The scale-up of AI models for analyzing largescale power systems necessitates a thorough understanding of their scaling properties. Existing studies on these properties provide only partial insights, showing that loss function decreases predictably with increased model scales; yet no scaling law for power system AI models has been established, and model performance remains unpredictable due to “emergent abilities”. This study pioneers the discussion on the emergent abilities of graph neural network (GNN) for analyzing large-scale power systems, revealing that model performance improves dramatically once model scale exceeds a threshold. Furthermore, we introduce an empirical power-law formula to quantify the relationship between this threshold and the power system size. Our theory accurately predicts the threshold for the appearance of emergent ability in large-scale power systems, including a synthetic 10,000- bus and a real-world 19,402-bus systems.