Generator capability curve (P-Q chart)
Five constraints — armature, field, prime-mover, stability, under-excitation — bounding the safe operating envelope.
Step 1 — The P-Q operating plane: every generator runs at one (P, Q) point
Reference notes
Use Next → on the narrator above to build up the synchronous-generator capability curve one constraint at a time, then see how the constraints interact to define the safe operating envelope.
The P-Q operating plane
Every synchronous generator on a grid sits, at any instant, at a single point in the P-Q plane:
- P (MW) — real power being delivered to the grid (controlled by the prime mover via fuel/steam).
- Q (MVAR) — reactive power being delivered (positive = supplying lagging VAR; controlled by field current).
The operator's job is to keep this point inside a region bounded by physical constraints. That region is called the capability curve (or capability chart).
Constraint 1: armature-current limit (heating)
The stator winding can carry up to its rated current Ia,rated before insulation overheats. Apparent power S = V·I, so the limit is:
This is a circle of radius Srated centred at the origin. Operating points outside it would cook the stator.
Constraint 2: field-current limit (rotor heating)
The rotor field winding is DC; too much If means too much Ef, which heats the rotor and the slip-ring brushes. From the synchronous-machine equations (cylindrical rotor, Ra neglected):
Eliminate δ to get the constraint shape:
This is a circle centred at (0, −V²/Xs) — well below the origin — with radius proportional to Ef,max. It clips the operating region on the LAGGING side (Q > 0 side).
Constraint 3: prime-mover limit (rated MW)
The turbine (or diesel, or hydro penstock) has a maximum mechanical power output. This shows up as a vertical line at P = Prated. Above it, you're asking the prime mover for more shaft power than it can deliver.
Constraint 4: steady-state stability limit
From the power-angle equation, theoretical pull-out is at δ = 90°. Operators don't run there — too small a disturbance would push past the peak and the machine slips a pole. A practical margin (say δ ≤ 70°) defines the steady-state stability limit on the LEADING side. Beyond it, synchronism is at risk.
Constraint 5: under-excitation limit (UEL)
At very low field current, the stator end-iron heats (because the magnetic flux pattern changes) and stability margin shrinks. A minimum Ef is enforced as a lower bound — a circle of smaller radius, also centred at (0, −V²/Xs):
The safe operating envelope
Intersect all five constraints. The result is a shrunk lens-shaped region in the P-Q plane. Operators dispatch the generator inside this envelope:
- Bottom-right corner (high P, modest lagging Q) — typical industrial operating regime.
- Top of the envelope (limited by field current) — when supplying heavy lagging VAR to compensate for industrial load.
- Bottom of the envelope (limited by under-excitation or stability) — when absorbing leading VAR to support voltage on a lightly-loaded transmission line.
Why this matters
- Grid operators use the capability curve as the master document for dispatch: which units can supply how much MW and MVAR, under what conditions.
- A generator's capability is a function of cooling — hydrogen-cooled units have larger curves than air-cooled. The same generator gets a bigger envelope if you raise the H₂ pressure.
- When an automatic voltage regulator (AVR) commands more field current to support grid voltage, the operating point may try to exit the field-limit arc — the protection system has to catch this before the rotor cooks.
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