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
. 75
~89
I
S
SN
: 208
8-8
7
0
8
¶
75
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
Congestion Management in Hybr
id Ele
c
tr
ic
ity Ma
rke
t
s w
i
th
FACTS Devices with Loadability Limits
Ch
aran Se
kh
ar, As
hwani
K
u
mar
Departm
e
nt o
f
E
l
ec
tric
al
Engin
e
e
r
ing, Na
tion
a
l In
stitute
of
Te
chno
log
y
, Kurukshet
r
a, Ind
i
a
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Oct 15
th
, 201
1
R
e
vi
sed Feb
5
th
, 201
2
Accepte
d Fe
b
11
th
, 201
2
Congestion management (CM) is one of
the most important ch
allenging tasks
of the Ind
e
pend
ent S
y
stem Operator (ISO)
in th
e
deregul
ated
environment. In
this paper
,
generators’ r
e
sch
e
duli
ng b
a
sed
CM approach
to manage
transmission line congestion con
s
idering
loadability
limit has been presented
for h
y
brid b
a
sed
electricity
mark
et mode
l. Th
e main contribu
tion o
f
the paper
is
(i) to obtain
s
ecure tr
ans
act
i
ons
fo
r hy
b
r
id
m
a
rket m
odel, (
ii) optim
al
rescheduling of generators with loadab
i
lit
y li
m
its taken into accoun
t wit
h
secure tr
ansact
io
ns, (iii) and im
p
act of FACTS devic
e
s on transm
ission line
congestion man
a
gement. Th
e I
S
O ensures
secure bilateral trans
actions in
a
h
y
brid market model and CM is
mana
ged with minimum preferred schedule
to obtain minim
u
m congestion cost. The
r
e
sults have been obtain
e
d for IEEE
24 bus test s
y
stem.
Keyword:
Gene
rato
r re-
d
ispatch
C
o
n
g
est
i
o
n m
a
nagem
e
nt
Pool and
Hybrid electric
ity
mar
k
et
B
i
d f
unct
i
o
n
Lo
ad
ab
ility li
mit
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
:
Ash
w
a
n
i K
u
m
a
r,
Depa
rt
m
e
nt
of
El
ect
ri
cal
Engi
neeri
n
g
,
Natio
n
a
l
In
stitu
te of Techn
o
l
o
g
y
,
Ku
ru
ks
hetra
,
Hary
a
n
a, In
dia
.
Em
a
il: ash
w
aks@g
m
ail.co
m
1.
INTRODUCTION
W
i
t
h
growing dem
a
nd of electricity, the transm
i
ssion network also
needs expansion to trans
f
er
po
we
r. T
h
e t
r
ansm
i
ssi
on ne
t
w
o
r
k
wi
t
h
gr
owi
ng
co
ncer
ns
of e
n
vi
ro
n
m
ent
,
ri
g
h
t
o
f
way
p
r
o
b
l
e
m
s
, a
n
d
p
r
essure fo
r effectiv
e u
s
e
o
f
ex
istin
g
faciliti
es in
co
m
p
etiti
v
e
env
i
ro
n
m
ent can
cau
se to
v
i
o
l
ate its p
h
y
sical
li
mits to
carry
m
o
re p
o
wer
wh
ich
lead
s t
o
t
h
e co
ng
estio
n
i
n
th
e tran
sm
iss
i
o
n
n
e
twor
k. Th
is co
ng
estion
in
th
e
net
w
or
k m
a
y
ham
p
er m
a
rket
effi
ci
ency
fo
rci
n
g
t
h
e c
u
st
om
ers t
o
bac
k
do
w
n
po
wer
co
ns
u
m
pti
on
due
t
o
r
i
se i
n
electricity p
r
ices. Thu
s
, it is th
e u
t
m
o
st d
u
t
y o
f
th
e
ISO to
mit
i
g
a
te co
ng
estio
n
u
tilizin
g
d
i
fferen
t
tech
niq
u
e
s
m
a
y
be co
st
f
r
ee o
r
c
o
st
base
d
[1]
.
T
h
e
basi
c t
r
an
sm
i
ssi
on di
s
p
at
ch
an
d
con
g
est
i
o
n m
a
nagem
e
nt
m
odel
fo
r
con
g
est
i
o
n m
a
nagem
e
nt
i
s
p
r
esent
e
d
[
2
]
.
T
h
e
basi
c c
onc
e
p
t
s
of
t
r
an
sm
i
s
si
on
m
a
nagem
e
nt
,
di
spat
c
h
m
odel
,
an
d ro
le
o
f
t
h
e
ISO and
its m
o
d
e
l are presen
ted
in th
e
p
a
p
e
r.
Th
e
ISO can
u
tilize co
rrectiv
e m
easu
r
es t
o
m
a
n
a
g
e
cong
estion
b
y
u
til
izin
g
tran
sformer tap
s
, rerou
tin
g of
lines, and t
h
e
outa
g
e
of c
o
ngested lines.
Howeve
r, t
h
e
out
age
of lines
ca
n furt
her aggra
v
ate the
problem
of
co
ng
estion
.
These so
lu
tion
s
may n
o
t
h
e
lp
t
h
e ISO
for C
M
an
d
t
h
e ISO u
tilizes o
t
h
e
r
mark
et b
a
sed
so
lu
tions
to
m
a
n
a
g
e
th
e
co
ng
estion
m
o
re effectiv
ely.
Tech
ni
q
u
es ba
sed o
n
pri
ces,
resc
hed
u
l
i
n
g of
ge
nerat
o
r
s
,
zo
nal
based
m
e
t
hods
, se
nsi
t
i
v
i
t
y
based
app
r
oaches
, fi
nanci
a
l
t
r
a
n
sm
i
ssi
on ri
g
h
t
s
,
and
FAC
T
S a
ppl
i
cat
i
o
ns t
o
con
g
est
i
o
n m
a
nagem
e
nt
has
been
p
r
esen
ted [3-
2
6
]
. Fang
an
d
D
a
v
i
d
[
3
-
4
]
pro
p
o
s
ed
a tr
an
smissio
n
d
i
sp
at
ch
m
e
th
o
d
o
l
ogy as an ex
ten
s
io
n
of
sp
o
t
p
r
icing
t
h
eory in
a
p
o
o
l
and
b
ilateral as we
ll as
m
u
ltilateral
t
r
an
saction
m
o
d
e
l. Prio
ritizatio
n
of
electricity transactions
an
d willingness
-to-pay for m
i
nim
u
m
curtailm
ent
strategies has been investigat
ed as a
p
r
actical alternativ
e to
d
eal
with
th
e con
g
e
stio
n
.
A
u
t
h
ors
i
n
[
5
]
pr
o
p
o
s
ed F
A
C
T
S
bas
e
d cu
rt
ai
lm
ent
base
d
str
a
teg
y
based
o
n
[4
] fo
r congestio
n
m
a
n
a
g
e
men
t
. H
a
r
r
y
Si
n
g
h
et
al. [6
]
pr
opo
sed appr
oach
es
f
o
r
con
g
estio
n
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
. 2,
Fe
br
uar
y
20
1
2
:
7
5
– 89
76
man
a
g
e
m
e
n
t
b
a
sed
o
n
OPF,
wh
ich
u
tilizes DC lo
ad
flow
m
o
d
e
l to
m
i
n
i
mize th
e co
ngestio
n
co
st fo
r
p
o
o
l
co
m
odel
and bi
l
a
t
e
ral
m
odel
.
The n
odal
p
r
i
c
i
ng t
h
e
o
ry
has been a
ppl
i
e
d i
n
t
h
e p
ool
m
odel
whe
r
eas a m
e
t
h
o
d
base
d on c
o
n
g
e
st
i
on cost
al
l
o
cat
i
on
has be
en su
gge
st
ed f
o
r bi
l
a
t
e
ral
m
o
del
.
A
n
opt
i
m
al
powe
r
fl
o
w
based
app
r
oach
usi
n
g
no
dal
co
n
g
est
i
on
pri
ce si
gnal
s
fo
r com
put
i
n
g t
h
e
opt
i
m
al
p
o
we
r o
u
t
p
ut
o
f
gene
rat
o
rs
has
bee
n
pr
o
pose
d
i
n
[7]
.
