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Power grids, due to their lack of network redundancy and structural interdependence, are particularly vulnerable to cascading failures, a phenomenon where a few failed nodes—having their loads exceeding their capacities—can trigger a widespread collapse of all nodes. Here, we extend the cascading failure (Motter-Lai) model to a more realistic perspective, where each node’s load capacity is determined to be nonlinearly correlated with the node’s centrality. Our analysis encompasses a range of synthetic networks featuring small-world or scale-free properties, as well as real-world network configurations like the IEEE bus systems and the US power grid. We find that fine-tuning this nonlinear relationship can significantly enhance a network’s robustness against cascading failures when the network nodes are under attack. Additionally, the selection of initial nodes and the attack strategies also impact overall network robustness. Our findings offer valuable insights for improving the safety and resilience of power grids, bringing us closer to understanding cascading failures in a more realistic context.