Payment Mechanisms for Electricity Markets with Uncertain Supply

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1 Paymet Mechasms for Electrcty Markets wth Ucerta Supply Rya Cory-Wrght a, Ady Phlpott b, Golbo Zaker b a Operatos Research Ceter, Massachusetts Isttute of Techology b Electrc Power Optmzato Cetre, The Uversty of Aucklad Abstract Ths paper provdes a framework for dervg paymet mechasms for termttet, flexble ad flexble electrcty geerators who are dspatched accordg to the optmal soluto of a stochastc program that mmzes the expected cost of geerato plus devato. The frst stage correspods to a pre-commtmet decso, ad the secod stage correspods to real-tme geerato that adapts to dfferet realsatos of a radom varable. By takg the Lagraga ad decouplg dfferet ways we study two paymet mechasms wth dfferet propertes. Keywords: Electrcty Markets, Stochastc Programmg, Wd Power, Locatoal Prcg 1. Itroducto I respose to pressure to reduce CO2 emssos ad creases the peetrato of reewables, electrcty pool markets are procurg creasg amouts of electrcty from termttet sources such as wd ad solar. Ths pressure ca cause dffcultes for the depedet system operator (ISO) whe dspatchg geerators. These dffcultes arse because may thermal geerators are flexble ad eed to be formed of ther geerato oblgatos before the output of reewables s kow. I markets wth large amouts of flexble geerato, geerators ca be provded wth a pre-commtmet setpot before the geerato output of reewables s kow. The setpot s determed by assumg that reewables geerate ther expected output capacty. The ISO the dspatches geerators at the commecemet of the trade perod, wth the objectve of mmzg the cost of geerato plus devato from the setpot, order to maage fluctuatos betwee the expected ad the realsed output capacty of termttet reewables. We refer to ths dspatch mechasm as the tradtoal mechasm. As oted by Chao et al [3] ths approach ca be suboptmal maxmsg cosumer welfare. The expected costs of correctve actos arsg from surpluses or shortfalls are ot prced to the market clearg problem, because the system operator does ot cosder ucertaty. Dspatches derved va the tradtoal mechasm therefore may have a small fuel cost ad a large expected cost of correcto. Wth small quattes of termttet reewable peetrato the prce of falg to cosder ucertaty s small, as flexble geerators ca be re-dspatched at a low cost to meet shortfalls. However, as termttet reewables crease ther peetrato, the cost of correctve actos becomes sgfcat. Lmts o flexble plats mea that expesve ad flexble plats eed to be re-dspatched to meet large shortfalls wd or solar geerato. Therefore, whe the peetrato of termttet reewables s hgh, a ew dspatch mechasm s requred. The frst alteratve to the tradtoal dspatch mechasm was proposed by Bouffard ad Galaa [2], whch volves clearg a co-optmzed pool ad reserve market usg a twostage stochastc program. The frst stage s a capacty market whch s cleared before radom evets pertag to the avalablty of plats ad trasmsso les take o a realsato. The secod stage s a market clearg problem where oly uts commtted the frst stage may be dspatched. Subsequetly, several authors cludg [9] [7] [4] [5] [10] have proposed explctly modellg ucertaty the geerato output of termttet reewables as a radom varable. Exstg acllary market servces are supplemeted wth a realtme market, whch s cleared after the geerato capacty of reewables s realsed. Ths ca be acheved by castg the market clearg problem wth a two-stage stochastc programmg framework, where the frst stage correspods to a precommtmet setpot, ad the secod stage correspods to a real-tme dspatch uder a partcular realsato of the geerato output of reewables. We refer to ths dspatch mechasm as the Stochastc Dspatch Mechasm, or SDM. I SDM, the pre-commtmet setpot ca be terpreted as the output level at whch geerators prepare to geerate before wd geerato s realsed. The real-tme dspatch quatty s the geerato quatty whch the geerator produces throughout the trade perod, after the geerato of reewables s kow. A devato to crease or decrease a geerators realtme dspatch output from ts day-ahead setpot causes the geerator to cur a cost, the form of e.g. wear-ad-tear o geerato equpmet. To replcate the dspatch foud by SDM ad maxmze expected total socal welfare, t s ecessary to pay market partcpats a maer whch causes ther optmal behavour to alg wth the optmal dspatch foud by a system operator whe solvg SDM. I the lterature (see [9] [7] [4] [10]) ths s acheved by compesatg ad chargg market partcpats wth a sgle-settlemet paymet mechasm, where the Preprt submtted to Operatos Research Letters May 23, 2017

