UNIT 1 INTRODUCTION 1. What are the advantages of per unit ...

Vel Tech High Tech Dr.Rangarajan Dr.Sakunthala Engg. College EE6501 – Power System Analysis (R-2013)

Prepared by Prabaakaran K AP/EEE Page No.1

UNIT – 1 INTRODUCTION

1. What are the advantages of per unit computation? (A/M 2017)

Manufacturers usually specify the impedance of a device or machine in per unit on the base

of the nameplate rating.

(ii) The p.u values of widely different rating machines lie within a narrow range, even

though the ohmic values has a very large range.

(iii) The p.u. impedance of circuit element connected by transformers expressed on a proper

base will be same if it is referred to either side of a transformer.

The p.u. impedance of a 3-phase transformer is independent of the type of winding

connection (Y or ∆).

2. A Y connected generated rated at 300 MVA, 33kV, has a reactance of 1.24 p.u. Find the

ohmic value of the reactance. (A/M 2017)

Z p.u. = 𝑍𝑎𝑐𝑡

𝑍𝑏𝑎𝑠𝑒 = > 1.24 = Zact / Zbase => Zact = 1.24 x Zbase.

Zbase = KV2 / MVA => 332 / 300 = 3.63

Zact = 1.24 x 3.36 = 4.5 Ω

3. How are the loads represented in impedance and reactance diagram. (N/D 2016)

The resistive and reactive loads can be represented by any one of the following representation.

i. Constant power representation.

Load power, S = P + JQ.

ii. Constant current representation.

Load current. , I = √𝑃2+ 𝑄2

|𝑉|∠𝛿 − 𝜃

iii. Constant impedance representation.

Load current, Z = |𝑉|2

𝑃−𝑗𝑄

4. Define per unit value of an electrical quantity and write the equation for the base

impedance for a three phases power system. (A/M 2016)

The per unit value of any quantity is defined as the ratio of the actual value of the quantity to the

base value expressed as a decimal. The base value is an arbitrary chosen value of the quantity.

Per unit Value = 𝐴𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒

𝐵𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒

Base Impedance

The equation for base impedance is given as follows

𝐵𝑎𝑠𝑒𝐼𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 = (𝐵𝑎𝑠𝑒𝐾𝑉)2

𝐵𝑎𝑠𝑒𝑀𝑉𝐴



5. Write the equation for per unit impedance if change of base occurs. (A/M 2016)

The equation for converting per unit impedance form one base to another can be given as

follows.

Zp.u. new = Zpu,oldx𝑘𝑉𝑏,𝑜𝑙𝑑

2

𝑘𝑉𝑏,𝑛𝑒𝑤2 x

𝑀𝑉𝐴𝑏,𝑛𝑒𝑤

𝑀𝑉𝐴𝑏,𝑜𝑙𝑑

6. Draw the simplified model of a generator. (N/D 2015)

7. Distinguished between impedance and reactance diagram. (N/D 2015)

Impedance Diagram :

The power system components are represented by their approximate or simplified equivalent

circuits. It is used in load flow analysis

Reactance Diagram :

The power system components are represented by their reactance. The reactance diagram is

obtained from impedance diagram if all the resistive components are neglected. The reactance

diagram is used for fault analysis.

8. What is single line diagram? (N/D’11,A/M’10,N/D’07)

A Single line diagram is diagrammatic representation of power system in which the

components are represented by their symbols and the interconnections between them are shown by

a single straight line (even though the system is 3- phase system). The ratings and the

impedances of the components are also marked on the single line diagram.

9. What is the need for power system analysis in planning and operation of power

system?(M/J’07)

The system operation engineers at the various points like area, space, regional and national load

dispatch deals with the dispatch of power for maintaining the power balance effective and

efficient manner.

Power balance equation is

PD=∑ PGiNi=1 , i=1,2,3……….N

Total demand = Sum of the real power generation.

i.e, the generation should be in such a way that it meet out the required demand. When this

relation is satisfied, it gives good economy and security.



10. Name the diagonal and off diagonal elements of Bus Admittance and Impedance Matrix.

(N/D 2013)

Bus Admittance Matrix.

The diagonal elements of bus admittance matrix are called self admittances of the Buses and off

diagonal elements are called mutual admittances of the buses.

Bus Impedance Matrix.

The diagonal elements of bus impedance matrix are called driving point impedances of the buses

and off diagonal elements of bus impedance matrix are called transfer Impedances of the buses.

11. What are the functions of power system analysis? (N/D 2013)

To monitor the voltage at various buses, real and reactive power flow between buses.

To design the circuit breakers.

To plan future expansion of the existing system

To analyze the system under different fault conditions

To study the ability of the system for small and large disturbances (Stability studies)

12. Draw the impedance diagram of the given system. (M/J 2014)

13. List out the application of Y-Bus. (N/D’12)

The application of Y-bus matrix are as follows.

Y-bus is used in solving load flow problems.

It has gained applications owing to the simplicity in data preparation.

It can be easily formed and modified for changes in the network.

It reduces computer memory and time requirements because of sparse matrix

.

14. Why bus admittance matrix is preferred in load flow?

Bus admittance matrix is preferred in load flow problem because,

It is easy to formulate.

No need of taking inverse.

Computation time is less.

Matrix is symmetric, so calculation of upper or lower triangular matrix is enough.

Each bus is connected to ony a few nearby buses.so many diagonal elements are zero.



15. A generator rated at 30MVA, 11KV has a reactance of 20%.Calculate its per unit

reactance for a base of 50 MVA and 10KV.

MVA new = 50 ; KV new = 10 ; MVA old = 30 ; KV old = 11

X p.u = 20% = 20/100 = 0.2 p.u

𝑋𝑝.𝑢,𝑛𝑒𝑤 = 𝑋𝑝.𝑢,𝑜𝑙𝑑 × [𝐵𝑎𝑠𝑒𝐾𝑉𝑜𝑙𝑑

𝐵𝑎𝑠𝑒𝐾𝑉𝑛𝑒𝑤]

2

× [𝐵𝑎𝑠𝑒𝑀𝑉𝐴𝑛𝑒𝑤

𝐵𝑎𝑠𝑒𝑀𝑉𝐴𝑜𝑙𝑑]

𝑋𝑝.𝑢,𝑛𝑒𝑤 = 𝑗0.2 × [11

10]

2

× [50

30] = 𝑗0.4033 𝑝. 𝑢

16. What is the need for base values?

The components of various sections of power system may operate at different Voltage and

power levels. It will be convenient for analysis of power system if the voltage, power, current

and impedance ratings of power system components are expressed with reference to a

common value called base value. Then all the voltages, power, current and impedance ratings of

the components are expressed as a percent or per unit of the base value.

17. What is the need for short circuit study.

To determine the current interrupting capacity of the circuit breakers so that the

faulted equipments can be isolated.

To establish the requirements and settings to detect the fault and cause the circuit breaker to

operate when the current flowing through it exceeds the max. value.

UNIT – 2 POWER FLOW ANALYSIS

1. Compare Newton Raphson and Gauss Seidal methods of load flow solutions. (A/M 2017)

Gauss Seidal method Newton Raphson method

Variable is expressed in rectangular

coordinates.

Variables are expressed in polar

coordinates.

Computation time per iteration is less Computation time per iteration is more.

It has linear convergence

characteristics.

It has quadratic convergence characteristics.

The number of iterations required for

convergence increases with size of the

system.

The numbers of iterations are independent of

the size of the system.

The choice of slack bus is critical. The choice of slack bus is arbitrary.

2. Write the quantities that are associated with each bus in a system.(A/M 2017)

Each bus in a power system is associated with four quantities and they are

i. real power

ii. reactive power

iii. magnitude of voltage, phase angle of voltage.



3. What is Jacobian Matrix? (N/D 2016)

The matrix formed from the first derivatives of load flow equation is called jacobian

matrix and it is denoted by J.

The elements of jacobian matrix will change in every iteration .the elements of the jacobian

matrix are obtain matrix are obtained by partially differentiating the load flow equation with respect

to a unknown variable and then evaluating the first derivative as using the solution of previous

iteration .

4. Write the need for slack bus in load flow analysis. (N/D 2016)

The slack bus is needed to account for transmission line losses. In a power

system the total power generated will be equal to sum of power consumed by loads and losses. In a

power system only the generated power and load power are specified for buses. The slack bus is

assumed to generate the power required for losses. Since the losses are unknown the real and reactive

power are not specified for slack bus. They are estimated through the solution of load flow equations.

5. What is the need for load flow analysis? (A/M 2016)

The study of various methods of solution to power system network is referred to as load flow

study. The solution provides the voltages at various buses, power flowing in various lines and line-

losses.

The load flow study of a power system is essential to decide the best operation of existing

system and for planning the future expansion of the system. It is also essential for designing a new

power system.

6. Mention the various types of buses in power system with specified quantities for each bus.

(A/M 2016)

The buses of a power system can be classified into three types based on the

quantities being specified for the buses, which are as follows:

Load bus or PQ bus (P and Q are specified)

Generator bus or voltage controlled bus or PV bus (P and V are specified)

Slack bus or swing bus or reference bus (|V| and δ are specified)

7. Write the static load flow equation. (N/D 2015)

Pi = ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗|𝑐𝑜𝑠(𝜃𝑖𝑗 + 𝛿𝑗𝑁𝑗=1 −𝛿𝑖)

Qi = ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗|𝑠𝑖𝑛(𝜃𝑖𝑗 + 𝛿𝑗𝑁𝑗=1 −𝛿𝑖)

8. State the role of acceleration factor in GS method. (N/D 2015)

In Gauss Seidal method, the number of iteration required for convergence can be reduced if the

voltage computed at each iteration is multiplied by a factor greater than unity called acceleration factor

to bring the voltage closer to the value to which it is converging. The range of 1.3 to 1.6 is found to be

satisfactory for the typical systems.

𝑉𝑖𝑛𝑒𝑤 = Vi

old + ∝ [𝑉𝑖𝑛𝑒𝑤 − Vi

old ]

Where Viold = Voltage value obtained in previous iteration



𝑉𝑖𝑛𝑒𝑤 = New value of Voltage value obtained in current iteration

∝ = Acceleration factor

9. Why the load flow studies are important for planning the existing system as well as its

future expansion?

The load flow studies are very important for planning, economic scheduling, control and

operation of existing systems as well as planning its future expansion depends upon knowing the

effect of interconnections, new loads, new generating stations, or new transmission lines, etc.

before they installed.

10. What is swing bus? (N/D 2013)

A bus is called swing bus when the magnitude and phase for bus voltage are specified for it.

The swing bus is the reference bus for load flow solution and it is required for accounting line

losses. Usually one of the generator bus selected as the swing bus.

Swing bus is also called as Slack bus.

11. When the generator buses are treated as load bus? (N/D’13, M/J’14)

If the reactive power constraints of a generator bus violates the specified limits then the generator

is treated as load bus.

If Qi

> Qi(max), substitute Qi = Qi(max)

If Qi cal < Qi(min) , substitute Qi= Qi(min)

12. What is meant by flat voltage start?

In iterative methods of load flow solution, the initial voltages of all buses exept slack bus are

assumed as 1+j0 pu. This is referred as flat voltage profile.

V = |𝑉𝑠𝑝𝑒𝑐| 0 for slack bus

V = |𝑉𝑠𝑝𝑒𝑐| 0 for generator bus

V = 1 0 for load bus

13. What are the works involved in a load flow study? ( Or) How a load flow study is

performed?

The following work has to be performed for a load flow study.

Representation of the system by single line diagrams.

Determining the impedance diagram using the information in single line diagram.

Formulation of network equations.

Solution of network equations.

14. List the quantities specified and the quantities to be determined from load flow study

for various types of buses.



The following table shows the quantities specified and to be obtained for various types of

buses.

Bus type Quantities specified Quantities to be obtained

Load bus P,Q V, δ

Generator bus P, V Q, δ

Slack bus V, δ P,Q

15. Why bus admittance matrix is used in load flow instead of bus impedance matrix? (M/J

2014)

Using bus admittance matrix is amenable to digital computer analysis, because it could be

formed and modified for network changes in subsequent cases. Bus admittance matrix is used in Gauss

seidal method because of the following reasons.

It requires less computation time

Less memory allocation

16. What are the advantages of Gauss seidal method?

The advantages of Gauss seidal method are as follows

i. Calculations are simple and so the programming task is less.

ii. The memory requirement is less.

iii. Useful for small systems

17. What are the disadvantages of Gauss seidal method?

The disadvantages of Gauss seidal method are listed as follows

i. Requires large no. of iterations to reach converge

ii. Not suitable for large systems.

iii. Convergence time increases with size of the system

18. What are the advantages of Newton-Raphson method?

The advantages of Newton-Raphson method are,

i. This load flow method is faster, more reliable and he results are accurate.

ii. Requires less number of iterations for convergence.

iii. The number of iterations are independent of the size of the system.

iv. Suitable for large size systems.

19. What are the disadvantages of Newton-Raphson method? (N/D’11)

The disadvantages of Newton-Raphson method are,

i. Programming is more complex.

ii. The memory requirement is more.

iii. Computational time per iteration is higher due to larger number of

calculations per iteration.



20. How the disadvantages of N-R method are overcome?

The disadvantage of large memory requirement can be overcome by decoupling the weak

coupling between P-δ and Q-V. (i.e., using decoupled load flow algorithm). The

disadvantage of large computational time per iteration can be reduced by simplifying the

decoupled load flow equations. The simplifications are made based on the practical operating

conditions of a power system.

UNIT 3 – SYMMETRIC FAULT ANALYSIS

1. What is the significance of sub transient reactance and transient reactance in short

circuit studies? (A/M 2017)

The sub-transient reactance can be used to estimate the initial value of fault current

immediately on the occurrence of the fault. The maximum momentary short circuit

current rating of the circuit breaker used for protection or fault clearing should be less

than this initial fault current.

