Hedging Strategies for Load-Serving Entities in Wholesale Electricity Markets
HHedging Strategies for Load-Serving Entitiesin Wholesale Electricity Markets
Datong P. Zhou ∗† , Munther A. Dahleh † , and Claire J. Tomlin (cid:63) Abstract — Load-serving entities which procure electricityfrom the wholesale electricity market to service end-users facesignificant quantity and price risks due to the volatile nature ofelectricity demand and quasi-fixed residential tariffs at whichelectricity is sold. This paper investigates strategies for loadserving entities to hedge against such price risks. Specifically,we compute profit-maximizing portfolios of forward contractand call options as a function of uncertain aggregate userdemand and wholesale electricity prices. We compare the profitto the case of Demand Response, where users are offeredmonetary incentives to temporarily reduce their consumptionduring periods of supply shortages. Using smart meter dataof residential customers in California, we simulate optimalportfolios and derive conditions under which Demand Responseoutperforms call options and forward contracts. Our analysissuggests that Demand Response becomes more competitive aswholesale electricity prices increase.
I. I
NTRODUCTION
Historically, electricity was supplied by vertically inte-grated entities which maintained full functional control overthe entire supply chain, including generation, transmission,and distribution assets. This static structure constituted animpediment for new energy providers on both the supplyand retail end to participate in the energy market. In theUnited States, the Federal Energy Regulatory Commissionissued Orders 888 and 889 in April 1996 to remove suchbarriers of entry in an attempt to promote competition andmarket efficiency [1], [2]. The result of this market designprocess was a combination between a central electricitypool operating day-ahead, overseen by Independent SystemOperators (ISOs), and bilateral trading between generatingcompanies and electric utilities, which supplanted the tradi-tional, vertically integrated entities.As a consequence of the restructuring process, generatorsand utilities in the electricity market started facing price andquantity risks ensuing from the inelasticity of user demand,the steep supply curve due to the slowly changing natureof power plants’ output adjustment, and prohibitive costof energy storage. These factors allow small increases ordecreases of demand to result in a price boom or bust,respectively. Furthermore, despite the fact that the economicconsensus calls for passing along varying electricity pricesto end-users in order to increase economic efficiency [3], ∗ Department of Mechanical Engineering, University of California, Berke-ley, USA. [email protected] † Laboratory for Information and Decision Systems, MIT, Cambridge,USA. [datong,dahleh]@mit.edu (cid:63)
Department of Electrical Engineering and Computer Sciences, Univer-sity of California, Berkeley, USA. [email protected]
This work has been supported in part by the National Science Foundationunder CPS:FORCES (CNS-1239166) and CEC Grant 15-083. [4], [5], policymakers have retained quasi-fixed electricitytariffs, e.g. Time-of-Use pricing. In conjunction with theobligation of utilities to service end-users with electricityat all times, risks associated with sudden price spikes areborne by the utility. This market situation has resulted inseveral crises. For instance, unseasonably warm climatein the summer of 2000 resulted in California’s wholesaleelectricity prices to rise to average prices of more than140 USD/MWh, leading to the bankruptcy of Pacific Gas& Electric, California’s largest utility, and high profits ofelectricity generators [6]. Similar crises occurred in Texas(2004) and in the Midwestern United States (1998).These crises resulted in the following notable develop-ments. Firstly, electric utilities and generating companiesstarted to hedge against price fluctuations through contractson different scales of time, ranging from short-term forwardcontracts to long-term contracts, thereby locking in a fixedprice and quantity to be delivered over a contractually spec-ified period of time. Secondly, Demand-Side Management(DSM), which aims to affect consumer behavior duringperiods of peak demand, emerged as a viable tool to par-tially relay price risks to end-users. For instance, companieslike
OPOWER provide Demand Response (DR) services toutilities, allowing them to offer monetary rewards to end-users in exchange for a reduction in electricity consumptionduring hours of peak demand.Motivated by these shortcomings, a large body of research,particularly in operations research, has studied optimal hedg-ing contracts, most often from the utility perspective, includ-ing [7], [8], where the authors construct an optimal one-stephedging portfolio with standard power options, or [9], whichfinds an optimal energy procurement policy with stochasticprogramming over a specified period. [10] analyzes hedginginstruments against price volatility for industrial customers.[11] investigates hedging strategies for electricity generators.While there exists a large body of literature on operationaland algorithmic aspects of DR (e.g. load scheduling andshifting [12], [13], [14]), significantly less research hasfocused on the role of DR programs as an alternative wayof hedging. Notable examples are [15], where the authorsdiscuss interruptible service contracts, and [16], which esti-mates the economic value of DR programs for commericalcustomers by adapting models used to value energy options.To the best of our knowledge, no significant research hasinvestigated the option value of residential DR programs. Toclose this gap, we derive a stylized model for the utility’sprofit under such DR schemes and determine its optimalprofit. The methodology we use is closest in spirit with [17], a r X i v : . [ c s . S Y ] M a r here the authors determine the optimal bidding volumeof wind generators in a conventional energy market. Wecompare the profit under Demand Response to the caseof forward contracts and call options by incorporating theconditional value at risk [18] measure. Using smart meterdata of residential customers in California, we find thatDR can yield higher expected profits than under forwardcontracts and call options, especially in the presence of highwholesale electricity prices.The remainder of this paper is organized as follows:In Section II, we describe the interactions between theparticipants in energy markets. Section III introduces for-ward contracts, options, and Demand Response as hedginginstruments for the Demand Response Provider. The effectof uncertainty in the user demand on the expected profit ofthe Demand Response Provider is investigated in Section IV.We compute optimal, profit-maximizing portfolios for load-serving entities in Section V and simulate decision bound-aries between them in Section VI. Section VII concludes.All proofs are relegated to the Appendix. Notation
Let E [ · ] denote the expectation of a random variable. Let [ · ] + denote the hinge function, i.e. [ x ] + = max(0 , x ) .II. M ARKET P ARTICIPANTS
Figure 1 illustrates the interaction between generatingcompanies, load-serving entities (utilities), the wholesaleelectricity market, and end-users of electricity.
