1

Design of Ancillary Service Markets.

Shmuel S. Oren

(Preliminary Draft)

March 20, 2000

1. Background

A common aspect shared by the various designs of restructured electricity markets in

the US and around the world is the designation of a system operator that is responsible for

the reliable real time control of the transmission system that enables operation of a

competitive energy market. In California as in PJM, NYPP and New England this role is

performed by an Independent System Operator (ISO) that is responsible for real time load

balancing, congestion management and provision of ancillary services that include voltage

support, AGC and reserves with varying levels of response. The precise definition of

ancillary services varies across different restructured systems and so is the design of

markets for competitive procurement and provision of such services. In California separate

ancillary service products have been defined which are procured by the ISO through an

auction on behalf of customers and scheduling coordinator that do not self-provide their

required share of such services to meet NERC reliability standards.

The proper product definition and design of ancillary service markets are the primary

determinants of efficiency and liquidity in these markets that in turn influences system

reliability. This lesson was learned in California through the poor performance of its

ancillary service market that necessitated price caps and other forms direct intervention by

the ISO. A similar experience in New England during the summer of 1999 resulted in the

suspension of market operations. In California, a market reform is under way whose

primary objective is to improve liquidity and efficiency in the ancillary service markets.

Key elements of that reform have been to move from a sequential to a simultaneous

2

procurement auction for the various services and to allow the ISO to exploit the

substitutability among the different services, particularly reserves.

Specifically, the California ISO needs to procure four types of reserves: Regulation

(RG) for AGC, Spinning reserves (SP) that are synchronized and available within 10

minutes, Non-Spinning reserves (NS) that are not synchronized but can be made available

within 10 minutes and Replacement reserves (RS) that can be made available within 60

minutes. These products are, for practical purposes, hierarchically substitutable. Regulation

resources can also provide SP, NS and RS. Likewise spinning reserves can provide NS and

RS whereas non-spinning reserves can provide RS. In the initial market design the auctions

for the four reserve products was conducted sequentially allowing generators to rebid

uncommitted resources at new prices that could exploit thin markets in the lower quality

reserves. The reform attempts to stabilize prices by permitting generator to submit a single

bid specifying reserve type quantity and price whereas the ISO can use any of the procured

resources to meet demand for any of the reserve products that a resource can fulfill. So for

instance the ISO may procure spinning reserves and use them to meet its needs for

replacement reserves.

Within the framework defined by the above reform there are several alternatives for

organizing the auction, which define how winning bids are selected and how payoffs are

determined. These alternatives have diverse efficiency and distributional implications,

which have been recognized and debated by the ISO board. The scheme that has been

adopted employs a "rational buyer" bid selection criterion that minimizes total procurement

cost and pays to winning bids of each reserve type (as declared by the bidder) a uniform

market clearing price set by the highest accepted bid of that type. We will reexamine the

implications of that design and of other design options some of which have not been

considered in the ISO board's deliberation. Our analysis will focus particularly on issue of

incentive compatibility that has received little attention in these deliberations. We will

employ a game theoretic framework in order to analyze bidders' strategic behavior that is

affected by the auction design and by the opportunities available to the generation owners.

In addition to the auction design on the procurement side we also address the question

of how to price the ancillary service products to the users (i.e. those who do not self-

3

provide and must purchase them from the ISO). Again several options are available with

diverse equity and efficiency implications. A key issue in that regard is whether customers

of one product (say RG) could be penalized (pay more) by the fact that the product they

buy is also used to meet the need for a lower quality product in order to reduce total

procurement cost or improve total efficiency. If one views the rational buyer paradigm as

though the ISO is acting on behalf of a coalition of different ancillary service users then

stability of the coalition as well as equity considerations would dictate that the benefits of

the joint procurement be allocated among all the customers so that no one is made worse

off. A related concern is the revenue adequacy of the pricing scheme, i.e., does the total

payment by customers match the procurement cost or will there be a shortfall or surplus.

The current pricing rule adopted by the ISO board does not meet the revenue adequacy

criterion nor does it assure that non of the customers is made worse off by the joint

procurement. We explore alternative pricing schemes (to the customers of ancillary

services) and attempt to evaluate their implication with respect to the incentives for self-

provision of ancillary services.

2. Review of relevant literature

The bidding literature addressing multipart and multi-product auctions is quite young.

A significant body of work in this area was spurred by the FCC spectrum auction and the

multitude of interesting questions they have raised. The two major camps in this area are

those advocating combinatorial auctions that employ combinatorial optimization algorithms

vs. those favoring simultaneous multiround auctions with activity rules for the individual

components. The later approach, which has been adopted in the early phases of the FCC

spectrum auction, is justified on the grounds of the computational complexity and lack of

transparency involved in a combinatorial auction. However, as illustrated by Rothkoph,

Pekee and Harstad [1998] these objectionable aspects of combinatorial auctions may be

irrelevant when special structure can be exploited. Backround information and discussions

as well as theoretical and empirical studies related to the FCC auction have been reported

in Millgrom [1998] MacMillan [1994], Cramton [1995], MacMillan and McAfee [1996]

Ausubel and Cramton [1996] Issac and James [1998].

Literature focusing on bidding in the context of electricity markets initially addressed

4

PURPA auctions for IPP contracts. Work examining electric power auctions by Kahn,

Rothkopf, Eto and Nataf [1990] examined efficiency aspects of the acquisition process but

did not treat cases where bids were curtailable. Stoft and Kahn [1991] have examined

questions concerning "bias" in scoring of curtailable bids but there has not been an explicit

treatment of trade-off between fixed and variable price component. Bushnell and Oren

[1993], [1994] addressed the question of whether bidders in the Biennial Resource

Planning Update (BRPU) auction mandated by the California Public Utility Commission

(CPUC) for procurement of IPP capacity by the utilities, will be induced to reveal their

"true" fixed and marginal cost of generation. Such information was essential for efficient

procurement and dispatch of procured resources. Their analysis predicted that the proposed

CPUC [1991] scoring rule is likely to induce understatement of marginal generation cost

(which indeed happened). The outcome of that auction, which was eventually voided by

FERC due to "gaming behavior" by the bidders are discussed in an article by Gribik [1995].

Bushnell and Oren [1993] also propose a Vickery auction type scheme for which true

revelation off marginal costs is a dominant strategy for all bidders.

Von der Fehr and Harbord [1993] is the first paper we are aware of that analyzed a

multi unit auction for electricity in the context of the UK power pool. Work by Green and

Newberry [1992], Newberry [1995] and by Patrick and Wolak [1997] provide further

insights and analysis of the UK bidding rules and their implications. The UK design like

the later market designs adopted in PJM, NYPP and New England have preserved the

central unit commitment aspect of the vertically integrated utilities by means of a multipart

auction in which bidders provide the necessary inputs for central unit commitment

optimization. Unfortunately multipart auctions are not well understood and with few

exceptions that we will mention below have limited theoretical foundation that would

enable a practical incentive compatible design. Johnson, Oren and Svoboda [1997]

illustrated some inherent difficulties arising from central unit commitment in a competitive

environment with dispersed generation ownership. The difficulties are due to

indeterminacies of the optimal unit commitment solution and basic incompatibility between

competitive behavior and truth telling due to integer affects and nonconvexities. In a recent

paper, Hobbs et al. [1998] studied an incentive compatible multi-part electricity auction

based on the Vickery-Clarke-Grove mechanism. Unfortunately, as noted by the authors,

5

this approach has limited practical value due to revenue deficiency and the discriminatory

nature of the payment scheme. A special type of multi-part auction arises in ancillary

service markets were generators compete for the provision of regulation reserves by

submitting a two-part bid consisting of a capacity price bid and an energy bid. For that

special case, Chao and Wilson [1999] were able to develop an incentive compatible design.

In their proposed scheme generators submit separate capacity and energy bids. The energy

bids are used in case it is necessary to call the reserves to supply energy and all dispatched

energy is remunerated uniformly at a price set by the highest dispatched energy bid. They

proved that under this setup a scoring rule that ranks reserves bids based only on their

capacity component and pays all accepted bids a capacity price set to the highest accepted

capacity bid is incentive compatible, i.e., bidders will be induced to reveal their true

marginal energy cost as well as their capacity cost (that may be an opportunity cost for

forgone profit from sale of energy.)

