Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
11,
No.
2,
April
2020,
pp.
1186
1199
ISSN:
2088-8708,
DOI:
10.11591/ijece.v11i2.pp1186-1199
r
1186
Obser
v
er
-based
tracking
contr
ol
f
or
single
machine
infinite
b
us
system
via
flatness
theory
Mohammad
P
ourmahmood
Aghababa
1
,
Bogdan
Marinescu
2
,
Flor
ent
Xa
vier
3
1,2
Ecole
Centrale
de
Nantes,
LS2N,
France
3
R
´
eseau
de
T
ransport
d’Electricit
´
e
(R
TE),
France
Article
Inf
o
Article
history:
Recei
v
ed
Dec
11,
2019
Re
vised
Jul
31,
2020
Accepted
Aug
17,
2020
K
eyw
ords:
Critical
clearing
time
Flat
system
Input
constraints
State
observ
er
T
ransient
stability
ABSTRA
CT
In
this
research,
we
aim
to
use
the
flatness
control
theory
to
de
v
elop
a
useful
control
scheme
for
a
single
machine
connected
to
an
infinite
b
us
(SMIB)
system
taking
into
account
input
magnitude
and
rate
saturation
constraints.
W
e
adopt
a
fourth-order
non-
linear
SMIB
model
along
an
e
xciter
and
a
turbine
go
v
ernor
as
actuators.
According
to
the
flatness-based
control
strate
gy
,
first
we
sho
w
that
the
adopted
nominal
SMIB
model
is
a
flat
system.
Then,
we
de
v
elop
a
full
linearizing
state
feedback
as
well
as
an
outer
inte
gral-type
loop
to
e
nsure
suitable
tracking
performances
for
the
po
wer
and
v
oltage
as
well
as
the
angular
v
elocity
outputs.
W
e
assume
that
only
the
angular
v
e-
locity
of
the
generator
is
a
v
ailable
to
be
measured.
So,
we
pro
vide
a
linear
Luenber
ger
observ
er
to
estimate
the
remaining
states
of
the
system.
Also,
the
saturation
nonlinear
-
ities
are
transferred
to
the
linear
part
of
the
system
and
the
y
are
canceled
out
using
their
estimations.
The
ef
ficienc
y
and
usefulness
of
the
proposed
observ
er
-controll
er
ag
ainst
f
aults
are
illustrated
using
simulation
tests
in
Eurostag
and
Matlab
.
The
results
sho
w
that
the
clearing
critical
time
of
the
introduced
methodology
is
lar
ger
than
the
classical
control
approaches
and
the
proposed
observ
er
-based
flatness
controller
e
xhibits
o
v
er
much
less
control
ener
gy
compared
to
the
classic
IEEE
controllers.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Bogdan
Marinescu
Ecole
Centrale
de
Nantes
LS2N,
Nantes,
France
Email:
Bogdan.Marinescu@ec-nantes.fr
1.
INTR
ODUCTION
No
w
adays,
the
electricity
has
become
as
an
important
and
vit
al
component
of
the
life
and
industry
.
So,
the
electrical
po
wer
netw
orks
should
be
in
a
secure
operation
with
a
reas
onable
stability
mar
gin
to
produce
the
demanded
electricity
.
The
ability
of
a
po
wer
system
in
maintaining
in
the
machines
synchronous
operation
point
after
occurrence
of
a
disturbance
and/or
f
ault
is
usually
interpreted
as
its
transient
stability
concept.
T
o
retain
the
po
wer
system
stability
in
a
suitable
limit
in
the
e
v
ent
of
uncertainties,
f
aults
and
disturbances,
it
is
necessary
to
add
control
actions,
such
as
e
xciters
and
go
v
ernors,
to
the
system
to
impro
v
e
dynamics
till
the
circuit
break
er
opening
and
reclosing
times
[1].
The
critical
clearing
time
(CCT)
is
one
useful
and
applied
f
actor
to
measure
the
transient
stability
mar
gin
of
a
po
wer
machines.
The
CCT
stands
for
the
maximum
time
during
which
a
f
ault
can
be
applied
without
missing
the
system’
s
stability
.
Such
a
stability
mar
gin
depends
on
the
design
of
the
controls
of
generators
connected
to
the
grid
[2].
T
o
enhance
the
CCT
of
a
grid,
some
control
de
vices
should
be
designed
and
implemented
in
the
netw
ork.
T
o
synthesis
and
analyze
the
performance
of
the
controllers
on
the
CCT
of
a
po
wer
grid,
a
single
J
ournal
homepage:
http://ijece
.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1187
machine
connected
to
an
infinite
b
us
(SMIB)
po
wer
syst
em
model
is
usually
adopted
to
a
v
oid
the
unnecessary
comple
xity
of
the
po
wer
system
in
the
control
design
phase.
Generally
speaking,
there
are
tw
o
main
controller
classes
for
this
case:
(i)
standard
controllers
and
(ii)
adv
anced
control
techniques.
The
first
class
belongs
to
the
well-kno
wn
standard
IEEE
controllers.
More
details
about
the
classic
IEEE
controllers
can
be
found
in
[3,
4].
Although
the
classic
IEEE
controllers
are
simple,
their
parameters
are
needed
to
be
appropriately
adjusted
and
their
stabil
ity
re
gions
are
limited.
In
the
w
orks
[5–7]
some
intelligent
heuristic
optimization
methods
ha
v
e
been
proposed
for
finding
the
suitable
parameters
of
the
re
gulators.
Ho
we
v
er
,
since
their
approaches
requires
implementation
of
some
iterati
v
e
numerical
algorithms
,
their
practi
cal
implementations
will
be
dif
ficult
in
online
and
f
ast
response
needed
situations.
On
the
other
hand,
the
approaches
in
the
second
class
use
some
so-called
adv
anced
control
strate
gies
to
enhance
the
transient
stability
of
the
po
wer
machines.
There
are
se
v
eral
e
xamples
in
the
literature
for
this
cate
gory
which
include
sliding
mode
control
[8],
fuzzy
control
[9],
nonlinear
control
[10],
dynamic
in
v
ersion
control
[11]
and
optimal
control
[12],
etc.
