TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 10, Octobe
r 20
14, pp. 7114
~ 712
2
DOI: 10.115
9
1
/telkomni
ka.
v
12i8.642
1
7114
Re
cei
v
ed
Jun
e
25, 2014; Revi
sed
Jul
y
1
9
, 2014; Acce
pted Jul
y
28,
2014
Optimal Design of a 3-Phase Core Type Distribution
Transformer Using Modified Hooke and Jeeves Method
Raju Basak
1
,
Arabinda
Das*
2
, Amar Nath San
y
al
3
1,2
Electrical En
gin
eeri
ng D
epa
rtment, Jadavp
u
r Univ
ersit
y
,
188, Ra
ja S.C. Mallick R
o
a
d
, Kolkata - 70
00
32, India
3
Calcutta Institute of Engi
neer
ing a
nd Ma
nag
ement, Kolkat
a
- 70004
0, Indi
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: basak.raj
u@
ya
ho
o.com
1
, adas_
e
e
_
ju
@
y
a
hoo.com
2
,
ansa
n
y
al
@
y
a
h
oo.co.in
3
A
b
st
r
a
ct
Hook
e a
nd J
e
eves
meth
od
i
s
de facto
a p
a
ttern se
arch t
e
chn
i
qu
e, w
h
ic
h can
be
e
m
p
l
oyed for
getting
a
n
o
p
ti
ma
l so
luti
on. I
n
this
pa
per th
e
meth
od, i
n
a
modifi
ed
for
m
, has
be
en
ap
pl
ied f
o
r the
des
i
g
n
opti
m
i
z
at
ion
of a distrib
u
tio
n
transformer. It is a c
onstra
i
ned multi-var
i
a
b
le opti
m
i
z
at
io
n
pro
b
le
m.
T
h
e
soluti
on
is o
b
ta
ine
d
by
ch
oosi
ng
an
initi
a
l
po
i
n
t in
the w
o
rl
d
ma
p
of the k
e
y
varia
b
l
e
s a
n
d
by
maki
ng
a
lo
cal
search
(
e
xpl
o
r
a
tory
in all dir
e
ctions in
th
e hyper
s
u
rfa
c
e
fo
rm
ed
b
y
the va
ri
ab
les. After recogni
z
i
ng the
pattern, its adv
antag
e is taken
by movi
ng tow
a
rds a low
e
r
co
st point, using
an acce
lerati
on
factor for faster
conver
genc
e. T
he step le
ngt
h is ad
juste
d
a
s
w
e
pr
ocee
d to exp
edite
i
m
p
r
ove
m
e
n
t. T
he meth
od
has b
e
e
n
app
lie
d to tw
o
different cost functions: the
cost
of production a
nd th
e cost agai
nst producti
on pl
u
s
capita
li
z
e
d
ru
n
n
in
g loss
es. In
both the
cases
,
the prob
le
m
h
a
s conv
erge
d t
o
a so
lutio
n
a
n
d the res
u
lts ar
e
both inter
e
stin
g and i
llu
min
a
ti
ng.
Ke
y
w
ords
: dis
t
ributio
n transformer, cost function, des
i
gn v
a
ria
b
les, patter
n
search, const
r
aints
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
There are fo
ur differe
nt a
ppro
a
che
s
fo
r so
lvin
g a d
e
sig
n
proble
m
[1] viz. analytical
desi
gn, synth
e
tic
d
e
si
gn, optimal de
sig
n
an
d
stan
da
rd de
sign. T
here
i
s
no loop or feedb
a
ck
from the
re
sults obtai
ned
in the an
al
ytic pro
c
ed
ure. Hen
c
e th
ere i
s
n
o
provision fo
r a
n
y
c
o
ns
traint satis
f
ac
tion in this
method.
T
he synth
e
tic
desi
gn i
s
bett
e
r tha
n
the a
nalytic de
sig
n
as
it provide
s
for con
s
trai
nt sa
tisfaction.
Ho
wever,
thi
s
m
e
thod give
s o
n
ly a feasi
b
le
solutio
n
not t
he
best po
ssible
one. Techni
cal pe
rson
s
aim at optim
al desi
g
n
-
it gives the b
e
s
t possibl
e o
u
t o
f
different fea
s
i
b
le sol
u
tion
s, satisfying gi
ven
con
s
train
t
s. Standard
desi
gn metho
d
s a
r
e follo
wed
by the bul
k
manufa
c
turers which a
r
e b
a
se
d on
sel
e
ction
of stan
d
a
rd stampi
ng
s,
stan
da
rd core
size etc. All these method
s are a
pplie
d to transfo
rme
r
desi
gn.
2.
The 3-p
h
as
e Core
-ty
pe Oil-immersed Distribu
tion
Trans
f
ormer
The 3
-
limbe
d core con
s
tru
c
tion is empl
oyed fo
r 3
-
ph
ase distri
bution
transfo
rme
r
s
as it is
more e
c
on
om
ic com
pared to shell type [1, 2]. T
hese are invari
ably
of the oil-immersed type with
natural
o
r
fo
rce
d
cooli
n
g
dep
endin
g
on the
si
ze
.
The
co
re
is made
by
stacking
varni
s
hed
lamination
s
of high g
r
ad
e sili
con
ste
e
l. Either co
pper or
alu
m
inium i
s
u
s
ed
as
co
nd
uctor
material. The
core
-coil stru
cture i
s
pla
c
e
d
on a
soft bed in the oil-filled tank havi
ng a protrudi
ng
con
s
e
r
vator along with
a breath
e
r.
