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E E E Transactions on Power Delivery, Vol. 10,No. 2,April 1995NE W DEVELOPMENTS ON THE COMBINED APPLICATION OF CHARGE SIMULATION AND

NUMERICAL METHODS FOR THE COMPUTATION O F ELECI'RIC FIELDS

J . N. Hoffm ann, IEEE MemberCompanh ia Pa ranaense de Ene rg ia

80.020 - C u r i t i b a , P a r a n& , B ra zi l(COPEL) - P.O.BOX 31 8

Abstract - A new mathematical treatm ent i s proposed to thehybr id method of Charge Simulation & Finite Differences forthe computation of electric fields, entirely applicable t oth e simila r hybrid method of Charge Simulation & FiniteElements. The resulting syst em of linea r equations is solvedby making use of th e fixed point theory, t he QRdecomposition (by using t he Modified Gram-Schmidt method)and the Conjugate Gradients Squared method with apreconditioning technique. New procedures are suggested forth e discre tizati on of th e boundary conditions, which lead t ores ult s with higher precision. Case studi es are included.Key words - Electric fields, numerical methods, high voltageengineering, electrical engineering computing, digital simu-lation.

1. INTRODUCTIONThe Charge Simulation Method (CSM), the Finite Difference

Method (FDM) and the Finite Element Method ( F EM ) are verycommonly used for field analysis of HV insulation systems.The CSM is suitable for unbounded problems, but becomescomplicated for problems including dielectrics. On the otherhand, the F DM and F EM are suitable for multi-dielectriccases, but require truncation of the domain f or unboundedproblems. Thus, a hybrid method may be a promising tool fo runbounded problems including dielectrics, where the FD M orthe FEM is applied inside a limited arbitrary region, andth e CSM is applied t o th e unbounded exterior. Along thecoupling surface, continuity conditions are imposed.

Early research work 11, 2, 31 have presented practicalexamples of thi s hybrid method, pointing out its advantagesand disadvan tages. Eit her t he hybrid CSM & FD M or th e CSM &FEM require the solution of a similar system of linearequations, whose matrix of coefficients (square inprinciple) is composed by full submatrices (resulting fromth e CSM) and spars e submatrices (resul ting from t he FDM orFEM). The present work particularly deals with theformulation of adequate procedures for t he solution of thislinear system, which results in a significant smallercomputer storage, smaller CPU computer time and results withhigher precision, if compared to [1,2,31. In addition, th econvergence of th e proposed it era tiv e method is notdependent on the arbitrary initial conditions (electricpotentials) as 111 does.

The application of t he CSM to the unlimited ex ter iorregion as it was previously considered ([l, 2, 311, lead s to

94 SM 483-8 FWRD A paper recommended and approvedby th e IEEE Transmission and Distr ibut ion Committeeof t h e IEEE Power Engineering Society for p r e s e n t a t -i o n a t t h e IEEE/PES 1994 Summer Meeting, SanFranc i sco , CA, J u l y 2 4 - 28 , 1994. Manuscriptsubmi t t ed Ju ly 14, 1993; made avai lable f o r p r i n t i n gAp ri l 20, 1994.

P . Pu l inoUnivel' s i da de Estadual de Campinas

13.081 - Campinas , S%o Paulo, Brazil(UNICAMP) - P.O.Box 6065

1105

full s quare su bma tric es and demands an excessive amount ofcomputer memory. It is sh om in sect ion 3 t ha t it ispossible to significantly reduce th e required computermemory, by choosing a number of simulated charges which issmaller than th e number of contour points. This leads t o aleast squares problem. The least squares problem is solvedby means of the QR decomposition 14.51 of the resulting fullrecta ngular submatrix, using the Modified Gram-Schmidtmethod. This procedu re only demands from 30% to 50% of thecomputer memory initially required by the original problem,without loss of precision.

