A Comparative Analysis of Machine Learning Algorithms in Predicting the Performance of a Combined Radiant Floor and Fan Coil Cooling System

Abstract

Machine learning algorithms have proven to be practical in a wide range of applications. Many studies have been conducted on the operational energy consumption and thermal comfort of radiant floor systems. This paper conducts a case study in a self-designed experimental setup that combines radiant floor and fan coil cooling (RFCFC) and develops a data monitoring system as a source of historical operational data. Seven machine learning algorithms (extreme learning machine (ELM), convolutional neural network (CNN), genetic algorithm-back propagation (GA-BP), radial basis function (RBF), random forest (RF), support vector machine (SVM), and long short-term memory (LSTM)) were employed to predict the behavior of the RFCFC system. Corresponding prediction models were then developed to evaluate operative temperature (T_op) and energy consumption (E_h). The performance of the model was evaluated using five error metrics. The obtained results showed that the RF model had very high performance in predicting T_op and E_h, with high correlation coefficients (>0.9915) and low error metrics. Compared with other models, it also demonstrated high accuracy in E_h prediction, yielding maximum reductions of 68.1, 82.4, and 43.2% in the mean absolute percentage error (MAPE), mean squared error (MSE), and mean absolute error (MAE), respectively. A sensitivity ranking algorithm analysis was also conducted. The obtained results demonstrated the importance of adjusting parameters, such as the radiant floor supply water temperature, to enhance the indoor comfort. This study provides a novel and effective method for evaluating the energy efficiency and thermal comfort of radiant cooling systems. It also provides insights for optimizing the efficiency and thermal comfort of RFCFC systems, and lays a theoretical foundation for future studies integrating machine learning algorithms in this field.

1. Introduction

In 2018, almost 37% of total energy consumption in China was attributed to the construction and operation of buildings [1,2]. It has been shown that 40–50% of the energy consumption of a building is attributed to the operation of HVAC systems that regulate indoor thermal environments. In addition, CO₂ emissions from building operations account for 42% of total carbon emissions [3]. Air conditioning systems are essential for establishing a comfortable indoor thermal environment, given that individuals spend 60–90% of their time indoors [4]. This widespread application emphasizes their significance. Air conditioning systems prove effective in mitigating overall energy consumption. Thus, many studies have been conducted on the optimization of their energy efficiency [5,6,7,8]. It has been shown that radiant cooling systems are able to significantly reduce energy consumption compared with conventional convective air conditioning systems [9,10,11].

Radiant floor cooling has been widely studied due to its low operating noise, space-saving design, low energy consumption, and thermal comfort [12,13,14,15]. Liu et al. [16] conducted a comparative analysis of three case studies and concluded that hybrid radiant cooling systems were particularly effective in extremely hot and humid conditions, prevalent during summer. They also deduced that these systems could significantly reduce energy consumption compared with other cooling systems. Ren et al. [17] studied many optimized control strategies for radiant cooling systems aiming to maximize energy savings while maintaining indoor thermal comfort. Seo et al. [18] conducted a simulation study on radiant floor cooling systems using different ventilation systems. They deduced that combining a ventilation system with radiant floor cooling was essential for efficient and energy-saving operation. Similarly, Srivastava et al. [19,20] conducted experiments and simulations to explore the combined application of roof-mounted radiant systems with various ventilation systems. Their results showed that radiant cooling systems could efficiently operate across various climatic regions. Similar to traditional air conditioning systems, employing suitable operational approaches is crucial for reducing energy consumption in combined radiant floor and fan coil cooling (RFCFC) systems [21]. Cui et al. [22] studied the efficiency of four control strategies for radiant cooling systems within office buildings situated across various climatic regions in China. They deduced that an appropriately engineered control system could effectively mitigate the influence of external climatic factors on indoor thermal comfort, which resulted in decreasing carbon emissions and energy usage. Zhu et al. [23,24] demonstrated that RFCFC systems had very high energy utilization efficiency and comfort levels in different environments. Atienza et al. [25,26] deduced that combining radiant floor systems with ventilation systems ensures indoor thermal comfort and provides a high potential for energy savings.

