After a
disturbance, due usually to a network fault, the synchronous machine’s
electrical loading changes and the machines speed up (under very light loading
conditions they can slow down). Each machine will react differently depending
on its proximity to the fault, its initial loading and its time constants.
This means
that the angular positions of the rotors relative to each other change. If any
angle exceeds a certain threshold (usually between 140° and 160°) the machine
will no longer be able to maintain synchronism. This almost always results in
its removal from service.
Early work
on transient stability had concentrated on the reaction of one synchronous
machine coupled to a very large system through a transmission line. The large
system can be assumed to be infinite with respect to the single machine and
hence can be modeled as a pure voltage source. The synchronous machine is
modeled by the three phase windings of the stator plus windings on the rotor
representing the field winding and the eddy current paths.
These are
resolved into two axes, one in line with the direct axis of the rotor and the
other in line with the quadrature axis situated 90° (electrical) from the
direct axis. The field winding is on the direct axis. Equations can be
developed which determine the voltage in any winding depending on the current
flows in all the other windings.
A full set
of differential equations can be produced which allows the response of the machine
to various electrical disturbances to be found. The variables must include
rotor angle and rotor speed which can be evaluated from knowledge of the power
from the turbine into, and power to the system out of the machine.
The great
disadvantage with this type of analysis is that the rotor position is
constantly changing as it rotates. As most of the equations involve
trigonometrical functions relating to stator and rotor windings, the matrices
must be constantly reevaluated. In the most severe cases of network faults the
results, once the dc transients decay, are balanced.
Further, on
removal of the fault the network is considered to be balanced. There is thus
much computational effort involved in obtaining detailed information for each
of the three phases which is of little value to the power system engineer. By
contrast, this type of analysis is very important to machine designers.
However,
programs have been written for multi-machine systems using this method. Several
power system catastrophes in the U.S. and Europe in the 1960s gave a major
boost to developing transient stability programs. What was required was a
simpler and more efficient method of representing the machines in large power
systems.
Initially,
transient stability programs all ran in the time domain. A set of differential
equations is developed to describe the dynamic behavior of the synchronous
machines. These are linked together by algebraic equations for the network and
any other part of the system that has a very fast response, i.e., an
insignificant time constant, relative to the synchronous machines. All the
machine equations are written in the direct and quadrature axes of the rotor so
that they are constant regardless of the rotor position.
The network
is written in the real and imaginary axes similar to that used by the load flow
and faults programs. The transposition between these axes only requires
knowledge of the rotor angle relative to the synchronously rotating frame of
reference of the network.
Later work
involved looking at the response of the system, not to major disturbances but
to the build-up of oscillations due to small disturbances and poorly set
control systems. As the time involved for these disturbances to occur can be
large, time domain solutions are not suitable and frequency domain models of
the system were produced.
Lyapunov
functions have also been used, but good models have been difficult to produce.
However, they are now of sufficiently good quality to compete with time domain models
where quick estimates of stability are needed such as in the day to day
operation of a system.
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