Steady state stability and synchronizing coefficient

Stability of a power system is its ability to remain in synchronism (all machines running at the same electrical speed) under normal operation and to return to a stable operating state after a disturbance.

Steady state stability

Steady state stability is the ability of the system to remain in synchronism when subjected to small and gradual changes in load. For a generator connected to an infinite bus through reactance $X$:

$$P_e = \frac{EV}{X}\sin\delta = P_{max}\sin\delta$$

 Pe
  ^          Pmax
  |        .-''-.
  |  Pm  .'  |   '.
  |----a'----|----b'.--
  |   /|     |      \
  |  / |stable  unstable
  +-+--+-----+--------+--> d
    0  d0   90       180

• At point $a$ ($\delta_0 < 90^\circ$): if load increases slightly, $\delta$ increases and $P_e$ also increases, matching the load. The operation is stable.
• At point $b$ ($\delta > 90^\circ$): an increase in $\delta$ reduces $P_e$, the rotor accelerates further and synchronism is lost.
• If $P_m$ is raised slowly, $\delta$ increases until $\delta = 90^\circ$, where $P_e = P_{max}$. This maximum is the steady state stability limit:

$$P_{ss} = \frac{EV}{X}$$

Ways to raise it: increase $E$ (excitation) or $V$, reduce $X$ (parallel lines, series compensation).

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.

Importance:

• Stability criterion: steady state stability requires $P_s > 0$, i.e. $0 < \delta_0 < 90^\circ$. At $\delta_0 = 90^\circ$, $P_s = 0$ (limit).
• Stability margin: a larger $P_s$ means a stiffer system and a larger margin.
• Natural frequency of oscillation: $f_n = \frac{1}{2\pi}\sqrt{P_s\pi f/H}$, used to study rotor oscillations and damping.
• Machines operated at light load (small $\delta_0$) have large $P_s$ and are more stable.