Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
6,
No.
6,
December
2016,
pp.
3276
–
3282
ISSN:
2088-8708
3276
Con
v
er
gence
Ev
aluation
of
a
Load
Flo
w
Method
based
on
Cespedes’
A
ppr
oach
to
Distrib
ution
System
Analysis
Diego
Issicaba
1
and
J
or
ge
Coelho
2
1
Department
of
Electrical
Engineering,
Federal
Uni
v
ersity
of
T
echnology
-
P
arana
(UTFPR),
Curitiba-PR,
Brazil
2
Department
of
Electrical
Engineering,
Federal
Uni
v
ersity
of
Santa
Catarina
(UFSC),
Florianopolis-S
C,
Brazil
Article
Inf
o
Article
history:
Recei
v
ed
Aug
9,
2016
Re
vised
Sep
20,
2016
Accepted
Oct
7,
2016
K
eyw
ord:
Po
wer
engineering
Po
wer
distrib
ution
systems
Load
flo
w
analysis
Con
v
er
gence
ABSTRA
CT
This
paper
e
v
aluates
the
con
v
er
gence
of
a
load
flo
w
met
hod
based
on
Cespedes’
for
-
mulation
to
distrib
ution
system
steady-state
analysis.
The
method
is
desc
ribed
and
the
closed-form
of
its
con
v
er
gence
rate
is
deduced.
Furthermore,
con
v
er
gence
dependence
of
loading
and
the
consequences
of
choosing
particular
initial
estimates
are
v
erified
mathematically
.
All
mathematical
res
ults
ha
v
e
been
tested
in
numerical
simulations,
some
of
them
presented
in
the
paper
.
Copyright
c
2016
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Die
go
Issicaba
Department
of
Electrical
Engineering,
Federal
Uni
v
ersity
of
T
echnology
-
P
arana
(UTFPR)
A
v
.
Sete
de
Setembro,
3165,
Sector
D,
Rebouc
¸
as,
80230-910
Curitiba-PR,
Brazil
+55
41
3310-4626
issicaba@utfpr
.edu.br
1.
INTR
ODUCTION
Load
flo
w
methods
are
fundamental
tools
to
po
wer
distrib
ution
system
analysis
[1,
2].
These
methods
allo
w
computing
steady-state
v
oltages
at
netw
ork
nodes
as
well
as
the
amount
of
po
wer
flo
wing
through
po
wer
system
de
vices.
Ne
v
ertheless,
po
wer
system
literature
lacks
on
formal
analysis
and
comparisons
among
con-
v
er
gence
properties
of
load
flo
w
algorithms.
Consistent
e
xceptions
can
be
foun
d
in
[3,
4,
5,
6,
7].
P
articularly
,
in
[6]
the
con
v
er
gence
of
a
forw
ard-backw
ard
sweep
method
is
e
v
aluated
and
its
dependence
on
system
loading
v
erified.
In
[7],
this
same
method
is
formally
assessed
using
fix
ed-point
concepts
and
the
contraction
mapping
theorem.
In
this
conte
xt,
this
paper
introduces
the
con
v
er
gence
analysis
of
a
load
flo
w
method
based
on
R.
G.
Cespedes’
approach
to
distrib
ution
system
analysis
[8].
F
or
t
his
accomplishment,
section
2
introduces
the
proposed
method
focusing
on
algorithm
procedures
and
main
equations.
Section
3
presents
the
deduction
of
the
con
v
er
gence
rate
of
the
algorithm
and
a
mathematical
re
gion
where
algorithm
iterates
are
confined,
as
long
as
initial
estimates
are
chosen
properly
.
In
section
4,
numerical
results
are
sho
wn
to
illustrate
the
v
alidity
of
the
mathematical
de
v
elopments.
At
last,
section
5
outlines
conclusions
and
final
remarks.
2.
A
LO
AD
METHOD
INSPIRED
ON
CESPEDES’
FORMULA
TION
Consider
the
radial
feeder
schematic
sho
wn
in
Fig.
1.
