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.
230~
2
4
1
IS
S
N
: 2088-
86
94,
D
O
I
:
10.11
59
1
/ij
ped
s
.
v10
.
i
1.pp
2
30-
24
1
230
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
A novel self-tuni
ng fractiona
l order PID co
ntrol based on
optimal mod
e
l reference
adapti
ve system
Moh
a
med
. A
. Sh
amseldin
1
,
Mo
ha
med
Sa
llam
2
,
A
.
M
.
B
a
s
s
i
u
n
y
3
, A
.
M
.
A
bd
el G
han
y
4
1
D
e
part
men
t
o
f M
echat
ron
i
cs
E
n
g
in
eerin
g, F
utu
r
e Un
iversit
y
i
n
Eg
y
p
t
,
C
ai
ro
, E
gypt
2,
3
D
e
p
a
rtm
e
n
t
of M
echani
cal E
ngin
eeri
n
g
,
H
elwa
n
Un
iv
ers
ity, Cairo
,
Egy
p
t
4
Depart
m
e
nt of E
l
ect
rical Eng
ineer
ing
, Oc
t
ob
e
r
6
Unive
rsi
t
y
(He
l
wan Un
i
v
ers
i
ty
O
ri
g
i
nally),
C
air
o
,
Egypt
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
Re
ce
i
v
e
d
Ju
l
1
2,
201
8
Re
vise
d N
ov
1
9
,
201
8
A
c
c
e
pte
d
D
ec 13,
2
0
1
8
Th
is
p
aper
p
rese
n
t
s
a
nov
el
s
elf
-
tu
nin
g
f
ractio
nal
o
r
der
P
I
D
(F
OPID)
cont
rol
bas
e
d o
n
opt
im
al
M
od
el Ref
erence
Adap
t
i
ve C
o
n
t
r
o
l
(
MR
A
C
).
T
he p
ropo
sed
con
t
ro
l
techn
i
q
u
e
has
s
u
b
j
ected
t
o
a
th
ird
ord
e
r
sys
t
e
m
case
st
ud
y
(po
w
er
syste
m
l
oa
d
fr
e
q
ue
n
c
y
c
o
ntro
l).
T
h
e
m
o
de
l
reference
des
c
ribes
t
he
requ
irem
ent
s
o
f
d
e
sig
n
er.
It
can
b
e
f
i
rst
o
r
s
eco
nd
o
rd
er
s
y
s
t
e
m.
T
h
e
p
a
r
a
m
e
t
e
r
s
o
f
M
R
A
C
h
a
v
e
o
b
t
a
i
n
e
d
u
s
i
n
g
t
h
e
h
a
r
m
o
n
y
s
e
a
r
c
h
(
H
S
)
op
timizati
o
n
tech
ni
que
t
o
a
c
h
i
ev
e
th
e
op
timal
p
erf
o
rm
a
n
ce.
S
o
m
e
ti
me
s,
t
he
tu
ni
ng
o
f
th
e
f
i
v
e
p
ara
m
et
ers
o
f
F
O
P
ID
c
o
n
trol
o
n
l
i
n
e
at
s
a
m
e
m
om
ent
con
s
u
m
es
m
ore
calcu
lat
i
on
t
im
e
an
d
m
o
re
p
rocess
i
n
g
.
S
o
,
th
is
s
t
u
dy
pro
p
o
s
es
t
hree
m
e
th
ods
f
or
s
elf-tu
n
in
g
FOP
I
D
cont
rol
.
T
h
e
f
i
r
st
m
et
ho
d
has
been
i
m
p
l
e
m
e
n
t
ed
t
o
tu
ne
t
h
e
t
wo
i
n
t
eg
ral
and
d
e
riv
a
tiv
e
p
a
ram
e
ters
only
and
th
e
res
t
o
f
param
e
ters
a
re
f
i
x
ed
.
Th
e
s
e
c
o
nd
m
eth
o
d
has
bee
n
de
sign
e
d
t
o
adj
u
s
t
t
he
p
rop
o
r
t
iona
l
,
i
ntegra
l
deri
vative
paramet
e
rs
w
hil
e
t
he
o
the
r
f
r
acti
ona
l
param
e
t
e
rs
a
re
c
on
st
ant
.
T
h
e
l
as
t
m
e
thod
h
a
s
d
ev
elo
p
ed
t
o
adju
st
th
e
fi
ve
p
aram
eters
o
f
F
O
P
ID
c
on
trol
s
im
u
l
t
a
neou
sl
y
.
T
h
e
s
im
u
l
a
ti
on
r
e
s
u
l
t
s
illus
t
rate
t
hat
t
h
e
third
met
h
od
o
f
sel
f
-
t
uning
F
OPI
D
c
ontrol
c
an
accom
m
o
d
a
te
t
h
e
s
u
dden
di
stu
r
b
a
n
ce
co
m
p
ared
t
o
ot
her
tech
ni
ques
.
Also,
it
can ab
s
o
r
b t
h
e sys
t
em u
ncertai
n
ty b
ett
e
r th
an t
he o
th
er con
tro
l
t
e
c
hn
iq
ue
s.
K
eyw
ord
s
:
F
O
P
I
D
contro
l
H
a
r
m
ony
resea
r
ch
(
HS
)
Mo
de
l re
fer
e
nce
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.
M
. Abdel G
h
any,
D
e
pa
rtme
nt
o
f
El
e
c
t
rica
l
Eng
i
ne
eri
ng,
O
c
to
ber
6 U
n
ive
r
sity
(
H
e
lw
a
n
U
ni
versi
t
y O
r
i
g
ina
l
ly),
Ca
i
r
o, Egy
pt.
