Effects of Line Dynamics on Stability Margin to Hopf Bifurcation in Grid-Forming Inverters

TL;DR

This work addresses oscillatory instability in grid-forming inverters caused by Hopf bifurcations and investigates how line dynamics affect the stability margin. It develops a normal-vector-based analytical framework to compute the sensitivity of the Hopf boundary to system parameters, and separates parameters into uncontrollable and controllable sets to guide tuning. The key findings show that line dynamics uniformly reduce the stability margin, with generally small effects except for the voltage–reactive power droop gain $K_Q$, while the feedforward gain of the voltage control loop $K^F_{VC}$ is identified as the most effective parameter for improving robustness. The approach enables efficient margin estimation and provides actionable guidance for parameter tuning in GFM inverter networks, with potential extension to larger networks and dynamic-line modeling decisions depending on application needs.

Abstract

This paper studies the parameter sensitivity of grid-forming inverters to Hopf bifurcations to address oscillatory instability. An analytical expression for the sensitivity of the stability margin is derived based on the normal vector to the bifurcation hypersurface. We identify the most effective control parameters through comprehensive analysis. In particular, the impacts of line dynamics on the stability margin to Hopf bifurcation are investigated. The results indicate that the feedforward gain in the voltage control loop is the most effective parameter for enhancing the stability margin. Furthermore, it is observed that line dynamics introduce a uniform reduction in the stability margin across all parameters. However, the reduction is generally small for most parameters except for the voltage-reactive power droop gain, which shows a more pronounced effect.

Figures (7)

• Figure 1: Geometry of the bifurcation hypersurface, normal vectors, and the stability margin.
• Figure 2: GFM inverter connected to the grid.
• Figure 3: Detailed control architecture of the droop-based GFM inverter.
• Figure 4: Trajectories of the active power output ($p$) under various parameter values of $K^F_{VC}$ with static (upper figure) and dynamic (lower figure) lines. The dashed lines correspond to Hopf bifurcation, while the solid lines correspond to stable equilibria.
• Figure 5: Trajectories of the filter output voltage ($v_c$) under various parameter values of $K^F_{VC}$ with static (upper figure) and dynamic (lower figure) lines. The dashed lines correspond to Hopf bifurcation, while the solid lines correspond to stable equilibria.

Theorems & Definitions (1)

• Definition 1