
o
r
d
s
:
A
l
g
o
r
ith
m
Op
ti
m
al
r
ea
cti
v
e
p
o
w
er
tr
an
s
m
is
s
io
n
lo
s
s
P
ar
titi
o
n
o
f
s
p
ac
es
T
h
is i
s
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
Kan
a
g
asab
ai
L
e
n
i
n
,
Dep
ar
t
m
en
t o
f
E
lectr
ical
an
d
E
lectr
o
n
ics E
n
g
i
n
ee
r
in
g
,
P
r
asad
V.
P
o
tlu
r
i Sid
d
h
ar
th
a
I
n
s
ti
tu
te
o
f
T
ec
h
n
o
lo
g
y
,
Kan
u
r
u
,
Vij
a
y
a
w
ad
a,
An
d
h
r
a
P
r
ad
esh
-
5
2
0
0
0
7
,
I
n
d
ia.
E
m
ail:
g
k
len
i
n
@
g
m
ai
l.c
o
m
1.
I
NT
RO
D
UCT
I
O
N
Op
ti
m
al
r
ea
cti
v
e
p
o
w
er
p
r
o
b
l
e
m
h
a
s
b
ee
n
k
e
y
p
r
o
b
le
m
in
p
o
w
er
s
y
s
te
m
,
s
i
n
ce
it
p
la
y
s
m
aj
o
r
r
o
le
in
s
ec
u
r
e
an
d
ec
o
n
o
m
ic
o
p
er
atio
n
o
f
t
h
e
p
o
w
er
s
y
s
te
m
.
Ma
n
y
co
n
v
e
n
tio
n
al
m
et
h
o
d
s
[1
-
6
]
h
av
e
b
ee
n
ap
p
lied
f
o
r
s
o
l
v
i
n
g
o
p
ti
m
al
r
ea
ctiv
e
p
o
w
e
r
p
r
o
b
lem
.
B
u
t
m
an
y
d
r
a
w
b
ac
k
s
h
a
v
e
b
ee
n
f
o
u
n
d
in
th
e
co
n
v
en
t
io
n
al
m
et
h
o
d
s
an
d
m
ain
l
y
d
if
f
ic
u
lt
y
in
h
a
n
d
l
in
g
t
h
e
in
eq
u
a
lit
y
co
n
s
tr
ai
n
ts
.
L
ast
t
w
o
d
ec
ad
es
m
a
n
y
e
v
o
lu
tio
n
ar
y
al
g
o
r
ith
m
s
[7
-
18
]
c
o
n
tin
u
o
u
s
l
y
ap
p
lied
t
o
s
o
lv
e
th
e
p
r
o
b
lem
.
I
n
th
i
s
p
ap
er
,
p
ar
titi
o
n
o
f
s
p
ac
es
alg
o
r
ith
m
i
s
p
r
o
p
o
s
ed
to
s
o
lv
e
t
h
e
r
ea
cti
v
e
p
o
w
er
p
r
o
b
le
m
.
I
n
th
i
s
ap
p
r
o
ac
h
,
elev
ate
d
q
u
alit
y
a
n
d
ca
p
ab
le
p
o
in
ts
o
f
th
e
ar
ea
i
s
ta
k
en
.
State
s
p
ac
e
ar
e
id
en
tif
ied
an
d
d
iv
i
d
ed
in
to
s
u
b
s
p
ac
es
iter
ativ
el
y
a
n
d
s
ea
r
ch
h
a
s
b
ee
n
m
ad
e
m
o
r
e
co
m
p
r
e
h
en
s
iv
e
l
y
.
P
er
f
o
r
m
an
c
e
o
f
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
is
ev
alu
ated
in
s
ta
n
d
ar
d
I
E
E
E
1
1
8
,
3
0
0
b
u
s
s
y
s
te
m
s
s
i
m
u
lat
io
n
r
es
u
lts
s
h
o
w
s
th
e
b
etter
p
er
f
o
r
m
a
n
ce
o
f
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
i
n
r
ed
u
ctio
n
o
f
r
ea
l p
o
w
er
lo
s
s
.
2.
