Reactive power injection and generation limits

Role of reactive power injection or consumption at a bus

Bus voltage depends mainly on the net reactive power at that bus, since $\Delta V \approx QX/V$ for a line with $X \gg R$.

• Injecting Q at a bus (generator over-excited, capacitor, condenser) reduces the reactive current drawn over the lines, so the voltage rises.
• Consuming Q at a bus (inductive load, reactor, under-excited generator) increases the drop across line reactance, so the voltage falls.

So Q injection/absorption is the main tool to hold the bus voltage at its specified value; this is why a PV bus has its Q left free (within limits) to keep $|V|$ fixed.

Generator on an infinite bus

For a cylindrical-rotor generator with internal emf $E_f$, terminal voltage $V$ (fixed by the infinite bus), synchronous reactance $X_s$ and load angle $\delta$:

$$P = \frac{E_fV}{X_s}\sin\delta,\qquad Q = \frac{E_fV\cos\delta - V^2}{X_s}$$

With P fixed by the turbine, Q is set by the excitation $E_f$:

• $E_f\cos\delta > V$ (over-excited): Q > 0, generator supplies Q (lagging pf).
• $E_f\cos\delta < V$ (under-excited): Q < 0, generator absorbs Q (leading pf).

Maximum reactive power limit

The upper limit is set by field (rotor) heating. The maximum field current gives $E_{f,max}$, so

$$Q_{max} = \frac{V}{X_s}\sqrt{E_{f,max}^2 - \left(\frac{PX_s}{V}\right)^2} - \frac{V^2}{X_s}$$

On the P–Q capability chart this is a circle of radius $VE_{f,max}/X_s$ centred at $(0, -V^2/X_s)$. At high P, the armature current limit (circle of radius $VI_{a,max}$ about the origin) may set the bound instead.

Minimum reactive power limit

When the generator is under-excited:

1. Steady-state stability limit: as $E_f$ is reduced, $\delta$ increases for the same P; at $\delta = 90^\circ$ the machine loses synchronism. A practical margin (e.g. $\delta \le 70^\circ$) defines the limit.
2. Stator end-region heating in under-excited operation, which further restricts absorption.
3. A minimum excitation limit set by the exciter.

        Q (supplied)
        ^    field-heating limit
        |  .---.
        | /     \  armature-current limit
  ------+--------+-----> P
        | \     /  stability / end-heating limit
        |  '---'
        v Q (absorbed)

So in load-flow studies a PV bus has $Q_{min} \le Q_G \le Q_{max}$; if a limit is hit, Q is fixed at that limit and the bus is treated as a PQ bus.