Optimization Based Bidding Strategies in the Deregulated Market

Transcription

1 Optimization ased idding Strategies in the Dereguated arket Daoyuan Zhang Ascend Communications, nc 866 North ain Street, Waingford, C 0649 Abstract With the dereguation of eectric power systems, market participants are facing an important task of bidding energy to an ndependent System Operator (SO his paper presents a mode and a method for optimization-based bidding and sef-scheduing where a utiity bids part of its energy and sef-schedues the rest as in New Engand he mode considers SO bid seections and uncertain bidding information of other market participants With appropriatey simpified bidding and SO modes, cosed-form SO soutions are first obtained hese soutions are then pugged into the utiity s bidding and sef-scheduing mode which is soved by using agrangian reaxation esting resuts show that the method effectivey soves the probem with reasonabe amount of CPU time Keywords: idding Strategies, agrangian Reaxation ntroduction idding, Sef-Scheduing, and iterature Review With the dereguation of eectrica power systems, market participants bid energy to an ndependent System Operator (SO n the daiy market, participants submit bids to the SO who then decides energy cearing prices (ECP and houry generation eves of each participant over a 4-hour period he reationship between SO and participants is shown in Figure n regions such as New Engand, a utiity bids part of the energy and sef-schedues the rest, whereas an independent power producers (PP bids a its energy his paper focuses on the daiy bidding and scheduing of a utiity For each participant, bidding strategies ideay shoud be seected to maximize its profit Game theory is a natura patform to mode such an environment [,, 3] n the iterature, matrix games have been used for its simpicity, and bidding strategies are discretized, such as bidding high, bidding ow, or bidding medium With discrete bidding strategies, payoff matrices are constructed by enumerating a possibe combinations of strategies, and an equiibrium of the bidding game can be obtained t is difficut to incorporate sef-scheduing in the method odeing and soving the bid seection process by the SO have aso been discussed n [4], bids are seected to Yajun Wang, Peter uh Department of Eectrica Engineering University of Connecticut, Storrs, C Generation eve & ECP SO idding Strategies Generation eve & ECP Participant Participant Participant Figure he reationship between SO and participants minimize tota system cost, and the ECP is determined as the price of the highest accepted bid n [5], a bid-cearing system in New Zeaand is presented Detaied modes are used, incuding network, reserve, and ramp-rate constraints, and the probem is soved by using inear programming n [6], the dynamics of pricing is considered in market power assessment n [7], an optima bidding strategy is proved to be a unit s true cost under the assumption that the bid of each unit does not effect the ECP his assumption is reaistic since units with significant capacities wi affect ECP Overview of the Paper he purpose of the paper is to present a mode and a method for the bidding and sef-scheduing probem from the viewpoint of a utiity, say Participant o obtain effective soutions with acceptabe computation time, bids are represented as quadratic functions of power eves For Participant, these parameters are to be optimized For other participants, the parameters are assumed to be avaiabe as discrete distributions ased on bids submitted, the SO is to minimize the tota system cost he probem for Participant is then formuated to minimize its expected cost, incuding generation costs and payment to the market he detaied modes are presented in Section n Section 3, the SO probem is first soved with cosedform soutions he soutions are pugged into Participant s mode with detaied modeing of units, and the probem is soved by using agrangian reaxation as presented in Section 4 esides the subprobems in traditiona hydrotherma scheduing, an additiona bidding subprobem is constructed to optimize bidding parameters his subprobem is stochastic because of the uncertainty of the market, and is optimized by a gradient method t is shown that this subprobem is inherenty degenerated with an infinite number of equivaent soutions Numerica testing presented in Section 5 shows that the method produces good bidding and sef-scheduing resuts for practica probems Compared with a mean method, this new method reduces system cost and effectivey handes uncertainties with a sma increase in CPU time

