Optimisation of an alternating current multi-conductor system

Calin Munteanu a,1, Vasile Topa a, Emil Simion a, Gheorghe Mates a,

Laura Grindei a, Gilbert De Mey b,*, Marius Purcar c

a Electrotechnics Department, Technical University of Cluj-Napoca, Baritiu 26-28, 400027 Cluj-Napoca, Romaniab ELIS Department, University of Gent, Sintpietersnieuwstraat 41, 9000 Gent, Belgium

c TW-ETEC, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium

Received 22 March 2004; received in revised form 23 February 2006; accepted 23 February 2006

Available online 2 May 2006

Abstract

In this paper, we consider a multi conductor system for the transport of electric energy. Not only the electric field between the conductors, but

also the internal field distribution inside the conductors is taken into account. Whereas the first one is governed by the Laplace potential equation,

the latter one is governed by the AC diffusion equation in order to include the eddy current effects. Both sets of equations are solved using the

Boundary Element Method. At last the geometrical layout of the multi conductor system will be optimized using a genetic algorithm. The criterion

for the optimization process is the minimization of the global electric losses of the multi conductor system.

q 2006 Elsevier Ltd. All rights reserved.

Keywords: Multi-conductor systems; Steady-state AC regime; BEM; Multi-objective optimal design; Losses minimisation

1. Introduction

Let us consider a multi conductor system for the transport of

electric energy. Several criteria can be put forward to find an

optimal placement of the conductors. Due to the high voltages

involved in present day systems, one should avoid too high

electric field strengths between neighbouring conductors. Due

to the corona effect, a leakage current can flow between these

conductors giving rise to so called parallel losses. In this

contribution we are dealing with serial losses, caused by the

resistivity of the cable materials. Due to the eddy currents

induced in any cable by all the other ones, the current density

and hence the losses will be increased. However, eddy currents

are also influenced by the geometrical layout of the conductors.

It is our intention to minimise these losses using a genetic

algorithm.

First of all, the potential problem has to be solved. The

potential between the conductors and the internal potential

(the component of the vector potential in the direction of the

conductors is used here) have to be calculated numerically. For

this purpose the Boundary Element Method (BEM) is used

0955-7997/$ - see front matter q 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enganabound.2006.02.006

* Corresponding author. Tel.: C32 9 264 3386; fax: C32 9 264 8961.

E-mail addresses: calin.munteanu@et.utcluj.ro (C. Munteanu), demey@

elis.ugent.be (M. Purcar).1 Tel./fax: C40 264 592903.

here. The Boundary Element Method offers the advantage that

one can easily handle different cross sectional geometries for

the cables. Moreover, the cables are put in the free air, which is

modelled as an infinite non-conducting medium. With the

Boundary Element Method, an infinite medium offers no

particular problems. Also the diffusion equation, used inside

the conductors, is solved using the Boundary Element

approach. The coupling between the interior and the exterior

domain is easily handled with the Boundary Element Method

as will be explained in the next sections. The final result of this

calculation is that the global loss of all the conductors is a

known quantity.

In the second part of the paper, a BEM-based multi-

objective optimisation problem based on genetic algorithms is

investigated. The global losses of the multi conductor system

will be minimized by searching for an optimal placement of the

conductors.

The optimisation problem of cables has been the subject of

several contributions in Refs. [1–3]. Also a genetic algorithm

has been used as well for the optimisation problem [4].

Although it must be stressed again that in this contribution the

emphasis is put on the optimal placement of the cables imposed

by the series resistance of the materials.

2. Basic equations

The basic equation in the study of the electromagnetic field

diffusion in conductive media, in terms of the vector magnetic

Engineering Analysis with Boundary Elements 30 (2006) 582–587

www.elsevier.com/locate/enganabound

C. Munteanu et al. / Engineering Analysis with Boundary Elements 30 (2006) 582–587 583

potential formulation, is given by [5,6]:

V!1

mV! �A

� �ZKs

v �A

vtCVF

� �(1)

where �A is the vector magnetic potential,F is the scalar electric

potential, s is the electric conductivity of the conductor and m

is the absolute magnetic permeability of the conductor.

