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v
alu
a
ted
b
y
n
u
m
er
ical
s
t
u
d
y
in
[
1
6
]
.
Fu
r
t
h
er
s
t
u
d
y
w
a
s
also
s
u
g
g
ested
in
[
1
7
-
1
9
]
,
h
o
w
ev
er
,
t
h
ese
ap
p
r
o
ac
h
es
r
eq
u
ir
ed
cu
r
r
en
t
o
r
a
v
o
ltag
e
r
ef
er
en
ce
f
o
r
co
n
tr
o
l
la
w
s
y
n
t
h
esi
s
an
d
ca
n
lead
to
a
lac
k
o
f
r
o
b
u
s
tn
e
s
s
to
o
p
er
atio
n
co
n
d
i
tio
n
s
.
I
n
[
2
0
]
th
er
e
is
n
o
n
ec
es
s
i
t
y
to
h
a
v
e
a
r
ef
er
en
ce
v
al
u
e
s
in
ce
th
e
s
li
d
in
g
s
u
r
f
ac
e
g
u
ar
an
tee
s
t
h
e
MP
P
w
h
e
n
it
is
eq
u
al
to
ze
r
o
.
B
u
t
th
e
ch
at
ter
in
g
p
h
e
n
o
m
e
n
o
n
,
o
r
ig
in
ated
b
y
th
e
in
ter
ac
tio
n
b
et
w
ee
n
p
ar
asit
e
d
y
n
a
m
ic
an
d
f
i
n
ite
-
f
r
eq
u
e
n
c
y
s
w
itc
h
i
n
g
c
o
n
tr
o
l
is
th
e
m
ai
n
d
is
ad
v
a
n
tag
es
o
f
t
h
is
tec
h
n
iq
u
es
o
f
co
n
tr
o
l
[
2
1
,
2
2
]
.
T
o
m
in
i
m
ize
ch
at
ter
in
g
p
h
e
n
o
m
en
a
s
o
m
e
m
eth
o
d
s
w
er
e
p
r
o
p
o
s
ed
[
2
3
-
2
5
]
.
T
o
p
r
eser
v
e
th
e
m
ai
n
ad
v
a
n
ta
g
es
o
f
th
e
s
lid
in
g
m
o
d
e
tech
n
iq
u
e
an
d
to
r
ed
u
ce
th
e
ch
atter
in
g
p
h
en
o
m
en
o
n
,
a
n
o
v
el
class
o
f
SMC
alg
o
r
ith
m
,
ca
lled
s
ec
o
n
d
-
o
r
d
er
SMC
alg
o
r
ith
m
(
2
-
SM
C
)
h
as b
ee
n
p
r
o
p
o
s
ed
in
[
2
6
,
2
7
]
.
I
n
S
ah
ar
ao
u
i
et
al.
[
2
8
]
,
a
2
-
SMC
w
a
s
ap
p
lied
w
i
th
a
t
w
o
-
lo
o
p
co
n
tr
o
l
ap
p
r
o
ac
h
.
A
s
i
m
u
la
tio
n
s
tu
d
y
b
y
Yati
m
i
et
al.
,
[
2
9
]
p
r
esen
ts
a
r
o
b
u
s
t
s
lid
i
n
g
m
o
d
e
m
e
th
o
d
f
o
r
a
p
h
o
to
v
o
lta
ic
en
er
g
y
s
to
r
ag
e
s
y
s
te
m
.
An
o
th
er
m
eth
o
d
to
tr
ac
k
th
e
MP
P
g
iv
en
i
n
Mo
j
allizad
eh
et
al.
,
[
3
0
]
,
th
e
p
r
o
p
o
s
ed
s
ch
e
m
e
i
s
b
ased
o
n
th
e
s
ec
o
n
d
-
o
r
d
er
f
u
zz
y
s
lid
i
n
g
m
o
d
e
co
n
tr
o
l
la
w
o
f
p
h
o
to
v
o
ltaic
p
o
w
er
g
en
er
atio
n
s
y
s
te
m
s
w
i
th
a
t
w
o
-
lo
o
p
co
n
tr
o
l.
Mo
r
eo
v
er
,
in
Kck
ao
u
et
al.
,
[
3
1
]
o
f
f
er
a
s
ec
o
n
d
-
o
r
d
er
s
lid
in
g
MP
P
T
co
n
tr
o
l
f
o
r
p
h
o
to
v
o
ltaic
ap
p
licatio
n
.
T
h
e
m
ai
n
o
b
j
ec
ti
v
e
o
f
th
is
w
o
r
k
is
t
h
e
u
s
e
o
n
e
lo
o
p
tech
n
iq
u
e
o
f
2
-
SMC
b
as
ed
o
n
s
u
p
er
t
w
i
s
ti
n
g
alg
o
r
ith
m
to
e
x
tr
ac
t
MP
P
,
r
e
d
u
ce
th
e
c
h
atter
i
n
g
p
h
en
o
m
e
n
o
n
a
n
d
r
ea
l
-
ti
m
e
i
m
p
le
m
e
n
t
a
tio
n
s
t
u
d
y
u
n
d
er
d
if
f
er
e
n
t
o
p
er
atin
g
s
ce
n
ar
io
s
.