In recent years, various artificial intelligence algorithms have been proposed to describe and predict the operation behavior of air conditioning systems in buildings [27]. These algorithms employ different methods for the development of models. The development of machine learning (ML) models allows the efficient optimization of energy-efficient automation systems [28,29]. Li et al. [30] combined the Extreme Gradient Boost (XGBoost), CatBoost, and other novel gradient boosting algorithms with different optimization techniques in order to predict the electricity load using twenty years of historical data from a Turkish transmission company. Their results demonstrated that machine learning algorithms had high accuracy in energy consumption forecasting, showing very high predictive performance. Zhang et al. [31] developed an intelligent algorithm based on dynamic time entropy to conduct a case study in green building projects. Their results demonstrated the high applicability and accuracy of this algorithm. Su et al. [32] applied computational fluid dynamics to develop a geometric model and used an artificial neural network (ANN) to evaluate the condensation risk associated with a radiant floor cooling system. They also employed a back propagation (BP) neural network to estimate the required pre-dehumidification duration and the associated energy consumption. Their results showed that the ANN performs effective predictions, successfully averting the risk of condensation. Aparicio-Ruiz et al. [33] employed eight variables, including the indoor globe temperature, indoor air temperature, mean radiant temperature, and wind speed, in order to predict indoor thermal comfort in Mediterranean climates. They aimed to optimize thermal comfort while reducing energy consumption. Tang et al. [34] proposed a control framework for managing the indoor temperature of radiant ceiling cooling systems, leveraging a deep-reinforcement learning model. They also conducted a comparison with conventional control strategies. Their results showed that their proposed approach could achieve a 10.5% reduction in energy consumption. Yang et al. [35] proposed a multi-objective adaptive data-driven predictive control system that uses artificial neural network models for the prediction of indoor dynamics and thermal comfort. This approach allows the balancing of energy costs and thermal comfort. Hou et al. [36] developed an extreme learning machine model optimized using the grey wolf optimization algorithm in order to predict thermal comfort and CO₂ concentration in air-conditioned rooms. Alaraj et al. [37] proposed a method for predicting and optimizing electricity consumption of the conditioning system of an office building for the next year using ML. Their results demonstrated that this model had a higher performance than existing traditional models. Pandey et al. [38] developed a deep neural network with three hidden layers to predict the window state of student dormitories and evaluate the energy losses associated with windows. Zhou et al. [39] applied the k-nearest neighbor, support vector classification, random forest, logistic regression, and extreme gradient boosting algorithms to predict the operation behavior of an open-space office air conditioning system. Their results demonstrated that all these algorithms have high performance. Peng et al. [40] optimized control strategies for air conditioning systems in different types of office buildings, resulting in a reduction of energy consumption in the range of 40–52% compared with conventional time-based scheduling systems. Liu et al. [41] proposed a fast method for predicting the cooling capacity of a radiant cooling floor. The method used CFD to develop a three-dimensional geometric model and employed a backpropagation neural network to learn and optimize operation behavior of the radiant cooling system. Mun et al. [42] applied support vector machines, random forest, and logistic regression algorithms to predict the operation behavior of air conditioning systems in residential buildings. Their results demonstrated that the random forest model outperforms the other two models.

At present, many HVAC systems do not accurately perform automation control. In addition, the systems that adopt automation mostly rely on traditional control methods [4,43]. These methods can meet basic adjustment requirements. However, they cannot guarantee the optimal operational efficiency of the equipment or perform coordinated load distribution and control optimization at the system level. Practical applications often do not require HVAC systems to operate at full capacity. Partial load operation is usually sufficient to meet user demands [44]. Moreover, running systems at full capacity when unnecessary results in significant energy wastage.

To solve this problem, this study proposes a machine learning-based modeling approach that uses historical operational data and outdoor climate conditions to predict the thermal comfort and energy consumption of radiant HVAC systems. This data modeling approach provides a clear visualization of system operation patterns, which allows to eliminate the guesswork. Through data observation and analysis, the optimal timing for equipment replacement can be planned, and practical adjustments can be performed based on the underlying model. This approach accurately describes equipment upgrade status and cost savings, mitigates uncertainties, and reduces the probability of erroneous decisions. It has been shown [45,46] that energy savings in the range of 15–40% can be achieved by adopting comprehensive intelligence, digitization, and integrated management of HVAC systems. Precise control over thermal comfort and energy consumption significantly reduces HVAC energy consumption and carbon emissions during building operation.

ML algorithms have been widely applied in various fields, yielding satisfactory results. However, studies on the prediction of energy consumption and thermal comfort of RFCFC systems using ML algorithms are still limited. In addition, there are few that compare the performances of different ML algorithms, and thus the evaluation of their advantages and disadvantages is not straightforward. Therefore, this study applied the following ML algorithms to predict the thermal comfort and energy consumption of RFCFC systems: extreme learning machine (ELM), convolutional neural network (CNN), genetic algorithm-back propagation (GA-BP), radial basis function (RBF), random forest (RF), support vector machine (SVM), and long short-term memory (LSTM). The performance and effectiveness of these algorithms in predicting the behavior of combined cooling systems were then compared using commonly employed metrics. Furthermore, the sensitivity of all the input variables was discussed using the RF and SVM systems and the BP neural network. The obtained results build a foundation for modeling thermal comfort and energy consumption of RFCFC systems using ML algorithms.

2. Materials and Methods

The technical framework of this study is presented in Figure 1. An RFCFC experimental platform, which serves as the data source, was first designed and constructed in Jinan, China. A custom-built detection system was then developed for experimental data collection. This system was equipped to capture both indoor and outdoor environmental variables, as well as data pertaining to inhabitant behavior. Afterwards, a case study was conducted on the experimental platform to evaluate the efficiencies of all the ML algorithms. The underlying assessment focused on their predictive abilities regarding indoor thermal comfort and the associated energy consumption of the system. Finally, based on the aforementioned study, the sensitivity of all the input variables was discussed using the RF, SVM, and BP sensitivity ranking models.

2.1. Experimental Setup

Field measurements were conducted on a bespoke experimental platform designed for RFCFC systems. The experimental room was situated in Jinan, China, characterized by its cold climate. Figure 2 shows this experimental room and a photograph of the experimental edifice. This one-story structure had a length, width, and height of 4, 2.8, and 3 m, respectively. Its exterior walls had aerated concrete composite panels, characterized by a U-value of 0.536 W/(m²·K) and a thickness of 250 mm. The southern façade was equipped with aluminum frame double-glazed windows covering an area of 2.42 m², with a U-value of 2.4 W/(m²·K).

The radiant floor coil contained a dual-circuit configuration using dry-buried pipes, as shown in Figure 3a. The sequential assembly, from bottom to top, was composed of floor, insulation, buried pipe, and filling layers, each with a thickness of 150, 18, 15, and 45 mm, respectively, as well as a topping 12 mm surface layer, which included a decorative floor covering of the room and a leveling layer. A radiation cooling module was formed by integrating the insulation and buried pipe layers. The floor radiation heat exchange coil had an outer diameter, wall thickness, and spacing between coils of 12, 2, and 60 mm, respectively. It was constructed from polypropylene random copolymer. Figure 3b illustrates this basic structure.

