Download TRANSIENT STABILITY -OBJECTIVE TYPE QUESTIONS The

TRANSIENT STABILITY -OBJECTIVE TYPE QUESTIONS
1.
The critical clearing angle of a given power system for a certain fault is
a.
b.
c.
proportional to the Inertia Constant M
proportional to the Inertia Constant M
independent of M.
Ans.: ©
1.
Equal area criterion of stability is applicable to
a.
b.
c.
a machine infinite bus system only
both to a machine-infinite bus and a two-machine system
a multi-machine system.
Ans.: (b)
1.
In a two-machine power system, machine A delivers power to machine B. A 3-phase fault occurs at the
terminals of machine A. Initial acceleration of machine A is
(a) positive
(b) zero
a.
negative
Ans.: (a)
1.
Which one of the following enhances the transient stability of a system the most
a.
b.
c.
proper choice of make and break capabilities of the circuit breakers
installation of 3-pole auto-reclose circuit breakers
installation of single pole auto-reclose circuit breakers
Ans.: (c )
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Inertia Constant H:
Inertia constant H is different from inertia constant M. For a synchronous machine inertia constant H is frequently
specified. It is defined as the ratio of the stored K.E at rated speed to the rated apparent power of the machine, i.e.
H= Stored K.E in MJ at synchronous speed / machine rating in MVA (5)
Swing equation (4) reduces to the form
(2H/ws) d2 /dt2= Pm -Pe
where Pm and Pe are pu powers,  and ws should have consistent units
Linearization of swing equation
For small perturbations, the dynamic behaviour of the system can be studied by linearising the swing
equations around the nominal operating point. The linearised swing equation is
d2  /dt2 +  s . S.   = 0
2H
where   = small change in nominal operating angle 
S = Synchronizing power coefficient = d Pe (at  =  o)
d
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a synchronous generator?
Equal area criterion (EAC) of stability
To determine whether a power system is stable after a disturbance, it is necessary, in general, to plot and inspect the
swing curves. If these curves show that the angle between any two machines tends to increase without limit, the
system of course is unstable. If, on the other hand, after all the disturbances including switching have occurred, the
angles between the two machines of every possible pair reach maximum values and thereafter decrease, it is
probable, although not certain, that the system is stable. Occasionally in a multi-machine system one of the
machines may stay in step on the first swing and yet go out of step on the second swing because the other machines
are in different positions and react differently on the first machines.
In a two-machine system, under the usual assumptions of constant input, and constant voltage behind transient
reactance, the angle between the machines either increases indefinitely or else, after all disturbances have occurred,
oscillates with constant amplitude. In other words the two machines either fall out of step on first swing or never.
Under these conditions the observation that the machines come to rest with respect top each other may be taken as
the proof that the system is stable. There is a simple graphical method of determining whether the machines come to
rest with respect to each other. This method is known as the equal area criterion of stability. When this criterion is
applicable its use wholly or partially eliminates the need of computing swing curves and thus saves considerable
amount of computation time. It is applicable to any two-machine system for which the assumptions stated above
may be made.
Application of EAC
Consider a machine infinite bus system. The swing equation of the system is
M d2 /dt2= Pm -Pe = Pa
Multiplying both sides of the equation by (2/M) d /dt, we get
2 d /dt .d2 /dt2= (2Pa/M) d /dt (6)
Since d /dt. (d /dt)2 = 2 d /dt .d2 /dt2,
Equation (6) reduces to
d /dt. (d /dt) 2 = (2Pa/M) d /dt
Next, multiply each side by dt, obtaining
d[. (d /dt) 2]= (2Pa/M) d
Integrating this equation, we get

(d /dt) 2 = (2/M) ƒ Pa d
o

(d /dt) 2 =  [(2/M) ƒ Pa d ]
o
When the machine comes to rest with respect to the infinite bus- a condition, which may be taken to indicate
stability-requiring that
m
ƒ Pa d =0
o
This integral may be integrated graphically (Fig.9) as the area under a curve of Pa plotted against  between limits 
o, the initial angle, and  m, the final angle. Area A1 and A2 may be interpreted in terms of kinetic energy gained
/lost by the synchronous generator.
Illustrate the application of equal area criterion by applying it (to the following two simple cases) for a synchronous
generator connected to infinite bus through a double-circuit line.


Sustained fault
Line fault cleared after the lapse of a certain time by the simultaneous opening of the circuit breakers at
both ends of the line.
Transient stability limit
Of a two-machine system is defined as the maximum power that can be transmitted from one machine to the other
without loss of synchronism for a specified, sudden, severe, unrepeated shock.
Illustrate the concept of transient stability limit using equal area criterion of stability.



Obtain an equivalent one-machine infinite bus system of a two finite machines system
Obtain an expression for the critical clearing angle
What is the effect of location and duration of fault on the transient stability limit
Numerical technique for solution of swing equation
The transient stability analysis requires the solution of a system of coupled non-linear differential equations. In
general, no analytical solution of these equations exists. However, techniques are available to obtain approximate
solution of such differential equations by numerical methods and one must therefore resort to numerical computation
techniques 9commonly known as digital simulation0. Some of the commonly used numerical techniques for the
solution of the swing equation are:




