Electricity Tracing in Systems With and Without Circulating Flows: Physical Insights and Mathematical Proofs

This paper provides new insights into the electricity tracing methodology, by representing the inverted tracing upstream and downstream distribution matrices in the form of matrix power series and by applying linear algebra analysis. The n th matrix power represents the contribution of each node to power flows in the other nodes through paths of length exactly n in the digraph of flows. Such a representation proves the link between graph-based and linear equation-based approaches for electricity tracing. It also makes it possible to explain an earlier observation that circulating flows, which result in a cyclic directed graph of flows, can be detected by appearance of elements greater than one on the leading diagonal of the inverted tracing distribution matrices. Most importantly, for the first time a rigorous mathematical proof of the invertibility of the tracing distribution matrices is given, along with a proof of convergence for the matrix power series used in the paper; these proofs also allow an analysis of the conditioning of the tracing distribution matrices. Theoretical results are illustrated throughout using simple network examples.