Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
Vol.
4, No. 6, Decem
ber
2014, pp. 868~
881
I
S
SN
: 208
8-8
7
0
8
8
68
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 Secure-Coordinated Expansion
Planning of Generation and
Transmi
ssion Usi
n
g Gam
e
Th
eory and Minimum Singular
Value
Mehdi Z
a
reian Jahr
omi,
Mohsen
Tajdini
a
n,
Mojtab
a
Jalalp
our
Amirkabir Univ
ersity
of
Tecnolog
y
(Tehr
a
n Poly
techn
i
c)
No. 424, Hafez
Avenue, Tehr
an
1591634311, Ir
an
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
J
u
l 15, 2014
Rev
i
sed
Sep
15
, 20
14
Accepted Oct 10, 2014
A In this p
a
per
a novel method
have b
een
propo
sed for
expansio
n plann
i
ng
of genera
tion
an
d trans
m
is
s
i
on that cons
id
ered s
t
ati
c
s
ecur
i
t
y
of
the s
y
s
t
em
s
u
ch as
voltag
e
s
ecurit
y
m
a
rgin
And lo
adabi
lit
y
lim
it. In
the sa
m
e
stud
y
o
f
expansion plann
i
ng Security
con
s
traints of
th
e s
y
s
t
em
ar
e neg
l
e
c
ted
.
In th
is
stud
y
at th
e first step minimum singular
value technique is
used to
evaluate
voltag
e
securit
y
m
a
rgin and loadabolit
y l
i
m
it, in order to select b
e
st bus for
load in
crimination. After it, in or
der to Supply
the lo
ad, coordinated
expansion plann
i
ng of generatio
n and transmission is needed
, therefor th
e
strateg
i
c int
e
ra
ction betwe
e
n
tran
smission
compan
y
(TransCo) and
genera
tion
com
p
an
y (GenCo)
f
o
r Tran
smission expansion
plan
ning (TEP)
and genera
tion e
xpansion planni
ng (GEP) in a com
p
etitive
ele
c
tr
icit
y m
a
rke
t
is proposed usin
g Game Th
eor
y
(GT).
Keyword:
Gam
e
theory
Gene
rat
i
o
n e
x
p
a
nsi
o
n
pl
an
ni
n
g
M
i
nim
u
m
si
ng
ul
ar
val
u
e
Tran
sm
i
ssi
on expa
nsi
o
n
pl
an
ni
n
g
Vo
ltag
e
secu
rit
y
m
a
rg
in
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
:
Mo
j
t
ab
a Jalalpo
u
r,
Am
i
r
kabi
r
Uni
v
ersi
t
y
o
f
Tec
n
ol
o
g
y
(Te
h
ran
Pol
y
t
echni
c
)
N
o
.
4
2
4
,
H
a
f
e
z Av
enu
e
, Teh
r
an
1
591
634
311
,
I
r
a
n
Em
a
il: Jalalp
u
r
@au
t
.ac.ir
1.
INTRODUCTION
Po
wer
sy
st
em
rest
r
u
ct
uri
n
g
a
n
d
de
re
gul
at
i
o
n, i
n
t
r
od
uce
n
e
w
defi
ni
t
i
ons
and
m
e
t
hods t
o
t
h
e
p
o
w
er
sy
st
em
pl
anni
n
g
.
I
n
t
h
e m
ono
pol
y
p
o
we
r m
a
rket
,
t
h
e
deci
si
on
m
a
ker i
s
j
u
st
o
n
e
or
ga
ni
zat
i
on
w
h
i
c
h
de
ci
des
fo
r t
h
e
Ge
nera
t
i
on E
xpa
nsi
o
n
Pl
anni
n
g
(
G
E
P
) a
nd
Tra
n
sm
i
ssi
on E
x
pansi
on
Pl
an
ni
n
g
(
T
EP) al
t
o
get
h
e
r
.
Du
e
to
em
erg
e
n
c
e
o
f
co
m
p
etitio
n in
t
h
e
po
wer
mark
et, th
e
d
e
cisio
n
m
a
k
e
rs
o
f
GEP and
TEP
b
eco
m
e
sep
a
rated
suc
h
that the transm
ission com
p
any (Trans
Co) deci
de
s for TEP and the gene
ration com
p
any (GenC
o
) deci
des
fo
r GE
P. I
n
s
u
ch an e
nvi
ro
n
m
ent
,
t
h
e co
or
di
nat
i
o
n bet
w
e
e
n t
h
ese t
w
o o
r
ga
ni
zat
i
ons
be
com
e
s
m
o
re cruci
a
l
as
capacity expansion of each orga
nizati
on affects the other side capacity e
xpa
nsi
on and
as a conseque
nce the
profit of each
com
p
any is affected
.
