Chapter-7

Power system Stability

Introduction

Power System Stability of a power system is its ability to return to normal or

stable operating conditions after having been subjected to some form of
disturbances.

Classification

For analysis the synchronous stability of a power system can be classified into
3 categories.

• Steady state stability.

• Transient stability.

• Dynamic stability.

Steady state stability

• The steady-state stability of a power system is defined as the ability of

the system to return to normal or stable configuration after having been
subjected to some form of disturbance.

• Study of steady state stability is basically concerned with the

determination of upper limit of machines loading before losing
synchronism, provided the loading is increased gradually at a slow rate.

Transient Stability of a Power System:

 Transient stability of a power system refers to the ability of the system to

reach a stable condition following a large disturbance in the network
condition.

• Large disturbance: sudden application or removal of load, switching

operations, line faults or loss due to excitation the transient stability of
the system comes into play.

• Transient stability is characterized by the highest magnitude of power

flow just prior to the transient disturbance for which the system can
remain in synchronism once the transient fault is cleared.

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  Transient Stability can be improved by Increasing the system voltage,
Increase in the X/R ratio, Using high speed circuit breakers, Using
Auto Re-closing, employing lightning arresters, high neutral grounding
impedance, single pole switching, quick Automatic Voltage Regulators
(AVRs).

Dynamic Stability:

• Dynamic Stability is the ability of a system to reach its stable condition

after a very small disturbance (disturbance occurs only for 10 to 30
seconds). It is also known as small signal stability.

• It occurs mainly due to the fluctuation in load or generation level.

• Dynamic stability can be improved by using power system stabilizers.

Rotor Dynamics and the Swing Equation

The transient stability of the system can be determined by the help of the swing
equation. It describes the relative motion of the rotor with respect to the stator
field as a function of time.

Let Tm – mechanical torque input

Te – Electromagnetic torque developed in N-m which opposes Tm

Therefore net torque which causes acceleration of the motor or accelerating

torque,

Ta = Tm – Te ……..(1)

When the machine is operating in steady state, Ta = 0, Tm = Te

From the law of mechanics,

If I= moment of inertia, kg-m2

α = d2θ/dt2 = angular acceleration, degree/sec2

θ =angular displacement, degree

Then I. α =Ta i.e

I. d2θ/dt2 = Ta ……..(2)

Since the angular position θ of the rotor is continuously varying with time,
then the value of is θ given by

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 θ = ωt + δ …….(3)

Where, ω = synchronous speed in electrical degree/sec,

δ = Angular displacement between the rotor and reference axis in electrical

degree.

Relation between θ, ω and δ

Swing equation:

Differentiating equation (3)

Again differentiating equation (5),

Substituting equation (6) in equation (2), we have

Multiplying equation (7) by rotor speed ω,

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Where, M= I = Angular momentum

Pm = Tm = mechanical power input,

Pe = Te = Electrical power output

Pa = Accelerating power = difference between input and output power

Equation (9) is known as the swing equation.

Inertia constant: It is defined as the ratio of energy stored in mega joules to

the rating of the machines in MVA.

But Kinetic energy stored by the rotating body is given by

Comparing equation (11) and (12)

Further considerations of the Swing equations

Case-a: Two machine systems (Not swing together or swing non-

coherently)

Consider the equivalent circuit of a two machine system.

Let M1 = Angular momentum of 1st machine

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M2 = Angular momentum of 2nd machine

= Power angle of 1st machine

= Power angle of 1st machine

= Relative angle between two machines

Using the swing equation,

For 1st machine,

For 2nd machine,

We know that relative angle between two machines, i.e., angle between two
rotor axis,

Differentiating equation 21 w.r.to t, we get,

Again differentiating equation (22)

Using equation 19 and 20 in 23 we have,

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Multiplying both sides by

Where,

Similarly,

It is clear from equation 27 that inertia constant of equivalent machine is less

than the inertia constant of individual machines.

Case-b: Two machine systems (Swing together or swing coherently)

Swing equation for 1st machine

Swing equation for 2nd machine

Adding equation (31) and (32), we have

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Where, M = M1 + M2, Pa = Pa1+ Pa2

Similarly, H = H1 + H2

Therefore it is clear that the inertia constant of equivalent machine is the sum
of inertia constant of individual machines.