The problems associated with the California ancillary service market that led to the

"rational buyer" reform are documented in the Annual report of the California ISO market

surveillance committee [June 1999] and discussed by Laura Brien [1999]. Some of the

deliberations concerning the ISO ancillary service market reform have been documented in

unpublished memoranda. An analysis of alternative market designs for ancillary services is

contained in the Griffes [1999] report to the California PX that is considering establishing

its own internal ancillary service market for the purpose of self-provision.

3. Design options and allingment of buyer's, seller's and social objectives

As discussed earlier, an important aspects of ancillary services is their hierarchical

nature that allows substitution of a high quality reserve for a lower quality one. Both

social efficiency and rational procurement behavior dictate that such substitution should

be allowed. In a perfectly competitive market such substitution would occur naturally in a

sequential auction (from high to low quality) since bidders would rebid their rejected bids

in the subsequent auctions for which their resources are eligible. In principle, assuming

bidders bid and rebid their true cost, such a sequential auction would lead to socially

efficient procurement. In the absence of market power uniform market clearing prices in

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each auction will indeed induce bidders to bid their true cost. Unfortunately, a sequential

auction with independent uniform market clearing prices in each round may result in

price reversal, i.e., the market clearing price for a high quality resource (e.g. regulation)

may be lower than that for a lower quality resource (e.g. spinning reserves). Indeed such

price reversals have been observed in California and in New England ancillary service

markets.

Price reversals pose serious incentive compatibility problems since even price taking

generators anticipating such reversal may be induced to understate their capability and

wait for a later round of the sequential auction that is expected to fetch a higher market

clearing price. With market power the situation is exacerbated as losing bids in early

rounds may raise their bids in subsequent round when they perceive potential scarcity of

bids. That type of low liquidity and high prices for low quality reserves has been

observed in California and has motivated the proposed reforms. The following example is

used to illustrate the possible price reversal described above. For simplicity we will only

consider two type of reserves (RG and SP) with demand of 500MW for each reserve type

and assume that bidders have no market power so that in a uniform price auction they bid

their true cost. Let us assume that the following bids have been submitted in a sequential

uniform price auction and the rejected RG bids in the first round are rebid in the second

round at the same price:

RG 600 MW at $10/MW 100MW at $15/MW

SP 200MW at $5/MW, 300MW at $20/MW

In the first round (RG auction) the ISO procures 500MW of RG at $10/MW

which sets the market clearing price for the RG auction at $10. The remaining 100MW

RG at $10 and 100MW RG at $15 are rebid for SP at the same price.

In the second round (SP auction) the ISO take 200MW of SP resources at $5 then

takes the rebid 100MW RG at $10 and 100MW RG at $15 and finally 100MW SP at

$20. The market clearing price for SP is therefore $20 which is paid to all the reserve

capacity procured in the second round. The resulting total procurement cost $15000 and

the social cost (assuming bid reveal true cost) is $10500. While the procurement

resulting from this approach is socially efficient we notice that the inferior reserves (SP)

are paid twice as much as the superior reserves (RG). Furthermore, the losing bids in the

7

first round end up being paid more than the winning bids. Clearly, this price reversal

creates perverse incentives for withholding bids or raising prices in the RG auction. This

is true even for resources that have no market power to affect market clearing prices but

simply try to take advantage of intermarket arbitrage opportunity by selling their product

in the auction that will fetch the best price. In reality bidders in the California ancillary

service market also exploited market power due to low liquidity and raised their bids in

the subsequent auctions.

The proposed reforms to the California ancillary service markets have two key elements

• The sequential auction with separate bidding in each stage will be changed to

a simultaneous auction with each resources submitting a single bid specifying

reserve type capacity bid and energy price if called.

• The ISO is allowed to use higher quality resources to meet demand for lower

quality resources (Rational Buyer)

Within this framework there are still several degrees of freedom in the market design as

follows:

• The objective function used by the rational buyer

o Minimum procurement cost

o Minimum social cost

• Settlement rule

o Pay uniform price based on bid type (demand substitution)

o Pay uniform price based on usage (product substitution)

o Pay as bid

• Pricing of the product to buyers of ancillary services

o Set product price to highest accepted bid of that type

o Set product price to highest price paid to meet product demand

o Set product price so that no purchaser is made worse of by rational

buyer procedure

Much of the deliberations in the California ISO focused on the choice of objective

function for implementing the rational buyer approach. The implication of that choice

depend on the settlement rule. Paying resources based on usage (product substitution) is a

natural choice in sequential auctions where rejected resources in one auction can rebid or

8

are carried over to the next. In such a case the payments are based on the clearing price of

the auction in which a resource wins. Under such a settlement rule, however, if a rejected

bid in an early auction is accepted in a later one (assuming no bid change) it follows that

the clearing price in the later auction (for lower quality reserves) will be higher than the

predecessor auction in which the bid was rejected. This implies price reversal where

lower quality products are paid more than the higher quality ones and as discussed above

such reversal creates perverse incentives for misrepresentation of quality. Avoiding such

situation is one of the main arguments in favor of a simultaneous auction. However, a

simultaneous auction in which resources are paid based on usage is de facto equivalent to

a sequential auction in which rejected bids are carried forward. In view of the above the

California ISO decision to exclude product substitution is justified and we will not

consider that option any further.

The two remaining settlement rules are uniform payment based on bid type and

pay as bid. The California ISO adopted the first option without even considering the

second. Given that choice, the two possible rational buyer objectives can lead to different

dispatch results and in particular procurement cost minimization may result in a socially

inefficient dispatch and in price reversal. We illustrate this possible outcome using the

same example employed above for the sequential auction illustration.

a. Simultaneous-Uniform price based on bid type-Selection based on min social cost:

The uniform price rule incents bidders with no market power to bid true cost. Hence,

in the absence of market power social cost is minimized by minimizing the cost as bid

of the dispatched resources. This would lead to the following procurement:

RG - 600MW at $10/MW +100MW at $15/MW (200MW used to meet SP demand)

SP - 200MW at $5/MW + 100MW at $20/MW

The market clearing price for RG is $15/MW whereas the market clearing price for

SP is $20/MW. The corresponding social cost is $10500 while the procurement cost

is $16500. In other words the procurement cost has gone up as compared with the

sequential auction whereas the social cost is still the same. We also note that price

reversal can still occur so there are still perverse incentives for bidders to understate

the quality of their product. In order to avoid the price reversal and guarantee

incentive compatibility one would have to set the uniform price of each reserve type

9

to its marginal value which is either its opportunity cost or the avoided social cost

resulting from increasing the supply of that reserve type by one unit. In the above

example that would imply paying all RG bids $20/MW which would raise the

procurement cost to $20,000.

A simultaneous auction design with demand substitution, minimization of social

cost as revealed by the bids and uniform clearing prices for each reserve type set to its

marginal value can be proved to be efficient and incentive compatible. The optimal

procurement in such a setup can be obtained by solving a simple linear program and

the market clearing prices for each reserve type are set to the shadow price

corresponding to the demand constraint for that type. Unfortunately, as in classic

Vickery auctions the efficiency and incentive compatibility in such an auction is

achieved at the expense of a transfer from buyers to sellers. Critics of that approach

would argue that this policy achieves incentive compatibility by giving away to the

sellers all they could hope to get through cheating. These considerations have led the

California ISO to reject this approach.

b. Simultaneous-Uniform price based on bid type-Selection based on min procurement

cost (Rational Buyer)

Under this design the optimal procurement is as follows:

RG - 600MW at $10/MW

SP - 200MW at $5/MW + 200MW at $20/MW

The resulting market clearing price for RG is $10/MW whereas the market-clearing price

for SP is $20/MW. The corresponding social cost is $11,000 while the procurement cost

is $14,000. We note that the outcome is less efficient than the sequential auction and the

social cost-minimizing auction but the procurement cost is lower then in the other two

designs. There is a clear tradeoff here between efficiency and procurement cost. While it

would be more efficient to use 100MW RG at $15/MW to replace 100MW of SP at

$20/MW, such substitution would raise the market-clearing price of RG from $10/MW to

$15/MW resulting in a net increase of $2,500 in total procurement cost. Therefore, the

rational buyer protocol forgoes such efficiency improvement in order to lower

procurement cost. We also note that this approach may lead to price reversal which again

may incent sellers to understate the quality of their resources and that may lead to

10

shortages of high quality reserves. Unfortunately, we are not aware of any systematic

analysis of the equilibrium that would result when "rational sellers" act strategically in

revealing their reserves quality in a Rational Buyer auction. In spite of the shortcomings

and uncertainties the California ISO has embraced the Rational Buyer approach in its

proposed reform of the ancillary service market.

