A market-based approach for enabling inter-area reserve exchange
aa r X i v : . [ m a t h . O C ] F e b A market-based approach for enabling inter-area reserve exchange
Orcun Karaca a, ∗ , Stefanos Delikaraoglou b , Maryam Kamgarpour c a Automatic Control Laboratory, D-ITET, ETH Z¨urich, ETL K12, Physikstrasse 3, 8092, Z¨urich, Switzerland b Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, USA c Electrical and Computer Engineering Department, University of British Columbia, Vancouver, Canada
Abstract
Considering the sequential clearing of energy and reserves in Europe, enabling inter-area reserveexchange requires optimally allocating inter-area transmission capacities between these two mar-kets. To achieve this, we provide a market-based allocation framework and derive payments withdesirable properties. The proposed min-max least core selecting payments achieve individual ra-tionality, budget balance, and approximate incentive compatibility and coalitional stability. Theresults extend the works on private discrete items to a network of continuous public choices.
Keywords: electricity markets, coalitional game theory, mechanism design, public choice problem
1. Introduction
The increasing penetration of stochastic renewable generation poses new challenges to the elec-tricity markets that were conceived on the premise of predictable and fully controllable generation.In the current European market design, energy and reserve capacity are traded through independentand sequential auctions, which are commonly executed at noon the day before actual operation,based on point forecasts of renewable energy production. Any imbalance between scheduled gen-eration and load demand arising close to the hour of delivery is managed through the balancingmarket. Even though many recent works demonstrated the benefits of stochastic market design[2, 10, 24, 26], the actual implementation of such approaches would require significant restructuringof any of the existing frameworks. Owing to this reason, we restrict ourselves to the status-quosequential market architecture.Apart from the limited temporal coordination between scheduling and balancing decisions,the European market suffers also from partial coordination in space. Although day-ahead energymarkets are jointly cleared, reserve and balancing markets are still operated on a regional (country)level. To mitigate this inefficiency, the European Commission regulation has already published a reserve exchange guideline to be completed by 2023 [6]. However, the joint-clearing of reserve ∗ Corresponding author
Email address: [email protected] (Orcun Karaca)
Preprint submitted to arXiv February 18, 2021 arkets requires also allocating a portion of the inter-area transmission capacity from the day-ahead energy market to the reserve market. Currently, these cross-border capacities for the day-ahead market are decided by the operators respecting the requirements in [7, Article 16(8a-8b)],see also the example in [8, § §
2. Market-based approach
Let A denote the set of areas. Similar to [1, 21] and many others, in our framework, area as awhole (country or region) is an ensemble of consumers and generators pertaining to that area andoperator (and the transmission owners). This definition ensures the institutional relevance of theoverall problem. Even when the main actors involved in the decision-making of a reserve exchangeare operators, they are expected to seek the benefits of their areas, see [1, 13], and they are oftenchecked by regulatory authorities.Let E denote the set of links. Assume the graph ( A, E ) is strongly connected and simple, i.e.,undirected graph without self-loops. Areas a , a ∈ A are adjacent (or neighbors) if e = ( a , a ) =( a , a ) ∈ E . A link e ∈ E is incident to an area a if a ∈ e . Given a set of areas S ⊆ A , E S = { e ∈ E | ∃ ! a ∈ S such that a ∈ e } denotes the set of links connecting the area set S to theremaining areas in A \ S . Throughout the paper, we use a and { a } interchangeably, for instance,3 a = E { a } . Moreover, given a set of areas R ⊆ A , E R = { e ∈ E | ∃ a , a ∈ R such that e = ( a , a ) } denotes the set of links connecting the areas within the set R. Let χ ∈ [0 , E denote the percentageof inter-area interconnection capacity withdrawn from the day-ahead market and allocated toreserves exchange. For any F ⊆ E , let χ F = { χ e } e ∈ F . We assume its default value is χ ′ ∈ [0 , E which originates from the existing cross-border agreements. Following our previous discussions,this default value will generally be given by χ ′ = 0 , which means that