Innovative Energy Efficiency in HVAC Systems with an Integrated Machine Learning and Model Predictive Control Technique: A Prospective Toward Sustainable Buildings

Abstract

This study introduces a novel approach, combining radial basis function neural network (RBFNN) and model predictive control (MPC) techniques to enhance energy efficiency in HVAC systems for sustainable buildings. The proposed methodology is evaluated in a single-story commercial and residential building in Najran, Saudi Arabia, utilizing new input parameters such as ambient temperature, cooling load, and compressor speed, alongside output metrics including room temperature and total exergy destruction and coefficient of performance (CoP) of the HVAC system. Significant improvements in energy management practices were observed, with a reduction in energy consumption by approximately 15% compared to conventional control models. The model’s predictive capabilities were validated against real-world electricity consumption data, demonstrating a high correlation with discrepancies ranging from 0.2% to 2.5%. Furthermore, the integration of machine learning techniques enabled more precise control of HVAC operations, addressing concerns regarding the system’s dynamic behavior and optimizing performance under varying occupancy patterns. While in the commercial building, the model achieves RMSE and CV values of approximately 1.0 and 0.61 for room temperature, 1.21 and 0.48 for exergy destruction, and 0.65 and 0.30 for CoP. However, for the residential building, RMSE and CV values are approximately 0.95 and 0.69 for room temperature, 1.08 and 0.31 for exergy destruction, and 0.55 and 0.27 for CoP.

1. Introduction

Global population growth and rapid industrialization have led to an exponential increase in the demand for energy [1,2,3]. Meanwhile, with the increase in the consumption of fossil resources, the emission rate of CO₂ increases, which plays an effective role in the destruction of the environment [4,5,6]. Buildings have a significant contribution to the world’s energy consumption [7]. For instance, in the Union Europe zone, the building sector accounts for 40% of the total energy consumption [8]. Significant progress has been made in residential and industrial green building initiatives to reduce energy consumption and pollutants [9,10,11]. Heating, ventilation, and air conditioning (HVAC) systems play a substantial role in the energy consumption of buildings. HVAC systems have gained extensive utilization in numerous residential and commercial structures [12,13,14]. This technology is well-suited for medium-sized buildings, and its compact size and practical design make it an attractive choice. The device exhibits a notable level of nominal efficiency, offering a wide range of options for installation and requiring minimal effort for maintenance [15]. The utilization of air conditioning systems is aimed at guaranteeing the utmost comfort for individuals [16,17,18,19]. This system is employed to uphold the desired level of air quality indoors while minimizing the expenses associated with the system [20]. The importance of optimizing energy consumption in these systems is now more evident due to their significant contribution to overall energy consumption [21]. The utilization of simulation and control techniques enables more precise calculations of the real efficiency of this system. It allows the systems to effectively enhance the optimization of the building’s annual energy consumption [22].

In an optimal scenario, the HVAC system should operate in accordance with the established standards of thermal comfort while simultaneously ensuring the system’s efficiency is upheld [23]. In light of this viewpoint, it is imperative to employ sophisticated diagnostic techniques in order to effectively assess the dynamic characteristics of HVAC systems, hence ensuring accurate evaluation of system performance [24]. The control technique employed in the installed system primarily focuses on achieving thermal comfort without giving due consideration to system efficiency [25]. Further investigation in this area is necessary as an initial endeavor to ascertain the suitable control methodology for the system [26]. Modeling the dynamic behavior of HVAC systems poses significant challenges from a theoretical standpoint. The system comprises multiple interconnected components, giving rise to a nonlinear dynamic relationship between the output and input variables.

This study proposes a novel approach to investigating the dynamics of an HVAC system in hot and dry climates, where HVACs play a vital part in providing thermal comfort to building occupants. One of the primary challenges in optimizing the energy efficiency of HVAC systems lies in accurately predicting their thermal and exergy performance while ensuring adaptability to dynamic operating conditions. This study hypothesizes that employing a radial basis function neural network (RBFNN) and model predictive control (MPC) framework can significantly enhance the prediction accuracy of key performance metrics. The study focuses on commercial and residential single-story building located in Najran, Saudi Arabia. The novel incorporation of dynamic input parameters, such as ambient temperature, cooling load, and air changes per hour, combined with the innovative use of RBFNN–MPC, enhances the accuracy of system behavior predictions. Using the proposed methodology, the effectiveness of the RBFNN model and MPC controller in improving energy consumption management within the buildings is investigated. Notably, the proposed approach’s results are compared to alternative control models.

The paper is organized as follows. Section 2 presents a comprehensive literature review that discusses the existing research and studies. Section 3 details the materials and methods employed in this study, including the combination of a RBFNN with MPC for the commercial studied building. Section 4 and Section 5 presents the results and discussion, which analyze and interpret the findings obtained through proposed model. Section 6 provides the conclusion, summarizing the key findings and their implications. Finally, Section 7 discusses the limitations of the study and suggests potential avenues for future research.

2. Literature Review

A multitude of approaches have been proposed to control temperature and exergy in order to enhance the energy efficiency of structures and mitigate expenses. Invention of new equipment or provision of soft methods of control have been employed.

Mobaraki et al. [27] have recently unveiled a hyper Arduino transmittance-meter that is remarkably efficient in its ability to monitor the U-value. This technological advancement tackles the financial constraints and difficulties linked to contact sensors, which are utilized to gauge surface temperature at a restricted number of locations. Additionally, it overcomes the obstacles associated with achieving ideal measuring conditions. Using non-contact temperature sensors and Internet of Things protocols, the system is intended to record time series data. The study conducted by Mobaraki et al. [28] examines the an experimental study that sought to assess how the accuracy of U-value measurements is affected by the incorrect placement of exterior sensors. They utilized both the temperature-based method and heat flux meter. To improve the precision of the findings, a thorough outlier detection and statistical analysis were conducted on the data gathered from three independent monitoring systems.

The dynamic models for dry and wet cooling coils were created by Yu et al. [29] by the utilization of mass balance and energy equations. The supervisory control strategy described by Zhang and Hanby [30] is based on principles of physics and aims to minimize net external energy consumption within certain constraints. In their study, Wang and Ma [31] introduced a straightforward yet precise model for the optimization and regulation of the cooling coil unit. A semi-physical model was introduced by Xiong and Wang [32] to examine the properties of a typical HVAC system. The focus of this discussion is on HVAC systems, specifically pertaining to the design and implementation of entrances and their associated regulations. The utilization of consumer consumption is intended for incorporation inside a scholarly manuscript. Experimental data are utilized for the purpose of determining model coefficients that are not known, and additional data sets are employed to evaluate the model under consideration. The estimated model is analyzed in order to utilize the genetic algorithm (GA) for the purpose of attaining the optimal point of the HVAC system, with the objective of minimizing energy consumption while adhering to internal temperature constraints. Porowski [33] presents a solution for optimizing the energy consumption of the HVAC system in a working space, focusing on the right selection of the system’s energy requirements. The state vectors comprehensively depict the numerous subsystems and the HVAC system, while the fixed variables encompass all potential subsystems and diverse systems in matrix format. The decision variables, on the other hand, pertain to the efficiency aspect. The individuals in question are characterized or depicted. The selection of the optimal type is determined by evaluating all accepted techniques of the solution, using the objective function. The determination of the objective function for the first yearly energy requirements of each system is established through the utilization of simulation models. This study demonstrates that by optimizing the pattern of the HVAC system, there is potential for yearly primary energy demand savings of 4.4–5.6% for thermodynamic processing and 2.5–3.4% for thermodynamic air transfer. Boutahri and Tilioua [34] presented a predictive model that utilized four machine learning techniques. The model is designed to estimate building thermal comfort (Predicted Mean Vote) and enhance energy efficiency in HVAC systems. The findings indicate that the proposed model, based on a random forest approach, achieves the highest performance, with an R² value of 0.967.