Em
ail:
gha
n
y
g
h
an
y
@
h
o
t
ma
i
l
.c
om
1.
I
N
TR
OD
U
C
TI
O
N
The
P
I
D
con
t
rol
has
bee
n
a
p
p
l
i
e
d
t
o
proc
es
s
c
o
nt
r
o
l
i
n
m
o
s
t
of
e
n
gi
nee
r
i
ng
a
p
p
l
ic
a
t
i
o
ns
f
or
d
ec
ade
s
[1].
T
he
P
ID
c
on
tro
l
h
a
s
s
im
ple
str
u
c
t
ure
and
li
nea
r
b
e
h
a
v
i
o
r.
A
l
so,
it
g
i
ves
ac
cep
t
a
ble
perf
orma
nce
for
sever
a
l
i
n
d
u
s
t
r
i
a
l
a
p
p
lica
t
io
n
s
[
2]
.
The
r
e
ar
e
seve
ral
me
t
h
ods
t
o
s
e
l
e
c
t
t
he
p
roper
val
u
es
f
or
P
ID
c
ontro
ller
para
me
ters
[
3]
.
The
tra
d
it
io
n
a
l
m
e
t
h
o
d
s
f
o
r
selec
tin
g
t
h
es
e
pa
r
am
eter
s
s
u
c
h
a
s
try
a
n
d
er
ror
and
Zie
g
le
r-
N
i
ch
ol
s
w
h
ic
h
w
e
re
b
e
cam
e
ina
p
propr
iate
t
o
a
c
h
ie
ve
a
g
o
od
per
f
o
rm
ance
[
4].
So,
the
rese
arc
h
ers
h
a
ve
ten
d
e
d
t
o
use
a
l
t
e
r
n
at
i
v
e
me
thods
s
uc
h
as
o
p
t
i
m
iz
at
i
on
te
ch
ni
q
u
es
(
Gen
e
t
i
c
Al
g
o
r
ith
m
(
G
A)
,
P
a
rt
i
c
l
e
S
wa
rm
Op
ti
mi
za
t
i
on
(
P
S
O)
,
An
t
Co
lo
ny
O
pt
i
m
i
z
at
io
n
(ACO)
and
Harmo
n
y
S
ea
rch
(
H
S
)
)
w
h
i
c
h
are
try
i
n
g
t
o
rea
c
h
the
op
tim
al
s
o
l
ut
i
o
n
f
o
r
c
o
n
t
r
o
l
l
e
r
p
ar
am
eters
[5]
.
S
til
l
,
t
he
b
e
h
a
vi
o
r
o
f
P
I
D
c
o
nt
rol
i
s
l
in
ea
r
and
c
a
nnot
d
e
a
l
w
ith
t
he h
i
gh d
i
s
t
urba
nc
e a
n
d hi
g
h
no
n
line
a
ri
ty o
f
com
p
l
i
c
a
t
ed
sys
t
em
s [
6
]
,
[
7].
T
h
e
f
r
a
c
t
i
o
n
a
l
o
r
d
e
r
P
I
D
(
F
O
P
I
D
)
c
o
n
t
r
o
l
h
a
s
b
e
e
n
w
i
d
e
l
y
u
s
e
d
i
n
c
ontr
o
l
en
g
i
nee
r
i
n
g
i
n
r
e
c
e
n
t
dec
a
de
s
[8].
T
he
F
O
P
ID
c
ons
ider
s
t
h
e
no
n
l
i
n
ea
r
co
p
y
o
f
P
I
D
c
o
n
t
ro
l
whe
r
e
two
m
o
re
p
ar
am
eters
(the
fra
c
ti
o
n
a
l
i
n
t
e
g
ral
an
d
de
ri
v
a
ti
v
e
)
ad
ded
to
t
he
P
ID
c
on
tro
l
p
a
ra
m
e
ters
[
9].
H
e
nc
e,
t
he
t
a
s
k
of
d
e
s
i
gner
selec
t
i
n
g
t
h
e
p
r
oper
va
lue
s
f
o
r
t
he
f
i
v
e
para
me
ters
o
f
the
F
O
P
I
D
contr
o
l
[1
0].
The
F
O
PID
con
t
rol
ca
n
so
l
v
e
the
no
n
line
a
ri
t
y
p
ro
b
l
em
b
u
t
i
t
can
no
t
de
a
l
w
ith
t
he
s
udde
n
d
i
s
t
u
rban
c
e
du
e
to
i
t
s
p
a
r
amet
ers
wh
i
c
h
stil
l fi
xe
d
[1
1].
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
A nov
e
l
sel
f
-tu
n
i
ng
frac
ti
on
a
l
order
PID
c
ont
r
o
l b
a
se
d
on o
p
t
i
m
a
l
m
ode
l..
.
(Moham
e
d
. A
.
Sham
se
ldi
n
)
23
1
Differe
n
t
tec
h
ni
q
u
es
h
a
v
e
b
een
p
ro
p
o
se
d
to
t
u
n
e
t
h
e
fi
ve
p
a
r
am
e
te
rs
o
f
FO
PID
contr
o
l
o
n
l
i
n
e
bu
t
most of the
s
e
t
e
c
h
n
i
que
s ar
e
ba
se
d
o
n
the f
u
zzy l
o
g
i
c
c
on
trol [
12],
[1
3].
The
f
u
zz
y l
ogic
con
t
ro
l ca
n s
o
lve
t
h
e
u
n
c
ert
a
int
y
p
rob
l
e
m
a
n
d
sudde
n
di
st
u
r
b
a
n
c
e
b
u
t
i
t
s
d
esi
g
n
de
p
e
n
ds on the exper
i
e
n
ce
w
h
i
ch some
tim
es
i
s not
ava
ila
b
l
e
for
some
s
ys
t
e
ms
[
14]-[
16].
