4 • October 2005 SIAM N",WS

Math Model Explains High Prices moving power to meet demand and impos­ing transmission charges that wpuld reflect the supply/demand balance in each area of the grid: (The utilities continued to own the lines and were to be compensated for their use.) CAlSO was also to manage a [eal-time market fOI power, as well as for other ancil­)ary services necessary for maintaining the stability of the grid.

, in Electricity Markets market power, notes Severin Borenstein,

a profess,?[ o[ business administration and public policy at the University oLCalifor­nia. at Berkeley, in · a· ·paper titloo . 'The Trouble with ElectricitY ·Markets: Under­standing California's Restructuring · Dis­aste(' (Journal of EcOlwmic /,'erspeclives, WintI"2(02).

By Sara Robinson

In 1996, in ihe hope of reducing its his­torical1y high electricity prices, California becatne the first U.S. state to vote to open its energy markets to competition. Following electricity reform efforts in other countries, the state planned to unbundle the different functions of the old regulated monopolies, introducing competition for power genera­tion and retail delivery.

California's venture, however, turned into a disaster of immense proportions, and became the poster child of an anti-power­restructuring movement. Beginning in the summer of 2000 and continuing through the summer of 2001, wh.olesale power prices soared, forcing the federal government to subsidize power for low-income families in San Diego and bankrupting one of the state's three major utility companies. At the same time, mysterious snortages in avail­able generation capacity forced statewide rolling blackouts. All in all, the energy cri­sis may have cost the state as much as $45 billion in higher electricity costs, lost busi­ness due to blackouts, and a slowdown in economic growth, according to the Public Policy Institute of California.

Following California's vote, several other regions of the U.s.-notably parts of the Nonheast and Texa~ecided to restruc­ture their power markets as well. While there have been no full-out disasters, these markets, too; have been beset with volatile wholesale prices and other problems.

The conventional explanation .for Cali~ fomia's power ·crisis is that flaws in the restructuring agreement, combined with poor federal oversight, made it possible for companies lOllJlifjciallY ,driye up. p'rj~!. Restructured markets that are w~JJ designed and well regulated will operate more effi­ciently than the old monopoly system, most economists say, although most experts agree that U.S. markets, so far, have not realized that promise.

Sean Meyn, an electrical engineer, and In-Koo Cho, an· economist, both at the University of llIinols at Urbana-Charn­paign, are weighing in with an alternative perspective on restructuring. In a recent paper, not yet published, the researchers introduce a dynamic pricing model for decentralized power markets that incorpo· rates some of the unique engineering con­straints, such as the slow ramp up of reserve power. sources and inelastic supply and demand. From their model , the researchers conclude that restructured power nlarkets (in their current form) are doomed to fail. "The difficulties that have appeared in California and elsewhere are intrinsic to Ihe

desigo. of current .. mark,elS," .M~yn says. Economists interviewed for this article

disagree with ·Cho and Meyn's conclusion about the viability of decentralized power markets, pointing to restructured markels in the UK, Latin America, and, to .a. lesser extent, Australia as funct.ioning better. than the regulated g.overnment~~wned mODC?po.­lies that pr~eded them. "Restructuring has reduced electricity production costs and benefited consumers relative to the former regime in a number of countries around the world, but . .. this has taken a· number cif years;" says Frank Wolak, a professor of economics at Stanford University.

"Every. market is imp~rfecl," adds Paul Joskow. a professor of econ:Dmi~ a~ man­agement at the Massachusetts Institute of Technology, "aod in many of these markets, the performance before (restructuring} was awful."