If the diffusion equation is approached by a 2D plan-parallel

application in the conductor cross-sections plane, the Coulomb

gauge is automatically satisfied [5,6] due to the fact that in this

case the magnetic vector potential has only one component,�AZAðx;yÞ �k. Thus, considering a sinusoidal steady-state

regime, Eq. (1) becomes

K1

mD�AZKjus

�AKsV

�F: (2)

where �A is the complex value of the vector magnetic potential

on the 0z-axis direction and u is the pulsation.

The electric scalar potential F(z) along a very long and

straight conductor of arbitrary cross section represents a

gradient field and therefore it satisfies the Laplace equation

DF(z)Z0 [5]. If�Js is the conductor’ source current density

�JsZKsVF, the governing Eq. (2) leads to the final system of

equations

D�Ak

tKja2k �

Ak

tZ 0; kZ 1; N ; (3)

where N is the number of the current carrying conductors from

the problem,�Ak

tZ

�AkC

�Ak

sis the total vector potential inside

the kth conductor, a2kZukmksk and �Ak

sZ jmk�

JksaK2k .

The external infinitely extended dielectric space is governed

by the Laplace equation D�A0Z0. Therefore, the interface

conditions must be considered on each conductor boundary Gk:

�A0;k

Z�Ak Z

�Ak

tK

�Ak

s

v�A0;k

vnZK

1

mrk

v�Ak

vnZK

1

mrk

v�Ak

t

vn

:

8>>>><>>>>:

(4)

In addition, the total current flowing through each conductor

must satisfy the Ampere’s theorem. Using the vector magnetic

potential integral formulation, it can be expressed as:

ðGk

vA0;k

vndGZm0�

Ik: (5)

The global conductor’s electric per-unit-length parameters

Rk, Lk can be computed numerically using the following

relations [5,6]:

Rk ZKuk

m0I2k

Re j

ðGk

�Ak

t

v�A�

0;k

vndG

8><>:

9>=>;; (6)

ukLk ZKuk

m0I2k

Im j

ðGk

�Ak

t

v�A�

0;k

vndG

8><>:

9>=>;: (7)

It is our intention to minimise the series resistance (6) by

finding the optimal placement of the conductors.

3. The boundary elements formulation

The governing equations for the interior and exterior

problems lead to the boundary integral formulations [7,8]

ci�Aki

tC

ðGk

�Ak

t

v�Gk

i

vnZ

ðGk

�G

k

i

v�Ak

t

vn; kZ 1; N ; (8)

ci�Ai

0C

ðG

�A0

vGe

vnZ

ðG

Ge

v�A0

vn; (9)

where GZgNkZ1Gk:

The fundamental solutions involved in (8), (9) are [6,7]

Gki ðrÞZ

1

2p½kerðakrÞC j keiðakrÞ�; (10)

GeðrÞZ1

2pln

1

r; (11)

if the proximity effects problem is approached in the case of a

multiple arbitrary cross-section conductor system placed in

outer infinitely-extended dielectric space, where ker and kei are

the Kelvin functions.

The problem can be solved in the most general case, thus the

currents flowing through these conductors may have different

effective values, different initial phases and also different

working frequencies.

If the problem has symmetries with respect to the 0y-axis,

where in this case either zero Dirichlet (identical currents in

opposite senses in corresponding conductors—case 1) or zero

Neumann (currents in the same sense in corresponding

conductors—case 2) boundary conditions might be set up,

the problem’s solution can be simplified by using the following

equivalent fundamental solutions [6]

Gcase 1e Z

1

2pln

r2r1

; (12)

Gcase 2e Z

1

2plnðr1r2Þ; (13)

where r1 refers to the distance from the source point to

the computation point, while r2 refers to the distance from the

Fig. 1. The three-phased system used for the BEM numerical analysis

convergence test.

C. Munteanu et al. / Engineering Analysis with Boundary Elements 30 (2006) 582–587584

source point to the image with respect to the 0y-axis of the

computation point.