The efficiency difference between the minimum procurement cost criterion and

minimum social cost criterion is a consequence of the payment rule that sets a uniform

price for each bid reserve type to the highest accepted bid of that type. That discrepancy

would disappear under a pay-as-bid (PAB) rule. Under such a rule bidders will no longer

have an incentive to misrepresent their reserve type but do have an incentive to bid above

marginal cost. However, under a rational expectations assumption (which is not

unreasonable in a daily repeated market) rational bids in a PAB auction are

monotonically increasing in true costs and hence the least social cost objective and the

least procurement cost objective are aligned in a PAB auction. The idea of using a PAB

rule in the ancillary service market was not considered by the California ISO board but it

is definitely worth investigating at least as a benchmark. The New Electricity Trading

Arrangements (NETA) in the UK adopted a PAB approach in the balancing market

(starting 3.5 hours before delivery) as a way of dealing with the heterogeneity of

generation resources close to real time. A PAB approach may be advantageous as we get

closer to real time since it eliminates the need to differentiate the energy commodity (by

location, ramp rate, flexibility, etc.) and create separate markets that are settled at

uniform market clearing prices. Instead, each resource is dealt with as unique but

partially substitutable and dispatched so as to minimize total cost subject to operational

constraints. It is up to the bidders to figure out who they are competing with and how

much they can charge which is not an unreasonable task in a market that is repeated

daily.

Attachment B. Is a self-contained working paper in which we analyze a stylized

model comparing the PAB approach to the social cost minimization and Rational Buyer

approach. For simplicity we assume two reserve types (RG and SP) with linear supply

functions and uncertain demand. At this phase we assume that bidders reveal their true

type (reserve quality) in all cases. That working paper represents work in progress and

11

further work is under way to add realism to the model and to incorporate the possibility

of strategic understatement of reserve quality in a uniform price rational buyer auction.

4. Solving the bid selection problem and pricing the products to the customers.

Selecting bids by minimizing cost as bid can be formulated as a simple linear

programming whether it is done in the context of social cost minimization with uniform

price payments to each resource type or in the context of minimizing procurement cost in

PAB auction. To simplify notation we will use an index i=1,2,3,4 to denote regulation,

spinning reserves, non spinning reserves and replacement reserves, respectively. We

assume that each resource is bid as a quantity and price pair into a particular market.

Thus, it is possible to rank all bids for a particular resource type i according to price and

obtain a supply function ( )i ip q specifying the market clearing price (or marginal price)

that will enlist capacity iq of resource type i. This function will have the form of an

increasing stair case rising vertically at the maximum capacity level bid for the resource

type.

We introduce the additional following notation:

Q qi jj

i

==

∑1

- Cumulative acquired capacity of resource type i or beter

di - Demand for ancillary service i

D di jj

i

==

∑1

- Cumulative demand for ancillary service i or higher in the hierarchy

The minimum cost problem can be written in the simple mathematical form;

1 2 3 4

4

, , , 0 1 0

( )

. for 1,2,3,4

iq

i i iq q q q i

i i

p d

s t Q D i

Min ϕ ϕ≥ =

≥ =

∑∫

In the incentive compatible socially efficient approach the uniform price paid to

each resource type is given by the shadow price corresponding to the demand constraint

for that resource. This shadow price would also be the socially efficient price charged to

consumers of that reserve type. In the PAB scheme charging consumers the marginal bid

price for each resource type is still the efficient price from the point of view of inducing

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economically efficient self-provision. However, such pricing implies that users of

regulation for instance are penalized by the fact that some regulation is being diverted to

meet demand for spinning reserves. An alternative scheme that is more equitable is to

charge users of regulation the price of the marginal regulation MW needed to serve

demand for regulation, charging users of spinning reserves the price of the most

expansive MW of regulation or spinning reserves needed to meet spinning reserves

demand etc. The later approach is Pareto Superior to the sequential auction uniform

pricing approach in the sense that no customer is made worse off by shifting high quality

resources to lower quality use. In both cases pricing reserves at marginal procurement

cost by resource type or usage will result in excess revenues. These excess revenues can

be used for reducing uplift charges, be reimbursed to buyers of reserves as a fixed rebate

or be returned to customers on a prorated basis by charging them average rather than

marginal costs (averaging can be done either over reserve type or usage type).

When bid selection is based on the Rational Buyer protocol, the selection problem

is nonconvex and requires combinatorial optimization. Indeed the algorithm adopted by

the California ISO for that purpose is based on enumeration. Fortunately, the hierarchical

nature of the constraints enables a Dynamic Programming (DP) formulation of the

problem, which can be easily solved in a spreadsheet.. Attachment A provides the

technical details of the DP formulation along with an illustrative example. Again there

are several options for pricing the products to the buyers. The approach adopted by the

ISO is to set the price of each product type to the procurement price of that product type.

Under that pricing scheme buyers of regulation will end up paying more if regulation is

also used for spinning reserves, thus raising the market clearing price for regulation.

Furthermore, if there is no price reversal, the scheme is revenue deficient in the sense that

the total sales revenue might not cover the total procurement cost. For instance, if

regulation reserves are used for spinning reserves and the market clearing price for

procured spinning reserve resources is lower than that of regulation (as one would hope)

then the ISO will have a revenue shortfall. This shortfall is due to the fact that the more

expansive regulation was used for spinning reserves (in order to replace even more

expansive spinning reserves resources) but they are sold to the spinning reserves

13

customers for the lower spinning reserves price. Thus, the ISO may have to make up the

revenue shortfall through uplift charges.

An alternative pricing approach is based on viewing the Rational Buyer protocol

as cooperative procurement by a coalition of customers demanding the four different

reserves. The two guiding principles of such a coalition that is represented by the

"rational buyer" are:

a. Since the purpose of the Rational Buyer is to reduce total procurement cost no

reserve type should be sold to customers for more than if it was procured

separately (in a sequential auction)

b. The coalition should break even i.e. total procurement cost should equal sales

revenues from customers.

The DP solution provides the ingredients for such a pricing scheme. The technical details

and illustrative calculation are contained in Attachment A.

REFERENCES

Ausubel, L., and P. Cramton, "Demand Reduction and Inefficiency in Multi-unit Auctions", Working Paper

No. 96-07, University of Maryland (1996). Brien, L. (1999), \Why the Ancillary Services Markets in California Don't Work

Brien, Laura, Why the Ancillary Services Markets in California Don't Work and What to do About It," The Electricity

Journal, 12(5): 38-49. (1999)

Bushnell, J. and S. S. Oren, "Bidder Cost Revelation in Electric Power Auctions," Journal of Regulatory

Economics, V. 6, No 1 (1994), pp. 1-22.

Bushnell, J. and S. S. Oren, "Two-dimensional Auctions for Efficient Franchising of Independent Power

Generation" Proceedings of the 29th Annual Conference of the Operational Research Society of New

Zealand, , Auckland University, New Zealand, (August 1993).

