Grid Influenced Peer-to-Peer Energy Trading

1 Grid Influenced Peer -to-Peer Ener gy T rading W ayes T ushar , Senior Member , IEEE, T apan Kumar Saha, F ello w , IEEE, Chau Y ue n, Senior Member , IEEE, Tho mas Morstyn, Member , IEEE , Nahid-Al-Masoo d, Senior Member , IEE E, H. V incent Poor, F ello w , IEEE and Richard Bean Abstract — This paper propose s a peer -to-pe er (P2P) energy trading scheme that can help the centraliz ed power system to reduce the total electr icity demand of its customers at the peak hour . T o do so, a cooperat iv e Stackelber g game is formula ted, in which the centralized power system acts as the leader that needs to decide on a price at the peak demand period to ince ntiviz e prosumers to not se eking any ener gy from it. The prosumers, on the other hand, act as fo llowe rs and respond to the leader’ s decision by formi ng suitable coalitions with neighboring prosume rs in order to participat e in P 2P ener gy trading to meet their ener gy demand. The properti es of the pr oposed Stack elber g game are studied. It is s h own that the game has a unique and stable Stackel berg equilibr ium, as a result of the stability of prosume rs’ coalitions. At the equilibr ium, the leader chooses its strategy using a deriv ed closed- f orm expression, while the prosumers choose their equilibrium coalitio n structur e. A n algorithm is proposed that enables the centraliz ed powe r system and the prosume rs to reach the equilibrium solution. Numerical case studies demonstra te the beneficial proper ties of the proposed sche me. Index T e rms —Peer -to-pee r , ener gy trading, game theory , prosumer , auction, coalition fo rmation. N O M E N C L A T U R E N T otal number of participating prosumers. T T otal number of time slots. S Set of seller prosumers. B Set of buyer prosumers. B a Set of buyer prosumers who trade with auction price. S a Set of seller prosumers who trade with auction price. N Set of participating prosumers. n Ind ex of each prosumer . t Index of time slot. e n,d ( t ) Energy demand of prosumer n at t . e n,g ( t ) Energy that n buys from CPS at t . e n,p ( t ) Energy that n buys from another peer at t . E n,b ( t ) Bidding energy by buying prosumer n at t . E n,s ( t ) Bidd ing energy by selling prosumer n at t . E S ( t ) T otal energy demand by all customers at t . E D ( t ) T otal energy demand by prosumers at t . E O ( t ) T otal energy demand by customers other than prosumers at t . E T ( t ) Threshold set by the CPS at t . E G ( t ) Energy supply capacity of the CPS at t . W . Tusha r and T . K. Saha are with the School of Information T e chnology and Electrical Engineering of the Univ ersity of Queensland , Brisbane, QL D 4072, Australia (e-mail: wayes.tushar .t@ieee.org; saha@it ee.uq.edu.au). C. Y uen is with the Enginee ring Product De velo pment Pillar of the Singapore Univ ersity of T ec hnology and Design (SUTD), 8 Somapah Road, Singapore 487372. (e-mail: yuenchau@sutd.edu .sg). T . Morstyn is with the Oxford Martin School of the Uni versit y of Oxford, Oxford, UK. (e-mail: thomas.morstyn@eng.ox.ac.uk) N. Masoo d i s wit h the Department of EEE, Ba ngladesh Uni v ersity of Engi- neering and T echnol ogy , Dhaka, Bangladesh. (e-mail: nahid@ee e.buet .ac.bd) H. V . Poor is with the Department of E lectrical Engineering at Princeton Uni versi ty , Princeto n, NJ 08544, USA. (e-mail: poor@prin ceton.edu) . R. Bean is with the Redback T echnologies, Indooroolippy , QLD 4068, Australia . (e-mail: richar d@redback tech.com). This work is supported in part by the Queensland Govern ment under the Adv ance Queensla nd Researc h Fellowshi p A QRF11016-17R D2, in part by NSFC 61750110529, and in part by the U.S. National Science Foundation under Grant ECCS-1824710. p g,b ( t ) Buying price per unit of energy set by the CP S at t . p g,s ( t ) Selling price per unit of energy set by the CPS at t . p p 2 p ( t ) P2P trading price at t . p n,b ( t ) B idding price of n at t . p auc Auction price. p max Highest reserv ation price. p FiT Feed-in-tariff price. p mid Mid-marke t price. U n,s ( t ) Utility of n for selling energy at t . U n,b ( t ) Utility of n for buying energy at t . J c ( t ) Cost to the CPS at t . a, b Design parameters for the CPS. α n ( t ) Preference parameter of n at t . β Design parameter . ( · ∗ ) Equilibrium v alue of the parameter ( · ) . Γ Stackelber g game. D hp Stability index of Γ . ν V alue of a coaliti on. η n Burden to prosumer n . I . I N T R O D U C T I O N Recent adv ancements in cryptocurrencies and blockchain hav e led to a proliferation of peer-to-peer (P 2P) energy trading schemes [1]– [3]. Essentially , P2P trading is a next-generation energy management mechanism for the smart grid that enables a customer of the network to i ndependently participate in energy trading with other prosumers and the grid [4]. P otential benefits of P2P ener gy trading include rene wable energy usage maximization, electricit y cost reduction, peak load shaving, prosumer empowerme nt, and network operation and in ve stment cost minimization. T o realize some of the above-men tioned benefits, a number of studies hav e been conducted in P2P trading in recent years. These studies can be di vided i nto three general categories. The first catego ry of studies, such as [5]–[12], focus on the financial modeling of P2P energy trading market. In [5], the authors present the concept of a blockchain-based microgrid energy mark et without the need for central intermediaries. Blockchain-base d energ y trading to reduce the cost of establishment in energy limited Industrial Internet of T hings nodes is also discussed in [6]. A multi-agent simulation framew ork and a consensus-b ased app roach for P2P ener gy trading in a microgrid are used in [7] and [8] respectively . The role of battery flexibility on P2P trading i s discussed in [9] and their controls for P2P energy sharing using a two-stage aggregated battery control are presented in [10]. Finally , game theory based contracts between sellers and b uyers of the P2P market are de veloped in [ 11] and [12]. The second catego ry of studies focuses on t he challenges of transferring energy ov er the physical layer [5] of the distribution network. For examp le, [13 ] proposes a methodology to assess the impact of P2P transactions on the network and to guarantee an excha nge of energy that does not violate network constraints. A real- time attri bution of po wer losses to each P2P transaction inv olving one generator and one l oad node is done by defining some suitable indices in [14]. A similar study of energ y blockchain in microgrids can be found in [15]. I n [16 ], the authors propose a graph-based loss 2 allocation framew ork for P2P market i n unbalanced radial distr i bu- tion networks. F inally , [17] shows ho w to achie ve decentralization using P2P frame works as underlying control structures, and t hen, implement a pure P2P to eliminate single points of failure. The final category of studies deals with engaging prosumers in P2P energy trading. Examples of such studies include [18], [19] and [20]. A ne w concept of energy classes, allowing energy to be treated as a heterogeneou s product, based on attributes of its source which are percei ved by prosumers to have value is introduced in [18]. The authors in [19] design a P2P energy trading platform to incentivize prosumers to form federated power plants. F inally , a motiv ational psycholog y