Supplementary Materials for: Resilient Optimization of Data Center-Renewable Energy Coupled Systems under Compound Extreme Weather Events – A Critical Review of Stochastic Programming and Distributionally Robust Models

This supplementary material accompanies the manuscript "Resilient Optimization of Data Center-Renewable Energy Coupled Systems under Compound Extreme Weather Events: A Critical Review of Stochastic Programming and Distributionally Robust Models". It provides comprehensive theoretical proofs, empirical data processing details, and algorithmic implementation specifications that support the numerical experiments and methodological innovations presented in the main text.

Appendix A – Theoretical Proofs establishes the mathematical foundations of the Adaptive DRO-CVaR-N1 framework. It proves the Kantorovich duality theorem for Wasserstein distances and demonstrates the convexity properties that ensure computational tractability of the distributionally robust optimization problem. The appendix further provides rigorous derivations transforming probabilistic N-1 security constraints into deterministic mixed-integer second-order cone constraints via Lagrangian duality and Big-M reformulation. Non-anticipativity constraints in multi-stage settings are formalized to ensure causal consistency of recourse decisions.

Appendix B – Supplementary Empirical Data documents the data preprocessing and robustness checks underlying the Guizhou case study. It specifies ERA5-Land reanalysis data extraction parameters, including spatial/temporal resolution and variable selection, and describes the encoding methodology for karst microclimate features such as quasi-stationary front persistence and freezing rain identification. Sensitivity analysis of the Wasserstein radius ε (0.01, 0.05, 0.10) demonstrates the Pareto trade-off between operational cost and resilience coverage. A comparison of extreme value distribution specifications (GEV versus Gumbel) validates the necessity of heavy-tailed models for capturing compound extreme weather risks.

Appendix C – Algorithmic Implementation Details ensures computational reproducibility of the numerical experiments. It specifies the Python/Gurobi implementation environment (version 10.0), MISOCP solver parameters, and column generation stopping criteria. The appendix also documents the Deep Q-Network baseline configuration, including network architecture, exploration strategy, and prioritized experience replay parameters. Training dynamics over 5,000 episodes reveal the sample efficiency crisis inherent to reinforcement learning under non-ergodic conditions, empirically validating the theoretical safety concerns discussed in the main text.

All meteorological data were extracted from the ERA5-Land reanalysis dataset (Copernicus Climate Change Service, 2000–2024). Processed data and model outputs are available from the corresponding author upon reasonable request. The supplementary materials are provided under a Creative Commons Attribution 4.0 International License (CC BY 4.0) to facilitate open science and reproducibility.