Hypergraph and Superhypergraph Approaches in Electronics: A Hierarchical Framework for Modeling Power-Grid Hypernetworks and Superhypernetworks

Abstract

Graphs are fundamental tools for modeling pairwise relationships in complex systems. However, many realworld infrastructures—such as energy systems—demand more expressive frameworks to represent multinode and hierarchical interactions. A hypergraph generalizes classical graphs by allowing edges, known as hyperedges, to connect multiple vertices simultaneously. Building on this, a superhypergraph introduces recursively nested powerset constructions, enabling the representation of layered and self-referential relationships among nodes and edges. In the context of network science, these extensions lead to hypernetworks and superhypernetworks, which generalize conventional networks to capture higher-order connectivity patterns. Graph theory is also used in electrical engineering. A power-grid network structurally represents an electrical system, where nodes correspond to generation stations, substations, or consumers, and edges represent physical transmission lines. This study presents rigorous mathematical definitions for two extended network structures specifically tailored to the energy domain: the Power-Grid HyperNetwork and the Power-Grid SuperHyperNetwork. We provide detailed examples, including the Four-Bus Power System and the Antenna System, to illustrate how these models can effectively capture the complexity of real-world power systems. Future research is expected to advance the computational analysis and practical applications of these models in smart grid optimization, simulation, and planning.