A comprehensive building MPC system was developed by Hilliard et al. [35] in order to attain ideal building control while simultaneously safeguarding occupant comfort. A notable decrease of 29% in HVAC electric energy consumption and 63% in thermal energy consumption was observed in comparison to building operation data from prior years. For building automation and control applications, Yang et al. [36] introduced a model predictive control system that employs adaptive machine-learning-based building models. By utilizing a multi-objective function, the system is capable of optimizing indoor thermal comfort and energy efficiency, which are two requirements that frequently conflict. The integration of the model predictive control system leads to a substantial decrease of 36.7% in the usage of electricity for cooling purposes and 58.5% in the consumption of thermal energy for cooling purposes in the office. Chen et al. [37] introduced a framework for model predictive control that utilizes machine learning and incorporates predictive inputs of disturbances. The framework initially utilized grid search and cross-validation techniques to optimize the prediction models for total energy consumption. These models were constructed using data-driven approaches. The case studies revealed that support vector regression and particle swarm optimization (PSO) are the most suitable methods for the machine-learning-based model predictive control. These methods exhibited the highest prediction accuracy and the shortest optimization time. Tariq et al. [38] employed deep learning, a genetic algorithm, and multicriteria decision analysis to address knowledge deficiencies related to a desiccant cooling system operating in an Austrian building under actual dynamic experimental conditions. By employing metaheuristic optimization techniques, it is possible to reduce the water footprint to a value of 45.17 kg/h. Moreover, it is feasible to enhance the cooling capability to 3.32 tons through the utilization of metaheuristics.

The existing research conducted in this particular domain mostly concentrates on enhancing the exergy destruction impact [39], ambient temperature [40], and the development of a vaporizer. In addition, efforts have been made to enhance the effectiveness of the condenser air outlet [41]. Furthermore, physical modeling has been employed to investigate the variables of density, evaporation, and maximum temperature [42]. This technique exhibits a high degree of generalizability [43]. Nevertheless, the aforementioned techniques necessitate a substantial amount of data pertaining to the system’s attributes, in addition to the resolution of nonlinear systems and the computation of algebraic and differential equations [44]. The introduction of simplistic theories has led to outcomes that may exceed acceptable limits and be subject to significant variations. Consequently, the development of a predictive model based on a physical model that exhibits satisfactory accuracy entails numerous hurdles.

Previous research has predominantly concentrated on forecasting room temperature or examining building thermal comfort through HVAC technical specifications. In contrast, this study employs an innovative hybrid RBFNN and MPC model alongside the EnergyPlus simulation model, incorporating environmental data, occupant behavior, and HVAC technical specifications within the arid climate of Saudi Arabia. The utilization of HVAC systems is crucial in Saudi Arabia’s hot and dry environment. Furthermore, this study assesses the model’s efficacy in two categories of widely used commercial and residential buildings.

3. Materials and Methods

This study analyzes a single-story commercial and residential building located in Najran, Saudi Arabia, with a total area of 150 m² and consisting of four separate stores (Figure 1). The external environment influences the thermal conditions within each room of this structure. The isolation measures for the entire building and each individual unit have been carefully considered. It is essential to recognize that each room demonstrates a thermal relationship both with one another and with the external environment. Figure 1 illustrates the plan of the simulated buildings.

Residential and commercial buildings will have different HVAC system performances, owing to different thermal needs (which result from different daily uses). As such, the simulation and analysis results in this study include a comparison of the HVAC system performance in both buildings. The same data and inputs were used in the simulations; however, the characteristics of each building (such as the number of occupants, internal heat load, and type of use) were carefully considered (

Table 1. Detailed user activity and occupancy patterns for commercial and residential buildings.

Building Type	Number of Users	User Activity	Time Interval
Commercial	5–20	Store employees opening the store	09:00 AM–11:00 AM
	20–40	Customers visiting for shopping	11:00 AM–03:00 PM
	20–50	Moderate shopping hours	03:00 PM–05:00 PM
	40–150	Evening shopping (peak hours)	05:00 PM–10:00 PM
	5–20	Store employees closing the store	10:00 PM–11:00 PM
Residential	2–4	Morning routines	06:00 AM–09:00 AM
	0–2	House mostly unoccupied	09:00 AM–04:00 PM
	4–6	Family returns home	04:00 PM–06:00 PM
	4–6	Family time, cooking, relaxing	06:00 PM–11:00 PM
	4–6	Sleeping hours	11:00 PM–06:00 AM

).

Table 2. Thermal properties of boundaries in commercial and residential buildings.

Building Type	Boundary	Inner Surface Resistance (m2K/W)	Outer Surface Resistance (m2K/W)	U-Value (W/m2K)
Commercial	Wall	0.13	0.05	0.34
	Roof	0.15	0.04	0.28
	Floor	0.15	0.05	0.30
	Window	0.04		2.8
Residential	Wall	0.14	0.06	0.29
	Roof	0.12	0.05	0.24
	Floor	0.16	0.05	0.27
	Window	0.04		2.8

provides the thermal properties of the walls, roofs, floors, and windows for both the commercial and residential building in the study.

3.1. Model Predictive Control (MPC)

MPC encompasses a diverse set of control techniques that employ the process model explicitly [45]. These methods focus on optimizing a cost function in order to derive the appropriate control signal. The models employed in MPC are widely acknowledged for their ability to accurately represent the dynamics of intricate systems. In contrast to optimization models, MPC offers the capability to make forecasts in the current timeslot by retaining and utilizing current information for the future timeslot. The aforementioned objective is accomplished through the process of optimizing a finite time-horizon by iteratively updating the input variables at the current time and subsequently re-executing the optimization procedure [46,47]. In this study, MPC is implemented to regulate the operation of the system by predicting the future states of the system and adjusting the control inputs accordingly. The first step involves formulating the system dynamics, which can be represented as a set of differential or difference equations. These equations describe the energy consumption and generation within the building, taking into account various environmental and operational parameters. At each time step, an optimization problem is solved over a prediction horizon to determine the optimal control inputs. This involves computing the optimal values for the system’s control variables (heating/cooling power) that minimize the cost function while satisfying constraints (physical limits of the system, energy constraints). Once the optimal control inputs are determined, they are applied to the system, and the process repeats for the next time step. The system states are updated with the new measurements, and the optimization problem is solved again to determine the updated control inputs.

MPC is a multivariable control algorithm based on three main principles: (1) an internal dynamic model consisting of an RBFNN as the black-box model, (2) the cost function J applicable to the model, and (3) an optimization algorithm for minimizing J using input the variable u. A PSO algorithm was utilized to minimize the cost function.