In
[13],
a
po
wer
system
stabilizer
has
been
proposed
for
synchronous
machines
based
on
con
v
en-
tional
fuzzy-PID
and
type-1
fuzzy
controller
combined
with
a
sliding
mode
control
strate
gy
.
The
w
ork
[14]
has
proposed
an
adapti
v
e
w
a
v
elet
netw
ork-based
nonlinear
e
xcitation
control
for
po
wer
systems
without
consider
-
ing
the
go
v
ernor
dynamics.
T
o
impro
v
e
the
stability
of
the
v
oltage
re
gulation
and
to
enhance
the
damping
of
lo
w
frequenc
y
po
wer
system
oscillations
of
SMIB
systems,
an
e
xtended
reduced-order
observ
er
along
with
an
automatic
v
oltage
re
gulator
has
been
de
v
eloped
in
[15].
In
[16],
the
relationship
between
transient
stability
=
instability
and
conca
vity
=
con
v
e
xity
of
the
phase-plane
tr
ajectory
has
been
found
and
a
transient
instability
cri-
terion
has
been
deri
v
ed
for
real-time
instability
detection
and
the
SMIB
system
has
been
stabilized.
Rout
et
al
[17]
ha
v
e
sho
wed
that
the
SMIB
system
can
possess
chaotic
and
oscillatory
dynamics
when
the
system
parame-
ters
f
all
into
a
certain
area.
Accordingly
,
the
y
ha
v
e
designed
an
adapti
v
e
controller
based
on
LaSalle’
s
in
v
ariant
principle
to
mak
e
the
system
oscillations
damped.
The
paper
[18]
has
in
v
estig
ated
the
problem
of
transient
sta-
bility
and
v
oltage
re
gulation
for
a
SMIB
system
via
a
modified
backstepping
control
design
method.
Ho
we
v
er
,
most
of
the
pre
vious
w
orks
either
ha
v
e
not
considered
the
ef
fects
of
input
saturations,
the
y
ha
v
e
been
designed
for
some
simplified
linear
and/or
nonlinear
models
of
the
SMIB,
there
are
usually
steady
state
errors
on
the
outputs
of
their
methods
or
the
y
ha
v
e
assumed
that
all
the
states
of
the
system
are
a
v
ailable
to
be
measured.
The
concept
of
dif
ferentially
flat
nonlinea
r
systems
w
as
first
introduced
by
Fliess
et
al
[19,
20].
The
scheme
is
an
e
xtension
from
the
input-output
scheme
with
zero
internal
dynamics.
A
system
is
con-
sidered
to
be
dif
ferentially
flat
if
all
its
state
v
ariables
and
its
control
inputs
can
be
e
xpressed
as
functions
of
one
single
algebraic
v
ariable
which
is
the
so-ca
lled
flat
output,
and
also
as
functions
of
the
flat-output’
s
deri
v
a-
ti
v
es.
The
dif
ferential
flatness
property
enables
the
transformation
of
the
nonlinear
system’
s
dynamics
into
the
linear
canonical
form
and
the
design
of
a
state
feedback
controller
through
the
application
of
pole
placement
techniques
in
the
linearized
equi
v
alent
model
of
the
system.
The
construction
of
the
feedback
la
w
is
done
by
a
simple
in
v
ersion
of
system
equations
with
respect
to
the
system
input.
Although
this
technique
has
been
applied
to
se
v
eral
nonlinear
and
linear
mechanical
systems
[21–23],
its
application
to
the
control
of
po
wer
systems
has
been
limited
to
a
fe
w
w
orks
[24–26]
and
[27].
Ho
we
v
er
,
the
pre
vious
w
orks
ha
v
e
not
focused
on
the
transient
stability
mar
gin
and
the
y
either
ha
v
e
not
carried
out
the
ef
fects
of
the
actuator
saturations
or
the
y
ha
v
e
assumed
that
all
the
states
of
the
synchronous
machines
are
a
v
ailable
to
be
measured.
In
this
research,
i
nspired
by
the
flatness
control
theory
,
we
propose
a
full
linearizing
state
feedback
for
the
system
to
cancel
out
the
nonlinearities
of
the
system
and
to
obtain
a
linear
canonical
(Bruno
vsk
y)
form
for
it.
Then,
we
transfer
the
input
saturation
nonlinearities
of
the
system
to
the
adopted
linear
part
of
the
model.
T
o
mak
e
the
obtained
linear
system
to
be
controllable
and
observ
able,
some
modifications
are
done
on
the
nonlinear
feedback
control
to
modify
the
linear
matrix
of
the
system.
Subsequently
,
a
full
order
state
linear
observ
er
is
designed
for
the
system
to
est
imate
the
states
of
the
model
using
the
only
a
v
ailable
output,
i.e.
the
angular
v
elocity
.
T
o
pro
vide
tracking
of
v
oltage
and
po
wer
reference
commands,
i
nte
gral
outer
loops
are
added
to
the
linear
part
of
the
system
.
The
Kalman
filter
and
the
linear
quadratic
re
gulator
approaches
are
adopted
to
deri
v
e
the
linear
observ
er
and
controller
,
respecti
v
ely
,
and
to
construct
the
frame
w
ork
of
the
final
augmented
system.
Finally
,
the
ef
ficienc
y
and
applicability
of
the
proposed
observ
er
-based
state
feedback
controller
are
re
v
ealed
using
some
illustrati
v
e
e
xamples
si
mulated
in
the
presence
of
short
circuit
f
aults
and
magnitude
and
rate
saturations
of
input
signals,
where
the
superiority
of
the
introduced
controller
is
re
v
ealed
for
its
rob
ustness
ag
ainst
unw
anted
f
aults
increasing
CCT
of
the
machines
and
its
lo
w
implementation
costs
reducing
the
control
ener
gy
required
for
the
stabilization
of
the
system
compared
to
the
classic
IEEE
control
schemes.
