TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.5, May 2014, pp
. 3444 ~ 34
5
9
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i5.5362
3444
Re
cei
v
ed
No
vem
ber 1
4
, 2013; Re
vi
sed
De
cem
ber 7,
2013; Accept
ed De
cem
b
e
r
27, 2013
A Lyapunov Approach to Control De
sign for
Grid-connected Inverters
Vu Tran
,
Mufeed Mah
D
*
Dep
a
rtment of Electrical E
ngi
neer
ing,
Un
iver
sit
y
of Massac
husetts, Lo
w
e
ll
Ball H
a
ll 3
21, 1
Unviersit
y
Ave
.
Lo
w
e
l
l
, MA, 0185
4, USA
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: mufeed mah
d
@uml.e
du
A
b
st
r
a
ct
T
h
is pa
per
de
velo
ps a
Lya
p
unov
ap
proac
h
to c
ontro
l d
e
s
ign for
gri
d
c
onn
ected
inv
e
rter. T
h
e
control o
b
j
e
ctiv
e is to track a referenc
e curr
ent w
h
ich is pr
oporti
ona
l to th
e fund
a
m
enta
l
har
mo
nic of th
e
grid vo
ltag
e. By using th
e inte
rnal
mo
del
prin
ciple,
the
grid v
o
ltag
e an
d the
refe
renc
e curre
nt are descr
ibe
d
as the outputs
of an autonom
ous linear oscillatory system
. The state of this oscill
atory sy
stem
contains
all
the infor
m
atio
n
for the har
mo
nics of
the
grid
voltag
e a
nd is
estimated v
i
a
an o
b
server w
i
t
z
e
ro
esti
mati
o
n
error at steady
state. A
state
space descr
iption for the whole system
is
obt
ained
by comb
ining the state
of
the inv
e
rter cir
c
uit and t
hat
of the oscillat
o
ry syst
em
for
the grid voltage.
Based on the state spac
e
descri
p
tion, a
Lyapu
nov ap
proac
h is dev
elo
ped to des
ign a state-fe
edb
ack control
l
er for trackin
g
a
referenc
e curr
ent w
i
th
mi
ni
mal tracki
ng
err
o
r. T
he
desi
g
n
pro
b
le
m is c
a
st into
an
opti
m
i
z
at
io
n pr
obl
e
m
,
w
h
ich ca
n
be
effectively
so
lved
w
i
th li
ne
ar
matrix
in
eq
uality
(LMI) to
olb
o
x i
n
M
a
tla
b
. T
he
Lya
p
u
n
o
v
appr
oach
e
n
su
res inter
n
a
l
sta
b
ility
an
d
mak
e
s efficie
n
t use
of the structur
a
l
infor
m
atio
n, such
as the
tota
l
har
mo
nic
disto
r
tion (T
HD)
of
the
gri
d
vo
lta
ge, a
n
d
the
magn
itud
e/phas
e of t
he r
e
fere
nce c
u
rrent. T
h
e
effectiveness
o
f
the
Lyap
un
ov
ap
pro
a
ch
is v
a
lid
ated
vi
a S
i
mPow
er
si
mu
l
a
tion. A
re
al
ci
rcuit is
b
u
ilt
usi
n
g
micr
ocontr
o
ll
er
e
z
D
SP2
83
35,
the outp
u
t cur
r
ent obta
i
n
ed i
s
in p
hase w
i
t
h
the gr
id v
o
lt
age
an
d has s
m
a
l
l
T
HD, as w
e
expected.
Ke
y
w
ords
:
grid-co
n
n
e
cted
inverter, state-space d
e
scripti
on, total har
mo
nic distorti
on (
T
HD), linear
matrix
ine
qua
lities, Ly
apu
nov a
ppro
a
c
h
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
As the
dem
a
nd for po
we
r is i
n
crea
sin
g
si
gnificantl
y
, rene
wabl
e
ene
rgy
sou
r
ce
s h
a
ve
recently recei
v
ed a l
o
t of at
tention a
s
an
alternativ
e
s
way of ge
nerating di
re
ctly el
ectri
c
ity. Usin
g
rene
wa
ble e
nergy
syste
m
s can elimi
nate ha
rmfu
l
emission
s from polluting
the environ
ment
while al
so of
fering inexh
a
u
stible resou
r
ce
s of prim
ary ene
rgy. There are m
any sou
r
ces of
rene
wa
ble
en
ergy, such a
s
sol
a
r
ene
rgy, win
d
turbine
s
, water tu
rbi
nes,
and
geot
herm
a
l en
erg
y
.
Ho
wever,
sol
a
r an
d
wind
energy syste
m
make u
s
e
of advan
ced
power el
ect
r
o
n
ics. Mo
st of the
rene
wa
ble e
nergy te
chno
logie
s
pro
d
u
c
e di
re
ct
current (DC) po
wer
and h
e
n
c
e inve
rters are
requi
re
d to
convert th
e
DC to th
e
alternatin
g
cu
rrent (A
C) po
wer.
The
r
e
a
r
e t
w
o
kin
d
s of
inverters: Sta
nd-al
one
(i
sla
nd) and
g
r
id-con
ne
cted. T
hese two typ
e
s
have
sev
e
ral
simil
a
riti
es,
but a
r
e
different in
term
s of
co
ntrol
functio
n
. A
stan
d-al
one
inverte
r
i
s
use
d
in
off
grid
appli
c
ation
s
.
