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Impedance and Admittance Calculations of a Three-

Core Power Cable by the Finite Element Method

Angelo. A. Hafner, Mauricio V. Ferreira da Luz, Walter P. Carpes Jr.

Abstract--The analytical modeling of a three-core cable system

is challenging because of the non-concentric configuration of the

components involved. Given these limitations, a 2D finite element

modeling of the cable is developed in order to obtain the values of

the self, mutual and sequence impedances and admittances. To

calculate the series impedance, a magnetic vector potential

magnetodynamic formulation is used and for the calculation of

the parallel admittance, an electric scalar potential electrostatic

formulation is applied. By calculating the series impedance of the

inner cables, the influence of the mutual impedances in all

metallic elements involved is shown. The methodology is applied

to a typical cable of 300 mm² - 18/30 kV. The numerical results

are compared with analytical ones and with values supplied by

the manufacturer for each phase, validating the numerical

modeling.

Keywords: Submarine power cable, 2D finite element method,

impedance, admittance.1

I. NOMENCLATURES

I0 ( ) Bessel functions of first kind and order 0.

I1 ( ) Bessel functions of first kind and order 1.

Electric conductivity, in S/m.

Electric resistivity, in

m.

Electric permittivity, in F/m.

Magnetic permeability, in H/m.

Magnetic reluctivity, in m/H.

SC Semiconductor

M Matrix composed by matrixes.

m Matrix composed only by scalars.

V

v

Vector.

( . , . ) Volume integral in

of products of scalar or

vector fields.

< . , . > Surface integral on

of products of scalar or

vector fields.

Im( . ) Function that returns only the imaginary part of a

complex number.

Electrical angular frequency.

FEM Finite Element Method.

Angelo A. Hafner is PhD student at the Department of Electrical Engineering,

Federal University of Santa Catarina, Florianópolis, Brazil (e-mail:

angelo.hafner@posgrad.ufsc.br)

Mauricio V. Ferreira da Luz is Professor at the Department of Electrical

Engineering, Federal University of Santa Catarina, Florianópolis, Brazil (e-

mail: mauricio.luz@ufsc.br)

Walter P. Carpes Jr is Professor at the Department of Electrical Engineering,

Federal University of Santa Catarina, Florianópolis, Brazil (e-mail:

walter.carpes@ufsc.br)

Paper submitted to the International Conference on Power Systems Transients

(IPST2015) in Cavtat, Croatia June 15-18, 2015

II. INTRODUCTION

HE expansion of submarine transmission systems

represents a major trend due to the growth of the oil and

offshore wind energy industry. The deployment of these

systems at large distances from the shore and in deep water

requires kilometric stretches of submarine power cables.

Fed equipment or systems, as well as cables, need to be

adequately protected in case of short circuits, overloads and

transients. An accurate cable model is needed to accurately

represent the waveforms of voltage and current on the load

and the transmission line providing technical support for the

choice of the most suitable protection to be adopted for each

situation.

When one considers the cable as single-core, the phases

distributed impedances and admittances of the cable for a

certain range of frequencies can be calculated analytically

applying classical analytical formulae ((4), (6), and (11)).

However, in three-core cables, even at 50/60 Hz, the

following aspects should be taken into account when

modeling: (i) proximity effect generated by the currents of the

central conductor; and (ii) current induced in the sheath and its

effects on the central conductor impedance [1].

The non-concentric configuration of the trefoil formation

(Fig 1) hampers the analytical modeling. Thus, a 2D finite

element model is developed to obtain the values of the cable

series impedance and parallel admittance.

Some studies about cable modeling are presented in [1-6]

and [16-22].

III. BASIC CHARACTERISTICS OF A SUBMARINE POWER CABLE

The physical constitution of submarine power cables is

very similar to underground power cables.

