UNIT - IV
POWER SYSTEM STEADY STATE STABILITY
ANALYSIS
Stability of power system is its ability to return to normal or stable operating condition
after been subjected to some of disturbance. Instability means a condition representing
loss of synchronism or fall out of step.
The instability of power system is divided into two parts
1. Steady state stability
2. Transient stability
Increase in load is a kind of disturbance to power system. If the increase in load takes
place gradually and slowly in small steps and the system withstand this change in load
and operates satisfactorily then this system phenomena is said to be STEADY STATE
STABILITY.
Cause of transient disturbances
1. Sudden change of load.
2. Switching operation.
3. Loss of generation.
4. Fault.
Due to the following sudden disturbances in the power system, rotor angular difference,
rotor speed and power transfer undergo fast changes whose magnitude are dependent
upon the severity of disturbances.
If the disturbance is so large that the angular difference increases so much which can
cause the machine out of synchronism. This kind of instability is denoted as transient
instability. It is a very fast phenomenon it occurs within one second for the generating
unit closer to the disturbance.
Dynamics Of A Synchronous Machine
The kinetic energy of the rotor at synchronous machine is
1
KE J 2 106 MJ ( 1)
sm
2
J =rotor moment of inertia in kg-m2
g =machine rating(base)in mva(3-phase)
h =inertia constant in mj/mva or mw-s/mva
so,
GH
2GH MJ-s/elect.rad ( 4)
M
s f
GH
MJ-s/elect.rad ( 5)
180 f
Taking G as base, the inertia constant in pu is
H
M
s2/elect.rad ( 6)
f
H
M s2/elect.degree ( 7)
180 f
Swing Equation
The differential equation that relates the angular momentum M, acceleration power Pa and the rotor
angle is known as swing equation. Solution of swing equation shows how the rotor angle changes with
respect time following a disturbance. The plot Vs t is known as swing curve. The differential equation
governing the rotor dynamics can then be written as.
Fig. 1 Electrical and mechanical power flow in motor
While the rotor undergoes dynamics as per Equation (9), the rotor speed changes by
insignificant magnitude for the time period of interest (1s)
Equation ( 8) can therefore be converted into its more convenient power form by assuming
the rotor .speed (ωsm). Multiplying both sides of Equation ( 8) by ωsm we can write
d 2 6
Jsm m
10 Pm Pe MW ( 9)
2
dt
Where,
Pm= mechanical power input in MW
Pe=electrical power output in MW; stator copper loss is assumed neglected.
Rewriting Equation ( 9)
Where
ϴe =angle in rad.(elect.)
As it is more convenient to measure the angular position of the rotor with respect to a
synchronously rotating frame of reference.
Let us
assume,
e st ( 13)
δ is rotor angular displacement from synchronously rotating reference frame, called
Torque Angle/Power Angle.
From Equation ( 9)
d 2 d 2
e
( 14)
dt2 dt2
Hence Equation ( 11) can be written in terms of as
d 2
M Pm PeMW ( 15)
dt2
Using Equation ( 11) we can also write
GH d 2
2
f dt
Dividing throught by G, the MVA rating of the machine
d 2
Pm Pe ( 17)
M ( pu)
dt2
Wher
e H H d 2
M (pu) , Pm Pe pu
f f dt2
Equation ( 17) is called as swing equation and it describes the rotor dynamics for a
synchronous machine (generating/motoring). It is a second-order differential equation
where the damping term (proportional to d dt ) is absent because of the assumption of a
loss less machine and the fact that the torque of damper winding has been ignored. Since
the electrical power Pe depends upon the sine of angle the swing equation is a non-linear
second-order differential equation.
Multi-Machine System
In a multi-machine system a common system base must be
chosen Let
Gmach=machine rating
(base) Gsystem=system base
Equation(18) can then be written as
Since the machine rotors swings together (coherently or in unison)
1 2
Adding Equation ( 20) and ( 21)
H eq d 2
Pm - P ( 22)
e
f dt
2
Wher
e
Pm Pm1 Pm2
Pe Pe1 Pe2
Heq H1 H2
The two machines swinging coherently are thus reduced to a single machine as in
Equation ( 22), the equivalent inertia in ( 22) can be written as
G H 2mach G2mach
H eq H1mach 1mach ( 23)
G system G
syste
m
The above results are easily extendable to any number of machines swinging coherently.
To solving the swing equation (Equation ( 23), certain simplifying assumptions are usually
made. These are:
1. Mechanical power input to the machine (Pm) remains constant during the period of
electromechanical transient of interest. In other words, it means that the effect of the
turbine governing loop is ignored being much slower than the speed of the transient. This
assumption leads to pessimistic result-governing loop helps to stabilize the system.
2. Rotor speed changes are insignificant-these have already been ignored in formulating
the swing equation.
3. Effect of voltage regulating loop during the transient is ignored, as a consequence the
generated machine emf remains constant. This assumption also leads to pessimistic
results- voltage regulator helps to stabilize the system.
Before the swing equation can be solved, it is necessary to determine the dependence of the
electrical power output (Pe) upon the rotor angle.
Simplified Machine Model
For a non-salient pole machine, the per-phase induced emf-terminal voltage equation
under steady conditions is.
E V jX d I d X q ( 24)
jX q I q ; X d
Fig. 3 Simplified machine model.
The machine model corresponding to Eq. ( 26) is drawn in Fig. ( 3) which also applies to a
/
X d X q X s (transient synchronous reactance).
