Privacy Impact on Generalized Nash Equilibrium in Peer-to-Peer Electricity Market
PPrivacy Impact on Generalized Nash Equilibrium in Peer-to-Peer Electricity Market
Ilia Shilov a,b,1 , H´el`ene Le Cadre b , Ana Busic a a Inria Paris, DI ENS, CNRS, PSL University, France b VITO/EnergyVille, Thorpark 8310, Genk, Belgium
Abstract
We consider a peer-to-peer electricity market, where agents hold private information that they might not want to share. The problemis modeled as a noncooperative communication game, which takes the form of a Generalized Nash Equilibrium Problem, where theagents determine their randomized reports to share with the other market players, while anticipating the form of the peer-to-peermarket equilibrium. In the noncooperative game, each agent decides on the deterministic and random parts of the report, such that(a) the distance between the deterministic part of the report and the truthful private information is bounded and (b) the expectation ofthe privacy loss random variable is bounded. This allows each agent to change her privacy level. We characterize the equilibrium ofthe game, prove the uniqueness of the Variational Equilibria and provide a closed form expression of the privacy price. In addition,we provide a closed form expression to measure the impact of the privacy preservation caused by inclusion of random noise anddeterministic deviation from agents’ true values. Numerical illustrations are presented on the 14-bus IEEE network.
Keywords:
Peer-to-peer market, communication game, generalized Nash equilibrium, variational equilibrium, privacy
1. Introduction
The large-scale integration of Distributed Energy Resources(DERs), the increasing share of Renewable Energy Source (RES)- based generators in the energy mix and the more proactiverole of prosumers, have led to the evolution of electricity mar-kets from centralized pool-based organizations to decentralizedpeer-to-peer market designs [20]. Within this peer-to-peer elec-tricity market, agents negotiate their energy procurement seek-ing to minimize their costs with respect to both individual andcoupling constraints, while preserving a certain level of privacy[1]. The problem is modeled as a generalized Nash equilibriumproblem (GNEP), parametrized in the privacy level, chosen bythe agents.Information sharing in the peer-to-peer market can improveagents’ performance, but also may violate their privacy, leadingto the disclosure of agent’s private information [10]. This callsfor the design of new communication mechanisms that capturethe agents’ ability to define the information they want to share(their report) with the other market participants, while preserv-ing their privacy [4]. In many applications, this problem is usu-ally addressed by including noise to the reports that the agentssubsequently use to compute the market equilibrium [1]. How-ever, this approach does not include the ability of the agentsto act strategically on the values of their report. Moreover, thequestion of the optimal noise distribution is crucial in such aframework [11].To analyse the market in presence of shared coupling con-straints, we employ Generalized Nash Equilibrium (GNE) as
Email address: [email protected] (Ilia Shilov ) Corresponding author solution concept [16], and a refinement of it, called VariationalEquilibria (VE), assuming the shadow variables associated withthe shared coupling constraints are aligned among the agents.In our proposed framework, agents compute GNE with respectto the constraints that bound (a) the distance between the deter-ministic deviation from the true values of the private informa-tion and (b) the Kullback-Leibler divergence, that measures theeffect of the additive random noise included in the reports.Game theoretic approaches integrating the prosumers’ strate-gic behaviors in the peer-to-peer trading are considered in [1],[19]. The economic dispatch in energy communities under dif-ferent structures of communications is analysed in [14], [15].The impact of privacy on an energy community was analyzedin the literature, e.g. in [1], where the sensitive informationand the noise added to agents’ reports were considered as ex-ogenous parameters. Using a prediction model, Fioretto et al.provide a privacy-preserving mechanism, to protect the infor-mation exchanged between the different market operators whileguaranteeing their coordination [4].The anticipation of the actions of the agents in our modelis represented by the common knowledge of the form of thesolution . This anticipation will be used in strategic behaviorframework to compute the prosumers’ optimal deviation in theirprivate information reports.Various definitions of privacy have been introduced in thedata science literature [13]. Several information metrics: e.g.,mutual information, entropy, Kullback-Leibler divergence, andFisher information are used to quantify information release [3],[11]. Differential privacy (DP) was recently successfully ap-plied to multi-energy market operations [4] and dynamical sys-tems [11]. DP relies on adding noise to the reports from pre-determined distributions. In our model we also use the additive a r X i v : . [ c s . G T ] J a n oise, but we relax the assumptions of DP mechanism and focuson the prosumers’ ability to determine their noise distribution.It is done by bounding the the expectation of the privacy lossrandom variable [2], which constitutes exactly the Kullback-Leibler divergence for the introduced privacy-preserving ran-domized mechanism.To analyse the market in presence of shared coupling con-straints, we employ Generalized Nash Equilibrium (GNE) assolution concept [16], and a refinement of it, called VariationalEquilibria (VE) [16, 17]. We focus on a certain properties of thegame, such as aggregative and potential structure. Differentalgorithms using such properties as strong/strict-monotonicityof the game operator for computing VE using decentralized orsemi-decentralized structure for multi-agent equilibrium prob-lems in generalized aggregative games has recently gained highresearch interest [5], [6]. We relate the notion of privacy preservation resulting fromthe non-disclosure of the nominal demands and RES-based gen-erations of the prosumers in [1], to the privacy mechanism withthe additive Gaussian noise, that allows each agent to controlher privacy level. It is done, firstly, by choosing the determin-istic value to report to other agents; secondly, by using the ran-dom noise added to that value. We quantify the impact of pri-vacy on the prosumers’ costs and provide an analytical expres-sion of the market equilibria. In addition, we allow each agentto change her level of privacy and show the existence of the in-centives for the prosumers to deviate from their true sensistiveparameter values. We rely on the notion of strong monotonic-ity to prove the existence and uniqueness of the solution to ourproblem. Using Kullback-Leibler divergence, we measure thecost of privacy, caused by inclusion of the random noise. All thetheoretical results are illustrated on the 14-bus IEEE network.The organization of the rest of this paper is as follows: inSection 2 we first describe the peer-to-peer electricity tradingproblem in Subsections 2.1 and 2.2, which constitutes a basisfor our communication game, that will be defined in Subsection2.4. In Section 3 we provide the analytical expression of theGNE, prove the uniqueness of the VE of the game and providean expression for the utility gap, caused by the introduction ofthe privacy. Theoretical results are illustrated on the 14-busIEEE network in Section 4.
