Stochastic Model Predictive Control of Charging Energy Hubs with Conformal Prediction

Technical Summary: Stochastic Model Predictive Control of Charging Energy Hubs with Conformal Prediction

Problem Statement
The rapid electrification of the transport sector necessitates infrastructure capable of managing large fleets of electric vehicles (EVs). Energy hubs, which integrate renewable generation (photovoltaics), battery energy storage systems (BESS), and grid connections, are critical for meeting this demand. However, operating these hubs cost-effectively is challenging due to the uncertainty inherent in EV charging loads, solar generation, and day-ahead electricity prices. Existing energy management systems (EMS) often rely on rule-based methods, heuristics, or deterministic optimization that neglects uncertainty or assumes specific data distributions, limiting their robustness and optimality.

Methodology
The authors propose an online energy management system based on a scenario-based Stochastic Model Predictive Control (MPC) framework. The core methodology integrates three main components:

1. System Modeling: The energy hub is modeled as a power flow system balancing grid power ($P_g$), BESS power ($P_b$), solar generation ($P_{pv}$), and EV demand ($P_{ev}$). The BESS dynamics include quadratic loss modeling relaxed into second-order cone constraints. The control objective is to minimize operational costs (buying/selling electricity) over a 24-hour horizon while satisfying power balance and battery capacity constraints.
2. Probabilistic Forecasting via Conformal Prediction: To address uncertainty without relying on distributional assumptions, the authors employ a machine learning pipeline.
   • Base Models: Gradient-boosted trees are used to generate point forecasts for 24-hour-ahead EV load, solar generation, and electricity prices.
   • Uncertainty Quantification: The Conformal Prediction (CP) algorithm, specifically EnbPI (Ensemble of Prediction Intervals), is applied to the base models. This generates calibrated, distribution-free confidence intervals (prediction intervals) for the forecasts.
   • Scenario Generation: Scenarios are constructed by selecting specific percentiles (e.g., 5th, 95th, and mean) from the prediction intervals. These scenarios are combined into a stochastic tree to represent the joint uncertainty of the variables, assuming independence between them.
3. Control Strategies: The paper evaluates two stochastic MPC variants against two deterministic benchmarks:
   • Stochastic MPC: Optimizes the expected cost across all generated scenarios simultaneously, enforcing a non-anticipativity constraint where control actions are identical across scenarios within the current time step.
   • Recourse MPC: A variant that relaxes the non-anticipativity constraint for future steps, allowing adjustments based on scenario realization.
   • Deterministic Benchmarks: One uses perfect (omniscient) forecasts, and the other uses point estimates without uncertainty quantification.

Key Contributions

• Integration of CP with MPC: The paper introduces a probabilistic prediction module for EV load, PV generation, and electricity prices that leverages state-of-the-art machine learning models combined with Conformal Prediction. This provides calibrated, distribution-free uncertainty intervals, a feature often missing in standard EMS implementations.
• Scenario-Based Optimization: The authors devise a scenario-based stochastic MPC scheme that utilizes these probabilistic forecasts to optimize battery (dis)charging strategies.
• Comprehensive Evaluation: The system is evaluated in a closed-loop simulated environment over 280 days, comparing the proposed stochastic approaches against deterministic alternatives and an omniscient benchmark.

Results
The evaluation was conducted using historical data from California (EV load and PV) and the Netherlands (electricity prices and CO2 emissions) across four seasons.

• Forecasting Performance: The Gradient-boosted models achieved low Normalized Mean Absolute Error (nMAE) for EV load and PV generation. However, electricity price forecasting showed higher error and lower prediction interval coverage in Autumn 2021, attributed to market regime changes driven by gas prices.
• Operational Cost: Compared to the Omniscient (perfect forecast) implementation, the proposed Stochastic MPC resulted in operational costs that were approximately 13% higher.
• Comparison with Deterministic Control: The Stochastic MPC outperformed the Deterministic MPC (using point forecasts) by reducing operational costs by 1%.
• CO2 Emissions: The stochastic approaches also yielded a slight reduction in grid CO2 emissions (0.3% improvement over the deterministic version).
• Computational Efficiency: The computational time for the Stochastic MPC was approximately 0.74 minutes per episode, well below the 15-minute time resolution ($\Delta T$) of the system, confirming its feasibility for real-time implementation.

Significance and Claims
The authors claim that their proposed Stochastic and Recourse MPCs represent the best overall performing controllers for the specific case study, offering a slight but consistent performance improvement over deterministic counterparts across all seasons. The work demonstrates that incorporating calibrated, distribution-free uncertainty quantification via Conformal Prediction into an MPC framework can yield cost-optimal operations without incurring prohibitive computational costs.

The paper modestly concludes that while point-estimate predictions yield "acceptable performance," the stochastic approaches provide a measurable benefit in cost and emissions. It highlights that the computational overhead of the stochastic methods remains negligible relative to the system's time resolution. The authors note that future work will focus on jointly optimizing charging power levels and schedules for individual vehicles, rather than the aggregated approach used here.