PSOC Class Note(Module-IV).pdf jajannanans

Module-IV
Power System Stability

Introduction
Stability of a power system is its ability to return to normal or stable operating
conditions after having been subjected to some form of disturbance. Conversely,
instability means a condition denoting loss of synchronism or falling out of step.
Though stability of a power system is a single phenomenon, for the purpose of
analysis, it is classified as Steady State Analysis and Transient Stability.
Increase in load is a kind of disturbance. If increase in loading takes place
gradually and in small steps and the system withstands this change and
performs satisfactorily, then the system is said to be in STADY STATE
STABILITY. Thus the study of steady state stability is basically concerned with
the determination of upper limit of machine’s loading before losing synchronism,
provided the loading is increased gradually at a slow rate.

In practice, load change may not be gradual. Further, there may be sudden
disturbances due to
i) Sudden change of load
ii) Switching operation
iii) Loss of generation
iv) Fault
Following such sudden disturbances in the power system, rotor angular
differences, rotor speeds, and power transfer undergo fast changes whose
magnitudes are dependent upon the severity of disturbances. For a large
disturbance, changes in angular differences may be so large as to cause the
machine to fall out of step. This type of instability is known as TRANSIENT
INSTABILITY. Transient stability is a fast phenomenon, usually occurring within
one second for a generator close to the cause of disturbance.

Short circuit is a severe type of disturbance. During a fault, electrical powers
from the nearby generators are reduced drastically, while powers from remote
generators are scarily affected. In some cases, the system may be stable even
with sustained fault; whereas in other cases system will be stable only if the fault
is cleared with sufficient rapidity. Whether the system is stable on the occurrence
of a fault depends not only on the system itself, but also on the type of fault,
location of fault, clearing time and the method of clearing.
Transient stability limit is almost always lower than the steady state limit and
hence it is much important. Transient stability limit depends on the type of
disturbance, location and magnitude of disturbance.
Review of mechanics
Transient stability analysis involves some mechanical properties of the machines
in the system. After every disturbance, the machines must adjust the relative
angles of their rotors to meet the condition of the power transfer involved. The
problem is mechanical as well as electrical.

The kinetic energy of an electric machine is given by
K.E. = 2
ω
2
1
I Mega Joules (1)
where I is the Moment of Inertia in Mega Joules sec.2
/ elec. deg.2
ω is the angular velocity in elec. deg. / sec.
Angular Momentum M = I ω; Then from eqn. (1), K.E. can be written as
K.E. = ω
M
2
1
Mega Joules (2)
The angular momentum M depends on the size of the machine as well as on its
type.
The Inertia constant H is defined as the Mega Joules of stored energy of the
machine at synchronous speed per MVA of the machine. When so defined, the
relation between the Angular Momentum M and the Inertia constant H can be
derived as follows.

Relationship between M and H
By definition H =
MVA
in
rating
s
Machine'
MJ
in
energy
Stored
Let G be the rating of the machine in MVA. Then
Stored energy = G H MJ (3)
Further, K.E. = ω
M
2
1
MJ = M
2
1
(2 π f) MJ = M x π f MJ (4)
From eqns. (3) and (4), we get
G H = M x π f; Thus
M =
f
π
H
G
MJ sec. / elec. rad. (5)
If the power is expressed in per unit, then G = 1.0 per unit and hence
M =
f
π
H
(6)
While the angular momentum M depend on the size of the machine as well as on
its type, inertia constant H does not vary very much with the size of the machine,
The quantity H has a relatively narrow range of values for each class of machine.

Swing equation
The differential equation that relates the angular momentum M, the acceleration
power Pa and the rotor angle δ is known as SWING EQUATION. Solution of swing
equation will show how the rotor angle changes with respect to time following a
disturbance. The plot of δ Vs t is called the SWING CURVE. Once the swing curve
is known, the stability of the system can be assessed.
The flow of mechanical and electrical power in a generator and motor are shown
in Fig. 1.
Consider the generator shown in Fig. 1(a). It receives mechanical power Pm at the
shaft torque Ts and the angular speed ω via. shaft from the prime-mover. It
delivers electrical power Pe to the power system network via. the bus bars. The
generator develops electromechanical torque Te in opposition to the shaft torque
Ts. At steady state, Ts = Te.
Fig. 1
Pe
Te
ω
Ts
Pm
Pe
Te
ω
Ts
Pm
Generator Motor
(b)
(a)

Assuming that the windage and the friction torque are negligible, in a generator,
accelerating torque acting on the rotor is given by
Ta = Ts – Te (7)
Multiplying by ω on both sides, we get
Pa = Ps – Pe (8)
In case of motor
Ta = Te – Ts (9)
Pa = Pe – Ps (10)
In general, the accelerating power is given by
Pa = Input Power – Output Power (11)

Pa = Ta ω = I α ω = M α = M 2
2
dt
θ
d
Thus M 2
2
dt
θ
d
= Pa (12)
Here θ = angular displacement (radians)
ω =
dt
dθ
= angular velocity (rad. / sec.)
α =
dt
dω
= 2
2
dt
θ
d
= angular acceleration
Now we can see how the angular displacement θ can be related to rotor angle δ.
Consider an object moving at a linear speed of vs ± Δv. It is required to find its
displacement at any time t. For this purpose, introduce another object moving
with a constant speed of vs. Then, at any time t, the displacement of the first
object is given by
x = vs t + d
where d is the displacement of the first object wrt the second as shown in Fig. 2.
vs + Δv
vs
x
d
Fig. 2

