Multi-Dimensional Forward Contracts under Uncertainty for Electricity Markets

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1 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 1 Muti-Dimensiona Forward Contracts under Uncertainty for Eectricity Markets Hamidreza Tavafoghi and Demosthenis Teneketzis Abstract We consider mechanism design probems for strategic agents with muti-dimensiona private information and uncertainty in their utiity/cost function. We show that the optima mechanism with firm aocation can be impemented as a noninear pricing scheme, and the optima mechanism with random aocation can be impemented as a menu of noninear pricing schemes. We provide two exampes to demonstrate the resuts: an optima energy procurement mechanism from a strategic seer with renewabe (random) generation, and the design of an optima demand response program for a network of heterogeneous oads. Index Terms renewabe energy, demand response, contract under uncertainty, eectricity market I. INTRODUCTION In recent years, eectricity markets have undergone profound structura changes both in the generation and the demand side. The traditionay monopoistic governmentreguated markets reformed toward iberaized eectricity markets in order to introduce competition and increase efficiency in generation [19]. Privatey-owned generators and utiity companies possess private information about their cost/utiity, behave strategicay, and seek to maximize their profit. Moreover, the deveoping network of smart grids aims to utiize the avaiabe fexibiity on the demand side to increase the efficiency of the grid. To invove the demand side activey into the operation of the grid, one needs to design appropriate mechanisms that incentivize the demand to exercise fexibiity in its consumption behavior. Long-term contracts, as an agreement between strategic parties with private information, is one of the main trading mechanisms used in eectricity markets. Generators and utiity companies sign ong-term contracts to hedge themseves against the risk of pooing markets. In fact, it has been suggested that ong-term contracts are necessary aong with the existing pooing markets to ensure the stabiity and reiabiity of eectricity markets [2]. A preiminary version of this paper appeared in the proceedings of the 52th Aerton Conference on Communication, Contro, and Computation, 214 (see [15]). This work was supported in part by the NSF under Grant CNS The authors woud ike to thank Gaina Schwartz for her hepfu comments on the paper. H. Tavafoghi and D. Teneketzis are with the Department of Eectrica Engineering and Computer Science, University of Michigan, Ann Arbor, MI 4819 USA (e-mai: tavaf,teneket@umich.edu) Contracts have been considered as one of the main mechanisms to induce a desired behavior on the demand side of smart grids. In comparison to rea time pricing or direct market participation, contracts with incentive payments resut in a direct contro of resources, and thus, give reiabiity and stabiity guarantees [11]. Furthermore, contracts with incentive payments are simper to impement and more appeaing to smaer market participants (e.g. househods) [16]. In this paper, we study a genera contract design probem for eectricity markets in a principa-agent (buyerseer(s)) setup. We assume that both the buyer and the seer sides have muti-dimensiona private information that and genera utiity/cost functions. Furthermore, we expicity consider a genera uncertainty in our probem formuation; uncertainty is becoming a critica issue in eectricity markets. As the share of intermittent generation from renewabe generation increases, the uncertainty in the avaiabe generation wi increase. Furthermore, the added fexibiity on the demand side in smart grids aso means a higher uncertainty on demand; such uncertainty shoud be propery managed through appropriatey designed incentives. In genera, both the buyer and the seer may have uncertainty, either in their cost/utiity functions, or the avaiabiity of the resources being traded between them. By expicity incuding uncertainty into our probem formuation we capture these facts and can address the probem of commitment (ex-post vountary participation), risk sharing, and forward contracts with random aocation. The probems formuated in this paper enabe us to capture and anayze interaction between energy consumers and renewabe energy generators, as we as interactions between an aggregator and a network of a demand popuation participating in the demand response program. We provide exampes for each of these scenarios so as to iustrate our resuts. A. Reated Literature There is a growing iterature on contract design for eectricity with information asymmetry and strategic behavior. A contract design probem for demand management with one-dimensiona private information and inear utiity has (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

2 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 2 been studied in [6]. The work in [1] addresses the probem of contract design for deferrabe demands with constant margina utiity for demand. The work in [4] considers a mechanism design probem for the forward reserved market assuming that the participants have constant margina cost and no market power. Athough the private information in [1], and [4] is muti-dimensiona, the simpifying assumption of constant margina cost/utiity enabes the authors to rank different types, and is critica to the soution approaches they provide. The specific structures of utiity/cost functions assumed in [1], [4], and [6] enabe the authors to provide soutions that are inspired by the soution methodoogy of the one-dimensiona screening probem. Contract design probem for demand response with quadratic cost functions is investigated in [7] by numerica methods. The work in [14] considers a mechanism design probem for energy procurement with a genera utiity/cost function and uncertainty and appies a Vickery- Cacks-Goves (VCG) based mechanism. However, the VCG mechanism is suboptima for the probem formuated in [14] when the cost function cannot be parameterized by ony a one-dimensiona type (see [8], Ch. 14). From the economics point of view, the probem we formuate in this paper beongs to the cass of screening probems. In economics, the one-dimensiona screening probem has been we-studied with both inear and noninear utiity functions [3]. However, the extension to the muti-dimensiona screening probem is not straightforward and no genera soution is avaiabe. The authors in [9] study a genera