Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
6,
No.
3,
June
2016,
pp.
1344
–
1352
ISSN:
2088-8708,
DOI:
10.11591/ijece.v6i3.10083
1344
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
Rotational
Load
Flo
w
Method
f
or
Radial
Distrib
ution
Systems
Diego
Issicaba
*
and
J
or
ge
Coelho
**
*
Department
of
Electrical
Engineering,
Federal
Uni
v
ersity
of
T
echnology
-
P
arana
(UTFPR),
Curitiba-PR,
Brazil
**
Department
of
Electrical
Engineering,
Federal
Uni
v
ersity
of
Santa
Catarina
(UFSC),
Florianopolis-SC,
Braz
il
Article
Inf
o
Article
history:
Recei
v
ed
Feb
5,
2016
Re
vised
Mar
24,
2016
Accepted
Apr
5,
2016
K
eyw
ord:
Po
wer
engineering
Po
wer
distrib
ution
systems
Load
flo
w
analysis
Radial
netw
orks
Coordinate
rotation
ABSTRA
CT
This
paper
introduces
a
modified
edition
of
classical
Cespedes’
load
flo
w
method
to
radial
distrib
ution
system
analysis.
In
the
de
v
eloped
approach,
a
distrib
ution
netw
ork
is
mod-
eled
in
dif
ferent
comple
x
reference
systems
and
reduced
to
a
set
of
connected
equi
v
alent
subnetw
orks,
each
without
resistance,
while
graph
topology
and
node
v
oltage
solution
are
preserv
ed.
Acti
v
e
po
wer
losses
are
then
not
dissipated
in
the
modeled
subnetw
orks
and
ac-
ti
v
e
po
wer
flo
ws
can
be
obtained
as
a
consequence
of
radiality
.
Thus,
the
proposed
method
preprocesses
a
series
of
v
ariable
transformations
concomitant
to
an
iterati
v
e
algorithm
using
a
forw
ard-backw
ard
sweep
to
arri
v
e
at
the
load
flo
w
solution.
The
proposed
approach
has
been
tested
using
literature
and
actual
distrib
ution
netw
orks,
and
ef
ficienc
y
impro
v
ements
are
v
erified
in
comparison
to
Cespedes’
load
flo
w
method.
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,
Rebouas,
80230-910
Curitiba-PR,
Brazil
+55
41
3310-4626
issicaba@utfpr
.edu.br
1.
INTR
ODUCTION
Load
flo
w
analysis
[1,
2]
pro
vides
the
steady-state
condition
for
po
wer
systems
and
is
one
of
the
most
im-
portant
numerical
tools
to
system
planning
and
designing.
In
order
to
design
solutions
to
po
wer
distrib
ution
system
real-time
operation
[3],
modeling
and
analysis
might
tak
e
into
account
unbalanced
operation
and
detailed
features
of
each
component
connected
to
the
netw
orks.
On
the
other
hand,
in
long-term
planning
stages,
some
h
ypotheses
and
simplifications
can
be
assumed
(e.g.
balanced
operation,
constant
po
wer
load)
due
to,
for
instance,
uncertainties
re
g
arding
future
load
profiles
and
distrib
uted
generation
productions.
This
is
the
case
in
adequac
y
e
v
aluations
[4,
5],
where
hundreds
of
thousands
of
load
flo
w
analysis
may
be
e
x
ecuted,
while
performance
indices
are
estimated
by
modeling
f
ailure/repair
c
ycles
of
components
and
load/generation
profiles
as
stochastic
processes.
Detailed
distrib
ution
system
modeling
and
analysis
is
a
well
established
topic,
co
v
ered
in
books
such
as
[6].
Simplified/single-phase
load
flo
w
modeling
and
analysis
dates
mostly
to
the
80’
s
and
90’
s
and
can
be
di
vided
into
tw
o
groups:
the
first
group
comprises
Ne
wtonian
based
methods
adjusted
to
distrib
ution
system
analysis
[7,
8]
and
the
second
group
includes
methods
based
on
iterati
v
e
forw
ard–backw
ard
sweep
processes
[9,
10,
11,
12,
13,
14].
The
sweep
based
techniques
are
kno
wn
by
their
ef
ficienc
y
and
tak
e
adv
antage
of
the
f
act
that
distrib
ution
netw
orks
usually
are
radially
operated.
F
orw
ard–backw
ard
sweep
methods
emplo
y
the
follo
wing
general
procedures:
(a)
assuming
a
flat
start
or
an
approximate
solution,
current
or
po
wer
do
wnstream
each
node
is
estimated
in
a
backw
ard
sweep
(f
rom
end
nodes
to
w
ards
the
substation
node);
(b)
using
esti
mates
from
the
pre
vious
step,
node
v
oltages
are
updated
in
a
forw
ard
sweep
(from
the
substation
node
to
w
ards
end
nodes);
(c)
these
t
w
o
steps
are
repeated
up
to
the
con
v
er
gence
of
node
v
oltages.
Among
the
se
v
eral
v
ariations
of
sweep
techniques,
the
one
p
r
op
os
ed
by
Baran
and
W
u
[11,
12]
stands
out
by
emplo
ying
a
set
of
equations,
kno
wn
as
Distflow
br
anc
h
equations
,
that
recursi
v
ely
relates
acti
v
e
po
wer
,
reacti
v
e
po
wer
and
v
oltage
magnitudes.
