Required heating energy in buildings: a comparative analysis of mathematical models based on standard HRN EN ISO 52016-1, neural networks, grey box, and measured data

Required heating energy in buildings: a comparative analysis of mathematical models based on standard HRN EN ISO 52016-1, neural networks, grey box, and measured data ☆ Juraj ˇ Cukelj a , Marino Grozdek a , Tomislav Veliki b a Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia b Department for Mechanical Engineering, University North, Varaˇ zdin, Croatia ARTICLE INFO Keywords: Energy Building Renovation Standard HRN EN ISO 52016–1 User Impact on Energy Consumption Artificial Neural Network Grey Box Modeling ABSTRACT Energy efficiency upgrades in public buildings are commonly evaluated using design phase projections, which often overestimate savings and contribute to a persistent performance gap between expected and actual energy consumption. Performance gap complicates post renovation assessments, particularly in cases financed through energy service models that rely on accurate savings to finance investment. To address this challenge, the presented study evaluates the performance of three modeling approaches for predicting delivered heating energy in a renovated building. These include a standardized physics-based model defined by HRN EN ISO 52016–1, a simplified resistance capacitance grey box model, and a data driven black box model implemented as a feedforward artificial neural network. All models were calibrated using real operational data collected through environmental and gas consumption meter installed throughout the building. The study examines each model’s ability to reflect real world energy usage patterns and explores their practical advantages and limitations. Results show that the standardized model can achieve a level of accuracy comparable to that of the more complex data driven approach, while offering greater transparency and ease of use. The grey box model emerged as a particularly effective compromise between complexity, adaptability, and performance. These findings highlight the potential of validated simplified models to serve as reliable tools for verifying true energy savings in public buildings and reducing uncertainty in energy performance contracting. 1. Introduction The building sector in the European Union is responsible for a substantial portion of energy consumption and greenhouse gas emissions, accounting for 40 % of total energy use and 36 % of emissions. Notably, 80 % of the energy used in buildings is devoted to heating, cooling, and domestic hot water preparation. In response to the urgent threat of climate change, the European Council committed in December 2020 to reduce greenhouse gas emissions by 55 % by 2030, compared to 1990 levels, with the aim of achieving climate neutrality by 2050 [1]. However, the EU faces significant challenges in the global energy market, which has prompted the implementation of the REPowerEU plan, designed to save energy, produce clean energy, and diversify energy supply [2]. As part of this broader effort, National Energy and Climate Plans (NECPs) were introduced under the Regulation on the Governance of the Energy Union and Climate Action (EU) 2018/1999, which was agreed upon as part of the Clean Energy for All Europeans package adopted in 2019. These NECPs outline how member states intend to contribute to the Energy Union’s goals, particularly in improving energy efficiency and sustainability [3]. A key strategy in reducing energy consumption is the deep renovation of buildings, which focuses on comprehensive upgrades to building envelopes, energy systems, and control systems. This long-term strategy aims to renovate the national building stock by 2050, with the goal of significantly reducing energy consumption for heating, cooling, and hot water [4]. However, it has been observed that partial renovations, such as those focusing solely on the building envelope or energy systems, often yield only modest energy savings [5]. In some cases, these partial renovations may even lead to an increase in energy consumption, highlighting the need for more comprehensive renovation approaches. Reasons for that could be due to reduction of indoor air quality [6], specifically increase in the concentration of carbon dioxide and volatile organic compounds [7]. One of the significant challenges in building ☆ This article is part of a special issue entitled: ‘SDEWES 2024_ECM’ published in Energy Conversion and Management. E-mail addresses: [email protected] (J. ˇ Cukelj), [email protected] (M. Grozdek), [email protected] (T. Veliki). Contents lists available at ScienceDirect Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman https://doi.org/10.1016/j.enconman.2025.120182 Energy Conversion and Management 343 (2025) 120182 Available online 14 July 2025 0196-8904/© 