Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
4, N
o
. 2
,
A
p
r
il
201
4, p
p
.
16
9
~
17
8
I
S
SN
: 208
8-8
7
0
8
1
69
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
Power S
y
st
em St
at
e Esti
mation with W
e
ighted L
i
near L
e
ast
Square
Seyed
Mah
d
i
Mahaei
*,
Mohamm
ad
Rez
a
Navayi*
*Azarbai
j
an
Reg
i
onal
El
ect
ric
Co
m
p
an
y
,
T
a
briz
, I
r
an
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ja
n 27, 2014
Rev
i
sed
Feb
27
, 20
14
Accepte
d
Mar 6, 2014
Power s
y
stem state estimation w
ith
c
onventional method, Weigh
t
ed
Linear
Leas
t (W
LS
),
is
perform
ed in tw
o s
t
eps
.
In th
e fi
rs
t s
t
age th
e obs
ervabi
lit
y of
s
y
stem is done and then is sy
stem is
ob
served state estimation is
carried out.
Otherwise estimator is d
i
srupted
and is
no
t ab
le
to est
i
m
a
te
the
states.
Th
e
another
estim
at
or, W
e
ight
ed le
ast Absolute Va
lue (W
LAV) ha
s presented
which is
able
to
es
tim
ate th
e s
y
s
t
em
s
t
at
es in a
ll
situations but
thi
s
estim
ator
have
auxiliar
y
v
a
riab
le which r
e
duces th
e conv
er
gence rate
estimator. In th
is
paper, a n
e
w method, Weighted
Linear
Least Square (WLLS)
,
is proposed
for power sy
stem state
estimation. The proposed
method, WLLS, has fewer
variab
les th
an
WLAV. The ob
jective function
of th
e proposed method is
line
a
r ther
efore
t
h
is
es
tim
ator is
able
to es
tim
at
e
the s
t
at
es
in uno
bs
ervabil
i
t
y
situations. Case studies on IEEE 14
bus s
y
stem show WLL
S
has good
accur
a
c
y
and
speed and is
abl
e
t
o
es
tim
at
e the
st
ates in all
condi
t
i
ons.
Keyword:
Measurem
ent function
No
rm
al distribution
Ob
serv
ab
ility
State estim
a
tion
Weigh
t
ed
least
Ab
so
l
u
te
Value
Weighted Line
ar
Least
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Sey
e
d M
a
hdi
M
a
haei
Azarbaija
n
Re
gional Electric
Com
p
any
E
m
a
m
S
t
.
,
Kh
ag
a
n
i cor
n
er
,
T
a
b
r
iz
, Ir
a
n
Em
a
il: me.
m
ah
aei@azrec.co
.i
r
1.
INTRODUCTION
In
power syste
m
s sta
t
e estimator c
o
m
p
lete recei
ve
d
data from
the SCAD
A system
. Powe
r system
state estim
a
tion is not re
quired, if t
h
e recei
ved
data
from SCADA a
r
e
adequate a
nd
correct. T
h
ere
f
ore i
n
po
we
r sy
st
em
s Ener
gy
M
a
na
gem
e
nt
Sy
st
em
(EM
S
) was creat
ed w
h
i
c
h
fi
rst
l
y
preser
ve
t
h
e sy
st
em
i
n
no
rm
al
conditions and seconda
ry operate in optim
um
econom
ic
co
nditions. A series
data of system
are
require
d
to
accom
p
lish this goal. This
data ar
e tra
n
smitted by SC
ADA system
, t
o
EMS. State
esti
m
a
tion of powe
r
syste
m
s is the
interface
of
SCADA and
E
M
S. Howe
ve
r, before state e
s
tim
a
tion an e
s
tim
a
tor shoul
d
be test
th
e ob
serv
ab
ility o
f
system
. Th
en
estim
ate
states with
m
i
n
i
m
u
m
erro
r.
Re
m
a
rk
ab
ly
ob
serv
ab
ility o
f
po
wer
sy
st
em
depen
d
s
o
n
t
h
e
num
ber a
n
d
l
o
cat
i
o
n
of
i
n
st
al
l
e
d m
e
t
e
rs.
St
at
e est
i
m
a
t
i
on m
e
t
hods can
be di
vi
de
d i
n
t
o
t
w
o g
r
o
u
p
s.
The fi
rst
gr
o
u
p
i
s
based
on m
a
t
h
em
ati
cal
m
e
thods a
nd t
h
e second group is ba
sed
on intelligent
m
e
thods.