Cespedes
[9]
steps
ahead
by
addressing
the
use
of
a
biquadratic
equation
to
relate
node
v
oltages
as
functions
of
do
wnstream
acti
v
e
and
reacti
v
e
po
wer
flo
ws
and
immediate
upstream
node
v
oltages.
This
paper
brings
out
the
f
act
that,
although
Cespedes’
method
is
ef
ficient
and
rob
ust
to
distrib
ution
system
analysis,
it
in
v
olv
es
a
non-ne
gligible
amount
of
unnecessary
calculations
associated
to
updating
acti
v
e
and
reacti
v
e
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
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A
d
v
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ine
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i
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w
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i
a
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o
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o
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Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1345
po
wer
flo
ws
in
netw
ork
branches.
In
f
act,
a
finite
and
relati
v
ely
small
number
of
standardized
conductor
types
are
utilized
in
each
distrib
ution
feeder
,
composing
subnetw
orks
with
the
same
r
+
j
x
=
km
and
r
=
x
ratio.
As
consequence,
gi
v
en
a
possible
iterati
v
e
load
flo
w
solution,
there
is
a
constant
relation
between
acti
v
e
and
reacti
v
e
losses
at
each
subnetw
ork
branch,
making
the
concomitant
ef
fort
of
updating
both,
unnecessary
.
The
proposed
approach,
of
the
Rotational
Load
Flo
w
Method,
goes
be
yond
Cespedes’
formulation
by
per
-
forming
generalized
comple
x
reference
system
rotations
and
by
taking
adv
antage
of
radiality
to
impro
v
e
the
ef
ficienc
y
of
sweep
procedures.
The
approach
has
been
tested
using
academi
c
and
actual
electrical
distrib
ution
netw
orks,
such
that
ef
ficienc
y
impro
v
ements
are
v
erified
in
comparison
to
Cespedes’
load
flo
w
method.
The
paper
is
or
g
anized
into
fi
v
e
sections
as
follo
ws.
Section
2
and
3
present
mathematical
background
and
the
proposed
method,
respecti
v
ely
.
Section
4
sho
ws
numerical
results
for
dif
ferent
test
systems
and
an
actual
system.
Conclusions
and
final
remarks
are
outlined
in
section
5.
2.
MA
THEMA
TICAL
B
A
CKGR
OUND
This
section
presents
a
brief
re
vie
w
of
the
load
flo
w
formulation
de
v
eloped
by
Cespedes
[9].
A
discussion
about
comple
x
reference
system
rotation,
focused
on
distrib
ution
system
analysis
and
applications,
is
also
approached.
2.1.
Cespedes’
Load
Flo
w
F
ormulation
Consider
the
radial
distrib
ution
system
schematic
sho
wn
in
Fig.
1,
where
distrib
ution
lines
are
modeled
as
series
impedances
Z
i
=
z
i
\
i
=
r
i
+
j
x
i
while
comple
x
v
oltages
and
comple
x
load
demands
are
represented,
respecti
v
ely
,
by
V
i
=
v
i
\
i
and
S
L
i
=
s
L
i
\
L
i
=
P
L
i
+
j
Q
L
i
,
8
i
=
1
;
:::;
N
.
Note
that
the
inde
x
i
is
uti
lized
to
identify
the
node
and
the
line
upstream
this
node,
depending
on
the
v
ariables
in
v
olv
ed.
In
addition,
u
i
denotes
the
node
upstream
node
i
.
q
q
?
S
L
i
q
?
S
L
i
+1
q
?
S
L
N
1
q
?
S
L
N
V
u
i
V
i
V
i
+1
V
N
1
V
N
Z
i
Z
i
+1
Z
N
S
i
-
S
i
|{z}
branch
N
Figure
1.
Radial
distrib
ution
netw
ork
schematic.
The
comple
x
po
wer
flo
w
injected
at
node
i
through
an
upstream
branch,
denoted
by
S
i
=
s
i
\
i
,
can
be
written
as
a
function
of
do
wnstream
node
v
oltages,
loads
and
line
losses.
This
po
wer
flo
w
can
be
also
interpreted
as
an
accumulated
po
wer
at
node
i
gi
v
en
by
S
i
=
S
L
i
+
X
k
2
i
S
L
k
+
X
k
2
i
Z
k
s
k
v
k
2
(1)
or
,
separating
in
real
and
imaginary
terms,
P
i
=
P
L
i
+
X
k
2
i
P
L
k
+
X
k
2
i
r
k
P
2
k
+
Q
2
k
v
2
k
;
Q
i
=
Q
L
i
+
X
k
2
i
Q
L
k
+
X
k
2
i
x
k
P
2
k
+
Q
2
k
v
2
k
(2)
where
i
denotes
the
set
of
nodes
do
wnstream
node
i
and
the
set
of
lines
do
wnstream
node
i
,
depending
on
the
v
ariables
in
v
olv
ed.
Once
accumulated
acti
v
e
and
reacti
v
e
po
wers
at
node
i
are
kno
wn,
the
v
oltage
magnitude
at
node
i
can
be
obtained
by
solving
the
biquadratic
equation
[9]
v
4
i
+
A
i
v
2
i
+
B
i
=
0
(3)
where
A
i
=
2
(
P
i
r
i
+
Q
i
x
i
)
v
2
u
i
;
B
i
=
P
2
i
+
Q
2
i
r
2
i
+
x
2
i
(4)
Similarly
,
the
angle
at
node
i
can
be
obtained
through
the
follo
wing
equation.