P
RO
B
L
E
M
F
O
R
M
UL
AT
I
O
N
T
h
e
k
e
y
ob
j
ec
tiv
e
o
f
th
e
r
e
ac
tiv
e
p
o
w
er
p
r
o
b
lem
is
to
m
i
n
i
m
ize
th
e
s
y
s
te
m
r
ea
l
p
o
w
er
lo
s
s
an
d
g
i
v
en
,
P
l
o
s
s
=
∑
g
k
(
V
i
2
+
V
j
2
−
2
V
i
V
j
c
o
s
θ
ij
)
n
k
=
1
k
=
(
i
,
j
)
(
1
)
V
o
ltag
e
d
ev
iatio
n
m
a
g
n
it
u
d
es
(
VD)
is
s
tated
as f
o
llo
w
s
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2252
-
8792
I
n
t J
A
p
p
l P
o
w
er
E
n
g
,
Vo
l.
9
,
No
.
1
,
A
p
r
il 2
0
2
0
:
1
–
5
2
Min
i
m
ize
VD
=
∑
|
V
k
−
1
.
0
|
nl
k
=
1
(
2
)
L
o
ad
f
lo
w
eq
u
alit
y
co
n
s
tr
ain
ts
:
P
Gi
–
P
Di
−
V
i
∑
V
j
nb
j
=
1
[
G
ij
c
os
θ
ij
+
B
ij
s
in
θ
ij
]
=
0
,
i
=
1
,
2
…
.
,
nb
(
3
)
Q
Gi
−
Q
Di
−
V
i
∑
V
j
nb
j
=
1
[
G
ij
s
in
θ
ij
+
B
ij
c
os
θ
ij
]
=
0
,
i
=
1
,
2
…
.
,
nb
(
4
)
I
n
eq
u
alit
y
co
n
s
tr
ain
t
s
ar
e:
V
Gi
m
i
n
≤
V
Gi
≤
V
Gi
m
ax
,
i
∈
ng
(
5
)
V
Li
m
i
n
≤
V
Li
≤
V
Li
m
ax
,
i
∈
nl
(
6)
Q
Ci
m
i
n
≤
Q
Ci
≤
Q
Ci
m
ax
,
i
∈
nc
(
7
)
Q
Gi
m
i
n
≤
Q
Gi
≤
Q
Gi
m
ax
,
i
∈
ng
(
8
)
T
i
m
i
n
≤
T
i
≤
T
i
m
ax
,
i
∈
nt
(
9
)
S
Li
m
i
n
≤
S
Li
m
ax
,
i
∈
nl
(
1
0
)
3.
P
ARTI
T
I
O
N
O
F
SPAC
E
S
AL
G
O
RI
T
H
M
I
n
th
is
al
g
o
r
ith
m
,
f
o
r
f
in
d
i
n
g
th
e
o
p
ti
m
al
s
o
lu
tio
n
b
ased
o
n
t
h
e
co
n
ce
n
tr
atio
n
o
f
elev
ated
q
u
alit
y
a
n
d
ca
p
ab
le
p
o
in
ts
in
s
p
ec
if
ic
ar
ea
is
co
n
s
id
er
ed
.
W
ith
eq
u
al
s
izes
th
e
s
tate
s
p
ac
e
ar
e
al
ien
ated
in
to
s
o
m
e
s
u
b
s
p
ac
es.
I
n
t
h
e
s
tate
s
p
ac
e
u
n
i
f
o
r
m
l
y
,
p
o
in
t
s
ar
e
g
en
er
a
ted
ar
b
itra
r
y
m
o
d
e
an
d
tar
g
et
f
u
n
ctio
n
v
al
u
e
i
s
ca
lcu
lated
.
P
r
o
m
is
i
n
g
p
o
in
ts
ar
e
ch
o
s
en
w
i
th
ea
ch
s
u
b
s
p
a
ce
is
d
eter
m
i
n
ed
.
A
l
w
a
y
s
t
h
e
ch
an
ce
s
o
f
f
i
n
d
i
n
g
th
e
o
p
ti
m
al
s
o
l
u
tio
n
ar
e
h
ig
h
e
r
w
h
e
n
p
r
o
m
is
in
g
p
o
in
ts
ar
e
c
o
n
s
id
er
ed
as
th
e
p
r
o
m
is
i
n
g
s
u
b
s
p
ac
es
.
T
h
e
d
etails
o
f
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
ar
e
as f
o
ll
o
w
s
:
a.
