A Theory of Solvability for Lossless Power Flow Equations—Part II: Conditions for Radial Networks

This two-part paper details a theory of solvability for the power flow equations in lossless power networks. In Part I, we derived a new formulation of the lossless power flow equations, which we call the fixed-point power flow. The model is parameterized by several graph-theoretic matrices—the power network stiffness matrices—which quantify the internal coupling strength of the network. In Part II, we leverage the fixed-point power flow to study power flow solvability. For radial networks, we derive parametric conditions which guarantee the existence and uniqueness of a high-voltage power flow solution, and construct examples for which the conditions are also necessary. The conditions imply convergence of the fixed-point power flow iteration, and unify recent results on the solvability of decoupled power flow. These results directly generalize the textbook two-bus system results, and provide new insights into how the structure and parameters of the grid influence power flow solvability.