Nonparametric Kernel Regression for Coordinated Energy Storage Peak Shaving with Stacked Services

1. Problem Statement

The paper addresses the challenge of optimizing behind-the-meter battery energy storage systems (BESS) in commercial buildings. The primary objective is to coordinate two competing services:

1. Peak Shaving: Reducing the monthly peak demand charge (which is often the largest component of electricity bills) by discharging the battery during high-demand periods.
2. Energy Arbitrage: Buying electricity when prices are low and selling (discharging) when prices are high to maximize revenue.

Key Challenges:

• Uncertainty: Peak shaving requires decision-making over a monthly horizon, but demand and electricity prices are uncertain.
• Computational Complexity: Traditional methods relying on long-horizon forecasting (e.g., neural networks, regression) or stochastic optimization are computationally intensive and often impractical for real-time control.
• Coordination: Balancing the need to reserve capacity for peak shaving (a conservative, safety-first approach) with the desire to use that capacity for arbitrage (a profit-driven approach) is difficult without explicit forecasts.

2. Methodology

The authors propose an end-to-end, two-stage framework that decouples the optimization problem, allowing for lightweight, real-time control without relying on explicit demand forecasts.

A. Theoretical Decomposition

The authors prove that the combined optimization problem can be decomposed into two sequential stages under the condition that the peak demand penalty ($\kappa$) is significantly larger than energy costs ($c, \lambda$).

1. Stage 1 (Peak Shaving): Determine the minimum State-of-Charge (SoC) reserve trajectory required to meet peak shaving targets. This stage prioritizes minimizing the peak demand ($p$) and establishes a "safety buffer" of energy.
2. Stage 2 (Arbitrage): Utilize the remaining battery capacity (total capacity minus the Stage 1 reserve) to perform energy arbitrage.

• Theoretical Guarantee: The paper provides a proof (Proposition 1) showing that as the peak penalty $\kappa \to \infty$, solving these two stages sequentially yields the same optimal solution as the original combined problem.

B. Nonparametric Kernel Regression (Stage 1)

Instead of using forecasting models to predict future demand, the authors use a data-driven kernel regression approach to predict the necessary SoC reserve and peak targets directly from historical patterns.

• Input Features: Historical demand look-back windows (e.g., the last $T$ time steps), time-of-day features (sine/cosine vectors), and current demand.
• Algorithm:
  • K-Nearest Neighbors (KNN): Identifies the $K$ most similar historical demand patterns to the current state using Euclidean distance.
  • Gaussian Kernel Weighting: Assigns weights to these neighbors based on similarity.
  • Confidence-Level Adjustment ($\alpha$): To handle uncertainty, the model does not predict the mean. Instead, it calculates the $\alpha$-quantile of the historical SoC reserves associated with the nearest neighbors. This allows the operator to tune the conservativeness (e.g., ensuring the reserve is sufficient 95% of the time).
  • Peak Target Prediction: The peak demand target is predicted as a weighted average of historical peaks.

C. Real-Time Control (Stage 2)

A real-time controller uses the predicted SoC reserve trajectory ($\hat{e}_{t+1}$) and peak targets ($p_{t+1}$) to manage the battery:

• Peak Shaving Logic: If demand exceeds the target, the battery discharges to meet the reserve requirement. If demand is low, the battery charges up to the reserve limit.
• Arbitrage Logic: Any capacity remaining after satisfying the peak shaving reserve is allocated to arbitrage using a non-anticipatory policy (from prior work [12]).

3. Key Contributions

1. Decoupled Control Formulation: Theoretical proof that the complex stacked service problem can be solved via a two-stage decomposition (Peak Shaving $\to$ Arbitrage) without loss of optimality under realistic cost structures.
2. Forecast-Free Prediction: Development of a kernel regression algorithm that jointly predicts SoC reserve requirements and peak targets directly from historical data, eliminating the need for computationally expensive demand forecasting models.
3. Confidence-Based Conservativeness: Introduction of an $\alpha$-confidence parameter in the regression model, allowing operators to explicitly tune the trade-off between battery utilization and the risk of failing to shave a peak.
4. End-to-End Framework: Integration of the regression model with a real-time controller and arbitrage strategy, validated on a large-scale commercial building dataset.

4. Simulation Results

The method was tested using 6 years of demand data (2019–2024) from a 35-story commercial building in New York City.

• Performance vs. Benchmarks:
  • Compared against a Deterministic Perfect Foresight model (upper bound) and a Demand Forecasting model (XGBoost-based).
  • The proposed Kernel Regression method achieved 1.3 times better performance than the state-of-the-art forecasting-based method.
  • In terms of cost savings relative to the perfect foresight benchmark, the Kernel Regression method captured 38.1% – 58.5% of the potential savings, whereas the forecasting method only captured 33.0% – 39.8%.
• Peak Shaving Effectiveness: The method successfully reduced monthly peak demand, outperforming the forecasting model in most months.
• Battery Cycling & Wear:
  • The Deterministic model (perfect foresight) cycled aggressively (~346–366 cycles/year) to maximize savings.
  • The Kernel Regression model exhibited moderate cycling (~76–126 cycles/year), balancing cost savings with battery longevity.
  • The Forecasting model cycled the least (~57–95 cycles/year) but achieved lower savings, indicating it was too conservative or inaccurate.
• Robustness: The method performed well across various battery power ratings (0.1 MW to 1.0 MW) without requiring retraining or complex parameter tuning for different scenarios.

5. Significance

• Practicality for Commercial Buildings: The method offers a computationally lightweight solution suitable for smaller businesses and non-profits where the cost of complex optimization or high-fidelity forecasting is prohibitive.
• Reliability without Forecasts: By relying on historical pattern matching rather than prediction, the system avoids the "garbage in, garbage out" problem associated with inaccurate demand forecasts, which is critical for monthly peak shaving where a single missed peak can be costly.
• Scalability: The two-stage decomposition allows for the easy addition of other stacked services (e.g., frequency regulation) in the future without re-engineering the core peak shaving logic.
• Economic Impact: The study demonstrates that significant cost reductions (up to ~58% of theoretical maximum) can be achieved with a simple, data-driven approach that extends battery life by avoiding unnecessary aggressive cycling.