Market Clearing Mechanisms under Uncertainty

Transcription

1 Submtted to Operatons Research Market Clearng Mechansms under Uncertanty Javad Khazae Department of Operatons Research and Fnancal Engneerng, Prnceton Unversty, Prnceton NJ 08544, Golbon Zaker Department of Engneerng Scence, Unversty of Auckland, Auckland, New Zealand, Shmuel S. Oren Department of Industral Engneerng and Operatons Research, Unversty of Calforna Berkeley, Berkeley CA 9470, Electrcty markets face a substantal amount of uncertanty. Whle tradtonally ths uncertanty has been due to varyng demand, wth the ntegraton of larger proportons of volatle renewable energy, added uncertanty from generaton must also be faced. Conventonal electrcty market desgns cope wth uncertanty by runnng two markets: a market that s cleared ahead of tme, followed by a real-tme balancng market to reconcle actual realzatons of demand and avalable generaton. In such mechansms, the ntal clearng process does not take nto account the dstrbuton of outcomes n the balancng market. Recently an alternatve so-called stochastc settlement market has been proposed see e.g. Prtchard et al. 010 and Bouffard et al In such a market, the ISO clears both stages n one sngle settlement market. In ths paper we consder smplfed models for two types of market clearng mechansms. Frst a market clearng mechansm utlzed n New Zealand, whereby frms offer n advance and are notfed of a clearng quantty and prce gude based on an estmate of demand. Then n real tme frms are dspatched accordng to realzed demand and prces are determned. We refer to ths as NZTS. Secondly we consder a smplfed stochastc programmng market clearng mechansm. We demonstrate that under the assumpton of symmetry, our smplfed stochastc programmng market clearng s equvalent to a two perod sngle settlement TS system that takes count of devaton penaltes n the second stage. These however dffer from NZTS. We prove that ths stochastc settlement models results n better socal welfare than the NZTS under the assumpton of symmetry. Our models are targeted towards analyzng mperfectly compettve markets. We wll construct Nash equlbra of the resultng games for the ntroduced market clearng mechansms and compare them under the assumptons of symmetry and n an asymmetrc example. Key words : Uncertanty, Stochastc Programmng, Optmzaton, Equlbrum 1

2 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 1. Introducton Electrcty markets face two key features that set them apart from other markets. The frst s that electrcty cannot be stored, so demand must equal supply at all tmes. Ths s partcularly problematc gven that demand for electrcty s usually uncertan. Second, electrcty s transported from supplers to load over a transmsson network wth possble constrants. The combnaton of these two features means that n almost all electrcty markets today an Independent System Operator ISO sets dspatch centrally and clears the market. Generators and demand-sde users can make offers and bds, and the ISO wll choose whch are accepted accordng to a pre-determned settlement system. The classc settlement system used n almost all exstng electrcty markets s one where the ISO sets dspatch to maxmze socal welfare. Effectvely the ISO matches supply to meet the uncertan demand at every moment whle maxmzng welfare. Ths becomes partcularly dffcult n the short-run up to 4 hours before actual market clearng as some types of generator e.g. steam turbnes and to some extent gas turbnes need to ramp up ther generaton slowly, and t s costly to change ther output rapdly. Dfferent markets have approached ths problem n dfferent ways. One approach used s to run a determnstc two-perod market clearng model see e.g. Kamat and Oren 004. In many jursdctons such as the PJM, ths amounts to a day ahead followed by a real tme market wth separate, fnancally bndng settlements for each market. In New Zealand, however, the generators can place offers for a gven half hour perod up to gate closure whch occurs two hours pror to a desgnated perod. At gate closure these offers are locked n. Estmates of dspatch quantty and prce are provded to the ndustry partcpants based on forecast demand n the perods leadng up to real tme. We refer to these as pre-dspatch quanttes and prces. 1 At the start of the desgnated perod an accurate measure of demand s avalable to the ISO and the generators are dspatched accordngly. In the New Zealand electrcty market NZEM, the ISO redspatches the generators every fve mnutes durng a half hour perod, usng updated demand nformaton but accordng to the same offer curves, locked n at gate closure. In the NZEM there s only a sngle settlement as the pre-dspatch quanttes and prces are not fnancally bndng. In ths determnstc two perod sngle-settlement NZTS market, expected demand s used to clear the pre-dspatch quanttes and the ISO has no explct measure of any devaton costs for a generator. 1 In so far as usng these estmates to make an offer, the nformaton s only useful up to gate closure as thereafter offers can not be changed. The current fnancal settlement n the NZEM s based on ex-post prces that are computed wth average demand over a perod. Constraned on and off payments are used to ensure suffcent payment s made to the generators. However the Electrcty Authorty has taken real tme prcng under consultaton wth the stakeholders.

3 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 3 An alternatve to determnstc settlement systems s to use a stochastc settlement process to deal wth varable demand. We wll consder demand as gross demand net ntermttent renewable generaton such as wnd, whch n the NZEM s forced to offer at zero prce. In a stochastc settlement, the ISO can choose both pre-dspatch and short-run devatons for each generator to maxmze expected socal welfare n one step. We mght then expect a stochastc settlement system to do better on average than a determnstc two perod system. The dea of a stochastc settlement can be attrbuted to Bouffard et al. 005, Wong and Fuller 007 and Prtchard et al In these two-stage, sngle settlement models, the pre-dspatch clears wth nformaton about the future dstrbuton of uncertantes n the system e.g. demand and volatle renewable generaton, and nformaton about devaton costs for each generator. These models assume that each frms offers and devaton costs are truthful. In an mperfectly compettve market, ths assumpton s not vald. The queston then remans: can the stochastc settlement aucton gve better expected socal welfare when frms are behavng strategcally? That s the queston explored by ths paper. In order to answer ths queston, we should frst construct an equlbrum model of these two market mechansms. There are several studes that analyse supply functon equlbrum SFE models n the lterature. Klemperer and Meyer 1989b were the frst authors who consdered uncertanty n demand for SFE models. In ther model, frms are uncertan about demand and should cater for dfferent demand scenaros, as the spot prce s determned after demand realzaton and s dependent on ther offered supply functon. They state that havng uncertanty n the model decreases the number of equlbra dramatcally as generators need to come up wth supply functons that are optmal for dfferent cases of resdual demand functons. There are also several studes that consder determnstc two settlement electrcty markets. Allaz and Vla 1993, Wllems 005 and Gans et al show that frms are desrous of partcpatng n a forward market, even though t decreases prces. Allaz 199 and Allaz and Vla 1993 consder a two settlement Cournot-Nash electrcty market for the frst tme. Ther model llustrates how frms are exposed to Prsoner s dlemma leadng them to contract n a forward market and ultmately reduce spot prces and ther respectve profts. Haskel and Powell 1994 extended Allaz and Vla s analyss for general conjectural varatons. Gans et al also upheld the concluson that a contract market can ncrease competton n the spot market and so leads to lower prces. They used a two-way contract for ther model. Wllems 005 replaced two-way contracts n Allaz and Vla s model wth call optons and compared the consequences of these assumptons, such as market effcency. Later, Bushnell 007 extended Allaz and Vla s model to an olgopoly and nvestgated the effects of forward contractng on the spot market. Su 007