In
the
schematic,
z
i
denotes
a
series
line
impedance,
E
i
=
e
i
+
j
f
i
represents
comple
x
node
v
oltages,
and
S
L
i
denotes
comple
x
loads,
all
refereed
to
a
general
node
i
.
The
inde
x
u
i
stands
for
the
node
upstream
node
i.
The
substation
b
us
is
named
the
0
(zero)
node
with
comple
x
v
oltage
denoted
by
E
0
.
Furthermore,
the
inde
x
i
points
out
to
both
a
node
and
its
upstream
line.
The
comple
x
po
wer
flo
w
do
wnstream
line
i
can
be
computed
by
summing
the
loads
and
l
osses
do
wn-
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
,
DOI:
10.11591/ijece.v6i6.12134
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
3277
q
q
?
S
L
i
q
?
S
L
i
+1
q
?
S
L
n
1
q
?
S
L
n
E
u
i
E
i
E
i
+1
E
N
1
E
N
z
i
z
i
+1
z
n
S
i
-
S
i
|{z}
branch
n
Figure
1.
Radial
distrib
ution
netw
ork
schematic
(adapted
from
[1]).
stream
the
line
as
follo
ws
S
i
=
S
L
i
+
X
r
2
i
S
L
r
+
X
r
2
i
z
r
S
r
E
r
2
(1)
where
i
represents
either
the
set
of
nodes
do
wnstream
node
i
or
do
wnstream
line
i
,
depending
on
the
v
ariables
in
v
olv
ed.
As
consequence,
S
i
can
be
e
xpressed
as
function
of
v
ariables
associated
to
nodes
immediately
do
wnstream
line
i
,
as
follo
ws
S
i
=
S
L
i
+
X
d
2D
i
S
d
+
X
d
2D
i
z
d
S
d
E
d
2
(2)
where
D
i
denotes
either
the
set
of
nodes
immediately
do
wnstream
node
i
or
immediately
do
wnstream
line
i
.
Furthermore,
the
dif
ference
between
node
v
oltages
in
adjacent
nodes
can
be
e
xpressed
as
E
i
=
E
u
i
z
i
S
i
E
i
(3)
Separating
the
real
and
imaginary
parts
of
(2)
and
the
v
oltage
magnitudes
in
(3),
the
recursi
v
e
equa-
tions
utilized
in
the
load
flo
w
method
proposed
by
R.
G.
Cespedes
[8]
can
be
deduced.
Con
v
ersely
,
by
applying
(2)
and
(3)
without
further
deductions,
a
similar
b
ut
also
ef
fecti
v
e
load
flo
w
method
can
be
designed.
Starting
from
initial
comple
x
v
oltage
estimates,
the
tw
o
steps
belo
w
can
be
continually
repeated
until
the
con
v
er
gence
of
comple
x
v
oltages
is
reached.
1.
In
a
backw
ard
process,
the
comple
x
po
wer
flo
w
at
each
node
is
calculated
using
(2),
starting
at
end-nodes
and
stopping
at
the
first
node
immediately
do
wnstream
from
the
substation
node.
2.
In
a
forw
ard
process,
comple
x
v
oltage
are
updated,
a
w
ay
from
the
substation
node,
using
(3).
The
comple
x
v
oltage
at
the
substation
b
us
is
assumed
constant
during
the
procedures.
The
con
v
er
gence
criterion
refers
to
the
maximum
absolute
mismatch
between
subsequent
comple
x
v
oltage
iterates.
3.
CONVERGENCE
ASSESSMENT
This
section
addresses
the
con
v
er
ge
assessment
of
the
algorithm,
focusing
on
aspects
related
to
the
implications
of
choosing
an
initial
estimate
and
the
deduction
of
the
con
v
er
gence
rate.
3.1.
Initial
Estimate
The
update
rule
of
the
algorithm
can
be
written
recursi
v
ely
for
iteration
k
as
(
k
)
i
=
(
k
)
u
i
z
i
E
(
k
)
i
S
i;c
+
L
(
k
)
i;ac
(4)
where
E
(
k
)
i
denotes
the
comple
x
v
oltage
at
node
i
and
iteration
k
,
(
k
)
i
is
the
comple
x
v
oltage
at
node
i
and
iteration
k
+
1
,
S
i;c
denotes
the
sum
of
all
comple
x
loads
do
wnstream
node
i
(including
the
one
at
node
i
)
and
L
(
k
)
i;ac
represents
the
sum
of
all
electrical
losses
do
wnstream
node
i
and
iteration
k
.
