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

1. Introduction

Reducing carbon dioxide emissions and achieving global climate targets require the large-scale integration of renewable energy sources into modern power grids. However, the intermittent and variable nature of renewable generation poses significant challenges for maintaining grid stability and balancing supply and demand [1]. To address these challenges, energy management on the demand side has gained increasing importance, as flexible consumers can adapt their operation to support grid stability and reduce overall energy costs [2]. Traditionally, energy management systems have relied on classical heuristics, proportional-integral (PI) controllers, or model predictive control (MPC) approaches based on MILP and related optimization techniques [3]. Although MILP-based MPC formulations can guarantee global optimality under given model assumptions [4], their computational complexity often hinders their deployment in real-time and large-scale control applications, necessitating trade-offs between optimality and computational efficiency [5].

In contrast, RL has shown promise for adaptive process control and demand response in energy systems [6], yet RL models typically require extensive training [7] and can destabilize physical systems during training, making direct deployment in safety-critical environments challenging [8]. Additionally, both optimization-based and learning-based approaches can face limited robustness under process uncertainty and coordination-related stability issues in closed-loop operation. On the other hand, heuristic or rule-based control strategies, though computationally efficient, tend to underutilize the available operational flexibility and rarely achieve near-optimal economic performance [9,10].

To overcome these limitations, two-stage or hierarchical control architectures have been introduced, combining the advantages of rule-based and optimization-based strategies to achieve improved performance and system stability [11,12]. In a two-stage control architecture, the lower stage operates in real time using simplified rule-based or heuristic logic to ensure fast and stable system response, while the upper stage performs computationally intensive optimization over longer horizons. Advanced methods such as MILP or RL are often applied at the upper level to generate optimal setpoints or scheduling targets for the lower controller. This hierarchical structure effectively combines the real-time feasibility of classical control with the decision quality of optimization and learning-based approaches, achieving a balance between responsiveness and overall energy efficiency [12,13].

In parallel, Transformer architectures [14] have revolutionized sequence modeling by efficiently capturing long-range temporal dependencies through self-attention mechanisms. Compared to recurrent neural networks and LSTM models, Transformers excel in learning context-dependent relationships across multiple time scales, properties that are highly desirable in energy management and process optimization tasks. Recent applications demonstrate their superiority in load and renewable forecasting [15,16,17,18,19], yet their application to real-time setpoint trajectory generation within two-stage control architectures for industrial energy management under dynamic pricing has received limited attention.

1.1. Related Works

In energy management systems operating under dynamic electricity pricing, MILP has proven to be a well-established and effective optimization approach. Kepplinger et al. [21] implemented a MILP-based model predictive control strategy combined with a physics-based system model derived from transient energy balances to enable load shifting of smart water boilers under dynamic pricing, achieving cost savings of around 12%. Similar MILP-based MPC approaches have been successfully investigated for energy management of electric vehicles [22,23], net-zero energy buildings [24], energy communities [25], and industrial manufacturing [26,27,28,29], reporting cost savings typically around 30% for electric vehicle charging, up to 40–50% for net-zero energy buildings, and around 10–25% for industrial applications, depending on system flexibility and pricing volatility. In addition to simulation-based studies, selected works have further validated MILP-based energy management strategies in real-world field tests, demonstrating their practical feasibility under realistic operating conditions [30,31].

In recent years, RL has also been explored for energy management problems in dynamic pricing environments. Afroosheh et al. [32] proposed a deep Q-network (DQN)–based home energy management system that directly controls the scheduling of household loads and a battery energy storage system under dynamic electricity pricing while respecting user comfort constraints, achieving energy cost reductions of about 12%. In Muriithi et al. [33], a DQN–based energy management system is proposed for a grid-tied PV–battery microgrid operating under dynamic electricity price signals. The RL agent directly controls battery charging and power exchange with the grid based on the observed system state. Depending on the considered use case, the proposed approach achieves energy cost reductions between 15% and 26%.

