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A Multi-model Combination Approach for Probabilistic Wind Power Forecasting

You Lin, Ming Yang, Can Wan, Jianhui Wang, Yonghua Song

arXiv:1702.03613v1cs.LGstat.AP

TL;DR

Wind’s stochastic prediction errors make a universal probabilistic forecaster difficult, while accurate forecasting matters for power-system operation. This paper combines SBL, KDE, and BDE forecasts in an MMC model; with further optimization, it achieved the best probabilistic forecasting performance among the benchmarks and better balanced reliability and sharpness.

  • Problem

    Complicated wind-power prediction errors make it difficult to develop a universal probabilistic forecasting model, although accurate forecasts matter for power-system operation and control.

    Source: §I · p. 1, §V · p. 8

  • Method

    The MMC approach combines SBL, KDE, and BDE probabilistic forecasts, whose member models use different mechanisms and PDF forms.

    Source: §I · p. 2, B. Member Models · p. 3

  • Results

    The optimized MMC achieved the best probabilistic forecasting performance among the benchmarks and better balanced reliability and sharpness.

    Source: PECTATIONS (%) · p. 8

  • Takeaways & Limitations

    The proposed MMC model can be beneficial for optimal operation and control of power systems with high wind-power penetration.

    Source: PECTATIONS (%) · p. 8

Abstract

from arXiv · show

Short-term probabilistic wind power forecasting can provide critical quantified uncertainty information of wind generation for power system operation and control. As the complicated characteristics of wind power prediction error, it would be difficult to develop a universal forecasting model dominating over other alternative models. Therefore, a novel multi-model combination (MMC) approach for short-term probabilistic wind generation forecasting is proposed in this paper to exploit the advantages of different forecasting models. The proposed approach can combine different forecasting models those provide different kinds of probability density functions to improve the probabilistic forecast accuracy. Three probabilistic forecasting models based on the sparse Bayesian learning, kernel density estimation and beta distribution fitting are used to form the combined model. The parameters of the MMC model are solved based on Bayesian framework. Numerical tests illustrate the effectiveness of the proposed MMC approach.

I. INTRODUCTION

Wind power’s intermittency and uncertain prediction errors complicate power-system operation, while probabilistic forecasts quantify uncertainty for operational decisions. Because individual models and existing combinations have limitations, the paper motivates combining probabilistic models with different distribution forms.

I. INTRODUCTION

  • Wind-power intermittency and uncertainty pose challenges for power-system operation and control.Source: §I · p. 1
  • Probabilistic wind forecasts quantify uncertainty and can inform reserve setting, unit commitment, and market trading.Source: §I · p. 1
  • Individual probabilistic models may not suit all wind farms, while prior combination approaches used point-forecast errors or member distributions of the same type.Source: §I · p. 1
  • The paper proposes extending model averaging to combine probabilistic forecasting models with different PDF forms, including SBL, KDE, and BDE.The MMC parameters are estimated through an expectation-maximization procedure and further optimized for distribution forecast performance.Source: §I · pp. 1–2

A. Prediction Uncertainty

  • Model misspecification and data uncertainty contribute to prediction errors, and wind power’s nonlinear, non-stationary behavior further complicates forecasting.Source: A. Prediction Uncertainty · p. 2

B. Basics of Bayesian Model Averaging

A. Formulations of MMC

The MMC framework combines predictive PDFs from member models using model weights. It trains the members and estimates and further optimizes combination parameters in separate stages.

A. Formulations of MMC

  • MMC combines member-model predictive PDFs using model weights to form a combined predictive PDF.Source: A. Formulations of MMC · p. 3
  • Member models may use different forecasting approaches and need not use the same explanatory variables.Source: A. Formulations of MMC · p. 3
  • Training proceeds in two stages: train member models on one dataset, then estimate MMC parameters on an optimization dataset and further optimize their distribution forecast effect.Source: A. Formulations of MMC · p. 3

B. Member Models

The MMC uses three probabilistic member models with distinct distributional approaches: SBL, KDE, and BDE.

B. Member Models

  • The combined model uses SBL for conditional Gaussian PDFs, conditional KDE without a predetermined distribution form, and BDE for Beta-distribution PDF estimation.The selected models have different mechanisms and probabilistic forecasting performance.Source: B. Member Models · p. 3

1) Sparse Bayesian Learning

Sparse Bayesian learning uses a kernel-based probabilistic model to produce conditional forecasts. Its parameters are estimated by maximum likelihood, with Bayesian rules used to estimate weights.

1) Sparse Bayesian Learning

2) Kernel Density Estimation

Kernel density estimation (KDE) estimates a target variable’s probability density function without assuming its distribution type. For wind generation, it estimates a conditional PDF from historical observations using kernels whose bandwidths control smoothness.