The main difference is that in the first, there are additional

protections for water entry (Fig. 1 and Table 1). In this

section, we briefly describe the constituent parts of the

submarine power cable as well as the aspects related to the

calculation of its series impedance. Each part of the inner

cable is described in Table 1. Small variations may occur from

one manufacturer to another.

The cable of this study is composed of a set of three power

inner cables in trefoil formation, as shown in Fig. 1. Parts 10-

11, 11-12 and 12-13 consist of insulating material, conductor,

and insulating, respectively. The conductive layer, called

armor, has the main function of mechanically protecting the

set.

T

Fig. 1. Set of inner cables in trefoil formation of a three-core submarine

power cable.

TABLE 1

PARTS OF A POWER INNER CABLE

Item Component Material

1 Core Copper Stranded Wires

2 Water-blocking tape Humidity absorber SC tape

3 Conductor Shield SC tape

4 Insulation XLPE

5 Insulation Shield Humidity absorber SC tape

6 Water-blocking tape Humidity absorber SC tape

7 Sheath Copper wires

8 Water-blocking tape Humidity absorber SC tape

9 Jacket Polyethylene

IV. HOMOGENIZATION OF THE CORE

Because the materials are composed of several parts,

homogenization techniques are applied to model them as

solids used for both numerical and analytical approaches.

Homogenization in the core due to the natural gaps of the

stranded conductor is made by the correction (increase) in

resistivity since the core is now considered massive. This is

done by applying:

ccc

k

(1)

where

ρ’

c is the corrected resistivity of the central conductor,

kc is the area correction factor (kc =

rc2/An), rc is the core

radius, and An is the nominal area provided by the

manufacturer’s catalog.

Homogenization in sheath also depends on the composition

of the material used. Therefore, its corrected resistivity is

given by:

s

ss

k

(2)

V. HOMOGENIZATION OF THE INSULATION

Between the core and insulation and sheath and insulation,

there are semiconductor tapes which have the function of

uniformly distributing the electric potential. These three

materials are homogenized as one. For this, a correction must

be applied to the insulation electric permittivity given by:

ln

ln

s

c

cs cs

rr

ba

(3)

where

cs

istheoriginalinsulationpermittivity,and a and b

are the inner and outer radii of the insulation, respectively.

VI. SERIES IMPEDANCE

The analytical modeling for non-concentric conductors,

like the case of the three-core cables, is a very complex task.

A full analytical computation of a submarine cable can be

found using the pipe-type cable formulas from [3]. As

explained in [4] there are some approximation in this, because

of the representation of the armor.

Subsection A presents a way to calculate the phase

impedance for a single-core cable, where only the impedance

in a single inner cable can be considered without mutual

couplings with any other metallic part of the cable. However,

in the numerical approach, all couplings involved are

regarded.

A. Analytical Approach

According to [3], [2], and [4], the impedance of each core

per unit of length is given by:

0

1

I

2I

cc

cc

c

ccc

r

zrr

, (4)

where

c is the intrinsic medium impedance, given by:

ccc

j

. (5)

The core impedance zc have one real part that represents

the core resistance; and other part imaginary, that represents

the internal core inductance times the electric angular

pulsation.

The phase inductance is given by the sum of internal and

external inductance. For a single core conductor, the external

inductance is given by:

_1 ln

2

cs c

ext C

s

r

lr

. (6)

However (6) considers the current of central conductor

returning by sheath. For three-core cables, the current returns

by other phases, creating a bigger area for the magnetic flux,

that now is the area between two phases. So (6) becomes:

_3 ln

2

c

ext C

r

lD

, (7)

where D is the distance between cores in the trefoil formation.

For other formations, the geometric mean must be applied.

Equation (7) is accurate and frequently used for impedance

calculation at 50/60 Hz. For bigger frequencies, it is necessary

to consider the sheath effects as covered by [5][6].

B. Numerical Approach

In order to find the self and mutual impedances of all

metallic parts of the cable, a 1 A current is applied at one

metallic element and measured the voltage in this and others.

