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Volume 8 Number 4

December 2015

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■ Electricity futures prices: time-varying sensitivity to fundamentals Stein-Erik Fleten, Ronald Huisman, Mehtap Kiliç, Enrico Pennings and Sjur Westgaard

■ Calculation of a term structure power price equilibrium with ramping constraints Miha Troha and Raphael Hauser

■ Approximation of the price dynamics of heating degree day and cooling degree day temperature futures Fred Espen Benth and Sara Ana Solanilla Blanco

■ Facilitating appropriate compensation of electric energy and reserve through standardized contracts with swing Deung-Yong Heo and Leigh S. Tesfatsion

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The Journal of Energy MarketsEDITORIAL BOARD

Editor-in-ChiefDerek W. Bunn London Business School

Associate EditorsRené Aïd EDF R&D Finance for Energy

Market Research Centre Rüdiger Kiesel University Duisburg-EssenKostas Andriosopoulos ESCP Europe Valery Kholodnyi Verbund Trading AG

Business School Carlo Lucheroni University of CamerinoCarlos Henggeler Antunes University Sophie Meritet Paris Dauphine University

of Coimbra, Portugal Robert Pindyck Massachusetts Institute ofFred Espen Benth University of Oslo TechnologyAntonio J. Conejo The Ohio State Andrea Roncoroni ESSEC Business

University School Paris–SingaporeHelyette Geman Birkbeck, University of Ehud I. Ronn University of Texas at Austin

London & ESSEC Business School Geoffrey Rothwell Stanford UniversityChristopher Harris RWE npower retail Leigh Tesfatsion Iowa State UniversityBenjamin F. Hobbs The Johns Hopkins Rafał Weron Wrocław University of

University, Baltimore TechnologyRonald Huisman Erasmus School of Sjur Westgaard Norwegian University

Economics of Science and TechnologyTakashi Kanamura Kyoto University Ramteen Sioshansi The Ohio State University

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The Journal of

Energy Markets

The journalEnergy markets are one of the fastest growing and most complex market sectors. Fromthe basic role that oil has in the global economy to the essential services that gas andelectricity provide, energy is an area of geopolitical concern as well as financialactivity. The Journal of Energy Markets serves as a major research outlet for newempirical and model-based work in this sector, publishing original papers on theevolution and behavior of electricity, gas, oil, carbon and other energy markets, bothwholesale and retail.

The Journal of Energy Markets considers submissions in the form of researchpapers on, but not limited to, the following topics.

� Econometric analyses of prices, volatilities and across particular energymarkets.

� Model-based simulation of price and investment behavior.� Theoretical and applied analyses of energy derivatives.� High-frequency nonlinear models of price formation.� Longer-term geopolitical analyses of energy market globalization.� Forward curve and risk premiums.� Strategic behavior by companies.� Financial aspects of new investment.� The relationship between energy and carbon markets and climate change

policies.� Renewable energy financing and policy analysis.




The Journal of Energy Markets Volume 8/Number 4

CONTENTS

Letter from the Editor-in-Chief vii

RESEARCH PAPERSElectricity futures prices: time-varying sensitivity to fundamentals 1Stein-Erik Fleten, Ronald Huisman, Mehtap Kiliç,Enrico Pennings and Sjur Westgaard

Calculation of a term structure power price equilibriumwith ramping constraints 23Miha Troha and Raphael Hauser

Approximation of the price dynamics of heating degree day andcooling degree day temperature futures 69Fred Espen Benth and Sara Ana Solanilla Blanco

Facilitating appropriate compensation of electric energy and reservethrough standardized contracts with swing 93Deung-Yong Heo and Leigh S. Tesfatsion

Editor-in-Chief: Derek W. Bunn Subscription Sales Manager: Aaraa JavedPublisher: Nick Carver Global Head of Sales: Michael LloydJournals Manager: Dawn Hunter Information and Delegate Sales Director: Michelle GodwinEditorial Assistant: Carolyn Moclair Composition and copyediting: T&T Productions LtdMarketing Executive: Giulia Modeo Printed in UK by Printondemand-Worldwide

©Copyright Incisive Risk Information (IP) Limited, 2015. All rights reserved. No parts of this publicationmay be reproduced, stored in or introduced into any retrieval system, or transmitted, in any form or by anymeans, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of thecopyright owners.

Marketing Manager: Rainy Gill




LETTER FROM THE EDITOR-IN-CHIEF

Derek W. BunnLondon Business School

Effective risk management in energy markets often requires extensive analysis offorward price dynamics and how they relate to spot prices. Electricity is particularlychallenging in this respect, with high spot volatility, lack of storage options and com-plex fundamental drivers. This issue of The Journal of Energy Markets includes fourpapers, each of which makes distinct and substantial contributions to this importantarea of research and practice.

The issue’s first paper, “Electricity futures prices: time-varying sensitivity to funda-mentals” by Stein-Erik Fleten, Ronald Huisman, Mehtap Kiliç, Enrico Pennings andSjur Westgaard, provides useful insights into the time-varying relationship betweenelectricity futures prices and fundamentals. The authors recognize that the supplyfunctions that generators offer to the market are not constant over time, and to researchthis they apply a model that relates electricity futures prices to the marginal costs ofproduction and calculates time-varying coefficients. The model is specified in state-space form and estimated via a Kalman filter to observe the dynamics. It is appliedto historical prices of futures contracts with different delivery periods (calendar yearand seasons, peak and off peak) from Germany and the United Kingdom. The resultsconfirm that analysts should choose a time-varying specification to relate the pricesof power futures to the prices of underlying fundamentals.

In our second paper, “Calculation of a term structure power price equilibriumwith ramping constraints”, Miha Troha and Raphael Hauser propose a quadratic pro-gramming formulation for calculating the equilibrium term structure of electricityprices. This is a competitive game-theoretic model with players divided into produc-ers and consumers, all of whom seek to maximize their own mean–variance utilityfunctions subject to production (eg, capacity, ramp-up/ramp-down times, etc) andconsumption constraints. The model incorporates information about over-the-counterand exchange-traded block electricity contracts, transaction costs and liquidity con-siderations. The authors’ numerical simulations examine the properties of the termstructure and its dependence on various parameters of the model. They apply theirmodel to the equilibrium term structure of electricity prices in the United Kingdom,with the model representing the entire power grid, consisting of several hundredpower units, within which the impacts of ramp-up and ramp-down constraints areevaluated. It is unusual, but increasingly relevant, to include so much operationaldetail in power price formation modeling and this paper therefore makes an importantcontribution.

www.risk.net/journal




The third paper in the issue, “Approximation of the price dynamics of heating degreeday and cooling degree day temperature futures” by Fred Espen Benth and Sara AnaSolanilla Blanco, focuses on temperature. To the extent that underlying temperatureis an important driver of energy consumption, fundamental analysis of futures pricesmay involve explicit models for heating degree days (HDDs) and cooling degree days(CDDs). In this paper the authors propose an approximation that makes the pricedynamics of HDD and CDD temperature futures linearly dependent on the underlyingtemperature. Specifically, their analysis is based on continuous-time, autoregressive,stochastic dynamics for the time evolution of temperature in a given location, and it isfitted to temperature data collected in New York over a long time period. The authorsthen derive a simple version of the Black-76 formula for pricing a call option on CDDand HDD futures, and in so doing they provide a link to energy risk managementthrough temperature hedging.

Finally, in the issue’s fourth paper, “Facilitating appropriate compensation of elec-tric energy and reserve through standardized contracts with swing”, Deung-YongHeo and Leigh S. Tesfatsion emphasize three key issues that have arisen for cen-trally managed wholesale electricity power markets in Europe and the United Statesas a consequence of the increased penetration of variable energy resources. First,rigid definitions for energy and reserve products make it difficult to ensure appropri-ate compensation for flexibility in start-up times, ramp rates, power dispatch levelsand duration. Second, participation restrictions hinder the achievement of an evenplaying field for potential providers of flexible services. And third, reliance on out-of-market compensation for the provision of some valued services encourages strategicmanipulation. This paper examines the possibility of addressing these three issuesthrough the introduction of standardized energy and reserve contracts with swing(flexibility) in their contractual terms. Concrete examples are used to demonstratehow the trading of these standardized contracts can be supported by linked forwardmarkets in a manner that permits efficient real-time balancing of net load subject tosystem and reserve-requirement constraints. Comparisons with existing wholesaleelectricity power markets are given and key policy implications are highlighted. Evi-dently, market design and policy need to coevolve to further facilitate the efficientrisk management of electricity price risk.

Journal of Energy Markets 8(4)




Journal of Energy Markets 8(4), 1–22

Research Paper

Electricity futures prices: time-varyingsensitivity to fundamentals

Stein-Erik Fleten,1 Ronald Huisman,2,3 Mehtap Kiliç,2

Enrico Pennings2 and Sjur Westgaard1

1Norwegian University of Science and Technology, NO-7491 Trondheim, Norway;emails: [email protected]; [email protected] School of Economics, PO Box 1738, 3000 DR, Rotterdam, The Netherlands;emails: [email protected]; [email protected]; [email protected] d’Economia de Barcelona, Facultat d’Economia i Empresa, Universitat de Barcelona,c/ John Maynard Keynes 1-11, 08034 Barcelona, Spain

(Received September 23, 2014; revised April 17, 2015; accepted June 15, 2015)

ABSTRACT

This paper provides insight into the time-varying relation between electricity futuresprices and fundamentals in the form of contract prices for fossil fuels. As supplycurves are not constant and different producers have different marginal costs of pro-duction, we argue that the relation between the prices of electricity futures and thoseof underlying fundamentals such as natural gas, coal and emission rights varies overtime. We test this view by applying a model that linearly relates electricity futuresprices to the marginal costs of production, and calculate the loglikelihood of differenttime-varying and constant specifications of the coefficients. To do so, we formulate themodel in state-space form and apply the Kalman filter to observe the dynamics of thecoefficients. We analyze historical prices of futures contracts with different delivery

Corresponding author: M. Kiliç Print ISSN 1756-3607 j Online ISSN 1756-3615Copyright © 2015 Incisive Risk Information (IP) Limited

1




2 S.-E. Fleten et al

periods (calendar year and seasons, peak and off-peak) from Germany and the UnitedKingdom. The results indicate that analysts should choose a time-varying specifica-tion to relate the futures price of power to the prices of underlying fundamentals.

Keywords: electricity futures prices; prices of fossil fuels; time-varying coefficients; state-spacemodel.

1 INTRODUCTION

Electricity can be sold via different marketplaces; which one a generation plant decidesto use depends on its risk perception. If it is sold on the day-ahead market, the profit forthe power plant will be uncertain, as the day-ahead prices and fuel costs are variableand thus difficult to predict far into the future.1 If it trades on the futures market2

in addition to the day-ahead market,3 the power generation plant has an opportunityto make its future income less uncertain (more predictable): with a futures contract,the power plant fixes the selling price for part of its output during a future deliveryperiod, thereby making its revenues more certain. The uncertainty that remains is thecost of the fuels needed (and emission rights if applicable), since profits decline onthe volume sold against a fixed price when fuel costs rise. For example, to deal withthis risk, market makers (or traders) in equity futures contracts4 purchase the stockdirectly after they sell the futures contract and store the stock in their portfolio untildelivery. This eliminates the risk that the stock price could rise between the momentof sale and moment of delivery (meaning that the stock consequently would have tobe purchased at a higher price than the futures price). The purchasing cost to eliminatesuch a risk is equal to the stock price plus financing costs. By knowing the costs of therisk-eliminating strategy, a risk averse market maker will charge a futures price that at

1 There is extensive literature that documents the dynamics of day-ahead power prices, eg, seasonal-ity, mean reversion, time-varying volatility and sudden price spikes. We refer the reader to Huisman(2009), Janczura and Weron (2010) and Paraschiv et al (2015).2 We assume no differences between forward and futures contracts in this paper. Futures contractsare traded on an exchange where participants are required to post collateral to cover potential losses.Futures prices change intermittently throughout each trading day, and at the end of the day there isa marking to market of the contract position, whereby participants must cover any losses or maywithdraw any profit in excess of their initial margin requirement. Following the date on which thecontract stops trading, a final marking-to-market adjustment is made. However, in our analysis thiswill not be taken into account.3 Day-ahead contracts can be seen as one-day futures contracts, but we apply the European conven-tion here and see one-day futures as day-ahead contracts and to define futures contracts as contractsthat deliver into periods farther away than one day.4 Or market makers in any other financial assets, eg, currencies or interest rates products.

Journal of Energy Markets www.risk.net/journal




Electricity futures prices: time-varying sensitivity to fundamentals 3

least equals the purchasing costs. When the stock market is competitive and perfectlyliquid the market futures price equals the costs of the risk-eliminating strategy, asotherwise a risk-free arbitrage opportunity emerges.

This thinking is based on the theory of storage as originally proposed by Kaldor(1939), Working (1948), Telser (1958) and Brennan (1958). The theory relates theprice of a futures contract on an asset to the costs of holding inventories of the assetto eliminate risks (storage and financing costs) and benefits (also called convenienceyield) from holding the asset (such as dividends in the case of a stock). Suppose thata power plant converts a (fossil) fuel into electricity and that futures contracts aretraded on that specific fuel for the same delivery period as the sold electricity futurescontract. This applies, for instance, to coal- and natural-gas-fired power plants, asrelatively liquid futures markets for coal and gas exist. The power plant can almosteliminate its risk from selling an electricity forward by purchasing the appropriateamount of fuel and emission rights contracts. After doing so, the owner is almostfree of risk as they have sold power at a fixed price, purchased the fuel and emissionrights against a fixed price and have the plant to convert the fuel into power during thedelivery period.5 Based on this argument, one would expect a direct relation betweenthe futures price of electricity and the futures prices of the fuel and emission rights.

In electricity markets, the underlying asset, electricity, cannot be stored (at least, notin an economically efficient way) and power plants compete in conversion technology.Power plants can use natural gas or coal to convert steam into power and differ inefficiency (the amount of fuel needed to produce one unit of power). In addition, someproducers, such as solar and wind power plants, have no fuel costs at all.6 Furthermore,since it takes time to increase or decrease the production volume of a fossil-fuel-firedpower plant (ramping times), the owner might be willing to sell against losses in somehours in order to make profits in others (must-run situations). The dynamics of pricesetting in power futures markets differ from equity and other financial markets forthat reason. Consequently, the futures price of electricity may not directly relate tothe price of a specific fuel: this is what is found in different studies about the relationbetween electricity futures prices and futures prices of underlying fuels.

Emery and Liu (2002) show evidence for a relation between electricity futuresprices and futures prices of fuels in terms of a co-integration relationship between theUS gas and electricity futures prices of the California–Oregon Border and Palo Verdemarkets. Mohammadi (2009) examines long-term relations and short-run dynamicsbetween electricity prices and prices for coal, natural gas and oil using annual US

5 The power plant owner is not perfectly free of risk, as the plant may break down. This risk, however,is manageable through maintenance.6 The marginal costs of hydropower depend on the reservoir levels and the option to delay production(see Huisman et al 2013).

www.risk.net/journal Journal of Energy Markets





data covering the period 1960–2007. Similarly to Emery and Liu (2002), Moham-madi examines these relations by testing for co-integration and using a vector error-correction model and only finds significant long-term relations between coal andelectricity prices and an unidirectional short-run causality from coal and natural gasprices to electricity prices. Redl et al (2009) examine the relationship between risk pre-miums of fuel markets and electricity using the German European Energy Exchange(EEX) and Nord Pool futures contracts. In this model, the futures price of electricityis a function of primary fuel costs (gas or coal) and the costs for carbon emissions.The EEX electricity prices show a higher correlation with gas and coal than thoseof Nord Pool. This can be explained by the fact that gas and coal are more often themarginal fuels for generating electricity for EEX than they are for Nord Pool, whereelectricity is mainly generated by hydropower. This was confirmed by Povh and Fleten(2009), who modeled the relationship between long-term futures contract prices onfuels (such as oil, coal and natural gas), the price of emission allowances, importedelectricity and the long-term price of electricity forwards for the Nord Pool market.Their co-integration analysis reveals a long-term relationship between all variablesexcept for natural gas. The mutual interactions of electricity, gas and carbon pricesin the United Kingdom were quantified by Fezzi and Bunn (2009). Energy producersvary in the technology of energy supply, and the prices of energy futures contractsrelate to the prices of these different technologies.

The literature on pricing electricity forwards contracts has developed in twostreams. Within the first stream, futures prices are obtained from a stochastic multi-factor process mostly derived from the Schwartz (1997) stochastic models for com-modity prices. Lucia and Schwartz (2002) is a direct application to power futuresprices (among others). In this stream, futures prices are seen as stochastic, consistingof different stochastic factors such as long- and short-term price developments andconvenience yields. Prices do not directly relate to underlying fundamentals such asfuels or the market structure, although the stochastic processes somehow reflect thesefundamentals.

We focus in this paper on the second stream. Within this stream forward electricityprices relate to fundamentals. Deng (2000) relates fuel and electricity prices in order tomodel the value of electricity generating and transmission assets. Carmona et al (2013)propose a structural model for spot and derivative electricity prices using a stochasticmodel of the bid stack. The model has a multi-fuel setting such that each fuel canset the market price and become the marginal fuel. Dong and Liu (2007) use storablefuels (natural gas and coal) in their model for electricity spot prices, and futures pricesare derived through a Nash bargaining process. According to Falbo et al (2010), thevalue of a futures contract is equal to the sum of the expected marginal productioncost and the spread option embedded in spot selling. Pirrong and Jermakyan (1999,2000) model the equilibrium price as a function of two state variables: electricity






demand and the futures price of the marginal fuel. Routledge et al (2001) derive theequilibrium futures prices by explicitly considering the conversion option of gas andother fuels to electricity.7 Bessembinder and Lemmon’s (2002) equilibrium modelimplies that the relationship between forward power price and the future spot priceis a function of both expected demand and demand variance. As a consequence,the futures price will generally be a biased forecast of the future spot price, withthe forward premium positively related to the skewness of the wholesale price andnegatively related to the variance of the wholesale price. Suenaga and Williams (2005)extend the Bessembinder and Lemmon (2002) model with fuel prices.

All these studies price electricity forwards by seeing futures prices as a biasedpredictor of future spot prices; they assume that the supply stack during the tradingperiod of a futures contract is constant, or that all producers have the same supplyfunction. They need these assumptions to derive futures price models. The objectiveof this paper is not to derive electricity futures price formulas but to examine the priceformation process during the lifetime of an electricity futures contract seen from therisk reduction strategies of power producers. We focus on the relation between thepower futures price and prices of fuel and emission forwards, assuming that differentpower producers use different conversion technologies.

The idea of time variation in the relation between electricity prices and explanatoryvariables is not new. Karakatsani and Bunn (2008) show, for the British market, thata model explaining changes in day-ahead (a one-day futures contract) electricityprices with market fundamentals (supply and demand variables) and time-varyingcoefficients exhibits the best predictive performance for day-ahead prices. We extendthis thinking from day-ahead prices to futures prices and examine the relation betweenelectricity futures prices and futures prices of underlying fuels and emission rights. Weassume a linear relation between electricity futures prices and the prices of underlyingfuel futures prices and compare the fits of different specifications of the model in termsof allowing coefficients to be time-varying or not. By doing so, we test the hypothesisthat a model with constant coefficients best explains variation in electricity futuresprices against the alternative that allowing at least one time-varying parameter is thebest explanation. Section 2 discusses the methodology that we apply and how weformulate and test different hypotheses.

2 METHODOLOGY AND DATA

We expect that the price of a power futures contract relates in a time-varying mannerto the prices of underlying fuel futures contracts. We test this view as follows. Let

7 This model is in fact the Routledge et al (2000) approach for pricing commodity futures contractsadapted to deal with electricity market specifics.






Fp;t;T be the price of a power futures contract at time t for delivery of 1 MW duringthe future period of time T . Let Mc;t;T be the value at time t of a portfolio that containsthe appropriate amount of coal futures contracts and emission rights contracts neededfor producing power with a coal-fired plant during delivery period T . We call thisthe marginal cost of future production for a coal-fired power plant. Similarly, Mg;t;T

is the marginal cost of future production for a gas-fired power plant. To determinethe appropriate amounts, we assume an average coal plant with an efficiency of 0.38(one unit of fuel generates 0.38 units of power) that emits 0.971 tonnes of CO2 permegawatt hour of power produced (net).8 The marginal cost of future production isgiven by

Mc;t;T D�

Fc;t;T =29:31

0:2777

��1

0:38

�C 0:971Fe;t;T (2.1)

for the average coal producer. Fc;t;T is the price of a coal futures contract and Fe;t;T isthe price of a futures contract that allows carbon emission (both prices are observed attime t and deliver during period T ). The numbers 29.31 and 0.2777 convert the coalfutures contract from tonnes (metric tons) into megawatts. For an average natural gasfired power plant, the marginal cost of future production is given by

Mg;t;T D�

Fg;t;T

0:5

�C 0:404Fe;t;T : (2.2)

Fg;t;T is the price of a futures contract that delivers gas during period T . The numbers0.5 and 0.404 for efficiency in (2.2) apply to an average plant and are obtained fromBloomberg. We relate the price of a power futures contract linearly to the averagecoal and gas plant marginal cost of future production:

Fp;t;T D at C btMc;t;T C ctMg;t;T C vt ; (2.3)

where vt is an error term and at , bt and ct are parameters. Our goal is to test whetheror not the price of a power futures contract relates in a time-varying manner to theprices of underlying fuel futures contracts. To do so, we test the null hypothesis thatthe parameters at , bt and ct are constant against the alternative that at least one ofthe parameters is time-varying:

(H0) at , bt and ct are constant

versus

(H1) at least one of at , bt , ct is time-varying.

8 These efficiencies should be noted as being averages for coal- and natural-gas-fired power plantsobtained from Bloomberg.






We have not yet discussed what we mean by time-varying. The coefficients at , bt andct are unobservable and we have to assume their dynamics. One way to test this isto apply rolling regressions in order to observe whether the coefficients change overtime. We prefer a different approach that makes it possible to better compare the fits ofdifferent specifications of the model. It is convenient to represent the model in statespace9 to capture the dynamics of the observed Fp;t;T in terms of the unobserved.3 � 1/ state vector �t D .at ; bt ; ct /

0. As is common in time-varying parameterregression models (see, for example, Nyblom 1989), we assume the parameters varyslowly over time, as independent random walks. The following equation describesthe dynamics of the state vector:

�tC1 D �t C wtC1; (2.4)

where the .3 � 1/ vector wt is taken to be independent and identically distributed(iid) N.0; Q/ with Q being a .3 � 3/ covariance matrix. The matrix Q could benondiagonal, capturing commonality in the parameter movements. This commonalitymay be present because of the dependence of the power futures price process on thesupply stack function. Once the generation capacity limit of the fuel with the lowestmarginal cost is reached, the electricity price gets decoupled and will be determinedby the price of the next marginal fuel in line. Therefore, the correlation betweenthe marginal cost coefficients bt for coal and ct for natural gas may be negative.However, we prefer to avoid this correlation, given the lack of a priori information onthis issue and for simplicity. We leave it to future research to examine the structure ofQ. Therefore, we constrain the covariance matrix (Q) to be diagonal to specify thatthe errors are independent and assume that the coefficients are mutually independent,and thus uncorrelated contemporaneously and over time. The observed variable Fp;t;T

is presumed to be related to the state vector through the observation equation:

Fp;t;T D Ht�t C vt ; (2.5)

where Ht is the .1 � 3/ vector Ht D .1; Mc;t;T ; Mg;t;T / and vt is the iid N.0; R/

measurement error. Having defined these, we apply the Kalman filter to obtain esti-mates for the unobserved coefficients in the vector. We estimate Q, R and the initial�0 using maximum likelihood.

We test the null hypothesis against the alternative by comparing the loglikelihoodof the constant-parameters model with various specifications of the time-varyingparameters model. Likelihood ratio tests then help us to observe whether the time-varying-parameters model, consistent with H1, fits better than the constant-parameters

9 We follow Hamilton (1994) in describing the model in state-space form.






model consistent with H0.10 We then test the hypothesis H0 against an alternative, bycomparing the likelihood under H0 against the likelihood of a specification under H1,where at least one of the variances in Q is set to zero.

2.1 Sample selection

To observe whether our findings are consistent over contract types and countries, weanalyze prices of peak load and off-peak load futures contracts in Germany and theUnited Kingdom. We selected these countries because their power is produced bycoal and natural gas (among other sources), and an active futures markets exists forcoal and natural gas. Secondly, the German and UK power markets are not (directly)connected, so we may assume that the supply and demand conditions in Germany varyindependently from those in the United Kingdom and vice versa (apart dependencyon coal and gas). By examining two different markets, we can compare the resultsbetween the two countries to conclude whether our results are consistent.

2.2 Data

To estimate the parameters, we use the complete history of prices of the German (EEX)calendar year 2013 base load11 and peak load12 contracts and the UK (IntercontinentalExchange (ICE)) October 2013 and April 2013 base load and peak load seasonalcontracts.13 The delivery period of the base load contract overlaps the delivery periodof the peak contract, as peak delivery takes place during the peak part of the dayand base delivery is for the whole day. We use the base load and peak load prices tocalculate the implied off-peak price14 to observe the price of two nonoverlappingdelivery periods (peak and off-peak), consistent with market practice.15 We thenexamine the nonoverlapping peak and off-peak prices.

The EEX calendar year futures contract starts trading approximately six yearsbefore delivery. The sample period for the calendar year 2013 contract that we exam-ine is from July 2, 2007 through December 5, 2012, yielding 1369 daily closingprice observations (€/MWh). The natural gas futures prices in €/MWh are from theNetConnect Germany (NCG) futures contract traded on the EEX. The coal prices

10 Likelihood ratio tests are suitable, as the constant-parameters specification is in fact a restrictedversion of the time-varying specification because the parameters, assumed to be constant, have zerovariance in the transition equation (2.4); that is, we set the diagonal element in Q, which containsthe variance of the parameter to be held constant, to zero.11 Delivering 1 MW during any hour of the day.12 Delivering 1 MW from Monday to Friday between 08:00 and 20:00.13 Seasons always comprise a strip of April to September or October to March.14 Delivering 1 MW from Monday to Friday outside the 08:00–20:00 period.15 The implied off-peak prices are calculated as .24 � base load price � 12 � peak load price/=12.






FIGURE 1 The behavior of power futures prices and marginal fuel costs.

0

100

50

150

2007 2008 2009 2010 2011 2012

€/M

Wh

20

30

40

50

60

70

2010 2011 2012 2013

£/M

Wh

20

40

60

80

2010 2011 2012 2013 2014

£/M

Wh

(a)

(b)

(c)

Peak (Fp,t,T) Off peak (Fp,t,T)Mc,t,T Mg,t,T

(a) Germany 2013. (b) United Kingdom April–September 2013. (c) United Kingdom October 2013–March 2014.Theplots show the behavior of the power peak and off-peak futures prices (Fp;t;T ) for (a) the German Cal 2013 fromJuly 2, 2007 through December 5, 2012; (b) United Kingdom April–September 2013 from February 16, 2010 throughMarch 27, 2013 and (c) United Kingdom October 2013–March 2014 from February 16, 2010 through September 26,2013 and the marginal costs of future production with coal (Mc;t;T ) and natural gas (Mg;t;T ).






TABLE 1 Descriptive statistics for power futures prices and marginal cost of futureproduction with coal and natural gas.

(a) Germany 2013

Peak Off-peak Coal Natural gas

Mean 80.350 39.431 42.313 58.905SD 18.350 4.177 7.993 9.208Observations 1384 1384 1384 1384

(b) United Kingdom April 2013



(c) United Kingdom October 2013



Descriptive statistics for the futures prices of the German Cal 2013 (July 2, 2007 to December 5, 2012), UnitedKingdom April–September 2013 (February 16, 2010 to March 27, 2013) and United Kingdom October 2013–March2014 (February 16, 2010 to September 26, 2013) contract prices for the peak and off-peak period (Fp;t;T ), the futurecost of marginal production with coal (Mc;t;T ) and natural gas (Mg;t;T ). SD denotes standard deviation.

in US$/kt16 and the emission rights derivative prices in €/t are obtained from theyearly Amsterdam–Rotterdam–Antwerp (ARA) coal futures contract and the Euro-pean Carbon Future (ECF) futures contract traded at the EEX. The ICE seasonalfutures contract starts trading approximately seven to eight consecutive seasons beforedelivery. The price series for both seasonal contracts range from February 16, 2010to March 27, 2013 for the April 2013 seasonal futures contract and to September 26,2013 for the October 2013 seasonal futures contract, having 805 and 935 daily closingprice observations, respectively (£/MWh). The natural gas futures contract prices in

16 1 kt D 1000 tonnes.






