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ELECTRICAL-EE POWER SYSTEM

ENGINEERS INSTITUTE OF INDIA28-B/7, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016. Ph. 011-26514888. www.engineersinstitute.com

1

ELECTRICAL ENGINEERING

STUDY MATERIAL

POWER SYSTEM



2

C O N T E N T

1. TRANSMISSION LINE MODEL AND PERFORMANCE …………………… 03-16

2. PERFORMANCE OF LINE ……………………………………………………… 17-37

3. PER UNIT SYSTEM ……………………………………………………………... 38-43

4. TRANSIENTS IN POWER SYSTEMS ……………………………….………… 44-55

5. SYMMETRICAL COMPONENT/FAULT ANALYSIS ………………………. 56-78

6. POWER SYSTEM STABILITY ………………………………………………… 79-104

7. LOAD FLOWS/ECONOMIC LOAD DISPATCH /

LOAD FREQUENCY CONTROL ……………………………………………… 105-131

8. CIRCUIT BREAKER & POWER SYSTEM PROTECTION ……………….. 132-154

9. CORONA / INSULATORS/CABLES ………………………………………….. 155-181

10. GENERATION ………………………………………………………………….. 182-220

11. PARCTICE SET IES WITH SOLUTION …………………………………….. 221-237

12. GATE QUESTION SET WITH SOLUTION …………………………………. 238-269



3

CHAPTER-1

TRANSMISSION LINE MODEL ANDPERFORMANCE

INTRODUCTION

An electric transmission line can be represented by a series combination of resistance, Inductance and

shunt combination of conductance and capacitance. These parameters are symbolized as R, L, G and

C respectively. Of these R and G are least important in the sense that they do not affect much the total

equivalent impedance of the line and hence the transmission capacity.

The effective resistance is equal to the d.c. resistance of the conductor only if the current is uniformly

distributed throughout the section of the conductor. The loss on the overhead line is due to

(i) ohmic loss in the power conductors, (ii) corona loss and (iii) leakage at the insulators

Magnetic Flux Density:

A current carrying conductor produces a magnetic field which is in the form of closed circular loops

around the conductor.

BH

H : Magnetic field intensity

B : Magnetic flux density/magnetic field

Inductors and Inductance:

An inductor is a device which stores energy in a form of magnetic field. By definition, the inductance

L of an inductor is the ratio of its total magnetic flux linkages to the current I through the inductor or

mNL

I I

Where Flux linkage in weber- turn

Above relationship is valid for a medium for which the permeability is constant. The permeability of

ferrous media is not constant and for such cases the inductance is defined as :

dL

dI

Magnetic Field Intensity due to a long current Carrying Conductor

The current is uniformly distributed across the section of the conductor. The flux linkages here will be

both due to internal flux and external flux. Cylinder with radius r < R

Cylinder with radius r < R Current flowing through the cylinder of ‘R’ is proportional to area of crossection is

2I R



4

Current flowing per unit area

2'

II

R

Current flowing in small cylinder

2

2rr

I IR

From ampere circuit law we have

.r rH dr Ir rH dr I

2r rH r I

2

2 22 2 2r

rI IrI r

Hr r R R

22r

IrH

R

Inductance of two-wire (1 – ) Transmission Line

Figure: Magnetic field due to one conductor of a 1 – transmission line

Internal flux linkages

22

IrH

R

B = H = o H=2

.2

oIr

R

(as r = 1) for conductors

d = B. Area normal to flux density B = B.dr.l

L= length of wire (Assume L = 1metre)



4


2'

II

R


2

2rr

I IR


.r rH dr Ir rH dr I

2r rH r I

2

2 22 2 2r

rI IrI r

Hr r R R

22r

IrH

R




22

IrH

R

B = H = o H=2

.2

oIr

R






4


2'

II

R


2

2rr

I IR


.r rH dr Ir rH dr I

2r rH r I

2

2 22 2 2r

rI IrI r

Hr r R R

22r

IrH

R




22

IrH

R

B = H = o H=2

.2

oIr

R






5

Flux linkages =flux no of turns

2

2

2

2 22o

rd d

R

I rd rdr

R R

Total internal flux linkages:

0

Rd 3

4 0 82

Ro oI Ir dr

R

External flux linkages

R r < D

The total external flux linkages due to current flow in one conductor

Limits are decided on the basis of distances taken between the surfaces of the conductors.