Figure 4 illustrates the components of the RFCFC system, which include a refrigeration unit, a thermostatic water tank, a circulating water pump, a mixing water pump, a stabilizing component, and an air-conditioning terminal. The RFCFC system initiates its process by chilling water from the air-source heat pump, which then enters the thermostatic water tank. The circulating water pump then pressurizes the water to mitigate the resistance within the pipeline. As the water advances through the system, it traverses sensors that measure the flow and temperature, which ensures that the parameters are within the desired range before reaching the air-conditioning terminal.

The circulating pipeline was constructed from polyvinyl chloride plastic hoses, and the exterior was insulated with cotton to increase thermal efficiency. This configuration limits the frequent cycling of the main unit, which ensures a steady supply of chilled water to the fan coil unit and the thermostatic water tank. This design facilitates the extraction of chilled water from the tank, as required. A smart socket was installed to measure the system power of the RFCFC system. In addition, in the experiment, two dummies of 124 W and 138 W power were placed indoors to simulate the heat release from indoor occupants. The size of the mannequins was based on the measurements of an adult male. Their dissipated heat represented the sensible heat released by individuals engaged in light physical activity in the indoor laboratory. The specifications of the experimental apparatus are presented in

Table 1. Parameters of experimental equipment.

Experimental Equipment	Description of Parameters
Air-source heat pump	Cooling capacity, 6 kW Operating flow rate, 1.8 m3/h Dimensions, 1294 mm × 485 mm × 730 mm Integrate partial load value, 2.81
Thermostatic water tank	Water capacity, 60 L Water tank dimensions, 470 mm × 600 mm Foam thickness, 50 mm Material, 304 Stainless Steel
Fan coil unit	Cooling capacity, 2.7 kW Fan velocity, 260/390/510 m3/h Dimensions, 820 mm × 760 mm × 785 mm Weight, 19 kg
Mixing water pump	Pump lift, 6 m Dimensions, 85 mm × 292 mm × 205 mm Permissible water temperature range, 0–80 °C Applicable area, 0–120 m2
Pressure regulating pipe	Capacity, 40 L Pressure resistance, ≤1.25 Mpa Dimensions, 350 mm × 570 mm

.

A custom-built experimental data detection system was used for the RFCFC. This system comprised a flow meter and water temperature sensor installed along the circulation pipeline. It was also equipped with a paperless recorder that operated in coordination with the flow meter and temperature sensor, allowing for direct readings and data uploading to be facilitated. In addition, the indoor environment was equipped with a multi-channel temperature and humidity measurement instrument, thermocouples, an indoor thermal comfort meter, and an outdoor solar radiation meter. Temperature assessments within the indoor space were conducted using the multi-channel instrument, while the PT-100 thermocouples were used for surface temperature measurements. The detailed specifications of all the deployed experimental instruments are presented in

Table 2. Parameters of measurement instrument.

Parameter	Range	Resolution	Accuracy	Instruments
Indoor air temperature	−20–80 °C	0.1 °C	±0.5 °C
Indoor relative humidity	0–100%	0.1%	±3%
Outdoor air temperature	0–100%	0.1 °C	±2.5 °C
Outdoor relative humidity	−40–100 °C	0.1%	±2.5%
Floor surface temperature	−200–1372 °C	0.1 °C	±0.5 °C
Water temperature	−200–1372 °C	0.1 °C	±0.5 °C
Wall and ceiling temperature	−100–300 °C	0.1 °C	±0.5 °C
Ventilation air temperature	−10–60 °C	0.1 °C	±0.3 °C
Ventilation air relative humidity	5–95%	0.1%	±3%
Water flow rate	0–3.0 m3/h	1 m3/h	±1 m3/h
Solar radiation	0–1300 W/m2	1.25 W/m2	±10 W/m2

.

The outdoor ambient air temperature, outdoor ambient air relative humidity, and outdoor solar radiation were measured using HOBO temperature and humidity meters and solar radiation meters, which allowed the determination of the local climatic conditions in the laboratory. A protective cover that allowed outdoor air circulation was also installed to prevent adverse outdoor environmental conditions from affecting the measurement data of the outdoor solar radiation meter.

2.2. Algorithms

CNN, ELM, GA-BP, RBF, RF, SVM, and LSTM ML algorithms were adopted to develop models for predicting the performance of the RFCFC system. Each of these algorithms has its unique characteristics. They were selected according to their suitability for the characteristics of the dataset used and their common usage in the field of energy consumption and occupant behavior prediction in buildings.

(1)

CNN

It represents a subclass of feedforward neural networks that are characterized by their use of convolutional operations and deep architectural constructs [47]. CNNs are well known for their crucial role in computer vision applications. They form the foundational architecture for contemporary deep learning approaches, such as generative adversarial networks (GANs). A typical CNN comprises convolutional layers for feature extraction, pooling layers for feature aggregation, and fully connected layers for generating probability distributions from input images. It significantly reduces the redundancy of information, and it is able to adapt to sudden events during model training, which results in very high performance even when dealing with complex datasets [48].

(2)

ELM

It is an enhanced and more efficient iterative approach derived from the conventional feedforward perceptron neural network [49]. It is used to train single-hidden-layer feedforward neural networks (SLFNs). In contrast to traditional training algorithms for SLFNs, ELM employs a random selection process for input layer weights and hidden layer biases. It then calculates the output layer weights by minimizing a loss function that comprises the training error term and the regularization term associated with these weights. Afterwards, the solution is determined through analytical computation based on the Moore–Penrose generalized inverse matrix theory. Theoretical studies demonstrated that ELM exhibits the universal approximation capacity of SLFN, even when dealing with randomly generated hidden-layer nodes [50].

