# Reduction to Simple System

With a number of generators connected to the same busbar the inertia constant (H) of the equivalent machine is:

where Si... S_{n} — MVA of the machines and S_{b} — system base MVA.

For example, consider six identical machines connected to the same busbar, each having an *H* of 5MWs/MVA and rated at 60 MVA. Making the system base MVA equal to the combined rating of the machines (360 MVA), the inertia constant of the equivalent coherent machine is:

It is important to remember that the inertia of the spinning loads must be included; normally, this will be the sum of the inertias of the induction motors and their mechanical loads.

Two synchronous machines connected by a reactance may be reduced to one equivalent machine feeding through the reactance to an infinite busbar system. The properties of the equivalent machine are found as follows.

The equation of motion for the two-machine system is:

where d is the relative angle between the machines. Note that

where *P _{0}* is the input power and

*P*is the maximum transmittable power.

_{m}Consider a single generator of *M _{e}* with the same power transmitted to the infinite busbar system as that exchanged between the two synchronous machines. Then,

therefore,

This equivalent generator has the same mechanical input as the first machine and the load angle d it has with respect to the busbar is the angle between the rotors of the two machines.

Often, the maximum powers transferable before, during, and after a fault need to be calculated from the system configuration reduced to a network between the relevant generators. The use of network reduction by nodal elimination is most valuable in this context; it only remains then for the transfer reactances to be calculated, as any shunt impedance at the reduced nodes does not influence the power transferred.

With unbalanced faults more power is transmitted during the fault period than with three-phase short circuits and the stability limits are higher.

# Effect of Automatic Voltage Regulators and Governors

These may be represented in the equation of motion as follows, where:

*K** _{d} =* damping coefficient;

*P**mech =* power input;

*DP** _{mech} =* change in input power due to governor action;

*P**eiec =* electrical power output modified by the voltage regulator.

Equation (8.6) is best solved by digital computer.