2. Statement of the problem
In distributed control systems there is a usual trade-off be-tween privacy and cost: to obtain a better solution, each agentrelies on the information of the other agents in the system, whichthey might not have incentives to provide.Consider a single-settlement market for peer-to-peer elec-tricity trading made of a set N of N agents, each one of thembeing located in a node of a communication network, that ismodeled as a graph G := ( N , E ) where E ⊆ N × N is theset of communication links between the players. Let Ω n be the set of nodes, player n wants to trade electricity with. Being theinterface node between the local electricity market and at thedistribution level and the transmission power network, node 0can communicate with any other nodes in Ω := N \ . Thegraph G does not necessarily reflect the distribution power net-work constraints.In this paper we focus on the privacy issues that arise aftersolving the peer-to-peer electricity trading problem, consideredin [1].Each agent n chooses independently her bilateral trades q n with agents she wants to trade electricity with, self-generation G n and flexible demand D n , in order to minimize her cost func-tion Π n : Π n ( D n , G n , q n ) := 1 / · a n G n + b n G n + d n (cid:124) (cid:123)(cid:122) (cid:125) C n ( G n ) ++ ˜ a n ( D n − D ∗ n ) − ˜ b n (cid:124) (cid:123)(cid:122) (cid:125) U n ( D n ) + (cid:88) m ∈ Ω n ,m (cid:54) = n c nm q mn (cid:124) (cid:123)(cid:122) (cid:125) ˜ C n ( q ) , (1)where a n , b n , d n , ˜ a n , ˜ b n > and D ∗ n denotes the nominal de-mand of agent n [1]. Thus, the vector of agent n ’s decision vari-ables is ( D n , G n , q n ) , where q n := ( q mn ) m ∈ Ω n is the vectorof the quantities exchanged between n and m in the directionfrom m to n , q mn , for all m ∈ Ω n \ { n } . We use the followingconvention: if q mn ≥ , then n buys q mn from m , otherwise( q mn < ) n sells − q mn to m . We let Q n denote the net import of agent n : Q n := (cid:80) m ∈ Ω n q mn .Each agent computes trading cost ˜ C n ( q ) using c nm whichmight represent preferences measured through product differ-entiation prices [1], [12] on the possible trades with the neigh-bors, or taxes. The following condition on agent’s trades called trading reciprocity constraint couples the decisions of two neigh-boring agents, ensuring for every node m ∈ Ω n that q mn + q nm = 0 . Note, that this formulation of the coupling con-straints differs from the one presented in [1], as we use equalityconstraint in our model, instead of the inequality. That meansthat the energy surplus is not allowed in the electricity tradingmodel. Let κ nm ∈ [0 , + ∞ ) be the equivalent trading capac-ity between node n and node m , such that κ nm = κ mn and ∀ m ∈ Ω n . This equivalent trading capacity is used to boundthe trading flows such that q mn ≤ κ mn .Local supply and demand should satisfy the following bal-ance equality in each node n in N : D n = G n + ∆ G n + (cid:80) m ∈ Ω n q mn , where ∆ G n is the renewable energy sources (RES)-based generation at node n , assumed to be non-flexible. As it was discussed in the introduction, each agent holdssome private information that takes the form of nominal de-mand D ∗ n and RES-based generation ∆ G n , which she does notdesire to reveal to the other agents in the system. In the furtheranalysis, we assume y n := D ∗ n − ∆ G n to be the private infor-mation of agent n . We assume that the agents desire to solvethe electricity trading problem endowed with the set of coupled2onstraints while not allowing the other agents to infer their val-ues of y n . We denote x n := ( D n , G n , q n ) to be the vector thatcontains agent n ’s decision variables and x − n is the vector ofthe other agents’ actions. We recall the optimization problemformulated in [1] for the clearing of the peer-to-peer electricitymarket. In the peer-to-peer setting the problem of the electricitytrading takes the form of generalized Nash equilibrium problemi.e., a game where the feasible sets of the players depend on theother players’ actions. With the notation introduced above, itmeans that each agent solves the following optimization prob-lem: min x n Π n ( x n ) , (2a) s.t. G n ≤ G n ≤ G n ( µ n , µ n ) (2b) D n ≤ D n ≤ D n ( ν n , ν n ) (2c) q mn + q nm = 0 ( ζ nm ) (2d) q mn ≤ κ mn ( ξ nm ) (2e) D n = G n + ∆ G n + (cid:88) m ∈ Ω n q mn ( λ n ) , (2f)where the corresponding dual variables are placed in blue at theright of each constraint. Note that in (2) the feasible set of theagent n can be rewritten in a more compact form C n ( x − n ) = { x n | (2b) − (2f) hold } . This notation will be used later in thepaper.We introduce the following assumption to guarantee thatthe interface trading capacities are big enough to supply trad-ing needs of all the agents and that differentiation prices aresymmetric for trading with the root node. Assumption 1.