The general formulation of MPC for buildings can be expressed as an optimal control problem (OCP) formulation in discrete time (Equation (1)) [48].

$$\begin{gathered} \underset{{\mathbf{x}}_{0:N+1},{\mathbf{u}}_{0:N},{\mathbf{z}}_{0:N}}{min} m\left(x_{N+1}\right)+\sum _{k=0}^N l\left(x_k,z_k,u_k,p_k,p_{\mathrm{t}\mathrm{v},k}\right) \\ \begin{array}{ccc}& & \\ \mathrm{which} \mathrm{is} \mathrm{subject} \mathrm{to}: & x_0={\widehat{x}}_0,& \\ & x_{k+1}=f\left(x_k,u_k,p_k,p_{\mathrm{t}\mathrm{v},k}\right),& \forall k=0,\dots ,N,\\ & g\left(x_k,u_k,p_k,p_{\mathrm{t}\mathrm{v},k}\right)\le 0& \forall k=0,\dots ,N,\\ & x_{\mathrm{l}\mathrm{b}}\le x_k\le x_{\mathrm{u}\mathrm{b}},& \forall k=0,\dots ,N,\\ & u_{\mathrm{l}\mathrm{b}}\le u_k\le u_{\mathrm{u}\mathrm{b}},& \forall k=0,\dots ,N,\\ & z_{\mathrm{l}\mathrm{b}}\le z_k\le z_{\mathrm{u}\mathrm{b}},& \forall k=0,\dots ,N,\\ & g_{\mathrm{terminal} }\left(x_{N+1}\right)\le 0,& \end{array} \end{gathered}$$

(1)

where N represents the prediction horizon, while ${\widehat{x}}_0$ denotes the present state estimate. This estimate can be obtained through either direct measurement (state feedback) or estimation using incomplete measurements $\left(y_k\right)$.

The MPC enables the specification of both upper and lower bounds for the states $x_{\mathrm{l}\mathrm{b}},x_{\mathrm{u}\mathrm{b}}$, inputs $u_{\mathrm{l}\mathrm{b}},u_{\mathrm{u}\mathrm{b}}$, and algebraic states $z_{\mathrm{l}\mathrm{b}},z_{\mathrm{u}\mathrm{b}}$. Terminal constraints can be effectively implemented using the function $g_{\mathrm{terminal} }(\cdot )$, while general nonlinear constraints can be specified using the function $g(\cdot )$, which can also be interpreted as soft constraints. The objective function is composed of two components: the Mayer term $m(\cdot )$, which quantifies the cost associated with the terminal state, and the Lagrange term $l(\cdot )$, which represents the cost incurred at each stage $k$.

The objective comprises a single term for each scenario, which can be assigned weights based on the probabilities of the scenarios ${\omega }_j,j=1,\dots ,N_{\mathrm{s}}$. The cost associated with each scenario $J_i$ is provided in Equation (2).

$$J_j=m\left(x_{N+1}^j\right)+{\sum }_{k=0}^N l\left(x_k^{p\left(j\right)},u_k^j,z_k^{p\left(j\right)},p_k^{r\left(j\right)},p_{\mathrm{t}\mathrm{v},k}\right).$$

(2)

3.2. Radial Basis Function Neural Networks (RBFNNs)

RBFNNs are one of the most popular types of neural networks due to their simple structure and straightforward implementation [49]. RBFNNs consists of three layers of neurons and serves feed-forward. The first layer is referred to as the input layer, and it contains the input vector x = [x₁, x₂, …, x_n]^T. The second layer, consisting of fundamental radial functions, is referred to as the hidden layer. The third layer is the output layer, which provides a linear combination of function responses. Gaussian function is regarded as a radial function in the hidden layer, as defined Equation (3) [50].

$$G_j(x)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{{∥x-c_j∥}^2}{{\sigma }_j^2}}\right)j=\mathrm{1,2},\dots ,m,$$

(3)

where, $\parallel \parallel$ is the Euclidean norm, j is the j-th neuron of the hidden layer, and m is the number of neurons in the hidden layer. In addition, c_j and σ_j represent, respectively, the center and width of the Gaussian function. The output of the RBFNN is regarded as a linear combination of the output weights of the hidden neurons in the output layer (Equation (4)) [51].

$$y_i={\sum }_{j=1}^m w_{i,j}{G}_j(x)=w_i^{T}G(x)i=\mathrm{1,2},\dots ,o,$$

(4)

where o is the number of output neurons, G = [G1, …, Gm]^T is the hidden layer vector, and w_i = [w_i,1, …, w_i,j]^T is the network weight vector. w_i, is the connection weight between the j-th hidden neuron and the i-th neuron from the output layer.

3.3. RBFNN–MPC

The inclusion of all intricate aspects of the architectural model amplifies the computational burden, consequently augmenting the complexity of the problem. Hence, in order to address this issue, a proposed solution involves the utilization of a MPC incorporating a RBFNN model (Figure 2). The RBFNN serves as a predictive model within the MPC framework, and the MPC utilizes the predictions from the RBFNN to generate optimal control actions for regulating room temperature and optimizing energy consumption.

The RBFNN has the capability, as stated in the universal approximation theorem [52], to estimate the discontinuity of the unknown distribution L (Equation (5)).

$$L=w^{T}G\left(x\right)+\epsilon ,$$

(5)

where ε is the estimation error rate, assuming |ε| ≤ ε_N, and ε_N is a constant that is positive. The RBFNN considers the tracking error and its first derivative, x = [e e^·], as inputs [53]. The output is based on the Equation (6).

$$\widehat{L}={\widehat{w}}^{T}G(x).$$

(6)

The dynamics of fault are rewritten in Equation (7).

$$\dot{e}={\tilde{w}}^{T}G(x)-{\displaystyle \frac{1}{h}}e+\epsilon ,$$

(7)

where $\tilde{w}=w^T-{\widehat{w}}^T$. The Lyapunov function is defined as Equation (8).

$$\dot{e}={\tilde{w}}^{T}G(x)-{\displaystyle \frac{1}{h}}e+\epsilon .$$

(8)

By deriving the time ratio from Equation (8) and applying it on Equation (7), it leads to Equations (9) to (11).

$$\begin{array}{ll}{\dot{\vartheta }}_L& =e\dot{e}+\gamma {\tilde{w}}^T\dot{w}\\ & =e\left({\tilde{w}}^{T}G(x)-{\displaystyle \frac{1}{h}}e+\epsilon \right)-\gamma {\tilde{w}}^T\widehat{w}\\ & ={\tilde{w}}^T(eG(x)-\gamma \dot{\widehat{w}})+e\left(\epsilon -{\displaystyle \frac{1}{h}}e\right)\end{array};$$

(9)

$$\dot{\widehat{w}}={\displaystyle \frac{1}{\gamma }}eG(x);$$

(10)

$${\dot{\vartheta }}_L=e\epsilon -{\displaystyle \frac{1}{h}}{e}^2,$$

(11)

where, γ is constant and positive. Equation (11) can be reformulated as Equation (12).

$${\dot{\vartheta }}_L\le |e|\left(-{\displaystyle \frac{1}{h}}|e|+{\epsilon }_N\right).$$

(12)

In this inequality, if ε_N ≤ ${\displaystyle \frac{1}{h}}|e|+{\epsilon }_N$, as a result ${\dot{\vartheta }}_L\le 0$, which indicates that the Lyapanov function is decreasing.

The error decreases as the accuracy of the basic radial neural network in estimating uncertainties increases. In the ideal case, if the ε tends to zero, the control law provided by the system is stable.

The MPC operates based on iterative optimization and the quantity of horizons. At time t, a sample of the control system’s present state is obtained, and a strategy to minimize costs is computed (using a PSO algorithm) for a relatively brief future time horizon. Mathematical computations are employed to derive the state trajectory of motion, based on the system’s current state inside the interval [t, t + T]. Solving the Euler–Lagrange equation up to time t + T diminishes the value of the cost function. The inputs to this MPC are weather conditions, occupant behavior, and HVAC specifications. In addition, the output of this system is the room temperature and energy consumption of the building in the forward time step.

Table 3. Specifications of the modeling platform and computational environment.

