TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 8, August 201
4, pp. 5827 ~ 5846
DOI: 10.115
9
1
/telkomni
ka.
v
12i8.555
6
5827
Re
cei
v
ed
Jan
uary 1, 2014;
Re
vised Ap
ril
8, 2014; Accepted Ma
y 4, 2014
An Overview of Electrical Tree Growth in Solid
Insulating Material with Emphasis of Influencing
Factors, Mathematical Models and Tree Suppression
M.H. Ahm
a
d
*
, N. Bashir, H. Ahm
a
d, A
.
A. Abd Jam
i
l, A.A. Suleim
an
Institute of Hig
h Voltag
e an
d High C
u
rre
nt, Univers
i
ti T
e
knolo
g
i Mal
a
ysia
P06, F
a
cult
y
of Electrical En
gi
neer
ing,
8
131
0
,
Johor Bahru,
Johor, Mal
a
ysi
a
.
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: mohdh
afizi@
fke.utm.my
A
b
st
r
a
ct
Now
adays, t
h
e
most w
i
de
ly
u
s
ed
insu
lati
ng
mater
i
als
i
n
hi
gh v
o
lta
ge
eq
u
i
p
m
e
n
t suc
h
a
s
cab
l
es
are
poly
m
eric
i
n
sul
a
tions
d
u
e
to th
e n
u
m
er
o
u
s
mer
i
ts they
possess
w
i
th r
egar
ds to
el
ect
r
ical
perfor
m
an
ce
compar
ed t
o
pap
er i
n
su
lati
ons. H
o
w
e
ver
,
electric
al
tr
eei
ng,
one
of
the
di
electri
c
pre-
break
do
w
n
phe
no
me
na,
h
a
s be
en c
ons
i
dere
d
as
a
ma
jor co
ntrib
u
tion
to the fa
ilur
e
of insu
lati
ng
p
o
ly
meric
mater
i
als.
T
hus, this p
a
p
e
r prov
id
es a
n
overvi
ew
on t
he factors
affe
cting th
e i
n
itiati
on a
n
d
pro
p
a
g
a
tion
of e
l
ectri
c
al
tree. Discuss
i
ons o
n
p
a
ra
meters that affe
ct the gr
ow
th
of electric
al tr
eei
ng suc
h
as
app
lie
d vo
lta
ge,
electric fiel
d e
nha
nce
m
e
n
t, partial disc
har
g
e
, frequen
cy o
f
appli
ed volta
ge an
d tempe
r
ature are giv
e
n
.
Some d
i
scussi
ons o
n
the var
i
ous
mo
de
ls ar
e als
o
hi
ghl
ig
h
t
ed. T
he tree r
e
late
d
mod
e
ls
bei
ng d
i
scuss
e
d
here
i
n
incl
ud
e
W
e
ib
ull,
Lo
g
nor
mal,
Joh
n
s
on SB,
Di
elec
tric Break
dow
n Mo
del,
Disc
harg
e
-Aval
anc
h
e
Mode
l, an
d Fie
l
d-Driv
en Tre
e
Grow
th. In add
ition, d
i
scu
ssi
o
n
s on
the
use
of na
no-si
z
e
d f
illers
in
po
ly
me
ric
insulating materi
al to in
hib
i
t el
ectric treein
g
a
r
e hig
h
li
ghte
d
.
Ke
y
w
ords
: ele
c
trical treei
ng, math
e
m
atic
al
mo
de
ls, partial
disch
arge, tree
suppress
i
on
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
Polymeric insulation mat
e
rial
s are
sti
ll wi
dely
used by power
ut
ilities as the base
material fo
r t
heir
pro
d
u
c
tion such a
s
n
a
tural
ga
s is
still availabl
e
in ab
unda
nce. Furthe
rmo
r
e,
power utilities are replaci
n
g their impregnated insu
lated cabl
es with
polymeri
c
insulation, namely
cro
s
s-li
nked
polyethylene
(XLPE) an
d
gene
ric m
a
te
rials
su
ch a
s
epoxy re
sin,
silico
ne rub
ber,
ethylene pro
p
ylene dien
e
mono
mer
(EPDM) and
etcetera due
to
thei
r su
perio
r ele
c
tri
c
al
perfo
rman
ce
s su
ch a
s
hig
h
magnitud
e
of
dielect
r
ic
strength in MV/cm, very
low di
electri
c
lo
sse
s
,
high ten
s
ile strength a
nd resi
stan
ce to electri
c
al d
e
g
r
adatio
n phe
nomen
a [1]. Ho
wever in t
h
e
long
run,
pol
ymeric mate
rials
experi
e
n
c
e
agei
ng
an
d finally di
ele
c
tri
c
b
r
ea
kd
o
w
n. O
ne
of t
h
e
main ca
uses of breakd
o
wn in polymeri
c
insul
a
tion
s is elect
r
ical treein
g
[2]. And in this view,
contin
ual efforts a
r
e ma
de to und
erstand a
nd
chara
c
te
rize e
l
ectri
c
al tre
e
i
ng me
cha
n
isms.
Treei
ng is o
b
s
erve
d to ori
g
inate at poi
nts wh
ere im
puritie
s, gas
voids, mecha
n
ical d
e
fect
s, or
con
d
u
c
ting projectio
n
s
cau
s
e excessive
local el
ectri
c
a
l
field stresse
s
within
small
region
s of the
diele
c
tric. Au
ckl
and
et al. [
3
] ch
ara
c
te
rized ele
c
tri
c
al t
r
eein
g
a
s
la
b
y
rinthi
ne stru
cture
s
of
na
rrow
gas-filled tub
u
les which created by
loca
lized pa
rtial d
i
scharge a
c
tivity. Barclay et al. [4] divided
the p
r
ocess o
f
elect
r
ical
tre
e
in i
n
sulatio
n
sy
stem
s
int
o
three pha
ses whi
c
h
are ince
ption pha
se,
prop
agatio
n pha
se and
compl
e
tion
p
hase.
Th
ey descri
bed
th
e
in
ceptio
n pha
se as
th
e
undete
c
tabl
e
damage a
c
cumul
a
ted at
pre-existing
defects
whi
c
h in
cre
a
ses the electri
c
field
locally. At the
pro
pag
ation
pha
se, a
bran
chin
g det
e
r
io
ration
stru
cture o
c
curs fro
m
the defe
c
t a
n
d
spread o
u
t acro
ss the die
l
ectri
c
. The completion ph
ase o
c
curs whe
n
the ga
p betwee
n
the
electrode
s is
bridg
ed by the electri
c
al tr
ee due to the
enlargem
ent
of the discharge.
Furthe
rmo
r
e,
Zhen
g et
al. [5] mention
e
d
t
hat tree p
r
opag
ation i
s
categ
o
ri
zed
i
n
thre
e
prima
r
ily sta
g
e
s,
initiation,
stag
nation
a
nd
rapi
d
p
r
op
agating
ph
ases. If initiatio
n
ph
ase i
s
v
e
ry
active,
the si
ngle bra
n
ch
t
r
ee will pro
p
agate,on
the other han
d
if
this pha
se
i
s
wea
k
th
en the
bush tree wil
l
occu
r more
easily. Meanwhil
e
, Wu e
t
al.
