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[
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2
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.
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m
a
t
r
i
x
t
o
d
e
t
e
r
m
i
n
e
t
h
e
o
p
t
i
m
a
l
DG
p
l
a
c
e
m
e
n
t
.
H
o
w
e
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r
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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t J Po
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&
Dr
i Sy
s
t
I
SS
N:
2088
-
8
6
9
4
A
n
ew a
p
p
r
o
a
c
h
fo
r
o
p
tima
l si
z
in
g
a
n
d
a
llo
ca
tio
n
o
f d
is
tr
ib
u
ted
g
en
era
tio
n
i
n
…
(
Hu
d
efa
h
A
lka
s
h
a
s
h
n
eh
)
1599
D
u
l
ă
u
e
t
a
l
.
[
7
]
u
s
e
d
t
h
e
N
e
w
t
o
n
-
R
a
p
h
s
o
n
p
o
w
e
r
f
l
o
w
m
e
t
h
o
d
t
o
c
a
l
c
u
l
a
t
e
t
h
e
a
c
t
i
v
e
p
o
we
r
l
o
s
s
es
a
t
e
a
c
h
b
u
s
w
h
i
l
e
d
et
e
r
m
i
n
i
n
g
t
h
e
b
e
s
t
DG
a
l
l
o
ca
t
i
o
n
i
n
t
h
e
el
e
c
t
r
i
c
a
l
s
y
s
t
e
m
.
S
h
r
i
v
a
s
t
a
v
a
et
a
l
.
[
8
]
em
p
l
o
y
e
d
l
o
a
d
f
l
o
w
a
n
a
l
y
s
is
t
o
d
e
t
e
r
m
i
n
e
t
h
e
o
p
t
i
m
a
l
s
i
z
e
a
n
d
l
o
c
a
t
i
o
n
o
f
D
G
;
t
h
e
p
r
o
p
o
s
e
d
a
p
p
r
o
a
c
h
w
a
s
a
p
p
l
i
e
d
t
o
a
n
I
E
E
E
1
2
-
b
u
s
r
a
d
i
a
l
p
o
we
r
s
y
s
t
e
m
.
T
h
e
p
r
i
m
a
r
y
o
b
j
e
c
t
i
v
e
o
f
t
h
e
w
o
r
k
w
a
s
t
o
e
n
h
a
n
c
e
t
h
e
v
o
l
t
a
g
e
p
r
o
f
i
l
e
a
n
d
r
e
d
u
c
e
o
v
e
r
a
l
l
a
c
t
i
v
e
p
o
w
e
r
l
o
s
s
es
.
Fu
r
t
h
e
r
i
m
p
r
o
v
e
m
e
n
t
s
i
n
o
v
e
r
a
l
l
s
y
s
t
e
m
s
t
a
b
il
i
t
y
a
n
d
r
e
d
u
c
t
i
o
n
s
i
n
s
y
s
t
e
m
l
o
s
s
e
s
w
e
r
e
d
is
c
u
s
s
e
d
i
n
[
9
]
.
G
h
o
s
h
e
t
a
l
.
[
1
0
]
e
m
p
l
o
y
e
d
t
h
e
c
u
c
k
o
o
s
e
a
r
c
h
al
g
o
r
i
t
h
m
(
C
SA
)
t
o
i
d
e
n
t
i
f
y
t
h
e
o
p
ti
m
al
l
o
c
a
t
i
o
n
s
a
n
d
s
i
z
es
f
o
r
D
G
u
n
i
t
s
i
n
a
r
a
d
i
a
l
d
is
t
r
i
b
u
ti
o
n
s
y
s
te
m
.
T
h
e
I
E
E
E
3
3
-
b
u
s
s
y
s
te
m
h
a
s
b
e
e
n
s
t
u
d
ie
d
t
o
m
i
n
i
m
i
z
e
a
ct
i
v
e
p
o
w
e
r
l
o
s
s
e
s
,
e
n
h
a
n
c
e
q
u
a
l
i
t
y
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n
d
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li
a
b
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li
t
y
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a
n
d
i
m
p
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t
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g
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p
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i
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u
e
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o
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a
l
l
o
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a
ti
o
n
o
f
D
G
.
A
s
i
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il
a
r
a
p
p
r
o
a
c
h
w
a
s
i
m
p
l
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m
e
n
t
e
d
i
n
[
1
1
]
,
w
h
e
r
e
a
m
u
l
t
i
-
l
ea
d
e
r
p
a
r
t
i
c
l
e
s
w
a
r
m
o
p
t
i
m
i
z
a
ti
o
n
t
e
c
h
n
i
q
u
e
w
as
u
s
e
d
t
o
e
v
a
l
u
a
t
e
t
h
e
r
e
d
u
c
ti
o
n
i
n
p
o
w
e
r
l
o
s
s
e
s
a
n
d
m
i
n
i
m
i
z
e
e
m
i
s
s
i
o
n
p
o
l
l
u
t
i
o
n
d
u
e
t
o
t
h
e
a
l
l
o
c
a
t
i
o
n
o
f
D
G
i
n
t
h
e
e
l
e
c
t
r
i
c
al
s
y
s
t
e
m
.
T
h
e
s
a
m
e
m
e
t
h
o
d
o
l
o
g
y
w
a
s
a
l
s
o
t
es
t
e
d
o
n
t
h
e
I
E
E
E
3
3
-
b
u
s
r
a
d
i
a
l
d
is
t
r
i
b
u
ti
o
n
s
y
s
t
e
m
,
as
d
e
t
a
i
le
d
i
n
[
1
2
]
.
Y
a
n
g
et
a
l
.
[
1
3
]
a
p
p
l
i
e
d
m
u
l
ti
-
o
b
j
e
c
t
iv
e
p
a
r
t
i
c
l
e
s
w
a
r
m
o
p
t
i
m
i
z
a
ti
o
n
(
M
O
PS
O
)
t
o
t
h
e
I
E
E
E
3
3
-
b
u
s
a
n
d
I
E
E
E
6
9
-
b
u
s
s
y
s
t
e
m
s
t
o
d
e
t
e
r
m
i
n
e
t
h
e
o
p
t
im
a
l
a
l
l
o
c
at
i
o
n
a
n
d
s
i
z
i
n
g
o
f
DG
u
n
i
t
s
.
Pra
k
ash
an
d
L
a
k
s
h
m
in
ar
a
y
an
a
[
1
4
]
em
p
lo
y
ed
p
ar
ticle
s
war
m
o
p
tim
izatio
n
(
PS
O)
to
d
et
er
m
in
e
th
e
o
p
tim
al
s
izin
g
an
d
lo
ca
tio
n
o
f
DG
u
n
its
,
aim
in
g
to
m
in
im
ize
ac
tiv
e
p
o
wer
lo
s
s
es.