4 Author: Market Clearng Mechansms under Uncertanty 4 Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! also extended Allaz and Vla s model to a case of non-symmetrc frms and concluded that forward market ncreases market effcency. Whle these models are nterestng, they fall short of analysng a stuaton lke what occurs n New Zealand. Our focus for NZTS s to address the effect of costs due to dspatch devatons, resultng from short term a few tradng perods varatons n net demand, to generator offers. There are also a number of models n the lterature that consder uncertanty n the context of two settlements. von der Fehr and Harbord 199 used a two settlement model wth multple demand scenaros and capacty constrants. They explaned how spot prces could be reduced as an effect of contract forward markets. They argued that wllngness to ncrease contract quantty and to be dspatched up to ther full capacty n most scenaros causes frms to take part n the contract market. Also on ths matter, Batstone 000 examned the problem of uncertanty n cost of generaton along wth rsk-neutral generators and rsk-averse consumers and found that generators are wllng to take advantage of the uncertanty of consumers towards the spot market. The strategc behavour of generators ncreases the uncertanty and rsk of the spot market for consumers. Shanbhag 006 nvestgates a two settlement stochastc market for a two-node model. The model he consders n hs nvestgaton s a Nash-Cournot model. Zhang et al. 010 propose a two stage olgopoly stochastc Nash-Cournot equlbrum problem wth equlbrum constrants. Ths means that ther forward market equlbrum s constraned to the spot market equlbrum as a complementarty problem. In ther model, they use ratonal rsk-neutral generators that are nvolved n a two-way contract market. The spot market s assumed to be a Nash-Cournot market. They demonstrate the exstence and unqueness of equlbrum, and use some numercal examples to show these characterstcs n a market mplementaton. None of these models caters for the recently proposed stochastc programmng market clearng model that we nvestgate n ths paper. We start by ntroducng a smplfed verson of the NZTS market currently operated n New Zealand. We wll then ntroduce a smplfed verson of the stochastc programmng mechansm for clearng electrcty markets. The frst smplfcaton n our models s to use affne supply functon offers. Green see Green 1996 used ths restrcted form of supply functons to assess the effects of polces on enhancng competton n the England and Wales market. He showed that n presence of lnear demand functons, one supply functon equlbrum s always an equlbrum over affne supply functons. Subsequently, Baldck et al. 004 extended Green s work to pecewse lnear supply functons. Day et al have also used lnear supply functons but wth specfc forms of conjectural varaton see Day et al. 00. We wll establsh that the stochastc program reduces to a two perod sngle settlement model slghtly dfferent from the NZTS model were devaton penaltes are explctly consdered by the

5 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 5 ISO. We refer to ths market clearng mechansm as ISOSP. We wll present exstence results of equlbra for the smplfed NZTS and derve an analytcal expresson for a symmetrc equlbrum. We then establsh the key result that reduces the smplfed stochastc market clearng mechansm ISOSP, to a NZTS type model, but wth explct devaton penaltes. Here agan we construct analytcal expressons for symmetrc equlbra. Fnally we compare the symmetrc equlbra of NZTS and ISOSP settlements and show that the ISOSP settlement wth explct devaton costs performs better n terms of expected socal welfare. When the assumpton of symmetry s relaxed, analytcal expressons for an equlbrum become ntractable for ether model. We wll therefore present numercal results for a number of cases and compare equlbra of the two market clearng systems. We fnd that n all cases ISOSP performs better than the NZTS. In secton 6., we repeat the symmetrc experments that construct and compare the equlbra of NZTS and ISOSP mechansms, but for the restrcted verson of lnear supply functons where the ntercept s set to zero. Here we fnd that there s a unque symmetrc equlbrum for the game. For ths case, we present an example where the ISOSP welfare s n fact less than the NZTS mechansm. Ths result s n contrast to the prevous set of results where generators have another degree of freedom n ther bds n the form of an ntercept. Secton 7 concludes the paper.. The Market Envronment In ths paper, we am to compare dfferent market desgns for electrcty. We begn by presentng assumptons that are common to all markets we consder, features of what we call the market envronment. These nclude such consderatons as the number of frms, the costs frms face, the structure of demand and so forth. Assumpton 1. The market envronment may be defned by the followng features. Electrcty s traded over a network wth no transmsson constrants and no lne losses, thus we may consder all tradng as takng place at a sngle node. 3 Demand for electrcty s uncertan, and may realze n one of s {1,, S} possble outcomes scenaros, each wth probablty θ s. Demand n state s s assumed to be lnear, and defned by the nverse demand functon p s = Y s ZC s, where C s s the quantty of electrcty and p s s the market prce, n scenaro s. Wthout loss of generalty, assume Y 1 < Y <... < Y S. We wll denote the expected value of Y s by Y = θ s sy s. The dstrbuton of demand s common knowledge to the agents. There are n symmetrc frms wshng to sell electrcty. 3 Ths assumpton may also shed lght on any scenaro where lne capactes do not bnd, even f n other scenaros they do bnd.