framework for a deterministic mutidimensiona screening probem with inear utiities. They discuss two genera approaches, the parametric-utiity approach and the demand-profie approach. The methodoogy we use to sove the probem formuated in this paper is simiar to the demand-profie approach. We consider a muti-dimensiona screening probem under uncertainty with noninear utiities. The presence of noninearities and uncertainty resuts in additiona compications that are not present in [9] where the utiities are inear and there is uncertainty 2. presented in [1], [4], and [6] does not extend to our probem. The generaity of our mode enabes us to capture many instances of probems arising in eectricity markets. Two such instances are discussed in Section IV and VI. Second, we expicity incorporate a genera uncertainty in the reaized cost/utiity of the buyer and the seer. The presence of uncertainty aong with the noninearity of the utiities resut in probems where the methodoogy used in previous works ( [1], [4], [6]) cannot be appied, as in these works the utiities are inear and any uncertainty can be repaced by its expected vaue. The incusion of uncertainty is crucia in the modeing and anaysis of emerging eectricity markets because: (1) the share of renewabe generation increases; (2) the existing demand becomes ess shieded from the market outcome and more eastic; and (3) new resources/oads (e.g. storage, pug-in eectric vehices) enter the market. Due to uncertainty, firm forward contracts (a priori fixed aocation and fixed payment) do not appear to be an appropriate form of contract for emerging eectricity markets. Moreover, in the presence of uncertainty, interim vountary participation (defined in Section III) of the seer does not necessariy impy ex-post vountary participation of the seer (defined in Section V). Therefore, additiona considerations are needed to ensure the commitment of the agents to the contract for every reaization of the uncertainty. We show that, in genera, the optima mechanism for the probem formuated in this paper is a menu of noninear pricing schemes. We prove that by aowing the payment to depend on the uncertainty, we can achieve ex-post vountary participation of the seer, and a desired risk-sharing (associated with the uncertainty) between the buyer and the seer. To the best of our knowedge, our resuts present the first optima forward contract under uncertainty for eectricity markets where the buyer and the seer have genera utiity/cost functions parameterized by muti-dimensiona private information. We iustrate our resuts by providing two exampes from eectricity markets: an optima demand response contract for anciary service; and a biatera trade between a buyer and a renewabe energy generator. B. Contribution The contribution of this paper is two-fod. First, we consider an optima contract design probem for eectricity markets with utiity/cost functions that are more genera than those considered in the iterature ( [1], [4], [6], [7]). The nature of utiity/cost functions with muti-dimensiona private information is such that the soution methodoogy 2 When a probem is inear, expectation of any random variabe can be repaced by its expected vaue and reduce the probem to a deterministic one. C. Organization The paper is organized as foows. We introduce the mode in Section II. In Section III, we formuate and anayze an optima forward contract with deterministic aocation, and address the probem of risk sharing between the buyer and the seer. We iustrate the resut via an exampe for a contract design probem for demand response program in section IV. In Section V, we formuate and anayze an optima forward contract with random aocation that depends on the uncertainty, and address the probem of the seer s imperfect commitment (ex (c) 215 IEEE. 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3 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 3 post vountary participation). We provide an exampe of a biatera trade between a buyer and a renewabe energy generator in Section VI. We discuss our resuts in Section VII. We concude in Section VIII. In the Appendix we present the proofs of emmas and coroaries appearing in the paper. II. MODEL A buyer wants to design a mechanism to procure energy/resource from a seer. 3 Let q be the amount of energy/resource the buyer procures, and t be his payment to the seer. The buyer s tota profit is given by V(q) t, where V(q) is his utiity by receiving q amount of energy/resource. The function V( ) is the buyer s private information and V() =. The seer s provision cost is given by C(q, x, w), convex and increasing in q, x = (x 1, x 2,, x n ) χ R n is the seer s type, and w denotes the reaization of a random variabe W (uncertainty) with a probabiity distribution F W (w) that is common knowedge. We assume that C(, x, w) (zero-provision cost) does not depend on the reaization of random variabe w and is equa to x 1, i.e. C(, x, w)=c(, x)=x 1. The seer s utiity is given by her tota expected revenue E W {t C(q, x, W )}. The seer s type x is her private information, the set χ is common knowedge, and there is a prior probabiity distribution F x over χ which is common knowedge between the buyer and the seer. Let c(q, x) := E W {C(q,x,W )} q denote the expected margina cost for the seer s type x. We assume that m, 1 <m n, such that c(q, x) is increasing in x i for 1 i m, and decreasing in x i for m<i n. 4 Moreover, there exists x χ (the seer s worst type) such that x i x i and x j x j for a x χ, 1 i m and m<j n. Definition 1. We say the seer s type x is better (resp. worse) than the seer s type ˆx if c(q, x) c(q, ˆx) for a q (resp. c(q, x) c(q, ˆx)) with strict inequaity for some q. Therefore, the seer s type x is better than the seer s type ˆx if and ony if x i ˆx i for 1 i m, and x i ˆx i for m<i n with strict inequaity for some i. The foowing exampe iustrates such ordering. Exampe 1. Consider an energy seer with a wind turbine and a gas generator. The generation from the wind turbine is free and given by γw 3, where γ is the turbine s technoogy and w is the reaized weather. The gas generator has 3 From now on, we refer to the buyer as he and to the seer as she. 4 Note that for a genera cost function C(q, x, W ) if the corresponding c(q, x) changes sign for ony finite number of times, one can expand the type space χ and reorder its dimensions so that it satisfies the assumption on the existence of m. a fixed margina cost θ c. There is a fixed cost c which incudes the start-up cost for both pants