MODELING THE ELECTRICITY SPOT MARKETS
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1 .... MODELING THE ELECTRICITY SPOT MARKETS Özgür İnal Rice University Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 1/27
2 . Motivation Modeling the game the electricity generating firms play Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 2/27
3 . Motivation Modeling the game the electricity generating firms play Understanding how firms in Texas electricity spot market bid - is there room for improvement? Predicting how the supply functions that firms provide relate to their costs and other factors Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 2/27
4 . Supply function equilibrium in the literature S. Grossman (1981) Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 3/27
5 . Supply function equilibrium in the literature S. Grossman (1981) Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs No uncertainty (implies too many equilibria) Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 3/27
6 . Supply function equilibrium in the literature S. Grossman (1981) Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs No uncertainty (implies too many equilibria) P.D. Klemperer & M.A. Meyer (1989) Supply Function Equilibria in Oligopoly Under Uncertainty Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 3/27
7 . Supply function equilibrium in the literature S. Grossman (1981) Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs No uncertainty (implies too many equilibria) P.D. Klemperer & M.A. Meyer (1989) Supply Function Equilibria in Oligopoly Under Uncertainty Introduce uncertainty Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 3/27
8 . Supply function equilibrium in the literature S. Grossman (1981) Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs No uncertainty (implies too many equilibria) P.D. Klemperer & M.A. Meyer (1989) Supply Function Equilibria in Oligopoly Under Uncertainty Introduce uncertainty Symmetric firms, convex costs, concave demand, Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 3/27
9 . Supply function equilibrium in the literature S. Grossman (1981) Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs No uncertainty (implies too many equilibria) P.D. Klemperer & M.A. Meyer (1989) Supply Function Equilibria in Oligopoly Under Uncertainty Introduce uncertainty Symmetric firms, convex costs, concave demand, Show the existence of a family of SFE Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 3/27
10 . Supply function equilibrium in the literature S. Grossman (1981) Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs No uncertainty (implies too many equilibria) P.D. Klemperer & M.A. Meyer (1989) Supply Function Equilibria in Oligopoly Under Uncertainty Introduce uncertainty Symmetric firms, convex costs, concave demand, Show the existence of a family of SFE Sufficient conditions for uniqueness Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 3/27
11 . Supply function equilibrium in the literature S. Grossman (1981) Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs No uncertainty (implies too many equilibria) P.D. Klemperer & M.A. Meyer (1989) Supply Function Equilibria in Oligopoly Under Uncertainty Introduce uncertainty Symmetric firms, convex costs, concave demand, Show the existence of a family of SFE Sufficient conditions for uniqueness R.J. Green & D.M. Newbery (1992) Competition in the British Electricity Spot Market Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 3/27
12 . Supply function equilibrium in the literature S. Grossman (1981) Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs No uncertainty (implies too many equilibria) P.D. Klemperer & M.A. Meyer (1989) Supply Function Equilibria in Oligopoly Under Uncertainty Introduce uncertainty Symmetric firms, convex costs, concave demand, Show the existence of a family of SFE Sufficient conditions for uniqueness R.J. Green & D.M. Newbery (1992) Competition in the British Electricity Spot Market Symmetric duopoly case, Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 3/27
13 . Supply function equilibrium in the literature S. Grossman (1981) Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs No uncertainty (implies too many equilibria) P.D. Klemperer & M.A. Meyer (1989) Supply Function Equilibria in Oligopoly Under Uncertainty Introduce uncertainty Symmetric firms, convex costs, concave demand, Show the existence of a family of SFE Sufficient conditions for uniqueness R.J. Green & D.M. Newbery (1992) Competition in the British Electricity Spot Market Symmetric duopoly case, Multiplicity of equilibria, but capacity constraints and threat of entry restrict their range. Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 3/27
14 . Key papers R. Baldick, R. Grant & E. Kahn (2004) Theory and Application of Linear Supply Function Equilibrium in Electricity Markets A. Hortaçsu & S. Puller (2008) Understanding Strategic Bidding in Multi-Unit Auctions - A Case Study of the Texas Electricity Spot Market F. N. Fritsch & R. E. Carlson (1980) Monotone Piecewise Cubic Interpolation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 4/27
15 . Key papers R. Baldick, R. Grant & E. Kahn (2004) Theory and Application of Linear Supply Function Equilibrium in Electricity Markets Asymmetric firms bidding piecewise affine supply functions, Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 5/27
16 . Key papers R. Baldick, R. Grant & E. Kahn (2004) Theory and Application of Linear Supply Function Equilibrium in Electricity Markets Asymmetric firms bidding piecewise affine supply functions, Piecewise affine case fits the empirical data better. Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 5/27
17 . Piecewise linear SFE Supply functions are piecewise affine Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 6/27
18 . Piecewise linear SFE Supply functions are piecewise affine Capacity constraints pose a problem Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 6/27
