Numerical method for determining potential coefficients matrix for multiconductor transmission line Текст научной статьи по специальности «Математика»

Numerical Method for Determining Potential Coefficients Matrix for Multiconductor Transmission Line Patiuk V.

Institute of Power Engineering Chisinau, Republic of Moldova

Abstract. Consider the propagation of electromagnetic energy through multiconductor three-phase high-voltage transmission line with arbitrary number of conductors. The mathematical formulation of the problem represents the system of partial differential equations known as transmission line equations. When constructing the capacitance matrix of an electric line, it is necessary to solve the boundary problems for the Laplace equation with nonlocal boundary conditions. The aim of this research is to develop a new method for constructing a potential coefficients matrix, which allows us to express the voltage vector in the wires through the charge vector. The application of this method leads to the necessity of solving a boundary value problem for Laplace equations with nonlocal boundary conditions. A new type of non-local boundary conditions has been identified, which has not been previously investigated. The boundary conditions of this type are represented by the contour integral over the boundary of the region from an unknown potential. In the general case, the obtained problem does not have a unique solution. The theorem of existence and uniqueness of the solution for the boundary value problem for the Laplace equation with nonlocal boundary conditions is proved. However, the requirement that the potential on the wire surface is constant allows us to prove the existence and uniqueness of the solution for considered problem. In order to illustrate the application of the developed approach we have solved numerically the problem for the sector-shaped cable with three cores. The values of potential and capacitive coefficients obtained by calculation are given. The error of the values of the diagonal elements of the matrices of potential and capacitive coefficients is estimated as less than 0.1%.

Keywords: telegraph equations, linear capacitance matrix, Laplace equation, nonlocal boundary conditions, electrical cable. DOI: 10.5281/zenodo.2222388

Metoda numerica pentru determinarea matricei coeficientilor potentiali a liniei de transmisie cu multe conductoare Patiuc V.

Institutul de Energetica Chi§inau, Republica Moldova Rezumat. in lucrare se considera problema de propagare a energiei electromagnetice prin intermediul unei linii de transmisie de inalta tensiune cu un numar arbitrar de conductoare. Formularea matematica a problemei reprezinta sistemul de ecuatii diferentiale cu derivate partiale, cunoscut ca ecuatii de linie de transmisie. Valorile capacitatilor liniare ale liniilor cu mai multe conductoare sunt determinate folosind valorile sarcinii electrice §i tensiunilor conductoarelor liniilor. Atunci cand se construie§te matricea de capacitate a unei linii electrice cu multe conductoare, este necesara rezolvarea problemelor la limita pentru ecuatia Laplace cu conditii la limita nelocale. Scopul acestei lucrari consta in dezvoltarea unei noi metode de construire a matricei coeficientilor potentiali, care permite sa exprimam vectorul tensiunii in conductoarele electrice prin intermediul vectorului de sarcina electrica. Aceasta matrice este inversa la matricea capacitatilor lineica, care este utilizata in ecuatiile telegrafi§tilor. Abordarea propusa permite compararea valorilor exacte ale coeficientilor matricei cu valorile aproximative obtinute fara a lua in considerare interactiunea potentialelor pe suprafata conductoarelor. Aplicarea acestei metode conduce la necesitatea rezolvarii unor probleme la limita pentru ecuatiile Laplace cu conditii de limita nelocale. Acest tip de conditii de limita nelocale nu a fost investigat anterior. Ele se reprezenta de un integral pe contur al potentialului necunoscut. S-a demonstrat teorema existentei §i unicitatea solutiei privind problema valorii limita pentru ecuatia Laplace cu conditii de limita nelocale prin utilizarea ipotezei, ca potentialul pe suprafata conductorului este constant. Pentru a ilustra aplicarea abordarii dezvoltate, a fost rezolvata numeric problema pentru cablul cu trei faze de tip AAB^, conductoarele caruia prezinta in sectiune sectoare curbliniare. Sunt date valorile coeficientilor potentiali §i capacitivi obtinuti prin calcul. Eroarea valorilor coeficientilor diagonali ai matricelor coeficientilor potentiali §i capacitivi este estimata ca fiind mai mica de 0.1%.