Th
e
con
s
e
r
va
to
r take
s
care
of the expan
sion of oil u
n
der
loadin
g
and t
he breathe
r i
s
u
s
ed to
sto
p
the ing
r
e
s
s of moistu
re i
n
to the oil ta
nk. Coolin
g tube
s
or
radi
ators a
r
e to
be
ad
de
d to
kee
p
the
tempe
r
atu
r
e
rise
of
oil
within
statutory l
i
mits. Fo
r la
rge
rating, forced air or forced oil-
cooli
ng
has to be augmented.
Othe
r auxiliaries f
o
r
protection
like
Buchh
o
ltz rel
a
y, indicato
rs etc. are a
dde
d. The
const
r
uction, p
r
in
ci
ple and
de
sig
n
con
s
id
erations
for dist
ributio
n and p
o
wer
transfo
rme
r
s
have bee
n el
ucid
ated in
several text-b
o
o
ks on el
ect
r
i
c
al
machi
ne de
si
gn [2-4].
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Desi
gn of a 3-Pha
s
e Core T
y
p
e
Distrib
u
tion
Tran
sfo
r
m
e
r Usi
ng… (Raj
u Basa
k)
7115
2.1. Design
Cons
ideration
s
The l
oad
fa
ctor of di
stri
bution t
r
an
sf
or
me
rs i
s
m
u
ch
le
ss th
an that
of
a po
we
r
transfo
rme
r
. So it is desi
gned for ma
ximum effici
e
n
cy at its probabl
e load
factor (0.4
- 0.6),
keeping iron loss relatively
less. So a lower flux
-d
en
sity is used co
mpared
to that for the power
transfo
rme
r
.
CRS
-type
co
res a
r
e inva
riably used fo
r all
appli
c
ati
ons. Alu
m
ini
u
m is u
s
ed
as
conductor i
n
distribution t
r
ansformers up to a si
ze of
about
500 KVA, for economic
reasons.
But
cop
per i
s
a
f
a
r
better mat
e
rial
for large
r
rating,
parti
cula
rly if the
r
e be
con
s
trai
nt on
the
bul
k of
the transfo
rm
er as i
s
usual
in densely populate
d
urb
an pla
c
e
s
. As the voltage regulatio
n has to
be kept at a low value for
a distrib
u
tion transfo
rme
r
, the gap bet
we
en L.T. and H.T. coils i
s
kept
at its minim
u
m allo
wable
value an
d th
e win
d
o
w
hei
ght: width
rat
i
o is
kept
at a rel
a
tively large
value co
mpa
r
ed to the po
wer t
r
an
sformer to redu
ce the lea
k
ag
e rea
c
tan
c
e [
3
, 4]. Admissible
values of de
si
gn variabl
es
are obtai
ned f
r
om data
-
bo
o
k
[5].
2.2. Optimal Design o
f
Tr
ansformers
- v
a
r
i
ous Methods
The first foot-step
s
to com
puter-aid
ed d
e
sig
n
of ele
c
t
r
ical
ma
chin
e
s
sta
r
ted l
ong
before
in fifties. The con
c
e
p
t of optimization wa
s esta
b
lish
e
d
long before in history but i
t
s appli
c
ation
to
machi
ne de
si
gn cam
e
in a much late
r st
age. The opti
m
izing p
r
o
g
ra
ms develo
p
e
d
much late
r
on.
One su
ch me
thod has b
e
e
n
repo
rted by
O.W. Ander
son [6] in 1967. Since the
n
the work is in
prog
re
ss- se
veral autho
rs have propo
sed new
er an
d newe
r
tech
nique
s and a
d
vanced pap
ers
towards reali
z
ation
of o
p
timal de
sig
n
. J.C. Olivares
et a
l
have
de
scrib
ed
a te
ch
nique
for
opti
m
al
desi
gn of
she
ll-type tran
sfo
r
mers [7]. He
has al
so hi
g
h
lighted
on
core l
a
minatio
n sel
e
ctio
n a
nd
choi
ce
of
con
ducto
r m
a
teri
als fo
r
distri
b
u
tion tran
sformers [8,
9]. Pavlos
et al
have p
r
op
ose
d
a
heuri
s
tic
solu
tion to cost-o
ptimization p
r
oblem
s
for transfo
rme
r
de
sign [10]. Bre
s
lin and
Hu
rl
e
y
have
propo
sed
a web
-
ba
sed de
sign o
f
transfo
rme
r
taking
help
from the inte
rnet [11]. So
me
authors have
used recent
ly devel
oped
soft-co
mputi
ng tech
nique
s for re
achin
g
the optima
l
solutio
n
[12].
He
rma
nde
z and
Au
rora
have
devel
oped
an
inte
lligent a
s
sist
ant for de
sig
n
ing
distrib
u
tion transfo
rme
r
[1
3]. Subrani
an
and Pa
dma
have propo
se
d a meth
od f
o
r optimi
z
atio
n of
transfo
rme
r
d
e
sig
n
usi
ng b
a
cteri
a
l fora
gi
ng algo
rithm
[14]. A. K. Jadav has
adva
n
ce
d a meth
od
for optimi
z
in
g
the d
e
si
gn
of po
wer tra
n
sf
orme
r u
s
in
g
simulated
ann
ealing
[15, 1
6
]. R.A. Jab
r
h
a
s
applie
d geom
etric p
r
og
ram
m
ing to tran
sforme
r de
sign
[17].
3. Procedure
for Optim
i
za
tion
For re
aching
an optimal so
lution, one h
a
s to
formula
t
e the proble
m
at first, choose the
desi
gn varia
b
l
es, fix up the con
s
traints and frame t
he obje
c
tive function [18].
