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8792
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1
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3
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h
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ro
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s
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lar
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p
ti
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(
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a
lg
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to
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lv
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o
p
ti
m
a
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c
ti
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e
r
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ro
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lem
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ro
p
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d
a
lg
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m
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th
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se
d
f
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m
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ich
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le
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g
m
e
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n
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g
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stin
g
p
lac
e
,
ti
m
e
to
a
w
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k
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e
tc.
se
c
o
n
d
lev
e
l
is
x
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th
o
se
a
c
ts
w
h
e
n
th
e
re
is
n
e
e
d
o
f
su
b
stit
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te
i
n
f
irst
c
a
s
e
.
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h
e
n
x
γ
b
e
a
s
f
in
a
l
lev
e
l
o
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th
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c
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T
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ry
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d
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te
s
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n
o
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ts
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rk
in
a
b
in
a
ry
sp
a
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e
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t
h
e
p
o
siti
o
n
m
o
d
e
rn
ize
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a
c
c
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rd
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g
ly
.
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d
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m
h
a
s
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e
e
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in
sta
n
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rd
IEE
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1
4
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3
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5
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1
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b
u
s
tes
t
sy
ste
m
s
a
n
d
sim
u
latio
n
re
su
l
ts
sh
o
w
th
e
p
r
o
jec
ted
a
lg
o
rit
h
m
s
re
d
u
c
e
d
th
e
re
a
l
p
o
w
e
r
lo
ss
c
o
n
si
d
e
ra
b
ly
.
K
ey
w
o
r
d
s
:
P
o
lar
w
o
l
f
R
ea
cti
v
e
p
o
w
er
T
r
an
s
m
is
s
io
n
lo
s
s
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
R
ea
cti
v
e
p
o
w
er
p
r
o
b
lem
p
la
y
s
an
i
m
p
o
r
tan
t
r
o
le
in
s
ec
u
r
e
an
d
ec
o
n
o
m
ic
o
p
er
atio
n
s
o
f
p
o
w
er
s
y
s
te
m
.
Nu
m
er
o
u
s
t
y
p
e
s
o
f
m
et
h
o
d
s
[
1
-
6
]
h
av
e
b
ee
n
u
tili
ze
d
to
s
o
lv
e
th
e
o
p
ti
m
al
r
ea
ct
iv
e
p
o
w
er
p
r
o
b
le
m
.
Ho
w
e
v
er
m
an
y
s
cien
tific
d
i
f
f
icu
ltie
s
ar
e
f
o
u
n
d
w
h
ile
s
o
lv
i
n
g
p
r
o
b
lem
d
u
e
to
a
n
a
s
s
o
r
t
m
en
t
o
f
co
n
s
tr
ain
t
s
.
E
v
o
lu
tio
n
ar
y
tec
h
n
iq
u
es
[
7
-
1
6
]
ar
e
ap
p
lied
to
s
o
lv
e
th
e
r
ea
ctiv
e
p
o
w
er
p
r
o
b
lem
.
T
h
is
p
ap
er
p
r
o
p
o
s
es
p
o
lar
w
o
l
f
o
p
ti
m
izatio
n
(
P
W
O)
al
g
o
r
ith
m
to
s
o
lv
e
t
h
e
o
p
tim
a
l
r
ea
ctiv
e
p
o
w
er
p
r
o
b
lem
.
PW
O
en
th
u
s
ed
f
r
o
m
ac
tio
n
s
o
f
p
o
lar
w
o
lv
es.
I
n
th
e
m
o
d
eli
n
g
s
o
cial
h
ier
ar
ch
y
i
s
d
ev
elo
p
ed
to
d
is
co
v
er
th
e
m
o
s
t
e
x
ce
l
len
t
s
o
lu
tio
n
s
ac
q
u
ir
ed
s
o
f
ar
.
T
h
en
th
e
en
cir
clin
g
m
et
h
o
d
is
u
s
ed
to
d
escr
ib
e
cir
cle
-
s
h
ap
ed
v
icin
it
y
ar
o
u
n
d
ev
er
y
ca
n
d
id
ate
s
o
lu
tio
n
s
.
T
h
e
h
u
n
ti
n
g
tec
h
n
i
q
u
e
as
s
is
t
s
ca
n
d
id
ate
s
o
l
u
tio
n
s
to
tr
ac
e
t
h
e
p
r
e
y
[
1
7
]
.
