  Given a flow network with variable suppliers and fixed consumers, the minimax
flow problem consists in minimizing the maximum flow between nodes, subject to
flow conservation and capacity constraints. We solve this problem over acyclic
graphs in a distributed manner by showing that it can be recast as a consensus
problem between the maximum downstream flows, which we define here for the
first time. Additionally, we present a distributed algorithm to estimate these
quantities. Finally, exploiting our theoretical results, we design an online
distributed controller to prevent overcurrent in microgrids consisting of loads
and droop-controlled inverters. Our results are validated numerically on the
CIGRE benchmark microgrid.
