Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
Vol
.
5
,
No
. 3,
J
une
2
0
1
5
,
pp
. 51
8~
52
4
I
S
SN
: 208
8-8
7
0
8
5
18
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
A Spanning Tree Approach in
Placing Multi-channel and
Minimum Channel PMU’s for Power System Observability
Sri
h
ari
Ma
nd
av
a,
V
a
ni
shr
e
e
J,
R
a
mes
h
V
School of
Electr
i
cal Engin
eer
ing,
VIT University
,
India
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Nov 18, 2014
Rev
i
sed
Ap
r 8, 20
15
Accepted Apr 25, 2015
S
y
nchroni
zed ph
as
or m
eas
urem
ents
have b
ecom
e
the m
eas
urem
en
t te
chniqu
e
of choic
e
for el
e
c
tri
c
power s
y
st
e
m
s. The
y
provid
e
positive sequ
e
n
ce volt
a
ge
and curren
t
measurements s
y
n
c
hroni
zed to w
ithin a microsecond. Th
e
objec
tive is
to u
s
e the s
p
anning tree appro
ach an
d tree s
ear
ch te
c
hnique for
optim
al pl
ac
em
ent of m
u
ltichannel and
minimum channel sy
nchron
ized
phasor measurement units (PMUs) in or
der to
have full observability
of
Power S
y
stem.
The nov
el
concept of d
e
pt
h o
f
observability
is
used and its
im
pact on the n
u
m
b
er of P
M
U
plac
em
ents
is
explain
e
d. Th
e s
p
anning tre
e
approach is
us
ed
for the power sys
t
em
graphs
an
d a tree s
ear
ch t
echniqu
e is
used for find
ing
the op
timal
location of
P
M
U
s
.
T
h
i
s
i
s
t
e
s
t
e
d
o
n
I
E
E
E
-
1
4
a
n
d
IEEE-30 bus s
y
stem. The same techniqu
e is modified to optimally
p
l
ace
m
i
nim
u
m channel P
M
Us on the sam
e
IEEE-14
and IEEE
-30 bus sy
stem
s.
Matlab
tool
has
been used
for
ful
f
illing
th
e obj
ec
t
i
ve.
Keyword:
M
i
n
i
mu
m c
h
a
n
n
e
l
Mu
ltich
a
nn
el
Ob
serv
ab
ility
PMU
Spa
nni
ng
t
r
ee
Tree sea
r
ch
Copyright ©
201
5 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
:
Srih
ari Mand
av
a,
School
of Elec
trical Engineering
VIT Un
iv
ersity, Ind
i
a
Em
a
il: man
d
a
vasrih
ari@yahoo
.co
.
in
1.
INTRODUCTION
Th
e
p
r
esen
t
wo
rl
d
is m
o
stly
u
s
ing
SC
ADA (Su
p
e
rv
isory Co
n
t
ro
l an
d
Data Acqu
isition
system
) fo
r
m
easuri
n
g
po
wer sy
st
em
dat
a
for
k
n
o
wi
ng
t
h
e st
at
e est
i
m
at
i
on. The
d
a
t
a
m
easured
f
r
om
di
ffe
rent
part
s
o
f
powe
r syste
m
network
by SC
ADA are
not s
y
nchronized a
nd
provide
a
n
inaccurate state of the system
during
the dy
nam
i
c events
on the syst
e
m
like load change
.
Po
sitiv
e sequ
en
ce m
easu
r
emen
ts [1
] are used
to d
e
fin
e
the state o
f
th
e
po
wer system
at
an
y instan
t.
Th
e
u
s
e of the po
sitiv
e sequ
en
ce vo
ltag
e
of th
e n
e
t
w
ork is ex
p
l
ain
e
d
well in
[2
]. Syn
c
h
r
on
ized ph
asor
measu
r
em
en
t u
n
its m
easu
r
es po
sitiv
e seq
u
e
n
ce
vo
ltage
s and
cu
rren
ts syn
c
h
r
on
ized
in
less th
an a
microsecond
with the use of Global
Positioning System
(GPS
). T
h
e measured sync
hronized data is the
n
sam
p
l
e
d and
p
r
oces
sed
usi
n
g
t
h
e t
ech
ni
q
u
es
de
vel
o
ped
f
o
r
com
put
er
rel
a
y
i
ng a
ppl
i
cat
i
o
ns
[3]
.
I
n
a
ddi
t
i
on
t
o
these, t
h
e PM
Us als
o
m
easure
local
fre
quency, rate
o
f
cha
nge
of fr
eque
ncy
whi
c
h hel
p
s
t
o
m
easure
harm
onics,
ne
gative a
n
d zero se
que
nce
quantities. Recent
spate of spec
tacular blac
kouts on
powe
r syste
m
s
t
h
r
o
u
g
h
o
u
t
t
h
e
wo
rl
d
has
p
r
ov
i
d
ed a
n
a
d
ded
i
m
pet
u
s t
o
wi
de
scal
e de
pl
oy
m
e
nt
o
f
PM
Us.
Th
e m
a
in
adv
a
n
t
ag
e
of using PMUs i
n
p
o
wer system
s is t
h
eir cap
a
b
ility
o
f
d
i
rectly
m
e
asu
r
i
n
g th
e
state of the sys
t
e
m
. Therefore
,
sync
hronize
d
m
easurem
ents
p
r
ov
id
ed
b
y
PMU
s
is gr
eat help
to
pow
er
syste
m
co
n
t
ro
l systems esp
ecially in in
terco
n
n
ected
n
e
two
r
k
s
.