£/thm17 are from the National Balancing Point (NBP) seasonal futures contracts tradedon the ICE. The coal prices in US$/kt and the emission rights in €/t futures pricesare obtained by the yearly ARA coal futures contract and the EU Allowances (EUA)futures contract traded at the ICE. The currency conversion is made by using theEUR/USD exchange rate provided by Reuters. All data is obtained from Bloomberg,Thomson Reuters Datastream and Montel databases.

Figure 1 on page 9 shows the price history of the marginal cost of future productionwith coal and natural gas plants (as in (2.1) and (2.2)) and the power futures prices.Table 1 on the facing page provides the summary statistics.

3 RESULTS

Table 2 on the next page shows the loglikelihoods of the different parameter specifi-cations for model (2.3). It shows the results for the peak and off-peak load contractsfor delivery during 2013 in Germany and during two seasons in 2013 in the UnitedKingdom. The first row of results in the table shows the loglikelihoods for that spec-ification in which both at and bt are assumed to be constant and ct is set to zerousing peak load contracts. This specification relates the futures price of electricitylinearly to a constant term and the marginal cost of future production for a coal-firedplant with constant (ie, not time-varying) coefficients. Using all the prices during thelifetime of the futures contracts, we calculated the loglikelihood that the model fitsthe data; for Germany this loglikelihood equals �4414:796. For the UK contracts, theloglikelihoods equal �1585:742 for the April–September 2013 delivery contract and�1748:653 for the October 2013–March 2014 delivery contract. The loglikelihoodsare meaningless by themselves, but help us to compare the fits of different specifi-cations. For instance, when we consider the second row (that includes the marginalcost of future production with a gas plant instead of a coal plant with constant param-eters), we observe that the loglikelihood is lower for Germany (�4544:917 insteadof �4414:796 for the specification in row 1) but higher for the UK contracts. Thehigher the loglikelihood, the more likely it is that the model fits the data. Hence, weconclude that the model that consists of the marginal cost of future production with agas plant fits the data better for the UK contracts than for the German contract. Whenwe include the marginal cost of future production with both a coal plant and a gasplant (the third row), we observe that the loglikelihoods are higher than in the twoprevious rows, meaning that out of these three specifications this one is most likely

17 The following formula is used to convert the gas price from £/thm into £/MWh:

Fg;t � 3:6.GJ=MWh/

0:1055.GJ=thm/

1

100I

one therm (thm) is equal to approximately 0.1055 GJ or 0.0293 MWh.






TABLE 2 Loglikelihoods for different specifications of the model Fp;t;T D at C bt Mc;t;T Cct Mg;t;T C �vt . [Table continues on next page.]

(a) Peak load futures

Coal Natural gas Germany UK UKat bt ct 2013 Apr 2013 Oct 2013

1 Const Const — �4414.796 �1585.742 �1748.6532 Const — Const �4544.917 �954.169 �1204.9523 Const Const Const �4378.618 �365.641 �664.4354 Const TV — �584.879 123.304 201.3125 Const — TV �609.527 541.150 750.2066 Const TV TV �558.130 612.183 786.4887 Const TV Const �554.462 616.265 790.8378 Const Const TV �554.900 541.291 750.6909 TV Const — �689.815 113.679 176.444

10 TV — Const �714.980 560.397 744.24111 TV Const Const �648.493 561.245 744.84812 TV TV Const �542.726 615.640 790.11313 TV Const TV �538.383 557.236 750.17814 TV TV — �573.140 109.950 215.71915 TV — TV �590.058 556.508 749.69416 TV TV TV �546.286 611.712 785.998

to describe the data for all the contracts that we examine. When we consider onlyconstant parameters, we want to include the marginal cost of future production withboth a coal- and a gas-fired power plant to fit the data.

Likelihoods dramatically increase when we allow one or more parameters to varyover time.All the rows with a time-varying (TV) parameter have higher loglikelihoodsthan the constant-parameter specifications. This holds for all the contracts that weexamine, for peak and off-peak load, for Germany and the United Kingdom and forcalendar year and seasonal contracts. Without assessing the significance of this resultfor now, it is clear that allowing at least one of the parameters to vary over timemakes the model more likely to fit the data. This is in line with our view that theprice of a power futures contract is expected to relate in a time-varying manner to theprices of underlying fuel futures contracts. To test this more formally, we comparethe loglikelihood of a specification under our null hypothesis that parameters areconstant with a specification under the alternative hypothesis that at least one of theparameters is time-varying. Using the likelihood ratio test, we then assess whether theloglikelihood under the alternative hypothesis is significantly higher than that under






TABLE 2 Continued.

(b) Off-peak load futures

Coal Natural gas Germany UK UKat bt ct 2013 Apr 2013 Oct 2013

1 Const Const — �2665.872 �1491.563 �1557.5242 Const — Const �2638.203 �601.486 �1046.6763 Const Const Const �2604.051 �378.624 �720.0374 Const TV — �642.181 88.927 76.6195 Const — TV �674.112 444.732 487.1286 Const TV TV �633.377 516.262 548.5707 Const TV Const �629.935 520.218 552.6648 Const Const TV �634.934 453.471 492.9039 TV Const — �771.892 60.978 24.359

10 TV — Const �809.168 468.451 495.38911 TV Const Const �762.445 475.789 501.76412 TV TV Const �630.118 519.679 552.12213 TV Const TV �635.589 471.628 497.51414 TV TV — �642.361 94.425 83.57015 TV — TV �674.312 464.298 491.14416 TV TV TV �633.519 515.860 548.219

# observations 1383 804 934

Loglikelihoods for the different specifications of the state-space model applied to the futures prices of the German Cal2013, United Kingdom April–September 2013 and United Kingdom October 2013–March 2014 contract prices forthe peak and off-peak period (Fp;t;T ).The parameter at measures the spread between the electricity (futures) priceand the fuel costs; bt is the coefficient of the future cost of marginal production with coal (Mc;t;T ); the coefficient ct

is the future cost of marginal production with natural gas (Mg;t;T ). Const denotes constant.TV denotes time-varying.

the null hypothesis. For instance, let us focus on peak load contracts and compare theloglikelihoods in rows 3 and 16 for the German contract. That is, we focus on a modelthat includes a constant term and the marginal cost of future production with a coal-and a natural-gas-fired power plant and compare the fits of the specification in whichall parameters are assumed to be constant (null hypothesis) with the specification thatall parameters are time-varying (the alternative hypothesis). The loglikelihood underH0 is �4414:796 and the loglikelihood of H1 is �633:519. The test statistic is givenby

D D �2 � .LLH0� LLH1

/ D �2 � .�4414:796 � �633:519/ D 7562:554:

The statistic D is chi-squared distributed with degrees of freedom equal to the differ-ence in the number of free parameters between the specifications. Under the alterna-tive hypothesis, we have three more parameters in this case, as all �a, �b and �c are






free under the alternative hypothesis and restricted to zero under the null hypothesis.Hence, the chi-squared distribution has three degrees of freedom. The value of thetest statistic D is so large that the p-value (the probability that we falsely reject thenull that the parameters are constant) equals zero. This also holds when we comparethe other constant-parameters specifications in rows 1 and 2 for all the contracts thatwe examine. We therefore find compelling support for rejecting the null hypothesis.This supports our view that we expect time variation, as the dependence of electricityfutures prices on the prices of underlying fuels varies over time as demand for futurescontracts progresses over the supply curve.

In Table 2 on page 12 we have printed the most likely specifications – the ones withthe highest loglikelihoods – in boldface. These reveal how the electricity futures pricesrelate to the marginal cost of future production. We find one dominant specification,which applies to all contracts except for the German peak load contract. The dominantspecification includes the marginal cost of future production with both a coal plantand a gas plant as explanatory variables and has a time-varying coefficient for the coalplant while the others are held constant. As stated earlier, the forward power price willbe determined by the marginal technology that depends on the fuel stack, the levelof demand and the capacity available for hedging purposes. Due to the supply stackfunction, which depends on the level of the “to be hedged” capacity of the powerplants that run on different fuels, we would expect the coefficients of the explanatoryvariables to be inversely time-varying. At the start of the trading period of a forwardcontract the price should be determined by the marginal fuel with the lowest cost.When the “to be hedged” capacity of the plants with the lowest marginal cost isalmost met, the forward price should be determined by the next marginal fuel in line.Therefore, an inverse relationship between the marginal cost of future production withcoal and natural gas is expected. However, finding only one time-varying coefficientof the marginal cost of future production, which can also be seen as a time-varyinghedge ratio, implies that this marginal fuel is significant in determining the forwardpower price set by the power plants running on this fuel with different efficiency rates.Therefore, such a fuel has a significant effect on the forward price variability. Figure 2on page 16, Figure 3 on page 17 and Figure 4 on page 18 show the behavior of thecoefficients over time for these contracts.

Let us take part (a) of Figure 3 on page 17 as an example for discussion. It shows theresults for the UK peak contract for delivery from April through September 2013. Thetop graph shows that the price of the electricity futures contract exceeds the marginalcost of future production with gas and coal for most of the time. The second graphshows the value of at , which is 16.242 and remains constant by assumption, as themost likely specification is one for which at is constant. This parameter estimate, andall others for the most likely specifications, are listed in Table 3 on the facing page.The third graph shows the value for bt , the coefficient for the marginal cost of future






TABLE 3 Parameter estimates and the loglikelihood-maximizing specification of the modelFp;t;T D at C bt Mc;t;T C ct Mg;t;T C vt .

Germany UK UKyear 2013 Apr–Sep 2013 Oct 2013–Mar 2014‚ …„ ƒ ‚ …„ ƒ ‚ …„ ƒ

Peak Off-peak Peak Off-peak Peak Off-peak

a0 60.583 13.523 16.242 3.168 18.502 0.212pQa 0.223 — —- — — —b0 0.419 0.249 �0.016 0.070 0.056 0.102pQb — 0.000 0.006 0.007 0.005 0.006c0 0.060 0.156 0.937 0.946 0.893 0.973pQc 0.011 — — — — —pR 0.000 0.023 0.032 0.054 0.083 0.116

LL �538.383 �629.935 616.265 520.218 790.837 552.664

# observations 1383 804 934

Parameter estimates of the loglikelihood-maximizing specification from the state-space model applied to the futuresprices of the German Cal 2013, United Kingdom April–September 2013 and United Kingdom October 2013–March2014 contract prices for the peak and off-peak periods .Fp;t;T / with coal .Mc;t;T / and natural gas .Mg;t;T / as thefuture marginal fuels.

pQa is the square root of the first diagonal element in Q;

pQb and

pQc are the square

roots of the second and third elements in Q. LL denotes loglikelihood.

production with coal, which value is estimated to be �0:016 at the start of the sample(see b0 in Table 3) and varies over time with a standard deviation

pQb equal to 0.006

per day (ie, a very low standard deviation).18 The fourth graph shows the value for thecoefficient ct , which is the one for the marginal cost of future production with naturalgas. Its value is 0.937 (see Table 3) and remains constant by assumption. Part (a)shows that the coefficient bt declines over time, which makes sense as the electricityfutures price seems to follow the marginal cost of future production with gas with anapparent constant spread reflected by the constant at and the almost unitary estimatefor the constant ct and the marginal costs of coal deviating more and more from themarginal costs of gas over time. From this we conclude that this specification capturesthe dynamics in the relation between electricity futures prices and the marginal costof future production over time.

A different case is the German peak load contract for delivery in 2013. The char-acteristics are plotted in part (a) of Figure 2 on the next page. From the top graph,

18 We chose to report the standard deviationsp

Q instead of the variances Q as the numbers aresmall and standard deviations have a clear interpretation in the case of a normal distribution, eg,68% of the observations lie in a one-standard deviation interval around the mean.






FIGURE 2 German power 2013 futures prices, marginal costs and coefficients for themost likely model according to Table 2 on page 12.

0 1000200 400 600 800 1200 1400

10050

150

0 1000200 400 600 800 1200 1400

0 1000200 400 600 800 1200 1400

0 1000200 400 600 800 1200 1400

0 1000200 400 600 800 1200 1400

0 1000200 400 600 800 1200 1400

0 1000200 400 600 800 1200 1400

0 1000200 400 600 800 1200 1400

40

6050

0

1

0

0.4

100

20

60

14

13

0.2

0.4

0

–1

1

(a)

(b)

b(t)

c(t)

a(t)

b(t)

c(t)

(a) Peak load. (b) Off-peak load. From top to bottom the plots show the behavior of the power futures contract prices(Fp;t;T ; dot-dashed line), marginal costs of fuel and coefficients at , bt and ct over time.The parameter at measuresthe spread between the electricity (futures) price and the fuel costs, bt is the coefficient of the future cost of marginalproduction with coal (Mc;t;T ; dashed line) and the coefficient ct represents the future cost of marginal productionwith natural gas (Mg;t;T ; solid line).

we observe that the electricity price converges to the marginal costs of gas over time.The spread declines and this behavior is apparent from the dynamics of at in thesecond graph. This declining spread occurs after observation 600 and is probably






FIGURE 3 United Kingdom April 2013–September 2013 futures prices, marginal costsand coefficients for the most likely model according to Table 2 on page 12.

0 200 400 600 800100 300 500 700 900

0 200 400 600 800100 300 500 700 900

0 200 400 600 800100 300 500 700 900

0 200 400 600 800100 300 500 700 900

0 200 400 600 800100 300 500 700 900

0 200 400 600 800100 300 500 700 900

0 200 400 600 800100 300 500 700 900

0 200 400 600 800100 300 500 700 900

204060

16

17

0–0.1

0

2

1

30

50

2

4

3

0–0.1

0

2

1

(a)

(b)

a(t)

b(t)

c(t)

a(t)

b(t)

c(t)

(a) Peak load. (b) Off-peak load. From top to bottom the plots show the behavior of the power futures contract prices(Fp;t;T ; dot-dashed line) and future marginal costs of fuel and coefficients at , bt and ct over time.The parameter at

measures the spread between the electricity (futures) price and the fuel costs, bt is the coefficient of the future costof marginal production with coal (Mc;t;T ; dashed line) and the coefficient ct represents the future cost of marginalproduction with natural gas (Mg;t;T ; solid line).

caused by the increase of photovoltaic and wind power in the German supply curve.By assumption, the influence of coal remains constant with its coefficient bt equalto 0.419 (see Table 3 on page 15). The coefficient ct for gas varies by assumption.






FIGURE 4 United Kingdom October 2013–March 2014 futures prices, marginal costs andcoefficients for the most likely model according to Table 2 on page 12.

0 1000200 400 600 800100 300 500 700 900

0 1000200 400 600 800100 300 500 700 900

0 1000200 400 600 800100 300 500 700 900

0 1000200 400 600 800100 300 500 700 900

0 1000200 400 600 800100 300 500 700 900

0 1000200 400 600 800100 300 500 700 900

0 1000200 400 600 800100 300 500 700 900

0 1000200 400 600 800100 300 500 700 900

20406080

18

19

0

–0.1

0.1

0

2

1

20

40

60

0

1

0

–0.2

0

2

1

(a)

(b)

a(t)

b(t)

c(t)

a(t)

b(t)

c(t)

(a) Peak load. (b) Off-peak load. From top to bottom the plots show the behavior of the power futures contract prices(Fp;t;T ; dot-dashed line) and future marginal costs of fuel and coefficients at , bt and ct over time.The parameter at

measures the spread between the electricity (futures) price and the fuel costs, bt is the coefficient of the future costof marginal production with coal (Mc;t;T ; dashed line) and the coefficient ct represents the future cost of marginalproduction with natural gas (Mg;t;T ; solid line).

It starts at 0.06 and changes daily with a standard deviation of 0.011 (Table 3). Onaverage, the coefficient ct is not trending and converges to about 0.1 at the end of thesample.






The difference in the price evolution of the German peak power contract and theUK contracts is that the spread at declined, probably due to the change in the supplycurve in Germany owing to an increase in renewables. The UK contracts show thedeclining influence of the marginal costs of coal over time, while keeping at constant.For the contracts that we examined, time variation is observable either in at or in oneof the marginal costs coefficients bt or ct . Again, we conclude that assuming timevariation in one of the parameters is more likely than assuming all coefficients areconstant. But, as the exact parameter that needs to be time-varying differs betweencontracts (and perhaps over sample periods as well), we cannot say ex ante which ofthe parameters should be time-varying. That implies that one cannot make a consistentchoice of which parameters to hold constant and which to allow to vary over time.Returning to the results of Table 2 on page 12, we observe that the loglikelihoods forthose specifications that allow all the parameters to be time-varying (in row 16) donot deviate too much from the most likely specifications. Consider, for instance, theGerman peak contract. The optimal specification yields a loglikelihood of �538:383;the specification in row 16 yields a loglikelihood of �546:286, a difference of abouteight. These differences are significantly different from zero (according to likelihood-ratio tests), but the deviation from the most likely specification is much less than whenwe assume constant parameters. The constant-parameters specifications in rows 1–3all yield much lower (more negative) loglikelihoods than those in row 16. This holdsfor all contracts. We therefore conclude that if a practitioner has to choose ex ante thebest specification, they should choose the one in which all parameters are allowed tovary over time.

4 CONCLUSIONS

Electricity is a derived commodity that is generated from the conversion of variousforms of fossil fuels or other energy sources. This paper focuses on how changesin market-determined prices for future delivery of underlying fuels affect the corre-sponding prices for future delivery of electricity. We find evidence of a time-varyingrelation between electricity futures prices and fundamentals, ie, the marginal produc-tion cost based on the fossil fuel contract prices. We argue that the reason for this isthat supply curves are not constant and different producers have different marginalcosts of production (gas versus coal). For contracts with different delivery periods(calendar year and seasons, peak and off-peak) from Germany and the United King-dom, we conclude that one has to choose a time-varying specification to relate thefutures price of power to the prices of underlying fundamentals.

Our paper supports the view that one can better relate the price of a power futurescontract to the futures prices of underlying commodities such as coal, natural gas andemission rights in a time-varying way. We leave it for future research to determine






the exact impact of making the wrong assumption of constant instead of time-varyingcoefficients, but we take the opportunity to discuss where we see that this could havean impact. In the energy sector, natural objects of analysis include spreads, such as theclean spark spread (the difference between the price of electricity and the marginalproduction costs of a natural gas producer) or the clean dark spread (the coal plantcounterpart of the clean spark spread). “Clean” refers to the inclusion of emissioncosts. These spreads reflect the profits that power plants can lock in, and differentderivatives such as spread options and swaps are traded to hedge the risk of changesin spark spreads. Thinking about spread option pricing, one can make a serious mistakeif the option valuation model were to assume a relation between the price of electricityand the underlying commodities. Option pricing models that allow for a time-varyingrelation are then needed.

One can also think about a risk manager measuring the risk of a portfolio of energycontracts. Our results indicate that the correlation between the prices of electricity andunderlying commodities varies over time, and thus should be taken into account inorder to correctly measure the amount of portfolio risk. This related to cross-hedgingissues in which one offsets price risk in one energy commodity by taking an oppositeposition in an appropriate number of contracts in another energy commodity. Thisappropriate number is likely to vary if the relationship between the two commoditiesis time-varying.

In this paper, we define time variation in a simple way. We do not, for instance,relate changes in the supply curve directly in the coefficients, although one wouldexpect that a change in the supply curve would immediately affect the relation betweenelectricity prices and underlying commodities in some way. We leave this issue forfuture research.

DECLARATION OF INTEREST

The authors report no conflicts of interest. The authors alone are responsible for thecontent and writing of the paper.

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Research Paper

Calculation of a term structure power priceequilibrium with ramping constraints

Miha Troha and Raphael Hauser

Mathematical Institute, Oxford University, Andrew Wiles Building,Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK;emails: [email protected], [email protected]

(Received November 6, 2014; revised May 7, 2015; accepted June 15, 2015)

ABSTRACT

In this paper, we propose a tractable quadratic programming formulation for calculat-ing the equilibrium term structure of electricity prices. We rely on a competitive gametheoretic model, with players divided into producers and consumers who all seek tomaximize their own mean–variance utility functions, subject to production (eg, capac-ity, ramp-up/down times, etc) and consumption constraints. The model incorporatesinformation about over-the-counter and exchange-traded block electricity contracts,transaction costs and liquidity considerations. Our numerical simulations examinethe properties of the term structure and its dependence on various parameters of themodel. The proposed quadratic programming formulation is applied to calculate theequilibrium term structure of electricity prices in the United Kingdom by modeling theentire power grid, consisting of a few hundred power plants. The impact of ramp-upand ramp-down constraints is also studied.

Keywords: term structure; quadratic programming; competitive equilibrium; mean–variance;optimization; Karush–Kuhn–Tucker (KKT) conditions.

Corresponding author: M. Troha Print ISSN 1756-3607 j Online ISSN 1756-3615Copyright © 2015 Incisive Risk Information (IP) Limited

23




24 M. Troha and R. Hauser

1 INTRODUCTION

Since the deregulation of the electricity markets in the 1990s, the modeling of elec-tricity prices has attracted a lot of attention. A first approach, nowadays called thenonstructural approach, attempted to model electricity prices by using the standardtechniques that had already been developed for modeling the prices of other financialsecurities. The seminal paper of Lucia and Schwartz (2000) studied the suitability ofone-factor and multifactor Ornstein–Uhlenbeck processes to modeling the spot price.As pointed out in this work, Gaussian distributions alone cannot be used for modelingspikes, and thus the Ornstein–Uhlenbeck process was combined with a pure jumpprocess in Hambly et al (2009). An extension to more general Levy processes hasbeen described in Meyer-Brandis and Tankov (2008). However, as argued in Benthand Meyer-Brandis (2009), such approaches cannot be directly used for pricing elec-tricity derivatives, because they neglect the nonstorable nature of power. They all relyon the nonarbitrage principle and thus implicitly assume that a buy-and-hold strategyis possible. Such an assumption is clearly not realistic for the electricity market. Thus,other models that try to capture some of the physical properties of electricity havebeen proposed.

The seminal work of Barlow (2002) uses the supply-and-demand stack to calculatethe electricity prices. The model was extended in Howison and Coulon (2009) andCarmona et al (2013), where the supply stack was modeled as a function of theunderlying fuels (eg, gas, coal, oil, etc) used to produce the electricity. A closed-formsolution for electricity forward contracts and for spark and dark spread options wasderived. These models capture some of the physical properties of the power markets,and in the literature they are referred to as structural approaches. However, as argued inRobinson (2005), models that include the ramp-up/down constraints of power plantsand long-term contracts are needed in order to understand and prevent catastrophicevents and market manipulation, such as that seen in California in 2001.

This is the motivation behind a third, competitive equilibrium approach, whichmodels the physical properties and decisions of market participants more closely.The seminal work in this line of research was produced by Bessembinder and Lem-mon (2002), where a dependency between a forward and a spot price in a two-stagemarket is studied. The market consists of producers and consumers, who each aim tooptimize a mean–variance utility function. Cavallo and Termini (2005) used this workto study the impact of derivatives in the power market. Bühler and Müller-Merbach(2009) and Bühler (2009) have extended it to a multistage setting and derived a for-mula for a dynamic equilibrium. The mean–variance objective function was replacedby a general convex risk measure in De Maere d’Aertrycke and Smeers (2013). Trohaand Hauser (2014) extended the work of Bühler (2009) to a setting with more thanone producer and consumer, who optimize their mean–variance objective functions. In





Calculation of a term structure power price equilibrium with ramping constraints 25

contrast to other competitive equilibrium approaches, the capacity and ramp-up/downconstraints of power plants are included. By modeling the profit of power plants asa difference between the power price and fuel costs, together with emissions obliga-tions, this work also incorporates ideas from the structural approach. As in Clewlowand Strickland (1999a,b), the model is consistent with observable fuel and emissionprices. Troha and Hauser (2014) focuses mainly on the description of the model andon the proof of the existence and uniqueness of the solution. In this paper, we extendthe model of Troha and Hauser (2014) to include transaction costs, liquidity con-straints and block electricity and fuel contracts. We also propose a tractable quadraticprogramming formulation to solve it numerically. The numerical results show thedependence of the term structure of electricity prices on various parameters of themodel. The algorithm is applied to a realistic setting by incorporating the entire UKpower grid, which consists of a few hundred power plants.

The competitive equilibrium approach deals with price-taking market participants.In a setup where the participants have the power to strategically influence prices, oneis interested in Cournot competition (see Allaz and Vila 1993).

From our model, it might not be immediately clear why market participants do notexecute all trades at the beginning of the planning horizon. One reason for spreadingout the trading activity is the availability of contracts and their liquidity: contractswith a delivery far into the future are much less liquid than, for example, day-aheadcontracts. Liquidity is included in our model through increased costs of trading (seeSection 5.3). Another reason is delayed cashflows: when trading is done throughfuture contracts, some players might be interested in entering into a position later,thus delaying the associated cashflows. In this paper, we mainly postulate the useof forward contracts. A simple extension to future contracts (assuming a constantinterest rate) is presented in Section 5.1. A third reason for delayed trading is theexploitation of a trend-following effect in the term structure: due to the risk aversionof most players, the term structure of forward or future contracts with fixed deliverydate is usually slightly upward sloping. A fourth reason is transaction costs: whenlarge trades, such as the hedging of a whole power plant, must be executed, they needto be spread over a longer period of time to decrease transaction costs and marketimpact.

Our paper is organized as follows. To keep the paper self-contained, we introducethe main components of the term structure power price model presented in Troha andHauser (2014) in Section 3. A quadratic programming formulation that can be usedto calculate the competitive price equilibrium is described in Section 4. In Section 5,we continue with various realistic extensions of the model. In Section 6, we illustratethe forecasting power of our model through numerical experiments by modeling theentire system of UK power plants. In the first part of this section, we study the termstructure of electricity prices on a simple example. In the second part, we calculate






the equilibrium electricity price for block as well as spot contracts. In the third part,we study a closed-form relation among equilibrium electricity prices. We concludethe paper in Section 7.

2 ORGANIZATION OF THE UK ELECTRICITY MARKET

Electricity supply and demand must match continuously in real time, which makeselectricity grids very difficult to manage. On a high-level scale, the UK electricitymarket can be divided into two different modes of operation. The first mode deals withelectricity contracts with more distant delivery periods (ranging from entire seasonsto a single hour). The second mode is performed by the system operator (called theNational Grid in the United Kingdom), which is responsible for the micromanagementof the grid in the last hour before the delivery of electricity.

In the first mode, electricity is traded through forward or future contracts. Forwardcontracts start to be traded up to four years before delivery. This period is usuallyreferred to as a liquid period. In the beginning of a liquid period, only seasonalcontracts are available. Seasonal forward contracts are agreements between a buyerand a seller that the seller will deliver a certain fixed amount of electricity every halfhour during the season of interest at a price decided upon today.As we get closer to thedelivery, electricity contracts with greater granularity appear. We can find quarterly,monthly, weekly, daily and intraday contracts. Intraday contracts can cover blocksof twelve hours, four hours, two hours, one hour and (the smallest granularity) halfan hour (see the APX power exchange for details).1 Further, various combinations ofthe above, such as month-ahead peak contracts, which cover all half hours between07:00 and 19:00 in the following month, are traded. One could use such contractsto hedge the production of a solar power plant, for example. As we can see, thereexists a huge variety of different contracts. Some of them are traded through anexchange, while others are traded over-the-counter (OTC). In the wholesale marketreport from Energy UK, we can see that a large majority of contracts is traded OTC.2

When choosing the right contract to trade, one also has to take into considerationliquidity and transaction costs. The wholesale market reports from Energy UK showthat contracts with a delivery after two or more years tend to be very illiquid. Baseloadcontracts are much more liquid than peak (ie, 07:00–19:00) or off-peak (ie, 19:00–07:00) contracts. The bid–ask spread for different types of contracts between theyears 2008 and 2011 is available in the Ofgem report.3 Besides trading electricity

1 See www.apxgroup.com/trading-clearing/apx-power-uk.2 Report for July 2014 is available at www.energy-uk.org.uk/publication.html?task=file.download&id=3291.3 See www.ofgem.gov.uk/ofgem-publications/39661/summer-2011-assessment.pdf.






through future and forward contracts, a significant amount of electricity is also tradedthrough day-ahead auctions. According to the wholesale market report from EnergyUK, between 15% and 20% of power, roughly, is traded through day-ahead auctionsoffered by APX4 and N2EX.5 Since the auctions are not explicitly included in ourmodel, we will not study them in further detail. An optimal bidding strategy for theauction market is discussed in Anderson and Xu (2002, 2005).

