In
te
r
n
ation
a
l Jou
rn
al
o
f Po
we
r
Elec
tron
ic
s an
d
D
r
ive S
y
stem
(IJ
PED
S
)
V
o
l.
10, N
o.
1, Mar
ch 20
19,
p
p.
137~
1
5
0
IS
S
N
: 2088-
86
94,
D
O
I
:
10.11
59
1
/ij
ped
s
.
v10
.
i
1.pp
1
37-
15
0
137
Jou
rn
a
l
h
o
me
pa
ge
:
ht
tp:
//i
a
e
score
.
com
/
j
o
u
r
na
l
s
/
i
n
d
e
x
.
p
hp/IJ
PED
S
Ou
tput feed
back nonlinear
co
n
trol of three-phase
grid-connected PV
generator
A
.
Y
ah
ya,
H. El
F
ad
il,
M.
O
ul
c
a
i
d
S
y
s
t
e
m
s
En
g
i
neerin
g
L
a
bo
ratory
(
LGS),
Nat
i
onal School o
f
Appl
ie
d S
c
i
e
nces
(
E
N
S
A
), Ibn
Tof
ail
Un
iv
ersi
ty
,
M
o
ro
cco
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
R
e
c
e
i
v
e
d
M
ay
1
6
,
2
018
Re
vise
d A
ug
18,
201
8
A
c
c
e
pte
d
S
ep 3,
2018
Th
is
p
ap
er
a
dd
res
s
es
t
h
e
p
ro
bl
em
o
f
co
ntro
llin
g
t
h
ree-p
h
ase
g
r
i
d
c
o
nn
ec
te
d
P
V
s
y
s
t
e
m
i
n
v
o
lvin
g
a
P
V
a
rrays,
a
v
o
lt
age
so
urce
in
vert
e
r
,
a
g
ri
d
filter
and
an electri
c
gri
d
.
Th
is
p
ap
er
p
res
e
nt
s
t
h
ree
m
a
i
n
c
o
n
t
r
ol object
i
v
es:
i)
e
nsur
i
n
g
th
e
Max
i
m
u
m
pow
er
p
oi
nt
t
racki
ng
(M
PP
T)
i
n
th
e
si
de
o
f
P
V
p
anel
s,
i
i)
gu
arant
e
ei
ng
a
p
ower
f
acto
r
u
nit
in
t
he
s
ide
o
f
t
h
e
g
ri
d,
i
ii)
a
n
d
e
n
suring
t
he
asy
m
ptoti
c
s
tab
i
l
i
t
y
o
f
the
clo
s
ed
l
oo
p
system
.
Int
e
restin
g
l
y
,
th
e
p
r
e
s
ent
st
ud
y
f
eatu
r
e
s
t
he
a
ch
ievem
e
n
t
o
f
th
e
ab
ov
e
en
er
geti
c
g
o
al
w
it
h
out
r
es
orti
ng
to
s
ens
o
rs
o
f
curren
ts
o
f
t
h
e
grid.
To
t
his
end
,
a
n
o
u
t
p
u
t
-f
eed
b
ack
c
on
trol
stra
te
gy
c
om
b
i
ning
a
s
ta
te
o
bse
r
v
e
r
a
n
d
a
n
o
n
lin
e
a
r
c
on
trol
l
a
w
s
is
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e
lo
ped.
T
he
p
rop
o
s
e
d
ou
tp
ut-f
ee
d
b
ack
c
on
tr
ol
s
t
r
ategy
is
b
ac
ke
d
by
a
f
o
r
m
al
analysis
s
h
owi
n
g
that
all
c
o
nt
rol objectives are actuall
y
ach
iev
e
d.
K
eyw
ord
s
:
MPP
T
N
o
n
line
a
r
con
t
rol
N
o
n
line
a
r
obse
r
ver
O
u
t
p
u
t
f
ee
dbac
k
c
on
tro
l
Re
new
a
b
l
e
ene
r
gy
St
a
b
ili
t
y
a
n
a
lysi
s
Co
pyri
gh
t © 2
019 In
stit
u
t
e
of Advanced
En
gi
neeri
n
g
an
d
S
c
ien
ce.
All
rights
res
e
rv
ed.
Corres
pon
d
i
n
g
Au
th
or:
A
bde
lha
f
i
d
Y
ahya
,
S
y
stem
s Eng
i
n
eer
ing
Labora
t
or
y (LG
S
),
N
a
ti
ona
l S
c
ho
ol o
f
A
p
p
l
i
e
d S
c
ience
s
(EN
S
A
), Ibn
Tofa
il U
n
ive
r
si
ty
,
14
0
00 K
é
n
itra
, Moroc
c
o
.
Em
ail:
yaab
ha
2
@
gma
il.c
o
m
1.
I
N
TR
OD
U
C
TI
O
N
I
t
w
a
s
v
er
y
c
l
e
a
r
fr
om
r
e
cent
stud
ies
an
d
d
o
c
um
enta
t
i
o
n
t
h
a
t
fo
ssi
l
f
ue
ls
w
ould
las
t
o
n
l
y
a
few
m
o
re
dec
a
de
s.
T
he
c
ost
of
f
oss
i
l
fu
els
ha
s
b
e
c
o
m
e
a
m
ajor
c
hal
l
en
ge
f
o
r
a
ll
o
f
h
uma
n
it
y.
N
ot
o
nly
t
h
e
ec
on
o
m
i
c
val
u
e
b
u
t
als
o
t
he
e
n
v
i
r
onm
enta
l
im
pac
t
s
of
f
ossil
fue
l
s
have
c
lea
r
ly
p
ush
e
d
us
t
ow
ards
a
l
t
er
na
t
i
ve
s.
T
he
bi
gge
s
t
a
l
t
erna
t
i
ves
tha
t
can
r
ea
lly
m
a
k
e
a
d
i
ffe
r
enc
e
t
o
sus
t
a
i
n
abi
l
i
t
y
,
su
ch
a
s
red
u
ci
n
g
g
r
e
e
nh
ou
se
g
as
em
issi
on
s
a
nd
th
e
l
o
n
g
-term
e
c
on
om
y,
a
re
r
ene
w
able
e
ne
r
gy
so
urc
e
s
(
R
E
S
)
s
uch
a
s
w
in
d
a
nd
s
o
la
r
e
n
erg
y
[1],
[
2].
P
V
s
ystem
i
s
i
ncr
eas
in
g
a
s
a
r
e
n
ew
a
b
l
e
s
ourc
e
du
e
t
o
its
a
d
v
an
ta
ges of l
i
t
tl
e
m
a
in
te
nanc
e,
a
bse
n
ce o
f
movi
n
g
m
e
c
ha
nica
l
pa
rts,
no
no
ise
a
n
d
n
o
p
oll
u
ta
n
t
e
m
i
ss
i
on
[
3
]
.
F
u
r
t
he
rm
ore,
its
c
ost
i
s
d
ec
rea
s
in
g
o
v
er
t
he
nex
t
t
en
y
ea
rs
,
w
h
i
l
e
th
e
de
pl
oym
ent
o
f
P
V
system
s
c
ont
i
nue
s
t
o
i
nc
r
e
a
s
e
r
a
pi
d
l
y.