2. For a fault at a given location, rank the various fault in the order of severity.

(A/M 2017)

Types of fault Relative frequency of occurrence of faults

Three phase fault 5%

Double line to ground fault 10%

Line to Line fault 15%

Line to ground fault 70%

3. How shunt and series fault are classified ? (N/D 2016)

Shunt faults - due to short circuits in conductors

Series faults - due to open conductors.

The shunt fault is also called as short circuit faults.

4. What is the need for short circuit study? (N/D 2016)

The short circuit studies are essential in order to design or develop the protective

schemes for various parts of the system. The protective scheme consists of current &

voltage sensing devices, protective relays and circuit breakers. The selection (or proper

choice) of these device mainly depends on various currents that may flow in the fault

conditions.

5. State and explain symmetrical fault. (A/M 2016)

The fault on the power system which gives rise to symmetrical fault currents (i.e. equal

fault current in the lines with 1200 displacement) is called as symmetrical faults. The

symmetrical fault occurs when all the three conductors of phase are brought together

simultaneously in short circuit condition.



6. What is bolted fault or solid fault? (A/M 2016)

A Fault represents a structural network change equivalent with that caused by the

addition of impedance at the place of a fault. If the fault impedance is zero, the fault is

referred as bolted fault or solid fault.

7. Define short circuit capacity. (N/D 2015)

Short circuit capacity or short circuit MVA of fault level at a bus is defined as the

product of the magnitudes of the perfect bus voltage and the post fault current.

SCC = |𝐸𝑇ℎ||𝐼𝑓| = |𝐸𝑇ℎ|2

𝑋𝑇ℎ p.u. MVA

8. What is meant by a fault?

A fault in a circuit is any failure which interferes with the normal flow of current. The

faults are associated with abnormal change in current, voltage and frequency of the power

system. The faults may cause damage to the equipments if it is allowed to persist for a long

time. Hence every part of a system has been protected by means of relays and circuit

breakers to sense the faults and to isolate the faulty part from the healthy part in the event

of fault.

9. Why fault occurs in a power system?

In a power system the fault may occur due to the following reasons.

Insulation failure of equipments.

Flashover of lines initiated by a lightning stroke.

Switching surges.

Sudden releasing of heavy loads at same instant of time.

Due to accidently faulty operation of power system operators.

10. What are the objectives of short circuit analysis.

To check the MVA ratings of the existing circuit breaker’s, when new generation are

added into system.

To select the rating for fuses, circuit breaker’s and switch gear in addition to set up of

protective relays.

To determine the magnitudes of currents flowing through out the power system at

various time intervals after a fault occurs.

11. State the applications of short circuit analysis.

Short circuit studies are used

For proper relay setting and co ordination.

To obtain the rating of the protective switch gears.

To select the circuit breakers.

To perform whenever system expansion is planned.



12. What are symmetrical components?

An unbalanced system of N related vectors can be resolved into N systems of balanced vectors

called symmetrical components. The various symmetrical components are as follows.

Positive sequence components

Negative sequence components

Zero sequence components

13. Why delta connected load will not have any zero sequence components

The current in the neutral is three times the zero sequence line current. A delta connected

load provides no path to neutral and hence the line currents flowing to a delta connected load

contain zero component. Hence the delta connected load will not have any zero sequence

components.

14. What is symmetrical components of three phase system.

Symmetrical components of three phase system is given as

For voltage

[𝑉𝑎

𝑉𝑏

𝑉𝑐

] = [1 1 11 𝑎2 𝑎1 𝑎 𝑎2

] [𝑉𝑎0

𝑉𝑎1

𝑉𝑎2

]

For current.

[𝐼𝑎

𝐼𝑏

𝐼𝑐

] = [1 1 11 𝑎2 𝑎1 𝑎 𝑎2

] [𝐼𝑎0

𝐼𝑎1

𝐼𝑎2

]

15. What is meant by doubling effect?

If a symmetrical fault occurs when the voltage wave is going through zero then the

maximum momentary short circuit current will be double the value of maximum

symmetrical short circuit current. This effect is called doubling effect.

16. What is synchronous reactance?

The synchronous reactance is the ratio of induced emf and the steady state rms current (i.e.

it is the reactance of a synchronous machine under steady state condition). It is the sum of

leakage reactance and the reactance representing armature reaction.

It is given by,

Xs = Xl + Xa

Where,

Xs = Synchronous reactance

Xl = Leakage reactance Xa = Armature reaction reactance.

17. What is the reason for transients during short circuits?

The faults or short circuits are associated with sudden change in currents. Most of the

components of the power system have inductive property which opposes any sudden change

in currents and so the faults (short circuit) are associated with transients.



18. What are the advantages of symmetrical components.

This method is simple.

This method leads to accurate prediction of system behavior.

Unsymmetrical faults on the transmission system are studied by the method of

symmetrical components.

They suitable to determine the currents and voltages in all parts of the power system

after the occurrence of fault.

19. What are the needs of sequence network in power system.

Sequence network in power system are very much useful in computing unsymmetrical

faults at different points of the power system network.

Positive sequence network is necessary foe load flow studies.

Negative and zero sequence networks are for stability studies involving unsymmetrical

fault.

20. What are sequence impedances and sequence networks?

The sequence impedances are the impedances offered by the devices or

components for the like sequence component of the current.

The single phase equivalent circuit of a power system consisting of impedances

to current of any one sequence only is called sequence network.

UNIT 4 – UNSYMMETRICAL FAULT

1. Express the unbalance voltage in terms of symmetrical components. (A/M 2017)

0

0

2

2

2

, , .

, , .

1 1 1

1

1

1 120 1 240

a b c

a a a

a a

b a

c a

Let I I I betheunbalanced phasecurrents

Let I I I bethe symmetrical components of phase

I I

I a a I

I a a I

where a and a

2. Draw the zero sequence network of Y - ∆ transformer with neutral ungrounded. (A/M

2017)



3. Why the neutral grounding impedance Zn appears as 3Zn in zero sequence equivalent

circuit? (N/D 2016)

Voltage drop is ZN*IN=ZN*3*I0=I0*3ZN. Therefore, to obtain the correct voltage drop

seen in the neutral conductor with a flow of only, model the zero sequence impedance as 3Zn.

4. What are symmetrical components of a 3ph system? (M/J 2016)

One of the most powerful tools for dealing with unbalanced polyphase circuits is the

method of symmetrical components. An unbalanced system of n related phasors can be resolved

into n systems of balanced phasors called symmetrical components. Symmetrical components

are positive, negative and zero sequence components.

5. Write down the equation to determine symmetrical currents from unbalance current .

(M/J 2016)

0

2

2

0

2

2

,

1 1 11

13

1

1

3

1

3

1

3

a a

a b

a c

a a b c

a a b c

a a b c

The symmentrical componets of currents are

I I

I a a I

I a a I

I I I I

I I aI a I

I I a I aI

6. State Fortescue theorem. (N/D 2015)

Fortescue's Theorem for a Three-Phase Unbalanced System. In 1918, C. L.

Fortescue stated his theorem: unbalanced phasors can be represented by systems of

balanced phasors. Sequence components were created to facilitate calculations in unbalanced

circuits and systems.

7. Which type of fault is very common in power system? (N/D 2015)

Single line to ground fault is most common occurrence fault in power system.

It occurs about nearly 70 % in fault conditions.

8. What are the features of zero sequence current? May/june 2014

It consists of three phasors equal in magnitude and with zero phase displacement from each

other.

Zero sequence phasors a,b,c can be written as 0 0 0

a b cI I I

Where 0 0 0, ,a b cI I I are the sequence components of ,a b cI I and I



9. What are the observation made from the analysis of various fault ? Nov/Dec 2013

Finding out the value of Fault current under different fault conditions

Zero sequence is not presence in LL fault conditions

Positive and negative sequence component of currents are opposite and equal to each

other in LL fault condition

Positive and negative sequence component of voltage are equal in LLG fault condition

10. Write the boundary condition for the single line to ground fault. Nov/Dec 2013

If the generator is solidly grounded, Zn = 0 and for bolted fault or solid fault,

Zf = 0.

If the neutral of the generator is ungrounded, the zero sequence network is open

circuited. 0 0a a a fI I I and I

11. Define negative sequence impedance. May/June 2013

Impedance offered to the flow of negative sequence current is known as negative sequence

impedance and it is denoted by Z-. The negative sequence impedance is occurred in all the fault

condition and it is important to find the fault current.

12. What is positive, negative and zero sequence impedance?

Impedance offered to the flow of positive sequence current is known as positive

sequence impedance and it is denoted by Z+

Impedance offered to the flow of negative sequence current is known as negative

sequence impedance and it is denoted by Z-

Impedance offered to the flow of zero sequence current is known as Zero sequence

impedance and it is denoted by Z0

13. What is the assumption made in short circuit studies?

Representing machines by reactance namely X’’, X’ and X

Prefault load current are neglected

Transformer tap are assumed to be nominal

Shunt capacitance and series resistance are ignored and neglected

14. Find the fault current whose impedance are j0.2 & j015 in series and in parallel with j0.35

if prefault voltage at the fault point is 0.97 p.u.?

Ans:

j0.2 and j0.15 are in series

Its becomes, j0.2 + j0.15 = j0.35

j0.35 is in parallel with j0.15.

0.35 0.150.105 . .

0.5 0.15

0.979.238 . .

0.105

Th

Thf

Th

j jZ j p u

j j

VFault current I j p u

Z j



15. Name the method of reducing short circuit capacity and uses of short circuit capacity.

By introducing artificial series reactors

By providing neutral reactance

By introducing a large value of shunt reactance between buses.

16. What are the reactances used in synchronous machines? "

'

1. tan

2. tan

3. tan .

d

d

d

Subtransient reac ce X

Transient reac ce X

Synchronous reac ce X

17. Define transient reactance.

It is the ration of induced emf on no-load and the transient symmetrical rms current.

It is given by,

'| | 1

tan ,1 1| ' |

tan

tan

tan

g

d l

a f

l

f

a

ETransient reac ce X X

I

X X

where X Leakage reac ce

X Field winding reac ce

X Armature reaction reac ce

18. What is the significant of transient reactance in short circuit studies?

The subtransient reactance can be used to estimate the initial value of fault current immediately

on the occurrence of the fault. The maximum momentary short circuit current rating of the

circuit breaker used for protection or fault clearing should be less than this initial fault current.

19. Difference between Load flow and short circuit studies.

Load flow Studies Short circuit studies

The resistance and reactance are considered The resistance are neglected

To solve load flow analysis, the bus

admittance matrix is used

To solve short circuit analysis, the bus

impedance matrix is used

It is used to determine the exact voltage and

currents

Prefault voltage are assumed to be1 p.u. and

the prefault current can be neglected.

20. Why synchronous impedance more than transient reactance?

The short circuit is of larger magnitude than the current measured half a second later. This mean

that magnitude of Z will have a minimum value immediately following the fault and then grow.

This change of the short circuit current with time is due to the characteristics of the synchronous

generators. Thus the synchronous impedance is more than transient reactance.



UNIT 5 – STABILITY ANALYSIS IN POWER SYSTEM

1. Define swing curve. What is the use of Swing curve? (A/M 2017)

2

( . ) ( . )2 m p u e p u

H dP P

f dt

The graphical display of δ versus t is called the swing curve. The plot of swing curves of all

machines tells us whether machines will remain in synchronism after a disturbance.

2. State equal area criterion. (A/M 2017)

The equal area criterion for stability states that the system is stable if the area under Pa – δ

curve reduced to zero at some value of δ.

This is possible if the positive (acceleration) are under Pa – δ curve is equal to the negative

(deceleration) area under Pa – δ curve for a finite change in δ. Hence the stability criteria is

called equal area criterion.

3. Define voltage stability. (N/D 2016)

Ability of power system to maintain steady acceptable voltages at all buses in power system

under normal operating conditions and after being subjected to a disturbance.

Reactive power balance

4. State few techniques to improve the stability of the power system. (N/D 2016)

Increase of the system voltage and use of AVR

Use of high speed excitation systems.

Reduction in system transfer reactance

Use of high speed reclosing

5. Define transient stability of a power system. (A/M 2016)

It is the ability of the system to bring it to a stable condition after a large disturbance. Large

disturbance can occur due to the occurrence of fault, sudden outage of a line, sudden loss of

excitation, sudden application or removal of loads, etc.

6. How are power system stability classified? (N/D 2015)

Power system stability can be classified into as follows

Rotor angle stability

Voltage stability

Frequency stability

7. Define critical clearing angle. (N/D 2015)

For any given initial load in the case of a fault clearance on a synchronous machine connected

to an infinite bus bar, there is a critical clearing angle. If the actual clearing angle is greater than the



critical value, the system is unstable, otherwise the system is stable. Maximum allowable angle for

a system to remain stable.

8. What is steady state stability limit? (Nov/Dec- 14)

The gradual increase of the generator power output is possible until the maximum

electrical power is transferred. This maximum power is referred to as the steady state stability

limit.

9. Define dynamic stability with an example. (May/June -14)

It is the ability of the system to remain in synchronism when it is subjected to sustained

oscillations followed by transient state.

10. Differentiate between voltage stability and rotor angle stability. (N/D 2013)

Voltage Stability Rotor angle stability

Ability of power system to maintain

steady acceptable voltages at all

buses in power system under

normal operating conditions and

after being subjected to a

disturbance.


Ability of inter – connected

synchronous machines of a power

system to remain in synchronism.

Torque balance of synchronous

machines

11. Define swing curve? What is the use of this curve. (N/D 2013) 2

( . ) ( . )2 m p u e p u

H dP P

f dt

The graphical display of δ versus t is called the swing curve. The plot of swing curves of all

machines tells us whether machines will remain in synchronism after a disturbance.

12. What are the causes of oscillatory and non-oscillatory instabilities in power systems?

(A/M 10)

Oscillatory :

Due to insufficient damping torque

Due to unstable control action

Non – oscillatory :

Due to insufficient synchronous torque

13. Write the power angle equation of a synchronous machine connected to an infinite bus

and also the expression for maximum power transferable to the bus.

