Swing equation

 ( Equations ) 

A power system consists of a number of alternator 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.

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Derivation A synchronous generator is driven by a prime mover. The equation governing the rotor motion is given by: $$J\backslash frac\{d^2\{\backslash theta\_\backslash text\{m\}\}\}\{dt^2\}\; =\; T\_a\; =\; T\_\backslash text\{m\}\; -\; T\_\backslash text\{e\},$$ where:

• $J$ is the total moment of inertia of the rotor mass in kg-m²
• $\backslash theta\_\backslash text\{m\}$ is the angular position of the rotor with respect to a stationary axis in radians (rad)
• $t$ is time in seconds (s)
• $T\_a$ is the net accelerating torque, in N-m
• $T\_\backslash text\{m\}$ is the mechanical torque supplied by the prime mover in N-m
• $T\_\backslash text\{e\}$ 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 $T\_a$. 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 $\backslash omega\_s$ in rad/s. The electric torque $T\_\backslash text\{e\}$ corresponds to the net air-gap power in the machine and thus accounts for the total output power of the generator plus $|I|^2\; R$ losses in the armature winding.

The angular position $\backslash theta$ is measured with a stationary reference frame. Representing it with respect to the synchronously rotating frame gives: $$\backslash theta\_\backslash text\{m\}\; =\; \backslash omega\_\backslash text\{sm\}t\; +\; \backslash delta\_\backslash text\{m\},$$ where the mechanical power angle $\backslash delta\_m$ is the angular position with respect to the synchronously rotating reference frame. The derivative of the above equation with respect to time is: $$\backslash frac\{d\backslash theta\_\backslash text\{m\}\}\{dt\}\; =\; \backslash omega\_\backslash text\{sm\}\; +\; \backslash frac\{d\backslash delta\_\backslash text\{m\}\}\{dt\}.$$ The above equations show that the rotor angular speed is equal to the synchronous speed only when $d\backslash delta\_m\; /dt$ is equal to zero. Therefore, the term $d\backslash delta\_m\; /dt$ represents the deviation of the rotor speed from synchronism in rad/s.

By taking the second order derivative of the above equation it becomes: $$\backslash frac\{d^2\backslash theta\_\backslash text\{m\}\}\{dt^2\}\; =\; \backslash frac\{d^2\backslash delta\_\backslash text\{m\}\}\{dt^2\}.$$ Substituting the above equation in the equation of rotor motion gives: $$J\backslash frac\{d^2\{\backslash delta\_\backslash text\{m\}\}\}\{dt^2\}\; =\; T\_a\; =\; T\_\backslash text\{m\}\; -\; T\_\backslash text\{e\}.$$ Multiplying both sides by the angular velocity of the rotor, given by $$\backslash omega\_\backslash text\{m\}\; =\; \backslash frac\}\{dt\},$$ results in $$J\backslash omega\_\backslash text\{m\}\backslash frac\{d^2\{\backslash delta\_\backslash text\{m\}\}\}\{dt^2\}\; =\; P\_a\; =\; P\_\backslash text\{m\}\; -\; P\_\backslash text\{e\},$$ where $P\_a$, $P\_\backslash text\{m\}$ and $P\_\backslash text\{e\}$ respectively are the accelerating, mechanical and electrical (active) power in Watt (W). Intuitivley, the equation can also be derived by taking the time derivative of the rotational energy.

The coefficient $J\; \backslash omega\_\{\backslash text\{m\}\}$ is the angular momentum of the rotor at synchronous speed $\backslash omega\_\{\backslash text\{sm\}\}$. In machine data supplied for stability studies this coefficient is often denoted by $M$ and called the inertia constant of the machine. In practice, $\backslash omega\_\{\backslash text\{m\}\}$ does not differ significantly from synchronous speed when the machine is in steady state $\backslash omega\_\{\backslash text\{sm\}\}$; allowing for another constant of inertia: $$H\; =\; \backslash frac\{\backslash frac\{1\}\{2\}J\backslash omega\_\backslash text\{sm\}^2\}\{S\_\{\backslash text\{rated\}\}\}=\; \backslash frac\{\backslash frac\{1\}\{2\}M\; \backslash omega\_\backslash text\{sm\}\}\{S\_\{\backslash text\{rated\}\}\}=\; \backslash frac\{\backslash text\{stored\; kinetic\; energy\; in\; mega\; joules\; at\; synchronous\; speed\}\}\{\backslash text\{machine\; rating\; in\; MVA\}\},$$ where $S\_\{\backslash text\{rated\}\}$ is the three phase rating of the machine in Volt-ampere. Substituting in the above equation $$2H\backslash frac\{S\_\{\backslash text\{rated\}\}\}\{\backslash omega\_\backslash text\{sm\}^2\}\backslash omega\_\backslash text\{m\}\backslash frac\{d^2\{\backslash delta\_\backslash text\{m\}\}\}\{dt^2\}\; =\; P\_\backslash text\{m\}\; -\; P\_\backslash text\{e\}\; =\; P\_a.$$ Since $P\_\backslash text\{m\}$, $P\_\backslash text\{e\}$ and $P\_a$ in the machine data are given in MW, dividing them by the generator MVA rating gives these quantities in per unit. Dividing the above equation on both sides by $S\_\{\backslash text\{rated\}\}$ gives \frac{d^2{\delta}}{dt^2} = P_\text{m} - P_e = P_a per unit |cellpadding= 5 |border |border colour = #0073CF |bgcolor=#F9FFF7 with the electrical power angle and electrical angular velocity given by $$\backslash delta\; =\; \backslash frac\{p\}\{2\}\backslash delta\_m,\; \backslash qquad\; \backslash omega\; =\; \backslash frac\{p\}\{2\}\backslash omega\_m,\; \backslash qquad\; \backslash omega\_s\; =\; \backslash frac\{p\}\{2\}\backslash omega\_\{sm\},$$ where $p$ is the number of poles of the synchronous machine.

The above equation describes the behaviour of the rotor dynamics and hence is known as the swing equation. The angle $\backslash delta$ is that of the internal EMF $(E)$ of the synchronous generator and dictates the amount of power that can be transferred. This angle is therefore called the power angle. Neglecting the machine's resistive losses, the corresponding power angle equation is: $$P\_e\; =\; P\_\{max\}\; \backslash sin\; (\backslash delta)=3\backslash frac$$

{X}\sin (\delta), where $X$ is the machine reactance and $V$ the system (i.e. grid) voltage. The angle $\backslash delta$ is also referred to as the torque angle as the electrical torque $T\_e$ can be derived from this equation as $$T\_e\; =\; \backslash frac\{P\_e\}\{\backslash omega\_m\}=3\backslash frac$${\omega_m X}\sin (\delta). Hence, for synchronous machines the swing equation is a non-linear function of $\backslash delta$ and can be solved numerically using, e.g., the fourth-order Runge-Kutta algorithm. When $\backslash delta$ is small, the equation can be linearized as $P\_e\; \backslash approx\; P\_\{max\}\backslash delta$.

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See also

• Synchronverter

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Notes

• Stephen J. Chapman (2025). 9780073529547, McGraw-Hill Education. ISBN 9780073529547
• (1994). 9780070612938, McGraw-Hill. ISBN 9780070612938
• Bhag S. Guru (2025). 9780195138900, Oxford University Press. ISBN 9780195138900
• Hadi Saadat (1999). 9780070122352, McGraw-Hill Companies. ISBN 9780070122352
• (2025). 9780470510278, Wiley. ISBN 9780470510278
• Stephen D. Umans (2025). 9780073380469, McGraw-Hill Education. ISBN 9780073380469

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