Uncertainty Quantification via Polynomial Chaos Expansion – Methods and Applications for Optimization of Power Systems

Abstract

Fossil fuels paved the way to prosperity for modern societies, yet alarmingly, we can exploit our planet’s soil only so much. Renewable energy sources inherit the burden to quench our thirst for energy, and to reduce the impact on our environment simultaneously. However, renewables are inherently volatile; they introduce uncertainties. What is the effect of uncertainties on the operation and planning of power systems? What is a rigorous mathematical formulation of the problems at hand? What is a coherent methodology to approaching power system problems under uncertainty? These are among the questions that motivate the present thesis that provides a collection of methods for quantification for (optimization of) power systems. We cover power flow (PF) and optimal power flow (OPF) under (as well as specific derivative problems). Under uncertainty---we view uncertainty as continuous random variables of finite variance---the state of the power system is no longer certain, but a random variable. We formulate PF and OPF problems in terms of random variables, thusly exposing the infinite-dimensional nature in terms of L2-functions. For each problem formulation we discuss a solution methodology that renders the problem tractable: we view the problem as a mapping under uncertainty; uncertainties are propagated through a known mapping. The method we employ to propagate uncertainties is called polynomial chaos expansion (PCE), a Hilbert space technique that allows to represent random variables of finite variance in terms of real-valued coefficients. The main contribution of this thesis is to provide a rigorous formulation of several PF and OPF problems under in terms of infinite-dimensional problems of random variables, and to provide a coherent methodology to tackle these problems via PCE. As numerical methods are moot without numerical software another contribution of this thesis is to provide PolyChaos.jl: an open source software package for orthogonal polynomials, quadrature rules, and PCE written in the Julia programming language.