In the recent years there has been a growing interest in electric systems which integrate communication, control, and sensing technologies to efficiently shape the electricity consumption also known as smartgrids. It has been shown that if such a system is organized into a network of interconnected microgrids there is a vast number of positive effects. The basic idea of this approach is to separate the electrical grid into smaller, highly independent subsections (microgrids). This approach has resulted in novel types of typologies for electrical grids and new aspects of such systems that should be considered. In this way many problems can be localized. For example when a high level of renewable energy sources are added to the system the fluctuations in the voltage and frequency that occur, can be to, a certain extent, isolated from the main grid. The new topology has made it crucial to optimize several important properties like the self-adequacy, reliability, supply-security and the potential for self-healing. Many of these practical issues can be modeled using graphs. Previous research has shown that the problem of the maximal partitioning of graphs with supply and demand (MPGSD) is closely related to electrical distribution systems, especially in the context of interconnected microgrids. The advantage of using this type of graph model is the possibility of solving large scale problems in reasonable time. The use of MPGSD is essential in analyzing the optimal division of the whole grid into microgrids. Here the term optimized is used for the case when there is a minimum of power exchange between the connected microgrids, which is generally referred to as the maximization of self-adequacy. Another important property of such interconnected systems is the fault resistance. In this work we present a new version of the MPGSD, suitable for maximizing failure resistance in such systems. To be exact we develop a model that attempts to maximize the self-adequacy but with an additional constraint of providing that the system maintains stability even in the case of some failures. To accomplish this, the original problem has been extended in two directions. The first one corresponds to fault tolerance in individual microgrids. In this case a new constraint is added that to each of the subgraphs (microgrids) must be Hamiltonian. By doing so it is guaranteed that no islanding will occur inside a microgrid even if some connections brake. The second adaptation is used to maximize the resistance of the entire distribution system to failures of entire microgrids. To be more precise we wish to guaranty that the system as a whole will be stable even if some of the microgrids fail. In practice we are minimizing the number of articulation points of the graph in which each of the subgraphs represents a node. For the proposed problem a mathematical model is developed that makes it possible to find optimal solution for small systems. These results are used to develop a heuristic method for finding near optimal solutions for large scale problem instances. We also explore the relation between the maximization of failure resistance on the level of individual microgrids and the whole system.


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