Online Convex Optimization for Coordinated Long-Term and Short-Term Isolated Microgrid Dispatch

Technical Summary: Online Convex Optimization for Coordinated Long-Term and Short-Term Dispatch Control of Isolated Microgrids

Problem Statement
Isolated microgrids in remote regions face significant challenges in ensuring reliable power supply due to their reliance on intermittent renewable energy sources (RES) and lack of connection to a main grid. These systems typically rely on a mix of flexible resources: diesel generators (DG) and battery energy storage (BES) for rapid response, as well as long-duration energy storage (LDES) such as hydrogen systems for seasonal energy conversion. The core difficulty lies in coordinating dispatch strategies across multiple time scales (daily versus seasonal) under uncertainty.

Existing approaches have specific limitations:

1. Non-anticipatory constraints: Real-time dispatch must operate without knowledge of future uncertainties, rendering conventional multi-period optimizations (which assume perfect foresight) impractical.
2. Temporal coupling: LDES introduces strong temporal coupling constraints via State-of-Charge (SoC) dynamics and long-term contractual obligations (e.g., target SoC values at period end), which are difficult to manage in short-term, myopic dispatch.
3. Non-convexity: LDES models, particularly electrochemical dynamics in electrolyzers and fuel cells, are inherently non-convex. Using traditional piecewise linear approximations requiring binary variables leads to high computational complexity.
4. Forecast dependence: Many methods (e.g., Model Predictive Control) rely heavily on forecast accuracy, which is often low in isolated microgrids, leading to suboptimal decisions or load shedding.

Methodology
The article proposes a non-anticipatory coordination framework for long-term and short-term dispatch control that integrates offline training with online control. The methodology consists of four main components:

1. Convex Hull Approximation for LDES:
   To address the non-convex electrochemical dynamics of LDES, the authors introduce a convex hull approximation model. Instead of using piecewise linear models with binary variables, this approach selects discrete samples from the non-convex power-hydrogen curve to form the vertices of a convex hull. Decision variables are represented as convex combinations of these vertices. This reformulation preserves convexity, eliminates binary variables, and ensures compatibility with gradient-based optimizations while maintaining high accuracy.

2. Offline Training with Hindsight-Optimal Trajectories:
   The framework utilizes historical or synthetic data to generate "hindsight-optimal" SoC trajectories and net loads by solving the Oracle Multi-Period Economic Dispatch (OED) problem offline. These trajectories serve as ground truth for training the online reference learning module.

3. Online SoC Reference Learning via Kernel Regression:
   In the online stage, the system dynamically updates the LDES SoC reference using a non-parametric conditional imitation strategy based on kernel regression.

   • Inputs: The model uses a rolling window of normalized net loads and historical SoC values.
   • Mechanism: It calculates dynamic weights based on the similarity between current online observations and historical scenarios using a Gaussian kernel.
   • Advantage: Unlike neural networks, this method offers low computational complexity, requires minimal hyperparameter tuning, and adapts to natural changes in system state without retraining.

4. Adaptive Online Convex Optimization (OCO) Algorithm:
   Short-term dispatch is formulated as a trajectory-guided single-period economic dispatch (TED). This includes a quadratic penalty term for tracking the long-term SoC reference generated by kernel regression.

   • Expert Tracking: The OCO algorithm employs an adaptive, virtual queue-based approach with "expert tracking." Multiple experts operate with different step sizes.
   • Decision Making: Decisions are updated based on surrogate losses and weighted averages of expert decisions, enabling the system to adaptively optimize step sizes and mitigate the myopia of single-period dispatch without reliance on future forecasts.

Main Contributions

1. Convex LDES Modeling: Development of a scalable, accurate convex hull approximation for LDES that avoids binary variables and preserves tractability for gradient-based methods.
2. Non-Anticipatory Coordination: A novel framework coordinating long-term and short-term dispatch without future information by utilizing hindsight-optimal trajectories for offline training and kernel regression for online reference updates.
3. Theoretical Guarantees: Rigorous proof that both long-term and short-term strategies achieve sublinear dynamic regret bounds. Regret improves with more regression scenarios, stronger tracking penalties, and finer convex approximations.
4. Resilient Online Control: An adaptive OCO algorithm integrating reference tracking and expert tracking to effectively handle temporal coupling constraints and system uncertainties.

Results
The proposed method was validated on an isolated microgrid in Alaska (100 kW wind, 100 kW solar, 150 kW load, 50 kW diesel, 50 kW/200 kWh BES, and 100 kW/1000 kg LDES) over a one-year hourly simulation.

• Cost Reduction: The proposed method reduced annual operating costs by 73.4% compared to state-of-the-art methods without reference tracking.
• Elimination of Load Loss: Reference tracking successfully eliminated load losses (0 kWh), whereas methods without tracking exhibited load losses exceeding 26 MWh.
• Algorithmic Improvement: The adaptive OCO algorithm achieved an additional 2.4% cost savings compared to standard MPC approaches.
• Scalability: Benefits scaled with longer LDES durations; with an LDES capacity of 2000 kg, the optimality gap decreased to 19.89% compared to 34.83% at 500 kg.
• Robustness: The method demonstrated resilience against poor forecasts (outperforming MPC by up to 17.88% at 30% MAPE) and unexpected system disturbances (wind/DG failures), with offline trajectories regenerated within minutes to update online references.

Significance and Claims
The article claims to provide a near long-term optimal, forecast-free, and computationally tractable online dispatch tool for microgrids. Its significance lies in:

• Bridging Time Scales: Effective coordination of seasonal (long-term) and daily (short-term) operations under uncertainty without the need for accurate long-term forecasts.
• Theoretical Rigor: Provision of sublinear regret bounds for both long-term and short-term strategies, a feature often lacking in heuristic or learning-based approaches.
• Practical Applicability: Delivery of a solution robust to system disturbances and forecast errors, making it suitable for real isolated microgrids where reliability is paramount.
• Addressing Non-Convexity: Successful handling of the computational intractability of non-convex LDES models via convex hull approximation, enabling the use of efficient gradient-based solvers.

The authors emphasize that the method requires no future information (0-lookahead), making it suitable for real-time implementation where uncertainty is realized sequentially.