ln ( ; )2

D R

R

o

d

I DD R D R D

R

Total flux linkages due to one conductor = Total internal flux linkage + Total external flux linkages

8 2o oI I D

InR

Total flux linkage due to both the conductors = 28 2

o oI I DIn

R

Inductance L per unit length =4

o o DIn

R

Henry/metre

Since 74 10o ,

71 4 10D

L InR

Henry/meter

7

1/4

7 1/4 7 71/4

14 10 ln /

4

ln 1/ 4

4 10 ln ln 4 10 ln 4 10 ln /Re '

DHenry metre

R

Since e

D D DL e Henry metre

R R

' 0.7788 .R R

Flux Linkages of one conductor in a group of conductors:

Find out the flux linkages of one conductor due to current flowing in the conductor self and the

current flowing in the other conductors. It is assumed here that the sum of the currents in various

conductors is zero. Assume here that P is a point very far from the group of the conductors. The

objective here is to calculate the flux linkages of say conductor 1 due to the current I1, carried by the

conductor itself and flux linkage to conductor 1 due to the current carried by conductors 2, 3,……n.



6

Figure: Cross-sectional view of a group of n conductors

Point P is remote from the group of conductors.

Due to the current I1

1

10 1 0 11

18 2p

p

DI IIn

R

7 11 '

1

2 10 lnD p

IR

Due to current in conductor 2

2

271 2

12

2 10 . pp

DI In

D

Since I1 + I2 + ……..+In=0,

The net flux linkages 1p

71 1 2'

12 11

1 1 12 10 ........... /p n

n

I In I In I In wb turns metreD DR

Inductance of 3 – unsymmetrically spaced transmission line:

Figure: 3- unsymmetrically spaced transmission line:

a b c and each has a radius of R metres

7 1 1 12 10a a b cI In I In I In

R c b

7 1 1 12 10b a b cI In I In I In

c R a

7 1 1 12 10

'c a b cI In I In I Inb a R



6




1

10 1 0 11

18 2p

p

DI IIn

R

7 11 '

1

2 10 lnD p

IR


2

271 2

12

2 10 . pp

DI In

D

Since I1 + I2 + ……..+In=0,


71 1 2'

12 11

1 1 12 10 ........... /p n

n






R c b


c R a

7 1 1 12 10




6




1

10 1 0 11

18 2p

p

DI IIn

R

7 11 '

1

2 10 lnD p

IR


2

271 2

12

2 10 . pp

DI In

D

Since I1 + I2 + ……..+In=0,


71 1 2'

12 11

1 1 12 10 ........... /p n

n






R c b


c R a

7 1 1 12 10




7

aI As reference2

2

&

where ( 0.5 0.8666), ( 0.5 0.866)

b a c aI K I I KI

K j K j

In case the transmission line is transposed i.e., each conductor takes all the three position of the

conductors

3a b cL L L

L

7 71 1 1 3 32 10 3 ln1 2 10 ln Henry/metre

3 2

abcIn In j

R abc R

L 72 10 lnd

R

Henry/metre

CONCEPT OF GEOMETRIC MEAN DISTANCE

Geometric mean distance is a mathematical concept used for the calculation of inductance.

On the circle is 1 2 3 4 5GMDp=5 D D D D D

Numbers of points on the circle are increased to infinity, the distance between the point P and centre

of the circle.

The GMD between two circular areas will be the distance between the centres of the two areas and so

on. For voltages in excess of 230 kV. It is preferable to use more than one conductor per phase which

is known as bundling of conductors. A bundle conductor is a conductor made up of two or more sub-

conductors and is used as one phase conductor.

The advantages in using bundle conductors

(i) Reduced reactance

(ii) Reduced voltage gradient

(iii) Reduced corona loss

(iv) Reduced radio interference

(v) Reduced surge impedance

The self GMD of the conductors is increased.

Reactance=K InGMD

GMR

Since the voltage gradient is reduced by using bundled conductors, the radio interference is also

reduced.

Surge impedance = /L C . Since by bundling, the self GMD is increased, the inductance is reduced

and capacitance increased, as a result the surge impedance is reduced. The maximum power that can

be transmitted is increased.

