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1、 April 2024 OIES Paper:EL 53 Hedging and Tail Risk in Electricity Markets Farhad Billimoria,Visiting Research Fellow,OIES,and Director,S&P Global Power Markets Jacob Mays,Assistant Professor,Cornell University Rahmatallah Poudineh,Head of Electricity Research,OIES ii The contents of this paper are t
2、he authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of
3、its members.Copyright 2024 Oxford Institute for Energy Studies(Registered Charity,No.286084)This publication may be reproduced in part for educational or non-profit purposes without special permission from the copyright holder,provided acknowledgement of the source is made.No use of this publication
4、 may be made for resale or for any other commercial purpose whatsoever without prior permission in writing from the Oxford Institute for Energy Studies.ISBN 978-1-78467-240-9 iii The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford
5、Institute for Energy Studies or any of its Members.Acknowledgements We wish to acknowledge without implicating Malcolm Keay,Dr.David Robinson,and Prof.Paul Simshauser for insightful comments and feedback on earlier versions of this paper,and Dr.Darryl Biggar for helpful discussions on the context an
6、d content of the work.Abstract A concern persistent in scarcity-based market designs for electricity over many years has been the illiquidity of markets for long-term contracts to hedge away volatile price exposures between generators and consumers.These missing markets have been attributed to a ran
7、ge of factors including retailer creditworthiness,market structure and the lack of demand side interest from consumers.Using a stochastic equilibrium model and insights from insurance theory,we demonstrate the inherent challenges of hedging a legacy thermal portfolio that is dominated by volatile fa
8、t-tailed commodities with significant tail dependence.Under such conditions the price required for generators to provide such hedges can be multiples of the expected value of prices.Our key insight is that when the real-world constraints of credit and financing are considered,the volatility of therm
9、al fuels and their co-dependence under extremes may be a key reason as to why electricity markets have been incomplete in terms of long-term hedging contracts.Counterintuitively,in the context of the energy transition,our results show that,ceteris paribus,increasing the penetration of low carbon res
10、ources like wind,solar and energy storage,can add tail-diversity and improve contractability.iv The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.Contents Acknowledgements.iii Ab
11、stract.iii Contents.iv Figures.iv Tables.v Introduction.1 1.Risk hedging in electricity markets.3 1.1 Insurance markets and the Unholy Trinity.4 1.2 The Unholy Trinity in Electricity Markets.5 2.Case Study and Results.10 2.1 Assumptions and Data Sources.10 2.2 Calibration and Price Statistics.13 2.3
12、 Results.16 Conclusions and Policy Implications.20 Appendix A:Models of Risk Hedging in Electricity Markets.22 A.1 Decision-making models for agents.22 A.2 Economic dispatch.23 A.3 Equilibrium search.24 Appendix B:Percentile Scatter and Conditional Probability Plots.26 Appendix C:Calculation of Mini
13、mum Hedge Contract Price under Full Contracting.28 Appendix D:Derivation of Total Inframarginal Rents and Contribution Factor .29 References.31 Figures Figure 1:Graphical analyses of tail risks in wholesale electricity prices for New South Wales,on a CPI-adjusted daily load-weighted basis from FY200
14、2-2022.6 Figure 2:Graphical analyses of tail risks in histograms and mean-excess plots for fuel commodities coal,natural gas,and diesel.7 Figure 3:Histograms and mean-excess plots for historical unavailability of the generation fleet in the NEM coal and natural gas(OCGT and CCGT).8 Figure 4:Histogra
15、ms and mean-excess plots for wind and solar*unavailability long-term historical backcast.9 Figure 5:Statistical Parameters and Density Plot of Wind and Solar Availability.14 Figure 6:Periods of Lowest Average Availability.15 Figure 7:Minimum Contract Price for Full Volumetric Hedge(Case 1 Thermal Do
16、minated).16 Figure 8:Proportion of Scenarios with Negative Surplus (Case 1 Thermal Dominated).17 Figure 9:Consumer Volumetric Appetite for Long-Term Hedges under low(=.)and high (=.)generator risk aversion.17 Figure 10:Minimum Contract Price for Resource Mix Cases 1-4,for four levels of supplier ris
17、k-aversion(=.,.,.,.).18 Figure 11:Equilibrium Contract Volumes and Prices under Thermal and Low-Carbon Cases.19 Figure 12:Portfolio Surplus Attribution Worst 20 Scenarios,Cases 1 and 3.19 Figure 13:Resource Surplus Attribution,Scenario 224,Cases 1 and 3.20 v The contents of this paper are the author
18、s sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.Figure B1:Percentile scatter plots for fuel commodities coal,natural gas,and diesel.26 Figure B2:Percentile scatter plots for demand(y-axis)against coal,natural gas,and
19、variable renewables*(x-axis).26 Figure B3:Percentile scatter plots for gas generation unavailability(y-axis)against coal and variable renewables*(x-axis).26 Figure B4:Conditional probability plot for gas prices against coal and diesel prices(daily).26 Figure B5:Conditional probability plot for gas p
20、rices against coal and diesel prices(monthly aggregation).27 Figure B6:Conditional probability plot for demand against gas,coal and variable renewables availability(dispatch interval).27 Figure B7:Conditional probability plot for gas availability against coal and variable renewables availability(dis
21、patch interval).27 Tables Table 1:Resource Cost and Technical Assumptions.11 Table 2:Resource Capacity(GW)by Case.13 Table 3:Price Statistics.15 1 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or
22、any of its Members.Introduction This paper seeks to address the issue of the viability of long-term hedges in transitioning electricity markets.The fundamentals of modern electricity market design are based on seminal works(Joskow&Schmalensee,1983;Schweppe et al.,1988)that advocate a canonical desig
23、n based on a centrally cleared,security-constrained economic dispatch(SCED),with generation and load executed in real-time and settled on the basis of locationally and temporally granular spot prices for electricity(locational marginal prices,or LMP).1 The volatility brought on by full-strength pric
24、e formation is intended as a feature rather than a bug of the canonical design.Dynamic prices provide theoretically efficient short-term signals for generation,storage,and load resources.It also provides incentives for hedging and risk-trading based on the given risk preferences of consumers and ele
25、ctricity providers.Nevertheless,in full-strength markets,there are concerns commonly raised as to the sufficiency of hedge markets and the capability to secure long-term contracts to underpin investment in capital-intensive generation plants(ACCC,2022).The lack of liquidity and depth of contracting
26、markets is a common refrain,especially for smaller or non-integrated market participants.Relatedly,the costs of hedging are also seen as high,and in some cases prohibitive,leading to under-hedging.In exchange-traded and over-the-counter(OTC)derivative markets,in addition to the hedge risk premia,par
27、ticipants must also incur expenses related to margining and collaterisation requirements.The market for long-term contracts,which are important to securing financing for generation investment,is an area of particular concern.Liquidity in wholesale derivative markets typically only extends to 1-3 yea