We assume that there are large trading capac-ities from and to node 0 – that is ξ n = ξ n = 0 ∀ n ∈ N and c n = c n for all n ∈ N . The following subsection describes the computation of ˜ C n ( q ) in the different setting for the differentiation prices. Under the conditions of Assumption 1, Proposition 8 in [1]states that:
Proposition 2.
For any couple of nodes n ∈ N, m ∈ Ω n , m (cid:54) = n with asymmetric preferences (such as c mn > c nm or c mn 1. All c nm are homogeneous. That means that c nm = c forall n, m ∈ N . This case reflects the interpretation of c nm as the taxes for energy trading, that should be naturally non-discriminating among agents. In this case bilateral trade cost isgiven by: ˜ C n ( q n ) = c · Q n (3) 2. All c nm for m, n (cid:54) = 0 are heterogeneous. This frameworkrepresents the case, when all c nm , m, n (cid:54) = 0 are drawn fromsome continuous distribution (e.g. uniform). Under the As-sumption 1 we are able to obtain the expressions for ˜ C n ( q n ) inthis framework for agent n . Again using Proposition 2 we havethat q n = Q n − (cid:80) m ∈ Ω n ,m (cid:54) =0 κ nm sgn( c mn − c nm ) , where Q n is obtained by combining (2f) and expressions for D n , G n .Thus, we are able to obtain the cost expressions for each agents n directly: Proposition 3. Bilateral trade costs for any agent n ∈ N inthe network except root node 0 are given by ˜ C n ( q n ) = c n (cid:2) Q n − (cid:88) m ∈ Ω n ,m (cid:54) =0 κ nm sgn( c mn − c nm ) (cid:3) + (cid:88) k ∈ Ω n ,k (cid:54) =0 c nk κ nk sgn( c kn − c nk ) . (4) Bilateral costs for node 0 are expressed as ˜ C ( q ) = (cid:88) n ∈ Ω c n (cid:104) (cid:88) m ∈ Ω n ,m (cid:54) =0 κ nm sgn( c mn − c nm ) − Q n (cid:105) (5) Remark 4. We do not impose any condition on the ratio be-tween the values of the coefficients c nm . Choosing c n < c mn , ∀ m, n ∈N , we can ensure the preference for the local trades.3. Intermediate case. To demonstrate the difficulties arising inthe general case for computing bilateral trades, we consider theintermediate case, in which there exists one additional symmet-ric relation c n (cid:48) m (cid:48) = c m (cid:48) n (cid:48) for m (cid:48) , n (cid:48) (cid:54) = 0 . Thus, for this pair ofnodes we have that Q n (cid:48) = q n (cid:48) + q m (cid:48) n (cid:48) + (cid:88) k (cid:54) = m (cid:48) ∈ Ω n (cid:48) κ n (cid:48) k sgn( c kn (cid:48) − c n (cid:48) k ) Q m (cid:48) = q m (cid:48) + q n (cid:48) m (cid:48) + (cid:88) k (cid:54) = n (cid:48) ∈ Ω m (cid:48) κ m (cid:48) k sgn( c km (cid:48) − c m (cid:48) k ) , where q m (cid:48) n (cid:48) = − q n (cid:48) m (cid:48) , which gives us a system of two equa-tions with three unknown variables q n (cid:48) , q m (cid:48) , q m (cid:48) n (cid:48) . Writingthe similar equation for every node k (cid:54) = m (cid:48) , n (cid:48) , , we get N − equations with N − unknowns and adding the expression for Q we obtain linear system with N independent equations and N unknown variables. It follows that adding even one sym-metric relation leads to the system of N equations with N + 1 unknowns. It is shown in [1], that at the VE, agent n ’s decision vari-ables x ∗ n depend on the dual variable λ n , which, under the As-sumption 1 is aligned across agents: λ n = λ , ∀ n ∈ N , where λ is the uniform market clearing price . The equilibrium ex-pressions, provided in [1], also hold for our model with equalityconstraint (2d). λ depends on the private information y n of theagents. Formally, λ is given by: λ = (cid:80) n y n + (cid:80) n b n a n (cid:80) n (cid:16) a n + a n (cid:17) (6)3nd the decision variables D n and G n are given at the equi-librium by the following expressions: D n ( y ) = D ∗ n − a n λ , G n ( y ) = − b n a n + a n λ . The expression for Q n is obtainedfrom the supply demand equality condition (2f): Q n ( y ) = D ∗ n + b n a n − ( a n + a n ) λ − ∆ G n .Thus, to solve (2) each agent needs to compute the uniformmarket clearing price λ , which requires a knowledge of all the ( y n ) n in the system. It leads to a question for each agent n of how to determine the report of her private information, sothat it has the minimal impact on her cost, while guaranteeingthat the certain level of privacy is met. That is, each agent n anticipates the form of the solution of the electricity tradingproblem at the equilibrium and determines the report ˜ y n of herprivate information, that she submits to the other agents in thesystem.In order to do so, each agent n minimizes the differencebetween the cost of the problem with the modified values andthe optimal solution of the problem (2) with the truthful reports Π ∗ n : min ˜ y n E (cid:104) Π n (˜ y n , ˜ y − n ) − Π ∗ n (cid:105) ,s.t. x ∗ n ( ˜ y ) ∈ C n ( x ∗− n (˜ y )) , (7)where the expectation is taken in order to account for both ran-domized and deterministic cases. Note that x ∗ n depends on ˜ y because in the expressions for