An Energy Sharing Mechanism Achieving the Same Flexibility as Centralized Dispatch
JJOURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 1
An Energy Sharing Mechanism Achieving the SameFlexibility as Centralized Dispatch
Yue Chen, Wei Wei, Han Wang, Quan Zhou, and Jo˜ao P. S. Catal˜ao
Abstract —Deploying distributed renewable energy at the de-mand side is an important measure to implement a sustainablesociety. However, the massive small solar and wind generationunits are beyond the control of a central operator. To encourageusers to participate in energy management and reduce the depen-dence on dispatchable resources, a peer-to-peer energy sharingscheme is proposed which releases the flexibility at the demandside. Every user makes decision individually considering onlylocal constraints; the microgrid operator announces the sharingprices subjective to the coupling constraints without knowingusers’ local constraints. This can help protect privacy. We provethat the proposed mechanism can achieve the same disutilityand flexibility as centralized dispatch, and develop an effectivemodified best-response based algorithm for reaching the marketequilibrium. The concept of “absorbable region” is presentedto measure the operating flexibility under the proposed energysharing mechanism. A linear programming based polyhedralprojection algorithm is developed to compute that region. Casestudies validate the theoretical results and show that the proposedmethod is scalable.
Index Terms —distributed renewable energy, uncertainty, flex-ibility, absorbable region, energy sharing mechanism. N OMENCLATURE
A. Indices, Sets, and Functionsi ∈ I
Consumer i in set I . j ∈ J Prosumer j in set J . l ∈ L Line l in set L . f k ( . ) Disutility function of user k ∈ I ∪ J .ˆ D k Local constraint for user k ∈ I ∪ J .˜ P Coupling constraint for all users. W Dc Absorbable region under centralized dispatch. W Ds Absorbable region under energy sharing.
B. ParametersI
Number of consumers. J Number of prosumers. d fi Fixed demand of consumer i ∈ I . d fj Fixed demand of prosumer j ∈ J . a Sharing market sensitivity. π wkl , π dkl Line flow distribution factors. F l Power flow limit of line l ∈ L . C. Decision Variablesd i Elastic demand of consumer i ∈ I . d j Elastic demand of prosumer j ∈ J . w j Renewable output of prosumer j ∈ J . p outk Net demand of user k ∈ I ∪ J . q k Sharing amount of user k ∈ I ∪ J . λ k Sharing price of user k ∈ I ∪ J . b k Bid of user k ∈ I ∪ J .I. I NTRODUCTION T HE penetration of distributed renewable generation hasbeen growing rapidly in recent decades [1], which helpspave the way towards a green and sustainable smart grid.However, the inherent volatility and intermittency of wind andsolar power also threaten the security of power system energymanagement [2]. How to maintain system reliability undera high share of renewable energy becomes a crucial topic.Related research can be roughly classified into two categories:one aims to come up with an optimal and reliable operatingstrategy given the uncertainty model of renewable output, suchas samples [3] or an uncertainty set [4]. Typical techniquesinclude stochastic optimization [3], [5], robust optimization[4], [6], and distributionally robust optimization [7], [8]. Theother endeavors to quantify the system’s ability to accommo-date renewable energy, by the region-based geometric method[9] or various metrics [10].For the region-based approaches, a pioneering one is the do-not-exceed limit (DNEL) proposed in [9]. The DNEL refersto the minimum and maximum output levels of renewableenergy within which the system constraints can be satisfied.The DNEL given by an optimization model maximizingthe weighted total renewable output variation range is thenembedded into a flexible dispatch model. Three alternativeapproaches were developed to derive dispatch strategies. Ex-tensive works have been carried out to further improve theDNEL based dispatch method by taking into account historicaldata [11], [12], corrective topology control [13], [14], anddifferent kinds of supporting sets [15]. Another well-knownconcept is the dispatchable region [16], which contains a setof renewable output scenarios under which the DC power flowmodel has at least one feasible solution. It was further extendedto AC power flow models [17], and multi-energy systems [18].The above works provide profound techniques for evalu-ating the system’s flexibility under a centralized manner. Itmeans all generators and users are controlled by a centraloperator. However, with the prevalence of distributed energyresources, the number of participants increases dramaticallyand their locations become more dispersed. In such circum-stances, the centralized approach is difficult to implement. Forone reason, massive participants add up to the difficulty of dataacquisition and privacy protection [19]; for example, the costcoefficients, preference, and capacity limits are known only toeach user but not the operator. For another, the consumptionof each user is beyond the control of a central authority. a r X i v : . [ m a t h . O C ] F e b OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 2
To overcome the obstacle of centralized operation, vari-ous market mechanisms were designed to manage massiveparticipants in a distributed manner [20], [21]. Coalitionalgame-based approaches were used for coordinating distributedstorages [22] and microgrids [23]. A trading mechanism withShapley value was presented in [24] and compared with threeclassic mechanisms, i.e., bill sharing, mid-market rate, andsupply-demand ratio. This Shapley value can be estimated bya coalitional stratified random sampling method [25]. The scal-ability of prosumers’ cooperative game was improved by K-means clustering [26]. A Nash bargaining model was adoptedin [27] to address the charge sharing among electric vehicles.For non-cooperative game-based approaches, distributed peer-to-peer energy exchange was modeled as a generalized Nashgame in [28] whose set of variational equilibria coincides withthe set of social optimum. A generalized demand functionbased energy sharing mechanism was proposed in [29] withproofs of several properties of its equilibrium. A practicalbidding algorithm was given in [30]. A decentralized algorithmwas proposed in [31] to provide renewable predictions toconsumers in a virtual power plant (VPP) with the conditionof its convergence. The above works investigate the strategicbehaviors of participants and the market equilibrium, but theflexibility of a certain market mechanism in accommodatingrenewable energy has not been well explored yet.To this end, this paper proposes an energy sharing schemeand quantifies its flexibility. The contributions are two-fold:1)
Mechanism Design . A mechanism that coordinates thepeer-to-peer energy sharing among massive users is presented.Each user makes a decision subject to its local constraints, andthe operator decides the energy sharing prices solely accordingto the system-wide coupling constraint without users’ privatedata. The sharing market can be described by a generalizedNash game. The existence of a generalized Nash equilibrium(GNE) which can achieve social optimum is proved. We alsodevelop a modified best-response based algorithm for reachingthe sharing market equilibrium with proof of its convergence.2)
Flexibility Characterization . We generalize the conceptof “dispatchable region” in [16] under centralized dispatch tothe “absorbable region” under any given scheme to character-ize the flexibility of the proposed energy sharing mechanismgeometrically. We prove that the absorbable region underenergy sharing is exactly the same as that under centralizeddispatch, meaning that the system’s flexibility is retained. Alinear programming based polyhedral projection algorithm isdeveloped to generate the absorbable region.The rest of this paper is organized as follows. A peer-to-peer energy sharing mechanism is proposed in Section II andthe existence and efficiency of its equilibrium are proved; Amodified best-response based algorithm is developed to reachthe sharing market equilibrium. To quantify the flexibility ofenergy sharing, the concept of “dispatchable region” undercentralized operation is extended to the “absorbable region”in Section III, and a linear programming based polyhedronprojection algorithm is established to compute that region.Case studies in Section IV validate the theoretical outcomes.Conclusions are drawn in Section V. II. E
NERGY S HARING M ECHANISM
To deal with the practical issues in centralized operation,an energy sharing mechanism for managing massive users isdeveloped in this section. The users in the sharing marketplay a generalized Nash game. We prove that the marketequilibrium always exists and is socially optimal. A modifiedbest-response based algorithm is presented to reach the marketequilibrium, and its convergence condition is given.
