Problem $4.(10$ points$).$

Consider a small system with $3$ generators with the following parameters $($all units in MW$)$:

Unit $1$: $C_1(p_1)=300+8p_1+0\mathrm{.}2p_1^2,P_1^{\text{m}\text{i}\text{n}}=200MW$ and $P_1^{\text{m}\text{i}\text{n}}=600MW.$

Unit $2$: $C_2(p_2)=500+5p_2+0\mathrm{.}4p_2^2,P_2^{\text{m}\text{i}\text{n}}=250MW$ and $P_2^{\text{m}\text{i}\text{n}}=500MW.$

Unit $3$: $C_3(p_3)=50+12p_3+0\mathrm{.}5p_3^2,P_3^{\text{m}\text{i}\text{n}}=120MW$ and $P_3^{\text{m}\text{i}\text{n}}=400MW.$

Each generator can be tuned on or off, i$.$e$.$ binary decisions on their status must be modeled. Together, the three generators must satisfy a load of $500$ MW $.$ Find the optimal $($least cost$)$ combination of the unit status and the generation of each unit.

Compare the previously found optimal dispatch to the case where in addition to $500$ MW of load, the gree generators must provide $100$ MW of reserve. Explain the physical and economic difference in primal and dual variables between these two cases.