1. Introduction

Modern power and energy systems are undergoing rapid transformation due to the rise of renewable energy and distributed infrastructure (Xu et al., 2021). This project addresses two critical challenges:

• Long-term investment planning in multi-carrier energy markets (electricity, gas, thermal) under regulatory incentives and strategic competition (Valinejad et al., 2017).
• Short-term uncertainty quantification in stochastic economic dispatch (SED) with high-dimensional wind power data (Valinejad et al., 2020).

Through a fusion of multi-level optimization, Bayesian machine learning, and manifold learning techniques, this work provides an integrated, computationally efficient solution for resilient, adaptive energy system operations and planning (Hu et al., 2020).

2. Key Questions Addressed

• How can strategic GENCOs plan investments across electricity, gas, and CHP units considering market regulations and competition?
• What are the optimal incentive mechanisms (e.g., capacity payments, firm contracts) to drive efficient infrastructure development?
• How can we model and forecast the uncertainty of renewable generation for real-time market operations?
• Can Gaussian Process Emulators and manifold learning replace Monte Carlo simulations in uncertainty propagation?
• How do these tools integrate into practical grid planning, operation, and policy analysis?

3. Problem Scope

This project targets two distinct but connected layers of power system challenges:

A. Strategic Investment in Multi-Energy Markets

• Tri-Level Optimization: Captures GENCO investment strategy, market offerings, and system-level welfare maximization.
• Regulatory and Financial Incentives: Models policy impacts like tax credits, firm contracts, and capacity payments.
• Stochastic Load Modeling: Uses Markov chains to simulate multi-stage demand uncertainty.

B. Uncertainty Quantification in Stochastic Economic Dispatch

• High-Dimensional Renewable Inputs: Hourly wind farm outputs across multiple locations, modeled as spatiotemporal random fields.
• Curse of Dimensionality: Addressed with nonlinear dimensionality reduction (Isomap).
• Surrogate Modeling: GPE replaces time-intensive Monte Carlo methods, enabling efficient inference.

4. Why It Matters

• For Policymakers: Quantifies the effectiveness of financial incentives for clean energy expansion (Valinejad et al., 2019).
• For System Operators: Enables reliable real-time decision-making under renewable variability.
• For Industry: Provides an integrated framework for investment, operation, and forecasting in future grid infrastructure.
• For Researchers: Demonstrates how advanced ML tools can solve hard optimization and simulation problems in power systems.

5. Methodological Highlights

5.1. Multi-Level Market Optimization

• Tri-level structure: Investment → Offering → Social Welfare
• Mathematical Tools: Mixed-Integer Linear Programming (MILP), Karush–Kuhn–Tucker (KKT) conditions
• Scenario Analysis: Demand uncertainty simulated using branching Markov chains
• Case Study: Realistic energy hub with electricity, gas, and heat networks (without IEEE benchmark dependence)

5.2. Bayesian Uncertainty Quantification (UQ) Framework

• Dimensionality Reduction: Isomap captures latent structure in wind data better than traditional PCA/KLE
• Gaussian Process Emulators (GPEs): Serve as nonparametric surrogate models for SED
• Bayesian Inference: Uses MLE and quadratic basis priors for accurate mean/variance estimation
• Sampling: Latin Hypercube Sampling (LHS) in latent space allows efficient scenario generation
• KDE-Based PDF Estimation: Models non-Gaussian latent distributions without parametric assumptions

6. Key Results

• 35% improvement in cost-optimized investment strategies with proper incentive modeling
• ~0.1% relative error in cost estimates using only 100 GPE training samples (vs. 8,000 MC samples)
• Computational time reduced from hours (MC) to seconds (GPE-based UQ)
• Dimensionality reduction from 72D to 9D while preserving statistical accuracy
• Cross-season validation proves robustness in both winter and summer wind scenarios

7. Tools and Technologies

• Optimization: MILP, KKT, duality, tri-level programming
• ML & Stats: GPE, KDE, Isomap, MLE, Bayesian inference
• Sampling: Latin Hypercube Sampling (LHS), scenario generation
• Programming: Python, NumPy, SciPy, Scikit-learn, GAMS
• Data Science: Surrogate modeling, statistical forecasting, probabilistic modeling

8. Future Work

• Scale UQ methods to national-level or real-time control applications
• Combine this framework with multi-modal data (e.g., solar, grid status, price signals)
• Investigate deep GP and neural surrogate models for more expressive representations
• Develop APIs or lightweight inference engines for integration into energy platforms
• Extend market planning framework to include demand response, storage, and microgrids

9. Conclusion

This unified project leverages data science, optimization, and statistics to solve two fundamental challenges in the energy domain:

• How to plan infrastructure investments strategically in complex, regulated markets, and
• How to make robust, fast operational decisions under uncertainty using statistical surrogates.

By bridging market optimization with Bayesian uncertainty modeling, it provides a blueprint for modern, intelligent energy systems capable of balancing investment, reliability, and renewables integration at scale.