Power loss change on doubling line length

With the same load density (kW per km) and the same conductor, doubling the length of a uniformly loaded line doubles both its total load and its resistance.

Original line (length $l$, uniform load density $\lambda$ kW/km, sending-end current $I \propto \lambda l$, resistance $R = rl$):

$$P_{loss,1} = 3 \times \frac{I^2 R}{3} = I^2 r l$$

(for a uniformly distributed load the loss is one-third of the end-loaded value).

Doubled line (length $2l$): total load $= 2\lambda l$, so current $I_2 = 2I$; resistance $R_2 = 2rl$:

$$\begin{aligned} P_{loss,2} &= (2I)^2 \times r(2l) = 8 I^2 r l \\ \frac{P_{loss,2}}{P_{loss,1}} &= 8 \end{aligned}$$

Result:

• Power loss increases 8 times ($\propto l^3$).
• The load supplied only doubles, so the percentage power loss increases 4 times ($\propto l^2$).
• Similarly, voltage drop ($\propto I \times l$) increases 4 times.

This is why LT feeders must be kept short; in a large area it is better to use more transformers placed near load centres than to extend long LT lines.