The basic difference between a composite conductor and bundled conductor is that the sub-

conductors of a bundled conductor are separated from each other by a distance of almost 30 cms or

more and the wires of a composite conductor touch each other.



8

Inductance of Composite Conductors:

The current is assumed to be equally divided amongst the strands. One group of conductors act as a

'go' conductor for the single-phase line and the other as the 'return'. The current per strand is I/m

ampere in one group and I/n ampere in the other.

Figure: Inductance of composite conductor -1- transmission line

1 2 ...... mav

L L LL

m

Since all the strands of conductor A are electrically parallel, the inductance of conductor will be

1 22

......av mA

L L L LL

m m

Substituting the values of L1, L2, ………, Lm in equation

2

11 12 1 21 22 22 1 27

12 13 1 21 23 2 1 2

( ' ' ..... )( ' ' ..... ' ).....( ' ' ..... ' )2 10 ln

( ' ..... )( ' ..... ).....( ' ..... )

m nn m m mn

A mm m m m mn

D D D D D D D D DL

R D D D R D D D R D D D

The mnth root of the product of the mn distances between m strands of conductor A and n strands of

conductor B is called geometric mean distance (GMD) and is denoted as Dm and the m2th root of m2

distance i.e., the distance of the various strands from one of the strands and the radius of the same

strand, geometric mean radius (GMR) or self GMD.

72 10 /mA

s

A B

DL In Henry metre

D

L L L

INDUCTANCE OF DOUBLE CIRCUIT 3– LINE

Conductors and are electrically parallel and constitute one phase. The conductors of two

phases are placed diagonally opposite rather than in the same horizontal plane, in all the three

positions. By doing this the self GMD of the conductors is increased whereas the GMD reduced,

thereby the inductance per phase in lowered.

Transposition of Power Lines

Figure: Transposition of conductors



8






1 2 ...... mav

L L LL

m


1 22

......av mA

L L L LL

m m


2

11 12 1 21 22 22 1 27

12 13 1 21 23 2 1 2

( ' ' ..... )( ' ' ..... ' ).....( ' ' ..... ' )2 10 ln

( ' ..... )( ' ..... ).....( ' ..... )

m nn m m mn

A mm m m m mn

D D D D D D D D DL






72 10 /mA

s

A B

DL In Henry metre

D

L L L










8






1 2 ...... mav

L L LL

m


1 22

......av mA

L L L LL

m m


2

11 12 1 21 22 22 1 27

12 13 1 21 23 2 1 2

( ' ' ..... )( ' ' ..... ' ).....( ' ' ..... ' )2 10 ln

( ' ..... )( ' ..... ).....( ' ..... )

m nn m m mn

A mm m m m mn

D D D D D D D D DL






72 10 /mA

s

A B

DL In Henry metre

D

L L L










9By transposition of conductors is meant the exchanging of position of the power conductors at regular

intervals along the line, so that each conductor occupies the original position of every other conductor

over an equal distance.

If the spacing is unsymmetrical even though the system operates under balanced condition, voltage

drops of different magnitude will be there in the three conductors due to unequal inductance of the

three phases. The magnetic field external to the conductors is not zero, thereby causing induced

voltages in adjacent electrical circuits, particularly telephone lines, that may result in telephone

interference.

It is enough to transpose either power line or the communication lines. Under balanced operating

condition, the magnetic field linking an adjacent telephone line is shifted 120° in time phase with

each rotation of the conductor positions in the net voltage induced in the telephone line is zero as it is

the sum of three induced voltage which are displaced by 120° in time phase.

The transposition, however, maybe effected at the intermediate switching station.

Composite Conductors

For transmission lines operating at high voltages normally stranded conductors are used. These

conductors are known as composite conductors as they compose of two or more elements or strands

electrically in parallel. By using different proportion of steel and aluminium strands different tensile

and current carrying capacity conductors can be obtained.

