Swing equation and swing curve

The swing equation relates the rotor angle $\delta$ of a synchronous machine to the difference between mechanical input and electrical output power; it is the basic equation of power system stability studies.

Derivation

Consider the rotor of a synchronous generator. Let

• $J$ = moment of inertia of rotor and turbine (kg·m²)
• $\theta_m$ = angular position of the rotor with respect to a stationary axis (mech. rad)
• $T_m$ = mechanical (shaft) torque, $T_e$ = electromagnetic torque (N·m)

Step 1: Newton's law of rotation. Accelerating torque = moment of inertia × angular acceleration:

$$J\frac{d^2\theta_m}{dt^2} = T_a = T_m - T_e$$

Step 2: Measure angle from a synchronously rotating frame. The rotor angle $\delta_m$ is the angle of the rotor relative to a reference axis rotating at synchronous speed $\omega_{sm}$:

$$\theta_m = \omega_{sm}t + \delta_m \;\Rightarrow\; \frac{d\theta_m}{dt} = \omega_{sm} + \frac{d\delta_m}{dt}, \quad \frac{d^2\theta_m}{dt^2} = \frac{d^2\delta_m}{dt^2}$$

so

$$J\frac{d^2\delta_m}{dt^2} = T_m - T_e$$

Step 3: Convert torque to power. Multiply by the rotor speed $\omega_m$ (power = torque × speed):

$$J\omega_m\frac{d^2\delta_m}{dt^2} = P_m - P_e$$

$M = J\omega_m$ is the angular momentum (inertia constant). Since the speed changes very little during a swing, $\omega_m \approx \omega_{sm}$ and $M$ is taken as constant:

$$M\frac{d^2\delta_m}{dt^2} = P_m - P_e$$

Step 4: Introduce the inertia constant $H$. $H$ is the stored kinetic energy at synchronous speed per MVA rating $S$:

$$H = \frac{\tfrac12 J\omega_{sm}^2}{S}\ \ \text{(MJ/MVA or s)} \;\Rightarrow\; M = J\omega_{sm} = \frac{2HS}{\omega_{sm}}$$

Substituting and dividing by $S$:

$$\frac{2H}{\omega_{sm}}\frac{d^2\delta_m}{dt^2} = \frac{P_m - P_e}{S} = P_m - P_e\ \ \text{(pu)}$$

Step 5: Use electrical angle. For a machine with $p$ poles, $\delta = \frac{p}{2}\delta_m$ and $\omega_s = \frac{p}{2}\omega_{sm}$, so $\delta/\omega_s = \delta_m/\omega_{sm}$:

$$\boxed{\frac{2H}{\omega_s}\frac{d^2\delta}{dt^2} = P_m - P_e} \qquad \text{or} \qquad \frac{H}{\pi f}\frac{d^2\delta}{dt^2} = P_m - P_e$$

with $\delta$ in electrical radians and $\omega_s = 2\pi f$. If $\delta$ is in electrical degrees, $\dfrac{H}{180f}\dfrac{d^2\delta}{dt^2} = P_m - P_e$.

For a machine connected to an infinite bus through reactance $X$, $P_e = \dfrac{EV}{X}\sin\delta = P_{max}\sin\delta$, so

$$\frac{H}{\pi f}\frac{d^2\delta}{dt^2} = P_m - P_{max}\sin\delta$$

This is the swing equation. It is a second-order non-linear differential equation; its solution $\delta(t)$ describes how the rotor swings after a disturbance. If damping is included, a term $D\frac{d\delta}{dt}$ is added on the left.

Swing curve

A swing curve is the plot of rotor angle $\delta$ against time $t$, obtained by solving the swing equation (usually numerically, e.g. point-by-point or Runge-Kutta method) for a given disturbance and clearing time.

 delta
  ^        unstable (fault cleared late)
  |                 /
  |              _/
  |   .--.     _/       stable: delta peaks
  |  /    \  _/         and swings back
  | /  .-. \/  .-.
  |/  /   \___/   \__
  +--+-------------------> t
  d0 tc

Information supplied by the swing curve

1. Stable or unstable: if $\delta$ reaches a maximum and then decreases (oscillates about a new value), the system is stable; if $\delta$ keeps increasing, the machine loses synchronism.
2. Maximum rotor swing $\delta_{max}$ and the margin from the critical angle.
3. Critical clearing time: by plotting swing curves for different clearing times, the largest time that still gives a stable curve is found; this sets the required speed of relays and breakers.
4. Frequency and damping of oscillations of the rotor after the disturbance.
5. In multi-machine systems, relative angles between machines show which machines swing together and which separate.