Wh
ile PMUs are no
t yet fou
nd in
ev
ery
sub
s
t
a
tio
n
,
th
eir u
tilizatio
n
in
su
b
s
tation
s
fo
r
p
r
o
t
ectio
n
and
con
t
ro
l fun
c
tio
ns is rap
i
d
l
y in
creasi
n
g. As they beco
m
e
avai
l
a
bl
e i
n
l
a
rge
n
u
m
b
ers t
h
ey
can p
r
ovi
de val
u
abl
e
i
n
f
o
rm
at
i
on fo
r ene
r
gy
m
a
nagem
e
nt
sy
st
em
applications as well. There wi
ll be a great improvem
ent
i
n
the st
at
e est
i
m
a
t
i
on of
po
wer s
y
st
em
net
w
o
r
k
wi
t
h
the im
ple
m
entation of these
PMUs. T
h
e great adva
ntage
of PM
U is tha
t
,
the PMU yields the sync
hronized
measurem
ents across the
power syst
em
. Initial work
on PMU placem
ent is
base
d on the assum
p
ti
on t
h
at
PMUs
will h
a
ve in
fi
n
ite nu
m
b
er of ch
an
n
e
ls t
o
m
o
n
ito
r
ph
aso
r
curren
t
s
o
f
all b
r
an
ch
es
th
at
are in
ci
d
e
n
t
to th
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 3,
J
u
ne 2
0
1
5
:
51
8 – 5
2
4
51
9
b
u
s
wh
ere a PMU
will b
e
i
n
stalled
[3
,
4
]
.
Wh
ile t
h
er
e are m
a
n
u
f
acturers th
at
p
r
od
u
c
e PMUs wit
h
sev
e
ral
chan
nel
s
t
o
m
e
asure
p
h
as
or
c
u
r
r
ent
s
a
n
d
vol
t
a
ges, t
h
e n
u
m
b
er
of ch
ann
e
ls is typ
i
cally li
mited
.
Also, no
te th
at
PMUs ca
pture
sam
p
les of
phase
voltages
and currents
t
h
at are
recei
ve
d
from
the ins
t
rum
e
nt trans
f
orm
e
rs
co
nn
ected
t
o
a
b
u
s
or a breaker in
th
e su
bstatio
n
.
Th
e sam
p
led
th
ree
p
h
a
se sig
n
a
ls are con
v
e
rted
i
n
to
positiv
e
sequence
pha
s
ors
at re
gul
ar intervals a
nd t
h
e
n
tele
meter to
th
e p
h
a
sor
d
a
ta co
ncen
trat
o
r
s. Ph
asor
measurem
ents are
used by
va
rious a
pplication functions
a
t
energy c
ont
rol cente
rs. One suc
h
a
p
plication is
state esti
m
a
tio
n
wh
ich
n
o
t
only p
r
ov
ides th
e b
e
st estim
ate
o
f
t
h
e system
state b
u
t
also
acts as filter for
g
r
o
ss
errors
in a
n
al
og and
digita
l
m
easurem
ents. Moreover
, res
u
lts of st
ate estim
a
tion are
use
d
by
m
a
n
y
appl
i
cat
i
o
ns as
i
n
p
u
t
s
an
d t
h
eref
ore
have a
si
gni
fi
ca
nt
i
m
pact
on t
h
e ove
ral
l
per
f
o
r
m
a
nce of t
h
e
ener
gy
m
a
nagem
e
nt
sy
st
em
s. Som
e
of t
h
e a
ppl
i
c
at
i
ons t
h
at
rel
y
on st
at
e est
i
m
a
ti
on res
u
l
t
s
i
n
cl
ude
real
-t
im
e
co
n
ting
e
n
c
y an
alysis, vo
ltage stab
ility ass
e
ssm
en
t, tran
sien
t stab
ility
assessm
en
t, real-ti
m
e p
o
w
er flo
w
,
security constrained
optim
a
l
po
we
r fl
ow, load
forecasting
as we
ll as t
h
e
market applications.
1.
1
Co
mplete Observ
ability
Sy
st
em
i
s
sai
d
t
o
be
com
p
l
e
t
e
l
y
obse
r
va
bl
e i
f
vol
t
a
ge
an
d c
u
r
r
ent
p
h
as
or
o
f
e
v
ery
bu
s i
n
net
w
or
k i
s
kn
o
w
n
.
Fi
g
u
re
1 sh
ows a com
p
l
e
t
e
l
y
obser
ve
d sy
st
em
. The
vol
t
a
ge at
b
u
se
s B
,
E and H ar
e di
rect
l
y
m
easure
d
by
PM
U
-
1,
P
M
U-
2 a
n
d
PM
U-
3
res
p
ect
i
v
e
l
y
,
whi
l
e
v
o
l
t
a
ges at
b
u
ses
A
,
C
,
D,
F,
G
a
n
d
I
can
be
cal
cul
a
t
e
d
usi
n
g t
h
e
m
e
asure
d
v
o
l
t
a
ges
an
d l
i
n
e c
u
r
r
e
nt
s. Let
b
u
se
s B
,
E,
H a
r
e
defi
ned
as P
M
U b
u
ses
w
h
ere b
u
s
v
o
ltag
e
s and
lin
e cu
rren
ts are d
i
rectly
m
eas
ure
d
. Buse
s A, C, D, F, G and I are de
fine
d as calculated
bus
es
because thei
r
voltages a
r
e cal
culated
from
the PM
U m
easure
m
ents of the
buses linke
d to them
.
Fig
u
re
1
.