L
ea
d
er
s
o
f
a
p
ac
k
ar
e
d
ef
in
ed
as
Alp
h
a
a
n
d
it
m
a
k
es
all
v
i
tal
d
ec
is
io
n
s
i
n
th
e
g
r
o
u
p
ab
o
u
t
d
a
y
to
d
a
y
ac
ti
v
it
y
.
Alp
h
a
w
i
ll
b
e
s
u
p
p
o
r
ted
b
y
B
eta
i
n
all
a
s
p
ec
ts
an
d
p
ar
ticu
lar
l
y
p
er
f
o
r
m
in
g
ac
tio
n
s
.
Delta
ex
ec
u
tes
its
d
u
t
y
as
s
co
u
t
s
,
g
u
ar
d
,
co
n
cier
g
e.
E
x
p
lo
r
atio
n
an
d
e
x
p
lo
itatio
n
ar
e
b
alan
ce
d
in
it
er
atio
n
s
.
Mo
d
er
n
izin
g
m
ec
h
a
n
is
m
o
f
w
o
l
v
es
is
f
u
n
ctio
n
o
f
t
h
r
ee
v
ec
to
r
s
p
o
s
itio
n
;
X
1
,
X
2
,
X
3
w
h
ic
h
en
d
o
r
s
e
e
v
e
r
y
w
o
l
f
to
r
ea
ch
th
r
ee
m
o
s
t
ex
ce
lle
n
t
s
o
lu
tio
n
s
.
Fo
r
t
h
at
t
h
e
a
g
e
n
ts
w
ill
w
o
r
k
in
a
b
in
ar
y
s
p
ac
e.
P
r
o
p
o
s
ed
P
W
O
alg
o
r
ith
m
h
as
b
ee
n
tes
ted
i
n
s
tan
d
ar
d
I
E
E
E
1
4
,
3
0
,
5
7
,
1
1
8
,
3
0
0
b
u
s
test
s
y
s
te
m
s
an
d
s
i
m
u
latio
n
r
es
u
lt
s
s
h
o
w
s
t
h
at
ac
tiv
e
p
o
w
er
lo
s
s
h
a
s
b
ee
n
r
ed
u
ce
d
.
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
.
2
,
A
u
g
u
s
t 2
0
2
0
:
1
0
7
–
1
1
2
108
2.
P
RO
B
L
E
M
F
O
R
M
UL
AT
I
O
N
Ob
j
ec
tiv
e
o
f
th
e
p
r
o
b
lem
i
s
to
r
ed
u
ce
th
e
tr
u
e
p
o
w
er
lo
s
s
:
F
=
P
L
=
∑
g
k
k
∈
Nbr
(
V
i
2
+
V
j
2
−
2
V
i
V
j
c
os
θ
ij
)
(
1
)
Vo
ltag
e
d
ev
iatio
n
g
i
v
e
n
as
f
o
llo
w
s
:
F
=
P
L
+
ω
v
×
Vol
ta
ge
De
via
tion
(
2
)
Vo
ltag
e
d
ev
iatio
n
g
i
v
e
n
b
y
:
Vol
ta
ge
De
via
tion
=
∑
|
V
i
−
1
|
N
p
q
i
=
1
(
3
)
C
o
n
s
tr
ain
t (
e
q
u
ali
t
y
)
P
G
=
P
D
+
P
L
(
4
)
C
o
n
s
tr
ain
t
s
(
i
n
eq
u
a
lit
y
)
P
g
s
l
ack
m
i
n
≤
P
g
s
l
ack
≤
P
g
s
l
ac
k
m
ax
(
5
)
Q
gi
m
i
n
≤
Q
gi
≤
Q
gi
m
ax
,
i
∈
N
g
(
6
)
V
i
m
i
n
≤
V
i
≤
V
i
m
ax
,
i
∈
N
(
7
)
T
i
m
i
n
≤
T
i
≤
T
i
m
ax
,
i
∈
N
T
(
8
)
Q
c
m
i
n
≤
Q
c
≤
Q
C
m
ax
,
i
∈
N
C
(
9
)
3.
