PaperlayerView exact revision on arXiv

Source-linked AI summary

A Bayesian Adaptive Spectral Surrogate Model for Efficient Probabilistic Optimal Power Flow Evaluation

Xiaoting Wang, Xiaozhe Wang, Gregory Kish, Yunwei, Li

arXiv:2609.00547v1eess.SY

TL;DR

Uncertainty in renewable generation and demand makes the probabilistic characteristics of AC-OPF solutions important for power-system operations. The paper proposes ASSE, combining adaptive domain partitioning with Bayesian compressive sensing to estimate these characteristics and production-cost bounds, and reports accurate evaluations across IEEE test systems, especially for skewed, heavy-tailed, or multimodal outputs.

  • Problem

    Renewable generation and demand create uncertainty, while probabilistic AC-OPF requires predicting complete distributions of optimal decisions for operational use.

  • Method

    ASSE uses adaptive domain partitioning and Bayesian compressive sensing-based coefficient calculation to estimate probabilistic AC-OPF solutions and map generator-output distributions to confidence-interval production-cost bands.

  • Results

    ASSE accurately estimated probabilistic AC-OPF characteristics on modified IEEE 9-bus and IEEE 118-bus systems and remained more accurate for skewed, heavy-tailed, or multimodal output distributions than representative surrogate models.

  • Takeaways & Limitations

    The method provides practical upper and lower decision-making bounds for generator outputs and operating costs under uncertainty.

Abstract

from arXiv · show

This paper presents an adaptive stochastic spectral embedding (ASSE) method to solve the probabilistic AC optimal power flow (AC-OPF), a critical aspect of power system operation. The proposed method can efficiently and accurately estimate the probabilistic characteristics (e.g., mean, variance, median, and quantile-based metrics) of AC-OPF solutions while minimizing power losses. Based on estimated AC-OPF decisions (i.e., generator outputs), the confidence interval (CI)-based production cost index can be determined. Specially, an adaptive domain partition strategy is adopted to guide refinement domain selection and partition. The Bayesian compressive sensing-based coefficient calculation algorithm is integrated to enhance its performance. Numerical studies on modified IEEE 9-bus and IEEE 118-bus systems demonstrate that the proposed ASSE method offers accurate and fast evaluations compared to Monte Carlo simulations. Comparisons with a sparse polynomial chaos expansion, Gaussian process regression, and deep neural networks, further illustrate its efficacy in accurately assessing the responses with strongly localized behavior and non-symmetric distributions, providing practical decision-making bounds for generator outputs and operating costs under uncertainty.

Nomenclature

The paper frames probabilistic AC-OPF as a distribution-estimation problem under renewable and load uncertainty, and introduces ASSE to make these evaluations more efficient and targeted. Its notation covers system variables, stochastic inputs, local polynomial expansions, and adaptive domain refinement.

  • The stochastic input vector ζ represents quantities such as wind speed, solar irradiation, and load, while Z denotes the resulting stochastic power-system response through Z = F(ζ).
  • Polynomial Chaos and Domain Decomposition: Local polynomial chaos expansions approximate responses on subdomains indexed by α = (k, d), where k is the expansion level and d identifies the subdomain.
  • AC-OPF supports power-system operations, while renewable generation and electric-vehicle adoption create variable generation and demand profiles that deterministic formulations may not adequately manage.
  • Probabilistic OPF estimates full distributions of optimal decision variables, whereas conventional analyses typically require extensive Monte Carlo simulations and repeated OPF runs.
  • ASSE combines local stochastic spectral surrogates with Bayesian compressive sensing, modified K-fold cross-validation, and Sobol’-guided adaptive partitioning.

II. The Probabilistic AC-OPF Problem

The paper formulates probabilistic AC-OPF as uncertainty-aware power-loss minimization and builds an adaptive stochastic spectral surrogate to estimate resulting responses across the input domain.