T
h
e
w
o
r
k
in
[
1
5
]
f
o
c
u
s
ed
o
n
u
s
in
g
PS
O
to
en
h
an
ce
v
o
l
tag
e
s
tab
ilit
y
an
d
r
ed
u
ce
p
o
w
er
lo
s
s
es,
wh
ich
was
te
s
ted
i
n
an
I
E
E
E
1
4
-
b
u
s
s
y
s
tem
.
El
-
E
la
et
a
l.
[
1
6
]
e
m
p
l
o
y
ed
b
o
th
PS
O
an
d
p
ar
allel
ca
t
s
war
m
o
p
tim
izatio
n
(
PC
SO)
to
d
eter
m
i
n
e
DGs'
o
p
tim
al
s
izin
g
an
d
p
lace
m
en
t
.
Kar
u
n
ar
ath
n
e
et
a
l
.
[
1
7
]
c
o
n
d
u
cted
a
s
tu
d
y
aim
ed
at
r
e
d
u
cin
g
ac
tiv
e
p
o
wer
lo
s
s
es,
im
p
r
o
v
in
g
v
o
ltag
e
p
r
o
f
iles
,
m
in
im
izin
g
g
en
e
r
atio
n
co
s
ts
,
an
d
d
ec
r
ea
s
in
g
C
O2
e
m
is
s
io
n
s
ass
o
ciate
d
with
DGs.
E
lat
tar
an
d
E
ls
ay
ed
[
1
8
]
u
s
ed
v
o
ltag
e
s
o
u
r
ce
in
v
er
ter
s
(
VSI
)
an
d
PS
O
as
o
p
ti
m
izatio
n
tech
n
iq
u
es
to
ac
h
iev
e
o
p
tim
al
s
izin
g
an
d
allo
ca
tio
n
o
f
DGs.
T
h
e
m
o
d
if
i
ed
m
o
th
f
lam
e
o
p
tim
izatio
n
(
MM
FO)
tech
n
iq
u
e
was
ap
p
lied
in
[
1
9
]
,
wh
ile
th
e
an
t
co
lo
n
y
alg
o
r
ith
m
(
AC
A)
was
u
s
ed
in
[
2
0
]
.
Var
io
u
s
in
d
ices,
in
clu
d
in
g
th
e
v
o
ltag
e
d
ev
iatio
n
in
d
e
x
(
V
DI
)
,
v
o
ltag
e
s
tab
ilit
y
in
d
e
x
(
VSI
)
,
an
d
in
d
ex
v
ec
t
o
r
m
eth
o
d
(
I
VM
)
,
wer
e
co
n
s
id
er
ed
t
o
m
in
im
ize
to
tal
ac
tiv
e
p
o
wer
lo
s
s
es
an
d
en
h
an
ce
th
e
v
o
ltag
e
p
r
o
f
ile,
as
in
tr
o
d
u
ce
d
in
[
2
1
]
.
Similar
o
b
jectiv
es
u
s
in
g
th
e
d
r
ag
o
n
f
ly
alg
o
r
ith
m
wer
e
p
r
e
s
en
ted
in
[
2
2
]
.
Aza
d
et
a
l.
[
2
3
]
a
d
d
r
ess
ed
DGs'
ir
r
eg
u
lar
a
n
d
u
n
s
tab
le
o
u
tp
u
t.
T
h
e
o
p
tim
al
s
izin
g
o
f
DGs
was
ac
h
iev
ed
th
r
o
u
g
h
Dif
f
er
en
t
ial
E
v
o
lu
tio
n
(
DE
)
,
as
in
tr
o
d
u
ce
d
in
[
2
4
]
,
wh
er
e
th
e
VSI
was
u
tili
ze
d
to
id
e
n
tify
th
e
o
p
tim
al
lo
ca
tio
n
s
f
o
r
DGs.
Fin
ally
,
th
e
o
p
tim
al
lo
ca
tio
n
a
n
d
s
izin
g
o
f
DGs
with
m
in
im
u
m
ac
tiv
e
p
o
wer
lo
s
s
es
wer
e
d
eter
m
i
n
ed
u
s
in
g
t
h
e
b
at
alg
o
r
ith
m
(
B
A)
b
ased
o
n
th
e
weig
h
ted
s
u
m
m
eth
o
d
(
W
SM)
in
[
2
5
]
.
T
h
is
p
ap
er
u
s
es
s
en
s
itiv
ity
a
n
aly
s
is
to
id
en
tify
o
p
tim
al
ca
n
d
id
ate
b
u
s
es
f
o
r
th
e
p
lace
m
e
n
t
o
f
DG.
T
h
is
an
aly
s
is
f
o
cu
s
es
o
n
b
u
s
es
wh
er
e
th
e
im
p
ac
t
o
f
r
ea
l
p
o
wer
lo
s
s
es,
attr
ib
u
ted
to
ac
tiv
e
p
o
wer
f
lo
w,
h
as
a
m
ax
im
u
m
v
alu
e.
T
h
e
o
p
tim
al
s
izin
g
o
f
th
e
DG
is
th
en
d
et
er
m
in
ed
u
s
in
g
a
n
ew
ap
p
r
o
ac
h
b
ased
o
n
r
an
d
o
m
n
u
m
b
er
g
en
er
atio
n
,
f
ac
ilit
atin
g
r
ap
id
an
d
ef
f
ec
tiv
e
co
n
v
e
r
g
en
ce
.
T
h
is
m
eth
o
d
ac
co
u
n
ts
f
o
r
b
o
th
th
e
v
o
ltag
e
p
r
o
f
ile
an
d
th
e
r
ed
u
ctio
n
o
f
p
o
wer
lo
s
s
es
in
t
h
e
elec
tr
ical
d
is
tr
ib
u
tio
n
s
y
s
tem
.
