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Based on coupled networks, taking node load, capacity and load redistribution between two networks into consideration, we propose functionally identical coupled networks, which consist of two networks connected by interlinks. Functionally identical coupled networks are derived from the power grid of the United States, which consists of many independent grids. Many power transmission lines are planned to interconnect those grids and, therefore, the study of the robustness of functionally identical coupled networks becomes important. In this paper, we find that functionally identical coupled networks are more robust than single networks under random attack. By studying the effect of the broadness and average degree of the degree distribution on the robustness of the network, we find that a broader degree distribution and a higher average degree increase the robustness of functionally identical coupled networks under random failure. And SF-SF (two coupled scale-free networks) is more robust than ER-ER (two coupled random networks) or SF-ER (coupled random network and scale-free network). This research is useful to construct robust functionally identical coupled networks such as two cooperative power grids. [ABSTRACT FROM AUTHOR]