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies. renovations is the performance gap between predicted and actual energy consumption post-renovation. This gap can arise from various factors, including meteorological conditions, user behavior, discrepancies between design and as-built conditions, and variations in building usage [8]. For instance, one study found that while calculated energy savings might range between 59 % and 71 %, the actual savings achieved were as low as 33 %, due to atypical building usage patterns and changes in user behavior following renovations [9]. Moreover, changes in user behavior post-renovation can further increase this gap [10]. Given the importance of accurately assessing energy savings from building renovations, reliable mathematical models are crucial. These models help quantify the impact of renovations and address the performance gap by considering variable factors such as external and internal climatic conditions, building usage, and user behavior [11]. Since all these factors are hard to model and predict, it is crucial to have a valid method and guidelines for baseline model [12]. Using data from Indoor Environmental Quality (IEQ) sensors is advised to use in correlation with energy performance [13] contracts to minimize performance issues and manage energy performance tradeoffs of IEQ against energy performance that could lead to occupants’ health consequences [14]. Recent studies have investigated the discrepancy between predicted and actual energy savings, emphasizing the importance of validating models with empirical data [15]. Others have focused on optimizing energy use in institutional buildings, highlighting the role of tailored modeling strategies to improve prediction accuracy [16]. Research has also addressed the barriers and drivers in achieving energy efficiency through renovation, stressing the need for reliable measurement frameworks to support policy and investment decisions [17]. Mathematical models for energy consumption are generally divided into white, black, and gray box models. White box models are based on detailed physical laws and heat transfer mechanisms, offering transparency but requiring significant expertise and computational resources. Black box models, in contrast, rely on statistical methods and historical data, offering high execution speed but with less interpretability [18]. Gray box models combine elements of both approaches, providing a balance between transparency and accuracy [19]. Comparative analyses have shown that while black box models may offer higher prediction accuracy, their applicability is limited due to their complexity and data requirements [20]. On the other hand, gray box models have shown promise in balancing prediction accuracy and computational efficiency [21], making them suitable for more complex building systems. These models could also be applied in implementing Model Predictive Control (MPC) [22] to achieve additional energy savings and potentially decrease the return on investment for energy contractors. MPC has demonstrated potential for achieving significant energy savings and reducing peak loads, although its implementation in real-world scenarios remains limited [23]. The development and application of robust mathematical models are crucial for accurately assessing energy savings in renovated buildings. These models play a vital role in understanding and mitigating the performance gap between predicted and actual energy. However, despite advancements in model development, existing models often become overly complex [24], tailored to specific buildings, and difficult to apply universally, making them time-consuming and resource intensive. To address these challenges, this paper focuses on the application of the HRN EN ISO 52016–1 standard, which is complex enough to capture building dynamics and usage patterns while remaining practical for broader applications. To enhance the model’s accuracy and applicability, standardized coefficient values and assumptions were refined through real life testing. Two specific tests were conducted: the Pseudo Random Binary Sequence (PRBS) test, which examines changes in indoor temperature in response to random patterns of heating source Nomenclature Cint;ztc Internal thermal capacity of the zone [J/K] ΔtTime step [s] θint;a;ztc;tIndoor air temperature [◦C] θint;a;ztc;t−1Indoor air temperature in the previous time interval (t-Δt) [s] Aeli Surface area of element eli [m 2 ] hci;eli Internal convective heat transfer coefficient of element eli [W/m 2 K] θpln;eli;tIndoor surface temperature of element eli [◦C] Hve;vei;tTotal ventilation heat exchange coefficient for element eli [W/K] θsup;vei;tSupply air temperature [◦C] θe;a;tOutdoor air temperature [◦C] Htr;tb;ztc Total heat transfer coefficient for thermal bridges [W/K] fint Convective fraction of internal gains [-] fsol