Similarly in the case of angular movement, the angular displacement θ , at any
time t is given by
θ = ωs t + δ (13)
where δ is the angular displacement of the rotor with respect to rotating
reference axis which rotates at synchronous speed ωs. The angle δ is also called
as LOAD ANGLE or TORQUE ANGLE. In view of eqn.(13)
dt
dθ
= ωs +
dt
dδ
(14)
2
2
dt
θ
d
= 2
2
dt
δ
d
(15)
From equations (12) and (15), we get
M 2
2
dt
δ
d
= Pa (16)
The above equation is known as SWING EQUATION

In case damping power is to be included, then eqn.(16) gets modified as
M 2
2
dt
δ
d
+ D
dt
dδ
= Pa (17)
Swing curve, which is the plot of torque angleδ vs time t, can be obtained by
solving the swing equation. Two typical swing curves are shown in Fig. 3.
Swing curves are used to determine the stability of the system. If the rotor angle
δ reaches a maximum and then decreases, then it shows that the system has
transient stability. On the other hand if the rotor angleδ increases indefinitely,
then it shows that the system is unstable.
. 0
dt
dδ

t
t
δ
s
δ
0
δ
0
δ
δ
(a) (b)
Fig. 3

We are going to study the stability of (1) a generator connected to infinite bus
and (2) a synchronous motor drawing power from infinite bus.
We know that the complex power is given by
P + j Q = V I * i.e. P – j Q = V *
I Thus real power P = Re {V *
I}
Consider a generator connected to infinite bus.
V + j X I = E
Internal voltage E leads V by angle δ.
Thus E = δ
E 
Current I = ]
V
δ
sin
E
j
δ
cos
E
[
X
j
1


Electric output power Pe = Re [ I
V ] = δ
sin
X
V
E
= Pmax sin δ
V is the voltage at infinite bus.
E is internal voltage of generator.
X is the total reactance
Taking this as ref. V =
0
0
V 
phasor dia. can be obtained as
E
δ
V
I
j X I
G
I V
E
Xd XT

Consider a synchronous motor drawing power from infinite bus.
V – j X I = E
Internal Voltage E lags the terminal voltage V by angle δ.
Thus E = δ
E 
 Current I = ]
)
δ
sin
E
j
δ
cos
E
(
V
[
X
j
1


Electric input power Pe = Re [ I
V ] = δ
sin
X
V
E
= Pmax sin δ
Thus Swing equation for alternator is
M 2
2
dt
δ
d
= Pm – Pmax sin δ
Swing equation for motor is
M 2
2
dt
δ
d
= Pmax sin δ - Pm
Notice that the swing equation is second order nonlinear differential equation
I
M
V
E
Xd XT
- j X I
E
I
δ
V

Equal area criterion
The accelerating power in swing equation will have sine term. Therefore the
swing equation is non-linear differential equation and obtaining its solution is not
simple. For two machine system and one machine connected to infinite bus bar,
it is possible to say whether a system has transient stability or not, without
solving the swing equation. Such criteria which decides the stability, makes use
of equal area in power angle diagram and hence it is known as EQUAL AREA
CRITERION. Thus the principle by which stability under transient conditions is
determined without solving the swing equation, but makes use of areas in power
angle diagram, is called the EQUAL AREA CRITERION.
From the Fig. 3, it is clear that if the rotor angle δ oscillates, then the system is
stable. For δ to oscillate, it should reach a maximum value and then should
decrease. At that point
dt
dδ
= 0. Because of damping inherently present in the
system, subsequence oscillations will be smaller and smaller. Thus while
δ changes, if at one instant of time,
dt
dδ
= 0, then the stability is ensured.

Let us find the condition for
dt
dδ
to become zero.
The swing equation for the alternator connected to the infinite bus bars is
M 2
2
dt
δ
d
= Ps – Pe (18)
Multiplying both sides by
dt
dδ
, we get
M 2
2
dt
δ
d
dt
dδ
= (Ps – Pe)
dt
dδ
i.e. 
2
)
dt
dδ
(
dt
d
M
2
1
(Ps – Pe)
dt
dδ
(19)
Thus
M
)
P
(P
2
dδ
dt
)
dt
dδ
(
dt
d e
s
2 
 ; i.e.
M
)
P
(P
2
)
dt
dδ
(
dδ
d e
s
2 
 On integration
2
dt
dδ
)
( = 

δ
δ
e
s
0
M
dδ
)
P
(P
2
i.e.
dt
dδ
= 

δ
δ
e
s
0
M
dδ
)
P
(P
2
Before the disturbance occurs, 0
δ was the torque angle. At that time
dt
dδ
= 0. As
soon as the disturbance occurs,
dt
dδ
is no longer zero and δ starts changing.
(20)

Torque angle δ will cease to change and the machine will again be operating at
synchronous speed after a disturbance, when
dt
dδ
= 0 or when


δ
δ
e
s
0
dδ
M
)
P
(P
2
= 0 i.e.
 
δ
δ
e
s
0
dδ
)
P
(P = 0 (21)
If there exist a torque angle δ for which the above is satisfied, then the machine
will attain a new operating point and hence it has transient stability.
The machine will not remain at rest with respect to infinite bus at the first time
when
dt
dδ
= 0. But due to damping present in the system, during subsequent
oscillation, maximum value of δ keeps on decreasing. Therefore, the fact that
δ has momentarily stopped changing may be taken to indicate stability.

