Ch4 Power System Slide07

BEE3133 Electrical Power Systems

Chapter 3Transmission Line Parameters

Rahmatul Hidayah Salimin

INTRODUCTION

• All transmission lines in a power system exhibit the electrical properties of resistance, inductance, capacitance and conductance.

• Inductance and capacitance are due to the effects of magnetic and electric fields around the conductor.

• These parameters are essential for the development of the transmission line models used in power system analysis.

• The shunt conductance accounts for leakage currents flowing across insulators and ionized pathways in the air.

• The leakage currents are negligible compared to the current flowing in the transmission lines and may be neglected.

RESISTANCE

• Important in transmission efficiency evaluation and economic studies.

• Significant effect– Generation of I2R loss in

transmission line.– Produces IR-type voltage drop

which affect voltage regulation.

RESISTANCE

• The dc resistance of a solid round conductor at a specified temperature is

Where :ρ = conductor resistivity (Ω-m),

l = conductor length (m) ; and

A = conductor cross-sectional area (m2)

dc

lR

A

RESISTANCE

• Conductor resistance is affected by three factors:-

• Frequency (‘skin effect’)

• Spiraling

• Temperature

RESISTANCE

Frequency – Skin Effect• When ac flows in a conductor, the

current distribution is not uniform over the conductor cross-sectional area and the current density is greatest at the surface of the conductor.

• This causes the ac resistance to be somewhat higher than the dc resistance. The behavior is known as skin effect.

RESISTANCE

• The skin effect is where alternating current tends to avoid travel through the center of a solid conductor, limiting itself to conduction near the surface.

• This effectively limits the cross-sectional conductor area available to carry alternating electron flow, increasing the resistance of that conductor above what it would normally be for direct current

RESISTANCE

RESISTANCE

• Skin effect correction factor are defined as

Where R = AC resistance ; and

Ro = DC resistance.

O

R

R

RESISTANCE

Spiraling• For stranded conductors, alternate

layers of strands are spiraled in opposite directions to hold the strands together.

• Spiraling makes the strands 1 – 2% longer than the actual conductor length.

• DC resistance of a stranded conductor is 1 – 2% larger than the calculated value.

RESISTANCETemperature• The conductor resistance increases as

temperature increases. This change can be considered linear over the range of temperature normally encountered and may be calculated from :

Where: R1 = conductor resistances at t1 in °C R2 = conductor resistances at t2 in °C T = temperature constant (depends on the conductor material)

22 1

1

T tR R

T t

RESISTANCE

• The conductor resistance is best determined from manufacturer’s data.

• Some conversion used in calculating line resistance:-

1 cmil = 5.067x10-4 mm2 = 5.067x10-

6 cm2

= 5.067x10-10 m2

Resistivity & Temparature Constant of Conductor Metals

Material

ρ20ºC T

Resistivity at 20ºC Temperature Constant

Ωm×10-8 Ωcmil/ft ºC

Copper

Annealed 1.72 10.37 234.5

Hard-drawn 1.77 10.66 241.5

Aluminum

Hard-drawn 2.83 17.00 228

Brass 6.4 – 8.4 38 – 51 480

Iron 10 60 180

Silver 1.59 9.6 243

Sodium 4.3 26 207

Steel 12 – 88 72 – 530 180 – 980

RESISTANCE

• Example:-A solid cylindrical aluminum conductor 25km long has an area of 336,400 circular mils. Obtain the conductor resistance at (a) 20°C and

(b) 50°C.

The resistivity of aluminum at 20°C is

ρ = 2.8x10-8Ω-m.

RESISTANCE

• Answer (a)

25

8 3

4

6

2.8 10 25 10

336,400 5.076 10

4.0994 10

l km

lR

A

RESISTANCE

• Answer (b)

5050 20

20

6

6

228 504.0994 10

228 20

4.5953 10

CC C

C

T tR R

T t

RESISTANCE

• Exercise 1A transmission-line cable consists of 12 identical strands of aluminum, each 3mm in diameter. The resistivity of aluminum strand at 20°C is 2.8x10-8Ω-m. Find the 50°C ac resistance per km of the cable. Assume a skin-effect correction factor of 1.02 at 50Hz.

