Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
2, N
o
. 1
,
Febr
u
a
r
y
201
2,
pp
. 57
~67
I
S
SN
: 208
8-8
7
0
8
¶
57
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Modelin
g FACTS Devices in Powe
r System State Estimation
S. M. M
a
h
a
ei*, M.T
a
r
a
fd
ar
Ha
gh
*
*
, K.
Z
a
re*
*
* Azarb
a
ij
an Re
gional
El
ect
ric
C
o
m
p
an
y
** Departmen
t
o
f
Power Eng
i
neering,
Univ
ersity
of Tabr
iz,
Tabriz, Ir
an
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Oct 29
th
, 201
1
R
e
vi
sed Dec 2
6
th
, 2
011
Accepte
d Ja
n
07
th
, 2
012
In this paper is modeled differ
e
nt ty
p
e
s of contr
o
l devices inclu
d
ing various
kinds of FACTS devices based on power
s
y
s
t
em
s
t
ates
. Als
o
, the
im
pact of
each d
e
vic
e
on t
h
e am
ount of injec
tion a
c
tiv
e or react
ive powers
as well as
act
ive and r
e
a
c
t
i
ve power flow
will be inv
e
stig
ated
. Based on
the t
y
p
e
of
these devices which can b
e
in p
a
rallel, in series or
in series–shunt in power
s
y
stems, proposed models are consider
ed diff
erently
.
Accordingly
,
case
studies will be p
e
rform
ed for three diffe
rent t
y
p
e
s of control devi
ces instal
led
in series
, in
shunt and
in series-
s
hunt
fashions. State estim
ation results
based
on Weighted Least Square
not onl
y
conf
irm the proposed models’
effec
tiven
es
s
in accur
a
t
e
l
y
s
t
at
e es
tim
ating of th
e s
y
s
t
em
and m
eas
urem
ent
values
but
als
o
s
hows
that the
es
tim
ated
valu
es
can b
e
obt
ain
e
d from
the
s
t
ates
of
th
e c
ont
rol dev
i
ces
.
Keyword:
FACTS
De
vise
s
Measurem
ent Function
State Esti
m
a
tio
n
WLS Estim
a
t
o
r
Copyright @
20
12 Insitute of Ad
vanced
Engin
e
eering and Scien
c
e.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
S. M. Mah
aei,
Azarbaija
n
Re
gional Electric
Com
p
any
Em
am
Street, Tabriz city,
Ira
n.
me.
m
ahaei@gmail.co
m
1.
INTRODUCTION
In rece
nt
years
,
the role of
power
syste
m
state estim
a
tors in estab
lis
hing
a real-tim
e control system
for en
erg
y
con
t
ro
l cen
ters h
a
s b
e
en
recog
n
i
zed
b
y
m
o
re an
d
m
o
re u
tilities. State
esti
m
a
to
rs are v
e
ry
n
ecessary in
estab
lish
i
ng
a com
p
le
te, reliab
l
e d
a
tab
a
se
for
powe
r system
real-tim
e co
m
put
er a
p
pl
i
cat
i
ons
[1]
.
Di
ffe
re
nt
al
g
o
r
i
t
h
m
s
have
bee
n
pres
ent
e
d t
o
est
i
m
a
te the state of
power s
y
ste
m
s. The al
gorithm
s
are
d
i
v
i
d
e
d
i
n
to
t
h
e two
m
a
j
o
r in
tellig
en
t and
m
a
th
e
m
at
ic-b
ased
m
e
th
o
d
s. In
co
n
t
rast
to
th
e m
a
th
ematical
m
o
d
e
ls, in
telli
g
e
n
t
m
o
d
e
ls
h
a
v
e
a reason
ab
l
e
sp
eed
[2
-6
],
h
o
wev
e
r,
d
u
e
to
th
e
d
i
fficu
lty in
trai
n
i
ng
i
n
tellig
en
t
m
o
d
e
ls in
th
e d
i
fferen
t
network
situ
ati
o
n
s
, th
eir accu
racy is less th
an
th
e
math
e
m
atica
l
m
o
d
e
ls.
Co
n
s
equ
e
n
tly,
d
e
sp
ite th
e ad
van
ces i
n
th
e i
n
tellig
en
t m
o
d
e
ls, th
e m
a
th
em
a
tical
m
e
th
o
d
s are still u
s
ed in
state
estim
a
tion
of powe
r system
s.
In m
a
t
h
em
ati
c
al
m
e
t
hod, t
h
e
aim
of st
at
e estim
ati
on i
s
t
o
fi
nd t
h
e est
i
m
at
e x of t
h
e t
r
ue st
at
e x whi
c
h
best
fi
t
s
t
h
e m
e
asurem
ent
s
z r
e
l
a
t
e
d t
o
x t
h
r
o
ug
h t
h
e
no
nl
i
n
ear m
odel
[
7
]
:
e
x
h
z
+
=
)
(
(1
)
Whe
r
e:
z
:
m
-
dim
e
nsi
onal
m
easurem
ent
vect
or
x
:
n-
di
m
e
nsi
o
n
a
l
st
at
e vect
o
r
of
v
o
l
t
a
ge m
a
gni
t
ude
s a
n
d
p
h
a
se an
gl
es
e
:
m
-
di
m
e
nsi
onal
er
ro
r
vect
o
r
h(
x)
: v
ecto
r
wi
th
non
-lin
ear
fu
n
c
tion
s
Evaluation Warning : The document was created with Spire.PDF for Python.
¶
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
2,
No
. 1,
Fe
br
uar
y
20
1
2
:
5
7
– 67
58
The st
at
e est
i
m
at
i
on
pr
oce
d
u
r
e i
n
v
o
l
v
e
s
fi
nd
i
ng t
h
e
n-
di
m
e
nsi
o
nal
st
at
e
v
ect
or
x
res
u
l
t
i
ng
fr
om
e
m
i
nim
i
zat
i
on.
