[Paper Review] Impacts of Grid Structure on PLL-Synchronization Stability of Converter-Integrated Power Systems

Linbin Huang, Huanhai Xin+2 more|arXiv (Cornell University)|Mar 13, 2019

Microgrid Control and Optimization18 citations

TL;DR

This paper investigates how power grid topology and network admittances affect the small-signal stability of multi-converter systems using phase-locked loops (PLLs). By applying Kron reduction to model the network and analyzing the resulting grounded Laplacian matrix, the authors show that the smallest eigenvalue of this matrix dominates the PLL-synchronization stability margin, enabling targeted stability improvements through network reconfiguration or PLL retuning.

ABSTRACT

Small-signal instability of grid-connected power converters may arise when the converters use a phase-locked loop (PLL) to synchronize with a weak grid. Commonly, this stability problem (referred as PLL-synchronization stability in this paper) was studied by employing a single-converter system connected to an infinite bus, which however, omits the impacts of power grid structure and the interactions among multiple converters. Motivated by this, we investigate how the grid structure affects PLL-synchronization stability of multi-converter systems. By using Kron reduction to eliminate the interior nodes, an equivalent reduced network is obtained which contains only the converter nodes. We explicitly show how the Kron-reduced multi-converter system can be decoupled into its modes. This modal representation allows us to demonstrate that the smallest eigenvalue of the grounded Laplacian matrix of the Kron-reduced network dominates the stability margin. We also carry out a sensitivity analysis of this smallest eigenvalue to explore how a perturbation in the original network affects the stability margin. On this basis, we provide guidelines on how to improve the PLL-synchronization stability of multi-converter systems by PLL-retuning, proper placement of converters or enhancing some weak connection in the network. Finally, we validate our findings with simulation results based on a 39-bus test system.

Motivation & Objective

• To understand how grid structure influences PLL-synchronization stability in multi-converter power systems, moving beyond single-converter infinite-bus models.
• To identify the physical and mathematical mechanisms through which network topology and admittance affect stability margins.
• To develop analytical guidelines for improving stability via converter placement, network reinforcement, or PLL retuning.
• To validate the theoretical findings using a 39-bus test system with nonlinear time-domain simulations.

Proposed method

• Applying Kron reduction to eliminate non-converter buses and obtain a reduced network model containing only converter nodes.
• Deriving a modal decomposition of the multi-converter system dynamics based on the Kron-reduced network, enabling decoupled analysis of individual modes.
• Formulating the stability margin as a function of the smallest eigenvalue of the grounded Laplacian matrix of the reduced network.
• Performing sensitivity analysis of this smallest eigenvalue with respect to changes in line susceptances to assess impact on stability.
• Using linearized small-signal models and eigenvalue analysis to quantify stability margins under varying network conditions.
• Validating theoretical predictions with time-domain simulations on a 39-bus system under various network perturbations.

Experimental results

Research questions

• RQ1 How does the topology and admittance distribution of the power grid affect the PLL-synchronization stability of multi-converter systems?
• RQ2 What is the dominant network-level factor governing the stability margin in a multi-converter PLL-based system?
• RQ3 How do perturbations in specific network branches influence the system's stability margin?
• RQ4 What are the practical design guidelines for improving PLL-synchronization stability through network reconfiguration or PLL parameter tuning?

Key findings

• The smallest eigenvalue of the grounded Laplacian matrix of the Kron-reduced network is the primary determinant of the PLL-synchronization stability margin in multi-converter systems.
• A decrease in the smallest eigenvalue below 2.25 leads to linear instability, as confirmed by simulation showing sustained oscillations when $ B_{32,39} < 30.95 $.
• Increasing the susceptance of a critical line, such as $ B_{6,9} $, from 0 to 50 increases the smallest eigenvalue from 3.3118 to 3.7393, significantly improving stability.
• The stability margin is highly sensitive to changes in certain weak or critical branches (e.g., $ B_{32,33} $), while largely insensitive to others (e.g., $ B_{17,18} $), as shown by simulation and sensitivity analysis.
• Time-domain simulations confirm that higher damping ratios and stable convergence occur when the smallest eigenvalue remains above the critical threshold.
• PLL bandwidth should be reduced proportionally to the decrease in the smallest eigenvalue to maintain stability, providing a practical tuning guideline.