W
it
h
i
n
c
r
ea
sin
g
P
V
pene
tr
at
ion
on the
grid, eve
nt
ual
l
y re
ac
h
i
n
g
hun
dr
eds o
f
g
i
g
aw
a
t
t
s (
G
W) of in
ter
c
o
nnec
t
e
d
ca
p
ac
it
y, a varie
t
y
of
m
e
t
h
o
d
s
mus
t
b
e
co
ns
i
d
ere
d
a
nd
imp
l
em
ented
at
d
i
f
fere
n
t
s
ca
l
e
s
f
or
a
r
elia
b
l
e
an
d
c
o
st
-
e
ff
ecti
v
e
con
n
ec
tio
n
int
o
pow
e
r
gr
i
d.
D
i
ffere
n
t
c
o
nt
r
o
l
stra
teg
y
f
or
t
hre
e
-p
ha
se
g
ri
d
c
o
n
n
ec
te
d
o
f
P
V
m
odules
h
as
b
ee
n
la
rgel
y
dea
lt
w
ith
in
t
he
s
pec
i
alis
t
litera
ture
i
n
the
las
t
f
ew
y
ea
rs
[
4]-[9].
N
ev
e
r
t
h
el
e
ss,
g
ood
i
n
t
e
g
r
at
io
n
o
f
m
e
d
i
u
m
o
r
l
a
r
g
e
P
V
syste
m
i
n
t
h
e
g
r
id
m
ay
t
here
fore
r
e
qui
r
e
a
d
d
iti
ona
l
fu
nct
i
o
n
al
i
ty from
t
h
e
in
verte
r
,
such
a
s
con
t
ro
l
o
f
r
e
acti
v
e
power
.
More
o
v
er,
the
i
n
cre
a
se
i
n
the
a
v
era
g
e
si
z
e
o
f
a
P
V
s
yste
m
may
l
e
a
d
t
o
ne
w
stra
teg
i
es
s
uc
h
a
s
elimi
n
a
t
in
g
t
h
e
D
C
-D
C
c
o
n
v
erter
,
w
hi
c
h
i
s
u
s
ua
lly
p
la
ced
b
e
t
w
e
e
n
the
P
V
g
e
n
era
t
or
a
nd
i
nver
t
e
r
,
and
mo
v
i
ng
t
he
M
PP
T
t
o
t
h
e
i
nve
rt
e
r
,
wh
i
c
h
l
ead
s
to
i
n
c
re
a
s
e
d
s
i
m
pl
ic
i
t
y, ove
ral
l
e
ffic
i
e
nc
y
a
n
d
a
c
o
s
t
r
e
duc
t
i
o
n
.
Th
e
s
e
t
w
o
fe
atu
r
es
a
re
p
re
se
n
t
i
n
t
h
e
th
re
e-ph
a
s
e
inv
e
rt
er
t
h
a
t
i
s prese
n
te
d i
n
t
h
i
s pa
per
, w
it
h
the
a
d
dit
i
on of a
P
e
rturb a
nd O
b
ser
v
e (
P
&O) M
P
P
T
a
lgori
t
h
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
13
7 –
15
0
13
8
The
presen
t
p
a
pe
r
is
t
he
n
fo
c
u
sin
g
o
n
t
h
e
pro
b
l
e
m
of
c
o
n
t
r
oll
i
n
g
thr
ee
p
h
ase
grid-
c
o
nnec
t
e
d
P
V
power
g
e
n
er
at
i
o
n
system
s.
T
he
c
o
n
t
r
o
l
o
b
j
e
ctive
s
a
r
e
t
hr
e
e
fo
l
d
:
(
i)
g
l
o
bal
a
s
ymp
t
ot
i
c
s
ta
bi
lit
y
of
t
h
e
w
ho
le
clo
s
ed-l
o
op
c
o
ntr
o
l
s
y
st
em
;
(ii)
achie
vem
e
nt
o
f
the
M
P
PT
f
or
t
h
e
P
V
a
r
r
a
y
;
a
n
d
(
i
i
i
)
e
n
s
u
r
i
n
g
a
g
r
i
d
con
n
ec
tio
n
w
i
th
u
ni
t
y
P
ow
er
F
actor
(
PF
).
T
hese
o
b
j
ec
t
i
ve
s
sho
u
l
d
be
a
ch
i
e
ve
d
in
s
pite
t
he
c
l
i
ma
t
i
c
v
a
r
i
ab
les
(tem
pera
ture
a
nd
r
a
d
i
a
tio
n)
c
han
g
es.
T
o
t
h
i
s
e
nd,
a
n
o
n
l
i
n
e
ar
c
ont
r
o
l
l
er
i
s
deve
l
ope
d
us
in
g
L
y
a
p
u
n
o
v
d
e
s
ig
n
tech
n
i
q
u
e.
A
t
heore
tica
l
a
na
l
y
s
i
s
is
d
eve
l
op
e
d
t
o
sh
ow
t
ha
t
t
h
e
c
o
n
t
rol
l
er
a
ctua
l
l
y
me
ets
i
t
s
ob
jec
t
i
v
es
a
f
act
tha
t
is c
onf
i
r
me
d by s
i
m
u
l
a
ti
on.
The
pa
p
e
r
i
s
o
rga
n
ize
d
a
s
fol
l
ow
s:
t
he
t
h
r
ee
-phase
g
ri
d
c
o
n
n
e
c
t
ed
P
V
system
i
s
describe
d
and
mode
le
d
i
n
S
ec
tio
n
2.
S
e
c
tion
3
is
d
ev
o
t
ed
t
o
c
ontro
l
l
e
r
d
e
sig
n
a
nd
a
n
al
y
s
i
s
.
Th
e
con
t
rol
l
e
r
t
r
ack
i
n
g
perform
ance
s ar
e ill
ust
r
a
t
ed b
y
nume
r
ica
l
s
i
m
ulat
ion
i
n
S
e
c
ti
o
n
4
.
2.
RESEARCH
M
ETH
O
D
2.1.
Syst
em
overv
i
e
w
The
m
a
i
n
c
irc
u
it
o
f
t
h
ree
-
phas
e
g
ri
d-c
o
n
n
ect
ed p
ho
tov
o
lta
i
c
s
ys
t
e
m
a
s
s
h
o
w
n
i
n
F
i
g
u
r
e
1
.
I
t
c
o
n
s
i
s
t
s
of
a
P
V
a
rra
ys
;
a
D
C
l
i
nk
c
a
p
a
c
i
t
or
C
;
a
th
re
e
phase
i
n
v
e
r
ter
(
incl
ud
ing
si
x
pow
er
s
e
m
i
c
on
d
u
c
t
ors)
t
ha
t
i
s
base
d
u
p
o
n
t
o
e
n
s
u
r
e
a
D
C-
A
C
pow
er
c
on
ver
s
i
o
n
a
n
d
u
n
i
t
y
p
ow
e
r
f
a
c
t
or;
a
n
i
nd
uc
tor
filte
r
L
w
i
th
a
s
e
r
ies
resista
n
ce
r
,
and
an
e
l
e
c
t
ric
g
r
i
d
.
The
co
ntr
o
l
i
npu
ts
o
f
the
s
ys
t
e
m
are
a
PWM
si
gna
ls
,
and
t
aki
ng
val
u
es
i
n
the
set
{0,
1
}.
The
grid v
o
lta
ge
s
,
and
c
onsti
t
u
t
e
a
t
hree
-phase
b
a
l
anc
e
d s
y
stem
.
2.2.
PV
A
rray
mod
e
l
A
n
e
qu
i
v
ale
n
t
circ
u
i
t
f
o
r
a
PV
c
ell
as
s
how
n
i
n
F
i
gur
e
2.
I
ts
c
ur
rent
c
h
a
rac
t
erist
i
c
ca
n
be
f
o
und
in
ma
ny plac
es [1
0
]-[1
3
]
and
presen
ts the
fo
l
l
o
w
i
n
g
e
xpre
ssi
o
n
.