P
O
L
AR
WO
L
F
O
P
T
I
M
I
Z
A
T
I
O
N
P
W
O
alg
o
r
ith
m
e
n
th
u
s
ed
f
r
o
m
ac
tio
n
s
o
f
p
o
lar
w
o
l
v
es.
L
ea
d
er
’
s
w
o
l
v
es
w
h
ic
h
d
e
n
o
ted
as
x
α
ar
e
ac
co
u
n
tab
le
f
o
r
tak
i
n
g
j
u
d
g
m
en
t
o
n
h
u
n
tin
g
,
r
esti
n
g
p
lace
,
ti
m
e
to
a
w
a
k
e
n
etc.
s
ec
o
n
d
lev
el
is
x
β
th
o
s
e
ac
t
s
w
h
e
n
th
er
e
is
n
ee
d
o
f
s
u
b
s
tit
u
te
in
f
ir
s
t
ca
s
e.
T
h
en
x
γ
b
e
as
f
i
n
al
lev
el
o
f
th
e
w
o
l
v
es.
I
n
t
h
e
m
o
d
eli
n
g
s
o
cial
h
ier
ar
ch
y
is
d
ev
elo
p
ed
to
d
is
co
v
er
th
e
m
o
s
t
ex
ce
l
len
t
s
o
lu
t
i
o
n
s
ac
q
u
ir
ed
s
o
f
ar
.
T
h
en
t
h
e
en
cir
cli
n
g
m
et
h
o
d
is
u
s
ed
to
d
escr
ib
e
cir
cle
-
s
h
ap
e
d
v
icin
it
y
ar
o
u
n
d
ev
er
y
ca
n
d
id
ate
s
o
lu
tio
n
s
.
T
h
e
h
u
n
t
in
g
te
ch
n
iq
u
e
a
s
s
i
s
t
s
ca
n
d
id
ate
s
o
lu
tio
n
s
to
tr
ac
e
th
e
p
r
e
y
.
E
x
p
lo
r
atio
n
a
n
d
ex
p
lo
itatio
n
ar
e
b
alan
ce
d
in
ea
c
h
iter
atio
n
.
Mo
d
er
n
izin
g
m
ec
h
a
n
i
s
m
o
f
w
o
l
v
es
is
f
u
n
ctio
n
o
f
t
h
r
ee
v
ec
to
r
s
p
o
s
itio
n
-
1
,
2
,
3
w
h
ic
h
en
d
o
r
s
e
ev
er
y
w
o
l
f
to
r
ea
ch
th
r
ee
m
o
s
t e
x
ce
l
len
t
s
o
l
u
tio
n
s
.
Fo
r
th
at
t
h
e
ag
e
n
ts
w
i
ll
w
o
r
k
i
n
a
b
in
ar
y
s
p
ac
e
.
̅
=
|
̅
̅
(
)
−
̅
(
)
|
(
1
0
)
̅
(
+
1
)
=
̅
(
)
−
⃗
∙
⃗
⃗
⃗
(
1
1
)
⃗
=
2
⃗
.
1
⃗
⃗
⃗
⃗
−
⃗
(
1
2
)
⃗
=
2
.
2
⃗
⃗
⃗
⃗
(
1
3
)
⃗
=
2
−
∗
2
m
ax
(
1
4
)
T
h
e
s
tate
o
f
w
o
l
v
es a
r
e
ad
j
u
s
t
ed
b
y
,
⃗
⃗
⃗
⃗
⃗
⃗
=
|
1
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
−
⃗
|
(
1
5
)
⃗
⃗
⃗
⃗
⃗
=
|
2
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
−
⃗
|
(
1
6
)
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
o
la
r
w
o
lf
o
p
timiz
a
tio
n
a
lg
o
r
i
th
m
fo
r
s
o
lvin
g
o
p
tima
l rea
cti
ve
p
o
w
er p
r
o
b
lem
(
K
a
n
a
g
a
s
a
b
a
i Len
in
)
109
⃗
⃗
⃗
⃗
⃗
=
|
3
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
−
⃗
|
(
1
7
)
w
h
er
e
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
s
y
m
b
o
lize
t
h
e
lo
ca
ti
o
n
s
o
f
,
,
,
1
⃗
⃗
⃗
⃗
⃗
=
⃗
⃗
⃗
⃗
⃗
−
1
⃗
⃗
⃗
⃗
⃗
.
