60301Line & Cable Modelling 1.2

8/2/2019 60301Line & Cable Modelling 1.2

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Line, Cable and Load Modelling

1.1


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Goals

To understand the basic electrical parameters of anoverhead line or a cable

1.2

length of the system affects the way in which it ismodelled

To understand the types of loads on the system andtheir response to variations in voltage


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1.3


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1.4

Common features metallic conductor & dielectric (air in the case of

the overhead line, XLPE in the case of the cable)


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1.5

erm na ower

Transition from

OHL to cable


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Introduction To Basic Line Model

We need a model that will represent the propertiesof an overhead line / cable

For a conductor we need to consider resistance

1.6

and inductance

For the dielectric, we need to consider capacitanceand conductance

We will see that these properties can vary as afunction of temperature and frequency


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Introduction To Basic Line Model

Line / cable model must be able to function in a number ofdifferent applications.

The following are examples of potential uses for a line / cable

1.7

Volt drop estimation

Power flow studies

Temporary overvoltage studies (e.g. ferroresonance)

Transient overvoltage studies (e.g. lightning)

The requirements of these simulation types are very different.Generally, the faster the event being simulated, the morecomplex the model.


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Equivalent of Line Or Cable

Typical model of line of cable uses lumped sections of resistance,

inductance, capacitance and conductance

How many of these sections are used in a cable model dependson the modelling need

1.8

on both the length of the cable and the intended use of the model

CONDUCTOR

INSULANT


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Resistance

Resistance per metre of a conductor is:

20

-1

1.9

A20


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Resistance

Variation in conductor resistance with temperature must beconsidered

[ ])20(1 2020 += tRRt m-1

1.10

Skin effect and proximity effect reduce the usable area ofconductor in practice

Tabulated values from manufacturers data sheets reflectimpact of these effects

Resistance is frequency dependent increases as frequencyincreases


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Resistance

Coefficients below mean that a 50K shift in temperature means a19.5% increase in resistance for copper

How should this be included in modelling?

1.11

Material Resistivity / m@ 20C

TemperatureCoefficient per K

Copper 1.72 x 10-8 3.9 x 10-3

Aluminium 2.83 x 10-8

4.0 x 10-3

Lead 21.4 x 10-8 4.0 x 10-3

Steel 13.8 x 10-8 4.5 x 10-3


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Inductance

Line and cable inductance made up from interaction ofmagnetic field with current

Both self and mutual inductances are present

For line mutual inductances with all other conductors

1.12

(earth assumed zero resistance and line currents arebalanced)

For screened cable, mutual inductance with sheath /armour

Frequency dependent falls slightly for increase infrequency (self-inductance dependent)

Overall geometries are more important than conductor

type


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Inductance

Self Inductance

1

8

= HmL oself

1.13

bundletheofradiusmeangeometrictheisDand

bundlesconductorbetweendistancemeangeometrictheiswhere

ln2

s

01

eq

s

eq

D

D

DL

=


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Inductance

Example below shows the positive sequence impedance calculatedfor three types of overhead line

Size of conductor has little impact on reactive part

In contrast, the use of a twin conductor as opposed to a single

1.14

As expected, resistance is proportional to cross-sectional area

Line voltage / Conductor details Positive sequence impedance (ohms/km)

132kV, 1 x 175mm2 0.0177+j0.402

132kV, 2 x 175mm2 0.0089+j0.293

132kV, 1 x 400mm2 0.0076+j0.379


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Inductance And Capacitance

For full derivation of inductance and capacitancesee:

1.15

-


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Capacitance Of An Overhead Line Equivalent capacitance to earth of a line made up of a number of

capacitances in parallel / series

Not frequency dependent (at frequencies of interest for powersystems)

Capacitance to earth of a two conductor line is:

1.16

For a multiple conductor line (fully transposed) this becomes:

ln

== Fm

rDv

Cii

bundleainconductorsbetweendistancedandradiusconductorr

bundles,conductorbetweendistancemeangeometricDwhere

ln2

eq

10

==

=

= Fm

rd

DC eq


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Capacitance For a screened cable, capacitance to earth is:

insulationofradiusouterRwhere

ln

2 10

=

==

Fm

r

Rv

qC

r

1.17

Cable capacitance to earth is higher per metre than OHconductor

This results from higher relative permittivity and smaller

spacing between conductor and earth Frequency dependent but only significant in transient studies

capacitance generally falls as a function of frequency

conductorofradiusouterr =


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Comparison Of Line/Cable Parameters

