cs.LGJan 31, 2026

CoCopula: Conformal Calibration and Copula Aggregation for Reliable Renewable Energy Forecasting

Authors: Alireza Moradi, Mathieu Tanneau, Reza Zandehshahvar, Pascal Van Hentenryck

Organizations: NSF Artificial Intelligence Institute for Advances in Optimization H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology, Atlanta, GA, USA

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Abstract

The rapid growth of renewable energy penetration has intensified the need for reliable probabilistic forecasts to support grid operations at aggregated (fleet or system) levels. In practice, however, system operators often lack access to fleet-level probabilistic models and instead rely on site-level forecasts produced by heterogeneous third-party providers. Constructing coherent and calibrated fleet-level probabilistic forecasts from such inputs remains challenging due to complex cross-site dependencies and aggregation-induced miscalibration. Motivated by a real-world operational challenge identified through collaboration with an industry partner, this paper presents a probabilistic aggregation and calibration framework that directly converts site-level probabilistic forecasts into reliable fleet-level forecasts. The paper introduces \method~ (Conformal Calibration and Copula Aggregation), which integrates Copula-based dependence modeling to capture cross-site correlations followed by Context-Aware Conformal Prediction (CACP) to mitigate miscalibration at the aggregated level. This combination enables dependence-aware aggregation while improving empirical coverage and maintaining sharp prediction intervals. Experiments on large-scale solar generation datasets from three system operators in the United States including MISO, ERCOT, and SPP demonstrate that the proposed \method{} approach achieves near-nominal empirical coverage and consistently lower Winkler scores than competing aggregation-based baselines across different target coverage levels. Importantly, \method{} achieves performance comparable to directly calibrated system-level forecasts, demonstrating its practical value when dedicated fleet-level forecasting models are unavailable.

Figures & tables

Fig. 1: Conceptual overview of the proposed CoCopula framework. Site-level probabilistic forecasts are first combined using copula-based aggregation to capture cross-site dependencies and construct a fleet-level predictive distribution. Context-aware conformal prediction (CACP) then calibrates the aggregated prediction intervals using historical observations and contextual similarity, producing reliable and sharp fleet-level forecasts without requiring a dedicated system-level forecasting model.
Fig. 2: Spatial distribution of solar generation sites across MISO, ERCOT, and SPP, illustrating the geographic diversity of the systems evaluated in this work.
Fig. 3: Coverage–sharpness trade-off for the MISO, ERCOT, and SPP systems. The figure shows the relationship between empirical coverage and average interval width for different aggregation and calibration strategies, for target coverage levels from 50% to 95%. CoCopula achieves the lowest interval width among the aggregation-based methods for a given coverage and closely tracks NREL+CACP.
Fig. 4: Hourly empirical coverage for the MISO, ERCOT, and SPP systems at target coverage 1−α=80%1-\alpha=80\% . Solid lines denote system-level coverage for different methods, while the semi-transparent bars indicate the corresponding site-level coverage.
ISOMethod(1−α)=90%(1{-}\alpha)=90\%(1−α)=80%(1{-}\alpha)=80\%(1−α)=70%(1{-}\alpha)=70\%(1−α)=60%(1{-}\alpha)=60\%
  PICP ↑\uparrowAIW ↓\downarrowWS ↓\downarrowPICP ↑\uparrowAIW ↓\downarrowWS ↓\downarrowPICP ↑\uparrowAIW ↓\downarrowWS ↓\downarrowPICP ↑\uparrowAIW ↓\downarrowWS ↓\downarrow
ERCOTNREL †0.75010.18530.34880.69100.15850.25290.62490.13500.21070.55410.11320.1826
NREL+CQR †0.87600.21650.32300.77360.16600.24790.68640.13790.20790.57970.11240.1806
NREL+CACP †0.92090.23170.28780.80800.16780.23050.68630.13080.19770.56620.10240.1735
Ind0.14490.01940.97190.11980.01510.50950.09260.01230.35010.07460.01000.2685
Copula0.60940.13010.47340.53470.10170.32800.47120.08230.26280.40560.06620.2224
TABLE I: Fleet-level forecasting performance across MISO, ERCOT, and SPP at four target coverage levels, averaged over weekly test windows. PICP, AIW, and WS denote empirical coverage, average interval width, and Winkler score, respectively. Lowest WS among the aggregation-based methods in blue , second lowest underlined; † NREL-based system-level forecasts are included as reference benchmarks and excluded from ranking.
Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

SettingValue
Kernel ψ\psi{RBF, Laplacian}
Bandwidth γ\gamma{0.5, 1, 2}
Validation setLast 7 days before each test window
Monte Carlo samples SS1000
Appendix
TABLE II: Hyperparameters of the copula aggregation and CACP.
Fig. 5: Example day-ahead probabilistic forecast for the MISO, ERCOT, and SPP systems on 1 July of the test year (2019 for MISO and SPP, 2018 for ERCOT). The figure compares prediction intervals produced by Copula and CoCopula at the 5%–95% level. In MISO, the CACP calibration step notably reduces interval width during peak generation hours while maintaining coverage; in ERCOT and SPP, it widens the intervals, most visibly near sunrise and sunset, where Copula misses.
Fig. 6: Mean width of the 90% prediction interval by hour of day over the test period for MISO, ERCOT, and SPP. The figure compares CoCopula with the directly calibrated NREL+CACP forecasts; CoCopula gives narrower intervals during the midday peak-generation hours.
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