SIXTH INTERNATIONAL SYMPOSIUM NIKOLA TESLA October 18 – 20, 2006, Belgrade, SASA, Serbia

An Improvement of Equivalent Electrodes Method for Calculation of Three-Core Cable Capacitances

Miodrag S. Stojanović1, Dragan S. Tasić2, Predrag D. Rančić3

Abstract – An improvement of equivalent electrode method is proposed in this paper. The accuracy of results of equivalent electrode method depends on the number of used equivalent electrodes. However, the certain problems in calculations may appear by increasing number of equivalent electrodes. Improvement proposed in this paper enables accurate calculation using small number of equivalent electrodes. Namely, instead of one calculation with large number of equivalent electrodes, two calculations with small number of electrodes are accomplished. Proposed improvement is verified on the examples of capacitance per unit length calculation of three-core power cables. Two different geometries of power cables are comprehended.

Keywords – Equivalent electrodes method, three-core cables, capacitance calculation.

I. INTRODUCTION

Equivalent Electrodes Method (EEM) [1] is numerical method for approximate solving of non-dynamic electromagnetic fields and other potential fields of theoretical physics. Afterwards, using EEM, good results were obtained in computations of the electrostatic field, in the theory of low-frequency grounding systems, for static magnetic field solving, as well as in transmission lines analysis. The basic idea of EEM is that an arbitrary shaped electrode can be replaced (represented) by finite system of equivalent electrodes (EEs). The EEs are placed on the electrode surface. Depending on problem geometry, for EEs can be used flat or oval strips, spherical bodies and toroidal electrodes. The radius of the EEs is equal to the equivalent radius of the electrode part that is substituted. The potential and charge of the EE and substituted real electrode part are also equal. Using boundary condition that the electrode surface is equipotential, -----------

1Miodrag S. Stojanović is with the Faculty of Electronic Engineering, Aleksandra Medvedeva 14, 18000 Nis, Serbia, E-mail: miodrag@elfak.ni.ac.yu

2Dragan S. Tasić is with the Faculty of Electronic Engineering, Aleksandra Medvedeva 14, 18000 Nis, Serbia, E-mail: dtasic@elfak.ni.ac.yu

3Predrag D. Rančić is with the Faculty of Electronic Engineering, Aleksandra Medvedeva 14, 18000 Nis, Serbia, E-mail: prancic@elfak.ni.ac.yu

it is possible to form system of linear equations, with EE charges as unknowns.

In the formal mathematical presentations, EEM is similar to the moment method form, but it is very important to emphasize difference in the physical fundaments and in the matrix forming procedure. Also, EEM application does not require numerical integration of any kind, in contrast to the moment method application, which always requires numerical integration, and thus produces some problems in the numerical solving of non-elementary integrals having singular subintegral function. As numerical method, EEM is applicable to the transmission lines analysis for all system geometries. In this paper EEM method is applied for calculating of capacitances of medium voltage power cables.

Calculation of capacitances of single-core cables is very simple, as well as of three-core cables with individual metal sheath over each conductor. Namely, there is an analytical expression for capacitance calculations of these cables. On the other hand, problem of capacitance calculations (operating and zero-phase sequence) of three-core cables with one common metal sheath or electrical screen is more difficult. Unfortunately, there are no analytical expressions for calculation of these capacitances which give satisfied results. Therefore, the numerical methods are used for this problem solving.

In this paper, EEM is applied for calculation of three-core cables capacitances. The method is adopted for capacitance calculation on the way that only phase conductors are replaced by EEs while the metal sheath is comprehended using images in the cylindrical conducting mirror. Instead of one calculation with large number of EEs, operating and zero-phase sequence capacitances of the cables are calculated based on two calculations with quite small number of EEs.

Verification of the method is made on two cable geometries. The first one is three-core cable with individual metal sheath over each phase conductor for which exact value of capacitance is known from analytical expression. The second geometry is three-phase paper insulated cable with common lead sheath and sector-shaped cross section of phase conductors. For this cable geometry, beside EEM, the finite element method is also used for operating and zero-phase sequence capacitances calculation. It is shown that results obtained for both cable geometries, using proposed approach

with only two EEM calculations with quite small number of equivalent electrodes, are very accurate.

