This paper presents a convex optimization model for the power management of an active distribution network operating in islanded mode when there is a lack of generation. The proposed model minimizes the total unattended load considering the generation capacity and bounds of voltage and frequency. In this proposal, the distributed generation units consider inverse droops, commonly used for the primary control in grids with a high <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r$</tex> / <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x$</tex> ratio. The proposed model guarantees global optimum, uniqueness of the solution, and convergence of the algorithm.