In a c
o
m
p
etitive power m
a
rket w
ith ope
n access to the transmission
syste
m
, th
e g
e
n
e
ration
co
m
p
an
y is exp
ected
to
sup
p
l
y the lo
ad
with
ou
t
an
y con
g
e
stion
in
th
e t
r
ansmissio
n
lin
es. Tran
sm
i
ssio
n
co
m
p
an
ies are ob
lig
ed
to
prov
id
e a
c
o
nge
stion-free, reliable
an
d
non
-d
iscrim
in
ativ
e p
a
t
h
fo
r t
h
e
ge
ne
rat
i
on c
o
m
p
ani
e
s
t
o
t
h
e
co
ns
um
ers
of
el
ect
ri
ci
t
y
. There
f
ore
,
t
h
e t
r
a
n
sm
i
ssi
o
n
net
w
or
ks m
u
st
b
e
reg
u
l
a
t
e
d so t
h
at
opt
im
al
operat
i
on o
f
p
o
w
e
r
sy
st
em
i
s
perf
orm
e
d. In a rest
r
u
ct
ure
d
p
o
we
r m
a
rket
, Genc
o
decides
on capacity, place and tim
e
of
c
ons
truction of
ne
w power
plant
s
at
its own di
scretion. T
h
e c
a
pacity
expa
nsi
o
n
st
rat
e
gy
use
d
by
G
e
nC
o i
n
vol
ves
Tran
sC
o a
n
d
p
r
o
d
u
ces
unce
r
t
a
i
n
t
y
and
chal
l
e
nge
s t
o
TEP.
On
t
h
e
o
t
h
e
r h
a
n
d
, th
ere is n
o
ab
so
l
u
te certain
ty th
at a tran
sm
i
ssi
on net
w
o
r
k ca
n pr
o
v
i
d
e
suffici
ent capacity for ne
w
gene
rat
i
o
n ca
p
aci
t
y
const
r
uct
e
d
by
a
GenC
o
[
1
-
5
]
.
Thi
s
i
n
t
e
ra
ct
i
o
n bet
w
een
Gen
c
o an
d Tra
n
C
o
l
eads t
o
a ne
w
m
e
t
hod o
f
G
E
P and
TEP t
h
at
consi
d
ers
b
o
t
h
en
tities’ p
r
o
f
it. In
su
ch an
en
v
i
ron
m
e
n
t, Gam
e
th
eo
ry see
m
s to
b
e
a u
s
efu
l
m
e
th
od
to
pred
ict th
e
st
rat
e
gi
es o
f
Genc
o a
nd T
r
ansC
o
fo
r ge
n
e
rat
i
on e
x
pans
i
on ca
paci
t
y
(GEC
) a
nd t
r
a
n
sm
i
ssi
on exp
a
nsi
o
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJECE Vol. 4, No. 6, D
ecem
ber 2014
:
868 – 881
8
69
cap
acity (TEC) an
d
ev
alu
a
t
e
th
e p
o
wer
mark
et eq
u
ilibriu
m
[6
-10
]
. TEP and
GEP h
a
v
e
b
een
stu
d
i
ed
sep
a
rately in
man
y
articles
b
u
t
th
e in
teractio
n
b
e
t
w
een
t
h
em
in
a d
e
reg
u
l
ated
m
a
rk
et
is stu
d
i
ed
in
a few
pape
rs.
In [
1
1]
i
n
or
der t
o
st
udy
t
h
e rel
a
t
i
o
nshi
p bet
w
een
gene
rat
i
on a
n
d t
r
ansm
i
ssi
on
i
nvest
m
e
nt
, a t
h
ree-
st
age m
odel
i
s
pre
s
ent
e
d.
A st
u
d
y
i
n
[1
2]
sh
ow
s t
h
at
i
n
a de
reg
u
l
a
t
e
d p
o
w
e
r m
a
rket
, t
h
e de
g
r
ee o
f
co
m
p
etitio
n
b
e
tween
d
i
fferen
t
g
e
n
e
rat
o
rs is
d
e
p
e
nd
en
t
on
t
h
e cap
acity o
f
tran
sm
issio
n
lin
es.
