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n
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d
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n
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c
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t
u
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y
.
2.
M
E
T
H
O
D
2
.
1
.
Sy
s
t
e
m
des
cr
iptio
n a
nd
m
o
delin
g
I
n
th
is
s
ec
tio
n
,
we
will
ex
p
lo
r
e
th
e
d
escr
ip
tio
n
an
d
m
o
d
elin
g
o
f
an
HVDC
s
y
s
tem
,
f
o
cu
s
in
g
s
p
ec
if
ically
o
n
th
e
d
y
n
a
m
ics
o
f
th
e
v
o
ltag
e
s
o
u
r
ce
co
n
v
er
t
er
(
VSC
)
.
Fig
u
r
e
1
illu
s
tr
ates
a
two
-
lev
el
VSC
-
HVDC
tr
an
s
m
i
s
s
io
n
s
y
s
tem
,
wh
ich
in
clu
d
es
an
AC
g
r
id
c
o
n
n
ec
ted
to
a
c
o
n
v
e
r
ter
th
r
o
u
g
h
a
p
h
ase
r
ea
cto
r
.
T
h
is
s
y
s
tem
co
m
p
r
is
es
two
c
o
n
v
er
ter
s
:
o
n
e
f
u
n
ctio
n
i
n
g
as
a
r
ec
tifie
r
a
n
d
t
h
e
o
t
h
er
as
an
in
v
er
te
r
.
T
h
ese
co
n
v
er
ter
s
ar
e
in
ter
co
n
n
ec
ted
v
ia
a
len
g
th
y
HVDC c
ab
le.
Fig
u
r
e
1
.
VSC
-
HVDC tr
an
s
m
i
s
s
io
n
s
y
s
tem
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
I
n
teg
r
a
l b
a
ck
s
tep
p
i
n
g
co
n
tr
o
l
d
esig
n
fo
r
en
h
a
n
ce
d
s
ta
b
ilit
y
a
n
d
d
y
n
a
mic
…
(
C
h
a
ima
a
La
k
h
d
a
ir
i
)
257
2
.
1
.
1
.
M
a
t
hem
a
t
ic
a
l m
o
del
T
h
e
d
-
q
tr
a
n
s
f
o
r
m
atio
n
is
em
p
lo
y
ed
in
o
r
d
er
to
s
tr
ea
m
lin
e
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e
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aly
s
is
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d
co
n
tr
o
l
o
f
t
h
e
d
y
n
am
ics
o
f
th
e
VSC
-
HVDC
s
y
s
tem
[
2
1
]
.
T
h
e
m
o
d
el
o
f
t
h
e
r
ec
tifie
r
s
tatio
n
(
VSC
1
)
in
th
e
r
o
tatin
g
d
-
q
r
ef
e
r
en
ce
f
r
am
e
is
ex
p
r
ess
ed
as
(
1
)
an
d
(
2
)
.
1
=
−
1
1
1
+
1
1
+
1
1
1
−
1
1
1
(
1
)
1
=
−
1
1
1
−
1
1
+
1
1
1
−
1
1
1
(
2
)
T
h
e
m
ath
em
atica
l r
ep
r
esen
tatio
n
o
f
th
e
in
v
e
r
ter
s
tatio
n
(
VS
C
2
)
in
th
e
d
-
q
r
e
f
er
en
ce
f
r
am
e
is
as
(
3
)
an
d
(
4
)
.
2
=
−
2
2
2
+
2
2
+
1
2
2
−
1
2
2
(
3
)
2
=
−
2
2
2
−
2
2
+
1
2
2
−
1
2
2
(
4
)
T
h
e
eq
u
iv
ale
n
t e
lectr
ical
m
o
d
el
o
f
th
e
HVDC c
ab
le
an
d
its
m
ath
em
atica
l r
ep
r
esen
tatio
n
is
d
escr
ib
ed
b
y
:
1
=
1
.
1
.