Maximum a
n
d
minimum bo
und
s are imp
o
se
d on the desi
gn vari
a
b
l
es by the experie
nced de
sign
er an
d the
optimal soluti
on is
sou
ght in the wo
rld m
ap of the
vari
able
s
, either
by classi
cal t
e
ch
niqu
es o
r
by
recently dev
elope
d intelli
gent techniq
ues. T
her
e a
r
e
several te
chni
que
s to reach an
opti
m
al
solutio
n
, for a
con
s
trai
ned
or an
un
con
s
trained
de
sig
n
pro
b
lem
by the cla
s
sical
method. Th
e
s
e
are broadly classified into me
thods based on: i)
exhaustive search, ii)
random search,
iii)
pattern
se
arch, iv) g
r
adie
n
t se
arch [1
9, 20]. The
exhau
stive search i
s
sim
p
le b
u
t it is t
i
me-
con
s
umi
ng, p
a
rticul
arly if there
be a la
rge
num
ber
o
f
variable
s
a
nd ch
osen
step length
s
a
r
e
small. Th
e random
search gives only
a qu
asi
-
opt
i
m
al solution,
not the o
p
tima. Gradient
or
pattern
sea
r
ch techni
que
s
are bette
r ma
thematical
to
ols which ca
n
be use
d
efficiently to find
out
the glo
bal
op
tima in
a m
u
ch l
e
ss
no.
o
f
step
s. Th
e
con
s
trai
nts can b
e
a
c
cou
n
ted fo
r a
nd
the
step-l
ength
can be va
ried
as the p
r
o
b
le
m conve
r
g
e
s
to its final solution. The
r
e are
a variet
y of
techni
que
s
b
a
se
d o
n
p
a
ttern
or gradie
n
t se
arch
.
Hooke a
nd
Je
eves m
e
thod
is
one
amo
ngst
them [18-20]
. It is a direct metho
d
based on
p
a
ttern sea
r
ch, appli
c
able
to multivari
able
probl
em
s.
3.1.
Hook an
d Jeev
es Method of Pa
tte
r
n Search
The metho
d
use
s
a set of sea
r
ch direction
s
which spa
n
s the
entire search spa
c
e
defined
by the bounds
of
the design v
a
riabl
es [18],[20]. In an
n
-dimensi
onal problem,
the
r
e
must be
n
numbe
r of lin
early in
depe
ndent
sea
r
ch directio
ns.
The
s
e di
re
ction
s
an
d t
he
corre
s
p
ondin
g
ste
p
le
ngth
s
a
r
e
to b
e
j
u
dicio
u
sly
ch
o
s
en
in
order t
o
rea
c
h th
e
solution
by sm
aller
no. of iterati
ons. In the
Hoo
k
e a
nd
Jeeves
m
e
tho
d
, a com
b
ina
t
ion of exploratory move
and
heuri
s
tic p
a
ttern sea
r
ch is used. Firstly, an init
ial point is ch
ose
n
in the sea
r
ch spa
c
e fro
m
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 711
4
– 7122
7116
desi
gne
r’s
experie
nce or
b
y
consulting
a desi
gn d
a
ta-bo
o
k. Th
en
a local
sea
r
ch is
made i
n
all
the dire
ction
s
to find out th
e best poi
nt aroun
d the ch
o
s
en p
o
int.
3.2.
Algorith
m
The al
gorith
m
[20, 21]
ha
s b
een f
r
ame
d
with
s
lig
ht
modificatio
n
over the
ori
g
i
nal Hoo
k
and
Jeeve
s
method, in
order to
en
su
re
conve
r
g
e
n
c
e
.
Let the b
a
se point
be
0
x
, where
x
is a
n-
vector; n i
s
the num
ber
of desi
gn va
riable
s
. Thi
s
has to be j
udici
ou
sly ch
ose
n
for fa
st
er
conve
r
ge
nce. Let the va
ria
b
le
i
x
(for
ea
ch
iteration
)
b
e
pertu
rbe
d
to
(1
)
ii
x
, where
is the
step len
g
th. The step
s to b
e
followe
d are as given b
e
l
ow:
Step 1: read
n
= numb
e
r of
desi
gn varia
b
l
es;
= convergen
ce facto
r
for the obje
c
ti
ve function;
ma
x
k
= maximum n
u
mbe
r
of iterations;
ma
x
c
= maximum numb
e
r of iterations f
o
r
Cha
nging ste
p
length
Step 2: for i
=
1
to n
Step 3: read
i
x
,
i
,
i
‘
x
desig
n varia
b
le,
s
t
ep length;
accel
e
ra
tion factor
Step 4 end for
Step 5: set
0
k
‘ex
p
lorato
ry
mov
e
Step 6: find
()
k
ii
f
fx
‘obtaine
d fro
m
transfo
rme
r
desi
gn sub
r
outine
Step 7: if
k
= 0 then go to step 16
Step 8:
If
1
0
kk
ii
ff
then step 1
5
Step 9:
for i =
1
to n
Step 10:
ii
s
‘red
uction of ste
p
length
Step 11: end
for
Step 12:
1
cc
Step 13; if
ma
x
cc
then go to step
24
Step 14:
1
kk
: g
o
to s
t
ep 6
Step 15: if
1
kk
ii
ff
then go to step
22
Step 16: For i=1 to n
Step 17: Find
[(
1
)
]
ii
i
ff
x
Step 18: Find
()
/
kk
ii
i
i
Gf
f
f
Step 19: end
for
Step 20: set
ii
i
i
i
xx
G
‘pattern mov
e
Step 21:
1
kk
: i
f
ma
x
kk
then go to step 23 el
se to step 6
Step 22: prin
t “Success- the solutio
n
ha
s co
nverg
ed.” print out re
su
lts: go to step
24
Step 23: prin
t “Failure, the
solution i
s
no
t obtained wit
h
in
ma
x
k
no. of iterations.”
