Capability curve of a generator

A capability curve (P–Q chart) of a synchronous generator shows the region of active power $P$ and reactive power $Q$ in which the generator can operate continuously without exceeding thermal and stability limits. Operators use it to set loading and VAr output.

 Q (lagging, over-excited)
  ^
  |-------._    field heating limit
  |         '.
  |           \
  |            |  <- turbine limit
  |            |     (P = Pmax)
  +------------+---> P
  |            |  <- stator heating
  |           /      limit (circle S)
  |   ______.'
  |  under-excitation /
  |  stability limit
  v
 Q (leading, under-excited)

Limits that form the curve

1. Stator (armature) current heating limit: $P^2 + Q^2 = (V I_a)^2 = S^2$; a circle centred at the origin with radius equal to rated MVA.
2. Field (rotor) current heating limit: maximum excitation emf $E_{max}$ gives a circle centred at $(0, -V^2/X_s)$ with radius $E_{max}V/X_s$. This limits the lagging (over-excited) region.
3. Prime mover (turbine) limit: maximum turbine output, a vertical line $P = P_{max}$.
4. Under-excitation limits: in the leading region, the steady-state stability limit ($\delta$ approaching 90°, with practical margin) and stator end-core heating limit the VAr absorption.
5. Minimum load limit: some turbines (e.g. Francis) cannot run stably below about 40% load.

Use

• Rated point lies where the stator circle meets the field circle at rated power factor (e.g. 0.85 lagging).
• Shows how much reactive power the unit can supply or absorb at a given $P$, needed for grid voltage control and for setting AVR limiters (over-excitation and under-excitation limiters).