and the capita cost for the seer. Therefore, the seer s type has n = 3 dimensions. The generation cost for the seer is given by C(q, w, x) = c + θ c max { q γw 3, }. (1) The seer s type x = (c, θ c, γ) is better than the seer s type ˆx = (ĉ, ˆθ c, ˆγ) if and ony if c ĉ, θ c ˆθ c, and γ ˆγ, with one of the above inequaities being strict. Note that in the one-dimensiona screening probem, the cost of production induces a compete order among the seer s types, which is crucia to the soution of the optima mechanism design probem. However, in mutidimensiona screening probems, the expected cost of production induces, in genera, ony a partia order among the seer s types. We assume that the buyer has a the bargaining power; thus, he can design the mechanism/set of rues that determines the agreement for the procurement quantity q, and the payment t. After the buyer announces the mechanism for procurement and the seer accepts it, both the buyer and the seer are fuy committed to foowing the rues of the mechanism. As a consequence of the assumption on the buyer s bargaining power and the fact that the seer s utiity does not directy depend on the buyer s private information (private vaue), the soution of the probem formuated in this paper does not depend on whether the buyer s utiity V( ) is private information or common knowedge 5. In the rest of the paper we formuate two contract design probems. In section III, we assume that the buyer can ony accept an a priori fixed energy deivery and formuate a forward contract design probem with deterministic aocation. In Section V, we assume that the buyer can toerate intermittency in the deivered energy by utiizing his existing storage/reserve resources, and formuate a forward contract design with random aocation. III. FORWARD CONTRACTS WITH DETERMINISTIC ALLOCATION In this section we consider a probem of forward contract design where the aocation q is deterministic and is decided in advance at the time of contract signing. Biatera trades with conventiona generators and demand response (DR) contracts for direct oad contro are forms of such a contract. A. Probem Formuation Let (M, h) be the mechanism/game form (see [1], Ch. 23) for energy procurement designed by the buyer. In this 5 This becomes more cear by ooking at the resut of Theorem (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

4 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 4 game form, M describes the message/strategy space for the buyer and the seer, respectivey, and h determines the outcome function; h : M R + R. For every message m M the outcome function h specifies the amount q of the procured energy/resource and the payment t made to the seer, i.e. h(m) = (q(m), t(m)). 6 The objective is to determine a mechanism (M, h) so as to maximize (M,(q( ),t( ))) E x,w {V(q(m )) t(m )}, (2) where m M is a Bayesian Nash equiibrium (BNE) of the game induced by the mechanism (M, h). We want the seer to vountariy participate in the procurement process. The vountary participation (VP) (or individua rationaity) for each type of the seer can be written as interim VP: E W {t(m ) C(q(m ), x,w )}, x χ (3) That is, at equiibrium m of the induced game the mechanism the seer must have an expected (with respect to the uncertainty W ) non-negative payoff. We ca the requirement expressed by (3) an interim vountary participation constraint. We ca the above probem (P1). B. Anaysis & Resuts We prove that the optima procurement mechanism is a pricing scheme that the buyer offers to the seer and the seer chooses a quantity according to her type. In such a pricing scheme we have M = χ, q : χ R +, and the payment function t( ) can be defined indirecty as a function of the quantity q(x), i.e. t(q(x)). We characterize the optima procurement mechanism by the foowing theorem, which reduces the origina functiona maximization probem (P1) to a set of equivaent pointwise maximization probems. Theorem 1. Under a certain concavity condition, stated in Lemma 3 beow, the optima mechanism (q( ), t( )) for the buyer is a noninear pricing scheme given by p(q) = t(q) = q(x) = arg max {P [x χ ˆp c(q, x)] (V (q) ˆp)},(4) q ˆp p()d + C(, x), (5) arg max R + {t () E W {C(, x, W )}} (6) where V (q) := dv(q) dq and M = χ. The assertion of Theorem 1 is estabished via severa steps. Beow we present these steps and the key ideas 6 Note that we use q (resp. t) to denote both the quantity vaue (resp. payment vaue) and the quantity outcome function (resp. payment function) of mechanism (M, h). behind each step. The proofs of the emmas and coroaries appearing in theses steps can be found in the appendix. In the seque, we omit the argument of the functions q( ) and t( ) whenever such an omission causes no confusion. Step 1. We set message space M = χ and formuate the foowing probem (P2) that is equivaent to probem (P1): maximize (q( ),t( )) subject to E x,w {V(q(x)) t(x)} (7) IC : x=arg max E W [t(x ) C(q(x ), x, W )], x χ(8) x interim V P : E W [t(x) C(q(x), x, W )], x χ, (9) where q : χ R + and t : χ R. The equivaence foows from the reveation principe [5]. By invoking the reveation principe, without oss of optimaity, we restrict attention to direct mechanisms (where M = χ) that are incentive compatibe and individuay rationa. Incentive compatibiity (IC) for a direct mechanism requires that truth-teing must be an optima strategy for the seer. Step 2. We show that for any incentive compatibe mechanism (q, t) the seer s worst type x gets the minimum utiity among a of the seer s types. We utiize the partia order among the seer s different types to rank her utiity for her different types (Lemma 1), and reduce the VP constraint (13) for a the seer s types to the VP constraint ony for the seer s worst type (Coroary 1). Lemma 1. For a given incentive compatibe mechanism (q, t), a better type of the seer gets a higher utiity. That is, et U(x) := E W {t(x) C(q(x), x, W )} denote the expected profit of the seer with type x. Then, 1) U, 1 i m, 2) U, m < i n. A direct consequence of Lemma 1, is that the seer s worst type x receives the minimum utiity among a the seer s types. Coroary 1. The vountary participation constraint is ony binding for the worst type x. That is, the genera VP constraint (13) can be reduced to U(x) := E W {t(x) C(q(x), x, W )}. (1) Step 3. We show, via Lemma 2 beow, that the optima mechanism (q, t) is a pricing scheme. That is the payment function t(x) can be defined indirecty as a function of q as t(q(x)). Lemma 2. For any pair of functions (q, t) that satisfies the IC constraint, we can rewrite t(x ) as t (q(x )) (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

