Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 5
,
O
c
tob
e
r
201
6, p
p
. 2
048
~206
3
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
5.1
025
9
2
048
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
Decomp
ositi
o
n-
Coordin
a
ting Method for Parall
el Solution of a
Multi Area Combined Economic
E
m
ission Dispat
ch Problem
Senthi
l
Kri
s
h
n
amur
th
y,
R
a
yni
t
c
h
k
a
T
z
on
eva
Center
for Substation
Automatio
n and
Energ
y
M
a
nagement S
y
st
ems, Department
of Electr
i
ca
l, Electronics and
Co
mputer
Engineering, Cape Peni
nsula Un
iversity
of
Tech
nolog
y
,
South
Africa
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Feb 4, 2016
Rev
i
sed
Jun
30,
201
6
Accepte
d
J
u
l 20, 2016
Multi-ar
ea Com
b
ined Econom
ic Em
ission Dispa
t
ch (MACEED) problem
is
an optimization
task in
power s
y
stem
operation f
o
r allocating
the amount o
f
genera
tion to
th
e com
m
itted uni
ts within
the s
y
s
t
em
areas
. Its ob
jec
tive
is to
m
i
nim
i
ze
th
e
fu
el cos
t
and
th
e quantit
y of em
is
s
i
ons
s
ubject to
the
power
balan
c
e
,
gen
e
ra
t
o
r lim
its,
trans
m
ission line a
n
d tie
-lin
e const
r
aints.
The
solutions of th
e MACEED problem in
th
e co
nditions of d
e
regulation
ar
e
difficu
lt, due to the model size,
nonlin
ear
ities, and
the big
number of
inter
c
onnections
, and require intensiv
e computations in real-time. High-
Performance Co
mputing (HPC) gives possi
bili
ti
es for th
e redu
c
tion of
the
problem complexity
and the time for
calculation b
y
th
e use of
parallel
processing techn
i
ques for
runnin
g
advan
ced
app
l
ication
programs
efficien
tly
,
reli
abl
y
and qui
ckl
y
.
Th
es
e app
l
ica
tions
ar
e con
s
idered
as
ver
y
new in t
h
e
power s
y
s
t
em
c
ontrol c
e
nt
ers
b
ecaus
e
ther
e
are
not av
ail
a
bl
e o
p
tim
izat
ion
methods and sof
t
ware based on them
that can solve the MACEED problem
in par
a
ll
el,
pa
yi
ng att
e
nti
on to
t
h
e ex
is
tenc
e of
the power
s
y
s
t
e
m
areas
and
the ti
e-lin
es
bet
w
een them
. A deco
mposition-co
ordinating meth
od based on
Lagrang
e
’s function is dev
e
lo
ped in
this pa
per.
Investig
atio
ns of the
performance of
the method
ar
e
done
using
I
E
EE ben
c
hmark p
o
wer s
y
stem
models.
Keyword:
Decom
posi
t
i
o
n
-
co
or
di
nat
i
n
g
m
e
t
hod
Interc
o
nnecte
d
p
o
we
r sy
stem
s
Lagra
n
ge’s
m
e
t
h
o
d
Mu
lti-area com
b
in
ed
economic
em
i
ssi
on di
s
p
a
t
ch p
r
obl
em
Parallel co
m
p
utin
g
Power system
d
e
regu
latio
n
Copyright ©
201
6 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
:
Sent
hi
l
Kri
s
hn
am
urt
h
y
,
C
e
nt
er
fo
r S
u
b
s
t
a
t
i
on
Aut
o
m
a
t
i
on a
n
d
E
n
er
g
y
M
a
nagem
e
nt
Sy
st
em
s,
Depa
rtem
ent of Electrical, El
ectroni
cs an
d
C
o
m
put
er E
ngi
neeri
n
g
,
C
a
pe Pe
ni
ns
ul
a U
n
i
v
e
r
si
t
y
of
Tech
n
o
l
o
gy
,
P.O.Box
:
1
906, Bellv
ille, South
Africa
-7
535
.
kse
n
t
h
i
l
v
pm
@gm
a
il
.com
1.
INTRODUCTION
Dere
g
u
l
a
t
i
on
of
t
h
e
el
ect
ri
ci
t
y
i
ndust
r
y
h
a
s bee
n
de
pl
o
y
e
d i
n
m
a
ny
cou
n
t
r
i
e
s
t
o
i
m
prove t
h
e
eco
no
m
i
c efficien
cy o
f
po
wer system
o
p
e
ratio
n
[1
]. Electric u
tility syst
e
m
s are in
terco
n
n
ected
t
o
ach
i
ev
e
high operating efficiency and to
pr
od
uce
cheap el
ect
ri
ci
t
y
wi
t
h
m
i
n
i
m
u
m
pro
duct
i
on c
o
st
, m
a
xim
u
m
reliab
ility, an
d
b
e
tter
o
p
e
ratin
g
cond
itio
n
s
[2
]. Th
e
term
m
u
lti-area p
o
wer syste
m
stan
d
s
fo
r th
e
in
terconn
ected p
o
wer
syste
m
. In
a m
u
lti-area p
o
wer
syst
em
, g
e
n
e
ratio
n
an
d
l
o
ads are co
ord
i
n
a
ted
with
each
ot
he
r t
h
r
o
ug
h
t
h
e t
i
e
-l
i
n
es am
ong t
h
e area
s. A l
o
a
d
cha
n
ge i
n
any
one
of t
h
e a
r
eas i
s
t
a
ken care
of
by
al
l
g
e
n
e
rators in
all areas. Simila
rly, if a g
e
n
e
rato
r is lo
st
i
n
o
n
e co
nt
r
o
l
area
, g
ove
r
n
i
n
g act
i
on
fr
om
gener
a
t
o
r
s
in
all con
n
ected
areas
will in
crease
g
e
n
e
ration
o
u
t
p
u
t
s to
m
a
k
e
-up
th
e m
i
s
m
atch
[3
].
Th
e obj
ectiv
e
o
f
M
u
lti-Area Co
m
b
in
ed
Eco
n
o
m
ic
E
m
issio
n
Disp
atch
(MACEED) pro
b
l
em
is to
determ
ine the am
ount
of the
powe
r ge
nerat
e
d by each
ge
nerator i
n
a system
and the power tra
n
sfer be
tween
th
e areas so
as to
min
i
m
i
ze
t
h
e to
tal g
e
n
e
ratin
g
co
st
with
ou
t v
i
o
l
ating
the li
mitatio
n
s
on
th
e
p
o
wer
pro
d
u
c
ed
by
t
h
e ge
ner
a
t
o
rs a
n
d o
n
t
h
e
am
ount
of t
i
e
l
i
n
e po
we
r t
r
an
sfer
. I
n
a m
u
l
t
i
-area
po
we
r sy
st
em
, t
h
e act
ual
l
o
cal
gene
rat
i
o
n m
a
y
not
be bal
a
n
ced wi
t
h
t
h
e l
o
cal
dem
a
nd d
u
e t
o
t
h
e im
port
an
d ex
p
o
rt
of p
o
we
r i
n
vari
ous
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Decomp
o
s
ition Coo
r
d
i
n
a
ting
Meth
o
d
f
o
r Para
llel So
lu
tion
o
f
a Mu
lti Area .... (
S
en
t
h
il Krish
namu
r
t
h
y)
2
049
areas.
Areas
of individual powe
r system
s
are interc
o
n
n
e
cted
to
o
p
e
rate with
m
a
x
i
mu
m
reliab
ili
ty, reserve
sh
ari
n
g, i
m
p
r
ov
ed
stab
ility a
n
d
less pro
d
u
c
tio
n
co
st th
an
wh
en
th
ey op
erate as iso
l
ated
areas. Con
s
ideration
of the t
r
ansm
ission capacity a
m
ong the
areas
in t
h
e m
u
lti-
area power
system
is one
of the
im
portant
problem
s
in
th
e op
eration
of th
e
po
wer
syste
m
, wh
ile so
lv
i
n
g
t
h
e MACEE
D
proble
m
.
Tie
line
p
o
wer tran
sfer li
mits an
d
po
we
r dem
a
nd
bet
w
ee
n a
r
ea’
s are c
o
nsi
d
e
r
e
d
as a
d
di
t
i
onal
co
nst
r
ai
nt
s i
n
t
h
e M
A
C
E
E
D
pr
o
b
l
e
m
.
Henc
e t
h
e
M
A
C
EED
i
s
c
onsi
d
ere
d
as
a
l
a
rge scal
e
o
p
t
i
m
i
sati
on
pr
ob
l
e
m
.
