Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
6,
No.
6,
December
2016,
pp.
3229
–
3237
ISSN:
2088-8708
3229
Ev
aluation
of
the
F
orward-Backward
Sweep
Load
Flo
w
Method
using
the
Contraction
Mapping
Principle
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-SC,
Br
azil
Article
Inf
o
Article
history:
Recei
v
ed
May
19,
2016
Re
vised
Jul
12,
2016
Accepted
Aug
1,
2016
K
eyw
ord:
Po
wer
engineering
Po
wer
distrib
ution
systems
Load
flo
w
analysis
Con
v
er
gence
ABSTRA
CT
This
paper
presents
an
assessment
of
the
forw
ard-backw
ard
sweep
load
flo
w
method
to
distrib
ution
system
analysis.
The
method
is
forma
lly
assessed
using
fix
ed-point
con-
cepts
and
the
contraction
mapping
theorem.
The
e
xistence
and
uniqueness
of
the
load
flo
w
feasible
solution
is
supported
by
an
alternati
v
e
ar
gument
from
those
obtained
in
the
literature.
Also,
the
closed-form
of
the
con
v
er
gence
rate
of
the
method
is
deduced
and
the
con
v
er
gence
dependence
of
loading
is
assessed.
Finally
,
boundaries
for
error
v
alues
per
iteration
between
iterates
and
feasible
solution
are
obtained.
Theoretical
results
ha
v
e
been
tested
in
se
v
eral
numerical
simulations,
some
of
them
presented
in
this
paper
,
thus
fostering
discussions
about
applications
and
future
w
orks.
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
widely
utilized
in
po
wer
system
analysis
and
applications
[1,
2,
3].
One
reasonable
w
ay
to
understand
and
compare
load
flo
w
approaches
i
s
through
their
mathematical
con
v
er
gence
analysis.
The
challenges
behind
analyzi
ng
the
con
v
er
gence
of
iterati
v
e
methods
be
gin
with
the
characterization
of
the
problem
solution
itself.
Some
researches
ha
v
e
studied
the
nature
and
multiplicity
of
stable
load
flo
w
solutions
for
transmission
systems
[4,
5,
6,
7],
and
a
fe
w
researches
ha
v
e
studied
the
load
flo
w
solution
for
po
wer
distrib
ution
systems.
H.
D.
Chiang
and
M.
E.
Baran
[8]
sho
wed
the
e
xistence
and
uniqueness
of
the
feasible
load
flo
w
solution
for
balanced
distrib
ution
systems,
while
J.
F
.
Chen
and
W
.
M.
W
ang
[9]
v
erified
the
e
xistence
and
feasibility
of
the
load
flo
w
solution
by
using
a
formulation
based
on
the
DistFlo
w
equations
[10,
11].
In
[12],
K.
N.
Miu
and
H.
D.
Chiang
pro
vided
a
useful
contrib
ution
by
e
xtending
the
w
ork
in
[9]
to
the
three-phase
case
with
detailed
netw
ork
modeling.
Since
an
analytical
solution
does
not
e
xist
for
the
load
flo
w
problem,
e
v
en
gi
v
en
some
kno
wledge
about
the
solution,
it
is
necessary
to
formulate
iterati
v
e
procedures
for
the
load
flo
w
calculation.
The
aforemen-
tioned
procedures
might
guarantee
a
f
ast
con
v
er
gence
to
w
ards
a
solution,
gi
v
en
an
initial
estimate
and
a
fix
ed
tolerance.
Hence,
aiming
at
analyzing
the
con
v
er
gence
of
load
fl
o
w
algorithms
for
transmission
systems,
J.
Meisel
and
R.
D.
Barnard
[13]
presented
a
vie
w
of
the
Gauss-Seidel
method
and
the
Ne
wton-Raphson
method
in
terms
of
a
fix
ed-point
formulation.
K.
Ganesan
et
al.
[14]
studied
the
con
v
er
gence
of
the
Ne
wton-Raphson
method
by
using
the
Kantoro
vich
theorem,
and
F
.
W
u
[15]
pro
v
ed
the
con
v
er
gence
and
demonstrated
the
strong
con
v
er
gence
dependence
of
netw
ork
r
=
x
ratios
for
the
F
ast
Decoupled
method.
Re
g
arding
distrib
ution
system
analysis,
there
is
a
lack
of
con
v
er
gence
formalization
for
load
flo
w
methods.
In
f
act,
these
methods
are
not
usually
based
on
Jacobian
e
v
aluations
and
generally
emplo
y
sweep
procedures
in
which
currents
[16]
(or
po
wers
[17,
18])
are
accumulated
from
end
nodes
to
w
ards
the
substation
b
us.
In
this
area,
E.
Bompard
et
al.
[19]
contrib
uted
significantly
by
studying
the
con
v
er
gence
of
a
sweep
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
,
DOI:
10.11591/ijece.v6i6.11303
Evaluation Warning : The document was created with Spire.PDF for Python.