Convective fraction of solar radiation [-] fH/CConvective fraction of heating/cooling system [-] Φint;ztc;tTotal internal heat gains [W] Φsol;ztc;tDirect solar radiation into the zone [W] ΦHC;ztc;tHeating or cooling load in the zone [W] Aelk Surface area of element elk in zone ztc [m 2 ] Atot Surface area of all elements elk =1…eln [m 2 ] θpli;eli;tTemperature of node pli [◦C] θpli−1;eli;tTemperature of node pli-1 [◦C] θint;a;ztc;tIndoor air temperature in the zone [◦C] hpli−1;eli Conductivity between nodes pli and pli-1 [W/mK] κpli;eli Surface heat capacity of node pli [J/m 2 K] hci;eli Heat transfer coefficient of the internal surface [W/m 2 K] hri;eli Radiative heat transfer coefficient of the internal surface [W/m 2 K] θpli;eli;t−1Temperature of node pli in the previous time interval (t-Δt) θpli;eli;t−1Temperature in node pli +1 [◦C] hpli;eli Conductivity between nodes pli +1 and pli [W/mK] θe;tOutdoor temperature [◦C] hce;eli Heat transfer coefficient of the external surface [W/m 2 K] hre;eli Radiative heat transfer coefficient of the external surface [W/m 2 K] α sol;pli;eli Solar energy absorption coefficient on the external surface [-] Isol;dif;eli;tDiffuse solar radiation [W/m 2 ] Isol;dir;eli;tDirect solar radiation [W/m 2 ] Fsh;obst;eli;tShading factor of element eli due to external obstructions [-] Φsky;eli;tThermal radiation towards the sky [W] TeEnvelope temperature [◦C] TiIndoor temperature [◦C] ToOutdoor temperature [◦C] RiThermal resistance between indoor air temperature and the envelope [K/W] RoThermal resistance between outdoor air temperature and the envelope [K/W] ΦhHeating power [W] ΦsGlobal horizontal solar irradiance [W/m 2 ] AeEnvelope solar gain coefficient [m 2 ] AiIndoor solar gain coefficient [m 2 ] CeHeat capacitance of envelope [J/K] CiHeat capacitance of interior [J/K] J. ˇ Cukelj et al. Energy Conversion and Management 343 (2025) 120182 2 operation, and a ramp-up test, where the indoor temperature is raised to a stationary value and after turning off heat source the subsequent temperature decrease is observed. These tests were performed under controlled conditions, such as when the building was unoccupied (no internal gains from occupants) and solar gains were negligible (e.g., at night or during very cloudy days), allowing for the determination of internal gains from equipment, specific heat capacity of zones, and infiltration rates. This approach aims to balance model complexity with practical applicability, ultimately leading to more reliable predictions of energy savings post-renovation. In addition to the white box model based on HRN EN ISO 52016–1, this paper also explores the use of an RC thermal model (gray box) and an Artificial Neural Network model (black box). By comparing these three common approaches to thermal modeling, the paper provides valuable insights into their respective strengths and limitations, offering a comprehensive perspective on building energy modeling. Artificial Neural Networks has been recognized in the literature as a relatively simple yet sufficiently precise approach for predicting building energy use. While white-box models are often criticized for their complexity and demand for significant resources and expert knowledge, the standardized model used in this study offers a streamlined but still accurate representation of building thermal behavior. The RC model, commonly applied in gray-box modeling, appears across the literature in varying levels of complexity depending on the modeling objective. A key novelty of this research lies in the empirical validation of these models using real world measurements obtained from a monitored public building. The use of controlled testing protocols [25], such as Pseudo Random Binary Sequence (PRSB) test and a natural cooling test conducted during periods of no occupancy and closed windows, enabled precise characterization of internal heat gains and thermal inertia. This experimental setup allowed for a robust assessment of the models’ ability to capture buildings thermal dynamics and better reflect actual energy use conditions. 2. Materials and methods During the project “Program energetske obnove zgrada javne namjene” from 2014 to 2020, a deep energy renovation of the building complex (UNIN1, UNIN2, and UNIN3) at the University North in Varaˇ zdin was carried out. The energy renovation was financed using the ESCO (Energy Service Company) model, which allows the initial investment to be repaid based on the energy savings achieved after the renovation. Under this model, the energy service provider is compensated from the difference between pre-renovation and post-renovation energy costs, with no additional costs incurred by the user. After the contractual obligation ends, the user benefits from paying their actual, now reduced, energy costs. However, it has been observed that the delivered energy often differs from calculated consumption due to various factors such as user behavior, building usage, and external environmental conditions like outdoor air temperature, ventilation, and solar radiation. These variations make the University North complex an ideal live laboratory for studying energy efficiency in buildings. 