1
−
(
5
)
2
=
−
+
(
6
)
=
−
−
1
+
2
(
7
)
,
,
an
d
,
ar
e
th
e
VS
C
in
p
u
t
v
o
ltag
e
in
th
e
Par
k
t
r
an
s
f
o
r
m
a
n
d
th
e
AC
n
etwo
r
k
v
o
ltag
e
(
k
=1
.
2
)
,
r
esp
ec
tiv
ely
.
an
d
ar
e
th
e
AC
n
etwo
r
k
cu
r
r
en
ts
in
th
e
Par
k
t
r
an
s
f
o
r
m
(
k
=
1
.
2
)
.
C
is
th
e
eq
u
i
v
alen
t
ca
p
ac
ito
r
o
f
th
e
DC
lin
k
,
wh
il
e
an
d
ar
e
th
e
r
esis
tan
ce
an
d
i
n
d
u
ctan
ce
o
f
t
h
e
p
h
ase
r
ea
cto
r
,
r
esp
ec
tiv
ely
(
k
=1
.
2
)
.
an
d
ar
e
th
e
r
esis
t
an
ce
an
d
in
d
u
ctan
ce
o
f
th
e
DC
lin
e,
r
esp
ec
tiv
ely
.
C
o
n
s
id
er
in
g
th
at
th
e
r
ef
er
en
ce
o
f
th
e
s
y
s
tem
d
q
0
is
ch
o
s
en
s
o
th
at
its
d
ax
is
is
in
p
h
ase
with
th
e
v
o
ltag
e
,
=
0
.
2
.
1
.
2
.
Av
er
a
g
e
mo
del
T
h
e
in
s
tan
tan
eo
u
s
m
o
d
el
p
r
o
v
id
es
a
d
etailed
r
ep
r
esen
tatio
n
o
f
th
e
s
y
s
tem
'
s
b
eh
av
io
r
at
e
v
er
y
in
s
tan
t.
Ho
wev
er
,
th
is
ap
p
r
o
ac
h
is
n
o
t
co
n
d
u
civ
e
to
d
ir
ec
tly
d
esig
n
i
n
g
a
co
n
tr
o
l
law.
Fo
r
th
at
p
u
r
p
o
s
e,
th
e
co
m
m
o
n
ap
p
r
o
ac
h
is
to
r
eso
r
t
to
av
er
a
g
e
m
o
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els,
wh
er
e
t
h
e
av
er
a
g
e
v
alu
es
o
f
th
e
v
ar
iab
les
r
ep
lace
t
h
eir
in
s
tan
tan
eo
u
s
v
alu
es
in
th
e
m
o
d
el.
T
h
e
s
tate
v
ar
iab
les
ar
e
d
ef
in
ed
as
f
o
ll
o
ws:
[
1
,
1
,
1
,
2
,
2
]
=
[
1
,
2
,
3
,
4
,
5
]
.
T
h
e
av
er
a
g
e
r
ep
r
esen
tatio
n
o
f
th
e
s
y
s
tem
,
o
b
tain
ed
f
r
o
m
th
e
s
y
s
tem
m
o
d
el,
is
d
escr
ib
ed
b
y
(
8
)
-
(
1
2
)
.
̇
1
=
1
.
1
2
−
1
(
8
)
̇
2
=
−
2
+
1
3
+
1
1
−
1
1
(
9
)
̇
3
=
−
3
−
1
2
−
1
1
(
1
0
)
̇
4
=
−
4
+
2
5
−
1
2
+
1
2
(
1
1
)
̇
5
=
−
5
−
2
4
+
1
2
(1
2
)
2
.
2
.
Co
ntr
o
ller
des
ig
n
I
n
th
is
p
ap
er
,
o
u
r
o
b
jectiv
e
is
to
d
ev
elo
p
a
n
o
n
lin
ea
r
co
n
tr
o
l
s
tr
ateg
y
b
ased
o
n
b
ac
k
s
tep
p
in
g
co
n
tr
o
l
f
o
r
VSC
-
HVDC
tr
an
s
m
is
s
io
n
s
y
s
tem
s
.
