11

Lumpy Electric Auction Lumpy Electric Auction with Credit Constraintswith Credit Constraints

Richard O’NeillRichard O’NeillChief Economic AdvisorChief Economic Advisor

Federal Energy Regulation CommissionFederal Energy Regulation Commission

richard.oneill@ferc.govrichard.oneill@ferc.gov

DIMACS Workshop on Computational DIMACS Workshop on Computational Issues in Auction DesignIssues in Auction Design

Rutgers Univ.Rutgers Univ.October 7, 2004October 7, 2004

Views expressed are not necessarily those of the Views expressed are not necessarily those of the CommissionCommission

22

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.

33

Two-Sided Two-Sided AuctionsAuctions

When multi-produce two-sided auctions are When multi-produce two-sided auctions are used as a means of exchange in an used as a means of exchange in an economic system, we want the auction economic system, we want the auction to: to:

have an efficient allocation rule; have an efficient allocation rule;

have a pricing rule that creates a have a pricing rule that creates a ’stable’ outcome; ’stable’ outcome;

be revenue adequate; and, be revenue adequate; and,

impart meaningful economic prices to impart meaningful economic prices to market participants for each commodity. market participants for each commodity.

44

Who said this?Who said this? ““All exchanges regulate in great detail the activities of All exchanges regulate in great detail the activities of

those who trade in these markets those who trade in these markets these exchanges often used by economists as examples these exchanges often used by economists as examples

of a perfect competition, of a perfect competition, It suggests … that for anything approaching perfect It suggests … that for anything approaching perfect

competition to exist, an intricate system of rules and competition to exist, an intricate system of rules and regulations would be normally needed. regulations would be normally needed.

Economists observing the regulations of the exchange Economists observing the regulations of the exchange often assume that they represent an attempt to exercise often assume that they represent an attempt to exercise monopoly power and to aim to restrain competition. monopoly power and to aim to restrain competition.

an alternative explanation for these regulations: that an alternative explanation for these regulations: that they exist in order to reduce transaction costs they exist in order to reduce transaction costs

Those operating in these markets have to depend, Those operating in these markets have to depend, therefore, on the legal system of the State."therefore, on the legal system of the State."

55

The Lumpy Auction with Credit The Lumpy Auction with Credit Constraints Constraints (LACC) can be (LACC) can be

defined as: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.

66

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.

77

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’

88

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}

99

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.

1010

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}

1111

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)

1212

Figure 1. Production Technologies for Vohra Ex. 4.1

0

2

4

6

8

10

12

14

16

18

20

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

Input y11 and y21

Out

put y

12 o

r Y2

3

firm 2 firm 1

y23 +2y21 + 10 0

y12 + 1.5y11 0

y12 + y11 - 2 0

1313

1. formulate, MIPh, where h = 0, initially.2. Set p = 0, w = 0 and solve MIPh to obtain (xh, yh). 3. form an LP adding yjk = yjk

h for the integer var. 4. Solve the LP for (xh , yh). Obtain prices, ph, 5. Set p = ph in the budget constraints of MIPh. 6. Solve to obtain (x*, y*) for prices ph.7. form an LP adding yjk = yjk* for integer var. 8. Solve the LP. Obtain p*, 9. 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.

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.

1414

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.

1515

We set y24 = 0 and u > u*, and then resolve the MIP

x11 = (9, 12, 0, 0, .5), u1

1 = 16.5,

x12 = (19.5, 0, 1.5, 0, .5), u2

1= 61,

y11 = (-11.5, 13.5, 0, 0, 1),

y12 = (0, 0, 0, 0, 0).

Solving for prices, p1 = (1, 1, 1, 0, -2).

1616

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).

1717

Looking at this process from a Looking at this process from a decentralized bargaining point decentralized bargaining point

of viewof 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.

1818

Computational considerationsComputational considerations“perennial gale of creative destruction” “perennial gale of creative destruction”

SchumpeterSchumpeter 1996: LMP in NZ 1996: LMP in NZ

300 nodes 300 nodes transmission constraints are manualtransmission constraints are manual

1990s: linear programs improved by 101990s: linear programs improved by 1066

10103 3 in hardwarein hardware 10103 3 in softwarein software

2000s: mixed integer programs already 102000s: mixed integer programs already 1022

Hardware: parallel processors and 64 bit FPHardware: parallel processors and 64 bit FP Software: ?Software: ?

New modeling capabilities in MIP New modeling capabilities in MIP 2006: 30000 nodes2006: 30000 nodes

10000+ transmission constraints10000+ transmission constraints 1000 generators with n-part bids1000 generators with n-part bids

“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