## Online optimization for power networks

#### Steven LowCalifornia Institute of Technology

We are interested in the optimization and control of a large network of distributed energy resources in a plug-and-play environment of the future. This involves solving power flow equations which is well-known to be hard. The grid, however, solves them in real-time exactly at scale, and we propose to exploit it explicitly to carry out part of our optimization algorithm. This approach not only improves scalability, but also naturally adapts to evolving network conditions. In this talk, we present two examples.

The first example presents an online algorithm to solve an optimal power flow problem at a slow timescale on a radial network where the controllable devices continuously interact with the network that implicitly computes a power flow solution given a control action. Collectively these devices and the network implement a gradient projection
algorithm in real time. We prove that the proposed algorithm converges to a set of
local optima and provide sufficient conditions under which it converges to a global optimum. We derive an upper bound on the suboptimality gap of any local optimum.
This bound suggests that any local optimum is almost as good as any strictly feasible point. In the second example, the online algorithm integrates primary frequency regulation, secondary frequency regulation, and congestion management at a fast timescale. The algorithm is distributed in that it requires communication only between neighboring buses. Collectively, the controllable devices and the swing dynamics of the network implement a primal-dual algorithm to rebalance power, restore nominal frequency and inter-area flows, and enforce line limits at a minimum control cost. We prove sufficient conditions under which the algorithm converges to a global optimum.
(Joint work with Changhong Zhao, Enrique Mallada, Lingwen Gan)

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