  • The Probabilistic AC-OPF Problem: Probabilistic AC-OPF minimizes total transmission loss over generator decisions and bus states while enforcing power-flow and operational inequality constraints under uncertain inputs.Uncertainty includes renewable-generation and load variability; constraints cover generation, voltage, and branch-flow limits.
  • The Probabilistic AC-OPF Problem: Mean, variance, median, distributions, and confidence intervals of losses, decisions, and power flows support uncertainty-aware production-cost calculations.The production-cost bounds use generator-output statistics, including 5% and 95% quantiles for lower and upper cost estimates.
  • The SSE Representation: ASSE approximates the stochastic AC-OPF mapping by sequentially expanding residuals across progressively smaller stochastic subdomains.The response may include decision variables, state variables, the objective function, or power flows, assuming finite second-order moments.
  • Partition Strategy: Classical SSE increases the number of subdomains exponentially with expansion level, whereas ASSE targets refinement through its adaptive partition strategy.The partition example uses maximum level K = 2 and Dk = 4 subdomains.
  • Residual Approximation: Adaptive Bayesian PCE models approximate local residuals, allowing the surrogate to adjust its expansion parameters across subdomains.The resulting enhanced stochastic spectral embedding is called adaptive SSE, or ASSE.
  • Partition Strategy: ASSE partitions the stochastic domain by selecting terminal regions for refinement and using sensitivity information to guide splitting directions.Terminal domains form the union of the entire input domain; classical SSE indexes expansion levels by k and subdomains by d.

B. The Adaptive Bayesian PCE-based Residual Expansions

The method approximates local residual expansions with adaptive polynomial chaos bases and Bayesian compressive sensing, using modified cross-validation to control complexity and small-sample overfitting.

  • Adaptive polynomial bases: PCE approximates residual expansions using multivariate polynomial bases assembled from univariate orthonormal polynomials and standard or sparse truncation.Local inner products use the restricted density and sample-based quadrature within each subdomain.
  • Adaptive polynomial bases: Dedicated orthonormal bases accommodate uncertain inputs from different distribution families, including wind speed, solar irradiance, and load.The bases are combined into multivariate functions for each local subdomain.
  • Bayesian coefficient calculation: BCS computes sparse PCE coefficients and selects effective polynomial bases for high-dimensional problems with limited data.The coefficient matrix is obtained by solving a Bayesian compressive-sensing optimization problem on each subdomain.
  • Training data: The training data pair input samples ζED with corresponding AC-OPF responses ZED to construct local residual surrogates.NED denotes the initial domain sample size, while Nα is the number of samples in subdomain Dαζ.
  • Modified cross-validation: A modified K-fold CV error penalizes overly complex local models by incorporating model complexity and sample-size effects.The correction factor increases with retained basis functions relative to available samples, approaches 1 as sample size grows, and excludes models with M ≥ Nα.

C. Partition Strategy

ASSE adaptively selects terminal subdomains with the greatest error contribution and partitions each along its most influential input direction using Sobol’ indices.

  • Refinement domain selection: Each iteration selects one terminal subdomain for splitting from the current unsplit-domain set.A minimum sample requirement Nref governs whether local residual approximations can be constructed.
  • Refinement domain selection: The refinement score combines modified CV error with empirical probability mass, prioritizing dense subdomains with higher prediction error.This targets nonlinear responses, generator-limit activations, and poorly fitted tail regions that contribute more to distributional error.
  • Refinement domain selection: Only subdomains with refinement score χk,s > τthr remain candidates for splitting, and the largest-score candidate is selected.Refinement terminates when sample or split limits are reached, or when no qualifying candidate remains.
  • Sobol’ index-based partitioning: The selected subdomain is bisected into two equal-probability-mass parts along the input direction with the largest first-order Sobol’ index.The split uses the theoretical or empirical median, while other input dimensions remain unchanged.
  • Sobol’ index-based partitioning: For independent inputs, Sobol’ indices derive directly from adaptive Bayesian PCE coefficients; correlated inputs use a covariance-based PCE approach.Tied indices are resolved by choosing the smallest dimension index.
  • Relation to SSLE: Unlike SSLE’s likelihood approximation for Bayesian inversion, ASSE refines OPF-response subdomains with large modified CV error and non-negligible probability mass.ASSE targets forward uncertainty propagation and decision-relevant distributional quantities.

D. Closed-Form Representation of Moments

Once the ASSE surrogate is constructed, moments of the stochastic AC-OPF response can be represented in closed form from the terminal-domain expansions.

  • Closed-form moments: The estimated mean and variance of Z are represented analytically using the PCE expansions over terminal domains.The constant PCE term contributes to the mean representation.

IV. The Proposed ASSE Framework for Probabilistic AC-OPF

The ASSE framework trains a surrogate from sampled uncertain inputs and AC-OPF responses, adaptively refines local models, and evaluates probabilistic outputs and costs on additional samples.

