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Combining Probabilistic Load Forecasts

Yi Wang, Ning Zhang, Yushi Tan, Tao Hong, Daniel Kirschen, Chongqing Kang

arXiv:1803.06730v1stat.AP

TL;DR

Probabilistic load forecasting offers richer uncertainty information, but formal combination of probabilistic forecasts remains limited. The paper proposes CQRA, which estimates constrained quantile-combination weights by minimizing pinball loss in a linear program. Across ISO-NE and Irish smart-meter data, the ensemble reduces pinball score by 4.39% on average versus the best individual forecast and outperforms nine benchmark ensembles.

  • Problem

    Forecast combination has not been formally adopted for combining probabilistic load forecasts, despite the value of quantile-based uncertainty information.

  • Method

    CQRA combines individual quantile forecasts by solving a linear program that minimizes pinball loss subject to nonnegative weights summing to one.

  • Results

    4.39% average pinball-score reduction versus the best individual probabilistic forecast, with superior performance over nine other benchmark ensembles.

  • Takeaways & Limitations

    The CQRA ensemble improves probabilistic load-forecasting performance over individual forecasts and benchmark ensembles in the reported case studies.

  • Takeaways & Limitations

    The reported method is evaluated on load datasets, while extension to probabilistic renewable-energy forecasting remains future work.

Abstract

from arXiv · show

Probabilistic load forecasts provide comprehensive information about future load uncertainties. In recent years, many methodologies and techniques have been proposed for probabilistic load forecasting. Forecast combination, a widely recognized best practice in point forecasting literature, has never been formally adopted to combine probabilistic load forecasts. This paper proposes a constrained quantile regression averaging (CQRA) method to create an improved ensemble from several individual probabilistic forecasts. We formulate the CQRA parameter estimation problem as a linear program with the objective of minimizing the pinball loss, with the constraints that the parameters are nonnegative and summing up to one. We demonstrate the effectiveness of the proposed method using two publicly available datasets, the ISO New England data and Irish smart meter data. Comparing with the best individual probabilistic forecast, the ensemble can reduce the pinball score by 4.39% on average. The proposed ensemble also demonstrates superior performance over nine other benchmark ensembles.

I. INTRODUCTION

Probabilistic forecasts represent future load uncertainty more comprehensively than single-valued forecasts, while combining such forecasts remains comparatively underdeveloped. The paper proposes CQRA, an LP-based quantile forecast combination method evaluated on ISO-NE and Irish household loads.

  • Motivation: Probabilistic forecasts express future load uncertainty through intervals, densities, or quantiles rather than a single estimated value.Uncertainty arises across planning horizons and is especially volatile for small areas or individual households.
  • Forecast combination: Forecast combination integrates multiple forecasting models because no individual method performs best across all datasets.Heterogeneous ensembles combine forecasts from different techniques, while homogeneous ensembles vary inputs, features, or targets within one algorithm.
  • Research gap: The literature on combining probabilistic load forecasts remains limited despite prior density-forecast and pinball-score-based approaches.Existing studies include sequential density combination and weighted kernel-density or quantile-regression methods.
  • Proposed method: CQRA combines quantile forecasts by estimating optimal weights through a linear program that minimizes pinball loss.The method targets individual quantiles and is evaluated using ISO-NE and CER Irish household load datasets.
  • Evaluation: The proposed ensemble outperforms individual forecasts and nine benchmark ensembles most of the time.The paper evaluates the approach on zonal and system-level ISO-NE loads and household-level Irish loads.

III. PROBABILISTIC LOAD FORECASTING MODEL GENERATION

The paper generates diverse probabilistic load forecasts using quantile regression models and varied hyperparameters or input data. Pinball loss provides the optimization and evaluation basis for these quantile forecasts, including neural-network models.

  • Model generation: The model-generation stage produces diverse quantile forecasts using three typical regression-model families and varied hyperparameters or input data.The families are quantile regression neural networks, quantile regression random forests, and quantile regression gradient boosting.
  • Quantile regression: Quantile regression is formulated as an optimization problem that minimizes pinball loss for a selected quantile.The pinball loss is described as a comprehensive index of forecast reliability, sharpness, and calibration.
  • Quantile regression: Pinball loss uses a weighted absolute error whose weight depends on whether the forecast is above or below the observed value.The weighting is asymmetric according to the target quantile q ∈ (0, 1).
  • Model diversity: The regression functions differ across models, including ANN, GBRT, and RF, allowing distinct quantile forecasting functions.The paper integrates these regression methods with pinball loss for load forecasting.
  • QRNN: The ANN models use hidden-layer weights, bases, output weights, and output biases to produce forecast values.The paper chooses a sigmoid activation function and adds weight-decay regularization to reduce ANN overfitting.

