Energy and Power Engineering, 2013, 5, 1273-1276
doi:10.4236/epe.2013.54B241 Published Online July 2013 (http://www.scirp.org/journal/epe)
Electrical Tree Simulation Based on the
Self-Organization Criticality
H. M. Hu, Y. Yang, W. Lu, G. P. Zhao
China Three Gorges University, Yi Chang, China
Received April, 2013
ABSTRACT
So far much effort has been made to understand the development of electrical treeing. For the simulation based study of
electrical treeing, the most common method is to apply DBM stochastic model to simulate the growing of electrical
treeing patterns. Previou s simulation results showed that this stochastic model is capable of simulating the real electrical
treeing patterns in a point-to-plane electrode system. However, th is model only allo ws the tree chan nels to prop agate on
equipotential lines prop ortional to local electrical field. Therefore, it is necessary to develop a novel stochastic model to
simulate the electrical patterns in order to get a good agreement with experimental results.
Keywords: Fractal; Electrical Tree simulation; Self-Organization Criticality
1. Introduction
Electrical treeing is a common pre-breakdown phenome-
non for solid insulating materials when materials are
subjected to high electrical stress. Electrical tree grows in
a pattern resembling the branches of a tree. The branch-
like structures are conductive channels formed through
partial discharge activities. Experience from field and
factory has shown that electrical treeing accounts for one
of the main sources of long-term deterioration of insulat-
ing materials. Therefore, it is of great importance to
study the electrical treeing phenomena in insulating ma-
terials for the improvement of reliability and safety of
power system.
One of the major causes for polymer insulation degra-
dation in a long time is electrical tree growth, the phe-
nomenon of electrical tree growth contains discharge
channels presented tubular random branching growth and
partial discharge, which causes damage resulting in new
formations of plasma channel, vice versa discharge
structures determine the trajectory and the parameters of
partial discharge, so electrical tree and partial discharge
are the self consistent process of insulation degradation.
Dynamic process of electrical tree growth and character-
istics of partial discharge depend on the applied vo ltage’s
amplitude, frequency, electrode shape, physical proper-
ties of the medium, temperature and so on.
2. The Traditional Method
Niemeyer, Pietronero and Wiesmann Build a model
when they study Flash discharge along surface smooth,
short for NPW model [1]. In the model, electric field
control indirectly discharge development but impact it
through stochastic process, introducing discharge branch
development probability in statistical sense, namely dis-
charges take place at points where development prob-
ability is largest rather than electric field, which change
the whole process from deterministic to stochastic. The
relationship between the probability and electric field
reflect that the overall discharge structure regulate the
micro mechanism of discharge process, voltage drop
along the discharge channel is less than the applied volt-
age, thus the formation of the channel is approximately
close to potential structure, electric field distributes in
symmetry of the circular and external boundar y is quite a
long way from discharge graphics.
The NPW model provide that discharge pattern grows
stepwise from the cathode to the anode, electric potential

of each grid unit is calculated by Laplace’s equation
with the boundary conditio ns as following[2]:
=0center dot
=1round edge





（）
（）
In the grid square, the poor score of unit is calculated
approximately in finite difference method througy fol-
lowing fomula:
,1,1,,1,
1()
4
xyxy xy xyxy

 

1
(1)
the probability P is weight function to select candidate
units according to electric potential, potential difference
between candidate units and electrical tree points is asso-

H. M. HU ET AL.
1274
ciated with the development probability:
''
''
, ''
,
()
(,, )()
ik
ik
piki k





 (2)
The essence of the model is that the probability of
electrical tree growth depend on local field determined
by discharge pattern, the key part is to solve Laplace eq-
uation in each process step by step growth at a time,
which reflects the impact of the new unit potential[5].
3. Self-Organization Criticality
The growth of electrical tree has the characteristics of
self-organization critical phenomenon and self organiza-
tion critical problem was put forward by Bake [3, 4] in
the research of sand of cellular automata model. The
state consisting of a square piece of sand is imagined that
slope distribution decision. Slope of random point in the
area is of increased to simulate the increasing number of
grains of sand. This will lead to ch ange around a few points
of the slope when it is more than the threshold value of
grains of sand can slide to a lower position. However, the
redistribution of the slope is caused by the sand move-
ment and further collapse of sand. As the grains of sand
sliding, sand pile will tend to a dormant state at this time
of infinite slope near threshold .Gradually increasing
number of grains of sand will lead to different scale of
sand pile collapse. But th e system is still in the stationary
state near collapse because size cannot be thought of system
is in critical state. Main characteristic of self-organiza-
tion critical system is developing to a critical state spon-
taneously under the influence of external factors. In this
state there is no specific measurement, space or time and
constant input produced the intermittent output as well as
conforming to no specific proportion of the power law
distribution.
From the perspective of self-organized criticality,
along the electrical tree distribution channel of the charge
is a spatial extension of dynamic system and presents a
spatial extension of dynamic systems. Partial discharge is
the output of the whole system as well as external factors
as the applied voltage. It is necessary to consider about
non-locality, nonlinear contact of space charge and re-
striction of electrical tree in charge movement and so on.
4. The Simulation Principle
For any kind of insulation degradation mechanism, the
dielectric damage is caused by partial discharge the en-
ergy released and the local field strength. Therefore the
degradation mechanism of rules need to consider the
above two aspects. Discharge channel formation rules of
the judge need to consider medium damage accumulation
and random electrical tree growth. Probability distribu-
tion is used to describe the random growth, and then it
based on the experimental data analysis of physical me-
chanism of insulation damage, it calculates the probabil-
ity of discharge channel development and discharge en-
ergy, the relationship between electric field strength.
When local damage energy or local field strength is low-
er than the critical value, the development of the prob-
ability is zero. And when the damage energy reaches
critical value and the local field strength more than the
critical value, the existin g electrical p oints in the tree will
form a new discharge channel. Many a branching pattern
of one dimensional point is used to simulate electrical
tree and point d of the connection between the distances
determines the minimum space simulation ratio. Discrete
time constant t

