Universitat Politècnica de Catalunya (UPC) - BarcelonaTech

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1 Solving electricity market quadratic problems by Branch and Fix Coordination methods. F.-Javier Heredia 1,2, C. Corchero 1,2, Eugenio Mijangos 1,3 1 Group on Numerical Optimization and Modeling, Universitat Politècnica de Catalunya (UPC) - BarcelonaTech 2 Department of Statistics and Operations Research Universitat Politècnica de Catalunya (UPC) - BarcelonaTech 3 Department of Applied Mathematics, Statistics and Operations Research Universidad del Pais Vasco (UPV/EHU) Project DPI , Ministry of Science and Innovation, Spain F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

2 1 Iberian Electricity Market F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

3 1 Iberian Electricity Market Iberian Electricity Market (MIBEL) Day-Ahead Market (DAM) in the MIBEL GenCo s optimal DAM bid problem 2 Iberian Electricity Market (MIBEL) Day-Ahead Market (DAM) in the MIBEL GenCo s optimal DAM bid problem 3 4 F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

4 Iberian Electricity Market (MIBEL) Iberian Electricity Market (MIBEL) Day-Ahead Market (DAM) in the MIBEL GenCo s optimal DAM bid problem The MIBEL (created in 2007) joins Spanish and Portuguese electricity system. The spot market operator (day-ahead, reserve and intraday markets), called OMIE, is located in Spanish.( This Derivatives Market has its own market operator called OMIP, located in Portugal ( The quality, security and reliability of the electric supply is guaranteed by the Independent System Operator (ISO). F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

5 Markets in the MIBEL Organized markets -Virtual PowerPlantsauctions(EPE) - Distribution auctions(sd) - International Capacity Interconnection auctions - International Capacity Interconnection nomination Iberian Electricity Market (MIBEL) Day-Ahead Market (DAM) in the MIBEL GenCo s optimal DAM bid problem Medium/ Long Term DayD-1D-1 Bilateral Contracts Derivatives Market Day-Ahead Market DayD Intradaily Markets Non organized markets Derivatives Market Bilateral Contracts Day-Ahead Market - National BC beforethe spot market - InternationalBC beforethe spot market -NationalBC afterthespotmarket Physical Futures Contracts Financial and Physical Settlement. Positions are sent to OMEL s Mercado Diario for physical delivery. Financial Futures Contracts OMIClear cash settles the differences between the Spot Reference Price and the Final Settlement Price Organized markets -Virtual PowerPlantsauctions(EPE) - Distribution auctions(sd) - International Capacity Interconnection auctions - International Capacity Interconnection nomination Non organized markets - National BC beforethe spot market - InternationalBC beforethe spot market -NationalBC afterthespotmarket Organized markets -Virtual PowerPlantsauctions(EPE) - Distribution auctions(sd) - International Capacity Interconnection auctions - International Capacity Interconnection nomination Non organized markets - NationalBC beforethe spot market - InternationalBC beforethe spot market - NationalBC afterthe spot market Physical Futures Contracts Financial and Physical Settlement. Positions are sent to OMEL s Mercado Diario for physical delivery. Financial Futures Contracts OMIClear cash settles the differences between the SpotReference Price and the Final Settlement Price Day-Ahead Market Hourly action. The matching procedure takes place 24h before the delivery period. Physical futures contracts are settled through a zero price bid. Day-Ahead Market Physical Futures Contracts Hourly F.-Javier action. Heredia, The matching Cristina procedure Corchero, takes Eugenio place Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

6 Day-Ahead Market mechanism Derivatives physical products Bilateral contracts execution Buyers and sellers offers VPP options GenCo bids DAY-AHEAD MARKET ANCILLARY SERVICES INTRADAY MARKETS International conexion use rights Distribution contracts auction Buyers offers Buyers and sellers offers Iberian Electricity Market (MIBEL) Day-Ahead Market (DAM) in the MIBEL GenCo s optimal DAM bid problem The Day-Ahead Market (DAM) is the most important part of the electricity market (78% of the total system demand traded through DAM on 2009). The objective of this market is to carry out the energy transactions Technical Restrictions for the next day by means of the selling and buying offers presented by the market agents. The DAM is formed up by twenty-four hourly auctions that are cleared simultaneously between 10:00 and 10:30am of day D-1. F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