Steel cored, reinforced aluminium conductors (ACSR) which combine the lightness, electrical

conductivity and restlessness of aluminum with the high tensile strength of steel

1. Aluminium conductors steel reinforced cheaper

2. The superior mechanical strength

3. A reduction in the number of supports

4. The increase in span length

5. Corona losses reduced

STRANDED CONDUCTOR

Steel is used at the centre to increase the tensile mechanical strength. Spiraling has done to increase the mechanical strength due to spiraling the length of the conductor

increases so that resistance also increases as

lR

A

Note: 2. ' ' 3 3 1no of Strands N n n For n = 1N = 3 – 3 + 1 = 1



10

D = dFor n = 2N = 3 4 – 3 2 + 1 = 7

D = (2 2 – 1) = 3dWhere D = Conductor overall diameter

d = diameter of each conductorn = layer number

and (2 1)D n d

SKIN AND PROXIMITY EFFECT

When direct current flows in the conductor, the current is uniformly distributed across the section of

the conductor whereas flow of alternating current is non-uniform, with the outer filaments of the

conductor carrying more current than the filaments closer to the centre.

A higher resistance to alternating current than to direct current and is commonly known as skin effect.

This effect is more; the more is the frequency of supply and the size of the conductor.

The flux linkages per ampere to inner strands is greater than those of outer strands. Hence the

inductance/impedance of the inner strands is greater than those of outer strands which results in more

current in the outer strands as compared to the inner strands. This non-uniformity of flux linkage is

the main cause of skin effect. The alternating magnetic flux in a conductor caused by the current

flowing in a neighboring conductor gives rise to circulating currents which cause an apparent

increased in the resistance of a conductor. This phenomenon is called proximity effect. In a two-wire

system more lines of flux link elements farther apart than the elements nearest each other. Therefore,

the inductance of the elements farther apart is more as compared to the elements near each other and

the current density is less in the elements farther apart than the current density in the elements near

each other. The effective resistance is, therefore, increased due to non-uniform distribution of current.

The proximity effect is pronounced in case of cables where the distance between the conductors is

small whereas for overhead lines with usual spacing the proximity effect is negligibly small.



11Explanation of Skin effect

From figure ‘b’. As we know that.N

LI

and flux at the centre is due to the current flowing

in each small cylindrical conductor so that flux is maximum at the centre i.e.

max at centre

maximum at centreL

Since max max2 2L LX L L X at centre and currentL

IX

i.e. current is minimum

at the centre.

So the effective area is reduces and thus ACl

RA

i.e. resistance is increases and

2

ACd f

R

Whered = diameter of conductorf = frequency of supply

= permeability and resistivity

Thus the current flows in the effective area as observed by figure ‘a’ i.e. d2 so area reduced to

2 2( ).D d

CAPACITANCE OF TRANSMISSION LINES

PRODUCTION

The flow of current through a conductor gives rise to a magnetic field and charging of conductor

results in an electric field. A charge if brought in the vicinity of this electric field experiences a force

as electric field intensity E. Newton per coulomb or volts per metre.

02L

rEr

2

0 12L r

V Inr



12

CAPACITANCE OF 1 – TRANSMISSION LINE

The charge L coulomb/metre is distributed on the surface of the conductor which is non-uniformly

distributed over the surface such that it has higher density on the adjacent sides of the conductors.

Operating voltage V, distance of separation h and radius of the equipotential surface r.

0

0 //

L hV In

r

C F metreln h r

Equation for inductance contains a constant term corresponding to the internal flux linkages whereassince charges reside on the surface of the conductor, similar term is absent in the capacitanceexpression.The concept of self GMD is applicable for inductance calculation and not for theCapacitance.The capacitance between one conductor and a neutral point

Figure

022

lnan abC C

h

r

Capacitance of A 3-phase unsymmetrically spaced transmission line

For an untransposed line the capacitances between conductor to neutral of the three conductors are

unequal. In transposed lines the average capacitance of each conductor to neutral is the same as the

Figure: Unsymmetrically spaced transposed 3-phase transmission line

capacitance to neutral of any other phase.

3-phase balanced system, a reference charge,

'

0

1ln ln ln

2a b c

a a b cD D D

Vr c b

"

0

0

1ln ln ln

2

1ln ln ln

2

b c aa a b c

c a ba a b c

D D DV

r a c

D D DV

r b a



12





0

0 //

L hV In

r

C F metreln h r


Figure

022

lnan abC C

h

r







'

0

1ln ln ln

2a b c

a a b cD D D

Vr c b

"

0

0

1ln ln ln

2

1ln ln ln

2

b c aa a b c

c a ba a b c

D D DV

r a c

D D DV

r b a



12





0

0 //

L hV In

r

C F metreln h r


Figure

022

lnan abC C

h

r







'

0

1ln ln ln

2a b c

a a b cD D D

Vr c b

"