For t
h
i
s
p
u
r
p
o
s
e, several
m
e
tho
d
s ha
ve bee
n
pr
o
v
i
d
e
d
i
n
c
l
udi
n
g
t
h
e wei
ght
e
d
l
east
squ
a
re (
W
L
S
)
,
the weighted l
east absol
u
te value
(
W
LA
V
)
,
no
n-
q
u
ad
rat
i
c
est
i
m
a
t
o
rs and t
h
e l
east
m
e
di
an s
qua
res.
Am
on
g
t
h
ese m
e
t
hods,
t
h
e fi
rst
t
w
o
m
odel
s
have
g
o
o
d
acc
uracy
[
8
,
9]
. Al
s
o
, t
h
e
fi
rst
m
odel
ha
s bet
t
e
r s
p
ee
d t
h
an t
h
e
secon
d
m
o
d
e
l an
d will b
e
used
m
u
ch
m
o
re [1
0,
1
1
]
.
WLS
-
ba
sed
st
at
e est
i
m
a
ti
on
o
b
ject
i
v
e f
u
nct
i
o
n
i
s
gi
ve
n as
(
2
)
[
12]
.
[]
[]
)
(
)
(
)
(
x
h
z
W
x
h
z
x
J
T
−
−
=
(2
)
Whe
r
e
W=R
-1
z
i
s
a di
agonal
m
a
t
r
i
x
whose
el
em
ent
s
are
t
h
e i
nve
rse o
f
t
h
e cova
ri
anc
e
m
a
t
r
i
x
of
measurem
ents (
R
z
).
In f
u
nct
i
o
n ex
p
r
esse
d i
n
(
2
),
W
and
Z
a
r
e co
nst
a
nt
f
o
r a
net
w
o
r
k i
n
a gi
ve
n st
at
e, but
h(
x)
alm
o
st
is a
n
o
n
lin
ear fu
n
c
tio
n
and
its
math
em
at
ical
mo
d
e
l will b
e
differen
t
b
a
sed
o
n
m
easu
r
em
e
n
t typ
e
an
d
n
e
twork
st
ruct
u
r
e.
Net
w
o
r
k st
ruct
ure
and t
h
us
h(
x)
is d
e
p
e
nd
on th
e in
stalled
co
n
t
rol de
vice
s such as ca
pacitors,
reactors, phase
shifters and FACTS de
vices
and
h(
x)
shou
l
d
be m
odel
e
d
based
on
dev
i
ce t
y
pe. In [8
]
t
h
e
influe
nce
of c
a
pacitors
, reac
tors a
nd
p
h
as
e shi
f
t
e
rs
ha
v
e
been st
udi
e
d
, a
nd i
n
t
h
i
s
wo
rk
, t
h
e i
m
pact
o
f
di
ffe
re
nt
t
y
pes of F
A
C
T
S d
e
vi
ces i
n
m
odel
i
ng
h(
x)
a
n
d
thu
s
p
o
wer syste
m
state
esti
m
a
tio
n
will b
e
in
v
e
stig
ated
.
2.
MO
DELIN
G
MEAS
U
R
ME
NT
F
U
N
C
TI
ON
Measurem
ents can
be
of differe
n
t types.
Exis
ting m
eas
urem
ents are
usually active and reacti
v
e
powe
r of lines
, injecte
d
active and reactive
powe
r as
wel
l
as vol
t
a
ge m
a
gni
t
u
de o
f
b
u
s
-
ba
rs m
easurem
ent
s
,
also system
states are voltage
m
a
gnitude a
nd a
n
gle of
bus-b
ars, it also
sh
ou
l
d
b
e
no
ted
th
at th
e
vo
ltag
e
of
refe
rence
bus
-
bar is consi
d
ered ze
ro, the
r
efore
the state
ve
ctor ca
n
be
wri
tten:
(3
)
]
.
.
.
.
.
.
[
3
2
2
1
θ
θ
θ
n
n
T
V
V
V
x
=
In
w
h
ich:
x
: state vector
V
n
:
vol
t
a
g
e
m
a
gni
t
u
de o
f
nt
h bus
-
b
ar
θ
n
:
vol
t
a
ge
an
g
l
e of
n
th
bu
s-b
a
r
In a
state in
whic
h the
r
e is
no control device
in
th
e network, fo
r each
of th
e afo
r
e
m
en
tio
n
e
d
measurem
ents, h(x) ca
n
be
ca
lculated as
follows
.
2.1. I
n
jec
t
ion
measure
ments
Injection active and
reactive
powe
rs in
i
th
bus a
r
e as
follow:
(4
)
∑
+
=
=
n
j
ij
ij
ij
ij
j
i
i
B
G
V
V
p
1
)
sin
cos
(
θ
θ
(5
)
∑
−
=
=
n
j
ij
ij
ij
ij
j
i
i
B
G
V
V
Q
1
)
cos
sin
(
θ
θ
Whe
r
e,
θ
ij
, G
ij
and
B
ij
are
obt
a
ined
us
in
g (6
) to
(
8
)
.
(6
)
θ
θ
θ
j
i
ij
−
=
(7
)
)
(
Re
Y
al
G
ij
ij
=
(8
)
)
(
Im
Y
ag
B
ij
ij
=
Y
ij
is th
e
n
e
twork ad
m
ittan
ce matrix
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
¶
Modeling
FAC
TS Devices
i
n
Power Syst
em
St
at
e Est
i
m
at
i
o
n (
S
.M.