One hour before delivery, at the event called a “gate closure”, trading of future andforward contracts ceases. This is when the system operator takes over the managementof the power grid. All market participants inform the system operator about theirpositions in future and forward contracts. They also submit their bids and offers. Abid is a volume–price pair that tells the system operator at what price a producer canincrease production. Similarly, the offer tells the system operator what compensationthe producer is willing to accept from the operator if asked to decrease the production.Consumers with a flexible consumption also submit bids and offers. Equipped withthis information, the system operator first compares the demand forecast with thesubmitted number of traded contracts, and adjusts the difference by accepting someof the bids or offers. While doing so, the system operator must also take into accountthe capacity constraints of the transmission lines in the network. After the delivery ofelectricity, the system operator compares the actual physical production/consumptionof electricity by each market participant with the contracted volumes adjusted by theaccepted bids and offers. The imbalance volume is then calculated as the differencebetween the two. If the imbalance volume helps the system operator match supplyand demand, then a fair market index price is used to calculate the imbalance cashflowto/from each player. On the other hand, if the imbalance volume hampers the systemoperator in matching supply and demand, then a worse price is applied to calculatethe cashflow.6 Roughly speaking, this price is calculated as the average of the mostexpensive 500 MWh of accepted bids or offers adjusted by transmission losses.

The two modes of operation described above are very different. The first mode ofoperation can be considered as a competitive market without a central agent, whilethe second is controlled by the system operator who is responsible for matching theelectricity supply and demand by choosing the cheapest actions. The first mode ofoperation can be seen as relatively independent of the second mode; thus, this workfocuses on the first mode of the operation only.A coupling of both modes is somethingthat we would like to investigate in the future.

4 See www.apxgroup.com/trading-clearing/auction/.5 See www.n2ex.com/.6 For a detailed calculation of this price, see www.elexon.co.uk/wp-content/uploads/2015/11/Imbalance_pricing_guidance_v9.0.pdf.






3 PROBLEM DESCRIPTION

In this section, we provide a detailed description of a model that we use for the purposeof modeling the term structure of electricity prices. The model belongs to a class ofcompetitive equilibrium models. Market participants are divided into consumers andproducers. A set of consumers is denoted by C and has cardinality 0 < jC j < 1.Similarly, a set of producers is denoted by P and has cardinality 0 < jP j < 1. Eachproducer owns a portfolio of power plants that can have different characteristics, suchas capacity, ramp-up and ramp-down constraints, efficiency and fuel type. The set ofall fuel types is denoted by L. Sets Rp;l denote all power plants owned by producerp 2 P that run on fuel l 2 L. A set Rp;l may be empty, since each producer typicallydoes not own all possible types of power plants. Moreover, this allows us to includenonphysical traders, such as banks or speculators, who do not own any electricitygeneration facilities and are without a physical demand for electricity, as producersp 2 P with Rp;l D f�g for all l 2 L.

As we will see in Section 3.4, it is useful to introduce the concept of an electricitymarket. The electricity market ensures that the term structure of the electricity price issuch that the market clearing condition is satisfied for all electricity forward contracts.

We are interested in delivery times Tj , j 2 J D f1; : : : ; T 0g, where power for eachdelivery time Tj can be traded through numerous forward contracts at times ti , i 2 Ij .The electricity price at time ti for delivery at time Tj is denoted by ˘.ti ; Tj /. Sincecontracts with trading time later than delivery time do not exist, we require tmaxfIj g DTj for all j 2 J . The number of all forward contracts, ie,

Pj 2J jIj j, is denoted by N .

Uncertainty is modeled by a filtered probability space .˝; F ; F D fFt ; t 2 I g; P/,where I D

Sj 2J Ij . The � -algebra Ft represents information available at time t .

The exogenous variables that appear in our model are

(a) aggregate power demand D.Tj / for each delivery period j 2 J ,

(b) prices of fuel forward contracts Gl.ti ; Tj / for each fuel l 2 L, delivery periodj 2 J and trading period i 2 Ij ,

(c) prices of emissions forward contracts Gem.ti ; Tj /, j 2 J , i 2 Ij .

Electricity prices and all exogenous variables are assumed to be adapted to thefiltration fFtgt2I and have finite second moments.

Let vk 2 Rnk , nk 2 N, k 2 K and K D f1; : : : ; jKjg be given vectors. Forconvenience, we define a vector concatenation operator as

kk2Kvk D ŒvT1 ; : : : ; vT

jKj�T:






3.1 Producer

Each producer p 2 P participates in the electricity, fuel and emission markets.Forward as well as spot contracts are available on all markets. Electricity prices, fuelprices and emission prices are denoted by ˘.ti ; Tj /, Gl.ti ; Tj /, where l 2 L andGem.ti ; Tj /, respectively. To simplify the notation, we introduce

� electricity price vectors

˘.Tj / D ki2Ij˘.ti ; Tj / and ˘ D kj 2J e�OrTj ˘.Tj /;

where Or 2 R is a constant interest rate;

� fuel price vectors

G.ti ; Tj / D kl2LGl.ti ; Tj /; G.Tj / D ki2IjG.ti ; Tj /

andG D kj 2J e�OrTj G.Tj /I

and

� emission price vectors

Gem.Tj / D ki2IjGem.ti ; Tj / and Gem D kj 2J e�OrTj Gem.Tj /:

A producer may participate in the market by buying and selling forward and spotcontracts. The number of electricity forward contracts that producer p 2 P buysat trading time ti , i 2 Ij , for delivery at time Tj , j 2 J , is denoted by Vp.ti ; Tj /.Similarly, the number of fuel and emission forward contracts that producer p 2 P

buys at trading time ti , i 2 Ij , for delivery at time Tj , j 2 J , is denoted by Fp;l.ti ; Tj /,l 2 L, and Op.ti ; Tj /, respectively. Producers own a generally nonempty portfolioof power plants. The actual production of electricity from power plant r 2 Rp;l atdelivery time Tj , j 2 J , is denoted by Wp;l;r.Tj /.

The notation is greatly simplified if the decision variables are concatenated into

� electricity trading vectors

Vp.Tj / D ki2IjVp.ti ; Tj / and Vp D kj 2J Vp.Tj /I

� fuel trading vectors

Fp.ti ; Tj / D kl2LFp;l.ti ; Tj /; Fp.Tj / D ki2IjFp.ti ; Tj /

andFp D kj 2J Fp.Tj /I






� emission trading vectors

Op.Tj / D ki2IjOp.ti ; Tj / and Op D kj 2J Op.Tj /I

� electricity production vectors

Wp;l.Tj / D kr2Rp;l Wp;l;r.Tj /; Wp.Tj / D kl2LWp;l.Tj /

and

Wp D kj 2J Wp.Tj /I

and, finally, vp D ŒV Tp ; F T

p ; OTp ; W T

p �T. Similarly, we concatenate all price vectors as

�p Dh˘T; GT; GT

em; 0; : : : ; 0„ ƒ‚ …dim.Wp/

iT;

where the number of zeros matches the dimension of the vector Wp .Producer p 2 P is not able to arbitrarily choose their decision variables, since

there are some inequality and equality constraints that limit their feasible set. Thechange in production of each power plant from one delivery period to the next islimited by the ramp-up and ramp-down constraints. For each j 2 f1; : : : ; T 0 � 1g,where T 0 denotes the last delivery period, l 2 L and r 2 Rp;l , these constraints canbe expressed as

4 NW p;l;rmin 6 Wp;l;r.Tj C1/ � Wp;l;r.Tj / 6 4 NW p;l;r

max ; (3.1)

where 4 NW p;l;rmax and 4 NW p;l;r

min represent maximum rates for ramping up and down,respectively. The ramping rates greatly depend on the type of power plant. Some gaspower plants can increase production from zero to the maximum in just a few minutes,while the same action may take days or weeks for a nuclear power plant.

The capacity constraints for each power plant r 2 Rp;l can be expressed as

0 6 Wp;l;r.Tj / 6 NW p;l;rmax ; (3.2)

where NW p;l;rmax denotes the maximum production.

There are also equality constraints that connect power plant production with elec-tricity, fuel and emission trading. For each j 2 J , the electricity sold in the forwardand spot market together must equal the actual produced electricity, ie,

�Xi2Ij

Vp.ti ; Tj / DXl2L

X

r2Rp;l

Wp;l;r.Tj /: (3.3)






Each producer p 2 P has to make sure that a sufficient amount of fuel l 2 L hasbeen bought to cover the electricity production for each delivery period j 2 J . Sucha constraint can be expressed as

X

r2Rp;l

Wp;l;r.Tj /cp;l;r DXi2Ij

Fp;l.ti ; Tj /; (3.4)

where cp;l;r > 0 is the efficiency of power plant r 2 Rp;l .The carbon emission obligation constraint can be written as

Xj 2J

Xi2Ij

O.ti ; Tj / DXj 2J

Xl2L

X

r2Rp;l

Wp;l;r.Tj /gp;l;r ; (3.5)

where gp;l;r > 0 denotes the carbon emission intensity factor for power plant r 2Rp;l . This constraint ensures that enough emission certificates have been bought tocover the electricity production over the whole planning horizon.

Any producer’s goal is to maximize its expected profit, subject to a risk budget. Inthis paper, we assume that the risk budget is expressed in a mean–variance framework.The profit Pp.vp; �p/ of producer p 2 P can be calculated as

Pp.vp; �p/ DXj 2J

e�OrTj

� Xi2Ij

Pti ;Tj

p .vp; �p/

�; (3.6)

where the profit Pti ;Tj

p .vp; �p/ for each i 2 Ij and j 2 J can be calculated as

Pti ;Tj

p .vp; �p/

D �˘.ti ; Tj /Vp.ti ; Tj / � Gem.ti ; Tj /Op.ti ; Tj / �Xl2L

Gl.ti ; Tj /Fp;l.ti ; Tj /:

(3.7)

Under a mean–variance optimization framework, producers are interested in themean–variance utility

�p.vp/ D EPŒPp.vp; �p/� � 12�p varPŒPp.vp; �p/�

D �EPŒ�p�Tvp � 12�pvT

pQpvp;

where �p > 0 is their risk preference parameter and

Qp WD EPŒ.�p � EPŒ�p�/.�p � EPŒ�p�/T�

is an “extended” covariance matrix. Their objective is to solve the optimizationproblem

p D maxvp

�p.vp/; (3.8)

subject to (3.1)–(3.5).






3.2 Consumer

We make the assumption that demand is completely inelastic and each consumerc 2 C is responsible for satisfying a proportion pc 2 Œ0; 1� of the total demand D.Tj /

at time Tj , j 2 J . Since pc is a proportion, we clearly have thatP

c2C pc D 1.The number of electricity forward contracts that consumer c 2 C buys at trading

time ti , i 2 Ij , for delivery at time Tj , j 2 J , is denoted by Vc.ti ; Tj /. Like forproducers, we can simplify the notation by introducing electricity trading vectorsVc.Tj / D ki2Ij

Vc.ti ; Tj / and Vc D kj 2J Vc.Tj /.Consumers are responsible for satisfying the electricity demand of end users. The

electricity demand is expected to be satisfied for each Tj , ie,Xi2Ij

Vc.ti ; Tj / D pcD.Tj /: (3.9)

At the time of calculating the optimal decisions, consumers assume that they knowthe future realization of demand D.Tj / precisely. If the knowledge about the futurerealization of the demand changes, then players can take recourse actions by recal-culating their optimal decisions with the updated demand forecast. Consumers mayassume that they will be able to execute the recourse actions, because it is the job ofthe system operator to ensure that a sufficient amount of electricity is available on themarket.

Consumers would like to maximize their profit subject to a risk budget. In a similarway to the model we introduced for producers, we assume that the risk budget canbe expressed in a mean–variance framework. The profit of consumer c 2 C can becalculated as

Pc.Vc ; ˘/ DXj 2J

e�OrTj

� Xi2Ij

�˘.ti ; Tj /Vc.ti ; Tj / C pcsc.Tj /D.Tj /

�; (3.10)

where Or 2 R denotes a constant interest rate and sc.Tj / 2 R denotes a time-dependentcontractually fixed price that consumer c 2 C receives for selling the electricity onto end users (eg, households, businesses, etc). Note that the contractually fixed pricesc.Tj / only affects the optimal objective value of consumer c 2 C , and not theiroptimal solution. Since we are primarily interested in optimal solutions, we simplifythe notation and set sc.Tj / D 0 for all j 2 J . The correct optimal value can alwaysbe calculated via post-processing when an optimal solution is already known. Thismay be needed for risk management purposes.

Under a mean–variance optimization framework, consumers are interested in themean–variance utility

�c.Vc/ D EPŒPc.Vc ; ˘/� � 12�c varPŒPc.Vc ; ˘/�

D �EPŒ˘�TVc � 12�cV T

c QcVc ;






where �c > 0 is their risk preference and

Qc WD EPŒ.˘ � EPŒ˘�/.˘ � EPŒ˘�/T�

is a covariance matrix. Their objective is to solve the optimization problem

˚c D maxVc

�c.Vc/; (3.11)

subject to (3.9).

3.3 Matrix notation

The analysis of the problem is greatly simplified if a more compact notation isintroduced.

The equality constraints of producer p 2 P can be expressed as

Apvp D 0;

and the inequality constraints can be expressed as

Bpvp 6 bp

for some Ap 2 RjJ j.jLjC1/C1�dim vp , Bp 2 Rnp�dim vp and bp 2 Rnp , where np

denotes the number of the inequality constraints of producer p 2 P . Define a feasibleset

Sp WD fvp W Apvp D ap and Bpvp 6 bpg:

It is useful to investigate the inner structure of the matrixes. By considering equalityconstraints (3.3)–(3.5), we can see that

Ap D"

OA1 0 OA3;p

0 OA2OA4;p

#; (3.12)

where

OA1 2 RjJ j�N ; OA2 2 R.jJ kLjC1/�N.jLjC1/;

OA3;p 2 RjJ j�dim Wp ; OA4;p 2 R.jJ kLjC1/�dim Wp :

One can see that matrixes OA1 and OA2 are independent of producer p 2 P , and matrixesOA3;p and OA4;p depend on producer p 2 P . One can further investigate the structure

of OA1 and see

OA1 D

264

11 0

: : :

0 1jJ j

375 ; (3.13)






where 1j , j 2 J , is a row vector of ones of length jIj j. Similarly,

OA2 D

266664

OA1 � � � 0 0:::

: : ::::

:::

0 � � � OA1 0

0 � � � 0 1jN j

377775

; (3.14)

where the number of rows in the block notation above is jLj C 1. The first jLj blockrows correspond to (3.4), and the last block row corresponds to (3.5).

The profit of producer p 2 P can be written as

Pp.vp; �p/ D ��Tp vp:

In a compact notation, the mean–variance utility of producer p 2 P can be calculatedas

�p.vp; EPŒ˘�/ D EPŒ��Tp vp � 1

2�pvT

p.�p � EPŒ�p�/.�p � EPŒ�p�/Tvp�

D �EPŒ�p�Tvp � 12�pvT

pQpvp;

whereQp WD EPŒ.�p � EPŒ�p�/.�p � EPŒ�p�/T�: (3.15)

The inner structure of matrix Qp is

Qp D

264

OQ1OQ2 0

OQT2

OQ3 0

0 0 0

375 (3.16)

where

OQ1 2 RN �N ;

OQ2 2 RN �.dim BpCdim Op/ D RN �N.jLjC1/;

OQ3 2 RN.jLjC1/�N.jLjC1/:

One can see that OQ1, OQ2 and OQ3 do not depend on producer p 2 P . The size ofthe larger matrix Qp depends on producer p 2 P , because different producers havedifferent numbers of power plants.

Producer p 2 P attempts to solve the following optimization problem:

p.EPŒ˘�/ D maxvp2Sp

�EPŒ�p�Tvp � 12�pvT

pQpvp:

The equality constraints can be expressed as

AcVc D ac ;






where Ac D OA1 and ac 2 RjJ j. Define a feasible set

Sc WD fVc 2 RN W AcVc D acg:

To keep the notation for producers and consumers consistent, we also define an emptymatrix of inequality constraints for consumers Bc 2 R0�N and bc 2 R0.

The profit of consumer c 2 C can be written as

Pc.Vc ; ˘/ D �˘TVc :

Note that we set sc.Tj / D 0 for all j 2 J , without loss of generality. In a compactnotation, the mean–variance utility of a consumer c 2 C can be calculated as

�c.Vc ; EPŒ˘�/ D EPŒ�˘TVc � 12�cV T

c .˘ � EPŒ˘�/.˘ � EPŒ˘�/TVc�

D �EPŒ˘�TVc � 12�cV T

c QcVc ;

whereQc WD EPŒ.˘ � EPŒ˘�/.˘ � EPŒ˘�/T�: (3.17)

Moreover, note that Qc D OQ1 for all c 2 C . Consumer c 2 C attempts to solve thefollowing optimization problem:

˚c.EPŒ˘�/ D maxVc2Sc

�EPŒ˘�TVc � 12�cV T

c QcVc :

3.4 Electricity market

Given the expected electricity EPŒ˘�, fuel EPŒG� and emission EPŒGem� price vec-tors, together with the correlation matrixes Qk , k 2 P [ C , each producer p 2 P

and consumer c 2 C can calculate their optimal electricity trading vectors Vp andVc by solving (3.8) and (3.11), respectively. However, the players are not necessarilyable to execute their calculated optimal trading strategies because they may not finda counterparty with which to trade. In reality, each contract consists of a buyer anda seller, which imposes an additional constraint (also called the market clearing con-straint) that matches the number of short and long electricity forward contracts foreach i 2 Ij and j 2 J as

Xc2C

Vc.ti ; Tj / CXp2P

Vp.ti ; Tj / D 0: (3.18)

The electricity market is responsible for satisfying this constraint by matching buyerswith sellers. The matching is done through sharing the price and order book informa-tion among all market participants. If at the current price there is more demand forlong contracts than short contracts, it means that the current price is too low, and asks






will start to be submitted at higher prices. The converse occurs if there is more demandfor short contracts than long contracts. The electricity price at which the number oflong and short contracts matches is found through dynamic adjustment. At such aprice, (3.18) is satisfied “naturally”, without explicitly requiring the players to satisfyit. They do so because it is in their best interest, ie, because their mean–varianceobjective functions are jointly maximized. Note that the same mechanism is at workto find the equilibrium expected forward prices when ti lies in the future.

We can see that, by affecting the expected electricity price, the electricity marketchanges the electricity price process. It is not immediately clear how to construct sucha stochastic process, or that such a stochastic process exists at all. We refer the readerto Troha and Hauser (2014), where a constructive proof of the existence is given. Theproof is based on the Doob decomposition theorem, where we allow the electricitymarket to control an integrable predictable term of the process while keeping themartingale term of the process intact.

For further argumentation, we define vP D kp2P vp and v D ŒvTP ; V T

C �T.

3.5 Competitive equilibrium

We are interested in finding a competitive equilibrium defined as follows.

Definition 3.1 (Competitive equilibrium (CE)) Decisions v� and EPŒ˘�� 2 RN

constitute a competitive equilibrium if the following conditions are met.

(1) For every producer p 2 P , v�p 2 Sp is a strategy such that

�p.vp; EPŒ˘��/ 6 �p.v�p ; EPŒ˘��/ (3.19)

for all vp 2 Sp .

(2) For every consumer c 2 C , V �c 2 Sc is a strategy such that

�c.Vc ; EPŒ˘��/ 6 �c.V �c ; EPŒ˘��/ (3.20)

for all Vc 2 Sc .

(3) For each i 2 Ij and j 2 J ,

0 DXc2C

Vc.ti ; Tj / CXp2P

Vp.ti ; Tj / (3.21)

must hold.






From Definition 3.1, it is not clear whether a CE for our problem exists and whetherit is unique. This problem was thoroughly investigated in Troha and Hauser (2014).Roughly speaking, it was shown that if the demand of the end users can be covered bythe available system of power plants, then a CE exists. Moreover, if the power plantsare similar enough (if there are no big gaps in the efficiency of the power plants),then one can show that the CE is also unique. On the other hand, if power plants aresimilar enough, then the expected equilibrium price of each electricity contract mightbe an interval instead of a single point.

In this paper, we focus on the numerical calculation of the CE under the assump-tion of the existence of a solution. Mathematically, we rely on the followingassumption.

Assumption 3.2 For all p 2 P , there exists a vector vp such that Apvp D ap

almost surely and Bpvp < bp almost surely; for all c 2 C , there exists a vector Vc

such that AcVc D ac almost surely and BcVc < bc almost surely; and the vectors Vp

and Vc can be chosen so that (3.21) is satisfied.

4 QUADRATIC PROGRAMMING FORMULATION

The traditional approach to solving competitive equilibrium optimization prob-lems in a linear programming setup is through shadow prices (see, for example,De Maere d’Aertrycke and Smeers 2013; Milano et al 2006). In Troha and Hauser(2014), it was shown how to extend this approach to the quadratic programmingsetup. In this section, we describe a variation of a shadow price approach, which ismathematically equivalent to the approach presented in Troha and Hauser (2014) butprovides some additional insight into the problem.

A naive approach for solving competitive equilibrium optimization problems wouldbe to choose an expected price vector EPŒ˘� and then calculate optimal solutions foreach producer p 2 P and consumer c 2 C . If at such a price k

Pc2C Vc C

Pp2P Vpk

is close to zero, then the solution is found and EPŒ˘� is an equilibrium expectedprice vector. Otherwise, we have to adjust the expected price vector and repeat theprocedure. We can see that such an algorithm is costly, because it requires us to solvea large optimization problem (ie, to calculate the optimal solutions of each producerand each consumer) multiple times. In the section below, we show that we can domuch better than the naive approach. Using the reformulation we propose, the largeoptimization problem must be solved only once.

The following are necessary and sufficient conditions for all vk , k 2 P [ C ,and EPŒ˘� to constitute a CE, due to the fact that Assumption 3.2 implies Slater’s






condition:

�EPŒ�k�T � �kQkvk � BTk �k � AT

k�k D 0;

�Tk.Bkvk � bk/ D 0;

Bkvk � bk 6 0;

Akvk � ak D 0;

�k > 0;Xc2C

Vc CXp2P

Vp D 0:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

(4.1)

The last equation corresponds to the market clearing constants (3.21).We can now interpret (4.1) as the Karush–Kuhn–Tucker (KKT) conditions of one

large optimization. To see this, we join all decision variables into one vector

x WD ŒvT; EPŒ˘�T�T

and rewrite

� the equality constraints as Ax D a with

a WDhaT

p1; : : : ; aT

pP; aT

c1; : : : ; aT

cC; 0; : : : ; 0„ ƒ‚ …

N

iT;

where the number of ending zeros is equal to N , and

A WD

2666666666664

Ap10 0

0: : : 0

:::

0 ApP0 0

0 Ac10 0

0: : : 0

:::

0 AcC0

Mp1� � � MpP

I � � � I 0

3777777777775

;

where Mp 2 RN � Rdim vp is a matrix defined as

Mp D

264diag.1; : : : ; 1„ ƒ‚ …

N

/

ˇˇˇˇ

0 � � � 0:::

: : ::::

0 � � � 0

375 I






� the inequality constraints as Bx 6 b with b WD ŒbTp1

; : : : ; bTpP

; bTc1

; : : : ; bTcC

�T,and

B WD

26666666664

Bp10 0

0: : : 0

:::

0 BpP0 0

0 Bc10 0

0: : : 0

:::

0 BcC0

37777777775

I

� the objective function as ��Tx � 12xTQx, with

� WDhEPŒ�0;p1

�T; : : : ; EPŒ�0;pP�T; 0; : : : ; 0„ ƒ‚ …

.jC jC1/N

iT;

where �0;p is �p with elements of ˘ set to zero, and

Q WD

2666666666664

�p1Qp1

0 M Tp1

0: : : 0

:::

0 �pPQpP

0 M TpP

0 �c1Qc1

0 I

0: : : 0

:::

0 �cCQcC

I

Mp1� � � MpP

I � � � I 0

3777777777775

I (4.2)

� the dual variables as � WD Œ�Tp1

; : : : ; �TpP

; �Tc1

; : : : ; �TcC

� and � WD Œ�Tp1

; : : : ;

�TpP

; �Tc1

; : : : ; �TcC

; �TM �.

In this setting, we can reformulate the KKT conditions (4.1) as follows:

�� Qx � BT� � AT� D 0;

�T.Bx � b/ D 0;

Bx � b 6 0;

Ax � a D 0;

� > 0;

�M D 0:

9>>>>>>>>>=>>>>>>>>>;

(4.3)

Lemma 4.1 Q � 0 for all x that satisfy the market clearing constraint (3.21).






Proof We have

xTQx D vT

26666666664

�p1Qp1

0

0: : : 0

0 �pPQpP

0

0 �c1Qc1

0

0: : : 0

0 �cCQcC

37777777775

v

C 2EPŒ˘�T� X

c2C

Vc CXp2P

Vp

�

DXp2P

�pvTpQpvp C

Xc2C

�cvTc Qcvc

> 0;

since �p > 0, �c > 0, Qc � 0 and Qp � 0 for all p 2 P and c 2 C . �

Proposition 4.2 The KKT conditions (4.3) are equivalent to the KKT conditions(4.1).

Proof We will consider each equation separately. Let us start with Ax � a D 0.Writing the equation in matrix form

2666666666664

Ap10 0

0: : : 0

:::

0 ApP0 0

0 Ac10 0

0: : : 0

:::

0 AcC0

Mp1� � � MpP

I � � � I 0

3777777777775

2666666666664

vp1

:::

vpP

vc1

:::

vcC

EPŒ˘�

3777777777775

�

2666666666664

ap1

:::

apP

ac1

:::

acC

0

3777777777775

D

2666666666664

0:::

0

0:::

0

0

3777777777775

leads to 2666666666664

Ap1vp1

:::

ApPvpP

Ac1vc1

:::

AcCvcCP

p2P Vp CP

c2C Vc

3777777777775

�

2666666666664

ap1

:::

apP

ac1

:::

acC

0

3777777777775

D

2666666666664

0:::

0

0:::

0

0

3777777777775

;

as required.






