UNIT II

TRANSMISSION LINE PARAMETERS

OBJECTIVES

• To become familiar with different arrangements of conductors of a three phase single and double circuit transmission lines and to compute the GMD and GMR for different arrangements.

• To compute the series inductance and shunt capacitance per phase, per km

of a three phase single and double circuit overhead transmission lines with solid and bundled conductors.

• To become familiar with per phase equivalent of a three phase short and

medium lines and to evaluate the performances for different load conditions.

• (a) To become familiar with the theory of long transmission line and study

the effect of distributed parameters on voltage and currents, along the line, (b) calculate the surge impedance and surge impedance loading.

INTRODUCTION The Electric power generated in the generating station is transmitted with the help of transmission line which are normally overhead. The parameter associated with these transmission lines are inductance, capacitance, resistance and conductance. The performance of the transmission line is dependent on all these parameters. These parameters are uniformly distributed along the length of transmission line. The efficiency and voltage regulation whether it is good or poor is determined from these constants. For good electric design of transmission line, a sound knowledge about all these parameters is essential. Conductance between conductors or between conductors and the grounds account for the leakage current at the insulators of over head lines and through the insulation of cables. The leakage at the insulators of over head lines is negligible, the conductance of the over head line is assumed to be zero.

Line Parameters

Transmission line has four electrical parameters - resistance, inductance, capacitance and conductance. The inductance and capacitance are due to the effect of magnetic and electric fields around the conductor. The shunt conductance characterizes the leakage current through insulators, which is very small and can be neglected. The parameters R, L and C are essential for the development of the transmission line models to be used in power system analysis both during planning and operation stages. While the resistance of the conductor is best determined from manufactures data, the inductances and capacitances can be evaluated using formula. The student is advised to read chapter 4 of ref [1] or any other text book before taking up the experiment. The transmission lines are represented by an equivalent circuit model with approximate circuit parameters on per phase basis. This model can be used to compute voltages, currents, power flows, efficiency and regulation etc. Normally the lines are classified into short, medium and long lines for the purpose of modeling. 2 Inductance

The inductance is computed from flux linkage per ampere. In the case of the three phase lines, the inductance of each phase is not the same if conductors are not spaced equilaterally. A different inductance in each phase results in unbalanced circuit. Conductors are transposed in order to balance the inductance of the phases and the average inductance per phase is given by simple formulas, which depends on conductor configuration and conductor radius. General Formula

The general formula for computing inductance per phase in mH per km of a transmission is given by L = 0.2 lnDm/Ds (1.1)

where

Dm = Geometric Mean Distance (GMD)

Ds = Geometric Mean Radius (GMR)

The expression for GMR and GMD for different arrangement of conductors of the

transmission lines are given in the following section.

I. Single Phase - 2 Wire System

D

Fig. 1.1. Conductor arrangement

GMD = D (1.2)

GMR = re-1/4 = r’ (1.3) r = radius of conductor

II. Three Phase - Symmetrical Spacing:

GMD = D (1.4) GMR = re-1/4 = r’ (1.5) r = radius of conductor III. Three Phase - Asymmetrical Transposed:

A

DAB

DCA B C DBC

Fig. 1.3. Conductor Arrangement

GMD = Geometric mean of the three distances of the unsymmetrically placed conductors 3

= √ DAB DBC DCA (1.6)

Fig. 1.2. Conductor Arrangement

D

D D

. .

1-2

GMR = re-1/4 = r’ (1.7) r = radius of conductors Composite Conductor Lines

Composite conductor is composed of two or more elements or strands electrically in parallel. The expression derived for the inductance of composite conductors can be used for the stranded and bundled conductors and also for finding GMD and GMR of parallel transmission lines. Fig 1.4 shows a single phase line with two composite conductors.