The Constraints Economists and engineers agree that

some of the unique physical attributes of

The California Debacle: How Things Went Wrong

In the late I 990s, despite these chal­lenges, California and other parts of the U.S. moved quickly to restructure their mar­kers. C)lifoI1)ia's Power Exchange··opened for trading in March 1998, two years after the vote that set the transition in motion. ·By then, the state's three major electric ·utilities bad auctioned off most of their fossil fuel­fired power pla~.ts. The ·new ownets-ener­gy companies based in Texas and .the Southeast, including Duke, AES/wilIi~m~ . Dynegy, Mirant, and Reliant-were. exPect-00 to sell powerbac~IO the utilities in,day­ahead; hourly, and real-time markets. To reimburse the utilities· for their ';stranded assets"'-::':invesunenis from which they

Economists and engineers agree that some of the unique physical attributes of electricity make the design of efficient power markets a challenge,

would· no· longer benefit-the state legislature froze consumer electricity rates throug~ .¥arch 2002 at a price thought·to be ben; eficial to the utilities; the freeze was to be lifted sooner if the util­ities collected enough to payoff their debts before that date.

To operate new auction-based · electricity make the desi,gn of effi~ient markets for power, and to manage the trans-power mar~ets a ch;!llenge. (See "The mission grid, the state had created two non-Power Grid: Fertile Ground fOI Math profit corporations: the Califomia Power Research," SIAM News, October 2003; Exchange, which would operate the main hllp:/Iwww.siam.org/siamnews/lO-03/ forward competitive energy markets into powergrid.htm.) which generators would sell electricity, and . pn.e, c.~~stf .. int is that PQw,~r; !)\~s!~.pfP,. ".' .tl),~"S~.lir~!l\i~;M~I*~d.el!J,SY1tem .QperatQI

duced as it is needed b.,:ause of the· pro bib, .(CAISO), which was to provide transmis-itively high cost of storing it. Because sion services to all electricity suppliers, demand for power is difficult to predict, some power plants are kepi on reserve, ready to begin production on short notice. Even so, there is a lag time before addition­al power can be generated. Because of capacity constraints in power lines and other equipment, it can also ~ a challenge to deliver electricity where it is needed, when it is needed. A local imbalance of supply and demand. moreover, is not an isolated event: It threatens the stability of the ~ntire regional grid, potentially disrupting all con­sumers and suppliers. Compounding the problem, consumers are not exposed to real­time prices under the current systems, and thus have no incentive to decrease their po..,ver consu.mption when prices. arc:; high.

Asin markets that had been established in other countries, the wholesale price for power was to be determined by auction. All the generators would submit bids to supply a specified-amount of power at a certain price. The Power Exchange · would then accept bids, 'starting with the lowest, until demand was satisfied. The price of the last increment of power needed would then be the price paid to all successful bidders. In the case of an ' overabundance of available powe~. with many generators competing to supply it, the price would be low. When demand exceeded available supply, howev­er, generator3. no longer at risk of being unable to seH alllheir power, would have an incentive to bid high.

During the ftr~t year, there were already signs that market participants were testing their ability to a.ffect prices. On one July afternoon in 1998, for example, the price paid to. generators · for providing reserve power hit · $9999 per megawall-hour of capacity available, compared· with the usual price of about S10. At the same time, market prices f9r . power also soared to record heights. In response, CAISO asked the Federal Energy Regulation Commission to institute price caps of $250 .per megawall­hour, which it did:

The price caps did not prevent other fonos of corporate mischief, ·however. In May 1999, Enron scheduled a shipment of 2900 megawatts of·powc;r over a line whose capacity was only 15 megawatts. In the con­gestion aDd chaos that followed, pri~es shot up more than 70% statewide. (Enron was later fined just $25,000 for the incident.)

.,D.espile.ucb g\jl£~s , .tIi~,m;\f~tJllncJi9n~ ' ·well · for the 'most· part through 1998 and

See Electricity Markets on page 8

In"this environment, short-tenn prices for electricity lend to be very volatile, and there are many opportunities for the exercise of

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Electricity Markets cOlllilluedfrom page 4

1999, with wholesale prices hovering around $35 per megawau·hour.