We continue with the expression �� Qx � BT� � AT� D 0. We first focus on�� Qx only. Writing the equation in matrix form

�

26666666666664

EPŒ�0;p1�

:::

EPŒ�0;pP�

0:::

0

0

37777777777775

�

26666666666664

�p1Qp1

0 M Tp1

0: : : 0

:::

0 �pPQpP

0 M TpP

0 �c1Qc1

0 I

0: : : 0

:::

0 �cCQcC

I

Mp1� � � MpP

I � � � I 0

37777777777775

26666666666664

vp1

:::

vpP

vc1

:::

vcC

EPŒ˘�

37777777777775

D �

26666666666664

EPŒ�0;p1�

:::

EPŒ�0;pP�

0:::

0

0

37777777777775

�

26666666666664

�p1Qp1

vp1C M T

p1EPŒ˘�

:::

�pPQpP

vpPC M T

pPEPŒ˘�

�c1Qc1

vc1C EPŒ˘�

:::

�cCQcC

vcCC EPŒ˘�P

p2P Vp CP

c2C Vc

37777777777775

and noting that �EPŒ�0;p� � M Tp EPŒ˘� D �EPŒ�p� for all p 2 P leads to

�� Qx D �

26666666666664

�p1Qp1

vp1C EPŒ�p1

�:::

�pPQpP

vpPC EPŒ�pP

�

�c1Qc1

vc1C EPŒ˘�

:::

�cCQcC

vcCC EPŒ˘�P

p2P Vp CP

c2C Vc

37777777777775

:






Similarly, writing the remainder �BT� � AT� in matrix form leads to

�

26666666666664

BTp1

0

0: : : 0

0 BTpP

0

0 BTc1

0

0: : : 0

0 BTcC

0 0 0 0

37777777777775

26666666664

�p1

:::

�pP

�c1

:::

�cC

37777777775

D �

26666666666664

BTp1

�p1

:::

BTpP

�pP

BTc1

�c1

:::

BTcC

�cC

0

37777777777775

and

�

26666666666664

ATp1

0 M Tp

0: : : 0

:::

0 ATpP

0 M Tp

0 ATc1

0 I

0: : : 0

:::

0 ATcC

I

0 � � � 0 0 � � � 0 0

37777777777775

26666666666664

�p1

:::

�pP

�c1

:::

�cC

�M

37777777777775

D �

26666666666664

ATp1

�p1

:::

ATpP

�pP

ATc1

�c1

:::

ATcC

�cC

0

37777777777775

;

where �M D 0 was used. Writing it all together

�

26666666666664

�p1Qp1

vp1C EPŒ�p1

�:::

�pPQpP

vpPC EPŒ�pP

�

�c1QC vc1

C EPŒ˘�:::

�cCQC vcC

C EPŒ˘�Pp2P Vp C

Pc2C Vc

37777777777775

�

26666666666664

BTp1

�p1

:::

BTpP

�pP

BTc1

�c1

:::

BTcC

�cC

0

37777777777775

�

26666666666664

ATp1

�p1

:::

ATpP

�pP

ATc1

�c1

:::

ATcC

�cC

0

37777777777775

D

26666666666664

0:::

0

0:::

0

0

37777777777775

gives the required conditions.The proofs for �T.Bx � b/ D 0, Bx � b 6 0 and � > 0 are trivial. �

Since the additional constraints �M D 0 on the dual variables of (4.3) cannotbe handled by most of the available quadratic programming solvers, we have toreformulate the problem in a dual form. We start by formulating the optimization






problem out of the KKT conditions (4.3) as

maxx

��Tx � 12xTQx

such that Ax D a;

Bx 6 b;

�M D 0;

(4.4)

and by defining the Lagrangian as

L.x; �; �/ D(

�12xTQx � �Tx � .Ax � a/T� � .Bx � b/T� if � > 0;

�1 otherwise:

By the virtue of Lemma 4.1, Q � 0, and L.x; �; �/ is therefore a smoothand convex function. The unconstrained minimizer can be determined by solvingDxL.x; �; �/ D 0. By calculating

DxL.x; �; �/ D �Qx � � � AT� � BT�

and inserting � back into the Lagrangian, an equivalent formulation is obtained asfollows:

L.x; �; �/

D(

12xTQx C aT� C bT� if � > 0 and �Qx � � � AT� � BT� D 0;

�1 otherwise:

Relating the latter to a maximization optimization problem, the following formulationis obtained:

maxx;�;�

�12xTQx � �Ta � �Tb

such that Qx C AT� C BT� C � D 0;

� > 0;

�M D 0:

(4.5)

Problem (4.5) is equivalent to (4.4), but it can be solved using any quadratic program-ming algorithm. The numerical results in this paper are calculated using Gurobi (seeGurobi Optimization 2014).

5 EXTENSIONS OF THE MODEL

In this section, we describe a few realistic extensions of the model from Section 3that are needed to make the model applicable in practice. Note that all extensions canbe incorporated into the quadratic programming framework; thus, the reformulationof Section 4 still applies.






5.1 Future contracts

In the model above, we assumed that the market participants trade only forwardcontracts. A more realistic market setting would also include future contracts. In thissection, we explain how the model can be extended in this regard.

Let K and J denote the prices of forward and future contracts, respectively. Ano-arbitrage argument allows one to calculate EPŒ J � as a solution of the followingtriangular system of linear equations for each j 2 J :

EPŒ K .ti ; Tj /�

D

8ˆ<ˆ:

EPŒ J .ti ; Tj /� i D maxfIj g;

EPŒ J .ti ; Tj /�

CPmaxfIj g�1

kDi

�ŒEPŒ J .tkC1; Tj /� � EPŒ J .tk; Tj /��

e�rtkC1

e�rTj

�i < maxfIj g:

(5.1)

Recall from the introduction that tmaxfIj g D Tj .

5.2 Block contracts

In the previous sections, we have implicitly assumed that each contract covers onlyone delivery period. As described in Section 2, this is clearly not true in reality, ascontracts often cover a longer delivery period. For example, season-ahead contractscover the delivery over every half hour in the next season. Similarly, a month-aheadcontract covers every half hour in the next month. One can also distinguish betweenpeak contracts, which cover only delivery periods from 07:00 to 19:00, and off-peakcontracts, which cover delivery periods from 19:00 to 07:00. Other blocks are alsopossible.

A contract that covers many delivery periods J 0 � J and is traded at ti , i 2Tj 02J 0 Ij 0 can be incorporated into our model by enforcing

EPŒ˘.ti ; Tj 0/� D EPŒ˘.ti ; Tj 00/� (5.2)

and

Vk.ti ; Tj 0/ D Vk.ti ; Tj 00/ (5.3)

for all k 2 P [ C and j 0; j 00 2 J 0.The same reasoning could also be applied to forward and future fuel (ie, gas, coal,

oil, etc) and emission contracts.






5.3 Costs of trading

In the previous sections, we assumed that players can change their position withoutpaying any fees. The reality is, of course, different. Cost of trading is an importantcomponent involved in the trading of power. The precise formulation of the tradingcosts depends on the market microstructure, and it is a research area itself. In thispaper, we adapt the view of Almgren and Chriss (2001), who propose that the cost oftrading one unit of power can be approximated by a linear function

h.Vp.ti ; Tj // D �ij sign.Vp.ti ; Tj // C �ij Vp.ti ; Tj /: (5.4)

The first term represents the fixed costs of trading that occur due to the bid–ask spreadand trading fees. A good estimate for �ij is the sum of a half of the bid–offer spreadand the trading fees. The second term approximates the microstructure of the orderbook. Selling a large volume Vp.ti ; Tj / exhausts the supply of liquidity, which causesa short-term decrease in the price. The factor �ij , therefore, represents a half of thechange in price caused by selling volume Vp.ti ; Tj /. We assume that the decrease inprice is only temporary and that the price returns the equilibrium level at the nexttrading time tiC1.

Equation (5.4) represents the costs of trading a unit of power. The costs of tradinga volume Vp.ti ; Tj / are then

Vp.ti ; Tj /h.Vp.ti ; Tj // D �ij jVp.ti ; Tj /j C �ij Vp.ti ; Tj /2: (5.5)

See Section 2 for a discussion of the bid–ask spread in the UK electricity market.

5.4 Recourse

The literature distinguishes between dynamic (ie, stochastic) and static models. Indynamic models, players adapt to changing environments (ie, fuel prices, demand,etc) by adapting their decisions. In each stage of their decision making, they determinethe optimal decisions they have to make now and also the optimal decisions they willmake in the future, when faced with all possible changes in the environment. Note thatthese changes include forecasts as well. Since future decisions affect present decisions,this is computationally very demanding. In static models, on the other hand, playersassume that they know the future state of the environment and can thus stick to aninitial plan about future decisions, regardless of the changes in the environment. Suchapproaches are computationally much more tractable, but they do not reflect realityvery well. The model we described in Section 3 can be seen as a hybrid of bothapproaches. The initial optimization problem is static, where players determine alltheir optimal decisions and assume they will not alter them in the future. However,as the environment changes, players may take recursive action by calculating new






optimal decisions, taking into account the new state of the environment as well as thedecisions made under the previous state of the environment.

Let OVk.Tj / denote the number of forward electricity contracts that player k 2 P [C

has already traded before taking the recourse. Then, for every p 2 P and j 2 J ,(3.3) must be replaced by

�Xi2Ij

Vp.ti ; Tj / � OVp.Tj / DXl2L

X

r2Rp;l

Wp;l;r.Tj /:

Similarly, for every c 2 C , (3.9) must be replaced byXi2Ij

Vc.ti ; Tj / C OVc.Tj / D pcD.Tj /:

Fuel constraints (3.4) and emission constraints (3.5) must be altered in a similarfashion for every producer p 2 P .

Objective functions of producers and consumers must be altered to reflect the profitand loss that has occurred or will occur due to the trading that happened before takingthe recourse. Mathematically, this can be understood as adding a constraint to theobjective function, which does not have any impact on the optimal solution. Thus,any alteration of the objective functions can be done via post processing, if the correctoptimal value is needed.

6 NUMERICAL RESULTS

In this section, we describe numerical results. First, we investigate the calibration ofpower plant parameters from historical production. Then, we study the term structureof electricity prices in a simple example. We then extend the example to include blockcontracts and multiple spot contracts as well. Finally, we study a closed-form relationamong equilibrium electricity prices.

6.1 Calibration

If our model is applied in practice, one has to estimate the physical characteristics ofthe power plants, such as capacity, ramp-up and ramp-down constraints, efficiencyand carbon emission intensity factor.

In the United Kingdom, all power plants are required to submit their availablecapacity as well as ramp-up and ramp-down constraints to the system operator on ahalf-hourly basis. This data is publicly available at the ELEXON website (see www.bmreports.com/).

A more challenging problem is to estimate the efficiency and the carbon emissionintensity factor of each power plant. As we explained in Section 2, all market partici-pants submit their expected production/consumption to the system operator one hour






before delivery. Our goal is to find the efficiency and the carbon emission intensityfactor that best describe the historical production of each power plant.

We first normalized the historical production of each power plant by calculatingthe ratio between the production and the available capacity of a power plant in eachhalf hour. We denote the normalized historical production by QWp;l;r.Tj / for eachdelivery period j 2 J and power plant r 2 Rp;l . Normalization makes sure thatQWp;l;r.Tj / 2 Œ0; 1�. For the purpose of calibration, we assume that producers are risk

neutral and set �p D 0 for all p 2 P . Further, we neglect the ramp-up and ramp-down constraints (3.1). With these simplifications, we can see that a power plant willproduce at time Tj if and only if the income from selling electricity at the spot price isgreater or equal to the costs of purchasing the required fuel and emission certificatesat the current spot price. In other words, for a power plant that runs on fuel l 2 L andproduces electricity at time Tj ,

˘.Tj ; Tj / � cp;l;rGl.Tj ; Tj / � gp;l;rGem.Tj ; Tj / > 0 (6.1)

must hold for production to take place.It is immediately clear why (6.1) must hold when only spot contracts are available.

Let us investigate why (6.1) holds also if forward and future electricity contractsare available on the market. At any trading time ti , i 2 Ij , a rational producer couldenter into a short electricity forward contract and simultaneously into a long fuel andemission forward contract if

˘.ti ; Tj / � cp;l;rGl.ti ; Tj / � gp;l;rGem.ti ; Tj / > 0: (6.2)

At delivery time Tj , this producer has two options.

� The first is to acquire the delivery of the fuel and emission certificates bought attrading time ti and produce electricity. In this case, they observe the followingprofit:

OP1.Tj / D ˘.ti ; Tj / � cp;l;rGl.ti ; Tj / � gp;l;rGem.ti ; Tj /: (6.3)

� The second is to produce no electricity and instead close the forward electricity,fuel and emission contracts. In this case, they observe the following profit:

OP2.Tj / D Œ˘.ti ; Tj / � ˘.Tj ; Tj /� � cp;l;r ŒGl.ti ; Tj / � Gl.Tj ; Tj /�

� gp;l;r ŒGem.ti ; Tj / � Gem.Tj ; Tj /�: (6.4)

Power plant r 2 Rp;l will run at Tj if and only if

OP1.Tj / > OP2.Tj /: (6.5)

With some reordering of the terms, it is easy to see that (6.5) is equivalent to (6.1).






To adjust (6.1) for the neglected risk premium and trading costs, we add anotherterm, Qcp;l;r > 0; then, (6.1) reads

�.cp;l;r ; gp;l;r ; Qcp;l;r I Tj / > 0; (6.6)

where

�.cp;l;r ; gp;l;r ; Qcp;l;r I Tj /

D ˘.Tj ; Tj / � cp;l;rGl.Tj ; Tj / � gp;l;rGem.Tj ; Tj / � Qcp;l;r : (6.7)

In the last step of the calibration, we use �.cp;l;r ; gp;l;r ; Qcp;l;r I Tj / in the logisticregression. The efficiency cp;l;r and carbon emission intensity factor gp;l;r that bestexplain the historical production of power plant r 2 Rp;l are found as an optimalsolution to

mincp;l;r ;gp;l;r ;Qcp;l;r

Xj 2J

�QWp;l;r.Tj /� 1

1 C exp.��.cp;l;r ; gp;l;r ; Qcp;l;r I Tj //

�2

: (6.8)

For each power plant, we used over 5000 training samples obtained from betweenJanuary 1, 2012 and January 1, 2013.

Figure 1 on the facing page shows part of the calibration results for coal and gaspower plants. We can see that power plants indeed base their production decision onthe value of �.cp;l;r ; gp;l;r ; Qcp;l;r I Tj /, since they produce more when their profitis higher and less when the profit is lower. However, it is also clear that the valueof �.cp;l;r ; gp;l;r ; Qcp;l;r I Tj / does not contain all of the information that is used todetermine the production schedule in reality. In our model in Section 3, we havealready seen that power plants have ramp-up and ramp-down constraints, which werenot taken into account in the calibration exercise. These constraints prevent the powerplants from immediately responding to price signals. Another important factor nottaken into account in this model is the start-up cost. If a power plant is turned off,then significant costs occur when it is turned on again. Thus, power plants sometimesproduce even when it is not profitable for them to do so, but production helps to avoidstart-up costs. Similarly, a power plant will only start production if the operatorsbelieve that the profit will exceed the costs of starting the power plant. In a similarway to ramp-up and ramp-down constraints, start-up costs also prevent power plantsfrom responding to price signals immediately. This explains the green horizontal linesin Figure 1 on the facing page. This effect is more visible for coal power plants, sincethey have tighter ramping constraints and higher start-up costs.

In this section, we apply our model to the entire system of UK power plants. Wefocus on coal, gas and oil power plants, because these adapt their production to coverthe changes in demand and are thus responsible for setting the price. Nuclear power






FIGURE 1 Goodness-of-fit for three coal and three gas power plants.

1.2

1.0

0.8

0.6

0.4

0.2

0

–0.2

1.2

1.0

0.8

0.6

0.4

0.2

0

–0.2

1.2

1.0

0.8

0.6

0.4

0.2

0

–0.2

1.2

1.0

0.8

0.6

0.4

0.2

0

–0.2

1.2

1.0

0.8

0.6

0.4

0.2

0

–0.2

1.2

1.0

0.8

0.6

0.4

0.2

50 100 150 200

0

0–0.2

–50

50 100 150 2000–50 50 100 150 2000–50

50 100 150 2000–50

50 100 150 200 2500–50 50 100 150 200 2500–50

FitIncreasedDecreasedUnchanged

(a) (b)

(c)

(f)(e)

(d)

The goodness-of-fit for three coal (namely, T_COTPS-4, T_RATS-3 and T_WBUPS-4, parts (a), (b) and (c)) andthree gas (namely, T_BARKB2, T_GRAI-6 and T_LAGA-1, parts (d), (e) and (f)) power plants.The x-axis on all partsrepresents the value of �.cp;l;r ; gp;l;r ; Qcp;l;r I Tj /, as defined in (6.7), and the y-axis represents QWp;l;r .Tj /.Eachpoint corresponds to one delivery period Tj . Colors are used to denote the value of QWp;l;r .Tj C1/. If QWp;l;r .Tj / DQWp;l;r .Tj C1/, then green is used. If QWp;l;r .Tj / < QWp;l;r .Tj C1/, then red is used. Otherwise, blue is used.






plants do not have to be modeled explicitly because their ramp-up and ramp-downconstraints are so tight that their production is almost constant over time. They usuallydeviate from the maximum production only for maintenance reasons. Renewablesources and interconnectors are not modeled explicitly because they require a differenttreatment not covered in this paper. In this subsection, we define demand D.Tj / forall j 2 J as

D.Tj / WD Dact.Tj / � Prenw.Tj / � Pinter.Tj /; (6.9)

where Dact.Tj / denotes the actual demand in the UK power system, Prenw.Tj / denotesthe production from all renewable sources including wind, solar, biomass, hydro andpumped storage, and Pinter.Tj / denotes the flow of power into the UK power systemthrough interconnectors. To make this model useful in practice, one has to model eachof these terms, but that exceeds the scope of this paper.

We calibrate the model, and for each power plant r 2 Rp;l , l 2 L and p 2 P , weestimate the efficiency cp;l;r , the carbon emission intensity factor gp;l;r , the maximumcapacity NW p;l;r

max , the ramp-up rate 4 NW p;l;rmax and the ramp-down rate 4 NW p;l;r

min using thehistorical production as described above. The covariance matrix was also estimatedfrom the historical data using the shrinkage approach described in Ledoit and Wolf(2003).

6.2 Term structure of the price

In this subsection, we study the term structure of the electricity price. We focus on asimple problem with only one delivery period and five trading periods. Our goal is tocalculate the electricity term structure with the information available on February 11,2013. We are interested in a half-hourly delivery period fromApril 4, 2013 at 00:00:00to April 4, 2013 at 00:00:30. The trading periods labeled with a number from oneto five correspond to a two-month-ahead, month-ahead, week-ahead, day-ahead andthe spot price, respectively. We use future prices of coal, gas and oil as available onFebruary 11, 2013. Since the historical demand forecast is not available, we use therealized demand instead, which is standard practice in the literature. To implementthis model in practice, one could use a demand forecast from the ELEXON website ordevelop a new approach. Since we do not have information about the ownership of thepower plants, we assume that there is only one producer who owns all power plantsconnected to the UK grid, and only one consumer that is responsible for satisfyingthe demand of the end users. In reality, market participants have more informationabout the ownership, which can be incorporated into the model. To keep the effects ofvarious parameters of the model on the term structure of the electricity price clearlyvisible, we set the risk-free interest rate Or to zero.

As described in the previous section, the covariance matrix is estimated from thehistorical data. However, the contracts used in this example cover only one delivery






period; thus, apart from spot contracts, they are not traded in the market. The covari-ance matrix in this example is estimated from the corresponding, actually traded baseload contracts instead.

From the historical data, we know that the spot price on April 4, 2013 at 00:00:00was £40.33/MWh, which broadly determines the ranges of the parameters studied.

Figure 2 on the next page shows how the term structure of the electricity price andthe number of contracts traded depends on the risk aversion of the players involved.We can see that the market is usually in normal backwardation (ie, the term structureis upward sloping) and that the slope as well as the price increase when the riskaversion of the players increases. This increase in the price is due to the increasedrisk premium of the producers. The risk premium for risky contracts is higher thanthe risk premium of less risky contracts.

Part (b) of Figure 2 on the next page shows that most of the trading happens inthe first two periods through the two-month-ahead and month-ahead contracts. Thisis due to the fact that monthly contracts have much smaller volatility and are thuspreferred by risk-averse investors. However, we can see that the volume traded in thethird trading period is slightly smaller than the volume traded in the fourth and fifthtrading periods. This seems counterintuitive, because the volatility of a week-aheadcontract is smaller than the volatility of a day-ahead or spot contract. A more detailedinvestigation shows that it is a consequence of a small (statistically nonsignificant)negative correlation detected between the week-ahead electricity contract and thetwo-month-ahead gas contract. Even though the correlation is small, it still has avisible impact on the number of contracts traded. Note that the number of contractstraded is shown from the perspective of a consumer. The number of contracts tradedfrom the producer’s perspective is the same in absolute value but has the oppositesign.

In Figure 3 on page 53, we study a case where only the risk aversion of the producersis increased, while the risk aversion of the consumers is kept constant. As we cansee, the risk aversion of the producers does not have any significant impact on theshape of the term structure of the price. However, the price of all contracts increasedsignificantly due to the increased risk premium of the producers.

In Figure 4 on page 54, we study a case where only the risk aversion of the consumersis increased, while the risk aversion of the producers is kept constant. We can see thatthis has a small impact on the term structure of the price. In contrast to the producers,the increased risk premium of the consumers is not reflected in an increased price.Thus, the consumers can only maintain their profitability by increasing the fixed pricethey charge the end users (ie, they increase the end-user price sc.Tj / defined in (3.10)).

One must increase the consumers’ risk aversion parameter �c significantly toobserve a visual impact. Such a hypothetical case is depicted in Figure 5 on page 55.We can see that by increasing the risk aversion of consumers, the term structure of






FIGURE 2 The impact of risk preferences on the term structure of electricity prices.

1 2 3 4 540.0

40.5

41.0

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λ=1x10–6

λ=5x10–6

λ=5x10–5

λ=1x10–4

λ=1x10–5

λ=1x10–6

λ=5x10–6

λ=5x10–5

λ=1x10–4

λ=1x10–5

(a) (b)

�i1 D 0, �i1 D 0 for all i 2 f1; : : : ; 5g, and increasing �k for all k 2 P [ C . The trading periods labeled from 1 to 5correspond to a two-month-ahead, month-ahead, week-ahead, day-ahead and the spot price.

electricity prices changes from the normal backwardation to contango. An increasedrisk aversion increases the consumers’ interest in less risky contracts and decreasesthe interest in risky contracts. Since the contracts that are closer to delivery are usu-ally more risky, this causes the term structure to flip from normal backwardation tocontango. A small kink at the fourth trading period is a consequence of a calibrationof the covariance matrix on historical values.

It may not be immediately clear why the change in risk aversion of the producersand consumers has such a different impact on the term structure of the electricityprice. The level of the price is defined by producers only. Let us consider a simplifiedsetting, where there is only one trading and one delivery period. Moreover, we havejust one producer and one consumer. In such a setting, the profit of the consumer canbe calculated as

Pc.Vc ; ˘/ D e�OrT1.�˘.t1; T1/ C sc.T1//D.T1/; (6.10)

where the constraint Vc.t1; T1/ D D.T1/ is used. We can see that the profit functiondoes not depend on the consumer’s decision variables Vc . Thus, the consumer doesnot have any power to influence the price. They have to buy a sufficient amount of






FIGURE 3 The impact of risk preferences of producers on the term structure of electricityprices.

1 2 3 4 540.0

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λp=1x10–6

λp=5x10–6

λp=1x10–5

λp=5x10–5

λp=1x10–4

λp=1x10–6

λp=5x10–6

λp=1x10–5

λp=5x10–5

λp=1x10–4

(a) (b)

�i1 D 0, �i1 D 0 for all i 2 f1; : : : ; 5g, �c D 10�6 for all c 2 C , and increasing �p for all p 2 P .The trading periodslabeled from 1 to 5 correspond to a two-month-ahead, month-ahead, week-ahead, day-ahead and the spot price.

electricity to satisfy the demand of end users, regardless of the electricity price. Onthe other hand, the producer has much more flexibility. Their profit can be calculatedas

Pp.Vp; ˘/ D e�OrT1 Œ�˘.t1; T1/Vp.t1; T1/ � Gem.t1; T1/Op.t1; T1/

�Xl2L

Gl.t1; T1/Fp;l.t1; T1/�: (6.11)

An obvious feasible solution to their optimization problem (3.8) is

Vp.t1; T1/ D Op.t1; T1/ D Fp;l.t1; T1/ D 0 (6.12)

for all l 2 L. This leads to the objective value p D 0. Thus, for any given price˘.t1; T1/, the objective value satisfies p > 0. A producer will only turn on thepower plants if the electricity price is high enough to cover all production costs,trading costs and the risk premium. A similar reasoning can also be extended to amultiperiod setting.






FIGURE 4 The impact of risk preferences of consumers on the term structure of electricityprices.

1 2 3 4 540.412

40.414

40.416

40.418

40.42

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40.428

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0

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λc=1x10–6

λc=5x10–6

λc=1x10–5

λc=5x10–5

λc=1x10–4

10 000

λc=1x10–6

λc=5x10–6

λc=1x10–5

λc=5x10–5

λc=1x10–4

(a) (b)

�i1 D 0, �i1 D 0 for all iR

f1; : : : ; 5g, �p D 10�6 for all p 2 P , and increasing �c for all c 2 C . The trading periodslabeled from 1 to 5 correspond to a two-month-ahead, month-ahead, week-ahead, day-ahead and the spot price.

In Figure 6 on page 56, Figure 7 on page 57, Figure 8 on page 58 and Figure 9on page 59, we study the impact of liquidity on the term structure of the electricityprice. In Figures 6 and 7, the impact of the market microstructure (ie, the quadraticterm �) is examined, while in Figures 8 and 9 the impact of the bid–ask spread andof the trading fees (ie, the linear term �) is depicted.

Increasing � for all the contracts simultaneously (see Figure 6 on page 56) increasesthe price without changing the shape of the term structure. It has a large impact on theoptimal trading strategy of the players. When � is large, players spread the number ofcontracts traded in each trading period equally among all the trading periods available.

When � is changed only for the first trading period (ie, the two-month-ahead con-tract), this significantly changes the term structure of the price. As we can see inFigure 7 on page 57, a large increase in � changes the term structure from nor-mal backwardation to contango (ie, it becomes downward sloping). As expected, theplayers also trade a much smaller number of contracts in the first trading period. Asmall negative trading position is a consequence of a small (statistically nonsignifi-cant) negative correlation detected between the week-ahead electricity contract and






FIGURE 5 The impact of large changes in risk preferences of consumers on the termstructure of electricity prices.

1 2 3 4 5

40.35

40.36

40.37

40.38

40.39

40.40

40.41

40.42

40.43

40.44

Trading period

Pric

e (

£/M

Wh)

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0

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λc=1λc=10λc=102

λc=103

λc=1λc=10λc=102

λc=103

(a) (b)

�i1 D 0, �i1 D 0 for all i 2 f1; : : : ; 5g, �p D 10�6 for all p 2 P , and increasing �c for all c 2 C .The trading periodslabeled from 1 to 5 correspond to a two-month-ahead, month-ahead, week-ahead, day-ahead and the spot price.

the week-ahead gas contract, which was already mentioned in the beginning of thissection.

An effect of a simultaneous change in the linear trading costs � for all the contractsis depicted in Figure 8 on page 58. We can see that the linear trading costs haveno impact on the trading strategy or the shape of the term structure. However, theprice of all contracts is increased to cover the losses of the producers, caused by theincreased linear cost of trading. The consumers cover losses caused by the increasedlinear cost of trading through the increased electricity prices they charge to end users.The simultaneous change in linear trading costs � for all contracts does not affect theoptimal trading strategy of the players.

When � is changed only for the first trading period (ie, the two-month-ahead con-tract), this, in contrast to the changes in � , significantly affects only the price in the firsttrading period. The price of the two-month-ahead contract is increased to cover thelosses of the producers caused by the increased linear cost of trading. If � is increasedsignificantly, this makes the two-month-ahead contract so unattractive that no con-tracts are traded (see part (b) of Figure 9 on page 59). As already mentioned, a smallnegative trading position is a consequence of a small (statistically nonsignificant)






FIGURE 6 The impact of quadratic trading costs on the term structure of electricity prices.

1 2 3 4 540.0

40.5

41.0

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υ=10–6υ=0

υ=10–5

υ=10–4

υ=10–3

υ=10–6υ=0

υ=10–5

υ=10–4

υ=10–3

(a) (b)

�k D 10�6 for all k 2 P [ C , �i1 D 0 and increasing �i1 for all i 2 f1; : : : ; 5g. The trading periods labeled from 1to 5 correspond to a two-month-ahead, month-ahead, week-ahead, day-ahead and the spot price.

negative correlation detected between the week-ahead electricity contract and theweek-ahead gas contract.

