In this package: 1. Calculation of inductance for ... Lecture 12 Summary of Inductance Calculations...

21-Oct-11

1Lecture 12 Power Engineering - Egill Benedikt Hreinsson

In this package:1. Calculation of inductance for multiconductor/multiphase transposed

High voltage transmission lines2. Underground cables3. Artistically designed high voltage towers

21-Oct-11


Voltage Drop and Reactance for Multiphase Multi-conductor Lines

}0

11 1 1 111 12 1

2 312 13 1

1 1 11 1 1ln ln ln2

1 1 1ln ln ln

n

mm

V j I I Ig d dn n n

I I ID D D

μω

π⎧⎡ ⎤Δ = + + +⎨⎢ ⎥⎣ ⎦⎩

+ + + +

n = number of conductors per phase (n = 1,2, 3 or 4)m = number of phases (m = 3)Ij = current in phase # jdij = distance between conductor # i and j within a phaseDij = distance between phases # i and jg11 = GMR of conductor # 1 in phase # 1ΔVij= Voltage drop per unit length of line in conductor #i in phase #j Then the voltage drop in conductor #1 in phase # 1 per

unit length will be:

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3Lecture 12 Power Engineering - Egill Benedikt HreinssonThe Voltage Drop for an Unsymmetrical 3 Phase Line

{ }011 1 2 3

11 12 131 1 1ln ln ln

2V j I I IG D D

μω

πΔ = + +

G11 is the GMR of phase # 1

11 22 33 11 12 13 1n

nG G G g d d d= = = ⋅ ⋅ ⋅ ⋅

g11 is the GMR of conductor #1 in phase # 1

For a m=3 phase line:

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Transposition of Transmission LinesObjective: To obtain phase symmetry with the approximate same average inductance and capacitance between phases and to earth for the total line length

1 32Position:

1 32Position:

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5Lecture 12 Power Engineering - Egill Benedikt HreinssonThe Voltage Drop for a Symmetrical Transposed 3 Phase Line (1)

a,b,c,: subscripts denote electrical phases along the line1,2,3,: subscripts denote geographical position along the line

“1”: the left position, “2”: the center position, “3”: the right positionI,II,III: subscript for the section of the line

Change in notation:

{ }0,

12 131 1 1ln ln ln

2a I a b caa

V j I I IG D Dμωπ

Δ = + +For section I we get per unit line length:

{ }0,

23 121 1 1ln ln ln

2a II a b caa


Δ = + +For section II we get

{ }0,

13 231 1 1ln ln ln

2a III a b caa


Δ = + +For section III we get

The distance from phase a current Ia to the current Ib is D12, D13 and D23, respectively depending on section. Similar for current Ic

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6Lecture 12 Power Engineering - Egill Benedikt HreinssonThe Voltage Drop for a Symmetrical Transposed 3 Phase Line (1)

These 3 formulas are be added and divided by 3 to take the average values

0

13 12 23

13 12 23

11 1 1 1ln ln ln ln2 3

1 1 1 1ln ln ln3

a a baa

c

V j I IG D D D

I D D D

μω

π⎧ ⎡ ⎤Δ = + + +⎨ ⎢ ⎥⎣ ⎦⎩

⎫⎡ ⎤+ + + ⎬⎢ ⎥⎣ ⎦⎭

I I Ia b c+ + = 0We then use the fact that the sum of the currents for a symmetrical system:

( )0

12 13 23

11 1 1 1ln ln ln ln2 3a a b c

aaV j I I IG D D D

μω

π⎧ ⎡ ⎤Δ = + + + +⎨ ⎢ ⎥⎣ ⎦⎩

, , ,

3a I a II a III

aV V V

VΔ + Δ + Δ

Δ =

21-Oct-11


The Voltage Drop for a Symmetrical Transposed 3 Phase Line (1)

b c aI I I+ = −

0

12 13 23

1 11ln ln2 3a a

aaV j I G D D D

μω

π⎧ ⎫⎡ ⎤⎪ ⎪Δ = −⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

The current balance gives:

..and combine the last 3 logarithm factors:...

