Quadratic performance of primal-dual methods with application to secondary frequency control of power systems

Primal-dual gradient methods have recently attracted interest as a set of systematic techniques for distributed and online optimization. One of the proposed applications has been optimal frequency regulation in power systems, where the primal-dual algorithm is implemented online as a dynamic controller. In this context however, the presence of external disturbances makes quantifying input/output performance important. Here we use the H₂ system norm to quantify how effectively these distributed algorithms reject external disturbances. For the linear primal-dual algorithms arising from quadratic programs, we provide an explicit expression for the H2 norm, and examine the performance gain achieved by augmenting the Lagrangian. Our results suggest that the primal-dual method may perform poorly when applied to large-scale systems, and that Lagrangian augmentation can partially (or completely) alleviate these scaling issues. We illustrate our results with an application to power system frequency control by means of distributed primal-dual controllers.