In the remainder of this section, we investigate the effect of the term structure offuel and emission prices on the term structure of electricity prices. Initially, we set

EPŒG1.t1; T1/� D EPŒG1.t2; T1/� D � � � D EPŒG1.t5; T1/� D 69:30Œ£/therm�;

EPŒG2.t1; T1/� D EPŒG2.t2; T1/� D � � � D EPŒG2.t5; T1/� D 57:87Œ£/tonne�;

EPŒGem.t1; T1/� D EPŒGem.t2; T1/� D � � � D EPŒGem.t5; T1/� D 3:883Œ£/tonne�;

(6.13)

where l D 1 and l D 2 denote gas and coal prices, respectively. The electricity priceis quoted in £/MWh.

We conducted two types of experiments. In the first type, we performed parallelshifts of the term structure of fuel/emission prices. In the second type, we changedthe shape of the term structure of fuel/emission prices. Since results for all fuels aswell as for emissions are very similar, we present them for gas only.

Figure 10 on page 60 depicts the effect of parallel shifts in the term structure of gasprices. Expectedly, an increase/decrease in gas prices causes an increase/decrease inelectricity prices.






FIGURE 7 The impact of quadratic trading costs of the twomonth- ahead forward contracton the term structure of electricity prices.

1 2 3 4 5

40.42

40.43

40.44

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40.46

40.47

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Trading period

Pric

e (£

/MW

h)

1 2 3 4 50.5

0

0.5

1.0

1.5

2.0

2.5x 104

Trading period

Num

ber

of c

ontr

acts

trad

ed

υ=10–6υ=0

υ=10–5

υ=10–4

υ=10–3

υ=10–6υ=0

υ=10–5

υ=10–4

υ=10–3

(a) (b)

�i1 D 0 for all i 2 f1; : : : ; 5g, �k D 10�6 for all k 2 P [ C , and increasing �11. For the other trading periodsi 2 f2; : : : ; 5g, �i1 D 0. The trading periods labeled from 1 to 5 correspond to a two-month-ahead, month-ahead,week-ahead, day-ahead and the spot price.

More interesting is the second experiment, where we calculated the term structureof electricity prices for three different shapes of the term structure of gas prices. InFigure 11 on page 61, we examined a constant term structure of gas prices, normalbackwardation and contango. We can see that the term structure of gas prices hasa large impact on the term structure of electricity prices. In the example where theterm structure of gas prices is in contango, we can see that the term structure ofelectricity prices also changes to contango (a small kink at the fourth trading periodis a consequence of the calibration of the covariance matrix using historical values).Note that most trading with gas contracts in this example occurs in the later tradingperiod. Since gas contracts in these trading periods are highly correlated with theelectricity contracts from the same trading periods, this lowers the electricity pricesin these trading periods. Remember that producers are short on electricity contractsand long on gas contracts, so high correlation between gas and electricity prices hasthe effect of reducing the portfolio risk. This is also visible in Preposition 6.1 ofSection 6.4.

We can see that the change of the term structure of gas prices from constant to normalbackwardation does not have a visible impact on the term structure of electricity prices.






FIGURE 8 The impact of linear trading costs on the term structure of electricity prices.

1 2 3 4 5

40.6

40.8

41.0

41.2

41.4

41.6

41.8

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/MW

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1 2 3 4 5

0100020003000400050006000700080009000

10 000

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Num

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ontr

acts

trad

ed

ε=0.0ε=0.1ε=0.5ε=1.0

ε=0.0ε=0.1ε=0.5ε=1.0

(a) (b)

�k D 10�6 for all k 2 P [ C , �i1 D 0 and increasing �i1 for all i 2 f1; : : : ; 5g. The trading periods labeled from 1to 5 correspond to a two-month-ahead, month-ahead, week-ahead, day-ahead and the spot price.

This is expected, as the change does not have an impact on the fuel trading strategy. Inboth cases, most of the gas buying is done in the early periods, since these contractsare less risky (in the case of normal backwardation, they are also cheaper) and thuspreferable for risk-averse producers.

6.3 Block and spot contracts

Our goal is to calculate the electricity spot price with the information available onFebruary 11, 2013. We are interested in a delivery period from April 4, 2013 at00:00:00 to April 8, 2013 at 00:00:00. We assume that there are two types of powercontract available. The first is a month-ahead contract traded on March 15, 2013 at17:00:00 and covers the delivery over all four days. The second type is a spot contractthat requires an immediate delivery and is traded for each half hour separately. Weuse future prices of coal, gas and oil, as available on February 11, 2013. Since thehistorical demand forecast is not available, we used the realized demand instead,which is standard practice in the literature. To implement this model in practice, onecould use a demand forecast from the ELEXON web page or develop a new approach.Since we do not have information about the ownership of the power plants, we assumethat there is only one producer who owns all power plants connected to the UK grid,and only one consumer that is responsible for satisfying the demand of the end users.






FIGURE 9 The impact of linear trading costs of the two-month-ahead forward contract onthe term structure of electricity prices.

1 2 3 4 5

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40.6

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ε=0.0ε=0.1ε=0.5ε=1.0

ε=0.0ε=0.1ε=0.5ε=1.0

(a) (b)

�i1 D 0 for all i 2 f1; : : : ; 5g, �k D 10�6 for all k 2 P [ C , and increasing �11. For the other trading periodsi 2 f2; : : : ; 5g, �i1 D 0. The trading periods labeled from 1 to 5 correspond to a two-month-ahead, month-ahead,week-ahead, day-ahead and the spot price.

In reality, market participants have more information about the ownership, which canbe incorporated into the model.

The numerical results in Figures 12–15 are all calculated using �k D 10�5 for allk 2 P [C , and �ij D 0:1 for all contracts. Since the forward contracts are less liquidthan the spot contracts, we use �ij D 10�4 and �ij D 10�3 for spot contracts andthe forward contract, respectively. Part (a) of Figure 12 depicts the calculated energymix between coal and gas power plants, while part (b) depicts the actual observedenergy mix. Both parts also contain the spot price calculated by our model and theactual observed spot price.

Figure 12 on page 61 shows that our model predicts the energy mix very closely.Moreover, the daily pattern of the electricity price predicted by our model is similar tothe actual observed one. Figure 13 on page 62 shows the optimal number of forwardand spot contracts traded. We can see that approximately the same amount of elec-tricity is delivered through forward and spot contracts. This happens because forwardcontracts have smaller volatility but higher transaction costs than spot contracts, sothe contribution of the two factors approximately cancels out. An equilibrium priceof the month-ahead forward contract is £58.31/MWh.






FIGURE 10 The impact of absolute levels of gas prices on the term structure of electricityprices.

1 2 3 4 538

39

40

41

42

43

44

Trading period

Pric

e (£

/MW

h)

1 2 3 4 564

66

68

70

72

74

76

Trading period

Pric

e (p

/ther

m)

v1v2v3

v1v2v3(a) (b)

Effect of parallel shifts in the term structure of (b) gas prices on the term structure of (a) electricity prices.The tradingperiods labeled from 1 to 5 correspond to a two-month-ahead, month-ahead, week-ahead, day-ahead and the spotprice.

Figure 14 on page 63 shows how the results change when we tighten the ramp-upand ramp-down constraints by 20%. We can see that the price is more serrated andslightly higher, because more expensive power plants must be turned on to cover thechanges in demand. This change has a very small impact on the number of contractstraded, so the results from Figure 13 on page 62 still apply. An equilibrium price ofthe month-ahead forward contract is £58.32/MWh.

Similarly, Figure 15 on page 63 shows how the results change when we tighten theramp-up and ramp-down constraints by 50%. This change has a very small impact onthe number of contracts traded, so the results from Figure 13 on page 62 still apply.An equilibrium price of the month-ahead forward contract is £58.36/MWh. FromFigures 12–15, we can see that tighter ramping constraints lead to the slightly higherequilibrium price for the forward contract.

Figure 12 on the facing page reveals a few problems with our model if we use itfor the forecasting of electricity spot prices. First, we can see that the daily variationin the calculated price is smaller than in the observed one. Second, two spikes in theobserved price are not captured in our model. Our hypothesis is that the first problemoccurs because our model does not incorporate the start-up costs of the power plantscorrectly. The inclusion of start-up costs exceeds the scope of this paper and will






FIGURE 11 The impact of the shape of the term structure of gas prices on the termstructure of electricity prices.

1 2 3 4 540.40

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/MW

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e (p

/ther

m)

v1v4v5

v1v4v5(a) (b)

Dependence of the term structure of (a) electricity prices on the shape of the term structure of (b) gas prices. Thetrading periods labeled from 1 to 5 correspond to a two-month-ahead, month-ahead, week-ahead, day-ahead andthe spot price.

FIGURE 12 The energy mix and the predicted spot price for the base model.

200

1

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3

4x 104

Half hour

Dem

and

(MW

)

60 100 140 180

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x 104

Coal (observed)Gas (observed)Calculated priceObserved price

Coal (calculated)Gas (calculated)Calculated priceObserved price

(a) (b)

�k D 10�7 for all k 2 P [ C , � D 0.1, � D 10�4 for spot contracts and � D 10�3 for the forward contract. Part (a)depicts the calculated energy mix of coal and gas power plants, while part (b) depicts the observed energy mix. Bothfigures also contains the spot price calculated by our model as well as the observed spot price.






FIGURE 13 The optimal number of contracts traded for the base model.

20 40 60 80 100 120 140 160 1800

0.5

1.0

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2.5x 104

Half hour

Num

ber

of c

ontr

acts

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ForwardSpot

�k D 10�7 for all k 2 P [ C , � D 0.1, � D 10�4 for spot contracts and � D 10�3 for the forward contract. Thisimage depicts the optimal number of forward and spot contracts traded.

be addressed separately. Our hypothesis for the explanation of the second problemis the error in the demand forecast. Spikes in the spot price occur when there is anunexpected change in demand. Since only a few (usually rather inefficient open cyclegas turbine) power plants are flexible enough to cover the demand, they require ahigh electricity price to be turned on. Thus, the spikes cannot be forecast two monthsbefore delivery. An investigation of the predictive power of our model to forecastspikes closer to delivery is left for future work.

6.4 Closed-form solutions

In this section, we derive a closed-form solution for calculating the term structure ofelectricity prices from the spot price. It is possible to establish the following uniqueclosed-form relation among the prices EPŒ˘.ti ; Tj /�, i 2 f1; 2; : : : ; ng, for any fixedj 2 J .

Proposition 6.1 Let EPŒ˘� denote competitive equilibrium prices and letEPŒ˘.tn/� be a given spot price.Then, other competitive equilibrium prices EPŒ˘.ti /�,






FIGURE 14 The energy mix and the predicted spot price if the ramping constraints aretightened by 20%.

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e (£

/MW

h)

x 104



(a) (b)

Ramp-up and ramp-down constraints are tightened by 20% from their initial value. Part (a) depicts the calculatedenergy mix of coal and gas power plants, while part (b) depicts the observed energy mix. Both parts also containthe spot price calculated by our model as well as the observed spot price.

FIGURE 15 The energy mix and the predicted spot price if the ramping constraints aretightened by 50%.

20 60 100 140 1800

1

2

3

4x 104

Half hour

Dem

and

(MW

)

0

20

40

60

80

Pric

e (£

/MW

h)

20 60 100 140 180

0

1

2

3

4x 104

Half hour

Dem

and

(MW

)

0

20

40

60

80

100

Pric

e (£

/MW

h)

100



(a) (b)

Ramp-up and ramp-down constraints are tightened by 50% from the initial value. Part (a) depicts the calculatedenergy mix of coal and gas power plants, while part (b) depicts the observed energy mix. Both parts also containthe spot price calculated by our model as well as the observed spot price.






i 2 f1; : : : ; n � 1g, can be derived using the following equality

EPŒ˘.ti /�

D EPŒ˘.tn/� � ��1 cov

�˘.ti / � ˘.tn/;

Xp2P

CpŒkp; Fp; G� �� X

c2C

scpc

�D

�;

(6.14)

where

��1 DPX

pD1

�1

�p

�C

CXcD1

�1

�c

�and kp D

nXiD1

Vp.ti /:

Cp.kp; F; G/ denotes the costs of producing kp units of electricity with the fuel tradingstrategy F at fuel prices G.

Proof To simplify the notation, we introduce QVk D ŒVk.t1/; : : : ; Vk.tn�1/�T for allk 2 P [ C and Q D Œ˘.t1/; : : : ; ˘.tn�1/�T.

The profit of a producer p 2 P can in this setting be written as

Pp.Vp; Fp; ˘/ D �nX

iD1

˘.ti /Vp.ti / � CpŒkp; Fp; G�; (6.15)

where

kp DnX

iD1

Vp.ti /: (6.16)

A profit of a consumer c 2 C can be written as

Pc.Vc ; ˘/ D scpcD �nX

iD1

˘.ti /Vc.ti /:

We can see that for any producer p 2 P there are no constraints on the small changesof Vp.ti /, but there is a capacity constraint on changes of kp . Thus, by keeping kp fixed,(3.1) and (3.2) cannot be violated by small changes in volume Vp.ti /. By expressingVp.tn/ from (6.16), inserting (6.15) into a mean–variance objective function andtaking a derivative with respect to QVp , we obtain the following expression:

@�p.Pp.Vp; Fp; ˘//

@ QVp

D EPŒ˘.tn/�e � EPŒ Q � � �pQQ QVp

C �p cov.˘.tn/e � Q ; kp˘.tn/e C CpŒkp; Fp; G�e/;

where e is a column vector of ones of length n � 1 and QQ is defined as

QQ WD EPŒ. Q � EPŒ Q �/. Q � EPŒ Q �/T�I






cov is applied element-wise. By setting @�p.Pp.Vp; Bp; ˘//=@ QVp D 0 and express-ing QVp , we obtain

QVp D 1

�p

QQ�1.EPŒ˘.tn/�e � EPŒ Q �/

C QQ�1 cov.˘.tn/e � Q ; kp˘.tn/e C CpŒkp; Fp; G�e/: (6.17)

Since QQ is a covariance matrix and none of the electricity contracts can be written asa linear combination of the others almost surely, QQ is invertible. Similarly, for eachconsumer c 2 C ,

QVc D 1

�c

QQ�1.EPŒ˘.tn/�e �EPŒ Q �/C QQ�1 cov.˘.tn/e � Q ; kc˘.tn/e �scpcDe/;

(6.18)where kc D

PniD1 Vc.ti /. The market clearing condition (3.21) requires

0 DXp2P

QVp CXc2C

QVc : (6.19)

Inserting expressions (6.17) and (6.18) into (6.19), we obtain the result

EPŒ˘.ti /�

D EPŒ˘.tn/� � ��1 cov

�˘.ti / � ˘.tn/;

Xp2P

CpŒkp; Fp; G� �� X

c2C

scpc

�D

�;

where � DPP

pD1.1=�p/ CPC

cD1.1=�c/. �

The analytical expression obtained in Proposition 6.1 matches the expression inBessembinder and Lemmon (2002), but extended to a multistage setting, and theexpressions in Bühler and Müller-Merbach (2009) and Bühler (2009), but with resultsextended to more than one producer and consumer. In contrast to these previous works,it also allows for capacity constraints.

By examining (6.14), one can study the impact of the ramp-up and ramp-downconstraints on the term structure of the electricity price. These constraints affect theenergy mix and consequently

Pp2P CpŒkp; Fp; G�. Since coal power plants have

tighter ramping constraints than gas power plants, one can conclude that during peri-ods of large increases in demand, there will be more gas production than one wouldpredict solely on costs. Similarly, during the periods of large decreases in demand,there will be more coal production than one would predict solely on costs. From(6.14), one can see that the correlation between the electricity price and the energymix plays an important role in the determination of the term structure of the electric-ity price. Thus, during periods of large demand increases (decreases), the correlation






between gas (coal) and electricity prices plays a more important role than one wouldpredict without taking ramping constraints into account.

Relation (6.14) can also be used to explain the numerical example in Figure 11 onpage 61.

7 CONCLUSIONS

In this paper, we proposed a tractable quadratic programming reformulation for cal-culating the equilibrium term structure of electricity prices in a market with multipleconsumers and producers. Reformulation can be used for solving quadratic competi-tive equilibrium optimization problems that include inequality constraints, and it canbe seen as an extension of the shadow price approach.

We have extended the term structure electricity price model proposed in Troha andHauser (2014) to a more realistic setting by taking into account transaction costs andliquidity considerations as well as realistic electricity contracts.

The section on numerical results first shows how to calibrate various parameters ofthe model. We investigated how different parameters affect the equilibrium electricityprice. Interestingly, we found that consumers have very little power to influence theelectricity price. They have to satisfy the demand of the end users, regardless of theprice level. To maintain their profitability, they propagate all increased costs to the endusers. Producers, on the other hand, have much more power to affect the electricityprices. They have a large impact on the price level and a slightly smaller impact onthe term structure itself. The market microstructure and liquidity of the contracts cansignificantly affect the term structure of the electricity price. In an extreme case, thechanges in the liquidity even altered the term structure from normal backwardationto contango. The term structure of electricity prices is also considerably affected bythe term structure of fuels and emissions.

We investigated the predictive power of our model when applied to the realisticsystem of UK power plants. The results show that our model predicts the prices quitewell and that the predicted price exhibits the main features of the electricity price.Numerical examples show the effect of tightened ramp-up and ramp-down constraintson spot as well as month-ahead forward contracts.


Miha Troha was supported through grants from the Slovene human resources devel-opment and scholarship fund and the Oxford-Man Institute. Raphael Hauser wassupported through grant EP/H02686X/1 from the Engineering and Physical SciencesResearch Council of the United Kingdom.






ACKNOWLEDGEMENTS

We thank the Oxford-Man Institute for providing the historical prices used to calibrateour model, and ELEXON for providing historical data about the balancing mechanismused to determine the physical characteristics of the power plants connected to theUK power grid.

REFERENCES

Allaz, B., andVila, J.L. (1993).Cournot competition, forward markets and efficiency.Journalof Economic Theory 59(1), 1–16.

Almgren, R., and Chriss, N. (2001). Optimal execution of portfolio transactions. Journal ofRisk 3, 5–39.

Anderson, E., and Xu, H. (2002). Necessary and sufficient conditions for optimal offers inelectricity markets. SIAM Journal on Control and Optimization 41(4), 1212–1228.

Anderson, E., and Xu, H. (2005). "-optimal bidding in an electricity market with discon-tinuous market distribution function. SIAM Journal on Control and Optimization 44(4),1391–1418.

Barlow, M. T. (2002). A diffusion model for electricity prices. Mathematical Finance 12(4),287–298.

Benth, F. E., and Meyer-Brandis, T. (2009). The information premium for non-storablecommodities. Journal of Energy Markets 2(3), 111–140.

Bessembinder, H., and Lemmon, M. L. (2002). Equilibrium pricing and optimal hedging inelectricity forward markets. Journal of Finance 57(3), 1347–1382.

Bühler, W. (2009). Risk premia of electricity futures: a dynamic equilibrium model. In RiskManagement in Commodity Markets, pp. 61–80. Wiley.

Bühler, W., and Müller-Merbach, J. (2009).Valuation of electricity futures: reduced-form vsdynamic equilibrium models. Working Paper 2007-07, July, Mannheim Finance.

Carmona, R., Coulon, M., and Schwarz, D. (2013). Electricity price modeling and assetvaluation: a multi-fuel structural approach. Mathematics and Financial Economics 7(2),167–202.

Cavallo, L., andTermini,V. (2005).Electricity derivatives and the spot market in Italy.Mitigat-ing market power in the electricity market.Working Paper 70, April, Centre for Economicand International Studies.

Clewlow, L., and Strickland, C. (1999a). A multi-factor model for energy derivatives.Research Paper Series 28, December, Quantitative Finance Research Centre, Univer-sity of Technology Sydney.

Clewlow, L., and Strickland, C.(1999b).Valuing energy options in a one factor model fitted toforward prices. Research Paper Series 10, April, Quantitative Finance Research Centre,University of Technology Sydney.

De Maere d’Aertrycke, G., and Smeers, Y. (2013). Liquidity risks on power exchanges: ageneralized Nash equilibrium model. Mathematical Programming 140(2), 381–414.

Gurobi Optimization (2014). Gurobi Optimizer reference manual. Gurobi Optimization Inc.Hambly, B., Howison, S., and Kluge, T. (2009). Modelling spikes and pricing swing options

in electricity markets. Quantitative Finance 9(8), 937–949.






Howison, S., and Coulon, M. C. (2009). Stochastic behaviour of the electricity bid stack:from fundamental drivers to power prices. Journal of Energy Markets 2(1), 26–69.

Ledoit, O., and Wolf, M. (2003). Improved estimation of the covariance matrix of stockreturns with an application to portfolio selection. Journal of Empirical Finance 10(5),603–621.

Lucia, J. J., and Schwartz, E. (2000). Electricity prices and power derivatives: evidencefrom the Nordic power exchange. Review of Derivatives Research 5(1), 5–50.

Meyer-Brandis, T., and Tankov, P. (2008). Multi-factor jump-diffusion models of electricityprices. International Journal of Theoretical and Applied Finance 11(5), 503–528.

Milano, F., Canizares, C. A., and Conejo, A. J. (2006). Sensitivity-based security-con-strained OPF market clearing model. In Power Systems Conference and Exposition,PSCE ’06 – 2006 IEEE PES, pp. 418–427. Institute of Electrical and ElectronicsEngineers.

Robinson, S. (2005). Math model explains high prices in electricity markets. SIAM News38(8). URL: www.siam.org/pdf/news/165.pdf.

Troha, M., and Hauser, R. (2014). The existence and uniqueness of a power priceequilibrium. ArXiv e-print, August.





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RNET16-AD156x234-NEWSLETTER.indd 1 21/03/2016 09:27


Research Paper

Approximation of the price dynamics ofheating degree day and cooling degree daytemperature futures

Fred Espen Benth and Sara Ana Solanilla Blanco

Department of Mathematics, University of Oslo, PO Box 1053, Blindern, 0316 Oslo, Norway;email: [email protected]

(Received March 3, 2014; revised October 29, 2014; accepted November 7, 2014)

ABSTRACT

We propose an approximation that makes the price dynamics of heating degree day(HDD) and cooling degree day (CDD) temperature futures linearly dependent onthe underlying temperature. The approximation is analyzed both theoretically andempirically. We base our analysis on a continuous-time autoregressive stochasticdynamics for the time evolution of temperature in a given location. The model isfitted to temperature data collected in New York over a long time period. We applyour results to derive a simple version of the Black-76 formula for pricing a call optionon CDD and HDD futures.

Keywords: temperature futures; autoregressive processes; call option; Black-76 formula; Taylorexpansion.

1 INTRODUCTION

The Chicago Mercantile Exchange (CME) organizes trade in futures contracts writtenon temperature indexes measured in several cities worldwide. Typical indexes are theso-called heating degree day (HDD) and cooling degree day (CDD) indexes, which are

Corresponding author: F. E. Benth Print ISSN 1756-3607 j Online ISSN 1756-3615Copyright © 2015 Incisive Risk Information (IP) Limited

69




70 F. E. Benth and S. A. Solanilla Blanco

used for cities in the United States. The CDD index measures the difference betweenthe observed daily average temperature and a threshold of 65 ıF over a given timeperiod. In the CME marketplace, the period over which the CDD index is measured istypically a month in the summer season. The index provides a measure of the demandfor air conditioning cooling, and futures written on this index can be used for hedgingsuch demand by power producers or retailers. On the other hand, the HDD indexmeasures the demand for heating during the winter season.

The CDD and HDD indexes are nonlinear functions of the underlying temperature.In fact, the two indexes can be viewed as strips of call and put (respectively) payoffstructures on the underlying temperature. As a result, the futures price will be nonlin-early dependent on the temperature as well (see Benth and Šaltyte Benth 2012). TheCME organizes a market for plain vanilla call and put options on temperature futures,and the pricing of these derivatives will in fact become very complex due to the non-linear nature of the underlying futures. The objective of this paper is to propose somemethods to linearize the futures price dynamics, which will pave the way for simpleoption pricing formulas. In fact, we can price call and put options using a variantof the Black-76 formula, avoiding highly inefficient Monte Carlo based pricing (seeBenth and Šaltyte Benth 2012). The suggested approximative price dynamics willalso significantly simplify the statistical analysis of futures prices.

Our approximation is based on a linearization of a function appearing in the theoret-ical futures price formula based on arbitrage-free pricing. The function in question isclosely connected to the cumulative distribution function and possesses nice analyt-ical properties, which enables us to perform an error analysis of the approximativefutures price. We discuss the approximation using several numerical examples basedon temperature observations from New York. As it turns out, our proposed approxi-mation works well in several cases; however, there are also examples where it doesnot perform well and care must be taken.

Our analysis of CDD and HDD futures prices is based on a continuous-time autore-gressive (CAR) temperature dynamics. Several empirical studies of temperature datahave shown that CAR models explain the statistical properties of the dynamics verywell (see Härdle and Lopez Cabrera (2012) and Benth and Šaltyte Benth (2012) andthe references cited therein). We fit a long series of daily average temperature dataobserved in NewYork to a CAR process, and this forms the reference point for severalof our numerical and empirical examples. CAR models have been applied in severalmarkets, such as power markets (see Garcia et al 2010), oil markets (see Paschke andProkopczuk 2010) and fixed-income markets (see Andresen et al 2014). The generalclass of continuous-time autoregressive moving average (CARMA) processes hasbeen extensively studied by Brockwell and co-authors (see Brockwell (2001) for anoverview).





Approximation of the price dynamics of HDD and CDD temperature futures 71

Our results are presented in the following way. In Section 2 the stochastic modelfor the temperature dynamics is presented, and we include an empirical analysis ofthe New York data. We also present the theoretical arbitrage-free prices of HDD andCDD futures here. Section 3 presents and analyzes the linearization of HDD andCDD futures prices, with several empirical examples given to illustrate the theory.In Section 4 we go on to give an analytical formula for the price of a call written ontemperature futures using its linearized dynamics. We conclude in Section 5.

2 TEMPERATURE MODELING AND FUTURES PRICING

In this section we introduce a stochastic dynamics for the time evolution of temper-atures measured in a given location. Introduce a complete filtered probability space.˝; F ; fFtgt>0; P /. Let T .t/ denote the temperature in the location at time t > 0,and assume that

T .t/ D �.t/ C Y.t/: (2.1)

Here, � is a deterministic function measuring the mean temperature at time t , andY.t/ is some stochastic process modeling the random variations around this mean. Weassume that � is a smooth function with growth that is at least linear. We frequentlyrefer to Y as the deseasonalized temperature.

As a model for Y , we introduce the class of CAR.p/ processes for p 2 N. To thisend, let X.t/ be a vector-valued stochastic process with values in Rp , having thedynamics

dX.t/ D AX.t/ dt C �.t/ep dB.t/ (2.2)

for a one-dimensional Brownian motion B.t/. Here, ei , i D 1; : : : ; p, are the canon-ical unit vectors in Rp , �.t/ is a bounded, real-valued deterministic function, and A

is the p � p matrix

A D

26666664

0 1 0 � � � 0

0 0 1 � � � 0:::

::::::

: : ::::

0 0 0 � � � 1

� p � p�1 � p�2 � � � �˛1

37777775

: (2.3)

The constants ˛i , i D 1; : : : ; p, are all assumed to be strictly positive. Moreover,we assume that the eigenvalues of A have negative real part, which implies that theprocess X.t/ has a stationary distribution (see Benth and Šaltyte Benth 2012).

A CAR.p/ process is defined as

Y.t/ D e01X.t/; (2.4)






that is, the first coordinate of the vector X.t/. We use the notation x0 to denote thetranspose of a vector (or matrix) x. We remark in passing that one may generalizeto so-called continuous-time autoregressive moving average processes by mixing theq � 1 6 p (q being a natural number or zero) first coordinates of X.t/ (we refer toBenth and Šaltyte Benth (2012) for more on this).

From the dynamics of X.t/, we see that it is a vector-valued Ornstein–Uhlenbeckprocess, with a particular “mean-reversion” matrix A and the stochastic evolutiondriven by a one-dimensional Brownian motion. Hence, it has simple analytical prop-erties that allow for reasonably explicit expressions for the dynamics of various tem-perature futures prices. We briefly recall some results from Benth and Šaltyte Benth(2012).