The complex integral Eqs. (8) and (9) together with the

interface relations (4) and Ampere theorems (5) lead to the

global matrix Eq. (14).

HR KHI SR KSI 0 0

HI HR SI SR 0 0

H0 0 KS0 0 K1 0

0 H0 0 S0 0 K1

0 0 LM 0 0 0

0 0 0 LM 0 0

266666666664

377777777775

ARt

AIt

DAR0

DAI0

ARs

AIs

266666666664

377777777775Z

0

0

0

0

m0IR

m0II

26666666664

37777777775: (14)

In the global matrix equation, HR, HI, SR, SI are the real

and imaginary components of the metric term matrices,

related to the interior problem of the N conductors’ system,

while H0, S0 are the metric term matrices related to the

exterior problem. All these matrices have n!n size, where n

is the total number of boundary elements used in the

application. LM are the transposed vectors of the boundary

elements length, while A-type terms and DA-type terms

represents the vectors of the real and imaginary parts of the

magnetic potential vectors and their derivatives, respectively.

IR and II are the vectors of the real and imaginary parts of the

complex currents flowing through current carrying

conductors.

In this way, a (4nC2N)!(4nC2N) size global system

of shape A$XZB to be solved using the Gauss elimination

algorithm [6] has been implemented. Once the boundary

unknowns are computed, the conductor electric parameters

can be evaluated by implementing (6) and (7) in the

numerical computation. Applying afterwards (8) and (9) for

the domain points computations, the distributions of the

current densities through conductors as well as the

magnetic potential values in all the sub-domains can be

determined.

Table 1

General settings for the BEM numerical convergence analysis application

Conductor

1

Conductor

2

Conductor

3

Conductivity (Cu) (UmK1!106) 55 55 55

Permeability m0 m0 m0Current effective value (A) 10 10 10

Current initial phase (8) 90 210 330

Current frequency (Hz) 50 50 50

Coordinates of the center (m) (0,

0.01733)

(K0.015,

K0.00853)

(0.015,

K0.00853)

Conductor radius (m) 0.001 0.001 0.001

No. of boundary elements 18 18 18

4. Numerical solution convergence

In order to test the BEM numerical analysis module

computation accuracy the problem described in Fig. 1 was

adopted. It consists of a three-phased conductor system whose

parameters are emphasized in Table 1. The analytical

expression of the magnetic vector potential value in a point

of space can be therefore computed by:

Ak Z1

2p

X3kZ1

mkIk ln1

rk

� �(15)

where Ak, mk, Ik are the vector magnetic potential along the 0z-

axis, the magnetic permeability and the carried current in the

kth conductor of the system considered, respectively.

In these conditions, the magnetic potential values computed

for the test point (0.04, 0.04, 0) are AanalyticalZ6.38799!10K5

(Tm) (computed using MATHEMATICA software) and

AnumericalZ6.38783!10K5 (Tm) using 18 constant boundary

elements on each conductor, which means a relative error

3[%]Z0.0025%. This result confirms that the numerical

computation using the BEM analysis module developed is

very accurate.

The instantaneous magnetic field distribution correspond-

ing to the analytical solution determined using the

MATHEMATICA software is presented in Fig. 2 for the

time moment tZ0. The instantaneous magnetic field

distribution determined using the BEM numerical analysis

module is presented in Fig. 3.

5. Multi-objective optimisation

Many problems involve simultaneous optimisation of

several objectives. Usually, there is no single optimal solution,

but there is a set of possible solutions. These solutions are

optimal if no other solutions in the search space are better

when all objectives are considered. These solutions are called

Pareto-optimal solutions. Mathematically, this concept can be

defined as follows [9]:

–0.04 -0.02 0 0.02 0.04–0.04

–0.02

0

0.02

0.04

Fig. 2. The analytical solution of the magnetic field distribution.

C. Munteanu et al. / Engineering Analysis with Boundary Elements 30 (2006) 582–587 585

Considering a multi-objective maximization problem with

m parameters (decision variables) and n objectives:

Minimize : yZ f ðxÞZ ðf1ðxÞ;f2ðxÞ;.;fnðxÞÞ (16)

where xZ(x1,x2,.,xm)2X, yZ(y1,y2,.,yn)2Y.

