The National Academy of Engineering has classified widespread electrification as one of the greatest engineering achievements of the 20th century, but a 2006 study estimated that the national annual cost of power interruptions was about $79 billion dollars. The power flow equations are at the heart of all tools used by power system engineers to maintain reliable, economically operated power systems. These equations have variability, and our team of engineers and mathematicians will address in this project how this affects their solutions.
These equations model the nonlinear relationship between voltages and active and reactive power injections in a power system and can be represented by multivariate quadratics. The goal now is to introduce uncertainty into the coefficients of the polynomials (the susceptances) and to investigate how their number of real solutions varies, bound it, and obtain practical algorithms for finding all these solutions under that uncertainty. The distribution of numbers of real solutions is known for general polynomial systems under assumptions such as that the coefficients are iid Gaussian but the power flow systems appear to have fewer real solutions than those results would predict. The group intends to generalize this work to our special systems, understand the phenomena, and then apply them to actual power systems.