Ma
h
a
e
i
)
59
2.2. Power flow
measure
me
nts
Activ
e an
d reactiv
e po
wers i
n
i-j
line a
r
e as
follow:
(9
)
)
sin
cos
(
2
θ
θ
ij
ij
ij
ij
j
i
ij
i
ij
b
g
V
V
g
V
p
+
−
=
(1
0)
)
cos
sin
(
)
(
2
θ
θ
ij
ij
ij
ij
j
i
ij
si
i
ij
b
g
V
V
b
b
V
Q
−
−
+
−
=
Whe
r
e:
g
ij
: series electrical conductance of tra
n
sm
iss
i
on line
b
ij
: series
su
scep
tan
ce
of tran
smissio
n
lin
e
b
si
: shun
t suscep
tan
ce
of tran
smissio
n
lin
e
2.
3. V
o
l
t
me
ter
Voltm
eter itself is a
state vari
able. T
h
us:
(1
1)
V
V
i
est
=
Whe
r
e:
V
i
: m
easu
r
ed
vo
ltag
e
m
a
g
n
itud
e
V
est
: esti
m
a
ted
v
o
ltag
e
m
a
g
n
itu
d
e
Wh
en
FACTS
d
e
v
i
ces are
u
tilized
in
th
e
n
e
t
w
ork, th
ese d
e
v
i
ces wh
ich
are in
th
ree typ
e
s o
f
p
a
rallel,
s
e
r
i
e
s
an
d s
e
r
i
es
-
s
hun
t c
a
n
b
e
u
s
ed
to c
h
ang
e
h(
x)
.
2.
4. SV
C an
d ST
AT
C
O
M
SVC a
n
d ST
ATCOM act as t
h
e ca
pacitor and reactor
with the
varia
b
le impe
da
nce
or rea
c
tive powe
r
source.
2.
4.
1.
SV
C
or
ST
AT
C
O
M
a
s
fi
xe
d i
m
pe
d
a
nce i
n
st
al
l
e
d
i
n
the
i
th
bus
By in
stallin
g
a su
scep
tan
c
e
b
sh
in
th
e
i
th
bus, t
h
e am
ount of i
n
jecte
d
reactive powe
r at the
bus
will be
change
d as
follows:
(1
2)
V
b
B
G
V
V
Q
i
i
sh
n
j
ij
ij
ij
ij
j
i
i
2
,
1
)
cos
sin
(
−
∑
−
=
=
θ
θ
Whe
r
e
b
sh,i
is th
e su
scep
tan
c
e o
f
SVC o
r
STATCOM in
stal
led
in
i
th
b
u
s
.
It sh
ou
ld
b
e
no
ted
th
at th
e
sign of
b
sh
is
negative
for i
n
ductive state.
2.
4.
2.
SV
C
or
ST
AT
C
O
M
a
s
fi
xe
d re
acti
v
e
pow
er i
n
st
al
l
e
d i
n
t
h
e i
th
bu
s
By installing a reactive powe
r source,
Q
sh
, in th
e
i
th
bus, the am
ount of inje
cted reactive powe
r at the
b
u
s
will
b
e
ch
an
g
e
d
as fo
llows:
(1
3)
Q
B
G
V
V
Q
i
sh
n
j
ij
ij
ij
ij
j
i
i
,
1
)
cos
sin
(
+
∑
−
=
=
θ
θ
In w
h
ich,
Q
sh,i
is reactiv
e
p
o
wer of SVC or STATC
O
M in
st
alled
in
i
th
bu
s.
It
sh
oul
d b
e
n
o
t
e
d t
h
at
t
h
e
sign of
Q
sh
is
neg
a
tiv
e
for capacitiv
e state.
2
.
5
.
TC
SC
,
GC
SC
a
n
d
TS
SC
Accord
ing
to
th
e Fig. 1
,
a
GCSC is co
n
s
tru
c
ted
o
f
a fi
x
e
d
cap
acitor in
sh
un
t with
a
bid
i
rectio
n
a
l
switch
,
GTO,
an
d it can
b
e
seen fro
m
Fig
.
2 th
at
aTSSC is
b
u
ilt
o
f
a fi
x
e
d
cap
a
cito
r in sh
un
t
with a
b
i
d
i
rection
a
l switch
,
th
yrist
o
r, and
TCSC stru
cture is d
e
picted
in
Fig
.
3
as a fix
e
d
cap
a
cito
r in
shun
t with
a
TCR. Consequently, these three de
vi
ces, from
the viewpoi
nt of the
powe
r
system
, are variable im
pedance in
series
with tra
n
smission line,
and as it ca
n
be seen fr
om
t
h
e (
1
4
)
, i
t
l
ead
s t
o
c
h
an
ge t
h
e l
i
ne i
m
pedance
.
Evaluation Warning : The document was created with Spire.PDF for Python.
¶
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
2,
No
. 1,
Fe
br
uar
y
20
1
2
:
5
7
– 67
60
Fi
gu
re
1.
El
ect
ri
cal
m
odel
of
GC
SC
C
T
1
i
c
T
2
a) T
S
SC
C
T
1
i
c
T
2
b) TCSC
L
Fi
gu
re
2.
El
ect
ri
cal
m
odel
s
of
TSSC
a
n
d TC
SC
(1
4)
b
b
b
ij
new
ij
′
′
+
=
Whe
r
e:
b”
: s
e
ries s
u
sc
eptance
of TSSC, TCSC
or GCSC
TCSC cou
l
d be also
resu
lted
in
ch
ang
i
ng
n
e
twork ad
m
ittan
ce m
a
trix
and
i
n
j
ection
po
wers.
(1
5)
b
j
Y
Y
ij
new
ij
′
′
−
=
(1
6)
b
j
Y
Y
ji
new
ji
′
′
−
=
(1
7)
b
j
Y
Y
ii
new
ii
′
′
+
=
(1
8)
b
j
Y
Y
jj
new
ijj
′
′
−
=
2.