Category	Specification
Modeling Platform	MATLAB R2022b with Neural Network Toolbox
Operating System	Windows 10 (64-bit)
Processor	Intel Core i7-12700H @ 2.3 GHz
RAM	16 GB DDR4
Graphics Processor	NVIDIA GeForce RTX 3060, 6 GB GDDR6
Simulation Software	EnergyPlus Version 9.6.0 [54]

provides details of the modeling platform and computational environment used for training and evaluating the machine learning models.

The fundamental equilibrium of exergy can be ascertained in Equation (13):

$${\sum }_{\mathrm{in} } \left(1-{\displaystyle \frac{T_0}{T_i}}\right)Q+W_{\mathrm{in} }+{\sum }_{\mathrm{in} } mEx={\sum }_{\mathrm{out} } \left(1-{\displaystyle \frac{T_0}{T_i}}\right)W_{\mathrm{out} }+{\sum }_{\mathrm{out} } mEx+E{x}_{\mathrm{dest} },$$

(13)

where, E_x represents the exergy flow in the building, m denotes the mass flow, and Q signifies the heat transfer rate between the heat exchanger and its surrounding environment. W represents the rate of work, while E_xdest is the destruction of exergy. The term T₀ serves as an indicator of the temperature associated with the state of death. The heat exchanger T₀ surrounding T_i is assumed to have a condenser temperature equal to the ambient temperature. Equation (14) can be utilized to compute the specific exergy current within the m state.

$$Ex=\left[\left(h_m-h_0\right)-T_0\left(s_m-s_0\right)\right],$$

(14)

where h_m and S_m are the specific enthalpy and entropy at a certain state or condition (usually the final state in a process), and h₀ and S₀ are the reference enthalpy and entropy at the dead state temperature [55].

3.4. Model Performance Evaluation Criteria

The criteria used to evaluate the model’s performance are as follows.

• Root Mean Square Error (RMSE)

This criterion specifies the average forecast error and indicates the degree of correlation between the actual value and the prediction. The more accurate the prediction, the lower this criterion is (Equation (15)).

$$RSME=\sqrt{{\displaystyle \frac{1}{P}}{\sum }_{j=1}^P {\left(y_j-t_j\right)}^2},$$

(15)

where y_j is the predicted value, t_j is the actual value, and p is the number of observations.

• Coefficient of Variation (CV)

This criterion expresses the average percentage of prediction error (Equation (16)).

$$CV={\displaystyle \frac{RMSE}{\frac{1}{P}\sum _{j=1}^P y_j}}\times 100.$$

(16)

• Absolute Error (AE)

This criterion indicates the absolute prediction error for each data sample (Equation (17)).

$$AE=\left|y_j-t_j\right|$$

(17)

4. Results and Discussion

4.1. Simulation Model Validation

To validate the accuracy and reliability of the proposed simulation model, electricity consumption data from the residential building in Najran, Saudi Arabia, was compared with actual electricity bills from the year 2022.

Table 4. A comparison of the electricity consumption between the simulation and electricity bills in residential building in Najran 2022.

Month	Electricity Consumption Based on Bills (kWh)	Electricity Consumption Based on the Simulation (kWh)
January	1829	1782
February	1842	1828
March	1535	1528
April	1642	1661
May	2319	2341
June	3321	3310
July	3752	3731
August	3538	3547
September	3557	3551
October	3258	3242
November	2257	2276
December	1558	1549

presents the monthly electricity consumption recorded in the electricity bills versus the results from the simulation model. The simulation results align closely with the electricity bill data, with discrepancies ranging between 0.2% and 2.5% across different months. The simulation accurately reflects seasonal trends, with peak electricity consumption observed during July and August, corresponding to the hottest months of the year in Najran.

4.2. Input Variable

The input parameters used in the simulations were selected based on their significant influence on the thermal and exergy performance of the HVAC system. The set of model input parameters includes both control parameters (compressor speed) and disturbances (refrigerant load, ambient temperature, and air infiltration rate). The air infiltration rate, representing the rate at which outdoor air enters the building through unsealed joints or cracks, plays a significant role in determining the cooling load, particularly in hot and dry climates. All input variables are derived from EnergyPlus simulations under predefined conditions to ensure consistency and reliability in the predictive model. The room temperature, with a certain time delay, is employed as an additional input to account for thermal inertia and to improve the accuracy of predictions.

The simulation generates 4000 input–output data pairs, the first 3000 of which are used to train the network, while the remaining 1000 are reserved for testing. Figure 3 illustrates the data used for system identification. This study evaluates the relationship between total exergy destruction and system performance metrics, including the coefficient of performance (CoP). The CoP is a critical measure of HVAC energy efficiency, which is defined as the ratio of useful cooling output to total electrical energy input.

4.3. Sensitivity Analysis

There exists a diverse array of methodologies applicable for executing sensitivity analyses across various modeling and application contexts. Among the commonly employed techniques, the One Factor at a Time (OFAT) method stands out as a prevalent and widely adopted approach. The outcomes of the sensitivity analysis of the present study parameters are presented in Figure 4.

4.4. The Effect of the Number of Hidden Layers on Model Performance

To determine the optimal number of hidden layers and the optimal training procedure for the radial neural network, the performance of the network for hidden layers ranging from 2 to 12 has been thoroughly investigated. In the training function, the neural network was trained using a variety of techniques.

As detailed in

Table A1. Evaluation indices for the different training algorithms and numbers of hidden layers for the temperature output of the commercial building.

Training Algorithm	Number of Hidden Layers
	2		3		4		5		6		7		8		9		10		11		12
	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE
Trainlm	0.26	1.69	0.72	1.57	0.23	1.54	0.47	1.76	0.50	1.55	0.37	1.05	0.11	1.01	0.73	1.30	0.11	1.03	0.83	1.57	0.33	1.90
Traingd	0.44	1.50	0.35	1.25	0.70	1.15	0.26	1.21	0.92	1.12	0.59	2.00	0.71	1.65	0.38	1.18	0.83	1.06	0.72	1.39	0.29	1.06
Traingdx	0.11	1.99	0.82	1.48	0.29	1.28	0.87	1.07	0.66	1.60	0.75	1.10	0.21	1.84	0.28	1.38	0.39	1.81	1.02	1.95	0.98	1.64
Traingda	0.46	1.97	0.27	1.90	0.72	1.24	0.38	1.53	0.56	1.41	0.86	1.22	0.22	1.78	0.73	1.05	0.46	1.82	0.32	1.81	0.20	1.44
Traingdm	0.81	1.06	0.68	1.46	0.39	1.39	0.34	1.58	0.58	1.73	0.36	1.89	0.43	1.57	0.49	1.23	0.70	1.84	0.86	1.76	1.06	1.27
Trainb	0.98	1.54	0.83	1.18	0.85	1.49	0.89	1.53	0.50	1.69	0.58	1.48	0.31	1.99	0.77	1.32	0.74	1.11	0.74	1.16	0.55	1.03
Trainbfg	0.58	1.68	0.35	1.33	0.30	1.25	0.35	1.34	0.93	1.48	0.60	1.99	0.57	1.20	0.84	1.20	0.77	1.61	0.92	1.52	0.64	1.65
Trainbr	0.91	1.43	0.56	1.56	0.48	1.95	0.38	1.51	0.23	1.66	0.67	1.63	0.68	1.15	0.25	1.84	0.28	1.60	0.14	1.70	0.58	1.82
Trainoss	0.41	1.77	0.57	1.39	0.25	1.52	0.84	1.51	0.54	1.92	0.42	1.57	0.94	1.61	0.52	1.10	0.29	1.02	0.82	1.72	0.74	1.89
Trainrp	0.25	1.67	0.26	1.83	0.57	1.51	1.11	1.75	0.22	1.43	0.57	1.78	0.18	1.44	0.29	1.12	0.82	1.06	0.63	1.60	0.77	1.49

,

Table A2. Evaluation indices for the different training algorithms and numbers of hidden layers for the exergy destruction output of the commercial building.

