Deadline Differentiated Pricing of Deferrable Electric Loads
11 Deadline Differentiated Pricing of DeferrableElectric Loads
Eilyan Bitar † and Yunjian Xu †† Abstract —A large fraction of total electricity demand iscomprised of end-use devices whose demand for energy isinherently deferrable in time. Of interest is the potential to usethis latent flexibility in demand to absorb variability in powersupplied from intermittent renewable generation. A fundamentalchallenge lies in the design of incentives that induce the desiredresponse in demand. With an eye to electric vehicle charging,we propose a novel forward market for deadline-differentiatedelectric power service, where consumers consent to deferredservice of pre-specified loads in exchange for a reduced pricefor energy. The longer a consumer is willing to defer, the lowerthe price for energy. The proposed forward contract provides aguarantee on the aggregate quantity of energy to be deliveredby a consumer-specified deadline. Under the earliest-deadline-first (EDF) scheduling policy, which is shown to be optimalfor the supplier, we explicitly characterize a non-discriminatory,deadline-differentiated pricing scheme that yields an efficientcompetitive equilibrium between the supplier and consumers. Wefurther show that this efficient pricing scheme, in combinationwith EDF scheduling, is incentive compatible (IC) in that everyconsumer would like to reveal her true deadline to the supplier,regardless of the actions taken by other consumers.
Index Terms —Demand response, electricity markets, renew-able energy, game theory, mechanism design.
I. I
NTRODUCTION
As the electric power industry transitions to a greater relianceon intermittent and distributed energy resources, there emergesa need for flexible resources that can respond dynamically toweather impacts on wind and solar photovoltaic output. Theserenewable generation sources have limited controllability andproduction patterns that are intermittent and uncertain. Suchvariability represents one of the most important obstacles tothe deep integration of renewable generation into the electricitygrid. The current approach to renewable energy integrationis to balance variability with dispatchable generation. Thisworks at today’s modest penetration levels, but it cannotscale, because of the projected increase in reserve generationrequired to balance the variability in renewable supply [7]. Ifthese increases are met with combustion fired generation, theywill both be counterproductive to carbon emissions reductionsand economically untenable.
This work was supported in part by NSF ECCS-1351621, NSF CNS-1239178, NSF IIP-1632124, PSERC under sub-award S-52, US DoE underthe CERTS initiative, and the MIT-SUTD International Design Center (IDC)Grant IDG21400103. Both authors contributed equally to the submitted work. † E. Bitar is with the School of Electrical and Computer Engineering,and the School of Operations Research and Information Engineering, CornellUniversity, Ithaca, NY, 14853, USA. Email: [email protected] †† Y. Xu is with the Engineering Systems and Design Pillar,Singapore University of Technology and Design, Singapore. Email: [email protected]
As wind and solar energy penetration increases, how mustthe assimilation of this variable power evolve, so as tominimize these integration costs, while maximizing the netenvironmental benefit? Clearly, strategies which attenuate theincrease in conventional reserve requirements will be an essen-tial means to this end. One option is to harness the flexibility indemand-side resources. As such, significant benefits have beenidentified by the Federal Energy Regulatory Commission [11]in unlocking the value of coordinating demand-side resourcesto address the growing need for firm, responsive resources toprovide balancing services for the bulk power system.
A. The Current Approach to Demand Response
There is an opportunity to transform the current operationalparadigm, in which supply is tailored to follow demand, toone in which demand is capable of reacting to variabilityin supply – an approach which is generally referred to asdemand response (DR) [1]. The primary challenge is the reliable extraction of the desired response from participatingdemand resources on time scales aligned with traditional bulkpower balancing services.The majority of DR programs in place today are limitedto peak shaving and contingency-based applications. Thetwo most common paradigms for customer recruitment andcontrol are: (1) direct load control where a load aggregatoror utility procures the capability of direct load adjustmentthrough a forward market or transaction (e.g. call optionsfor interruptible load) [8], [20], [25], [37] and (2) indirectload control where consumers or devices adjust their energyconsumption in response to dynamic price signals [12], [14],[17], [18]. While dynamic pricing has the potential to improvethe economic efficiency of electricity markets [5], [33], [15],it subjects consumers to the risk of paying high peak prices,and can have the counterproductive effect of increasing thevariability in demand [19], [29]. And, of particular relevanceto this paper’s emphasis on electric vehicle (EV) charging isa recent empirical study [31] that indicates dynamic pricingmay perform worse than a flat-rate tariff for EV charging interms of generation costs and emissions impacts. In short,demand response implemented through dynamic pricing maynot provide the level of assurance required to avoid the use ofconventional generation to manage the electric power system.
B. A Deadline Differentiated Energy Service Approach
In the following paper, we propose a novel market frame-work to enable the direct control of deferrable loads, with aparticular emphasis on electric vehicle (EV) charging. Broadly, a r X i v : . [ m a t h . O C ] A ug the proposed market centers on the provisioning of deadlinedifferentiated energy services to customers possessing theinherent ability to delay their consumption (up to a point)in time. From the consumer’s perspective, the longer she iswilling to delay the receipt of a specified quantity of energy,the less that customer pays (per-unit) for said energy. Thesupplier, on the other hand, implicitly purchases the rightto manage the real-time delivery of power to participatingconsumers by offering a discount on energy with longerdeadlines on delivery. And, the longer the consumer-specifieddeadlines, the more flexibility the supplier has in meeting thecorresponding energy requirements. Put simply, the suppliercan extract flexibility from the demand-side through directload control, all while providing firm guarantees on deliveryby consumer-specified deadlines. In this way, the supplier canalign its operational requirements with the vast heterogeneityin end-use customer needs.The general concept of electric power service differentiationis not new [25], [24]. Many have studied the problem ofcentralized control of a collection of loads for load-followingor regulation services – all while ensuring the satisfaction of apre-specified quality-of-service (QoS) to individual resources[6], [13], [9], [21], [22], [26], [28], [36], [39]. There has, how-ever, been little work in the way of designing market mecha-nisms that endogenously price the flexibility being offered bythe demand side, while incentivizing consumers to truthfullyreveal their preferences to the operator – for example, theability to delay energy consumption in time. Along these lines,the authors in [20] propose a forward market for duration-differentiated energy services. Several classic [8], [37] andmore recent [2] papers have explored the concept of reliability-differentiated pricing of interruptible electric power service,where consumers take on the risk of supply interruption inexchange for a reduction in the nominal energy price. Beyondthe difficulty in auditing such markets and the apparent issuesof moral hazard, the primary drawback of such approachesstems from the explicit transferal of quantity or price risk tothe demand side. This amounts to requiring that consumersplan their consumption in the face of uncertain supply or prices– a nontrivial decision task.With the aim of alleviating the aforementioned challenges,we propose and analyze a novel forward market for deadlinedifferentiated energy services , where consumers consent todeferred supply of energy in exchange for a reduced per-unit energy price. Such a market would naturally complementthe proliferation of plug-in electric vehicles (EV) in the UStransportation fleet. Example 1 (Electric Vehicle Charging) . With this motivationin mind, we now illustrate several basic questions that mightarise in the design and operation of markets for deadline dif-ferentiated energy services. To frame the discussion, considera scenario involving the operation of a parking infrastructureequipped with an array of EV charging stations and local The concept of deadline differentiated pricing was first proposed in aconference paper [3] and subsequently empirically evaluated in [30]. photovoltaic (PV) generation. The EVs being charged in suchfacilities are naturally modeled as having the ability to delaytheir receipt of energy in time. Accordingly, upon connectingan EV to a charging station, the vehicle’s owner is presentedwith a menu of prices – each of which stipulates a per-unit price for energy and a corresponding delivery deadline.Faced with such choices, how does the EV owner decide uponwhich bundle of energy-deadline pairs to purchase given theinherent ability to delay consumption? On the supply side, theoperator of the parking infrastructure is required to meet allenergy requests by their corresponding deadlines. How shouldthe operator set the menu of deadline differentiated prices toinduce the desired demand from a population of EV owners?What if the available energy supply is random, as is the casewith PV generation? In order to address such questions, wefirst require a description of the underlying market, which weinformally describe below to orient the reader. (cid:3)
Market Operation.