Weighted Linea
r
Lea
s
t (W
LS
),
We
ighte
d
least Ab
so
lu
te Valu
e
(WLAV) and
Estim
a
tio
n
wit
h
N
o
n
-
Fi
xe
d
E
r
r
o
r (
M
-Est
im
at
or) are
fam
ous pr
esent
e
d
m
a
t
h
em
at
i
c
al
m
e
t
hods
[1
-
1
0
]
. Am
ong i
n
t
e
l
l
i
g
ent
t
ech
ni
q
u
e
s can
be p
o
i
n
t
out
F
u
zzy
I
n
f
e
rence
Sy
st
em
(FI
S
)
,
st
at
e est
i
m
a
t
i
on base
d
on
Ne
ural
N
e
t
w
or
k (
N
T) a
n
d A
d
a
p
t
i
v
e Ne
ur
o
n
Fu
zzy
Infe
re
nce
Sy
st
em
(ANF
I
S
) [
1
1-
15]. Intelligent
m
e
thods
of m
a
them
a
tical
m
e
thods
for e
s
tim
a
ting re
quir
ed less tim
e,
but they
have not
sufficient accurate. The acc
uracy and spee
d of
WLS
and
W
L
AV are
better tha
n
the
other m
a
thematical
m
e
t
hods
[
1
6
-
1
7
]
.
Acco
r
d
i
n
gl
y
,
i
n
t
h
i
s
pa
per ar
e st
udi
ed t
w
o
m
e
t
hods f
o
r p
o
we
r sy
st
em
st
at
e est
i
m
a
ti
on
,
W
L
S a
n
d
WL
AV
. The
n
a new m
e
t
hod,
W
e
i
g
ht
ed Li
n
ear Least
Sq
ua
re (
W
LLS
), i
s
pr
o
pose
d
. T
h
e
pr
o
pose
d
m
e
t
hod
has
fewer v
a
riab
les th
an
W
L
AV an
d
h
a
s m
o
re p
e
rform
a
n
ce ran
g
e
th
an
th
e
WLS So
th
at ev
en
in
u
nob
serv
ab
ility
of
syste
m
this estim
a
tor can
be es
tim
a
te power system
state
s
. Meanwhile
,
accuracy a
n
d s
p
eed of the
propos
ed
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. 2, A
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20
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9 – 1
7
8
17
0
m
e
thod is reas
ona
ble and acceptable. Case s
t
udy on I
EEE 14 bus
system
show
the
acc
uracy and pe
rform
a
nce
range of
the propose
d
m
e
thod is accepta
ble.
2.
STATE ESTIMATION WITH WLS
St
at
e est
i
m
a
t
i
o
n fu
nct
i
o
n i
s
the m
easured am
ount
s l
i
k
ene
ss t
o
corre
sp
o
ndi
ng cal
cul
a
t
e
d am
ount
s.
Seve
ral
m
e
t
h
o
d
s
have
p
r
o
p
o
se
d f
o
r c
r
ea
t
i
ng st
at
e est
i
m
at
i
on fu
nct
i
o
n
.
M
o
st
of
t
h
em
are based
o
n
p
r
ob
ab
ilistic meth
od
s so
t
h
us, th
e m
easu
r
emen
t error is assu
m
e
d
to
h
a
v
e
a certain
p
r
ob
abilit
y d
i
stribu
tion
.
In
power system
s
th
e
m
easu
r
e
m
en
t error is g
e
n
e
rally assu
m
e
d
as n
o
r
mal p
r
ob
ab
ility
d
i
stribu
tio
n
according t
o
e
quation
(1), which is
also quit
e close t
o
re
ality.
e
Z
pdf
Z
est
Z
meas
est
)
(
2
2
1
2
1
)
(
)
1
(
Whe
r
e:
Z
meas
: m
easured
value
Z
est
: The estimated val
u
e
(expected
Z=
E
(
Z
)
)
σ
:
st
anda
rd
de
v
i
at
i
on o
f
Z
Th
e aim
is to
m
a
x
i
m
i
ze th
e
p
r
ob
ab
ility
pd
f(Z
est
)
in
stat
e esti
m
a
t
i
o
n
by p
r
o
b
a
b
ilistic
m
e
th
o
d
. In
g
e
n
e
ral, aim
is max
i
m
i
zin
g
the m
easu
r
e
m
en
ts m
u
ltip
lied
pd
f
s according t
o
equation
(2).
