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Probabilistic electrical power demand forecasting with uncertainty quantification

Mahesh Neupane, Pragya Dhungana, Pradip Khatri, Swechhya Baskota, Hariom Dhungana

arXiv:2609.34120v1cs.LGcs.AI

TL;DR

Renewable variability and changing consumer behavior increase uncertainty in power-system demand and operation, motivating probabilistic forecasting. The study compares Bayesian, MC Dropout, NGBoost, and GPR models; across the evaluated zones, NGBoost achieves the lowest deterministic errors and strong probabilistic scores with high interval coverage.

  • Problem

    Renewable variability and dynamic consumer behavior introduce significant uncertainty into power-system demand and operation, complicating reliable planning.

    Source: §I · p. 1

  • Method

    The study compares Bayesian, MC Dropout, NGBoost, and GPR probabilistic forecasting models and evaluates temporal and meteorological inputs.

    Source: §II · p. 2, B. Monte Carlo (MC) Dropout · p. 3, §IV · p. 6

  • Results

    Across zones, NGBoost achieves the lowest MAE and RMSE, with PICP ≈0.95–0.96 and reasonably narrow prediction intervals.

    Source: §IV · p. 5

  • Takeaways & Limitations

    Among the compared models, NGBoost provides the most reliable probabilistic forecasts for the considered power-consumption zones.

    Source: §V · p. 8

Abstract

from arXiv · show

The majority of research on electricity consumption forecasting has focused on deterministic approaches, which generate a single point estimate for each time step in the forecasting horizon. However, the increasing penetration of renewable energy sources and the growing complexity of modern smart grids have introduced greater variability and uncertainty into power-system demand and operation. Consequently, probabilistic forecasting, which quantifies the uncertainty and variability associated with future electricity demand, is becoming increasingly important for reliable power-system planning and operation. This study presents an empirical comparison of four contemporary probabilistic forecasting models for electricity consumption, highlighting their respective strengths and limitations. We have performed comparision on real-world power systems related datasets. Across all power-consumption zones, NGBoost demonstrates superior probabilistic forecasting performance, achieving the lowest MAE and RMSE while providing well-calibrated uncertainty estimates with high prediction-interval coverage and reasonably narrow intervals. These results indicate that NGBoost offers a more accurate and reliable forecasting framework than Bayesian, Monte Carlo (MC) Dropout, and Gaussian Process Regression (GPR) models for the considered electricity consumption data.

I. INTRODUCTION

Renewable variability and changing consumer behavior add uncertainty to power-system demand and operation, increasing the value of forecasts that quantify possible outcomes. The introduction contrasts deterministic point forecasts with probabilistic approaches and presents the paper’s comparison of forecasting techniques.

I. INTRODUCTION

  • Renewable variability and dynamic consumer behavior introduce uncertainty into grid operation, making reliable forecasts important for planning and management.The passage links forecasting to resource allocation, system stability, cost reduction, and long-term energy planning.Source: §I · p. 1
  • Deterministic methods provide a single point prediction, whereas probabilistic methods characterize multiple possible outcomes and their probabilities.Uncertainty quantification describes prediction reliability and variability, but may add computational cost and depends on model quality and accurate characterization of uncertainty sources.Source: §I · p. 1
  • Quantile Regression estimates conditional demand quantiles rather than only the mean, enabling prediction intervals and explicit uncertainty quantification.Multiple quantiles characterize the conditional distribution; the passage also notes robustness to outliers and non-Gaussian noise.Source: §I · p. 1
  • The introduction identifies missing IoT measurements and variable selection as forecasting concerns, noting that excessive inputs can challenge neural networks sensitive to feature scale.Weather, holidays, and economic data are among the external factors considered for forecasting.Source: §I · pp. 1–2
  • The paper compares load-forecasting techniques and highlights uncertainty quantification, risk-informed decisions, and proper-score evaluation such as CRPS as uses of probabilistic forecasts.The stated comparison concerns methodological differences, strengths, and limitations.Source: §I · p. 2

II. PROBABILISTIC FORECASTING

The paper evaluates probabilistic demand forecasts using temporal and meteorological inputs, and compares four models spanning distinct probabilistic frameworks. Forecasts represent distributions that express both expected values and uncertainty.