The self impedance is found by dividing the induced voltage

by the current at the element that the current is applied. The

mutual impedances are found by dividing the induced voltage

at the elements that have no current by the current that

produced this induction.

C. Cable Impedances

Fig. 2 presents a three-core cable impedance diagram

where: (i) the letters a, b, and c represent each core; (ii) the

numbers 1, 2, and 3 represent each sheath; (iii) and g

represents the armor.

The voltage drop from g to g' on conductors are:

'123

'123

'123

1 1 ' 1 1 1 11 12 13 1

2 2 ' 2 2 2 21 22 23 2

33' 33331

'

ag a g aa ab ac a a a ag

bg b g ab bb bc b b b bg

cg c g ac bc cc c c c cg

gg abc g

g g abc g

gg abc

gg

VV z z z z z z z

VV z z z z z z z

VV z z z z z z z

VV z z z z z z z

VV z z z z z z z

VV z z z z

VV

1

2

32 33 3 3

123

a

b

c

g

ag bg cg g g g gg g

I

I

I

I

I

zzzI

zzzzz zz I

(8)

Fig. 2. Representation of three-core cable impedances.

In [5] it is proved that the sequence impedances for

interconnected sheaths are:

12

11 12

12

0

11

2

2

2

22

aa

aa ab

aa

aa ab

a

zz

zzzz zz

zz

zz z zz

(9)

If the sheaths are not interconnected at both terminals, or

are interconnected only at one point (grounded or not), there is

not circulation current and the sequence impedances become

[1]:

02

aa ab

aa ab

zzzz

zz z

(10)

VII. PARALLEL ADMITTANCE

There are three kinds of admittances on three-core cables:

(i) core-sheath, (ii) sheath-sheath, and (iii) sheath-armor. Only

the core-sheath armor is analytically feasible, given by:

12

2

ln

cs cs

sc

j

yrr

(11)

where

cs is the insulation conductivity and

cs is the

corrected permittivity of the insulation. Because of the high

resistivity of the insulating materials, only the capacitances on

them are considered (see Section VIII-B). The three-core

cable capacitance diagram is shown in Fig. 3.

In addition, a numerical approach is performed and

compared with the analytical results for this capacitance.

However, for sheath-sheath and sheath-armor capacitances,

only numerical results are considered due to the non-

concentricity between these parts. The leakage currents on

insulations from g to g’ in phases are:

11

22

33

00 00

0000

00 00

00

00

00

cc cs ag

aa

cc cs bg

bb

cc cs

cc

cs ss ss ss

cs ss ss ss

cs ss ss ss

yy

V

II

yy

V

II

yy

II

II yyyy

II yyyy

II yyyy

1

2

3

cg

g

g

g

V

V

V

V

(12)

where ycc, ycs, yss and yss are, respectively, the core self-

admittance, core-sheath mutual admittance, sheath self-

admittance and sheath-sheath mutual admittance.

Fig. 3. The three-core cable capacitance diagram.

By using the technique presented in [7] it is possible to get:

1cc cs a

yyy

, (13)

11 12 311 1ass g

yyyyyy

(14)

12 13ss

yyy

(15)

VIII. NUMERICAL MODELING USING FEM

To perform the numerical modeling, the software Gmsh [8]

and GetDP [9] are used. Gmsh is the pre and post-processor

and the GetDP is the solver. The problem is implemented in

the software by two codes: one that defines the geometries and

the mesh of the structure (“.geo” file) and other that defines

the physical proprieties of the materials, the constraints and

the formulation to be used (“.pro” file).

The electrostatic formulation used to calculate of the

parallel admittance is given by:

grad ,grad , ,

D

V

v

VV nDV V

VF

(16)

where V is the electric scalar potential, V' is the test function

for scalar potential,

V is the volume charge density, n

is the

unit normal vector exterior to , and nD

is a constraint on

the electric flux density associated with nonfixed potential

boundaries D of the domain , e.g. on floating potential

boundaries f [14].

