Optimization of an MPC Controller Based on a Hybrid Cooling Load Prediction Model and Experimental Validation in HVAC Systems

Abstract

The high energy intensity of public buildings, especially those with HVAC systems, calls for advanced control strategies such as Model Predictive Control (MPC) to balance energy efficiency and thermal comfort. However, the performance of MPC relies critically on the accuracy and robustness of building cooling and heating load calculations, which remain challenging, particularly for buildings with complex dynamic characteristics. This study proposes a simplified modeling-based MPC approach and investigates the influence of three different load calculation methods on controller performance: a physics-driven white-box model, a data-driven black-box model, and a novel Closed-Loop Load Grey Model (CLLGM). Under identical outdoor conditions during summer cooling operation, the three controllers exhibit distinct performance disparities: although the proposed CLLGM-based controller only reduces the load prediction MAPE by 0.63% compared with the black-box model, it improves the temperature control stability index (TDI) by 80.43% and increases the comprehensive score from the MPC multi-objective optimization function by 16.55%. Its key advantage is that it can use on-site temperature measurements as feedback to correct the cooling load, making it better suited for simulation and computation in MPC.

1. Introduction

Public buildings constitute a critical component of urban energy consumption, with the 2025 China Conservation Annual Development Research Report revealing that these structures account for 38% of total building energy use, despite occupying only 18.1% of the national building floor area [1]. This disproportionate energy intensity is particularly pronounced in HVAC (heating, ventilation, and air conditioning) systems, which account for 40% of public building energy consumption [2]—a share that continues to rise amid intensifying climate change impacts. Against this backdrop, traditional “on–off control” or “proportional–integral–derivative (PID) control” for HVAC systems can no longer meet the dual requirements of energy saving and thermal comfort, due to their poor adaptability to dynamic changes in building loads [3].

Model Predictive Control (MPC) has emerged as a promising solution to this challenge [4]. By predicting future system states (e.g., indoor temperature, cooling/heating load) over a finite horizon and optimizing control actions based on a predefined objective function (e.g., minimum energy consumption, maximum thermal comfort), MPC can significantly improve the energy efficiency of HVAC systems while ensuring indoor environmental quality [5]. Field experiments have verified the effectiveness of MPC: Yang et al. demonstrated in a laboratory that MPC reduced energy consumption by 33.58% compared with basic control strategies under high-load conditions in summer [6]. Sturzenegger et al. [7] implemented MPC in a Swiss office building and achieved a 15–20% reduction in HVAC energy consumption compared to traditional control. In addition, MPC implementations in commercial buildings [8], educational buildings [9], hospitals [10], and pharmaceutical plants [11] have achieved 10–20% energy savings, validating its potential for large-scale applications. However, despite such remarkable performance, MPC still faces severe industrial penetration barriers due to its complex and non-transparent implementation process [12], and poor adaptability to actual system dynamic changes [13]—a dilemma that is fundamentally rooted in the accuracy and timeliness of cooling/heating load calculation, which serves as the “decision-making basis” for MPC’s predictive optimization and directly determines the rationality of control strategies (e.g., chiller set temperature, fan speed adjustment) [14]. A critical review of recent studies shows that, despite significant progress, four interrelated challenges have yet to be addressed.

(i)

Load prediction for MPC. The accuracy of load prediction directly determines the MPC’s ability to balance energy efficiency and thermal comfort [15]. Zhao et al. demonstrated that in MPC-based heat pump systems, accurate load calculation is crucial to avoid “over-control” and “under-control” [16]. Afram et al. quantified this relationship, showing that in MPC systems integrated with an artificial neural network (ANN)-based load predictor, each 10% increase in load-prediction error results in a 5–8% decrease in the achieved energy-savings rate [17]. However, existing prediction methods have inherent limitations. Although physical models are precise, they are computationally intensive and sensitive to parameter values [18]. Data-driven models are fast but lack generalization ability and require large volumes of high-quality datasets [19]. The fundamental challenge is that predictions used for control require not only accuracy but also adaptability to varying operating conditions [20].

(ii)

Robustness of MPC under model mismatch. Even complex controllers will fail when their underlying models are inaccurate. HVAC systems are confronted with continuous fluctuations in load-driven factors, including outdoor meteorological conditions and varying indoor occupancy density [21]. Oldewurtel et al. emphasized that MPC inherently relies on real-time load calculation to capture these dynamic changes; otherwise, the system will experience control lag [22]. Physical models are particularly susceptible to parameter errors—Mui et al. achieved a prediction error of 5–8% in a controlled environment [23], but this accuracy degrades rapidly when actual building parameters deviate from design values. In real-world buildings, such model mismatch is not an exception but the norm, yet most MPC methods assume that the model is perfect.

(iii)

Hybrid or gray-box correction methods. To mitigate these limitations, researchers have developed hybrid models (also referred to as “gray-box models”), which combine simplified physical models with data-driven techniques for error correction. Cipriano reduced computational time by 60% while maintaining prediction error below 10% [20], and Peng et al. applied a similar method to mixed-mode buildings [24]. However, a critical limitation emerges upon careful examination: the correction mechanisms in these methods are inherently reactive. Instead of proactively suppressing disturbances within the prediction horizon, they use data-driven components to correct steady-state deviations after the prediction error has been observed. Therefore, their improvements in dynamic disturbance suppression remain limited, and they cannot fundamentally resolve the instability issue in MPC load prediction when facing rapidly changing conditions.

(iv)

Challenges in Field Validation. The disparity between simulated performance and real-world implementation remains a persistent barrier [25]. Field studies of cloud-based MPC retrofit projects have shown that inaccurate load calculations are a primary cause of the gap between simulated and actual energy savings. Bird et al. observed in an MPC project for a commercial building that the deviation between the predicted and actual cooling load during peak periods exceeded 20%, resulting in an actual energy savings rate 12% lower than the simulated value [26]. Almatared et al. validated their approach in commercial buildings under the extreme climatic conditions of Saudi Arabia, achieving a 9.7% reduction in operating costs, while also demonstrating that the performance of different models varies significantly across different climates [27]. These implementation gap demonstrates that laboratory success does not guarantee reliability in practical applications, and even field-validated models face cross-scenario generalization challenges.

