Ch5 Power System -Transmission Line Parameters

EEE3233 Power Systems

Chapter 5Transmission Line Parameters

Nur Diyana Kamarudin

INTRODUCTION• All transmission lines in a power system

exhibit the electrical properties of resistance, inductance, capacitance, and conductance. The inductance and capacitance are due to the effects of magnetic and electric field around the conductor.

• The selection of an economical voltage level for the transmission line is based on the amount of the power and the distance of transmission. The voltage choice together with the selection of the conductor size is mainly a process of weighting I2R losses, audible noise, and radio interference against fixed charge investment.

• Conductor manufacturers provide the characteristics of the standard conductors with conductor sizes expressed in circular mils (cmil). 1 mil equals 0.001 inch, and for a solid round conductor the area in circular mils is defined as the square of diameter in mils.

• At voltage above 230kV, it is preferable to use bundling of conductors. Bundling increases the effective radius of the line's conductor and reduces the electric field strength near the conductors, which reduces corona power loss, audible noise and radio interference.

INTRODUCTION

INTRODUCTION

Transmission line is non ideal, therefore cannot be assumed that the wires that transmit electricity are loss less.

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.

a model of a transmission line as a resistive element.

INTRODUCTION

• 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 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 :A 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.

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.

LI

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

INDUCTANCE :A SINGLE CONDUCTOR

• 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 CONDUCTOR• 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 :A SINGLE CONDUCTOR• INDUCTANCE DUE

TO EXTERNAL FLUX LINKAGE– External

inductance between two points D2 and D1 can be express as follows:

7 2

1

2 10 ln /ext

DL H m

D

INDUCTANCE :A SINGLE PHASE LINES

• 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 :A SINGLE PHASE LINES

• 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

A single phase line

• The flux beyond D links a net current of zero and does not contribute to the net magnetic flux linkages in the circuit. Thus, to obtain the inductance of conductor 1 due to the net external flux linkage D1 = r1 to D2 = D

--------L1

• Similarly for conductor 2. If the 2 conductors are identical, r1=r2=r and L1=L2=L, and the inductance per conductor per meter length of the line is ------ L2

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

SELF AND MUTUAL INDUCTANCES

• 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

SELF AND MUTUAL INDUCTANCES

• Flux linkage of conductor i

ijD

Ier

Ixn

j iji

iii

1

ln1

ln1021

25.07

Inductance of single phase line• 1st term is only function of the

conductor radius which is considered as the inductance due to both the internal flux and the external to conductor 1 to a radius of 1m.

• 2nd term is dependent only upon conductor spacing (inductance spacing factor)

• Term r' is known as self-geometric mean distance of a circle with radius r and is abbreviated by GMR (Geometric Mean Radius) and designated by Ds.

• (where r'1 = r1e-1/4)

Thus, the inductance per conductor in millihenries per kilometer,

Inductance of 3 Phase Transmission Lines

Symmetrical Spacing

Symmetrical spacing

Balanced 3 phase currents,

Ia + Ib + Ic = 0 Total flux linkage of phase a conductor is

Since Ib + Ic = -Ia

Because of symmetry, λb = λc = λa and the inductance per phase per kilometer length is

For a solid round conductor, Ds = re-0.25

And for stranded conductor Ds

can be evaluated from equation * Inductance per phase for a 3

phase circuit with equilateral spacing is the same as for 1 conductor of single phase circuit.

Asymmetrical Spacing

Practical transmission lines cannot maintain symmetrical spacing of conductors.

In asymmetrical spacing, with balanced currents the voltage drop due to line inductance will be unbalanced.

• For balanced 3 phase currents with Ia as reference,

Ib = IaL240o = a2Ia

Ic = IaL120o = aIa

• Where a =1L120o and a2 = 1L240o

Transpose Line

• Per phase model of the transmission line is required in power system analysis.

• Transposition consists of interchanging the phase configuration every 1/3 the length so that each conductor is moved to occupy the next physical position in a regular sequence.

• The inductance per phase can be obtained by finding the average value of

• Since a + a2 = 1L120o + 1L240o =

-1, the average becomes

• where

• GMD (Geometric mean distance) is equivalent conductor spacing.

• The error introduced as a result of this transposition is very small and suitable for modelling purpose.

Inductance of Composite Conductors• Solid round conductors were

considered before this.

• In practical transmission lines, stranded conductors are used

• A single phase line consisting of 2 composite conductors x and y

• The current is assumed to be equally divided among the subconductors (I/n in x and I/m in y) and refer to

• The inductance of subconductor n in x is

• The inductance of conductor x is

• Where

*

• where

Daa = Dbb……= Dnn = r'x

 • The inductance of conductor y

can also be similarly obtained with same GMD and GMRx ≠ GMRy

• Extra-high voltage transmission lines are usually constructed with bundled conductors.

• Typically, this conductors consist of 2, 3 or 4 subconductors symmetrically arrange and are separated by spacer dampers.

• Bundled configuration and the GMR, Ds of the equivalent single conductor is

• for 2 subconductor bundle

• for 3 subconductor bundle

• For 4 subconductor bundle

Advantages

• Relative immunity to short circuits caused by external forces (wind, fallen branches), unless they abrade the insulation.

• Can stand in close proximity to trees and will not generate sparks if touched.

• Simpler installation, as crossbars and insulators are not required.

• Less cluttered appearance.

• Can be installed in a narrower right-of-way.

• Reduced transmission losses, due to closer spacing of the conductors.

• Reduced corona loss, radio interference and surge impedance.

• At junction poles, insulating bridging wires are needed to connect non-insulated wires at either side.

Disadvantages

• Additional cost for the cable itself.• Insulation exposed to the sun

degrades. (The critical insulation between the wires is somewhat shielded from the sun.)

• Insulation thickness makes this economical only for low voltage power lines.