Rotational
Load
Flow
Method
for
Radial
Distrib
ution
Systems
(Die
go
Issicaba
and
J
or
g
e
Coelho)
Evaluation Warning : The document was created with Spire.PDF for Python.
1346
ISSN:
2088-8708
i
=
i
1
+
tan
1
P
i
x
i
Q
i
r
i
P
i
r
i
+
Q
i
x
i
+
v
2
i
(5)
Thus,
assuming
that
the
substation
node
comple
x
v
oltage
V
0
=
v
0
\
0
o
is
specifed
and
node
v
oltage
magni-
tudes
are
ini
tiated
using
a
flat
start
or
an
approximate
solution,
the
load
flo
w
solution
can
be
found
through
iterati
v
e
forw
ard–backw
ard
sweeps
as
follo
ws.
In
a
backw
ard
process,
accumulated
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.
In
a
forw
ard
process,
v
oltage
magnitudes
are
updated,
a
w
ay
from
the
substation
node,
using
(3)–(4).
These
procedures
are
continually
repeated
until
the
con
v
er
gence
of
v
oltage
magnitudes
is
reached.
The
backw
ard
process
is
the
most
time
consuming
part
of
the
algorithm
and
in
v
olv
es
the
calculation
of
po
wer
losses
and
po
wer
accumulation
through
netw
ork
branches.
Ho
we
v
er
,
a
finite
and
relati
v
ely
small
number
of
standardized
conductor
types
are
installed
in
a
distrib
ution
netw
ork,
each
one
with
a
specified
r
=
x
ratio.
These
ratios
can
be
mathematically
controlled
through
a
proper
reference
system
rotation,
as
it
will
be
e
xposed
in
the
follo
wing
subsection.
2.2.
Complex
Refer
ence
System
Rotation
Assume
an
impedance
Z
i
represented
in
an
alternati
v
e
comple
x
reference
system
rotated
by
'
,
as
illustrated
in
Fig.
2.
ℑ
ℜ
x
ϕ
i
r
ϕ
i
r
i
x
i
ℑ
ϕ
ℜ
ϕ
Z
i
=
r
i
+
j
x
i
Z
ϕ
i
=
r
ϕ
i
+
j
x
ϕ
i
ϕ
Figure
2.
Impedance
represented
into
tw
o
dif
ferent
comple
x
reference
systems
phased
by
a
'
angle.
The
comple
x
v
alue
of
the
impedance
in
the
alternati
v
e
reference
system
can
be
obtained
by
Z
'
i
=
Z
i
e
j
'
=
r
'
i
+
j
x
'
i
(6)
where
r
'
i
=
r
i
cos
'
x
i
sin
';
x
'
i
=
r
i
sin
'
+
x
i
cos
'
(7)
and
the
updated
r
=
x
ratio
gi
v
en
by
r
'
i
x
'
i
=
r
i
cos
'
x
i
sin
'
r
i
sin
'
+
x
i
cos
'
for
x
'
i
6
=
0
(8)
Analogously
,
a
comple
x
apparent
load
in
the
alternati
v
e
reference
system
is
gi
v
en
by
S
'
L
i
=
S
L
i
e
j
'
=
P
'
L
i
+
j
Q
'
L
i
(9)
where
P
'
L
i
=
P
L
i
cos
'
Q
L
i
sin
';
Q
'
L
i
=
P
L
i
sin
'
+
Q
L
i
cos
'
(10)
Once
all
impedances
and
loads
are
rotated
by
(6)
and
(9),
the
system
model
is
represented
in
the
alternati
v
e
reference
system.
Hence,
the
comple
x
po
wer
injected
at
node
i
through
an
upstream
branch
is
also
rotated
by
'
,
such
that
S
'
i
=
S
i
e
j
'
=
P
'
i
+
j
Q
'
i
(11)
IJECE
V
ol.
6,
No.
3,
June
2016:
1344
–
1352
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1347
where
P
'
i
=
P
i
cos
'
Q
i
sin
';
Q
'
i
=
P
i
sin
'
+
Q
i
cos
'
(12)
Since
the
system
model
is
represented
in
the
al
ternati
v
e
reference
system,
the
load
flo
w
v
oltage
solution
is
also
rotated
by
'
,
though
v
oltage
magnitudes
remain
unaltered.
In
f
act,
using
(3)–(4),
the
biquadratic
v
oltage
equation
can
be
re
written
in
the
alternati
v
e
reference
system
as
(
v
'
i
)
4
+
A
'
i
(
v
'
i
)
2
+
B
'
i
=
0
(13)
where
A
'
i
=
2
(
P
'
i
r
'
i
+
Q
'
i
x
'
i
)
(
v
'
u
i
)
2
;
B
'
i
=
(
P
'
i
)
2
+
(
Q
'
i
)
2
(
r
'
i
)
2
+
(
x
'
i
)
2
(14)
By
substituting
(7)
and
(12)
in
(14),
as
well
as
by
assuming
v
'
u
i
=
v
u
i
,
we
obtain
that
A
'
i
=
A
i
and
B
'
i
=
B
i
.
As
consequence
v
'
i
=
v
i
,
causing
that
v
oltage
magnitude
solutions
are
preserv
ed.