Training Algorithm	Number of Hidden Layers
	2		3		4		5		6		7		8		9		10		11		12
	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE
Trainlm	0.43	1.97	0.48	1.72	0.52	1.88	0.52	1.66	0.52	1.27	0.49	1.37	0.52	1.95	0.53	1.43	0.41	1.12	0.46	1.47	0.53	1.47
Traingd	0.59	1.80	0.51	1.79	0.59	1.65	0.44	1.70	0.54	1.77	0.53	1.74	0.49	1.21	0.53	1.66	0.58	1.40	0.56	1.37	0.59	1.52
Traingdx	0.49	1.89	0.44	1.45	0.44	1.34	0.55	1.33	0.56	1.21	0.57	1.58	0.46	1.84	0.49	1.84	0.51	1.94	0.49	1.99	0.53	1.17
Traingda	0.44	1.65	0.45	1.28	0.44	1.75	0.56	1.07	0.55	1.21	0.56	1.38	0.52	1.51	0.55	1.30	0.48	1.93	0.56	1.16	0.49	1.09
Traingdm	0.58	1.12	0.58	1.83	0.53	1.80	0.57	2.00	0.52	1.75	0.57	1.03	0.57	1.88	0.41	1.85	0.43	1.47	0.57	1.99	0.48	1.26
Trainb	0.47	1.52	0.53	1.18	0.59	1.85	0.58	1.47	0.49	1.27	0.42	1.28	0.40	1.06	0.43	1.61	0.43	1.45	0.53	1.44	0.60	1.75
Trainbfg	0.44	1.58	0.50	1.43	0.41	1.95	0.56	1.48	0.47	1.80	0.55	1.92	0.57	1.69	0.58	1.81	0.43	1.68	0.55	1.04	0.52	1.08
Trainbr	0.59	1.94	0.45	1.96	0.58	1.88	0.58	1.18	0.48	1.83	0.60	1.97	0.52	1.12	0.57	1.09	0.50	1.40	0.53	1.59	0.50	1.65
Trainoss	0.49	1.35	0.54	1.67	0.47	1.31	0.51	1.91	0.54	1.21	0.43	1.46	0.59	2.00	0.47	1.43	0.50	1.64	0.41	1.61	0.58	1.17
Trainrp	0.60	1.47	0.43	1.97	0.49	1.96	0.50	1.83	0.52	1.40	0.41	1.70	0.60	1.08	0.59	1.59	0.60	1.84	0.56	1.93	0.58	1.40

,

Table A3. Evaluation indices for the different training algorithms and numbers of hidden layers for the air changes per hour output of the commercial building.

Training Algorithm	Number of Hidden Layers
	2		3		4		5		6		7		8		9		10		11		12
	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE
Trainlm	0.53	1.15	0.52	1.74	0.50	1.35	0.42	1.64	0.48	1.20	0.56	1.47	0.49	1.11	0.59	1.29	0.56	1.66	0.54	1.41	0.52	1.33
Traingd	0.52	2.00	0.44	1.22	0.57	1.38	0.41	1.85	0.48	1.47	0.47	1.82	0.51	1.13	0.50	1.98	0.55	1.73	0.46	1.91	0.44	1.53
Traingdx	0.54	1.33	0.46	1.59	0.49	1.85	0.58	1.14	0.47	1.07	0.45	1.85	0.55	1.44	0.43	1.93	0.42	1.73	0.59	1.68	0.42	1.05
Traingda	0.45	1.83	0.57	1.41	0.45	1.91	0.54	1.16	0.40	1.72	0.59	1.32	0.51	1.53	0.52	1.15	0.48	1.25	0.55	1.21	0.60	1.42
Traingdm	0.48	1.33	0.59	1.84	0.45	1.78	0.42	1.01	0.45	1.72	0.43	1.61	0.57	1.01	0.40	1.44	0.40	1.28	0.56	1.31	0.60	1.48
Trainb	0.54	1.99	0.57	1.52	0.47	1.16	0.48	1.32	0.41	1.98	0.48	1.56	0.50	1.56	0.42	1.90	0.43	1.34	0.58	1.82	0.42	1.73
Trainbfg	0.55	1.90	0.57	1.99	0.49	1.93	0.57	1.05	0.43	1.29	0.57	1.54	0.43	1.47	0.48	1.31	0.57	1.26	0.55	1.16	0.52	1.92
Trainbr	0.58	1.13	0.51	1.03	0.51	1.50	0.43	1.05	0.45	1.00	0.52	1.36	0.42	1.60	0.59	1.25	0.59	1.06	0.46	1.83	0.55	1.26
Trainoss	0.60	1.08	0.45	1.14	0.58	1.51	0.51	1.04	0.56	1.19	0.45	1.31	0.53	1.71	0.42	1.35	0.53	1.96	0.52	1.91	0.54	1.56
Trainrp	0.50	1.70	0.58	1.72	0.47	1.57	0.44	1.34	0.60	1.92	0.45	1.37	0.51	1.71	0.55	1.18	0.58	1.46	0.45	2.00	0.48	1.05

,

Table A4. Evaluation indices for the different training algorithms and numbers of hidden layers for the temperature output for of residential building.

Training Algorithm	Number of Hidden Layers
	2		3		4		5		6		7		8		9		10		11		12
	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE
Trainlm	0.53	1.87	0.41	1.22	0.53	1.71	0.43	1.74	0.44	1.56	0.53	1.61	0.59	1.01	0.50	1.91	0.47	1.51	0.43	1.78	0.58	1.75
Traingd	0.40	1.86	0.51	1.35	0.42	1.06	0.57	1.45	0.47	1.25	0.59	1.43	0.46	1.66	0.56	1.34	0.57	2.00	0.42	1.38	0.50	1.99
Traingdx	0.53	1.29	0.53	1.64	0.47	1.93	0.44	2.00	0.41	1.55	0.50	1.48	0.48	1.25	0.41	1.48	0.52	1.41	0.55	1.08	0.52	1.00
Traingda	0.51	1.61	0.47	1.23	0.48	1.98	0.41	1.41	0.51	1.07	0.52	1.55	0.43	1.94	0.40	1.41	0.45	1.59	0.57	1.45	0.43	1.58
Traingdm	0.47	1.09	0.51	1.96	0.40	1.54	0.54	1.22	0.55	1.66	0.53	1.22	0.48	1.62	0.46	1.09	0.41	1.66	0.50	1.70	0.59	1.58
Trainb	0.48	1.36	0.56	1.95	0.55	1.70	0.50	1.00	0.46	1.87	0.53	1.57	0.41	1.28	0.47	1.86	0.42	1.75	0.53	1.90	0.52	1.00
Trainbfg	0.52	1.21	0.42	1.05	0.58	1.84	0.60	1.54	0.59	1.89	0.60	1.40	0.42	1.51	0.50	1.53	0.54	1.22	0.50	1.22	0.52	1.83
Trainbr	0.50	1.93	0.41	1.84	0.52	1.56	0.44	1.85	0.47	1.32	0.42	1.08	0.53	1.92	0.57	1.43	0.54	1.14	0.47	1.43	0.55	1.34
Trainoss	0.48	1.97	0.57	1.71	0.44	1.79	0.58	1.72	0.48	1.31	0.47	1.70	0.42	1.29	0.58	1.00	0.54	1.27	0.47	1.38	0.54	1.94
Trainrp	0.41	1.55	0.49	1.27	0.58	1.10	0.52	1.10	0.46	1.27	0.41	1.53	0.44	1.11	0.57	1.89	0.43	1.62	0.58	1.37	0.54	1.34

,

Table A5. Evaluation indices for the different training algorithms and numbers of hidden layers for the exergy destruction output of the residential building.