© Patiuk V.I., 2018

Cuvinte cheie: ecuatii telegrafice, matricea capacitâtilor liniare, ecuatia Laplace, conditii la limita nelocale, cablu electric.

Численный метод определения матрицы потенциальных коэффициентов для многопроводной линии Пацюк В.И.

Институт Энергетики Кишинэу, Республика Молдова Аннотация. Рассматривается задача о распространении электромагнитной энергии по многопроводной трехфазной высоковольтной линии электропередачи с произвольным количеством проводников. Математическая постановка задачи представляет собой систему уравнений с частными производными, описывающих распределение напряжения и тока по времени и расстоянию в линии и известную как телеграфные уравнения. Значения погонных емкостей многопроводных линий определяются, используя значения заряда и напряжений на проводах линий. При построении матрицы емкостей многопроводной электрической решены краевые задачи для уравнения Лапласа с нелокальными граничными условиями. Целью данной работы является разработка нового метода построения матрицы погонных потенциальных коэффициентов, которая позволяет выразить вектор напряжений в проводах через вектор зарядов. Эта матрица является обратной к матрице погонных коэффициентов электростатической индукции, которая используется в телеграфных уравнениях. Поставленная цель достигается путем решения задачи Дирихле с нелокальными граничными условиями. Предлагаемый подход позволяет проводить сравнение точных значений коэффициентов матрицы с приближенными, полученными без учета взаимовлияния величины потенциалов на поверхности проводов. Применение данного метода приводит к необходимости решения краевой задачи для уравнений Лапласа с нелокальными граничными условиями. Рассмотрен новый тип нелокальных граничных условий, который не был ранее исследован. Граничные условия этого типа представляют собой контурный интеграл по границе рассматриваемой области от неизвестного потенциала. В общем случае, полученная задача не имеет единственного решения. Наложение требования постоянства потенциала на поверхности проводов кабеля позволяет доказать теорему существования и единственности решения поставленной задачи. Приведено численное решение сформулированной задачи для кабеля типа ААБл с тремя жилами секторной формы. Приведены значения потенциальных и емкостных коэффициентов, полученных расчетным путем. Погрешность значений диагональных коэффициентов матриц потенциальных и емкостных коэффициентов оценивается, как меньше 0.1 %. Ключевые слова: телеграфные уравнения, матрица погонных емкостей, уравнение Лапласа, нелокальные граничные условия, электрический кабель.

INTRODUCTION

In this paper, we propose a new method for calculating the coefficients of the matrix p of linear capacities. The conventional method consists in applying the relation Q = pu, which expresses the dependence of the charge vector Q through the voltage vector U in the wires. This approach is described in [1-4]. In [1] a physical experiment is proposed to determine the coefficients of the matrix p, in [2, 3] well-known commercial software products ELKUT and QuickField are used for this purpose, in [4] the initial problem is reduced to solving integral equations. We propose to determine the values of linear capacities from potential coefficients matrix a that allows expressing the vector of potentials U through the charge vector. In this case, the exact values of the coefficients au can be

compared with the approximate values obtained in [1] using analytical formulas:

2л e l R

1 r

an =-ln-^, i Ф j.

j 2л e l r„

j

Such a method for constructing this matrix requires solving some boundary value problems for the Laplace equation with nonlocal boundary conditions. In [5-10], the problems with different types of nonlocal boundary conditions were investigated, but the problem presented in this paper was not reflected.

I. MATHEMATICAL MODEL

The mathematical model that describes the propagation of electromagnetic energy over a multiconductor three-phase high-voltage transmission line is represented by the system of differential equations for voltage u(x, t) and current i( x, t) vectors:

T Si du

L---1--+ R ■ i = 0,

St Sx

Su di C--H--+ G ■ u = 0.

St Sx

(1.1)

(1.2)

These equations are derived from Maxwell's equations for a multiconductor power transmission line [15, 16].