Step 24: stop
Step 25: end
Step 26: print
“Cha
nge in
step length do
es not ma
ke
any improve
m
ent. Initialize once agai
n”: go
to step 24
4.
The
De
sign Variables
The sol
u
tion
of an optimization pro
b
le
m starts
by identificatio
n of design va
ri
able
s
[18,
19]. The obj
ective fun
c
tio
n
may be hi
ghly sen
s
itive to ce
rtain
variable
s
. Th
ese
are th
e
key
variable
s
. For some othe
r variabl
es, the
sen
s
itivit
y may be less. Th
ey are given l
e
ss impo
rtan
ce.
The key varia
b
les to b
e
ch
ose
n
to optim
ize a d
e
si
gn
probl
em de
p
end on th
e o
b
jective fun
c
tion-
wheth
e
r it is the co
st of produ
ction or a
wei
ghted
co
mbination of the co
st of produ
ction and
the
lost ene
rgy u
n
its du
ring it
s op
eratin
g life or someth
ing else. The
variable
s
m
a
y be de
cisi
on
variable
s
o
r
continuo
us variable
s
. For a tran
sform
e
r, the variabl
es
have bee
n id
entified:
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TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Desi
gn of a 3-Pha
s
e Core T
y
p
e
Distrib
u
tion
Tran
sfo
r
m
e
r Usi
ng… (Raj
u Basa
k)
7117
a. De
cisi
on
vari
able
s
:
(1)
The choi
ce of
core mate
ria
l
- co
stlie
r CRGOS may be
more
econo
mic than
ch
e
aper
CRNOS
con
s
iderin
g the over-all co
st incl
uding that of cop
per.
(2)
The choi
ce of
con
d
u
c
tor m
a
terial
s-
co
stli
er copp
er m
a
y have to be
use
d
con
s
ide
r
ing
over-all perfo
rman
ce a
nd cost, parti
cula
rly if
there be
spa
c
e
con
s
traints.
b. Contin
uou
s
variabl
es:
(1)
The emf co
nstant
K
(in eqn.
=
t
E
KS
, where
t
E
=
emf per turn,
S
=
KVA rating).
(2)
The ratio of windo
w height
to windo
w wi
dth:
/
ww
w
R
HW
(3)
The maximu
m flux-den
sity
m
B
(4)
The maximum current
-density,
(5)
The ratio of iron loss to co
pper lo
ss:
/
ic
PP
4.1. The Bou
nds on De
sign Variables
From
de
sign
er’s expe
rien
ce
and
from
de
sign
data
boo
k [1
9] th
e follo
wing v
a
lue
s
of
desi
gn va
ria
b
les have
be
en
sugg
este
d for
a 3
-
ph
ase
core
type di
stributio
n
tran
sform
e
r
with
cop
per a
s
co
ndu
ctor mate
rial:
E.M.F.
const
ant,
K
= 0.45 (somewhat sm
aller for Alumi
n
ium)
Window
height/width,
w
R
:3.0-4.0 for dis
t
ribution trans
f
ormer
The followi
ng
choi
ce of ma
terials h
a
s b
e
en re
comm
en
ded:
Core material: CRNOS
for smalle
r ratin
g
s
, CR
GOS
for larger ratings
.
Con
d
u
c
tor m
a
terial
s: Aluminium for sm
a
ller ratin
g
s, Coppe
r for larg
er ratin
g
s
After ch
oo
sin
g
the
con
d
u
c
tor an
d the
core
materi
al j
udici
ou
sly, our ta
sk is to
cho
o
se
su
ch v
a
lue
s
of
,,
&
wm
KR
B
which giv
e
s minimality
of the obj
ective function without violating
the desi
gn co
nstrai
nts. Parallelly, we ha
ve
to check the iron lo
ss: ohmic lo
ss ra
tio
/
ic
PP
.
4.2. Cons
trai
nts
The de
sig
n
con
s
trai
nts a
ppea
r du
e to st
atuto
r
y rules im
po
se
d by the re
gulatory
authoritie
s o
r
by the custo
m
er. Th
e foll
owin
g co
n
s
traints h
a
ve be
en ide
n
tified for a
distri
buti
on
trans
former [1-2], [4]:
(1)
The efficien
cy should n
o
t fall belo
w
the spe
c
ified limit
(2)
The voltage regulatio
n sh
o
u
ld be kept with
in the sp
ecified limit - so their lea
k
age
rea
c
tan
c
e sh
ould be relati
vely low.
(3)
The m
a
ximu
m allo
wabl
e
temperature
rise
mu
st n
o
t be
exce
ede
d- thi
s
ha
s t
o
be
accompli
sh
ed
by using
co
oling tube
s/
radiators. Fo
rced
co
oling
may have to
be
adde
d.
The de
sign v
a
riabl
es
sho
u
l
d be ch
osen
with a loo
k
to these p
o
ints.