(
⃗
⃗
⃗
⃗
⃗
⃗
)
(
1
8
)
2
⃗
⃗
⃗
⃗
⃗
=
⃗
⃗
⃗
⃗
⃗
−
2
⃗
⃗
⃗
⃗
⃗
.
(
⃗
⃗
⃗
⃗
⃗
)
(
1
9
)
3
⃗
⃗
⃗
⃗
⃗
=
⃗
⃗
⃗
⃗
⃗
−
3
⃗
⃗
⃗
⃗
⃗
.
(
⃗
⃗
⃗
⃗
⃗
)
(
2
0
)
̅
(
+
1
)
=
1
+
2
+
3
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
⃗
3
(
2
1
)
I
n
o
r
d
er
to
ag
en
ts
w
o
r
k
in
a
b
i
n
ar
y
s
p
ac
e,
th
e
p
o
s
it
io
n
m
o
d
er
n
izi
n
g
ca
n
b
e
cu
s
to
m
ized
b
y
,
+
1
=
{
1
(
1
+
2
+
3
3
)
≥
0
ℎ
(
2
2
)
(
)
=
1
1
+
−
10
(
−
05
)
(
2
3
)
1
,
2
,
3
ar
e
u
p
d
ated
b
y
,
1
=
{
1
(
+
)
≥
1
0
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(
2
4
)
2
=
{
1
(
+
)
≥
1
0
ℎ
(
2
5
)
3
=
{
1
(
+
)
≥
1
0
ℎ
(
2
6
)
w
h
er
e
,
,
=
{
1
,
,
≥
0
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(
2
7
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,
,
=
1
1
+
−
10
(
1
,
,
−
0
.
5
)
(
2
8
)
P
o
s
itio
n
s
an
d
v
elo
citie
s
ar
e
m
o
d
er
n
ized
to
i
m
p
r
o
v
e
th
e
p
er
f
o
r
m
a
n
ce
o
f
t
h
e
ex
p
lo
r
atio
n
an
d
ex
p
lo
itatio
n
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n
t
h
e
p
r
o
j
ec
t
ed
alg
o
r
ith
m
.
v
i
k
+
1
=
ω
∗
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i
k
+
c
1
.
r
1
.
(
x
1
−
x
i
k
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+
c
2
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2
.
(
x
2
−
x
i
k
)
+
c
3
.
r
3
.
(
x
3
−
x
i
k
)
(
2
9
)
x
i
k
+
1
=
x
d
t
+
1
+
v
i
k
+
1
(
3
0
)
ω
=
(
ω
m
ax
−
ω
m
i
n
)
.
(
t
m
ax
−
t
)
t
m
ax
+
ω
m
i
n
(
3
1
)
E
x
p
lo
r
atio
n
an
d
ex
p
lo
itatio
n
h
as b
ee
n
co
n
tr
o
lled
b
y
t
h
e
i
n
er
t
ia
w
ei
g
h
t
"
ω"
;
⃗
⃗
⃗
⃗
⃗
⃗
=
|
1
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
−
ω
∗
⃗
|
(
3
2
)
⃗
⃗
⃗
⃗
⃗
=
|
2
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
−
ω
∗
⃗
|
(
3
3
)
⃗
⃗
⃗
⃗
⃗
=
|
3
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
−
ω
∗
⃗
|
(
3
4
)
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
.
2
,
A
u
g
u
s
t 2
0
2
0
:
1
0
7
–
1
1
2
110
I
n
itializatio
n
o
f
p
ar
a
m
eter
s
n
”
w
o
l
v
es p
o
s
itio
n
s
ar
e
in
itia
l
ized
ar
b
itra
r
ily
∈
[
1
,
0
]
,
,
So
lu
tio
n
s
ar
e
attain
e
d
b
y
t
h
e
f
i
tn
es
s
f
u
n
ctio
n
v
al
u
e
Ag
e
n
t’
s
f
itn
e
s
s
v
al
u
es a
r
e
ca
lc
u
lated
by
(
3
5
)
:
⃗
⃗
⃗
⃗
⃗
⃗
=
|
1
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
−
ω
∗
⃗
|
;
⃗
⃗
⃗
⃗
⃗
=
|
2
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
−
ω
∗
⃗
|
;
⃗
⃗
⃗
⃗
⃗
=
|
3
⃗
⃗
⃗
⃗
⃗
,
⃗
⃗
⃗
⃗
⃗
−
ω
∗
⃗
|
(
3
5
)
w
h
ile
(t
<
Ma
x
i
m
u
m
_
iter
atio
n
s
)
.