A. Problem description
We consider the optimal dispatch of a group of massiveusers, including consumers indexed by i ∈ I : = { , , ..., I } andprosumers indexed by j ∈ J : = { , , ..., J } , in a stand-alonemicrogrid. Consumer i ’s fixed demand is d fi and its elasticdemand is d i . Each prosumer j ∈ J is equipped with renewableunits whose total output is w j . Its fixed demand is d fj andelastic demand is d j . In this paper, we consider a specific typeof demand response, where users can adjust their demand tominimize their disutility as in [32], [33]. Under a centralizedmanner, the microgrid operator solves problem (1) to maintainpower balancing with the lowest total user disutility.min d k , ∀ k ∈ I ∪ J ∑ k ∈ I ∪ J f k ( d k ) (1a)s.t. ∑ k ∈ I ∪ J p outk = p outk = (cid:40) d fk + d k , ∀ k ∈ I d fk + d k − w k , ∀ k ∈ J : λ k (1c) d k ∈ ˆ D k , ∀ k ∈ I ∪ J (1d) p out ∈ ˜ P (1e)Here, f k ( d k ) characterizes the disutility of user k ∈ I ∪ J caused by adjusting its elastic demand, which is a strictlyconvex and twice differentiable function. Constraints (1b) and(1c) represent the power balancing between the total outputof renewable units and the total demand. Constraint (1d) isthe local feasible set of each user k ∈ I ∪ J , e.g., the rangelimit for each responsive load, represented by a closed convexset ˆ D k . Constraint (1e) collects the global coupling constraints,e.g., the network power flow limit, for the net demand (definedin (1c)), represented by a closed convex set ˜ P .Throughout the paper, we assume A1 . { d : s.t. (1b) − (1e) are satisfied. } (cid:54) = /0.The centralized dispatch performs well in some rural orisolated microgrids, where the operator can get all the infor-mation needed for operation. Nevertheless, with the growingscale of microgrids and power sector decentralization, themarketization of microgrid has become a prevalent trend [34].Under this circumstance, how to protect users’ informationprivacy and reduce the possible market power exploitation dueto information asymmetry is getting increasingly important.To be specific, in problem (1), we assume that the operatorknows all related constraints, but in practice, the ˆ D k , thedisutility function f k ( . ) , and fixed demand d fk are usuallyprivate information only available to user k ∈ I ∪ J . Askingfor such information may jeopardize users’ privacy. The usermay even have the incentive to misrepresent this informationto lower its disutility. Besides, each user k may not have access OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 3
Fig. 1. Sketch of the energy sharing market. to the coupling constraint ˜ P , which is known to the operatoronly. Thus, the centralized model (1) will encounter somechallenges for microgrid management in practice, and a newapproach that can perform with local information and reducethe impact of information asymmetry is desired. Remark:
We further distinguish three similar expressionsrelated to the information structure, i.e. information privacy,information asymmetry, and information scarcity, for a betterunderstanding. By saying information privacy , we mean thatsome information is available to a specific party while otherparties cannot get access to it. The concept of informationasymmetry stems from contract theory and economics. Thoughit also refers to the case where different parties have differentinformation, it highlights more on the situation that the partywith more information may misrepresent their private infor-mation to gain more profit, resulting in an imbalance marketpower or even a market failure [35]. For example, in orderto solve problem (1), the central coordinator needs to collectinformation about f k ( . ) for all k ∈ I ∪ J , which are privateto users. Therefore, the user may deliberately misreport thesedata so as to lower its disutility. An example of how thisasymmetric information structure may influence the outcomesof the system is given in Section IV-A. Information scarcity indicates that a party does not know other parties’ strategies,the only information it has access to is the outcome [36]. Inthis paper, we assume that all users and the microgrid operatorknow others’ strategies when making their own decisions.
B. Mechanism design
To solve problem (1), the microgrid operator needs toknow ˆ D k , f k ( d k ) and d fk , which could vary among differentend-users [37]. Therefore, the centralized operation may beimpracticable due to the privacy protection requirement andpossible speculative behavior under information asymmetry.To overcome these problems, a peer-to-peer energy sharingmechanism is proposed: Each user k ∈ I ∪ J determines itsown demand d k , and meanwhile, can share energy q k withother prosumers at a price λ k to maintain self-power balancing.Instead of deciding on the λ k or q k directly, each user offers abid b k to the operator considering its local constraint ˆ D k . Withall b k , ∀ k ∈ I ∪ J , the microgrid operator clears the sharingmarket subject to the coupling constraint ˜ P and determines theprice λ k , ∀ k ∈ I ∪ J ; the transactive energy q k : = − a λ k + b k istransferred via the power network, where a > q k > k buys energy from the market, and when q k < k ∈ I ∪ J is:min d k , b k f k ( d k ) + λ k ( − a λ k + b k ) (2a)s.t. (cid:40) − a λ k + b k = d fk + d k , ∀ k ∈ I w k − a λ k + b k = d fk + d k , ∀ k ∈ J (2b) d k ∈ ˆ D k (2c)where f k ( d k ) is its disutility in monetary terms, and λ k ( − a λ k + b k ) is its payment to the energy sharing market (or revenuewhen this term is negative). Constraint (2b) is the powerbalancing condition, and (2c) is the local limit. Note that whenmaking a decision, each user k ∈ I ∪ J needs not know thecoupling constraint ˜ P . We will prove latter in Proposition 1that the energy sharing prices λ k , ∀ k ∈ I ∪ J equal to the dualvariables of (1b) at the optimal point, which is also the value oflocational marginal prices [38]. When there are massive users,if we change the b k of one user, the λ k , ∀ k ∈ I ∪ J obtainedby solving (3) will remain nearly unchanged. Therefore, theimpact of each b k on λ k is negligible, and in problem (2) λ k is seen as an exogenously given constant. This is a commonassumption in economics [39].Given all users’ bids b k , ∀ k ∈ I ∪ J , the microgrid operatorsolves the following problem to clear the sharing market anddetermine the energy sharing prices.min λ k , ∀ k ∈ I ∪ J ∑ k ∈ I ∪ J λ k (3a)s.t. ∑ k ∈ I ∪ J ( − a λ k + b k ) = ( − a λ + b ) ∈ ˜ P (3c)First, the sharing market needs to be cleared, which meansthe energy bought equals the energy sold. Therefore, we have ∑ k ∈ I ∪ J q k = q k , ∀ k ∈ I ∪J will be transferred via network, and shouldmeet the network constraints (3c). The objective function aimsto enhance market fairness by minimizing the variance ofenergy sharing prices, i.e. ∑ k ∈ I ∪ J (cid:32) λ k − I ∑ k ∈ I ∪ J λ k (cid:33) = ∑ k ∈ I ∪ J (cid:32) λ k − aI ∑ k ∈ I ∪ J b k (cid:33) = ∑ k ∈ I ∪ J λ k − a I (cid:32) ∑ k ∈ I ∪ J b k (cid:33) (4)which is equivalent to minimizing (3a) since the second termin (4) is a constant. Moreover, if all network constraintsin (3c) are not binding, it degenerates to the case withoutnetwork constraints in [30] with a unified price λ given by λ = ∑ k ∈ I ∪ J b k / ( aI ) . After clearing the market, the microgridoperator returns the price λ k back