1

Page 1Cigré SCB1

CABLE SYSTEMS ELECTRICAL CHARACTERISTICS

Convener: Christian RoyerSecretary: Eric Dorison

TUTORIAL B1.30 – TB 531

Put specific WG picture here

Put specific WG picture here

Put specific WG picture here

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Page 2Cigré SCB1

Copyright

DISCLAIMEROwnership of a CIGRE publication, whether in paper form or on electronic support only infers right of use for personal purposes. Are prohibited, except if explicitly agreed by CIGRE, total or partial reproduction of the publication for use other than personal and transfer to a third party; hence circulation on any intranet or other company network is forbidden.

2

Page 3Cigré SCB1

Contents

• Needs for modelling

• Definition of cable system electrical characteristics:o Basic cable impedances

o Sequence impedances

o Surge impedances

• Formulae from the literature

• Models’ applicability

• Areas for improvement in cable modelling

• Parameters needed for modelling

Page 4Cigré SCB1

Introduction

• There is an increasing demand for cable integration in transmission networks.

• Underground/submarine cables have different electrical characteristics than overhead lines.

• This must be taken into account during cable system planning, design, and operation.

• The Brochure provides a state of the art in cable electrical modelling.

3

Page 5Cigré SCB1

Studies for cable installation

• System planning, which determines where new lines are needed, the voltage and current ratings, the need for shunt compensation.

• System impact, which determines the impact of a cable vs. overhead choice on the rest of the power system.

• Equipment design, which establishes detailed protection and operating procedures.

According to CIGRE WG C4.502 “Power system technical performance issues related to the application of long HVAC cables”

Page 6Cigré SCB1

Needs

Sequence Impedances

Surge Impedance

Modal Surge

Impedances

System Planning

System Impact

Equipment Design

Cable characteristics Link arrangement

Elementary Impedances

depending on softwares for transients studies

power frequency

high frequency

Basic Impedances

4

Page 7Cigré SCB1

Telegrapher’s equations

The telegrapher’s equations which link the voltages and currents at a distance z along the cable system may be expressed as:

Z and Y are square matrices of the series impedances and shunt impedances (or admittances).

The size of these matrices is equal to the number of conductors in the system: cable conductors, metal screens, armours, pipes…

VYz

IIZ

z

V··

Page 8Cigré SCB1

Basic impedances

The basic impedances are composed of elementary impedances which are derived through solving Maxwell’s equations.

ic1 is1

ic2

is2

ic3

is3

Matrices of series and shunt impedances

of the link

3300

0220

0011

332313

232212

131211

Y

Y

Y

Y

ZZZ

ZZZ

ZZZ

Z

ijzijzijzijz

ijZ

3

2

1

3

2

1

I

I

I

I

V

V

V

V

cssgcs

cscsii

sscs

csccii

yyy

yyY

zz

zzZ

Matrices of series and shunt impedances of cable i

si

cii

si

cii i

iI

v

vV

Voltages and currents in cable i

Matrix of mutual impedances

between cables i and j

5

Page 9Cigré SCB1

Symmetrical component circuit analysis technique

• To simplify the analysis of three-phase systems under both normal and abnormal conditions.

• Transforms the complex system of telegrapher’s equations involving coupling between conductors into several easy-to-handle systems without coupling between conductors.

• Provides one-line diagrams with sequence impedances.

• Basis for the design of protective relays used to detect faults and trip circuit breakers in order to protect electrical systems.

Page 10Cigré SCB1

Equivalent circuit for long links

• Starting point of studies dealing with the transmission capability as a function of the link length.

• The critical length may be derived through this model.

Zs

Ys/2

Ys/2

Link Length L

YZ

L

L

YYsL

LshZZs

.

2

.2

.tanh

..

..

6

Page 11Cigré SCB1

Modal analysis

Telegraphers’equations link voltages and currents : Modal voltages and currents are defined by : where ² is a diagonal matrix Modes propagate themselves independently one from another :

mImv ITIVTV ..

VYz

IIZ

z

V..

zmk

zmkmkcmk

zmk

zmkmk

kk

kk

eveviz

evevv

.

.

...

..