Aut
h
o
r
s i
n
[
8
]
pr
op
ose
d
c
o
m
b
i
n
ed z
ona
l
and
Fi
xe
d
Tran
sm
i
ssi
on R
i
ght
(
F
TR
)
schem
e
fo
r
co
ng
estion
m
a
n
a
g
e
m
e
n
t
h
a
s
b
een
p
r
op
osed. Th
e co
m
b
in
ed sch
e
m
e
h
a
s b
een
u
tilized
with
lo
cation
a
l
marg
i
n
al
pri
ces (
L
M
P
s)
t
o
defi
ne zo
nal
bo
u
nda
ri
es ap
pr
o
p
ri
at
el
y
.
An OPF a
p
proach based
on DC l
o
ad
flow as we
ll as
AC
l
o
ad
fl
o
w
has bee
n
f
o
r
m
ul
at
ed t
o
m
i
ni
m
i
ze t
h
e net
cost
of re
-
d
i
s
pat
c
h t
o
m
a
nage i
n
t
e
rz
ona
l
an
d
in
trazon
a
l cong
estion
[9
]
.
A
no
vel
Lag
r
an
gi
an R
e
l
a
xat
i
o
n base
d al
g
o
ri
t
h
m
for area
decom
posi
t
i
o
n
OPF,
m
i
nim
i
zi
ng t
h
e con
g
est
i
o
n c
o
st
o
f
re
-di
s
pa
t
c
h i
n
o
r
der t
o
deal
wi
t
h
t
h
e
m
u
lt
i
-
zone c
o
n
g
est
i
o
n m
a
nagem
e
nt
,
has
been
p
r
o
p
o
se
d i
n
[
1
0]
.
B
o
t
h
i
n
t
e
r-z
on
al
and i
n
t
r
a-z
onal
c
o
n
g
est
i
o
n m
a
nagem
e
nt pr
o
b
l
e
m
has bee
n
fo
rm
ul
at
ed. Fa
st
LP al
g
o
ri
t
h
m
t
o
m
a
nage con
g
est
i
o
n
by
r
e
sche
dul
i
n
g
ge
nerat
i
o
n i
n
C
h
i
n
ese el
ect
ri
ci
t
y
m
a
rket
i
s
present
e
d i
n
[1
1]
. A
n
au
gm
ent
e
d Lagr
angi
a
n
R
e
l
a
xa
t
i
on base
d al
g
o
ri
t
h
m
has been pr
o
p
o
s
ed i
n
[1
2]
.
B
o
m
p
ard et
al
. [1
3]
de
vel
o
pe
d a u
n
i
f
i
e
d f
r
a
m
ewor
k f
o
r
mathem
atical re
prese
n
tation
of the m
a
rket dispatch
and re-
d
i
s
pat
c
h
p
r
obl
em
s,
w
h
i
c
h
i
s
base
d on
C
o
nges
t
i
o
n
M
a
nagem
e
nt
(C
M
)
sc
hem
e
s and t
h
e ass
o
c
i
at
ed
pri
c
i
n
g m
echani
s
m
s
. A
uni
fi
ed
fram
e
wor
k
has
bee
n
use
d
t
o
de
vel
o
p m
eani
n
gf
ul
m
a
t
r
i
ces t
o
c
o
m
p
are t
h
e
vari
ous
C
M
a
p
pr
oac
h
es s
o
a
s
t
o
assess
their
efficiency a
n
d
effective
n
ess
of
th
e m
a
rk
et sig
n
a
ls prov
i
d
ed to
t
h
e
mark
et p
a
rticipan
ts.
Kum
a
r et
al
. pro
p
o
sed c
o
m
p
rehen
s
i
v
e s
u
r
v
e
y
of co
n
g
est
i
o
n m
a
nagem
e
nt
m
e
t
hods a
nd
cat
ego
r
i
z
e
d
t
h
ese m
e
t
hods
base
d
o
n
t
h
ei
r m
odel
s
f
o
r
C
M
[1
4]
.
A c
o
n
g
est
i
o
n m
a
nagem
e
nt
app
r
o
ach
base
d
o
n
r
eal
a
n
d
react
i
v
e
po
we
r
co
ng
est
i
o
n
di
st
ri
but
i
o
n
fact
ors
ba
sed
zo
n
e
s an
d
ge
nerat
o
r
’
s
resc
hed
u
l
i
n
g
wa
s
pr
o
pos
ed i
n
[1
5]
.
Kum
a
r et
al
. pr
o
p
o
s
e
d
di
st
ri
b
u
t
i
o
n
fact
o
r
s
base
d
ge
nerat
o
rs
’ r
e
sche
dul
i
n
g
f
o
r C
M
[
16]
.
FAC
T
S
depl
oy
m
e
nt
i
n
t
h
e t
r
ansm
i
ssi
on net
w
o
r
k p
r
o
v
i
d
es
p
o
we
r fl
ow c
o
nt
r
o
l
an
d
hel
p
s t
o
m
a
na
ge co
n
g
est
i
o
n
i
n
t
h
e
n
e
two
r
k
.
Many au
tho
r
s u
tilized
FACTS
for co
ng
esti
on m
a
n
a
g
e
m
e
n
t
[18-25
]. C
o
ng
estion
m
a
n
a
ge
m
e
n
t
co
nsid
eri
n
g vo
ltag
e
stab
ility co
n
s
train
t
s
h
a
v
e
b
e
en
in
co
rpo
r
ated
in
[2
3
]
. FA
CTS
b
a
sed
m
o
d
e
l for re-
di
spat
c
h
i
n
g i
s
prese
n
t
e
d i
n
[
2
4
-
25]
. H
o
w
e
ver
,
t
h
e con
g
e
s
t
i
on m
a
nagem
e
nt
m
e
t
hod
s
have bee
n
ap
pl
i
e
d fo
r
po
ol
m
a
rket
m
odel
.
Som
e
of t
h
e a
u
t
h
ors
ha
ve t
a
ke
n
b
i
l
a
t
e
ral
m
odel
i
n
t
o
acc
ou
nt
,
ho
we
ver
,
t
h
e
opt
i
m
al
bi
l
a
t
e
ral
t
r
ansa
ct
i
ons
ha
ve
not
bee
n
e
n
s
u
re
d
du
ri
n
g
c
o
ngest
i
on m
a
nagem
e
nt
st
u
d
y
.
In t
h
e
pre
s
ent
wo
rk
, ge
nera
t
i
on resc
he
dul
i
ng
ba
se
d co
n
g
est
i
o
n m
a
nagem
e
nt
appr
oac
h
has
bee
n
form
u
l
ated
alon
g with th
e
voltag
e
stab
ility co
nstrain
t
taken
as l
o
adab
ility p
a
ram
e
ter. Th
e ap
pro
ach has b
e
en
also
ap
p
lied in
a po
o
l
+b
ilateral
m
i
x
m
a
rk
et mo
d
e
l
wh
ere
bil
a
teral transacti
ons
are
en
su
re
d
opt
i
m
al
by
t
h
e IS
O
bef
o
re
di
spat
c
h
i
n
g t
h
e ge
ne
rat
o
r
s
. T
h
e m
a
i
n
co
nt
ri
b
u
t
i
o
n o
f
t
h
e pa
pe
r i
s
t
o
pr
o
pos
e (i
) secu
re bi
l
a
t
e
ral
tran
saction
s
mo
d
e
l in
p
o
o
l
+mix
m
a
rk
et for con
g
e
sti
o
n
man
a
g
e
m
e
n
t
e
n
suring
vo
ltage stab
ility
li
mi
t. (ii) to
pr
o
pose t
h
e i
m
pact
of
FAC
T
S de
vi
ces vi
z,.
STATC
O
M
,
I
PFC
, an
d
UPF
C
i
n
t
h
e m
odel
fo
r o
b
t
a
i
n
i
n
g
opt
i
m
al
re-
d
i
s
pac
h
i
n
g
of
ge
nerat
o
rs
wi
t
h
m
i
nim
u
m
co
ngest
i
o
n c
o
st
. A
n
o
p
t
i
m
al
po
we
r fl
ow
pr
obl
em
usi
n
g
n
o
n
-
l
i
n
ea
r
pr
o
g
ram
m
i
ng app
r
oach has been
sol
v
ed
us
i
ng
C
O
N
O
PT
sol
v
e
r
of GA
M
S
wi
t
h
M
A
TLAB
i
n
t
e
rfac
i
ng [2
7
-
2
8
]
. Th
e resu
lts h
a
v
e
b
e
en
o
b
tain
ed
for
IEEE 24
bu
s Reliab
ility Test Syste
m
[29
]
.
2.