to each user; then theuser will adjust its bid according to (2) and submit it to theoperator again. This happens until convergence. We will provelater in Proposition 1 that the market outcome at equilibriumsatisfies all local constraints ˆ D k spontaneously. During the OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 4 market clearing process, local constraints ˆ D k are not exposedto the operator, which is one advantage of our mechanism.Thinking of the microgrid operator as a special participant,problem (2) for all k in I ∪ J and (3) constitute a generalizedNash game [40]. The game consists of the following elements:1) a set of players K , including I consumers indexed by k = , ..., I , J prosumers indexed by k = ( I + ) , ..., ( I + J ) anda microgrid operator indexed by k = I + J + X k ( λ k ) : = { ( d k , b k ) | (2b) − (2c) are satisfied . } for all k = , ..., ( I + J ) , X I + J + ( d , b ) : = { λ | (3b) − (3c) are satisfied . } , and action space X = ∏ k X k ;3) cost functions Γ k ( λ k ) : = f k ( d k ) + λ k ( − a λ k + b k ) for all k = , ..., ( I + J ) , and Γ I + J + : = ∑ k ∈ I ∪ J λ k .The strategies of each user k ∈ I ∪ J are its elastic demand d k and bid b k ; the strategy of the microgrid operator is thesharing prices λ k , ∀ k ∈ I ∪ J . When making their decisions,the operator only needs to know the bids b k , ∀ k ∈ I ∪ J submitted by users; Each user only needs to know the price λ k given by the operator. Under a market environment, it isreasonable to assume that the microgrid operator can get thesebids and the users can get the prices [41]. Since the transferredinformation, i.e. the bids b k and prices λ k , are scalar numbersinstead of complex messages, the communication is efficient.For simplicity, we use G ( w ) = {K , X , Γ } to denote the sharinggame in an abstract form. Then we define a generalized Nashequilibrium (GNE) as below. Definition 1. (Generalized Nash Equilibrium) A profile ( d ∗ , b ∗ , λ ∗ ) ∈ X is a generalized Nash equilibrium (GNE) ofthe energy sharing game G ( w ) = {K , X , Γ } if ( d ∗ k , b ∗ k ) = argmin d k , b k { Γ k ( λ ∗ k ) , ∀ ( d k , b k ) ∈ X k ( λ ∗ k ) } (5)holds for k = , ..., ( I + J ) , and λ ∗ = argmin λ { Γ I + J + , ∀ λ ∈ X I + J + ( d ∗ , b ∗ ) } (6)Different from a standard Nash game, in G ( w ) = {K , X , Γ } ,every player’s action set depends on other players’ strategies.For example, the action set for user k = , ..., ( I + J ) , the X k ,depends on the operator’s strategy λ k ; the action set for theoperator, the X I + J + , depends on all prosumers’ strategies b .This complicated coupling makes it difficult to search for andanalyze the equilibrium. C. Properties of the GNE
The proposed energy sharing market can be used in com-mercial and industrial microgrids [34], campus microgrids[42], and community microgrids [43]. In the following, wediscuss its performance. Proposition 1 shows the existence ofa market equilibrium with social optimal disutility, while thesystem’s flexibility will be quantified in Section III.
Proposition 1.
The game G ( w ) = {K , X , Γ } has at least oneGNE. Moreover, the triplet ( d ∗ , b ∗ , λ ∗ ) is an GNE if and onlyif d ∗ is the unique optimal solution of (1), and λ ∗ k , ∀ k ∈ I ∪ J equal the corresponding dual variables, with b ∗ k = d fk + d ∗ k + a λ ∗ k for all k in I , b ∗ k = d fk + d ∗ k − w k + a λ ∗ k for all k in J .The proof of Proposition 1 can be found in AppendixA. It shows the equivalence of the GNE of G and the optimal solution of problem (1), indicating that the proposedenergy sharing mechanism is economically efficient: the GNE ( d ∗ , b ∗ , λ ∗ ) for accommodating w has the lowest social totaldisutility ∑ k ∈ I ∪ J f k ( d ∗ k ) . Remark on relevant works:
While both this paper and [44]provide a centralized counterpart for computing the marketequilibrium, the problems considered and model configura-tions are different. Reference [44] investigated a case wheredifferent agents (price-setting agent, producer, and consumer)have different information on the probability distribution ofrenewable uncertainty. Even in the equilibrium problem (2) of[44], there is still such kind of information asymmetry. In thispaper, we consider a case where different agents have accessto different constraint sets. To ensure information privacy, anenergy sharing mechanism is proposed, resulting in an equilib-rium problem where each agent (microgrid operator or user)makes decisions according to its private information, whichis known exactly to it. Reference [45] proposes a proximalminimization based algorithm to deal with the distributedconstrained optimization. The constraint X i is imposed on eachlocal participant i ; each agent receives information only fromits out-neighbours. In this paper, we take into account thecoupling constraints ˜ P which is imposed on all users. Besides,instead of exchanging information with neighbors, user onlycommunicates with the microgrid operator. Remark on possible extensions: (i) Although the aboveanalyses are based on the example of massive users in a stand-alone microgrid, the proposed model and mechanism can beextended to various cases with the main properties remained.For example, if we allow d k to be negative, the model canincorporate the case with electricity market. To be specific,when d k > k ∈ I ∪ J adjusts its elastic demand to d k or when d k exceedsthe upper bound of elastic demand, it also sells electricity tothe power market; when d k < k adjusts its elastic demand to zero, and besides buys − d k from the power market. (ii) Although there might be somecomplex units with nonconvex constraints in the microgrid,our approach can still be applied via some convex relaxationtechniques, such as those for AC power flow models [46], [47]and for storage-like devices [48]. D. Modified best-response based algorithm
We prove that the GNE under the proposed mechanism hasan appealing property. How to reach such an equilibrium isanother important issue. In this paper, a modified best-response(BR) based algorithm (Algorithm 1) is developed. We choosethe BR based algorithm as it is one of the most fundamentalmethod in game theory [49]. This approach iteratively solvesuser’s problem (2) or the modified market clearing problem(3) with a modified objective function (7) given other players’strategies until convergence.The modified objective function (7) in Algorithm 1 cansmooth the fluctuation of market prices during the biddingprocess. The value of the market sensitivity a will influence theperformance of the modified BR based algorithm. We provein Appendix B that when Condition C1 holds, the biddingprocess converges. OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 5
Algorithm 1:
Modified Best-Response
Input: parameters a , d fk , ∀ k ∈ I ∪ J , w k , ∀ k ∈ J ,disutility functions f k ( . ) , ∀ k ∈ I ∪ J Output: generalized Nash equilibrium ( d ∗ , b ∗ , λ ∗ ) Initialization: n = b = repeat Operator : given all bids b nk , ∀ k ∈ I ∪ J , solve problem (3)with a modified objective function:min λ k , ∀ k ∈ I ∪ J ∑ ∀ k ∈ I ∪ J λ k + ∑ ∀ k ∈ I ∪ J ( λ k − λ nk ) (7)to update the price λ n + k , ∀ k ∈ I ∪ J . for User k ∈ I ∪ J do given λ n + k , solves problem (2) to get d n + k and b n + k . end Iteration : n + + until | b n − b n − | ≤ ε ; Fig. 2. Economic intuition behind Condition C1.
