Discrimination on isomorphism and equivalence of topologies of power electronics converters

In the process of deriving topologies for power electronic converters, researchers often use topology equivalence or topology isomorphism to identify topologies with different structures but identical performance, thereby avoiding redundant studies. However, the connotations of topology equivalence and topology isomorphism differ significantly, which can easily lead to confusion. To address this issue, this paper aims to clarify the distinctions and connections between the two concepts and propose a method for accurately identifying topologies with identical performance. First, it is deduced that a necessary condition for topology isomorphism is that their adjacency matrices have the same determinant. Furthermore, it is derived that the necessary and sufficient condition for topology equivalence is that the components are identical and the fundamental loops correspond one-to-one. Subsequently, an analysis from the perspective of topology subgraphs reveals that topology isomorphism is a sufficient but not necessary condition for topology equivalence, while topology equivalence is a necessary and sufficient condition for identical performance. Based on this, the paper recommends prioritizing topology equivalence over topology isomorphism in practical applications to identify topologies with identical performance. The theoretical results are validated through case studies demonstrating their correctness and feasibility. Additionally, this paper proposes a method based on the depth-first search algorithm to automatically determine equivalent topologies, enabling the rapid and accurate identification of converter topologies with identical performance.