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Engineering, Technology & Applied Science Research Vol. 5, No. 4, 2015, 818-824 818

www.etasr.com Gong et al.: Optimum Shape Design of Metal-Enclosed 550 kV Disconnectors Based on Response…

Optimum Shape Design of Metal-Enclosed 550 kV
Disconnectors Based on Response Surface Method

and Finite Element Analysis

Ruilei Gong
School of Electrical
Engineering, Xi’an
Jiaotong University,

Shaanxi, Xi’an, 710049,
China

Shuhong Wang
School of Electrical
Engineering, Xi’an
Jiaotong University,

Shaanxi, Xi’an, 710049,
China

Xianjue Luo
School of Electrical
Engineering, Xi’an
Jiaotong University,

Shaanxi, Xi’an, 710049,
China

Michael G. Danikas
Department of Electrical

and Computer
Engineering, Democritus

University of Thrace,
67100 Xanthi, Greece

Abstract— In this paper, the optimum shape design of 550 kV
disconnectors in Gas Insulated Switchgears (GIS) are firstly
presented employing the Finite Element Method (FEM) for
electric field analysis coupled with an optimal design method. For
effective analysis, the FEM is conducted in transient quasistatic
electric field, using a finite element FORTRAN code. The
structure parameters of disconnectors that provide the required
electric field strength are obtained by the Response Surface
Method (RSM) and the optimal values are presented by the
variation in maximal electric field strength. The RSM and
optimal design methods are also conducted by FORTRAN codes.
The optimal result reveals that a uniform electric field
distribution is achieved in 550 kV disconnectors. Additionally,
the optimal result of disconnectors is verified by the proposed
disconnector undertaken power frequency withstanding voltage
of 740 kV for 1 minute, lightening impulse of 1675 kV, and
operating impulse of 1300 kV, respectively.

Keywords—Disconnectors; response surface method (RSM);
optimization; structure design; finite element method (FEM)

I. INTRODUCTION

Disconnectors in Gas Insulated Switchgears (GIS) are a
switching device without an arc extinguishing device. An
opened state should be obvious (visible) and a breakdown is
not allowed under any circumstances, in order to ensure the
safety of the maintenance personnel. Furthermore,
disconnectors in the closed state must reliably carry normal
and/or short circuit currents. Because of the special role of
disconnectors, there is the need to consider its two working
states: the opened state, to meet the insulation design, and the
closing state, to meet the through flow capacity design and
insulation design. Therefore, the insulation of disconnectors
design is essential.

The insulation performance of disconnectors mainly
depends on the electric field distribution in the breaking. The
electric field distribution in the breaking state is directly related
with the structure shape of shielding. Therefore, the
optimization of disconnectors mainly concerns the optimization
of the shielding structure. Because of the structural complexity

of disconnectors, it is difficult to determine the field
distribution in the inside, and therefore a Finite Element
Method (FEM) is employed [1-2]. 3D modeling is more
realistic but due to the axial symmetry, a simplified two-
dimensional calculation can be considered not to affect the
analysis results. Herein, a commercial software is used and it is
compared with a simplified two-dimensional calculation and
three dimensional calculation and analysis. The result reveals
that the simplified model meets the accuracy required. In
addition, the model of disconnectors employs a two-
dimensional model of transient quasistatic electric field
considering the conductivity and permittivity of materials in
order to obtain more accurate results.

At present, several optimization methods, e.g. genetic
algorithms and the simulated annealing method, have been
used in combination with FEM, but they usually suffer from
low efficiency. The Response Surface Method (RSM) has been
successfully combined with FEM as an efficient way to realize
the optimization of structure parameters of different electric
devices [3-5]. RSM uses discrete points obtained according to
some experimental design rules and constructs the optimal
approximate function to attain optimal results. RSM combined
with FEM can effectively reduce the calculation and workload,
and also obtain high accuracy results. In this paper, this is the
first time that RSM and FEM are successfully applied to the
optimization of a high-voltage disconnector in GIS.
Additionally, in order to realize optimization analysis for
automatic calculation, RSM, FEM and subsequent optimization
algorithm programs are written using the FORTRAN language.

II. TRANSIENT QUASISTATIC ELECTRIC MODEL

Analysis of the internal electric field of disconnectors, due
to the lightning impulse voltage, is a function of time as well as
of the conductivity of the insulating medium. Therefore, this
paper adopts the quasi-static electric field model for transient
analysis. A transient electric analysis determines the effects of
time-dependent current or voltage excitation in electric devices.
For this reason, the time-varying electric and magnetic fields

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are uncoupled, and the electromagnetic field can be treated as
quasistatic [2, 6-8].