Sudden load increase on Synchronous motor
Let us consider a synchronous motor connected to an infinite bus bars.
I
j X
E
-
M E = V – I ( jX )
+
V
A2
A1
m
δ
Changed output
Initial output
e
d
c
b
a
s
δ
0
δ
δ
P
P0
Ps
Input power
Pe= Pmax sin δ
Fig. 4
δ
V
E
I
Pe = δ
sin
P
δ
sin
X
V
E
max


The following changes occur when the load is increased suddenly.
Point a Initial condition; Input = output = P0; ω = ωs; δ = δ0
Due to sudden loading, output = Ps; output > Input;
ω decreases from ωs; δ increases from δ0.
Between a-b Output > Input; Rotating mass starts loosing energy resulting
deceleration; ω decreases; δ increases.
Point b Output = Input; ω = ωmin which is less than ωs; δ = δs
Since ω is less than ωs, δ continues to increase.
P
A2
A1
m
δ
Changed output
Initial output
e
d
c
b
a
s
δ
0
δ
δ
P0
Ps
Input power
Pe= Pmax sin δ
Fig. 4
δ
V
E
I

Between b-c Input > output; Rotating masses start gaining energy;
Acceleration; ω starts increasing from minimum value but still less
than ωs; δ continues to increase.
Point c Input > output; ω = ωs; δ = δm; There is acceleration; ω is going
to increase from ωs; hence δ is going to decrease from δm.
Between c-b Input > output; Acceleration; ω increases and δ decreases.
Point b Input = output; ω = ωmax ; δ = δs. Since ω is greater than ωs,
δ continues to decrease.
Between b-a Output > input; Deceleration; ω starts decreasing from ωmax ; but
still greater than ωs; δ continues to decrease.
Point a ω = ωs; δ = δ0; Output > Input; The cycle repeats.
P
A2
A1
m
δ
Changed output
Initial output
e
d
c
b
a
s
δ
0
δ
δ
P0
Ps
Input power
Pe= Pmax sin δ
δ
V
E
I

Because of damping present in the system, subsequent oscillations become
smaller and smaller and finally b will be the steady state operating point.
Interpretation of equal area
As discussed earlier (eqn. 21), the condition for stability is
 
δ
δ
e
s
0
dδ
)
P
(P = 0 i.e. 
δ
δ
e
0
dδ
P = 
δ
δ
s
0
dδ
P
From Fig. 4, 
m
0
δ
δ
e dδ
P = area δ0 a b c δm
and 
m
0
δ
δ
s dδ
P = area δ0 a d e δm
Thus for stability,
area δ0 a b c δm = area δ0 a d e δm
Subtracting area δ0 a b e δm from both sides of above equation, we get A2 = A1.
Thus for stability,
A2 = A1 (22)
A2
A1
m
δ
Changed output
Initial output
e
d
c
b
a
s
δ
0
δ
δ
P
P0
Ps
Input power
Pe= Pmax sin δ
Fig. 4

Fig. 5 shows three different cases: The one shown in case a is STABLE. Case b
indicates CRITICALLY STABLE while case c falls under UNSTABLE.
Note that the areas A1 and A2 are obtained by finding the difference
between INPUT and OUTPUT.
s
δ
m
δ
δ
Ps
Ps
Ps
P
P
δ δ
A2
A1
A1
A2
P0
P0
0
δ
0
δ
Case c
Case b
Case a
Fig. 5
5
P
A2
A1
P0
m
δ
s
δ
0
δ

POWER SYSTEM STABILITY
LESSON SUMMARY-1:-
1. Introduction
2. Classification of Power System Stability
3. Dynamic Equation of Synchronous Machine
Power system stability involves the study of the dynamics of the power system
under disturbances. Power system stability implies that its ability to return to
normal or stable operation after having been subjected to some form of
disturbances.
From the classical point of view power system instability can be seen as loss of
synchronism (i.e., some synchronous machines going out of step) when the system
is subjected to a particular disturbance. Three type of stability are of concern:
Steady state, transient and dynamic stability.
Steady-state Stability:-
Steady-state stability relates to the response of synchronous machine to a
gradually increasing load. It is basically concerned with the determination of the
upper limit of machine loading without losing synchronism, provided the loading
is increased gradually.
Dynamic Stability:-
Dynamic stability involves the response to small disturbances that occur on the
system, producing oscillations. The system is said to be dynamically stable if
theses oscillations do not acquire more than certain amplitude and die out quickly.
If these oscillations continuously grow in amplitude, the system is dynamically
unstable. The source of this type of instability is usually an interconnection
between control systems.
Transient Stability:-
Transient stability involves the response to large disturbances, which may cause
rather large changes in rotor speeds, power angles and power transfers. Transient
stability is a fast phenomenon usually evident within a few second.