6. SSS
C
As shown i
n
F
i
g. 3, SSSC is
com
posed
of a
voltage
s
o
urce
in series with
th
e transm
ission line. T
h
is
vol
t
a
ge
so
urc
e
i
n
ject
s t
h
e
v
o
l
t
a
ge wi
t
h
t
h
e
m
a
gni
t
ude
an
d
angl
e,
Vq
and
θ
q
,
resp
ectiv
ely, wh
ich
is i
n
sa
me
alig
n
m
en
t with th
e tran
sm
issi
o
n
lin
e
v
o
ltag
e
drop
, th
erefo
r
e (2
0) is
ju
stified
for SSSC.
Figure 3.
Electri
cal m
odel of
SSSC
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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ECE
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8
¶
Modeling
FAC
TS Devices
i
n
Power Syst
em
St
at
e Est
i
m
at
i
o
n (
S
.M.
Ma
h
a
e
i
)
61
(1
9)
)
sin
sin
(
)
cos
cos
(
ˆ
ˆ
θ
θ
θ
θ
θ
θ
j
j
i
i
j
j
i
i
j
j
i
i
j
i
V
V
j
V
V
V
V
V
V
−
+
−
=
∠
−
∠
=
−
(2
0)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
=
−
θ
θ
θ
θ
θ
j
j
i
i
j
j
i
i
q
V
V
V
V
cos
cos
sin
sin
tan
1
Converting t
h
e series voltage sour
ce i
n
to
the shunt current source
,
ac
cor
d
i
n
g t
o
t
h
e
Fi
g.
4, t
h
e
in
j
e
c
t
ed
p
o
w
e
r of
sh
un
t cu
rr
en
t
so
urce is cal
cu
lated
as fo
llows:
Fi
gu
re
4.
Si
m
p
l
i
f
i
e
d el
ect
ri
cal
m
odel
of
SS
S
C
(2
1)
)
(
ˆ
b
j
g
V
I
ij
ij
q
q
q
+
∠
=
θ
(2
2)
[]
[]
θ
θ
θ
θ
iq
ij
iq
ij
q
i
iq
ij
iq
ij
q
i
q
i
SSSC
ij
b
g
V
V
j
b
g
V
V
I
V
S
cos
sin
sin
cos
ˆ
ˆ
ˆ
*
,
−
+
+
=
=
(2
3)
[
]
θ
θ
iq
ij
iq
ij
q
i
SSSC
ij
b
g
V
V
P
sin
cos
,
+
=
(2
4)
[
]
θ
θ
iq
ij
iq
ij
q
i
SSSC
ij
b
g
V
V
Q
cos
sin
,
−
=
In
w
h
ich:
(2
5)
θ
θ
θ
q
i
iq
−
=
I
t
shou
ld
b
e
noted
th
at:
(2
6)
P
P
P
SSSC
ij
ij
new
ij
,
+
=
(2
7)
Q
Q
Q
SSSC
ij
ij
new
ij
,
+
=
In th
e sam
e
way can
b
e
writing
:
(2
8)
[
]
θ
θ
jq
ij
jq
ij
q
j
SSSC
ji
b
g
V
V
P
sin
cos
,
+
=
(2
9)
[
]
θ
θ
jq
ij
jq
ij
q
j
SSSC
ji
b
g
V
V
Q
cos
sin
,
−
=
(3
0)
P
P
P
SSSC
ji
ji
new
ji
,
+
=
(3
1)
Q
Q
Q
SSSC
ji
ji
new
ji
,
+
=
The i
n
jection
powe
rs m
a
y be
change
d as
follows:
(3
2)
P
P
P
SSSC
ij
i
new
i
,
+
=
(3
3)
Q
Q
Q
SSSC
ij
i
new
i
,
+
=
Evaluation Warning : The document was created with Spire.PDF for Python.
¶
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
2,
No
. 1,
Fe
br
uar
y
20
1
2
:
5
7
– 67
62
(3
4)
P
P
P
SSSC
ji
j
new
j
,
+
=
(3
5)
Q
Q
Q
SSSC
ji
j
new
j
,
+
=
2.
7. T
C
P
A
R
TC
PAR
i
s
a
p
h
a
se an
gl
e re
g
u
l
a
t
o
r,
an
d as
s
h
ow
n i
n
Fi
g.
5 i
s
pl
ace
d i
n
seri
es wi
t
h
t
h
e t
r
an
sm
i
ssi
on
lin
e.
Fi
gu
re
5.
El
ect
ri
cal
m
odel
of
TC
PAR
Th
us, i
t
i
s
e
n
o
u
g
h
t
o
m
a
ke ch
ange
i
n
t
h
e e
q
u
a
t
i
ons
P
ij
,
Q
ij
,
P
ji
,
Q
ji
,
P
i
,
P
j
,
Q
i
an
d
Q
j
u
s
e of
fo
llow
e
d
vari
a
b
l
e
.
(3
6)
θ
θ
θ
Δ
+
=
i
new
i
2
.
8
.
TC
VAR
As illu
strated
i
n
Fi
g
.
6
,
TCVAR is a
vo
ltage reg
u
l
at
or
p
l
aced
in series with
th
e tran
sm
i
ssio
n
lin
e.
Fi
gu
re
6.
El
ect
ri
cal
m
odel
of
TC
VAR
Thu
s
to calcu
late th
e inj
ection and
tran
sm
itte
d
p
o
wers, v
a
riab
le
ch
ang
e
is utilized
.
(3
7)
V
V
V
i
new
i
Δ
+
=
2.