Generators
Wholesale Market
Utility End Users λ f , r d, h ( r )[ d − ¯ q ] + λ s Supply Payment ¯ q ¯ λ, P Fig. 1: Energy Market Participants and their Interactions
The electric utility can strike one-to-one contracts withgenerating companies to purchase a fixed amount of elec-tricity ¯ q at a locked-in price ¯ λ to be delivered at some a-priori specified time in the future. P denotes the premiumfor each reserved unit of electricity. The utility provides end-users with electricity at a fixed unit rate λ f and is obligatedto cover the random demand d at all times. The rate λ f isexogenously set by the Public Utilities Commission. How-ever, the utility can use DR to incentivize users to temporarilyreduce their demand. This is achieved by offering the reward r to end-users, which elicits a demand reduction h ( r ) . If thedemand d exceeds ¯ q , i.e. the purchased amount of electricity through one-to-one contracts with generators, the utility hasto procure [ d − ¯ q ] + units of electricity from the wholesalemarket at uncertain wholesale price λ s per unit. The marketclearing price λ s , reflected by Locational Marginal Prices(LMPs), is a random variable and depends on the ratio ofenergy supply by generators, the total demand [ d − ¯ q ] + ,operational constraints, as well as congestion of the grid.The interactions between generators and the utility aswell as between end-users and the utility are instruments tohedge utilities against high prices λ s . If the utility expectshigh wholesale prices λ s , then it has an incentive to reducecustomer demand d by engaging in DR, or to procure cheaperelectricity through contracts with generating companies. Wemake the following assumptions: Assumption 1.
The utility is risk-neutral.
Assumption 2.
The utility is price-taking.
Assumption 2 is a natural assumption, stating that theutility cannot influence prices by exerting market power.Together with Assumption 1, the question we seek to answerin the remainder of this paper is how the utility maximizesits expected profit in the presence of the random variables d and λ s and hedging instruments. For simplicity, we focus ona single load zone to avoid spatial heterogeneity of LMPs.III. O PTIMAL H EDGING S TRATEGIES
Let λ s and d be random variables with cumulative distri-bution functions (CDF) G ( · ) and F ( · ) , respectively. G and F are assumed to have support [0 , ∞ ) and [ d min , d max ] , respec-tively, where ≤ d min ≤ d max . We assume the absence ofany energy storage capabilities and focus on a single-periodsetting, where the LSE can purchase hedging instruments attime 0, possessing only an estimate of consumer demand d and real-time spot price λ s at time 1. At time 1, the randomvariables d and λ s materialize and the LSE’s profit Π asa function of the hedging instruments purchased at time 0is determined. Figure 2 illustrates the hedging process. TheLSE aims to maximize its expected profit E [Π] by decidingon its portfolio of hedging instruments at t = 0 . ¯ λ, P, h ( r ) are announced.Utility purchases hedginginstruments to maximize E [Π] as function of random d, λ s . t = 0 d and λ s materialize.Profit Π is determined.Settlements between generators,LSE, and end users take place. t = 1 Fig. 2: Timeline of Hedging
In the following, we analyze the cases for (a) no hedginginstruments, (b) forward contract, (c) call option, and (d) DRand derive explicit expressions for the optimal contracts andcorresponding profits for cases (b)-(d).