The application of t he FDM or FEM t o the limited regionleads to a sparse linear system of the Ax = b type. It ispossible to use th e Conjugate Gradients method (CC) 141, aKrylov Subspace method (KS) 161 or the Conjugate GradientsSquared (CGS) [71 fo r the solution of t his system. Thesemethods have advantages over the traditional Gauss-Seidel,SOR and others (used by [1,2,31), since they reach the exactsolution in a number of steps at most equal t o th e dimensionof the matrix. The CGS with a preconditioning technique issuggested for the solution of the discretised region, sinceit demands less CPU computer time and less computer memorythan other methods (s ee section 4).

The hybrid method requires a discretization for thenormal component of the electric flux density vector D (i.e.n * D ) all over th e arbi trary rectangular surface, as well asall over each surface of discontinuity on the electricalpermittivity E (boundary conditions). This le ads to adiscre tizati on of the normal component of t he derivative ofthe scalar electric potential function (since D = -E Val.The error of such discretization as computed by 11, 2, 31for rectangular surfaces is of the order o h (i.e. O(h)),where h is the grid step (distance between two consecutivenodes). It is shown (section 5.1) by the use of the Taylorseries expansion, that it is possible to obtain a simplediscretization with an error of O(h2). A discretization forcurvilinear surfaces is also presented (section 5.2). beingof inter est when a discontinuity on E exists. Thesemethodologies for the discretization of n*V@ ead to resultswith high er precision if compared to tho se used by 11,2,31.

Finally, adequate procedures for the solution of thecombined system of linear equations tha t resu lts fro m thehybrid method are stated. A direct solution is notrecommended due to the irregular struc ture of th ecoefficient matri x. Reference 111 proposes an itera tivescheme which depends on a good initial estimate. Inaddition, it may not reach the solution fo r certain cases.The approach used here (section 6) is based on the fixedpoint theory of linear systems [4,51. The utilization of anarbitrary but rather predictable parameter (e ) leads toconvergence in a significantly great er c lass of problems (ifnot all), not depending on the initial estimates fo r th eelectric potentials. Some examples are included and theresults are analysed (section 7).

Considering the points above, it is supposed that thepresent work may represent a new motivation for theutilization of th e hy brid CSM & FDM or CSM & FEM on th ecomputation of two-dimensional o r three-dimensional withaxial symmetry electric fields, as frequently found in theHV engineering.

0885-8977/95/$04.00 1994 EEE


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1106g r i d w i t h Mn o d e s ( @ = @ acontour

nodes 1 I

I I I I I

V contourp o i n t s

e 1

K s i m u l a t e dc h a r g e sa n 2 1 L s i m u l a t e d

( @ = a 2 1 c h a r g e sb ou nd ar y w i t h d i s c o n t i -n u i t y o n e , w i t h W nodes

e a r t h s u r f ac e ( @ = 0 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / / / / / / I / / / / / / / / / / / / / / / / /

Figure 1. General problem tr eat ed with th e CSM & FDM

2. DEFINITION OF THE COMBINEDSYSTEM OF LINEAR EQUATIONSThe hybrid CSM & FDM applied t o the g eneral two-dimensional problem of Figure 1 will be considered on thefollowing sections.Suppose that it is needed to calculate the electric fielddistrib ution n ear elect rode A, surroun ded by a region ( Q z )

with electrical permittivity EZ . An arbi trar y rectangularboundary asli involving A and 8 2 is defined, inside which afinite difference grid with M nodes is placed. This alsowill define N nodes on a B and W nodes on aQ2. Adiscretization with V contour points is made on the boundaryai 3 of the external electrode, and K+L charges are simulatedas indicated in Figure 1.

It was already shown by [1,2,31 tha t, fo r two-dimensionalor three-dimensional problems with axial symmetry, theapplica tion of th e CSM & FDM (o r the CSM & FEM) t o Figure 1leads to a system of linear equations, which is representedhere by the following equivalent system with submatrices:

so S I 000 Sz D

whereThe matrix equations PI q - I '$1 = 0 and Pz q =resu lt from the CSM, andPI = full matrix, dimensions N x (K+L)P2 = full matrix, dimensions V x (K+L)I = identity matrix, dimensions N x Nq = vector of simulated charges, dimension K+L01 = vector of electri c potentials on ani , dimension N@E = vector of electric potentials on a h , dimension VPI and Pz are the Maxwell electric potential coefficimatrices

