  In this paper, we study phase-amplitude multivariable dynamics in
converter-based power systems from a complex-frequency perspective. Complex
frequency represents the rate of change of voltage amplitude and phase angle by
its real and imaginary parts, respectively. This emerging notion is of
significance as it accommodates the multivariable characteristics of power
networks where active and reactive power are inherently coupled with both
voltage amplitude and phase. We propose the notion of complex-frequency
synchronization to study the phase-amplitude multivariable stability issue in a
power system with dispatchable virtual oscillator-controlled (dVOC) converters.
To achieve this, we separate the system into linear fast dynamics and
approximately linear slow dynamics. The linearity property makes it tractable
to analyze fast complex-frequency synchronization and slower voltage
stabilization. From the perspective of complex frequency and complex-frequency
synchronization, we provide novel insights into the equivalence of dVOC and
complex-power-frequency droop control, stability analysis methods, and
stability criteria. Our study offers a practical solution to address
challenging stability issues in converter-based power systems.