Fo
r
ea
ch
p
o
p
u
latio
n
m
o
d
er
n
ize
th
e
v
elo
cit
y
b
y
:
v
i
k
+
1
=
ω
∗
v
i
k
+
c
1
.
r
1
.
(
x
1
−
x
i
k
)
+
c
2
.
r
2
.
(
x
2
−
x
i
k
)
+
c
3
.
r
3
.
(
x
3
−
x
i
k
)
M
o
d
er
n
ize
th
e
ag
e
n
t
’
s
p
o
s
itio
n
in
to
a
b
in
ar
y
p
o
s
itio
n
b
y
:
x
i
k
+
1
=
x
d
t
+
1
+
v
i
k
+
1
E
n
d
Mo
d
er
n
ize
A
,
a
,
C
a
n
d
w
T
h
r
o
u
g
h
o
b
j
ec
tiv
e
f
u
n
ct
io
n
as
s
ess
all
p
ar
ticle
s
Mo
d
er
n
ize
th
e
p
o
s
itio
n
s
o
f
,
,
; t=
t+1
E
n
d
w
h
ile
4.
SI
M
UL
AT
I
O
N
R
E
S
UL
T
S
A
t
f
ir
s
t
i
n
s
ta
n
d
ar
d
I
E
E
E
1
4
b
u
s
s
y
s
te
m
[
1
8
]
th
e
v
alid
it
y
o
f
th
e
p
r
o
p
o
s
ed
P
W
O
alg
o
r
ith
m
h
a
s
b
ee
n
test
ed
.
T
ab
le
1
s
h
o
w
s
t
h
e
c
o
n
s
tr
ain
ts
o
f
co
n
tr
o
l
v
ar
iab
le
s
.
T
ab
le
2
s
h
o
w
s
t
h
e
li
m
i
ts
o
f
r
ea
cti
v
e
p
o
w
er
g
en
er
ato
r
s
,
an
d
co
m
p
ar
is
o
n
r
esu
lt
s
ar
e
p
r
esen
ted
in
T
ab
le
3
.
T
ab
le
1
.
C
o
n
s
tr
ain
t
s
o
f
co
n
tr
o
l
v
ar
iab
les
V
a
r
i
a
b
l
e
s
M
i
n
i
m
u
m
(
P
U
)
M
a
x
i
m
u
m
(
P
U
)
G
e
n
e
r
a
t
o
r
v
o
l
t
a
g
e
0
.
9
5
1
.
1
T
r
a
n
sf
o
r
me
r
t
a
p
0
.9
1
.
1
V
A
R
so
u
r
c
e
0
0
.
2
0
T
ab
le
2
.
C
o
n
s
tr
ain
s
o
f
r
ea
ctiv
e
p
o
w
er
g
en
er
ato
r
s
V
a
r
i
a
b
l
e
s
Q
M
i
n
i
mu
m
(
P
U
)
Q
M
a
x
i
mu
m
(
P
U
)
1
0
10
2
-
40
50
3
0
40
6
-
6
24
8
-
6
24
T
ab
le
3
.
Sim
u
latio
n
r
esu
lts
o
f
I
E
E
E
-
1
4
s
y
s
te
m
C
o
n
t
r
o
l
v
a
r
i
a
b
l
e
s
B
a
se
c
a
se
M
P
S
O
[
1
9
]
P
S
O
[
1
9
]
EP [
1
9
]
S
A
R
V
A
[
1
9
]
P
W
O
R
e
d
u
c
t
i
o
n
i
n
P
L
o
ss
0
9
.
2
9
.
1
1
.
5
2
.
5
2
6
.
0
7
T
o
t
a
l
P
L
o
ss (M
W
)
1
3
.
5
5
0
1
2
.
2
9
3
1
2
.
3
1
5
1
3
.
3
4
6
1
3
.
2
1
6
1
0
.