@B

en t

The matrix equation F q + So @I + SI 00 = 0 resultsfrom the continuity condition n * D on an i , andFSo = sparse matrix, dimensions N x N= full matrix, dimensions N x (K+L)

SI = sparse matrix, dimensions N x M00 = vector of electric potentials on Q l, Qz , and QA,F is the Maxwell electric field coefficient matrix

dimension M

c) The matrix equation Sz 01 + D @o = c results from theFDM (disc ret izat ion of the Laplacean oper ator [3,91),andS z = sparse matrix, dimensions M x ND =sp ar se matrix, dimensions M x Mc = sparse vector which depends on the given electricpotential vector @A of the electrode A.The form (1) is adequate for a problem with only one

dielectric. A small modification is suggested for thegeneral problem o f Figure 1 (with two or more dielectrics),since it is more convenient to treat QI and Qz separately,taking an additional vector 0z for the electric potentialson aQz. The problem can, th erefor e, be formulated as follows:On the region 81 and its boundary:

A @ = 0 o n R i0 = 01 on am0 = @z on a mn * D = g, i.e., - ~ i w an = g on 8622where A is the Laplacean operatorand g is an auxiliary function

On the region Qz and its boundary:A @ = 0 o n RzQ! = @A on am@ = 82 on a mn * D = -g, i.e., -EZ aWan = -g on aQz

(2 )

(3 )

Joining (2 ) and (3 ) together, the following problem isobtained,

a. A @ = 0 on 81 U 622 (4 )b. a = +A on amC. 0 = @I on a81d. @ = 02 on aQ2e. ~i swan + E2 a w a n = o on as22


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1108where NTOT is the tot al number of non-zero eleme nts of A

It is easy to see in this algorithm that no ordination isnecessary to the elements of A when generating the vectorsILIN, JCOL, and VAL. T his pro per ty is specially suitable toproblems of the type which we are intending to study.

Therefore, any method that doesnt require operationsmore expensive than the product matrix-vector can be usedfor the solution of ( 9 ) . in principle. To this class ofmethods belong the Conjugate Gradients and some derivations.Besides this simple way of storage, these methods have theadvantages of reaching the exact solution af te r a finitenumber of steps (theoretically at most equal to the matrixdimension), and for any initial estimate. The following wasconsidered :a ) Conjugate Gradients (t radit ional ) (CG) l4,51 withoutb) Krylov Subspace method (KS) [61 without pi-econdi-c ) Conjugate G radie nts Squared (CGS) 171 without precondi-

preconditioning and with a diagonal preconditioning.tioning and with a tridiagonal preconditioningtioning and with a tridiagonal preconditioning.The CG requires A to be a symmetric and positive definite

matrix. Since this is not the general case, it only can beapplied to the modified normal system AT A 9 = ATb (since AAis symmetric and positive definite).The KS requires the calculation of a vector basis t o theKrylov subspace (which dimension is arbit rary) . This vector

basis may demand a signi fican t additional memory of thecomputer.The CGS doesnt require the product AA a s the CG nor theadditional storage as the KS and, when used with apreconditioning technique, i t needs less CPU computer time

(see section 7) . Therefore, the CGS is suggested for thesolution of (9). Appendix I includes some detail s a s well asthe algorithm used for the CGS.

5. CONTINUITY CONDITIONS ON BOUNDARlESAn adequate application of the Maxwell equations givesthe n ecessary continuity conditions ([1,2,31), i.e ., 9 andn-D are continuous across ~ R I nd aRz . The condition n *D de-mands a specific equation to be applied at any contour node.

