Market Clearing Mechanisms under Demand Uncertainty

Transcription

1 Market Clearng Mechansms under Demand Uncertanty Javad Khazae, Golbon Zaker, Shmuel Oren June, 013 Abstract Electrcty markets face a substantal amount of uncertanty. Tradtonally ths uncertanty has been due to varyng demand. Wth the ntegraton of larger proportons of volatle renewable energy, ths added uncertanty from generaton must also be faced. Conventonal electrcty market desgns cope wth uncertanty by runnng two markets: a day ahead or pre-dspatch 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 markets, the day ahead market 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. [5] and Bouffard et al. []. In such a market, the ISO co-optmzes pre-dspatch and spot n one sngle settlement market. In ths paper we consder smplfed models for three types of market clearng mechansms. 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 a TS model that dspenses wth devaton penaltes and has been and contnues to be n use n New Zealand NZTS. 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. 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. 1

2 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 common approach used s to run a determnstc two-settlement model. In the frst perod, usually run about 4 hours before clearng, generators make offers, and the ISO chooses a pre-dspatch. Note that n the remander of ths paper, we use the terms generator and frm nterchangeably. Ths frst market s run based on an estmate of what demand s expected to be, then a second balancng market s run soon before the market actually clears. In ths second market, new sets of offers are submtted and upon market clearng the dspatches can devate from pre-dspatch levels. Both perod s markets are based on that descrbed above - maxmzng socal welfare, but they are run separately, and the result of one s not ted to the other. However the two markets are fnancally bndng hence the term two settlement. The results of the frst pertan to pre-dspatch prces and quanttes, whle the results of the second are used for balancng prces and quanttes. Another opton s used n New Zealand. Here generators can place offers for a gven half hour perod up to two hours pror to a desgnated perod. Durng the actual half hour, the ISO wll then run an optmzaton problem every fve mnutes, usng the same bds each tme, to fgure out dspatch. Any generator may then be asked to devate at 5 mnutes notce. Note that n ths case, the same offer curves are used n the pre-dspatch phase as well as the actual half hour n queston. The predspatch market n New Zealand provdes generators wth an dea of what they mght expect to be producng but t s not fnancally bndng. Hence there s only a sngle settlement n the New Zealand system. 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. An alternatve to determnstc settlement systems s to use a stochastc settlement process. 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. By co-optmzng both together, we mght 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., Wong and Fuller, and Prtchard et al. [, 5, 6]. 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. 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. We wll establsh that the stochastc program reduces to a two

3 perod sngle settlement model slghtly dfferent from the NZTS model were devaton penaltes are explctly consdered by the ISO. We refer to ths market clearng mechansm as ISOSP. We wll present results pertanng to the exstence 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 that s the 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. Secton 6 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. 1 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 = s θ sy s. There are n symmetrc frms wshng to sell electrcty. For a gven frm n scenaro s, we wll denote the pre-dspatch quantty by q, and any short-run 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}. 1 Ths assumpton may also shed lght on any scenaro where lne capactes do not bnd, even f n other scenaros they do bnd. 3

4 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. 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 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. In the New Zealand market, frms bd a step supply functon for a gven half hour perod. The bd s made at least two hours n advance. The market wll then be cleared sx tmes, every fve mnutes durng the gven half hour perod. We smplfy the stuaton by assumng the market clears only twce; once after the bds are submtted, but before demand s realzed. Ths we call the pre-dspatch settlement whch tells the generators approxmately how much they should produce. Once 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. 4

5 demand s realzed, the same bds 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. 3.1 Mathematcal Model Our smplfed mathematcal model for the NZTS market has two dstnct stages; pre-dspatch and spot. Each generator bds a supply functon a + b q before the pre-dspatch market to represent ther quadratc costs. At ths pont, demand s uncertan. 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 pre-dspatch market determnes the predspatch 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 a q + b q s.t. n q Q = 0 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. 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 a y,s + b y,s Ys C s Z C s s.t. n y,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. [4, 1] n order to enable the analytc computaton of equlbrum supply offers. Frm s proft n scenaro s n ths market s then gven by 5

6 u TS,s q, x,s = fq + p s y,s q 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 3.1 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 A and B = n 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 Usng equaton 5 we can rewrte equaton 6 as Y ZQ = a + b q 7 Now summng over all we obtan q = 1 b Y ZQ a b 8 Note that B = 1 b and A = A. Ths together wth equatons 8 and 4 yelds Q = BY A ZB + 1. Now substtutng Q from the above nto equaton 5, we obtan f = Y + ZA ZB