(3)

GA-BP

BP neural networks have distinct input, hidden, and output layers. These layers are fully interconnected with no interneuronal connections within the same layer. The functioning of BP neural networks is facilitated by a two-phase process: the forward propagation of the input signal and the subsequent backpropagation of the error which allows the adjustment of the underlying weights [51]. The genetic algorithm is a model that simulates the natural selection and genetic mechanisms of biological evolution in Charles Darwin’s theory. By performing mathematical calculations and conducting computer simulations, it translates the problem-solving process into operations, including chromosome gene crossover and mutation, allowing for the simulation of biological evolution. In order to solve the problem of BP neural networks (i.e., easily falling into local optima), a hybrid approach combining the genetic algorithm and Levenberg–Marquardt algorithm is implemented to optimize the BP neural network and thus determine the global optimal solution. The fundamental concept of the genetic algorithm consists in optimizing the weights and thresholds of the BP neural network [52].

(4)

RBF

It is a three-layer feedforward network with a single hidden layer. It is able to rapidly approximate any nonlinear function with arbitrary precision [53]. The input layer consists of nodes that signify the sources of input signals. The second layer and third layer are considered as the hidden and output layers, respectively. The transformation from the input space to the hidden layer space is characterized by nonlinear mapping, while the progression from the hidden layer space to the output layer space is characterized by linear mapping. The hidden units use radial basis functions as their transformation functions, while the neurons in the output layer use linear units. Compared with other feedforward networks, RBF neural networks have higher generalization ability, and they do not face the problem of local minima [54].

(5)

RF

It is an ensemble of multiple decision trees, which mitigates overfitting and increases the prediction accuracy compared with single decision trees [55]. It consists in partitioning the underlying data into out-of-bag and in-bag subsets, based on ensemble learning with random sampling. Subsequent resampling techniques generate distinct training sample sets that are used to train individual learners within the ensemble and to facilitate the data processing [56].

(6)

SVM

It is a machine-learning algorithm based on the statistical learning theory. It is well-suited for small sample sizes [57]. It mainly employs linear and radial basis function kernels. The least squares SVM variant reformulates the inequality constraints of the standard SVM as equality constraints and modifies the error term to a sum of squared errors. This method transforms the quadratic programming challenge into a set of linear equations by representing the empirical loss of the training set with a loss function. This allows for a significant increase in the efficiency and convergence accuracy of the problem-solving stage. SVM is widely applied in various practical applications due to its very high generalization performance [58].

(7)

LSTM

It is an improved version of recurrent neural networks (RNNs), designed to mitigate the challenge of handling long-term dependencies. It consists of two fundamental components: gates and cell [47]. Compared with RNN, LSTM introduces an innovative “gating” mechanism, including forget gates, input gates, and output gates. These gated structures allow the model to selectively retain or discard information, which significantly enhances the learning of long-term dependencies [59]. In addition, it adds connections between the hidden layer nodes of the RNN structure and filters past states. It outperforms the ordinary RNN models when dealing with continuous and well-structured long-term sequences. The key feature of LSTM is the cell state. It carries and updates relevant information through cell state propagation, and input gate, output gate, and forgetting gate mechanisms. As a result of these mechanisms, LSTM networks excel in capturing long-term dependencies and overcoming the gradient vanishing issue commonly encountered in traditional RNNs, making them widely used in tasks related to time series data and sequence prediction [60].

2.3. Feature Parameter Selection

The data was collected from a self-built RFCFC system. Since this study focuses on the summer cooling season, the data were selected from 1 August to 31 August 2021, and 1 August to 31 August 2023. An interval of 10 min was adopted to ensure high accuracy of the data. The measured data were divided into two parts: 70% were randomly selected for the ML algorithm and 30% were used for verification.

Data preprocessing techniques were applied to address the presence of abnormal and missing data. Interpolation methods were also adopted to fill in the gaps for missing data. When encountering abnormal data, outlier values were first identified using the 3σ criterion, and interpolation methods were then applied to determine suitable replacement values. In the data collection process, the instances of long-term repetition or omission were considered anomalies, and thus they were excluded from the data [61].

Several determinants affect indoor thermal comfort and system energy consumption, including climatic conditions, characteristics of the HVAC system, building attributes, and societal factors. In addition, the climatic variables—(i.e., temperature, relative humidity, precipitation, and solar radiation)—affect the indoor cooling load. In particular, when the temperature increases, the cooling load also gradually increases. The solar radiation significantly affects the cooling load. More precisely, it increases the indoor temperature and consequently increases the cooling load demand. Some other characteristics of the RFCFC system also have a significant effect, such as the supply/return water temperature, supply air temperature, and supply water flow rate. Building-related factors, including the envelope thermal performance, geographical location, functional use, insulation quality, and number of residences, also have an obvious effect. Social factors, including the population size, population growth rate, and lifestyle habits of the residents, may affect thermal comfort and system energy consumption. However, their impact is modest and thus it is often not accounted for in short-term forecasting. In addition, once a building is operational, the impact of its intrinsic factors on the cooling load is usually considered negligible.

In summary, through a preliminary analysis and a review of relevant literature on the factors affecting the thermal comfort and system energy consumption, ten features were determined as input variables for the prediction model: the fan coil supply water temperature (T_cs), radiant floor supply water temperature (T_fs), fan coil supply water flow rate (V_cs), radiant floor supply water flow rate (V_fs), fan coil supply air temperature (T_ca), outdoor ambient air temperature (T_out), outdoor ambient air relative humidity (φ_out), outdoor solar radiation (SR), radiant floor surface temperature (T_f), and indoor heat source (Q_c). Moreover, the operative temperature (T_op) and energy consumption (E_h) were considered as the output variables.