F
i
gur
e 1.
Thr
ee-
ph
a
s
e gr
id
c
o
nnec
t
e
d
P
V system
sh
s
s
sat
ph
R
IR
V
AkT
IR
V
q
I
I
I
1
exp
(
1
)
Whe
r
e
298
1
1
exp
0
3
T
K
I
I
T
T
Ak
qE
T
T
I
I
I
phr
ph
r
G
r
satr
sat
(
2
)
Th
e
me
an
in
g
an
d
t
ypi
ca
l
v
a
lue
s
o
f
t
h
e
p
a
ra
me
t
e
rs
g
i
v
en
b
y
(1
)
a
nd
(2)
can
b
e
fou
n
d
i
n
m
a
ny
pla
c
e
s
(see
e
.g.
[14
]
,
[15],
[
16]).
A
i
s
di
od
e
i
d
ea
l
fac
t
o
r
,
k
i
s
Bo
l
t
z
m
ann
c
ons
ta
n
t
,
T
i
s
t
e
mp
era
t
u
r
e
on
a
b
s
o
l
ut
e
sca
l
e
in
K
,
q
is
e
le
c
t
ro
n
char
ge
a
n
d
λ
i
s
the
ra
d
i
a
tio
n
i
n
k
W/m
2
,
I
ph
r
i
s
the
shor
t
-
circ
ui
t
curre
nt
a
t
2
9
8
K
a
nd
1
kW/m
2
,
K
I
=
0.00
1
7
A
/
K
is
t
he
c
u
r
rent
t
e
m
per
a
ture
c
oe
ff
i
c
ie
n
t
a
t
I
ph
r
,
E
G0
i
s
t
h
e
ba
nd
ga
p
for
sili
con,
i
s
reference
tem
p
erature,
I
sa
t
r
i
s
c
e
ll
sat
u
rat
i
on
curr
ent
a
t
T
r
.
P
V
a
rray
cons
ists
o
f
N
s
c
el
ls
i
n
series
f
orm
e
d
t
h
e
pane
l
a
nd
o
f
N
p
p
a
n
el
s
in
p
a
r
al
l
e
l
ac
co
rdin
g
to
t
h
e
r
a
t
e
d
p
o
w
er
r
e
q
ui
red
.
T
h
e
o
u
t
p
ut
v
o
lta
ge
a
n
d
c
urre
nt
c
an
be
g
ive
n
by t
h
e
fol
l
o
w
i
ng
(3)
and (
4
).
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
P
o
w
Elec
&
D
r
i
S
y
st
I
S
S
N
:
2088-
86
94
Ou
t
p
ut
f
eed
ba
ck n
o
n
li
n
e
a
r
c
o
n
t
rol
of
t
h
re
e
-
ph
a
s
e g
r
id
-c
onne
ct
e
d
PV
g
e
n
e
ra
to
r (
A
.Ya
h
y
a
)
13
9
I
R
V
N
v
s
d
s
pv
(
3
)
I
N
i
p
pv
(
4
)
Fi
g
u
r
e 2
.
So
l
ar ce
l
l
ci
rc
ui
t
di
ag
ram
Th
e
p
hot
ovo
l
t
a
i
c
g
e
n
e
rato
r
c
o
nsid
e
r
ed
i
n
t
h
i
s
p
a
p
e
r
c
o
n
si
st
s
o
f
s
e
v
er
a
l
N
U
-
18
3E1
modu
les.
T
he
c
o
rre
spo
n
d
i
n
g
ele
c
tr
ical
c
h
a
ra
cter
i
s
t
i
cs
o
f
PV
m
odu
les
ar
e
sh
ow
n
in
T
abl
e
1
.
T
h
e
associ
at
ed
p
o
w
e
r
-v
o
l
t
a
g
e
(
P
-V
)
c
h
ara
c
t
e
ri
st
i
c
s
und
er
c
h
a
ngi
ng
c
l
i
ma
ti
c
con
d
iti
on
s
(
t
emp
e
r
a
tur
e
a
n
d
r
adia
t
i
on)
a
r
e
s
how
n
in
F
i
g
ur
es
3
an
d
4
.
T
h
i
s
hi
gh
lig
ht
s
th
e
M
a
x
i
mu
m
Po
wer
Po
i
n
t
(
M
P
P
)
M
1
t
o
M
5
,
w
h
ose
c
o
or
d
i
na
te
s
ar
e
g
i
ven
i
n
T
ab
le
3
.
The
da
ta
i
n
Ta
bl
e
1
to
T
ab
le
3
w
il
l
be
u
se
d
for
simu
la
ti
on.
Tab
l
e
2
sh
ow
s
t
h
e
m
a
in
c
ha
r
a
c
t
er
is
t
i
c
s
o
f
t
h
e
P
V
a
r
r
a
y,
d
esigne
d
us
ing
S
h
ar
p
N
U
-
183
E1
m
od
u
l
es
c
o
n
n
ec
ted
i
n
a
p
ro
pe
r
se
ries-
p
a
r
al
lel,
m
ak
in
g
up
a
pea
k
i
n
s
ta
ll
e
d
p
ower
o
f
7
1
k
W.
A
s
ther
e
i
s
n
o
DC-DC
c
o
n
v
erter
betw
ee
n
the
PV
g
e
n
e
r
at
or
a
n
d
t
he
i
nve
r
t
er,
the
P
V
a
rr
a
y
c
on
fi
g
u
r
a
tio
n
s
h
ou
ld
b
e
c
hose
n
s
uc
h
tha
t
t
h
e
o
u
tp
ut
v
ol
ta
g
e
o
f
th
e
ph
ot
ovo
l
t
a
i
c
g
en
e
r
at
o
r
i
s
ad
ap
t
e
d
t
o
the
r
e
q
u
i
r
e
m
e
nts
o
f
t
he
i
nver
t
er
.
I
n
t
his
case
a
38
0V
g
r
i
d
h
a
s
bee
n
c
h
o
sen,
s
o
the
in
ve
r
t
er
w
ou
l
d
n
e
e
d
a
t
le
as
t
5
70V
D
C
bus
i
n
or
der
t
o
b
e
a
b
le
t
o
o
p
er
at
e
c
o
r
r
e
ctly.
The
m
i
ni
m
u
m
num
ber
of
m
o
d
u
l
e
s
co
nnec
t
e
d
i
n
se
r
i
es
s
h
o
u
l
d
b
e
de
t
e
r
m
i
n
e
d
by
the
v
a
lue
of
t
he
m
i
ni
m
u
m
D
C
bus
v
olta
ge
a
n
d
t
he
w
or
st-
c
a
s
e
c
l
im
at
ic
c
o
n
d
i
tio
ns.
The
P
V
a
r
r
a
y
w
a
s
fo
u
nd
t
o
r
eq
ui
r
e
28
se
r
i
es
c
onne
c
t
e
d
m
odu
les
per
st
r
i
ng.
Ta
b
l
e
1.
E
lec
t
r
i
cal
s
pe
ci
f
i
ca
ti
ons
f
or
t
he
s
o
l
a
r
m
odu
le
N
U
-
183
E
1
P
a
r
a
m
e
t
e
r
Sym
bol
V
a
l
ue
Ma
xi
m
u
m
P
o
we
r
P
m
183W
S
hor
t
c
i
rc
uit
c
u
rr
e
n
t
I
sc
r
8
.
48A
O
p
e
n
c
i
r
c
u
it
volt
a
ge
V
oc
30.
1V
Ma
xi
m
u
m
powe
r
volta
g
e
V
m
23.
