Simulation of Heat Pump with Heat Storage and PV System—Increase in Self-Consumption in a Polish Household

Abstract

The use of renewables in heat production requires methods to overcome the issue of asynchronous heat load and energy production. The most effective method for analyzing the intricate thermal dynamics of an existing building is through transient simulation, utilizing real-world weather data. This approach offers a far more nuanced understanding than static calculations, which often fail to capture the dynamic interplay of environmental factors and building performance. Transient simulations, by their nature, model the building’s thermal behavior over time, reflecting the continuous fluctuations in temperature, solar radiation, and wind speed. Leveraging actual meteorological data enables the simulation model to faithfully capture system dynamics under realistic operational scenarios. This is crucial for evaluating the effectiveness of heating, ventilation, and air conditioning (HVAC) systems, identifying potential energy inefficiencies, and assessing the impact of various energy-saving measures. The simulation can reveal how the building’s thermal mass absorbs and releases heat, how solar gains influence indoor temperatures, and how ventilation patterns affect heat losses. In this paper, a household heating system consisting of an air source heat pump, PV, and buffer tank is simulated and analyzed. The 3D model accurately represents the building’s geometry and thermal properties. This virtual representation serves as the basis for calculating heat losses and gains, considering factors such as insulation levels, window characteristics, and building orientation. The approach is based on the calculation of building heat load based on a 3D model and EN ISO 52016-1 standard. The heat load is modeled based on air temperature and sun irradiance. The heating system is modeled in EBSILON professional 16.00 software for the calculation of transient 10 min time step heat production during the heating season. The results prove that a buffer tank with the right heat production control system can efficiently increase the auto consumption of self-produced PV electric energy, leading to a reduction in environmental effects and higher economic profitability.

1. Introduction

In the face of growing global energy demand and increasing awareness of environmental sustainability, renewable energy technologies are becoming an essential component of modern energy systems. As global energy demand continues to rise, driven by factors such as population growth, urbanization, industrialization, and an expanding middle class, the need for sustainable energy solutions has never been more urgent. Simultaneously, there is a heightened awareness of the environmental impacts of traditional energy sources, particularly fossil fuels, which contribute significantly to greenhouse gas emissions and global warming. As a result, renewable energy technologies have emerged as a critical element in the transition to cleaner, more sustainable energy systems worldwide. The integration of renewable energy into the global energy mix presents both challenges and opportunities. While renewable energy technologies have made significant strides in recent years, they still face obstacles related to cost, efficiency, infrastructure, and policy support. Nevertheless, the rapid technological advancements and falling costs associated with solar, wind, and battery storage technologies are making renewables increasingly competitive with conventional energy sources. In many parts of the world, renewables are already proving to be a cost-effective and reliable source of energy, providing power to homes, businesses, and industries, and offering the potential to reshape the future of energy production and consumption [1]. Renewable energy technologies harness energy from natural processes that are replenished over time, such as sunlight, wind, rain, tides, geothermal heat, and even biomass. Unlike conventional energy sources, such as coal, oil, and natural gas, which are finite and cause environmental degradation, renewable energy sources are abundant and environmentally friendly. The widespread adoption of renewable energy technologies is not only seen as a way to reduce dependence on fossil fuels but also as a strategy to mitigate climate change, promote energy security, and create economic opportunities. Among these technologies, heat pumps, energy storage systems, and photovoltaic (PV) panels play a crucial role in enhancing energy efficiency, reducing carbon emissions, and ensuring energy independence [2].

In this paper, a household heating system consisting of an air source heat pump, PV, and buffer tank is simulated and analyzed. This paper investigates the performance of an integrated system utilizing renewable energy sources. We develop and employ a dynamic simulation model to analyze a household heating system configured with an air source heat pump (ASHP) as the primary heat generator, photovoltaic (PV) panels for on-site electricity generation, and a thermal buffer tank for energy storage and operational flexibility. The simulation assesses the intricate dependencies arising from variable solar irradiance, dynamic thermal loads, and the heat pump’s performance parameters. The analysis focuses on quantifying the system’s energy efficiency, maximizing the self-consumption of PV-generated electricity by the heat pump, assessing the role of the buffer tank in decoupling heat generation from demand, and evaluating the overall potential for reducing reliance on conventional energy sources and associated costs. The results of this research will inform the optimal design, control strategies, and economic evaluation of integrated renewable heating systems for contemporary homes.

