Optimal Prediction Intervals of Wind Power Generation

The paper addresses the challenge of producing accurate and reliable forecasts for wind power, a variable and uncertain resource increasingly important to power systems. Traditional point forecasts are limited due to nonstationarity and lead to significant errors. Probabilistic forecasting, particularly prediction intervals (PIs), can better quantify uncertainty and support operational decisions such as reserve allocation, unit commitment, storage control, and market bidding. Existing PI methods often rely on statistical inference and distributional assumptions of forecast errors. The research question is how to directly construct optimal PIs of wind power that ensure both reliability (coverage close to nominal confidence) and sharpness (narrow intervals), without assuming an error distribution. The authors propose a hybrid intelligent algorithm combining Extreme Learning Machine (ELM) and Particle Swarm Optimization (PSO) to directly optimize PI bounds for multiple confidence levels simultaneously, targeting improved reliability and sharpness across various look-ahead horizons relevant to system and market operations.

The paper reviews probabilistic wind power forecasting approaches: ensemble meteorological forecasts to derive predictive distributions (e.g., Horns Rev studies), approaches linking wind speed forecast errors and non-linear power curves, quantile regression to estimate conditional quantiles, and nonparametric methods combining probability forecasts with adaptive resampling. Other methods include radial basis function neural networks for quantile forecasts and conditional kernel density estimation for full predictive densities. Neural-network-based PI construction methods often assume normally distributed errors, which may not hold and are incompatible with ELM training dynamics. The LUBE (Lower Upper Bound Estimation) method has been applied to loads and wind power but suffers from traditional NN limitations (overtraining, computational burden) and evaluation issues (coverage width-based criterion). Many prior wind PI methods separate PI construction from performance assessment and often rely on point forecast errors and distributional assumptions. The authors aim to integrate construction and assessment via direct optimization, aligning with decision-theoretic views on optimal interval estimation.

• Model: An Extreme Learning Machine (ELM) for single-hidden-layer feedforward networks is used to map inputs (e.g., past wind power, optionally weather predictors) to lower and upper PI bounds at multiple nominal confidence levels simultaneously. ELM randomly sets input weights and hidden biases; output weights are solved via least-squares (Moore–Penrose pseudoinverse), enabling fast initialization and strong generalization.
• PI formulation: For each input xi and target ti, a PI I^(α)(xi) = [L^(α)(xi), U^(α)(xi)] is produced with nominal confidence 100(1−α)%. The framework generates multiple PIs for α = [α1, α2, ..., αn] in one model.
• Evaluation criteria: Reliability measured by PI Coverage Probability (PICP) and Average Coverage Error (ACE = PICP − PINC), with ACE ideally near zero. Sharpness assessed via the interval score, which penalizes width and non-coverage; the average absolute interval score over the test set summarizes overall skill (lower magnitude implies sharper intervals, conditional on similar reliability).
• Objective function: Multi-objective optimization directly on ELM output weights to balance reliability and sharpness across n confidence levels: minimize sum over i of γi·ACE_i + λi·|S̄_i|_norm, where S̄_i is the average interval score at level i normalized between 0 (perfect) and 2α (most conservative). Constraints enforce valid and nested PIs across confidence levels (lower ≤ upper; and bounds are ordered across α).
• Optimizer: Particle Swarm Optimization (PSO) minimizes the non-differentiable objective over the ELM’s output weights. PSO maintains a swarm of candidate solutions updated via inertia, cognitive, and social components, with boundary handling.
• Hybrid Intelligent Algorithm (HIA) steps:
  1. Prepare dataset Dt = {(xt, tt)}. Create initial training targets for upper and lower bounds by scaling tt by ±ρ% to initialize ELM for interval outputs.
  2. Initialize ELM with random input weights and biases; compute initial output weights β_int.
  3. Initialize PSO particles around β_int (positions and velocities).
  4. Iterate PSO: evaluate objective on Dt using ELM with candidate β; update personal and global bests; update velocities and positions; enforce constraints and bounds; loop until convergence or max iterations.
  5. Use optimized β to generate PIs; evaluate on test data.
• Data/use case: Hour-ahead PIs generated using only wind power series as inputs for computational efficiency; framework is extensible to include NWP for longer horizons. Multiple confidence levels are produced in a single optimization (90%, 95%, 99%).

