IEEE Transactions on Electrical Insulation Vol. EI-22 No.1, February 1987

OPTIMIZING FIELD STRESS ON HIGH-VOLTAGE INSULATORS

M. Abdel-Salam and E. K. Stanek

Department of Electrical EngineeringMichigan Technological University

Houghton, MI

ABSTRACT

A method is described for optimizing the field stress on HVinsulators by modifying their profile, seeking a uniform dis-tribution of the tangential field among the insulator surface.This results in an increase of the onset voltage for surfaceflashover and in a significant saving of the space of the HVinstallation. The optimization process was achieved by an

algorithm developed for calculating the tangential field com-

ponent and mathematical expressions of the profile to be cor-

rected through an iterative procedure. The algorithm was

based on a modified charge-simulation technique to satisfy a

better matching of the boundary conditions to the electrodeand insulator surfaces involved in the HV installation. Thealgorithm is expanded to study the effect of contamination on

the tangential field distribution. It is found that the higherthe conductivity of the contamination layer, the higher is thefield uniformity along the insulator surface.

I NTRODUCT ION

HV insulators usually serve as a support/spacer of HVelectrodes with respect to grounded frames (e.g. ingas-insulated systems) or the ground plane (in air).In practice, there are many applications where the HVinsulators are surrounded by gaseous dielectrics andstressed between two electrodes depending on the ar-

rangement. Flashover takes place along the insulatorsurface if the tangential field is high enough to sus-

tain a discharge.

To keep the tangential field below the limit requiredfor a sustained discharge, the insulator length has tobe a minimum value for a given voltage rating with a

subsequent minimum cost of installation. Another ef-ficient approach is to design the insulator with an op-

timized profile with a resulting shorter length, thusminimizing the space needed for installation.

To optimize a HV insulator, the distribution of thetangential field component along its surface should beuniform, thus increasing the onset voltage of surfaceflashover. This was achieved [1,22] by correcting theprofile of the insulator step by step. At any point on

the insulator surface, the tangential field is derivedfrom the potential difference between two test points inthe vicinity of the point under consideration. If thetangential field exceeds the demanded value, either thepotential difference [1] is decreased, or the distancebetween the test points [2] is increased. In this paper,an algorithm developed for calculating the tangential

field component was applied to mathematical expressionsof the profile to be corrected. Then an iterative pro-cedure was used to modify some parameters involved inthese expressions to achieve an acceptable degree offield uniformity after a few iterations.

The algorithm developed for calculating the tangentialfield along an insulator supporting a HV electrode abovethe ground plane, was based on a modified charge sim-ulation technique. The HV electrode was simulated byfictitious lumped charges. The dielectric boundary wasalso simulated by lumped charges as opposed to othermethods [1,2] that used surface (area) charges. Gen-erally, lumped charges more accurately simulate curvedprofiles than area charges which approximate the profileby straight-line segments. Images of the simulationcharges with respect to the ground plane are consideredin the algorithm. The magnitudes of the simulationcharges were determined by minimizing a novel error-function formulated over the HV electrode and insulatorsurfaces.

To study the effect of contamination on the insulatorupon the tangential field, the algorithm was expandedto treat capacitive-resistive fields also. The sim-ulation charges were replaced by complex charges. Thecontamination layer was simulated by a uniform surfaceresistivity along the insulator. The magnitudes of thesimulation charges (real and imaginary components) weredetermined by the aforementioned concept of error-func-tion minimization.

0018-9367/87/0200-0047$01.00 @ 1987 IEEE

47

IEEE Transactions on Electrical Insulation Vol. EI-22 No.1, February 1987

First, the algorithm developed for calculating thetangential field is explained. Then, the method of op-timization is discussed and the effect of contaminationon the calculated tangential field distribution is in-vestigated showing how the developed algorithm will ac-commodate this effect. Finally, the obtained resultsare discussed.

PROPOSED ALGORITHM

Defining Equations

Fig. 1 shows an insulator supporting a HV electrodeabove the ground plane. In the non-contaminated case

z1

surface. These charges are simulated by fictitiouslumped ring charges to match the axial symmetry of thearrangement. The number of simulation charges in theinsulator is Ni, the same as that in air, Na.

Along the z-direction, the rings are uniformly distri-buted as shown in Fig. 1. The z-coordinates Z(i) forrings in air are the same as those in the insulator.

The Az difference in level of two successive rings is

Az = H/(Nil+) (2)

where H is the insulator height in m.

The radii of the ring charges R(i) in the insulatorare chosen to satisfy the equality

R2-R(i) = Az, i=Ne+l, Ne+2,.. .,Ne+Ni (3)

ElectrodeRadius = B1

Air

c1~~~~~O

);_

~i-z--~ Radius=B2

I Ground I

( j-1

Plane

---H Ring ChargeX Boundary point

Fig. 1: Charge representation of an insuZatorsupporting a HV eZectrode above the ground plane.

the field distribution is a purely capacitive one. Dueto the axial symmetry of the arrangement, the surfa-cecharge on the electrode is simulated by Ne fictitiousring charges, whose radii are given by

R(i) = l + RF sin6(i) (1)

where 6(i)=(i-1)7T/(Ne-l) for i-1,2,... Ne. RF is a fic-titious radius ranging from 0.2Rl to 0.5R1 where R1 isthe radius of curvature (m) of the electrode as shownin Fig. 1.