1 1 2 2eq

b b

M S M SM

S S

1 11

12

max

| ' || || ' | sin

sin

e

e C

E VP P E G

X

P P P

This equationis called power angleequation

Expression for Maximum Power transfer:-

max

12

| ' || |sin

E VP

X



14. Define infinite bus in a power system. (M/J 13)

The substation bus voltage and frequency is assumed to remain constant. This is called as

infinite bus, since its characterized do not change regardless of the power supplied or consumed

by any device connected to it.

15. What is an infinite bus?

The substation bus voltage and frequency is assumed to remain constant. This is called

as infinite bus, since its characterized do not change regardless of the power supplied or

consumed by any device connected to it.

16. What are the methods used to solve swing equation? (M/J 09)

Modified Euler’s method

Runge-Kutta method.

17. Give the expression for critical clearing time. (N/D 07)

Critical time margin = critical clearing time – clearing time specified

( )cr critical spec

spec

t t

where

t Specified clearing time

18. Distinguish between small disturbance and large disturbance.

Small disturbance Large disturbance

The disturbances that cause small

and gradual infinitesimal variation

in system variables such as rotor

angle, voltage, etc., are classified as

small disturbance.

Any disturbance that causes a

change in the admittance matrix

encountered in stability is classified

as large disturbance

Eg – Fault occurring, removal of

line

19. What is meant by synchronism and damping torques?

The component of electrical torque proportional to rotor angle deviation from initial value is

referred to as synchronizing torque and that proportional to speed deviation from initial value is

called damping torque.

Assume initially, the machine is in steady state, then

Machine speed = Rated speed = synchronous speed

Then Initial speed deviation = 0

20. Define power or torque angle.

Under normal operating conditions, the relative position of the rotor axis and the resultant

magnetic field are fixed. The angle between the two is known as the power angle or torque

angle.

Vel Tech High Tech Dr.Rangarajan Dr.Sakunthala Engg. College EE6501 – Power System Analysis

Prepared by Prabaakaran K AP/EEE Page |1

UNIT – 2 POWER FLOW ANALYSIS

1. Compare Newton Raphson and Gauss Seidal methods of load flow solutions. (A/M 2017)

Gauss Seidal method Newton Raphson method

Variable is expressed in rectangular

coordinates.

Variables are expressed in polar

coordinates.

Computation time per iteration is less Computation time per iteration is more.

It has linear convergence

characteristics.

It has quadratic convergence characteristics.

The number of iterations required for

convergence increases with size of the

system.

The numbers of iterations are independent of the size

of the system.

The choice of slack bus is critical. The choice of slack bus is arbitrary.

2. Write the quantities that are associated with each bus in a system.(A/M 2017)

Each bus in a power system is associated with four quantities and they are

i. real power

ii. reactive power

iii. magnitude of voltage, phase angle of voltage.

3. What is Jacobian Matrix? (N/D 2016)

The matrix formed from the first derivatives of load flow equation is called jacobian

matrix and it is denoted by J.

The elements of jacobian matrix will change in every iteration .the elements of the jacobian matrix

are obtain matrix are obtained by partially differentiating the load flow equation with respect to a

unknown variable and then evaluating the first derivative as using the solution of previous iteration.

4. Write the need for slack bus in load flow analysis. (N/D 2016) A/M 2017, 2018

The slack bus is needed to account for transmission line losses. In a power

system the total power generated will be equal to sum of power consumed by loads and losses. In a power

system only the generated power and load power are specified for buses. The slack bus is assumed to

generate the power required for losses. Since the losses are unknown the real and reactive power are not

specified for slack bus. They are estimated through the solution of load flow equations.

5. What is the need for load flow analysis? (A/M 2016) N/D 2017

The study of various methods of solution to power system network is referred to as load flow study.

The solution provides the voltages at various buses, power flowing in various lines and line-losses.

The load flow study of a power system is essential to decide the best operation of existing system

and for planning the future expansion of the system. It is also essential for designing a new power

system.

6. Mention the various types of buses in power system with specified quantities for each bus. (A/M

2016) N/D 2017

The buses of a power system can be classified into three types based on the

quantities being specified for the buses, which are as follows:

Load bus or PQ bus (P and Q are specified)

Generator bus or voltage controlled bus or PV bus (P and V are specified)

Slack bus or swing bus or reference bus (|V| and δ are specified)

7. Write the static load flow equation. (N/D 2015)

Pi = ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗|𝑐𝑜𝑠(𝜃𝑖𝑗 + 𝛿𝑗𝑁𝑗=1 −𝛿𝑖)

Qi = ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗|𝑠𝑖𝑛(𝜃𝑖𝑗 + 𝛿𝑗𝑁𝑗=1 −𝛿𝑖)



8. State the role of acceleration factor in GS method. (N/D 2015) A/M 2018

In Gauss Seidal method, the number of iteration required for convergence can be reduced if the

voltage computed at each iteration is multiplied by a factor greater than unity called acceleration factor to

bring the voltage closer to the value to which it is converging. The range of 1.3 to 1.6 is found to be

satisfactory for the typical systems.

𝑉𝑖𝑛𝑒𝑤 = Vi

old + ∝ [𝑉𝑖𝑛𝑒𝑤 − Vi

old ]

Where Viold = Voltage value obtained in previous iteration

𝑉𝑖𝑛𝑒𝑤 = New value of Voltage value obtained in current iteration

∝ = Acceleration factor

9. Why the load flow studies are important for planning the existing system as well as its future

expansion?

The load flow studies are very important for planning, economic scheduling, control and operation

of existing systems as well as planning its future expansion depends upon knowing the effect of

interconnections, new loads, new generating stations, or new transmission lines, etc. before they

installed.

10. What is swing bus? (N/D 2013)

A bus is called swing bus when the magnitude and phase for bus voltage are specified for it. The

swing bus is the reference bus for load flow solution and it is required for accounting line losses.

Usually one of the generator bus selected as the swing bus.

Swing bus is also called as Slack bus.

11. When the generator buses are treated as load bus? (N/D’13, M/J’14)

If the reactive power constraints of a generator bus violates the specified limits then the generator is

treated as load bus.

If Qi

> Qi(max), substitute Qi = Qi(max)

If Qi cal < Qi(min) , substitute Qi= Qi(min)

12. What is meant by flat voltage start?

In iterative methods of load flow solution, the initial voltages of all buses exept slack bus are assumed

as 1+j0 pu. This is referred as flat voltage profile.

V = |𝑉𝑠𝑝𝑒𝑐| 0 for slack bus

V = |𝑉𝑠𝑝𝑒𝑐| 0 for generator bus

V = 1 0 for load bus

13. What are the works involved in a load flow study? ( Or) How a load flow study is

performed?

The following work has to be performed for a load flow study.

Representation of the system by single line diagrams.

Determining the impedance diagram using the information in single line diagram.

Formulation of network equations.

Solution of network equations.



14. List the quantities specified and the quantities to be determined from load flow study for

various types of buses.

The following table shows the quantities specified and to be obtained for various types of buses.

Bus type Quantities specified Quantities to be obtained

Load bus P,Q V, δ

Generator bus P, V Q, δ

Slack bus V, δ P,Q

15. Why bus admittance matrix is used in load flow instead of bus impedance matrix? (M/J 2014)

Using bus admittance matrix is amenable to digital computer analysis, because it could be formed

and modified for network changes in subsequent cases. Bus admittance matrix is used in Gauss seidal

method because of the following reasons.

It requires less computation time

Less memory allocation

16. What are the operating Constrains imposed in load flow studies?

The operating constraints imposed in load flow studies are reactive power limits for generator buses

and allowable change in magnitude of voltage for load buses.

17. What is infinite bus?

A bus is called infinite bus if its voltage remains constant and does not altered by any

changes in generator excitation.

18. Define Voltage control bus?

A bus is called voltage controlled bus if the magnitude of |V| and real power (P) are specified for it. In a

voltage controlled bus the magnitude of the voltage is not allowed to change. The other names for voltage

controlled bus are generator bus and PV bus.

19. Why do we go for iterative methods to solve load flow problems?

The load flow equations are nonlinear algebraic equation and so explicit solution is not possible. The

solution of nonlinear equations can be obtained only by iterative numerical techniques.

20. How the disadvantages of N-R method are overcome?

The disadvantage of large memory requirement can be overcome by decoupling the weak coupling

between P-δ and Q-V. (i.e., using decoupled load flow algorithm). The disadvantage of large

computational time per iteration can be reduced by simplifying the decoupled load flow equations. The

simplifications are made based on the practical operating conditions of a power system.

PART – B

1. Derive the development of load flow model in complex variable form and polar variable form.

(13) (M/J’14)

The power flow or load flow model in complex form is obtained by writing one complex power

matching equation at each bus for the figure shown below. (4)

Net power injected into the bus i.

Si = SGi – SDi

= PGi + jQGi – (PDi + jQDi)

= Pi + jQi



We know,

Pi + jQi = Vi Ii*

Consider two bus system with I1 and I2 as net current entering into bus 1 and 2.

[I]= [Y][𝑉]

[I1I2

] = [𝑌11 𝑌12

𝑌21 𝑌22] [

𝑉1

𝑉2] (4)

𝑌11 = 𝑦10 + 𝑦12

𝑌22 = 𝑦20 + 𝑦21

𝑌12 = 𝑌21 = - 𝑦21

In general, Yij = |𝑌𝑖𝑗|∠𝜃𝑖𝑗

I1 = 𝑌11 𝑉1+ 𝑌12𝑉2

I2 = 𝑌21 𝑉1+ 𝑌22𝑉2

In general , the net current entering into ith bus

Ii = Yi1V1 + Yi2V2+………………………………..+ YiNVN = ∑ Y𝑖𝑗𝑉𝑗𝑁𝑗=1

Substituting the value of Ii in power flow eqn we get.

Si = Pi + jQi = Vi Ii*

Si = Pi - jQi = Vi Ii*

Pi - jQi = Vi* ∑ Y𝑖𝑗𝑉𝑗

𝑁𝑗=1 where i = 1,2,3…………..N (4)

There are N complex variable equations for which the N unknown complex variables V1,V2…….VN can be

determined.

Substituting Yij from the above eqn, we get.

Pi - jQi = Vi* ∑ |𝑌𝑖𝑗|∠𝜃𝑖𝑗 𝑉𝑗

𝑁𝑗=1

Where Vi = |𝑉𝑖|∠𝛿𝑖 , Vi* = |𝑉𝑖|∠ − 𝛿𝑖,

Vj = |𝑉𝑗|∠𝛿𝑗

Therefore net power equations can be written as

Pi - jQi = ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗|∠(𝜃𝑖𝑗 + 𝛿𝑗 − 𝛿𝑖 ) 𝑁𝑗=1

Equating real and reactive parts, (4)

Pi = ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗| 𝑐𝑜𝑠 (𝜃𝑖𝑗 + 𝛿𝑗 − 𝛿𝑖 ) 𝑁𝑗=1

Qi = - ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗| 𝑠𝑖𝑛 (𝜃𝑖𝑗 + 𝛿𝑗 − 𝛿𝑖 ) 𝑁𝑗=1

We can write the above equation as

Pi = |𝑉𝑖|2 |𝑌𝑖𝑖| 𝑐𝑜𝑠 𝜃𝑖𝑖 ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗| 𝑐𝑜𝑠 (𝜃𝑖𝑗 + 𝛿𝑗 − 𝛿𝑖 )

𝑁𝑗=1

Qi = - |𝑉𝑖|2 |𝑌𝑖𝑖| 𝑠𝑖𝑛 𝜃𝑖𝑖 ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗| 𝑐𝑜𝑠 (𝜃𝑖𝑗 + 𝛿𝑗 − 𝛿𝑖 )

𝑁𝑗=1

The above equations are called as polar form of the power flow equations.

2. Write the step-by-step algorithm for load flow solution by Gauss-seidal method. (13)

Step 1 : form Y-Bus (4)

Step 2 : Assume Vk = Vk(spec) ∠00 at all generator buses.

Step 3 : Assume Vk = 1∠00 = 1+j0 at all load buses.

Step 4 : set iteration count = 1 (iter = 1)

Step 5 : let bus number i = 1 (4)

Step 6 : If ‘i’ refers to generator bus go to step no. 7, otherwise go to step 8.

Step 7(a) : If ‘i’ refers to slack bus go to step no. 9, otherwise go to step 7(b).

Step 7(b) :compute Qi using

Qi cal = - Im [ ∑ 𝑁

𝑗=1 𝑉𝑖∗ 𝑌𝑖𝑗 Vj ]



QGi = Qi cal + QLi

Check for Q limit violation (6)

If Qi(min) < QGi < Qi(max), then Qi(spec) = Qi cal

If Qi(min) < QGi, then Qi(spec) = Qi(min) - QLi

If Qi(max) < QGi, then Qi(spec) = Qi(max) - QLi

If Qlimit is violated, then treat this bus as P-Q bus till convergence is obtained.

Step 8 : Compute Vi using eqn.

𝑉𝑖𝑛𝑒𝑤=

1

𝑌𝑖𝑖 [

𝑃𝑖− 𝑗𝑄𝑖

𝑉𝑖𝑜𝑙𝑑∗ − ∑

𝑗−1𝑖=1 𝑌𝑖𝑗 Vj

new - ∑ 𝑁𝑖=𝑗+1 𝑌𝑖𝑗 Vj

old

Step 9 : If i is less than number of buses, increment I by 1 and go to step 6.

Step 10 : Compare two successive iteration values for Vi

If 𝑉𝑖𝑛𝑒𝑤 - 𝑉𝑖

𝑜𝑙𝑑 < tolerance, go to step 12.

Step 11 : Update the new voltage as

Vnew = Vold + ∝ (Vnew - Vold)

Vnew = Vold

Iter = iter +1; go to step 5.