2) Kernel Density Estimation

  • KDE estimates the target variable’s PDF without pre-assuming its distribution type.The paper describes this non-parametric approach as more adaptable than parametric estimation.Source: 2) Kernel Density Estimation · p. 4
  • Historical observations are used to estimate the conditional PDF of a target value for a new input through the law of total probability.The estimate uses joint and marginal PDFs of the input and target variables.Source: 2) Kernel Density Estimation · p. 4
  • Gaussian kernels are weighted, and the target-variable bandwidth controls the smoothness of the estimated PDF.A bandwidth matrix controls smoothness for the explanatory-variable vector; this paper selects the KDE bandwidth using Silverman’s rule of thumb.Source: 2) Kernel Density Estimation · p. 4

3) Beta Distribution Estimation

The beta distribution is used to represent normalized wind generation uncertainty because it is bounded between 0 and 1. Its shape parameters are obtained from estimated expectation and variance.

3) Beta Distribution Estimation

  • The beta distribution is used to describe wind-generation uncertainty because normalized generation lies in [0, 1].The passage states that the distribution is bounded by 0 and 1.Source: 3) Beta Distribution Estimation · p. 4
  • The beta distribution’s shape parameters α and β are positive, and its expected value and variance are related to those parameters.The paper gives expressions for α and β in terms of μ and σ^2.Source: 3) Beta Distribution Estimation · p. 4
  • Estimating expectation and variance from samples yields the beta distribution’s shape parameters and PDF.The paper states that the expectation and variance determine α and β.Source: 3) Beta Distribution Estimation · p. 4

C. Parameter Estimation

The paper estimates the multi-model combination parameters using training data, with separate training for the SBL and KDE members and expectation-and-variance estimation for the beta member. It uses EM for maximum likelihood estimation and further refines parameters against CRPS to address possible local optima.

C. Parameter Estimation

  • SBL and KDE members are trained independently, while the beta distribution’s shape parameters are estimated from its expectation and variance.An SVM spot-forecast model estimates the beta member’s expectation, and its variance is estimated separately.Source: C. Parameter Estimation · p. 4
  • EM is used to estimate member-model weights and the beta distribution variance by maximizing the training-data likelihood.The paper adopts EM because direct analytical or numerical maximization of the log-likelihood is difficult.Source: C. Parameter Estimation · p. 4
  • The EM procedure iterates expectation and maximization steps until parameter changes meet preset tolerances, then returns the maximum-likelihood estimator.The M step updates parameters by maximizing the expected log-likelihood from the E step.Source: C. Parameter Estimation · p. 4
  • Because EM maximum-likelihood estimates may converge to local optima, the parameters are further refined to optimize distribution accuracy using CRPS.The paper describes CRPS as assessing forecast-PDF calibration and sharpness simultaneously; lower CRPS indicates a better distribution forecast.Source: C. Parameter Estimation · p. 4

ASE STUDIES

The case studies evaluate MMC probabilistic forecasting performance against individual and benchmark models, including forecast errors, reliability, sharpness, and predictive intervals. They also examine wind–power relationships, component weights, performance across look-ahead times, and seasonal examples.

ASE STUDIES

  • The study examines autocorrelation and cross-correlation, reporting significant relationships between wind speed and direction and wind generation, especially for wind speed.The passages point to Fig. 3 for autocorrelation and Table II for cross-correlation results.Source: Data Descr · p. 5, Figure 3 · p. 5

WIND PARAMETE

  • In a 16-hour-ahead example, the observation falls within the 80% nominal coverage interval of the MMC predictive PDF.The MMC PDF is described as a weighted sum of member-model PDFs.Source: WIND PARAMETE · p. 5
  • MMC component weights reflect component contributions and vary over time and space.The passages refer to weights of the components and their variation across wind farms.Source: WIND PARAMETE · p. 6
  • MMC has lower MAE and RMSE than the individual models across the five wind farms.The passages identify accuracy results for five wind farms and state that MMC is more accurate; no numerical values are supplied.Source: WIND PARAMETE · p. 6

PECTATIONS (%)

  • Predictive-interval reliability is evaluated by comparing observed coverage rates with nominal coverage at different interval levels.The section describes coverage-rate assessment for predictive PDFs and compares different nominal coverage levels.Source: PECTATIONS (%) · p. 6
  • Reliability results indicate that the MMC improves forecast reliability across different look-ahead times.Source: PECTATIONS (%) · p. 6
  • MMC CRPS values are much smaller than those of the individual models except at the first three look-ahead times, with its advantage growing at longer lead times.Source: PECTATIONS (%) · p. 7
  • Winter and summer 24-hour examples show that 10%-90% predictive intervals cover observations well and the distributions capture time-varying heteroscedasticity.Source: PECTATIONS (%) · p. 7
  • The optimized MMC achieves the best probabilistic forecasting performance by balancing reliability and sharpness better than benchmarks, while EM optimization maintains high-quality predictive distributions.Source: PECTATIONS (%) · p. 8

V. CONCLUSIONS

The proposed MMC combines probabilistic forecast models with different distribution forms, and numerical studies report that it outperforms the adopted individual models.

V. CONCLUSIONS

  • Numerical studies demonstrate significant superiority of the proposed MMC over the adopted individual models.Source: §V · p. 8
  • The MMC forms a weighted combination of individual probabilistic forecast models with different parametric and nonparametric distribution forms.Source: §V · p. 8
  • Member-model weights are first estimated by EM and then optimized for comprehensive model performance, addressing both deterministic and probabilistic forecasts.Source: §V · p. 8
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