In a quasistatic electric model, the circulation of the electric
field is essentially zero for any path; this allows the electric
field E


to be represented by a scalar potential function  as:

E  


(1)

The potential function  is continuous in the domain, i.e.
there are no charge double layers. From the Ampere-Maxwell
equation follows that:

( )
0

D
J

 
  






(2)

where J


is the conduction current density and D


is the electric
displacement.

We assume that at 0t  there are no ‘free charges’ in the
system. The electric displacement in the linear isotropic
materials is simply related to the electric field:

D E
 

(3)

In the nonconducting regions, from (1), (2) and (3) follows
that the Laplace equation for the potential holds:

0( ) 0j      (4)

where
0

 is the permittivity of free space and j is the
relative permittivity of the insulating material.

In the materials considering the conductivity, the current
density and the electric field are related by:

( )J E E
 

(5)

where  E is a field dependant conductivity. In the
regions considering conductivity, from (1), (2), (3) and (5) it
follows that the dynamics of the electric potential is described
by a diffusion-like equation:

0[ ( ) ] [ ( )]jE
t

    


     


(6)

In order to analyze the field distribution of the conductive
materials, (6) can be discretized by applying the backward
Euler method:

2 2

0
( ) ( )

[ ( ( )) ( )] j
t t t

E t t
t

 
   

   
   


(7)

where t is the time and t is a suitable small time interval.
Equation (7) can be rewritten as:

2
0

0
( )

[ ( ( ( )) ) ( )]
j

j
t t

E t t
t t

  
   

  
     

 
(8)

Summarizing, (4) and (8) can be written as:

[ ( ) ] 0c f     (9)

where

0
( )

( ( ))

j
c

E t t

in insulating materials
in conductive materials

 


  


 

 

0
( )

( ( ))

j
c

E t t

in insulating materials
in conductive materials

 


  


 

 

2
0

( )j
f

t t t

in insulating materials

in conductive materials  


 

    

The electric field distribution in disconnectors is symmetric.
By adopting a cylindrical reference system with the z-axis
coincident with the axis of the conductor of disconnector, the
variational problem of (9) is shown as:

( ) 2 ( ( ( ) ) )ES S

F c E dE rdrdz f rdrdz   
 

      




 
(10)

The corresponding finite element equation is:

( ) ( )K P   (11)

where  K  is the n n nonlinear matrix,  is the n column
vector of the nodal potentials, and  P  is the sum of the n
column vector produced by dealing with Dirichlet boundary
conditions and produced by equivalent charge density.

Equation (11) can be iteratively solved by means of the

Newton-Raphson method. Starting from a guess n of the

solution, an improved approximation 1n  is obtained by
solving the series linearized problem. This iterative algorithm
is repeated until the condition

1n n

 




 
 (12)

is verified.  is a prescribed small value, and the

corresponding value 1n  is assumed as the solution of the
system at time instant t . The solution at time instant t t 
could be finished through solving (11). And last, in all time
domains the solution can be attained.

This numerical model represents an improvement of the
quasistatic electric field approach presented in [9] for two main

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reasons. The first one concerns that the model can finish the
transient electric field analysis. And the second is that the
model considers the conductivity of the material, thus
providing more accurate results..

III. RESPONSE SURFACE METHOD

The response surface method (RSM) is a powerful tool to
build up a macro model for an approximation of the desired
system response. The main advantage in this macro model
would be the significant reduction of the needed number of
simulations to characterize the electric insulation performance
of disconnectors, and thus make the electric field optimization
of disconnectors feasible and efficient. The macro model
generally comprises a simple, mathematical expression that
describes the desired objective or response, obtained from
either measurements or simulations, as a function of the
specified design parameters, which are the input variables to
the simulations. The expression is usually one, two or higher
degrees of the design parameters. The technique engaged in
model fitting is originally stemmed from the design of
experiments techniques. The computational cost needed for
deriving the expression is orders of magnitude much less than
that needed for simulations. Once this expression is obtained, it
can be used to replace the simulations, and furthermore, it
makes feasible for the search of the optimal combinatorial set
of design parameters that minimize the desired goal.