2.1. Measurement and building description The University North complex, located in Varaˇ zdin, consists of three main buildings: UNIN 1, UNIN 2, and UNIN 3. UNIN 1 and UNIN 2 primarily house classrooms and laboratories, while the administrative offices of the University are situated in UNIN 3. The UNIN 1 building, located at 104th Brigade Street 3, is a four-level structure comprising a basement, ground floor, first floor, and attic. However, only the rooms on the ground floor and first floor are furnished and operational, providing a total heated usable area of 1,570 m 2 and a heated volume of 7,949 m 3 . The building is equipped with sensors and measurement devices to monitor temperature (◦C), relative humidity (%), CO2 concentration (ppm), and Volatile Organic Compounds (VOC) concentration (ppb) in each room. Each room is equipped with at least one sensor, while larger rooms are equipped with multiple sensors (Figs. 1 and 2) to ensure accurate data collection across the space. In addition to indoor environmental monitoring, meteorological data are continuously recorded. This data includes wind speed, wind direction, external relative humidity, solar radiation, and outdoor temperature, all measured by a weather station located within the University North complex. The indoor microclimatic parameters and outdoor environmental data are recorded at 15-minute intervals. The gas consumption of the building’s heating system is measured and logged hourly for most of the period, during PRBS test gas consumption was measured and logged at 5-minute intervals. Gas consumption data is converted to delivered energy using a conversion factor of 9.2607 kWh/ Sm3, where 1 Sm3 is the standard cubic meter of natural gas under standard conditions (pressure of 101 325 Pa and temperature of 288.15 K). Continuous monitoring of indoor and outdoor parameters, along with the delivered natural gas quantity, was conducted from October 1, 2021, to August 1, 2024. The recorded data was processed to align with full-hour intervals. In cases where data was not recorded exactly on the hour, values were linearly interpolated. Solar radiation data, however, represents the sum of measurements over the preceding hour. The accuracy of the measurement devices used in the study is provided in Table 1a. To capture the building’s dynamic thermal response, a Pseudo Random Binary Sequence (PRBS) test was performed. The PRBS test introduced varying heating power inputs over time, allowing observation of the indoor temperature’s response to these fluctuations. This test was crucial for accurately calibrating both the grey box and white box models, enabling them to capture the building’s thermal dynamics effectively. Additionally, a temperature ramp-up test was conducted by setting the heating system to its maximum capacity to reach a stationary indoor temperature. Once the stationary temperature was achieved, the heating system was turned off, and the resulting temperature drop was recorded. This data was used to fit the zone’s thermal capacity and internal gains, providing essential input for the grey and white box model’s thermal parameters. 2.2. Mathematical models This study uses three different types of models to analyze the energy usage in the building: a white box model based on thermodynamic laws, a black box model driven purely by data, and a grey box model that integrates both data-driven approaches and physical principles. Each of these models was developed using open-source programming language Python. 2.2.1. White box model – HRN EN ISO 52016–1 White box models stand out for their transparency, providing a clear understanding of the background of the process. On the other hand, their complexity requires highly qualified personnel and a significant amount of time during development. The calculation of the energy required for heating a building is based on the HRN EN ISO 52016–1 standard [26], which provides guidelines for calculating the necessary energy for heating buildings. The process of calculating the required heating energy involves gathering data on the building’s properties, determining climatic conditions, zoning the space, identifying heat losses, and calculating the required energy for heating. When gathering data on the building, it is necessary to consider the building’s area, volume, the surface area of the exterior walls, the type and thickness of insulation, and the type of windows and doors. The climatic conditions affecting the building’s heating include outdoor temperature, relative humidity, wind