Problem $2$: Networked electricity market

The data files p$1$Bus$1$Data.csv and p$1$Bus$2$Data.csv contain two sets of bids and offers. For

each file, the csv includes a matrix with four columns:

The first column includes the quantities of the bids. Denote the $k$ th element of this

column by $p_k^b.$

The second column includes the prices of the bids. Denote the $k$ th element of this

column by ${}_k^b.$

The third column includes the quantities of the offers. Denote the $k$ th element of this

column by $p_k^o.$

The fourth column includes the prices of the offers. Denote the $k$ th element of this

column by ${}_k^0.$

Each row of the matrix include the data for a bid and an offer. The first file contains $30$

rows and the second file contains $50$ rows.

Consider a two bus electricity network, connected with a transmission line. Suppose the

bids and offers in p$1$Bus$1$Data.csv are from suppliers and consumers connected to bus $1$ and

the bids and offers in p$1$Bus$2$Data.csv are from suppliers and consumers connected to bus

$($a$)$ Suppose the transmission line does not exist so there are two separate pool markets,

one for each bus. Calculate the market equilibrium price and quantity for the market

corresponding to each bus. Note: this can be formulated as an optimization as shown

in lectures.

$($b$)$ Under the condition of $($a$),$ denote the equilibrium price and quantity for the market

at bus $i$ by ${}_i^{eq}$ and $p_i^{eq}.$ For each bid $k,$ also calculate the supplier surplus defined as

follows:

$({}_i^{ea}-{}_k^b)x_k^b,$

where $x_k^b$ is the cleared$/$accepted quantity for bid $k$ such that $0x_k^b{p}_k^b.$ For each

offer, calculate the consumer surplus defined as follows:

$({}_k^o-{}_i^{eq})x_k^o,$

where $x_k^o$ is the cleared$/$accepted quantity for offer $k$ such that $0x_k^o{p}_k^o.$ For

each market, plot the histogram for supplier surplus and the histogram for consumer

surplus. Bus$1$: $41\mathrm{.}702200479\mathrm{.}83468338310\mathrm{.}2334428811\mathrm{.}4745973$

$72\mathrm{.}0324493442\mathrm{.}110762541\mathrm{.}4055987894\mathrm{.}94892587$

$0\mathrm{.}01143748295\mathrm{.}7889530269\mathrm{.}4400157744\mathrm{.}99121335$

$30\mathrm{.}2332572653\mathrm{.}316528541\mathrm{.}4179269557\mathrm{.}83896144$

$14\mathrm{.}6755890869\mathrm{.}18771144\mathrm{.}99534589540\mathrm{.}81368028$

$9\mathrm{.}23385947731\mathrm{.}551563153\mathrm{.}5896405923\mathrm{.}70269802$

$18\mathrm{.}6260211468\mathrm{.}6500927766\mathrm{.}3794645290\mathrm{.}33795206$

$34\mathrm{.}556072783\mathrm{.}4625671951\mathrm{.}4889112157\mathrm{.}36794867$

$39\mathrm{.}676747421\mathrm{.}82882773494\mathrm{.}45947560\mathrm{.}287032703$

$53\mathrm{.}881673475\mathrm{.}0144314958\mathrm{.}6555040561\mathrm{.}71449136$

$41\mathrm{.}9194514498\mathrm{.}8861088990\mathrm{.}3401915332\mathrm{.}66449018$

$68\mathrm{.}5219500474\mathrm{.}8165654413\mathrm{.}7474704152\mathrm{.}70581023$

$20\mathrm{.}4452249728\mathrm{.}0443992113\mathrm{.}9276347388\mathrm{.}59420993$

$87\mathrm{.}8117436478\mathrm{.}9279328580\mathrm{.}7391288735\mathrm{.}726976$

$2\mathrm{.}7387593210\mathrm{.}3226006639\mathrm{.}767683790\mathrm{.}85351509$

$67\mathrm{.}0467510244\mathrm{.}7893526216\mathrm{.}5354197162\mathrm{.}33601158$

$41\mathrm{.}7304802490\mathrm{.}8595503192\mathrm{.}750858041\mathrm{.}582124285$

$55\mathrm{.}8689828429\mathrm{.}3614148434\mathrm{.}7765859792\mathrm{.}94372337$

$14\mathrm{.}0386938628\mathrm{.}7775338675\mathrm{.}0812103169\mathrm{.}08969175$

$19\mathrm{.}8101489113\mathrm{.}0028572172\mathrm{.}5997985499\mathrm{.}73228505$

$80\mathrm{.}074456871\mathrm{.}93669578788\mathrm{.}3306091217\mathrm{.}23405083$

$96\mathrm{.}8261575767\mathrm{.}8835532962\mathrm{.}3672207113\mathrm{.}71357496$

$31\mathrm{.}3424178221\mathrm{.}162811675\mathrm{.}094243493\mathrm{.}2595463$

$69\mathrm{.}2322615726\mathrm{.}5546659434\mathrm{.}889834269\mathrm{.}68181615$

$87\mathrm{.}6389152349\mathrm{.}1573159326\mathrm{.}992789186\mathrm{.}600017272$

$89\mathrm{.}460666355\mathrm{.}33625451289\mathrm{.}5886218275\mathrm{.}54630526$

$8\mathrm{.}50442113757\mathrm{.}4117605542\mathrm{.}8091189975\mathrm{.}38761885$

$3\mathrm{.}90547832314\mathrm{.}6728574996\mathrm{.}4840047192\mathrm{.}30245355$

$16\mathrm{.}9830419658\mathrm{.}9305536966\mathrm{.}3441497871\mathrm{.}15247586$