Power system stability mainly concerned with rotor stability analysis. For this
various assumptions needed such as:
 For stability analysis balanced three phase system and balanced disturbances
are considered.
 Deviations of machine frequencies from synchronous frequency are small.
 During short circuit in generator, dc offset and high frequency current are
present. But for analysis of stability, theses are neglected.
 Network and impedance loads are at steady state. Hence voltages, currents
and powers can be computed from power flow equation.
Dynamics of a Synchronous Machine :-
The kinetic energy of the rotor in synchronous machine is given as:
KE=1/2Jws
2
x10 -6
MJoule……………………...………...… (1)
Where J= rotor moment of inertia in kg-m2
ws = synchronous speed in mechanical radian/sec.
Speed in electrical radian is
wse = (P/2) ws = rotor speed in electrical radian/sec…….....….(2)
Where P = no. of machine poles
From equation (1) and (2) we get
KE= MJ………………..……….. (3)
or KE= MJ
Where M = = moment of inertia in
MJ.sec/elect. radian……..... (4)
We shall define the inertia constant H, such that
GH = KE = MJ…………………….…….…………. (5)
Where G = three-phase MVA rating (base) of machine
H = inertia constant in MJ/MVA or MW.sec/MVA

From equation (5), we can write,
M = MJ.sec/elect. radian……....………… (6)
or M = MJ.sec/elect. degree……………….…….…………(7)
M is also called the inertia constant.
Assuming G as base, the inertia constant in per unit is
M(pu) = Sec2
/elect.radian………………….………………(8)
or M(pu) = Sec2
/elect.degree………………….……………(9)
LESSON SUMMARY-2:-
1. Swing equation
2. Multi machine system
3. Machines swinging in unison or coherently
4. Examples
Swing Equation:-
GENERATOR
Pe
Tm Ws
Te
Pm
(Fig.-1 Flow of power in a synchronous generator)
Consider a synchronous generator developing an electromagnetic torque Te(and
a corresponding electromagnetic power Pe) while operating at the synchronous
speed ws. If the input torque provided by the prime mover, at the generator shaft is
Ti, then under steady state conditions (i.e., without any disturbance).

Te = Ti ……………………………………….. (10)
Here we have neglected any retarding torque due to rotational losses. Therefore
we have
Te ws= Ti ws……………………………………….. (11)
And Te ws - Ti ws = Pi - Pe = 0……………...…………………… (12)
When a change in load or a fault occurs, then input power Pi is not equal to Pe.
Therefore left side of equation is not zero and an accelerating torque comes into
play. If Pa is the accelerating (or decelerating) power, then
Pi- Pe = …………………...………… (13)
Where D = damping coefficient
θe = electrical angular position of the rotor
It is more convenient to measure the angular position of the rotor with respect
to a synchronously rotating frame of reference. Let
δ = θe -ws.t ……………………………..……………… (14)
So …………………………...……………… (15)
Where δ is power angle of synchronous machine.
δ
θe
Ws
Rotor Field
Reference rotating
axis
Reference axis
(Fig.2 Angular Position of rotor with respect to reference axis)
Neglecting damping (i.e., D = 0) and substituting equation (15) in equation (13)
we get

MW……………………………….. (16)
Using equation (6) and (16), we get
MW……………………………… (17)
Dividing throughout by G, the MVA rating of the machine,
pu………………………….... (18)
Where …………………………………………. (19)
or pu………………….…………… (20)
Equation (20) is called Swing Equation. It describes the rotor dynamics for a
synchronous machine. Damping must be considered in dynamic stability study.
Multi Machine System:-
In a multi machine system a common base must be selected. Let
Gmachine = machine rating (base)
Gsystem = system base
Equation (20) can be written as:
………………… (21)
So pu on system base…………...…… (22)
Where ………………………….(23)
= machine inertia constant in system base
Machines Swinging in Unison (Coherently) :-
Let us consider the swing equations of two machines on a common system
base, i.e.,
………………………...……. (24)

…………..…………...…. (25)
Since the machines rotor swing in unison,
δ 1 = δ 2 = δ …………………….……………..(26)
Adding equations (24) and (25) and substituting equation (26), we get
…………………………...… (27)
Where Pi = Pi1 + Pi2
Pe = Pe1 + Pe2
Heq = H1 + H2
Equivalent inertia Heq can be expressed as:
…….. (28)
Example1:-
A 60 Hz, 4 pole turbo-generator rated 100MVA, 13.8 KV has inertia constant
of 10 MJ/MVA.
(a) Find stored energy in the rotor at synchronous speed.
(b)If the input to the generator is suddenly raised to 60 MW for an electrical
load of 50 MW, find rotor acceleration.
(c) If the rotor acceleration calculated in part (b) is maintained for 12 cycles,
find the change in torque angle and rotor speed in rpm at the end of this
period.
(d)Another generator 150 MVA, having inertia constant 4 MJ/MVA is put in
parallel with above generator. Find the inertia constant for the equivalent
generator on a base 50 MVA.
Solution:-
(a) Stored energy = GH
= 100MVA x 10MJ/MVA
= 1000MJ
(b)Pa = Pi-Pe = 60-50 = 10MW
We know, M = = = MJ.sec/elect.deg.

Now
 = 10
 = = 108 elect.deg./sec2
So, α = acceleration = 108 elect.deg./sec2
(c) 12 cycles = 12/60 = 0.2sec.
Change in δ = ½ α.(Δt)2
= ½.108.(0.2)2
=2.16 elect.deg
Now α = 108 elect.deg./sec2
= 60 x (108/360ᵒ) rpm/sec
= 18 rpm/sec
Hence rotor speed at the end of 12 cycles
=
= ( ) rpm
= 1803.6 rpm.
(d)Heq = =32MJ/MVA
LESSON SUMMARY-3:-
1. Power flow under steady state
2. Steady-state Stability
3. Examples
Power Flow under Steady State:-
Consider a short transmission line with negligible resistance.
VS = per phase sending end voltage
VR = per phase receiving end voltage
Vs leads VR by an angle δ
x = reactance of per transmission line
I
jx
+ +
- -
Vs
VR
(Fig.3-A short transmission line)