While the aforementioned studies apply RL directly at the power and scheduling level, other works employ hierarchical control structures, in which RL operates at a supervisory level. Jiang et al. [34] applied a DQN for building HVAC control under a time-of-use electricity tariff with demand charges, where the RL agent adjusts temperature setpoints at a supervisory level, while a conventional lower-level controller translates these setpoints into actual power consumption. Simulation results show total cost reductions of approximately 6–8% compared to a rule-based baseline. Vetter et al. [35] investigate multi-energy optimization in an energy community under real-time electricity pricing, comparing a MILP formulation with a RL approach. While the MILP controller is implemented with direct power control, the DQN-based method follows a two-stage control structure, in which high-level decisions are translated into low-level control actions. As a result, the MILP approach exhibits slightly higher flexibility and achieves marginally higher cost reductions (about 10%) compared to the RL controller (approximately 9%). In an industrial context, a supervisory energy management framework compared MILP and DQN as RL approach for load shifting in a food processing plant under dynamic electricity pricing [36]. Both approaches operate at a supervisory control level by optimizing control setpoints that are implemented by a lower-level controller, with MILP serving as an optimization benchmark and RL as a data-driven alternative. The results demonstrate energy cost reductions of 18% for the RL-based approach and 19% for the MILP-based strategy, indicating that RL can achieve performance close to optimization-based control in industrial demand response applications.

These energy management systems typically require load or renewable generation forecasts. Supervised learning (SL) models, particularly Transformer-based architectures, are therefore widely used for load and renewable forecasting tasks. Liu et al. [15] apply Transformer-based SL to forecast solar radiation, photovoltaic power, wind speed, and wind power, demonstrating the effectiveness of sequence models for renewable energy forecasting. Tian et al. [16] present a day-ahead probabilistic load forecasting framework that combines convolutional neural networks with Transformer architectures, achieving high accuracy even with limited historical data, particularly for weekend load profiles. Moosbrugger et al. [17] investigate electrical load forecasting for energy communities using transfer learning, benchmarking LSTM, Transformer, and other machine learning models, with the forecasts subsequently applied in MILP-based energy management. Similarly, two studies by Wohlgenannt et al. [18,19] investigate electrical and thermal load forecasting in the food processing industry as inputs for downstream MPC via MILP or RL.

While most studies employ SL for forecasting to support downstream optimization, other works explore SL-based approaches that are used directly as controllers. For example, Novak and Dragičević [37] apply supervised IL to approximate finite-set MPC in power electronic converters, reducing computational effort while maintaining control performance. In the energy domain, Gao et al. [38] propose an IL–based approach for online optimal power scheduling of a microgrid, where a neural network approximates a MILP policy under uncertain prices and renewable generation. Dinh and Kim [39] propose a MILP-based IL approach for HVAC control, where a deep neural network is trained on optimal control actions generated by an MILP solver. The learned controller operates without explicit load or weather forecasts, using real-time data and day-ahead electricity prices to minimize energy costs while maintaining thermal comfort. Simulation results show near-optimal performance compared to forecast-based MILP, MPC, and RL approaches.

1.2. Contributions

Despite significant progress in optimization- and learning-based energy management, several practically relevant limitations remain insufficiently addressed. First, many strategies depend on explicit load or generation forecasts, which increases modeling effort and sensitivity to prediction errors, and complicates the overall learning pipeline since forecasting and decision-making must be trained and tuned separately rather than learned end-to-end. Second, while SL has been widely adopted in control and automation, its use for price-driven energy management is still limited and predominantly targets direct power scheduling rather than controller-compatible setpoint manipulation, which better aligns with practice. Third, most learning-based approaches are designed as single-stage controllers, whereas their integration into two-stage architectures, where a supervisory layer optimizes setpoints for embedded low-level controllers, is rarely investigated. Finally, experimental evaluations often focus on comparisons to non-optimized baselines, while systematic benchmarking against rule-based and heuristic strategies commonly applied in real settings remains scarce. Motivated by these gaps, this work makes the following contributions:

• A Transformer-based IL model predicts optimal refrigeration setpoint trajectories directly from historical price and temporal data, removing the need for intermediate load or production forecasts.
• Viability of IL-based prediction is proven within a systematic comparison to the optimal trajectory.
• Comprehensive evaluation of sequence, non-sequence, and linear SL models highlights the importance of temporal and non-linear modeling.
• Combined with a lower-level P controller, the model ensures stable operation while achieving cost-optimal setpoints, facilitating practical industrial implementation.