5 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 5 With some abuse of notation we assume that the payment function t : R R refers to the indirecty defined function t(q(x)) (non-inear pricing scheme) and we denote t(q(x)) by t(q). Lemma 2 impies that the VP constraint (1) can be written as U(x) := E W {t(q(x)) C(q(x), x, W )}. (11). Step 4. We show that under a certain quasi-concavity condition, stated in Lemma 3 beow, we can define indirecty the aocation function q(x) as a function of the payment function t() by utiizing the incentive compatibiity constraint. We define the foowing probem (P3), that is equivaent to probem (P2), in terms of the margina price p() = dt() and the minimum payment t(): max p( ),t() d subject to interim VP: E W { P[x χ p() c(, x)](v () p())d t()(12) t()+ q(x) p()d C(q(x),x,W ) }. (13) The equivaence is estabished in two steps. First, consider an arbitrary incentive compatibe mechanism (q, t). The optima quantity q (x) for each type x of the seer is given by q (x) = arg max E W {t () C(, x, W )}. (14) Incentive compatibiity then requires that the seer must te the truth to achieve this optima vaue, and cannot do better by ying, i.e. q(x) = q (x) for a x χ. For any function t( ), this ast equaity can be taken as the definition for the associated function q( ). Thus, the IC constraint can be eiminated by defining q( ) := q ( ) and the probem of designing the optima direct reveation mechanism (q, t) can be reduced to an equivaent probem where we determine ony the optima payment function t( ) subject to the vountary participation constraint for the worst type. Next, using Lemma 3, stated beow, we rewrite the buyer s expected utiity in terms of the margina price p(q):= t(q) q and the minimum payment t() (which aong with p( ) uniquey determines the payment function t( )). Lemma 3. Assume that the seer s probem defined by (14) is continuous and quasi-concave 7. Then, the buyer s 7 This is a standard assumption in economics iterature, e.g. see [9] and [18]. Basicay, it can be seen as a situation where the seer can decide for each margina unit of production independenty. Thus, in genera, there is no guarantee that the seer s independent decisions about each margina unit of production resuts in a continuous and pausibe tota production quantity q. Therefore, the continuity of the resut must be checked a posteriori for each type of the seer. expected utiity can be expressed in terms of p(.) and t() as E x [V(q (x))] E x [t(q (x))]= P (x χ q (x) ) V ()d t() P (x χ q (x) ) p()d, (15) where P (x χ q (x) ) = P [x χ p() c(, x)]. (16) Using (15) and (16), we can rewrite the objective of probem (P2) and obtain the equivaent probem (P3) given by (12) and (13) Equation (16) states that the seer is wiing to produce the margina quantity at if the resuting expected margina profit is positive, i.e. the margina price p() exceeds the margina expected cost of generation c(, x). Equation (15) expresses the buyer s tota expected utiity in term of an integra of his tota margina utiity V () p() at quantity, times the probabiity that the seer s production exceeds, minus the minimum payment t(). Step 5. We prove that the seer s worst type produces the minimum quantity among a the seer s types, i.e. q(x) = min x χ q(x). As a resut, we show that probem (P3) is equivaent to the foowing probem (P4): max p( ) subject to iterim VP: C(, x)+ P [x χ p() c(, x)] (V () p()) d (17) q (x) p()d E W [C(q (x), x, W)].(18) We estabish the equivaence by providing a ranking for the seer s optima decision q (x) based on the partia order among the seer s types. Lemma 4. For a given mechanism specified by (t( ), q( )), a better type of the seer produces more. That is, the optima quantity q (x) that the seer with true type x wishes to produce satisfies the foowing properties: a) b) q (x), 1 i m, q (x), m < i n. As a consequence of Coroary 1 and Lemma 4 we can then simpify the VP constraint (13) as foows. Coroary 2. The interim VP constraint is satisfied if t() = C(,x) and the seer s worst type payment is equa to her expected production cost, i.e. t(q (x)) = E W {C(q (x), x, W )}. The equivaence of probems (P3) and (P4) foows from Coroary 2 and by repacing the VP constraint (13) by (18). Note that we aso dropped the constant term (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

6 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 6 t() = C(,x) (from Coroary 2) in the objective of probem P4 given by (17). Probem (P4) is in terms of the margina price p() and requires that the payment the seer s worst type receives is equa to her cost of production. Step 6. We show that the soution of probem (P4) is given by p() = arg max {P [x χ ˆp c(, x)](v () ˆp)} ˆp To prove the caim of Step 6 we consider a reaxed version of (P4) without the VP constraint (18). The unconstrained probem can be soved pointwise at each quantity to determine the optima p() as p()=arg max {P [x χ ˆp c(, x)](v () ˆp)}, (19) ˆp which is the same as (4). From Coroary 2 and the fact that the worst type has the highest expected margina cost, we can simpify (19), for q (x), as p() = c(, x), for q (x). (2) Note that for q (x) we have P [x χ ˆp c(, x)] = 1 from Lemma 4. Therefore, the minimum margina price p() that ensures a the seer s type are wiing to produce more than q (x) is equa to the margina expected cost for the seer s worst type c(, x). Therefore, the soution to the unconstrained version of probem (P4) satisfies constraint (18) of probem (P4), and therefore, (19) is aso the optima soution of probem (P4). We compete now the proof of Theorem 1. Using Caim 4 aong with Coroary 2, the optima payment function (noninear pricing) can be written as t(q) = q p()d + C(, x) which is the same as (5). From (14) we determine the optima energy procurement function, q(x) = arg max E W {t () C(, x, W )} which is the same as (6). The specification of t( ) and q( ) competes the proof of theorem 1 and the soution to probem (P1). In essence, Theorem 1 states that at each quantity, the optima margina price p() is chosen so as to maximize the expected tota margina utiity at, which is given by the tota margina utiity (V () p()) times the probabiity that the seer generates at east. Remark 1. In probem (P1), we assume that there exists a seer s worst type which has the highest cost at any quantity among a the seer s types, and we reduce the VP constraint for a of the seer s type to ony the VP