Fv () denotes the function space defined on , which

contains the basis and test functions for both scalar potentials

V and V' [14]. At the discrete level, Fv () is approximated

with nodal finite elements.

The harmonic magnetodynamic formulation used to

calculate of the series impedance is given by:

a

curl , curl , ,

grad , , 0, F

Hc

c

s

s

vA A nHA jAA

VA J A A

(17)

where

is the electric conductivity defined on conducting

parts c of ,

A

is the magnetic vector potential,

A

is the

test function for vector potential,

s

J

is the source electric

current density defined in s, and

s

nH

is a constraint on the

magnetic field associated with boundary H of the domain

[15]. Fa () denotes the function space defined on which

contains the basis and test functions for both vector potentials

A

and

A

.

IX. METHODOLOGY

The physical and geometric data cable is obtained from

manufacturer’s catalog [10], for a three-core cable in trefoil

formation.

Due to the complexity of the cable geometry, some

simplifications like homogenization are required. In addition,

it is imperative that some correction factors be applied before

starting the simulation as explained in Section II.

A. Physical and geometry constants used in the model

At the central conductor, the copper resistivity is

considered (17.24 nΩ m). It is then corrected for a temperature

of 90°C followed by an equivalent area (homogenization)

resulting in a resistivity

ρ

c of 23.57 nΩ m.

The transversal magnetic permeability used for all

materials is considered μ0, even the armor, because it is

composed of wires that are not in direct contact [11].

The cable is considered as totally surrounded by seawater

with a conductivity of 5 S/m [12].

B. Simplified diagram of the cable

The parts considered for the cable model (analytic and

numeric) are all solids and represented by Table 2 and Fig. 4.

C. The Finite Element Approach Implementation

Initially it is necessary to implement the surfaces (.geo file)

from the model as shown in Table 2 and Fig. 4. In the same

file, the mesh density factors must be inserted set in each point

of the geometric figure.

Based on the geometry file, a mesh is built by Gmsh.

Fig. 5(a) shows the mesh of the whole calculation domain

while Fig. 5(b) shows the mesh of the lower left inner cable of

the three-core cable, respectively, for the calculation of series

impedance.

Both figures are shown with the aim of highlighting the

mesh density utilized. Region A is a necessary region in order

to avoid domain truncation errors, where the magnetic vector

potential on its outer circle is zero. The physical constants

values in this region are the same values as region B.

When calculating the parallel capacitance the electric scalar

potential at the armor is set to zero.

TABLE 2

PARTS OF A POWER INNER CABLE

Item Radius [mm]

1 10.20

2 20.60

3 20.72

4 24.00

5 51.75

6 53.75

7 57.95

8 61.95

Fig. 4. Set of inner cables in trefoil formation of a three-core submarine

power cable.

(a) (b)

Fig. 5. Diagram and mesh for calculation of the series impedance. In (a) is the

domain of calculation and in (b) the mesh detail of one power inner cable.

D. Obtaining the mutual and self-impedances and

admittances

In order to find the self and mutual impedances of all

metallic parts of the cable, the circuit presented in Fig. 6 is

implemented and the technique explained in Section IV-B is

applied.

The sequence impedances are also obtained where three

short-circuited cores are fed by a 1 V / 50 Hz three-phase

sinusoidal source (Fig. 7). For this implementation, two

considerations are made: (i) with the sheath and armor

opened; and (ii) with all sheaths interconnected at both ends

and these connected to the respective armor end. The armor

potential is considered floating (Fig. 7).

The representation of the diagrams shown in Fig. 6 and 7

illustrates as the electrical circuits are considered in GetDP.

However, the modeling is carried out in two dimensions.