IIVv TZYTTYZT ...... 1112

I

I

I I

I I

I

2.I

Mode 1 : zero-sequence coaxial Mode 3 : interwires coaxial

I

I

I

I I

I I

2.I

3.I

Mode 2 : zero-sequence screen Mode 4 : interwires screen Mode 6 : two-wires screen

Mode 5 : two-wires coaxial

I

I

I

Page 12Cigré SCB1

Transient studies

Various methods have been worked out to define approximations of the characteristic impedances, propagation constants and transfer matrix elements, to make easier the transfer from the frequency domain to the time domain.

Model name Characteristics

Bergeron / Dommel Surge impedance and propagation velocity at a fixed frequency in modal domain

Semlyen / J. Marti Frequency dependent except for transformation matrix in modal domain

Noda Frequency dependent in phase domain

L. Marti Frequency dependent including transformation matrix in modal domain

Gustavsen (Universal Line)

Frequency dependent in phase domain

7

Page 13Cigré SCB1

Comparison OHL / UGC

Sequence impedances of EHV transmission lines

• 400 kV system

• OHL: bundled conductor, 3 ACSR subconductors per phase

• UGC: 2500 mm² Cu - XLPE

Unit OHL UGC Operating temperature °C 75 90 AC Resistance at operating temperature r mOhm/km 23,1 13,3 Inductance mH/km 0,858 0,576 Conductance g nS/km 10 52 Capacitance c nF/km 13,3 234 Characteristic impedance zc Ohm 254,5 -0,042 rad 49,68 -0,04 rad Propagation constant 1/km 1,110-3 1,53 rad 3,710-3 1,53 rad

Page 14Cigré SCB1

From Maxwell to telegraphers

Assuming perfect materials

HBt

BE .x

EDt

DH .x

z

HEj

HjE

z

E z

.

.

11

2

21

11

221

..2

..2

Izj

r

rLn

VV

Ir

rLnjVV

z

INTEGRATION

Maxwell’s equations Telegraphers’equations

dEVV

r

r

.2

1

21

00 21 rErE zz

H E

r1 r2 V 1

V2

I1

I2

21IH

(,)

8

Page 15Cigré SCB1

A cable is a wave guide

..2

.1

2

IH

r

rLn

VE

Conductor

Metal screenInsulation

V

E

H

Energy flow

I

VYIz

IZVz

.

.

L

C / 2 C / 2

...

0.

22

22

2

YZh

VhVz

1

2..2

.r

rLnjZ

1

2

..2..

r

rLn

jY

vztj

eGG..

.0

rr

cv

.

Page 16Cigré SCB1

Schelkunoff’smodel

)(

)(

)(22

)()()()(

)()()()(

)(2

)()()()(

1.

)(2

1

)()()()(

)()()()(

)(2

2

)(

)(

)(2

1

41

40

47

3

46

331231331231

330231330231

333

35

33123133123133.324

331231331231

230331230331

332

33

1

22

111

110

11

1

11

rmK

rmK

rjg

mz

r

rLn

jz

rmKrmIrmIrmK

rmKrmIrmIrmK

jgr

mz

rmKrmIrmIrmKjgrrz

rmKrmIrmIrmK

rmKrmIrmIrmK

jgr

mz

r

rLn

jz

rmI

rmI

jg

m

rz

s

s

ss

s

I1

I1

I1 + I2 V2

V1 – V2

z3

z1

z5

z7

I1

I2

I1 + I2

V2

V1 - V2 z2

z4

z6

r1r2

r3

r4

).()(

..

..

.

2174

215143

214132

111

IIzrE

IIzIzrE

IIzIzrE

IzrE

z

z

z

z

765321

2

1

4

44

2

1 .2

zzzzzzzz

I

I

zzz

zzzzz

V

V

z

outin

outout

outoutin

9

Page 17Cigré SCB1

Electromagnetic field in the ground

X axis

Y axis

Ground surfaceAir

Ground

Wire carrying current I

r

y x

Laying depth h

22222

+

- 2s

2

a

s

2s

2

00

2

'...

)..exp(

m+

m+h)+(y-exp '.

..2,

yhxRyhxRgjm

dxjRmKRmKg

ImyxE

sss

sss

sz

No closed form for the integral term.