POOL+BILATERAL MARKET
MODEL
Th
e con
cep
t
u
al m
o
d
e
l o
f
b
ilateral d
i
sp
atch
i
s
th
at sellers an
d bu
yers en
ter in to tran
sactio
n
s
wh
ere
th
e q
u
a
n
tities t
r
ad
ed
and
th
e trad
e prices are at
th
e d
i
sc
retio
n
o
f
th
ese p
a
rties an
d
n
o
t
a matter o
f
ISO.
Th
ese
transactions a
r
e the
n
brought
to t
h
e
IS
O
with a
re
quest tha
t
transm
issi
on
facilities for the releva
nt am
ount
of
po
we
r be
pr
o
v
i
ded.
If t
h
ere i
s
no
vi
ol
at
i
o
n
of st
at
i
c
an
d
dy
nam
i
c securi
t
y
, t
h
e ISO si
m
p
ly
di
spat
ch
es al
l
requested tra
n
s
actions a
n
d c
h
arges
for the
s
e
rvice. T
h
e
b
ilateral
con
c
ep
t can
b
e
g
e
n
e
rali
zed
t
o
th
e
m
u
lt
i-no
d
e
case whe
r
e t
h
e
sel
l
e
r, fo
r exa
m
pl
e a generat
i
on c
o
m
p
any
,
m
a
y
i
n
ject
po
wer at
seve
ral
no
des a
nd t
h
e
buy
e
r
also
d
r
aw
lo
ad at sev
e
ral
n
odes.
U
n
lik
e pool d
i
sp
atch
, t
h
ere w
ill b
e
a tran
saction
p
o
w
e
r
b
a
lan
ce i
n
that th
e
aggre
g
ate inj
e
c
tion equals the aggre
g
ate dra
w
off for
each transaction.
A m
u
l
tilate
ral
transaction differs from
th
is
m
u
lti-n
o
d
e b
ilateral
m
o
d
e
l in
th
at it e
n
v
i
sag
e
s th
e activ
ity o
f
p
o
wer b
r
ok
er. Th
e co
n
c
ep
t o
f
a b
r
ok
er is
th
at o
f
a
firm
wh
ich
en
ters i
n
to
p
u
rch
a
se
& sales agreemen
ts with
several
bu
yer
s
and
seller
s
, a
gr
ou
p. In
th
is
case the
powe
r balance
c
onst
r
aints are t
h
at t
h
e
broker’s
a
g
gre
g
at
e
pu
rc
ha
ses f
r
om
al
l
ge
nerat
o
rs
at
any
t
i
m
e
eq
u
a
l aggreg
at
e sales to
all
th
e b
r
o
k
e
r’s
b
u
y
ers. Th
at is
, all
th
e tran
saction
s
con
s
titu
tin
g a g
r
oup
n
e
ed
ed
to
b
e
bal
a
nce
d
[2
4]
.
The m
o
st likely arrangem
ents whic
h
will em
erge in practical system
s
in
the fut
u
re is that a pool will
ex
ist sim
u
ltan
e
o
u
sly with
b
i
lateral an
d mu
ltilateral tr
ansactio
n
s
. Th
e
sig
n
i
fican
t
d
i
fferen
ce
b
e
tween
th
is
m
odel
& p
ool
m
odel
i
s
t
h
at
t
r
ansm
i
ssi
on se
ct
or i
s
u
n
b
u
ndl
ed in to a
“m
arket” sector a
nd a “sec
urity” s
ector.
Thi
s
m
odel
i
s
sho
w
n i
n
Fi
gu
re
1.
4. I
n
t
h
e m
a
rket
sector, t
h
ere a
r
e
m
u
ltiple separate energy m
a
rkets,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
¶
C
o
n
g
est
i
o
n M
a
na
ge
ment
i
n
H
y
bri
d
El
ect
ri
ci
ty Mar
kets
with FACTS
Devices w
i
t
h
…
.
(
C
h
a
ra
n
Sek
h
ar)
77
cont
ai
ni
ng a
p
ool
m
a
rket
t
a
k
e
n care
of
by
t
h
e Po
wer
Exc
h
an
ge a
nd
bi
l
a
t
e
ral
cont
ract
s
est
a
bl
i
s
hed
b
y
t
h
e
sche
dul
i
n
g c
o
o
r
di
nat
o
r
s
. T
h
e
ISO i
s
res
p
o
n
s
i
bl
e fo
r sy
st
em
ope
rat
i
o
n an
d
gua
ra
nt
ees sy
st
em
securi
t
y
and i
n
o
p
e
ration
a
l m
a
tters h
o
l
d
s
a su
p
e
rior po
sition
ov
er t
h
e
PX an
d
SCs. Th
e ex
isten
ce of
a p
o
wer
po
o
l
is n
o
t
man
d
a
tory in
th
is m
o
d
e
l b
u
t
will in
v
a
riab
ly
b
e
th
e case.
Mark
et p
a
rticip
an
ts m
a
y n
o
t
o
n
l
y b
i
d
i
n
to
th
e po
o
l
but
al
s
o
m
a
ke bi
l
a
t
e
ral
co
nt
ra
ct
s wi
t
h
eac
h
ot
he
r. T
h
e
r
efore, th
is m
o
d
e
l p
r
ov
id
es m
o
re flex
i
b
le op
tion
s
for
transm
ission access. A California
m
odel is represe
n
tativ
e
of this category. The
Nordic
m
odel and the New
Zeelan
d Mod
e
l also
fall in to
th
is
catego
r
y
with
so
m
e
mo
d
i
ficatio
n
s
. Oth
e
r m
o
d
e
ls such
as t
h
e
New Yo
rk
Power P
o
ol (NYPP) a
n
d the Pennsy
lv
an
ia New Jersey Marylan
d
(PJM)
m
o
d
e
l fall so
m
e
wh
ere in
b
e
tween
these three cat
egories.
A tran
saction
matrix
h
a
s b
e
en
tak
e
n
a co
llectio
n
o
f
t
r
a
n
sa
ct
i
ons bet
w
een
Genc
os (G
),
Di
scos (
D
).
Th
e tran
saction
m
a
trix
can
be rep
r
esen
ted as:
[]
[
]
T
DG
GD
=
(1
)
Each elem
ent of GD,
nam
e
ly GD
ij
,
represents a b
ilateral con
t
ract
b
e
tween
a supp
lier (P
gi
) of
row
i
wi
t
h
a c
o
ns
um
er (
P
dj
)
of
col
u
m
n
j. F
u
rt
herm
ore
,
t
h
e
sum
of
r
o
w i
re
prese
n
t
s
t
h
e t
o
t
a
l
po
w
e
r
pr
od
uce
d
by
g
e
n
e
rator i and th
e su
m
o
f
co
lu
m
n
j
represents th
e to
tal
p
o
wer co
n
s
u
m
ed
at lo
ad j.
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
≡
nd
ng
ng
nd
nd
GD
GD
GD
GD
GD
GD
GD
,
1
,
,
2
1
,
2
,
1
1
,
1
...
...
...
(2
)
whe
r
e:
n
g
=
n
u
m
b
er o
f
ge
nerat
o
rs
, a
n
d
n
d
=
num
b
er of
loa
d
s.
In
ge
neral
,
t
h
e
con
v
e
n
t
i
onal
l
o
ad
fl
ow
va
ri
abl
e
s,
ge
nerat
i
o
n
(P
g
)
and l
o
ad (P
d
)
v
ect
or
s, a
r
e
no
w e
x
pan
d
e
d i
n
t
o
two
d
i
m
e
n
s
io
nal
tran
saction
matrix
as:
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
d
g
T
g
d
u
u
GD
GD
P
P
0
0
(3
)
Vector u
g
and
u
d
are
col
u
m
n
vect
o
r
s
of
o
n
e
s
wi
t
h
t
h
e
di
m
e
nsi
o
ns
of
n
g
an
d n
d
res
p
ectively. There
are
some
in
trin
sic p
r
op
erties
asso
ciated
with
th
is tran
saction
m
a
tri
x
GD. Th
ese ar
e co
lu
m
n
r
u
le,
ro
w ru
le,
rang
e ru
le,
and
fl
ow
r
u
l
e
.
These
p
r
o
p
ert
i
es ha
ve
bee
n
e
xpl
ai
ne
d i
n
[3
0-
3
1
]
.