2.4. Objective Evaluation

2.4.1. Error Indicators

Five indicators were used to evaluate the accuracy of the prediction model: the coefficient of determination (R²), mean absolute error (MAE), root mean squared error (RMSE), mean squared error (MSE), and mean absolute percentage error (MAPE). These indicators were calculated using the following equations:

$$R^2=1-\frac{\sum _{\mathrm{i}=1}^n{(x_{\mathrm{i}}-y_i)}^2}{\sum _{\mathrm{i}=1}^n{(x_{\mathrm{i}}-\overline{y})}^2}$$

(1)

$$MAE=\frac{1}{n}\sum _{\mathrm{i}=1}^n\left|x_{\mathrm{i}}-y_{\mathrm{i}}\right|$$

(2)

$$RMSE=\sqrt{\frac{1}{n}\sum _{\mathrm{i}=1}^n{(x_{\mathrm{i}}-y_{\mathrm{i}})}^2}$$

(3)

$$MSE=\frac{1}{n}\sum _{\mathrm{i}=1}^n{\left(x_{\mathrm{i}}-y_{\mathrm{i}}\right)}^2$$

(4)

$$MAPE=\frac{100\%}{n}\sum _{\mathrm{i}=1}^n\left|\frac{x_{\mathrm{i}}-y_{\mathrm{i}}}{y_{\mathrm{i}}}\right|$$

(5)

where x_i is the i-th simulated predicted value, y_i is the i-th actual experimental measurement value, $\overline{y}$ is the average of the actual experimental measurement values, and n is the number of samples in the underlying dataset.

Note that small MAE, RMSE, MSE, and MAPE values denote a small difference between the actual measured and predicted values, which indicates that the method has higher accuracy. R² is used to evaluate the regression model in terms of its goodness of fit. It ranges between 0 and 1, where high values (approaching 1) denote a high accuracy in fitting the regression model.

2.4.2. Operational Indicators

Radiant floor systems mainly engage in heat exchange with the human body through radiation. Thus, the operational parameters of the RFCFC system should not only depend on the indoor air temperature. The T_op concept is then introduced to address this issue. The operative temperature is a comprehensive indicator of the combined effect of the indoor air temperature (T_a) and average radiant temperature (T_r) on the human body. It is crucial for analyzing the sensible heat balance between the human body and the environment, and thus it is an appropriate operational indicator. The T_op was calculated using the following equation [62]:

$$T_{op}=\frac{h_a{T}_a+h_r{T}_r}{h_c+h_r}=a_c{T}_a+a_r{T}_r$$

(6)

where T_a is the surrounding air temperature (°C), T_r is the average radiant temperature (°C), and a_r and a_c represent the proportions of radiation and convection, respectively.

The T_r could be calculated by weighting the six surface radiant temperatures. However, the coefficients used for the calculations differ based on whether the individuals are in a standing or sitting position. Since the occupants mainly sit indoors, the calculation formula corresponding to the seated position was adopted. It was calculated using the following equation [62]:

$$T_r=\frac{0.18\left(T_U+T_D\right)+0.22\left(T_L+T_R\right)+0.30\left(T_F+T_B\right)}{2\times \left(0.18+0.22+0.30\right)}$$

(7)

where T_U, T_D, T_L, T_R, T_F, and T_B represent the surface radiative temperatures of the six indoor faces.

The energy consumption (E_h) of the RFCFC system was monitored using instruments that were able to track the energy usage in real time. At the end of the experiment, the recorded data facilitated the calculation of the average energy consumption of the system over time. The analysis of these data allowed the evaluation of the energy efficiency of the system, providing empirical support for future studies and optimization. Continuous monitoring of energy consumption yields critical insights for the enhancement of the existing strategies to increase the energy efficiency of the system.

3. Results

3.1. Testing Conditions

The study was conducted in August, during the summer cooling season in Jinan, China. Basic statistical methods were applied to evaluate the performance of the RFCFC system. The experiment involved recording outdoor temperatures varying between 22.6 °C and 34.9 °C. When the system reached a stable state, it maintained a consistent indoor operative temperature between 25.5 °C and 27 °C. Figure 5a illustrates the correlation between the outdoor and indoor temperatures during the test period in 2023. It can be seen that the indoor temperature adhered to residential comfort standards. The indoor humidity (φ_in) fluctuated between 59% and 96%, which denotes significant variations during the initial test period. However, after 0–3 h of system operation, it gradually decreased and reached a stable state.

The predicted mean vote (PMV) index quantifies the thermal comfort sensations experienced by most individuals in a given environment. However, differences among individuals exist, and thus the PMV may not necessarily represent the sensations of all the individuals. The predicted percent dissatisfied (PPD) index denotes the proportion of people likely to be dissatisfied with a particular thermal environment. The PMV and PPD indexes are widely used to evaluate and describe indoor thermal comfort. According to ISO7730, the recommended PMV range and PPD values are −0.5–+0.5 and ≤10%, respectively. The PMV data during the 2023 test period is shown in Figure 5b. The PMV value within the room was found to fluctuate between −0.1 and 0.3, with a mean value of 0.16. Although in some cases, the PMV slightly exceeded the recommended range, the overall comfort level was still acceptable. The RFCFC system ensured indoor thermal comfort. The PMV was computed as in [63], using the following equation:

$$\mathrm{P}\mathrm{M}\mathrm{V}=(0.303e^{-0.036M}+0.0275)·f\left(W, M, t_a, P_a, I_{cl}, t_{age}, v\right)$$

(8)

where W is the mechanical work (W/m), M is the metabolic rate (W/m²), t_a is the surrounding air temperature (°C), P_a is the surrounding air vapor pressure (kPa), I_cl is the garment thermal resistance (m²·K/W), t_age is the indoor average radiative temperature (°C), and v is the air velocity in the room (m/s).