The global imperative to transition towards sustainable energy systems has placed significant emphasis on the integration of renewable energy sources and energy-efficient technologies within the residential sector. In this context, the synergistic deployment of heat pumps, energy storage systems, and photovoltaic (PV) installations emerges as a promising strategy to achieve substantial reductions in greenhouse gas emissions and enhance energy autonomy in residential buildings. Decentralized energy systems are rapidly gaining traction as a means of reducing reliance on traditional power grids and mitigating the environmental impact of residential buildings. The growing environmental concerns, coupled with escalating energy costs, have spurred a paradigm shift in the design and operation of residential energy systems [3]. Conventional fossil fuel-based heating and cooling solutions are increasingly being replaced by more sustainable alternatives. Among these, heat pumps have gained prominence due to their high efficiency and ability to utilize renewable heat sources such as air, ground, or water. When combined with on-site PV generation, heat pumps can significantly reduce the reliance on grid electricity, particularly during peak demand periods [4]. In response to the growing demand for efficient and ecological heating systems in Poland, this publication presents a detailed analysis of the integration of heat pumps (e.g., air source), energy storage (e.g., thermal in the form of buffer tanks), and photovoltaic installations within a typical Polish residential building The study evaluated system performance under changing conditions, paying special attention to PV energy self-consumption and heat pump efficiency. Additionally, its energy efficiency as a complete system was examined. The publication aims to provide a comprehensive overview of the key technical and energy-related aspects associated with the implementation of these integrated systems, taking into account the specifics of the Polish climate, building standards, and regulatory environment, serving as a valuable source of information for homeowners and professionals [5]. Poland’s energy sector has historically been dominated by coal-fired power plants, resulting in high greenhouse gas emissions and environmental pollution. However, the country has made significant strides in diversifying its energy mix and promoting renewable energy sources. In recent years, Poland has experienced a surge in the adoption of PV systems, driven by favorable government policies, declining technology costs, and growing public awareness of environmental issues. The integration of heat pumps and energy storage systems is also gaining momentum, driven by the need to enhance energy efficiency and reduce reliance on fossil fuels. The integration of heat pumps, energy storage systems, and PV offers a synergistic approach to achieving sustainable and energy-efficient residential buildings. Heat pumps provide an efficient and environmentally friendly alternative to traditional heating systems, while energy storage systems enable the storage of excess PV-generated electricity for later use, maximizing self-consumption and reducing reliance on the grid. PV systems generate clean and renewable electricity, further reducing the carbon footprint of residential buildings [6].

Heat pumps utilize ambient heat from the air, ground, or water to provide heating and cooling, offering significant energy savings compared to traditional heating systems. In Poland, ground-source and air-source heat pumps are the most commonly used types, depending on the specific site conditions and heating requirements. The selection and sizing of heat pumps should be carefully considered to ensure optimal performance and energy efficiency.

Energy storage systems, such as batteries, play a crucial role in maximizing the benefits of PV systems. By storing excess PV-generated electricity, energy storage systems enable homeowners to increase self-consumption, reduce reliance on the grid, and enhance energy independence. In addition, energy storage systems can contribute to grid stability by providing ancillary services, such as frequency regulation and voltage support.

PV systems convert sunlight into electricity, providing a clean and renewable energy source for residential buildings. The performance of PV systems depends on various factors, including solar irradiance, temperature, and shading. In Poland, the southern regions offer the highest solar irradiance, making them particularly suitable for PV installations. The selection and sizing of PV systems should be based on the building’s energy consumption and available roof space.

The economic viability of integrated systems depends on various factors, including technology costs, energy prices, and government incentives. In Poland, various financial support schemes are available for homeowners who invest in renewable energy technologies. The environmental benefits of integrated systems include reduced greenhouse gas emissions, improved air quality, and enhanced energy security.

This publication includes case studies of successful implementations of integrated systems in residential buildings in Poland, showcasing the practical aspects of design, installation, and operation. These case studies provide valuable insights and best practices for homeowners and professionals considering the adoption of these technologies [7]. The future of sustainable and energy-efficient residential buildings in Poland will be shaped by ongoing advancements in renewable energy technologies, the development of smart grids, and the implementation of supportive policies. Challenges remain in terms of grid integration, energy storage costs, and public awareness. However, the long-term prospects for the widespread adoption of integrated systems are promising, driven by the growing need to address climate change and ensure a sustainable energy future [8].