The Power angle equation (2017)

Power angle curve

It is defined as the variation of steady state real power with power angle for
both generator and motor action for constant values of E, V and Xd.

Consider a synchronous generator having direct axis synchronous reactance Xd

connected to infinite bus through a transmission line having reactance Xt as
shown in the fig. below.

Let

The complex power output of a generator is given by

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But S= P+jQ

Equating the real parts on both side of equation

Equating the imaginary parts on both side of equation

From equation (43) & (45), it is clear that both active and reactive power
depends on Generator voltage,(E), Bus bar voltage, (V), Total reactance, (X) and
Power angle, δ

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From equation (43), By taking δ in X-axis and real power P in Y-axis, a curve is
shown in the fig is obtained. This curve is called power angle curve.

For positive value of , E leads V applies to generator action.

For negative value of , E lags V applies to generator action.

Synchronizing power coefficients (Ps)

Let the generator is working under steady state conditions.

If δ =Power angle, Δδ =increase in power angle by a small amount, then

increase in synchronizing power is given by

Therefore the quantity Ps is known as Synchronizing power coefficient.

Equal area criterion for stability

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The Equal area criterion is a “graphical technique used to examine the
transient stability of the machine systems (one or more than one) with an
infinite bus”.

Here, the stability conditions are stated by equating the two area segments
which is present in the power angle diagram.

The power angle curve is considered which is shown in fig.1. Imagine a system
delivering ‘Pm’ power on an angle of δ0 (fig.2) is working in a steady state. When
a fault occurs; the circuit breakers opened and the real power (Pe) is decreased
to zero. But the Pm will be stable. As a result, accelerating power,

The power differences will result in rate of change of kinetic energy stored
within the rotor masses. Therefore, due to the stable influence of non-zero
accelerating power, the rotor will accelerate. Consequently, the load angle (δ)
will increase.

Now, we can consider an angle δc at which the circuit breaker re-closes. The
power will then come back to the usual operating curve. At this moment, the
electrical power will be higher than the mechanical power. But, the accelerating
power (Pa) will be negative. Therefore, the machine will get decelerate. The load
power angle will still continue to increase because of the inertia in the rotor
masses. This increase in load power angle will stop in due course and rotor of
the machine will start to decelerate or else the synchronization of the system
will get lose.

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If both the area of accelerating power and decelerating power is same i.e. A1 =
A2. Then the system will come back to stable state. This is called equal area
criterion.

Mathematical interpretation of equal area criterion

The Swings equation is given by

Multiplying both sides of the swing equation by 2(dδ/dt), we get

Multiply both side by ‘dt’, we have

Integrating both sides, we have

Where =Load angle

The condition of stability can come when

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Critical Clearing Angle

The critical clearing angle is defined as the maximum change in the load angle
curve before clearing the fault without loss of synchronism.

Or

Critical clearing angle is “the load angle at which the fault will be clear and the
system becomes stable”.

Factors affecting transient stability

The post-disturbance system reactance as seen from the generator: The

weaker the post-disturbance system, the lower the Pmax will be.

The duration of the fault-clearing time: The longer the fault is applied, the
longer the rotor will be accelerated and the more kinetic energy will be gained.
The more energy that is gained during acceleration, the more difficult it is to
dissipate it during deceleration.

The inertia of the generator:. The higher the inertia, the slower the rate of
change of angle and the lesser the kinetic energy gained during the fault.

The generator internal voltage: (determined by excitation system) and infinite

bus voltage (system voltage). The lower these voltages, the lower the Pmax will
be.

The generator loading before the disturbance: The higher the loading, the
closer the unit will be to Pmax, which means that during acceleration, it is more
likely to become unstable.

The generator internal reactance: The lower the reactance, the higher the
peak power and the lower the initial rotor angle.

The generator output during the fault: This is a function of faults location
and type of fault.

Further application of the equal area criterion

The equal area criterion is very useful means for analyzing stability of a system
of two machines or of a single machine supplied from an infinite bus. However
the computer is the only practical way to determine the stability of a large
system. Because the equal area criterion is so helpful to determine transient
stability, we continue to examine it briefly before discussing the determination
of swing curves by the computer approach.