3230
ISSN:
2088-8708
method
and
v
erifying
that
its
con
v
er
gence
properties
can
be
depreci
ated
for
high
loadings.
Thi
s
observ
ation
w
as
first
brought
out
by
R.
P
.
Broadw
ater
in
his
discussions
of
[16].
In
this
conte
xt,
this
paper
presents
the
assessment
the
forw
ard-backw
ard
sweep
load
flo
w
method
proposed
by
D.
Shirmohammadi
et
al.
[16]
using
fix
ed-point
concepts
and
the
contraction
mapping
theorem.
Also,
an
alternati
v
e
ar
gument
sho
wing
the
e
xistence
and
uniqueness,
un
de
r
certain
conditions,
of
the
load
flo
w
solution
is
presented.
At
last,
a
set
of
error
boundaries
per
iteration
limiting
the
v
ectorial
distance
between
a
v
oltage
iterate
v
alues
and
the
solution
is
deduced.
The
paper
is
or
g
anized
as
follo
ws.
Section
2
presents
a
brief
mathematical
background
of
the
contrac-
tion
mapping
theorem
and
the
load
flo
w
method
under
assessment.
In
Section
3,
the
aforementioned
mathe-
matical
contrib
utions
are
demonstrated
for
a
general
distrib
ution
system.
Numerical
simulations
are
sho
wn
in
Section
4
aiming
at
v
alidating
the
theoretical
results.
Finally
,
Section
5
outlines
conclusions
and
future
w
orks.
2.
MA
THEMA
TICAL
B
A
CKGR
OUND
2.1.
The
Contraction
Mapping
Theor
em
The
fix
ed-point
theorems
compose
a
set
of
theorems
applied
in
se
v
eral
areas
such
as
ph
ysics,
mathe-
matics,
economics
and
engineering.
W
e
can
define
a
fix
ed-point
x
/
of
a
function
:
X
!
C
n
(
X
C
n
)
as
an
y
point
x
/
2
X
such
that
(
x
/
)
=
x
/
.
Some
of
the
most
important
fix
ed-point
theorems
are
the
Brouwer
theorem,
the
Knaster-T
arski
theorem,
the
Lefschetz
theorem
and
the
contraction
mapping
theorem
utilized
in
this
w
ork.
Fix
ed-point
theorems
ha
v
e
an
important
role
in
the
analysis
of
nonlinear
problems,
as
well
as
the
analysis
of
algorithms
associated
to
the
solution
of
these
problems.
W
ith
the
aim
of
introducing
the
contraction
mapping
theorem,
let
us
define
a
general
nonlinear
prob-
lem
com
p
os
ed
by
n
nonlinear
equations
f
i
(
x
1
;
:
:
:
;
x
n
)
=
y
i
,
for
all
i
=
1
;
:
:
:
;
n
.
F
or
simplicity
,
note
that
this
problem
can
also
be
represented
in
the
v
ector
form
F
(
x
)
=
y
,
where
x
=
[
x
i
]
n
and
y
=
[
y
i
]
n
.
By
formu-
lating
iterati
v
e
methods
to
solv
e
this
system
of
equations,
we
search
for
an
iteration
scheme
x
(
k
+1)
=
(
x
(
k
)
)
aiming
at
generating
a
sequence
of
iterates
f
x
(0)
;
x
(1)
;
:
:
:
g
that
con
v
er
ges
if
and
only
if
F
(
x
)
=
y
.
One
of
the
reasonable
choices
for
this
iteration
scheme
is
(
x
(
k
)
)
=
x
(
k
)
W
(
x
(
k
)
)
h
F
(
x
(
k
)
)
y
i
(1)
where
W
(
x
)
is
a
n
n
nonsingular
matrix
function
of
x
.
In
the
s
imple
iteration
scheme
abo
v
e,
the
con
v
er
gence
is
obtained
at
a
fix
ed-point
and
x
/
is
a
fix
ed-point
of
if
and
only
if
F
(
x
)
=
y
.
W
e
emphasize
that
once
an
iteration
scheme
is
chosen,
by
starting
with
a
point
x
(0)
the
sequence
of
iterates
must
con
v
er
ge
to
a
fix
ed-point
x
/
,
i.e.
lim
k
!1
x
(
k
)
x
/
=
0
,
in
a
reasonable
amount
of
time.
There
also
e
xists
an
interesting
property
in
which
the
mismatches
between
tw
o
subsequent
iterates
x
(
k
+1)
x
(
k
)
geometrically
decrease
throughout
the
iterati
v
e
process.
In
mathematical
terms
this
prop-
erty
can
be
written
as
(
x
(
k
+1)
)
(
x
(
k
)
)
c
x
(
k
+1)
x
(
k
)
(2)
for
some
c
2
R
such
that
0
c
<
1
.
In
a
general
sense,
if
a
function
satisfies
the
inequality
jj
(
x
)
(
x
0
)
jj
c
jj
x
x
0
jj
,
8
x
;
x
0
2
X
,
for
some
c
2
R
such
that
0
c
<
1
,
then
is
a
contraction
mapping
with
contraction
constant
c
.