9V
Ma
xi
m
u
m
powe
r
c
urre
nt
I
m
7
.
66A
Nu
m
b
e
r
of
pa
r
a
ll
e
l
m
odule
s
N
p
1
Nu
m
b
e
r
o
f
s
e
r
i
e
s
m
odul
e
s
N
s
48
Figure
3.
(
P-
V)
c
har
acte
r
ist
i
cs
o
f The
P
V
Genera
t
or
(
N
P
=
1
4
A
nd
N
S
=
2
8)
w
ith
c
o
n
s
t
an
t
te
mpe
r
at
ur
e
and
va
r
y
i
n
g
r
a
dia
t
i
o
n
F
i
gur
e
4.
(
P
-
V)
c
har
a
c
t
e
r
istics
of
T
he
P
V
G
e
ner
a
t
o
r
(
N
P
=
14
A
nd
N
S
=28)
w
ith
c
ons
tan
t
r
adia
t
i
o
n
a
nd
va
r
y
in
g
tem
p
er
a
t
ur
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
13
7 –
15
0
14
0
Tab
l
e 2
: P
V
Ar
r
ay
s
pe
cifica
t
i
ons u
si
n
g
shar
p
N
U
-
18
3E1
Tab
l
e
3.
M
ax
i
m
um
pow
e
r
poin
t
s (MP
P
): in
F
i
gure
3
and F
i
gur
e
4
Pa
r
a
m
e
t
e
r
Sym
bol
V
alue
T
o
ta
l pea
k
powe
r
P
tm
71
kW
N
u
m
b
e
r
o
f
s
e
r
i
e
s
s
trings
N
S
28
Nu
m
b
e
r
o
f
pa
r
a
l
l
e
l
N
P
14
Nu
m
b
er
o
f
P
V
p
an
els
N
3
9
2
V
o
lta
g
e
i
n m
a
xi
mum
pow
e
r
V
m
6
64V
C
u
rre
nt
p
e
a
k
I
m
1
07A
MPP
V
m
[
V
]
P
m
[ K
W
]
M
1
6
64.
2
71
M
2
6
61
57.
2
M
3
6
48.
8
35.
1
M
4
6
64.
2
71.
5
M
5
5
84
61.
7
2.3.
Mo
d
e
ling
o
f
t
h
r
ee-
p
ha
se
g
rid-
co
nnecte
d
PV
sy
stem
The
s
t
a
t
e-spa
c
e
m
ode
l
of
a
t
hr
ee-
phase
g
r
i
d-
co
nnec
t
e
d
p
ho
tov
o
l
t
a
i
c
sys
t
em
s
h
o
wn
i
n
F
i
gur
e
1
ca
n
be
obta
i
ne
d b
y
the
d
y
n
am
ic e
q
u
at
ion
s
d
escri
b
ed
i
n (5a
)-(5d)
:
ga
c
b
a
pv
a
a
v
L
u
u
u
L
v
i
L
r
i
dt
d
1
2
3
(
5
a
)
gb
c
b
a
pv
b
b
v
L
u
u
u
L
v
i
L
r
i
dt
d
1
2
3
(
5
b
)
gc
c
b
a
pv
c
c
v
L
u
u
u
L
v
i
L
r
i
dt
d
1
2
3
(5
c
)
c
c
b
b
a
a
pv
pv
i
u
i
u
i
u
C
i
C
v
dt
d
1
1
(
5d)
W
h
ere:
on
K
off
K
off
K
on
K
u
iL
iH
iL
iH
i
:
;
:
0
:
;
:
1
A
p
p
l
y
i
ng
t
he
C
onc
o
r
d
i
a
t
r
an
sform
a
t
i
o
n
t
o
(5a-
d),
the
i
n
sta
n
t
a
ne
ou
s
m
o
de
l
in
s
ta
t
i
onary
c
oor
di
na
te
s
is
g
i
v
e
n
s
as
(
6a)-(6b):
g
pv
v
L
u
L
v
i
L
r
i
dt
d
1
(6
a
)
g
pv
v
L
u
L
v
i
L
r
i
dt
d
1
(
6b)
pv
pv
i
C
i
u
i
u
C
v
dt
d
1
1
(6
c
)
wh
er
e
abc
o
abc
o
i
i
;
gabc
o
abc
o
g
v
v
;
abc
o
abc
o
u
u
(
7
a
)
A
nd t
h
e
trans
f
orm
a
ti
on m
a
tri
x
o
abc
is
g
iv
e
n
b
y
:
2
1
2
1
2
1
2
3
2
3
0
2
1
2
1
1
3
2
o
abc
(
7
b
)
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
I
S
S
N
:
2088-
86
94
O
u
t
p
u
t
fee
d
b
a
c
k
non
l
i
ne
ar c
o
ntro
l o
f
three
-
p
has
e gri
d
-c
on
n
e
cted
PV ge
ne
rat
o
r (
A
.Y
ahy
a)
14
1
A
c
cordi
n
g
to
t
he
t
hre
e
-p
ha
s
e
i
ns
ta
nta
n
e
o
u
s
acti
v
e
an
d
rea
c
ti
ve
pow
e
r
t
he
ory,
t
he
i
ns
tan
t
a
n
e
o
u
s
ac
t
i
ve
p
ow
e
r
P
a
nd
t
h
e
ins
t
a
n
tane
o
u
s
rea
c
t
i
ve
pow
er
Q
c
a
n
b
e
der
i
ve
d
in
t
hre
e
-pha
se
abc
c
oor
di
na
tes
fo
r
the
three
-
p
h
ase
p
h
o
to
vo
l
t
ai
c
g
r
id-
c
on
nec
t
e
d
i
nv
erte
r:
c
gc
b
gb
a
ga
i
v
i
v
i
v
P
(
8
a
)
3
/
b
ga
gc
a
gc
gb
c
gb
ga
i
v
v
i
v
v
i
v
v
Q
(
8b)
In
t
h
e
αβ
c
oor
dina
tes,
t
he
i
nsta
n
t
a
n
eo
us
a
c
t
i
v
e
p
o
w
e
r
P
a
nd
the
i
n
sta
n
tan
e
ou
s
re
acti
v
e
p
o
w
e
r
Q
ca
n
be
e
xpre
s
se
d as
f
ol
low
i
ng
:
i
v
i
v
P
g
g
(
9
a
)
i
v
i
v
Q
g
g
(
9
b
)
3.
OUT
P
U
T
FEEDB
AC
K
CO
N
T
ROLLER DESIGN
The
m
o
del
(6a
-
c)
i
s
usef
u
l
t
o
bu
il
d
an
a
cc
u
r
ate
sim
u
l
a
tor
for
t
he
s
t
u
d
i
e
d
s
ys
tem
h
o
w
e
v
e
r
i
t
i
s
not
a
d
eq
u
a
t
e
t
o
el
ab
o
r
at
e
a
co
nt
in
uou
s
c
ont
ro
lle
r
a
s
i
t
i
nvo
l
v
es
a
bi
na
ry
i
np
ut
u
a
nd
u
.