)
(
.
.
).
(
max
,
1
,
Z
pdf
Z
pdf
f
m
est
est
)
2
(
Whe
r
e:
m
:
num
b
er of measurem
ents
Equ
a
tio
n (2
) can
b
e
rewrite as a log
a
rim
i
c fu
n
c
tion
fo
r
simp
lifyin
g
th
e
opti
m
izatio
n
.
]
log
2
log
2
)
[(
2
1
)
(
log
log
max
1
1
2
1
,
,
1
,
m
i
i
m
i
m
i
i
est
m
Z
Z
Z
pdf
f
i
i
est
meas
)
3
(
Th
e secon
d
and
th
i
r
d sen
t
en
ces
are con
s
tan
t
th
en
t
h
e
fin
a
l
ob
j
ective fun
c
tio
n will b
e
as eq
u
a
tion
(4
).
m
i
i
Z
i
est
Z
meas
J
1
2
)
,
1
,
(
min
)
4
(
In power system
s
Z
est
is d
e
p
e
n
d
e
n
t
on
state
v
a
riab
les
o
f
the system
, an
d
t
h
e
ob
j
ectiv
e fun
c
tio
n
will
depe
nd on
the state
varia
b
le
s
according t
o
e
quation
(6).
)
ˆ
(
,
x
h
Z
i
i
est
)
5
(
m
i
i
x
h
i
Z
meas
x
J
1
2
)
)
ˆ
(
1
,
(
)
ˆ
(
)
6
(
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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Pow
e
r
Syst
em
St
at
e Est
i
m
at
i
o
n w
i
t
h
Wei
ght
e
d
Li
nea
r
Le
ast
Sq
u
a
re (
S
eye
d
Ma
hdi
M
a
haei
)
17
1
Whe
r
e t
h
e
h
i
(x)
is called
th
e i
th
m
easurem
ent
f
unct
i
o
n.
I
f
Z
me
a
s
and
h(
x)
are written
as th
e
matrix
J(x)
can
b
e
sim
p
lified
as equ
a
tion
(7).
)]
ˆ
(
[
)]
ˆ
(
[
)
ˆ
(
1
x
h
Z
R
x
h
Z
x
J
T
)
7
(
Whe
r
e we ha
v
e
:
Z
Z
Z
Z
m
.
.
2
1
)
8
(
So i
n
stead of
Z
meas
,
i
,
Z
i
is
use
d
.
)
,
.
.
.
,
(
.
.
)
,
.
.
.
,
(
)
,
.
.
.
,
(
)
ˆ
(
2
1
2
1
2
2
1
1
x
x
x
h
x
x
x
h
x
x
x
h
x
h
n
m
n
n
)
9
(
2
2
2
2
1
.
.
m
R
)
10
(
For o
p
t
i
m
i
zat
ion of f
unct
i
o
n
J(x
)
exi
s
t
several
m
e
t
hod
w
h
i
c
h t
h
e
us
ual
m
e
t
hod i
s
Ga
uss-
Ne
wt
o
n
(GN). Th
is
o
p
t
i
m
izat
io
n
m
e
th
o
d
is
b
a
sed on
iterativ
e m
e
th
od
an
d states m
a
trix
is
o
p
tim
ize
d
as equ
a
tion
(1
1).
)
(
)
(
1
1
x
g
x
k
G
x
x
k
k
k
)
11
(
Whe
r
e
k
is k
th
repeating stepsa
and
g(
x
k
)
and
G(
x
k
)
are obt
ai
ned
f
r
om
equat
i
ons
(
1
2
)
t
o
(
1
4)
.
)
(
)
(
)
(
)
(
1
x
H
R
x
H
x
x
g
x
G
k
k
T
k
k
)
12
(
))
(
(
)
(
)
(
1
x
h
Z
R
x
H
x
g
k
k
T
k
)
13
(
x
x
h
x
H
)
ˆ
(
)
ˆ
(
)
14
(
G(
x)
is called
t
h
e
g
a
in
m
a
trix
.
There
f
ore
po
w
e
r sy
st
em
st
at
e est
i
m
a
t
i
on wi
t
h
W
L
S al
go
ri
t
h
m
and GN
op
t
i
m
i
zat
i
on can prese
n
t
su
c
h
as figure (1).
Evaluation Warning : The document was created with Spire.PDF for Python.