Observer
-based
tr
ac
king
contr
ol
for
...
(Mohammad
P
ourmahmood
Aghababa)
Evaluation Warning : The document was created with Spire.PDF for Python.
1188
r
ISSN:
2088-8708
2.
SYSTEM
MODELING
AND
PR
OBLEM
FORMULA
TION
Figure
1
sho
ws
a
simple
representation
of
a
po
wer
system
consisting
of
a
single
machine
connected
to
an
infinite
b
us
through
a
transmission
line
with
impedance
Z
=
r
e
+
j
x
ep
.
In
the
sequel,
we
will
present
the
one-axis
model,
named
also
flux
decay
model,
of
the
SMIB
po
wer
system
where
the
reader
may
refer
to
[1]
for
more
details.
T
r
an
s
m
is
s
io
n
lin
e
=
+
~
(
,
)
G
I
n
f
in
ite
b
u
s
∞
~
(
,
)
Figure
1.
A
SMIB
po
wer
system
2.1.
One-axis
model
of
a
SMIB
The
mechanical
dynamics
of
the
generator
,
which
correspond
to
the
rotor’
s
relati
v
e
angle
(
)
and
the
angular
v
elocity
(
!
),
are
described
by:
_
=
(
!
!
s
)
!
0
(1)
_
!
=
!
s
2
H
(
T
m
T
f
w
T
e
)
(2)
where
!
s
is
the
rated
synchronous
speed
in
p.u,
!
0
is
the
rated
synchronous
speed
in
rad/s,
H
is
the
machine’
s
inertia,
T
m
represents
the
mechanical
torque
applied
to
the
shaft,
T
f
w
=
D
(
!
!
s
)
is
a
friction
windage
torque
with
a
damping
coef
ficient
D
and
T
e
is
the
electrical
torque
defined
by
the
follo
wing
e
xpression
T
e
=
i
q
e
0
q
+
(
x
q
x
0
d
)
i
d
i
q
(3)
where
x
q
is
the
synchrounous
reactance
of
the
generator
in
quadratic
axis
and
x
0
d
is
its
transient
reactance
in
direct
axis.
The
mechanical
po
wer
,
denoted
P
m
,
is
defined
as:
P
m
=
T
m
!
(4)
Inserting
in
2)
T
e
gi
v
en
by
(3)
and
T
m
gi
v
en
by
(4)
and
assuming
D
=
0
,
one
gets
_
!
=
1
2
H
(
T
m
i
q
e
0
q
(
x
q
x
0
d
)
i
d
i
q
)
(5)
The
dynamic
of
the
induced
v
oltage
on
the
q
-axis,
denoted
e
0
q
,
is
gi
v
en
by:
_
e
0
q
=
1
T
0
d
0
e
0
q
(
x
d
x
0
d
)
i
d
+
E
f
d
(6)
where
E
f
d
represents
the
v
oltage
across
the
rotor
field
coil
,
which
is
considered
as
an
input
for
the
SMIB
model
to
be
designed.
The
mathematical
equation
of
the
go
v
ernor
dynamics
is
gi
v
en
belo
w
.
_
P
sv
=
1
T
g
(
T
cc
P
sv
+
K
g
(
!
!
s
))
(7)
in
which
T
cc
is
a
constant
to
be
designed
as
a
reference
for
mechanical
torque
and
P
sv
is
the
steam
v
alv
e
position
and
it
represents
the
second
input
for
the
SMIB
model.
The
turbine
dynamical
equation
is
also
gi
v
en
by:
_
T
m
=
1
T
ch
(
P
sv
T
m
)
(8)
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
2,
April
2020
:
1186
–
1199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1189
The
currents
in
dq
-coordinates
(
i
d
,
i
q
)
are
obtained
by
solving
a
set
of
algebraic
equations,
whic
h
represents
the
interaction
of
the
machine
with
the
po
wer
grid:
i
d
=
c
1
+
c
2
e
0
q
i
q
=
c
3
+
c
4
e
0
q
(9)
with
c
1
=
(
r
s
+
r
e
)
sin
(
)
+
(
x
q
x
ep
)
cos
(
)
(
r
s
+
r
e
)
2
+
(
x
0
d
+
x
ep
)(
x
q
+
x
ep
)
V
s
c
2
=
x
q
+
x
ep
(
r
s
+
r
e
)
2
+
(
x
0
d
+
x
ep
)(
x
q
+
x
ep
)
c
3
=
(
x
0
d
+
x
ep
)
sin
(
)
(
r
s
+
r
e
)
cos
(
)
(
r
s
+
r
e
)
2
+
(
x
0
d
+
x
ep
)(
x
q
+
x
ep
)
V
s
c
4
=
r
e
+
r
s
(
r
s
+
r
e
)
2
+
(
x
0
d
+
x
ep
)(
x
q
+
x
ep
)
Assuming
r
s
=
r
e
=
0
(i.e.,
o
v
erhead
lines
with
ne
glectable
resistances
compared
to
reactances),
the
currents
i
d
,
i
q
are
simplified
as
follo
ws:
i
q
=
V
s
x
ep
+
x
d
sin
(
)
i
d
=
1
x
ep
+
x
0
d
e
0
q
V
s
x
ep
+
x
0
d
cos
(
)
(10)
The
abo
v
e
dynamics
(1)-(10)
can
be
re
grouped
in
the
follo
wing
fifth-order
state
form:
_
x
(
t
)
=
f
(
x
(
t
)
;
u
(
t
))
;
x
(
t
0
)
=
x
0
(11)
where
x
1
=
,
x
2
=
!
,
x
3
=
e
0
q
,
x
4
=
T
m
,
u
1
=
E
f
d
and
u
2
=
P
sv
.
It
is
noted
that
in
this
w
ork,
we
will
include
the
go
v
ernor
dynamics
(7)
in
an
outer
control
loop.