In order to find the parallel capacitance we apply the

Maxwell Capacitance Matrix concept [7]. Firstly, a 1 V

potential is applied on the core and zero on all other parts. The

result is the core self-capacitance. After that it is applied a 1 V

on the sheath and zero on all other parts (Fig. 8). From this

measurement we find the sheath’s self-capacitance which is

the sum of sheath-core, sheath-sheath (2 times), and sheath-

armor capacitances.

Fig. 6. Circuit diagram implemented in GetDP to determine the core self and

mutual impedances.

Fig. 7. Circuit diagram implemented in GetDP to determine the phase

sequence impedances.

Fig. 8. Circuit diagram implemented in GetDP to determine the sheath self-

capacitance.

To find the sheath 1-sheath 2 capacitance, is imperative, for

instance, to apply 1 V to core a, sheath 1, core c, sheath 3, and

armor, and zero on all other parts. The capacitance sheath 1-

sheath 2 is obtained with basis on the electric flux that goes

out from surface sheath 1. A similar procedure is applied to

find core-sheath and sheath-armor capacitances.

Finally, the numerical results are compared with the

analytical ones and also with the values supplied by the

manufacturer for each phase, validating the numerical

modeling.

X. RESULTS AND VALIDATION

The presentation of results is divided into two parts:

(i) analysis of impedances, and (ii) analysis of admittances.

Validations are made by comparison with analytical methods,

when possible, and with manufacturer’s catalog [10] for the

frequency of 50 Hz.

A. Series impedance

At 50 Hz when we apply a current of 1150 A / 50 Hz to

the core a (Fig. 9), we obtain the induced voltages shown in

Table 3. Dividing the induced voltage in each metallic part of

the cable by the current (imposed on core a) that originated

them; we obtain the core self-impedance and the mutual

impedance between the respective conductive part and the

core (Table 4). Because it is a cable in trefoil formation

(symmetric configuration), the same values are repeated when

current is applied only in the core b or c.

TABLE 3

INDUCED VOLTAGE IN ALL CABLE CONDUCTIVE PARTS

WHEN CORE A IS FED BY 1150 A / 50 HZ

Voltage at: Modulus

[mV/km]

Angle

[]

Core a 414.1 -70.5

Core b 294.7 -60.0

Core c 294.7 -60.0

Sheath 1 347.2 -60.4

Sheath 2 294.7 -60.0

Sheath 3 294.7 -60.0

Armor g 285.2 -60.0

TABLE 4

CORE SELF-IMPEDANCE, MUTUAL BETWEEN CORES, MUTUAL CORE-SHEATH,

AND MUTUAL CORE-ARMOR

Impedance Resistance

[m/km]

Inductance

[H/km]

zaa 75.7 1295.9

zab 0.0 938.1

zac 0.0 938.1

za1 2.4 1105.2

za2 0.0 938.1

za3 0.0 938.1

zag 0.0 907.9

One notices a great similarity in the values of mutual

impedances between cores and between core and sheaths of

other cores. In other words (zab = zac) (za2 = za3), as described

in Section 3.3 of [1].

The same process is repeated but now the current is applied

to sheath 1 and the induced voltages in all the metallic

elements of the cable are calculated. From this process Table 5

is formulated for sheath’s self and mutual impedances.

Finally, the calculation is repeated applying current at the

armor and calculating the other induced voltages, resulting in

Table 6.

As expected, independent of where the current is applied,

the mutual impedances are always the same as evidenced in

Tables 4, 5, and 6.

(a) (b)

Fig. 9. (a) Current density [A/mm2] and (b) Magnetic flux [Wb/m] used for

the calculation of core self and mutual sequence impedances via FEM.