The forward market for deadline dif-ferentiated energy service operates according to a three-stepprocess. In step 1, the supplier announces a mechanism. Instep 2, the consumers simultaneously report their demand.In step 3, the mechanism is executed; namely, prices areset and the requested demand is delivered to each consumer.Time is assumed to be discrete with periods indexed by k = 0 , , , . . . , N . Step 1 (Mechanism Design). At the beginning of period k = 0 , the supplier announces a market mechanism ( π, κ ) consisting of both a scheduling policy π (cf. its formal defi-nition in Section III), and pricing scheme κ = ( κ , . . . , κ N ) that maps the aggregate demand bundle x (cf. its definitionin Step 2) into a menu of deadline-differentiated prices , p k = κ k ( x ) , k = 1 , . . . , N. (1)The price menu stipulates a per-unit price p k ($/kWh) forenergy guaranteed delivery by period k . At the heart of themechanism design is the restriction that prices are nonincreas-ing in the deadline. Namely, the longer a customer is willingto defer her consumption, the less she is required to pay. Wewill use P = { p ∈ R N + | p ≥ p ≥ . . . ≥ p N } to denote theset of feasible deadline-differentiated price bundles. Step 2 (Consumer Reporting). Each consumer then reportsa bundle of deadline-differentiated energy quantities a =( a , . . . , a N ) (cid:62) ∈ R N + . Here, the quantity a k (kWh) denotesthe amount of energy guaranteed delivery by the deadline k .It follows that said consumer will receive at least (cid:80) kt =1 a t amount of energy by deadline k . The aggregate demand bundleis the sum of all individual consumer bundles, which wedenote by x = ( x , . . . , x N ) (cid:62) ∈ R N + . Step 3 (Pricing and Energy Delivery). Given an aggregatedemand bundle x , the deadline differentiated prices are setaccording to p = κ ( x ) . Pricing is non-discriminatory, in that Of particular relevance to our development is the growing multitude ofcompanies offering turn-key products that integrate EV charging infrastructurewith solar photovoltaic canopies [10], [32]. all customers are charged according to the same menu ofprices. Thus, a customer requesting a bundle a pays p (cid:62) a .The supplier must also deliver the aggregate demand bundle x according to the previously announced scheduling policy π . Essentially, a scheduling policy is said to be feasible ifit delivers each consumer’s requested energy bundle by itscorresponding deadline. The supplier is assumed to have twosources of electricity from which she can service demand: intermittent and firm . • Intermittent supply: an intermittent supply modeled as adiscrete time random process s = ( s , s , . . . , s N − ) with known joint probability distribution. Here, s k ∈ S (kWh) denotes the energy produced during period k and S ⊂ R + the feasible supply interval. The intermittentsupply is assumed to have zero marginal cost. • Firm supply: a firm supply with a fixed price of c > .The price c can be interpreted as the nominal flat ratefor electricity set by the local utility.While stylized in nature, our models of supply and demandare meant to reflect the essential features of an emerging gridinfrastructure that enables the direct coupling of deferrableelectric loads with variable renewable supply. C. Summary of Main Results
The primary contribution of our paper is the design of amarket mechanism for deadline differentiated energy services,which implements truth-telling at a dominant strategy equilib-rium across the population of consumers, while maximizingsocial welfare at a competitive equilibrium between a price-taking supplier and a continuum of infinitesimally smallconsumers. We provide here a roadmap of the paper togetherwith a summary of our main results. • We provide a stylized, yet descriptive, model of both de-ferrable electricity demand and a supplier with both firmand intermittent supply in Sections II and III, respectively. • We formulate the supplier’s scheduling problem asa constrained stochastic optimal control problem andprove average-cost optimality of the earliest-deadline-first (EDF) scheduling policy. As a corollary, this resultenables the explicit characterization of the supplier’smarginal cost curve in Theorem 2. • It is reasonable to expect that the supplier cannot observethe true deadline of each individual consumer. The pres-ence of asymmetric information may lead to significantwelfare loss, if consumers misreport their true deadlines.Somewhat surprisingly, we show under mild assumptionson consumers’ utility functions that a mechanism consist-ing of an EDF scheduling policy, together with a uniformmarginal cost pricing scheme implements consumers’ truth-telling behavior in dominant strategies. • We show in Section IV-B that marginal cost pricing, incombination with EDF scheduling, induces an efficient A price-taking supplier cannot control market prices, because of govern-ment regulation or perfect competition. Instead, it seeks to maximize its profitby scheduling the purchase of its firm supply. competitive equilibrium that simultaneously maximizesthe (price-taking) supplier’s profit and the social welfare.A preliminary version of this paper appeared in [4], wherewe proved the existence of a truth-telling Nash equilibrium.In the present paper, we establish a stronger incentive com-patibility result. Namely, it is a dominant strategy for eachconsumer to be truth-telling, regardless of the actions takenby other consumers. All formal proofs of our stated resultscan be found in the Appendix of this paper.
D. Related Work
There have recently emerged several papers concernedwith the design of incentives for deferrable electric loads.The authors of [16] propose an idea similar to this paper,where consumers are offered a discounted electricity price inexchange for the delay of their energy consumption. How-ever, the focus of [16] is not on pricing, but rather on theproblem of optimal scheduling faced by the operator, whois assumed to have full information about consumers – forexample, knowledge of their true deadlines. Closer to thepresent paper, [27] proposes a greedy online mechanism forelectric vehicle charging, which is shown to be incentivecompatible and achieve a bounded (worst-case) competitiveratio. These theoretical results require, however, a VCG-type(discriminatory-price) payment scheme, as opposed to theuniform-price scheme proposed in this paper. More strongly,they impose an additional assumption that consumers cannotreport false deadlines exceeding their true underlying dead-lines. Such assumption substantially simplifies the problemof designing an incentive compatible pricing scheme whenconsumers are permitted to report arbitrary deadlines – thesetting considered in the present paper.II. M
ODEL OF D EMAND
We consider a model involving a continuum of infinitesimalconsumers, indexed by i ∈ [0 , . Since each individualconsumer’s action has no influence on the price, she will act asa price taker. Such an assumption is reasonable and commonlyemployed in the context of retail electricity markets [8], [23],[35], [37], where it is not uncommon for an electric powerutility to service to customers – a setting in whicheach consumer herself is too small to influence the price.We propose a consumer utility model yielding a preferenceordering on deadlines, where the longer the delay in con-sumption, the smaller the utility derived from consumption.In particular, we assume that each consumer has a singledeadline preference. More precisely, a consumer with deadlinepreference k incurs no loss of utility by deferring consumptionuntil deadline k and derives zero utility for any consumptionthereafter. This assumption is reasonable for electric loads suchas plug-in electric vehicles (PEVs), dish washers, and laundrymachines, as customers commonly require only that such loadsfully execute before a specific time. Such intuition lends itselfto the following definition of consumer type . Definition 1 (Consumer type) . The type of consumer i isdefined as a triple q i R i q i Energy consumption by deadline, y U t ili t y , U ✓ i ( y ) (a) q i R i q i Energy consumption by deadline, y (b) q i R i q i Energy consumption by deadline, y (c) q i R i q i Energy consumption by deadline, y (d)Fig. 1. Four examples of utility functions satisfying Assumption 1. θ i = ( k i , R i , q i ) , which consists of her deadline k i ∈ N , marginal utility R i ∈ R + , and maximum demand q i ∈ R + .Consumer i ’s utility function depends only on her type θ i andis assumed to satisfy the following conditions. Assumption 1.