A. Basic Principles

Constructing the response surface model requires an
iterative process. In this study, numerical simulation that adopts
the FE analysis is applied to carry out the maximum of electric
field strength associated with selected design points, which are
needed to fit all the coefficients in the regression model. First
define  0,1, ,j j k   the coefficients of the regression
model and  the error. By further assuming that there are N
numerical experiments  N k , the variance of the error  is
equal to 2 and each

 is a random variable, the relation
between the dependent response T and the independent

variables  1 2ˆ , , , kX x x x x  can be denoted as:

X̂ T   . (13)

By further performing the minimization of the least square
error norm of the random error vector, the regression model
can be derived as:

ˆY Xb (14)

where Y is the estimated response and b the estimated
solution of the regression coefficients to this minimization
problem, denoted as:

1ˆ ˆ ˆ( )T Tb X X X T (15)

where Y is the estimated response and b the estimated
solution of the regression coefficients to this minimization
problem, denoted as:

  1ˆ ˆ ˆT Tb X X X T (16)
Note that the mathematical expression can be linear or

nonlinear. The choice of the degree of the assumed polynomial
model would be dependent of the number of simulations to be
performed and the desired accuracy. In fact, it would not be
practical in real applications to have a polynomial degree
greater than 2 because of the significant increasing of the
number of simulations. For one degree of polynomial
expression, it is presented as

0
1

j j
j

y x  


   (17)

where y is the response,  1, 2, ,jx j k  the
independent variables,  0,1, ,j j k   the coefficients of
the regression model, which are to be determined using
regression analysis techniques, and  the random error. As the
numerical simulators are deterministic, the error would only
emanate from the fitting inaccuracy. Furthermore, when the
linear regression model is incapable of accurately describing
the response as function of the specified design variables, the
nonlinear regression model can be then attempted. A quadratic
regression model with k independent variables is denoted as

0
1 1 1

k k k

j j ij i j
j i j

y x x x   
  

      (18)

The experimental design plan used in this work is the
Central Composite Design (CCD) proposed by Box and Wilson
[10]. It is essentially an orthogonal one such that it would allow
a better estimation of the coefficients of the regression model.
It consists of all the two-level factorial points, the central point

and the axial points or star points. In total, there are 2k
factorial points, 2k axial points and a central point in a CCD
model, where k is the number of independent design variables.
Basically, the model is formed by two parts:

1) two-level factorial points with an added central point;

2) the symmetrical star points aligned in the axes of the
factors and the central point.

Note that the first part can provide the estimation of the first
order and two-factor interaction polynomial terms while the
second part can allow the fit of the quadratic terms. Once the
important design parameters are determined, a complete CCD
plan for design parameters can be then defined. The
experimental design plan used for a 2-independent-variable
system is specifically listed in Table I where 1x and 2x are the

independent variables,  1, 2, ,
i

y i n  the desired response,

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and “+1” is the upper limit, “  1” the lower limit, and “0” the
central point of the independent variables.

B. Statistical Tests

It would be generally required to check the validity of the
mathematical expression constructed from the regression
analysis, and also the importance of the included factors. One
of the possibilities is to examine the relative and absolute errors
between the exact analysis and the responses based on the
mathematical expression. Other can be sought through
statistical tests. In this context, two simple statistical
hypothesis-testing procedures, including F-test and R2 test, are
used to get a basic indication of the validity of the constructed
model. The F-test that basically adopts the analysis of variance
examines the significance of the regression model while the R2
test provides an informal indication of how well the estimated
regression model describes the relationship between the
independent and dependent variables. In other words, the R2
gives the fraction of variation accounted for by the regression
model fit to the observed data. Moreover, the verification of the
regression model is also tested by a new selected design point
that is not included in the fit.

In F statistics, the statistical index F0 is defined as

 0 1
SSR SSE

F
k N k

  
       

(19)

where N is the total selected design points in the experimental
design plan, k denotes the number of independent design
variables in the model, SSR expresses the sum of squares due
to regression, and SSE is the sum of squares due to residual.
Accordingly, the total sum of squares SST can be expressed as:

i
iT

SST SSR SSE T T
N



 
 

     (20)

where iy is the i -th actual response. If 0 , , 1k N kF F   ,
where  is the specified significance level in the
F distribution, the null hypothesis, 0H is shown in the
following two-sided hypotheses:

0 1 2: 0kH      

and

1 : 0 { 1, 2, , }jH for some j in j k    (21)

is rejected, implying that at least one independent design
variable is significant to estimated response.