C1.
The Hessian Matrix of ∑ k ∈ I ∪ J f k ( d k ) − a ∑ k ∈ I ∪ J ( d k + D k ) is definite, where D k = d fk , ∀ k ∈ I , D k = d fk − w k , ∀ k ∈ J .Condition C1 can be equivalently written as 1 / a < min { ∂ f k / ∂ d k , ∀ k ∈ I ∪ J } . Specially, if the disutility func-tion is quadratic, i.e. f k ( d k ) = α k ( d k ) + α k ( d k ) with givenconstants α k , α k >
0, Condition C1 holds if and only if a > max { α k , ∀ k ∈ I ∪ J } . We try to give an economic inter-pretation of Condition C1 in Fig. 2. ∂ f k / ∂ d k is the marginaldisutility of user k ∈ I ∪ J and can be regarded as the supplycurve of load adjustment, where ∂ f k / ∂ d k is the slope of thissupply curve. The demand for load adjustment comes from thesharing market, and since q k = − a λ k + b k , the slope of demandcurve is − / a . From Fig. 2, we can find that the biddingprocess converges if and only if ∂ f k / ∂ d k > / a , ∀ k ∈ I ∪ J ,which is also known as the cobweb model in economics.Furthermore, When a is small (1 / a is large), the sharing priceresponses more quickly to the changing bids, and therefore,the algorithm can reach the equilibrium faster.III. F LEXIBILITY UNDER E NERGY S HARING
In this section, we show that energy sharing has the sameflexibility as centralized dispatch. The “absorbable region” isproposed to characterize the flexibility.
A. Dispatchable region of centralized operation
In the centralized operation problem (1), the renewableoutputs w j , ∀ j ∈ J are volatile and uncertain. In real-time, themicrogrid operator adjusts the d k , ∀ k ∈ I ∪J to ensure supply-demand balance. One critical issue is the system’s abilityto accommodate uncertainty, which can be quantified by the“ dispatchable region ” proposed in [16]. Fig. 3. Illustrative diagram of the dispatchable region.
Definition 2. (Dispatchable Region [16]) The dispatchableregion under centralized operation is a set of renewable outputssuch that at least one feasible dispatch solution exists: W Dc = { w | ∃ d : (1b) − (1e) are satisfied . } Depicting the flexibility of a power grid under centralizedoperation with the dispatchable region has been widely studied[50], [51]. An illustrative diagram is given in Fig. 3. Thehorizontal axis of the figure is the uncertain factor w , and thevertical axis is the elastic demand d . The light purple region isthe feasible region of problem (1) characterized by constraints(1b)-(1e). Projecting it onto the horizontal axis we can obtain arange of w , the “dispatchable region” W Dc . For any given point w within W Dc , we can always find a corresponding feasiblerange for d , i.e. [ d , d ] . The operator minimizes the objectivefunction ∑ k ∈ I ∪ J f k ( d k ) over [ d , d ] at the red point.In recent years, innovative approaches emerge for runningthe power system in a smarter and more scalable way, whichcalls for methods to quantify the flexibility under thoseapproaches. In this paper, we generalize the conventionaldispatchable region to the “absorbable region” under anycertain schemes for characterizing the flexibility under ourenergy sharing mechanism. B. Absorbable Region Under Energy Sharing
The dispatchable region is a useful tool in describing thesystem’s flexibility but limited to centralized operation. First,we raise the concept of “absorbable region” which generalizesthe “dispatchable region” for flexibility evaluation.