T
h
e
c
o
n
tr
i
b
u
tio
n
o
f
th
is
p
a
p
er
is
as
f
o
llo
ws:
i
)
A
J
ac
o
b
ia
n
m
atr
ix
-
b
ased
f
o
r
m
u
latio
n
to
an
aly
ze
DG
s
ize
im
p
ac
t
o
n
p
o
wer
f
lo
w
in
tr
an
s
m
is
s
io
n
lin
es
;
ii
)
A
s
y
s
te
m
atic
f
r
am
ewo
r
k
is
d
ev
elo
p
e
d
to
f
in
d
b
o
th
o
p
tim
al
lo
ca
tio
n
an
d
o
p
tim
al
s
ize
o
f
DG
in
th
e
p
o
wer
g
r
id
; iii
)
A
d
escen
d
in
g
o
r
d
er
r
an
k
in
g
o
f
th
e
b
est f
it o
f
o
p
tim
al
s
ize
o
f
DG
an
d
lo
ca
tio
n
o
f
th
e
o
p
tim
al
b
u
s
f
r
o
m
th
e
p
o
wer
lo
s
s
es
o
n
tr
an
s
m
is
s
io
n
lin
es
p
o
in
t
o
f
v
iew
is
p
r
esen
ted
,
i.e
.
f
r
o
m
h
ig
h
e
r
r
e
d
u
ctio
n
ac
h
iev
ed
o
f
p
o
wer
lo
s
s
es
to
th
e
lo
west
r
ed
u
ctio
n
.
T
o
o
u
r
k
n
o
wled
g
e,
n
o
o
th
er
p
a
p
er
i
n
th
e
liter
atu
r
e
h
as
p
r
o
v
id
e
d
th
is
r
an
k
in
g
s
y
s
te
m
.
T
h
is
r
an
k
i
n
g
s
y
s
tem
o
f
b
u
s
es
is
ess
en
tial
co
n
s
id
er
in
g
th
e
p
o
s
s
ib
ilit
y
o
f
u
n
av
ailab
ilit
y
o
f
th
e
o
p
tim
al
b
u
s
in
r
ea
l
-
wo
r
ld
s
ce
n
a
r
io
s
; o
th
er
o
p
tio
n
s
m
a
y
n
ee
d
to
b
e
f
u
r
t
h
er
ex
p
l
o
r
ed
.
T
h
e
r
est o
f
th
is
p
ap
er
is
s
tr
u
ct
u
r
ed
as f
o
llo
ws:
Sectio
n
2
p
r
esen
ts
th
e
p
r
o
b
lem
f
o
r
m
u
latio
n
,
s
u
p
p
o
r
ted
b
y
m
ath
e
m
atica
l
eq
u
atio
n
s
.
S
ec
tio
n
3
p
r
esen
ts
th
e
s
o
l
u
tio
n
f
o
r
o
p
tim
al
l
o
ca
tio
n
a
n
d
s
izin
g
,
alo
n
g
with
th
e
ass
o
ciate
d
alg
o
r
ith
m
.
I
n
s
ec
tio
n
4
,
n
u
m
er
ical
ex
am
p
les
ar
e
p
r
o
v
id
e
d
to
d
em
o
n
s
tr
ate
th
e
ef
f
ec
tiv
en
ess
o
f
th
e
p
r
o
p
o
s
ed
ap
p
r
o
ac
h
.
Sectio
n
5
in
clu
d
es
an
an
aly
s
is
an
d
d
is
cu
s
s
io
n
o
f
t
h
e
r
esu
lts
.
E
v
en
tu
all
y
,
th
e
co
n
clu
s
io
n
is
p
r
esen
ted
in
s
ec
tio
n
6
.
2.
P
RO
B
L
E
M
F
O
R
M
U
L
AT
I
O
N
I
n
teg
r
atin
g
DG
in
to
an
e
x
is
tin
g
p
o
we
r
s
y
s
tem
in
f
lu
en
ce
s
p
o
wer
f
lo
w
a
n
d
im
p
ac
ts
f
ac
t
o
r
s
s
u
ch
as
o
v
er
all
p
o
wer
lo
s
s
es,
v
o
ltag
e
p
r
o
f
iles
,
an
d
s
y
s
tem
s
tab
ilit
y
[
2
6
]
.
Min
im
izin
g
ac
tiv
e
p
o
wer
lo
s
s
es
p
lay
s
a
cr
u
cial
r
o
le
in
en
h
a
n
cin
g
th
e
p
o
wer
s
y
s
tem
'
s
ef
f
icien
cy
,
s
tab
ilit
y
,
r
eliab
ilit
y
,
an
d
ec
o
n
o
m
ic
v
iab
ilit
y
.
As
a
r
esu
lt,
lo
wer
ac
tiv
e
p
o
wer
lo
s
s
es
d
ir
ec
tly
co
r
r
elate
with
r
ed
u
ce
d
p
er
-
k
ilo
watt
-
hour
p
r
o
d
u
ctio
n
c
o
s
ts
.
T
h
is
s
tu
d
y
aim
s
to
m
in
im
ize
to
tal
ac
tiv
e
p
o
wer
lo
s
s
es
wh
ile
en
s
u
r
in
g
th
at
v
o
ltag
e
le
v
els
at
s
y
s
tem
b
u
s
es
r
em
ain
with
in
ac
ce
p
tab
le
lim
its
.
Ma
th
em
atica
lly
,
th
is
o
b
jectiv
e
ca
n
b
e
f
o
r
m
u
lated
as (
1
)
.
.
{
(
)
}
(
1
)
W
h
er
e
is
th
e
to
tal
ac
tiv
e
p
o
w
er
lo
s
s
es [
MW].
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
6
9
4
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
,
Vo
l.
16
,
No
.
3
,
Sep
tem
b
er
20
25
:
1598
-
1
6
0
7
1600
Su
b
ject
to
eq
u
ality
c
o
n
s
tr
ain
ts
as d
ef
in
ed
in
(
2
)
an
d
(
3
)
.
(
,
)
=
0
(
2
)
(
,
)
=
0
(
3
)
W
h
er
e
is
th
e
ac
tiv
e
p
o
wer
v
alu
e
at
b
u
s
[
MW],
is
th
e
r
ea
ctiv
e
p
o
wer
v
alu
e
at
b
u
s
[
MV
Ar
]
an
d
th
e
in
eq
u
ality
co
n
s
tr
ain
ts
as in
(
4
)
an
d
(
5
)
.
|
|
<
|
|
<
|
|
(
4
)
≤
(
)
(
5
)
W
h
er
e
|
|
is
th
e
v
o
ltag
e
m
ag
n
itu
d
e
at
b
u
s
[
V]
,
|
|
is
th
e
m
in
i
m
u
m
lim
it
o
f
th
e
v
o
ltag
e
m
a
g
n
itu
d
e
at
b
u
s
[
V]
,
|
|
is
th
e
m
ax
im
u
m
li
m
it
o
f
th
e
v
o
ltag
e
m
ag
n
itu
d
e
at
b
u
s
[
V]
,
(
)
is
allo
wed
ac
tiv
e
p
o
wer
v
alu
e
o
f
DG
ca
n
b
e
in
s
t
alled
at
a
s
p
ec
if
ic
b
u
s
[
MW].