speed, and solar radiation, which are collected at the weather station located on the building’s roof. The energy required for heating the building is calculated using formulas that consider heat losses and climatic conditions, and the enJ. ˇ Cukelj et al. Energy Conversion and Management 343 (2025) 120182 3 ergy equation for a zone describes the heat exchange between the building and its environment. By following these steps, it is possible to accurately calculate the required energy for heating a building in accordance with the HRN EN ISO 52016–1 standard. Energy equation for zone is written in equation (1) with following equations for layer of wall in contact with zone (2), equation for inside layers of wall (3) and layer of wall with contact with outside environment (4). −hpli;eli⋅θpli−1;eli;t+[κpli;eli Δt+hpli;eli +hpli−1;eli]⋅θpli;eli;t−hpli;eli⋅θpli+1;eli;t =κpli;eli Δt⋅θpli;eli;t−1(3) (κpli;eli Δt+hce;eli +hre;eli +hpli;eli)⋅θpli;eli −hpli;eli⋅θpli+1;eli;t=κpli;eli Δt⋅θpli;eli;t−1 +(hce;eli +hre;eli)⋅θe;t+ α sol;pli;eli⋅(Isol;dif;eli;t+Isol;dir;eli;t⋅Fsh;obst;eli;t) −Φsky;eli;t (4) Standard HRN EN ISO 52016–1 requires a lot of input data and most of them are shown in Table 2. The ones that are not mentioned in table and are in calculation procedure of the standard are values specified in standard and are not changed. From November 17, 2022, to November 21, 2022, a natural cooling test (“Free floating temperature”) of the building was conducted. This test begins by heating the building to stationary indoor air temperature conditions, which are significantly higher than the usual indoor air temperatures, using the boiler at maximum power. Once stationary conditions are achieved, the boilers are turned off, and no further thermal energy is supplied to the rooms. The indoor air temperature naturally decreases due to the lower outside temperature and air Fig. 1. Ground floor UNIN1. Fig. 2. First floor UNIN1. [Cint;ztc +∑ eln eli (Aeli •hci,eli)+∑ ven vei=1 Hve;vei;t+Htr;tb;ztc ]•θint;a;ztc;t−∑ eln eli=1(Aeli⋅hci;eli⋅θpln;eli;t) =Cint;ztc Δt⋅θint;a;ztc;t−1+∑ ven vei=1(Hve;vei;t⋅θsup;vei;t)+Htr;tb;ztc⋅θe;a;t+fint⋅Φint;ztc;t+fsol⋅Φsol;ztc;t+fH/C⋅ΦHC;ztc;t(1) −(hpli−1;eli⋅θpli−1;eli;t)+[κpli;eli Δt+hci;eli +hri;eli⋅∑ eln elk=1(Aelk Atot)+hpli−1;eli ]⋅θpli;eli;t−hci;eli⋅θint;a;ztc;t−∑ eln elk=1(Aelk Atot ⋅hri;eli⋅θpli;elk;t) =κpli;eli;t−1 Δt⋅θpli;eli;t−1+1 Atot ⋅[(1−fint;c)⋅Φint;ztc;t+(1−fsol,c)⋅Φsol;ztc +(1−fH/C,c)⋅ΦHC;ztc;t](2) J. ˇ Cukelj et al. Energy Conversion and Management 343 (2025) 120182 4 infiltration through the building envelope, until a new stationary point and indoor air temperature are reached. During the test conducted from November 17 to November 21, the stationary temperature after cooling the building was not achieved, as the heating of the spaces began when the building was put back into use. The aim of this test was to determine internal gains from equipment, such as computers, lighting, and other equipment used in the building during the test period. Another goal was to verify whether the properties of the building envelope correspond to the design values, which was examined with known infiltration determined from the drop in CO 2 concentration, measured during the test. 2.2.2. Black box model – Acyclic neural network Machine learning is a branch of artificial intelligence. Within the domain of machine learning, neural networks are the most prominent. An artificial neural network is a data processing system designed based on the example of neurons in biological systems, specifically their interconnections. Each neuron in the network performs calculations and is interconnected by “weighted” connections that are determined during the training of the neural networks with known data. In this work, a feedforward artificial neural network is used, which, like most, is composed of an input layer, hidden layers, and an output layer, which in this case has only one output value. The number of neurons in each of Table 1b Accuracy of the measurement devices. Measurement Location Device Type Device Description Measurement range Accuracy Gas Measurement Itron ACD G10 Diaphragm Gas Meter 0.10–16 m3/h Class 1.5 Weather Station Davis Vantage PRO2 Meteorological Station for Measuring External Temperature, Relative Humidity, Wind Speed and Direction, and Solar Radiation Temperature: −40 to + 150 ◦C Relative Humidity: 1–100 % Wind Speed: 0–809 m/s Wind Direction: 0-360◦ Solar Radiation: 0–1800 W/m 2 Temperature: +/- 0,3◦C Relative Humidity: +/- 2 % Wind Speed: 1 m/s Wind Direction: 3◦Solar Radiation: 5 % of full scale In room sensors Enless AMB 600–023 Sensor for Temperature, Relative Humidity, CO2 Concentration, and Volatile Organic Compounds Temperature: −40 to + 125 ◦C Relative Humidity: 0–100 % CO2 Concentration: 0–5000 ppm VOC Concentration: 0–60000 ppb Temperature: +/- 0,2◦C Relative