Definition 3. (Absorbable Region) The absorbable region of amicrogrid under a certain scheme is a set of renewable outputssuch that at least one feasible strategy exists under that scheme.The “absorbable region” differs and improves from the“dispatchable region” in two ways: Firstly, it is designed for
OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 6 any given scheme instead of just the centralized operation.Secondly, the renewable output is “absorbable” if and only ifthere exists a strategy, which could not only be a dispatch orderbut also a market equilibrium: When the “certain scheme”refers to centralized operation, the absorbable region degener-ates to the dispatchable region. If under “certain scheme” everyuser makes decision individually, then the renewable output isabsorbable when there is a feasible self-sufficient strategy, i.e. ( w k − d fk ) ∈ ˆ D k , ∀ k ∈ J . When the “certain scheme” refers to amarket mechanism, “absorbable” is equivalent to the existenceof a market equilibrium. Specially, the absorbable region underthe proposed energy sharing mechanism is given by W Ds = { w | an GNE ( d ∗ , b ∗ , λ ∗ ) for the game G ( w ) exists . } The proposed absorbable region can accommodate variousscenarios, however, it may also encounter complicated situ-ations, such as conflicting interests among stakeholders andtime or spatial coupling constraints. Though fruitful workshave been conducted for computing the dispatchable regionunder centralized operation, the proposed methods cannot beapplied directly to the characterization of the absorbable regionin general cases due to the above complexity. In the following,we show that our proposed energy sharing mechanism isflexibility-retained, which facilities the computation of itsabsorbable region:According to Proposition 1, given a w , if problem (1) isfeasible, meaning that w ∈ W Dc , then we can always constructa GNE for the game G ( w ) , so that w ∈ W Ds . Conversely, if w ∈ W Ds , i.e. there exists a GNE for the game G ( w ) , then the d ∗ is exactly the optimal solution (of course also a feasible one) ofthe problem (1), indicating w ∈ W Dc . Thus, we have W Dc = W Ds .The proposed energy sharing mechanism can achieve the sameflexibility as the centralized dispatch. We can use the simplerequivalent model (1) for computing the absorbable region ofthe proposed energy sharing mechanism. C. Linear programming based projection algorithm
Here, we use a common power system model with capacityconstraints ( ˆ D k is chosen as [ d k , d k ] ) and network constraints( ˜ P is chosen as the DC power flow limits) as an example. Insuch a case, constraints (1b)-(1e) are ∑ k ∈ J w k = ∑ k ∈ I ∪ J d k + ∑ k ∈ I ∪ J d fk (8a) d k ≤ d k ≤ d k , ∀ k ∈ I ∪ J (8b) − F l ≤ ∑ k ∈ J π wkl w k − ∑ k ∈ I ∪ J π dkl ( d fk + d k ) ≤ F l , ∀ l ∈ L (8c)where L is the set of lines, F l is the power flow limit forline l ∈ L , and π wkl , π dkl are the line flow distribution factors.Since all constraints in (8) are linear, it can be represented ina compact form: ψ ( w ) = { d | Ad + Bw ≤ c } (9)Here, d is a collection of d k , ∀ k ∈ I ∪ J , and w is a collectionof w k , ∀ k ∈ J . According to Proposition 1, the absorbableregion is defined as W Ds = W Dc = { w | ψ ( w ) (cid:54) = /0 } (10) Algorithm 2:
Linear Programming Based Projection
Input: initial set W temp = { w | w ≥ } Output: output dispatchable region W Ds Update vert( W temp ); for w ∈ vert(W temp ) do solve problemmax u u T ( c − Bw ) s.t. u ∈ U and denote the optimal solution as u ∗ , the optimalvalue as r ∗ . Let r max = max { r max , r ∗ } and update u max as the corresponding u ∗ . end if r max = then let W Ds = W temp ; else add a cutting plane ( u max ) T Bw ≥ ( u max ) T c in W temp ; go to 1; end For a given w , we can check whether ψ ( w ) is empty by thefollowing problem:min d , z T z (11a)s.t. Ad − Iz ≤ c − Bw : u (11b) z ≥ z is a slack variable and u is the dual variable. It is easyto see that ψ ( w ) (cid:54) = /0 if and only if the optimal value of (11)is zero. The dual problem of the linear program (11) ismax u u T ( c − Bw ) (12a)s.t. A T u = , − ≤ u ≤ U : = { u | A T u = , − ≤ u ≤ } , then ψ ( w ) (cid:54) = /0 isequivalently to u T ( c − Bw ) ≤ , ∀ u ∈ U (13)The set U is a closed convex set, so the above condition canbe further simplified to u T ( c − Bw ) ≤ , ∀ u ∈ vert ( U ) (14)Consider w as the variable, polyhedron (14) is the conditionwhich w must meet in order to ensure the non-emptiness of ψ ( w ) , i.e.: W Ds = { w | u T ( c − Bw ) ≤ , ∀ u ∈ vert ( U ) } (15)Though W Ds can be represented as an explicit polyhedron,it is still difficult to locate all the vertices of U with ahigh-dimension. The projection algorithm (Algorithm 2) isdeveloped to generate the absorbable region W Ds efficiently, inorder to exhibit that the distributed method possesses the sameflexibility as the centralized method. This projection algorithmis not actually executed in the market clearing procedure andconsumes real-time computational resources. OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 7
The intuition behind Algorithm 2 is as follows. We knowthat w ∈ W Ds if and only if (14) holds; w / ∈ W Ds indicates that ( u max ) T ( c − Bw ) > u max ∈ vert ( U ) . Therefore, wefirst initiate a large enough polyhedron W temp which contains W Ds , and then test if (13) is met by solving the linear programin step 3. If not, the hyperplane ( u max ) T c = ( u max ) T Bw (16)gives the boundary of W Ds , which is added to W temp . Thealgorithm converges once (13) is met. Algorithm 2 is moreefficient than the mixed-integer programming based algorithmin [16] and does not need a big-M parameter. Remark on convergence of Algorithm 2 : The convergencecriteria of Algorithm 2 is that (13) is met for all w in W Ds .The region W Ds is a polyhedron with a finite number of planefaces and the closed-form in (15) based on enumerating thevertices of U . However, when the dimension of U is high, itis challenging to list all its extreme points. Actually, most ofthese extreme points only exert redundant constraints in W Ds which can be removed. Algorithm 2 avoids these redundantconstraints by adaptively selecting the vertices in U and addsa hyperplane to give the boundary of W Ds in each iteration.Since there are a finite number of vertices of U , the algorithmcan always converge. When there are many constraints in theproblem (8), the polyhedron W Ds is likely to have many facets,and thus needs more iterations to characterize. Nonetheless, asthe procedures in the Algorithm 2 only require solving linearprograms, the algorithm is efficient.IV. C ASE S TUDIES
Numerical examples are tested to validate the theoreticalresults and show the effectiveness of the proposed algorithm.