The remainder of this study is organized as follows. The system, the IL approach, and the case study design are described in Section 2. The case study results are presented and discussed in Section 3. Finally, conclusions are drawn in Section 4.

2. Methods

This study investigates a two-stage optimal control approach for energy management under dynamic pricing in an industrial refrigerated warehouse. The cooling demand is supplied by the refrigeration system, whereas the aggregated thermal mass of stored goods, building structure, and equipment constitutes an intrinsic thermal energy storage. Allowing a temperature band between 0 °C and 5 °C provides flexibility that can be exploited for load shifting and energy cost optimization. Our control framework follows a two-stage structure as can be seen in Figure 1. In the upper layer, different optimization strategies determine the optimal temperature setpoint trajectory. This setpoint serves as the reference for a lower-level P controller, which regulates the refrigeration system to maintain the desired warehouse temperature. This hierarchical setup ensures both optimized operation and system stability.

Various control strategies are investigated for the upper-level optimization. MILP, RL, and heuristic control are used as benchmark methods, while a novel IL approach is proposed. All control strategies aim to shift the electrical load of the refrigeration plant to periods with lower electricity prices under hourly real-time pricing.

Figure 2 illustrates the overall architecture and highlights the structural difference between the different methods. MILP explicitly incorporates the physical model, price signal, and load forecast to compute an optimal setpoint trajectory, which is then translated into cooling power by the P-controller. The resulting historical optimal trajectories serve as training targets for the IL model. The IL models are trained on historical MILP optimization results to learn optimal control behavior from prior data. Once trained, the IL model directly maps price signal and time information to a setpoint trajectory, thereby approximating the MILP solution without solving an optimization problem online. This enables a computationally efficient surrogate of the original optimization-based controller. RL is used as a closed-loop benchmark, learning a control policy through direct interaction with the environment by maximizing cumulative economic reward, while heuristic strategies provide rule-based baseline controllers without optimization or learning. A constant setpoint temperature of 2.5 °C, representing the current operational practice, is used as a reference case. Hyperparameters of all IL and RL algorithms were tuned using Bayesian optimization to achieve optimal performance on the training set. Further details on the individual upper-layer control strategies are provided in the following subsections.

2.1. Data Acquisition and Processing

The case study is based on data from previous work [36]. Hourly data from May 2020 to April 2023 obtained from an industrial refrigerated warehouse is used. The dataset includes production data (mass flow of products entering the warehouse, ${\dot{m}}_{\mathrm{in}}$), ambient temperature ($T_{\infty }$), the energy efficiency ratio of the refrigeration system ($\beta$), and estimated heat gains from the ambient (${\dot{Q}}_{\mathrm{gain}}$). While production data and thermal losses are available in the dataset, they are only used for benchmarking the proposed IL approach and are not provided as direct inputs to the forecasting model. Hourly spot market prices from the Energy Exchange Austria (EXAA) [44] served as the dynamic price signal for the optimization process. The electricity price was assumed to be available on a day-ahead basis, as it is published at noon for the following day by the EXAA. The dataset was divided into two subsets: two years were used for training, while the remaining year was reserved for testing and performance evaluation of IL against MILP, RL, and heuristic control as benchmarks. All simulations and analyses were implemented in Python and are executed on a MacBook Pro M2 from 2023.

2.2. Industrial Warehouse Model

Figure 3 shows a schematic of the industrial warehouse of a food processing plant, which was developed and validated in previous work [36], in combination with the two-stage controller.