19 . Piecewise linear SFE Supply functions are piecewise affine Capacity constraints pose a problem Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 6/27
20 . Key papers, cont d A. Hortaçsu & S. Puller (2008) Understanding Strategic Bidding in Multi-Unit Auctions - A Case Study of the Texas Electricity Spot Market More general then SFE model - S i now is a function of the forward position as well. Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 7/27
21 . Key papers, cont d A. Hortaçsu & S. Puller (2008) Understanding Strategic Bidding in Multi-Unit Auctions - A Case Study of the Texas Electricity Spot Market More general then SFE model - S i now is a function of the forward position as well. S i (p, QC i ) = f i (p) + g i (QC i ) = uncertainty shifts the residual demand; does not rotate it. Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 7/27
22 . Empirical strategy of Hortaçsu & Puller Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 8/27
23 . Empirical strategy of Hortaçsu & Puller Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 8/27
24 . Empirical strategy of Hortaçsu & Puller Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 8/27
25 . Empirical strategy of Hortaçsu & Puller Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 8/27
26 . Empirical strategy of Hortaçsu & Puller Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 8/27
27 . Empirical strategy of Hortaçsu & Puller Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 8/27
28 . Empirical strategy of Hortaçsu & Puller Residual demand shifts in a parallel fashion, thanks to the additive separability assumption. Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 9/27
29 . Key papers, cont d F. N. Fritsch & R. E. Carlson (1980) Monotone Piecewise Cubic Interpolation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 10/27
30 . A stylized model N = {1, 2,..., n}: set of firms, D(p): market demand, differentiable, C i (q i ): firm i s cost function, convex and differentiable, S i (p): supply function of i RD i (p) = D(p) S j (p): residual demand facing firm i j i Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 11/27
31 . A stylized model N = {1, 2,..., n}: set of firms, D(p): market demand, differentiable, C i (q i ): firm i s cost function, convex and differentiable, S i (p): supply function of i RD i (p) = D(p) S j (p): residual demand facing firm i j i Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 11/27
32 . A stylized model N = {1, 2,..., n}: set of firms, D(p): market demand, differentiable, C i (q i ): firm i s cost function, convex and differentiable, S i (p): supply function of i RD i (p) = D(p) S j (p): residual demand facing firm i j i Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 11/27
33 . A stylized model N = {1, 2,..., n}: set of firms, D(p): market demand, differentiable, C i (q i ): firm i s cost function, convex and differentiable, S i (p): supply function of i RD i (p) = D(p) S j (p): residual demand facing firm i j i Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 11/27
34 . A stylized model N = {1, 2,..., n}: set of firms, D(p): market demand, differentiable, C i (q i ): firm i s cost function, convex and differentiable, S i (p): supply function of i RD i (p) = D(p) S j (p): residual demand facing firm i j i Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 11/27
35 . A stylized model N = {1, 2,..., n}: set of firms, D(p): market demand, differentiable, C i (q i ): firm i s cost function, convex and differentiable, S i (p): supply function of i RD i (p) = D(p) S j (p): residual demand facing firm i j i Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 11/27
36 . A stylized model N = {1, 2,..., n}: set of firms, D(p): market demand, differentiable, C i (q i ): firm i s cost function, convex and differentiable, S i (p): supply function of i RD i (p) = D(p) S j (p): residual demand facing firm i j i Each firm tries to maximize π i (p) = prd i (p) C i (RD i (p)) = p ( D(p) S j (p) ) ( C i D(p) S j (p) ) j i j i Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 11/27
37 . A stylized model π i (p) = prd i (p) C i (RD i (p)) = p ( D(p) S j (p) ) ( C i D(p) S j (p) ) j i j i FOC: S i (p) = ( p C {S i (p)} )( S j (p) D (p) ), i = 1, 2,...n j i Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 12/27
38 . A stylized model FOC: π i (p) = prd i (p) C i (RD i (p)) = p ( D(p) S j (p) ) ( C i D(p) S j (p) ) j i j i S i (p) = ( p C {S i (p)} )( S j (p) D (p) ), {S 1 (p),..., S n (p)} is an equilibrium provided that S i are nondecreasing. j i i = 1, 2,...n Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 12/27
39 . A simple 2-firm example Suppose D(p) = 80 p and i believes that j will bid the following price-quantity pairs: (0, 0), (5, 20), (10, 35), (15, 45), (20, 50), (40, 60) Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 13/27
40 . A simple 2-firm example Suppose D(p) = 80 p and i believes that j will bid the following price-quantity pairs: (0, 0), (5, 20), (10, 35), (15, 45), (20, 50), (40, 60) 4p p 5 3p + 5 p 10 S j (p) = 2p + 15 p 15 p + 30 p p 20 p 2 Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 13/27
41 . A simple 2-firm example Then, the residual market for i is MR i (q) = q q q q q q q q q q 80 Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 14/27
42 . A simple 2-firm example Then, the residual market for i is MR i (q) = q q q q q q q q q q 80 Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 14/27
43 . A simple 2-firm example Then, the residual market for i is MR i (q) = q q q q q q q q q q 80 Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 14/27
44 . Our approach Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 15/27
45 . Our approach Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 15/27
46 . Our approach F. N. Fritsch & R. E. Carlson (1980) Monotone Piecewise Cubic Interpolation gives Necessary & sufficient conditions for a cubic polynomial to be monotone on an interval, Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 16/27