7 Smlarly substtutng Q nto 7 yelds Ths equaton smplfes to q = B Y Z BY A ZB + 1 A. q = fb A, and we obtan the expressons n the statement of the proposton. Proposton 3. 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 = Y s + ZA ZB + 1 = p s B A where A, B, A and B are defned above n proposton 3.1. Proof Problems and 1 are structurally dentcal, therefore the smple proof of proposton 3.1 apples agan here. Remark 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. 9 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 3.3 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. Proof Note that for any x, y D x D y, fx, y fx y, y by the assumpton on x y D x. Furthermore fx y, y fx y, y. Clearly then fx, y fx y, y for any x, y D x D y. 7

8 3..1 Frm s computatons In ths secton we wll focus on frm s expected proft functon. Note that usng equaton 9 we obtan u TS = E s [u TS,s ] = θ s p s y,s αy,s + β y,s + δ y,s q. Usng propostons 3.1 and 3., we can re-wrte u TS as a functon of a and b. In order to fnd a maxmum of u TS for a fxed set of compettor offers we appeal to a transformaton that wll yeld concavty results for u TS. We consder u TS to be a functon of A and B nstead of a and b. Note that the transformaton A = a B = 1 b s a one-to-one transformaton. Proposton 3.4 Let all compettor lnear supply functon offers be fxed. The followng maxmzes u TS and s therefore frm s best response. b, B = A 1 + ZB Z + β + δ + Zβ + δb where A = j A j and B = j B j. = α + B Zα δ Y + ZA + ZαB Z + β + ZβB Proof We can show that u TS s a concave functon of A, assumng B s a fxed parameter. Here we have dspensed wth the expresson for u TS as a functon of A and B as t s long and rather complcated. Ths expresson can be found n the onlne techncal companon [3]. We note that u TS s a smooth functon of A and B and that u TS A = 1 + ZB Z + β + ZβB 1 + ZB 0 Let B be arbtrary but fxed. As u TS s a concave functon of A the frst order condton yelds an expresson for A B, the value of A that maxmzes u TS for the fxed B. A B = 1+ZB Y +α ZA +ZαB +B ZY +ZA +Zα+βY +ZβA ZB +1 1+ZB Z+β+ZβB. We can embed A B nto u TS and fnd the maxmzer n terms of B. Lemma 3.3 then can be appled to demonstrate that the end result delvers the maxmum of u TS After embeddng ths value of A nto the proft functon, the dervatve wth respect to B s of u TS du = Y s θ sys 1 + Z + β + δb + Z 1 + β + δb B. db 1 + ZB 3. 8

9 1+ZB B = 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 sys 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 S-EQM: B = or alternatvely A = n Z + β + δ + n Z + nzβ + δ + β + δ 10 α + nzα Y δb, Z + β + n 1Zβ + δb 11 S-EQM: b = n Z + β + δ + n Z + nzβ + δ + β + δ 1 αb + nzα Y δ a = Zb + βb + n 1Zβ + δ, 13 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 [3]. Proposton 3.5 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 1 and 13. Proof For the proof please consult the techncal companon [3]. 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 9

10 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. 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 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. 14 Mathematcally, the stochastc optmzaton problem solved by the ISO can be represented as follows. 3 3 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. 10

11 ISOSP: mn z = S θ n [ ] s 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} [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. 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 4.1 In the stochastc settlement market clearng, the expected devaton of frm from pre-dspatch quantty q s zero, that s, the optmal soluton to ISOSP wll always satsfy θ s x,s = 0. s 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 s θ sx j,s 0. The optmal objectve value of ISOSP s then gven by θ s a k,s + b k,s + d θ s x,s + Y s k,s Z k,s. 15 s s Note that as s θ sx j,s 0, the term s θ s d x,s s postve. Now, for a fxed and k,s gven from above, consder the problem mn w = d q,x,s θ s x,s Ths problem clearly reduces to the unvarate problem q + x,s = k,s s. 16 mn q w = θ s k,s q, 11

