  We use a power grid model with $M$ generators and $N$ consumption units to
optimize the grid and its control. Each consumer demand is drawn from a
predefined finite-size-support distribution, thus simulating the instantaneous
load fluctuations. Each generator has a maximum power capability. A generator
is not overloaded if the sum of the loads of consumers connected to a generator
does not exceed its maximum production. In the standard grid each consumer is
connected only to its designated generator, while we consider a more general
organization of the grid allowing each consumer to select one generator
depending on the load from a pre-defined consumer-dependent and sufficiently
small set of generators which can all serve the load. The model grid is
interconnected in a graph with loops, drawn from an ensemble of random
bipartite graphs, while each allowed configuration of loaded links represent a
set of graph covering trees. Losses, the reactive character of the grid and the
transmission-level connections between generators (and many other details
relevant to realistic power grid) are ignored in this proof-of-principles
study. We focus on the asymptotic limit and we show that the interconnects
allow significant expansion of the parameter domains for which the probability
of a generator overload is asymptotically zero. Our construction explores the
formal relation between the problem of grid optimization and the modern theory
of sparse graphical models. We also design heuristic algorithms that achieve
the asymptotically optimal selection of loaded links. We conclude discussing
the ability of this approach to include other effects, such as a more realistic
modeling of the power grid and related optimization and control algorithms.