On the per phase basis power on sending end,
SS = PS + j QS =VSI*…………………….………….. (29)
From Fig.3 I is given as
I =
or I* = ………….………………………. (30)
From equation (29) and (30), we get
SS = …………………………………. (31)
Now VR =|VR| so, VR = VR
*
= |VR|
VS =|VS| =|VS|
Equation (31) becomes
SS = PS + jQS = δ δ
So PS = δ……………………...…………….. (32)
and Qs= ………………...………...…..………(33)
Similarly, at the receiving end we have
SR = PR + j QR = VRI*………………………………. (34)
Proceeding as above we finally obtain
QR = .................................................. (35)
PR = ……………………………..…….. (36)
Therefore for lossless transmission line,
PS = PR = ………………………...…….. (37)
In a similar manner, the equation for steady-state power delivered by a lossless
synchronous machine is given by

Pe = Pd =
= δ………………………………… (38)
Where |Eg| is the rms internal voltage, |Vt| is the rms terminal voltage, xd is the
direct axis reactance (or the synchronous reactance in a round rotor machine) and δ
is the electrical power angle.
Steady-state Stability:-
The steady state stability limit of a particular circuit of a power system defined
as the maximum power that can be transmitted to the receiving end without loss of
synchronism.
Now consider equation (18),
……………………………… (39)
Where
And Pe = = δ………………………. (40)
Let the system be operating with steady power transfer of with torque
angle . Assume a small increment ΔP in the electric power with the input from
the prime mover remaining fixed at causing the torque angle to change to
( . Linearizing the operating point ( we can write
…………………………………...… (41)
The excursions of are then described by
…………………...….. (42)
or …………………………….. (43)
or [Mp2
+ ……………………..……….. (44)
Where p =

The system stability to small changes is determined from the characteristic
equation
Mp2
+ = 0…………………………….. (45)
Where two roots are p = …………………………….. (46)
As long as is positive, the roots are purely imaginary and conjugate
and system behavior is oscillatory about . Line resistance and damper windings
of machine cause the system oscillations to decay. The system is therefore stable
for a small increment in power so long as δ
When δ is negative, the roots are real, one positive and the other
negative but of equal magnitude. The torque angle therefore increases without
bound upon occurrence of a small power increment and the synchronism is soon
lost. The system is therefore unstable for δ
is known as synchronizing coefficient. This is also called stiffness of
synchronous machine. It is denoted as Sp. This coefficient is given by
δ ………………………… (47)
If we include damping term in swing equation then equation (43) becomes
δ
or δ
or δ
or δ …………………….. (48)

Where and …………………………… (49)
So damped frequency of oscillation, ……………..……….. (50)
And Time Constant, T = = …………………………….. (51)
Example2:-
Find the maximum steady-state power capability of a system consisting of a
generator equivalent reactance of 0.4pu connected to an infinite bus through a
series reactance of 1.0 p.u. The terminal voltage of the generator is held at1.10 p.u.
and the voltage of the infinite bus is 1.0 p.u.
Solution:-
Equivalent circuit of the system is shown in Fig.4.
(Fig.4 Equivalent circuit of example2)
δ ……………………………….. (i)
I = …………………………….. (ii)
Using equation (i) and (ii)
δ θ
δ θ θ θ
δ θ θ ⁡…………….. (iii)

Maximum steady-state power apab l ty rea hed he δ 9 ᵒ, i.e.,
real part of equation is zero. Thus
θ
9
9 p
9
p
LESSON SUMMARY-4:-
1. Transient Stability-Equal area criterion
2. Applications of sudden change in power input
3. Examples
Transient Stability-Equal Area Criterion:-
The transient stability studies involve the determination of whether or not
synchronism is maintained after the machine has been subjected to severe
disturbance. This may be sudden application of load, loss of generation, loss of
large load, or a fault on the system.
A method known as the equal area criterion can be used for a quick prediction
of stability. This method is based on the graphical interpretation of the energy
stored in the rotating mass as an aid to determine if the machine maintains its
stability after a disturbance. This method is only applicable to a one-machine
system connected to an infinite bus or a two-machine system. Because it provides
physical insight to the dynamic behavior of the machine.
Now consider the swing equation (18),
or
or ………………………………………….. (52)

As shown in Fig.5, in an unstable system, δ increases indefinitely with time and
machine looses synchronism. In a stable system, δ undergoes oscillations, which
eventually die out due to damping. From Fig.4, it is clear that, for a system to be
stable, it must be that = 0 at some instant. This criterion ( = 0) can simply be
obtained from equation (52).
(Fig. 5 A plot of δ (t))
Multiplying equation (52) by , we have
……………………………………. (53)
This upon integration with respect to time gives
dδ ………………………...………. (54)
Where = accelerating power and δ is the initial power angle
before the rotor begins to swing because of a disturbance. The stability ( = 0)
criterion implies that
dδ ………………………………..……… (55)
For stability, the area under the graph of accelerating power versus δ must be
zero for some value of δ; i.e., the positive (accelerating) area under the graph must
be equal to the negative (decelerating) area. This criterion is therefore know as the
equal area criterion for stability and is shown in Fig. 6.