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When a generator is supplying power to an infinite bus over parallel
transmission lines, opening one of the lines may cause the generator to lose
synchronism even though the load could be supplied over the remaining line
under steady state conditions. If a three phase short circuit occurs on the bus
to which two parallel lines are connected, no power can be transmitted over
either lines. However if the fault is at the end of one of the lines, opening
breakers at both ends of the line will isolate the fault from the system and
allow power to flow through the other parallel line. When a three phase fault
occur at some point on a double circuit line other than on the paralleling buses
or at the extreme ends of the lines, there is some impedances between the
paralleling buses and the fault. Therefore some power is transmitted while the
fault is still on the system.

Multi machine Stability Studies; Classical representation

Multi machine equations can be written similar to the one machine connected
to infinite bus bar. In order to reduce the complexity of the transient stability
analysis, some assumptions must be made, the assumptions are:

 Each synchronous machine is represented by a constant voltage source

behind the direct axis transient reactance.
 The governors actions are neglected and the input powers are assumed
to remain constant during the entire period of simulation.
 Using the prefault bus voltage, all loads are converted to equivalent
admittances to ground and are assumed to remain constant.
 Damping or asynchronous powers are ignored.
 The mechanical rotor angle of each machine coincides with the angle of
the voltage behind the machine reactances.
 Machines belonging to the same stations swing together and are said to
be coherent. A group of machines are represented by one equivalent
machine.

Steps for determining multimachine stability

• From the prefault load data determine voltage behind transient

reactances for all generators. This establishes the generator emf
magnitude which remain constant during the study and initial rotor
angle. Also record prime mover input to generators.

• Augment the load flow network by the generator transient reactances.

Shift network buses behind the transient reactances.

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 • Find the bus admittance matrix for various network condition during
fault, post fault and after line reclouser.

• For faulted mode, find generator outputs from power angle equations and
solve the swing equations by point by point method.

• Repeat the above steps for post fault mode and after the line reclouser
mode.

• Examine δ(t) plots of all the generators and establish the answer to the
stability questions.

Short Questions
1. State swing equation of a generator.

Where, M= I = Angular momentum

Pm = Tm = mechanical power input,
Pe = Te = Electrical power output
Pa = Accelerating power = difference between input and output power
2. What are the assumptions made in solving swing equation?
1) Mechanical power input to the machine remains constant during
the period of electromechanical transient of interest.
2) Rotor speed changes are insignificant that had already been
ignored in formulating the swing equations.
3) Effect of voltage regulating loop during the transient are ignored.
2. Define the per unit inertia constant of an alternator.
It is defined as the ratio of energy stored in mega joules to the rating of
the machines in MVA.

3. Define the transient stability.

Transient stability of a power system refers to the ability of the system to
reach a stable condition following a large disturbance in the network
condition.
In all cases related to large changes in the system like sudden
application or removal of load, switching operations, line faults or loss
due to excitation the transient stability of the system comes into play.
4. What is transient stability limit?
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 When the load on the system is increased suddenly, maximum power
that can be transmitted without losing synchronism is termed as
transient state stability limit. Normally, steady state stability limit is
greater than transient state stability limit.
5. Write any two methods for improvement of transient stability of
power system.
Different methods for improvement of transient stability are
a. Increase of system voltages
b. Use of high speed excitation systems.
c. Reduction in system transfer reactance
d. Use of high speed reclosing breakers
6. Write down four factors that effects transient stability in a power
system.
The transient stability is generally affected by two factors namely, (1)
Type of fault (2) Location of fault.

7. Define dynamic stability of a system.

A system is said to be Dynamically stable, if the oscillation of the system
do not acquire more than certain amplitude and die out quickly i.e. the
system is well damped.

8. Difference between stability and loss of synchronism.

Stability of power system: It is defined as a condition at which the
various synchronous machines operating in parallel in the system
remain in synchronism or in step with each other.
Loss of synchronism: During the swing, the rotor angle, δ becomes
greater than maximum rotor angle, δm the input becomes greater than
output. Hence rotor will again accelerate causing δ to increase further. If
δ swings past δm, the stability will be lost. This is called loss of
synchronism.
9. What is transfer reactance?
It is defined as the total reactance which directly connects the two emf
sources is known as transfer reactance. It has an important effect on
power angle curve maximum power transfer is inversely proportional of
the transfer reactance.

