Mathematical models are employed across various fields, including engineering, ecology, agriculture, medicine, and economics, primarily to forecast and regulate the operation of systems in real-world applications (Homod, 2013). Mathematical modelling is, for example, extensively used in construction and civil engineering (Addis, 2020), because replicating large-scale structures like buildings, bridges, or dams in a laboratory is impractical and costly, while modelling provides accurate insights into structural behaviour under various conditions. In the field of building energy management, creating precise and efficient mathematical models of a building’s components is a vital process (Pan et al., 2023). These components include the building envelope, orientation, and building services. Such models are crucial for predicting and optimising performance, especially in energy-intensive services like lighting and Heating, Ventilation, and Air Conditioning (HVAC) systems (Jang et al., 2019). Accurately predicting building service behaviour is crucial for energy-efficient operations. A reliable mathematical model supports optimal decision-making, helping to optimize energy use while ensuring performance.

HVAC systems are responsible for a significant portion of building energy use, often consuming up to 50% of a building’s total energy (Simpeh et al., 2022). They contribute substantially to greenhouse gas emissions, mainly when they operate on fossil fuels, which can harm the environment and exacerbate climate change. According to the International Energy Agency (IEA), energy consumption for air conditioning is projected to increase 4.5 times by 2050 in countries outside the Organisation for Economic Co-operation and Development (OECD), compared to a 1.3 times increase within OECD countries (Goetzler et al., 2016). This increase in demand is driven by higher incomes and better access to air conditioning, especially in developing countries. As a result, the need for HVAC system installations is likely to grow further in the coming years, making these systems even more vital for meeting future energy needs. Consequently, it is imperative to explore methods to decrease energy usage and enhance the efficiency of HVAC systems, thereby diminishing their carbon footprint and contributing to environmental conservation (Kathiravel et al., 2024).

To reduce energy use in buildings, indirect approaches such as passive design strategies are often recommended (Palmero-marrero & Oliveira, 2010). These strategies include features like improved insulation, shading devices, and green roofs, which help minimise energy demand for heating and cooling (Castleton et al., 2010). Additionally, cooling methods, including enhanced insulation, green roofs, and photovoltaic systems, have been explored to improve energy efficiency (Touqan et al., 2017). Retrofitting buildings to improve energy efficiency and sustainability is another important focus in building design (Ma et al., 2012). Despite these efforts to reduce energy consumption, HVAC systems remain essential in maintaining high levels of thermal comfort and air quality (Asim et al., 2022), which cannot be fully achieved through passive cooling strategies alone.

HVAC systems are inherently complex due to their multicomponent structure and strong interdependencies among elements such as evaporators, compressor and condensers, making their modelling and control a challenging task (Seyam, 2018; Touqan and Ameer, 2024). Inadequate control design can lead to increased energy consumption, frequent maintenance, poor performance, and user dissatisfaction (Angel, 2011). Enhancing system efficiency requires the integration of multidisciplinary knowledge through Systems Engineering, which applies structured methodologies to optimise performance (Subbaram Naidu and Rieger, 2011). Central to this process is the development of accurate and dynamic mathematical models that reliably represent system behaviour, forming the basis for effective control strategies (Li et al., 2010; Whalley and Abdul-Ameer, 2011). Such models enable the precise regulation of system outputs, thereby reducing energy waste and improving operational efficiency in real-time scenarios.

To contextualise the investigation of HVAC systems’ performance, they are vital for comfort in extreme-heat social housing, where high energy bills strain low-income residents. In the Arabian Gulf (>40°C), inefficient HVAC units worsen energy poverty and emissions; thus, despite passive design and retrofit efforts, dynamic modelling and control remain essential for maximizing efficiency without compromising comfort.

Hence, the motivation for this study is to derive a simplified and reliable mathematical model. While prior works validated a high-order state-space model (Yao et al., 2013), its complexity challenges real-time control applications. Computational efficiency is paramount for adaptive HVAC control in dynamic environments, yet most MIMO models either oversimplify coupling effects or lack experimental validation. A simplified two-input, two-output model that preserves essential dynamics can address this gap, facilitating deployment in resource-constrained contexts such as social housing.