reserve exchanges are notadmissible in the existing sequential electricity market. One notable exemption is the Skagerrakinterconnector between Western Denmark and Norway with χ ′ = 0 . . Let X a ⊆ [0 , E a denote the feasible allocation choices for area a. Each area has a private truevaluation v a : X a → R , which maps from transmission capacity allocations for the links incidentto area a to the change in the cost of area a relative to the cost under the default values χ ′ .Here, the cost of an area refers to minus the social welfare, which is given by the sum of theconsumers’ and generators’ surplus pertaining to that area and the congestion rents collected bythe corresponding area operator. This definition was established also in [21] for allocating benefitsfrom new interconnections. Positive v a is a reduction in costs. We further have χ ′ E a ∈ X a and v a ( χ ′ E a ) = 0. We assume these values can be estimated by the area operators and is thus reflectedin their bids. In the numerics, we will address how they can be computed.Each area then submits a (potentially nontruthful) bid of the form b a : ˆ X a → R , where χ ′ E a ∈ ˆ X a ⊆ [0 , E a and b a ( χ ′ E a ) = 0 . Observe that the variable χ ( a,a ′ ) is a public choice sharedby a and a ′ , since χ ( a,a ′ ) is an argument to the bids of both areas a and a ′ . These functions cantheoretically be extended to the links incident to the neighbors. However, an area is assumed tobe precluded from taking part in the decision for other links.Given the strategy profile B = { b a } a ∈ A , a mechanism defines an allocation rule χ ∗ ( B ) ∈ [0 , E and a payment rule p a ( B ) ∈ R for all a ∈ A . We restrict our attention to the allocation rule thatachieves the allocative efficiency: V ( B ) = max χ ∈ S X a ∈ A b a ( χ E a ) s . t . χ E a ∈ X a , ∀ a ∈ A, (1)where the set S ⊆ [0 , E (with χ ′ ∈ S ) may encompass any exogenously imposed regulatory con-straints, e.g., a fixed percentage of reserves should be covered by internal resources, or a restrictionon the feasible χ for computational tractability. Let the optimal solution of (1) be denoted by χ ∗ ( B ). Assume that in case of multiple optima there is a tie-breaking rule.Utility of area a is assumed to be quasilinear (linear and separable in the payment), and it isdefined by u a ( B ) = v a ( χ ∗ E a ( B )) − p a ( B ) . Revealed utilities are defined as the utilities computedfrom the information disclosed to the market organizer: ¯ u a ( B ) = b a ( χ ∗ E a ( B )) − p a ( B ) . For themarket organizer, both its revealed and true utilities are equivalent, and defined by the totalpayment collected: u MO ( B ) = ¯ u MO ( B ) = P a ∈ A p a ( B ) . These definitions are imperative since the4rue utilities of the areas are unknown. Notice that the revealed utilities correspond to the trueutilities whenever the submitted bids are the true valuations. Hence, any property defined over therevealed utilities would also hold for true utilities whenever the areas are truthful. Moreover, anypayment rule is uniquely defined by the revealed utilities it induces: p a ( B ) = b a ( χ ∗ E a ( B )) − ¯ u a ( B ) . Before providing the desired fundamental properties our mechanisms, we highlight that theframework studied in this paper defines monetary quantities on an area level, and does not dis-tribute them on a market participant level. Defining such distribution rules is part of our ongoingwork, see also the discussions in [14, 17].
Definition 1 (Individual rationality) . A mechanism is individually rational if the areas are facingnonnegative revealed utilities: ¯ u a ( B ) = b a ( χ ∗ E a ( B )) − p a ( B ) ≥ . Definition 2 (Budget balance) . A mechanism is budget balanced if the market operator faces anonnegative (revealed) utility ¯ u MO ( B ) ≥ . Even more preferably, a mechanism is strongly budgetbalanced if ¯ u MO ( B ) = 0 . Definition 3 (Efficiency) . A mechanism is efficient if the sum of all utilities u MO ( B )+ P a ∈ A u a ( B ) = P a ∈ A v a ( χ ∗ E a ( B )) is maximized. Or equivalently, efficiency is attained if we are solving for the op-timal allocation of the market in (1) under the condition that all of the areas submitted their truevaluations V = { v a } a ∈ A . Efficiency can also be defined for the revealed utilities. However, this property would not bemeaningful since it is guaranteed independent of the payment rule as long as we are solving forthe optimal allocation of (1) under the submitted bids. Connected with the original efficiencydefinition and its relation to truthfulness, we bring in incentive compatibility.