F
o
r
contro
l
de
s
i
g
n
pur
pose,
i
t
is
m
or
e
c
o
n
v
en
ie
nt
t
o
c
ons
i
d
er
t
he
f
ol
l
o
w
i
ng
a
ve
rag
e
d
m
od
e
l
,
o
b
ta
i
n
ed
b
y
ave
r
ag
in
g
t
h
e
m
ode
l
(6a-
c)
over
one
s
w
itch
i
ng pe
ri
od [1
7].
g
v
L
L
x
x
L
r
x
1
3
1
1
(
1
0
a
)
g
v
L
L
x
x
L
r
x
1
3
2
2
(
10b)
pv
i
C
x
x
C
x
1
1
2
1
3
(
1
0
c
)
wh
ere
1
x
,
2
x
,
3
x
,
a
nd
d
e
n
ote
t
h
e
ave
r
age
val
u
es
o
f,
r
espe
c
t
i
v
e
l
y
i
,
i
,
pv
v
,
u
a
nd
u
.
T
h
e
s
i
g
n
a
l
s
a
n
d
,
ca
ll
e
d
d
u
t
y
ra
ti
os
w
h
i
ch
b
el
on
g
t
o
1
,
0
,
a
r
e
c
o
n
s
id
e
r
ed
a
s
t
h
e
i
npu
t
s
o
f
t
h
e
system
(
10a-
c
)
.
O
n
t
he
b
as
is
o
f
(1
0a-c
),
t
he
n
ex
t
tw
o
s
ubs
ec
t
i
o
n
s
w
i
l
l
b
e
de
vo
t
ed
t
o
t
h
e
ob
se
rv
er
d
e
s
i
g
n
a
n
d
the
n
onl
in
e
a
r c
o
nt
ro
l
l
e
r d
e
si
gn
.
3.1.
Observer
desi
gn
Le
t
us
now
c
o
n
s
i
de
r
i
n
(
1
0
a-
c
)
t
ha
t
is
t
he
o
nl
y
m
easura
b
le
v
a
r
i
a
b
le.
T
h
en
t
he
f
o
l
low
i
ng
n
on
lin
ea
r
obs
erve
r is pr
opos
e
d
:
g
v
L
L
x
x
L
r
x
1
ˆ
ˆ
ˆ
3
1
1
(
1
1
a
)
g
v
L
L
x
x
L
r
x
1
ˆ
ˆ
ˆ
3
2
2
(
11b)
)
ˆ
(
1
ˆ
ˆ
1
ˆ
3
3
2
1
3
x
x
i
C
x
x
C
x
pv
(
1
1
c
)
Wh
ere
1
ˆ
x
,
2
ˆ
x
,
3
ˆ
x
re
prese
n
t
t
h
e
e
s
tim
ates of t
h
e s
t
at
e varia
b
le
s an
d
0
be
i
n
g
the
o
bs
erve
r
de
s
i
g
n
parameter.
Let
us
i
ntroduce
the
esti
m
a
tion
errors
1
1
1
ˆ
x
x
z
,
2
2
2
ˆ
x
x
z
and
3
3
3
ˆ
x
x
z
. The
n, fr
o
m (10a-
c
)
and
(1
1a-c
), on
e
h
as:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
13
7 –
15
0
14
2
L
z
z
L
r
z
3
1
1
(
1
2
a
)
L
z
z
L
r
z
3
2
2
(
12b)
3
2
1
3
1
z
z
z
C
z
(
1
2
c
)
F
r
om
t
he
e
rror syste
m
(
12a-
c
),
one
c
a
n
state
t
he
fo
l
l
o
w
i
ng:
Prop
ositi
o
n
:
C
ons
ide
r
the
est
ima
t
i
o
n e
rror system
(12a
-c), obta
i
ned
b
y
c
o
m
b
i
n
ing
t
h
e
sys
t
em
(10a
-c) and
the
no
n
l
i
n
ea
r
ob
se
rver
(11a
-c).
The
n,
0
,
0
,
wh
at
e
v
e
r
t
h
e
i
niti
al
c
ond
iti
on
s, t
h
e
s
ta
te e
st
i
m
ati
o
n
e
r
ro
r
T
z
z
z
z
,
,
3
2
1
c
onver
g
e
s
e
xp
one
n
tia
l
l
y
to z
e
r
o.
Proof:
Consi
d
er
the
f
ol
low
i
n
g
q
ua
drat
ic
L
y
a
pu
n
ov fu
nc
ti
o
n
:
3
1
2
2
1
i
i
o
z
V
(
1
3
)
Its der
i
va
t
i
ve
,
u
s
in
g (
12a-c
), ca
n
b
e
ob
t
a
ine
d
as
fol
l
ow
s:
3
2
2
3
1
1
2
3
2
2
2
1
-
-
-
z
z
z
z
z
z
L
r
z
L
r
V
o
(
1
4
a
)
W
ith
C
L
C
L
1
1
,
1
1
2
1
(
14b)
U
s
ing
Y
o
u
ng’s
I
n
e
qua
li
ty
(
14
a)
bec
ome
s
2
3
2
2
2
2
2
2
3
1
2
1
1
1
2
3
2
2
2
1
2
2
1
2
2
1
z
z
z
z
z
z
L
r
z
L
r
V
o
(
15)
wher
e
1
an
d
2
bei
n
g any
rea
l
p
osit
ive
co
ns
t
a
nts.
As
1
,
0
a
nd
1
,
0
(dut
y r
a
ti
o fu
nc
ti
o
n
s),
it
f
ol
l
o
w
s
t
ha
t
0
1
a
nd
0
2
w
h
e
r
e
C
L
1
1
0
(
1
6
)
F
r
om
(
15)
b
ecom
e
s,
using
(16)
2
3
3
2
2
2
2
1
1
z
k
z
k
z
k
V
o
(
1
7
a
)
Whe
r
e
1
0
1
2
L
r
k
2
0
2
2
L
r
k
2
1
0
3
2
k
(
17b)
Let
1
and
2
to
b
e c
h
o
s
en a
s fo
l
l
o
w
s:
r
L
2
0
1
a
nd
r
L
2
0
2
.
Then,
the
obse
r
ver
des
i
g
n
pa
r
am
et
er
ca
n be
c
ho
sen
so
t
hat
0
where:
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
I
S
S
N
:
2088-
86
94
O
u
t
p
u
t
fee
d
b
a
c
k
non
l
i
ne
ar c
o
ntro
l o
f
three
-
p
has
e gri
d
-c
on
n
e
cted
PV ge
ne
rat
o
r (
A
.Y
ahy
a)
14
3
2
1
0
0
2
(
1
8
)
Therefore
0
1
k
,
0
2
k
and
0
3
k
.
It
f
ol
l
o
w
s
t
ha
t
(17a)
c
a
n be
re
w
ritt
e
n
a
s
fo
ll
ow
s:
o
o
kV
V
(
1
9
)
wher
e
3
2
1
,
,
min
2
k
k
k
k
(
2
0
)
Fr
om
(
19), it is
ea
s
y to sh
o
w
tha
t
:
)
(
)
(
0
)
(
0
t
V
e
t
V
o
t
t
k
o
(
2
1
)
w
h
ic
h yie
l
ds t
h
a
t
)
(
t
V
o
is
e
x
p
o
n
e
n
t
i
a
lly
v
a
n
i
s
hi
ng.
It foll
ow
s tha
t
0
)
(
lim
t
z
t
, w
hi
c
h
in t
u
rn
s
h
o
w
s
t
ha
t the
est
i
ma
t
e
s co
nv
e
r
ge
t
ow
a
r
d th
e
i
r
true
v
al
ues.
T
his en
ds
t
he
pro
o
f
o
f
t
he
P
rop
o
si
ti
o
n
. I
n the
next s
ubsec
t
i
on,
w
e
foc
u
s
o
n
ela
bo
r
a
ting
a
con
t
ro
l
l
e
r
tha
t
st
a
b
il
iz
e
s
t
he sys
tem
.
3.2.