Th
is
s
t
u
d
y
p
rese
nt
s
a
no
v
e
l
t
e
c
h
n
i
q
u
e
t
o
tu
ne
t
he
F
O
P
I
D
c
o
n
t
r
o
l
para
me
ters o
nline
ba
sed
o
n
o
pt
ima
l
mode
l
r
e
fer
e
nce
a
d
a
p
t
i
ve
c
on
tro
l
(
M
R
A
C
).
I
t i
s
k
n
o
w
n
t
hat
the
M
R
A
C
is
hi
gh
r
a
nk
i
n
g
a
d
ap
t
i
ve
c
o
n
tro
l
w
here
it
forc
es
t
he
overa
ll
s
ys
te
m
to
f
o
l
l
o
w
t
h
e
be
ha
vi
or
o
f
pr
ese
l
ec
t
e
d
mode
l
refere
nce
[1
7].
The
pr
esele
c
t
ed
m
ode
l
c
a
n
be
f
irst
o
r
sec
o
nd
o
r
der
sys
t
e
m
a
ccor
d
i
n
g
to
t
he
p
o
i
nt
o
f
vi
ew
t
he
des
i
g
n
er
a
nd c
o
mpl
i
ca
t
e
d deg
r
ee
of the
sys
t
e
m
[18].
Th
e
t
a
sk
o
f
mod
e
l
re
f
e
ren
c
e
ad
apt
i
v
e
co
nt
rol
i
s
a
d
j
u
s
ti
ng
t
he
F
O
P
ID
c
ontrol
parameters
o
nline.
T
he
mode
l
r
e
fe
re
n
c
e
con
t
a
i
ns
t
h
e
d
e
s
i
r
ed
p
erf
o
rm
ance
w
hi
c
h
c
an
s
at
i
sfy
the
de
sig
n
e
r
.
More
o
v
e
r
,
t
o
guara
nt
e
e
hi
gh
pe
rfor
ma
n
c
e
the
pa
ram
e
ter
s
o
f
mo
de
l
r
e
fe
renc
e
o
p
tim
ize
d
u
s
in
g
t
h
e
har
m
on
y
sea
r
ch
(
H
S
)
op
ti
miz
a
t
i
o
n
t
e
c
hni
qu
e
a
c
c
ord
i
n
g
to
a c
ert
a
i
n
cos
t
fu
n
c
tion
.
The
pr
op
ose
d
t
echn
i
que
w
il
l
be
s
u
b
j
e
c
t
ed
t
o
a
th
ird
orde
r
syste
m
a
s
c
ase
st
ud
y
(p
ow
e
r
s
yste
m
lo
a
d
fre
que
nc
y
c
o
nt
rol).
A
l
so,
the
pro
pose
d
t
ech
ni
q
u
e
ha
s
bee
n
i
m
p
le
me
nt
ed
w
i
t
h
dif
f
e
r
ent
me
t
h
od
s.
T
he
f
i
r
st
me
tho
d
h
as
b
e
e
n
de
sig
n
e
d
t
o
t
h
e
MRA
C
w
il
l
tune
t
he
t
w
o
i
nte
g
ral
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nd
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er
iva
t
i
v
e
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me
ters
o
n
l
y
a
n
d
the
rest
o
f
parameters
a
re
f
ixed.
Th
e
se
cond
m
et
h
o
d
h
a
s
b
een
d
e
v
elo
pe
d
t
o
a
d
j
ust
the
pro
port
i
ona
l
,
i
nte
g
ra
l
deri
va
ti
ve
p
ar
am
eters
w
h
i
l
e
the
o
t
he
r
fra
c
t
i
ona
l
par
a
m
e
ter
s
a
re
c
on
sta
n
t.
T
he
l
as
t
me
tho
d
f
a
b
rica
te
d
to
a
dj
ust
t
h
e
fiv
e
p
a
r
amet
ers
of
F
OPID
c
o
n
t
r
ol
a
t
th
e
sa
me
t
i
m
e.
T
h
e
p
ap
e
r
ha
s
orga
n
i
z
e
d
a
s
f
ol
low
s
,
firs
tl
y
,
t
he
syste
m
m
ode
l
is
p
rese
nte
d
.
S
e
c
ondl
y,
t
h
e
p
r
o
pose
d
c
on
tro
l
t
e
c
h
n
ique
s
a
r
e
dem
ons
tra
t
e
d
.
T
h
i
r
dly,
t
he
si
m
u
lat
i
on re
su
lt
s a
r
e
i
l
l
u
st
r
a
t
e
d.
F
ina
l
l
y
t
he
c
onc
lus
i
o
n
is d
iscussed.
2.
T
H
IRD
O
R
D
E
R
CA
SE S
TUDY
M
O
D
EL
S
e
ve
ral
n
o
n
l
i
n
e
a
r
models
f
or
l
arge
pow
er
s
ystem
s
h
a
s
t
o
be
e
s
t
a
b
l
ishe
d
b
u
t
a
l
l
a
re
c
om
pl
i
c
a
t
e
d
mode
l
s
.
S
o
,
the
li
near
i
z
e
d
m
ode
l
ha
s
be
e
n
u
se
d
i
n
o
ur
w
ork.
T
he
pow
er
s
ys
t
e
m
re
prese
n
t
e
d
by
a
n
e
q
u
i
va
le
n
t
tur
b
ine
,
g
o
v
ern
o
r
a
n
d
ge
ne
rat
o
r
sy
stem
[
1
9
]
,
[
20].