M
A
C
EED i
s
a c
o
m
p
l
e
x
pr
obl
em
wi
t
h
m
a
ny
d
i
fferen
t
formu
l
atio
n
s
an
d
it h
a
s b
een
co
nsid
ered
fo
llo
win
g
th
e d
e
v
e
lop
m
en
t o
f
th
e co
m
p
etitiv
e ele
c
tricity
m
a
rket
s an
d t
h
e sm
art
gri
d
t
e
chn
o
l
o
gi
es.
Ne
w f
o
rm
ul
at
i
on of
M
A
C
E
E
D
p
r
o
b
l
e
m
i
s
pro
p
o
se
d i
n
t
h
i
s
pa
per
.
The
recent m
e
thods to so
l
v
e
the m
u
lti-area powe
r system
proble
m
s
are based
on dec
o
mposition
of
th
e ov
erall power system
p
r
ob
lem
in
to
su
bprob
lem
s
c
o
rres
ponding to the
powe
r
syste
m
areas and a
co
ord
i
n
a
tor wh
ich
are so
lv
ed
in
an
iterativ
e way. Cu
rren
t
ly av
ailab
l
e d
eco
m
p
o
s
itio
n
tech
n
i
q
u
e
s assume th
at
t
h
e m
odel
s
an
d co
nt
r
o
l
o
b
j
e
c
t
i
v
es of t
h
e ar
eas are f
o
rm
ul
at
ed t
o
be
n
o
n
-
o
v
erl
a
ppi
ng a
s
i
t
was m
e
nt
ione
d i
n
[4
], i.e., th
e
bord
e
r of
o
n
e
area is at th
e same ti
m
e
al
so
th
e b
o
rd
er
o
f
a
n
e
ig
hb
oring
area. Th
is typ
e
o
f
m
u
l
ti-
area system
model is
cons
id
ered in
t
h
e
p
a
p
e
r.
Multiarea Econom
ic Dispatch (MAED) problem
is
reviewed in t
h
is section according to the
m
e
t
hods
use
d
f
o
r
i
t
s
sol
u
t
i
o
n
.
They
are
b
r
oad
l
y
cl
assi
fi
ed i
n
t
o
cl
assi
cal
a
n
d
heu
r
i
s
t
i
c
s m
e
t
h
ods
.
The
pape
rs [
5
]
-
[
9
]
use
d
cl
ass
i
cal
m
e
t
hods t
o
s
o
l
v
e t
h
e M
A
ED
p
r
o
b
l
e
m
.
The
pape
r [
5
]
,
[6]
an
d
[1
0]
use
d
La
gra
nge
’s al
g
o
ri
t
h
m
f
o
r M
A
E
D
pr
o
b
l
e
m
sol
u
t
i
on.
The
pa
pers
[
5
]
and
[6]
,
p
r
o
p
o
se
d an
ap
pr
o
ach t
o
in
corpo
r
ate power
con
t
racts in
to
m
u
lti-area u
n
it co
mmit
m
en
t an
d
so
lve th
e econo
m
i
c d
i
sp
atch
so
lu
tio
ns
usi
n
g an ada
p
t
i
v
e Lag
r
an
g
i
an
m
e
t
hod.
C
o
m
b
i
n
ed Econ
om
i
c
Em
i
s
si
on Di
s
p
at
ch
(C
EED
) p
r
o
b
l
e
m
for
i
n
t
e
rco
n
n
ect
ed
gri
d
s i
s
i
n
vest
i
g
at
ed i
n
[8]
.
E
m
i
ssi
ons an
d c
o
st
s o
f
gene
rat
i
on a
r
e o
b
j
ect
i
v
e f
u
nct
i
ons
, a
nd t
h
e
interconnected
gri
d
is divide
d into
seve
ral sub-gri
d
s. T
h
e calculation of
each subarea
problem is performed in
parallel. A strategy is propose
d that each sub-a
r
ea
set
s
diffe
rent m
u
lti-obj
ec
tive function acc
ording to
di
ffe
re
nt
co
n
d
i
t
i
ons/
c
o
n
st
rai
n
t
s
usi
n
g
a Li
n
ear
Wei
g
ht
ed
Sum
M
e
t
hod
(
L
W
S
M
)
.
L
W
S
M
i
s
use
d
t
o
c
h
an
ge
th
e m
u
lti-o
b
j
ectiv
e prob
lem
in
to
sing
le
o
b
j
ect
iv
e pro
b
l
em
an
d
it g
i
v
e
s a
preferen
ce
d
e
gree
o
f
d
ecision
m
a
k
e
rs
to the
objective functions.
The direct
sea
r
ch m
e
thod is
ext
e
nde
d t
o
facilitate econom
i
c s
h
ari
ng
of ge
ne
rati
on
an
d
reserv
e acro
ss areas and
m
i
n
i
mize th
e to
tal g
e
n
e
ratio
n
co
st in
t
h
e m
u
lti-area reserv
e con
s
t
r
ain
e
d
econom
i
c dispatch in [11]. Jacobia
n
base
d algorithm
is
use
d
to calculate penalty factors for a
n
are
a
in a
m
u
lt
i
-
area p
o
w
e
r sy
st
em
i
n
[7
]
.
The pa
pe
r [
9
]
devel
o
pe
d a
m
u
lt
i
-
area ge
n
e
rat
i
on sc
he
d
u
l
i
ng sc
hem
e
t
h
at
can
provide
proper unit c
o
mmitm
ent in each
area, a
n
d effe
ctively prese
r
ve the tie-
line constraints usi
n
g
an
im
pro
v
ed
dy
na
m
i
c pro
g
ram
m
i
ng m
e
t
hod
.
The
researc
h
p
a
pers
[
9
]
,
[
1
2]
–[
1
9
]
use
d
heu
r
i
s
t
i
c
m
e
t
hods
t
o
sol
v
e t
h
e
M
A
ED
p
r
obl
em
. The
pa
pe
r
[12
]
, p
r
esen
ts
a d
eco
m
p
o
s
ition
app
r
o
a
ch
to
m
u
l
tiarea
g
e
n
e
ratio
n
sch
e
du
lin
g
p
r
o
b
l
em
u
s
in
g
ex
p
e
rt system
.
It
proposes
a large-scale m
i
xed intege
r-n
on
linear op
timizatio
n
pro
cess to
so
lv
e the prob
l
e
m
u
s
in
g
a t
w
o
-
layer
decom
posi
t
i
o
n
t
echni
que
.
In
t
h
e fi
rst
l
a
y
e
r o
f
dec
o
m
posi
t
i
on, t
h
e
pr
o
b
l
e
m
i
s
di
vi
d
e
d i
n
t
o
se
vera
l
sub
-
pr
o
b
l
e
m
s
duri
ng t
h
e co
nsi
d
e
r
ed
peri
od
. Th
e i
n
fo
rm
at
i
on t
h
at
t
h
e p
r
o
b
l
e
m
sends t
o
eac
h su
b-
p
r
o
b
l
e
m
i
s
t
h
e
load dem
a
nds
of all areas
at
the c
o
rresponding tim
e peri
od
an
d th
e ou
tpu
t
of
t
h
e sub
-
pr
ob
lem
is th
e syste
m
ope
rat
i
o
n c
o
st
at
t
h
at
t
i
m
e
. The c
o
o
r
di
nat
i
o
n
fact
or
of t
h
i
s
l
a
y
e
r o
f
dec
o
m
posi
t
i
on i
s
t
h
e o
p
erat
i
o
n c
o
s
t
of
t
h
e
syste
m
in
th
e g
i
v
e
n
p
e
riod
,
wh
ich
shou
ld
b
e
min
i
m
u
m
.
Th
e second
layer o
f
d
e
co
mp
o
s
ition
d
i
v
i
des th
e
pre
v
ious
sub-problem
s
furthe
r acc
or
ding to
the control a
r
e
a
s in t
h
e
powe
r pool.
T
h
e su
b-problem
for
each
area receives syste
m
Lagra
n
ge
m
u
ltiplier
λ
an
d
ret
u
rn
s th
e area
m
u
ltip
lie
r
λ
. Th
e coo
r
din
a
tio
n
at th
is lev
e
l
uses the
difference bet
w
een t
h
e system
m
u
ltiplier
λ
and
th
e area m
u
lt
ip
lier
λ
w
h
i
c
h s
h
o
u
l
d be zer
o e
x
ce
pt
fo
r
areas that
reac
h thei
r
gene
ration lim
its.