5.1 Discret ization of n- D on a R iSince c is constant on both sides of ~ R I , he continuityof n*D is equivalent to the continuity of n. 89. As alreadyshown by [2,31, the CSM applied to n.8 9 leads t o theexpression

n.VQ(Pi) = - f i j q j , i = l . .N (10)Kr= lwhere fi j are the Maxwell electric field coefficien ts

References [2,31 use only one internal node of RI fo r theapplication of th e FDM ( or the FEM) to t he continuity ofn* W, which result in an expression with an err or of O(h)only. A discretization with an error of O ( h Z ) can beobtained using the Taylor series expansion as exposed inAppendix 11. Then

n.VQ(Pi) = aNPi ) + 13Np1-1)+ ~ ~ ( P L - z ) (11)where PI lies on a R i and Pi-1, Pi-z ar e inter nal nodes of Q i.(10) = (11) . o rThe final coupling equation is a result of taking

f ij qj + [ae(Pi) + pa(Pi-1) + 79(Pi-2)1 = 0 , i =i ...j = 1

which has the equivalent matrix f orm F q + SO@I Si90 = 0,also included in (7). I t is easy to see that F is a fullmatr ix , So is a diagonal matrix and SI is a sparse matrixwith only two non-zero eleme nts per line.

5.2 Discret izat ion of n -D on aR zIt is seen from (4e) that thi s continuity conditionrequires the discretization of aW an in two stages

a) ci a W a n , using nodes of R Ib) cz N/an , using nodes of RzThe method proposed by [81 uses the node on the boundaryand other three internal nodes, and is considered in somedetails in Appendix 111. Using the proposed discretization,

(4e) takes the form

where Pi lies on 8 2 2 ,and Plj,PZj ar e inte rnal nodes of R i , R z respectivelyThe matrix form ( 6 ) is obtained applying (12) to i= 1 . ..W ,where E is a diagonal matrix and S4 is a sparse matrix with6 non-zero elements per line.

6. SOLUTION OF THE COMBINED SYSTEM OF LI NEAR EOUATIONSIt was already shown tha t the system of linear equationsth at r epres ents the hybrid CSM & FDM has th e m atri x form (71.In order to facilit ate our analysis, it is written as

Since (13) is composed by diff ere nt kind of s ubma trice sthat came from distinct problems (i .e. the full rectangularsubmatrices from the CSM and a spar se squa re submatri x fromthe FDM). it is suggested to solve this system using anite rat ive method. Doing so , each part of (13) can be treate dseparately in an optimized way.Notice tha t once the vector of cha rges ( q ) is known, thewhole problem is solved. Theref ore, an it erati ve method canbe derived by using the general ex press ion of th e fixedpoint theory [41

where T is a mat rix of dimensions K+L x K+L,U is a vector of dimension K+L, andn is the iteration number.

According t o the fixe d point theor y applied to lineal.systems [41, (14) converges whenever o(T) < 1, where d T ) isthe spe ctral radius (t he largest absolute value in the setof eigenvalues) of T, fo r any initial estim ate qO. T isalso known as the iteration matrix.It is possible to show (Appendix I V ) that an iterativemethod can be derived so that an expression for T results i n

(15)[ iA-IZZ - SoIp1= e I + (l-e)R-QTwhere Q and R rise from the decomposition [ = QRe is a paramet er to be chosen f or every problem inorder to make o(T) < 1, and so allowing (14) toconverge. I t holds in general that 0 5 e < 1 .

Obviously d T ) and T do not need t o be explicitlycalculated. Figure 4 includes a flowchart for the iterativemethod which generates T precisely as expression (15)


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1 1 ) E s t i m a t e O ! I12 ) C a l c u l a t e q o w i t h CSMl

I&4 3 ) S o lv e w i t h C G S : A O n = z - ZzO?]1 4 ) D e t e r m i n e f" = S o@ ? + Z i O " )eyeso l e r a n c e 17 ) D e t e r m i n e q " + ' = @ q" + ( i - e l r " ]

Figure 4. Iterative method fo r the solution of (13)Observations:a) The iterative method (14) is not dependent on the initialestimate qo. However, it is $ossible to obtai n a goodestimate for qo by choosing 81 (step 1 in Figure 4) andsolving once with the QR decomposition (step 21 th efollowing least squ ares problem, taken from (7)

b) The given electrode potentials are included in vectors OBand z , which do not appear in expression (15). So theconvergence to the solution (which according to the fixedpoint the ory depends on the eigenvalues of T ) does notdepend on the given electric potentials.c) In view of th e complexity of (151, the par am ete r Ica nonly be estimated by experience. For most of the testedproblems we had convergence with e I .6 to 0.9.d) The iterat ive method proposed by [I1 corresponds somewhatto the particular case of e = 0, which doesn't mean that(14) will always converge. The possibility of a choice fo re # 0 permits the solution of a gre ate r class of problems.

