Title: Dynamic Complex-Frequency Control of Grid-Forming Converters

Abstract: Complex droop control, alternatively known as dispatchable virtual oscillator control (dVOC), stands out for its unique capabilities in synchronization and voltage stabilization among existing control strategies for grid-forming converters. Complex droop control leverages the novel concept of ``complex frequency'', thereby establishing a coupled connection between active and reactive power inputs and frequency and rate-of-change-of voltage outputs. However, its reliance on static droop gains limits its ability to exhibit crucial dynamic response behaviors required in future power systems. To address this limitation, this paper introduces \textit{dynamic complex-frequency control}, upgrading static droop gains with dynamic transfer functions to enhance the richness and flexibility in dynamic responses for frequency and voltage control. Unlike existing approaches, the complex-frequency control framework treats frequency and voltage dynamics collectively, ensuring small-signal stability for frequency synchronization and voltage stabilization simultaneously. The control framework is validated through detailed numerical case studies on the IEEE nine-bus system, also showcasing its applicability in multi-converter setups.