2 Probem Formuation Representation of ids A bid consists of price offers and the amount of oad to be satisfied by the market for each hour Price offers specify a stack of W eves and the corresponding prices as iustrated in Figure y integrating a staircase price offer curve, the bidding cost function is piecewise inear he amount of oad to be satisfied by the market is denoted as p ock 0 ock ock Rate eve $/W W $/W W (p Figure An Exampe of ids o reduce the number of parameters associated with a bid, the piece-wise inear bidding cost function is approximated by a quadratic C (p A (often done in scheduing probems [8]: C (pa a pa + b pa,,,, ( where is the participant index, t the hour index, p A the accepted eve by SO, a and b are nonnegative bidding parameters, and the number of participants he p A is non-negative, and is upper bounded by a maxima vaue, ie, 0 pa p A ( Participant does not have exact bids of others, but has probabiity distributions of {a, b and p },,, based on market information and experiences he distributions are represented by J discrete sets of bids: j { a j j j, b, p,, }, j,, J (3 he probabiity of j j J j is p with p For Participants j, a, b and p depend on its unit characteristics and others bids, and are to be optimized he SO ode he SO decides houry generation eves of participants to satisfy the tota submitted oad at the minimum cost over 4 hours ids of a participants are avaiabe to the SO, and the deterministic SO probem is J SO min C (p A, (4 pa t subject to pa p, (5 where p is the amount of oad that wi be satisfied by the market for Participant at hour t Constraints ( shoud be satisfied for a participants For simpicity of derivation, however, they are ony required to be satisfied for Participant but are ignored for others 3 he ode of Participant Participant is to decide the generation eves of each unit and a bidding strategy to maximize its profit, or to minimize its costs whie satisfying various constraints he costs incude generation costs and payment to the market, with the payment equa to ECP mutipied by (p - p A Ony therma units are considered to simpify presentation, however, there is no difficuty in incorporating hydro and pumped-storage units he probem is therefore p E t i min,a,b,p J, with J { C (p + λ ( p p } ti ti ti n the above, is the number of hours, the number therma units, Cti the cost function of therma unit i, p ti the generation eve of unit i at hour t, and λ the ECP at hour t he expectation is taken with respect to uncertain bidding parameters refected through λ and p A For each hour, the oad baance constraint requires that p ti + E(p pa pd (7 i n the foowing, the of SO scheduing wi be soved first, foowed by bidding and sef-scheduing 3 he SO Scheduing From SO s point of view, its probem is deterministic When Participant soves the SO probem, however, it ony has distributions of parameters, and has to sove the SO probem for every set of bidding parameters Soution of the SO probem for the jth set j is derived as foows he index j is omitted as appropriate for simper presentation agrangian mutipiers λ are used to reax (5, and π and π to reax ( he resuting agrangian is: SO C (pa + λ p pa t πpa π ( pa pa (8 t t n (8, π, π, and p A satisfy π p A 0, and π p A 0 he three cases of SO soutions are presented beow 3 Case :Accepted eve in ound ( 0 < p A < p A With the given {λ }, (8 can be decomposed into subprobems he subprobem for participant is A (6