Consider a temperature futures contract settled on the CDD index measured in agiven location over the period Œ�1; �2�, �1 < �2. Typically, this measurement periodis a month in the warm season of the year, which is the period between April andOctober for US cities. The index is defined as

CDD.�1; �2/ DZ �2

�1

max.T .t/ � c; 0/ dt; (2.5)

where c is 65 ıF (or 18 ıC). As we see, the CDD index aggregates temperatures abovethe threshold c over the measurement period, and as such it provides a measure of thedemand for air conditioning cooling, say, in the period Œ�1; �2�. In the actual market,the index is measured as the sum of the average daily temperatures, where the averageis calculated as the average of the recorded minimum and maximum temperatures.We apply the definition for notational convenience. The HDD index is analogouslydefined as

HDD.�1; �2/ DZ �2

�1

max.c � T .t/; 0/ dt; (2.6)

and it measures the demand for heating in the cold season, which ranges from Octoberto April in the US market.

A CDD futures contract with measurement period Œ�1; �2� is financially settled onthe CDD index CDD.�1; �2/. The settlement takes place at time �2, the end of themeasurement period, at which point one gets a payment equal to US$20 per indexpoint. To simplify notation, we assume that one simply gets the index value in dollars.The arbitrage-free futures price FCDD.t; �1; �2/ at time t 6 �2 is defined as

FCDD.t; �1; �2/ D EQŒCDD.�1; �2/ j Ft �; (2.7)

where EQŒ� j Ft � is the conditional expectation operator with respect to some proba-bility Q � P . The probability Q is some pricing measure modeling the risk premiumin the market charged by the actors for the inability to hedge the underlying index.We have the obvious analogous expression for FHDD.t; �1; �2/.






We let Q be given by a Girsanov transform such that the process

dW.t/ D � �.t/

�.t/dt C dB.t/ (2.8)

is a Q-Brownian motion. Here, � is a bounded deterministic function, and in order tohave this Girsanov transform validated, we must assume that �.t/ is bounded belowby a constant strictly bigger than zero. The parameter function � is referred to as themarket price of risk, and it is an implicit parameterization of the risk premium in themarket. We note that the Q-dynamics of X.t/ becomes

dX.t/ D .�.t/ep C AX.t// dt C �.t/ep dW.t/:

We find the following result, which is a slight extension of Proposition 5.4 in Benthand Šaltyte Benth (2012, p. 121).

Proposition 2.1 It holds for t 6 �2 that

FCDD.t; �1; �2/

DZ max.t;�1/

�1

max.T .s/ � c; 0/ ds CZ �2

max.t;�1/

˙.t; s/�

�m� .t; s; X.t// � c

˙.t; s/

�ds;

where, for a vector x 2 Rp ,

m� .t; s; x/ D �.s/ C e01eA.s�t/x C

Z s

t

e01eA.s�u/ep�.u/ du;

˙2.t; s/ DZ s

t

.e01eA.s�u/ep/2�2.u/ du:

Further, �.x/ D x˚.x/ C ˚ 0.x/, with ˚ being the cumulative standard normaldistribution function.

Proof First, if t 6 �1, the result follows from Proposition 5.4 in Benth andŠaltyte Benth (2012). Now let �1 6 t 6 �2. Then, as

Z t

�1

max.T .s/ � c; 0/ ds

is Ft -measurable, we have

FCDD.t; �1; �2/ DZ t

�1

max.T .s/ � c; 0/ ds C EQŒCDD.t; �2/ j Ft �:

The second term is the price of a CDD futures contract at time t with measurementperiod Œt; �2�, which we again find from Proposition 5.4 in Benth and Šaltyte Benth(2012). Hence, the proof is complete. �






We remark that m� .t; s; X.t// is the mean of X.s/ conditional on X.t/, s > t ,while ˙2.t; s/ is its conditional variance. Note in Proposition 2.1 that eAt , for t > 0,is the matrix exponential defined as

eAt D1X

nD0

Antn

nŠ:

Further, we recall ˚ 0.x/ to be �.x/ D .2�/�1=2e�x2=2, the standard normal densityfunction (see Proposition 2.1).

There is a similar expression for the HDD futures price, which may be conve-niently derived from the so-called CDD–HDD parity. From Corollary 5.1 in Benthand Šaltyte Benth (2012, p. 119) we find the following corollary.

Corollary 2.2 It holds that

FCDD.t; �1; �2/ � FHDD.t; �1; �2/ D EQ

� Z �2

�1

T .s/ ds

ˇˇ Ft

�� c.�2 � �1/:

The expectation on the right-hand side is in fact the price of a so-called CAT futurescontract at time t (where CAT stands for cumulative average temperature). From aslight generalization of Proposition 5.6 in Benth and Šaltyte Benth (2012, p. 123) wehave the following.

Proposition 2.3 It holds for t 6 �2 that

FHDD.t; �1; �2/

DZ max.t;�1/

�1

max.c � T .s/; 0/ ds CZ �2

max.t;�1/

˙.t; s/�

�c � m� .t; s; X.t//

˙.t; s/

�ds;

where � , m� and ˙ are defined in Proposition 2.1.

Observe that both the HDD and CDD futures prices depend nonlinearily on thestate of the vector X.t/ in the temperature dynamics. The nonlinearity stems from thefunction �.x/ defined in Proposition 2.1 combined with an integration. For example,this will make parameter estimation a complex matter, as the dynamics of the futuresprices will have a state-dependent volatility that is nonlinear.Also, it seems impossibleto derive analytic formulas for call and put option prices on the CDD and HDDfutures (which are the actual products traded at the CME). Accurate pricing must relyon Monte Carlo simulation, which is slow, or on numerical solution of certain partialdifferential equations. The latter approach is complicated by the fact that the associatedpartial differential equation will be defined on a p-dimensional domain with diffusionin only one direction, while having (strong) gradients in all directions. Further, theintegration over the measurement period gives complex boundary conditions.






In the next section we propose ways to linearize the dynamics of the HDD andCDD futures based on approximating the function � . Such linearizations will yieldanalytically tractable dynamics for HDD and CDD futures. As we will demonstrateempirically and theoretically, the approximation performs well in many cases, and itis therefore an attractive alternative to exact pricing formulas.

We end this section with an empirical case study of the CAR dynamics for dailytemperatures recorded in NewYork. The study will demonstrate the validity of our pro-posed dynamics for the stochastic time evolution of temperatures, as well as providingus with a basis for later examples that illustrate the results in this paper.

A time series of daily average temperatures (DATs) observed in New York fromJanuary 1, 1960 to April 20, 2013 is available. The DAT is calculated as the averageof the minimum and maximum temperatures recorded for that day. The temperaturesare given in degrees Fahrenheit. The DATs from February 29 in each leap year aredeleted from the data set in order to equalize the length of all years. In total, we havea time series of 19 455 observations.

We estimate the parameters in our stochastic model for the temperature dynamicsbelow. We let the observed DATs be measurements of the dynamics T .t/ defined in(2.1). We let Ti be the temperature observed at day i for i D 0; 1; 2; : : : , with i D 0

being January 1, 1960, and we assume that

Ti D �i C yi :

The seasonal function �.t/ is assumed to have the form

�.t/ D a0 C a1t C a2 cos.2�.t � a3/=365/;

where at C a1t is a linear trend capturing the possible influence of global warmingand urbanization, say, and the cosine term models the yearly seasonal cycle of theDATs. The parameters a0; : : : ; a3 are all constants.

Fitting �.t/ to the observed DATs by the least squares method resulted in theestimates Oa0 D 54:19, Oa1 D 0:0001, Oa2 D 22:15 and Oa3 D 204:81. In Figure 1 onthe next page, a snapshot of the observed DATs from the first ten years of the data setis depicted together with the fitted seasonality function �.t/.

Next we move to the deseasonalized temperatures yi D Ti � �.i/. In part (a) ofFigure 2 on the next page we have plotted the autocorrelation function (ACF) of thetime series, which decays in a seemingly exponential manner to zero. This is a clearsign of stationary autoregressive dynamics. This is further emphasized by looking atthe partial ACF (PACF) shown in part (b) of the figure, where it is suggested that thedata follows an autoregressive time series dynamics of order 3, denoted AR.3/.






FIGURE 1 Observed DATs in New York together with the fitted seasonal function �.t/.

0 500 1000 1500 2000 2500 3000 35000

20

40

60

80

100

Time

Dai

ly a

vera

ge te

mpe

ratu

re

DATs Lambda

A snapshot of the first ten years of the data set, starting from January 1, 1960.

FIGURE 2 (a) The ACF and (b) the PACF of the deseasonalized temperature seriesTi � �.i/.

0 5 10 15 20 25 30 35 40 45 50–0.2

00.20.40.60.81.0

Lag

AC

F

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15–0.2

0

0.2

0.4

0.6

0.8

Lag

Par

tial a

utoc

orre

latio

n

(a)

(b)






FIGURE 3 The empirical density of the normalized residuals from the estimated AR.3/

time series together with the standard normal density (dashed line).

–5 –4 –3 –2 –1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

Pro

babi

lity

dens

ity fu

nctio

n

ResidualsStandard normal distribution

We suppose that yi follows an AR.3/ dynamics, which means that for i D0; 1; 2; : : : ,

yiC3 D3X

j D1

bj yiC3�j C ��i

for constant autoregression parameters bj , j D 1; 2; 3, where � is the constantvolatility function and f�igiD0;1;::: are independent standard normally distributed ran-dom variables. By using standard techniques for estimating the parameters of AR.p/

processes, we find Ob1 D 0:8382, Ob2 D �0:2869 and Ob3 D 0:1123, and O� D 5:25.In Figure 3 we have plotted the empirical density of the normalized residuals of the

estimated AR.3/ time series along with the standard normal distribution. Although itis not perfect, we obtain a reasonable fit to the standard normal distribution. To obtaina better fit, one might introduce a seasonally varying volatility �.t/, as advocatedin Benth and Šaltyte Benth (2012) and empirically observed in many cities. As ourpurpose here is to study an approximation of futures prices, we refrain from furthergenerality of the temperature model as this is not likely to influence the approximationitself.

The link between the coefficients of an AR.p/ process and a CAR.p/ process Y.t/

defined in (2.4) is established in Lemma 10.2 in Benth and Šaltyte Benth (2012). Basedon the estimated bj s, we find the coefficients in the matrix A of the process X.t/ in(2.2) defining the CAR.3/ process to be O1 D 2:1618, O2 D 1:6105 and O3 D 0:3364.The eigenvalues of the matrix A become �1 D �0:34, �2;3 D �0:91 ˙ 0:40i. Hence,Y is a stationary CAR.3/ process as the eigenvalues have negative real part. We applythis model as the basic case study in an empirical investigation of the approximationof HDD and CDD futures prices. We introduce this next.






3 APPROXIMATION OF THE DYNAMICS OF HDD AND CDDFUTURES PRICES

In this section we propose a linearization of HDD and CDD futures prices. This isdone by analyzing the function �.x/ defined in Proposition 2.1. Let us start withsome initial considerations on this function.

As the derivative of �.x/ is

� 0.x/ D ˚.x/ 2 .0; 1/;

where ˚ is the cumulative standard normal distribution function, we find that � ismonotonically increasing. Also, we find that �.x/ tends to infinity as x ! 1 and ittends to zero as x ! �1. In fact, since

limx!1

�.x/

xD 1;

we have that �.x/ � x for large x. This is of course a reflection of the fact thatmax.x � c; 0/ D x � c when x is larger than c. In the following lemma we quantifythe convergence rate.

Lemma 3.1 For x > 0 it holds that

j�.x/ � xj 6�

x

2C 1p

2�

�e�x2=2:

Proof By the triangle inequality we find that

j�.x/ � xj 6 x.1 � ˚.x// C ˚ 0.x/:

But, after a change of variables,

1 � ˚.x/ D 1p2�

Z 1

x

e�y2=2 dy

D 1p2�

Z 1

0

e�.zCx/2=2 dz

D 1p2�

Z 1

0

e�z2=2e�zx dz e�x2=2:

After noting that e�zx 6 1 for z; x > 0, the lemma follows. �

From the lemma we see that �.x/ converges to x at a rate xe�x2=2 as x tends toinfinity. This is very rapid convergence, indicating that x does not need to be largebefore we have �.x/ � x.






One could also consider a Taylor approximation of � . Doing a Taylor expansionof � of order 1 with remainder around zero gives

�.x/ D �.0/ C � 0.0/x C 12� 00.z/x2

D 1p2�

C 12x C �.z/x2;

since � 0.x/ D ˚.x/, and hence � 00.x/ D �.x/, where we recall that �.x/ is thedensity function of the standard normal distribution. Note that jzj 6 jxj. A reasonableapproximation of �.x/ around zero is therefore

�.x/ � 1p2�

C 12x:

Since �.z/ 6 .2�/�1=2, the error in this approximation will beˇˇ 1p

2�C 1

2x � �.x/

ˇˇ 6 1p

2�x2:

We see that the error is of order x2 close to the origin.In Figure 4 on the next page we have plotted the function � together with the two

linear approximations discussed above.We see that x is a good approximation of �.x/ for values of x above 1, and that

the Taylor approximation works well around zero. In Figure 5 on page 81 we haveplotted the relative error (in percent) of the two approximations.

For jxj . 0:25 the relative error of the Taylor approximation is below 5%;5% relative error is achieved for the other linear approximation for x & 1:2.

We compute the approximative CDD futures price using �.x/ � x. Substitutingx for �.x/ in the formula in Proposition 2.1 yields

FCDD.t; �1; �2/

DZ max.t;�1/

�1

max.T .s/ � c; 0/ ds CZ �2

max.t;�1/

˙.t; s/�

�m� .t; s; X.t// � c

˙.t; s/

�ds

�Z max.t;�1/

�1

max.T .s/ � c; 0/ ds CZ �2

max.t;�1/

˙.t; s/m� .t; s; X.t// � c

˙.t; s/ds

DZ max.t;�1/

�1

max.T .s/ � c; 0/ ds CZ �2

max.t;�1/

m� .t; s; X.t// � c ds

DZ max.t;�1/

�1

max.T .s/ � c; 0/ ds CZ �2

max.t;�1/

�.s/ � c ds

C e01A�1.eA.�2�t/ � eA.max.t;�1/�t//X.t/

CZ �2

max.t;�1/

Z s

t

e01eA.s�u/ep�.u/ du ds:






FIGURE 4 The function �.x/ defined in Proposition 2.1 together with its Taylorapproximation and the function x (dotted lines).

1.0 0.5 0 0.5 1.0 1.5 2.0 2.5 3.0–1.0

–0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

x

y(x)

In conclusion, we find that

FCDD.t; �1; �2/ �Z max.t;�1/

�1

max.T .s/�c; 0/ ds C�x.t; �1; �2/Cax.t; �1; �2/X.t/

(3.1)for

ax.t; �1; �2/ D e01A�1.eA.�2�t/ � eA.max.t;�1/�t// (3.2)

and

�x.t; �1; �2/ DZ �2

max.t;�1/

�.s/�c ds CZ �2

max.t;�1/

Z s

t

e01eA.s�u/ep�.u/ du ds: (3.3)

Hence, the approximative CDD futures price can be represented as a linear functionin the coordinates of X.t/.

Recall the seasonal function estimated on the NewYork temperature data. From this,we can obtain mean temperatures for NewYork over a given month. For example, in thesummer period the mean temperature for June is 72:47 ıF, for July it is 76:85 ıF andfor August it is 75:33 ıF. The estimated monthly summer means are all significantlyhigher than the threshold value of c D 65 ıF in the CDD futures, indicating that theapproximation �.x/ � x should work well.






FIGURE 5 The relative error (%) between � and (a) its Taylor approximation and (b) x.

0.5 0 0.5 1.00

5

10

15

20

25

x

Rel

ativ

e er

ror

(%)

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.00

1

2

3

4

5

6

7

8

x

Rel

ativ

e er

ror

(%)

(a)

(b)

In Figure 6 on the next page we have plotted the exact CDD price from the expres-sion in Proposition 2.1 together with the approximated CDD price in (3.1). The mea-surement month of the CDD index is chosen to be August, and we compute the pricesbased on observations of New York temperatures. Note that the first coordinate of thevector X.t/ 2 R3 in the CAR.3/ dynamics is the deseasonalized temperature at timet , the second is the derivative and the third is the second derivative (see Benth andSolanilla Blanco 2014). To find the first and second derivatives, we apply numericaldifferentiation to the daily observations of deseasonalized temperatures. The CDDprices are computed from March 3, 2011 to July 31, 2011, with the market price ofrisk being equal to zero. As we see from Figure 6 on the next page, the approximativeprices (the solid line) are very close to the exact ones. In fact, the maximum relativeerror is less than 0.5%.






FIGURE 6 Forward prices for CDD contracts from March 3, 2011 to July 31, 2011(measurement period: August 2011).

20 40 60 80 100 120 140320

330

340

350

360

370

380

390

400

410

Time t (days), March 3–July 31, 2011

CD

D p

rice

(US

$)

Theoretical prices (dashed line) versus approximated prices (solid line).

Let us investigate the approximation a bit more closely. In the derivation of theapproximative futures price we use that �.x/ � x for x equal to

x D m� .t; s; X.t// � c

˙.t; s/:

We let � be constant (as in the New York data example), which yields

˙2.t; s/ WD ˙2.s � t / D �2

Z s�t

0

.e01eAuep/2 du:

If the random variable X.t/ is stationary, we have

X.t/ D �

Z t

�1eA.t�s/ep dW.s/

as the stationary representation. Hence, X.t/ becomes a p-variate Gaussian variablewith zero mean and variance–covariance matrix

Cov.X.t// D �2

Z 1

0

eAuepe0peA0u du:

Hence, for s > t we have that .m� .t; s; X.t//�c/=˙.s � t / is a normally distributedrandom variable with mean

EQ

�m� .t; s; X.t// � c

˙.s � t /

�D

�.s/ � c CR s

te0

1eA.s�u/ep�.u/ du

˙.s � t /(3.4)






and variance

VarQ

�m� .t; s; X.t// � c

˙.s � t /

�D �2

e01eA.s�t/

R 10

eAuepe0peA0u du eA0.s�t/e1

˙2.s � t /: (3.5)

If s � t ! 1, we have that ˙2.s � t / tends to a constant since A is assumed to haveeigenvalues with negative real part. By the same argument it therefore follows thatthe variance of .m� .t; s; X.t// � c/=˙.s � t / converges to zero as s � t ! 1. Fromthis we conclude that when s is significantly bigger than t ,

m� .t; s; X.t// � c

˙.s � t /�

�.s/ � c CR s

te0

1eA.s�u/ep�.u/ du

˙.s � t /:

On the other hand, if s � t # 0, then the variance tends to 1 and the expected valueto ˙1.

In Figure 7 on the next page we have plotted the expected value in (3.4) for themonth of June, based on the parameters estimated from the New York data and amarket price of risk � D 0. We have chosen t D �1 (the start of the measurementmonth) to be June 1. The mean (the thick solid line) starts around 5, decays to around1, and then increases toward a value slightly below 2.5. There is a lot of uncertaintyfor the first 1–5 days within the measurement period, after which the uncertaintybecomes effectively zero. This is due to the stationarity of the model, of course. Atthe start of the measurement period, the approximation �.x/ � x will work very wellby looking at only the mean; however, the large variations may induce a large error.From around ten days, when the mean is above 1.5 and we have basically no error, theapproximation will be good. There are some days in between when the mean valuewill be around 1 and the approximation may be less good. We integrate over the entiremeasurement period to obtain the CDD price, so large errors at the start may not beas influential as this makes up only a small part of the total price.

Taking t < �1 will make the uncertainty of the expected value smaller, and in factif t � �1, we are basically considering the case with no uncertainty. We note that theexpected value in (3.4) varies with s mostly because the volatility function ˙.t � s/

scales with the seasonality function less the threshold c.The CDD prices for the months of July and August indicate much better agreement

with the approximation �.x/ � x. In Figure 8 on the next page we observe theexpected value in (3.4) for a July contract to be slightly below 2.5 for most of themeasurement period. For CDD prices measured in August, the expected value decays(see Figure 9 on page 85) but is above 2.5 throughout the measurement period. Bothmonths start out with expected values much higher than 2.5 for the first 1–5 days.

Let us turn our attention to HDD futures contracts. For New York, these are tradedin during the period October–April. Based on the estimated seasonal function �.t/






FIGURE 7 The expected value function (solid line) in (3.4) for the month of June as afunction of s � �1, where we have chosen t D �1 D June 1.

5 10 15 20 25–15

–10

–5

0

5

10

15

20

25

Exp

ecte

d va

lue

s – t1 (days)

In addition, we have inserted the bounds for ˙1 standard deviation (dashed line) and ˙2 standard deviations (dottedline).

FIGURE 8 The expected value function (solid line) in (3.4) for the month of July as afunction of s � �1, where we have chosen t D �1 D July 1.

5 10 15 20 25 30

0

5

10

15

20

25

30

Exp

ecte

d va

lue

s – t1 (days)







FIGURE 9 The expected value function (solid line) in (3.4) for the month of August as afunction of s � �1, where we have chosen t D �1 D August 1.

5 10 15 20 25 300

5

10

15

20

25

30

35

Exp

ecte

d va

lue

s – t1 (days)


for New York, we find the monthly mean temperatures to be 46:51 ıF in November,37:59 ıF in December, 33:34 ıF in January, 34:89 ıF in February and 41:62 ıFin March. For the “border months” we have 52:05 ıF for October and 57:76 ıF forApril. As we can see, all the relevant months for HDD futures have mean temperaturessignificantly below the threshold of c D 65 ıF, and it is highly reasonable to applythe approximation �.x/ � x. The approximative HDD futures price based on this istherefore

FHDD.t; �1; �2/ �Z max.t;�1/

�1

max.c �T .s/; 0/ ds C�x.t; �1; �2/Cax.t; �1; �2/X.t/

(3.6)for

ax.t; �1; �2/ D e01A�1.eA.�2�t/ � eA.max.t;�1/�t// (3.7)

and

�x.t; �1; �2/ DZ �2

max.t;�1/

c ��.s/ ds CZ �2

max.t;�1/

Z s

t


CDD futures are also traded for the months of October and April, where we haveseen that the mean temperature is “far” below the threshold c D 65ı. Hence, itis not reasonable to expect the approximation �.x/ � x to work well for CDD






futures measured in these two months. On the other hand, the HDD futures can beapproximated rather well, and by resorting to the CDD–HDD parity in Corollary 2.2we can work out an alternative approximation of the CDD futures for these two monthsas well.

There are months for which the first-order Taylor expansion of �.x/ may providea useful approximation of the futures prices. In the months May and September theaverage daily temperatures are 62:8 ıF and 68:4 ıF, respectively. For these monthsthe daily average temperatures will evolve in the close vicinity of the threshold c D65 ıF. As May and September are months for which CDD futures are traded, it isreasonable to look at an approximation of the CDD futures price based on the Taylorexpansion of �.x/ around zero. The same approximation analysis as above exceptusing �.x/ � 1=

p2� C 0:5x yields

FCDD.t; �1; �2/ �Z max.t;�1/

�1

max.T .s/ � c; 0/ ds C �taylor.t; �1; �2/

C ataylor.t; �1; �2/X.t/ (3.9)

for

ataylor.t; �1; �2/ D 12e0

1A�1.eA.�2�t/ � eA.max.t;�1/�t// (3.10)

and

�taylor.t; �1; �2/ DZ �2

max.t;�1/

12.�.s/ � c/ C 1p

2�˙.t; s/ ds

C 1

2

Z �2

max.t;�1/

Z s

t


In Figure 10 on the facing page we have plotted the exact CDD price from theexpression in Proposition 2.1 together with the approximated CDD price in (3.9). Themeasurement month of the CDD index is chosen to be September, and we computethe prices based on observations of New York temperatures. The plot shows that boththe approximated and exact prices move similarly, but that they are quite far from eachother. The approximation does not work satisfactorily in this case. One could possiblymove to a second-order Taylor expansion of �.x/ to obtain a better approximation.

We remark that our analysis of the accuracy of the approximations is based ontemperature evolution in New York, and the results may be different for other cities.The mean temperatures are obviously different for different locations. As the CDDand HDD futures are traded for locations all over the United States (and, in fact, forother places around the world), the linearizations that we have proposed may fail, orwork, for different months or periods over the year.






FIGURE 10 Forward prices for CDD contracts from March 3, 2011 to August 31, 2011(measurement period September 2011).

20 40 60 80 100 120 140 160 180110

120

130

140

150

160

170

180

CD

D p

rice

(US

$)

Time t (days), March 3, 2011–August 31, 2011

Theoretical prices (dashed line) versus approximated prices (solid line).

Finally, we include an empirical example where we have estimated the marketprice of risk in the theoretical and approximative model that best fits observed for-ward prices from CME. The prices of a NewYork HDD contract with a measurementperiod of March 2012 were available to us. In Figure 11 on the next page we haveplotted (with a solid line) the time series of prices between March 3, 2011 and Febru-ary 27, 2012. Based on nonlinear fitting we obtained a market price of risk O� D �0:102

for the theoretical model, while we found O� D �0:095 for the approximative model.Not unexpectedly, the calibrated market price of risk for the approximative modelis reasonably close to the corresponding value for the theoretical price, although weconsider an HDD contract that is settled in a “border” month in which the approx-imation may fail to work. On the other hand, neither the approximative prices northe theoretical prices (seen as dotted lines in Figure 11 on the next page) explain theobserved forward prices well, as we are near the start of the settlement period. Fur-ther away from the start of the period, we have a good match between the model andmarket prices, but it seems that market prices are much more sensitive to variations inthe underlying temperature than the model is. One may also expect weather forecaststo play a significant role, and these are not accounted for in our theoretical model.One may therefore argue for much more sophisticated pricing measures Q than theones we have introduced in this paper. Another aspect to consider is liquidity, which






FIGURE 11 Forward prices for HDD contracts from March 3, 2011 to February 27, 2012(measurement period March 2012) for an estimated market price of risk.

0 50 100 150 200 250500

550

600

650

700

750

800

March 3, 2011–February 28, 2012

FH

DD

(t, M

arch

201

2)

RealApproximated (θ = 0.1020)Theoretical (θ = 0.0946)

is low for these weather derivatives compared with other financial assets like stocksand commodities.

4 APPLICATION TO THE PRICING OF PLAIN VANILLA OPTIONS

As an application of our approximation of CDD and HDD futures, we include adiscussion on pricing of call options on these futures. Such options are offered fortrade at the CME, and it is therefore highly relevant to have efficient pricing methodsavailable.

To this end, we consider a call option with strike K at exercise time � , written ona CDD futures contract with measurement period Œ�1; �2�. We suppose that exercisetakes place prior to the measurement period, so that � 6 �1, which is the typicalsituation for options traded at the CME. The arbitrage-free price at time t 6 � is

C.t; �; �1; �2; K/ D e�r.��t/EQŒmax.FCDD.�; �1; �2/ � K; 0/ j Ft �; (4.1)

where the constant r > 0 is the risk-free interest rate. We recall from (3.1) and (3.9)that the dynamics of the CDD futures price can be approximated by dynamics of theform

QFCDD.t; �1; �2/ D �.t; �1; �2/ C a.t; �1; �2/X.t/ (4.2)

for t 6 �1. Here, � and a are generic notation referring to �x; �taylor and ax; ataylor,respectively. We find the following.






Proposition 4.1 The price of a call option at time t with strike K and exercise� > t written on a CDD futures contract with measurement Œ�1; �2�, � 6 �1, thathas approximative dynamics FCDD.t; �1; �2/ � QFCDD.t; �1; �2/ defined in (4.2) isQC .t; �; �1; �2; K; QFCDD.t; �1; �2//, where

QC .t; �; �1; �2; K; x/ D e�r.��t/S.t; �; �1; �2/�

�d.t; �; �1; �2; x/ � K

S.t; �; �1; �2/

�

with

d.t; �; �1; �2; K; x/ D x C �.�; �1; �2/ � �.t; �1; �2/ CZ �

t

�.s/a.s; �1; �2/ep ds

and

S2.t; �; �1; �2/ DZ �

t

�2.s/.a.s; �1; �2/ep/2 ds:

We recall that �.x/ D x˚.x/C˚ 0.x/, with ˚ being the cumulative standard normaldistribution function.

