Flow-based Polynomial Chaos Expansion for Uncertainty Quantification in Power System Dynamic Simulation

Abstract

The large-scale integration of renewable energy sources introduces significant operational uncertainty into power systems. Although Polynomial Chaos Expansion (PCE) provides an efficient tool for uncertainty quantification (UQ) in power system dynamics, its accuracy depends critically on the faithful representation of input uncertainty, an assumption that is oftern violated in practice due to correlated, non-Gaussian, and otherwise complex data distributions. In contrast to purely data-driven surrogates that often overlook rigorous input distribution modelling, this paper introduces flow-based PCE, a unified framework that couples expressive input modelling with efficient uncertainty propagation. Specifically, normalising flows are employed to learn an invertible transport map from a simple base distribution to the empirical joint distribution of uncertain inputs, and this map is then integrated directly into the PCE construction. In addition, the Map Smoothness Index (MSI) is introduced as a new metric to quantify the quality of the learned map, and smoother transformations are shown to yield more accurate PCE surrogates. The proposed Flow-based PCE framework is validated on benchmark dynamic models, including the IEEE 14-bus system and the Great Britain transmission system, under a range of uncertainty scenarios.

Figures (6)

• Figure 1: A normalizing flow can be visualized as a sequence of bijective transformations. A sample drawn from the base distribution $\mathbf{z}_0$ is mapped to the complex target distribution $\bm{\xi}$ through $K$ invertible functions $T_1, \dots, T_K$. Since each $T_i$ is bijective, the transformations are reversible, illustrating that the flow operates both as a generative model (from latent to data space) and as an exact mapping method (from data to latent space) without information loss.
• Figure 2: Pair plot visualizing the joint distribution learned by the spline-based normalising flow on the Gaussian copula dataset.
• Figure 3: Comparison of the PCE surrogate outputs for the Gaussian copula test case.
• Figure 4: Comparison of learned distributions on the bimodal dataset. The classical copula model (a) fails to capture the two distinct modes of the data, representing it as a single unimodal distribution. The normalising flow model (b) successfully learns the bimodal structure.
• Figure 5: Comparison of PCE surrogate outputs for the multimodal test case.