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l
e
[
1
]
,
t
h
e
a
u
t
h
o
r
s
p
r
o
p
o
s
e
a
n
o
n
-
i
t
e
r
a
t
i
v
e
m
et
h
o
d
f
o
r
d
e
t
e
r
m
i
n
i
n
g
t
h
e
b
o
u
n
d
a
r
i
e
s
o
f
s
t
at
i
c
a
p
e
r
io
d
i
c
s
t
a
b
il
i
t
y
.
In
r
esear
ch
[
7
]
,
p
o
wer
s
er
ies
ar
e
u
s
ed
to
d
ef
i
n
e
th
e
r
eg
i
o
n
s
o
f
ex
is
ten
ce
o
f
th
e
E
PS
m
o
d
e
.
Ar
ticle
[
8
]
p
r
esen
ts
th
e
r
esu
lts
o
f
an
an
al
y
s
is
o
f
th
e
s
ta
te
o
f
th
e
E
PS
b
ased
o
n
th
e
tr
o
p
ical
g
eo
m
etr
y
o
f
p
o
wer
b
alan
ce
eq
u
atio
n
s
o
v
er
c
o
m
p
lex
m
u
lti
p
o
les.
T
h
is
cir
cu
m
s
tan
ce
is
v
er
y
im
p
o
r
tan
t
w
h
en
s
tu
d
y
in
g
s
u
ch
s
y
s
tem
s
.
T
h
e
s
tu
d
y
[
9
]
p
r
esen
ts
th
e
r
esu
lts
o
f
s
tu
d
ies o
f
s
tatic
s
tab
ilit
y
m
ar
g
in
s
in
s
y
s
tem
s
with
DC
tr
an
s
m
is
s
io
n
lin
es.
T
h
e
s
o
lu
tio
n
to
th
e
p
r
esen
te
d
p
r
o
b
lem
s
is
esp
ec
ially
r
elev
an
t
in
m
o
d
er
n
p
o
wer
s
y
s
t
em
s
with
d
is
tr
ib
u
ted
g
e
n
er
atio
n
(
DG)
i
n
s
tallatio
n
s
[
1
0
]
,
[
1
1
]
,
wh
ich
ar
e
lo
ca
ted
in
clo
s
e
p
r
o
x
im
ity
to
co
n
s
u
m
er
s
.
Ar
ticle
[
1
2
]
p
r
o
p
o
s
es
a
m
eth
o
d
f
o
r
ass
ess
in
g
th
e
s
tab
ilit
y
o
f
E
PS
with
d
is
tr
ib
u
ted
g
e
n
er
atio
n
u
n
d
er
c
o
n
d
itio
n
s
o
f
r
a
p
id
ly
c
h
an
g
in
g
lo
a
d
s
.
I
n
p
ar
ticu
lar
,
DG
p
lan
ts
ca
n
b
e
im
p
lem
en
te
d
o
n
th
e
b
asis
o
f
n
o
n
-
tr
ad
itio
n
al
r
en
ewa
b
le
en
e
r
g
y
s
o
u
r
ce
s
(
R
E
S)
[
1
3
]
-
[
1
5
]
u
s
in
g
elec
tr
icity
s
to
r
ag
e
d
e
v
ices
[
1
6
]
,
[
1
7
]
.
T
h
e
r
en
ewa
b
le
e
n
er
g
y
s
o
u
r
ce
s
em
p
lo
y
ed
ca
n
b
e
lo
ca
ted
f
ar
f
r
o
m
c
o
n
s
u
m
p
tio
n
ce
n
ter
s
,
wh
ich
“
n
ar
r
o
ws”
th
e
SAS
r
eg
io
n
s
a
n
d
h
as
a
s
ig
n
if
ican
t
im
p
ac
t
o
n
em
er
g
en
cy
co
n
tr
o
l
th
at
e
n
s
u
r
es
s
tab
ilit
y
in
p
o
s
t
-
em
er
g
en
cy
c
o
n
d
itio
n
s
o
f
th
e
E
PS
[
1
8
]
.
T
h
e
s
tu
d
y
p
r
esen
ted
in
[
1
9
]
ex
am
in
es
th
e
p
r
o
b
lem
s
o
f
th
e
s
tab
ilit
y
m
ar
g
in
o
f
elec
tr
i
ca
l
n
etwo
r
k
s
with
d
is
tr
ib
u
ted
g
en
e
r
atio
n
p
la
n
ts
.
C
h
en
et
a
l.
[
2
0
]
f
o
c
u
s
o
n
p
r
ed
ictin
g
th
e
s
tab
ilit
y
o
f
elec
tr
ic
p
o
wer
s
y
s
tem
s
with
a
lar
g
e
s
h
ar
e
o
f
win
d
g
en
er
ati
n
g
p
lan
ts
.
C
h
an
g
in
g
th
e
p
o
wer
o
f
th
e
DG
p
lan
t
g
en
er
ato
r
s
to
en
ab
le
th
e
E
PS
o
p
er
atin
g
co
n
d
itio
n
s
t
o
m
ee
t
th
e
s
tab
ilit
y
r
eq
u
ir
em
e
n
ts
s
h
o
u
ld
b
e
ca
r
r
ie
d
o
u
t
with
an
ac
ce
p
t
ab
le
q
u
ality
o
f
d
y
n
am
ic
p
r
o
c
ess
es.
T
h
is
ca
n
b
e
ac
h
iev
ed
th
r
o
u
g
h
t
h
e
u
s
e
o
f
a
u
to
m
atic
r
eg
u
lato
r
s
o
f
e
x
citatio
n
(
AE
R
)
an
d
s
p
ee
d
(
ASR
)
o
f
th
e
s
y
n
ch
r
o
n
o
u
s
g
en
er
ato
r
r
o
to
r
s
o
f
th
e
DG
p
la
n
ts
.
I
t is also
wo
r
th
n
o
tin
g
th
at
o
p
tim
al
co
n
tr
o
l r
eq
u
ir
es a
d
ju
s
tin
g
th
e
s
ettin
g
s
o
f
th
e
AE
R
an
d
ASR
o
f
th
e
D
G
p
lan
ts
in
th
e
e
v
en
t
o
f
s
ig
n
if
ican
t
ch
a
n
g
es
in
o
p
e
r
atin
g
co
n
d
itio
n
s
.
T
h
ese
r
eq
u
ir
em
e
n
ts
ca
n
b
e
m
et
b
y
u
s
in
g
in
tellig
en
t
co
n
tr
o
l
alg
o
r
ith
m
s
,
f
o
r
ex
a
m
p
le
u
s
in
g
g
en
etic
alg
o
r
ith
m
s
[
2
1
]
o
r
f
u
zz
y
in
f
er
e
n
ce
s
y
s
tem
s
[
2
2
]
,
[
2
3
]
.
Similar
s
tu
d
ies
[
2
4
]
-
[
2
7
]
co
n
s
id
er
th
e
is
s
u
es
o
f
an
aly
zin
g
s
tatic
s
tab
ili
ty
an
d
ass
ess
in
g
th
e
s
tab
ilit
y
m
a
r
g
in
o
f
E
PS
with
in
th
e
f
r
am
e
wo
r
k
o
f
th
e
u
s
e
o
f
in
tellig
en
t
co
n
tr
o
l
alg
o
r
ith
m
s
an
d
ac
tiv
e
n
etwo
r
k
elem
en
ts
.
Fo
r
ex
am
p
le,
in
ar
ticle
[
2
4
]
it
is
p
r
o
p
o
s
ed
to
e
n
s
u
r
e
th
e
s
tab
ilit
y
o
f
a
m
u
lti
-
m
ac
h
in
e
E
PS
u
s
in
g
a
s
tatic
s
y
n
ch
r
o
n
o
u
s
c
o
m
p
e
n
s
ato
r
c
o
n
tr
o
lled
b
ased
o
n
th
e
an
t
co
lo
n
y
alg
o
r
ith
m
.
I
n
[
2
5
]
,
th
e
s
u
s
tain
ab
ilit
y
o
f
th
e
E
PS
i
s
in
cr
ea
s
ed
b
y
cr
ea
tin
g
an
in
t
ellig
en
t
h
y
b
r
id
win
d
-
s
o
lar
p
o
wer
p
lan
t
as
a
s
tatic
co
m
p
en
s
ato
r
.
I
n
ar
ticle
[
2
6
]
,
t
h
e
au
th
o
r
s
co
n
s
id
er
th
e
p
r
o
b
le
m
s
o
f
in
cr
ea
s
in
g
th
e
s
tatic
ap
e
r
io
d
ic
s
tab
ilit
y
in
a
two
-
m
ac
h
in
e
E
PS
b
y
u
s
in
g
a
f
u
el
ce
ll
as
a
ST
AT
C
OM
.
R
e
al
-
tim
e
ass
e
s
s
m
en
t
o
f
v
o
ltag
e
s
tab
ilit
y
in
a
lar
g
e
-
s
ca
le
E
PS
b
ased
o
n
s
p
ec
tr
u
m
ass
es
s
m
en
t
o
f
v
ec
to
r
m
ea
s
u
r
em
en
t
u
n
it
d
ata
is
g
iv
en
i
n
[
2
7
]
.
