Therefore, the power L ² i is more than the demand j Req i j ð L ² i 9 j Req i jÞ . L ² i is the solution of following quadratic equation: L i ¼ P i 0 þ j Req i j ¼ R i 0 L 2 i U 2 0 þ ±L i ± Req i : (4) For a given Req i , three possible solutions of eq. (4) exist, namely none (zero), one, and two solutions. Because we want to minimize the value of Req i , if eq. (4) has two roots, the smaller one is to be used. For the cases that eq. (4) has no solution, we assume that the root is the same as eq. (4) having a single root, which is L ² i ¼ ð 1 ± ± Þ U 2 0 2 R i 0 . Because in the noncooperative case each MG can be regarded as a coalition, the payoff of MG is equal to that of coalition. Thus, we are able to define the noncooperative payoff (utility) of each MG i as the total power loss due to the power transfer, as follows: u f i g ð Þ ¼ ± w 2 P i 0 ; (5) where w 2 is the price of a unit power in MS. Because the objective is to minimize u ðf i gÞ , the negative sign is able to convert the problem into a problem of seeking the maximum. 3.2 Cooperative Coalition Model In the remainder of this section, the cooperative coalition model is considered for managing the MGs acting as ‘‘buyers’’ and ‘‘sellers’’. Also, the functions of power loss and utility in the cooperative case along with how to form the coalitions are proposed. Toward the end of the section, the concept of ‘‘Shapley’’ function is presented. Besides exchanging power with the MS, the MGs can exchange power with others. Because the power loss during transmission among the neighbouring MGs are always less than that between the MS and a MG, the MGs Fig. 1. Construction of micro grids. WEI ET AL.: GAME THEORETIC COALITION FORMULATION STRATEGY 2309
can form cooperative groups, referred to as coalitions throughout this paper, to exchange power with others, so as to alleviate the power loss in the main smart grid and maximize their payoffs in eq. (5). Before formally studying the cooperative behaviour of the MGs, the framework of coalition game theory is firstly introduced in the work in . A coalition game is defined as a pair ðN ; v Þ . The game comprises three parts, namely the set of players N , the strategy of players, and the function v : 2 N ! R . In this game, v is a function that assigns for every coalition S ³ N a real number representing the total profits achieved by S . We divide any coalition S ³ N into two parts: the set of ‘‘sellers’’ denoted by S s & S and the set of ‘‘buyers’’ represented by S b & S . S s and S b satisfy that S s [ S b ¼ S . Therefore, for a MG i 2 S s , Req i 9 0 and it means that MG i wants to sell power to others. On the other hand, an arbitrary MG j 2 S b having Req j G 0 indicates that MG j wants to buy power from others. It is obvious that any coalition S ³ N should have at least one seller and one buyer.
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