Along the z-direction, the rings are distributed uni-formly over the electrode thickness, see Fig. 1.

In the insulator, dipoles are aligned by the appliedelectric field and compensate each other through thevolume of the insulator, leaving net charges only on the

The radii for the rings R(i) in air are chosen to satis-fy the equality

R(i)-R2 = Az, i=Ne+Ni+l,. ..,Ne+Ni+Na (4)

To simulate the effect of the ground plane, images ofthe simulation charges are considered.

The potential p1(r,z) at any point (r,z) on the elec-trode/air boundary is the algebraic sum of the potentialsat this point, produced by the ring charges (and theirimages) belonging to the electrode and the insulator,

Ne+Nip1 (r,z) = .

i=1

Q(i) 2 K (k1) K(kJ]4TT0 Tr7 a1 a2

(5)

where Q(i) is the value of the ith ring charge in C,

a,=[{r+R(i)}2 + {z-Z(i)}2] 0 5, o2= [{rr+R(i) }2 +

+ {z+Z(i)}2] 05, k1-2[rR(i)05 a1, k =2[rR(i)] 0.5 a2, K isthe complete elliptic integral of tge first kind [3],and c, is the permittivity of free space.

Similarly, the potential W2(r,z) at any point on theelectrode/insulator boundary is the algebraic sum of thepotentials at this point produced by the ring charges(and their images) belonging to the electrode and theair;

Ne-14~2(r,z) =jI 4Trc 0

i~=1

Ne+Ni+Na

+iE1>Ne+Ni+l

2 K(kl) K(k2)ITr a1i a2

(6)

Q(i) 2 K(k1) K(k2)47r 0 Tr a1 a2

Keep in mind that Ni=Na. Of course, these potentialsf (r,z) and W2(r,z) must be equal to the applied voltageV, i.e.

f1(r,z) - V = 0

(7)

q2(r,z) - V = 0

The potential 1,(R2,z) at any point (R2,z) along theinsulator surface is the algebraic sum of potentials at

this point due to the ring charges (and their images)belonging to the electrode and the insulator if the pointis seen from the air side. If the point is seen from the

insulator side, the potential p2(R2,z) is the algebraic

H

I

^1InsulatorI 1 s~~~~~

EI r I

I

II -~ ~~~~~~ag,! A1 r-, ,

!9Dv

48

z -/ z z z-z z Z/ /-Z./ Z /r,

Abdel-Salam and Stanek: Optimizing field stress on HV insulators

sum of the potentials due to the ring charges (andtheir images) belonging to the electrode and the air.

Of course the potentials 4,(R2,z) and 42(R2,z) areequal, i.e.

¢1(R2,z) - W2(R2,z) = 0 (8)

The radial electric field Erl(R2,z) at any point(R23z) along the insulator is the vector sum of theradial field components at this point due to the ringcharges (and their images) belonging to the electrodeand the insulator if the point is seen from the airside. If the point is seen from the insulator side,the radial electric field Er2(R2,z) is the vector sumof the radial field components due to the ring charges(and their images) belonging to the electrode and theair, i.e.

Er, (R2 z) i 4ir iNN X

2 2 . 2 2 ~~~~~~~(9)X R2 (iJ-.R2 (Z-Z (iJ J ]Ci(klj-$l kill2at22

and[2 (iJ-2f (zfZ (iJ J 2] i Qk2) -_ 62 (2

2 £cyO2Ne

Er2(R2, Z) = i Q(i) 1xi=1 4'fhc0 1TR2

[R29 -R2+(z-Z(i))2] 2kl)-6J K(k )x

11 2

1 1

2 ~~~~~2[,R2 (-)-R9+ (zfZ(i) J2] i(k,g)-62K(k_g,2

N-Ni.+Na+ I Q(i) 1

i=Ne+N +14lTro TrR2

[R2(i)-R2+(z-Z(i) J2] x (kl - lK(kl)x i -

a1

(10)

air or insulator sides. However, the equality of poten-tial as calculated by Eq. 8 cannot easily be satisfiedfor every point on the insulator surface for the assumedsimulation charges.

Similarly, the radial components of the field in theair and in the insulator at any point on the insulatorsurface are related through the relative permittivityEr of the insulator. However, the radial field relation-ship given by Eq. 11 cannot easily be satisfied forevery point on the insulator surface for the assumedsimulation charges. This is explained further in thefollowing paragraph.

