A new primal-dual framework for European day-ahead electricity auctions

Transcription

1 A new primal-dual framework for European day-ahead electricity auctions Mehdi Madani*, Mathieu Van Vyve** *Louvain School of management, ** CORE Catholic University of Louvain Mathematical Models and Methods for Energy Optimization Workshop, Budapest University of Technology and Economics, September 2014

2

3 1 Background information Day-ahead Electricity Markets Equilibrium with block orders? : Examples Primal Program (welfare maximizing dispatch) Duality and Linear Equilibrium Prices 2 Get rid of the subset of compl. constr.? Three ways Linearise each of them... Equality of objective functions + linearisation of quadratic terms New primal-dual framework... 3 Algorithmic application - Strengthened (locally valid) Benders cuts 4 Minimizing opportunity costs, Maximizing the traded volume, etc 5 Numerical results Maximizing welfare, new approach Opportunity costs vs Welfare 6 Conclusions and Extensions

4 Introduction Day-ahead Markets: 24 periods (23 or 25 once a year) several areas/locations for bids + network constraints Both demand and offer bids (elastic demand) We are interested in uniform/linear prices. Here: only one location but everything holds for any linear transmission network representation (Flow-based, ATC, etc)

5 Main kinds of bids - CWE region (EPEX, APX), Nordpool - : Hourly bids - (Monotonic) DEMAND / SUPPLY bid curves Block bids, i.e. binary bids Span multiple periods and fill-or-kill condition : bid must be entirely accepted or rejected.

6 Well-behaved, continuous setting:

7 Well-behaved convex context (convexity of feas. set & welfare function): Market equilibrium Welfare maximization (True more generally when considering spatially separated markets, and remarkable according to Paul Samuelson in [a])

8 Equilibrium with block orders?... Welfare maximizing solution Traded volume maximizing solution Welfare maximizing solution Opportunity costs minimizing solution

9 Matching in the convex case as a simple LP: max x i 0 (100 15)x 1 + (50 10)x 2 (120 5)x 3 (30 12)x 4 s.t. x i 1 i {1, 2, 3, 4} [s i ] (1) 100x x 2 = 120x x 4 [p m ] (2) welfare optimal solution = 1100 with x 1 = 1, x 2 = 2 5, x 3 = 1, x 4 = 0 p m gives the equilibrium market clearing price

10 Primal (welfare maximizing) program: max x i,y j Q i P i x i + i j Q j P j y j subject to: x i 1 i I [s i ] (3) y j 1 j J [s j ] (4) Q i x i + Q j y j = 0 [p m ] (5) i j x i, y j 0, y j Z. (6) Q < 0 for sell orders and Q > 0 for buy orders!

11 Classical MPCC formulation Of European market rules Primal constraints (feasible dispatch) Walrasian equilibrium Dual program (prices) Paradoxically rejected block orders allowed Related compl. constraints

12

13 Primal (welfare maximizing) program with blocks fixed: max x i,y j subject to: Q i P i x i + Q j P j y j i x i 1 i I [s i ] Q i x i = Q j y j [p m ] i j x i 0, j Dual with these blocks fixed: min s i,p m i s i p m ( j Q j y j ) subject to: s i + Q i p m Q i P i i I [x i ] s i 0 Complementarity Constraints: s i (1 x i ) = 0 i I x i (s i + Q i p m Q i P i ) = 0 i I

14

15 1 Background information Day-ahead Electricity Markets Equilibrium with block orders? : Examples Primal Program (welfare maximizing dispatch) Duality and Linear Equilibrium Prices 2 Get rid of the subset of compl. constr.? Three ways Linearise each of them... Equality of objective functions + linearisation of quadratic terms New primal-dual framework... 3 Algorithmic application - Strengthened (locally valid) Benders cuts 4 Minimizing opportunity costs, Maximizing the traded volume, etc 5 Numerical results Maximizing welfare, new approach Opportunity costs vs Welfare 6 Conclusions and Extensions

16 First way: linearise all three kinds of compl. constraints real instances: more than hourly bids aux. bin. vars.!! Not tractable for real large-scale instances! s i (1 x i ) = 0 x i (s i + Q i p m Q i P i ) = 0 y j (s j + Q j p m Q j P j ) = 0 With big-m s and z i, v i {0, 1}, i.e. 2 Hourly bids auxiliary bin. vars.: 0 (1 x i ) Mz i 0 s i M(1 z i ) 0 x i Mv i 0 (s i + Q i p m Q i P i ) M(1 v i ) 0 s j + Q j p m Q j P j M(1 y j ) (since y j {0, 1})

17 Second way: linearise quad. terms in equality of objectives Proposition of Zak. et al. [c]

18 Second way: linearise quad. terms in equality of objectives Proposition of Zak. et al. [c] Again, not tractable for real large-scale instances (cf. paper) Well-known trick: To linearise p m y j, replace it by z j, adding constraints with big-ms: z j My j z j p m z j p m M(1 y j ) If n coupled markets, n prices, and n block orders aux. vars. e.g in the CWE region: 24 periods, 4 counties 96 submarkets and about 700 block orders auxiliary vars.