California ISO Market Surveillance Committee Annual Report (June 1999)

Chao, H. P., and R. B. Wilson, "Multi-Dimensional Procurement Auction for Power Reserves: Incentive-

Compatibility Evaluation and Settlement Rules", Presened at the UCEI POWER Conference, Berkeley

California, March 1999 (Revised October 1999)

Cramton, P. "Money Out of Thin Air: The Nationwide Narrowband PCS Auction," Jornal of Economics and

Management Strategy, Vol. 4, No, 2 pp. 267-343 (1995)

14

Elmaghrabi Wedad and Shmuel S. Oren, "Efficiency of Multi-Unit Electricity Auctions", Energy Journal, 20,

4, pp. 89-115 (1999).

Gribik,Paul:“Learning from California's QF Auction,” PublicUtilities Fortnightly, (April 15, 1995)

Griffes, Peter, "A report to the California Power Exchange: The benefits of a simultaneous vs. sequential PX

market for energy and ancillary services", (March 1999)

Isaac, R., D. James, "Robustness of the Incentive Compatible Combinatorial Auction," Maryland Auction

Conference, (May 29-31, 1998).

Kahn, E., Rothkopf, M., Eto, J., and Nataf, J.M., "Auction for PURPA Purchases: a Simulation Study,"

Journal of Regulatory Economics," Vol. 2, pp. 129-149, (1990).

McAfee, P. and J. McMillan, "Analyzing the airwave Auction," Journal of Economic Perspective, Vol. 10,

No. 1, pp. 159-175, (1996)

McMillan, J., "Selling Spectrum Rights," Journal of Economic Perspective, Vol. 8, No. 3, pp. 145-162,

(1994)

Millgrom, P. "Putting Auction Theory to Work: The Simultaneous Ascending Auction," Maryland Auction

Conference, (May 29-31, 1998)

Rothkopf, M. "Daily Repetition: A neglected Factor in Analyzing Electricity Auctions," Working Paper

Presented at the International INFORMS Meeting, Tel Aviv, Israel (July 1998)

Rothkopf, M. A. Pekee, and R. Harstad, "Computationally Managable Combinatorial Auctions,"

Management Science, Vol. 44, No. 8, pp. 1131-1147, (1998)

Stoft, S., and Kahn, E., "Auction Markets for Dispatchable Power: How to Score the Bids," Journal of

Regulatory Economics, Vol. 3, pp.. 275-286 (1991) .

Wilson , R. "Activity Rules for The California PX Electricity Auction", Presented at the UCEI POWER

Conference, Berkeley, California, (March 1998).

15

ATTACHMENT A.

16

A Dynamic Programming Algorithm for the ISO Rational Buyer

Procedure.

Shmuel S. Oren

This note describes a Dynamic Programming solution for the Rational Buyer problem adopted by the ISO for acquisition of ancillary services. The procedure described below assumes that the objective function adopted for this procedure is to minimize total payment for ancillary services. It does not address the issue of incentive compatibility and may therefore result in prices that are not monotone, i.e. the price paid for higher quality resources could be lower than those paid to lower quality resources. The following formulation assumes that resources are hierarchical so that a resource bid for regulation can provide any of the ancillary services, a resource bid for spinning reserves can provide spinning, non spinning and replacement reserves, etc. The formulation also assumes that all accepted bids of the same type will be paid the same market clearing price for that resource type regardless of their actual use by the ISO (e.g., resources bid into the regulation market will be paid the same market clearing price regardless of whether they are used for regulation or spinning reserves). The procedure can be easily modified to accommodate a protocol that pays based on the resource ultimate use.

For simplicity of notation we will use an index i=1,2,3,4 to denote regulation, spinning reserves, non spinning reserves and replacement reserves, respectively. We assume that each resource is bid as a quantity and price pair into a particular market. Thus, it is possible to rank all bids for a particular resource type i according to price and obtain a supply function p qi i( ) specifying the market clearing price that will enlist capacity qi of resource type i. This function will have the form of an increasing stair case rising vertically at the maximum capacity level bid for the resource type. We introduce the additional following notation: P q q p qi i i i i( ) ( )= - Total payment to resource type i as a function of acquired capacity of that type

Q qi jj

i

==

∑1

- Cumulative acquired capacity of resource type i or beter

di - Demand for ancillary service i

D di jj

i

==

∑1

- Cumulative demand for ancillary service i or higher in the hierarchy

The rational buyer problem can be written in the simple mathematical form;

q q q qi i

i

i i

Min P q

s t Q D i1 2 3 4 0 1

4

1 3 4, , ,

( )

. ,2, ,≥ =

∑≥ = for

Because of the discrete nature of the supply functions, this problem is combinatorial in nature and may have multiple local solutions. Hence the problem is not easily solved by standard linear or nonlinear programming algorithms. Fortunately, however, the special

17

structure of the problem allows us to formulate it the simple Dynamic Programming problem below whose solution guarantees a global minimum.

D Q D

Q Q D

Q Q D

MinMin

Min

P D Q

P Q Q

P Q Q

P Q

4 3 3

3 2 2

2 1 1

4 4 3

3 3 2

2 2 1

1 1

≥ ≥

≥ ≥

≥ ≥

−

+

−

+−

+

LNM

OQP

L

N

MMMM

O

Q

PPPP

L

N

MMMMMM

O

Q

PPPPPP

b gb g

b g( )

This is a four stage deterministic dynamic program that will determine the optimal quantities of each of the resource types that will minimize total acquisition cost to the ISO for meeting the demands for the four ancillary services. The stages in this DP formulation are the subsequent resource types in the hierarchy while the states are the cumulative amount of resources acquired at each stage. The resource quantities are discretized using an appropriate increment for the required precision. . The computation time will depend on that discretization. The solution involves one forward and one backward pass. In the forward pass we start with the most inner minimization first computing the cost of acquiring any feasible quantity of regulation capacity in the range between the demand for regulation and the combined demand for all services. Next we compute the least cost feasible mix of regulation and spinning reserves resources for any total amount of the two in the range between the combined demand for regulation and spinning reserves and the demand for all services combined, subject to the constraint that the regulation capacity exceeds the demand for regulation. Similarly we then compute the least cost feasible mix of the first two resources and non spinning reserves for each possible total amount of the three and so on. In the backward pass we start with the total amount of the four ancillary services and trace back the least cost path from which we can extract the optimal procured quantity of each resource type. Once we have these quantities we can use the supply functions to determine the corresponding market clearing prices.

The solution to the above dynamic program can be implemented in a simple spreadsheet even for realistic size problems. For example, if the combined demand for spinning, non-spinning and replacement reserves is 6000MW and we measure quantities in 1 MW increments, the calculations can be contained in a 6000 rows by 3 column spreadsheet which is well within the capability of EXCELL (the amount of regulation capacity required does not affect the computational complexity of the problem). The fill of the spreadsheet depends on the demands for the individual services. Suppose for example that the requirement for each of the reserves is 2000MW then the solution involves the computation of 12,000 quantities involving simple comparison operations. The worse case fill occurs when there is no demand for spinning and non-spinning reserves so all the resources compete to provide replacement reserves. In the spreadsheet is full and 18,000 entries need to be computed. Pareto-superior pricing of ancillary services The Rational Buyer procedure described above determines the market clearing prices for each resource type. Now we need to determine a set of selling prices to be charged to the users of the four ancillary services. These prices must meet the following properties:

18

1. The price for any service should not exceed the "stand-alone" price that would be charged if each service was provided exclusively by resources bid for that service and the price set to the market clearing price of these resources.

2. Total revenue should equal total cost. 3. Each service should be assigned the savings due to rational buying of resources

for that service but it should bear the incremental cost imposed on higher level service if such procurement raises the market clearing prices of the higher quality resources.