RESISTANCE

• Exercise 2:-A solid cylindrical aluminum conductor 115km long has an area of 336,400 circular mils. Obtain the conductor resistance at:(a) 20°C

(b) 40°C

(c) 70°C

The resistivity of aluminum at 20°C is

ρ = 2.8x10-8Ω-m.

RESISTANCE

• Exercise 3A transmission-line cable consists of 15 identical strands of aluminum, each 2.5mm in diameter. The resistivity of aluminum strand at 20°C is 2.8x10-8Ω-m. Find the 50°C ac resistance per km of the cable. Assume a skin-effect correction factor of 1.015 at 50Hz.

INDUCTANCE :A SINGLE CONDUCTOR

• A current-carrying conductor produces a magnetic field around the conductor.

• The magnetic flux can be determined by using the right hand rule.

• For nonmagnetic material, the inductance L is the ratio of its total magnetic flux linkage to the current I, given by

where λ=flux linkages, in Weber turns.L

I

INDUCTANCE : A SINGLE CONDUCTOR

• For illustrative example, consider a long round conductor with radius r, carrying a current I as shown.

• The magnetic field intensity Hx, around a circle of radius x, is constant and tangent to the circle.

2x

x

IH

x


• The inductance of the conductor can be defined as the sum of contributions from flux linkages internal and external to the conductor.

Flux Linkage


INDUCTANCE :A SINGLE PHASE LINES

INDUCTANCE :3-PHASE TRANSMISSION LINES


What and How to Calculate:-• Lint , Lext @ L?

• L1 , L2 @ L?

• L11 , L12 @ L22?

• GMR?• GMD?


• INTERNAL INDUCTANCE– Internal inductance can be express as

follows:-

– Where

µo = permeability of air (4π x 10-7 H/m)

– The internal inductance is independent of the conductor radius r

70int

110 /

8 2L H m


• INDUCTANCE DUE TO EXTERNAL FLUX LINKAGE– External

inductance between to point D2 and D1 can be express as follows:

7 2

1

2 10 ln /ext

DL H m

D


• A single phase lines consist of a single current carrying line with a return line which is in opposite direction. This can be illustrated as:


• Inductance of a single-phase lines can be expressed as below with an assumption that the radius of r1=r2=r.

7 7 2int

1

7 7 7

17 74

1

4

70.25

110 2 10 ln /

2

1 110 2 10 ln / 2 10 ln /

2 4

12 10 ln ln / 2 10 ln ln /

2 10 ln /

ext

DL L L H m

D

D DH m H m

r r

D De H m H m

r re

DH m

re

SELF AND MUTUAL INDUCTANCES • The series inductance per phase can be

express in terms of self-inductance of each conductor and their mutual inductance.

• Consider the one meter length single-phase circuit in figure below:-

– Where L11 and L22 are self-inductance and the mutual inductance L12

SELF AND MUTUAL INDUCTANCES

Dx

DxL

DxL

erxL

ILLID

xer

xIL

ILL

ILL

mHD

xer

xL

mHD

xer

xL

1ln102

1ln102

1ln102

1ln102

1ln102

1ln102

/1

ln1021

ln102

/1

ln1021

ln102

7712

712

25.01

711

1121117

25.01

7111

222212

112111

725.0

2

72

725.0

1

71


• L11, L22 and L12 can be expressed as below:-

711 0.25

1

722 0.25

2

712 21

12 10 ln

12 10 ln

12 10 ln

Lre

Lr e

L LD


• Flux linkage of conductor i

ijD

Ier

Ixn

j ijj

iii

1

ln1

ln1021

25.07


• Symmetrical Spacing– Consider 1 meter length of a three-phase

line with three conductors, each radius r, symmetrically spaced in a triangular configuration.


• Assume balance 3-phase current

Ia+ Ib+ Ic = 0

• The total flux linkage of phase a conductor

• Substitute for Ib + Ic=-Ia

DI

DI

erIx cb

aaa

1ln

1ln

1ln102

25.07

25.07

25.07 ln102

1ln

1ln102

er

DIx

DI

erIx

aaa

aaa

INDUCTANCE : 3-PHASE TRANSMISSION LINES

• Because of symmetry, λa=λb=λc

• The inductance per phase per kilometer length

kmmHre

Dx

IL /ln102

25.07


• Asymmetrical Spacing– Practical transmission lines cannot maintain

symmetrical spacing of conductors because of construction considerations.