19 Third way: yields a useful primal-dual framework *upper bound* on the actual loss of executed order j

20 Core European market model, new formulation: No auxiliary variables at all & tractable for large-scale instances max Q i P i x i + Q j P j y j x i,y j,p m,s i,s j i j subject to: x i 1 i I [s i ] (7) y j 1 j J [s j ] (8) Q i x i + Q j y j = 0 [p m ] (9) i j x i, y j 0, y j Z (10) s i + Q i p m Q i P i i I [x i ] (11) s j + Q j p m Q j P j M j (1 y j ) j J [y j ] (12) Q i P i x i + Q j P j y j s i + s j (13) i j i j s i, s j 0, param. M j >> 0 (14)

21 Third way: yields a useful primal-dual framework Consider a block order selection J = J 0 J 1 Primal: subject to: max x i,y j Q i P i x i + i j Q j P j y j x i 1 i I [s i ] (15) y j 1 j J [s j ] (16) y j0 0 j 0 J 0 J [d 0j0 ] (17) y j1 1 j 1 J 1 J [d 1j1 ] (18) Q i x i + Q j y j = 0, [p m ] (19) i j x i, y j 0 (20)

22 Dual: min s i + i j subject to: s j j 1 J 1 d 1j1 s i + Q i p m Q i P i i I [x i ] (21) s j0 +d 0j0 + Q j 0 p m Q j 0 P j 0 j 0 J 0 [y j0 ] (22) s j1 d 1j1 + Q j 1 p m Q j 1 P j 1 j 1 J 1 [y j1 ] (23) s i, s j, d j0, d j1, u m 0 (24) Complementarity constraints s i (1 x i ) = 0 i I (25) s j0 (1 y j0 ) = 0 j J (26) s j1 (1 y j1 ) = 0 j J (27) x i (s i + Q i p m Q i P i ) = 0 i I (28) y j0 (s j0 +d 0j0 + Q j 0 p m Q j 0 P j 0 ) = 0 j J 0 (29) y j1 (s j1 d 1j1 + Q j 1 p m Q j 1 P j 1 ) = 0 j J 1 (30) y j0 d 0j0 = 0, (1 x j1 )d 1j1 = 0 j 0 J 0, j 1 J 1 (31)

23 With primal, dual and complementarity constraints: d 0j d 1j is an *upper bound* on the opportunity cost of order j is an *upper bound* on the actual loss of (executed) order j Block order selection J = J 0 J 1 not known ex ante. Decide a selection J = J 0 J 1 according to some objective: Using equality of objective functions to enforce complementarity conditions... and using a dispatcher

24 *upper bound* on the actual loss of executed order j Strong duality <-> relaxed complementarity constraints

25 Core European market model, new formulation: No auxiliary variables at all & tractable for large-scale instances max Q i P i x i + Q j P j y j x i,y j,p m,s i,s j i j subject to: x i 1 i I [s i ] (32) y j 1 j J [s j ] (33) Q i x i + Q j y j = 0 [p m ] (34) i j x i, y j 0, y j Z (35) s i + Q i p m Q i P i i I [x i ] (36) s j + Q j p m Q j P j M j (1 y j ) j J [y j ] (37) Q i P i x i + Q j P j y j s i + s j (38) i j i j s i, s j 0, param. M j >> 0 (39)

26 1 Background information Day-ahead Electricity Markets Equilibrium with block orders? : Examples Primal Program (welfare maximizing dispatch) Duality and Linear Equilibrium Prices 2 Get rid of the subset of compl. constr.? Three ways Linearise each of them... Equality of objective functions + linearisation of quadratic terms New primal-dual framework... 3 Algorithmic application - Strengthened (locally valid) Benders cuts 4 Minimizing opportunity costs, Maximizing the traded volume, etc 5 Numerical results Maximizing welfare, new approach Opportunity costs vs Welfare 6 Conclusions and Extensions