Carson’sapproximation

Page 18Cigré SCB1

WedepohlWilcox’s model

se

sm

sa

eieaimei

eimaiaai

zzzz

zzzzz

zzzzzzzz

IzIzz

V

IzIzz

V

65

465

654321 .2

..

..

ejajijei

ejajijai

IIzz

V

IIzz

V

.

.

z1

356.0

+ )..(0,777 .2 2

1111

11

1

rgrmcoth

rg

m

z2

1

2

2 r

rLn

j

z3

...g2.

1- .m .

r 2

1.

32233

23

3

rrrcoth

g

m

z4 3323

3

m

1.

.

1.

shrrg

m

z5

...g2.

1.m

r 2

1.

32333

33

3

rrr.coth

g

m

z6

3

42 r

rLn

j

z7

hm

r

DLn

js

s ..3

4.

.2

..

4

zij

jis

ij

s hhmd

DLn

j..

3

2.

.2

..

r3

r1

r4

r2

23 rr

hi hj

dij

Groundlevel

)constant sBessel'(7811.1

.

.2 5,0

sm

eD

10

Page 19Cigré SCB1

A model for power frequency concerns

ei

ai

ei

ai

V

V

yyy

yy

I

I

z 211

11

ej

aj

ji ijij

ijij

ei

ia

em

ma

ei

ai

I

I

zz

zz

I

I

zz

zz

V

V

z.

insulation

metal screen

outersheath r4

r1

r3

r2

spacing S

Vai

Iei

Iai

Vei

cable i

conductor

3

4

442

'1

'2

221

.2y

.2y

r

rLn

jg

r

rLn

jg

Admittances of the insulation and the outersheath

1a 4

1.

21.

8z

r

DLnjYYR psa

ee r

DLnjR .

28ze

er

DLnj .

28zm

i jd

DLnj .

28zij

Conductor self impedance

Screen self impedance

Mutual impedances conductor-screen and between cables

Skin effect factor Ys Proximity effect factor Yp (three-phase link)

.

.2 21

sg

eD

D.C. resistances of the conductor and the screen

sa

ss

ss k

Rx

x

xY .

.

.8,0192 4

4

pa

p

p

pp

pp k

Rx

x

xS

r

S

r

x

xY .

.

.

27,0.8,0192

18,1.2.312,0

.2.

.8,01924

4

2

1

2

1

4

4

Depth of earth return path : Bessel’s constant (1.7811)

22

233

211

1

rr.π.gR

r.gπ.

kR e

ca

Page 20Cigré SCB1

Models’ applicability

When applying this model to actual underground links, some concerns have to be dealt with:

• Conductor designs are various and may not be considered as solid conductors, without care.

• Semi-conductive layers have to be taken into account.

• Use of an equivalent sheath for some screen designs such as bundle of wires or tapes is not fully correct.

• Armours involving steel wires or tapes may not be regarded as non-magnetic sheaths.

• Proximity effects.

This modelling is quite correct for unarmoured single-core cables, with a solid conductor and a lead sheath.

11

Page 21Cigré SCB1

Conductors (1/4)

• The real part of the impedance of the conductor is the AC resistance R.

• R’ is the DC resistance of conductor at operating temperature.

• Ys and Yp are respectively the skin effect factor and the proximity effectfactor, depending on the conductor design.

ps YYRR 1'.

Page 22Cigré SCB1

Conductors (2/4)

• The DC resistance should not be calculated, using the classical formula, as the ratio of the resistivity to the cross-section.

• This formula does not hold, using standard value of the electrical resistivity, as given in IEC 60287-1-1, one reason being the stranding of the wires.

• The DC resistance may be found in IEC 60228 for usual designs.

• Alternatively, a corrected value of the resistivity may be used, leading to a better estimate of the resistance from the nominal cross-section.

in Ω.m at 20 °C

12

Page 23Cigré SCB1

Conductors (3/4)

• The relationship between the nominal cross-section and the geometric radius of the conductor suitable for solid conductors has to be corrected for other conductor types to account for stranding and compacting effects.

• When modelling the core, the cross-section may be based on an equivalent radius deduced from the DC resistance according to:

• Alternatively, if the conductor radius is known, a conductor “corrected” resistivity may be used, as:

20

20

. DCRr

202

20 .. DCRr

Page 24Cigré SCB1

Conductors (4/4)

Skin and proximity effect coefficients

• For solid conductors, simple formulae have been derived, through approximations of involved Bessel’s functions.