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7
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2
Figure
1.
W
L
S state estim
a
tio
n algorithm
The val
u
e of
ε
determ
ines ac
curacy of
state estim
a
tion.
Whateve
r
ε
is litt
le the results are accurate
and calculate time increase.
3.
STATE ESTIMATION WITH WLAV
WLAV state esti
m
a
t
i
o
n
is si
milar WLS m
e
th
od
with
t
h
is
d
i
fferen
ce th
at
in
WLAV m
e
th
od
is
u
s
ed
ab
so
lu
te erro
r
in
stead squ
a
r
e
er
ro
r.
m
i
i
i
i
x
h
Z
w
x
f
1
)
ˆ
(
)
ˆ
(
)
15
(
Whe
r
e
w
i
is the rev
e
rse of
σ
i
.
R
w
W
i
i
1
1
)
16
(
Op
tim
iz
in
g
t
h
e fu
n
c
tion
f(x)
is
linear s
o
t
h
at t
h
e Taylor a
p
proxim
a
tion
h(
x)
at
t
h
e
poi
nt
x
0
is
u
s
ed
.
e
x
H
Z
)
17
(
Whe
r
e we ha
v
e
:
)
(
0
x
h
Z
Z
)
18
(
|
)
(
0
x
x
x
x
h
H
)
19
(
x
x
x
0
)
20
(
It will assu
m
e
ab
so
lu
te erro
r
(|
e
i
|
)
i
s
l
e
ss t
h
a
n
a cert
a
i
n
am
ount
, s
u
c
h
as
ε
i
:
i
i
e
)
21
(
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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8-8
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8
Pow
e
r
Syst
em
St
at
e Est
i
m
at
i
o
n w
i
t
h
Wei
ght
e
d
Li
nea
r
Le
ast
Sq
u
a
re (
S
eye
d
Ma
hdi
M
a
haei
)
17
3
The a
b
ove
eq
u
a
t
i
on ca
n c
o
nv
ert
t
o
t
w
o
eq
ua
t
i
ons
by
a
ddi
n
g
t
w
o
n
o
n
-
neg
a
t
i
v
e sl
ack
va
ri
abl
e
s.
i
i
i
l
e
)
22
(
i
i
i
k
e
)
23
(
Wh
ich
we
can
write:
v
u
e
i
i
i
)
24
(
So
t
h
at:
l
u
i
i
2
1
)
25
(
k
v
i
i
2
1
)
26
(
Whate
v
er
ε
i
is sm
a
ller, it is l
i
k
e
th
e
e
i
is minim
i
zed. So t
h
e objective function
f(x)
has chan
ge
d t
o
ob
ject
i
v
e
f
unct
i
on
f
΄
(x)
.
v
u
H
t
s
x
f
x
x
Z
v
u
w
w
k
k
k
m
i
i
i
i
i
m
i
i
)
(
.
)
)
ˆ
(
1
1
(
)
27
(
The fu
nct
i
o
n
f
΄
(x
)
can be
optim
i
zed by lin
ear
program
m
ing (L
P). T
h
er
efore the alg
o
rith
m
fo
r
o
p
tim
izat
io
n
fun
c
tio
n
f
΄
(x
)
can be
presente
d a
s
follows
.
Figure
2.
W
L
AV state estim
ation al
gorithm
|
ˆ
ˆ
)
ˆ
(
0
x
x
x
x
h
H
)
ˆ
(
ˆ
ˆ
x
h
Z
Z
x
x
x
0
x
ma
x
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
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08
IJEC
E V
o
l
.
4, No
. 2, A
p
ri
l
20
14
:
16
9 – 1
7
8
17
4
Op
tim
iz
in
g
th
e
f
΄
(x
)
b
y
lin
ear p
r
og
rammin
g
,
it is n
ecessary th
e ab
so
lu
te is eli
m
in
ated
from o
b
j
ectiv
e
fun
c
tio
n.
To el
i
m
i
n
at
e absol
u
t
e
i
n
WL
AV
m
e
t
hod t
h
e ne
w
vari
a
b
l
e
s u
an
d
v
was a
dde
d t
o
t
h
e
o
b
j
ect
i
v
e f
unct
i
o
n
.
4.
STATE ESTIMATION WL
LS
Alth
oug
h
t
h
e
WLAV
It is op
ti
m
i
zed
b
y
li
n
ear
p
r
og
ram
m
in
g
bu
t add
i
n
g
t
h
e n
e
w aux
iliary v
a
riab
les
to objective
function, the num
ber of
varia
b
les increases.