Proof First we note that

X.�/ D eA.��t/X.t/ CZ �

t

�.s/eA.��s/ep ds CZ �

t

�.s/eA.��s/ep dW.s/

for t 6 � . Thus,

QFCDD.�; �1; �2/ D �.�; �1; �2/ C a.�; �1; �2/X.�/

D �.�; �1; �2/ C a.�; �1; �2/eA.��t/X.t/

CZ �

t

�.s/a.�; �1; �2/eA.��s/ep ds

CZ �

t

�.s/a.�; �1; �2/eA.��s/ep dW.s/

D �.�; �1; �2/ C a.t; �1; �2/X.t/

CZ �

t

�.s/a.s; �1; �2/ep ds

CZ �

t

�.s/a.s; �1; �2/ep dW.s/

D QFCDD.t; �1; �2/ C �.�; �1; �2/ � �.t; �1; �2/

CZ �

t

�.s/a.s; �1; �2/ep ds

CZ �

t

�.s/a.s; �1; �2/ep dW.s/;






since a.�; �1; �2/eA.��s/ D a.s; �1; �2/ for t 6 s 6 � . Since the Ito integral in the lastterm above is independent of Ft , we find by the Ft -measurability of QFCDD.t; �1; �2/

that

EQŒmax. QFCDD.�; �1; �2/ � K; 0/ j Ft �

D EQ

�max

�d.t; �; �1; �2; QFCDD.t; �1; �2//

CZ �

t

�.s/a.s; �1; �2/ep dW.s/ � K

� ˇˇ Ft

�

D EŒmax.d.t; �; �1; �2; x/ � K C S.t; �; �1; �2/Z; 0/�jxD QFCDD.t;�1;�2/

for a standard normally distributed random variable Z. Here we used the factthat

R �

t�.s/a.s; �1; �2/ep dW.s/ is normally distributed with variance given by

S2.t; �; �1; �2/. A straightforward calculation using properties of the normal distribu-tion completes the proof. �

We observe that the option price becomes explicitly dependent on the current(approximative) CDD futures price. The pricing formula will be a version of thefamous Black-76 formula (see Black 1976) for the price of a call option on a futurescontract when the underlying dynamics is a linear Brownian model rather than a geo-metric Brownian motion. Further, we also allow for time-dependent volatility, leadingto the term S.t; �; �1; �2/ in the pricing formula. Due to the very complex nature ofthe CDD futures price FCDD.�; �1; �2/, it is hard to derive an analytical pricing for-mula for call options, and one must resort to numerical procedures to find a price.The approximative formula that we have derived in the above proposition thereforeprovides an attractive alternative for efficient pricing.

If we consider the approximative case �.x/ � x, we have that �.t; �1; �2/ D�x.t; �1; �2/. A straightforward computation of the involved integrals shows that

d.t; �; �1; �2; x/ D x C �x.�; �1; �2/ � �x.t; �1; �2/ CZ �

t

ax.s; �1; �2/ep�.s/ ds:

The Taylor case becomes slightly more involved, and we leave the derivation of thecorresponding expression to the interested reader.

5 CONCLUSIONS AND OUTLOOK

We have analyzed a linear approximation of the HDD and CDD temperature futuresprice, and we have demonstrated both theoretically and empirically that such approxi-mations work well in several cases. Our investigations are based on a continuous-timeautoregressive model with seasonal mean estimated for temperature data observed in






New York. For this city we find a satisfactory fit of the approximative CDD pricesfor the summer months, while for autumn months there appears to be a larger error.Our results tell us that one may in many cases price call and put options by resortingto the approximative linear price dynamics, for which we compute a “Black-76-like”pricing formula. We therefore avoid numerical pricing, and the option’s Greeks areeasily available.

When comparing the theoretical and approximative prices with real data, weobserve a large difference, which may be explained by liquidity issues, price sen-sitivity to weather forecasts and more complex market-price-of-risk structures. Webelieve that liquidity is a serious issue in this market, and we argue that our analysishas validity for benchmarking purposes.

In future studies we would like to study more general pricing measures Q that willbe more sensitive to variations in underlying temperature variations (see the generalchange of measure in Benth and Šaltyte Benth (2012)). We believe that this will resultin better calibration to actual observed prices, while still preserving the validity ofthe approximation studies discussed in this paper. By introducing weather forecasts,we can obtain an even better fit to actual futures prices.


Financial support from the project “Energy Markets: Modelling, Optimization andSimulation” (EMMOS), funded by the Norwegian Research Council, is gratefullyacknowledged.

REFERENCES

Andresen, A., Benth, F. E., Koekebakker, S., and Zakamoulin, V. (2014). The CARMAinterest rate model. International Journal of Theoretical & Applied Finance 17(2),1450008.

Benth, F. E., and Šaltyte Benth, J. (2012). Modelling and Pricing in Financial Markets forWeather Derivatives. World Scientific.

Benth, F. E., and Solanilla Blanco, S. A. (2014). Forward prices in markets driven bycontinuous-time autoregressive processes. In Recent Advances in Financial Engineer-ing: Proceedings of the International Workshop on Finance 2012, Tokyo MetropolitanUniversity, Takahashi, A., Muromachi,Y., and Shibata, T. (eds), pp.1–24.World Scientific.

Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics 3,167–179.

Brockwell, P.J. (2001).Continuous-time ARMA process. In Handbook of Statistics:Stochas-tic Processes, Theory and Methods, Rao, C. R., and Shanbhag, D. N. (eds), pp. 249–276.Elsevier.

Garcia, I., Klüppelberg, C., and Müller, G. (2010). Estimation of stable CARMA models withan application to electricity spot prices. Statistical Modelling 11(5), 447–470.






Härdle, W., and Lopez Cabrera, B. (2012).The implied market price of weather risk.AppliedMathematical Finance 19(1), 59–95.

Paschke, R., and Prokopczuk, M. (2010). Commodity derivatives valuation with autore-gressive and moving average components in the price dynamics. Journal of Bankingand Finance 34(11), 2741–2752.






Research Paper

Facilitating appropriate compensation ofelectric energy and reserve throughstandardized contracts with swing

Deung-Yong Heo1 and Leigh S. Tesfatsion2

1Korea Institute of Local Finance, 14, Gukhoe-daero 76ga-gil, Yeongdeungpo-gu, Seoul,07239, Korea; email: [email protected] of Economics, 260 Heady Hall, Iowa State University, 518 Farmhouse Lane,Ames, IA 50011-1070, USA; email: [email protected]

(Received January 26, 2015; revised March 29, 2015; accepted June 15, 2015)

ABSTRACT

Three key issues have arisen for centrally managed wholesale electric power marketsin Europe and the United States as they attempt to handle an increased penetrationof variable energy resources. First, rigid definitions for energy and reserve productsmake it difficult to ensure appropriate compensation for important required flexibilityin start-up times, ramp rates, power dispatch levels and duration. Second, participationrestrictions hinder the achievement of an even playing field for potential providersof flexible services. Third, reliance on out-of-market compensation for the provi-sion of some valued services encourages strategic manipulation. This study examinesthe possibility of addressing these three issues through the introduction of standard-ized energy and reserve contracts with swing (flexibility) in their contractual terms.Concrete examples are used to demonstrate how the trading of these standardized con-tracts can be supported by linked forward markets in a manner that permits efficient

Corresponding author: L. S. Tesfatsion Print ISSN 1756-3607 j Online ISSN 1756-3615Copyright © 2015 Incisive Risk Information (IP) Limited

93




94 D.-Y. Heo and L. S. Tesfatsion

real-time balancing of net load subject to system and reserve-requirement constraints.Comparisons with existing wholesale electric power markets are given, and key policyimplications are highlighted.

Keywords: electric power markets; variable energy resources; standardized contracts; swing(flexibility); energy and reserve co-optimization; linked forward markets.

1 INTRODUCTION

European and US electricity sectors have undergone substantial restructuring overthe past twenty years. They have devolved from highly regulated systems operatedby vertically integrated utilities to relatively decentralized systems based more fullyon market valuation and allocation mechanisms.

As part of this restructuring, oversight agencies have been established at severaldifferent levels to encourage cooperation and coordination. The European Networkof Transmission System Operators for Electricity (ENTSO-E), founded in 2008, cur-rently consists of forty-one transmission system operators (TSOs) from thirty-fourEuropean countries; its primary task is to promote the coordinated management of theEuropean power grid (ENTSO-E 2015). The US Federal Energy Regulatory Com-mission (FERC) oversees the activities of six of the seven US independent systemoperators (ISOs), established since the mid-1990s, that manage power system oper-ations in electric energy regions comprising approximately 60% of US generatingcapacity (US Energy Information Administration 2015).1

These restructuring efforts have been driven by a desire to ensure efficient energyproduction and utilization, reliable energy supplies, affordable energy prices andeffective rules and regulations for environmental protection. In keeping with the lattergoal, a dramatic change is taking place in energy mixes: namely, a rapid penetration ofvariable energy resources combined with a movement away from traditional thermalgeneration.

Variable energy resources (VERs) are renewable energy resources, such as windand solar power, whose generation cannot be closely controlled to match changes inload or to meet other system requirements. Consequently, the integration of VERstends to increase the volatility of net load (ie, load minus as-available generation)as well as the frequency of strong ramp events. Flexibility in service provision byother types of resources then becomes increasingly important in order to maintain thereliability and efficiency of power system operations.

1 One US ISO, the Electric Reliability Council of Texas (ERCOT), is not under FERC jurisdictionbecause its grid has been deliberately designed to avoid interstate commerce transactions that wouldsubject it to US Federal jurisdiction (Spence and Bush 2009).





Compensation of energy and reserve through swing contracts 95

To accommodate increased VER penetration, TSOs and ISOs have introducedmajor changes in their market rules and operational procedures (Ela 2011; ENTSO-E2014; Hand et al 2012; Henry et al 2014). These changes have included new productsto enhance net load following capability (eg, ramping products), revised market eli-gibility requirements to encourage greater VER participation, and the introduction ofcapacity markets in an attempt to ensure sufficient thermal generation as a backstopfor the intermittency of VER generation.

Nevertheless, several key issues arising from increased VER penetration still needto be resolved. One is that energy and reserve products are variously defined andcompensated across the different energy regions (see, for example, Ellison et al 2012).This makes it difficult to compare and evaluate the efficiency and fairness of systemoperations across these regions.

A second key issue is appropriate compensation for flexibility in service provision.TSO/ISO product definitions are specified in broad rigid terms (eg, capacity, energy,ramp rate, regulation, non-spinning reserve) that do not permit resources to be furtherdifferentiated and compensated on the basis of additional valuable flexibility in serviceprovision, such as an ability to ramp up and down between minimum and maximumvalues over very short time intervals.

A third key issue is that attempts to accommodate new products have led to theintroduction of out-of-market (OOM) compensation processes. In 2011 the FERCissued Order 755 to address OOM payment problems for one particular product cate-gory in US ISO-managed wholesale power markets: namely, regulation with differingabilities to follow electronic dispatch signals (FERC 2011). However, given its lim-ited scope, Order 755 does not fully eliminate the need in these markets to resort toOOM processes. As stressed by Bushnell (2013), the additional complexity result-ing from OOM compensation processes provides increased opportunities for marketparticipants to gain unfair profit advantages through strategic behaviors.

In response to these issues, a group of researchers sponsored by Sandia NationalLaboratories prepared a report (Tesfatsion et al 2013) recommending that energy andreserve contracts be standardized in firm and option forms permitting separate pricingfor service availability and for real-time service performance, and that the trading ofthese contracts be supported by a linked sequence of forward markets whose designis also standardized. This report builds on important earlier work by Bidwell (2005),Bunn (2004), Chao and Wilson (2002) and Oren (2005), who stress the relevance ofoptions and two-part pricing for electricity markets.

The current study uses concrete numerical examples to explore the policy impli-cations of the recommendations in Tesfatsion et al (2013). In Section 2 we present ageneral template for a standardized contract (SC) with swing (flexibility) in its con-tractual terms, together with an illustrative SC example. We also outline in broad termshow the trading of SCs can be supported by linked centrally managed day-ahead and






real-time markets. In Sections 3 and 4 we present our main results, namely, examplesdemonstrating how our proposed SC system, implemented via linked day-ahead andreal-time markets, permits efficient real-time balancing of net load subject to systemand reserve-requirement constraints.

Comparisons of our proposed SC system with existing European and US wholesalepower market operations, standardized power contracts, pricing mechanisms andVERinitiatives are provided in Sections 5.1–5.4. In Section 5.5 we discuss how our SCsystem provides a robust-control approach to the handling of uncertain net load thatavoids the need to specify detailed scenarios with associated probabilities, a commonrequirement of standard stochastic control approaches. In Section 5.6 we conjecturehow our proposed SC system, extended to longer-term forward markets, could helpto provide better incentives for thermal generation capacity investment as a backstopfor the intermittency of VER generation by facilitating the resolution of merit-orderand missing-money problems.

Throughout Sections 2–5 the following key policy implications of our proposedSC system are highlighted.

(1) It permits full market-based compensation for availability and performance.

(2) It facilitates a level playing field for market participation.

(3) It facilitates co-optimization of energy and reserve markets.

(4) It supports forward-market trading of energy and reserve.

(5) It permits service providers to offer flexible service availability.

(6) It provides system operators with real-time flexibility in service usage.

(7) It facilitates accurate load forecasting and following of dispatch signals.

(8) It permits resources to internally manage unit commitment and capacityconstraints.

(9) It permits the robust-control management of uncertain net load.

(10) It eliminates the need for OOM payment adjustments.

(11) It reduces the complexity of market rules.

The concluding section (Section 6) provides a concise summary discussion of eachof these policy implications.






FIGURE 1 Hierarchical structure of contracts.

Swing OC

Swing FC Fixed OC

Fixed FC

Block energy

2 PROPOSED STANDARDIZED CONTRACT SYSTEM

2.1 General form of a standardized contract

“Energy” refers to the actual generation of electrical energy, whereas “reserve” refersto generation-capacity availability. Four standardized contracts to facilitate energyand reserve trading are proposed in Tesfatsion et al (2013): firm contracts (FCs) andoption contracts (OCs) taking either fixed or swing form.

An FC is a noncontingent contract that requires specific performance from bothcounterparties. It obligates the holder to procure services from the issuer, and theissuer to deliver these services, under the contractually specified terms of the FC. Incontrast, an OC gives the holder the right, but not the obligation, to procure servicesfrom the issuer under contractually specified terms. The right can be activated byexercise of the OC at a contractually permitted exercise time. Once exercised, an OCimposes specific performance obligations on both counterparties. That is, as for anFC, an exercised OC obligates the holder to procure services from the issuer, and theissuer to deliver these services, under the contractually specified terms of the OC.

An FC or OC is a fixed contract if each of its contractual terms is designatedas a single possible value. An FC or OC is a swing contract if at least one of itscontractual terms is designated as a set of possible values, thus permitting somedegree of flexibility in its implementation. A fixed FC is a block-energy contract ifits contractual terms obligate the issuer to maintain a specified constant power levelduring a specified time interval. As depicted in Figure 1, fixed/swing OCs, fixed/swing FCs and block-energy contracts are all special cases of swing OCs.

Hereafter, this study focuses on standardized contracts in swing-OC form for theflexible provision of energy and reserve services. For concreteness, we next present atemplate for a standardized contract (SC) that provides seven basic types of servicesfor a particular operating hour: delivery location, down/up direction, exercise time,






power-begin time, power-end time, down/up ramp rate, and power level. We illustrateswing in five of these service types by depicting their sets of possible values asintervals.2

2.1.1 Template for a standardized contract

SC D Œk; d; Tex; Tpb; Tpe; RC; PC; ��; (2.1)

where

� k D the location in which service delivery is to occur;

� d D the direction (down or up);

� Tex D Œtminex ; tmax

ex � is the range of possible exercise times tex;

� Tpb D Œtminpb ; tmax

pb � is the range of possible power-begin times tpb;

� Tpe D Œtminpe ; tmax

pe � is the range of possible power-end times tpe;

� RC D Œ�rD; rU� is the range of possible down/up ramp rates r ;

� PC D Œpmin; pmax� is the range of possible power levels p;

� � is the performance payment method for real-time service performance.

The down/up limits �rD and rU for the ramp rates r (MW/min) are assumed tosatisfy �rD 6 0 6 rU. The lower bound pmin for the power levels p (MW) is assumedto be nonnegative. The direction (down or up) of an SC determines whether thesepower levels describe power curtailments or absorptions (down) or power injections(up). The time points tex, tpb and tpe denote specific calendar times expressed at thegranularity of minutes.

The presence of swing in the contractual terms of an SC permits it to function asboth an energy product and a reserve product. Actual real-time service performanceunder such an SC cannot be determined until after the end of the operating hour H

even if the SC is a firm (nonoptional) contract. Consequently, the contractual terms ofan SC include a performance payment method � to be used to determine the ex postpayment to the SC issuer for real-time service performance (if any).

The performance payment method � can take a wide variety of forms. For example,as illustrated in Section 3, � might denote a prespecified price (US$/MWh) for deliv-ered down/up energy. More generally, � could denote a contingent price for delivered

2 SCs can take much more general forms than illustrated in the current study. For example, SCs caninclude other types of services, such as voltage control, reactive power support and energy storagecapacity; swing can be present in any of these services; swing possible value sets do not need to bein interval form; the operating period does not need to be an hour.






down/up energy that depends on market conditions (eg, fuel prices) at the time ofthe delivery. Alternatively, � could provide for the compensation of delivered powermeasured as mileage, ie, as the sum of absolute-value up and down power movementsover the real-time dispatch interval, a metric now being used for regulation serviceperformance in many energy markets to meet the requirements of FERC Order 755(Beacon Power 2014).

In order for an SC to be implementable, its contractual terms must satisfy certainbasic requirements. For example, tmin

pb cannot exceed tmaxpe . In this study it is presumed

that an SC issuer is responsible for ensuring that it can feasibly implement the terms ofany SC it offers. Realistically, however, penalties and eligibility requirements mightneed to be introduced to help ensure that the issuers of cleared SCs accurately followreal-time dispatch instructions, and that these instructions are in accordance with thecontractual terms of the cleared SCs. These contract enforcement mechanisms couldconstitute part of the performance payment method � included within each SC, orthey could be instituted at the level of the power system as a whole.

2.2 Illustrative example of a standardized contract

The illustrative up-energy SC depicted in Figure 2 on the next page provides a com-bination of fixed and swing attributes. The delivery location (bus k) and direction(up) are specified as single values, as are the exercise time tex, the power-begin timetpb and the power-end time tpe. On the other hand, the down/up ramp rate r and thepower level p are swing attributes that can be varied over a range of values.

The darker area within the resulting corridor of contractually admissible powerdispatch paths depicted in Figure 2 on the next page is the up-energy injection thatresults from one such path. Any actual up-energy injection is compensated ex postin accordance with the performance payment method � included among the SC’scontractual terms. An example of a down-energy SC can be obtained from Figure 2by considering a 180ı rotation of the depicted figure around the time axis.

The SC depicted in Figure 2 on the next page can be concretely interpreted as anup-energy SC offered by a demand response resource (DRR) into an ISO-managedday-ahead market (DAM) on day D � 1 for a particular operating hour H on day D,as follows. Consider a load serving entity (LSE) functioning as a load aggregator fora large distribution feeder connected to the transmission grid at a particular bus k.Residential households on this feeder have smart meters for their heating, ventilationand air conditioning (HVAC) loads in wireless communication with the LSE thatpermits the LSE to make adjustments to these loads. The LSE has permission fromeach of these households to make small adjustments in their HVAC energy usage inreturn for an agreed monthly lump-sum compensation. The LSE can participate in the






FIGURE 2 Example of an SC for up-energy with ramp-rate swing and power-level swingthat is offered at bus k by a generator with a maximum capacity of 70 MW.

MW

pmax = 50

pmin = 10

tex tpb tpe

70

Up

Powersystem

gridBus k'

Bus k

Capacity

Timeφ

r U

r D

Contractprocurement

time

DAM as a DRR either by offering up-energy implemented via HVAC load reductionsor by offering down-energy implemented via HVAC load increases.

Suppose the LSE participates in the DAM on day D � 1 as a DRR by offering thefollowing up-energy SC at some offer price v for hour H of day D, where hour H isthe time interval between 13:00 EST and 14:00 EST.

� Delivery location D bus k.

� Direction D up.

� Tex D exercise time tex D 09:00 EST on day D.

� Tpb D power-begin time tpb D 13:00 EST on day D.

� Tpe D power-end time tpe D 14:00 EST on day D.

� RC D Œ�1:3 MW/min; C1:4 MW/min� D range of possible down/up ramprates r .

� PC D Œ10 MW; 50 MW� D range of possible power levels p.

� � D payment method for compensation of delivered power mileage, includinga penalty payment adjustment for deviations between instructed and actualpower mileage.






Suppose also that this SC is cleared by the ISO. The ISO is then obligated to ensure thatthe DRR receives in compensation its offer price v as payment for making availablefor hour-H operations on day D the services included in this SC. In turn, the ISO hasthe right, but not the obligation, to exercise this SC at 09:00 EST on day D.

If the SC is exercised, the DRR must be ready to follow any electronic dispatchsignal on day D, starting at time tpb D 13:00 EST and ending at time tpe D 14:00 EST,that calls for the DRR to provide a path of power injections lying within its offeredrange PC of power levels that can feasibly be achieved without violating the DRR’soffered range RC of down/up ramp rates. In turn, the ISO is obligated to ensure thatthe DRR is compensated ex post for the mileage of this controlled power path inaccordance with the terms of the performance payment method �.

2.3 Support of SC trading via linked forward markets

As in Tesfatsion et al (2013), we propose that SC trading be supported by a sequenceof linked centrally managed forward markets whose planning horizons can range fromminutes to years. For concreteness, however, we focus in this study on the supportof SC trading by means of linked day-ahead and real-time markets that are centrallymanaged by a nonprofit ISO (see Figure 3 on the next page).

The non-ISO participants in our proposed day-ahead market (DAM) and real-timemarket (RTM) include

(i) load-serving entities (LSEs), who submit SC demand bids in the form of blockenergy contracts on behalf of retail energy customers;

(ii) dispatchable generation companies (GenCos), demand response resources(DRRs) and energy storage devices (ESDs) who submit SC supply offers; and

(iii) nondispatchable VERs whose as-available generation is treated as negativeload.3

The requirement that LSE SC demand bids be in block-energy form avoids the needfor LSEs to exercise load-balancing discretion in the implementation of SCs withswing or option exercise times.

Participation in our proposed DAM/RTM processes is not meant to preclude elec-tricity traders from procuring physical and financial instruments in power exchangesand over-the-counter power markets to hedge their price and volume risks. However,

3 As discussed in Section 5.4, our proposed SC system could be generalized to allow designatedtypes of VERs to offer their generation as “dispatchable intermittent resources” in DAM/RTMoperations, as permitted in MISO (2011). However, this would raise a number of issues best leftfor future studies, eg, should VERs be charged or penalized the same as ordinary dispatchablegeneration for deviations from their cleared dispatch offers?






FIGURE 3 Proposed ISO-managed day-ahead and real-time markets.

Day-aheadmarket(DAM)

Real-timemarket(RTM)

LSEs SC block-energybids

Disp. GenCos,DRRs and ESDs

LSE SC bids;Disp. GenCo/

DRR/ESDSC offers;

ISO SC bids

Security-constrainedunit commitment

(SCUC) and security-constrainedeconomic dispatch

(SCED)

Disp. GenCo/DRR/ESDSC offers;

ISO SC bidsSCED

Disp. GenCos,DRRs and ESDs

SC offers

SC offers

Nondisp. VERs

Nondisp. VERs

ISO

ISO

SC bids

SC bids

physical instruments whose terms require the use of transmission line facilities mustbe self-scheduled and cleared in the DAM or RTM to ensure transmission availabilityand overall system reliability.

The ISO managing the DAM undertakes security-constrained unit commitment(SCUC) and security-constrained economic dispatch (SCED) conditional on LSE SCdemand bids, ISO SC demand bids (for reserve procurement only) and SC supplyoffers from dispatchable GenCos, DRRs and ESDs. To retain the ISO’s nonprofitstatus, all costs incurred by the ISO for SC procurement must be passed through tomarket participants.

This cost pass-through could simply require all procurement costs to be allocatedto the LSEs in proportion to their share of real-time loads. However, the presenceof performance payment methods � in SC bids/offers permits more sophisticatedarrangements. For example, an LSE’s cost allocation could be based in part on itsforecasting performance, measured ex post by comparing its cleared SC demand bidsagainst the actual real-time loads of its customers; and an SC supplier’s cost allocationcould be based in part on the accuracy of its service performance, measured ex postby examining how well it was able to follow real-time dispatch instructions.






The ISO’s DAM SCUC/SCED objective is to minimize the expected total net costof ensuring that sufficient generation is available to balance next-day forecasted netloads with suitable local and system-wide reserve buffers. Dispatchable generationavailability is determined from dispatchable GenCo, DRR and/or ESD supply offers.Next-day net load forecasts for power-balance purposes are determined from LSE SCdemand bids and forecasted VER generation. Reserve buffers are ensured by ISO SCdemand bids.

As usual, the DAM SCUC/SCED is subject to unit commitment (UC) conditions,generation-capacity limits, power-balance constraints, transmission-line limits andboth local and system-wide reserve-requirement constraints. However, the impositionof the UC conditions and generation-capacity limits occurs through the contractualterms of the DAM SC supply offers rather than through ISO-imposed constraints.

We also propose an ISO-managed RTM that runs a SCED every five minutes.Dispatchable GenCos, DRRs and ESDs can offer SCs into the RTM. Only the ISOis permitted to procure these SCs, for balancing and reserve procurement purposes;and all ISO RTM procurement costs must be passed through to market participantsin order to preserve the nonprofit status of the ISO.

The ISO’s RTM SCED objective is to minimize the expected total cost of ensur-ing that adequate generation is available to balance ISO-forecasted real-time netloads with suitable local and system-wide reserve buffers, given the existing inven-tory of previously cleared SCs. This RTM SCED is subject to generation-capacitylimits, power-balance constraints, transmission-line limits and both local and system-wide reserve-requirement constraints. The imposition of the generation-capacity lim-its occurs through the contractual terms of the RTM SC supply offers rather thanthrough ISO-imposed constraints.

SCs can provide diverse services through their contractual terms. As discussedin greater detail in Section 5.3, appropriate compensation for these diverse servicesrequires a flexible pricing mechanism. Our DAM and RTM are therefore formulatedas discriminatory-price auctions in which participants pay (or are paid) their bid/offerprices for cleared SCs. These bid/offer price payments are compensations for serviceavailability. Any real-time service performance rendered through these cleared SCs iscompensated ex post in accordance with the performance payment methods appearingamong the contractual terms of the cleared SCs.

Finally, SCs with swing in their contractual terms can function as both energy andreserve, and SCs in option form can also function as reserve even if their contractualterms are fixed. Consequently, our proposed DAM and RTM intrinsically involve aco-optimization of energy and reserve.

The next two sections use concrete examples to demonstrate how SC trading canbe supported by means of our proposed linked DAM and RTM processes in a way






FIGURE 4 ISO-forecasted net load profile for hour H of day D at the start of the RTM.

0

500

1000

1500

2000

2500

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Time (min)

Pow

er (

MW

)

that ensures optimal balancing of real-time net loads subject to system and reserve-requirement constraints.

3 REAL-TIME MARKET ILLUSTRATIVE EXAMPLE

3.1 Overview

Sections 3.2–3.7 present a numerical example illustrating how SC trading can be sup-ported by means of an RTM in the absence of transmission congestion and withoutconsideration of linkages to earlier DAM processes. The handling of RTM transmis-sion congestion is addressed in Section 3.8, and linkages with earlier DAM processesare considered in Section 4.

3.2 Basic assumptions

Suppose an RTM takes place immediately prior to a particular operating period forwhich no congestion is anticipated. For concreteness, we assume this operating periodis a particular hour H on a particular day D, expressed at the granularity of minutes.