2 amout charged to each partcpat uder each possble realsato of reewable geerato s made kow apror, ad the actual charge to each partcpat occurs after reewable geerato s realsed. I sgle-settlemet paymet mechasms, two desrable attrbutes are reveue adequacy ad cost recovery. A paymet mechasm s reveue adequate f ad oly f the system operator collects at least as much from cosumers as t pays geerators. A paymet mechasm exhbts cost recovery f ad oly f geerators are pad at least as much as ther fuel costs. The frst paper to study ths aspect of prcg was by Wog ad Fuller [9] who troduced a two-stage dspatch mechasm ad suggested several paymet mechasms to compesate geerators that were reveue adequate ad recovered costs o average. A smlar approach was take by Prtchard et al [7] who focused o wd termttecy. The dspatch mechasm of [7] was mproved by Zaker et al [10], who observed that welfare could be ehaced by removg the frst-stage costrat o supply meetg demad, ad suggested a alteratve paymet mechasm, whch s reveue adequate every scearo ad provdes geerator cost recovery o average. I ths paper, we revst the dspatch mechasm of [10] a classcal Lagraga framework for modellg perfectly compettve partal equlbrum. We defe a prce of formato, the amout whch a geerator eeds to be pad to eforce oatcpatvty o ts frst-stage geerato. The prce of formato leads to a dscrmatory varato o the prcg mechasm of [10], gvg cost recovery every scearo, expected reveue adequacy ad expected reveue equvalece for all agets compared to [10] s mechasm, assumg that agets behave the rsk-eutral prce-takg maer assumed [7] ad [10]. Ths dscrmatory varato explctly detfes the uplft paymet requred to make all geerators whole, ad esures that the expected uplft paymet to each geerator s zero. Followg [7] ad [10], we cosder a SDM based o the DC-Load Flow model of a costraed electrcty trasmsso etwork. We follow [1] lettg ucertaty be represeted by the scearo Ω, whch occurs wth probablty P(). We assume that a realsato of prescrbes all ucertaty due to termttet geerato a electrcty pool market, ad that the sample space Ω s a fte set cotag all possble future realsatos of. Although the real dstrbuto of wd s ot fte, Ω ca be vewed as a approxmato, obtaed for example by samplg from the true dstrbuto f ths s avalable. The dspatch models obtaed are the Sample Average Approxmatos for whch oe ca derve asymptotc covergece results as the sample sze creases (see [8] for a geeral theory, ad [10] for asymptotc results the cotext of SDM). We ote that ay computatoal dspatch model must work wth a fte Ω, ad sce our focus ths paper s o dfferet paymet mechasms for these models, we restrct atteto to ths case. Our otato closely follows that of [7] ad [10] whch determstc varables are deoted by lower case Roma symbols, radom varables by upper case Roma symbols, ad prces by lower case Greek symbols. The sets ad dces the Stochastc Program (SP) solved to obta the day-ahead setpot SDM are defed as follows: s the dex of a geerator. We assume perfect competto, so the owershp of geerato does ot affect the soluto. Ths meas that each aget ca be thought of as operatg a sgle ut. N s the set of all odes the trasmsso etwork. j() s the ode j N where geerator s located. a = 1 whe aget s located at ode ad 0 otherwse. T () s the set of all offers at ode. F s a set costrag the flows the etwork to meet thermal lmts ad the DC-Load Flow costrats mposed by Krchhoff s Laws. We assume that 0 F. The decso varables SP are defed as follows: x s the day-ahead setpot level whch geerator s advsed to prepare to produce before the geerato capacty of termttet reewables s kow. X () s the real-tme dspatch produced by geerator scearo. U () ad V () are the amouts whch geerator devates up/dow by scearo. That s, U () =max(x () x, 0), ad V () =max(x X (), 0). F() F s the vector of brach flows the etwork scearo. τ (F()) s the et amout of eergy flowg from the grd to ode scearo. We assume that τ s a cocave fucto of F wth τ (0) = 0, N. 2. The Stochastc Dspatch Mechasm 2 The shadow prces for the costrats SP are defed as follows: λ () s the margal cost of geeratg oe addtoal ut of electrcty at ode scearo uder SP. As the costrat o supply meetg demad s a equalty, we requre that λ () 0, N. ρ () s the margal cost of aget geeratg oe addtoal ut of electrcty scearo uder SP. λ = E [λ ()] s the expected value of the shadow prce λ () SP. The problem data SP are defed as follows: c s the margal cost curred by geerator. D () s the cosumer demad at ode scearo. r u, ad r v, are the margal costs curred by geerator for devatg up or dow ts geerato. We requre that r u, > c > r v, to avod provdg geerator wth arbtrage opportutes whe t s choosg whether to ramp up or dow.