It
should
be
also
noted
that
due
to
ph
ysical
limi
tations,
the
control
inputs
ha
v
e
to
v
erify
the
follo
wing
constraints:
Magnitude
limitation:
The
e
xciter
may
generate
positi
v
e
and
ne
g
ati
v
e
v
alues
where
symmetric
bounds
are
generally
considered:
j
u
1
j
u
1
(12)
Ho
we
v
er
,
the
steam
v
alv
e
position
is
limited
between
0
(completely
closed)
and
1
(completely
open):
0
u
2
u
2
(13)
Rate
limitation:
Beside
the
magnitude
saturation,
the
v
ariation
of
steam
v
alv
e
position
has
to
respect
some
limitations:
1
T
c
_
u
2
1
T
o
(14)
where
T
o
(
T
c
)
is
the
necessary
time
to
pass
from
a
completely
closed
(open)
position
to
a
completely
open
(closed)
position.
Remark
1:
One
can
use
the
first-order
deri
v
ati
v
e
approximation
_
u
=
u
(
t
)
u
(
t
)
(where
is
a
small
constant)
to
con
v
ert
the
rate
saturation
to
a
magnitude
saturation
as
follo
ws:
1
T
c
_
u
2
1
T
o
)
1
T
c
u
2
(
t
)
u
2
(
t
)
1
T
o
)
T
c
+
u
2
(
t
)
u
2
(
t
)
T
o
+
u
2
(
t
)
(15)
Observer
-based
tr
ac
king
contr
ol
for
...
(Mohammad
P
ourmahmood
Aghababa)
Evaluation Warning : The document was created with Spire.PDF for Python.
1190
r
ISSN:
2088-8708
No
w
,
defining
u
2
M
in
=
T
c
+
u
2
(
t
)
and
u
2
M
ax
=
T
c
+
u
2
(
t
)
and
noting
to
0
u
2
u
2
,
we
will
ha
v
e
u
2
L
u
2
(
t
)
u
2
H
(16)
with
u
2
L
=
max
f
0
;
u
2
M
in
g
and
u
2
H
=
min
f
u
2
;
u
2
M
ax
g
.
2.2.
SMIB
Model
in
faulty
mode
operation
The
appearance
of
a
temporary
short
circuit
in
the
transmission
line
af
fects
considerably
the
system’
s
stability
.
Let
us
c
o
ns
ider
the
case
of
tw
o
parallel
transmission
lines
with
the
same
impedance
Z
0
=
2(
r
e
+
j
x
ep
)
.
This
allo
ws
us
to
preserv
e
the
same
impedance
of
the
pre
vious
t
ransmission
line
Z
.
The
short
circuit
will
happen
at
ti
me
t
c
at
the
middle
of
the
second
transmission
line.
This
f
aulty
mode
af
fects
the
SMIB
model
by
changing
its
parameters
(
r
e
,
x
ep
)
by
2
3
(
r
e
,
x
ep
)
and
V
s
by
1
3
V
s
as
reported
in
[28].
Notice
that
in
practice
the
short
circuit
may
happen
at
an
y
point
of
the
line.
It
has
been
e
xamined
here
in
the
middle
for
the
sak
e
of
simplicity
.
F
or
e
xample,
if
the
short
circuit
appears
at
the
terminal
of
the
generator
,
V
s
will
be
replaced
by
zero
in
the
dynamics
(1)-(10).
In
this
case,
the
machine’
s
speed
increases,
with
respect
to
the
dynamic
(17),
as
long
as
the
short
circuit
is
present
and
depending
on
its
durat
ion
(denoted
t
)
the
machine
may
lose
synchronism.
_
!
=
!
s
2
H
T
m
(17)
2.3.
Contr
ol
objecti
v
es
The
main
control
objecti
v
e
for
the
synchronous
machine
is
to
operate
at
synchronous
speed
!
s
,
main-
tain
a
constant
modulus
of
the
terminal
v
oltage
V
r
ef
and
achie
v
e
a
desired
mechanical
po
wer
P
r
ef
under
control
inputs
constraints.
Accordingly
,
we
will
try
to
mimic
the
standard
classic
IEEE
go
v
ernor
model
(7)
in
the
closed-loop
system.
The
modulus
of
the
terminal
v
oltage
of
the
synchronous
generator
,
denoted
v
t
,
is
gi
v
en
by:
v
t
=
q
v
2
d
+
v
2
q
(18)
where:
v
d
=
r
e
i
d
x
ep
i
q
+
V
s
sin
(
x
1
)
v
q
=
r
e
i
q
+
x
ep
i
d
+
V
s
cos
(
x
1
)
(19)
3.
OBSER
VER-B
ASED
NONLINEAR
CONTR
OLLER
DESIGN
In
this
sect
ion,
first
we
sho
w
that
the
nonlinear
model
SMIB
system
is
flat.
Then,
we
will
c
ompensate
the
ef
fects
of
the
input
saturations.
Finally
,
a
linear
state
observ
er
is
designed
for
the
system
to
b
uilt
the
final
flatness-based
controller
.
3.1.
Checking
out
flatness
of
SMIB
Consider
a
general
case
of
the
nonlinear
system
(11)
with
x
(
t
)
2
R
n
and
u
(
t
)
2
R
m
.
It
is
said
to
be
dif
ferentially
flat
if
and
only
if
there
e
xists
a
flat
output
z
(
t
)
2
R
m
of
the
form
:
z
=
h
(
x;
u;
_
u;
:::;
u
(
r
)
)
(20)
such
that
x
=
(
z
;
_
z
;
:::;
z
(
q
)
)
u
=
(
z
;
_
z
;
:::
;
z
(
q
+1)
)
(21)
The
components
of
the
flat
output
z
are
dif
ferentially
independent.
These
equations
yield,
that
for
e
v
ery
gi
v
en
trajectory
of
the
flat
output
t
!
z
(
t
)
,
the
e
v
olution
of
all
other
v
ariables
of
the
system
t
!
x
(
t
)
and
t
!
u
(
t
)
are
also
gi
v
en
without
inte
gration
of
the
system
of
dif
ferential
equations
Proposition
1
:
The
one-axis
model
of
the
single
machine
infinite
b
us
po
wer
system
gi
v
en
by
(11)
is
dif
ferentially
flat
with
the
output
z
=
[
x
1
x
4
]
T
.