A. Base Case (No Hedging)
If the LSE does not buy any options at stage 0, its expectedprofit at time 1 is simply E [Π] = ( λ f − E [ λ s ]) · E [ d ] . (1)We will compare the profit of this base case to the forwardcontract, call option, and DR in the following. . Forward Contract A forward contract is a one-on-one agreement betweenthe LSE and an electricity generator, which obligates thegenerator (at time 0) to deliver a fixed amount of electricity ¯ q at a locked-in price ¯ λ F to the LSE at some point in thefuture (time 1). Forward contracts possess high flexibilityand are traded as over-the-counter products. The LSE seeksto sign such a contract if it has reason to believe the expectedwholesale price at the time of delivery to exceed ¯ λ F , and thegenerator will do so in the opposite case. If, at time 1, ¯ q > d ,the LSE has purchased too much volume at time 0, and so ¯ q − d are wasted. Conversely, if ¯ q < d at time 1, d − ¯ q unitsof electricity have to be bought at real-time spot price λ s .The profit Π F under a forward contract of volume ¯ q atunit price ¯ λ is therefore expressed as Π F = λ f d − ¯ λ F ¯ q − λ s [ d − ¯ q ] + . (2) Theorem 1 (Optimal Forward Contract) . With E [ λ s ] > ¯ λ F ,the optimal contract volume ¯ q ∗ and the optimal expectedprofit E [Π ∗ F ] become ¯ q ∗ = F − (cid:18) − ¯ λ F E [ λ s ] (cid:19) , (3a) E [Π ∗ F ] = λ f E [ d ] − E [ λ s ] (cid:90) ∞ F − (cid:16) − ¯ λF E [ λs ] (cid:17) xf ( x ) dx. (3b) C. Call Option
Similar to fixed forward contracts, the LSE can strike one-on-one deals with a counterparty over an agreed volume ¯ q atstrike price ¯ λ C . The key difference is that the LSE can, butis not obliged to, exercise the call option if ¯ λ C < λ s at time1. Typically the buyer of the call option pays a premium P for each unit of the call option.The profit Π C under a call option with volume ¯ q at strikeprice ¯ λ C at the premium P per unit can thus be written as Π C = λ f d − λ s [ d − ¯ q ] + − P ¯ q − min(¯ λ C , λ s ) · min( d, ¯ q ) . (4)The last term of (4) encodes the fact that the LSE cancover up to ¯ q units at the cheaper of the strike price ¯ λ C or the wholesale price λ s . The remainder [ d − ¯ q ] + has to bepurchased from the spot market at price λ s . Theorem 2 (Optimal Call) . With E [ λ s ] > P + ¯ λ C − (cid:82) ¯ λ C G ( y ) dy , the profit-maximizing call volume ¯ q ∗ and thecorresponding optimal expected profit E [Π ∗ C ] are ¯ q ∗ = F − (cid:32) − P E [ λ S ] − ¯ λ C + (cid:82) ¯ λ C G ( y ) dy (cid:33) , (5a) E [Π ∗ C ] = (cid:32) λ f − ¯ λ + (cid:90) ¯ λ C G ( y ) dy (cid:33) E [ d ] (5b) − (cid:32) E [ λ s ] − ¯ λ C + (cid:90) ¯ λ C G ( y ) dy (cid:33) (cid:90) ∞ ¯ q ∗ xf ( x ) dx. D. Demand Response
We model the effect of demand response as a shift in thedistribution of the consumer towards zero, induced by themonetary reward r ∈ R + transferred from the LSE to theconsumer as a lump sum. Note that the real reduction of theconsumer in response to the DR signal has to be estimated byconstructing the counterfactual consumption in the absenceof the DR signal, whose estimation is beyond the scope ofthis paper. The interested reader is referred to [19], [20].Let f ( d ) denote the probability density function of d in theabsence of any reward with support [ d min , d max ] . Let F ( d | r ) denote the cumulative distribution function of the randomvariable d , given the reward level r . Then the distributionshift is modeled as F ( d | r ) = (cid:40) , if d < d min F ( d + h ( r )) , if d ≥ d min (6)where h ( r ) is a concave, increasing function representingthe elasticity of the user in response to reward r , i.e.the relative reduction of consumption as a function of r . h ( r ) is equivalent to the shift of the location parameter ofdistribution f ( · ) . We make the following Assumption 3.
The reward r ≥ induces a linear shift, i.e. h ( r ) = αr, α > . (7)With Assumption 3 and the definition of the distributionshift, it becomes clear that the distribution f ( ·| r ) , given areward r > , has support [ d min , d max − h ( r )] with discretemass (cid:82) d max d max − h ( r ) f ( x ) dx at d min .Assumption 3 is necessary for analytical tractability ofthe DR hedging case. We note that the linearity of h ( r ) is unrealistic, since it implies that for large enough rewardlevels r , the user consumes zero with probability 1. However,for small reward levels, a linear price elasticity of demand h ( r ) can be justified.The LSE’s profit Π DR with Demand Response is Π DR = ( λ f − λ s ) d ( r ) − r. (8)From (8), it immediately follows that DR only makes sensein the presence of large expected spot prices E [ λ s ] at time1 which exceed the fixed contractual price λ f . Then theoptimal profit Π ∗ DR is the minimal expected loss of the LSE. Theorem 3 (Optimal Demand Response) . With E [ λ s ] > λ f ,the profit-maximizing reward r ∗ and the optimal expectedprofit E Π ∗ DR are r ∗ = (cid:40) α F − (cid:16) − α · ( E [ λ s ] − λ f ) (cid:17) , if α < E [ λ s ] − λ f , otherwise (9a) E Π ∗ DR = (cid:40) ( λ f − E [ λ s ]) (cid:82) ∞ αr ∗ xf ( x ) dx, if α < E [ λ s ] − λ f ( λ f − E [ λ s ]) E [ d ] , otherwise (9b)The condition α > ( E [ λ s ] − λ f ) − for the optimal rewardmeans that the ability to shift, /α , must be greater than thenverse of the expected price difference ( E [ λ s ] − λ f ) − tomake DR profitable. The higher the expected price difference E [ λ s ] − λ f , the less stringent the requirement on α , whichagrees with intuition. Theorem 4 (Diversified Portfolios) . For general demanddistributions, the optimal portfolio can either consist of aunique option or a combination of call and forward contractoptions, but never of a combination of DR and either call orforward contract options. For the special case of a uniformdemand distribution, the optimal portfolio always consists ofa unique option, i.e. diversified portfolios consisting of morethan one option are always suboptimal.