The domain of the solution is the rectangle D = {(X, t): x e (0,l), t e (0,T^)}, where l is the

length of the transmission line and Tmax is the maximal time of calculating for vectors u (x, t) and i(x, t) . In a «-conductor line the vector functions u (x, t) and i(x, t) have n components each, but L, C, R and G in (1.1), (1.2) are symmetrical matrices (n x n) of linear

inductances, capacitances, wire resistances and conductivities of insulation (vector objects are denoted by boldface)

u( x, t ) =

( u (x, t u2 ( x, t)

Vun( X, t)y

C C

Cii Ci;

i(x, t) =

i ( x, t) i ( x, t)

C =

C12 C22

Cin C2

V in ( X, t) y

Cin

C

C

The system (1.1), (1.2) is a system of partial differential equations of hyperbolic type.

To obtain a unique solution, we must add the initial (when t — 0 ) and boundary (when x — 0 and x — l ) conditions. We assume that at the initial time t — 0 there are no voltages and currents in the line

u(x,0) — i(x, t) — 0, x e[0, l], (1.4)

at the input of the line, at x — 0 , voltages are given, and at the output for x — l we have a load

with resistance R

S

u(0,t) = U0(t), u(l,t) = RSi(l,t). (1.5)

II. MATRIX OF COEFFICIENTS OF ELECTROSTATIC INDUCTION AND POTENTIAL COEFFICIENTS MATRIX

The matrix C in (1.2) is called the matrix of coefficients of electrostatic induction and in [1] it is noted by ß. For multiconductor line, this matrix allows to express the charge vector Q through the vector of potentials U in each conductor as follows

Q = ßU ;

Q = ßiUi HßiU2 +...+ ßnUn Q2 =ß2iUi +ß22U2 + ... + ß2nUn

Qn =ßnUi +ßn2U 2 +... + ßnnUn

(2.1)

L =

Lii Li2 Li2 L22

V L n L2 n

R =

f R i 0 0 Rnn

00

fGii 0

G =

L n

0 Ï 0

R

0 ^

0 G22 ... 0

v 0 0 ... Gnn

In [1, §25.2] a method for calculating the (1.3) exact values of the coefficients /?ÉJ- is proposed. To this end, it is required to solve n Dirichlet problems with the following boundary conditions. Consider the domain D, which is the outer region for n wires (Fig. 1)

Fig. 1. Domain D, where the Dirichlet problems must be solved.

When solving the problem with the number k, as the boundary conditions on the surface of all wires, except the wire A, a zero potential is

given, and on the wire A - a nonzero potential

of the value Uk is given. On the outer boundary

and on the surface of the earth the potential is considered to be zero. Thus, the following Dirichlet problems are to be solved:

Au = + = 0,(x,y) e D, (2.2) dx dy

u( x, y) = Ui =

\Uk * 0,i = k 0, i * k ,

Pik = Q, i, k = 1, n (2.6)

Uk

At the same time in [1, §25.3] it is proposed numerical method for calculating the potential coefficients matrix a, which allows to express the vector of potentials U through the charge vector Q in each conductor by formulas

U = aQ ;

U1 =ai1Q1 +ai2Q2 + - + a1nQn U2 =a21Q1 +a22Q2 + - + a2nQn

(2.7)

Un = an1Q1 +an2Q2 + - + annQn

(x, y) e ri, i = i, n, (2.3)

As follows from (2.1) and (2.7) the matrices a and are mutually inverse. To determine the

coefficients a¡^, i = 1, n, it is assumed that all

u(x y) _ O^-^ y) e re • (2 4) charges except the charge with the number k are

Here by ri the surface of the 7-th wire is denoted and re is the outer boundary of the

_ __a

domain D , D = D[j = Te U •

7=1

After solving the Dirichlet problem and determining the potential distribution u(x,y). for

each wire the charge Qi, i = 1, n is calculated

through the electric field strength vector E applying the Gauss theorem

Qi=s0li (E, n)dl,

E = - grad u = —

(du ^ dx du

\.dy j

(2.5)

Here e = 8,85418782 • 10~l2[F/m\ is the electrical constant, rt is the contour surrounding

Then the coefficients P^ can be calculated by

formulas the wire Ai, n = outer normal to contour r .