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
2,
April
2020
:
1186
–
1199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1191
Proof:
W
e
should
pro
v
e
that
the
remaining
states
x
2
=
!
and
x
3
=
e
0
q
as
well
as
the
control
inputs
u
1
=
E
f
d
and
u
2
=
P
sv
can
be
written
as
a
function
of
the
flat
outputs
and
their
successi
v
e
deri
v
ati
v
es.
First,
from
(2),
one
gets
!
=
_
w
0
+
!
s
(22)
So,
x
2
=
!
is
stated
by
the
flat
output
.
From
(2),
one
obtains
the
follo
wing
relation
for
e
0
q
.
e
0
q
=
T
m
2
H
w
0
w
•
AAV
s
i
q
(1
+
AA
)
i
q
(23)
where
AA
=
x
q
x
0
d
x
ep
+
x
0
d
.
The
ne
xt
step
is
to
e
xpress
the
control
inputs
as
functions
of
the
outputs
and
their
deri
v
ati
v
es.
T
o
this
end,
in
what
follo
ws,
we
replace
the
control
inputs
u
1
(
t
)
and
u
2
(
t
)
with
S
at
(
u
1
)
and
S
at
(
u
2
)
,
respecti
v
ely
to
include
the
corresponding
saturation
nonlinearities.
As
a
result,
using
(6)
and
(23),
one
can
obtain
S
at
(
u
1
)
=
T
d
0
_
e
0
q
+
e
0
q
+
(
x
d
x
0
d
)
i
d
(24)
where
_
e
0
q
=
h
_
T
m
2
H
w
0
w
s
•
+
AAV
s
(
_
iq
cos
(
)
sin
(
)
i
q
)
i
(
i
q
(1+
AA
))
F
(
X
)
(
i
q
(1+
AA
))
2
with
F
(
X
)
=
_
iq
(1
+
AA
)(
T
m
2
H
w
0
w
_
+
AAV
s
cos
(
)
i
q
)
and
_
iq
=
V
scos
(
)
x
ep
+
xq
.
And,
according
to
the
(8)
one
can
easily
obtain
S
at
(
u
2
)
=
T
ch
_
T
m
+
T
m
(25)
So,
the
proof
is
completed
and
the
selected
v
ariables
z
=
[
x
1
x
4
]
T
are
the
flat
outputs
of
the
SMIB
system.
3.2.
Flatness-based
linearizing
state
feedback
Here,
we
consider
the
ne
w
state
v
ector
=
[
1
2
3
4
5
]
T
with
1
=
x
1
=
,
2
=
_
1
,
3
=
•
1
,
4
=
x
4
=
P
m
and
5
=
_
P
m
.
In
what
follo
ws,
we
conclude
a
proposition
to
highlight
the
equi
v
alence
between
the
nonlinear
model
of
a
SMIB
po
wer
system
and
a
linear
controllable
one.
Before
proceeding
to
the
proposition,
we
first
re
write
the
saturation
function
as
follo
ws:
S
at
(
u
i
)
=
u
i
+
u
i
;
i
=
1
;
2
:
(26)
where
u
i
is
defined
as
follo
ws:
u
i
=
8
<
:
u
H
i
u
i
;
u
i
u
H
i
0
;
u
L
i
u
i
u
H
i
u
L
i
u
i
;
u
i
u
L
i
(27)
Therefore,
based
on
(26)
and
defining
v
1
=
•
and
v
2
=
_
T
m
,
the
control
inputs
u
1
(24)
and
u
2
(25)
are
reformed
as
(28)
with
v
1
=
w
s
w
0
u
1
2
H
and
v
2
=
u
2
T
ch
.
u
1
=
T
d
0
[
_
T
m
2
H
w
0
w
s
(
v
1
+
v
1
)+
AAV
s
(
_
iq
cos
(
)
sin
(
)
i
q
)](
i
q
(1+
AA
))
T
d
0
F
(
X
)
(
i
q
(1+
AA
))
2
+
T
m
2
H
w
0
w
•
AAV
s
i
q
(1+
AA
)
i
q
+
(
x
d
x
0
d
)
i
d
u
2
=
T
ch
(
v
2
+
v
2
)
+
T
m
(28)
Proposition
2
:
Under
the
mapping
=
(
x
)
gi
v
en
by:
(
x
)
=
0
B
B
@
x
1
x
2
!
s
!
0
(
x
3
i
q
(1+
AA
)+
x
4
+
AAV
s
i
q
)
w
0
w
s
2
H
x
4
1
C
C
A
(29)
Observer
-based
tr
ac
king
contr
ol
for
...
(Mohammad
P
ourmahmood
Aghababa)
Evaluation Warning : The document was created with Spire.PDF for Python.
1192
r
ISSN:
2088-8708
and
the
state
feedback
gi
v
en
in
(28),
the
nonlinear
model
of
the
SMIB
is
equi
v
alent
to
the
follo
wing
semi-linear
(i.e.
linear
system
with
nonlinear
uncertainties)
controllable
system
presented
in
(30).
_
=
A
+
B
(
v
+
v
)
(30)
where:
v
=
v
1
v
2
;
v
=
v
1
v
2
;
A
=
2
6
6
4
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
3
7
7
5
;
B
=
2
6
6
4
0
0
0
0
1
0
0
1
3
7
7
5
(31)
Proof:
By
considering
the
flat
v
ariables
x
1
and
x
5
,
the
corresponding
Bruno
vsk
y
form
has
a
full
rank
controllability
matrix,
which
means
that
one
can
propose
a
state
feedback
that
can
linearize
the
SMIB
model
(11)
[20,
29].
Using
the
abo
v
e
dif
feomorphism
and
replacing
the
control
e
xpression
(28)
in
the
dynamics
of
the
ne
w
coordinates
yields
the
abo
v
e
linear
controllable
system
(30).