Therefore, the research objectives can be derived from addressing the following research question: How can a simplified multivariable model enhance control efficiency and energy performance in HVAC systems for social housing? So, the objectives of the study are:

i. Deriving a reduced-order transfer function matrix from (Yao et al., 2013) validated state-space framework, focusing on refrigerant and chilled water flow rates as inputs. A transfer function, obtained through Laplace transforms, turns time-domain dynamics into an algebraic input-output relation. For the chiller it maps refrigerant-flow changes to cooling capacity, capturing delays and response shape.

This study aims to develop and validate a simplified two-input, two-output transfer function model of a vapour compression chiller, derived from a high-order state-space system. The model maintains key dynamic behaviour, improves computational efficiency, and supports real-time control. It facilitates advanced feedback strategies to boost stability, energy efficiency, and performance, especially in extreme climates. This model underpins cost-effective, resilient HVAC solutions aligned with energy and comfort goals in sustainable social housing.

Given the focus on energy-efficient HVAC operation and control-oriented modelling, the literature review situates the research within existing scholarly work. It outlines key modelling methods for HVAC, highlighting their advantages, limitations, and suitability for streamlined control models. This forms the basis for the methodological choices in this study.

Some researchers, especially those in (Homod, 2013; Afram and Janabi-Sharifi, 2014), have reviewed three distinct approaches to mathematical modelling. The first is the physics-based model, also known as the “white box model.” This model requires a deep understanding of the underlying processes. In the case of HVAC systems, physics-based models usually rely on equations related to mass and energy conservation, as well as principles of heat transfer. This approach necessitates comprehensive knowledge of the system being analysed. A significant benefit of physics-based modelling is its capacity to predict the system’s behaviour and estimate various outputs accurately. For instance, the research in (Ali, 2019) developed a mathematical load model for major household appliances, particularly air conditioners (ACs), based on physical and operational characteristics. Physics-based models, grounded in thermodynamic principles, offer high prediction accuracy and detailed system representation (Jorissen et al., 2021).

However, their complexity can limit real-time control applications (Silvestri et al., 2023; Zhao et al., 2021). These models are computationally demanding and require significant development effort (Y. Li et al., 2022). Moreover, while the model is built on fundamental physical relationships and validated against real data, it lacks a transparent methodology for parameter identification and uncertainty quantification, which are key requirements in white box modelling as emphasised by (Bacher and Madsen, 2011). Meanwhile, these models often encounter challenges in adapting to real-time control applications due to their inherent complexity and rigidity. Furthermore, they exhibit limited robustness when applied to systems with varying operating conditions, thereby reducing their practicality for adaptive HVAC control strategies necessary in dynamic environments such as social housing.

The second modelling approach is data-driven, often called a “black box model”. Studies such as those by (Ma et al., 2015; Zlatanović et al., 2011) have applied this method for HVAC systems, which are inherently complex with numerous parameters that make physics-based modelling difficult. Data-driven models can model complicated behaviour system (Zhou et al., 2017) and (Zhou et al., 2016). These models are beneficial for managing nonlinearities and are simpler to compute than traditional physical-based methods. However, the accuracy of these models can be significantly impacted by poor data quality, which is prevalent in large databases. Despite their advantages, data-driven methods face challenges such as poor interpretability and conservative predictions in extreme cases (Wu and Xue, 2024). In the same context, the study by (Zhang, 2023) provides a comprehensive critique of machine learning-driven building control systems, highlighting that although machine learning demonstrates potential in optimising energy consumption, considerable challenges persist, particularly concerning data quality, model interpretability, and the ability to generalise to new operational conditions. This highlights an important challenge in current data-driven methods, which often work as mysterious “black boxes” and can be difficult to validate in many real-world settings.

