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1 Lumpy Electric Auction with Credit Constraints Richard O’Neill Chief Economic Advisor Federal Energy Regulation Commission richard.oneill@ferc.gov DIMACS Workshop on Computational Issues in Auction Design Rutgers Univ. October 7, 2004 Views expressed are not necessarily those of the Commission

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2 Retail sales of electric power are between $200 and $250 b/yr in the US and about a trillion worldwide. A good portion is traded in multi-product auction markets. Minor gains in auction efficiency are measured in millions With the financial collapse of independent generators and traders, credit issues have become more important. credit limits are difficult to implement in multi-product two-sided auctions. Here we propose to internalize the credit limits.

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3 Two-Sided Auctions When multi-produce two-sided auctions are used as a means of exchange in an economic system, we want the auction to: have an efficient allocation rule; have a pricing rule that creates a ’stable’ outcome; be revenue adequate; and, impart meaningful economic prices to market participants for each commodity.

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4 Who said this? Ä “All exchanges regulate in great detail the activities of those who trade in these markets Ä these exchanges often used by economists as examples of a perfect competition, Ä It suggests … that for anything approaching perfect competition to exist, an intricate system of rules and regulations would be normally needed. Ä Economists observing the regulations of the exchange often assume that they represent an attempt to exercise monopoly power and to aim to restrain competition. Ä an alternative explanation for these regulations: that they exist in order to reduce transaction costs Ä Those operating in these markets have to depend, therefore, on the legal system of the State."

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5 The Lumpy Auction with Credit Constraints (LACC) can be defined as: Market participant j (j = 1,…, J) with a credit limit, c j > 0, submits a multi-product x j (i = 1,…, I) bid: b j (x j ) subject to B j (x j ) <= h j x ij є {integers} for i є j’ {1,…, I} where -b j (x j ) and B j (x j ) are convex for fixed values of the integers.

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6 max ∑b j (x j ) (bid functions) Subject to: B j (x j ) <= h j (constraints on j’s bid ) px j + p’x ij <= c j (budget constraint for j) x ij є {integers}i є j’ {1,…, I} H(x 1,…, x J ) <= h 0 (market clearing) H(x 1,…, x J ) is convex for fixed values of the integers.

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7 LLACC: max ∑b j x j Subject to: dual variables B j x j <= h j (p j )(constraints on j) px j + p’x ij <= c j (budget constraint for j) ∑H j x j <= b 0 (p 0 )(market clearing) x ij = x ij *(p’) p = p 0 - p’

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8 For incentive compatibility, x j * must be an optimal solution to mp i. Assuming the bid function is a market participants value or cost function, each bidder solves the following problem: for prices p, mp i : Max b j x j - px j Subject to: B j x j <= b j (constraints on j) px j <= c j (budget constraint for j) some x ij є {integers}

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9 If x* is an optimal solution to LACC. x* is efficient and px j * is revenue adequate. The payments, px j, are called make whole payments in ISO markets.

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10 GEMIP: Max ∑ w j b j (x j )bid functions Subject to: dual variables ∑x j <= h 0 (p 0 )(market clearing) B j (x j ) <= h j (p j )(constraints on j ) px j <= c j (budget constraint for j) x ij є {integers} i є j’ {1,…, I}

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11 Max μ 1 (.5x 11 + x 12 + x 13 )+ μ 2 (3x 21 + x 22 + x 23 ) Preference constraints for consumer 1: x 11 20; x 12 + x 13 12; Preference set for consumer 2: x 21 19.5; Production technology for firm 1: 1.5y 11 + y 12 0; y 11 + y 12 – 2y 15 0; y 15 {0, 1} Production technology for firm 2: 2y 21 + y 23 + 10y 24 0; 2 y 21 + 10 y 24 0; Balancing accounts: x ij + x ij - y ij = h 0 ; (p j )

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13 1. 1. formulate, MIP h, where h = 0, initially. 2. 2. Set p = 0, w = 0 and solve MIP h to obtain (x h, y h ). 3. 3. form an LP adding y jk = y jk h for the integer var. 4. 4. Solve the LP for (x h, y h ). Obtain prices, p h, 5. 5. Set p = p h in the budget constraints of MIP h. 6. 6. Solve to obtain (x*, y*) for prices p h. 7. 7. form an LP adding y jk = y jk * for integer var. 8. 8. Solve the LP. Obtain p*, 9. 9. If x* i is not an optimal for i, increase μ i so that x* i is no longer an optimal solution to MIP h. Set p = p h = p*, and go to step 5. Otherwise, go to Step 10. 10. 10. If x* i is optimal for all i, we have a WE. Add u i (x i ) u i (x i *) for all i, to MIP h to create MIP h+1. Find another set of integer values, y jk, feasible to MIP h+1 If there is one, go to Step 3. If each feasible integer solution has been searched and no Pareto superior solution has been found, (x*, y*, p*) is a POWE.

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14 x* 1 = (9, 0, 12, 1, 0), u* 1 = 16.5, x* 2 = (19.5, 0, 1, 0), u* 2 = 59.5, y* 1 = (0, 0, 0, 0, 0), y* 2 = (-11.5, 0, 13, 1, 0), and p* = (2, 1, 1, 10, 0), this is a Walrasian equilibrium.

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15 We set y 24 = 0 and u > u*, and then resolve the MIP x 1 1 = (9, 12, 0, 0,.5), u 1 1 = 16.5, x 1 2 = (19.5, 0, 1.5, 0,.5), u 2 1 = 61, y 1 1 = (-11.5, 13.5, 0, 0, 1), y 1 2 = (0, 0, 0, 0, 0). Solving for prices, p 1 = (1, 1, 1, 0, -2).

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16 If we set y 24 = 0 and without u > u* and we solve the MIP, result is x* 1 = (8, 12, 0, 0, 0), u 1 * = 16, x* 2 = (19.5, 0, 2.5, 0, 1), u 2 *= 61, y* 1 = (-12.5, 14.5, 0, 0, 1), y* 2 = (0, 0, 0, 0, 0). Solving for prices, p* = (1, 1, 1, 0, -2).

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17 Looking at this process from a decentralized bargaining point of view consumer 1 has a better negotiation position because by itself. Consumer 1 will accept no deal where u 1 < 16.5. The best consumer 2 can do with consumer 1 out of the market is x 2 = (19.5, 0,.75, 0, 1), u 2 = 58.5 +.75 = 59.25. Therefore, consumer 2 must offer consumer 1 a deal which he cannot refuse (i.e. one with u 1 ≥ 16.5). The prices derived from the algorithm support this equilibrium and it is Pareto optimal.

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18 Computational considerations “perennial gale of creative destruction” Schumpeter 1996: LMP in NZ 300 nodes transmission constraints are manual 1990s: linear programs improved by 10 6 10 3 in hardware 10 3 in software 2000s: mixed integer programs already 10 2 Hardware: parallel processors and 64 bit FP Software: ? New modeling capabilities in MIP 2006: 30000 nodes 10000+ transmission constraints 1000 generators with n-part bids

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“Almost every generally accepted view was once deemed eccentric or heretical.” Everett Mendelson, Stephen Jay Gould, Gerald Holton and other leading scholars in a Supreme Court brief

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