• For other designs, the formulae may be used, as far as a corrective factoris introduced (ks for skin effect, kp for proximity effect) into the leading parameter, e.g.

• A study from CIGRE concluded that the recommendations in IEC 60287-1-1 are not suitable for Milliken conductors in extruded cables. CIGRE TB 272 gives more realistic values of skin and proximity coefficients.

'.

..

Rkx ss

13

Page 25Cigré SCB1

Insulation properties

• Generally, the electrical conductivity gi is expressed as:

• Values of the power dissipation factor as given in IEC 60287 are unlikely to be actual ones, specially for VHV extruded cables.

• The standard value for XLPE is much larger than values determined through testing.

• Assuming a constant value for any frequency is probably an approximation.

• Note that many models assume dielectric losses negligible.

Taniig ..

Page 26Cigré SCB1

Semi-conductive layers

• Works from Weeks, Ametani, Gustavsen provide solutions to integrate the semi-conductive layers.

• A transient voltage is attenuated more and its oscillating period becomes greater than those on a cable with no semi-conductive layer.

• The effect of semi-conductive layers on the propagation constants is rather minor.

• Semi-conductive layers may be modelled as admittances in serieswith the admittance of the insulation.

14

Page 27Cigré SCB1

Metal screens (1/2)

• Metal screens are modelled through an equivalent smooth sheath.

• Same mean diameter and cross-section (to get the same DC resistance).

Page 28Cigré SCB1

Metal screens (2/2)

Wires’ bundleOne has to take into account:

• the lay length ⇒ DC resistance

• the solenoid effect ⇒ inductance

Composite screensmay be modelled as an single equivalent component.

Component 2 carrying Icp2

Component 1carrying Icp1

conductor

Single equivalent component carrying Icp1+ Icp2

15

Page 29Cigré SCB1

Needs for improvements: armour modelling

• Formulae based on a theoretical approach.

• Semi-empirical coefficients for steel wires or tapes.

• Armour wires are assumed to be laid straight as well as the conductors.

• Where the conductors are twisted, the losses in the armour are probably over-estimated.

Conductor

Insulation

Lead sheath

Steel wires

Page 30Cigré SCB1

Needs for improvements: proximity effects

For power frequency concerns, several papers tackle this question, at a sufficient level of accuracy for solid conductors and metal sheaths, but the extension to other designs is questionable.

Ie1

r2 r3

Ia1

Ie2

Ia2

Ie3

Ia3

H3 H2

CONDUCTOR

METAL SCREEN

THREE-PHASE

TRANSMISSION

LINES

16

Page 31Cigré SCB1

Parameters needed for modelling purpose

• The accuracy of the calculation results will depend on the accuracy of any assumptions made in the values chosen for different parameters at design stage.

• WG C4.502 has addressed this particular issue in its technical brochure by performing a sensitivity analysis to highlight the more important cable parameters and their influence.

• The importance of three parameters is stressed: the conductor radius, the permittivity and the thickness of the insulation.

Page 32Cigré SCB1

Cable data sheet from manufacturers

Physical data

• Cable data sheets usually include all physical dimensions but they are subject to manufacturing tolerances.

Electrical data• Electrical data found in cable data sheets should be DC and AC resistance

of conductor and sometimes screen and armour, capacitance, inductance and sequence impedances.

• Resistance values are usually nominal or maximum values.

• Most other values are obtained by calculations using nominal dimensional data.

• Calculation hypothesis are usually not clearly indicated.

• For example, sequence impedances may not be representative of the actual installation.

17

Page 33Cigré SCB1

Production tests

IEC requirements appropriate for standard electrical modelling needs, except that:

• A construction check, associated with dimensional measurements, is advisable on each shipping length.

• An assessment of the AC resistance of the conductor should be introduced for constructions designed to get a reduced skin effect.

Page 34Cigré SCB1

On site measurements

Measurement of sequence impedance• IEC 60909-4 provides general considerations on the measurement.

• On site measurement of impedance can lead to a different result than calculations, due to nearby buried installation.