Synthesizing these four research directions, a clear research gap emerges: existing load calculation methods—whether physics-based, data-driven, or hybrid—fail to concurrently deliver the accuracy, dynamic disturbance robustness, and computational efficiency required for reliable MPC operation in real-world buildings. Physics-based methods offer high precision but suffer from slow computation and parameter sensitivity, rendering them unsuitable for real-time control. Data-driven methods enable fast computation but lack generalization, making them prone to failure when confronted with novel disturbances. Conventional grey-box methods provide a compromise, yet their reactive correction mechanisms leave them vulnerable to the dynamic disturbances that characterize real-world building operation.

What is therefore required is a fundamentally different approach—one that embeds disturbance suppression directly into the load estimation process to proactively correct predictions in real time, rather than merely reacting after errors have occurred.

To address this gap, this paper proposes a novel Closed-Loop Load Gray Model (CLLGM) for MPC-based HVAC control. Its core innovation lies in using a calibrated, fast-running building simulation as a “virtual sensor” to guide real-time load prediction. Unlike conventional grey-box methods that apply data-driven corrections after error occurrence, the CLLGM operates as follows: (1) a computationally efficient data-driven predictor generates an initial load baseline; (2) a simplified physical model simulates the resulting indoor temperature trajectory; (3) a closed-loop feedback controller dynamically adjusts the load baseline to minimize the error between the simulated temperature and the desired set point during trajectory evolution. This closed-loop mechanism proactively compensates for disturbances and model mismatch within the prediction horizon, yielding continuously corrected, inherently more robust load predictions.

The main contributions of this study are threefold:

Development of a novel methodological framework: We introduce the CLLGM, an innovative hybrid approach that integrates data-driven prediction with physics-based simulation into a closed-loop feedback loop for real-time load correction. By embedding disturbance suppression directly into the load estimation process rather than applying ex post correction, this mechanism establishes a clear technical distinction from conventional grey-box models.

Experimental validation on a physical testbed: We implement and validate the CLLGM within the MPC framework on a real-world HVAC laboratory system, providing empirical evidence for its performance under actual operating conditions.

Rigorous comparative performance analysis: We conduct an intensive comparative study to benchmark the proposed CLLGM against the conventional physics-based method (the heat balance method) and the standard data-driven method (Extreme Gradient Boosting, XGBoost). The results demonstrate that the CLLGM-enhanced MPC achieves superior control accuracy and robustness.

2. Methodology

The overall research flow of this study is illustrated in Figure 1, which is divided into four main steps: building load calculation methods, development of the MPC controller framework, establishment of the laboratory simulation model, and design of MPC control experiments. Additionally, the construction of the evaluation index system and the presentation of result statistics are also included. Section 2 presents the theoretical foundations and formulas for the methods employed throughout the flow. Specifically, Section 2.1 introduces three distinct calculation principles for building loads, Section 2.2 presents the specific structural configuration of the MPC controller, and Section 2.3 outlines the evaluation indices used throughout the study.

2.1. Load Calculation Models for MPC

Building load is a key input for MPC in all building HVAC systems, and its accuracy determines the stability of the overall MPC control. Therefore, this study adopts two classical building load calculation methods, namely white-box models and black-box models, and further proposes a gray-box calculation model by integrating the simulation model of the HVAC system based on these two approaches. Specifically, for the white-box model, EnergyPlus (EP) (NREL, Golden, CO, USA), a widely recognized professional software for building simulation and load calculation, is used to develop the model. By entering weather conditions and building maintenance parameters, the room load is calculated, serving as the pre-input condition for the subsequent MPC. For the black-box model, it is trained using historical data, in which the building load is calculated by quantifying the room’s heat change and the water system’s heat exchange capacity. In actual load prediction, the black-box model uses the current weather conditions and the load at the previous time step as input to calculate the current room load. However, the standalone application of these two models has inherent limitations: the white-box model is costly to obtain accurate simulation setup parameters, while the black-box model lacks interpretability and scalability due to its data-driven characteristics [3,13]. To address the challenges in building load prediction accuracy associated with black-box and white-box models, Valenzuela et al. proposed a model predictive control (MPC) strategy based on real-time data feedback, which was verified on the simulation experimental platform. The results show that introducing real-time data feedback can effectively improve the control accuracy of air conditioning systems [28].

Based on this control strategy, this study further proposes a CLLGM for practical MPC applications, aiming to integrate the advantages of white-box and black-box models and address their deficiencies. The proposed CLLGM builds on the HVAC system simulation model and introduces a closed-loop feedback mechanism to dynamically correct the building load calculated by the black-box model; the corrected, accurate load is then used as the key input to the MPC optimizer.

It can not only circumvent the problems of high cost for high-precision load calculation and cumulative errors caused by simplified approximations in traditional methods [15], but also leverage closed-loop feedback to continuously optimize the MPC load input.

The specific procedure is illustrated in Figure 2:

• Current indoor/outdoor environmental parameters and device control signals are input into the simulation model.
• The initial load estimate—provided by a data-driven predictor trained on historical data—is used to generate three candidate load values $W$, $W_{-1\varphi }$ and $W_{+1\varphi }$ via correction coefficients.
• If the current measured temperature lies within the range of model-predicted temperatures corresponding to these three loads, the final predicted load is generated according to the distance ratios between the measured temperature and the three predicted temperatures.
• If the measured temperature lies outside this range, the value of W is adjusted in the corresponding direction to $W_{+1\varphi }$ or $W_{-1\varphi }$. Three new candidate load values are then regenerated based on the updated W, and the judgment is repeated. This process continues until the measured temperature falls within the range of the three predicted temperatures. The final predicted load is determined according to the distance ratio between the measured temperature and the three predicted temperatures.

Figure 2. Overall workflow of the proposed CLLGM, encompassing data-driven heat flux prediction, correction coefficient generation, FMU model-based temperature coupling, and temperature range judgment for corresponding load value output.

Figure 2. Overall workflow of the proposed CLLGM, encompassing data-driven heat flux prediction, correction coefficient generation, FMU model-based temperature coupling, and temperature range judgment for corresponding load value output.