Remark
2:
It
should
be
noted
that
although,
we
ha
v
e
not
used
the
deri
v
ati
v
es
of
the
saturation
func-
tions
of
the
control
inputs
in
proposition
1
to
pro
v
e
the
flatness
property
of
the
SMIB
system,
the
a
v
ailable
discontinuities
in
the
u
may
mak
e
the
proposed
dif
feomorphism
mapping
to
be
local
from
the
equi
v
alenc
y
point
of
vie
w
.
Ho
we
v
er
,
transferring
the
discontinuous
term
u
to
the
semi-linear
side
as
the
term
v
and
noting
to
this
f
act
that
the
discontinuities
of
v
are
e
xactl
y
equal
to
those
of
u
(due
t
o
this
f
act
that
v
includes
the
e
xact
u
),
will
a
v
oid
this
problem
and,
therefore,
the
proposed
dif
feomorphism
will
be
global.
Remark
3:
It
should
be
noted
that
although,
we
ha
v
e
not
used
the
deri
v
ati
v
es
of
the
saturation
func-
tions
of
the
control
inputs
in
Proposition
1
to
pro
v
e
the
flatness
property
of
the
SMIB
sys
tem,
the
a
v
ailable
discontinuities
in
the
u
mak
e
the
proposed
dif
feomorphism
mapping
to
be
local
from
t
he
equi
v
alenc
y
point
of
vie
w
.
In
f
act,
when
the
control
inputs
hit
the
saturation
limits,
there
will
not
be
a
one-by-one
mapping
between
the
semi-linear
Bruno
vsk
y
system
and
the
corresponding
nonlinear
SMIB
model
an
ymore.
Ho
we
v
er
,
e
v
en
in
this
case
,
one
can
design
proper
linear
control
inputs
v
(
t
)
to
stabili
ze
the
system
lik
e
an
output
feedback
linearization
scheme.
3.3.
Linear
state
obser
v
er
design
From
flatness-control
point
of
vie
w
,
one
of
the
main
practical
significance
of
the
flat
outputs
of
a
dynamical
system
is
that
if
the
flat
outputs
are
measurable,
then
all
the
system
v
ariables
required
for
feedback
can
be
directly
computed
without
inte
grating
an
y
dif
ferential
equations.
In
our
case,
the
rotor’
s
relati
v
e
angle
and
the
mechanical
po
wer
are
selected
as
the
system
flat
outputs.
Ho
we
v
er
,
in
practice,
these
v
alues
cannot
be
easily
measured.
And,
in
most
practical
situations,
the
angular
v
elocity
of
the
machine
is
the
only
a
v
ailable
output
to
be
measured.
Accordingly
,
if
we
w
ant
to
implement
the
abo
v
e-designed
controller
,
we
should
fist
de
v
elop
an
observ
er
for
the
system
to
estimate
the
states
of
the
system
using
the
angular
v
elocity
output.
In
what
follo
ws,
the
observ
er
design
procedure
for
the
SMIB
system
is
presented.
Instead
of
directly
de
v
eloping
an
observ
er
for
the
SMIB
system
gi
v
en
in
(1)-(10),
we
use
the
equi
v
alent
semi-linear
system
(31)
to
propose
an
ef
ficient
linear
Luenber
ger
observ
er
.
T
o
this
end,
we
first
select
the
output
matrix
of
the
system
as
y
=
[0
1
0
0]
T
and
check
the
observ
ability
property
of
the
system
(31).
After
checking
the
rank
of
the
observ
ability
matrix,
it
is
re
v
ealed
that
it
is
singular
(it
is
rank
ef
ficient)
and,
therefore,
the
system
(31)
is
not
observ
able.
T
o
solv
e
this
problem,
we
propose
an
alternati
v
e:
modify
the
linear
matrix
A
in
(31)
using
modifying
the
linear
control
inputs
v
1
and
v
2
to
ob
t
ain
an
observ
able
system.
In
this
line,
we
add
T
m
and
to
the
control
inputs
u
1
and
u
2
in
(28),
respecti
v
ely
as
formulated
in
(32).
u
1
=
T
d
0
[
_
T
m
2
H
w
0
w
s
(
v
1
T
m
|
{z
}
v
1
n
+
v
1
+
T
m
|
{z
}
v
1
n
)+
AAV
s
(
_
iq
cos
(
)
sin
(
)
i
q
)](
i
q
(1+
AA
))
T
d
0
F
(
X
)
(
i
q
(1+
AA
))
2
+
T
m
2
H
w
0
w
•
AAV
s
i
q
(1+
AA
)
i
q
+
(
x
d
x
0
d
)
i
d
u
2
=
T
ch
(
v
2
|
{z
}
v
2
n
+
v
2
+
|
{z
}
v
2
n
)
+
(32)
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
2,
April
2020
:
1186
–
1199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1193
The
abo
v
e
modifications
are
equi
v
alent
to
put
a
1
in
the
third
ro
w-forth
column
of
the
matrix
A
and
another
1
in
the
firth
ro
w-fifth
column
of
the
matrix
A
as
follo
ws:
A
=
2
6
6
4
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
3
7
7
5
(33)
No
w
,
the
pair
(
A;
C
)
is
observ
able
and
the
pair
(
A;
B
)
is
controllable
(their
corresponding
observ
abil-
ity
and
controllability
matrices
are
full
rank).
The
linear
Luenber
ger
observ
er
is
designed
using
the
follo
wing
dynamics:
_
b
=
A
b
+
B
V
+
L
(
y
C
)
(34)
where
V
=
[
v
1
n
v
2
n
]
T
is
the
linear
control
input
to
be
designed
later
.
The
observ
er
g
ain
matrix
L
can
be
easily
obtained
using
the
Kalman
command
of
the
Matlab
softw
are.
So,
the
estimated
states
b
will
be
used
to
construct
the
final
controller
(including
linear
controls
v
1
n
;
v
2
n
and
nonlinear
ones
u
1
;
u
2
).
T
able
1.