Th
e in
teractio
n
bet
w
ee
n t
r
a
n
s
m
i
ssi
on an
d
ge
nerat
i
o
n e
x
pan
s
i
on
pl
a
nni
ng
usi
n
g
gam
e
t
h
eory
, i
s
st
u
d
i
e
d
i
n
a t
h
ree-
b
u
s
sy
st
em
in
[13
]
. To
stud
y th
e strateg
i
c in
teractio
n
between
Ge
nC
o
and Tra
n
sC
o,
a si
ngl
e-st
a
g
e det
e
rm
i
n
i
s
t
i
c
m
odel
i
s
pr
o
pose
d
i
n
[
1
4]
. The ex
pa
ns
i
on be
ha
vi
o
r
s of b
o
t
h
GenC
o
and Tra
n
sC
o are sim
u
l
a
t
e
d usi
n
g C
o
ur
not
m
odel
.
Th
e equ
ilib
ri
um in
th
e p
o
w
er m
a
rk
et in
[14
]
is o
b
t
ain
e
d
u
s
ing
Mix
e
d
Co
m
p
le
m
e
n
t
arity Pro
b
l
em
ap
p
r
o
a
ch
and
t
h
e
p
r
o
p
o
s
e
d m
odel
i
s
a
p
pl
i
e
d t
o
a t
h
ree
-
b
u
s sy
st
em
and t
h
e
IE
EE
1
4
-
bus
sy
st
em
.
On
t
h
e
o
t
h
e
r
h
a
nd
, power syste
m
secu
rity wh
ich
is th
e
ab
ility o
f
th
e
p
o
wer system
to
with
stan
d
di
st
ur
ba
nces a
g
ai
nst
a
n
y
vi
ol
at
i
on i
n
sy
st
em
op
erat
i
n
g c
o
n
d
i
t
i
ons
sh
o
u
l
d
be c
onsi
d
ere
d
i
n
t
h
i
s
new
m
e
tho
d
o
f
capaci
t
y
expa
nsi
o
n. T
h
e a
f
orem
ent
i
one
d
st
udi
es i
n
t
h
e
fi
el
d o
f
gene
rat
i
on a
n
d t
r
a
n
sm
i
ssi
on ex
p
a
nsi
o
n
pl
an
ni
n
g
ha
ve
n’t
c
o
nsi
d
e
r
ed
po
we
r sy
st
em
securi
t
y
.
H
o
we
ver
i
n
t
h
i
s
pa
p
e
r,
fi
rst
t
h
e l
o
a
d
pat
t
e
rn
of
a s
i
x-
bus
p
o
wer syste
m
is i
m
p
r
o
v
e
d
an
d
th
en
th
e
b
e
st bu
s for lo
ad
in
crem
en
t is d
e
ter
m
in
ed
u
s
ing
a sensitiv
ity
characte
r
istic of
ANN.
Afte
rwa
r
ds t
h
e strategic in
t
e
ract
i
on
bet
w
ee
n t
r
ansm
i
ssi
on co
m
p
any
(Trans
C
o
) an
d
g
e
n
e
ration
com
p
an
y (Gen
C
o
)
for TEP and
GEP in
a
com
p
et
itiv
e elect
ricity
m
a
rk
et i
s
p
r
op
osed
u
s
i
n
g
Gam
e
The
o
ry
.
It
s
h
o
u
l
d
be
t
a
ke
n i
n
t
o
c
onsi
d
erat
i
o
n t
h
at
t
h
e
l
o
a
d
di
scuss
e
d
i
n
t
h
i
s
pa
per
i
s
a m
a
nagea
b
l
e
l
o
ad
w
h
i
c
h
can
be inc
r
eas
ed or
decrea
se
d using re
ward or
pe
nalty.
The pa
per is
orga
nized as
follows.
In section
II, t
h
e
lo
ad
ab
ility
li
mit as a
secu
rity in
d
e
x
is in
tro
d
u
c
ed
. Th
e
neu
r
al n
e
twork
u
s
ed
for th
e im
p
r
o
v
e
m
e
n
t
o
f
lo
ad
p
a
ttern
is p
r
esen
ted
in
sectio
n III. Ap
p
lication
o
f
Co
ur
n
o
t
m
odel
of
d
u
o
p
o
l
y
f
o
r
TEP
a
n
d
GEP
i
s
di
scu
ssed i
n
sect
i
on I
V
. Se
ct
i
on V p
r
o
p
o
s
e
s Gam
e
Theo
ry
for s
o
l
v
i
ng
TEP an
d GEP
pr
o
b
l
e
m
.