A consumer of type θ i derives a utility thatdepends only on the total energy consumed by her truedeadline k i . The utility function U θ i : R + → R + is assumedto be non-negative and non-decreasing over [0 , q i ] with U θ i ( y ) ≤ yR i , for all y ∈ [0 , q i ] , where R i = U θ i ( q i ) /q i . We also assume that U θ i ( y ) = U θ i ( q i ) for all y ≥ q i . (cid:3) Note that the marginal utility R i associated with a consumertype θ i is defined as the ratio of the maximum utility U θ i ( q i ) to the maximum demand q i . It is also worth mentioningthat Assumption 1 prevents the treatment of general concaveutility functions, as incentive compatibility may fail to holdfor certain concave utility functions. We refer the reader toRemark 3 for a more detailed discussion. Example 2 (Consumer utility functions) . It is worth empha-sizing that Assumption 1 accommodates a large family ofutility functions. Indeed, every utility function that lies belowthe piecewise affine function U θ i ( y ) = R i min { y, q i } satisfiesAssumption 1. Figure 1 depicts several such utility functions.The utility function depicted in Figure 1(b) characterizeselectric loads with ‘all or nothing’ utility characteristics. Forexample, in the context of electric vehicle charging, q i > can be interpreted as the minimum amount of battery chargerequired by the consumer in order to fulfill her next trip.Naturally, the consumer obtains zero utility if the battery levelis below this threshold. Many job-oriented appliances, suchas dishwashers, are similarly modeled. The utility function inFigure 1(c) is a natural generalization of that in 1(b), and couldcapture the utility function of a consumer having multipleall-or-nothing jobs requiring completion before a commondeadline. Finally, the utility function in Figure 1(d) is a generalnonlinear utility function that satisfies Assumption 1. (cid:3) We let Θ denote the set of all possible consumer types ,which is assumed to be finite. Let ρ : Θ → [0 , denotethe distribution of consumer types over the space Θ . In otherwords, for every θ ∈ Θ , there is a ρ ( θ ) fraction of consumersof type θ . It follows that (cid:80) θ ∈ Θ ρ ( θ ) = 1 . Definition 2 (Consumer action) . The action of a consumeris a vector a = ( a , . . . , a N ) (cid:62) ∈ R N + , where a k denotes theamount of energy that is guaranteed delivery by deadline k .The maximum amount of energy any consumer can request is Q = max ( k,R,q ) ∈ Θ { q } . Hence, each consumer’s action space is restricted to A = (cid:8) a ∈ R N + | (cid:80) k a k ≤ Q (cid:9) .It follows from the above definition that q ≤ Q for everytype θ = ( k, R, q ) . In other words, it is feasible for everyconsumer to request her maximum demand q . Given a fixedscheduling policy and pricing scheme, a consumer’s strategy ϕ : Θ → A maps her type into an action. In the followinganalysis, we will be concerned with identifying conditions onboth the scheduling policy and pricing scheme that lead toefficient allocations, while inducing consumers to truthfullyreveal their underlying deadline preferences. A consumer oftype θ = ( k, R, q ) is defined to be truth-telling if she requests q units of energy at her true deadline k , and nothing else. Wemake this notion precise in the following definition. Definition 3 (Truth-telling) . A consumer of type θ = ( k, R, q ) is defined to be truth-telling if her strategy a ∗ = ϕ ∗ ( θ ) satisfies a ∗ j = 0 for all j (cid:54) = k and a ∗ k = q .Indeed, Definition 3 is equivalent to the standard definitionof truth-telling in which a consumer is required to report herexact type. In Corollary 1, we show that social welfare ismaximized at a competitive equilibrium, even if the supplierdoes not have knowledge of the consumers’ utility functions(including the parameter R ), as long as R is no less thanthe marginal cost of firm supply c (cf. Assumption 3).In other words, a consumer of type θ = ( k, R, Q ) onlyneeds to report her true deadline k and maximum demand q to the supplier, and the profit maximization problem of a(price-taking) supplier naturally yields social optimality. InDefinition 3, we require that the consumer requests the entiretyof her demand q to be delivered by her true deadline k .This is without loss of generality, because the consumer’sutility depends only on the total energy consumed by her truedeadline k (cf. Assumption 1).It is worth mentioning that under an arbitrary schedulingpolicy and pricing scheme, it is indeed possible that a con-sumer’s best response is untruthful (cf. Remark 2 and thediscussion following Definition 6). Our aim is to provide anexplicit characterization of a scheduling policy and pricingscheme that implement truth-telling as a dominant strategyfor every consumer i .Given the collection of consumer types θ = { θ i } i ∈ [0 , and a strategy profile ϕ = { ϕ i } i ∈ [0 , , the aggregate demandbundle x is given by the mapping x = d ( θ , ϕ ) = (cid:90) i ∈ [0 , ϕ i ( θ i ) η ( di ) , (2)where η is the Lebesgue measure defined over [0 , , and d = ( d , . . . , d N ) maps ( θ , ϕ ) into an N -dimensional non-negative vector. Under the truth-telling strategy profile ϕ ∗ = { ϕ ∗ i } i ∈ [0 , specified in Definition 3, the aggregate demandbundle simplifies to x ∗ j = d j ( θ , ϕ ∗ ) = (cid:88) θ ∈ Θ q · ρ ( θ ) · { j = k } (3)for all j = 1 , . . . , N , where θ = ( k, R, q ) . Here, {·} denotesthe indicator function for the event in the subscript. A. Consumer Surplus
We are now in a position to characterize the expectedsurplus derived by a consumer. It depends on: (i) her own typeand strategy, (ii) the remaining consumers’ types and strategyprofile, (iii) the pricing scheme, and of course, (iv) the schedul-ing policy employed by the supplier. Before proceeding, werequire the definition of pertinent notation. We define therandom variable ω πk,i ( x , a ) to denote the amount of energydelivered to consumer i by stage k given her requested bundle a and aggregate demand bundle x . This random variable,which naturally depends on the scheduling policy π employedby the supplier, is formally defined in Appendix A. We require that the requested quantities are always suppliedby their corresponding deadlines and the total quantity deliv-ered to a consumer never exceeds said consumer’s total de-mand. More formally, we require for each consumer i ∈ [0 , taking action a ∈ A that (cid:88) kt =1 a t ≤ ω πk,i ( x , a ) ≤ (cid:88) Nt =1 a t (4)with probability one, for all aggregate demand bundles x and k = 1 , . . . , N .In order to formally define and analyze incentive com-patibility of our proposed market mechanism, we require adefinition of the expected surplus derived by a consumer undera particular strategy. Given a scheduling policy π and a pricingscheme κ (that maps every aggregate demand bundle x intoa menu of deadline differentiated prices p ) employed by thesupplier, and all consumer types θ = { θ i } i ∈ [0 , , consumer i receives an expected surplus (payoff) under a strategy profile ϕ = { ϕ i } i ∈ [0 , that is given by v πi ( θ , ϕ , κ ) = E (cid:110) U θ i (cid:16) ω πk i ,i ( x , a ) (cid:17)(cid:111) − κ ( x ) (cid:62) a . (5)Here, k i is consumer i ’s true deadline, x = d ( θ , ϕ ) is theaggregate demand bundle, a = ϕ i ( θ i ) is the action taken by Note that we have implicitly assumed that for every θ , the function ϕ = { ϕ i ( θ i ) } i ∈ [0 , is Lebesgue integrable in i . This assumption holds, forexample, under a symmetric strategy profile according to which all consumersof the same type take the same action. Note that we have allowed the supply ω πk,i ( x , a ) to depend explicitly onthe consumer index i , as the supplier may employ a scheduling policy thatdepends on the consumer index. consumer i , and κ ( x ) is the price bundle set according to thepricing scheme κ at the aggregate demand x . Expectation istaken with respect to the random variable ω πk i ,i ( x , a ) .Clearly, a scheduling policy π together with a pricingscheme κ defines a game for the consumer population, witheach individual consumer’s payoff expressed in (5). We notethat this is an aggregative game , where the payoff functionof each player depends on the population’s strategy profileonly through the sum of their actions – the aggregate demandbundle x = d ( θ , ϕ ) . We can therefore rewrite consumer i ’spayoff in (5) in a form that depends on other players’ strategiesonly through their sum x . Namely, V πi ( θ i , ϕ i , x , κ ) = E (cid:110) U θ i (cid:16) ω πk i ,i ( x , a ) (cid:17)(cid:111) − κ ( x ) (cid:62) a , (6)where a = ϕ i ( θ i ) . Moreover, it follows from the serviceconstraint in (4) that under the truth-telling strategy ϕ ∗ i , thepayoff derived by consumer i simplifies to the deterministicquantity V πi ( θ i , ϕ ∗ i , x , κ ) = U θ i ( q i ) − κ k i ( x ) q i , (7)where θ i = ( k i , R i , q i ) . It is important to note that theexpression in (7) does not depend on the types and strategiesof the other consumers, or the scheduling policy used by thesupplier, as long as the service constraint in (4) is respected.Bayesian Nash equilibrium may not be a plausible solutionconcept to explore for this game, as it requires each individ-ual consumer to have information regarding the distributionof other consumers’ types, as well as knowledge of theprobability distribution of ω πk i ,i ( x , a ) , which in turn dependson the distribution of the intermittent supply process s . Wecircumvent such informational assumptions by focusing ouranalysis around a stronger solution concept – namely, thedominant strategy equilibrium of the game. A strategy ϕ i is a dominant strategy for consumer i of type θ i , if it maximizesher expected payoff regardless of the actions taken by the otherconsumers. We have the following definition. Definition 4 (Dominant strategy) . A strategy ϕ i is a dominantstrategy for a consumer i of type θ i if V πi ( θ i , ϕ i , x , κ ) ≥ V πi ( θ i , ϕ (cid:48) i , x , κ ) , ∀ ϕ (cid:48) i , ∀ x ∈ R N + . We note that in Definition 4, a change of an individualconsumer i ’s strategy has no influence on the aggregatedemand bundle x , because each consumer is assumed to beinfinitesimal in size, and the aggregate demand bundle is givenby Eq. (2). In Definition 6, we define a mechanism to be incentive compatible if it implements truth-telling in dominantstrategies. Surprisingly, we show in Section IV that a mecha-nism consisting of an earliest-deadline-first (EDF) schedulingpolicy in combination with a marginal cost pricing schemeis incentive compatible and induces an efficient competitiveequilibrium between the supplier and consumers.III. M ODEL OF S UPPLY
As one of the primary thrusts of this paper is the character-ization of a competitive equilibrium between the supplier andconsumers and its efficiency properties, we now consider the behavior of a price-taking supplier , whose aim is to maximizehis expected profit given a predetermined price bundle. Theexpected profit derived by a supplier equals the revenuederived from the sale of a bundle of deadline differentiatedenergy quantities less the expected cost of firm supply requiredto service said bundle. In determining his supply curve underprice taking behavior, the supplier’s objectives are two-fold: • Scheduling.
Determine an optimal scheduling policy tocausally allocate the intermittent supply across the dead-line differentiated consumer classes, in order to minimizethe expected cost of firm supply required to ensure thatdemand bundles are served by their respective deadlines. • Pricing.
Given the optimal scheduling policy, determinea marginal cost supply curve that specifies the bundle ofenergy he is willing to supply at every price bundle.Essentially, the specification of the supplier’s marginal costcurve requires the explicit characterization of the gradient ofthe supplier’s minimum expected cost of firm supply. This, inturn, requires the specification of supplier’s optimal schedulingpolicy, which is proven to have an earliest-deadline-first (EDF)structure in Theorem 1. With the optimal scheduling policy inhand, we derive a closed-form expression for the supplier’smarginal cost curve in Theorem 2.