n

V y J

is the vector of

equal to zero

Qi =

\Qk * 0, i = k

It is

0, i ^ k

assumed also that the field of the charged wire will be the same as the field of a single wire stretched above the earth (since the distortion of the field due to the existence of other wires can be neglected because of the smallness of their cross sections and of the absence of the charges on them). Then, using the exact solution for one

wire, the values a^ and akp, k, p = 1, n (fig. 2)

can be calculated. Here s is absolute permittivity, are the length and the radius of the wire, r £ is the distance between wires with numbers p

and k, rp,k is the distance between the wire

numbered by k and the mirror image of the wire numbered by p. It follows from the foregoing that in order to determine the exact values of the coefficients of the matrix a it is necessary to solve problems analogous to problems (2.2) -(2.4), only at the boundaries ri, (on the surfaces of the wires) the known value of the charges \Qk * 0, i = k.t —

Qi I 0, i * k

(not the potential values).

i, k = 1, n must be setted

1

"■kk

, 2hk -ln—-

2n e l Rt

1 —

ïp ln, p * k. (2.8)

2^ e l —

pk

Fig. 2. Wires stretched above the earth and their mirror images.

Dirichlet condition (2.3) by the boundary condition (2.9) or (2.10), which contains integrals over the boundary of the domain from the values of the unknown function u(x, y). Such problems are called problems with nonlocal boundary conditions [5-10]. Let prove the solvability of this problem through the solution of the classical Dirichlet problem for the Laplace equation (2.2) - (2.4). It is known that the exact solution of problem (2.2) - (2.4) exists, is unique [11-16] and can be obtained using the Green's function. The Green's function G(M,M0) for problem (2.2) - (2.4) is the solution of the following problem for each fixed point M0 e D .

AMG(M, M0) =S(M, M0),

M, M0 e D,

(2.12)

Hence these problems are formulated as follows. In the domain D = D U r, it is required to determine the field of the potential u(x, y) that

inside of the domain D satisfies the Laplace equation (2.2), on the outer boundary re satisfies the condition (2.4), and on the wire surface potential takes the constant value and satisfies the conditions

<j> (E,n)dl = -i -dl = Q- (2.9)

Jri Jri dn £n

or

I

du дх

■n

du

ду

-и,

w=-Q

(2.10)

G(P,M0) = 0,P e r = 5D. (2.13)

Here M, M0 are two-dimensional points belonging to the domain D, S(M,M0) is the

n

Dirac delta function, r = Ts Uj]^ is the

i=1

boundary of the domain D , AM is the Laplace operator, where variables x and y are the coordinates of the point M (x, y).

If the Green's function is known, then the solution of problem (2.2) - (2.4) can be represented in the form

u(M) = -j)rU(p)

dG^M)

дп„ p

After solving the problem (2.2), (2.4), (2.9), the exact values of the coefficients aik are calculated as the ratio between the found values of the potentials Uk on the wire surface and the

specified charge values Q by formulas

= —, i, k = i, n.

Q

(2.11)

The problem (2.2), (2.4), (2.9) differs from the classical Dirichlet problem for the Laplace equation. The difference consists in replacing the

M e D, P e Г.

(2.14)

Here U(p) are the given values of the potential on the boundary r of the domain D, n is the vector of the outer normal to the

contour r. Taking into account the form of the boundary conditions (2.3), (2.4), formula (2.14) can be rewritten as follows

n i=1

dG(PuM)

r 8np

dl

Pi '

M e D, P e Г,.

(2.15)

a

ik

To obtain the solution of the problem (2.2), (2.4), (2.9), we substitute (2.15) in (2.9) and obtain a system of algebraic equations for determining the unknown values of the potential on the wire surface U in terms of the known

values of the charges Q

YuA -A. cf

dG(P„M)

r,- dnf

= Qk-, k = 1, n.

dl

Pi

dlk =

(2.16)

Comparing the obtained system (2.16) with (2.1), it is easy to note that the coefficients of the system (2.16) coincide with the values of the coefficients of the matrix of electrostatic induction /?

cnPk \

ÔG(P„M)

A

■dl,

rî dnP Pl V li J

dl

Pk '

k, i = 1, n.

(2.17)

Thus, to solve the problem (2.2), (2.4), (2.9) it is required to prove that there exists a unique solution of the system of equations (2.16). This affirmation will be proved in the form of a theorem.