T
h
e
to
tal
ac
ti
v
e
p
o
wer
l
o
s
s
es
an
d
th
e
p
e
r
m
is
s
ib
le
DG
p
o
we
r
at
a
g
iv
e
n
b
u
s
ca
n
b
e
ca
lcu
l
ated
u
s
in
g
(
6
)
an
d
(
7
)
,
r
esp
ec
tiv
ely
.
(
)
=
∑
=
1
(
6
)
(
)
≤
.
.
(
7
)
W
h
er
e
is
th
e
ac
tiv
e
p
o
wer
lo
s
s
es
at
th
e
b
u
s
[
MW],
is
a
f
ac
to
r
th
at
v
a
r
ies
b
etwe
en
0
an
d
1
,
d
eter
m
in
in
g
th
e
allo
wab
le
ca
p
ac
ity
o
f
d
is
tr
ib
u
te
d
g
en
er
ati
o
n
r
elativ
e
to
th
e
m
a
x
im
u
m
d
esig
n
ed
ca
p
ac
ity
at
b
u
s
,
s
u
b
ject
to
th
e
lo
ca
l
r
e
g
u
latio
n
s
in
ea
ch
co
u
n
tr
y
.
.
is
th
e
m
ax
im
u
m
p
o
wer
d
esig
n
ed
ca
p
ac
ity
o
f
th
e
p
o
wer
s
y
s
tem
[
MW].
Fo
r
an
y
p
o
wer
s
y
s
tem
,
th
e
to
tal
ac
tiv
e
p
o
wer
lo
s
s
ca
n
b
e
d
eter
m
in
ed
as
th
e
alg
eb
r
aic
s
u
m
o
f
th
e
lo
s
s
es a
t e
ac
h
b
u
s
,
f
o
r
m
u
lated
in
(
8
)
.
=
∑
∆
2
=
1
.
(
8
)
T
h
e
v
o
ltag
e
d
r
o
p
s
b
etwe
en
an
y
two
b
u
s
es,
d
en
o
ted
as
an
d
,
f
o
r
an
y
in
d
u
ctiv
e
lo
ad
is
ill
u
s
tr
ated
in
Fig
u
r
e
1
[
2
7
]
-
[
2
9
]
an
d
ca
n
b
e
m
ath
em
atica
lly
f
o
r
m
u
lated
u
s
i
n
g
th
e
c
o
s
in
e
th
eo
r
em
as sh
o
wn
in
(
9
)
.
∆
2
=
2
+
2
−
(
2
.
.
.
(
−
)
)
(
9
)
Su
b
s
titu
tin
g
(
9
)
in
t
o
(
8
)
y
ield
s
th
e
ac
tiv
e
p
o
wer
lo
s
s
es a
t
th
e
b
u
s
as in
(
1
0
)
.
=
∑
=
1
.
(
2
+
2
−
(
2
.
.
.
c
os
(
−
)
)
)
(
1
0
)
W
h
er
e
∆
is
th
e
v
o
ltag
e
d
r
o
p
v
e
cto
r
b
etwe
en
b
u
s
es
an
d
,
∆
=
∆
+
∆
,
[
V]
,
∆
is
th
e
v
o
ltag
e
v
ec
to
r
ac
r
o
s
s
th
e
r
esis
tan
ce
b
etwe
en
b
u
s
es
an
d
[
V]
,
∆
is
th
e
v
o
lt
ag
e
v
ec
to
r
ac
r
o
s
s
th
e
in
d
u
cta
n
ce
b
etwe
en
b
u
s
es
an
d
[
V]
,
is
th
e
co
n
d
u
ctan
ce
b
etwe
en
b
u
s
es
an
d
,
[
1
/
]
,
,
ar
e
v
o
ltag
e
p
h
ase
an
g
l
e
at
b
u
s
es
an
d
,
r
esp
ec
tiv
ely
,
[
V]
,
is
th
e
n
u
m
b
er
o
f
b
u
s
es in
th
e
p
o
wer
s
y
s
tem
.
Fig
u
r
e
1
.
Ph
aso
r
d
iag
r
am
o
f
t
h
e
v
o
ltag
e
d
r
o
p
b
etwe
en
two
b
u
s
es
3.
P
RO
B
L
E
M
SO
L
U
T
I
O
N
T
h
e
s
o
lu
tio
n
to
th
e
p
r
o
b
le
m
co
m
p
r
is
es
two
m
ain
c
o
m
p
o
n
en
ts
.
T
h
e
f
ir
s
t
co
m
p
o
n
e
n
t
in
v
o
lv
es
id
en
tify
in
g
th
e
o
p
tim
al
ca
n
d
id
ate
b
u
s
f
o
r
DG
allo
ca
tio
n
.
T
h
e
s
ec
o
n
d
co
m
p
o
n
e
n
t
f
o
cu
s
es
o
n
d
eter
m
in
in
g
th
e
o
p
tim
al
s
izin
g
o
f
DG
at
th
at
b
u
s
wh
ile
co
n
s
id
er
i
n
g
l
o
ca
l
r
eg
u
latio
n
s
th
at
d
ictate
t
h
e
allo
w
ab
le
p
er
ce
n
tag
e
o
f
DG
p
o
wer
th
at
ca
n
b
e
in
s
ta
lled
in
an
y
ar
ea
,
s
u
b
s
tatio
n
,
o
r
r
eg
io
n
r
elativ
e
to
th
eir
m
ax
im
u
m
d
esig
n
e
d
ca
p
ac
ity
.
T
h
is
f
ac
to
r
is
r
ef
er
r
e
d
to
as
.
T
h
e
m
ath
e
m
atica
l
ca
l
cu
latio
n
s
ass
u
m
e
th
at
th
e
ad
m
ittan
ce
s
,
v
o
ltag
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
I
SS
N:
2088
-
8
6
9
4
A
n
ew a
p
p
r
o
a
c
h
fo
r
o
p
tima
l si
z
in
g
a
n
d
a
llo
ca
tio
n
o
f d
is
tr
ib
u
ted
g
en
era
tio
n
i
n
…
(
Hu
d
efa
h
A
lka
s
h
a
s
h
n
eh
)
1601
m
ag
n
itu
d
es,
a
n
d
l
o
ad
s
o
f
th
e
t
h
r
ee
-
p
h
ase
lin
es
ar
e
id
e
n
tical
an
d
th
at
t
h
e
eq
u
ality
co
n
s
tr
ain
ts
[
3
0
]
s
p
ec
if
ied
i
n
(
2
)
an
d
(
3
)
a
r
e
s
atis
f
ied
.