based game theoretic approach is proposed in [20] to attract prosumers to participate in P2P energy trading. More surve y of P2P energy trading schemes can also be f ound in [21] and [22]. As can be seen from the above discussion, studies relat ed to P2P energy trading are plentiful and have significantly contributed to prosumer-focused energy management. Ho we ver , as identified in [5], integrating the P2P ener gy trading mechanism into the current energy polic y requires P2P trading to fit and complemen t the existing energy system. Otherwise, P2P trading markets would be difficult to implement in grid-connected systems. One potential way t o establish the suitability of a P 2P energy trading scheme to i mplement in the traditional energ y system is to sho w ho w such a scheme can benefit the centralized po wer system (CPS). For examp le, by effecti vely sup- porting the outdated electri cal grids at Br ooklyn, Brooklyn microgrid is activ ely working closely with utilities and is currently on it s way to being licensed as an energ y retailer [5]. Nonetheless, such benefits of P 2P ener gy trading to the traditional po wer system are yet to be established. T o this end, this pape r proposes a comprehensi v e analytical frame work of a dynamic price based P2P energy trading scheme that can help the CPS reducing its cost of energy production and supply to t he prosumers at the peak demand period. In particular , we propose a cooperativ e Stackelber g game that comprises a C PS as t he leader and the prosumers as followers. At the peak hour , the CPS strategically chooses the sell ing price per unit of energ y such that supply of energy to prosumers (consequently , the cost of excess generation or reserve) reduces to zero. In response to the choice of price made by the CPS, prosumers, as follo wers of the game, use the double auction to participate in a coalition formation game. The objective of the followers is to form suitable coalitions, based on their submitted bids, and participate in P2P energ y trading with the neighboring peers of the same coalition with the purpose of meeting the demand of energy without interacting wit h the CPS. W e analyze various properties of the resulting game. In particular , it is sho wn that, due to the strategy proof property of the auction and stability of the formed coaliti ons, the proposed game possesses a unique cooperati ve Stackelber g equilibrium. W e deriv e a closed- form pricing function for the CPS and propose an algorithm that the CPS and prosumers can use to reach the solution of the game. Further , w e sho w that the resultant solution is also prosumer-centric. Using numerical simulation, we assess the properties of t he proposed scheme. W e stress t hat the Stackelberg game has been extensi vely used in the literature such as in [23]–[25] and [26] for designin g energy trading schemes for the smart grid. Ho we ver , in these studies, the fr amework of a noncooperati ve Stackelberg game is used, in which foll owers, in response to the l eader’ s strategy , participate in a noncoop erativ e game such as a N ash game, G eneralized Nash game, or best response strate gies. In this paper , on the other hand, follo wers’ responses are captured via a coalition formation game considering its capacity to dynamically consider the i mpact of en vironmental changes on prosumer’ s decision on cooperati ve interaction wi th one another for P2P trading. Thus, the proposed game is a cooperativ e Stackelber g game. Conse quently , the studied properties, the design of the algorithm, and the resulting solutions are substantially different than the existing studies. W e further note that price-based coordi- nation of distri buted energy resources to support the grid is also a topic of interest in the lit erature related t o virtual po wer plant (VPP). Ho wev er , the coordination strategies of D E Rs for VPP and P2P are differe nt from one another [19]. The rest of the paper is organized as follows. The problem is formulated in Secti on II and an analytical framew ork of the cooper - ativ e S tackelber g game is proposed in Section III. The properties of the game is studied in Section IV. W e provide some results from numerical case studies in S ecti on V, which is followed by some concluding remarks in Section VI. I I . P R O B L E M F O R M U L A T I O N T o formulate the problem, we assume an energy network t hat consists of a CP S and a large number of customers. Among t he customers, N are prosumers, where N = |N | and N is the set of all prosumers. In this paper , customers refer to energy entit ies in t he network that buy en ergy from the CPS. Prosumers, on the other han d, hav e energy producing capability , can participate in P2P t r ading, and sell its energ y to the CPS or any other third party . Both the CPS and prosumers hav e two-way communication and po wer flow facilities [27] and are parts of the P2P network through a blockcha in based platform [5]. Each prosumer n ∈ N is a rational individua l, which is equipped with a rooftop solar panel wi th or without a battery i n a grid- connected system. At any time t , a prosumer n meets its energy demand e n,d ( t ) from its o wn solar system and, if there is any surplus after consumption and storage, i t can sell the excess energ y either t o the grid or t o other energy entities within t he system at the next time slot. Similarly , if a prosumer determines an deficiency at the end of t , it can buy its required energy either from t he grid or f rom other entities within the system at t + 1 . The CP S, on the other han d, alw ays need s to meet the total demand E S ( t ) of its customers. If the total demand E S ( t ) of the customers is very high, for example, during peak hours, the network could be ov erloaded and the CP S might need to start a new generation unit or alway s maintain reserve to meet the extra demand of its customers. This increases t he cost to the CPS significantly [28]. T o this end, we assume that the CP S has a contract wi t h N prosumers, in which the CPS may instruct the prosumers t o not demand any energ y from the CPS at the peak hour when their total demand E D ( t ) is higher t han a threshold E T ( t ) . Here, E T ( t ) is set to ensure that the total demand E S ( t ) from the customer do not exceed the supply E G ( t ) from the CPS. Now , E S ( t ) can be defined as E S ( t ) = E D ( t ) + E O ( t ) , (1) where, E O ( t ) is the total demand of the customers exclud ing t he prosumers in set N . No w , when E D ( t ) > E T ( t ) , where the value of E T ( t ) : E S ( t ) ≤ E G ( t ) can be determined through suitable artificial intelligence technique, such as in [29], based on the information of E O ( t ) , E D ( t ) and E G ( t ) , the CPS sends a very high pricing signal to t he prosumers under contract to encourage them to manage their o wn energy among themselves without getting any supply from the CPS at t . The objecti ve of the CPS is to make E D ( t ) = 0 through its choice of price p g,s ( t ) per unit of energy for t he selected prosumers at the respecti ve peak demand time slot. Thus, on the one hand, the CPS meets the deman d of its other customers w i thout raising the cost of production and supply of energy significantly . On the other hand, this enables the prosumers to participate in P2P trading among one another to meet their demands of that particular time period. 