GATE 2022 EE Q.43: solution and answer

GATE 2022 · EEQ.43 · 2 marks · MCQ

The fuel cost functions in rupees/hour for two 600 MW thermal power plants are given by Plant 1: $C_{1} = 350 + 6 P_{1} + 0.004 P_{1}^{2}$ ; Plant 2: $C_{2} = 450 + a P_{2} + 0.003 P_{2}^{2}$ , where $P_{1}$ and $P_{2}$ are power generated by plant 1 and plant 2, respectively, in MW and $a$ is constant. The incremental cost of power ( $λ$ ) is 8 rupees per MWh. The two thermal power plants together meet a total power demand of 550 MW. The optimal generation of plant 1 and plant 2 in MW, respectively, are

(A)
200, 350
(B)
250, 300
(C)
325, 225
(D)
350, 200

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Answer and solution

ANSWER

The optimal generation is 250 MW from plant 1 and 300 MW from plant 2, option (B).

Official answer: BSource: Official answer key, GATE 2022 EE (IIT Kharagpur)

With transmission losses ignored, economic dispatch requires equal incremental cost at every plant: $\frac{d C _{1}}{d P _{1}} = \frac{d C _{2}}{d P _{2}} = λ = 8$ rupees per MWh.
Plant 1 has no unknown constant, so apply the condition to it directly: $\frac{d C _{1}}{d P _{1}} = 6 + 0.008 P_{1} = 8$ .
Solving gives $P_{1} = \frac{8 - 6}{0.008} = 250 MW$
The power balance is $P_{1} + P_{2} = 550 MW$ , so $P_{2} = 550 - 250 = 300 MW$ .
Check with plant 2: $\frac{d C _{2}}{d P _{2}} = a + 0.006 P_{2} = 8$ gives $a = 8 - 0.006 \times 300 = 6.2$ . This is a valid constant, so the data are consistent.
Both outputs are below the 600 MW rating, so no plant limit is active. Hence $(P_{1}, P_{2}) = (250, 300)$ MW, which is option (B).

KEY IDEA

Without losses, economic dispatch requires equal incremental cost at both plants. Use the plant whose cost function is fully known to fix the common value of $λ$ , then use the power balance for the other plant.

WATCH OUT FOR

Trying to find $a$ before $P_{2}$ is known, or splitting the 550 MW demand equally between the plants. Forgetting that $\frac{d}{d P} (0.004 P^{2}) = 0.008 P$ and using $0.004 P$ also gives a wrong $P_{1}$ .

From the GATE 2022 EE paper. Every question of this paper · Download the full paper · All 9 Dispatch & control questions, with notes

GATE 2022 EE Q.43: solution and answer

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