Therefore, the magnitudes of the simulation chargesQ(i)J, i=l,2.. . ,Ne+Ni+Na are so chosen that:

(1) The calculafed potentials (given by Eqs. 5 and 6)at a large number of boundary points ne>Ne on the elec-trode surface deviate slightly from the actual potentialV.

(2) The calculated value of the equality (given byEq. 8) at a large number of boundary points on the in-sulator surface n>Ni deviate slightly from the zerovalue.

(3) The calcul,ated value of the equality (given byEq. 11) at the same large number n of boundary points onthe insulator surface deviate slightly from the zerovalue.

To accomplish this, an error function has to be formu-lated over the electrode and insulator surfaces. Thedesired simulation charges are those which minimize theerror function.

The boundary points are located at gradually decreas-ing distances on approaching a contact point (insulator/HV electrode or insulator/ground plane).

Error Function

The deviation 6, at any point on the electrode surfaceis formulated using Eq. 7 as

61 = ¢1 (r, Z) - V = 42 (r, z) - V (12)

The deviation 62 at any point on the insulator surfaceis formulated using Eq. 8 as

62 = ¢1i(R2,z) - W2(R2,z)[R2 (i) -R2+ (Z+Z (i) J 2 i(kc2 -2K (k2J2-

°U26

where B -[{R R(i)}2+{z_Z(i)}2] 0'5,V2=-[{R2-R(i)}2 ++fz+Z(i3}2 °I, and i is the complete elliptic integralof the second kind [3]. The radial fieids Erl (R2,z)and Er2(R2,Z) are related to each other through therelative permittivity cr of the insulator

£rEr2 (R2-, z) - Eri (R2, z) = 0 (11)

Boundary Conditions

The actual potential at every point on the electrodesurface is V, the applied voltage. However, the poten-tial q(r,z) as calculated by Eqs. 5 and 6 cannot easilyequal V for every point on the electrode surface forthe assumed simulation charges.

Also, the potential at every point on the insulatorsurface is the same whether the point is seen from the

(13)

Also, along the insulator surface the deviation 63 isformulated according to Eq. 11 as

63 = crEr2 (R2, Z) - Erl (R2, z) (14)

The criterion for potential matching at ne boundarypoints on the electrode, potential matching at n boundarypoints on the insulator surface, and the radial fieldmatching at the same n points, is to minimize the summa-tion of the squared deviations over all the boundarypoints. In other words, the goal is to minimize the so-called error function f expressed as

ne n 5+2n1=12+ 1 (62+62)ji1 .i=i1

(15)

The criterion of matching all the boundary conditions ismathematically formulated as

'f = 0 i=1,2,.. .,Ne+N +N5 (16)

49?

fL1-

t

TCCC7>^%n+1nnFon Electrical Insulation Vol.. EI-22 No.l1. February 1987so 1 CLt_ X: I r n 1 l *= Lj a "

This technique was found to improve the stability ofpotential and field matching considerably [4,5].

From the simultaneous Eq. 16 the simulation chargesQ(i) could be evaluated using a computer program.

It is quite clear that Eqs. 7, 8, and 11, applied atthe chosen ne+2n boundary points, represent a set ofn,+2n equations. The number of equations is much great-er than that of the unknown simulation charges(Ne+Ni+Na). The matrix defining this set of equationsis sparse, it has several zero terms as shown in Fig.2. However, the number of equations in the set givenby Eq. 16 is equal to the number of unknowns. Thematrix defining the last set is free from zero terms.

SimlationCharge Electrode ielectric Air

Boun dary (j)Point Q(i) Q(i) Q(i)(i) _

on Electrode InALr Zero

(EquatiLon 7a)On Electrode In

Dielectric Zero(Equation 7b)

oni Equation 8 ZeroBoundaryAir/ Equation 11

Oielectric

Fig. 2: Zero terms of8, and 11.

the matrix defined by Eqs. 7,

when moving along the axis of the insulator, the profileradius has to be increased in that direction.

(2) Exponential mathematical expressions which definea smooth enlargement of the profile radius are used inthis work.

On formulating these expressions, some constraintshave to be considered at the contact points of the in-sulator with the electrodes involved (e.g. the HV elec-trode and the ground plane in the arrangement shown inFig. 1). It has been realized [7,8] that a dielectrichas to contact an electrode at right angles to avoid ex-cessive field intensification, the so-called "EmbeddingEffect". Some investigators [7] found that the fieldintensity may reach extremely high values if the contactangle at the insulator/electrode junction differs from900. Therefore, the expressions proposed for the pro-file have to satisfy the right-angle contact criterionfor the insulator with the electrodes.

For the geometry shown in Fig. 3, the profile radiuswas proposed to have the following expressions:

(18)r - RS for Oz<O.05H

r = R2+AR{exp[0.693(z-O.05H/(O.9H)]-1} for 0.05H<z<O.95H

r = R2+AR for O.95H<z.H

where AR is the ultimate value of the enlargement in theprofile radius (through the exponential increase ofradius along the z-direction from z=O.05H to z=0.95H).