3 Here, it is important to note that prosumers can participate in P2P trading at any t ime, as studied in [12], [20]. Ho we ver , in this paper , the main objectiv e is to show ho w P 2P trading can be used as a tool to help the CPS to reduce its cost of oversupply of ener gy during periods of peak demand. Therefore, we assume t hat prosumers only participate in P2P energy trading when t he grid price is very high at the peak hours. Otherwise, the prosumers exchange their energy with the CPS as a rooftop solar o wner in a traditional energ y market. Now , to capture the benefit and cost of energ y trading to the prosumers and CPS respectiv ely , we define a suitable utility function for the prosumer and a cost function of the CPS in the follo wing sub sections. A. Utility Function of the Prosumer W e consider that the utility to a prosumer consists of two parts: 1) the utility of energy usage (consumption and supply), which can be captured by a logarithm function log( · ) as proposed in [30]. 2) The rev enue and cost of trading with another party (the CPS or another prosumer), which is a function of the traded energy amount and the price per unit of energ y . T o t his end, we propose to use t he follo wing two functions (2) and (3) U n,s ( t ) = α n ( t ) log 2 (1 + e n,g ( t ) + e n,p ( t )) + p g,b ( t ) e n,g ( t ) + p p 2 p ( t ) e n,p ( t ) (2) and U n,b ( t ) = α n ( t ) log 2 (1 + e n,g ( t ) + e n,p ( t )) − p g,s ( t ) e n,g ( t ) − p p 2 p ( t ) e n,p ( t ) (3) to capture t he utilities U n,s ( t ) and U n,b ( t ) that the prosumer n obtains when it sells and buys energy respectiv ely . Here, α n ( t ) is a preference parameter of the prosumer n that captures the satisfaction lev el of the prosumer for using a unit of energy at t , and e n,g ( t ) and e n,p ( t ) are the energy amount that the prosumer n tr ades with the CPS and other peers respectiv ely . p g,s ( t ) , p g,b ( t ) and p p 2 p ( t ) are the grid’ s selling price, grid’ s buying price and the P2P trading price respecti vely . Clearly , as the system is defined, when a prosume r is participating in energy trading with the CPS, it does not perform P2P trading with another prosume r , i.e., e n,p ( t ) = 0 , when e n,g ( t ) > 0 , and vice versa. No w , when e n,g ( t ) > 0 , the utility to a prosumer n for buying its energy from t he C PS under such conditions can be expressed as U n,b ( t ) = α n ( t ) log 2 (1 + e n,g ( t )) − p g,s ( t ) e n,g ( t ) . (4) From ( 4), the utility attains its maximum value when δU n,b ( t ) δe n,g ( t ) = 0 , and hence e n,g ( t ) = α n ( t ) p g,s ( t ) ln 2 − 1 . (5) Indeed, from (5), the amount of energy that a prosumer buys from the CPS is affected by the price set by the CPS. Hence, if the price is very high, prosumers may decide to l ook for alt ernate venues for their energy demand, which is the main aven ues that i s explored in this study . B. Cost Function of the CPS W e assume that there is a net cost J c ( t ) to the CPS when it trades i ts energy with the prosumers. J c ( t ) has a cost component and a rev enue component. The re venu e component p g,s ( t ) E D ( t ) is the rev enue that the CPS obtains by selling the total energy E D ( t ) to the prosumers at a price p g,s ( t ) per unit of energy . Meanwh ile, the cost component refers to the equi v alent cost t hat the CPS needs to pay for meeting the demand of prosumers beyond its predefined threshold, for example, for violating the network constraint or for starting an additional generator to meet the excess demand. Giv en this contex t, t he cost function of the CPS is proposed to be J c ( t ) = a ([ E D ( t ) − E T ( t )] + ) 2 + b [ E D ( t ) − E T ( t )] + − p g,s ( t ) E D ( t ) , (6) where [ · ] + = max( · , 0) and a, b > 0 . In (6), we note that the cost to CPS only occurs when the CPS needs to meet a total demand bey ond its defined threshold. T o minimize the cost in such cases, the CPS needs to choose a suitable price that would encourage prosumers to purchase their required energy from other alternativ e venues, instead of buying from t he CP S . As such, the C PS sets a suitable price p g,b ( t ) when E D ( t ) > E T ( t ) with the purpose of minimizing i ts cost. Thus, by setting δJ c ( t ) δE D ( t ) = 0 , we obtain the price p g,s ( t ) as p g,s ( t ) = 2 a ( E D ( t ) − E T ( t )) + b. (7) No w , based on the total demand E D ( t ) and the threshold E T ( t ) , the CPS can choose a suitable price by setting the v alues of a and b in order to alter the demand of the prosumers for secured an d su stainable operation of energy tr ading. Giv en this conte xt, follo wing [30], replacing the v alue e n,g ( t ) from (5) to ( 4) , and then taking the first deriv ati ve of U b,n ( t ) with respect to e n,g ( t ) , we find the maximum price p n, max that the prosumer n needs to pay the CPS for purchasing its required energy at the peak hour as p n, max = α n ( t ) ln 2 . (8) From (7) and (8), to influence a prosumer n not to buy any energy at the peak hour , the CPS needs to set its price p g,s ( t ) such that p g,s ( t ) > α ( t ) ln 2 , (9) where α ( t ) = max( α 1 ( t ) , α 2 ( t ) . . . . , α N ( t )) . Therefore, for a predefined value of a , to keep the total demand E D belo w E T , the CPS sets its pricing parameters b in (7) such that b > α ( t ) − 2 a ln 2( E D ( t ) − E T ( t )) ln 2 . (10) No w , once the price is set by the CPS, the prosumers require to respond in a way such that they do not need to rely on the CPS’ s energy , and at the same time, are capable of meeting their energ y demand. T o assist the CPS in reducing its peak demand through P2P , application of such a model wil l potentially be found in the future energy markets that will eit her engage prosumers with existing solar panels (and batteries) with the CPS under an Energy Performance Contract (EPC) [31] or recei ve in v estment from the local gov ernment or the CPS to i nstall solar panels ( and batteries) at the prosumers’ premises to receiv e such services (for example, see [32 ] ) . No w , to capture the decision making process of each prosumer , when E D ( t ) > E T ( t ) , we propose an ener gy management scheme based on a cooperati ve Stackelber g game in the next section. I I I . C O O P E R A T I V E S TAC K E L B E R G G A M E T o study the decision making process of the CPS and the partici- pating prosumers i n the proposed grid instructed P2P energ y trading scheme, we use the frame work of a Stackelber g game Γ , which is formally defined in its strategic form as Γ = { ( N ∪ CPS ) , U n ∈N ( t ) , J c ( t ) , e n,p ( t ) , p g,s ( t ) , p p 2 p ( t ) } . (11) In (11), Γ consists of the following components: i) The prosumers in set N that act as follo wers and cooperate with one another to choose their strategies in response to the price set by the CPS at the peak hour . 