Depending on the properties of the demand distribution F ( · ) , a mixed portfolio of call and forward contract optionscan exist, but is impossible to obtain in closed form forgeneral distributions. This is consistent with the approach in[7] where the authors replicate the optimal portfolio (whichwould be continuous) with a finite set of options. Due toTheorem 4, we restrict our attention to optimal portfoliosconsisting of a unique option in the remainder of this paper.IV. T HE E FFECT OF U NCERTAINTY
For a better understanding of the optimal profits under thedifferent contracts Π ∗ F , Π ∗ C , Π ∗ DR introduced in the previoussection, we relate these quantities to properties of the con-sumption distribution F ( · ) . A. Influence of Distribution Tail
By incorporating the
Conditional Value-at-Risk (CVaR)measure [18], we can relate the optimal profits to the tailproperties of the consumption density f ( · ) . The CVaR atconfidence level α ∈ (0 , of a random variable X withCDF F ( · ) representing loss is formally defined asCVaR α ( X ) = E [ X | X ≥ F − ( α )] (10)and can be interpreted as the expected loss attained in theworst (1 − α ) · of cases or the expectation of the (1 − α ) probability tail of X . With this definition, the optimalexpected profits under the different options Π ∗ F , Π ∗ C , and Π ∗ DR are reformulated in Proposition 1. Proposition 1.
With α > ( E [ λ s ] − λ f ) − and the definitionof CVaR, the optimal expected profits under the forward con-tract E [Π ∗ F ] , the call option E [Π ∗ C ] , and Demand Response E [Π ∗ DR ] can be expressed as follows: E [Π ∗ F ] = λ f E [ d ] − ¯ λ F E [ d | d ≥ F − (1 − ¯ λ F / E [ λ s ])]= λ f E [ d ] − ¯ λ F · CVaR α F ( d ) (11a) E [Π ∗ C ] = (cid:32) λ f − ¯ λ C + (cid:90) ¯ λ C G ( y ) dy (cid:33) E [ d ] (11b) − P · CVaR α C ( d ) E [Π ∗ DR ] = − α · CVaR α DR ( d ) (11c) where we used the definitions α F = 1 − ¯ λ F E [ λ s ] (12a) α C = 1 − P E [ λ s ] − ¯ λ C + (cid:82) ¯ λ C G ( y ) dy (12b) α DR = 1 − α · ( E [ λ s ] − λ f ) (12c)From Proposition 1, it follows that the optimal profitdecreases as the conditional expectation of the tail increases,that is, the more heavy-tailed the consumption distribution f ( · ) becomes. It is illustrative to analyze the optimal deci-sions and corresponding optimal expected profits for perfectinformation of d , which are given in the following: ¯ q ∗ F | d = d, ¯ q ∗ C | d = d, r ∗ | d = d/α E [Π ∗ F | d ] = ( λ f − ¯ λ F ) · d (13a) E [Π ∗ C | d ] = (cid:32) λ f − ¯ λ C + (cid:90) ¯ λ C G ( y ) dy − P (cid:33) d (13b) E [Π DR | d ] = − d/α (13c) ¯ q ∗ F | d and ¯ q ∗ C | d denote the optimal forward contract and callvolume, respectively. r ∗ | d signifies the optimal DR reward. B. Influence of Statistical Dispersion
In this section, we attempt to construct a relationshipbetween the statistical dispersion of the consumption distri-bution F ( · ) and the optimal expected profit. Intuitively, themore spread out the distribution F ( · ) , the lower the expectedprofit. While many measures for statistical dispersion exist inthe literature, such as interquartile ranges, absolute deviation,variance-to-mean-ratio, etc., we express the optimal expectedprofits E [Π ∗ F ] , E [Π ∗ C ] , and E [Π ∗ DR ] in terms of the standarddeviation σ for the special case of a uniform distributionwith support [ d min , d max ] for expositional ease and analyticaltractability. Proposition 2.