the decision variables D n ( · ) , G n ( · ) and q n ( · ) we use the reports ˜ y instead of the true values y asthe input. Also note that Π ∗ n is a constant as it is calculated us-ing true values of y , thus it can be omitted from the objectivefunction. Remark 5. In (7) we assume that the form of the electricitytrading problem is known by all the agents in the system. Itenables each agent to anticipate the form of the solution x ∗ n ( · ) , for all n ∈ N and thus, based on this form to decide on theoptimal information ˜ y n , ∀ n ∈ N to report to the other agents before they actually obtain the solution of the electricity tradingproblem. Note that it differs from [4], as we take the form of thesolution x ∗ of the GNEP as given.2.4. Communication game The report of the agent n takes the form ˜ y n = ˆ y n + ε n .The first part of the report captures the ability of agent n to actstrategically on her report by determining the deterministic part ˆ y n that solves the cost minimization problem. In the second partof the report, each agent implements a randomized mechanism M ( · ) by choosing the noise ε n to add to ˆ y n in order to preservea certain level of privacy. First, we define an upper bounded distance as a symmet-ric adjacency relation y n (cid:39) ˆ y n for agent n : y n (cid:39) ˆ y n ⇐⇒ d ( y n , ˆ y n ) ≤ α n , where α n is chosen beforehand and reflectsthe amount of information agent n desires to preserve [22]. Definition 6 (Privacy loss). Given a randomized mechanism M ,let p M ( y n ) ( z ) denote the density of the random variable Z = M ( y n ) . The privacy loss function of M ( · ) on a pair of y n (cid:39) ˆ y n is defined as l M,y n , ˆ y n ( z ) = log (cid:0) p M ( y n ) ( z ) /p M (ˆ y n ) ( z ) (cid:1) Theprivacy loss random variable L M,y n , ˆ y n := l M,y n , ˆ y n ( Z ) is thetransformation of the output random variable Z = M ( y n ) bythe function l M,y n , ˆ y n . We assume that each agent samples a Gaussian noise ε n ∼N (0 , σ n ) , thus obtaining the report ˜ y n ∼ N (ˆ y n , σ n ) . Whenthe Gaussian isotropic random noise is added to the determin-istic value of the input, it is well-known that the privacy lossrandom variable is also Gaussian: Lemma 7 ([2]). The privacy loss L M,y n , ˆ y n of a Gaussian out-put perturbation mechanism follows a distribution N ( η, η ) ,with η = D / σ , where D = || y n − ˆ y n || .2.4.2. A randomized mechanism for information reporting We aim to allow agents to be able to decide on the opti-mal noise added to their private information, by choosing theoptimal variance V n . For simplicity of notations, we denote V n := σ n .First, each agent chooses the neighboring input ˆ y n (cid:39) y n ,on which she later implements M ( · ) . It is reflected in the con-straint (8d). In the constraint (8e), the expectation of the privacyloss random variable measures the expected privacy loss of themechanism M ( y n ) on the fixed private information y n , ˆ y n . Inother words, it shows, how much information can be extractedfrom the report ˜ y n . Note, that it is exactly the Kullback-Leiblerdivergence (or the relative entropy) between M ’s output distri-butions on y n and ˆ y n .Thus, to decide on the optimal value of the report ˜ y n , eachagent needs to solve the following optimization problem: min ˆ y n ,V n E ε n ∼N (0 ,V n ) (cid:104) Π n ( ˆ y , ε ) (cid:105) (8a) s.t. G (cid:48) n ≤ E ε n ∼N (0 ,V n ) (cid:2) G n ( ˜ y ) (cid:3) ≤ G (cid:48) n ( µ n , µ n ) (8b) D (cid:48) n ≤ E ε n ∼N (0 ,V n ) (cid:2) D n ( ˜ y ) (cid:3) ≤ D (cid:48) n ( ν n , ν n ) (8c) (ˆ y n − y n ) ≤ α n ( γ n , γ n ) (8d) E (cid:2) L M,y n , ˆ y n (cid:3) ≤ A n ( β n , β n ) (8e)where G (cid:48) n = G n + ω G n , G (cid:48) n = G n − ω G n and D (cid:48) n = D n + ω D n , D (cid:48) n = D n − ω D n , in which ω D n , ω G n > are introduced in or-der to account for the strictly feasible solutions of problem (2).In the numerical experiments we set ω D n , ω G n to be a small,e.g. − .As it is shown below, the only term depending on the vari-ance in the utility function of the agent n is B n B (cid:80) m V m . Inthe special case y n = ˆ y n for some n , where the constraint (8e) (ˆ y n − y n ) V n ≤ A n holds for any < V n < ∞ . The possibleconvention could be to exclude this constraint from the consid-eration, when y n = ˆ y n and set V n = 0 .The condition for the uniform market clearing price λ tohave a form given in (6) is to have zero total net import, i.e. (cid:80) n Q n = 0 . In the case a fully coordinated mechanism is im-plemented, i.e the local MO has an access to all the constraints4nd parameters of the agents and solves the problem in a cen-tralized way, it is possible to oblige agents to align their reports ˜ y such that E (cid:104) (cid:80) n Q n ( ˜ y ) (cid:105) = (cid:80) n ( y n − ˆ y n ) = 0 . It followsthat (cid:80) n ˆ y n = (cid:80) n y n . So, when we compute λ using ˜ y