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.
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