For the uniform distribution F ( · ) with sup-port [ d min , d max ] , the optimal expected profits under theconditions E [ λ s ] > max (cid:16) ¯ λ F , P + ¯ λ C − (cid:82) ¯ λ C G ( y ) dy (cid:17) and α > ( E [ λ s ] − λ f ) − are expressed as follows: E [Π ∗ F ] = λ f E [ d ] − ¯ λ F d min − √ E [ λ s ](1 − α F ) σ (14a) E Π ∗ C = (cid:32) λ f − ¯ λ C + (cid:90) ¯ λ C G ( y ) dy (cid:33) E [ d ] − P d min (14b) − √ (cid:32) E [ λ s ] − ¯ λ C + (cid:90) ¯ λ C G ( y ) dy (cid:33) (1 − α C ) σ E [Π ∗ DR ] = − d min /α − √ E [ λ s ] − λ f )(1 − α ) σ (14c)For the case of perfect information, i.e. σ = 0 and d min = d max = d , the equations for the optimal expectedprofit under perfect information (13a)-(13c) are recovered.Equations (14a)-(14c) explain that the optimal expectedprofit for each case decreases linearly in σ , giving rise tothe notion that more “spread out” distributions diminish thexpected profit. The rate of decrease depends on case-specificparameters, whose relation to each other determines whichhedging option is profit-maximizing for a particular case. Asconsumption distributions typically are plagued by a largeamount of uncertainty (large σ ), improved load predictionsto decrease σ have a direct economic benefit to the utility.V. C HOOSING THE B EST O PTION
We now derive conditions on the random variables λ s and d with distributions G ( · ) and F ( · ) and the option parameters ¯ λ F , ¯ λ C , P , and α announced at time 0 to determine the besthedging strategy consisting of a unique option. For analyticaltractability, we make the following assumptions: Assumption 4.
The real-time spot price λ s is uniformly dis-tributed with support [0 , s max ] , i.e. G ( y ) = s max ≤ y ≤ s max . Assumption 5.
The consumption is uniformly distributed in [0 , d max ] , i.e. F ( x ) = d max ≤ x ≤ d max . Theorem 5.
Under Assumptions 4 and 5 and E [ λ s ] > λ f ,the forward contract is preferred over the call option, if ¯ λ F ≤ E [ λ s ] − E [ λ s ] − ¯ λ C + ¯ λ C / (4 E [ λ s ]) − P (cid:113) − ¯ λ C − ¯ λ C / (4 E [ λ s ]) E [ λ s ] . (15) DR is preferred over the forward contract, if α ≤ ( E [ λ s ] − λ f ) (cid:34) − (cid:115) E [ λ s ] E [ λ s ] − λ f (cid:18) − ¯ λ F E [ λ s ] (cid:19)(cid:35) . (16) Finally, DR is preferred over the call option, if α ≤ ( E [ λ s ] − λ f ) (cid:34) − (cid:115) L ( E [ λ s ] − λ f ) (cid:18) − PL (cid:19)(cid:35) . (17) with L = ( E [ λ s ] − ¯ λ C + ¯ λ C ) / (4 E [ λ s ]) and where ¯ λ F and ¯ λ C denote the unit price for each reserved unit of electricityunder the forward contract and the call option, respectively. VI. S
IMULATIONS
Assumptions 4 and 5 admitted a closed form solutionto the best hedging instrument, stated in (15)-(17). For amore elaborate analysis, we now repeat this exercise byapproximating the demand distribution F ( · ) as well as thedistribution of spot prices G ( · ) with real data from Californiato approximate decision boundaries for which the expectedprofits under different hedging instruments are identical.Since closed-form solution under this more realistic scenariodo not exist, we plot these optimal decision boundaries as afunction of the hedging parameters P, ¯ λ F , ¯ λ C , and α . A. Empirical Distribution of Demand
We use hourly smart meter data from residential customersin California from the utilities Pacific Gas & Electric, SanDiego Gas & Electric, and Southern California Edison tocreate a demand distribution for different sizes of useraggregations. The observations are restricted to hourly con-sumptions between 4-5 pm and 5-6 pm. Figure 3 shows the empirical PDFs and CDFs for different sizes of useraggregations. We approximate both functions as follows: ˆ f ( x ) = a ( x − d min ) e − cx , a, c ∈ R + , x ∈ [ d min , d max ] (18a) ˆ F ( x ) = ac ( cd min − cx − e − cx + γ, γ ∈ R (18b)With the constraints ˆ F ( d min ) = 0 and ˆ F ( d max ) = 1 , theparameters a and γ can be found as a function of the decayparameter c . It can be seen that the approximations (18a) and(18b) fit the observed data reasonably well.
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Aggregate Consumption [kWh] (250 users)
Empirical CDF
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MeanFitEmpirical100 150 200 250 300
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MeanFitEmpirical
Aggregate Demand: Empirical Distributions
Fig. 3: Distribution of Aggregate Hourly Consumption for Varying Aggre-gation Sizes, 4-6 pm. Top: 250 Users, Middle: 150 Users, Bottom: 50 Users.