6 Author: Market Clearng Mechansms under Uncertanty 6 Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! For a gven frm n scenaro s, we wll denote the pre-dspatch quantty by q, and any shortrun change n producton by x,s. Thus a generator s actual producton n scenaro s s equal to q + x,s, whch we denote by y,s. Each frm s long-run cost functon s αq + β q, where q s the quantty produced by frm, and β > 0. Each frm s short-run cost functon s α q + x,s + β q + x,s + δ x,s, where q s the expected dspatch of frm, and q + x,s s the actual short-run dspatch and δ > 0. As mnmum margnal cost of generaton should not be more than maxmum prce of electrcty, we assume α Y s s {1,..., S}. There s an Independent System Operator ISO who takes bds and determnes dspatch and prces accordng to the gven market desgn. All the above assumptons are common knowledge to all partcpants n the market. We assume that the strategy space of each partcpant s defned by ther choce of lnear supply functon parameters dscussed under each market clearng mechansm. The payoff for each partcpant s determned by revenues earned from the spot sales of electrcty less the cost of producton defned above. Our assumptons on generators cost functons are partcularly crtcal to the analyss that follows, and deserve further explanaton. Generators face two dstnct costs when generatng electrcty. If gven suffcent advance notce of the quantty they are to dspatch, the generator can plan the allocaton of turbnes to produce that quantty most effcently. Ths s what we mean by a long-run cost functon. The nterpretaton of ths s the lowest possble cost at whch a generator can produce quantty q. In electrcty markets, however, demand fluctuates at short notce, and the ISO may ask a generator to change ts dspatch at short notce. In ths case, generators may not have enough tme to effcently reallocate ts turbnes. For example, many thermal turbnes take hours to ramp-up. Most lkely, the generator wll have to adopt a less effcent producton method, such as runnng some turbnes above ther rated capacty whch also ncreases the wear on the turbnes. Thus there s some nherent cost n devatng from an expected pre-dspatch n the short-run. Ths cost can be ncurred even f the requested devaton s negatve. We assume that the generator wll be unable to revert to the most effcent mode of producng ths quantty q,s + x,s n the short-run, so pays a penalty cost. Note that ths mposes a postve penalty cost upon the generator for makng the short-run change, even f the change s negatve. Ths penalty cost s addtvely mposed on top of the effcent cost of producng at the new level. We call ths

7 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 7 cost the devaton cost. Note that we assume the symmetrc case n whch cost of generaton and devaton s determned through the same constant parameters α, β, δ. Our goal s to compare the outcomes of dfferent markets mposed upon ths envronment. To be able to draw comparsons n dfferent paradgms, we need to examne the steady state behavour of partcpants under the dfferent market clearng mechansms. To ths end, we need to compute equlbra arsng under the dfferent market clearng mechansms. In order to make the computatons tractable, we wll restrct the frms to offer lnear supply functons n the followng sectons of ths paper. 3. Determnstc Two Perod Settlement NZTS Model In ths secton we wll ntroduce a determnstc two perod market whch s nspred by the market clearng mechansm as t operates currently n New Zealand. As explaned n the ntroducton, n the NZEM frms bd a step supply functon for a gven half hour perod. The bd s made at least two hours n advance. Once gate closure occurs two hours n advance of any gven perod, the supply functon offers can not be changed. The market wll then be cleared sx tmes, every fve mnutes durng the gven half hour perod. Each fve mnute re-dspatch s computed wth real tme demand, but wth the supply offer stacks that have been submtted pror to gate closure. 4 We smplfy the stuaton by assumng the market clears only twce; once after the offers are submtted, but before demand s realzed. Ths we call the pre-dspatch phase whch tells the generators approxmately how much they should produce. Once demand s realzed, the same offers wll be used to determne actual dspatch n what we call the spot settlement. The dfference between pre-dspatch and spot dspatch s a generator s short-run devaton, whch s subject to potentally hgher costs as we descrbed earler however the ISO has no knowledge of ths cost and t s not explctly stated n the generators bds. Ths cost can be ndrectly reflected n the supply functons the generators bd n Mathematcal Model Our smplfed mathematcal model for the NZTS market has two dstnct stages; pre-dspatch and spot. Followng the large body of lterature on affne supply functons see e.g. Green 1996, Baldck et al. 004 we wll work wth a lnear demand curve and frame generator supply offers as lnear functons. Explctly, each generator bds a supply functon a + b q before the pre-dspatch market to represent ther quadratc costs. Ths supply functon s requred to be ncreasng,.e. the 4 Wthn ths fve mnute perod a frequency keepng generator wll match any small changes n demand. We gnore ths aspect of the market, as frequency-keepng s purchased through a separate market and untl recently was procured through annual contracts.