constraint for this worst type. As a resut, we pin down the optima payment function by setting t() = C(, x) to ensure the vountary participation of the worst type, which consequenty impies the vountary participation for a of the seer s types. In absence of the assumption on the existence of the seer s worst type, the argument used to reduce the VP constraint is not vaid anymore and we cannot pin down the payment function and specify t() a priori. Assuming that a types of the seer participate in the contract, their decision on the optima quantity q ony depends on the margina price p(q), and therefore, the optima margina price p(q), given by (19), is sti vaid without the assumption on the existence of the worst type. To pin down the payment function t( ), we find the minimum payment t() a posteriori so that a types of the seer vountariy participate. That is, [ ] t()=max x χ E W {C(q(x), x, W )} q (x) p(ˆ)dˆ, (21) where the optima decision of type x is given by [ ] q (x) = arg max p(ˆ)dˆ E W {C(, x, W )}. (22) Remark 2. In a setup with a positive zero-provision cost for the seer, it might not be optima for the buyer to require a the seer s types to vountariy participate in the procurement process, since t() depends on the zeroprovision cost of the seer s worst type C(, x). In such cases, it might be optima for the buyer to excude some ess efficient types of the seer from the contract, seect an admissibe set of the seer s types, and then design the optima contract for this admissibe set of the seer s types 8. Note that this is not the case for setups without a zero-provision cost. In such setups, if it is not optima for some type x to be incuded in the optima contract, it is equivaent to set q(x) = in a contract menu that considers a types of the seer. C. Risk Aocation In the optima mechanism/contract menu presented by Theorem 1, the buyer faces no uncertainty, and he is guaranteed to receive quantity q(x), and a the risk associated with the reaization of W is taken by the seer. We wish to modify the mechanism to reaocate the abovementioned risk between the buyer and the seer. To do so, we modify the payment function so that the risk is reaocated between the buyer and the seer. Consider the foowing modified payment function with α [, 1], 8 To find the optima admissibe set, the optima contract can be computed for different potentia admissibe sets. Then, the resuting utiities can be compared to find the optima admissibe set (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

7 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 7 ˆt(x, w) = t(q(x)) + α [C(q(x), x, w) E W {C(q(x), x, W )}]. (23) From (23) it foows that E W {ˆt(x, W ) } =t(q(x)). Therefore, the strategic behavior of the seer does not change and the seer chooses the same quantity under the modified payment function ˆt( ) as under the origina payment function t(q) given by (5). Note that for α = we have the same payment as t(q). For α = 1, the seer is competey insured against any risk and a the risk is taken by the buyer. The parameter α determines the aocation of the risk between the buyer and the seer; the buyer undertakes α and the seer undertakes (1 α) share of the risk. The resut of Theorem 1 is iustrated by Exampe 1 in section IV. IV. EXAMPLE - DEMAND RESPONSE (DR) We consider a contract design for DR program. There is a oad aggregator that offers contracts with incentive payments to a heterogeneous popuation of oads who are wiing to yied the direct contro of their oad to the aggregator given that they are offered an appropriate incentive payment. The aggregator participates in an anciary service market and ses the aggregated resources to the reserve market at exogenous margina price p r. 9 Formay, there are I types of oads with a popuation distribution f over different types. Each oad of type i has a maximum controabe oad L i. Let q i L i denote the quantity that each oad of type i yieds its contro to the aggregator to be dispatched. We assume that each oad of type i has a quadratic cost (increasing margina cost) given by C i = α i + α 1 i q i + α 2 i q 2 i. (24) Therefore, the oad s type is x = (L i, α i, α1 i, α2 i ). Let t i denotes the incentive payment to each oad of type i for yieding the contro of oad q i. Then, the tota utiity of each oad of type i is given by U i = t i ( α i + α 1 i q i + α 2 i q 2 i ). (25) there exists no worst type; at ower quantities smaer oads (e.g. type (b)) have a ower margina cost whie at higher quantities arger oads (e.g. type (e)) have ower cost. Via Theorem 1 we determine the optima menu of contracts the aggregator offers to the heterogeneous popuation of oads (Tabe II). The optima menu of contracts can be interpreted aso as a noninear pricing that the aggregator offers to oads (Fig. 1). The optima choice and the resuting payoff for each type of oad are summarized in Tabe III. We note that, unike one-dimensiona contracts, a type with a higher quantity does not necessariy get a higher payoff. type L i (kw h) αi ( ) α1 i ( kw h ) α2 i ( kw h 2 ) i (a) (b) (c) (d) (e) TABLE I: Different types of oads Quantity q( ) (kw h) Payment t( ) ( ) TABLE II: Options menu offered by the aggregator Type Quantity Payment Cost Profit (a) (b) (c) (d) (e) TABLE III: Optima contract and the resuting outcome The aggregator participates in the anciary service market and provides capacity q = f i q i at a given uniform price p r. Therefore, the aggregator s revenue is given by p r (fi q i ) (f i t i ). (26) We consider I = 5 types of oads described in Tabe I aong with a normaized popuation distribution f with fi = 1, and set p r = 2 /kw h. Note that no compete ordering can be defined based on their margina cost and 9 If p r is not exogenous, the aggregator s interactions with the reserve market on one hand and the demand popuation on the other hand become couped. In this case these interactions must be studied simutaneousy. Fig. 1: The optima pricing scheme for DR program V. FORWARD CONTRACTS WITH RANDOM ALLOCATION In some instances of the probem considered in this paper, the buyer has a reserve resource [17] or wants to suppy deferrabe oads [1] that gives him the fexibiity to accept a random aocation q(x, W ) that depends on the (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