[6] characteri
ze
d the
developme
n
t
o
f
electri
c
al tree
ing as tre
e
ini
t
iation and tre
e
gro
w
th.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 582
7 –
5846
5828
The ele
c
tri
c
al
tree shap
es
can
be rough
ly
characte
rized a
s
bran
ch
, bush
-
b
r
an
ched an
d
bush-sha
ped
structu
r
e
s
. Ding et al. [7] visualiz
ed e
l
ectri
c
al tree
prop
agatio
n as a process of
intense locali
zed el
ectri
c
al
degradatio
n, whi
c
h lea
d
s t
o
the creation
of new
tree
chann
els, by the
formation of sub
-
mi
cro
s
co
pic voids. Du
ring this
ph
a
s
e, elect
r
ical tree gro
w
s in
the function of
time by following the dire
ction of elect
r
ic fiel
d. Bre
a
kd
own occu
rs qui
ckly after the tree h
a
s
bridged the ins
u
lation [8].
The insulatio
n
lifetime depend
s upo
n the elect
r
ical tree growth. In the laboratory, the
needl
e-pl
ate electrode i
s
u
s
ually u
s
ed t
o
investigat
e
the tree g
r
o
w
th in insul
a
tin
g
material. T
he
needl
e is used to enha
n
c
e the ele
c
tri
c
field aro
u
n
d
the needl
e
tip region. This cau
s
e
s
the
electri
c
al
tre
e
to g
r
ow fro
m
the n
eedl
e ti
p (lo
c
al
st
re
ssed
re
gion
) t
o
wa
rd
s the
o
ppo
site el
ectrode
and he
nce the tree failure time is dete
r
m
i
ned a
s
the in
sulatio
n
lifetime [9].
Thus, d
ue to severe effect
s of elect
r
ical
tr
ee on the i
n
sul
a
tion sy
stem
, state-of
-art study
has b
een p
e
rformed to u
n
derstand
cle
a
r
ly the tree m
e
ch
ani
sms
a
nd this p
ape
r has ta
ken in
to
accou
n
t to
re
view
som
e
of
the fa
ctors that infl
ue
nce
the ele
c
tri
c
al
tree i
ndu
ction
and
g
r
o
w
th i
n
polymeri
c
in
sulating mate
rials an
d exa
m
ines the
e
ffects of ap
pli
ed voltage, local el
ectri
c
f
i
eld
enha
ncement
, partial
discharg
e
, fre
q
u
ency
of ap
pl
i
ed voltag
e, a
nd temp
eratu
r
e o
n
el
ect
r
ical
treeing. Be
sid
e
s, several m
a
thematical
model
s h
a
ve
been
revie
w
e
d
in thi
s
pa
pe
r. The
s
e i
n
clu
de
Weib
ull m
o
d
e
l, logn
orm
a
l
mod
e
l, Joh
n
so
n SB m
o
del, Di
ele
c
tri
c
Brea
kdo
w
n
Mod
e
l
(DB
M
),
Disch
a
rg
e A
v
alanche M
o
del (DAM
) a
nd Fiel
d-Driv
en T
r
ee
Gro
w
th Mo
del
(FDTG
)
.
De
spite
nume
r
ou
s
st
udie
s
o
n
ele
c
tri
c
al tree,
a full u
nde
rstanding
of th
is p
hen
omen
on i
s
yet to
be
achi
eved d
u
e
to complexit
y
and va
riou
s factors. In
vi
ew
of the fo
regoin
g
, this
p
aper attempt
s
to
give a gene
ra
l understan
di
ng of electri
c
al tree studi
e
s
throu
gh this review.
2. Electrical Tree Forma
ti
on Mecha
n
is
ms
In literature, electri
c
al t
r
ee
is initiated
b
y
cha
r
ge
ca
rriers inje
ction
and extra
c
tio
n
from
embed
ded
el
ectro
d
e
into t
he p
o
lymer.
The
ch
arg
e
carri
ers
are
cl
assified
as el
ectro
n
, h
o
le
and
ions. T
h
e
s
e
cha
r
ge
carrie
rs
are recog
n
ize
d
a
s
“sp
a
ce
charge”
from the
con
t
ext of dielectric
material. T
h
e
y
are al
so
kn
own to
move
arou
nd
on th
e diele
c
tri
c
m
a
terial
by the
electri
c
fiel
d
and
become
trap
ped i
n
th
e b
u
lk
of
materi
al
[10].
Howe
ver, electro
n
is con
s
id
ered
or assume
d
as
cha
r
ge
carrie
r in
the m
o
st
situatio
ns.
Und
e
r
neg
ative half
cycle
of AC volta
ge, a
num
be
r of
electron
s will
be inje
cted
into the poly
m
er mate
rial
from an el
e
c
trod
e for
a sho
r
t dista
n
ce.
Meanwhile, during the posi
tive half cycle of AC vo
ltage, the electron will
be extracted backwards
into the poly
m
er mate
rial.
This inj
e
ctio
n and ex
tracti
on process
will be re
peate
d
duri
ng n
e
g
a
tive
and po
sitive half
cy
cle of AC
voltag
es. Duri
ng
th
e po
sitive o
r
n
ega
tive half cy
cle
,
these
ele
c
trons
gain suffici
en
t energy to form polyme
r
decom
po
sition or to attack p
o
lymer chain in ord
e
r to
initiate chem
ical re
actio
n
to cause p
o
lyme
r de
gradation. The
polymer de
grad
ation co
uld
eventually
form
a hollo
w chann
el
that will result
in
the
ga
se
s di
scharge
s,
a
s
a tree
sta
r
ts to
initiate [11].
More
over, el
ectri
c
al tre
e
can be initiate
d by mech
ani
cal fatigue d
u
e
to Maxwell
stre
ss.
In detail
s
, a
crackin
g
of th
e
polyme
r
is p
r
odu
ce
d
due
to
the high M
a
xwell com
p
ressive stre
ss a
t
the electrode
tip under AC
voltage. Thus, electrical tre
e
will start to initiate from this crack du
e to
gas di
scharg
e
. On th
e oth
e
r h
and,
ele
c
trical t
r
ee
ca
n be
initiated
by small
cav
i
ty or void
wh
ich
coul
d eventually leads to the PD occurrence. As
a result, this void
will
attract the fi
eld
enha
ncement
to locally stress on it
sel
f
becau
se it
is filled with gases
whi
c
h ha
s a lo
wer
permittivity compa
r
ed
with
the
polymer
material
which
ha
s high
er permittivity.
The
el
ect
r
ic
fie
l
d
will ioni
ze the gases and thereby fo
rces the void to discharge [12].
3. Factor
s Affec
t
ing the G
r
o
w
th o
f
Electrical Tre
e
ing
This se
ction
discusse
s the
factors
su
ch a
s
the applie
d voltage, ele
c
tric field
enha
ncement
, partial di
scharg
e
, freq
ue
ncy, and te
m
peratu
r
e th
at enha
nce the
elect
r
ical
tre
e
formation me
cha
n
ism
s
.
3.1. Applied Voltage
Nume
ro
us studie
s
have shown
t
hat the initiation a
nd growth of
electri
c
al tre
e
ing is
affected
by the mag
n
itude
of the voltage
applie
d to
th
e polyme
r
ic
material
wh
ether i
n
DC, A
C
o
r
impulse volta
ge [13-20]. T
h
is i
s
pri
m
aril
y beca
u
se
when hi
gh volt
age i
s
ap
plie
d to insulatio
n
, the
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Ove
r
vi
ew
of Electrical T
r
ee G
r
o
w
th in
Solid Ins
u
lating Material with… (M.H. Ahmad)
5829
local ele
c
tri
c
field near the defe
c
t site is enha
nce
d
. If
the ord
e
r of magnit
ude exceed
s the
maximum ele
c
tri
c
field strength of the
material,
initi
a
l dama
ge would sta
r
t to indu
ce ne
ar t
he
defec
t s
i
te [21].