B. Quantile Regression Random Forests (QRRF)

QRRF uses bootstrap-trained decision trees to produce multiple estimates, from which quantile distributions and forecasts are derived. Four QRRF variants are generated by varying the randomly sampled variables.

  • Random Forests: Random Forests are decision-tree-based learning methods in which each tree is trained on a bootstrap-sampled subset of the training data.The trees use CART-style training and provide individual forecast results.
  • Quantile regression random forests: A QRRF obtains a forecast distribution and quantiles by computing them from the multiple estimates produced by its decision trees.Although RF is itself an ensemble regression method, it serves here as a base probabilistic load forecasting model.
  • Model variants: Four QRRF models are trained by setting MRF to 1/2, 1/3, 1/4, and 1/5 of the whole training dataset.The paper uses NRF = 500 trees to ensure each input row is predicted at least a few times.

C. Quantile Regression Gradient Boosting (QRGB)

QRGB builds gradient-boosted quantile forecasts iteratively by fitting new trees to previous residuals. The paper varies the number of trees to create four QRGB models for ensemble construction.

  • Gradient boosting: Gradient boosting differs from RF by adding each new decision tree to fit residuals from previously generated trees.The final regression model is trained iteratively.
  • Gradient boosting: Each gradient-boosting iteration attempts to minimize the residuals left by earlier trees.The method uses a learning rate and terminal-node estimates in its tree-based updates.
  • Model variants: The number of boosting iterations, NGB, is varied from 70 to 100 in increments of 10 to obtain four QRGB models.NGB corresponds to the total number of trees fitted.
  • Model generation: The complete model-generation stage contains thirteen quantile regression models spanning QRNN, QRRF, and QRGB variants.The variants combine homogeneous models with different hyperparameters and heterogeneous model families.

IV. CQRA BASED MODEL COMBINATION

The paper combines quantile forecasts by estimating either shared or quantile-specific nonnegative weights, while using sequential time-series splits to reduce overfitting during model integration and evaluation.

  • Pinball loss provides a comprehensive probabilistic-forecast score and matches the quantile-regression objective.
  • The shared-weight formulation uses identical individual-model weights across all quantiles.
  • The quantile-specific formulation allows different model weights for different quantiles and solves Q separate optimization problems.
  • T1 trains individual models, T2 validates them for hyperparameter tuning, T3 supplies forecasts for model combination, and T4 evaluates the final ensemble.
  • The paper roughly chooses time-series split points by number of years rather than optimizing the split strategy.

C. Model Combination

The combination optimization is converted into a linear program using auxiliary variables for pinball loss, while quantile crossing is handled by naive rearrangement rather than additional joint constraints.

  • The method approximates the quantile of a weighted sum by the weighted sum of corresponding individual quantiles.
  • Auxiliary variables represent the maximum of the two pinball-loss expressions, enabling the optimization to be rewritten as a linear program.
  • Removing the final equality constraint yields a relaxed linear program whose optimum is also optimal for the original model.
  • The alternative joint integration strategy would add variables and constraints to prevent quantile crossing, increasing model complexity and computation time.
  • The paper instead applies naive rearrangement because separate combination models performed well and only few quantile crossings occurred in the case studies.
  • Compared with Bayesian model averaging, the method uses pinball loss rather than posterior probabilities and can combine models without explicit probability structures.

V. EVALUATION INDEXES AND COMPETING METHODS

The evaluation uses pinball loss as the primary index and compares the proposed approach with nine alternative weight-determination or quantile-combination methods.

  • The study introduces nine competing ensemble methods for comparison.The competitors include naive sorting, median-based selection, simple averaging, weighted averaging, three QRA methods, and two CQRA methods.
  • Pinball loss evaluates probabilistic forecasts comprehensively across reliability, sharpness, and resolution-related performance.
  • The naive sorting method estimates a new quantile sequence by sorting the N × Q observed quantiles in descending order.
  • The median-based method selects the median among the N forecasts for each quantile as the final quantile.