The
contraction
mappings
ha
v
e
tw
o
interesting
properties,
both
presented
in
the
follo
wing
theorem.
Theor
em
1
(The
Contr
action
Mapping
Theor
em)
[20]:
Let
X
C
n
be
a
closed
subset
and
:
X
!
X
be
a
map
such
that
jj
(
x
)
(
x
0
)
jj
c
jj
x
x
0
jj
(3)
for
some
contraction
constant
0
c
<
1
,
c
2
R
,
8
x
;
x
0
2
X
.
Then
has
an
unique
fix
ed-point
in
X
.
Additionally
,
the
sequence
of
iterates
f
x
(0)
;
(
x
(0)
)
;
(
(
x
(0)
))
;
:
:
:
g
con
v
er
ges
to
the
fix
ed-point
for
an
y
x
(0)
2
X
.
This
theorem
will
be
used
to
analyze
some
important
properties
of
the
forw
ard-backw
ard
sweep
load
flo
w
method.
It
is
important
to
emphasize
that
the
nonlinear
iteration
schem
e
in
(1)
is
the
basis
in
which
classical
methods
solv
e
load
flo
w
equations
for
high
v
oltage
transmission
systems.
F
or
instance,
if
we
tak
e
W
(
x
(
k
)
)
to
be
the
in
v
erse
Jacobian
matrix
of
the
function
F
at
x
(
k
)
,
then
(1)
becomes
the
update
rule
of
the
Ne
wton-Raphson
m
ethod.
On
the
other
hand,
in
case
of
distrib
ution
system
analysis,
the
update
rules
of
the
methods
change
considerably
depending
upon
the
chosen
approach,
although
it
is
possible
to
establish
some
similarities
to
the
nonlinear
iteration
scheme
presented
in
(1).
IJECE
V
ol.
6,
No.
6,
December
2016:
3229
–
3237
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
3231
2.2.
The
F
orward-Backward
Sweep
Method
W
ith
the
aim
of
mathematically
describing
D.
Shirmohammadi
et
al.
’
s
load
flo
w
method
[16],
let
us
consider
the
radial
schematic
depicted
in
Fig.
1,
where
lines
are
modeled
as
series
impedances
z
i
while
comple
x
b
us
v
oltages
and
comple
x
load
demands
are
modeled,
respecti
v
ely
,
by
E
i
=
e
i
+
j
f
i
and
S
i
,
8
i
=
1
;
:::;
n
.
The
substation
b
us
is
named
the
0
(zero)
b
us
with
comple
x
v
oltage
denoted
by
E
0
.
Observ
e
that
the
inde
x
i
is
used
to
identify
the
b
us
and
the
line
upstream
this
b
us,
depending
on
the
v
ariables
in
v
olv
ed.
q
q
?
S
i
q
?
S
i
+1
q
?
S
n
1
q
?
S
n
E
i
1
E
i
E
i
+1
E
n
1
E
n
z
i
z
i
+1
z
n
-
I
i
|{z}
line
n
?
I
L
n
Figure
1.
Radial
distrib
ution
netw
ork
schematic.
In
distrib
ution
systems
load
flo
w
analysis,
the
substation
b
us
is
usually
assumed
to
be
the
slack
b
us,
with
a
constant
real
v
oltage
E
0
.
Under
this
assumption,
the
algorithm
be
gins
with
an
initial
solution
for
all
b
uses
and
performs
three
basic
steps
until
a
con
v
er
gence
criterion
is
satisfied.
1.
The
current
injection
I
(
k
+1)
L
i
at
b
us
i
and
iteration
k
+
1
is
calculated
as
I
(
k
+1)
L
i
=
S
i
=
E
(
k
)
i
,
where
E
(
k
)
i
is
the
comple
x
v
oltage
at
b
us
i
calculated
during
the
k
th
iteration;
2.
Starting
from
the
end
branches
and
mo
ving
to
w
ards
the
branch
connected
t
o
the
substation
b
us,
the
current
at
branch
i
can
be
c
alculated
by
I
(
k
+1)
i
=
I
(
k
+1)
L
i
+
P
r
2
i
I
(
k
+1)
L
r
=
P
r
2
i
I
(
k
+1)
L
r
,
where
i
denotes
the
set
of
do
wnstream
b
uses
of
b
us
i
,
and
i
denotes
the
set
of
elements
of
i
including
the
i
th
b
us,
i.e.
i
,
i
S
f
i
g
;
3.
The
b
us
v
oltages
are
updated
in
a
forw
ard
sweep
starting
from
the
first
branch
and
mo
ving
to
w
ards
end
branches
by
E
(
k
+1)
i
=
E
(
k
+1)
u
i
z
i
I
(
k
+1)
i
,
where
u
i
represents
the
upstream
b
us
of
the
i
th
b
us.
One
basic
con
v
er
gence
criterion
is
the
maximum
absolute
po
wer
mismatch
for
all
b
use
s.