The model represents the warehouse as a lumped thermal capacity $C_{\mathrm{H}}$ that includes both the building structure and the stored food. Heat gains from the ambient environment (${\dot{Q}}_{\mathrm{gain}}$) and mass flows of incoming (${\dot{m}}_{\mathrm{in}}$) and outgoing food products (${\dot{m}}_{\mathrm{out}}$) are combined into a simplified heat flow ${\dot{Q}}_{\mathrm{heat}}$. Cooling is provided by an industrial chiller controlled via a discrete-time P controller with saturation. The set point temperature $T_{\mathrm{set}}$ can be adjusted within a predefined band $[T_{\mathrm{lb}},T_{\mathrm{ub}}]$, providing thermal flexibility that can be exploited for load shifting. The simplified warehouse temperature dynamics are described by the discrete-time energy balance:

$$\begin{array}{c}\hfill T_{\mathrm{H},i+1}=T_{\mathrm{H},i}+{\displaystyle \frac{\left({\dot{Q}}_{heat,i}-P_{\mathrm{el},i}\beta \right)}{C_{\mathrm{H}}}}\Delta t\end{array}$$

(1)

The model parameters are listed in Appendix A. This model has been used as the environment for both RL and MILP-based optimization in previous work [36], and is reused here for evaluation.

2.3. Mixed-Integer Linear Programming

MILP provides the global optimal temperature setpoint trajectory, making it a strong benchmark for control performance. In this study, MILP is solved over a 36-hour prediction horizon, of which the first 24 hours are applied; the problem is then re-optimized in a rolling-horizon fashion. MILP outperforms other methods not only because perfect knowledge of the load is assumed, but also because it guarantees the global optimum, whereas IL, RL, or heuristic controllers can only approximate it. On the downside, MILP can become computationally expensive for problems with many binary variables and must be solved repeatedly for real-time operation. The full MILP formulation can be found in Appendix B. The MILP is formulated and solved using Gurobi [46]. Here, MILP is used as one reference benchmark to evaluate the performance of the IL models.

2.4. Supervised Imitation Learning Models

In this work, imitation learning is implemented in the form of supervised behavior cloning, where optimal MILP-generated setpoint trajectories serve as expert demonstrations. The learning problem is therefore formulated as a supervised regression task that directly maps electricity price and temporal features to optimal setpoint trajectories. The models considered in this study include sequence-based models (LSTM, Transformer), tree-based models (DecisionTree, RandomForest, ExtraTrees, GradientBoosting, AdaBoost, XGBoost, LightGBM), instance-based models (KNN), kernel-based models (SVR, KernelRidge), and linear models (LinearRegression, LinearSVR, Ridge, Lasso, ElasticNet, BayesianRidge, SGD, Huber, PassiveAggressive, PLS). The control policy is learned end-to-end without an explicit load forecasting step. The model maps electricity prices and temporal features directly to control setpoints, avoiding forecast-induced errors and additional model maintenance. The input features include:

• Electricity price at the current timestep.
• Hour of the day encoded as sine and cosine functions to capture daily periodicity.
• Weekday encoded as a one-hot vector to represent weekly patterns.
• Day of the year encoded as sine and cosine functions to capture seasonality.

All feature values are standardized based on the mean $\mu$ and the standard deviation $\sigma$ from the training dataset via z-transform:

$$\begin{array}{cc}\hfill X^{*}& ={\displaystyle \frac{X-\mu }{\sigma }}\hfill \end{array}$$

(2)

In addition, the electricity price is first min-max normalized for each day before standardization:

$$\begin{array}{cc}\hfill X_{\mathrm{minmax}}& ={\displaystyle \frac{X-X_{min}}{X_{max}-X_{min}}}\hfill \end{array}$$

(3)

The target of the prediction is the P-controller setpoint, constrained between 0 °C and 5 °C. All target values are standardized based on the training dataset. The optimal setpoint trajectories used as training targets were obtained by solving the corresponding MILP problems on the training data.