T
h
e
co
n
tr
o
l
cir
cu
it
u
s
e
a
n
a
lg
o
r
ith
m
to
ad
ap
t
th
e
d
u
t
y
c
y
cle
o
f
th
e
s
w
itc
h
co
n
tr
o
l
o
f
th
e
DC
-
D
C
co
n
v
e
r
ter
to
s
ea
r
ch
MPP
tr
ac
k
in
g
as
a
f
u
n
ctio
n
o
f
ev
o
lu
tio
n
o
f
th
e
p
o
w
er
in
p
u
t
.
T
h
i
s
p
a
p
e
r
c
o
n
s
is
ts
o
f
f
o
u
r
s
e
c
t
i
o
n
s
,
in
cl
u
d
i
n
g
th
e
in
t
r
o
d
u
c
t
i
o
n
.
S
ec
t
i
o
n
2
m
a
t
e
r
i
a
ls
an
d
m
e
th
o
d
s
,
an
d
s
e
c
t
i
o
n
3
r
esu
lt
s
an
d
d
is
c
u
s
s
io
n
.
Fi
n
all
y
,
th
e
co
n
clu
s
io
n
s
o
f
t
h
e
s
t
u
d
y
ar
e
g
iv
e
n
in
s
ec
tio
n
4
.
2.
M
AT
E
RIAL
S AN
D
M
E
T
H
O
DS
2
.
1
.
P
ho
t
o
v
o
lt
a
ic
s
y
s
t
e
m
s
2
.
1
.
1
.
M
a
t
he
m
a
t
ica
l
m
o
delli
ng
a
nd
s
i
m
u
la
t
io
n
T
h
e
p
h
y
s
i
c
a
l
b
eh
av
i
o
u
r
o
f
th
e
p
a
n
e
l
h
as
c
o
n
v
en
t
i
o
n
al
ly
b
e
en
s
tu
d
i
e
d
b
y
r
e
p
r
e
s
e
n
t
in
g
i
t
a
s
a
n
e
q
u
iv
al
en
t
elec
tr
ical
cir
cu
i
t
co
m
p
o
s
ed
o
f
li
n
ea
r
an
d
n
o
n
-
li
n
e
ar
co
m
p
o
n
en
ts
.
So
lar
ce
ll
(
SP
V)
is
t
h
e
ele
m
e
n
tar
y
co
m
p
o
n
e
n
t
w
h
ich
co
n
v
er
ts
t
h
e
en
er
g
y
o
f
li
g
h
t
d
ir
ec
tl
y
in
to
elec
tr
icit
y
b
y
th
e
P
V
ef
f
ec
t.
P
V
ar
r
ay
s
ar
e
b
u
ilt
u
p
w
it
h
co
m
b
i
n
ed
s
er
ies
/p
ar
allel
co
m
b
i
n
atio
n
s
o
f
SP
V
[
3
2
,
3
3
]
.
E
ac
h
ce
ll
is
t
y
p
icall
y
a
p
-
n
j
u
n
ct
io
n
.
T
h
er
e
ar
e
v
ar
io
u
s
cir
c
u
it
s
ch
e
m
es
f
o
r
a
p
h
o
to
v
o
ltaic
ce
ll
i
n
liter
at
u
r
e.
A
s
i
n
g
le
d
io
d
e
m
o
d
el
i
s
co
n
s
id
er
ed
as
t
h
e
eq
u
iv
ale
n
t
p
h
o
to
v
o
ltaic
ce
l
l
i
n
t
h
e
p
r
ese
n
t
p
ap
er
[
3
3
]
.
T
h
e
b
asic
m
o
d
el
f
o
r
a
p
h
o
to
v
o
lt
aic
ce
ll
i
s
s
h
o
w
i
n
Fig
u
r
e
1.
Fig
u
r
e
1
.
Si
m
p
li
f
ied
eq
u
i
v
alen
t c
ir
cu
it P
V
m
o
d
el
T
h
e
o
n
e
d
io
d
e
e
q
u
iv
ale
n
t
cir
cu
it
d
eter
m
in
e
s
th
e
I
-
V
ch
ar
ac
ter
is
tic
o
f
th
e
ce
ll
is
d
escr
ib
ed
b
y
th
e
f
o
llo
w
i
n
g
(
1
)
:
I
=
I
ph
−
I
0
[
e
(
V
+
I
R
s
V
t
)
−
1
]
−
V
+
I
R
s
R
sh
(
1
)
w
h
er
e
I
is
t
h
e
ce
ll
o
u
tp
u
t
cu
r
r
en
t
(
A
)
,
V
is
t
h
e
ce
ll
o
u
tp
u
t
v
o
ltag
e
(
V)
,
I
ph
i
s
t
h
e
p
h
o
to
cu
r
r
en
t,
f
u
n
ctio
n
o
f
th
e
ir
r
ad
iatio
n
le
v
el
(
G)
an
d
j
u
n
ct
io
n
te
m
p
er
atu
r
e,
I
0
is
t
h
e
r
ev
er
s
e
s
at
u
r
atio
n
c
u
r
r
en
t
o
f
d
io
d
e,
V
t
=
aKT
c
/q
is
th
e
t
h
er
m
al
v
o
ltag
e,
q
is
t
h
e
el
ec
tr
o
n
ch
ar
g
e
(
1
.
6
0
2
×1
0
-
19
C
)
,
K
is
th
e
B
o
ltz
m
an
n
co
n
s
tan
t (
1
.
3
8
×1
0
-
23
J
/K)
,
a
is
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
.