Al
ternati
v
ely
,
the
con
v
er
gence
can
be
tested
by
other
criteria
lik
e
the
maximum
absolute
dif
ference
between
subsequent
v
oltage
iterates.
3.
CONTRA
CTION
MAPPING
ASSESSMENT
3.1.
Con
v
er
gence
Rate
Let
us
e
xamine
the
con
v
er
gence
of
comple
x
v
oltages
using
the
iterati
v
e
scheme
of
the
algorit
hm.
Observ
e
that
the
general
update
rule
can
be
written
recursi
v
ely
for
iteration
k
as
i
(
E
(
k
)
i
)
=
u
i
(
E
(
k
)
u
i
)
z
i
X
r
2
i
S
r
E
(
k
)
r
;
i
(
E
(
k
)
i
)
=
E
(
k
+1)
i
(4)
No
w
let
p
i
be
the
set
of
lines
between
the
substation
b
us
and
b
us
i
.
Also,
let
o
ir
be
the
intersection
of
sets
p
i
and
p
r
.
Then,
e
xpression
(4)
can
be
re
written
as
i
(
E
(
k
)
i
)
=
E
0
n
X
r
=1
z
ir
S
r
E
(
k
)
r
;
z
ir
=
X
t
2
o
ir
z
t
:
(5)
As
a
matter
of
consequence,
the
same
rul
e
can
no
w
be
written
for
iteration
k
+
1
,
such
that
the
dif
ference
between
subsequent
v
oltage
iterates
can
be
recursi
v
ely
calculated
as
follo
ws
(
k
+1)
i
=
n
X
r
=1
z
ir
S
r
E
(
k
+1)
r
E
(
k
+1)
r
E
(
k
)
r
!
(6)
Evaluation
of
the
F
orwar
d-Bac
kwar
d
Sweep
Load
Flow
Method
using
...
(Die
go
Issicaba
and
J
or
g
e
Coelho)
Evaluation Warning : The document was created with Spire.PDF for Python.
3232
ISSN:
2088-8708
By
rearranging
the
terms
of
the
e
xpression
abo
v
e,
we
can
re
write
(6)
in
the
con
v
enient
form
i
(
k
+1)
=
n
X
r
=1
d
(
k
+1)
ir
E
(
k
+1)
r
;
d
(
k
+1)
ir
=
z
ir
S
r
E
(
k
+1)
r
E
(
k
)
r
(7)
As
consequence,
(7)
can
be
e
xpressed
in
their
matrix
forms
(
k
+1)
=
D
(
k
+1)
E
(
k
+1)
;
D
(
k
+1)
=
ZS
F
(
k
+1)
(8)
where
(
E
(
k
)
)
is
a
n
1
v
ector
whose
elements
are
the
comple
x
v
oltages
obtained
by
the
update
rule
at
the
k
th
iteration,
(
k
+1)
,
(
E
(
k
+1)
)
(
E
(
k
)
)
,
E
(
k
)
is
a
n
1
v
ector
whose
elements
are
the
comple
x
v
oltages
at
the
k
th
iteration,
E
(
k
+1)
,
E
(
k
+1)
E
(
k
)
,
Z
is
the
impedance
matrix
of
the
distrib
ution
netw
ork
with
size
n
n
,
S
is
a
diagonal
matrix
with
size
n
n
whose
elements
are
the
conjug
ated
comple
x
loads,
and
F
(
k
+1)
is
a
diagonal
matrix
with
size
n
n
whose
elements
are
reciprocals
of
the
product
of
subsequent
conjug
ated
comple
x
v
oltage
iterates.
By
applyi
n
g
the
norm
operator
(in
C
n
)
and
their
pr
o
pe
rties
in
(8)
we
ha
v
e
(
k
+1)
D
(
k
+1)
E
(
k
+1)
=
D
(
k
+1)
E
(
k
+1)
At
this
point,
dif
ferent
norms
can
be
used
to
e
v
aluate
the
con
v
er
gence
of
the
algorithm.
In
particular
,
we
can
e
v
aluate
D
(
k
+1)
using
the
infinity
norm
(or
Chebyshe
v
norm)
as
follo
ws.
D
(
k
+1)
=
max
1
i
n
n
X
r
=1
d
(
k
+1)
ir
=
n
X
r
=1
z
~
r
S
r
E
(
k
+1)
r
E
(
k
)
r
;
~
=
ar
g
max
1
i
n
(
n
X
r
=1
z
ir
S
r
E
(
k
+1)
r
E
(
k
)
r
)
(9)
Hence,
inequality
(9)
can
be
con
v
eniently
written
as
(
k
+1)
c
(
k
+1)
E
(
k
+1)
;
c
(
k
+1)
=
n
X
r
=1
z
~
r
S
r
E
(
k
+1)
r
E
(
k
)
r
(10)
Therefore,
by
using
the
inequalities
abo
v
e,
the
e
xistence
as
well
as
the
uniqueness
of
the
fe
asible
solution
can
be
analyzed.