The g
ene
rat
ed p
o
wer f
r
o
m
re
ne
wabl
e
ene
rgy i
s
d
e
livered
to lo
ads,
or ca
n
be
store
d
in batterie
s
. But th
at kind of system
s
req
u
ires complexit
y
and high maintena
nce, such
that recharg
eable b
a
tteri
es. That al
so increa
se
s
the size and
cost for th
e system. G
r
id-
con
n
e
c
ted in
verters ove
r
come thi
s
limit
ation. Fo
r g
r
i
d
co
nne
ct
ed
inv
e
rt
er
s,
t
h
e
y
must
f
o
llo
w
t
h
e
voltage and
the frequ
en
cy characte
ristics of
the u
t
ility generat
ed po
we
r prese
n
ted on t
he
distrib
u
tion li
ne. The main
advantage o
f
this system
is that no battery is requi
red for stori
n
g
the
energy from rene
wable
so
urces, wh
ich redu
ce
s the size and
co
st of
the system
. Moreover, it is
easi
e
r to
cre
a
te a
po
rtabl
e inve
rter du
e to th
e
com
pact
si
ze
of
t
he
sy
st
em.
Investigatio
ns of
different
confi
guratio
ns an
d
co
ntrol
meth
ods fo
r
gri
d
-conne
cted
inv
e
rters
are
bei
ng d
e
velope
d
in
recent years. A comp
rehe
n
s
ive revie
w
of singl
e
-
ph
ase
grid-co
nne
ct
ed inverte
r
s [
1
] has covere
d
some
of the
standards th
at
inverter
s fo
r g
r
id ap
plicatio
ns m
u
st b
e
fu
lfilled, su
ch a
s
the
stand
ards
EN610
00
-3-2
, and the
U.S Nation
al Ele
c
tri
c
al
Code
(NEC) 69
0. It also
provid
ed
a cla
s
sificati
on
of the inverte
r
s
reg
a
rding t
he sta
ge
(si
n
gle, dual
sta
g
e
, and m
u
lti-string
inverte
r), tra
n
sfo
r
me
rs
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Lyapu
no
v Appro
a
ch to Control De
sig
n
for Grid
-con
necte
d Inve
rters
(Vu Tran)
3445
and types of
Intercon
ne
ctions, and typ
e
s of grid
int
e
rfaces (li
n
e
-
comm
utated curre
n
t-sou
r
ce
inverter a
nd
self-comm
u
ta
ted voltage-source in
ve
rte
r). Three
-
Pha
s
e G
r
id-Co
n
n
e
cted inve
rte
r
s
were present
ed in [2, 3]. The com
b
in
ation
of isla
nd mode a
n
d
grid
-conne
cted mod
e
wa
s
investigate
d
i
n
[4, 5]. In grid-co
nne
cted
mode, inve
rter ope
rate
s
in co
nne
ction
with the g
r
i
d
.
Whe
n
discon
necte
d from the grid, the inverter
sho
u
ld be automat
ically discon
n
e
cted fro
m
the
utility grid a
n
d
chan
ged
to
isla
nd
mode
in o
r
d
e
r to
e
n
su
re t
he
co
ntinuou
s
po
wer
sup
p
ly for
the
load from the
inverter [6]. A main p
r
obl
em whic
h ne
eds
mu
ch eff
o
rt in p
o
wer
inverters the
s
e
days i
s
reducing the
harmonics. The IEEE 929 standard
st
at
ed that the
Total Harmonic
Disto
r
tion (T
DH) of voltage and current shoul
d be lowe
r than 5% in normal op
eration.
Harmoni
cs are not desi
r
abl
e beca
u
se they cau
s
e ove
r
heatin
g, decrea
s
ed
volta
m
pere cap
a
ci
ty,
increa
sed l
o
sses, di
stort
ed voltage a
nd current
waveform, etc.
There are two
sou
r
ces
of
harm
oni
cs:
o
ne i
s
from th
e inve
rters
(d
ue to
t
he
pul
se wi
dth m
odu
lation a
n
d
the
switchi
n
g
)
, a
nd
the other is f
r
om load
s o
r
grid. These harm
oni
c cu
rrents the
n
ca
use di
stortio
n
in the voltag
e
becau
se of t
he imp
edan
ces in
the di
st
ribution
net
work an
d in
sid
e
the voltag
e
sou
r
ce. And
the
harm
oni
cs in
voltage can
cau
s
e ha
rmo
n
ics in cu
rre
nt as well. Several re
se
arche
s
have b
een
prop
osed to
redu
ce the vol
t
age T
HD
of inverters. Fo
r
example, t
he
repetitive con
t
rol theo
ry ha
s
been
su
cce
s
sfully appli
e
d
to PWM inv
e
rters [7
-14],
active filters [15-1
7
],
dead
-beat control [18,
19], to redu
ce THD. Ha
rm
onic
dro
op
co
ntrol techni
q
u
e
[20] is al
so
pre
s
ente
d
. Repetitive co
ntrol
has an ex
cell
ent ability in
eliminating periodi
c di
st
urbances, however, in prac
tical, this technique
is limite
d
in
slo
w
dyn
a
mi
cs,
poo
r t
r
a
c
king
a
c
curacy, and
poo
r
perfo
rman
ce
to non
-p
erio
d
i
c
disturban
ce
s.