Humidity: +/- 2 % CO2 Concentration: −VOC Concentration: − Fig. 3. Visualization of Acyclic Neural Network. Table 1a Statistical analysis of results. R 2 [-] MAE [kWh] MAPE [%] RMSE [kWh] CV (RMSE) [%] MBE [%] White box (hourly) 0.29 8.81 60.64 13.73 78.51 5.18 White box (daily) 0.85 2.29 12.53 2.89 16.50 5.13 Black box 0.26 6.48 36.13 7.93 38.47 1.46 Grey box 0.99 0.027 0.13 0.86 0.0054 −0.00017 J. ˇ Cukelj et al. Energy Conversion and Management 343 (2025) 120182 5 the hidden layers is determined experimentally, i.e., by trial and error, with a focus on ensuring that the network is not overly complex, as this could jeopardize its applicability to input data sets different from those on which it was trained, leading to the phenomenon of “overfitting.”. The input data consists of measured data from all rooms within the building and external environmental parameters, as described previously. In a later chapter, the chosen architecture of the neural network, which consists of an input layer, four hidden layers with the following number of neurons: 250, 200, 100, 50 and an output layer composed of one neuron representing the desired value, i.e., the delivered energy for heating the building. Fig. 3 shows the scheme of the feedforward neural network where the first layer is the input layer, followed by four hidden layers, and finally, the output layer. The activation functions used in the hidden layers are Rectified Linear Unit (ReLU), while the output layer uses a linear activation function. The model was trained using the Adam optimizer, with early stopping applied to avoid overfitting. To validate model performance and ensure generalizability, 5-fold cross-validation was conducted on the training dataset. Regularization techniques such as dropout were also applied during training. The training and testing of the neural network were done using measured data normalized to the full hour, with gas consumption transformed into delivered energy. Meteorological data and indoor thermal comfort parameters were measured during the periods from October 1, 2022 – June 1, 2023. The data used to train and later test the neural network were randomly divided in a 70:30 ratio, with 70 % of the available data used for training the network and the remaining 30 % for testing. The measured data, as input variables in the input layer, were supplemented with information about the hour of the day and the day of the week for each measured value. 2.2.3. Grey box model – Two-resistor, two-capacitor thermal model Grey box modeling represents a hybrid approach that combines the strengths of both white-box (physics-based) and black-box (data-driven) models. In the context of building energy simulations, grey box models aim to capture the essential physical processes while also allowing for data-driven tuning to improve accuracy and robustness. One commonly used grey box model shown in Fig. 4a, Fig. 4b. is the 2-resistor, 2capacitor (2R2C) model, which simplifies the thermal dynamics of a building into two primary elements: resistance and capacitance. The resistances (R) represent the thermal resistance, or the difficulty with which heat flows between different parts of the building or between the building and its surroundings. The capacitances (C) represent the thermal mass, or the ability of building materials to store heat. The 2R2C model divides a building into two main thermal nodes: the indoor air temperature and the envelope temperature (associated with the building’s structure, such as walls and roof). The resistances and capacitances in this model are used to describe how heat transfers between these nodes and between the building and the external environment. The core of the 2R2C model consists of a set of differential equations that govern the heat exchange processes. These equations consider factors such as internal heating (e.g., from HVAC systems), solar gains (e.g., through windows), and heat losses to the environment. The two main equations (5) and (6) describe the change in indoor air temperature and the envelope temperature over time, incorporating the effects of thermal resistance and capacitance. T(t+1) i=T(t) i+dt Ci(1 Ri (Te−Ti) + Φh+AiΦs)(t) (5) T(t+1) e=T(t) e+dt Ce(1 Ri (Ti−Te) + 1 Ro (To−Te) + AeΦs)(6) For example, the change in indoor temperature (T i ) is influenced by the heat flow from the envelope to the interior, the heating power applied inside the building, and solar radiation. Similarly, the change in envelope temperature (T e ) depends on the heat exchange between the interior and exterior, as well as between the envelope and the external environment. These equations can be represented in a state-space form, which is common in control theory and allows for systematic analysis and simulation. The state vector in this model includes the indoor and envelope temperatures, and the input vector typically includes external conditions like outdoor temperature, solar radiation, and heating power. 