Conductor X-with n strands Conductor Y with m strands

Fig. 1.4. Single Phase Line With Composite Conductor The inductance of composite conductor X., is given by

Lx = 0.2 ln GMD/GMRx (1.8)

where mn

GMD = √ (Daa’ Dab’ … Dam’) …… (Dna’ Dnb’….. Dnm’) (1.9)

n2

n2 GMRx = √ (Daa Dab … Dan) …… (Dna Dnb….. Dnn) (1.10) r’a = rae-1/4

The distance between elements are represented by D with respective subscripts and r’a , r’b and r’n have been replaced by Daa, Dbb …… and Dnn respectively for symmetry. Stranded Conductors:

The GMR for the stranded conductors are generally calculated using equation (1.10). For the purpose of GMD calculation, the stranded conductors can be treated as solid conductor and the distance between any two conductors can be taken as equal to as center-to-center distance between the stranded conductors as shown in Fig 1.5, since the distance between the conductors is high compared to the distance

c

b

a n

b’

a'

m'

c’

1-3

between the elements in a stranded conductor. This method is sufficiently accurate.

Fig. 1.5. Three Phase Line with Stranded Conductors

Bundle Conductors:

EHV lines are constructed with bundle conductors. Bundle conductors improves power transfer capacity and reduces corona loss, radio interference and surge impedance.

The GMR of a bundle conductor is normally calculated using (1.10). GMR for two subconductor Ds

b = √ Ds x d

GMR for three subconductor Dsb = (Dsxd2)1/3

GMR for four subconductor Dsb = 1.09 (Dsxd3)1/4

Where Ds is the GMR of each subconductor and d is the bundle spacing For the purpose of GMD calculation, the bundled conductor can be treated as a solid conductor and the distance between any two conductors can be taken as equal to center-to-center distance between the bundled conductors as shown in

DAB

DBC

DAC

A

B

C

d

d

d d d

d

d

d

Fig. 1.6. Examples of bundles

1-4

Fig 1.7, since the distance between the conductors is high compared to bundle spacing. A B C

Fig.1.7. Bundled conductor arrangement Three phase - Double circuit transposed: A three-phase double circuit line consists of two identical three-phase circuits. The phases a, b and c are operated with a1-a2, b1-b2 and c1-c2 in parallel respectively. The GMD and GMR are computed considering that identical phase forms a composite conductor, For example, phase a conductors a1 and a2 form a composite conductor and similarly for other phases. a1 S11 c2

H12 b1 S22 b2

H23 S33 a2

c1

Fig. 1.8. Conductor Arrangement

Relative phase position a1b1c1 –c2b2a2.

It can also be a1b1c1 – a2b2c2.

The inductance per phase in milli henries per km is

L = 0.2 ln (GMD/GMRL) mH/km.

(1.11)

DAB DBC

DAC

where

GMRL is equivalent geometric mean radius and is given by

GMRL = (DSA DSB DSC)1/3

(1.12)

where

DSA DSB and DSC are GMR of each phase group and given (refer 1.10) by

DSA = 4√( Dsb Da1a2 )2 = [Ds

b Da1a2]1/2

DSB = 4√ (Dsb Db1b2 )2 = [Ds

b Db1b2]1/2

(1.13)

DSC = 4√ (Dsb Dc1c2 )2 = [Ds

b Dc1c2]1/2

where

Dsb = GMR of bundled conductor if conductor a1, a2 …. are bundle conductor.

Dsb= ra1’ = rb1’ = rc1’ = ra’2 = rb’2 = rc’2 if a1, a2 ….. are not bundled conductor.

GMD is the “equivalent GMD per phase” & is given by

GMD = [DAB DBC DCA]1/3

(1.14)

where

DAB, DBC, & DCA are GMD between each phase group A-B, B-C, C-A which are

given by

DAB = [Da1b1 Da1b2 Da2b1 Da2b2]1/4 (1.15)

DBC = [Db1c1 Db1c2 Db2c1 Db2c2]1/4 (1.16)

DCA = [Dc1a1 Dc2a1 Dc2a1 Dc2a2]1/4

(1.17)

1.4.3 Capacitance

A general formula for evaluating capacitance per phase in micro farad per km of a

transmission line is given by

C = 0.0556/ln (GMD/GMR) µF/km

(1.18)

1-5

where

GMD is the “Geometric Mean Distance” which is the same as that defined for

inductance under various cases.

GMR is the Geometric Mean Radius and is defined case by case below:

(i) Single phase two wires system (for diagram see inductance):

GMD = D

GMR = r (as against r’ in the case of L)

(ii) Three phase - symmetrical spacing (for diagram see inductance):

GMD = D

GMR = r in the case of solid conductor

=Ds in the case of stranded conductor to be obtained from manufacturers

data.