4 ii) U n ( t ) ∈ { U n,s ( t ) , U n,b ( t ) } is the utility of prosumer n for trading its energy at t , either as a seller or a buyer , with other prosumers and the CPS. iii) J c ( t ) is the cost to the CP S at t . iv ) e n,p ( t ) and p p 2 p ( t ) are the strategies, that is tr aded energy and the price per unit of traded energy in P2P market, of t he follo wers while participating in P2P trading. v) p g,s ( t ) is the strategy of the CP S when E D ( t ) > E T ( t ) . In the proposed Γ , as discussed pre vious ly , the objective of the CP S is to set a price that reduces its energy trading with the prosumers to zero when E D ( t ) > E T ( t ) . The aims of the prosumers with energy deficiency , on the other hand, are to buy the necessary energy from prosumers of the community with energy surplus via forming suitable coalitions for P 2P energy trading. A suitable solution of the proposed Γ is the cooperativ e Stackelberg equilibrium (CSE) . At the CSE, neither the CPS nor any prosumer will hav e any i ncenti ve to choose alternati ve str ategies to improve their benefits in terms of achie ved cost and utility respecti vely . Definition 1. Consider the game Γ defined in (11) , wher e U n,s ( t ) , U n,b ( t ) and J c ( t ) ar e determined by (2) , (3) and (6) r espectively . Now , a set of strate g ies ( p ∗ g,s ( t ) , e ∗ ( t ) , p ∗ p 2 p ( t )) constitutes a C SE of the pr oposed Γ , i f J c ( p ∗ g,s )( t ) = 0 (12) and the followers’ strate gies ( e ∗ , p ∗ p 2 p ) in r espon se to p ∗ g,b ( t ) estab- lish a D hp stable coalition structur e. Definition 2. A gr oup of coalitions is said to be in D hp stable if, at a given time slot, no player has an interest to split fr om one coa lition and form another new coalition for a better utility . A. Leader’ s Strate gy Based on our discussion in Section II-B and the utility of the CPS considered in (9), clearly the CSE strategy of the CPS is p ∗ g,s ( t ) = 2 a ∗ ( E D ( t ) − E T ( t )) + b ∗ . ( 13) Thus, at the peak hour , t o influence the prosumers to buy their required energy through participating in P2P trading, the CPS sets p g,s ( t ) = p ∗ g,s ( t ) , according to (13). In (13), the v alue of a ∗ and b ∗ can be obtained either based on a tabular database, in which the various pre-defined v alues of a ∗ and b ∗ are stored for dif ferent scenarios based on the historical cases, or by forming and ex ecuting an additional optimization technique that pro vides the v alues of a ∗ and b ∗ for t he considered scenario at t . Nonetheless, in both cases, the v alues of a ∗ and b ∗ will need to satisfy condition (10). No w , according to Definition 1, to reach the CSE solution, the follo wers need to find their str ategies to form a coalition structure in response to p ∗ g,s ( t ) , which is D hp stable. Giv en this context, we formulate a coalition formation game among the prosumers in the next section. B. F ollowers’ Response In response to the leader’ s strategy , prosumers, as the followers of Γ , participate in a coalition formation game to decide on their own strategies. Coa lition formation game is a branc h of cooperati ve game, in which the main purpo se is to an alyze the formation o f the coa lition structure through players’ interaction, and study its properties and its adaptability to the en vironmental v ariation [33]. It can be f ormally defined as a pair ( N , ν ) , where N is the set of all participating players and ν is the value of coalition. Essentially , ν is a real number assigned to ev ery coalition of prosumers S ⊂ N for P2P trading. In the proposed coalition formation game, a prosumer n can independe ntly choose other prosumers wi th whom it wants to form coalition to trade energy . T his decision is motiv ated by the individual benefit U n that a prosumer n can obtain by cooperating with the selected prosumers. Thus, in a coalition formation game, choosing a particular coalition i s based upon the i ndi vidual benefit t o the prosumer , r ather t han the coalition valu e ν . Of course, a prosumer may also choose not to participate in the P2P trading, rather buy its energy from the CPS despite the price p ∗ g,s ( t ) . Such behaviour is referred to as the non-cooperati ve behavior of the prosumer i n this paper . Nonetheless, to determine how a prosumer chooses other neighboring prosumers to form a coalition in order to participate in P2P trading, we assume that prosumers adopt a double auction scheme, as described in the following section. 1) Double auction between pr osumer s: A double auction market in vo lves a number of buyers and sellers t hat interact with o ne another with their reserv ation prices and bids to decide 1) the price of P2P trading and 2) how much energy each prosumer can trade with one another . Thus, due to the interactive and independent nature of the decision making process of each participant, a double auction i s chosen f or the proposed P2P energ y trading between the prosumers. In general, the proposed auction scheme consists of three key elements: • Seller: The prosumers in S ⊂ N t hat hav e surplus energ y to sell. • Buyer: The prosumers in B ⊂ N that need to buy energy to meet their needs. • Auctioneer: An auctioneer could either be a third party such as the distri bution system operator (DSO) that can participate in the trading through a digital trading platform such as the Elecbay , as proposed in [11], or an automated and secured information system such as the blockchain [1], [34], which is able to accurately decide the auction price and trading energy amount follo wing some well defined rules based on the information provid ed by all prosumers. W e assume that the auction between prosumers are conducted fol- lo wing two steps. These steps are explained as follo ws. Step 1- Determination of auction price and participating pro- sumers: In this step of the proposed auction process, the auctioneer determines the number of prosumers that intend to form a coalition group t o trade their energy based on the outcomes determined by the auction process. T o do so, fi rst, at each time slot, each sell er n ∈ S submits its reservation price p n,s and energy E n,s that it is interested to sell to the auctioneer . S i milarly , each buy er n ∈ B submits i t s bidding price p n,b and ener gy E n,b that it is interested to buy from the auctioneer . S econd, once all the ordered i nformation is receiv ed by the auctioneer , it arranges the reservation prices in ascending order and the bidding prices in descending order [35]. That is, p n,s , ∀ n ∈ S and p n,b , ∀ n ∈ B are ordered without loss of generality as p 1 ,s < p 2 ,s < . . . < p S,s and p 1 ,b > p 2 ,b > . . . > p B,b (14) respecti vely , where B = |B | and S = |S | . Third, the auctioneer generates the aggregated supply and demand curves using (14) to determine the intersection point of the two curves, as shown in Fig. 1. Clearly , the price p max at the intersection point in Fig. 1 refers to the highest reservation price for the sell ers. No w , once p max is determined, there are differen t mechanisms avail- 5 10 12 14 16 18 20 22 24 26 28 30 Price per unit of energy (cents/kWh) 5 10 15 20 25 30 35 40 45 50 Energy to be traded (kWh) Seller Buyer !"#$%&%'()*)(+",$ -.'/($0) 1$02()3 /($0) 4&3)(*'/"(,$0$/",$.5'$.' 676',8(-&58'"&0,$-. 9)::)(*'/"(,$0$/",$.5'$.' 676',8(-&58'"&0,$-. 