TABLE 5

CORE-SHEATH MUTUAL IMPEDANCE, SHEATH SELF,

AND MUTUAL SHEATH-ARMOR

Impedance Resistance

[m/km]

Inductance

[H/km]

z1a 2.4 1105.2

z1b 0.0 938.1

z1c 0.0 938.1

z11 1731.3 1104.9

z12 0.0 938.1

z13 0.0 938.1

z1g 0.0 907.9

TABLE 6

CORE-ARMOR MUTUAL, SHEATH-ARMOR MUTUAL,

AND ARMOR SELF-IMPEDANCE

Impedance Resistance

[m/km]

Inductance

[H/km]

zga 0.00 907.9

zgb 0.00 907.9

zgc 0.00 907.9

zg1 0.00 907.9

zg2 0.00 907.9

zg3 0.00 907.9

zgg 616.8 905.4

If the sheaths are interconnected only at one of the ends

(whether grounded or not), only the mutual impedances

between cores influence the phase positive sequence

impedance, which for the inductance can be obtained

from (10):

1295.9 938.1 0.3578 mH kmL (18)

The series inductance value of the cable in the

manufacturer’s catalog [10] is 0.36 mH/km, which validates

the accuracy of the method used.

Similarly we obtain the value of the positive sequence

resistance:

75.7 0.0 75.7 m kmR

. (19)

The positive sequence impedance are also determined

when three balanced voltages are applied, displaced 120

degrees from each other, with the three cores short-circuited

and the sheaths and armor opened (Fig. 10). Values equal to

those found in (18) and (19) are obtained.

(a) (b)

Fig. 10. (a) Current density [A/mm2] and (b) Magnetic Flux [mWb/m] used

for the calculation of positive sequence impedance directly via FEM.

Finally, an analytical approach is made by applying of (4)

and (7).

Table 7 presents a comparison of the results obtained for

the positive sequence resistance and inductance between the

adopted approaches. The error is found by taking the

reference value provided by the manufacturer. The

manufacturer did not provide the distributed cable resistance

value.

If the sheaths are connected at both ends, the distributed

positive sequence series resistance and inductance would be:

R+ = 77.3 m/km and L+ = 357.83 H/km.

TABLE 7

INDUCTANCE PROVIDED BY THE MANUFACTURER, CALCULATED

ANALYTICALLY, AND CALCULATED VIA FEM

R+

[m/km]

L+

[mH/km]

L+ Error

[%]

Manufacturer --- 0.360 ---

Analytical 73.2 0.359 0.171

Numeric 77.3 0.358 0.603

The increase in resistance occurs because, when the sheaths

are interconnected, a circulation path is created for the

induced currents. The introduction of an effect in the core

current distribution is therefore due to the sheath's current

increasing the proximity effect in the respective core

compared to the case where the sheaths are not

interconnected. As the frequency increases, this effect is

increased [13].

Knowing the resistances and inductances (selves and

mutual) found for all conductive parts of the cable, the

impedance matrix can be mounted:

core core sheath core armor

core sheath sheath sheath armor

core armor sheath armor armo r

z

z

zz z

zz

Zz z (20)

where, zcore, zsheath, zcore-sheath, zarmor, zcore-armor, zsheath-armor, are, in

/m:

75.7 407 295 295

295 75.7 407 295

295 295 75.7 407

core

jj j

jjj

jj j

z (21)

1731 347 295 295

295 1731 347 295

295 295 1731 347

sheath

jj j

jjj

jj j

z (22)

2.39 347 295 295

295 2.39 347 295

295 295 2.39 347

core sheath

jj j

jjj

jj j

z

(23)

616 285

295

295

armor

core armor

sheath armor

zj

zj

zj

(24)

B. Parallel admitance

Because the insulating material has a high resistivity, the

branch that represents the parallel conductance can be

neglected, which can be seen already at 50 Hz by applying

(11) to the cable under consideration (XLPE insulation),

where a = 11.4 mm and b = 19.4 mm.

12 0.0098 82.184 nS myj (25)

The core-sheath capacitance is Im (y12) /

= 261.60 pF/m,

very close to the value provided by the manufacturer’s catalog

[10], which is 0.26 F/km.

