  The dynamic response of power grids to small disturbances influences their
overall stability. This paper examines the effect of topology on the linear
time-invariant dynamics of electricity networks. The proposed framework
utilizes ${\cal H}_2$-norm based stability metrics to study the optimal
selection of transmission lines on existing networks as well as the topology
design of new networks. The design task is first posed as an NP-hard
mixed-integer nonlinear program (MINLP) that is exactly reformulated as a
mixed-integer linear program (MILP) using McCormick linearization. To improve
computation time, a cutting plane generation procedure is put forth that is
able to interject the MILP solver and augment additional constraints to the
problem on-the-fly. Moreover, graph-theoretic properties are exploited to
derive valid inequalities (cuts) and tighten bounds on the continuous
optimization variables to significantly accelerate the solver run times. The
efficacy of our approach in designing optimal grid topologies is demonstrated
through numerical tests on the IEEE 39-bus network.