De
ad
-beat
and slid
i
ng-mode
control
s
h
a
ve ex
cel
l
ent dynami
c
pe
rform
a
n
c
e in
control of o
u
tput voltage,
but t
hese te
chniqu
es
suffe
r from
co
mp
l
e
xity, sensitivity, and stea
dy-
state e
r
rors
In order to
elim
inate th
e
cu
rrent
dist
ortion,
so
m
e
cu
rrent
cont
rol m
e
thod
s
are
prop
osed, su
ch that pro
p
o
r
tional resona
nt controll
er a
nd multi-reso
nant co
ntrolle
rs in [21], acti
ve
power filters
in [17, 22].
A promi
s
in
g
control te
chn
i
que
s in
grid
-con
ne
cted in
verter i
s
o
u
tput
curre
n
t tracki
ng. The inve
rter’
s
current
polarit
y mu
st be taken
care of, to m
a
tch the volt
age
polarity of the grid. Vario
u
s syn
c
h
r
oni
zation me
th
o
d
s are su
mm
arized in [23
-
25]. The cu
rrent
hystereti
c
co
mpari
s
o
n
co
ntrol meth
od
, timing
co
ntrol of
curre
n
t instantan
eo
us
com
pari
s
on
method an
d
the triangle
wave comp
arison cont
ro
l method of timing tracking cu
rre
nt are
prop
osed in [26]. In [27], algorithm
s of current
de
cou
p
ling are deri
v
ed for perfo
rming the re
active
power
contro
l of grid-con
necte
d invert
er. Throug
h zero-crossin
g
detecting
circuit in [28] a
nd
[29], the inverter is controll
ed so a
s
to g
enerate t
he o
u
tput cu
rre
nt in pha
se with
the grid volta
ge.
A current co
ntrol employi
ng intern
al model pr
i
n
ci
ple in [30] is pro
p
o
s
ed
to supp
re
ss
the
harm
oni
c currents inje
cte
d
into the
grid. Although
most exi
s
ting
co
ntrolle
rs
give satisfa
c
tory
results, the theory behi
nd the dynam
i
cs
and pe
rform
a
nce
s
is not cl
early de
scrib
ed. The purp
o
se
of this
pap
er i
s
to
develo
p
a sy
stemati
c
state spa
c
e
a
ppro
a
ch so
th
at the dyn
a
mi
cs of th
e
who
l
e
system
can b
e
more
clea
rl
y understood.
A simple fee
dba
ck la
w is
desi
gne
d to track a refe
ren
c
e
current with
minimal tracking error. T
h
e design
problem will
be investigated using
advanced
nonlin
ear con
t
rol system th
eory and line
a
r matrix in
e
quality (LMI) optimizatio
n tech
niqu
e, as in
[31-35]. The
probl
em of tracking curren
t error
w
ill be
caste
d
into Lyapun
ov framewo
r
k, whi
c
h
ensure
s
inte
rnal sta
b
ility and the total
harm
oni
cs di
stortion (T
HD)
re
quireme
nts.
The pape
r
is
orga
nized
as follows. Se
ction II de
scri
bes the
ope
n loo
p
de
scri
ption for the
inverte
r
circuit,
followin
g
is the state-sp
a
c
e de
scriptio
n fo
r the grid voltage an
d an observ
e
r, and co
ntrol
objec
tive. Sec
t
ion III reviews the main
tool to
be used in this
paper
- Lyapunov approac
h
to
evaluation
of
the t
r
ackin
g
erro
r.
Sectio
n IV casts the probl
em
of tracki
ng erro
r into Lyapun
ov
frame
w
ork
an
d convert
s
th
e de
sig
nprobl
em into
th
e L
M
I optimization. Sectio
n V
uses SimPo
w
er
in MATLAB t
o
sim
u
late
an
d verify the
re
sults. Se
ction
VI use
s
expe
riment
re
sults to validate
th
e
desi
gn metho
d
. Section VII con
c
lud
e
s th
e pape
r.
2. State Spa
ce De
scripti
on of the Inv
e
rter a
nd Co
ntrol Objec
t
i
v
e
2.1. Open-lo
op Des
c
ripti
on for th
e Inv
e
rter Circuit
Figure 1 is t
he equival
e
n
t
circuit of a grid-co
nne
ct
ed inverte
r
, whe
r
e
vg
is the
grid
voltage. The input voltage
u
is actually the output of the tran
si
sto
r
bridg
e
, whi
c
h
is driven by a
pulse-width
-modulate
d
(P
WM)
sign
al. For si
mplicity, the transi
s
tor bridge i
s
not inclu
ded.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3444 – 34
59
3446
Figure 1. Equivalent Circuit
for a Grid
-co
nne
cted Inverter
Let the duty cycle of the PWM si
gnal b
e
d
and the
DC voltage
su
pply be
Vs
. Under th
e
assumptio
n
o
f
ideal switchi
ng, wh
en
PWM sign
al is i
n
the ON
state
,
uON
=
Vs
, and whe
n
PWM
sign
al is in the OFF stat
e,
uOFF
=
−
Vs
. Unde
r th
e assumptio
n
of high freq
uen
cy, the input
sign
al
u
can
be average
d from
uON
and
uOF
F
over one switchin
g peri
od,
u
=
dVs
+ (
1
−
d
)(
−
Vs
) =
(2
d
−
1)
Vs
. Th
u
s
we call the
simplified
ci
rcuit of
the ave
r
age
d mo
del
for the inve
rter,
and
u
can be
con
s
id
ere
d
a
s
the input.