4.3. The Objectiv
e Function
The n
e
xt step
is to
fram
e t
he o
b
je
ctive functio
n
[18,
1
9
]
in terms of
the d
e
si
gn v
a
riabl
es
and othe
r pa
ramete
rs. If the co
st of produ
ction of
the tran
sform
e
r is taken a
s
the obje
c
tive
function, th
e i
r
on
loss and
cop
per lo
ss a
r
e
kept
at their maximum possible
values. Accordingl
y,
the flux density and the current den
sit
y
are kept a
t
their maximum po
ssibl
e
values
wit
hout
violating the desi
gn co
nst
r
aint
s. Only the emf con
s
ta
nt
K
and windo
w height: widt
h ratio
w
R
are
con
s
id
ere
d
to
be
key
varia
b
les. B
u
t if we take into
consi
deration t
he ove
r
-all e
c
on
omy of th
e
cu
stome
r
an
d the ma
nufa
c
ture
r the
n
th
e ru
nnin
g
co
st towa
rd
s lo
st ene
rgy u
n
i
t
s mu
st also
be
inclu
ded in th
e objective fu
nction. The
r
e
f
ore, the
flux
den
sity and the cu
rre
nt de
nsity are also
to
be ch
osen a
s
desig
n varia
b
les to find th
e minimalit
y condition
s for the ch
osen ob
jective functio
n
.
In this pap
er,
we have ta
ke
n both types
of objecti
ve f
unctio
n
s a
nd
made two ca
se-studi
es to
get
a clea
r pictu
r
e.
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 711
4
– 7122
7118
5. Results a
nd Discu
ssi
ons
The co
st optimal desi
gn o
f
a 3-phase oil-f
illed dist
ri
bution tran
sf
orme
r ha
s be
en taken
up usi
ng mod
i
fied Hoo
k
e a
nd Je
eves m
e
thod. Two
case
-stu
die
s
h
a
ve been ma
de viz.
a)
Optimizatio
n
of the cost of
produ
ction, based on the
current
market price of material
s an
d
labou
r.
b)
Dual
optimi
z
ation
with a
look to the
inte
re
st of th
e custom
er
and th
e ma
n
u
facturer- th
e
obje
c
tive fun
c
tion i
s
a
weighted
com
b
ination
of t
he
co
st of p
r
odu
ction
an
d the
pri
c
e f
o
r
annu
al ene
rg
y loss.
The co
mmon
element
s in the two case-studie
s
are gi
ven belo
w
:
Specifications:
KVA-rating of
the machin
e = 100
0
Nomin
a
l po
wer facto
r
= 0.
8; Nominal freque
ncy = 5
0
Hz.
Rated lin
e voltage in L.T. /
H.T.: 433 V/
1100
0 V
Con
n
e
c
tion: Delta/ Star; Condu
ctor m
a
terial: Co
ppe
r
No. of taps =
5; % turns be
tween tap
s
=
2.5
Material
s:
Conduc
tor material: Copper
Heli
cal win
d
in
g has b
een
chosen for L.
T
.
and cross-o
v
er windi
ng for the H.T.
Core materi
al
: Laser-Core;
Stacking fa
ct
or =0.
9
2
3-ste
ppe
d co
re has b
een u
s
ed.
Con
s
trai
nts:
Ef
fic
i
enc
y
0.98
No-loa
d current
1%
Voltage
re
gul
ation
4%
T
e
mp
er
a
t
ur
e
r
i
s
e
40
o
C
The sp
ecifi
c
cost of materia
l
s/BOT unit:
Co
st of copp
er =
Rs. 600/
- per Kg
Co
st of iron = Rs. 150/
- per Kg
Co
st of steel tank
= Rs. 90/- per Kg
Co
st of oil = Rs. 80/
- per li
ter
Co
st of BOT unit = Rs. 4/-
Cas
e
-I
:
I
n
t
h
i
s
ca
se,
t
h
e
c
o
st
f
u
n
c
t
i
on
i
s
t
h
e
s
e
lling
co
st
w
h
ic
h in
clud
es
t
he di
r
e
ct
co
st
for mate
rial
s and
lab
o
u
r
and th
e in
direct
co
st to
ward
s
overh
e
a
d
s. T
w
o
key varia
b
les
which
affect the cost function h
a
ve bee
n identi
f
ied. They are: i) the emf
con
s
tant,
K
; ii) the wi
ndo
w
height: width
ratio,
w
R
. The
minimality co
ndition is
obt
ained fo
r the
following va
lues of
key
variable
s
:
The EMF-
co
nstant,
K
= 0.49366 the
win
dow h
e
ight:
width ratio,
w
R
= 3.881
6. Th
e
followin
g
values have b
een
cho
s
en for t
w
o othe
r de
si
gn variabl
es:
Maximum flux-den
sity in the La
ser-core = 1.55 Te
sl
a
Curre
n
t den
si
ty in the copp
er co
ndu
cto
r
= 3.0 A/mm
2
Higher the values
of these
tw
o vari
ables lower
will be
the co
st of
production. Therefore,
maximum po
ssi
ble value
s
have be
en ch
ose
n
for the
s
e two vari
abl
es
without vio
l
ating the de
sign
con
s
trai
nts. Result
s obtai
n
ed on convergen
ce are:
Dim
ensio
ns:
Curre
n
t in Primary/ Secon
dary, A: 30.303 / 1333.4
Cro
s
s se
ction
of primary/ Seco
nda
ry, mm
2
: 10.101 / 444.46
Numb
er of no
minal turn
s of
the primary
= 704
Numb
er of ad
ditional turn
s
of the primary
for tapping = 36
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Desi
gn of a 3-Pha
s
e Core T
y
p
e
Distrib
u
tion
Tran
sfo
r
m
e
r Usi
ng… (Raj
u Basa
k)
7119
Total numb
e
r of turns of the prima
r
y = 7
40
Numb
er of no
minal turn
s of
the second
ary = 16
Net are
a
of core iron = 4.5
407E-02 m
2
Gro
s
s are
a
of core i
r
on
= 4
.