T
h
is
ap
p
r
o
ac
h
aim
s
to
s
im
u
ltan
eo
u
s
ly
ac
h
iev
e
th
e
f
o
llo
win
g
two
co
n
tr
o
l
o
b
jectiv
es:
i)
c
o
n
tr
o
lli
n
g
th
e
DC
v
o
ltag
e
to
tr
ac
k
its
r
ef
er
en
ce
an
d
th
e
ac
tiv
e
p
o
w
er
at
s
tatio
n
2
an
d
ii)
e
n
s
u
r
in
g
a
s
in
u
s
o
id
al
g
r
id
c
u
r
r
en
t
th
at
is
i
n
p
h
ase
with
t
h
e
g
r
id
v
o
ltag
e
b
y
s
tr
ateg
ically
a
d
d
r
ess
in
g
r
ea
ctiv
e
p
o
wer
at
b
o
th
s
tatio
n
s
.
B
y
ad
d
r
ess
in
g
th
ese
o
b
jectiv
es,
th
e
p
r
o
p
o
s
ed
n
o
n
lin
ea
r
co
n
tr
o
l
s
tr
ateg
y
is
ex
p
ec
ted
to
p
r
o
v
id
e
a
m
o
r
e
e
f
f
ec
tiv
e
a
n
d
e
f
f
icien
t
s
o
lu
tio
n
f
o
r
VSC
-
HVDC
tr
an
s
m
is
s
io
n
s
y
s
tem
s
,
im
p
r
o
v
in
g
th
eir
o
v
er
all
p
er
f
o
r
m
an
ce
an
d
s
tab
ilit
y
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
1
4
,
No
.
2
,
J
u
n
e
20
2
5
:
255
-
26
3
258
2
.
2
.
1
.
I
nte
g
ra
l
ba
ck
s
t
ep
pin
g
co
ntr
o
l
I
n
teg
r
al
b
ac
k
s
tep
p
in
g
co
n
tr
o
l
is
a
h
ig
h
ly
ef
f
ec
tiv
e
n
o
n
lin
e
ar
co
n
tr
o
l
tech
n
iq
u
e
th
at
co
m
b
in
es
th
e
b
ac
k
s
tep
p
in
g
m
eth
o
d
with
i
n
teg
r
al
ac
tio
n
in
o
r
d
er
to
attain
r
o
b
u
s
t
tr
ac
k
in
g
p
er
f
o
r
m
an
ce
an
d
elim
in
ate
s
tead
y
-
s
tate
er
r
o
r
s
[
2
2
]
,
[
2
3
]
.
T
h
e
f
u
n
d
am
e
n
tal
co
n
ce
p
t
b
eh
in
d
t
h
is
tech
n
iq
u
e
is
to
au
g
m
en
t
th
e
p
lan
t
d
y
n
am
ics
with
an
in
teg
r
al
s
tate,
wh
ich
in
cr
ea
s
es
th
e
v
ec
to
r
r
elativ
e
d
eg
r
ee
an
d
n
ec
ess
itat
es
ad
d
itio
n
al
b
ac
k
s
tep
p
in
g
s
tep
s
[
2
4
]
.
T
h
is
m
eth
o
d
em
p
lo
y
s
L
y
a
p
u
n
o
v
-
b
ased
d
esig
n
p
r
in
cip
les
to
g
u
a
r
an
tee
s
y
s
tem
s
tab
ilit
y
an
d
p
er
f
o
r
m
an
ce
,
w
h
ile
also
d
is
p
l
ay
in
g
r
e
m
ar
k
a
b
le
ab
ilit
y
in
h
a
n
d
lin
g
u
n
ce
r
tain
ties
,
d
is
tu
r
b
a
n
ce
s
,
an
d
p
a
r
am
eter
v
ar
iatio
n
s
with
in
s
y
s
tem
s
.
