RMS-Based PLL Stability Limit Estimation Using Maximum Phase Error for Power System Planning in Weak Grids

The increasing interconnection of inverter-based resources (IBRs) with low short-circuit current has weakened grid strength, making phase-locked loops (PLLs) susceptible to instability due to accumulated phase-angle error under current limiting. This study defines such instability as IBR instability induced by reduced grid robustness and proposes a root-mean-square (RMS) model-based screening method. After fault clearance, the residual q-axis voltage observed by the PLL is treated as a disturbance signal and, using the PLL synchronization equations, is analyzed with a standard second-order formulation. The maximum phase angle at which synchronization fails is defined as θpeak, and the corresponding q-axis voltage is defined as Vq,crit. This value is then mapped to a screening metric Ppeak suitable for RMS-domain assessment. The proposed methodology is applied to the IEEE 39-bus test system: the stability boundary and Ppeak are obtained in Power System Simulator for Engineering (PSSE), and the results are validated through electromagnetic transient (EMT) simulations in PSCAD. The findings demonstrate that the RMS-based screening can effectively identify operating conditions that are prone to PLL instability in weak grids, providing a practical tool for planning and operation with high IBR penetration. This screening method supports power system planning for high-penetration inverter-based resources by identifying weak-grid locations that require EMT studies to ensure secure operation after grid faults.