A key tool for power grid operators in ensuring reliable operation is the simulation of the power grid under possible contingency conditions. Central to these simulations are the power flow equations, a set of complex quadratic equations whose solution provides the voltage levels around the network. Modern power flow solvers typically use some variation of Newton's method, which is efficient for small to medium sized problems. However, the advent of the "smart grid" has brought with it a desire for increasingly detailed simulations. As problem sizes grow correspondingly larger, Newton's method appears less attractive.

Multigrid methods are a set of recursive techniques for solving linear systems that turn cheap, parallelizable methods such as the Jacobi iteration into highly effective solvers for large problems. In this talk, I will extend the multigrid framework to complex quadratic problems, and show how to use it to solve the power flow problem in a highly parallel manner. While this research is ongoing, initial experiments indicate that it scales favorably with problem size, making it a potentially useful approach for large-scale simulations.