A case st
udy
i
s
pre
s
ent
e
d
in
section
VI. Fin
a
lly,
con
c
lusio
n
s
are
prese
n
ted i
n
section
VII.
2.
LOAD
ABILI
T
Y LIM
I
T A
N
D
V
O
LTAG
E
SEC
URIT
Y
M
A
R
G
IN
A
S
A
SEC
U
R
I
TY I
NDE
X
In
o
r
d
e
r t
o
stu
d
y
th
e static v
o
ltag
e
stab
ility o
f
th
e power syste
m
, lo
adab
ility l
i
m
it o
f
syste
m
i
s
p
r
op
o
s
ed
as
vo
ltag
e
stab
ility
in
dex fo
r syste
m
secu
rity evalu
a
tio
n
.
Lo
adab
ility li
mit o
f
a
p
o
wer system
is
defi
ned as the m
a
xim
u
m
load, whic
h can be
im
posed on
buses of a system
without loss
of voltage stability.
t
h
e m
i
nim
u
m
si
ng
ul
ar
val
u
e t
echni
que i
s
us
ed t
o
e
v
al
uat
e
vol
t
a
ge
secu
ri
t
y
m
a
rgi
n
f
o
r e
ach st
at
e of l
o
adi
n
g.
The m
i
nim
u
m si
ng
ul
ar val
u
e of t
h
e l
o
a
d
fl
o
w
Jaco
bi
an
m
a
t
r
i
x
i
s
obt
ai
ned f
r
om
sol
v
i
ng t
h
e l
o
ad
fl
o
w
equations
that
are s
h
own in (1) and
(2):
1
cos
n
ii
j
i
j
i
j
i
j
j
PV
V
Y
(1
)
1
sin
n
ii
j
i
j
i
j
i
j
j
QV
V
Y
(2
)
Whe
r
e:
: Activ
e power transfer fro
m
b
u
s
I;
: Reactiv
e po
wer tran
sfer
from
b
u
s
i;
:
V
o
l
t
a
ges
of
bus
i
a
n
d
b
u
s
j;
:
Ad
m
i
ttan
ces fro
m
b
u
s
i to
b
u
s
j
;
:
a
n
gl
e o
f
i
m
peda
nces
f
r
om
bus
i
t
o
b
u
s
j;
:
Ad
m
i
ttan
ces fro
m
b
u
s
i to
b
u
s
j
;
:
angl
e o
f
vol
t
a
ge
at
bu
s
i
a
n
d
bus
j.
An
d:
12
34
JJ
P
JJ
QV
(3
)
P
: Sm
a
ll d
e
v
i
atio
n in
active
p
o
wer;
Q
: Sm
a
ll d
e
v
i
atio
n in
reactiv
e
p
o
wer;
i
P
i
Q
,
ij
VV
ij
Y
ij
ij
Y
,
ij
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
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:
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8-8
7
0
8
A S
e
cu
re-
C
oo
rd
ina
t
ed
Expansio
n Plan
n
i
n
g
o
f
Gen
e
ra
tion
a
n
d
Tran
smissio
n
Usin
g
…
(M
ojtaba Ja
la
lpou
r)
87
0
14
J
J
: Jacobian m
a
trix elem
ents
of l
o
ad fl
o
w
e
q
ua
t
i
ons;
: Sm
a
ll d
e
v
i
atio
n
s
in vo
ltag
e
an
g
l
e;
V
: Sm
a
ll d
e
v
i
atio
n
s
in vo
ltag
e
mag
n
itu
d
e
.
3.
M
I
NIMUM
SIN
G
U
L
AR
VA
LU
E
One
of t
h
e ef
f
ect
i
v
e
m
e
t
hods
of
det
e
rm
i
n
at
ion
of
v
o
l
t
a
ge c
o
l
l
a
pse p
o
i
n
t
i
s
m
i
nim
u
m
si
n
gul
a
r
val
u
e
of
t
h
e l
o
ad
fl
o
w
Jac
o
bi
an m
a
t
r
i
x
.