3 { a p + b p λ p A } min A A (9 p A t he soution for (9 is λ b p A a (0 n (0, a is assumed to be non-zero f it is zero, a quadratic function with a sma a is used to approximate the inear function foowing the idea of [9] With an anaytica soution for each subprobem, it is not necessary to iterativey update λ at the high eve Substituting (0 into (5, one obtains the ECP λ : b + P a λ a ( Substituting ( into (0, one obtains the accepted eve for Participant at hour t as b b p + a p A + a a ( he energy cearing price λ and the accepted eve p A for Participant are functions of a, b and p o simpify ( and (, et b c0 p +, and a (3 c a (4 hen ( and ( can be rewritten as: λ (c0 + pa + b, and c a (5 + c0 / + p cb / p A (6 ca + Participant may be a buyer or a seer depending on the sign of (p - p A, where p is the soution for p to be derived ater 3 Case : Zero Accepted eve (p A 0 n this case, the bidding prices of Participant are high, causing p A 0 he derivation is simiar to Case with p c0 λ, and (7 c + p A 0 (8 33 Case 3: aximum Accepted eve ( p A p A n this case, Participant s bidding price is ow, resuting in p A p A and the foowing energy cearing prices b p a + λ a (9 he derivation is simiar to Case with ECP determined by other participants bids 4 idding Strategy and Sef-Scheduing For Participant s probem, using mutipiers λ to reax demand constraints (7 the agrangian can be written as E t i [ ( C (p + λ p p (t ] ti ti A + λ pd p ti E( p p A (0 t i he RHS of (0 is separabe for a given {λ }, and can be decomposed into individua therma unit subprobems and a bidding subprobem A two-eve agorithm is deveoped, where at the ow eve, individua subprobems are soved, and at the high eve, {λ } is updated 4 Soutions of herma Subprobems A herma Subprobem is min ti,with ti min { C ti (p ti λp ti } pti pti t his minimization is subject to individua unit constraints With {λ } given, the subprobem is deterministic, and can be soved by using the method presented in [0] 4 he Soution of the idding Subprobem he bidding subprobem is { λ [ min, with E p p A ] p,a,b t λ [ p p A ]} ( n (, λ and p A are obtained from SO scheduing as presented in Section 3, and depend on bids of participants his bidding subprobem is therefore stochastic n the foowing, the deterministic version of the subprobem wi be soved first he stochastic version can be simiary soved except that the expectation of is optimized Foowing the derivations of Section 3, three cases wi be considered, ie, 0 < p A < p A, p A 0, and p A pa ( t

4 Case : Accepted eve in ound ( 0 < p A < p A he degeneracy of the bidding subprobem ( wi be anayzed first, and then a numerica method to obtain a soution is presented Soution Degeneracy Anaysis From the SO oad baance constraints (5, the net energy exchange between Participant and the market is p p A [p A p ] ( y substituting ( into (, can be rewritten as [ λ λ ] [ p A p ] (3 t y substituting (0, (3 and (4 into (3, becomes { c λ [ c λ + c 0 ] λ + λc0 } t t is separabe in time o obtain its minimum, the partia derivatives with respect to a, b and p are set to zeros, ie, P where a λ b λ λ λ 0, a λ 0, b λ P 0, λ c0 + p cb, a [c a + ] λ, and b ca + λ a p ca + (4 (5 (6 λ t is cear that cannot be zero, therefore, the b simutaneous equations (4, (5 and (6 degenerate to c λ [c λ + c0 ] 0 (7 λ With three variabes and one constraint (7 for each t, the bidding subprobem has an infinite number of soutions he degeneracy can be seen from Figure 3 with the case of two participants Suppose that ine is the bidding price curve of Participant, and ine is an optima bidding strategy of Participant with optima a, b, p A, p and λ Another bid of Participant is an equivaent soution if it satisfies 0 < b < λ and p p - (p A - p A because it resuts in the same energy cearing price λ and the net energy exchange (p - p A λ ($/W λ p A p A p A p A (W Fig 3 Degeneracy of the Deterministic idding Subprobem Obtaining a Soution Having shown the degeneracy of ( above, it is straightforward to obtain one of its soutions Substituting λ in (5 and p A in (6 into (, the subprobem cost at hour t can be written as c0 a + p a + b λ c a + (8 c0 / + p cb / p ca + o minimize, any two of its three decision variabes a, b, are p are fixed first, and the third one is optimized by a gradient method Case: Zero Accepted eve (p A 0 n this case, the accepted eve for Participant is zero, therefore the bidding subprobem cost is obtained by substituting (7 and (8 into (: c0 p + λ P (9 c c inimizing (9, the soution is p [ λc - c0 ] / 4 (30 Participant purchases p from the market at the energy cearing price of the hour Case 3: aximum Accepted eve ( p A p A n this case, Participant s accepted eve p A reaches the maximum, and the ECP is determined by bids of others A soution to this bidding subprobem is to set a equa to 0, and b equa to Participant s margina cost without bidding his eads to p A p A if Participant s generation cost is ow