– Consider one meter length of three-phase line with three conductors, each with radius r. The conductor are asymmetrically spaced with distances as shown.


– The flux linkages are:-

231325.0

7

231225.0

7

131225.0

7

1ln

1ln

1ln102

1ln

1ln

1ln102

1ln

1ln

1ln102

DI

DI

reI

DI

DI

reI

DI

DI

reI

bacc

cabb

cbaa

INDUCTANCE : 3-PHASE TRANSMISSION LINES

– For balanced three-phase current with Ia as reference, we have:-

ao

ac

ao

ab

aIII

IaII

120

240 2


• Thus La, Lb and Lc can be found using the following equation:-

23

225.0

12

7 1ln

1ln

1ln102

Da

reDa

IL

b

bb

1312

225.0

7 1ln

1ln

1ln102

Da

Da

reIL

a

aa

25.0

2313

27 1ln

1ln

1ln102

reDa

Da

IL

c

cc

INDUCTANCE : 3-PHASE TRANSMISSION LINES• Transpose Line

– Transposition is used to regain symmetry in good measures and obtain a per-phase analysis.

INDUCTANCE :3-PHASE TRANSMISSION LINES• This consists of interchanging the phase

configuration every one-third the length so that each conductor is moved to occupy the next physical position in a regular sequence.

• Transposition arrangement are shown in the figure


• Since in a transposed line each phase takes all three positions, the inductance per phase can be obtained by finding the average value.

0.2512 13

7

0.2523 12

0.2513 23

7

0.2512

3

1 1 1ln 1 240 ln 1 120 ln

2 10 1 1 1ln 1 240 ln 1 120 ln

3

1 1 1ln 1 240 ln 1 120 ln

2 10 1 1 13ln ln ln

3

a b cL L LL

re D D

re D D

re D D

re D D

23 13

312 23 137

0.25

1ln

2 10 ln

D

D D D

re

• Since in a transposed line each phase takes all three positions, the inductance per phase can be obtained by finding the average value.

3cba

a

LLLL

• Noting a + a2 = -1

• Inductance per phase per kilometer length

25.0

3

1

1323127

3

1

132312

25.07

13231225.0

7

ln102

1ln

1ln102

1ln

1ln

1ln

1ln3

3

102

re

DDD

DDDre

DDDreL

kmmH

re

DDDL /ln2.0

25.0

3

1

132312

What and How to Calculate:-• Lint , Lext @ L?

• L1 , L2 @ L?

• L11 , L12 @ L22?

• GMR?• GMD?

Inductance of Composite Conductors

In evaluation of inductance, solid round conductors were considered. However, in practical transmission lines, stranded conductors are used.

Consider a single-phase line consisting of two composite conductors x and y as shown in Figure 1. The current in x is I referenced into the page, and the return in y is –I.


Conductor x consist of n identical strands or subconductors, each with radius rx. Conductor y consist of m identical strands or subconductors, each with radius ry.

The current is assumed to be equally divided amon the subconductors. The current per strands is I/n in x and I/m in y.


b d'

a

c

d

n

b'

a'

c'

m'

x y

nncnbnax

mnmncnbnan

n

nanacabx

mamacabaaa

a

nanacabx

mamacabaa

a

amacabaa

anacabxa

DDDr

DDDDn

nIL

DDDr

DDDDn

nIL

DDDr

DDDDI

or

DDDDm

I

DDDrn

I

...'

...ln102

/

...'

...ln102

/

...'

...ln102

1ln...

1ln

1ln

1ln102

1ln...

1ln

1ln

'

1ln102

'''7

'''7

'''7

'''

7

7

'...