A. Feasible Scheduling Policies
We now offer a precise formulation of the supplier’s classof feasible scheduling policies given an aggregate demand.When considering the problem of scheduling, it is importantto distinguish between intra-class and inter-class scheduling.An inter-class scheduling policy (denoted by σ ) represents asequence of scheduling decisions, which causally allocate theintermittent and firm supply across the deadline-differentiateddemand classes. In addition to inter-class scheduling, thesupplier must also determine as to how the available supplyis allocated between customers within a given demand class.As such, we let the intra-class scheduling policy (denotedby φ ) represent a sequence of scheduling decisions, whichdictates how the intermittent and firm supply made available toa given demand class is allocated across customers within saidclass. Finally, we let π = ( σ, φ ) denote the joint inter-classand intra-class scheduling policy employed by the supplier. Inwhat follows, we formally define the space of feasible inter-class and intra-class scheduling policies.
1) Inter-Class Scheduling Policies:
We characterize theoptimal inter-class scheduling policy as a solution to a con-strained stochastic optimal control problem. First, we definethe system state at period k as the pair ( z k , s k ) ∈ R N + × R + ,where the vector z k denotes the residual demand requirementof the original aggregate demand bundle x after having beenserviced in previous periods , , · · · , k − . Define as the control input the vectors u k , v k ∈ R N + , which denote (element-wise in j ) the amount of intermittent and firm supply allocatedto demand class j at period k , respectively. Naturally then, thestate of residual demand evolves according to the discrete time state equation : z k +1 = z k − u k − v k , k = 0 , . . . , N − , (8) where the process is initialized with z = x . The deliverydeadline constraints manifest in a sequence of nested con-straint sets R N + ⊃ Z ⊇ Z ⊇ · · · ⊇ Z N = 0 convergingto the the origin, where the set Z k characterizes the feasiblestate space at stage k . More precisely, Z k = { z ∈ R N + | z j = 0 , ∀ j ≤ k } . In other words, the feasible state space is such that eachdemand class is fully serviced by its corresponding deadline.We define as the feasible input space at stage k the set of allinputs belonging to the set U k ( z , s ) = { ( u , v ) | (cid:62) u ≤ s and z − u − v ∈ Z k +1 } , which ensures one-step state feasibility and that the totalallocation of renewable supply does not exceed availability atthe current stage. In characterizing the feasible set of causalscheduling policies, we restrict our attention to those policieswith Markovian information structure , as opposed to allowingthe control to depend on the entire history. This is withoutloss of optimality, since Markovian policies are capable ofperforming as well as the optimal oracle policy. We describethe scheduling decision at each stage k by the functions u k = µ k ( z , s ) and v k = ν k ( z , s ) , where µ k : Z k ×S → R N + and ν k : Z k ×S → R N + . A feasibleinter-class scheduling policy is any finite sequence of schedul-ing decision functions σ = ( µ , . . . , µ N − , ν , . . . , ν N − ) such that ( µ k , ν k )( z , s ) ∈ U k ( z , s ) , ∀ ( z , s ) ∈ Z k × S and time periods k = 0 , . . . , N − . We will occasionally writethe state and control process as { z σk } , { u σk } , and { v σk } toemphasize their dependence on the inter-class policy σ , unlessotherwise clear from the context.Given an aggregate demand bundle x , we denote by Σ( x ) the space of all feasible inter-class scheduling policies avail-able for use by the supplier.
2) Intra-Class Scheduling Policies:
Recall that an intra-class scheduling policy φ determines the allocation of supplyto specific consumers within each deadline-differentiated de-mand class, where the supply available to each demand classis determined by the inter-class policy. Given an inter-classscheduling policy σ ∈ Σ( x ) , we denote by Φ( σ ) the space ofall feasible intra-class scheduling policies . It is important to note that the supplier’s expected profit de-pends only on the inter-class scheduling policy being used, andis invariant under the family of feasible intra-class schedulingpolicies. This follows from the supplier’s cost indifference tosupply allocation between consumers within a given demandclass. Therefore, in characterizing the optimal scheduling pol-icy for the supplier, we restrict our attention to the characteri-zation of optimal inter-class policies. The distinction betweeninter-class and intra-class scheduling is, however, important, As the formal definition of feasible intra-class policies is relevant onlyto the derivation of technical proofs, we defer their precise specification toAppendix A in order to maintain continuity in exposition. as the choice of intra-class scheduling policy φ can affectthe probability distribution of each consumer’s random supply ω πk i ,i ( x , a ) . This can, in turn, influence consumer purchasedecisions. B. Optimal Scheduling Policy
We define the expected profit J ( x , p , σ ) derived by a sup-plier as the revenue derived from an aggregate demand bundle x less the expected cost of servicing said demand bundleunder a feasible inter-class scheduling policy σ ∈ Σ( x ) . Moreprecisely, let J ( x , p , σ ) = p (cid:62) x − Q ( x , σ ) , where Q denotes the expected cost of firm generation incurredservicing x under a feasible policy σ ∈ Σ . It follows that Q ( x , σ ) = N − (cid:88) k =0 E (cid:8) c (cid:62) v σk (cid:9) , (9)where c = ( c , . . . , c ) . We wish to characterize inter-classscheduling policies, which lead to a minimal expected cost offirm supply. Accordingly, we have the following definition ofoptimality. Definition 5 (Optimal policy) . The inter-class schedulingpolicy σ ∗ ∈ Σ( x ) is defined to be optimal if Q ( x , σ ∗ ) ≤ Q ( x , σ ) , ∀ σ ∈ Σ( x ) . (10)We denote by Q ∗ ( x ) = Q ( x , σ ∗ ) the minimum expected costof firm supply. The following result provides an explicit characterization ofan optimal inter-class scheduling policy.
Theorem 1 (Earliest-Deadline-First) . Given an aggregate de-mand bundle x ∈ R N + , the optimal scheduling policy σ ∗ ∈ Σ( x ) is given by: µ j, ∗ k ( z , s ) = min (cid:110) z j , s − (cid:80) j − i =1 µ i, ∗ k ( z , s ) (cid:111) ,ν j, ∗ k ( z , s ) = ( z j − µ j, ∗ k ( z , s )) · { k = j − } , for j = 1 , . . . , N , k = 0 , . . . , N − , and ( z , s ) ∈ Z k × S .Theorem 1 is intuitive. The optimal inter-class policy σ ∗ ∈ Σ( x ) is such that the intermittent supply s k available ateach period k is allocated to those unsatisfied demand classeswith earliest-deadline-first (EDF), while the firm supply isallocated to a demand class only when the EDF allocationof intermittent supply is insufficient to ensure its deadlinesatisfaction. In other words, the firm supply is used only as alast resort. For the remainder of the paper, we refer to σ ∗ asthe EDF scheduling policy.
Remark 1 (Implementation requirements) . An attractive prop-erty of the EDF scheduling policy σ ∗ is that it is distribution-free . Namely, it can be implemented by the supplier withoutrequiring explicit knowledge of the underlying probabilitydistribution of the intermittent supply process. On the other We omit a formal proof of Theorem 1, given its immediacy in derivationupon examination of scheduling problem’s dynamic programming equations. hand, a potential limitation of the proposed scheduling policyis the centralization of information exchange it entails, as it’simplementation requires periodic communication between thesupplier and each consumer-specific device. Specifically, atevery time period k , the supplier must transmit a control signalto each device specifying its energy consumption level in thatperiod. These control signals are determined according to theintra-class scheduling policy φ ∈ Φ( σ ∗ ) being used by thesupplier. In return, each device must transmit a measurementsignal to the supplier summarizing its state (e.g., residualenergy requirement). Naturally, the time scale at which suchcontrol schemes can be implemented will depend on the num-ber of devices being coordinated and the available bandwidthof the underlying communication network being used. C. Marginal Cost Pricing
Given the EDF characterization of the optimal inter-classscheduling policy in Theorem 1, we are now in a position tocharacterize the supplier’s optimal supply curve under pricetaking behavior. We define the residual process induced bythe EDF scheduling policy σ ∗ as ξ k +1 ( x , s ) = max { , ξ k ( x , s ) } + s k − x k +1 , (11)for k = 0 , . . . , N − , where ξ = 0 . We denote the entireresidual process by ξ = ( ξ , . . . , ξ N ) , omitting its dependencyon x and s when it is clear from the context. A positiveresidual ( ξ k > ) represents the amount of intermittent supplyleftover after having serviced demand class k by its deadline k , according to the EDF inter-class scheduling policy σ ∗ .A negative residual ( ξ k ≤ ) represents the amount bywhich the intermittent supply fell short – or, equivalently,the amount of firm supply required to ensure satisfaction ofthe demand class k . Using this newly defined process, wehave the following characterization of the minimum expectedcost of firm supply under EDF scheduling. First, we require atechnical assumption. Assumption 2.