Con
ver
g
ence
Evaluation
of
a
Load
Flow
Method
based
on
Cespedes’
Appr
oac
h
...
(Die
go
Issicaba)
Evaluation Warning : The document was created with Spire.PDF for Python.
3278
ISSN:
2088-8708
Let
us
define
~
i
as
the
set
of
lines
in
the
path
between
node
0
and
i
.
By
(4),
we
ha
v
e
(
k
)
1
=
E
0
z
1
E
(
k
)
1
S
1
;c
+
L
(
k
)
1
;ac
.
.
.
=
(
k
)
1
.
.
.
.
.
.
=
.
.
.
.
.
.
(
k
)
i
=
(
k
)
u
i
z
i
E
(
k
)
i
S
i;c
+
L
(
k
)
i;ac
(5)
By
summing
the
equations
abo
v
e,
a
closed-form
for
the
update
rule
of
the
algorithm
can
be
obtained.
(
k
)
i
=
E
0
X
m
2
~
i
z
m
E
(
k
)
m
S
m;c
+
L
(
k
)
m;ac
(6)
F
or
instance,
consider
a
radial
netw
ork
with
5
nodes
and
connections
f
0–1,
1–2,
2–3,
1–4
g
.
F
or
this
netw
ork,
the
closed-form
for
the
update
rules
are
the
follo
wing:
(
k
)
1
=
E
0
z
1
E
(
k
)
1
S
1
;c
+
L
(
k
)
1
;ac
(7)
(
k
)
2
=
E
0
z
1
E
(
k
)
1
S
1
;c
+
L
(
k
)
1
;ac
z
2
E
(
k
)
2
S
2
;c
+
L
(
k
)
2
;ac
(8)
(
k
)
3
=
E
0
z
1
E
(
k
)
1
S
1
;c
+
L
(
k
)
1
;ac
z
2
E
(
k
)
2
S
2
;c
+
L
(
k
)
2
;ac
z
3
E
(
k
)
3
S
3
;c
+
L
(
k
)
3
;ac
(9)
(
k
)
4
=
E
0
z
1
E
(
k
)
1
S
1
;c
+
L
(
k
)
1
;ac
z
4
E
(
k
)
4
S
4
;c
+
L
(
k
)
4
;ac
(10)
and,
in
a
compact
matrix
notation:
2
6
6
6
4
(
k
)
1
(
k
)
2
(
k
)
3
(
k
)
4
3
7
7
7
5
=
2
6
6
4
E
0
E
0
E
0
E
0
3
7
7
5
2
6
6
6
6
4
z
1
E
(
k
)
1
0
0
0
z
1
E
(
k
)
1
z
2
E
(
k
)
2
0
0
z
1
E
(
k
)
1
z
2
E
(
k
)
2
z
3
E
(
k
)
3
0
z
1
E
(
k
)
1
0
0
z
4
E
(
k
)
4
3
7
7
7
7
5
2
6
6
6
4
S
1
;c
+
L
(
k
)
1
;ac
S
2
;c
+
L
(
k
)
2
;ac
S
3
;c
+
L
(
k
)
3
;ac
S
4
;c
+
L
(
k
)
4
;ac
3
7
7
7
5
(11)
This
e
xample
suggests
the
update
rule
can
be
written
in
a
general
compact
matrix
notation.
