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Fixed and Variable Factors of Production

The increment...

Question

The incremental cost characteristic of two plant systems are $I C_{1} = 1.0 P_{1} + 85 R s / M W h r$ $I C_{2} = 1.2 P_{2} + 72 R s / M W h r$
where $P_{1}$ and $P_{2}$ are in MW. The loss coefficient matrix in MW is $(\begin{matrix} 0.012 & - 0.001 - 0.001 & 0.04 \end{matrix}]$ the optimum scheduling for $λ = 150 R s / M W h r$ will be

P_{1} = 14.54 M W, P_{2} = 10.18 M W

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P_{1} = 14.54 M W, P_{2} = 6.24 M W

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P_{1} = 24.32 M W, P_{2} = 6.24 M W

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P_{1} = 24.32 M W, P_{2} = 10.18 M W

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Solution

The correct option is B $P_{1} = 14.54 M W, P_{2} = 6.24 M W$
Given, $I C_{1} = 1.0 P_{1} + 85 R s / M W h r$

$I C_{2} = 1.2 P_{2} + 72 R s / M W h r$

$(\begin{matrix} B_{11} & B_{12} B_{21} & B_{22} \end{matrix}] = (\begin{matrix} 0.012 & - 0.001 - 0.001 & 0.04 \end{matrix}]$

Total transmission loss,

$P_{L} = B_{11} P_{1}^{2} + 2 B_{12} P_{1} P_{2} + B_{22} P_{2}^{2}$

$\frac{\partial P_{L}}{\partial P_{1}} = 2 B_{11} P_{1} + 2 B_{12} P_{2} = 0.024 P_{1} - 0.002 P_{2}$

$\frac{\partial P_{L}}{\partial P_{2}} = 2 B_{12} P_{1} + 2 B_{22} P_{2} = - 0.002 P_{1} + 0.08 P_{2}$

The penalty factor for the plant-1

$L_{1} = \frac{1}{1 - \frac{\partial P_{L}}{\partial P_{1}}} = \frac{1}{1 - (0.024 P_{1} - 0.002 P_{2})}$

The penalty factor for the plant-2

$L_{2} = \frac{1}{1 - \frac{\partial P_{L}}{\partial P_{2}}} = \frac{1}{1 - (- 0.002 P_{1} + 0.08 P_{2})}$

We know, $I C_{1} L_{1} = I C_{2} L_{2} = λ$

$\frac{P_{1} + 85}{1 - 0.024 P_{1} + 0.002 P_{2}} = \frac{1.2 P_{2} + 72}{1 + 0.002 P_{1} - 0.08 P_{2}} = 150$

Using above relation,
$4.6 P_{1} - 0.3 P_{2} = 65$

$- 0.3 P_{1} + 13.2 P_{2} = 78$

$P_{1} = 14.537 M W$

$P_{2} = 6.239 M W$

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Similar questions

\frac{d C_{1}}{d P_{G 1}} = 0.04 P_{G 1} + 20 R s / M W h

\frac{d C_{2}}{d P_{G 2}} = 0.02 P_{G 2} + 16 R s / M W h

λ

I C_{1} = 20 + 0.3 P_{1}

I C_{2} = 30 + 0.4 P_{2}

I C_{3} = 30

_{i}

\frac{d C_{1}}{d P_{G 1}} = 0.10 P_{G 1} + 20.0; \frac{d C_{2}}{d P_{G 2}} = 0.12 P_{G 2} + 15

P_{G}

_{D}

_{1}

_{2}

_{1}

_{G 1}

_{2}

_{G 2}

_{l o s s (p u)}

_{G 1}^{2} (p u)

F_{1} = 20 P_{1} + 0.05 P_{1}^{2} + 50

F_{2} = 16 P_{2} + 0.1 P_{2}^{2} + 40

20 \leq P_{2} \leq 60 M W;

40 \leq P_{1} \leq 150 M W

P_{1}

P_{2}

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