The AC power flow equations describe the flow of energy through a synchronous grid, and are the foundation for all power systems planning, operations, optimization, and control. The existence (or non-existence) of solutions to these nonlinear equations is intimately related to stability of grid dynamics, network transmission limits, and the feasibility of optimal power flow. Despite decades of interest and investigation, the space of solutions remains poorly understood, and the theoretical basis for power flow solvability remains underdeveloped.
In this talk we present a theory of solvability for power flow equations in lossless networks. We begin by formulating a new model of power flow, termed the fixed-point power flow (FPPF), which is parameterized by three graph matrices quantifying the internal coupling strength of the network. The FPPF leads immediately to an explicit approximation of the high-voltage solution, which is found to be quite accurate in standard test cases. For acyclic networks, we leverage the FPPF to derive necessary and sufficient parametric conditions for the existence of a unique power flow solution. The conditions (i) imply exponential convergence of the FPPF iteration, (ii) natually generalize the classic textbook result for a two-bus system, and (iii) unify recent results in the research literature. We conclude with an application to power system stability, and comment on open problems.
John W. Simpson-Porco is an Assistant Professor of Electrical and Computer Engineering at the University of Waterloo. His research focuses on the control and optimization of complex dynamic networks, with an emphasis on modernized electric power grids. John received his B.Sc. degree in Engineering Physics from Queen's University in 2010, his PhD in Mechanical Engineering from the University of California, Santa Barbara in 2015, and was a visiting scientist at ETH Zurich in Fall 2015. He is a recipient of the Automatica Best Paper Prize and the Center for Control, Dynamical Systems and Computation Outstanding Scholar Fellowship and Best Thesis Award.