Examples below refer to lines/cables of similar ratings

Lines and cables have similar series inductance values

Cables have a much higher shunt capacitance

Resistance is dependent on conductor size

1.18

un con uc ance no s gn can n mos sys ems u may nee

to be considered for some DC applications

Cable Line

275kV 315nF/km 30nF/km

400kV 330nF/km 34nF/km

Cable Line

275kV 0.7mH/km 1.1mH/km

400kV 0.7mH/km 0.9mH/km

InductanceCapacitance


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Short Line Model

Simple model of resistance and inductance. Capacitance neglected

For short overhead lines (approx. 50 miles) where shuntcapacitance is not important / significant

Not suitable for transient studies

1.19

Vrs

I

XLR

)( LRS jXRIVV ++=


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Medium Line Model

For medium length line, shunt capacitance must be included

Middle ground model, not fully realistic but better than short line

For medium length overhead lines (approx. 150 miles)

Not suitable for transient studies

1.20

Computationally straightforward

VrVs

IrXLR

-jXC/2-jXC/2

Is


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Medium Line Model

Lr

rrs

c

rs

rs

jXRZZYV

IVV

jXY

YVYV

II

+=

++=

=++=

)where(

)/1where(22

1.21

VrVs

IrXLR

-jXC/2- XC/2

Is I

rrs

rrs

IZYZY

YVI

VZY

ZIV

++

+=

++=

1

24

1

12

or


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Medium Line Model

VrVs

IrXLR

- XC/2- XC/2

Is I

1.22

+==

+==

+=

+=

411

2

ZYYCZB

ZYDA

DICVI

BIAVV

rrs

rrs


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Long Line Model

For long length line, distributed parameter model mustbe used

Can represent potential drop along the line and

1.23

production/ consumption along line

Accurate model - also used for transient analysis

xx+x

lx rx

gx cx v

i

v+v

i+i

vr

ir

vs

is


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Long Line Model

Long line model introduces surge impedance and propagationconstant familiar from travelling wave analysis

Propagation constant leads to existence of time delay

In travelling wave studies, complications exist from frequencydependence of z and y.

1.24

impedancesticcharactericonstant,npropagatiodistance,

sinhsinhcosh

===

==

====

+=

+=

c

c

c

c

rrs

rrs

Zl

zyy

zZ

Z

lClZBlDA

DICVI

BIAVV


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Cable Modelling Example A 275kV cable has the following parameters (per phase):

R: 0.06/km Xl: j0.2/km

C: 0.17F/km

1.25

The thermal rating of the cable is 1000A. Using the short andmedium length line models, calculate the sending end current,voltage and injected power based on the following loads beingapplied to the end of a 75km length of cable:

125MVA @ 0.98pf lagging (per phase), 1pu receiving end voltage 125MVA @ 0.98pf leading (per phase), 1pu receiving end voltage


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Cable Modelling Example Answers Use of short cable model shows no difference between receiving and start

end current

In both cases, medium length model sees capacitive flows at start of cable.Not the case for the short model

Significant differences in calculated current values smaller difference involtage drops

1.26

Vsend Isend Ssend

Leading Medium (155+j13.7)kV 0.98pu

(744+j785)A 1.08pu

(126-j111)MVA

Leading Short (160+j12.3)kV

1.01pu

(771+j157)A

0.79pu

(125-j16)MVA

Lagging Medium (160+j12.3)kV 1.01pu

(747+j481)A 0.89pu

(125-j68)MVA

Lagging Short (164+j10.9)kV

1.04pu

(771-j156)A

0.79pu

(125+j34)MVA


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Cable Modelling Example Answers Long length model and medium length models give similar results for sending

end current

Medium length model not accurate for use in sending end voltage estimation

Computer software can always implement long line models by hand this isnot so straightforward (exponentials of complex numbers required)

Errors will reduce as system length reduces

1.27

Vsend Isend Ssend

Leading Long (159+j25.1)kV 1.01pu

(758+j791)A 1.1pu

(140-j107)MVA

Leading Medium (155+j13.7)kV

0.98pu

(744+j785)A

1.08pu

(126-j111)MVA

Lagging Long (168+j22.4)kV 1.07pu

(759+j482)A 0.9pu

(138-j64)MVA

Lagging Medium (160+j12.3)kV

1.01pu

(747+j481)A

0.89pu

(125-j68)MVA


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Impact Of System Capacitance Example below shows results of calculations performed with long and

medium length models on a cable and line (lagging pf). 1000A base retained

for conversion of current to per-unit. Line capacitance around one order of magnitude less than the cable