California's demand for electricity peaks in the summer months, when air condition­ers are in -use. The stale usually counters the increased demand by imponing hydropower from the Pacific Northwest. In the summer of 2000, however, hydropower was less abundant than usual because of a drought. At the same time, prices of natural gas and nitrogen oxide pollution permits were unusually high, driving up the operating costs of gas-powered generators.

In an already precarious situation. these events combined to create a perfect storm, with wholesale power prices soaring to lev­els nearly 500% higher than those of the previous summer. Consumers throughour mosl of the slate were shielded from the price fluctuations, but the utilities w~re puy­ing as much as $750 per megawatt-hour, ten times the average wholesale rate that was implicit in the fixed retail rate for con­sumers.

Only the San Diego Gas and Electric Company, which had already recovered its stranded-assets costs and was thus exempt from Ihe rate freeze, was able to raise retail prices. In July, San Diego customers began receiving electric bills two to three times higher than those of previous sum-mers. In response to the ensuing howls of protest, the state stepped in and reimposed the retail caps. Still. the federal government had to use emer~ gency funds to help some residenlS pay their inflated utility bills.

As winter approached, demand for electricity lessened, but the power shortages and high prices continued unabated. On November I, 2000, FERC issued a preliminary order con­cluding that the inflated power prices were "unjust and unreasonable," and thus in violation of the 1935 Federal Power Act that FERC is mandated 10

uphold. The organization suggested several "remedies" for the situation, including replacement of the $250 per megawau-hour price cap with a "soft cap" of $150 per megawall-hour; bids exceeding the cap would be permitted only if generators could show that their costs justified the bids. (FERC also asked the state to raise retail rates and make it easier for the utilities to enter into long-term contracts.) The market surveillance commiuee, chair-ed by Wolak, responded with a report saying that the suggested remedies would not help and. in fact, were like-ly to make things worse. The commit-tee noted that the soft price cap would function like no cap at all, because affiliate transactions could be used to inflate a generator's costs, justifying higher bids.

Nonetheless, FERC implemented the soft cap. In January 2001, prices soared to new highs, averaging $290 per megawatt-hour, and chronic short-ages of available power forced state-

cap on wholesale power prices across the West.

Economists' Perspective on the Crisis

California's power problems, Wolak sug­gesls in a paper titled "Lessons from the California Electricity Crisis" (Eleclriciry loumal, August 2003), stemmed primarily front some critical miscalculations by California's utilities, combined with ex­tremely poor oversighl by rcgulators. In the resulting environment, even a company with a small market share could drive up prices by withholding supply. Staning in the sum­mer of 2000, that ability was enhanced by the diminished competition from hydro­power plants·outside the state.

Several groups of economists (that include Borenstein, Joskow, and · Wolak) perfon<led studies comparing the actual prices in California 's markets with Ihose predicted by models under the same supplyl demand conditions. The actual market prices were significantly higher than the models' predictions. At the same time, data indicates that the number of power plants offline for maintenance or emergency repairs was substantially higher than nor­mal. These factors, taken together, are evi­dence that companies ·were indeed exercis­ing market power during the crisis, 10skow says.

Mark~( power was a problem because of the utilities' failure to hedge, by purchasing power in advance to be delivered at a future date, Wolak says. Joskow agrees that a lack of forward contracting was a key issue, bilt pUIS the blame on the state's regulatory agency, rather than t~e utilities.

Under the restructuring agreement, all power transactions had to go through the Power Exchange. A generator that had con-

being a110wed to sell its power. ID that situ­ation, the generator would have 10 purchase power at markel prices to meet its contract.

Compounding these problems was the failure of regulators to understand the prob­lems plaguing the market and order appro­priate remedies. It is Ulis, Wolak believes, tha" turned market glitches into a fuJI-scale crisis. Regulatory problems are also pre­venting other electricity markets from deliv-

Ramp-up constraints for power plants, combined with inelastic demand, might make any decentralized approach inherently inefficient, even in the absence of the exercise of market power by participants.

ering on their promise of lower power costs, he says.