28、rs ahead,while retail contracts too have embedded optionality for retailers to reprice and consumers to switch.The repercussions on the system,often felt after extreme conditions in the market,include retailer bankruptcies and insufficiently resilient generation resources to support extreme load and
29、 weather conditions(Mays et al.,2022).The 2022 energy crisis brought concerns around long-term hedging and price formation to the forefront of attention(Gabel,2022).Of relevance in this crisis was the integrated nature of international commodity supply across fuels and geography.The almost simultane
30、ous record spikes in coal,gas,and oil prices(all three fuels being integral to legacy power systems)were a stark demonstration of the linkages between these partial substitutes.Global supply chain integration also meant that events affecting one part of the world could and did constrain availability
31、 in many other otherwise geographically disparate locations.Further complicating this has been the trend towards spot pricing in fuel markets.Legacy fixed-price contracts for coal and gas roll off and are replaced by shorter-term or spot-indexed contracts(though oil-price escalation in gas contracts
32、 has been common for some time)and shorter-term forward derivative products(Losz et al.,2023.This has been a particular challenge in regions with tight supply balances such as the UK,Europe,and Australia(Lewis Grey Advisory,2023).This paper seeks to introduce a novel perspective on the market for lo
33、ng-term hedges by framing the problem as one of insurability.That is,while the inherent risk attitude of market participants is relevant for the desire to contract,we incorporate the requirement for hedge providers to be able to financially deliver on the hedge during normal and extreme scenarios.Th
34、is provides a unique insight into why hedging may continue to pose a challenge for systems without truly diversified tail exposures.With respect to diversification,a critical question is whether concerns around long-term hedging in electricity markets will persist,exacerbate,or soften with the trans
35、ition to low-carbon sources of supply,such as wind,solar,and energy storage.Indeed,while recent studies point to the higher volatility of renewable-dominated portfolios,the question of whether this extends to the hedgeability of such portfolios is still open.Our scope is restricted to system margina
36、l costs in the context of fuel costs and resource 1 Many regions,particularly in the US and Europe,also augment the real-time market with central short-term forward markets,typically cleared in day-ahead or intraday timeframes.In the US a security-constrained unit commitment process will accompany a
37、 day-ahead SCED incorporating non-convexities associated with certain plant(minimum generation levels,minimum/maximum run times,startup costs etc).Other regions such as Australia and New Zealand have long operated real-time only markets(with decentralised participant self-commitment).2 The contents
38、of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.availability we defer consideration of other contributing issues(for example,network reliability,system security,or market power)to future wo
39、rk.An insurance business model depends upon being able to financially support any insurance products that are sold(Rees&Wambach,2008).This means they must have access to enough capital or reinsurance to compensate losses even under the extremes of the probability distribution.Thus,insurers are often
40、 managed to meet solvency or reserve constraints,as determined by regulation,rating agencies,or internal firm decisions.While there are limited formal capital adequacy and regulatory reserving provisions in electricity markets,the concepts are transferrable.Hedge providers manage exposures through o
41、ffsetting contracts,or through the operation of assets in the organized spot market(as in,defending the contract).For example,a gas peaker selling a call option(or cap)at a$300 strike will seek to run when prices exceed the strike to offset hedge exposures.Margin and credit requirements in exchange
42、and OTC markets also apply to ensure that the hedge provider has sufficient creditworthiness to perform on the hedge contract(Simshauser,2021).Climate change has become a major challenge for the insurance sector,with the increasing frequency and severity of losses and disruptions from extreme weathe
43、r.Climate change is considered among the top risks,if not the top risk exposure,for insurers and re-insurers(Swiss Re,2017;Wagner,2022).In recent times,major insurers have stopped offering common insurance lines(for instance,home insurance)in certain regions.Kousky and Cooke provide an underlying ra
44、tionale for this through the idea of the unholy trinity in insurance markets:three phenomena fat tails,tail dependence,and micro-correlations which can make the traditional insurance of such risks in an era of climate change not just expensive,but unfeasible(Kousky&Cooke,2012).Applying the concept t
45、o the electricity markets,we provide an underlying basis for the challenge of contracting for long durations in thermal dominated systems.We describe key drivers of market incompleteness as being the fat-tailed nature of underlying coal and gas fuel markets and the tail dependence between them.Under
46、 such loss distributions,small changes in risk perception can result in large changes in consumer contracting appetite.We posit this as a potential rationale for why long-term hedging contracts between consumers and energy suppliers have been challenging to obtain and execute in practice.Going forwa
47、rd,we show how the underlying economics of zero-marginal cost resources may alter the long-term hedgeability of resource portfolios as systems progress through the energy transition.Our key insight is that while spot markets may become more volatile,low-carbon portfolios can benefit from a more dive
48、rsified exposure to tail-correlated risks.This then shifts the focus of risk management towards understanding potential tail and common-mode risks in renewables and storage-heavy grids.This should include inter-alia a consideration of policy uncertainty and political intervention under extrema,and o
49、f tail-resilient market designs.We illustrate the role of the unholy trinity in insurance and financial markets and extend this concept to electricity markets using extreme value theory(EVT).This provides an understanding of the nature and extremity of risk that has been inherent to electricity mark
50、ets.We then construct an instance of a stochastic equilibrium model and demonstrate how such tail risks may shift over the energy transition.In our case study,the value of tail diversity from renewable additions to the portfolio more than offsets variability risk.The rest of the paper is structured
51、as follows.Section 1 reviews recent research findings as it relates to electricity risk hedging.It also outlines the unholy trinity of risks to insurability and extends this concept to electricity markets.Section 2 sets out the results of the modelled case study(Methods and the formulation of the mo
52、del are explained in Appendix A).The final section discusses critical policy implications and concludes.3 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.1.Risk hedging in elec
