Part $2$: Optimizing Profit from Solar Energy Over a $10-$Year Lifetime

Background:

In addition to the cost of building the solar panel system, we now consider the revenve and profit

over the lifetime of the system, which is set at $10$ years. The total energy produced by the solar

panels over $10$ years is proportional to the panel area $A,$ the energy per area, $\frac{E}{A},$ and the total

number of sunlight hours per day $S,$ with was determined in Part $1.$ The total energy production

over $10$ years is:

$E_{10yr}(A)=A*E*365*10$

where $E$ is the energy per area per day used in Part $1.$

The revenue from selling the energy over $10$ years is:

$R_{10yr}(A)=p_{\text{e}\text{n}\text{e}\text{r}\text{g}\text{y}}*E_{10yr}(A)$

where:

$p_{\text{e}\text{n}\text{e}\text{r}\text{g}\text{y}}=0\mathrm{.}10$ USD $\frac{?}{kWh}$ is the price at which energy is sold.

$A$ is the area of the solar panels in $m^2.$

$E_{\text{a}\text{r}\text{e}\text{a}}$ is the energy per area $($from part $1)$

The total cost of the solar panel installation includes the fixed and variable costs:

$C_{\text{t}\text{o}\text{t}\text{a}\text{l}}(A)=C_{\text{f}\text{i}\text{x}\text{e}\text{d}}+C_{\text{p}\text{a}\text{n}\text{e}\text{l}}*A+C_{\text{s}\text{c}\text{a}\text{l}\text{e}}*A^2$

where:

$C_{\text{f}\text{i}\text{x}\text{e}\text{d}}=10,000$ USD is the fixed installation cost.

$C_{\text{p}\text{a}\text{n}\text{e}\text{l}}=50US\frac{D}{m^2}$ is the linear cost per square meter of solar panel

$C_{\text{s}\text{c}\text{a}\text{l}\text{e}}=0\mathrm{.}0005$ USD $\frac{?}{m^6}$ is the nonlinear cost component that increases cubically as the

area grows, reflecting overhead costs.

The total profit over $10$ years is the difference between the total revenue and total cost:

$P_{10yr}(A)=R_{10yr}(A)-C_{\text{t}\text{o}\text{t}\text{a}\text{t}}(A)$

Expanding the expression:

$P_{10yr}(A)=(p_{\text{e}\text{n}\text{e}\text{r}\text{g}\text{y}}*(E*A*365*10))-(C_{\text{f}\text{i}\text{x}\text{e}\text{d}}+C_{\text{p}\text{a}\text{n}\text{e}\text{l}}*A+C_{\text{s}\text{c}\text{a}\text{l}\text{e}}*A^3)$

Task:

Use an optimization method to find the optimal solar panel area A that maximizes the total profit

$P_{10yr}(A)$ over the $10-$year system lifetime. Instructions for Part $2$:

Implement an optimization method to optimize the solar panel area $A,$ such that it

maximizes the total profit $P_{10yr}(A).$

Use an initial interval of $(50,500)m^2$ for $A.$

Use a stopping criterion corresponding to $3$ significant figures, ${}_s=0\mathrm{.}05\%$

Output the optimal solar panel area $A,$ the maximum profit $P_{10yr}(A),$ and the number of

iterations.