The maximum number of iterations is set to 10. If convergence is not achieved within this limit, a calculation error is declared. In subsequent experiments, the controller will maintain the original control command and report an error. Each iteration involves three Functional Mock-up Unit (FMU) evaluations, resulting in a maximum computational burden of 30 FMU runs.

The calculation methods for $W_{-1\varphi }$ and $W_{+1\varphi }$ are shown in Equations (1) and (2), where ${\varphi }_{Dynamic Calibration}$ is determined based on the error of the load predicted by the data-driven model for load prediction.

$$W_{-1\varphi }=\mathrm{W}\times \left(1+{\varphi }_{Dynamic Calibration}\right)$$

(1)

$$W_{+1\varphi }=\mathrm{W}\times \left(1-{\varphi }_{Dynamic Calibration}\right)$$

(2)

In Section 3.3, the Mean Absolute Percentage Error (MAPE) values for all four models exceed 90%. However, the 95th percentile of the absolute percentage error (APE) across these models is 4.63%, and the maximum APE is 7.97%, both of which are below 10%. Therefore, the dynamic calibration coefficient ϕ is set to 10% in the prediction structure for $W_{-1\varphi }$ and $W_{+1\varphi }$. This setting can cover 100% of the prediction errors of the data-driven models. If the maximum APE of the data-driven model exceeds 10% in future applications, it is recommended to round $\varphi$ to the nearest multiple of 5% to ensure iterative convergence while avoiding over-correction.

2.2. MPC Controller Structure

A classical MPC system can be divided into three core components, which correspond to the structural layers shown in Figure 3: the Device Layer, the Connection Layer, and the Computational Layer. Specifically, the Device layer serves as the execution layer, directly controlling the target objects within the entire control system, which consists of cooling/heating sources, water pumps, terminal fans, and sensors. Functioning as a hardware transmission module, the Connection layer uploads operational data and sensing data from all HVAC equipment and sensors to the cloud via programmable logic controllers (PLC), while enabling data exchange between remote computers and on-site control devices. It thus acts as a bridge connecting the virtual and physical domains. The Computational layer integrates a simulation-computing module and an algorithm-optimization module, enabling simulation-based predictive control of the overall system through predictive optimization.

In the computation layer, the load calculation model serves as the primary model for calculating load inputs throughout the entire simulation framework. In subsequent experiments, the three models introduced in Section 2.1 will be used as control groups, all implemented in Python. The simulation model is exported as an FMU file, which is then integrated with the load calculation model in the Python environment. This integration enables a seamless connection from predictive inputs to control outputs, forming the complete simulation and calculation workflow. Finally, this workflow is incorporated into the optimization algorithm to search for the optimal solution. The MPC controller in this paper employs a particle swarm optimization (PSO) algorithm. An objective function is constructed with temperature and energy consumption as variables, and the optimization goal is to maximize this function, which is expressed as follows:

$$\mathrm{f}\left(\mathrm{x}\right)=100-\left|T\left(x\right)-26\right|\times {\phi }_T+\left(W_{mean}\left(T_{out}\right)-P\left(\mathrm{x}\right)\right)\times {\phi }_P$$

(3)

In the equation, $\mathrm{x}$ represents the set of control commands for all devices; $T\left(x\right)$ denotes the average indoor temperature of the controlled environment after 15 min, output by the FMU simulation model; $T_{out}$ is the current outdoor temperature; $W_{mean}$ stands for the average power consumption from historical data when the outdoor temperature is $T_{out}$ and the control target is 26 °C; $P\left(\mathrm{x}\right)$ is the total power consumption of all devices calculated by the simulation model; ${\phi }_T$ and ${\phi }_P$ are correction coefficients for thermal comfort and energy consumption in the objective function, respectively, whose values are determined based on control objectives and task requirements.

In this experiment, ${\phi }_T$ = 35 when T(x) < 26 °C, and ${\phi }_T$ = 50 when T(x) > 26 °C; ${\phi }_P$ = 1/200. A higher weight is assigned to ${\phi }_T$ to stabilize the target temperature at 26 °C, facilitating a clearer evaluation of the controller performance. In practical HVAC system control, these coefficients can be adjusted to modify control preferences.

The actual prediction step of the MPC controller is 15 min, and the control step is also 15 min. Meanwhile, the controller uses the average of the previous 15 min of sampled data as input, and the sampling method is consistent with the model training process described in Section 3.2.

2.3. Evaluation Metric

To evaluate the prediction accuracy of the three models in the experiments, a set of evaluation metrics is employed to assess the MPC control outcomes of the different load calculation models. The evaluation of subsequent experimental results focuses on three key aspects: the prediction accuracy of building loads, the accuracy of MPC control outcomes, and the overall performance of the MPC controller.

The prediction accuracy of building loads is evaluated by comparing the predicted load at a specific time with the actual load calculated from post-experiment data. Thus, direct prediction accuracy metrics are employed to assess and compare the results of the control experiments using the three load models, including the dimensioned Mean Absolute Error (MAE), the dimensionless MAPE, and the APE for each individual prediction sample. The formulas for MAE, MAPE, and APE are presented as follows:

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|y_i-{\widehat{y}}_i\right|$$

(4)

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|{\displaystyle \frac{y_i-{\widehat{y}}_i}{y_i}}\right|\times 100\%$$

(5)

$$\mathrm{A}\mathrm{P}\mathrm{E}=\left|{\displaystyle \frac{y_i-{\widehat{y}}_i}{y_i}}\right|\times 100\%$$

(6)

where ${\widehat{y}}_i$ denotes the predicted value, and $y_i$ denotes the actual value.

