Load angle, active power and reactive power

The generator phasor model, the power–angle curve and excitation.

A source behind reactance

Represent a round-rotor generator by a constant internal EMF E∠δ behind X. An infinite bus holds V∠0 and fixed frequency. Resistance and losses are omitted. Positive current flows from the generator toward the grid.

Power at the grid bus

Positive Q means reactive-power export to the grid. If X includes a line and transformer, these are powers at the remote grid bus, not stator terminals. Reactive power at the internal source exceeds bus reactive power by X|I|².

For E = 1.2, V = 1, X = 0.6 and P = 0.8, δ = arcsin(0.4) = 23.578°, Q = 0.1664 pu and |I| ≈ 0.8171 pu. At 100 MVA this is 80 MW and 16.64 Mvar.

Excitation and the transfer limit

At fixed active power, increasing E reduces the stable angle and usually raises Q export. Here E is set manually and stays constant during a run; there is no automatic voltage regulator. The branch 0 ≤ δ₀ < 90° has positive synchronizing stiffness. The other equilibrium π − δ₀ is unstable. The margin Pmax − P₀ does not replace current, thermal or transient-stability checks.

Independent exercises

1. Find Pmax for E = 1.2, V = 1 and X = 0.6.

2. For E = V = 1 and δ = 0, find current and reactive power.

Practical work

Compare E = 1.0, 1.2 and 1.4 at P₀ = 0.8 with no disturbance. Record initial angle, Q and current. Identify reactive-power absorption and explain it using the formula.

Key takeaways

Active power depends on the sine of load angle; reactive power depends on the projection of internal EMF. Current and thermal limits may bind before the angle limit.