Tangential Field CaZculations

Once the simulation charges are determined, the tan-gential field Elz(R2,z) at any point (R2,z) on the in-sulator surface is expressed as

Ne+=i {zZi:}hakji=i 47Tco Tr -y

(17)

{z+Z (i)1Uk(}2)_2

if the point is seen from the air side.

If the point is seen from the insulator side, the tan-gential field E2,(R2,z) is the vector sum of the z-fieldcomponents at this point due to the ring charges (andtheir images) belonging to the electrode and the air.

Of course, the tangential field EZ(R2,z) at a givenpoint can be determined from the difference of potential(R2, z) between two points in the vicinity of the point

under consideration [6].

INSULATOR OPTIMIZATION

The following is the procedure adopted for optimizingthe insulator profile, i.e. to search for a profilewhere the tangential field along it is uniform.

(1) Having calculated the tangential field distribu-tion for the non-optimized profile, define whether theinsulator profile will be enlarged on going up or downalong its axis. If the field distribution increases

Fig. 3: Proposed approach for profiZe optimization.

In Eq. 18, the profile has a radius equal to R2 closeto the ground plane and a radius equal to R2+AR close tothe HV electrode in order to satisfy normality of itscontact with the electrode and ground plane.

(3) For the profile described by Eq. 18, the coordi-nates of the simulation charges in air and in the insula-tor have to be changed accordingly (Eqs. 3 and 4 whereR2 is to be replaced by r given by Eq. 18). Of course,the n boundary points along the insulator surface haveto be chosen along the profile given by Eq. 18.

(4) Iteration is carried out over the value AR toachieve an acceptable degree of tangential field uni-formity along the insulator surface.

51Abdel-Salam and Stanek: Optimizing field stress on HV insulators

EFFECT OF CONTAMINATION

To study the effect of contamination of the insula-tors upon the tangential field distribution, the algo-rithm is expanded to treat capacitive-resistive fieldsalso. The simulation charges were replaced by complexcharges. The contamination layer was simulated by auniform surface resistivity along the insulator.

Capacitive fields discussed above are not frequencydependent as long as V is the instantaneous value ofthe applied voltage. However, capacitive-resistivefields are frequency dependent. If the applied voltageis dc, the surface resistance will determine the fielddistribution along the insulator surface. For ac ap-plied voltages, the capacitive and resistive (due tothe resistivity of the contamination layer) componentswill determine the tangential field distribution alongthe insulator surface.

Several papers [9-12] have been published on the anal-ysis of capacitive-resistive fields. It has been real-ized [12] that the charge simulation technique is moreaccurate compared to other techniques, e.g. finite-difference, finite-element, Monte Carlo, and surface-charge simulation techniques.

The modified charge simulation technique explainedabove for capacitive fields is extended to treat capac-itive-resistive fields as described below.

The following are the basic equationsformulate the error function.

necessary to

With complex simulation charges, the potential¢(r,z) and the field components Er(r,z) and Ez(r,z)become complex.

Therefore, Eqs. 7, 8, and 11 take the form

[¢1 (r, z)] - V = 0real

[12(r,z)] - V = 0real

[¢j(r,z)]imag

[f2(r,z)]imag

where I, is the surface current flowing at the jth pointand Sj is the surface area of the insulator correspond-ing to the jth point.

For 60 Hz applied voltage, the potential and field in-tensity change with the angular frequency w throughoutthe passive space, exponentially in the form exp[iwt][12]. The surface charge density is given as

[Pj.i(R2,z)]1, ,aComp

RjconrpZex S3 iw [

[kj(R23z)]comp

- [4j(R2,z)]comp

(23)- [¢jA+(R2,z)]camrO1

where Rj=Azl(27R2P.) 0 where p. is thevity in Q and Azl is the difference inFig. 2.

To formulate the error function withtion charges, the following deviationshave to be defined.

surface resisti-z-level shown in

complex simula-611, 612, @@ @632

Using Eq. 19, the deviations 61, and 612 are expressedas

(24)611 = [41(r,z)] -V = [c2(r,z)] - Vreal reaZ

and

(25)612 = [,1 (r, z)] = [2 (r, z)]imag imag

Also, by using Eq. 20, the deviations 621 and 622 areexpressed as

(26)621 = [1,(R2Pz)] - [f2 (R2, z) IreaZ reaZ

and

(19)

= 0

(27)622 = [p1(R2,z) ] - [p2(R2,z)]imag imag

Similarly, by using Eq. 21, the deviations 631 and 632are expressed as

631 = Er[Er2(R2,z)] - [Erl (R2,z)] - a reaZreaZ reaZ= 0

[¢1(R2,Z)] - 12(R2,Z)I = 0real real

[f, (R2' Z) I - [2(R2PZ)]imag imag

632 = Er [Er2(R2,Z)] - [Er(R23Z)] - a gimag imag

(28)