676',(";$.5'/($0)'(".5 ) ! "#$ Fig. 1: A demonstr ation of ho w t he auction price is determined i n the propos ed scheme. able in the literature to decide t he auction price 1 . For example, in the V ickre y auction mechan ism [37 ], the auction price (also kno wn as V ickre y price p vic ) is considered to be the second highest reservation price. In [27], the authors propose a Stackelber g game and show that the auction price corresponds to the trading price at the S tackelber g equilibrium. Nevertheless, without loss of generality , in this work, the auction price p auc is assumed t o be as same as p max , that is, p auc = p max . Once the auction price p auc is calculated, then determine the number of buy ers |B a | = B a ≤ B and sellers |S a | = S a ≤ S that will participate in P2P trading based on the auction price. For examp le, according to Fig. 1, clearly all t he prosumers ∀ n ∈ B a ∪ S a that satisfy t he condition { p auc : p n,s ≥ p n,b } will trade their energy among themselves using P2P trading. Step 2- Allocation of ener gy: Once p auc is determined, the amount of energy e n,s that each seller n ∈ { 1 , 2 , . . . , S a } sells in the auction market is influenced by the demand of each buyer m ∈ { 1 , 2 , . . . , B a } . In particular , e n,s = ( E n,s if P S a n =1 E n,s ≤ P B a n =1 E n,b ( E n,s − η n ) + if P S a n =1 E n,s > P B a n =1 E n,b . (15) Clearly , each seller prosumer can sell their reserved energ y to the buye rs when the deficiency is at l east equal to the surplus. If the surplus is higher , howe v er , each seller n experiences a burde n of η n that they cannot sell to the buyers. There could be different ways to determine the burden to each sell er. Examples of such techniques include proportionate distribution and equal distribution . In this study , we propose to use the equal distribution scheme for deciding the burde n of each p rosumer . This is due to the fact that equal distrib ution has been pro ven to be more suitable to deliv er a strategy proof auction mechanism [38]. T o this end, we define η n as η n = 1 S a S a X n =1 E n,s − B a X n =1 E n,b ! . (16) 1 Note that while such dynamic pricing is not being used for residential elect ricity rates in most parts of the world at present, with the emerge nce of P2P energy trading, it is being used in ex isting P2P litera ture [27] and pilot trials [36]. !"#$%&'(&$)$%*# +$,,$%&'(&$)$%*# -)('%./01 ')& ,/#$% 2134./%5$ 0& 6%171)* !"#$%!&'('%)#%*#+,%-./0,-#+!,('1'+!(0#')# 232#(,!4')5#6'!#!.1('%)# 7$%!&'('%)#8" 9"#$%!&'('%)#%*#+,%-./0,-#+!,('1'+!(0#')# 232#(,!4')5#:'(;%.(#!.1('%)# 7$%!&'('%)#3" <:'(1;')5#%*#+,%-./0,- #90(:00)# 1%!&'('%)-#!1,%--#4'** 0,0)(#('/0#-&%(- Fig. 2: This figure demonstra tes ho w a prosumer may choose diffe rent coalitions to participate in P2P trading at differ ent time slots of the day . 2) F ormation of differ ent coalition: As the auction is designed, clearly , prosumers, for which the condition { p auc : p n,s ≥ p n,b } is not satisfied, will not trade their energy using the auction price. Hence, we consider that the prosumers that cannot participate in P2P trading based on p auc form another coalition among themselves to trade their energy in a P2P fashion at the market clearing price. In fact, mid-market rate is currently being used in a diffe rent P 2P pilot project 2 [36]. In this work, we define the mid-market price selli ng p mid as a function of p auc and feed-in-tariff (F iT) price p FiT , that is p mid = p auc + p FiT 2 , ( 17) and the mid-market buying price as (1 + β ) p mid . Here, β p mid , β > 0 is the price that buy er needs to pay as a subscription fee for using the network for P2P trading [30 ]. Here, it i s important to note the mid- market price always brings lo wer bene fit to both sellers and buyers of the respective coalition compared t o trade wit h p auc . Further, unlike the proposed auction process, prosumers cannot directly choose the trading price under mid-market rate. Thus, it is r easonable to state that prosumers wil l alwa ys be interested to rebid when instructed by the CPS for P2P trading in the next available ti me slot . 3) Coalition formation algorithm: The formation of a coalition between different prosumers for P2P trading, as sho wn in Fig. 2, is influenced by a prosumer’ s decision to socially interact with one another for trading its energy in selected time slots (when p g,s = p ∗ g,s ), giv en t he fact that it cannot trade its energy with the grid. Therefore, in each time slot, when prosumers are instructed for P2P trading, first, a prosumer decides on an energy amount that it wants to trade in the auction market and put a reservation price for that. Second, based on all the reservation prices and submitted energ y amounts, auction price p auc is determined by the auctioneer follo wing the process e xplained i n Section III-B1 and subsequently t he members of the coalition B a ∪ S a that will trade energ y with one another based on p auc are confirmed. F inally , once B a ∪ S a is confirmed for the selected time slot, t he rest of t he prosumers N \ B a ∪ S a decides to form a second coalition to trade their energy amon g themselves at the mid-market pricing schemes. The detail of the algorithm is sho wn in Algorithm 1. T he assumption of an accurate forecast of total demand E D ( t ) in Algorithm 1 is motiv ated by the unprecedented performance improv ement demonstrated by the recently proposed state estimation algorithms [39]. Once t he coalition st r ucture is formed following Algorithm 1, the 2 For example, in the Horizon P2P-SmartT est community m icrogrid. 6 Algorithm 1: Algorithm to f orm a stable coalition structure for P2P trading. 1: Set a n and b n of each prosumer n . 2: for Each time t ∈ { 1 , 2 , . . . , T } do 3: The CPS accurately forecasts the total demand E T ( t ) from prosumers. 4: if E T ( t ) < E D ( t ) then 5: Coalition formation algorithm terminates. 6: else 7: The CPS sets p g,s ( t ) = p ∗ g,s ( t ) according to (13). 8: Prosumers receiv e p ∗ g,s ( t ) from the CPS . 9: Coalition formation algorithm 10: Each seller prosumer n ∈ S submits its bid { p n,s ( t ) , E n,s ( t ) } to the auctioneer . 11: Each buyer prosumer n ∈ B submits i ts bid { p n,b , E n,b } to the auctioneer . 12: The auctioneer determines p auc and consequen tly B a and S a follo wing Section III-B1. 13: The mid-market price p mid is calculated via (17). 14: The prosumers in B a ∪ S a form coalition 1 and the prosumers in N \ ( B a ∪ S a ) form coalition 2 for P2P trading with p auc and p mid respecti vely . 15: A stabl e coalition structure is formed at time slot t . 16: end if 17: end fo r prosumers t r ade their energy with one another within each respecti ve coalition. Prosumers can adopt any standard av ailable techniques, such as the technique proposed in [40] or a pool based auction, to choose partners for P2P trading within each coalition. For the coalition wi t h the auction price, the t raded energy amount by each prosumers is decided by (15). As for the coalition with mid-market price, t he P2P trading is completed following the process described in [20]. Remark 1. Indeed, it is possible that, at any selected t ime slot of P2P trading , the total deficiency is gr eater than the total surplus within a coalition and vice versa. In that case, the prosu mers with ener gy deficiency can either r eschedu le their activit i es to another time slot or trade the ener gy from a t hir d party , such as a lar ge neighborhood stora ge , at a differ ent third party price. I V . P R O P E RT I E S O F T H E S TAC K E L B E R G G A M E A. Existence of A Stable CSE T o study the properties of the proposed game, we note that the strategy chosen by the CPS in (13) delivers a unique outcome to the CPS for any v alue of b that satisfies (10). Thus, to determine whether the proposed Γ has a unique and st able CSE, it is suf ficient to only in ve stigate whether the coalition structure decided by the prosumers through the proposed auction process is D hp stable. No w , to determine the stability of the proposed coalition structure, first, we define the str ategy-proof property of an auction mechanism, which ensu res that n o prosume