Mrinal K Pal. Power system stability
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SYNCHRONOUS MACHINE STABILITY BASICS
3-15
Division of Suddenly Applied Load (or Generation Loss) Among Generators in the System
A suddenly applied load (or a sudden generation loss) creates an unbalance between generation
and load and a transient results. The system passes through an oscillatory state before settling to
a new steady-state condition. Following the impact, the amount of load picked up by each
generator at various instants of time can be estimated from a knowledge of the system and the
various control parameters. Although the following analysis is approximate, it throws much light
on the behavior of the system immediately following a load (or generation) impact. For the
purpose of our analysis we assume that the generators can be represented by the classical model
(i.e., by constant voltage magnitude behind transient reactance) and that all the non-generator
buses except the one where the load impact occurs have been eliminated.
Referring to the network shown in Figure 3.6, where a load P
L
is suddenly applied at bus n, the
equation for power into each node is obtained from
()
niBGEEP
n
j
ijijijijjiei
,,2,1sincos
1
L=+=
∑
=
δδ
(3.63)
It is assumed that the applied load P
L
has a negligible reactive component.
Fig. 3.6 Network showing load impact at bus n.
Since the impact load P
L
is small compared with the total load, and we are concerned with the
change in the loading of the system generators (load pick-up), we can linearize the system
equations. Linearizing (3.63),
ni
P
P
j
n
j
j
ei
ei
,,2,1
1
L=∆
∂
∂
=∆
∑
=
δ
δ
(3.64)
where the
j
ei
P
δ
∂
∂
's are the synchronizing power coefficients given by, neglecting line
conductances,
∑
≠
=
=
∂
∂
n
ij
j
ijijji
i
ei
BEE
P
1
cos
δ
δ
(3.65)
and
ijijji
j
ei
BEE
P
δ
δ
cos−=
∂
∂
(3.66)
Also
enL
PP
∆
−
=
(3.67)
SYNCHRONOUS MACHINE STABILITY BASICS
3-16
From equations (3.64) through (3.67), it follows that
∑
=
=∆
n
i
ei
P
1
0 (3.68)
or
∑
−
=
=∆−=∆
1
1
n
i
Lenei
PPP (3.69)
Load pick-up at t = 0
+
At the instant immediately following the load impact, i.e., at t = 0
+
, ∆
δ
i
= 0 for all generators,
since the rotor angles cannot change instantaneously because of the rotor inertias. The only
change in the angle at t = 0
+
is that of the load bus voltage. Therefore, from (3.64),
1,,2,1cos −=∆−=∆
∂
∂
=∆ niBEE
P
P
nininnin
n
ei
ei
L
δδδ
δ
(3.70)
From equations (3.69) and (3.70), we have
∑∑
−
=
−
=
∆
∂
∂
==∆
1
1
1
1
n
i
n
i
n
n
ei
Lei
P
PP
δ
δ
(3.71)
and
1,,2,1
1
1
−=
∂
∂
∂
∂
=∆
∑
−
=
ni
P
P
P
P
n
i
n
ei
L
n
ei
ei
L
δ
δ
(3.72)
Therefore, the load impact P
L
at bus n at t = 0
+
is shared by the synchronous generators according
to their synchronizing power coefficients with respect to the bus n. Thus, the machines
electrically close to the point of impact will pick up the greater share of the load regardless of
their size. At t = 0
+
, the source of energy supplied by the generators is the energy stored in their
magnetic field.
Due to the sudden increase in the output power the generators will start decelerating. From the
swing equations, it is apparent that the initial deceleration of a particular machine i will be
dependent on the synchronizing power coefficient
n
ei
P
δ
∂
∂
and the inertia constant H
i
. Thereafter,
the machines follow an oscillatory motion governed by the swing equations.
Average behavior prior to governor action
As the machines decelerate the individual machine speeds will drop and governor action will be
initiated following a time lag. We now estimate the system behavior during the period from t =
0
+
to the time when the governor action becomes appreciable. During this period the system as a
whole will be retarding. Although the individual machines will initially be retarding at different
rates, if the system remains stable, they will acquire the same mean retardation after the initial
transients have decayed.
