Characterization and Application of Nanomaterials (2019) Volume 2 Issue 1 
doi:10.24294/can.v2i1.541

1

Improved mathematical model of polluted insulators nonlinear 
behaviour under ac voltage based on experimental tests

Mousalreza Faramarzi palangar1, Mohammad Mirzaie2

1,2Faculty of Electrical and Computer Engineering, Babol University of Technology, Babol, Iran.
r_faramarzi@stu.nit.ac.ir

ABSTRACT
In this paper, an improved mathematical model for flashover behavior of polluted insulators is proposed based on

experimental tests. In order to determine the flashover model of polluted insulators, the relationship between
conductivity and salinity of solution pollution layer of the insulator is measured. Then, the leakage of current amplitude
of four common insulators versus axial, thermal conductivity and arc constants temperature was determined. The
experimental tests show that top leakage distance (TLd) to bottom leakage distance (BLd) ratio of insulators has a
significant effect on critical voltage and current. Therefore, critical voltage and current were modeled by TLd to BLd
ratio Index (M). Also, salinity of solution pollution layer of the insulators has been applied to this model by resistance
pollution parameter. On the other hand, arc constants of each insulator in new model have been identified based on
experimental results. Finally, a mathematical model is intended for critical voltage against salinity of solution pollution
layer of different insulators. This model depends on insulator profile. There is a good agreement between the
experimental tests of pollution insulators obtained in the laboratory and values calculated from the mathematical models
developed in the present study.
Keywords:flashover; mathematical model; polluted insulators; salinity
1. Introduction

Pollution phenomena constitute a serious problem, which must be taken into account in the design and the
operation of HV insulating devices, under difficult environmental conditions, a pollution layer is entrusted on the
insulator surface and when humidity is present due to frost sediment, fog or rain, a humid conducting layer is formed
and a leakage current starts to stream[1-23]. The curtailment of the discharge current leads to the formation of dry tape on
the surface which results in arcing. The arcs may intrusively extend among the humid surface, leading to a total
flashover[1-18].

Some researchers have done very good work on the problem of polluted flashover. Several models to calculate
the flashover of polluted insulators have been proposed[1,3.6-9,12,13]. The expansion of models for the flashover of
contaminated insulators which could take into account the real empirical situation is not an easy labor since of the
intricacy of the phenomena which can happen[6-14]. However, many efforts have been made to produce models letting
one to forecast almost the critical specifications of the contaminated insulators [6]. For this order, the supposition that
there is a lone major arc, a uniform contamination distribution and monotone wetting have been made. So, the arc is
supposed to move alone on the surface of the insulator. But there is still no consensus as to what amounts of N and n
should be used in instance of different polluted atmosphere. In fact, the optimal combination of A and n may be
dependent on the chemical nature of the pollutants[16]. The experimental results allowed a great exact evaluation of the
discharge constants of the insulator by means of mathematical method. It has been ascertained that the calculated values
of the arc constants are autonomous of the insulator kind and the type of pollution[17]. The performance of polluted
insulation has been the subject of extensive research and development, and the mechanism of flashover has been
elucidated qualitatively. A detailed account of this work is given in[1-6], but the mechanism can be described briefly

Copyright © 2019 Mousalreza Faramarzi palangar et al.
doi: 10.24294/can.v2i1.541
EnPress Publisher LLC.This work is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). 
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2

as follows. notwithstanding the complication of operations evolved in the flashover mechanism, several test and
theoretical studies have been assigned to the comprehension of the phenomena leading to flashover of contaminated
insulators in order to expansion models letting one to predict the critical voltage and critical current[1-8].

Prevalent feature between all these models is a simple display of an arc promotion arc included of a partial arc in
series with the resistance of the cross section of contaminated layer. Often of these models are confined to the static
state of the phenomenon and do not include to the dynamic state of some particular parameters of the flashover. In
addition to, the few available dynamic models reported in literature[20]. The above discussion manifests that agreement
has not been reached on the value of n, the differences of n when AC function, respectively, the relationship between n
and degree of pollution, and the relationship between n and pollution level. On the other, often works offered in
literature uses communication giving the critical flashover voltage deduced from the Simplified equivalent electrical
circuit in[13]. In these relationships, the critical voltage dependent two experimental discharge characteristic constants n
and N, the quantities of which is different from one author to another[1-12].