5 he Stochastic idding Subprobem and Soution Now consider the stochastic version Foowing ( and (8, the subprobem is changed to J j j min E[ ] p (3 j n the above, j is simiar to in (8, and E[ ] is a function of a, b and p With a derivation simiar to that for the deterministic case, it can be shown that (5 and (6 degenerate to one as can be seen from Figure 4 Suppose that Participant has two possibe bidding prices, ine (bidding at ow prices and H (bidding at high prices, and ine is an optima bidding strategy for Participant Another bid of Participant is an equivaent soution if it satisfies 0 < b < λ, a a and p p - (p A - p A he difference between this and the deterministic case is that a is required to be equa to a so that ine is parae to ine, and p aso satisfies p p - (p AH - p AH to resut in the same expected net energy exchange n soving the subprobem, b is fixed, and p and a are optimized using the gradient method λ ($/W λ H H λ p A pah p AH pa pa p AH pa (W Figure 4 Degeneracy of the Stochastic idding Subprobem 43 Update of utipiers at the High eve utipiers are updated to maximize dua function φ(λ : ax φ( λ λ,with φ( λ p in,a,b,p 0 ti, (3 where is defined in (0 With a given set of subprobem soutions obtained at the ow eve, this is a deterministic probem he subgradient of φ(λ is a vector g λ, and the t-th eement is g λ pd p ti ( p p A (33 i he dua probem is usuay soved by a subgradient method [,, 3], and is soved in this paper by the bunde trust region method (R presented in [4, 5] to improve the convergence R is a kind of bunde method that accumuates subgradients obtained thus far in a bunde, and use a convex combination of these subgradients to find a search direction Obtaining the convex combination coefficients invoves quadratic programming that is recursivey soved in R to reduce time requirements 5 Numerica Resuts he method has been impemented in C++ based on our hydrotherma scheduing code [8, 9, and 0] A data set provided by Northeast Utiities (NU is used to demonstrate the capabiities of the method in handing various market situations o simpify testing, a other market participants are aggregated as Participant with three possibe bidding strategies, bidding ow (, bidding medium (, and bidding high (H each with the same probabiity /3 he High vaue for a is 009, edium 005, and ow 00 Participant s b and p are varied to represent different market situations With Case as the base, four additiona cases are created where b as a percentage of Participant s origina margina cost (without bidding and p as a percentage of Participant s origina oad are isted in abe he method is compared with the mean method which considers Participant s bidding mode as deterministic with each parameter set to its mean vaue Comparison of testing resuts for Participant based on 00 simuation runs is presented in abe Case represents a voatie market with a arge variance on p, and the saving of the stochastic method over the mean method is increased as compared with the base case Case 3 aso represents a voatie market with arge variance on b, and the saving is increased as compared with the base case Cases and 3 therefore show that the method works better than the mean method in voatie markets Case 4 represents a more expensive market with the mean vaue of b increased by 0%, and the saving over the mean method is 038% Case 5 represents a cheap market with the mean vaue decreased by 40%, and the saving is 05% Cases 4 and 5 therefore show that the method works we in both expensive and cheap market situations he average CPU time for the mean method is 70 seconds, and that of the stochastic method is about 95 seconds he CPU time requirements are cose because both methods sove the bidding subprobem using a gradient method, and there is ony one stochastic bidding subprobem abe idding Parameters of Participant b (% p (% Case H H b (%: b as a percentage of Participant s margina cost without bidding; p (%: p as a Percentage of Participant s oad