In evaluating MPC control performance for comfort HVAC systems, the agreement between predicted temperatures and actual operating temperatures serves as a core reference metric. Actual operating temperature data are collected and aggregated from real-time monitoring at each return air vent. Considering human perceptual characteristics of indoor temperature, a ±1 °C fluctuation range typically does not compromise occupant thermal comfort. Based on this practical scenario, cases where the actual temperature falls within the ±1 °C interval of the predicted temperature are classified as effective control. At the same time, deviations exceeding this range are taken into account in the assessment of control performance. To comprehensively reflect the long-term stability of control effectiveness across different load prediction models, it is necessary to quantify the cumulative effect of temperature deviations beyond the reasonable range and their duration—specifically, through Temperature Deviation Integral (TDI)—to capture the extent of deviation between the MPC optimal control outputs and the actual operating conditions. Furthermore, the Maximum Deviation Error (MDE) is employed to quantify the severity of control deviations. The formulas for TDI and MDE are presented as follows:

$$\mathrm{T}\mathrm{D}\mathrm{I}={\int }_0^T\max (\left|T_{act}\left(t\right)-T_{pred}\left(t\right)\right|-1,0)dt$$

(7)

$$\mathrm{M}\mathrm{D}\mathrm{E}={\mathit{max}}_{t\in \left[0,T\right]}\left(\left|T_{act}\left(t\right)-T_{pred}\left(t\right)\right|-1,0\right)$$

(8)

where $T_{act}\left(t\right)$ denotes the average temperature of the controlled environment at time t, and $T_{pred}\left(t\right)$ denotes the predicted temperature of the controlled environment at time t.

The MPC controller constructed in this study employs the PSO algorithm for multi-objective optimization in practical computations. The control performance of the three controllers is evaluated using Equation (9), which is derived from the PSO objective function (Equation (3)). In both equations, a higher score is obtained when the temperature approaches 26 °C with lower energy consumption. Therefore, a higher evaluation score indicates better controller control performance.

$$\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}=100-\left|T_{act}\left(t\right)-26\right|\times {\phi }_T+\left({W_{mean}\left(T_{out}\right)-W}_{act}\left(t\right)\right)\times {\phi }_P$$

(9)

where $W_{act}\left(t\right)$ denotes the filtered real-time power at time t.

3. Experimental

This chapter presents a detailed description of the specific configurations and relevant details of the MPC control experiments. Section 3.1 focuses on the particular characteristics of the comfort HVAC system employed in the laboratory. Section 3.2 describes the Modelica-based laboratory simulation model and the corresponding load calculation methods. Section 3.3 presents the prediction inputs and outputs of the data-driven models, along with their training process, test performance, and final model selection. Section 3.4 outlines the experimental design of the MPC control comparison experiments. All experimental setups, dataset specifications, and hyperparameter configurations of the data-driven models employed in this study are summarized in

Table 1. Reproducibility Summary of Experimental Setup and Data-Driven Models (Section 3).

Category	Specific Information
Dataset Period	Model calibration/validation: 1 week of historical operational data of the laboratory HVAC system under fully randomized parameters, focusing on thermally stable periods (10:00–17:00 daily). Data-driven model training: Historical operational data from the laboratory HVAC system under randomized parameter experiments MPC control experiments: 9 consecutive days in July 2025 (3 independent test days for each controller variant).
Raw Sampling Rate	1 min per sample (1 Hz) for all sensor and equipment operational data
Data Aggregation Rule	Arithmetic mean over a 15 min time window for all raw data to generate experimental input parameters; time step Δt = 900 s (balanced for engineering control stability and computational feasibility)
Effective Sample Size	Model validation: 28 samples per day (7 h thermal stability period with 15 min aggregation), 196 validation samples in total MPC control experiments: 32 control samples per day (8 h test period 09:30–17:00 with 15 min control step), 96 samples per controller, 288 experimental samples in total Data-driven model training: 7-dimensional input features with laboratory historical operational data
Feature Preprocessing	Data-driven models (ANN/RF/XGBoost/SVR): Employed 7-dimensional inputs (indoor/outdoor temperatures and HVAC equipment control commands) and a 1-dimensional output (cooling load prediction). Experimental input: 1 min raw data aggregated by 15 min arithmetic mean, directly used as model/controller input with no additional preprocessing (no missing values in field-measured data)
Data-Driven Model Hyperparameter	ANN: Network structure (64-ReLU-Dropout(0.2)→32-ReLU-Dropout(0.2)→16-ReLU→1-linear); optimizer (Adam, learning rate = 0.001); early stopping (monitor = val_loss, patience = 10, restore_best_weights = True); training epochs = 100, batch size = 32, validation split = 0.2 RF: n_estimators = 100, max_depth = 10, random_state = 42 XGBoost: n_estimators = 100, max_depth = 5, learning_rate = 0.1, random_state = 42 SVR: kernel = rbf, C = 100, gamma = 0.1, epsilon = 0.1
Model Evaluation Split	Data-driven models: Random split with training set: test set = 8:2, random_state = 42 FMU simulation model validation: Time-series direct validation on 1-week aggregated historical data, no train/test split MPC controller evaluation: Independent comparative testing with no data split; three controllers tested under identical experimental conditions for 3 days each, performance evaluated by experimental result comparison
Key Experimental Setup & Parameters	Training environment: Python 3.9 Simulation model: Modelica model built in Dymola(France) 2023, exported as FMU standard file for co-simulation with Python Optimization algorithm: PSO as the multi-objective optimizer for MPC Control temporal parameters: Prediction horizon = control step = sampling period = 15 min Hardware transmission: Cloud-edge data interaction via PLC for remote device control and data acquisition

for reproducibility and auditability.

3.1. Experimental Setup and Laboratory Model

This paper focuses on the air-source heat pump air-conditioning system in an enterprise’s HVAC laboratory, which was retrofitted in an aged building. Detailed parameters of the building envelope structure are unavailable due to the building’s renovation. The laboratory test area measures 6.90 m in length, 6.16 m in width, and 2.7 m in height, with an area of 42.5 m² and a volume of 114.76 m³, as illustrated in Figure 4.

The laboratory’s air-conditioning system utilizes an air-source heat pump as the cooling/heating source, with a rated cooling capacity of 9.5 kW and a rated heating capacity of 11 kW. It is equipped with two fan coil units, each with a rated air supply volume of 1170 m³/h, and a fresh air handling unit with a rated air supply volume of 1500 m³/h. All devices in the laboratory are interconnected via PLC to a local server, with operational data uploaded to the cloud. This setup enables remote control of device switches and parameter settings through an interface. The controllable precision and operational ranges for each device are specified in

Table 2. Remote control objects and intervals of laboratory equipment.

Equipment Name	Control Content	Controlling Section
Air-source Heat Pump	Host Outlet Water T	7–15 °C
Air Handling Unit	Unit Operating Frequency	35–50 Hz
Fan Coil Unit 1#	Coil Gear Stage	Gear 1, Gear 2, Gear 3
Fan Coil Unit 2#	Coil Gear Stage	Gear 1, Gear 2, Gear 3
Water Pump	Water Pump Frequency	35–50 Hz

.