Once
the
same
v
oltage
v
alue
is
chosen
for
t
he
substation
node
in
both
reference
systems,
i.e.
v
'
0
=
v
0
,
the
condition
v
'
u
i
=
v
u
i
is
al
w
ays
satisfied
because
of
the
order
in
which
the
nodes
are
visited
in
the
ladder
iterati
v
e
proce-
dures.
Therefore,
a
netw
ork
represented
in
the
alternati
v
e
reference
system
and
other
represented
in
the
con
v
entional
reference
system
present
both
the
same
load
flo
w
v
oltage
magnitudes,
independently
of
the
rotation
angle
chosen.
In
addition,
the
load
flo
w
procedures
will
present
the
same
v
oltage
iterates
in
both
reference
systems.
One
important
application
of
reference
system
rotation
is
that
a
distrib
ution
netw
ork
can
no
w
be
represented
in
a
coordinate
system
where
the
r
=
x
ratios
assume
lo
w
figures,
impro
ving
the
Ne
wtonian
based
load
flo
w
al
gorithms
in
terms
of
rob
ustness
and
ef
ficienc
y
.
The
proposed
approach
goes
be
yond
such
concept
by
introducing
generalized
comple
x
reference
system
rotations
to
tak
e
adv
antage
of
distrib
ution
netw
ork
radiali
ty
in
Cespedes’
ladder
iterati
v
e
formulation.
3.
R
O
T
A
TION
AL
LO
AD
FLO
W
METHOD
This
section
concerns
the
proposed
approach
and
is
or
g
anized
for
didactic
reasons
into
tw
o
cases:
a
case
comprising
netw
orks
with
a
single
conductor
type
and
a
general
case
with
dif
ferent
conductor
types
installed
through
distrib
ution
netw
orks.
3.1.
Single
cable
type
case
Consider
a
netw
ork
with
only
one
conductor
type
installed.
By
h
ypothesis,
comple
x
impedances
present
the
same
r
=
x
ratio
and
angle.
As
e
xposed
pre
viously
,
r
=
x
ratios
can
be
controlled
by
a
comple
x
reference
system
rotation.
F
or
the
general
schematic
sho
wn
in
Fig.
1,
by
con
v
eniently
choosing
the
rotation
angle
'
=
2
and
by
Kirchhof
f
’
s
current
la
w
we
obtain
V
u
i
V
i
=
Z
'
i
S
'
i
V
i
=
j
z
i
S
'
i
V
i
(15)
Moreo
v
er
,
using
(1)
and
(11),
the
rotated
accumulated
comple
x
po
wer
at
node
i
can
be
computed
by
S
'
i
=
S
L
i
e
j
'
+
X
k
2
i
S
L
k
e
j
'
+
X
k
2
i
Z
k
e
j
'
s
k
v
k
2
=
S
'
L
i
+
X
k
2
i
S
'
L
k
+
X
k
2
i
j
z
k
s
k
v
k
2
(16)
Separating
the
real
and
imaginary
terms,
and
denoting
L
'
Q
i
as
the
rotated
reacti
v
e
netw
ork
losses
at
branch
i
,
we
ha
v
e
P
'
i
=
P
'
L
i
+
X
k
2
i
P
'
L
k
(17)
Q
'
i
=
Q
'
L
i
+
X
k
2
i
Q
'
L
k
+
L
'
Q
i
(18)
where
L
'
Q
i
=
X
k
2
i
z
k
s
k
v
k
2
(19)
Using
the
v
ariable
transformation,
the
accumulated
acti
v
e
po
wer
at
node
i
can
be
directly
obtained
summing
rotated
acti
v
e
load
demands
do
wnstream
from
node
u
i
.
Therefore,
there
will
not
be
an
y
acti
v
e
losses
found
and
acti
v
e
Rotational
Load
Flow
Method
for
Radial
Distrib
ution
Systems
(Die
go
Issicaba
and
J
or
g
e
Coelho)
Evaluation Warning : The document was created with Spire.PDF for Python.
1348
ISSN:
2088-8708
po
wer
flo
w
calculation
through
netw
ork
branches
becomes
strai
g
ht
forw
ard.
Consequently
,
load
and
netw
ork
data
can
be
stored
in
a
con
v
enient
coordinate
reference
system
and
the
sweep
algorithm
can
no
w
be
formulated
to
concern
only
the
calculation
of
reacti
v
e
losses,
as
described
in
Fig.
3.
1:
Read
node
and
netw
ork
data,
both
rotated
by
'
=
2
,
using
(6)
and
(11);
2:
Initiate
node
v
oltages
assuming
v
i
=
v
0
;
8
i
=
1
;
:::;
N
;
3:
Compute
accumulated
acti
v
e
po
wer
using
(17)
at
all
netw
ork
nodes;
4:
while
Changes
in
v
i
tolerance,
8
i
do
5:
f
or
all
i
,
follo
wing
the
backw
ard
direction
do
6:
Calculate
accumulated
reacti
v
e
po
wer
at
netw
ork
nodes
using
(18);
7:
end
f
or
8:
f
or
all
i
,
follo
wing
the
forw
ard
direction
do
9:
Obtain
node
v
oltage
magnitudes
starting
from
the
first
node
using
(21)–(20);
10:
end
f
or
11:
end
while
12:
Print
result
reports.