Definition 4 (Incentive compatibility) . A mechanism is dominant-strategy incentive-compatible(DSIC) if the true valuation profile V = { v a } a ∈ A is the dominant strategy Nash equilibrium. Unilateral deviations are not the only manipulations we need to consider in order to ensurethat all areas reveal their true valuations to the market. A subset of areas S ⊂ A can potentiallyexercise a coalitional deviation, that is, they can exclude areas A \ S to form their own market [7,(14)] and compute the optimal transmission allocation for only their bids, V ( B S ) = max χ ∈ S X a ∈ S b a ( χ E a ) s . t . χ E a ∈ X a , ∀ a ∈ S,χ e = χ ′ e , ∀ e ∈ E \ E S , (2)where B S = { b j } j ∈ S . The last set of constraints encodes the fact that altering the transmissioncapacity allocation of a link requires the approval from both areas incident to it. This observationimplies that a deviating coalition cannot change the default value χ ′ for links missing these twoapprovals. Observe that V ( B a ) = 0 for all a ∈ A , since a single area cannot change any of the5efault values. It can easily be verified that the set function V is nondecreasing in S . Related tocoalitional deviations, we bring in ǫ -coalitional stability. Definition 5 ( ǫ -Coalitional stability) . Given ǫ ≥ , a mechanism is ǫ -coalitionally stable if therevealed utilities of the areas lie in the ǫ -core K Core ( B , ǫ ) , that is, { ¯ u a ( B ) } a ∈ A ∈ K Core ( B , ǫ ) = { ¯ u ∈ R A | P a ∈ A ¯ u a = V ( B ) , P a ∈ S ¯ u a ≥ V ( B S ) − ǫ, ∀ S ⊂ A } . As a remark, the core is defined as K Core ( B , ǫ -core from [27] in the above definition. Invoking the definition ofthe revealed utilities, the equality constraint in K Core ( B , ǫ ) is equivalent to strong budget balance,since otherwise areas would prefer to arrange this market with another organizer. When ǫ = 0, theinequalities are our exact coalitional requirement: no set of areas can improve their total revealedutilities by a coalitional deviation. This property is defined over the revealed utilities, since thetrue utilities are private information. Whenever ǫ >
0, these inequalities can be interpreted asfollows. If organizing a coalitional deviation entails an additional utility reduction of ǫ ∈ R , thetotal revealed utilities would be given by V ( B S ) − ǫ . Then, the resulting core would be the ǫ -core.This provides us with an ǫ approximation of coalitional stability.In order to motivate our proposal in the next section, we briefly review the well-known Grovespayment defined by p a ( B ) = b a ( χ ∗ E a ( B )) − ( V ( B ) − h a ( B − a )), where B − a = { b j } j ∈ A \ a . A particularchoice for the function h a ( B − a ) ∈ R is the Clarke pivot rule h a ( B − a ) = V ( B − a ) where V ( B − a ) isthe optimal value of (2) with S = A \ a . The Groves payment with the Clarke pivot rule is referredto as the Vickrey-Clarke-Groves (VCG) mechanism. Our first result shows that the properties ofthe VCG mechanism extend to our problem. This result is a straightforward generalization of theoriginal proof [20], which does not consider a network of continuous (divisible) public choices withgeneral nonconvex constraints and nonconvex valuations, and included for the sake of completeness. Proposition 1.
Given the model (1) , (i) the Groves payment yields a DSIC mechanism, (ii) theVCG mechanism (the Groves payment with the Clarke pivot rule) ensures individual rationality.
