  We propose an optimization approach to design cost-effective electrical power
transmission networks. That is, we aim to select both the network structure and
the line conductances (line sizes) so as to optimize the trade-off between
network efficiency (low power dissipation within the transmission network) and
the cost to build the network. We begin with a convex optimization method based
on the paper ``Minimizing Effective Resistance of a Graph'' [Ghosh, Boyd \&
Saberi]. We show that this (DC) resistive network method can be adapted to the
context of AC power flow. However, that does not address the combinatorial
aspect of selecting network structure. We approach this problem as selecting a
subgraph within an over-complete network, posed as minimizing the (convex)
network power dissipation plus a non-convex cost on line conductances that
encourages sparse networks where many line conductances are set to zero. We
develop a heuristic approach to solve this non-convex optimization problem
using: (1) a continuation method to interpolate from the smooth, convex problem
to the (non-smooth, non-convex) combinatorial problem, (2) the
majorization-minimization algorithm to perform the necessary intermediate
smooth but non-convex optimization steps. Ultimately, this involves solving a
sequence of convex optimization problems in which we iteratively reweight a
linear cost on line conductances to fit the actual non-convex cost. Several
examples are presented which suggest that the overall method is a good
heuristic for network design. We also consider how to obtain sparse networks
that are still robust against failures of lines and/or generators.