(Fig.6 Power angle characteristic)
Application to sudden change in power input:-
In Fig. 6 point ‘a’ corresponding to the δ is the initial steady-state operating
point. At this point, the input power to the machine, , where is the
developed power. When a sudden increase in shaft input power occurs to , the
accelerating power , becomes positive and the rotor moves toward point ‘b’
We have assumed that the machine is connected to a large power system so that
|Vt| does not change and also xd does not change and that a constant field current
maintains |Eg|. Consequently, the rotor accelerates and power angle begins to
increase. At point and δ =δ1. But is still positive and δ overshoots ‘b’,
the final steady-state operating point. Now is negative and δ ultimately reaches a
maximum value δ2 or point ‘c’ and swing back towards point ‘b’. Therefore the
rotor settles back to point ‘b’, which is ultimate steady-state operating point.
In accordance with equation (55) for stability, equal area criterion requires
Area A1 = Area A2
or δ dδ δ dδ ………….… (56)
or
…………………………. (57)
But
Which when substituted in equation (57), we get
………..………….…… (58)

On simplification equation (58) becomes
…………….…....... (59)
Example 3:-
A synchronous generator, capable of developing 500MW power per phase,
operates at a power angle of 8ᵒ. By how much can the input shaft power be
increased suddenly without loss of stability? Assume that Pmax will remain
constant.
Solution:-
Initially, = 8ᵒ
δ =69.6MW
(Fig. 7 Power angle characteristics)
Let δm be the power angle to which the rotor can swing before losing
synchronism. If this angle is exceeded, Pi will again become greater than Pe and the
rotor will once again be accelerated and synchronism will be lost as shown in Fig.
7. Therefore, the equal area criterion requires that equation (57) be satisfied with
δm replacing δ2.
From Fig. 7 δm = π-δ1. Therefore equation (59) becomes
………………… (i)
Substituting = 0.139radian in equation (i) gives
99 …………………………… (ii)
Solving equation (ii) we get, δ1 = 50ᵒ
Now δ
Initial power developed by machine was 69.6MW. Hence without loss of
stability, the system can accommodate a sudden increase of

9 MW per phase
= 3x313.42 = 940.3 MW (3-φ) of input shaft power.
LESSON SUMMARY-5:-
1. Critical clearing angle and critical clearing time
2. Application of equal area criterion
a) Sudden loss of one parallel line
Critical Clearing Angle and Critical Clearing Time:-
If a fault occurs in a system, δ begins to increase under the influence of
positive accelerating power, and the system will become unstable if δ becomes
very large. There is a critical angle within which the fault must be cleared if
the system is to remain stable and the equal area criterion is to be satisfied.
This angle is known as the critical clearing angle.
(Fig. 8 Single machine infinite bus system)
Consider a system as shown in Fig. 8 operating with mechanical input at
steady angle . as shown by point ‘a’ on the power angle diagram as
shown in Fig. 9. Now if three phase short circuit occur at point F of the outgoing
radial line , the terminal voltage goes to zero and hence electrical power output of
the generator instantly reduces to zero i.e., and the state point drops to ‘b’.
The acceleration area A1 starts to increase while the state point moves along b-
c. At time tc corresponding clearing angle δc, the fault is cleared by the opening of
the line circuit breaker. tc is called clearing time and δc is called clearing angle.
After the fault is cleared, the system again becomes healthy and transmits
power Pe = Pmax sinδ, i.e., the state point shifts to‘d’ on the power angle
curve. The rotor now decelerates and the decelerating area A2 begins to increase
while the state point moves along d-e. For stability, the clearing angle, δc, must
be such that area A1 = area A2.

(Fig. 9 δ characteristics)
Expressing area A1 =Area A2 mathematically we have,
δ δ dδ
δ δ δ dδ δ δ
δ δ δ δ δ δ
δ δ δ δ …………....……. (60)
Also …………………………………. (61)
Using equation (60) and (61) we get,
δ δ δ δ
δ δ δ δ …………………… (62)
Where δ = clearing angle, δ = initial power angle, and δ = power angle to
which the rotor advances (or overshoots) beyond δ
For a three phase fault with Pe =0,
………………………………………….. (63)
Integrating equation (63) twice and utilizing the fact that and t = 0 yields
δ t δ …………………..……… (64)

If t is the clearing time corresponding to a clearing angle δ , then we obtain
from equation (64),
δ t δ
So t …………………………… (65)
Note that δ can be obtained from equation (62). As the clearing of faulty line is
delayed, A1 increases and so does δ to find A2=A1 till δ δ as shown in Fig.
10.
(Fig. 10 Critical clearing angle)
For a clearing angle (clearing time) larger than this value, the system would be
unstable. The maximum allowable value of the clearing angle and clearing time for
the system to remain stable are known as critical clearing angle and critical
clearing time respectively.
From Fig. 10, . Substituting this in equation (62) we have,
δ δ δ δ
δ δ δ δ
δ δ δ
δ δ δ
δ δ δ ……….…...……. (66)
Using equation (65) critical clearing angle can be obtained as
t ……….…………...………. (67)

Application of the Equal Area Criterion:-
(1)Sudden Loss of One of parallel Lines:-
Pi Infinite
Bus
|V|∟0ᵒ
(a)
Pi
|V|∟0ᵒ
Switched Off
Eg∟δ
Xd
X1
X2
(b)
(Fig. 11 Single machine tied to infinite bus through two parallel lines)
Consider a single machine tied to infinite bus through parallel lines as shown in
Fig. 11(a). The circuit model of the system is given in Fig. 11(b).
Let us study the transient stability of the system when one of the lines is
suddenly switched off with the system operating at a steady load. Before switching
off, power angle curve is given by
δ δ
Immediately on switching of line 2, power angle curve is given by
δ δ
In Fig. 12, wherein as . The system is
operating initially with a steady state power transfer at a torque angle δ
on curve I.