A. Simple case with two groups of prosumers
First, we verify Proposition 1 by a simple example with twogroups of prosumers. The first group consists of prosumers k ∈ J = { , ..., } , and the second group consists of pro-sumers k ∈ J = { , ..., } . Each group is made up ofidentical prosumers. We adopt quadratic disutility functions f k ( d k ) : = α k ( d k ) + α k d k , ∀ k ∈ J ∪ J with the parametersgiven in Table I. Two groups of prosumers are connected bya line whose flow limit is 10 kW. The sensitivity a is set to 1kW/$. The modified best response based algorithm is used forreaching the GNE, and the changes of energy sharing pricesand elastic demands are plotted in Fig. 4. TABLE IP
ARAMETERS OF PROSUMERS
Prosumer α k α k w k d fk d k group ($ / kW ) ($ / kW) (kW) (kW) (kW)1 0.30 0.42 1.25 1.00 [0.2,0.5]2 0.60 0.72 1.75 1.30 [0.1,0.6] From Fig. 4 we can find that both the energy pricesand prosumers’ strategies converge. At GNE, we have d = . d = . E n e r gy s h a r i ng p r i ce ( $ / k W ) Iteration E l a s ti c d e m a nd ( k W ) prosumer in group 1prosumer in group 22 4 6 8 10 12 14-1.5-1-0.50 prosumer in group 1prosumer in group 2 Fig. 4. Sharing prices and optimal strategies in each iteration. of problem (1) with the same parameters. So the proposedenergy sharing mechanism achieves social optimum.We further reveal the impact of information asymmetry bytesting how the misrepresentation of α k and α k will influencethe users’ disutilities and social total disutility. We change α and α from 0.2 to 4 times of their original values¯ α = . , ¯ α = . F l = F l = . α and 1 . α , the total disutilityremains unchanged; in (d), user 1 has no incentive to misreportbecause its lowest disutility is with ¯ α and ¯ α . The impact ofinformation asymmetry on the total disutility can be lessenedvia the proposed energy sharing scheme. B. Five-bus system for flexibility characterization
Next, the effectiveness of Algorithm 2 for computing theabsorbable region is tested via a five-bus system, whosetopology and parameters are given in Fig. 6. There are threeelastic demands, whose range is marked in red; the power flowlimits are in blue; the fixed demands are in green. We randomlychoose renewable output scenarios ( w , w ) , and mark thosewith which the problem (1) is feasible (thus ( w , w ) ∈ W Ds )in Fig. 7(a). The one provided by Algorithm 2 is shown inFig. 7(b). Both regions are exactly the same, demonstratingthat Algorithm 2 can successfully identify the boundary ofthe actual absorbable region. Any renewable output profile ( w , w ) inside the grey region is absorbable .Let w =
250 kW, w =
400 kW, and a =
100 kW/$, thesequence of the elastic demands generated by Algorithm 1 isshown in Fig. 8(a), and we project it onto d − d , d − d and d − d planes, respectively, and get Fig. 8(b)-(d). Again, OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 8 proportion to the original value D i s u tilit y ( $ ) Proportion to the original value D i s u tilit y ( $ ) Proportion to the original value D i s u tilit y ( $ ) Proportion to the original value D i s u tilit y ( $ ) (a) Centralized scheme & F l =10kW (b) Energy sharing scheme & F l =10kW(c) Centralized scheme & F l =50kW (d) Energy sharing scheme & F l =50kW Disutility of user 1 Disutility of user 1 Total disutility
Fig. 5. Impact of information asymmetry.
A B C DE G G G G . M W . M W A B C DE G G G G [50, 200]MWa = 0.00240b = 12.2399 150MW 0.0304250MW 0.0297 M W . M W . W W W W A B C DE W W Fig. 6. Topology and parameters of the five-bus system. the optimal strategies of three elastic demands converge to d = .
26 kW, d = .
03 kW and d = .
46 kW, which coincidewith the optimal dispatch in (1). Another two renewable outputscenarios are tested with results in Fig. 9. For the renewableoutputs inside the absorbable region, a GNE exists and can bereached by the modified BR based method.
C. Practicability of the proposed mechanism and algorithm
A larger case with a modified 38-bus microgrid [52] anda modified 69-bus microgrid [53] are tested to show thescalability of our method. The topology of the test systemsare in Fig. 10 and Fig. 11, respectively, with other parametersin [54]. Algorithm 2 is used to generate the absorbable region.The result for 38-bus system is presented in Fig. 12 togetherwith some intermediate results. At the beginning, a largeenough box that contains the absorbable region is used asthe initial polyhedron; then the points outside the absorbableregion are gradually removed by the cutting planes in eachiteration. We then generate the absorbable region of the 69-bussystem with the flow limits equal to F l and 2 F l , respectively,as shown in Fig. 13. We check that the output absorbableregions are the same as the actual ones, which validates theproposed method. With looser flow limits, the absorbableregion enlarges. (a) Actual Absorbable Region (b) Generated by Algorithm 1 w (kW)w1 (kW) w ( k W )
150 200 250 300250300350400450500550600
150 200 250 300250300350400450500550600
150 200 250 300250300350400450500550600 w ( k W ) Fig. 7. Absorbable region: actual (a) v.s. generated by Algorithm 2 (b).
Elastic demand 1 (kW)Elastic demand 2 (kW) E l a s ti c d e m a nd ( k W ) Elastic demand 1 (kW) E l a s ti c d e m a nd ( k W ) Elastic demand 2 (kW) E l a s ti c d e m a nd ( k W ) Elastic demand 2 (kW) E l a s ti c d e m a nd ( k W ) (a) change of decisions of d ,d ,d (b) change of decisions of d ,d (c) change of decisions of d ,d (d) change of decisions of d ,d Fig. 8. Sequence of elastic demands during iterations in the five-bus system.
We choose two wind output scenarios within the aboveobtained absorbable region for 38-bus system, and let w =[ . , . , . ] (p.u.) and w = [ . , . , . ] (p.u.), respectively.The change of prosumers’ strategies on elastic demands arerecorded in Fig. 14. Under both cases, prosumers’ strategiesconverge within 10 iterations. In addition, the output strategiesare [ . , . , . ] (p.u.) and [ . , . , . ] (p.u.), whichis the optimal solution of problem (1). Similarly, for the 69-bussystem, let w = [ , , ] kW, the change of end users’strategies when implementing the modified BR algorithm isgiven in Fig. 15, and all of them converge in 40 iterations.We list the time needed for reaching equilibrium for the sim-ple case in Section IV.A, the five-bus case in Section IV.B, andthe 38-bus and 69-bus cases in Section IV.C in TABLE II. Wecan find that all cases converge within 300 seconds, while thescale of the system and the number of consumers/prosumersdo not matter much. Our proposed energy sharing mechanismfocuses on the real-time market [55], which will be conductedhourly. Therefore, the time cost of the algorithm is acceptablefor its implementation. D. Factors influencing the performance of algorithms
First, we test the impact of a on the performance of themodified BR algorithm by the 69-bus system. Here, ConditionC1 is satisfied when a > .
84. Let a equals to 0.1, 0.5, 1, 5, OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 9
TABLE IIC
OMPUTATIONAL TIME ( S ) FOR DIFFERENT CASES .Case Simple 5-bus with 5-bus with 5-bus with 38-bus with 38-bus with 38-bus with 38-bus withcase w = [ , ] w = [ , ] w = [ , ] w = [ . , . , . ] w = [ . , . , . ] w = [ , , ] w = [ , , ] Time 24.85 238.32 65.85 47.55 23.61 32.20 143.14 132.83
Elastic demand 1 (kW)Elastic demand 2 (kW) E l a s ti c d e m a nd ( k W ) Energy sharing price 2 ($/kW) Energy sharing price 1 ($/kW) E n e r gy s h a r i ng p r i ce ( $ / k W ) (a) Demand under w = [200,450]kW (b) Price under w = [200,450]kW Elastic demand 1 (kW)Elastic demand 2 (kW) E l a s ti c d e m a nd ( k W ) E n e r gy s h a r i ng p r i ce ( $ / k W ) Energy sharing price 2 ($/kW) Energy sharing price 1 ($/kW) (c) Demand under w = [300,600]kW (d) Price under w = [300,600]kW -2 -1 0 1-3-2-10-2.5-2-1.5-1-0.50 -5 0 5 10-15-10-50-40-30-20-100
Fig. 9. Results under two scenarios for the five-bus system.
