Transformer Overload Analysis in Open Delta Operation

This question examines the overload condition of transformers when a Delta-Delta ($\Delta$-$\Delta$) three-phase bank operating at rated load experiences the removal of one transformer, leading to an open-delta configuration.

Understanding the Initial $\Delta$-$\Delta$ Setup

In a standard Delta-Delta connection supplying a balanced three-phase load, each of the three single-phase transformers carries a specific portion of the total load. Let's define the key parameters:

• Let $S_{rated}$ represent the rated apparent power (in kVA) of each individual transformer in the bank.
• The system operates with a line voltage $V_L$ and draws a line current $I_L$.
• In a $\Delta$ connection, the voltage across each transformer winding (phase voltage) is equal to the line voltage: $V_{ph} = V_L$.
• The current flowing through each transformer winding (phase current) relates to the line current by: $I_{ph} = \frac{I_L}{\sqrt{3}}$.
• Using these values, the rated kVA of each transformer can be expressed as:

  $$S_{rated} = V_{ph} \times I_{ph} = V_L \times \frac{I_L}{\sqrt{3}}$$

• The total apparent power supplied by the complete three-transformer bank is the sum of the power handled by each unit:

  $$S_{total} = 3 \times S_{rated} = 3 \times \left( V_L \times \frac{I_L}{\sqrt{3}} \right) = \sqrt{3} V_L I_L$$

Operation in Open-Delta Configuration

When one transformer is removed from the $\Delta$-$\Delta$ bank, the remaining two transformers must supply the original total load ($S_{total}$) but now operate in an open-delta configuration.

• The connected load continues to demand the same line voltage $V_L$ and line current $I_L$.
• In the open-delta connection, the two remaining transformers are connected across two lines of the three-phase system.
• The voltage across the windings of each of these two transformers remains the line voltage, $V_L$.
• However, the current that flows through the windings of each of the two operating transformers increases to become equal to the line current: $I_{actual\_ph} = I_L$.
• Consequently, the actual apparent power handled by each transformer in this open-delta setup is calculated as:

  $$S_{actual} = V_L \times I_{actual\_ph} = V_L \times I_L$$

Determining the Overload Factor

The overload factor quantifies how much the actual power handled by a transformer exceeds its rated power. It is calculated as the ratio of the actual apparent power ($S_{actual}$) to the rated apparent power ($S_{rated}$):

$$ \text{Overload Factor} = \frac{S_{actual}}{S_{rated}} $$

By substituting the derived expressions for $S_{actual}$ and $S_{rated}$ into the formula:

$$ \text{Overload Factor} = \frac{V_L \times I_L}{V_L \times \frac{I_L}{\sqrt{3}}} $$

Simplifying this mathematical expression gives:

$$ \text{Overload Factor} = \frac{I_L}{\frac{I_L}{\sqrt{3}}} = \sqrt{3} $$

Quantifying the Overload Magnitude

The calculated overload factor is $\sqrt{3}$. The numerical value of $\sqrt{3}$ is approximately $1.732$. This result indicates that each of the two transformers operating in the open-delta configuration is forced to handle approximately $1.732$ times its rated apparent power (kVA) to meet the demands of the original full load.