The present work deals with the development of a mathematical model to predict flashover voltage of polluted
insulators energized with AC voltages. The objective is to involve the major parameters, namely, profile of the insulator,
pollution salinity, the critical current and the environmental conditions such as humidity and temperature. An analytical
model letting one to predict the critical voltage, critical current and arc length for contaminated insulators. Also
constants arc are dependent insulators profile and type of pollution, given the different insulators profile and different
types of pollution, each insulator has its own constant. In this paper, is determined the arc constants using programs
written. For each insulator is determined an arc constant in which case the answer to experimental result have many
similarities and the results are satisfactory. The experimental results allowed a highly exact estimation of the arc
constants of the insulator by using a mathematical method.

Background
Model[12] among the first flash overs models is presented for contaminated insulators. The equivalent circuit model of

contaminated insulators composed of two resistance together series one the contaminated layer resistance pR and a

dry bond resistance aR and supplied by a voltage U (Figure 1). Thus, one can display arc voltage aV :
V =ra a aXI (1)

a p aU V R I 

Where aI is the creepage current, whereas n and N are the arc constants and pR is the pollution layer and

supplied by a voltageU . pR indicative comprehensive segment of the contamination layer:
( )p pR r L X 

Where pr resistance per unit length and L is the total creepage length of insulator. On the other, the electrical field

discharge is aE . According to [6, 20] is such as:
na

a a
V

E NI
x

 

Where N and n are the arc constants. The values of these constants depend upon the environment (i.e., the
experimental conditions) in which the discharge is burning [20].

By integrating relations (1), (2) and (3):
( )n

p aU XNI r L X I  

The purpose of scrutiny is development an analytical model that allowing to predict the critical voltage levels.



3

Figure 1, (a), contaminated insulation layer and dry band. (b), Model of discharge according to obenaus.
The critical length, critical current are determined using the following relations [17, 18]:

0U
X





0U
I






Using equations (5), (6) and (7), one can display the critical length of the arc: cX , the critical current cI and the

critical voltage cV [1, 6, 8, and 20]:

1c
LX
n



1
1( ) nc

p

NI
r



1
1 1

n
n nc pV LN r 

Arc propagation can occur when a pE E
(Ep is Electrical field contaminated layer) .

The proposed model
3.1 Determine the electrical conductivity and polluted linear resistance

In order to determine the surface conductivity, according to the severity of pollution salinity, experiments
have been performed in laboratory nano Chemistry Department babol University of technology. The amount of salt
needed after measuring its weight by digital scales in 1000 ml of distilled water dissolved and then by conductivity
meter, electrical conductivity of the solution was measured. This experiment were carried out for different amounts of
salt in the range [0-100] (g) and 40(g) kaolin were in the range of all kinds of pollution intensity (light, moderate, heavy
and very heavy) will be covered. In figure 2 measured values for conductivity in salinity at these experiments is given.
Using data obtained and programs written in MATLAB, mathematical linear relationship for conductivity according to
pollution salinity is determined:

1.6 30p S   
(11)

Where S and p are the salinity in (g/l) and surface conductivity in ( s ).

0 10 20 30 40 50 60 70 80 90 100
0

50

100

150

200

salinity (g/l)

co
nd

uc
tiv

ity
 (u

s)

 

 

experimental data
eq. 11

Figure 2: curve fitting experimental data of conductivity

According[8] to the total resistance of the pollution layer, it is defined by the following equation:



4

p
p

FR



(12)

where F is the form factor of the insulator:

0
( ( ))

L
dlF
D l




(13)
( )D l is the diameter of the insulator, varying across its leakage length L .

Using equations (11) and (12) one can define the linear resistance per unit pollution layer:

p
p

Fr
L


(14)

3.2 Investigated insulators and testing of pollution
Different type of insulator is currently used in Iran distribution network. Standard cap & pin glass and porcelain

insulators are widely used in most of the old system. Some 4 different types of insulators were investigated for the
purposes of this paper. The characteristics and a view of Picture these insulators are presented in Table 1 and figure 3.

