Mathematical modelling and dynamic optimization of phase-locked loop systems using hybrid PSO-gradient descent approach

Abstract

This paper presents a novel mathematical framework for modelling and optimizing Phase-Locked Loop (PLL) dynamics in grid-connected systems using a hybrid optimization approach. The proposed model combines a state-space representation of PLL dynamics with an innovative dual-optimization algorithm integrating Particle Swarm Optimization (PSO) and Gradient Descent (GD). A comprehensive mathematical model is developed, incorporating the nonlinear dynamics of the PLL system through differential equations and transfer functions. The hybrid optimization framework is formulated as a constrained optimization problem, where PSO provides global search capabilities while GD ensures local convergence. Numerical simulations demonstrate the model's superior performance compared to conventional approaches including SRF-PLL, DDSRF-PLL, and MSOGI-PLL, achieving 40% faster convergence and maintaining phase tracking errors below 3 degrees during severe grid disturbances. The framework offers a systematic method for analyzing and optimizing dynamical systems in power electronics.