Ar
ticle
[
2
8
]
p
r
esen
ts
a
h
y
b
r
i
d
m
eth
o
d
f
o
r
ca
lcu
latin
g
th
e
v
o
ltag
e
s
tab
ilit
y
m
ar
g
in
i
n
th
e
p
o
wer
s
u
p
p
l
y
s
y
s
tem
,
tak
in
g
in
to
ac
co
u
n
t
th
e
u
n
ce
r
tain
ty
o
f
lo
a
d
ch
an
g
es
an
d
p
o
s
s
ib
le
u
n
f
o
r
e
s
ee
n
m
o
d
es.
T
o
ass
ess
th
e
v
o
ltag
e
s
tab
ilit
y
o
f
an
elec
tr
ical
n
etwo
r
k
,
ar
ticle
[
2
9
]
p
r
o
p
o
s
es to
u
s
e
th
e
E
u
clid
ea
n
d
is
tan
ce
b
etwe
en
b
u
s
v
o
ltag
e
v
ec
to
r
s
.
T
o
im
p
r
o
v
e
th
e
s
tab
ilit
y
o
f
t
h
e
p
o
wer
s
y
s
tem
u
n
d
er
r
an
d
o
m
d
is
tu
r
b
an
ce
s
[
3
0
]
,
s
tr
ateg
ie
s
b
ased
o
n
ad
ap
tiv
e
co
n
tr
o
l
[
3
1
]
,
f
r
e
q
u
e
n
cy
d
o
m
ain
a
p
p
r
o
ac
h
[
3
2
]
,
a
n
d
p
a
r
ticle
s
war
m
o
p
tim
izati
o
n
m
eth
o
d
s
[
3
3
]
a
r
e
u
s
ed
.
Ass
ess
in
g
s
tab
ilit
y
in
t
h
e
tr
an
s
ien
t
p
r
o
ce
s
s
f
o
r
E
PS
with
DG
i
n
s
tallatio
n
s
is
an
im
p
o
r
tan
t
task
in
co
n
n
ec
tio
n
with
p
o
s
s
ib
le
f
lu
ct
u
atio
n
s
in
th
e
p
o
wer
o
f
s
o
u
r
ce
s
.
Pre
v
io
u
s
s
tu
d
ies
[
3
4
]
-
[
3
6
]
,
t
h
e
au
th
o
r
s
p
r
o
p
o
s
e
a
m
eth
o
d
f
o
r
esti
m
atin
g
th
e
s
tab
ilit
y
m
ar
g
in
d
u
r
in
g
tr
a
n
s
ien
t
p
r
o
ce
s
s
es
o
n
lin
e.
T
h
e
p
r
o
p
o
s
ed
m
eth
o
d
u
s
es
a
n
eu
r
al
n
etwo
r
k
to
d
e
p
ict
th
e
r
elatio
n
s
h
ip
b
etwe
en
s
tead
y
-
s
tate
p
o
wer
f
lo
w
an
d
g
en
er
at
o
r
s
tab
ilit
y
in
d
ices
u
n
d
er
a
n
ex
p
ec
ted
s
et
o
f
u
n
ex
p
ec
ted
co
n
d
itio
n
s
.
I
n
ar
ticle
[
3
7
]
,
a
m
eth
o
d
f
o
r
an
aly
zin
g
th
e
tr
an
s
ien
t
s
tab
ilit
y
o
f
E
PS
is
p
r
o
p
o
s
ed
,
wh
ich
al
lo
ws
illu
s
tr
atin
g
th
e
s
tr
u
ctu
r
a
l
ch
ar
ac
ter
is
tics
o
f
th
e
d
y
n
am
ic
s
tab
ilit
y
r
eg
io
n
.
T
h
e
wo
r
k
also
p
r
esen
ts
r
esu
lts
th
at
allo
w
d
is
p
lay
in
g
t
h
e
b
o
u
n
d
ar
y
o
n
th
e
g
l
o
b
al
p
h
ase
p
lan
e.
I
t
is
v
er
y
im
p
o
r
tan
t th
at
th
e
s
tab
ilit
y
b
o
u
n
d
ar
y
an
d
p
h
ase
p
o
r
tr
aits
ar
e
s
tu
d
ied
in
th
r
ee
-
d
im
en
s
io
n
al
s
p
ac
e
[
3
8
]
.
Pap
er
[
3
9
]
p
r
esen
ts
a
h
y
b
r
id
an
aly
tical
ap
p
r
o
ac
h
co
m
b
in
in
g
th
e
d
ir
ec
t
L
y
a
p
u
n
o
v
s
tab
ilit
y
m
eth
o
d
an
d
tim
e
d
o
m
ain
m
o
d
elin
g
f
o
r
r
a
p
id
an
aly
s
is
o
f
p
o
wer
s
y
s
tem
tr
an
s
ien
ts
.
Pap
er
[
4
0
]
p
r
esen
ts
a
p
r
e
d
ictiv
e
co
n
tr
o
l
m
o
d
el
b
ased
o
n
s
en
s
itiv
ity
an
aly
s
is
th
at
aim
s
to
im
p
r
o
v
e
tr
an
s
ien
t
s
tab
ilit
y
b
y
r
ec
o
n
f
i
g
u
r
in
g
th
e
tr
an
s
m
is
s
io
n
s
y
s
tem
.
Me
th
o
d
s
f
o
r
d
ete
r
m
in
in
g
lim
itin
g
m
o
d
es
ca
n
b
e
d
iv
i
d
ed
in
t
o
th
r
ee
g
r
o
u
p
s
[
3
]
-
[
5
]
:
d
is
cr
ete
weig
h
tin
g
,
c
o
n
tin
u
o
u
s
weig
h
tin
g
,
an
d
th
o
s
e
b
ased
o
n
lim
itin
g
m
o
d
e
eq
u
atio
n
s
.
T
h
e
r
esu
lts
o
f
th
eir
co
m
p
ar
is
o
n
b
ased
o
n
tab
u
lar
a
n
aly
s
is
ar
e
g
iv
en
b
el
o
w.
T
h
e
p
r
esen
te
d
r
esear
ch
aim
s
t
o
d
ev
el
o
p
e
f
f
ec
tiv
e
m
eth
o
d
s
f
o
r
en
a
b
lin
g
t
h
e
o
p
er
atin
g
co
n
d
itio
n
s
o
f
E
PS
with
D
G
p
lan
ts
to
m
ee
t
t
h
e
r
eq
u
ir
em
e
n
ts
o
f
s
tatic
ap
er
io
d
ic
s
tab
ilit
y
an
d
to
test
th
e
ef
f
ec
tiv
en
ess
o
f
th
e
f
u
zz
y
co
n
tr
o
l
s
y
s
tem
u
s
ed
to
ad
ju
s
t
th
e
s
ettin
g
s
o
f
a
u
to
m
at
ic
r
eg
u
lato
r
s
o
f
DG
p
lan
ts
to
en
s
u
r
e
ap
p
r
o
p
r
iate
d
y
n
am
ic
b
eh
a
v
io
r
o
f
th
e
E
P
S.
T
h
is
ef
f
icien
c
y
lies
in
im
p
r
o
v
in
g
th
e
q
u
ality
o
f
r
e
g
u
la
tio
n
o
f
E
PS
m
o
d
e
p
ar
am
eter
s
.
T
h
u
s
,
th
e
m
ai
n
r
esu
lt
o
f
th
is
wo
r
k
is
th
e
ap
p
l
icatio
n
o
f
m
eth
o
d
s
f
o
r
in
t
r
o
d
u
cin
g
th
e
o
p
er
atin
g
m
o
d
e
o
f
E
PS
with
DG
p
lan
ts
in
to
th
e
r
eg
io
n
o
f
s
tatic
ap
er
io
d
ic
s
tab
ilit
y
b
ased
o
n
th
e
ca
lcu
latio
n
o
f
lim
itin
g
m
o
d
es,
as
well
as
th
e
u
s
e
o
f
a
f
u
zz
y
lo
g
ic
in
f
er
e
n
ce
s
y
s
tem
to
co
n
tr
o
l
th
e
r
e
g
u
lato
r
s
o
f
g
en
er
ato
r
s
o
f
DG
p
lan
ts
an
d
en
s
u
r
i
n
g
h
ig
h
-
q
u
a
lity
d
y
n
am
ic
tr
an
s
itio
n
at
s
h
u
td
o
wn
s
o
f
lo
ad
ed
n
etwo
r
k
elem
en
ts
.
T
h
e
p
ap
er
p
r
esen
ts
th
e
f
o
r
m
u
latio
n
o
f
t
h
e
p
r
o
b
lem
in
d
etail,
d
escr
ib
es
alg
o
r
ith
m
s
f
o
r
e
n
ter
in
g
th
e
E
PS
m
o
d
e
in
to
th
e
s
tab
ilit
y
r
eg
io
n
,
as
well
as
m
e
th
o
d
s
f
o
r
s
o
lv
in
g
th
e
“
f
ar
b
o
u
n
d
ar
y
”
p
r
o
b
lem
wh
e
n
en
ter
i
n
g
th
e
m
o
d
e
i
n
to
th
e
s
tab
ilit
y
r
eg
io
n
.