Th
e
tra
n
sfer
f
un
c
t
i
o
n
o
f
a
s
in
gle
ar
ea
pow
er
s
ys
tem
mode
l
in a
c
lo
sed
lo
o
p
form
as
s
ho
w
n
in F
i
g
u
r
e
1.
F
i
gure
1.
B
l
o
c
k
d
i
a
gram
of a
si
ngle
ar
ea
pow
er
syst
e
m
Where
is
u
t
he
c
on
tro
l
a
ct
ion,
d
i
s
t
h
e dist
urb
a
nce
vec
t
or
(
ΔP
d
)
,
T
p
is
t
he pla
n
t
m
ode
l
ti
m
e
c
onsta
n
t
,
Tt
i
s
th
e
t
u
r
b
i
n
e
t
i
m
e
co
nsta
n
t
,
Tg
i
s
th
e
go
v
e
rno
r
t
i
m
e
c
o
n
s
t
a
nt
,
Kp
i
s
t
h
e
pl
ant
g
a
i
n
,
R
i
s
t
he
s
p
eed
regu
la
ti
on
d
u
e
to
gove
rn
or
a
cti
o
n,
x1
is
t
h
e
c
han
g
e
in
s
y
s
t
e
m
f
r
e
q
u
enc
y
,
x2
is
t
he
i
n
c
re
me
nt
a
l
c
ha
nge
s
i
n
gene
ra
tor
ou
t
p
ut
a
nd
x3
is
t
h
e
g
o
v
ern
o
r
va
lve
pos
i
tio
n.
T
able
1
p
re
se
n
t
t
he
v
a
l
ue
s
of
s
y
s
te
m
par
a
m
e
te
rs
t
ha
t
use
d
in
th
is w
or
k.
T
a
b
l
e 1.
S
yste
m P
a
ra
m
e
ter
s
P
a
ra
m
e
te
rs
V
a
l
u
e
K
p
120
pu
Tp
20
s
Tt
0
.3
s
Tg
0
.
08
s
R
2.
4
H
z
/
p
u
.
M
W
3.
SELF-
T
U
N
I
N
G
FO
PID
CONT
ROL
The
Mo
de
l
Re
fer
e
nce
A
d
a
p
t
i
ve
C
o
n
t
r
ol
(
M
R
A
C
)
i
s
h
ig
h-r
a
nk
i
ng
a
d
a
pt
iv
e
c
ontr
o
l
l
er
[
2]
,
[5].
I
t
m
a
y
be
r
e
g
ard
e
d
a
s
a
n
ada
p
t
i
ve
s
er
vo
s
y
stem
i
n
w
h
i
c
h
t
h
e
de
sire
d
pe
rfo
rm
anc
e
i
s
e
x
p
r
essed
in
t
er
ms
o
f
a
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
:
23
0 –
24
1
23
2
refere
nce
m
o
d
e
l.
I
n
th
is
w
o
r
k
t
h
e
F
O
PID
c
o
nt
r
o
l
par
a
m
e
ters
w
il
l
b
e
a
d
ju
ste
d
o
n-l
i
ne
u
si
n
g
t
he
m
odel
refere
nce
tech
n
i
q
u
e
as show
n
i
n
F
ig
ure
2.
F
i
gure
2.
T
he
overa
ll sy
stem
w
it
h se
l
f
-
t
un
in
g F
O
P
I
D
ba
sed on
mo
de
l
reference
technique
To
a
d
j
us
t
para
me
ters usin
g
MIT
r
ule
w
h
ic
h use
t
h
e fo
l
l
o
w
in
g lo
s
s f
u
nction
(
1
)
To ma
k
e j
sma
l
l
,
i
t is r
easo
n
a
b
le
t
o cha
n
ge t
he
par
am
eters in t
he dire
c
t
i
on of
t
h
e
ne
g
at
ive
gra
d
ie
n
t
o
f
j, that
is
,
(
2
)
wher
e
s
tand
for
the
a
d
ap
ta
ti
on
ga
in
w
hile
i
s
t
h
e
centro
i
d
vec
t
or
o
f
th
e
ou
tpu
t
m
em
bersh
i
p
fu
nc
t
i
o
n
.
T
h
e
trans
f
e
r
func
t
i
o
n
o
f F
O
P
I
D c
ont
r
o
l
ca
n be
d
e
s
cr
i
b
e
d
as
fol
l
o
w
s.
(3
)
(
4
)
Assum
e
t
ha
t
t
h
e plan
t
can
b
e
sim
p
l
i
fie
d
t
o a
firs
t orde
r syst
em
a
s obv
i
ous
i
n
t
h
e f
o
ll
ow
i
n
g (5)
.
(5
)
wher
e
a
re
unkn
ow
n
para
me
ters.
A
l
so,
assum
e
t
ha
t
the
m
odel
re
fer
e
n
c
e
t
a
k
e
s
a
f
o
r
m
f
i
r
s
t
o
r
d
e
r
syste
m
a
s the
fol
l
o
w
i
ng
rela
ti
ons
h
i
p.
(6
)
Where
ar
e
se
l
e
c
ted
b
y
des
i
g
ner.
Fr
om (4-6)
c
a
n
conc
lu
de
t
hat
(7
)
⎯
⎯
⎯
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
A nov
e
l
sel
f
-tu
n
i
ng
frac
ti
on
a
l
order
PID
c
ont
r
o
l b
a
se
d
on o
p
t
i
m
a
l
m
ode
l..
.
(Moham
e
d
. A
.
Sham
se
ldi
n
)
23
3
1
.
.
.
.
.
.
(8
)
(9
)
.
.
(
10)
3.1.