in-stead of y , we obtain E (cid:2) λ ( ˜ y ) (cid:3) = B ( (cid:80) n ˆ y n + (cid:80) n b n a n ) = B ( (cid:80) n y n + (cid:80) n b n a n ) Thus, the final market clearing price doesnot depend on the reports of the agents, which is formalized inthe following statement: Proposition 8. When the prosumers align their reports ˜ y sothat the condition E (cid:2) (cid:80) n Q n ( ˜ y n ) (cid:3) = 0 is met, then the uniformmarket clearing price λ depends only on the true values oftheir initial parameters y . In the case a peer-to-peer communication mechanism is im-plemented, the sum of the net imports at each node might notbe equal to zero. Indeed, agents might have incentives to vi-olate this condition in order to decrease their costs. Thus, thecondition (cid:80) n Q n = 0 might not hold.On the market level it is necessary for the condition of zerototal net import to hold such that supply and demand balanceeach other in problem (2) [15]. Also, note that non zero totalnet import (cid:80) n Q n (cid:54) = 0 implies that there exists at least onepair of agents ( n, m ) with q nm + q mn (cid:54) = 0 . Besides, this mightcause the violation of the capacity condition q nm ≤ κ nm . Itmeans that the local Market Operator (MO) has to compensatethe difference E (cid:2) (cid:80) n Q n ( ˜ y n ) (cid:3) caused by the lack of coordina-tion in the agents’ reports. In the case E (cid:2) (cid:80) n Q n ( ˜ y n ) (cid:3) ≤ ,there is an energy surplus in the system, which can be sold bythe MO (by the intermediate of an aggregator) to the wholesalemarket at price p . If E (cid:2) (cid:80) n Q n ( ˜ y n ) (cid:3) ≥ , then the MO (bythe intermediate of an aggregator) has to buy the energy on thewholesale market at price p , which depends on the wholesalemarket price, in order to supply the system demand.When the constraints and the private information of the agentsare not shared, the MO only knows the aggregate deviation (cid:80) n ( y n − ˆ y n ) thus penalties imposed on the agents depend onit and not on the personal deviation y n − ˜ y n of the agent n . Remark 9. For prosumers, imports/exports of energy from/tothe community manager are possible at prices p − / p + respec-tively such that p + ≤ p ≤ p − . To avoid non-differentiabilityin the utility function, we let p + = p = p − . To compensate for the cost of buying the lack of energy atthe local market level from the wholesale market, the MO im-poses penalty to each prosumer that takes the form P ( ˜ y ) = p N (cid:80) n ( y n − ˜ y n ) . Note that in case of the excess of the pro-duction on the local market level, the prosumers will be equallyreimbursed based on the surplus produced. The division by N is introduced in order to equally split the burden of the non zerototal net import and mitigate the possible volatility of the price p . Assumption 10. A local MO ensures the compensation for thenonzero total net import. This implies that the formula for λ in (6) is used by all the prosumers to compute their decisionvariables. Proposition 11. Dual variables β n , β n for the constraint (8e) can be interpreted as the privacy price for agent n and are com-puted by the formula ( β n + β n ) = B n (ˆ y n − y n ) B Proof. Constraint (8e) can be rewritten as follows, when weconsider V n (cid:54) = 0 for all n ∈ N : E (cid:2) L M,y n , ˆ y n (cid:3) ≤ A n ⇐⇒ (ˆ y n − y n ) ≤ V n A n . In the following analysis, we denote B n := a n + a n and B := (cid:80) n B n . The objective function(8a) of the agent n depends linearly on the V n , thus attainingthe minimum with respect to this decision variable on the lowerboundary of the feasible region. The lower boundary is givenby the constraint (8e), from which we can conclude that V n = (ˆ y n − y n ) A n for any given value of the decision variable ˆ y n forany agent n . From the KKT conditions we have that V n = B A n B n ( β n + β n ) , from which we obtain the expression for ( β n + β n ) .First, from the complementarity conditions we know thateither of β n , β n equals 0. Clearly, it is non-negative term thatappears in the utility of the agent n at the equilibrium. Thus,we can view β n , β n as a privacy price . Remark 12. The privacy price increases with respect to thedistance between the truthful ( y n ) and biased ( ˆ y n ) values ofagent n ’s private information. 