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Lumpy Electric Auction with Credit Constraints

This paper explores the implementation of credit limits in multi-product two-sided auctions to address credit issues in the electric power trading market.

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Lumpy Electric Auction with Credit Constraints

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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

  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.

  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.

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

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

  6. max ∑bj(xj) (bid functions) Subject to: Bj(xj) <= hj (constraints on j’s bid ) pxj + p’xij <= cj (budget constraint for j) xij є {integers} i є j’  {1,… , I} H(x1 ,…, xJ)<= h0 (market clearing) H(x1 ,…, xJ) is convex for fixed values of the integers.

  7. LLACC: max ∑bjxj Subject to: dual variables Bjxj <= hj (pj) (constraints on j) pxj + p’xij <= cj (budget constraint for j) ∑Hjxj <= b0 (p0) (market clearing) xij = xij* (p’) p = p0- p’

  8. For incentive compatibility, xj* must be an optimal solution to mpi. Assuming the bid function is a market participants value or cost function, each bidder solves the following problem: for prices p, mpi: Max bjxj- pxj Subject to: Bjxj <= bj (constraints on j) pxj <= cj (budget constraint for j) some xij є {integers}

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

  10. GEMIP: Max ∑ wjbj(xj) bid functions Subject to: dual variables ∑xj <= h0 (p0) (market clearing) Bj(xj) <= hj (pj) (constraints on j ) pxj <= cj (budget constraint for j) xij є {integers} i є j’  {1,… , I}

  11. Max μ1(.5x11 + x12 + x13)+ μ2(3x21 + x22 + x23) Preference constraints for consumer 1: x11 20; x12 + x13  12; Preference set for consumer 2: x21  19.5; Production technology for firm 1: 1.5y11 + y12 0; y11 + y12 – 2y15 0; y15  {0, 1}  Production technology for firm 2: 2y21 + y23 + 10y24 0; 2 y21 + 10 y24  0; Balancing accounts: xij + xij - yij = h0; (pj)

  12. formulate, MIPh, where h = 0, initially. • Set p = 0, w = 0 and solve MIPh to obtain (xh, yh). • form an LP adding yjk = yjkh for the integer var. • Solve the LP for (xh , yh). Obtain prices, ph, • Set p = ph in the budget constraints of MIPh. • Solve to obtain (x*, y*) for prices ph. • form an LP adding yjk = yjk* for integer var. • Solve the LP. Obtain p*, • If x*i is not an optimal for i, increase μi so that x*i is no longer an optimal solution to MIPh. Set p = ph = p*, and go to step 5. Otherwise, go to Step 10. • If x*i is optimal for all i, we have a WE. Add ui(xi)  ui(xi*) for all i, to MIPh to create MIPh+1. Find another set of integer values , yjk, feasible to MIPh+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.

  13. 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.

  14. We set y24 = 0 and u > u*, and then resolve the MIP • x11 = (9, 12, 0, 0, .5), u11 = 16.5, • x12 = (19.5, 0, 1.5, 0, .5), u21= 61, • y11 = (-11.5, 13.5, 0, 0, 1), • y12 = (0, 0, 0, 0, 0). • Solving for prices, p1 = (1, 1, 1, 0, -2).

  15. If we set y24 = 0 and without u > u* and we solve the MIP, • result is x*1 = (8, 12, 0, 0, 0), u1* = 16, • x*2 = (19.5, 0, 2.5, 0, 1), u2*= 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).

  16. 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 u1 < 16.5. • The best consumer 2 can do with consumer 1 out of the market is x2 = (19.5, 0, .75, 0, 1), u2 = 58.5 + .75 = 59.25. • Therefore, consumer 2 must offer consumer 1 a deal which he cannot refuse (i.e. one with u1 ≥ 16.5). • The prices derived from the algorithm support this equilibrium and it is Pareto optimal.

  17. Computational considerations“perennial gale of creative destruction” Schumpeter • 1996: LMP in NZ • 300 nodes • transmission constraints are manual • 1990s: linear programs improved by 106 • 103 in hardware • 103 in software • 2000s: mixed integer programs already 102 • 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

  18. “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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