A
d
aptat
i
on
la
w
of
pa
ra
met
e
r
Th
is
s
u
b
-
s
ect
i
on
sh
ow
s
the
ste
p
s
of
d
es
i
g
n
of
t
he
a
da
pt
ati
on
la
w
for
pr
o
porti
ona
l
g
a
in
p
ara
m
e
t
er
(
. By de
ri
vin
g
(
10)
r
espec
t
to
the
pr
op
ort
i
o
n
a
l
g
ai
n (
to
obta
i
n
t
h
e
fo
ll
ow
i
n
g
re
l
a
tions
h
i
p.
.
.
.
(
11)
(
11) c
an be
rew
r
itte
n
;
.
.
.
(
12)
.
(
13)
.
.
(
14)
F
r
om (12)
and
(14);
.
(
!
5
)
To
a
c
h
ie
ve t
he
d
es
ired
p
erfor
m
ance
, the
fol
l
o
w
i
ng
c
o
n
d
i
t
i
o
n
mus
t be
h
ol
d
.
.
1
1
(
16)
(
17)
F
r
om the
M
I
T
rule
ca
n o
b
ta
i
n
the
f
o
l
l
o
w
i
n
g
r
elat
io
ns
hip
.
.
(
18)
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
:
23
0 –
24
1
23
4
.
.
(
19)
.
(
20)
0
(
21)
Where
0
is
t
h
e
i
n
it
ial
va
l
u
e of
p
ro
port
i
ona
l ga
i
n
.
3.2.
A
d
aptat
i
on
la
w
of
pa
r
a
m
e
ter
Th
is
s
ub-
se
c
t
i
on show
s
t
h
e ste
p
s
of des
ig
n
of
t
he ada
pta
t
i
o
n
la
w
for
in
tegr
a
l
g
ai
n para
me
t
e
r (
. By
deri
vi
n
g
(8)
respec
t
t
o
the
i
n
t
e
gra
l
g
a
i
n (
t
o
ob
t
a
in t
he f
oll
o
w
i
n
g
re
l
at
i
o
ns
hi
p.
.
.
.
(
22)
Fro
m
(2
2
)
c
an
b
e rewri
t
t
en;
.
.
.
(
23)
.
(
24)
.
.
(
25)
F
r
om (23)
and
(25);
.
(
26)
To
a
c
h
ie
ve t
he
d
es
ired
p
erfor
m
ance
, the
con
di
t
i
o
n
m
ust be
h
o
l
d
in (
1
6
).
(
27)
F
r
om the
M
I
T
rule
ca
n o
b
ta
i
n
the
f
o
l
l
o
w
i
n
g
r
elat
io
ns
hip
.
.
(
28)
.
.
(
29)
.
(
30)
0
(
31)
Where
0
i
s
t
h
e
i
n
i
tia
l
v
a
l
u
e
of
p
ro
por
ti
ona
l ga
in
.
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
A nov
e
l
sel
f
-tu
n
i
ng
frac
ti
on
a
l
order
PID
c
ont
r
o
l b
a
se
d
on o
p
t
i
m
a
l
m
ode
l..
.
(Moham
e
d
. A
.
Sham
se
ldi
n
)
23
5
3.3.
A
d
aptat
i
on
Law of
Param
e
ter
Th
is
s
u
b
-se
c
t
i
on
i
l
l
u
s
t
r
a
tes
the
ste
p
s
o
f
d
e
s
i
gn
o
f
t
he
a
d
a
pta
t
ion
law
for
de
riva
t
i
ve
g
a
i
n
para
me
ter
(
.
By
d
eri
v
i
n
g
(10)
r
espe
ct to
the
der
i
va
tive
gai
n
(
to
obta
i
n
t
h
e
fol
l
ow
in
g re
l
a
t
i
o
n
s
h
i
p
.
(
32)
(
33)
(
34)
(
35)
A
l
so,
from
(
33) a
nd (35)
.
.
(
36)
.
.
(
37)
.
.
.
.
(
36)
.
.
(
37)
.
.
.
(
38)
0
(
39)
Where
0
i
s
the i
n
i
tia
l
v
a
l
u
e
of
d
e
r
i
v
at
ive
ga
in
.
3.4.
A
d
aptat
i
on
Law of
Pa
ra
m
e
te
r
Th
is
s
u
b
-sec
ti
on
i
l
l
ustra
t
e
s
t
he
s
te
ps
o
f
de
sign
o
f
the
a
d
apta
t
i
o
n
l
a
w
for
fra
c
t
i
o
na
l
i
n
te
gral
g
a
i
n
parameter
(
. By de
ri
vin
g
(
10) r
espec
t
t
o t
h
e
frac
tio
na
l
inte
gr
al ga
i
n
(
t
o o
b
t
ai
n t
h
e
fo
l
l
ow
i
ng
rela
ti
o
n
sh
i
p
.
(
40)
(
41)
(
42)
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
:
23
0 –
24
1
23
6
(
43)
A
l
so,
from
(
41) a
nd (43)
;
(
44)
.
(
45)
.
.
.
(
46)
.
.
(
47)
.
.
0.
(
48)
0
(
49)
Where
0
is
t
h
e
i
n
it
ial
va
l
u
e of fr
acti
ona
l i
n
te
gr
a
l
g
ai
n
.
3.5.
A
d
aptat
i
on
la
w
of
param
e
te
r
Th
is
s
ub-
se
c
t
i
on
dem
ons
t
r
ates
t
h
e
s
t
e
ps o
f
de
sign of
t
he
a
da
pta
ti
o
n
l
a
w
fo
r
f
r
ac
ti
on
al
d
eriv
a
t
i
v
e
g
a
i
n
parameter
(
.