C
o
n
s
eq
u
en
tly
,
it c
an
b
e
class
if
ied
as a
v
er
s
atile
an
d
r
eliab
le
co
n
tr
o
l a
p
p
r
o
ac
h
.
2
.
2
.
2
.
T
he
re
ct
if
ier
s
t
a
t
io
n c
o
ntr
o
l
T
h
e
r
ec
tifie
r
s
tatio
n
s
er
v
es
a
p
iv
o
tal
f
u
n
ctio
n
with
in
th
e
VSC
-
HVD
C
s
y
s
tem
,
f
ac
ilit
at
in
g
s
tab
le
o
p
er
atio
n
th
r
o
u
g
h
t
h
e
s
im
u
lta
n
eo
u
s
m
a
n
ag
em
e
n
t
o
f
DC
v
o
l
tag
e
an
d
r
ea
ctiv
e
p
o
wer
.
I
ts
p
r
im
ar
y
o
b
jectiv
e
is
to
m
ain
tain
alig
n
m
en
t
with
th
e
r
ef
er
en
ce
v
alu
e
o
f
th
e
DC
v
o
ltag
e
V
DC
,
th
er
eb
y
en
s
u
r
in
g
th
e
d
esire
d
s
tab
ilit
y
o
f
th
e
s
y
s
tem
ac
r
o
s
s
v
ar
y
in
g
o
p
er
atio
n
al
c
o
n
d
itio
n
s
.
Fu
r
th
er
m
o
r
e,
t
h
e
co
n
tr
o
l
o
f
t
h
e
r
ec
tifi
er
s
tatio
n
en
h
an
ce
s
o
v
er
all
p
o
wer
q
u
ality
b
y
co
r
r
ec
tin
g
th
e
p
o
wer
f
ac
to
r
,
a
cr
it
ical
co
m
p
o
n
en
t
f
o
r
th
e
ef
f
icie
n
t
tr
an
s
m
is
s
io
n
o
f
elec
tr
ical
en
er
g
y
.
a)
DC
v
o
ltag
e
co
n
tr
o
l
T
o
im
p
lem
e
n
t
th
e
b
ac
k
s
tep
p
in
g
s
tr
ateg
y
f
o
r
co
n
tr
o
llin
g
DC
v
o
ltag
e,
f
o
llo
w
th
ese
s
t
ep
s
wh
ile
u
tili
zin
g
(
8
)
-
(
1
2
)
.
˗
Step
1
:
First,
we
tak
e
in
to
co
n
s
id
er
atio
n
th
e
tr
ac
k
i
n
g
er
r
o
r
1
=
1
−
1
,
(
with
1
=
dc
an
d
its
d
er
iv
ativ
e
,
wh
ich
r
em
ain
s
co
n
s
tan
t,
eq
u
als ze
r
o
)
.
T
h
e
d
e
r
iv
a
tiv
e
o
f
1
is
g
iv
en
b
y
(
1
3
)
.
1
̇
=
̇
1
−
̇
1
=
1
.
1
2
−
1
−
̇
1
(
1
3
)
In
(
1
3
)
,
th
e
ter
m
2
is
id
en
tifie
d
as
a
v
ir
tu
al
co
n
tr
o
l
s
ig
n
al.
A
s
s
u
m
in
g
th
is
to
b
e
th
e
ac
t
u
al
co
n
tr
o
l
s
ig
n
al
tem
p
o
r
ar
ily
,
we
p
r
o
ce
e
d
to
c
o
n
s
id
er
th
e
L
y
a
p
u
n
o
v
f
u
n
ctio
n
ca
n
d
id
ate.
1
=
1
2
1
2
⇒
1
̇
=
1
1
̇
(
1
4
)
T
h
e
tim
e
d
er
iv
ativ
e
o
f
1
ca
n
b
e
tr
an
s
f
o
r
m
ed
in
to
a
n
eg
ativ
e
d
e
f
in
ite
f
u
n
ctio
n
o
f
1
,
as:
1
̇
=
1
(
1
.
