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The widespread installation of distributed generation systems is crucial for making optimal use of renewable energy. However, local distribution networks face voltage fluctuation problems if numerous photovoltaic (PV) systems are connected. Recently, energy storage systems that can be installed at commercial customers have been developed. This paper proposes a concept that solves the voltage fluctuation problem in distribution networks with high penetration of PV systems by using customer-side energy storage systems. The distribution network operator (DNO) is allowed to control the output of the energy storage systems of customers during a specific time period in exchange for a subsidy covering a portion of the initial cost of the storage system. The cost effectiveness of the cooperative operation for both customer and DNO is discussed by numerical simulations based on minute-by-minute solar irradiation data. Our results have clarified the possibilities of making voltage management more economical in distribution networks.

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We define the cobordism group of Morse functions on manifolds and show that it is an infinite cyclic group for dimension two in the oriented case. We also give an explicit Morse function which gives a generator of the group. Introduction.The purpose of this paper is to determine the structure of the 2-dimensional oriented cobordism group of Morse functions.Let us recall a brief history of the cobordism theory of smooth maps. Thom [18] described the cobordism groups of embeddings in terms of homotopy groups of certain spaces, using the so-called Pontrjagin-Thom construction. Wells [19] defined the cobordism groups of immersions and studied them again by using a Pontrjagin-Thom type construction. Eliashberg [3] generalized Wells' result to cobordism groups of smooth maps satisfying certain di¤erential relations of order one. Cobordism groups of smooth maps with a given set of local and global singularities were introduced and studied by Rima ´nyi and Szu ˝cs [14], who showed that these groups are isomorphic to the homotopy groups of certain ''universal spaces'', where the isomorphisms are obtained again by a Pontrjagin-Thom type construction. Recall that they considered only the non-negative codimension case, i.e., the case where the dimension of the target is greater than or equal to that of the source.In this paper, as a simple but important example for the strictly negative codimension case, we study the cobordism group Mð2Þ of Morse functions on oriented surfaces and show that it is an infinite cyclic group, using a totally di¤erent method. Note that the group Mð2Þ corresponds to the 2-dimensional oriented cobordism group for fold singularities of codimension À1 in a sense similar to that of [14].Recently Ando [1], [2] studied the n-dimensional oriented cobordism group for fold singularities of codimension zero. Our method is totally di¤erent from Ando's and uses the notion of the Stein factorization. The second author [15], [16] studied the cobordism group of Morse functions with only minima and maxima as their critical points using the Stein factorizations, which are manifolds for such functions. Our situation admitting critical points of any index is more complicated, since the Stein factorizations are no longer manifolds.

We show that Rubinstein-Scharlemann graphics for 3-manifolds can be regarded as the images of the singular sets (: discriminant set) of stable maps from the 3-manifolds into the plane. As applications of our understanding of the graphic, we give a method for describing Heegaard surfaces in 3-manifolds by using arcs in the plane, and give an orbifold version of Rubinstein-Scharlemann's setting. Then by using this setting, we show that every genus one 1-bridge position of a nontrivial two bridge knot is obtained from a 2-bridge position in a standard manner.