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Multi-Port Thévenin--Norton Equivalents With Internal Complex Power: A Circuit-Theoretic Framework
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DOI：10.1109/tcsi.2026.3683814.png)
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Abstract
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Classical Thévenin and Norton equivalents preserve terminal voltages and currents but do not preserve internal active and reactive power. This paper presents a circuit-theoretic framework for a power-aware multi-port equivalent that augments the affine port relation <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {V}=\boldsymbol {V}_{\mathrm {th}}-\boldsymbol {Z}_{\mathrm {th}}\boldsymbol {I}$ </tex-math></inline-formula> with an exact quadratic internal complex-power model <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{\mathrm {int}}(\boldsymbol {I})=S_{\mathrm {int,OC}}+\boldsymbol {I}^{H}\boldsymbol {K}\boldsymbol {I}+\boldsymbol {p}^{H}\boldsymbol {I}+\boldsymbol {I}^{H}\boldsymbol {q}$ </tex-math></inline-formula>. The construction is derived from modified nodal analysis (MNA) and applies to any chosen subset of internal elements at a fixed frequency. When the subset coincides with the full network, the formulation recovers known quadratic relations for total internal power; for proper subsets, the coefficients <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\boldsymbol {K},\boldsymbol {p},\boldsymbol {q})$ </tex-math></inline-formula> encode internal-power information not contained in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\boldsymbol {Z}_{\mathrm {th}},\boldsymbol {V}_{\mathrm {th}})$ </tex-math></inline-formula> alone. The framework generalizes the DC one-port power-equivalent results of Barbi and Corradini to arbitrary AC multi-port networks and admits a homogeneous quadratic representation in the augmented current variable. The approach is validated on a distribution feeder, where subset matrices are extracted for a line-loss element and a shunt capacitor bank, and the reduced model is used to solve a voltage-constrained port-current dispatch problem that minimizes monitored losses with minimum port-current effort.
Keywords:
Thévenin equivalents
Norton equivalents
multi-port networks
internal power
reactive power
modified nodal analysis
homogeneous circuit models
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