Module 6
Power system stability
6.1
Introduction
In general terms, power system stability refers to that property of the power system which enables
the system to maintain an equilibrium operating point under normal conditions and to attain a
state of equilibrium after being subjected to a disturbance. As primarily synchronous generators
are used for generating power in grid, power system stability is generally implied by the ability
of the synchronous generators to remain in ’synchronism’ or ’in step’. On the other hand, if the
synchronous generators loose synchronism after a disturbance, then the system is called unstable.
The basic concept of ’synchronism’ can be explained as follows.
In the normal equilibrium condition, all the synchronous generators run at a constant speed and
the difference between the rotor angles of any two generators is constant. Under any disturbance, the
speed of the machines will deviate from the steady state values due to mismatch between mechanical
and electrical powers (torque) and therefore, the difference of the rotor angles would also change.
If these rotor angle differences (between any pair of generators) attain steady state values (not
necessarily the same as in the pre-disturbance condition) after some finite time, then the synchronous
generators are said to be in ’synchronism’. On the other hand, if the rotor angle differences keep on
increasing indefinitely, then the machines are considered to have lost ’synchronism’. Under this ’out
of step’ condition, the output power, voltage etc. of the generator continuously drift away from the
corresponding pre-disturbance values until the protection system trips the machine.
The above phenomenon of instability is essentially related with the instability of the rotor angles
and hence, this form of instability is termed as ’rotor angle instability’. Now, as discussed above,
this instability is triggered by the occurrence of a disturbance. Depending on the severity of the
disturbance, the rotor angle instability can be classified into two categories:
• Small signal instability: In this case, the disturbance occurring in the system is small. Such
kind of small disturbances always take place in the system due to random variations of the loads
and the generation. It will be shown later in this chapter that under small perturbation (or
disturbance), the change in the electrical torque of a synchronous generator can be resolved into
two components, namely, a) synchronizing torque (Ts ) - which is proportional to the change in
the rotor angle and b) damping torque (Td ), which is proportional to the change in the speed
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of the machine. As a result, depending on the amounts of synchronizing and damping torques,
small signal instability can manifest itself in two forms. When there is insufficient amount of
synchronizing torque, the rotor angle increases steadily. On the other hand, for inadequate
amount of damping torque, the rotor angle undergoes oscillations with increasing amplitude.
These two phenomena are illustrated in Fig. 6.1. In Fig. 6.1(a), both the synchronizing
and damping torques are positive and sufficient and hence, the rotor angle comes back to a
steady state value after undergoing oscillations with decreasing magnitude. In Fig. 6.1(b), the
synchronizing torque is negative while the damping toque is positive and thus, the rotor angle
envelope is increasing monotonically. Fig. 6.1(c) depicts the classic oscillatory instability in
which Ts is positive while Td is negative.
Figure 6.1: Influence of synchronous and damping torque
In an integrated power system, there can be different types of manifestation of the small signal
instability. These are:
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a. Local mode: In this type, the units within a generating station oscillate with respect to
the rest of the system. The term ’local’ is used because the oscillations are localized in a
particular generating station.
b. Inter-area mode: In this case, the generators in one part of the system oscillate with
respect to the machines in another part of the system.
c. Control mode: This type of instability is excited due to poorly damped control systems
such as exciter, speed governor, static var compensators, HVDC converters etc.
d. Torsional mode: This type is associated with the rotating turbine-governor shaft. This
type is more prominent in a series compensated transmission system in which the mechanical system resonates with the electrical system.
• Transient instability: In this case, the disturbance on the system is quite severe and sudden
and the machine is unable to maintain synchronism under the impact of this disturbance. In
this case, there is a large excursion of the rotor angle (even if the generator is transiently stable).
Fig. 6.2 shows various cases of stable and unstable behavior of the generator. In case 1, under
the influence of the fault, the generator rotor angle increases to a maximum, subsequently
decreases and settles to a steady state value following oscillations with decreasing magnitude.
In case 2, the rotor angle decreases after attaining a maximum value. However, subsequently,
it undergoes oscillations with increasing amplitude. This type of instability is not caused by
the lack of synchronizing torque; rather it occurs due to lack of sufficient damping torque in the
post fault system condition. In case 3, the rotor angle monotonically keeps on increasing due
to insufficient synchronizing torque till the protective relay trips it. This type of instability, in
which the rotor angle never decreases, is termed as ’first swing instability’.
Figure 6.2: Illustration of various stability phenomenon
Apart from rotor angle instability, instability can also occur even when the synchronous generators
are maintaining synchronism. For example, when a synchronous generator is supplying power to an
induction motor load over a transmission line, the voltage at the load terminal can progressively
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reduce under some conditions of real and reactive power drawn by the load. In this case, loss of
synchronism is not an issue but the challenge is to maintain a stable voltage. This type of instability
is termed as voltage instability or voltage collapse. We will discuss about the voltage instability issue
later in this course.
Now, for analysing rotor angle stability, we have to first understand the basic equation of motion
of a synchronous machine, which is our next topic.
