Problem $3$: Economic dispatch with network constraints

In this problem, you will implement a solver to solve the economic dispatch problem with a

general power network.

For a power network with $n$ buses and $m$ lines, the inputs of your solver include:

Network data:

Number of buses $n,$ number of lines $m$

Susceptances $bin{R}^m$

Node$-$edge incidence matrix $Ain{R}^{nm}$

Flow capacities: $\frac{?}{\text{b}\text{a}\text{r}}(f)in{R}^m$

Generator data:

For convenience, we assume there is a generator at every bus. For a bus where

there is not actually a generator, we can simply set the corresponding generation

capacity to be zero.We work with quadratic generator costs:

$C_i(p_i)={}_i{p}_i^2+{}_i{p}_i.$

We are given $in{R}_{+}^n$ and $in{R}^n.$

Minimum and maximum allowed generation level: $p^{\text{m}\text{i}\text{n}}in{R}^n$ and $p^{\text{m}\text{a}\text{x}}in{R}^n.$

Again, for buses without a generator, we have $p_i^{\text{m}\text{i}\text{n}}=p_i^{\text{m}\text{a}\text{x}}=0.$

Load data:

We work with inelastic loads and are given $lin{R}^n.$

Your solver should solve the economic dispatch problem with network constraints and

produce the outputs listed below:

Solver status: solved, infeasible or unbounded

Depending on the data, the problem may not have a well$-$defined solution. If it

has a finite solution $J^{***}=-n{R}^{n}in{R}^{n}fin{R}^m{J}^{***}2,3,4,5,6,9,10,11,12,13,14==1(6,12)-,$ the solver status reported should $be$ solved.

$If$ the problem does not have any feasible point, the solver status reported should

$be$ infeasible. $If$ the problem $is$ unbounded below $so{J}^{***}=-,$ the solver status

reported should $be$ unbounded.

Generator dispatch: $n{R}^n$

Phase angles: $in{R}^n$

Line flows: $fin{R}^m$

Optimal cost: $J^{***}$

$(a)$ Implement the solver described above $in$ Matlab $(or$ any other language$).$

$(b)$ Consider the IEEE $14-$bus test case shown below. The generator buses are $1,2,3,6,$

The load buses are $2,3,4,5,6,9,10,11,12,13,14.$ Suppose $==1$; all

generator minimum generation level $is0$ and maximum generation level $is10$; all loads

are $1$ ; all lines have susceptance $-1$ except line $(6,12)$ which has susceptance $-2$ ; all line

capacities are $2.$ Solve the economic dispatch problem with this set $of$ inputs using

your solver and report the outputs. Need help determining these inputs in the code code: Ain $=$ np$.$zeros$([])$ # Node$-$edge incidence matrix $($replace with actual data$)$

b $=$ np$.$array$([])$ # Flow capacities $($replace with actual data$)$

c $=$ np$.$array$([])$ # Generator costs $($replace with actual data$)$

p$\_$min $=$ np$.$zeros$($n$)$ # Minimum generation level

p$\_$max $=$ np$.$array$([10]*$ n$)$ # Maximum generation level

l $=$ np$.$array$([1]*10+[-1]*10)$ # Load data