Carbon potential solution using bidirectional power and rank-adaptive pseudoinverse for transmission-distribution-microgrid systems

Nodal carbon potential quantifies the carbon emissions embodied in each unit of electricity at a bus. It links active-power flow, branch carbon flow, and load-side carbon responsibility. In a coordinated transmission-distribution-microgrid system, bidirectional power exchange causes the carbon intensity at the point of common coupling (PCC) to vary with operating conditions. Reverse flows and near-zero active-power-flow branches alter the effective carbon-flow paths and may render the nodal-carbon-potential equations ill-conditioned or rank deficient. This paper combines a state-dependent PCC boundary, effective propagation topology, and rank-based solution for coordinated transmission–distribution–microgrid systems. Time-series power-flow results are first used to identify the PCC exchange state and update the carbon intensity of a dynamic boundary virtual source. Branch directions are then corrected according to actual active-power flows, and near-zero-flow branches are removed to construct a directed effective carbon-flow topology. Finally, singular-value decomposition diagnoses equation degeneration and determines the effective rank, after a rank-adaptive Moore-Penrose pseudoinverse gives the nodal-carbon-potential solution. The proposed method updates the PCC carbon potential with the power-exchange state and the carbon-flow paths with actual branch flows. It also maintains numerical stability at degenerate operating points.