Calculations of Capacitance for Transposed Bundled ... · PDF fileCalculations of Capacitance...

28-Oct-11

1Lecture 14 Power Engineering - Egill Benedikt Hreinsson

Calculations of Capacitance for Transposed Bundled Conductor

Transmission Lines

28-Oct-11


Multi-conductor Lines. An example with a 2 conductor bundle

h1

h1

+q1/2 +q1/2D13

D23

D'23

D'13

D'12

D12h2

h2

h3

h3

+q2/2 +q2/2

+q3/2 +q3/2

-q1/2 -q1/2

-q2/2 -q2/2

-q3/2 -q3/2

1 1 2,1,

12

1 1 1 1ln ln ln2 2 2 2a I

q q qVr d Dπε

⎧ ⎫= + + +⎨ ⎬

⎩ ⎭

1 1,2,

1 1 1ln ln2 2 2a I

q qVr dπε

⎧ ⎫= + +⎨ ⎬⎩ ⎭

r: conductor radius, d: distance between conductors of the same phaseExample: Va,2,II = Voltage of conductor # 2 in phase “a”, in section IIWe calculate the voltage drop of the 1st conductor in phase “a”:

We then calculate the voltage drop of the 2nd conductor in phase “a”:

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Multi-conductor Lines (2)

h1

h1

+q1/2 +q1/2D13

D23

D'23

D'13

D'12

D12h2

h2

h3

h3

+q2/2 +q2/2

+q3/2 +q3/2

-q1/2 -q1/2

-q2/2 -q2/2

-q3/2 -q3/2

,1, ,2,, 2

a II a IIa II

V VV

+=

,1, ,2,, 2

a III a IIIa III

V VV

+=

,1, ,2,, 2

a I a Ia I

V VV

+=

, , ,

3a I a II a III

a

V V VV

+ +=

For phase “a” in a transposed line with sections I, II and III, we calculate the “average voltage”for each section and then the average voltage drop for the whole line (all sections, phase “a”)

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4Lecture 14 Power Engineering - Egill Benedikt HreinssonGeometric Mean of Heights - Bundled Conductors

ha hb

DaDb

+qi/2 +qi/2

-qi/2 -qi/2

Phase # "i"

Image of phase # "i"

422

babai

DDhhh ⋅⋅⋅=

We get a combination of factor with logarithms that for instance lead to roots as follows

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Geometric Mean Distances to Images

Da

D'a

Db

D'b

Dc

D'c

Dd

D'd

+qi/2+qi/2

-qj/2 -qj/2

phase # "i"

Image of phase # "j"

+qj/2 +qj/2

phase # "j"

4dcbaij DDDDD ⋅⋅⋅=

4dcbaij DDDDD ′⋅′⋅′⋅′=′

Similarly for a combination of distances between phases These lead to roots as follows:

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Effective Radius - Bundled Conductors

24 32R r d d d R r d R r d= ⋅ ⋅ ⋅ = ⋅ = ⋅

d

d

-qj/4-qj/4

A phase with 4 conductors



-qj/4-qj/4 d

dd

-qj/3

-qj/3

-qj/3d

-qj/2 -qj/2 -qj

R r=

Radius of each conductor = rThe effective radius of each

conductor bundle = R. Compare with the GMR

A phase with 1 conductor

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Summary of capacitance calculations

22ln

'

rCh D

R D

πε=

⎛ ⎞⋅⎜ ⎟⎝ ⎠

312 23 31D D D D= 3

12 23 31' ' ' 'D D D D=

31 2 3h h h h=

24 32R r d d d R r d R r d= ⋅ ⋅ ⋅ = ⋅ = ⋅

2

lnrC

DR

πε=With earth’s

influence:Without earth’s influence:

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Capacitance - Inductance Relation

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Capacitance - Inductance Relation• Transposing of lines allows us to form a symmetric

circuit model or single phase equivalent which is identical for all phases both regarding reactance and capacitance

• We remember that the earth is conductive while it is not ferromagnetic

• Therefore 3-phase transmission lines with equi-distant conductors (located at the corner of a triangle with equal sides) will ensure a symmetric model regarding inductance while the earth will influence its capacitance.

• This is because the conductor closest to the ground has a different geographical relation than the other conductors - to the earth but not to the other conductors

d

dd

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Inductance Matrix

Δ

ΔΔ

Δ

u

uu

u

j L

II

In n

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

ω

1

2

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

nn

n

n

rD

DrD

DDr

L

1ln1ln

1ln1ln1ln

1ln1ln1ln

2

1

2221

1121

0

πμ

Inductance for a system of parallel conductors without considering the internal inductance

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Capacitance Matrix - Beta Matrix

1

1

111−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡= β

nnn

n

CC

CCC

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡== −

nnn

n

Cββ

βββ

1

1111

We can now compare the previous matrices regarding both inductance and capacitance. In both cases these matrices can not exist physically, although mathematically there is no problem. This is because each element in these matrices is a logarithm of a factor which has a dimension of m !!