C
o
n
s
eq
u
en
tly
,
th
e
ca
lcu
latio
n
s
a
r
e
co
n
d
u
cte
d
f
o
r
a
s
in
g
le
p
h
ase,
as
th
e
ca
lcu
latio
n
s
f
o
r
th
e
r
em
ain
in
g
p
h
ases
will b
e
th
e
s
am
e.
3
.
1
.
Sens
it
iv
it
y
-
b
a
s
ed
f
o
r
o
ptim
a
l D
G
lo
c
a
t
io
n
Sen
s
itiv
ity
an
aly
s
is
[
3
1
]
is
u
s
ed
to
id
en
tif
y
th
e
o
p
tim
al
lo
c
atio
n
f
o
r
DG.
T
h
is
s
y
s
tem
atic
p
r
o
ce
d
u
r
e
id
en
tifie
s
th
e
n
o
d
es
with
th
e
m
o
s
t
s
ig
n
if
ican
t
im
p
ac
t
o
n
r
e
al
p
o
wer
lo
s
s
es
r
esu
ltin
g
f
r
o
m
ac
tiv
e
p
o
wer
f
l
o
w.
T
h
is
h
as b
ee
n
m
ath
em
atica
lly
p
r
o
v
e
n
in
r
e
f
er
en
ce
s
[
6
]
,
[
7
]
,
[
1
8
]
an
d
is
p
r
esen
ted
i
n
(
1
1
)
.
[
]
=
[
|
|
|
|
]
−
1
·
[
|
|
]
=
[
]
−
1
·
[
|
|
]
=
[
1
2
3
4
]
−
1
·
[
|
|
]
(
1
1
)
W
h
er
e
,
−
1
ar
e
J
ac
o
b
ian
m
atr
ix
an
d
th
e
J
ac
o
b
ian
m
atr
ix
in
v
er
s
e
o
f
th
e
New
to
n
–
R
ap
h
s
o
n
p
o
wer
f
lo
w
ar
e
m
atr
ices
with
(
(
−
1
)
(
−
1
)
)
elem
en
ts
,
[
]
,
[
]
ar
e
p
ar
tial
d
er
iv
ativ
es
o
f
th
e
r
ea
l
p
o
wer
lo
s
s
es;
f
o
r
th
e
ac
tiv
e
an
d
r
ea
ctiv
e
p
o
wer
f
lo
w,
th
ey
ar
e
v
ec
to
r
s
with
(
(
−
1
)
1
)
elem
en
ts
,
[
]
,
[
|
|
]
ar
e
p
ar
tial
d
er
iv
ativ
es
o
f
t
h
e
r
ea
l
p
o
wer
f
o
r
b
u
s
v
o
ltag
e
an
g
le
an
d
m
ag
n
itu
d
e,
th
e
y
ar
e
m
at
r
ices
with
(
(
−
1
)
(
−
1
)
)
elem
en
ts
,
[
]
,
[
|
|
]
ar
e
p
ar
tial
d
e
r
iv
ati
v
es
o
f
th
e
r
e
ac
tiv
e
p
o
wer
to
b
u
s
v
o
ltag
e
a
n
g
le
an
d
m
ag
n
itu
d
e;
th
ey
ar
e
m
atr
ices w
ith
(
(
−
1
)
(
−
1
)
)
elem
en
ts
.
T
h
e
p
ar
tial
d
er
iv
ativ
e
|
|
in
(
1
1
)
ca
n
b
e
o
b
tain
e
d
b
y
d
i
f
f
er
en
tiat
in
g
(
1
0
)
with
r
esp
ec
t
to
th
e
v
o
ltag
e
m
ag
n
itu
d
e
a
n
d
v
o
ltag
e
an
g
le,
r
esp
ec
tiv
ely
.
T
h
e
r
esu
ltin
g
p
r
o
d
u
cts ar
e
ex
p
r
ess
ed
in
(
1
2
)
a
n
d
(
1
3
)
.
|
|
=
∑
=
1
.
[
2
(
|
|
−
(
|
|
.
(
−
)
)
]
(
1
2
)
=
∑
=
1
.
[
2
.
|
|
.
|
|
.
s
in
(
−
)
]
(
1
3
)
Af
ter
s
o
m
e
m
at
h
em
atica
l
ar
r
a
n
g
em
en
t
o
f
(
1
1
)
,
a
v
ec
to
r
with
(
(
−
1
)
1
)
elem
en
ts
ar
e
o
b
tain
ed
,
r
ep
r
es
en
tin
g
th
e
r
ea
l p
o
wer
lo
s
s
es d
u
e
to
ac
tiv
e
p
o
wer
f
l
o
w,
as in
(
1
4
)
.
[
]
=
[
]
.
[
]
+
[
|
|
]
.
[
|
|
]
(
1
4
)
T
h
e
elem
en
ts
with
th
e
m
ax
im
u
m
v
alu
e
in
(
1
4
)
in
d
icate
th
e
b
u
s
n
u
m
b
er
with
t
h
e
h
ig
h
est
im
p
ac
t
o
n
ac
tiv
e
p
o
wer
lo
s
s
es
f
r
o
m
ac
t
iv
e
p
o
wer
f
lo
w.
I
n
o
t
h
er
w
o
r
d
s
,
in
s
tallin
g
DG
o
n
th
is
b
u
s
will
s
ig
n
if
ican
tly
r
ed
u
ce
ac
tiv
e
p
o
wer
lo
s
s
es.
C
o
n
s
eq
u
en
tly
,
th
is
v
ec
to
r
ca
n
b
e
r
ea
r
r
a
n
g
ed
in
d
escen
d
in
g
o
r
d
er
to
id
en
tify
th
e
b
u
s
es
with
th
e
h
ig
h
est
s
u
b
s
tan
tial
ef
f
ec
t
o
n
ac
tiv
e
p
o
wer
l
o
s
s
r
ed
u
ctio
n
.
