Synchronizing power coefficient and application

Synchronizing power coefficient $P_s$ is the rate of change of electrical power output with rotor angle at the operating point:

$$P_s = \left.\frac{\partial P_e}{\partial\delta}\right|_{\delta_0} = \frac{EV}{X}\cos\delta_0 = P_{max}\cos\delta_0\ \ \text{(pu power/rad)}$$

It gives the change in power that tends to pull the rotor back to synchronism for a small change in rotor angle: $\Delta P_e = P_s\,\Delta\delta$. It is also called the stiffness of the coupling between the machine and the system.

Derivation

For a machine connected to an infinite bus, $P_e = P_{max}\sin\delta$. If the rotor angle increases from $\delta_0$ by $\Delta\delta$:

$$\Delta P_e = P_{max}\sin(\delta_0 + \Delta\delta) - P_{max}\sin\delta_0 \approx (P_{max}\cos\delta_0)\Delta\delta = P_s\Delta\delta$$

With $P_m$ constant, an increase of $\Delta\delta$ makes $P_e > P_m$ when $P_s > 0$; the rotor decelerates and returns to $\delta_0$. So $P_s$ acts like the spring constant of the electrical coupling.

 Ps = Pmax cos(d0)
  ^
 Pmax.
  |    '.
  |      '.
  +--------'.------> d0
  0          90 deg (Ps = 0, limit)

Applications

1. Steady state stability check: the operating point is stable only if $P_s > 0$, i.e. $\delta_0 < 90^\circ$; the limit is $P_s = 0$.
2. Measure of stability margin/stiffness: larger $P_s$ (lower loading, lower reactance, higher excitation) means a more stable machine.
3. Natural frequency of rotor oscillation: from the linearised swing equation $M\frac{d^2\Delta\delta}{dt^2} + P_s\Delta\delta = 0$,

$$f_n = \frac{1}{2\pi}\sqrt{\frac{P_s}{M}}, \quad M = \frac{H}{\pi f}$$

used for small-signal (dynamic) stability studies and design of damper windings and power system stabilisers. 4. Parallel operation and synchronising of alternators: $P_s$ gives the synchronising power that keeps paralleled machines in step and shares load changes. 5. Hunting analysis: used to find the period of oscillation of a machine after a small disturbance.

Example: $P_{max} = 2$ pu, $P_m = 1$ pu → $\delta_0 = 30^\circ$, $P_s = 2\cos30^\circ = 1.732$ pu/rad.