8 Author: Market Clearng Mechansms under Uncertanty 8 Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! offered b must satsfy b ε > 0, where ε s the machne epslon. Note that unlke Green 1996, we do not assume that the ntercept a = 0. Followng our man analyss, we wll present a bref specal case where a = 0 s requred. When generators lock n ther offers demand s uncertan n New Zealand ths pont n tme s referred to as gate closure. The ISO wll then use the generator s bd twce: once to clear the pre-dspatch market, and once agan after demand s realzed to clear the spot market. The predspatch market determnes the pre-dspatch quanttes each generator s asked to dspatch, and the spot market determnes the fnal quanttes the generators are asked to dspatch. As n realty, n both the pre-dspatch and spot markets, the ISO ams to maxmze socal welfare, assumng generators are bddng ther true cost functons. Snce demand s unknown n pre-dspatch, the ISO wll nomnate and use an expected demand and wll not consder the dstrbuton of demand. mn z = n =1 a q + b n s.t. q =1 Q = 0 q Y Q Z Q [f] 1 From ths frst settlement, the ISO can extract a forward prce f equal to the shadow prce on the expected demand balance constrant. f s not used for any settlements as the pre-dspatch quntty and prces are merely a gude at ths stage. Recall that the pre-dspatch quantty for generator s denoted by q. After pre-dspatch s determned, true demand s realzed, and the ISO then clears the spot market usng the specfc demand scenaro that has been realzed to maxmze welfare by solvng. mn z = n =1 a y,s + b,s y Ys C s Z C s n s.t. y =1,s C s = 0 [p s ] Here agan the ISO computes a spot prce p s as the shadow prce on the constrant. Note that we can elmnate the constrant and substtute C s n the objectve, however mposng ths constrant enables the easy ntroducton of the prce as the shadow prce attached to the constrant. The generator s not permtted to change ts bd after pre-dspatch, but does face the usual addtonal devaton cost δ for ts short-run devaton. Note that n both ISO optmzaton problems 1, we have dspensed wth non-negatvty constrants on the pre-dspatch and dspatch quantty respectvely. We wll demonstrate that the resultng symmetrc equlbra of our NZTS market model wll always have assocated non-negatve pre-dspatch and dspatch quanttes. We have elmnated the non-negatvty constrants followng the conventon of supply functon equlbrum models see e.g. Klemperer and Meyer 1989a, Bolle 199 n order to enable the analytc computaton of equlbrum supply offers.

9 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 9 Frm s proft n scenaro s n ths market s then gven by u T S,s q, x,s = p s y,s αy,s + β y,s + δ y,s q Equlbrum Analyss of the Determnstc Two Perod Market In ths secton we wll present equlbra of the NZTS market model. We wll frst compute the optmal dspatch quanttes from the ISO s optmal dspatch problems 1 and for any number of players. We wll then embed these quanttes n each generator s expected proft functon and allow the generators to smultaneously optmze over ther lnear supply functon parameters to obtan equlbrum offers. Proposton 1. Problem 1 s a convex program wth a strctly convex objectve. Its unque optmal soluton and the correspondng optmal dual f are gven by f = Y + ZA ZB + 1 q = fb A where A = a b, B = 1 b, A = n =1 A and B = n =1 B. Proof. Note that problem 1 has a sngle lnear constrant and that ts objectve s a strctly convex quadratc as we have assumed that b > 0 and Z > 0. The problem therefore has a unque optmal soluton delvered by the frst order condtons provded below. Q q = 0 4 f Y + ZQ = 0 5 f + a + b q = 0 6 Algebrac manpulaton of equatons 5 6 wll provde the results. We note that ths s smlar to Green s analyss n Green 1996, although we allow for an ntercept parameter as well. Proposton. For each scenaro s, problem s a convex program wth a strctly convex objectve. Its unque optmal soluton and the correspondng optmal dual p s, are gven by p s = Y s + ZA ZB + 1 y,s = p s B A where A, B, A and B are defned above n proposton 1.

10 Author: Market Clearng Mechansms under Uncertanty 10 Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! Proof. Problems and 1 are structurally dentcal, therefore the smple proof of proposton 1 apples agan here. Remark 1. Note from the above that the pre-dspatch prce and quantty are equal to the expected spot market prces and quanttes respectvely. That s f = θ s p s. 7 We wll now compute the lnear supply functons resultng from the equlbrum of the TS market game lad out n 1. Before we begn wth the frm computatons, we wll establsh a techncal lemma that we utlze n establshng the equlbrum results. Lemma 1. Assume that functon fx, y : R R s defned on a doman D x D y wth D x, D y R. Furthermore assume that x y D x, maxmzes fx, y for any arbtrary but fxed y. Also assume gy = fx y, y s maxmzed at y D y. Then, fx, y s maxmzed at x y, y Frm s computatons In ths secton we wll focus on frm s expected proft functon. Note that usng equaton 7 we obtan u T S = E s [u T S,s ] = Usng propostons 1 and, we can re-wrte u T S a maxmum of u T S θ s p s y,s αy,s + β y,s + δ y,s q. as a functon of a and b. In order to fnd for a fxed set of compettor offers we appeal to a transformaton that wll yeld concavty results for u T S. We consder u T S b. Note that the transformaton A = a b, to be a functon of A and B nstead of a and B = 1 b s a one-to-one transformaton. Proposton 3. Let all compettor lnear supply functon offers be fxed. The followng maxmzes u T S and s therefore frm s best response. B = where A = j A j and B = j B j. Proof. We can show that u T S 1 + ZB Z + β + δ + Zβ + δb A = α + B Zα δ Y + ZA + ZαB Z + β + ZβB Here we have dspensed wth the expresson for u T S s a concave functon of A, assumng B s a fxed parameter. as a functon of A and B as t s long and rather complcated. Ths expresson can be found n the onlne techncal companon Khazae et al. 013a. We note that u T S s a smooth functon of A and B and that

11 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 11 u T S A Let B be arbtrary but fxed. As u T S = 1 + ZB Z + β + ZβB 1 + ZB 0 s a concave functon of A the frst order condton yelds an expresson for A B, the value of A that maxmzes u T S for the fxed B. A B = 1+ZB Y +α ZA +ZαB +B ZY +ZA +Zα+βY +ZβA ZB ZB Z+β+ZβB We can embed A B nto u T S and fnd the maxmzer n terms of B. Lemma 1 then can be appled to demonstrate that the end result delvers the maxmum of u T S. u T S After embeddng ths value of A s B = nto the proft functon, the dervatve wth respect to B of du = Y s θ sys 1 + Z + β + δb + Z 1 + β + δb B. db 1 + ZB 3 1+ZB Z+β+δ+Zβ+δB, s the zero of ths dervatve. Recall that Y = s θ sy s, therefore Jensen s nequalty mples Y s θ sy s 0. Thus, du db 0, when B < B, and du db 0, when B > B. In other words, u s a quas-concave functon of B and s maxmzed at B = B. Note that evaluatng A at B yelds A = α + B Zα δ Y + ZA + ZαB Z + β + ZβB. From the above, we can obtan the equlbrum of the NZTS model by solvng all best responses smultaneously. Ths gves the unque and symmetrc soluton or alternatvely S-EQM: B = n Z + β + δ + 8 n Z + nzβ + δ + β + δ A = α + nzα Y δb Z + β + n 1Zβ + δb, 9 S-EQM: b = n Z + β + δ + n Z + nzβ + δ + β + δ 10 αb + nzα Y δ a = Zb + βb + n 1Zβ + δ, 11 As we dscussed earler, these equlbrum offers yeld non-negatve pre-dspatch and dspatch quanttes. Below we formally state ths result, however the computatons to show the nonnegatvty of these quanttes can be found n the techncal companon Khazae et al. 013a.