In
t
h
i
s
pa
per
,
m
i
nim
u
m
si
ng
ul
ar
val
u
e
t
echni
q
u
e
i
s
u
s
ed t
o
sel
ect
i
v
e
best
bus
f
o
r l
o
adi
n
g.
On
Thi
s
wa
y
l
o
ad i
n
c
r
ease
s
i
n
desi
red
bus when l
o
ad is
fixe
d at another bus t
h
en cal
culate
v
o
ltag
e
security
m
a
rg
in
with
ap
pro
ach
techn
i
qu
e.
Wh
en
vo
ltag
e
security
m
a
rg
in
is calcu
lated
fo
r all of th
e
state, the state with m
a
xim
u
m VSM is
t
h
e
be
st
st
at
e for
l
o
a
d
i
n
g. Fl
o
w
cha
r
t
of t
h
i
s
p
r
oce
s
s has
bee
n
s
h
o
w
n i
n
figu
re
1
.
Equ
a
tio
n
(3
) can
be written
b
y
eq
u
a
tion
(4
). Un
d
e
r no
rm
al o
p
e
rating
cond
itio
n
,
ap
p
licatio
n
of
sin
g
u
l
ar
v
a
lue
d
eco
m
p
o
s
ition
is app
lied
to Jaco
b
i
an
m
a
tr
ix
th
at h
a
s
b
een sho
w
n
i
n
eq
u
a
tion
in (5
) to
(9
):
P
J
P
(4
)
2(
1
)
1
n
T
jj
j
j
J
uv
(5
)
2(
1
)
11
1
n
T
jj
j
j
PP
JV
U
V
(6
)
1
2(
1
)
2(
1
)
2
(
1
)
T
nn
n
P
Vu
V
(7
)
Let
2(
1
)
n
P
U
(8
)
The
n
2(
1
)
2(
1
)
n
n
V
V
(9
)
Whe
r
e:
n
:
nu
m
b
er
o
f
bu
ses i
n
th
e pow
er n
e
t
w
or
k,
u
j
,v
j
: singu
lar
v
ectors th
at
u
j
and
v
j
ar
e
th
e
j
th
co
lu
m
n
s of
un
itary m
a
trix
,
: Po
sitiv
e
real sin
g
u
l
ar
v
a
lu
es.
Ab
o
v
e anal
y
s
i
s
cl
earl
y
shows
where t
h
e m
i
nim
u
m
si
ngul
ar val
u
e o
f
t
h
e l
o
ad fl
o
w
Jaco
bi
an m
a
t
r
i
x
i
s
al
m
o
st zero
,
this su
itab
l
e in
d
i
cato
r
d
e
tects th
e clo
s
en
ess of p
o
wer system o
p
e
ratin
g
con
d
ition
to
th
e v
o
ltag
e
co
llap
s
e po
in
t.
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I
S
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:
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08
IJECE Vol. 4, No. 6, D
ecem
ber 2014
:
868 – 881
8
71
Fi
gu
re
1.
Fl
o
w
chart
of
be
st
b
u
s Sel
ect
i
o
n
fo
r l
o
a
d
i
n
g
4.
APPLICATION OF COURNOT
DUOPOLY
MODE
L TO TEP
AND GEP
PROBLEM
In t
h
e C
o
u
r
n
o
t
du
o
pol
y
m
odel
,
t
w
o c
o
m
p
ani
e
s pr
od
uce a h
o
m
ogen
o
u
s
pr
od
uct
an
d wi
t
h
out
k
n
o
wi
n
g
t
h
e deci
si
o
n
o
f
t
h
e ot
he
r o
n
e
, t
h
ey
m
u
st
deci
de h
o
w
m
u
ch
pr
od
uct
t
h
ey
sho
u
l
d
pr
od
uce t
o
o
b
t
a
i
n
t
h
e
m
a
xim
u
m
pro
f
i
t
[1
8]
. A
s
t
h
e
C
o
ur
not
m
o
d
e
l
i
s
som
e
how
si
m
i
l
a
r t
o
t
h
e
TEP a
n
d
GE
P be
ha
vi
o
r
s, i
t
can
b
e
ap
p
lied to th
e
p
r
ob
lem
o
f
TEP and GEP i
n
th
e restru
ct
ure
d
po
we
r m
a
rk
et. The
fi
rst si
m
ilarity
betwe
e
n t
h
e
co
urn
o
t
m
o
d
e
l an
d
TEP and
GEP
p
r
o
b
l
em
is th
e ex
p
a
n
s
ion
cap
acity o
f
TEP and
GEP, wh
ich
is qu
antity
in
C
o
u
r
n
o
t
m
ode
l
.