• Even if the new link is not energized yet, possible induced voltages have to be taken into account (safety precautions, accuracy).

• Three examples of impedances measurement procedures as part of commissioning test are shown in appendix C.

Measurement of the wave velocity• Using an impulse generator, this is recommended as the result may be used

later for fault localization.

18

Page 35Cigré SCB1

Conclusion

• The electrical modelling of cable system has been addressed for about 150 years.

• The state of the art provided by the Brochure shows that, satisfactory models are now available to perform the studies needed by the integration of cables into the network.

• For power frequency concerns, simple formulae are displayed.

• Measurement techniques of some electrical parameters are described and case studies show that a good agreement may be reached between calculated and measured electrical parameters.

• Nevertheless, some areas are not fully covered and improvements are needed, especially regarding the modelling of magnetic armours in submarine cables.

Page 36Cigré SCB1

TB content (1/2)

1. Modelling needs required by the studies to be carried out during cable system planning and design.

2. Definition of the various cable system electrical characteristics, mainly basic impedances, sequence impedances, characteristic and surge impedances.

3. Details of the many parameters that can have influence on the cable systems electrical characteristics.

4. Formulae from the literature.

5. Sources of information where data required for modelling purpose can be collected.

6. Introduction to 3 case studies to illustrate some of the difficulties involved in the calculation and measurement of cable system electrical characteristics.

19

Page 37Cigré SCB1

TB content (2/2)

Appendix A: Typical cable designs and installation

Appendix B : Mathematical models

Schelkunoff’s model, modelling of 3-core cables, symmetrical components, sequence impedances of double circuits.

Appendix C: Test procedures to measure the sequence impedances.

Appendix D: Case studies.

69 kV HPFF Cable

225 kV kV HPFF Cable

Page 38Cigré SCB1

Appendix: a focus on sequence impedances

Contents

• Symmetrical component circuit analysis technique

• Application to underground cables

• Earth modelling

• Self and mutual impedances

• Zero-sequence concerns

• Admittances

• Formulae

20

Page 39Cigré SCB1

Fortescue sequences

Positive sequence The phase conductor currents are equal in magnitude and 120° out of phase.

Representative of normal operation conditions.

Negative sequenceAs the positive sequence, except that the phase sequence is reversed.

Zero sequenceThe phase conductor currents are equal in magnitude and phase.

Positive sequence

idii IzImzVz

IIIIII

..

.. 32

21

zd is the positive sequence impedance Negative sequence

iiii IzImzVz

IIIIII

..

.. 2321

zi is the negative sequence impedance Zero sequence

ihii IzImzVz

IIIIII

...2

321

zh is the zero sequence impedance.

I1

I3

I2

I1

I3

I2

I1

I3

I2

Page 40Cigré SCB1

Symmetrical components background

Eigenvalues:

zmm

mzm

mmz

Z

mzzmzzmzz hid .2

2

3.

2

1

2

3.

2

1

1

1

1112

2

2 jjFFF hid

Possible eigenvectors:

111

1

1

.3

1

1

1

1112

2

1

2

2

FF

Fortescue matrix

and its inverse:

Diagonalisation of the impedance matrix

21

Page 41Cigré SCB1

Application to underground cables

• An elementary length of cable system may be modelled as a π equivalent circuit.

• One series and one shunt impedances, under positive, negative and zero sequence conditions.

To get this diagram, the screen currents have to be determined as a function of the currents flowing in the phase conductors, taking into account the screen bonding technique.

Z

Y/2

Y/2

Elementary length dz

Page 42Cigré SCB1

Basic impedances to derive sequence impedances

In line with formulae included in:

• the IEC standard 60287-1-3 “Current sharing between parallel single-core cables and calculation of circulating current losses”,

• the IEC Technical Report 60909-2 “ Short-circuit currents in three-phase AC systems – Part 2: data on electrical equipment for short-circuit current calculations”,

• the CIGRE Technical Brochures 283 and 347 which bring some comments on their background,

• the Underground Transmission System Reference Book by EPRI.

22

Page 43Cigré SCB1

Earth modelling – CIGRE approach

The currents flowing in the phase conductors return to their sources through the earth which can be modelled as a conductor with a resistance per unit length R’E located at a depth DE (provided that the soil is uniform and homogeneous).