9355E-02 m
2
Diamete
r
of the co
re ci
rcle
= 0.2734
6 m
Length of the
core sid
e
s in
mm: 247 / 193 / 116
Area of the wi
ndo
w = 0.11
6
63 m
2
Wind
ow h
e
ig
ht/width, m:
0.6728
3 / 0.17334
Dista
n
ce bet
wee
n
co
re ce
nters
= 0.420
82 m
Width/hei
ght of yoke, m: 0.2474
8 / 0.199
43
Total length o
f
core
= 1.14
11 m ; Total
height of co
re
= 1.0717 m
Mean len
g
th of turn of Primary/ se
con
d
a
ry, m: 0.96802 / 1.2403
Re
sista
n
ce of Primary/ Seconda
ry,
Ω
: 1.4168 / 9.376
3
E
-04
The tan
k
leng
th * width * height: 0.561 *
1.414 * 1.22
2
The num
ber
of tubes (50
mm dia.) re
q
u
ired
= 186
Perform
a
n
c
e eval
uation
:
Iron loss
= 26
74 W / % Iron loss= 0.26
74
Cop
per lo
ss = 890
4 W ;
% Coppe
r lo
ss = 0.89
04 ;
Total % loss = 1.1578
Efficiency at full load & 0.8 laggin
g
p.f = 0.9857
3
Maximum efficien
cy of 0.99034 o
c
curs
at a load of 54.8 %
The mag
netizing cu
rrent =
0.5882 %;
Co
re loss
curren
t = 0.26739
%
No loa
d
cu
rre
n
t = 0.6461
4 %
Lea
kag
e
rea
c
tance
= 3.461
8 %
Voltage regul
ation at rated
power & p.f.
= 2.7894 %
Cos
t:
The wei
ght/ cost of tank: 454.57 Kg / Rs. 40911/-
The volume/
co
st of oil: 0 .9690
8 liter / Rs. 77
526/-
Volume of iro
n
= 0.195
28
m
3
; Weight of iron = 14
93.
9 Kg
Co
st of iron = Rs. 224
083/-
Volume of co
pper
= 4.816
734E-02 m
3
; Weig
ht of cop
per = 4
28.69
Kg
Co
st of copp
er =
Rs. 25
72
14/-
Dire
ct co
st all
o
win
g
25 % labou
r ch
arge
= Rs. 74
966
7/-
Selling co
st a
llowing
35% overhe
ad =
Rs. 1012
051/-
Average lo
ad
is a
s
sumed
to be 100% f
o
r 6 h
ours, 7
5
% for 12 h
o
u
rs
and
50%
for 6
hours. Fo
r a life-sp
an of 7
years, the co
st
of lost
units = Rs. 1952
61
3/-
The selli
ng cost plu
s
the cost of lost uni
ts = Rs. 2964
664/-
Table 1. Step
s in Pattern S
earch (val
ue
s at every 10
th
step)
No Cost
K
Gr
-K
R
Gr
-R
Gr
-
B
m
Gr
-B
m
1 2769591/-
0.55
-7.375e
-4
3
-2.757e
-4
2.3
-3.102e
-4
1.4
-5.291e
-4
11
2758500/-
0.574
-3.369e
-4
3.064
-2.016e
-4
2.327
-2.866e
-4
1.419
-2.974e
-4
21
2754217/-
0.5864
-1.258e
-4
3.123
-1.570e
-4
2.356
-2.394e
-4
1.431
-1.597e
-4
31
2752643/-
0.5904
-5.204e
-5
3.163
-1.369e
-4
2.376
-2.002e
-4
1.437
-1.040e
-4
41
2751484/-
0.5921
-1.208e
-5
3.206
-1.203e
-4
2.396
-1.575e
-4
1.442
-6.733e
-5
51
2750777/-
0.5923
2.908e-6
3.240
-1.097e
-4
2.410
-1.254e
-4
1.445
-4.853e
-5
61
2750305/-
0.5920
8.272e-6
3.269
-1.020e
-4
2.421
-1.018e
-4
1.447
-3.781e
-5
71
2749937/-
0.5916
1.045e-5
3.296
-9.500e
-5
2.429
-8.255e
-5
1.448
-3.000e
-5
81
2749613/-
0.591
1.164e-5
3.323
-8.819e
-5
2.437
-6.519e
-5
1.449
-2.328e
-5
91
2749356/-
0.5904
1.146e-5
3.349
-8.822e
-5
2.442
-5.147e
-5
1.450
-1.855e
-5
101
2749147/-
0.5899
1.109e-5
3.373
-7.648e
-5
2.447
-4.056e
-5
1.451
-1.464e
-5
111
2748975/-
0.5893
1.037e-5
3.395
-7.148e
-5
2.451
-3.210e
-5
1.452
-1.173e
-5
121
2748832/-
0.5889
9.822e-6
3.416
-6.657e
-5
2.454
-2.537e
-5
1.452
-9.095e
-6
131
2748712/-
0.5884
9.095e-6
3.435
-6.239e
-5
2.456
-2.001e
-5
1.453
-7.276e
-6
140
2748640/-
0.5881
8.823e-6
3.446
-5.930e
-5
2.457
-1.692e
-5
1.453
-6.185e
-6
141
2748630/-
0.5880
8.459e-6
3.450
-5.921e
-5
2.457
-1.674e
-5
1.453
-6.003e
-6
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 711
4
– 7122
7120
Cas
e
-II
: In this ca
se the co
st function ha
s been ta
ken
as the selli
ng co
st plus capi
talized
co
st for the l
o
st BOT
unit
s
fo
r an
anti
c
ipated life
of
7 years. T
he
key vari
able
s
whi
c
h
affect
the
co
st function
have bee
n id
entified. They are:
i) emf co
nsta
nt,
K
ii) windo
w heig
h
t:widt
h
ratio,
w
R
iii) flux-density,
m
B
iv)
c
u
rr
e
n
t
de
ns
ity,
.