  • Inputs and initialization: Random inputs include wind speed, solar irradiance, and load power, obtained from historical data or prescribed distributions.Correlated-input studies use vine copulas, while domain-wise marginals are treated as independent for polynomial-basis construction.
  • Inputs and initialization: The workflow solves AC-OPF for training samples, passes input-response pairs to ASSE, and initializes a full-domain adaptive Bayesian PCE surrogate.The initial residual and refinement score are then computed before adaptive partitioning begins.
  • Adaptive refinement: The adaptive loop selects and splits refinement subdomains, rebuilds local residual PCEs, updates residuals, and recomputes refinement scores.Splitting follows the Sobol’-index strategy and continues while sample and expansion-level conditions permit.
  • Adaptive refinement: The surrogate is accepted when its prescribed accuracy is reached; otherwise, the training set is enriched and the construction cycle repeats.The stopping measure uses an ASSE error based on probability-weighted cross-validation errors relative to response variance.
  • Probabilistic evaluation: After construction, many additional random samples are evaluated with ASSE to estimate means, variances, PDFs, CDFs, medians, quantiles, and distributional metrics.Performance is assessed with validation error, pinball loss, Kolmogorov–Smirnov distance, and Wasserstein-1 distance.
  • Parameter effects: The minimum local sample size Nref trades finer adaptation of asymmetric or localized behavior against greater variance and runtime.Larger Nref stabilizes estimates and shortens training but may oversmooth tail behavior and introduce bias.
  • Parameter effects: Kmax guarantees termination and bounds worst-case refinement, while τthr controls whether a candidate subdomain is split.The BCS procedure adaptively changes PCE order and truncation based on modified CV error.

V. Numerical Studies

The numerical studies validate ASSE on modified IEEE 9-bus and IEEE 118-bus systems across varied size, dimensionality, dependence structure, and data sources. Comparisons use MC as benchmark and SPCE, GPR, and, in selected cases, DNN surrogates under shared datasets.

  • Five case studies vary system size, input dimensionality, dependence structure, and random-input data source.
  • MC simulations using MATPOWER provide the benchmark, with default settings for Case 1 and linked network data for Cases 2–5.
  • ASSE is compared with adaptive SPCE and GPR across all cases, while DNN is included for the higher-dimensional and real-data cases.
  • SPCE, GPR, and DNN use specified polynomial, Matérn-5/2, and multilayer-perceptron configurations, respectively.
  • All surrogate models are trained and evaluated on the same datasets to ensure a fair comparison.

A. Case 1: The Modified 9-Bus System with 5 Independent Inputs

Case 1 evaluates ASSE on a modified 9-bus system with five independent random inputs. ASSE closely reproduces MC distributions and tail quantiles, while achieving favorable distributional accuracy and showing diminishing computational returns as training samples increase.

  • Five independent inputs comprise wind speed, solar irradiation, and three Gaussian random loads with 5% base-power variations.
  • Computational Cost: The study reports validation-error evaluation over NED and Nref settings and summarizes runtime components for ASSE with Nref = 20.
  • ASSE, SPCE, GPR, and MC distributions are compared for generator output PG1 and minimum-loss objective Ploss using CDFs and 5%/95% quantiles.
  • ASSE closely aligns with MC for Ploss and provides more accurate tail estimates; it also yields the lowest WD values for both PG1 and Ploss.
  • For the 95% quantile of Ploss, ASSE stays within 0.07% of MC, whereas SPCE deviates by as much as 0.20%.On a 40 GW system, the cited SPCE error corresponds to an unnecessary 80 MW reserve, while ASSE corresponds to roughly one-third that amount.
  • Computational Cost: Increasing NED from 350 to 500 increases refinement and construction time, but accuracy improvement becomes marginal.

B. Case 2: The Modified 118-Bus System with 12 Independent Inputs

Case 2 extends ASSE to a modified IEEE 118-bus system with 12 independent renewable inputs and constant loads. The method achieves stronger accuracy for skewed and localized generator-output distributions while reducing construction time relative to SPCE.

  • The 12 independent inputs are six 60 MW wind farms and six 40 MW solar plants, while loads remain at base power.
  • ASSE construction uses adaptive refinement with NED from 60 to 960 and selects Nref = 24 for most generator-output estimates.
  • ASSE, SPCE, GPR, and MC are compared using generator statistics, validation errors, distributional metrics, quantile bounds, and computation times.
  • Decisions under Uncertainties: The 5% and 95% generator-output quantiles define lower and upper decision bounds that feed directly into production-cost calculations.
  • ASSE yields smaller normalized bound errors, especially for skewed PG5, and remains superior to SPCE for highly localized PG19 behavior.
  • Computational Cost and Sample Efficiency: For comparable PG19 accuracy, SPCE construction takes approximately 954.04 s versus approximately 33.33 s for ASSE, roughly ten times longer.