In the 2R test, the coefficient of multiple determinations
2R is defined as:

2 1
SSR SSE

R
SST SST

   (22)

Note that 20 1R  . If  1, 2, , 0j k  , then 2 0R  .
Furthermore, if all responses derived from the exact numerical
simulations are fully equivalent to the estimated responses,

then 2 1R  , suggesting that the fit of the least square line to
the data points is perfect.

TABLE I. EXPERIMENTAL DESIGN PLAN OF TWO-INDEPENDENT-
VARIABLE SYSTEM

1x 2x Result

1 +1 +1
1y

2 +1  1
2y

3  1 +1
3y

4  1  1
4y

5 +1.414 0
5y

6  1.414 0
6y

7 0 +1.414
7y

8 0  1.414
8y

9 0 0
9y

IV. THE MODEL OF DISCONNECTORS

The structural model of disconnectors in GIS is shown in
Figure 1, where the horizontal break is of disconnectors, and
the vertical break is of the earthing switch. The analysis model
in disconnectors can be simplified to a two-dimensional one
without affecting the calculation results. The disconnector
model for the simplified two-dimensional analysis is shown in
Figure 2, and the key parts of disconnectors are marked. The
relative permittivity and conductivity of each part are given in
Table II, in order to finish finite element calculation under the
lightning impulse voltage. Lightning impulse voltage
waveform applied on the conductor is shown using the
following function [11].

1 2( ) 1675 ( )
t t

T Te t e e
 

   (23)

where 1 68T s , 2 0.4T s 。

V. THE ELECTRIC FIELD ANALYSIS IN DISCONNECTORS

In Figure 2, several parameters, which may have an effect
on the electric field distribution, have been described. The
original parameters of disconnectors are listed in Table III.

According to relative standards, lightning impulse
withstand voltage with 1675 kV peak value should be applied
on one side of the break in the disconnectors, the opposite peak
value of power frequency withstand voltage 450 kV applied on
the other side. The disconnector model with original
parameters was solved using a FORTRAN. The potential
distributions and electric field distributions are depicted in
Figure 3.

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TABLE II. THE RELATIVE PERMITTIVITY AND CONDUCTIVITY OF EACH
PART

Conductor and
shield

Metal part insulation SF6

j   3.3 1

  s m 3.5×10
7 3.5×107 1×10-13 0

Fig. 1. Structural model of disconnectors

Fig. 2. Simplified model of disconnectors (1,4-the conductor and the
shield, 2-the metal part, 3-the insulation, 5-SF6)

TABLE III. THE DESIGN PARAMETERS OF DISCONNECTORS (MM)

parameters L h t 1R 2R

Initial value 180 111 20 25 95

Fig. 3. The potential distribution and field distribution in the model

of disconnectors

As shown in Figure 3, the maximal electric field strength is
29.591 kV/mm with the reference value being 24 kV/mm in
[12] (when the gauge pressure value of SF6 is 0.4 MPa, the
reference value can be computed). The results indicate that the
design of disconnectors should be modified because the
maximum electric field strength value has exceeded the
allowable value.

In Figure 2 and Table III the several main design parameters
are shown. Next, the present study focuses on the change of
electric field by altering the design parameters. The relative
results are listed in Table IV to VI.

a) Altering parameter t , which is the distance between the
contact and insulation (Table IV). The results show that
parameter t has almost no effect on the maximal electric field
strength in disconnectors.

TABLE IV. THE RELATION BETWEEN PARAMETER t AND THE MAX
ELECTRIC FIELD STRENGTH

Parameter t (mm)
maxE （kV/mm）

20 29.59
10 29.56
5 29.56

b) Altering parameter L , which is the distance between the
contacts (Table V). The results demonstrate that parameter L
has almost no effect on the maximal electric field strength in
disconnectors.

TABLE V. THE RELATION BETWEEN PARAMETER L AND THE MAX
ELECTRIC FIELD STRENGTH

Parameter L (mm)
maxE (kV/mm)

180 29.59
190 29.33
200 29.20

c) Structural parameters h , 1R and 2R of the shield are

rather significant to the electric field distributions, so these
three parameters are considered together. In Table VI the
electric field distributions in disconnectors can be more
uniform owing to the change of these three parameters.

TABLE VI. RELATION BETWEEN PARAMETERS h , 1R AND 2R AND THE

MAXIMAL ELECTRIC FIELD STRENGTH maxE

No. h （mm） 1R （mm） 2R （mm） maxE （kV/mm）

1 121 60 95 24.67
2 121 60 150 24.72
3 111 25 95 29.59

Based on the aforementioned analysis, it is concluded that
parameters t and L have almost no effect on the field
distributions, but parameters h , 1R and 2R are vital, namely,
the structural parameters of the shield have more effect on the

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field distributions. So, parameters h , 1R and 2R should be
selected as design variables.

