Making short-line regulation zero

The voltage regulation of a short line is zero when the receiving-end voltage at full load equals the sending-end voltage, i.e. $|V_S| = |V_R|$. This happens only for a suitable leading power factor at the receiving end.

Derivation

Short line: $V_S = V_R + I(R + jX)$. Take $V_R$ as reference and the current at angle $\phi$ (lagging taken positive):

$$I = I\cos\phi - jI\sin\phi$$ $$\begin{aligned} V_S &= V_R + (I\cos\phi - jI\sin\phi)(R + jX) \\ &= (V_R + IR\cos\phi + IX\sin\phi) + j(IX\cos\phi - IR\sin\phi) \end{aligned}$$

The quadrature part changes $|V_S|$ very little, so

$$|V_S| \approx V_R + IR\cos\phi + IX\sin\phi$$ $$\%VR \approx \frac{I(R\cos\phi + X\sin\phi)}{V_R}\times100$$

For a leading load the angle is negative, so $\sin\phi$ changes sign. Writing $\phi$ from here on for the magnitude of the leading angle:

$$\%VR \approx \frac{I(R\cos\phi - X\sin\phi)}{V_R}\times100$$

Condition for zero regulation

$$\begin{aligned} R\cos\phi - X\sin\phi &= 0 \\ \tan\phi &= \frac{R}{X} \\ \cos\phi &= \frac{X}{\sqrt{R^2 + X^2}} = \frac{X}{|Z|}\ \text{(leading)} \end{aligned}$$

Equivalently, the load angle $\phi = 90^\circ - \theta$, where $\theta = \tan^{-1}(X/R)$ is the line impedance angle.

         I (leads Vr by phi)
        /      Vs (same length as Vr)
       /     .
      /   .   IZ (perpendicular to Vr+Vs bisector)
     / .
    O--------------> Vr

How it is achieved in practice

1. Connect a shunt capacitor bank or synchronous condenser at the receiving end so that the net receiving-end pf becomes $X/|Z|$ leading.
2. Required leading reactive power (approx.): $Q_{net} = -P\,R/X$, so capacitor rating $Q_C = P\tan\phi_{load} + P\,R/X$.
3. If the load is already more leading than this (negative regulation), a shunt reactor is added instead.

Example: $R = 4\ \Omega$, $X = 8\ \Omega$: zero regulation needs $\cos\phi = 8/\sqrt{80} = 0.894$ leading.

(For an exact value, solve $|V_R + Z(P - jQ)/V_R| = |V_R|$ for $Q$; it gives a slightly more leading pf than the approximate formula.)