3. Equilibrium problems Note that as λ depends on the sum of (cid:80) n (cid:0) D ∗ n − ∆ G n (cid:1) ,the objective function in (8) has an aggregative game structure,i.e. it depends on player n ’s decision ˆ y n and on the aggregateof the other agents’ decisions.Below we provide the computations of the objective func-tion of agents Π n (˜ y n , ˆ y − n ) both in (i) the fully coordinatedmechanism and (ii) the peer-to-peer coordination mechanism.To arrive to this closed form expression, we observe that ˜ y n ∼ N (ˆ y n , V n ) . The sum of normal variables is a normal vari-able itself: (cid:80) n ˜ y n ∼ N ( (cid:80) n ˆ y n , (cid:80) n V n ) , from where it fol-lows that ( (cid:80) n ˜ y n + (cid:80) n b n a n ) ∼ N ( (cid:80) n ˆ y n + (cid:80) n b n a n , (cid:80) n V n ) .Using a formula for the second moment of the normal distribu-tion, expression (3) and Proposition 3, we obtain the expressionfor the utility of the agents in cases (i) and (ii).In the homogeneous differentiation price case c nm = c , thecost function of agent n is given by E (cid:2) Π n ( ˜ y ) (cid:3) = B n B (cid:104)(cid:16) (cid:88) m ˆ y m + (cid:88) m b m a m (cid:17) + (cid:88) m V m (cid:105) + c (cid:2) D ∗ n − ∆ G n + b n a n − B n B (cid:16) (cid:88) m ˆ y m + (cid:88) m b m a m (cid:17)(cid:3) + p N (cid:88) m ( y m − ˆ y m ) − b n a n + d n − ˜ b n , Expression for the utility in the case when c nm are heteroge-neous is similar, except that for ˜ C n ( q n ) , we use the expressionsfrom Proposition 3.5 .2. GNE computation3.2.1. c nm are homogeneous From the computations of the KKT conditions, we obtainthat B n B (cid:80) m ∈N ˆ y m + M (cid:48) n = 0 , ∀ n ∈ N , where M (cid:48) n := B n B (cid:80) m b m a m − p N − cB n B + a n B ( µ n − µ n ) + a n B ( ν n − ν n ) + γ n − γ n + β n − β n . c nm are heterogeneous for m (cid:54) = 0 Analogously, first order stationarity conditions for agents n ∈ N are given by B n B (cid:80) m ∈N ˆ y m + M (cid:48)(cid:48) n = 0 , with M (cid:48)(cid:48) n := B n B (cid:80) n b n a n − c n B n B − p N + a n B ( µ n − µ n ) + a n B ( ν n − ν n ) + γ n − γ n + β n − β n and M (cid:48)(cid:48) = B B (cid:80) n b n a n + a B ( µ − µ ) + (cid:80) n (cid:54) =0 c n B n B − p N + a B ( ν − ν ) + γ − γ + β − β . Definition 13. An operator F : K ⊆ R n → R n is stronglymonotone on the set ˆ K ⊆ K with monotonicity constant α > if ( F ( x ) − F ( y )) (cid:62) ( x − y ) ≥ α || x − y || , ∀ x, y ∈ ˆ K. Theoperator is monotone if α = 0 In order to show the uniqueness of the VE of the problem (8),we check if the operator F ( ˆ y , V ) := (cid:2) ∇ n E (cid:0) Π n ( ˆ y , V ) (cid:1)(cid:3) Nn =1 (9)is strongly monotone. To do so, we use the following lemma: Lemma 14 ([6]). A continuously differentiable operator F : K ⊆ R n → R n is α -strongly monotone with monotonicityconstant α (resp. monotone) if and only if ∇ x F ( x ) (cid:23) αI (resp. ∇ x F ( x ) (cid:23) ) for all x ∈ K . Moreover, if K is compact, thenthere exists α > such that ∇ x F ( x ) (cid:23) αI for all x ∈ K if anonly if ∇ x F ( x ) (cid:31) for all x ∈ K . For homogeneous differentiation price c nm = c , F ( ˆ y , V ) writesas follows: F ( ˆ y , V ) = col (cid:40)(cid:32) B i B (cid:32)(cid:88) m ˆ y m + (cid:88) m b m a m (cid:33) −− c B i B − p N , V i B i B (cid:33) N − i =0 (cid:41) (10)When differentiation prices c nm are heterogeneous for m (cid:54) = 0 ,operator F ( ˆ y , V ) is obtained similarly, but for the expressionsof ˜ C n ( q n ) , we take expressions from Proposition 3. Lemma 15. Operator F ( ˆ y , V ) defined in (10) is strongly mono-tone. Proof. The proof can be found in the Appendix. Proposition 16. By the strong monotonicity of F ( ˆ y , V ) , VE ofthe game (8) is unique [16]. 3.4. Generalized Potential Game extension Assumption 17. Assume that ∀ i, j ∈ N : B i B (cid:39) B j B , i.e. eachagent n ’s contribution B n to the sum B is relatively small. De-note H := B n B ∀ n ∈ N . Proposition 18. Under Assumption 17, the game (8) is a Gen-eralized Potential Game. Proof. The proof can be found in the Appendix.Generalized Potential Games constitute a subclass of gamesfor which the convergence of the BR algorithms is established[7] in the deterministic case. Taking into account that the BRscheme is suited for our private framework, an interesting di-rection of the research would be to establish the convergence ofthe BR algorithm for the stochastic NE of the GPG. 4. Numerical Results In the paper [21], authors employ the penalized individualcost functions to deal with coupled constraints and provide athree stochastic gradient strategies with constant step-sizes inorder to approach the Nash Equilibrium. In order to estab-lish their results, authors consider the model with the opera-tor F ( ˆ y, V ) to be strongly-monotone and Lipschitz continuous ,which holds for our case. We consider the scheme, called bythe authors as Diffusion Adapt-then-Penalize : (cid:40) ψ kν = ˆ y k − ν − µ ∇ ˆ y ν Π ν (ˆ y ν , ˜ y − ν )ˆ y kν = ψ kν − µR ∇ ˆ y ν ( θ ν ( ψ kν )) , where µ denotes the step-size, R - penalty parameter and θ ν ( · ) -penalty function for the coupling constraints, which we chooseto be the sum over all the constraints of the form x ≤ , of allfunctions p ν ( x ) such that p ν ( x ) = (cid:80) I x ≥ · x / . We consider the IEEE 14-bus network system, which is de-picted in Figure 1, where each bus (node) of the network cor-responds to a prosumer in our model. We consider the system,which consists of the agents with