53、tricity markets The literature on price formation and hedging in electricity markets is extensive but tends to fall into one of two categories.One set considers price formation in the context of archetypes of electricity market design these works can be either qualitative or have simulated outcomes(
54、generally as an output of capacity expansion planning models,equilibrium models or agent-based models);while the second set undertakes empirical and statistical analyses of historical prices.There have been relatively few attempts to reconcile them.The volatility of electricity prices is a necessary
55、 component in the theory of competitive electricity markets(Boiteux,1960;Caramanis et al.,1982;Chao,1983;Harvey&Hogan,2019).Getting the prices right for an inelastic good,such as electricity,will invariably involve volatility or price spikes during scarcity(Hogan,2014).To manage the risks associated
56、 with spot prices,participants can hedge,or trade risk based on their individual preferences(Biggar&Hesamzadeh,2022).Derivative products(including forwards,swaps,and options)have evolved to allow generators and retailers to exchange volatile spot exposures for more stable cashflows(Deng&Oren,2006),i
57、ncluding a suite of products catered towards variable renewables(Billimoria,2021;Lucy&Kern,2021).However,obtaining contract of long tenor has been a concern,with a set of literature arguing there are missing markets for long-term contracts,which are seen as necessary to support capital-intensive gen
58、eration investment when participants are risk-averse(Abada et al.,2019;Neuhoff&De Vries,2004;Newbery,2016;Roques&Finon,2017;Simshauser,2019).This is often linked to broader notions of the incompleteness of markets(de Maere dAertrycke et al.,2017;Mays et al.,2022;Willems&Morbee,2010).The nature of th
59、e hedging challenge can vary significantly among resource types.For renewable projects,multi-year commercial power purchase agreements can be common(Gohdes et al.,2022;Simshauser,2020).Interestingly such demand appears driven not only by retailers,but also by corporate and smaller institutional ener
60、gy consumers driven by price and decarbonisation imperatives.Non-traditional risk-traders with diverse market exposures,such as insurance companies and hedge funds,have also entered long-term contracts for difference as net energy buyers(Billimoria,2021).The assumption of market completeness is impo
61、rtant in characterising how literature integrates the issue of long-term hedging into electricity market design.Under complete markets,hedging can be considered secondary to market design,given that participants are best placed to design,price,and execute risk-hedging decisions,in the presence of ap
62、propriate scarcity incentives(Hogan,2022).In this context,Biggar and Hesamzadeh integrate dispatch and risk-averse hedging under a set of qualifications complete markets,symmetric risk-preferences,and a hedge price that approximates the expected value of hedge cashflows(Biggar&Hesamzadeh,2022).The d
63、iagnosis of incompleteness and implications for hedging is an active area of research.Mays et al.,2022 set out underpinning factors for market incompleteness that include fuel market curtailment regulations,and the uncompensated value of the consumer forced outage hedge.Schittekatte,Batlle et al att
64、ribute much to the lack of demand-side interest in hedging driven by implicit(and explicit,in the case of the recent European energy crisis)pricing support and assistance from central governments(Batlle et al.,2023;Schittekatte T,2023).Another stream of literature points to structural protections in
65、 retail tariffs and lack of retail creditworthiness(Neuhoff&De Vries,2004).Integrating retail with generation operations to provide a physical hedge(also known as integrated generator-retailers,or gentailers)has been a response to this issue in full-strength markets(Simshauser,2021;Simshauser et al.
66、,2015).Despite vertical integration extending to the fuel source itself,long-term demand has not been forthcoming given the risk of being undercut by new entrant retailers with short-dated portfolios(Simshauser,2018).Yet the question of whether these factors are a cause or symptom of an underlying p
67、roblem would assist in clarifying whether these issues will persist as the electricity system transitions to low-carbon resources.In considering this,a common focus is upon the impact of the changing supply mix on spot prices.Higher penetrations of zero-marginal cost resources are expected to increa
68、se the frequency of very low prices(when renewables are abundant),but also the frequency of very high prices(when 4 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.renewables a
69、re unavailable,and prices are set by load or firming resources)(Hogan,2019,2022;Mallapragada et al.,2023).Taken to an extreme this would result in a bid-curve that is“L-shaped”,and a bi-modal distribution of prices.Recently,Mays(2023)invites some scepticism of this notion in the context of sequentia
70、l market clearing under uncertainty.In terms of risk-hedging implications,a surprising result from Mays and Jenkins(2023)is that overall investment risk may be lower in systems dominated by variable renewables due to reduced exposure to fuel price uncertainty.A current gap in the research relates to
71、 the consideration of hedging under a granular resolution of tail cases for energy systems.Under such events,risk is inherently asymmetric,normal relationships between resources can break down and traditional measures of correlation or co-movement tend to have less relevance.Importantly,risk-hedging
72、 should be understood from a range of tail risk parameters,not just a single set point because risk preferences across the market(including for both resources and consumers)are not transparent;and the assessment of tail exposure requires a degree of comprehension of the risk(Leslie et al.,2022).1.1
73、Insurance markets and the Unholy Trinity To provide a new perspective on this issue we draw from other risk hedging markets,and most specifically insurance.A central principle of risk management is aggregation.Firms hold not one contract,but a portfolio of contracts,diversified across location,custo
74、mer type and time.Holding such bundles offers diversification benefits and stabilizes losses.This is common in risk management in many sectors for example,insurance companies will cover claims across a range of loss lines,diversified by region,customer,and timeframe.However,climate change has been a
75、rgued to pose significant challenges for insurability.In many regions,especially where physical risks are increasing,insurers have struggled to provide insurance at viable rates,and consumers have had consistently low penetration rates of certain coverage lines many of which are related to catastrop
76、hes or extreme events(Kousky,2023).Looking to the underlying reason for such under-insurance,Kousky and Cooke(2009,2012)seek to explain the reasons for why consumers persistently fail to hedge against extreme risks,by not purchasing catastrophe insurance.They argue that there are three factors:fat t
77、ails,tail dependence and micro-correlations(which together they call the unholy trinity)that challenge traditional risk management.“With fat-tailed losses,the probability declines slowly,relative to the severity of the loss.Tail dependence is the propensity of dependence to concentrate in the tails,