Re
sults
of pattern sea
r
ch
at intermedi
ate step
s ha
ve been
sho
w
n in T
able
1 to sho
w
th
e
probl
em conv
erge
s to a sol
u
tion (G
r-
sta
nds for
gra
d
ie
nts).
The initial values for the d
e
sig
n
variabl
es have be
e
n
cho
s
en eith
er from de
sig
ner’
s
experi
e
nce
or from
de
sig
n
data-boo
k.
It may be noted that the
cost differe
ntia
l come
s d
o
wn rapi
dly at first
and then
slo
w
ly. The sam
e
is true ab
ou
t the individual gradi
ents.
After
140
th
step, ch
ang
es
have be
co
me
very sm
all.
T
he cost
differential ha
s
co
me do
wn
belo
w
Rs. 10
/-. So this point has be
en take
n as
the
point of conv
erge
nce.
At converg
e
n
c
e, we
get the followi
ng variabl
es f
o
r the de
sign
variable
s
:
K
= 0.588;
w
R
= 3.
452;
= 2.45
8 A/mm
2
;
m
B
= 1.453 Te
sla
Based o
n
the
s
e value
s
of variabl
es, the
desi
gn
detail
s
of the optimal machin
e are
given belo
w
:
Dim
ensio
ns:
Numb
er of no
minal turn
s of
the primary
= 572
Numb
er of ad
ditional turn
s
of the primary
for tapping = 28
Total numb
e
r of turns of the prima
r
y = 6
00
Numb
er of no
minal turn
s of
the second
ary = 13
Curre
n
t in Primary/ Secon
dary: 30.303
/ 1333.4 A
Cho
s
e
n
cu
rre
n
t density = 2
.
4575 A/mm
2
Cro
s
s se
ction
of primary/ seco
nda
ry, mm
2
: 12.331 /
542.6
Net are
a
of core iron = 5.9
6078
8E-0
2 m
2
Gro
s
s are
a
of core i
r
on
= 6
.
47911
8E-0
2 m
2
Diamete
r
of the co
re ci
rcle
= 0.3133
2 m
Length of the
core sid
e
s in
mm: 284; 222; 133
Area of the wi
ndo
w = 9.63
9
534E-02 m
2
Wind
ow h
e
ig
ht/width ratio
= 3.452
Wind
ow h
e
ig
ht/width, m:
0.5768 / 0.16
71
Dista
n
ce bet
wee
n
co
re ce
ntres
= 0.450
67 m
Width/hei
ght of yoke, m: 0.28355 / 0.22
850
Total length/ height of co
re
, m: 1.2444 /
1.0338 m
Inside/o
u
tsid
e diamete
r
of L.T. winding,
m: 0.319 / 0.375
Inside/o
u
tsid
e diamete
r
of H.T. windin
g
, m: 0.405 / 0.455
Mean len
g
th of turn of pri
m
ary/ se
con
d
a
ry, m: 1.089324 1.35
183
2
Re
sista
n
ce of Primary/ Seconda
ry: 1.0611
; 6.8015E-04
The tan
k
leng
th, width * height: 0.591 * 1.512 * 1.184
The num
ber
of tubes (50
mm dia.) re
q
u
ired
= 141
Perform
a
n
c
e eval
uation:
Iron loss
= 28
54.4 W; % Iron loss = 0.28
54
Cop
per lo
ss = 655
0.4 W; % Coppe
r lo
ss = 0.65
51 ;
Total % loss
= 0.940
5
Efficiency at full load and
0.8 lagging p.f. = 0.9884
Maximum efficien
cy of 0.9914 o
c
curs at a % load of 66.01
The mag
netizing cu
rrent =
0.5466 %; Th
e
core loss current = 0.28
54 %;
The num
ber l
oad current = 0.6167 %
The % leakag
e rea
c
tan
c
e = 2.886
The % voltage regul
ation a
t
rated power
& p.f = 2.2557
Cos
t:
The wei
ght / co
st of tank: 360.6 / Rs. 3
2454/-
The volume / co
st of oil: 1.0569 / Rs. 84
554/-
Volume of iro
n
= 0.251
5 m
3;
Weight of iron = 19
24 Kg
; Cost of iron
= Rs. 25012
0 /-
Volume of co
pper
= 5.278
595E-02 m
3;
Weig
ht of cop
per =
469.8 kg.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Optim
a
l Desi
gn of a 3-Pha
s
e Core T
y
p
e
Distrib
u
tion
Tran
sfo
r
m
e
r Usi
ng… (Raj
u Basa
k)
7121
Co
st of copp
er =
Rs. 28
18
77/-
Dire
ct co
st all
o
win
g
25 % labou
r ch
arge
= Rs. 811
25
6/-
Selling co
st a
llowing
35 % overhe
ad =
Rs. 1095
196/-
Average lo
ad
is a
s
sumed
to be 100% f
o
r 6 h
ours, 7
5
% for 12 h
o
u
rs
and
50%
for 6
hours.