12 Author: Market Clearng Mechansms under Uncertanty 1 Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! Proposton 4. The equlbrum pre-dspatch and spot producton quanttes of the frms n the NZTS market are non-negatve,.e.q 0, and y,s 0, s where q and y,s are the optmal solutons to problems 1 and respectvely usng the equlbrum parameters from 10 and 11. Proof. For the proof please consult the techncal companon Khazae et al. 013a. 4. Stochastc Settlement Market 4.1. ISOSP Model We now ntroduce the market model we wll use to analyze a stochastc settlement market. As dscussed n the ntroducton, the stochastc settlement market contans only a sngle stage of bddng, but the market clearng procedure takes nto account the dstrbuton of future demand when determnng dspatch. The market works as follows. When the market opens, demand s uncertan. Frms are allowed to bd ther normal cost functons the cost of producng a gven output most effcently and a penalty cost functon that they would need to be pad to devate n the short-run. Snce frms have quadratc cost functons, they can bd ther actual costs by submttng a lnear supply functon. Each frm chooses a and b to bd the lnear supply functon a + b q, and d to bd the margnal penalty cost d q. For a margnal devaton penalty, we look for a functon that s zero when the dspatch s not changed from the pre-dspatch quantty, as well as postve to the rght and negatve to the left of the pre-dspatch. One of the smplest form ths functon can take s the lnear form we have assumed. Whle the true margnal cost of devaton for a staton may be non-lnear, t s expected to be smooth as t relates to engneerng attrbutes such as flow of water through an aperture. Therefore to the frst order, t can be approxmated by a lnear functon. Note that as wth the NZTS model, these bds a, b, d need not be ther true values α, β, δ. The offered b s requred to be postve and d should be non-negatve. After generators have placed ther bds, the ISO computes the market dspatch accordng to the stochastc settlement model outlned below. At ths pont demand s stll uncertan. The ISO chooses two key varables. The frst s the pre-dspatch quantty for each frm. Ths s the quantty the ISO asks each frm to prepare to produce, namely the pre-dspatch quanttes q defned n Secton. The second s the short-run devaton for generator under each scenaro s. Ths devaton s the varable x,s defned n Secton, representng the adjustment made to frm s predspatch quantty n scenaro s. The ISO can choose both pre-dspatch and short-run devatons smultaneously, whle amng to maxmze expected socal welfare. The ISO assumes that generators have bd ther true costs. In the fnal stage, demand s realzed, and the ISO wll ask generators to modfy ther pre-dspatch quantty accordng to the short-run devaton for the partcular scenaro. Each generator ends up

13 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 13 wth producng q + x,s. Two prces are calculated durng the course of optmzng welfare. The frst s the shadow prce of the pre-dspatch quanttes. We wll denote ths by f. The second are the prces of each of the devatons, for each of the scenaros. We wll denote these by p s for scenaro s. Each generator s pad f per unt for ts pre-dspatch quantty q, and p s for ts devatons x,s. Thus n realzaton s, generator makes proft equal to u SS,s q, x,s = fq + p s x,s α q,s + x,s + β q,s + x,s + δ x,s. 1 Mathematcally, the stochastc optmzaton problem solved by the ISO can be represented as follows. 5 ISOSP: mn z = S θ n [ s =1 a q + x,s + b q ] + x,s + d x,s Ys C s Z C s s.t. q Q = 0 [f] Q + x,s C s = 0 s {1,..., S} [θ s p s ] Q and C s stand for the total contracted or pre-dspatched quantty and total consumpton n scenaro s respectvely. Note that we could have elmnated the two equalty constrants. However, ther dual varables are the market prces f and p s respectvely, so for clarty we have left them n. Note that n the ISOSP, an estmate of a future dstrbuton s used. Khazae et al. 013b examne the effectveness of ISOSP n an emprcal compettve settng. 4.. Characterstcs of the Stochastc Optmzaton Problem We begn by presentng a seres of results that smplfy the set of solutons to the ISOSP problem. We start by establshng techncal lemmas that enable us to prove that out ISOSP s equvalent to a two perod market clearng mechansm smlar to NZTS, wth the essental dfference that now a devaton penalty s present n the ISO s dspatch n real tme. These results drastcally smplfy the subsequent analyss of frms behavour n equlbrum. Lemma. In the stochastc settlement market clearng, the expected devaton of frm from predspatch quantty q s zero, that s, the optmal soluton to ISOSP wll always satsfy θ s x,s = 0. s 5 Ths s a modfed verson of Prtchard et al. s problem. There s only one node and thus no transmsson constrants, and demand s elastc.