312 13 230 ln

2a aaa

D D DV j I

Gμωπ

Δ =..This is the final formula for the voltage drop:

21-Oct-11


312 13 230 ln

2a a a aaa

D D DV j L I jXI j I

Gμ

ω ωπ

Δ = = =

The Voltage Drop for a Symmetrical Transposed 3 Phase Line (2)

….. this formula is valid for inductance, reactance and voltage drop calculations

The voltage drop now only depends on the current in the same phase. Due to symmetry the phases are independent!Therefore a one phase inductance equivalent circuit can be established!!

21-Oct-11


GMR for Phases with Different Number of Conductors

d

d

A phase with 4 conductors



1 d

dd

d

Radius of each conductor: r

14 34 2aaG re d

−= ⋅ ⋅

13 24

aaG re d−

= ⋅1

2 4aaG re d

−= ⋅

14

aaG re−

=

A phase with 1 conductor

2d

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10Lecture 12 Power Engineering - Egill Benedikt HreinssonExample of Reactance Calculations for a Line with 4 Conductors Per Phase

1 2

3 4

D=15m D=15m

d 14

d 12

d 13

2R

3312 13 230 0

1 34 4

2ln ln2 2 e 2aa

D D D DXG R d

μ μω ωπ π −

⋅= =

⋅ ⋅

14 34411 12 13 14

33312 13 23

e 2

2 2aaG g d d d R d

D D D D D D D

−= ⋅ ⋅ ⋅ = ⋅ ⋅

= ⋅ ⋅ = ⋅

12 14 20cmd d d= = =

13 2d d= ⋅

1.5cmR =

360 15 30 15 18.8992 3.1416 50 ln 50 1.26 10 ln

2 3.1416 0.10722 0.10722X μ −⋅ ⋅= ⋅ ⋅ ⋅ = ⋅ ⋅

⋅

14 340.015 e 0.2 2 0.10722maaG−

= ⋅ ⋅ ⋅ =

( ) [ ] [ ]6 663 10 ln 176.26 325.8 10 / m 0.325 / kmX − −= ⋅ = ⋅ Ω = Ω

D12 =D D23 =D

D13=2D

15mD =

21-Oct-11

11Lecture 12 Power Engineering - Egill Benedikt HreinssonSummary of Inductance Calculations for Multi-phase Transmission Lines

• Internal and external inductances have been calculated for a single phase line.

• These calculation have been extended to a system of parallel phases and conductors

• The models above have been applied to 3 phase transposed transmission lines with bundled conductors

• (Magnetic potential calculations can also be applied to multiple parallel conductors with a general cross section to define GMR and GMD mathematically. Can be skipped)

21-Oct-11


Underground cables

21-Oct-11


145 kV underground cables• The 3 phase 145 kV

cables are laid as 3 individual cables

• The trench is about 1.5m deep and the width is also 1.5 m

• 2 parallel cables often mean 2 separate trenches

Source: Landsnet

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145 kV underground cables

Source: Landsnet

21-Oct-11


145 kV Underground cablesEnglish translation:

• Location is appraised with respect to earthquakes and geothermal activity

• Cable characteristics and power quality

• Reliability and availability• Transfer capability in the power

system• Environmental impact• Lifecycle cost assessment

Source: Landsnet

21-Oct-11


145 kV underground cables

Earth wire

Sand

3 m wide right of way

Concrete filling

Source: Landsnet

21-Oct-11


1 phase underground cables

Single-phase high-voltage cable with a solid dielectric

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3 phase underground cables

ConductorConductor shield

PEX-Insulation

Insulation shieldFiller

Copper screenPVC-sheet

Three-phase distribution cable with solid dielectric.

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Cross section of a double circuit cable

R S T T S R

350 350 350 35033001450

Tvöfaldur 2500 mm2 jarðstrengur230-400 kV

(öll mál í mm)

A double circuit 2500mm2 cable for 230-400 KV

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Artistically designed high voltage power

lines

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Artistically designed power lines

21-Oct-11

22Lecture 12 Power Engineering - Egill Benedikt HreinssonArtistically designed power lines (2)

21-Oct-11


Artistically designed power lines (3)

Source: Pylon Design Competition of British architects: http://www.flickr.com//photos/deccgovuk/sets/72157627544769507

21-Oct-11


Artistic Power lines in Finland?

21-Oct-11


References• Photographs:

– Sigurjón Svavarsson [[email protected]]

In this package: 1. Calculation of inductance for ... Lecture 12 Summary of Inductance Calculations...

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