Net load for hour H consists of aggregate load minus aggregate VER as-availablegeneration. The net load profile for hour H that the ISO forecasts at the start of theRTM takes the form given in Figure 4 on the facing page. The objective of the ISOmanaging the RTM is to ensure that this forecasted net load profile is balanced bygeneration with an appropriate reserve buffer, keeping costs to a minimum. The ISOattempts to achieve this objective by procuring a suitable combination of SCs fromdispatchable generation suppliers participating in the RTM.

These dispatchable suppliers are assumed to consist of three GenCos with thefollowing ramp-rate attributes and generation-capacity attributes, expressed in Sec-tion 2.1 notation:

G1 W rD1 D rU

1 D 120 MW/min; Capmin1 D 0 MW; Capmax

1 D 600 MW;

G2 W rD2 D rU


2 D 700 MW;

G3 W rD3 D rU


3 D 900 MW:

Each of these GenCo offers into the RTM a collection of portfolios, called GenPorts,together with associated GenPort offer prices. A GenPort consists of one or more SCswhose terms the GenCo could simultaneously fulfill during hour H if called upon todo so by the ISO. The ISO can clear at most one GenPort from each GenCo in theRTM.

The offer price vi;j for GenPorti;j is the payment requested by Gi for guaranteeingit will be available in hour H to fulfill the terms of the SCs included in GenPorti;jif signalled to do so. Thus, vi;j compensates Gi for service availability costs, suchas avoidable fixed costs and lost opportunity costs. In addition, assuming GenPorti;j

is cleared by the ISO, Gi will also receive performance payments for any servicesit renders during hour H under the contractual terms of the SCs in GenPorti;j . Anysuch performance payments will be determined in accordance with the performancepayment methods � included among the contractual terms of the SCs in GenPorti;j .For the example at hand, each of these performance payment methods � is assumedto take the form of a prespecified price (US$/MWh) for delivered down/up energy.4

As clarified in subsequent sections, this two-part pricing scheme permits theGenCos to ensure the recovery of their expected total costs through a market pro-cess, taking into account their local attributes and conditions. It also permits the ISOto closely tailor the cleared RTM GenPorts to real-time needs for net load balancingsubject to system and reserve-requirement constraints.

The ISO is permitted to clear at most one GenPort from each GenCo in the RTM.The resulting cleared GenPorts can thus be represented in the following ISO portfolio

4 For example, each SCi;j;m in GenPorti;j could correspond to a distinct generation unit m ownedby Gi , and the performance payment method �i;j;m for SCi;j;m could be a down/up energy price(US$/MWh) given by the expected next-day marginal dispatch cost for unit m.






(ISOPort) form:

ISOPort D fGenPort1; GenPort2; GenPort3g; (3.1)

where no procurement from a GenCo Gi (GenPorti D none) is possible.

3.3 RTM supply offer specifications

A GenCo’s RTM supply offer is a collection of GenPorts together with associatedGenPort offer prices. Suppose each GenCo offers up-energy in firm contract form,ie, exercise time tex D tmin

ex D tmaxex D contract procurement time. Suppressing

location (k), direction (up), the exercise time tex and measurement units from SCrepresentations for ease of exposition, the RTM supply offers of GenCos G1, G2 andG3 are assumed to take the following form.

G1’s supply offer consists of two GenPorts, each with one SC:

GenPort1;1 D fSC1;1g at offer price v1;1;

SC1;1 D Œtpb D 0; tpe D 60; jr j 6 100; 0 6 p 6 500; � D 100�I (3.2)

GenPort1;2 D fSC1;2g at offer price v1;2;

SC1;2 D Œtpb D 0; tpe D 60; jr j 6 120; 0 6 p 6 500; � D 105�: (3.3)

G2’s supply offer consists of three GenPorts with multiple SCs:

GenPort2;1 D fSC2;1;1; SC2;1;2g at offer price v2;1;

SC2;1;1 D Œtpb D 10; tpe D 20; jr j 6 200; 0 6 p 6 600; � D 135�;

SC2;1;2 D Œtpb D 30; tpe D 60; jr j 6 200; 0 6 p 6 600; � D 130�I (3.4)

GenPort2;2 D fSC2;2;1; SC2;2;2; SC2;2;3g at offer price v2;2;







SC2;3;3 D Œtpb D 30; tpe D 60; jr j 6 200; 0 6 p 6 700; � D 135�: (3.6)






G3’s supply offer consists of two GenPorts with multiple SCs:








SC3;2;3 D Œtpb D 44; tpe D 54; jr j 6 200; 0 6 p 6 400; � D 150�: (3.8)

3.4 Power-balance constraints for ISOPorts

Any ISOPort cleared by the ISO in the RTM must permit the achievement of azero-balance gap (ZBG), ie, an exact balancing of RTM-cleared generation againstthe ISO’s forecasted hour-H net load profile in Figure 4 on page 104. As demon-strated in Heo and Tesfatsion (2015), each of the following three ISOPorts enablesthe achievement of a ZBG:

ISOPort1 D fGenPort1;1; GenPort2;2; GenPort3;1g; (3.9)

ISOPort2 D fGenPort1;1; GenPort2;3; GenPort3;1g; (3.10)

ISOPort3 D fGenPort1;2; GenPort2;3; GenPort3;2g: (3.11)

For example, the achievement of a ZBG by ISOPort2 in (3.10) is depicted in Figure 5on the next page. Each color in the figure indicates the dispatch of generation froma particular GenPort for a particular GenCo, and different shades of the same colorindicate the dispatch of generation from distinct SCs within a particular GenPort.

3.5 Expected total cost of a power-balanced ISOPort

Consider any ISOPort D .GenPort1; GenPort2; GenPort3/ that achieves a ZBG forhour H . The expected total cost of this ISOPort is the sum of payments arising fromtwo sources:

(i) the portfolio offer prices fv1; v2; v3g that must be paid to GenCos G1–G3 forthe procurement of GenPort1, GenPort2 and GenPort3; and

(ii) the total performance payments the ISO expects it will have to make to G1–G3for down/up energy delivery during hour H under the contractual terms of theseconstituent GenPorts in order to achieve the ZBG.






FIGURE 5 Zero-balance gap achieved by ISOPort2 for hour H of day D.

3 7 11 15 19 23 27 31 35 39 43 47 51 55 590

500

1000

1500

2000

2500

Pow

er (

MW

)

Time (min)

SC1,1

SC2,3,2

SC3,1,1SC3,1,2 SC3,1,3

SC2,3,3

GenPort1,1 GenPort2,3 GenPort3,1

For example, to calculate the expected total performance payments (ii) implied bythe ZBG implementation of ISOPort2 depicted in Figure 5, first measure the energy(MWh) for each of the areas in the figure with a distinct color shading; each such areacorresponds to a distinct SC implementation. Next, multiply each of these energyamounts by the performance price � (US$/MWh) included among the contractualterms of the corresponding SC. Finally, add up all of these amounts.

3.6 Reserve inherent in a power-balanced ISOPort

The achievement of a ZBG by an ISOPort implies that the generation available throughthis ISOPort is capable of balancing the ISO’s forecasted hour-H net load profile.However, if the SCs constituting this ISOPort include swing, then the ISOPort canalso achieve a ZBG for a range of hour-H net load profiles that deviate from theISO’s forecasted hour-H net load profile. Hereafter, this range will be referred to asthe reserve range (RR) of the ZBG ISOPort.

The RR of a ZBG ISOPort with swing in its contractual terms is a robust-controldevice for ensuring net load balancing, eliminating the need for the ISO to considerdetailed net load scenarios and scenario probabilities. However, its exact form dependsin a complicated manner on the particular attribute specifications of the SCs that






FIGURE 6 Reserve range RR for ISOPort2 during hour H of day D.

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59Time (min)

0

500

1000

1500

2000

2500

Pow

er (

MW

)

LoadRRmax RRmin

constitute the ISOPort as well as on the minute-by-minute operating state of theGenCo suppliers, ie, the GenCos that have offered these SCs. Consequently, in anypractical application the RR will have to be approximated.

For example, Figure 6 plots an approximate RR for ISOPort2 in (3.10) obtained byassuming that the GenCo suppliers at the start of each minute M are at their ZBG-generation levels (see Heo and Tesfatsion (2015, Section 3.6) for a detailed derivation).The depicted approximate RR is conditional on the ISO’s forecasted hour-H net loadprofile shown in Figure 4 on page 104 and on the ISO’s hour-H ZBG implementationfor ISOPort2 shown in Figure 5 on the facing page.

3.7 Practical determination of optimal ISOPorts

Let L D fLM j 1 6 M 6 60g denote the ISO-forecasted aggregate net load profilefor hour H depicted in Figure 4 on page 104, expressed at the granularity of minutesM . Suppose the ISO’s system-wide requirements for down/up reserve during H canbe expressed in terms of the following restrictions on the lower and upper bounds ofthe reserve range RR corresponding to any ZBG ISOPort cleared to balance L, where˛� D .˛D�; ˛U�/ > 0: for each minute M of hour H ,

RRminM 6 Œ1 � ˛D��LM 6 Œ1 C ˛U��LM 6 RRmax

M : (3.12)






Suppose at least one feasible ISOPort achieves a ZBG for H . Then, as detailed inHeo and Tesfatsion (2015, Section 3.7), the ISO can formulate its RTM optimizationproblem as a multi-criteria optimization problem with three lexicographically orderedobjectives:

(i) to ensure a ZBG;

(ii) to ensure system-wide RR reliability at level ˛�, ie, satisfy condition (3.12) forthe aggregate net load profile L; and

(iii) to minimize the expected total cost of ensuring (i) and (ii).

3.8 Incorporation of transmission-line limits

Until now, our RTM illustrative example has assumed the absence of transmissioncongestion. This simplification has permitted us to focus solely on the economicdispatch problem of ensuring a balance between aggregate dispatched generation andISO-forecasted aggregate net load, subject to a system-wide RR˛� constraint (3.12).

In Heo and Tesfatsion (2015, Section 3.8) we extend this RTM example to permitcongested transmission lines. To ensure reliability, we assume the ISO imposes aZBG constraint at each bus, referred to as a local ZBG constraint.5 We also assumethe ISO imposes a reserve-requirement constraint (3.12) at each bus, conditional onthe ISO’s forecasted net load for that bus, referred to as a local RR˛� constraint.6 Wethen consider how the ISO would conduct a bid/offer-based RTM SCED optimizationsubject to constraints that include local ZBG and reserve-requirement constraints ateach bus.

4 LINKAGES BETWEEN THE REAL-TIME AND THE DAY-AHEADMARKETS

The operations of an RTM that supports SC trading for a particular hour H in theabsence of a previously accumulated SC inventory were illustrated in Section 3. Inthis section we consider an extension of the example in which the operations of theRTM are conditioned on an SC inventory acquired in the prior operations of a DAM,as depicted in Figure 7 on the facing page. The operations of this DAM are assumedto be in accordance with the general DAM description provided in Section 2.3.

5 Ignoring losses, the local ZBG constraint at each bus k is an equation ensuring that the total powerinjected at k equals the total power withdrawn at k plus the total power flowing out from k to otherbuses.6 In practice, local reserve-requirement constraints are imposed at the level of reserve zones. In Heoand Tesfatsion (2015) reserve zones are assumed to consist of singleton buses for ease of exposition.






FIGURE 7 Illustrative time-line for DAM/RTM linkages.

Day D – 1 Day D

Day-aheadmarket(DAM)

Real-timemarket(RTM)

Hour H

A key distinction between the DAM on day D � 1 and the RTM on day D isthat the DAM power-balance constraints are based on LSE demand bids, not on theISO’s own forecasts for LSE-customer loads. Nevertheless, the ISO has a fiduciaryresponsibility to balance actual real-time net loads to ensure grid reliability.

Consequently, we assume the ISO is permitted to bid for SCs in the DAM on dayD � 1 to ensure that reserve-requirement constraints are met, where these constraintsare informed by the ISO’s own next-day net load forecasts.7 The ISO then matchesand clears DAM-submitted SC bids and offers to achieve a least-cost ZBG subjectto system and reserve-requirement constraints. The ISO subsequently enters into theRTM on day D with a record of all DAM-cleared SCs and conducts RTM operationsconditional on this SC inventory.

In Heo and Tesfatsion (2015, Sections 4.1–4.2) we discuss at length, with illus-trative examples, how RTM operations for hour H are affected by SC inventoryconditioning when reserve-requirement constraints are imposed entirely for regula-tion (load-balancing) purposes. In Heo and Tesfatsion (2015, Section 4.3) we considerthis same issue for an augmented set of reserve-requirement constraints that includesconstraints for contingency reserve.

5 DISCUSSION

5.1 Comparison with real-world TSO/ISO operations

Our ISO-managed DAM/RTM design for the support of SC trading is structurallysimilar to existing European and US wholesale power market designs. European

7 As in Section 2.3, we require all costs arising from the ISO’s DAM SC procurement to be chargedto market participants in order to preserve the ISO’s nonprofit status.






wholesale power markets include “spot” (day-ahead) and intraday markets for energyand reserve managed by TSOs operating on a nonprofit-making basis (ENTSO-E2015; European Power Exchange 2015). US wholesale power markets include day-ahead and real-time markets for energy and reserve managed by nonprofit ISOs (USEnergy Information Administration 2015; Heo and Tesfatsion 2015, Table 1).

Moreover, the idea of permitting resources to offer options into TSO/ISO-managedwholesale power markets is not new. For example, Moriarty and Palczewski (2014)demonstrate how a small electricity storage unit could advantageously be permittedto offer American call options into a centrally managed real-time imbalance marketto facilitate load balancing.

On the other hand, our SC system differs sharply from current TSO/ISO operationsin other regards. SCs with swing function as intrinsically combined energy and reserveproducts that permit the provision of a wide range of flexibly provided services. Also,rewards and penalties can be included in SC performance payment methods to encour-age good service performance, eg, accurate load forecasting and/or accurate followingof dispatch instructions, where the rewards and penalties are assessed ex post basedon actual performance. This inclusion could be required at the SC system level. Alter-natively, SC suppliers could voluntarily undertake this inclusion as a way to signalthe quality of their offered services to potential SC buyers.

Moreover, our SC system functions as a two-part pricing system under whichall payments are compensations for value rendered, with no additional market orout-of-market payment adjustments required. Service availability compensation (inthe form of SC offer-price payments) becomes obligatory at the commencement ofservice availability, ie, as soon as SC supply offers are cleared. In contrast, serviceperformance compensation (through SC performance payment methods) does notbecome obligatory until services have been performed in real time.

This two-part pricing system contrasts sharply with the locational marginal pricing(LMP) systems currently implemented in US ISO-managed wholesale power markets.Schweppe et al (1988) conceptualized LMP systems for true spot markets in whichthere is no separation in time between payment and delivery, rather than for forwardmarkets such as DAMs and RTMs. Currently, DAM LMP payment commitments aremade in advance for the anticipated real-time dispatch of DAM-cleared generation,that is, in advance of value received. They are then subsequently adjusted throughRTM LMP payments to account for any deviations between DAM and RTM scheduleddispatch levels.

Moreover, DAM/RTM LMP payments do not necessarily provide adequate com-pensation for the costs incurred by resources to provide service availability. Theperceived need to cover such costs more fully has led to the institution of capacitymarkets and various out-of-market uplift payments.






5.2 Comparison with existing standardized power contracts

The restructuring of European and US electricity sectors, together with their increasedreliance on VER generation, has resulted in increased price and volume risks forutilities and independent power producers as prices and net loads have become morevolatile and difficult to forecast (Lemming 2004). Financial and physical instrumentsare now heavily traded in Europe and the United States on exchanges and in over-the-counter markets as a means for hedging exposure to these risks (Äid 2015; Dengand Oren 2006; European Energy Exchange 2015; NYMEX 2015).

In Europe, standardized power contracts have been developed by theAgency for theCooperation of Energy Regulators (ACER; ACER 2014). In the United States, stan-dardized power contracts have been developed by the Edison Electric Institute (EEI;Edison Electric Institute 2014) and the Western Systems Power Pool (WSPP; West-ern Systems Power Pool 2014). These widely used contracts are negotiated bilateralcontracts between two counterparties.

Our proposed SCs differ in three important ways from ACER, EEI and WSPPcontracts. First, SCs are bids/offers for submission to an ISO-managed wholesalepower market for possible clearing against other submitted offers/bids. In contrast, anACER, EEI or WSPP contract is a private agreement between two counterparties; itis subsequently self-scheduled in a TSO/ISO-managed wholesale power market onlyif fulfillment of the terms of the contract requires the use of power transmission lines.

Second, although the services provided through the contractual terms of SCs cancover the full range of product attributes included in ACER, EEI and WSPP contracts,SC services are not rigidly separated into product types (capacity, reserve and energy).Rather, SC services can be used to fulfill capacity requirements (general availability),reserve requirements (designated availability) and/or energy requirements (scheduledreal-time dispatch) as appropriate.

Third, SCs permit swing (flexibility) in all of the services included in their contrac-tual terms. In contrast, swing in ACER, EEI and WSPP contracts is limited to optionexercise dates in contracts taking an option form (ACER 2014, Table 1, p. 131; EdisonElectric Institute 2014; Western Systems Power Pool 2014).

5.3 Discriminatory versus uniform pricing of contracts

A market is said to exhibit market efficiency if the total net surplus extracted fromthe market by the market participants is at a maximum. Total net surplus is measuredin practice as the sum of the differences between the buyers’ maximum willingnessto pay and the sellers’ minimum acceptable payment for each successively tradedcommodity unit (see Stoft 2002; Tesfatsion 2009).

In order for market efficiency to hold, all valued attributes of a market-tradedcommodity must be properly priced and compensated at the margin. In a day-ahead






energy market organized as a bid/offer (double) auction, market efficiency can beachieved by means of a locally uniform pricing mechanism that assigns the sameprice to all energy units (MWh) being traded at a particular location for delivery atthis location at a particular later time (see Li and Tesfatsion 2011; Tesfatsion 2009).This is because the units of the traded product, characterized by physical type (energy),delivery location and delivery time, are homogeneous.

However, a uniform pricing mechanism applied to a traded product does not neces-sarily result in market efficiency if the units of this product are not homogeneous. Inparticular, in a market for which buyers and sellers are submitting bids and offers fordifferentiated products (referred to as a “monopolistically competitive market” withineconomics), the buyers and sellers must be permitted to bid and offer differentiatedprices for units of these differentiated products in order for these prices to reflect thetrue value of these units to buyers and sellers at the margin, a necessary prerequisitefor market efficiency.

As discussed in previous sections, the SCs traded in our proposed DAM and RTMcan be highly differentiated products. First, SCs can differ in terms of the types ofservices they offer. Second, even if two SCs offer the same types of services, thetwo SCs can differ in terms of the amount of swing included in the specification ofthese services. Consequently, our DAM and RTM are monopolistically competitivemarkets. The most appropriate pricing mechanism for SCs in our DAM and RTM isthus a discriminatory pricing mechanism in which SC sellers are permitted to offerdifferentiated prices for the sale of their differentiated products and SC buyers arepermitted to bid differentiated prices for the purchase of these differentiated products.

5.4 Comparison with existing VER initiatives

A major development in European and US TSO/ISO-managed wholesale power mar-kets is that increased VER penetration is increasing the volatility of net load (ie, loadminus as-available generation). Some TSOs/ISOs are revising their market rules andproduct definitions to accommodate this development.

For example, as discussed by Navid and Rosenwald (2013) and Xu and Trethe-way (2014), Midcontinent ISO (MISO) and California ISO (CAISO) have each pro-posed the introduction of “flexible ramping” products. Also, as discussed by Seligaet al (2013), ISO New England (ISO-NE) has introduced a major rule change calledEnergy Market Offer Flexibility. In addition, some ISOs are exploring innovativeways to incorporateVERs more fully into DAM/RTM operations. For example, MISOhas introduced a new resource category called Dispatchable Intermittent Resource,designed primarily for its wind resources (MISO 2011).

Our proposed SC system is not in conflict with the above market developments.On the contrary, as detailed in previous sections, SC trading would provide additional






types of flexibility to both market participants and system operators that complementand extend these developments.

5.5 Robust-control management of uncertain net load

A key requirement of standard two-stage stochastic SCUC formulations is the needto specify probability-weighted load scenarios with sufficient accuracy that a switchfrom currently used deterministic SCUC formulations can be justified in terms ofimproved performance. For example, as shown in Krishnamurthy et al (2015, Sec-tion V), given a simulated “true” load distribution and an approximate set S of loadscenarios, a deterministic SCUC formulation can result in lower energy costs than astochastic SCUC formulation based on S if reserve requirements for the former areset within a “sweet spot” range of values.

The rapidly growing reliance on VERs, resulting in increased net load uncertaintyand volatility, has encouraged efforts to develop improved stochastic SCUC formula-tions based on net load scenarios (see, for example, Morales et al 2009; Papavasiliouet al 2011; Vrakopoulou et al 2013). However, these approaches rely on having anaccurate modeling of the stochastic behavior of net load, a goal that has not yetbeen attained for as-available generation such as wind and solar power. In addition,to ensure tractability, they require the application of scenario reduction techniquescapable of retaining the essential features of the net load scenarios derived from theoriginal stochastic net load modeling.

Our proposed SC system offers an alternative robust-control approach to the man-agement of uncertain net load. As detailed in Section 3, under this system the ISOconsiders in advance of an operating period how much swing (flexibility) will beneeded in cleared SCs to cover a suitably wide corridor around an expected net loadprofile for this operating period. Consequently, a detailed specification of net loadscenarios is not required.

5.6 Amelioration of merit-order and missing-money issues

As noted in Section 5.4, centrally managed wholesale power markets such as MISOare attempting to integrateVERs into the operations of their DAMs by permitting theseresources to submit DAM supply offers based on generation forecasts. VERs tend tohave relatively low marginal dispatch costs. Hence, increasedVER participation tendsto decrease the profits of thermal generators by reducing day-ahead energy prices, anoutcome referred to in the power systems literature as the merit-order effect (Sensfußet al 2008). On the other hand, increased VER penetration requires an increase inflexibly controllable generation to handle the resulting increased volatility of netload. Given the current state of electric energy storage development, this increase inflexibly controllable generation must largely come from thermal generation.






The problem is then as follows. How can an adequate amount of flexibly control-lable thermal generation be ensured for matching the increased volatility of net loadresulting from an increased penetration of VERs when the latter penetration reducesthermal generation profits and hence the incentive to invest in and maintain thermalgeneration?

This problem can be ameliorated by guaranteeing that thermal generators receivefull compensation for all of the valuable services they provide, including flexiblycontrollable generation. Our SC system permits this full compensation.

Specifically, under our SC system a thermal generator can offer a GenPort (ie, aportfolio of SCs) that accurately expresses the types of services it can provide as wellas the degree of flexibility (swing) with which each of these types of services canbe provided. The generator should offer this GenPort at a price that fully covers thecosts it would incur to ensure the availability of these services, including capital andlost opportunity costs. If the GenPort is cleared, the generator receives an immediatecompensation commitment for service availability equal to the GenPort’s offer price.The generator also receives ex post compensation for any real-time services performedunder the terms of the GenPort, where this ex post compensation is determined bythe performance payment methods appearing in the SCs that comprise the GenPort.

Another problem arising in centrally managed wholesale power markets is missingmoney. Cramton and Ockenfels (2012) characterize this problem as follows:

In “normal” periods, when there is no shortage of capacity, prices are below the levelneeded to cover operating and capital costs of new capacity, and in scarcity events,prices are unlikely to accurately reflect the scarcity.

For concreteness, the current paper focuses on the support of SC trading throughrelatively short-horizon DAM and RTM operations. More generally, however, SCtrading could be supported by a sequence of linked forward markets that includeslonger-term forward markets with planning horizons spanning a year or more. In theselonger-term forward markets, the two-part pricing of SCs would permit investors toreceive availability and performance payments that fully cover their capital costs,lost opportunity costs and operating costs, thus helping to resolve the missing-moneyproblem.

6 CONCLUSION: ENERGY POLICY IMPLICATIONS

The key policy implications of our proposed market-supported trading of standardizedcontracts (SCs) permitting swing (flexibility) in their contractual terms are notedthroughout Sections 1–5. These policy implications are concisely summarized below.

(i) The SC system permits separate full market-based compensation for serviceavailability and service performance. SCs can function both as standardized






instruments for the procurement of service availability in forward markets andas standardized blueprints for the procurement of service performance in real-time system operations. Thus, SC trading supports the goals of FERC Order 755(FERC 2011), but this support is for a much broader array of services thanenvisioned in this order.

(ii) The SC system facilitates a level playing field for market participation. TheSC system focuses on service provision capability rather than on the physicalcharacteristics of resources. This should permit and encourage the participationof a wider array of resources in wholesale power markets.

(iii) The SC system facilitates co-optimization of energy and reserve markets. SCswith swing intrinsically function as both energy and reserve products, elim-inating the need to provide separate eligibility requirements and settlementprocesses for energy versus reserve services.

(iv) The SC system supports forward-market trading of energy and reserve. Theoffer price of an SC, determined through market processes, compensates theSC issuer for a guarantee of service availability. In contrast, the performancepayment method of an SC, appearing among its contractual terms, determineshow the SC issuer is to be compensated ex post for actual services rendered inreal-time operations.

(v) The SC system provides a fair way for all potential service providers to offerflexible service availability. SCs with swing permit the providers of these con-tracts to be appropriately compensated for flexibility in offered services, such asoffered exercise times, begin-times, end-times, down/up ramp rates and down/up power levels. Moreover, the ability of one or more resources to offer servicesin the combined form of an SC portfolio (GenPort) can enhance the ability ofresources to obtain appropriate compensation for the full value of their services.

(vi) The SC system provides system operators with real-time flexibility in serviceusage. SCs with swing permit system operators who procure these SCs toimplement the services offered in these SCs in a flexible manner during real-time operations.

(vii) The SC system encourages accurate load forecasting and the accurate followingof real-time dispatch instructions. Rewards and/or penalties can be incorporatedinto the performance payment methods � appearing among the contractualterms of SC demand bids to encourage LSEs and other wholesale intermediarieswho bid for services on behalf of retail customers to submit bids that accuratelyreflect the service needs of these customers. Similarly, rewards and/or penalties






can be incorporated into the performance payment methods � appearing amongthe contractual terms of SC supply offers to encourage service suppliers tofollow real-time service performance instructions with high accuracy.

(viii) The SC system permits resources to internally manage unit commitment andgeneration-capacity constraints. By offering an SC for a particular operat-ing period, a resource is guaranteeing that it can feasibly perform the ser-vices represented in this SC during this period. For generators, this feasibilityincludes the assurance that power generation units with suitable capacities willbe synchronized to the grid as necessary to perform these services.

(ix) The SC system permits robust-control management of uncertain net load. Underthe SC system, the ISO considers in advance of an operating period how muchswing (flexibility) will be needed in cleared SCs to cover a suitably wide corri-dor around an expected net load profile for this operating period. The SC systemthus provides a robust-control alternative to standard stochastic formulationsfor SCUC/SCED requiring detailed specifications of net load scenarios andscenario probabilities.

(x) The SC system eliminates the need for payment adjustments. SC offer pricesfor service availability and SC performance payments for service performanceprovide full compensation for all rendered value, without need for additionalmarket or out-of-market payment adjustments.

(xi) The SC system properties (i)–(x) reduce the complexity of power market rules,and hence the opportunity for market participants to game these rules for theirown advantage.


The work reported in this study has been supported in part by Sandia National Labo-ratories (Contract 1163155) and the US Department of Energy/ARPA-E (Award DE-AR0000214).

ACKNOWLEDGEMENTS

This study is a shortened revised version of Heo and Tesfatsion (2015). Versions ofthis work have been presented at the 2014 FERC Technical Conference (Washington,DC, June 23–25, 2014), the GridWise Architecture Council Meeting and Workshop(CAISO, September 10–11, 2014) and the 2015 IEEE Power and Energy SocietyGeneral Meeting (Denver, CO, July 26–30, 2015). The authors are particularly grate-ful for helpful comments received from J. Ellison, T. Heidel, M. Ilic, D. Kirschen,S. Lence, N. Yu, R. Zimmerman, the editor and an anonymous referee.






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