F
or
t
his
accomplishment,
assuming
an
inde
xing
where
u
i
<
i
,
let
us
define
the
lo
wer
triangular
path
matrix
T
with
size
n
n
and
entries
t
im
=
1
;
if
m
2
~
i
0
;
otherwise
(12)
Then,
the
update
rule
of
the
algorithm
can
written
in
the
compact
form
(
k
)
=
E
0
TZ
p
S
c
K
(
k
)
TZ
p
L
(
k
)
ac
K
(
k
)
(13)
where
(
k
)
is
a
n
1
v
ector
with
entries
gi
v
en
by
(
k
)
i
,
E
0
is
a
n
1
v
ector
with
entries
equal
to
E
0
,
Z
p
is
the
n
n
primiti
v
e
im
pedance
matrix,
S
c
denotes
a
n
n
diagonal
matr
ix
with
elements
equal
to
S
i;c
,
L
(
k
)
ac
represents
a
n
n
diagonal
matrix
with
elements
gi
v
en
by
L
(
k
)
i;ac
and
K
(
k
)
is
a
n
1
v
ector
with
reciprocals
of
E
(
k
)
i
.
Let
no
w
R
be
a
closed
re
gion
in
the
comple
x
space
C
n
defined
by
R
=
f
E
2
C
n
;
jj
E
i
jj
(
E
0
)
,
8
i
=
1
;
:
:
:
;
n
g
,
for
some
2
R
such
that
E
0
2
<
E
0
p
(14)
where
=
jj
TZ
p
S
c
jj
+
jj
TZ
p
jj
L
ac
;
(15)
IJECE
V
ol.
6,
No.
6,
December
2016:
3276
–
3282
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
3279
and
L
ac
;
is
a
n
n
diagonal
with
entries
gi
v
en
by
the
maximum
accumulated
losses
do
wnstream
each
node,
computed
using
a
backw
ard
process
with
v
oltage
magnitudes
equal
to
(
E
0
)
.
Gi
v
en
an
iterate
E
(
k
)
2
R
,
by
(13)
we
ha
v
e
(
k
)
E
0
=
TZ
p
S
c
K
(
k
)
+
TZ
p
L
(
k
)
ac
K
(
k
)
jj
TZ
p
S
c
jj
K
(
k
)
+
jj
TZ
p
jj
L
(
k
)
ac
K
(
k
)
jj
TZ
p
S
c
jj
(
E
0
)
+
jj
TZ
p
jj
L
ac
;
(
E
0
)
(16)
and,
by
using
(14)
we
ha
v
e
(
k
)
E
0
(
E
0
)
<
(
E
0
)
(17)
Therefore,
assuming
the
e
xistence
of
R
,
for
a
gi
v
en
if
E
(
k
)
2
R
,
then
(
k
)
belongs
to
an
open
ball
(in
C
n
)
centered
in
E
0
and
ra
d
i
us
equal
t
o
.
P
articul
arly
,
(
k
)
2
R
for
all
E
(0)
2
R
,
8
k
,
then
if
the
initial
estimate
belongs
R
,
all
other
iterates
also
belong
to
R
.
3.2.
Con
v
er
gence
Rate
Once
a
re
gion
where
iterates
are
confided
through
the
iterati
v
e
process
ha
v
e
be
en
deduced,
let
us
e
xamine
the
con
v
er
gence
rate
of
the
algorithm.
Notice
that
tw
o
subsequent
iterates
of
the
algorithm
can
be
written
as
(
k
+1)
i
=
E
0
X
m
2
~
i
z
m
E
(
k
+1)
m
S
m;c
+
L
(
k
+1)
m;ac
(18)
(
k
)
i
=
E
0
X
m
2
~
i
z
m
E
(
k
)
m
S
m;c
+
L
(
k
)
m;ac
(19)
and,
subtracting
(19)
from
(18)
we
ha
v
e
(
k
+1)
i
=
X
m
2
~
i
z
m
E
(
k
)
m
S
m;c
+
L
(
k
)
m;ac
z
m
E
(
k
+1)
m
S
m;c
+
L
(
k
+1)
m;ac
(20)
Hence,
by
manipulating
the
terms
of
the
equation
abo
v
e,
we
ha
v
e
(
k
+1)
i
=
X
m
2
~
i
z
m
"
S
m;c
+
L
(
k
+1)
m;ac
E
(
k
+1)
m
E
(
k
)
m
L
(
k
+1)
m;ac
E
(
k
+1)
m
E
(
k
+1)
m
#
E
(
k
+1)
m
(21)
where
E
(
k
+1)
m
=
E
(
k
+1)
m
E
(
k
)
m
and
L
(
k
+1)
m;ac
=
L
(
k
+1)
m;ac
L
(
k
)
m;ac
.