It will increas
e the optim
i
zing tim
e
so that am
ong
WL
AV a
n
d
W
L
S,
WLS
has
bet
t
e
r s
p
eed
[
1
6-
1
7
]
.
O
n
t
h
e
ot
he
r ha
n
d
s
du
e t
o
t
h
e
n
o
n
-
l
i
n
eari
t
y
of t
h
e o
b
j
ect
i
v
e
fu
nct
i
o
n,
WLS
i
n
som
e
sy
st
em
s such as sy
st
em
s wi
t
h
di
f
f
use
d
m
easure
m
ent
s
can n
o
t
fi
nd
re
verse
o
f
gai
n
matrix
. In
o
t
h
e
r word
s, there i
s
no
t inv
e
rse
o
f
H m
a
trix
in
some syste
m
s an
d
it is lead to
WLS is d
i
srup
t
e
d
.
Acco
r
d
i
n
gl
y
,
t
h
i
s
pa
per
pr
o
p
o
ses
WLLS m
e
t
h
o
d
t
h
at
i
t
com
b
i
n
es
W
L
A
V
an
d
WLS
m
e
t
hods
fo
r
opt
i
m
i
zati
on. I
n
W
L
L
S
, acc
or
di
n
g
eq
uat
i
o
n (2
8
)
t
h
e ob
j
ect
i
v
e fu
nct
i
o
n i
s
sam
e
W
L
S ob
ject
i
v
e f
u
nct
i
on
whi
c
h
has a
p
pl
i
e
d s
o
m
e
chan
ges.
m
i
i
x
h
i
Z
i
w
x
f
1
2
)
ˆ
(
)
ˆ
(
)
28
(
W
i
t
h
placem
ent
e
i
instead m
e
asurem
ent error we
have:
m
i
i
e
i
w
x
f
1
2
)
ˆ
(
)
29
(
Wh
it lin
earing
h(
x)
in
arou
nd
th
e po
in
t
x
0
, we
can
write:
x
x
H
x
x
H
x
h
Z
x
H
Z
e
i
i
i
i
i
0
0
0
0
)
(
)
(
)
(
)
30
(
The e
q
uation (24) is
re
placed in
equation (29):
m
i
i
m
i
i
x
x
H
x
x
H
x
h
Z
w
x
H
Z
w
x
g
i
i
i
i
1
2
1
2
0
0
0
0
)
(
)
(
)
(
)
ˆ
(
)
31
(
Accordingly,
C and
d are
as
follows:
)
(
0
x
H
C
)
32
(
Z
x
h
x
x
H
d
i
i
)
(
)
(
0
0
0
)
33
(
Th
erefo
r
e t
h
e
fin
a
l ob
j
e
ctiv
e fu
n
c
tion
will b
e
as eq
u
a
tion
(34
)
:
m
i
i
d
x
C
w
x
f
1
2
ˆ
)
ˆ
(
)
34
(
Th
e abo
v
e
fun
c
tio
n
was obtain
e
d
b
y
lin
earing
H m
a
trix
so
it sh
ou
ld b
e
o
p
tim
ized
b
y
iteratio
n
m
e
t
hod.
St
at
e
est
i
m
a
ti
on al
g
o
r
i
t
h
m
based
on
W
L
L
S
ca
n
be
con
s
i
d
ere
d
a
s
f
i
gu
re (
3
).
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Pow
e
r
Syst
em
St
at
e Est
i
m
at
i
o
n w
i
t
h
Wei
ght
e
d
Li
nea
r
Le
ast
Sq
u
a
re (
S
eye
d
Ma
hdi
M
a
haei
)
17
5
Figure
3.
W
L
L
S
state estim
a
t
ion algorithm
5.
CASE ST
UDY
Acco
r
d
i
n
g t
o
r
e
fere
nces [
1
6-
17]
WLS m
e
tho
d
i
s
fast
er t
h
an t
h
e
W
L
A
V
m
e
t
hod. S
o
i
n
t
h
i
s
case
study
WLS a
n
d
WLLS is c
o
m
p
ared. is se
lected. The
IE
EE 14 bus sy
ste
m
data for
state estim
a
tion has
p
r
esen
ted in
[5].