Co
mp
lete ob
serv
ab
i
lity u
s
in
g
PMU’s
The p
r
obl
em
of st
rat
e
gi
c pl
ac
em
ent
of PM
Us i
n
po
wer s
y
st
em
t
o
have
a ful
l
y
obs
er
v
a
bl
e sy
st
em
,
has
receive
d a
lot of attention
from
researc
h
ers
.
[4],
[5] inve
stigate the
problem
of optim
a
l
placem
e
n
t of
PM
Us t
o
obs
e
r
ve t
h
e
net
w
o
r
k
usi
n
g i
n
t
e
g
e
r p
r
o
g
ram
m
ing
.
I
n
[
6
]
,
aut
h
o
r
s
pr
op
ose
an ex
ha
ust
i
v
e
search
approach t
o
de
termine the minim
u
m
nu
m
b
er and optim
al
placem
ent of PM
Us
for
state esti
m
a
tion. Although
t
h
e p
r
op
ose
d
m
e
t
hod i
n
t
h
i
s
pa
per y
i
el
ds
t
h
e gl
obal
o
p
t
i
m
al
sol
u
t
i
o
n
,
t
h
e m
e
t
hod i
s
y
e
t
com
put
at
ional
l
y
in
ten
s
iv
e
fo
r l
a
rg
e
syste
m
s. In
[3
], au
t
h
ors p
r
op
ose a m
e
th
od
to
i
d
en
tify th
e strateg
i
c lo
catio
n fo
r
PM
U
installation in t
h
e system
based on t
h
e dual s
earch m
e
t
hod
usi
n
g a m
odi
fi
ed bi
sect
i
n
g se
arch a
n
d a si
m
u
l
a
t
e
d
annealing m
e
thod. T
h
e m
o
dified bisecting search
fi
x
e
s
th
e nu
m
b
er o
f
PMUs fo
r
wh
ich
th
e sim
u
lated
anneal
i
n
g
bas
e
d m
e
t
hod
t
h
en at
t
e
m
p
t
s
to
fi
n
d
a
pl
ac
em
ent
set
t
h
a
t
m
a
kes t
h
e
sy
st
em
t
opol
o
g
i
cal
l
y
o
b
s
erv
a
b
l
e.
A g
r
aph
th
eoretic ap
pro
a
ch
fo
r
p
l
acin
g
PM
Us
b
a
sed
on
in
com
p
le
te o
b
s
ervab
ility is
p
r
opo
sed
in
[7]
w
h
er
e si
m
u
l
a
t
e
d an
neal
i
ng m
e
t
hod i
s
use
d
t
o
s
o
l
v
e
t
h
e p
r
agm
a
t
i
c
com
m
uni
cat
ion
-
c
onst
r
ai
ne
d
PM
U
placem
ent problem
.
In [8] a
u
thors
re
prese
n
t a m
e
thod
t
o
identify the
optim
a
l pl
acem
e
nt of PM
U
for powe
r
sy
st
em
st
at
e est
i
m
a
t
i
on base
d
on t
h
e m
i
nimum
condi
t
i
on
num
b
er of t
h
e norm
alized
m
e
a
s
urem
ent m
a
trix.
[9]
Sh
ows
a t
e
c
h
n
i
que
o
f
i
d
e
n
t
i
f
y
i
ng t
h
e
opt
i
m
al
PM
U
pl
ace
m
e
nt
usi
n
g t
h
e
ge
net
i
c
al
g
o
ri
t
h
m
(GA
)
.
A
u
t
h
o
r
s i
n
[10], use t
h
e particle swarm
optim
i
zation in power syste
m
,
to obtain
t
h
e optim
a
l
PMU placem
ent for full
o
b
s
erv
a
b
ility o
f
th
e system
. In
[11
]
a b
i
n
a
ry p
a
rticle swarm o
p
t
i
m
izatio
n
b
a
sed
m
e
th
od
is u
s
ed
to
m
i
n
i
mize
the num
b
er of re
quire
d
PMUs and
maxim
i
ze
th
e
m
easurem
ent redunda
n
cy. In case of syste
m
I
PMU
H
G
D
PMU2
E
F
A
B
C
PMU1
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A Spa
nn
ing
Tree App
r
o
a
c
h
i
n
Pla
c
ing
Mu
lti-ch
ann
el
&
Min
i
mum ch
ann
el
PMU’s fo
r …
(S
riha
ri Man
dava
)
52
0
unobserva
b
ility, a
m
e
thod to test the sys
t
em
observability
, as well as identifying the
obs
ervable islands is
p
r
esen
ted in
[12
]
.
Although, m
a
ny resea
r
chers
ha
ve investigated
the
proble
m
of strate
gic placem
ent of PMUs
on
powe
r system
s
,
to ac
hieve
full obse
r
vability, all their st
udies is base
d
on a
n
unrealistic assum
p
tion that
each
PM
U i
s
ca
pa
bl
e of
m
easuri
n
g
t
h
e
phas
o
r
vol
t
a
ge o
f
a
b
u
s
a
n
d
the phas
or curr
ent
of all branc
h
es i
n
cide
nt t
o
that bus
. Since
num
ber of bra
n
che
s
incide
nt to each bu
s is diffe
re
nt, sol
v
ing the
optim
i
z
a
tion problem
base
d
on the ass
u
m
p
tion t
h
at each
PMU can m
easure phas
or curre
nt of all branc
h
es incide
nt to t
h
e
bus
where
PMU
is in
stalled
is
m
o
re o
f
an
i
d
ealistic assu
m
p
t
i
o
n
.