B. Empirical Distribution of Wholesale Prices
To obtain the price distribution G ( · ) , we convert 5-minutelocational marginal prices (LMPs) λ s set by the CaliforniaIndependent System Operator into an hourly format. Thedistribution G ( · ) of “high” LMPs is obtained by fitting adensity function to the normalized histogram of those LMPsfor which the two previous LMPs exceed the threshold ξ > ,i.e. we consider all { λ s | λ s,t − ≥ ξ, λ s,t − ≥ ξ } for differentthresholds ξ . We approximate the density function with alog-normal distribution: N (ln x ; µ, σ ) = 1 σ √ π exp (cid:18) − (ln x − µ ) σ (cid:19) (19)which has support [0 , ∞ ) , that is, we disregard negativeLMPs. Figure 4 shows the observed data and the approx-imations for thresholds ξ = 80 , , USDMWh . C. Pairwise Comparison of Hedging Instruments
We now compute decision boundaries of equal expectedprofit for all 3 pairs of hedging instruments with Newton’smethod, using the demand and price distributions derived in(18a), (18b), and (19).
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LMP | (LMP( t − , LMP( t − > Empirical CDF
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LMP | (LMP( t − , LMP( t − > Empirical CDF
LMP | (LMP( t − , LMP( t − > Empirical PDF
MeanFitEmpirical
Empirical Distributions of LMPs
Fig. 4: Distributions of CAISO LMPs conditional on previous pricesexceeding threshold ξ for ξ ∈ { USDMWh , USDMWh , USDMWh } . W h o l e s a l e P r i ce s
60 70 80 90 100110120130140 F W C o n t r a c t P r i ce s ¯ λ F E l a s t i c i t y α DR vs. Forward Contract (250 users)
60 70 80 90 100 110 120 130 140
Wholesale Prices E [ λ s ] F W C o n t r a c t P r i ce s ¯ λ F . . . . . . . Contour Plot for α Fig. 5: Boundaries and contours of equal expected profit for forward optionand DR, 250 users, λ f = 0 . USDkWh .
1) DR vs. Forward Contract:
Figure 5 shows the decisionboundary of elasticity α above which the optimal expectedprofits under DR is greater than under the forward contract,that is, E [Π DR ] ≥ E [Π F ] , for different expected spot prices E [ λ s ] and forward contract prices ¯ λ F , assuming λ f ≤ E [ λ s ] .It is observed that α decreases as ¯ λ F or the expectedwholesale price E [ λ s ] increase. The negative correlation of α with ¯ λ F is consistent with expectations as a higher ¯ λ F makesforward contracts more expensive. The fact that decreasingwholesale prices E [ λ s ] make DR more competitive thanforward contracts can be explained by comparing (3b) to(9b), which states that the entire demand d has to be coveredat price λ s in the DR case, compared to only [ d − ¯ q ] + in theforward contract case. Also shown in Figure 5 is the lowerbound on α (gray transparent surface) below which DR isnon-profitable, i.e. { ( E [ λ s ] − λ f ) − | ≤ E [ λ s ] ≤ } ,where we set the residential tariff to λ f = 0 . USD/kWh.
2) DR vs. Call:
Figure 6 shows the decision boundaryof α for different call strike prices λ C and premium levels P above which E [Π DR ] ≥ E [Π C ] with ξ = 80 . As thepremium and strike price for the call option increase (andhence the call option becomes less attractive), DR becomesmore profitable because α decreases.
