GATE EE 2019 Set 1 — Question 51

Given:
Generator internal reactance $X_d = 0.25$ pu
Transformer reactance $X_{L1} = 0.4$ pu
Line reactance $X_{L2} = 0.4$ pu
Infinite bus voltage $V = 1 \angle 0^\circ$ pu
Real power delivered $P = 0.8$ pu
Power factor $PF = 0.8$ laggingFirst, calculate the total reactance between the generator internal voltage and the infinite bus:
$X_{total} = X_d + X_{L1} + X_{L2} = 0.25 + 0.4 + 0.4 = 1.05$ puNext, calculate the reactive power $Q$ delivered by the generator.
For lagging power factor, $Q = P \tan(\cos^{-1}(PF))$.
$\cos^{-1}(0.8) = 36.87^\circ$
$\tan(36.87^\circ) = 0.75$
$Q = 0.8 \times 0.75 = 0.6$ puThe complex power delivered to the infinite bus is $S = P + jQ = 0.8 + j0.6$ pu.
The current delivered to the infinite bus is $I = \left(\frac{S}{V}\right)^* = \left(\frac{0.8 + j0.6}{1 \angle 0^\circ}\right)^* = (0.8 + j0.6)^* = 0.8 - j0.6$ pu.The generator internal voltage $E_g$ can be found using the equation:
$E_g = V + jI X_{total}$
$E_g = 1 \angle 0^\circ + j(0.8 - j0.6)(1.05)$
$E_g = 1 + j(0.8 \times 1.05) - j^2(0.6 \times 1.05)$
$E_g = 1 + j0.84 + 0.63$ (since $j^2 = -1$)
$E_g = 1.63 + j0.84$ puTo find the power angle $\delta$, we need to convert $E_g$ to polar form:
$|E_g| = \sqrt{(1.63)^2 + (0.84)^2} = \sqrt{2.6569 + 0.7056} = \sqrt{3.3625} \approx 1.8337$ pu
$\delta = \tan^{-1}\left(\frac{0.84}{1.63}\right) = \tan^{-1}(0.5153) \approx 27.26^\circ$Alternatively, the power angle $\delta$ is the angle of $E_g$ with respect to $V$. The power delivered is given by:
$P = \frac{|E_g||V|}{X_{total}} \sin\delta$
$0.8 = \frac{1.8337 \times 1}{1.05} \sin\delta$
$0.8 = 1.7464 \sin\delta$
$\sin\delta = \frac{0.8}{1.7464} \approx 0.4580$
$\delta = \sin^{-1}(0.4580) \approx 27.26^\circ$The question asks for the power angle of the generator in degrees. The power angle is $\delta$.
Rounding off to one decimal place, $\delta = 27.3^\circ$.Note on discrepancy: The calculated power angle is $27.3^\circ$. However, the specified answer range is $0.45$ to $0.47$. This suggests the question might be asking for the power angle in radians, or there is a significant discrepancy in the expected answer. If the answer is $0.46$ (within the range), it would correspond to $0.46 \times \frac{180}{\pi} \approx 26.35^\circ$. This is close to the calculated value, suggesting the answer might be in radians. However, the question explicitly asks for degrees. Given the strict instruction to match the provided answer, we acknowledge this discrepancy. If the answer is $0.46$, it implies a different interpretation or a flaw in the problem statement/given answer.