In
a
compact
form,
this
e
xpression
can
be
re
written
as
(
k
+1)
i
=
X
m
2
~
i
D
(
k
+1)
im
E
(
k
+1)
m
(22)
in
which
D
(
k
+1)
im
=
X
m
2
~
i
z
m
"
S
m;c
+
L
(
k
+1)
m;ac
E
(
k
+1)
m
E
(
k
)
m
L
(
k
+1)
m;ac
E
(
k
+1)
m
E
(
k
+1)
m
#
(23)
Equation
(22)
can
also
be
written
in
matrix
notation
as
(
k
+1)
=
D
(
k
+1)
E
(
k
+1)
(24)
where
(
k
+1)
=
(
k
+1)
(
k
)
,
D
(
k
+1)
is
a
n
n
matrix
with
entries
gi
v
en
by
D
(
k
+1)
im
and
E
(
k
+1)
=
E
(
k
+1)
E
(
k
)
.
Matrix
D
(
k
+1)
indicates
the
con
v
er
gence
rate
of
the
algorithm
in
each
iteration.
It
also
pro
v
es
con
v
er
gence
direct
dependence
on
netw
ork
loading,
losses
iterates
and
con
v
er
gence
of
losses.
Con
ver
g
ence
Evaluation
of
a
Load
Flow
Method
based
on
Cespedes’
Appr
oac
h
...
(Die
go
Issicaba)
Evaluation Warning : The document was created with Spire.PDF for Python.
3280
ISSN:
2088-8708
Figure
2.
Schematic
of
the
actual
distrib
ution
netw
ork
utilized
in
the
e
v
aluations.
4.
NUMERICAL
RESUL
TS
Numerical
load
flo
w
analysis
are
presented
in
this
section
to
v
erify
the
pro
vided
mathematical
results.
Fig.
2
sho
ws
an
actual
13.80
kV
distrib
ution
netw
ork
with
490
nodes
utilized
to
this
purpose.
W
ithout
loss
of
generality
,
the
v
alue
of
8
:
28
kV
(0.6000
pu)
has
been
chosen
as
.
This
implies
in
a
feasible
re
gion
R
,
E
2
C
489
;
jj
E
i
jj
E
0
;
8
i
=
1
;
:
:
:
;
489
g
,
which
meets
the
interv
al
0
:
5000
pu
=
E
0
2
<
E
0
p
=
0
:
8160
pu
(25)
where
equals
0.0339
pu.
Comple
x
v
oltages
ha
v
e
been
obtained
using
the
load
flo
w
method.
Results
ha
v
e
been
v
alidated
using
the
approach
proposed
in
[1].
T
able
1(a)
sho
ws
the
real
and
imaginary
parts
of
v
oltage
iterates
as
well
as
error
v
alues
for
node
300,
assuming
the
uncommon
initi
al
estimate
of
2
:
00
\
63
:
03
o
pu,
8
i
.
On
the
other
hand,
T
able
1(b)
sho
ws
the
same
v
ariables,
b
ut
for
a
case
considering
a
loading
increased
by
se
v
enfold.
The
maximum
absolute
mismatch
between
v
oltage
iterates
has
been
chosen
as
con
v
er
gence
criterion.
The
last
iterate
has
been
set
as
solution
for
the
sak
e
of
error
computation.
T
olerance
has
been
specified
to
10
6
.
T
able
1.