Step 12: Compute relevant quantities. (2)

Slack bus power, S1 = V*I = Vi* ∑ 𝑁

𝑗=𝑖 𝑌𝑖𝑗 Vj

line flow losses, Sij = Pij + jQij

Real power loss, Ploss = Pij + Pji

Reactive power loss, Qloss = Qij + Qji

Step 13 : Stop the execution.



3. Write the Procedure for load flow solution by Newton Raphson method. (13) (N/D’08, M/J’12,

N/D’14)

Step 1 : Form Y-Bus (8)

Step 2 : Assume flat start voltage solution

𝛿𝑖0 = 0, for i = 1…..N

|𝑉𝑖0|=1.0,

|𝑉𝑖| = |𝑉𝑖|spec

Step 3 : for load buses, calculate Pi cal and Qi

cal

Step 4 : for PV bus, Check for Q limit violation

If Qi(min) < Qi cal < Qi(max), the bus acts as PV bus

If Qi cal

> Qi(max), then Qi(spec) = Qi(max)

If Qi cal < Qi(min) , then Qi(spec) = Qi(min) , the PV bus will act as PQ bus.

Step 5 : Compute mismatch vector using. (8)

∆Pi = Pi(spec) - Pi cal

∆𝑄i = Qi(spec) - Qi cal

Step 6 : Compute ∆Pi(max) = max |∆P𝑖| i = 1,2……….N except slack

∆Qi(max) = max |∆𝑄𝑖| i = M+1……….N

Step 7 : compute jacobian matrix using J = [

𝜕𝑃𝑖

𝜕𝛿

𝜕𝑃𝑖

𝜕|𝑉|

𝜕𝑄𝑖

𝜕𝛿

𝜕𝑄𝑖

𝜕|𝑉|

]

Step 8 : Obtain state correction vector [∆V∆𝛿

] = [J]-1 [∆𝑃∆𝑄

]

Vnew = Vold +∆V

𝛿 new = 𝛿 old +∆𝛿

Step 9 : Update state vector using .

Vnew = Vold +∆V

𝛿 new = 𝛿 old +∆𝛿

Step 10 : This procedure is continued until

|∆P𝑖| < 휀 and |∆𝑄𝑖| < 휀, otherwise go to step 3.



4. Derive the load flow equation using Newton Raphson method. (13)

The most widely used method for solving simultaneous non linear algebraic equation is the Newton

Raphson method. (4)

from the fig the complex power balance at bus i is given by



PIi + jQIi = Pi + jQi ………………………………………………………..…(1)

Complex power injection at the ith bus PIi + jQIi

= (PGi - PDi ) + (jQGi - jQDi) (4)

Since the bus generation and demand are specified, the complex the complex power injection is a specified

quantity and is given by,

PIi(spec) + jQIi(spec) = [PGi(spec) - PDi(spec)] +j[QGi(spec) - QDi(spec)] ………………(a)

The current entering bus i is given by,

Ii = ∑ Y𝑖𝑗𝑉𝑗𝑁𝑗=1

In polar form

Ii = ∑ |𝑌𝑖𝑗||𝑉𝑖|∠(𝜃𝑖𝑗 + 𝛿𝑗𝑁𝑗=1 ) ………………………………………………….(b)

[Y𝑖𝑗 = |𝑌𝑖𝑗|∠𝜃𝑖𝑗 ; 𝑉𝑗 = |𝑉𝑖|∠𝛿𝑖]

Complex power at bus i

Pi - jQi = Vi* Ii = Vi

*∑ Y𝑖𝑗𝑉𝑗𝑁𝑗=1 (4)

Substituting from eqn (b), we get

Pi - jQi = |𝑉𝑖|∠−𝛿𝑖 ∑ |𝑌𝑖𝑗||𝑉𝑗|∠(𝜃𝑖𝑗 + 𝛿𝑗𝑁𝑗=1 )

= ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗|∠(𝜃𝑖𝑗 + 𝛿𝑗𝑁𝑗=1 −𝛿𝑖)

Equating the real and imaginary parts

Pi = ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗|𝑐𝑜𝑠(𝜃𝑖𝑗 + 𝛿𝑗𝑁𝑗=1 −𝛿𝑖)………………………(c) (4)

Qi = ∑ |𝑉𝑖||𝑌𝑖𝑗||𝑉𝑗|𝑠𝑖𝑛(𝜃𝑖𝑗 + 𝛿𝑗𝑁𝑗=1 −𝛿𝑖) ………………………………………….(d)

The above eqn constitute a set of nonlinear algebraic equations in terms of the independent variables.

Substitute eqn (a),(c),(d) in (1) we get power balance eqns.

Pi (δ,V) - PIi(spec) = 0

Qi (𝛿,V) - QIi(spec) = 0

5. For the system shown in Fig., determine the voltages at the

end of the first iteration by Gauss – seidel method and also

find the slack bus power, line flows, transmission line loss.

Assume base MVA as 100. (13) (N/D’13)

Solution:

Bus No. Voltage Generator Load Qmin

MVAR

Qmax

MVAR P Q P Q

1. 1.050 p.u. - - - - - -

2. 1.02 p.u. 0.3 p.u. - - - -10 100

3. - - - 0.4 p.u. 0.2 p.u. - -



Step – 1: From Y-bus. (4)

Ybus =

[

1

Y11

−1

Y12

−1

Y13

−1

Y21

1

Y22

−1

Y23

−1

Y31

−1

Y32

1

Y33]

=

[

1

j0.4+

1

j0.3

−1

j0.4

−1

j0.3

−1

j0.4

1

j0.4+

1

j0.2

−1

j0.2

−1

j0.3

−1

j0.2

1

j0.3+

1

j0.2]

Ybus = [

−j5.8333 j2.5 j3.3333j2.5 −j7.5 j5

j3.3333 j5 −j8.3333]

Step – 2: Initialize bus voltages.

V1old = 1.050 p.u. [Bus 1 is a slack bus i.e., V and is specified]

V2old= 1.020 p.u. [Bus 2 is a PV bus i.e., P and V is specified]

V3old= 1.0 p.u. [Bus 3 is a load bus i.e., P and Q is specified]

For Slack bus, the specified voltage will not change in any iteration.

For generation bus, calculate Vinew using the formula and write

Vinew = Vspecifiedcalculated value

Step 3 : Calculate Q value for all generator buses. (4)

Qical = - Im Vi

old*[∑ Yij Vjnewi−1j=1 + ∑ Yij VjoldN

j=i ]

Q2cal = - Im V2

old*[Y21V1new +Y22V2

old + Y23V3old]

Q2cal = - Im 1.020 *[j2.51.050 +(-j7.51.020) + j5 10]

Q2cal = - Im 1.020 *j2.625-j7.65+j5]

Q2cal = 0.025 p.u.

Now Q2(min) Q2calQ2(max)

i.e., Q2cal is within the specified limit.

Step – 4: Calculate Vinew .

Vinew = 1.050 p.u.

Vinew =

1

Yii

[Pi−jQi

Viold*

− ∑ Yij Vjnewi−1

j=1 − ∑ Yij VjoldN

J=i+1 ]

V2new =

1

Y22

[P2−jQ2

V2old*

− Y21V1new − Y23 V3

old]

P2 = 0.3 p.u. (Given); Q2= 0.025 p.u.

V2new =

1

−j7.5[0.3−j0.025

1.020− j2.5 X 1.050 − j5 X 10]

= 1.0199+j0.0392

V2new = 1.02072.2

V2new = V2(spec)2

cal = 1.022.2 = 1.0192 + j0.0392

P3 = PG3 – PL3 = 0- 0.4 = -0.4 p.u.

Q3= QG3 – QL3 = 0 – 0.2 = -0.2 p.u.

V3new =

1

Y33

[P3−jQ3

V3old*

− Y31V1new − Y32 V2

new]

V3new =

1

−j8.3333[−0.4+j0.2

1.00− j3.3333 X 1.050 − j5 X 1.022.2]

= 1

−j8.3333[−0.4 + j0.2 − j3.4999 − j5.096 + 0.196]

= 1.0075 – j0.0244 = 1.0078 -1.39

V3new = 1.0075 – j0.0244 = 1.0078 -1.39



6. Perform two iteration of Newton Raphson load flow method and determine the power flow

solution for the given system. Take base MVA as base 100. (13)

Solution:

Line Data:

Line Bus R(p.u.) X(p.u.) Half line charging

admittance (p.u.) From To

1 1 2 0.0839 0.5183 0.0636

Bus Data:

Bus PL QL

1 90 20

2 30 10

Step – 1: Ybus = [0.3044 − j1.816 −0.3044 + j1.88−0.3044 + j1.88 0.3044 − j1.816

] (4)

Ybus = [1.842 − 1.405 1.9041.73141.9041.7314 1.842 − 1.405

] Note: Use in rad mode

Step – 2: Assume the initial value i.e., =0, V=1.0

[X] = [2

V2] = [

01.0

]

Step – 3: Calculate P2cal, Q2

cal, P2 and Q2.

P2cal = |V2||V1||Y2| cos(12 + 2 − 1) + |V2||Y22|cos (22 + 2 − 2)

= 1.0[1.05 X 1.904 cos(1.7314) + 1.842 cos(-1.405)]

= 1.05 X 1.904(-0. 15991) + 1.842(0.16503)

= -0.015 p.u.

P2(spec) = PG2 – PL

= 0 - 30

100 = - 0.3 p.u.

P2(spec) = - 0.3 p.u.

P2 = P2(spec) – P2cal

= -0.3 – (-0.015)= - 0.285

Q2cal = - V2|V1||Y21| sin(12 + 1 − 2) + |V2||Y22| sin(22 + 2 − 2)

= - 1.0[1.05 X 1.904 sin(1.7314)+ 1.0 X 1.842 sin(-1.405)]

= - 0.157 p.u.

Q2 = Q2(spec) – Q2cal = - 0.1 – ( -0.157)

= 0.057

Step – 4: Form Jacobian matrix (4)

[

∂P2

∂2

∂P2

∂V2

∂Q2

∂2

∂Q2

∂V2

] [2

V2] = [

P2

Q2]

∂P2

∂2

= |V2||V1||Y12| sin(12 + 1 − 2) + |V2|2|Y22|X 0

= 1.0 X 1.05 X1.904 sin(1.7314)

= 1.973

∂P2

∂V2

= |V1||Y21| cos(12 + 1 − 2) + 2|V2||Y22| cos(22)

= 1.05 X 1.904 cos(1.7314) + 2 X 1.842 cos(-1.405)

= 0.289



-1

-1

∂Q2

∂2

= |V2||V1||Y21| cos(12 + 1 − 2) − |V2|2|Y22|X 0

= 1.05 X 1.904 cos(1.7314)

= - 0.3197 ∂Q2

∂V2

= −|V1||Y21| sin(12 + 1 − 2) − 2|V2||Y22| sin(22)

= - 1.05 X 1.904 sin(1.7314) – 2 X 1.842 sin(-1.405)

= 1.66

Step – 5: Compute x, (4)

[2V2

] = [

∂P2

∂2

∂P2

∂V2

∂Q2

∂2

∂Q2

∂V2

] [P2

Q2]

= [1.973 0.289

−0.3196 1.66] x [

−0.285 0.057

]

= [0.493 −0.0860.0949 0.586

] [−0.285 0.057

]

[2V2

] = [−0.1450.0064

]

X = X + X = [0

1.0] + [

−0.1450.0064

] = [−0.1450.0064

]

UNIT 3 – SYMMETRIC FAULT ANALYSIS

1.What is the significance of sub transient reactance and transient reactance in short circuit studies?

(A/M 2017)

The sub-transient reactance can be used to estimate the initial value of fault current immediately

on the occurrence of the fault. The maximum momentary short circuit current rating of the circuit

breaker used for protection or fault clearing should be less than this initial fault current.

2. For a fault at a given location, rank the various fault in the order of severity.(A/M 2017)

Types of fault Relative frequency of occurrence of faults

Three phase fault 5%

Double line to ground fault 10%

Line to Line fault 15%

Line to ground fault 70%

3.How shunt and series fault are classified ? (N/D 2016)

Shunt faults - due to short circuits in conductors

Series faults - due to open conductors.

The shunt fault is also called as short circuit faults.

4.What is the need for short circuit study? (N/D 2016)

The short circuit studies are essential in order to design or develop the protective schemes for

various parts of the system. The protective scheme consists of current & voltage sensing devices, protective

relays and circuit breakers. The selection (or proper choice) of these device mainly depends on various

currents that may flow in the fault conditions.

5.State and explain symmetrical fault. (A/M 2018, 16)

The fault on the power system which gives rise to symmetrical fault currents (i.e. equal fault current

in the lines with 1200 displacement) is called as symmetrical faults. The symmetrical fault occurs when all

the three conductors of phase are brought together simultaneously in short circuit condition.



6.What is bolted fault or solid fault? (N/D 2017, A/M 2016)

A Fault represents a structural network change equivalent with that caused by the addition of

impedance at the place of a fault. If the fault impedance is zero, the fault is referred as bolted fault or solid

fault.

7.Define short circuit capacity. (N/D 2015)

Short circuit capacity or short circuit MVA of fault level at a bus is defined as the product of the

magnitudes of the perfect bus voltage and the post fault current.

SCC = |𝐸𝑇ℎ||𝐼𝑓| = |𝐸𝑇ℎ|2

𝑋𝑇ℎ p.u. MVA

8.What is meant by a fault?

A fault in a circuit is any failure which interferes with the normal flow of current. The faults are

associated with abnormal change in current, voltage and frequency of the power system. The faults may

cause damage to the equipments if it is allowed to persist for a long time. Hence every part of a system has

been protected by means of relays and circuit breakers to sense the faults and to isolate the faulty part from

the healthy part in the event of fault.

9.Why fault occurs in a power system?

In a power system the fault may occur due to the following reasons.

Insulation failure of equipments.

Flashover of lines initiated by a lightning stroke.

Switching surges.

Sudden releasing of heavy loads at same instant of time.

Due to accidently faulty operation of power system operators.