Also,
con
v
er
gence
properties
of
the
load
flo
w
algorithm
can
be
assessed
and
error
boundaries
per
iteration
specified.
3.2.
Existence
and
Uniqueness
of
the
F
easible
Solution
The
e
xistence
and
uniqueness
of
the
load
flo
w
feasible
solution
in
closed
subset
is
an
important
result
for
system
analysis.
Once
this
result
is
v
erified,
it
is
guaranteed
that
studies
such
as
reacti
v
e
po
wer
compensa-
tion,
distrib
ution
automation,
and
netw
ork
reconfiguration,
all
usually
endorsed
by
load
flo
w
calculations,
will
lead
to
the
steady
state
obtained
in
practice,
gi
v
en
the
uncertainties
of
loading
and
netw
ork
data.
Aiming
at
v
erifying
the
e
xistence
and
uniqueness
of
the
feasible
load
flo
w
solution,
let
R
be
a
closed
subset
of
C
n
gi
v
en
by
R
,
f
E
2
C
n
;
jj
E
i
jj
E
0
g
,
8
i
=
1
;
:
:
:
;
n
,
where
E
0
2
<
E
0
p
jj
ZS
jj
.
Also,
let
c
be
a
real
constant
defined
as
c
=
jj
ZS
jj
=
(
E
0
)
2
.
Observ
e
that
by
construction
(
E
0
)
2
>
jj
ZS
jj
and,
as
consequence,
c
<
1
.
Also,
notice
that
(4)
can
be
written
in
the
matrix
form
as
follo
ws.
(
E
(
k
)
)
=
E
0
ZS
K
(
k
)
(11)
where
K
is
a
n
1
v
ector
with
elements
gi
v
en
by
reciprocals
of
comple
x
v
oltage
iterates
at
iteration
k
and
E
0
is
a
n
1
v
ector
with
entries
equal
to
E
0
.
Gi
v
en
a
comple
x
v
oltage
E
(
k
)
in
R
,
the
comple
x
v
oltages
obtained
throughout
update
rule
can
be
analyzed
as
follo
ws.
(
E
(
k
)
)
E
0
=
ZS
K
(
k
)
jj
ZS
jj
K
(
k
)
<
jj
ZS
jj
(
E
0
)
<
(
E
0
)
Consequently
,
if
E
(
k
)
2
R
then
E
(
k
+1)
=
(
E
(
k
)
)
belongs
to
an
open
ball
(in
C
n
)
centered
in
E
0
and
a
radius
equal
to
,
8
k
=
0
;
:
:
:
;
N
iter
.
Therefore,
(
E
(
k
)
)
belongs
to
R
and
the
inequality
belo
w
holds
8
k
.
IJECE
V
ol.
6,
No.
6,
December
2016:
3229
–
3237
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
3233
0
D
(
k
+1)
=
ZS
F
(
k
+1)
jj
ZS
jj
F
(
k
+1)
jj
ZS
jj
(
E
0
)
2
=
c
(12)
By
using
(10)
and
(12),
we
obtain
that
0
c
(
k
+1)
c
<
1
;
8
k
,
as
well
as
(
k
+1)
c
E
(
k
+1)
(13)
By
construction,
the
function
:
R
!
R
is
a
contraction
mapping
(in
R
)
with
contraction
constant
c
and
unique
fix
ed-point.
In
f
act,
suppose
E
?
and
E
??
are
tw
o
dif
ferent
elements
of
R
,
then
jj
(
E
?
)
(
E
??
)
jj
c
jj
E
?
E
??
jj
(14)
Suppose
no
w
that
E
?
and
E
??
are
both
load
flo
w
solutions.
Then,
by
the
h
ypothesis
we
obtain
jj
E
?
E
??
jj
=
jj
(
E
?
)
(
E
??
)
jj
c
jj
E
?
E
??
jj
If
E
?
6
=
E
??
then
jj
E
?
E
??
jj
6
=
0
.
Hence,
we
can
di
vide
the
inequality
abo
v
e
by
jj
E
?
E
??
jj
obtaining
that
c
1
,
which
is
an
inconsistent
conclusion.
Therefore
we
ha
v
e
that
E
?
=
E
??
,
i.e.,
the
e
xistence
and
uniqueness
of
the
load
flo
w
feasible
solution
has
been
v
erified,
limited
to
conditions
related
to
v
oltage
magnitude
at
the
substation
b
us,
netw
ork
and
loading.
3.3.
P
o
wer
Mismatch
Con
v
er
gence
Using
the
contraction
mapping
theorem,
it
is
v
erified
that
since
:
R
!
R
is
a
contraction
mapping
(in
R
),
the
sequence
of
iterates
f
E
(0)
;
(
E
(0)
)
;
(
(
E
(0)
))
;
:
:
:
g
con
v
er
ges
to
the
load
flo
w
feasible
solu-
tion,
8
E
0
2
R
.