The third approach to modelling is the hybrid method, also known as the “grey box model,” which combines both data-driven and physics-based techniques. In this method, physics-based equations are used to develop the core model, while actual data helps to identify key parameter values. These data can be sourced from system specifications provided by manufacturers or from the system commissioning process. For example, in the study (Jin et al., 2011), a grey box model was created for cooling coil units, incorporating heat transfer equations and energy and mass balance principles. The model streamlined the calibration process by focusing on six parameters that represented the unit’s nonlinear characteristics. These parameters were derived from manufacturer data or experimental findings. The grey box approach faces challenges in parameter calibration and balancing physical insight with computational efficiency. Calibration is sensitive to data quality and quantity, with longer, diverse datasets improving model generalisation and accuracy (Murphy et al., 2021). The process requires substantial domain expertise to manage effectively. While grey-box models can provide physical interpretations of parameters, they may not fully capture parameter interactions or uncertainties (Pavlak et al., 2014). Despite these limitations, grey-box modelling can significantly reduce labour input compared to white-box approaches (Murphy et al., 2021). Overall, grey-box modelling remains the most effective modelling approach, offering a compromise between physical insight and data-driven efficiency. Still, it requires careful consideration of calibration strategies and data quality.

Traditionally, HVAC research has centred on single-input, single-output (SISO) models that isolate components such as compressors, cooling coils, air-handling units, or building envelopes (Li et al., 2014; Lapinskiene and Martinaitis, 2013). However, real HVAC systems are inherently multivariable, featuring several interacting inputs and outputs. Given the high energy consumption of these systems, it is more practical and representative to model them as multiple-input, multiple-output (MIMO) systems. This approach, where a change in one input affects multiple outputs, better captures the system’s complexity and energy demands, as discussed in (Hassan and Al-Samarraie, 2022), offering significant potential to optimize performance by manipulating various system parameters.

Recent research underscores the importance of MIMO modeling for HVAC systems, as it provides a more comprehensive representation of their complex internal interactions. MIMO modelling has been shown to provide a more comprehensive representation of the complex interactions within HVAC systems (Abdo-Allah et al., 2018; Abdo-Allah et al., 2017). State-space approaches have been effectively utilized to develop MIMO models for HVAC systems, facilitating more robust control system design and analysis (Abdo-Allah et al., 2018). Furthermore, advanced control strategies, including Direct Nyquist Array (DNA) and PID controllers, have been successfully implemented in MIMO HVAC models, leading to improved system performance and effective decoupling of system outputs (Touqan et al., 2022).

Most HVAC studies still rely on SISO models that overlook the strong inter-variable couplings of real systems. Although grey-box MIMO models offer richer dynamics, existing work is too computationally demanding for real-time control, highlighting a research gap. This research addresses this by extracting a validated high-order chiller model into a simplified grey-box 2 × 2 formulation, enabling practical multivariable control that reduces energy use, capturing the key dynamical system behaviour and enhances comfort in high-energy consumption systems.

This study builds upon the rigorously validated state-space model of a vapour compression chiller system established by (Yao et al., 2013). It will be developing a more concise, two-input, two-output multivariable model; this research accurately reflects the system’s dynamic behaviour. The model, which integrates essential components such as the condenser and evaporator based on fundamental principles of mass and energy conservation and heat transfer, aims to provide a computationally efficient yet realistic representation of chiller operations. It will indeed derive a simplified reduced-order model that still captures the same transient and steady-state behaviour of the original high-order model. Instead of presenting the detailed equations from the original work, this research focuses specifically on the sets of state equations and output equations only provided by (Yao et al., 2013), avoiding redundancy and duplication of effort.

This section evaluates the performance and significance of the derived open-loop chiller system model. Particular emphasis will be placed on the model’s behavior in response to perturbations in system inputs within the open-loop configuration and under various operational condition changes.

This section provides comprehensive interpretations of the system responses illustrated in Figures 3–8, to explain the system’s behaviour within the open-loop configuration. These insights aim to augment the understanding of system engineers, thereby facilitating the proposal of suitable control strategies.

In Figure 3 where the system is subjected to initial conditions which are basically ( t c w , E = 29.5   ℃ ; G c w , E = 1.16   k g / s ; t e w , E = 24.8   ℃ ; G e w , E = 1.45   k g / s , ; G r m = 0.116   k g / s ), the first chiller output t e w , E t is approaching a steady state value of 21 ℃ coupled with the second output Q c t approaching almost 1.02 k W .

