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We are tasked with finding the maximum length of a transmission line for a three-phase, wye-connected system with the following specifications:

- Load Power: \( P = 20 \, \text{MW} \)
- Power factor: \( \text{pf} = 0.866 \) (lagging)
- Voltage: \( V = 138 \, \text{kV} \)
- Line resistance per phase: \( R = 0.7 \, \Omega \)
- Line losses should not exceed 5% of the load.

### Step-by-step Solution:

1. **Calculate the total current in the system**:
   The power equation for a three-phase system is:
   \[
   P = \sqrt{3} \cdot V_L \cdot I_L \cdot \text{pf}
   \]
   Rearranging to solve for the line current \( I_L \):
   \[
   I_L = \frac{P}{\sqrt{3} \cdot V_L \cdot \text{pf}}
   \]
   Substituting the known values:
   \[
   I_L = \frac{20 \times 10^6}{\sqrt{3} \cdot 138 \times 10^3 \cdot 0.866}
   \]

2. **Calculate the allowable power loss**:
   The maximum allowable loss is 5% of the load:
   \[
   P_{\text{loss}} = 0.05 \cdot P = 0.05 \cdot 20 \, \text{MW} = 1 \, \text{MW}
   \]

3. **Use the power loss to find the line length**:
   The power loss in a three-phase system due to resistance is given by:
   \[
   P_{\text{loss}} = 3 \cdot I_L^2 \cdot R_{\text{total}}
   \]
   The total resistance \( R_{\text{total}} \) is the resistance per phase multiplied by the length \( L \):
   \[
   R_{\text{total}} = 2 \cdot R \cdot L
   \]
   The factor of 2 comes from the fact that current travels to the load and back.

   Rearranging to solve for \( L \):
   \[
   L = \frac{P_{\text{loss}}}{3 \cdot I_L^2 \cdot 2 \cdot R}
   \]

After computing these steps, we will have the maximum line length \( L \). Let me calculate the exact values for you.The line current, \( I_L \), is approximately 96.62 A. 

The maximum length of the transmission line is about 25.5 km.

Would you like further clarification or additional details on any part of the solution?

Here are some related questions you might find useful:
1. How does the power factor affect the current and losses in a transmission line?
2. What would happen to the maximum line length if the resistance per phase were reduced?
3. How can the power losses in a transmission line be minimized?
4. How is the voltage drop related to the line length and resistance?
5. What are some practical ways to improve power transmission efficiency?

**Tip:** To reduce power losses, increasing the transmission voltage significantly reduces the current, which in turn reduces the power losses due to resistance.

A three-phase, wye-connected, 20-MW load of power factor 0.866 lagging is to be supplied by a transmission line at 138 kV. It is desired that the line losses not exceed 5 percent of the load. If the per-phase resistance of the line is 0.7 Ω, what is the maximum length of the line?

Find the maximum length of a transmission line for a 20 MW, 138 kV, three-phase, wye-connected system with a power factor of 0.866 lagging. This electrical engineering problem involves calculating power loss, current, and resistance per phase.

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