Supervised Imitation Learning for Optimal Setpoint Trajectory Prediction in Energy Management Under Dynamic Electricity Pricing

Energy management systems operating under dynamic electricity pricing require fast and cost-optimal control strategies for flexible loads. Mixed-integer linear programming (MILP) can compute theoretically optimal control trajectories but is computationally expensive and typically relies on accurate load forecasts, limiting its practical real-time applicability. This paper proposes a supervised imitation learning (IL) framework that learns optimal setpoint trajectories for a conventional proportional (P) controller directly from electricity price signals and temporal features, thereby eliminating the need for explicit load forecasting. The learned model predicts setpoint trajectories in an open-loop manner, while a lower-level P controller ensures stable closed-loop operation within a two-stage control architecture. The approach is validated in an industrial case study involving load shifting of a refrigeration system under dynamic electricity pricing and benchmarked against MILP optimization, reinforcement learning (RL), heuristic strategies, and various machine learning models. The MILP solution achieves a cost reduction of 21.07% and represents a theoretical upper bound under perfect information. The proposed Transformer model closely approximates this optimum, achieving 19.33% cost reduction while enabling real-time inference. Overall, the results demonstrate that the proposed supervised IL approach can achieve near-optimal control performance with substantially reduced computational effort for real-time energy management applications.