Model training was conducted using the historical MILP solutions as targets and the mean absolute error (MAE) as loss function, which is defined as follows:

$$\begin{array}{c}\hfill \mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n|y_i-{\widehat{y}}_i|\end{array}$$

(4)

The training dataset covers two years of operation and the remaining year is reserved for testing.

Hyperparameters for all models were tuned using Bayesian optimization with 50 trials, employing 5-fold cross-validation on the training set and a parameter grid specified in Appendix D. This procedure was applied individually for each model to maximize prediction accuracy on the validation folds.

The performance was evaluated on four levels. First, the economic impact of the predicted setpoint trajectories was quantified using the industrial warehouse model. The trajectories were implemented in the simulation environment, and the resulting operational costs were compared to a non-optimized reference scenario in which a conventional proportional P-controller operates with a constant temperature setpoint. This comparison directly quantifies the achievable cost reductions under dynamic electricity pricing.

Second, the quality of the predicted setpoint trajectories was evaluated using standard regression metrics, including MAE, root Mean Square Error (RMSE), $R^2$, and symmetric mean absolute percentage error (sMAPE), computed on the test set. These metrics assess the accuracy of the predicted setpoints independently of their economic effect. The error metrics are defined as:

$$\begin{array}{cc}\hfill \mathrm{RMSE}& =\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill R^2& =1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \mathrm{sMAPE}& ={\displaystyle \frac{100}{n}}\sum _{i=1}^n{\displaystyle \frac{|y_i-{\widehat{y}}_i|}{(\left|y_i\right|+\left|{\widehat{y}}_i\right|)/2}}\hfill \end{array}$$

(7)

Third, computational efficiency was evaluated by measuring both training time and inference time, providing insight into the practical applicability of each model for real-time operation and edge deployment.

Fourth, the robustness of the proposed approach was assessed by verifying whether each method consistently fulfills all temperature constraints throughout the entire test horizon.

Classical SL models were implemented using scikit-learn [47], while sequence-based models were implemented in PyTorch [48]. Hyperparameters for all models were tuned using Optuna [49].

2.5. Deep Reinforcement Learning

A DQN was implemented as a benchmark following the approach described in previous studies [19,35,36]. The network structure illustrates the interaction between the agent and the environment through states, actions, and rewards. The DQN framework comprises a policy network for action selection and a target network for Q-value estimation. To enhance learning efficiency and stability, experience replay with mini-batch sampling is employed, and soft updates are applied to the target network. The implementation builds on the original deep Q-learning algorithm introduced by Mnih et al. [40] and incorporates improvements proposed by Van Hasselt et al. [42], namely the separation of policy and target networks. Soft target updates, as described by Lillicrap et al. [45], further stabilize training, while experience replay ensures more efficient and stable learning, consistent with the original DQN methodology [40]. In contrast to the previous work [36], as first modification the planning horizon has been extended to 36 hours; however, at each decision point (0:00), only the first 24 hours of the optimized trajectory are applied. The state, action, and reward definitions follow [36], with the second modification that the load forecast was removed from the state representation to enable a fair comparison with the IL models. Consequently, the RL agent must implicitly capture and predict the underlying load behavior through its learned policy, rather than relying on an external forecasting module. Optimized hyperparameters were adopted from the earlier study [19]. The DQN was implemented using PyTorch [48].

2.6. Heuristic Approaches

Simple heuristic methods were implemented as baseline approaches to provide additional benchmarks for evaluating the performance of the IL models. The heuristics considered in this study include a day-night strategy and price-based approaches using daily mean, median, or min-max normalization. The day-night heuristic applies a fixed daily pattern, with lower setpoints during the night (0–6 h and 18–24 h) and higher setpoints during the day (6–18 h), independent of the electricity price. For each day, the mean or median price is computed; hours with prices above this threshold are assigned a high setpoint (5 °C), and hours below are assigned a low setpoint (0 °C). This encourages higher temperatures during low-price periods and lower temperatures during high-price periods. The min-max heuristic normalizes the daily electricity price to the range [0,1] and linearly scales the setpoint to a defined range (0–5 °C). Hours with the lowest prices are mapped to the lowest setpoints, and hours with the highest prices are mapped to the highest setpoints, providing a continuous price-dependent setpoint trajectory.