Let the state of the circuit as:
Define:
The ci
rcuit ca
n be de
scribe
d as:
(
1
)
Whe
r
e
u
is th
e control inpu
t and
vg
can
be co
nsi
dere
d
as an external distu
r
ba
n
c
e.
2.2. State
-
sp
ace Descrip
tion for the G
r
id Voltage a
nd an Obse
r
v
er
The grid volta
ge is peri
odi
c with frequen
cy
50Hz or 6
0
Hz. The freq
uen
cy may subje
c
t to
some
pe
rturb
a
tion but
ca
n
be m
e
a
s
ure
d
. Let the fu
ndame
n
tal freque
ncy b
e
β
0 (rad/seco
nd).
Acco
rdi
ng to [36], the grid voltage
vg
(
t
)
can be expressed a
s
a Fou
r
ier se
rie
s
:
(
2
)
The ma
gnitud
e
bk
and
the
pha
se
ϕ
k
fo
r
each ha
rmo
n
i
c
can b
e
eval
uated
with a
ban
k of
resona
nt filters [21], [37-38]
, or a comp
osite obs
e
r
ver [
39, 40]. The reso
nant filters are d
e
scrib
ed
with tra
n
sfe
r
f
unctio
n
s,
whil
e the
com
p
o
s
ite
observe
rs are de
scribe
d
via
state
-
spa
c
e equatio
ns
.
They a
r
e
all
based
on th
e
internal m
o
d
e
l p
r
inci
ple
i
n
[41, 42].
He
re we a
dopt t
he m
a
in id
ea
s in
[40] to descri
be
vg
via sta
t
e spa
c
e eq
u
a
tions a
nd th
en co
nstruct
an ob
serve
r
to estimate th
e
state.
The a
d
vantage of usin
g
t
he
sta
t
e spa
c
e
de
scriptio
n i
s
th
at the dynam
ics
of the
wh
ole
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TELKOM
NIKA
ISSN:
2302-4
046
A Lyapu
no
v Appro
a
ch to Control De
sig
n
for Grid
-con
necte
d Inve
rters
(Vu Tran)
3447
system
can b
e
simply de
scribed by
stacking u
p
the st
ate equatio
n for
vg
and tha
t
for the circui
t,
i.e., (1). Th
e resultin
g state
equatio
n for
the wh
ol
e
system ma
kes it
very co
nveni
ent to stu
d
y the
intera
ction
be
tween
the g
r
i
d
voltage
an
d the dyn
a
mi
cs of L
C
L filt
er. Fu
rthe
rm
ore, it fa
cilita
t
es
analysi
s
of system perfo
rmance via advanced tool
s
develop
ed
in recent yea
r
s, su
ch a
s
the
Lyapun
ov ap
proa
ch a
nd th
e linear-mat
ri
x-inequ
ality (LMI) ba
sed o
p
timization.
W
e
ma
y c
o
ns
id
er
vg
(
t
) a
s
a di
stu
r
ban
ce with
kno
w
n
freq
uen
cy fo
r the
ha
rmon
ics but
uncertain
ma
gnitude
and
pha
se
s. Thi
s
type of di
stu
r
ban
ce
s can
b
e
mod
e
led
as the outp
u
t of
a
linear time in
variant syste
m
, as in [31] and [41]. Let:
Denote:
Then
vg
1(
t
) is the output of the followin
g
2
nd
orde
r line
a
r sy
stem.
(
3
)
We m
a
y incl
u
de mo
re
harmonics
by si
mply increa
si
ng the
si
ze of
S with m
o
re
diago
nal
blocks of the form
and in
creasi
ng the di
mensi
on of
wg
.
Define:
Whe
r
e 0’
s in the above mat
r
ix are all 2 b
y
2 blocks. Also defin
e:
Then
vg
is th
e output of the followin
g
au
tonomou
s lin
ear o
scill
atory system:
Whe
r
e
wg
∈
R
2
N
. A prom
inent feature of the matrix
Sg
is that
Beca
use of this,
we have
wg
(
t
)
Twg
(
t
) =
wg
(0)
Tw
g
(0)
=
∥
wg
(0)
∥
2 for all
t
. This kin
d
of state-sp
ace d
e
scri
ptions
for peri
odi
c si
gnal
s ha
s be
en wid
e
ly used in the out
put reg
u
lation
literature fo
r tracking
peri
o
dic
referen
c
e
s
or rejectio
n of perio
dic di
stu
r
ban
ce
s
[31
-
[33], [41], where the line
a
r system (4
)
wa
s
referred to a
s
the exogeno
us sy
stem, or simply, exosystem.
The state
wg
∈
R
2
N
can b
e
decompo
se
d as:
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046
TELKOM
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KA
Vol. 12, No. 5, May 2014: 3444 – 34
59
3448
w
h
er
e
wgk
∈
R
2,
k
= 1,
. . . ,
N
. By the
stru
ctu
r
e of
Γ
g
, we have
.