Th
e
o
b
j
ectiv
e
fun
c
tion
in
WLS is op
timized
after 109
iteratio
n
s
(
ε
=10
-7
). Th
e
resu
lted esti
m
a
ted
val
u
es
ha
ve
pr
esent
e
d
i
n
t
a
bl
e (
1
).
Tabl
e 1.
E
s
t
i
m
a
t
e
d
val
u
es by
WLS
o
n
IEEE
14
b
u
s sy
st
em
Actual values
Esti
m
a
t
e
d
values
Measured
values
Measure
m
ents
pu
Pu
pu
1.
06
1.
0613
1.
05
V
1
1.
01
1.
0136
1
V
3
1.
02
1.
0237
1.
04
V
5
1.
09
1.
09
1.
09
V
8
1.
056
1.
0569
1.
05
V
9
1.
057
1.
0588
1.
057
V
11
1.
05
1.
0414
1.
03
V
13
1.
5688
1.
4441
1.
4276
P
1,2
0.
4152
0.
3809
0.
4072
P
2,5
0.
4409
0.
4153
0.
4076
P
5,6
0.
1775
0.
1665
0.
1714
P
6,13
0
0
0
P
7,8
0.
0161
0.
0303
0.
0158
P
12,13
0.
0564
0.
0517
0.
0518
P
13,14
-
0
.
7091
-
0
.
6632
-
0
.
6648
P
3,2
|
ˆ
ˆ
)
ˆ
(
0
x
x
x
x
h
H
x
x
x
0
x
ma
x
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
4, No
. 2, A
p
ri
l
20
14
:
16
9 – 1
7
8
17
6
Actual values
Esti
m
a
t
e
d
values
Measured
values
Measure
m
ents
pu
Pu
pu
-
0
.
5445
-
0
.
5081
-
0
.
5043
P
4,2
-
0
.
2807
-
0
.
2544
-
0
.
2581
P
7,4
-
0
.
1608
-
0
.
1505
-
0
.
1597
P
9,4
-
0
.
073
-
0
.
0629
-
0
.
0661
P
11,6
-
0
.
0521
-
0
.
0463
-
0
.
0495
P
10,9
0.
0117
0.
0131
0.
0108
Q
2,5
-
0
.
0162
-
0
.
0147
-
0
.
015
Q
10,11
0.
0175
-
0
.
0004
0.
016
Q
13,14
0.
0223
0.
0118
0.
022
Q
5,1
0.
0302
0.
0185
0.
0288
Q
4,2
-
0
.
142
-
0
.
1251
-
0
.
1332
Q
5,4
0.
1138
0.
0956
0.
1032
Q
7,4
0.
0173
0.
0112
0.
0159
Q
9,4
-
0
.
0344
-
0
.
025
-
0
.
0313
Q
11,6
-
0
.
0235
0.
0117
-
0
.
0221
Q
12,6
-
0
.
0336
-
0
.
0491
-
0
.
0316
Q
14,9
0.
183
0.
1792
0.
1674
P
2
-
0
.
478
-
0
.
4737
-
0
.
4763
P
4
-
0
.
112
-
0
.
0991
-
0
.
1021
P
6
-
0
.
295
-
0
.
2933
-
0
.
2939
P
9
-
0
.
09
-
0
.
084
-
0
.
0833
P
10
-
0
.
061
-
0
.
0554
-
0
.
0601
P
12
0.
3086
0.
2938
0.
3042
Q
2
0.
0608
0.
0393
0.
0571
Q
3
0.
0523
0.
0554
0.
051
Q
6
0.
1762
-
0
.
1648
-
0
.
1606
Q
9
-
0
.
05
-
0
.
0478
-
0
.
048
Q
14
IEEE
14 bus s
y
ste
m
is assum
e
d with the
pre
v
io
us
para
m
e
ters for com
p
aring the
WLS a
n
d
WL
LS
esti
m
a
to
rs. The resu
lted
estimated
v
a
lu
es
with
th
e sam
e
p
r
ev
iou
s
erro
r
(
ε
=1
0
-7
) by
WLLS
are pre
s
e
n
t
e
d
i
n
tab
l
e (2
).
Tabl
e 2.
E
s
t
i
m
a
t
e
d
val
u
es by
WLLS
o
n
IEE
E
1
4
b
u
s
sy
st
e
m
Actual values
Esti
m
a
t
e
d
values
Measured
values
Measure
m
ents
pu
Pu
pu
1.
06
1.
0519
1.
05
V
1
1.