VI. OPTIMUM SHAPE DESGIN OF DISCONNECTORS

The shape design optimization mathematical equations of
disconnector model can be described as follows.

1 2 3 maxmin ( , , )f x x x E (24)

where maxE is the maximal electric field strength value, 1x ,

2x and 3x represents the parameters h , 1R , and 2R in

Figure 2, respectively.
The regions of the design variables are listed as follows

according to the relative working experience.

110 150

20 60

60 120

 


 
  

(25)

The values of design variables and the code of central
composite design are listed in Table VII and VIII respectively.
The  value is 1.216.

TABLE VII. THE INITIAL VALUES OF DESIGN VARIABLES

Design
variable

Code -  -1 0 +1 + 

h (mm) 1x 117.8 120 130 140 142.2

1R (mm) 2x 27.8 30 40 50 52.2

2R (mm) 3x 77.8 80 90 100 102.2

TABLE VIII. THE CODES OF CENTRAL COMPOSITE DESIGN COMPRISING
THREE DESIGN VARIABLES

Design points Code points
No.

h /mm 1R (mm) 2R (mm) 1x 2x 3x

1 120 30 80 -1 -1 -1
2 120 30 100 -1 -1 +1
3 120 50 80 -1 +1 -1
4 120 50 100 -1 +1 +1
5 140 30 80 +1 -1 -1
6 140 30 100 +1 -1 +1
7 140 50 80 +1 +1 -1
8 140 50 100 +1 +1 +1
9 130 40 90 0 0 0
10 117.8 40 90 -  0 0
11 142.2 40 90 +  0 0
12 130 27.8 90 0 -  0
13 130 52.2 90 0 +  0
14 130 40 77.8 0 0 - 
15 130 40 102.2 0 0 + 

The results can be obtained by calculating the design values
using the FEM in turn. Based on the second order model, the
response surface model can be obtained as follows:

1 2 3 1 2 3 1 2

2 2 2
1 3 2 3 1 2 3

( , , ) 26.81 1.78 0.62 0.55 0.05

0.13 0.13 0.21 0.3 0.25

f x x x x x x x x

x x x x x x x

    

    
(26)

Equation (26) can be solved using sequential quadratic
programming, and the optimal results are shown in Table IX
The electric potential distributions and electric field
distributions in the disconnector with optimum structural
parameters have been depicted in Figure 4 using FEM.

TABLE IX. THE CONTRAST BETWEEN THE INITIAL AND OPTIMAL RESULTS

Parameters h /mm 1R /mm 2R /mm maxE (kV·mm
-1)

Optimal value 138.16 33.96 86.76 22.90
Initial value 111 25 95 29.59

Fig. 4. The potential and electric field distributions in the
disconnector with optimum structural parameters

As shown in Figure 4 and Table Ⅸ the maximum electric
field strength is 22.9kV/mm, lower than the allowable electric
field strength 24.0kV/mm. So the optimal results meet the
design requirements, increase the withstanding voltage, and
improve the reliability of disconnectors.

VII. CONCLUSION

In this paper, RSM coupled with FEM are successfully
applied, for the first time, for the optimization of 550kV
disconnector in GIS. The electric field distributions in 550kV
disconnectors have been analyzed and calculated based on
FEM in transient quasistatic electric field, through a
FORTRAN code. Then the optimum shape design in 550kV
disconnectors was completed based on the RSM coupled with
FEM. The optimal results indicate that the electric field
distributions in the optimized disconnector model are more
uniform than those of the initial model. So, setting the optimal
structural parameters in electric devices is feasible and highly
efficient using RSM coupled with FEM. The proposed
disconnector model with optimum structure parameters has
undertaken all withstanding voltage tests. The work discussed
in this paper can be employed to improve the insulation
performance and operational reliability of 550kV disconnectors
in gas insulated switchgears.

REFERENCES
[1] G. Zhang, Finite Element Method, Beijing, Mechanical Industry Press,

1991

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www.etasr.com Gong et al.: Optimum Shape Design of Metal-Enclosed 550 kV Disconnectors Based on Response…

[2] J. Sheng, Numerical analysis of electromagnetic field. Xi’an, Xi’an
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