Electrical Tree Simulation Based on the Self-Organization Criticality

doi:10.4236/epe.2013.54B241

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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

Email: [email protected]

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()

xyxy xy xyxy



 



(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:

()

(,, )()

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

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]:

4rr

ji ij

jij

 





 







(3)



is the vacuum dielectric constant,



is the relative

dielectric constant, i



,



 is the position vector, n



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,



is expressed in

Spherical charge formula:





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

H. M. HU ET AL. 1275

accordance with development probability P that is ex-

pressed as following [7]:

12 c

()(d)(W )

PZE W

 



  (5)

()Z





 is normalization factor, summation is

made over all possible attachment positions providing



 , ()



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 .

()

nij

, (0)W



()

)k(

nij

(0)

(

nij

W

) 0



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]:

() ()

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



:

() ()

ijres

kkE



d (7)

The bond state at time step k + 1 is expressed by the

step function ()



as following:

,,,

(1)()( ())

(1())(())

nnn

ijijij res

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)()

ijij ij

qk Snk



 

(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

qkqkq k 

The charge movement will release damage energy to

form new plasma channel the specific damage energy

(1)



at time step k+1 can be expressed as:

(1)

()( 1)()/

nn n

iij ij

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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