The decrease in t e w , L t temperature from 24.8 ℃ to 21 ℃ indicates that the evaporator is successfully absorbing heat from the cold carrier fluid, causing its temperature to drop. This temperature reduction is typical of an evaporator’s operation, where heat from the fluid is transferred to the refrigerant, cooling the cold carrier fluid as it passes through. The system reaches a stable cooling output, meaning that the heat transfer in the evaporator has balanced out with the conditions imposed by the inputs (e.g., refrigerant flow rate G r m t ). The cooling capacity of 1.02   k W reflects the rate at which the chiller removes heat from the cold carrier fluid. This steady-state value indicates that the system has reached an equilibrium where the refrigerant is effectively absorbing heat from the cold carrier fluid at a rate of 1.02   k W . The cooling capacity value shows that the chiller is operating at a relatively low load, typical for initial conditions with moderate refrigerant flow rates. As the cooling demand increases or the system parameters change (e.g., G e w , E t , Q c t would adjust accordingly.

Figure 3 will be the benchmark for analysing the performance of the system under perturbations from other inputs.

Figure 4 shows the system behaviour when cold carrier mass flow rate (first input, G e w , E t ) is a step increase to 1.95   k g / ⁡ sec , while the other inputs remain unaltered and as per their initial conditions. The decrease in exit cold carrier temperature t e w , L t from 24.8 ℃ to 22.8 ℃ is caused by the increase of G e w , E t to 1.95  kg / ⁡ sec . The difference is less than the previous case, when it was from 24.8 ℃ to 20.8 ℃ . As more fluid is passing through the evaporator, the same amount of released heat is spread across a larger mass of fluid. This means that the cold carrier fluid releases less heat per unit of mass, causing a slower temperature reduction in the cold carrier as it moves through the evaporator. In the same time, as more fluid is passing through the evaporator, the same amount of released heat is spread across a larger mass of fluid so the cooling capacity Q c t increases to approximately 1.15   k W , up from the previous steady-state value of 1.02 kW . The increase in G e w , E t does not only affect t e w , L t , but also impacts Q c t , demonstrating an inherent coupling between these two outputs.

In Figure 5 when the refrigerant mass flow rate (second input, G r m t ) is a step increase to 0.216   k g / ⁡ sec the exit cold carrier temperature t e w , L t is decreasing to approximately 17.3 ℃ with coupling effect on the cooling capacity output increasing to 2.35  kW . By increasing G r m t , the amount of refrigerant flowing through the evaporator increases. This allows the evaporator to absorb more heat from the cold carrier fluid (chilled water) causing a larger reduction in the temperature of the cold carrier fluid as it passes through the evaporator. The cooling capacity Q c t increases to 2.35 kW, up from the previous value of approximately 1.02 kW. The greater refrigerant flow rate allows the system to absorb more heat, which directly translates into a higher Q c t (cooling capacity). The system is now removing heat from the cold carrier at a higher rate to maintain the desired temperature decrease.

Similarly, in Figure 6, a step disturbance input on the inlet temperature of the condenser coolant t c w , E t is increasing to 40 ℃ while all other inputs remain equal to their initial conditions, the exit cold carrier temperature t e w , E t is approaching 21.5 ℃ while the cooling capacity is approaching − 0.75  kW . The increase in t c w , E t (inlet coolant temperature to the condenser) reduces the condenser’s ability to reject heat. This results in higher refrigerant pressures and temperatures within the system, leading to a less effective heat exchange process in the evaporator. The temperature t e w , L t rises to 21.5 ° C , up from its previous steady-state value of 21 ° C . With the evaporator unable to effectively absorb heat from the cold carrier fluid, the outlet temperature of the cold carrier fluid t e w , L t increases slightly, reflecting a reduced cooling effect. A negative Q c t indicates that instead of removing heat from the cold carrier, the chiller system is now adding heat. This happens because the condenser cannot reject enough heat due to the elevated t cw , E t , causing a thermal imbalance in the refrigerant cycle. With the refrigerant unable to condense efficiently in the condenser, the evaporator’s cooling capacity is severely compromised, leading to the system effectively transferring heat into the cold carrier rather than removing it.