Figure
3.
Rotational
Load
Flo
w
algorithm
–
Single
conductor
type
case
with
constant
po
wer
load
modeling.
The
biquadratic
equation
becomes
simplified
by
remo
ving
resistances
from
the
formulation,
as
follo
ws.
v
i
=
A
'
i
+
(
A
'
i
)
2
4
B
'
i
1
=
2
2
!
1
=
2
(20)
where
A
'
i
=
2
x
'
i
Q
'
i
v
2
u
i
;
B
'
i
=
(
P
'
i
)
2
+
Q
'
i
)
2
(
x
'
i
)
2
(21)
By
eliminating
all
these
calculations,
non-ne
gligible
CPU
time
can
be
sa
v
ed.
3.2.
General
case
Consider
a
general
distrib
ution
netw
ork
with
lateral
branches
and
v
oltage
dependent
load
demands.
Notice
that
this
netw
ork
can
be
di
vided
into
a
set
of
connected
subnetw
orks
with
the
same
conductor
type.
These
subnetw
orks
can
be
represented
in
a
con
v
enient
coordinate
system
to
eliminate
the
line
resistances
from
modeling,
as
illus
trated
in
Fig.
4.
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
conductor
type
I
conductor
type
II
conductor
type
IV
conductor
type
III
)
)
)
s
q
q
q
q
q
q
q
s
q
q
q
q
q
q
q
q
q
s
q
q
q
q
q
q
q
q
q
s
q
q
q
q
q
q
q
e
j
'
I
e
j
'
I,IV
e
j
'
I,II
e
j
'
I,III
Angle
marks
Figure
4.
Rotational
Load
Flo
w
principle
–
General
case.
By
preserving
the
topological
structure
and
load
flo
w
v
oltage
solution,
the
po
wer
will
flo
w
through
the
sub-
netw
orks
assuming
dif
ferent
representations
for
each
reference
system.
Hence,
the
load
flo
w
formulation
can
be
simplified
to
eliminate
the
acti
v
e
losses
from
modeling
by
mathematically
handling
the
dif
ferences
of
coordinate
reference
systems
among
subnetw
orks.
This
accomplishment
is
obtained
through
a
generalized
compl
e
x
reference
system
rotation
algorithm,
sho
wn
in
Fig.
5,
in
which
stored
data
is
changed
and
connections
between
subnetw
orks
are
assigned
by
angle
mar
ks
.
The
rotation
procedures
are
identical
for
the
case
of
modeling
other
series
components
by
series
impedances.
IJECE
V
ol.
6,
No.
3,
June
2016:
1344
–
1352
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1349
1:
Read
node
and
netw
ork
data;
2:
S
'
i
=
S
i
;
8
i
=
1
;
:::;
N
;
3:
counter
0
;
4:
Start
a
list
=
0
=
f
1
g
with
the
first
node
immediately
do
wnstream
of
the
substation;
5:
while
counter
N
do
6:
f
or
all
node
i
2
do
7:
'
i
=
2
i
;
S
'
i
S
'
i
e
j
'
i
;
Z
'
i
Z
'
i
e
j
'
i
;
8:
f
or
all
k
2
i
do
9:
S
'
k
S
'
k
e
j
'
i
;
Z
'
k
Z
'
k
e
j
'
i
;
10:
end
f
or
11:
counter
counter
+
1
;
12:
end
f
or
13:
f
m
,
such
that
m
2
i
,
8
i
2
g
;
14:
end
while
Figure
5.
Generalized
comple
x
reference
system
rotation.
1:
Read
rotation
angles,
as
well
as
the
node
and
netw
ork
data,
both
obtained
by
the
the
algorithm
sho
wn
in
Fig.
5;
2:
Initiate
node
v
oltages
assuming
v
i
=
v
0
;
8
i
=
1
;
:::;
N
;
3:
while
Changes
in
v
i
tolerance,
8
i
do
4:
Compute
load
demands
using
an
appropriate
v
oltage
dependent
model;
5:
f
or
all
i
,
follo
wing
the
backw
ard
direction
do
6:
Calculate
reacti
v
e
losses
at
branch
i
using
(19);
7:
if
'
i
=
0
then
8:
Calculate
accumulated
acti
v
e
and
reacti
v
e
po
wers
through
(17)
and
(18);
9:
else
10:
Obtain
accumulated
acti
v
e
and
reacti
v
e
po
wers
using
(17)
and
(18),
as
well
as
a
proper
coordinate
rotation
through
(12);
11:
end
if
12:
end
f
or
13:
f
or
all
i
,
follo
wing
the
forw
ard
direction
do
14:
Obtain
node
v
oltage
magnitudes
starting
from
the
first
node
using
(20)–(21);
15:
end
f
or
16:
end
while
17:
Print
result
reports.
Figure
6.
Rotational
Load
Flo
w
algorithm
–
General
case.
The
Rotational
Load
Flo
w
method
for
radial
distrib
ution
netw
orks
is
presented
in
Fig.
6.
The
approach
utilizes
a
sweep
algorithm
and
performs
comple
x
reference
system
rotations
of
accumulated
po
wers
whereas
a
subnet-
w
ork
connections
is
assigned.
In
the
forw
ard
procedure,
rotations
are
not
required
and
the
basic
biquadrati
c
equation
is
simplified
with
the
remo
ving
of
line
resistances
from
modeling.
