Performance Limits and Tradeoffs of Power Systems Synchronization Recovery in Complex-Frequency Representation

Power-system synchronization studies mainly ask whether synchronization can be achieved, whereas a complementary question is how well post-disturbance synchronization recovery can be shaped. In inverter-dominated systems, this recovery involves coupled phase-angle and voltage-magnitude dynamics, for which complex frequency provides a unified representation. This paper establishes a conservation law for synchronization recovery in this representation. For a given system and its initial state, the logarithmic integral of the proposed recovery residual is fixed by the initial contraction rate and the right-half-plane (RHP) zero--pole structure. Hence, synchronization recovery performance cannot be improved uniformly over frequencies and variables unless the budget set by these terms changes; controller tuning that preserves them can only redistribute the residual response across frequencies. The conservation law applies to both linear and nonlinear recovery responses satisfying the mild analyticity conditions. It further yields finite-frequency peak--bandwidth limits and a lower bound on the time-domain integral of squared complex frequency recovery. A system-level extension recovers classical Bode-type sensitivity conservation as an LTI special case. Studies on a heterogeneous IEEE 39-bus system verify the theory.