Positive synchronizing power coefficient

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.

Small-signal analysis

Let the machine operate at $\delta_0$ with $P_m = P_{max}\sin\delta_0$, and let the angle change by a small amount: $\delta = \delta_0 + \Delta\delta$. Then

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

Substituting in the swing equation $M\dfrac{d^2\delta}{dt^2} = P_m - P_e$ (with $M = H/\pi f$):

$$M\frac{d^2\Delta\delta}{dt^2} + P_s\,\Delta\delta = 0$$

The characteristic equation is $Ms^2 + P_s = 0$, so

$$s = \pm\sqrt{-\frac{P_s}{M}}$$

• $P_s > 0$: roots are imaginary, $s = \pm j\omega_n$ with $\omega_n = \sqrt{P_s/M}$. $\Delta\delta$ oscillates with constant amplitude (and decays once damping is included). The rotor is pulled back: the operating point is stable.
• $P_s < 0$: one root is real and positive, $\Delta\delta$ grows exponentially; the machine falls out of step: unstable.
• $P_s = 0$ ($\delta_0 = 90^\circ$): the steady state stability limit.

The frequency of natural oscillation is

$$f_n = \frac{1}{2\pi}\sqrt{\frac{P_s}{M}} = \frac{1}{2\pi}\sqrt{\frac{\pi f P_s}{H}}\ \text{Hz}$$

Physical reasoning

 Pe
  ^         .-''-.
  |  Pm   .' Ps<0 '.
  |-----a'---------b.---
  |  Ps>0           \
  +-+---+-----+------+--> d
    0   d0    90    180

At point $a$ ($\delta_0 < 90^\circ$, $P_s > 0$) a small increase in $\delta$ raises $P_e$ above $P_m$, decelerating the rotor back. At point $b$ ($\delta > 90^\circ$, $P_s < 0$) the same increase lowers $P_e$, the rotor accelerates further and synchronism is lost. Hence a positive synchronizing power coefficient is necessary for a stable equilibrium.