Maruyam
a
et al. [22] reported that el
ectr
i
c
al tree
cha
nge
s fro
m
bran
ch
-type to the
den
se bu
sh
-t
ype tree with
the increa
se
of appli
ed v
o
ltage. In ad
dition, Den
s
l
e
y [23], Noto and
Yoshim
ura
[2
4]observed
th
at the b
r
an
ch
-type tre
e
s grew
at lo
wer voltage
s
while
the b
u
sh-typ
e
trees g
r
e
w
at
high
er volta
g
e
s. Be
side
s,
Du
et
al. [25]
perfo
rmed
ex
perim
ents on
sili
con
e
rubb
er
(SiR) to i
n
ve
stigate el
ectri
c
al tre
e
ph
en
omeno
n. Based on th
eir
result
s, AC vo
ltage was fo
u
n
d
eage
r to initi
a
te the el
ect
r
ical
tree
co
mpared
wi
th
DC volta
ge.
Electri
c
al tre
e
grew
at lo
nger
length u
nde
r
elevated AC
voltage while
unde
r DC,
sh
ort tree
pro
p
a
gation
wa
s o
b
se
rved. Fig
u
r
e
1 sho
w
s the
stron
g
co
rrel
ation b
e
twe
e
n
ele
c
tri
c
al
tree le
ngth
an
d ap
plied
voltage i
n
n
eat
room
temperature vulcani
ze
d
(RTV) silicone
rubbe
r.
Figure 1. Gro
w
th Ch
ara
c
te
ristics of Electrical
T
r
ee in
SiR unde
r Dif
f
erent Type
s of Voltage [25]
In details, du
ring the po
sit
i
ve and nega
tive hal
f cycles of AC voltage, the ch
arges a
r
e
injecte
d
and
extracted into
the polymer, thereby lead
to the initiation of electri
c
al tree. At some
portion
s, the i
n
jecte
d
cha
r
g
e
s a
r
e
retain
ed in
deep
traps
du
ring th
e su
bsequ
ent
half-cycle
wh
ich
the co
alesce
nce of the l
u
mine
sc
ent
gives ri
se to
the elect
r
ol
umine
s
cen
c
e
.
The amou
nt of
injecte
d
cha
r
ges is an
in
d
i
cator of th
e
intensit
y of
el
ectrol
umin
escen
c
e
an
d th
e light i
n
ten
s
i
t
y
cha
nge
s with
the values
of applied vo
ltage. M
ean
while, un
der
DC voltag
e, the indu
ction
of
electri
c
al
tre
e
is
ca
use
d
b
y
the hom
ocharg
e
s
which
mode
rate
th
e stress
enh
ancement
at
the
injectio
n poin
t. In
addition
, the electro
n
avalanche
cau
s
e
s
the tree to pro
p
a
gate insid
e
the
polymer un
d
e
r
DC voltag
e ap
plication.
In othe
r
wo
rds,
the i
n
je
cted an
d extracted
ele
c
tro
n
s
gene
rate a first ch
ann
el. Heterocha
r
ge
which is de
posite
d
on the cha
nnel wall is swept out
thereby
tran
sferrin
g
m
a
ximum
stre
ss t
o
the
chan
ne
l re
gion
which may
gen
erate an
othe
r
new
tree ch
ann
els by electron
avalan
che re
petition.
Und
e
r impul
se voltage
s whi
c
h have risen
and
fall times of a few to several hu
ndred
micro
s
e
c
on
ds, more ele
c
trolu
m
ine
scence pulses
are
emitted. Thi
s
is d
ue to
inje
ction
of ele
c
tron d
u
ri
n
g
the
neg
ative imp
u
lse
into th
e
polymer with
the
combi
nation
of trappe
d charg
e
s. Th
u
s
, more li
gh
ts being
em
itted as a result of mo
re
electrolumi
n
e
s
cen
c
e. In ad
dition, repe
ated impul
se
of
the same pol
arity allows the spa
c
e
cha
r
ge
to being di
ssipated fro
m
the tree
ch
an
nel tips. Thi
s
woul
d re
sult
in the tree
prop
agatio
n
by
activating all the tree
chan
n
e
l tips and th
ereby
could d
e
velop the tre
e
bran
ch
es [1
2].
3.2. Electric Fields Implication
Diverg
ent hig
h
fields have
been re
po
rted as o
ne of
the cau
s
e
s
of electri
c
al t
r
ee [26].
Maso
n [27] reporte
d that t
he ele
c
tri
c
fie
l
d, E
depe
nd
s on th
e ap
pli
ed voltage, pi
n-tip radiu
s
a
nd
pin-pl
ane
sep
a
ration di
stan
ce a
c
cordi
ng
to the followin
g
formula;
0
20
40
60
80
10
0
12
0
0
0.
2
0.
4
0.
6
0.
8
1
G
r
o
w
t
h
T
i
me,
mi
n
El
ec
t
r
i
c
a
l
Tr
ee Le
n
g
t
h
,
m
m
6kV
,
50
H
z
8kV
,
50
H
z
10
kV
, DC
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 582
7 –
5846
5830
2
4
ln
V
E
d
r
r
(
1
)
Whe
r
e,
d
i
s
t
he di
stan
ce
from the
nee
dl
e tip to
plan
e
electrode,
r
i
s
the n
eedl
e ti
p radiu
s
and
V
a
s
the
applie
d volta
ge. Th
e m
a
ximum el
ect
r
ic
field was calculated
by con
s
ide
r
ing
r
a
nd
d
as de
picte
d
in Figure 2.
The
size of
n
eedle
tip radi
us
plays a
si
gnifica
nt influ
ence o
n
the
i
n
itiation of
el
ectri
c
al
treeing
due to
electri
c
field
enha
ncement
[28]. Bas
ed
on Ma
son’
s f
o
rmul
a (Equ
a
t
ion (1
)), ele
c
t
r
ic
field is inversely pro
porti
onal to the needl
e tip
ra
dius. The
r
efo
r
e acco
rdin
g
to Equation (1),
smalle
r tip ra
dius would re
sult in highe
r electri
c
fi
eld whi
c
h is sufficient to initiate the electri
c
al
treeing.
Figure 2. Simple Dia
g
ra
m of Needl
e Ti
p
with its Co
rre
spo
ndin
g
Tip Radi
us
Cham
pion et al. [29] report
ed that the incept
io
n of electri
c
al tree d
epen
ds p
r
im
arily on
the ele
c
tri
c
fi
eld at th
e
pin
-
tip, the
qualit
y of
the
poly
m
er
ele
c
trod
e inte
rface a
nd the
di
strib
u
tion
of the discrete ch
arg
e
tran
sfer
sites
at the pin
su
rface. In addition,
No
skov
et al. [30] pointed
out
that the ele
c
t
r
ical
tree
gro
w
th is gove
r
n
ed by
the
ele
c
tri
c
field a
n
d
the dam
age
accumul
a
tion
in
the dielect
r
ic material. Th
e damag
e value
s
we
re found to be
prop
ortio
nal to the amoun
t of
energy being
dissipate
d
in the tree
chan
n
e
l.