I
n
ad
d
itio
n
,
t
h
e
r
esu
lts
o
f
m
o
d
elin
g
em
er
g
en
cy
co
n
tr
o
l
o
f
DG
p
lan
ts
w
ith
en
s
u
r
in
g
h
ig
h
-
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2088
-
8
6
9
4
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
,
Vo
l.
16
,
No
.
1
,
Ma
r
c
h
20
25
:
138
-
150
140
q
u
ality
d
y
n
am
ic
tr
a
n
s
itio
n
ar
e
p
r
esen
ted
.
T
h
e
s
tr
u
ct
u
r
e
o
f
th
e
ar
ticle
co
n
s
is
ts
o
f
th
e
s
tatem
en
t
o
f
th
e
p
r
o
b
lem
,
m
eth
o
d
s
f
o
r
en
s
u
r
i
n
g
th
e
s
tab
ilit
y
o
f
E
PS
,
m
eth
o
d
s
f
o
r
s
o
lv
in
g
th
e
p
r
o
b
lem
o
f
th
e
f
ar
b
o
u
n
d
ar
y
wh
e
n
in
tr
o
d
u
cin
g
th
e
E
PS
m
o
d
e
in
t
o
th
e
s
tab
ilit
y
ar
ea
,
m
o
d
elin
g
r
esu
lts
o
f
th
e
p
r
o
p
o
s
ed
em
er
g
en
cy
co
n
tr
o
l
s
y
s
tem
f
o
r
d
is
tr
ib
u
te
d
g
en
e
r
atio
n
in
s
tallatio
n
s
with
en
s
u
r
in
g
a
h
i
g
h
-
q
u
ality
d
y
n
am
ic
tr
an
s
itio
n
an
d
a
co
n
clu
s
io
n
wh
er
e
th
e
m
ain
c
o
n
clu
s
io
n
s
ar
e
p
r
esen
ted
.
2.
P
RO
B
L
E
M
ST
A
T
E
M
E
NT
E
q
u
ilib
r
iu
m
s
tate
o
f
an
au
to
n
o
m
o
u
s
s
y
s
tem
o
f
d
if
f
er
en
tial
eq
u
atio
n
s
is
asy
m
p
to
tically
s
t
ab
le.
T
h
is
s
tatem
en
t
is
b
ased
o
n
th
e
L
y
a
p
u
n
o
v
s
tab
ilit
y
th
e
o
r
em
if
th
e
lin
ea
r
s
y
s
tem
o
f
th
e
f
ir
s
t
a
p
p
r
o
x
im
atio
n
.
W
h
er
e
=
−
0
;
0
–
co
o
r
d
in
ates
co
r
r
esp
o
n
d
i
n
g
to
th
e
eq
u
atio
n
s
(
10
,
20
,
.
.
.
,
0
)
=
0
;
=
1
.
.
.
.
L
in
ea
r
izatio
n
ca
n
b
e
d
o
n
e
u
s
in
g
f
u
n
ctio
n
e
x
p
an
s
io
n
(
1
,
2
,
.
.
.
,
)
,
=
1
.
.
.
;
(
1
,
2
,
.
.
.
,
)
=
(
10
,
20
,
.
.
.
,
0
)
+
∑
(
)
|
=
0
+
=
1
1
2
!
∑
∑
(
2
)
=
1
=
1
+
.
.
.
an
d
eq
u
atin
g
n
o
n
lin
ea
r
ter
m
s
to
ze
r
o
,
i.e
.
1
2
!
∑
∑
(
2
)
=
1
=
1
+
.
.
.
≈
0
.
T
h
e
s
o
lu
tio
n
to
(
1
)
an
d
(
2
)
is
s
tab
le
if
th
e
r
ea
l
p
ar
ts
o
f
all
r
o
o
ts
o
f
th
e
c
h
ar
ac
ter
is
tic
eq
u
atio
n
a
r
e
n
eg
ati
v
e,
as in
(
3
)
.
=
(
1
,
2
,
.
.
.
,
)
;
=
1
.
.
.
(
1
)
=
∑
(
)
=
1
|
=
0
;
=
1
.
.
.
(
2
)
(
)
=
(
−
)
=
0
(
3
)
W
h
er
e
is
th
e
J
ac
o
b
ian
m
atr
ix
o
f
(
)
:
=
[
1
1
…
1
…
1
…
]
.
W
h
en
an
aly
zin
g
s
tead
y
-
s
tate
E
PS
m
o
d
es,
L
y
ap
u
n
o
v
s
tab
ilit
y
is
ca
lled
s
tatic
.
I
n
th
is
ca
s
e,
two
ty
p
es
o
f
in
s
tab
ilit
y
ar
e
d
is
tin
g
u
is
h
ed
:
ap
er
io
d
ic
an
d
o
s
cillato
r
y
.
T
h
e
f
ir
s
t
ty
p
e
a
r
is
es
in
th
e
p
r
ese
n
ce
o
f
r
ea
l
p
o
s
itiv
e
r
o
o
ts
>
0
,
th
e
s
ec
o
n
d
-
wh
en
c
o
m
p
lex
r
o
o
ts
=
+
with
a
p
o
s
itiv
e
r
ea
l
p
ar
t
ap
p
ea
r
,
i.e
.
>
0
.
T
h
e
m
eth
o
d
s
an
d
alg
o
r
ith
m
s
p
r
esen
ted
b
elo
w
ar
e
ap
p
licab
l
e
to
th
e
an
aly
s
is
o
f
o
n
ly
s
tatic
ap
er
io
d
ic
s
tab
ilit
y
.
C
h
ar
ac
ter
is
tic
(
3
)
ca
n
b
e
p
r
ese
n
ted
in
ex
p
an
d
e
d
f
o
r
m
as in
(
4
)
.
(
)
=
+
−
1
−
1
+
.
.
.
+
0
=
0
(
4
)
I
n
o
r
d
e
r
f
o
r
th
is
eq
u
atio
n
n
o
t
to
h
av
e
p
o
s
itiv
e
r
ea
l
r
o
o
ts
,
it
is
n
ec
ess
ar
y
an
d
s
u
f
f
icien
t
th
a
t
all
it
s
co
ef
f
icien
ts
b
e
g
r
ea
ter
th
a
n
0
.
I
n
p
r
ac
tice,
th
e
p
o
in
ts
co
r
r
es
p
o
n
d
in
g
to
th
e
b
o
u
n
d
ar
y
o
f
a
p
er
io
d
ic
s
tab
ilit
y
in
th
e
p
ar
am
eter
s
p
ac
e
ar
e
d
eter
m
in
ed
b
y
weig
h
tin
g
th
e
in
itial
s
tab
le
m
o
d
e.
I
n
th
is
ca
s
e,
y
o
u
d
o
n
o
t
n
ee
d
to
m
o
n
ito
r
t
h
e
s
ig
n
s
o
f
all
co
ef
f
icien
ts
.
T
h
is
is
d
u
e
to
t
h
e
f
ac
t
th
at
th
e
f
r
ee
ter
m
will
b
e
th
e
f
ir
s
t
0
to
c
h
an
g
e
s
ig
n
.
T
h
is
co
n
clu
s
io
n
f
o
llo
ws
=
0
f
r
o
m
r
elatio
n
s
(
3
)
a
n
d
(
4
)
at
(
5
)
.
0
=
(
−
1
)
=0
(
5
)
T
h
u
s
,
if
th
e
s
ig
n
o
f
a
r
ea
l
r
o
o
t
ch
an
g
es
f
r
o
m
n
eg
ativ
e
to
p
o
s
itiv
e,
a
s
ig
n
ch
an
g
e
o
cc
u
r
s
0
.
T
h
is
p
r
o
p
er
ty
f
o
r
m
s
th
e
b
asis
f
o
r
m
eth
o
d
s
u
s
ed
in
p
r
ac
tice
f
o
r
d
eter
m
in
i
n
g
th
e
lim
itin
g
m
o
d
es
o
f
s
tatic
ap
er
io
d
ic
s
tab
ilit
y
,
i.e
.
,
m
o
d
es
th
at
m
ee
t
th
e
co
n
d
itio
n
=
0
.
T
h
e
s
tead
y
-
s
tate
co
n
d
itio
n
s
o
f
th
e
E
PS
ar
e
d
ef
i
n
ed
b
y
n
o
n
lin
ea
r
eq
u
atio
n
s
o
f
th
e
f
o
r
m
as in
(
6
)
.
(
,
)
=
0
(
6
)
W
h
er
e
=
[
1
2
.
.
.
]
–
v
ec
to
r
f
u
n
ctio
n
o
f
p
o
wer
o
r
cu
r
r
e
n
t
b
alan
ce
eq
u
at
io
n
s
in
elec
tr
ical
n
etwo
r
k
n
o
d
es
.
=
[
1
2
.
.
.