)...)...(...(

)...)...(...(

/ln102

2

''''

7

xnnbbaa

nnnnbnaanabaax

mnnmnbnaamabaa

xx

rDDD

where

DDDDDDGMR

DDDDDDGMD

where

mHGMR

GMDL

GMR of Bundled Conductors

d

d d

d

d

d

d

d

Extra high voltage transmission lines are usually constructed with bundled conductors. Bundling reduces the line reactance, which improves the line performance and increases the power capability of the line.

GMR of Bundled Conductors

4 316 42/1

3 29 3

09.1)2(

)(

dDdddDD

bundleorsubconductfourthefor

dDddDD

bundleorsubconductthreethefor

ssbs

ssbs

dDdDD

bundleorsubconducttwothefor

DDDDDDGMR

ssbs

nnnnbnaanabaax

4 2)(

)...)...(...(2

Inductance of Three-phase Double Circuit Lines

A three-phase double-circuit transmission line consists of two identical three-phase circuits. To achieve balance, each phase conductor must be transposed within it group and with respect to the parallel three-phase line.Consider a three-phase double-circuit line with relative phase positions a1b1c1-c2b2a2.


c1

a1

a2

b1 b2

c2S11

S22

S33

GMD between each phase group

422122111

422122111

422122111

cacacacaAC

cbcbcbcbBC

babababaAB

DDDDD

DDDDD

DDDDD


The equivalent GMD per phase is then

3ACBCAB DDDGMD

Similarly, GMR of each phase group is

214 2

21

214 2

21

214 2

21

)(

)(

)(

ccb

ccb

SC

bbb

bbb

SB

aab

aab

SA

DDDDD

DDDDD

DDDDD

ss

ss

ss

where is the geometric mean radius of bundled conductors.

bsD


The equivalent GMR per phase is then

3SCSBSAL DDDGMR

The inductance per-phase is

mHGMR

GMDL

Lx /ln102 7


• Question 4

A three-phase, 50 Hz transmission line has a reactance 0.5 Ω per kilometer. The conductor geometric mean radius is 2 cm. Determine the phase spacing D in meter.


• Question 4

A three-phase, 60 Hz transmission line has a reactance 0.25Ω per kilometer. The conductor geometric mean radius is 5 cm. Determine the phase spacing D in meter.


• Question 4

A three-phase, 50 Hz transmission line has

Xc = 0.5 Ω per kilometer. The conductor geometric mean radius is 2 cm. Determine the phase spacing D in meter.

CAPACITANCE

• Transmission line conductors exhibit capacitance with respect to each other due to the potential difference between them.

• The amount of capacitance between conductors is a function of conductor size, spacing, and height above ground.

• Capacitance C is:-

qC

V

LINE CAPACITANCE

• Consider a long round conductor with radius r, carrying a charge of q coulombs per meter length as shown.

• The electrical flux density at a cylinder of radius x is given by:

2

q qD

A x

LINE CAPACITANCE• The electric field intensity E is:-

Where permittivity of free space, ε0 = 8.85x10-12 F/m.

• The potential difference between cylinders from position D1 to D2 is defined as:-

The notation V12 implies the voltage drop from 1 relative to 2.

0 02

D qE

x

212

0 1

ln2

q DV

D

CAPACITANCE OF SINGLE-PHASE LINES • Consider one meter length of a single-

phase line consisting of two long solid round conductors each having a radius r as shown.

• For a single phase, voltage between conductor 1 and 2 is:-

120

ln /q D

V F mr

CAPACITANCE OF SINGLE-PHASE LINES

• The capacitance between the conductors:-

012 /

lnC F m

Dr


• The equation gives the line-to-line capacitance between the conductors

• For the purpose of transmission line modeling, we find it convenient to define a capacitance C between each conductor and a neutral line as illustrated.