r can de viate or cheat from its re vealed strategy during the auction process without af fecting the str at egy of other prosumers. Finally , considering the strategy-proof property of the proposed auction mechanism, we prov e the st ability of the proposed coalition formation framew ork, which subsequently prov es the existenc e of t he CS E . Definition 3. An auction mechanism is said to be strate gy-pr oof , if the participants r ev eal their true stra te gies during the auction pr o cess and do not cheat and dev iate fro m their chosen strate gies during the trading . Theorem 1. The pr opose d auction mechanism followed by the pr osumer s to decide on their r espective coalition and subsequ ent P2P trading is strate gy-pr oof . Pr oof. Let us assume that the energ y amount t hat each prosumer n rev eal to trade via auction is E ∗ n,s and E ∗ n,b for a seller and buyer respecti vely . No w , according to (16) , if the total av ailable supply and demand are X n ∈S a E ∗ n,s and X n ∈B a E ∗ n,b respecti vely , the burden shared by each participating seller n ∈ S a is η ∗ n = 1 S a S a X n =1 E ∗ n,s − B a X n =1 E ∗ n,b ! . (18) No w , let us assume that one prosumer i , which is a seller , cheats during t he energy trading and trades E ′ n,s instead of E ∗ n,s . Then, the burde n at the solution is η ∗ n = 1 S a   X n ∈ S a ,n 6 = i E ∗ n,s + E ′ i ∈S a ,s − X n ∈ B a E ∗ n,b   , (19) which is impossible. This is due to the fact that, as the scheme is proposed, the burden , which is shared equally by all seller prosumers only possess the value η ∗ n only if all ∀ n ∈ S a stick to E ∗ n,s for trading, as they rev ealed during the auction. Similarly , by considering that one buy er j ∈ B a chooses to buy E ′ n,b instead of E ∗ n,b during the t r ading process, i t can be shown that the buyers also need to stick to their revea led strategies and cannot cheat i n the proposed auction. Hence, no prosumer would cheat and de viate f r om their chosen strategies without affecting others, and this subsequently proves the strategy-proof property of the proposed auction scheme. Theorem 2. At any given time slot, the network structur e or partitions r esulting f rom the followers respon se to the decision made by the CPS is D hp stable. Pr oof. T o prove this theorem, first, we note that t he prosumers that participate in the P2P trading via auction price are decided by the proposed auction mechanism as discussed in the previous section. Since the auction process is strategy-proof, as prov en in Theorem 1, no participating prosumers in the auction wo uld leave the coalition and become a part of another new coalition or act noncooperati vely during that particular time slot. Second, as the scheme is designed, prosumers that cannot partici- pate in the auction form another new coalition among themselves for P2P trading at t he mid-market price set by (17). Here it is important t o note that, indeed prosumers may also decide to act noncooperati v ely instead of forming another coalition or form multiple joint coalitions. Nonetheless, it is sho wn in [20] that at a mid market pricing for P2P trading, prosumers alway s benefit more by forming a single coalition with one another instead of acting noncooperati vely or forming multiple disjoint coalitions. Therefore, it is reasonable to consider that, as a rational entity , each prosumer in N \ ( S a ∪ B a ) will alw ays choose to form a coalition among t hemselves without acting noncoope rativ ely . Thus, at any given each time slot of P 2P energy trading, the coalition formation game among prosumers always results in two coalitions with set of players ( S a ∪ B a ) and N \ ( S a ∪ B a ) , in which no player has any incentiv e to leave its own coaliti on for a greater benefit. This establishes the fact that the network structure or partitions resulting fr om the followers response to the decision made by the CPS is D hp stable. 7 No w , based on the above discussion, clearly , in response to the unique decision made by the CPS in (13), the prosumers form a stable group of partitions for P 2P trading. Consequently , the following corollary holds true. Corollary 1. The pr opose d cooper ative Stack elber g game Γ between the CPS and the pr osumer s possesses a unique and stable CSE. B. Pro sumer-Centric Prop erty While the proposed Γ is prov en to be effectiv e for reducing the cost to the CPS to zero at the peak hours through P2P trading, it is also important that the scheme does not compromise the benefit to t he prosumers. Prosumer-centric can be defined as the properties of a technology that can potentially motiv ate people to participate in or accept the technology [20]. Motiv ation al psychology is an ideal way to demonstrate this property [ 41 ]. This is particularly due to the fact t hat a prosumer might not always be motiv ated through econo mic incenti ve only . Now , we prov ide a summary of motiv ational psycholog y properties and then show whether t he proposed scheme possesses these characteristics. T o this end, what follo ws is a summary of properties that a prosumer-centric scheme needs to satisfy: • T r ansitivity: Tran sitivity refers to the property that establishes the fact t hat only one decision of the prosumer would maximize its utili t y or minimize its cost compared to other alternativ e decisions. As a consequen ce, as a r ational prosumer , it makes the rational choice [42]. • Dominance: If an action chosen by a prosumer is better than another option in one state and at least as good in all other states, the dominant option is chosen by the prosumer [ 42]. • Rational-economic: If an action economically benefits a pro- sumer , the prosumer is most likely to take that action. That is, monetary incentiv e is the ke y motiv ator for practicing a technology [43]. • P ositive r einfor cemen t: The property of positiv e r einforcement assumes that an equi v alent positi ve outcome of the choice made by a prosumer would motiv ate the prosumer to choose the same action in the future to fulfill his relev ant objecti ve [44]. • Elaboration likelihood : According t o this property , a technique needs to hav e the capability t o communicate its benefit to the prosumers follo wing either a central path or a peripheral path [45]. The central path is assumed to be suitable when an individual cares about the issue and the peripheral path is more appropriate when the issue may be subjected to con ve y unfa vorab le thoughts due t o the ambiguity of the message. No w , based on the discussion in the previous section, the proposed P2P energy trading scheme clearly possesses the prosumer-centric property due to following reasons: • Once a prosumer receiv es a signal from the CPS to participate in P 2P trading, it can either cooperate with other prosumers in the network or it can continue t rade i ts energy with the CPS at a higher price. N ow , as stated in Corollary 1, the cooperation between prosumers results in a stable solution. Therefore, no prosumer would have any incentiv e to choose an alternativ e strategy other than P 2P trading, other than cooperate with one another . This satisfies the transitivity and dominance properties of the scheme. • Since the choice of a coalition and the trading prices produce a unique and stable CSE, an d due to the fact that the price to tr ade energy with the grid is very high, clearly trading in P2P energy network benefits the prosumers economically . Consequen tly , t he P2P trading satisfies the rational-economic property . 