T
h
e
p
r
o
ce
d
u
r
e
f
o
r
d
eter
m
i
n
in
g
th
e
o
p
tim
al
b
u
s
es
f
o
r
allo
ca
tio
n
[
3
0
]
,
[
3
2
]
ca
n
b
e
s
u
m
m
a
r
ized
as
f
o
llo
ws:
i)
I
n
p
u
t
th
e
s
y
s
tem
d
ata
an
d
ex
ec
u
te
th
e
p
o
wer
f
lo
w
p
r
o
g
r
am
;
ii)
C
alcu
late
th
e
J
ac
o
b
ian
m
atr
ices
[
1
,
2
,
3
,
4
]
;
iii)
Dete
r
m
in
e
[
|
|
]
an
d
[
]
u
s
in
g
(
1
0
)
an
d
(
1
1
)
;
iv
)
C
o
m
p
u
te
[
]
b
ased
o
n
(
1
4
)
,
ex
clu
d
in
g
ca
n
d
id
ate
b
u
s
es
wh
er
e
DG
in
s
tallatio
n
is
im
p
r
ac
tical
f
o
r
v
ar
io
u
s
r
ea
s
o
n
s
(
e.
g
.
,
p
r
o
x
im
ity
t
o
b
u
ild
in
g
s
an
d
r
o
a
d
s
)
;
an
d
v
)
Select
th
e
b
u
s
with
th
e
h
ig
h
est
v
alu
e
o
f
[
]
as
th
e
f
ir
s
t
ca
n
d
id
ate
f
o
r
DG
allo
ca
tio
n
[
3
3
]
,
[3
4
]
,
f
o
llo
wed
b
y
th
e
b
u
s
with
th
e
s
ec
o
n
d
-
h
ig
h
est
v
alu
e,
a
n
d
s
o
o
n
.
3
.
2
.
O
pti
m
a
l sizin
g
o
f
DG
T
h
e
am
o
u
n
t
o
f
p
o
wer
f
r
o
m
th
e
DG
in
jecte
d
at
a
n
y
ca
n
d
id
at
e
b
u
s
(
)
is
th
e
s
tan
d
a
r
d
ized
s
i
ze
o
f
(
7
)
,
d
en
o
ted
as
(
)
,
an
d
ca
n
b
e
d
ete
r
m
in
ed
u
s
in
g
a
n
ew
alg
o
r
ith
m
b
ased
o
n
r
a
n
d
o
m
n
u
m
b
e
r
g
en
er
atio
n
with
r
ap
id
co
n
v
er
g
en
ce
,
as
(
1
5
)
.
=
2
+
(
)
(
1
5
)
W
h
e
r
e
i
s
a
s
e
t
o
f
1
0
0
d
i
s
t
i
n
c
t
r
e
a
l
n
u
m
b
e
r
s
r
a
n
d
o
m
ly
ch
o
s
e
n
f
r
o
m
th
e
in
t
e
r
v
a
l
[
−
10
,
2
(
(
)
)
]
.
T
h
e
v
a
lu
e
2
−
10
d
en
o
te
s
th
e
m
in
i
m
u
m
n
u
m
b
e
r
n
e
c
e
s
s
a
r
y
t
o
m
e
e
t
t
h
e
s
p
ec
i
f
i
ed
t
o
l
er
a
n
c
e
r
eq
u
ir
e
m
e
n
t
,
(
P
D
G
S
i
)
i
s
t
h
e
i
n
i
t
i
a
l
D
G
c
a
p
ac
i
t
y
a
t
t
h
e
o
n
s
e
t
o
f
th
e
o
p
t
i
m
i
z
a
t
i
o
n
p
r
o
c
e
s
s
,
m
e
a
s
u
r
ed
in
m
e
g
a
w
a
t
t
s
(
MW
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
6
9
4
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
,
Vo
l.
16
,
No
.
3
,
Sep
tem
b
er
20
25
:
1598
-
1
6
0
7
1602
Step
s
f
o
r
d
eter
m
in
i
n
g
o
p
tim
al
s
izin
g
u
s
in
g
a
r
an
d
o
m
n
u
m
b
er
g
en
er
atio
n
-
b
ased
alg
o
r
ith
m
:
i)
Gen
er
ate
a
s
et
o
f
r
a
n
d
o
m
n
u
m
b
er
s
r
ep
r
esen
ted
b
y
.
ii)
Fo
r
ea
ch
α
,
ad
d
th
e
ac
tiv
e
p
o
wer
,
as
d
ef
in
ed
in
(
1
5
)
,
to
a
s
elec
ted
b
u
s
(
o
p
tim
al
lo
ca
tio
n
)
id
en
tifie
d
in
s
ec
tio
n
3
.
1
.
iii)
Up
d
ate
th
e
J
ac
o
b
ian
m
atr
ix
f
o
r
ea
ch
u
s
in
g
th
e
N.
R
m
eth
o
d
.
iv
)
C
alcu
late
th
e
[
]
f
o
r
all
g
e
n
er
ate
d
an
d
f
o
r
ev
er
y
b
u
s
in
th
e
s
y
s
tem
to
f
in
d
th
e
s
en
s
itiv
ity
o
f
DG
at
th
e
s
elec
ted
.
T
h
is
is
f
o
r
m
u
lated
in
th
e
f
o
llo
win
g
m
atr
ix
:
[
11
⋯
⋯
1
⋮
⋱
⋱
⋮
1
⋯
⋯
]
(
∗
)
v)
Fin
d
th
e
m
ax
v
alu
e
o
f
[
]
in
th
e
a
b
o
v
e
m
atr
i
x
,
d
e
n
o
ted
(
*
)
an
d
d
en
o
te
it a
s
.
v
i
)
U
p
d
a
t
e
t
h
e
(
*
)
m
a
t
r
i
x
a
f
t
e
r
a
d
d
i
n
g
t
h
e
n
e
w
i
n
j
ec
t
e
d
p
o
w
e
r
f
r
o
m
D
G
a
t
t
h
e
o
p
t
i
m
al
l
o
ca
t
i
o
n
f
r
o
m
s
t
e
p
5
.
I
f
t
h
e
n
e
w
(
*
)
m
a
t
r
i
x
h
a
s
a
d
i
f
f
e
r
e
n
t
r
o
w
i
n
d
e
x
f
r
o
m
t
h
e
m
i
n
v
a
l
u
e
o
f
[
]
(
)
t
h
a
n
i
n
s
t
e
p
5
,
i
.
e
.
,
p
o
i
n
t
i
n
g
t
o
a
d
i
f
f
e
r
e
n
t
o
p
t
i
m
a
l
l
o
c
a
ti
o
n
(
o
p
t
i
m
a
l
b
u
s
)
,
t
h
e
n
u
p
d
a
t
e
t
h
e
o
p
t
i
m
a
l
l
o
c
at
i
o
n
i
n
f
o
b
a
s
e
d
o
n
t
h
e
n
e
w
.
v
i
i
)
T
h
e
v
a
l
u
e
s
o
f
a
n
d
a
r
e
t
a
k
e
n
a
s
a
r
e
f
e
r
e
n
c
e
t
o
r
e
g
e
n
e
r
at
e
a
n
ew
r
a
n
d
o
m
v
a
l
u
e
,
a
s
s
h
o
w
n
i
n
(
1
6
)
.