3.2. Comparison between Seven Models

The models were trained by adopting the measured data from the RFCFC system in August 2021 and August 2023, using MATLAB 2022, MathWorks, MA, USA. Based on the trained data, the optimal input parameters of all the models are summarized as follows: (1) CNN: the SGDM algorithm was used for optimization with a maximum iteration count of 1200 and an initial learning rate of 0.01. After performing 800 iterations, the learning rate decreased to 0.01 × 0.1. (2) ELM: the Sig algorithm was used for optimization with 50 nodes in the hidden layer. (3) GA-BP: the iteration count, learning rate, population size, crossover parameter, selection parameter, and critical error threshold were set to 1000, 0.01, 5, 2, 0.09, and 10⁻⁶, respectively. (4) RBF: the radial basis function expansion rate was set to 50. (5) RF: the minimum leaf node size for each tree and the number of decision trees were set to 5 and 100, respectively. The model was trained using the regRF_train function. (6) LSTM: the Adam optimizer was used with a maximum iteration count of 1200 and an initial learning rate set to 0.01. After 800 iterations, the learning rate decreased to 0.01 × 0.5. (7) SVM: the RBF was used as a Kernel function. The penalty factor (C), loss function (p), and radial basis function were set to 4, 0.01, and 0.8, respectively.

3.2.1. Prediction of Thermal Comfort

In the process of thermal comfort prediction, CNN, ELM, GA-BP, RBF, RF, LSTM, and SVM were used to predict the T_op. The MSE, RMSE, MAE, R², and MAPE were used to evaluate the accuracy of these models.

Table 3. Results of operative temperature prediction obtained by all models.

Model	MSE	RMSE	MAE	R2	MAPE (%)
CNN	0.0310	0.1760	0.1199	0.9824	0.445
ELM	0.0810	0.2846	0.2056	0.9530	0.768
GA-BP	0.0597	0.2443	0.1759	0.9656	0.657
RBF	0.0624	0.2497	0.1811	0.9640	0.676
RF	0.0159	0.1261	0.0823	0.9915	0.307
LSTM	0.1104	0.3323	0.2397	0.9389	0.902
SVM	0.0357	0.1889	0.1147	0.9797	0.428

shows the obtained error metrics for each model applied to the training and testing datasets. It could be seen that all the models has MAPE values less than 1%, which indicates high T_op prediction performance. The RF and CNN models had the lowest error indicators, with MAPE values below 0.5%, MSE values below 0.035, and MAE values below 0.15. In addition, these models had minimal deviation between the experimental results and the simulated prediction values.

The T_op prediction performances of the seven ML algorithms were compared using the robust Taylor diagram. The obtained results are shown in Figure 6. It could be seen that the results obtained by the models and the target variable were consistent. The proximity to the observed data, represented by the centered RMSE, served as a measure. A model can be considered the most optimal if it closely approximates the observed values, reflected by a high correlation value (approaching 1), a low RMSE, and a minimal variance. During the training stage, the RF algorithm outperformed the other models (e.g., CNN and SVM) in terms of the T_op prediction, denoted by the high correlation value (>0.9869) and low RMSE (<0.1080). SVM ranked second, with a correlation value greater than 0.9694 and RMSE less than 0.1651. Figure 7a shows the predictions made by RF on the training set. Based on their predictive abilities, the models were ranked in the following order: RF > CNN > SVM > GA-BP > RBF > ELM > LSTM. In the process of prediction for testing phases, RF also outperformed the other models, exhibiting a high correlation value (>0.9702) and low RMSE (<0.1607). CNN ranked second, with a correlation value greater than 0.9528 and RMSE less than 0.1990. Figure 7b shows the prediction results obtained by SVM on the training set. The models were ranked based on their predictive ability as follows: RF > CNN > SVM > RBF > GA-BP > ELM > LSTM. It could be deduced from the comparative analysis of the two phases that the RF algorithm outperforms the other models. Figure 7c shows the prediction results obtained by the RF model on the training and testing results. Figure 7d presents the error curves during the training and testing processes. The error was mainly controlled within the range of −0.4–0.4. The maximum relative error was 2.87%, and most of the relative errors were less than 0.8%, which indicates a high accuracy.

Comparative analysis of the two phases indicated that the RF algorithm surpasses the performance of other models. Figure 7c illustrates the predictive outcomes of the RF model on both the training and testing datasets.

3.2.2. Prediction of the Energy Consumption

In this section, the seven models were employed to predict the system’s energy consumption. The error metrics calculated on the results obtained by all the models are presented in

Table 4. Results of energy consumption prediction obtained by all the models.

Model	MSE	RMSE	MAE	R2	MAPE (%)
CNN	0.0017	0.0407	0.0270	0.9502	10.34
ELM	0.0048	0.0691	0.0494	0.8463	21.09
GA-BP	0.0042	0.0647	0.0462	0.8670	17.02
RBF	0.0042	0.0645	0.0453	0.8675	18.62
RF	0.0009	0.0306	0.0174	0.9742	7.24
LSTM	0.0051	0.0716	0.0520	0.8364	22.73
SVM	0.0033	0.0576	0.0333	0.8959	13.48

. The ELM, GA-BP, RBF, and LSTM models had MAPE values greater than 15%. Consequently, the prediction performance of these four models is low, and they are not recommended for energy consumption forecasting. On the contrary, the RF model had lower evaluation metrics, with an MAE value less than 0.02, MSE value less than 0.001, and MAPE value less than 10%, followed by the CNN and SVM models. Furthermore, the RF model outperformed these four models, reaching maximum reductions of 68.1, 82.4, and 43.2% in the MAPE, MSE, and MAE values, respectively.