47 . Our approach F. N. Fritsch & R. E. Carlson (1980) Monotone Piecewise Cubic Interpolation gives Necessary & sufficient conditions for a cubic polynomial to be monotone on an interval, An algorithm to construct a monotone and piecewise interpolant to monotone data. Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 16/27
48 . 2-firm example, cont d j bids {(0, 0), (5, 20), (10, 35), (15, 45), (20, 50), (40, 60)} P Q 1 10 p p3 + 4p p 5 9 S j (p) = 2 p 1 10 p2 p p 3 5 p p3 p p 1 16 p p3 p 40 Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 17/27
49 . 2-firm example, cont d Spline approximation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 18/27
50 . 2-firm example, cont d Spline approximation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 18/27
51 . 2-firm example, cont d Spline approximation Affine approximation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 18/27
52 . Simulation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 19/27
53 . Simulation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 19/27
54 . Simulation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 19/27
55 . Simulation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 19/27
56 . Simulation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 19/27
57 . Simulation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 19/27
58 . Simulation Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 19/27
59 . Problem with marginal revenue Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 20/27
60 . Problem with marginal revenue Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 20/27
61 . Problem with marginal revenue Rarity, or, should we expect to see it frequently? Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 20/27
62 . Problem with marginal revenue. Lemma... If an invertible function is decreasing and convex, so is its inverse.... Lemma.. A C 2 function is convex on I iff its second derivative is nonnegative. on I... Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 21/27
63 . Problem with marginal revenue. Lemma... If an invertible function is decreasing and convex, so is its inverse.... Lemma.. A C 2 function is convex on I iff its second derivative is nonnegative. on I... Let: D, inelastic S i (p) C 2, concave in p D S i (p) convex in p, hence ( D S i (p) ) 1 = RD(q) is convex by the first lemma Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 21/27
64 . Problem with marginal revenue MR(q) = d dq ( qrd(q) ) = RD(q) + qrd (q) Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 22/27
65 . Problem with marginal revenue MR(q) = d dq ( qrd(q) ) = RD(q) + qrd (q) Slope of the marginal revenue is given by: d dq MR(q) = 2RD (q) + qrd (q). Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 22/27
66 . Problem with marginal revenue MR(q) = d dq ( qrd(q) ) = RD(q) + qrd (q) Slope of the marginal revenue is given by: 2RD (q) + qrd (q) > 0 d dq MR(q) = 2RD (q) + qrd (q). 2 > qrd (q)/rd (q) 2 > drd (q) / RD (q) = drd (q) dq q RD (q) /dq q. Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 22/27
67 . Problem with marginal revenue MR(q) = d dq ( qrd(q) ) = RD(q) + qrd (q) Slope of the marginal revenue is given by: 2RD (q) + qrd (q) > 0 d dq MR(q) = 2RD (q) + qrd (q). 2 > qrd (q)/rd (q) 2 > drd (q) / RD (q) = drd (q) dq q RD (q) /dq q. Result Marginal revenue will be increasing if quantity elasticity of the residual demand is less than -2 (P. Coughlin, 84). Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 22/27
68 . Increasing marginal revenue Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 23/27
69 . What s next? Tests with data from ERCOT Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 24/27
70 . What s next? Tests with data from ERCOT Analytic solution: Is that possible? Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 24/27
71 . What s next? Tests with data from ERCOT Analytic solution: Is that possible? Learning (Linear case studied by A. Rudkevich, 99, 05) Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 24/27
72 . What s next? Tests with data from ERCOT Analytic solution: Is that possible? Learning (Linear case studied by A. Rudkevich, 99, 05) Effect of transmission constraints on bidding behavior. Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 24/27
73 . Concluding remarks Mimics reality better, since we can approximate supply and demand with arbitrary precision with cubic polynomials, Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 25/27
74 . Concluding remarks Mimics reality better, since we can approximate supply and demand with arbitrary precision with cubic polynomials, This method gives us smooth functions, Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 25/27
75 . Concluding remarks Mimics reality better, since we can approximate supply and demand with arbitrary precision with cubic polynomials, This method gives us smooth functions, Resolves some of the uncertainty resulting from discontinuity: Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 25/27
76 . Concluding remarks Mimics reality better, since we can approximate supply and demand with arbitrary precision with cubic polynomials, This method gives us smooth functions, Resolves some of the uncertainty resulting from discontinuity: Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 25/27
77 . Concluding remarks Mimics reality better, since we can approximate supply and demand with arbitrary precision with cubic polynomials, This method gives us smooth functions, Resolves some of the uncertainty resulting from discontinuity: Do not ignore the possibility of multiple profit-maximizing equilibria. Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 25/27
78 Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 26/27
79 . Özgür İnal MODELING THE ELECTRICITY SPOT MARKETS 27/27
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