TY - JOUR

T1 - Electricity tracing in systems with and without circulating flows

T2 - Physical insights and mathematical proofs

AU - Achayuthakan, Chira

AU - Dent, Chris J.

AU - Bialek, Janusz W.

AU - Ongsakul, Weerakorn

PY - 2010/5

Y1 - 2010/5

N2 - 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.

AB - 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.

KW - Power system economics

KW - Power transmission economics

UR - http://www.scopus.com/inward/record.url?scp=77951667182&partnerID=8YFLogxK

U2 - 10.1109/TPWRS.2009.2037506

DO - 10.1109/TPWRS.2009.2037506

M3 - Article

AN - SCOPUS:77951667182

VL - 25

SP - 1078

EP - 1087

JO - IEEE Transactions on Power Systems

JF - IEEE Transactions on Power Systems

SN - 0885-8950

IS - 2

M1 - 5418852

ER -