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Linear power flow formulations enable the integration of power flow constraints to linearly constrained power system optimization problems in transmission and distribution systems. Typical examples include the application of DC power flow and DistFlow to distribution system reconfiguration and restoration. While non-linear power flow equations and its associated optimal power flow have been developed in the literature, there is no existing linear power flow formulation for islanded droop-controlled microgrids to the best of the authors' knowledge. To address this need, this paper presents two linear power flow approximations for three-phase islanded droop-controlled microgrids. The first approximation was derived based on the current injection method and the second approximation was derived from the three-phase DistFlow formulation. The two approximations were verified using detailed time-domain simulations at steady-state in PSCAD with resulting mean errors of the nodal voltages being under 1%. To demonstrate the application of the proposed power flow formulations, the two approximations were expanded for integration in optimal power flow formulations for islanded droop-controlled microgrids to minimize cost. Unlike the existing non-linear models, the linear power flow formulations presented in this paper can be solved easily using non-iterative linear matrix operation and its optimal power flow extensions are linearly constrained.