Eac
h
c
ont
ract
has
t
o
ran
g
e
fr
om
zero t
o
a
maxim
u
m
allowable value,
GD
ij
ma
x
. Thi
s
m
a
xi
m
u
m
val
u
e is bo
u
nde
d by
t
h
e val
u
e of c
o
r
r
esp
o
ndi
ng P
gi
ma
x
or
P
dj
wh
ich
e
v
e
r is sm
al
ler. The
rang
e
ru
le
satisfies:
(
)
dj
gi
ij
ij
P
P
GD
GD
,
min
0
max
max
≤
≤
≤
(4
)
It is also possi
ble for s
o
m
e
c
ont
racts to
be
firm
so that GD
ij
0
is equ
a
l to
GD
ij
ma
x
[30]
.
Ac
cor
d
ing t
o
fl
ow r
u
le
th
e
lin
e flows o
f
th
e n
e
two
r
k
can
b
e
ex
pressed
as
fo
llo
ws:
[
]
d
g
line
P
P
DF
P
−
=
(5
)
Th
e m
a
trix
DF is th
e
d
i
stribu
t
i
o
n
factors m
a
t
r
ix
[31
]
.
If th
e
represen
tatio
n
s
of th
e P
g
an
d
P
d
are
sub
s
titu
ted
b
y
u
s
i
n
g
th
e d
e
fin
itio
n of
GD
as g
i
v
e
n in
(29), t
h
e lin
e
flows can b
e
exp
r
essed
i
n
an
alternativ
e as fo
llows:
[]
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
=
1
1
M
T
line
GD
GD
DF
P
(6
)
The ge
neral
pr
obl
em
form
ul
at
i
on fo
r det
e
rm
i
n
at
i
on o
f
se
cure tran
saction
matrix
for
h
ybrid
m
a
rk
et m
o
d
e
l can
be rep
r
ese
n
t
e
d as:
A.
Obj
ectiv
e fu
n
c
tio
n
M
i
nim
i
zati
on
of
de
vi
at
i
ons
f
r
o
m
t
h
e pr
o
pos
ed t
r
a
n
sact
i
o
ns
G
D
ij
0
:
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
. 2,
Fe
br
uar
y
20
1
2
:
7
5
– 89
78
Min
()
2
0
⎩
⎨
⎧
−
∑∑
ij
ij
ij
ij
GD
GD
b
(7
)
B.
Ope
r
at
i
n
g c
ons
t
r
ai
nt
s
i) Equ
a
lity constrain
t
s:
Power
flow
bal
a
nce e
quati
ons
at each
bus a
r
e:
()
(
)
[]
b
j
i
ij
j
i
ij
j
N
j
i
di
gi
i
n
i
B
G
V
V
P
P
P
b
K
,
2
,
1
sin
cos
1
=
∀
−
+
−
=
−
=
∑
=
δ
δ
δ
δ
(8
)
()
(
)
[]
b
j
i
ij
j
i
ij
j
N
j
i
di
gi
i
N
i
B
G
V
V
Q
Q
Q
b
K
,
2
,
1
cos
sin
1
=
∀
−
−
−
=
−
=
∑
=
δ
δ
δ
δ
(9
)
Po
wer
bal
a
nce
equat
i
o
ns
f
o
r
d
e
m
a
nd a
n
d
ge
n
e
rat
i
o
n
f
o
r
hy
b
r
i
d
m
a
rket
m
odel
u
s
i
n
g
bi
l
a
t
e
ral
dem
a
nd m
a
t
r
i
x
GD are:
∑
=
i
ij
GD
db
P
,
∑
=
j
ij
GD
gb
P
(1
0)
gp
gb
g
P
P
P
+
=
,
dp
db
d
P
P
P
+
=
(1
1)
Power
flow equ
a
tio
ns for
h
ybrid
m
o
d
e
l:
)
(
db
gb
fb
P
P
P
−
=
DF
(1
2)
)
(
dp
gp
fp
P
P
P
−
=
DF
(1
3)
fp
fb
f
P
P
P
+
=
(1
4)
Eq
uat
i
ons
(
1
2)
an
d
(1
3)
re
pre
s
ent
s
t
h
e
real
a
n
d
react
i
v
e
p
o
w
er
fl
o
w
i
n
ject
i
on at
a
n
y
bu
s i
.
E
quat
i
o
ns
(
1
3
)
t
o
(1
4)
re
pre
s
ent
t
h
e
po
we
r fl
ow
bal
a
nce
eq
uat
i
ons
f
o
r
hy
bri
d
m
odel
.
ii) In
equ
a
lity co
n
s
t
r
ain
t
s:
Real and reacti
v
e
powe
r
gene
ration for ge
ne
rators:
max
min
g
g
g
P
P
P
≤
≤
(1
5)
max
min
g
g
g
Q
Q
Q
≤
≤
(1
6)
Transactio
n limit b
e
tween
seller bu
s-i and
bu
yer
bu
s
j
:
(
)
dj
gi
ij
ij
ij
P
P
GD
GD
GD
,
min
max
max
min
≤
≤
≤
(1
7)
Lim
i
t
s
on v
o
l
t
a
ge m
a
gni
t
u
de a
n
d
an
gl
e:
max
min
i
i
i
V
V
V
≤
≤
(1
8)
max
min
i
i
i
δ
δ
δ
≤
≤
(1
9)
M
VA p
o
we
r flow
lim
i
t:
max
ij
ij
S
S
≤
(2
0)
Eq
uat
i
ons
(
1
5)
t
o
(2
0)
re
pr
es
ent
t
h
e
i
n
e
qual
i
t
y
const
r
ai
nt
s
fo
r
real
p
o
w
er
gene
rat
i
o
n,
rea
c
t
i
v
e p
o
we
r
gene
ration, and bilateral transactions, limits on the vo
ltage m
a
gnitudes
,
voltage a
ngl
es at each bus
in the
syste
m
, an
d
MVA
flow li
m
it.
Th
e vo
ltag
e
li
mit, p
o
w
e
r
a
ngl
e l
i
m
i
t
has been c
o
nsi
d
e
r
e
d
bet
w
een
1.
0
5
p
.
u
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
¶
C
o
n
g
est
i
o
n M
a
na
ge
ment
i
n
H
y
bri
d
El
ect
ri
ci
ty Mar
kets
with FACTS
Devices w
i
t
h
…
.
(
C
h
a
ra
n
Sek
h
ar)
79
an
d 0.95 p.u.,
-30
d
e
g
r
ee to
+3
0 d
e
gree,
resp
ectiv
ely.
Secu
re b
ilateral tran
saction
m
a
trix
u
tilizin
g equ
a
tions
(7
) t
o
(
2
0)
ha
v
e
bee
n
obt
ai
ne
d
usi
n
g
GAM
S
C
O
N
O
P
T
s
o
l
v
er
wi
t
h
M
A
T
L
AB
i
n
t
e
rfaci
n
g
.
3.
STATIC
M
O
DEL OF
FA
C
T
S DE
VICE
S
In
th
is section
,
it is ex
p
l
ain
e
d
th
e resu
l
t
s o
f
research an
d
at th
e sam
e
ti
me
is
g
i
v
e
n
the
com
p
rehe
nsi
v
e
di
scus
si
o
n
. R
e
sul
t
s
can
be
pr
esent
e
d i
n
fi
gu
res,
gra
p
hs, t
a
b
l
es and
ot
hers t
h
at
m
a
ke t
h
e r
eade
r
un
de
rst
a
n
d
eas
i
l
y
[2]
,
[
5
]
.
T
h
e di
sc
ussi
o
n
ca
n
be m
a
de i
n
s
e
veral
s
u
b-c
h
a
p
t
e
rs.
3.
1.
M
o
del
o
f
ST
AT
C
O
M
STATC
O
M co
n
s
ists
of a co
nv
erter,
c
o
up
l
i
ng t
r
a
n
s
f
o
r
m
e
r, a
n
d a
DC
capaci
t
o
r. T
h
e m
a
i
n
f
unct
i
on
o
f
con
v
e
r
t
e
r i
s
t
o
gene
rat
e
a f
u
ndam
e
nt
al
out
put
vol
t
a
ge w
a
vef
o
rm
wi
t
h
t
h
e dem
a
nde
d
m
a
gni
t
ude
an
d
pha
se
an
g
l
e in
sy
n
c
hron
ism
with
th
e AC syste
m
. Sin
ce, static
com
p
ensat
o
r ca
nn
ot
ge
ne
rat
e
or a
b
s
o
r
b
real
po
we
r
(assum
i
ng
n
o
e
n
er
gy
st
ora
g
e
f
o
r
ST
ATC
O
M
)
,
p
o
we
r t
r
ans
m
i
ssi
on
of
t
h
e
sy
st
em
i
s
affec
t
ed i
n
di
rect
l
y
b
y
t
h
e
v
o
ltag
e
con
t
ro
l
.
