Propagating Uncertainty in Power Flow with the Alternating Direction Method of Multipliers

In this paper, we develop an optimization-based method to propagate input and parametric uncertainty to the power flow solution. The approach is based on maximizing and minimizing quadratic approximations of the power-flow states as a function of the uncertainties subject to inequality constraints that capture all possible values the uncertain elements can take. A major computational bottleneck in such an approach is that the formulation of the quadratic approximations requires the solution of sensitivities (up to second order) from algebraic equations that are derived from the power flow equations. We demonstrate how decoupling assumptions based on the form and function of power networks can be applied to facilitate computations in this regard. The formulated quadratic programs are non-convex in general, and we adopt the Alternating Direction Method of Multipliers to solve them. Conditions for convergence in this non-convex setting are established leveraging recent advances in optimization theory. Numerical simulations for the matpower 1354-bus test system are provided to validate the accuracy and demonstrate the scalability of the approach.