1
2
−
1
−
̇
1
)
=
−
1
1
2
(
1
5
)
b
y
d
ef
i
n
in
g
2
=
δ
1
wh
er
e
:
δ
1
=
.
1
1
(
1
−
1
1
)
(
1
6
)
1
r
ep
r
esen
ts
a
p
o
s
itiv
e
d
esig
n
p
ar
am
ete
r
.
Sin
ce
th
e
ac
tu
al
co
n
tr
o
l
in
p
u
t
is
n
o
t
δ
1
.
We
d
ef
in
e
a
n
ew
tr
ac
k
in
g
er
r
o
r
:
2
=
2
−
δ
1
(
1
7
)
˗
Step
2
: Stab
ilizin
g
th
e
s
y
s
tem
(
1
,
2
).
I
t f
o
llo
ws f
r
o
m
(
1
7
)
th
at
th
e
d
er
iv
ativ
e
o
f
2
is
(
1
8
)
.
2
̇
=
̇
2
−
δ
̇
1
=
−
2
+
1
3
+
1
1
−
1
1
−
δ
̇
1
(
1
8
)
T
h
e
p
o
s
itiv
e
L
y
ap
u
n
o
v
f
u
n
ctio
n
2
is
u
s
ed
.
W
e
u
tili
ze
th
e
p
o
s
itiv
e
L
y
ap
u
n
o
v
f
u
n
ctio
n
2
=
1
+
1
2
3
2
+
1
2
1
1
2
.
T
h
e
tim
e
d
e
r
iv
ativ
e
o
f
2
ca
n
b
e
m
ad
e
a
n
e
g
ativ
e
d
e
f
in
ite
f
u
n
ctio
n
o
f
2
.
I
t is ex
p
r
ess
ed
as
(
1
9
)
.
2
̇
=
1
̇
+
2
̇
2
+
1
1
2
=
−
2
2
2
(
1
9
)
W
h
er
e
1
is
th
e
in
teg
r
al
ter
m
o
f
2
as
1
=
∫
2
.
I
n
(
1
8
)
,
δ
1
is
th
e
ac
tu
al
co
n
tr
o
l
in
p
u
t
f
o
r
th
e
VSC
1
s
tatio
n
.
T
o
s
tab
ilize
th
e
s
y
s
tem
(
1
,
2
)
.
W
e
s
u
g
g
est th
e
f
o
llo
wi
n
g
co
n
t
r
o
l la
w:
1
=
1
[
2
2
+
1
.
1
1
−
1
1
2
+
1
3
+
1
1
1
−
δ
̇
1
+
1
1
]
(
2
0
)
wh
er
e
2
,
1
ar
e
an
y
p
o
s
itiv
e
d
esig
n
p
ar
am
eter
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Ap
p
l Po
wer
E
n
g
I
SS
N:
2252
-
8
7
9
2
I
n
teg
r
a
l b
a
ck
s
tep
p
i
n
g
co
n
tr
o
l
d
esig
n
fo
r
en
h
a
n
ce
d
s
ta
b
ilit
y
a
n
d
d
y
n
a
mic
…
(
C
h
a
ima
a
La
k
h
d
a
ir
i
)
259
b)
R
ea
ctiv
e
p
o
wer
co
n
tr
o
l
T
h
e
o
b
jectiv
e
o
f
t
h
is
s
u
b
s
ec
ti
o
n
is
to
en
s
u
r
e
th
at
th
e
r
ea
cti
v
e
p
o
wer
f
o
llo
ws
a
s
p
ec
if
ied
r
ef
er
en
ce
s
ig
n
al.
T
o
ac
h
iev
e
th
is
,
we
in
tr
o
d
u
ce
a
n
ew
tr
ac
k
in
g
er
r
o
r
3
=
3
−
3
r
ef
,
(
with
3
=
1
)
.