T
ypical
parameters
of
a
SMIB
po
wer
system
in
p.u
P
arameter
!
s
V
s
T
0
d
0
x
d
x
q
x
0
d
H
T
ch
r
s
r
ep
D
x
ep
V
alue
1
1
9.6
2.38
1.21
0.336
5
5
0
0
0
0.4
3.4.
Output
tracking
loop
The
ne
w
v
ector
of
control
V
has
to
guarantee
the
stability
of
the
SMIB
system
and
achie
v
e
the
control
requirements
discussed
before.
Thus,
the
state
v
ector
has
to
be
dri
v
en
to
a
desired
state
r
ef
,
which
corre-
sponds
to
a
desired
relati
v
e
angle,
a
zero
relati
v
e
speed,
a
zero
acceleration
and
a
desired
mechanical
torque.
Ho
we
v
er
,
in
addition
to
the
dri
ving
the
states
of
the
semi-linear
system
to
the
desired
the
reference
v
alues,
for
better
tracking
purposes
and
good
rob
ustness
properties,
we
add
an
output
inte
grator
loop
to
the
system
to
mak
e
it
for
ef
ficient
satisf
action
of
the
control
objecti
v
es.
On
the
other
hand,
one
needs
to
cancel
out
the
ef
fects
of
the
semi-linear
ter
ms
v
in
the
system.
So,
the
output
tracking
loop
includes
thre
e
parts:
i)
an
acti
v
e
control
for
cancelling
the
semi-linear
parts,
ii)
a
feedforw
ard
tracking
scheme
for
pro
viding
the
tracking
conditions
of
linear
state
space
systems,
and
iii)
an
inte
grator
loop
for
including
the
output
errors
in
the
obtained
equv
alent
linear
system.
These
items
are
e
xplained
in
the
ne
xt
subsections.
3.4.1.
Acti
v
e
contr
ol
Here,
we
aim
to
cancel
out
the
ef
fects
of
the
semi-linear
parts
v
1
n
and
v
2
n
.
Noting
to
the
defi-
nitions
v
1
n
=
w
s
w
0
u
1
2
H
+
T
m
and
v
2
n
=
u
2
T
ch
+
,
it
is
re
v
ealed
that
these
terms
are
kno
wn.
So,
we
easily
use
the
so-called
acti
v
e
control
method
[30]
to
directly
remo
v
e
them
by
a
feedback
control.
Ho
we
v
er
,
it
should
be
noted
that
we
cannot
use
t
he
states
of
the
nonlinear
system;
ins
tead,
we
use
the
outputs
of
the
deri
v
ed
observ
er
to
b
uild
estimations
for
v
1
n
and
v
2
n
.
Accordingly
,
the
estimated
semi-linear
parts
will
be
added
to
the
control
input
v
n
to
cancel
out
the
ef
fects
of
the
corresponding
terms.
In
this
line,
one
can
define
a
ne
w
linear
control
input
as
follo
ws:
v
n
=
v
f
v
(35)
where
v
f
is
the
ne
w
linear
control
input
and
v
=
[
v
1
n
v
2
n
]
T
.
Based
on
the
abo
v
e
formulation,
the
equi
v
alent
linear
system
for
the
nonlinear
model
of
the
SMIB
becomes:
_
=
A
+
B
v
f
y
=
C
(36)
Observer
-based
tr
ac
king
contr
ol
for
...
(Mohammad
P
ourmahmood
Aghababa)
Evaluation Warning : The document was created with Spire.PDF for Python.
1194
r
ISSN:
2088-8708
3.4.2.
F
eedf
orward
tracking
scheme
Here,
we
incorporate
the
reference
input
into
the
observ
er/controller
state
feedback.
This
method
is
useful
for
tracking
of
slo
wly
changing
references.
Consider
the
state
space
system
(36).
1.
W
e
first
design
a
state
feedback
g
ain
K
such
that
A
B
K
is
stable
(with
poles
at
nice
locations);
2.
Suppose
that
r
ef
is
the
reference
input.
W
e
w
ould
associate
a
steady
state
v
ector
ss
=
N
r
ef
for
an
y
constant
reference
input
r
ef
.
No
w
we
define
the
control
to
be:
V
f
=
K
(
ss
)
+
v
ss
(37)
where
V
ss
=
N
v
f
r
ef
is
the
steady
state
control
input
to
maintain
at
ss
.
3.
T
o
define
N
v
f
and
N
,
we
consider
the
desired
steady
state
relationships:
_
ss
=
A
ss
+
B
v
f
ss
=
AN
+
B
N
v
f
r
ef
=
0
y
ss
=
C
ss
=
C
N
r
ef
=
r
ef
(38)
T
o
mak
e
this
w
ork
for
all
r
ef
,
we
need
to
solv
e:
A
B
C
0
N
N
v
f
=
0
1
(39)
4.
Once
N
and
N
v
f
ha
v
e
been
found,
we
can
re
write
the
control
la
w
to
be:
V
f
=
K
(
N
r
ef
)
+
N
v
f
r
ef
V
f
=
K
+
N
r
ef
(40)
where
N
=
N
v
f
+
K
N
.
3.4.3.
Integrator
loop
T
o
t
rack
constant
references
for
v
t
and
T
m
without
steady-state
error
,
system
(30)
should
be
aug-
mented
by
intermediate
states
X
=
[
_
T
e
v
e
w
]
T
where
e
v
=
K
v
(
v
t
V
r
ef
)
with
K
v
0
as
a
constant
g
ain
and
e
w
=
w
w
r
ef
are
gi
v
en
by
e
v
=
g
1
(
)
e
p
=
g
2
(
)
;
(41)
where
g
1
(
)
and
g
2
(
)
are
nonlinear
functions
that
can
be
found
using
equations
(18)
and
(4).
In
our
case,
these
functions
are
linearized
around
an
operating
point
to
get
a
local
approximation
in
the
form
e
v
=
C
1
(
)
+
O
(
2
)
e
T
=
C
2
(
)
+
O
(
2
)
:
(42)
Ho
we
v
er
,
to
inspire
the
standard
classic
IEEE
go
v
ernor
model
(7)
in
the
closed-loop
syst
em,
we
modify
the
second
inte
grator
as
follo
ws:
e
w
=
1
T
g
(
T
r
ef
P
sv
+
K
g
(
!