7 Iberian Electricity Market (MIBEL) Day-Ahead Market (DAM) in the MIBEL GenCo s optimal DAM bid problem The GenCo s optimal DAM bid problem The GenCo s optimal DAM bid problem considers a Price-Taker generation company with: Derivatives physical products Bilateral contracts execution GenCo bids DAY-AHEAD MARKET Buyers offers A set of thermal generation units, I, with quadratic generation costs, start-up and shut-down costs and minimum operation and idle times. Each generation unit can submit sell bids to the T = 24 auctions of the DAM. A set of S scenarios for the spot price of auction t, λ s t. A set of physical futures contracts, F, of energy L FC j j F. A pool of bilateral contracts B of energy L BC k, k B. F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

8 Objectives Iberian Electricity Market The objective of the study is to decide: Iberian Electricity Market (MIBEL) Day-Ahead Market (DAM) in the MIBEL GenCo s optimal DAM bid problem the optimal economic dispatch of the physical futures and bilateral contract among the thermal units the optimal bidding at Day-Ahead Market abiding by the MIBEL rules the optimal unit commitment (binary on/off state) of the thermal units maximizing the expected Day-Ahead Market profits taking into account futures and bilateral contracts. F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

9 1 Iberian Electricity Market 2 Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results 3 4 F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

10 Variables Iberian Electricity Market Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results First stage variables: for each time period t T and thermal unit i I Unit commitment: u it {0, 1}, c u it, cd it Zero price offer bid : q it Scheduled energy for futures contract j: f itj Scheduled energy for bilaterals contract: b it j F Second stage variables: for each t T, i I and scenario s S Matched energy: p M,s it Total generation: p s it F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

11 Physical Future Contracts modeling Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results A base load Futures Contract j F consists in a pair (L FC, λ FC Lj FC : amount of energy (MWh) to be procured each interval of the delivery period by the set U j of generation units. λj FC : price of the contract (ce/mwh). Physical future contract contraints: f itj = L FC j, j F, t T i U j f itj 0, j F, i I, t T j j ) F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

12 Bilateral Contracts modeling Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results A base load Bilateral Contract k B consists in a pair (L BC k, λbc k ) L BC k : amount of energy (MWh) to be procured each interval t of the delivery period. λ BC k : price of the contract (ce/mwh). Bilateral contract constraints: b it = i I k B L BC k, t T 0 b it P i u it, i I, t T F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

13 Day-Ahead Market bidding model Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results The energies L FC j and L BC k should be integrated in the MIBEL s day-ahead bid respecting the following bidding rules: (BR1) To guarantee its inclusion in the operational programming, any committed unit i (u it = 1) would bid its minimum generation level P i at zero price (instrumental price bid q it ). (BR2) If generator i contributes with f itj MWh at period t to the coverage of the FC j, then the energy f itj must be included into the instrumental price bid q it. (BR3) If generator i contributes with b it MWh at period t to the coverage of the BCs, then the energy b it must be excluded from the bid to the DAM. Unit i can offer its remaining production capacity P i b it to the pool. F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

14 Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results Day-ahead market bidding model: constraints Matched energy: (BR3) p M,s P i u it b it, i I, t T, s S it (BR1) p M,s q it, i I, t T, s S Instrumental price bid: it (BR1+3) q it P i u it b it, i I, t T (BR2) q it j i U j f itj, t T, i I Total energy generation: pit s = b it + p M,s, t T, i I, s S it F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

15 Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results Start-up/shut-down costs and up/down time constraints Start-up and shut-down costs constraints: cit u ci on [u it u i,(t 1) ] t T \ {1}, i I cit d ci off [u i,(t 1) u it ] t T \ {1}, i I Minimum up/down time constraints: where: u X (t on, t off ) ci on and ci off are the constant start-up and shut-down costs, respectively, of thermal unit i. t on i (resp. ti off ) is minimum operation (iddle) time.. F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