Fig. 10. Topology of the 38-bus test system. and 10, respectively, and the changes of user’s demand andthe sharing price over iterations are plotted in Fig. 16 (takethe user at bus 32 as an example). We can find that, witha smaller a , the modified best response algorithm convergesfaster; however, when a is too small, oscillation may occur.Moreover, when a = . a , our algorithm reachesequilibrium within 180 seconds.We then test the performance of Algorithm 2 using the 38-bus and 69-bus cases with different settings in TABLE III,where F l refers to the original flow limit. Generally, it takes alonger time and more iteration to output the absorbable regionwhen the system constraints are more stringent. But under allcases, the algorithm terminates in less than a minute, whichis acceptable for online use. Fig. 11. Topology of the 69-bus test system.
Original Region Region at 6-th iterationRegion at 18-th iteration Absorbable Region
Fig. 12. Results when applying Algorithm 2 to the 38-bus system.
V. C
ONCLUSION
With the mushrooming of distributed renewable energyat the demand side, new energy management schemes withexplicit characterization on flexibility are in great need. Thispaper proposes an energy sharing mechanism that encouragesenergy exchange among end-users. A generalized Nash gamemodel is proposed to formulate the equilibrium of the sharingmarket. The energy transaction is coordinated via price sig-nals which meet the requirement of information privacy. Weprove that the generalized Nash equilibrium achieves socialoptimum, and leads to the same flexibility as the centralizeddispatch in the sense of the absorbable region. The linear pro-gramming based projection algorithm can efficiently generatethe boundaries of the absorbable region. Future research mayuse more accurate power flow models and consider multipleperiods and temporal correlations.
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Absorbable Region under F l w (kW)w (kW) w ( k W ) w (kW)w (kW) w ( k W ) Absorbable Region under 2 F l Fig. 13. Absorbable region under different power flow limits. (a) w= [1.6, 1.6, 1.5] (p.u.) (b) w= [1.0, 1.0, 2.1] (p.u.)
Elastic demand 2 (p.u.) E l a s ti c d e m a nd ( p . u . ) Elastic demand 1 (p.u.)
Elastic demand 2 (p.u.) E l a s ti c d e m a nd ( p . u . ) Elastic demand 1 (p.u.)
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Encyclope-dia of Optimization , vol. 2, pp. 106–114, 2001. A PPENDIX
A. Proof of Proposition 1
Denote p outk : = − a λ k + b k , and let Y be the feasible regionof p outk , ∀ k ∈ I ∪ J characterized by (3b) and (3c) (or (1b) and(1e)). Obviously, Y is also a closed convex set.Then problem (3) can be equivalently written asmin p outk , ∀ k ∈ I ∪ J ∑ k ∈ I ∪ J ( p outk − b k ) (A.1a)s.t. ∑ k ∈ I ∪ J p outk = p out ∈ ˜ P (A.1b)Suppose ( d ∗ , b ∗ , λ ∗ ) is a GNE of the game G ( w ) , then forproblem (2) we have f k ( d k ) − f k ( d ∗ k ) + ( d k − d ∗ k ) λ ∗ k ≥ , ∀ d ∈ ˆ D k , ∀ k ∈ I ∪ J (A.2)Note that b k is eliminated by substituting constraint (2b) intothe objective function (2a). For problem (3) we have ∑ k ∈ I ∪ J ( p outk − p out ∗ k )( p out ∗ k − b ∗ k ) ≥ , ∀ p out ∈ Y (A.3)For problem (1), its Lagrangian function is L ( d , p out , λ ) = ∑ k ∈ I ∪ J f k ( d k ) + ∑ k ∈ I λ k ( d fk + d k − p outk )+ ∑ k ∈ J λ k ( d fk + d k − w k − p outk ) (A.4)which is defined on Ω : = ∏ k ∈ I ∪ J ˆ D k × Y × R ( I + J ) . Let ( ˆ d , ˆ p out , ˆ λ ) be a saddle point of the Lagrangian function, then ( ˆ d , ˆ p out , ˆ λ ) ∈ Ω and it satisfies ∀ ( d , p out , λ ) ∈ Ω [56]: (cid:104) f k ( d k ) − f k ( ˆ d k ) + ( d k − ˆ d k ) ˆ λ k (cid:105) ≥ , ∀ k ∈ I ∪ J (A.5a) − ∑ k ∈ I ∪ J ( p outk − ˆ p outk )( ˆ λ k ) ≥ ∑ k ∈ I ( λ k − ˆ λ k )( d fk + ˆ d k − ˆ p outk ) OURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, FEB. 2019 12 + ∑ k ∈ J ( λ k − ˆ λ k )( d fk + ˆ d k − w k − ˆ p outk ) ≤ Existence . When Assumption A1 holds, suppose ˆ d is theoptimal solution of (1) and ˆ λ is the corresponding dualvariable. Let d ∗ = ˆ d , λ ∗ = ˆ λ , p out ∗ k = d fk + ˆ d k for all k in I , p out ∗ k = d fk + ˆ d k − w k for all k in J , and b ∗ k = a ˆ λ k + p out ∗ k , ∀ k ∈I ∪ J , then it is easy to check that (A.2) and (A.3) are met.Thus, we have constructed a GNE ( d ∗ , b ∗ , λ ∗ ) . Uniqueness . Given a GNE ( d ∗ , b ∗ , λ ∗ ) , when k ∈ I , wehave b ∗ k = d fk + d ∗ k + a λ ∗ k ; when k ∈ J , we have b ∗ k = d fk + d ∗ k − w k + a λ ∗ k . Let ˆ d = d ∗ , ˆ λ = λ ∗ , and ˆ p out = − a λ ∗ + b ∗ ,then it is easy to check that ( ˆ d , ˆ p out , ˆ λ ) satisfies (A.5), so ˆ d is the optimal solution of (1) and ˆ λ is the corresponding dualvariable. Since the objective function is strictly convex, andthe constraint sets ˆ D k , ∀ k ∈ I ∪ J and ˜ P are all closed convexsets, problem (1) has a unique solution [57], so ˆ d is unique. B. Convergence of Modified Best-Response based Algorithm