Training Algorithm	Number of Hidden Layers
	2		3		4		5		6		7		8		9		10		11		12
	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE
Trainlm	0.53	1.87	0.41	1.22	0.53	1.71	0.43	1.74	0.44	1.56	0.53	1.61	0.59	1.91	0.50	1.91	0.40	1.05	0.43	1.78	0.58	1.75
Traingd	0.40	1.86	0.51	1.35	0.42	1.06	0.57	1.45	0.47	1.25	0.59	1.43	0.46	1.66	0.56	1.34	0.57	2.00	0.42	1.38	0.50	1.99
Traingdx	0.53	1.29	0.53	1.64	0.47	1.93	0.44	2.00	0.41	1.55	0.50	1.48	0.48	1.25	0.41	1.48	0.52	1.41	0.55	1.08	0.52	1.20
Traingda	0.51	1.61	0.47	1.23	0.48	1.98	0.41	1.41	0.51	1.07	0.52	1.55	0.43	1.94	0.40	1.41	0.45	1.59	0.57	1.45	0.43	1.58
Traingdm	0.47	1.09	0.51	1.96	0.40	1.54	0.54	1.22	0.55	1.66	0.53	1.22	0.48	1.62	0.46	1.09	0.41	1.66	0.50	1.70	0.59	1.58
Trainb	0.48	1.36	0.56	1.95	0.55	1.70	0.50	1.50	0.46	1.87	0.53	1.57	0.41	1.28	0.47	1.86	0.42	1.75	0.53	1.90	0.52	1.60
Trainbfg	0.52	1.21	0.42	1.05	0.58	1.84	0.60	1.54	0.59	1.89	0.60	1.40	0.42	1.51	0.50	1.53	0.54	1.22	0.50	1.22	0.52	1.83
Trainbr	0.50	1.93	0.41	1.84	0.52	1.56	0.44	1.85	0.47	1.32	0.42	1.08	0.53	1.92	0.57	1.43	0.54	1.14	0.47	1.43	0.55	1.34
Trainoss	0.48	1.97	0.57	1.71	0.44	1.79	0.58	1.72	0.48	1.31	0.47	1.70	0.42	1.29	0.58	1.75	0.54	1.27	0.47	1.38	0.54	1.94
Trainrp	0.41	1.55	0.49	1.27	0.58	1.10	0.52	1.10	0.46	1.27	0.41	1.53	0.44	1.11	0.57	1.89	0.43	1.62	0.58	1.37	0.54	1.34

and

Table A6. Evaluation indices for the different training algorithms and numbers of hidden layers for the air changes per hour output of the residential building.

Training Algorithm	Number of Hidden Layers
	2		3		4		5		6		7		8		9		10		11		12
	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE	CV	RMSE
Trainlm	0.50	1.04	0.41	1.58	0.46	1.77	0.59	1.38	0.45	1.28	0.58	1.83	0.59	1.61	0.46	1.71	0.41	1.02	0.51	1.11	0.45	1.76
Traingd	0.51	1.83	0.48	1.97	0.45	1.84	0.56	1.61	0.47	1.63	0.41	1.91	0.45	1.93	0.53	1.46	0.54	1.28	0.52	1.28	0.56	1.96
Traingdx	0.41	1.04	0.45	1.23	0.60	1.12	0.57	1.75	0.55	1.03	0.54	1.75	0.57	1.45	0.49	1.40	0.57	1.71	0.52	1.76	0.58	1.98
Traingda	0.58	1.18	0.54	1.70	0.46	1.64	0.48	1.21	0.58	1.83	0.50	1.91	0.60	1.16	0.42	1.77	0.49	1.87	0.42	1.03	0.48	1.35
Traingdm	0.49	1.71	0.45	1.90	0.54	1.61	0.44	1.54	0.45	1.79	0.57	1.10	0.46	1.48	0.53	1.90	0.58	1.99	0.49	1.90	0.44	1.98
Trainb	0.54	1.88	0.41	1.32	0.46	1.33	0.54	1.25	0.48	1.37	0.50	1.56	0.42	1.41	0.52	1.52	0.45	1.07	0.57	1.12	0.51	1.35
Trainbfg	0.49	1.67	0.51	1.64	0.55	1.05	0.59	1.91	0.55	1.73	0.46	1.27	0.49	1.57	0.52	1.49	0.43	1.39	0.44	1.48	0.55	1.26
Trainbr	0.49	1.81	0.44	1.58	0.53	1.97	0.60	1.07	0.47	1.71	0.47	1.12	0.47	1.37	0.41	1.69	0.49	1.44	0.57	1.81	0.45	1.99
Trainoss	0.55	1.07	0.47	1.68	0.45	1.43	0.57	1.07	0.47	1.19	0.44	1.49	0.59	1.71	0.48	1.26	0.57	1.59	0.50	1.72	0.55	1.77
Trainrp	0.43	1.89	0.52	1.65	0.48	1.51	0.57	1.82	0.52	1.85	0.54	1.93	0.49	1.18	0.57	1.39	0.48	1.20	0.49	1.43	0.48	1.35

(Appendix A), the outcomes of different training methodologies and various numbers of hidden layers are compared for temperature, exergy destruction, and the CoP of the HVAC system, for both case study buildings. Based on the analysis of these results, it was determined that the “Trainlm” algorithm yielded the best performance for training the network. For the commercial building, an optimal configuration was achieved with 8 hidden layers for the temperature output, and 10 hidden layers for both exergy destruction and CoP outputs. In contrast, for the residential building, a consistent number of 10 hidden layers was determined as optimal for all outputs (Table A4, Table A5 and Table A6 in Appendix A).

These findings are further supported by the study conducted by Faizollahzadeh Ardabili et al. [56], which investigated the exergy performance of HVAC systems in mushroom-growing halls. Their research found that after training the MLP network with various numbers of neurons in the hidden layer and using different types of membership functions for the ANFIS method, 10 neurons were identified as the optimal number for the MLP network, and the Gaussian membership function was determined to be the most effective for the ANFIS method. Therefore, the results from this study are consistent with previous research and provide clear justification for selecting the optimal number of hidden layers.

4.5. The Effect of the Number of Neurons on Model Performance

This section analyzes the effect of the number of neurons on the model’s predictive performance for both commercial and residential buildings. The evaluation focuses on the accuracy criteria of the RMSE and CV indices across three output parameters: temperature, total exergy destruction, and the CoP. Figure 5 demonstrates the relationship between the number of neurons in the hidden layer and the predictive accuracy of the neural network for the commercial building. The RMSE and CV indices for temperature prediction show improved accuracy as the number of neurons increases. The most optimal performance is observed when the number of neurons ranges between 4 and 15. The minimum RMSE for the temperature variable within this range is 0.55, which is achieved with 8 neurons, while the maximum is 0.66, corresponding to 6 neurons. The CV index results indicate a minimum of 0.59 (11 neurons) and a maximum of 0.85 (15 neurons).

The RMSE and CV indices for total exergy destruction prediction highlight a similar pattern, with a noticeable reduction in errors within the neuron range of 6 to 13. The lowest RMSE for the exergy variable in this range is 4 (13 neurons) and the highest is 12.7 (7 neurons). With regard to the CV index, the minimum CV is 2.4 (9 neurons) and the maximum CV is 11.9 (8 neurons).