3
,
Dec
em
b
er
2
0
2
0
:
284
–
296
286
th
e
id
ea
l
f
ac
to
r
,
T
c
is
th
e
te
m
p
er
atu
r
e
o
f
th
e
ce
ll,
R
s
a
n
d
R
sh
th
e
s
er
ial
an
d
p
ar
allel
r
esi
s
t
an
ce
s
r
esp
ec
ti
v
el
y
an
d
I
d
,
ca
lled
d
i
o
d
e
(
D)
cu
r
r
en
t o
r
d
ar
k
cu
r
r
en
t
.
T
h
e
p
h
o
to
cu
r
r
en
t I
ph
ca
n
b
e
ass
es
s
ed
w
i
th
t
h
e
(
2
)
:
I
ph
=
I
S
T
C
G
G
S
T
C
[
1
+
α
(
T
c
−
T
c
,
S
T
C
)
]
(
2
)
w
h
er
e
I
S
TC
is
t
h
e
s
h
o
r
t
cir
cu
it
cu
r
r
en
t
at
s
tan
d
ar
d
test
co
n
d
itio
n
(
ST
C
)
,
w
h
ile
G
STC
an
d
T
c,
S
TC
ar
e
th
e
ir
r
ad
iatio
n
an
d
te
m
p
er
atu
r
e
o
f
th
e
P
V
ce
ll
at
S
T
C
,
r
esp
ec
tiv
el
y
;
α
is
t
h
e
cu
r
r
en
t
te
m
p
er
atu
r
e
co
ef
f
icie
n
t.
W
ith
r
eg
ar
d
to
th
e
r
ev
er
s
e
s
a
tu
r
atio
n
c
u
r
r
en
t
I
0
p
ar
a
m
eter
,
its
v
al
u
e
c
h
an
g
es
w
it
h
ce
ll
t
e
m
p
er
atu
r
e
at
ST
C
co
n
d
itio
n
s
a
n
d
ca
n
b
e
f
o
u
n
d
b
y
u
s
in
g
t
h
e
f
o
llo
w
i
n
g
(
3
)
.
I
0
=
I
rs
(
T
c
T
c
,
S
T
C
)
3
e
q
E
g
(
1
T
c
−
1
T
c
,
S
T
C
)
Ka
(
3
)
w
h
er
e
I
rs
is
th
e
r
e
v
er
s
e
s
atu
r
at
io
n
cu
r
r
en
t
at
ST
C
co
n
d
itio
n
s
,
E
g
i
s
t
h
e
b
an
d
-
g
ap
e
n
er
g
y
o
f
t
h
e
m
ater
ial.
In
th
i
s
w
o
r
k
f
o
r
R
s
an
d
R
sh
th
e
s
a
m
e
r
elatio
n
s
i
n
[
3
4
]
ar
e
u
s
ed
as (
4
)
an
d
(
5
)
.
R
sh
=
R
sh
,
S
T
C
G
G
S
T
C
(
4
)
R
s
=
R
s
,
S
T
C
(
5
)
w
h
er
e
R
s,
S
TC
an
d
R
sh,
S
T
C
ar
e
th
e
s
er
ial
r
esis
ta
n
ce
an
d
p
ar
allel
r
esis
ta
n
ce
at
ST
C
co
n
d
it
io
n
s
,
r
esp
ec
tiv
el
y
.
I
n
(
1
)
is
v
alid
f
o
r
a
s
o
lar
ce
ll.
Fo
r
th
e
ex
ac
t
ap
p
licati
o
n
o
f
t
h
i
s
eq
u
a
tio
n
f
o
r
P
V
m
o
d
u
le,
th
e
ter
m
o
f
(
+
)
is
r
ep
lace
d
b
y
(
+
)
.
T
o
d
eter
m
in
e
t
h
e
f
i
v
e
p
ar
a
m
eter
s
ex
is
t
i
n
(
1
)
,
w
h
ic
h
ar
e:
ℎ
,
,
ℎ
,
0
an
d
a,
y
o
u
ca
n
s
ee
[
3
5
,
3
6
]
.
T
y
p
icall
y
N
s
ce
lls
ar
e
co
n
n
ec
ted
in
s
er
ies
to
g
et
th
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r
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u
is
ite
v
o
ltag
e
o
f
P
V
m
o
d
u
le.
A
ll
t
h
e
ce
lls
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r
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f
o
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ce
d
to
ca
r
r
y
th
e
s
a
m
e
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u
r
r
en
t
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lled
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er
ies
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el.
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n
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s
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r
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al
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le
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tili
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ed
,
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ell
So
lar
S7
5
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T
h
e
elec
tr
ical
p
a
r
am
e
ter
s
o
f
th
e
m
o
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u
l
e
u
n
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er
ST
C
f
o
r
m
m
an
u
f
ac
t
u
r
er
ar
e
lis
ted
in
T
ab
le
1
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T
ab
le
1
.
Data
o
f
ex
p
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im
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tal
P
V
m
o
d
u
le
s
S
i
l
i
c
o
n
t
y
p
e
S
h
e
l
l
s
o
l
a
r
S
7
5
O
p
e
n
c
i
r
c
u
i
t
v
o
l
t
a
g
e
(
V
oc
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2
1
.
6
V
S
h
o
r
t
-
c
i
r
c
u
i
t
c
u
r
r
e
n
t
(
I
sc
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4
.
7
A
M
a
x
i
m
a
l
v
o
l
t
a
g
e
(
V
mp
)
1
7
.