16 Objective function Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results Maximization of the day-ahead market clearing s benefits ( max cit u cit d + p,q,f,b t T i I + [ ] ) P s λ s tp M,s it (citu b it + ci l pit s + c q i (pit) s 2 ) s S Incomes from Futures and bilateral contracts (constant): Futures contracts: ( ) t T j J λ FC j λ t L FC t Bilateral contracts: T k B λbc k LBC k F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

17 Summary of the model Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results Problem DAM-FBC (Day-Ahead Market bid with Futures and Bilateral Contracts) Max E[Profit from the Day-ahead market] s.t. Physical future and bilateral contracts coverage Day-ahead market rules Start-up/shut-down costs and minimum up/down time (Two-stage mixed integer quadratic stochastic programming problem) F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

18 Results: optimal bidding curve Spot price (c/kwh) Thermal Unit 6 Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results "! $# Energy(x1000kWh) Optimal bidding curve for thermal unit 6 at interval 18 F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

19 Branch and Fix Coordination / Perspective 350 Cuts 160 Unit commitment, qit 563 and 250 b it Energy(x1000kWh) Energy(x1000kWh) Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results Unit 1 Unit 2 Unit 3 Unit 4 Unit Unit Hour Unit 7 b* q* 313 F.-Javier Heredia, 110 Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Unit Berlin, 8 September /33 (b) (b) Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9

20 Variables Futures and Bilateral contracts model Day-Ahead Market bidding model Thermal units operation model Objective function Problem DAM-FBC Results Economic dispatch of each futures contracts f Weekly contract MWh Monthly contract MWh Yearly contract MWh Unit 1 Unit 3 Unit 2 Unit 4 Unit 5 Unit 6 Unit 3 Unit 7 Unit 8 Unit 9 0 F.-Javier Heredia, Cristina Corchero, Eugenio2 Mijangos 4 6- gnom.upc.edu th 16 IFIP 18 TC7 20 Conference, Berlin, September /33 (a) itj

21 1 Iberian Electricity Market 2 QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results 3 QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results 4 F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

22 QBFC method Iberian Electricity Market QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results Model DAM-FBC can be rewritten as the so-called Deterministic Equivalent Model (DEM) minimize c t δ + ω Ω p ω q ω (x, y ω ) subject to : [ ] δ l a A u a, x δ T ω x ut ω, ω Ω, l ω t y ω x 0, y y ω y, ω Ω, δ {0, 1} n δ, where δ = u, x = (c u, c d, b, f, q), y = (p M, p), q(x, y) = b t xx + b t y y+ +y t Q yy y, and Q yy being a diagonal matrix. F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

23 QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results The compact representation DEM can be written as a splitting variable representation (Escudero et al. (2009)). (MIQP) minimize p ω (c t δ ω + q ω (x ω, y ω )) subject to : ω Ω l a A Nonanticipativity constraints: l ω t [ δ ω x ω T ω ] δ ω x ω y ω u a, ω Ω, ut ω, ω Ω, x ω 0, y y ω y, δ ω {0, 1} n δ, ω Ω, (NAC δ ) δ ω δ ω = 0, ω, ω Ω : ω ω, (NAC x ) x ω x ω = 0, ω, ω Ω : ω ω, F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

24 QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results In this method DEM is solved by using a Branch-and-Fix-Coordination scheme (BFC), an special application of the branch cut algorithm to solve coordinately the problem associated to each scenario ω Ω in suach a way that the NAC constraints are implicitly satisfied by the Twin-Node-Families concept (TNF) (Alonso-Ayuso et al. (2003)). A similar approach to that suggested by Escudero et al. (2009) is used in this method to coordinate the selection of the branching node and branching variable for each scenario-related BF tree. More details in the contributed talk An algorithm for two-stage stochastic quadratic problems (CT7 Stochastic Optimization I, Thursday). F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