3.2. Simulation Model of MPC Controller

In this study, a summer cooling simulation model for the HVAC laboratory was a real-world engineering project using the Dymola platform. To enhance practicality and computational efficiency for control-oriented applications, the lumped method is used to calculate the cooling load of the conditioned space, assuming it is a well-mixed air volume [21].

This simplification, however, creates a critical dependency: the accuracy of the entire simulation model is now highly sensitive to the precision of this input load value. An inaccurate load estimate would lead to significant errors in predicting the room’s thermal dynamics, thereby compromising any controller reliant on this model. This need for highly accurate, real-time load input motivates the development and comparison of the three load calculation models (white-box, black-box, and grey-box).

The equipment models were built using the Modelica Standard Library and the Building 8.0 library, with parameters initialized from nameplate data and later fine-tuned through calibration with historical operational data. The overall model structure is depicted in Figure 5. Since the laboratory load could not be measured directly by sensors, it was calculated for model calibration purposes using Equations (10)–(12) based on historical system operation data.

$$W_{load}={\displaystyle \frac{Q_{Hvac\_cold}}{{∆}_t}}+{\displaystyle \frac{{∆Q}_t}{{∆}_t}}$$

(10)

$$Q_{Hvac\_cold}=C_{water}\ast M_{water}\ast (T_{return}-T_{supply})$$

(11)

$${∆Q}_t=\sum _t^{t+1}{C}_{air}\times M_{la{b}_{air}}\times (T_{env-t+1}-T_{env-t})$$

(12)

where $W_{load}$ refers to the load of the lab room, $Q_{Hvac\_cold}$ refers to the heat exchange capacity of the entire HVAC system’s chilled water system, ${∆Q}_t$ refers to the heat change in the laboratory air per unit time, $M_{water}$ refers to the mass flow rate of the chilled water system in the laboratory, $T_{return}$ refers to the return temperature of the chilled water in the HVAC system, $T_{supply}$ refers to the supply temperature of the chilled water in the HVAC system, $C_{air}$ refers to the specific heat capacity of the laboratory air, $M_{la{b}_{air}}$ refers to the mass of the laboratory air, $T_{env-t+1}$ refers to the temperature of the environment at time t + 1, and $T_{env-t}$ follows the same notation.

After model construction, the model was exported as an FMU standard file. Model error validation was conducted in a Python environment using historical operational data. The historical dataset, sampled at 1 min intervals, contained complete records from all sensors in the air conditioning system. To align with practical control cycles and enhance computational efficiency, the validation process adopted a 15 min analysis window (${∆}_t$ = 900 s) with the following data/simulation handling principles:

• Input parameter generation: Control parameters (e.g., chiller water outlet temperature set points) were calculated as the arithmetic mean of 15 sampling points within each 15 min window to form model inputs.
• Simulation initialization: At the start of each test window, the model’s initial system temperature was calibrated to match the actual measured temperature at the first minute of the period.
• Simulation execution: Based on the initialized state and averaged input parameters, a continuous 900 s (15 min) simulation was performed to generate temperature prediction sequences.

The validation used historical records from a laboratory-scale air conditioning system operating under fully randomized parameter settings for 1 week, with a focus on thermally stable operating periods (10:00–17:00 daily). The adoption of 15 min averaged values instead of minute-level actual measurements as control inputs was based on two engineering considerations:

• Control system stability requirements: Frequent actuator movements (e.g., valves, compressors) in large-scale air conditioning systems may induce equipment oscillations and wear. Practical engineering typically employs control cycles of at least 15 min to ensure system stability [22,23,24].
• MPC computational feasibility: A 15 min cycle provides sufficient solution time for optimization algorithms while maintaining control command update frequency, which has been widely adopted as the computational step size in building MPC studies [16].

In summary, this validation methodology simulates practical control sequences through temporal window averaging and ensures state continuity via segmented initialization, effectively balancing model accuracy verification with engineering practicality.

As illustrated in Figure 6, the blue curve shows the 15 min-ahead temperature predictions from the FMU model. In contrast, the green curve shows the actual temperature values recorded in historical data for the same 15 min interval. Both curves exhibit strong temporal correlation, with the predicted values remaining close to the measured values. Quantitative evaluation reveals a MAPE of 1.42% and a MAE of 0.381 °C. Although minor discrepancies exist, considering the inherent multidimensional complexity of practical application scenarios. These results confirm the model’s sufficient reliability for subsequent MPC experiments, provided that an accurate load input is supplied.

3.3. Load Prediction and Data-Driven Models for MPC Controllers

In the domain of data-driven load forecasting, various algorithms, including ANN, Random Forests (RF), XGBoost, and Support Vector Regression (SVR), have been extensively applied [14,25]. This study employs four models—ANN, RF, XGBOOST, and SVR—to predict building load using historical datasets.

The prediction inputs and outputs of the four data-driven models are summarized in

Table 3. The inputs and outputs of the prediction model.

Parameter Name	Unit	Category	Description
Host Outlet Water Temperature	°C	input	Chilled water supply temperature set by the chiller; the chiller automatically adjusts its load rate based on the actual supply temperature.
Water Pump Frequency	Hz	input	Water pump frequency: It regulates the flow rate of the entire chilled water system.
Coil#1 Gear Stage		input	Fan Coil Unit (FCU) speed setting: This setting adjusts the FCU’s airflow rate, with options for Low, Medium, and High.
Coil#2 Gear Stage		input
Unit Operating Frequency	Hz	Input	Air Handling Unit frequency: it regulates the supply air volume of the entire unit.
Outdoor temperature	°C	Input	Outdoor average temperature
Indoor temperature	°C	Input	Indoor average temperature
Predict Load	W	Output	Predicted Load for the next 15 min

below. Except for the predicted load, all other parameters are collected from on-site equipment and sensors. The specific calculation formula for the expected load is provided in Section 3.2. Each set of input and output data is averaged over 15 min to ensure the stability required by engineering practice.