6
V
M
a
x
i
m
a
l
c
u
r
r
e
n
t
(
I
mp
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4
.
2
6
A
M
a
x
i
m
a
l
p
o
w
e
r
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P
mp
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7
5
W
N
u
mb
e
r
o
f
c
e
l
l
s (N
s
)
36
Fig
u
r
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2
s
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d
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p
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iatio
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r
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t
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g
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er
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ch
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Fig
u
r
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2
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Fig
u
r
e
3
r
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e
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im
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em
a
o
f
a
o
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e
d
i
o
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e
m
o
d
e
l
o
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th
e
PV
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a
n
e
l
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M
o
r
e
d
e
t
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il
s
ca
n
b
e
f
o
u
n
d
in
[
3
7
]
.
Fig
u
r
e
2
.
Si
m
u
latio
n
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d
ex
p
e
r
i
m
en
tal
r
esu
lts
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n
t
h
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tes
t set
u
p
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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2252
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8792
R
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m
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ar
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ter
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2
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1
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2
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Dy
na
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ic
m
o
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f
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DC
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o
s
t
co
nv
er
t
er
I
n
o
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o
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ce
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e
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V
p
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el
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n
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s
a
t
t
h
e
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PT
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w
e
p
r
esen
t
th
e
p
r
i
n
cip
le
o
f
t
h
e
D
C
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DC
b
o
o
s
t
co
n
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er
ter
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h
is
ty
p
e
o
f
co
n
v
e
r
ter
u
s
e
in
d
u
cto
r
s
an
d
ca
p
ac
ito
r
s
to
co
n
tr
o
l
th
e
en
er
g
y
f
lo
w
th
e
P
V
m
o
d
u
le
to
lo
ad
b
y
co
n
ti
n
u
o
u
s
l
y
o
p
en
in
g
an
d
clo
s
in
g
a
s
w
itc
h
(
K)
[
3
8
]
.
T
h
e
s
w
itc
h
is
g
en
er
all
y
a
n
elec
tr
o
n
ic
d
ev
ice
(
Mo
s
f
et
o
r
I
GB
T
tr
an
s
is
to
r
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.
I
t
is
d
r
iv
e
n
b
y
a
p
u
ls
e
w
id
t
h
m
o
d
u
latio
n
(
P
W
W
)
s
ig
n
al
w
it
h
a
f
ix
ed
f
r
eq
u
en
c
y
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d
an
ad
j
u
s
tab
le
d
u
t
y
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y
cle
D
(
0
<D
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.
Fig
u
r
e
4
s
h
o
w
s
a
DC
-
D
C
b
o
o
s
t
co
n
v
er
ter
.
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h
e
r
elatio
n
b
et
w
ee
n
th
e
o
u
tp
u
t v
o
lta
g
e
an
d
i
n
p
u
t
v
o
ltag
e
in
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C
-
DC
b
o
o
s
t c
o
n
v
er
ter
is
g
i
v
en
b
y
(
6
)
:
0
=
1
1
−
(
6
)
th
e
d
y
n
a
m
ic
o
f
t
h
e
b
o
o
s
t c
o
n
v
er
ter
is
g
iv
e
n
b
y
:
{
C
1
d
V
pv
dt
=
I
pv
−
I
L
L
d
I
L
dt
=
V
pv
−
(
1
−
D
)
V
0
C
2
d
V
0
dt
=
−
I
0
+
(
1
−
D
)
I
L
(
7
)
w
h
er
e
V
p
v
an
d
I
pv
:
ar
e
t
h
e
v
o
ltag
e
an
d
cu
r
r
e
n
t
o
f
th
e
P
V
m
o
d
u
le,
I
L
:
t
h
e
i
n
d
u
cto
r
cu
r
r
en
t
o
f
th
e
D
C
-
DC
c
o
n
v
e
r
t
e
r
.
V
0
:
t
h
e
D
C
-
D
C
c
o
n
v
e
r
t
e
r
o
u
t
p
u
t
v
o
l
t
a
g
e
,
D
:
t
h
e
d
u
t
y
c
y
c
l
e
,
L
:
t
h
e
f
i
l
t
e
r
i
n
d
u
c
t
o
r
,
C
1
a
n
d
C
2
:
t
h
e
f
i
l
t
e
r
c
a
p
a
c
i
t
o
r
,
R
:
t
h
e
n
o
m
i
n
a
l
r
e
s
i
s
t
a
n
c
e
o
f
l
o
a
d
.
B
y
co
m
b
i
n
g
t
h
e
d
if
f
er
en
t
eq
u
atio
n
s
d
escr
ib
in
g
t
h
e
s
y
s
te
m
[
2
9
]
,
g
lo
b
al
d
y
n
a
m
ic
m
o
d
el
ca
n
b
e
w
r
itte
n
as
f
o
llo
w
s
:
{
I
=
I
ph
−
I
0
[
e
(
V
+
I
N
s
R
s
V
t
N
s
)
−
1
]
−
V
+
I
N
s
R
s
N
s
R
sh
d
V
pv
dt
=
I
pv
C
1
−
I
L
C
1
d
V
pv
dt
=
V
pv
L
−
(
1
−
D
)
L
V
0
d
V
0
dt
=
−
I
0
C
2
+
(
1
−
D
)
C
2
I
L
(
8
)
I
n
(
8
)
ca
n
b
e
w
r
it
ten
i
n
co
m
p
a
ct
f
o
r
m
o
f
t
h
e
no
n
li
n
ea
r
ti
m
e
i
n
v
ar
ia
n
t s
y
s
te
m
;
{
X
1
̇
=
I
pv
C
1
−
1
C
1
X
2
X
2
̇
=
1
L
X
1
−
u
L
X
3
X
3
̇
=
−
1
R
C
2
X
3
+
u
C
2
X
2
(
9
)
w
h
er
e
X
1
=
V
pv
;
X
2
=
I
L
;
X
3
=
V
0
;
u
=
(
1
−
D
)
;
I
0
=
V
0
R
⁄
.
