AMPS: Topological disturbance for power flow equations through the lens of convex and algebraic geometry

Power grids are experiencing unprecedented transformations under the pressure of population growth, urbanization, and the introduction of renewable sources. Our ability in controlling and designing increasingly complicated power grids will no doubt be constrained by our understanding of the complex and nonlinear interactions among their components. The power flow equations are nonlinear equations that are ubiquitous in all power system studies. They describe the intricate balancing conditions on the active and reactive power injections, and their solutions give operating points for the underlying power network. There can be more than one potential operating point due to the inherent nonlinearity of the power flow equations. The problem of finding some or all of them has been an active research topic, and it is the focus of this project. It is well known that there is a deep connection between the connectivity structure of a network and its power flow solutions. Changes to a network naturally induce qualitative changes to its power flow solutions. Yet, a rigorous understanding of connection disturbances on power flow solutions remains elusive. This research attacks this problem from a new viewpoint. This project includes educational component and will lead to the development of software packages. This project revolves around the rigorous and global analysis of the effect of “topological disturbances” (changes in network topology) on the full set of power flow solutions. Based on the theoretical foundation and promising results established by the PIs’ recent works, the proposed research aims to develop a deep understanding of the impact of topological disturbances on power networks from the viewpoint of convex geometry. This will be done through the machinery of toric deformations, which links the study of convex polytopes with power flow solutions. The concrete goals include developing theoretical underpinnings of the geometry of the full set of power flow solutions, establishing a global understanding of how these solutions are influenced by topological disturbances, and implementing new solvers for power flow equations based on this understanding. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.