SYNCHRONOUS MACHINE STABILITY BASICS
3-17
The incremental swing equations governing the motion of the machines are given by
ei
ii
P
dt
dH
∆−=
∆
ω
ω
o
2
1,,2,1
−
=
ni L (3.73)
Assuming that the individual machines have acquired the same mean retardation
dtd /
ω
∆
,
equation (3.73) for the individual machines can be added together to give
L
n
i
ei
n
i
i
PP
dt
d
H
−=∆−=
∆
∑
∑
−
=
−
=
1
1
o
1
1
2
ω
ω
(3.74)
Substituting (3.74) into (3.73) we have
i
n
i
i
L
ei
H
H
P
P
∑
−
=
=∆
1
1
(3.75)
Equation (3.75) shows that the (mean) increase in output of the individual generators in response
to an impact load is proportional to their inertia constants.
Equation (3.73) can also be written as
eiii
o
PH
dt
d
∆−=∆ )(
2
ω
ω
1,,2,1
−
=
ni L (3.76)
Adding the equations corresponding to the individual machines, we have
L
n
i
ei
n
i
ii
PPH
dt
d
−=∆−=
∆
∑∑
−
=
−
=
1
1
1
1
o
2
ω
ω
(3.77)
Equation (3.77) can be written as
L
n
i
i
n
i
ii
n
i
i
P
H
H
dt
d
H
−=
∆
∑
∑∑
−
=
−
=
−
=
1
1
1
1
o
1
1
2
ω
ω
(3.78)
Comparing equation (3.78) with equation (3.74), we can define the mean angular speed deviation
as the weighted mean
∑
∑
−
=
−
=
∆
=∆
1
1
1
1
n
i
i
n
i
ii
H
H
ω
ω
(3.79)
Similarly, we can define a mean angle deviation
SYNCHRONOUS MACHINE STABILITY BASICS
3-18
∑
∑
∆
=∆
i
ii
H
H
δ
δ
(3.80)
In the derivation of equation (3.75), which shows the change in electrical output of each machine
to be proportional to their respective inertias, the assumption was made that the individual
machines in the system reached the same mean retardation a short time after the load impact
occurred and before appreciable governor action took place. In a large system the time needed to
reach a uniform mean retardation may be considerable. Generators electrically remote from the
point of load impact may not "feel" the effect of the impact and, therefore, experience any speed
change in order to participate in the inertial load pick-up, until after an appreciable length of
time. By then, the governor action would start modifying the mechanical input of the local
generators in response to their speed change that has been sustained sufficiently long to
overcome the time lag. It follows, therefore, that in a large interconnected system, the generation
pick-up following a sudden load addition (or generator loss) will never be strictly in accordance
with the inertial distribution of the entire system. In estimating (inertial) pick-up during the brief
period immediately following a load impact, inertias of generators electrically remote from the
point of load impact should be excluded, since these generators will not participate significantly
in the inertial pick-up during this period. Governor action will start modifying system response
well before the inertias of the remote machines can exert any appreciable effect in the generation
pick-up.
In order for the governor of a generator to respond, the speed deviation has to exceed the
governor dead band. Since the generators electrically close to the bus at which the load impact
occurred would experience the largest speed deviation initially, these will respond first, by
overcoming the governor dead band and valve set point nonlinearities. The generators further
away would experience a more gradual drop in speed, and in the event the speed change does not
exceed the governor dead band, governor action will not be initiated. The actual pick-up by
individual generators following the initiation of governor action will depend on the total system
capacity, the amount of load addition (or lost generation), the relative locations of the generators
and their governor-turbine characteristics. The change in mechanical input of a particular
generator due to governor action is obtained from
R
P
P
i
mi
o
ω
ω
∆
=∆
(3.81)
where
P
i
= MW capacity of generator i
=R system regulation in pu(= 0.05)
=
o
ω
nominal system frequency in rad/sec
If ∆f (Hz) is the governor dead band, i.e., the governor is insensitive to frequency deviation of
less than ∆f Hz, the condition for all the generators in the interconnected system to respond due
to governor action is
R
P
f
f
P
i
L
∑
∆
≥
o
(3.82)
where P
L
is the amount of load addition (or lost generation).