12 and whch s optmzed at Defne ˆq and xˆ,s by ˆq = q = θ s k,s. { q, j S θ sk j,s otherwse, { 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 stasfyng 16. 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 15, as we have already establshed that s θ s d x,s > 0. Ths yelds the contradcton that proves the result. Corollary 4. In the stochastc problem ISOSP, f q + x,s 0 s satsfed s {1,..., S} then q 0 wll hold. Proof In Lemma 4.1 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 4.1 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. 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. see [5] 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. 1

13 Lemma 4.3 Problem ISOSP s equvalent to the followng optmzaton problem whch s separable n the pre-dspatch and the spot market varables z = + a q + b q b + d Y q + Z q θ s x,s θ s Y s x,s + Z θ s x,s. Proof Substtutng for C s from constrants nto the objectve functon of ISOSP yeld z = θ s a q + x,s + b q + x,s + d Y s q + x,s + Z q + x,s x,s Rearrangng the above we obtan: z = + a q + b q b + d a Y q + Z q θ s x,s θ s x,s + q b θ s Y s x,s + Z θ s x,s + θ s x,s θ s Z q x j,s j=1 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. z = + a q + b q b + d a Y q + Z q θ s x,s θ s x,s + q b 13 θ s Y s x,s + Z θ s x,s + Z θ s x,s q j=1 θ s x j,s

14 Recall from Lemma 4.1 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 R R n Secton 3. We wll use the followng transformaton H : R + {0} R + {0} R + R + {0} whch s a bjecton A B R = H a b d := a /b 1/b 1/b + d If we further defne A = A, B = B and R = R, ISOSP reduces to mnmzng the followng: A z = q + 1 q Y q + Z q B B + θ s 1 x,s R Y s Y x,s + Z. x,s. Note that as before Lemma 4.3, 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 4.4 If q, x, f, p represents the soluton of ISOSP, then we have q x,s = Y + ZAB 1 + ZB A 17 = Y s Y R 1 + ZR f = Y + ZA 1 + ZB = Y + ZA 1 + ZB + Y s Y 1 + ZR p s 18 14

15 Proof For dervaton of the expressons for the optmal soluton above from frst order condtons please refer to the techncal companon [3]. 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. Prce π s = b + d x,s = 1 R x,s π s = 1 R x,s π s πs = Y s Y Z x,s x,s x,s Quantty Fgure 1: Market clearng of the spot market usng frms supply functons as an equvalent representaton of ISOSP problem Corollary 4.5 In the soluton of ISOSP, forward prce s equal to the expected spot market prce. Proof Ths s smply observed from proposton 4.4. The fact that the contract prce f s equal to the expected spot market prce, mples that there s no systematc arbtrage. 4.3 Equlbrum Analyss of the Stochastc Settlement Market In Secton 4.1 we presented frm s proft under scenaro s n equaton 14. 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, 19 15

16 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 α θ s x,s βq x,s θ s x,s Note that from Lemma 4.1, 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 We can use the expressons obtaned from proposton 4.4 to wrte u as follows. u = 1 βa + A ZA + α + ZBα + ZAβB + Y 1 + βb 1 + ZB ZB 1 + ZR x,s. 1 + ZR ZA + Y ZA + Y 1 + ZBαB 1 + ZR ZA + Y βb +1 + ZB R + β + δr Y s θ s Y s 0 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 dr 1 + ZR 3. 1 Recall that R = j R j. For verfcaton of ths dervatve term see the techncal companon [3]. 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. 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. 16

17 Equaton 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 can be found n the techncal companon [3]. Holdng B fxed, note that d g da = 1 + ZB Z + β + ZβB 1 + ZB. 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 To fnd the optmal value for g, we can now appeal to Lemma 3.3 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 depcts g. 3 Fgure : Two vews of the functon g. Note that the optmal value of g s obtaned along a contnuum, for any value of B. To uncover the ntuton behnd ths feature of g, we can offer the followng mathematcal explanaton. We observe 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 and that dg = Y + ZA db 1 + ZB. dg. da Therefore, statonary condtons enforced n A wll also mply statonarty n 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. The followng analyss on h wll explan how optmal R s constraned by the value of B. 17