8 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 8 uncertainty W, and compensate the randomness in the aocation utiizing the existing fexibiity. In this section, we formuate and anayze a forward contract design probem with random aocation. We assume that the reaization of the random variabe W is common knowedge between the buyer and the seer. A. Probem Formuation Let e(x) denote the forward schedued quantity (deterministic) by the buyer and q(x, w) denote the random deivered quantity by the seer with type x. Let C R (e(x) q(x, w)) denote the cost incurred by the buyer to compensate the rea-time deviation e(x) q(x, w) from the forward schedue e(x). Then, for a given set of contract menus (q(x, w), t(x, w)), the buyer s optima schedue e(x) for the seer s type x is defined by e(x)=arg maxe W {V(ê) t(x,w) C R (ê q(x,w))}, (27) ê and the buyer s expected utiity is given by E W,x {V(e(x)) t(x, W ) C R (e(x) q(x, W ))}. (28) The buyer wants to design a mechanism (q(x, w), t(x, w)) so as to maximize his expected utiity given by (28), subject to the vountary participation of the seer. Formay, the contract design probem with random aocation for the buyer, caed (Q1), can be stated as foows: maximize E W,x {V(e(x)) t(x,w) C R (e(x) q(x,w))} (29) {q(, ),t(, )} subject to E W {t(x, W ) C(q(x, W ), x, W )}, x X. (3) B. Anaysis & Resuts We show, via Theorem 2 beow, that the optima forward contract with random aocation is a menu of pricing schemes, one for each type of the seer. Theorem 2. The optima forward contract with random aocation for probem (Q1) is a menu of pricing schemes given by e(x) = q(x) (31) q(x, w) = q(x) q R (x, w), (32) t(x, w) = t(x) C R (q R (x, w)), (33) where { q(x), t(x) } denotes the optima soution to the optimization probem maximize { q( ), t( )} E { W,x V( q) t } (34) subject to E W,x { t(x) C( q(x), } x, W ), (35) and C(q, x, w):=min {C(, x, w) + C R ( q )}, (36) q R (x, w):=arg min {C( q(x), x, w)+c R ()}. (37) Proof. Consider the foowing contract design probem where the seer s cost function is defined as C( q, x, w) = min {C(, x, w) + C R ( q )}, where C(,, ) is the seer s cost function in (Q1), and the buyer s utiity is defined as E W,x { V( q) t( q) }. (38) The optima contract design probem for the defined environment above, caed (Q2), can be stated as { maximizee x,w V( q) t } (39) { q, t} subject to IC: x=arg max E W { t(x ) C( q(x } ),x,w ), x χ (4) x interim VP: E W { t(x) C( q(x), } x, W ), x χ (41) where q and t denote the quantity and payment function for the defined probem above. By construction, probem (Q2) is the same as probem (P2). Let { q(x), t(x) } denote the optima contract for probem (Q2) obtained via Theorem 1. Note that through the cost function C( q(x), x, w), defined by (36), we absorb the optima schedue choice e(x), given by (27), and internaize the compensation cost C R (e(x) q(x, W )) for the random deviation e(x) q(x, W ) in probem (Q1) into the seer s cost function. Therefore, the optima schedued quantity e(x) for probem (Q1) is equa to the optima function q(x) for probem (Q2), i.e. e(x) = q(x). Consequenty, one can reconstruct the optima contract {q(x, w), t(x, w)} for probem (Q1) using the optima contract { q(x), t(x) } for the equivaent probem (Q2) as where q(x, w) = q(x) q R (x, w), t(x, w) = t(x) C R (q R (x, w)), q R (x, w):=arg min {C( q(x), x, w)+c R ()}, denotes the random reserve quantity required to compensate the random aocation q(x, w). Theorem 2 has the foowing interpretation. The (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

9 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 9 buyer offers different pricing schemes (quantity-payment curves), and each type of the seer chooses one based on her private information and expectation about W. Then, in rea time as W is reaized, based on the reaization w, one point from the chosen pricing scheme is seected and the payment t and the energy deivery q are determined. C. Imperfect Commitment and Ex-post Vountary Participation The vountary participation constraint imposed in probem (Q1) is interim. That is, the expected profit with respect to W must be non-negative for each type of the seer. Up unti now (probem (P1) and (Q1)) we have assumed that once the seer agrees to sign the contract (such an agreement takes pace before the reaization of random variabe W ) she is fuy committed to foowing the agreement, even if the reaized profit is negative (due to some reaization w) 1. Therefore, it woud be desirabe to modify the contract in order to ensure a positive payoff for the seer for every reaization of W and fu commitment without any outside enforcement. To ensure that the seer s reaized profit is non-negative for every reaization w, we impose an ex-post vountary participation constraint and repace the interim VP constraint (3) by Ex-post VP: t(x) C(q, x, w), w, x χ. (42) To satisfy the ex-post vountary participation constraint, we modify the payment function of the mechanism given by Theorem 2 as foows: ť(x, w)= E W {t(x, W )} E W {C(q(x, W ), x, W )} +C(q(x, w), x, w). (43) We have E W {ť(x, W ) } = EW {t(x, W )}, and therefore, the seer aways chooses the same quantity q under the modified payment function ť as under the origina payment function t given by (33). Furthermore, we have ť(x, w) C(q(x, w), x, w) = E W {t(x, W )} E W {C(q(x, W ), x, W )} for a w, x, where the ast inequaity is true since {q, t} satisfies the interim VP constraint (3). Therefore, under the modified payment function ť, { q, ť } satisfies the expost VP constraint (42). VI. EXAMPLE - FORWARD BILATERAL TRADE Consider a forward biatera trade between a buyer and a seer with wind generation. The buyer has an (amost ineastic) energy demand curve given by Fig. 2. The seer has a wind farm and (possiby) a reserve generator/storage that can be used to compensate for wind 1 Since the seer s reserved utiity is zero by not participating (outside option), we can aways think of the seer waking away from the agreement for these negative profit reaizations and not foowing the mechanism rues. Fig. 2: The buyer s demand curve Fig. 3:The wind turbine generation curve g(w,v ci,v r,v co,γ) Fig. 4: The wind forecast F W generation intermittency. The seer s wind generation is given by g(w, v ci, v r, v co, γ) as in Fig. 3, where w denotes the wind speed and (v ci, v r, v co, γ) denotes the specification of the wind turbine. The wind speed is random and the wind forecast f W is given by Fig. 4, which is a Weibu distribution with shape parameter k = 3 and average wind speed of 5m/s. We assume that the wind forecast f W as we as the wind reaization w are common knowedge between the buyer and the seer. The wind generation has a margina operationa cost θ w. The seer (possiby) has a reserve generator/storage with capacity r and a margina cost θ r that can be utiized if needed. The seer has a zeroproduction cost c which accounts for her capita cost and the start-up cost of her faciities. Therefore, the seer s private information is as x = (c, θ w, θ r, v ci, v r, v co, γ, r). We assume that the buyer has a reserve generator/deferrabe oad that can be utiized to compensate the rea-time random energy deivery by the seer. We assume that deviation from the schedued energy has an increasing margina cost for the buyer given by b + b 1 q. We consider 4 types for the seer as in Tabe IV, and set b = 1.4 $ kwh and b 1 =.5 $ kw h. 2 The optima forward contract menu for the buyer is given by Fig. 5. Since the energy demand considered in this exampe is amost ineastic, the schedued quantity e(x), and therefore, the quantity-demand curves are aso (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