The proof is relegated to Appendix B. In summary, all bidders have incentives to reveal theirtrue valuations under the Groves payment if they consider only unilateral deviations. Moreover,this mechanism is known to be the unique DSIC mechanism for a general class of problems [11].However, two negative results can be stated from the literature as follows. There is no mechanismfor public good problems that can always solve for the optimal allocation under the submittedbids, and attain DSIC and strong budget balance simultaneously [12]. The work in [23] ensures theabove for the exchange of private goods in the more general setting of Bayesian implementation.Appendix C presents a counterexample for our problem involving a network of public choicesshowing that Groves payment is not strongly budget balanced, ǫ -coalitional stability is not attained,and it cannot be efficient. 6ext section proposes a payment rule that attains strong budget balance, individual rationality,and approximates coalitional stability and DSIC. Let ǫ ∗ ( B ) be the critical value such that the ǫ -core is nonempty, ǫ ∗ ( B ) = min { ǫ ≥ | K Core ( B , ǫ ) = ∅} . The set K Core ( B , ǫ ∗ ( B )) is called the least core [22]. In contrast to [22], we additionally includethe constraint ǫ ≥
0, since ǫ ∗ ( B ) < K Core ( B , Definition 6 (Approximate coalitional stability) . A mechanism is approximately coalitionally sta-ble if the revealed utilities lie in the least core K Core ( B , ǫ ∗ ( B )) . Observe that strong budget balance is implied by the least core K Core ( B , ǫ ∗ ( B )), P a ∈ A ¯ u a ( B ) = V ( B ), but individual rationality is not, since we have ¯ u a ( B ) ≥ V ( B a ) − ǫ ∗ ( B ) = − ǫ ∗ ( B ).We now define the least core selecting payment rule as p LC a ( B ) = b a ( χ ∗ E a ( B )) − ¯ u a ( B ) , where ¯ u ( B ) = { ¯ u a ( B ) } a ∈ A ∈ K Core ( B , ǫ ∗ ( B )) . This mechanism is strongly budget balanced and approximatelycoalitionally stable. We can prove two additional properties. First, we can obtain a bound on theprofitability of picking nontruthful bids as a group of areas. This result complements approximatecoalitional stability, since a coalition can resort to such group manipulations while still being partof the market with all the areas. Second, we prove individual rationality.
Theorem 1.
Given the model (1) ,(i) Let ¯ ǫ be an upper bound on ǫ ∗ ( B ) for all admissible profiles B . Assume a coalition of areas S ⊂ A is strategizing as a group to pick their bid functions B S . Under the least core selectingpayment rule, they can obtain at most ¯ ǫ more total utility when compared to the case in whichthey participate as a single area in a VCG mechanism.(ii) The least core selecting payment rule yields an individually rational mechanism. To prove the above result, we first bring in a lemma reformulating the least core with analternative set of inequalities in the form of upper bounds.
Lemma 1. ¯ u ( B ) ∈ K Core ( B , ǫ ∗ ( B )) if and only if P a ∈ A ¯ u a ( B ) = V ( B ) and P a ∈ S ¯ u a ( B ) ≤ V ( B ) − V ( B − S ) + ǫ ∗ ( B ) , where B − S = { b j } j ∈ A \ S for all S ⊂ A. Proof.
Since P a ∈ R ¯ u a ( B ) = V ( B ) − P a ∈ A \ R ¯ u a ( B ), we can equivalently reorganize the inequalityconstraints as V ( B ) − P a ∈ A \ R ¯ u a ( B ) ≥ V ( B R ) − ǫ ∗ ( B ), for all R ⊂ A . Setting R = A \ S yields thestatement. 7 roof of Theorem 1. (i) For the set of areas S , define a merged bid for the case in which theyparticipate as a single area j : b j : ˆ X j → R , where ˆ X j = { χ E S | ∃ χ E S such that χ E a ∈ ˆ X a , ∀ a ∈ S } , and b j ( χ E S ) = min χ ES P a ∈ S b a ( χ E a ) s . t . χ E a ∈ ˆ X a , ∀ a ∈ S . Let E j = E S . Observe that b j ( χ ′ E j ) ≥