The prediction results, as illustrated in Figure 7, reveal that RF, XGBOOST, and SVR exhibit comparable error margins, whereas ANN shows significantly larger deviations. The APEs of the four models are presented in Figure 8. The mean APE is 1.3745%, the 95th percentile APE is 4.3998%, and the maximum APE is 7.9679%. Quantitative evaluation using standard statistical metrics (

Table 4. Statistical indicators of the four models.

Model Name	MAPE	MAE	MSE	R2
ANN	2.21%	56.42	5318.04	0.9425
RF	1.46%	37.19	2665.89	0.9712
XGBOOST	1.29%	33.37	2085.33	0.9775
SVR	1.37%	35.09	2392.78	0.9741

) confirms the superior performance of SVR and XGBOOST over the other models. Notably, XGBOOST yields the lowest mean APE and achieves the best overall error metrics, making it the preferred choice for subsequent load prediction validation in this experimental framework.

3.4. Experimental Arrangement

This section details the implementation of the MPC controllers and the design of the experimental campaign conducted to evaluate their performance.

3.4.1. Controller Implementation and Variants

The MPC optimization framework was established within a Python 3.9 environment. The high-fidelity simulation model, constructed in Modelica and compiled into an FMU via Dymola 2023, was integrated as the core predictive plant model within this framework. This co-simulation approach ensures the resultant MPC controller is portable and can be conveniently deployed across standard computational environments.

The core of this study’s experimental design involves altering the source of the critical load prediction input to the MPC. By substituting the load calculation module within this unified MPC structure, we developed three distinct controller variants for comparison:

The overall architectures of these three controllers are summarized in

Table 5. MPC controller information.

MPC Name	Hardware Transmission	Load Calculation Model	Laboratory Simulation Model	Multi-Objective Optimization Model
Controller 1	Cloud-connected PLC with field sensors	Thermal equilibrium theoretical model	Modelica model	Particle swarm optimization algorithm
Controller 2	Cloud-connected PLC with field sensors	XGBoost	Modelica model	Particle swarm optimization algorithm
Controller 3	Cloud-connected PLC with field sensors	CLLGM	Modelica model	Particle swarm optimization algorithm
MPC Name	Hardware Transmission	Load Calculation Model	Laboratory Simulation Model	Multi-Objective Optimization Model

. This setup enables a direct, fair comparison of control performance attributable solely to differences in load calculation methodology.

3.4.2. Experimental Procedure and Evaluation Metrics

A comprehensive experimental campaign was conducted in a laboratory setting in Wuhan, Hubei Province, China, during July 2025 to evaluate the three controllers under typical summer conditions. Each controller variant was deployed for a dedicated three-day testing period to ensure statistical reliability.

The daily operational procedure was standardized as follows: the laboratory system was activated at 08:00 under fixed initial parameters. The formal experimental data collection commenced at 09:30, with control calculations executed and the resulting commands applied every 15 min. All equipment was deactivated at 17:00 to conclude daily operations. The overall experimental procedure is illustrated in Figure 9.

4. Results and Discussion

The experiments were conducted over nine days. The following sections analyze: (a) Outdoor conditions, (b) Load prediction accuracy, (c) Controlled temperature stability, and (d) Comprehensive controller performance.

4.1. Outdoor Conditions

As occupant-induced disturbances were absent throughout the experiments, meteorological parameters constituted the primary driver of indoor load variations. Figure 10, Figure 11 and Figure 12 present the outdoor temperature profiles recorded at the laboratory site over the nine experimental days, grouped by the three MPC controllers.

Outdoor temperatures across all nine days exhibited characteristic diurnal patterns: rising from dawn, peaking in the afternoon, and declining toward dusk. Pronounced temperature fluctuations occurred during the afternoon (12–18 h) for Controller 3 (e.g., heightened amplitude and frequency on Day 2), posing a significant challenge to regulation stability and potentially compromising control quality. In contrast, fluctuations under Controllers 1 and 2 remained moderate. Minor inter-day variations in mean temperature (e.g., Controller 1: 35.92 °C, 35.83 °C, 36.02 °C) further highlighted the dynamic environmental conditions.

To ensure the fairness of the performance comparison, we conducted a one-way ANOVA to test for statistically significant differences in the daily mean outdoor temperatures across the three controller groups. The results (

Table 6. Outdoor temperature conditions.

Metric	Controller 1	Controller 2	Controller 3
Day 1 Mean Temperature (°C)	35.92	35.55	36.30
Day 2 Mean Temperature (°C)	35.83	36.15	36.93
Day 3 Mean Temperature (°C)	36.02	36.09	36.09
Overall Mean ± SD (°C)	35.92 ± 0.08	35.93 ± 0.27	36.44 ± 0.36
95% Confidence Interval (°C)	[35.79, 36.05]	[35.53, 36.33]	[35.93, 36.95]
ANOVA F-statistic	2.5345
ANOVA p-value	0.1593
Pairwise t-test p-values	All > 0.05

) show an F-statistic of 2.5345 and a p-value of 0.1593 (p > 0.05), indicating no significant difference in outdoor temperature conditions between the groups. Pairwise t-tests further confirmed that no significant differences existed between any two controller groups (all p > 0.05). The overall mean temperatures were 35.92 ± 0.08 °C for Controller 1, 35.93 ± 0.27 °C for Controller 2, and 36.44 ± 0.36 °C for Controller 3, with 95% confidence intervals of [35.79, 36.05] °C, [35.53, 36.33] °C, and [35.93, 36.95] °C, respectively. These statistical findings confirm that the weather conditions were sufficiently comparable across all testing scenarios, validating the experimental design for evaluating controller performance.

4.2. Load Prediction Accuracy

The load-prediction results for the three controllers are illustrated in Figure 13, Figure 14 and Figure 15, and their performance differences can be analyzed from the dimensions of trend fitting, error characteristics, and disturbance robustness:

• Controller 1 (Thermal equilibrium theoretical model)

While it captures the general trend of base-load, Controller 1 exhibits significant deficiencies in predicting abrupt changes, nearly failing to track sudden load variations. In terms of error metrics, the MAPE exceeds 25% on two out of three prediction days, indicating a lack of robustness to disturbances.

2.

