Generator reactive power at constant terminal voltage

A synchronous generator on an infinite bus has its terminal voltage $V_t$ and frequency fixed by the bus. With constant mechanical input (constant P), it can still be made to supply or absorb any reactive power within limits by changing its field excitation.

Equations

With synchronous reactance $X_s$ (R neglected), internal emf $E_f$ and load angle $\delta$:

$$E_f = V_t + jX_sI_a,\qquad P = \frac{E_fV_t}{X_s}\sin\delta,\qquad Q = \frac{E_fV_t\cos\delta - V_t^2}{X_s}$$

• P depends on the turbine input; since $V_t$ and $X_s$ are fixed, $E_f\sin\delta$ = constant. So as $E_f$ changes, the tip of $E_f$ moves along a line parallel to $V_t$.
• $I_a\cos\phi$ (active current) also remains constant.
• Q depends on $E_f\cos\delta - V_t$, which changes with excitation.

Three cases

     locus of Ef tip (P constant)
   --------x-------x--------x-------
          /       /        /
     Ef1 /   Ef2 /    Ef3 /     Ef1 < Ef2 < Ef3
        /       /        /
       O--------------------> Vt
   Ef1: under-excited (absorbs Q)
   Ef3: over-excited (supplies Q)

Explanation

• Increasing field current raises $E_f$; $\delta$ decreases and $I_a$ swings to lagging. The generator sends Q into the bus.
• Decreasing field current lowers $E_f$; $\delta$ increases and $I_a$ becomes leading. The generator draws Q from the bus.
• Terminal voltage remains constant because the infinite bus holds it.

Limits

Maximum Q is set by field heating (maximum $E_f$) and armature current; minimum Q (maximum absorption) by the steady-state stability limit ($\delta$ near 90°) and stator end heating. This is how generators are used for voltage/Q support in a real grid.