Swing Equation

The transient stability of the system can be determined by the help of the swing equation. Let θ be the angular position of the rotor at any instant t. θ  is continuously changing with time, and it is convenient to measure it with respect to the reference axis shown in the figure below. The angular position of the rotor is given by the equation

Where,

θ – angle between rotor field and a reference axis
w_s – synchronous speed
δ – angular displacement

Differentiation of equation (1) gives

Differentiation of equation (2) gives

Angular acceleration of rotor

Power flow in the synchronous generator is shown in the diagram below. If the damping is neglected the accelerating torques, T_a in a synchronous generator is equal to the difference of mechanical input shaft and the electromagnetic output torque, i.e.,

Where,

T_a – accelerating torque
T_s – shaft torque
T_e – electromagnetic torque

Angular momentum of the rotor is expressed by the equation

Where,

w- the synchronous speed of the rotor
J – moment of inertia of the rotor
M – angular momentum of the rotor

Multiplying both the sides of equation (5) by w we get

Where,

P_s – mechanical power input
P_e – electrical power output
P_a – accelerating power

But,

Equation (7) gives the relation between the accelerating power and angular acceleration. It is called the swing equation. Swing equation describes the rotor dynamics of the synchronous machines and it helps in stabilizing the system.