(29)

(20) Then, the error function expressed before by Eq. 15 takesthe form= 0

6r[Er2(R2,z)] - [Erl(R2,Z)] = areal real

£r [Er2 (R23 z)] - [Erl(R2,z)] = aimag imag

real(21)

imag

where areaZ and aimaa are the real and imaginary com-

ponents of the surface charge density, a. The surfacecharge density a- at the jth point on the insulator sur-face is expressed [12] as

1t

[ai] = J dt (22)orpZe; " O

ne 2 2 n 222 22f = 1 (6 + 6) + (621+622+631+632)31i 11 12 j=1

(30)

and the criterion of matching the boundary condition ap-plied to the electrode and insulator surfaces takes theform

(31)aq(k) = 0 k=1,2,..., 2Ne+2Nl+2Na

where q(k), k=1,2,. . . ,Ne+Ni+Na represent the real com-ponents of the simulation charges Q(i), i=1,2, ....Ne+Ni+Na, while q(k), k-fnee+Ni+Na+1, . . .,2Ne+2Ni+2Na repre-sent the respective imaginary components of Q(i).

].~~~~~~~~

IEEE Transactions on Electrical Insulation Vol. EI-22 No.1, February 1987

The solution of the set of Eq. 31 determines the realand imaginary components of the simulation charges.

It is quite clear that Eqs. 19, 20, and 21, appliedat the chosen ne+2n boundary points, represent a set of2ne+4n equations. The number of equations is muchgreater than that of the unknowns 2Ne+2N1i+2Na. Thematrix defining this set of equations is quite sparse,it has numerous zero terms, see Fig. 4. However, thenumber of equations in the set given by Eq. 31 is equalto the number of unknowns. The matrix defining thelast set is full; it is free from zero terms.

Simulation Real Component of Q(i)Charge

Boundary (1)Point In In In(1) Electrode Dielectric Air Ele

EluatLon 19a Zero 2CO Electrode

In AiLrEkuation 19c Zero Zero Zero

Ekluation 19b Zero 2CO ElectrodeIn Dielectric

Equation 19d Zero Zero Zero

muation 20a Zero 2

Co Boundary E4uation 20b Zero Zero Zero 2Air/Dielectric

E3quation 21a

E4wuation 21b Zero

Fig. 4: Zero terms of the matrix defined by

Wlth contaminated insulators, the tangential fieldalong the surface has real and imaginary components. IncreasingSuch components can be expressed by an equation similar sometimes le;to Eq. 17 by replacing Q(i) by the respective component of producingof the charge. geometry [14

by the preseiRESULTS AND DISCUSSION the two-diel

1).<CSST versus MCST

In the MCSCompared to the proposed modified charge simulation are chosen ii

technique (MCST),CCST (conventional charge simulation The number o:technique) has to limit the number of boundary points stant while Ito be equal to the number of unknowns. For capacitive creased in tiand capacitive-resistive fields, the matrix defining the passive spac(set of equations describing thepassive space has numer- number of boious zero terms as shown by Figs. 2 and 4. Subsequently, the boundarythe matrix has a wide variation in its elements ranging surfaces. A'from very high values to zero values. This reflects it- number of sinself in the "matrix condition", when solved by Gauss or All of this iCrout decomposition algorithms [13]. On the other hand, the matrix olthe set of equations involving the unknowns are formu- (Eq. 16 for clated in MCST to match the given boundary conditions to resistive fiEa large number of boundary points on the surface of theelectrode and the insulator. Then, through the error- Accuracy offunction minimization, the equations are reduced innumber to equal that of the unknowns. For both capac- For capacititive and capacitive-resistive fields, the matrix to be Ne, Ni, and Asolved is not only free from zero terms but also has no spectively.wide variation in its elements as in CCST. points on th

are chosen ec

The values of the unknown charges obtained in theCCST satisfy the boundary conditions in a satisfactorymanner only with a careful choice of both the number ofsimulation charges and the coordinates of these charges.Therefore, the simulation accuracy depends strongly onthe assumptions concerning the simulation charges inboth number and coordinates [14]. These assumptionsare usually made on the basis of experience which maydiffer from one investigator to another. As the geo-metry increases in complexity by involving more than onedielectric, the experience of the investigator may failto achieve the boundary conditions with an acceptable

accuracy. This explainswhy investigators [12]

Imginary Component o(i) used their expertise tochoose the coordinates for

In In in their simulation charges-ctrode Dielectric Air in a non-systematic manner

to achieve the boundaryZero Zero Zero conditions. For capaci

tive and capacitive-re-sistive fields, the number

zero of simulation charges waslarge but not stated ex-

Zero Zero Zero plicitly [12]. However,other investigators [6] do

Zero not mention how theychoose the coordinates of

3ero zero Zero their simulation chargesfor CCST to analyze capac-itive fields. They foundZero that with an increase inthe number of simulation

Zero charges (to increase thenumber of boundary pointsinvolved in the matrixformulation), the resultsvary and a critical number

Eqs. 19-20. is reached where the com-puted results becameacceptable.

ithe number of simulation charges with CCSTads to bad conditioning of solutions insteadbetter accuracy for a single dielectric

]. This agrees with the findings observednt authors when they tried to apply CCST forectric installation they investigated (Fig.