1-6

(iii) Three-phase – Asymmetrical - transposed (for diagram see Inductance):

GMD = [DAB DBC DCA]1/3 (1.19)

GMR = r ; for solid conductor

GMR = Ds for stranded conductor

= rb for bundled conductor

where

rb = [r*d]1/2 for 2 conductor bundle

rb = [r*d2]1/3 for 3 conductor bundle (1.20)

rb = 1.09 [r*d3]1/4 for 4 conductor bundle

where

r = radius of each subconductor d = bundle spacing

(iv) Three phase - Double circuit - transposed (for diagrams see inductance):

C = 0.0556 / ln (GMD/GMRc) µF/km

GMD is the same as for inductance as equation (1.14).

GMRc is the equivalent GMR, which is given by

GMRc = [rA rB rC ]1/3 (1.21)

where

rA, rB and rC are GMR of each phase group obtained as

rA = [rb Da1a2]1/2

rB = [rb Db1b2]1/2

rC = [rb Dc1c2]1/2 (1.22)

where rb GMR of bundle conductor 1.4.4 Line Modelling and Performance Analysis

The following nomenclature is adopted in modelling: z = series impedance per unit length per phase y = shunt admittance per unit length per phase to neutral. L = inductance per unit length per phase C = capacitance per unit length per phase r = resistance per unit length per phase l = length of the line Z = zl = total series impedance Y = yl = total shunt admittance per phase to neutral. Short line Model and Equations (Lines Less than 80km)

The equivalent circuit of a short transmission line is shown in Fig.1.9 R X IR

Is IR

Vs VR Vs VR

Fig. 1.9 Short Line Model Fig.1.10 Two port representation of a Line In this representation, the lumped resistance and inductance are used for modelling and the shunt admittance is neglected. A transmission line may be represented by a two port network as shown in Fig 1.10 and current and voltage equations can be written in terms of generalised constants known as A B C D constants.

A B C D

Constants

For the circuit in Fig.1.9 the voltage and currents relationships are given by

Vs = VR + Z IR (1.23)

Is = IR (1.24)

In terms of A B C D constants VS A B VR (1.25) IS = C D IR

where

A = 1, B = Z, C = 0 D= 0. |VR (NL)| - |VR (FL) | Percentage regulation = -------------------------------------------- x 100 (1.26) |VR( FL)| Transmission efficiency of the line = Receiving end power in MW = PR (3Φ) (1.27) Sending end power in MW Ps (3Φ) Medium Line Model and equations (Lines above 80km):

The shunt admittance is included in this model. The total shunt admittance is divided into two equal parts and placed at the sending and receiving end as in Fig.1.11 Z=R+jX IR Is Vs Y/2 Y/2

VR

Fig.1.11 Nominal π Model The voltage current relations are given by Vs = (1+ ZY ) VR + ZIR (1.28) 2 Is = Y(1+ZY) VR + (1+ZY) IR (1.29)

4 2 In terms of ABCD constants Vs A B VR = (1.30) Is C D IR where

A = (1+ZY ), B=Z 2

C = Y(1+ ZY) , D=(1+ZY ) 4 2 The receiving end quantities can be expressed in terms of sending end quantities as VR D -B Vs (1.31)

= IR -C A Is Long line Model and Equations (lines above 250 km):

In the short and Medium lines, lumped line parameters are used in the model. For accurate modelling, the effect of the distributed line parameter must be considered. The voltage and current at any specific point along the line in terms of the distance x from the receiving end is given by V(x) =( VR + ZcIR) eγx ( VR - ZcIR ) e-γx 2 2 (1.32) I(x) = ((VR / Zc) + IR) eγx ((VR/ZC) - IR) e-γx (1.33)

2 2 In term of Hyperbolic functions

V(x) = VRcosh γx + ZcIR sinh γx

(1.34)

I(x) = (1/Zc) VRsinh γx + IR cosh γx (1.35)

where

Zc = √ z/y is called characteristic impedance

γ = √zy is called propagation constant

+

-

= α + jβ = √zy = √ (γ + jωL) (g + jωc)

α – is called attenuation constant

β – is called phase constant

The relation between sending and receiving end quantities is given by

Vs = VR cosh γl+ Zc IRsinh γl

IS = (VR/Zc) sinh γl + IRcosh γl (1.36)

The equivalent π model of the long line is given in Fig. 1.12.