Theorem. The system (2.16) and therefore the problem (2.2), (2.4), (2.9) has a unique solution.

Proof. We prove the theorem by using the Fredholm alternative for a system of algebraic equations, i.e. we prove that the corresponding homogeneous system for (2.16) does not have nonzero solutions. We carry out the proof by contradiction. Suppose that the system has a nonzero solution. This means that the problem (2.2), (2.4), (2.9) has a nonzero solution for given zero charges on all wires. It is known that the solution of the Laplace equation (2.2) satisfies the strong maximum principle, i. e. the solution of the Dirichlet problem for the Laplace equation attains its maximum and minimum values on the boundaries of the solution domain, and within the interior of the domain the solution value is strictly less than the maximum value and strictly greater than the minimum value. In our

case, the solution is different from the constant, since we assumed that the solution of the homogeneous problem (2.2), (2.4), (2.9) is nonzero and on the outer boundary re it accepts

the zero condition (2.4). It means that the maximum positive value and the minimum negative value of the potential are attained at the boundaries ri(i = 1, n Let the maximum positive value of the constant potential Umav > 0 is attained on the wire . Then the

max

derivative of potential with respect to normal takes only strictly negative values at all points of du

the contour r , i.e. —

dn

< 0. Then the charge

of this wire, which is calculated by the formula

(2.9), Q = —£0(b —dl is strictly positive,

J rp dn

that contradicts the assumption that all charges in the homogeneous problem are zero. A similar argument can be drawn for a wire on which a minimum negative value of the potential is attained.

So, the original assumption of the existence of a nonzero solution for a homogeneous problem is incorrect. Hence, by the Fredholm alternative, the nonhomogeneous problem (2.2), (2.4), (2.9) has a unique solution.

III. CALCULATION THE POTENTIAL COEFFICIENTS MATRIX FOR A THREE-PHASE CABLE

Numerical experiments have shown that for overhead power line with real values R, r ^

and r ,k, approximate formulas (2.8) for

determining the potential coefficients give values with high accuracy. Therefore, the determination of the potential coefficients by numerical solution of problems (2.2), (2.4), (2.9) or (2.2), (2.3), (2.4) does not make sense. At the same time, there are no acceptable approximate formulas [17] for determination of the potential coefficients of power cables, therefore, below we give the results of calculating a specific cable.

As an example we have chosen the sector-shaped cable with three cores (fig. 3). In the figure, the metallic parts of the cable (3 cores and aluminum screen) are light gray.

Since the zero potential is given on the screen, the domain D of the solution to the

e

0

r

problem (2.2), (2.4), shown in fig. 4.

(2.9) will be the region

Fig. 3. Cable with three triangular cores.

Fig. 4. Domain of the solution: domain D -unpainted inner part of the circle

The domain D is represented by unpainted inner part of the circle, being determined by the following radii values:

r = 15 mm, r2 = 16,9 mm, r3 = 18,45 mm.

Each sector-shaped core is surrounded by insulation and all three cores are surrounded by another layer of insulation. In general, these insulations can have different permittivity. Therefore, to the relations (2.2), (2.4), (2.9), we

must add the conjugation conditions on the interface between two media

du dn

du+ dn

(3.1)

Here, signs "+" and "-" denote values from one and the other side of the media interface.

The problem (2.2), (2.4), (2.9), (3.1) was solved numerically by the finite volume method. Constructed grid consists of 30853 triangular elements (fig. 5).

Based on the results of calculations, using formulas (2.11), a matrix a was constructed and the matrix /? was calculated:

a ■

3.9689 1.0822 1.0822 1.0829 3.9689 1.0822 1.0822 1.0829 3.9689

J

109[1/ F ],

(3.2)

ß = al =

( 0.2853 -0.0611 -0.0612^

-0.0612 0.2853 -0.0611

-0.0611 -0.0612 0.2853

[nF ]. (3.3)

From the geometry of the domain D it is clear that the matrices should have the same values on the diagonal and all the off-diagonal elements should be the same. From the presented results it is seen that the diagonal elements coincide, and there is a difference in the fourth sign after the comma in the off-diagonal elements. The difference is about 0.06% and is caused by numerical method error.