4.
RESUL
TS
AND
DISCUSSIONS
The
proposed
approach
w
as
implemented
in
MA
TLAB
using
an
old
computer
(1,66
GHz,
Core
Duo),
and
it
has
been
e
xtensi
v
ely
tested
with
distrib
ution
netw
orks
obtained
from
literature
and
with
actual
distrib
ution
netw
orks.
F
or
all
simulations,
tolerance
w
as
set
up
in
0.000001
and
initial
v
oltage
magnitudes
were
assumed
1
pu.
V
oltage
dependent
loads
were
modeled
using
the
polynomial
form
P
L
i
=
P
0
L
i
P
+
P
v
i
+
P
v
2
i
;
Q
L
i
=
Q
0
L
i
Q
+
Q
v
i
+
Q
v
2
i
(22)
where
P
0
L
i
and
Q
0
L
i
are
the
rated
acti
v
e
and
reacti
v
e
loads,
respecti
v
ely
.
The
coef
ficients
(
P
;
P
;
P
)
and
(
Q
;
Q
;
Q
)
were
set
in
(0
:
8
;
0
:
1
;
0
:
1)
.
Cespedes’
method
and
the
Rotational
Load
Flo
w
approach
were
applied
to
solv
e
load
flo
w
problems
in
a
12-
node
netw
ork
[15],
27-node
netw
ork
[15],
29-node
netw
ork
[9],
32-node
netw
ork
[12]
and
a
69-node
netw
ork
[11].
Rotational
Load
Flow
Method
for
Radial
Distrib
ution
Systems
(Die
go
Issicaba
and
J
or
g
e
Coelho)
Evaluation Warning : The document was created with Spire.PDF for Python.
1350
ISSN:
2088-8708
T
able
1
sho
ws
simulation
results
achie
v
ed
with
these
comparati
v
e
case-studies.
T
able
1.
Simulation
results
for
the
rotational
and
Cespedes’
methods
using
distrib
ution
netw
orks
obtained
in
the
literature
Runtime
(ms)
Sa
ving
Ref.
N
CM
RLFM
Iter
.
(%)
[15]
11
56.9101
45.4312
4
0.1666
25.27
[15]
27
58.4912
48.7431
5
0.2222
20.01
[12]
32
56.4634
47.1671
5
0.7188
19.71
[9]
29
57.9068
48.5995
6
0.9655
19.15
[11]
69
61.8391
50.1512
5
0.1884
23.31
CM:
Cespedes’
method.
RLFM:
Rotational
Load
Flo
w
method.
:
Ratio
between
the
number
of
changes
on
conductor
types
and
the
number
of
netw
ork
nodes.
The
comparati
v
e
studies
indicate
that
the
proposed
approach
pro
vides
superior
result
s
in
term
of
ef
ficienc
y
when
compared
to
Cespedes’
technique.
In
f
act,
calculations
of
acti
v
e
losses
ha
v
e
been
eliminated
in
the
proposed
approach
and
sweep
procedures
were
adapted
to
handle
dif
ferences
of
comple
x
reference
system
representations.
Node
v
oltages
are
equally
updated
in
both
methods
follo
wing
that
the
number
of
iterations
necessary
for
con
v
er
gence
are
als
o
preserv
ed
in
the
proposed
approach.
Also,
it
w
as
not
found
an
y
sacrifice
in
solution
quality
by
using
the
rotation
technique.
A
similar
case
study
w
as
de
vised
for
an
actual
distrib
ution
netw
ork,
sho
wn
in
Fig.
7.
This
netw
ork
is
composed
of
236
nodes,
1337
kilometers
of
lines
and
5
conductor
types
installed.
Simulation
results
for
this
netw
ork
are
presented
in
T
able
2.
Figure
7.
An
actual
radial
structured
distrib
ution
netw
ork.
T
able
2.
Simulation
results
for
the
rotational
and
Cespedes’
methods
using
an
actual
distrib
ution
netw
ork
Runtime
(ms)
Sa
ving
N
CM
R
LFM
Iter
.
(%)
236
70.0321
55.4044
5
0.1525
26.40
CM:
Cespedes’
method.
RLFM:
Rotational
Load
Flo
w
method.
:
Ratio
between
the
number
of
changes
on
conductor
types
and
the
number
of
netw
ork
nodes.
Extensi
v
e
simulations
pointed
out
runtime
impro
v
ements
depend
upon
the
ratio
between
the
number
of
changes
in
conductor
types
across
the
netw
ork
and
the
number
of
netw
ork
nodes.
This
f
act
w
as
e
xpected
since
a
lar
ge
number
of
changes
in
conductor
type
might
increase
the
number
of
rotations
during
the
backw
ard
procedures,
as
formalized
in
Appendix
A.
All
outcomes,
either
using
literature
or
actual
netw
orks,
indicate
that
the
proposed
method
pro
vides
impro
v
ed
results
in
comparison
to
Cespedes’
method
in
term
of
ef
ficienc
y
.
IJECE
V
ol.
6,
No.
3,
June
2016:
1344
–
1352
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IJECE
ISSN:
2088-8708
1351
5.
CONCLUSIONS
AND
FIN
AL
REMARKS
Adequac
y
e
v
aluations
usually
require
a
lar
ge
number
of
load
flo
w
computations
to
estimate
a
small
set
of
performance
indices.