In Figure 7, the exit cold carrier temperature t e w , E t is approaching 32.5 ℃ while the cooling capacity is approaching 2.25   k W associated with some oscillations. The increase in the inlet cold carrier temperature t e w , E t to 40 ℃ imposes a significantly higher thermal load on the evaporator, which can be understood as social housing occupancy dynamics. In high-density residential environments, such as social housing in extreme climates, occupant density and intermittent usage patterns can cause significant fluctuations in thermal loads. The 40°C disturbance models scenarios like peak occupancy or severe weather events, where internal heat gains increase due to higher occupant activity. As a result, the evaporator absorbs more heat from the cold carrier fluid, but it cannot reduce the temperature as effectively due to the larger thermal input. The exit temperature t e w , L t stabilizes at 32.5 ℃ , reflecting the system’s new equilibrium under the higher inlet temperature. The increase in t e w , E t raises the amount of heat the evaporator needs to absorb to maintain the cold carrier’s flow temperature. This causes the cooling capacity Q C t to increase significantly, as the evaporator works harder to remove the additional heat from the cold carrier fluid. The increase to 2.67  kW reflects the system’s response to handle the higher thermal load introduced by the elevated inlet temperature.

The cooling capacity exhibits oscillations before reaching its steady-state value of 2.67  kW . The evaporator’s thermal response is dynamic, and a sudden increase in t ew , E t creates an imbalance in the heat transfer process. This imbalance induces transient oscillations as the system attempts to reach a new equilibrium. Any overshoot in heat absorption by the evaporator leads to fluctuations in cooling capacity before stabilization. The chiller system’s physical components, such as the refrigerant cycle and heat exchanger, exhibit thermal and fluidic inertia. These inertial effects delay the system’s ability to settle immediately after the disturbance, resulting in oscillatory behavior. As t e w , E t increases, the heat absorbed by the evaporator rises, leading to simultaneous increases in t e w , L t and Q C t . This coupling is inherent in the chiller system’s design and reflects the interconnected nature of its heat transfer processes.

The results illustrated in Figure 8 due to disturbance applied on the system output show that no significant change in the exit cold carrier temperature ending with steady state value of t e w , L t = 21 ℃ but the cooling capacity is approaching steady state value of Q c t = 0.55   k W . Despite the disturbance, the cold carrier fluid’s exit temperature is unaffected, reducing the chilled water temperature from 24.8 ℃ to 21 ℃ . This indicates that the evaporator, though partially fouled, still manages to absorb sufficient heat to maintain the temperature difference required to cool the cold carrier fluid to 21 ℃ . The absence of a significant temperature change could result from the reduced heat transfer being compensated by the evaporator’s thermal inertia or by the refrigerant still operating within its effective range despite the fouling. On the other hand, the fouling in the evaporator reduces its ability to transfer heat effectively, directly decreasing the cooling capacity. A reduction of 500   W corresponds to the disturbance magnitude, which accounts for the drop in Q c t to 0.55   k W . This highlights the evaporator’s sensitivity to fouling and its direct impact on the system’s overall cooling performance. The reduced cooling capacity reflects a loss of energy efficiency, as the chiller is no longer operating at its full potential due to fouling. This could lead to higher operational costs and reduced performance in meeting cooling demands. This situation represents fouling or scaling in the evaporator, a common issue in social housing due to limited maintenance budgets and ageing infrastructure. Fouling reduces heat transfer efficiency, as evidenced by the drop in cooling capacity Q c t to 0.55   k W without affecting outlet temperature t e w , L t . This scenario underscores the importance of low-maintenance design and robust control in social housing, where infrequent servicing can degrade system performance. By quantifying fouling’s impact, the model supports the development of predictive maintenance strategies or fault-tolerant controls, critical for cost-sensitive and resource-constrained environments.