δ0 δ1 δ2 π
δ
(Fig. 12 Equal area criterion applied to the opening of one of the two lines in
parallel)
On switching off line2, the electrical operating point shifts to curve II (point b).
Accelerating energy corresponding to area A1is put into rotor followed by
decelerating energy for δ > δ1. Assuming that an area A2 corresponding to
decelerating energy (energy out of rotor) can be found such that A1 = A2, the
system will be stable and will finally operate at c corresponding to a new rotor
angle is needed to transfer the same steady power.
If the steady load is increased (line Pi is shifted upwards) a limit is finally
reached beyond which decelerating area equal to A1 cannot be found and therefore,
the system behaves as an unstable one. For the limiting case, δ1 has a maximum
value given by
δ δ δ
LESSON SUMMARY-6:-
1. Sudden short circuit on one of parallel lines
a) Short circuit at one end of line
b) Short circuit at the middle of a line
2. Example
Sudden Short Circuit on One of Parallel Lines:-
(1)Short circuit at one end of line:-
Let us a temporary three phase bolted fault occurs at the sending end of one of
the line.

X1
X2
Xd
|Eg|∟δ
Infinite
Bus
|V|∟0ͦᵒ
Pi
(a)
Xd
|Eg|∟δ
Pi
X1
X2
Infinite
Bus
|V|∟0ͦᵒ
(b)
(Fig.13 Short circuit at one of the line)
Before the occurrence of a fault, the power angle curve is given by
δ δ
This is plotted in Fig. 12.
Upon occurrence of a three-phase fault at the generator end of line 2 , generator
gets isolated from the power system for purpose of power flow as shown Fig. 13
(b). Thus during the period the fault lasts.
The rotor therefore accelerates and angles δ increases. Synchronism will be
lost unless the fault is cleared in time. The circuit breakers at the two ends of the
faulted line open at time tc (corresponding to angle δc), the clearing time,
disconnecting the faulted line. The power flow is now restored via the healthy line
(through higher line reactance X2 in place of ( ), with power angle curve
δ δ

(Fig. 14 Equal area criterion applied to the system)
Obviously, . The rotor now starts decelerate as shown in Fig 14.
The system will be stable if a decelerating area A2 can be found equal to
accelerating area A1before δ reaches the maximum allowable value δ . As area
A1 depends upon clearing time tc (corresponding to clearing angle ), clearing
time must be less than a certain value (critical clearing time) for the system to be
stable.
(2)Short circuit at the middle of a line:-
When fault occur at the middle of a line or away from line ends, there is some
power flow during the fault through considerably reduced. Circuit model of the
system during the fault is shown in fig. 15 (a). This circuit reduces to fig. 15 (c)
through one delta-star and star-delta conversion.
The power angle curve during fault is given by
δ δ
Xa
G V
X’d
Xb
X2/2
Xc
X1
X2/2
(a)

Xa
G V
X’1 Xb
Xc
(b)
XII
G V
(c)
(Fig.15 Circuit Model)
and as in Fig. 12 and as obtained above are all plotted in Fig. 16.
(Fig. 16 Fault on middle of one line of the system with δc < δcr)
Accelerating area A1 corresponding to a given clearing angle δ is less in this
case. Stable system operation is shown in Fig. 16, wherein it is possible to find an
area A2 equal to A1 forδ δ . As the clearing angle δ is increased, area A1
increases and to find A2 = A1, δ increases till it has a valueδ , the maximum
allowable for stability. This case of critical clearing angle is shown in Fig. 17.

(Fig. 17 Fault on middle on one line of the system)
Applying equal area criterion to the case of critical clearing angle of Fig. 17, we
can write
δ dδ δ dδ
Where
δ ………………..……..…….. (68)
Integrating we get
δ δ δ δ
or δ δ δ δ δ δ
δ δ
δ
δ δ δ δ
This critical clearing angle is in radian. The equation modifies as below if the
angles are in degree
δ
δ δ δ δ
Example 4:-
Find the critical clearing angle for the system shown in Fig. 18 for a three phase
fault at point P. The generator is delivering 1.0 pu. Power under prefault
conditions.

|V|=1.0∟0ᵒ
|E|=1.2pu
j0.25
j0.28
j0.14
j0.14
j0.15
j0.15
j0.15
j0.15
Infinite
Bus
j0.15
P
(Fig. 18)
Solution:-
1. Prefault Operation:- Transfer reactance between generator and infinite bus
is
XI = 0.25+0.17 + = 0.71
δ 9 δ
The operating power angle is given by
9 δ
or δ rad
2. During Fault:- The positive sequence reactance diagram during fault is
presented in Fig. 17.
j0.15
j0.15 j0.15
j0.14 j0.14 j0.15
j0.28
j0.17
j0.25
|E|=1.2pu
|V|=1.0∟0ᵒ
(a)Positive sequence reactance diagram during fault
j0.0725
j0.145 j0.145 j0.17
j0.25
|E|=1.2pu |V|=1.0∟0ᵒ
(b) Network after delta-star conversion