• Voltage to neutral is half of V12 and the capacitance to neutral is C=2C12 or:-

02/

lnC F m

Dr

Potential Difference in a Multiconductor configuration

• Consider n parallel long conductors with charges q1, q2,…,qn coulombs/meter as shown below.

ki

kjn

kkij D

DqV ln

2

1

10

• Potential difference between conductor i and j due to the presence of all charges is

qjqi

q1

q3q2

qn

CAPACITANCE OF THREE-PHASE LINES

• Consider one meter length of 3-phase line with three long conductors, each with radius r, with conductor spacing as shown below:

qc

qb

qa

D13

D23

D12


For balanced 3-phase system, the capacitance per phase to neutral is:

1/3

12 23 13

2F/m

ln

a o

an

qC

V D D D

r


1/3

12 23 13

0.0556F/km

ln

CD D D

r

The capacitance to neutral in µF per kilometer is:

Effect of bundling

mF

rGMD

C

b

/ln

2 0

• The effect of bundling is introduce an equivalent radius rb. The radius rb is similar to GMR calculate earlier for the inductance with the exception that radius r of each subconductor is used instead of Ds.

Effect of bundling

• If d is the bundle spacing, we obtain for the two-subconductor bundle

drrb

• For the three-subconductor bundle

3 2drrb

• For the four-subconductor bundle

4 309.1 drrb

Capacitance of Three-phase Double Circuit Lines

mF

GMRGMD

C

c

/ln

2 0

• The per-phase equivalent capacitance to neutral is obtained to

• GMD is the same as was found for inductance calculation

422122111

422122111

422122111

cacacacaAC

cbcbcbcbBC

babababaAB

DDDDD

DDDDD

DDDDD

Capacitance of Three-phase Double Circuit Lines

• The equivalent GMD per phase is then

3ACBCAB DDDGMD

• The GMRC of each phase is similar to the GMRL, with the exception that rb is used instead of

bsD

• This will results in the following equ…

21

21

21

ccb

C

bbb

B

aab

A

Drr

Drr

Drr

3CBAC rrrGMR

EFFECT OF EARTH ON THE CAPACITANCE • For isolated charged conductor the

electric flux lines are radial and orthogonal to cylindrical equipotential surfaces, which will change the effective capacitance of the line.

• The earth level is an equipotential surface. Therefore flux lines are forced to cut the surface of the earth orthogonally.

• The effect of the earth is to increase the capacitance.

EFFECT OF EARTH ON THE CAPACITANCE

• But, normally, the height of the conductor is large compared to the distance between the conductors, and the earth effect is negligible.

• Therefore, for all line models used for balanced steady-state analysis, the effect of earth on the capacitance can be negligible.

• However, for unbalance analysis such as unbalance faults, the earth’s effect and shield wires should be considered.

MAGNETIC FIELD INDUCTION

• Transmission line magnetic fields affect objects in the proximity of the line.

• Produced by the currents in the line.

• It induces voltage in objects that have a considerable length parallel to the line (Ex: telephone wires, pipelines etc.).

MAGNETIC FIELD INDUCTION

• The magnetic field is effected by the presence of earth return currents.

• There are general concerns regarding the biological effects of electromagnetic and electrostatic fields on people.

ELECTROSTATIC INDUCTION

• Transmission line electric fields affect objects in the proximity of the line.

• It produced by high voltage in the lines.

• Electric field induces current in objects which are in the area of the electric fields.

• The effect of electric fields becomes more concern at higher voltages.

ELECTROSTATIC INDUCTION

• Primary cause of induction to vehicles, buildings, and object of comparable size.

• Human body is effected to electric discharges from charged objects in the field of the line.

• The current densities in human cause by electric fields of transmission lines are much higher than those induced by magnetic fields!

CORONA

• When surface potential gradient exceeds the dielectric strength of surrounding air, ionization occurs in the area close to conductor surface.

• This partial ionization is known as corona.

• Corona generate by atmospheric conditions (i.e. air density, humidity, wind)

CORONA

• Corona produces power loss and audible noise (Ex: radio interference).

• Corona can be reduced by:– Increase the conductor size.– Use of conductor bundling.

Review

• Transmission Line Parameters:– Resistance

• Skin effect

– Inductance• Single phase line• 3 phase line equal & unequal spacing

– Capacitance• Single phase line• 3 phase line equal & unequal spacing

– Conductance• Neglected• Corona

Review

• Effect of Earth on the Capacitance

• Magnetic Field Induction

• Electrostatic Induction

• Corona

Ch4 Power System Slide07

Documents

conductor resistivity

solid conductor

conductor resistances

conductor material22

line resistance

solid round conductor

dc resistance

long round conductor