The above principles are satisfied by the following pricing rule. Price for service i = (Least cost of meeting cumulative demand for service i or better

-Least cost of meeting cumulative demand for services better than i)/ Demand for service i

Assume for instance that the demand for regulation is 100MW and the regulation clearing price is $8/MW while the demand for spin is 100MW with a stand alone clearing price of $15/MW. Suppose that the least cost of regulation plus spin (using the rational buyer rule) is $2250 which is achieved by purchasing and additional 50MW of regulation at $10/MW for spin and reducing the spin purchase to 50MW. The additional purchase of regulation has increased the clearing price to $10 and that price is paid to all the 150MW of regulation resources. Under the above formula the Pareto superior prices charged to the users will be $8/MW for regulation and (2250-800)/100=$14.5/MW for spin. Non of the users is worse off and the total revenue collected covers the total cost (14.5x100+8x100=2250) The Dynamic Programming formulation of the rational buyer problem produces as a by product the information needed to compute the Pareto superior prices defined above. To describe the above pricing formula more rigorously we introduce the additional notation:

C Q D Di i i( ,.., )1 1− - Least cost of serving a cumulative quantity of Qi MW of ancillary service i or better given the demand for ancillary services 1,...,i-1.

With this notation, the Dynamic Programming formulation can be expressed in the form: C Q D D Min P Q Q C Q D D C Q P Qi i i

Q Q Di i i i i i

i i i

( ,.., ) ( ) ( ,.. ) ( ) ( )1 1 1 1 1 1 2 1 1 1 11 1

− ≥ ≥ − − − −= − + =− −

,

The forward pass of the DP produces the least cost function C Q D Di i i( ,.., )1 1− and these values can be used to calculate the Pareto superior prices $pi for each of theservices using the formula: $ ( ,.., ) ( ,.., ) /p C D D D C D D D di i i i i i i i= −− − − −1 1 1 1 1 2 for i=2,3,4 $ ( ) /p P D D1 1 1 1= Revenue neutrality: This is easily verified as follows:

$ [ ( , , ) ( , )] [ ( , ) ( )]

[ ( ) ( )] ( ) ( , , )

p d C D D D D C D D D C D D D C D D

C D D C D C D C D D D D

i ii

= − + −

− ==∑

1

4

4 4 3 2 1 3 3 2 1 3 3 2 1 2 2 1

2 2 1 1 1 1 1 4 4 3 2 1

+

In other words total payments for the services equals the total least cost of acquisition.

19

Pareto superiority: Here we need to show that no buyer of services is worse off as compared to the "stand alone" sequential acquisition. The total charge to users of ancillary service i based on stand alone acquisition of each service is simply P di i( ) while the comparable cost based on rational buying and Pareto superior pricing is: $ ( ,.., ) ( ,.., )p d C D D D C D D Di i i i i i i i= −− − − −1 1 1 1 1 2 From the DP recursion it follows that: C Q D D Min P Q Q C Q D D P D Q C D D Di i i

Q Q Di i i i i i i i i i i i

i i i

( ,.., ) ( ) ( ,.. ) ( ) ( ,.. )1 1 1 1 1 1 2 1 1 1 1 21 1

− ≥ ≥ − − − − − − − −= − + ≤ − +− −

and in particular if we set Q Di i= we have: C D D D P D D C D D Di i i i i i i i i( ,.., ) ( ) ( , .. )1 1 1 1 1 1 2− − − − −≤ − + From which it follows that: C D D D C D D D P di i i i i i i i( ,.., ) ( ,.., ) ( )1 1 1 1 1 2− − − −− ≤

Illustrative Example: The following spreadsheet illustrates the DP calculation of the Rational Buyer protocol, the corresponding acquisition quantities and market clearing prices and the computed Pareto-superior prices charged to the users of the different services.

20

EX

AM

PL

E O

F R

ATI

ON

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Uni

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nit 6

Dem

and

(MW

)C

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MW

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MW

MW

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W$/

MW

MW

$/M

WM

W$/

MW

MW

$/M

WR

eg10

05

100

815

015

0S

pin

405

7010

100

250

Non

spin

607

5030

0R

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100

1140

340

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507

507

350

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Rep

lace

men

t40

1140

1144

040

11

Pay

men

ts $

800

400

400

600

350

440

Tota

l cos

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99

0

Ra

tio

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al

Bu

yer

Reg

100

850

820

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ulat

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1200

/150

= 8

(2

100-

1200

)/100

= 9

(2

450-

210)

/50

=

7

(282

0-24

50)/4

0

= 9.

25

21

Cu

mu

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3015

015

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300

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660

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035

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.

22

ATTACHMENT B.

Design of a Hierarchial Products Auction for Ancillary Service

Markets

Rajnish Kamat and Shmuel S. Oren

March 21, 2000

1 Introduction

We analyze three simultaneous auction formats for a two product ancillary services market:

1. Pay-as-bid auction (PAB).

2. Uniform price auction minimizing revealed social cost (MSC).

3. Uniform price auction minimizing ISO's cost (Rational Buyer, RB).

The two products are assumed to be hierarchially substitutable e.g regulation and spin-

ning reserves. Regulation capacity can be used as spinning reserves but not vice-versa.

Bid-functions and expected revenues of generators in each auction format is calculated.

We aim to compare revenues from a pay-as-bid auction to the two uniform price auctions.

The rational buyer protocol which minimizes the ISO's procurement costs can result in price

reversals that may violate incentive compatibility in this auction. High type generators may

want to hide their true type. We �rst study the Rational Buyer uniform price auction under

the constraint that generator types are veri�able and the ISO enforces truthful revelation of

types. The pay-as-bid auction does not have this incentive problem as each generator is paid

its bid if selected for service regardless of the type of service provided. The Uniform Price

auction also results in monotone prices and thus always satis�es incentive compatibility.

In further research we intend to relax the true type revelation assumption. If generators

have incentive to misrepresent their types the resulting equilibria will have to be computed

numerically.

2 Pay-as-bid Auction

We begin by specifying expected pro�ts of a generator with q1 MW above it in the merit

order (i.e. with lower costs.) If this generator bids b (with s1(b) MW above it in the merit

1

order), its expected pro�t is:

�1(b) = (b� c1(q1))�(1� F1(s1(b))) +Z s1(b)

0(1� F2 (s1(b) + s2(b)� s)) dF1(s)

�(1)

where,

c1(q1) is cost of a type 1 generator at q1 in the merit order.

s1(b) is the amount of type 1 bidders bidding less than b.

s2(b) is the amount of type 2 bidders bidding less than b.

F1(s) is the cumulative distribution of type 1 demand.

F2(s) is the cumulative distribution of type 2 demand.

The �rst term is the pro�t margin that a generator with cost c1(q1) will receive when

selected for service at bid b. The �rst term in brackets after the pro�t margin term is the

probability that this generator is selected for type 1 service i.e. that demand is greater than

s1(b). The second term is the probability that the generator is selected for type 2 service

i.e. that type one demand is lower than s1(b), but type two demand is greater than what

is available at bid b, s1(b)� s + s2(b) .

The expected pro�t of a type 2 generator with q2 MW above it in the merit order and

bidding b is:

�2(b) = (b� c2(q2))� Z s1(b)

0(1� F2 (s1(b) + s2(b)� s)) dF1(s)

+ (1� F1(s1(b))) (1� F2(s2(b)))�

(2)

where,

c2(q2) is cost of a type 2 generator at q2 in the merit order.

The probability of selection of a type 2 generator depends on the demand for type 1

service. When this demand is low, more low cost type 1 generators will be moved over to

the type 2 auction. This will reduce the probability of selection of a type 2 generator with a

bid more the �rst type 1 generator moved over. The �rst term in brackets is the probability

that a type 2 generator is selected for service given that type 1 demand was such that some

generators with bids lower than b were moved over to the type 2 auction. The second term

is the probability that a type 2 generator is selected for service given that type 1 demand

was such that only generators with bids above b (and thus, not competing with our type 2

generator) were moved over to the type 2 auction.