7. APPLICATIONSWe used th e hybrid CSM & FDM fo r the calculati on of theelectric field distribution close to the surface of aninfinite cylinder above earth, since this case have an exact(anal ytic al) solution. Det ails ar e reporte d in [111, and

indicate that for this case the electric potentialscalculated with the hybrid CSM & FDM have an err or of le ssthan 1 % .

In a more general case, we used the hybrid CSM & FDM fo rthe calculation of the electric field distribution in thetwo-dimensional, unbounded and multi-dielectric problem asindicated in Figure 5.

Submatrices P I , P 2 and F of (7 ) include the Maxwellcoefficients, calculated as indicated by [ l, 2, or 31. Inthe FDM region, a rec tangu lar g rid wit h equally spaced nodesof h = 0,75 was placed, defining M = 15x31 = 465, N = 96 andW = 72. It was also chosen K = V = 4.Table 1 shows the maximum percentile deviation of theelectric potentials calculated on 8 2 1 (vector 811,considered as a function of th e relation between the number

CSM region

an1 I IEl I IIDM regiontlr 14I

Figure 5. Two-dimensional example

of simulated charges inside the FDM region (L ) and thenumber of contour points on aR i (NI. The values wereobtained by comparison with results computed with L / N 0.75(this relation is assumed to give insignificant error for81). As a result it may be suggested that L/N I 0% t o 50%is a good choice.

Ta b l e 1E r r o r o n Cp

0. 6

T a b l e 2C P U t i m e ( s )M e t h o d l 1 I z

4 5 3 . 7 1 7 . 8CG S 3 6 . 7 9 . 60.201 1 . 3 3 1 - wi t h o u t v r e c o n d .0 . 1 0 1 k 20 .0 z - w i t h p r e co nd .

Table 2 shows the consumed CPU time for the methods CG ,KS and CGS (with L / N E 33x1 in an IBM 3090-300s computer.The CGS with a tridiagonal preconditioning was shown to bethe fas tes t .

A s an illustration, Figure 6 shows the computedequipotential lines inside the FDM region, con sideringEZ = 2.0 an d e3 = 5.0 .

Figure 6. Equipotential lines

Figure 7 shows the electric potential distribution alongthe direction 1 of Figure 5, as a function of EZ and E3.It illustrates the effect of materials with differentelectrical permittivities on the electric fielddistribution.


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1110

1 0 0

90.

BO

70

60

5 0

t potential

// ,",",.0 2.02.0 5. 0' 4 / 2 . 0 / 1 0 0 distance-e -4 0 4 B 12 206

Figure 7. Electric potential a s a function of

Additional comments:a ) The charges simulat ed inside the FDM region must belocated at least approximately 1,0 to 2,0 times the gridstep (h) distant from ani. Charges located very close to aR 1lead to a severe loss of precision in the resu lts (a s alsorepo rted by 1211, due t o the si ngul arit ies of t heexpressions for the Maxwell coefficients.b) When a simulated charge is located very close t o another,the matr ix Q approximates of a rank deficient matrix (see[4,51), which may cause some diff icul ties in th e convergenceof (14). This corresponds to the case of nearly singularmatrices in the traditional CSM.c) It was considered an absolute err or of 10 for theconvergen ce of the CG, KS and CGS, and an er ro r of 0.1% fo rthe elements of the simulated charges vector q of (14) (2 2iterations was necessary).d) I t is easy to see that all the mathematical approachexposed in thi s work applies t o three-dimensional problemswith axial symmetry as well (which in fact become reducedt o a two-dimensional problem). This is feasi ble once thecorresponding Maxwell coefficients f or submat rices Pi, P2and F (see [1,2,31) and the corresponding discretiz ation fo rthe Laplacean oper ator of equati on (4) (se e 1101) ar econsidered. Thus, problems like the electric potentialdistribution across an insulator chain, or across aninsulati ng column of a HV equipment can be solved.Three-dimensional problems without any symmetry ar e alsopossible in principle.e) Problems with t he dimensions of the example above may berun on a personal computer as well.