1
0
1
7
N
o
t
e
:
N
R
*
-
N
o
t
r
e
p
o
r
t
e
d
T
h
en
th
e
p
r
o
p
o
s
ed
P
W
O
al
g
o
r
ith
m
h
as
b
ee
n
te
s
ted
in
I
E
E
E
3
0
b
u
s
s
y
s
te
m
.
T
ab
le
4
s
h
o
w
s
th
e
c
o
n
s
tr
ain
ts
o
f
co
n
tr
o
l
v
ar
iab
les.
T
ab
le
5
s
h
o
w
s
th
e
l
i
m
i
ts
o
f
r
ea
cti
v
e
p
o
w
er
g
e
n
er
ato
r
s
,
an
d
co
m
p
ar
is
o
n
r
esu
lt
s
ar
e
p
r
esen
ted
in
T
ab
le
6
.
T
ab
le
4
.
C
o
n
s
tr
ain
t
s
o
f
co
n
tr
o
l
v
ar
iab
les
V
a
r
i
a
b
l
e
s
M
i
n
i
m
u
m
(
P
U
)
M
a
x
i
m
u
m
(
P
U
)
G
e
n
e
r
a
t
o
r
v
o
l
t
a
g
e
0
.
9
5
1
.
1
T
r
a
n
sf
o
r
me
r
t
ap
0
.9
1
.
1
V
A
R
s
o
u
r
c
e
0
0
.
2
0
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
o
la
r
w
o
lf
o
p
timiz
a
tio
n
a
lg
o
r
i
th
m
fo
r
s
o
lvin
g
o
p
tima
l rea
cti
ve
p
o
w
er p
r
o
b
lem
(
K
a
n
a
g
a
s
a
b
a
i Len
in
)
111
T
ab
le
5
.
C
o
n
s
tr
ain
s
o
f
r
ea
ctiv
e
p
o
w
er
g
en
er
ato
r
s
V
a
r
i
a
b
l
e
s
Q
M
i
n
i
mu
m
(
P
U
)
Q
M
a
x
i
mu
m
(
P
U
)
1
0
10
2
-
40
50
5
-
40
40
8
-
10
40
11
-
6
24
13
-
6
24
T
ab
le
6
.
Sim
u
latio
n
r
esu
lts
o
f
I
E
E
E
-
3
0
s
y
s
te
m
C
o
n
t
r
o
l
v
a
r
i
a
b
l
e
s
B
a
se
c
a
se
M
P
S
O
[
1
9
]
P
S
O
[
1
9
]
EP [
1
9
]
S
A
R
G
A
[
1
9
]
P
W
O
R
e
d
u
c
t
i
o
n
i
n
P
L
o
ss (%)
0
8
.
4
7
.
4
6
.
6
8
.
3
2
0
.
0
5
T
o
t
a
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5.
CO
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SI
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r
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m
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h
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g
tech
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s
ca
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d
id
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tio
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s
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tr
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d
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itatio
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d
in
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s
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th
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in
a
b
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y
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s
.
RE
F
E
R
E
NC
E
S
[1
]
K.
Y.
L
e
e
,
Y.
M
.
P
a
rk
a
n
d
J.
L
.
Ortiz,
"
F
u
e
l
-
c
o
st
m
in
im
isa
ti
o
n
f
o
r
b
o
th
re
a
l
-
an
d
re
a
c
ti
v
e
-
p
o
w
e
r
d
i
sp
a
tch
e
s,"
IEE
Pro
c
e
e
d
in
g
s C
-
Ge
n
e
ra
ti
o
n
,
T
r
a
n
s
miss
io
n
a
n
d
Distrib
u
ti
o
n
,
v
o
l.
1
3
1
,
n
o
.
3
,
p
p
.
8
5
-
9
3
,
1
9
8
4
.
[2
]
N.
I.
De
e
b
,
"
A
n
e
ff
icie
n
t
tec
h
n
iq
u
e
f
o
r
re
a
c
ti
v
e
p
o
w
e
r
d
isp
a
tch
u
sin
g
a
re
v
is
e
d
li
n
e
a
r
p
ro
g
ra
m
m
in
g
a
p
p
ro
a
c
h
,
"
El
e
c
tric P
o
we
r S
y
ste
m R
e
se
a
rc
h
,
v
o
l
.
15
,
n
o
.
2
,
p
p
.
1
2
1
-
1
3
4
,
1
9
8
8
.
[3
]
M
.