!
s
))
(43)
Ne
glecting
high
order
terms
in
(42),
the
linear
augmented
system
is
_
X
=
A
a
X
+
B
a
_
v
;
(44)
where
A
a
=
2
4
A
0
4
2
C
1
0
1
2
C
2
0
1
2
3
5
;
B
a
=
B
0
2
2
;
(45)
The
pair
(
A
a
,
B
a
)
is
pro
v
ed
controllable
in
our
case
where
the
follo
wing
rank
condition
is
v
erified:
r
ank
B
a
A
a
B
a
A
2
a
B
a
A
3
a
B
a
A
4
a
B
a
A
5
a
B
a
=
6
(46)
Int
J
Elec
&
Comp
Eng,
V
ol.
11,
No.
2,
April
2020
:
1186
–
1199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1195
Hence,
there
e
xists
a
stabilizing
state
feedback:
_
v
=
K
a
X
;
with
K
a
=
[
K
K
v
K
T
]
(47)
where
K
2
R
2
4
,
K
v
2
R
2
1
and
K
T
2
R
2
1
.
Inte
grating
the
abo
v
e
equation,
the
outer
-loop
has
the
follo
wing
form:
v
(
t
)
=
K
(
t
)
K
v
Z
t
0
e
v
(
)
d
K
T
Z
t
0
e
p
(
)
d
(48)
It
includes
proportional
and
inte
gral
terms,
which
achie
v
e
zero
steady-state
error
in
po
wer
and
v
olt
age
responses.
One
of
the
interesting
approaches
to
design
the
g
ain
K
a
is
the
LQR
(Linear
Quadratic
Re
gulator)
method.
The
g
ain
K
a
is
computed
by
solving
a
Riccati
equation
and
minimizes
the
follo
wing
inde
x
J
=
Z
1
0
X
T
(
)
QX
(
)
+
_
v
T
(
)
R
_
v
(
)
d
(49)
where
Q
and
R
are
the
weight
matrices.
4.
SIMULA
TION
RESUL
TS
In
order
to
illustrate
the
performance
of
the
proposed
flatness
controller
in
impro
ving
the
transient
sta
-
bility
of
the
SMIB
system,
some
computer
simulations
with
Matlab
and
Eurostag
are
pro
vided
here.
F
or
comparison
purposes,
classic
controllers
[3,
4]
ha
v
e
also
been
implemented
and
tested,
which
are
described
by
the
follo
wing
dif
ferential
equations:
_
u
1
=
1
T
a
(
u
1
+
K
a
(
v
t
V
r
ef
))
(50)
_
u
2
=
1
T
g
(
u
2
+
K
g
(
!
s
!
)
+
T
r
ef
)
(51)
T
o
enhance
the
performance
of
the
classic
controller
,
we
add
a
PID
controller
for
the
A
VR.
This
PID
controller
will
ensure
an
e
xact
tracking
scheme
for
the
term
inal
v
oltage
and
will
speed
up
the
response
time
of
it.
Also,
the
classic
go
v
ernor
controller
,
generating
u
2
,
is
equipped
with
a
F
ast
V
alving
Scheme
(FVS),
which
will
enhance
the
transient
stability
[31].
Once
a
se
v
er
disturbance
is
detected,
the
steam
v
alv
e
is
closed
with
a
f
ast
dynamic
respecting
the
rate
limitation
(
_
u
2
=
1
=T
c
).
This
situation
is
maintained
for
a
time
T
2
and
after
that
the
v
alv
e
position
is
released
to
meet
the
dynamic
(51)
and
respect
the
rate
constraints.
The
parameters
of
the
controllers
are
gi
v
en
in
T
able
2
.
T
able
2.
P
arameters
of
the
adopted
controllers
P
arameter
u
1
K
p
K
I
K
d
K
a
T
a
K
g
T
g
K
v
V
r
ef
w
r
ef
T
c
T
o
V
alue
3
10
1
0
1
0.1
25
0.3
5
1
1
0.3
3
W
e
consider
a
short
circuit
in
the
middle
of
the
second
transmission
line.
Figure
2
illustrates
the
machine
speed,
mechanical
torque
and
terminal
v
oltage
obtained
from
Eurostag
softw
are.
The
CCT
in
this
case
is
585
ms.
It
is
clear
that
the
proposed
approach
sho
ws
appropriate
rob
ustness
ag
ainst
hard
f
aults
occurred
in
the
netw
ork.
On
the
other
hand,
the
classic
controllers
re
v
eal
a
maximum
CCT
equal
to
528
ms.
This
means
that
the
suggested
flatness
control
algorithm
has
o
v
er
10%
impro
v
ement
in
CCT
compared
to
the
IEEE
common
controllers.
Also,
Figure
3
to
Figure
5
illustrate
dif
ferent
state
e
v
olutions
for
the
proposed
controller
ag
ainst
the
classic
IEEE
controllers
which
ha
v
e
been
obtained
from
Matlab
.
One
can
see
that
the
proposed
controller
w
orks
well.
In
f
a
ct,
the
CCT
obtained
by
the
proposed
controller
is
lar
ger
than
that
of
the
classic
IEEE
controllers.
The
applied
control
inputs
Efd
and
Psv
are
illustrated
in
Figures
6
and
7,
respecti
v
ely
.
One
can
see
that
the
control
inputs
of
the
proposed
scheme
are
feasible
in
practice.
Also,
the
proposed
controller
has
the
ef
fect
of
f
ast
v
alving
as
it
can
be
observ
ed
in
Figure
7,
which
is
an
interresting
feature
that
enhances
the
transient
stability
of
the
po
wer
system.
Finally
,
Figure
8
depicts
the
dedicated
control
ener
gies
for
both
Observer
-based
tr
ac
king
contr
ol
for
...
(Mohammad
P
ourmahmood
Aghababa)
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