T
h
e
d
er
iv
ativ
e
o
f
3
is
d
ef
in
ed
as
(
2
1
)
.
3
̇
=
̇
3
−
̇
3
r
ef
=
−
1
1
3
−
1
2
−
1
1
1
−
̇
3
r
ef
(
2
1
)
W
e
u
tili
ze
th
e
p
o
s
itiv
e
L
y
a
p
u
n
o
v
f
u
n
ctio
n
3
=
1
2
3
2
+
1
2
2
2
2
.
T
h
e
tim
e
d
er
iv
ativ
e
o
f
3
ca
n
b
e
m
a
d
e
a
n
eg
ativ
e
d
ef
i
n
ite
f
u
n
ctio
n
o
f
3
.
I
t is ex
p
r
ess
ed
as
(
2
2
)
.
3
̇
=
3
̇
3
+
2
2
3
=
−
3
3
2
(
2
2
)
W
h
er
e
2
is
th
e
q
u
an
tity
th
at
is
o
b
tain
e
d
b
y
in
teg
r
atin
g
3
,
as
ex
p
r
ess
ed
in
th
e
eq
u
ati
o
n
2
=
∫
3
.
T
h
er
ef
o
r
e,
b
y
em
p
lo
y
in
g
t
h
e
d
er
iv
ativ
es o
f
(
2
1
)
a
n
d
(
2
2
)
,
we
ca
n
estab
lis
h
th
e
in
p
u
t c
o
n
tr
o
l
lo
w.
1
=
1
[
3
3
−
1
1
3
−
1
2
−
1
1
1
−
̇
3
r
e
f
+
2
2
]
(
2
3
)
W
h
er
e
3
,
2
ar
e
an
y
p
o
s
itiv
e
d
esig
n
p
ar
a
m
eter
s
.
2
.
2
.
3
.
T
he
inv
er
t
er
s
t
a
t
io
n c
o
ntr
o
l
T
h
e
in
v
er
te
r
s
tatio
n
is
r
esp
o
n
s
ib
le
f
o
r
r
e
g
u
latin
g
b
o
th
ac
tiv
e
an
d
r
ea
cti
v
e
p
o
we
r
with
in
t
h
e
s
y
s
tem
.
B
y
m
eticu
lo
u
s
ly
tr
ac
k
in
g
t
h
eir
r
esp
ec
tiv
e
r
e
f
er
en
ce
v
alu
es,
it
g
u
ar
a
n
tees
p
r
ec
is
e
p
o
wer
d
eliv
er
y
to
t
h
e
in
ter
co
n
n
ec
ted
AC
g
r
id
o
r
lo
a
d
.
T
h
is
co
n
tr
o
l
s
tr
ateg
y
n
o
t
o
n
ly
o
p
tim
izes
th
e
ef
f
icien
c
y
o
f
p
o
wer
tr
an
s
m
is
s
io
n
b
u
t a
ls
o
co
n
tr
i
b
u
tes to
g
r
i
d
s
tab
ilit
y
b
y
r
esp
o
n
d
in
g
ad
e
p
tly
to
d
y
n
am
ic
s
y
s
tem
r
e
q
u
ir
em
e
n
ts
.
a)
Activ
e
p
o
wer
co
n
tr
o
l
T
o
im
p
lem
en
t
th
e
b
ac
k
s
tep
p
i
n
g
s
tr
ateg
y
f
o
r
ac
tiv
e
p
o
wer
co
n
tr
o
l
an
d
tr
ac
k
its
r
ef
er
en
c
e
v
alu
e,
we
f
o
llo
w
th
ese
s
tep
s
.
W
e
d
ef
in
e
a
tr
ac
k
in
g
er
r
o
r
4
=
4
−
4
r
ef
,
(
with
4
=
2
)
.
W
h
o
s
e
d
er
iv
ativ
e
is
ex
p
r
ess
ed
as
(
2
4
)
.