The core-sheath capacitance is also calculated through

finite element technique, obtaining the value of 261.60 pF/m,

which is exactly the value found by the analytical method

(also very close to the value provided by the manufacturer).

Fig. 11 shows the electric field in the region under analysis (as

well as in sheath-sheath and sheath-armor regions).

Fig. 11. Electric field [V/m] lines when a 1 V potential is applied at the

sheath 1 and 0 V to all other metallic parts of the cable, to obtain the sheath

self-capacitance.

Finally, the capacitance sheath-sheath and sheath-armor by

the finite element method are calculated. Table 8 shows the

cables’ capacitances values between core and sheath, sheath

and sheath and sheath and armor, as also the error of

measurement, considering the value of the manufacturer [10]

as reference.

According to (12), the parallel capacitance matrix is:

j

YC, (26)

where C, in pF/m, is:

262 0 0 262 0 0

0 262 0 0 262 0

0 0 262 0 0 262

262 0 0 510 54.4 54.4

0 262 0 54.4 510 54.4

0 0 262 54.4 54.4 510

C (27)

TABLE 8

CAPACITANCE OF THE THREE-CORE CABLE IN STUDY

Region Numeric

[F/km]

Analytical

[F/km]

Manuf.

[F/km]

Error

[%]

Core-Sheath 0.2616 0.2616 0.26 0.006

Sheath-Sheath 0.0544 --- --- ---

Sheath-Armor 0.1401 --- --- ---

XI. CONCLUSIONS

Space in manufacturer’s cable catalogs is typically

dedicated only for distributed positive sequence inductance

and capacitance values at industrial frequency (50 or 60 Hz).

In [13] it is presented for the same cable of present study, the

behavior of positive sequence impedances for a frequency

range from 20 Hz to 20 kHz.

In the present work, it was considered 50 Hz, with the

improvement to find the parallel admittance and the specificity

in relation to mutual coupling between phases, thereby

allowing to get the sequence impedances.

Similar to what was done in reference [13], in future works

the goal will be to evolve the work presented in this paper by

examining the frequency range from 20 Hz to 20 kHz. We

will intend: (i) to simulate underground cables with grounded

ends; and (ii) to apply the same modeling of this paper without

the application of homogenization techniques (such as those

applied in the central conductor and sheath in Section III). It is

expected that through this study, an increase in the accuracy

of the model’s response, especially at high frequencies, may

be achieved. Moreover, field measurements have to be made

for validation.

XII. ACKNOWLEDGEMENT

The authors would like to gratefully acknowledge the

PETROBRAS for the financial support to this research effort.

XIII. REFERENCES

[1] F. F. Da Silva and C. L. Bak, Electromagnetic Transients in Power

Cables. Springer London, Limited, 2013.

[2] S. A. Schelkunoff, “The Electromanetic Theory of Coaxial Transmission

Lines and Cylindrical Shields,” Bell Syst. Tech. J., p. 47, 1934.

[3] A. Ametani, “A General Formulation of Impedance and Admittance of

Cables,” IEEE Trans. Power Appar. Syst., vol. PAS-99, no. 3, pp. 902–

910, May 1980.

[4] A. Pagnetti, “Cable Modeling for Electromagnetic Transients in Power

Systems,” Universite Blaise Pascal - Clermont II, 2012.

[5] T. Aloui, F. Ben Amar, and H. H. Abdallah, “Modeling of a three-phase

underground power cable using the distributed parameters approach,” in

Systems, Signals and Devices (SSD), 2011 8th International Multi-

Conference on, 2011, pp. 1–6.

[6] W. A. Lewis and G. D. Allen, “Symmetrical-Component Circuit

Constants and Neutral Circulating Currents for Concentric-Neutral

Underground Distribution Cables,” Power Appar. Syst. IEEE Trans.,

vol. PAS-97, no. 1, pp. 191–199, Jan. 1978.

[7] E. Di Lorenzo, “The Maxwell Capacitance Matrix,” no. March, pp. 1–3,

2011.