A decision vector a2X is said to dominate a decision vector

b2X (also written as a3b) if:

ci2f1; 2;.; ng: fiðaÞ% fiðbÞodj2f1; 2;.; ng: fjðaÞ

! fjðbÞ (17)

Definition 1. : Let A4XThe function A1ðAÞ gives the set of

non-dominated decision vectors in A:

Fig. 3. The numerical solution of the magnetic field distribution using BEM.

A1ðAÞZ fx2Aj o e a2A : a3xg:

In this light, a Strength Pareto Evolutionary Algorithms

(SPEA) [10] was implemented. It consists of the following

main steps [10].

The initial set, P0, will be populated with N indivi-

duals whose decision variables are generated randomly within

the domain. Considering T2N the number of generations:

ct2f1; 2;.;TgPtZ:; �PtZ:

† Step 1. Find the best non-dominated set A1ðPtÞ of Pt. Copy

these solutions to �Pt or perform �PtZ �PtgA1ðPtÞ, where �Pt

is the external population at iteration t;

† Step 2. Find the best non-dominated solutions A1ð �PtÞ of the

modified population �Pt and delete all dominated solutions

or perform �PtZA1ðPtÞ;

† Step 3. If j �PtjO �N, where �N is the bounding limit of the

external population size, use a clustering technique to

reduce the size to �N. Otherwise, keep �Pt unchanged. The

resulting population is the external population �PtC1 of the

next generation;

† Step 4. Assign fitness to each elite solution i2 �PtC1 by

using the strength function S:X/R S(i)Zn(i)/(NC1),

where n:X/N n(i) is the number of the current popu-

lation members that an external solution i dominates and

N is the size of the population P. Then, assign fitness

to each population member j2Pt by using

F:X/R FðjÞZ1CP

i2�Ptoi3jSðiÞ. The addition of 1

makes the fitness of any current population member Pt

to be more than the fitness of any external population

member �Pt. In this way, a solution with smaller fitness is

the best;

† Step 5. Apply a binary tournament selection with these

fitness values (in a minimisation sense), a crossover and a

mutation operator to create the new population PtC1 of

size N from the combined population ð �PtC1gPtÞ.

† Step 6. If tRT or another stopping criterion is satisfied,

then the final population is PtC1, else go to Step 1 where

tZtC1.

For the optimisation module has being used simulated

binary crossover (SBX) and Polynomial Mutation [10].

As the name suggests, the SBXoperator simulates theworking

principle of the single-point crossover operator on binary strings.

Let us consider two decision vectors xZ(x1,x2,.,xm)2X, yZ(y1,y2,.,yn)2X, which we call parent solutions. The results of

the crossover operator are the offspring’s v and w which we

compute as follows [10]ci2{1,2,.,m}

† Step 1. Choose a random number ui2(0,1).

† Step 2. Calculate bqi using the following equation:

bqi Z

ð2uiÞ1=ðhcC1Þ; if ui%0:5

1

2ð1KuiÞ

0@

1A1=ðhcC1Þ

; otherwise:

8>>>><>>>>:

(18)

Table 3

General settings and the main results of optimisation process

C 1 C 2 C 3

Conductivity (Cu) (UmK1!106) 55 55 55

Permeability m0 m0 m0Current value (kA) 1 1 1

Current initial phase (8) 0 90 180

Frequency (Hz) 50 50 50

Conductor radius (cm) 1 1 1

No. of boundary elements 24 24 24

Coordinates of the fixed conduc-

tor centres (cm)

(8, 8) (K8, 8) (K8, K8)

Initial coordinates of the mobile

conductor centres (cm)

– – –

Optimal coordinates of the

mobile conductor centres (cm)

– – –

Bounds of the mobile conductor

centres (cm)—0x-axis

– – –

Bounds of the mobile conductor

centres (cm)—0y-axis

– – –

Initial values of the conductor

resistances (U!10K5)

5.994 5.990 5.999

Optimal values of the conductor

resistances (U!10K5)

6.002 5.989 6.002

Resistances variations (%) C0.13 K0.02 C0.05

Table 2

MOOP general settings

Population size 40

Number of generations 25

Crossover probability 0.8

Mutation probability 0.1

External population size 10

Fig. 4. The multi-conductor system at the beginning of the optimisation

process.