9. UPF
C
UPFC, as
de
picted in Fig. 7
consists of a s
e
ries
voltage s
o
urce and a s
h
unt
cu
rren
t so
urce with
the
co
nd
itio
n th
at
th
e inj
ected
activ
e po
wer
o
f
series
v
o
ltag
e
so
urce an
d th
e shun
t curren
t so
urce is sam
e
.
Modeling the
voltage
source
is identi
cal to the SSSC m
o
deling with t
h
e
di
ffe
re
nce that
the condition of (20)
is eli
m
inated and re
placed by
this condition.
Fi
gu
re
7.
El
ect
ri
cal
m
odel
of
UPFC
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
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:
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8-8
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0
8
¶
Modeling
FAC
TS Devices
i
n
Power Syst
em
St
at
e Est
i
m
at
i
o
n (
S
.M.
Ma
h
a
e
i
)
63
(3
8)
P
P
SSSC
ij
UPFC
ij
,
,
=
(3
9)
Q
Q
SSSC
ij
UPFC
ij
,
,
=
(4
0)
P
P
SSSC
ji
UPFC
ji
,
,
=
(4
1)
Q
Q
SSSC
ji
UPFC
ji
,
,
=
(4
2)
P
P
P
UPFC
ij
ij
new
ij
,
+
=
(4
3)
Q
Q
Q
UPFC
ij
ij
new
ij
,
+
=
(4
4)
P
P
P
UPFC
ji
ji
new
ji
,
+
=
(4
5)
Q
Q
Q
UPFC
ji
ji
new
ji
,
+
=
Sh
unt
c
u
rre
nt
s
o
u
r
ce c
h
a
nges
onl
y
t
h
e
i
n
ject
i
o
n
p
o
w
er
o
f
i
th
bus
.
(4
6)
[]
θ
θ
sh
i
sh
i
sh
i
sh
i
sh
i
j
I
V
I
V
S
,
,
*
,
sin
cos
ˆ
+
=
=
(4
7)
θ
sh
i
sh
i
sh
i
I
V
P
,
,
cos
=
(4
8)
θ
sh
i
sh
i
sh
i
I
V
Q
,
,
sin
=
It is i
m
p
o
r
tan
t
to
n
o
t
e th
at acco
rd
ing
to
the aforem
en
tio
n
e
d
cond
itio
n
P
i,sh
and
P
ij
s
h
o
u
l
d be eq
ual
,
thus i
n
jecte
d
a
c
tive powe
r of
i
th
b
u
s
is ex
actly eq
u
a
l to
t
h
e in
j
ected
activ
e
p
o
wer,
P
i,sh
.
Injected reactive powe
r
of
i
th
bus
as
well as injected
active and reac
tive powe
r
of
j
th
b
u
s
will b
e
ch
ang
e
d
acco
r
d
i
n
g
to
th
e fo
llowing
equat
i
o
n,
t
o
o.
(4
9)
P
P
P
sh
i
i
new
i
,
+
=
(5
0)
Q
Q
Q
Q
sh
i
UPFC
ij
i
new
i
,
,
+
+
=
(5
1)
P
P
P
UPFC
ji
j
new
j
,
+
=
(5
2)
Q
Q
Q
UPFC
ji
j
new
j
,
+
=
2.
10
. IP
FC
IPFC is alm
o
st the sa
m
e
as S
SSC and
UPFC except
t
h
at
as i
t
can be seen fr
om
Fi
g. 8, i
t
has t
w
o
v
o
ltag
e
so
urces in
series with
two
o
r
m
o
re circu
it’s
tran
sm
issio
n
lin
es; and
th
e su
mm
ed
injecte
d
active powe
r
by the
voltage
sources
are ze
ro.
P
Fi
gu
re
8.
El
ect
ri
cal
m
odel
of
IPFC
fo
r t
h
e t
w
o
ci
rc
ui
t
’
s t
r
a
n
sm
i
ssi
on l
i
n
es
Thus, i
n
jected
and tra
n
sm
itted
po
we
rs t
h
rough eac
h line
are cha
n
gi
ng like the
SSSC
if t
h
e c
o
nditi
on
of (20)
will be
rem
oved a
n
d replaced
by
t
h
e
condition of
pr
evious para
gra
p
h.
3.
CASE ST
UDY
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¶
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
2,
No
. 1,
Fe
br
uar
y
20
1
2
:
5
7
– 67
64
For ca
se studies, IEEE
14-bus system
is s
e
lected. As
s
h
own in Fi
g.
9, this
syste
m
h
a
s a variety of
lin
e po
wer
flow m
easu
r
em
en
ts, vo
ltag
e
m
a
g
n
itu
d
e
m
easure
m
ents and
power injection m
e
asurem
ents.
Fi
gu
re
9.
IEE
E
1
4
-
b
us sy
st
em
In
th
e first mo
d
e
, it is
assu
m
e
d
th
at th
e
r
e ar
e no c
o
n
t
rol
devi
ces i
n
t
h
e net
w
or
k
.
W
L
S st
at
e
est
i
m
a
ti
on
resu
l
t
s
are be
o
b
t
a
i
n
ed
an
d
gi
ve
n i
n
Ta
bl
e
1.
