Log(v) 3LPF: A Linear Power Flow Formulation for Unbalanced Three-Phase Distribution Systems

In this work, we introduce Log(v) 3LPF, a linear power flow solver for unbalanced three-phase distribution systems. Log(v) 3LPF uses a logarithmic transform of the voltage phasor to linearize the AC power flow equations around the balanced case. We incorporate the modeling of ZIP loads, transformers, capacitor banks, switches and their corresponding controls and express the network equations in matrix-vector form. With scalability in mind, special attention is given to the computation of the inverse of the system admittance matrix, Ybus. We use the Sherman-Morrison-Woodbury identity for an efficient computation of the inverse of a rank-k corrected matrix and compare the performance of this method with traditional LU decomposition methods using Big-\mathcal {O} notation. We showcase the solver for a variety of network sizes, ranging from tens to thousands of nodes, and compare the Log(v) 3LPF with commercial-grade software, such as OpenDSS.