The
algorithm
presents
a
geometric
con
v
er
gence
to
w
ards
the
solution
with
a
geometric
rate
e
v
aluated
by
c
(
k
+1)
.
In
addition,
since
c
(
k
+1)
depends
upon
system
loading,
the
con
v
er
gence
characteristic
is
also
dependent
on
loading.
No
w
,
let
S
(
k
)
i
be
the
comple
x
po
wer
mismatch
at
b
us
i
and
iteration
k
.
In
addition,
let
y
i
be
the
series
line
admittance
upstream
b
us
i
.
Notice
that
S
(
k
)
i
=
E
(
k
)
i
E
(
k
)
i
E
(
k
)
u
i
y
i
+
E
(
k
)
i
P
r
2
i
S
r
E
r
+
S
i
,
and
by
conjug
ating
this
e
xpression
we
ha
v
e
S
(
k
)
i
=
E
(
k
)
i
y
i
E
(
k
)
i
E
(
k
)
u
i
+
z
i
X
r
2
i
S
r
E
(
k
)
r
!
By
adding
the
null
element
E
(
k
)
i
y
i
(
(
k
)
u
i
(
k
)
u
i
)
in
(15)
we
ha
v
e
S
(
k
)
i
=
E
(
k
)
i
y
i
E
(
k
)
i
(
k
)
u
i
+
z
i
X
r
2
i
S
r
E
(
k
)
r
!
+
E
(
k
)
i
y
i
(
k
)
u
i
E
(
k
)
u
i
=
E
(
k
)
i
y
i
E
(
k
)
i
(
k
)
i
E
(
k
)
i
y
i
E
(
k
)
u
i
(
k
)
u
i
Assuming
E
(
k
)
i
y
i
6
=
0
,
8
i
=
1
;
:
:
:
;
n
,
then
lim
k
!1
i
(
E
(
k
)
i
)
E
(
k
)
i
=
0
(
)
lim
k
!1
S
(
k
)
i
=
0
;
8
i
(15)
which
indicates
that
v
oltages
con
v
er
ge
if
and
only
if
po
wer
mismatches
con
v
er
ge
to
zero.
3.4.
Dependence
of
Initial
Estimate
A
flat
start
or
a
s
o
l
ution
estimate
must
be
specified
in
t
he
be
ginning
of
the
load
flo
w
algorithm.
Hence,
it
is
important
to
highlight
that
the
load
flo
w
update
rule
will
be
a
contraction
in
a
trajectory
where
c
<
1
,
and
thus
it
will
con
v
er
ge
to
w
ards
the
solution
if
an
initial
estimate
is
chosen
in
a
re
gion
where
the
contraction
mapping
theorem
is
v
alid.
Re
gion
R
w
as
chosen
to
ease
the
deductions,
though
in
theory
other
re
gions
could
be
set
to
pro
v
e
the
same
mathematical
results.
Evaluation
of
the
F
orwar
d-Bac
kwar
d
Sweep
Load
Flow
Method
using
...
(Die
go
Issicaba
and
J
or
g
e
Coelho)
Evaluation Warning : The document was created with Spire.PDF for Python.
3234
ISSN:
2088-8708
Therefore,
it
is
note
w
orth
y
to
mention
that
other
uncommon
initial
estimates
might
lead
to
the
solution
as
well.
F
or
instance,
if
quite
high
v
oltages
are
chosen
as
initial
estimate,
contraction
should
lead
v
oltages
at
the
first
iteration
to
be
located
into
an
open
ball
centered
in
E
0
with
radius
.
In
f
act,
high
v
oltages
lead
to
reduced
currents
in
the
first
iteration,
caus
ing
v
oltage
iterates
to
be
clos
e
to
E
0
.
On
the
other
hand,
if
v
ery
lo
w
v
oltages
are
chosen
as
initial
estimates,
v
oltage
errors
might
increase
considerably
leading
t
he
comple
x
iterates
into
re
gion
R
.
Therefore,
the
load
flo
w
method
is
sho
wn
to
be
rob
ust
to
a
lar
ge
v
ariety
of
initial
estimates.
3.5.
Err
or
Boundaries
Since
:
R
!
R
is
a
contraction
mapping
(in
R
),
with
contraction
constant
c
,
the
follo
wing
relation
holds
by
the
contraction
mapping
theorem
E
/
E
(
k
)
c
k
1
c
(
E
(0)
)
E
(0)
(16)
E
/
E
(
k
)
c
1
c
E
(
k
)
E
(
k
1)
(17)
Therefore,
by
using
(16)
and
(17)
we
can
e
v
aluate
boundaries
for
the
error
gi
v
en
by
the
dif
ference
between
the
computed
iterate
v
alue
and
the
load
flo
w
solution.
4.
NUMERICAL
RESUL
TS
This
section
presents
some
numerical
load
flo
w
analysis
to
v
alidate
the
theoretical
results
deduced
in
the
paper
.
The
con
v
er
gence
criterion
w
as
chosen
to
be
the
maximum
absolute
dif
ference
between
subsequent
v
oltage
iterates,
with
tolerance
gi
v
en
by
1E-06.