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 10. What is critical clearing angle?
The critical clearing angle δcc is the maximum allowable change in the
power angle δbefore clearing the fault , without loss of synchronism.
11. Define critical clearing time.
The critical clearing time , tcc can be defined as the maximum time delay
that can be allowed to clear a fault without loss of synchronism . The time
corresponding to the critical clearing angle is called critical clearing time
tcc.
12. Define Stability.
The stability of a system is defined as the ability of power system to return
to a stable operation in which various synchronous machines of the system
remain in synchronism, when it is subjected to a disturbance.
13. Define steady state stability.
The steady state stability is defined as the ability of a power system to
remain stable i.e., without losing synchronism for small disturbances.
14. What is meant by steady state stability limit?
When the load on the system is increased gradually, maximum power that
can be transmitted without losing synchronism is termed as steady state
stability limit. In steady state, the power transferred by synchronous
machine of a power system is always less than the steady state stability
limit.
15. Write any three assumptions upon transient stability.
a. Rotor speed is assumed to be synchronous. In fact, it varies
insignificantly during the course of the stability study.
b. Shunt capacitances are not difficult to account for in a stability study.
c. Loads are modeled as constant admittances.
16. Define swing curve. What is the use of swing curve?
The swing curve is the plot or graph between the power angle δ, and time, t.
It is usually plotted for a transient state to study the nature of variation in δ
for a sudden large disturbance. From the nature of variations of δ, the
stability of a system for any disturbance can be determined.
17. State equal area criterion.
In a two machine system under the usual assumptions of constant input ,
no damping and constant voltage being transient reactance , the angle
between the machines either increases or else, after all disturbances have
occurred oscillates with constant amplitude. There is a simple graphical
method of determining whether the system comes to rest with respect to
each other. This is known as equal area criterion
18. What are various faults that increase severity of equal area
criterion?
The various faults that increases severity of equal area criterion are,
A Single line to ground fault
A Line to line fault
A Double line to ground fault
A Three phase fault

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 19. What are the assumptions that are made in order to simplify
the computational task in stability studies?
The assumptions are,
• The D.C offset currents and harmonic components are neglected. The
currents and voltages are assumed to have fundamental component alone.
• The symmetrical components are used for the representation of unbalanced
faults.
• It is assumed that the machine speed variations will not affect the generated
voltage.
20. What is Multimachine stability?
If a system has any number of machines, then each machine is listed for
stability by advancing the angular position, δ of its internal voltage and
noting whether the electric power output of the machine increases (or)
decreases. If it increases, i.e if ∂Pn / ∂δn > 0. then machine n is stable. If
every machine is stable, then the system having any number of machine is
stable.
21. What is meant by an infinite bus?
The connection or disconnection of a single small machine on a large system
would not affect the magnitude and phase of the voltage and frequency.
Such a system of constant voltage and constant frequency regardless of the
load is called infinite bus bar system or infinite bus.
22. List the assumptions made in multimachine stability studies.
The assumptions made are,
• The mechanical power input to each machine remains constant during the
entire period of the swing curve computation
• Damping power is negligible
• Each machine may be represented by a constant transient reactance in
series with a constant transient voltage.
• The mechanical rotor angle of each machine coincides with δ , the electrical
phase angle of the transient internal voltage.
23. 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.
24. Why transient stability limit is lower than the steady state stability
limit.
Steady state stability limit is the maximum flow of power through a particular
point in the power system without loss of stability when small disturbances

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occur. Transient stability limit is the maximum flow of power through a
particular point in the power system without loss of stability when large and
sudden disturbances occur. A system can never be operated higher than its
steady state stability limit but it can operate beyond the transient stability
limit.
25. Does reactive power have an effect on stability? Explain.

Long question:
1. Write short notes on “Role of Automatic voltage regulator in improving
stability” (2017)

The automatic voltage regulator is used to regulate the voltage. It takes the
fluctuate voltage and changes them into a constant voltage. The fluctuation in
the voltage mainly occurs due to the variation in load on the supply system.
The variation in voltage damages the equipment of the power system. The
variation in the voltage can be controlled by installing the voltage control
equipment at several places likes near the transformers, generator, feeders,
etc., The voltage regulator is provided in more than one point in the power
system for controlling the voltage variations.