3) Forward Contract vs. Call:
Lastly, Figure 7 shows thedecision surface for ¯ λ F as a function of the call optionparameters P and ¯ λ C above which the forward contract P r e m i u m L e v e l P C a ll S t r i k e P r i ce ¯ λ C E l a s t i c i t y α Call vs. DR, E [ λ s ] | (LMP( t − , LMP( t − >
10 12 14 16 18 20 22 24
Premium Level P C a ll S t r i k e P r i ce ¯ λ C . . . . . . Contour Plot for α Fig. 6: Boundaries and contours of equal expected profit for DR and calloption, 250 users, λ f = 0 . USDkWh . is more profitable in expectation, i.e. E [Π F ] ≥ E [Π C ] . Asexpected, the forward contract becomes more attractive aseither the premium P or the call strike price ¯ λ C increase. P r e m i u m L e v e l P C a ll S t r i k e P r i ce ¯ λ C F o r w a r d C o n t r a c t P r i ce ¯ λ F Call vs. FW, E [ λ s ] | (LMP( t − , LMP( t − >
10 12 14 16 18 20 22 24
Premium Level P C a ll S t r i k e P r i ce ¯ λ C . . . . . . . Contour Plot for FW Contract Prices ¯ λ F Fig. 7: Boundaries and contours of equal expected profit for forward andcall option, 250 users, λ f = 0 . USDkWh . D. Evaluation
Assuming a residential tariff of . USDkWh , a lower bound onthe elasticity α of approximately . MWhUSD = 20 kWhUSD at firstglance seems to be an unachievable goal. However, note thatwholesale prices can spike at up to
USDMWh , which is faroutside the range of our calculations. Further, we disregardedtransmission losses and capacity costs inherent to generatorsand utilities, which make the delivery of electricity under theforward contract and the call option more expensive, therebylowering the bound on α .VII. C ONCLUSION
We analyzed hedging instruments for load-serving entitiesto mitigate price risks associated with volatile energy supplyand demand. Hedging against such risks is motivated bythe fact that load-serving entities are obligated to meetenergy demand of customers under contract instantaneously,which, in the absence of any hedging instruments, has tobe procured in its entirety from the wholesale electricitymarket (at potentially high prices). Forward contracts andcall options between load-serving entities and generatingcompanies as well as Demand Response programs for end-users are methods to share this risk with other marketparticipants. We formulated the optimal hedging strategy asa profit maximization problem which is random in the ag-gregate demand and wholesale electricity price. The optimalexpected profit under each hedging instrument was foundto be monotonely decreasing in the statistical dispersionof the demand distribution, and linearly decreasing for thepecial case of a uniform distribution. Using smart meterconsumption data and locational marginal prices in Califor-nia, we compared the optimal expected profits between thehedging methods in a pairwise fashion to generate decisionboundaries of equal profit.Our results can be extended in several regards. Firstly,a more involved analysis that takes into account operationalconstraints of the smart grid, e.g. transmission capacities andgrid congestion, would add credibility to the suggestions ofthis paper. Secondly, analyzing how the optimal expectedprofit increases as a function of diminished uncertainty inelectric wholesale prices and aggregate consumer demanddue to forecasting is interesting from the perspective ofprofit maximization. Lastly, forgoing Assumptions 1 and2 to allow utilities or generating companies to exercisemarket power calls for a game-theoretic formulation of theprofit-maximization problem from the perspective of bothgenerating companies and utilities, where each player seeksbids from the other in a mechanism design framework.R
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Lemma 1 (Leibniz Integral Rule) . For a function f ( x, t ) with both f ( x, t ) and ∂f∂x continuous in t ∈ [ a ( x ) , b ( x )] and x ∈ [ x , x ] , where a ( x ) and b ( x ) are continuous in x ∈ [ x , x ] , for x ∈ [ x , x ] : ddt (cid:32)(cid:90) b ( t ) a ( t ) f ( x, t ) dx (cid:33) = (cid:90) b ( t ) a ( t ) ∂f∂t dx + f ( b ( t ) , t ) · b (cid:48) ( t ) − f ( a ( t ) , t ) · a (cid:48) ( t ) Proof of Theorem 1