The joint probability distribution of the inter-mittent supply process s is assumed to be absolutely continu-ous and have compact support. (cid:3) Lemma 1.
Suppose that Assumption 2 holds. The minimumexpected recourse cost Q ∗ ( x ) derived under an aggregate de-mand bundle x ∈ R N + and EDF scheduling policy σ ∗ ∈ Σ( x ) satisfies Q ∗ ( x ) = E (cid:40) c N (cid:88) k =1 − min { , ξ k } (cid:41) , (12)and is convex and differentiable in x over [0 , ∞ ) N .Lemma 1 admits a natural interpretation. Namely, the mini-mum expected cost of firm supply is equivalent to the amountby which the intermittent supply is expected to fall short foreach demand class under EDF scheduling. Moreover, it followsreadily that the expected profit J ( x , p , σ ∗ ) derived under EDFscheduling is also differentiable and concave in x . As such,any allocation x satisfying the first order condition, ∇ x J ( x , p , σ ∗ ) (cid:62) ( x − y ) ≥ for all y ∈ R N + , (13) is profit maximizing for the supplier given a price bundle p ∈P . Accordingly, we provide an explicit characterization of thesupplier’s marginal cost supply curve in the following theorem. Theorem 2 (Marginal cost supply curve) . Suppose that As-sumption 2 holds. An allocation x is profit maximizing for agiven price bundle p if p = ζ ( x ) , where the pricing scheme ζ : R N + → P satisfies, ζ k ( x ) c = P ( ξ k ≤
0) + N (cid:88) t = k +1 P ( ξ k > , . . . , ξ t − > , ξ t ≤ (14)for k = 1 , . . . , N .The marginal cost pricing scheme ζ specified in Equation(14) maps every aggregate demand bundle in R N + into amenu of deadline differentiated prices in P , which reflectsthe marginal cost of the supplier. Moreover, one can readilyinterpret such pricing scheme as setting the price p k for energywith deadline k equal to (up to a constant factor c ) theprobability that firm supply will be required to service thebundle x at any subsequent time period t ≥ k − under theoptimal inter-class scheduling policy σ ∗ . Naturally, the largerthe probability of shortfall, the larger the price. It is readilyverified that the pricing scheme p = ζ ( x ) yields, in general,prices that are nonincreasing in the deadline. More precisely, c ≥ p ≥ p ≥ · · · ≥ p N for all x ∈ R N + . This property of price monotonicity isconsistent with our initial criterion for constructing such amarket system. Namely, the longer a customer is willing todefer her consumption in time, the less she is required to payper unit of energy. And the price of deferrable energy is nogreater than the nominal flat rate for electricity, c .IV. I NCENTIVE C OMPATIBILITY AND E FFICIENCY
We now establish several important properties of the marketmechanism developed in Section III. In particular, underan additional mild assumption on each consumer’s marginalvaluation on energy, we show in Sections IV-A and IV-B that amechanism consisting of earliest-deadline-first (EDF) schedul-ing in combination with the marginal cost pricing schemein (14) is indeed incentive compatible , and achieves socialoptimality at a competitive market equilibrium, respectively.
A. Incentive Compatibility
With market efficiency considerations in mind, it is im-portant to understand when a consumer has incentive tomisreport her underlying deadline preference. A mechanism ( π, κ ) consisting of a feasible scheduling policy π = ( σ, φ ) and a pricing scheme κ is said to be incentive compatible (IC)for consumers of a particular type, if it is a dominant-strategyfor consumers of this type to be truth-telling. Definition 6 (Incentive compatibility) . A mechanism ( π, κ ) is incentive compatible for every consumer i of type θ i , if V πi ( θ i , ϕ ∗ i , x , κ ) ≥ V πi ( θ i , ϕ i , x , κ ) , ∀ ϕ i , ∀ x ∈ R N + , where ϕ ∗ i is the truth-telling strategy defined in Definition 3.It is worth mentioning that the above definition of dominantstrategy incentive compatibility is strong. It requires that aconsumer would like to reveal her true type regardless ofthe types and actions of other consumers and the probabilitydistribution on the intermittent supply process. Since theoptimal price schedule is non-increasing in the deadline, anddemand is guaranteed to be met before the requested deadline,a consumer i does not have an incentive to request anyquantity of energy before her true deadline k i . However, ifthe price of energy associated with later deadlines is lowenough, said consumer may have an incentive to report afalse later deadline if early delivery is likely (i.e., with highprobability) under the specified scheduling policy. Intuitively,a consumer i will have incentive to misreport its deadlineif the reduction in total expenditure derived by requestingenergy with later deadlines exceeds the expected loss of utilityincurred by a shortfall in the amount of energy delivered byher true deadline k i . Surprisingly, we show in Theorem 3 thatEDF scheduling in combination with marginal cost pricingprecludes this possibility.For the remainder of the paper, we denote by ( π ∗ , ζ ) themarket mechanism defined by EDF scheduling and marginalcost pricing. More precisely, the scheduling policy π ∗ =( σ ∗ , φ ) consists of the EDF inter-class scheduling policy σ ∗ and an arbitrary feasible intra-class policy φ ∈ Φ( σ ∗ ) . And, ζ denotes the marginal cost pricing scheme in Equation (14). Theorem 3 (Incentive compatibility) . Suppose that Assump-tions 1-2 hold. The mechanism ( π ∗ , ζ ) is incentive compatible for all consumers of a type that satisfies R ≥ c .We further establish in Corollary 1 that the mechanism ( π ∗ , ζ ) also maximizes social welfare at a unique market equi-librium between the supplier and consumers, if the condition R ≥ c holds for every consumer type θ ∈ Θ . Remark 2.
We note that the requirement R ≥ c is reasonablefor electricity consumers, as their marginal valuation on elec-tricity consumption is commonly higher than the nominal flatrate for electricity, which in our model is denoted by c . Onthis basis, electricity demand is generally modeled as inelastic[34], [38], especially in the short term [40]. Remark 3.
We also note that incentive compatibility mayfail to hold if certain assumptions regarding a consumer’sutility function are violated. First, one can readily show thatincentive compatibility may fail to hold for a consumer i oftype θ i = ( k i , R i , q i ) , if the marginal cost of firm supplyexceeds her marginal valuation of energy – namely, R i < c .Second, we note that the result in Theorem 3 fails to hold forarbitrary concave utility functions. This is intuitive. Consider aconsumer i having a highly concave utility function. Becauseof the large underlying concavity in her utility, said consumerwill prefer to report a false deadline that is later than her true deadline k i , if she can obtain a fraction of her demand beforestage k i with high probability and at a low price. B. Market Equilibrium and Efficiency
In this section, we show that mechanism consisting ofmarginal cost pricing together with EDF scheduling, resultsin an efficient market equilibrium at which social welfare (thesum of aggregate consumer surplus and supplier profit) ismaximized. First, we make an assumption under which wewill operate for the remainder of the paper.
Assumption 3.
We assume that every consumer type ( k, R, q ) ∈ Θ satisfies R ≥ c . (cid:3) See Remark 2 for a discussion on Assumption 3. It followsfrom Assumption 3 and Theorem 3 that under the mechanism ( π ∗ , ζ ) , it is a dominant strategy for every consumer tobe truth-telling. The aggregate demand resulting from thetrue-telling population, which we denote by x ∗ , is given byEquation (3). Theorem 2 shows that it is profit-maximizingfor a supplier to meet the aggregate demand x ∗ at the pricebundle p = ζ ( x ∗ ) . In what follows, we will show that theresulting quantity-price pair ( x ∗ , ζ ( x ∗ )) constitutes a marketequilibrium that is social welfare maximizing. We first offer adefinition of market equilibrium . Definition 7 (Market equilibrium) . Let ( π, κ ) be a marketmechanism consisting of a scheduling policy π and pricingscheme κ . Given the types of all consumers θ , a quantity-price pair ( x , p ) is a market equilibrium under the mechanism ( π, κ ) if the following two conditions hold.(i) It is a dominant strategy for every consumer to be truth-telling under the mechanism ( π, κ ) , and the resultingaggregate demand bundle is x .(ii) The aggregate demand bundle x together with thescheduling policy π maximize a price-taking supplier’sexpected profit at price bundle p = κ ( x ) .In Definition 7, the first condition ensures that under themechanism ( π, κ ) , the aggregate demand (resulting from atruth-telling non-atomic population) is x . The second conditionrequires that given the price bundle p = κ ( x ) (which isdetermined according to the pricing scheme κ and aggregatedemand x ), a price-taking supplier would like to employ thescheduling policy π and supply the bundle x . We thereforehave supply equal to demand. We show in the following corol-lary that the EDF scheduling policy π ∗ = ( σ ∗ , φ ) togetherwith the marginal cost pricing scheme ζ constitute a marketmechanism that induces a unique market equilibrium, whichmaximizes the social welfare. Corollary 1.
