A hydroelectric power plant situated at a distance of 20 km from a town. Whose energy demand is 100 MW at 240 V. At hydro electric power plant, the water pressure head is at a height of 400 m and the water flow available is \( 100\,\mathrm{m^3/s} \). Electrical energy generated in power plant is transmitted to the town with the use of transformers voltage output of the AC generator at power plant is stepped up and is there transmitted over long distances to a sub-station near the consumers in town. Sub-stations and utility poles steps down the voltage finally the 240 V which finally reaches the houses.
Resistance of the two wires transmitting the power is \( 1\,\Omega/\mathrm{km} \). If the 50,000 – 5000 V step down transformer is used at a sub-station in town, the power loss in transmission line in the form of heat is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 160 MW
The heat loss in the line is \( I^2R \), where \( I \) is the current in the line and \( R \) the resistance of both wires. The line works at the primary voltage of the step-down transformer, 50,000 V, and carries the 100 MW demanded by the town.
The two wires each run 20 km, so the total length is 40 km:
\( R = 1\,\Omega\,\mathrm{km^{-1}} \times 40\,\mathrm{km} = 40\,\Omega \)
\( \displaystyle I = \frac{P}{V} = \frac{100\times10^{6}\,\mathrm{W}}{50000\,\mathrm{V}} = 2\times10^{3}\,\mathrm{A} \)
\( P_{\mathrm{loss}} = I^2 R = (2\times10^{3}\,\mathrm{A})^2 \times 40\,\Omega \)
\( P_{\mathrm{loss}} = 4\times10^{6} \times 40 = 1.6\times10^{8}\,\mathrm{W} = 160\,\mathrm{MW} \)