2.7. Ablation Study

An ablation study was conducted to assess the contribution of individual feature groups used by the Transformer model. In each experiment, one complete feature block was removed from the input while keeping all other model settings unchanged. Interdependent features, such as sine and cosine encodings of temporal variables, were always removed jointly to ensure consistency.

2.8. Exploration of Additional Feature Candidates

An exploratory analysis was conducted to examine whether incorporating additional system-level information beyond electricity prices and temporal encodings improves model performance. Candidate features were added individually to the baseline Transformer model and evaluated using the same training and validation procedure as in the main experiments. The following system-level feature candidates were considered:

• Ambient temperature (current and lagged)
• Thermal load (current and lagged)

3. Results and Discussion

3.1. Cost Savings

Figure 4 summarizes the achieved cost reductions for all evaluated control strategies relative to the non-optimized baseline, grouped by model family and represented by their best-performing model per class. The optimized hyperparameters are described in Appendix C. The MILP solution provides the global optimum, achieving a cost reduction of 21.07% and serving as a theoretical upper bound, since it is computed under perfect foresight of system states, cooling demand, and external heat gains. Among IL approaches, sequence-based models clearly outperform other model families. The Transformer achieves a cost reduction of 19.33%, while the LSTM attains a similar improvement, highlighting that capturing temporal dependencies is essential for cost-optimal control under dynamic electricity pricing. Reinforcement learning (DQN) achieves a slightly higher cost reduction of 19.69%, closely approaching the MILP optimum. This performance is enabled by its closed-loop formulation but comes at the cost of substantially higher training effort and system complexity, and is therefore used primarily as a benchmark in this study. Tree-based, kernel-based, and instance-based models form a second performance tier, achieving cost reductions around 17%. While clearly inferior to sequence models, they significantly outperform heuristic approaches. Price-based heuristics such as Mean, Median, and MinMaxPrice already achieve moderate cost reductions (13–15%) with minimal effort. In contrast, the Day–Night heuristic increases total costs, illustrating that rigid, time-based strategies can be counterproductive under flexible electricity pricing. This highlights the necessity of price-aware and temporally adaptive control strategies. Linear and generalized linear models achieve less than 10% cost reduction and are consistently outperformed even by simple heuristics, indicating that linear modeling assumptions are insufficient to capture the nonlinear and temporal structure of the problem. For completeness, the full numerical results for all evaluated models are summarized in Appendix E.

While aggregated results compare economic performance across model families, a detailed temporal analysis is required to assess control quality. Figure 5 therefore presents system trajectories over three representative days for the MILP reference and the Transformer. The Transformer closely follows the overall structure of the MILP solution, resulting in similar power trajectories and cost accumulation, with minor deviations reflecting its prediction inaccuracies. Note that the MILP assumes perfect knowledge of the system behavior, the future thermal load, and disturbances and therefore represents a theoretical upper bound.

3.2. Prediction Quality

Figure 6 relates prediction accuracy, measured by sMAPE, to the achieved cost reduction for all evaluated approaches. A clear overall trend can be observed: models with lower prediction errors generally achieve higher cost savings. Sequence-based models, particularly the Transformer and LSTM, combine low prediction errors with high economic performance. Tree-based and instance-based models show moderate prediction accuracy and form a dense cluster with intermediate cost reductions, while linear and generalized linear models exhibit the highest errors and the weakest economic performance. RL constitutes a notable exception to this general trend: despite exhibiting the highest prediction errors with respect to the MILP reference trajectory, it achieves the greatest cost savings. This apparent contradiction can be explained by two factors. First, due to saturation effects of the lower-level P controller, different setpoint values can lead to identical cooling power, meaning that deviations from the MILP setpoint do not necessarily translate into different electrical load behavior. Second, unlike imitation learning (IL) models that aim to replicate the MILP trajectory in an open-loop fashion, the DQN follows a closed-loop control strategy and directly optimizes the economic reward through interaction with the environment. Consequently, although its temperature trajectory deviates from the MILP reference, it induces comparable load-shifting behavior and achieves competitive - even superior - economic performance. Overall, this observation underscores that prediction accuracy alone is not a sufficient indicator of economic performance. Different control strategies can yield similar cost reductions while exhibiting fundamentally different operational behaviors.

