Voltage regulation of 2 MVA Δ-Y transformer

Regulation is worked out per phase. The primary is delta (phase voltage = line voltage) and the secondary is star (phase voltage = line voltage/$\sqrt{3}$).

Step 1: Per-phase voltages and turns ratio

$$\begin{aligned} V_{1ph} &= 33000\ \text{V}, \quad V_{2ph} = \frac{6600}{\sqrt{3}} = 3810.5\ \text{V} \\ K &= \frac{V_{2ph}}{V_{1ph}} = \frac{3810.5}{33000} = 0.11547 \end{aligned}$$

Step 2: Equivalent resistance referred to primary (per phase)

$$\begin{aligned} R_2' &= \frac{R_2}{K^2} = \frac{0.08}{0.11547^2} = 6.0\ \Omega \\ R_{01} &= R_1 + R_2' = 8 + 6 = 14\ \Omega \end{aligned}$$

Step 3: Percentage resistance and reactance

Full-load primary phase current:

$$I_{1ph} = \frac{2 \times 10^6}{3 \times 33000} = 20.2\ \text{A}$$ $$\begin{aligned} \%R &= \frac{I_{1ph} R_{01}}{V_{1ph}} \times 100 = \frac{20.2 \times 14}{33000} \times 100 = 0.857\% \\ \%Z &= 7\% \ (\text{given}) \\ \%X &= \sqrt{7^2 - 0.857^2} = 6.947\% \end{aligned}$$

Step 4: Regulation at 0.75 pf lagging

$\cos\phi = 0.75$, $\sin\phi = \sqrt{1 - 0.75^2} = 0.6614$.

$$\begin{aligned} \%\text{Reg} &\approx \%R\cos\phi + \%X\sin\phi \\ &= 0.857 \times 0.75 + 6.947 \times 0.6614 \\ &= 0.643 + 4.595 = 5.24\% \end{aligned}$$

If the more exact formula is used, adding the second-order term $\frac{(\%X\cos\phi - \%R\sin\phi)^2}{200}$ gives about 5.35%.

Answer: Full-load voltage regulation at 0.75 pf lagging is about 5.24% (approximate formula).