Th
e reactiv
e
o
u
t
p
u
t
po
wer
(cap
acitiv
e or i
n
du
ctiv
e)
o
f
t
h
e co
m
p
en
sator is v
a
ried
t
o
co
n
t
ro
l
t
h
e vol
t
a
ge at
gi
ve
n t
e
rm
i
n
al
of t
r
a
n
sm
i
ssi
o
n
net
w
or
k so a
s
t
o
m
a
i
n
t
a
i
n
the desi
re
po
we
r fl
o
w
un
de
r p
o
ssi
bl
e
sy
st
em
di
st
urb
a
nces a
n
d c
ont
i
nge
nci
e
s [
3
2]
.
0
)
*
sh
I
sh
Re(
V
=
Fi
g.
1. M
o
del
o
f
ST
ATC
O
M
Fo
r th
e
p
o
wer flow an
alysis, STATCOM
will b
e
rep
r
esen
ted
b
y
a
syn
c
hrono
us vo
ltag
e
so
urce with
m
a
gni
t
ude
V
sh
and
a
ngl
e
δ
sh
with
its
in
tern
al im
p
e
d
a
n
ce
Z
se
ap
p
lied
in
an
y bu
s
i
, sh
ow
n i
n
Fi
g 1. T
h
en t
h
e
real
and reactive
power inj
ection a
t
any bus
i of t
h
e STATC
O
M
are:
)]
sin(
)
cos(
[
2
s
h
i
s
h
B
s
h
i
s
h
G
s
h
V
i
V
s
h
G
i
V
c
i
P
δ
δ
δ
δ
−
+
−
+
=
(2
1)
)]
cos(
)
sin(
[
2
s
h
i
s
h
B
s
h
i
s
h
G
s
h
V
i
V
s
h
B
i
V
c
i
Q
δ
δ
δ
δ
−
−
−
+
−
=
(2
2)
Op
eration
a
l con
s
train
t
of th
e
STATC
O
M (real p
o
wer ex
ch
an
g
e
v
i
a
DC link
)
can b
e
written
as:
0
)
Re(
=
∗
=
sh
I
sh
V
exchange
P
or
0
)]
sin(
)
cos(
[
2
=
−
−
−
+
s
h
i
s
h
B
s
h
i
s
h
G
s
h
V
i
V
s
h
G
i
V
δ
δ
δ
δ
(2
3)
w
h
er
e
1/
Z
sh
=G
sh
+jB
sh
3.
2.
M
o
del
o
f
IPFC
IPFC ca
n
be
m
odeled as multiple SSSC c
o
nnected
via
c
o
mm
on DC link. An
IPFC
with com
b
ining two or
m
o
re series connecte
d
converters working toget
h
er at
thei
r DC links. In
addition to
providing se
ries reactive
co
m
p
en
satio
n, an
y co
nv
erter can
b
e
co
n
t
ro
lled to
real
p
o
we
r t
o
t
h
e
com
m
on DC
l
i
nk
fr
om
i
t
s
o
w
n
tran
sm
issio
n
lin
e.
For sim
p
lest form
o
f
th
e
IPFC con
s
ists
o
f
two
co
nv
ert
e
rs in series with
two tran
smissio
n
lin
es. Th
is can co
n
t
ro
l th
e power fl
o
w
o
f
t
h
e two
lin
es.
T
h
e equi
val
e
nt
ci
rcui
t
o
f
t
h
e IP
FC
consi
s
t
i
n
g
of t
w
o
cont
rol
l
a
bl
e
se
ri
es i
n
ject
ed
v
o
l
t
a
ge so
urce
s i
s
sh
ow
n i
n
Fi
g.
3 S
u
m
of
real
po
we
r e
x
cha
n
g
e
sh
o
u
l
d
be ze
r
o
.
Accord
ing
to t
h
e equ
i
v
a
len
t
circu
it of
IPFC sho
w
n in
Fig
.
3
,
t
h
e i
n
j
ected
Power equ
a
t
i
o
n
s
can
b
e
written
as
[3
4]
:
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
. 2,
Fe
br
uar
y
20
1
2
:
7
5
– 89
80
∑
+
+
=
−
+
−
+
∑
h
ih
se
ih
ih
se
ih
ih
se
h
ih
ih
ih
ih
h
i
ii
i
i
i
B
i
G
V
i
V
B
G
V
V
G
V
P
)]
sin(
)
cos(
[
,
,
,
2
)]
sin(
)
cos(
[
δ
δ
δ
δ
δ
δ
(2
4)
∑
+
−
+
−
=
−
−
−
∑
h
ih
se
i
ih
B
ih
se
i
ih
G
ih
se
V
i
V
h
ih
ih
ih
ih
h
i
ii
i
i
B
G
V
V
B
V
Q
)]
,
cos(
)
,
sin(
[
,
)]
cos(
)
sin(
[
2
δ
δ
δ
δ
δ
δ
(2
5)
)]
sin(
)
cos(
[
)]
sin(
)
cos(
[
,
,
,
2
ih
se
h
ih
ih
se
h
ih
ih
se
h
hi
ih
hi
ih
h
i
hh
h
hi
B
G
V
V
B
G
V
V
G
V
P
δ
δ
δ
δ
δ
δ
−
+
−
+
+
=
+
(2
6)
)]
sin(
)
cos(
[
)]
sin(
)
cos(
[
,
,
,
,
2
ih
se
i
ih
h
se
i
h
i
h
se
i
ij
ij
ij
ij
h
i
hh
h
hi
B
G
V
V
B
G
V
V
B
V
Q
δ
δ
δ
δ
δ
δ
−
−
−
−
+
−
=
+
(2
7)
Ope
r
at
i
n
g c
ons
t
r
ai
nt
s o
f
IPFC
, real
p
o
we
r e
x
chan
ge
vi
a c
o
m
m
on DC
l
i
n
k
sh
oul
d
be ze
ro
.
0
)
Re(
,
=
=
∑
∗
h
hi
ih
se
exchange
I
V
P
(2
8)
0
)]]
sin(
)
cos(
[
)]
sin(
)
cos(
[
[
,
,
,
,
=
−
−
−
+
−
−
−
∑
h
ih
se
h
ih
ih
se
h
ih
se
h
ih
se
i
ih
ih
se
i
ih
se
i
B
G
V
V
B
G
V
V
δ
δ
δ
δ
δ
δ
δ
δ
(2
9)
Whe
r
e,
∑
=
h
ih
ii
G
G
;
∑
=
h
ih
ii
B
B
whe
r
e h=
j, k...etc.
Controllable i
n
jected
voltage
so
ur
ce bo
und
co
n
s
t
r
ain
t
s:
max
,
,
min
,
ih
se
ih
se
ih
se
V
V
V
<
<
max
,
,
min
,
ih
se
ih
se
ih
se
δ
δ
δ
<
<
(3
0)
i
j
jQ
P
j
i
+
ki
ki
jQ
P
+
0
)
*
ki
I
sei
k
V
*
ji
I
se
i
j
Re(
V
=
+
Fi
g.
2. M
o
del
o
f
I
PFC
3.
3
Mo
del o
f
UPF
C
UPFC
ca
n
be
d
i
vi
ded
i
n
t
o
t
w
o
FAC
T
S c
ont
r
o
l
l
e
rs,
fi
rst
on
e is series con
t
ro
ller an
d seco
nd
o
n
e
sh
un
t
cont
roller. S
e
ries controller is equivale
nt to th
e SSSC
and s
h
unt controller is STATCOM.
Whe
n
the
STATC
O
M
a
nd t
h
e S
SSC
ope
rat
e
as st
a
ndal
one
FAC
T
S co
nt
r
o
l
l
e
rs
, t
h
ey
exc
h
a
n
ge al
m
o
st
excl
usi
v
el
y
reactive power at their ter
m
inals. Du
ring the
stand-alone oper
ations, t
h
e
SSSC injects a voltage i
n
qua
d
rat
u
re
with the line curren
t, the
r
e
b
y e
m
ulating an inductive and c
a
pac
itive reactance at the poi
nt of com
p
ensation in
series with t
h
e
line, and the
STATC
O
M injects a reactive
curren
t, there
b
y also em
ulating a
reactance
at the
p
o
i
n
t
of
co
m
p
en
sation
in shunt w
ith
th
e lin
e
[3
2-3
3
]
.