̇
4
=
̇
4
−
̇
4
r
ef
=
−
2
2
4
+
2
5
−
1
2
2
+
1
2
2
−
̇
4
r
ef
(
2
4
)
T
h
e
p
o
s
itiv
e
L
y
ap
u
n
o
v
f
u
n
cti
o
n
4
=
1
2
4
2
+
1
2
3
3
2
is
u
s
ed
.
T
h
e
tim
e
d
e
r
iv
ativ
e
o
f
4
ca
n
b
e
f
o
r
m
u
lated
as
a
n
eg
ativ
e
d
e
f
in
ite
f
u
n
ctio
n
o
f
4
,
p
r
esen
ted
as
(
2
5
)
.
̇
4
=
4
̇
4
+
3
3
4
=
−
4
4
2
(
2
5
)
W
h
e
r
e
3
i
s
t
h
e
q
u
a
n
t
i
t
y
o
b
t
ai
n
ed
b
y
i
n
t
e
g
r
a
t
i
n
g
4
,
a
s
3
=
∫
4
.
B
y
u
t
i
l
i
zin
g
t
h
e
d
e
r
i
v
a
t
i
v
e
s
o
f
(
2
4
)
a
n
d
(
2
5
)
,
we
s
u
g
g
est th
e
f
o
llo
win
g
co
n
t
r
o
l la
w:
2
=
2
[
−
4
4
+
2
2
4
−
2
5
+
1
2
2
+
̇
4
r
ef
−
3
3
]
(
2
6
)
4
,
3
r
ep
r
esen
ts
a
p
o
s
itiv
e
d
esig
n
p
ar
am
eter
.
b)
R
ea
ctiv
e
p
o
wer
co
n
tr
o
l
I
n
t
h
i
s
s
e
c
t
i
o
n
,
t
h
e
o
b
j
e
c
t
i
v
e
i
s
t
o
e
n
s
u
r
e
t
h
a
t
t
h
e
r
e
a
c
t
i
v
e
p
o
w
e
r
a
l
i
g
n
s
w
i
t
h
a
s
p
e
c
i
f
i
e
d
r
e
f
e
r
e
n
c
e
s
i
g
n
a
l
.
W
e
d
e
f
i
n
e
a
t
r
a
c
k
i
n
g
e
r
r
o
r
5
=
5
−
5
r
ef
,
(
w
i
t
h
5
=
2
)
.
W
h
o
s
e
t
i
m
e
d
e
r
i
v
a
t
i
v
e
i
s
e
x
p
r
e
s
s
e
d
a
s
(
2
7
)
.
̇
5
=
̇
5
−
̇
5
r
ef
=
−
2
2
5
−
2
4
+
1
2
2
−
̇
5
r
ef
(
2
7
)
W
e
em
p
lo
y
th
e
p
o
s
itiv
e
L
y
a
p
u
n
o
v
f
u
n
ctio
n
5
=
1
2
5
2
+
1
2
4
4
2
.
I
t
is
tim
e
d
e
r
i
v
ativ
e
5
is
m
ad
e
a
n
e
g
ativ
e
d
ef
in
ite
f
u
n
ctio
n
an
d
its
d
er
iv
a
tiv
e
is
g
iv
en
b
y
(
2
8
)
.
̇
5
=
5
̇
5
+
4
4
5
=
−
5
5
2
(
2
8
)
W
h
er
e
3
is
th
e
q
u
an
tity
o
b
tain
ed
b
y
in
teg
r
atin
g
5
,
as
4
=
∫
5
.
Hen
ce
,
b
y
u
s
in
g
t
h
e
d
e
r
iv
ativ
es
o
f
(
2
7
)
an
d
(
2
8
)
.
T
h
en
we
g
et
th
e
co
n
tr
o
l la
w
o
f
th
e
in
v
er
ter
s
tatio
n
as
(
2
9
)
.
2
=
2
[
−
5
5
+
2
2
5
+
2
4
+
̇
5
r
ef
−
4
4
]
(
2
9
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
7
9
2
I
n
t J Ap
p
l Po
wer
E
n
g
,
Vo
l.