18 Proposton 4.6 Suppose that R s fxed. Then h s optmzed at R = mn{b, 1 + ZR Z + β + α + Zβ + δr } Proof Note that at The dervatve dh dr ˆR = 1 + ZR Z + β + α + Zβ + δr 4 = du dr vanshes. Also recall from Jensen s nequalty that Y s θ sy s. It can therefore be seen from 1 that ths dervatve s postve for R < ˆR and negatve for R > ˆR. Recall further that the defnton of B and R requre R B. Therefore, n optmzng h, we need to enforce ths constrant and we obtan R = mn{b, 1 + ZR Z + β + α + Zβ + δr }. 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 4.6 readly delvers the optmal R. We have therefore proved the followng theorem. Theorem 4.7 The best response of frm, holdng compettor offers fxed, s to offer n A = 1 + ZB α ZA + ZαB Y + Y + ZA Z + β + ZβB B, 1 + ZB Z + β + ZβB and any choce of B where R = 1 + ZR Z + β + α + Zβ + δr, B 1 + ZR Z + β + α + Zβ + δr. Theorem 4.7 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 B as a system parameter. Ths parameter s dentcal for and known to all partcpants. Ths also provdes the ISO wth the opportunty to choose B n a way to obtan a preferable equlbrum.e. an equlbrum that yelds hgher socal welfare. Proposton 4.8 The unque symmetrc equlbrum quanttes of the stochastc settlement market are as follows. a = α Y + B ZY n n 1α + Y β + Zn 1Znα + Y βb B Zn β + Y n 1Zn + βb 5 d = max{0, Zn + β + δ + Z n + Znβ + δ + β + δ 1 B } 6 18

19 Proposton 4.9 Let Zn + β + δ + Z ˆb n + Znβ + δ + β + δ =. In a stochastc settlement market wth b ˆb and large number of frms, frms tend to offer ther true cost parameters. In other words, lm a = α n lm b + d = β + δ. n When the fxed parameter b s chosen equal to β, lm n d = δ. Proof The equatons are smply derved from the equlbrum values of a and d gven n proposton 4.8. Proposton 4.9 shows that our 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. Theorem 4.10 Let q,x represent the equlbrum of the stochastc settlement market, then the followng nequaltes hold., s : q + x,s 0 : q 0 The proof of the above theorem s contaned n the techncal companon [3]. 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 [3]. Theorem 4.11 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 [3] for the proof of ths theorem. 19

20 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 choose the margnal cost parameters 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 = ˆR, we obtan an optmal soluton subject to boundary condtons of course. Whle n the context of our computatons, due to the natural decomposton of u, t was natural to treat b 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 pckng d = d 0. If we thnk of the ISO choosng d, announcng d 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 for the range where 0 d ˆb. Here observe that f ISO selects d = 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 = 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 θ s y,s Z y,s αy,s + β y,s + δ y,s q. 7 Note that the dfferent socal welfare values W SS for the ISOSP and W NZTS for the NZTS mechansm are found through the same formula, however wth the dfferent equlbrum y,s varables. Recall that followng Theorem 4.7 the choce of B equvalently the choce of b, was delegated to the ISO. The next proposton establshes that when frms are bddng strategcally, the stochastc settlement market domnates the NZTS market provded the ISO chooses the slope of the supply functon suffcently low. 0

21 Proposton 5.1 The socal welfare of ISOSP s hgher than the NZTS market provded the parameter b s chosen less than the threshold value ˆb, where Zn + β + δ + Z ˆb n + Znβ + δ + β + δ =. Proof To prove the proposton, we show when b = ˆb, we can conclude W SS = W NZTS. Then, we demonstrate W SS s a decreasng functon of b, and therefore, W SS W NZTS, when b ˆb note that W NZTS s a constant and does not change wth b. When b = ˆb, equatons 5, 11, and 10 yeld that the equlbrum quanttes are dentcal n the stochastc settlement and determnstc two perod settlement markets. That s 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 3.1, 3., and 4.4 to obtan q SS y SS,s = q NZTS = y NZTS,s = Y B A 1 + ZB = Y sb A 1 + ZB Therefore socal welfare of these models equaton 7 are the same provded b = ˆb. We can rewrte the socal welfare expresson 7 as W = θ s Y s y,s Z y,s αy,s + β y,s + δ x,s. 8 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,s SS vs y,s NZTS etc. Furthermore, note that x SS,s s ndependent of b, and therefore, dw SS db = 1 b,s dw SS dy SS,s dy SS,s db. 9 On the other hand, takng the dervatve of y SS,s wth respect to B see the techncal companon [3] we obtan dy SS,s db = Y αn 1Z Z + nz + β + n 1ZnZ + βb 0. 1