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
.
3
,
Dec
em
b
er
2
0
2
0
:
284
–
296
288
Fig
u
r
e
4
.
DC
-
D
C
b
o
o
s
t c
o
n
v
e
r
ter
2
.
2
.
Seco
nd
o
rder
s
lid
ing
m
o
de
a
pp
ro
a
ch
As
th
e
s
u
p
p
lied
b
y
t
h
e
p
h
o
to
v
o
ltaic
en
er
g
y
d
ep
en
d
s
o
n
o
u
td
o
o
r
c
o
n
d
itio
n
s
,
an
i
m
p
o
r
tan
t
a
cc
o
u
n
t
i
n
th
e
d
esi
g
n
o
f
ef
f
ic
ien
t
P
V
s
y
s
te
m
s
is
to
ex
tr
ac
t
MP
P
co
r
r
ec
tl
y
.
T
h
e
p
u
r
p
o
s
e
o
f
MP
PT
is
to
m
o
v
e
t
h
e
o
u
tp
u
t
p
o
w
er
o
p
er
atin
g
clo
s
e
to
th
e
MPP
u
n
d
er
v
ar
y
i
n
g
o
u
td
o
o
r
c
o
n
d
itio
n
s
.
T
h
e
2
-
SMC
is
estab
lis
h
ed
f
o
r
th
e
s
y
s
te
m
s
w
it
h
r
elati
v
e
d
eg
r
ee
t
w
o
(
r
=2
)
an
d
d
o
es
n
o
t
s
u
f
f
er
f
r
o
m
ch
at
ter
in
g
w
h
ile
m
ai
n
tai
n
in
g
th
e
r
o
b
u
s
t
n
e
s
s
o
f
t
h
e
ap
p
r
o
ac
h
.
T
h
er
e
ar
e
s
ev
er
al
al
g
o
r
ith
m
s
to
r
ea
lize
2
-
SM
C
i
n
liter
atu
r
e.
Fo
r
ex
a
m
p
le,
s
u
b
-
opt
im
a
l
alg
o
r
it
h
m
,
th
e
ter
m
i
n
al
s
lid
in
g
m
o
d
e
al
g
o
r
ith
m
,
t
h
e
t
w
i
s
ti
n
g
alg
o
r
it
h
m
,
t
h
e
s
u
p
er
-
t
w
i
s
ti
n
g
alg
o
r
it
h
m
.
Hen
ce
,
th
e
ST
A
alg
o
r
it
h
m
i
s
c
u
r
r
en
tl
y
p
r
ef
er
ab
le
o
v
er
th
e
cl
ass
ical
s
lid
in
g
m
o
d
e
co
n
tr
o
l.
2
.
2
.
1
.
Sh
o
rt
re
v
ie
w
o
f
2
-
S
M
C
T
h
e
SMC
co
n
s
is
ts
o
f
t
w
o
p
h
a
s
e:
f
ir
s
t,
w
e
d
eter
m
i
n
e
a
s
lid
in
g
s
u
r
f
ac
e
S
(
X
)
u
p
o
n
w
h
ich
t
h
e
co
n
tr
o
l
o
b
j
ec
tiv
es
ar
e
r
ea
lis
ed
.
Nex
t,
w
e
d
er
iv
e
a
co
n
tr
o
l
la
w
i
n
o
r
d
er
to
b
r
in
g
th
e
s
tate
tr
aj
ec
to
r
y
to
t
h
is
o
u
tp
u
t
an
d
m
ai
n
tai
n
it
th
er
e
at
all
ti
m
e
[
3
9
]
.
I
n
th
e
s
it
u
atio
n
t
h
e
d
if
f
icu
lt
is
to
g
en
er
ate
a
2
-
S
M
o
n
an
ap
p
r
o
p
r
iately
ch
o
s
en
s
lid
i
n
g
s
u
r
f
ac
e
an
d
,
th
u
s
,
to
co
n
s
tr
ai
n
t
h
e
tr
aj
ec
to
r
ies
s
y
s
te
m
to
e
v
o
lv
e
i
n
f
i
n
ite
ti
m
e
o
n
S
=
{
X
:
S
=
S
̇
=
0
}
.
Ho
w
e
v
er
,
th
e
in
cr
ea
s
in
g
in
f
o
r
m
atio
n
d
e
m
a
n
d
i
n
te
r
m
s
o
f
t
h
e
f
ir
s
t
d
er
i
v
ati
v
e
o
f
th
e
s
lid
in
g
s
u
r
f
ac
e
is
t
h
e
m
ai
n
d
if
f
ic
u
lt
i
n
t
h
e
i
m
p
le
m
e
n
tatio
n
o
f
t
h
e
2
-
SM
C
,
t
h
e
s
u
p
er
t
w
is
t
in
g
s
l
id
in
g
m
o
d
e
co
n
tr
o
l
s
c
h
e
m
e
is
a
m
o
d
if
ied
2
-
SM
C
s
c
h
e
m
e
t
h
at
d
o
es
n
o
t
r
eq
u
ir
e
an
y
k
n
o
w
led
g
e
o
f
th
e
d
er
iv
ati
v
e
o
f
th
e
s
lid
i
n
g
v
ar
iab
le
S
̇
.
