current coalition. Hence, a merge or split decision by Pareto order will ensure that all the involved MGs agree on it (namely, merge-and-split proof). Because the MGs act as players of a cooperative game, we propose a coalition formation algorithm called Game Theoretic Coalition Formulation Strategy (GT-CFS) by exploiting the merge and split operations as shown in Alg. 1. First, in our envisioned algorithm, each MG could obtain information of others (e.g., position, neighbour MGs, and so on) by using the communication infrastruc- ture or communication technology of smart grid (i.e., smart meters). Second, the MGs will produce the power, meet the demands of the users, and decide to buy or sell the power. Third, the forming coalition stage starts when the merge process occurs as follows. Given a partition S ¼ f S 1 ; . . . S k g , each coalition S i 2 S will communicate to its neighbours. Using these negotiations, the coalitions will exchange the information with others. MGs want to find the best partners MGs to form coalitions, so as to get more profits (payoff). The rules of merge and split will help them to deal with it. The coalitions or MGs calculate their payoffs by employing eqs. (8) and (9), find that the payoffs of all of them will increase, and this is the Pareto order in eq. (12), if they can form a coalition. They will do it with the rule of the merge operation. For example, consider that there is a MG, which is able to sell power. Assume that the power loss between it and the MS is 0.2. If it can find a coalition, which needs power, and the power loss between them is 0.15, the MG will join the coalition. But during the next time interval, the surrounding circumstances of the MG may change, such as the coalition does not want to buy power from the MG, or there exists another coalition for the buyer such that the power loss is lower than that in the current coalition. Therefore, the MG will leave this coalition to find a new one so as to alleviate its power loss. For any MG, the decision of merge and split is a distributed operation, and it is not be affected by other MGs or the MS. Most importantly, a MG is able to make it individually by following Alg. 1. After the merge and split iterations, the network will compose of disjoint coalitions, and no coalitions may have any incentive to perform further merge or split operation. Upon such convergence, the MGs within each formed coalition will start its power transfer stage. In next section, a proof on the stability, convergence, and optimality of proposed GT-CFS algorithm is presented. 5 P ROOF OF S TABILITY , C ONVERGENCE , AND O PTIMALITY OF GT-CFS It is important to show that our proposal is stable regardless of the environmental changes in the grid. Furthermore, it is also important to prove that it converges to an optimal solution. We begin our proof by providing a definition of stability followed by two theorems. The proof of each of the theorems is provided separately.
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