A Particle Swarm
Op
ti
m
i
sat
i
o
n
(PSO) algorith
m
to
so
l
v
e
vari
ous t
y
pes
of ec
o
nom
i
c
di
spat
ch
(E
D)
p
r
ob
lem
s
in
p
o
w
er system
s
su
ch
as, m
u
lti
-area ED w
ith tie l
i
n
e
li
mit
s
, ED with
m
u
ltip
le fu
el optio
n
s
,
com
b
i
n
ed e
nvi
ro
nm
ent
a
l
econom
i
c
di
spat
ch
, an
d t
h
e E
D
o
f
ge
ne
rat
o
rs
wi
t
h
p
r
o
h
i
b
i
t
e
d
o
p
erat
i
n
g z
o
nes
usi
n
g
bot
h t
h
e
PS
O
m
e
t
hod a
n
d t
h
e C
l
assi
cal
Ev
ol
ut
i
o
nary
Pr
o
g
ram
m
i
ng (C
E
P
) a
p
pr
oach
i
s
desc
ri
be
d i
n
[
13]
.
I
n
th
e PSO m
e
th
o
d
, th
ere is
on
ly o
n
e
pop
u
l
atio
n in
each
iteratio
n
th
at m
o
ves to
ward
s t
h
e
g
l
ob
al op
tim
al
po
in
t
.
Thi
s
i
s
u
n
l
i
k
e t
h
e C
E
P m
e
t
hod,
whi
c
h has t
o
deal
wi
t
h
t
w
o p
o
pul
at
i
o
ns,
t
h
e pa
rent
s a
n
d
t
h
e chi
l
d
ren
,
i
n
ea
c
h
i
t
e
rat
i
on.
Thi
s
m
a
kes t
h
e
PS
O
m
e
t
hod
com
p
ut
at
i
onal
l
y
fast
er i
n
com
p
ari
s
on
t
o
t
h
e C
E
P
m
e
t
hod.
In
[
14]
a
n
d
[
15]
,
M
A
C
E
E
D
pr
o
b
l
e
m
i
s
i
nvest
i
g
at
e
d
t
o
ad
dres
s t
h
e
e
nvi
ro
nm
ent
a
l
i
ssue
o
f
t
h
e
eco
no
m
i
c d
i
sp
atch
p
r
ob
lem
.
Th
e MACEED prob
lem
is
first form
u
l
at
ed
and
th
en
an
Im
p
r
ov
ed
Mu
lti-
ob
ject
i
v
e
Part
i
c
l
e
Swa
r
m
Op
t
i
m
i
zat
i
on (M
OPS
O
)
al
g
o
ri
t
h
m
i
s
devel
o
p
e
d t
o
deri
ve
a
set
of
Pa
ret
o
-o
pt
im
al
so
lu
tion
s
.
Each
Pareto
-
op
timal so
lu
tio
n
is a trad
eo
ff
bet
w
een
o
p
erat
i
o
nal
cost
a
nd
p
o
l
l
u
t
a
nt
em
i
ssions
. T
h
e
p
a
p
e
r [16
]
rev
i
ews and
co
m
p
ares so
m
e
ev
o
l
u
tio
n
a
ry
techn
i
q
u
e
s fo
r Mu
lti Area Econ
o
m
i
c
Disp
atch
(M
AED)
p
r
ob
lem so
lu
tio
n
s
u
s
ing
i) Classical Differen
tial Ev
o
l
u
tion
(DE), ii) Classical Particle
Swarm
Op
timi
zatio
n
(PSO), a
n
d iii) An im
proved
PSO w
ith a
pa
ram
e
ter automation strate
gy
havi
ng Tim
e
Varying
Accel
eration
Coefficients (PSO_TVAC).
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
204
8
–
20
63
2
050
Fu
zzified
Particle Swarm
Op
t
i
m
i
zat
io
n
(FPSO) al
g
o
rith
m
fo
r so
l
v
ing
th
e
secu
rity-co
n
s
t
r
ain
e
d
m
u
lti-
area econom
i
c
dispatch
proble
m
for an
int
e
rconnected
powe
r system
is i
nvest
i
g
at
e
d
i
n
[1
7]
an
d [
1
8]
. The
FPSO algo
rithm is b
a
sed
o
n
th
e co
m
b
in
ed ap
p
lication
of fu
zzy log
i
c st
rateg
y
in
co
rporated
in th
e
p
a
rticle
swarm
o
p
tim
iz
atio
n
algorith
m.
M
A
C
EED
an
d S
A
C
E
E
D
p
r
o
b
l
e
m
i
s
sol
v
ed
usi
n
g
PS
O m
e
t
hod i
n
[1
9]
an
d
[2
0]
res
p
ect
i
v
el
y
.
SAC
E
E
D
p
r
o
b
l
em
wi
t
h
val
v
e poi
nt
i
s
co
nsi
d
ere
d
i
n
[
21]
.
The gravati
ona
l search m
e
th
od
is u
s
ed
in
[
2
2
]
and
[23
]
to
so
l
v
e t
h
e econo
m
i
c d
i
sp
atch
p
r
o
b
l
em with
m
u
ltip
le g
e
n
e
rator syste
m
s with
co
n
s
id
ering
th
e
p
r
oh
ib
ited
ope
rat
i
n
g z
one
s.
In
de
re
gul
at
ed
p
o
we
r
sy
st
em
en
vi
r
onm
ent
[
24]
a
n
d
[2
5]
i
t
i
s
esse
nt
i
a
l
t
o
f
o
rm
ul
at
e and
sol
v
e t
h
e
eco
no
m
i
c d
i
sp
atch
p
r
ob
lem
f
o
r m
u
lti-area cases with
tie
lin
e co
n
s
t
r
ain
t
s. Th
e
n
e
w stru
cture of th
e p
o
wer
sy
st
em
requi
re
s new o
p
t
i
m
i
s
at
i
on m
e
t
hod
s b
a
sed o
n
t
h
e sp
eci
fi
cs of t
h
i
s
st
ruct
u
r
e an
d pr
ovi
di
n
g
fast
, re
l
i
a
bl
e
and
rel
e
vant
t
o
t
h
e
re
qui
re
m
e
nt
s of
t
h
e
po
we
r sy
st
em
as a
whole a
n
d to its
elements s
o
lutions. These
sol
u
t
i
o
ns can
gi
ve dee
p
er
i
n
si
g
h
t
i
n
t
o
t
h
e dy
nam
i
cs
of t
h
e sy
st
em
. M
e
t
h
o
d
s usi
ng
vari
ous t
y
pes o
f
decom
posi
t
i
o
n
and c
o
or
di
na
t
i
on ap
p
r
oac
h
es i
n
sol
u
t
i
o
n
of t
h
e p
o
w
er
di
spat
c
h
p
r
o
b
l
e
m
s
are capabl
e
t
o
r
e
spon
d to
t
h
e
ab
ov
e
r
e
q
u
i
r
e
men
t
s.
Th
e ex
istin
g
situ
atio
n
is th
at
m
o
st o
f
t
h
e
co
nv
en
tio
n
a
l
grad
ien
t
and
heu
r
istic m
e
th
o
d
s are time
co
nsu
m
in
g
and
still u
s
e a seq
u
e
n
tial m
e
th
od
to
so
lv
e
th
e
MACEED
prob
lem
[1
4
]
-[17
], [2
6
]
, and
[27]. Th
e
t
r
adi
t
i
onal
he
u
r
i
s
t
i
c
m
e
t
hods f
o
r M
A
C
E
E
D
p
r
o
b
l
e
m
do not
al
way
s
gua
rant
ee gl
obal
be
st
sol
u
t
i
o
ns;
t
h
ey
oft
e
n
achi
e
ve a fa
st
and
near
gl
o
b
a
l
opt
i
m
al
sol
u
t
i
on
[1]
,
[2]
,
[1
4]
-
[
1
9
]
.
R
e
searc
h
es have c
o
nst
a
nt
l
y
obse
r
ve
d t
h
at
al
l
t
h
ese m
e
t
hods
very
qui
c
k
l
y
fi
nd a
g
o
o
d
l
o
ca
l
sol
u
t
i
o
n b
u
t
get
st
uc
k t
h
er
e
fo
r a n
u
m
b
er
of i
t
e
rat
i
o
ns
w
i
t
hout
fu
rt
he
r i
m
prov
em
ent
,
som
e
t
i
m
e
s causi
n
g
pr
em
at
ure co
n
v
e
r
ge
nce.