Table
1.State e
s
tim
a
tion in the norm
al
m
ode for
IEEE
14-bus system
1
Me
a
sure
me
nts
2
A
ctual data
(pu)
3
M
eas
ured
data (pu)
4
E
s
tim
ated
data (pu)
5
Me
a
sure
me
nts
6
A
ctual data
(pu)
7
M
eas
ured
data (pu)
8
E
s
tim
ated
data (pu)
V
1
1.0600
1.1108
1.0600
Q
10
,
11
-0.0162
-0.0161
0.0298
V
3
1.0100
1.0220
1.0100
Q
13
,
14
0.0175
0.0182
0.0154
V
4
1.0180
1.0636
1.0180
Q
5
,
1
0.0223
0.0224
0.0243
V
5
1.0200
1.0071
1.0200
Q
4
,
2
0.0302
0.0289
0.0316
V
8
1.0900
1.0603
1.0900
Q
5
,
4
-0.1420
-0.1462
-0.1383
V
9
1.0560
1.0632
1.0560
Q
7
,
4
0.1138
0.1169
0.1146
V
11
1.0570
1.0634
1.0470
Q
9
,
4
0.0173
0.0173
0.0168
V
13
1.0500
1.0270
1.0500
Q
11
,
6
-0.0344
-0.0328
-0.0767
P
1
,
2
1.5688
1.6272
1.5695
Q
12
,
6
-0.0235
-0.0246
-0.0242
P
2
,
5
0.4152
0.4296
0.4141
Q
14
,
9
-0.0336
-0.0327
-0.0324
P
4
,
5
-0.6116
-0.6207
-0.6118
P
2
0.1830
0.1903
0.1795
P
5
,
6
0.4409
0.4508
0.4408
P
4
-0.4780
-0.4755
-0.4760
P
6
,
13
0.1775
0.1734
0.1785
P
6
-0.1120
-0.1071
-0.0905
P
7
,
8
0
0
0
P
7
0
0
-0.0008
P
12
,
13
0.0161
0.0155
0.0166
P
9
-0.2950
-0.2911
-0.2946
P
13
,
14
0.0564
0.0573
0.0554
P
10
-0.0900
-0.0901
-0.0709
P
3
,
2
-0.7091
-0.6975
-0.7085
P
12
-0.0610
-0.0603
-0.0607
P
4
,
2
-0.5445
-0.5509
-0.5435
Q
2
0.3086
0.2932
0.3047
P
7
,
4
-0.2807
-0.2884
-0.2811
Q
3
0.0608
0.0621
0.0588
P
9
,
4
-0.1608
-0.1645
-0.1608
Q
4
0.0390
0.0384
0.0381
P
11
,
6
-0.0730
-0.0747
-0.0924
Q
6
0.0523
0.0531
0.0970
P
10
,
9
-0.0521
-0.0498
-0.0527
Q
9
-0.1660
-0.1605
-0.1713
Q
2
,
5
0.0117
0.0114
0.0091
Q
14
-0.0500
-0.0506
-0.0468
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
¶
Modeling
FAC
TS Devices
i
n
Power Syst
em
St
at
e Est
i
m
at
i
o
n (
S
.M.
Ma
h
a
e
i
)
65
In
t
h
e n
e
x
t
step
, it is assu
m
e
d
th
at a
p
a
rall
el d
e
v
i
ce
h
a
v
i
n
g
cap
acitiv
e i
m
p
e
d
a
n
ce of
0.2
p
u
to
b
e
i
n
st
al
l
e
d i
n
b
u
s
-1
4.
I
n
t
h
i
s
m
ode t
h
e
devi
ce
i
s
m
odel
e
d ac
cor
d
i
n
g (
1
3) a
s
a react
i
v
e
po
wer s
o
urce
. T
h
e
W
L
S
st
at
e
est
i
m
a
ti
on resul
t
s
a
r
e gi
ven
i
n
Tabl
e 2.
Tabl
e
2.St
at
e e
s
t
i
m
a
ti
on r
e
ga
r
d
i
n
g a
sh
u
n
t
de
vi
ce f
o
r
IE
EE
14
-
bus
sy
st
em
9
Me
a
sure
me
nts
10
A
ctual data
(pu)
11
M
eas
ured
data (pu)
12
E
s
tim
ated
data (pu)
13
Me
a
sure
me
nts
14
A
ctual data
(pu)
15
M
eas
ured
data (pu)
16
E
s
tim
ated
data (pu)
V
1
1.0600
1.0600
1.0601
Q
13
,
14
-0.0883
-0.0883
-0.0884
V
3
1.0100
1.0100
1.0101
Q
5
,
1
0.0294
0.0294
0.0291
V
4
1.0200
1.0200
1.0201
Q
4
,
2
0.0443
0.0443
0.0441
V
5
1.0210
1.0210
1.0211
Q
5
,
4
-0.1640
-0.1640
-0.1640
V
8
1.0900
1.0900
1.0900
Q
7
,
4
0.1419
0.1419
0.1419
V
9
1.0710
1.0710
1.0712
Q
9
,
4
0.0425
0.0425
0.0422
V
11
1.0630
1.0630
1.0632
Q
11
,
6
-0.0025
-0.0025
-0.0027
V
13
1.0610
1.0610
1.0608
Q
12
,
6
-0.0024
-0.0024
-0.0025
P
1
,
2
1.5696
1.5696
1.5692
Q
14
,
9
0.0926
0.0926
0.0924
P
2
,
5
0.4144
0.4144
0.4142
P
2
0.1830
0.1830
0.1829
P
4
,
5
-0.6217
-0.6217
-0.6217
P
4
-0.4780
-0.4780
-0.4780
P
5
,
6
0.4306
0.4306
0.4303
P
6
-0.1120
-0.1120
-0.1120
P
6
,
13
0.1763
0.1763
0.1761
P
7
0
0
0.0001
P
7
,
8
0
0
-0.0003
P
9
-0.2950
-0.2950
-0.2949
P
12
,
13
0.0120
0.0120
0.0119
P
10
-0.0900
-0.0900
-0.0899
P
13
,
14
0.0514
0.0514
0.0512
P
12
-0.0610
-0.0610
-0.0610
P
3
,
2
-0.7088
-0.7088
-0.7088
Q
2
0.2863
0.2863
0.2863
P
4
,
2
-0.5463
-0.5463
-0.5461
Q
3
0.0469
0.0469
0.0468
P
7
,
4
-0.2878
-0.2878
-0.2875
Q
4
0.0390
0.0390
0.0390
P
9
,
4
-0.1654
-0.1654
-0.1652
Q
6
-0.0946
-0.0946
-0.0947
P
11
,
6
-0.0684
-0.0684
-0.0683
Q
9
-0.1660
-0.1660
-0.1661
P
10
,
9
-0.0567
-0.0567
-0.0567
Q
14
-0.0500
-0.0500
-0.0500
Q
2
,
5
0.0029
0.0029
0.0030
b
sh
,
14
0.2
-
0.1999
Q
10
,
11
0.0158
0.0158
0.0159
To ve
rify the
effective
n
ess
of series
device
s
m
ode
ling
,
it is assu
m
e
d
th
at a series d
e
v
i
ce with
th
e
im
pedance o
f
0.