/ /
cylindrical rotor machine
where
Power Angle Curve
For the purposes of stability studies E , transient emf of generator motor remains constant
or is
the independent variable determined by the voltage regulating loop but V, the generator
determined terminal voltage is a dependent variable. Therefore, the nodes (buses) of the
stability study network to the ernf terminal in the machine model as shown in Fig. 4, while
the machine
reactance ( X d is absorbed in the system network as different from a load flow study.
) Further,
the loads (other than large synchronous motor) will be replaced by equivalent static
admittances (connected in shunt between transmission network buses and the reference
bus).
Fig. 4 Simplified Machine studied Network
Fig 5 Power Angle Curve
This is so because load voltages vary during a stability study (in a load flow study, these
remain constant within a narrow band). The simplified power angle equation is
Pe Pmax sin ( 27)
Wher
e
E1 E 2
P
max ( 28)
X
The graphical representation of power angle equation ( 28) is shown in Fig. 5. The swing
equation ( 27) can now be written as
H d 2
Pm Pe pu ( 29)
f dt 2
It is a non linear second-order differential equation with no damping.
Machine Connected to Infinite Bus
Figure 6 is the circuit model of a single machine connected to infinite bus through a line of
reactance Xe. In this simple case
X transfer X d X e
From Eq ( 30) we get
Pe
E V sin Pmax sin
( 30)
X transfer
Fig. 6 Machine connected to infinite bus bar
The dynamics of this system are described in Eq. ( 15 ) as
H d 2
Pm Pe pu ( 31)
f dt 2
Two Machine Systems
The case of two finite machines connected through a line (Xe) is illustrated in Fig. 5
where one of the machines must be generating and the other must be motoring. Under
steady condition, before the system goes into dynamics and the mechanical input/output of
the two machines is assumed to remain constant at these values throughout the dynamics
(governor action assumed slow).During steady state or in dynamic condition, the electrical
power output of the generator must be absorbed by the motor (network being lossless).
Fig. 7 Two machine system
Thus at all
time
Pm1 Pm2 Pm ( 32)
Pe1 Pem 2 Pe ( 33)
Steady State Stability
The steady state stability limit of a particular circuit of a power system is defined as
the maximum power that can be transmitted to the receiving end without loss of
synchronism.
Consider the simple system of Fig. 7 whose dynamics is described by equations
d 2
M e
Pm PeMW ( 40)
2
dt
H
M
in pu system ( 41)
f
And,
Pe E V sin Pmax sin ( 42)
Xd
For determination of steady state stability, the direct axis reactance (Xd) and, voltage
behind Xd
are used in the above equations. Let the system be operating with steady power transfer of
Pe0=Pm with torque angle 0 as indicated in the figure. Assume a small increment P in the
electric power with the input from the prime mover remaining fixed at Pm (governor
response is slow compared to the speed of energy dynamics), causing the torque angle
to change to ( 0 ) .
d 2
Pm (Pe0 Pe ) Pe
M
dt 2
or
d 2 P
M e 0 ( 43)
dt 2 0
or
Pe 0
Mp 2
0
Where
d
p
dt
The system stability to small change is determined from the characteristic equation.
pe
Mp2 0
0
Its two roots are
1
pe 2
p
M
As long as p e 0it positive, the roots are purely imaginary and conjugate and the
system
behaviour is oscillatory about 0 . Line resistance and damper windings of machine, which
have been ignored in the above modelling, cause the system oscillations to decay. The
system is therefore stable for a small increment in power so long as
pe
0 ( 44)
0
When pe 0 , 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 (disturbance) and the synchronism is soon lost. The system is therefore
unstable for
pe 0
( 45)
0
pe is known as synchronizing coefficient. This is also called stiffness (electrical) of
0
synchronous machine.
Assuming |E| and |V| to remain constant, the system is unstable, if
EV
cos 0
0
X
0 90 ( 46)
The maximum power that can be transmitted without loss of stability (steady state) occurs
for
0 90 ( 47)
EV
Pmax ( 48)
X
If the system is operating below the limit of steady stability condition (Eq. 48), it
may continue to oscillate for a long time if the damping is low. Persistent oscillations are a
threat to system security. The study of system damping is the study of dynamical stability.
The above procedure is also applicable for complex systems wherein governor
action and excitation control are also accounted for. The describing differential equation is
linerized about the operating point. Condition for steady state stability is then determined
from the corresponding characteristic equation (which now is of order higher than two).
It was assumed in the above account that the internal machine voltage |E| remains
constant (i.e., excitation is held constant). The result is that as loading increases, the
terminal voltage |Vt| dips heavily which cannot be tolerated in practice. Therefore, we must
consider the steady state stability limit by assuming that excitation is adjusted for every
load increase to keep
|Vt| constant. This is how the system will be operated practically. It may be understood
that we are still not considering the effect of automatic excitation control.
Some Comment on Steady State Stability
Knowledge of steady state stability limit is important for various reasons. A system
can be operated above its transient stability limit but not above its steady state limit. Now,
with increased fault clearing speeds, it is possible to make the transient limit closely
approach the steady state limit.
As is clear from Eq. ( 50), the methods of improving steady state stability limit of a
system are to reduce X and increase either or both |E| and |V|. If the transmission lines are
of sufficiently high reactance, the stability limit can be raised by using two parallel lines
which incidentally also increases the reliability of the system. Series capacitors are
sometimes employed in lines to get better voltage regulation and to raise the stability limit
by decreasing the line reactance. Higher excitation voltages and quick excitation system are
also employed to improve the stability limit.