8. CONCLUSIONNew developments on the combined applica tion of th e

Charge Simulation method with numerical methods (FiniteDifferences or Finite Elements) were shown.The application of the Charge Simulation method as aleast squares problem with the QR decomposition results inan economy of more tha n half of th e original amount of

computer memory, without loss of precision.The use of th e CGS method with a tridiagon al pre-

conditioning for the solution of the discretised problemresults in a significant economy in the CPU computer time,wit h the consequence of co sts reduction and reduced erro rpropagation in the computer.

New procedures for the discretization of the boundaryconditions were suggested, leading to results with higherprecision.

9. REFERENCES

It was presented an iterative method for the solution ofthe combined problem by making use of the f ixed pointtheory of linear systems, which allows convergence for agre ate r class of electrostat ic problems. Another featur e ofthis new iterative method is the non-dependence on theinitial estimate for electric potentials.

Finally, case studies show that the suggested innovationsar e effective in the ma themati cal formulat ion of the CSM &FDM or t he CSM & FEM, and they may represent a new motiva-tion for the application of these hybrid methods on thecomputation of electric fields for unbounded problems.

I

H. Steinbigler, "Combined application of finiteelement method and charge simulation method for thecomputation of electric fields," 3rd InternationalSymposium on High Voltage Engineering, section 11-11,August 1979H. Okubo, M. Ikeda, M. Honda, and T. Yanari, "Electricfield analysis by combination method," IEEETransactions on Power Apparatus and Systems, vol. 101,no 10, pp 4039 - 4048, October 1982M. Abdel-Salam and M. T. El. Mohandes, "Combinedmethod based on fin ite diffe renc es and chargesimulation for calculating electric fields, " IEEETransac tions on Industry Applications, vol. 25,no 6, pp 1060-1066, November/December 1989G. H. Golub and C. F. Van Loan, Matrix computati ons,Baltimore, The John Hopkins University Press, 1985B. Noble and J. Dan iel , Applied Linear Algebra, NewJersey, Prentice Hall, 1977Y. Saad, "Krylov subspace methods on supercomputers,"J. Soc. Industrial and Applied Mathematics, Vol. 10,no 6, pp 1200-1232, November 1989P. Sonneweld, "CGS, a f as t Lanczos-type solver fo rnonsymmetric linear systems ," Siam J. Sci. Stat.Comput., Vol. 10, no 1, pp 36-52, Janu ary 1989J . H. Brambl e and B. E. Hubbard , "Apro ximation ofsolution s of mixed boundary value problems fo rPoisson's equation by fin ite differen ces," Journal ofth e Associat ion fo r Computing Machinery, Vol. 12, no1, pp 114-123, Jan uar y 1965G. E. Forsythe and W. R. Wasow, Finite-differencemethods fo r parti al dif f erential equations, JohnWilley & Sons, 1960J. T. Stor ey and M. J. Billings, "Generaldigital-computer program fo r the determination of3-dimensional electros tatic axially symm etric fields,"Proc . IEE, Vol. 114, no 10, pp 1551-1555, October 1967J. N. Hoffmann, Numerical simulations for the electricfiel d calcul atio n of unbounded domains, M.S. The sis(in Portuguese), University of Campinas (UNICAMP),IMECC Dept., Campinas/SP, Brazil, 261 pp. April 1993.

APPENDIX ICGS algorithm with preconditioning

Consider the CGS method with tridiagonal preconditioningfor the solution of the system of linear equations A x = b.For the applicati on of the preconditioning technique, t hissystem is modified a s

(L U)-'(A x ) = (L U)- ' bwhere L,U = matrices resulting from the LU decompositionof the tridiagonal part of A 14,51


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1111In this(171) :

10

case, the algorithm for the CGS is the followinge s t i m a t e x os o l v e ( LU) r o = ( b - A xo )qo= p-,= 0 ; p - l = 1 ; n=Oi f ( r n < t o l e r a n c e ) e n d

q n + l = U n - a n V nVn = a n ( u n + q n + l )X n + l = Xn + Vns o l v e ( LU) r n + l = ( b - A -+I) ; n=n+lgo t o 10