Like
wi
se, in
[31] the auth
o
rs
have
me
ntioned th
at
the ele
c
tri
c
field was re
qu
ired to
initiate void discha
rge
s
which m
a
y ca
use
elec
t
r
ical
tree to initia
te and the
r
e
b
y prop
agate
in
insul
a
tion bul
k materi
als. In ca
se of el
ectri
c
al tr
e
e
ince
ption, the
inceptio
n time of electri
c
al
treeing
de
pe
nds upo
n the
inten
s
ity of the lo
cali
zed
electri
c
fiel
d
whi
c
h i
s
exp
e
rien
ce
d at t
he
centre of the
needl
e tip su
rface [3
2]. In anothe
r
stu
d
y
by [33], it
wa
s found
ou
t that prolon
g
e
d
exposure
to
electri
c
fields of value
s
hig
her t
han
5 to
10kV/mm
wo
uld le
ad to
fo
rmation
of voi
d
s
or mi
cro
c
aviti
e
s in th
e pol
ymer (PE).
With time
, th
ese mi
cro
c
av
ities may unit
e
to form bi
g
ger-
sized
cavitie
s
in
whi
c
h P
D
wo
uld
occu
r and
initiate t
he el
ect
r
ical
t
r
eein
g
. Howe
ver, an
ele
c
tric
field with
value mo
re
than
200
kV/mm i
s
requi
re
d to
initiate an
ele
c
tri
c
al tree
which
this leve
l is
calle
d light in
ceptio
n level.
The a
pplied
electri
c
field
gives
sufficie
n
t ene
rgy to the fre
e
ele
c
trons
insid
e
the
voi
d
s
whi
c
h
may
ca
use th
e m
a
terial
to io
ni
ze
due
to i
m
p
a
ct o
n
th
e
wa
lls of
the voi
d
s.
The e
nergeti
c
ele
c
tron
s capture
the th
ermal
ele
c
tro
n
s
whi
c
h
re
sult in the fo
rmation of fre
e
radi
cal
and
t
hus lea
d
to
the
chai
n
sci
ssion.
Th
e
r
efo
r
e, mo
re
mi
crovoids
would
form
an
d
co
uld
coal
esce to form large
r
voids which lead
to
the PD occurre
n
ce and
tree to be ind
u
ce
d [12].
3.3. Partial Discharge Im
plication
A partial discharg
e
is defin
ed as
“localized ele
c
trical discha
rge tha
t
only partially bridg
e
s
the in
sulatio
n
between
con
ducto
rs an
d
whi
c
h m
a
y o
r
may not
o
c
cur a
d
ja
cent to
a
con
d
u
c
tor” by
referring to I
E
C 602
70: 2
00 sta
nda
rd
(BS EN 60
27
0:2001
). It appea
rs
as
a
pulse which
has
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Ove
r
vi
ew
of Electrical T
r
ee G
r
o
w
th in
Solid Ins
u
lating Material with… (M.H. Ahmad)
5831
duratio
n le
ss
than 1µ
s. PD is
norm
a
lly classified a
s
coron
a
, surfa
c
e, and
intern
al di
scharge
s. In
definition, corona di
scha
rg
e is
a form
o
f
PD t
hat o
c
curs in g
a
seo
u
s m
edia
aro
und
con
d
u
c
to
rs
whi
c
h a
r
e
re
mote from
so
lid or li
quid
in
sulatio
n
[34].
Surface di
scharg
e
i
s
defi
ned a
s
a type of
PD that occu
rs o
n
the
su
rface
of a diele
c
tri
c
ma
teri
al
and inte
rnal
discha
rge i
s
a disch
a
rg
e that
occurs from the cavities o
r
inclu
s
ion
s
which
origi
nally
exist insid
e
a diele
c
tri
c
m
a
terial. It is al
so
in the fo
rm
of
ch
ann
el
crea
ted du
e to
hig
h
ele
c
tri
c
fiel
d
stressin
g. Electri
c
al
treei
n
g
is con
s
ide
r
ed
as an inte
rnal
discharge o
c
currin
g insi
de
an insul
a
tion
material [35].
The initiatio
n
of elect
r
ical tree sta
r
ts fro
m
the
void i
n
the insulation
bulk.
Thu
s
, it attract
s
the electri
c
field to enhan
ce at itself beca
u
se
it is filled with the
gase
s
whi
c
h have a lower
permittivity over the
re
st
o
f
the diel
ect
r
i
c
. It then
wo
uld le
ad to
th
e ioni
zatio
n
o
f
the ga
se
s a
n
d
cau
s
e the voi
d
to discha
rg
e. However, i
f
the
locali
ze
d avalan
che
s
do not ca
use bre
a
kdo
w
n
it
woul
d cau
s
e
partial
discha
rge to
occu
r.
The el
ect
r
ic
field for the
ince
ption of
p
a
rtial di
scha
rge
depe
nd
s o
n
t
he void
si
ze
a
nd it
rang
es from
~3
kV/mm
to mo
re th
an
1MV/mm fo
r
void si
ze
s fro
m
1mm to micrometre
-si
z
e
d
voids [12]. This ele
c
tri
c
field value
s
are in line with
the electri
c
field
values
whi
c
h
are requi
red t
o
gene
rate el
ectri
c
al
tre
e
in
g mentione
d in the previou
s
su
bsectio
n
.
No
skov et al. [30] rep
o
rted
t
hat partial
di
scharge a
c
tivity in
fluence
d
the
formatio
n of
new
tree ch
ann
els due to the enhan
cem
ent of local ele
c
tric fields ne
ar the chann
els. Based on the
PD mea
s
u
r
e
m
ent, it wa
s found th
at the
partial
disch
a
rge
mag
n
itu
de (in
p
C
) in
cre
a
sed
with
the
increa
se
of the tre
e
len
g
th and
the n
u
m
ber
of
tree
bran
ch
es.
Ch
ara
c
teri
stic fe
ature
s
b
a
sed
on
partial di
scha
rge a
nalysi
s
and opti
c
al o
b
se
rvat
ion
co
uld identify the type and
the seve
rity o
f
treeing [36].
Electri
c
al tre
e
is visibl
e i
n
the t
r
an
sp
arent
materi
als
su
ch
a
s
epoxy re
sin,
sili
con
e
rubb
er, p
o
lye
s
ter
re
sin, p
o
lyuretha
ne,
P
MMA
and polyca
r
bo
nat
e.
The
tree gro
w
th can be
monitored u
n
der
micro
s
co
pic vide
o ima
ges.
Ho
weve
r, for
opaq
ue
materi
als li
ke XLPE, LDP
E
,
HDPE, EVA,
PVC and PE, electri
c
al tree inception v
o
ltage
can
be measured by
measuri
ng t
h
e
PD occu
rrin
g
at the tree gro
w
th incept
ion [37]. T
hus, simultan
eo
us ob
se
rvatio
ns of ele
c
trical
tree g
r
o
w
th
s
and th
e corre
s
po
ndin
g
PD
occurre
n
ces
are th
e p
opul
ar a
nd
perha
ps m
o
re p
r
e
c
i
s
e
method
to
study the
initiation a
nd
prop
agation
of
el
e
c
tri
c
al tree
are bei
ng
used
no
wad
a
ys [3
8]–
[44].