C
o
n
s
id
er
a
s
y
s
te
m
w
h
o
s
e
d
y
n
a
m
ics i
s
g
iv
en
b
y
:
X
̇
=
f
(
X
,
t
)
+
g
(
X
,
t
)
u
y
=
g
(
X
,
t
)
(
1
0
)
w
h
er
e:
X
∈
ℛ
n
is
th
e
s
y
s
te
m
s
ta
te
v
ar
iab
le,
u
∈
ℛ
is
th
e
co
n
tr
o
l,
f
,
g
ar
e
s
u
f
f
icie
n
tl
y
s
m
o
o
th
v
ec
to
r
f
iel
d
s
.
S
=
S
(
X
,
t
)
∈
ℛ
i
s
t
h
e
o
u
t
p
u
t
f
u
n
c
ti
o
n
,
c
al
l
ed
s
l
i
d
in
g
v
a
r
i
a
b
l
e
.
B
y
d
i
f
f
e
r
en
t
ia
t
in
g
S
w
it
h
r
es
p
e
c
t
t
o
t
im
e
,
t
,
w
e
h
av
e:
S
̈
=
φ
A
(
t
,
S
,
S
̇
)
+
∅
(
t
,
S
,
S
̇
)
u
̇
(
1
1
)
T
h
e
co
n
tr
o
l
u
is
b
o
u
n
d
ed
f
u
n
ctio
n
|
u
|
≤
U
m
ax
.
T
h
e
d
y
n
a
m
ic
s
in
(
1
1
)
ar
e
ass
u
m
ed
to
s
ati
s
f
y
t
h
e
f
o
llo
w
i
n
g
b
o
u
n
d
in
g
co
n
d
itio
n
s
[
2
6
]
:
0
<
k
m
≤
|
∅
(
t
,
S
,
S
̇
)
|
≤
K
M
|
φ
(
X
,
t
)
|
≤
β
0st
(
1
2
)
T
h
e
s
et
{
t
,
X
,
u
:
|
S
(
t
,
X
)
|
<
S
0
}
is
th
e
li
n
ea
r
r
eg
io
n
,
w
h
er
e
k
m
,
K
M
an
d
β
0
s
t
ar
e
s
o
m
e
p
o
s
itiv
e
co
n
s
tan
ts
.
T
h
e
alg
o
r
ith
m
i
n
cl
u
d
es
t
w
o
co
n
tin
u
o
u
s
ter
m
s
th
at,
a
g
ain
,
d
o
n
o
t
d
e
p
en
d
u
p
o
n
th
e
f
ir
s
t
ti
m
e
d
er
iv
ati
v
e
o
f
s
lid
in
g
v
ar
iab
le.
T
h
e
alg
o
r
ith
m
ca
n
b
e
d
ef
i
n
ed
b
y
th
e
f
o
llo
w
i
n
g
co
n
tr
o
l la
w
:
u
st
=
u
1
+
u
2
(
1
3
)
w
h
er
e
{
u
1
̇
=
−
α
1
s
ign
(
S
)
u
2
=
α
2
|
S
|
ρ
s
ign
(
S
)
(
1
4
)
w
it
h
:
α
1
,
α
2
an
d
ρ
v
er
if
y
i
n
g
t
h
e
f
o
llo
w
i
n
g
in
eq
u
alit
y
[
2
7
]
an
d
[
4
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
R
o
b
u
s
t m
a
ximu
m
p
o
w
er p
o
in
t
tr
a
ck
in
g
co
n
tr
o
l fo
r
p
h
o
to
vo
lt
a
ic
s
ystem
b
a
s
ed
o
n
s
ec
o
n
d
o
r
d
er
…
(
A
.
F
ezz
a
n
i
)
289
{
α
1
=
β
0st
k
m
α
2
2
=
4
β
0st
K
m
(
α
1
+
β
0st
)
K
M
2
k
m
(
α
1
−
β
0st
)
0
<
≤
0
(
1
5
)
T
h
e
ch
o
ice
ρ
=
0
.
5
en
s
u
r
es
t
h
at
h
e
m
ax
i
m
al
p
o
s
s
ib
le
f
o
r
2
-
Sli
d
i
n
g
m
o
d
e
co
n
tr
o
l
r
ea
lizatio
n
r
ea
l
s
lid
in
g
o
r
d
er
t
w
o
is
ac
h
ie
v
ed
.
Usi
n
g
2
-
SM
C
ass
u
r
es t
h
e
f
in
ite
t
i
m
e
co
n
v
er
g
en
ce
.
2
.
2
.
2
.
Ro
bu
s
t
2
-
S
M
C
M
P
P
T
co
ntr
o
l a
pp
ro
a
ch
I
n
th
i
s
w
o
r
k
t
h
e
ST
A
h
as
b
ee
n
d
esi
g
n
ed
to
s
ea
r
ch
MP
P
.
