The price of everything

In order to be able to simulate the Nord Pool market, amethod should

to beused that can, given thecorrect data, calculate the prices,

produc-tion,consumption, export andimport for each market. With such atool

developed, adeviation from the production cost function can be used to

simulatemarketpower.

12.1.1 Eltra's method

Eltra has developed aprogram to calculate the correct market prices on

eachmarket,givencertaindemandandsupplycurves.Eltrapresentsthisas

anintegeroptimizationproblem,wherethetotalsocialsurplusisoptimized.

This surplus is composed of producer surplus, consumer surplus and the

gridorthebottlenecksurplus. Thegridsurplusistheprotderivedfrom

buying aunit of electricityfrom acheap market and selling it to amore

expensiveoneandisadministratedbytherelevantsystemoperators. This

maximizationprovidesthecorrectresultsassellingaunit totheonewho

is ready to pay the most for it, is always optimal and gives the correct

quantityofnetexport andimport,giventhetransmissionlimitations.

Example12.1 If aunitwith aproduction costof NOK190 couldsatisfy

aSwedish buyer ready topay NOK 200, the combinedconsumer and

pro-ducersurplus fromthat sale would beNOK 10. However, ifinsteadof the

Swedish buyer, a Norwegian buyer would be ready to buy that unit for up

toNOK 220, the consumersand producer surplus withthe additionof the

grid surpluswouldbe 30.

Thus,ifthecheapestpowerunit,fromallthemarkets,isalwaystakenand

soldto thebuyer whois ready to paythe mostfor it and is ablebuy it,

giventhatthiscanonlyhappenwhenthereisadirectconnectionbetween

markets, we will end up with the optimalsolution. This may sometimes

meanthat units willbereturned when thenextunit issold from market

AtomarketB,ifBhadpreviouslybeenthenetexportertoA.

ThebeautybehindEltra'sapproachisthatforeachmarket,duetothe

sup-ply and demand functions,the correct surpluses,given price,import and

export, canalwaysbecalculated. However,thisneedsalittledata

prepa-ration as the cumulative consumers surplus (

CCS

^{), and the cumulative}

producersurplus (

CP S

^{)functions,mustbecreated. wecanassumethat}

theproductioncostofthelastunitproducedwillbeeitherthesameasor

closetothemarketprice. Iftheproductioncostoftherstthreeunitsis1,

2and3respectively,usingtheaboveassumption,thecumulativeproducer

surplusfunctionwouldbe0,1and3(

1 −1, 2 ×2 −(1+2), 3 ×3 −(1+2+3)

^).

Therecanalsoonlybeapricedierencebetweenareaswheretransmission

is at its maximum, as otherwise there would be the same (or practically

thesame)price.

Therefore,if

t j,i

^{istheexportfrommarket}

i

^tomarket

j, Capacity i,j

^isthe

maximumexport capacityfrom market

i

^to

j

^,

BN S i,j

^{isthe gridsurplus}

fromexportingfrommarket

i

^to

j

^and

S

^and

D

respectivelyarethesupply anddemandfunctions:

BN S i,j = (D j (x j ) − D i (x i )) × Capacity i,j . . . ∀i, j

^(12.4)

This is a linear integer problem, with two multidimensional variables,

x

which is theconsumption in each market and

t

^{which is the}transmission betweenmarkets. Although

y

^{ispresentedasavariable,itissimplythesum}

ofthe othertwo. This is,in fact, notEltra'spresentation oftheproblem

butasimpleronetounderstand,althoughbuiltonthesameprinciplesand

requiresmorecomputerpowerfortheoptimization.

Eltrabeginsbycalculatingthepricesand surplusesforeach market when

thereisnointernationaltransmissionandthencreatestheCCSandCPSas

functionsofnetimportorexportaswellasthepricefunction

P

^{. Therefore,}

when there is import, CCSincreases but CPS decreases. More

precalcu-lations are required, asdemand and supply must be compared for every

possiblemagnitudeofimport/exportforeach market.

x i = X

Here, there is only a single multidimensional variable,

t

^{, as}

x i

^{, now net}

exportfrommarket

i

^{,isonlythesumofsomeoftheelementsof}

t

^.

Eltra'salgorithmiswrittenin theoptimization programlanguageGAMS.

12.1.2 IMM's verication algorithm

Toverify whetherEltra's GAMSalgorithm wasgivingthecorrect results

giventhedata, amatlabalgorithmhadbeendevelopedatIMM.

1

As with Eltra's algorithm, the data has to be prepared, and vectors for

supplyanddemand havetobecreated. Thevector'sindex numberis the

volume and the vector's value is the price. Interpolation is used to ll

theover26thousandlongvectorsforeachmarket,whichisthemaximum

volume oered onthe largestmarket,Sweden. Therefore, theproblem is

solveddiscreetlywith each MWh/hasthelowest unit,which seemstobe

satisfactoryasthelowestequilibriumisaround5000MWh/h.

TheIMM'salgorithmsolvedtheproblemin thefollowingsteps:

1. Calculate pricesfor each market by comparing where there is least

dierencebetweenthesupplyanddemand vectors.

2. FindallpossiblelegaltransmissionsofoneMWhfromanymarketto

another.

3. TransferoneMWh fromthe cheapest market tothemostexpensive

marketthancanreceivetransmissionfromit.

4. Repeatfrom step1untilthereisnolegaltransmissionavailable.

The algorithm did conrm Eltra's results, but the calculation time, four

daysforeachtradinghour,onIMM'sserverSunre,madeitratherlimited

toolforanalysis. [27]

12.1.3 Revision

Inordertospeedupthecalculations,ImadesomechangestoIMM's

algo-rithm. Thefollowingaretheforemost:

•

^{Allowed morethan1 unit to be} transmitted eachtime; now the al-gorithmnowusuallybeginswith1000units.

•

^{Usedpointers (asinC++)insteadof changingthelongvectors.}

•

^{Exploited thefact that when x units are sent from market A to B,}

thenewequilibriumislessthanorequaltoxunitsawayfromtheold

equilibrium. Therefore,onlyasmallpartofthesupplyanddemand

vectorshadtobecompared.

1

TheDepartmentofInformaticsandMathematicalModellingattheTechnical

Uni-•

^{Skipped a numberof} unnecessary repeated calculations asprevious resultscouldoftenbekeptandreused.

Ialso madeitpossibleto useaunit largerorsmallerthan1MWh asthe

basic unit. However, I do use the 1MWh as the basic unit. Although,

there is not much left of the original algorithm, the four steps listed in

section12.1.2arethereasbefore,exceptthatmorethanoneunitcannow

be transmitted. Afterathorough revision,eachcycle ofcalculationsnow

takesapprox.0.05secondsonSunre,withoutthedatapreparations. This

isaconsiderableimprovementandmakesthisalgorithm thefastest ofthe

three mentionedandwill thereforebeused hereafter. Thenewalgorithm

canbefoundin appendix B.1andhasthefollowingmain steps:

1. Calculate pricesfor each market by comparing where there is least

dierencebetweenthesupplyanddemand vectors.

2. FindallpossiblelegaltransmissionsofoneMWhfromanymarketto

another.

3. Transfer

x

^{MWh from the cheapest market to the most expensive}

marketthancanreceivetransmissionfromit.

4. Repeat from step 1 until no legal transmission available, otherwise

reduce

x

^.

5. Repeat from step1 until there is nolegal transmission available or

x < 1

^.

In document IMM YGBY2003ESAESRETR.2003 46 (Sider 117-121)