Non
l
in
ear
con
t
rol
l
er
de
sign
W
i
t
h
t
he
a
im
o
f
des
i
gn
a
n
a
ppr
opria
te
c
o
n
t
ro
l
for
the
m
o
del
(10
a-
c)
d
es
cribe
d
i
n
prev
i
ous
s
ec
ti
o
n
,
the c
o
n
t
r
o
l
o
b
j
ecti
v
es, t
h
e co
ntr
o
l
de
si
g
n
a
n
d
s
ta
bi
l
ity an
a
l
y
s
is w
il
l
be in
v
e
stiga
t
e
d
i
n
t
h
i
s
Se
c
tio
n, t
a
k
ing int
o
a
c
c
o
unt
t
he
n
o
n
li
n
e
ar
f
e
a
t
u
re
a
n
d
t
h
e
mult
i
-
i
npu
t
mu
lti
-o
utp
u
t
(
M
IMO
)
a
spec
t
o
f
t
he
s
ys
tem
.
I
n
or
der
t
o
defi
ne
t
he
c
o
n
t
ro
l
stra
t
e
g
y
,
the
firs
t
st
e
p
i
s
to
e
sta
b
l
i
sh
c
on
tro
l
o
b
j
e
c
t
i
v
es,
w
h
ic
h
ca
n
be
s
um
ma
rize
d
as
fo
l
l
ow
s
:
a.
Ma
ximum
pow
e
r
poi
nt tra
c
k
i
n
g (MP
P
T)
o
f P
V
a
rra
y
s,
b.
U
n
i
t
y
p
o
w
e
r
fa
ctor (
PF
) in
t
he
grid,
c.
A
s
ympt
o
tic sta
bi
l
ity
o
f the
w
h
ole s
y
stem
.
I
t
i
s
w
o
rt
h
no
t
i
ng
t
h
a
t
a
ll
o
b
j
e
c
t
i
v
e
s
m
ust
be
a
c
h
ie
ve
d
w
i
tho
u
t
s
en
si
ng
a
l
l
v
a
ria
b
le
s.
T
he
P
V
vo
l
t
a
g
e
3
x
is
t
h
e
o
n
l
y
m
easur
ab
le va
r
ia
b
l
e.
Th
e
fi
rst
co
nt
ro
l
obj
ec
tiv
e
i
s
t
o
e
n
f
o
rce
t
h
e
re
al
p
o
w
er
P
t
o
trac
k
the
m
a
xim
u
m
pow
er
p
o
i
nt
P
M
.
It’s
alre
ad
y
poi
n
t
out
;
tha
t
t
h
i
s
p
o
w
er
c
an
b
e
c
o
n
t
ro
ll
e
d
b
y
t
h
e
α
-axis
c
urr
e
nt
i
α
a
n
d
β
-a
xi
s
curre
n
t
i
β
(
se
e
(9
a)
).
I
n
th
i
s
p
a
p
er
t
he
M
P
P
T
a
lg
ori
t
hm
b
a
s
ed
o
n
the
P
e
rt
urb
a
nd
O
b
se
rve
(
P
&O)
technique
[18]
i
s
res
o
rted
t
o
gene
ra
te
t
h
e
c
oeffi
c
i
e
n
t
(
in
vo
lve
d
i
n
(
22)
)
so
t
ha
t
t
h
e
a
c
ti
ve
p
ow
e
r
P
t
r
a
ck
s
it
s
max
i
m
u
m
v
a
l
u
e
i
.
e.
M
P
P
.
The
se
co
n
d
c
o
n
tr
ol
o
b
j
ec
t
i
ve
m
ea
ns
t
ha
t
the
gr
id
c
ur
rents,
a
i
,
b
i
a
nd
c
i
s
h
o
u
l
d
b
e
s
i
n
u
s
o
i
d
a
l
a
n
d
i
n
pha
se
w
it
h
t
h
e
A
C
g
ri
d
v
o
l
ta
ge
gb
ga
v
v
,
a
n
d
gc
v
r
espe
cti
v
e
l
y.
T
o
thi
s
e
nd
t
h
e
rea
c
ti
v
e
pow
er
h
a
v
e
to
b
e
nu
l
l
.
To
ach
ie
ve
t
h
i
s
o
b
j
ec
t
i
ve
s
i
t
s
u
f
f
i
c
e
s
to
e
nforce
t
he
α
-
a
xi
s
c
u
rrent
i
a
nd
β-a
x
i
s
c
urr
e
nt
i
t
o
t
r
a
c
k
refere
nce
signa
ls,
say
*
1
x
and
*
2
x
,
of the
f
o
l
l
o
w
i
n
g
f
orm
s
:
g
v
x
*
1
(
2
2
)
g
v
x
*
2
(
2
3
)
W
ith
i
s
an
y
re
a
l
p
os
iti
v
e
p
a
r
amet
e
r
(
al
th
oug
h
t
r
an
si
ent
time-v
a
ri
at
io
ns
a
re
a
ll
ow
ed).
I
ts
g
e
n
e
r
at
io
n
w
ill
be
seen
l
a
t
e
r
u
si
n
g
t
he
M
P
P
T
algor
it
hm
(
see
secti
o
n
4).
O
n
c
e
t
he
c
o
n
tr
ol
o
bjec
t
i
v
e
s
a
r
e
c
l
ear
ly
d
e
f
ine
d
,
as
t
he
MIMO
s
y
s
t
e
m
i
s
h
i
ghly
no
nli
n
e
a
r
,
a
Ly
a
pun
ov
b
a
s
ed
n
onli
n
e
a
r
c
on
t
r
o
l
i
s
prop
ose
d
[
17]
.
G
i
ve
n
t
h
e
ob
ser
v
er
guara
nt
e
e
s
t
h
a
t
t
he
e
rror
s
1
1
1
ˆ
x
x
z
a
n
d
2
2
2
ˆ
x
x
z
con
v
er
ge
t
o
ze
ro,
t
h
e
fol
l
o
w
i
n
g
c
on
t
r
o
lle
r
desi
g
n
w
i
ll
be
b
ase
d
o
n
t
h
e e
s
t
i
m
a
te
v
ari
a
ble
s
i
nstea
d
o
f
the
i
r true
va
l
ue
s
.
The
r
efore,
the
f
o
l
l
o
w
i
n
g
e
rror
s
a
r
e
intro
duce
d
:
*
1
1
4
ˆ
x
x
z
(
2
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
13
7 –
15
0
14
4
*
2
2
5
ˆ
x
x
z
(
2
5
)
In
o
rder to
a
c
h
ieve
t
he ob
j
e
c
t
ive
s
:
MP
P
T
a
nd
pow
e
r
f
ac
tor un
i
t
,
o
ne
c
a
n
s
e
e
k t
h
a
t
t
he
e
rr
ors
4
z
a
n
d
5
z
a
r
e
van
i
s
h
i
n
g.
T
o
the
s
e
e
n
ds
t
he
d
y
n
am
i
c
s
of
4
z
and
5
z
ha
ve
t
o
be
c
lear
ly
d
e
f
i
n
e
d
.