25 QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results In order to gain computational efficiency we can take scenario clusters; i.e., instead of solving a submodel for each scenario ω Ω we can solve a submodel for each scenario cluster p = 1,..., p. (MIQP p ) minimize subject to : p ω (c t δ p + q ω (x p, y ω )) ω Ω p [ ] l a A l ω t δ p x p T ω δ p x p y ω u a, ut ω, ω Ω p x p 0, y y ω y, ω Ω p, Submodels are linked by NAC δ p δ p = 0 and x p x p = 0, p, p {1,..., p}, p p. δ p {0, 1} n δ F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

26 Perspective Cuts Iberian Electricity Market QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results Problem (MIQP p ) is a Mixed-Integer Quadratic Program (MIQP), which is difficult to solve efficiently, especially for large-scale instances. A possibility is to approximate the quadratic functions q ω (x p, y ω ) by the outer polyhedral approximation defined by the so called perspective cuts (Frangioni and Gentile, 2006), so that this problem can be solved as a Mixed-Integer Linear Program (MILP) by general-purpose MILP solvers. F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

27 The PC formulation of problems MIQP p are: { ( n ) } (MILP p ) min bxx t + v ω s.t.: ω Ω p p ω vi ω i=1 QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results i αyi ω + βδi ω, (α, β) Ci ω, i {1,..., n}, ω Ω p ( ) [ ] l a A l ω t δ p x p T ω δ p x p y ω u a, ut ω, ω Ω p x p 0, y ω [y, y], ω Ω p, and δ p {0, 1} n δ where the linear inequalities ( ) define the perspective cuts approximation to q ω (x p, y ω ). F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

28 QBFC s implementation QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results These two algorithmic alternatives have been considered: QBFC: coordination of δ in the TNF of the BF trees for clusters p {1,..., p} without using PCs. QBFC-PC: coordination of δ in the TNF of the BF trees for clusters p {1,..., p} using PCs. Both algorithmic alternatives has been compared using a set of 15 instances of the DAM-FBC problem. These methods have been implemented in C++ with the help of Cplex 12.1 to solve both the (MIQP p ) and (MILP p ) subproblems. The tests have been performed on HP with Intel(R) Core(TM)2 Quad CPU Q GHz 4 CPU under SUSE Linux Enterprise Desktop 11 (x86 64). F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

29 QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results Prob. S T # var QBFC # var QBFC PC # bin # constr P P P P P P P P P P For every problem F = BC = 2 and I = 4. If we use PCF, the problem increases the number of variables in m = T I S and the number of constraints in 2 m. F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

30 QBFC method QBFC with Perspective Cuts (PC) QBFC s implementation and results QBFC PC Prob. p QBFC (sec.) QBFC-PC (sec.) QBFC ( ) # PC P P P P P P P P P P ( ) the running time with PC is a 30% of the running time without PC (average). F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

31 1 Iberian Electricity Market F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

32 Iberian Electricity Market We have presented an Optimal Bidding Model for a price-taker generation company operating both in the MIBEL Derivatives and Day-Ahead Electricity Market (DAM-FBC). The model developed gives the producer: The optimal bid for the spot market. The optimal allocation of the physical futures and bilateral contracts among the thermal units The unit commitment following in detail the MIBEL rules. The (DAM-FBC) has been solved both with the standard BFC method and with a PC variation which reduces the running time to a 30% on the average. F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33

33 Solving electricity market quadratic problems by Branch and Fix Coordination methods. F.-Javier Heredia 1,2, C. Corchero 1,2, Eugenio Mijangos 1,3 1 Group on Numerical Optimization and Modeling, Universitat Politècnica de Catalunya (UPC) - BarcelonaTech 2 Department of Statistics and Operations Research Universitat Politècnica de Catalunya (UPC) - BarcelonaTech 3 Department of Applied Mathematics, Statistics and Operations Research Universidad del Pais Vasco (UPV/EHU) Project DPI , Ministry of Science and Innovation, Spain F.-Javier Heredia, Cristina Corchero, Eugenio Mijangos - gnom.upc.edu 25th IFIP TC7 Conference, Berlin, September /33