There
f
ore, t
h
i
s
pape
r
devel
ops a
n
e
ffi
ci
ent
al
g
o
rith
m
i
n
d
ealing
with
a larg
e-scal
e MACEED
pr
o
b
l
e
m
usi
ng
t
h
e dec
o
m
posi
t
i
on a
p
pr
oach
. I
n
c
o
m
p
ari
s
on t
o
l
a
m
bda, di
re
ct
, dy
nam
i
c, Jacobi
a
n
a
nd
he
u
r
i
s
t
i
c
search
t
ech
ni
q
u
es,
t
h
e
Lag
r
a
nge
’s
g
r
adi
e
nt
m
e
t
hod i
s
m
o
re
p
o
w
er
ful
t
ool
f
o
r
cal
cul
a
t
i
on
of
t
h
e
co
m
p
l
e
x
optim
isation
problem
s.
This m
e
thod gene
ra
lly begins
with an initial feasible sol
u
tion a
n
d re
fines t
h
e s
o
lution
rep
eated
ly un
til th
e op
tim
al s
o
lu
tion
is
found
.
Lagra
n
ge’s decom
position-coor
dinating m
e
thod a
nd
algorithm
are develope
d for m
u
lti-area
eco
no
m
i
c d
i
sp
atch
prob
lem
s
o
lu
tion
in
Parallel MATLA
B
envi
r
o
nm
ent
usi
n
g a C
l
ust
e
r o
f
C
o
m
put
er
s. Th
e
fu
nct
i
o
n
of
La
gra
n
ge i
s
dec
o
m
posed i
n
a
n
u
m
b
er o
f
s
u
b-
f
unct
i
o
ns
o
f
La
gra
n
ge acc
or
di
ng
t
o
t
h
e
n
u
m
b
er
o
f
areas b
y
u
s
ing
the v
a
lu
es o
f
th
e Lagrang
e
v
a
riab
le
s as coo
r
d
i
n
a
tin
g on
es. Th
en th
e i
n
itial p
r
o
b
le
m
is
decom
pose
d
i
n
areas
su
b
-
p
r
o
b
l
e
m
s
and a
co
or
di
nat
i
n
g
su
b-
pr
o
b
l
e
m
.
The
opt
i
m
al
sol
u
t
i
on
of
t
h
e
co
or
di
nat
i
n
g
sub
-
pr
obl
em
det
e
rm
i
n
es t
h
e opt
i
m
al
sol
u
t
i
ons
of t
h
e are
a
s sub
-
pr
obl
e
m
s and t
h
e o
p
t
im
al
sol
u
t
i
on
fo
r t
h
e
in
itial
m
u
lt
i-area p
r
ob
lem
.
A fou
r
area three g
e
n
e
rator
an
d
a
four area fou
r
g
e
n
e
rat
o
r IEEE b
e
n
c
h
m
a
rk
m
o
d
e
ls are
u
s
ed
to test and
valid
ate th
e resu
lts ob
tain
ed
by th
e dev
e
l
o
p
e
d
so
ft
ware in
t
h
e M
A
TLAB
Clu
s
ter
of
C
o
m
put
ers.
Thi
s
pa
per
f
o
r
m
ul
at
es t
h
e M
A
C
E
E
D
p
r
o
b
l
e
m
i
n
sect
i
on 2, La
gra
n
ge'
s
decom
posi
t
i
o
n
-
co
or
di
nat
i
n
g
m
e
t
hod an
d al
go
ri
t
h
m
for so
l
u
t
i
on o
f
t
h
e M
A
C
EED
pr
o
b
l
e
m
are descr
i
bed i
n
sect
i
o
n
3 and 4 res
p
e
c
t
i
v
el
y
,
sin
g
l
e area and
m
u
lti-area CEED
p
r
ob
lem
so
lu
tion
s
for
4 area 10
g
e
n
e
rato
r system
an
d
4
area
1
2
g
e
n
e
rat
o
r
syste
m
s are presented in secti
o
n 5. T
h
e c
o
nc
l
u
si
o
n
i
s
gi
ve
n
i
n
sect
i
o
n
6.
2.
MAT
H
EM
AT
ICAL
FO
R
M
ULATIO
N
O
F
THE
MA
C
EED PR
OBL
E
M
A sch
e
m
a
tic d
i
ag
ram
o
f
a
mu
ltiarea p
o
wer syste
m
is sh
own
in
Fi
gu
re 1. Th
e system
h
a
s M areas.
Every
area
has
i
t
s
own set
of
gene
rat
o
r
s
m
N,
m
1
,
M
. The areas are interconnecte
d
by tie-lin
es as s
h
own
in
Fi
gu
re 1.
Fig
u
re
1
.
Model o
f
a Mu
lti-Area
p
o
wer syste
m
with
tie-line po
wer tran
sfer
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Decomp
o
s
ition Coo
r
d
i
n
a
ting
Meth
o
d
f
o
r Para
llel So
lu
tion
o
f
a Mu
lti Area .... (
S
en
t
h
il Krish
namu
r
t
h
y)
2
051
Sin
g
l
e criterion
or m
u
lti-criteria d
i
sp
atch
prob
lem
s
can
be form
u
l
at
ed
fo
r ev
ery area.
Mu
lti-criteria
com
b
ined
dispatch problem
is consid
er
ed
in th
e
p
a
p
e
r
f
o
r
ev
e
r
y ar
e
a
:
Tw
o typ
e
s
o
f
cr
ite
r
i
a
a
r
e
con
s
id
er
ed
:
T
y
pe
one
:
O
p
erati
on
al
cos
t
Th
e to
tal o
p
e
ratio
n
a
l co
st is th
e su
m o
f
th
e gen
e
ration
co
st
an
d
th
e co
st o
f
tran
sm
issio
n
o
f
th
e p
o
wer
betwee
n the
areas, as
follows:
The
operati
o
nal cost for
power
production i
n
all areas
2
11
m
N
M
C
m
nm
n
m
nm
n
m
n
mn
FP
a
P
b
P
c
(1
)
whe
r
e
M is the
num
ber
of t
h
e inte
rc
onnected area
s
N
m
i
s
t
h
e n
u
m
b
er
of
ge
nerat
o
r
s
bel
o
n
g
i
n
g t
o
area
m1
,
M
an
d co
mmitted
to
th
e power produ
ctio
n in
th
is area
P
mn
i
s
t
h
e
real
po
we
r pr
od
uce
d
by
t
h
e
n
th
ge
nerat
o
r i
n
t
h
e
m
th
area
1
2
3
.
.
.
.
.
.
m
T
m
m
m
m
m
N
P
P
P
P
P
is
the vector of
the r
eal power produced in t
h
e
m
th
ar
e
a
,
and
1,
mM
12
3
...
...
T
M
PP
P
P
P
is
the vector of
the real
power
produced in t
h
e whole
powe
r
syste
m
a
mn
,b
mn
,c
mn
a
r
e
the cost coe
ffic
i
ents fo
r the
power produced
by the
n
th
ge
ne
rat
o
r
i
n
t
h
e
m
th
area.
The
operationa
l cost for tra
n
s
m
ission of t
h
e
r
eal power t
h
rough t
h
e tie-line
s
is gi
ven as:
11
MM
TT
m
j
T
m
j
j
m
T
j
m
mj
jm
FP
q
P
q
P
(2
)
W
h
er
e
Tm
j
P
is the real
power fl
ow from
the area
m
t
o
the
area
j
and
Tj
m
P
is the real
power fl
ow from
the area
j
to
th
e ar
e
a
m
,1
,
2
,
3
,
..
...
.
,
1
,
T
Tm
Tm
m
T
m
m
Tm
m
T
m
M
PP
P
P
P
m
M
is
the vector of the real
po
w
e
r t
r
ansm
i
ssi
on bet
w
ee
n t
h
e
m
th
area
and all
othe
r a
r
ea.
12
3
,
1
...
..
.
T
TT
T
T
T
M
PP
P
P
P
is
the vector of
the real
power
tran
sm
issio
n
between
all areas.
Ty
pe
two
:
Emissio
n
quantity
An add
ition
a
l criterion
expressing
m
i
n
i
misatio
n
o
f
th
e po
llu
tan
t
em
issio
n
is co
n
s
id
ered
.
