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.