3 G () s the maxmum output capacty of geerator scearo. The day-ahead setpot s foud by solvg the followg stochastc program for x, as defed by Zaker et al [10]: SP: m c x + P()(ru U() + rv V()) s.t. T () X () + τ (F()) D (),, Ω, [P()λ ()], x + U() V() = X(), Ω, [P()ρ()], F() F, Ω, 0 X() G(), U(), V(), x 0, Ω. max c x P()λ ()D () + P()λ j() ()X () + P()λ ()τ (F()) P() ( r u, U () + r v, V () ) s.t. x + U() V() = X(), Ω, 0 X () G (), U (), V (), x 0, F() F. Observe that we have matched each costrat wth a correspodg shadow prce square brackets. Sce λ () s the margal cost of meetg addtoal demad at ode scearo, ad ρ () s geerator s margal cost of geerato scearo, each of these must be weghted by probablty P(). After the optmal day-ahead setpot x s foud, the termttet geerato scearo = ˆ s realsed, ad the ISO follows the dspatch defed by ( X ( ˆ), U ( ˆ), V ( ˆ), F ( ˆ) ). I a compettve market settg, agets respod to prce sgals, so the remuerato from these prces for each aget s dspatch provdes cetves to respod as the dspatch dctates. To model ths, we decouple SP usg a Lagraga approach. 3. Paymet Mechasms Observe that ths problem ca be decoupled by geerato aget. Gve prces λ j() () each geerato aget ca determe ts ow dspatch by solvg the followg stochastc program. SP1: max P() ( c x + λ j() ()X () r u, U () r v, V () ) s.t. x + U () V () = X (), Ω, 0 X () G (), U (), V (), x 0. The objectve fucto of SP1 also specfes the remuerato for each aget. Here f scearo ˆ s observed the purchasers ode pay λ ( ˆ)D ( ˆ) ad geerator s pad λ j() ( ˆ)X ( ˆ). Ths s the paymet scheme studed by Zaker et al. [10]. As demostrated by [10] the amout collected from purchasers s always at least eough to cover the paymets beg made to geerators. Ths result s called reveue adequacy. Assume that SP satsfes a costrat qualfcato such as a Slater codto (see [1]). The we ca solve SP by mmzg a Lagraga. We choose frst to use multplers oly o eergy balace costrats to yeld: L = c x ( + P() ru, U () + r v, V () ) ( + P() D () a X () τ (F()) ) λ (), whch s to be mmzed, subject to the followg costrats: x + U () V () = X (),, Ω, 0 X () G (), U (), V (), x 0, F() F. All Lagrage multplers should be oegatve. Observe that a λ () = λ j() (). Rearragg L ad expressg as a maxmzato we obta: 3 Defto 3.1. A paymet mechasm s reveue adequate f ad oly f every scearo Ω, clearg the market does ot leave the system operator a facal defct. As show by Phlpott ad Prtchard [6], reveue adequacy s equvalet to the followg statemet: λ ()τ (F()) 0, Ω. Proposto 1. If (x, X (), U (), V (), F ()) solves SP, the payg λ j() ()X () to geerator ad chargg λ ()D () to demad aget results reveue adequacy every scearo. Proof. See [10]. It s ot hard to see that some crcumstaces a geerator mght ot be compesated for the short-ru costs of ts frststage dspatch. For example, f x > 0 ad X ( ˆ) = 0 the realzed scearo the o reveue wll be eared to cover the cost c x +r v,v ( ˆ). If ths s the case, the there s o cetve for geerator to partcpate the market, uless the paymets ca be made whole some maer.