In th
is
wo
rk
,
a n
e
w typ
e
o
f
PMU is co
nsid
ered
.
Th
is typ
e
o
f
PM
U is cap
ab
le
o
f
measu
r
i
n
g th
e
pha
so
r v
o
l
t
a
ge
of t
h
e
bu
s, as
wel
l
as t
h
e p
h
a
s
or c
u
r
r
e
n
t
of
onl
y
som
e
bra
n
che
s
(i
.e.
de
p
e
nd
s u
p
o
n
t
h
e
num
ber
o
f
ch
an
n
e
ls
wi
th
PMU) in
ciden
t
to
th
e
bu
s
wh
ere PM
U
is in
stalled
,
th
ese typ
e
s of
PM
Us will b
e
referred
to
as “min
i
m
u
m
ch
ann
e
l” PM
Us and
n
u
m
b
e
r of th
em
are alread
y in
st
alled
in
u
tility co
m
p
an
ies syste
m
.
Diffe
re
nt
m
e
thods a
r
e us
ed
by researche
r
s t
o
test the
observability of the
system
.
List of the
m
o
st com
m
only
use
d
m
e
thod
for obse
rva
b
ility check
in
power system
s is as follows:
Num
e
ri
cal
m
e
tho
d
base
d o
n
n
odal
va
ri
abl
e
f
o
rm
ul
at
i
o
n
Num
e
ri
cal
m
e
tho
d
base
d o
n
b
r
anc
h
vari
a
b
l
e
fo
rm
ul
at
i
on
Top
o
l
o
g
i
cal
o
b
serv
ab
ility ch
eck
m
e
th
o
d
In th
is
p
a
per, th
e sp
an
n
i
n
g
tree app
r
o
a
ch
is
u
s
ed
to ch
eck
t
h
e
o
b
serv
ab
ility o
f
th
e
system
wh
ich
co
m
e
s und
er
th
e th
ird
m
e
th
od
m
e
n
tio
n
e
d
ab
ov
e.
1.
2
Tree Searc
h
P
l
acement Tec
hnique
The objective is to place PMUs suc
h
that the entire syste
m
is obser
va
ble.
The envisione
d
technique
consists of a
se
ries of “wal
ks”
along bra
n
che
s
of a
spa
n
ning tree and que
r
ies are
m
a
de on each node if a
PMU
placem
ent is possible.
Th
e search
p
r
oced
ure starts at
a roo
t
nod
e and
g
o
e
s
d
o
wn
the tree
u
n
til it reach
es a term
in
al n
o
d
e
.
At
t
h
i
s
p
o
i
n
t
,
i
t
bac
k
t
r
ac
ks a
n
d sea
r
che
s
f
o
r
anot
her
r
out
e.
Ide
n
t
i
f
y
a
wal
k
by
a
b
u
s
pai
r
f
r
om
bus
-t
o
b
u
s
.
A roo
t
nod
e is sp
ecified
arb
itrarily
; th
e search
fo
r
PMU lo
cation
s
is termin
ated
wh
en th
e
p
r
o
c
ed
ure
retu
rn
s to
t
h
is ro
o
t
no
d
e
.
At th
is tim
e, th
e sp
ann
i
ng
tree h
a
s b
e
en
fu
lly
search
ed
.
The resulting PMU placem
e
n
t
strategy gua
r
antees
t
h
e
e
x
istence
of a
com
p
le
tely observable c
o
ndition for the
sp
an
n
i
n
g
tree.
It lik
ewise
gu
aran
tees t
h
e com
p
le
te o
b
s
erv
a
b
ility fo
r t
h
e
paren
t
g
r
aph
.
2.
ILLUSTR
A
T
I
ON OF SP
A
NNI
NG
TRE
E
APP
R
O
A
C
H
Th
e PMU
p
l
ace
m
en
t tech
n
i
qu
e (o
r altern
ativ
ely th
e tree search) is illu
strated
first b
e
fore th
e form
al
al
go
ri
t
h
m
i
s
present
e
d.
C
o
ns
i
d
er t
h
e s
p
a
n
n
i
ng t
r
ee i
n
Fi
g
u
re
2
t
h
at
i
s
c
o
m
posed
o
f
1
4
no
des
an
d
wi
t
h
20
b
r
an
ch
e
s
.
Jum
p
-start the
placem
ent proc
ess by a
r
bitrari
l
y designating
Node
5 a
s
the
root
node
.
Fi
rst
PM
U
pl
acem
e
nt
sho
u
l
d
be o
n
e b
u
s a
w
ay
fr
om
t
h
e root
no
de
5 at
No
de
6 so as t
o
o
b
se
rve t
h
e r
o
o
t
n
o
d
e
. Th
en
, we tak
e
a ser
i
es
o
f
fo
rw
ard
m
o
v
e
s along
a cho
s
en
p
a
t
h
d
e
f
i
n
e
d b
y
th
e nodal seq
u
e
n
ce
6
-
1
1
-
10-9-7-8 eac
h t
i
m
e
querying
for possible
PM
U
placem
ent.
The
next l
ogic
a
l placem
ent
will be at Node 9,
whic
h
m
a
kes nodes 10,7obse
rva
b
le with
de
pth-of-ze
r
o
unobservability .Note t
h
at the PMUs
are physically separated 3
buses
from each other along the chosen
pat
h
.
We can
no
w po
se t
h
e
fo
llowi
n
g
ru
le th
at:
giv
e
n
a
d
e
sired
d
e
p
t
h
o
f
uno
bserv
a
b
ility,
th
e n
e
x
t
cand
i
d
a
te
PMU
placem
e
n
t node m
u
st be of
distance
[13].