]
–
v
ec
to
r
o
f
a
d
ju
s
tab
le
p
ar
a
m
eter
s
,
wh
ich
ar
e
u
s
ed
as
ac
tiv
e
an
d
r
ea
ctiv
e
p
o
wer
s
o
f
g
en
er
ato
r
s
an
d
lo
ad
s
,
as
well
as
v
o
ltag
e
m
o
d
u
les
f
ix
ed
in
in
d
iv
id
u
al
n
etwo
r
k
n
o
d
es
.
=
[
1
2
.
.
.
]
–
v
ec
to
r
o
f
u
n
r
eg
u
lated
p
a
r
am
et
er
s
,
wh
ich
ar
e
tak
en
as
r
ea
l
a
n
d
im
ag
i
n
ar
y
co
m
p
o
n
en
ts
o
r
m
o
d
u
les
an
d
p
h
ases
o
f
n
o
d
al
v
o
ltag
es.
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
Meth
o
d
s
fo
r
en
s
u
r
in
g
s
ta
b
ilit
y
o
f o
p
era
tin
g
co
n
d
itio
n
s
o
f a
n
elec
tr
ic
p
o
w
er sy
s
tem
w
ith
…
(
I
liya
I
liev
)
141
T
h
e
lim
itin
g
m
o
d
es
o
f
E
PS
in
ter
m
s
o
f
s
tatic
ap
er
io
d
ic
s
tab
ilit
y
ar
e
t
h
e
m
o
d
es
th
at
co
r
r
esp
o
n
d
to
t
h
e
p
o
in
ts
,
o
f
th
e
p
ar
am
eter
s
p
ac
e
=
∪
in
wh
ich
(
6
)
an
d
co
n
d
itio
n
in
(
5
)
a
r
e
s
atis
f
ied
.
T
h
e
ex
p
r
ess
io
n
f
o
r
0
ca
n
b
e
o
b
tain
ed
d
ir
ec
tly
f
r
o
m
th
e
s
tead
y
s
tate
eq
u
atio
n
s
as in
(
7
)
.
(
,
)
=
0
(
7
)
W
h
ich
ar
e
wr
itten
tak
in
g
in
to
ac
co
u
n
t
t
h
e
c
h
ar
ac
ter
is
tics
o
f
E
PS
elem
en
ts
u
n
d
e
r
s
m
all
d
is
tu
r
b
an
ce
s
.
I
t
s
h
o
u
ld
b
e
n
o
ted
th
at
to
s
o
lv
e
p
r
ac
tic
al
p
r
o
b
lem
s
wh
en
d
eter
m
in
i
n
g
0
,
y
o
u
ca
n
u
s
e
(
6
)
.
I
n
th
is
ca
s
e,
p
ar
am
eter
0
ca
n
b
e
d
e
f
in
ed
as
in
(
8
)
.
0
=
(
−
1
)
(
8
)
I
n
Y
s
p
ac
e,
p
o
in
ts
Y
L
f
o
r
m
a
h
y
p
er
s
u
r
f
ac
e
L
F
,
as
s
ee
n
in
Fig
u
r
e
1
,
lim
itin
g
th
e
r
eg
io
n
o
f
s
tatic
s
tab
ilit
y
.
I
n
th
e
p
r
o
ce
s
s
o
f
m
an
ag
in
g
m
o
d
es,
it is
n
ec
ess
ar
y
to
en
s
u
r
e
th
at
th
e
p
o
in
ts
o
f
th
e
cu
r
r
e
n
t m
o
d
es
ar
e
lo
ca
ted
with
in
th
e
s
tab
ilit
y
r
eg
io
n
.
Ap
p
r
o
ac
h
in
g
th
e
L
F
b
o
u
n
d
ar
y
is
p
o
s
s
ib
le
at
a
d
is
tan
ce
d
eter
m
i
n
ed
b
y
th
e
s
tan
d
ar
d
v
alu
e
o
f
th
e
s
tab
ilit
y
m
ar
g
i
n
.
i
y
j
y
=
0
d
et
X
F
F
L
(
)
1
LP
Y
(
)
2
LP
Y
(
)
3
LP
Y
(
)
n
LP
Y
0
det
X
F
0
det
X
F
F
i
g
u
r
e
1
.
P
r
o
j
e
ct
i
o
n
s
o
f
t
h
e
r
e
g
i
o
n
s
o
f
s
t
a
b
i
l
it
y
a
n
d
e
x
i
s
te
n
c
e
o
f
o
p
e
r
a
t
i
n
g
c
o
n
d
i
t
i
o
n
s
o
n
t
o
t
h
e
c
o
o
r
d
i
n
a
t
e
p
l
a
n
e
3.
M
E
T
H
O
DS F
O
R
E
N
SUR
I
N
G
T
H
E
ST
AB
I
L
I
T
Y
O
F
T
H
E
E
P
S O
P
E
RAT
I
NG
CO
ND
I
T
I
O
N
S
E
n
s
u
r
in
g
s
tatic
s
tab
ilit
y
o
f
p
o
s
t
-
em
er
g
en
cy
c
o
n
d
itio
n
s
(
PA
C
)
is
o
n
e
o
f
th
e
m
ain
task
s
o
f
em
er
g
en
c
y
co
n
tr
o
l
o
f
E
PS
.
T
h
e
in
p
u
t
o
f
t
h
e
PAC
in
to
th
e
s
tab
ilit
y
r
eg
io
n
is
ca
r
r
ied
o
u
t
alo
n
g
a
ce
r
ta
in
tr
ajec
to
r
y
i
n
th
e
s
p
ac
e
o
f
ad
ju
s
tab
le
p
ar
am
eter
s
Y
[
3
]
,
wh
ich
is
ch
o
s
en
to
b
e
lin
ea
r
.
T
h
is
tr
ajec
to
r
y
ca
n
b
e
s
et
b
ased
o
n
p
r
elim
in
ar
y
ca
lcu
latio
n
s
;
co
r
r
esp
o
n
d
t
o
th
e
s
h
o
r
test
d
is
tan
c
e
to
th
e
L
F
b
o
u
n
d
a
r
y
o
r
b
e
s
el
ec
ted
b
y
m
in
im
izin
g
th
e
d
am
ag
e
ass
o
ciate
d
with
d
is
co
n
n
ec
tin
g
elec
tr
icity
co
n
s
u
m
er
s
.
T
o
d
eter
m
in
e
lim
itin
g
m
o
d
es
with
o
u
t
u
s
in
g
m
u
lti
-
s
tep
co
m
p
u
tatio
n
al
p
r
o
ce
d
u
r
es,
a
n
d
also
to
a
v
o
i
d
d
if
f
icu
lties
ass
o
ciate
d
with
s
o
lv
in
g
p
o
o
r
ly
-
co
n
d
itio
n
e
d
s
y
s
tem
s
o
f
lin
ea
r
eq
u
atio
n
s
,
o
n
e
ca
n
u
s
e
lim
itin
g
m
o
d
e
in
[
3
]
,
b
ased
o
n
r
e
p
lacin
g
co
n
d
itio
n
(
2
)
with
an
eq
u
iv
ale
n
t r
elatio
n
s
h
i
p
,
wh
ich
ca
n
b
e
wr
itten
in
two
f
o
r
m
s
as in
(
9
)
an
d
(
10
)
.
=
=
0
(
9
)
=
=
0
(
10
)
W
h
er
e
V
is
an
n
-
d
im
en
s
io
n
al
v
ec
to
r
f
u
n
ctio
n
;
=
[
1
2
.
.
.
]
[
1
2
.
.
.
]
ar
e
eig
e
n
v
ec
to
r
s
o
f
m
atr
ices
,
(
)
th
at
co
r
r
esp
o
n
d
to
ze
r
o
eig
en
v
alu
e.
Un
d
er
th
e
em
er
g
en
c
y
o
p
er
atin
g
co
n
d
itio
n
s
o
f
th
e
E
PS
,
f
o
r
e
x
am
p
le,
wh
en
,
as a
r
esu
lt o
f
a
p
o
wer
lin
e
s
h
u
td
o
wn
,
th
e
s
tab
ilit
y
r
eg
io
n
“n
ar
r
o
ws”
an
d
it
is
n
ec
ess
ar
y
to
en
s
u
r
e
th
at
th
e
o
p
er
atin
g
co
n
d
itio
n
s
ar
e
with
in
th
e
s
tab
ilit
y
r
eg
io
n
,
as
s
h
o
wn
in
Fig
u
r
e
2
,
s
u
ch
a
p
r
o
ce
d
u
r
e
ca
n
b
e
p
er
f
o
r
m
ed
alo
n
g
th
e
g
iv
en
p
ath
s
.
I
n
th
is
ca
s
e,
th
e
eq
u
atio
n
s
o
f
lim
it c
o
n
d
itio
n
s
ar
e
u
s
ed
,
as in
(
1
1
)
.
[
,
(
)
]
=
0
;
[
,
,
(
)
]
=
(
)
(
)
=
}
}
(1
1
)
W
h
er
e
=
0
+
is
a
v
ec
to
r
o
f
co
n
tr
o
lled
p
ar
am
eter
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2088
-
8
6
9
4
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
,
Vo
l.
