Distinguishing between topological isomorphism and topological equivalence of power electronic converters

In the process of deducing the topology of power electronic converters, scholars often use topological equivalence or topological isomorphism to identify topologies with different structures but the same performance to avoid repeated research. However, the connotations of topological equivalence and topological isomorphism are not the same. This paper will clarify the method for accurately identifying topologies with the same performance by distinguishing the differences and connections between the two. First, it is derived that the necessary condition for topological isomorphism is that the determinants of their adjacency matrices are equal; then, it is derived that the necessary and sufficient conditions for topological equivalence are that their component composition is the same and their simple circuits correspond one to one; finally, the above two conditions are analyzed from the perspective of topological subgraphs, and it is found that topological isomorphism is a sufficient but not necessary condition for topological equivalence, and topological equivalence is a necessary and sufficient condition for the same topological performance. Therefore, in practice, equivalence rather than isomorphism should be used to identify topologies with the same performance. This paper verifies the correctness of the theoretical analysis through a case analysis. In addition, this paper also introduces a method for automatically determining equivalent topologies based on the depth-first search algorithm to help quickly and accurately identify converter topologies with the same performance.