  We analyze a distributed approach for automatically reconfiguring
distribution systems into an operational radial network after a fault occurs by
creating an ordering in which switches automatically close upon detection of a
downstream fault. The switches' reconnection ordering significantly impacts the
expected time to reconnect under normal disruptions and thus affects
reliability metrics such as SAIDI and CAIDI, which are the basis for
regulator-imposed financial incentives for performance.
  We model the problem of finding a switch reconnection ordering that minimizes
SAIDI and the expected reconnection time as Minimum Reconnection Time (MRT),
which we show is a special case of the well-known minimum linear ordering
problem from the submodular optimization literature, and in particular the Min
Sum Set Cover problem (MSSC). We prove that MRT is also NP-hard.
  We generalize the kernel-based rounding approaches of Bansal et al. for Min
Sum Vertex Cover to give tight approximation guarantees for MSSC on c-uniform
hypergraphs for all c. For all instances of MSSC, our methods have a strictly
better approximation ratio guarantee than the best possible methods for general
MSSC.
  Finally, we consider optimizing multiple metrics simultaneously using local
search methods that also reconfigure the system's base tree to ensure fairness
in service disruptions and reconnection times and reduce energy loss. We
computationally validate our approach on the NREL SMART-DS Greensboro synthetic
urban-suburban network. We evaluate the performance of our reconfiguration
methods and show significant reductions compared to single-metric-based
optimizations.