10.What are the objectives of short circuit analysis.

To check the MVA ratings of the existing circuit breaker’s, when new generation are added

into system.

To select the rating for fuses, circuit breaker’s and switch gear in addition to set up of

protective relays.

To determine the magnitudes of currents flowing through out the power system at various

time intervals after a fault occurs.

11.State the applications of short circuit analysis.

Short circuit studies are used

For proper relay setting and co ordination.

To obtain the rating of the protective switch gears.

To select the circuit breakers.

To perform whenever system expansion is planned.

12.What are symmetrical components?

An unbalanced system of N related vectors can be resolved into N systems of balanced vectors called

symmetrical components. The various symmetrical components are as follows.

Positive sequence components

Negative sequence components

Zero sequence components



13.Why delta connected load will not have any zero sequence components

The current in the neutral is three times the zero sequence line current. A delta connected load

provides no path to neutral and hence the line currents flowing to a delta connected load contain zero

component. Hence the delta connected load will not have any zero sequence components.

14. What is symmetrical components of three phase system.

Symmetrical components of three phase system is given as

For voltage

[𝑉𝑎

𝑉𝑏

𝑉𝑐

] = [1 1 11 𝑎2 𝑎1 𝑎 𝑎2

] [𝑉𝑎0

𝑉𝑎1

𝑉𝑎2

]

For current.

[𝐼𝑎𝐼𝑏𝐼𝑐

] = [1 1 11 𝑎2 𝑎1 𝑎 𝑎2

] [𝐼𝑎0

𝐼𝑎1

𝐼𝑎2

]

15.What is meant by doubling effect?

If a symmetrical fault occurs when the voltage wave is going through zero then the maximum

momentary short circuit current will be double the value of maximum symmetrical short circuit current.

This effect is called doubling effect.

16. What is synchronous reactance? N/D 2015

The synchronous reactance is the ratio of induced emf and the steady state rms current (i.e. it is the

reactance of a synchronous machine under steady state condition). It is the sum of leakage reactance and the

reactance representing armature reaction.

It is given by, Xs = Xl + Xa

Where,Xs = Synchronous reactance, Xl = Leakage reactance, Xa = Armature reaction reactance.

17.What is the reason for transients during short circuits?

The faults or short circuits are associated with sudden change in currents. Most of the

components of the power system have inductive property which opposes any sudden change in currents

and so the faults (short circuit) are associated with transients.

18. What are the advantages of symmetrical components.

This method is simple.

This method leads to accurate prediction of system behavior.

Unsymmetrical faults on the transmission system are studied by the method of symmetrical

components.

They suitable to determine the currents and voltages in all parts of the power system after

the occurrence of fault.

19.What are the needs of sequence network in power system.

Sequence network in power system are very much useful in computing unsymmetrical faults

at different points of the power system network.

Positive sequence network is necessary foe load flow studies.

Negative and zero sequence networks are for stability studies involving unsymmetrical fault.

20.What are sequence impedances and sequence networks?

The sequence impedances are the impedances offered by the devices or

components for the like sequence component of the current.

The single phase equivalent circuit of a power system consisting of impedances

to current of any one sequence only is called sequence network.



21. What is meant by fault calculation? A/M 2018

Fault calculations are one of the most common types ofcalculation carried out during the design and

analysis of electrical systems. These calculations involve determining the current flowing through circuit

elements during abnormal conditions – short circuits and earthfaults.

22. What are all the assumption to be made to simplify the short circuit study? A/M 2018

Representing each machine by constant voltage source behind proper reactance.

Prefault load currents are neglected.

Transformer taps are assumed to be nominal.

Shunt elements & core loss in transformer are neglected.

A symmetric three phase power system is considered.

Shunt capacitance & Series resistance are neglected.

23. How the shunt & series fault are classified? N/D 2016

Shunt : LLLG, LG, LL, LLG Fault

Series : Open conductor fault , Two open conductor fault.

24. Draw the oscillogram of short circuit current when an unloaded generator is subjected to

symmetrical fault. A/M 2015

25. A motor is drawing 10000 kW at 0.85 pf leading and a terminal voltage of 10kV. Determine the

load current.

Load Current, IL = 10000

√3 ×0.85 ×12 ∠cos-1 0.85 = 566 ∠31.8o

26. What is circuit breaker?

A circuit breaker is an automatically operated electrical switch designed to protect an electrical

circuit from damage caused by excess current from an overload or short circuit. Its basic function is to

interrupt current flow after a fault is detected. Unlike a fuse, which operates once and then must be

replaced, a circuit breaker can be reset (either manually or automatically) to resume normal operation.

27. What are the main factors to be considered for selection of circuit breaker?

Normal working power level specified as rated interrupting current or rated interrupting MVA.

The fault level specified as either rated short circuit interrupting current or rated short circuit

interrupting MVA.

Momentary current rating

Normal working voltage

Speed of circuit breaker.



28. Why the circuit breaker interrupting current is asymmetrical?

The interrupting current of circuit breaker is the sum of symmetrical short circuit current and dc –

offset current. The presence of dc – offset makes the interrupting current as asymmetric.

PART – B

1. Algorithm & Flowchart for symmetrical fault analysis using Zbus

Step 1 : Start.

Step 2 : Read line impedance,

generator impedance and fault

impedance.

Step 3 : Form the Zbus matrix using

step by step assembly using 4 types

of modifications.

Step 4 : Set all bus voltages as

10 p.u.

Step 5 : Connect Zf in series with

faulted bus.

Step 6 : Calculate fault current for

ith bus.

Ii(F) = 𝑉𝑖(𝑜)

𝑍𝑖𝑖+𝑍𝑓

Step 7 : Calculate the generator

current IG = Zeq IGi

Step 8 : Calculate change in

voltage ∆𝑉𝑖 and Vi(F)

Vi(F) = 𝑉𝑖(𝑜) + ∆𝑉𝑖

Step 9 : Calculate Vj(F) = 𝑉𝑗(𝑜) = -∑ 𝑍𝑖𝑗𝐼𝑖(𝐹)𝑛𝑗=1≠𝑖

Step 10 : Calculate Iij(F) = 𝑉𝑖(𝐹)−𝑉𝑗(𝐹)

𝑍𝑖𝑗

Step 11 : Print the result.

Step 12 : Stop.

2. A symmetrical fault occurs on bus 4 of system shown in fig. Determine the fault current, post fault

voltages and line currents. G1, G2 : 100 MVA, 20 KV, X+=15%. Transformer: Xleak=9%,

L1, L2: X+=10%.



Solution.

Step 1: draw reactance diagram

Step 2: form Z-bus using building algorithm.

Zbus = [j0.15]

Zbus =[j0.15 j0.15j0.15 j0.24

]

Zbus =[

j0.15 j0.15 j0.15j0.15 j0.24 j0.24j0.15 j0.24 j0.29

]

Zbus =[

j0.15 j0.15 j0.15 j0.15j0.15 j0.24 j0.24 j0.24j0.15 j0.24 j0.29 j0.29j0.15 j0.24 j0.29 j0.38

]

Zbus =

[ j0.15 j0.15 j0.15 j0.15 j0.15j0.15 j0.24 j0.24 j0.24 j0.24j0.15 j0.24 j0.29 j0.29 j0.29j0.15 j0.24 j0.29 j0.38 j0.38j0.15 j0.24 j0.29 j0.38 j0.53]

Eliminate node 5 using the relation.

Zij new = Zij old - 𝑍𝑖(𝑛+1)𝑍(𝑛+1)𝑗

𝑍(𝑛+1)(𝑛+1)



= [

j0.1075 j0.172 j0.068 j0.0424j0.172 j0.13 j0.108 j0.068j0.068 j0.108 j0.13 j0.082j0.0424 j0.13 j0.082 j0.075

]

Step 3: fault current

If = 𝑉0

𝑍𝑞𝑞+𝑍𝑓 =

1∠00

𝑗0.1075 = 9.3∠ − 900 p.u

If = 9.3∠ − 900 p.u

Actual current in KA = p.u value x base current.

= 9.3∠ − 900 x 𝑀𝑉𝐴

√3 𝑥 𝐾𝑉

= 9.3∠ − 900 x 100

√3 𝑥 20 = 26.85 KA

Actual current in KA = 26.85 KA

Step 4 : Post fault bus voltages:

𝑉1𝑓 = 𝑉0 - 𝑍14𝑍𝑓

= 1∠00- j0.042 x 9.3∠ − 900 = 0.6056 p.u

𝑉2𝑓 = 𝑉0 - 𝑍24𝑍𝑓

= 1∠00- j0.068 x 9.3∠ − 900 = 0.3686 p.u

𝑉3𝑓 = 𝑉0 - 𝑍34𝑍𝑓

= 1∠00- j0.082 x 9.3∠ − 900 = 0.2374 p.u

𝑉4𝑓 = 𝑉0 - 𝑍44𝑍𝑓

= 1∠00- j0.1075 x 9.3∠ − 900 = 0 p.u

𝑉1𝑓 = 0.6056 p.u

𝑉2𝑓 = 0.3686 p.u

𝑉3𝑓 = 0.2374 p.u

𝑉4𝑓 = 0 p.u

Step 5 : Post fault line currents.

𝐼𝑖𝑗𝑓

= 𝑉𝑖

𝑓−𝑉𝑗

𝑓

𝑍𝑖𝑗 𝑠𝑒𝑟𝑖𝑒𝑠 ; 𝐼12

𝑓 =

𝑉1𝑓−𝑉2

𝑓

𝑍12 =

0.6056−0.3686

𝑗 0.09

|𝐼12𝑓

| = 2.634 p.u

𝐼23𝑓

= 𝑉2

𝑓−𝑉3

𝑓

𝑍23 =

0.3686−0.2374

𝑗 0.05

|𝐼23𝑓

| = 2.63 p.u.

𝐼34𝑓

= 𝑉3

𝑓−𝑉4

𝑓

𝑍34 =

0.2374−0

𝑗 0.09

|𝐼34𝑓

| = 2.637 p.u.



3. A 3 phase transmission line operating at 33kv and having resistance and reactance of 5 ohms and

15 ohms respectively is connected to the generating station bus-bar through a 5000 KVA step up

transformer which has a reactance of 0.05 p.u. Connected to the bus-bars are two alternators, are

10,000 KVA having 0.08 p.u. reactance and another 5000 KVA having 0.06 p.u. reactance. Calculate

the KVA at a short circuit fault between phases occurring at the high voltage terminals of the

transformers.

Solution :

Base MVA = 10 MVA

Base KV = 33KV

Transmission line :𝑍𝑝.𝑢𝑛𝑒𝑤 =

𝑍 𝑝.𝑢𝑜𝑙𝑑

𝐾𝑉𝑏2 x MVA

= 5+𝑗15

332 x10 = 0.0459 +j0.1377 p.u

𝑍𝑝.𝑢𝑛𝑒𝑤 = 0.0459 +j0.1377 p.u

Transformer : 𝑘𝑉𝑏,𝑛𝑒𝑤 = 33KV

Zp.u. new = Zpu,given x [ 𝑘𝑉𝑏,𝑔𝑖𝑣𝑒𝑛

𝑘𝑉𝑏,𝑛𝑒𝑤 ]2 x [


𝑀𝑉𝐴𝑏,𝑔𝑖𝑣𝑒𝑛 ]

Zp.u. new =j 0.05 x(33/33)2 x(10/5)

Znew = j 0.1 p.u

Generator 1: 𝑘𝑉𝑏,𝑛𝑒𝑤 = 11KV





= j 0.08 x(33/33)2 x(10/10)

Znew = j 0.08 p.u






= j 0.06 x(33/33)2 x(10/5)

Znew = j 0.12 p.u



Reactance diagram:

a. Total impedance upto the fault F1, Z1 p.u = - j0.148 p.u

Short circuit KVA fed into the fault at F1 = |𝐾𝑉𝐴1 𝑆.𝐶| = |𝐾𝑉𝐴𝑏|

|𝑍1 𝑝.𝑢|

= 10,000

0.148 = 67567.56 KVA

= 67.568 MVA

Short circuit KVA fed into the fault at F1 = 67.568 MVA

a. Total impedance upto the fault F2, Z2 p.u = 0.0459 +j0.2857 p.u

|𝑍2 𝑝.𝑢| = 0.289 p.u

Short circuit KVA fed into the fault at F2 = |𝐾𝑉𝐴2 𝑆.𝐶| = |𝐾𝑉𝐴𝑏|

|𝑍2 𝑝.𝑢|

= 10,000

0.289 = 34602 KVA

= 34.602 MVA

Short circuit KVA fed into the fault at F2 = 34.602 MVA

4. For the radial network shown in the figure, three phase fault occurs at F. Determine the fault

current and the line voltage at 11KV bus under fault conditions.



Solution.