4.1.
A
T
w
o
Bus
Case
Study
Consider
a
tw
o
b
us
distrib
ution
feeder
with
substation
b
us
v
oltage
of
11
kV
as
well
as
line
param-
eter
and
comple
x
load
gi
v
en
by
1
:
35309
+
j
1
:
32349
and
5
+
j
3
MV
A,
respecti
v
ely
.
From
the
theoret-
ical
basis
presented
in
this
paper
,
con
v
er
gence
is
assured
and
the
update
rule
is
a
contr
action
inside
re
gion
R
,
f
E
1
2
C
;
jj
E
1
jj
E
0
g
,
for
an
y
such
that
0
:
50000
<
0
:
69800
pu.
W
ithout
loss
of
general-
ity
,
we
ha
v
e
chosen
to
be
0
:
60000
pu
leading
to
a
contraction
constant
c
of
0
:
57007
.
The
last
iterate
w
as
considered
the
solution
for
the
sak
e
of
error
computation.
T
able
1(a)
sho
ws
the
real
and
imaginary
parts
of
v
oltage
iterates,
contraction
iterates,
error
v
alues
and
error
boundaries
computed
using
(17),
assuming
the
uncommon
initial
estimate
of
E
(0)
1
=
4
:
00
\
0
o
pu.
Once
E
(0)
1
belongs
to
R
,
contraction
mapping
w
as
assigned
from
the
v
ery
first
i
teration,
and
error
boundaries
indeed
limited
the
iterate
errors.
As
e
xpected,
contraction
iterates
did
not
e
xceed
the
contraction
constant
c
,
i.e.
0
<
c
(
k
)
c
=
0
:
57007
<
1
.
T
able
1.
T
w
o
b
us
case
study
(a)
E
(0)
1
=
4
:
00
\
0
o
pu
k
e
1
(pu)
f
1
(pu)
c
(
k
)
Error
Boundary
0
4.00000
0.00000
-
3.09E+00
-
1
0.97782
-0.00529
0.02332
7.84E-02
9.31E+00
2
0.90915
-0.02113
0.10257
8.11E-03
2.17E-01
3
0.90192
-0.02098
0.11118
9.03E-04
2.23E-02
4
0.90113
-0.02114
0.11216
1.01E-04
2.48E-03
5
0.90104
-0.02114
0.11227
1.14E-05
2.78E-04
6
0.90103
-0.02114
0.11228
1.26E-06
3.12E-05
7
0.90103
-0.02114
0.11229
1.28E-07
3.50E-06
8
0.90103
-0.02114
0.11229
-
3.93E-07
(b)
E
(0)
1
=
0
:
02
\
0
o
pu
k
e
1
(pu)
f
1
(pu)
c
(
k
)
Error
B
oundary
0
0.02000
0.00000
-
8.81E-01
0.00E+00
1
-3.43633
-1.05710
1.26849
4.46E+00
1.11E+01
2
1.02186
0.01288
0.02483
1.26E-01
1.41E+01
3
0.91345
-0.02178
0.09768
1.24E-02
3.51E-01
4
0.90237
-0.02082
0.11060
1.38E-03
3.43E-02
5
0.90119
-0.02115
0.11210
1.54E-04
3.79E-03
6
0.90105
-0.02114
0.11226
1.73E-05
4.25E-04
7
0.90103
-0.02114
0.11228
1.92E-06
4.77E-05
8
0.90103
-0.02114
0.11229
1.95E-07
5.36E-06
9
0.90103
-0.02114
0.11229
-
6.01E-07
Alternati
v
ely
,
the
near
zero
initial
estim
ate
of
E
(0)
1
=
0
:
02
\
0
o
pu
w
as
chosen.
V
oltage
and
contraction
iterates
obtained
in
this
analysis
are
sho
wn
in
T
able
1(b).
In
this
case,
one
can
notice
that
e
v
en
by
choosing
a
nonrealistic
solution
as
initial
estimate,
the
algorithm
con
v
er
ged
t
o
the
feasible
solution.
In
f
act,
the
contraction
iterate
c
(
k
)
is
greater
than
the
unit
near
the
initial
estimate,
where
the
update
rule
is
not
a
contraction
mapping
IJECE
V
ol.
6,
No.
6,
December
2016:
3229
–
3237
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
3235
(dilation).
As
a
consequence,
v
oltage
mismatches
increased
and
v
oltage
iterates
mo
v
ed
to
w
ards
re
gion
R
,
where
con
v
er
gence
is
guaranteed
by
the
mathematical
results.
4.2.
A
27-b
us
Case
Study
The
load
flo
w
algorithm
w
as
applied
to
the
27-b
us
system
gi
v
en
in
[21].
The
mathematical
anal
ysis
sho
wed
that
con
v
er
gence
is
assured
and
the
update
rule
is
a
contraction
into
the
re
gion
R
,
E
2
C
27
;
jj
E
i
jj
E
0
;
8
i
=
1
;
:
:
:
;
27
g
,
for
an
y
such
that
0
:
50000
<
0
:
71270
.