Th
i
s
cri
t
e
ri
on
refe
rs
onl
y
t
o
t
h
e
po
we
r ge
nerat
i
on. T
h
e t
i
e
-
lines are
not consi
d
ere
d
in t
h
is case beca
use the
tran
sm
issio
n
of th
e po
wer does n
o
t
create ch
em
ical
p
o
llu
tio
n. Qu
an
tity of th
e po
llu
tan
t
e
m
issio
n
cau
sed
b
y
th
e pow
er pr
odu
ctio
n is ex
pr
essed
as:
2
11
m
N
M
Em
n
m
n
m
n
m
n
m
n
mn
FP
d
P
e
P
f
(3
)
Whe
r
e:
d
mn
,e
mn
,f
mn
are t
h
e em
ission coefficients
of the
n
th
g
e
nerator in
th
e
m
th
a
r
ea.
Th
e co
m
b
in
ed
so
lu
tion
of th
e real p
o
w
er d
i
sp
atch
prob
lem
for th
e
m
u
lti-area syste
m
d
e
t
e
rm
in
es th
e
opt
i
m
al
power
t
o
be
p
r
o
d
u
ced
by
t
h
e
ge
nerat
o
rs i
n
e
v
er
y a
r
ea and the
optimal values of t
h
e
power trans
f
erred
b
e
tween
th
e areas th
rou
g
h
t
h
e tie-lin
es.
The m
a
t
h
em
atical
form
ul
at
i
on o
f
t
h
e M
A
C
EED
pr
obl
em
cri
t
e
ri
on i
s
bas
e
d o
n
i
n
t
r
o
duct
i
on
of a p
r
i
c
e
penalty fact
ors
to convert t
h
e
e
m
ission
v
a
l
u
es of th
e criterion
(3) in
to cost v
a
lu
es and
th
e
d
i
sp
atch
pro
b
l
e
m
to
b
e
d
e
scr
i
b
e
d by a sing
le cr
iter
i
on
FP
.
Vari
ous
t
y
pes
of
p
r
i
ce
penal
t
y
fact
o
r
s a
r
e
pr
op
ose
d
i
n
[2
8]
.
Deri
vat
i
o
ns i
n
t
h
e paper are
base
d on t
h
e
m
i
n/m
a
x penal
t
y
fact
or,
as
fol
l
ows:
m
N
M
2
mn
m
n
,
m
i
n
mn
m
n
,
m
i
n
mn
mn
2
m
n
m
n
,
m
a
x
mn
mn
,
m
a
x
mn
m1
n1
aP
bP
c
h[
$
/
k
g
]
dP
e
P
f
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
204
8
–
20
63
2
052
W
h
er
e
m
mm
1
m
2
m
N
h
h
h
....
h
i
s
t
h
e vect
or
of
t
h
e penal
t
y
fac
t
ors f
o
r
t
h
e
th
m
are
a
,
m1
,
M
12
M
h
h
h
....
h
i
s
t
h
e
vect
or
of
t
h
e
penal
t
y
fac
t
ors
f
o
r t
h
e
wh
ol
e p
o
w
er
sy
st
em
.
,m
i
n
mn
P
and
,m
a
x
mn
P
a
r
e
t
h
e
mi
n
i
mu
m a
n
d
t
h
e
ma
x
i
mu
m r
e
a
l
po
wer t
h
at
can be
pr
o
duce
d
by
t
h
e
n
th
ge
ne
rat
o
r
in
th
e
m
th
a
r
ea.
Th
en
th
e MAC
EED
p
r
ob
lem
i
s
fo
rm
u
l
ated
in th
e fo
llowing
way: Find
th
e
v
ector
o
f
t
h
e real p
o
wer
P
an
d th
e v
ect
o
r
o
f
th
e
po
wer t
r
an
sm
it
ted
th
rou
g
h
t
h
e tie-lin
es
P
T
s
u
ch
t
h
at
t
h
e c
r
i
t
e
ri
o
n
(4
)
i
s
m
i
nim
i
sed un
de
r
th
e fo
llowing
co
n
s
t
r
ain
t
s
for
2
11
1
2
1
m
m
N
M
mn
mn
m
n
m
n
mn
mj
Tmj
j
m
T
jm
M
nj
jm
m
N
m
n
mn
mn
mn
mn
mn
n
aP
b
P
c
q
P
q
P
FP
h
d
Pe
Pf
(4
)
i.
M
i
n
i
m
u
m an
d
ma
x
i
mu
m
r
e
a
l po
we
r
p
r
o
d
u
c
ed
by
ev
er
y
g
e
n
e
ra
to
r
,m
i
n
,
m
a
x
,1
,
,
1
,
mn
mn
mn
m
PP
P
n
N
m
M
(5
)
ii. Minimum
and m
axim
u
m
acti
ve power s
e
nt
thr
o
ugh
the tie-lines
,m
i
n
,
m
a
x
,1
,
Tm
Tm
Tm
PP
P
m
M
(6
)
Th
ese lim
its are v
a
lid
fo
r t
h
e t
w
o d
i
rectio
n
s
o
f
th
e
po
wer
flo
w
and
can
b
e
written
as
,m
i
n
,
m
a
x
,m
i
n
,m
a
x
1,
,
,
1,
1,
,
,
1,
Tmj
T
mj
Tmj
Tjm
T
jm
T
j
m
PP
P
j
M
j
m
m
M
PP
P
j
M
j
m
m
M
(7
)
iii. Po
wer ba
lance
The bal
a
nce be
t
w
een t
h
e
po
w
e
r pr
o
duct
i
on a
nd t
h
e
p
o
we
r d
e
m
a
nd f
o
r t
h
e
m
th
area and for the whole
sy
st
em
i
s
gi
ve
n
by
E
quat
i
o
n
(8
),
11
1,
1
,
m
N
M
mn
Dm
Lm
T
m
j
j
m
T
j
m
nj
jm
PP
P
P
P
m
M
(8
)
W
h
er
e
00
0
11
1
,1
,
mm
m
NN
N
L
m
mn
mn
r
m
r
m
n
m
n
m
nr
n
PP
B
P
B
P
B
m
M
(9
)
W
h
er
e
00
0
,,
mn
r
m
n
m
BB
B
are the
tra
n
sm
ission loss
coe
f
ficients
o
f
th
e
in
t
e
r
c
o
n
n
ected powe
r
system
j
m
is th
e fraction
a
l lo
ss
rate fro
m
th
e area
j
to the area
m
Lm
P
is the real
power los
s
of t
h
e
n
th
tran
sm
issio
n
lin
e in
the
m
th
area.
Dm
P
is the real
power
dem
a
nd in t
h
e
m
th
area
.
Tmj
P
is the real
power fl
ow from
the area
m
t
o
the
area
j
Tjm
P
is the real
power fl
ow from
the area
j
to
th
e ar
e
a
m
Th
en
eq
u
a
tion
(8) can
b
e
written
as fo
llo
ws:
00
0
11
1
1
1
1,
1
,
mm
m
m
NN
N
N
M
mn
D
m
mn
mn
r
m
r
m
n
m
n
m
T
m
j
j
m
T
j
m
nn
r
n
j
jm
PP
P
B
P
B
P
B
P
P
m
M
(1
0)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Decomp
o
s
ition Coo
r
d
i
n
a
ting
Meth
o
d
f
o
r Para
llel So
lu
tion
o
f
a Mu
lti Area .... (
S
en
t
h
il Krish
namu
r
t
h
y)
2
053
The
fo
rm
ul
at
ed
pr
o
b
l
e
m
gi
ven
by
E
quat
i
o
ns
(
1
t
o
1
0
)
i
s
c
h
aract
eri
s
ed
as:
A co
m
b
in
ed
op
ti
m
i
satio
n
d
i
sp
at
ch
problem
for e
v
ery a
r
ea.
The
pri
ce
pe
na
l
t
y
fact
ors a
r
e i
n
t
r
od
uce
d
f
o
r e
v
ery
ge
nerat
o
r
separat
e
l
y
.
The a
r
eas a
r
e i
n
terc
onnected
by tie-lines
w
ith two
direction of real
powe
r t
r
ans
f
er.
Th
e
op
ti
m
i
sati
o
n
d
i
sp
atch pro
b
l
em
fo
r th
e in
te
rco
n
n
ected
p
o
wer system
i
s
m
u
lticriteria
l
o
n
e
.
Di
m
e
nsi
on of t
h
e i
n
t
e
rc
on
nec
t
ed pr
obl
em
depen
d
s o
n
t
h
e n
u
m
b
er of areas
, num
ber o
f
t
h
e gener
a
t
o
r
s
an
d
th
e tie-lin
es.