4 Defto 3.2. A paymet mechasm exhbts cost recovery f ad oly f every scearo Ω, all geerators recover ther short-ru (fuel ad devato) costs. That s, R () c x () r u, U () r v, V () 0,, Ω, where R () s geerator s reveue scearo. The paymet mechasm of SP1 does ot exhbt cost recovery. We say that a market clearg mechasm exhbts expected cost recovery f all geerators recover ther geerato ad rampg costs expectato. Ths was show to be the case for SP1 by [10]. Proposto 2. Payg λ j() ()X () to geerato aget scearo results expected cost recovery. Proof. See [10]. It s ot surprsg that the paymet mechasm of SP1 mght result some geerators ot recoverg ther costs. I SP1 each aget s preseted wth scearo prces ad solves a stochastc program to maxmze ther expected proft. The costrats ca be rewrtte x + U () V () = X (), Ω, x () + U () V () = X (), Ω, x () = x, Ω, where the secod set of costrats are called oatcpatvty costrats. Cost recovery for each geerator would be possble f there were oly oe scearo, or f the geerator s oatcpatvty costrats were relaxed, so that the frst-stage decso x ca vary wth scearo (becomg x ()). Ths eables a geerator to use perfect foresght choosg x () ad ga from ths. Eforcg a oatcpatvty costrat curs a cost from the loss of formato, ad so to esure a oegatve proft, each geerator should be compesated wth a ex-post formato paymet scearos where t makes less tha ts expected proft ad charged a ex-post formato ret scearos where t makes a proft greater tha ts expected proft. The values of these paymets ad charges ca be made explct by applyg a Lagraga relaxato of oatcpatvty costrats. Equvaletly, oe ca troduce Lagrage multplers for the costrats lkg x ad X () SP. Ths gves the followg Lagraga. ˆ L = c x ( + P() ru, U () + r v, V () ) ( + P() D () a X () τ (F()) ) λ () + P() ρ () ( ) V () U () + X () x, 4 whch s to be mmzed, subject to the followg costrats: 0 X () G (), U (), V (), x 0, F() F. Rearragg Lˆ ad expressg as a maxmzato yelds: max ( c + P()ρ ())x + P() ( λ j() () ρ () ) X () + P()λ ()τ (F()) P()λ ()D () + P() ( (ρ () r u, )U () + ( ρ () r v, )V () ) s.t. 0 X () G (), U (), V (), x 0, F() F. Ths ca be decoupled by geerato aget. Geerato aget solves the followg problem: SP2: max P() ( (ρ () c )x + (λ j() () ρ ())X () +(ρ () r u, )U () + ( ρ () r v, )V () ) s.t. 0 X () G (), U (), V (), x 0. The objectve fucto coeffcet for the frst-stage decso ca be rewrtte as ρ c, where: ρ = P()ρ (). Observe that SP2 the decouples to the optmzato of a acto x, ad optmal actos for each scearo. Whe ρ > c, x wll be fte, so we requre ρ c, ad (c ρ )x = 0. Ths meas x wll be o zero oly whe geerator s pad exactly ts margal cost c. Smlarly for fteess we requre ρ () r u, ad ( ρ () r v, ), yeldg (ρ () r u, )U () = 0 ad ( ρ () r v, )V () = 0. We collect these results the followg lemmas. Lemma 3. For every geerato aget, the optmal dspatch polcy (x, X (), U (), V ()) satsfes: ( ρ c )x + (ρ () r u, )U () + ( ρ () r v, )V () = 0. Lemma 4. For every geerato aget wth X () > 0, t follows that ρ () λ j() (). Proof. We take the cotrapostve. Suppose ρ () > λ j() (). The sce X () 0 t follows that the optmal choce of X () s X () = 0. We use Lemmas 3 ad 4 to yeld the followg result:

5 Proposto 5. For every, f aget has made the optmal choce of x the payg: ρ x + (λ j() () ρ ())X () + ρ ()U () ρ ()V (), results cost recovery every scearo. Proof. Assume that geerato aget has see the realsato of ad s makg ts secod-stage decso. The aget s proft scearo s: φ () = max ( ρ c )x + (λ j() ρ ())X () X (),U (),V () +(ρ () r u, )U () + ( ρ () r v, )V () s.t. 0 X () G (), U (), V () 0. Now, by Lemma 3, we kow that: ( ρ c )x + (ρ () r u, )U () + ( ρ () r v, )V () = 0. Therefore, φ () = (λ j() ρ ())X (). Furthermore, we kow from Lemma 4 that (λ j() ρ ())X () 0. Therefore, () 0, Ω. φ Proposto 6 shows that each geerato aget s pad exactly the same as the frst paymet scheme except they are compesated by ( ρ ρ ())x for the frst-stage dspatch. The expected value of ths compesato wll be zero, as ths compesato wll be egatve some scearos, ad so expectato geerators wll receve the same amout uder both paymet schemes. A market clearg mechasm s sad to exhbt reveue adequacy expectato f the retal collected by the ISO s oegatve expectato. The frst paymet scheme satsfes ths because t s reveue adequate every scearo, ad the dscusso above demostrates that the secod paymet scheme addtoally pays a prce of formato whch has a expected value of zero, meag that the secod scheme s reveue adequate expectato. We formalze ths as corollares to Proposto 6. Corollary 7. If (x, X (), U (), V (), F ()) solves SP, the payg each geerato aget ( ρ ρ ())x +λ j()()x () results reveue adequacy expectato. Corollary 8. A suffcet codto for reveue adequacy scearo ˆ s: (ρ( ˆ) ρ) x 0. Proposto 5 gves cost recovery for the paymet mechasm from SP2 because ρ = c wheever x > 0. Ths dscrmatory mechasm meas each geerator that s dspatched the frst stage has ther exact costs for ths dspatch pad by the ISO. Ths mght requre the ISO to suffer a egatve ret f a partcular scearo s realzed, but as show Corollary 7 below the ISO retal wll be postve expectato. It follows that the ISO wll ot cur a defct the log ru f we assume that the radom outcomes each dspatch stace are..d. The paymet mechasm Proposto 5 ca be rewrtte as follows: Proposto 6. For every, f geerato aget makes the optmal choce of x, X (), U () ad V () the payg: ρ x + (λ j() () ρ ())X () + ρ ()U () ρ ()V (), gves the same proft as payg aget : ( ρ ρ ())x + λ j() ()X (). Proof. Aget s actos satsfy: It follows that: x + U () V () = X (), Ω. ρ x + (λ j() () ρ ())X () + ρ ()U () ρ ()V () = ρ x + λ j() ()X () + ρ ()(U () X () V ()) = ρ x + λ j() ()X () ρ ()x = ( ρ ρ ())x + λ j() ()X (). 5 Corollary 9. If all agets act as rsk-eutral prce takers, the the paymet mechasm of chargg cosumer the amout λ ()D () ad payg geerator the amout λ j() X () results the same expected proft for each geerator, each cosumer ad the ISO as the paymet mechasm of chargg cosumer the amout λ ()D () ad payg geerator the amout ( ρ ρ ())x () + λ j() X (). 4. A Sx Node Example To llustrate the dffereces betwee the two paymet mechasms, we aalyse the payoffs to agets that would occur the sx ode example descrbed by [7]. The trasmsso etwork s depcted Fgure 1, where there are two flexble thermal geerators who caot devate the secod stage, two flexble hydro geerators who ca ramp up or dow at costs of 35 ad 20 per ut, ad two termttet wd geerators. The geerato capacty of each geerator ad cost per ut geerato are dcated by X@$Y. The wd geerators depedetly produce oe of the followg amouts wth equal probablty: {30, 50, 60, 70, 90}, resultg 25 scearos each havg probablty There s a sgle cosumer who requres 264 uts of geerato each scearo ad the secod stage a trasmsso costrat dctates that at most 150 uts ca be trasmtted from ode A to ode B or vce versa. The reactaces of all les are assumed to be detcal, meag 5 6 of the power geerated by Thermal 1 flows va the costraed le, ad 2 3 of the power geerated by Wd 1 flows va the costraed le. I order to prevet dual degeeracy, we mpose quadratc losses o all trasmsso les, wth a loss coeffcet of That s, τ ( f ) = k( f k 10 8 fk 2 )+ k( f k 10 8 fk 2 ), where f k s a flow to ode from ode k, ad f k s a flow out of ode to ode k.

