Transformer location for three villages (B)

The distribution transformer should be located at the load centre (centre of gravity) so that the sum of load × distance, and hence LT drop and loss, is minimum.

Assumption: the villages are 0.6 km from each other, i.e. at the corners of an equilateral triangle of side 0.6 km. B is taken as the origin.

              C (60 kW)
             /\
        0.6 /  \ 0.6
           / T  \
          /______\
   B (40 kW) 0.6  A (40 kW)

Coordinates (B at origin, BA along x-axis)

Load centre

$$\begin{aligned} \bar x &= \frac{40(0)+40(0.6)+60(0.3)}{140} = \frac{42}{140} = 0.30\ \text{km}\\ \bar y &= \frac{60(0.5196)}{140} = 0.2227\ \text{km}\\ d_B &= \sqrt{0.30^2+0.2227^2} = 0.374\ \text{km}\\ \theta &= \tan^{-1}\frac{0.2227}{0.30} = 36.6^\circ \end{aligned}$$

Because A and B carry equal loads, the load centre lies on the perpendicular bisector of AB (x = 0.3 km), shifted towards C, the heaviest load.

Result

• The transformer is about 0.374 km from B, at 36.6° from BA towards C (0.30 km along BA and 0.223 km perpendicular towards C).
• Practically it is placed at an accessible point near this location; total load 140 kW suggests a standard 200 kVA transformer allowing for growth.

Note: if the villages were instead in a straight line A–B–C with 0.6 km spacing, the load centre would be $(−40\times0.6 + 60\times0.6)/140 = 0.086$ km from B towards C.

Answer: Transformer at (0.30 km, 0.223 km) from B, i.e. 0.374 km from B at 36.6° to BA towards C.