(a)

(1) (2)

(3) (4)
(b)

Figure 3, a; 3D profile of insulator- b: Glass and porcelain insulators

Number of insulator 1 2 3 4

D(mm) 255 280 280 255

H(mm) 146 146 170 127

L(mm) 440 442 370 320

TLd(mm) 320 297 240 190

BLd (mm) 120 145 130 115

Material porcelain Glass porcelain Glass

Table 1. characteristics value of glass and porcelain insulators



5

3.3 Laboratory setup
The experimental arrangements have been prepared according to IEC60507 as shown in Fig. 4. All experiments

were carried out in a fog chamber in the high voltage laboratory of Babol University of Technology with a volume of
(200cm × 200cm × 200cm). The main power supply is a single phase, (220 V/100 KV), 5 kVA, and 50 Hz transformer.
Supplied voltage by tables controlling and Feeders by a cable with the appropriate dielectric Durability, to testing
insulators in the fog chamber is applied. (Figure.4)

Figure 4: Laboratory investigation setup (right: HV Transformer and regulator – left: fog chamber and fog generator).

Figure 5 illustrates the schematic diagram of the experimental setup, which consists of a 500 kV ac high voltage
system, a capacitance divider (C1=100 PF and C2=2500 PF) for applied voltage measurements, a shunt resistor (R =510
Ω) to measure the leakage current flowing through the test object, a data acquisition system, and a vertical circulation
climate room, making possible not only very realistic simulation of different types of pollutions, but also the collection
of natural frost, fog in particular.

Figure 5: Circuit of the laboratory test setup
3.4 Determining the leakage current and arc resistans

For an arc burning in air and water vapour at atmospheric pressure, radiation losses are small and the
majority of the electrical dissipation heats the surrounding gas. By considering that the energy P is dissipated
only by thermal condition in the arc[11], one obtains.

a a
dQP E I
dt

 

Q being the quantity of heat. According to Fourier’s law[11]:
dQ TA
dt n

 
 



Where  , A , n and T are respectively the thermal conductivity, the isothermal surface element, the
normal in the direction of the temperature gradient and the temperature of the medium[11].

By using an appropriate assumption (assuming the isothermal surfaces of the arc channel to be hemispherical
and the temperature propagation equation to be one-dimensional), the energy balance can be expressed as[11]:



6

a a avE I T

Where T , the axial temperature is required to provide the necessary amount of thermal ionization and
av is the thermal conductivity. The thermal conductivity of such a gas mixture is not readily calculated. Some

available expressions have been used to give an approximate value according to[11].

(1 )1

v
i

av
i ii a

i

A v
v












Where i , iv and iA are the thermal conductivity, volume fraction and a kinetic gas coefficient respectively for
each constituent; indices a and v are related to air and water vapour respectively [11].

On the other hand, according to Ohm's law:
n

a a a aE r I NI  

By combining (17) and (19), it yields
1

(1 )
( )

nav
a

T
I

N
 



Resistance per unit of the discharge is function of temperature based on relations (1) and (4):
( 1)n

a ar NI  

3.5 The new model of critical current arc length and voltage
At the critical condition to wit a circle that before the complete flashover, the leakage current is called critical

current. Due to the influence of the insulator geometry on the critical parameters in the most recent article of this

effect is not seen in the critical current. In others, such as the [15] constant factor (
( 1)2( )

1.3
nK  

) was considered. In
this paper, according to the impact of insulator geometry and contamination on the leakage current, a
mathematical relation is presented to have a good agreement with experimental results.

1
1( )nc

p

MNI
r



Where M is:
TLdM
BLd


(23)

Where TLd is equivalent Top leakage path per shed and BLd is equivalent Bottom leakage path per shed. For arc
length according to [1-15]:

1c
LX
n




hence, the critical voltage is obtained by substituting cI and cX into equation (5). Thus, the critical flashover voltage
will be:

1 1
1 1 1 ( )

1

n
nn n nc p

LV N r M M n
n

   


The relationship 25 by approximation (n<1, 1nM   ) is modified to relationship 26. To obtain this
relationship, the approximation does not greatly influence the critical voltage, and numerical value calculated
through these two equations.