Base MVA = 10 MVA

Base KV = 11KV






= j 0.15 x(11/11)2 x(10/10)

Znew = j 0.15 p.u






= j 0.125 x(11/11)2 x(10/10)

Znew = j 0.125 p.u

Transformer 1: primary side 𝑘𝑉𝑏,𝑛𝑒𝑤 = 11KV

Zp.u. new =j 0.1 x(11/11)2 x(10/10)

Znew = j 0.1 p.u

Transmission line: 𝑘𝑉𝑏,𝑛𝑒𝑤 = 11 x (33/11) = 33 KV

Z = 𝑍𝑎𝑐𝑡𝑢𝑎𝑙

𝑍𝑏𝑎𝑠𝑒 =

𝑍𝑎𝑐𝑡𝑢𝑎𝑙

𝐾𝑉𝑏2 x 𝑀𝑉𝐴𝑏

𝑍𝑎𝑐𝑡𝑢𝑎𝑙 = 0.27 + j0.36 Ω/km = (0.27 + j0.36) x 30 Ω

𝑍𝑛𝑒𝑤 = 0.27+𝑗0.36

332 x 10 x 30 = 0.0744 +j0.0992 p.u

𝑍𝑛𝑒𝑤 = 0.0744 + j0.0992 p.u

Transformer 2 : 𝑘𝑉𝑏,𝑛𝑒𝑤 = 33 KV (PY)

Zp.u. new =j 0.08 x(33/33)2 x(10/5)

Znew = j 0.16 p.u

Cable : 𝑍𝑎𝑐𝑡𝑢𝑎𝑙 = (0.135 + j0.08) Ω/km = (0.135 + j0.06) x 3 Ω

Z = 𝑍𝑎𝑐𝑡𝑢𝑎𝑙


𝑍𝑎𝑐𝑡𝑢𝑎𝑙

𝐾𝑉𝑏2 x 𝑀𝑉𝐴𝑏 =

0.135 +𝑗0.08

6.62 x 3 x 10 = 0.093 +j0.0055 p.u



Fault current If = 𝐸𝑇ℎ

𝑍𝑇ℎ =

1∠00

0.1674 + 𝑗0.4822 = 0.6425 - j1.85

= 1.959 ∠−70.850 p.u

Fault current If = 1.959 ∠−70.850 p.u

Base current, IB = 𝑀𝑉𝐴𝑏

√3 𝑥 𝐾𝑉𝑏 =

10 𝑥 103

√3 𝑥 6.6 = 874.77 Amp

IB = 874.77 Amp

[𝑘𝑉𝑏 for cable because fault point at F]

If = 1.959 ∠−70.850 p.u. x IB

= 1.959 ∠−70.850 x 874.77 =1714 Amp

If = 1714 Amp

Voltage at 11 KV bus : ZAF = j0.1 + 0.0744 +j0.0992 + j0.16 + 0.093 + j0.55

= 0.1674 + j0.4142

= 0.447 ∠67.990 p.u

Voltage at 11 KV bus = ZAF p.u x If p.u

= 0.447 ∠67.990 x 1.959 ∠−70.850

= 0.874 – j0.0436 = 0.875 p.u

= 0.875 x Base voltage

= 0.875 x11 = 9.615 KV

Voltage at 11 KV bus = 9.615 KV

5. A 25 MVA, 11 KV, 3φ generator has a subtransient reactance of 20 %. The generator supplies two

motors over a transmission line with transformers at both ends as shown in fig. The motors have rated

input of 15 and 7.5 MVA, both 10 KV and 20% subtransient reactance. The 3φ transformers are both

rated 50 MVA, 10.8/121 KV with delta-star connection with leakage reactance of 10% each. The series

reactance of the line is 80 Ω. Draw the impedance diagram of the system with reactances marked in

p.u. when symmetrical fault occurs at bus 2 and calculate fault current. (16)

Generator values:

G = 25 MVA; 11KV;𝑿𝒅′′=20% M1 = 15 MVA; 10KV; 𝑿𝒅

′′=20% M2 = 7.5 MVA; 10KV; 𝑿𝒅′′=20%

Transformer rating:

X=50MVA, 10.8/121 KV;Xd =10%

Transmission line rating : X = 80 Ω, MVAbnew = 25, MVAb =11KV



Generator





= j 0.2 x (11/11)2 x(25/25)

Znew = j 0.2 p.u

Transformer 1: (referred to primary)

KVb =11KV





= j 0.1 x (10.8/11)2 x(25/50)

Znew = j 0.0482 p.u

Transmission line:

KVb,new = KVb old x 𝐻.𝑇 𝑟𝑎𝑡𝑖𝑛𝑔 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑒𝑟 1

𝐿.𝑇 𝑟𝑎𝑡𝑖𝑛𝑔 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑒𝑟 1

= 11 x 121

10.8 =123.24 KV

KVb,new =123.24 KV

Zp.u. new = 𝑍𝑎𝑐𝑡𝑢𝑎𝑙


𝑗 80

𝐾𝑉𝑏2 x MVAb =

𝑗 80 𝑥 25

123.242 = j0.1316 p.u

Zp.u. new = j0.1316 p.u

Transformer 2: (referred to primary)





= j 0.1 x (121/123.4)2 x(25/50)

Znew = j 0.482 p.u

Motor 1 :

KVb,new = KVb old x 𝐿.𝑇 𝑟𝑎𝑡𝑖𝑛𝑔 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑒𝑟 2

𝐻.𝑇 𝑟𝑎𝑡𝑖𝑛𝑔 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑒𝑟 2

= 123.24 x 10.8

121 =11 KV

Zp.u. new = j 0.2 x (10/11)2 x(25/11)

= j0.2755 p.u

Zp.u. new = j0.2755 p.u

Motor 2 :

Zp.u. new = j 0.2 x (10/11)2 x(25/50)

= j0.551 p.u

Zp.u. new = j0.551 p.u



Impedance diagram

Fault occurs at bus 2, the circuit becomes.

Find ZTh.

ZTh = 𝑗0.2482 𝑥 [𝑗0.6136+𝑗0.1837]

𝑗0.2482 + 𝑗0.6136+𝑗0.1837 = j0.1893 p.u

ZTh = j0.1893 p.u

If = 𝐸𝑇ℎ

𝑍𝑇ℎ+𝑍𝑓 =

1∠00

𝑗0.1893 = -j5.28 = 5.28 ∠ − 900 p.u

If = 5.28 ∠ − 900 p.u

Base current Ibase = 𝑀𝑉𝐴𝑏

√3 𝑥 𝐾𝑉𝑏 =

25 𝑥 106

√3 𝑥 123.24 𝑥 103

[𝐾𝑉𝑏 for secondary of transformer 1 or transmission line]

= 117.119 Amp.

Base current Ibase = 117.119 Amp.

Actual fault current = If x Ibase = 5.28 x 117.119 = 618.388 Amp.

Actual fault current = 618.388 Amp.



6. A 3-phase, 5 MVA, 6.6 kV alternator with a reactance of 8% is connected to a feeder of series

impedance of 0.12+j0.48 ohms/phase per km. The transformer is rated at 3 MVA, 6.6kV/3.3 kV and

has a reactance of 5%. Determine the fault current supplied by the generator operating under no-load

with a voltage of 6.9kV, when a 3-phase symmetrical fault occurs at a point 15km along the feeder.

Solution:

The single line diagram of the power system and the prefault reactance diagram are shown in fig

respectively. Let the point F be the point where fault occurs.

Let us choose generator rating as base values.

MVAb = 5 MVA and kVb = 6.6 kV

To find Eg and Vpf:

Here the generator is not delivering any load current and so the induced emf of the generator will be

same as operating voltage.

Actual value of induced emf, Eg = 6.9 kV

p.u. value of induced emf = 𝐴𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒

𝐵𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒 =

6.9

6.6 = 1.0455 p.u.

p.u. value of induced emf = 1.0455 p.u.

The prefault voltage, Vpf at the fault point F can be obtained using the transformer ratio as below.

Actual value of prefault voltage, Vpf = 6.9× 33

6.6 = 34.5 kV

The base kV referred to HT side of transformer = 6.6× 33

6.6 = 33 kV

The p.u. value of the prefault voltage, Vpf = 𝐴𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒

𝐵𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒 =

34.5

33 = 1.0455 p.u.

Vpf = 1.0455 p.u.

Note: The open circuit p.u. value of voltage is same at every point in a series path irrespective of their actual

voltages.

To find generator reactance:

Since the rating of the generator is chosen as base value, the p.u. reactance of the generator will be

same as the specified value.

p.u. reactance of the generator, Xd = 8% = 0.08 p.u.

Xd = 8% = 0.08 p.u.

To find transformer reactance:

The transformer reactance can be computed with respect to the new base using the following

equation.

Xpu,new = Xpu,old × (𝑘𝑉b old

𝑘𝑉b new)2 ×

𝑀𝑉𝐴b,new

𝑀𝑉𝐴b,old

Here Xpu,old = 5% =0.05 p.u.

p.u. reactance of transformer, XT = 0.05× 66

6.6 ×

5

3= 0.0833 p.u.

XT = 0.0833 p.u.



To find feeder reactance:

The base impedance, Zb = (𝑘𝑉b)2

𝑀𝑉𝐴b =

(33)2

5 = 217.8 /phase.

Actual impedance of the feeder for a length of 15km, Zfeed = impedance/km length

Zfeed = (0.12+j0.48)15 = 1.8+j7.2/phase

p.u. value of the impedance of the feeder, Zfeed = 𝐴𝑐𝑡𝑢𝑎𝑙 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒

𝐵𝑎𝑠𝑒 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 =

1.8+𝑗7.2

217.8

Zfeed = 0.0083+j0.0331 p.u.

To find fault current:

The prefault reactance diagram can be represented by Thevenin’s equivalent circuit shown. Here the

Thevenin’s voltage is the prefault voltage is the prefault voltage at fault point F and the Thevenin’s

impedance is the sum of three impedances in series.

Zth = jXd + jXT + Zfeed = j0.08 + j0.0833+ j0.0083 + j0.0331

= 0.0083 + j0.1964 p.u. = 0.196687.6 p.u.

Zth = 0.196687.6 p.u.

The fault in the feeder can be represented by a short circuit as shown. Now the current If through the

short circuit is the fault current.

p.u. value of fault current, If = 𝑉th

𝑍th =

1.04550

0.196687.6 = 5.3179-87.6 p.u.

Note: The solution can also be attempted by neglecting the resistance of the feeder. In this case,

Zth = j0.1964 = 0.196490 p.u. and

If = 5.3233-90 p.u.

It is important to note that the error in neglecting the resistive component is negligible.

Base current, Ib = 𝑘𝑉𝐴b

√3𝑘𝑉b =

𝑀𝑉𝐴b ×1000

√3𝑘𝑉b =

5×1000

√3 ×33 = 87.4773 A.

Ib = 87.4773 A.

Actual value of fault current, If = p.u. value of If Ib = 5.3179-87.6 87.4773

If= 465.2-87.6 amps.

Result:

Fault current = 5.3179-87.6 p.u. or 465.2-87.6 A.



UNIT IV FAULT ANALYSIS – UNBALANCED FAULTS

PART - A

1. What is meant by symmetrical fault? (A/M 2018)

The fault is called symmetrical, if the fault current is equal in all the phases.

i.e. Ia = Ib = Ic

2. Explain the concept of sequence impedance and sequence networks. (A/M 2018)

One of the most powerful tools for dealing with unbalanced polyphase circuits is the method of

sequential components. An unbalanced system of n related phasors can be resolved into n systems of

balanced phasors called sequential components. Sequential newtorks are positive, negative and zero

sequence components.

3. What are the features of zero sequence current? N/D 2017, M/J 2014

It consists of three phasors equal in magnitude and with zero phase displacement from each other.

Zero sequence phasors a,b,c can be written as 0 0 0

a b cI I I

Where 0 0 0, ,a b cI I I are the sequence components of ,a b cI I and I

4. Write down the equation to determine symmetrical currents from unbalance current.

N/D 2017, M/J 2016

0

2

2

0

2

2

,

1 1 11

13

1

1

3

1

3

1

3

a a

a b

a c

a a b c

a a b c

a a b c

The symmentrical componets of currents are

I I

I a a I

I a a I

I I I I

I I aI a I

I I a I aI

5. Express the unbalance current in terms of symmetrical components. (A/M 2017)

0

0

2

2

2

, , .

, , .

1 1 1

1

1

1 120 1 240

a b c

a a a

a a

b a

c a

Let I I I betheunbalanced phasecurrents

Let I I I bethe symmetrical components of phase

I I

I a a I

I a a I

where a and a

6. Draw the zero sequence network of Y - ∆ transformer with neutral ungrounded. (A/M 2017)



7. Why the neutral grounding impedance Zn appears as 3Zn in zero sequence equivalent circuit?

(N/D 2016)

Voltage drop is ZN*IN=ZN*3*I0=I0*3ZN. Therefore, to obtain the correct voltage drop seen

in the neutral conductor with a flow of only, model the zero sequence impedance as 3Zn.

8. What are symmetrical components of a 3ph system? (M/J 2016)

One of the most powerful tools for dealing with unbalanced polyphase circuits is the method

of symmetrical components. An unbalanced system of n related phasors can be resolved into n systems

of balanced phasors called symmetrical components. Symmetrical components are positive, negative

and zero sequence components.

9. State Fortescue theorem. (N/D 2015)

Fortescue's Theorem for a Three-Phase Unbalanced System. In 1918, C. L. Fortescue stated his

theorem: unbalanced phasors can be represented by systems of balanced phasors. Sequence

components were created to facilitate calculations in unbalanced circuits and systems.

10. Which type of fault is very common in power system? (N/D 2015)

Single line to ground fault is most common occurrence fault in power system.

It occurs about nearly 70 % in fault conditions.

11. What are the observation made from the analysis of various fault ? Nov/Dec 2013

Finding out the value of Fault current under different fault conditions

Zero sequence is not presence in LL fault conditions

Positive and negative sequence component of currents are opposite and equal to each other in

LL fault condition

Positive and negative sequence component of voltage are equal in LLG fault condition

12. Write the boundary condition for the single line to ground fault. Nov/Dec 2013

If the generator is solidly grounded, Zn = 0 and for bolted fault or solid fault,

Zf = 0.

If the neutral of the generator is ungrounded, the zero sequence network is open circuited. 0 0a a a fI I I and I

13. Define negative sequence impedance. May/June 2013

Impedance offered to the flow of negative sequence current is known as negative sequence impedance

and it is denoted by Z-. The negative sequence impedance is occurred in all the fault condition and it

is important to find the fault current.

14. What is positive, negative and zero sequence impedance?

Impedance offered to the flow of positive sequence current is known as positive sequence

impedance and it is denoted by Z+

Impedance offered to the flow of negative sequence current is known as negative sequence

impedance and it is denoted by Z-

Impedance offered to the flow of zero sequence current is known as Zero sequence impedance

and it is denoted by Z0

15. What is the assumption made in short circuit studies?







16. Find the fault current whose impedance are j0.2 & j015 in series and in parallel with j0.35 if

prefault voltage at the fault point is 0.97 p.u.?