Furthe
rmo
r
e
is exa
c
tly the
kt
h
h
a
rm
on
ic. Let
C
1 be
a 1 by 2
N
row vecto
r
whose first el
e
m
ent i
s
o
ne
a
nd the
rest
are all
ze
ro. T
h
en the
first ha
rmoni
c, d
enot
ed
vg
1, is
vg
1(
t
) =
C
1
wg
(
t
).
It is
eas
y to
verify that the s
y
s
t
em
(4
), in parti
cula
r, the pair
(
Γ
g,
Sg
), is ob
se
rvable.
Thus a
n
o
b
server can
be
con
s
tru
c
ted
t
o
e
s
timate th
e state
wg
, an
d
he
nc
e
(
bk,
ϕ
k
) fo
r all
k
.
Let
the state of the observe
r be
wz
. We have:
(
5
)
Whe
r
e,
L i
s
the o
b
serve
r
gai
n
whi
c
h
ca
n b
e
d
e
signed
via va
riou
s a
p
p
r
oa
che
s
. A
sim
p
le
approa
ch i
s
t
o
choo
se th
e
de
sire
d p
o
le
s at
−α
} j
k
β
0
, k
= 1
,
・
・
・
N
and
use the
pol
e
placement fu
nction in MA
TLAB. The n
u
mbe
r
α
ca
n
be adju
s
ted
via simulatio
n
for satisfa
c
tory
conve
r
ge
nce rate.
If the freque
ncy
β
0 for th
e ob
se
rver i
s
exactly the
same
a
s
the
freque
ncy
of the gri
d
voltage, then
the ob
serve
r
error
wz
(
t
)
−
wg
(
t
) will
go
the 0 a
s
ymp
t
otically and
we
can
use the
es
timated s
t
ate
wz
for va
rious
pu
rpo
s
e
s
. If the gri
d
freque
nc
y i
s
subj
ect to
pe
rturb
a
tion, thi
s
freque
ncy ca
n be mea
s
u
r
ed on line a
n
d
use
d
for th
e observe
r. Due to ro
bu
stness, the sa
me
gain
L
sho
u
ld
be
stabili
zin
g
for a
ce
rtai
n ra
nge
of
β
0 an
d
Sg
(
β
0). The
discrete-time ve
rsi
o
n of
the observe
r are u
s
ually u
s
ed in p
r
a
c
tice. More detai
ls ca
n be fou
nd in [40]. With the estimat
e
d
state
wz
, the first harmoni
c of
vg
is estim
a
ted as
C
1
wz
.
2.3. The Con
t
rol Objec
t
iv
e and Augm
ente
d Exos
y
s
tem
Ideally, we
would li
ke
to fe
ed the
g
r
id
a
sinu
soi
dal
cu
rre
nt
ig
,
whi
c
h is in
ph
ase
with
vg
.
The m
agnitu
de of
ig
can
be va
ried
de
pendi
ng
on t
he n
eed
of t
he g
r
id
and
t
he lo
cal
en
ergy
stora
ge d
e
vice
s. Thi
s
ob
jective can b
e
stat
ed
as
a refe
ren
c
e
tracking
prob
lem wh
ere th
e
referen
c
e for
the grid curre
n
t is given as:
(
6
)
Whe
r
e,
r
is a
positive num
ber that can b
e
cha
nge
d. Recall that
vg
1
is the first ha
rmoni
c of
vg
.
If
r
is fixed, then the
refe
ren
c
e
ig;
r
ef
can
be
con
s
i
dere
d
a
s
an
other o
u
tput
for the
exosystem
(4
) alo
n
g
s
ide
vg
. If
r
is va
ri
able, it wo
uld
be mo
re
co
nvenient to i
n
trodu
ce
ano
ther
exosystem:
(
7
)
Whe
r
e,
wr
∈
R
2. The
co
nd
ition (6
) can b
e
sati
sfied if
wr
(0) =
rw
g
1(0). Since
w
˙
g
1 =
S
0
wg
1, we
have
wr
(
t
) =
rwg
1(
t
). The
r
e is some redun
dan
cy introdu
cin
g
(7
). The purpo
se is to ma
ke it
easi
e
r to han
dle
r
.
With the
reference
cu
rre
nt
spe
c
ified, th
e
co
ntrol
obj
ective is to
mini
mize
the m
a
g
n
itude
of the trackin
g
error.
(
8
)
At steady
sta
t
e. Note
that
ig
can
be
con
s
ide
r
ed
a
s
an
output
to the
inverte
r
syst
em (1):
ig
= [0 0 1]
xc
. Since
B
and
E
in (1) are not aligned, i.e., the contro
l input
u
and the disturban
ce
vg
a
r
e not in
the same
chann
el, the trackin
g
erro
r cann
ot be completely eli
m
inated. In most
works o
n
o
u
tput re
gulatio
n
(e.g., [32
-
33]
, [41-42]
),
si
m
ilar cont
rol problem
s we
re con
s
id
ere
d
,
b
u
t
unde
r the a
s
sumption that the co
ntrol in
p
u
t and the
di
sturban
ce
ente
r
the sy
stem from the
same
cha
nnel. Und
e
r this a
s
sum
p
tion, part of the co
ntrol ca
n be used to can
c
el the di
sturban
ce.
