Need matlab code: Problem $2$: Storage control

We consider the use of a storage device $($say$,$ a battery$)$ to reduce the total cost of electricity

consumed over one day. We divide the day into $T$ time periods, and let $(t)$ denote the

$($positive$,$ time$-$varying$)$ electricity price ${?}^1,$ and $l(t)$ denote the $($nonnegative$)$ load, in period

$t,$ for $t=1,$dots,$T.$ Without the use of a battery, the total cost is ${}^{TT}l.$

Let $x(t)$ denote the $($nonnegative$)$ energy stored in the battery in period $t.$ For simplicity,

we neglect energy loss $($although this is easily handled as well$),$ so we have $x(t+1)=$

$x(t)+u(t),t=1,$dots,$T,$ where $u(t)$ is the charging of the battery in period $t$ if $u(t)0$;

$u(t)<mn\; id="EditorElement\_698768">0</mn>$ means the battery is discharged. With the battery operating, the net consumption

in period $t$ is $l(t)+u(t)$; we require this to be nonnegative $($i$.$e$.,$ in this case, the utility does

not allow selling back to the grid$).$ The total cost is then ${}^{TT}(l+u).$

The battery is characterized by three parameters: The capacity $\stackrel{}{x},$ where $0x(t)\stackrel{}{x}$;

the maximum charge rate $\frac{?}{\text{b}\text{a}\text{r}}(u^c),$ where $u(t)\frac{?}{b}ar(u{)}^c$; and the maximum discharge rate $\frac{?}{\text{b}\text{a}\text{r}}(u{)}^d,$ where

$u(t)\frac{-}{b}ar(u{)}^d.($The parameters $\stackrel{}{x}\frac{,}{b}ar(u{)}^c,$ and $\frac{?}{\text{b}\text{a}\text{r}}(u{)}^d$ are nomegative.$)$

The initial storage level is $0.$ Since we are not dealing with generator dispatch directly,

it's convenient to work with energy $($in $kWh)$ instead of power $($in $kW).$ Hence all units in

this problem is $kWh,$ including the units of load and storage control $u(t).$ The unit of price

is $\frac{\text{\$}}{k}Wh.$

$($a$)$ Formulate an optimization to find the charging profile $uin{R}^T($and associated stored

energy profile $xin{R}^{T+1})$ that minimizes the total cost, subject to the constraints.

$($b$)$ Solve the problem instance with data $$ and $l$ given in storage$\_$data.m$,\stackrel{}{x}=35,$ and

$\frac{?}{\text{b}\text{a}\text{r}}(u{)}^{}\frac{?}{b}=ar(u{)}^d=3.$ Plot $l(t),(t),u(t),$ and $x(t)$ versus $t.$

$($c$)$ Plot the minimum total cost versus the storage capacity $\stackrel{}{x},$ using $$ and $l$ from storage$\_$data.m$,$

and charge$/$discharge limits $\frac{?}{\text{b}\text{a}\text{r}}(u{)}^c\frac{?}{b}=ar(u{)}^d=3.$ Repeat for charge$/$discharge limits $\frac{?}{\text{b}\text{a}\text{r}}(u{)}^c\frac{?}{b}=ar(u{)}^d=$

$($Put these two trade$-$off curves on the same plot.$)$ Give an interpretation of the

endpoints of the trade$-$off curves.