Transformer location for three villages (A)

The distribution transformer should be placed at the load centre (centre of gravity of the loads), because this minimises the load moment ($\sum P_i d_i$), and so the LT voltage drop and loss.

Since the villages are 0.6 km from each other, they form an equilateral triangle of side 0.6 km.

              C (30 kW)
             /\
        0.6 /  \ 0.6
           / T  \
          /______\
   A (20 kW) 0.6  B (50 kW)

Coordinates (A at origin, AB along x-axis)

Load centre

$$\begin{aligned} \bar x &= \frac{\sum P_ix_i}{\sum P_i} = \frac{20(0)+50(0.6)+30(0.3)}{100} = \frac{39}{100} = 0.39\ \text{km}\\ \bar y &= \frac{\sum P_iy_i}{\sum P_i} = \frac{30(0.5196)}{100} = 0.156\ \text{km} \end{aligned}$$

Distance from A:

$$d = \sqrt{0.39^2+0.156^2} = 0.42\ \text{km},\quad \theta = \tan^{-1}\frac{0.156}{0.39} = 21.8^\circ$$

Engineering judgement

• The transformer is located about 0.42 km from A, at 21.8° from line AB towards C, i.e. pulled towards B, the largest load.
• In practice it is placed at the nearest accessible point (road side, public land) to this centre. The demand of 100 kW needs about 118–125 kVA (at 0.85–0.8 p.f.), which is more than a 100 kVA unit can carry, so the next standard size (160 or 200 kVA, allowing for growth) would be chosen.

Answer: Transformer at (0.39 km, 0.156 km) from A, i.e. 0.42 km from A at 21.8° to AB towards C.