II. CAPACITANCES CALCULATION OF THREE-CORE POWER CABLES

Three main geometries of cables are used in medium and high voltage power networks. Single-core cables and three-core cables with individual metal sheath or electrical screen over each phase conductor (e.g. XHP 48, IPZO) have coaxial structure (Fig 1). Therefore, capacitances of these cables can be calculated very simply using analytical expression [2], [3]:

)/ln(2 0 pr raC επε=′ (1)

Fig. 1. Single-core power cable

where a is inner radius of metal sheath, rp is radius of phase conductor, and rε is relative permittivity of insulation.

For the third cable geometry, three core cables with one common metal sheath and circular or sector-shaped cross section of phase conductors, analytical expression for capacitances calculation does not exist. For this type of cables operating and zero-phase sequence capacitances are used in power system analyses. Reference [2] gives expressions for calculation of operating and zero-phase sequence capacitances derived under assumption of "thin" phase conductors which can be derived using theory of images in the cylindrical conducting mirror. Starting from the geometry shown in Fig. 2 where s is distance between conductor and center of metal sheath, a is inner radius of metal sheath and sad 2= , and assuming that all phase conductors are equipotential ( QQQQ ′=′=′=′ 321 ), the following expression for zero-phase sequence capacitance is derived:

prassa

C

32

660

3ln

2−

πε=′ . (2)

Fig. 2. Illustration for derivation of capacitance expressions

Similarly, assuming that sum of the phase conductor charges is equal to zero ( 0321 =′+′+′ QQQ ), as well as sum of conductor voltages is equal to zero, expression for operating capacitance can be derived:

)()(3ln

4

662

2322

sarssa

C

p −−πε

=′ . (3)

If partial capacitance between two phase conductors is labelled with cC′ (Fig. 4), the capacitance per one phase of three-core cable is:

cCCC ′+′=′ 30 . (4)

Fig. 3. Partial capacitances of three-core cable with

common metal sheath

III. EQUIVALENT ELECTRODES METHOD

The basic idea of EEM is that an arbitrary shaped electrode can be replaced (represented) by finite system of EEs. Applicability of EEM for capacitance calculation will be illustrated on three-core cable with one common metal sheath example. Each of phase conductors will be replaced with n EEs located on the conductor surface. The metal sheath can be comprehended on two different ways: replaced by certain number of EEs or comprehended using the images theory in cylindrical conducting mirror. The second one is used in this paper. The zero-phase sequence capacitance of three-core cable with common metal sheath is calculated forming the system of equations with assumption that all EEs are on the same potential (e.g. 1V). The potential of the metal sheath is assumed to be zero. The charges of EEs can be obtained solving previously formed system of equations. Sum of calculated EEs charges divided by their assumed potential gives tripled value of zero-phase sequence capacitance i.e.:

nn CC 01 3 ′=′ (5) The index n in capacitances subscripts denotes that each

phase conductor is replaced with n EEs. In order to obtain operating capacitance one more system of linear equations must be formed. Now, two of three phase conductors are assumed to be on the some potential (e.g. 1V). The third conductor and metal sheath are on the zero potential. Solving this new system of equations, and dividing sum of calculated charges of the first two conductors (i.e. EEs which replace these conductors) by assumed potential we obtain capacitance (see Fig. 3):

)(2 02 ncnn CCC ′+′=′ . (6) Operating capacitance can be obviously calculated using

capacitances nC1′ and nC2′ :

nnn CCC 21 23

32 ′+′−=′ . (7)

The main problem in EEM application is requirement of large number of the EEs for obtaining results with a satisfying accuracy. However, analyses of numerical results show that the error of EEM is inverse proportional to selected number of EEs. Having in mind this fact, we can make very accurate estimation of cable capacitances using results of two calculations with quite small number of EEs (e.g. 50 and 100).