78、such that severe losses are more likely to happen together.Micro-correlations are negligible correlations between risks which may be individually harmless,but very dangerous when aggregated.These three phenomena types of catastrophic and dependent risks undermine traditional approaches to risk manag
79、ement.”Kousky and Cooke(2009).Fat tails are a statistical concept used to describe the distributions where the tails decline very slowly.2 The precise mathematical definition of fat tails is a probability distribution where the tail of the distribution degrades in line with a power law(i.e.,the prob
80、ability that a random variable X exceeds x is where,0)3.This is a particularly extreme form of tail degradation,for our purposes the focus is less upon a direct empirical fit than an understanding of the loss outcomes at distribution extremities.Fat tails can be diagnosed using a range of indicators
81、 and metrics in extreme value theory(EVT).They are particularly problematic in risk markets due to the high likelihood of extreme events at the very ends of the distribution;and a higher proportion of events concentrated around the median.Without an 2 For the normal distribution,an event larger than
82、 3 standard deviations from the mean occurs only 0.13 per cent of the time.For a typical fat tailed distribution(say a Pareto distribution with a tail parameter equal to three)a 3 times standard deviation event occurs with probability 1.5 per cent,more than ten times as often a normal distribution.3
83、 Parameters and:These constants shape the distributions tail.Specifically:can be viewed as a scaling parameter that affects the overall level of the probability.A higher would generally mean a higher probability for X to exceed x,all else being equal.is a shape parameter that controls the rate at wh
84、ich the probability decreases as x increases.A higher means the probability P(Xx)declines more rapidly as x grows.5 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.appreciation
85、 of the importance of extreme events for such distributions,risk can often be underestimated.However,when this type of risk is incorporated into insurance analyses the tail outcomes can often skew the characterisation and pricing of risk(Kousky&Cooke,2009).The challenge of extreme outcomes extends t
86、o financial markets more broadly,where the failure to properly account for fat tails has resulted in large scale private losses and systemic risk.4 Tail dependence relates to the tendency of random variables to co-occur in the extremes;that is,the random variables will be concentrated in the tails(C
87、ooke et al.,2010).The failure to consider tail dependence can result in a significant underestimate of insurance loss exposure,as it can undermine traditional measures of portfolio diversification such as(Pearsons)correlation.Quoting from Kousky and Cooke(2009),the upper tail dependence of a set of
88、variables X and Y is defined as the limit(if it exists)of the probability that X exceeds its r-percentile,given that Y exceeds its r-percentile,the conditional probability as r goes to 100(Kousky&Cooke,2009).Percentile scatter plots provide one means of visualising such dependence.Micro-correlations
89、 are small positive correlations between variables that can be amplified when claims(or exposures)are aggregated.This is of evident concern to risk managers,as it will compromise risk management based on diversification(Cooke et al.,2010).The example is provided in Kousky&Cooke(2009)of the link betw
90、een crop insurance indemnities and flood insurance claims which can demonstrate low correlations with each other,but when groups of claimants are aggregated the exposure can be magnified.Such correlations can often go undetected and can be counter-intuitive to traditional claim diversification strat
91、egies.The key conclusion of Kousky and Cooke(2012)is that under conditions of fat tails,tail-dependence and micro-correlation(or the unholy trinity as it is named),the benefits of aggregation and portfolio diversity tend to fall away.When insuring risks with loss distributions characterised by the u
92、nholy trinity insurers need to charge a price that is many times the expected loss in order to ensure solvency.At this price,homeowners with budget constraints may thus rationally forgo such insurance if their budgets do not allow for it,notwithstanding the potential utility of such insurance.Yet th
93、is framework may have applications beyond the insurance sector and could potentially be applied to any hedging instrument that seeks to provide protection against extrema.To this end,our paper seeks to make a novel application of the theory of the unholy trinity to provide an underlying rationale fo
94、r why markets in long-term risk trading have been missing in the electricity sector.1.2 The Unholy Trinity in Electricity Markets This section provides empirical observations on the application of extreme value theory to electricity markets,with a focus upon price and resource availability variables
95、 in the National Electricity Market of Australia(the NEM).Our focus is upon the three elements of the unholy trinity of risk namely the potential for fat tails,tail dependence,and aggregative risk.Our Online Companion also provides a full set of results for all regions,time periods and seasonal aggr
96、egations(Billimoria et al.,2024).Fat Tails Several studies have looked at the statistical properties of electricity prices in a range of markets.Many have found evidence of heavy and fat tails via the application of EVT(Boothe&Glassman,2003;Bystrm,2005;Huisman&Huurman,2003;Weron,2005),supported by f
97、indings of significant higher order moments,such as positive skewness5 and high kurtosis(Knittel&Roberts,2005).These studies are predominantly conducted on thermal dominated systems.Empirical and EVT analyses of electricity prices in the NEM are consistent with this theme6.It is important to note th
98、at aggregation matters here,4 See for example Jorion(2000);Taleb(2007);Taleb&Martin(2012).5 Positive skewness in electricity prices indicates that the distribution of prices is asymmetric,with a longer right tail,representing infrequent but substantial spikes in prices above the average,rather than
99、frequent small increases.6 The Australia National Electricity Market(NEM)has traditionally been an energy-only design,with wide administrative settings-a market price cap of$16,600/MWh,a cumulative price cap of$1.49 million,and a price floor of-$1000/MWh.A rule change(currently under consideration)r
100、ecommends an increase in the market price cap to$21,500/MWh and the cumulative price cap 6 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.because the prices that consumers ult
101、imately see and are sensitive to are typically weighted aggregations across time,such as the periodicity at which electricity bills come due.Figure 1 sets out a panel of common graphical tools for diagnosing fat tails as applied to monthly load-weighted wholesale prices in the Victoria region of the
102、 NEM,as an example.Panel A shows the histogram of load-weighted prices where the distribution appears highly skewed and tail events(load-weighted prices above$400/MWh are observable).In Panel B the analysis is complemented with a mean excess function plot,which is an intuitive way to understand tail
103、 behaviour.If a random variable is possibly fat tailed,its mean excess function(u)=|should grow linearly in u,at least above a certain threshold,which is as observed via the increasing trend.7 The presence of a fat tail is also supported by the quantile-quantile plot in Panel C,where the exponential