Detailed prediction metrics for all evaluated models are provided in Appendix F.

3.3. Computational Efficiency

Figure 7 illustrates the trade-off between achieved cost reduction and inference time for all evaluated approaches. The MILP solution requires optimization times on the order of seconds and is therefore probably not the best choice for real-time operation, particularly on edge devices with limited computational resources. In contrast, all IL models and RL achieve inference times several orders of magnitude lower, ranging from nanoseconds for linear models to microseconds for tree-based, kernel-based, and sequence models. Among IL approaches, the Transformer achieves near-optimal cost reductions while maintaining sub-microsecond inference times. Tree-based and kernel-based models also provide fast inference but achieve lower cost savings, while linear and generalized linear models, despite their minimal computational cost, perform poorly in terms of economic outcomes.

Figure 8 relates cost reduction to training time. All IL models require only seconds for training (up to 18 s), which is substantially faster than solving a single MILP instance for one day. RL requires significantly longer training times, on the order of tens of minutes, as the agent must learn the control policy through interaction and exploration rather than direct supervision. Note that while the MILP optimizer itself does not require training, it relies on an external load forecast. If this forecast is generated using deep learning, it introduces an additional training pipeline and effort that is comparable to training the IL models. A potential limitation of IL is that training assumes the availability of historical MILP reference trajectories; if these had to be generated online, training time would increase accordingly, whereas inference time would remain unaffected. Overall, all IL models are well suited for real-time deployment on edge devices with limited computational resources, with the Transformer offering the most favorable balance between training effort, inference speed, and achieved cost reduction.

Detailed training and inference times for all evaluated models are reported in Appendix G.

3.4. Temperature Constraint Violations

The temperature control over the test year was analyzed for all models, with deviations beyond the target setpoint plus a small tolerance (±0.25 °C) counted as violations. The results, summarized in Table 1, indicate that MILP, RL, and the heuristic approaches remained almost perfectly within the allowable range, exhibiting zero violations, with the base model showing only a minimal spread of 0.21 °C. Sequence models also closely tracked the setpoint. The Transformer model, which achieved the highest cost savings, exhibited zero violations, while the LSTM recorded only two rare violations, with a maximum temperature of 5.29 °C, just 0.04 °C above the upper bound, caused by the proportional gain of the controller rather than model extrapolation. In contrast, linear models such as LinearRegression, Ridge, Lasso, ElasticNet, BayesianRidge, and LinearSVR were more prone to violations due to their extrapolative behavior, with LinearSVR reaching up to 157 violations. Kernel-based methods exhibited even more extreme deviations. For example, KernelRidge recorded 205, 27, and 232 violations for different metrics, with maximum and minimum temperatures of 6.69 °C and -1.68 °C, respectively, highlighting its strong tendency to exceed the allowable bounds. Tree-based and boosting models were generally stable, with only a few exceptions observed for GradientBoosting, XGBoost, and LightGBM. Overall, the results show that IL can reliably maintain the temperature within the allowed bounds. Using a two-stage approach, where IL predicts the setpoint for the P controller, the Transformer model, for example, delivers fast and accurate predictions, achieves substantial cost savings, and robustly respects all temperature limits. Models prone to extrapolation could be further constrained by explicitly limiting setpoints to the 0–5 °C range. This approach clearly outperforms methods that directly control the power, which are more likely to violate operational constraints.

