# Presentation: 190594

## 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

## 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.

## 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.

## 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."

## 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.

## 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.

## 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’

## 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}

- 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.

## 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}

## 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)

- 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.

- 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),

- 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).

- 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).

## 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.

## 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

## “Almost every generally accepted view was once deemed eccentric or heretical.”	

- “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