Calculating Synchronous Generator Impedance with Base Change

This problem involves converting the per unit (pu) impedance of a synchronous generator from its original base values (rating) to a new set of base values (different MVA and voltage rating). This is a common task in power system analysis.

Understanding Per Unit Impedance Base Change

Per unit impedance represents the impedance relative to a specified base impedance. When the base MVA or base voltage changes, the per unit impedance value also changes accordingly. The actual impedance value (in Ohms) remains constant, but its representation in per unit changes.

Formula for Base Change

The formula used to convert per unit impedance from an old base to a new base is:

$Z_{\text{pu, new}} = Z_{\text{pu, old}} \times \left( \frac{V_{\text{base, old}}}{V_{\text{base, new}}} \right)^2 \times \left( \frac{S_{\text{base, new}}}{S_{\text{base, old}}} \right)$

Where:

• $Z_{\text{pu, new}}$ is the new per unit impedance.
• $Z_{\text{pu, old}}$ is the old per unit impedance.
• $V_{\text{base, old}}$ is the old base voltage.
• $V_{\text{base, new}}$ is the new base voltage.
• $S_{\text{base, new}}$ is the new base MVA.
• $S_{\text{base, old}}$ is the old base MVA.

Given Values

From the question, we have the following values:

• Old per unit impedance, $Z_{\text{pu, old}} = 0.2$ pu
• Old base voltage, $V_{\text{base, old}} = 11$ kV
• Old base power, $S_{\text{base, old}} = 50$ MVA
• New base voltage, $V_{\text{base, new}} = 22$ kV
• New base power, $S_{\text{base, new}} = 150$ MVA

Step-by-Step Calculation

Now, we substitute these values into the base change formula:

1. Calculate the voltage ratio squared: $\left( \frac{V_{\text{base, old}}}{V_{\text{base, new}}} \right)^2 = \left( \frac{11 \text{ kV}}{22 \text{ kV}} \right)^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4} = 0.25$
2. Calculate the power ratio: $\left( \frac{S_{\text{base, new}}}{S_{\text{base, old}}} \right) = \left( \frac{150 \text{ MVA}}{50 \text{ MVA}} \right) = 3$
3. Multiply the old per unit impedance by these ratios: $Z_{\text{pu, new}} = 0.2 \text{ pu} \times (0.25) \times (3)$
4. Perform the final multiplication: $Z_{\text{pu, new}} = 0.2 \times 0.75 = 0.15$ pu

Result

Therefore, the impedance of the synchronous generator referred to the new base of 22 kV, 150 MVA is 0.15 pu.