The lost BOT
units/ann
um = 590
77
For a life-spa
n
of 7 years, the co
st of lost units = Rs. 1654
168/-
The co
st fun
c
tion = Rs. 27
4936
4/-
A comparison of the cost functions of ca
se-I and case-II, reveals
that dual opti
m
ization
is more gainf
ul.
6. Comparison
w
i
th other Methods
There a
r
e
se
veral m
e
thod
s fo
r o
p
timizi
ng a
d
e
si
gn
probl
em. T
h
e
ea
sie
s
t met
hod i
s
based on ex
hau
stive sea
r
ch. It is simpl
e
and st
rai
g
h
t
-forward
but it takes la
rge
comp
uter ru
n-
time, particul
a
rly if the
steps
ch
ose a
r
e sm
all.
The
r
e are meth
o
d
s b
a
sed
on
ran
dom
se
a
r
ch,
gradi
ent
sea
r
ch
and
patte
rn
sea
r
ch. T
he rand
om
search te
ch
ni
que d
o
e
s
n
o
t
give the ex
ac
t
s
o
lution; it gives
a
s
o
lution c
l
os
e to it.
The
c
l
osene
ss de
pen
ds
on
no of iteratio
ns. Th
e gradi
ent
serach te
chn
i
que
s a
r
e
go
od; they yiel
d the
sol
u
tio
n
in
small
e
r no
of ste
p
s. Ho
wever with
gradi
ent
sea
r
ch, the
r
e
is a
ch
an
ce
of b
e
ing t
r
app
ed
in lo
cal
mini
ma, provided
the
hypersu
rface
in the
serach
spa
c
e
is
no
t con
c
ave. T
he g
r
adi
ent
serach meth
ods
are effe
ctive but a
r
e
not
comp
utationa
lly efficient. On the oth
e
r hand, t
he di
rect se
ra
ch method
u
s
e
s
an
optimi
z
at
ion
algorith
m
ba
sed on o
n
ly the functio
n
values, not
the
gradi
ents. T
h
e evol
utiona
ry method, the
simplex
sea
r
ch meth
od, Hook a
nd
Jeev
es patte
rn
se
arch metho
d
, Powell’
s co
n
j
ugate di
re
ction
method et
c. falls in thi
s
ca
tegory. The
s
e method
s
are co
mputatio
nally more ef
ficient compa
r
ed
to the gradie
n
t
-based meth
ods.
7. Conclu
sion
This p
ape
r h
a
s de
alt with
desig
n opti
m
ization
based on mo
dified Ho
ok a
n
d
Jeeve
s
method.
T
h
is
metho
d
i
s
based on pa
ttern sea
r
ch
appli
ed
to
a
p
r
ope
rly chosen obje
c
t
i
ve
function. It i
s
, in e
s
sen
c
e,
a combin
atio
n of an
explo
r
atory m
o
ve
and
a patte
rn
move to
qui
ckly
rea
c
h
t
he o
p
timality
(in this ca
se mi
nimality) crit
erion.
The
d
e
sig
n
vari
abl
es are cho
s
en
according to t
he obj
ective f
unctio
n
an
d b
ound
s a
r
e im
posed o
n
it to define the
se
arch spa
c
e. A
n
initial point i
s
cho
s
e
n
in t
he sea
r
ch sp
ace. In th
e e
x
plorato
r
y m
o
ve, a lo
cal
sea
r
ch i
s
ma
de
arou
nd thi
s
p
o
int to find o
u
t
the be
st poi
nt aro
und th
e
cu
rre
nt point
. Two
su
ch
p
o
ints a
r
e
use
d
in
the pattern m
o
ve in the original wo
rk of Ho
o
k
and
Je
eves, but we
have use
d
on
ly one base
d
on
gradi
ent alo
n
g
with an a
c
cele
ration fa
ctor for fa
ste
r
conve
r
ge
nce.
Prov
ision
ha
s bee
n kept for
redu
cin
g
step
-length fo
r exactly rea
c
hin
g
the minimal
point.
In this
work
, two case-s
tu
dies
have be
en mad
e
on
the sam
e
tra
n
sformer
de
sign. The
machi
ne i
s
an oil-filled di
stribution transform
er of rating 11000/433 V,
50 Hz.,
1000 KVA, with
5% additio
nal
turn
s fo
r ta
p
p
ing’
s o
n
the
H.V. sid
e
.
In
the first case, the o
b
je
ctive functio
n
i
s
th
e
co
st of pro
d
u
c
tion a
nd in t
he second it i
s
a wei
ghte
d
sum of the
cost of
produ
ction and the l
o
st
energy unit
s
.
In the
first
case, th
e mi
ni
mal
co
st of
prod
uctio
n
h
a
s
bee
n fou
n
d
out
to b
e
Rs.
1012
051/-. It
is le
ss th
an
that for the
se
con
d
which is
Rs. 1
0
9
5196/-. But t
he p
r
ice
for l
o
st
energy units for an estima
ted life of 7
years i
s
Rs. 1952
613/- for the first case. This is mu
ch
more tha
n
that in the second case whi
c
h is
Rs. 16
5
4168/-. Fo
r a
saving of a small am
ount
of
Rs. 83
145/- i
n
the co
st of prod
uctio
n
, the additional
cost of lost en
ergy units i
s
Rs. 29
844
5/-. So
the advantag
e of lower
co
st of prod
ucti
on is tota
lly offset due to additional lo
sses. Thi
s
prov
es
that the obje
c
tive function
sho
u
ld be fra
m
ed for d
ual
optimizatio
n
of cost of p
r
o
ductio
n
and l
o
s
t
energy units.
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ces
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w
n
e
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l
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c
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i
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