=
+
(
−
)
(
1
6
)
v
iii)
R
ep
ea
t th
e
ca
lcu
latio
n
as in
(
1
5
)
u
n
til
th
e
to
ler
an
ce
b
etwe
en
an
d
h
as th
e
s
am
e
v
alu
e
(
≅
)
o
r
u
n
til th
e
ca
lcu
latio
n
o
b
tai
n
s
th
e
r
eq
u
ir
e
d
to
ler
a
n
ce
,
i.e
.
∆
(
)
=
1
×
10
−
4
.
ix
)
T
h
e
alg
o
r
ith
m
is
s
to
p
p
ed
if
th
e
allo
wed
ac
tiv
e
p
o
wer
is
ac
h
iev
ed
,
as
in
(
7
)
.
Oth
er
wis
e,
p
r
o
ce
ed
to
s
tep
2
in
s
ec
tio
n
3
.
1
(
n
ew
o
p
tim
al
l
o
ca
tio
n
f
o
r
th
e
u
p
d
ated
s
y
s
te
m
af
ter
ad
d
in
g
th
e
DG,
as
o
u
tlin
ed
in
th
e
p
r
ev
io
u
s
s
tep
s
)
.
T
h
e
p
r
o
p
o
s
ed
a
p
p
r
o
a
ch
's s
o
lu
tio
n
alg
o
r
ith
m
is
illu
s
tr
ated
in
a
f
lo
wch
ar
t sh
o
wn
in
Fig
u
r
e
2
.
Fig
u
r
e
2
.
Allo
ca
tio
n
an
d
s
izi
n
g
o
f
DG
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
I
SS
N:
2088
-
8
6
9
4
A
n
ew a
p
p
r
o
a
c
h
fo
r
o
p
tima
l si
z
in
g
a
n
d
a
llo
ca
tio
n
o
f d
is
tr
ib
u
ted
g
en
era
tio
n
i
n
…
(
Hu
d
efa
h
A
lka
s
h
a
s
h
n
eh
)
1603
4.
NUM
E
RIC
AL
E
XA
M
P
L
E
T
h
e
p
r
o
p
o
s
ed
m
eth
o
d
is
test
ed
o
n
two
s
tan
d
a
r
d
p
o
wer
s
y
s
tem
s
:
th
e
I
E
E
E
1
4
-
b
u
s
an
d
I
E
E
E
3
0
-
b
u
s
s
y
s
tem
s
[
3
5
]
,
f
o
r
wh
ich
d
ata
wer
e
o
b
tain
ed
f
r
o
m
[
3
6
]
,
[3
7
]
,
r
esp
ec
tiv
ely
.
T
h
e
s
in
g
le
-
lin
e
d
iag
r
am
o
f
th
e
I
E
E
E
14
-
b
u
s
s
y
s
tem
is
s
h
o
wn
in
Fig
u
r
e
3
(
a)
,
w
h
ile
Fig
u
r
e
3
(
b
)
illu
s
tr
ates
th
e
s
in
g
le
-
lin
e
d
iag
r
a
m
o
f
th
e
I
E
E
E
3
0
-
b
u
s
s
y
s
tem
.
T
h
e
ca
lcu
latio
n
s
p
r
esen
ted
b
el
o
w
ar
e
s
p
ec
if
ic
to
th
e
I
E
E
E
1
4
-
b
u
s
s
y
s
tem
;
h
o
wev
er
,
th
e
s
am
e
ca
lcu
latio
n
s
ar
e
also
co
n
d
u
cte
d
f
o
r
th
e
I
E
E
E
3
0
-
b
u
s
s
y
s
tem
.
T
h
e
test
an
aly
s
es
o
f
th
e
I
E
E
E
1
4
-
b
u
s
s
y
s
tem
ar
e
p
r
esen
ted
as f
o
llo
ws
.
A
t
e
s
t
i
s
c
o
n
d
u
c
t
e
d
b
a
s
e
d
o
n
s
e
n
s
i
t
i
v
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16
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1598
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5.
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I
SS
N
:
2
0
8
8
-
8
6
9
4
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
,
Vo
l.
16
,
No
.
3
,
Sep
tem
b
er
20
25
:
1598
-
1
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1606
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1
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[
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[
6
]
D
.
H
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[
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P
o
we
r
En
g
i
n
e
e
rin
g
fr
o
m
th
e
Un
iv
e
rsit
y
o
f
S
c
ien
c
e
a
n
d
Tec
h
n
o
lo
g
y
-
AG
H,
Cra
c
o
w,
P
o
lan
d
,
in
1
9
8
9
a
n
d
1
9
9
7
,
re
sp
e
c
ti
v
e
l
y
.
He
jo
i
n
e
d
Jo
rd
a
n
P
h
o
sp
h
a
te
M
in
e
s
C
o
.
(JP
M
C)
i
n
1
9
8
9
a
s
a
m
a
in
te
n
a
n
c
e
p
lan
n
e
r
a
n
d
p
re
v
e
n
ti
v
e
m
a
in
ten
a
n
c
e
e
n
g
in
e
e
r
i
n
th
e
p
ro
jec
ts
d
e
p
a
rtme
n
t
,
wh
e
re
h
e
wo
rk
e
d
u
n
ti
l
2
0
0
0
.
F
ro
m
2
0
0
0
to
2
0
0
4
,
h
e
s
e
rv
e
d
a
s
th
e
m
a
n
a
g
e
r
o
f
th
e
S
tu
d
ies
a
n
d
De
sig
n
De
p
a
rtme
n
t
a
t
Alfa
n
a
r
Co
.
,
Riy
a
d
h
,
Kin
g
d
o
m
o
f
S
a
u
d
i
Ara
b
ia.
In
2
0
0
4
,
h
e
re
tu
r
n
e
d
to
JPM
C
,
wh
e
re
h
e
fo
c
u
se
d
o
n
m
e
g
a
p
ro
jec
ts.
He
h
a
s
b
e
e
n
a
m
e
m
b
e
r
o
f
se
v
e
ra
l
c
o
m
m
it
tee
s
in
v
o
l
v
e
d
in
b
o
t
h
c
o
n
v
e
n
t
io
n
a
l
a
n
d
u
n
c
o
n
v
e
n
ti
o
n
a
l
e
n
e
rg
y
i
n
it
iati
v
e
s
in
Jo
rd
a
n
.