Dp
=U+
3
Whe
r
e
- D
p
is
the
num
b
er of buses
a
w
ay
from
the curre
nt PMU placem
ent the
next
one
will be;
-U is
th
e d
e
sired
lev
e
l of u
nobserv
a
b
ility
(For co
m
p
lete o
b
s
erv
a
b
ility U
=
0)
The
ne
xt m
ove is to t
h
e term
inal node
8
but
PMU at
this
place observes
only
one
bus
s
o
the
ne
xt PM
U
lo
catio
n
will be at lo
cation
7
.
B
y
pl
aci
ng PM
Us 6,
9, 7 t
h
e
ent
i
r
e spa
n
5,
6, 1
1
,
10
, 9,
7,
8 obse
r
vabl
e
and al
s
o
t
h
e n
ode
s co
nnect
e
d
t
o
the PM
U
buse
s
(6,
9, 7)
i.e.,12,
13,
14, 4. Is al
so
observable. Bac
k
trac
k is
done
until it reach a
node
whe
r
e
a
n
u
n
o
b
s
erva
bl
e pat
h
c
a
n be
t
a
ke
n.
The
n
m
ove t
o
no
des 1
,
2
,
3
whe
r
e he
re 1
act
s as sub r
oot
n
o
d
e an
d
we agai
n ap
pl
y
span
ni
n
g
t
r
e
e
al
go
ri
t
h
m
by
pl
aci
ng PM
U at
no
de
di
rect
l
y
con
n
ect
ed
t
o
r
o
ot
n
ode
i
.
e.,
2
whi
c
h m
a
kes u
n
o
b
se
rva
b
l
e
p
a
t
h
1,
2,
3
o
b
se
rva
b
l
e
.
So by
pl
aci
ng
PM
U’
S
at
2, 6,
7,
9
ent
i
r
e
net
w
o
r
k
i
s
obs
er
v
a
bl
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 3,
J
u
ne 2
0
1
5
:
51
8 – 5
2
4
52
1
To ens
u
re m
i
ni
m
u
m
num
ber of PM
U pl
a
c
em
ent
s
, i
t
i
s
necessa
ry
t
o
p
e
rf
orm
anot
he
r searc
h
fr
om
a
di
ffe
re
nt
r
oot
n
ode
s i
.
e.
, f
r
o
m
1 t
o
1
4
.
Co
m
p
are all solu
tio
n
s
fo
r on
e
with
least
n
u
m
b
e
r of PM
Us
Figure 2.
IEE
E
-
14 bus
system
3.
M
I
NIMUM
CHAN
N
EL PMU
Manufacturer
specifies cha
n
nel num
ber of PMU i.
e. ei
ther single channel or two channel or
m
u
l
tich
a
n
n
e
l.
Th
e co
nv
en
tion
a
l tech
n
i
q
u
e
s
assu
m
e
PMU as
m
u
lti-ch
annel PMU bu
t these tech
n
i
q
u
e
s can
no
t
be a
p
pl
i
e
d
whe
n
t
h
e
r
e i
s
o
n
l
y
si
ngl
e c
h
a
nnel
or
t
w
o c
h
an
nel
PM
Us
are
p
r
e
s
ent
.
Su
p
pose
t
h
ere
are
o
n
l
y
s
i
ngl
e
ch
ann
e
l
o
r
two
ch
ann
e
l PM
Us are
p
r
esen
t
d
e
p
t
h
o
f
ob
serv
ab
ility co
nd
itio
n
an
d ob
serv
ab
le
bu
ses m
a
trix
in
program
shoul
d
be c
h
ange
d.
In this case
PMUs
placem
en
t in net
w
ork is
obtaine
d a
s
fol
l
ows.
In case there i
s
on
ly sing
le ch
ann
e
l the
d
e
pth
of
ob
serv
abilit
y co
nd
itio
n
Dp
=U+3
ch
an
g
e
s to Dp
=U+2 as s
h
ow
n i
n
4
.
1
.
M
o
re
ove
r,
fo
r si
n
g
l
e
chan
nel
pl
ac
i
ng PM
U st
art
s
fr
om
root
n
o
d
e i
t
s
el
f as i
t
pro
v
i
d
e
l
e
ss num
ber of
PM
Us t
h
an pl
aci
ng at
one
b
u
s away
fr
om
ro
ot
no
de. I
f
t
h
ere i
s
o
n
l
y
singl
e cha
n
nel
PM
U,
o
b
s
erv
a
b
l
e
b
u
ses in
th
is con
d
ition
is ju
st
th
e en
tire span
of trav
erse, b
u
t
n
o
t
the
o
t
h
e
r bu
ses t
h
at also
connected to PMU buses
as t
h
at
i
n
ca
se
of
m
u
lt
i
c
hannel
.
In
case th
ere i
s
on
ly two ch
an
n
e
l t
h
e
d
e
p
t
h o
f
ob
serv
ab
ility co
nd
itio
n
Dp
=U+3
is
same. If th
ere is
o
n
l
y two
ch
ann
e
l PMU, ob
serv
ab
le bu
ses in th
is co
n
d
ition
is j
u
st th
e en
tire sp
an
of traverse, bu
t no
t th
e o
t
h
e
r
bus
es that als
o
connected to PMU buse
s
as t
h
at
i
n
ca
se
of
m
u
lt
i
c
hannel
.
3.