Figure 8 presents a graphical comparison between the prediction performances of E_h obtained by the ML algorithms on the training and testing sets, using the robust Taylor diagram. It could be seen that during the training phase, RF had the highest correlation (>0.9591) and the lowest RMSE (<0.0260), followed by CNN (correlation > 0.9180 and RMSE < 0.0375). The remaining five models had correlations less than 0.9 and RMSE values greater than 0.05. In addition, it could be clearly seen that RF and CNN had higher energy consumption prediction abilities compared to the other five algorithms. Figure 9a shows the prediction results obtained by RF on the training set. Based on their prediction accuracies, the ranking of the seven algorithms is as follows: RF > CNN > SVM > RBF > GA-BP > ELM > LSTM.

In the testing phase, RF outperformed the other models, with a higher correlation value (>0.9117) and a lower RMSE value (<0.0260). Figure 9b illustrates the prediction results obtained by RF on the training set. Based on their prediction abilities, the ranking of the models was ranked as follows: RF > CNN > SVM > GA-BP > RBF > ELM > LSTM. The analysis and comparison of the two phases showed that in general, the ranking of the prediction abilities of all the algorithms was consistent. RF had high prediction performance, and it was suitable for the prediction of system energy consumption. Figure 9c presents the prediction results obtained by RF on the training and testing datasets. Figure 9d shows the error curves obtained during the training and testing processes. It can be seen that most of the relative errors are within 10%, which indicates high prediction accuracy.

3.3. Sensitivity Analysis of Parameters

The Pearson correlation coefficient quantifies the linear relationship between two variables, A and B, by dividing their covariance by the product of their respective standard deviations [64]:

$${\gamma }_{\alpha \beta }=\frac{cov\left(A,{\rm B}\right)}{{\sigma }_A·{\sigma }_{{\rm B}}}$$

(9)

where cov (A, B) is the covariance of A and B, σ_A and σ_B are the standard deviations of A and B, respectively.

This can be written in a more detailed form as:

$${\gamma }_{\alpha \beta }=\frac{n·{\sum }_{i=1}^n{\alpha }_i·{\beta }_i-\left({\sum }_{i=1}^n{\alpha }_i\right)·\left({\sum }_{i=1}^n{\beta }_i\right)}{\sqrt{n·{\sum }_{i=1}^n{\alpha }_i^2-{\left({\sum }_{i=1}^n{\alpha }_i\right)}^2}·\sqrt{n·{\sum }_{i=1}^n{\beta }_i^2-{\left({\sum }_{i=1}^n{\beta }_i\right)}^2}}$$

(10)

where α_i represents the i-th data point in variable A, β_i represents the i-th data point in variable B, and n is the number of samples in the dataset.

The range of correlation coefficient values lies between −1 and 1. A value of zero indicates a lack of correlation between the two variables. A value between 0 and 1 indicates a positive correlation, while values closer to 1 indicate a higher degree of correlation. On the contrary, a value between −1 and 0 indicates a negative correlation, while values closer to −1 indicate a higher degree of negative correlation. In addition, an absolute value closer to 1 indicates a higher correlation [65].

Table 5. Interpretation of Pearson correlation coefficient.

Interpretation	Negligible	Weak	Moderate	Strong	Very Strong
Absolute magnitude of the Pearson	0.00–0.10	0.10–0.39	0.40–0.69	0.70–0.89	0.90–1.00

presents an example of interpretation of the Pearson correlation coefficients [66].

The Pearson correlation was used to describe the correlation between 10 input variables and two output variables as shown in Figure 10. T_op was highly correlated with T_f, T_fs, and T_ca, while E_h was highly correlated with Q, T_fs, and T_out. This demonstrates that the temperature of the floor supply water temperature is a crucial parameter for maintaining the indoor thermal comfort and high energy efficiency of the RFCFC system. To minimize energy consumption and regulate indoor thermal comfort, the floor supply water temperature should be optimized through accurate adjustments.

The RF, SVM, and BP neural network sensitivity ranking methods were employed to evaluate the sensitivity of each input variable to T_op and E_h. Figure 11 presents the results of the sensitivity of all the input variables to T_op, where the horizontal axis represents the results of the sensitivity ranking, with higher values indicating higher sensitivity. It was observed that even within the same experimental room, the sensitivity rankings of the 10 influencing factors differed between the algorithms.

The importance ranking varied between the three sorting methods. In RF and SVM, T_f was the most sensitive parameter, while in BP, T_ca was the most sensitive one. The top five importance parameters according to the RF, BP, and SVM sensitivity ranking methods were (T_f, T_fs, T_ca, T_cs, and φ_out), (T_ca, SR T_f, φ_out, and T_fs), and (T_f, T_ca, SR, T_out, and T_fs). T_f, T_ca, SR, φ_out, and T_fs frequently appeared but in different orders. Considering the combined results of the three sensitivity ranking methods, T_f and T_ca were consistently ranked among the top four important parameters. In addition, T_f was considered the most significant parameter. The floor surface temperature indirectly reflected the floor supply water temperature. Therefore, in the system adjustment process, the floor supply water temperature should be considered a crucial parameter for improving indoor environmental comfort.