Ans:

j0.2 and j0.15 are in series

Its becomes, j0.2 + j0.15 = j0.35

j0.35 is in parallel with j0.15.

0.35 0.150.105 . .

0.5 0.15

0.979.238 . .

0.105

Th

Thf

Th

j jZ j p u

j j

VFault current I j p u

Z j

17. Name the method of reducing short circuit capacity and uses of short circuit capacity.

By introducing artificial series reactors

By providing neutral reactance

By introducing a large value of shunt reactance between buses.

18. What are the reactances used in synchronous machines? "

'

1. tan

2. tan

3. tan .

d

d

d

Subtransient reac ce X

Transient reac ce X

Synchronous reac ce X

19. Define transient reactance.

It is the ration of induced emf on no-load and the transient symmetrical rms current.

It is given by,

'| | 1

tan ,1 1| ' |

tan

tan

tan

g

d l

a f

l

f

a

ETransient reac ce X X

I

X X

where X Leakage reac ce

X Field winding reac ce

X Armature reaction reac ce

20. What is the significant of transient reactance in short circuit studies?

The subtransient reactance can be used to estimate the initial value of fault current immediately on the

occurrence of the fault. The maximum momentary short circuit current rating of the circuit breaker

used for protection or fault clearing should be less than this initial fault current.

21. Difference between Load flow and short circuit studies.

Load flow Studies Short circuit studies

The resistance and reactance are considered The resistance are neglected

To solve load flow analysis, the bus admittance

matrix is used

To solve short circuit analysis, the bus

impedance matrix is used

It is used to determine the exact voltage and

currents

Prefault voltage are assumed to be1 p.u. and the

prefault current can be neglected.

22. Why synchronous impedance more than transient reactance?

The short circuit is of larger magnitude than the current measured half a second later. This mean that

magnitude of Z will have a minimum value immediately following the fault and then grow. This

change of the short circuit current with time is due to the characteristics of the synchronous generators.

Thus the synchronous impedance is more than transient reactance.



PART – B

1. Explain the detail derivation for Sequence networks analysis of single line to ground(LG Fault)

N/D 2017, M/J 2016

Suppose a line to ground fault is occurs on phase a connected to ground through impedance Zf.

0

2

2

0

0 (1)

1 1 11

1 (2)3

1

0,

1 1 11

13

a f a

b c

f a

a a

a b

a c

b c

a

a

a

V Z I

I I

I I

Symmetrical componets ofcurrents are

I I

I a a I

I a a I

Substitute for I I symmetrical components of currents are

I

I a a

I

2

2

0 (3)

1 0

aI

a a

0

0

(3),

3

3

3 3

(4)3

aa

aa

faa

aa a a

Fromeq we get

II

II

III

II I I



0

2

2

1 1 1

1 (6)

1

a a

b a

c a

The phasevoltage are givenby

V V

V a a V

V a a V

0

0

.(6) ,

a a a a

a f a

a a a f a

Fromeq we get

V V V V

FromtheconditionV Z I

V V V Z I

0

0

,

(7)3

33

3

aa

KK KK KK f

af a a

KK KK KK f

Substituting the symmetrical components we get

EI

Z Z Z Z

The fault current is

EI I I

Z Z Z Z

2. Explain the derivation for Sequence networks analysis of line to line (LL Fault)

N/D 2016, A/M 2016

0( )

, 0, :

b c

a

b c f b c b f b

b c a

I I

I unloaded generator

V V Z I V V Z I

Substitute for I I I the symmetrical components of current are

0

2

2

1 1 11

13

1

a a

a b

a c

I I

I a a I

I a a I

0 0 0

0 0 0 0

,

0 0 0

0 0

0 0 0

(5)

a KK a

a a KK a

a KK a

a KK a KK a

a a KK a

a KK a KK a

From sequence network of generator symmetrical

voltages are givenby

V Z I

V E Z I

V Z I

V Z I Z I

V E Z I

V Z I Z I



0

2

2

1 1 1 01

13

1

a

a b

a b

I

I a a I

I a a I

2 2

,

( ) ( )

b

a a a f

a a

Substitutethevalueof I we get

a a V V a a I Z

V V

0

2

2

0

10 0

3

1

3

1

3

0

a b b

a b b

a b b

a a a

I I I

I aI a I

I a I aI

I I and I

0

2 2

2 2

2 2

2 2

1 1 1 1 1 1 0

1 1

1 1

0, ( )

( )

a a

b a a

c a a

a b a a a

c a a a b

The Phasecurrent are givenby

I I

I a a I a a I

I a a I a a I

I I a I aI I a a

I aI a I I a a I

2

2

0

2

2

1 1 1 0

1

1

0

a

b a

c a

a

a a a

b a a

c a a

The Phasevoltage are

V

V a a V

V a a V

V

V V V

V a V aV

V aV a V

2

,

( )

b c f b

b c

a a f b

FromtheconditionV V Z I

SubstitutingV and V we get

a a V V Z I

2 2

,

( ) ( )

b

a a a f

a a

Substitutethevalueof I we get

a a V V a a I Z

V V

0

, , ,

0

a a

a K K f a

aa

K K f

a a

a

SubstitueV V we get

E Z Z Z I

EI

Z Z Z

I I

I



0

2 2 2

2 2 2

2

min

1 1 1 0 0

1 0 ( ) ( )

1 0 ( ) ( )

( )

( 0.5 0.866 0.5 0

a a a a

b a a a

c a a a

b c a

Current phase do e

I I I I

I a a I a a I a a I

I a a I a a I a a I

The fault current is I I a a I

j j

.866) 1.732

3

a a

a

I j I

j I

,

3

a

af b

KK KK f

Substituting I we get

j EI I

Z Z Z

3. Explain the derivation for Sequence networks analysis of Double line to ground fault (LLG

Fault) N/D 2015, A/M 2015

A three phase generator with a fault on phases b and c through an impedance Zf to ground.

0

( ) (1)

:

a

b c f

b c f f f b c

I

I I I

V V Z I Z I I

The symmetrical components of voltage are

0

2

2

1 1 11

1 (2)3

1

a a

a b

a c

V V

V a a V

V a a V

Substitute Vb =Vc in equation (2), we get

0

2

2

1 1 11

13

1

a a

a b

a b

V V

V a a V

V a a V



0 1( 2 )

3

1( )

3

1( )

3

(3)

a a b

a a b

a a b

a a

V V V

V V V

V V V

V V

0

2

2

0

0 2

0 2

0 2 0 2

0 2 2

1 1 1

1

1

2 ( ) ( ) (4)

a a

b a

c a

a a a a

b a a a

c a a a

f b c a a a a a a

a a a

The Phasecurrent are givenby

I I

I a a I

I a a I

I I I I

I I a I aI

I I aI a I

I I I I a I aI I aI a I

I I a a I a a

0

0

0

( ) (5)

(5) (6),

3

, 3 (6)

a a a

b c a

b f a

I I I

substituting in we get

I I I

Fromthecondition V Z I

0

2

2

0

0 2

0 0

1 1 1

1

1

3 (7)

a a

b a

c a

a a a a

b a a a a a

b a a b f a

The Phasevoltage are

V V

V a a V

V a a V

V V V V

V V a V aV V V

V V V V Z I

0 0 0

(8)

a KK a

a a KK a

a KK a

The symmetrical components voltageis givenby

V Z I

V E Z I

V Z I

0

0

[ ](9)

3

a KK aa

KK f

E Z IThen I

Z Z

[ ](10)a KK a

a

KK

E Z II

Z

0

0

a a a

a a a

I I I

I I I



0

0

( 3 )

3

aa

KK KK f

KK

KK f KK

onby solving we get

EI

Z Z ZZ

Z Z Z

00

0

3

( 3 )3

3

KK af a

KK KK fKK f

KK

KK KK f

Z EI E

Z Z ZZ ZZ

Z Z Z

4. Determine the fault current and MVA at faulted bus for a line to ground (solid) fault at bus 4 as

shown in fig. A/M 2018, N/D 2017, A/M 2017

G1,G2: 100MVA, 11KV, X+= X-=15%, X0=5%, Xn=6%

T1,T2: 100 MVA, 11 KV/220KV, Xleak= 9%

L1,L2: X+= X-=10%, X0=10%, on a base of 100 MVA.Consider a fault at phase ‘a’

Sol :-

Step 1 : Positive sequence thevenin equivalent viewed from bus 4:

Step 2: Negative sequence thevenin equivalent viewed fron bus 4:



. 2

2

0.121

0.121100 0.1

11

0.1 0.1 .

0.1 0.10.05

0.1 0.1

new

p u

Actual value jFor transmissionline Z Base MVA

Basevalue Base KV

jj

j and j arein parallel

j jj

j j

0

0

0

0

44 44 44

.

1 0

1 0

0.1075 0.1075 0.1533

2.7152

3 3 2.7152

8.1455 .

( )

1008.1455 42.75

3 11

3

a

a a a

f

f a

f p u

prefault voltage E V

I I I

V

Z Z Z Z

j j j

j

I I j

j p u

Actual fault current KA I Basecurrent

KA

Fault MVA KV

3 11 42.75 814.55

KA



:

42.75

0

42.75

a f

a c

n a b c

Current in phase domain

I I KA

I I

I I I I KA

Assumption made in short circuit studies (N/D 2017)





5. Two 11 KV, 20 MVA, three phase star connected generators operate in parallel as shown in

figure. The positive, negative and zero sequence reactance’s of each respectively, j0.18, j0.15,

j0.10 p.u. The star point of one of the generators is isolated and that of other is earthed through

a 2 ohms resistor. A single line to ground fault occurs at the terminals of one of the generators.

N/D 2017

Estimate,

i) The fault current, ii) Current in grounding resistor and iii)The voltage across grounding

resistor,

Solution :-

Positive sequence network : (Generator G1 & G2 are in parallel)

) :

| | 2

3.07 2 6.14

r

iii Voltageacross the grounding resistor

I in Amp

KV

0.18 0.18

0.090.18 0.18

j jZ j

j j

Negative Sequence network :

0.15 0.150.075

0.15 0.15

j jZ j

j j

Zero Sequence network, 0

. 2

2

0

3

2

202 0.33 .

11

3

0.1 3 0.33

0.99 0.1

s n

bn p u

b

s n

Z Z Z

MVAZ

KV

p u

Z Z Z

j j

j

Sequence network for L-G fault:



0

0

1 0

0.09 0.075 0.99 0.1

1

0.99 0.265

a a a

a

I I I

E

Z Z Z

j j j

j

) . . 3

13

0.99 0.265

2.827 0.756 .

f ai Fault current I in p u I

j

j p u

3

) :

2.827 0.756 . .

| | 2.926 .

20 1010497

3 3 11

| | 2.926 10497 3.07

r

f

f

b

b

r

ii Current in the grounding resistor I

I j p u

I p u

MVABasecurrent A

KV

I in Amp KA

) :

| | 2

3.07 2 6.14

r

iii Voltageacross the grounding resistor

I in Amp

KV

6. A single line diagram of power system is shown in figure, determine the fault current and fault

MVA for a line to line fault occurs between phases b and c at bus 4 as shown in fig. (May-13)

G1:G2 : 100 MVA, 20KV, X+=X—15 %

T!,T2 : 100 MVA, 20/345 KV, Xleak=9 %: Line : X+=X—5 %

Solution:

Positive Sequence Thevenin equivalent:

Negative sequence thevenin equivalent:



0

2 2

2 2

2

min :

1 1 1 1 1 1 0

1 1 13.426

1 1 13.426

0

1 0 ( 13.426) ( 13.426) 23.254

23.254

0

a a

b a

c a

a

b

c b

n a b c

Current in phase do e

I I

I a a I a a j

I a a I a a j

I

I a j a j KA

I I KA

I I I I

Fa

3 ( ) 3 23.254 20

805.542

fult MVA I KA KV

0 1 0

1 04.651 . .

0.1075 0.1075

100| | | | 4.651 13.426

3 20

13.426, 13.426

a

a a

a a

a a

prefault voltage E V

I I j p uj j

I I KA

I j I j KA

0

2 2

2 2

2

min :

1 1 1 1 1 1 0

1 1 13.426

1 1 13.426

0

1 0 ( 13.426) ( 13.426) 23.254

23.254

0

a a

b a

c a

a

b

c b

n a b c

Current in phase do e

I I

I a a I a a j

I a a I a a j

I

I a j a j KA

I I KA

I I I I

Fa

3 ( ) 3 23.254 20

805.542

fult MVA I KA KV



UNIT 5 – STABILITY ANALYSIS IN POWER SYSTEM

1. Define Stability. A/M 2018

The stability of a system is defined as the ability of power system to return to stable (Synchronous) operation

when it is subjected to a distance.

2. What is the significance of sub-transient reactance and transient reactance in short circuit studies? A/M

2018

The sub - transient reactance is used to estimate the initial value of fault current immediately on the

occurrence of the fault. The maximum momentary short circuit current rating of the circuit breaker used for

protection or fault clearing should be less than this fault clearing value.

3. Define transient stability of a power system. (N/D 2017, A/M 2016)

It is the ability of the system to bring it to a stable condition after a large disturbance. Large disturbance

can occur due to the occurrence of fault, sudden outage of a line, sudden loss of excitation, sudden

application or removal of loads, etc.

4. State the cause of voltage instability. N/D 2017

A system enters a state of voltage instability when a disturbance, increase in load demand, or change in

system condition causes a progressive and uncontrollable drop in voltage. The main factor causing instability

of the power system to meet the demand for reactive power. The voltage drop that occurs when active power

and reactive power flow through inductive reactance associated with the transmission network.

5. Define critical clearing angle. (N/D 2015)

For any given initial load in the case of a fault clearance on a synchronous machine connected to an

infinite bus bar, there is a critical clearing angle. If the actual clearing angle is greater than the critical

value, the system is unstable, otherwise the system is stable. Maximum allowable angle for a system to

remain stable.