Calcu
l
atio
n
of th
e so
l
u
tion o
f
t
h
e m
u
lti
criterial in
terco
n
n
ected proble
m
is d
i
fficult an
d
time
con
s
um
i
ng.
If
t
h
e s
o
l
u
t
i
o
n
ha
s t
o
be
d
o
n
e
o
f
t
e
n a
n
d i
n
rea
l
-t
im
e, t
h
en
be
t
t
e
r com
put
at
i
onal
ap
pr
oac
h
es are
neede
d
.
O
n
e o
f
t
h
em
i
s
a paral
l
e
l
com
put
ing
of t
h
e area'
s su
b-
pr
o
b
l
e
m
s
and c
o
or
di
na
t
i
on o
f
t
h
e
obt
ai
ne
d
areas sol
u
tions
using a High
Perform
a
nce Parallel Co
m
put
i
ng (
H
P
P
C
)
en
vi
r
onm
ent
(C
l
u
st
er o
f
com
p
ut
ers
)
.
Fi
gu
re
2 s
h
o
w
s t
h
e st
r
u
ct
u
r
e
of t
h
e C
l
ust
e
r
of C
o
m
put
ers
wo
rki
n
g i
n
M
A
TL
AB
pa
ral
l
e
l
com
put
i
ng s
o
ft
war
e
en
v
i
ron
m
en
t u
s
ed
to
im
p
l
e
m
en
t th
e o
p
tim
is
atio
n
alg
o
rithm
s
.
Paralleliza
tio
n
of th
e so
l
u
tio
n
is don
e th
rou
gh
d
eco
m
p
o
s
ition o
f
th
e MAEED prob
lem
acc
o
r
d
i
ng
to
th
e
powe
r system interconnect
e
d
areas an
d co
or
di
nat
i
on
o
f
th
e
ob
tain
ed so
l
u
tio
n
s
fo
r
e
v
ery a
r
ea
by a
coordinator.
This a
p
proach
requires:
i.
A dec
o
m
position-c
o
ordi
nating m
e
thod to
be de
velope
d
in orde
r to obtain an algorithm
for pa
rallel
calculation.
ii.
Soft
ware
devel
opm
ent
based
on t
h
e a
b
o
v
e a
l
go
ri
t
h
m
a
llo
win
g
allo
cation
o
f
th
e sep
a
rate su
b-prob
lem
s
to
t
h
e st
r
u
ct
u
r
e
of
t
h
e
HPPC
en
v
i
ro
nm
ent
– A
C
l
ust
e
r
of C
o
m
put
ers.
iii.
Im
p
l
e
m
en
tat
i
o
n
of the d
e
v
e
lop
e
d software i
n
th
e
HPPC
env
i
ro
n
m
en
t, in
vestig
atio
n
s
o
f
th
e cap
a
b
ilities of
t
h
e de
vel
o
pe
d
al
go
ri
t
h
m
.
iv
.
Ev
alu
a
tion
and v
e
rification
of th
e o
b
t
ain
e
d
so
lu
tion
b
y
com
p
ariso
n
of the resu
lts ob
tain
ed
b
y
sequ
en
t
i
al
an
d p
a
rallel ones.
Fi
gu
re
2.
St
r
u
c
t
ure
of
t
h
e M
A
TLAB
Paral
l
e
l
C
o
m
put
i
n
g
To
ol
b
o
x
The proposed
m
e
thod is base
d on classical
Lagra
n
ge
’s
opt
im
isation [28]-[31]. The literature re
view
found that the
classical Lagrange
'
s
m
e
thod has
been
us
ed till now on
ly for se
que
n
tial solution
of the
MA
CEED
pr
ob
lem
in
[
5
],[
8
]
,
[1
3
]-[
18
], an
d [
2
0
]
. Th
e Lagr
ang
e
s m
e
th
o
d
is in
tr
odu
ced
in
ear
ly 19
72
fo
r
t
h
e
p
o
wer system
sp
inn
i
ng
reserv
e
d
e
term
in
ati
o
n in a m
u
lti syste
m
co
n
f
i
g
u
r
ation
[3
2
]
, an
i
n
terim
m
u
lt
i-area
econom
i
c dispatch [33], and t
w
o area
power
syste
m
s econom
ic dispat
ch
pr
ob
lem
in
[
3
4
]
.
Th
e pap
e
r in
t
r
odu
ces a p
a
rallel so
lu
tio
n o
f
th
e M
A
EED pro
b
l
em
b
y
con
s
id
ering
two
-
level
calcu
latio
n
stru
cture, Fi
g
u
re
3
,
wh
ere th
e i
n
itial o
p
t
i
m
isa
t
i
o
n
p
r
ob
lem
is
d
eco
m
p
o
s
ed
into
sub
-
prob
lems (fo
r
every
area
o
n
e
)
, s
o
l
v
e
d
o
n
t
h
e fi
rst
l
e
vel
,
a
nd t
h
e o
b
t
a
i
n
e
d
sol
u
t
i
o
ns are
coo
r
di
nat
e
d
b
y
a coo
r
di
nat
i
ng s
u
b-
pr
o
b
l
e
m
on t
h
e
seco
n
d
l
e
vel
-
i
n
o
r
der t
o
obt
a
i
n t
h
e
gl
o
b
al
s
o
l
u
t
i
on
o
f
t
h
e
w
hol
e
p
r
o
b
l
e
m
.
Im
p
l
e
m
en
tat
i
o
n
of th
e MAEED prob
lem
i
n
real-tim
e
is
d
o
n
e
in
th
e
follo
wing
way,
Fig
u
re 3
:
All
data from
the
separate a
r
eas
are sent
using
the comm
uni
cat
i
on sy
st
em
to t
h
e m
a
i
n
control center
where the
Clu
s
ter of co
mp
u
t
ers is lo
cat
ed
. Th
e
p
r
ob
lem is so
lv
ed
and
th
e
op
ti
m
a
l s
o
lu
tion
s
are sen
t
to
th
e areas
to
be
use
d
f
o
r c
ont
r
o
l
o
f
t
h
e
ge
ne
rat
o
r
s
po
we
r
p
r
o
d
u
ct
i
o
n
.
T
h
i
s
scenario ca
n be
re
peated through s
o
m
e
selected
peri
ods
of tim
e
,
for exam
ple e
v
ery
hour.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
204
8
–
20
63
2
054
1
m
1
P
opt
1
()
D
Pd
e
m
m
P
opt
D
m
P
D
M
P
M
P
opt
2
M
1
P
2
P
m
P
M
P
whe
r
e :
λ
1
– c
o
or
di
nat
i
n
g
va
ri
abl
e
;
(
opt
)
-
o
p
t
im
al
;
(dem
) -
dem
a
nd
Fi
gu
re
3.
R
eal
-
t
im
e im
pl
em
ent
a
t
i
on st
r
u
ct
u
r
e
o
f
t
h
e
M
A
EE
D
pr
obl
em
sol
u
t
i
o
n
3.
DEVELOP
M
ENT OF
LAG
R
A
N
GE
’S
DE
CO
MPO
S
ITI
O
N
-
C
O
O
RDI
NATI
N
G
ME
THOD
FO
R
SOLUTION OF
THE
MACEED PROBLEM
Devel
opm
ent
of t
h
e m
e
t
hod
i
s
base
d o
n
c
o
nst
r
uct
i
o
n
o
f
a
fu
nct
i
o
n
of L
a
gra
n
ge f
o
r
t
h
e m
u
l
t
i
-
area
di
spat
c
h
pr
obl
e
m
, as fol
l
o
w
s
:
22
11
11
1
1
00
0
11
1
1
1
1
mm
mm
m
m
NN
MM
M
M
m
n
m
n
mn
mn
mn
mn
m
n
m
n
mn
mn
mn
mj
Tmj
j
m
T
jm
mn
mn
m
j
jm
NN
N
N
M
m
m
n
D
m
m
n
m
n
r
mr
m
n
mn
m
T
m
j
j
m
T
j
m
nn
r
n
j
jm
aP
b
P
C
h
d
P
eP
f
q
P
q
P
L
PP
P
B
P
B
P
B
P
P
1
M
m
(1
1)
Whe
r
e:
12
..
..
.
.
.
.
mM
is th
e
v
ector
of th
e Lagran
g
e
’s m
u
ltip
liers.