In gen
eral,
th
ere
are two
comm
on te
ch
nique
s
of an
alyzing
PD d
a
ta which a
r
e wi
dely
use
d
the ph
a
s
e-re
solve
d
p
a
rtial di
scharge patte
rn (P
RPD) and
pu
lse
sequ
en
ce
analysi
s
(PS
A
)
techni
que
s.
Proba
bly, the
mo
st comm
only u
s
ed
type of
data fo
r
PD mo
nitorin
g
an
d in
sul
a
tion
diagn
osti
cs in
the ca
se of a
c
appli
ed voltage
s is
the p
hase-re
solve
d
partial di
scharg
e
pattern
(
φ
-
q-n
)
. It represents info
rmati
on in 3
D
cont
aining th
e
int
e
rrelation
s
hip
s
of a
c
voltag
e pha
se
of PD
occurre
n
ce (
φ
), disch
a
rg
e
magnitud
e
(
q
),
and disch
a
rge rate (
n
).
A typic
a
l PD pattern from an
electri
c
al t
r
ee
ing expe
rime
nt with an
alternatin
g
volta
ge of 50
Hz was a
pplie
d to
the pin el
ectrode
is
sho
w
n
in
Figure 3. Vol
t
age p
h
a
s
e
φ
is on th
e
absci
ssa, di
scha
rge
ma
gn
itude
q
is
on
the
ordin
a
te, and
dot den
sity in ea
ch wi
ndo
w re
pre
s
e
n
ts
the numb
e
r o
f
discharge
s
n
as d
epi
cted
in
the Figure 3. Positive PDs are m
o
stly
present
duri
ng the positi
v
e half-cycl
e
of the applied
voltage (
01
8
0
o
) whi
l
e the ne
gative PDs are m
o
stly
pre
s
e
n
t
durin
g the n
e
gative half-cycle
(
180
3
60
o
) [45].
Figure 3. Typical PD Pattern from an Ele
c
tri
c
al Treein
g
Experiment
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 582
7 –
5846
5832
Based o
n
ph
ase
-
resolved PD pattern (
φ
-q
-n
)
sho
w
n
in the Figure
3, the individual P
D
event is de
n
o
ted by dot
and it
co
ntai
ns the info
rmation of th
e pha
se of
occurre
n
ce (
φ
),
magnitud
e
(
q
), an
d n
u
mbe
r
of P
D
event
s
(
n
).
PDs a
r
e often
re
garded
as sto
c
h
a
stic p
r
o
c
e
s
ses
and thu
s
, sta
ndard stati
s
tical an
alysi
s
can be em
plo
y
ed to interpret and analy
z
e the larg
e PDs
data [46].
3.4. Frequen
c
y
Freq
uen
cy a
c
ts as repe
ated el
ec
t
r
ical
stre
ss in
vert
ical
dire
ction
of the
ele
c
tric fiel
d
arou
nd
the n
eedle
tip. Wh
en
a
centralized stre
ss
ex
ceed
s a
limit o
f
the di
ele
c
tri
c
stre
ngth
of
the
material, a
crack o
r
void
would
start to
prod
uce
which co
nsequ
ent
ly could i
n
itia
te the ele
c
tri
c
a
l
tree. When freque
ncy in
creases, the
m
e
ch
ani
cal
stress
unite
s
wi
th the ele
c
tri
c
al stress a
ppl
ied
at the needle
tip thereby e
nhan
cin
g
the initiation of
electri
c
al tre
e
a
nd re
du
ce
s the magnitu
de
of
tree in
ceptio
n
voltage a
s
well. Che
n
et a
l
. [47]
repo
rte
d
that high freque
ncy up t
o
500
Hz co
ul
d
accele
rate th
e b
r
ea
kdo
w
n
or complete
brid
ging
of
the polyme
r
i
c
in
sul
a
tion
due to
ele
c
trica
l
treeing
whi
c
h
is cau
s
e
d
fro
m
a gene
rati
on of highe
r
numbe
r of pa
rtial discha
rg
es. De
nsl
e
y [23]
in their work observed that
electri
c
al tre
e
s of
differe
n
t
shape
s g
r
o
w
und
er vari
o
u
s voltage
s a
nd
freque
nci
e
s. I
n
ad
dition, hi
gher fre
que
n
c
y a
c
celera
te
d time to
bre
a
kd
own. Du
et al. [25] fou
n
d
that the g
r
o
w
th rate a
n
d
the tre
e
structures
were
influen
ced
b
y
the ap
plied
frequ
en
cy. The
gro
w
th
rate
o
f
elect
r
ical tree in
crea
sed
with
t
he i
n
crease of
freq
u
ency
of a
ppli
ed voltag
e. T
he
length
of ele
c
trical
tree
al
so in
crea
sed
with the
in
cre
a
se
of voltag
e fre
que
ncy.
This is de
pict
ed in
Figure 4.
The re
sult
shows that th
e gro
w
th sp
eed of el
e
c
trical tre
e
in silicon
e rub
b
e
r
at the
freque
ncy of
4.5kHz was faster than
that at
powe
r
frequ
en
cy (50
H
z). The
silicone rubb
er
insul
a
tion
wa
s fully
brid
ge
d by th
e el
e
c
tri
c
al tr
ee
a
t
duration
of 90
minute
s
at freq
uen
cy
of
4.5kHz
whil
st
at freque
ncy
of 50Hz, the
tree ha
s ta
ken a time of
120 min
u
tes
to gro
w
up t
o
0.7mm.
A si
milar study o
n
sili
co
ne rub
ber wa
s ca
rri
ed o
u
t by
Nie
et al. [48]. T
hey re
po
rted
that
tree in
ceptio
n
voltage de
creased with th
e increa
se of
the freque
ncy from 50Hz
to 130kHz. At
low frequ
en
cy (50
-
50
0Hz),
ele
c
trical tre
e
grew in
th
e
form of
bra
n
ch-type a
nd
pi
ne-type t
r
ee
s.
At
middle f
r
equ
e
n
cy rang
e fro
m
1kHz to
10
kHz, the
tree
s ten
d
to fo
rm more in
th
e shap
e of pi
ne-
type and b
u
sh-type tre
e
s,
while
at frequ
ency a
bov
e 1
0
kHz, all th
e
trees form
ed
were bu
sh
-type
trees. Th
e finding
s by Nie et al. is in agreem
ent
with Che
n
’s work whe
r
eby freq
uen
cy of 500Hz
accele
rated t
he gro
w
th of electri
c
al tre
e
s
and a
c
celerated the bre
a
k
do
wn be
ca
u
s
e the tree was
a bran
ch
-typ
ed one
which has
single
bran
ch a
n
d
propa
gated
in a straig
ht line towa
rd
s the
grou
nd ele
c
trode.
Figure 4. Gro
w
th Ch
ara
c
te
ristics of Electric
al T
r
ee in
SiR unde
r AC Voltage with Different
Freq
uen
cie
s
[25]
3.5. Tempera
t
ure
Studies
con
d
u
cted by Ied
a
[49] reg
a
rdi
ng to t
he effe
ct of tempera
t
ure on el
ect
r
ical tre
e
gro
w
th ha
s shown that the tree ince
ption time
decreased whil
e the tree prop
agation rate has
increa
sed
wit
h
the i
n
crea
se of the
tem
peratu
r
e
ra
n
g
ing from
ro
om temp
erat
ure to
10
0°
C in
Polyethylene
(PE). The t
r
ee p
r
opa
gati
on rate is
someho
w d
e
termin
ed by t
he ma
gnitud
e
of
0
20
40
60
80
100
120
0
0.