II. PROBABILISTIC FORECASTING

  • The study evaluates demand forecasting with temporal features and meteorological variables, including temperature, humidity, wind speed, diffuse solar radiation, and global diffuse flows.The passage states that temporal features alone cannot fully represent weather-induced variability affecting heating and cooling loads.Source: §II · p. 2
  • Feature scaling uses training-set means and variances, then applies the fitted scaler unchanged to validation and test data to prevent leakage.StandardScaler transforms features to zero mean and unit variance.Source: §II · p. 2
  • A predictive distribution expresses the expected value as well as the forecast’s spread and tail behavior.The notation identifies the future value at horizon k and the information available at time t.Source: §II · p. 2
  • The four compared frameworks are Bayesian linear regression, Monte Carlo Dropout, Gaussian Process Regression, and Natural Gradient Boosting.They represent parametric Bayesian, approximate Bayesian deep-learning, non-parametric kernel-based, and distributional-boosting approaches, respectively.Source: §II · p. 2

A. Bayesian Linear Regression

Bayesian Linear Regression treats regression parameters as random variables and uses priors to represent parameter uncertainty. Its predictive distribution combines expected demand with uncertainty from data noise and the model.

A. Bayesian Linear Regression

  • Bayesian Linear Regression treats model parameters as random variables rather than fixed but unknown quantities.The model uses regression coefficients and Gaussian noise, with a commonly used multivariate Gaussian prior over the coefficients.Source: A. Bayesian Linear Regression · p. 2
  • Conjugacy yields a closed-form Gaussian posterior that explicitly captures parameter uncertainty.The posterior is over the weights given the data.Source: A. Bayesian Linear Regression · p. 2
  • For a new input, the predictive distribution is Gaussian, with its mean as the expected prediction and its variance reflecting both data noise and model uncertainty.The paper identifies this uncertainty estimation as useful in energy-consumption forecasting.Source: A. Bayesian Linear Regression · p. 2

B. Monte Carlo (MC) Dropout

MC Dropout estimates uncertainty without changing a neural network’s architecture by retaining dropout at inference and sampling predictions across stochastic forward passes.

B. Monte Carlo (MC) Dropout

  • MC Dropout estimates neural-network uncertainty without modifying the underlying architecture.Dropout layers remain active during inference, and repeated stochastic forward passes approximate Bayesian inference.Source: B. Monte Carlo (MC) Dropout · p. 3
  • For an input, MC Dropout estimates the predictive distribution by sampling T network outputs with dropout activated.The passage describes these outputs as stochastic forward-pass samples.Source: B. Monte Carlo (MC) Dropout · p. 3
  • Wt denotes the randomly dropped network weights used at the t-th forward pass.Source: B. Monte Carlo (MC) Dropout · p. 3

C. Gaussian Process Regression

Gaussian Process Regression (GPR) is a non-parametric approach that predicts distributions over functions consistent with observed data. For a new input, its predictive distribution is Gaussian, with a mean and variance.

C. Gaussian Process Regression

  • GPR predicts a distribution over functions consistent with observed data, providing mean predictions and uncertainty estimates.It defines a Gaussian process prior over the latent function.Source: C. Gaussian Process Regression · p. 3
  • The covariance function k(x, x′) encodes similarity between inputs, while the mean function m(x) is often assumed to be zero.Source: C. Gaussian Process Regression · p. 3
  • For a new input, GPR gives a Gaussian predictive distribution whose mean and variance depend on training-point covariances and noise variance.Source: C. Gaussian Process Regression · p. 3

D. Natural Gradient Boosting

NGBoost extends gradient boosting to estimate full predictive distributions by learning distribution parameters with natural gradients. At each iteration, it fits a base learner to the natural gradient of a proper scoring rule.

D. Natural Gradient Boosting

  • NGBoost models the conditional distribution p(y | x; θ) by parameterizing a probability distribution and iteratively learning its parameters with natural gradients.Source: D. Natural Gradient Boosting · p. 3
  • At each boosting iteration, a base learner, typically a decision tree, fits the natural gradient of a proper scoring rule such as negative log-likelihood.Source: D. Natural Gradient Boosting · p. 3
  • The parameter update uses the negative log-likelihood gradient, inverse Fisher information, and learning rate.Source: D. Natural Gradient Boosting · p. 3

A. Dataset

The study uses hourly electricity-demand data from three distribution zones in Tetouan, Morocco, and describes daily, weekly, monthly, and hourly consumption patterns. The dataset includes temporal, meteorological, and flow-related features.

A. Dataset

  • The dataset contains 52,416 observations from three Tetouan distribution zones, recorded every 10 minutes through 2017 and resampled hourly.It had no identified missing values, duplicates, outliers, or anomalous readings before resampling.Source: A. Dataset · p. 3
  • Zone 1 has the highest demand and greater variability, while Zones 2 and 3 follow similar daily and weekly patterns at lower, more stable levels.The weekly profile covers the first week of January 2017.Source: A. Dataset · p. 3, Figure 1 · p. 4
  • Average monthly consumption stays near 30,000 units, with a modest increase in June–August.Source: A. Dataset · p. 3, Figure 2 · p. 4
  • Average hourly consumption is lower from 02:00–07:00 and rises to an evening peak around 20:00–21:00.Source: A. Dataset · p. 4, Figure 2 · p. 4

B. Evaluation metric

The evaluation uses seven metrics to assess predictive accuracy and uncertainty quality. These include accuracy and distribution-fit measures, quantile and interval reliability measures, and interval sharpness.