Controller 2 (Data-Driven Model)

Compared with Controller 1, Controller 2 demonstrates a superior overall prediction trend. By comparing predicted and actual values, it is evident that Controller 2 exhibits a distinct trend-following ability for heat-flux changes in the laboratory. Although deviations exist between predicted and actual values, the expected load still shows a synchronous increasing trend during the actual rise phase. However, an MAPE of 18.34% on one test day indicates that indoor-outdoor disturbances significantly affect average prediction accuracy, with its stability constrained by the magnitude of these disturbances.

3.

Controller 3 (Closed-Loop Load Gray Model)

The overall error of Controller 3 is slightly lower than that of Controller 2, and it exhibits better tracking of sudden load changes than Controller 1. Compared with Controller 2, its prediction curve has lower fluctuation frequency and amplitude, with MAPE values concentrated in the 12–15% range over three days, indicating good consistency. Considering the outdoor temperature trend in Section 4.1, Controller 3 experiences the greatest fluctuations in outdoor temperature, yet its prediction accuracy remains high, demonstrating strong robustness to indoor-outdoor disturbances.

Table 7. Prediction of Load.

MPC Name	Day 1 (MAE/MAPE)	Day 2 (MAE/MAPE)	Day 3 (MAE/MAPE)	Mean (MAE/MAPE)
Controller 1	392.14W/15.70%	618.12W/26.15%	618.12W/26.15%	544.07W/22.04%
Controller 2	285.76W/11.58%	422.55W/16.63%	344.56W/13.05%	351.05W/13.77%
Controller 3	331.19W/13.25%	396.11W/13.85%	335.74W/12.79%	354.33W/13.14%

presents the prediction metrics of the three controllers, indicating that Controller 3 achieves a slightly higher overall accuracy than Controller 2, and both outperform Controller 1 significantly. The aforementioned images and indicator results reveal the differences in emphasis among the three types of models in load prediction:

Heat balance based theoretical model: Captures the general trend but lacks responsiveness to dynamic fluctuations;

Data-driven model: While capturing the general trend, it achieves more accurate predictions of dynamic fluctuations, but is less robust to external fluctuations.

Closed-Loop Load Gray Model: It significantly enhances the model’s robustness against external disturbances and ensures stable prediction performance.

4.3. Controlled Temperature Stability

The indoor temperatures recorded over the 9-day experimental period are presented in Figure 16, Figure 17 and Figure 18. In these figures, the solid lines denote the actual room temperatures recorded by sensors, the dashed lines represent the predicted temperatures generated by the MPC controller, optimized using the PSO algorithm in conjunction with the simulation model, and the colored regions indicate the ±1 °C error bounds around the predicted temperatures.

Analysis of overall trends reveals that Controller 1 exhibits the largest deviation from the predicted range in control performance, followed by Controller 2, whereas Controller 3 shows the smallest deviation. This ranking of deviation magnitudes is consistent with the model’s prediction accuracy reported in Section 4.2, indicating a strong correlation between the model’s prediction accuracy and its control performance.

Figure 16 shows that Controller 1 of the white-box model failed to accurately predict the building cooling load, resulting in excessively high temperatures across the entire controlled environment under its incorrect control. This poor prediction accuracy stems from several factors, including the lack of temperature sensors to measure temperature differences across interior walls, the roof, and the floor, as well as material heat transfer coefficients provided without detailed measurement. Although white-box models are theoretically the most accurate, in practice, due to cost constraints, it is not feasible to obtain detailed boundary temperatures or conduct precise parameter-measurement experiments, leading to deviations in their predictions.

Figure 17 shows that, after being trained on historical data from actual operations, the black-box model can provide accurate predictions to the controller, resulting in a significant improvement in control performance compared to Controller 1. However, there are still deviations in the overall control performance. This is because errors are inevitably present not only in the load prediction model but also in the actual HVAC simulation model, and the cumulative errors of the two models lead to relatively large control errors.

Figure 18 illustrates the CLLGM—after correction based on on-site temperature data and the simulation model itself—providing predictions more compatible with the controller’s simulation model. Compared to Controller 2, the controller adopting the corrected CLLGM exhibits a significant improvement in overall control performance, despite a few minor instances of exceeding the control range boundaries. The root cause of this improvement lies in the reduced matching error between the load prediction model and the simulation model. Specifically, while the prediction results still exhibit roughly the same deviation from the actual cooling load, the control errors induced by the simulation model’s inherent inaccuracies have been effectively compensated for.

This study employs the TDI and MDE metrics, as described in Section 2.3, to quantitatively evaluate the control performance of the three controllers. The corresponding statistical results of TDI and MDE are summarized in

Table 8. TDI and MDE statistical results.

Index	Controller 1	Controller 2	Controller 3	ANOVA
TDI Mean ± SD (°C·h)	4.86 ± 1.48	1.44 ± 0.20	0.28 ± 0.28	F = 21.99, p = 0.0017
TDI 95% CI (°C·h)	[2.31, 7.41]	[1.04, 1.84]	[−0.42, 0.98]
TDI vs. C3			94.19%/80.43%
TDI p (vs. C3)	p = 0.0296	p = 0.0059
MDE Mean ± SD (°C)	2.46 ± 0.68	2.11 ± 0.63	0.74 ± 0.36	F = 7.56, p = 0.0229
MDE 95% CI (°C)	[0.77, 4.14]	[0.55, 3.66]	[−0.17, 1.64]
MDE reduction vs. C3			70.02%/64.99%
p-value (vs. C3)	p = 0.0293	p = 0.0422

. One-way ANOVA indicates statistically significant differences in TDI among the three controllers (F = 21.9889, p = 0.0017 < 0.05). The proposed controller achieves a mean TDI of 0.28 °C·h, corresponding to reductions of 94.19% and 80.43% compared with the two baseline controllers. Similarly, significant differences are observed in MDE (F = 7.5581, p = 0.0229 < 0.05), with the proposed controller reducing MDE by 70.02% and 64.99%, respectively. Pairwise t-tests confirm that all these improvements are statistically significant (p < 0.05).

Notably, as reported in Section 4.2, the difference in load prediction accuracy between the black-box model and CLLGM is only 0.63% in terms of MAPE. Despite a small discrepancy in prediction performance, the proposed controller yields a dramatic 80.43% improvement in temperature stability, demonstrating its strong advantage in practical control performance.