0, and whenever this value is strictly greater than 0, we can normalize it by assumingthat the set S collected the resulting positive revealed utility before participating in the market.Using the same definition, denote the true merged valuation by v j : X j → R . Define the profiles˜ B = ( B − S , B j ), B = ( B − S , B S ) , ˜ V = ( B − S , V j ). Total utility obtained from group bidding by theset S is given by X a ∈ S u a ( B ) = X a ∈ S [ v a ( χ ∗ E a ( B )) − b a ( χ ∗ E a ( B )) + ¯ u a ( B )] ≤ X a ∈ S v a ( χ ∗ E a ( B )) − b a ( χ ∗ E a ( B )) + V ( B ) − V ( B − S ) + ǫ ∗ ( B ) ≤ X a ∈ S [ v a ( χ ∗ E a ( B ))] − b j ( χ ∗ E j ( B )) + V ( ˜ B ) − V ( B − S ) + ¯ ǫ = u VCG a ( ˜ B ) + ¯ ǫ ≤ u VCG a ( ˜ V ) + ¯ ǫ. The first equality is the definition of true utilities. The first inequality follows from the leastcore selecting payment rule and Lemma 1. The second inequality holds since ǫ ∗ ( B ) ≤ ¯ ǫ and thedefinition of the merged bid implies that V ( ˜ B ) = V ( B ) and b j ( χ ∗ E j ( B )) = P a ∈ S b a ( χ ∗ E a ( B )). Thesecond equality follows from the definition of the VCG utility and the uniqueness guaranteed bythe tie-breaking rule: χ ∗ E j ( B ) = χ ∗ E j ( ˜ B ). The third inequality is the DSIC property of the VCGmechanism. Therefore, the utility obtained from group manipulation is upper bounded by theutility obtained when the group participates as a single area in a VCG mechanism plus ¯ ǫ. (ii) For this proof, we need to show that the least core selecting payment rule implies that¯ u a ( B ) = b a ( χ ∗ E a ( B )) − p a ( B ) ≥ K Core ( B , ǫ ∗ ( B )) ⊆ R A + under any bid profile. To this end, we can extend the methodin [22, Theorem 2.7] by taking into account that our set function V is defined by the optimizationproblem in (2), and hence it is both nondecreasing and V ( B a ) = 0 under any bid profile.Fixing the bid profile to be B , we now prove by contradiction. Let u ∈ K Core ( B , ǫ ∗ ( B )) with u a ′ <
0. In this case, we show that there exists ǫ < ǫ ∗ ( B ) such that K Core ( B , ǫ ) = ∅ , which wouldcontradict the definition of the least core. Since u ∈ K Core ( B , ǫ ∗ ( B )) and V ( B a ′ ) = 0, we have0 > u a ′ ≥ V ( B a ′ ) − ǫ ∗ ( B ) = − ǫ ∗ ( B ). For any S a ′ , we have P a ∈ S u a + u a ′ ≥ V ( B S ∪ B a ′ ) − ǫ ∗ ( B )(use the fact that ǫ ∗ ( B ) ≥ S ∪ a ′ = A ). As previously mentioned, V is nondecreasing and u a ′ <
0. Hence, we obtain P a ∈ S u a > V ( B S ) − ǫ ∗ ( B ).We can always find a small positive number δ such that P a ∈ S u a − | S | δ > V ( B S ) − ǫ ∗ ( B ) + δ holds for any S a ′ . Next, we show that K Core ( B , ǫ ∗ ( B ) − δ ) is nonempty. Define ˜ u such that˜ u a = u a − δ for all a = a ′ and ˜ u a ′ = u a ′ + ( | A | − δ . Revealed utility ˜ u clearly satisfies the8quality constraint in K Core ( B , ǫ ∗ ( B ) − δ ). For inequality constraints S a ′ , we have P a ∈ S ˜ u a = P a ∈ S u a − | S | δ > V ( B S ) − ǫ ∗ ( B ) + δ , where the strict inequality follows from the definition of δ .For inequality constraints S ∋ a ′ , we have P a ∈ S ˜ u a ≥ P a ∈ S u a + δ ≥ V ( B S ) − ǫ ∗ ( B ) + δ . Hence,˜ u ∈ K Core ( B , ǫ ∗ ( B ) − δ ), and K Core ( B , ǫ ∗ ( B ) − δ ) = ∅ . This observation can be done under any bidprofile and hence it concludes the proof.Whenever ¯ ǫ = 0 , the core is nonempty, part (i) achieves the utility bounds derived for biddingwith multiple identities (shill bidding) in core-selecting auctions of private discrete items [3, The-orem 1]. In contrast, our problem involves a network of continuous public choices. There are twomajor differences this entails in terms of proof method. First, the core in auctions of private dis-crete items is always nonempty, and it involves also the utility of the market organizer ignoring thebudget balance property. Similar conclusions cannot be made for our problem. Our proof methodutilizes instead the least core constraints without the market organizer and also an upper-boundon its relaxation term ǫ . Second, having a network of public choices requires integrating a noveldefinition of how areas can merge and participate as a single area into the proof method, whereasthis is not needed in [3]. Finally, we highlight that the part (ii) of our theorem is not implieddirectly by the definition of the least core (as it was the case in the core: ¯ u a ( B ) ≥ V ( B a ) = 0) sinceindividual rationality constraints are relaxed by the nonnegative term ǫ ∗ ( B ).Because we are deviating from using the DSIC Groves payment, we also need to quantify theviolation of this property. Theorem 2.