6 Wd 1: W Thermal 2: C D E Wd 2: W Table 3: Statstcs for the paymet mechasm correspodg to SP1 Aget Expected Proft Std Devato % Negatve Proft M Proft Max Proft Thermal Wd Thermal Wd Hydro Hydro ISO Sum Agets Thermal 1: B F max = 150 A Cosumer Demad=264 F Hydro 1: +$35, $20 Hydro 2: +$35, $20 Fgure 1: The sx ode example from Prtchard et al [7] Table 4: Statstcs for the paymet mechasm correspodg to SP2 Aget Expected proft Std Devato % Negatve Proft M Proft Max Proft Thermal Wd Thermal Wd Hydro Hydro ISO Sum Agets Summary statstcs regardg the frst ad secod stage dspatches are avalable Tables 1 ad 2, ad payoffs of each partcpat uder the paymet mechasms cosdered ths paper are avalable Tables 3 ad 4. The Expected Proft umbers llustrate the equvalece of payoffs expectato, ad the Std Devato values dcate how the two mechasms allocate rsk to agets ad the ISO each case. Ideed, for a worst case rsk measure, Table 3 detfes a scearo where wd geerator 1 ca produce up to 90 uts, wd geerator 2 ca produce up to 90 uts, agets lose $5560 ad the ISO loses othg. I cotrast, the same scearo Table 4 leads to a ISO shortfall of $5560 ad all agets recoverg ther costs. Ths scearo also shows why t s ot possble a stochastc dspatch to have both reveue adequacy ad cost recovery every scearo. Here 114 uts of demad are satsfed by the flexble thermal geerators, leavg 150 uts of demad to be satsfed. However, 180 uts of wd geerato are avalable, meag that 30 uts of wd geerato must be shed, resultg a real-tme margal prce at each ode of zero. I ths scearo, ether the thermal geerators or the ISO must experece a shortfall. Refereces [1] J. Brge, F. Louveaux, Itroducto to Stochastc Programmg, Sprger New York, 2d edto, [2] F. Bouffard, F. Galaa, A electrcty market wth a probablstc spg reserve crtero, IEEE Trasactos o Power Systems 19 (2004) [3] H. Chao, H. Hutgto, Desgg Compettve Electrcty Markets, Sprger US, [4] J. Morales, A. Coejo, K. Lu, J. Zhog, Prcg electrcty pools wth wd producers, IEEE Trasactos o Power Systems 27 (2012) [5] J. Morales, A. Coejo, H. Madse, P. Pso, M. Zugo, Itegratg Reewables Electrcty Markets, 205, Sprger US, [6] A. Phlpott, G. Prtchard, Facal trasmsso rghts covex pool markets, Operatos Research Letters 32 (2004) [7] G. Prtchard, G. Zaker, A. Phlpott, A sgle-settlemet, eergy-oly electrc power market for upredctable ad termttet partcpats, Operatos Research 58 (2010) [8] A. Shapro, A. Ruszczysk, D. Detcheva, Lectures o Stochastc Programmg: Modelg ad Theory, SIAM, [9] S. Wog, J. Fuller, Prcg eergy ad reserves usg stochastc optmzato a alteratve electrcty market, IEEE Trasactos o Power Systems 22 (2007) [10] G. Zaker, G. Prtchard, M. Bjordal, E. Bjordal, Prcg wd: A reveue adequate cost recoverg uform prce for electrcty markets wth termttet geerato, NHH Dept. of Busess ad Maagemet Scece (2016). Table 1: Frst ad secod stage dspatches Aget Frst Stage Dspatch Mea 2d Stage Dspatch Std Devato 2d Stage Dspatch Thermal Wd 1 /a Thermal Wd 2 /a Hydro Hydro Sum Agets /a Table 2: Dspatch summary statstcs Aget % Not dspatched M 2d Stage Dspatch Max 2d Stage Dspatch Thermal Wd Thermal Wd Hydro Hydro Sum Agets