XII
|E|=1.2pu |V|=1.0∟0ᵒ
(c) Network after star- delta conversion
(Fig.19)
Converting delta to star, the reactance network is changed to that Fig. 19 (b).
Further upon converting star to delta, we obtain the reactance network of Fig.
19(c). The transfer reactance is given by
=2.424
δ 9 δ
3. Post fault operation(faulty line switched off):-
δ δ
With reference to Fig. 16 and equation (68), we have
δ rad
To find critical clearing angle, areas A1 and A2 are to be equated.
δ 9 δ dδ
And δ dδ δ
Now A1 =A2
or δ 9 δ dδ
= δ dδ δ
or 9 δ δ
or 9 99 δ
or δ

or δ 9
LESSON SUMMARY-7:-
1. Step by step solution of swing equation
2. Multimachine stability studies
3. Factors affecting transient stability
Step by Step Solution of Swing Equation:-
The swing equation is
δ ………………………… (69)
Its solution gives a plot of δ versus t. The swing equation indicates that δ starts
decreasing after reaching maximum value, the system can be assumed to be stable.
The swing equation is a non-linear equation and a formal solution is not feasible.
The step by step solution is very simple and common method of solving this
equation. In this method the change in δ during a small time interval Δt is
calculated by assuming that the accelerating power calculated at the beginning
of the interval is constant from the middle of the preceding interval to the middle
of the interval being considered.
Let us consider the nth time interval which begins at t = (n-1) Δt. The angular
position of the rotor at this instant is δn-1(Fig. 20 c). The accelerating power
and hence, acceleration αn-1 as calculated at this instant is assumed to be constant
from t = (n-3/2) Δt to (n-1/2) Δt.
During this interval the change in rotor speed can be written as
………………………… (70)
Thus, the speed at the end of nth interval is
…………...……………..… (71)
Assume the change in speed occur at the middle of one interval, i.e., t=(n-1)Δt
which is same the same instant for which the acceleration was calculated. Then the
speed is assumed to remain constant till the middle of the next interval as shown in
Fig. 18(b). In other words, the speed assumed to be constant at the value
throughout the nth interval from t = (n-1) Δt to t = n Δt.

(a)
(b)
(c)
(Fig. 20 Step by step solution of swing equation)
The change in angular position of rotor during nth time interval is
…………………………….……….. (72)
And the value of δ at the end of nth interval is

……………………………….…. (73)
This is shown in Fig. 20 (c). Substituting equation (70) into equation (71) and
the result in equation (72) leads to
……………………..... (74)
By analogy with equation (72)
…………………………………. (75)
Substituting the value of from equation (75) into equation (74)
…..…………………… (76)
Equation (76) gives the increment in angle during any interval (say nth) in
terms of the increment during (n-1) th interval.
During the calculations, a special attention has to be paid to the effects of
discontinuities in the accelerating power which occur when a fault is applied or
cleared or when a switching operation takes place. If a discontinuity occurs at the
beginning of an interval then the average of the values of before and after the
discontinuity must be used. Thus, for calculating the increment in occuring in the
first interval after a fault is applied at t=0, equation (76) becomes
……………………………………. (77)
Where , is the accelerating power immediately after occurrence of the fault.
Immediately before the occurrence of fault, the system is in steady state with
and the previous increment in rotor angle is zero.
Multimachine stability Studies:-
The equal-area criterion cannot be used directly in systems where three or more
machines are represented, because the complexity of the numerical computations
increases with the number of machines considered in a transient stability studies.
To ease the system complexity of system modeling, and thereby computational
burden, the following assumptions are commonly made in transient stability
studies:
1. The mechanical power input to each machine remains constant.
2. Damping power is negligible.
3. Each machine may be represented by a constant transient reactance in series
with a constant transient internal voltage.

4. The mechanical rotor angle of each machine coincides with δ.
5. All loads may be considered as shunt impedances to ground with values
determined by conditions prevailing immediately prior to the transient
conditions.
The system stability model based on these assumptions is called the classical
stability model, and studies which use this model are called classical
stability studies.
Consequently, in the multi-machine case two preliminary steps are required.
1. The steady-state prefault conditions for the system are calculated using a
production-type power flow program.
2. The prefault network representation is determined and then modified to
account for the fault and for the postfault conditions.
The transient internal voltage of each generator is then calculated using the
equation
……………………………. (80)
Where Vt is the corresponding terminal voltage and I is the output current. Each
load is converted into a constant admittance to ground at its bus using the equation
……………………………… (81)
Where the load and |VL| is is the magnitude of the corresponding bus
voltage. The bus admittance matrix which is used for the prefault power-flow
calculation is now augmented to include the transient reactance of each generator
and the shunt admittance of each load, as shown in Fig. 21. Note that the injected
current is zero at all buses except the internal buses of the generators.
(Fig. 21 Augmented network of a power system)
In the second preliminary step the bus admittance matrix is modified to
correspond to the faulted and post fault conditions. During and after the fault the
power flow into the network from each generator is calculated by the

corresponding power angle equation. For example, in Fig. 21 the power output of
generator 1 is given by
……...……. (82)
Where equals . Similar equations are written for and using
the elements of the 3X3 bus admittance matrices appropriate to the fault or
postfault condition. The expressions form part of the equations
i=1, 2, 3…………………… (83)
Which represent the motion of each rotor during the fault and post fault
periods. The solutions depend on the location and duration of the fault, and Ybus
resulting when the faulted line is removed.
Factors Affecting Transient Stability:-
Various methods which improve power system transient stability are
1. Improved steady-state stability
a) Higher system voltage levels
b) Additional transmission line
c) Smaller transmission line series reactance
d) Smaller transfer leakage reactance
e) Series capacitive transmission line compensation
f) Static var compensators and flexible ac transmission systems
(FACTs)
2. High speed fault clearing
3. High speed reclosuer of circuit breaker
4. Single pole switching
5. Large machine inertia, lower transient reactance
6. Fast responding, high gain exciter
7. Fast valving
8. Breaking resistor

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