These
e
v
aluations
are
utilized
in
long-term
planning
stages,
where
simplifications
such
as
bal-
anced
operation
and
constant
po
wer
load
demands
can
be
assumed,
while
ef
forts
are
directed
to
accurate
ly
modeling
the
f
ailure/repair
c
ycles
of
netw
ork
components
and
load/generation
profiles
as
stochastic
processes.
Aiming
at
achie
ving
impro
v
ed
ef
ficienc
y
in
adequac
y
e
v
aluations,
a
modified
v
ersion
of
Cespedes’
load
flo
w
method
to
radial
distrib
ution
systems
is
proposed.
The
approach
eliminates
unnecessary
calculations
in
Cespedes’
load
flo
w
me
thod
by
re
presenting
distrib
ution
systems
in
dif
ferent
and
con
v
eniently
chosen
coordinate
reference
systems.
F
or
this
accomplishment,
a
generalized
comple
x
reference
systems
rotation
algorithm
w
as
introduced
and
the
load
flo
w
formulation
w
as
adapted
to
handle
dif
ferences
in
coordinate
reference
system
representations.
Result
analysis
obtained
from
both
literature
and
actual
netw
orks
indicate
that
the
proposed
method
pro
vide
impro
v
ed
ef
ficienc
y
in
comparison
to
Cespedes’
method.
Future
w
orks
will
in
v
estig
ate
e
xtensions
of
the
proposed
approach
with
re
g
ard
to
unbalanced
distrib
ution
netw
orks,
dispersed
generation,
and
weakly-meshed
distrib
ution
sys-
tems.
A
CKNO
WLEDGMENTS
The
authors
w
ould
lik
e
to
ackno
wledge
the
financial,
technical
and
human
support
of
the
CNPq,
CAPES
and
INESC
P&D
Brasil.
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Maref
at
jou
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M.
Sarvi,
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uted
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impro
v
e
steady
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v
oltage
stability
of
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ution
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orks
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imperialist
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v
e
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J
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ower
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Ghaf
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arzadeh,
M.
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Khanjanzadeh,
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uted
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allocation
to
impro
v
e
steady
state
v
oltage
stability
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orks
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v
e
algorithm,
”
International
J
ournal
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Ap-
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P
ower
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APE)
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pp.
71–78,
2013.
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Issicaba,
A.
J.
S.
Costa,
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L.
Colombo,
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of
common
coupling
in
distrib
u-
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systems
through
state
estimation
and
geometric
tests,
”
IEEE
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Smart
Grid
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Jan.
2016.
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Issicaba,
J.
A.
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Lopes,
and
M.
A.
Rosa,
“
Adequac
y
and
security
e
v
aluation
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distrib
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with
distrib
uted
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”
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P
ower
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Aug.
2012.
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Issicaba,
M.
A.
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a
n
d
J.
A.
P
.
Lopes,
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ution
systems
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e
v
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islanded
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”
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of
the
P
ower
System
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Confer
ence
,
Stockholm,
Sweden,
2011.
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W
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H.
K
ersting,
Distrib
ution
Sys
tem
Modeling
and
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2nd
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The
Electrical
Engineering
Series,
L.
Grigsby
,
Ed.
CRC
Press,
Jul.
2011.
[7]
D.
Radicic
and
A.
Bose,
“
A
modification
to
the
f
ast
decoupled
po
wer
flo
w
to
netw
orks
with
high
r/x
ratios,
”
IEEE
T
r
ansactions
on
P
ower
Systems
,
v
ol.
3,
no.
2,
pp.
743–746,
May
1988.
[8]
F
.
Zhang
and
C.
S.
Cheng,
“
A
modified
ne
wton
method
for
radial
distrib
ution
system
po
wer
flo
w
analysis,
”
IEEE
T
r
ansactions
on
P
ower
Systems
,
v
ol.
12,
no.
1,
pp.
389–397,
Feb
.
1997.
[9]
R.
G.
Cespedes,
“
Ne
w
method
for
the
analysis
of
distrib
ution
netw
orks,
”
IEEE
T
r
ansactions
on
P
ower
Delivery
,
v
ol.
5,
no.
1,
pp.
391–396,
Jan.
1990.
[10]
D.
Shirmohammadi,
H.
W
.
Hong,
A.
Semlyen,
and
G.
X.
Luo,
“
A
compensation-based
po
wer
flo
w
method
for
weakly
meshed
distrib
ution
and
transmission
netw
orks,
”
IEEE
T
r
ansactions
on
P
ower
Systems
,
v
ol.
3,
no.
2,
pp.
753–762,
May
1988.
[11]
M.
E.
Baran
and
F
.
F
.
W
u,
“Optimal
sizing
of
capacitor
placed
on
a
radial
distrib
ution
system,
”
IEEE
T
r
ansaction
on
P
ower
Delivery
,
v
ol.
4,
no.
1,
pp.
735–743,
Jan.
1989.
[12]
——,
“Netw
ork
reconfiguration
in
distrib
ution
systems
for
loss
reduction
and
load
balancing,
”
IEEE
T
r
ansac-
tions
on
P
ower
Delivery
,
v
ol.
4,
no.
2,
pp.
1401–1407,
Apr
.
1989.
[13]
M.
H.
Haque,
“Load
flo
w
solution
of
distrib
ution
systems
with
v
oltage
dependent
load
models,
”
Electric
P
ower
Systems
Resear
c
h
,
v
ol.