01
1.
0036
1
V
3
1.
02
1.
0281
1.
04
V
5
1.
09
1.
09
1.
09
V
8
1.
056
1.
0534
1.
05
V
9
1.
057
1.
057
1.
057
V
11
1.
05
1.
0319
1.
03
V
13
1.
5688
1.
4351
1.
4276
P
1,2
0.
4152
0.
3394
0.
4072
P
2,5
0.
4409
0.
4192
0.
4076
P
5,6
0.
1775
0.
1626
0.
1714
P
6,13
0
0
0
P
7,8
0.
0161
0.
0428
0.
0158
P
12,13
0.
0564
0.
0527
0.
0518
P
13,14
-
0
.
7091
-
0
.
7068
-
0
.
6648
P
3,2
-
0
.
5445
-
0
.
4904
-
0
.
5043
P
4,2
-
0
.
2807
-
0
.
2564
-
0
.
2581
P
7,4
-
0
.
1608
-
0
.
1486
-
0
.
1597
P
9,4
-
0
.
073
-
0
.
0627
-
0
.
0661
P
11,6
-
0
.
0521
-
0
.
0393
-
0
.
0495
P
10,9
0.
0117
-
0
.
0345
0.
0108
Q
2,5
-
0
.
0162
-
0
.
0196
-
0
.
015
Q
10,11
0.
0175
-
0
.
0109
0.
016
Q
13,14
0.
0223
0.
0663
0.
022
Q
5,1
0.
0302
0.
0573
0.
0288
Q
4,2
-
0
.
142
-
0
.
1044
-
0
.
1332
Q
5,4
0.
1138
0.
0744
0.
1032
Q
7,4
0.
0173
-
0
.
0003
0.
0159
Q
9,4
-
0
.
0344
-
0
.
0233
-
0
.
0313
Q
11,6
-
0
.
0235
0.
0397
-
0
.
0221
Q
12,6
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Pow
e
r
Syst
em
St
at
e Est
i
m
at
i
o
n w
i
t
h
Wei
ght
e
d
Li
nea
r
Le
ast
Sq
u
a
re (
S
eye
d
Ma
hdi
M
a
haei
)
17
7
Actual values
Esti
m
a
t
e
d
values
Measured
values
Measure
m
ents
pu
Pu
pu
-
0
.
0336
-
0
.
063
-
0
.
0316
Q
14,9
0.
183
0.
1756
0.
1674
P
2
-
0
.
478
-
0
.
4751
-
0
.
4763
P
4
-
0
.
112
-
0
.
0978
-
0
.
1021
P
6
-
0
.
295
-
0
.
2933
-
0
.
2939
P
9
-
0
.
09
-
0
.
0866
-
0
.
0833
P
10
-
0
.
061
-
0
.
0519
-
0
.
0601
P
12
0.
3086
0.
2939
0.
3042
Q
2
0.
0608
-
0
.
0043
0.
0571
Q
3
0.
0523
0.
0586
0.
051
Q
6
0.
1762
-
0
.
1761
-
0
.
1606
Q
9
-
0
.
05
-
0
.
0511
-
0
.
048
Q
14
The optim
u
m
resul
t
s
o
f
o
b
ject
i
v
e f
u
nct
i
o
n
an
d i
t
e
rat
i
o
n
num
ber
ha
ve
pre
s
e
n
t
e
d i
n
t
a
bl
e
(
3
).
Tabl
e
3. T
h
e
o
p
t
i
m
u
m
J(x
)
and i
t
e
rat
i
o
n
num
ber
I
t
er
ation nu
m
b
er
J(x)
fi
na
l
Esti
m
a
tor
111
11.
729
0
WLS
114
26.
276
7
WL
LS
T
h
e
o
p
t
i
mu
m
J(x
)
in
W
LLS is m
o
re th
an th
e
o
p
tim
u
m
J(x
)
i
n
WLS bu
t it sh
ou
ld b
e
no
ted th
at:
T
h
e
o
p
t
i
mu
m
J(x
)
in
W
LLS is
accepta
ble by
consider
i
n
g the
num
b
er of m
e
asurem
ents.
In
so
m
e
si
tu
ati
o
n
s
,
W
L
S estimato
r
can
no
t find
rev
e
rse
o
f
g
a
in
m
a
trix
. On
e su
ch
situ
atio
n
is wh
en
the state varia
b
les are m
o
re
than
the number of
m
easure
m
ents
or
when the m
easure
m
ents are
d
i
ffu
s
ed
.