S
in
c
e
2
0
1
3
,
h
e
h
a
s
b
e
e
n
a
c
e
rti
fied
c
o
n
su
lt
a
n
t
i
n
El
e
c
tri
c
a
l
P
o
we
r
En
g
in
e
e
rin
g
a
n
d
P
ro
jec
ts
(
P
QA
C
-
JA
E
-
Jo
r
d
a
n
).
I
n
2
0
1
4
,
h
e
tran
siti
o
n
e
d
t
o
a
c
a
d
e
m
ia,
jo
in
i
n
g
th
e
De
p
a
rtme
n
t
o
f
El
e
c
tri
c
a
l
E
n
g
i
n
e
e
rin
g
a
t
P
h
i
lad
e
lp
h
ia
Un
i
v
e
rsity
.
Cu
rre
n
tl
y
,
h
e
is
a
n
As
so
c
iate
P
ro
fe
ss
o
r
in
th
e
Re
n
e
wa
b
le
En
e
rg
y
De
p
a
rtme
n
t
a
t
Ara
b
Am
m
a
n
Un
iv
e
rsity
,
Am
m
a
n
,
Jo
rd
a
n
.
He
c
a
n
b
e
c
o
n
tac
ted
a
t
e
m
a
il
:
a
.
a
g
h
a
@a
a
u
.
e
d
u
.
jo
.
Mo
h
a
m
m
e
d
Ba
n
i
y
o
u
n
is
wa
s
b
o
rn
i
n
Jo
r
d
a
n
.
He
o
b
tain
e
d
h
is
P
h
.
D.
fro
m
th
e
Un
iv
e
rsity
o
f
Ka
ise
rsla
u
tern
,
Ka
ise
rsla
u
tern
,
G
e
rm
a
n
y
,
in
2
0
0
6
.
He
h
a
s
S
tro
n
g
Kn
o
wle
d
g
e
a
n
d
e
x
p
e
rien
c
e
in
Re
-
En
g
in
e
e
rin
g
a
n
d
Re
v
e
rse
En
g
in
e
e
rin
g
,
d
e
leg
a
ted
th
r
o
u
g
h
sp
e
c
ifi
c
a
ti
o
n
s
a
n
d
v
e
rifi
c
a
ti
o
n
o
f
s
o
ftwa
re
a
n
d
h
a
rd
wa
re
sy
ste
m
s
u
sin
g
v
a
ri
o
u
s
t
o
o
ls
a
n
d
l
o
g
ic.
He
a
lso
h
a
s
e
x
p
e
rt
ise
in
th
e
Distrib
u
ted
a
n
d
C
o
n
c
u
r
re
n
t
S
y
ste
m
s
a
p
p
ro
a
c
h
fo
r
so
f
twa
re
m
o
d
e
li
n
g
,
d
e
si
g
n
,
a
n
d
d
e
v
e
lo
p
m
e
n
t.
He
wa
n
ts
to
a
p
p
l
y
th
e
se
m
e
th
o
d
s
a
n
d
S
o
f
twa
re
En
g
in
e
e
rin
g
tec
h
n
o
lo
g
ies
to
so
l
v
e
a
u
to
m
a
ti
o
n
a
n
d
c
o
n
tro
l
-
s
p
e
c
ifi
c
p
r
o
b
lem
s.
A
b
a
sic
u
n
d
e
rsta
n
d
i
n
g
o
f
e
lec
tri
c
a
l
m
a
c
h
in
e
s
a
ls
o
a
ss
ists
with
th
is
k
n
o
wle
d
g
e
.
Cu
rre
n
tl
y
,
h
e
se
rv
e
s
a
s
th
e
Ac
ti
n
g
De
a
n
o
f
th
e
En
g
in
e
e
ri
n
g
a
n
d
Tec
h
n
o
l
o
g
y
F
a
c
u
lt
y
a
t
P
h
il
a
d
e
l
p
h
ia
Un
i
v
e
rsity
in
Am
m
a
n
.
He
c
a
n
b
e
c
o
n
tac
ted
a
t
e
m
a
il
:
m
b
a
n
iy
o
u
n
is@p
h
il
a
d
e
lp
h
ia.ed
u
.
j
o
.
Wa
ss
e
e
m
Al
-
Ro
u
s
a
n
re
c
e
iv
e
d
B.
S
.
a
n
d
M
.
S
.
d
e
g
re
e
s
in
e
lec
tri
c
a
l
e
n
g
i
n
e
e
rin
g
fr
o
m
Ya
rm
o
u
k
U
n
iv
e
rsit
y
,
Irb
i
d
,
Jo
r
d
a
n
,
i
n
2
0
1
2
a
n
d
2
0
1
5
,
re
sp
e
c
ti
v
e
ly
.
A
n
d
a
P
h
.
D.
i
n
e
lec
tri
c
a
l
e
n
g
in
e
e
rin
g
fro
m
Way
n
e
S
tate
Un
iv
e
rsity
,
De
tr
o
it
,
M
I,
USA,
i
n
2
0
2
2
.
F
ro
m
S
e
p
tem
b
e
r
2
0
1
2
to
Au
g
u
st
2
0
1
6
,
h
e
wo
rk
e
d
a
s
a
n
El
e
c
tri
c
a
l
En
g
in
e
e
r
with
t
h
e
Na
ti
o
n
a
l
El
e
c
tri
c
P
o
we
r
Co
m
p
a
n
y
in
Am
m
a
n
,
Jo
rd
a
n
.
Cu
rre
n
tl
y
,
h
e
is
a
n
As
sista
n
t
P
ro
fe
ss
o
r
in
th
e
El
e
c
tri
c
a
l
En
g
in
e
e
ri
n
g
De
p
a
rtme
n
t
a
t
P
h
il
a
d
e
lp
h
ia
Un
i
v
e
rsity
,
Am
m
a
n
,
Jo
rd
a
n
.
His
c
u
rre
n
t
re
se
a
rc
h
in
tere
sts
in
c
lu
d
e
m
o
d
e
li
n
g
a
n
d
c
o
n
tro
l
o
f
p
o
we
r
sy
ste
m
s,
d
isc
re
te
-
e
v
e
n
t
sy
ste
m
s,
s
u
p
e
r
v
iso
ry
c
o
n
tro
l,
a
n
d
re
n
e
wa
b
le
e
n
e
rg
y
.
He
c
a
n
b
e
c
o
n
tac
ted
at
e
m
a
il
:
wa
lro
u
sa
n
@p
h
il
a
d
e
l
p
h
ia.e
d
u
.
jo
.
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