Derive a matlab code to solve the below: Auctions are common in economic theory. Prior to deregulation of the electricity industry, the system operator $($SO$)$ had access to the information about the actual cost of generation from all generators. In the deregulation industry, generators offer prices to supply energy in an auction, and consumers in some markets, may submit bid prices for the purchase of energy. Although economic theory suggests that in the presence of competition, the bid and offer prices approach marginal cost $($MC$)$ in the longer run, they need not be related to actual cost in every hour, particularly in the shorter term. The SO is the bid taker who chooses generators to supply energy and reconcile demand in a manner not dissimilar to that of an auction. The following problems related to the electricity industry illustrate the reason for our interest in auctions.

Formulation and discussion of the problem:

Table $2$ shows the resolution of a simple auction in which there are offers to sell and bids to buy products. Note that in a normal simple dispatch problem above, there were only offers to sell. In the table, price and quantity of offers and bids of different entities are shown.

Table $2$: Simple Auction Data

SIMPLE AUCTION DATA

UNSORTED

SORTED

Offers

Entity

MW

$$/$MW

Entity

MW

$$/$MW

Choice Variables

Product Q$*$$

A

$20$

$20$

B

$10$

$5$

$10$

$50$

A

$40$

$18$

A

$40$

$18$

$40$

$720$

A

$30$

$25$

A

$20$

$20$

$20$

$400$

B

$40$

$30$

A

$30$

$25$

$?$

$?$

B

$10$

$5$

B

$40$

$30$

$0$

$0$

B

$25$

$35$

B

$25$

$35$

$0$

$0$

$70$

$1170$

Bids

X

$10$

$12$

X

$5$

$100$

$5$

$500$

X

$20$

$80$

X

$20$

$80$

$20$

$1600$

X

$5$

$100$

Y

$15$

$40$

$15$

$600$

Y

$15$

$25$

Y

$30$

$30$

$30$

$900$

Y

$30$

$30$

Y

$15$

$25$

$0$

$0$

Y

$15$

$40$

X

$10$

$12$

$0$

$0$

Index

$70$

$3600$

$2430$

The offers and bids are first sorted in increasing order of price for offers and decreasing order of prices for bids. The sorted offers and bids represent an increasing and a decreasing curve. The search is for the point of intersection of the two curves $($equilibrium$).$ This is the goal in many other physical problems including that of finding the equilibrium of demand and supply curves. If the demand curve $($bids to buy$)$ is denoted by $1()$ and the supply $($offer to sell$)$ curve is denoted by $2(),$ then consider the expression,

UBFET$\_$EEM $613$ Labs $$ Dr D$.$ Setlhaolo Page $5$

$=1()=02()=0$

which denotes the area between $1$ and $2$ from the origion to $.$ Therefore in the above expression, D is the difference between the two areas which is a maximum at the equilibrium. The descrete form of the above expression is used in simulating auctions in a linear optimization program $($LP$).$

Draw the relationship between offers and bids. The objective function $($OF$)$ of the problem is to maximise the area between the two curves of the sorted offers and bids. In order to do so$,$ we set up choice variables, where we select quantities that are non$-$negative and less than or equal to the offered $($or bid$)$ quantity. The solver procedure is asked to select these choice variables subject to constraints. The product of the quantity selected and the price gives the area under the offer curve and the bid curve.

There is one more constraint: The total quantity of offers selected must be equal to the quantity of bids awarded.

Mathematical Statement of the Propblem

Given $$ offers to supply and $$ bids to buy with quantity price relationships, choose $,=[1,],$ and $,=[1,]$

Maximise $=\backslash$Sigma $==1\backslash$Sigma $==1,$

s$.$t;

$,>=0,=[1,]$; $=[1,]$;

$\backslash$Sigma $==1=\backslash$Sigma $==1$

$<=($offer$),=[1,],$

and

$<=($sell$),=[1,],$

Solution Methodoly

Use Matlab Linear Programming $($LP$)$ solver to derive the solution of this problem.