The sec
o
nd
sim
i
l
a
ri
ty
i
s
that
i
n
a
p
ool
m
a
rket
t
h
e
pr
od
uct
of eac
h
ge
nerat
i
o
n
un
i
t
i
s
a
hom
oge
no
us
p
r
o
d
u
ct
w
h
i
c
h
i
s
o
ffe
red
by
G
e
nC
o
at
eac
h s
u
p
p
l
y
bi
d.
The
C
o
u
r
n
o
t
m
odel
m
a
xim
i
zes t
h
e p
r
o
f
i
t
of eac
h c
o
m
p
any
an
d
defi
nes
t
h
e am
ount
o
f
t
h
ei
r
o
u
t
p
ut
s.
In t
e
rm
s of m
a
t
h
em
ati
cal
for
m
ul
as, t
h
e opt
im
al
q
u
a
n
tity p
a
ir
**
12
(,
)
qq
is th
e Cou
m
o
t
equ
ilib
riu
m
, if for fi
rm
1
,
*
1
q
so
lv
es (1
0):
*
11
2
ma
x
(
,
)
qq
(1
0)
Whe
r
e:
i
: Pro
f
it
for
firm
i; i=1
,
2
;
i
q
: Qu
an
tities produ
ced b
y
firm
i;
*
i
q
: Op
tim
al q
u
a
ntities p
r
od
uced b
y
firm
.
The pr
o
f
i
t
fu
nc
t
i
on fo
r fi
rm
1
can be rep
r
ese
n
t
e
d by
(1
1):
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I
J
ECE
I
S
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:
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8-8
7
0
8
A S
e
cu
re-
C
oo
rd
ina
t
ed
Expansio
n Plan
n
i
n
g
o
f
Gen
e
ra
tion
a
n
d
Tran
smissio
n
Usin
g
…
(M
ojtaba Ja
la
lpou
r)
87
2
11
2
1
2
1
11
(,
)
(
)
(
)
qq
p
q
q
q
cq
(1
1)
Whe
r
e:
(.)
p
: Mark
et
p
r
ice
for ag
greg
ate qu
an
tity;
(.
)
i
C
:cost functi
on for firm
i.
5.
APPL
YING
GAME THE
O
RY T
O
TE
P AND GEP
PROBLEM
5.
1.
Assum
p
ti
ons
In o
r
der to f
o
rm
ulate
the stra
tegic interac
tion betwee
n
GenC
o an
d T
r
ansC
o in the
expa
nsio
n
plan
nin
g
gam
e
, som
e
assum
p
tions
are c
o
nsid
ered
.
TransCo is owner of all transmission lines.
Power m
a
rket structure is Poolco
-type in which GenCo offers th
e price and
quantity bids to a
Inde
pende
n
t Syste
m
Operat
or (ISO). T
h
en, t
h
e
IS
O dispatc
h
es
th
e
chea
pest po
wer
co
nside
r
i
ng
operational constraints like transf
er capacity li
m
i
tations and en
ergy balance
constraints
.
Dem
a
nd is c
o
nsidere
d
as
a c
o
nstant l
o
ad.
Ex
pansi
on stra
tegies of Tra
n
s
C
o a
nd
Genc
o
are discrete. T
h
is
m
eans
that TransCo can ei
ther expa
nd its
line capacity or m
a
intain
the initial tran
sfer capacity of the l
i
ne.
Ex
pansi
o
n
be
h
a
vio
r
s
of
Tra
n
s
C
o a
n
d
Ge
nC
o
are
base
d
on
C
o
u
r
n
o
t m
odel.
5.
2.
Problem For
m
ulati
o
n
As p
r
ofit m
a
xim
i
zation is the m
a
in pur
p
o
se
of eac
h si
de o
f
the
gam
e
theo
ry
an
d
pr
ofit is the
diffe
re
nce bet
w
een reve
nues
and costs
,
the
reve
nue of
the TransC
o for each line is given by a congestion
charge of that line [19] which is
the differe
nce betwee
n local
m
a
rginal
prices (LM
P
s)
[20
-
21]
. The
sum
of
congestion c
h
a
r
ges
of eac
h line is
total revenue
of TransC
o.