1
Illus
t
ration of Spanning Tree A
ppr
oach to Si
ngle
Channel P
M
U
Th
e PMU
p
l
ace
m
en
t tech
n
i
qu
e (o
r altern
ativ
ely th
e tree search) is illu
strated
first b
e
fore th
e form
al
al
go
ri
t
h
m
i
s
present
e
d.
C
o
ns
i
d
er t
h
e s
p
a
n
n
i
ng t
r
ee i
n
Fi
g
u
re
2
t
h
at
i
s
c
o
m
posed
o
f
1
4
no
des
an
d
wi
t
h
20
b
r
an
ch
e
s
.
Jum
p
-start the
placem
ent proc
ess by a
r
bitrari
l
y designating
Node
12 as
the
root
node
.
For single c
h
a
nnel
PMU
first
PMU
placem
e
n
t should
be
root node
12. T
h
en,
we ta
ke a s
e
ries of
forwa
r
d
m
oves al
on
g a chose
n
pat
h
d
e
fi
ne
d by
t
h
e
no
dal
seq
u
e
n
c
e
12
, 1
3
,
14
,
9,
7, 8 eac
h t
i
m
e query
i
n
g f
o
r
pos
sible PM
U
placem
ent.
We can
n
o
w
po
se th
e fo
llowin
g
ru
le th
at: g
i
v
e
n
a
d
e
sired
d
e
p
t
h
of uno
b
s
erv
a
b
ility , th
e n
e
x
t
cand
i
d
a
t
e
PMU
placem
e
n
t node m
u
st be of
distance
[2]
D
p
=U+
2
Whe
r
e
- D
p
is the
num
b
er of buses
a
w
ay from
the curre
nt PMU placem
ent the
next
one
will be;
- U is th
e d
e
sired
lev
e
l
of
u
nob
serv
ab
ility (Fo
r
co
m
p
lete o
b
serv
ab
ility U=0
)
The ne
xt
m
o
ve is to place P
M
U at bus 14 accordi
n
g to D
p
=U+2
ru
le an
d
n
e
x
t
will b
e
7
b
y
app
l
ying
sam
e
ru
le. No
te th
at PM
U’s are sep
a
rated
b
y
two bu
ses.
By p
l
acin
g
PMU’s
12
,
1
4
,
7
th
e en
tire span
12
, 13
,
14
,
9
,
7
,
8
o
b
s
erv
a
b
l
e. Back
t
r
ack u
n
til we
reach
a
no
de
w
h
ere
an
un
o
b
ser
v
a
b
l
e
p
a
t
h
can
be
t
a
ke
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A Spa
nn
ing
Tree App
r
o
a
c
h
i
n
Pla
c
ing
Mu
lti-ch
ann
el
&
Min
i
mum ch
ann
el
PMU’s fo
r …
(S
riha
ri Man
dava
)
52
2
The
n
m
ove t
o
no
des 1,
2, 3
,
4, 5
,
6, 1
1
,
10
whe
r
e he
re
1 act
s as sub
root
n
o
d
e an
d we agai
n ap
pl
y
spa
nni
ng tree a
l
gorithm
by pl
acing
PMU at
sub root
node
and appl
ying a
b
ove
rule PM
U
placem
ent will
be
at
3, 5,
1
1
.
So
by
pl
aci
ng
PM
Us at
1,
3
,
5,
7,
1
1
,
1
2
,
1
4
ent
i
r
e
net
w
or
k
i
s
o
b
se
rva
b
l
e
.
To ens
u
re m
i
ni
m
u
m
num
ber of PM
U pl
a
c
em
ent
s
, i
t
i
s
necessa
ry
t
o
p
e
rf
orm
anot
he
r searc
h
fr
om
a
di
ffe
re
nt
r
oot
n
ode
s i
.
e.
, f
r
o
m
1 t
o
1
4
.
Co
m
p
are all solu
tio
n
s
fo
r on
e
with
least
n
u
m
b
e
r of PM
Us.
For t
w
o c
h
annel PMU placement will be a
t
locations
as give
n by Dp=
u
+3. Only buses prese
n
t i
n
spa
n
w
h
i
c
h
has
been t
r
ave
r
se
d
i
s
t
a
ken as
ob
s
e
rva
b
l
e
b
u
ses
but
not
bu
ses l
i
nke
d t
o
PM
U
bus
es as t
h
at
i
n
case
o
f
m
u
ltich
a
n
n
e
l. Th
e sam
e
ap
p
r
o
a
ch
is ap
p
l
i
e
d
to IEEE-30
b
u
s
system
als
o
.
4.
RESULTS
The s
p
a
nni
ng
t
r
ee a
p
p
r
oac
h
i
s
fi
rst
a
ppl
i
e
d t
o
m
u
lt
i-ch
ann
e
l
PMU an
d th
en
it is ap
p
lied to
min
i
m
u
m
channel PMU i
.
e to one cha
n
nel PMU a
nd t
w
o c
h
a
nne
ls P
M
U for
optim
um
place
m
e
nt
of PM
Us on IEEE-14
bus a
n
d IEEE
-
30
bu
s net
w
or
ks. T
h
e o
p
t
i
m
u
m
l
o
cat
i
on of
PM
Us i
s
pr
o
v
i
d
ed i
n
t
h
e t
a
bl
e 1 fo
r 1
4
b
u
s
sy
st
em
.