22 The rght hand sde s readly seen to be non-negatve as Y > α and n > 1. 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 9 and obtan As dyss,s db dw SS db = 1 b On the other hand, dfferentatng 8 yelds dy SS,s db,s dw SS. dy,s SS Hence, dw SS dy SS,s = θ s Ys α Zn + βy,s SS. s dw SS dy SS,s = Y α Zn + βq SS = bzy α Zn 1nZ + β + bn + 1Z + β 0. Therefore we can conclude that, dw SS db 0. Note that we can easly show that ˆb β + δ, and therefore, f the fxed b s chosen equal to β then W SS W NZTS. Example Consder a market wth two symmetrc generators as defned n table 1. Parameter Value α, β, δ 50, 1, 0.5 Y 1, Y, Z 100, 150, 1 θ 1, θ 0.5, 0.5 n Table 1: The market envronment for the example Fgure 3 shows how the socal welfare of the stochastc settlement mechansm s affected by the choce of b. It also demonstrates that for small enough bs, the stochastc settlement mechansm has a hgher equlbrum socal welfare n comparson wth the NZTS mechansm. For the rest of ths example, we assume the ISO chooses b = β = 1 whch ensures hgher

23 Socal Welfare d=b^ d=0 d=0 ISOSP NZTS b Fgure 3: The effect of b on the socal welfare of the stochastc settlement model and on how t compares to the determnstc two perod settlement mechansm Socal Welfare δ ISOSP NZTS Fgure 4: The SP mechansm yelds hgher socal welfare for dfferent δ values Producer Welfare ISOSP NZTS δ Fgure 5: The SP mechansm yelds lower producer welfare for a range of examples.e. dfferent δ values equlbrum socal welfare from the stochastc settlement n comparson wth the conventonal mechansm. 3

24 Consumer Welfare ISOSP NZTS δ Fgure 6: The SP mechansm yelds hgher consumer welfare for a range of examples.e. dfferent δ values Another nterestng experment s to nvestgate the effect of δ on these mechansms. Fgures 4, 5, and 6 compare the stochastc settlement and the NZTS mechansms for ths market, however for dfferent δ values. A frst observaton s the stochastc settlement mechansm ncreases socal and consumer welfare and decreases producer welfare n comparson wth the two settlement mechansm. Also, these effects are enhanced by ncreasng δ. Ths s an expected result that the strength of the stochastc settlement s bolder when cost of devaton s hgher. It s also nterestng to nvestgate the effect of competton on these mechansms. To do so, we can test the effect of number of frms on these mechansm. Socal Welfare ISOSP NZTS n Fgure 7: Socal welfare of the determnstc two perod settlement mechansm converges that of the stochastc mechansm when n ncreases. Competton ncreases wth a bgger market. Fgure 7 shows the dfference n the socal welfare of our two mechansm as a functon of n. It shows that when the number of generators ncrease, the performance of the stochastc and determnstc two perod settlement mechansms converges. 4

25 6 Concluson In ths paper, we set up a smple modellng envronment n whch we were able to compare the New Zealand nspred determnstc two perod sngle settlement market clearng mechansm aganst a stochastc settlement aucton whch reduces to another two perod sngle settlement aucton wth explct penaltes of devaton, therefore dfferent from the NZTS model. We were able to model frms best responses n these markets, and so fnd equlbrum behavour n each. We fnd that n our symmetrc models, the ISOSP aucton provably domnates the NZTS aucton when measurng expected socal welfare. References [1] F. Bolle. Supply functon equlbra and the danger of tact colluson. the case of spot markets for electrcty. Energy Economcs, 14:94 10, 199. [] F. Bouffard, F. D. Galana, and A. J. Conejo. Market-clearng wth stochastc securty-part : formulaton. Power Systems, IEEE Transactons on, 04: , 005. [3] J. Khazae, G. Zaker, and S. Oren. Techncal companon for the paper market clearng mechansms under uncertanty. [4] P. Klemperer and M. Meyer. Supply functon equlbra n olgopoly under uncertanty. Econometrca, 576: , [5] G. Prtchard, G. Zaker, and A. Phlpott. A Sngle-Settlement, Energy-Only electrc power market for unpredctable and ntermttent partcpants. Operatons Research, Apr [6] S. Wong and J. Fuller. Prcng energy and reserves usng stochastc optmzaton n an alternatve electrcty market. Power Systems, IEEE Transactons on, : ,