Minimal spanning tree and percolation on mosaics: graph theory and percolation

Abstract This paper proposes a minimal spanning tree (MST) algorithm to solve the networks’ reconfiguration problem in radial distribution systems (RDS). The paper focuses on power losses’ reduction by selecting the best radial configuration. The reconfiguration problem is a non-differentiable and highly combinatorial optimization problem. The proposed methodology is a deterministic Kruskal’s algorithm based on graph theory, which is appropriate for this application generating only a feasible radial topology. The proposed MST algorithm has been tested on an actual RDS, which has been split into subsystems.

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An upper bound for the probability of a union

Journal of Applied Probability ◽

10.1017/s0021900200104164 ◽

1976 ◽

Vol 13 (03) ◽

pp. 597-603 ◽

Cited By ~ 27

Author(s):

David Hunter

Keyword(s):

Graph Theory ◽

Upper Bound ◽

Spanning Tree ◽

Upper Bounds ◽

Minimal Spanning Tree ◽

Multivariate Normal ◽

Tail Probabilities ◽

Tree Algorithm ◽

New Family ◽

The Family

The problem of bounding P(∪ Ai ) given P(A i) and P(A i A j) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.

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An upper bound for the probability of a union

Journal of Applied Probability ◽

10.2307/3212481 ◽

1976 ◽

Vol 13 (3) ◽

pp. 597-603 ◽

Cited By ~ 179

Author(s):

David Hunter

Keyword(s):

Graph Theory ◽

Upper Bound ◽

Spanning Tree ◽

Upper Bounds ◽

Minimal Spanning Tree ◽

Multivariate Normal ◽

Tail Probabilities ◽

Tree Algorithm ◽

New Family ◽

The Family

The problem of bounding P(∪ Ai) given P(Ai) and P(AiAj) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.

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A Unified Proof for Finding a Minimal Spanning Tree

Proceedings of the 2020 3rd International Conference on Mathematics and Statistics ◽

10.1145/3409915.3409925 ◽

2020 ◽

Author(s):

Ray-Ming Chen

Keyword(s):

Spanning Tree ◽

Minimal Spanning Tree

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Optimal feeder reconfiguration in distributed generation environment under time-varying loading condition

SN Applied Sciences ◽

10.1007/s42452-021-04557-w ◽

2021 ◽

Vol 3 (6) ◽

Author(s):

Yanrenthung Odyuo ◽

Dipu Sarkar ◽

Lilika Sumi

Keyword(s):

Graph Theory ◽

Distributed Generation ◽

Spanning Tree ◽

Voltage Stability ◽

Electrical Network ◽

List Type ◽

Network Reconfiguration ◽

Time Varying ◽

Electrical Networks ◽

Tree Algorithm

Abstract The development and planning of optimal network reconfiguration strategies for electrical networks is greatly improved with proper application of graph theory techniques. This paper investigates the application of Kruskal's maximal spanning tree algorithm in finding the optimal radial networks for different loading scenarios from an interconnected meshed electrical network integrated with distributed generation (DG). The work is done with an objective to assess the prowess of Kruskal's algorithm to compute, obtain or derive an optimal radial network (optimal maximal spanning tree) that gives improved voltage stability and highest loss minimization from among all the possible radial networks obtainable from the DG-integrated mesh network for different time-varying loading scenarios. The proposed technique has been demonstrated on a multiple test systems considering time-varying load levels to investigate the performance and effectiveness of the suggested method. For interconnected electrical networks with the presence of distributed generation, it was found that application of Kruskal's algorithm quickly computes optimal radial configurations that gives the least amount of power losses and better voltage stability even under varying load conditions. Article Highlights Investigated network reconfiguration strategies for electrical networks with the presence of Distributed Generation for time-varying loading conditions. Investigated the application of graph theory techniques in electrical networks for developing and planning reconfiguration strategies. Applied Kruskal’s maximal spanning tree algorithm to obtain the optimal radial electrical networks for different loading scenarios from DG-integrated meshed electrical network.

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DEVELOPING PROXIMITY GRAPHS BY PHYSARUM POLYCEPHALUM: DOES THE PLASMODIUM FOLLOW THE TOUSSAINT HIERARCHY?

Parallel Processing Letters ◽

10.1142/s0129626409000109 ◽

2009 ◽

Vol 19 (01) ◽

pp. 105-127 ◽

Cited By ~ 87

Author(s):

ANDREW ADAMATZKY

Keyword(s):

Foraging Behavior ◽

Delaunay Triangulation ◽

Spanning Tree ◽

Physarum Polycephalum ◽

Minimum Spanning Tree ◽

Nearest Neighbour ◽

Minimal Spanning Tree ◽

Proximity Graphs ◽

Relative Neighborhood Graph ◽

Plasmodium Of Physarum Polycephalum

Plasmodium of Physarum polycephalum spans sources of nutrients and constructs varieties of protoplasmic networks during its foraging behavior. When the plasmodium is placed on a substrate populated with sources of nutrients, it spans the sources with protoplasmic network. The plasmodium optimizes the network to deliver efficiently the nutrients to all parts of its body. How exactly does the protoplasmic network unfold during the plasmodium's foraging behavior? What types of proximity graphs are approximated by the network? Does the plasmodium construct a minimal spanning tree first and then add additional protoplasmic veins to increase reliability and through-capacity of the network? We analyze a possibility that the plasmodium constructs a series of proximity graphs: nearest-neighbour graph (NNG), minimum spanning tree (MST), relative neighborhood graph (RNG), Gabriel graph (GG) and Delaunay triangulation (DT). The graphs can be arranged in the inclusion hierarchy (Toussaint hierarchy): NNG ⊆ MST ⊆ RNG ⊆ GG ⊆ DT . We aim to verify if graphs, where nodes are sources of nutrients and edges are protoplasmic tubes, appear in the development of the plasmodium in the order NNG → MST → RNG → GG → DT , corresponding to inclusion of the proximity graphs.

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