Suppose that Assumptions 1-3 hold. Given thetypes of all consumers θ and a market mechanism ( π ∗ , ζ ) ,there exists a unique market equilibrium ( x ∗ , ζ ( x ∗ )) thatmaximizes the social welfare. Here, x ∗ is the truth-tellingaggregate demand bundle specified in Equation (3).V. C ONCLUSION
To explore the flexibility of deferrable electricity loads,we propose a novel market for deadline-differentiated energy services that offers discounted (per-unit) electricity prices toconsumers in exchange for their consent to defer their electricpower consumption, and provides a guarantee on the aggregatequantity of energy to be delivered by a consumer-specifieddeadline. We provide a full characterization of the jointscheduling and pricing scheme that yields an efficient (compet-itive) market equilibrium between a (price-taking) supplier anda large consumer population. Somewhat surprisingly, we showthat this efficient mechanism is incentive compatible in thatevery consumer would like to reveal her true deadline to thesupplier, regardless of the actions taken by other consumers.There are several interesting directions for future research.First, it would be of practical interest to relax the assumptionrequiring a fixed price of firm supply to allow for time-varyingprices. Second, the market we have considered in our analysisis single shot, in the sense that it is cleared only once. As anatural extension, it would be of interest to analyze a dynamicanalog of our formulation in which the market is clearedon a dynamic basis over a finite or infinite horizon. Such ageneralization of our model would also serve to facilitate thetreatment of random consumer arrival times.A
PPENDIX AI NTRA - CLASS S CHEDULING P OLICIES
Formally, for a consumer i who purchases a bundle a =( a , . . . , a N ) , we let λ k,i ( x , a ) ∈ R N + denote (element-wise in j ) the amount of energy delivered (at period k ) to consumer i so as to satisfy her demand a j . We denote the intra-classscheduling policy by φ = { ( λ ,i , . . . , λ N − ,i ) | i ∈ [0 , } , where λ k,i : R N + × A → R N + for all i ∈ [0 , and k =0 , . . . , N − . Given an inter-class scheduling policy σ ∈ Σ ,an intra-class scheduling policy φ is feasible if and only if itsatisfies the following constraints.1. The intra-class scheduling policy should not deliver anysupply that is allocated to class j to consumers outsidethis class. That is, for each demand class j = 1 , . . . , N ,we have that λ jk,i ( x , a ) = 0 for every consumer i suchthat a j = 0 .2. At every period k , no energy is delivered to demandclass j with j ≤ k by the feasibility of the inter-classpolicy σ ∈ Σ . That is, λ jk,i ( x , a ) = 0 for every i ∈ [0 , ,and every ≤ j ≤ k ≤ N − .3. The total supply allocated to demand class j at timeperiod k must be fully utilized, i.e. (cid:90) [0 , λ jk,i ( x , a ) η ( di ) = µ jk + ν jk , ∀ ≤ k < j ≤ N − , where we use the Lebesgue integral (with respect toLebesgue measure η defined on [0 , ), and µ jk and ν jk denote the amount of intermittent and firm supplyallocated to demand class j at period k , according tothe inter-class policy σ .
4. Each consumer’s individual delivery commitments mustbe met: (cid:88) kt =1 a t ≤ ω πk,i ( x , a ) ≤ (cid:88) Nt =1 a t , k = 1 , . . . , N, where the total energy delivered to consumer i by it’strue deadline k i is given by ω πk i ,i ( x , a ) = k i − (cid:88) t =0 N (cid:88) j =1 λ jt,i ( x , a ) . (15)Notice that for any feasible inter-class scheduling policy σ ∈ Σ , it is always possible to ensure the satisfaction ofthe above constraint.We denote the feasible intra-class policy space by Φ( σ ) , whichis parameterized by a given inter-class policy σ ∈ Σ .A PPENDIX BP ROOF OF L EMMA Q ∗ in Eq. (12).For notational simplicity, we denote the sequence of optimalinputs by u ∗ k = µ ∗ k ( z k , s k ) and v ∗ k = ν ∗ k ( z k , s k ) for k =0 , . . . , N − . Under the optimal scheduling policy, firm supplyis deployed only as a last resort to ensure task satisfaction. Itfollows that Q ∗ ( x ) = c · N (cid:88) k =1 E (cid:110) v k, ∗ k − (cid:111) . To establish the desired result, it suffices to show that v k, ∗ k − = − min { , ξ k } . First, define the quantity δ k = k − (cid:88) j =0 s j − k (cid:88) (cid:96) =1 (cid:96) − (cid:88) j =0 u (cid:96), ∗ j , ∀ k = 1 , . . . , N, which denotes the maximum amount of intermittent supplyavailable to demand class k + 1 across the first k timeperiods under a sequence of EDF allocations u ∗ , . . . , u ∗ k − .Clearly, we have that δ k ≥ for all k , given feasibilityof the allocations u ∗ , . . . , u ∗ k − under the intermittent supplyavailability constraints. One can readily show via an inductiveargument that ξ k = δ k − v k, ∗ k − , ∀ k = 1 , . . . , N. Using this characterization of the residual process ξ , we havethat δ k > ⇒ v k, ∗ k − = 0 = ⇒ min { , ξ k } = 0 ,δ k = 0 = ⇒ v k, ∗ k − ≥ ⇒ min { , ξ k } = − v k, ∗ k − , which yields the desired result that min { , ξ k } = − v k, ∗ k − and establishes the form of Q ∗ in Eq. (12).We now establish convexity of the expected recourse cost Q ∗ ( x ) directly. Let x , x ∈ R + and denote the correspondingoptimal scheduling policies by σ ∗ ∈ Σ( x ) and σ ∗ ∈ Σ( x ) .Define the convex combination of demand bundles x λ = λ x + (1 − λ ) x , where λ ∈ [0 , . It follows that λQ ∗ ( x ) + (1 − λ ) Q ∗ ( x ) = E N − (cid:88) k =0 c (cid:62) ( λ v σ ∗ k + (1 − λ ) v σ ∗ k ) . And, it is not difficult to show that the convex combination ofthe constituent policies λσ ∗ + (1 − λ ) σ ∗ is admissible for theconvex combination of demand bundles, i.e., λσ ∗ +(1 − λ ) σ ∗ ∈ Σ( x λ ) . Convexity of Q ∗ follows. Differentiability of Q ∗ ( x ) over (0 , ∞ ) N follows immediately from the proof of Theorem2, in which we show that ∂Q ∗ ( x ) ∂x k = ζ k ( x ) , k = 1 , . . . , N, where the function ζ k ( x ) , defined in (14), is bounded andcontinuous over (0 , ∞ ) N for each k = 1 , . . . , N . (cid:4) A PPENDIX CP ROOF OF T HEOREM ( · ) − = min { , ·} . Fix a price bundle p ∈ R N + . We havepreviously shown in Theorem 1 that the EDF inter-classscheduling policy σ ∗ ∈ Σ( x ) is optimal for any demandbundle x ∈ R N + . Hence, it suffices to show that p = ζ ( x ) satisfies the first order condition for optimality (13). Takingthe gradient of the supplier’s expected profit with respect to x yields ∇ x J ( x , p , σ ∗ ) = p − ∇ x Q ∗ ( x ) . It remains toshow that ∂Q ∗ ( x ) ∂x k = ζ k ( x ) for k = 1 , . . . , N. Working the with the simplified form of Q ∗ established in Lemma 1, it follows readily that ∂Q ∗ ( x ) ∂x k = c · N (cid:88) (cid:96) = k ∂∂x k E (cid:8) ξ (cid:96) ( x , s ) − (cid:9) , (16)where we’ve truncated the summation from below at (cid:96) = k ,as ξ (cid:96) ( x , s ) is wholly independent of x k for all (cid:96) < k . Wetherefore restrict our attention to (cid:96) ≥ k for the remainderof the proof. The next step of the proof relies on the abilityto interchange the order of differentiation and expectation in(16). It is obvious from construction that ξ (cid:96) ( x , s ) − is both acontinuous function of ( x , s ) and piecewise affine in x k (witha finite number of linear segments) for each s . It follows that ξ (cid:96) ( x , s ) − is differentiable almost everywhere in x k ∈ R + andsatisfies (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x k ξ (cid:96) ( x , s ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ almost everywhere. Then, for each x k ∈ R + , we have that ∂ξ (cid:96) ( x , s ) − /∂x k is integrable in x k and by the dominatedconvergence theorem ∂∂x k E (cid:8) ξ (cid:96) ( x , s ) − (cid:9) = E (cid:26) ∂∂x k ξ (cid:96) ( x , s ) − (cid:27) . Finally, it is not difficult to see that ∂∂x k ξ (cid:96) ( x , s ) − = { ξ k ≤ } , (cid:96) = k, { ξ (cid:96) ≤ } · (cid:96) − (cid:89) t = k { ξ t > } , (cid:96) > k. And taking expectation, we have the desired result. (cid:4) A PPENDIX DP ROOF OF T HEOREM x be the aggregate demand of the other consumers (ex-cluding i ). Suppose that the supplier uses the EDF (inter-classscheduling) policy σ ∗ , and an arbitrary intra-class schedulingpolicy φ ∈ Φ( σ ∗ ) . We let π ∗ = ( σ ∗ , φ ) . We consider aconsumer i of some type θ i = ( k, R, q ) such that c ≤ R . Wewill show that the consumer, who faces an arbitrary aggregatedemand bundleh, x , requested by other consumers, would liketo take the truth-telling action specified in Definition 3. Forthe rest of this proof, we will use p = { p k } Nk =1 to denote theprice bundle induced by the pricing scheme ζ (cf. its definitionin Eq. (14)), at the aggregate demand x .If consumer i is truth-telling, she will request a quantity q before deadline k , and receive an expected payoff of V π ∗ i ( θ i , ϕ ∗ i , x , ζ )= U ( q ) − qp k = qR − qc (cid:104) P ( ξ k ≤ (cid:88) Nt = k +1 P ( ξ k > , . . . , ξ t − > , ξ t ≤ (cid:105) . (17)Since the optimal price schedule is nonincreasing in dead-line, and demand is guaranteed to be met before the requesteddeadline, the consumer has no incentive to request a positiveamount of electricity at some period t that is earlier than k . We can therefore assume that consumer i takes an action a (cid:48) = ϕ (cid:48) i ( θ i ) such that a (cid:48) t = 0 , t = 0 , . . . , k − . Since the consumer cannot increase its expected payoff(compared to being truth-telling) by reporting a (cid:48) k ≥ q , wefocus on the case where a (cid:48) k < q . We first write the consumer’sexpected payoff (achieved by the action a (cid:48) ) as V π ∗ i ( θ i , ϕ (cid:48) i , x , ζ ) = E (cid:110) U θ (cid:16) ω π ∗ k,i ( x , a (cid:48) ) (cid:17)(cid:111) − N (cid:88) t = k p t a (cid:48) t . Showing that V π ∗ i ( θ i , ϕ (cid:48) i , x , ζ ) is no more than the expectedpayoff in (17) is equivalent to Rq − ( q − a (cid:48) k ) c (cid:104) P ( ξ k ≤ (cid:88) Nt = k +1 P ( ξ k > , . . . , ξ t − > , ξ t ≤ (cid:105) ≥ E (cid:110) U θ (cid:16) ω π ∗ k,i ( x , a (cid:48) ) (cid:17)(cid:111) − (cid:88) Nt = k +1 p t a (cid:48) t . (18)We will derive an upper bound on the right hand side of (18),and show that this upper bound cannot exceed the left handside of (18). For notational convenience, we define η t = { ξ k > , . . . , ξ t − > , ξ t ≤ } for all t = k + 1 , . . . , N − and η k = { ξ k ≤ } , η N = { ξ k > , . . . , ξ N − > } . Note that these N − k + 1 events are mutually disjoint.Further, it is straightforward to see that ξ k ≤ implies that ω πk,i ( x , a (cid:48) ) = a (cid:48) k . While, on the other hand, ξ k > implies thatone of the (mutually disjoint) events { η t } Nt = k +1 must occur,i.e., P (cid:18)(cid:91) Nt = k η t (cid:19) = N (cid:88) t = k P ( η t ) . (19)Under the event η t , the amount of energy delivered by theEDF scheduling policy before her true deadline k , ω π ∗ k,i ( x , a (cid:48) ) ,cannot exceed (cid:80) tτ = k a (cid:48) τ , for t = k, . . . , N . It follows fromAssumption 1 that Rq − U θ ( x ) ≥ R ( q − x ) + , ∀ x ≥ , where ( · ) + = max {· , } . We then have Rq − E (cid:110) U θ (cid:16) ω π ∗ k,i ( x , a (cid:48) ) (cid:17)(cid:111) ≥ R (cid:88) Nt = k P ( η t ) · (cid:16) q − (cid:88) tτ = k a (cid:48) τ (cid:17) + . (20)We now argue that the right hand side of (20) is minimizedat some vector ˜ a such that (cid:80) Nt = k ˜ a t ≤ q . To see this,suppose that (cid:80) Nt = k a (cid:48) t > q . Let T be the smallest t such that (cid:80) tm = k a (cid:48) m > q . We define an alternative vector ˜ a ˜ a t = a (cid:48) t , t ≤ T − , ˜ a T = q − T − (cid:88) m = k a (cid:48) m , (21)and ˜ a t = 0 , for every t > T . We have (cid:80) Nt = k ˜ a t = q , and (cid:88) Nt = k P ( η t ) · (cid:16) q − (cid:88) tτ = k a (cid:48) τ (cid:17) + = (cid:88) Nt = k P ( η t ) · (cid:16) q − (cid:88) tτ = k ˜ a τ (cid:17) + . Note that if (cid:80) Nt = k a (cid:48) t ≤ q , then ˜ a = a (cid:48) . To validate (18), itsuffices to show that for any vector ˜ a defined in (21), R (cid:88) Nt = k P ( η t ) · (cid:16) q − (cid:88) tτ = k ˜ a τ (cid:17) ≥ − (cid:88) Nt = k +1 p t ˜ a t + ( q − ˜ a k ) c (cid:104) P ( ξ k ≤ (cid:88) Nt = k +1 P ( ξ k > , . . . , ξ t − > , ξ t ≤ (cid:105) , (22)where the left hand side is a lower bound on the loss of ex-pected utility due to the non-truthful action ˜ a (cf. the inequalityin (20)), and the right hand side is the difference between thepayment under the truth-telling action and the action ˜ a . Wewill prove the following inequality that is equivalent to (22) R (cid:88) Nt = k P ( η t ) · (cid:16) q − (cid:88) tτ = k ˜ a τ (cid:17) = R ( q − ˜ a k ) − R (cid:88) Nt = k +1 (cid:16) P ( η t ) · (cid:88) tτ = k +1 ˜ a τ (cid:17) = R ( q − ˜ a k ) − R (cid:88) Nt = k +1 (cid:18) ˜ a t · (cid:88) Nτ = m P ( η m ) (cid:19) ≥ ( q − ˜ a k ) c (cid:104) P ( ξ k ≤ N (cid:88) t = k +1 P ( ξ k > , . . . , ξ t − > , ξ t ≤ (cid:105) − (cid:88) Nt = k +1 p t ˜ a t , (23) where the first equality follows from (19), and the secondequality is obtained by rearranging terms.It follows from the characterization of supplier marginalcost in (14) that for t = k + 1 , . . . , N, ˜ a t p t = ˜ a t c N (cid:88) m = t P ( ξ t > , . . . , ξ m − > , ξ m ≤ ≥ ˜ a t c N (cid:88) m = t P ( ξ k > , . . . , ξ m − > , ξ m ≤ . (24)We then have, for t = k + 1 , . . . , N , R ˜ a t (cid:88) Nm = t P ( η m ) − ˜ a t p t = R ˜ a t (cid:18) − P ( ξ k ≤ − (cid:88) t − m = k +1 P ( η m ) (cid:19) − ˜ a t p t ≤ R ˜ a t (cid:18) − P ( ξ k ≤ − (cid:88) t − m = k +1 P ( η m ) (cid:19) − ˜ a t c (cid:88) Nm = t P ( ξ k > , . . . , ξ m − > , ξ m ≤ ≤ R ˜ a t (1 − P ( ξ k ≤ − ˜ a t c (cid:88) Nm = k +1 P ( ξ k > , . . . , ξ m − > , ξ m ≤ ≤ R ˜ a t − ˜ a t c (cid:16) P ( ξ k ≤ (cid:88) Nm = k +1 P ( ξ k > , . . . , ξ m − > , ξ m ≤ (cid:17) . (25)Here, the first inequality follows from (24); the second inequal-ity is true, because R ≥ c and η m = { ξ k > , . . . , ξ m − > , ξ m ≤ } , for m = k + 1 , . . . , t − ; the last inequalityfollows from R ≥ c . For any vector ˜ a with (cid:80) Nt = k ˜ a t ≤ q ,from (25) we have (cid:88) Nt = k +1 (cid:18) − p t ˜ a t + R ˜ a t (cid:88) Nm = t P ( η m ) (cid:19) ≤ (cid:88) Nt = k +1 ˜ a t (cid:104) R − c (cid:16) P ( ξ k ≤ (cid:88) Nm = k +1 P ( ξ k > , . . . , ξ m − > , ξ m ≤ (cid:17)(cid:105) . (26)Since R ≥ c , the right hand side of (26) is nondecreasingin (cid:80) Nt = k +1 ˜ a t . The desired result in (23) follows from thefact (cid:80) Nt = k +1 ˜ a t ≤ q − ˜ a k . Since the preceding analysisholds for any action a (cid:48) and any aggregate demand bundle x , we conclude that it is a dominant strategy for consumer i to be truth-telling, i.e., the pricing scheme (14) is incentivecompatible, in the sense of Definition 6. (cid:4) A PPENDIX EP ROOF OF C OROLLARY ( x ∗ , ζ ( x ∗ )) is a market equilibrium (inthe sense of Definition 7), and then argue that it is the uniquemarket equilibrium that maximizes the social welfare. Underthe mechanism ( σ ∗ , φ, ζ ) , it follows from Assumption 3 andTheorem 3 that it is a dominant strategy for every consumerto be truth-telling, and therefore the aggregate demand is x ∗ .Given the price bundle ζ ( x ∗ ) , the aggregate demand bundle x ∗ together with the EDF scheduling policy maximizes thesupplier’s expected profit (cf. Theorem 2). Hence, the pair, ( x ∗ , ζ ( x ∗ )) , constitutes a market equilibrium.Let ( x , p ) be a market equilibrium. The second conditionof Definition 7 requires that the given the price bundle p , thequantity x maximizes the (price-taking) supplier’s expectedprofit. We note that this implies that p is the supplier’smarginal cost to supply x . Since the supplier marginal costnever exceeds c , at any price bundle that represents suppliermarginal cost, a truth-telling consumer would request her max-imum demand by her true deadline, and therefore the aggregatedemand of a truth-telling consumer population is always x ∗ .The uniqueness of the truth-telling aggregate demand impliesthe uniqueness of a market equilibrium.It is straightforward to see that social welfare is maximizedat this market equilibrium, because under Assumption 3, it issocially optimal to fully serve the aggregate demand x ∗ , andfurther, the EDF scheduling policy σ ∗ minimizes the expectedcost of servicing x ∗ . (cid:4) R EFERENCES[1] Albadi, M. H., E. F. El-Saadany. 2007. Demand response in electricitymarkets: An overview.
