Adaptive Optimal PI Control for Grid-Forming Inverters With Provable Multitime-Scale Stability via Lyapunov-Constrained Learning

Modern inverter-based power systems require controllers that adapt to changing grid conditions while maintaining strict stability across nested control loops. Existing approaches face a fundamental tradeoff: classical proportional-integral (PI) controllers offer provable stability but are static, while learning-based methods can adapt but often violate the structural stability guarantees needed for safe operation, especially in systems with tightly coupled, multitime-scale dynamics. This work proposes a distributed adaptive control framework for grid-forming inverters that combines neural gain scheduling with provable Lyapunov stability across multiple time scales. Our approach reformulates nested Lyapunov stability conditions as explicit gain-ratio inequality constraints and embeds them into both: 1) an offline augmented-Lagrangian constrained optimization for base gain synthesis and 2) online neural schedulers that adapt controller parameters to changing operating conditions while preserving stability certificates. A key contribution is the introduction of per-converter dynamic adaptation states governed by Lyapunov-passivity constraints, enabling state-dependent gain scheduling within certified stability envelopes. Implemented within a physics-informed, differentiable simulation environment, controllers tuned with our method preserve stability under severe transients, including large load steps and setpoint changes, while improving closed-loop regulation. This framework retains the familiar PI control architecture while rigorously enforcing stability margins, offering a practical stability-certified learning paradigm for emerging power-electronic systems. The adaptive control performs algorithmic gain synthesis and scheduling as operating conditions change, implemented through learned neural schedulers operating at multiple timescales.