36,
no.
3,
pp.
151–156,
Mar
.
1996.
[14]
——,
“Ef
ficient
load
flo
w
method
for
distrib
ution
systems
with
radial
or
mesh
configuration,
”
in
IEE
Pr
oceedings
on
Gener
ation,
T
r
ansmission
and
Distrib
ution
,
v
ol.
143,
no.
1,
Jan.
1996,
pp.
33–38.
[15]
D.
Das,
H.
S.
Nagi,
and
D.
P
.
K
othari,
“No
v
el
method
for
solving
radial
distrib
ution
netw
orks,
”
in
IEE
Pr
oceed-
ings
on
Gen.,
T
r
an.
and
Distr
.
,
v
ol.
141,
no.
4,
Jul.
1994,
pp.
291–298.
Rotational
Load
Flow
Method
for
Radial
Distrib
ution
Systems
(Die
go
Issicaba
and
J
or
g
e
Coelho)
Evaluation Warning : The document was created with Spire.PDF for Python.
1352
ISSN:
2088-8708
APPENDIX
A.
RELA
TION
BETWEEN
NUMBER
OF
R
O
T
A
TIONS
AND
NUMBER
OF
NODES
The
proposed
approach
can
be
compared
to
Cespedes’
method
according
to
the
usage
of
basic
mathematical
operations
per
iteration
such
as:
multiplication/di
vision
(
[
),
sum/subtraction
(
[
)
and
square
root
(
p
:
).
Although
the
usage
of
mathematical
operations
may
v
a
ry
according
to
actual
implementation
and
the
number
of
operations
to
rotating
system
data
depends
on
the
system
itself,
it
is
possible
to
relate
the
operations
per
iteration
of
both
methods
in
order
to
identify
whether
the
proposed
approach
will
be
more
ef
ficient
for
a
gi
v
en
adequac
y
study
.
In
our
implementation,
Cespedes’
method
utilizes
in
each
iteration
the
calculations
listed
belo
w
.
I
2
i
P
2
i
+
Q
2
i
v
2
i
;
P
u
i
P
u
i
+
P
i
+
r
i
I
2
i
;
Q
u
i
Q
u
i
+
Q
i
+
x
i
I
2
i
)
N
times
A
i
2(
r
i
P
i
+
x
i
Q
i
)
v
2
u
i
;
B
i
P
2
i
+
Q
2
i
r
2
i
+
x
2
i
;
v
i
A
i
+
(
A
2
i
4
B
i
)
1
=
2
2
1
=
2
)
N
times
totalizing,
in
terms
of
number
of
operations,
18
N
(
[
)
+
11
N
(
[
)
+
2
N
(
p
:
)
(23)
Similarly
,
let
M
be
the
number
of
po
wer
flo
w
rotations
between
subnetw
orks,
the
basic
mathematical
opera-
tions
per
iteration
of
the
proposed
approach
are
the
follo
wing:
L
Q
i
x
i
P
2
i
+
Q
2
i
v
2
i
)
N
times
P
u
i
P
u
i
+
P
i
)
(
N
M
)
times
Q
u
i
Q
u
i
+
Q
i
+
L
Q
i
)
(
N
M
)
times
A
i
2
x
i
Q
i
v
2
u
i
)
N
times
B
i
P
2
i;ac
+
Q
2
i;ac
x
2
i
)
N
times
v
i
A
i
+
(
A
2
i
4
B
i
)
1
=
2
2
1
=
2
)
N
times
P
u
i
P
u
i
+
P
i
cos
'
(
Q
i
+
L
Q
i
)
sin
'
)
M
times
Q
u
i
Q
u
i
+
P
i
sin
'
+
(
Q
i
+
L
Q
i
)
cos
'
)
M
times
totalizing,
also
in
terms
of
number
of
operations,
(15
N
+
4
M
)(
[
)
+
(8
N
+
3
M
)(
[
)
+
2
N
(
p
:
)
(24)
By
using
(23)
and
(24),
we
conclude
that
the
proposed
approach
will
be
adv
antageous
in
terms
of
number
of
operations
per
iteration
whether
(3
N
4
M
)(
[
)
+
(3
N
3
M
)(
[
)
>
0
(25)
This
inequality
is
true,
for
instance,
if
M
<
0
;
75
N
,
which
is
a
condition
easily
satisfied
in
actual
netw
orks.
BIOGRAPHIES
OF
A
UTHORS
Diego
Issicaba
recei
v
ed
the
B.S.
and
M.S.
de
grees
in
Electrical
Engineering
from
the
Fe
deral
Uni-
v
ersity
of
Santa
Catarina
(UFSC),
Sa
nta
Catarina,
Brazil,
in
2006
and
2008,
respecti
v
ely
.
Fur
-
thermore,
he
r
ecei
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
Electri
cal
Engineering
at
the
Federal
Uni
v
ersity
of
Santa
Catarina,
Brazil,
since
March
1978.
His
research
interests
include
distrib
ution
systems
e
xpansion
and
operation
planning,
po
wer
systems
reliability
,
probabilistic
methods
applied
to
po
wer
systems,
and
po
wer
quality
.
IJECE
V
ol.
6,
No.
3,
June
2016:
1344
–
1352
Evaluation Warning : The document was created with Spire.PDF for Python.