16
,
No
.
1
,
Ma
r
c
h
20
25
:
138
-
150
142
T
h
e
m
o
d
elin
g
r
esu
lts
an
d
ca
lcu
latio
n
s
o
f
th
e
s
tab
ilit
y
m
ar
g
in
f
o
r
a
th
r
ee
-
n
o
d
e
s
ch
em
e,
as
s
h
o
wn
in
Fig
u
r
e
3
,
ar
e
p
r
esen
ted
i
n
T
ab
le
1
an
d
Fig
u
r
e
4
.
W
h
e
n
s
ettin
g
th
e
u
n
l
o
ad
in
g
d
ir
e
ctio
n
,
wh
ich
d
i
f
f
er
s
s
ig
n
if
ican
tly
f
r
o
m
th
e
s
h
o
r
tes
t
o
n
e
t
o
th
e
lim
it
s
u
r
f
ac
e,
it
b
ec
o
m
es
im
p
o
s
s
ib
le
to
r
ea
ch
th
e
s
tab
ilit
y
r
eg
io
n
p
ar
am
eter
s
.
I
n
th
is
ca
s
e,
th
e
u
n
lo
ad
in
g
p
o
in
t
in
s
p
ac
e
Y
c
an
b
e
ca
lc
u
lated
in
two
s
tag
es,
as
d
is
p
lay
ed
in
Fig
u
r
e
5
.
T
h
ey
a
r
e,
in
th
e
f
ir
s
t
s
tag
e,
th
e
lim
it
h
y
p
e
r
s
u
r
f
ac
e
is
r
ea
ch
ed
in
t
h
e
d
ir
ec
tio
n
Δ
Y
an
d
in
th
e
s
ec
o
n
d
s
tag
e,
u
n
lo
ad
in
g
is
p
er
f
o
r
m
e
d
in
th
e
d
ir
ec
tio
n
o
f
v
ec
to
r
R
,
wh
ich
co
in
cid
es with
th
e
d
ir
ec
tio
n
o
f
n
o
r
m
al
to
th
e
lim
it h
y
p
er
s
u
r
f
ac
e
.
Fig
u
r
e
2
.
“
Nar
r
o
win
g
”
o
f
th
e
s
tab
ilit
y
r
eg
io
n
wh
e
n
th
e
lin
e
is
d
is
co
n
n
ec
ted
1
2
3
D
G
1
Р
n
o
m
=
2
4
M
W
D
G
2
Р
n
o
m
=
2
4
M
W
E
P
S
O
hm
4
.
1
034
.
0
12
j
Z
+
=
Ohm
75
.
1
04
2
.
0
13
j
Z
+
=
O
h
m
1
.
2
05
.
0
23
j
Z
+
=
4
6
5
kV
6
=
1
U
kV
6
=
2
U
kV
3
.
6
=
3
U
Fig
u
r
e
3
.
Netwo
r
k
s
ch
em
e
Fig
u
r
e
4
.
E
n
s
u
r
in
g
t
h
e
o
p
e
r
atin
g
co
n
d
itio
n
s
with
in
th
e
s
tab
ilit
y
r
eg
io
n
al
o
n
g
th
e
g
iv
e
n
p
ath
s
T
ab
le
1
.
DG
p
o
wer
an
d
s
tab
ilit
y
m
ar
g
in
s
No
P
1
0
,
M
W
P
2
0
,
M
W
P
L1
,
M
W
P
L2
,
M
W
P
Z1
,
M
W
P
Z2
,
M
W
,
M
W
, %
1
15
15
9
.
2
3
9
.
2
3
8
.
3
5
8
.
3
5
1
.
2
4
15
2
15
15
1
0
.
4
0
8
.
1
0
9
.
7
5
7
.
1
3
1
.
1
7
15
3
15
15
8
.
1
0
1
0
.
4
0
7
.
1
3
9
.
7
5
1
.
1
7
15
N
o
t
e
:
ℑ
=
√
∑
(
−
)
2
2
=
1
;
ℑ
∗
=
100
√
∑
(
−
)
2
2
=
1
i
s s
t
a
b
i
l
i
t
y
mar
g
i
n
Vec
to
r
R
is
ca
lcu
lated
wh
en
s
o
lv
in
g
th
e
eq
u
atio
n
s
o
f
lim
i
t
co
n
d
itio
n
s
(
1
0
)
.
An
ex
a
m
p
l
e
o
f
s
u
ch
u
n
lo
ad
in
g
is
illu
s
tr
ated
in
Fi
g
u
r
e
5
.
I
t
is
co
n
v
e
n
ien
t
to
d
eter
m
in
e
th
e
m
ar
g
in
o
f
s
tatic
ap
er
io
d
ic
s
tab
ilit
y
th
r
o
u
g
h
th
e
n
o
r
m
o
f
th
e
v
ec
to
r
K,
co
r
r
esp
o
n
d
in
g
to
th
e
d
is
tan
ce
f
r
o
m
t
h
e
p
o
in
t
o
f
th
e
an
aly
ze
d
m
o
d
e
t
o
th
e
b
o
u
n
d
ar
y
L
F
(
1
2
)
:
ℑ
=
(
)
1
2
=
(
∑
2
=
1
)
1
2
=
(
∑
2
=
1
(
−
0
)
2
)
1
2
,
=
1
(1
2
)
*
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
Meth
o
d
s
fo
r
en
s
u
r
in
g
s
ta
b
ilit
y
o
f o
p
era
tin
g
co
n
d
itio
n
s
o
f a
n
elec
tr
ic
p
o
w
er sy
s
tem
w
ith
…
(
I
liya
I
liev
)
143
E
ac
h
d
ir
ec
tio
n
o
f
th
e
lo
a
d
in
c
r
ea
s
e
will
co
r
r
esp
o
n
d
to
its
v
alu
e
i
an
d
f
o
r
a
r
elia
b
le
ass
ess
m
en
t
o
f
th
e
s
tab
ilit
y
m
ar
g
in
,
it
is
n
ec
ess
ar
y
to
s
ea
r
ch
f
o
r
th
e
cr
itical
d
ir
ec
tio
n
o
f
lo
a
d
in
cr
ea
s
e
∗
(
)
=
0
+
∗
co
r
r
esp
o
n
d
in
g
to
th
e
least
len
g
th
ℑ
o
f
v
ec
to
r
K.
Stab
ilit
y
ca
n
b
e
p
r
o
v
id
ed
al
o
n
g
th
e
s
h
o
r
test
p
ath
b
y
m
o
d
if
y
in
g
th
e
eq
u
atio
n
s
o
f
lim
it
co
n
d
itio
n
s
.
T
h
is
m
o
d
if
ica
tio
n
aim
s
to
s
ea
r
ch
f
o
r
th
e
lim
it
co
n
d
itio
n
s
in
th
e
cr
itical
d
ir
ec
tio
n
o
f
lo
ad
in
c
r
e
ase
(
1
3
)
.
(
,
0
−
−
2
(
)
(
)
)
(
)
}
}
(1
3
)
W
h
er
e
0
is
th
e
v
alu
e
o
f
th
e
v
ec
to
r
o
f
co
n
tr
o
lled
p
ar
am
eter
s
in
th
e
o
p
er
atin
g
co
n
d
itio
n
s
;
=
[
1
2
…
]
is
th
e
v
ar
iab
le
in
cr
em
en
t
v
ec
to
r
0
en
ab
lin
g
th
e
o
p
er
atin
g
co
n
d
itio
n
s
to
r
ea
ch
th
e
lim
it
h
y
p
er
s
u
r
f
ac
e;
=
.
T
h
e
m
o
d
elin
g
s
h
o
ws
th
at
u
s
in
g
(
1
3
)
,
it
is
p
o
s
s
ib
le
to
en
s
u
r
e
th
e
E
PS
o
p
e
r
atin
g
co
n
d
itio
n
s
r
ea
c
h
in
g
th
e
s
tab
i
lity
r
eg
io
n
b
o
u
n
d
ar
y
alo
n
g
t
h
e
s
h
o
r
test
p
at
h
.
T
h
e
r
e
q
u
ir
ed
m
ar
g
in
ca
n
b
e
ac
h
iev
ed
b
y
ad
d
itio
n
al
u
n
l
o
ad
in
g
,
as d
is
p
lay
ed
i
n
Fig
u
r
e
6.
An
alg
o
r
ith
m
t
h
at
p
r
o
v
id
es
d
ir
ec
t
in
p
u
t
o
f
a
m
o
d
e
to
th
e
b
o
u
n
d
ar
y
o
f
an
ad
m
is
s
ib
le
r
eg
io
n
co
r
r
esp
o
n
d
in
g
to
a
g
iv
e
n
s
tab
ilit
y
m
ar
g
in
ca
n
b
e
im
p
lem
en
ted
b
ased
o
n
a
m
o
d
if
icatio
n
o
f
th
e
lim
it
m
o
d
e
eq
u
atio
n
s
as in
(
1
4
)
u
s
in
g
th
e
s
u
m
m
atio
n
o
f
th
e
d
ir
ec
t a
n
d
tr
an
s
p
o
s
ed
J
ac
o
b
ian
m
at
r
ices o
f
(
6
)
.