T, the coordinates of the simulation chargesn a systematic way as described before [1-4].f simulation charges (Ne+Ni+Na) is kept con-the number of boundary points (ne+2n) is in-he range 2 to 5x(Ne+Ni+Na) depending on thee to be analyzed. Of course, the larger theundary points, the better is the matching ofconditions to the insulator and electrode

s a result, the MCST does not have a criticalmulation charges whereas the CCST does [6].is attributed to the improved condition ofbtained from the error-function minimizationcapacitive fields and Eq. 31 for capacitive-elds).

the MCST

tive fields, the number of simulation chargesNa are chosen equal to 8, 12, and 12, re-However, the number ne of the boundarye electrode and n on the insulator surfact,qual 56 and 20, respectively.

Abdel-Salam and Stanek: Optimizing field stress on HV insulators

For capacitive-resistive fields, the number of simu-lation charges Ne, Ni, and Na are chosen as 6, 9, and9, respectively. However, the number of boundarypoints ne and n are chosen equal to 54 and 18. Here,each charge involves two unknowns (real and imaginarycomponents).

While one boundary condition is to be satisfied overthe electrode surface, there are two boundary condi-tions for the insulator surface. In other words, thenumber of equations written per point on the insulatorsurface is double that for a point on the electrodesurface. This is why the number of boundary points non the insulator surface is chosen smaller than thenumber of boundary points ne on the electrode surface.

For capacitive fields, shown in Fig. 1, Fig. 5 showscomputer results for the potential distribution along

0.02 0.04 0.C56 0.0 m

*Z

Fig. 5: PotentiaZ distribution along the insulatorsurface (in capacitive fields).

the insulator surface. At each point, the potential iscalculated when the point is seen from the air side41(R2,z) and when seen from the insulator side W2(R2,z).It is very satisfying to observe the precise equality of¢1(R2,z) and WR2,z) which differed by less than 1%.This indicates the accuracy of the proposed MCST.

Similarly, Fig. 6 shows computer results for the po-tential distribution along the insulator surface for a

capacitive-resistive field (i.e. for Fig. 1 with con-

tamination on the insulator).

The calculated potentials satisfy not only the bound-ary conditions on the insulator surface but also theboundary condition on the electrode where the deviationfrom the-applied voltage V did not exceed 1%.

Of course, the accuracy of the potential calculationalong the insulator surface reflects itself in the ac-curacy of the predicted values of the tangential fieldcomponent.

0.02 0.04

Fig. 6: Potential distribution aZong the insuZatorsurface (in capacitive-resistive fields).

InsuZator Optimization

While Fig. 7 shows the potential distribution alongthe insulator surface, Fig. 8 gives the tangential fielddistribution for the optimized and non-optimizedinsulators.

It is quite clear that the tangential field for non-optimized insulator is low near the ground plane and ishigh near the HV electrode. Such high field intensitymay be the origin of flashover on the insulator surface.On the other hand, the field is almost uniform along theoptimized insulator which makes the probability offlashover on the insulator lower for the same appliedvoltage.

As shown in Fig. 8, the line integral of. the tangen-tial field along the insulator surface equals the ap-plied voltage for both the optimized and non-optimizedprofiles. This is also a measure of the accuracy of theproposed MCST.

53

IEEE Transactions on Electrical Insulation Vol. EI-22 No.l, February 1987

Effect of Contamination

Fig. 9 shows how the potential on insulator surfacechanges its distribution with increased conductivity ofthe contamination layer. The higher the conductivity(i.e. the lower the ps) of the contamination layer, the

flV

d 0.03

higher is the field uniformity along the insulator sur-face and the higher the specific critical creepage dis-tance. The latter is defined as the critical creepagedistance per unit voltage (cm/kV) at which flashover oc-cures on a contaminated insulator [15]. The field uni-formity is attributed to the resistive field being moreand more predominant with an increase of the contamina-tion layer conductivity. At layer resistivities <106 Q,the resistive field becomes the predominating one andthe tangential field assumes a constant value along theinsulator surface.

Non-optimized, p = -o X X XOptimized, ps -On ------Non-optimized, ps = 10Si 0 0

1 000.02 0.04 0.06 0.08 m.

ph z

Fig. 7: Potential distribution aZong the profile ofoptimized and non-optimized insulators.

0I

0O. 0 = X)/

1- 0.11 X/ a/Dimensions in m. X /

/ 0. 6 0e /.

I'x ,

0 10 0

- 0.4

/0x ,IS

,.X1 2

I (b

'x,

01 1 1~~~~~~~~~~~~~~~~

0.02' .04 0.00

optimized

0

-10

I .02 00 0I .