2
0.
4
0.
6
0.
8
1
1.
2
1.
4
G
r
o
w
t
h
T
i
me
,
mi
n
El
e
c
t
r
i
c
a
l
Tr
e
e
Le
n
g
t
h
,
m
i
n
6k
V
,
5
0
H
z
8k
V
,
5
0
H
z
6k
V
,
4
.
5k
H
z
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Ove
r
vi
ew
of Electrical T
r
ee G
r
o
w
th in
Solid Ins
u
lating Material with… (M.H. Ahmad)
5833
discha
rge
an
d the extent
of the dam
ag
e forme
d
at
t
he chan
nel ti
p. The in
crea
se of tem
p
e
r
ature
may influence these fa
ctors
whi
c
h co
uld lead to
the increa
se
of tree pro
p
a
gation rate [
26].
Den
s
ley [23], Ieda and
Na
wata [50] fou
nd that at
temperature
greater tha
n
80
°C, only bu
sh
-type
trees g
r
e
w
(n
ot takin
g
i
n
to
acco
unt th
e
value of
voltage
stress)
whe
r
ea
s for tempe
r
ature l
e
ss
than 70°
C, o
n
ly the bran
ch-type
tree
s formed at voltages of 14
kV or less in PE sample
s. The
tree
pro
pag
ation
rate i
n
cre
a
se
d
with in
crea
se
in tem
p
eratu
r
e
due
to the
incre
a
se of
deg
rad
a
tion
prod
uced at t
he tree
ch
an
nel tip by a gi
ven discha
rg
e, sin
c
e the p
enetratio
n
of
the polymer
b
y
a
charged pul
se increases with te
mperature [26]. In ethylene-viny
l
acetate copolymer (EVA), i
t
wa
s found th
at the tree in
ceptio
n time and b
r
ea
kd
o
w
n time were sho
r
ter
at high temp
era
t
ure
(60°
C) com
p
ared
with a temperature of
20°
C [51].
Tempe
r
atu
r
e has al
so be
e
n
repo
rted to have an
effect on the tree sha
pe. In the ca
se of
room tem
p
e
r
ature vul
c
ani
zed
(RTV)
si
licon
e ru
bbe
r, different tre
e
ch
ara
c
te
ristics h
ad b
e
e
n
observed
by
Du
et al. [52].
In thei
r
work, three
differe
nt ambie
n
t te
mperature
s
were
selecte
d
t
o
investigate
th
e be
havior of
ele
c
trical tre
e
s. At 3
0
°C, the trees tend to fo
rm
in
branch-type t
r
e
e
s,
whe
r
ea
s whe
n
the temperature was in
crea
sed from 60°
C to 90°
C, the bush-ty
pe tree
s be
came
the domi
nan
t. This
wa
s be
cau
s
e
when th
e tem
peratu
r
e
wa
s in
crea
sed,
the d
e
g
r
ee
s of
vulcani
ze
d point also in
creased. Thu
s
, this led to
the formation o
f
a uniform vulca
n
ized net
and
increa
sed th
e numb
e
r of
cro
s
s-lin
ks there
b
y req
u
iring more en
ergy to brea
k mo
re bo
nd
s in
orde
r to
pro
d
u
ce th
e
cavity. In a simila
r work
carrie
d out by Zh
o
u
et al. [53]
on ele
c
tri
c
al t
r
ee
initiation in
Hi
gh Te
mpe
r
at
ure V
u
lca
n
ize
d
(HTV)
s
ili
co
ne rubb
er, te
mperature
of
25°
C resulted
in
the form
ation
of bran
ch-li
k
e an
d mo
nke
y
-puzzle
tre
e
s
o
n
ly. Mean
while
at 50
°C, bush-li
ke tree
became
domi
nant
comp
are
d
to the
mon
k
ey-pu
zzl
e
a
n
d
bran
ch
-like
trees. At
high
er tem
perature
(>1
00°
C), all
the tree
s b
e
came b
u
sh-li
k
e tree
s. Th
ese re
sult
s were simila
r to
re
sults
re
porte
d
by
Du
et al. [52]
. The rea
s
on
of the form
ation of
b
u
sh-li
k
e
tree
s at
hig
her
te
mpe
r
at
ure wa
s due to
vulcani
zation
effect whi
c
h i
s
enh
an
ced b
y
the increa
se in temperature.
In epoxy resi
n (BADGE)
system, ra
pid
gro
w
th
of
ele
c
tri
c
al tree h
a
s
been
ob
served
at
the temperature n
ear th
e resi
n
gla
s
s transitio
n temp
eratu
r
e,
T
g
which
wa
s 95
°
C
. This
wa
s
due
to the m
o
vement of m
o
lecular
c
hai
n or chain
mobilit
y has becom
e
more
active and thereby
led
to the enlarg
e
m
ent of electron diffusio
n
a
t
the needle e
l
ectro
de [54].
4. Statistical Models
The mo
st appro
p
ri
ate method of interp
reting
coll
ected data i
s
by using
statistical
analysi
s
sin
c
e differe
nt re
sults are obt
ained fo
r gi
v
en te
st co
nd
ition in e
a
ch
test sampl
e
s.
Extreme-valu
e statisti
cs
such
as
Weib
ull and lo
g-n
o
rmal di
stri
b
u
tions h
a
ve
been a
pplie
d
to
descri
be the failure of
solid insulation by fi
tting t
he exp
e
rim
e
ntal dat
a
to
these
statistical
distrib
u
tion
s via gra
phi
cal a
nd compute
r
-based te
c
hni
que
s. Thu
s
, it can
be
said t
hat Wei
bull a
n
d
logno
rmal
are commo
nly used in
dat
a interpretation of
HV fa
ilure
phe
nom
enon.
Re
cen
t
ly
Joh
n
son SB
has
also be
e
n
used in thi
s
field. A brief
review
of the
s
e mo
del
s is
pre
s
ente
d
in t
h
is
se
ct
ion.
4.1. Weibull Distribu
tion
Weib
ull is th
e most
widel
y used di
stri
bution fun
c
tion in hig
h
voltage en
gin
eerin
g to
descri
be the failure of solid insulation. In co
ntra
st-voltage
te
sts, time to
bre
a
kdo
w
n
and
brea
kd
own voltage di
strib
u
tion functio
n
s
are det
e
r
mi
ned, whi
c
h
are gene
rally a
pproxim
ated
by
two-p
a
ramete
r Weib
ull distribution. In ca
se of insu
latio
n
failure du
e to electri
c
al tree, Weib
ull has
been
applie
d
prin
cipally t
o
tree b
r
e
a
kdown time, tree b
r
ea
kd
o
w
n voltag
e
probl
em
s an
d
determi
ning t
he water an
d ele
c
tri
c
al tree len
g
ths in
solid
in
sulati
ng mate
rial
s. [55, 56]. T
h
e
probability density function (pdf) a
nd
cumulative distribution function (cdf) of
Weibull di
stribution
are given a
s
follow:
1
()
e
x
p
xx
fx
(2)
0
()
1
e
x
p
xx
Fx
(
3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 582
7 –
5846
5834
Whe
r
e,
α
= scale pa
ra
meter,
α
>0,
β
= sha
pe pa
rameter,
β
>0
or mea
s
u
r
e o
f
dispersio
n
,
0
x
= initial value of
x
or lo
cation
param
eter,
X
= mea
s
ured
variable,
F(x)
= pro
b
a
b
ility of failure at a voltage
or time less than or eq
ual
to x,
For the initial
value,
0
0
x
the derived two-p
a
ra
meter Weibull
is as follo
ws;
(
4
)
The scal
e a
nd shape
pa
ramete
rs
ca
n
be esti
m
a
te
d by usin
g l
east
squ
a
re
method,
maximum li
kelihoo
d e
s
tim
a
tion o
r
by
plotting the
prob
ability g
r
aph
on
Weib
ull pa
per.