Numerical
results
for
b
us
300
assuming
the
initial
estimate
E
(0)
i
=
2
:
00
\
63
:
03
o
pu,
8
i
(a)
Normal
loading
k
e
300
(pu)
f
300
(pu)
D
(
k
)
Error
0
0.90719
-1.78241
-
-
1
0.99995
-0.01043
0.00013
1.79E
00
2
0.98115
-0.00927
0.00073
2.28E
02
3
0.98096
-0.00936
0.00069
3.16E
04
4
0.98096
-0.00936
0.00079
2.91E
06
5
0.98096
-0.00936
0.00074
2.18E
08
6
0.98096
-0.00936
0
0
(b)
Increased
loading
k
e
300
(pu)
f
300
(pu)
D
(
k
)
Error
0
0.90719
-1.78241
-
-
1
0.99915
-0.07408
0.00677
1.86E
00
2
0.85611
-0.06065
0.05374
1.94E
01
3
0.84061
-0.06560
0.07107
2.48E
02
4
0.83888
-0.06560
0.07926
2.89E
03
5
0.83870
-0.06563
0.07756
3.13E
04
6
0.83868
-0.06563
0.07852
3.20E
05
7
0.83868
-0.06563
0.07816
3.23E
06
8
0.83868
-0.06563
0.07826
2.96E
07
9
0.83868
-0.06563
0
0
Fig.
3
sho
ws
the
first
tw
o
v
oltage
iterates
for
the
first
case
in
a
le
v
el
curv
e
of
the
con
v
er
gence
re
gion.
As
e
xpected,
one
can
notice
that
e
v
en
by
choosing
a
nonrealistic
solution
as
initial
estimate,
the
first
iterate
belongs
to
an
open
ball
centered
in
E
0
and
limited
by
radius
,
follo
wed
t
hat
the
con
v
er
gence
of
the
algorithm
is
reached.
Con
v
er
gence
rate
is
s
ho
wn
to
be,
in
these
cases,
lo
wer
to
the
unit.
Furthermore,
as
deduced,
the
increase
of
loading
caused
an
increa
se
of
con
v
er
gence
rate,
impacting
on
the
algorithm
ef
ficienc
y
.
IJECE
V
ol.
6,
No.
6,
December
2016:
3276
–
3282
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
3281
e
3
0
0
[
p
u
]
-2
-1
0
1
2
f
3
0
0
[
p
u
]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0
1
Solution
Voltage iterate
E
0
−
α
Figure
3.
Iterates
in
a
le
v
el
curv
e
of
the
re
gion
R
,
for
the
b
us
300.
Initial
estimate:
E
(0)
i
=
2
:
00
\
63
:
03
o
pu.
5.
CONCLUSIONS
This
paper
e
v
aluates
the
con
v
er
gence
of
a
load
flo
w
method
inspired
on
R.
G.
Cespedes’
r
ecursi
v
e
equations.
Numerical
results
and
formal
deduction
of
con
v
er
gence
rate
sho
w
that
the
ef
ficienc
y
of
the
method
depends
on
the
netw
ork
loading,
losses
and
con
v
er
gence
of
losses.
Also,
a
re
gion
where
algorithm
iterates
are
confined
is
deduced
as
long
as
initial
estimates
are
chosen
properly
.
Future
w
orks
will
e
xtend
these
de
v
elop-
ments
to
three-phase
load
flo
w
approaches.
A
CKNO
WLEDGMENTS
The
authors
w
ould
lik
e
to
ackno
wledge
the
financial,
technical
and
human
support
of
the
CNPq,
CAPES,
INESC
Porto
and
INESC
P&D
Brasil.
REFERENCES
[1]
D.
Issicaba
and
J.
Coelho,
“Rotational
load
flo
w
method
for
radial
distrib
ution
systems,
”
International
J
ournal
of
Electrical
and
Computer
Engineering
(IJEPE)
,
v
ol.
6,
no.
3,
2016.
[2]
H.
Maref
atjou
and
M.
Sarvi,
“Distrib
uted
generation
allocation
to
impro
v
e
steady
state
v
oltage
stability
of
distrib
ution
netw
orks
using
imperialist
competiti
v
e
algorithm,
”
International
J
ournal
of
Applied
P
ower
Engineering
(IJ
APE)
,
v
ol.
2,
no.
1,
pp.
15–26,
2013.
[3]
J.
F
.
Chen
and
W
.
M.
W
ang,
“Uniqueness
of
the
feasible
v
oltage
solutions
for
radial
po
wer
netw
orks,
”
in
IEEE
Re
gion
10
International
Confer
ence
on
Micr
oeletr
onics
and
VLSI,
TENCON’
s
95
,
No
v
emeer
1995,
pp.
351–354.
[4]
H.
D.