T
h
e
s
u
p
er
t
w
is
t
in
g
al
g
o
r
ith
m
is
e
s
tab
lis
h
ed
f
o
r
th
e
s
y
s
te
m
w
it
h
r
elativ
e
d
eg
r
ee
o
n
e
s
o
as
to
r
ed
u
ce
t
h
e
ch
atter
i
n
g
[
3
1
]
.
T
o
m
ak
e
s
u
r
e
th
at
th
e
s
y
s
te
m
s
tates
w
ill
h
it
th
e
s
lid
i
n
g
s
u
r
f
a
ce
an
d
p
r
o
v
id
es t
h
e
MP
P
o
u
tp
u
t,
w
e
c
h
o
o
s
e
t
h
e
s
lid
in
g
s
u
r
f
a
ce
as
g
iv
e
n
i
n
[
1
9
]
.
T
h
e
s
tate
(
9
)
ca
n
b
e
ex
p
r
ess
ed
b
y
:
X
̇
=
f
(
X
,
t
)
+
g
(
X
,
t
)
u
S
(
X
,
t
)
=
∂
P
pv
∂
V
pv
(
1
6
)
w
h
er
e
X
=
[
I
pv
V
0
]
T
,
u
=
[
0
1
]
.
T
h
e
s
lid
in
g
m
o
d
e
s
u
r
f
ac
e
S (
t)
is
d
ef
in
ed
as:
S
(
X
,
t
)
=
∂
P
pv
∂
V
pv
=
I
pv
+
V
pv
∂
pv
∂
V
pv
=
0
(
1
7
)
I
f
w
e
d
if
f
er
en
tia
te
th
e
s
lid
in
g
s
u
r
f
ac
e
S,
w
e
ca
n
w
r
ite
[
2
9
]:
S
̈
=
φ
A
(
t
,
S
,
S
̇
)
+
∅
(
t
,
S
,
S
̇
)
u
̇
(
1
8
)
w
it
h
φ
A
(
t
,
S
,
S
̇
)
=
(
∂
3
P
pv
∂
V
pv
3
)
(
∂
V
pv
∂
t
)
2
+
1
C
1
(
∂
2
P
pv
∂
V
pv
2
)
[
(
∂
I
pv
∂
V
pv
)
(
∂
V
pv
∂
t
)
−
V
pv
L
]
(
1
9
)
∅
(
t
,
S
,
S
̇
)
=
1
C
1
(
∂
2
P
pv
∂
V
pv
2
)
V
0
L
(
2
0
)
w
h
er
e
∂
2
P
pv
∂
V
pv
2
=
2
∂
I
pv
∂
V
pv
+
V
pv
∂
2
I
pv
∂
V
pv
2
∂
3
P
pv
∂
V
pv
3
=
3
∂
2
I
pv
∂
V
pv
2
+
V
pv
∂
3
I
pv
∂
V
pv
3
(
2
1
)
T
h
e
co
n
tr
o
l
o
f
th
e
b
o
o
s
t
co
n
v
er
ter
is
a
b
o
u
n
d
ed
f
u
n
ctio
n
(
0
<u
<1
)
.
W
e
ass
u
m
e
t
h
at
t
h
e
(
1
8
)
s
atis
f
y
co
n
d
itio
n
in
(
1
5
)
,
th
e
co
n
tr
o
l
la
w
g
u
ar
a
n
tees
t
h
e
f
i
n
ite
ti
m
e
co
n
v
er
g
e
n
ce
.
T
h
e
p
r
o
o
f
o
f
th
e
co
n
tr
o
l
la
w
alg
o
r
ith
m
ap
p
r
o
ac
h
is
p
r
esen
ted
in
th
e
ap
p
en
d
ix
.
W
e
ca
n
co
n
s
id
er
th
e
ap
p
lied
co
n
tr
o
l
la
w
an
d
D
ca
n
b
e
d
ed
u
ce
d
f
r
o
m
th
e
eq
u
atio
n
u
=1
-
D,
it
is
g
u
ar
an
teed
th
at
t
h
e
s
y
s
te
m
s
tate
w
ill
h
it
th
e
s
u
r
f
ac
e
an
d
p
r
o
d
u
ce
m
ax
i
m
u
m
p
o
w
er
o
u
tp
u
t p
er
s
is
ten
tl
y
.
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
h
e
s
tr
u
ctu
r
e
o
f
t
h
e
clo
s
ed
lo
o
p
s
y
s
te
m
f
o
r
MA
T
L
A
B
an
d
Si
m
u
li
n
k
,
is
s
h
o
w
n
in
F
ig
u
r
e
5
,
w
h
ic
h
in
cl
u
d
es
th
e
elec
tr
ical
cir
cu
it
o
f
th
e
p
h
o
to
v
o
ltaic
m
o
d
u
le
Sh
ell
So
lar
S7
5
,
w
h
o
s
e
ch
ar
ac
ter
is
tics
ar
e
s
h
o
w
n
i
n
T
ab
le
1
,
th
e
DC
-
D
C
co
n
v
er
ter
B
OOST
w
o
r
k
w
it
h
L
=1
3
0
m
H,
C
1
=1
0
0
0
µF
an
d
C
2
=5
0
0
µF,
lo
ad
R
=2
0
Ω
an
d
th
e
MP
PT
alg
o
r
ith
m
.
T
h
e
s
w
itc
h
i
n
g
f
r
eq
u
e
n
c
y
o
f
t
h
e
b
o
o
s
t
co
n
v
er
ter
is
s
et
to
2
5
KHz
.