I
n
itiall
y
w
h
o
le
s
tate
s
p
ac
e
is
m
ea
s
u
r
ed
as th
e
ca
p
ab
le
ar
ea
.
b.
Su
b
s
p
ac
es
ar
e
cr
ea
ted
f
r
o
m
th
e
s
tate
s
p
ac
e.
Un
til
f
i
n
all
y
a
g
r
id
co
n
tain
in
g
gi
1
×
gi
2
×
…
gi
n
s
u
b
s
ec
tio
n
s
in
d
th
d
i
m
e
n
s
io
n
(
1
≤
d
≤
n
)
an
d
is
d
iv
id
ed
in
to
as
m
an
y
as
g
d
eq
u
iv
ale
n
t
s
u
b
in
ter
v
al
s
.
T
h
en
,
Size
o
f
ea
ch
s
u
b
in
ter
v
al,
in
d
th
d
i
m
e
n
s
io
n
=
U
d
−
L
d
gi
d
(
1
1
)
c.
P
r
elim
i
n
ar
y
p
o
p
u
latio
n
ar
e
cr
ea
ted
ar
b
itra
r
ily
an
d
th
at
p
o
p
u
latio
n
is
co
n
s
id
er
ed
as
th
e
ex
is
ti
n
g
p
o
p
u
latio
n
.
d.
Fo
r
ev
er
y
p
o
in
t o
f
t
h
e
ex
i
s
ti
n
g
p
o
p
u
latio
n
o
f
f
u
n
c
tio
n
“f
”
is
c
o
m
p
u
ted
.
e.
Q1
-
%
o
f
p
o
in
ts
;
ex
is
ti
n
g
p
o
p
u
latio
n
ar
e
r
eg
ar
d
ed
as
ca
p
a
b
le
p
o
in
ts
,
w
it
h
th
e
lo
w
e
s
t
v
al
u
es
o
f
f
u
n
ctio
n
f,
f.
I
n
ea
ch
o
f
th
e
s
u
b
s
p
ac
es
t
h
e
n
u
m
b
er
s
o
f
ca
p
ab
le
p
o
in
ts
ar
e
d
eter
m
i
n
ed
an
d
in
d
icate
th
e
d
eg
r
ee
o
f
ca
p
ab
le
p
o
in
ts
in
th
e
s
u
b
s
p
ac
e
.
C
ap
ab
le
r
an
k
s
=
N
u
m
b
er
o
f
ca
p
ab
le
p
o
in
ts
in
“
s
”
(
1
2
)
g.
Q2
s
u
b
s
p
ac
es a
r
e
s
ea
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ed
m
o
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e
ac
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r
atel
y
an
d
co
m
p
r
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e
n
s
i
v
el
y
.
1)
E
x
tr
a
s
p
ec
if
ic
s
ea
r
ch
i
n
t
h
e
ca
p
ab
le
s
u
b
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p
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es:
s
m
aller
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u
b
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p
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e
m
ad
e
f
r
o
m
t
h
e
ca
p
ab
le
s
u
b
s
p
ac
es.
2)
T
o
s
ea
r
ch
s
u
b
s
p
ac
es
e
x
p
an
s
i
v
el
y
th
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m
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er
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t
s
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ated
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s
a
lin
ea
r
f
u
n
ctio
n
an
d
f
o
u
n
d
b
y
[
1
9
]
:
nu
m
be
r
os
p
oi
nts
ge
ne
ra
t
e
d
in
s
=
(
cap
a
b
le
r
a
nk
s
∑
cap
a
b
le
r
a
nk
k
×
p
o
p
u
lati
o
n
s
i
ze
k
∈
a
l
l
sub
s
p
a
ce
s
)
(
1
3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
A
p
p
l P
o
w
er
E
n
g
I
SS
N:
2252
-
8792
P
a
r
titi
o
n
o
f sp
a
ce
s
b
a
s
ed
a
lg
o
r
ith
m
fo
r
r
ed
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ctio
n
o
f rea
l p
o
w
er lo
s
s
(
K
a
n
a
g
a
s
a
b
a
i
Len
in
)
3
h.
Fo
r
ea
ch
p
o
in
t in
t
h
e
n
e
w
p
o
p
u
latio
n
t
h
e
v
alu
e
o
f
f
u
n
ct
io
n
f
is
ca
lcu
la
ted
.
i.