10 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 1 type c θ w θ c v ci v r v co γ r f i a b c d TABLE IV: Different types of the seer cose to each others. 11 Tabe V summarizes the optima energy schedue e(x), and the expected utiity U(x) for different types of the seer. Fig. 7: Payment t(x, w) in terms of wind w Fig. 5: The optima forward contract menus type a b c d e(x) U(x) TABLE V: The outcomes of the optima contract menus The energy q(x, w) deivered to the buyer, the payment t(x, w) made to the seer, and the seer s utiity u(x, w) in terms of wind w are given by Figures 6, 7, and 8, respectivey. For ow reaizations of wind speed, the deivered energy is ow and the seer may even incur some penaty for very ow energy deivery (Fig. 5). For higher reaization of wind speed, the energy deivery increases, and therefore, the payment and the reaized utiity increase. However, for very high reaization of wind speed that surpasses the cut-off speed v co (see Figure 3), the energy deivery, and consequenty the payment and the reaized utiity, drop. Fig. 6: Energy deiver q(x, w) in terms of wind w VII. DISCUSSION For the probem on energy/service procurement formuated in this paper the optima mechanism is a menu of 11 For a competey ineastic energy demand, we have e(x) fixed and independent of the seer s type x. Therefore, a the quantity-demand curves coincide and are equa to the quantity-cost curve for the worst type. Fig. 8: The seer s utiity u(x, w) in terms of wind w contracts/noninear pricing schemes. The noninearity is due to three factors. First, the buyer s utiity function V(q) is not inear in the quantity q. Second, for each type of the seer, the cost function is a noninear function of the quantity. Third, the seer has private information about her technoogy and cost (seer s type). The buyer has to pay information rent (monetary incentive) to the seer to incentivize her to revea her true type. Therefore, the payment the buyer makes to the seer incudes the cost of provision the seer incurs pus the information rent, which varies with the seer s type; the better the seer s type, the higher is the information rent. The optima forward contracts discovered in this paper can be impemented as foows: the buyer offers the seer a menu of contracts (noninear pricing schemes); the seer chooses one of these contracts based on her type. The optima forward contracts induce some incentives for investment in infrastructure and technoogy deveopment. From Lemma 1, the seer with the higher type has a higher utiity. Therefore, there is an incentive for the seer to improve her technoogy and decrease her cost of generation. It is we-known that in the presence of private information and strategic behavior, in genera, there exists no mechanism/contract that is (1) individuay rationa, (2) incentive compatibe, and (3) efficient (Pareto-optima) [13]. In the optima forward contract given by Theorems 1, and 2 the aocation for the seer s different types is not ex-post efficient (Pareto-optima) except for the seer s worst type who gets zero utiity. In this paper, we formuated the contract design probem in a principa-agent setup. Therefore, the resut can be appied to the contract design probem for a setup with (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

11 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 11 one buyer (principa) and a heterogeneous popuation of seers (agents), if the buyer has a inear utiity function, as in Exampe 1, or if the share of each individua agent is sma and their effect on the market is negigibe. However, if one considers a setup with noninear utiity for the buyer or market power for each individua agent, the associated probem for such setup with mutipe agents becomes equivaent to the design of optima muti-unit auctions in economics. It is known that there exist no cosed form soution to the genera probem of optima mutiunit auctions, and their soutions can ony be computed numericay or with approximation [8]. VIII. CONCLUSION We investigated the probem of optima forward contract design under uncertainty and muti-dimensiona private information. The consideration of muti-dimensiona private information and genera utiity/cost functions enabes us to capture many appications in eectricity markets as we as other discipines. We assume that the buyer and/or the seer has uncertainty in their utiity/cost function which is reaized after the time of contract signing. We considered froward contracts with random aocation that depends on the rea-time reaization of the uncertainty. We characterized the optima forward contract under uncertainty as a menu of contracts. We addressed the probem of commitment (ex-post vountary participation), and risk sharing in the presence of uncertainty. We demonstrated our resuts by two exampes; an optima contract design for a demand response program, and an optima forward biatera trade between a buyer and a seer with wind energy generation. APPENDIX - PROOFS Proof of emma 1. The given mechanism (q, t) is incentive compatibe, so we can rewrite U(x) as U(x) = max E W {t(x ) C(q(x ), x, W )} (44) x By appying the enveope theorem [12] on (44), we get U = E W {C(q(x ), x, W )}. (45) x i x =x The above equation aong with the assumption on the monotonicity of the margina expected cost c(q, x) with respect to x i (Section II.A) gives U, 1 i m (46) U, m < i n. (47) Proof of emma 2. The proof is by contradiction. Assume that there exist x, x χ such that q(x) = q(x ) but t(x ) > t(x). Then a seer with type x is aways better off by reporting x instead of her true type x, which contradicts the IC constraint. Proof of Lemma 3. Consider the buyer