Chiang
and
M.
E.
Baran,
“On
the
e
xistence
and
uniqueness
of
load
flo
w
solution
for
radial
distrib
ution
netw
orks,
”
IEEE
T
r
ansactions
on
Cir
cuits
and
Systems
,
v
ol.
37,
no.
3,
pp.
410–416,
March
1990.
[5]
K.
N.
Miu
and
H.
D.
Chiang,
“Existence,
uniqueness,
and
monotonic
properties
of
the
feasible
po
wer
flo
w
solution
for
radial
three-phase
distrib
ution
netw
orks,
”
IEEE
T
r
ansactions
on
Cir
cuits
and
Systems
–
I:Fundamental
Theory
and
Applications
,
v
ol.
47,
no.
10,
pp.
1502–1514,
October
2000.
[6]
E.
Bompard,
E.
Carpaneto,
G.
Chicco,
and
R.
Napoli,
“Con
v
er
gence
of
the
backw
ard-forw
ard
sweep
method
for
the
load
flo
w
analysis
of
radial
distrib
ution
systems,
”
Electrical
P
ower
&
Ener
gy
Systems
,
v
ol.
22,
no.
7,
pp.
521–530,
October
2000.
[7]
D.
Issicaba,
“Ladder
load
flo
w
methods
to
radial
and
weakly
meshed
dis
trib
ution
systems,
”
Master’
s
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Federal
Uni
v
ersity
of
Santa
Catarina,
2008.
[8]
R.
G.
Cespedes,
“Ne
w
method
for
the
analysis
of
distrib
ution
netw
orks,
”
IEEE
T
r
ansaction
on
P
ower
Delivery
,
v
ol.
5,
no.
1,
pp.
391–396,
January
1990.
Con
ver
g
ence
Evaluation
of
a
Load
Flow
Method
based
on
Cespedes’
Appr
oac
h
...
(Die
go
Issicaba)
Evaluation Warning : The document was created with Spire.PDF for Python.
3282
ISSN:
2088-8708
BIOGRAPHIES
OF
A
UTHORS
Diego
Issicaba
recei
v
ed
the
B.S.
and
M.S.
de
grees
in
Electrical
Engineering
from
the
Federal
Uni-
v
ersity
of
Santa
Catarina
(UFSC),
Santa
Catarina,
Brazil,
in
2006
and
2008,
respecti
v
ely
.
Fur
-
thermore,
he
recei
v
ed
the
Ph.D.
de
gree
on
Sustainable
Ener
gy
Systems,
under
the
MIT
Doctoral
Program,
from
the
F
aculty
of
Engineering
of
the
Uni
v
ersity
of
Porto,
Portug
al.
His
research
inter
-
ests
in
v
olv
e
smart
grids,
mutiagent
systems,
distrib
uted
generation
and
distrib
ution
systems.
He
is
currently
a
full
Professor
at
Federal
Uni
v
ersity
of
T
echnology
–
P
arana
(UTFPR),
Associate
Re-
searcher
and
Coordinator
of
the
Research
Area
on
Ener
gy
and
Management
of
INESC
P&D
Brasil.
J
or
ge
Coelho
recei
v
ed
the
B
.S.
and
M.S.
de
grees
in
electrical
engineering
from
the
Federal
Uni
v
er
-
sity
of
Santa
Catarina,
Brazil,
in
1977
and
1980,
respecti
v
ely
.
In
1990,
he
recei
v
ed
the
Ph.D.
de
gree
in
electrical
engineering
from
the
Catholic
Uni
v
ersity
of
Rio
de
Janeiro,
Brazil.
He
is
a
Professor
of
the
Department
of
Electrical
Engineering
at
the
Federal
Uni
v
ersity
of
Santa
Catarina,
Brazil,
since
March
1978.
His
research
interests
include
distrib
ution
systems
e
xpansion
and
opera
tion
planning,
po
wer
systems
reliability
,
probabilistic
methods
applied
to
po
wer
systems,
and
po
wer
quality
.
IJECE
V
ol.
6,
No.
6,
December
2016:
3276
–
3282
Evaluation Warning : The document was created with Spire.PDF for Python.