1
4
,
No
.
2
,
J
u
n
e
20
2
5
:
255
-
26
3
260
5
,
4
d
en
o
tes a
p
o
s
itiv
e
d
esig
n
p
ar
a
m
eter
.
B
ased
o
n
th
e
L
aSalle
-
Yo
s
h
izaw
a
th
eo
r
em
,
1
,
2
,
3
,
4
,
an
d
5
ar
e
b
o
u
n
d
ed
an
d
co
n
v
er
g
e
to
ze
r
o
a
s
t
ap
p
r
o
ac
h
es
in
f
in
ity
.
Sin
ce
1
=
1
−
1
r
ef
,
3
=
3
−
3
r
ef
,
4
=
4
−
4
r
ef
,
5
=
5
−
5
r
ef
,
an
d
(
1
,
3
,
4
,
5
)
ar
e
also
b
o
u
n
d
e
d
an
d
c
o
n
v
er
g
e
to
th
e
d
esire
d
e
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u
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2
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I
O
G
RAP
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AUTH
O
RS
Cha
im
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a
La
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iri
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ti
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tme
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m
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ss
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n
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iv
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o
f
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sa
b
l
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o
ro
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h
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sm
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ti
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ss
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ro
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h
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tac
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s.
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talk
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tern
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h
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a
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z
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a
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ste
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fro
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in
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t,
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ro
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,
h
e
re
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h
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D
.
i
n
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El
a
b
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d
c
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l
stra
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rid
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m
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in
th
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a
b
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ra
to
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ste
m
An
a
ly
sis
a
n
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fo
rm
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ti
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P
ro
c
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in
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f
ro
m
Un
i
v
e
rsity
Ha
ss
a
n
I.
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is
c
u
rre
n
tl
y
a
p
r
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ss
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r
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sic
s D
e
p
a
rtme
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t
a
t
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a
c
u
lt
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S
c
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c
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M
’S
ik
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v
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rsity
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ss
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n
II
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sa
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lan
c
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o
r
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c
o
.
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re
se
a
rc
h
in
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n
fo
rm
a
ti
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T
re
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tme
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o
ra
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sists
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th
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sy
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tem
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with
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se
o
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d
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n
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d
c
o
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tr
o
ll
e
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n
d
h
e
is
a
n
e
x
p
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rt
in
I
T
d
e
v
e
lo
p
m
e
n
t.
He
c
a
n
b
e
c
o
n
tac
te
d
a
t
e
m
a
il
:
h
b
a
h
ri.
i
n
f@g
m
a
il
.
c
o
m
.
Mo
h
a
m
e
d
Ta
lea
r
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c
e
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e
d
h
is
P
h
.
D.
d
e
g
re
e
in
P
h
y
sic
s
in
c
o
ll
a
b
o
ra
ti
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n
wit
h
P
o
it
iers
Un
iv
e
rsity
,
F
ra
n
c
e
i
n
2
0
0
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.
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e
o
b
tain
e
d
a
d
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to
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te o
f
h
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h
g
ra
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a
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stu
d
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d
e
g
re
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fr
o
m
Ha
ss
a
n
II
Un
iv
e
rsity
,
M
o
ro
c
c
o
,
i
n
1
9
9
4
.
C
u
rre
n
tl
y
,
h
e
is
a
p
ro
fe
ss
o
r
in
th
e
De
p
a
rtme
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t
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f
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h
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sic
s
a
t
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ss
a
n
II
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e
rsity
,
M
o
r
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c
c
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a
n
d
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e
is
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d
irec
to
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o
f
t
h
e
In
fo
rm
a
ti
o
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T
re
a
tme
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Lab
o
ra
to
ry
.
He
c
a
n
b
e
c
o
n
tac
ted
a
t
e
m
a
il
:
tale
a
m
o
h
a
m
e
d
@y
a
h
o
o
.
fr.
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