D
e
rivi
n
g
(
24)
a
nd
(2
5),
i
t
fo
l
l
ow
s
from
(
11a)
an
d
(11b) tha
t
:
*
1
3
1
4
1
ˆ
ˆ
x
v
L
L
x
x
L
r
z
g
(
2
6
)
*
2
3
2
5
1
ˆ
ˆ
x
v
L
L
x
x
L
r
z
g
(
2
7
)
The
g
o
a
l
,
now
,
is
t
o
ma
ke
4
z
a
nd
5
z
e
xp
one
n
t
ia
lly
v
a
n
is
hin
g
by
e
n
forc
ing
i
t
s
deriva
tiv
es
4
z
a
nd
5
z
t
o
beha
ve
a
s
fo
ll
o
w
s:
d
z
z
z
)
(
)
sgn(
4
1
4
1
4
(
2
8
)
d
z
z
z
)
(
)
sgn(
5
2
5
2
5
(
2
9
)
wher
e
0
>
1
,
0
2
,
0
1
a
n
d
0
2
b
e
i
n
g
d
e
s
i
g
n
p
a
r
a
m
e
t
e
r
s
a
n
d
i
s
a
s
i
g
n
u
m
f
u
n
c
t
i
o
n
.
I
t
i
s
w
o
r
t
h
no
ti
n
g
tha
t
t
he
i
nte
g
r
a
l
a
c
t
i
ons
a
re
i
ntr
o
d
u
c
e
d
i
n
(28)
a
n
d
(
2
9
)
to
a
l
low
a
g
o
o
d
robu
st
ness
of
t
he
c
o
n
tr
ol
l
e
r
a
g
ai
nst
unmo
d
e
l
e
d
d
y
n
am
ics a
n
d per
t
ur
bat
i
ons.
C
o
mbin
i
ng (2
6)
a
nd
(2
8) t
he
f
irs
t
c
on
t
r
ol
l
a
w
i
s
obta
i
ne
d:
d
z
x
v
L
x
L
r
z
x
L
g
)
(
1
ˆ
)
sgn(
ˆ
4
1
*
1
1
4
1
3
(
3
0
)
F
i
nal
l
y
,
com
b
ini
n
g (
27) a
nd (
29),
the
seco
nd
c
ontr
o
l law
is also obtained
d
z
x
v
L
x
L
r
z
x
L
g
)
(
1
ˆ
)
sgn(
ˆ
5
2
*
2
2
5
2
3
(
3
1
)
Since
t
h
e
c
o
n
t
r
o
l
laws
a
re
c
l
e
arly
d
efine
d
,
t
h
e
conc
er
n
n
o
w
i
s
to
i
nves
t
i
g
a
t
e
the
c
o
n
v
e
r
ge
nc
e
o
f
t
he
e
rro
rs
4
z
and
5
z
. To
this e
nd t
h
e
fol
l
ow
i
n
g qua
drat
ic
L
y
a
pu
n
ov fu
nc
ti
o
n
is c
o
ns
ide
r
ed
2
5
2
2
4
1
2
5
2
4
)
(
2
)
(
2
2
1
2
1
d
z
d
z
z
z
V
C
(
3
2
)
Its der
i
va
t
i
ve
i
s ob
tai
n
e
d
a
s fo
l
l
o
w
s
I
t
s
d
erivat
i
v
e
i
s
obta
i
n
e
d as
f
ol
low
s
:
d
z
z
d
z
z
z
z
z
z
V
C
)
(
)
(
5
5
2
4
4
1
5
5
4
4
(
3
3
)
w
h
ic
h,
u
sing
(28)
a
n
d
(
29
)
,
g
i
v
e
s
:
5
2
4
1
z
z
V
C
(
3
4
)
As
V
c
i
s
p
o
s
i
t
i
v
e
de
fin
ite
f
u
n
c
ti
o
n
a
nd
i
t
s
d
e
riva
ti
ve
(
34)
i
s
ne
gat
i
v
e
de
fi
nite
it
fo
ll
ow
s
tha
t
t
he
e
qu
i
l
i
b
ri
um
)
0
,
0
(
)
,
(
5
4
z
z
is gl
oba
l
l
y asy
m
ptot
ica
l
l
y
s
ta
ble
[1
8].
W
h
i
c
h,
i
n
t
u
r
n
, give
s
)
0
,
0
(
))
(
),
(
(
lim
5
4
t
z
t
z
t
The
ma
i
n
r
esul
t of t
he pr
o
pos
e
d
ou
t
put
f
ee
d
b
ack
c
o
n
t
ro
ll
e
r
is s
um
m
a
rize
d in the
f
o
l
l
o
w
i
n
g
t
he
orem
.
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
I
S
S
N
:
2088-
86
94
O
u
t
p
u
t
fee
d
b
a
c
k
non
l
i
ne
ar c
o
ntro
l o
f
three
-
p
has
e gri
d
-c
on
n
e
cted
PV ge
ne
rat
o
r (
A
.Y
ahy
a)
14
5
Th
e
o
r
e
m
:
C
o
n
sider
t
h
e
cl
os
ed-l
oo
p
sys
t
em
c
ons
isti
n
g
o
f
t
h
e
co
ntro
l
l
e
d
s
ys
tem
of
F
igur
e
1
r
e
pre
s
e
n
te
d
by
i
t
s
no
n
l
i
n
ea
r
m
o
d
e
l
(
10a
-c)
,
t
he
n
o
n
l
i
n
ea
r
obse
r
ver
(11a-c
)
an
d
th
e
c
o
n
t
ro
ll
e
r
c
om
pose
d
o
f
the
c
o
ntro
l
law
s
(
30)
and
(3
1).
Then,
one
h
as:
a.
The
cl
osed l
o
o
p
sys
tem
is G
AS
. It foll
o
w
s
th
a
t
all c
l
ose
d
l
o
op
sig
n
a
l
s a
r
e bo
u
nde
d.
b.
The
est
i
m
a
t
i
o
n
e
r
rors
T
z
z
z
z
,
,
3
2
1
conve
r
g
es
e
x
p
o
n
e
n
t
i
a
lly
t
o
ze
ro.
c.
The
t
r
ac
ki
ng
error
s
4
z
a
nd
5
z
c
on
v
e
r
g
e to
zer
o impl
y
i
n
g
MP
P
T achie
ve
me
nt
a
n
d
pow
e
r
fact
or
u
n
it.
Proof:
let us
cons
i
d
e
r
t
he
fol
l
o
w
i
ng q
u
a
d
ra
ti
c
Lyap
un
o
v
fu
n
ct
ion
:
C
O
V
V
V
=
5
1
2
2
1
i
i
z
+
2
5
2
2
4
1
)
(
2
)
(
2
d
z
d
z
(
35)
Its der
i
va
te,
using
(1
9)
a
nd (
3
4
)
,
can
b
e
ob
ta
i
n
e
d
as
fo
llow
s
:
5
2
4
1
2
3
2
2
2
1
2
z
z
z
z
z
k
V
V
V
C
O
(
3
6
)
W
h
i
c
h
cl
ea
rly
s
h
o
w
s
t
h
at
,
th
e
eq
uili
b
r
i
u
m
0
z
o
f
t
he
c
lose
d
l
oop
s
y
s
tem
w
i
t
h
t
he
s
ta
t
e
e
rror
ve
ct
or
T
z
z
z
z
z
z
5
4
3
2
1
,
,
,
,
i
s
g
l
oba
l
l
y
as
ympt
o
tic
al
l
y
s
ta
bl
e
.
I
t
f
o
l
l
o
w
s
t
ha
t
al
l
error
s
are
vani
sh
i
ng.
T
h
i
s
en
ds
t
h
e
pro
o
f
o
f
t
he
o
r
e
m
.
The
nex
t
s
e
c
t
i
on
de
v
o
t
e
d
t
o
t
he
p
erforma
n
ce
s
e
v
al
ua
tio
n
o
f
t
he
p
ro
p
o
sed
ou
t
p
u
t
fe
ed
bac
k
c
on
tr
ol
ler.
4.