14 Author: Market Clearng Mechansms under Uncertanty 14 Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! Proof. Let us assume q and x,s form ISOSP s optmal soluton. Let us defne for each and s the quantty k,s = q + x,s, the total producton of frm n scenaro s. Note that C s = q + x,s. Assume, on the contrary, that there exsts at least one frm j such that θ s sx j,s = 0. The optmal objectve value of ISOSP s then gven by s θ s a k,s + b k,s + Note that as s θ sx j,s = 0, the term gven from above, consder the problem d θ s x,s + Y s k,s Z k,s. 13 s θ s s d x,s s postve. Now, for a fxed and k,s mn w = d q,x,s θ s x,s Ths problem clearly reduces to the unvarate problem q + x,s = k,s s. 14 whch s optmzed at Defne ˆq and xˆ,s by mn q w = q = θ s k,s q, θ s k,s. { q ˆq =, = j S θ sk j,s otherwse, and { x xˆ,s =,s, = j k j,s ˆq j otherwse. By defnton, ˆq + xˆ,s = q + x,s for all and s. It s easy to see that the quanttes ˆq and xˆ,s yeld a feasble soluton to ISOSP satsfyng 14. Furthermore, the objectve functon evaluated at ˆq and xˆ,s s gven by θ s a k,s + b k,s + Y s k,s Z s k,s. Ths value s strctly less than the objectve evaluated at q and x,s gven by 13, as we have already establshed that θ s s d x,s > 0. Ths yelds the contradcton that proves the result. Corollary 1. In the stochastc problem ISOSP, f q + x,s 0 s satsfed s {1,..., S} then q 0 wll hold.

15 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 15 Proof. In Lemma we establshed that s θ sx,s = 0. Therefore there exsts a scenaro s such that x,s 0. Clearly then q + x,s 0 mples q 0. Dscusson Lemma s the crucal result that drves the rest of our characterzatons. Ths result hnges on the fact that we penalze quadratc devaton from the pre-dspatch quantty. In the proof of Lemma, we demonstrate that the second stage of ISOSP reduces to selectng a contract pont that mnmzes the quadratc devaton penalty functon whch s known to be the mean of any dstrbuton. For the quadratc penalty ths s rrespectve of the dstrbuton. It s possble to also use an absolute value based devaton penalty and requre a symmetrc demand dstrbuton. When the penalty functon s an absolute value, the pont of best estmate s the medan of the dstrbuton. For a symmetrc dstrbuton of course ths reduces agan to the mean. Ths model penalzes the devatons upward and downward dentcally. Therefore the predspatch pont s optmzed based on the mean demand scenaro. The reader may argue that allowng for dfferent upward and downward penaltes s more realstc. However as Prtchard et al. 010 show, such allowance of asymmetrc penaltes can lead to systematc arbtrage by the ISO, where a generator may be requred to devate upward n every scenaro smply to ncrease expected welfare. Ths s undesrable for a market clearng mechansm. We have therefore confned our attenton to the symmetrc upward and downward penalty case for ths paper, whch guarantees systematc arbtrage wll not occur. We now use the above results and ntuton to prove that the ISO s optmzaton problem can be vewed as a determnstc two perod settlement system where unlke NZTS, the devaton penaltes are explctly stated n the ISO s problem n the second perod. Lemma 3. The objectve functon of ISOSP s equvalent to the followng functon whch s separable n the pre-dspatch and the spot market varables n z = a q + b n n q Y q + Z q =1 =1 =1 n b + d n + θ s x,s θ s Y s x,s + Z n θ s x,s. =1 =1 =1 Proof. Substtutng for C s from constrants nto the objectve functon of ISOSP yeld n z = a q + b n n q Y q + Z q =1 =1 =1 n b + d n + θ s x,s θ s Y s x,s + Z n θ s x,s =1 =1 =1 n n n n + a θ s x,s + q b θ s x,s + θ s Z q x j,s =1 =1 =1 j=1

16 Author: Market Clearng Mechansms under Uncertanty 16 Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! We have splt the objectve n three parts above. Note that the frst part of the objectve above s exclusvely a functon of pre-dspatch quanttes q and the second only a functon of the spot dspatches x,s. We rewrte the thrd segment to make obvous that t s zero at optmalty. n z = a q + b n n q Y q + Z q =1 =1 =1 n b + d n + θ s x,s θ s Y s x,s + Z n θ s x,s =1 =1 =1 n n n n + a θ s x,s + q b θ s x,s + Z q θ s x j,s =1 =1 =1 j=1 Recall from Lemma that S θ sx j,s = 0 for the optmal choce of real tme dspatches. Therefore we can elmnate the part of the objectve. Ths completes the proof. Note: We have therefore establshed that ISOSP reduces to a determnstc two perod sngle settlement model very smlar to NZTS but wth penaltes d explctly present n the second perod. The rest of ths secton s devoted to dervng explct expressons for the soluton of ISOSP. In the next secton we wll use these expressons to arrve at best response functons for the frms and subsequently n constructng an equlbrum for the stochastc market settlement. In order to smplfy the equatons and arrve at explct solutons, we wll transform the space of the parameters of ISOSP.e. the frm decson varables, much n the same way that we dd n Secton 3. If we further defne R = 1 b +d and R = R, ISOSP reduces to mnmzng the followng: n A z = q + 1 n n q Y q + Z q B =1 B =1 =1 n + θ s 1 n n x,s Y s Y x,s + Z x,s. R =1 Note that as before Lemma 3, the problem s separable n q s and x,s s, we can therefore solve the two stages separately. Note also that the problem n each stage s a convex optmzaton problem, therefore the frst order condtons wll readly produce the optmal soluton. Proposton 5. If q, x, f, p represents the soluton of ISOSP, then we have =1 =1 q = Y + ZAB 1 + ZB A 15 x,s = Y s Y R 1 + ZR f = Y + ZA 1 + ZB p s = Y + ZA 1 + ZB + Y s Y 1 + ZR 16