is used to describe the transient proc-
ess of electrical tree growth as well as time step n corre-
sponding to instantaneous time n. The point i
corresponds to the charge and specific energy
and at each discrete time points in the electrode potential
will change. Existing electrical tree structure or the in-
crease of the electrode tip point is the simulation of elec-
trical tree growth which depends on the electric field
strength and the energy distribution of the damage.
However, partial discharge will lead to field strength,
failure energy, and charge distribution of change. The
process time lasted until the electrical tree structure come
into contact with the other electrode.
tnt
n
i
qn
i
W
Every point of the electric potential is equal to all the
charge voltage superposition，the i-th point on the po-
tential of a discrete time step n can be expressed as fol-
lowing[6]:
0
*
0
4rr
4rr
n
j nn
ij
ji ij
n
j
jij
q
q
 


 






π
π
(3)
o

is the vacuum dielectric constant,

is the relative
dielectric constant, i
r

,
j
r

 is the position vector, n
i

is
a part of point charge electric potential , relationship
lies on the size of the electric field distributed around
electrical tree channel, The minimum measure in space
adopts the size of space charge area,
n
i
q
n
i

is expressed in
Spherical charge formula:
0
2d
n
ni
i
q



π (4)
Specific damage energy of any i-th point at ex-
isting electrical trees reaches a critical value c, there is
a new point is attached to the structure. Several addi-
tional locations are randomly situated around i-th point,
the gaps between them and i-th point are not less than
bond length d, and the selection of lately add itional posi-
tion among certain number of impossible locations is in
n
i
WW

H. M. HU ET AL. 1275
accordance with development probability P that is ex-
pressed as following [7]:
12 c
()(d)(W )
n
ic
PZE W  

  (5)
2
()Z


 is normalization factor, summation is
made over all possible attachment positions providing
c
Ed

 , ()
x

is step function, () 0x

 if 0x

,
() 1x

 if . The first Growing point attach solely
to electrode at the beginning of the electrical tree growth.
0x
Assume that there is only a partial discharge at each
time step, so the time interval should be small enough so
as to distinguish the different partial discharge. Partial
discharge is the process in which all kinds of factors in-
fluence each other, involving charge, self-consistent cal-
culation of electrical field and the change of boundary
conditions [9], in each interactive time step k, growing
point is characterized with charge and specific
damage energy . Segment between i and j point
is characterized with status value ,, there are two
distinct states: conducting ,= 1 and non-con-
ducting ,=0, conducting state correspond to the
discharge. The electric potentials of point charge are
calculated according to equation (1), at the beginning of
the discharge the point states are determined by the pre-
vious time step, namly , all
bonds are in the non-conducting state .
()
ki
qk
()
nij
Sk
)k
, (0)W

(0
n
ij
S
()
ki
Wk
)k(
nn
ii
qq
nij
S
(0)
(
nij
S
1n
i
W
) 0
1n
i

,
The bond in the non-conducting state transits into the
conducting state if the absolute value of the potential
difference exceeds the inception voltage
,()
nij k
 
ign

[8]:
() ()
nn
ijign
kkE  d (6)
The bond in the conducting state transits into the non-
conducting state if the absolute value of the potential
difference falls below the residual voltage
,()
nij k
 
res

:
() ()
nn
ijres
kkE  d (7)
The bond state at time step k + 1 is expressed by the
step function ()
x

as following:
,,,
,,
(1)()( ())
(1())(())
nnn
ijijij res
nn
ijij ign
Sk Skk
Sk k
 
 
 
  (8)
Charge transfer takes place along the bonds in the
conducting state; charge value , is propor-
tional to the potential difference among points:
(1)
nij
qk
,,
(1)(1)()
nn
ijij ij
qk Snk   
,
n
(9)

is a numerical parameter that determines speed and
accuracy of the process, the new charge distribution is
given by:
,
(1) ()(1
nn n
ii ij
j
qkqkq k 
The charge movement will release damage energy to
form new plasma channel the specific damage energy
(1)
n
i
Wk

at time step k+1 can be expressed as:
,,
(1)
()( 1)()/
n
i
nn n
iij ij
Wk
Wkq kkd


 (11)
Partial discharge path is composed of contact state
between all of the above points, discharge process will be
terminated when all bonds are in non-conducting.
5. Conclusions
Traditional methods calculate the electric field distribu-
tion in means of Solving Laplace equation or Poisson’s
equation in Lattice squares, but the disadvantage of this
approach is the electrical tree growth is limited on the
grid of line and diagonal direction, electrical tree channel
diameter is equal to the size of grid line, the simulation
only probably reflects the rough trend of discharge. In
this paper a novel stochastic model based on analyses of
self-organized criticality of electrical treeing was applied
to simulate the formation of electrical tree. Fractal di-
mension and multi-fractal spectrum of simulated electri-
cal treeing patterns were calculated by using box-
ing-counting method. The relationship between the frac-
tal characteristics of the simulated electrical treeing pat-
terns and those of real electrical treeing patterns were
finally defined.
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