By
d
eri
v
i
ng
the
(
8
)
r
e
spe
c
t
to
t
he
f
r
a
cti
o
nal
de
ri
va
tive
ga
i
n
(
t
o
o
b
t
a
i
n
t
h
e
fo
l
l
ow
i
n
g
relat
i
o
n
s
h
i
p
.
.
.
.
.
(
50)
.
.
(
51)
.
.
(
52)
.
.
(
53)
A
l
so,
from
(
51) a
nd (53)
;
.
.
.
(
54)
.
.
.
(
55)
.
.
.
.
.
(
56)
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
A nov
e
l
sel
f
-tu
n
i
ng
frac
ti
on
a
l
order
PID
c
ont
r
o
l b
a
se
d
on o
p
t
i
m
a
l
m
ode
l..
.
(Moham
e
d
. A
.
Sham
se
ldi
n
)
23
7
.
.
(
57)
.
.
0.
.
.
0.
(
58)
0
(
59)
Where
0
i
s
the i
n
i
t
i
a
l
v
al
ue
o
f f
r
ac
t
i
ona
l
inte
gr
al ga
i
n
.
3.6.
Harm
on
y sear
ch
(HS)
op
timizat
i
on
The
h
a
rm
on
y
s
e
arc
h
h
a
s
b
e
e
n
p
rop
o
se
d
in
2
00
1.
I
t
is
u
se
d
late
l
y
i
n
se
vera
l
en
gi
nee
r
i
n
g
a
p
p
lic
at
io
ns
t
o
obt
ai
n
th
e
op
ti
mal
v
a
lu
e
s
o
f
c
ont
rol
p
a
ra
me
t
e
rs
.
In
t
h
i
s
stu
dy,
t
he
H
S
w
i
l
l
b
e
u
s
ed
t
o
ob
t
a
in
t
he
o
pt
im
a
l
para
me
ters
v
al
ues
o
f
m
ode
l
r
e
fe
renc
e
ada
p
ti
ve
c
on
tro
l
.
The
m
o
s
t
e
ffe
c
tive
para
me
t
e
r
of
M
RA
C
is
t
he
ada
p
ta
tio
n
ga
i
n
f
or
e
a
c
h
ada
p
t
a
t
i
o
n
l
aw
.
He
re,
the
har
m
ony
se
ar
c
h
t
un
i
ng
s
y
s
t
e
m
w
i
l
l
adj
u
s
t
t
he
p
ar
a
m
e
t
ers
of
m
od
el
r
efe
r
enc
e
a
dap
t
i
v
e
sys
t
em
(
,
,
,
a
ccor
d
in
g
to
t
he
o
bjec
ti
ve
f
u
n
c
t
io
n
as
show
n in (6
0
)
.
(
6
0
)
Whe
r
e
the o
ve
rsho
ot
o
f sys
t
e
m
r
espons
e i
s
,
i
s t
h
e
st
e
a
d
y
st
at
e
erro
r,
i
s the
se
t
t
li
ng ti
m
e
and
is the
rise
t
ime
.
A
lso,
t
h
i
s
ob
jec
t
i
v
e
f
unc
t
i
on
is
a
ble
to
c
om
pr
omise
the
de
signe
r
dem
a
n
d
b
y
the
w
e
i
g
h
tin
g
para
me
ter
va
l
u
e
(β).
T
he
p
ara
m
e
t
er
i
s
set
la
r
g
er
t
ha
n
0.7
to
r
e
d
u
c
e
ov
e
r
s
h
oot
a
nd
s
t
e
a
d
y
-
st
at
e
erro
r.
I
f
th
i
s
parameter
i
s
s
et sm
a
ller
t
h
a
n
0
.
7
the r
ise
t
i
me
and
set
tli
ng t
i
me
w
ill be
re
d
u
c
ed.
The
ini
t
i
a
l
p
o
p
u
l
a
t
i
o
n
of
H
arm
ony
Me
mor
y
(
H
M
)
is
p
ro
duce
d
r
a
n
d
o
m
l
y.
H
M
c
o
n
t
a
i
n
s
H
ar
mony
Mem
o
ry S
olut
i
on (H
M
S
) ve
ctors.
The
H
M is fil
le
d w
i
t
h
H
M
S
ve
c
t
o
r
s as
f
ollows:
⎣
⎢
⎢
⎢
⎢
⎡
,
,
,
,
,
,
,
,
,
,
.
.
...
.
.
...
.
.
...
,
,
,
,
,
⎦
⎥
⎥
⎥
⎥
⎤
(
6
1
)
Ta
b
l
e
2
dem
o
ns
trates
t
he
o
bta
i
ne
d
va
l
u
e
s
o
f
MRA
C
p
ar
a
m
e
t
e
r
s
aft
er
t
he
o
ff
l
i
ne
t
u
n
i
n
g
usi
n
g
t
h
e
harm
on
y se
arc
h
tu
n
i
n
g
s
y
s
te
m
.
Ta
bl
e
2
.
M
R
A
C
Para
me
t
e
rs
MR
A
C
Pa
r
a
m
e
t
e
rs
0.
234
0
.
678
0
.
456
0
.
568
0
.
704
3
S
o
me
ti
m
e
s,
s
om
e
system
s
do
n’t
nee
d
a
d
j
ust
the
f
i
ve
p
a
r
am
eter
s
o
f
F
O
P
I
D
contro
l
o
n
l
i
ne
a
t
sam
e
mom
e
nt w
h
i
c
h
w
ill sa
ve
the c
alc
u
l
a
tio
n tim
e and m
a
ke
t
he
overa
l
l sys
t
em
m
ore
re
ady t
o
d
eal w
ith the
s
ud
de
n
di
st
urba
nce.
S
o,
t
h
i
s
s
t
u
d
y
p
r
opos
e
s
t
hree
m
e
t
ho
ds
f
or
s
elf-
tu
n
ing
FOPID
c
o
n
t
rol
.
