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

1. Problem Statement

The integration of large-scale renewable energy sources (RES) and flexible demand (e.g., electric vehicles) has introduced significant, complex operational uncertainties into power systems. These uncertainties are characterized by:

• Correlation: Spatial and temporal dependencies between different generation sites and loads.
• Non-Gaussianity: Heavy-tailed distributions and multimodality (e.g., weather regime switching).
• Complexity: Traditional parametric models often fail to capture these high-dimensional, non-standard distributions.

The Core Challenge:
Uncertainty Quantification (UQ) in power system dynamics typically relies on Polynomial Chaos Expansion (PCE) for efficient propagation of uncertainty. However, standard PCE requires independent input variables to construct orthogonal polynomial bases.

• Current Limitation: Conventional approaches either ignore dependencies (leading to inaccurate statistics) or use Copula-based methods (e.g., Gaussian Copula) to model dependencies. Copulas often rely on restrictive parametric assumptions that fail to capture complex, multimodal, or high-dimensional empirical data distributions.
• The Gap: There is a disconnect between accurate input distribution modeling (often handled by deep generative models) and efficient uncertainty propagation (handled by PCE).

2. Methodology: Flow-based PCE

The authors propose a unified framework that couples Normalising Flows (a type of deep generative model) with Generalized Polynomial Chaos Expansion (gPC).

A. The Workflow

1. Input Modeling (Normalising Flow):

   • Instead of assuming a fixed parametric form (like a Gaussian Copula), the method uses a Normalising Flow to learn an invertible transport map, $T$, from a simple base distribution (e.g., standard Gaussian, $Z$) to the complex empirical joint distribution of the uncertain inputs ($\xi$).
   • $\xi = T(Z)$.
   • The flow is trained via maximum likelihood estimation to minimize the Kullback-Leibler (KL) divergence between the learned density and the data.

2. Transformation to Latent Space:

   • The learned inverse map, $Z = T^{-1}(\xi)$, transforms the correlated physical inputs into independent latent variables ($Z$).
   • This allows the use of classical gPC with standard orthogonal polynomial bases (e.g., Hermite polynomials for Gaussian latents) in the latent space.

3. Surrogate Construction:

   • A PCE surrogate is constructed in the latent space: $M(\xi) \approx \sum h_\alpha \Psi_\alpha(Z)$.
   • Coefficients ($h_\alpha$) are estimated via regularized regression (e.g., LASSO) using samples drawn from the base distribution, mapped to physical space, and evaluated by the power system simulator.

4. Post-Processing:

   • Statistical moments (mean, variance) and sensitivity indices (Sobol') are computed analytically from the PCE coefficients, avoiding the need for expensive Monte Carlo sampling.

B. Theoretical Analysis & Metrics

The paper introduces a theoretical error bound and a new metric to guide model selection:

• Error Decomposition: The total error is bounded by three components:
  1. Distribution Error: Related to the Wasserstein distance between the true data and the flow model.
  2. Truncation Error: Related to the polynomial degree and the smoothness of the transport map.
  3. Estimation Error: Related to sample size and sparsity.
• Map Smoothness Index (MSI): A novel metric defined as the root-mean-square Frobenius norm of the Jacobian of the transport map.
  • Significance: Theoretical analysis shows that a smoother map (lower MSI) leads to a tighter upper bound on the truncation error. Therefore, selecting a flow architecture that balances expressiveness with smoothness is critical for PCE accuracy.

3. Key Contributions

1. Unified Framework: Proposes the first framework that seamlessly integrates expressive, data-driven input modeling (Normalising Flows) with efficient spectral uncertainty propagation (PCE), eliminating the need for prescribed parametric distributions.
2. Theoretical Guidance: Derives a generalization error bound linking the Map Smoothness Index (MSI) to PCE convergence. This provides a theoretical basis for selecting flow architectures and polynomial degrees.
3. Scalability: Demonstrates that the method scales effectively to high-dimensional power system problems (up to 18+ uncertain inputs) where Copula-based methods degrade significantly.
4. New Metric: Introduces the MSI as a practical tool for model selection, showing that "smoother" transformations yield more accurate surrogates.

4. Results & Case Studies

The method was validated on two systems: the IEEE 14-bus system and a large-scale Great Britain (GB) transmission system (2224 buses).

• Gaussian Copula Data (Baseline):
  • When data perfectly matched a Gaussian Copula assumption, the Flow-based PCE achieved accuracy comparable to the classical Copula-PCE, despite having no prior knowledge of the structure.
• Multimodal Data (Realistic Scenario):
  • On bimodal (mixture) distributions, the classical Copula-PCE failed to capture the modes, resulting in high errors (NIRMSE $\approx 0.178$).
  • The Flow-based PCE successfully captured the multimodal structure, reducing the error by more than 16x (NIRMSE $\approx 0.0107$).
• Architecture Comparison:
  • Comparing Neural Spline Flows (NSF), Masked Autoregressive Flows (MAF), and NICE flows revealed that NSF (which uses smooth splines) produced the lowest MSI and the most accurate PCE surrogate.
  • This confirmed the theoretical link: smoother maps $\rightarrow$ lower truncation error $\rightarrow$ better PCE performance.
• Scalability (GB System):
  • As input dimension increased (6 to 18 variables), the Copula-based approach suffered from model misspecification, leading to large errors.
  • The Flow-based PCE maintained low error rates and high fidelity, demonstrating robustness in high-dimensional settings.

5. Significance

This paper addresses a critical bottleneck in modern power system analysis: the inability of traditional UQ tools to handle the complex, correlated, and non-Gaussian nature of renewable energy uncertainties.

• Practical Impact: It enables system operators to perform rigorous probabilistic dynamic assessments (e.g., transient stability) with higher accuracy and lower computational cost than Monte Carlo methods.
• Methodological Advance: It bridges the gap between deep learning (generative modeling) and classical uncertainty quantification (spectral methods), offering a theoretically grounded, data-driven approach that outperforms state-of-the-art parametric methods in realistic, complex scenarios.
• Guidance for Practitioners: The introduction of the MSI provides a concrete criterion for selecting generative models, moving beyond "black-box" deep learning to interpretable, performance-guaranteed surrogates.