SYNCHRONOUS MACHINE STABILITY BASICS
3-19
Therefore, if the total generation of the interconnected system (ΣP
i
) is greater than (f
o
/∆f) R P
L
,
then some of the generators electrically remote from the bus of load impact will not experience
the required frequency change of ∆f Hz in order for their governors to respond. Of course, some
of the generators in the system may not be under active governor control. This will have to be
accounted for in the above analysis.
It should be noted that the frequency change at some of the generator buses will always exceed
the governor dead band no matter how small the load impact is, since the mismatch between
generation and load created by the load impact, if left uncorrected, will continually increase the
frequency deviation from normal.
The transition from the inertial generation pick-up to the generation pick-up dictated by governor
action is oscillatory, the frequency of oscillation being a function of the machine inertia,
governor speed regulation and servomotor time constant. Eventually, the speed deviation and the
changes in the tie-line flows will be detected by the AGC system and the system generation
would be adjusted according to a set criterion.
Transient Stability by Equal Area Criterion
In power system stability studies the term transient stability usually refers to the ability of the
synchronous machines to remain in synchronism during the brief period following a large
disturbance. In a large disturbance, system nonlinearities play a dominant role. In order to
determine transient stability or instability following a large disturbance, or a series of
disturbances, it is usually necessary to solve the set of nonlinear equations describing the system
dynamics. Conclusions about stability or instability can then be drawn from an inspection of the
solution. Since a formal solution of the equations is not generally possible, an approximate
solution is usually obtained by a numerical technique.
For one and two machine systems, assuming that the machines can be represented by the
classical model, i.e., by constant voltage behind transient reactance, a simple graphical method
known as the equal area criterion can be utilized to assess transient stability. When a large
interconnected system is subjected to a large disturbance, as a rule, it splits into a small number
of groups of machines which swing against one another while the machines within each group
swing together. Frequently there are only two major groups and the general behavior of the
system is similar to that of a two machine system. In such situations, for the purpose of
approximate analyses, the large system may be replaced by an equivalent two machine system.
Consider a single synchronous machine connected to an infinite bus as shown in Figure 3.7.
Fig. 3.7 Single machine infinite bus system.
The swing equation of the machine is
ema
PPP
dt
dH
−==
2
2
o
2
δ
ω
(3.83)
which can be written as
SYNCHRONOUS MACHINE STABILITY BASICS
3-20
a
P
H
dt
d
2
o
2
2
ω
δ
= (3.84)
Multiplying both sides of (3.84) by
dt
d
δ
2 , we have
dt
d
P
H
dt
d
dt
d
a
δ
ω
δδ
2
2
2
o
2
2
=
or
dt
d
P
Hdt
d
dt
d
a
δ
ω
δ
o
2
=
or
δ
ω
δ
dP
Hdt
d
d
a
o
2
=
(3.85)
Integrating (3.85), we obtain
δ
ω
δ
δ
δ
dP
Hdt
d
o
a
∫
=
o
2
(3.86)
In equation (3.86), d
δ
/dt is the relative speed of the machine with respect to a synchronously
rotating reference frame. The initial value of d
δ
/dt is zero. Following a disturbance P
a
has a
positive value and the machine accelerates. For stability, P
a
must reverse sign and d
δ
/dt must
come to zero before P
a
reverses sign again. Therefore, for stability, there is a
δ
such that P
a
(
δ
)
≤ 0 and
0
o
=
∫
δ
δ
δ
dP
a
(3.87)
In the limit
0)(
max
=
δ
a
P and
0
max
o
=
∫
δ
δ
δ
dP
a
(3.88)
If the accelerating power is plotted as a function of
δ
, equation (3.88) can be interpreted as the
area under the curve between
δ
o
and
δ
max
. The net area under the curve adds to zero (the positive
and negative parts of the area are equal; hence the name -- equal area criterion).