104、 distribution is used as a benchmark.The concave behaviour observed in the plot is also indicative of sub-exponential decline and potential fat tails(Cirillo&Taleb,2016,2020).Finally in the log-log or Zipf plot in Panel D,possible fat tails can be identified in the presence of a linearly decreasing
105、behaviour of the curve.8 Importantly such trends appear consistent for all the mainland regions,across aggregation periods(daily,weekly,quarterly)and across seasons,with results available in the Online Companion(Billimoria et al.,2024).Table 3 provides statistical moments and percentiles on wholesal
106、e price distributions for two mainland regions in the NEM.Heavy skewing and kurtosis of prices are evident.Anderson-Darling tests are applied confirming the non-normality of the sample distributions.Figure 1:Graphical analyses of tail risks in wholesale electricity prices for New South Wales,on a CP
107、I-adjusted daily load-weighted basis from FY2002-2022 to$2.19 million.Though there have also been recent initiatives that have looked to supplement scarcity price formation,including triggered retailer reliability obligations,state-based revenue support hedges and the national Capacity Investment Sc
108、heme.Power market bidding is relatively unfettered,though subject to good faith obligations.7 Kousky and Cooke(2009)provides an example of the intuition of the mean-excess plot comparing a thin-tailed normal distribution,with a fat-tailed distribution.8 This is further supported by concentration plo
109、t profiles and Hill estimator results with goodness of fit statistics(see the Online Companion(Billimoria et al.,2024).7 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.A range
110、 of fundamental factors of electricity markets support the statistical assessment that fat tails are inherent to electricity markets.First,demand for electricity is relatively inelastic often due to consumers being implicitly protected via tariff structures and aspects of market design(Billimoria&Po
111、udineh,2019;Mays et al.,2022).Second,with electricity markets clearing on marginal prices,small changes in supply or demand can result in large shifts in prices9.In this respect,the exercise of market power,especially during scarcity,may also be relevant.Finally,an important underlying driver in fos
112、sil fuel-dominated electricity markets has been the volatility in the underlying fuel markets themselves.With fossil fuels dominating the supply mix over the last 20 years,understanding the fatness of tails for fuel prices could inform the historical patterns of price formation for electricity,and g
113、uide discussion on the future.The top row of Figure 2 shows histograms for the local gate or hub prices for natural gas,thermal coal,and liquid fuel(diesel).The bottom row shows mean excess plots for these commodities.The graphs demonstrate a sharply increasing trend above thresholds of$10/GJ for ga
114、s,$5/GJ for coal and$35/GJ for diesel.This is indicative of heavy-,and possibly fat-tailed behaviour.There are also levels beyond which the mean excess begins to decline again(at$25-30/GJ for gas,$10/GJ for coal and$40 for diesel),though given the sparsity,such data points are of less significance f
115、or EVT tails(Cirillo&Taleb,2016,2020).Thus,it would be reasonable to infer that the extended tail volatility of primary fuels,when unhedged,would reasonably have played a role in the intensity of tail exposure in intermediate wholesale electricity markets.Figure 2:Graphical analyses of tail risks in
116、 histograms and mean-excess plots for fuel commodities coal,natural gas,and diesel Source:Natural gas prices Short-Term Trading Market(STTM)Sydney Hub daily;Coal prices daily front-month Newcastle Coal futures contract(6000kcal/kg net calorific value,translated to AUD at the prevailing exchange rate
117、);AIP daily diesel terminal gate price for Sydney.9 Merit order curves in the NEM are highly non-linear and often have a knee point where above a certain quantity the offered prices increase dramatically.This can mean that the electricity bid curve jumps up in orders of magnitude over a few hundred
118、MWs 8 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.Examining the statistical profiles of generation unavailability is also important for this study(noting that we neglect co
119、nsideration of system security and network constraints in this study).Histograms and mean-excess plots for the unavailability of the coal and natural gas generation fleet in the NEM are shown in Figure 3(with gas broken into OCGT and CCGT generation).Actual historical unavailability is shown for coa
120、l and gas generation.For renewables,given the limited dataset of actual outcomes,a long-term backcast of renewable availability based on actual weather outcomes over the last 20 years in the NEM is adopted(further information is provided in the subsequent sections).Figure 4 shows histograms and mean
121、-excess plots for wind,solar(between the hours of 0500 and 2000),and a combined 70/30 wind-solar portfolio.As a comparator,a recently computed 80 year backcasted dataset for ERCOT from Gruber et al(2022)is also shown.By contrast with the results on fuel prices,the mean-excess plots of renewable and
122、thermal generation availability do not appear to have a consistently increasing trend even after a particular threshold.It does not immediately suggest that the statistical distributions are fat-tailed in nature.In terms of weather patterns of the historical data period,while geographical regions co
123、vered by the NEM have experienced periods of extreme heat and drought,extreme freezing events are less notable.Hence it is caveated that the applicability of this data set is limited for areas that have experienced major winter freeze events,for example ERCOT during Winter Storm Uri.Figure 3:Histogr
124、ams and mean-excess plots for historical unavailability of the generation fleet in the NEM coal and natural gas(OCGT and CCGT)9 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.
125、Figure 4:Histograms and mean-excess plots for wind and solar*unavailability long-term historical backcast *Solar data restricted to the hours 0500-2000.Tail Dependence and Aggregation The second factor in the unholy trinity of insurability relates to tail dependence.In electricity markets with price
126、s formed by primarily fossil fuel generation,a key issue relates to the tail dependence between the fuel commodity costs.There are physical factors to suggest that tail dependence could exist between fossil fuel commodities,which is to say the highest periods of pricing for such fuels tend to occur
127、at the same time.The integrated nature of commodity supply chains suggests that common-mode factors may impact the price of such fuels similarly.Fuels are partially substitutable(at least on an energy portfolio basis),and with adjacent physical supply and logistics infrastructure that may be commonl
128、y vulnerable to physical events.Furthermore,as evident in the energy crisis of 2022,there are common sources of supply that can be affected by geo-political and other events.How does this flow into the statistics?Based on Australian commodity data,the conditional probability of daily coal prices exc
129、eeding the 50th percentile(P50),when gas prices exceed their 50th percentile,is 0.71;at the 75th percentile a similar level of 0.67 is observed,while at the 95th percentile(P95)this increases to 0.81.Under a monthly aggregation of prices over the P50,P75 and P90 conditional probabilities are 0.76,0.