SIMU
L
A
TION
R
ESULT
S
The
t
h
eore
tica
l
p
erf
o
rma
n
ce
s
descr
i
be
d
b
y
t
he
t
he
ore
m
o
f
an
out
p
u
t
f
eedb
a
ck
c
ont
rol
l
e
r,
i
n
c
l
udi
ng
the
c
o
n
t
ro
l
la
w
s
(
30-3
1
)
a
n
d
the
no
n
line
a
r
obser
ve
r
(11
a
-c
),
d
e
s
i
gn
e
d
i
n
S
e
ct
i
o
n
3
,
a
re
n
o
w
i
l
l
u
st
rat
e
d
by
si
mul
a
tio
n
.
T
he
e
xp
eri
m
e
n
t
a
l
se
tup
,
d
e
s
crib
ed
b
y
Fi
gu
re
5
,
i
s
s
im
ulated
u
s
i
ng
M
ATLA
B
/
SIMULI
NK.
The
cha
r
ac
t
e
ris
tics
of
t
he
c
ontr
o
l
l
e
d
s
y
s
tem
are
lis
t
e
d
i
n
T
a
b
le
4
.
N
ote
t
h
a
t
t
he
c
on
tro
l
led
sys
t
em
i
s
si
m
u
late
d
usi
n
g.
F
i
gure
5.
S
im
ul
a
t
i
on be
nc
h of
the
p
ro
pose
d
t
hre
e
-pha
se
gri
d
con
n
e
ct
ed
syst
e
m
The
i
n
s
t
a
n
ta
ne
ous
t
hre
e
p
has
e
m
ode
l
g
i
ve
n
by
(5a
-
d).
T
he
m
odel
(
10a-
c
)
in
α
-β
a
x
i
s
is
o
n
l
y
use
d
i
n
the
c
o
n
t
ro
l
l
er
d
e
s
ig
n.
T
he
d
e
s
i
g
n
pa
ram
e
te
r
s
o
f
t
h
e
co
ntr
o
ller
a
r
e
gi
ven
val
u
es
o
f
Ta
b
l
e
5.
T
he
se
p
ara
m
e
t
ers
have
b
e
e
n
se
le
c
t
e
d
u
s
i
n
g
a
‘
trial-a
nd-er
ror’
sea
r
ch
m
etho
d
and
pro
v
ed
t
o
b
e
s
ui
tab
l
e.
I
t
w
o
rth
no
t
i
n
g
t
ha
t
th
e
parameter
i
n
v
o
l
v
e
d
i
n
(
2
2
)
a
nd
(
2
3)
i
s
ge
ner
a
te
d
us
i
n
g
P
e
rturb
and
O
b
se
r
ve
a
l
g
ori
t
h
m
w
ith
t
he
b
l
o
ck-
dia
g
ra
m
i
l
l
u
s
t
r
a
ted
b
y
F
ig
ure
6.
T
he
r
e
s
u
l
ti
ng
c
l
o
se
d
lo
op
c
on
t
ro
l
per
f
or
m
a
nce
s
a
re
i
l
l
u
s
t
r
a
ted
b
y
F
i
gure
7
to F
ig
ure
13.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N: 2
0
8
8
-
86
94
I
nt
J
P
ow
E
l
e
c
&
Dr
i
S
y
st,
Vol.
10,
N
o.
1
,
Mar
c
h
2
0
1
9
:
13
7
–
150
14
6
Tab
l
e
4.
C
ha
r
a
c
t
e
r
is
t
i
c
s
o
f
co
ntr
o
lle
d
sys
t
e
m
Pa
r
a
m
e
te
r
Sym
bol
V
a
l
ue
PV
a
rray
PV
powe
r
40 kW
DC
link
ca
p
a
c
itor
C
3300μF
Grid
f
ilte
r
induc
tor
L
r
3m
H
0.
2
Ω
P
W
M
S
w
itc
hing
fre
que
nc
y
10kHz
Grid
A
C
s
ource
L
i
ne
fre
que
n
c
y
220V
50Hz
F
i
g
u
r
e
6
.
P
&O
a
lgor
it
hm
i
mple
me
nta
t
i
on
i
n
M
a
t
l
a
b
/
S
i
mul
i
nk
so
f
t
w
ar
e
Tab
l
e
5.
C
on
tr
ol
l
e
r
pa
r
a
m
e
te
r
s
Pa
r
a
m
e
te
r
S
y
m
bol
V
a
l
ue
Des
i
g
n
p
a
r
a
m
et
er
s
1
3
10
2
3
10
4
1
1
.
0
2
1
.
0
5
10
5
P&
O
a
l
gor
i
t
hm
p
aram
e
t
e
r
s
Delay
time T
d
4
10
S
t
ep
v
a
l
u
e k
3
.
0
4.
1.
R
a
d
i
at
ion
c
h
an
ge
e
ff
e
c
t
Fi
g
u
r
e
7
sh
ows
t
h
e
p
e
rfe
ct
M
PPT
i
n
t
he
p
re
sen
c
e
of
r
ad
i
a
ti
on
s
t
e
p
c
h
a
ng
es
m
ea
n
w
hi
l
e
,
t
h
e
tem
p
er
at
ur
e
is kep
t
co
ns
t
a
n
t
,
e
qua
l t
o
298.
1
5
K
(
25
°C)
.
T
he
s
im
ul
a
t
e
d
r
adi
a
ti
o
n
pro
file is as fo
llow
:
a first ste
p
c
h
a
n
g
e
i
s
p
e
r
f
o
r
m
e
d
b
e
t
w
e
e
n
5
0
0
a
n
d
1
0
0
0
W
/
m
²
a
t
t
i
m
e
t
=
0
.
1
s
a
nd
t
h
e
sec
ond
o
n
e
b
e
tw
e
e
n
1
0
00
a
nd
80
0
W/
m
²
a
t
time
t
=
0.
2
s.
T
he
F
igur
e
8
s
how
s
tha
t
t
he
P
V
po
w
e
r
c
a
ptur
ed
v
ar
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35.
1
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7
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2
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h
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h
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w
s
i
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F
i
gur
e
3
t
o
m
ax
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p
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nt
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(
M
3,
M
1
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M2)
of
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i
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d
t
o
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o
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er
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d
r
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iat
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espe
c
tive
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.
T
he
F
i
gur
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8
a
l
s
o
s
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o
w
s
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t
the
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l
t
age
o
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t
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P
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n
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=
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=
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4
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and
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e
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et
ur
ns
t
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6
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,
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ic
h
c
o
r
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spo
n
d
v
e
r
y
w
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ll
to
t
h
e
o
pt
im
um
volta
ge
s.
F
igur
e
9.
s
how
s
t
h
e
i
njec
te
d
curr
ent
an
d
t
h
e
gr
i
d
vo
l
t
age
T
h
i
s
F
i
g
u
r
e
9
c
l
e
a
r
l
y
s
h
o
w
s
t
h
a
t
t
h
e
g
r
i
d
c
u
r
r
e
n
t
i
s
sinus
o
i
da
l
and
in
pha
se
w
ith
t
he
g
r
i
d
vo
l
t
age
,
pr
ov
in
g
t
h
at
t
he
pow
er
f
ac
t
o
r
un
it
i
s
a
c
h
i
e
ve
d.
T
he
a
lter
n
a
t
i
n
g
c
ur
r
e
nts
i
n
j
e
c
t
e
d
t
o
t
h
e
gr
i
d
a
r
e
il
l
u
s
t
r
a
t
e
d
by
F
i
g
u
r
e
10.
(a)
(b
)
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