(
,
0
+
)
=
0
;
(
+
)
=
−
1
=
0
,
}
(1
4
)
W
h
er
e
is
th
e
m
in
im
u
m
eig
en
v
alu
e
o
f
m
atr
ix
=
+
;
is
th
e
ei
g
en
v
ec
to
r
c
o
r
r
esp
o
n
d
in
g
to
th
e
eig
en
v
alu
e
o
f
m
atr
i
x
А.
Af
ter
s
im
p
le
tr
an
s
f
o
r
m
atio
n
s
,
we
ca
n
wr
ite
(
1
5
)
.
(
,
0
+
)
=
0
;
+
=
−
1
=
0
.
}
(
1
5
)
T
h
e
ca
lcu
latio
n
r
esu
lts
th
at
d
em
o
n
s
tr
ate
an
ex
a
m
p
le
o
f
en
s
u
r
in
g
s
tab
ilit
y
o
f
th
e
o
p
e
r
atin
g
co
n
d
itio
n
s
u
s
in
g
(
1
4
)
a
r
e
p
r
esen
ted
in
Fig
u
r
e
7
an
d
T
ab
le
2
.
T
ab
le
2
.
DG
p
o
wer
No
P
1
0
,
M
W
P
2
0
,
M
W
P
Z1
,
M
W
P
Z2
,
M
W
1
18
18
4
1
0
.
5
2
18
18
7
.
3
7
.
3
3
18
18
1
0
.
5
4
F
i
g
u
r
e
5
.
U
s
e
o
f
v
e
c
t
o
r
R
t
o
e
n
s
u
r
e
t
h
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e
r
a
t
i
n
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c
o
n
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i
t
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o
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r
e
wi
t
h
i
n
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h
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a
b
i
l
it
y
r
e
g
i
o
n
a
l
o
n
g
g
i
v
e
n
p
a
t
h
s
Fig
u
r
e
6
.
E
n
s
u
r
in
g
t
h
e
o
p
e
r
atin
g
co
n
d
itio
n
s
ar
e
with
in
th
e
s
tab
ilit
y
r
eg
io
n
al
o
n
g
th
e
s
h
o
r
test
p
ath
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2088
-
8
6
9
4
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
,
Vo
l.
16
,
No
.
1
,
Ma
r
c
h
20
25
:
138
-
150
144
Fig
u
r
e
7
.
E
n
s
u
r
in
g
t
h
e
o
p
e
r
atin
g
co
n
d
itio
n
s
ar
e
with
in
t
h
e
s
tab
ilit
y
r
eg
io
n
u
s
in
g
(
1
4
)
:
4.
M
E
T
H
O
DS
O
F
SO
L
V
I
NG
T
H
E
F
AR
B
O
UNDA
R
Y
P
RO
B
L
E
M
WH
E
N
E
NSU
RING
T
H
A
T
T
H
E
O
P
E
R
AT
I
NG
CO
NDI
T
I
O
NS
L
I
E
WI
T
H
I
N
T
H
E
ST
A
B
I
L
I
T
Y
R
E
G
I
O
N
T
o
o
b
tain
a
r
eliab
le
s
o
lu
tio
n
u
s
in
g
th
e
ab
o
v
e
-
d
escr
ib
e
d
alg
o
r
ith
m
s
,
th
e
in
itial
ap
p
r
o
x
im
a
tio
n
s
m
u
s
t
b
e
s
p
ec
if
ied
n
ea
r
th
e
lim
it
h
y
p
er
s
u
r
f
ac
e
th
at
b
o
u
n
d
s
th
e
SAS
ar
ea
.
I
n
itial
ap
p
r
o
x
im
atio
n
s
th
at
en
s
u
r
e
r
eliab
le
co
n
v
er
g
en
ce
o
f
iter
ativ
e
p
r
o
c
ess
es
ca
n
b
e
o
b
tain
ed
b
ased
o
n
th
e
s
tar
tin
g
alg
o
r
ith
m
s
d
escr
ib
ed
in
[
3
]
.
I
n
ad
d
itio
n
,
th
er
e
is
a
p
r
o
b
lem
with
r
ea
ch
in
g
th
e
“f
ar
b
o
u
n
d
ar
y
”
o
f
th
e
s
tab
ilit
y
r
e
g
io
n
:
th
e
p
o
i
n
t
(
)
in
Fig
u
r
e
s
8
(
a)
an
d
8
(
b
)
.
I
n
th
is
ca
s
e,
th
e
r
esu
ltin
g
s
o
lu
tio
n
,
w
h
ich
d
if
f
er
s
in
t
h
e
r
ev
er
s
e
o
f
th
e
s
ig
n
s
o
f
p
o
wer
in
jectio
n
s
,
ca
n
n
o
t
b
e
u
s
ed
in
p
r
ac
tice.
T
h
is
p
r
o
b
lem
is
s
o
lv
ed
b
y
th
e
m
eth
o
d
p
r
o
p
o
s
ed
in
[
2
]
,
wh
ich
is
ap
p
licab
le
o
n
l
y
in
s
p
ec
ial
ca
s
es.
(
a)
(
b
)
Fig
u
r
e
8
.
T
o
th
e
“f
a
r
b
o
u
n
d
ar
y
"
p
r
o
b
lem
:
(
a
)
w
h
e
n
d
eter
m
i
n
in
g
th
e
lim
it c
o
n
d
itio
n
s
an
d
(
b
)
w
h
en
en
s
u
r
in
g
th
at
o
p
er
atin
g
co
n
d
itio
n
s
ar
e
with
in
th
e
s
tab
ilit
y
r
eg
io
n
T
o
p
r
ev
e
n
t
th
e
co
m
p
u
tatio
n
al
p
r
o
ce
s
s
f
r
o
m
r
ea
c
h
in
g
t
h
e
“f
ar
b
o
u
n
d
ar
y
”,
s
tar
tin
g
alg
o
r
ith
m
s
ca
n
b
e
u
s
ed
th
at
tak
e
in
to
ac
co
u
n
t
th
e
n
o
n
lin
ea
r
ter
m
s
o
f
th
e
T
ay
l
o
r
s
er
ies
ex
p
an
s
io
n
o
f
th
e
r
esid
u
al
v
ec
to
r
f
u
n
ctio
n
o
r
ar
e
b
ased
o
n
m
in
im
izin
g
th
e
f
o
llo
win
g
f
u
n
ctio
n
al
as in
(
1
6
)
[
4
1
]
.
ℵ
(
)
=
(
,
0
)
(
,
0
)
(1
6
)
I
n
th
is
ca
s
e,
0
s
h
o
u
ld
b
e
s
et
s
o
t
h
at
th
e
co
r
r
esp
o
n
d
in
g
p
o
i
n
t
in
th
e
s
p
ac
e
o
f
c
o
n
tr
o
lled
p
ar
a
m
eter
s
lies
o
u
ts
id
e
th
e
s
tab
ilit
y
r
eg
io
n
.
T
h
is
co
n
d
itio
n
is
m
et
in
itially
wh
en
th
e
co
n
tr
o
l
ac
tio
n
s
o
f
em
e
r
g
en
c
y
co
n
tr
o
l
s
y
s
tem
s
ar
e
ch
o
s
en
,
a
s
s
ee
n
in
Fig
u
r
e
8
(
b
)
.
I
n
th
e
ca
s
e
s
h
o
wn
in
Fig
u
r
e
8
(
a
)
,
s
u
ch
a
p
o
in
t
0
(
∗
)
is
d
eter
m
in
ed
b
ased
o
n
th
e
f
o
llo
win
g
r
elatio
n
as in
(
1
7
)
.
0
(
∗
)
=
0
+
0
(
1
7
)
(
)
db
L
Y
(
)
nb
L
Y
i
y
j
y
0
Y
(
)
nb
L
Y
(
)
db
L
Y
i
y
j
y
0
Y
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
Meth
o
d
s
fo
r
en
s
u
r
in
g
s
ta
b
ilit
y
o
f o
p
era
tin
g
co
n
d
itio
n
s
o
f a
n
elec
tr
ic
p
o
w
er sy
s
tem
w
ith
…
(
I
liya
I
liev
)
145
T
h
e
u
s
e
o
f
p
a
r
a
m
et
e
r
s
X
c
alc
u
la
te
d
b
y
m
i
n
i
m
i
zi
n
g
t
h
e
f
u
n
cti
o
n
al
(
1
6
)
as
i
n
i
tial
a
p
p
r
o
x
i
m
at
io
n
s
t
o
s
o
l
v
e
t
h
e
eq
u
a
ti
o
n
s
o
f
li
m
i
t
c
o
n
d
iti
o
n
s
a
n
d
t
h
ei
r
m
o
d
i
f
i
ca
t
io
n
s
en
s
u
r
es
r
el
ia
b
l
e
c
o
n
v
er
g
en
ce
t
o
th
e
d
esir
e
d
p
o
i
n
ts
(
)
ly
i
n
g
o
n
t
h
e
"
n
e
ar
"
b
o
u
n
d
a
r
i
es
o
f
t
h
e
s
ta
b
i
lit
y
r
eg
io
n
,
as
s
ee
n
i
n
F
ig
u
r
e
2
.