130、66 and 0.88,respectively.Diesel to gas conditional probabilities are 0.48(P50),0.37(P75)and 0.80(P95)for daily prices,and 0.47(P50),0.37(P75)and 0.88(P95)for monthly aggregations(plots of the conditional probabilities in 5 per cent increments are shown in Appendix B,Figures B4-B7).Percentile scatter
131、 plots(Figure B1-B3)as shown in Appendix B lend support to this,and importantly also illustrate that such tail dependence is not restricted to the recent 2022 supply crisis.This issue may have been less concerning in less integrated markets where coal and gas could be contracted at fixed prices over
132、 multi-year timeframes,but it takes on an increasing relevance where contracts are increasingly shorter-term or indexed to spot.A similar analysis is conducted in relation to tail dependence between the availability of different forms of generation and with demand.Both the conditional probabilities
133、and scatter plots on NEM actual and backcasted(for renewables)data suggest limited tail dependence to date(see Appendix B)with the 10 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Me
134、mbers.conditional probabilities declining as the nth percentile increases.While events like the Winter Storms Uri and Elliot in the US are observations of the weather sensitivity of fossil generation(particularly natural gas),it is interesting to observe that in the NEM that higher availability tend
135、s to be associated with periods of higher demand.This is intuitive in a market where generators are not centrally committed,but self-commit into the spot market based on expectations of electricity scarcity and prices,noting that this includes scheduling of planned maintenance.As indicated above,the
136、re have been relatively few freezing events in the empirical dataset to date.Finally micro-correlations under aggregation are also relevant because of the time periods over which consumers perceive price risks.Given consumers pay electricity bills over aggregated periods,say a month,this would sugge
137、st that the willingness and ability to hedge is guided by perceptions of risk over similar periods,returned to in what follows.2.Case Study and Results We provide a mathematical framework for analysing risk hedging strategies in electricity markets amid the transition to low-carbon energy sources.Th
138、e model employs a non-cooperative game-theoretic approach to capture the decision-making processes of market participants,specifically focusing on two risk-averse agents:a consumer and an energy generation supplier.This approach allows for the examination of individual strategies within the markets
139、structure,considering the agents risk-aversions and market incompleteness.The model considers the conditional value-at-risk(CVaR)to model aversion to downside outcomes,recognising the asymmetry in risk perception between upside gains and downside losses.Furthermore,it introduces a stochastic equilib
140、rium model that integrates the operation of generation and storage assets in response to market dynamics and contractual arrangements.The detail of this model is presented in Appendix A.This section provides data,assumptions and results of a case study conducted to investigate the impacts of differe
141、nt electricity mix on long-term risk hedging.The case study is conducted on the National Electricity Market or NEM,which is the largest electricity system in Australia covering the states along the eastern seaboard.The study is restricted to the mainland regions of the NEM,excluding Tasmania.The NEM
142、 provides an interesting example of a grid in transition with already high levels of renewable penetration(recently recording its highest instantaneous penetration of 68 per cent),and significant additional renewables and storage deployment expected in the near term.The optimal decarbonisation pathw
143、ay identified by the Australian Energy Market Operator(AEMO)in its Integrated System Plan(ISP)provides useful datapoints as to resource capacity trajectories and cost structures.Further the market has a high degree of transparency and there is rich and granular dataset for the numerical experiment.N
144、EM has an energy-only style market design with high scarcity-based market settings;a market price cap of$16,600/MWh and market price floor of-$1,000/MWh,based on a minimum reliability standard of 0.002 per cent expected unserved energy(EUSE).To the extent that the reliability settings are expected t
145、o be breached,a range of additional measures apply to source resources in a quantity sufficient to meet the standard,including the Retailer Reliability Obligation(RRO),and the Reliability and Emergency Reserve Trader function.The retail market is contestable,with a high portion of the market served
146、by gentailers.Over much of its history the NEM has relied upon market participants to make hedging and investment decisions.However,most recently in November 2023,the Commonwealth government announced plans for a large-scale Capacity Investment Scheme,a state-initiated risk-hedge program for up to 3
147、2GW of resource capacity(Commonwealth Government,2023).2.1 Assumptions and Data Sources This section outlines the assumptions and sources of data underpinning this case study.The case study considers nine types of generation and storage resources including thermal generation,namely black coal,natura
148、l gas and liquid-fuelled(diesel)generators;variable renewable generation(wind and solar PV)and battery storage(of multiple durations).The technical assumptions for each form of resource(including ramp rates,auxiliary losses,heat rates,and storage efficiency),are based on the Integrated System Plan(I
149、SP)produced by the market operator(AEMO,2022a)and set out in Table 1.11 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.Table 1:Resource Cost and Technical Assumptions Resource
150、 ()*($/MWh)($/kW)*($/kW/yr)/(%/min)/(%)Heat Rate-HHV Aux Losses(%)Black Coal 40 0#55 0.3-9.0 6.7 Gas CCGT 55 1792 11 0.02-7.5 1.8 Gas OCGT 132 898 16 10-10.9 0.7 Liquid Fuel RE 375 1400 16 10-10.5 0.6 Solar PV 0 936 18 100-0.2 Wind 0 1959 26 100-0.3 1-hr BESS 0 706 7 100 0.9-2-hr BESS 0 859 11 100 0
151、.9-4-hr BESS 0 1220 17 100 0.9-8-hr BESS 0 1971 28 100 0.9-*The base variable cost(AEMO,2022a)is indexed by fuel indices as below.Figures in A$unless indicated.Black coal capital costs assumed to be wholly written down.The time-series data covers a period of approximately 20 years(from January 2004
152、to October 2023),which results in 237 scenarios each of approximately a month in duration10.To reflect the inherent correlations and co-movement between different time-variant parameters,the time-series data is sourced from actual historical data where available,and supplemented by a calibrated back
153、cast where historical data is not available.Generation availability for thermal plant is based on the historical availability of thermal plant in the NEM over using the NEMOSIS package and based on data from AEMOs NEMWEB repository(AEMO,2022c;Gorman et al.,2018).In the absence of long-term historica