From
literature, se
veral
resea
r
che
r
s have
applie
d an
d
modell
ed
Weib
ull di
stri
bution fo
r m
a
n
y
purp
o
ses
rela
ted to high voltage insulatio
n
.
Barcl
a
y et al.
[57] have a
p
p
lied the
two
-
pa
ramete
r
Weib
ull di
stri
bution to
sim
u
late the
failure p
r
o
bab
ility and to ex
amine th
e mo
del pa
ram
e
te
rs i
n
orde
r to
find its influe
n
c
e
s
on th
e tre
e
gro
w
th b
eha
viours. S
a
rat
h
i et al. [2
1] used
We
i
b
u
ll distri
bution
to un
dersta
nd the
severity
attributed to
electri
c
al
stre
ss a
nd a
nalysed the
ele
c
trical tree p
r
o
c
e
s
ses. Di
ssado [31] ap
pl
ied
the Weibull
functio
n
to
de
termine
a
co
ntinuou
s fu
nction for the
cumulative p
r
o
bability of fail
ure
from the num
ber of brea
kd
own
s
du
e to electri
c
al tree
ing.
Furthe
rmo
r
e,
Arainy
et a
l
. [58] have
appl
i
ed th
e
Wei
bull to
model
the
water-t
ree
popul
ation
an
d tre
e
le
ngth
s
at different t
e
mpe
r
atur
es.
Hu
uva et
al [
59] u
s
ed
2
-
p
a
ram
e
ter and
3-
para
m
eter
Weibull stati
s
tics to model th
e tree in
cepti
on field of L
D
PE with the
aid of MINITAB
softwa
r
e. In their stu
d
y, 90% conf
iden
ce
limit was ap
plied for fitti
ng. However, it was foun
d that
several p
o
int
s
h
ad fall
out
side
the
co
nfiden
ce l
e
vels.
More
over, th
e tree
in
cepti
on time
s a
nd
tree
prop
agatio
n times
were an
alysed u
s
in
g two-p
a
rame
te
r of Weib
ull statistics fo
r p
o
st-cu
r
e cro
s
s-
linke
d polye
ster re
sin [60
]. The 90% confid
ent
limit proved the
data wa
s fitted with two-
para
m
eter
Weibull stati
s
tics. In this ca
se, it
was foun
d that none o
f
the short-ti
me experim
e
n
tal
points fall out
side of the 90
% confiden
ce
limits.
4.2. Log-nor
mal Distribution
Logn
orm
a
l is
a stati
s
tical
di
stributio
n that
ha
s be
en
used to
rep
r
e
s
e
n
t the failu
re
data of
the insulation
system
s. Lo
g-no
rmal i
s
u
s
ed to m
odel
a failure
of co
mpone
nts’ lif
e whi
c
h i
s
du
e to
the fatigue-s
tress [61]. Bas
e
d on
the
e
x
perime
n
tal result
s, Cza
szejko
[62,
63]
has cl
arified
t
hat
the wate
r tre
e
length
dat
a point
s fit very well in
l
og-n
o
rm
al di
stributio
n pl
o
t
s. This re
sul
t
is
con
s
i
s
tent wit
h
the studi
es
done by Q
u
re
shi et
al. which have be
en
publi
s
hed i
n
[64-6
6
]. Base
d
on their st
udi
es, wate
r tree
populatio
ns
and tree le
ng
th distributio
n
s
we
re fit better to log-normal
model.
A variabl
e, Y
is te
rmed
log
-
norm
a
l
whe
n
the vari
able
p
r
odu
ce
d by
th
e tra
n
sfo
r
mati
on X
=
logY or X = lnY. The probability di
stributi
on function (PDF) of logn
ormal di
stribution is as follows;
2
11
()
e
x
p
2
2
ln
fx
x
x
(5)
Whil
st, CDF
of Logno
rmal
distrib
u
tion can be expressed a
s
follo
ws;
ln
()
x
Fx
(6)
Whe
r
e
is Laplace integral
whic
h is expressed a
s
follo
ws:
()
1
e
x
p
x
Fx
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Ove
r
vi
ew
of Electrical T
r
ee G
r
o
w
th in
Solid Ins
u
lating Material with… (M.H. Ahmad)
5835
2
2
0
1
2
x
t
dt
e
(7)
4.3. Johnson
SB Distribu
tion
Joh
n
son SB mostly has b
een appli
ed i
n
the fields
of forest scien
c
e [67], hydrology [68],
exposure
sci
ence a
n
d
env
ironm
ental
ep
idemiolo
gica
l studie
s
[69]. Re
cently, it h
a
s bee
n appli
ed
in high volta
ge engi
nee
ri
ng by Ahma
d et al. [68].
It was found
that after fitting pro
c
e
s
s u
s
ing
Anderso
n-Da
rling (A
D) g
o
odne
ss-of
-
fit (GOF
) test
s,
the tree in
cept
ion voltage of
silico
ne ru
bb
er
and e
poxy re
sin fitted well
with John
so
n
SB model
[7
0,
71].
They also sh
owed that
Joh
n
son SB
wa
s more fit than Wei
bull
and Lo
gno
rm
al by comp
aring the fitting erro
r. PDF
of John
so
n SB
distrib
u
tion is
given as follo
ws [72];
2
1
()
e
x
p
l
n
21
2
z
fx
z
(8)
Mean
while, CDF of Jo
hn
so
n SB distribut
ion is a
s
follo
ws [72];
()
l
n
1
z
Fz
z
(9)
Whe
r
e
δ
and
γ
are shap
e p
a
ram
e
ters,
ξ
i
s
a lo
cation
p
a
ram
e
ter,
λ
is a scale p
a
ra
meter,
and
is the CDF of the
sta
ndard no
rmal
rand
om
vari
able a
s
sho
w
n in Equation
(9).
Whe
r
ea
s x
is define
d
on
boun
ded ran
ge of and z i
s
referring to followin
g
tran
sformation;
x
z
(
1
0
)
All the John
son SB pa
rameters
can
be cal
c
ul
ated by usi
n
g
Maximum L
i
kelih
ood
Estimation m
e
thod (M
LE).
The Joh
n
so
n SB model wa
s found suitable for a
pplication in the
analysi
s
of el
ectri
c
al
tree
i
n
ce
ption volt
age
dat
a. T
h
i
s
wa
s
prove
d
by
cal
c
ulatin
g the
AD GO
F
test. Based o
n
GOF test, the fitting erro
r wa
s the sm
allest for
Joh
n
so
n SB rath
er than
Weib
ull
and Log
no
rm
al whi
c
h exhi
bit large
r
fitting error.