It is neces
sary
to find t
h
e
values of
,
,,
1
,
mn
T
m
j
T
j
m
m
P
P
P
and
m
M
su
ch
th
at
th
e fun
c
tio
n of Lag
r
ang
e
has an
optim
u
m
solution
which is
m
i
nim
u
m accordi
ng t
o
,
,
mn
T
m
j
T
j
m
PP
P
, and m
a
xim
u
m accordi
n
g to
m
,
un
de
r t
h
e
co
nst
r
ai
nt
s.
,m
i
n
,m
a
x
,1
,
,
1
,
mn
mn
mn
m
PP
P
m
M
n
N
(1
2)
,m
i
n
,m
a
x
,1
,
,
1
,
,
Tmj
T
mj
Tmj
PP
P
m
M
j
M
j
m
(1
3)
,m
i
n
,m
a
x
,1
,
,
1
,
,
Tjm
T
jm
Tjm
PP
P
m
M
j
M
j
m
(1
4)
The
opti
m
a
l solution is
fo
und
on the
ba
sis
of the
necessary
conditions for
optim
a
lit
y according t
o
th
e real
po
wers and
acco
r
d
i
ng
to
th
e
Lagran
ge’s m
u
ltip
liers as fo
llo
ws:
0
1
22
1
2
0
m
N
mn
mn
mn
mn
m
n
mn
m
n
mn
m
m
n
r
mr
m
n
mn
r
L
aP
b
h
d
P
h
e
B
P
B
P
(1
5)
Whe
r
e
1,
,
1
,
m
nN
a
n
d
m
M
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Decomp
o
s
ition Coo
r
d
i
n
a
ting
Meth
o
d
f
o
r Para
llel So
lu
tion
o
f
a Mu
lti Area .... (
S
en
t
h
il Krish
namu
r
t
h
y)
2
055
0,
1
,
,
1
,
,
mj
m
p
Tm
j
Tmj
L
qe
m
M
j
M
j
m
P
(1
6)
10
,
1
,
,
1
,
,
jm
m
j
m
p
T
j
m
Tj
m
L
qe
m
M
j
M
j
m
P
(1
7)
Whe
r
e
,
p
Tmj
e
and
p
Tjm
e
are
the val
u
es
of the corres
p
onding
deri
vatives.
00
0
11
1
1
1
10
mm
m
m
NN
N
N
M
mn
D
m
mn
m
n
r
m
r
m
n
m
n
m
T
m
j
j
m
T
j
m
m
m
nn
r
n
j
jm
L
PP
P
B
P
B
P
B
P
P
e
(1
8)
whe
r
e
m
e
is th
e v
a
l
u
e
o
f
th
e
d
e
ri
vativ
e,
1,
,
1
,
,
mM
j
M
j
m
Sol
u
t
i
o
n o
f
t
h
e sy
st
em
of Equat
i
o
ns (
1
5)
– (1
8
)
can be
achi
e
ved i
n
a hi
erarc
h
i
cal
cal
cul
a
t
i
on
st
ruct
u
r
e
usi
n
g
dec
o
m
posi
t
i
on a
p
p
r
oac
h
bas
e
d
on
sel
ect
i
o
n
of
co
or
di
nat
i
n
g
vari
abl
e
s
[
2
8
]
. These a
r
e se
l
ect
ed
to
b
e
,1
,
m
mM
It
i
s
s
u
p
p
o
sed
t
h
at
t
h
e
val
u
es
o
f
t
h
e
co
o
r
di
n
a
t
i
ng
vari
a
b
l
e
s
are
gi
ve
n
by
t
h
e c
o
o
r
di
nat
o
r
i
n
t
h
e
t
w
o
level hiera
r
c
h
ical structure
of
calcu
latio
n
s
, Fig
u
re
3
as
fo
llows:
,1
,
l
mm
mM
(19)
Whe
r
e
l
is t
h
e i
n
d
e
x
of th
e iteratio
n
s
on
th
e co
ord
i
n
a
ting
level
Whe
n
m
is k
n
o
wn
th
e Equ
a
tio
n (1
5) can
b
e
so
lv
ed
in
a d
e
cen
t
ralized
way
o
n
a first lev
e
l o
f
th
e
calculation structure and t
h
e
Equations
(1
6) to
(1
8) can
b
e
so
lv
ed
o
n
th
e
secon
d
lev
e
l for th
e
ob
tain
ed
o
n
t
h
e
first lev
e
l
v
a
riab
les
mn
P
Deri
vat
i
on
of t
h
e sol
u
t
i
on
of
equat
i
o
n (
1
5) i
s
as fol
l
o
ws:
Eq
uat
i
on
(1
5)
i
s
t
r
ansf
orm
e
d t
o
exp
r
ess
ex
p
licitly all te
rm
s o
f
mn
P:
0
1
22
2
2
1
0
m
N
mn
mn
m
m
n
n
mn
mn
m
n
mn
mn
mn
mn
m
m
n
r
mr
m
n
mn
r
rn
L
aP
B
P
b
h
d
P
h
e
B
P
B
P
(2
0)
If
0
m
an
d i
s
gi
ve
n
by
t
h
e c
o
or
di
n
a
t
o
r E
q
uat
i
o
n (
2
0
)
ca
n
be
wri
t
t
e
n as:
0
1
1
10
2
m
N
mn
mn
mn
mn
mn
mn
mnn
m
n
m
nr
m
r
m
n
mn
m
m
m
m
m
mn
ah
d
b
h
e
L
BP
B
P
B
P
(2
1)
From
here
0
1
1,
1
1,
2
1,
m
N
m
n
mn
mn
m
n
mn
m
n
mn
n
m
n
m
n
r
m
r
m
n
mm
r
m
rn
mM
ah
d
b
h
e
BP
B
P
B
nN
(2
2)
Equ
a
tio
n
(2
2) can
b
e
written
in
a v
ector
m
a
t
r
ix
form
in
Equ
a
tio
n
(23
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
204
8
–
20
63
2
056
11
1
11
1
11
12
1
0
1
1
22
2
22
2
21
22
2
0
2
2
12
..
1
..
.
1
1
2
:
::
:
:
:
..
1
m
m
m
m
m
mm
m
m
mm
m
mm
m
mm
m
N
m
m
m
m
mm
m
mm
m
mm
m
N
m
m
m
m
mN
mN
m
m
mN
m
m
mN
mN
n
N
N
mm
ah
d
bh
e
BB
B
B
P
ah
d
b
h
e
BB
B
B
P
P
ah
d
b
h
e
BB
B
0
,1
,
m
nN
mM
B
(2
3)
In a sh
ort form Equ
a
tion
(23
)
can
b
e
written
as
,1
,
mm
m
EP
D
m
M
(2
4)
I
f
th
e
v
a
lu
e
o
f
,1
,
l
m
mM
is k
nown, th
en th
e so
l
u
tion
for th
e activ
e
power
o
f
th
e
m
th
area for the l
th
iteratio
n
is
\,
1
,
ll
l
mm
m
PE
D
m
M
(2
5)
Sol
u
t
i
o
n
o
f
t
h
e
eq
uat
i
o
n
s
(
1
6)
an
d
(1
7)
can
b
e
d
one
by
g
r
ad
i
e
nt
p
r
oce
d
ures
as f
o
l
l
o
ws:
a)
In
itial v
a
l
u
es of
mm
qq
Tm
j
T
j
m
Pa
n
d
P
are gues
s
e
d
,1
,
1
mm
sg
Tm
j
T
m
j
Tjm
T
j
m
m
m
P
P
and
P
P
s
g
Whe
r
e
1,
mm
sk
and
1,
mm
gk
are t
h
e i
ndexe
s of
t
h
e gra
d
i
e
nt
p
r
oce
d
ures i
n
t
h
e
m
th
area for th
e tie-lin
e
po
we
rs
m
s
Tmj
P
and
m
g
Tjm
P
respectively,
,
1,
m
km
M
are the m
a
xim
u
m
nu
m
b
er
of i
t
eration
for t
h
e
th
m
area.
The value
of
m
k
can be diffe
re
nt for
eac
h of
the areas.
b) T
h
e im
proved
values
of the tie-lines real
powe
r are
1
mm
m
ss
s
Tm
j
T
m
j
Tm
j
P
T
m
j
PP
e
(2
6)
1
m
mm
PTj
m
g
gg
Tj
m
T
j
m
Tj
m
PP
e
(2
7)
Whe
r
e
Tm
j
and
Tjm
are
the steps
of t
h
e
gr
ad
ien
t
pr
o
ced
ur
es.
m
s
PTm
j
e
and
m
g
PTj
m
e
are gi
ve
n by
E
quat
i
o
ns (
1
6
)
a
n
d
(
1
7)
,
1,
,
1
,
,
mM
j
m
j
m
The
calc
u
lation of
the values
of
the
11
mm
sg
Tmj
T
j
m
Pa
n
d
P
stops when
1
m
PTmj
s
e
and
2
m
PT
j
m
g
e
(2
8)
Whe
r
e:
1
and
2
are v
e
ry sm
al
l p
o
sitiv
e nu
m
b
ers.