B
ased
o
n
th
e
tr
u
n
ca
tio
n
s
e
lecti
o
n
n
e
w
p
o
p
u
latio
n
ar
e
r
ep
lace
th
e
cu
r
r
en
t p
o
p
u
latio
n
.
j.
Q3
p
er
ce
n
t
o
f
th
e
p
o
in
ts
ar
e
ar
b
itra
r
ily
s
elec
ted
in
n
e
w
p
o
p
u
l
atio
n
an
d
cu
s
to
m
ized
b
y
ad
d
in
g
a
Gau
s
s
ian
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o
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e
to
th
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m
it
’
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al
ik
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to
m
u
t
atio
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o
p
er
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n
in
g
e
n
etic
al
g
o
r
ith
m
.
k.
E
v
alu
a
tio
n
o
f
t
h
e
s
to
p
co
n
d
iti
o
n
.
l.
T
h
e
o
u
tp
u
t
o
f
alg
o
r
ith
m
i
s
th
e
m
o
s
t
ex
ce
lle
n
t
s
o
lu
tio
n
g
e
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er
ated
s
o
f
ar
.
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r
ad
e
-
o
f
f
b
et
w
e
en
ex
p
lo
r
atio
n
an
d
ex
p
lo
itatio
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i
s
co
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tr
o
lled
b
y
t
h
e
v
a
lu
e
s
o
f
p
ar
a
m
eter
s
Q
1
,
Q2
,
an
d
Q3
.
P
ar
titi
o
n
o
f
s
p
ac
e
s
alg
o
r
it
h
m
f
o
r
s
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lv
in
g
r
ea
ctiv
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p
o
w
er
p
r
o
b
le
m
:
Exploration Space, f
{Inputs:
With boundaries the n
-
dimensional space
-
State Space
Function target denoted by “f”
Most excellent solution found by the algorithm is the output
Pop Size, Q1, Q2,
Q3, gd for 1 ≤ d ≤ n are initialized
When Pop0 = the initial population; i = 0;
points are generated.
Wh
ol
e
ex
pl
or
at
io
n
sp
ac
e
as
th
e
sk
il
le
d
ar
ea
at
th
e
co
mm
en
ce
me
nt
;
co
mp
et
en
t
Su
bs
pa
ce
s=
St
at
e
Space; 1 ≤ d ≤ n for any dimension d
State Space is Partitio
n into gid parts;
When end condition not satisfied
{
Values of
Good Points; Quality [,
... Pop Size] = the quality of all the points in Popi
Q1 points with the uppermost superiority = [1... Pop Size]
Number of superior points in each subspace Count =
[1
,
... Number of Subspaces]
co
u
nt
[
1
…
nu
m
b
e
r
os
s
u
b
s
p
a
ce
s
]
p
o
p
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=
c
a
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le
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a
n
k
[
1
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n
um
b
e
r
os
s
ub
spa
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e
s
]
Q2 percent of the subspaces with highest capable rank = Capable Sub spaces
Dimension
d
:
1
≤
d
≤
n
for
any
proficient
subs
paces
“s”
;
Promising
subspace
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is
partitioned into gd parts;
Fresh
generated subspaces
=
p
o
p
i
[
1
…
p
o
p
si
z
e
]
Most excellent points of
Popi
=
p
o
p
i
+
1
[
1
…
p
o
p
si
z
e
]
;
Gaussian no
ise is added to Q3 arbitr
arily and
chosen
Popi+1 ; i = i +1;
}
Superior
solution found so far will be the out put
Revert to Solution
}
4.
S
I
M
UL
AT
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R
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S
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P
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2
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1
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2252
-
8792
I
n
t J
A
p
p
l P
o
w
er
E
n
g
,
Vo
l.
9
,
No
.
1
,
A
p
r
il 2
0
2
0
:
1
–
5
4
Fig
u
r
e
1
.
R
ea
l p
o
w
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lo
s
s
co
m
p
ar
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o
n
Fig
u
r
e
2
.
A
cti
v
e
p
o
w
er
lo
s
s
co
m
p
ar
i
s
o
n
5.
CO
NCLU
SI
O
N
P
ar
titi
o
n
o
f
s
p
ac
es
al
g
o
r
ith
m
h
as
b
ee
n
ef
f
icie
n
tl
y
ap
p
lied
f
o
r
s
o
lv
i
n
g
r
ea
cti
v
e
p
o
w
er
p
r
o
b
l
e
m
.
