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304 HANDBOOK OF ELECTRICAL ENGINEERING

additional reactances (or impedances) due to the presence of cables, overhead lines and transformers.
The system will be assumed to be stable in the steady state.
In order to change the operating conditions of the system there must be a change in the load
(or loads). This may be due to starting a motor, switching in or out a cable or overhead line, changing
the load on a motor or changing a static load. When a load change occurs, the relative position of
the generator rotors will change, i.e. δ g of each generator will change. This angular change of rotor
position will be accompanied by an oscillatory movement of the rotors as they reposition themselves.
The amplitude and duration of the oscillatory motion is mainly determined by the mechanical inertia
and the damping characteristics of the generators and their prime-movers.
The inertia and damping characteristics can be represented by an accelerating power term
and a frictional or damping power term in a simplified second-order differential equation for each
generator. Also in the equation is a term for the electrical power generated. The right-hand side of
the equation represents the mechanical power that is applied to the shaft of the generator.
Each generator prime-mover unit can be thought to be rather like a mechanical spring/mass/
damper dynamic system. Once disturbed in any way, the mass will oscillate and eventually settle at
a new position. The static characteristic of the spring is analogous to the electrical power generated
and sent out from the generator. The inertia term includes all the rotating masses of the generator,
its prime-mover and a gearbox that may be used. The damping term consists of two parts; firstly the
damping due to eddy current induction in the rotor electrical circuits and, secondly, the damping due
to the friction, windage and governor action at the prime-mover.
The subject of electromagnetic damping within synchronous machines is a complicated one
and some of the earliest analytical work was recorded in the 1920s e.g. References 9 to 11 using
mechanical analogues. A later mechanical analogue was made by Westinghouse Electrical Corpora-
tion, Reference 7, Chapter 13, based on that given in Reference 9. A comprehensive summary of the
historical developments made in this subject, and automatic voltage regulation, from 1926 to 1973
can be found in Reference 12.

A typical set of system equations will now be described in their simpler form. There are many
variations on the general theme, depending upon the results being sought. The analysis of fast-acting
transients to match field tests would require very detailed modelling of all the dynamic components
of the machinery in the system. The starting of motors or the loss of generation would not require
such a detailed representation since the transients of interest take longer to manifest themselves, i.e.
20 seconds, instead of 1 second, are required to pass in order to reach a conclusion.

11.11.2.1 The equation of motion of one generator
The transient power balance equation of an individual generator prime-mover set may be written as:-

P a + P fw + P em + P elec = P mech
Where: P a = accelerating power for the polar moment of inertia.
P fw = friction and windage power.
P em = electromagnetic damping power.
P elec = electrical power delivered from the generator terminals.

P mech = mechanical power received by the generator at its coupling.