1 2 3 4 5 6 7 Different time slots 15 20 25 30 35 Demand of energy (kWh) 0 100 200 300 400 500 600 Price per unit of energy (cents/kWh) ! "# $%&'()$*'&+,&-+.-(/&#-$*&#-(&'(0+*('&#-/(,-"%' 123/$#+"*&"0&454&#/$'+ *.6 7,3$%&,(%%+*.&8/+9(& "0&#-(&:4; :4;< ,&8/+9(&'3/+*.& 454&#/$'+*. =>?@&#+)(,& -+.-(/ =5?A&#+)(,& -+.-(/ !-/(,-"%'&,(#&BC& #-(&:4; ! "# $%&'()$*'&"0& 8/",3)(/, Fig. 3: This figure demonstrates how the CPS sets selli ng prices per unit of ener gy at dif ferent time slots based on the t otal demand from prosumer s and the pre-d efined threshold, which influence prosumers to participate in P2P trading. • By demonstrating ho w the proposed P 2P energy t rading can benefit both the grid by reducing its peak demand and the pro- sumers by reducing their cost of energy purchase from the grid in each t ime slot of P2P trading, the proposed sch eme establishes a peripheral path t o communicate with the prosumers. Further , by exhib iting the same beneficial outcome for both the grid and prosumers in e very time slot of P2P trading, t he proposed energy trading sch eme also v alidates its positiv e reinforcement property . Thus, based on the abov e discussion, it is r easonable to state the follo wing Corollary . Corollary 2. The prop osed P2P ener gy trading scheme is a pr osumer -centric technique . V . C A S E S T U D I E S In this section, we sho w some results from numerical case studies to demonstrate ho w the proposed P2P energ y trading scheme benefits both the CPS and prosumers at the peak hour . For numerical case studies, we consider a r esidential network with 12 prosumers. Each prosumer is assumed to have a solar panel of 5 kWp. T he energy surplus and deficiency of each prosumer is randomly chosen fr om the range [2,9]. Howe v er , these values could be differen t for different generation and consumption patterns of prosumers at different loca- tions. T o trade the ener gy at each time slot 3 , the prosumers choose their bidding from the range [11,15], which is chosen such that the minimum bidd ing price is higher than the FiT price 10 cen ts per kWh and the highest bidding price is lo wer than the grid’ s selling price 28 cents per kWh at the off-peak hour . The residential solar data is provided by Redback T echnolog ies, Australia, which is a startup compan y based in Brisbane that provides smart in verter solutions to household s. A. Benefit to the CPS In order to sho w how the proposed approach benefits the C PS, we first show the strategy adopted by the CP S at dif ferent demand scenarios from it s contracted prosumers. In particular , in Fig. 3, the selling price per unit of energy chosen by the CPS is demonstrated at different selected time slots during a day . In this figure, we 3 Each time slot is assumed to have a duration of 30 minutes [11]. 8 2 4 6 8 10 12 14 16 18 20 22 Different time slots -1000 -500 0 500 1000 1500 2000 2500 Total cost to the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ig. 4: The strategy of the CPS reduc ing it s total cost when the total demand of prosumers is higher than the t hreshold. The ne gativ e cost refer s to the rev enue of the CPS from selling i ts energ y for the case when prosumers’ total demand is lower than the threshold. deliberately sho w only sev en time slots for clarity of illustration of ho w t he CP S sets it s price based on the demand and threshold at any gi ven time slot. According to Fig. 3, when the total demand of t he prosumers is belo w its threshold of that selected time slot, the grid sells its energy to t he prosumers at the general standard off peak rate of 28 cents per kWh. Ho wev er , as soon as the t otal demand becomes greater than the threshold, the CPS changes its trading price for the contracted prosumers to a higher value via (13). For exa mple, at time slot 3 and 4 the price per unit of energy i s set t o 19.6 times higher than the usual price, whereas at ti me slot 6, it is 12.5 times. In other instances, when the total demands from the prosumers is wit hin the limits, the CPS sells the en ergy at the usua l rate of 28 cents per kWh. According to the proposed scheme, such a choice of price by the CPS influences the prosumers to participate in P2P trading, instead of trading energ y with the CPS . In Fi g. 4, we show the cost to the CPS at dif ferent selected time slots of a day ( f rom 7 am to 5 pm). Based on this figure, • At times, when the total demand is l ower than the threshold, the CPS is able to make r evenu e by selling the electricity to the prosumers. For example, at time slots 1, 2, 5, 7–12, 14, 16–18, and 20–22 the rev enue to CPS is more than 500 cents. • At times, when the t otal demand is higher than the t hreshold, due to maintaining reserve and/or running ne w generation units, the cost to the CPS increases significantly . For instance, at time slot 6, the cost to the CP S i ncreases to around 600 cents f or meeting the demand of the prosumers. Si milar rises are also observ ed at t i me slots 4, 13, 15, and 19. • Ho wev er , for the proposed scheme, the price set by the CPS instructs the prosumers to participate in P 2P trading instead of buyin g the energy from the CPS. This subsequen tly reduces the cost to t he CPS to zero as no excess energy needs to be generated. B. Benefit to the pr osumers According to the proposed scheme, prosumers participate in P 2P energy trading only when they are influenced by the CPS with a very large electricity price signal. Ot herwise, they trade their energy wi t h the grid in usual manner . No w , t o sho w ho w participating in P2P ! " # $ % & ' ( ) !* !! !" # $ % & ' ( ) ! " !* !! !" +,-./01,.234562 171,892./,:;/. +,-./01,.234562 171,892<1=4>417>9 ?,-/:2! ?,-/:2" +@,54>4:@5 12472+"+2/.4782 5612@/>54-72:,4>1 +@,54>4:@5 12472+"+2/.4782 5612041 Fig. 5: Formation of coalitions by differe nt prosumers for P2P energy trading. energy scheme can help the prosumers, we sho w the results for a particular time slot. For ex ample, let us consider ti me slot 3 of Fig. 3 and Fig. 4. Clearly , due to the high price of the CP S , prosumers need to participate in P2P trading to balance t heir demand and surplus. No w , without a loss of generality and as sho wn in F ig. 5, we consider that prosumers 1, 2, 3, 4, 5 and 6 hav e surplus, whereas prosumers 7, 8, 9, 10, 11 and 12 need more energy to fulfil their demand. T o participate in P 2P trading, the prosumers adopt Algorithm 1 and based on the auction price, prosumers form t wo coalitions. Accord ing to Fig. 5, prosumers 3, 4, 5, 6, 7, 8 and 9 form coalition 1 to trade their energy at the auction price. T he rest of the prosumers, t hat is prosumers 1, 2, 10, 11 and 12, form a second coalition t o trade their energy at the mid-market rate, following the process explained in [20]. T ABLE I: Demonstration of seller prosumer s’ increase in their re ven ues for participating in P2P energy trading. !"##"$% &'(#)*)'+% ,"-"+."%/$'0% 1($*)2)1(*)+3%)+% 454%62"+*78% ,"-"+."%/$'0% 7"##)+3%*'%*9"%&4!% 62"+*78% ,"-"+."% )01$'-"0"+*% 6:8% !"#$%&'"()( *( +),-.( ./,0-( )1,-/( !"#$%&'"(+( *( 0+,02( )/,)0( )*,01( !"#$%&'"(0( *( 0+,02( )/,)0( )*,01( !"#$%&'"(1( *( 11,*+( +/,0-( .2,*2( !"#$%&'"(*( .( .-,00( .-,--( .,1( !"#$%&'"(.( .( )-,2)( )-,--( .,1( ;-"$(3"%$"-"+."