Sol
u
t
i
o
ns
(
2
5
)
,
(
2
6
)
a
n
d
(
2
7)
depe
n
d
on
,1
,
m
mM
.
Whe
n
t
h
e
opt
im
al value of
m
is ob
tain
ed,
th
e v
a
l
u
es
o
f
,
mn
T
m
j
PP
and
Tjm
P
will reach thei
r
opt
i
m
u
m
. A g
r
adi
e
nt
pr
oce
d
u
r
e i
s
use
d
t
o
ca
l
c
ul
at
e t
h
e
opt
i
m
al
val
u
e
of
,1
,
m
mM
, as
fo
llo
ws:
1
,1
,
ll
l
mm
m
m
em
M
(2
9)
Whe
r
e
,1
,
m
mM
are the steps
of t
h
e
gra
d
ient
proce
d
ure a
n
d
,1
,
l
m
em
M
i
s
gi
ven
by
e
quat
i
on
(
1
8
)
.
The
gra
d
i
e
nt
pr
oce
d
u
r
e
o
n
t
h
e
sec
o
n
d
l
e
vel
st
o
p
s
w
h
en
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Decomp
o
s
ition Coo
r
d
i
n
a
ting
Meth
o
d
f
o
r Para
llel So
lu
tion
o
f
a Mu
lti Area .... (
S
en
t
h
il Krish
namu
r
t
h
y)
2
057
33
,0
,
1
,
l
m
em
M
(
3
0)
Or w
h
en
l
reac
hes t
h
e m
a
xim
u
m
num
ber o
f
i
t
e
rat
i
ons
m
L
.
4.
ALGO
RITH
M OF
THE
LAG
R
A
N
G
E
’S
DECO
M
P
OSITIO
N
-
C
OOR
DI
NATI
N
G
METH
O
D
FOR CALCULATION OF
THE MAEED PROBLEM
Th
e step
s
o
f
the Lagrang
e
’s
deco
m
p
o
s
itio
n-co
ord
i
n
a
ting
meth
od
are:
1.
Val
u
es
o
f
al
l
c
o
ef
fi
ci
ent
s
are
gi
ve
n a
n
d
t
h
e
num
ber
of
i
t
e
r
a
t
i
ons
of
t
h
e s
econ
d
m
L
,1
,
m
km
M
are
gi
ve
n.
2.
In
itial v
a
lu
es
of th
e coo
r
d
i
n
a
ti
n
g
v
a
riab
les
,1
,
l
m
mM
are guesse
d, l=
1
3.
In
itial v
a
lu
es
of th
e tie-lin
es activ
e po
wers are gu
essed
,,
1
,
,
1
,
,
mm
Tm
j
T
j
m
sg
PP
m
M
j
M
j
m
4.
C
a
l
c
ul
at
i
on
on
t
h
e fi
r
s
t
l
e
vel
a
r
e
do
ne
fo
r e
v
e
r
y
su
bsy
s
t
e
m
1,
mM
usi
n
g E
q
uat
i
o
n
(
2
5
)
.
5.
Th
e so
l
u
tion
s
o
f
th
e fi
rst level
,1
,
,
1
,
l
mn
m
Pn
N
m
M
are chec
ke
d according to th
e constrai
nts (12), and
are se
nt to t
h
e
second level.
6.
On th
e
second
lev
e
l th
e
v
a
lu
es of th
e tie-
l
i
n
e
s
are cal
c
u
l
a
t
e
d
by
g
r
a
d
i
e
nt
p
r
oce
d
ures:
i.
Eq
uat
i
ons
(
2
6)
an
d
(1
3)
are
s
o
l
v
e
d
ii.
Eq
uat
i
ons
(
2
7)
an
d
(1
4)
are
s
o
l
v
e
d
Th
e grad
ien
t
p
r
o
c
ed
ures (26) and
(2
7) stop
wh
en
th
e con
d
ition
s
(28
)
are fu
lfilled
resp
ectiv
ely o
r
t
h
e
m
a
xim
u
m
num
ber
of i
t
e
rat
i
o
ns
,1
,
m
km
M
is reac
hed.
7.
Th
e so
lu
tion
s
o
f
th
e secon
d
lev
e
l for
l
Tm
j
P
and
l
Tjm
P
,1
,
,
1
,
,
mM
j
M
j
m
are chec
ked a
ccording to t
h
e
co
nstr
ain
t
s
(
13)
an
d (1
4)
.
8.
On t
h
e seco
n
d
l
e
vel
t
h
e im
pr
ove
d
val
u
es
of
t
h
e Lag
r
an
ge
’
s
vari
a
b
l
e
s are
det
e
rm
i
n
ed. E
quat
i
o
ns
(1
8
)
and
(29
)
are calculated
.
Th
e
grad
ien
t
pro
cedure (29
)
stop
s
wh
en
th
e cond
itio
n
(30
)
is fu
lfilled
o
r
t
h
e
m
a
xim
u
m
num
ber
of i
t
e
rat
i
o
n
s
m
L
is reached. In this cas
e the optim
al solution for
,1
,
m
mM
i
s
obt
ai
n
e
d
an
d th
e corresp
ond
ing
t
o
it op
ti
m
a
l so
lu
tio
ns of t
h
e
p
r
o
b
l
em
s o
n
th
e first
and
secon
d
lev
e
ls are
o
b
t
ained.
Th
e iteratio
n
s
o
n
th
e seco
nd
lev
e
l for calcu
latio
n
of
,1
,
m
mM
can stop before
reaching the
optim
u
m
so
lu
tion
if a m
a
x
i
m
u
m
n
u
m
b
e
r
o
f
iteration
m
L
is d
e
term
in
ed
.
The hie
r
arc
h
ic
al calculating structur
e i
s
sh
ow
n i
n
Fi
g
u
re
4. I
n
t
h
e a
b
ov
e al
go
ri
t
h
m
for o
n
e-st
e
p
o
f
t
h
e gra
d
i
e
nt
p
r
oced
u
r
e fo
r o
p
t
im
i
s
at
i
on of
m
on t
h
e
2
nd
lev
e
l,
m
k
m
a
xim
u
m
num
ber o
f
i
t
e
rat
i
ons o
n
t
h
e
seco
nd
l
e
vel
a
r
e pe
rf
orm
e
d t
o
o
b
t
a
i
n
t
h
e
opt
im
al
sol
u
tio
n
s
for th
e tie-lin
e
real powers and
o
n
e
calcu
latio
n
of
th
e g
e
n
e
rators
real power is do
n
e
on
t
h
e
first
lev
e
l.
1
1
1,
1
,
nn
N
P
m
M
,
1,
mn
m
Pn
N
,
1,
Mn
M
Pn
N
Fi
gu
re
4.
Hi
era
r
chi
cal
st
r
u
ct
u
r
e f
o
r t
h
e M
A
E
E
D
pr
o
b
l
e
m
sol
u
t
i
on
usi
n
g
t
h
e de
vel
o
ped
La
gra
n
ge’s
decom
posi
t
i
o
n
-
co
or
di
nat
i
n
g
a
l
go
ri
t
h
m
5.
STUDIES OF THE SINGL
E
ARE
A
AND
MU
LTI-AREA CEE
D
PR
OBLEM SOLUTIONS
Th
e pro
p
o
s
ed alg
o
r
it
h
m
is
tested
u
s
ing
t
w
o
b
e
n
c
h
m
ark
m
o
d
e
ls of sin
g
l
e an
d
m
u
lti-area power
syste
m
s.
They are:
(i)
Four
areas
with ten ge
ne
rators in eac
h a
r
ea
with
ou
t co
nsidering
th
e tran
smissio
n
lin
e l
o
sses
(ii)
Four area with three ge
nerat
o
rs
in
eac
h
a
r
ea w
ith
co
nsid
eri
n
g th
e t
r
an
sm
issio
n
lin
e lo
sses
The t
w
o
be
nc
h
m
a
rk m
odel
s
are a
ppl
i
e
d
f
o
r two consi
d
ere
d
scenari
o
s, they
are:
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