Fig. 5. A grid of triangular elements in the domain D. The red dots denote the boundary nodes of the grid

and the nodes on the media interfaces.

IV. CONCLUSIONS

1. A method for constructing the potential coefficients matrix is proposed.

2. The theorem of existence and uniqueness of the solution for the boundary value problem for the Laplace equation with nonlocal boundary conditions is proved.

3. The developed method is applied for determining the potential field distribution and the potential coefficients matrix in a three-core cable.

References

[1] Demirchyan K.S., Korovkin N.V., Chechurin V.L. Teoret7chesk7e osnovy elektrotehn7k7 [Theoretical foundations of electrical engineering]. Vol. 3, SPb: Piter, 2004. 364 p. (In Russian)

[2] Reutov, Y.Y., Gobov, Y.L. & Loskutov, V.E. Feasibilities of Using the ELCUT Software for Calculations in Nondestructive Testing Russian Journal of Nondestructive Testing (2002) 38: 425. https://doi.org/10.1023/A:1022174825844

[3] Power cable parameters calculation. https://quickfield.com/advanced/cable.htm (accessed 29.11.2018)

[4] W.T. Weeks. Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of a Dielectric Interface. IEEE Transact7ons on M7crowave Theory and Techn7ques, 1970, vol. MTT-18, no. 1, p.p. 3543

[5] Cannon J.R. The solution of the heat equation subject to the specification of energy. Quart. Appl. Mat, 1963, no. 21, p.p. 155-160

[6] Kamynin L.I. Ob odnoi kraevoi zadache teorii teploprovodnosti s neklassicheckimi granichny-mi usloviyami [On a boundary problem of the theory of heat conduction with non-classical boundary conditions]. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki - Journal of Computational Mathematics and Mathematical, 1964, vol. 4, no. 6, p.p. 1006-1024. (In Russian).

[7] A. L. Skubachevskii, Nonclassical boundary value problems. I, CMFD , 2007, Volume 26, 3132

[8] G. Avalishvili, M. Avalishvili, D. Gordeziani. On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation. Appl7ed Mathemat7cs Letters, 2011, vol. 24, p.p. 566-571

[9] Bashir Ahmad, Sotiris K. Ntouyas, Hamed H. Alsulami. Existence Theory for nth Order Nonlocal Integral Boundary Value Problems and Extension to Fractional Case. Abstract and Appl7ed Analys7s, vol. 2013, Article ID 183813, 12 pages

[10] Siraj-ul-Islam, Imran Aziz, Masood Ahmad. Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions. Computers and Mathematics with Applications, 2015 vol. 69, Issue 3, p.p. 180-205

[11] Sveshnikov A.G., Bogolyubov A.N., Kravtsov V.V. Lektsii po matematicheskoy fizike [Lectures on mathematical physics]. Moscow, 2004, 416 p. (In Russian)

[12] S. S. Bayin. Mathematical Methods in Science and Engineering. Wiley, 2006.

[13] Eyges, Leonard, The Classical Electromagnetic Field. Dover Publications, New York, 1972.

[14] A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations. 2nd edition, Chapman & Hall/CRC Press, Boca Raton, 2003

[15] W.T. Weeks. Multiconductor Transmission-line Theory in the TEM Approximation. IBM Journal of Research and Development, 1972, vol. 16, Issue 6, p.p. 604-611

[16] Paul Clayton R. Analysis of multiconductor transmission lines. 2-nd Ed. Wiley-IEEE Press, 2077, 780 p.

[17] Raschot kabeley iz pervykh printsipov [Calculation of cables from first principles] (In Russian) https://www.ruscable.ru/interactive/ forum/show msg-91744.html (accessed 29.11.2018)

Information about author.

Patiuc Vladimir. Doctor of Physics and Mathematics, Associate Professor at Moldova State University, Leading Researcher at the Institute of Power Engineering of the Academy of Sciences of Moldova. Areas of scientific interests: mathematical physics, numerical analysis, theoretical mechanics and theoretical electrical engineering. E-mail: [email protected]

i jj^jjj