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Tempered Christoffel-Weighted Polynomial Chaos Expansion for Resilience-Oriented Uncertainty Quantification

Mahsa Ebadat-Parast, Xiaozhe Wang

Abstract

Accurate and efficient uncertainty quantification is essential for resilience assessment of modern power systems under high impact and low probability disturbances. Data driven sparse polynomial chaos expansion (DDSPCE) provides a computationally efficient surrogate framework but may suffer from ill conditioned regression and loss of accuracy in the distribution tails that determine system risk. This paper studies the impact of regression weighting schemes on the stability and tail accuracy of DD-SPCE surrogates by introducing a tempered Christoffel weighted least squares (T-CWLS) formulation that balances numerical stability and tail fidelity. The tempering exponent is treated as a hyperparameter whose influence is examined with respect to distributional accuracy compared with Monte Carlo simulations. Case studies on distribution system load shedding show that the proposed method reduces 95th percentile deviation by 16%, 5th percentile deviation by 6%, and improves the regression stability index by over 130%. The results demonstrate that controlling the weighting intensity directly influences both stability index and the accuracy of tail prediction.

Tempered Christoffel-Weighted Polynomial Chaos Expansion for Resilience-Oriented Uncertainty Quantification

Abstract

Accurate and efficient uncertainty quantification is essential for resilience assessment of modern power systems under high impact and low probability disturbances. Data driven sparse polynomial chaos expansion (DDSPCE) provides a computationally efficient surrogate framework but may suffer from ill conditioned regression and loss of accuracy in the distribution tails that determine system risk. This paper studies the impact of regression weighting schemes on the stability and tail accuracy of DD-SPCE surrogates by introducing a tempered Christoffel weighted least squares (T-CWLS) formulation that balances numerical stability and tail fidelity. The tempering exponent is treated as a hyperparameter whose influence is examined with respect to distributional accuracy compared with Monte Carlo simulations. Case studies on distribution system load shedding show that the proposed method reduces 95th percentile deviation by 16%, 5th percentile deviation by 6%, and improves the regression stability index by over 130%. The results demonstrate that controlling the weighting intensity directly influences both stability index and the accuracy of tail prediction.
Paper Structure (17 sections, 36 equations, 3 figures, 1 table)

This paper contains 17 sections, 36 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Resilience-Oriented Monte Carlo-Based Optimization Modeling
  3. Model Description and Assumptions
  4. Optimization Formulation
  5. Proposed Approach
  6. Data-Driven Polynomial Chaos Expansion (DD-PCE)
  7. Moment-based orthogonal polynomial basis.
  8. Data-driven PCE representation and OLS formulation.
  9. Christoffel Function and Weighted Least Squares
  10. Proposed Tempered Christoffel-Weighted Regression (T-CWLS)
  11. Motivation.
  12. Tempered weighting formulation.
  13. Modeling Assumptions and Experimental Design
  14. Case study
  15. Simulation Setup
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Modified IEEE 34-bus distribution system integrated with three MGs
  • Figure 2: Comparison of absolute percent deviations in the (a) 5th and (b) 95th percentiles of the surrogate-predicted output distributions for different weighting configurations.
  • Figure 3: (a)Absolute percent deviation in the 95th percentile (P95%) for different $\alpha$ values, (b)Variation of $\mathrm{Score}_{LR}$ with tempering exponent $\alpha$.