In
o
r
de
r t
o
m
a
ke the
p
r
oblem
m
o
re sim
p
le, Tran
sco
total c
o
st is
not c
o
nsi
d
ere
d
.
S
o
,
pr
o
f
it of t
h
e T
r
ans
C
o ca
n
be e
x
p
r
esse
d a
s
(
1
2
)
:
()
Tj
i
i
j
ij
P
(1
2)
Whe
r
e:
T
: pr
ofit
of
the
Transco in
$/hr
;
i
: LM
P at
no
de
i in
$/M
W
hr;
j
: LMP at node
j
in $/M
W
hr;
ij
P
: Active
power flow
from
node i to
node
j in
M
W
.
As
GEP
is pe
rf
orm
e
d at n
o
d
e
i, The
Ge
nC
o
pr
ofit
fr
om
GEP is o
b
tained us
in
g (1
3)
:
()
(
)
Gi
G
G
Pc
P
i
G
(1
3)
Whe
r
e:
G
: pr
ofit
of
Ge
n
C
o in
$/
hr;
i
: LMP at node
i in $/M
W
hr;
G
P
: active power
generation in
M
W
;
()
G
cP
: qua
d
r
atic cost
f
unctio
n
of
act
ive p
o
w
er
ge
ne
ration
in
$/h
r
;
()
iG
: investm
e
nt co
st in term
s of
g
e
neratio
n e
x
pa
nsio
n ca
pacity
in $/
hr
.
5.
3.
Soluti
on
Me
th
od
olo
g
y
The
follo
win
g
notatio
ns a
r
e
u
s
ed i
n
the
sol
u
tion m
e
thod
ol
o
g
y
.
i
T
S
: i
th
expa
nsio
n
strategy
o
f
T
E
P
; i= 1,
2,
...
,n;
j
G
S
: j
th
ex
pa
nsio
n
strategy
o
f
GE
P;
j=1
,
2
,
….
.,
n;
T
:transm
ission expa
nsi
o
n
capac
ity (TEC) in MW
;
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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:
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088
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08
IJECE Vol. 4, No. 6, D
ecem
ber 2014
:
868 – 881
8
73
G
:
Gene
rat
i
o
n e
xpa
nsi
o
n
capac
i
t
y
(GEC
) i
n
M
W
;
0
T
: in
itial TEC in
M
W
;
0
G
: in
itial GEC in M
W
;
T
: in
crem
en
t in
TEC in
M
W
;
G
: in
crem
en
t in
GEC in M
W
;
T
:
ma
x
i
mu
m T
E
C
i
n
MW
;
G
:
ma
x
i
mu
m G
E
C
i
n
MW
.
Algo
rith
m
o
f
fin
d
i
ng
t
h
e
Nash
eq
u
ilibriu
m
a
m
o
n
g
all po
ssib
le co
m
b
in
atio
n
s
of ex
p
a
n
s
i
o
n strateg
i
es
is sh
own
i
n
flowch
art of Figure 7
and
th
e C
o
u
r
no
t eq
u
ilib
riu
m
is
fo
und
b
y
u
s
ing
an
iterat
i
v
e
search
pro
c
ed
ure
[2
2]
as s
h
ow
n i
n
Fi
gu
re
8.
6.
CASE ST
UDY
The
pr
o
pose
d
al
go
ri
t
h
m
i
s
appl
i
e
d t
o
a si
x
-
bus
sy
st
em
show
n i
n
Fi
g
u
r
e
9.
The
Dat
a
o
f
t
h
e si
x-
b
u
s
sy
st
em
i
s
gi
ven i
n
Tabl
es 1 t
o
3.
In t
h
e
pr
o
pos
ed m
e
t
hod,
fi
rst
t
h
e m
o
st
app
r
op
ri
at
e bus
fo
r l
o
ad i
n
c
r
e
m
ent
i
s
sel
ect
ed usi
n
g
m
i
nim
u
m
si
ngul
ar
val
u
e t
ech
ni
q
u
e. T
h
e m
i
nim
u
m
si
ngul
ar
val
u
e t
ech
ni
qu
e sho
w
s
b
u
s 2
i
s
t
h
e
best and the m
o
st suitable
bus for loa
d
inc
r
e
m
ental becau
se
this state has
maxim
u
m
VS
M, this issue is
clearly
sh
own
in
Figure 2
.