2

Pro�t maximizing generators face a trade o� between increased pro�ts from bidding

above costs and lower probability that they will be selected for service if they raise their

bids. The �rst order necessary conditions for an optimal bid, b, are:

d�1(b)

db= 0 = (1� F1(s1(b))) +

Z s1(b)

0(1� F2 (s1(b) + s2(b)� s)) dF1(s) +

(b� c1(q1))�� f1(s1(b))s

01(b) + (1� F2(s2(b)))f1s

01(b) +Z s1(b)

0(�f2 (s1(b) + s2(b)� s)) dF1(s)

�s01(b) + s02(b)

� �(3)

d�2(b)

db= 0 =

Z s1(b)

0(1� F2 (s1(b) + s2(b)� s)) dF1(s)

+ (1� F1(s1(b))) (1� F2(s2(b)))

+ (b� c2(q2))�(1� F2(s2) f1(s1(b))s

01(b)�Z s1(b)

0f2 (s1(b) + s2(b)� s) dF1(s)

�s01(b) + s01(b)

�

+ (1� F1(s1(b))��f2(s2(b)) s

02(b)

�+ (1� F2(s2(b)))

��f1(s1(b)) s

01(b)

��(4)

We assume that generators have linearly increasing costs given by:

c1(q1) = �1 + 1q1

c2(q2) = �2 + 2q2 (5)

This function can be inverted to give:

q1(c1) = ��1

1+

c1

1

q2(c2) = ��2

2+

c2

2(6)

We analyze the case where type 1 and type 2 demands are uncorrelated and uniformly

distributed on [r10; r10+ r1] and [r20; r20+ r2], respectively. In the pay-as-bid and uniform

price auctions the lower supports can be normalized to zero the as selection protocol does

not consider how much of each type has already been procured when selecting generators

for service. For the rational buyer protocol, lower supports become important as total

payments are considered and quantity already procured for type 1 may play a role whether

any type 1 genrators will be selected for type 2 service. Table 1 shows the data for the cost

structure and ranges used in the example (see Figure 1.)

As the cumulative distribution function (CDF) of a uniform random variable is non-

di�erentiable at the supports, we derive �rst order conditions accounting for this fact. Figure

2 gives the CDFs for the uniform demand uncertainty case. With our data, c1(r1) > c2(r2).

3

Parameter Value

�1 6

1 0.2

�2 5

2 0.08

r1 100

r2 200

Table 1: Data used in the examples

0 50 100 150 200Capacity -- MW

5

10

15

20

25

Cost--$

Type 1

Type 2

Figure 1: Cost of Generation for Type 1 and 2 Service

4

Thus there are 4 distinct regions of bids over which we integrate the resulting ODEs.

Part 1: As c1(r1) > c2(r2) there will be some type 1 generators that will never be selected

for type 2 service (those with cost greater than c2(r2).) These generators will see demand

as uniformly distributed in the range [q1(c2(r2)); r1]. In the pro�t function of the generator

the bid premium will be multiplied with only the �rst term in brackets. This is like a single

demand bidding problem (Green, 1999). The marginal generator at r1 will bid cost as it

cannot sustain a bid above cost in equilibrium. 1

0 50 100 150 200Demand -- MW

0.2

0.4

0.6

0.8

1

CDF

Type 1

Type 2

Figure 2: CDF for Uniformly Distributed Demands

Proposition 1 (Green 1999) Bidders with bids above c2(r2) will bid:

b(s1) =1

1� F1(s1)

Z r1

s1

(�1 + 1s) dF1(s) (7)

Proof: See appendix.

Part 2: At b = c2(r2) the total capacity available is greater than r2 (as s2(c2(r2)) = r2).

Thus, there will exist a b� such that:

s1(b�) + s2(b

�) = r2 (8)

1If the marginal bidder bids above cost, the best response of inframarginal bidders is to increase their bids

as this increases their pro�ts without decreasing the probability of being selected. However, the best response

of the marginal generator to this response is to under-cut an inframarginal bidder. Now the marginal bidder

has a positive probability of being selected and will make a strictly positive pro�t. One can see that this

cannot be sustained and in equilibrium the marginal bidder bids cost.

5

The pro�t functions for b� � b � c2(r2) are given by:

�1(b) = (b� c1(q1))�(1� F1(s1(b)))

+

Z s1(b)

maxfs1(b)+s2(b)�r2;0g

Z r2

minfs1(b)+s2(b)�d1 ;r2gdF1(d1)dF2(d2)

�(9)

�2(b) = (b� c2(q2))�Z s1(b)

maxfs1(b)+s2(b)�r2;0g

Z r2

minfs1(b)+s2(b)�d1;r2gdF1(d1)dF2(d2)

+ (1� F1(s1(b))) (1� F2(s2(b)))�

(10)

Whenever total capacity available at bid b for type 2 service (after deducting type 1

demand) exceeds r2, the probability that a generator who bids b will be selected for type

2 service is 0. Therefore, the amount of capacity available for type 2 service will be less

than r2 for d1 > s1(b)+ s2(b)� r2. One can derive �rst order conditions using this range of

(d1; d2).

Initial conditions for part 2 are:

s1(c2(r2)) = s�1

s2(c2(r2)) = r2 (11)

where,

s�1 is calculated from the bid function in part 1 at b = c2(r2).

Part 3: For b < b�, there is always a strictly positive probability that a type 1 generator

who is in a type 2 auction gets selected for service. The pro�t function in this case will

have 0 as the lower limit on the range of d1 and the lower limit on d2 is s1(b) + s2(b)� d1.

Initial conditions for part 3 are:

s1(b�) = s��1

s2(b�) = s��2 (12)

where,

s��1 and s��2 are calculated from the bid functions in part 2 at b = b�.

This set of ODEs is solved till s1(b) = 0. This is the minimum bid that will be submitted

by a type 1 generator, b1min.

Part 4: At prices below b1min for type 2 service, no type 1 generator will ever get selected

for type 2 service. The pro�t function for type 2 generators will therefore be independent

6

of type 1 demand and will resemble Part 1 of the analysis. Proceeding in a similar fashion

we can derive the bid function for type 2 generators in this region.

The initial condition for this set of ODEs is:

s2(b1min) = s���2 (13)

where,

s���2 is calculated from the bid functions in part 3 at b = b1min.

This is solved until s2(b) = 0. This is the minimum bid that will be submitted by a

type 2 generator, b2min. The bid functions (after inversion) are shown in Figure 3.

0 50 100 150 200Capacity -- MW

5

10

15

20

25

$

Type 1

Type 2

Figure 3: Bid functions for the pay-as-bid auction

One can now use the bid functions to calculate expected revenues paid out by the ISO

in a pay-as-bid auction. Figure 4 shows expected revenues of a generator as a function of

their place in the merit order.

3 Uniform Price Auction (MSC)

In a uniform price auction for procuring a single service type all generators bid cost in

equilibrium. The logic is that a small generator cannot a�ect the price it is paid by altering

its bid. However, increasing its bid above cost reduces the probability of selection. Also,

there is no equilibrium with any generator bidding below cost. Though, this does increase

the probability of service, it does not change prices the generator receives when demand is

greater than q1, its place in the merit order, while resulting in negative bid premiums when

demand is below this level.

7

0 50 100 150 200Capacity -- MW

2.5

5

7.5

10

12.5

15

ExpectedRevenue--$

Type 1

Type 2

Figure 4: Expected Revenues in a PaB Auction: Optimal Bid vs. Bidding Cost

This logic extends to the case of 2 service types. For part 1 and part 4 generators

who are in a single service type world, the above logic applies directly. For the remaining

type 1 generators, bidding above cost reduces the probability of selection for type 1 service.

Also, these generators will spill over to the type 2 auction at a higher bid and thus have

a lower probability of selection for type 2 service as well. The prices a small generator

receives cannot be changed in any case. Thus, bidding cost is an equilibrium strategy in a

simultaneous hierarchial uniform price auction.

We are now in a position to calculate expected revenues that the ISO pays out in a

uniform price auction format. As prices are decided by the marginal generator selected of

each type, irrespective of what service it provides, prices will depend on both demand levels.

This leads to Proposition 2.