17 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 17 Proof. For dervaton of the expressons for the optmal soluton above from frst order condtons please refer to the techncal companon Khazae et al. 013a. Observe from the expresson for f that ths forward prce pad on pre-dspatch quanttes s ndependent of any devaton costs n the spot market. Corollary. In the soluton of ISOSP, forward prce s equal to the expected spot market prce. Proof. Ths s smply observed from proposton 5. Note that Y = s θ sy s. The fact that the contract prce f s equal to the expected spot market prce, mples that there s no systematc arbtrage Equlbrum Analyss of the Stochastc Settlement Market In Secton 4.1 we presented frm s proft under scenaro s n equaton 1. In our market model, we assume that all frms are rsk neutral and therefore nterested only n maxmzng ther expected proft. Frm s expected proft s gven by u = fq + θ s p s x,s αq + x,s + β q + x,s + δ x,s The above expresson for u can be expanded and we can observe that u = fq αq + β q + θ s p s x,s β + δ x,s α θ s x,s βq θ s x,s, 17 Note that from Lemma, the generator would know that for any admssble bd, the correspondng expected devaton from pre-dspatch quanttes S θ sx,s = 0. Therefore the expected proft for the generator becomes u = fq αq + β q + θ s p s x,s β + δ x,s. We can use the expressons obtaned from proposton 5 to wrte u as follows. u = 1 βa + A ZA + α + ZBα + ZAβB + Y 1 + βb 1 + ZB ZB 1 + ZR 1 + ZR ZA + Y ZA + Y 1 + ZBαB 1 + ZR ZA + Y βb +1 + ZB R + β + δr Y s θ s Y s 18

18 Author: Market Clearng Mechansms under Uncertanty 18 Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! Although ths expresson of the expected proft for the generator s rather ugly, t does have the advantage that upon dfferentatng wth respect to R, all dependence on A and B drops and we are left wth du = Y s θ sys 1 + Z + β + δr + ZR 1 + β + δr. 19 dr 1 + ZR 3 Recall that R = j R j. For verfcaton of ths dervatve term see the techncal companon Khazae et al. 013a. The fact that ths dervatve s free of A and B ndcates that u s separable n R and A, B, that s u A, B, R = g A, B + h R. 0 Due to ths natural separablty, our equlbrum analyss wll focus on fndng best responses n terms of A, R and B, very smlar to the NZTS secton. Equaton 0 enables us to maxmze u by maxmzng g and h over A, B and R respectvely. Ths s helpful as we can establsh quas-concavty results for g and h separately. We start our nvestgatons by examnng g. The full expresson for g techncal companon Khazae et al. 013a. Holdng B fxed, note that d g da = 1 + ZB Z + β + ZβB 1 + ZB. can be found n the Ths demonstrates that g s concave n A for any fxed B. Furthermore, for any fxed B, we can use the frst order condtons to fnd A B,.e. the value of A that maxmzes g A, B for the fxed B. A B = 1 + ZB α ZA + ZαB Y + Y + ZA Z + β + ZβB B 1 + ZB Z + β + ZβB 1 To fnd the optmal value for g, we can now appeal to Lemma 1 and substtute the expresson for A B n g A B, B. Surprsngly, upon undertakng ths substtuton, t can be observed that g A B, B s a constant value. Fgure 1 depcts g. To uncover the ntuton behnd ths feature of g, we can offer the followng mathematcal explanaton. We observe that and that dg da = 1 + ZB Y α + ZA + Z + βa + ZB α + βa 1 + ZB + ZY + α + Y β + Y Zα + Y βb + ZA Z + β + ZβB B 1 + ZB dg db = Y + ZA 1 + ZB. dg da. Therefore, statonary condtons enforced n A wll also mply statonarty n B.

19 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 19 Fgure 1 Two vews of the functon g. Note that the optmal value of g s obtaned along a contnuum, for any value of B. As g A B, B s constant for any B > 0, for any value of B > 0, the tuple A B, B s an argmax of g for any postve B. Let D denote 1 d +ε. Recall that accordng to our ntal assumptons, we have b ε, thus b + d ε + d > b + d ε + d R D. The followng analyss on h wll explan how optmal R s constraned by the value of D. Proposton 6. Suppose that R s fxed. Then h s optmzed at R = mn{d, 1 + ZR Z + β + δ + Zβ + δr } Proof. Note that at ˆR = 1 + ZR Z + β + δ + Zβ + δr, the dervatve dh dr = du dr vanshes. Also recall from Jensen s nequalty that Y s θ sy s. It can therefore be seen from 19 that ths dervatve s postve for R < ˆR and negatve for R > ˆR. Recall further that the defntons of D and R requre R D. Therefore, n optmzng h, we need to enforce ths constrant and we obtan R = mn{d, 1 + ZR Z + β + δ + Zβ + δr }.

20 Author: Market Clearng Mechansms under Uncertanty 0 Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! We now return to u, the expected proft functon for frm. As u A, B, R = g A, B +h R, we can start by obtanng the maxmum value of g attaned at a pont A B, B for any postve B. Subsequently, we proceed to optmze h R. Proposton 6 readly delvers the optmal R. We have therefore proved the followng theorem. Theorem 1. The best response of frm, holdng compettor offers fxed, s to offer any d whch we have 1 + ZR D. Z + β + δ + Zβ + δr For any such d, optmal a and b can be computed from the followng equatons. for 1 + ZR R = Z + β + δ + Zβ + δr 1 B = 1 R d A = 1 + ZB α ZA + ZαB Y + Y + ZA Z + β + ZβB B 1 + ZB Z + β + ZβB Theorem 1 ndcates that the game has multple nfnte symmetrc equlbra. To prevent the problem of unpredctablty, caused by multple equlbra, from here on we assume that the ISO chooses d = d SO as a system parameter. Ths parameter s dentcal for and known to all partcpants. Ths also provdes the ISO wth the opportunty to choose d SO n a way to obtan a preferable equlbrum.e. an equlbrum that yelds hgher socal welfare. Proposton 7. The unque symmetrc equlbrum quanttes of the stochastc settlement market are as follows. b = max{ε, Zn + β + δ + Z n + Znβ + δ + β + δ d SO } 3 a = α Y + B ZY n n 1α + Y β + Zn 1Znα + Y βb B Zn β + Y n 1Zn + βb 4 The proof of the above proposton s contaned n the techncal companon Khazae et al. 013a. Let us defne ˆd = Zn + β + δ + Z n + Znβ + δ + β + δ. Theorem. In a stochastc settlement market wth d SO ˆd ε, as the number of partcpatng frms ncreases, they tend to offer ther true cost parameters. In other words, lm a = α n lm b + d SO = β + δ. n When the fxed parameter d SO s chosen equal to δ, lm n b = β.