Taking the expectation of (2) with respect to the randomvariables λ s and d yields: E [Π F ] = − ¯ q · ¯ λ F + λ f (cid:90) ¯ q xf ( x ) dx + λ f ¯ q (1 − F (¯ q ))+ ( λ f − E [ λ s ]) (cid:90) ∞ ¯ q ( x − ¯ q ) f ( x ) dx (20)With the Leibniz Integral Rule, its derivatives with respectto ¯ q are d E [Π F ] d ¯ q = − ¯ λ F + λ f (1 − F (¯ q ) + ( λ f − E [ λ s ])( F (¯ q ) − d E [Π F ] d ¯ q = − λ f f (¯ q ) + f (¯ q )( λ f − E [ λ s ]) < from which the optimal contract volume ¯ q ∗ (3a) follows.Plugging ¯ q ∗ back into (20) yields E [Π F ] = − ¯ λ F F − (cid:18) − ¯ λ F E [ λ s ] (cid:19) + λ f (cid:90) ¯ q xf ( x ) dx + λ f ¯ λ F E [ λ s ] F − (cid:18) − ¯ λ F E [ λ s ] (cid:19) , from which the optimal profit (3b) follows. Proof of Theorem 2
Similar to the previous proof, we take the expectation of(4) with respect to λ s and d : E [Π C ] = λ f E [ d ] − (cid:90) ¯ q xf ( x ) dx (cid:90) ¯ λ C yg ( y ) dy − P ¯ q − r − ¯ λ C (1 − G (¯ λ C )) (cid:90) ¯ q xf ( x ) dx − ¯ q (1 − F (¯ q )) (cid:90) ¯ λ C yg ( y ) dy − ¯ q (1 − F (¯ q ))(1 − G (¯ λ C ))¯ λ C − E [ λ S ] (cid:90) ∞ ¯ q ( x − ¯ q ) f ( x ) dx he first order optimality condition reads d E Π C d ¯ q = − P + E [ λ s ](1 − F (¯ q )) − (1 − F (¯ q )) (cid:34)(cid:90) ¯ λ C yg ( y ) dy + ¯ λ C (1 − G (¯ λ C )) (cid:35) , which yields (5a) at the optimum. To show that this is amaximum, we compute the second derivative: d E Π C d ¯ q = f (¯ q ) (cid:34)(cid:90) ¯ λ C yg ( y ) dy + ¯ λ C (1 − G (¯ λ C )) − E [ λ s ] (cid:35) , which is negative as we show below: (cid:90) ¯ λ C yg ( y ) dy + ¯ λ C (1 − G (¯ λ C )) ? < E [ λ s ]¯ λ C G (¯ λ C ) − (cid:90) ¯ λ C G ( y ) dy + ¯ λ C − ¯ λ C G (¯ λ C ) ? < E [ λ s ]0 ≤ ¯ λ C − (cid:90) ¯ λ C G ( y ) dy < ¯ λ C < E [ λ s ] Finally, the optimal expected profit E [Π ∗ C ] (5b) follows fromplugging (5a) back into the expectation of (4). Proof of Theorem 3
Taking the expectation of (8) with respect to λ s and r byperforming Lebesgue-Stieltjes Integration gives E [Π DR ] = ( λ f − E [ λ s ]) (cid:90) d max − h ( r ) d min xf ( x + h ( r )) dx − r + ( λ f − E [ λ s ]) d min (cid:90) d max d max − h ( r ) f ( x ) dx (21) = ( λ f − E [ λ s ]) (cid:90) d max d min + h ( r ) ( x − h ( r )) f ( x ) dx − r where we used the change of variables x + h ( r ) → x and thefact that F ( d max ) = F ( d max − h ( r )) = 1 . With the LeibnizIntegral Rule, its derivatives with respect to r read d E [Π DR ] dr = ( λ f − E [ λ s ])[1 − F ( h ( r ))]( − h (cid:48) ( r )) − d E [Π DR ] dr = ( λ f − E [ λ s ] (cid:124) (cid:123)(cid:122) (cid:125) ≤ )[ f ( h ) h (cid:48) + ( F ( h ) − h (cid:48)(cid:48) (cid:124) (cid:123)(cid:122) (cid:125) ≥ ] (cid:12)(cid:12)(cid:12) h = h ( r ) For the linear shift, i.e. h ( r ) = αr , first order optimalityyields (9a), which is valid only under the condition that α > ( E [ λ s ] − λ f ) − . The second derivative is negative due to theconcavity of h ( r ) , which results in h (cid:48)(cid:48) ( r ) ≤ . The optimalprofit Π ∗ DR follows from plugging r ∗ back into (21): E [Π ∗ DR ] = ( λ f − E [ λ s ]) (cid:90) αr ∗ + d max αr ∗ ( x − αr ∗ ) f ( x ) dx − r ∗ + ( λ f − E [ λ s ]) d min [ F ( d max ) − F ( d max − h ( r ))]= ( λ f − E [ λ s ]) (cid:90) ∞ F − (1 − α ( E [ λs ] − λf ) ) xf ( x ) dx Proof of Theorem 4
This theorem can be proved by showing that the deter-minant of the Hessian of the two-dimensional optimizationproblem is negative, and hence yields a saddle at each jointminimum of portfolios ( ( r ∗ , ¯ q ∗ C for DR + call, ( r ∗ , ¯ q ∗ F ) forDR + forward contract, (¯ q ∗ F , ¯ q ∗ C ) for call + forward contract).The objectives for each of these pairwise portfolios are Π FC = λ f d − ¯ λ F ¯ q F − P ¯ q C − ( d − ¯ q F − ¯ q C ) λ s d> ¯ q F +¯ q C − ( d − ¯ q F ) min( λ s , ¯ λ C ) ¯ q F ≤ d ≤ ¯ q F +¯ q C Π FD = ( λ f − λ s )[ d ( r ) − ¯ q F ] + − ¯ λ F ¯ q F − r + λ F d ( r ) d ( r ) ≤ ¯ q F + λ f ¯ q F d ( r ) > ¯ q F Π CD = λ f d − λ s [ d ( r ) − ¯ q C ] + − P ¯ q C − r + min(¯ λ C , λ s ) (cid:2) − d ( r ) d ( r ) ≤ ¯ q C − ¯ q C d ( r ) > ¯ q C (cid:3) where Π FC , Π FD , and Π CD denote the profit under the pair-wise portfolios (forward contract, call), (forward contract,DR), (call, DR), respectively. Taking the expectation w.r.tto the random variables d and λ s and the derivatives w.r.t.the decision variables yields the Hessian matrix, from whichfurther analysis proves the claim. Proof of Proposition 1
Using the definition of the conditional expectation forcontinuous random variables
X, Y E [ X | Y ] = (cid:90) x ∈ R p X | Y ( x | y ) dx, it follows that E [ d | d ≥ τ ] = (cid:82) d max τ xf ( x ) dx (cid:82) d max τ f ( x ) dx , d min < τ < d max . (22)Applying (22) on (3b), (5b), and (9b) with τ = α F (12a), τ = α C (12b), and τ = α DR (12c), respectively, yields thedesired expressions. Proof of Proposition 2