From
t
h
e resul
t
s prese
n
t
e
d i
n
t
h
e t
a
bl
e 1 bel
o
w
,
i
t
can be
said
th
at th
e n
u
m
b
er o
f
PMU lo
catio
ns in
a network
with
m
u
lt
ich
a
nn
el PMU is les
s
th
an
min
i
m
u
m ch
an
n
e
l PMU. Fo
r ex
am
p
l
e th
e n
u
m
b
e
r of PMU lo
cation
s
fo
r
IEEE-14
bu
s network
with
mu
ltich
a
nn
el
is
4
i.e. at b
u
s
es 2
,
6, 7
and
9
wh
ere as with
2
-
ch
ann
e
l PMU, t
h
e
n
u
m
b
e
r
o
f
PM
U lo
cation
s
are 5
i.e. at
b
u
s
es
2
,
4
,
6
,
9
an
d
1
3
fo
r co
m
p
let
e
o
b
serv
ab
ility o
f
t
h
e system
.
Also
th
e n
u
m
b
e
r o
f
PMU lo
catio
ns req
u
i
red
fo
r a g
i
v
e
n
n
e
t
w
ork will d
ecrease with
th
e in
crease in
th
e n
u
m
b
e
r o
f
chan
nel
s
of
P
M
U.
W
i
t
h
1-c
h
an
nel
PM
U,
f
o
r a
n
IEEE
-
14
bus
net
w
o
r
k t
h
e n
u
m
b
er o
f
P
M
U l
o
cat
i
o
ns
are 8
i
.
e
.
at
buses 1,
3, 5
,
7, 9,
11
, 12 a
nd
14 w
h
e
r
e as wi
t
h
2-c
h
an
ne
l
PM
U t
h
e nu
m
b
er of PM
U l
o
cat
i
ons are
5
i
.
e. at
bus
es 2, 4,
6
,
9
an
d 13
.
Fro
m
th
e resu
lts sh
own
in
the tab
l
e 1
,
it ca
n
b
e
ob
serv
ed
th
at th
ere are m
o
re th
an
so
lutio
n
for the
o
p
tim
u
m
lo
cati
o
n
s
o
f
PMUs fo
r th
e IEEE-14 b
u
s
and
IEEE-3
0
b
u
s
n
e
two
r
k
s
fo
r m
u
lti-ch
an
n
e
l and
m
i
n
i
m
u
m
ch
ann
e
l i.e.1-ch
ann
e
l an
d 2-ch
ann
e
l PM
Us. All th
e op
tim
u
m
so
lu
tio
n
s
are in bo
ld.
So
, th
e
PMUs can
b
e
lo
cated
w
ith
an
y o
n
e
o
f
th
e so
lu
tion
and
to
co
nsid
er
w
h
ich so
lu
tion
is b
e
st d
e
p
e
nd
s
u
pon th
e b
u
s
es
w
h
i
c
h
ar
e
i
m
p
o
r
tan
t
fo
r
o
b
s
erv
a
tio
n, the av
ailab
ility
an
d
t
h
e co
st of PMUs. In
ord
e
r to
op
timi
ze th
e co
st, the o
n
e
channel PMU
can re
place the
two cha
n
nels PMU in som
e
cases without the loss of
com
p
lete obse
r
vabi
lity of
t
h
e sy
st
em
. Th
i
s
can be e
x
pl
ai
ned
by
t
a
ki
n
g
t
h
e r
oot
no
de
3 f
o
r t
h
e IE
EE
-1
4 b
u
s
net
w
or
k wi
t
h
t
w
o c
h
a
nnel
s
PMU. Fro
m
ta
b
l
e 1
,
with
ro
o
t
n
o
d
e
3
th
e
op
ti
m
a
l p
l
ace
men
t
o
f
PM
U for com
p
le
te o
b
s
erv
a
b
ility o
f
th
e syste
m
i
s
2,
4,
7,
1
1
a
nd
1
3
bus
.
Her
e
, i
t
i
s
con
s
i
d
e
r
ed t
h
at
al
l
t
h
e
bu
ses i
.
e.
2,
4
,
7,
1
1
an
d
1
3
,
t
h
e PM
Us
are
of t
w
o
ch
ann
e
ls. Th
e
o
b
s
erv
a
b
ility of each
b
u
s
is as sho
w
n
i
n
tab
l
e 2
.
From
t
a
bl
e 2, i
t
i
s
obse
r
ve
d t
h
at
t
h
e b
u
s
9 i
s
obs
er
ved
by
t
w
o
PM
Us
pl
ac
ed at
b
u
ses
4 a
nd
7.
S
o
, t
h
e
t
w
o c
h
a
nnel
s
PM
U at
b
u
s
2
o
r
at
bu
s
7 ca
n
be
repl
ace
d
by
o
n
e
cha
n
ne
l
PM
U s
o
t
h
at
t
h
e t
o
t
a
l
cost
can
be
red
u
ce
d. T
h
e sam
e
t
h
i
ng can
be ap
pl
i
e
d f
o
r
ot
he
r ro
ot
n
o
d
e
s
of o
p
t
i
m
al sol
u
t
i
ons
. The el
apse
d t
i
m
e
for IEEE
-
14
b
u
s sy
st
em
usi
n
g spa
n
ni
n
g
t
r
ee al
g
o
ri
t
h
m
i
s
2.95
5
9
9
3
secon
d
s a
nd
f
o
r
IEEE
-3
0 b
u
s
sy
st
em
it
i
s
onl
y
5
.
2
705
14
secon
d
s
. Th
is is
v
e
ry m
u
ch
less th
an
o
t
h
e
r algo
rith
m
s
li
k
e
m
a
trix
m
a
n
i
pu
latio
n which
is
3
9
1
.
14
565
7 seco
nd
s [1
4
]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 3,
J
u
ne 2
0
1
5
:
51
8 – 5
2
4
52
3
Tabl
e 1. IEEE
14
b
u
s sy
st
em
Tab
l
e 2
.