%)01$'-"0"+*% 55% % Once the coalition is formed and the t rading of energy is taken place at respective coalitions, the sellers and buyers of energy hav e their o wn rev enue and costs. Now , we sho w ho w each prosumer benefits compared to t he case of t rading energy wi th the CPS or a third party (such as neighborhood storage) at the selected time slot. In T able I, we show the rev enue that t he seller prosumers in differe nt coalitions obtain for participating in P2P trading. B ased on T able I, following properties of the scheme can be summarized: 1) auction based P2P demonstrates greater r evenu e to the prosumer compared to the reven ues to the sellers trade t heir energy at the mid- market rate. T his property always encourages prosumers to rebid in the next av ailable t ime slot of P2P trading in order to get a better benefit; 2) it is always beneficial for the sellers to sell their energ y to other prosumers within the P2P network compared t o sell t hem to the CPS ; 3) for the considered system and selected time slot, each 9 prosumer achiev es a rev enue, which is 22% greater, on average, than the rev enue that it could have obtained by selling its surplus to the CPS. T ABLE II: Demonstration of buyer prosumer s’ saving s in their costs for participating in P2P energy trading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n T able II, we show the cost to each bu yer within each coalitions for participating in P2P trading. Clearly , P2P trading is the better option for prosumers to meet their demand. Otherwise, the very high price of the CPS 4 (Fig. 3) increases the cost to each buy er in both coalitions significantly . For example, if a prosumer decides to buy its required energy from the CPS despite the high price signal, i t s cost of buy ing the energy increases by about 97% compared to b uying the energy from the P2P energy trading market. Of course, as a rational entity , a prosumer may also decide to buy its energy from other alternativ e sources such as third party neighborhoo d storage [47], for examp le by participating in a non-cooperati ve game. In such a non- cooperati ve scenario, each prosumer acts as a rational player who can independe ntly decide on its strategy of whether it wants to buy its energy from the CPS or from the third party in response to the price signal sent by the CPS . Thus, the game framew ork can be referred to as a non-cooperati ve Stackelber g game, in which the CPS acts as the leader , which decides on its o wn strategy of selling price per unit of energy . Prosumers, on the other hand, act as followers, who decide on the venue from which it can buy its energy in response to the price set by the CPS. T he solution of the game is a non-cooperati v e Stackelber g equibrium, at which, due to the v ery high price of t he CPS, all prosumers start trading with the third party . Thus, both the CPS and prosumers reach an equilibrium state from where neither the CPS nor any prosumer want to de viate at the selected time slot. Ho wev er , the cost to the prosumers is still very high compared to the price in the P2P t rading market. In particular, as can be seen from T able II, participating non-cooperati v ely increases the cost to the prosumer by on av erage 50% com pared to the P2P ener gy trading market. Thus, when the CPS sends the high price signal to the prosumers, participating in the proposed P 2P energy trading scheme is the best option f or both sell ers and buy ers, instead of acting non- cooperati vely , considering t heir respecti ve re venu es and cost. T ABLE III: This t able sho ws ho w the change in t otal number of prosumer s impacts the total cost to the CPS and prosumers in P2P t rading. !"#$%&' ()' *&(+"#%&+' ,(-./' 0%#.10' 2(+-'-('234'567' ,(-./'2(+-'-(' *.&-898*.-81:'*&(+"#%&+' 567' 2(+-'&%0"9-8(1')(&' *.&-898*.-81:'81'3;3'567' !"#$%&'&% !"#$()#%&'&% !"#$%&'&% !"#$()#%&'&% 234' 3%&'3&(+"#%&' *+% ',% +% -.'/% ,.01% *+/.++% -.'/% *+.-'% *2% ,1% +% 2-.*'% 3.-,% 2'3.++% 2-.*'% -,.1,% '+% 1-% +% '-,.-1% *'.+2% *,/+.2+% '-,.-1% 0-.,'% '2% 10% +% '2/.'2% *'./0% *2+0.2+% '2/.'2% 23.0/% -+% 3'% +% /00.',% *1.0/% -3*+.++% /00.',% *'3.00% <=%&.:%'9(+-'&%0"9-8(1' ;>?@;?' AB@?>' ' Finally , in T able I I I, we show ho w an increase in the total number of prosumers impacts the total cost to the CPS and av erage cost 4 Example of such a high increa se in selli ng price can also be found in elect ricity spot markets in high rene wabl e penetrate d regions such as South Australia [46]. per participating prosumers. Based on t he results demonstrated in T able III, • The cost t o the CPS is alw ays 0 when prosumers participate in P2P in response to the C PS’ s price signal, which is consistent with the proposed scheme. Howe v er , if there is no participation in P2P , the cost to the CPS increases sub stantially as the number of prosumers increases, which is mainly due to the increase in total demand to the CPS . • The cost per prosumer 5 increases as the number of prosumers increases. Howe v er , due t o the higher selling price per unit of energy , the cost per prosumer is much higher for not participat- ing in P2P compared to the case when prosumers participate in P2P . On averag e, for all the considered prosumers number , the cost per prosumer shows a $61.58 reduction for participating in P2P for the considered parameter values in this case study . As for the CPS, the cost savings for t he proposed case is $285.25, on average, when prosumers follow the scheme proposed in this study . Thus, engag ing in P2P ene rgy trading by the prosumers following the proposed framew ork brings benefit to both the CP S and prosumers. Hence, the proposed scheme has the potential to attract the CPS and prosumers to participate in scenarios that are aligned with the assumptions outlined in the paper . Finally , we note that the comp utational complexity of the proposed algorithm falls within a category of that of a single leader multiple follo wer Stack elberg game, which has been sho wn to be reasonable in numerous studies such as in [24] and [25]. As suc h, the computational complex ity is feasible for adopting the proposed scheme. V I . C O N C L U S I O N In this paper , we have studied the analytical framew ork of a peer-to-pee r energy trading scheme that can help the energy grid to get rid of exces s demand from the prosumers at the peak hour , while, at the same t ime, confirmed a prosumer -centric solution. T o this end, we hav e proposed a coop erativ e Stackelberg game by assuming the cen tralized power system as the leader and prosu mers as follo wers. A closed form expression has been presented to capture the decision making process of the leader , whereas we hav e formulated a double-auction based coalition formation game to facilitate pro- sumers’ decisions of t he energy trading parameters. The properties of t he proposed cooperativ e S t ackelbe rg game have been studied and it has been shown that the game possesses a unique and stable Stackelber g equilibrium. Further , we hav e propose d an algorithm that enables the prosumers and the centralized po wer st ati on to reach the equilibrium of the game. Fi nally , we hav e presented some numerical case studies to sho w how the proposed scheme can ensure benefits to all participating energy entities in the P2P tr ading. A potential extension of the proposed work is the in vestiga tion of the impact of uncertainty of prosumers’ demand as well as their non- cooperati ve beha vior on the CP S’ s decision making process. 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