Testin
g
t
h
e fi
rst situ
atio
n on
IEEE 14
bu
s system sh
o
e
s
WLS
is u
n
a
b
l
e in
esti
m
a
tio
n
of
states wh
ile
WLLS can
b
e
est
i
m
a
te th
e state
s
in
t
h
is situ
atio
n.
6.
CO
NCL
USI
O
N
Power system
state estim
a
tion is done in two ways
.
Th
e
fi
rst m
e
th
o
d
invo
lv
es t
h
e estimatio
n
of th
e
state b
y
m
a
th
e
m
atical
m
e
th
od
s and
th
e m
e
t
h
od
s
b
a
sed
on
in
tellig
en
t techn
i
qu
es su
ch
as
fu
zzy log
i
c or n
e
ural
n
e
two
r
k
s
.
In
tellig
en
t
m
e
th
o
d
s n
eed
less ti
me th
an
m
a
th
e
m
atical
m
e
th
o
d
s
fo
r estim
a
tin
g
bu
t th
ey are no
t
accurate s
u
fficiently. The in
telligent
m
e
thods nee
d
m
a
ny
data sets to
train whic
h data
collection and
m
odel
l
earni
n
g
w
o
ul
d be nea
r
l
y
im
possi
bl
e i
n
real
sy
st
em
s.
Am
ong m
a
t
h
em
at
i
cal
m
e
t
h
o
d
s,
WLS has
sui
t
a
bl
e
accuracy a
n
d s
p
eed tha
n
othe
rs.
In
t
h
i
s
pa
per,
a
new
est
i
m
at
or,
W
e
i
ght
e
d
Li
near
Leas
t
Sq
uare
(
W
L
L
S),
ha
s
been
p
r
o
p
o
se
d
.
Th
e
p
r
o
p
o
s
ed
esti
m
a
to
r is b
e
tter th
an
WLS
esti
m
a
to
r so
that WLS can
not fin
d
t
h
e rev
e
rse o
f
g
a
in
m
a
trix
in
so
m
e
si
tu
atio
ns th
erefo
r
e it is d
i
stu
r
b
e
d
i
n
so
m
e
si
tu
a
t
i
ons
but
W
L
LS d
o
n
’t
use
gai
n
m
a
t
r
i
x
an
d t
h
ere
f
o
r
e i
t
can estim
ate th
e states in all si
tuations
.
IEEE
14 bus s
y
ste
m
was selected for t
h
e ca
se st
udy.
State estim
a
tion is
done by W
L
S and WL
LS.
The
results s
h
ow that the
propose
d
WLL
S
estim
a
tor
has
acceptable ac
curacy a
n
d
is
able to estim
ate the
states in
all con
d
ition
s
.
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NC
ES
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Mahaei, M Tarafd
ar Hagh,
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.
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-
8.
BIOGRAP
HI
ES OF
AUTH
ORS
S
e
yyed
Meh
d
i
Mah
a
ei
was bo
rn in Tabriz,
Iran, in
1984.
He
received
the B.Sc. and
M. Sc.
degrees (with First Class Honors) in power elect
rical
engineerin
g from University
of Applied
Science and
Technolog
y
,
Tabriz, Iran, in 2007
,
a
nd Islamic Azad
University
, Ahar
, Iran
,
in 2011,
respectively
.
Also he is project supervisor in
Azerbai
j
an R
e
gi
onal E
l
ec
tri
c
Co
m
p
an
y
,
Tabr
iz,
Iran. He h
a
s published more th
an 40 papers in
power s
y
stems and power electronics related
topics in journals and conferen
c
e
s
.
His
res
earch
interes
t
s
includ
e power s
y
s
t
em
operation and
power s
y
stem planning.
Mohammad Reza Navay
i
was born in Tabriz,
Iran, in 1976
. He received
the B
.
Sc. degr
ee in
power electr
i
cal
engineering fro
m University
of
Ta
briz,
Tabr
iz, I
r
an,
in 2000. Also he is project
s
upervis
or in Azerbai
j
an Region
al Ele
c
tr
ic Com
p
an
y, Tabr
iz
, Ira
n. His
res
earch interes
t
s
includ
e
power s
y
s
t
em
op
erat
ion an
d
pow
er s
y
s
t
em planning.
Evaluation Warning : The document was created with Spire.PDF for Python.