T
h
e
f
i
r
st
m
et
hod
con
s
i
d
ers
,
a
n
d
are
const
a
n
t
s
whi
l
e
a
nd
a
re
v
aryi
ng.
T
he
s
e
c
on
d
me
t
h
o
d
c
onsi
d
er
s
,
a
nd
ar
e
va
ryin
g
whi
l
e
a
nd
a
re
c
on
st
a
n
t
s
.
The
third
m
e
th
o
d
t
u
n
e
s
t
he
f
i
v
e
pa
ram
e
te
rs
o
f
F
OP
ID
c
on
tr
ol
o
n
l
i
n
e si
mu
lt
an
e
o
u
s
ly
.
4.
SIMU
L
A
TION
R
ESULT
S
Th
is
s
ec
ti
o
n
p
r
e
sents
the
si
m
u
lat
i
on
r
e
s
ul
ts
o
f
t
h
e
pro
p
o
se
d
d
i
f
f
er
en
t
ty
pes
of
s
el
f-
tun
i
ng
F
O
P
I
D
con
t
ro
l
a
l
g
o
r
i
t
h
m
s
b
ased
o
n
m
odel
r
e
fe
renc
e.
T
he
f
irs
t
m
etho
d,
the
ada
p
tive
m
ech
a
n
ism
w
ill
a
d
j
u
s
t
t
h
e
fra
c
ti
o
n
a
l
i
nte
g
r
a
l
a
nd
de
ri
va
tive
par
a
m
e
ters
o
n
l
y
w
h
ile
t
h
e
o
t
her
paramet
e
rs
a
r
e
f
ix
ed
.
Th
e
s
eco
nd
m
e
t
hod
,
the
fr
act
io
na
l
inte
g
r
al
a
nd
d
e
r
i
va
tive
para
me
t
e
rs
a
re
f
ix
ed
w
h
i
le
t
he
o
ther
t
hr
ee
pa
ra
m
e
ter
s
w
i
ll
be
t
u
n
e
d
us
i
n
g
the
a
d
a
p
tive
me
cha
n
is
m
.
T
he
t
h
i
rd
m
etho
d,
t
he
f
ive
para
me
t
e
rs
o
f
F
O
PID
c
o
ntr
o
l
w
i
l
l
b
e
a
d
jus
t
e
d
si
m
u
lta
ne
ous
l
y
.
F
i
gure
3
dem
ons
t
r
ate
s
t
he
S
im
uli
nk
di
agr
a
m
of
t
he
over
a
ll
s
yste
m
w
i
th
s
e
l
f-t
u
nin
g
F
O
P
ID
base
d
on o
p
tim
al
m
ode
l
refere
nce
tech
n
i
que.
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
:
23
0
–
241
23
8
F
i
gur
e
3.
T
he
s
im
uli
nk
d
i
a
g
r
a
m
of
t
he
ove
r
a
l
l
s
ys
tem
w
ith
s
elf-
tu
ni
n
g
F
OPI
D
b
a
s
ed
o
n
op
t
i
m
a
l
m
o
del
r
e
f
e
re
n
c
e
t
e
c
hni
qu
e
The
r
e
su
l
t
s
a
r
e
div
i
de
d
in
t
o
t
w
o
c
a
s
es.
T
h
e
nor
m
a
l
c
a
se
w
he
r
e
a
p
p
l
i
e
d
t
w
o
t
y
p
e
s
o
f
t
h
e
d
i
s
t
u
r
b
e
r
s
.
The
s
e
dist
ur
be
r
s
n
am
ely
P
d
1
as
s
h
o
w
n
i
n
F
i
gur
e
4
a
n
d
P
d2
a
s
ill
us
t
r
ate
d
i
n
F
i
g
u
r
e
5
.
The
par
a
m
e
tr
ic
unc
er
tai
n
ty
c
a
s
e
w
h
e
r
e
to
c
h
a
nge
t
he
o
per
a
tin
g
p
o
i
n
t
t
o
t
e
st
t
h
e
pow
er
f
u
l
of
t
he
p
r
o
p
o
se
d
me
thod
a
g
a
i
nst
the
i
r
cou
n
t
erpa
rts
F
i
gur
e
4.
P
ow
er
d
e
m
a
nd
var
i
a
tio
n
(
P
d1)
F
igur
e
5.
P
ow
er
d
e
m
a
nd
va
r
i
a
tio
n
(
Pd
2
)
4.
1.
No
r
m
a
l
c
a
s
e
Tw
o
e
xper
i
me
nt
s
w
e
r
e
d
one
a
t
n
o
r
m
al
o
p
e
r
a
t
i
ng
c
o
n
d
i
t
i
on.
T
he
d
y
na
mic
r
e
sponse
o
f
the
sys
t
em
f
r
e
que
n
c
y
Δ
F
w
ith
t
he
s
y
s
te
m
dr
i
v
e
n
b
y
ea
ch
t
y
p
e
o
f
t
h
e
p
r
opose
d
c
ontr
o
l
l
er
s.
I
n
e
x
p
e
r
i
m
e
n
t
n
um
ber
o
n
e
wh
ere
P
d1
is
a
pp
lie
d
to
t
he
s
ys
tem
.
F
igur
e
6
sh
ow
s
t
h
e
ti
me
r
espo
nse
of
t
he
s
ys
tem
.
A
lso,
t
he
out
p
u
t
sig
n
a
l
of
e
ac
h
c
o
n
t
r
o
ller
is
s
how
n
in
F
i
g
ur
e
7.
