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.
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