28-Oct-11


The Beta-Matrix

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

nn

n

n

rD

DrD

DDr

1ln1ln

1ln1ln1ln

1ln1ln1ln

21

1

2221

1121

πεβ

Therefore the Beta-matrix shown here is not physically possible since each element is a logarithm of the quantity 1/length where the length is measured in m !!

28-Oct-11


Capacitance - Inductance

1 2/ 2L C πε

μ π−= ⋅

2

1L C E Ec

με= ⋅ =

We now consider the product of these 2 matrices shown to the right. The result is that the product of the capacitance matrix and the inductance matrixis constant for a system of thin conductors

1 0

0 1is the unit matrixE

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

…

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Inductive Reactances (ohm/km)

The inductance for a power line lies in the range of 0.3-0.4 ohm/km

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ACSR Table Data

Inductance and Capacitance GMR

Code words

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16Lecture 14 Power Engineering - Egill Benedikt HreinssonTypical values of overhead linecharacteristics at 50 Hz

Typically for voltages below 60 kV line charging may be ignored. For extra high voltages (400 kV+) line charging must be carefully analyzed

28-Oct-11

17Lecture 14 Power Engineering - Egill Benedikt HreinssonTypical values for underground cablecharacteristics at 50 Hz

For underground cables SIL exceeds the thermal rating which means that underground cable connections are always net producers of reactive power

28-Oct-11


Capacitances (nF/km)

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28-Oct-11

Additional Transmission topics• Ground wires: Transmission lines are usually

protected from lightning strikes with a ground wire. This topmost wire (or wires) helps to attenuate the transient voltages/currents that arise during a lighting strike. The ground wire is typically grounded at each pole.

• Corona discharge: Due to high electric fields around lines, the air molecules become ionized. This causes a crackling sound and may cause the line to glow!

28-Oct-11


Resistance of transmission linesand transmission real losses

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Factor Influencing Line Resistance

• Skin effect (0-5%)• Temperature• Conductor winding (Spiraling effect ) (0-5%)

dcRAρ ⋅

=

Because ac current tends to flow towards the surface of a conductor, the resistance of a line at 60 Hz is slightly higher than at dc.Resistivity and hence line resistance increase as conductor temperature increases (changes is about 8% between 25°C and 50°C)

28-Oct-11


Real losses – an example

-8

-8

Line resistance per length, is given by

where is the resistivity

Resistivity of Copper = 1.68 10 Ω-m

Resistivity of Aluminum = 2.65 10 Ω-mExample: What is the resistance in Ω / mile of a

RAρ ρ=

×

×

( )22 23.1416 0.0127

1609 0.084-8

1" diameter solid aluminum wire (at dc)?

m

2.65 10 Ω-m

A r

mRA mile mile

π= = ×

× Ω= =

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Effective Resistance (ohm/km)

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Circuit models for short transmission lines

Transmission Capacity

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One phase equivalent model for a short line

C/2

R+jXi j

C/2

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Equivalent circuit of a short line.

One phase equivalent model for a short line (2)

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One phase equivalent model for a long line

Sendingend

Receivingend

···

Equivalent circuit for a long transmission line.

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One phase equivalent model for a long line (3)

Sendingend

Receivingend

···

Equivalent circuit for a long transmission line.

•Series L draws reactive power•QL=wLI2 – decreases V along line

•Line charging C generatesreactive power

•QC=wCV2 – increases V along line

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Voltage balance along the line• QL << QC

– Light load – voltage increases along line• QL >> QC

– Heavy load – voltage decreases

V1

V(x)

x

Light

Heavy

What happens if: QL = QC ?

28-Oct-11


Transmission capacity definitions• Thermal limits:

– With V being constant, I is the limiting factor (Imax)

• Steady State Stability Limits:

• Natural Loading – Surge impedance Loading

or (SIL)

1 2 sinV VPX

δ⋅

=

3 cosP V I φ= ⋅ ⋅

1 2max

V VPX⋅

=

cLZC

=2 2

* ( )SIL SILc

V VS PLZC

= = =

max max3 cosP V I φ= ⋅ ⋅

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Surge Impedance Loading (SIL)

• SIL is reached, when the generated reactive power equals the consumed power in the high voltage line.

• SIL is not maximum loading but a “characteristic loading”

2 2consumed LQ X I L Iω= =

2 22

1generatedc

V VQ C V

X Cω

ω= = =

2 2

generated consumedQ Q

L I C Vω ω

=

=

c

XLZC C

ω= =

22

2 cV L Z

CI= =

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Surge Impedance Loading• Surge Impedance:

– Also called characteristic impedance. this is the impedance with which you can insert a surge the sending end of the line and not get any reflection back at the receiving end.

• X is the reactance of the line

– (in Ohm/km or in Ohm)• B is the succeptance of the

line – (in Siemens/km or in

Siemens)

Surge impedance

Receiving end

Sending end

c

XL XZC C B

ω= = = (≅ 250 – 400 ohm)

C/2

R+jXi j

C/2

28-Oct-11

33Lecture 14 Power Engineering - Egill Benedikt HreinssonRactive power ballance of a transmission line

•For light loading the line produces more Mvars than it consumes

•For heavy loading the line is a net consumer of reactive power

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