s objective (7). For any function t( ), we can determine from (14) the cumuative distribution function for q, caed F q. Consequenty, we can rewrite the buyer s objective as E q [V(q ) t(q )] = (V() t()) df q () = (F q () 1) (V() t()) d (V() t()) + (1 F q ()) d. (48) d We have (F q () 1) (V() t()) = t() (49) because V() = by assumption, and (F q ( ) 1) =. Because of (49), we can rewrite (48) as E q [V(q ) t(q )] = P (q ) (V () p()) d t() (5) where V () = dv() d. We can rewrite P (q ) as { } P(q )=P [x χ argmaxe W t(ˆ) C(ˆ,x,W ) ]. (51) ˆ We impicity assume that the seer s probem given by (14) is continuous and quasi-concave, so that from the first order optimaity condition for (14) we obtain p(q (x)) = E W {C(, x, W )} = c(q (x), x). (52) q (x) Therefore, from the optimaity of q (x) and the quasiconcavity of (14), we must have p() > c(; x) and p() < c(; x) for < q (x) and > q (x), respectivey. That is, each type of the seer wishes to produce more than quantity if and ony if the margina price p(q) that she is paid at is higher than the expected margina cost of production c(, x) that she incurs at. Consequenty, combining (51) and (52) we obtain P (q ) = P [x χ p() c(, x)]. (53) Substituting (53) in (5), we obtain the foowing aternative expression for the buyer s objective E q [V(q ) t(q )]= P[x χ p() c(, x)] (V () p()) d t(). (54) (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

12 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI 1.119/TCNS , IEEE 12 Proof of emma 4. Let x, x χ, where x is a better type than x. From IC for seer s type x we have t(q(x)) E W {C(q(x), x, W )} t(q(x )) E W {C(q(x ), x, W )} (55) Simiary from IC for seer s type x we have t(q(x )) E W {C(q(x ), x, W )} Subtracting (56) from (55), we get t(q(x)) E W {C(q(x), x, W )}(56) E W {C(q(x), x, W )} E W {C(q(x ), x, W )} E W {C(q(x), x, W )} E W {C(q(x ), x, W )}(57) By assumption, de W{C(,x,W )} d de W{C(,x,W )} d if x is a better type than x. Therefore, (57) hods if and ony if q(x) q(x ). (58) Proof of coroary 2. Because of coroary 1, the VP constraint impies U(x) = t(q(x)) E W [C(q (x), x, W )] =, (59) which is equivaent to t() + q (x) p()d = E W [C(q (x), x, W )]. (6) Furthermore, from Lemma 4 it foows that if the worst type wishes to produce more than q (x), then a types produce more than q (x). Therefore, P [x χ p() c(, x)] = 1, for q (x). (61) Using (61), we can rewrite the objective function of probem (P3) as, ( ) t() + q (x) p()d + q (x) V ()d + P [x χ p() c(, x)] (V () p()) d.(62) q (x) The term t()+ q (x) p()d appears in both the objective (62) and the VP constraint (6). Therefore, without oss of optimaity, we can assume t() = C(, x), and set t(q(x)) = E W {C(q(x), x, W )}. REFERENCES [1] E. Bitar and Y. Xu. On incentive compatibiity of deadine differentiated pricing for deferrabe demand. In 52nd Annua Conference on Decision and Contro (CDC). IEEE, 213. [2] S. Borenstein. The troube with eectricity markets: Understanding caifornia s restructuring disaster. Journa of economic perspectives, pages , 22. [3] Timan Börgers. An introduction to the theory of mechanism design. Oxford University Press, 215. [4] H. Chao and R. Wison. Muti-dimensiona procurement auctions for power reserves: Robust incentive-compatibe scoring and settement rues. Journa of Reguatory Economics, 22. [5] P. Dasgupta, P. Hammond, and E. Maskin. The impementation of socia choice rues: Some genera resuts on incentive compatibiity. The Review of Economic Studies, pages , [6] M. Fahriogu and F. Avarado. Designing incentive compatibe contracts for effective demand management. IEEE Transactions on Power Systems, 15(4): , 2. [7] T. Haring, J. Mathieu, and G. Andersson. Decentraized contract design for demand response. In European Energy Market (EEM), 213 1th Internationa Conference on the, pages 1 8. IEEE, 213. [8] V. Krishna. Auction Theory. Academic press, 29. [9] L. Hansen M. Dewatripont and S. Turnovsky. Advances in Economics and Econometrics: Theory and Appications, Eighth Word Congress, voume 1. Cambridge University Press, 23. [1] A. Mas-Coe, M.D. Whinston, and J.R. Green. Microeconomic theory, voume 1. Oxford university press New York, [11] J. Mathieu, T. Haring, J. Ledyard, and G. Andersson. Residentia demand response program design: Engineering and economic perspectives. In European Energy Market (EEM), 213 1th Internationa Conference on the, pages 1 8. IEEE, 213. [12] P. Migrom and I. Sega. Enveope theorems for arbitrary choice sets. Econometrica, pages , 22. [13] R. Rosentha. Arbitration of two-party disputes under uncertainty. The Review of Economic Studies, 45(3):595 64, [14] W. Tang and R. Jain. Stochastic resource auctions for renewabe energy integration. In 49th Annua Aerton Conference on Communication, Contro, and Computing, pages IEEE, 211. [15] H. Tavafoghi and D. Teneketzis. Optima contract design for energy procurement. In Communication, Contro, and Computing (Aerton), 52th Annua Aerton Conference on. IEEE, 214. [16] Report to the US Congress U.S. Department of Energy. 21 smart grid system report. February 212. [17] M. Vrakopouou, J. Mathieu, and G. Andersson. Stochastic optima power fow with uncertain reserves from demand response. In 47th Hawaii Internationa Conference on System Sciences. IEEE, 214. [18] R. Wison. Noninear Pricing. Oxford University Press, [19] R. Wison. Architecture of power markets. Econometrica, 22. Hamidreza Tavafoghi received the Bacheor s degree in Eecterica Engineering from Sharif University of Technoogy, Tehran, Iran, 211, and the Master s degree in Eectricia Engineering from the University of Michigan, Ann Arbor, MI, USA, in 213. He is currenty a PhD student in Eectrica Engineering at the University of Michigan. His reaserach interests ie in game theory, mechanism design, and stochastic contro, and their appications to power systems. Demosthenis Teneketzis (M 87 SM 97 F ) received the dipoma in eectrica engineering from the University of Patras, Patras, Greece, and the M.S., E.E., and Ph.D. degrees, a in eectrica engineering, from the Massachusetts Institute of Technoogy, Cambridge, MA, USA, in 1974, 1976, 1977, and 1979, respectivey. He is currenty Professor of Eectrica Engineering and Computer Science at the University of Michigan, Ann Arbor, MI, USA. His research interests are in stochastic contro, decentraized systems, queueing and communication networks, stochastic scheduing and resource aocation probems, mathematica economics, and discrete-event systems (c) 215 IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.