0

0

1ln ln ln

2

1ln ln ln

2

b c aa a b c

c a ba a b c

D D DV

r a c

D D DV

r b a



13

Average' '' '''

3a a

aV a V V

V

0a b c

3

0

ln2

aa

abcV

r

0

ln2

a GMD

r

02/ metre

ln

a

a

C FGMDV

r

For a symmetrical spacing of the conductors,

a = b = c = h

02C

hIn

r

Effect of Earth on the Capacitance of Conductors

The electric flux lines and the equipotential lines are orthogonal to each other. The earth is considered

to be conducting and an equipotential plane of infinite extent. The positive charge on the conductor

induces negative charges on the earth surface. This distribution of charge on the surface of the earth

should be replaced by an equivalent charge for the calculation of electric field potential and other

related quantities due to this isolated charged conductor.

Since earth is an equipotential plane which is possible only if we assume the presence of an imaginary

conductor below the surface of the earth at a depth equal to the height of the actual conductor above

the surface of the earth.

Capacitance of single conductor

The single conductor with the earth is equivalent to a single-phase transmission line is :

0

ab aa

ab

CD D

Inr D

Ratio aa

ab

D

D

<1, the effect of earth on capacitance of the system is to increase it.



13

Average' '' '''

3a a

aV a V V

V

0a b c

3

0

ln2

aa

abcV

r

0

ln2

a GMD

r

02/ metre

ln

a

a

C FGMDV

r


a = b = c = h

02C

hIn

r












0

ab aa

ab

CD D

Inr D

Ratio aa

ab

D

D




13

Average' '' '''

3a a

aV a V V

V

0a b c

3

0

ln2

aa

abcV

r

0

ln2

a GMD

r

02/ metre

ln

a

a

C FGMDV

r


a = b = c = h

02C

hIn

r












0

ab aa

ab

CD D

Inr D

Ratio aa

ab

D

D




14The effect of earth is to increase the capacitance.

Explanation of Proximity Effect:The flux linkage due to adjacent conductor the current distribution is uneven or irregular in conductorthis effect is called proximity effect.

Due to proximity effect AC DCR R

Proximity and Skin Effect depends upon:1. Size of the conductor2. Supply frequency3. Distance between the conductor4. Conductivity ‘’5. Permeability ‘’

Note: Due to skin effect the effective area of cross section of the conductor decreasesNote: In case of cables the proximity effect is high as compare to the overhead transmission line.

Question-1:Find the G.M.D. and G.M.R. of the given symmetrical configurationSolution:

1/3. . . ( )

(0.7788 2) 1.56

(0.7788 2) 1.56

(0.7788 2) 1.56

R B

R

y

B

G M R GMR GMRyGMR

GMR

GMR

GMR

3 1/ 3[(0.7788 2) ] 0.7788 2 1.56GMR cm

And1/3( )R y BGMD GMD GMD GMD

1/ 2[6 6] 6RGMD 1/ 2[6 6] 6yGMD 1/ 2[6 6] 6BGMD



151/ 3[6 6 6] 6GMD m

Note: Self-GMD = Self GMD of a phase when all the phases have equal no of sub conductors withdistance between conductors and radius of the conductor being equal Self GMD of a phase is equal to self GMD of one sub-conductors of a phase when the distance

between sub-conductors is same and radius of sub-conductor is same. GMD depends upon the radius of conductor and independent of the distance between the phases GMD is independent upon the radius of conductor

Question: Find GMR and GMD

Solution:

1 2

1/2(0.7788 1.5) 80R RGMR GMR 1/2[0.7788 1.5 80] 9.66R Y BGMR GMR GMR cm

And mutual 1/3( )R Y BGMD MGMD MGMD MGMD 1/ 36 6 6 6 6 6 = 6m

Question: Ds is the GMR of each subconductor of a four subconductor bundle conductor and d is thebundle spacing. What is the GMR of the equivalent-single conductor?

[IES-2004]

(a) 31.09 sD d

(b) 3 31.09 sD d

(c) 3 341.09 sD d

(d) 341.09 sD d

Solutions:

As DS is the GMR of each sub conductor

Equivalent 1/ 4( )SGMR D d d d 341.09 SD d

electrical engineering - engineers · pdf fileconcept of geometric mean distance ... gmd gmr...

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