In
the
steady state operation, the
m
a
in
o
b
j
ectiv
e o
f
an
UPFC is to
co
ntro
l v
o
ltage and
po
wer flow. Th
e
equi
val
e
nt
ci
rc
ui
t
o
f
a
n
UPFC
i
s
sh
o
w
n
i
n
Fi
g.
3.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
¶
C
o
n
g
est
i
o
n M
a
na
ge
ment
i
n
H
y
bri
d
El
ect
ri
ci
ty Mar
kets
with FACTS
Devices w
i
t
h
…
.
(
C
h
a
ra
n
Sek
h
ar)
81
c
ij
c
ij
jQ
P
+
c
ji
c
ji
jQ
P
+
0
)
*
ji
I
se
V
sh
I
sh
Re(V
=
−
∗
Fi
g.
3.
E
qui
val
e
nt
ci
rc
ui
t
o
f
U
PFC
B
a
sed o
n
ci
rc
u
i
t
sho
w
n i
n
Fi
g.
3, t
h
e i
n
ject
ed active a
nd
reactive powe
r
equations at
bus
i
an
d bus
j
can
be
written
as:
)]
sin(
)
cos(
[
)]
sin(
)
cos(
[
)]
sin(
)
cos(
[
)
(
2
s
h
i
s
h
B
s
h
i
s
h
G
s
h
V
i
V
se
i
ij
B
se
i
ij
G
se
V
i
V
ij
ij
B
ij
ij
G
j
V
i
V
sh
G
ii
G
i
V
c
ij
P
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
−
+
−
+
−
+
−
+
+
+
+
=
(31)
)]
cos(
)
sin(
[
)]
cos(
)
sin(
[
)]
cos(
)
sin(
[
)
(
2
s
h
i
s
h
B
s
h
i
s
h
G
s
h
V
i
V
se
i
ij
B
se
i
ij
G
se
V
i
V
ij
ij
B
ij
ij
G
j
V
i
V
sh
B
ij
B
i
V
c
ij
Q
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
−
−
−
+
−
−
−
+
−
+
+
−
=
(3
2)
Op
erating
co
n
s
train
t
s is real
po
wer ex
ch
ang
e
v
i
a
DC link
can
b
e
written
as:
0
)]
sin(
)
cos(
[
2
)]
sin(
)
cos(
[
)]
sin(
)
cos(
[
=
−
−
−
+
+
−
−
−
+
−
−
−
sh
i
sh
B
sh
i
sh
G
sh
V
i
V
sh
G
i
V
se
j
ij
B
se
j
ij
G
se
V
j
V
se
i
ij
B
se
i
ij
G
se
V
i
V
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
δ
(33)
whe
r
e
1/
Z
sh
=G
sh
+jB
sh
;
G
ij
and
B
ij
are t
a
ke
n f
r
om
Y
bus
. The
po
we
r fl
o
w
e
q
uat
i
on
o
b
t
a
i
n
e
d
f
o
r
FAC
T
S c
a
n be
adde
d i
n
an
O
PF m
odel
t
o
i
n
cor
p
orat
e t
h
e
e
ffect
of
FAC
T
S de
vi
ces f
o
r r
e
sche
dul
i
n
g
of
ge
nerat
o
rs t
o
r
e
m
o
v
e
congestion.
For c
o
nge
st
i
o
n
m
a
nagem
e
nt
, Genc
os se
n
d
b
i
ds t
o
t
h
e I
S
O
al
on
g wi
t
h
t
h
ei
r m
a
xim
u
m
and m
i
nim
u
m
l
i
m
i
t
s
of
gen
e
rat
o
r re
sche
dul
i
n
g. T
h
e bi
d f
u
nct
i
o
n can be c
o
nst
a
n
t
bi
d o
r
l
i
n
ear
bi
d f
u
nct
i
o
n. I
n
t
h
i
s
w
o
rk
, l
i
n
ear b
i
d
fu
nct
i
o
n
has
b
een c
o
n
s
i
d
e
r
ed
. B
a
sed
on
t
h
e
q
u
al
i
f
y
i
ng
bi
d
s
, t
h
e
IS
O
sen
d
si
gnal
s
t
o
t
h
e Ge
nco
s
t
o
re
gul
at
e
their output during congestion hours
to m
itigate congestion for which th
e gene
rators are paid accordi
ng
t
o
t
h
ei
r q
u
al
i
f
i
e
d
bi
ds
. F
o
r t
h
e
gene
rat
o
rs t
o
r
e
sche
dul
e t
h
ei
r ge
nerat
i
on
u
p
/
d
ow
n, t
h
ei
r
base case
ge
n
e
rat
i
o
n
in
fo
rm
atio
n
is essen
tial. Th
is h
a
s b
e
en
ob
tain
ed
so
l
v
i
n
g o
p
t
i
m
a
l
power
f
l
ow p
r
obl
em
wi
t
h
m
i
nim
i
zat
i
on o
f
fuel cost. T
h
e
congestion m
a
nagem
e
nt
m
o
del has been
fo
r
m
ul
at
ed as t
h
e no
n l
i
n
ear p
r
o
g
ram
m
i
ng pr
o
b
lem
so
lv
ed
u
s
ing
GAMS C
O
NOPT so
l
v
er u
tilisin
g MATLAB
an
d GAMS i
n
t
e
rfacing
.
4.
CO
NGESTI
O
N
MA
N
A
GE
MENT MO
D
EL WITH LOADABILITY
LIMIT
In
t
h
i
s
sect
i
o
n,
i
t
i
s
expl
ai
ne
d t
h
e
resul
t
s
of
re
search
an
d at
t
h
e sam
e
t
i
m
e
is gi
ven
t
h
e c
o
m
p
rehensi
v
e
d
i
scu
ssi
on
. Resu
lts can b
e
presen
ted in
f
i
gur
es,
gr
aph
s
, tab
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
. 2,
Fe
br
uar
y
20
1
2
:
7
5
– 89
82
Min
()
λ
ξ
,
,
,
,
FACTS
p
u
x
F
(3
4)
Subject t
o
()
0
,
,
,
,
=
λ
ξ
FACTS
p
u
x
h
(3
5)
()
0
,
,
,
,
≤
λ
ξ
FACTS
p
u
x
g
(3
6)
F is an
o
b
j
ecti
v
e
fun
c
tion
,
wh
ich
is su
bj
ect
ed
to
p
o
wer fl
o
w
equ
a
lity co
n
s
t
r
ain
t
s
rep
r
esen
ted as
h
an
d all
in
equ
a
lity co
n
s
train
t
s rep
r
esen
ted
as g.
Vecto
r
x
represen
ts
state
v
a
riab
les,
u
rep
r
esen
ts co
n
t
ro
l
v
a
riab
les, and
p
re
pre
s
ent
s
fi
xed pa
ram
e
t
e
r
s
,
λ
ξ
,
FACTS
are th
e contro
l p
a
ram
e
ter
s
for FACVTS d
e
v
i
ces and
lo
ad
ab
ility
facto
r
as
vo
ltag
e
stab
ility li
mit.
Ob
ject
i
v
e
f
unc
t
i
on:
M
i
ni
m
i
ze co
nge
st
i
o
n
co
st
C
C
()
(
)
∑
∑
=
=
Δ
+
Δ
=
nd
d
down
g
nd
d
up
g
P
C
P
C
CC
1
1
(3
7)
Th
e co
m
p
on
ents o
f
th
e co
ng
estio
n
co
st CC are th
e su
m
o
f
t
h
e lin
ear b
i
d
fun
c
tio
ns o
f
th
e
d
e
m
a
n
d
su
b
m
itted
to
t
h
e IS
O f
o
r c
o
nge
st
i
on m
a
nagem
e
nt
based on ge
nerat
i
o
n
r
e
sche
dul
i
n
g.
bsm
v
a
i
s
t
h
e ba
se M
VA a
nd
R
g
up
and
R
g
down
are t
h
e
u
p
a
n
d
d
o
w
n
co
st
com
pone
nt
i
n
i
n
$/
hr
.