21 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 1 Proof. The equatons are smply derved from the equlbrum values of a and d gven n proposton 7. Theorem shows that the ISOSP market s behavng compettvely n the sense that when number of frms ncreases, they offer ther true cost parameters. One mportant feature of the equlbrum values are the non-negatvty of the pre-dspatch and dspatch. Ths s mportant, because we neglected the non-negatvty constrants n ISOSP n the frst place. Proposton 8. Let q, x represent an equlbrum of the stochastc settlement market, then the followng nequaltes hold., s : q + x,s 0 : q 0 The proof of the above proposton s contaned n the techncal companon Khazae et al. 013a. Though, the equlbrum pre-dspatch and dspatch are non-negatve, one mght rase an objecton that a game wthout the non-negatvty constrants embedded n the ISO s optmzaton problem, s dfferent from the orgnal game. Therefore, there s no assurance the found equlbrum s also the equlbrum of the orgnal game. The followng theorem states that the obtaned equlbrum values are also the equlbrum of the orgnal game wth non-negatvty constrants. The proof of ths theorem s qute lengthy and conssts of several techncal lemmas. Ths proof can be found n the techncal companon Khazae et al. 013a. Theorem 3. The equlbrum of the symmetrc stochastc settlement game wthout the nonnegatvty constrants n ISOSP s also the equlbrum of the stochastc settlement game wth the non-negatvty constrants. Proof. Please refer to the techncal companon Khazae et al. 013a for the proof of ths theorem. Thus far we establshed that under the assumpton of symmetry, the stochastc settlement ISOSP market s equvalent to a two perod determnstc settlement n whch the devaton penaltes are explctly present n the second perod DTS. We then proceeded to derve an analytcal symmetrc equlbrum expresson for ISOSP. In ths process, we enhanced the defnton of our game to avod multple equlbra and allow the ISO to set a devaton penalty for the symmetrc players n ths game. The ssue of multple equlbra arses as there are multple optmal solutons to the best response problem. Specfcally, for any choce of b and d, so long as R = 1 b +d = 1ˆd, we obtan an optmal soluton subject to boundary condtons of course.

22 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! Whle n the context of our computatons, due to the natural decomposton of u, t was natural to treat d as the free varable, our ntenton has been to compare the NZTS mechansm wth the ISOSP market clearng proposed. As we observed that ISOSP s equvalent to DTS, t would make sense to thnk of a game where the ISO mposes the devaton penalty on all partcpants by choosng d = d SO 0. If we thnk of the ISO choosng 0 d SO ˆd, announcng d SO to all partcpants and mposng ths value as the devaton penalty n the second stage, the resultng game, along wth ts symmetrc equlbrum, s equvalent to the game where the ISO selects b SO = ˆd d SO. Here observe that f ISO selects d SO = δ, then as the number of partcpants ncreases, n the symmetrc equlbrum we obtan b β and a α. Furthermore, t s clear that f d SO = 0, then the equlbra for NZTS are recovered. 5. Comparson of the Two Markets We are nterested n the performance of the two market clearng mechansms ISOSP and NZTS, under strategc behavour. Our crteron for comparng the two models s socal welfare. Socal welfare s defned as the sum of the consumer and producer welfare and n our market envronments ths reduces to W = θ s Y s n =1 θ s n =1 y,s Z n =1 y,s αy,s + β y,s + δ y,s q. 5 Note that the dfferent socal welfare values W SS for the ISOSP and W NZT S for the NZTS mechansm are found through the same formula, however wth the dfferent equlbrum y,s varables. Recall that followng Theorem 1 the choce of d SO was delegated to the ISO. The next theorem establshes that when frms are bddng strategcally, the stochastc settlement market domnates the NZTS market for any choce of d SO 0, ˆd. Theorem 4. The socal welfare of ISOSP s hgher than the NZTS market provded the parameter d s chosen less than or equal to the threshold value ˆd ε. Proof. To prove the proposton, we show when d SO = 0, then W SS = W NZTS. We then demonstrate W SS s a ncreasng functon of d SO for 0 d SO ˆd, and therefore, W SS W NZTS, when d SO ˆd note that W NZTS s a constant and does not change wth d SO. When d SO = 0, equatons 4, 9, and 8 yeld that the equlbrum quanttes are dentcal n the stochastc settlement and determnstc two perod settlement markets. That s

23 Author: Market Clearng Mechansms under Uncertanty Artcle submtted to Operatons Research; manuscrpt no. Please, provde the manuscrpt number! 3 B SS A SS R SS = B NZTS = A NZTS = B SS Here we can smplfy the expressons for y,s and q from propostons 1,, and 5 to obtan q SS = q NZTS = Y B A 1 + ZB y SS,s = y NZTS,s = Y sb A 1 + ZB Therefore socal welfare of these models equaton 5 are the same provded b = ˆb. We can rewrte the socal welfare expresson 5 as W = θ s Y s n =1 n y,s Z y,s =1 n =1 αy,s + β y,s + δ x,s. 6 Note that the expresson for socal welfare s the same for both models and only depends on the correspondng quanttes dspatched from each model.e. y SS,s vs y NZT S,s Note that, accordng to the results of proposton 7, for 0 d SO ˆd ε, we have b = ˆd d SO, and therefore R has a constant value of 1/ ˆd, whle B s a functon of d SO. Furthermore, note that x SS,s s ndependent of d SO, and therefore, dw SS dd SO = 1 ˆd d SO On the other hand, takng the dervatve of y SS,s,s dw SS dy SS,s etc. dy SS,s db. 7 wth respect to B we obtan dy,s SS Y αn 1Z = db Z + nz + β + n 1ZnZ + βb 0. The rght hand sde s readly seen to be non-negatve as Y > α and n > 1. As dyss,s db s ndependent of frm and scenaro s note that B s chosen by the ISO and fxed to a sngle parameter for all frms, we can re-arrange 8 and obtan dw SS dd = 1 dy,s SS SO ˆd d SO db On the other hand, dfferentatng 6 yelds,s dw SS. dy,s SS