• Data: Two Australian wind farms were used for case studies with 1-hour resolution: Challicum Hills (52.5 MW, 35×1.5 MW, Sep 2008–Aug 2010 data) and Starfish Hill (34.5 MW, 23×1.5 MW, Jan 2009–May 2010 data). Testing periods: Challicum Hills Mar–Aug 2010; Starfish Hill Jan–May 2010; remaining data for training.
• Benchmarks: Climatology, Constant (normal with mean/variance from observations), Persistence-based probabilistic (normal), Exponential Smoothing Method (ESM; normal), and Quantile Regression (QR).
• Reliability (PICP) near nominal and very small ACE for proposed method at all PINCs: • Challicum Hills: 90% PINC → PICP 90.80%, ACE 0.80%, score −6.61%; 95% → 95.50%, 0.50%, −3.94%; 99% → 98.48%, −0.52%, −1.14%. • Starfish Hill: 90% → 90.91%, 0.91%, −6.43%; 95% → 94.90%, −0.10%, −4.00%; 99% → 99.28%, 0.28%, −1.08%.
• Compared to benchmarks, the proposed method: • Achieves PICPs closest to nominal with |ACE| generally < 1% across all tested levels and both sites, outperforming or matching QR in reliability and clearly outperforming climatology, constant, persistence, and ESM, especially at higher confidences (≥95%). • Delivers the smallest magnitude interval scores across all cases, indicating the sharpest intervals while maintaining reliability (e.g., Starfish Hill 90%: score −6.43% best among all methods).
• Visual inspections (figures) show actual wind power well covered by the proposed PIs; abnormal predictive mass beyond feasible generation bounds was censored, yielding realistic predictive densities.
• Overall, the proposed HIA produces the best comprehensive PI performance (reliability + sharpness) across both wind farms and all studied confidence levels.

The findings demonstrate that directly optimizing PI bounds via ELM and PSO yields intervals that are both reliable (calibrated coverage near nominal) and sharp (narrow), addressing the core objective of probabilistic wind power forecasting without relying on distributional assumptions of forecast errors or prior point forecasts. The method integrates construction and evaluation within a single performance-oriented optimization, improving over traditional two-stage approaches. The ability to generate multiple confidence levels simultaneously adds practical value for operators needing different risk tolerances. Results across two distinct wind farms confirm robustness and generality for short-term (hour-ahead) applications. These calibrated intervals can enhance operational decision-making in reserve sizing, unit commitment, storage dispatch, and market bidding by providing quantified uncertainty. The framework’s flexibility suggests straightforward extension to include meteorological predictors for longer horizons, potentially further improving performance.

The paper introduces a hybrid intelligent algorithm that combines Extreme Learning Machine and Particle Swarm Optimization to directly construct optimal prediction intervals for wind power. A novel objective function balances reliability (ACE) and overall skill/sharpness (interval score), enabling simultaneous optimization of multiple confidence levels. Case studies on two Australian wind farms show highly calibrated coverage (|ACE| < 1% in most cases) and superior sharpness versus five benchmarks, including quantile regression. The approach offers a generalized, flexible framework for probabilistic wind power forecasting with strong potential for operational applications such as reserve determination, market participation, and wind farm control. Future work may include incorporating numerical weather prediction inputs for longer look-ahead horizons, extending to other renewable sources and time resolutions, and integrating the PIs directly into stochastic optimization tools for power system operations and planning.

• Validation is limited to two wind farms in Australia and hour-ahead forecasts using only historical wind power as inputs; generalization to other regions, time horizons, and input sets (e.g., NWP) should be tested.
• Benchmark comparisons focus on high confidence levels (90%, 95%, 99%); performance at lower confidences or different operating conditions is not reported.
• The optimization relies on PSO hyperparameters and initialization (including the initial ±ρ% bound generation), which may influence convergence and results; sensitivity analyses are not detailed.
• Some generated PIs required censoring to adhere to feasible generation limits, indicating the need for explicit constraints or post-processing in practical deployments.