()
R
P
up
g
up
g
up
g
bsmva
k
P
C
+
=
Δ
Δ
*
*
2
(3
8)
()
R
P
down
g
down
g
down
g
bsmva
k
P
C
+
=
Δ
Δ
*
*
2
(3
9)
k1 an
d
k
2
a
r
e
dem
a
nd c
o
st
coef
fi
ci
ent
s
of
a ge
nerat
i
o
n
sche
dul
i
n
g
bi
d
fu
nct
i
o
n s
u
b
m
i
t
t
e
d t
o
t
h
e
ISO
i
n
$/
M
W
h
.
(a
)
E
q
u
a
lity con
s
tra
i
n
t
s
Let co
m
p
lex
vo
ltag
e
s at
bu
s-i an
d bu
s-j are V
i
∠
δ
i
and
V
j
∠
δ
j
resp
ectiv
ely. Th
e power i
n
j
ectio
n eq
u
a
tion
s
at
each
bus ca
n
be written as:
()
(
)
[]
b
j
i
ij
j
i
ij
j
N
j
i
i
N
i
B
G
V
V
P
b
K
,
2
,
1
sin
cos
1
=
∀
−
+
−
=
∑
=
δ
δ
δ
δ
(4
0)
()
()
[]
b
j
i
ij
j
i
ij
j
N
j
i
i
N
i
B
G
V
V
Q
b
K
,
2
,
1
cos
sin
1
=
∀
−
−
−
=
∑
=
δ
δ
δ
δ
(4
1)
0
1
1
=
Δ
−
Δ
∑
∑
=
=
Ng
g
down
g
Ng
g
up
g
P
P
(4
2)
down
g
up
g
g
gni
P
P
P
P
Δ
−
Δ
+
=
(4
3)
d
gni
i
P
P
P
∗
−
=
ρ
(4
4)
di
gi
i
Q
Q
Q
−
=
(4
5)
(b
)
I
n
e
q
u
a
lity co
n
s
tra
i
n
t
s
(i
) U
p
/
d
o
w
n
d
e
m
a
nd l
i
m
i
t
s
f
o
r
dem
a
nd
m
a
nagem
e
nt
:
The
l
i
m
i
t
s
for u
p
and
do
w
n
dem
a
nd m
a
nagem
e
nt
are
gi
ve
n by
down
g
g
down
g
P
P
P
max
min
Δ
≤
Δ
≤
Δ
(4
6)
up
g
g
up
g
P
P
P
max
min
Δ
≤
Δ
≤
Δ
(4
7)
max
min
gn
gn
gn
P
P
P
≤
≤
(4
8)
max
min
g
g
g
Q
Q
Q
≤
≤
(4
9)
max
min
i
i
i
V
V
V
≤
≤
(5
0)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
¶
C
o
n
g
est
i
o
n M
a
na
ge
ment
i
n
H
y
bri
d
El
ect
ri
ci
ty Mar
kets
with FACTS
Devices w
i
t
h
…
.
(
C
h
a
ra
n
Sek
h
ar)
83
max
min
i
i
i
δ
δ
δ
≤
≤
(5
1)
(ii) Po
wer
flow li
m
i
ts
()
2
max
2
2
ij
ij
ij
Q
P
S
≤
+
(
5
2
)
Po
wer
bal
a
nce
equat
i
o
ns
f
o
r
d
e
m
a
nd a
n
d
ge
n
e
rat
i
o
n
f
o
r
hy
b
r
i
d
m
a
rket
m
odel
u
s
i
n
g
bi
l
a
t
e
ral
dem
a
nd m
a
t
r
i
x
GD are:
∑
=
i
ij
GD
db
P
,
∑
=
j
ij
GD
gb
P
(5
3)
gp
gb
g
P
P
P
+
=
(5
4)
p
b
P
P
P
d
d
d
+
=
(5
5)
max
min
ij
ij
ij
GD
GD
GD
≤
≤
(5
6)
whe
r
e
P
g
and P
gn
:
are t
h
e base case gene
rat
i
on
and
new sc
he
dul
e o
f
ge
ner
a
t
i
on o
b
t
a
i
n
ed
wi
t
h
dem
a
nd si
de
m
a
nagem
e
nt.
P
d
:
B
a
se case
po
w
e
r
dem
a
nd
up
g
P
Δ
:
u
p
sche
d
u
l
i
ng
o
f
gene
rat
o
r at
bus
-
i
f
o
r c
o
n
g
est
i
o
n m
a
nagem
e
nt
down
g
P
Δ
:d
own
r
e
sch
e
du
lin
g
of
g
e
n
e
r
a
to
r
at bu
s-
i
fo
r con
g
est
i
o
n
m
a
nagem
e
nt
5.
R
E
SU
LTS AN
D ANA
LY
SIS
In
th
is section
,
resu
lts h
a
v
e
been
ob
tain
ed
fo
r thr
ee di
ffe
r
e
nt
cases o
f
l
i
n
e co
n
g
est
i
on
wi
t
h
bi
d f
unct
i
on
su
b
m
itted
b
y
th
e GENC
Os t
o
th
e
ISO. The resu
lts
h
a
v
e
b
een
ob
tain
ed
fo
r
IEEE 24 RTS. Th
e cases fo
r
congestion in
transm
ission lines ha
ve
been consi
d
er
ed
a
s
s
u
mi
n
g
t
h
e
p
o
w
e
r
f
l
o
w
ma
x
i
mu
m r
a
t
i
n
g
i
n
t
h
e
co
rresp
ond
ing
lin
es b
e
l
o
w th
eir b
a
se
case
power flows
.
For creating the
c
o
ng
estion
,
th
e
fo
llowing
lin
es h
a
v
e
been
t
a
ke
n:
C
a
se 1:
Fo
r si
ngl
e l
i
n
e
(SL
)
con
g
est
i
o
n,
p
o
w
er
fl
o
w
rat
i
n
g o
f
23
rd
line c
o
nnected
bet
w
een buses
14 a
nd
16
has bee
n
t
a
ke
n as
2
.
6
0
p.
u.
co
m
p
ared
t
o
i
t
s
g
i
ven rat
i
n
g of
5
.
0
0
p
.
u
.
Case 2
:
Fo
r two
lin
e (2
L) cong
estion
case, th
e rating
o
f
1
8
th
l
i
n
e connect
e
d
bet
w
ee
n b
u
s
e
s 11 an
d 1
3
h
a
s bee
n
t
a
ken a
s
2.
25
p
.
u.
com
p
are
d
t
o
i
t
s
gi
ve
n rat
i
ng
o
f
5.
0
0
p
.
u
.
al
on
g
wi
t
h
pre
v
i
o
us c
o
ngest
e
d
l
i
n
e.
Case 3
:
Fo
r three lin
e
(3L) co
ng
estion
case, rating
o
f
11
th
l
i
n
e co
n
n
ect
ed
bet
w
ee
n
bus
es 7 a
n
d
8
has
be
e
n
tak
e
n
as
1
.
5
0
p.u
.
co
m
p
ar
ed
t
o
its
g
i
v
e
n
r
a
ti
n
g
of
1
.
75
p.u. alo
n
g
w
ith
p
r
ev
iou
s
t
w
o
congested
lin
es.
5.
1.
Gener
a
t
o
r Resc
hedul
i
n
g Wi
th
out
F
A
CT
S
Secu
re t
r
ansac
t
i
ons ha
ve bee
n
o
b
t
a
i
n
ed s
o
l
v
i
n
g
G
D
m
a
t
r
i
x
de
vi
at
i
on m
i
ni
m
i
zat
i
on as descri
bed i
n
sect
i
on I
I
. Th
e pro
p
o
se
d t
r
a
n
sact
i
o
ns an
d opt
i
m
al t
r
ansa
ct
i
ons are gi
v
e
n i
n
Tabl
e I
and I
I
. The
secure
t
r
ansact
i
o
ns
h
a
ve bee
n
i
n
c
o
rp
orat
e
d
cal
l
i
ng G
D
m
a
trix
in
GAMS from
M
A
TLAB
envi
ro
nm
ent
in C
C
m
i
nim
i
zat
i
on pr
o
b
l
e
m
as descri
be
d i
n
sect
i
on
II
I.