Let q k : = − a λ k + b k for all k ∈ I ∪ J . At the n -th iteration,given b n , the microgrid operator’s problem is equivalent to q n + = argmin { θ ( q ) − ( b n ) T qa + | q − q n + b n − − b n | a | q ∈ Q} (B.1a) λ n + = a ( b n − q n + ) (B.1b)where θ ( q ) : = a ∑ k ∈ I ∪ J q k and Q : = (cid:110) q | s.t. ∑ k ∈ I ∪ J q k = , q ∈ ˜ P (cid:111) . The users’ problems (2) are equivalent to: d n + = argmin { θ ( d ) + ( b n ) T da + | q n + − d − D | a | d ∈ ∪ k ∈ I ∪ J ˆ D k } (B.2a) b n + = b n − q n + + d n + + D (B.2b)where θ ( p ) : = ∑ k ∈ I ∪ J f k ( d k ) − a ∑ k ∈ I ∪ J ( d k + D k ) , D k = d fk , ∀ k ∈ I , D k = d fk , ∀ k ∈ J .If Condition C1 holds, both θ ( q ) and θ ( d ) are convex, andthe following function has a unique saddle point ( q ∗ , d ∗ , b ∗ ) . θ ( q ) + θ ( d ) − b T ( q − d − D ) a (B.3)By variational inequality technique [58] together with (B.1b)and (B.2b), (B.1a) is equivalent to for all q ∈ Q : θ ( q ) − θ ( q n + ) + ( q − q n + ) T (cid:26) − a b n + + a ( d n + − d n ) (cid:27) ≥ d ∈∪ k ∈ I ∪ J ˆ D k : θ ( d ) − θ ( d n + ) + ( d − d n + ) T (cid:26) a b n − a ( q n + − d n + − D ) (cid:27) ≥ t : = ( q , d ) and θ ( t ) : = θ ( q ) + θ ( d ) , (B.4)-(B.5) implies θ ( t ) − θ ( t n + ) + q − q n + d − d n + b − b n + T · − b n + / ab n + / aq n + − d n + − D + − I / a I / a ( d n − d n + )+ I / a I / a (cid:32) d n + − d n b n + − b n (cid:33) ≥ w : = ( q , d , b ) ∈ Q × ∪ k ∈ I ∪ J ˆ D k × R | I |×| J | and define amapping F ( w ) : = ( − b / a , b / a , q − d − D ) , which is indeedmonotone. Then we have θ ( t n + ) − θ ( t ∗ ) + ( w n + − w ∗ ) T F ( w n + ) ≥ θ ( t n + ) − θ ( t ∗ ) + ( w n + − w ∗ ) T F ( w ∗ ) ≥ (cid:32) d ∗ − d n + b ∗ − b n + (cid:33) T (cid:32) I / a I / a (cid:33) (cid:32) d n + − d n b n + − b n (cid:33) ≥ ( w n + − w ∗ ) T − I / a I / a ( d n − d n + )= a ( b n + − b n ) T ( d n − d n + ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) d n − d ∗ b n − b ∗ (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) d n + − d ∗ b n + − b ∗ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) d n − d n + b n − b n + (cid:12)(cid:12)(cid:12)(cid:12) (B.7)The sequence { ( d n , b n ) } is F´ejer monontone [59], with | ( d n − d ∗ ) T , ( b n − b ∗ ) T | decreasing in each iteration n by | ( d n − d n + ) T , ( b n − b n + ) T | . As a result, the sequence {| ( d n − d ∗ ) T , ( b n − b ∗ ) T | } converges and sequences { d n } and { b n } are bounded. With (B.7), the sequence { d n } ( { b n } ) only hasone cluster point. According to (B.5) we can get d n → d ∗ and b n → b ∗ . With (B.1) we know q n → q ∗ and λλ
Let q k : = − a λ k + b k for all k ∈ I ∪ J . At the n -th iteration,given b n , the microgrid operator’s problem is equivalent to q n + = argmin { θ ( q ) − ( b n ) T qa + | q − q n + b n − − b n | a | q ∈ Q} (B.1a) λ n + = a ( b n − q n + ) (B.1b)where θ ( q ) : = a ∑ k ∈ I ∪ J q k and Q : = (cid:110) q | s.t. ∑ k ∈ I ∪ J q k = , q ∈ ˜ P (cid:111) . The users’ problems (2) are equivalent to: d n + = argmin { θ ( d ) + ( b n ) T da + | q n + − d − D | a | d ∈ ∪ k ∈ I ∪ J ˆ D k } (B.2a) b n + = b n − q n + + d n + + D (B.2b)where θ ( p ) : = ∑ k ∈ I ∪ J f k ( d k ) − a ∑ k ∈ I ∪ J ( d k + D k ) , D k = d fk , ∀ k ∈ I , D k = d fk , ∀ k ∈ J .If Condition C1 holds, both θ ( q ) and θ ( d ) are convex, andthe following function has a unique saddle point ( q ∗ , d ∗ , b ∗ ) . θ ( q ) + θ ( d ) − b T ( q − d − D ) a (B.3)By variational inequality technique [58] together with (B.1b)and (B.2b), (B.1a) is equivalent to for all q ∈ Q : θ ( q ) − θ ( q n + ) + ( q − q n + ) T (cid:26) − a b n + + a ( d n + − d n ) (cid:27) ≥ d ∈∪ k ∈ I ∪ J ˆ D k : θ ( d ) − θ ( d n + ) + ( d − d n + ) T (cid:26) a b n − a ( q n + − d n + − D ) (cid:27) ≥ t : = ( q , d ) and θ ( t ) : = θ ( q ) + θ ( d ) , (B.4)-(B.5) implies θ ( t ) − θ ( t n + ) + q − q n + d − d n + b − b n + T · − b n + / ab n + / aq n + − d n + − D + − I / a I / a ( d n − d n + )+ I / a I / a (cid:32) d n + − d n b n + − b n (cid:33) ≥ w : = ( q , d , b ) ∈ Q × ∪ k ∈ I ∪ J ˆ D k × R | I |×| J | and define amapping F ( w ) : = ( − b / a , b / a , q − d − D ) , which is indeedmonotone. Then we have θ ( t n + ) − θ ( t ∗ ) + ( w n + − w ∗ ) T F ( w n + ) ≥ θ ( t n + ) − θ ( t ∗ ) + ( w n + − w ∗ ) T F ( w ∗ ) ≥ (cid:32) d ∗ − d n + b ∗ − b n + (cid:33) T (cid:32) I / a I / a (cid:33) (cid:32) d n + − d n b n + − b n (cid:33) ≥ ( w n + − w ∗ ) T − I / a I / a ( d n − d n + )= a ( b n + − b n ) T ( d n − d n + ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) d n − d ∗ b n − b ∗ (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) d n + − d ∗ b n + − b ∗ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) d n − d n + b n − b n + (cid:12)(cid:12)(cid:12)(cid:12) (B.7)The sequence { ( d n , b n ) } is F´ejer monontone [59], with | ( d n − d ∗ ) T , ( b n − b ∗ ) T | decreasing in each iteration n by | ( d n − d n + ) T , ( b n − b n + ) T | . As a result, the sequence {| ( d n − d ∗ ) T , ( b n − b ∗ ) T | } converges and sequences { d n } and { b n } are bounded. With (B.7), the sequence { d n } ( { b n } ) only hasone cluster point. According to (B.5) we can get d n → d ∗ and b n → b ∗ . With (B.1) we know q n → q ∗ and λλ n → λλ