[Paper Review] Applying Polynomial Chaos Expansion to Assess Probabilistic Available Delivery Capability for Distribution Networks with Renewables

Hao Sheng, Xiaozhe Wang|arXiv (Cornell University)|Apr 9, 2018

Probabilistic and Robust Engineering DesignDecision Sciences27 references4 citations

TL;DR

This paper proposes a sparse polynomial chaos expansion (SPCE) combined with the continuation method to efficiently and accurately assess probabilistic available delivery capability (ADC) in distribution networks with high renewable energy and load uncertainty. The method handles correlated, non-Gaussian random inputs using Stieltjes procedure and Copula-Nataf transformation, achieving 30x faster computation than Monte Carlo simulation with comparable accuracy, revealing a 4.51% reduction in ADC needed for 95% operational confidence.

ABSTRACT

Considering the increasing penetration of renewable energy sources and electrical vehicles in utility distribution feeders, it is imperative to study the impacts of the resulting increasing uncertainty on the delivery capability of a distribution network. In this paper, probabilistic available delivery capability (ADC) is formulated for a general distribution network integrating various RES and load variations. To reduce the computational efforts by using conventional Monte Carlo simulations, we develop and employ a computationally efficient method to assess the probabilistic ADC, which combines the up-to-date sparse polynomial chaos expansion (PCE) and the continuation method. Particularly, the proposed method is able to handle a large number of correlated random inputs with different marginal distributions. Numerical examples in the IEEE 13 and IEEE 123 node test feeders are presented, showing that the proposed method can achieve accuracy and efficiency simultaneously. Numerical results also demonstrate that the randomness brought about by the RES and loads indeed leads to a reduction in the delivery capability of a distribution network.

Motivation & Objective

• To formulate probabilistic available delivery capability (ADC) in distribution networks with diverse, correlated renewable energy sources (RES) and load variations.
• To address the high computational cost of Monte Carlo simulation (MCS) in probabilistic power flow analysis under uncertainty.
• To develop a method that efficiently handles a large number of correlated random inputs with different marginal distributions (e.g., Weibull, Beta, Gamma).
• To provide accurate statistical estimates of ADC distribution and its moments (mean, variance) under uncertainty.
• To quantify the impact of RES and load variability on system delivery capability, showing necessary reductions in ADC for secure operation.

Proposed method

• The method employs generalized polynomial chaos expansion (gPCE) with a sparse, basis-adaptive scheme to reduce computational cost.
• It uses the Stieltjes procedure to construct orthogonal polynomial bases for arbitrary marginal distributions of RES and load inputs.
• The Copula and Nataf transformation are applied to model the joint probability distribution of correlated random variables with different marginal types.
• The continuation method is integrated to trace the voltage collapse point and determine the overall ADC under probabilistic constraints.
• The approach enables non-intrusive, deterministic power flow solvers to be used within a probabilistic framework, preserving computational efficiency.
• The method is validated using IEEE 13- and 123-bus test feeders with realistic RES and load uncertainty scenarios.

Experimental results

Research questions

• RQ1 How can probabilistic available delivery capability (ADC) be accurately and efficiently assessed in distribution networks with high penetration of diverse renewable energy sources and variable loads?
• RQ2 What is the impact of correlated, non-Gaussian uncertainties in RES and load on the system's delivery capability and voltage stability?
• RQ3 Can sparse polynomial chaos expansion (SPCE) with adaptive basis selection outperform traditional Monte Carlo simulation in terms of accuracy and computational efficiency for ADC assessment?
• RQ4 How does the number of random inputs and their correlation structure affect the convergence and scalability of the SPCE-based ADC analysis?
• RQ5 What reduction in deterministic ADC is required to achieve a 95% confidence level of secure operation under uncertainty?

Key findings

• The proposed sparse PCE method achieves accuracy comparable to Monte Carlo simulation (MCS) with 10,000 samples, but reduces computation time by a factor of approximately 30.
• The mean and variance of the probabilistic ADC estimated by SPCE differ from MCS by less than 0.03% in absolute relative error across all test cases.
• The computational time of SPCE grows linearly with the number of random inputs, unlike MCS, which grows exponentially with sample size.
• The method converges to stable statistical estimates of ADC when the sample size reaches 2.5 times the number of random inputs (e.g., 335 samples for 134 inputs), indicating robust convergence.
• A 4.51% reduction in the deterministic ADC is required to ensure 95% confidence of avoiding thermal violations, highlighting the significant impact of uncertainty.
• The probabilistic ADC framework reveals that uncertainty from RES and load variability systematically reduces the effective delivery capability, necessitating conservative operational margins.