Ob
serv
ab
ility
o
f
each
b
u
s
PMU at Bus
Observed B
u
ses
2 2,
1,
3
4 4,
5,
9
7 7,
8,
9
11
11,
6,
10
13
13,
12,
14
5.
CO
NCL
USI
O
N
The spa
n
ning tree algorithm
i
s
applied for opti
m
u
m
placement of
PMUs for c
o
m
p
lete
observability
o
f
p
o
wer system n
e
twork. The PMUs
with
id
eal assu
m
p
tio
n
as m
u
ltich
a
n
n
e
l PMU is con
s
id
ered
and
also
t
h
e
realistic PMUs i.e th
e m
i
n
i
mu
m
ch
an
n
e
l PMUs (1
-ch
a
n
n
el and
2-c
h
a
n
nel) are c
onsi
d
e
r
ed a
n
d are a
p
plied
on
IEEE
-1
4
bu
s a
n
d
IEE
E
-
3
0
b
u
s
net
w
or
k.
T
h
i
s
m
e
t
hod ca
n
be a
ppl
i
e
d
o
n
a
n
y
sy
st
em
wi
t
h
m
o
re n
u
m
ber of
bus
es a
n
d
t
h
e c
o
m
put
at
i
on t
i
m
e re
qui
re
d i
s
fe
w sec
o
nds
l
i
k
e
3 t
o
5
sec
o
n
d
s.
REFERE
NC
ES
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chronized
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ach
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a
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l
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c, M. Begov
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eas
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ansactio
n on Power s
y
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[10]
Y.
Del Valle, G.
K.
Venay
a
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,
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M
ohagheghi,
J
.
C. Hernandez, R.G
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Harle
y
, “
P
arti
cle s
w
arm
optim
izat
ion: b
a
sic con
cepts
,
varian
ts, and
a
pplic
ati
ons in
powers
y
stem
s”,
IEEE
Transa
c
tion evolu
tion
a
r
y
computation
,
Vo
l. 12
, No
. 2
,
(Ap
r
2008.)
,
pp
. 171
-195.
Root Node
Multi Channel
One Channel
Two Channel
PMU bus
Observability
PMU bus
Ob
servability PMU
bus
Observability
1 2,
5,
7,
10,
13
Co
m
p
lete
1,
3,
5,
7,
9,
11,
12,
14
Co
m
p
lete
2,
5,
7,
10,
13
Co
m
p
let
e
2
1,
3,
6,
7,
9
Co
m
p
lete
1,
2,
4,
6,
7,
10,
12,
14
Co
m
p
lete 1,
3,
6,
7,
9,
13
Co
m
p
lete
3 1,
4,
7,
11,
13
Co
m
p
lete
1,
3,
5,
7,
9,
11,
12,
14
Co
m
p
lete
2,
4,
7,
11,
13
Co
m
p
let
e
4 3,
5,
7,
10,
13
Co
m
p
lete
1,
3,
4,
6,
7,
10,
12,
14
Co
m
p
lete
2,
5,
7,
10,
13
Co
m
p
let
e
5
2,
6,
7,
9 Co
m
p
let
e
1,
3,
5,
7,
9,
11,
12,
14
Co
m
p
lete
2,
4,
6,
9,
13
Co
m
p
let
e
6 2,
5,
7,
11,
13
Co
m
p
lete
1,
3,
5,
6,
7,
10,
12,
14
Co
m
p
lete
2,
5,
7,
11,
13
Co
m
p
let
e
7 2,
6,
7,
10,
12,
14
Co
m
p
lete
1,
3,
5,
7,
9,
11,
13
Co
m
p
let
e
2,
5,
7,
10,
13
Co
m
p
let
e
8 2,
6,
7,
10,
12,
14
Co
m
p
lete
1,
3,
5,
7,
9,
11,
13
Co
m
p
let
e
2,
5,
7,
10,
13
Co
m
p
let
e
9 2,
6,
7,
10,
12,
14
Co
m
p
lete
1,
3,
5,
7,
9,
11,
12,
14
Co
m
p
lete
2,
5,
7,
10,
14
Co
m
p
lete
10
2,
6,
7,
9,
13
Co
m
p
lete
1,
3,
5,
7,
10,
11,
13
Co
m
p
let
e
2,
5,
7,
9,
12,
14
co
m
p
lete
11
2,
6,
7,
10,
14
Co
m
p
lete
1,
3,
5,
7,
9,
11,
12,
14
Co
m
p
lete
2,
5,
7,
10,
13
Co
m
p
let
e
12
2,
5,
7,
10,
13
Co
m
p
lete
1,
3,
5,
7,
11,
12,
14
Co
m
p
let
e
2,
5,
7,
10,
13
Co
m
p
let
e
13
2,
6,
7,
10,
12,
14
Co
m
p
lete
1,
3,
5,
7,
9,
11,
12,
13
Co
m
p
lete
2,
5,
7,
10,
14
Co
m
p
let
e
14
2,
6,
7,
9,
13
Co
m
p
lete
1,
3,
5,
7,
11,
12,
14
Co
m
p
let
e
2,
5,
7,
9,
10,
13
Co
m
p
lete
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
A Spa
nn
ing
Tree App
r
o
a
c
h
i
n
Pla
c
ing
Mu
lti-ch
ann
el
&
Min
i
mum ch
ann
el
PMU’s fo
r …
(S
riha
ri Man
dava
)
52
4
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M
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P
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(Nov. 2011)
, 4577-1022
.
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