I
n
t
h
i
s
ap
p
r
o
ac
h
,
elev
ated
q
u
alit
y
a
n
d
ca
p
ab
le
p
o
in
ts
o
f
th
e
ar
ea
i
s
tak
e
n
.
State
s
p
ac
e
ar
e
id
en
tifi
ed
an
d
d
iv
i
d
ed
in
to
s
u
b
s
p
ac
es
i
ter
ativ
el
y
a
n
d
s
ea
r
ch
h
a
s
b
ee
n
m
ad
e
m
o
r
e
co
m
p
r
eh
en
s
iv
el
y
.
E
f
f
icien
c
y
o
f
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
is
ev
al
u
ated
in
s
tan
d
ar
d
I
E
E
E
1
1
8
,
3
0
0
b
u
s
s
y
s
te
m
s
an
d
s
i
m
u
lated
o
u
tco
m
e
g
i
v
es
b
etter
r
esu
lts
w
h
e
n
co
m
p
ar
ed
to
o
th
er
r
ep
o
r
ted
s
ta
n
d
ar
d
alg
o
r
ith
m
s
.
RE
F
E
R
E
NC
E
S
[1
]
O.
A
lsa
c
a
n
d
B.
S
to
tt
,
"
Op
ti
m
a
l
Lo
a
d
F
lo
w
w
it
h
S
tea
d
y
-
S
tate
S
e
c
u
r
it
y
,
"
in
IE
EE
T
ra
n
sa
c
ti
o
n
s o
n
Po
we
r
A
p
p
a
ra
t
u
s
a
n
d
S
y
ste
ms
,
v
o
l.
P
A
S
-
9
3
,
n
o
.
3
,
p
p
.
7
4
5
-
7
5
1
,
M
a
y
1
9
7
4
.
[2
]
K.
Y.
L
e
e
,
Y.
M
.
P
a
rk
a
n
d
J.
L
.
Ortiz,
"
A
Un
it
e
d
A
p
p
ro
a
c
h
to
Op
ti
m
a
l
Re
a
l
a
n
d
Re
a
c
ti
v
e
P
o
w
e
r
Disp
a
tch
,
"
in
IEE
E
T
ra
n
sa
c
ti
o
n
s
o
n
Po
we
r A
p
p
a
ra
tu
s
a
n
d
S
y
ste
ms
,
v
o
l.
P
A
S
-
1
0
4
,
n
o
.
5
,
p
p
.
1
1
4
7
-
1
1
5
3
,
M
a
y
1
9
8
5
.
[3
]
A
.
M
o
n
ti
c
e
ll
i,
M
.
V
.
F
.
P
e
re
i
ra
a
n
d
S
.
G
ra
n
v
il
le,
"
S
e
c
u
rit
y
-
Co
n
stra
in
e
d
Op
ti
m
a
l
P
o
w
e
r
F
lo
w
w
it
h
P
o
st
-
Co
n
ti
n
g
e
n
c
y
Co
rre
c
ti
v
e
Re
s
c
h
e
d
u
li
n
g
,
"
in
IEE
E
T
r
a
n
s
a
c
ti
o
n
s
o
n
P
o
we
r
S
y
ste
ms
,
v
o
l.
2
,
n
o
.
1
,
p
p
.
1
7
5
-
1
8
0
,
F
e
b
.
1
9
8
7
.
[4
]
N.
De
e
b
a
n
d
S
.
M
.
S
h
a
h
id
e
h
p
o
u
r,
"
L
in
e
a
r
re
a
c
ti
v
e
p
o
w
e
r
o
p
ti
m
iz
a
ti
o
n
i
n
a
larg
e
p
o
w
e
r
n
e
tw
o
rk
u
sin
g
th
e
d
e
c
o
m
p
o
siti
o
n
a
p
p
r
o
a
c
h
,
"
in
IEE
E
T
ra
n
sa
c
ti
o
n
s o
n
P
o
we
r S
y
ste
ms
,
v
o
l.
5
,
n
o
.
2
,
p
p
.
4
2
8
-
4
3
8
,
M
a
y
1
9
9
0
.
[5
]
E.
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