Proposition 2 Expected revenues in a uniform price auction are given by:

ERMSC1 (s1) =

Z r1

s1

Z minfd�2;r2g

0(�1 + 1d1) dF2(d2)dF1(d1)

= +

Z r1

s1

Z r2

minfd�2;r2g

(�3(d1) + 3d2) dF2(d2)dF1(d1)

+Z s1

0

Z r2

minfs1�d1+q2(�1+ 1s1);r2g(�3(d1) + 3d2) dF2(d2)dF1(d1) (14)

ERMSC2 (s2) =

Z r1

minfq1(�2+ 2s2);r1g

Z minfq2(�1+ 1d1);r2g

s2

(�2 + 2d2) dF2(d2)dF1(d1)

+Z r1

minfq1(�2+ 2s2);r1g

Z r2

minfq2(�1+ 1d1);r2g(�3(d1) + 3d2) dF2(d2)dF1(d1)

8

+

Z minfq1(�2+ 2s2);r1g

0

Z r2

minfq1(�2+ 2s2)�d1+s2;r2g(�3(d1) + 3d2) dF2(d2)dF1(d1)(15)

where,

d1 and d2 and demands for type 1 and type 2 service, respectively.

d�2 = q2(�1 + 1d1)

�3(d1) = 1 2 1+ 2

��1 1

+ �2 2

+ d1

�is the intercept of type supply when demand for type 1 ser-

vice is d1.

3 = 1 2 1+ 2

is the slope of the resulting type 2 supply schedule.

Proof: See appendix.

Figure 5 shows expected revenues in a MSC uniform price auction.

0 50 100 150 200Capacity -- MW

2.5

5

7.5

10

12.5

15

17.5

20

ExpectedRevenues--$

Figure 5: Expected Revenues in a MSC Uniform Price Auction.

4 Rational Buyer

Finally, we analyze the Rational Buyer (RB) protocol that was adopted by the ISO in

California. This is a simultaneous uniform price auction that minimizes the ISO's pro-

curement cost. Given bid functions and demand realizations, the ISO solves a minimum

cost-to-procure problem and determines optimal quantities to procure. We �rst examine the

auction under the assumption that types are veri�able and that the ISO enforces true type

disclosure. We aim to compare expected revenues paid to a bidder under this constrained

auction to the other auction formats studied so far.

In this case the ISO solves (assuming that generators bid cost) a minimum Cost-to-

Procure (CtP) program:

9

minq1;q2

CtP = (q1 + r10)(�1 + 1q1) + (q2 + r20)(�2 + 2q2)

q1 � d1

q1 + q2 � d1 + d2 (16)

where,

q1 and q2 are quantities, net of lower supports on the demand uncertainty, of the two service

types to be procured.

Using Karush-Kuhn-Tucker conditions one can solve for the optimal quantities to be

procured for given demand realizations.

Case 1: 1st constraint inactive; 2nd constraint active.

q1(d1; d2) =�2 � �1

2( 1 + 2)�

1r10 � 2r20

2( 1 + 2)+

2(d1 + d2)

1 + 2

q2(d1; d2) = ��2 � �1

2( 1 + 2)+

1r10 � 2r20

2( 1 + 2)+

1(d1 + d2)

1 + 2(17)

Case 2: Both constraints active.

q1(d1; d2) = d1

q2(d1; d2) = d2 (18)

In what follows we assume 1r10� 2r20 = 0 or that r20 = 2:5r10 for our data. Historical

data shows that average amount of spin and non-spin reserves (our type 2 service) demanded

is about 2.5 times the amount of regulation (our type 1 service). We begin by stating

Proposition 3.

Proposition 3 Bidding cost is an equilibrium in a constrained RB (cRB) auction.

Proof: See appendix.

One criticism of the Rational Buyer model is that under certain cost and demand con-

ditions it can lead to price reversals (i.e. low type generators are paid more than same

cost high type ones), thus making the auction violate incentive compatibility constraints.

One should remember, however, that when demand is uncertain, even if instances of price

reversals do occur, the auction will violate incentive compatibility only if expected pro�ts

for a type 1 generator under the constrained auction are lower than for a type 2 generator

with the same cost. The auction formats discussed in earlier sections do not su�er from

such incentive problems and even equivalence between these auction formats and the cRB

auction will suggest that there may better alternatives to the RB model. We are now in a

position to state Proposition 4.

10

Proposition 4 Expected revenues in a constrained Rational Buyer (cRB) uniform price

auction are:

ERcRB1 (s1) =

Z d�2

0

Z r1

s1

pcase21 (d1; d2)dF1(d1)dF2(d2) +

Z d���2

d�2

Z r1

s1

pcase21 (d1; d2)dF1(d1)dF2(d2)

+Z r2

d���2

Z r1

minfd�1(d2);r1g

pcase21 (d1; d2)dF1(d1)dF2(d2) +Z d��

1

0

Z r2

d��2(d1)

pcase11 (d1; d2)dF2(d2)dF1(d1)

+Z r1

d��1

Z r2

d��2(d1)

pcase11 (d1; d2)dF2(d2)dF1(d1) (19)

ERcRB2 (s2) =

Z r2

s2

Z r1

minfd�1(d2);r1g

pcase22 (d1; d2)dF1(d1)dF2(d2) +Z d��

1

0

Z r2

d��2(d1)

pcase12 (d1; d2)dF2(d2)dF1(d1)

+

Z r1

d��1

Z r2

minfd�2(d1);r2g

pcase12 (d1; d2)dF2(d2)dF1(d1) (20)

where,

pcaseji is the price for service type i, for Case j described in the ISO cost-to-procure program

above.

d�2 =�1��22 2

.

d���2 = �1��22 2

+ 1 2s1.

d�1(d2) =�2��12 1

+ 2 1d2.

d��2 (d1) =�1��22 2

+ ( 1+ 2) 2

s1 � d1.

d��1 is be derived by solving d�2(d1) = d��2 (d1), where d�2(d1) is the inverse of d

�1(d2).

Proof: See appendix.

Figure 6 shows expected revenues for the cRB auction.

5 Discussion and Future Research

We can now compare expected revenues across the auctions. Figure 7 shows expected

revenues for the three auction formats, while Table 2 shows the expected payments made

by the ISO.

Observe that the MSC and RB auctions are almost equivalent. Type 1 generators are

paid a little more in expected terms in the RB auction than in the MSC auction. The

reverse holds true for type 2 generators. This can be seen by comparing the prices that

these generators get paid when type 1 procurement is greater than type 1 demand (the

probability of selection is the same.) Under the RB auction:

pcase1;RB1 (d1; d2) = �1 + 1q1(d1; d2)

11

0 50 100 150 200Capacity -- MW

2.55

7.510

12.515

17.520

ExpectedRevenues--$

Figure 6: Expected Revenues in a constrained Rational Buyer Auction

0 50 100 150 200Capacity -- MW

2.5

5

7.5

10

12.5

15

17.5

20

ExpectedRevenue--$ MSC

RBPYB

Figure 7: Comparison of Expected Revenues across Auctions

Type 1 Type 2 Total

PYB 994.99 1508.04 2503.03

MSC 1085.05 1394.91 2479.96

cRB 1100.26 1379.39 2479.65

Table 2: Expected Payments by the ISO for the three auctions

12

= �1 + 1

��2 � �1

2( 1 + 2)+

2

1 + 2(d1 + d2)

�(21)

(22)

On the other hand, under the MSC auction:

pMSC1 (d1; d2) = �3(d1) + 3d2

= 1 2

1 + 2

��1

1+

�2

�2+ d1

�+

1 2

1 + 2d2 (23)

(24)

The term multiplying total demand is the same in both cases. The constant term is 5.643

for the RB auction for our data, while it is 5.286 for the MSC auction. As the probability

of selection is the same, the higher prices in the RB case result in the di�erence seen in

Table 2. The reverse holds true for type 2 generators.

The striking feature of the results is that in expected terms, the PYB auction pays a

signi�cantly lower sum to some type 1 generators as compared to the other two auction

formats. Again, the reverse holds true for type 2 generators. Expected payments increase

by 0.93 percent in the PYB auction as compared to the MSC auction. This is in contrast

to the single demand case results (see Green and McDaniel, 1999a) where the PYB auction

was shown to be equivalent to an MSC or uniform price auction.