the non-zero self-generationand demand parameters, thus we exclude one interim node 6,which sole purpose in the initial 14-bus system is to connectthe flows. Thus, in our model there are 13 buses (nodes). Asparameters of the algorithm, we set µ = 0 . and R = 700 .We first focus on the homogeneous differentiation price case c nm = 1 . [$/MWh], for all n, m ∈ N . The cost p , usedby local MO to trade with the wholesale market, is set to behigher than c and equals . [$/MWh]. The natural assump-tion is the homogeneity of the self-generation parameters ofthe prosumers, which we set to be a n = 0 . , b n = 6 . forall n ∈ N . Also, there are three nodes (2, 5, 7) that are addi-tionally equipped with a RES-based generation. Values on thelinks between the nodes on Figure 1 specify the trading capac-ity parameters κ nm . Recall from the Assumption 1 that thereare large trading capacities to and from node 0, thus we do not6 . . . . . . . . . . . . . . . . . . Grid connection w i nd ∆ G = . c o a l wind ∆ G =7 . D ∗ . s o l a r ∆ G = . D ∗ = . D ∗ = . D ∗ . D ∗ . D ∗ = . D ∗ . D ∗ = . D ∗ . D ∗ = . D ∗ = . D ∗ = . Figure 1: IEEE 14-bus network system specify them on the scheme. The nominal demands and RES-based generations in Figure 1 are given in [GWh]. All the pa-rameters that are used to calibrate the agents’ utility functionsare specified in Table 1. Table 1: Agents’ utility function parameters. Node ˜ a ˜ b d D G We measure the impact of our mechanism on the cost ofthe agents, measured through the agent’s utility gap E (cid:104) Π ∗ n − Π n (˜ y n , ˜ y − n ) (cid:105) for each agent n . In Figure 2, we plot the util-ity gap as a function of A n for all the agents. We observethat the nodes 3 and 8 decrease their costs the most among allthe prosumers and nodes 9 and 11 have, on the contrary, in-creasing costs. From Table 1 it can be seen that node 3 and 8 have the minimal flexible demand coefficients: ˜ a = 0 . and ˜ a = 0 . . Similarly, nodes 9 and 11 have the biggest flexibledemand coefficients: ˜ a = 4 . and ˜ a = 5 . . The cost ofthe demand flexibility affects the utility of the agents, i.e. smallcost allows them to adjust their demand such that they can de-crease their costs, while deviating from their true values.For the graphs shown below, we set α n = 3 . when weplot the dependance w.r.t. A n , and A n = 10 . when we plotthe dependance w.r.t. α n . For this choice of parameters, thecolor of the nodes in Figure 1 shows the privacy price β n , β n [$/MWh] from Proposition 11 in each n ∈ N . Light blue de-notes the lowest privacy price ( . . − [$/MWh]) and darkviolet denotes the highest ( . . − [$/MWh]).Figure 3 represents the dependence of the plot of the utilitygap on the parameter α n of the agents. It is shown, that whenthe maximal bound on the distance is low, the agents expect-edly deviate from their costs Π ∗ n . As soon as α n increases, thusproviding more possibility to deviate, agents tend to show thesimilar behavior as on the plot with respect to A n : nodes 3 and8 gain the most and nodes 9 and 11 have the increasing costs. Figure 2: Utility gap wrt. A n Figure 3: Utility gap wrt. α n Figures 4 and 5 depict the dependance of the social cost ofthe system w.r.t. A n and α n respectively. We compare threeinstances: peer-to-peer communication mechanism, fully coor-dinated communication mechanism and the social cost evalu-ated in the truthful reports. Note, that the latter one providesthe same cost when Proposition 8 holds.It can be seen that increase in A n affects the peer-to-peercommunication the most, which is caused by the decrease ofthe privacy price induced by the noise in the agents’ reports (re-call, that the variance is given by V n = B A n B n ( β n + β n ) ), thusallowing them to compute their decision variables more pre-cisely. Clearly, it affects the centralized communication mech-anism less. On the other hand, increase of the α n affects thecentralized communication mechanism the most, as it allowsthe local MO to find an optimal solution for each agent in thesystem, thus leading to the biggest decrease in the costs.The results above are shown in the homogeneous A n , α n and c nm case. Heterogeneity in the distance parameters α n and A n does not affect the behavior of the agents in the system de-scribed above. Nevertheless, setting parameters α n to be smallfor those who deviate the most (e.g. nodes 3 and 8) can boundtheir influence on the sum (cid:80) n ˜ y n , thus, bounding the deviationfrom the (cid:80) n y n .In the case of heterogeneous differentiation prices c nm for n, m (cid:54) = 0 , we compute the trading costs of the agents, usingthe expressions given in Proposition 3. Numerical experimentsshow the same behavior for all the agents in the system, while7 igure 4: Social cost decrease wrt. A n Figure 5: Social cost decrease wrt. α n distinguishing the node 0: in this setting it decreases its cost themost. An interesting research direction would be to adapt thepenalty for the prosumers in order to account for this behavior. 