Is IR

Z’ = Z sinh γl γl Y’/2 Y’/2 = (1/Zc) tanh (γl/2) Vs VR

Loss Less Line

For loss less line, the equations for the rms voltage and currents along the line is given by V(x) = VR cos βx + jZcIRsinβx (1.37) I(x) = j 1 VR sinβx + IRcosβx (1.38) Zc For open circuited line IR = 0 and the no load receiving end voltage is given by

VR(nl) = Vs / cos βl (1.39) For solid short circuit at the receiving end VR = 0, the equation (1.37) and (1.38)

reduces to

Vs = jZc IR sinβl

Is = IR cosβl

Fig. 1.12 Equivalent π model

1-10

For a loss less line the surge impedance (SIL)= √L/C

The load corresponding to the surge impedance at rated voltage is known as surge impedance loading (SIL) given by SIL = 3 VR IR* (1.40)

= 3 |VR|2 / Zc for lossless line Zc is purely resistive (1.41)

= |VL|2 MW

Zc (1.42)

VL = in k.v

1.5 EXERCISES:

1.5.1 A three-phase transposed line composed of one ACSR, 1,43,000 cmil, 47/7 Bobolink conductor per phase with flat horizontal spacing of 11m between phases a and b and between phases b and c. The conductors have a diameter of 3.625 cm and a GMR of 1.439 cm. The line is to be replaced by a three conductor bundle of ACSR 477,000-cmil, 26/7 Hawk conductors having the same cross sectional area of aluminum as the single-conductor line. The conductors have a diameter of 2.1793 cm and a GMR of 0.8839 cm. The new line will also have a flat horizontal configurations, but it is to be operated at a higher voltage and therefore the phase spacing is increased to 14m as measured from the centre of the bundles. The spacing between the conductors in the bundle is 45 cm.

(a) Determine the inductance and capacitance per phase per kilometer of the above two lines. (b) Verify the results using the available program.

(c) Determine the percentage change in the inductance and capacitance in (d) the bundle conductor system. Which system is better and why?

1.5.2 A single circuit three phase transposed transmission line is composed of four

ACSR 1,272,000 cmil conductor per phase with flat horizontal spacing of 14 m between phases a and b and between phases b and c. The bundle spacing is 45 cm. The conductor diameter is 3.16 cm.

a) Determine the inductance and capacitance per phase per kilometer of

the line.

b) Verify the results using available program.

1.5.3 A 345 kV double circuit three phase transposed line is composed of two ACSR,1,431,000 cmil, 45/7 bobolink conductors per phase with vertical conductor configuration as shown in Fig. 1.13. The conductors have a diameter of 1.427 in and the bundle spacing is 18 in.

a) Find the inductance and capacitance per phase per kilometer of the line.

b) Verify the results using the available program.

c) If we change the relative phase position to abc-a’b’c’, determine the inductance and capacitance per unit length using available program.

d) Which relative phase position is better and why?

a 11m c’ 7m 18” b 16.5m b’ 6.5m c 12.5m a’ Fig. 1.13

1.5.4 A 230 kV, 60 HZ three phase transmissions is 160 km long. The per phase resistance is 0.124Ω per km and the reactance is 0.497Ω per km and the shunt admittance is 3.30 x 10-6 ∟90o simens per km. It delivers 40MW at 220 KV with 0.9 power factor logging. Use medium line π model.

i. Determine the voltage and current at sending end and also compute

voltage regulation and efficiency.

ii. Verify the results using the available program

1.5.5 A three phase transmission line has a per phase series impedance of z=0.03 + j0.4 Ω per km and a per phase shunt admittance of y=j4.0 x 10-6 Simens per km. The line is 200 km long. Obtain ABCD parameters of the transmission line. The line is sending 407 MW and 7.833 MVAR at 350 kV. Use medium π model

i. Determine the voltage and current at receiving end and also compute

voltage regulation and efficiency.

ii. Verify the results using the available program

1.5.6 A three phase 50 Hz, 400 kV transmission line is 250 km long. The line parameters per phase per unit length are found to be

r = 0.032 Ω/km L = 1.06mH/km C = 0.011µF/km

Determine the following using the program available use long line model.

