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In many fields, including science, technology, engineering, and communication networks, the idea of domination is used. Double domination is a widely recognized idea of domination concept in graph theory. A prominent crucial factor in network architecture is integrity. A subset $S$ of $V(G)$ is called a double dominating set of $G$ if each vertex of $G$ is dominated at least twice by $S$. In this paper, an algorithm to compute the double domination integrity of graphs is given. Also, an application of double domination integrity in the optimal phasor measurement unit placement problem in a power system is given.