154、l data on wind and solar availability for modern turbines of the scale and form implemented in the NEM,a back-casting approach is adopted.We select sites from nine of the largest wind and solar generators in the NEM11,and then simulate wind and solar availability using Renewables.ninja(Pfenninger&St
155、affell,2016),which is derived from reanalysis models and satellite observations.The technical parameters of each wind farm(turbine model,hub height etc)were sourced from AEMO Generation Information(AEMO,2022b)and developers websites.For solar,the azimuth was set based on the latitude of each farm,sy
156、stem loss was set at 1 per cent,an azimuth of 180 with farms having single-axis tracking.Electricity demand is based on historical gross operational demand,which is adjusted for an assumed penetration of consumer energy resources(CER),in the form of rooftop solar.To do so,the historical demand is fi
157、rst grossed up for actual historical consumer energy generation to create a set of scenarios for native demand.To integrate CER,a rooftop solar generation time-series is created using Renewables.ninja.Scheduled demand is then calculated by netting off the CER generation from native demand based on a
158、n assumed penetration of rooftop solar in each NEM mainland region.To ensure appropriate calibration against actual data,we follow the approach in Gilmore et al(2022)to rescale the Renewables.ninja time-series for wind,solar and CER availability,which is calibrated against actual availability data f
159、rom 10 Adjusted to keep all scenarios with the same number of dispatch intervals.11 These generators comprise for wind Stockyard Hill Wind Farm,Coopers Gap Wind Farm,Hornsdale Wind Farms(1-3),Dundonnell Wind Farm,Moorabool Wind Farm,Gullen Range Wind Farm,Macarthur Wind Farm,Sapphire Wind Farm,Silve
160、rton Wind Farm,and Lincoln Gap Wind Farm;and for solar Darlington Solar Farm,Daydream Solar,Coleambally Solar Farms,Limondale Solar,Finely Solar Farm,Ross River Solar,Sunraysia Solar,Bungala One Solar Farm,and Nevertire Solar Farm.12 The contents of this paper are the authors sole responsibility.The
161、y do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.2021 and 2022 calendar years.Linear interpolation is used to convert from hourly to half-hourly trading intervals.Generation fuel costs over time are indexed based on actual historical monthly a
162、verages of fuel prices,with natural gas prices sourced from the Short-Term Trading Market(STTM)Sydney Hub,coal prices from the front-month Newcastle Coal futures contract(6000kcal/kg net calorific value,translated to AUD at the prevailing exchange rate),liquid fuel at the Australian Institute of Pet
163、roleum(AIP)diesel terminal gate price.Dispatch is cleared at half-hourly trading intervals,with the supplier and consumer maximising their risk-averse utility,based on approximately monthly12 aggregations of surplus.The monthly aggregation is adopted for two reasons.First,electricity consumers are b
164、illed on a monthly basis in the NEM,and thus is a natural point at which consumer would observe bill volatility.While some retailers also report metering over shorter periods a weekly or daily basis this is not consistent across the market.Second,the monthly period is also relevant for generators an
165、d gentailers.While debt service and financing covenant reporting typically occurs over quarterly(or longer)periods,the monthly reporting cycle is important in terms of corporate liquidity and credit assessment.As such,a month was considered an appropriate period over which consumers and generators m
166、ay assess and comprehend risk.In understanding supplier and consumer appetite for hedging,four resource cases are considered,reflecting different stages of the energy transition,based on the capacity mix trajectory outlined in the NEM ISP.This is intended to approximate an emissions trajectory progr
167、essing towards legally binding 2015 Paris Agreement targets.To approximate the NEMs legacy portfolio and the path towards net-zero,the model in Case 1 is initiated with an exogenous resource capacity for thermal,renewable,and storage resources for the 2017-18 year(as set out in the 2016 National Tra
168、nsmission Network Development Plan),with the capacity of OCGT firming resources sized to an expected or risk-neutral utility measure of zero.This additional resource capacity is required to supplement for hydro-generation and other generation resources,which are not modelled in this study.This resul
169、ts in a EUSE of 0.0013 per cent,which is well below the threshold of 0.002 per cent as per the NEMs reliability standard,thereby implying that existing market intervention measures to procure additional capacity(such as the Reserve and Emergency Trader function)would be unlikely to be utilised.For t
170、he remainder of the cases,the model is initiated with resource capacity that aligns with the relevant forecast year per the 2022 ISP(Low VRE:2023-24,Mid VRE:2029-30,and High VRE:2035-36).Firming resources are then adjusted to ensure that the EUSE in higher VRE scenarios is in line with the original
171、legacy system,(equating to 0.0013 per cent),noting this does imply higher levels of OCGT gas than that set out in ISP projections.For the VRE Cases(Cases 2-4),the investment costs of renewable generation and storage are discounted to reflect potential cost reductions and low-carbon subsidy schemes t
172、o drive low-carbon deployment to reach Australias net zero target.The level of discount is sized to an expected or risk-neutral utility measure of zero.While it is recognised that renewable generation has to date been subsided via a production-based credit,there are potential impacts associated with
173、 this type of structure on bidding and price formation.Thus,this paper abstracts from the question of the optimal subsidy form via the capital cost discount.12 The aggregation is 30.4 days rather to ensure that each scenario is of the same length,so all have the same number of dispatch intervals.13
174、The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.Table 2:Resource Capacity(GW)by Case Case Thermal Dominated 1 Low VRE 2 Mid VRE 3 High VRE 4 Black Coal 24700 21300 9000 3000 Ga
175、s CCGT 4400 4100 4100 2600 Gas OCGT*10300 11180 18700 25900 Liquid Fuel RE 700 700 700 700 Solar PV 200 8400 12200 18700 Wind 3700 11500 31500 42900 1-hr BESS-190 190 190 2-hr BESS-560 560 720 4-hr BESS-250 3000 4300 8-hr BESS-160 800 CER-Rooftop PV Peak MW 10100 10100 10100 10100 Renewable Gen Shar
176、e(%)6 28 61 79 Emissions(mil.kg-CO2eq/mo)11.4 8.5 4.0 1.9 ISP/NTNDP Base Yr 2017-18 2023-24 2029-30 2035-36*Denotes firming resource.2.2 Calibration and Price Statistics The statistical parameters of the estimated backcasted availability of wind and solar(over the hours 0500-2000)over the period of