Proc. of Power Engineering Society GeneralMeeting .[2] Bitar, E., P. Khargonekar, K. Poolla, R. Rajagopal, P. Varaiya, F.Wu. 2012. Selling random wind.
Proc. of 45th Hawaii InternationalConference on System Science (HICSS) .[3] Bitar, E., S. Low. 2012. Deadline differentiated pricing of deferrableelectric power service.
Proc. of the 51th IEEE Conference on Decisionand Control .[4] Bitar, E., Y. Xu. 2013. On incentive compatibility of deadline differenti-ated pricing for deferrable demand.
Proc. of the 52nd IEEE Conferenceon Decision and Control .[5] Borenstein, S. 2005. The long-run effciency of real-time electricitypricing.
The Energy Journal , vol. 26, no. 3, 93–116.[6] Callaway, D. S., I. A. Hiskens. 2011. Achieving controllability of electricloads.
Proc. of the IEEE , vol. 99, 184–199.[7] CAISO. 2010. Integration of renewable resources: operational require-ments and generation fleet capability at 20% RPS. California Indepen-dent System Operator Report.[8] Chao, H. P., R. Wilson. 1987. Priority service: pricing, investment, andmarket organization.
American Econ. Review , vol. 77, no. 5, 899–916.[9] Chen, S., T. He, L. Tong. 2011. Optimal deadline scheduling withcommitment.
Proc. of Allerton Conference on Communication, Control,and Computing .[10] Envision Solar. 2014. Envision Solar announces over two million dollarsin new orders. http://envisionsolar.com/category/press-releases[11] FERC. 2009. A national assessment of demand response potential. FERCStaff Report.[12] Fuller, J.C., K.P. Schneider, D. Chassin. 2011. Analysis of residentialdemand response and double-auction markets.
Proc. of IEEE Power andEnergy Society General Meeting .[13] Hao, H., W. Chen. 2014. Characterizing flexibility of an aggregation ofdeferrable loads.
Proc. of the 53nd IEEE Conference on Decision andControl
IEEE Tran.on Power Systems , vol. 26, no. 4, 1875–1884.[16] Kefayati, M., R. Baldick. 2011. Energy delivery transaction pricing forflexible electrical loads.
Proc. of IEEE International Conference onSmart Grid Communications (SmartGridComm) .[17] Li, N., L. Chen, S.H. Low. 2011. Optimal demand response basedon utility maximization in power networks.
Proc. of IEEE Power andEnergy Society General Meeting .[18] Li, S., W. Zhang, J. Lian, K. Kalsi. 2016. Market-based coordinationof thermostatically controlled loads – part I: a mechanism designformulation.
IEEE Tran. on Power Systems , vol. 31, no. 2. [19] Mathieu, J. L., D. S. Callaway, S. Kiliccote. 2011. Variability inautomated responses of commercial buildings and industrial facilitiesto dynamic electricity prices. Energy and Buildings , vol. 43, no. 12,3322-3330.[20] Nayyar, A., M. Negrete-Pincetic, K. Poolla, P. Varaiya. 2014. Duration-differentiated energy services with a continuum of loads.
Proc. of the53nd IEEE Conference on Decision and Control .[21] Negrete-Pincetic, M., S. P. Meyn. 2012. Markets for differentiatedelectric power products in a Smart Grid environment.
Proc. of IEEEPower and Energy Society General Meeting .[22] O’Brien, G., Rajagopal, R. 2015. Scheduling non-preemptive deferrableloads.
IEEE Tran. Power Systems , To Appear.[23] Oren, S. S., S. A. Smith, and R. B. Wilson. 1983. Competitive nonlineartariffs,
Journal of Economic Theory , vol. 29, no. 1, 49–71.[24] Oren, S. S., S. A. Smith. 1993.
Service Opportunities for Electric Util-ities: Creating Differentiated Products , Kluwer Academic Publishers:Boston.[25] Oren, S. S., S. A. Smith. 1992. Design and management of curtailableelectricity service to reduce annual peaks.
Oper. Res. , vol. 40, no. 2,213–228.[26] Papavasiliou, A., S. Oren. 2010. Supplying renewable energy to de-ferrable loads: Algorithms and economic analysis.
Proc. of IEEE Powerand Energy Society General Meeting .[27] Robu, V., S. Stein, E. H. Gerding, D. C. Parkes, A. Rogers, N. R.Jennings. 2012. An online mechanism for multi-speed electric vehiclecharging.
Auctions, Market Mechanisms, and Their Applications , vol.80, 100–112.[28] Roozbehani, M., A, Faghih, M. Ohannessian, M. A. Dahleh. 2011. Theintertemporal utility of demand and price-elasticity of consumption inpower grids with shiftable loads.
Proc. of the 50th IEEE Conference onDecision and Control and European Control Conference .[29] Roozbehani, M., M. A. Dahleh, S. K. Mitter. 2012. Volatility of powergrids under real-time pricing.
IEEE Tran. Power Systems , vol. 27, no.4, 1926–1940.[30] Salah, F., C. M. Flath. 2014. Deadline differentiated pricing in practice:marketing EV charging in car parks.
Computer Science-Research andDevelopment , 1–8.[31] Sioshansi, R. 2012. Modeling the impacts of electricity tariffs on plug-inhybrid electric vehicle charging, costs, and emissions.
Oper. Res. , vol.60, no. 3, 506–516.[32] Solaire Generation. 2014. Solaire Generation unveils premium solarcarport structure at GE campus. http://solairegeneration.com/category/press-releases/[33] Spees, K., L. B. Lave. 2007. Demand response and electricity marketefficiency.
The Electricity Journal , vol. 20, no. 3, 69–85.[34] Stoft, S. 2002.
Power System Economics: Designing Markets for Elec-tricity , IEEE Press, Piscataway, NJ.[35] Strauss, T., S. Oren. 1993. Priority pricing of interruptible electricservice with an early notification option,
The Energy Journal , 175–196.[36] Subramanian, A., M. Garcia, A. Dom´ınguez-Garc´ıa, D. Callaway, K.Poolla, P. Varaiya. 2012. Real-time scheduling of deferrable electricloads.
Proc. of American Control Conference (ACC) .[37] Tan C. W., P. P. Varaiya. 1993. Interruptible electric power servicecontracts.
Journal of Economic Dynamics and Control , vol. 17, 495–517.[38] Wilson, R. 2002. Architecture of power markets.
Econometrica , vol. 70,no. 4, 1299–1340.[39] Yang, I., D. S. Callaway, C. J. Tomlin. 2014. Risk-limiting dynamiccontracts for direct load control. http://arxiv.org/abs/1409.1994[40] Zachariadis, T., N. Pashourtidou. 2007. An empirical analysis of electric-ity consumption in Cyprus.
Energy Economics , vol. 29, no. 2, 183–198.
Eilyan Bitar currently serves as an Assistant Profes-sor and the David D. Croll Sesquicentennial FacultyFellow in the School of Electrical and ComputerEngineering at Cornell University, Ithaca, NY, USA.Prior to joining Cornell in the Fall of 2012, he wasengaged as a Postdoctoral Fellow in the departmentof Computing and Mathematical Sciences at the Cal-ifornia Institute of Technology and at the Universityof California, Berkeley in Electrical Engineering andComputer Science, during the 2011-2012 academicyear. His current research examines the operationand economics of modern power systems, with an emphasis on the design ofmarkets and optimization methods to manage uncertainty in renewable energyresources. He received the B.S. and Ph.D. degrees in Mechanical Engineeringfrom the University of California at Berkeley in 2006 and 2011, respectively.Dr. Bitar is a recipient of the NSF Faculty Early Career Development Award(CAREER), the John and Janet McMurtry Fellowship, the John G. MaurerFellowship, and the Robert F. Steidel Jr. Fellowship.