Some Optimization Problems in Smart Grids

Optimal power flow (OPF) is fundamental because it underlies numerous power system operation and planning problems.   In this talk, I will give a sample of optimization problems in the management of a large network of distributed energy resources.   The nonlinearity of power flow equations leads to the nonconvexity of OPF, one of the main computational challenges in power system applications.  We describe a method to deal with nonconvexity through semidefinite relaxation.  Semidefinite programs are hard to scale to large OPF problems.  We describe a highly scalable distributed solution based on ADMM.  These algorithms are offline in that they iterate until the computation has converged before applying the final solution to the grid, and are therefore not suitable for real-time optimization of distributed energy resources at scale. We describe realtime OPF that explicitly exploits the network as a power flow equation solver and characterize its performance in tracking changing network conditions.   Finally, in practice, not all network nodes have sensors or canbe controlled.  We characterize controllability and observability of power flow dynamics in terms of the spectral properties of its Laplacian matrix. This characterization can be used to optimize the placement of sensors and actuators in the grid.