6.2
Equation of motion of a synchronous machine
The equation of motion of a synchronous generator is based on the fact that the accelerating torque
is the product of inertia and its angular acceleration. In the MKS system, this equation can be
written as,
J
d2 θm
= Ta = Tm − Te
dt2
(6.1)
In equation (6.1),
J → The total moment of inertia of the rotor masses in Kg − m2
θm → The angular displacement of the rotor with respect to a stationary axis, in mechanical
radians
t → Time in seconds
Ta → The net accelerating torque, in N-m
Tm → The mechanical or shaft torque supplied by the prime mover less retarding torque due
to rotational losses, in N-m
Te → The net electrical or electromagnetic torque in N-m
Under steady state operation of the generator, Tm and Te are equal and therefore, Ta is zero.
In this case, there is no acceleration or deceleration of the rotor masses and the generator runs
at constant synchronous speed. The electrical torque Te corresponds to the air gap power of the
generator and is equal to the output power plus the real power loss of the armature winding.
Now, the angle θm is measured with respect to a stationary reference axis on the stator and
hence, it is an absolute measure of the rotor angle. Thus, it continuously increases with time
even with constant synchronous speed. However, in stability studies, the rotor speed relative to
the synchronous speed is of interest and hence, it is more convenient to measure the rotor angular
position with respect to a reference axis which also rotates at synchronous speed. Hence, let us
define,
θm = ωsm t + δm
(6.2)
In equation (6.2), ωsm is the synchronous speed of the machine in mechanical radian/sec. and
δm (in mechanical radian) is the angular displacement of the rotor from the synchronously rotating
reference axis. From equation (6.2),
dδm
dθm
= ωsm +
dt
dt
or,
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dδm dθm
=
− ωsm
dt
dt
(6.3)
d2 θm d2 δm
=
(6.4)
dt2
dt2
dδm
represents the deviation of the actual rotor speed
Equation (6.3) shows that the quantity
dt
from the synchronous speed in mechanical radian per second. Substituting equation (6.4) into
equation (6.1) one gets,
J
d2 δm
= Ta = Tm − Te
dt2
(6.5)
Now. let us define the angular velocity of the rotor to be ωm =
Jωm
dθm
. From equation (6.5) we get,
dt
d2 δm
= ωm Ta = ωm Tm − ωm Te
dt2
Or,
Jωm
d2 δm
= Pa = Pm − Pe
dt2
(6.6)
In equation (6.6), Pa , Pe and Pm denote the accelerating power, electrical output power and the
input mechanical power (less than the rotational power loss) respectively.
The quantity Jωm is the angular momentum of the rotor and at synchronous speed, it is known
as the inertia constant and is denoted by M . Strictly, the quantity Jωm is not constant at all
operating conditions since ωm keeps on varying. However, when the machine is stable, ωm does not
differ significantly from ωsm and hence, Jωm can be taken approximately equal to M . Hence, from
equation (6.6) we obtain,
M
d2 δm
= Pa = Pm − Pe
dt2
(6.7)
Now, in machine data, another constant related to inertia, namely H-constant is often encountered. This is defined as;
H=
stored kinetic energy in megajoules at synchronous speed
machine rating in MVA
Or,
1 Jωsm 2 1 M ωsm
H=
=
2 Smc
2 Smc
MJ/MVA =
1 M ωsm
2 Smc
sec.
(6.8)
In equation (6.8), the quantity Smc is the three phase MVA rating of the synchronous machine.
Now, from equation (6.8),
M=
2HSmc
ωsm
MJ/mech. rad
(6.9)
Substituting for M in equation (6.7), we get,
2H d2 δm Pa Pm − Pe
=
=
ωsm dt2
Smc
Smc
(6.10)
In equation (6.10), both δm and ωsm are in mechanical units. Now, the corresponding quantities
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in electrical units are given as,
ωs =
P
ωsm ;
2
δ=
P
δm ;
2
(6.11)
In equation (6.11), P is the number of pole in the generator, ωs is the synchronous speed of the
machine in electrical radian/sec. and δ (in electrical radian) is the angular displacement of the rotor
from the synchronously rotating reference axis. Substituting equation (6.11) in equation (6.10) we
get,
2H d2 δ
= Pa = Pm − Pe per unit
ωs dt2
(6.12)
Equation (6.12) is known as the swing equation of the synchronous machine. As this is a
second order differential equation, it can be written as a set of two first order differential equations
as below.
2H dω
= Pm − Pe per unit
ωs dt
(6.13)
dδ
= ω − ωs
dt
(6.14)
In equations (6.13) - (6.14), the quantity ω is the speed of the synchronous machine and is
expressed in electrical radian per second. Now, in the above two equations, no damping of the
machine is considered. If damping is considered (which opposes the motion of the machine), a term
proportional to the deviation of the speed (from the synchronous speed) is introduced in equation
(6.13). Therefore, the modified equation becomes;
2H dω
= Pm − Pe − d(ω − ωs ) per unit
ωs dt
(6.15)
In equation (6.15), d is called the damping co-efficient. However, in the presence of damping, equation
(6.14) does not change. Therefore, in the presence of damping, this pair of equations ((6.14) and
(6.15)) describe the motion of the synchronous machine.
With this introduction of motion of synchronous machine, we are now ready to address the various
stability issues. From the next lecture we will start with transient stability analysis.
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