I
n
e
xper
i
m
e
nt
num
ber
t
wo
,
Fi
gu
re
8
d
emon
st
ra
te
s
th
e
ti
me
r
e
s
p
o
n
s
e
of
t
he
s
ys
tem
w
h
e
n
P
d
2
is
a
p
p
lied
t
o
th
e sy
s
t
em. Fig
u
r
e
9
p
r
esents th
e
c
o
n
troll
e
r’
s
outpu
ts
s
i
g
nal
s
.
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
A nov
e
l
sel
f
-tu
n
i
ng
frac
ti
on
a
l
order
PID
c
ont
r
o
l b
a
se
d
on o
p
t
i
m
a
l
m
ode
l..
.
(Moham
e
d
. A
.
Sham
se
ldi
n
)
23
9
F
i
gure
6.
S
ystem
dyna
mic
re
spo
n
ses
at
P
d
1
F
i
gure
7. The
c
ontro
l
l
er's
o
utp
u
t s
i
g
n
a
l
s a
t
Pd
1
F
i
gure
8.
S
ystem
dyna
mic
re
spo
n
ses
at
P
d
2
F
i
gure
9. The
c
ontro
l
l
er's
o
utp
u
t s
i
g
n
a
l
s a
t
Pd
2
I
t
i
s
c
l
ear
ly
s
h
o
w
n
t
ha
t
T
h
e
pro
pose
d
t
h
i
r
d
m
etho
d
of
s
e
l
f-tu
n
i
n
g
F
O
P
ID
c
ontr
o
ller
g
i
ve
s
ba
t
t
er
y
perform
ance
w
ith
a
s
m
a
l
l
er
s
e
ttl
i
ng
t
i
m
e
a
nd
a
c
c
e
p
ta
bl
e
u
n
de
rs
h
o
o
t
b
ut
w
i
t
h
a
rela
tive
l
y
hi
g
h
er
e
f
f
ort
as
show
n
in
t
he
r
esp
onse
o
f
its
c
ontro
l
out
pu
t
r
e
spo
n
se
u
.
a
l
s
o
,
t
he
p
r
o
p
o
se
d
th
ird
t
ype
o
f
se
lf-tu
n
i
n
g
F
O
P
ID
con
t
ro
l
l
er
r
ec
over
to
z
ero
ste
a
dy
stat
e
e
r
ror
aft
e
r
a
sm
a
ller
t
ime
de
l
a
y
fr
om
t
he
a
p
p
l
i
c
a
t
ion
o
f
t
he
t
w
o
di
st
urba
nce
typ
e
s.
4.2.
Param
e
tri
c
u
nc
ertain
ty cas
e
I
n
t
h
i
s
s
u
bsec
ti
o
n
,
t
h
e
para
me
t
r
i
c
u
nc
erta
in
t
i
es
o
f
t
h
e
power
s
yst
e
m
ha
ve
t
o
be
c
on
si
dere
d.
A
c
c
o
rdi
n
g
to
l
oa
d
var
i
a
tio
n
a
n
d
p
o
w
e
r
sy
ste
m
c
o
n
f
i
g
urati
o
n,
t
he
o
pera
t
i
o
n
p
oi
n
t
s
o
f
t
he
s
ystem
w
i
l
l
b
e
cha
nge
d
ran
d
o
m
l
y
d
uri
n
g
a
dai
l
y
c
y
cle.
T
he
s
ystem
par
a
me
t
r
i
c
u
n
ce
rtai
nti
e
s
a
r
e
ob
tai
n
e
d
b
y
c
h
a
n
g
i
ng
para
me
ters
by
50
%
fr
om
t
h
e
i
r
nor
ma
l
val
u
e
s
a
c
c
ordi
n
g
t
o
Table
3
a
nd
un
der
the
pow
er
d
em
and
var
i
a
t
ion
P
d2.
I
n
th
is
e
x
p
er
i
m
en
t,
t
he
pow
e
r
s
yst
e
m
re
sponse
s
a
t
P
d
2
,
i
nc
l
udi
n
g
t
he
e
ffe
c
t
o
f
t
h
e
par
a
m
e
tric
unce
r
tai
n
t
i
e
s
i
s
prese
n
t
e
d in F
i
g
ur
e 10
an
d t
h
e
contr
o
l
l
ers
out
p
u
t
s
ar
e
show
n
i
n
F
ig
ure
11.
A
c
cordi
n
g
to
t
hese
r
e
s
ul
ts,
it
i
s
c
l
e
a
r
t
h
at
t
he
p
r
opose
d
t
h
i
r
d
m
et
h
od
o
f
s
e
l
f-
tu
nin
g
F
O
P
ID
c
on
t
r
o
l
show
s
the
bes
t
r
espon
se
c
om
pa
red
t
o
o
t
h
er
c
on
tro
llers.
A
l
s
o
,
th
e
resu
l
t
s
appr
o
v
e
the
e
f
fec
tive
n
ess
a
nd
t
h
e
abi
l
i
t
y
o
f
t
h
e
p
r
op
ose
d
c
on
t
r
o
l
le
r
aga
i
nst
t
h
e
para
me
t
r
i
c
u
n
cer
t
a
in
t
i
e
s.
M
ore
o
v
e
r,
t
h
e
u
n
d
e
r
sh
oo
t
at
1
s
ec
ond
and
ov
e
r
s
h
o
o
t
a
t
5
seco
n
d
o
f
t
h
ird
me
th
od
o
f
se
lf-t
uni
n
g
F
OP
ID
co
n
t
rol
a
r
e
si
g
n
i
fi
ca
nt
s
ma
l
l
co
mp
are
d
t
o
ot
her
me
t
h
o
d
s of sel
f-
tun
i
ng
F
O
P
I
D
c
ont
r
o
l
.
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