Even though
the cont
rol problem in this work
doe
s n
o
t fit into
the frame
w
ork of
output
regul
ation, we ca
n u
s
e th
e sa
me b
a
si
c idea of
i
n
ternal mo
del for the di
sturb
a
n
ce
and
usi
n
g an
observe
r to reco
nstruct th
e state for an
ex
osystem
which p
r
od
uce
s
the distu
r
ba
nce.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Lyapu
no
v Appro
a
ch to Control De
sig
n
for Grid
-con
necte
d Inve
rters
(Vu Tran)
3449
The two exosystems (4) a
nd (7)
can be
stacked up t
o
obtain a 2(
N
+ 1) -ord
er system.
Define:
And,
Then,
This sy
stem descri
b
e
s
all the dynamics of
the grid voltage and th
e refere
nce current.
Combi
ned
wit
h
the state
-
space de
script
ion of t
he L
C
L filter, the d
y
namics of th
e wh
ole
syst
em
can
be d
e
scri
bed. Before that, we n
eed
to provide
so
me u
s
eful p
r
o
pertie
s
ab
out
the exosy
s
te
m
and some im
portant impli
c
ations a
bout the initial co
nd
ition.
It is
clea
r that
the exo
s
yste
m (9
)
evolves all by
itself
and i
s
driven by
its initial
condition
w
(0
). Re
call t
hat
w
co
nsi
s
t
s
of the state
s
for all the h
a
rmo
n
ics of
vg
and for
ig;ref
, in partic
u
lar,
Since
, we have:
(
1
0
)
(
1
1
)
(
1
2
)
For all
t
. T
hus
∥
wgk
(0
)
∥
2
rep
r
e
s
ent
s the po
we
r of t
he
k
th
ha
rmo
n
ics of
vg
and
∥
w
(0)
∥
2 the
total powe
r
of the harmo
nics of
vg
plu
s
the po
wer of the refe
ren
c
e
curre
n
t.
Note that the
conditio
n
wr
(
t
) =
rw
g
1(
t
) i
m
plies that
ig;ref
is propo
rt
ional to the first ha
rmo
n
ic
of
vg
. In terms
of the c
o
mbined s
t
ate
w
, this can be
writte
n as:
(
1
3
)
This
conditio
n
will be u
s
ed
as a co
nst
r
ai
nt in an optimization p
r
obl
e
m
to be form
ulated.
3. L
y
apuno
v
Appro
ach to
Ev
aluation of the Tr
acki
ng Error
3.1. Ev
aluation of Trac
king Error for
Gener
a
l Exos
y
s
tem
As we
ca
n se
e from
(9), b
o
t
h the grid vol
t
age an
d the
referen
c
e
current a
r
e d
r
ive
n
by an
autonom
ou
s linear
system
. According to [43],
the state variable
of
an autonomou
s syste
m
contai
ns all the informatio
n that
determines its fut
u
re be
havior,
it can be effectively use
d
to
corre
c
t the
dy
namic be
havi
o
r of th
e
whol
e sy
stem.
In t
he
ca
se of th
e inverte
r
circuit, the state
w
can b
e
used to minimize the tracking e
r
ror of the grid
curre
n
t (8).
For the
invert
er
circuit in
Fi
gure
1, the
st
ate
of the
wh
ole sy
stem i
s
a co
mbin
atio
n of the
cir
c
uit
st
at
e
xc
and th
e exosyste
m stat
e
w
. They ca
n be eithe
r
measured o
r
estimated vi
a an
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02-4
046
TELKOM
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KA
Vol. 12, No. 5, May 2014: 3444 – 34
59
3450
observe
r, thu
s
state fee
d
b
a
ck i
s
fea
s
ibl
e
. In this
wo
rk,
we
attem
p
t to u
s
e
sta
t
e feedb
ack t
o
minimize the tracking e
r
ror.
The
pro
b
lem
boil
s
d
o
wn
to ho
w to
m
easur
e
the
magnitud
e
of
the trackin
g
erro
r at
steady
state
?
If the tracking e
r
ror
ca
n be effe
ctiv
ely evaluate
d
via a ce
rt
ain pe
rform
a
nce
measure, the next ste
p
woul
d b
e
minimizi
ng
t
h
is p
e
rfo
r
ma
nce
mea
s
u
r
e via a
ce
rtain
optimizatio
n algorith
m
.
Traditio
nal m
easure
s
b
a
se
d on tra
n
sfe
r
function m
a
y not be rea
d
ily applicabl
e to this
ca
se. Here
we note that th
e inverter
ci
rcuit ha
s two
exogen
ou
s in
puts, on
e is t
he gri
d
voltage
and the
othe
r one i
s
th
e re
feren
c
e
cu
rre
n
t. The g
r
id v
o
ltage
can
be
co
nsid
ered a
s
a
distu
r
b
a
n
c
e
who
s
e m
agni
tude is alm
o
st fixed, or varie
s
wi
thi
n
a small
ran
g
e
, but the magnitud
e
of the
referen
c
e current is varia
b
l
e
. Thus it is d
i
fficult
to use a certai
n inpu
t-output gain
to measu
r
e t
h
e
tracking
error as the outp
u
t
to
these two exogen
ou
s inputs. Anot
her
difficulty is that there
are
many harm
o
n
i
cs a
nd it is n
o
t easy to co
nsid
er thei
r combine
d
effects.
The Lyapu
n
o
v appro
a
ch
developed i
n
[31] seem
s to be tailored for thi
s
kind o
f
probl
em
s. It d
eals with
mo
re ge
neral
systems
(no
n
lin
e
a
r,
time
-varyi
ng) with pe
rio
d
ic excitation
s,
whi
c
h co
uld be
di
stu
r
ban
ce
o
r
referen
c
e.
