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 Swing equation

Inertia Constant H

Linearization of swing equation

Angular frequency of undamped oscillations

Equal area criterion(EAC) of stability

Application of EAC

Transient stability limit

Numerical technique for solution of swing equation

Improvement of power system stability

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HOME takes you to the start page after you have read these Topics. Start page has links to other topics.

1. a free Modelica Library for Power Systems, suitable for the simulation and analysis of transient and voltage stability in power systems.

2. TEFTS -free software- transient stability program to study transient energy functions and voltage stability issues- includes example & a brief tutorial


  1. The critical clearing angle of a given power system for a certain fault is

  1. proportional to the Inertia Constant M
  2. proportional to the Inertia Constant M
  3. independent of M.

Ans.: ©

  1. Equal area criterion of stability is applicable to

  1. a machine infinite bus system only
  2. both to a machine-infinite bus and a two-machine system
  3. 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

  1. negative

Ans.: (a)

  1. Which one of the following enhances the transient stability of a system the most

  1. proper choice of make and break capabilities of the circuit breakers
  2. installation of 3-pole auto-reclose circuit breakers
  3. installation of single pole auto-reclose circuit breakers

Ans.: (c )



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) d2d /dt2= Pm -Pe

where Pm and Pe are pu powers, d 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


d2D d /dt2 + w s . S. D d = 0


where D d = small change in nominal operating angle d

S = Synchronizing power coefficient = d Pe (at d = d o)



  • State the assumptions made in transient stability analysis
  • What is the range of undamped natural frequency of oscillation of 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 d2d /dt2= Pm -Pe = Pa

Multiplying both sides of the equation by (2/M) dd /dt, we get

2 dd /dt .d2d /dt2= (2Pa/M) dd /dt (6)

Since dd /dt. (dd /dt)2 = 2 dd /dt .d2d /dt2,

Equation (6) reduces to


dd /dt. (dd /dt) 2 = (2Pa/M) dd /dt

Next, multiply each side by dt, obtaining

d[. (dd /dt) 2]= (2Pa/M) dd

Integrating this equation, we get


(dd /dt) 2 = (2/M) Pa dd

d o



(dd /dt) 2 = Ö [(2/M) Pa dd ]

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


d m

Pa dd =0

d o

This integral may be integrated graphically (Fig.9) as the area under a curve of Pa plotted against d between limits d o, the initial angle, and d 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.



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.

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

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. First, compute the angular position d , and angular speed dd /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 d and dd /dt a the beginning of the interval
  2. 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 d during the nth time interval is

D d n =D d n-1 + [(D t) 2 /M] Pa(n-1)


D d n = change in angle during the nth time interval

D d n-1 = change in angle during the (n-1)th time interval

D 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:

D d 1 =[(D 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:

Discrete Controls:

Fast Valving

Discuss the effect of variation of M and auto -reclosure on the transient stability limit