Limits of the alternator capability curve

The capability curve of an alternator is a P–Q chart that shows the safe region of operation, i.e. the combinations of active power P and reactive power Q that the machine can deliver continuously at rated voltage without exceeding any limit.

Elements that limit the curve

1. Armature (stator) current heating limit: $S = \sqrt{P^2 + Q^2} = 3V_tI_a$. At rated $V_t$ this is a circle centred at the origin with radius equal to rated MVA.
2. Field (rotor) current heating limit: the maximum excitation $E_f$ allowed by rotor heating. It is a circle centred at $(0, -3V_t^2/X_s)$ with radius $3V_tE_f/X_s$. It limits operation at lagging (over-excited) power factor.
3. Prime mover (turbine) limit: maximum mechanical power, a vertical line $P = P_{max}$.
4. Steady-state stability limit: at leading power factor the load angle approaches $90^\circ$. A practical limit with margin is used.
5. Stator end-core heating limit: in under-excited operation leakage flux heats the stator ends and limits leading-var absorption.
6. Minimum excitation / minimum turbine load: for example cavitation zones in hydro turbines limit low load.

Typical capability curve

  Q lagging (MVAr)
   ^
   |~~~~~--..    <- field current limit
   |    A     `.
   |            o  <- rated MVA, rated pf
   |            |  <- stator current limit
   |            |  <- turbine (max P) limit
   +------------+------> P (MW)
   |            |
   |    B      /
   |       ..-`  <- stability / end-core limit
   |..--~~
   v  Q leading (MVAr)

Explanation

• Region A (over-excited, lagging pf): the field current limit is the binding curve from zero P up to the rated power factor point. Above rated pf (more P), the stator current limit governs.
• Rated point: where the field and stator limit circles meet, i.e. rated MVA at rated pf (e.g. 0.85 lagging).
• High P region: the stator circle and the turbine limit line cut off the curve.
• Region B (under-excited, leading pf): the steady-state stability limit (with margin) and stator end heating limit how much reactive power the machine can absorb.
• The operator must keep the operating point inside the enclosed area. The AVR limiters (OEL, UEL) are set from this curve.