177、the case study are compared with actual outcomes over dispatch intervals between 2020-21,showing good calibration of low order statistical moments.The density plots also are a reasonable comparison against the 2020-21 actuals,though we specifically call out a disparity in the distribution of actual
178、and backcast outcomes over higher availabilities.The backcast data tends to be more concentrated at approximately 0.7 with minimal availability predicted beyond 0.8,where the actual data tends be more smoothly distributed across such horizons.As such it is possible that the backcasted data for solar
179、 may represent a more conservative perspective on resource risk as it does not reach availability points as high as the actual data.14 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its M
180、embers.Figure 5:Statistical Parameters and Density Plot of Wind and Solar Availability Wind Solar(b/w 0500-2000hrs)Actuals 20-21 Backcast Actuals 20-21 Backcast Mean 0.29 0.29 0.36 0.37 Std.Dev 0.15 0.14 0.30 0.28 P10 0.11 0.11 0.00 0.00 P50 0.28 0.29 0.36 0.40 P90 0.49 0.49 0.80 0.73 Note:Solar ava
181、ilability between 0500-2000hrs.For the purposes of this paper,it is important to consider the calibration of the left tails of wind and solar over extended periods of low availability.In the absence of relevant long-term historical data we compare the periods of lowest average availability(Figure 6)
182、for the backcast estimates for wind,solar,and 70/30 and 60/40 wind-solar portfolios,with a recent long-term reanalysis study of renewable droughts in the NEM by Gilmore et al(2022).The backcast data shows the worst availability increasing from 0-0.2 over a single day to 0.4-0.6 over a 10-day period,
183、tapering out thereafter.These levels are consistent with Gilmore et al(2022),if not a slightly conservative estimate of the extended availability of wind and solar.15 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute fo
184、r Energy Studies or any of its Members.Figure 6:Periods of Lowest Average Availability Table 3 provides the statistical metrics of wholesale spot price outcomes under each of the four cases,compared with historical prices between 2002 and 2022 for two mainland regions,New South Wales(NSW)and Victori
185、a(VIC).As the modelling exercise aims to identify directional trends rather than recreate price outcomes,the price calibration is concerned primarily with orders-of-magnitude comparisons between the Thermal Dominated(Case 1)and historical outcomes.Differences can be attributed to a range of factors
186、including,but not limited,to the impacts of network and security constraints,bidding dynamics,and the extent of market competition.Table 3:Price Statistics Modelled Historical 2002-22 Case Thermal Dominated 1 Low VRE 2 Mid VRE 3 High VRE 4 NSW VIC Price Statistics($/MWh,Load Weighted Monthly Basis)M
187、ean 66 64 66 61 56 52 Std.Dev 63 60 58 75 35 38 Skew 4 4 5 9 3 3 Kurtosis 18 15 31 100 10 19 -VAR =0.5 47 46 49 45 49 40 =0.9 108 109 103 88 92 99 =0.95 174 193 151 114 105 111 =0.99 329 365 365 346 199 182 16 The contents of this paper are the authors sole responsibility.They do not necessarily rep
188、resent the views of the Oxford Institute for Energy Studies or any of its Members.2.3 Results This section provides the modelled results on the insurability of long-term hedges in electricity systems.First,we consider the willingness of the supplier to offer,and consumers to purchase long-term hedge
189、s under differing risk attitudes under a set of exogenously determined resource capacity mixes.Using the Thermal Dominated case as an illustration,Figures 10-12 show a set of density plots(heatmaps)under different risk parameters risk-aversion and tail risk thresholds.Figure 7 sets out the minimum c
190、ontract price required for a supplier to provide a full volumetric risk hedge for a range of risk attitudes 0.0 1.0 and 0.7 0.99.(the contract price at which risk-averse utility of the supplier is zero).The minimum contract price is provided on a relative basis as a proportion of the risk-neutral or
191、 expected value of the hedge cashflows.Figure 8 records the proportion of scenarios in the suppliers problem that have a negative surplus,for the same range of and values.It is observed that at low levels of risk-aversion and low tail risk thresholds(relatively low values of and,respectively),the su
192、pplier would be willing to execute contracts at,or close to the expected value of spot prices.However,for such risk parameters the supplier experiences negative surplus in a higher proportion of scenarios,where total revenues from the spot and contract markets in those scenarios are insufficient to
193、pay fixed and variable costs(for example,26 per cent in the risk-neutral case).The key implication for the deliverability of contracts relates to potential counterparty credit or solvency risk given insufficient revenue in a higher proportion of scenarios.At higher levels of risk-aversion and risk c
194、omprehension,solvency risk improves(with negative surplus in a much lower percent of scenarios,as per 1 and and do:5 (),(),solve()#estimate of at iteration k 5 ()=(1)+()(1)+#increment to estimate of()6 +1()=()+()#updated estimate of()7 +1 8 end After preliminaries in lines 1 and 2,to iteratively cal
195、culate the value of()we begin in line 3 by setting an initial value at()=0 and running,which obtains initial estimates of and.An incremental improvement to the estimate of()is calculated in lines 5 and 6.Successive iterations converge to the final value of(),which is confirmed when the zero-utility
196、condition is reached|.29 The contents of this paper are the authors sole responsibility.They do not necessarily represent the views of the Oxford Institute for Energy Studies or any of its Members.Appendix D:Derivation of Total Inframarginal Rents and Contribution Factor The dual of is calculated as
197、 below:max(+).+(+).+(D1a)+,+1,+,+1,=0 ,/1,(D1b)1 1+1 1,2,+,2,=0 ,=1,(D1c)+=0 ,=,(D1d)+=0 ,(D1e)+=0 ,(D1f)+=0 ,+(D1g)+,+1,=0 ,/,(D1h)+1=0 ,(D1i)The Karush-Kahn Tucker complementarity conditions of are:()0 ,(D2a)0 ,(D2b)0 ()0 ,(D2c),0 ,(D2d)0 (+(1)0 ,/1,(D2e)0 (1)+)0 ,/1,(D2f)(+)+0 ,(D2g)+0 ,(D2h)()0
198、,(D2i)0 ,(D2j)()0 ,(D2k)0 ,(D2l)By combining equations(D1b-d)with(D2a-f)it can be observed that the total gross margin of a generation resource across a scenario (the gross revenues net off variable costs)().is equivalent to.Thereby,the following term reflects the unit contribution of resource to to
199、tal supplier profits in scenario.=/=.+(B3)Similarly,for storage resources the equations(D1f-i)can be combined with(D2g-l)to indicate the unit contribution of a storage resource to total supplier profits in scenario as:30 The contents of this paper are the authors sole responsibility.They do not nece
200、ssarily represent the views of the Oxford Institute for Energy Studies or any of its Members.=/=+1 (B4)The dual of can be written as:min,(1 )+(B5a)=(B5b)=+(B5c)(1)()()=0 (B5e)The vector(1 )+represents the risk-weighted probability of each scenario (Shu&Mays,2023).Thus,the marginal contribution facto
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