The open-loop simulation results of the simplified mathematical model provide valuable insights into the dynamic behavior of the chiller system, highlighting its sensitivity to input changes and disturbances. The results show how variables like t e w , L t and Q c t respond passively, with phenomena such as coupling effects and oscillations, underscoring the system’s inability to maintain stability in some cases without feedback control. These findings emphasize the need for robust feedback control mechanisms to counteract disturbances, decouple interdependent outputs, and stabilize the system under varying conditions. Engineers can use this data along with the transfer function matrix in Equation 31 to design controllers, refine operational limits, and develop advanced strategies such as multivariable control techniques to enhance stability, reduce oscillations, and improve overall efficiency. By understanding these open-loop characteristics, system engineers are better equipped to optimize the chiller’s functionality and ensure reliable performance in real-world applications. The findings also highlight opportunities to integrate this model into sustainable social housing developments in the extreme climates areas, where adaptive HVAC systems can enhance energy resilience and occupant comfort.

This study presents a simplified multivariable state-space model of a vapor compression chiller system, emphasising the operation of key components such as the condenser and evaporator. The model expands upon an existing state-space mathematical framework, as detailed in (Yao et al., 2013), which has been validated through experimental data obtained from a chiller unit. While the original model in (Yao et al., 2013) involves a 5 × 7 multivariable structure with high-order dynamics, the current 2 × 2 model retains core thermal interactions while significantly reducing the state-space dimension, simplifying real-time control implementation. It delivers an accurate yet computationally straightforward mathematical model representation of the system’s behaviour. In control engineering, the mathematical model, represented through a state-space model, can be converted into a transfer function. The MIMO-related transfer function matrix plays a crucial role in the analysis of multivariable systems and controller design. It enables direct frequency-domain evaluation of input-output dynamics, such as system gain, phase lag, and coupling effects. For HVAC applications, especially vapor compression chillers, such a matrix is essential for designing decoupled or coordinated control strategies that regulate thermal outputs efficiently. The HVAC simplified mathematical model’s performance developed in this study is tested across different input scenarios using SIMULINK/MATLAB. The results are then plotted to validate the model’s ability to predict system responses. This includes scenarios such as operational changes, fouling, and thermal load variations. The open-loop simulation results demonstrate that the system reaches steady-state values after a certain response time, but the performance reveals significant interdependencies between outputs. For instance, the cooling capacity stabilizes at 1.02 kW, while the exit cold carrier temperature approaches 20.8°C under initial conditions, reflecting the system’s ability to reach equilibrium. The system reaches steady-state conditions in approximately 40 s for the exit cold carrier temperature and 70 s for the cooling capacity under nominal operating conditions. However, the simulation reveals strong input-output coupling, where a change in one input such as an increase in refrigerant mass flow rate affects both outputs simultaneously. For instance, a step increase in refrigerant flow from 0.116 kg/m3 to 0.216 kg/m3 not only lowers the cold carrier temperature to 17°C but also increases cooling capacity to 2.4 kW. Additionally, when subjected to a step disturbance in the condenser inlet temperature, the system exhibits oscillations in cooling capacity with a peak deviation of ±0.4 kW and a settling time exceeding 80 s. These dynamics highlight the system’s sensitivity to thermal and hydraulic disturbances, which would necessitate a multivariable control strategy that manages the system’s behaviour, aiming to enhance its performance while optimising its energy consumption. Therefore, implementing closed-loop multivariable control is essential. Such strategies can compensate for cross-coupling, dampen oscillations, and stabilize output responses in the presence of real-world disturbances and load variations, ensuring both performance and energy efficiency.

The novelty of this research lies in introducing a reduced-order multivariable model that retains essential dynamics while significantly lowering computational demands, making it suitable for cost-sensitive control applications. The simplified mathematical model can serve as the foundation for advanced feedback control designs that stabilise HVAC operation, decouple interacting outputs, and enhance energy efficiency, helping buildings meet sustainability targets without compromising comfort or reliability.

Future work could enhance the model’s accuracy by incorporating a secondary verification step to increase confidence. Since the model has been deemed accurate at the initial confidence level validated earlier with the Air Conditioning Experimental unit, future efforts might involve checking the simplified model’s accuracy at an additional confidence level. This can be achieved by testing the simplified mathematical model against a comparable experimental Air Conditioning unit, built based on the key parameters and components used in the model.