[8] C. Geuzaine and J.-F. Remacle, “Gmsh Reference Manual.” 2013.

[9] P. Dular and C. Geuzaine, “GetDP Reference Manual.” 2013.

[10] ABB, “XLPE Submarine Cable Systems.” Available at (15/april/2015):

http://www04.abb.com/global/seitp/seitp202.nsf/0/badf833d6cb8d46dc1

257c0b002b3702/$file/XLPE+Submarine+Cable+Systems+2GM5007+.

pdf.

[11] “IEC 287-1-1: ‘Electric cables-calculation of the current rating, part 1:

current rating equations (100% load factor) and calculation of losses,

section 1: general.’” 1994.

[12] L. Rossi and J.-P. Thibault, “Investigation of wall normal

electromagnetic actuator for seawater flow control,” J. Turbul., Aug.

2007.

[13] A. A. Hafner, M. V. F. da Luz, F. F. da Silva, W. P. Carpes Jr., and S. de

Lima, “Aplicaççao do Método de Elementos Finitos no Cálculo da

Impedância Distribuída em Cabos de Potência Tripolares,” in 16 SBMO

- Simpósio Brasileiro de Micro-ondas e Optoeletrônica e 11 CBMag -

Congresso Brasileiro de Eletromagnetismo (MOMAG 2014), 2014 (in

Portuguese).

[14] P. Dular, W. Legros, and A. Nicolet, “Coupling of local and global

quantities in various finite element formulations and its application to

electrostatics, magnetostatics and magnetodynamics,” IEEE Transactions

on Magnetics, vol. 34, no. 5, pp. 3078–3081, September 1998.

[15] M. V. Ferreira da Luz, "Desenvolvimento de um software para cálculo

de campos eletromagnéticos 3D utilizando elementos de aresta, levando

em conta o movimento e o circuito de alimentação", PhD Thesis (in

Portuguese), Federal University of Santa Catarina, Brazil, 2003.

[16] Multiwire shielded cable parameter computation. Kane, M.; Ahmad, A.;

Auriol, P. IEEE Transactions on Magnetics. Volume: 31, Issue: 3.

Publication Year: 1995, Page(s): 1646-1649.

[17] Yin, Y.; Dommel, H.W. Calculation of frequency-dependent impedances

of underground power cables with finite element method. IEEE

Transactions on Magnetics. Volume: 25, Issue: 4. Publication Year:

1989, Page(s): 3025-3027

[18] Xiao-Bang Xu; Guanghao Liu; Chow, P. A finite-element method

solution of the zero-sequence impedance of underground pipe-type

cable. IEEE Transactions on Power Delivery. Volume: 17, Issue: 1.

Publication Year: 2002, Page(s): 13-17

[19] Andreou, G.T.; Labridis, D.P. Electrical Parameters of Low-Voltage

Power Distribution Cables Used for Power-Line Communications. IEEE

Transactions on Power Delivery. Volume: 22, Issue: 2. Publication Year:

2007, Page(s): 879-886.

[20] Gustavsen, B.; Bruaset, A.; Bremnes, J.J.; Hassel, A. A Finite-Element

Approach for Calculating Electrical Parameters of Umbilical Cables.

IEEE Transactions on Power Delivery. Volume: 24, Issue: 4. Publication

Year: 2009, Page(s): 2375-2384.

[21] Hoidalen, H.K. Analysis of Pipe-Type Cable Impedance Formulations at

Low Frequencies. IEEE Transactions on Power Delivery. Volume: 28,

Issue: 4. Publication Year: 2013, Page(s): 2419-2427.

[22] de Arizon, Paloma; Dommel, H.W. Computation of Cable Impedances

Based on Subdivision of Conductors. IEEE Transactions on Power

Delivery. Volume: 2, Issue: 1. Publication Year: 1987, Page(s): 21-27.