GATE 2018 EE Q.30: solution and answer

GATE 2018 · EEQ.30 · 2 marks · MCQ

The per-unit power output of a salient-pole generator which is connected to an infinite bus, is given by the expression, $P = 1.4 sin δ + 0.15 sin 2 δ$ , where $δ$ is the load angle. Newton-Raphson method is used to calculate the value of $δ$ for $P = 0.8$ pu. If the initial guess is $3 0^{\circ}$ , then its value (in degree) at the end of the first iteration is

(A)
$1 5^{\circ}$
(B)
$28.4 8^{\circ}$
(C)
$28.7 4^{\circ}$
(D)
$31.2 0^{\circ}$

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Answer and solution

ANSWER

After one Newton-Raphson iteration from $3 0^{\circ}$ , $δ_{1} \approx 28.7 4^{\circ}$ , which is option (C).

Official answer: CSource: Official answer key, GATE 2018 EE (IIT Guwahati)

Write the equation as a root-finding problem: $f (δ) = 1.4 sin δ + 0.15 sin 2 δ - 0.8 = 0$ .
Differentiate, keeping the chain-rule factor of 2 on the second term: $f^{'} (δ) = 1.4 cos δ + 0.3 cos 2 δ$ .
Evaluate at $δ_{0} = 3 0^{\circ}$ : $f (δ_{0}) = 1.4 (0.5) + 0.15 (0.8660) - 0.8 = 0.7 + 0.1299 - 0.8 = 0.0299$ .
$f^{'} (δ_{0}) = 1.4 (0.8660) + 0.3 (0.5) = 1.2124 + 0.15 = 1.3624$ .
Newton-Raphson update: $δ_{1} = δ_{0} - \frac{f ( δ _{0} )}{f ^{'} ( δ _{0} )}$ . The correction is $Δ δ = - \frac{0.0299}{1.3624} = - 0.02195 rad$ . It is in radians because $f^{'}$ is the derivative with respect to $δ$ measured in radians.
Convert to degrees: $Δ δ = - 0.02195 \times \frac{180}{π} = - 1.25 8^{\circ}$ .
$δ_{1} = 3 0^{\circ} - 1.25 8^{\circ} = 28.7 4^{\circ}$ , which is option (C).

KEY IDEA

Newton-Raphson on $f (δ) = P (δ) - 0.8$ gives $δ_{1} = δ_{0} - f (δ_{0}) / f^{'} (δ_{0})$ . Differentiating the sine terms gives $f^{'}$ per radian, so the correction comes out in radians and has to be converted to degrees before you subtract it from $3 0^{\circ}$ .

WATCH OUT FOR

There are two traps. (1) Subtracting the radian correction $0.02195$ straight from $30$ gives $29.9 8^{\circ}$ . (2) Dropping the chain-rule factor, i.e. writing $\frac{d}{d δ} (0.15 sin 2 δ) = 0.15 cos 2 δ$ instead of $0.3 cos 2 δ$ , gives about $28.6 7^{\circ}$ . Neither result is among the options.

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GATE 2018 EE Q.30: solution and answer

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