Afterward
s
, in
o
r
d
e
r t
o
su
pp
ly th
e in
cre
m
en
ted
lo
ad
i
n
th
e selected
b
u
s
, Gam
e
Th
eo
ry is
u
s
ed
to stud
y th
e strateg
i
c in
t
e
ractio
n b
e
t
w
een
th
e TEP an
d GEP.
Fig
u
r
e
2
.
VSM and
Min Singu
lar
V
a
lu
e of
s
y
ste
m
when load inc
r
ease
at
bus
2
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISS
N
:
2088-8708
A
S
e
cu
re-
C
oo
rd
ina
t
ed
Expansio
n
Plan
n
i
n
g
o
f
G
e
n
e
ra
tion
a
n
d
Tran
sm
issi
o
n
U
s
i
n
g
…
(Mo
jta
ba
Ja
la
l
pou
r)
87
4
Fig
u
r
e
3
.
VSM and
Min Singu
lar
V
a
lu
e of
s
y
ste
m
when load inc
r
ease
at
bus
3
Fig
u
r
e
4
.
VSM and
Min Singu
lar
V
a
lu
e of
s
y
ste
m
when load inc
r
ease
at
bus
4
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJECE Vol. 4, No. 6, D
ecem
ber 2014
:
868 – 881
8
75
Fig
u
r
e
5
.
VSM and
Min Singu
lar
V
a
lu
e of
s
y
ste
m
when load inc
r
ease
at
bus
5
Fig
u
r
e
6
.
VSM and
Min Singu
lar
V
a
lu
e of
s
y
ste
m
when load inc
r
ease
at
bus
6
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I
J
ECE
I
S
SN:
208
8-8
7
0
8
A S
e
cu
re-
C
oo
rd
ina
t
ed
Expansio
n Plan
n
i
n
g
o
f
Gen
e
ra
tion
a
n
d
Tran
smissio
n
Usin
g
…
(M
ojtaba Ja
la
lpou
r)
87
6
0
GG
0
TT
TT
GG
TT
T
GG
G
(,
)
TG
SS
Figu
re 7.
Flo
w
chart of
the str
a
tegic
interacti
o
n bet
w
een Transc
o a
n
d Ge
nCo
Figu
re 8.
Flo
w
chart of
th
e C
o
urnot sol
u
tion
algorithm
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
IJECE Vol. 4, No. 6, D
ecem
ber 2014
:
868 – 881
8
77
Figure
9. Six-bus system
Table 1. Ge
ner
a
tor Data
Generator
Min
Generation
Po
wer (M
W)
Min
Generation
Po
wer (M
W)
G1 50
200
G2 37.
5
68
G3 45
73
Table 2.
Li
ne Data
From
Bus
To Bus
R (pu)
X (pu)
Transm
ission
Capacity
(MW
)
1 2
0.
1
0.
2
15.
28
1 4
0.
05
0.
2
40.
32
1 5
0.
08
0.
3
30.
63
2 3
0.
05
0.
25
20.
2
2 4
0.
05
0.
1
42.
6
2 5
0.
1
0.
3
31.
42
2 6
0.
07
0.
2
24.
6
3 5
0.
12
0.
26
28
3 6
0.
02
0.
1
50.
4
4 5
0.
2
0.
4
18.
1
5 6
0.
1
0.
3
21.
6
Table 3.
B
u
s D
a
ta
Bus
Number
Bus Type
V
o
ltage
(p
u V)
P
gen
(p
u M
W
)
P
l
oad
(p
u M
W
)
λ
( $
/
M
W
hr)
1 Swing
1.
05
-
-
12.
492
2 Gen.
1.
05
0.
5
0
11.
565
3 Gen.
1.
07
0.
6
0
11.
877
4 L
o
ad
-
0
70
15.
674
5 L
o
ad
-
0
70
12.
939
6 L
o
ad
-
0
70
12.
206
A.
Selection of
the best bus
for
load increment
As shown in
Figure 2 the
best loadability li
mit and
the
most appropriate bus
for load increm
ent, is
bus
2
because in t
h
is state l
o
adability li
mi
t (
ma
x
P
) an
d m
i
nim
u
m
eigenval
u
e
o
f
Jac
o
bean
m
a
trix (
mi
n
) of
po
we
r fl
ow
eq
uation
s
that is
prese
n
ted
in
(
1
4)
are t
h
e m
o
st secu
re state a
m
ong ot
her
states.
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