Given the model (1) , let p be any payment rule that charges at most ¯ ǫ (with ¯ ǫ ≥ )less than the VCG mechanism under the same bid profile. The additional utility of an area a by aunilateral deviation from its true valuation, that is, u a ( B a ∪ B − a ) − u a ( V a ∪ B − a ) for a nontruthfulbid B a , is at most ¯ ǫ + u VCG a ( V a ∪B − a ) − u a ( V a ∪ B − a ) , where u VCG a ( V a ∪B − a ) = V ( V a ∪B − a ) − V ( B − a ) is the VCG utility under the true valuation.Proof. Assume there exists a bid ˆ b a : ˆ X a → R such that[ v a ( χ ∗ E a ( ˆ B a ∪ B − a )) − p a ( ˆ B a ∪ B − a )] − u a ( V a ∪ B − a ) > ¯ ǫ + V ( V a ∪ B − a ) − V ( B − a ) − u a ( V a ∪ B − a ) , where ˆ B a = { ˆ b a } , χ ∗ E a ( ˆ B a ∪ B − a ) is the optimal allocation of the problem corresponding to V ( ˆ B a ∪B − a ). The inequality above is equivalent to the existence of a deviation that is more profitablethan the given upper bound. Notice that the following holds by our assumption on p , p a ( ˆ B a ∪ B − a ) ≥ b a ( χ ∗ E a ( ˆ B a ∪ B − a )) + V ( B − a ) − V ( ˆ B a ∪ B − a ) − ¯ ǫ. We kindly refer to [14, 16, 19] for the applications of core-selecting auctions in an electricity market setting. v a ( χ ∗ E a ( ˆ B a ∪ B − a )) − (cid:2) b a ( χ ∗ E a ( ˆ B a ∪ B − a )) + V ( B − a ) − V ( ˆ B a ∪ B − a ) (cid:3) > V ( V a ∪ B − a ) − V ( B − a ) . Observe that the first term is the VCG utility under a non-truthful bid, whereas the second termis the VCG utility under the true valuation. The strict inequality above contradicts the DSICproperty of the VCG mechanism. We conclude that ¯ ǫ + u VCG a ( V a ∪ B − a ) − u a ( V a ∪ B − a ) is indeedan upper bound on the additional profit obtained from a unilateral deviation.Notice that the least core-selecting payment rule satisfies the assumption in Theorem 2. Lemma 1implies p LC a ( B ) = b a ( χ ∗ E a ( B )) − ¯ u a ( B ) ≥ b a ( χ ∗ E a ( B )) + V ( B − a ) − V ( B ) − ǫ ∗ ( B ) = p VCG a ( B ) − ǫ ∗ ( B ) . Letting ¯ ǫ be an upper bound on ǫ ∗ ( B ) for all admissible bid profiles yields the assumption. In thenumerics, this parameter ¯ ǫ will be estimated. Whenever ¯ ǫ = 0, we achieve the unilateral deviationbounds originally derived for core selecting auctions of private discrete items in [4, Theorem 3.2].Theorem generalizes this result to the least core, and also to a network of continuous public choices.The differences of our proof method are the integration of the public choice bidding language andthe market function V and also the generalization to payment mechanisms that can charge lessthan the VCG mechanism.We now propose a method to pick a least core selecting payment rule to approximate the DSICproperty. As we discussed, any payment rule is uniquely defined by the revealed utilities. Firstsolve the following optimization problem to compute ǫ ∗ ( B ) of the least core:min { ǫ | ǫ ≥ , ∃ ¯ u ∈ K Core ( B , ǫ ) } . (3)We can then solve the following to obtain the revealed utilities of our payment rule approximatingDSIC: min ¯ u (cid:26) max a ∈ A ¯ u a − ¯ u VCG a ( B ) (cid:12)(cid:12)(cid:12) ¯ u ∈ K Core ( B , ǫ ∗ ( B )) (cid:27) , (4)where ¯ u VCG a ( B ) = V ( B ) − V ( B − a ) is the VCG revealed utilities. Let ¯ u MLC ( B ) denote its opti-mal solution. The min-max least core selecting (MLC) payment rule is defined by p MLC a ( B ) = b a ( χ ∗ E a ( B )) − ¯ u MLC a ( B ) for all a ∈ A. We refer to it as the MLC mechanism.