APPENDIX I1Discret izat ion fo r n*V@

A simple discretization for n-V@(Pi) is obtained usingthe Taylor series expansion, assuming existence of nodesarranged on a line with the same direction of n (Figure 8) .

hz I h i - n- -Pi-2 Pi-1 Pi

Figure 8. Di s c r e t i z a t i o n f o r a@/an a t PiThen

@(Pi-1)= @(PI)- hi@'(Pi) + h?@"(Pi)/Z + O(h3) (16)@(Pi-z)= @(Pl)-(hl+h~)@'Pi)+(hi+hz)'@"(Pi)/2+O(h3) (17)

Taking (h1+h2)~*(16) h?*(17),n*V@(Pi)= @' (Pi) = a@(Pi)+B@(Pi-i)+r@(Pi-z) O(h2)

where a = l/hi + l/(hi+hz)6 = 4 l h l + lAl2)r = l / h ~ l/(hl+h2)h = max {hi,hz)ha s an error of O(h2).So th e approximation n*V@(Pi) = a@(Pi)+BO(Pi-i)+rO(Pi-2)

APPENDIX 111General d iscret izat ion for n*V@

A general discretization fo r n*V@(Pi) on an is obtained,not requiring the existence of nodes arranged as those ofAppendix I1 (Figure 9).

A s descr ibed by I81 anda c c o r d i ng t o F i g u r e 9 ,a d i s c r e t i z a t i o n f o ra@ /an at Pi may be0 ~ j ( x j , y j ) w r i t t e n as1 y=-n

F i g u r e 9. D i s c r e t i z a t i o n w h e r e P i E a Rf o r a w a n at pi and PJ E R

The coefficients aj are determined using the geodesicnormal coordinates, wha t resu lts in the solution of thesimple system of equations (see 181 for details)

where K is the curvature of an a t P I .

APPENDIX IVProposed i tera t ive method

Suppose that qn of (14) is known. The CSM gives@pP = Pi qn. (18)

Thus, vector en ca n be calculated from the application ofthe CGS method t o the FDM problem. T heoreti cally, fr om (131,

Determine the auxiliary vector f", by using (19)

A new vector of charges rn can be derived fr om t he CSMwith lea st squ ares and the QR decomposition, obtained from(13)

Using (18) and (20) n the last expression, we obtain

Taking the combination qn+'= 0 qn + (1-0) rn , where 0 isan arbitrary parameter, and by using (211, the expression(15) for T is immediately obtained. This combination assurestha t at the end of th e iterative calculations (when rn = qn),we will have qn+'= qn = r , for any chosen value of 0 .

The flowchart of Figure 4 resumes the above procedure.

Jog0 Nelson Hoffmann was born in Lapa,state of Pa ra nl , Brazil, on April 22,1958. He received the B.S. degree inElectrical Engineering from Universi-dade Fe deral d o Paran & in 1981, andth e M.S. degree in Applied andComputational Mathematics from Univer-sidade Estadual de Campinas (UNICAMP)in 1993. In 1982 and 1983 he workedwith the design of transmission linesand sub statio ns. From 1984 to 1987 heworked with technical specifications

for HV equipments. Since 1988 he is with CompanhiaParanaense de Energia (COPEL). Curitiba, Paranl, Brazil,working with transmission line researches, specially foroverhead urban and compact lines, radio-interference andearthing problems.

Petronio Pulino was born in Bebedouro,s t a t e o f S%o Paulo, Brazil, on October14, 1956. He received the B.S. degr eein Mathematics, the M.S. degree inApplied Mathematics (working withfini te elements) and the Ph.D. degreein Electrical Engineering (workingwith numerical analysis applied tosemiconductor devices) from Universi-dade Estadual de Campinas (UNICAMP) in1980, 1983 and 1990, respectivel y. In1983 he joined UNICAMP as professor

of numerical analysis and linear algebra. Presently he is anassistant professor at the Department of Applied Mathematicsof UNICAMP, working specially with boundary value problemsof elliptic type and applied computational algebra.

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