Consider the case when P
m
is constant. For the system shown in Figure 3.7, P
e
is given by
δ
sin
21
X
EE
P
e
= (3.89)
The value of X would depend on the network condition, i.e. whether pre-fault, fault or post-fault.
The power angle curves corresponding to pre-fault, fault and post-fault conditions are shown in
Figure 3.8. By applying the criterion of equation (3.88), it can be seen that the system is stable
for a fault clearing angle of up to
δ
c
, when the areas A
1
and A
2
are equal. For a clearing angle
greater than
δ
c
the system is unstable. The angle
δ
c
is called the critical clearing angle and the
corresponding maximum rotor angle reached is
δ
m
.
SYNCHRONOUS MACHINE STABILITY BASICS
3-21
Fig. 3.8 Equal area criterion applied to the system of Fig. 3.7.
Problem
Show that the critical clearing angle
δ
c
shown in Figure 3.8 is given by
−+−
−
=
o12o
12
coscos)(
1
cos
δδδδδ
rr
P
P
rr
mm
M
m
c
where
P
M
= peak of the pre-fault power-angle curve
r
1
= ratio of the peak of the power-angle curve of the faulted network to P
M
r
2
= ratio of the peak of the power-angle curve of the post-fault network to P
M
The equal area criterion provides information on the critical clearing angle. However, the
corresponding clearing time is of primary importance. The clearing time must be obtained from a
time solution of the swing equation. A graphical solution of the swing equation is illustrated
below.
From equation (3.86), the relative angular speed of the machine is given by
∫
==
δ
δ
δ
ω
ω
δ
o
o
dP
Hdt
d
a
(3.90)
Graphically, the integral in equation (3.90) is the area under the curve of P
a
against
δ
, or between
the curves of P
m
and P
e
against
δ
. By rearranging equation (3.90) and integrating, we obtain
∫
∫
∫
==
δ
δ
δ
δ
δ
δ
δ
ω
δ
ω
δ
o
o
o
o
dP
H
dd
t
a
(3.91)
Evaluation of this integral gives t as a function of
δ
, which is the swing curve.
SYNCHRONOUS MACHINE STABILITY BASICS
3-22
In general, a formal solution of equation (3.91) is not possible. If P
a
is constant, a formal solution
can be obtained. Then
a
P
H
t
o
o
)(4
ω
δδ
−
=
(3.92)
The integral of equation (3.91) can be evaluated graphically. This may be done by plotting a
curve of 1/
ω
against
δ
and finding the area under the curve as a function of
δ
. Some difficulty
will be encountered in determining the area under the curve for values of
δ
near the initial value
δ
o
and the maximum value
δ
m
, because at these values of
δ
,
ω
is zero. This difficulty may be
avoided by assuming P
a
to be constant over a small range of
δ
(until the curve of 1/
ω
comes back
on scale) and by using equation (3.92) to compute the time for the machine to swing through this
range of
δ
.
Effect of damping
In the system model used in the previous section (equation (3.83)), damping was neglected. In a
stable system when the machine rotor reaches the maximum angle, d
δ
/dt = 0, and the net torque
on the rotor is retarding. The rotor will therefore start to swing backward and go past the post-
fault equilibrium point and continue until it reaches the point where d
δ
/dt = 0, and the total area
between the post-fault power-angle curve and the input line is zero, i.e., the area above the input
line is equal to the area below the input line (the same equal area criterion applies). At this point
the net torque on the rotor is accelerating and therefore the rotor will start to swing forward. In
the absence of damping the rotor will continue to swing forward and backward at constant
amplitude.
The presence of positive damping (see equation (3.28)) will provide a component of electrical
power (or torque) in proportion to speed, i.e., a component of power that is positive when the
angle is increasing and vice versa. This will modify the power-angle curve in accordance with
the damping term in (3.28). It can be seen that, including the effect of damping, the area between
the power-angle curve and the input line becomes smaller in each successive swing, thereby
allowing the rotor to eventually settle at the post-fault equilibrium point. Note that if the damping
is assumed to be negative, i.e., the component of electrical power decreasing with increase in
angle, the area between the power-angle curve and the input line will increase in each subsequent
swing, thereby causing instability.