(a) The sending end voltage, current and efficiency when the load at the receiving end is 640 MW at 0.8 power factor logging at 400 kV.

(b) The receiving end voltage, current, efficiency and losses when 480 MW and 320 MVAR are being transmitted at 400 kV from the sending end. (c) The sending end voltage, current and efficiency and losses when the receiving end load impedance is 230 Ω at 400 kV. (d) The receiving end voltage when the line is open circuited and is energized with 400kV at the sending end. Also, determine the reactance and MVAR of a three phase shunt reactor to be installed at the receiving end in order to limit the no load receiving end voltage to 400 kV.

(e) The MVAR and capacitance to be installed at the receiving end for the loading condition in (a) to keep the receiving end voltage at 400 kV when the line is energized with 400 kV at the sending end. (f) The line voltage profile along the line for the following cases: no load, rated load of 800 MW at 0.8 power factor at sending end at 400 kV, line terminated in the SIL and short circuited line.

Problems

A conductor is composed of seven identical copper strands, each having a radius r, as shown in fig.1. find the self GMD of the conductor.

fig.1

2. The outside diameter of the single layer of aluminum strands of an ACSR conductor shown in fig .2. is 5.04cm. The diameter of each strand is 1.68cm. Determine the 50 Hz reactance at 1 m spacing; neglect the effect of the central stand of the steel and advance reason for the same.

fig .2. 3. The arrangement of conductors of a single – phase transmission line is shown in fig 3. wherein the forward circuit is composed of there solid wire 2.5 mm in radius and the return circuit of two – wires of radius 5mm placed of each side of the line and that of the complete line.

fig 3.

4.A three – phase, 50 Hz, 15 km long line has four No. 4/0 wires (1 cm dia) spaced horizontally 1.5m apart in a plane. The wires in order are carrying current Ia, Ib and Ic, and the fourth wire. This is a neutral, carries zero current. The currents are:

The line is un transposed.

(a) from the fundamental consideration, find the flux linkages of the neutral. (b) Find the voltage drop in each of the three – phase wires.

5. A single – phase 50Hz power line is supported on a horizontal cross – arm. The spacing between the conductors is 3 m. a telephone line is supported symmetrically below the power line as shone in fig 4. Find the mutual inductance between the two circuits and voltage induced per kilometer in the telephone line if the current in power line is l00 A assume the telephone line current to be zero.

fig 4 6. Find the inductive reactance in ohms per kilometer at 50Hz of a three phase bundled conductor line two conductors per phase as shown in fig.5 all the conductor are ACSR with radii of 1.725cm.

7.Calculate the loop inductance per km of a single phase line comprising of 2 parallel conductors

1m apart and 1 cm in diameter, when the material of conductor is

(i) Copper

(ii) Steel of relative permeability 50

8.Derive the equation for capacitance of a 3phase unsymmetrical spaced overhead lines.

Short Questions:

1. Define - Self and mutual – G.M.D. 2. What is meant by inductive interference? 3. What is transposition of conductor? 4. Write the expression for Inductance between Two Single Phase Conductors. 5. Write the expression for Inductance of Single Phase Line with Composite Conductor. 6. State why transposition of line conductor are needed. 7. List the factors that governing the capacitance of a transmission line. 8. Write the expression for Inductance Three Phase - Asymmetrical Transposed line. 9. What is fictitious conductor radius? 10. Define capacitor. 11. Write the expression for Inductance Three Phase - Symmetrical Spacing. 12. What do you understand by inductive interference? 13. Explain changing current of a transmission line. 14. Draw equivalent circuit and phasor diagram for short transmission line. 15. Write the expression for Inductance of Balanced Three Phase Line 16. Write the expression for Capacitance of a Single Phase Line. 17. Write the expression for Capacitance of Balanced Three Phase Line. 18. Write the expression for Inductance Three phase - Double circuit transposed. 19. What is electrostatic effect? 20. What is Electromagnetic effect?