02
9
6
p.
u. t
o
be i
n
st
al
l
e
d i
n
l
i
n
e 1-
2. I
n
t
h
i
s
case, t
h
i
s
de
vi
ce chan
ges t
h
e im
pedan
ce of t
h
e
t
r
ansm
i
ssi
on l
i
n
e acc
or
di
n
g
t
o
(1
4)
. T
h
e
W
L
S st
ate estim
a
tio
n
resu
lts are
sh
own
in Tab
l
e 3
.
Tabl
e
3.St
at
e e
s
t
i
m
a
ti
on r
e
ga
r
d
i
n
g a
seri
es
d
e
vi
ce f
o
r
IE
EE
1
4
-
b
us sy
st
em
17
Me
a
sure
me
nts
18
A
ctual data
(pu)
19
M
eas
ured
data (pu)
20
E
s
tim
ated
data (pu)
21
Me
a
sure
me
nts
22
A
ctual data
(pu)
23
M
eas
ured
data (pu)
24
E
s
tim
ated
data (pu)
V
1
1.0600
1.0600
1.0599
Q
13
,
14
0.0179
0.0179
0.0176
V
3
1.0100
1.0100
1.0099
Q
5
,
1
-0.0182
-0.0182
-0.0183
V
4
1.0180
1.0180
1.0175
Q
4
,
2
0.0460
0.0460
0.0460
V
5
1.0200
1.0200
1.0195
Q
5
,
4
-0.1231
-0.1231
-0.1228
V
8
1.0900
1.0900
1.0900
Q
7
,
4
0.1142
0.1142
0.1138
V
9
1.0560
1.0560
1.0558
Q
9
,
4
0.0177
0.0177
0.0174
V
11
1.0570
1.0570
1.0566
Q
11
,
6
-0.0351
-0.0351
-0.0346
V
13
1.0500
1.0500
1.0499
Q
12
,
6
-0.0237
-0.0237
-0.0251
P
1
,
2
1.6876
1.6876
1.6876
Q
14
,
9
-0.0332
-0.0332
-0.0334
P
2
,
5
0.4668
0.4668
0.4668
P
2
0.1830
0.1830
0.1829
P
4
,
5
-0.5613
-0.5613
-0.5611
P
4
-0.4780
-0.4780
-0.4781
P
5
,
6
0.4375
0.4375
0.4377
P
6
-0.1120
-0.1120
-0.1122
P
6
,
13
0.1764
0.1764
0.1766
P
7
0
0
0.0000
P
7
,
8
0
0
-0.0004
P
9
-0.2950
-0.2950
-0.2951
P
12
,
13
0.0159
0.0159
0.0156
P
10
-0.0900
-0.0900
-0.0898
P
13
,
14
0.0552
0.0552
0.0544
P
12
-0.0610
-0.0610
-0.0612
P
3
,
2
-0.7262
-0.7262
-0.7261
Q
2
0.5736
0.5736
0.5736
P
4
,
2
-0.5805
-0.5805
-0.5805
Q
3
0.0599
0.0599
0.0601
Evaluation Warning : The document was created with Spire.PDF for Python.
¶
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
2,
No
. 1,
Fe
br
uar
y
20
1
2
:
5
7
– 67
66
17
Me
a
sure
me
nts
18
A
ctual data
(pu)
19
M
eas
ured
data (pu)
20
E
s
tim
ated
data (pu)
21
Me
a
sure
me
nts
22
A
ctual data
(pu)
23
M
eas
ured
data (pu)
24
E
s
tim
ated
data (pu)
P
7
,
4
-0.2828
-0.2828
-0.2826
Q
4
0.0390
0.0390
0.0392
P
9
,
4
-0.1620
-0.1620
-0.1620
Q
6
0.0522
0.0522
0.0519
P
11
,
6
-0.0709
-0.0709
-0.0708
Q
9
-0.1660
-0.1660
-0.1660
P
10
,
9
-0.0542
-0.0542
-0.0546
Q
14
-0.0500
-0.0500
-0.0500
Q
2
,
5
-0.0022
-0.0022
-0.0023
b
se
,
1
,
2
0.0296
-
0.0296
Q
10
,
11
-0.0168
-0.0168
-0.0167
Fo
r th
e
p
a
rall
el-series
d
e
v
i
ces, it is assumed
th
at a UPFC with
V
se
= 0.015
8
<9
5.49
º
and
I
sh
=0.408
<-
29
.3
4º
t
o
be i
n
st
al
l
e
d i
n
l
i
n
e 1-2
.
UPFC
, i
n
t
h
i
s
m
ode, i
s
m
odel
e
d based
on (
3
8) t
o
(
5
2
)
an
d WL
S
st
at
e est
i
m
a
ti
on i
s
d
one
. T
h
e
resul
t
s
a
r
e
obt
a
i
ned a
s
gi
ve
n i
n
Ta
bl
e
4.
