A power system consists of a number of synchronous machines operating synchronously under all operating conditions. Under normal operating conditions, the relative position of the rotor axis and the resultant magnetic field axis is fixed. The angle between the two is known as the power angle, torque angle, or rotor angle. During any disturbance, the rotor decelerates or accelerates with respect to the synchronously rotating air gap magnetomotive force, creating relative motion. The equation describing the relative motion is known as the swing equation, which is a non-linear second order differential equation that describes the swing of the rotor of synchronous machine. The power exchange between the mechanical rotor and the electrical grid due to the rotor swing (acceleration and deceleration) is called Inertial response.
Derivation A synchronous generator is driven by a prime mover. The equation governing the rotor motion is given by:
J d 2 θ m d t 2 = T a = T m − T e , {\displaystyle J{\frac {d^{2}{\theta _{\text{m}}}}{dt^{2}}}=T_{a}=T_{\text{m}}-T_{\text{e}},}
where:
J {\displaystyle J} is the total moment of inertia of the rotor mass in kg-m2
θ m {\displaystyle \theta _{\text{m}}} is the angular position of the rotor with respect to a stationary axis in radians (rad)
t {\displaystyle t} is time in seconds (s)
T a {\displaystyle T_{a}} is the net accelerating torque, in N-m
T m {\displaystyle T_{\text{m}}} is the mechanical torque supplied by the prime mover in N-m
T e {\displaystyle T_{\text{e}}} is the electrical torque output of the alternator in N-m Neglecting losses, the difference between the mechanical and electrical torque gives the net accelerating torque T a {\displaystyle T_{a}} . In the steady state, the electrical torque is equal to the mechanical torque and hence the accelerating power is zero. During this period the rotor moves at synchronous speed ω s {\displaystyle \omega _{s}} in rad/s. The electric torque T e {\displaystyle T_{\text{e}}} corresponds to the net air-gap power in the machine and thus accounts for the total output power of the generator plus | I | 2 R {\displaystyle |I|^{2}R} losses in the armature winding. The angular position θ {\displaystyle \theta } is measured with a stationary reference frame. Representing it with respect to the synchronously rotating frame gives:
θ m = ω sm t + δ m , {\displaystyle \theta _{\text{m}}=\omega _{\text{sm}}t+\delta _{\text{m}},}
where the mechanical power angle δ m {\displaystyle \delta _{m}} is the angular position with respect to the synchronously rotating reference frame. The derivative of the above equation with respect to time is:
d θ m d t = ω sm + d δ m d t . {\displaystyle {\frac {d\theta _{\text{m}}}{dt}}=\omega _{\text{sm}}+{\frac {d\delta _{\text{m}}}{dt}}.}
The above equations show that the rotor angular speed is equal to the synchronous speed only when d δ m / d t {\displaystyle d\delta _{m}/dt} is equal to zero. Therefore, the term d δ m / d t {\displaystyle d\delta _{m}/dt} represents the deviation of the rotor speed from synchronism in rad/s. By taking the second order derivative of the above equation it becomes:
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