3. Results and discussion The results are organized into three sections corresponding to the three models used in this study: the white box model, the grey box model, and the black box model. These sections present a detailed comparison of the models’ performance, focusing on indoor temperature predictions and heating energy consumption. 3.1. White box model – Indoor temperature and energy used for heating The white box model, developed according to the HRN EN ISO 52016–1 standard, was initially validated through a natural cooling test (Fig. 4a, Fig. 4b). The results demonstrated a high degree of accuracy in matching the calculated and measured indoor temperatures, achieved by fine-tuning the internal heat gains from equipment and adjusting the specific heat capacity of the zone. This test is a crucial method for validating the quality of the building envelope’s construction and design. The results provide essential data on the building’s thermal properties, including specific heat capacities of the zone as well as internal heat gains from equipment, which were determined to be 8 W/ m2. Two primary calculations were performed using the HRN EN ISO 52016–1 model. The first calculation involved predicting indoor temperature using the measured heating energy as an input, while the second calculation focused on determining the heating energy required to maintain a known indoor temperature. It is important to note that the calculation of energy needs differs from the measured energy due to factors such as distribution losses, heat emitters inefficiencies, and energy source losses. The model’s predictions were compared against measured data, starting from a period before February 11 to account for initial conditions across the wall layers. The results, shown in Fig. 5, indicate that the model closely follows the demand for heating energy, with delays that probably occur due to missing distribution and heat emitters models. The calculated energy consumption for the period was 4 656 kWh, while the measured consumption was 4 839 kWh, representing a 4 % difference. This discrepancy aligns well with expected energy losses due to distribution and heat transfer, validating the model’s accuracy. The temperature predictions on Fig. 6, averaged across all zones, also closely matched the measured temperatures, with minor deviations observed around March 19 when heating was off, and presumably the model did not capture solar gain spikes since the heating was turned off during that period, only solar gains could increase indoor temperature. Fig. 4a. 2R2C Thermal model. J. ˇ Cukelj et al. Energy Conversion and Management 343 (2025) 120182 6 3.2. Grey box model – Indoor temperature The grey box model, utilizing the 2R2C thermal model, demonstrated an even closer match to the measured indoor temperatures. This improvement is attributed to the data-driven nature of the model, which allows for more accurate parameter tuning. The coefficients used in the model are shown in Table 3. The grey box model’s ability to incorporate both physical principles and real-world data makes it a powerful tool for predicting indoor temperatures. The results in Fig. 6 indicate that the model captures the building’s thermal dynamics more effectively than the white box model. This is particularly evident around March 19th, where the grey box model accurately predicts a temperature spike despite the absence of heating. Using the temperatures acquired from the model energy consumption for heating is calculated and shown in Fig. 8. Fig. 8 illustrates the comparison between the actual heating power and the calculated heating power derived from the 2R2C grey-box model. The close agreement between the two curves across the entire simulation period ensures the model can capture the dynamics of heat transfer and energy consumption in the building. The model performs well even during transient events, such as sudden temperature changes or heating cycles during PRBS test. The grey-box approach integrates the benefits of physical modeling with data-driven techniques, ensuring both robustness and adaptability. As seen in Figs. 7 and 8, the model accurately calculates indoor air temperature and heating power, offering a powerful tool for energy management and control strategies. These results underline the potential of the 2R2C model for optimizing heating systems in buildings and maintaining thermal comfort. 