A three phase transmission line delivers a load of 5 MW at 0.8 power factor lagging. Resistance of each conductor is 0.5 ohm / km. Receiving end voltage is 33 kV. If the line loss is not to exceed 10%, determine the length of the line.

Explanation:

Three-Phase Transmission Line Problem

Given Data:

• Load Power (P_load) = 5 MW = 5000 kW
• Power Factor (PF) = 0.8 (lagging)
• Resistance of each conductor (R) = 0.5 Ω/km
• Receiving end voltage (V_R) = 33 kV = 33,000 V
• Maximum allowable line loss = 10%

To determine: The maximum length of the transmission line (L) such that the line loss does not exceed 10%.

Solution:

The transmission line loss is caused by the resistance of the conductors. To compute the maximum length of the line, we follow these steps:

The total power delivered to the load (P_load) is 5000 kW. The maximum allowable line loss is 10% of the load power:

P_loss = 10% × P_load = 0.1 × 5000 = 500 kW

The line current (I) can be calculated using the formula:

I = P_load / (√3 × V_R × PF)

Substitute the given values:

I = 5000 × 10³ / (√3 × 33,000 × 0.8)

I ≈ 109.07 A

The power loss in the transmission line is given by:

P_loss = 3 × I² × R_total

Here, R_total is the total resistance of the line, which depends on the length of the line (L):

R_total = 2 × R × L (since current flows through the forward and return paths of the line)

Substitute R_total into the formula for P_loss:

P_loss = 3 × I² × (2 × R × L)

P_loss = 6 × I² × R × L

Rearrange the formula to solve for L:

L = P_loss / (6 × I² × R)

Substitute the known values:

L = (500 × 10³) / (6 × (109.07)² × 0.5)

L ≈ 7 km

1. Calculate the total line loss:
2. Calculate the line current:
3. Relate the line loss to resistance and current:
4. Solve for the length of the line (L):

Final Answer: The maximum length of the transmission line is 7 km.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: 27.9 km

If the length of the line were 27.9 km, the power loss would exceed the allowable 10% limit. This is because the resistance of the line would increase significantly, resulting in a higher power dissipation (P_loss = 6 × I² × R × L). Thus, this option is incorrect.

Option 3: 56.8 km

A line length of 56.8 km is much longer than the maximum permissible length (7 km). At this length, the power loss would be excessively high, far exceeding the 10% limit. Hence, this option is also incorrect.

Option 4: 13.9 km

While 13.9 km is closer to 7 km than the other options, it still exceeds the maximum permissible length. At this length, the power loss would be greater than the 10% limit, making this option incorrect as well.

Conclusion:

From the calculations and analysis, the correct option is Option 1: 7 km. This is the maximum length of the transmission line for which the line losses do not exceed 10% of the total load power.