If following notation is used for zero-phase sequence and operating capacitances:

0C′ , C′ - exact values of capacitances per unit length,

10 nC′ , 1nC′ - capacitances calculated by EEM with n1 EEs,

20 nC′ , 2nC′ - capacitances calculated by EEM with n2 EEs.

and we assume that the error is inverse proportional to number of used EEs, then:

1000 1nKCC n +′=′ , 11

nKCCn +′=′ , (8)

2000 2nKCC n +′=′ , 22

nKCCn +′=′ , (9) where: K0 and K are corresponding factors:

12

00210

)(21

nnCCnn

K nn

−

′−′= ,

12

21 )(21

nnCCnn

K nn

−

′−′= . (10)

Substituting Eq. (10) into Eqs. (8) or (9) follows:

12

01020

12

nnCnCn

C nn

−

′−′=′ ,

12

12 12

nnCnCn

C nn

−

′−′=′ . (11)

The Eq. (11) can be used for very accurate calculation of zero-phase sequence and operating capacitances based on the EEM results that are obtained using quite small number of EEs (n1 and n2). Relative errors in percents of results calculated by EEM with n EEs are:

1000

00 Cn

KC n ′

=′δ , 100CnKCn ′

=′δ . (12)

IV. TEST EXAMPLE

Presented approach is applied for capacitance calculation of two types of three-core cables. The first one is 10 kV three-core power cable XHP48 3x150/25 6/10 [4]. Cross-section area of each copper conductor is 150 mm2. Insulation type is XLPE, and each core has individual cooper screen with cross-section area of 25 mm2. Therefore, each core of the cable can be separately analyzed from the aspect of capacitance calculation. Radius of phase conductors is

mm9.7=pr , insulation thickness mm4.3=δ ins and relative permittivity of XLPE is 3.2=ε r . Exact value of capacitance per unit length calculated using the Eq. (1) is

µF/km3573.0=′C . The results of capacitance per unit length calculation of this

cable for 6 different numbers of EEs are given in Table I. Substitution of the first two results ( 100,50 21 == nn ) into Eqs. (10) and (11) gives 46.0=K , µF/km3573.0=′C . Obviously, this result is more accurate then one obtained by EEM using 750 EEs. Fig. 4 shows the relative error versus the number of EEs.

The second example is 10 kV three-core cable IPO13-S 3x150 6/10 [5]. This is paper insulated cable with common lead sheath and sector-shaped phase conductors. Cross-section

TABLE I

CAPACITANCE OF XHP48 3X150/25 6/10 CABLE CALCULATED BY EEM

n nC ′ nC ′δ 50 0.3665 µF/km 2.59 % 100 0.3619 µF/km 1.28 % 150 0.3603 µF/km 0.85 % 250 0.3591 µF/km 0.51 % 500 0.3582 µF/km 0.25 % 750 0.3579 µF/km 0.17 %

area of each conductor is 150 mm2. The required data for capacitances calculation of this cable are: thickness of the insulation over conductor mm2.3=δ ins , thickness of insulation between phase conductor and metal sheath

mm7.3=δ ps , 5.3=εr , mm6.14=segR , mm6.12=H , mm2.12=B , mm3.20=D , mm3.5=F . The geometry of

sector-shaped conductor is shown in the Fig. 5, where all dimensions are labelled.

0 100 200 300 400 500 600 700 800 9000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

δC' 0 [%

]

n

EEM Equation (12), K=0.46

Fig. 4. Relative error versus number of EEs

Fig. 5. Sector-shaped conductor with dimension labels

Fig. 6. Locations of EEs on the sector-shaped conductor surface

TABLE II CALCULATED CAPACITANCES OF IPO13-S 3x150 6/10 CABLE (EEM) n N1; N2; N3; N4 nC ′ nC ′δ 54 6; 7; 7; 20 0.2458 µF/km 0.4498 µF/km102 10; 13; 13; 40 0.2437 µF/km 0.4453 µF/km150 14; 19; 19; 60 0.2430 µF/km 0.4438 µF/km198 18; 25; 25; 80 0.2426 µF/km 0.4429 µF/km246 22; 31; 31; 100 0.2424 µF/km 0.4424 µF/km306 30; 39; 39; 120 0.2422 µF/km 0.4420 µF/km354 38; 98; 98; 140 0.2421 µF/km 0.4417 µF/km420 50; 55; 55; 150 0.2421 µF/km 0.4415 µF/km