P
o
i
n
t
(
)
i
n
t
h
e
Fi
g
u
r
e
is
o
b
t
ai
n
e
d
u
s
i
n
g
a
"
f
l
at
s
ta
r
t
"
(
n
o
m
i
n
a
l
v
o
lta
g
es
a
n
d
ze
r
o
a
n
g
l
es)
.
Desi
r
e
d
p
o
i
n
t
(
)
is
f
o
u
n
d
b
y
s
o
l
v
i
n
g
th
e
(1
8
)
.
[
,
0
−
−
2
(
)
(
1
+
)
[
]
;
]
(
,
)
=
(
)
;
(
)
=
(
)
−
2
(
)
1
2
(1
8
)
W
h
e
n
u
s
i
n
g
i
n
i
tial
a
p
p
r
o
x
i
m
at
io
n
s
b
as
ed
o
n
t
h
e
m
i
n
i
m
i
za
t
io
n
o
f
t
h
e
f
u
n
c
ti
o
n
al
(
1
6
).
F
o
r
t
h
e
s
t
ar
t
in
g
al
g
o
r
it
h
m
,
y
o
u
c
an
u
s
e
t
h
e
al
g
o
r
it
h
m
o
f
V
.
A
.
Ma
tv
ee
v
,
w
h
i
ch
i
m
p
le
m
e
n
ts
t
h
e
f
o
ll
o
wi
n
g
i
te
r
at
iv
e
p
r
o
ce
d
u
r
e
(
1
9
)
.
(
+
1
)
=
(
)
−
(
)
[
(
(
)
)
]
−
1
(
(
)
)
(
19)
W
h
er
e
(
)
is
ca
lcu
lated
lik
e
th
is
(
2
0
)
.
(
)
=
{
1
,
>
1
1
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≤
1
;
=
1
2
|
(
(
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|
|
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∑
2
(
(
)
)
(
)
(
)
(
)
(
)
|
(
2
0
)
Pro
ce
d
u
r
e
(
1
9
)
en
s
u
r
es
th
e
c
alcu
latio
n
o
f
a
n
y
ex
is
tin
g
m
o
d
es
th
at
m
ee
t
th
e
co
n
d
itio
n
>
0
.
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h
en
tr
y
in
g
to
d
eter
m
in
e
u
n
s
tab
le
m
o
d
es
(
at
<
0
)
,
th
e
c
o
m
p
u
tatio
n
al
p
r
o
ce
s
s
“g
ets
s
tu
ck
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th
e
b
o
u
n
d
ar
y
p
o
i
n
t
L
F
,
c
o
r
r
esp
o
n
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in
g
to
th
e
g
iv
en
weig
h
tin
g
tr
aj
ec
to
r
y
a
n
d
co
n
d
itio
n
=
0
.
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r
th
e
s
tar
tin
g
alg
o
r
ith
m
,
y
o
u
ca
n
u
s
e
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ite
r
ativ
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r
o
ce
s
s
th
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o
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n
ea
r
ter
m
s
in
th
e
T
a
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r
s
er
i
es
ex
p
an
s
io
n
o
f
th
e
v
ec
to
r
f
u
n
ctio
n
=
(
)
in
v
er
s
e
to
(
)
;
in
th
is
ca
s
e,
th
e
v
ec
to
r
o
f
th
e
r
eq
u
ir
ed
p
ar
a
m
eter
s
is
r
e
p
r
esen
ted
in
th
e
f
o
r
m
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
:
2088
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8
6
9
4
I
n
t J Po
w
E
lec
&
Dr
i Sy
s
t
,
Vo
l.
16
,
No
.
1
,
Ma
r
c
h
20
25
:
138
-
150
146
J
ac
o
b
ia
n
m
a
tr
ix
o
f
s
y
s
te
m
s
(
1
1
)
,
(
1
3
)
a
n
d
(
1
5
)
a
t
th
e
s
o
l
u
t
i
o
n
p
o
i
n
t.
H
o
w
ev
e
r
,
a
cc
o
r
d
i
n
g
t
o
cr
ite
r
i
o
n
3
,
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t
is
in
f
e
r
i
o
r
t
o
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h
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d
is
cr
ete
w
ei
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n
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m
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h
o
d
.
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h
is
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d
u
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ct
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a
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n
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r
est
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ic
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1
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(
1
3
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,
(
1
5
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ca
n
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th
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n
u
m
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e
r
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f
ite
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n
s
a
n
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n
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.
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im
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n
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d
r
aw
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p
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s
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o
d
o
t
h
is
,
t
h
e
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im
it
m
o
d
e
is
ca
l
cu
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t
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n
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a
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m
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h
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d
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23
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r
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f
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d
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l
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ep
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r
o
b
l
em
.
(
m
in
)
≤
≤
(
max
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.
(
23
)
T
ab
le
3
.
R
esu
lts
o
f
co
m
p
a
r
ativ
e
an
aly
s
is
M
e
t
h
o
d
C
r
i
t
e
r
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a
1
2
3
D
i
scret
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w
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s
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s
No
C
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s we
i
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n
s
No
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Y
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s
5.
M
O
DE
L
I
NG
R
E
SU
L
T
S
F
O
R
E
M
E
RG
E
NCY
CO
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RO
L
SYS
T
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M
O
F
A
DI
ST
R
I
B
U
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D
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NE
R
AT
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N
P
L
ANT
I
N
P
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ST
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M
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RG
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NCY
CO
NDIT
I
O
NS
WI
T
H
AP
P
RO
P
RIAT
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DYNA
M
I
C
CO
ND
I
T
I
O
NS
E
NSUR
E
D
Nu
m
er
o
u
s
c
o
m
p
u
tatio
n
al
ex
p
er
im
en
ts
in
d
icate
th
at
b
ased
o
n
th
e
eq
u
atio
n
s
o
f
lim
it
co
n
d
itio
n
s
,
th
e
o
p
er
atin
g
co
n
d
itio
n
s
o
f
E
PS
with
DG
p
lan
ts
ca
n
r
ea
ch
th
e
b
o
u
n
d
ar
y
o
f
t
h
e
s
tab
ilit
y
r
e
g
io
n
.
T
o
ac
h
ie
v
e
th
e
r
eq
u
ir
ed
s
tab
ilit
y
m
ar
g
in
,
ad
d
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u
n
lo
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is
r
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.
Un
lo
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(
)
s
h
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ed
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p
tab
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y
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am
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p
r
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s
s
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wh
ich
ca
n
b
e
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with
AE
R
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d
ASR
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n
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s
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s
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f
th
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.
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m
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f
DG
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in
t
h
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-
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en
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n
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itio
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s
.
A
t th
e
s
am
e
tim
e,
f
o
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th
e
o
p
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atin
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itio
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to
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with
in
th
e
s
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eg
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,
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g
en
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s
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R
an
d
AS
R
ar
e
ad
ju
s
ted
.
T
h
e
two
-
s
tep
p
r
o
ce
d
u
r
e
u
s
in
g
th
e
v
ec
to
r
R
is
in
ten
d
ed
o
n
ly
to
f
i
n
d
th
e
p
o
in
t
(
)
.
T
h
e
d
y
n
a
m
ic
tr
an
s
itio
n
is
m
ad
e
d
ir
ec
tly
to
t
h
is
p
o
in
t
,
as
r
ev
ea
led
in
Fig
u
r
e
9
.
T
o
en
s
u
r
e
h
ig
h
q
u
ality
o
f
th
is
tr
an
s
itio
n
,
y
o
u
ca
n
u
s
e
o
p
tim
izatio
n
o
f
th
e
AE
R
an
d
ASR
s
e
ttin
g
s
[
2
3
]
.
T
h
e
m
o
d
elin
g
was
p
er
f
o
r
m
ed
f
o
r
th
e
n
etwo
r
k
s
ch
em
e
s
h
o
w
n
in
Fig
u
r
e
3
.
T
h
e
s
h
u
td
o
w
n
o
f
lin
e
1
–
3
was
co
n
s
id
er
ed
as
em
er
g
en
c
y
co
n
d
itio
n
s
.
E
n
s
u
r
in
g
th
e
s
tab
ilit
y
o
f
p
o
s
t
-
em
er
g
en
cy
o
p
er
atin
g
co
n
d
itio
n
s
alo
n
g
a
g
iv
e
n
p
ath
is
illu
s
tr
at
ed
in
Fig
u
r
es
1
0
(
a)
an
d
1
0
(
b
)
.
T
h
e
p
o
in
t
with
c
o
o
r
d
i
n
ates
0
=
[
20
20
]
is
co
n
s
id
er
ed
as
in
itial
lo
a
d
in
g
c
o
n
d
itio
n
s
o
f
DG
p
lan
t
g
en
e
r
ato
r
s
.
