Q2GATE 2023MCQ

The expressions of fuel cost of two thermal generating units as a function of the respective power generation $\mathrm{P}_{\mathrm{G} 1}$ and $\mathrm{P}_{\mathrm{G} 2}$ are given as $\mathrm{F}_{1}\left(\mathrm{P}_{\mathrm{G} 1}\right)=0.1 \mathrm{aP}_{\mathrm{G} 1}^{2}+40 \mathrm{P}_{\mathrm{G} 1}+120 \mathrm{Rs} /hour$ $0 \mathrm{MW} \leq \mathrm{P}_{\mathrm{G} 1} \leq 350 \mathrm{MW}$ $\mathrm{F}_{2}\left(\mathrm{P}_{\mathrm{G} 2}\right)=0.2 \mathrm{P}_{\mathrm{G} 2}^{2}+30 \mathrm{P}_{\mathrm{G} 2}+100 \mathrm{Rs} /hour$ $0 \mathrm{MW} \leq \mathrm{P}_{\mathrm{G} 2} \leq 300 \mathrm{MW}$ where $a$ is a constant. For a given value of a, optimal dispatch requires the total load of $290 \mathrm{MW}$ to be shared as $\mathrm{P}_{\mathrm{G} 1}=175 \mathrm{MW}$ and $\mathrm{P}_{\mathrm{G} 2}$ $=115 \mathrm{MW}$ . With the load remaining unchanged, the value of $a$ is increased by $10 \%$ and optimal dispatch is carried out. The changes in $\mathrm{P}_{\mathrm{G} 1}$ and the total cost of generation, $F\left(=F_{1}+F_{2}\right)$ in Rs/hour will be as follows.

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📖 Explanation

\begin{aligned} & \mathrm{IC}_{1}=0.2 \mathrm{aP}_{\mathrm{a}_{1}}+40 \\ & \mathrm{IC}_{2}=0.4 \mathrm{P}_{\mathrm{a}_{2}}+30 \end{aligned}

For optimal dispatch, $\mathrm{IC}_{1}=\mathrm{IC}_{2}$ $0.2 \mathrm{aP}_{\mathrm{G}_{1}}+40=0.4 \mathrm{P}_{\mathrm{G}_{2}}+30$ Given :

\begin{aligned} \mathrm{P}_{\mathrm{G}_{1}} & =175 \mathrm{MW} \text { and } \mathrm{P}_{\mathrm{G}_{2}}=115 \mathrm{MW} \\ \therefore 0.2 \mathrm{a} \times 175+40 & =0.4 \times 115+30 \\ \Rightarrow \quad \mathrm{a} & =1.0286 \end{aligned}

Economic scheduling, $\mathrm{P}_{\mathrm{G}_{1}}+\mathrm{P}_{\mathrm{G}_{2}}=290 \mathrm{MW}$ Now, if a increases by $10 \%, \mathrm{P}_{\mathrm{G}_{1}}$ should be decreases and total cost of generation increases.

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