The integration of heat pumps, energy storage systems, and photovoltaics offers a compelling pathway towards sustainable and energy-efficient residential buildings in Poland. By embracing these technologies, homeowners can reduce their carbon footprint, enhance energy independence, and contribute to a cleaner and more sustainable future. This publication serves as a comprehensive source of knowledge regarding the intricate technical and energy aspects involved in the practical implementation of these integrated systems (photovoltaic, heat pumps, and energy storage). It moves beyond basic concepts to provide detailed analysis, data, and insights crucial for understanding system design, operational dynamics, performance optimization, and potential integration hurdles. Simultaneously, storage significantly increases the operational flexibility of heat pumps. Heat pumps are major electrical loads, and their energy demand (especially for heating) often occurs when solar generation is low or non-existent (e.g., evenings, winter). Stored battery energy allows the heat pump to run efficiently using clean, self-generated solar power even during these times. This decoupling also enables smart control strategies, where the heat pump operation can be shifted to periods with lower grid electricity prices or higher renewable energy availability on the grid, further enhancing cost savings and environmental benefits. In essence, energy storage acts as the vital link that buffers intermittent solar supply and flexible heat pump demand, maximizing both economic value and environmental impact for the integrated system [9]. By storing excess solar energy during periods of high irradiance, energy storage systems enable the provision of heating and cooling services during periods of low or no solar generation. This not only increases the self-consumption of renewable energy but also enhances grid stability by reducing peak demand and mitigating voltage fluctuations. By moving beyond theoretical calculations and simulations to examine real-world applications and case studies, this work adopts a crucial evidence-based approach. This means analyzing actual, operational systems where technologies like solar panels, heat pumps, and energy storage are already integrated and functioning, potentially in diverse settings such as residential homes, commercial buildings, or community energy projects. Observing these systems in practice allows for the collection of tangible performance data, identification of unforeseen installation or operational challenges, assessment of real-world economic viability under current market conditions, and understanding of user experiences and maintenance requirements.

In the face of growing global energy demand and increasing environmental concerns, renewable energy technologies have become a cornerstone of modern energy systems. As solar, wind, geothermal, biomass, and other renewable sources continue to evolve, they offer the potential to reshape the global energy landscape, providing clean, reliable, and sustainable power for generations to come. While challenges remain in terms of cost, infrastructure, and policy, the rapid advancements in renewable energy technologies, combined with strong government support and private sector investment, offer a pathway to a more sustainable energy future. The ultimate goal is to provide actionable insights for engineers, policymakers, building designers, homeowners, and energy companies on how to effectively leverage these combined technologies. This contributes to the broader objective of sustainable energy management, which involves transitioning away from fossil fuels, improving energy efficiency, enhancing energy security, and building a resilient, low-carbon energy future. Through this detailed examination of practical implementations, this work aims to provide valuable, actionable insights that are often missed in purely theoretical analyses. These insights can clarify which system configurations perform best under specific conditions, highlight the most effective control strategies for maximizing self-consumption or minimizing costs, quantify the actual energy savings and carbon emission reductions achieved, and reveal practical lessons learned regarding reliability and integration complexities. Such findings are invaluable for engineers seeking proven designs, policymakers needing evidence for effective support schemes, investors assessing project risks, and end-users wanting realistic expectations. Ultimately, these empirically grounded insights contribute significantly to understanding the evolving role of renewable energy in actively shaping the future of sustainable energy management. This future is increasingly defined by the trends to decarbonize the energy supply, enhance energy security, manage variable generation from sources like solar and wind, and create more resilient and efficient energy systems. By demonstrating the practical feasibility, benefits, and challenges of integrating key renewable and enabling technologies at the point of use, this work provides crucial knowledge to guide strategic decisions, technological innovation, and policy development necessary to accelerate the transition towards a cleaner, more reliable, and sustainable energy infrastructure capable of meeting long-term climate goals.

