Linear Time-Invariant State-Space PLL Modeling Based on Dynamic Harmonics Balance for Performance Analysis in Power Grid Applications

Phase-Locked Loops (PLLs) play an essential role in a variety of power grid applications and measurement devices. Unfortunately, PLL performance analysis and design are often hindered by their inherent nonlinear nature that cannot be adequately described by standard linearized models. To address these issues, this paper presents a two-step modeling approach. First, a Linear and Time-Periodic (LTP) model is applied. Then, a variant of the harmonic balance method is applied to turn the LTP model into a Linear Time-Invariant (LTI) one, whose smallsignal dynamics can be potentially used to analyze the system behavior of terms other than the fundamental one. The proposed methodology is applied to an advanced three-phase PLL scheme based on an Orthogonal System Generator (OSG) followed by a pair of Mixed Second- and Third-Order Generalized Integrators (MSTOGIs), which provide not only harmonics attenuation, but also DC offset cancellation and imbalance mitigation. Some simulation results under different testing conditions, confirm that the obtained model is very accurate, thus paving the way to optimal PLL design in different power grid applications using standard linear system tools.