6 ab Cost Comparison of ean ethod and Stoch ethod Case ean eth ($ Stoc eth ($ Savings (% 06,69 05, % 00,930 00, % 3 0,40 0,43 097% 4 03,076 0,68 038% 5 05,334 04,800 05% 6 Concusions uit on an existing hydrotherma scheduing approach, an innovative mode and an efficient agrangian reaxationbased method are presented to sove the bidding and sefscheduing probem Numerica testing shows that the method effectivey handes various market situations Acknowedgment his work is supported in part by the Nationa Science Foundation under Grants ECS and a grant from Northeast Utiities Service Company References [] Owen, G, Game heory, hird Edition, Academic Press, 995 [] Krishna, V, V C Ramesh, nteigent Agents for Negotiations in arket Games, Part and Part, Proceedings of the 0 th nternationa Conference on PCA, ay 997, Coumbus, Ohio, pp [3] Ferrero, R W, S Shahidehpour, V C amesh, ransaction Anaysis in Dereguated Power Systems Using Game heory, EEE rans on Power Syst, Vo, No 3, August 997, pp [4] Hao, S, G A Angeidis, H Singh, A D Papaexopouos, Consumer Payment inimization in Power Poo Auctions, EEE rans on Power Syst, Vo 3, No 3, August 998, pp [5] Avey,, D Goodwin, X a, D Streiffert, and D Sun, A Security-Constrained id-cearing System for the New Zeaand Whoesae Eectricity arket, EEE rans on Power Syst, Vo 3, No, ay 998, pp [6] Avarado F, arket Power: A Dynamic Definition, [7] Gross, G, D Finay, Optima idding Strategies in Competitive Eectricity arkets, uiucedu/pubications/summary/ece/powerhtm [8] Guan, X, P uh, H Yan, and P Rogan, Optimization-based Scheduing of Hydrotherma Power Systems with Pumped-Storage Units, EEE rans on Power Syst, Vo 9, No, ay 994, pp [9] Guan, G, P uh and Zhang, Noninear Approximation ethod in agrangian Reaxation-ased Agorithms for Hydrotherma Scheduing, EEE rans on Power Syst, Vo 0, No, ay 995, pp [0] Guan, X, P uh, H Yan, and J A Amafi, An Optimization-ased ethod for Unit Commitment, Eectric Power & Energy Syst Vo 4, 99, pp 9-7 []Shaw, J, A Direct ethod for Security-Constrained Unit Commitment, EEE rans on Power Syst, Vo 0, No 3, August 995, pp [] Wang, S J, S Shahidehpour, D S Kirschen, S okhtari, G D risarri, Short-erm Generation Scheduing with ransmission and Environmenta Constraints Using an Augmented agrangian Reaxation, EEE rans on Power Syst, Vo 0, No 3, August 995, pp [3] Cohen, A, S H Wan, A ethod for Soving the Fue Constrained Unit Commitment Probem, EEE rans on Power Syst, Vo PWRS-, August 987, pp [4]Zhang D, P uh, and Y Zhang, A unde ethod for Hydrotherma Scheduing, to appear in EEE rans on Power syst [5] Schramm, H, J Zowe, A Version of the unde ethod for inimizing a nonsmooth Function: Conceptua dea, Convergence Anaysis, Numerica Resuts, SA J Optimization, Vo, No, February 99, pp -5 Daoyuan Zhang received his E degree in Contro Engineering from singhua University, P R China in 989 He had been working on power systems scheduing and microprocessor appications from 989 to 995 n 995, he joined the Department of Eectrica Engineering, University of Connecticut where he got his PhD degree in 998 He is currenty a software engineer at Ascend Communications Yajun Wang was born in Shenyang, P R China on Dec 6, 97 He received his S degree in Eectrica Engineering from singhua University, eijing, P R China in 995 Currenty he is a PhD candidate in the Department of Eectrica & System Engineering, University of Connecticut Peter uh (S'77-'80-S'9-F 95 received his S degree in Eectrica Engineering from Nationa aiwan University, aipei, aiwan, Repubic of China, in 973, the S degree in Aeronautics and Astronautics from, Cambridge, assachusetts, in 977, and the PhD degree in Appied athematics from Harvard University, Cambridge, assachusetts, in 980 Since 980, he has been with the University of Connecticut, and currenty is the Director for the ayor ooth Center for Computer Appications and Research, and a Professor in the Department of Eectrica and Systems Engineering His major interests incude schedue generation and reconfiguration for manufacturing systems and power systems Dr uh is a Feow of EEE, and an Editor of the EEE rans Robotics and Automation