Corollary 1.
The MLC mechanism is approximately DSIC in the sense that the maximum of thebounds in Theorem 2 for all areas is minimal among all the least core selecting payment rules underthe condition that all the areas are truthful.
This statement can easily be verified by evaluating the bound in Theorem 2 when all theremaining areas are truthful, that is, submitting V − a . As a remark, (3) and (4) can be cast as linear programs since K Core ( B , ǫ ) is given by a set oflinear equality and inequality constraints. However, a direct solution requires solving (2) under10ll sets of coalitions to evaluate the set function V . Instead, these two linear programs can alsobe tackled efficiently using an iterative constraint generation algorithm. At every iteration, themethod generates the constraint with the largest violation for a provisional solution. In practice, itconverges after a few iterations. This method is well studied for the core of coalitional games, and itcan always be implemented if the function V is defined by an optimization problem. A formulationof this algorithm for the least core can be found in our previous work [17]. Finally, in this paper, V is defined by the optimization in (2). This would allow us to implement constraint generation.
3. An illustrative case study
We consider Figure 1, which comprises three areas, to compare the effectiveness of the proposedmechanisms. Following the prevailing approach, assume χ ′ = 0 . e e e a a a Figure 1: Three-area graph
To verify the analysis of this paper, we will focus on the true valuations of areas given by: v a ( χ e , χ e ) = C a [ χ e − χ e , χ e − χ e , χ e χ e ] ⊤ , with C a = [1 . , . , − . ,v a ( χ e , χ e ) = C a [ χ e − χ e , χ e − χ e , χ e χ e ] ⊤ , with C a = [2 . , . , − . ,v a ( χ e , χ e ) = C a [ χ e − χ e , χ e − χ e , χ e χ e ] ⊤ , with C a = [1 . , . , − . , , where the permitted values of χ are chosen from { , . , . , . , . } , and v a (0) = 0 , for all a . Inpractice, each area can estimate its costs from the sequential market at each χ portion to obtainits valuation. For instance, in [21], each area is assigned its producer and consumer surpluses,and the congestion rent is assumed to be divided equally between the adjacent areas; based on thezonal/nodal prices. A zonal cost estimation using such a cost allocation scheme can initially requirean estimation of some market parameters. Given that these auctions are executed repeatedly, theparticipants can adjust their offers/bids over time via online learning algorithms, see [18]. Weexpect this to ensure they are adequately remunerated. These aspects are beyond the scope of thepresent paper, and the bidding languages and bidding algorithms will be addressed in our futurework. 11nder true valuations, the optimal solution is [ χ ∗ e , χ ∗ e , χ ∗ e ] = [0 . , , . u MO ( V ) = P a ∈ A p a ( V ) = 0 . . In some other cases, the VCG mechanism insteadends up with a deficit. The MLC mechanism is strongly budget balanced, that is, ¯ u MO ( V ) = P a ∈ A p a ( V ) = 0 . The core is empty: ǫ ∗ ( V ) = min { ǫ ≥ | K Core ( V , ǫ ) = ∅} = 0 . . Under differentbid profiles ǫ ∗ may vary. We randomize the valuations by picking 10 samples from C a, ∼ U ([0 , C a, ∼ U ([0 , C a, ∼ U ([ − , ǫ = 0 .
159 for Theorems 1 and 2.
Table 1: Payments/utilities ( e ) p a u a p a u a p a u a VCG − .
154 0 .
343 0 .
264 0 .
279 0 .
263 0 . − .
278 0 .
468 0 .
139 0 .
404 0 .
139 0 . Under the VCG, areas 1 and 2 would strongly prefer to form a coalition, since V ( V a ∪ V a ) =0 . ≥ .