1 1
1 1 1

n
n n nc pV LN r M  

(26)
The above equations ((22) to (26)) indicates that the critical current and the critical voltage depend to the Profile

insulators, type of contamination and temperature of discharge band.



7

The relationship (26) indicates that having the exact values of the arc constants can be polluted insulator behaviour
until complete flashover is studied.
3.6 Determining the arc constants

Using the analytical relations, which are based on the polluted insulator model, the most important agent, is
the determination of the arc constants N and n . the pervious literature illustrates s that the real values of
constants N and n , in the case of thin pollution layers, are in the range of (10-500) and (0-1),
respectively [6, 14, 15]. Constants arc are dependent insulators profile and type of pollution, given the different
insulators profile and different types of pollution, each insulator has its own constant. In this paper, to determine
the arc constants using programs written in the software MATLAB, for each insulator is determined by an arc
constant in which case the answer to experimental result the same insulator have many similarities and the results
are satisfactory. The experimental data ( cU , against S) and the geometric characteristics of the tested insulators
were put in equation (26), thus leading in relations with unknowns the discharge constants N and n . This
equation, with unknowns the arc constants N and n were solved using different method (Genetic algorithm,
newton-raphson and other statistical methods) that the minimum the difference between critical voltages tested
with the mathematical model presented in equation (26). In table 4, the results obtained by these tests are given.

(N, ) minimumcU f n  (27)
Insulator N N

1 66.9218 0.5293

2 64.0350 0.5402

3 67.5785 0.5279

4 65.2565 0.5579

Table 4. the arc constants N and n for tested insulators
Performance model

The accuracy of the model was verified by comparing the mathematical results with measurements found in the
literature besides the already presented experimental results. Fig. 6 shows a collation of the presented model and
experimental results. According to [17], it can be seen that there is a good relation and Accurate between the presented
model and the measured data at every contamination severity. This is of great concern in act especially for the
prediction of occurrence flashover. Consequently, the mathematical model developed in this work represents a good
relation for predicting the critical flashover voltage and the critical current.

0 10 20 30 40 50 60 70 80 90 100
1

1.5

2

2.5

3

3.5 x 104

Salinity(g/litter)

C
ri

tic
al

 v
ol

ta
ge

(v
ol

t)

 

 
Model (insulator: NO1)
[17]
Experimental data

Very
heavy

Heavy

Medium

Light

a.

0 10 20 30 40 50 60 70 80 90 100
1

1.5

2

2.5

3

3.5 x 104

Salinity(g/litter)

C
ri

tic
al

 v
ol

ta
ge

 (v
ol

t)

 

 
Model (insulator: NO2)
[17]
Experimental data

Very
heavy

Heavy

Medium

Light

b.



8

0 10 20 30 40 50 60 70 80 90 100
0.5

1

1.5

2

2.5

3

3.5 x 104

Salinity(g/litter)

C
ri

tic
al

 V
ol

ta
ge

(v
ol

t)

 

 
Model (insulator: NO3)
[17]
Experimental data

Heavy Very
heavy

Medium

Light

c.

0 10 20 30 40 50 60 70 80 90 100
0.5

1

1.5

2

2.5

3 x 104

Salinity(g/litter)

C
ri

tic
al

 V
ol

ta
ge

(v
ol

t)

 

 
Model (insulator: NO4)
[17]
Experimental data

Very
heavy

Medium

Light

Heavy

d.
Figure 6: Critical voltage model in compared experimental test and [17]

a: NO.1, b: NO.2, c: NO.3, d: NO4
Figure 7 displays changes critical voltage in the pollution resistance using the relationships (22) and (26) .

0 1000 2000 3000 4000 5000 6000
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

pollution resistance(ohm/cm)

C
ri

tic
al

 C
ur

re
nt

(A
)

 

 

Model(equation.29)

Figure 7: Critical current variation versus pollution resistance

5. Conclusion
In this paper, we presented a new mathematical model for critical parameters of polluted insulators based on

experimental tests. The validity of these models was verified by comparing the computed results of critical flashover
voltage based on experimental tests and theoretical data for different insulators. By using arc constants of each insulator
in different polluted condition it is possible to estimate the critical voltage close to real condition. It is expected that
intricate tests for the prediction of this behavior can be considerably decreased or even eliminated.

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