Explanation :
$ \begin{array}{l}\;\;\;\;{\mathrm C}_1=0.01\mathrm P_1^2+30{\mathrm P}_1+10\\\frac{{\mathrm{dC}}_1}{{\mathrm{dP}}_1}=2\times0.01{\mathrm P}_1+30\\\;\;\;\;\;\;\;\;=0.02\;{\mathrm P}_1+30\\\;\;\;{\mathrm C}_2=0.05\mathrm P_2^2+10{\mathrm P}_2+10\\\frac{{\mathrm{dC}}_2}{{\mathrm{dP}}_2}=0.1\;{\mathrm P}_2+10\end{array} $
For optimum incremental cost,
$ \begin{array}{l}\frac{{\mathrm{dC}}_1}{{\mathrm{dP}}_1}=\frac{{\mathrm{dC}}_2}{{\mathrm{dP}}_2}\\0.02{\mathrm P}_1+30=0.1{\mathrm P}_2+10\\0.02{\mathrm P}_1-0.1{\mathrm P}_2=-20\;\;\;\;...(\mathrm i)\end{array} $
Given, $ \;\;\;\;\;{\mathrm P}_1+{\mathrm P}_2=200\;\;...(\mathrm{ii}) $
Solving equation (i) and (ii),
$ \begin{array}{l}{\mathrm P}_2=200,{\mathrm P}_1=0\\\end{array} $
But the range of $ \begin{array}{l}{\mathrm P}_1\\\end{array} $ and $ \begin{array}{l}{\mathrm P}_2\\\end{array} $ are specified as
$ \begin{array}{l}100\;\mathrm{MW}\leq{\mathrm P}_1\leq150\;\mathrm{MW}\\100\;\mathrm{MW}\leq{\mathrm P}_2\leq180\;\mathrm{MW}\end{array} $
Hence, $ {\mathrm P}_2=200\;\mathrm{MW} $ is not valid.
So, both $ \begin{array}{l}{\mathrm P}_1\\\end{array} $ and $ \begin{array}{l}{\mathrm P}_2\\\end{array} $ are operating at
$ \begin{array}{l}\;\;{\mathrm P}_1=100,{\mathrm P}_2=100\Rightarrow{\mathrm P}_1+{\mathrm P}_2=200\\{\mathrm{IC}}_1=\frac{{\mathrm{dC}}_1}{{\mathrm{dP}}_1}=0.02\times{\mathrm P}_1+30=32\;\mathrm{Rs}/\mathrm{MW}\\{\mathrm{IC}}_2=\frac{\displaystyle{\mathrm{dC}}_2}{\displaystyle{\mathrm{dP}}_2}=0.01\times{\mathrm P}_2+10=20\;\mathrm{Rs}/\mathrm{MW}\end{array} $