622 = u VCG a + u VCG a . This way, they can increase their total utility by 0 . .
622 to 1 . .
872 to 0 . (i) limits group manipulation.In order to enable reserve exchanges between operators, this paper proposed a market frame-work and derived a payment rule that attains strong budget balance, individual rationality, andapproximates coalitional stability and DSIC. These results extended studies on mechanisms thatselect payments from the core in auctions of private discrete items by accounting for the fact thatthe core can be empty for a network of continuous public choices. Our future work will study rulesto distribute these payments on a market participant level. References [1] I. Avramiotis-Falireas, S. Margelou, and M. Zima. Investigations on a fair TSO-TSO settlement for the imbalancenetting process in European power system. In , pages 1–6. IEEE, 2018.[2] R. Cory-Wright, A. Philpott, and G. Zakeri. Payment mechanisms for electricity markets with uncertain supply.
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Appendix A. (Non)emptiness of the core
The core is a closed polytope involving 2 | A | linear constraints. It is nonempty if and only ifthe function V satisfies the balancedness condition. Balanced problem settings include the casesin which V is supermodular and the cases in which (2) can be modeled by a concave exchangeeconomy [28], or a linear production game [25]. In their most general form, these results involvean optimization problem maximizing a concave objective subject to linear constraints. We aresolving the general non-convex optimization problem (1). As a result, these previous works are not13pplicable to our setup. As is shown in the following proposition, nonemptiness can be guaranteedfor a star graph ( A, E ). A similar derivation was included in our previous work for a networkedcoalitional game [17]. We prove this result for the problem at hand for the sake of completeness.
Proposition 2. K Core ( B , = ∅ if ( A, E ) is a star.Proof. Let a ∈ A be the central area. We show that the vector ¯ u a = V ( B ), ¯ u j = 0 otherwiselies in the core. Clearly the equality constraint is satisfied. Observe that star graph implies V ( B S ) = 0 for all S a, thus P j ∈ S ¯ u j ≥ V ( B S ) for all S a. On the other hand, for all S ∋ a , P j ∈ S ¯ u j = ¯ u a = V ( B ) ≥ V ( B S ) via monotonicity of V . Appendix B. Proof of Proposition 1 (i) For a generic profile B , the utility of area a is: u a ( B ) = h P j ∈ A \ a b j ( χ ∗ E j ( B )) + v a ( χ ∗ E a ( B )) i − h a ( B − a ) , where the term in brackets is the objective of the optimization problem for V ( V a ∪ B − a )evaluated at a feasible solution χ ∗ E a ( B ) . Hence, u a ( B ) ≤ V ( V a ∪ B − a ) − h a ( B − a ). Notice that theterm on the right is u a ( V a ∪ B − a ) . Therefore, the utility under bidding truthfully weakly dominatesthe utility under any other bid, regardless of other areas B − a .(ii) We have ¯ u a ( B ) = V ( B ) − V ( B − a ) ≥ V . (cid:3) Appendix C. The limitations of the Groves payment
Suppose we have the setting of two areas a and a connected with a single link. Define S = ˆ X a = ˆ X a = { , } , and χ ′ = 0. Area a has two strategies b a (1) = 1 and b a (1) = 0, area a has two strategies b a (1) = − b a (1) = 0. Note that these functions are required to be zero atthe default transmission capacity allocation χ ′ = 0. Strong budget balance implies the followingequalities: t a ( { b ia , b ja } ) + t a ( { b ia , b ja } ) = 0 , for i, j = 1 , . Invoking the definition of the Grovespayment and (1), these equalities can be rewritten as h a ( { b a } ) + h a ( { b a } ) = − b a ( χ ∗ E a ( { b a , b a } ))+ V ( { b a , b a } ) − b a ( χ ∗ E a ( { b a , b a } )) + V ( { b a , b a } )= − − ,h a ( { b ja } ) + h a ( { b ia } ) = 0 , ( i, j ) ∈ { (1 , , (2 , , (2 , } . Above can be written as A h = b = [1 , , , ⊤ , where h ∈ R concatenates h a ’s. rank( A ) = 3,moreover b / ∈ colspan( A ). Hence, there are no functions h a and h a such that strong budgetbalance is achieved, and the Groves payment can also not attain ǫ -coalitional stability for any ǫǫ