6. Define swing curve. What is the use of Swing curve? (A/M 2017)

2

( . ) ( . )2 m p u e p u

H dP P

f dt

The graphical display of δ versus t is called the swing curve. The plot of swing curves of all machines

tells us whether machines will remain in synchronism after a disturbance.

7. State equal area criterion. N/D 2017, A/M 2017

The equal area criterion for stability states that the system is stable if the area under Pa – δ curve reduced

to zero at some value of δ.

This is possible if the positive (acceleration) are under Pa – δ curve is equal to the negative (deceleration)

area under Pa – δ curve for a finite change in δ. Hence the stability criteria is called equal area criterion.

8. Define voltage stability. (N/D 2016)

Ability of power system to maintain steady acceptable voltages at all buses in power system under normal

operating conditions and after being subjected to a disturbance.

Reactive power balance.



9. State few techniques to improve the stability of the power system. (N/D 2016)

Increase of the system voltage and use of AVR

Use of high speed excitation systems.

Reduction in system transfer reactance

Use of high speed reclosing

10. How are power system stability classified? (N/D 2015)

Power system stability can be classified into as follows

Rotor angle stability

Voltage stability

Frequency stability

11. What is steady state stability limit? (Nov/Dec- 14)

The gradual increase of the generator power output is possible until the maximum electrical power

is transferred. This maximum power is referred to as the steady state stability limit.

12. Define dynamic stability with an example. (May/June -14)

It is the ability of the system to remain in synchronism when it is subjected to sustained oscillations

followed by transient state.

13. Differentiate between voltage stability and rotor angle stability. (N/D 2013)

Voltage Stability Rotor angle stability

Ability of power system to maintain

steady acceptable voltages at all buses in

power system under normal operating

conditions and after being subjected to a

disturbance.


Ability of inter – connected

synchronous machines of a power

system to remain in synchronism.

Torque balance of synchronous

machines

14. Define swing curve? What is the use of this curve. (N/D 2013) 2

( . ) ( . )2 m p u e p u

H dP P

f dt

The graphical display of δ versus t is called the swing curve. The plot of swing curves of all machines tells

us whether machines will remain in synchronism after a disturbance.

15. What are the causes of oscillatory and non-oscillatory instabilities in power systems? (A/M 10)

Oscillatory :

Due to insufficient damping torque

Due to unstable control action

Non – oscillatory :

Due to insufficient synchronous torque

16. Write the power angle equation of a synchronous machine connected to an infinite bus and also the

expression for maximum power transferable to the bus.

1 1 2 2eq

b b

M S M SM

S S

1 11

12

max

| ' || || ' | sin

sin

e

e C

E VP P E G

X

P P P

This equationis called power angleequation

Expression for Maximum Power transfer:-

max

12

| ' || |sin

E VP

X



17. Define infinite bus in a power system. (M/J 13)

The substation bus voltage and frequency is assumed to remain constant. This is called as infinite bus,

since its characterized do not change regardless of the power supplied or consumed by any device

connected to it.

18. Give the expression for critical clearing time. (N/D 07)

Critical time margin = critical clearing time – clearing time specified

( )cr critical spec

spec

t t

where

t Specified clearing time

19. Distinguish between small disturbance and large disturbance.

Small disturbance Large disturbance

The disturbances that cause small and

gradual infinitesimal variation in system

variables such as rotor angle, voltage,

etc., are classified as small disturbance.

Any disturbance that causes a change in

the admittance matrix encountered in

stability is classified as large disturbance

Eg – Fault occurring, removal of line

20. What is meant by synchronism and damping torques?

The component of electrical torque proportional to rotor angle deviation from initial value is referred to as

synchronizing torque and that proportional to speed deviation from initial value is called damping torque.

Assume initially, the machine is in steady state, then

Machine speed = Rated speed = synchronous speed

Then Initial speed deviation = 0

21. Define power or torque angle.

Under normal operating conditions, the relative position of the rotor axis and the resultant magnetic field

are fixed. The angle between the two is known as the power angle or torque angle.

PART - B

1. Derive the swing equation from the basic principles. Why it is non-linear? (A/M ’18,17, N/D ’17)

Let Tm be the driving mechanical torque

Te be the electrical torque

The motor action and generator action is shown in figure.

For generator action, Tm and Te are positive

θm is positive

Under steady state with losses neglected.

Tm = Te

Acceleration torque Ta = Tm – Te = 0

i.e. no accelerating torque

Ta = Tm – Te

Let J be the moment of inertia of the prime mover and generator.



From Law’s of rotation, 2

2

2

2(1)

m

a

mm e

dAcceleration

dt

AccelerationtorqueT J

dJ T T

dt

Where θm is the angular displacement of the rotor with respect to the stationary reference axis on stator.

θm increase with time even at constant synchronous speed.

(2)

.

tan

m sm m

m

sm

t

where Angular displacement of rotor

before disturbancein mechancal radians

t Cons t angular velocity

. .(2), ,

(3)m mm sm

Diff eq wrt t we get

d dt

dt dt

2 2

2 2

. .(3), ,

m m

Diff eq wrt t rotor accelerationis

d d

dt dt

2

2

(1) ,

mm e

Substituting ineq we get

dJ T T

dt

2

2

,m

mm m m m e

Multiplying by onboth side

dJ T T

dt

Inertia constant

tanmM J istheinertiacons t

2

2

. . .

( ) (4)mm e

i e Angular momentumof the rotor at synchronous speed

dM P P P T

dt

p.u. Inertia Contant

21

2

,

. .

K mKinetic Energy of rotating masses W J

Stored kinetic energy in mega joules of turbine

alternator and exciter rotor at synchronous speedp u of H

Machinerating in MVA

2

2

1

2 sec

2

sm

B

Bsm

sm

J

HS

HSJ M

2

2

(4)

2(5)mB

m e

sm

Substituting ineq

dHSP P

dt



2

2(6)mB

m e

by solving we get

dHSP P

f dt

2

2

,

. .

. .

B

m m e

B B

m

B

e

B

Dividing by MVArating S onboth side we get

d P PH

f dt S S

Pp u mechanical power

S

Pp u electrical power

S

2

( . ) ( . ) ( . ) max2sin (7)m

m p u e p u m p u

dHP P P P

f dt

2

( . ) ( . ) ( . )2

( . ) ,

p u m p u e p u

p u

dM P P

dt

Hwhere M in radians

f

2

( . ) ( . )2

exp deg ,

180m p u e p u

If ressed inelectrical rees

H dP P

f dt

Theseequation arecalled as

Swing Equation

2. Describe the algorithm for modified Euler method of finding solution for power system stability

problem studies. (A/M’17)

Numerical integration techniques can be applied to obtain approximate solutions of non-linear

differential equations.

Consider a generator connected to an infinite bus through two parallel lines and a 3ϕ fault occurs at

the middle of line 2 as shown in figure.

Let Pm be the input power which is a constant.

Prefault Condition: Under steady state operation,

Power transfer from generator to an infinite bus,

Pe=Pm

0 max1 0

1

1

0 0

max1 max1

'sin sin

sin sin (1)

m

m m

E VP P

X

P P

P P

max1

1

1

'

tan

E Vwhere P

X

X Transfer reac ce for the prefault condition



0

0

,

2

.

. ., 0

The rotor is running at synchronous speed

f

Changein angular velocity is zero

i e

During the fault : Consider a 3ϕ fault occurs at the middle of one line 2 as shown in fig.

2 1 max 2

max 2

| ' || |sin sin

| ' || |

tan

e

II

II

II

E VP P

X

E Vwhere P

X

X Transef reac ce during the fault

2

max 22

. ,

sinmm a

the swing eq is givenby

d f fP P P

dt H H

(1)

2 (1)

2

. int var ,

(2)

(3)ma

theaboveeq canbetransformed othe state iable form

d

dt

d d fP

dt dt H

1 0

(1)

1

(1)

1

.

. (4)

. (5)

i

i

P

i i

P

i i

Computethe first estimate at t t t

dt

dt

dt

dt

Compute the derivatives : Using the predicted value, determine the derivatives at the end of iteration.

(2)

1 (6)Pi

P

i

d

dt

1

1

(2)

(7)Pi

Pi

a

d fP

dt H

Computing the final estimated corrected value,

1

1 (8)2

Pi iC

i i

d d

dt dtt

1

1 (9)2

Pi iC

i

d d

dt dtt



3. Derive the swing equation of a single machine connected to an infinite bus system and explain

the steps of solution by Runge-Kutta method (N/D ’17)

Runge – Kutta method

The following steps involved in Runge-Kutta method to determine stability.

I estimates :

1

'

1 ( )

(1)

(2)

i

i

m e i

i

dK t t

dt

d fl t P P t

dt H

1

12

2 ( ( /2))

:

(3)2

' (4)i

i

m e K

II estimates

lK t

fl P P t

H

2

23

3 ( ( /2))

:

(5)2

' (6)i

i

m e K

III estimates

lK t

fl P P t

H

3

4 3

4 ( ( )

:

( ) (7)

' (8)i

i

m e K

IV estimates

K l t

fl P P t

H

1

1 1 2 3 4

1 1 2 3 4

12 2 (9)

6

12 2 (10)

6

i i

i i

Final estimates at t t

K K K K

l l l l

4. Explain the methods of improving power system stability (N/D’16)

Methods of Improving transient Stabiltiy:-

Reducing in the disturbing influence by minimizing the fault severity and duration

Increasing the restorating synchronizing forces.

Reduction of acceleration torque through control of prime mover mechanical power.

Reduction of accelerating torque by applying artificial load.

Traditional approach to transient stability problems

Increase of system voltage by AVR

Use of high speed excitation systems

Reduction in system transfer reactance.

Use of high speed reclosing breakers

Single pole operation of reclosing circuit breakers

Use of bundled conductors

High speed fault clearing

Increasing no .of parallel lines between points

Regulated shunt compensation



Dynamic breaking

Single pole switching

Generator tripping.

Recent trends :-

HVDC links

Use of breaking resistors

Short circuit current limiters

Turbine fast valving of bypassing valve

Full load rejection technique.

5. A generator is operating at 50 Hz delivers 1 p.u. power to an infinite bus through a transmission

circuit in which resistance is ignored. A fault takes place reducing the maximum power

transferable to 0.4 p.u., whereas before the fault, this power was 1.6 p.u. and after the clearance

of the fault, it is 1.2 p.u. By the use of equal area criterion, determine the critical clearing angle.

(N/D’17)

The power angle curve is as shown in fig.

Pe1= 1.6 sin δ, Pmax1= 1.6 p.u., Pe2= 0.4 sin δ, Pmax2= 0.4 p.u. Pe3= 1.2 sin δ, Pmax3= 1.2 p.u.

Initial loading Pm = sin δ0 = 1/ 1.6

1

0

1sin 0.675

1.6rad

1

max

max3

1

sin

1sin 2.156

1.2

mP

P

rad

Applying equal area criterion,

Area A1 = Area A2

max

0

0 2 3 max

max max 2 0 max3 max

max3 max 2

1

( ) ( )

( ) cos coscos

1.0(2.156 0.675) 0.4cos0.675 1.2cos 2.156

1.2 0.4

cos 0.632

cos 0.632 0.887 50.82

cr

cr

m cr e e m cr

m crcr

cr

cr

P P d P d P

P P P

P P

rad

rad



6. A 150 MVA generator transformer unit having an overall reactance of 0.3 p.u. is delivering 150

MW to infinite bus bar over a double circuit 220 KV line having reactance per phase per circuit

of 100 ohms. A three phase fault occurs midway along one of the transmission lines. Calculate

the maximum angle of swing that the generator may achieve before the fault is cleared without

loss of stability.

Ans:

. . 2

2

Re tan( )

100150 0.31 . .

220

actualp u b

b

Xac ceof line X MVA

KV

j p u

Prefault condition : Impedance diagram for prefault condition is as shown in fig.

j0.31 is in parallel with j0.31

1

1

0

0

1

0

| ' || | 1.0 1.0sin sin 2.198sin

0.455

2.198sin 1.0

1.0sin 27.06 0.472

2.198

e

E VP

X

operating power angle is givenby

rad

During the fault : Positive sequence reactance diagram



Using Delta-star conversion, the circuit becomes

12 10

12 10 20

10 200

12 10 20

0.31 0.1550.0775 . .

0.31 0.155 0.155

0.155 0.1550.039 .

0.31 0.155 0.155

IN

N

Z Z j jZ j p u

Z Z Z j j j

Z Z j jZ j p u

Z Z Z j j j

Using star – Delta conversion,

0 0 2 212

0

0.3775 0.039 0.039 0.0775 0.3775 0.07751.2

0.039

IN N N N IN N

N

Z Z Z Z Z ZZ

Z

j j j j j jj

j

Power transfer during fault

2

12

| ' || |sin

1.0 1.0sin 0.833sin . .

1.2

e

E VP

X

p u

Post fault condition: Faulted line is removed by opening the circuit breaker at ends.

Impedance diagram for postfault is as shown in fig.

3

0.3 0.31 0.61 .

| ' || | 1.0 1.0sin sin 1.64sin . .

0.61

III

e

III

Z X p u

E VP p u

X

Power angle curve is as shown in fig.



1 1

max

max3

1sin sin 3.14 0.656 2.48

1.64

mPrad

P

Determining of critical clearing angle:

0

1 0 21.0cr

cr eArea A P d

max

2 3 max

cr

e m crArea A P d P

max

1.5sin 1.0 2.4

cr

crd

Applying equal area criteria A1 = A2 2.48

0.472

0.472 0.833sin 1.64sin (2.48 )cr

cr

cr cr

2.48

0.4720.472 0.833cos 1.64cos 2.48

cr

cr

0.472 0.833cos 0.393 1.294 1.64cos 2.48cr cr

cos (0.833 1.64) 1.294 2.48 0.472 0.393cr

0.807cos 0.321

cos 0.398

1.16

cr

cr

cr rad

UNIT 1 INTRODUCTION 1. What are the advantages of per unit ...

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