Ra+jXa

R’E

Rs+jXs

jXcs =j.Xs

dz

Ia

Ia + Ie

Ie

Core

Screen

Ground

Va

Ve

Vg

r1

r4

r1 rm

EE

E

E

eD

R

00

5.0

0

85,1

.

.2

8

.'

Page 44Cigré SCB1

Earth modelling – EPRI approach

• An alternative modelling is to consider a current return path with zero impedance (which thus is the voltage reference).

• The earth equivalent conductor resistance has then to be added to self impedances of all conductors and mutual impedances between conductors to get same values for voltage drops along the conductors.

R’E+Ra+jXc

R’E+Rs+jXs

R’E +j.Xs

dz

Ia

Ia + Ie

Ie

Core

Screen

Ground

Va

Ve

r1

r4

r1 rm

23

Page 45Cigré SCB1

Self impedance of a phase conductor with earth return

• Ra is the AC resistance of the phase conductor.

• GMRa is the geometric mean radius of the phase conductor, which may be expressed as a function of the outer radius r1.

1...2

..' rGMR

GMR

DLnXXjRRZ a

a

EaaaEa

Page 46Cigré SCB1

Self impedance of metal screen with earth return

• Rs is the resistance of the metal screen.

For power frequency application, skin effect may be ignored and the DC resistance may be used.

• rs is the mean radius of the metal screen.

s

EsssEe r

DLnXXjRRZ .

.2

..'

24

Page 47Cigré SCB1

Mutual impedances

• Mutual impedance between phase conductor and metal screen of a cable with earth return:

• Mutual impedance between cables with earth return:

s

EssEm r

DLnXXjRZ .

.2

..'

ij

EijijEij d

DLnXXjRZ .

.2

..'

This is the mutual impedance between the phase conductor or the metal screen of cable i and the phase conductor or the metal screen of cable j, with dij as the axial distance between these cables.

Page 48Cigré SCB1

Equivalent mutual impedances between cables

• Where the cables are in trefoil formation, the mutual impedances between the cables are equal.

• That is not true with other laying conditions. It is usual practice to consider that these configurations behave like a trefoil formation with a spacing equal to the geometric mean distance between cables.

d12 d13

d12 d13

d23

d23

3231312 .. dddGMD

Flat formation

SGMD .23

Trefoil formation

SGMD

25

Page 49Cigré SCB1

Zero sequence concerns (1/2)

The return current path impacts the zero-sequence impedance.

ia1

ia2

ia3

ie1

ie2

ie3

L

RrRl

igr

iai - igr

iai - iei - igr

iai - igl

igl

iai - iei - igl

Sources Fault Cables conductors

Cables screens

Page 50Cigré SCB1

Zero sequence concerns (2/2)

The short-circuit current returning to the left end shares between the ground on one side and, on the other side, the metal screens through the grounding resistances.

The short-circuit current returning to the left end, shares also between the ground and the metal screens, but, in this case, the path to the ground is less “attractive” because of the grounding resistance.

Overhead line without skywire

Overhead line with skywire

G1

G2

26

Page 51Cigré SCB1

Formulae

Single-core cables

POSITIVE-SEQUENCE ZERO-SEQUENCE

SOLID BONDING

XZ

XZXZZ

e

mad

2

LRXZ

LXXZXZXZZ

e

hmmah

.3.2

.3.2.2.2

SINGLE POINT BONDING

XZZ ad LRZ

LXZZXZZ

ct

hmtmtah /

..3.2

CROSS-BONDING

XZZ ad

LRXZ

LXXZZZZZZ

e

hmcmcah

.3.2

.3.2.2.2

3

.2 Lc ZZX

00 .31.

.31.

I

IR

I

IRX

gll

grrh

Page 52Cigré SCB1

Double circuits

The sequence impedances are nearly equal to half the sequence impedances of a link involving only one circuit…

• … for positive and negative sequences.

• … for zero sequence only if the grounding resistances are considered nil or the return current in the ground is assumed negligible.

T1 R2 S2 T2R1 S1

Circuit 1 Circuit 2

T1 R2S2 T2R1 S1 Symmetrical arrangement

Unsymmetrical arrangement

27

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