B. Evaluation metric

  • MAE measures average absolute error, RMSE evaluates point accuracy using the predictive mean, and NLL measures fit of the predicted distribution.Lower NLL is better.Source: B. Evaluation metric · p. 4
  • CRPS assesses probabilistic forecast accuracy, while pinball score evaluates quantile prediction accuracy across quantiles and the forecast horizon.CRPS captures reliability and sharpness; pinball score assesses calibration, reliability, and sharpness.Source: B. Evaluation metric · p. 4
  • PICP measures the share of true values inside predicted intervals, while MPIW measures interval width; the study uses 95% Gaussian prediction intervals.PICP near the nominal confidence level indicates better calibration, and narrower MPIW is better.Source: B. Evaluation metric · p. 4

IV. RESULTS AND DISCUSSIONS

Across the three power-consumption zones, NGBoost has the strongest reported overall performance, combining lower errors and strong probabilistic scores with high interval coverage. The results also examine model uncertainty, feature importance, and computational cost.

IV. RESULTS AND DISCUSSIONS

  • For XGBoost, hour is the most important feature, with mean SHAP value above 5000; month ranks second, followed by temperature.GeneralDiffuseFlows, day, DiffuseFlows, WindSpeed, and Humidity each have mean SHAP values below 1000, though their importance is non-zero.Source: §IV · pp. 4–5, Figure 3 · p. 5
  • GPR has substantially higher errors and PICP ≈0, indicating inaccurate mean predictions and severely miscalibrated uncertainty estimates.The passage attributes its difficulty to high dimensionality, non-stationarity, kernel and hyperparameter sensitivity, and cubic computational complexity that limits scaling.Source: §IV · p. 5
  • ≈75–89 s is GPR's training time across zones, compared with ≈0.01–0.05 s for Bayesian models and ≈22–63 s for MC Dropout and NGBoost.The passage relates GPR's higher cost to cubic-complexity kernel matrix inversion.Source: §IV · p. 5
  • Across zones, NGBoost achieves the lowest MAE and RMSE, strong probabilistic scores, and PICP ≈0.95–0.96 with reasonably narrow intervals.Table II compares models across zones using MAE, RMSE, NLL, CRPS, Pinball Loss, PICP, and MPIW.Source: §IV · p. 5, Table II · p. 6
  • Across zones, 95% intervals widen at high demand and contract at low demand; BLR bands are smoother, while NGBoost width fluctuates more locally.The passage identifies this demand-linked variation as heteroscedasticity and reports a positive association between uncertainty and load magnitude.Source: §IV · p. 6
  • NGBoost most effectively replicates the observed consumption distributions across zones, with sharp, well-calibrated 95% intervals compared with Bayesian and MC Dropout.The KDE discussion reports reasonable alignment for all models in Zone 1, including bimodal peaks around 20,000 and 35,000 units; NGBoost is slightly shifted right there.Source: §IV · p. 6, Figure 6 · p. 7

V. CONCLUSION AND FUTURE WORKS

Across multiple power-consumption zones, NGBoost outperforms the compared Bayesian, MC Dropout, and GPR models, combining lower errors with calibrated uncertainty estimates. The paper also proposes evaluating additional probabilistic models under a unified framework, with attention to calibration, robustness, and deployment trade-offs.

V. CONCLUSION AND FUTURE WORKS

  • Across multiple power-consumption zones, NGBoost outperforms Bayesian, MC Dropout, and GPR, with the lowest quantitative errors and well-calibrated uncertainty estimates, high prediction-interval coverage, and sharpness.The conclusion presents NGBoost as a robust and practical framework for accurate, reliable probabilistic power-demand forecasting.Source: §V · p. 8
  • Probabilistic forecasts provide richer, more informative predictions than deterministic approaches and support uncertainty-aware planning and operation in modern energy systems.The conclusion says the comparison offers guidance for selecting probabilistic models in real-world smart-grid and energy-management applications.Source: §V · p. 8
  • Future work will evaluate heteroscedastic neural networks and Bayesian deep learning in a unified framework using CRPS, NLL, reliability, and sharpness metrics.The proposed evaluation emphasizes calibration and robustness across diverse operating and weather conditions.Source: §V · p. 8
  • Future research will systematically assess trade-offs among predictive accuracy, uncertainty quality, and computational efficiency to inform practical deployment in real-world power systems.Source: §V · p. 8
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