These results confirm that the proposed controller achieves substantially higher control accuracy and significantly lower temperature deviation, validating its superiority over the baseline strategies.

4.4. Comprehensive Controller Performance

The energy consumption of each controller and the statistical scores calculated from the PSO objective function (Equation (9)) presented in Section 3.1 are illustrated in Figure 19, Figure 20 and Figure 21. Owing to the actual operating characteristics of the motor in the air-conditioning system, the measured energy consumption data show considerably high-frequency fluctuations. To better visualize the data trends, a moving-average filter with a window length of 5 is applied to the raw energy consumption data, where the arithmetic mean within each sliding window is used as the filtered value at the window center. Notably, this filtering process is used only for graphical visualization, while all statistical analyses and quantitative energy consumption results are computed from the original, unfiltered data to ensure accuracy and reliability. In these figures, the orange solid line represents the control score obtained from the actual experimental results, while the dashed line indicates the control score derived from the prediction model during MPC decision-making.

The overall statistical results of energy consumption and control score are summarized in

Table 9. Score of three types of controllers.

Index	Controller 1	Controller 2	Controller 3	ANOVA
Energy Mean ± SD (kW)	1356.60 ± 28.08	1473.00 ± 29.70	1359.40 ± 261.07	F = 0.5684
Energy 95% CI (kW)	[1286.83, 1426.37]	[1399.21, 1546.79]	[710.88, 2007.92]	p = 0.5942
Energy p (vs. C3)	p = 0.9869	p = 0.5303
Energy vs. C3 (%)	0.21%	−7.71%
Score Mean ± SD	38.72 ± 19.05	73.91 ± 0.77	86.15 ± 8.51	F = 12.5154
Score 95% CI	[−8.61, 86.05]	[71.99, 75.84]	[65.02, 107.28]	p = 0.0072
Score p (vs. C3)	p = 0.0338	p = 0.1292
Score vs. C3 (%)	122.49%	16.55%

. No statistically significant difference in energy consumption is observed among the three controllers, indicating that their overall energy usage is comparable. Nevertheless, the mean values reveal that Controller 2 consumes 7.71% more energy than Controller 3. In contrast, the final control scores exhibit a statistically significant difference across the three controllers. Specifically, Controller 3 achieves a substantial improvement over Controller 1, which is attributed to the insufficient calibration of the white-box model parameters without detailed experimental validation. Although Controller 3 improves the score by approximately 16.55% compared with Controller 2, this enhancement is not statistically significant because Controller 2 already achieves a high level of control performance, leaving limited room for further significant improvement.

5. Conclusions and Future Work

This study addresses the challenges of high energy consumption in public building HVAC systems and the limitations of existing control strategies, focusing on developing a simplified modeling-based MPC controller suitable for this purpose. The research aims to improve the practicality and control performance of MPC by addressing the complexity of load prediction and modeling in dynamic environments. To achieve this, three load calculation models were designed and integrated into MPC controllers: a thermal equilibrium theoretical model (Controller 1), a data-driven model (XGBOOST, Controller 2), and a Closed-Loop Load Gray Model (CLLGM, Controller 3). Experimental validation was conducted in an HVAC laboratory retrofitted in an aged building in Wuhan, China, over 9 days, evaluating performance in terms of load-prediction accuracy, temperature control stability, energy consumption, and comprehensive scores. Key findings are as follows:

1. Load prediction performance: Controller 3 (CLLGM) exhibited the strongest robustness to indoor-outdoor disturbances, with a mean MAPE of 13.14%, outperforming Controller 1 (22.04%) and showing better consistency than Controller 2 (13.77%). It effectively tracked sudden load changes, addressing the limitations of theoretical models (poor dynamic response) and data-driven models (instability under fluctuations).

2. Temperature control stability: Controller 3 outperforms Controllers 2 and 1 by achieving the smallest values for both key stability metrics—with a mean TDI of 0.28 °C·h (significantly lower than Controller 2 is 1.44 °C·h and Controller 1 is 4.86 °C·h) and a minimum MDE of 0.74 °C (compare to 2.11 °C for Controller 2 and 2.46 °C for Controller 1)—confirming its capability to maintain stable indoor temperatures.

3. Comprehensive performance: Based on the multi-objective optimization function, Controller 3 achieved the highest average score of 86.15, which is 16.55% higher than that of Controller 2 (73.91). Although this improvement is not statistically significant, given Controller 2’s already high performance, Controller 3 still exhibits better comprehensive performance. In contrast, Controller 1 shows relatively poor comprehensive performance, with a significantly lower score of 38.72, attributed to inaccurate white-box model parameters and insufficient control adjustments. In terms of energy consumption, no significant statistical difference is observed among the three controllers, while Controller 3 maintains a lower average energy consumption than Controller 2. Notably, Controller 3 demonstrates balanced performance in both energy efficiency and thermal comfort, further verifying its superiority in overall operational effectiveness.

This research has limitations that should be explicitly acknowledged. First, the experimental validation was conducted in a single HVAC laboratory retrofitted in an aged building in Wuhan, where detailed building envelope structural parameters were unavailable due to the retrofitting conditions. This lack of precise envelope parameters may have introduced minor biases to the physics-based load calculation model and the overall simulation framework. Second, the actual cooling load was derived entirely from indirect estimation via system operational data and heat transfer equations (Equations (10)–(12)) for model calibration and performance evaluation. This indirect estimation approach may have introduced a small degree of deviation to the load reference values, which serve as the benchmark for assessing model prediction accuracy. Third, the experimental results are limited to the typical summer cooling conditions in Wuhan, and the performance of the CLLGM under other climate zones, building types, or HVAC system configurations remains to be verified.

This research validates that a simplified modeling-based MPC controller can effectively improve HVAC control performance in buildings, with the CLLGM emerging as the best-performing load calculation method among the tested approaches under the specific experimental conditions of the retrofitted HVAC laboratory in Wuhan over the nine-day test period. Future work will focus on: (1) expanding the experimental scope to more building types and climate zones to verify generalizability; (2) optimizing the CLLGM algorithm to reduce computational overhead and improve real-time performance; (3) integrating more dynamic disturbance factors (e.g., occupant behavior) to enhance control adaptability further.