3) Simple Averaging (SA):

The benchmark methods combine quantile forecasts through equal weighting, accuracy-based weighting, or quantile regression applied to averaged quantiles.

  • Simple Averaging (SA): Simple averaging assigns equal weights to the different forecasting methods.
  • Weighted Averaging: Weighted averaging gives higher weights to methods with higher accuracy.
  • Quantile Regression Averaging: QRA-E applies linear quantile regression to the average of the individual models’ quantiles.
  • Quantile Regression Averaging: The parameters of QRA-E are estimated by minimizing the pinball loss function.

6) QRA-A:

The paper distinguishes QRA variants by the quantiles used as regressors and whether model weights are constrained. Its proposed method uses targeted quantiles and is identified as CQRA-T, while CQRA-A applies constrained regression to all quantiles.

  • QRA variants: QRA-A applies linear quantile regression to all averaged quantiles SAt to produce new quantiles.This distinguishes QRA-A from QRA-E, which uses targeted q-th quantiles as regressors.
  • Constrained variants: CQRA-A extends QRA-A by adding constraints to the regression weights.The CQRA-A formulation conducts constrained regression on all quantiles SAt.
  • Proposed method: The paper’s proposed method is called CQRA-T because it uses targeted quantiles as regressors, with weights ωq equal to βT q.The dimensions of the regression parameters differ because the QRA variants use quantiles of different dimensions.
  • Method comparison: QRA methods differ in both whether their weights are constrained and which quantiles they use.Table I summarizes the competing methods along these two dimensions.

VI. CASE STUDIES

Case studies evaluate probabilistic load-forecast combination across ISO-NE zones and total load, comparing pinball losses, quantile-level improvements, and selected model weights. The proposed combination achieves the lowest pinball loss across all nine load profiles, although improvements are not uniform across quantiles.

  • Dataset and evaluation: The ISO-NE case study evaluates eight zones and their total load using individual and combination probabilistic forecasting methods.The dataset covers zonal and system-level load profiles.
  • Overall performance: The proposed combining method has the lowest pinball loss for all nine load profiles, improving 4.39% on average over the best individual model.The best individual model is the individual forecast with the best performance.
  • Overall performance: Most combination methods outperform the best individual model, but the two QRA models for VT perform slightly worse than it.The results therefore do not show universal improvement from combination.
  • Overall performance: Adding nonnegative, sum-to-one constraints to QRA-E strongly worsens performance by restricting forecasts to a small interval bounded by the component quantiles.The constrained variant is denoted CQRA-E.
  • Quantile-level performance: Relative improvements vary across zones and quantiles, with some quantiles failing to improve and no clear relationship between improvement and quantile.Figure 4 reports relative improvements for different quantiles in different zones.
  • Model selection: For total load, 6 to 9 models are retained across quantiles, while Methods #12 and #13 are pruned for all quantiles because of poor performance.The pruned models need not be the individually worst-performing models, and weights need not change smoothly across adjacent quantiles.
  • Model selection: At the 90th quantile, different models are pruned across ISO-NE zones, indicating zone-specific model selection.Figure 6 presents the selected models for the 90th quantile across zones.

B. Hour-ahead Residential Load Forecasting

The study evaluates the proposed ensemble on volatile residential load profiles from ten Irish consumers. It generally performs best, but its advantage is not universal and further dataset refinement remains future work.

  • Data Description: The CER Irish dataset contains hourly profiles for ten selected residents from 2009-7-15 to 2010-12-31.The original 30-minute data were aggregated to one-hour load profiles.
  • Results: High volatility in consumer #1002's load profile prevents the 10-th and 90-th quantiles from effectively covering spikes.This contrasts with the system-level load profiles discussed in the results.
  • Results: The proposed combining method achieves the best performance for all selected consumers except consumer #1016.For consumer #1016, sample averaging, weighted averaging, QRA, and the proposed averaging do not outperform the best individual method.
  • Results: The proposed ensemble improves forecasting performance over individual models in terms of pinball loss.The ensemble problem is formulated as a series of linear programs, one for each quantile.
  • Future Work: Future work includes integrating pruning methods and evaluating the approach on additional load datasets and renewable energy forecasting.The method operates on quantile results and is not limited to load forecasting.
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