0.02 0.04 0.00) O.W" m.

Fig. 8: TangentiaZ fieZd distribution aZong theprofile for optimized and non-optimized insuZators.

Fig. 9: Effect of resistivity of contamination layeron the potential distribution along the insuZator.

The flashover on an insulator (with a resistivity psfor its contamination layer) originates when the elec-tric field exceeds the electric strength of the sur-rounding air [16], i.e., in the vicinity of the HVelectrode where the electric field is maximum. Then,sparkover occurs causing discharges along small portionsof the insulator. These discharges are maintained bycurrent through the discharge-free portion of the in-sulator surface. Owing to the heat generated at thedischarge roots, the contamination layer dries out intheir neighborhood, with a subsequent increase of re-sistivity, and ceases to conduct. To maintain conduc-tion, the discharge root must travel along the surfaceto a region whose resistivity is still equal to p5.The discharges, therefore, elongate and flashover occursif they span the distance between the HV electrode andthe ground plane. The flashover voltage is the criticalvalue of the applied voltage which results in dischargesspanning all the insulator profile length. The latteris defined as the length measured along the insulatorprofile between the HV electrode and the ground plane.Optimization of the insulator profile increases the

.v*,:)

- 0.4

0

X/

X//0

-20

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4-

E

U.

4-1

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54

_- 0.2

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t I I

Abdel-Salam and Stanek: Optimizing field stress on HV insulators

field uniformity along the insulator surface with a sub-sequent decrease of the field intensity in the vicinityof the HV electrode, Fig. 8, where the flashover orig-inates. Not only the field uniformity but also theprofile length increases by insulator optimization.This is why the flashover increases as a result of in-sulator optimization.

Practical Considerations

As discussed before, optimization of insulators' pro-files improves their electrical characteristics as thetangential field distribution becomes uniform alongthe insulator surface.

It is worthwhile to discuss quantitatively the opti-mization process from an economical point of view asregards the cost of the insulating material and thespace of installation. Does the optimized insulatorrequire more volume in comparison with the non-optimiz-ed one and how much installation space does it save?To answer these questions, a comparative study is donebetween the optimized insulator and an equivalent non-optimized one whose maximum tangential field is equalto the tangential field of the optimized insulator forthe same voltage. This is the criterion of comparison.

Different lengths of non-optimized insulators wereattempted to search for the length which satisfies thecriterion of comparison. It is found that a non-opti-mized insulator of length equal to 1.4 times the lengthof the optimized insulator satisfies the criterion ofcomparison, Fig. 10. This means that optimizationprocess results in 30% saving in the installation space.

4-)

C)

r-

C..---4

t

4-1C:C.

F--

- 20

1liOptinm-zed insulator ep

Length=H

- 10 Noni-optimizedeqLliva-tent-length 1.4H

55

This indicates a 50% increase of the volume of the in-sulating material for the optimized insulator relativeto the non-optimized insulator.

On the other hand, it is not easy to compare quanti-tatively the mechanical stresses for the optimized andthe equivalent non-optimized insulators. For outdoorinsulators, momental stresses due to wind are the mostimportant and depend upon the projected area and thelength of the insulator as well as the wind speed overthe projected area.

Evaluation of mechanical stresses is out of the scopeof the present paper. However, brief comments can bemade.

The optimized insulator has a length equal to 70% ofthe equivalent one with a subsequent reduction in themomental stresses due to wind. Also, the relative pro-jected area of the optimized insulator is 21.6 versus21.0 for the equivalent non-optimized insulator. There-fore, the projected area of the optimized insulator isalmost the same as that for the equivalent one. Fromthese figures, it appears that the optimized insulatorhas better mechanical characteristics than the equiva-lent non-optimized one.

TangentiaZ Field Similarity

One of the questions which faces the design engineeris related to how the dimensions of the installation (HVelectrode and insulators) will affect the simulationprocedure and its accuracy.

The dimensions of outdoor insulators may reach severalmeters while those for indoor applications may be frac-tions of a meter. The dimensions of the insulators alsochange widely with the voltage rating. Similarly, thesize of the HV electrodes increases with the appliedvoltage to keep them free from corona discharge. In-spection of the defining equations outlined before showsthat for capacitive fields a field-similarity does exist.In other words, the dimensions of a given installationwhen scaled-down by a specific ratio, the per-unit tan-gential field distribution when referred to the peakvalue remains the same. However, the absolute values ofthe field for the original installation are those forthe scaled-down case after also being scaled-down by thesame ratio. This is very helpful for the proposed sim-ulation technique, as a big installation can be treatedthrough an equivalent small one where a reasonably largenumber of boundary points results in a satisfactorymatching of the boundary conditions over the surfaces in-volved in the installation. Fig. 11 shows the similarityof the tangential field distribution in the capacitivefield on scaling-up the dimensions of the installationby a ratio of 10. On the other hand, inspection of Eqs.21-23 shows that the field-similarity does not hold forcapacitive-resistive fields.