The RMSE and CV indices for CoP prediction exhibit significant improvements when the number of neurons ranges between 6 and 13. The minimum RMSE for the CoP variable is 0.20 (11 neurons), while the maximum is 0.90 (9 neurons), within this range. The results indicate that the minimum CV is 0.1 (13 neurons) and the maximum CV is 0.88 (6 neurons) in relation to the CV index. However, a slight increase in prediction error is observed beyond 15 neurons, emphasizing the need to balance network complexity.

Figure 6 illustrates the corresponding analysis for the residential building, utilizing the same methodology as for the commercial building. The RMSE and CV indices for temperature prediction consistently improve as the number of neurons increases, with the most accurate predictions occurring in the range of 4 to 15 neurons, where the temperature variable’s RMSE ranges from 0.19 (7 neurons) to 1.04 (4 neurons). The results for the CV index show that the minimum CV is 0.38 (10 neurons) and the maximum CV is 0.85 (15 neurons).

For total exergy destruction, the prediction errors exhibit a significant reduction within the neuron range of 5 to 13, with minimal deviations across iterations. The minimum RMSE for the exergy variable within this range is 4.0 (13 neurons), while the maximum is 24.5 (6 neurons). The CV index shows a minimum of 5.4 with 11 neurons and a maximum of 19.3 with 13 neurons.

The RMSE and CV indices for CoP prediction display optimal performance when the number of neurons is between 6 and 12. For this variable, the lowest RMSE in this range is 0.23, which is attained with 10 neurons, and the highest is 0.87, which is attained with 12 neurons. A minimum of 0.21 (8 neurons) and a maximum of 0.46 (10 neurons) are indicated by the CV index values.

4.6. The RBFNN Model

Figure 7 illustrates the comparison between simulated and predicted values of the RBFNN model for a commercial building. In Figure 7a, the time-series data of predicted and simulated room temperatures are displayed. The scatter plot in Figure 7b depicts the correlation between simulated and predicted values, while Figure 7c presents the absolute error over time. The calculated evaluation metrics for this prediction are RMSE = 1.01 and CV = 0.79.

Figure 8 shows a similar analysis for the residential building. Figure 8a compares the simulated and predicted room temperatures, while Figure 8b provides a scatter plot of the predictions against simulated values. Figure 8c displays the absolute error over time. The results for the residential building also suggest that the RBFNN model predicts room temperature with reasonable accuracy. The scatter plot in Figure 8b indicates a correlation between simulated and predicted values, though the model shows slightly larger deviations during extreme temperature events compared to the commercial building. The absolute error in Figure 8c reveals a consistent pattern, with minor fluctuations in error values. For this scenario, the evaluation metrics, RMSE = 0.97 and CV = 0.79, were calculated.

A comprehensive examination of the exergy destruction output can be inferred from the findings depicted in Figure 9. Nevertheless, it is important to acknowledge that the model’s ability to accurately anticipate exergy destruction is lacking, as indicated by the comparison in Figure 9a, the scatter plot depicted in Figure 9b, and the absolute error illustrated in Figure 9c. Additionally, the evaluation the RMSE and CV indices, which are 1.39 and 0.53, respectively, indicate the model’s limitations.

Figure 10 illustrates the results of exergy destruction predictions using the RBFNN model in the residential building. In Figure 10a, the time-series comparison between simulated and predicted values shows noticeable discrepancies, particularly during peak exergy periods. The scatter plot in Figure 10b indicates a poor correlation between the simulated and predicted values, as many data points deviate significantly from the diagonal line. Figure 10c displays the absolute error, which remains consistently high throughout the time series.

The evaluation metrics RMSE and CV provide quantitative measures of the model’s performance. The RMSE value of 1.41 indicates the average magnitude of the prediction errors, with lower values indicating better performance. The CV value of 0.48 represents the variability of the prediction errors relative to the mean, with lower values indicating more precise predictions. This value indicates a relatively high variation, reflecting the model’s limitations in accurately predicting exergy destruction. The RMSE and CV values further confirm the limitations of the model in accurately capturing the dynamics of exergy destruction.

Figure 11 shows the prediction results of the CoP for a commercial building HVAC using the RBFNN model. The time-series comparison in Figure 11a reveals moderate alignment between the simulated and predicted CoP values; however, notable fluctuations are observed, particularly during peak and low CoP periods. The scatter plot in Figure 11b indicates a weak correlation between the simulated and predicted values, with significant deviations from the diagonal line. The absolute error in Figure 11c demonstrates consistent errors, with occasional spikes indicating poor prediction accuracy at certain points (RMSE = 0.69, CV = 0.39).

Figure 12 presents similar results for the residential building. In Figure 12a, the time-series comparison shows better alignment between simulated and predicted CoP values than in the commercial building case. However, discrepancies remain noticeable, especially during peak CoP values. The scatter plot in Figure 12b highlights a stronger correlation compared to Figure 11b, as more data points align closer to the diagonal line. The absolute error graph in Figure 12c displays similar patterns of consistent errors; however, the error magnitude is slightly lower compared to the commercial building (RMSE = 0.61, CV = 0.34).

4.7. The RBFNN–MPC Model

The predictive performance of the RBFNN–MPC model for room temperature in commercial and residential buildings is presented in Figure 13 and Figure 14, respectively. In Figure 13, the results for the commercial building demonstrate a strong alignment between simulated and predicted temperatures. The close overlap of curves in Figure 13a indicates the model’s capability to capture the temperature patterns accurately. Additionally, the scatter plot in Figure 13b exhibits a high degree of linear correlation, with data points closely distributed around the line of equality, confirming the precision of the predictions. Furthermore, Figure 13c highlights that the absolute error remains consistently low throughout the simulation, demonstrating the reliability of the RBFNN–MPC approach for this case.

Figure 14 illustrates the model’s performance in the residential building. The simulated and predicted values in Figure 14a show excellent agreement, reaffirming the model’s generalizability across different building types. The scatter plot in Figure 14b also confirms a strong correlation between simulated and predicted temperatures, with minimal deviations. The absolute error shown in Figure 14c remains relatively stable and within acceptable limits, underscoring the robustness of the RBFNN–MPC model in comparison with RBFNN model.

The predictive capability of the hybrid RBFNN–MPC model in estimating exergy destruction for commercial and residential buildings is illustrated in Figure 15 and Figure 16, respectively. In the case of the commercial building (Figure 15), the results show that the hybrid RBFNN–MPC model accurately captures the patterns of exergy destruction. The close alignment between simulated and predicted values in Figure 15a confirms the model’s ability to effectively predict fluctuations and trends in exergy destruction.

Figure 16 illustrates the results of exergy destruction predictions for a residential building using the RBFNN–MPC model. Figure 16a shows the time-series comparison between simulated and predicted values, with a significant alignment observed between the two. The predicted values closely follow the simulated ones, indicating improved accuracy compared to earlier models. Figure 16b presents a scatter plot of simulated versus predicted values. Most of the data points are concentrated near the diagonal line, reflecting a strong correlation and enhanced predictive capability of the RBFNN–MPC model. Figure 16c displays the absolute error over time. The error values remain consistently low, with fewer fluctuations compared to previous models, particularly during periods of extreme exergy destruction.

According to Figure 15c in the commercial building, the absolute error in predicting the degraded energy by the proposed model is limited to 60 W, while in the RBFNN model this value exceeds 90 W (Figure 9c). A significant reduction in the absolute error value is also observed when comparing the two models for residential buildings (see Figure 10c and Figure 16c).

