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

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 uncertainty quantification for (optimization of) power systems. We cover power flow (PF) and optimal power flow (OPF) under uncertainty (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 uncertainty 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

PolyChaos.jl - An open source Julia package for polynomial chaos expansion

Polynomial chaos expansion (PCE) is a Hilbert space technique for random variables that alleviates uncertainty propagation. Random variables are expanded in terms of polynomials that are orthogonal relative to a given probability density function. The applicability of PCE hinges on software that allows, among others, to construct orthogonal polynomials. We offer a package for (intrusive) PCE written in the Julia programming language, a trending programming language dedicated to scientific computing

Uncertainty Quantification for Optimal Power Flow Problems

The need to de‐carbonize the current energy infrastructure, and the increasing integration of renewables pose a number of difficult control and optimization problems. Among those, the optimal power flow (OPF) problem—i.e., the task to minimize power system operation costs while maintaining technical and network limitations—is key for operational planning of power systems. The influx of inherently volatile renewable energy sources calls for methods that allow to consider stochasticity directly in the OPF problem. Here, we present recent results on uncertainty quantification for OPF problems. Modeling uncertainties as second‐order continuous random variables, we will show that the OPF problem subject to stochastic uncertainties can be posed as an infinite‐dimensional L$_{2}$‐problem. A tractable reformulation thereof can be obtained using polynomial chaos expansion (PCE), under mild assumptions. We will show advantageous features of PCE for OPF subject to stochastic uncertainties. For example, multivariate non‐Gaussian uncertainties can be considered easily. Finally, we comment on recent progress on a Julia package for PCE

The Price of Uncertainty: Chance-constrained OPF vs. In-hindsight OPF

The operation of power systems has become more challenging due to feed-in of volatile renewable energy sources. Chance-constrained optimal power flow (ccOPF) is one possibility to explicitly consider volatility via probabilistic uncertainties resulting in mean-optimal feedback policies. These policies are computed before knowledge of the realization of the uncertainty is available. On the other hand, the hypothetical case of computing the power injections knowing every realization beforehand---called in-hindsight OPF(hOPF)---cannot be outperformed w.r.t. costs and constraint satisfaction. In this paper, we investigate how ccOPF feedback relates to the full-information hOPF. To this end, we introduce different dimensions of the price of uncertainty. Using mild assumptions on the uncertainty we present sufficient conditions when ccOPF is identical to hOPF. We suggest using the total variational distance of probability densities to quantify the performance gap of hOPF and ccOPF. Finally, we draw upon a tutorial example to illustrate our results.Comment: Accepted for publication at the 20th Power Systems Computation Conference (PSCC) in Dublin, 201

A Generalized Framework for Chance-constrained Optimal Power Flow

Deregulated energy markets, demand forecasting, and the continuously increasing share of renewable energy sources call---among others---for a structured consideration of uncertainties in optimal power flow problems. The main challenge is to guarantee power balance while maintaining economic and secure operation. In the presence of Gaussian uncertainties affine feedback policies are known to be viable options for this task. The present paper advocates a general framework for chance-constrained OPF problems in terms of continuous random variables. It is shown that, irrespective of the type of distribution, the random-variable minimizers lead to affine feedback policies. Introducing a three-step methodology that exploits polynomial chaos expansion, the present paper provides a constructive approach to chance-constrained optimal power flow problems that does not assume a specific distribution, e.g. Gaussian, for the uncertainties. We illustrate our findings by means of a tutorial example and a 300-bus test case

Distributed Power Flow and Distributed Optimization - Formulation, Solution, and Open Source Implementation

Solving the power flow problem in a distributed fashion empowers different grid operators to compute the overall grid state without having to share grid models-this is a practical problem to which industry does not have off-the-shelf answers. In cooperation with a German transmission system operator we propose two physically consistent problem formulations (feasibility, least-squares) amenable to two solution methods from distributed optimization (the Alternating direction method of multipliers (ADMM), and the Augmented Lagrangian based Alternating Direction Inexact Newton method (Aladin)); with Aladin there come convergence guarantees for the distributed power flow problem. In addition, we provide open source matlab code for rapid prototyping for distributed power flow (rapidPF), a fully matpower-compatible software that facilitates the laborious task of formulating power flow problems as distributed optimization problems; the code is available under https://github.com/KIT-IAI/rapidPF/. The approach to solving distributed power flow problems that we present is flexible, modular, consistent, and reproducible. Simulation results for systems ranging from 53 buses (with 3 regions) up to 4662 buses (with 5 regions) show that the least-squares formulation solved with aladin requires just about half a dozen coordinating steps before the power flow problem is solved.Comment: 35 pages, 6 figures, 7 tables, journal submissio

Analytical uncertainty propagation for multi-period stochastic optimal power flow

The increase in renewable energy sources (RESs), like wind or solar power, results in growing uncertainty also in transmission grids. This affects grid stability through fluctuating energy supply and an increased probability of overloaded lines. One key strategy to cope with this uncertainty is the use of distributed energy storage systems (ESSs). In order to securely operate power systems containing renewables and use storage, optimization models are needed that both handle uncertainty and apply ESSs. This paper introduces a compact dynamic stochastic chance-constrained DC optimal power flow (CC-OPF) model, that minimizes generation costs and includes distributed ESSs. Assuming Gaussian uncertainty, we use affine policies to obtain a tractable, analytically exact reformulation as a second-order cone problem (SOCP). We test the new model on five different IEEE networks with varying sizes of 5, 39, 57, 118 and 300 nodes and include complexity analysis. The results show that the model is computationally efficient and robust with respect to constraint violation risk. The distributed energy storage system leads to more stable operation with flattened generation profiles. Storage absorbed RES uncertainty, and reduced generation cost