Literature Review

The integration of heat pumps (HP), photovoltaic (PV) systems, and energy storage (both thermal and electrical) in residential buildings represents a key strategy to obtain an environmentally friendly building sector and enhance energy autonomy. Simulating these complex systems is crucial for understanding their dynamic interactions, optimizing performance, and maximizing the self-consumption of locally generated PV electricity. This review examines relevant literature, focusing on simulation studies that investigate the synergy between these technologies to increase PV self-consumption in households, a context directly applicable to scenarios like a Polish household [10]. Modeling the combined operation of HP, PV, and storage systems is complex due to the variable nature of solar generation, fluctuating household energy demands (thermal and electrical), and the dynamic performance characteristics of the components. Simulation tools and methodologies are therefore essential for analyzing system behavior and designing effective control strategies. The current best practices in modeling integrated systems, along with associated challenges, are often summarized in comprehensive reviews, which also establish a basic understanding of the utilized simulation approaches (e.g., TRNSYS, Modelica, custom MATLAB/Python models). Studies frequently employ dynamic simulations to capture time-dependent interactions [11]. For example, research has utilized dynamic simulation to evaluate smart grid-ready heat pumps coupled with PV and electrical storage, assessing their potential for demand-side management, which intrinsically involves optimizing self-consumption [12].

A primary driver for integrating these technologies is to increase the on-site consumption of PV-generated electricity, reducing reliance on the grid and potentially lowering energy costs. Research often focuses on developing and simulating advanced control strategies, such as model predictive control (MPC), to intelligently manage energy flows between the PV array, storage, heat pump, and the grid. These studies simulate scenarios comparing different control algorithms (e.g., rule-based vs. MPC) to quantify the potential increase in self-consumption and self-sufficiency achievable through smart operation. Similarly, often investigates how different system configurations (e.g., sizing of PV array, storage capacity) and control logic impact the self-consumption ratio, providing valuable insights into optimal system design based on simulation results [13]. Beyond simple integration, significant research focuses on optimizing the sizing of components (PV array, storage capacity, HP size) and developing sophisticated control strategies. Model Predictive Control (MPC) is frequently investigated for its potential to optimize system operation based on weather forecasts, electricity prices, and predicted loads, thereby maximizing self-consumption or minimizing costs. Reference [14] explored MPC for optimal operation of PV-HP systems with both TES and BESS [15]. Furthermore, the concept of energy flexibility, which these systems provide to the grid, is increasingly studied. The publication [16] assessed the flexibility potential offered by residential HPs with TES and PV, often quantified through simulation studies. Smart control strategies considering grid signals or dynamic tariffs are also explored to further enhance system value [17]. While this review seeks literature relevant to a Polish household context, many studies use simulation to compare system performance across various locations or climate zones. The findings from studies conducted in Central Europe or similar heating-dominated climates are particularly relevant [13]. The specific building characteristics, local electricity tariffs, and policy support mechanisms significantly influence system performance and economic viability [18]. Research considering building integration aspects and holistic performance within the building envelope provides further context [19]. Although simulations explicitly parameterized for Poland might be less common in top international journals, studies using representative climate data for similar regions offer valuable insights applicable to the Polish context.

Another publication details research into the application of Phase Change Materials (PCMs) integrated directly into the empty spaces created by inter-floor void formers used in the construction of concrete floor slabs. By incorporating PCMs, which absorb and release heat at specific temperatures, into these structural elements, the study investigates the potential to significantly increase the effective thermal mass and thermal energy storage capacity of the floor system. The work likely analyzes (potentially through simulation or experiment) the impact of this integration on building energy efficiency, indoor temperature stability, occupant thermal comfort, and peak load reduction for HVAC systems. It may also cover aspects like material selection, encapsulation methods for the PCM, and the practicalities of implementing such a system during construction [20]. Another publication in a similar vein presents a numerical study aimed at improving the thermal performance of a shell-and-coil ice storage enclosure designed for air conditioning systems. The study focuses specifically on the effect of adding spiral longitudinal fins to the internal surface of the refrigerant flowing coil. Using computational methods (such as CFD with phase change modeling), the study simulates the dynamic processes of ice formation (charge cycle) and ice melting (discharge cycle) inside the enclosure. The main objective is to quantify the extent to which these specialized fins improve the heat transfer rate, potentially reduce the time required for charge/discharge, and increase the overall performance of the latent heat storage system compared to a baseline configuration without fins or with simpler fin designs. The numerical approach allows for a detailed analysis of the temperature distributions, ice growth/decay, and fluid flow patterns, all of which are influenced by the fin geometry [21].