Ba
sed on the wo
rk do
ne by Ahmad et al. [70,
71, 73], th
e e
s
timated
valu
es
of tre
e
in
ception vo
lta
g
e
of
sili
cone
rubbe
r
and
ep
oxy re
sin
sa
mple
were cal
c
ul
ated by taking
inverse fun
c
tion of the Joh
n
son SB CDF. The i
n
verse functi
on
equatio
n is gi
ven in Equation (11
)
;
1
1
1
()
()
ex
p
()
1e
x
p
P
P
xF
P
(11)
Based
on
the
above
equ
ation, the va
ria
b
le x
was a
s
sume
d a
s
t
r
e
e
in
ception
voltage
at
prob
ability of
0.5. In
additi
on, the
value
of x
can
b
e
cal
c
ulate
d
g
r
aphi
cally by
plotting the
CDF
grap
h.
5. Electrical Models
Electri
c
al m
o
dels are diffe
rent compa
r
e
d
with
st
atisti
cal mod
e
ls in t
e
rm
of an
alysis, data
interp
retation,
calculation,
param
eters and appli
c
a
t
ion. In this se
ction, sev
e
ral mo
del
s are
discu
s
sed bri
e
fly in order t
o
differentiate
them
in term
s of con
c
e
p
t and ap
plication. The mod
e
ls
su
ch a
s
Die
l
ectri
c
Bre
a
kdown Model,
Discha
rg
e-Avalanch
e
Model
a
nd Field-Drive
n
Tree
Growth Model are c
o
vered in this
s
e
c
t
ion.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 582
7 –
5846
5836
5.1. Dielectri
c Brea
kdo
w
n Model
Diele
c
tri
c
bre
a
kd
own
p
hen
omeno
n occu
rs
by
n
a
rro
w
discha
rge
ch
annel
s
th
at
e
x
hibit
a
st
ron
g
t
e
n
den
cy
t
o
br
an
ch i
n
t
o
co
mpli
cat
ed st
o
c
h
a
sti
c
electri
c
al t
r
ee
pattern
s. Th
e ele
c
tri
c
al tree
discha
rge g
r
o
w
s to the poi
nts wh
ere the
electri
c
field is high
est.
Pietrone
ro et
al. [74, 75] propo
se
d a m
odel base
d
on the ide
a
of the local field
discha
rge p
a
ttern did not g
o
vern the tre
e
gro
w
th
dire
ctly but throu
gh a st
o
c
ha
st
ic proce
s
s. This
mean
s that th
e field p
a
ttern
did n
o
t ju
st g
r
ow at
the
poi
nt of maximu
m local field
but at thi
s
poi
nt
exhibits the hi
ghe
st prob
abi
lity of growing
.
The el
ectri
c
field is determined
by the global
stru
ctu
r
e
of the discha
rge patte
rn. The rel
a
tion betwe
en
pro
bability and field sh
ow
s the fact that the
microsco
pic
mech
ani
sm o
f
the pr
opaga
tion of the discha
r
ge i
s
mo
dul
ated by its global stru
ct
ure.
The el
ect
r
ic
field di
stributi
on i
s
o
b
tain
ed fr
o
m
the
Lapl
ace e
q
uation
by a
s
sumin
g
the
t
r
ee
cha
nnel
a
s
a
n
ele
c
trode
e
x
tension
o
r
v
o
ltage
dr
op
within t
r
eein
g
ch
ann
els.
Howeve
r, in
[7
6],
Wei
s
man
n
an
d Zelle
r modif
i
ed this m
ode
l by introdu
ci
ng a
critical field for
cha
n
n
e
l gro
w
th a
n
d
a
voltage d
r
op
along
the tre
e
chan
nels.
T
he
stoc
ha
stic model
was d
e
fined
on a
la
ttice an
d b
a
sed
on the equi
po
tential and co
nne
cted di
scharg
e
pattern
.
DBM re
prese
n
ts a recta
n
g
u
lar lattice where
ea
ch lat
t
ice poi
nt co
rresp
ond
s to a
point in
the dielect
r
ic. An electro
de gap con
s
ist
s
of 100
lattice units repre
s
e
n
tin
g
the dielectric
breakdown. The DBM
assu
mes that the electri
c
al
tree
will
grow
stepwise. The growth of
electri
c
al
tree
will
start at
n
eedle
ele
c
tro
de with
ele
c
trical p
o
tential,
ρ
=
0
and
en
d at the
co
unt
er
electrode
with electri
c
al p
o
tential,
ρ
= 1
. The prob
abi
lity,
P
of a tre
e
cha
nnel g
r
o
w
th at each site
of ele
c
trical t
r
ee
neig
hbo
u
r
hoo
d i
s
p
r
o
portion
al to
a po
we
r,
τ
o
f
electri
c
fiel
d,
E
at that
site.
Electri
c
field in the form of
ρ
is
written as follows
;
,
,
(,
,
)
ij
ij
Pi
j
i
j
(12)
Whe
r
e
i,j
and
i’
,j
’
are the discrete lattice positio
ns.
Th
e summ
ation
of denomin
ator refe
rs
to all of the p
o
ss
ible tree growth
s
i
te (
i’
,
j
’
)
which are adja
c
ent to the elect
r
ical tree chan
nel. By
solving La
pla
c
e eq
uation
with co
nsid
eration of
that
the tree stru
cture h
a
s el
e
c
tri
c
al potenti
a
l
equal to zero,
the electri
c
field,
E
is obtained. Then, th
e 2-dim
e
n
s
io
nal lattice can
be written a
s
;
,1
,
1
,
,
1
,
1
1
4
ij
i
j
i
j
ij
ij
(13)
After iteration
of this
equati
on, the el
ectri
c
pote
n
tial at
possibl
e tre
e
gro
w
th
site (i’
,
j’) ca
n
be obtain
ed.
5.2. Dischar
ge Av
alanche Model
Dissa
do et al. [77] proposed di
scha
rg
ed-av
al
an
che
model whi
c
h descri
bed
electri
c
al
tree p
r
op
agat
ion qu
antitatively. When t
he voltage
is
increa
se
d, it is difficult to determi
ne t
h
e
voltage d
epe
nden
ce
of tre
e
g
r
o
w
th
whe
r
eby th
e
sha
pe h
a
s
chan
g
ed from
bra
n
c
h to
b
u
sh type.
The shape
chang
es o
c
curred
with a re
ductio
n
in t
he length g
r
o
w
th at a fixed time arou
n
d
the
voltage. The total damag
e in bush tree
s
wa
s gre
a
te
r t
han bran
ch trees. Th
us, th
e elect
r
ical tree
damag
e wa
s
formulate
d
as:
t
d
b
L
S
L
(
1
4
)
Whe
r
e
S
i
s
th
e numb
e
r
of tree
cha
nnel
d
a
mage,
L
b
as averag
e of e
l
ectri
c
al tree l
ength,
L
a
s
the l
eng
th of a tree,
and
d
t
a
s
fr
acta
l d
i
me
ns
ion. W
h
er
e
,
d
t
=
1.2 to 1.8 fo
r bra
n
ched
tre
e
s
and 2.4 to
3 for bu
sh t
r
ee
s [26], [57]. Then, if the da
mage g
r
o
w
s i
n
a straight li
ne (
d
t
=
1), then
the numbe
r o
f
newly cre
a
ted ch
ann
els i
s
as follo
ws;
b
L
L
S
(15)
Evaluation Warning : The document was created with Spire.PDF for Python.