3.5. Ablation Study

Table 2 summarizes the results of the ablation study for the Transformer model. Removing the electricity price signal causes the strongest performance degradation, increasing operational costs by 4.20% relative to the reference model. Removing temporal encodings results in smaller cost penalties: omitting weekday information increases costs by 1.15%, while excluding day-of-the-year or hour-of-the-day features leads to penalties below 0.5%. Overall, temporal structure supports performance, but electricity price information is the dominant factor for economic optimization under dynamic pricing.

3.6. Exploration of Additional Feature Candidates

Table 3 reports the impact of additional system-level feature candidates on predictive accuracy and operational costs. None of the evaluated features consistently improves economic performance compared to the reference Transformer model.

Marginal improvements in sMAPE are observed for some feature configurations, but these do not translate into lower operational costs. In several cases, predicted costs are slightly higher than for the reference model, although the differences are not statistically significant. Even when all candidate features are combined, no further performance gains are achieved. Overall, the results indicate that the proposed price-driven approach already captures the most relevant information for generating cost-optimal setpoint trajectories. Incorporating additional system-level features increases model complexity and data requirements without providing measurable economic benefits.

3.7. Practical Applicability, Limitations and Future Research Directions

The proposed setpoint optimization framework is applicable to a wide range of energy systems that can be represented by setpoint-based control and storage-like dynamics. Beyond the investigated refrigeration system, potential application domains include heat pumps, hot water boilers, thermal storage systems, and battery energy storage systems operating under dynamic electricity pricing. The ability of the Transformer to generate entire setpoint trajectories directly from price and temporal signals enables effective load shifting without relying on explicit forecasts or continuous state feedback.

Despite the promising results, several limitations must be considered. The IL models rely on optimal MILP trajectories for training, which requires solving computationally expensive optimization problems under perfect information. However, this dependency does not preclude practical deployment, as MILP optimization can be performed periodically on centralized servers, while the trained IL models enable fast inference on resource-constrained edge devices.

Future research should therefore focus on three main directions. First, hybrid learning strategies that combine MILP with RL are promising, for example by injecting MILP-generated trajectories into the replay buffer of RL agents to accelerate training and improve convergence. Second, the development of generalized or so-called foundation models trained on representative load and storage profiles could enable privacy-preserving optimization without requiring feedback of system-specific measurements, while transfer learning could be used to adapt such models to individual systems when data sharing is permitted. Third, real-world validation under practical operating conditions, including disturbances, communication delays, and long-term system drift, remains essential to assess robustness and industrial applicability.

Addressing these aspects is key to transferring data-driven setpoint optimization from simulation studies to scalable, privacy-aware, and robust real-world energy management systems.

4. Conclusions

This study demonstrated the effectiveness of supervised IL to optimize the setpoints of an industrial refrigeration system. By predicting optimal setpoint trajectories directly with the help of historical price signals and temporal features, the proposed framework eliminates the need for intermediate load forecasts required for MILP or feedback loops required for RL. The results confirm that the Transformer model can replicate near-globally optimal behavior with high accuracy, achieving cost savings of 19.33%, thereby approaching the theoretical upper bound of 21.07% achieved by the MILP formulation. However, the MILP benchmark assumes perfect knowledge of both system dynamics and future load profiles, representing an idealized scenario that is not attainable in real-world applications. While RL achieved slightly higher savings (19.69%), the Transformer offers a distinct advantage in implementation simplicity and computational efficiency. With an inference time of just 526 ns, compared to over 22 seconds for a single MILP optimization, the model enables scalable, real-time control even on resource-constrained hardware. Unlike linear and kernel-based models, which frequently violated temperature constraints due to extrapolation errors, the Transformer model maintained the warehouse temperature strictly within the allowed bounds without a single violation throughout the test year. This demonstrates that sequence-based SL models can safely manage physical constraints when paired with a standard lower-level controller. In conclusion, the Transformer as supervised IL model represents a robust and cost-effective alternative to traditional MILP and RL approaches. It successfully bridges the gap between theoretical optimization and practical application, offering a scalable solution for data-driven energy management.