For type 1 generators below 75 MW in the merit order the three auctions pay the same in

expected terms. This is because these generators have zero probability of selection for type

2 service and are the three auctions are equivalent in a single demand case. For genrators

higher up in the merit order, say, the generator at 50 MW in the type 1 merit order the

MSC and RB auction pays more in expected terms than the PYB auction. This generator

bids $21 in the PYB auction and has a probability of selection of 0.5 (this generator is never

selected for type 2 service as s2(21) = 200.) Thus, our generator has expected revenues of

$10.5. In the MSC auction the probability of selection is greater than 0.5 (it can be selected

when demand for type 1 service is lower than 50 MW.) The average price it gets over the

range of selection in a PYB auction is greater than the average of its cost and the cost of the

most expensive generator (the one at 100 MW in the merit order), which is 21. Therefore

expected revenues are higher as seen in Figure 7.

5.1 Future Research

We had mentioned initially that under certain cost and demand conditions the RB protocol

could result in price reversals and thus would violate incentive compatibility. For the data

chosen for this study (and in general when type 1 cost schedules dominate type 2 cost

schedules and the term 1r10 � 2r20 = 0), this does not happen. We state this result as

Proposition 5.

13

Proposition 5 The RB auction satis�es incentive compatibility when type 1 costs dominate

type 2 costs and 1r10 � 2r20 = 0.

Proof: See appendix

In ongoing research we aim to solve for the equilibrium of an unconstrained RB (uRB)

auction. As noted earlier, the cRB auction will violate incentive compatibility under certain

cost and demand conditions. When the term above is non-zero, quantities procured for

the two service types are very di�erent from demands and thus comparison for same cost

generators becomes irrelevant. With the above assumptions, however, we conjecture that

in equilibrium a band of type 1 generators will want to misrepresent their types. This will

result in discontinuities in the type 1 cost schedule and in kinks in the type 2 cost schudule.

Thus, the equibrium will have to be solved numerically.

6 Appendix

Proof of Proposition 1: A generator with cost c(q1) will solve:

maxb

�1(b) = (b� c(q1)) (1� F1(s1(b))

) (b� c(q1))�1� f1(s1(b))s

01(b)

�+ (1� F1(s1(b))) = 0 (25)

This can be solved in inverse form by substituting s01(b) =1

b0(s1). This gives:

db

ds1= (b(s1)� (�1 + 1s1))

�f1(s1)

1� F1(s1)

�(26)

We can use an integrating factor, eRd log(1�F1(s1)) to get the formula in Proposition 1.

Proof of Proposition 2: A type 1 generator at s1 in the merit order will be selected for

type 1 service if demand is greater than s1. For a demand of d1 ( > s1, the price that this

generator will be paid will depend on whether any type 1 generators that spilled over to

the type 2 auction were selected for service. There will be a type 2 demand level, d�2, such

that if type 2 demand is less than d�2, no type 2 generator is selected for type 2 service. In

this case both types will be paid di�erent prices based on their marginal generators. At

demand levels > d�2 (if they exist), some type 1 generators are selected for type 2 service

and both types will be paid the same price. Finally, when type 1 demand is below s1 our

generator will be selected whenever type 2 demand is greater than s1 � d1 + q2(�1 + 1s1)

(if it is within its range). It will be paid the price of the marginal generator selected for

type 2 service. Figure 8 shows the portion of range over which a type generator at s1 in

14

the merit order is selected for service. Substituting the appropriate limits and prices when

taking expected values gives us the formula in the proposition.

For type 2 generators there will also be three cases. In the �rst case, type 1 demand is

greater than the quantity corresponding to this generators cost and type 2 demand is such

that no type 1 generator is selected for type 2 service. In the second case, there are some

type 1 genrators that are selected for type 1 service and both types are paid the same price.

In the third case, demand is less than the quantity corresponding to this generators cost,

and when this generator is selected some type 1 generators are also selected and both are

paid the same price. Figure 9 shows the portion of range over which a type generator at s2

in the merit order is selected for service. One can substitute these limits and prices in the

expected revenues formula and arrive at the formula in the proposition.

0 20 40 60 80 100Type1 Range -- MW

0

50

100

150

200

Tye2Range--MW

Figure 8: Range where a Type 1 Generator at s1 is Selected

Proof of Proposition 3: We �rst examine the best response of an arbitrary type 1

generator in an constrained RB auction when everyone else bids cost. Figures 10 and 11

show the ranges of demands where the two types are selected. One can see that in the

upper left corner, when type 1 demand is low and tyepe 2 demand is high, the ISO can

reduce cost by selecting more type 1 generators than type 1 demand. As one proceeds to the

lower-right corner, the optimal procurement involves meeting each demand with generation

of that type. As quantities are a linear function of the two demands the two regions will be

separated by a straight line at:

q1(d1; d2) =�2 � �1

2( 1 + 2)+

2(d1 + d2)

1 + 2= d1 (27)

15

0 20 40 60 80 100Type1 Range -- MW

50

100

150

200

Tye2Range--MW

Figure 9: Range where a Type 2 Generator at s2 is Selected

This can be written as:

d�2(d1) =�1 � �2

2 2+

2

2d1 (28)

Another boundary can be written when quantity of type 1 demand is lower than s1 and

Case 1 holds to make it equal to s1.

q1(d1; d2) =�2 � �1

2( 1 + 2)+

2(d1 + d2)

1 + 2= s1 (29)

This can be written as:

d��2 (d1) =�1 � �2

2 2+

1 + 2

2s1 � d1 (30)

Now a type 1 generator at s1 in the merit order will be selected for service whenever

demand for type 1 service is > s1. In this case altering its bid does not change the prices it

receives, while it reduces the probability of selection. When type 1 demand is < s1, there is

still a positive probability that this generator is selected for service. There is a region where

type 1 demand is su�ciently low and type 2 demand su�ciently high to justify procuring

more type 1 service than demanded. In this case too our generator cannot change prices

it receives, while bidding above cost reduces the probability of selection. Reducing bids

below cost is unpro�table in both cases. For demand realizations that this generator will

not be selected prices are below its cost and it rather not be selected for service if this does

not have any e�ect on outcomes in other regions. In the previous cases bidding below cost

16

0 20 40 60 80 100Type 1 Range -- MW

0

50

100

150

200

Type2Range--MW

Case 1

Case 2

Figure 10: Range where a Type 1 Generator at s1 is Selected (RB)

0 20 40 60 80 100Type 1 Range -- MW

50

100

150

200

Type2Range--MW

Case 1

Case 2

Figure 11: Range where a Type 2 Generator at s2 is Selected (RB)

17

is also not optimal for this generator because of similar reasoning as in the uniform price

auction minimizing social cost. One can follow this reasoning for a type 2 generator as well.

Therefore, bidding cost results in an equilibrium in the constrained RB auction.

Proof of Proposition 4: The formulae can be derived by substitution appropriate limits

according to the areas outlined in the proof of Proposition 3.

d�2 = d�2(0) =�1��22 2

.

d���2 can be derived by solving d�2(d1) = s1.

d�1(d2) is the inverse of d�2(d1).

d��1 is be derived by solving d�2(d1) = d��2 (d1).

Proof of Proposition 5: With the assumption made lower supports can be normalized

to zero. This means that whenever demand realizations are such that type 1 price will

be lower than type 2 price for a certain procurement policy, total payments can always

be reduced by procuring more type 1 generators. This is because the quantity of type 1

generators already procured will be smaller (as its cost dominate type 2 generators' costs).

Even at equal prices, it will be desirable to procure more type 1 generators. Therefore, in

the optimal solution to each of the ISO's cost-to-procure programs, type 1 prices will be

higher than type 2 prices.

7 Bibliography

1. Green R., 1999. "Rebidding in the balancing mechanism: An Economic Analysis,"

mimeo, Department of Applied Economics, University of Cambridge, Cambridge, UK.

2. Green R. and T. McDaniel, 1999a. "Expected Revenues in the Balancing Market:

Equivalence between Pay-as-bid and SMP," mimeo, Department of Applied Eco-

nomics, University of Cambridge, Cambridge, UK.

3. Green R. and T. McDaniel, 1999b. "Modelling Reta: A Model of Forward Trading and

the Balancing Mechanism," mimeo, Department of Applied Economics, University of

Cambridge, Cambridge, UK.

18