Bjelo
g
rli
c
,
M
.
S
.
Ca
l
o
v
ic,
P
.
Ristan
o
v
ic
a
n
d
B.
S
.
Ba
b
ic,
"
A
p
p
li
c
a
ti
o
n
o
f
Ne
w
to
n
'
s
o
p
ti
m
a
l
p
o
w
e
r
f
lo
w
in
v
o
l
tag
e
/rea
c
ti
v
e
p
o
w
e
r
c
o
n
tro
l,
"
I
EE
E
T
ra
n
sa
c
ti
o
n
s
o
n
Po
we
r
S
y
st
e
ms
,
v
o
l
.
5
,
n
o
.
4
,
p
p
.
1
4
4
7
-
1
4
5
4
,
1
9
9
0
.
[4
]
S
.
G
ra
n
v
il
le,
"
Op
ti
m
a
l
r
e
a
c
ti
v
e
d
isp
a
tch
th
r
o
u
g
h
i
n
terio
r
p
o
i
n
t
m
e
th
o
d
s,"
IEE
E
T
ra
n
s
.
o
n
P
o
we
r
S
y
ste
ms
,
v
o
l.
9
,
n
o
.
1
,
p
p
.
1
3
6
-
1
4
6
,
1
9
9
4
.
[5
]
N.
G
ru
d
in
i
n
,
"
Re
a
c
ti
v
e
p
o
w
e
r
o
p
ti
m
iza
ti
o
n
u
si
n
g
su
c
c
e
ss
iv
e
q
u
a
d
ra
ti
c
p
ro
g
ra
m
m
in
g
m
e
th
o
d
,
"
IE
E
E
T
ra
n
sa
c
ti
o
n
s
o
n
P
o
we
r S
y
ste
ms
,
v
o
l
.
1
3
,
n
o
.
4
,
p
p
.
1
2
1
9
-
1
2
2
5
,
1
9
9
8
.
[
6
]
W.
Y
a
n
,
J
.
Y
u
,
D
.
C
.
Y
u
a
n
d
K
.
B
h
a
t
t
a
r
a
i
,
"
A
n
e
w
o
p
t
i
m
a
l
r
e
a
c
t
i
v
e
p
o
w
e
r
f
l
o
w
m
o
d
e
l
i
n
r
e
c
t
a
n
g
u
l
a
r
f
o
r
m
a
n
d
i
t
s
s
o
l
u
t
i
o
n
b
y
p
r
e
d
i
c
t
o
r
c
o
r
r
e
c
t
o
r
p
r
i
m
a
l
d
u
a
l
i
n
t
e
r
i
o
r
p
o
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n
t
m
e
t
h
o
d
,
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I
E
E
E
T
r
a
n
s
.
o
n
P
o
w
.
S
y
s
t
.
,
v
o
l
.
2
1
,
n
o
.
1
,
p
p
.
6
1
-
6
7
,
2
0
0
6
.
[7
]
A
.
M
u
k
h
e
rjee
a
n
d
V
.
M
u
k
h
e
rjee
,
"
S
o
lu
ti
o
n
o
f
o
p
t
im
a
l
re
a
c
ti
v
e
p
o
we
r
d
isp
a
tch
b
y
c
h
a
o
ti
c
k
ril
l
h
e
rd
a
lg
o
rit
h
m
,
"
IET
Ge
n
e
ra
ti
o
n
,
T
ra
n
sm
issio
n
&
Distrib
u
ti
o
n
,
v
o
l.
9
,
n
o
.
1
5
,
p
p
.
2
3
5
1
-
2
3
6
2
,
2
0
1
5
.
[8
]
Z
.
Hu
,
X
.
W
a
n
g
a
n
d
G
.
T
a
y
lo
r
,
"
S
to
c
h
a
stic
o
p
ti
m
a
l
re
a
c
ti
v
e
p
o
w
e
r
d
isp
a
tch
:
F
o
rm
u
latio
n
a
n
d
s
o
l
u
ti
o
n
m
e
th
o
d
,
"
El
e
c
tr.
Po
we
r E
n
e
rg
y
S
y
st
.
,
v
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l
.
3
2
,
n
o
.
6
,
p
p
.
6
1
5
-
6
2
1
,
2
0
1
0
.
[9
]
M.
A
.
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