The
obj
ect
i
ve is to eval
uate the
ma
g
n
itude
of
cert
ain
output at stea
dy state, whic
h coul
d be th
e tracking e
r
ror.
To ap
ply the
Lyapun
ov a
ppro
a
ch, we
first n
eed t
he stat
e spa
c
e d
e
scriptio
n
for th
e
whol
e
system
, whi
c
h
ca
n
b
e
ea
sily o
b
tai
ned
by combi
n
ing th
e
state-spa
c
e
equ
a
t
ion (1) for th
e
circuit an
d the state-sp
ace
equation
(9
) for the
vg
and
ig;ref
. Since
ig
= [0 0 1]
xc
,
if we let
C
= [
0
0 1
−
Γ
2], then:
(
1
4
)
(
1
5
)
To red
u
ce the tracking e
r
ror, we ap
ply a simple
state feedba
ck:
(
1
6
)
Substitute the
feedba
ck la
w in (16) into (14), we h
a
ve the clo
s
ed
-lo
op syste
m
.
(
1
7
)
(
1
8
)
Re
call th
at th
e dime
nsi
on
of
xc
a
nd
w
are 3
a
nd 2
N
+2, respe
c
tively, the order
of the
whole
sy
st
em i
s
2
N
+5.
As
lo
ng
as
A
+
BK
1 i
s
stab
le, the solutio
n
for th
e ab
o
v
e system
wil
l
be b
oun
ded
and fo
r
any initial
co
ndition, the
solution
will
a
ppr
o
a
ch a
steady
state o
scill
ation. Sin
c
e
A
+
BK
1 is
stable, the effect of the initial con
d
ition o
f
xc
(0) will vanish asymptotica
lly. Thus t
he steady state
oscillation, i
n
pa
rticula
r
, th
e tra
c
king
error, d
epen
ds
only on
the
initial conditio
n
w
(0
). H
e
n
c
e, a
gain fro
m
the
norm of the
initial con
d
itio
n
to the ma
gnitude of
e
at
steady state
can be d
e
fin
ed. The mai
n
result of
[3
1] was ap
pli
ed to estimat
e
this gain via
a
quad
ratic Lya
punov fun
c
tio
n
.
Here we su
mmari
ze the
main re
sult
of the Lyapu
nov app
roa
c
h whe
n
ap
pli
ed to the
linear
system
(17). Denote:
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TELKOM
NIKA
ISSN:
2302-4
046
A Lyapu
no
v Appro
a
ch to Control De
sig
n
for Grid
-con
necte
d Inve
rters
(Vu Tran)
3451
Theorem 1:
For
γ
>
0, if there
exist a
p
o
sitive definit
e matrix
, and a
numbe
r
η
>
0 s
u
ch that:
(
1
9
)
(
2
0
)
W
h
er
e
I
2
N
+2
is an
ide
n
tity matrix
of di
mensi
o
n
2
N
+ 2,
then
for
any initial
co
ndition
xc
(0) and
w
(0
),
xc
(
t
)
and
e
(
t
) will
con
v
erge to
a bo
unde
d set. Moreove
r
, the t
r
ackin
g
e
rro
r
e
at ste
ady st
ate
is bou
nde
d b
y
.
The num
be
r
γ
sati
sfying T
heorem 1 i
s
calle
d a bou
nd on the
steady state g
a
in fro
m
to the tracki
ng erro
r
e
. F
o
r given
K
1,
K
2, this stea
dy state
gain
can
be eval
uated by
minimizi
ng
γ
satisfying the
LMI con
s
trai
n
t
s (19
)
and
(2
0), by using t
he LMI toolbo
x in Matlab.
Here we not
e that the no
rm of the initia
l con
d
ition,
, is clo
s
ely rel
a
ted to the
magnitud
e
of
vg
and
ig;
r
ef
. To be sp
ecific, recall that the state co
rre
sp
ond
ing to the
kt
h
harm
oni
c is
wg
k
. Its
norm
, where
bk
is the m
agni
tude of the
kth
harm
oni
c. Furtherm
o
re:
3.2. Impro
v
e
d
Ev
aluation b
y
Explorin
g Structural Information
In the p
r
evio
us
se
ction, th
e tra
cki
ng
error
wa
s eval
u
a
ted b
a
sed o
n
. Let
γ
>
0
satisfy (19
)
a
nd (20
)
,
the
n
at
stea
dy stat
e,
. Thi
s
i
neq
uality is valid
for all
types
of
w
(0) but
co
uld b
e
too
co
nse
r
vative for a p
r
a
c
ti
cal g
r
id voltage
wh
ose
first ha
rm
onic domi
nat
es
the high
er
order
harmoni
cs an
d for a referen
c
e
cu
rrent whi
c
h
is
prop
ortio
nal t
o
the first-o
r
d
e
r
harm
oni
c.
In practi
ce, th
e T
H
D of th
e
grid
voltage
i
s
b
e
lo
w a
certain level
and
the m
agnitu
d
e
of th
e
referen
c
e
cu
rrent i
s
within
a given
ra
n
ge. In th
is
s
e
ct
ion,
su
ch st
ru
ct
ural inf
o
rmatio
n will
be
effectively utilized to im
prove t
he eval
uation of
the
tracki
ng error.