In DC supply system the voltage can be controlled by using over compound

generators in case of feeders of equal length, but in the case of feeders of
different lengths the voltage at the end of each feeder is kept constant using
feeder booster. In AC system the voltage can be controlled by using the various
methods likes booster transformers, induction regulators, shunt condensers,
etc.,

Working Principle of Voltage Regulator

It works on the principle of detection of errors. The output voltage of an AC

generator obtained through a potential transformer and then it is rectified,
filtered and compared with a reference. The difference between the actual
voltage and the reference voltage is known as the error voltage. This error
voltage is amplified by an amplifier and then supplied to the main exciter or
pilot exciter.

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Thus, the amplified error signals control the excitation of the main or pilot
exciter through a buck or a boost action (i.e. controls the fluctuation of the
voltage). Exciter output control leads to the controls of the main alternator
terminal voltage.

Application of the Automatic Voltage Regulator

The main functions of an AVR are as follows.

1. It controls the voltage of the system and has the operation of the
machine nearer to the steady state stability.
2. It divides the reactive load between the alternators operating in parallel.
3. The automatic voltage regulators reduce the overvoltages which occur
because of the sudden loss of load on the system.
4. It increases the excitation of the system under fault conditions so that
the maximum synchronising power exists at the time of clearance of the
fault.

When there is a sudden change in load in the alternator, there should be a

change in the excitation system to provide the same voltage under the new load
condition. This can be done by the help of the automatic voltage regulator. The
automatic voltage regulator equipment operates in the exciter field and changes
the exciter output voltage, and the field current. During the violent fluctuation,
the ARV does not give a quick response.

For getting the quick response, the quick acting voltage regulators based on the
overshooting the mark principle are used. In overshoot mark principle, when
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the load increase the excitation of the system also increase. Before the voltage
increase to the value corresponding to the increased excitation, the regulator
reduces the excitation of the proper value.

2. Distinguish between steady-state stability and transient stability of a

power system? How to improve transient stability of a power
system?(2017)
Ans: Refer class notes
3. Derive The Power-Angle Equation. (2017)
Ans: Refer class notes
4. Write short notes on the application of swing equation in the study of
power system stability. (2017)
Ans: Refer class notes

5. Derive the swing equation of a single generator system.

Ans: Refer class notes

6. A 50 Hz, 4 pole turbo generator of rating 20 MVA, 13.2 kV has an

inertia constant of 9 kW-sec/kVA. Find the kinetic energy stored in the
rotor at synchronous speed. Find the acceleration, if the input less the
rotational loss is 26,800 hp and the electric power developed is 16 MW
equal at 115 kW. (2016, 14)
[5]

Solution:
No. of poles, N=4
Rating of Generator, G = 20 MVA
Rated voltage, V = 13.2 KV
Inertia constant, H = 9kW-sec/kVA20MVA
Electric power developed, Pe = 16 MW = 16000 kW
Mechanical power input,

Kinetic energy stored in the rotor at synchronous speed

Accelearating power,

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Again

7. Write short notes on Equal area criterion for power system stability analysis.(2016)
Ans: Refer class note.
8. A power station A consists of two synchronous consists of two
synchronous generators. The generator-1has a rating of 50 MVA, 50 Hz,
1500 rpm and has an inertia constant of 8MJ/MVA. The generator-2 has a
rating of 100MVA, 50 Hz, 3000 rpm and has inertia constant of 4
MJ/MVA.
i. Find the inertia constant for the equivalent generator on a base of
100MVA
ii. Another power station B has 4 generators two each of the above type.
Find the inertia constant for the equivalent generator on a base of
100MVA
iii. If the two power systems are connected through an inter connector,
find the inertia constant for the equivalent generator connected to
infinite bus bar. (2015) [10]
Ans:
Rating of Generatr-1=50MVA, 50Hz, 1500 rpm
Inertia constant = 8 MJ/MVA
Rating of Generator-2= 100MVA, 50Hz, 3000rpm
Inertia constant = 4 MJ/MVA
(i) Base=100 MVA
Inertia constant of generator-1 on a base of 100MVA,

Inertia constant of generator-1 on a base of 100MVA,

Therefore the inertia constant of the equivalent generator is given by

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(ii) Base=100 MVA
Power station B has 4 generators each having inertia constant of 4MJ/ MVA
Therefore the Inertia constant of B is HB = 4 X4 = 16 MJ/MVA
(iii) Inertia constant of equivalent generator is

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