Figure 12 presents the results of sensitivity to E_h for all of the input variables. In RF and BP, Q_c was the most sensitive parameter, while in SVM, T_fs was the most sensitive one. The top five importance parameters according to the RF, BP, and SVM sensitivity ranking methods were (Q_c, φ_out, T_f, T_out, and T_fs), (Q_c, T_out, T_fs, φ_out, and T_ca), and (T_fs, Q_c, T_out, T_f, and T_cs). Considering the three sorting methods, Q_c, T_out, and T_fs were the most important parameters, with Q_c being the most critical. Hence, the activation of the indoor heat source was essential in the E_h prediction.

4. Discussion

The collection of experimental data was one of the limited experiments conducted in China, where the RFCFC system was incorporated into a standalone outdoor building. This paper constructs a theoretical framework that supports the integration of combined cooling systems in the construction of intelligent buildings. Based on these obtained, it was demonstrated that seven ML algorithms had high performance in predicting the behavior of the RFCFC system. However, this study had some limitations, which are summarized as follows:

(1)

The seven ML algorithms had much higher predictive accuracy for operating temperatures than for energy consumption. This is due to the fact that the experimental instruments had long data time intervals, which resulted in some inaccuracies. In future work, we aim to study more accurate energy recording instruments to increase the accuracy of the measurement data.

(2)

In the conducted experiments, the outdoor weather conditions were limited. This restricts the richness of the datasets. Thus, a broader range of outdoor weather conditions should be considered to ensure that the RFCFC system can be applied to different outdoor weather conditions. The experiment was conducted for a limited number of days while considering the relatively short duration of the cooling season. Therefore, data from a longer time range and more frequent sampling should be used to increase the generalization ability of the models.

(3)

The ML algorithm models used in the operational predictions of the RFCFC system had high accuracy and performance. However, these developed models were not implemented in specific experimental cases. Future work will incorporate advancements towards implementing the operational patterns obtained from this ML algorithm into the system, provided that the underlying conditions allow for it. This will involve combining advanced model algorithms such as deep learning and reinforcement learning to further optimize the air conditioning system.

(4)

This study considered seven commonly used ML algorithms for predicting building energy consumption and indoor comfort. With the continuous development of artificial intelligence technologies, boosting techniques such as XGBoost and Catboost have been applied to the prediction of energy consumption [67,68]. In this study, XGBoost, Catboost, and LightGBM were used to establish predictive models for RFCFC systems. The performance metrics of the three algorithms and the two models having high performance are shown in Figure 13. The three boosting techniques had high prediction performance, with XGBoost having the highest performance. More precisely, in terms of thermal comfort, the R² of XGBoost was 2.29% lower than its RF. As for the energy consumption prediction, the difference between the two was 3.88%. Thus, in future work, we aim to develop and fine-tune novel models to increase the prediction performance of RFCFC systems. Moreover, integrating deep learning and ensemble methods, for increasing the prediction accuracy of RFCFC systems, is also of our interest.

Furthermore, the results of the conducted comparisons show that key influencing factors are crucial for increasing calculation accuracy. Increasing the number of input parameters may introduce interference, which decreases the accuracy of the results obtained [69,70]. Thus, the coupling relationship between different input quantities and algorithm performances will be further explored. In conclusion, future work will focus on overcoming the previously identified limitations and establishing a robust theoretical framework, facilitating the broader adoption and implementation of the RFCFC system.

5. Conclusions

This paper studied the performance of the RFCFC system, including energy consumption and thermal comfort. Seven models were employed to predict the T_op and E_h values. A sensitivity analysis on various input parameters was also conducted. The prediction accuracies of these seven models were then compared. The main contributions of this study, as well as the obtained results, are summarized as follows:

(1)

Seven ML models were developed to predict the thermal comfort and energy consumption of a radiant cooling system using historical operational data and outdoor climate conditions. Traditional control methods are unable to optimize the equipment operational efficiency and coordinate load distribution at the system level, which leads to the waste of energy. The proposed approach visually demonstrates system operational patterns, providing a solid foundation for achieving more efficient energy utilization and precise thermal comfort control.

(2)

The seven algorithms had similar thermal comfort prediction performance, with R² values greater than 0.93 and MAPE values less than 1%. A conducted comparative analysis of the training and testing results showed that all the ML methods had high performance, with RF having the highest one, making it the optimal choice.

(3)

The RF model had lower error metrics for energy consumption prediction compared to that of the other models. More precisely, it exhibited reductions of MAPE, MSE, and MAE of 68.1, 82.4, and 43.2%, respectively. The comparison between the two stages showed that RF outperformed the other models.

(4)

A sensitivity analysis was conducted using neural networks. The obtained results showed that the floor surface temperature was a crucial parameter affecting indoor thermal comfort, and the operating condition of indoor heat sources significantly affected energy consumption. Furthermore, a correlation existed between the floor surface temperature and the supply water temperature for the radiant floor cooling system. Therefore, in the process of RFCFC system regulation, the supply water temperature should be considered as an important control parameter in order to enhance indoor environmental comfort.

(5)

The number of input parameters was not positively correlated with the performance of the ML algorithms. In fact, the most efficient way to increase prediction accuracy is to incorporate key influencing factors as inputs. The increase in the number of input parameters does not necessarily lead to higher performance. In fact, it may result in overfitting. Therefore, the selection of input parameters should prioritize key factors related to the output variable for increasing the prediction accuracy.

This study comprehensively compared the performances of seven ML algorithms for predicting RFCFC system efficiency. The obtained results demonstrated the considerable significance of employing ML algorithms for predicting the operational behavior of the RFCFC system. The prediction results obtained by all the algorithms were consistent with the actual measured data related to thermal comfort and energy consumption. In future work, additional metrics will be used to evaluate the stability of the simulation results, and more human factors will be considered in order to increase the prediction accuracy.