Point by point method
Euler's method
Euler's modified method
Runge-Kutta method, etc.
Point -by -point method
Point by point solution, also known as step-by-step solution is the most widely used way of solving the swing
equation. The following two steps are carried out alternately.
1.
2.
First, compute the angular position  , and angular speed d /dt at the end of the time interval using the
formal solution of the swing equation from the knowledge of the assumed value of he accelerating power
and the values of  and d /dt a the beginning of the interval
Then compute the accelerating power of each machine from the knowledge of the angular position at the
end of the interval as computed in step 1.
There are two different point-by-point methods. Method 2 is more accurate compared to method 1.
Method 2
In this method the accelerating power during the interval is assumed constant at its value calculated for the middle of
the interval.
The desired formula for computing the change in  during the nth time interval is
  n =  n-1 + [( t) 2 /M] Pa(n-1)
where,
  n = change in angle during the nth time interval
  n-1 = change in angle during the (n-1)th time interval
 t= length of time interval
Pa(n-1)= accelerating power at the beginning of the nth time interval
Due attention is given to the effects of discontinuities in the accelerating power Pa which occur, for example, when a
fault is applied or removed or when any switching operation takes place. If such a discontinuity occurs at the
beginning of an interval, then the average of the values of Pa before and after the discontinuity must be considered.
Thus, in computing the increment of angle occurring during first interval after a fault is applied at t=0, the above
equation becomes:
  1 =[( t) 2 /M] Pa0+/2
where Pa0+ is the accelerating power immediately after the occurrence of the fault.
If the fault is cleared at the beginning of the mth interval, then for this interval,
Pa(m-1) = 0.5 [Pa(m-1)- + Pa(m-1) +]
Where Pa(m-1)- is the accelerating power before clearing and Pa(m-1) + is that immediately after clearing the fault..
If the discontinuity occurs at the middle of the interval, no special treatment is needed.
Improvement of power system stability
System parameters affecting stability:

Synchronous machine parameters





Transmission line parameters
Circuit breaker & relay characteristics
System layout
Excitation system and governor characteristics
Neutral grounding
Discrete Controls:








Auto -reclosing of circuit breakers
Single-pole switching
Resistor breaking
Generator dropping
Load shedding
Series capacitor switching
Shunt reactor/capacitor switching
Boosting power on D.C links
Fast Valving
Discuss the effect of variation of M and auto -reclosure on the transient stability limit
STEADY-STATE STABILITY- OBJECTIVE TYPE QUESTIONS
1. The power angle characteristic of a machine-infinite bus system is P= 2 sin  (pu).
The initial operating angle is 60 deg., inertia constant H= 5 sec. System frequency is 50 Hz. The angular frequency
of oscillation following small perturbation will be
a. sqrt(31.4) rad/sec
b. sqrt(15.7) rad/sec
c. sqrt(62.8) rad/sec
Ans.: (a)
2.
2.
In an interconnected power system ,the frequency of electro-mechanical modes of oscillation lies in the
range
(a) 0.5-2.5 Hz
(b) 1-10 Hz
(c) 30-60 Hz
Ans.: (a) Small and large perturbations
3.
3.
If the sending end and receiving end voltages for a 3-phase transmission line are each 33kV(line), and if
the reactance of the line is 13 ohms per phase, the maximum power transmitted per phase will be
a.
b.
c.
d.
a.
b.
c.
d.
60 MW
30 MW
29 MW
28 MW
Ans. d
4.
4.
The torque angle corresponding to the steady-state stability limit of a salient-pole alternator is
a.
b.
c.
a.
b.
c.
Less than 90 deg.
Greater than 90 deg.
Equal to 90 deg.
Ans. a
5.
5.
When the alternator stalls (near the stability limit) the armature current is
a.
b.
c.
a.
b.
c.
Less than the rated value
Greater than the rated value
Equal to the rated value.
Ans. a
Introduction
Power system stability is a term applied to alternating current electric power systems, denoting a condition in which
the various synchronous machines of the system remain in synchronism, or "in step" with each other. Conversely,
instability denotes a condition involving loss of synchronism, or falling "out of step".
The AIEE standard definition of stability is as follows" Stability when used with reference to a power system, is that
attribute of the system, or apart of the system which enables it to develop restoring forces between the elements
thereof, equal to or greater than the disturbing forces so as to restore a state of equilibrium between the elements".
Small and large perturbations
A power system is subjected to a variety of disturbances. These are classified into two categories:
 
Small perturbations
 
Large perturbations
Small Perturbations
Perturbations are characterized as small if the changes in system states are small due to these perturbations. The
magnitude of perturbation is small enough o allow the use of linearized state equations obtained by linearizing the
nonlinear differential equations around the operating point for studying he dynamics of the system. Random
changes in load which occur in the system continuously is a n example of small perturbation. Stability problem
associated with small perturbation is known as dynamic stability.
Large perturbations
A power system may be subjected to large perturbations such as :
 
Occurrence of faults on the line
 
Loss of large generating units
 
Loss of major transmission facilities
 
Loss of large loads
The stability problem associated with large perturbations is known as transient stability.
Derive and explain the power angle characteristic of
 
Machine infinite-bus system
 
Two machine system
Discuss the concepts of stable & unstable equilibrium points.
What is meant by steady state & transient stability of a system?
Voltage Stability-ModuleC(.xls file)
This module introduces the static voltage instability problem. The module uses a simple generator connected to a
load through two parallel transmission lines. The student can plot PV, QV, or PQ curves for different values of line
reactance, generator terminal voltage, and power factor. Parameters can be changed under the parameter column.
Different plots can be obtained from the Plot Command. Clicking the Introduction button provides student with
basic material to understand the concept of maximum power transfer. The information button includes the
instructions needed to plot various curves. The student is asked to calculate the reactive power required maintaining
a certain voltage Vr at the load bus for a given Pr, Vs, and Xl. The student is asked to calculate the reduction in
maximum power
at the load bus with one line out.