T
h
e
co
n
tr
o
ller
p
ar
am
eter
s
ar
e
s
et
to
α
1
=0
.
2
7
an
d
α
2
=0
.
0
5
.
T
h
e
p
r
o
p
o
s
e
d
MPPT
co
n
tr
o
l
is
ev
alu
ated
f
r
o
m
t
h
r
ee
s
ca
s
e
s
in
cl
u
d
in
g
f
i
x
ed
ir
r
ad
iatio
n
,
v
ar
y
in
g
ir
r
ad
iatio
n
an
d
te
m
p
er
atu
r
e.
Fu
r
t
h
er
m
o
r
e,
f
o
r
th
e
s
ak
e
o
f
co
m
p
ar
is
o
n
,
r
esp
o
n
s
es
o
b
tain
ed
w
it
h
2
-
S
MC
b
ased
o
n
s
u
p
er
t
w
is
tin
g
a
lg
o
r
ith
m
(
ST
A
)
ar
e
co
m
p
ar
ed
w
it
h
o
n
e
s
r
esu
lt
in
g
f
r
o
m
t
h
e
1
-
SM
C
(
f
i
x
ed
ir
r
ad
iatio
n
)
.
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
.
3
,
Dec
em
b
er
2
0
2
0
:
284
–
296
290
Fig
u
r
e
5
.
Si
m
u
li
n
k
s
tr
u
ct
u
r
e
o
f
th
e
clo
s
ed
lo
o
p
co
n
tr
o
l o
f
b
o
o
s
t c
o
n
v
er
ter
3
.
1
.
Sta
nd
a
rd
t
est
co
nd
it
io
ns
:
T
=
2
5
°C a
nd
G
=
1
0
0
0
W/
m
2
Fig
u
r
e
6
(
a)
d
is
p
la
y
s
t
h
e
w
a
v
ef
o
r
m
s
o
f
P
pv
,
V
pv
,
V
0
,
I
pv
an
d
D
w
it
h
t
h
e
1
-
SMC
as
an
MP
P
T
C
o
n
tr
o
ller
.
Fi
g
u
r
e
7
(
a)
an
d
d
i
s
p
la
y
s
th
e
w
a
v
ef
o
r
m
s
o
f
P
pv
,
V
pv
,
V
0
,
I
p
v
an
d
D
w
it
h
th
e
2
-
SMC
b
ased
o
n
ST
A
as
an
MP
PT
co
n
tr
o
ller
,
s
o
th
at
th
e
MP
P
is
lo
ca
ted
at
a
p
o
w
er
o
f
7
5
.
0
2
W
.
I
t
is
ea
s
ily
f
r
o
m
th
e
b
elo
w
r
es
u
lt
s
s
ee
n
t
h
at
th
e
s
y
s
te
m
r
ea
ch
e
s
th
at
P
V
m
o
d
u
le
p
o
w
er
an
d
s
h
o
w
s
a
f
ast
r
esp
o
n
s
e
a
n
d
a
g
o
o
d
tr
ac
k
in
g
p
er
f
o
r
m
a
n
ce
.
I
t
o
n
l
y
tak
e
s
m
illi
s
ec
o
n
d
s
to
tr
ac
k
MP
P
.
W
h
en
th
e
MP
P
is
r
ea
ch
ed
,
th
e
s
li
d
i
n
g
s
u
r
f
ac
e
co
n
v
er
g
e
s
to
ze
r
o
Fig
u
r
e
7
(
b
)
.
(
a)
(
b
)
Fig
u
r
e
6
.
Si
m
u
latio
n
r
esp
o
n
s
e
s
(
co
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[
4
1
]
:
S
(
X
,
t
)
=
∂
P
pv
∂
V
pv
(
A
.
1
)
I
f
S(X
,
t)
is
m
ad
e
eq
u
al
to
ze
r
o
,
th
en
th
e
m
ax
i
m
al
p
o
w
er
is
tak
en
.
T
h
e
co
n
tr
o
ller
s
teer
th
e
d
er
iv
ati
v
e
S
to
ze
r
o
,
b
y
ac
tio
n
o
n
t
h
e
d
u
t
y
c
y
cle
u
.
W
e
ca
n
w
r
ite:
S
(
X
,
t
)
=
∂
P
pv
∂
V
pv
=
I
pv
+
V
pv
∂
pv
∂
V
pv
=
0
(
A
.
2
)
0
5
10
15
20
25
30
30
40
50
60
70
80
90
100
110
P
o
w
e
r
M
o
d
u
le
T
im
e
(
s
e
c
)
P
o
w
e
r
M
o
d
u
l
e
(
W
)
0
5
10
15
20
25
30
0
.4
0
.5
0
.6
0
.7
0
.8
0
.9
D
u
t
y
C
y
c
le
T
im
e
(
s
e
c
)
D
u
t
y
C
y
c
l
e
0
5
10
15
20
25
30
35
40
30
40
50
60
70
80
90
100
110
P
o
w
e
r
M
o
d
u
le
T
im
e
(
s
e
c
)
P
o
w
e
r
M
o
d
u
l
e
(
W
)
G
=
1
0
0
0
W
/m
2
G=
5
0
0
W
/m
2
0
5
10
15
20
25
30
35
40
0
.5
5
0
.6
0
.6
5
0
.7
0
.7
5
0
.8
0
.8
5
0
.9
D
u
t
y
C
y
c
le
T
im
e
(
s
e
c
)
D
u
t
y
C
y
c
l
e
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