A new min‐cut problem with application to electric power network partitioning

Abstract

SUMMARYThe problem of partitioning a graph into two or more subgraphs that satisfies certain conditions is encountered in many different domains. Accordingly, graph partitioning problem has been studied extensively in the last 50 years. The most celebrated result among this class of problems is the max flow ¼ min cut theorem due to Ford and Fulkerson. Utilizing the modifications suggested by Edmonds and Karp, it is well known that the minimum capacity cut in the directed graph with edge weights can be computed in polynomial time. If the partition divides the node set V into subsets V 1 and V 2 , where V 1 contains one of the specified nodes s and V 2 contains the other specified node t, the capacity of a cut is defined as the sum of the edge weights going from V 1 to V 2 . In electrical power distribution networks, a slow-coherency-based islanding strategy is used as a prevention against the cascading failures. In this paper, we concentrate on the graph partition problems which are encountered in electric power distribution networks. In this environment, two different definitions of capacity of a cut are used. In the first definition, capacity of a cut is taken to be the difference of the edge weights going from V 1 to V 2 and from V 2 to V 1 . In the second definition, the capacity of a cut is taken to be the maximum of sum of the edge weights going from V 1 to V 2 and from V 2 to V 1 . Surprisingly, with slight change of the definition of the capacity of a cut, the computational complexity of the problem changes significantly. In this paper, we show that with the new definitions of the capacity of a cut, the minimum cut computation problem becomes NP-complete. We provide an optimal solution to the problems using mathematical programming techniques. In addition, we also provide heuristic solutions and compare the performance with that of the optimal solution.

A Tabu Search Algorithm for the Power System Islanding Problem

“…Therefore, they proposed a new islanding algorithm based on slow coherency [22] and angle-modulated particle swarm optimization (AMPSO) [23] for the large-scale power systems with up to 118 nodes. The work in [24] formulated the power network partitioning problem as the min-cut problems, where they used a new definition for the capacity of a cut, different from the traditional one. They solved their problems by simple heuristics that are built on the modified version of the approximation algorithm for the node-weighted Steiner tree problem [25].…”

“…The power system islanding problem is a graph partitioning problem that has to abide by the following two types of constraints. The first constraint is called the connectivity constraint (CC) [24], which requires that any pair of nodes in each resultant island is linked by at least one path. The second constraint is the generator coherency constraint (GCC).…”

“…We can get the initial solution for the 30-node power system as shown in Figure 7, where the boundary nodes are {4, 6, 7, 10, 17, 19, 20, 22, 24, 25}. In this solution, the transmission lines tripped are {(6, 7), (2,6), (4,6), (17,10), (19,20), (22,24), (24,25)}. …”

Two-Step Spectral Clustering Controlled Islanding Algorithm

“…There is no known algorithm that can efficiently solve problems of this type within polynomial time [1], [9].…”

“…Most existing algorithms overcome this challenge by using heuristic search methods, or by only solving the problem for a simplified network model, or a select subset of the original power system [1]- [9]. For example, when using an OBDD based method in online applications, the network model should be simplified to contain less than approximately 40 nodes [3], [11].…”