Design and Analysis of Enhanced Phase-Locked Loop: Methods of Lyapunov and Natural Gradient

The phase-locked loop (PLL) plays a crucial role in modern power systems, primarily for estimating line voltage parameters and tracking variations needed to synchronize and control grid-connected power converters. The enhanced phase-locked loop (EPLL) builds upon the standard PLL by tracking sinusoidal signal amplitude. While EPLL has been extensively explored in experimental applications, the theoretical modeling of a modified EPLL has received limited attention. Originally introduced by M. Karimi-Ghartemani et al. in IEEE Trans. Instrum. Meas., 61(4):930-940, 2012, the modified EPLL has not been fully explored in certain areas. In this work, we first address the limitations in existing gradient- and Hessian-based EPLL designs by examining stationary points in their autonomous forms. We then introduce two new derivations of the modified EPLL for single-phase power systems, which incorporate synthesized quadrature components of the input signal. These derivations are based on Lyapunov stability theory and natural gradient optimization. We comprehensively analyze convergence and stability by employing averaging theory and Poincaré maps to establish stability limits for the filter's proportional and integral gains. Additionally, we show that the design and tuning of the EPLL can be simplified by managing all three core equations with a single control parameter. Simulation results confirm that, within the derived gain limits, the EPLL effectively tracks sudden changes in amplitude, phase, and frequency without inducing double-frequency effects.