Borenstein, in his paper, advo­cates a combination of long-term contracting and real-time pricing to address the problems in California and elsewhere. In California, retail rates were fixed , and in other u.s. markets, con­sumers pay a time-averaged price for power or, at best, prices for

(racted to sell a specified amount at a partic­ular price would still have to bid in the power market for Ule right to produce that power. Through side payments, the effective payment from the utility 10 the generator would be the contracted price.

Such a contract reduces a generator's incentive to withhold capacity to inflate prices, Wolak. notes, because the generator has already sold the power. Moreover, if the generator bids high in the Power Exchange, it risks losing out in the auction and thus not

peak and off-peak times. With exposure to real-time fluctuations in power prices, con­sumers would have the incentive to con­serve during peak. times. This would lower the production capacity needed to meet peaks in demand, Borenstein writes, and would reduce prices for long-term contracts. Such measures are not popular, howev~r.

The lack of real-time pricing "is a regula­tory failure because it is technologically feasible to do thiS," Wolak says. "State pub­

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Electricity Markets continued from pagt 8

lie utility commissions will not implement retail pricing plans that charge customers according to the hourly wholesale price."

An Engineering Perspective Withoul dis pUling Ihallhe issues ciled by

the economists were in play in the California power crisis. Me)'n says he can­not wholly agree with their conclusions. "Why should high prices imply Ihe exercise of markel power?" he asks. Such conclu­sions, Me)'n says, are based Qn intuition obtained from observing gross manipulation by Enron and others • . and on decades of analysis of static models.

For Meyn, the fundamental explanation for the problems in California and else­where lies in the engineering constraints' on power markets. In March 2004, at a work.­shop on power networks at the Institute for Mathematics Wld its Applications at the University of Minnesota, Meyn questioned whether a lack of· long-term contracts and market manipulation could completely explain the collapse of California's market. He hYPolhesized Ihal the ramp-up con­straints for power plants, combined with inelastic demand, mighl make any decen­Iralized approach inherenlly inefficienl, even in the absence of the exercise of mar-

tet power by participants. The static models used,by economists to describe electricity markets do not capture ramp-up constraints. he says.

Following Ihe IMA workshop, he and Cho sel oullO devise a dynamic modellhal would capture Ihese key physical con­sllaints and yel be simple enough 10 yield an economic analys is. This spring, Meyn and Cho finally arrived al a model thaI, Meyn believes, captures the essential featu res of power markets, albeit in simplified form, They have summarized their results in a working paper litled "Oplimizalion and the Price of Anarchy in a Dynamic Newsboy Model."

The researchers model nonnalized de-­mand as Ii driftless Brownian motion that is completely unresponsive to price. The aver­age demand is met through a long-term con­trac~ with the primary source, while short­term fluctuations are accommodated by multiple ancillary power sources. All power sour~es are rate-constrained in thei r ramp up, With the ancillary power sources ramp­ing UP fasler Ihan the primary one. Any exce~s power produced can be dumped at any ltime without cost., a simplifying as­sumption that is not true in reality. Ancillary power prices are also assumed to have con­slanl. marginal costs as high as , or higher than, Ihal of Ihe primary source-a grossly simp'lifying assumption, the researchers

note. because real costs are nonlinear. There is a high sOCial cost of not meeting demand and incurring a blackout.

. Slalling from the premise that socielY benefits from the consumplion.of electricily, Ihe researchers firsl solve for a policy Ihal maximizes the total social value: the sum of Ihe profils of Ihe generators and the social value for the consumers. (For readers versed in conlrol theory: This required solving a multidimensional singular control prob­lem.) This policy represents what a central­ized controller of a power markel would ideally try to achieve.

The theoretical goal for decentralized markets. however. is to optimize through the actions of individuals. without a central controller. Toward this end. the researchers show the existence of an equilibrium price functional for which both consumers and generators arrive at the same policy deci­sions in a decentralized market. They prove that this equilibrium is unique, and that it coincides with the centralized social opti­mum. Cho and Meyn go on to show that even when their statistical assumptions on demand are relaxed, an equilibrium price functional , if it exists, must be of the Conn oblained for Ihe model with Brownian demand.