CONCLUS IONS

1. The conventional charge simulation technique is--- I I I modified to satisfy better matching of the boundary con-

0.02 0.04 0.06 0.0O m. 0.10 ditions to the electrode and insulator surfaces involved___________

7in the investigated installation.

Fig. 10: Tangential field distribution aZong theoptimized insuZator and the equivalent one.

The relative volume of the optimized insulator is 50versus 33 for the equivalent non-optimized insulator.

2. The modified technique is characterized by system-atic coordination of the simulation charges and by inde-pendency from a critical number of these charges as no-ticed with the conventional technique.

IEEE Transactions on Electrical Insulation Vol. EI-22 No.1. February 1987

3. An approach is suggested to optimize the profileof HV insulators to have a uniform tangential-field dis-tribution over their surfaces. The approach is simplein comparison with those reported in the literature.

4. Insulator optimization results in an increase ofthe onset voltage for surface flashover, a significantsaving of the space for the HV installation and an im-provement of its mechanical characteristics on the ex-pense of an increase in the volume of the insulatingmaterial.

Fig. 11: Tangential fieZd distribution along an

insulator and its scaled-up geometry.

5. For contaminated insulators, the higher the con-

ductivity of the contamination layer, the larger is the

degree of tangential-field uniformity over the surface.

6. The concept of field similarity does exist forcapacitive fields but not for capacitive-resistivefields.

REFERENCES

[1] H. Singer and P. Grafoner, "Optimization ofElectrode and Insulator Contours", Proc. 2ndInt. High Voltage Symp., Zurich, Switzerland,Sept. 1975, pp. 111-116.

[2] H. Gronewald, "Computer-Aided-Design of HVInsulators", Proc. of 4th Int. High VoltageSymp., Athens, Greece, Sept. 1983, paper No.11-01.

[3] H. Singer, H. Steinbigler and P. Weiss, "ACharge Simulation Technique for Calculationof High Voltage Fields", IEEE Trans., Vol.PAS-93, pp. 1660-1668, 1974.

[4] P. Lura, "Discussion Contribution", Proc. IEE,Vol. 120, p. 608, 1973.

[5] M. Khalifa, M. Abdel-Salam, F. Aly and M. Abou-Seada, "Electric Fields Around Conductor Bundlesof EHV Transmission Lines", IEEE PES paper No.A-75-563-7, 1975.

[6] P. K. Mukherjee and C. K. Roy, "Computation ofFields in and Around Insulators by FictitiousPoint Charges", IEEE Trans. Electrical Insula-tion, Vol. EI-13, pp. 24-31, 1978.

[7] P. Weiss, "Fictitious Peaks and Edges in Elec-tric Fields", Proc. 3rd Int. High Voltage Symp.,Milan, Italy, Aug. 1979, paper No. 11-21.

[81 K. J. Murtz and P. Weiss, "The Practical Im-portance of the Effect of Embedding an Electrodein a Dielectric Medium", Proc. 7th Int. Conf. onGas Discharges and Their Applications", London,England, Aug./Sept. 1982, pp. 478-481.

[9] A. DiNapoli and C. Mazzetti, "Electro-Static andElectromagnetic Field Computation for the HVResistive Divider", IEEE Trans., Vol. PAS-98,pp. 197-206, 1979.

[10] 0. W. Anderson, "Finite Element Solution ofComplex Potential Electric Fields", IEEE Trans.,Vol. PAS-96, pp. 1156-1161, 1977.

[11] B. Bachman, "Models for Computation Mixed Fieldswith the Charge Method", Proc. of 3rd Int. HighVoltage Symp., Milan, Italy, Aug. 1979, paperNo. 12-05.

[12] T. Takuma, T. Kawamoto and H. Fujinami, "ChargeSimulation Method with Complex Fictitious Chargesfor Calculating Capacitive Resistive Fields",IEEE Trans., Vol. PAS-100, pp. 4665-4671, 1981.

[13] M. L. James, G. M. Smith and J. C. Wolford,"Applied Numerical Methods for Digital Computa-tion", 3rd Edition, Harper and Row Publish. 1985,pp. 149-156.

[14] F. Youssef, "An Accurate Fitting-Oriented ChargeSimulation Method for Electric Field Calculation:,Proc. of 4th Int. High Voltage Symp., Athens,Greece, Sept. 1983, paper No. 11-13.

[15] F. Taher, "Choosing Insulators for Desert Environ-ments", Electrical Review Inter., Vol. 207, No.21, pp. 37-39, 1980.

[16] L. L. Alston and S. Zoledziowski, "Growth of Dis-charges on Polluted Insulation", Proc. IEE, Vol.110, pp. 1260-1266, 1963.

Manuscript was received on 12 ApriZ 1985 in finaZform 24 March 1986.

56