Power Reserve Procurement Considering Dependent Random Variables with PCE

TL;DR

The paper tackles probabilistic power reserve procurement under dependent uncertainty among distributed energy resources by marrying generalised polynomial chaos (gPC) with Gaussian copulas. It builds an orthonormal polynomial basis with respect to the joint pdf $p(\bm{\xi})$ using Gram–Schmidt, enabling chance-constrained optimization that accounts for dependencies. A Gaussian copula with correlation matrix $\Sigma$ decouples marginals from dependence and, together with a latent Gaussian transformation $z_i=\Phi^{-1}(u_i)$, enables tractable Gauss–Hermite quadrature and a convex reformulation. Numerical improvements—precomputing monomial inner products and reducing the effective integration dimension to $2\nu$—further enhance scalability. The approach is demonstrated on a probabilistic power reserve procurement use case, showing improved dependency handling over independent-variable methods and offering a scalable framework for risk-aware resource deployment in power systems.

Abstract

This paper presents an approach for the modelling of dependent random variables using generalised polynomial chaos. This allows to write chance-constrained optimization problems with respect to a joint distribution modelling dependencies between different stochastic inputs. Arbitrary dependencies are modelled by using Gaussian copulas to construct the joint distribution. The paper exploits the problem structure and develops suitable transformations to ensure tractability. The proposed method is applied to a probabilistic power reserve procurement problem. The effectiveness of the method to capture dependencies is shown by comparing the approach with a standard approach considering independent random variables.