5. Conclusion In our work we considered a peer-to-peer electricity market,in which agents have private information. The problem is mod-eled as a noncooperative communication game, which takes theform of a GNEP, where the agents determine their randomizedreports to share with the other market players, while anticipat-ing the form of the peer-to-peer market equilibrium. Agents de-cide on the deterministic and random parts of the report, suchthat the (a) the distance between the deterministic part of thereport and the truthful private information is bounded (b) ex-pectation of the privacy loss random variable is bounded. Thisallows them to act strategically on the values of the determinis-tic part and to choose the random noise included in their reports.We characterized the equilibrium of the problem and proved theuniqueness of the Variational Equilibria. We provided a closedform expression for the privacy price. The theoretical resultsare illustrated on the 14-bus IEEE network, using the stochasticgradient descent algorithm. We show the impact of the privacypreservation caused by inclusion of random noise and determin-istic deviation from agents’ true values.Since our problem has a potential form under mild assump-tions, as the next step, we will focus on the development of thedistributed learning algorithm for the stochastic NE of the Gen-eralized Potential Game. Another interesting research directionwould be to consider the decentralized communication mecha-nism, where agents do not have the ability to anticipate the formof the uniform market price. References [1] H. Le Cadre, P. Jacquot, C. 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First, note that for heterogeneous c nm , operator F ( ˆ y , V ) writes as follows for nodes i (cid:54) = 0 : F ( ˆ y , V ) = col (cid:40)(cid:32) B i B (cid:32)(cid:88) m ˆ y m + (cid:88) m b m a m (cid:33) + − c i B i B − p N , V i B i B (cid:33) N − i =1 (cid:41) , (11)and for node we can write it as follows: F ( ˆ y , V ) = (cid:32) B B (cid:32)(cid:88) m ˆ y m + (cid:88) m b m a m (cid:33) ++ (cid:88) m (cid:54) =0 c m B m B − p N , V B B (cid:33) . (12)We want to prove that the operator F ( ˆ y , V ) defined in (10) orin (11) and (12) is strongly monotone.We denote vector z to be z := (ˆ y , V , . . . , ˆ y N − , V N − ) .We need to investigate whether ∇ z F ( ˆ y , V ) is positive-definite.Denote F ( ˆ y , V ) i := B i B (cid:16)(cid:80) n ˆ y m + (cid:80) n b n a n (cid:17) − c B i B − p N forthe homogeneous c nm case, and F ( ˆ y , V ) i = V i B i B . Simi-larly, for the heterogeneous c nm case, denote F ( ˆ y , V ) i := B i B (cid:16)(cid:80) n ˆ y m + (cid:80) n b n a n (cid:17) − c i B i B − p N for i (cid:54) = 0 and F ( ˆ y , V ) := B i B (cid:16)(cid:80) n ˆ y m + (cid:80) n b n a n (cid:17) + (cid:80) m (cid:54) =0 c m B m B − p N . Thus we havethat ∂ F ( ˆ y , V ) i ∂ ˆ y j = B i B , ∀ j ∈ N and ∂ F ( ˆ y , V ) i ∂V i = B i B . All otherpartial derivatives are 0. Thus ∇ z F ( ˆ y , V ) is a matrix definedwith its entries to be ∇ z F ( ˆ y , V ) ij = B i +22 B if i, j are even B i +12 B if i, j are odd and i = j otherwiseSymmetric matrix A is positive definite on compact if its quadraticform is positive: x (cid:62) Ax > , ∀ x ∈ R n \ . Note, that non-symmetric matrix A is positive definite iff symmetric matrix (cid:0) A + A (cid:62) (cid:1) is. In our case the quadratic form is given by thefollowing expression: z (cid:62) (cid:0) ∇ z F ( ˆ y , V ) + ∇ z F ( ˆ y , V ) (cid:62) (cid:1) z == 1 B N (cid:88) i =1 B i ˆ y i + 12 B N (cid:88) i =1 (cid:88) j (cid:54) = i ( B i + B j )ˆ y i ˆ y j + 1 B N (cid:88) i =1 B i V i ≥ B N (cid:88) i =1 B i ˆ y i + 1 B N (cid:88) i =1 (cid:88) j (cid:54) = i (cid:112) B i B j ˆ y i ˆ y j + 1 B N (cid:88) i =1 B i V i = 1 B ( N (cid:88) i =1 (cid:112) B i ˆ y i ) + 1 B N (cid:88) i =1 B i V i , which is positive for all (ˆ y i , V i ) i in the feasible region. Proof. The notion of Generalized Potential Game is widelyused in the literature, see e.g. [7]. To verify that our game isindeed the potential game, consider function P ( ˆ y ) , defined asfollows: P ( ˆ y , V ) = N (cid:88) i =1 (cid:104) H ˆ y i B ( 12 (cid:88) n ˆ y n + (cid:88) n b n a n ) − p ˆ y i N − cH ˆ y i B + HB V i (cid:105) We can check that it is a potential function. Indeed, for all ˆ y − n ,and for all admissible x n , z n , x (cid:48) n , z (cid:48) n : Π n ( x n , ˆ y − n , x (cid:48) n , V − n ) − Π n ( z n , ˆ y − n , z (cid:48) n , V − n ) == H B (cid:104) ( x n − z n )( x n + z n + 2 (cid:88) m (cid:54) = n ˆ y m + 2 (cid:88) k ∈N b k a k ) (cid:105) − ( x n − z n ) cHB − p N ( x n − z n )++ HB ( x (cid:48) n − z (cid:48) n ) == P ( x n , ˆ y − n , x (cid:48) n , V − n ) − P ( z n , ˆ y − n , z (cid:48) n , V − n ))