C. Case 3: The Modified 118-Bus System with 30 Dependent Inputs

Case 3 evaluates ASSE on an IEEE 118-bus system with 30 dependent inputs. ASSE closely matches MC distributions, especially for skewed outputs, while requiring less data and computation than competing surrogates.

  • System and setup: Case 3 introduces nine additional wind generators and nine solar PV plants, creating a 30-dimensional dependent-input uncertainty space.
  • Distributional accuracy: ASSE achieves the lowest PBL, KSD, and WD for PG22, indicating the closest match to the MC benchmark across quantiles, CDFs, and distributional shape.
  • Decision bounds: ASSE provides lower and upper generator decision bounds that can be used to calculate corresponding production-cost bounds.
  • Distributional accuracy: ASSE closely follows MC PDFs for skewed generator outputs, whereas SPCE and DNN accuracy deteriorates noticeably under the same NED = 840 training size.
  • Computational cost: Comparable accuracy requires substantially more training time and samples for SPCE, and an even larger amount of data for DNN.

D. Case 4: The Modified 118-Bus System with 51 Dependent Inputs

Case 4 raises the dependent-input dimension to 51 in the IEEE 118-bus system. ASSE remains robust while GPR accuracy degrades and SPCE and GPR require more samples and construction time.

  • System and setup: Case 4 increases the uncertainty dimension to Min = 51 by combining 20 wind generators and 31 solar PV generators with dependent inputs modeled by a vine copula.
  • Benchmark comparison: ASSE, SPCE, GPR, and the MC benchmark achieve comparable accuracy in the reported comparison.
  • Evaluation setup: The Case 4 distribution comparison uses NED = 1000 for all surrogate models.
  • Accuracy and cost bounds: ASSE continues to perform robustly for production-cost bounds, whereas GPR accuracy degrades with increased input dimension.
  • Computational cost: SPCE and GPR require more training samples and longer construction time than ASSE to achieve comparable PG1 accuracy.

E. Case 5: Real-World Data Test Case

Case 5 tests ASSE with historical weather data on the modified IEEE 118-bus system. ASSE remains stable and produces the smallest production-cost-bound errors under empirical renewable uncertainty.

  • Real-world data: Case 5 constructs 12-dimensional uncertainty inputs from Open-Meteo historical wind and solar weather profiles rather than prescribed parametric distributions.
  • Results: ASSE remains stable under real-data conditions and yields the smallest errors in both lower and upper production-cost bounds relative to MC.
  • Practical implication: ASSE is presented as a tool for reliable decision-oriented cost bounds under empirical renewable uncertainty.
  • Distributional accuracy: ASSE maintains good accuracy for both Gaussian-like PG2 and skewed PG35, PG51, and PG52 output distributions.
  • Evaluation setup: The Case 5 evaluation uses NED = 840 and Nref = 72 for all surrogate models.
  • Scope and caveat: The paper reports that stressed settings can produce infeasible or non-convergent OPF samples, which are discarded from training and evaluation.

Appendix A The BCS Algorithm

The appendix describes Bayesian compressive sensing for sparse local PCE fitting and defines the metrics used to evaluate surrogate distributions. It also reports training-time scaling and identifies Gaussian likelihoods as a scope boundary.

  • BCS formulation: The BCS formulation converts PCE coefficient estimation into a Bayesian optimization problem supporting sparsity, probabilistic predictions, and automatic hyperparameter estimation.
  • Bayesian priors: Hierarchical priors on PCE coefficients and hyperparameters promote sparse representations, while a Gamma hyperprior is placed on λ.
  • Model interpretation: The Gaussian working likelihood does not assume Gaussian uncertain inputs, OPF outputs, or final estimated distributions.
  • Parameter selection: Fast Laplace updates γm and λ iteratively, while σ2 is selected by minimizing modified cross-validation error, avoiding case-specific manual tuning.
  • Limitation and extension: Heavier-tailed likelihoods such as Student-t models are proposed as future extensions when local regression residuals are strongly heavy-tailed.
  • Evaluation metrics: PBL, KSD, and WD measure quantile accuracy, maximum CDF mismatch, and overall distributional discrepancy, respectively.
  • Training-time scaling: The reported training-time scaling varies ASSE training time with Nref and pmax on the IEEE 9-bus system while unspecified parameters remain fixed.
Loading 2609.00547v1…