Voltage estimation from medium length line model now more accurate

Vsend Isend Ssend

1.28

Cable - Long (168+j22.4)kV 1.07pu

(759+j482)A 0.9pu

(138-j64)MVA

Cable - Medium (160+j12.3)kV 1.01pu

(747+j481)A 0.89pu

(125-j68)MVA

Line - Long (170+j21.8)kV 1.08pu

(770-j92)A 0.78pu

(129+j32)MVA

Line - Medium (164+j11.0)kV 1.04pu

(769-j92)A 0.77pu

(125+j23)MVA


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Reactive Losses Short line model only allows consumption of reactive power

even when system is lightly loaded

800.00

1000.00

1.29

-400.00

-200.00

0.00

200.00

400.00

600.00

0.00 200.00 400.00 600.00 800.00 1000.00 1200.00 1400.00 1600.00

Power / MW

ReactiveLosses/MVAR

50km 100km 200km Short Line - 200km


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Impact Of Capacitance On Performance

PI Model of a 275kV Cable, 630mm2

Let us plot possible combinations of real andimaginary current at receiving end

1.30

j0.2

0.062

-j38350 -j38350Vs

Vr

IrIs


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600

800

1000

1200

tA

tR

ecev

ingE

nd/

A

Reactive Power Flow To Sending EndReactive Power Flow From Sending End

Optimal power transferachieved for long cable when

50% of charging current

supplied from either end

allowin maximum real current

1.31

0

200

400

-1200 -800 -400 0 400 800 1200

Imaginary Current At Receiving End / A

R

ealC

urren

1km 50km 100km

Reactive power transfer to sending

end imposes thermal limit at sending

end where 860A of charging current

flows. Maximum of 200A reactivecurrent can flow at the receiving end

Inductive current at receiving end at

a maximum owing to thermal

constraint. Current at start of cable is

reduced owing to the contribution of

the capacitive current.


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Travelling Wave Theory

Travelling wave theory describes the propagation ofsurges along overhead lines and cables. It is importantto understand the behaviour of lightning/switchingsurges

1.32

Travelling waves: have a velocity that depends on the item of plant through

which they are propagating

are altered in magnitude and shape when they propagatethrough a piece of plant

are reflected at discontinuities in surge impedance


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Major Travelling Wave Equations

At any point on a lossless line, the voltage can beexpressed as:

reflectedincidentalter VVV +=min

1.33

Voltage and current are related by surge impedancereflecteincidentalter III +=min

c

reflected

c

incidentalter

Z

V

Z

VI =min


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Major Travelling Wave Equations

The surge impedance can be written as:

c

lZc =

1.34

where l and c are impedance and capacitance per unit length,

Surge impedances range from 400 for lines to 25 for cables

Waveforms propagate along the line with a velocity of:

Velocities range from near light speed (300m/s) in lines tohalf of this (150m/s) in cables

lcv

1=


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Behaviour At An Open Circuit

At an open circuit, transmitted current must be zero

reflectedincidentalter VVV +=minreflecteincidentalterIII +=min

1.35

incidenreflectedincidentcincidentalter

Cincidentcincidentalter

Creflectedcincidentalter

creflectedreflectedcincidentincident

VVVZIV

ZIZIV

ZIZIV

ZIVZIV

===

+=

=

==

22

and

min

min

min

reflectedincident

reflectedincident

II

II

=

+=0


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Behaviour At A Short Circuit

At a short circuit, transmitted voltage must be zero

reflectedincidentalter

ZVIZVI

III

==

+=

/and/

minreflectedincidentalter

VV

VVV

+=

+=

0

min

1.36

incidentreflectedincidentcincidentalter

Cincidentcincidentalter

Creflectedcincidentalter

IIIZVI

ZVZVI

ZVZVI

===

+=

=

22

//

//

min

min

min

reflectedincident VV =


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Behaviour At A Specific Impedance

creflectedcincidenttermterm

termtermtermcreflectedreflectedcincidentincident

reflectedincidentterm

ZVZVZV

ZVIZVIZVI

III

=

===

+=

///

/,/,/

1.37termc

term

term

incident

incident

term

cterm

termincidenttermcterm

termincidentincidenttermcterm

reflectedincidentterm

ZZ

Z

V

V

VZ

ZV

VVZZV

VVVZZVVVV

+=

=

+

=

+=+=

2

21

]2[)/(

)]([)/(g...rearranginsoand:Also


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Transmission / Reflection Coefficients Transmission and reflection coefficients (relating magnitude of

transmitted and reflected wave to incident) can be derived asfollows:

Voltagetransmission

VoltageReflection

CurrentTransmission

Current Reflection

BasicalterZ min2 alter ZZ min Z2

alter ZZ min

1.38

Note in all cases the values of transmitted and reflectedpulses when the terminal surge impedance is equal to the line

surge impedance Transmitted pulses have a magnitude of 1pu Surge impedance termination appears as an infinite line

Reflected pulses are all zero

t

alterZZ+ min

r

alter ZZ +min

t

alterZZ+ min

r

alter ZZ +min


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Transmission / Reflection Coefficients

Table below shows transmission and reflectioncoefficients for three main cases

Complex impedances behave differently as a

1.39

.

Capacitors act as a short circuit Inductors act as an open circuit

Line Terminated With: at=voltagetransmission

coefficient

ar=voltagereflection

coefficient

bt=currenttransmission

coefficient

br=currentreflection

coefficient

Line Surge Impedance 1 0 1 0Short Circuit 0 -1 2 1Open Circuit 2 1 0 -1


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A voltage surge of 95kV is propagating down a 400overhead line and is transmitted into a 20 cable

The current in the overhead line will be95kV/400=237.5A

Practical Example - Overhead Line To Cable

1.40

ccor ng o e vo age ransm ss on equa on on e

previous slide:

Reflected voltage wave given using VT=VF+VRequation on former slide. Must be 9kV-95kV, i.e. -86kV

kVkVZZ

ZVV

lineOHcable

cablelineOHcable 0.9

40020

4095

2=

+

=

+=


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Thevenin Equivalent To produce a Thevenin equivalent of a circuit:

Find voltage source with magnitude equal to open circuitresponse

Find impedance to give short circuit response

1pu voltage propagating into an open circuit willproduce a voltage of 2pu - this is the open circuit

1.41

response - voltage source of 2pu required

1pu current propagating into a short circuit will givea current of 2pu - this is the short circuit response -impedance equal to Zc required

LINE IMPEDANCE Z

2VincidentTERMINATIONIMPEDANCE

Vterminal


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Multi-Line Substation Multi-line substation an important case, many

substations have multiple overhead lines connected.

Thevenin equivalent for system with n-lines connectedhas impedance of Z/n

1.42

2VincidentZ/ n-1 Vterminal

n

V

nZZ

nZVV incidentincidentterm

2

)1/(

)1/(2 =

+

=


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Bewley Lattice Diagram Used to find voltages and current on points of finitelength systems

Transit times can be low in relation to waveform time

Firstly, all coefficients required must be calculated Cable Connection Z2Overhead Line Z1

1.43

Incoming Surge RVoltage

transmissionVoltage

ReflectionCurrent

TransmissionCurrent Reflection

BasicFormulae t

alter

alter aZZ

Z=

+ min

min2

r

alter

alter aZZ

ZZ=

+

min

min

t

alter

bZZ

Z=

+ min

2 r

alter

alter bZZ

ZZ=

+

min

min

OH Line to

Cable

21

22

ZZ

Z

+()

12

12

ZZ

ZZ

+

()

21

12

ZZ

Z

+

12

12

ZZ

ZZ

+

Cable -

ResistanceRZ

R

+2

2()

2

22

ZR

ZR

+

()

RZ

Z

+2

22

2

2

ZR

ZR

+

Cable to

OH Line21

12

ZZ

Z

+()

21

21

ZZ

ZZ

+

()

21

22

ZZ

Z

+

21

21

ZZ

ZZ

+


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BewleyLattice

Diagram 0

RZ2Z1

e

e

e

e

e

e

e

1.44

4L/v

6L/v

Time

22e

e

23e

22e

2e

e

2e

23e


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Further reading

There are many text books which have a classicaltreatment: See library shelves around 621.319

For example

1.45

,

Wiley

H Waddicor The Principles of Electric PowerTransmission

60301Line & Cable Modelling 1.2

Documents

Transcript of 60301Line & Cable Modelling 1.2