The
up
an
d do
w
n
gene
rat
i
o
n obt
ai
ned fo
r SL, 2L, 3
L
congestion cas
es are give
n in Table III
. In t
h
e t
a
bl
e base case opt
i
m
al po
wer ge
ne
rat
i
o
n
,
Pg an
d ne
w P
g
aft
e
r
rem
ovi
ng c
o
ng
est
i
on f
o
r al
l
con
g
est
i
o
n case
s
are al
so
gi
ve
n. T
h
e
gene
rat
o
rs
w
h
i
c
h a
r
e
part
i
c
i
p
at
i
n
g f
o
r t
h
e
con
g
est
i
o
n m
a
nagem
e
nt
wi
t
h
t
h
ei
r
up
an
d
do
w
n
gene
rat
i
o
n
,
P
g
,
ne
w P
g
are
al
so
gi
ve
n i
n
Tabl
e
II
I
fo
r al
l
lines conge
stion cases
. For t
w
o line a
n
d thre
e line cong
estion cases, t
h
e Pg,
ne
w Pg,
up
and down
ge
ne
ration
resche
d
u
l
i
n
g
h
a
s bee
n
gi
ve
n i
n
Ta
bl
e a
n
d
sh
ow
n i
n
Fi
gs.
4
f
o
r
3L
case.
Tab
l
e I Propo
sed
Bilateral Tran
saction
Matrix
es,
GD
ij
0
Val
u
e
o
f
t
r
a
n
sa
ct
i
on
bet
w
ee
n
gene
rat
o
r a
n
d
l
o
ad
b
u
s
(
p
.
u
)
G
(
1
,
1
)
=0
.5
G
D
(1
,2
)
=
0
.
3
G
D
(1
,3
)
=
0
.
3
G
D
(1
,1
5)
=0
.1
G
D
(1
,1
8)
=0
.4
G
D
(2
,1
0)
=0
.2
G
D
(2
,1
3)
=0
.3
G
D
(2
,1
5)
=0
.4
G
D
(2
,1
8)
=0
.5
G
D
(2
,1
9)
=0
.2
G
D
(7
,9
)
=
0
.
2
G
D
(7
,1
0)
=0
.2
G
D
(7
,1
3)
=0
.4
G
D
(7
,1
5)
=0
.5
G
D
(1
3,
18)
=1.5
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
. 2,
Fe
br
uar
y
20
1
2
:
7
5
– 89
84
Tab
l
e II
Secure Bilateral Tran
saction
Matri
x
Val
u
e
o
f
t
r
a
n
sa
ct
i
on
bet
w
ee
n
gene
rat
o
r a
n
d
l
o
ad
b
u
s
(
p
.
u
.
)
G
D
(
2
,18)
=.3
84 G
D
(1
3,
1
8
)=
1.
13
6
G
D
(2
1,8)
=.158
G
D
(
2
1
,
9
)
=.179
GD
(
2
1
,
13
)
=
.22
60
GD
(2
1,
1
5
)=
1.
46
GD
(2
1,
1
8
)=.
1
45
G
D
(2
2,1)
=.515
G
D
(2
2,2)
=.485
G
D
(
2
2
,
3
)
=.019
7
GD
(
2
2
,
4
)
=.019
7
GD
(
2
2
,
5
)
=.016
9
GD
(
2
2
,
6
)
=.016
9
GD
(
2
2
,
7
)
=.019
7
G
D
(2
2,8)
=.507
G
D
(
2
2
,
9
)
=.119
GD
(2
2,
1
0
)=.
5
56
GD
(2
2,
1
3
)=.
2
53
GD
(2
2,
1
5
)=.
1
24
GD
(2
2,
1
6
)=.
4
74
GD
(2
2,
1
9
)=.
1
74
GD
(
2
3
,
1
)
=.024
6
GD
(
2
3
,
3
)
=.088
0
GD
(
2
3
,
4
)
=.350
GD
(
2
3
,
5
)
=.338
1
G
D
(2
3,6)
=.663
G
D
(2
3,7)
=.605
G
D
(
2
3
,
8
)
=.189
G
D
(2
3,9)
=.576
G
D
(2
3,
1
0
)=.
4
20
GD
(2
3,
1
3
)=.
8
46
GD
(2
3,
1
4
)=.
9
70
GD
(
2
3
,
16
)
=
.02
63
GD
(2
3,
1
9
)=.
7
31
GD
(2
3,
2
0
)=.
6
40
T
ABLE
II
I
G
ENERA
T
ORS
UP A
ND DO
W
N
GENE
RA
TION FO
R CON
G
EST
I
ON MAN
AGEM
E
NT
case
SL
congestion case
2L
congestion case
3L
congestion case
Gen
.
1
1.
3524
1.
3524
0
0
1.
3524
1.
3524
0
0
1.
3524
1.
3524
0
0
2
0.
15
0.
3186
0.
1686
0
0.
15
0.
9343
0.
7843
0
0.
15
0.
95
0.
8
0
7
3
2.
9983
5
0
0.
0016
5
3
2.
9970
5
0
0.
0029
5
3
2.
7385
5
0
0.
2614
5
13
5.
91
5.
91
0
0
5.
91
5.
3474
9
0
0.
5625
1
5.
91
5.
1649
3
0
0.
7450
7
15
2.
15
2.
15
0
0
2.
15
2.
15
0
0
2.
15
2.
95
0.
8
0
16
1.
55
1.
55
0
0
1.
55
1.
3311
8
0
0.
2188
2
1.
55
0.
75
0
0.
8
18
4 4
0
0
4
4 0
0 4
3.
2
0
0.
8
21
1.
2613
5
1.
2613
5
0
0
1.
2613
51
1.
2613
5
0
0
1.
2613
5
0.
6678
7
0
0.
5934
8
22
3 2.
8330
8
0
0.
1669
2
3
3 0
0 3
3.
8
0.
8
0
23
6.
6
6.
6 0
0
6.
6
6.
6
0
0
6.
6 7.
4 0.
8
0
0
5
10
15
20
25
0
1
2
3
4
5
6
7
8
G
e
n
e
ra
t
o
r b
u
s
R
eal
pow
e
r
(
p
.
u
.
)
Pg
Pg
n
dp
gu
dp
gd
Fi
g. 4.
Ge
ne
rat
o
r
resc
he
dul
i
n
g fo
r 3L
c
o
nge
st
ed
case (
w
i
t
h
out
FAC
T
S)
For si
n
g
l
e
l
i
n
e con
g
est
i
o
n ge
n
e
rat
o
r at
bus
2
goe
s u
p
ge
nera
t
i
on an
d at
b
u
s
e
s 7 an
d 2
2
go
es do
w
n
ge
ner
a
t
i
o
n
.
For
2L c
o
nge
st
i
on,
ge
nerat
o
r
at
bus
2
goe
s u
p
ge
ne
rat
i
on
a
nd
at
bu
s
7
,
1
3
and 1
6
goes d
o
w
n
gene
rat
i
o
n. F
o
r
3L
c
o
nge
st
i
o
n
gene
rat
o
r
at
b
u
ses 2,
15
, 22
and
23
g
o
es
u
p
ge
nerat
i
o
n
a
n
d
at
bus
7, 1
3
, 1
6
,
1
8
a
n
d 21
g
o
e
s
do
w
n
gene
rat
i
o
n
.
T
h
e c
o
nges
t
i
on c
o
st
,
real
a
n
d
react
i
v
e
p
o
w
er
l
o
ss m
e
nt
i
one
d i
n
Ta
bl
e
VI
I
.
5.
2.
Gener
a
tor Resc
hedulin
g with
STAT
CO
M
The u
p
an
d d
o
w
n
gene
rat
i
o
n obt
ai
ne
d f
o
r si
ngl
e l
i
n
e, t
w
o l
i
n
es, t
h
re
e l
i
n
e con
g
est
i
o
n cas
es are gi
ve
n i
n
Tabl
e
IV.
I
n
t
h
e
t
a
b
l
e base case
o
p
t
i
m
a
l
powe
r
gene
rat
i
o
n,
Pg
an
d ne
w
Pg
aft
e
r rem
ovi
ng
co
ngest
i
o
n
f
o
r al
l
congestion cas
es are also given. The genera
tors which
are
p
a
rticip
ating
fo
r th
e cong
estio
n
m
a
n
a
g
e
m
e
n
t
with
t
h
ei
r
up
an
d
d
o
w
n
ge
nerat
i
o
n,
Pg
,
ne
w P
g
are al
s
o
s
h
o
w
n
i
n
Fi
g.
5
f
o
r
3
L
co
n
g
est
i
o
n.
For
t
w
o l
i
n
e
a
n
d
t
h
re
e
l
i
n
e co
nge
st
i
o
n
cases, t
h
e P
g
,
new
P
g
,
u
p
a
n
d
do
w
n
gene
ra
t
i
on
resche
d
u
l
i
n
g
are
al
so
gi
v
e
n i
n
Ta
bl
e.
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