Differential Equations and Differential Inequalities Governing Power Flow in Multiconductor Systems

ABSTRACT

Differential equations and differential inequalities governing the electric power flowing along a multiconductor system with constant parameters, in stationary (DC) or in quasi‐stationary sinusoidal steady‐state (AC) regimes, are derived. A second‐order matrix ordinary differential equation (ODE), involving the power matrix and its adjoint (or transpose ), is obtained, and a fourth‐order autonomous ODE is subsequently derived for alone. In the stationary regime, the AC formulation reduces to a description of the total scalar power while in the AC regime, the active power is identified as the trace of the Hermitian part of . By passing to a modal basis, both and are shown to satisfy spectral differential inequalities—exact in the lossless or stationary cases and approximate in the weakly lossy AC regime. Systems with space‐dependent parameters are also considered. It is demonstrated that if the per‐unit‐length parameters are polynomials of maximum degree , then the power matrix satisfies a closed‐form ODE of order . Finally, it is shown that lower‐order differential inequalities for the power flow can be established by retaining terms associated with local energy densities, providing a hierarchical bounding method for non‐uniform multiconductor systems.