3.3. Black box model – Predicted energy consumption for heating The black box model, powered by a feedforward artificial neural network (ANN), was tested using data collected prior to the PRBS test. The ANN was trained on data from the 2022/2023 heating season, covering the period from October 1, 2022, to June 1, 2023. The trained Fig. 4b. Natural cooling test. Fig. 5. Calculated and measured energy needed for heating – White box. J. ˇ Cukelj et al. Energy Conversion and Management 343 (2025) 120182 7 model was subsequently used to predict heating energy consumption for a two-day period between February 12 and February 14, 2024, with the results presented in Fig. 9. Measured energy consumption for this period was 763 kWh, while the predicted consumption was 818 kWh, a difference of 7 %. This level of accuracy shows that the trained network was not “overfitted” but still hour by hour results shows inaccurate results in some cases indicating model could benefit from general reduction in input data or even adding additional data that is crucial for energy consumption, e.g. solar irradiance or concentration of VOC where high VOC values could make occupants increase ventilation. Although the black box model showed highly accurate results, it is important to note that it lacks the physical interpretability of the white and grey box models. This limitation makes the black box model less reassuring for use in scenarios where a clear understanding of the underlying processes is essential. Possible advantages of black box model are detection in IEQ changes, system and sensors faults since all measured data is required. 3.4. Discussion The comparative analysis of the three modeling approaches white box, grey box, and black box reveals important insights into their respective strengths, weaknesses, and practical implications. The white box model, grounded in thermodynamic principles and standardized through HRN EN ISO 52016–1, offers high transparency and traceability, making it particularly suitable for regulatory compliance and certification purposes. However, it requires extensive input data and a detailed modeling process, which can be time-consuming and requires significant domain expertise. While this model achieved good agreement with measured data on a daily scale (R 2 =0.85), its performance on an hourly scale was lower due to the absence of system-level modeling such as heat distribution and emitter dynamics. The grey box model, implemented using a 2R2C thermal network, demonstrated the best overall performance with a high coefficient of determination (R 2 =0.99) and minimal error across all statistical metrics. This model provides a balance between computational simplicity and physical insight, making it a strong candidate for real-time energy management and model predictive control. Due to its reduced complexity and minimal input data requirements, the grey box model is also the most computationally efficient, making it well-suited for applications with limited processing capacity or data availability. The black box model, based on a feedforward artificial neural network, showed strong predictive ability with a 7 % deviation from measured energy use during the test period. However, its interpretability is limited, and its accuracy depends heavily on the quality and quantity of training data. Training such a model requires a relatively large dataset and considerable computational resources, including the use of a GPU with CUDA cores to optimize training performance. While effective for short-term forecasting and anomaly detection, its use in certification or regulatory processes is less advisable due to its “black box” nature. A statistical summary of the models’ performance is provided in Table 1b, showing key evaluation metrics including R2, MAE, MAPE, RMSE, CV(RMSE), and MBE. These models were designed with scalability in mind. All three approaches are modular and adaptable, allowing for application in buildings with different typologies and usage patterns. While the current case study focuses on a university building, the modeling techniques and calibration procedures are transferable to other public buildings, office Fig. 6. Calculated and measured indoor air temperature – White box. Table 2 HRN EN ISO 52016–1 input data. Description Symbol Value Unit of Measurement Number of air changes in hour n 0.8 1/h Zone height H 3.5 m Zona area A 1 570 m 2 Specific heat capacity of air and furniture κ int 524 600 J/m 2 K Specific heat capacity of walls κ m 1 069 692 J/m 2 K Thermal transmittance of walls U wall 0.22 W/m 2 K Thermal transmittance of windows U window 1.31 W/m 2 K Thermal transmittance of roof U roof 2.51 W/m 2 K Thermal transmittance the groundcontact wall U ground 0.19 W/m 2 K Internal gains Q int 8 W/m 2 Table 3 2R2C model coefficients. Coefficient R i R o C i C e A i A e Value 0.075 1.64 4.11e7 1.64e6 0.041 −0.11 J. ˇ Cukelj et al. Energy Conversion and Management 343 (2025) 120182 8 Fig. 7. Calculated and measured indoor air temperature – Grey box. Fig. 8. Calculated and measured heating power – Grey box. Fig. 9. Predicted and measured energy needed for heating – Black box. J. ˇ Cukelj et al. Energy Conversion and Management 343 (2025) 120182 9