Swing equation

A synchronous generator is driven by a prime mover. The equation governing the rotor motion is given by:

${\displaystyle J{\frac {d^{2}{\theta _{\text{m}}}}{dt^{2}}}=T_{a}=T_{\text{m}}-T_{\text{e}}}$ N-m

Where:

• ${\displaystyle J}$ is the total moment of inertia of the rotor mass in kg-m²
• ${\displaystyle \theta _{\text{m}}}$ is the angular position of the rotor with respect to a stationary axis in (rad)
• ${\displaystyle t}$ is time in seconds (s)
• ${\displaystyle T_{\text{m}}}$ is the mechanical torque supplied by the prime mover in N-m
• ${\displaystyle T_{\text{e}}}$ is the electrical torque output of the alternator in N-m
• ${\displaystyle T_{a}}$ is the net accelerating torque, in N-m

Neglecting losses, the difference between the mechanical and electrical torque gives the net accelerating torque T_a. In the steady state, the electrical torque is equal to the mechanical torque and hence the accelerating power is zero.[1] During this period the rotor moves at synchronous speed ω_s in rad/s. The electric torque T_e corresponds to the net air-gap power in the machine and thus accounts for the total output power of the generator plus I²R losses in the armature winding.

The angular position θ is measured with a stationary reference frame. Representing it with respect to the synchronously rotating frame gives:

${\displaystyle \theta _{\text{m}}=\omega _{\text{s}}t+\delta _{\text{m}}}$

where, δ_m is the angular position in rad with respect to the synchronously rotating reference frame. The derivative of the above equation with respect to time is:

${\displaystyle {\frac {d\theta _{\text{m}}}{dt}}=\omega _{\text{s}}+{\frac {d\delta _{\text{m}}}{dt}}}$

The above equations show that the rotor angular speed is equal to the synchronous speed only when dδ_m/dt is equal to zero. Therefore, the term dδ_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:

${\displaystyle {\frac {d^{2}\theta _{\text{m}}}{dt^{2}}}={\frac {d^{2}\delta _{\text{m}}}{dt^{2}}}}$

Substituting the above equation in the equation of rotor motion gives:

${\displaystyle J{\frac {d^{2}{\delta _{\text{m}}}}{dt^{2}}}=T_{a}=T_{\text{m}}-T_{\text{e}}}$ N-m

Introducing the angular velocity ω_m of the rotor for the notational purpose, ${\displaystyle \omega _{\text{m}}={\frac {d\theta _{\text{m}}}{dt}}}$ and multiplying both sides by ω_m,

${\displaystyle J\omega _{\text{m}}{\frac {d^{2}{\delta _{\text{m}}}}{dt^{2}}}=P_{a}=P_{\text{m}}-P_{\text{e}}}$ W

where, P_m , P_e and P_a respectively are the mechanical, electrical and accelerating power in MW.

The coefficient Jω_m is the angular momentum of the rotor: at synchronous speed ω_s, it is denoted by M and called the inertia constant of the machine. Normalizing it as

${\displaystyle H={\frac {\text{stored kinetic energy in mega joules at synchronous speed}}{\text{machine rating in MVA}}}={\frac {J\omega _{\text{s}}^{2}}{2S_{\text{rated}}}}}$ MJ/MVA

where S_rated is the three phase rating of the machine in MVA. Substituting in the above equation

${\displaystyle 2H{\frac {S_{\text{rated}}}{\omega _{\text{s}}^{2}}}\omega _{\text{m}}{\frac {d^{2}{\delta _{\text{m}}}}{dt^{2}}}=P_{\text{m}}-P_{\text{e}}=P_{a}}$.

In steady state, the machine angular speed is equal to the synchronous speed and hence ω_m can be replaced in the above equation by ω_s. Since P_m, P_e and P_a are given in MW, dividing them by the generator MVA rating S_rated gives these quantities in per unit. Dividing the above equation on both sides by S_rated gives

The above equation describes the behaviour of the rotor dynamics and hence is known as the swing equation. The angle δ is the angle of the internal EMF of the generator and it dictates the amount of power that can be transferred. This angle is therefore called the load angle.