27 New primal-dual formulation - [b] - M. Van Vyve & M. No auxiliary variables at all! Could then derive a powerful Benders decomposition with strengthened Benders cuts improving on Martin-Muller-Pokutta [d] (both propositions: projection on the space of primal variables) Martin-Muller-Pokutta: Branch-and-Cut with exact (globally valid) no-good cuts: (1 y j ) + y j 0 j y j =1 j y j =0 with the new stuff: recovering these globally valid cuts + stronger locally valid cuts: (1 y j ) 0 j y j =1 needed with piecewise linear bid curves ( quad. prog. setting, using convex quad. prog. duality) State-of-the-art

28 1 Background information Day-ahead Electricity Markets Equilibrium with block orders? : Examples Primal Program (welfare maximizing dispatch) Duality and Linear Equilibrium Prices 2 Get rid of the subset of compl. constr.? Three ways Linearise each of them... Equality of objective functions + linearisation of quadratic terms New primal-dual framework... 3 Algorithmic application - Strengthened (locally valid) Benders cuts 4 Minimizing opportunity costs, Maximizing the traded volume, etc 5 Numerical results Maximizing welfare, new approach Opportunity costs vs Welfare 6 Conclusions and Extensions

29 *upper bound* on the actual loss of executed order j Strong duality <-> relaxed complementarity constraints

30 Other optimization problems under European rules d 0j 0 *upper bound* on the opportunity cost of block order j... Minimizing opportunity costs? min d 0j Minimizing # PRBs? min z 0j s.t. Mz 0j d 0j & z 0j {0, 1}, j J Maximizing the traded volume? max Q i x i + Q j y j i Q i >0 j Q j >0

31 1 Background information Day-ahead Electricity Markets Equilibrium with block orders? : Examples Primal Program (welfare maximizing dispatch) Duality and Linear Equilibrium Prices 2 Get rid of the subset of compl. constr.? Three ways Linearise each of them... Equality of objective functions + linearisation of quadratic terms New primal-dual framework... 3 Algorithmic application - Strengthened (locally valid) Benders cuts 4 Minimizing opportunity costs, Maximizing the traded volume, etc 5 Numerical results Maximizing welfare, new approach Opportunity costs vs Welfare 6 Conclusions and Extensions

32 Numerical results: welfare optimization Real data of 2011, thanks to Apx-Endex and Epex Spot Belgium, France, Germany and the Nederlands, 24 time slots time limit: 10 min., about cont. vars, 600/700 bin. vars Branch-and-cut in AIMMS, using Cplex 12.5 with locally valid lazy constraints callbacks (not in Gurobi) platform: windows 7 64, i5 with GHz, 4 GB RAM Stepwise preference curves (linearisation): Solved instances Running time Final abs. gap Nodes Cuts (solved instances, sec) (unsolved instances) (solved - unsolved) instances (solved - unsolved) instances New MILP formulation 84% / Decomposition Procedure 72.78% Quadratic setting: Solved instances Running time Final abs. gap Nodes Cuts (solved instances, sec) (unsolved instances) (solved - unsolved) instances (solved - unsolved) instances Decomposition Procedure 70.41% Many blocks (almost binary orders only): Solved instances Running time Final abs. gap Nodes Cuts (solved instances, sec) (unsolved instances) (solved - unsolved) instances (solved - unsolved) instances New MILP formulation 100% 4.17 / / - Decomposition Procedure 78% / / 82497

33 Opportunity costs vs Welfare Maximization CWE Region, some instances from 2011 (see [e] for more details)

34 Short conclusion and Extensions Conclusions / Extensions Well-known nowadays: MIP formulation issues really matter... framework useful both for economic modelling (min. opportunity costs, max traded volume, etc) and algorithmically (feeding Cplex or e.g. Benders decomposition to derive new cuts) The same approach (whole MIP) extends to complex bids with a Minimum income condition! (WP available within a couple of days) Many other things to say e.g. about network/spatial equilibrium, etc

35 bibliography [a] Paul A. Samuelson. Spatial price equilibrium and linear programming. The American Economic Review, 42(3), 1952 [b] M. & Van Vyve, CORE Discussion Paper 2013/74 (online) + revised version available on request (published soon) [c] E.J. Zak, S. Ammari, and K.W. Cheung. Modeling price-based decisions in advanced electricity markets. In European Energy Market (EEM), th International Conference on the, May 2012 [d] Alexander Martin, Johannes C. Müller, and Sebastian Pokutta. Strict linear prices in nonconvex european day-ahead electricity markets. Optimization Methods and Software, 2014 [e] M. & Mathieu Van Vyve, Minimizing opportunity costs of paradoxically rejected block orders in European day-ahead electricity markets, European Energy Market (EEM), 2014 (see IEEE Xplore) contact: mehdi.madani@uclouvain.be