Tabl
e
4.St
at
e e
s
t
i
m
a
ti
on r
e
ga
r
d
i
n
g a
UP
FC
f
o
r
IEE
E
14
-b
u
s
sy
st
em
25
Me
a
sure
me
nts
26
A
ctual data
(pu)
27
M
eas
ured
data (pu)
28
E
s
tim
ated
data (pu)
29
Me
a
sure
me
nts
30
A
ctual data
(pu)
31
M
eas
ured
data (pu)
32
E
s
tim
ated
data (pu)
V
1
1.0600
1.0600
1.0599
Q
5
,
1
0.0562
0.0562
0.0564
V
3
1.0100
1.0100
1.0101
Q
4
,
2
0.0621
0.0621
0.0628
V
4
1.0260
1.0260
1.0258
Q
5
,
4
-0.2451
-0.2451
-0.2449
V
5
1.0250
1.0250
1.0250
Q
7
,
4
0.0927
0.0927
0.0926
V
8
1.0900
1.0900
1.0900
Q
9
,
4
0.0099
0.0099
0.0098
V
9
1.0600
1.0600
1.0601
Q
11
,
6
-0.0303
-0.0303
-0.0302
V
11
1.0590
1.0590
1.0588
Q
12
,
6
-0.0233
-0.0233
-0.0224
V
13
1.0510
1.0510
1.0510
Q
14
,
9
-0.0358
-0.0358
-0.0359
P
1
,
2
1.5421
1.5421
1.5424
P
2
0.1830
0.1830
0.1830
P
2
,
5
0.4468
0.4468
0.4474
P
4
-0.4780
-0.4780
-0.4779
P
4
,
5
-0.6849
-0.6849
-0.6851
P
6
-0.1120
-0.1120
-0.1120
P
5
,
6
0.4213
0.4213
0.4221
P
7
0
0
0.0003
P
6
,
13
0.1713
0.1713
0.1717
P
9
-0.2950
-0.2950
-0.2950
P
7
,
8
0
0
0.0002
P
10
-0.0900
-0.0900
-0.0897
P
12
,
13
0.0145
0.0145
0.0148
P
12
-0.0610
-0.0610
-0.0612
P
13
,
14
0.0488
0.0488
0.0497
Q
2
0.2211
0.2211
0.2215
P
3
,
2
-0.6892
-0.6892
-0.6888
Q
3
0.0135
0.0135
0.0136
P
4
,
2
-0.5109
-0.5109
-0.5111
Q
4
0.2390
0.2390
0.2392
P
7
,
4
-0.2928
-0.2928
-0.2935
Q
6
0.0179
0.0179
0.0180
P
9
,
4
-0.1680
-0.1680
-0.1684
Q
9
-0.1660
-0.1660
-0.1661
P
11
,
6
-0.0614
-0.0614
-0.0614
Q
14
-0.0500
-0.0500
-0.0501
P
10
,
9
-0.0636
-0.0636
-0.0635
|
V
se
,
4
,
5
| 0.0158
-
0.0158
Q
2
,
5
-0.0289
-0.0289
-0.0287
<V
se
,
4
,
5
95.49
º
-
95.49
º
Q
10
,
11
-0.0122
-0.0122
-0.0126
|
I
sh
,
4
| 0.408
-
0.408
Q
13
,
14
0.0150
0.0150
0.0150
<
I
sh
,
4
-29.34
º
-
-29.34
º
It
was
see
n
t
h
a
t
, al
l
t
h
e p
r
op
o
s
ed m
odel
s
a
r
e
abl
e
t
o
m
odel
di
ffe
re
nt
t
y
pes
of
co
nt
r
o
l
de
vi
ces suc
h
a
s
FAC
T
S
de
vi
ce
s. M
o
del
i
n
g t
h
ese de
vi
ces n
o
t
o
n
l
y
coul
d es
t
i
m
a
te the corre
ct values
of m
e
asurem
ents but also
is able to estimate these
devic
e
s’ stat
es acc
or
di
n
g
t
o
Ta
bl
es
2,
3 a
n
d
4.
4.
CO
NCL
USI
O
N
To
day
,
se
veral
t
y
pes o
f
c
o
nt
r
o
l
de
vi
ces s
u
c
h
as
FAC
T
S
d
e
vi
ces are
us
e
d
i
n
p
o
w
er sy
s
t
em
s. These
devi
ces m
a
y
be i
n
fl
ue
nce
d
st
at
e est
i
m
a
t
i
on i
n
p
o
we
r sy
stem
s, which the
correct state estim
a
tion of s
y
ste
m
req
u
i
r
e
d
m
odel
i
ng i
n
st
al
l
e
d cont
rol
de
vi
ces i
n
t
h
e sy
st
em
. In t
h
i
s
pape
r,
di
ffere
nt
t
y
pes of co
nt
r
o
l
de
vi
ce
s
suc
h
as pa
ral
l
e
l
,
seri
es an
d se
ri
es-s
hu
nt
F
A
C
T
S de
vi
ces w
e
re m
odel
e
d f
o
r p
o
w
e
r sy
st
e
m
st
at
e est
i
m
a
ti
on.
To
dem
onstrate the effective
n
ess
of t
h
e propos
ed m
odels,
a
shunt, a se
ries
and
a series
-shunt
devices
were
considere
d
on
the IEEE
14-bus system
separately and
state estim
a
tion we
re done
base
d on
WLS
.
The
results
have
sh
o
w
n t
h
at
t
h
e
pr
op
o
s
ed m
odel
s
n
o
t
o
n
l
y
coul
d effectively lead in
pr
oper state estim
a
tion
of
measurem
ents values
and syst
e
m
states but a
l
so is a
b
le
to
present the
estimated st
at
es
of t
h
e c
ont
rol
de
vi
ces.
Evaluation Warning : The document was created with Spire.PDF for Python.