Similar
to
the
pre
vious
case
study
,
an
v
alue
of
0
:
60000
w
as
chosen
and
the
last
iterate
w
as
considered
as
solution
for
error
computation
purposes.
The
contraction
constant
c
is
0
:
51580
for
this
case.
The
initial
estimate
w
as
chosen
to
be
E
(0)
i
=
0
:
05
\
120
:
32
o
pu,
8
i
=
1
;
:
:
:
;
27
.
T
abl
e
3
sho
ws
the
iterates
obtained
in
this
simulation.
Also,
the
first
tw
o
v
oltage
iterates
are
illustrated
in
a
le
v
el
curv
e
of
the
con
v
er
gence
re
gion
for
b
us
25
in
Fig.
2.
T
able
2.
The
27-b
us
case
study
(b
us
25).
Initial
estimate:
E
(0)
i
=
0
:
05
\
120
:
32
o
pu.
k
e
25
(pu)
f
25
(pu)
c
(
k
)
Error
Boundary
0
-0.02524
0.04316
-
1.01E-00
-
1
1.51846
1.56655
0.83438
1.70E-00
2.31E-00
2
0.97564
-0.03361
0.04262
6.54E-02
1.80E-00
3
0.91703
-0.01505
0.09096
5.19E-03
6.55E-02
4
0.91273
-0.01732
0.09637
4.30E-04
5.18E-03
5
0.91234
-0.01720
0.09679
3.56E-05
4.29E-04
6
0.91231
-0.01721
0.09683
2.94E-06
3.56E-05
7
0.91231
-0.01721
0.09683
2.29E-07
2.95E-06
8
0.91231
-0.01721
0.09683
-
2.44E-07
e
2
5
[
p
u
]
-1
-0.5
0
0.5
1
1.5
2
f
2
5
[
p
u
]
-1
-0.5
0
0.5
1
1.5
2
0
1
2
0
Solution
Voltage iterate
Contraction Region
E
0
−
α
Figure
2.
Three
v
oltage
iterates
for
the
27-b
us
case
study
(b
us
25).
Initial
estimate
E
(0)
i
=
0
:
05
\
120
:
32
o
pu.
As
e
xpected,
since
the
initial
estimate
has
lo
w
v
oltage
entries
the
v
oltage
mismatche
s
increased
caus-
ing
the
v
oltage
iterates
to
be
placed
in
R
,
where
the
con
v
er
gence
is
guaranteed
by
the
mathematical
analysis.
Also,
the
contraction
iterates
did
not
e
xceed
the
contraction
constant
in
R
and
the
error
boundaries
indeed
limited
the
iterate
error
v
alues.
5.
CONCLUSIONS
This
paper
e
v
aluates
the
forw
ard-backw
ard
sweep
load
flo
w
method
to
distrib
ution
s
y
s
tem
analysis
using
fix
ed-point
concepts
and
the
contraction
mapping
theorem.
The
con
v
er
gence
of
the
method
is
pro
v
en
gi
v
en
certain
conditions
related
to
substation
b
us
v
oltage,
netw
ork
data
and
system
loading.
The
e
xistence
and
uniqueness
of
the
feasible
load
flo
w
solution
is
also
e
v
aluated,
subjected
to
the
same
conditions.
Furthermore,
boundaries
for
error
v
alues
per
iteration
between
iterates
and
solution
are
obtained.
It
w
as
sho
wn
that
the
Evaluation
of
the
F
orwar
d-Bac
kwar
d
Sweep
Load
Flow
Method
using
...
(Die
go
Issicaba
and
J
or
g
e
Coelho)
Evaluation Warning : The document was created with Spire.PDF for Python.
3236
ISSN:
2088-8708
algorithm
e
xhibits
a
geometric
con
v
er
gence
to
w
ards
solution.
Also,
t
he
algorithm
is
rob
ust
to
a
lar
ge
v
ariety
of
initial
estimates.
All
the
mathematical
results
are
v
alidated
with
load
flo
w
simulations.
Future
w
orks
are
en
visioned
to
e
xtend
to
de
v
eloped
analysis
with
re
g
ard
to
three-phase
distrib
ution
netw
orks,
weakly-meshed
distrib
ution
netw
orks,
v
oltage
dependent
load
models
and
distrib
uted
generators.
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.
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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
Feder
al
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
Ele
ctrical
Engineering
at
the
Federal
Uni
v
ersity
of
Santa
Catarina,
Brazil,
since
March
1978.
His
research
interests
incl
ude
distrib
ution
systems
e
xpansion
and
operati
on
planning,
po
wer
systems
reliability
,
probabilistic
methods
applied
to
po
wer
systems,
and
po
wer
quality
.
Evaluation
of
the
F
orwar
d-Bac
kwar
d
Sweep
Load
Flow
Method
using
...
(Die
go
Issicaba
and
J
or
g
e
Coelho)
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