TABLE III

CALCULATED CAPACITANCES OF IPO13-S 3x150 6/10 CABLE (FEM)

nn nnC0′

nnC ′

387 0.2452 µF/km 0.4510 µF/km 635 0.2443 µF/km 0.4488 µF/km 1020 0.2432 µF/km 0.4452 µF/km 1673 0.2426 µF/km 0.4425 µF/km 3552 0.2422 µF/km 0.4416 µF/km 9518 0.2420 µF/km 0.4410 µF/km

If sector-shaped conductors are replaced with "thin" ones located in the centers of inscribed circles, we can calculate capacitances using Eqs. (2) and (3). The results obtained on this way are: µF/km2301.00 =′C , µF/km7557.0=′C .

For capacitance calculation of mentioned cable using EEM, equivalent electrodes will be located on the conductor surfaces according to Fig. 6. On the same figure N1, N2, N3, and N4 represent number of EEs located on the individual parts of sector-shaped conductor. Total number of EEs for each phase conductor is 4321 22 NNNNn +++= . The results of EEM for different numbers of EEs are given in Table II.

Substituting EEM results obtained for 54 and 102 EEs into Eqs. (10) and (11) gives: 24052.00 =K , 51867.0=K ,

F/km2413.00 µ=′C , F/km4402.0 µ=′C . The relative errors, shown in Figs. 7 and 8, are calculated using these values of capacitances as exact ones. The zero-phase sequence and operating capacitances per unit length of IPO13-S 3x150 6/10 cable, calculated using finite element (FEM) [6], are given in Table III for different number of nodes.

V. CONCLUSION

The approach proposed in this paper allows us to obtain very accurate results using EEM with quite small number of EEs. Verification of proposed approach is made on capacitance calculation of two geometries of three-core cables. Capacitance of three-core cable with individual electrical screen over each conductor, for which analytical expression exist, were very accurate obtained using two EEM

0 200 400 600 8000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

δC' 0 [%

]

n

EEM Equation (12) K0=0.24052

Fig. 7. Relative error of zero-phase sequence capacitance

versus number of EEs

0 200 400 600 8000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

δC' [

%]

n

EEM Equation (12), K=0.51867

Fig. 8. Relative error of operating capacitances versus number of EEs

calculations with small number of EEs (50 and 100). The result obtained on this way is more accurate then result of one EEM calculation with 750 EEs. The second geometry is three-core cable with common metal sheath. Zero-phase sequence and operating capacitances of this cable are also very accurate calculated using proposed approach with small number of EEs (54 and 102). Namely, increasing the number of EEs, values calculated by EEM tend to the ones obtained using proposed approach. Increasing number of nodes in calculations by finite element method, results also tend to the same values.

REFERENCES [1] Velickovic, D. M.: "Equivalent Electrodes Method", Scientific

review, Number 21-22 Belgrade, 1996, pp. 207-248. [2] L. Heinhold, Power Cables and Their Application. Berlin:

Siemens Aktiengesellschaft, 1990. [3] D. Tasić, Power Cables Technique. Niš: Faculty of Electronic

Engineering, 2001. (in Serbian) [4] Cable Factory Zajecar, “Medium voltage electrically protected

XLPE insulated and PVC sheathed three-core power cable”, Technical Specification. [Online] Available: http://www.fkz.co.yu/indexe.htm.

[5] Cable Factory Jagodina, "Cables with impregnated paper insulation" [Online] Available:

http://www.fks.co.yu/fkse/maticna/energet/papir/index.htm [6] D. Meeker, "Finite Element Method Magnetics-User’s

Manual", February 2004 [Online] Available:http://femm.berlios.de