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This paper presents fully parallel domain decomposition (DO) techniques for efficient simulation of large-scale linear circuits such as power grids. DD techniques that use non-overlapping and overlapping partitioning of power grids are described in this paper. Simulation results show that with the proposed parallel DD framework, existing linear circuit simulators can be extended to handle large-scale power grids. Results for circuits with more than four million nodes indicate that parallel DD with LU factorization is most suitable for power grid simulation. However, for densely connected power grids, parallel DD with additive Schwarz preconditioning offers maximum scalability and best performance.