Robust synchronization (phase locking) of power plants and consumers centrally underlies the stable operation of electric power grids. Despite current attempts to control large-scale networks, even their uncontrolled collective dynamics is not fully understood. Here we analyze conditions enabling self-organized synchronization in oscillator networks that serve as coarse-scale models for power grids, focusing on decentralizing power sources. Intriguingly, we find that whereas more decentralized grids become more sensitive to dynamical perturbations, they simultaneously become more robust to topological failures. Decentralizing power sources may thus facilitate the onset of synchronization in modern power grids.