Local voltage control in distribution networks: Game and variational inequalities

Inverter-based voltage regulation for distribution systems is gaining importance as the penetration of distributed renewable energy resources increases, bringing about frequent and rapid supply and voltage fluctuations which traditional voltage regulation methods cannot handle. While previous works proposed and analyzed several inverter-based local volt/var control algorithms in linearized power flow model with line loss ignored, we are still interested in analytical characterization of these algorithms in frameworks without linear approximation. In this work, we considered a nonlinear dynamical system with nonincremental local Volt/VAr control, reverse engineer the nonlinear Volt/VAr dynamics as a voltage control game, and leveraged the fixed-point theorem and defined the equivalent variational inequalities (VI) problem. We made convex relaxation of the VI problems and characterize the equilibrium of dynamics through the solution set of VI problems. We give sufficient convergence conditions through showing contraction mapping. Numerical examples are provided to complement the analytical results.