Specifical
ly, the struct
ural
informatio
n will
be exactly expre
s
sed
i
n
terms
of
qua
dratic ine
qual
ities a
nd i
n
co
rpo
r
ated
in th
e
Lyapun
ov ap
proa
ch to o
b
tain less rest
ri
ctive co
n
s
trai
nts, thus
red
u
cin
g
the mi
nimal value o
f
γ
for the optimi
z
ation p
r
obl
e
m
.
1) Qu
ad
ratic i
nequ
ality for THD
bou
nd:
Con
s
id
er the
grid voltag
e e
x
presse
d in (2). The
THD value is:
Suppo
se that
a kno
w
n bo
u
nd on the THD is
ε
0. Then
we have:
Rec
a
ll from (11)
for all
t
,
the above in
e
quality can b
e
expre
s
sed
as:
(
2
1
)
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02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3444 – 34
59
3452
In terms of the combi
ned
state
, this inequality can b
e
expre
s
sed
as:
(
2
2
)
Whe
r
e,
And 0
p
den
otes a
p × p
0 b
l
ock and oth
e
r
0’s have
co
mpatible dim
ensi
o
n
s
.
2) Q
uad
ratic inequ
ality for mag
n
itude
of referen
c
e
curre
n
t:
The referen
c
e current
i
s
prop
ortio
nal to the first ha
rmonic
vg
1 an
d is set as
i
2
;r
e
f
(
t
) =
rv
g
1(
t
). Suppose that
r
is b
ound
ed
by
rm
ax
. Then we have
. In terms
of
,
this
co
nst
r
aint
ca
n b
e
written
as:
(
2
3
)
Whe
r
e
Wrm
is given by:
3) Qua
d
ratic
equality for p
hase of reference cu
rrent:
The referen
c
e curre
n
t is in pha
se
with the first harm
oni
c
vg
1
.
This implie
s that the state
wr
is p
r
o
p
o
r
tional to the state
wg
1. Let
.
Then:
In terms of the whol
e state
,
this is equiv
a
lent to:
(
2
4
)
Whe
r
e
Wrp
i
s
given by:
No
w we ca
n
use th
e qua
d
r
atic in
equ
alities (22
)
,
(23
)
and (24) to i
m
prove th
e e
v
aluation of the
tracking e
r
ror.
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TELKOM
NIKA
ISSN:
2302-4
046
A Lyapu
no
v Appro
a
ch to Control De
sig
n
for Grid
-con
necte
d Inve
rters
(Vu Tran)
3453
Corollar
y
1:
Con
s
id
er the
clo
s
ed lo
op
system
(17
)
.
Suppo
se that
the THD of the gri
d
voltage is le
ss than
ε
0 an
d
ig;ref
(
t
) =
rv
g
1(
t
) with
r
≤
rm
ax
. For
γ
>
0, if there exist a po
si
tive
definite matri
x
P
=
PT
∈
R
(2
N
+5
)
_
(2
N
+5), and nu
mb
er
η
,
α
1
,
α
2
≥
0,
α
3
∈
R
s
u
c
h
that:
(
2
5
)
(26
)
W
h
er
e
I
2
N
+2
is an
ide
n
tity matrix
of di
mensi
o
n
2
N
+ 2,
then
for
any initial
co
ndition
xc
(0) and
w
(0
),
xc
(
t
)
and
e
(
t
) will
con
v
erge to
a bo
unde
d set. Moreove
r
, the t
r
ackin
g
e
rro
r
e
at ste
ady st
ate
is bou
nde
d b
y
:
Proof:
Due
to line
a
rity of t
he
system
(1
7),
we
ca
n
co
nsid
er
w
(0
) s
u
ch
that
.
We ne
ed to p
r
ove that
at
steady state. Note that
.
For
sim
p
licit
y, denote
. Co
nsi
der
t
he q
uad
ratic Lyapu
nov
function
. Its
γ
level set is:
Whi
c
h is a
n
e
llipsoid. Th
e condition (25
)
implies that [4
4]:
It suffice
s to
sho
w
that
un
der
, the stat
e
ξ
will converge
to
E
(
P,
γ
). This i
s
g
u
a
r
anteed
if
V
(
ξ
(
t
)) i
s
st
rict
ly
de
cre
a
s
i
ng,
i.
e.
,
˙
V
(
ξ
)
<
0, a
s
long
as
V
(
ξ
)
≥
γ
.
We u
s
e the
condition (26
)
to
prove this. Fo
r the system
(17), we h
a
ve:
Thus
condition (26) implies that:
By the structu
r
al inform
atio
n (22
)
-(24
) an
d
α
1
,
α
2
≥
0, we have:
Since
, we have
for all
t
, i
.
e.,
.
Therefore,
Whi
c
h implie
s that
V
(
ξ
(
t
))
is stri
ctly decreasi
ng a
s
lon
g
as
V
(
ξ
(
t
))
≥
γ
. This
completes
the proof.
The
co
nstrai
nt (2
6) i
s
l
e
ss
re
strictive t
han
th
e
co
rre
spo
ndin
g
con
d
ition (20
)
d
u
e to the
addition
al pa
ramete
rs
α
1
,
α
2
,
α
3 in th
e terms
−α
1
WT
HD,
−α
2
Wr
m
and
−α
3
Wr
p
, which result
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