Equal area criterion for a two-machine system
A two-machine system can be reduced to an equivalent one machine-infinite bus system. The
equivalent system can be derived as follows.
The swing equations of the two machines are
111
2
1
2
o
1
2
ema
PPP
dt
dH
−==
δ
ω
(3.93)
222
2
2
2
o
2
2
ema
PPP
dt
dH
−==
δ
ω
(3.94)
Equations (3.93) and (3.94) can be combined as
SYNCHRONOUS MACHINE STABILITY BASICS
3-23
2
2o
1
1o
2
12
2
22 H
P
H
P
dt
d
aa
ωω
δ
−= (3.95)
where
2112
δ
δ
δ
−
=
Equation (3.95) can be written as
21
2112
2
12
2
21
21
o
2
HH
PHPH
dt
d
HH
HH
aa
+
−
=
+
δ
ω
(3.96)
or
ema
PPP
dt
dH
−==
δ
ω
2
o
2
(3.97)
where
21
21
HH
HH
H
+
=
(3.98)
is the equivalent inertia constant, and
12
δ
δ
=
The equivalent mechanical input,
21
2112
HH
PHPH
P
mm
m
+
−
= (3.99)
and the equivalent electrical output,
21
2112
HH
PHPH
P
ee
e
+
−
=
(3.100)
As in the one-machine case, the expression for equal area criterion for the two-machine case is
given by
∫
=−=
12
o12
0)(
122112
21
o
2
12
δ
δ
δ
ω
δ
dPHPH
HHdt
d
aa
(3.101)
If network resistance is neglected
12 aa
PP
−
=
,
and equation (3.101) reduces to
0
12
o12
12
o12
121
o
121
21
21
o
2
12
==
+
=
∫∫
δ
δ
δ
δ
δ
ω
δ
ω
δ
dP
H
dP
HH
HH
dt
d
aa
(3.102)
Transient Stability by Liapunov’s Method
As has been pointed out in Chapter 2, the application of Liapunov’s method in multi-machine
transient stability studies involves mathematical compromises and, therefore, the method is not
suitable for practical use. However, in a simple one or two machine system, when the machines
SYNCHRONOUS MACHINE STABILITY BASICS
3-24
are represented by the classical model, the method can be used conveniently without sacrificing
mathematical rigor. Actually, using the transient energy as the Liapunov function, one would
arrive at a stability criterion which is the same as the equal area criterion.
With
δ
= x
1
and d
δ
/dt = x
2
, equation (3.28) can be written as
2
o
1
212
o
2
1
sin
2
x
K
x
X
EE
P
dt
dx
H
x
dt
dx
d
m
ωω
−−=
=
(3.103)
Equilibrium points are obtained by putting the right hand side of (3.103) equal to zero. Thus, one
equilibrium point is at
0,
21
== xx
θ
and a second one at 0,π
21
=
−
=
xx
θ
, where
21
1
sin
EE
XP
m
−
=
θ
As will be seen later, the first equilibrium point is stable, and the second one is unstable.
Shifting the origin to the stable equilibrium point by a change of variable, (3.103) can be written
as
2
o
1
212
o
2
1
)sin(
2
x
K
x
X
EE
P
dt
dx
H
x
dt
dx
d
m
ω
θ
ω
−+−=
=
(3.104)
The equilibrium points after shifting the origin are given by (i)
0,0
21
== xx and (ii)
0,2π
21
=−= xx
θ
.
We choose a Liapunov function
1
0
1
21
2
2
o
21
1
)}sin({
2
2
1
),( xdx
X
EE
Px
H
xxV
x
m
′′
++−+=
∫
θ
ω
(3.105)
Stationary points are given by
0,0
21
=
∂
∂
=
∂
∂
x
V
x
V
or
0,0)sin(
21
21
==++− xx
X
EE
P
m
θ
These are the same as the equilibrium states of the system given by (3.104).
The matrix
H of the second derivative of V is given by
+
=
o
1
21
2
)cos(
ω
θ
H
x
X
EE
H