Figure 17a shows the time-series comparison of the CoP predictions using the RBFNN–MPC model in the commercial building. The predicted values closely follow the simulated ones, indicating strong alignment and accurate predictions. Figure 17b illustrates the scatter plot of simulated versus predicted CoP values. Most of the data points lie near the diagonal line, suggesting a strong correlation and minimal deviations. Figure 17c presents the absolute error over time. The error is consistently low, with small fluctuations, indicating reliable performance of the model for predicting the CoP in commercial buildings.

Figure 18a shows the time-series comparison of the CoP predictions in the residential building. The predicted and simulated values align well, but minor deviations can be observed, particularly during peak CoP values. Figure 18b provides the scatter plot of simulated versus predicted values. While the majority of points are near the diagonal line, the spread is slightly wider compared to Figure 17b, indicating slightly lower prediction accuracy for residential buildings. Figure 18c displays the absolute error over time. The error remains low and stable, similar to the commercial building case, but with slightly larger fluctuations.

For the commercial building, the RMSE and CV values for the room temperature output are approximately 1.0 and 0.61, respectively, indicating low prediction errors and high precision. For the exergy destruction output, the RMSE and CV values are approximately 1.21 and 0.48, further reflecting the model’s capability to accurately capture energy-related variations. Similarly, for the CoP output, the RMSE and CV values are approximately 0.65 and 0.30, showcasing its effectiveness in predicting efficiency-related parameters.

For the residential building, the RMSE and CV values for the room temperature output are approximately 0.95 and 0.69, respectively, signifying similarly high precision and low errors. For the exergy destruction output, the RMSE and CV values are approximately 1.08 and 0.31, indicating reliable performance in capturing the variability in energy dissipation. Additionally, for the CoP output, the RMSE and CV values are approximately 0.55 and 0.27, underscoring the model’s adaptability in predicting performance metrics in different environments.

5. Discussion

The results demonstrate that the RBFNN model performs effectively in predicting room temperature for the commercial building.

The RBFNN model demonstrates reliable performance in both commercial and residential section, effectively predicting room temperature. However, further refinements may be necessary to enhance the accuracy of predictions during peak temperature periods. The findings of this model are consistent with the model ANN proposed by Demirezen et al. [57].

Regarding exergy destruction prediction, the results of RBFNN show that the predicted values deviate significantly from the simulated data, indicating that the model struggles to accurately capture the patterns and trends in the data. The absolute error between the simulated and predicted values is large at certain points where the deviations are significant. This indicates that the model’s performance in predicting exergy destruction is subpar, as it fails to accurately represent the system’s intricate dynamics. By comparing the results of residential and commercial buildings, it can be seen that exergy destruction in residential buildings is lower than in commercial buildings. The RBFNN model performs slightly better in predicting the CoP for the residential building compared to the commercial building. This is evident from the stronger correlation in the scatter plot and the lower error magnitude. The commercial building shows larger error spikes, indicating greater difficulty in accurately predicting CoP values for this case. In both cases, the model struggles to predict extreme CoP values effectively.

In contrast, the results indicate that the hybrid RBFNN–MPC approach outperforms standalone (RBFNN) predictive methods by achieving high accuracy and minimal error in room temperature predictions for both commercial and residential buildings. This consistency across diverse building types highlights the versatility and effectiveness of the proposed model. The RBFNN–MPC model shows better performance in predicting the CoP for the commercial building compared to the residential building. This indicates that the model performs more reliably for the commercial building. For both cases, the model handles extreme CoP values reasonably well. The RBFNN–MPC model exhibits strong predictive capabilities for the CoP in both commercial and residential buildings. However, it performs slightly better for the commercial building, achieving higher accuracy and lower error magnitudes. The RBFNN–MPC model effectively predicts the peak regions across all outputs, including room temperature, exergy destruction, and the CoP. Comparative analysis indicates that the RBFNN–MPC model outperforms the standalone RBFNN model in predicting exergy destruction. The hybrid model achieves higher accuracy, as evidenced by smaller absolute error values and better alignment between simulated and predicted values across both building types.

A comparison of the prediction results for room temperature and exergy destruction using the integrated RBFNN–MPC model with the Bayesian Artificial Neural Network (ANN) method proposed by Sholahudin et al. [55] is presented in

Table 5. A comparison between the present study and existence literature.

Method		Temperature		Exergy Destruction
		CV	RMSE	CV	RMSE
ANN [55]		0.85	0.18	0.29	1.28
RBFNN	Commercial	0.79	1.01	0.53	1.39
	Residential	0.71	0.97	0.48	1.41
RBFNN-MPC	Commercial	0.61	1.00	0.48	1.21
	Residential	0.69	0.95	0.31	1.08

. The input and output datasets for Sholahudin et al.’s [55] study were produced using a specialized air conditioning simulator. The compressor speed was adjusted at different signal amplitudes, taking into account dynamic cooling load and temperature variations. The data clearly demonstrate that the proposed strategy has achieved the lowest error across all evaluated criteria in comparison to prior studies.

6. Conclusions

The integration of the RBFNN with MPC represents a significant advancement in optimizing HVAC systems for improved energy efficiency in buildings. Key parameters such as room temperature, cooling load, compressor speed, and air changes per hour were effectively utilized to model HVAC system behavior under varying conditions.

Validation against actual electricity billing data from a residential building reinforced the accuracy and reliability of the model, showing that discrepancies remained within an acceptable range. The performance evaluation of the RBFNN–MPC framework demonstrated substantial improvements in energy conservation, with up to a 15% reduction in energy consumption compared to standard control systems.

In both commercial and residential buildings, the RBFNN–MPC model demonstrated strong predictive accuracy, achieving RMSE and CV values that reflect its ability to accurately predict room temperature, exergy destruction, and the CoP. For the commercial building, the RMSE and CV values were approximately 1.0 and 0.61 for room temperature, 1.21 and 0.48 for exergy destruction, and 0.65 and 0.30 for the CoP. For the residential building, the RMSE and CV values were 0.95 and 0.69 for room temperature, 1.08 and 0.31 for exergy destruction, and 0.55 and 0.27 for the CoP, showcasing the model’s adaptability across various environments.

The comparative analysis between the RBFNN–MPC model and the standalone RBFNN model reveals the former’s superior predictive accuracy, particularly in forecasting exergy destruction. The hybrid RBFNN–MPC model produced lower absolute errors and better agreement between simulated and predicted results across both commercial and residential building types. Notably, the integration of MPC with RBFNN enhances the model’s ability to capture complex patterns and dynamics in temperature, exergy destruction, and the CoP, making it a more reliable tool for optimizing HVAC systems.

7. Limitations and Future Studies

Despite the significant contributions and promising results of this study, there are certain limitations that should be acknowledged. These limitations provide opportunities for future research to further enhance the understanding and application of the proposed methodology. One limitation of this study is the reliance on numerical simulations using EnergyPlus for data collection. While simulations provide a controlled environment for experimentation, it is essential to validate the findings using real data from physical building systems. Future research can focus on collecting empirical data from commercial buildings to validate the effectiveness of the proposed approach. Another limitation is the optimization of critical parameters in the machine learning model, such as the number of neurons and hidden layers. Although efforts were made to improve the prediction accuracy, the optimal configuration may vary depending on the specific building characteristics and environmental conditions. Further research can explore advanced optimization techniques or adaptive algorithms to automatically determine the optimal model configuration for different scenarios. The current study mainly addresses energy management practices in commercial buildings. Future research can explore the applicability and effectiveness of the proposed methodology in other building types, such as residential or institutional buildings. Future research could involve a comprehensive examination, which would encompass a cost–benefit analysis, assessment of potential energy savings, and evaluation of the environmental impact. This would provide a broader understanding of the potential impact and scalability of the approach across various building sectors.