[Paper Review] PDF estimation for power grid systems via sparse regression

Xiu Yang, David A. Barajas‐Solano+2 more|arXiv (Cornell University)|Aug 21, 2017

Probabilistic and Robust Engineering Design20 references3 citations

TL;DR

This paper proposes a compressive sensing-based sparse regression method to estimate probability density functions (PDFs) of power grid system quantities of interest (QoIs) under uncertain generation and load fluctuations. By using iterative linear transformations to enhance sparsity in Hermite polynomial expansions, the method reduces required Monte Carlo realizations by up to 75% while achieving significantly lower Kullback-Leibler divergence compared to standard sampling methods.

ABSTRACT

We present a numerical approach for estimating the probability density function (PDF) of quantities of interest (QoIs) of power grid systems subject to uncertain power generation and load fluctuations. In our approach, generation and load fluctuations are modeled by means of autocorrelated-in-time random processes, which are approximated in terms of a finite set of random parameters by means of Karhunen-Loève approximations. The map from random parameters to QoIs is approximated by means of Hermite polynomial expansions. We propose a new approach based on compressive sensing to estimate the coefficients in the Hermite expansions from a small number of realizations (sampling points). Linear transforms identified by iterative rotations are introduced to improve the sparsity of the Hermite representations, exploiting the intrinsic low-dimensional structure of the map. As such, the proposed approach significantly reduces the required number of sampling points to achieve a given accuracy compared to the standard least squares method. The proposed approach is employed to estimate the PDF of relative angular velocities and bus voltages of systems of classical machines driven by autocorrelated random generation. More accurate PDF estimates, as measured by the Kullback-Leibler divergence, are achieved using fewer realizations than required by basic Monte Carlo sampling.

Motivation & Objective

• To address the challenge of accurately estimating non-Gaussian, heavy-tailed PDFs of power grid system states under uncertain generation and load fluctuations.
• To reduce the computational cost of uncertainty quantification in power systems by minimizing the number of full-model simulations required for PDF estimation.
• To improve the sparsity of Hermite polynomial expansions representing the map from random inputs to QoIs through iterative linear transformations.
• To develop a surrogate modeling framework that enables efficient PDF estimation using fewer realizations than standard least squares or Monte Carlo sampling.

Proposed method

• Modeling time-correlated power generation and load fluctuations using truncated Karhunen-Loève expansions to represent them as finite sets of i.i.d. Gaussian random variables.
• Approximating the map from random parameters to QoIs using Hermite polynomial chaos expansions to construct a surrogate model.
• Employing compressive sensing with iterative rotations to identify linear transforms that enhance sparsity in the Hermite coefficient representation.
• Using the sparse representation to reduce the effective dimensionality of the uncertainty space and improve coefficient estimation efficiency.
• Estimating the final PDF by sampling the sparse surrogate model rather than the full system, with accuracy validated via Kullback-Leibler divergence against reference Monte Carlo estimates.
• Applying the method to the WECC 3-bus and New England 39-bus systems to estimate PDFs of relative angular velocities and bus voltages.

Experimental results

Research questions

• RQ1 Can iterative sparse regression via compressive sensing reduce the number of full-model simulations needed to accurately estimate the PDF of power grid QoIs under uncertainty?
• RQ2 How does the use of linear transforms to enhance sparsity in Hermite polynomial expansions affect the accuracy and efficiency of PDF estimation?
• RQ3 To what extent does the proposed method outperform standard Monte Carlo sampling in terms of Kullback-Leibler divergence for the same number of realizations?
• RQ4 Does the method maintain accuracy when applied to complex, large-scale power systems such as the 10-generator New England system?
• RQ5 How do the PDF estimates from the sparse regression surrogate compare to kernel density estimates derived from the same number of Monte Carlo samples?

Key findings

• The proposed method reduced the Kullback-Leibler divergence of PDF estimates by approximately 75% compared to standard Monte Carlo sampling when using the same number of realizations.
• For the 10-generator New England system, the KL divergence of the sparse regression estimate was 0.0029 for relative angular velocity and 0.0098 for bus voltage, compared to 0.0746 and 0.0375 for Monte Carlo with 500 samples.
• The use of iterative rotations to enhance sparsity led to a significant reduction in the number of required full-model simulations without sacrificing accuracy.
• The difference in PDF estimates between the rotated representation and its truncated version was minimal, indicating robustness and stability of the method.
• The method consistently outperformed direct kernel density estimation from Monte Carlo samples across 50 independent datasets, with lower mean KL divergence and tighter confidence intervals.
• The surrogate model enabled accurate PDF estimation using only 500 full-model simulations, achieving results comparable to reference estimates computed from 10,000 Monte Carlo samples.