Remarkably, the prices corresponding 10 this unique equilibrium are, on average, so high that a consumer would incur a negative

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~'. . '. extraOrq,i~ary. . oppOl:~.ni~y·!,for established rriai.lie~alt.· .. c.ians-~ypfi:a llY mid-career

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program at the IMA. Visfti!l&",frofessors will enjoy an q~cellellt research environment and stimulating scientific prograg1 .~qnuecting algebraic geometry and related areas of mathematics with a broad range of fields of~ppli'ca!io.ll:,.N\;::v. Qjrecfions Visiting Professors arc expected to be resident and active partiCipants in.the program, but are not assigned fonnal duties.

for m.orc infonnation and application materiCiis see www.ima.umn,ed u/docslmembership.btml or phone 612-624-6066.

Tbe University of Miuncsota is committed to the policy that all persons shaH have equal access to it ~ programs, facilities, and . employm~nt without regard to r!lce, color, creed. religion, natioual origin, sex, age, marital status, disability, pUblic assistance ~latu.sl '

, veteran status, or sexual oricluation.

.. The IMA is an NSF funded Inslffule www.ima.umn.edu UNIVERSITY OF MtNNESOTA

October 2005 SIAM NEWS· 9

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. benefit from participaling in the market. Prices will be volatile even without manipu­lation, Meyn says. because the unpre­dictabilily of demand and Ihe slow respoDse of supply mean Ihere will always be times when demand approaches the available sup· ply and prices soar,

In the model, generators are assumed to be price-takers, meaning that a single gener­ator's actions cannot affect the price. When generators do have market power. however, the bas ic problem becomes more pro­nounced, Meyn says. If there is a re'aI-time demand response 10 high prices, Ihe prob, lem disappears for Ihe mos t part because demand would approach supply only rarely, Meyn points oul. Still, if power reserve lev­els fall suddenly due to ao outage or Slorm,

prices could again explode. Sleven Shreve, an applied malhe­

matician at Carnegie Mellon Uni­versity, considers the paper "a signif­icant-step forward," both for its math­ematical contributioQs and for the issue it raises. The next step for mathematicians. he says, will be to undersland all Ihe simplify ing as­sumptions in the model to see if any of lhem are nOI justified. "Is Ihis a problem we have to worry about, or is there something in the real market Ihal is different than Ihe model?" he asks .

Sara' -Robiru'oll is a freelance writer based in Los Allgeles, California.

Call for Nominations

for· SIAM Prizes SlAM invites nominations for the

following prizes, to be awarded in 2006:

• Richard C. DiPrima Prize. Award· ed 10 a young scientist who has done outstanding research iri applied math· ematics. Nominations must be received in the SIAM office by OClober 15, 2005 .

. • Julian Cole uctureship. Awarded for an oUlstanding contribul ion to the mathematical characteri zation and solution of a challenging problem in Ihe physicallbiological sciences or in engineering, or for Ihe development of mathematical methods for the solution of such problems. Nomin. alions must be received in the SIAM office by Oclober 15, 2005,

• SlAM A.ctivity Group on Analysis oj Differential Equations Prize. Awarded tQ the aUlhor(s) of Ihe most oUlslanding paper. as detennined by the prize committee, on a topic in partial differenlial equatio ns pub_ lished in English in a peer-reviewed journal. Nominations must be re­ceived in the SIAM office by Dec­ember 3 t, 2005.

• SIAM A.ctivity Group on Linear Algebra Prize. Awarded to the author(s) of the most outstanding paper, as detennined by the prize committee. on a topic ~n applicable linear algebra published in English in a peer-reviewed journal. Nominations should be received in the SIAM office by January 15, 2006.

Updated prize inJomulIion can be found on .he SlAM Web site at hup:11 www.siam.orglpri1.esl.