C. Munteanu et al. / Engineering Analysis with Boundary Elements 30 (2006) 582–587586

† Step 3. Compute the offspring by using equations:

vi Z 0:5½ð1CbqiÞxi C ð1KbqiÞyi�

wi Z 0:5½ð1KbqiÞxi C ð1CbqiÞyi�: (19)

The distribution index hc is any non-negative real number [6].

This distribution assures that vZ(v1,v2,.,vm)2X,

wZ(w1,w2,.,wn)2X.

For mutation, we used a Polynomial mutation. Considering

xZ(x1,x2,.,xm)2X and the result of mutation operator yZ(y1,y2,.,yn)2X which we compute as follows:

ci2f1; 2;.;mg: yi Z xi C xðUÞi KxðLÞi

� ��di;

where xðLÞi ; xðUÞi , are domain boundaries given as initial

parameters:xi2½xðLÞi ; xðUÞi �.

�di Zð2riÞ

1=ðhmC1ÞK1; if ri!0:5

1K½2ð1KriÞ�1=ðhmC1Þ; if riR0:5

:

(

The distribution index hm is any non-negative real number [6]

and ri2(0,1) is a random generated number.

6. Numerical example

An application of the BEM-based SPEA multi-objective

optimisation algorithm implemented is presented below. It

consists of the multi-conductor system described by Fig. 4.

There are seven conductors placed in an infinitely extended

space where four of them are considered fixed in space-those

placed in the corners of the geometry, numbered from (1) to

(4) and coloured by light-grey.

C 4 C 5 C 6 C 7

55 55 55 55

m0 m0 m0 m01 1 1 1

270 0 240 120

50 50 50 50

1 1 1 1

24 24 24 24

(8, K8) – – –

– (K4, 0) (0, 0) (4, 0)

– (K3.2, K0.7) (0.15, 4.7) (3.9, 3.2)

– (K5, K3) (K1, 1) (3, 5)

– (K8, 8) (K8, 8) (K8, 8)

5.993 6.118 6.130 6.070

5.988 6.002 5.992 6.03

K0.08 K1.9 K2.2 K0.6

Fig. 5. The optimal placement of the conductors at the end of the optimisation

process.

C. Munteanu et al. / Engineering Analysis with Boundary Elements 30 (2006) 582–587 587

The three-phased system made by the conductors numbered

from (5) to (7) and coloured by dark-grey is considered mobile

with respect to their position in space.

The aim of the optimisation process is to find out the best

solution of placement of the mobile conductors in order to get

the minimal values of the conductor’s resistances (the fitness in

this case).

In this light, the optimal design process consists of mZ6

parameters-the centres coordinates of the three mobile con-

ductors and nZ7 objectives-the resistance values of the seven

conductors.

The SPEA optimal design algorithm settings are given in

Table 2. The settings for the numerical analysis together with the

main results of the optimal design process are given in Table 3.

The positions of the mobile conductors at the end of the optimal

design process are shown in Fig. 5.

One can notice that at the end of the optimisation process the

resistances of the three-phased mobile conductors are signifi-

cantly reduced in comparison with the resistances of the four

fixed conductors whose values remained almost the same (very

small variations).

7. Conclusions

The paper proposes a BEM analysis module designed

for solving electromagnetic field diffusion applications in

steady-state regime. The algorithm’s computation accuracy

was tested in case of a three-phased conductor system. The

implemented analysis algorithm was afterwards included in an

SPEA multi-objective optimisation algorithm. The main

applications involve the reduction of the internal losses of

multi-conductor systems. The example considered in the paper

emphasises the efficiency of the proposed optimisation

algorithm.

Acknowledgements

The authors wish to thank the financial support from NATO

Collaborative Linkage Grant PST.CLG.978487.

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