E
n
s
u
r
in
g
th
at
t
h
e
o
p
er
atin
g
co
n
d
itio
n
s
r
ea
ch
th
e
b
o
u
n
d
ar
y
o
f
t
h
e
s
tab
ilit
y
r
eg
io
n
alo
n
g
a
g
iv
en
p
ath
is
r
ep
r
esen
ted
b
y
p
o
in
t
2
=
[
15
.
8
13
.
6
]
.
Dy
n
am
ic
p
r
o
ce
s
s
es
in
th
e
s
ch
em
e
u
n
d
e
r
s
tu
d
y
,
as
s
ee
n
in
Fig
u
r
e
3
ar
e
m
o
d
elled
i
n
th
e
MA
T
L
AB
s
y
s
tem
,
co
n
s
id
er
in
g
th
e
AE
R
an
d
ASR
,
wh
ich
ar
e
d
esc
r
ib
ed
in
[
2
3
]
.
Usi
n
g
th
e
m
et
h
o
d
o
f
c
o
o
r
d
in
ated
ad
ju
s
tm
en
t
o
f
th
e
AE
R
an
d
A
SR
s
ettin
g
s
[
2
3
]
,
th
eir
p
ar
am
e
ter
s
wer
e
d
eter
m
in
e
d
f
o
r
th
r
ee
k
in
d
s
o
f
o
p
er
atin
g
co
n
d
itio
n
s
o
f
th
e
g
en
e
r
ato
r
(
m
in
im
u
m
,
av
er
a
g
e,
an
d
m
a
x
im
u
m
lo
ad
)
.
B
ased
o
n
th
e
o
b
t
ain
ed
s
ettin
g
s
o
f
th
e
r
eg
u
lato
r
s
f
o
r
th
e
co
n
s
id
er
e
d
o
p
er
atin
g
co
n
d
itio
n
s
,
th
e
r
u
le
b
ase
o
f
th
e
f
u
zz
y
co
n
tr
o
l
s
y
s
tem
was
cr
ea
ted
[
2
3
]
.
I
n
th
e
ev
en
t
t
h
at
o
n
e
o
f
th
e
c
o
n
n
ec
tio
n
s
is
d
is
co
n
n
ec
ted
d
u
r
in
g
th
e
o
p
e
r
atio
n
o
f
DG
p
lan
ts
,
th
e
s
y
s
tem
lo
s
es
s
tab
ilit
y
.
T
o
en
s
u
r
e
th
e
PEOC
s
tab
ilit
y
,
it
is
n
ec
es
s
ar
y
to
u
n
lo
ad
th
e
g
en
e
r
ato
r
s
.
I
n
th
is
ca
s
e
th
e
co
o
r
d
in
ated
ad
ju
s
tm
en
t
o
f
th
e
AE
R
an
d
A
SR
s
ettin
g
s
an
d
th
eir
ch
an
g
e
b
y
f
u
zz
y
co
n
tr
o
l
s
y
s
tem
ca
n
i
m
p
r
o
v
e
th
e
q
u
ality
in
d
icato
r
s
o
f
tr
a
n
s
ien
ts
.
T
h
e
g
r
ea
test
ef
f
ec
t
is
o
b
s
er
v
ed
f
o
r
th
e
v
o
ltag
e
o
f
th
e
g
e
n
er
ato
r
s
,
an
d
f
o
r
th
e
p
ar
am
eter
s
o
f
th
e
g
e
n
er
ato
r
h
av
in
g
a
d
ir
ec
t
co
n
n
ec
tio
n
with
th
e
d
is
co
n
n
ec
ted
lin
e.
C
o
r
r
esp
o
n
d
in
g
o
s
cillo
g
r
am
s
o
f
v
o
ltag
e,
d
e
v
i
atio
n
s
o
f
r
o
t
o
r
s
p
ee
d
,
an
d
p
o
wer
o
f
th
e
DG
p
lan
t
g
en
e
r
ato
r
s
in
ca
s
e
o
f
a
s
h
o
r
t
cir
cu
it a
n
d
d
is
co
n
n
ec
tio
n
o
f
li
n
e
1
-
3
b
y
t
h
e
r
elay
p
r
o
tectio
n
ar
e
s
h
o
wn
in
Fig
u
r
es 1
1
(
a
)
an
d
1
1
(
b
)
.
Fo
r
th
e
v
o
ltag
e
o
f
th
e
g
en
er
a
to
r
o
f
th
e
DG1
p
lan
t,
th
e
u
s
e
o
f
a
f
u
zz
y
s
y
s
tem
f
o
r
co
n
t
r
o
llin
g
th
e
s
ettin
g
s
o
f
th
e
r
eg
u
lato
r
s
m
ad
e
it
p
o
s
s
ib
le
to
r
ed
u
ce
th
e
tim
e
o
f
th
e
tr
an
s
ien
t
p
r
o
ce
s
s
b
y
4
tim
es,
as
in
Fig
u
r
e
1
1
(
a)
,
an
d
f
o
r
th
e
DG2
p
la
n
t
-
b
y
2
.
3
tim
es,
as in
Fig
u
r
e
1
1
(
b
)
.
T
h
is
also
m
ad
e
it p
o
s
s
ib
le
to
elim
in
ate
v
o
ltag
e
f
lu
ctu
atio
n
s
o
f
DG
p
lan
ts
in
tr
an
s
ien
t
m
o
d
e.
Fo
r
DG1
,
th
e
m
ax
im
u
m
d
ev
iatio
n
o
f
th
e
g
e
n
er
ato
r
r
o
t
o
r
s
p
ee
d
f
r
o
m
t
h
e
s
tead
y
v
alu
e
is
r
ed
u
ce
d
d
u
e
to
th
e
u
s
e
o
f
f
u
zz
y
c
o
n
tr
o
l
o
f
t
h
e
co
n
tr
o
ller
s
ettin
g
s
.
Fo
r
th
e
p
o
wer
o
f
th
e
g
en
e
r
ato
r
o
f
th
e
DG1
p
lan
t,
a
d
ec
r
ea
s
e
in
th
e
am
o
u
n
t
o
f
o
v
e
r
r
eg
u
latio
n
b
y
3
0
%
is
als
o
o
b
s
er
v
ed
(
Fig
u
r
e
1
1
(
a)
)
wh
en
u
s
in
g
th
is
s
y
s
tem
.
T
h
e
r
esu
lts
o
b
tai
n
ed
d
ete
r
m
in
e
th
e
ef
f
ec
tiv
e
n
ess
o
f
u
s
in
g
a
f
u
zz
y
s
y
s
tem
co
n
tr
o
l
an
d
a
u
to
m
atic
co
n
tr
o
l sy
s
tem
s
o
f
g
en
er
ato
r
s
o
f
DG
p
lan
ts
,
wh
ich
p
r
o
v
id
es
a
b
etter
d
y
n
am
ic
tr
an
s
itio
n
wh
en
th
e
r
e
g
im
e
en
ter
s
th
e
s
ta
b
ilit
y
r
eg
io
n
.
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
Meth
o
d
s
fo
r
en
s
u
r
in
g
s
ta
b
ilit
y
o
f o
p
era
tin
g
co
n
d
itio
n
s
o
f a
n
elec
tr
ic
p
o
w
er sy
s
tem
w
ith
…
(
I
liya
I
liev
)
147
1
2
0
d
e
t
=
X
F
c
o
n
s
t
=
М
В
т
,
i
P
М
В
т
,
j
P
0
Y
(
)
1
L
Y
(
)
R
Y
Z
R
P
i
,
M
W
P
j
,
M
W
3
Fig
u
r
e
9
.
Ad
d
itio
n
al
u
n
lo
ad
in
g
in
th
e
d
ir
ec
tio
n
o
f
v
ec
to
r
R
(
a)
(
b
)
Fig
u
r
e
1
0
.
E
n
s
u
r
in
g
th
at
o
p
er
a
tin
g
co
n
d
itio
n
s
ar
e
with
in
th
e
s
tab
ilit
y
r
eg
io
n
: (
a)
b
o
u
n
d
ar
ie
s
o
f
th
e
s
tab
ilit
y
r
eg
io
n
a
n
d
(
b
)
in
c
r
ea
s
ed
s
ca
le
o
f
tr
an
s
itio
n
to
t
h
e
ar
ea
o
f
s
u
s
tain
ab
ilit
y
(
a)
(
b
)
Fig
u
r
e
1
1
.
Oscill
o
g
r
am
s
o
f
v
o
ltag
e,
r
o
to
r
s
p
ee
d
g
e
n
er
ato
r
,
a
n
d
p
o
we
r
o
f
DG
p
lan
ts
in
ca
s
e
o
f
a
s
h
o
r
t c
ir
cu
it
an
d
d
is
co
n
n
ec
tio
n
o
f
lin
e
1
-
3:
(a
)
w
ith
n
o
ch
an
g
es in
th
e
AE
R
an
d
ASR
s
ettin
g
s
(
f
o
r
DG1
)
an
d
(
b
)
f
u
zz
y
co
n
tr
o
l o
f
AE
R
an
d
ASR
s
etti
n
g
s
(
f
o
r
DG2
)
Evaluation Warning : The document was created with Spire.PDF for Python.