The reviewed literature highlights a strong research focus on simulating integrated PV, HP, and storage systems in residential buildings to enhance PV self-consumption. Simulation is established as a vital tool for understanding system dynamics, optimizing component sizing, and developing effective control strategies (including TES management and advanced MPC) [22]. Studies consistently show that adding thermal or electrical storage significantly boosts self-consumption and system flexibility. While context-specific analyses, such as for Polish households, are essential for precise local assessment, the methodologies and general findings from broader European or climate-specific studies provide a solid foundation and valuable benchmarks for such investigations. Future work often points towards refining control algorithms, exploring hybrid storage solutions, and analyzing grid interaction capabilities.

This review focuses on the importance of simulating integrated systems comprising heat pumps (HP), photovoltaic (PV), and energy storage (both thermal and electrical) in residential buildings. Such integration is key for decarbonizing the building sector and increasing energy autonomy, with a primary goal often being the maximization of self-consumed PV electricity.

The text highlights the complexity of modeling these systems due to the variability of solar energy, fluctuating household demands, and dynamic component performance. Simulation, using tools like TRNSYS, Modelica, or custom models, is presented as an essential methodology for understanding system behavior, optimizing component sizing (PV arrays, storage capacity), and designing effective control strategies.

Drawing primarily from literature published by Elsevier and MDPI, the review notes that numerous studies use simulation to evaluate different system configurations and control approaches, ranging from rule-based logic to advanced Model Predictive Control (MPC). These studies aim to quantify the potential increases in PV self-consumption and energy flexibility achievable through smart operation and storage integration.

While acknowledging that simulations specifically tailored to a Polish context might be less prevalent in top international journals, the review emphasizes that findings from studies in similar climates are highly relevant. It also briefly mentions related simulation research into novel thermal storage concepts like Phase Change Materials (PCMs) in building structures and enhanced ice storage designs.

In conclusion, the literature strongly confirms that simulation is a vital tool for understanding, optimizing, and developing control strategies for integrated HP, PV, and storage systems in homes, ultimately aiming to boost PV self-consumption and system efficiency.

2. Materials and Methods—Methodology

This case study describes how a heat pump with a buffer tank can increase the auto consumption of PV electric energy in a renovated building. The methodology used for calculations comprises heat pump heat production simulation based on historical meteorological data and the design load of an analyzed building. The transient calculation of the heat pump system is emphasized with an uncommon software approach. The model was developed in Ebsilon professional software. The calculations are based on a non-linear system of equations, solved using a Newton-like linearization and a matrix solution. Using this approach for a transient solution results in a series of quasi-steady states with a fixed time step.

Overall, the calculation procedure is as follows:

• Calculation of the total design heat load for the analyzed building. This step is based on a standardized procedure with the EN ISO 52016-1 standard.
• Determination of the building heat load for different temperatures and solar heat gains for the building. The standardized method is based on the steady-state conditions for each month. Those calculations are the basis for heat load correlated with ambient temperature and solar gains correlated with solar irradiance.
• Calculation of the parameters in the heating system in steady state conditions using the Ebsilon software model.
• Calculation of transient parameters in the Ebsilon model based on a 10 min time step and historical parameters of ambient temperature and solar irradiance.

This calculation method allows the creation of a simulation of the heat pump-based heating system during the whole heating season (Figure 1).

2.1. Modeling the Building Heat Load

Modeling the building heat load is a fundamental process in the design and optimization of energy-efficient buildings. It involves creating a digital representation of a building to analyze and predict its thermal behavior under various environmental conditions. This process is crucial for accurately sizing heating, ventilation, and air conditioning (HVAC) systems, assessing energy performance, and identifying opportunities for energy savings. The foundation of any heat load model lies in accurately representing the building’s geometry and material properties. This includes defining the dimensions of walls, roofs, floors, and windows, as well as assigning thermal characteristics such as thermal conductivity, heat capacity, and density to each building element. This detailed representation allows for precise calculations of heat transfer through the building envelope. Environmental factors play a significant role in determining a building’s heat load. Climate data, including temperature, solar radiation, wind speed, and humidity, are essential inputs for the model. These data can be obtained from local meteorological stations or integrated climate databases. The model then uses these data to simulate the building’s thermal response to varying weather conditions.

Heat transfer calculations are at the core of heat load modeling. These calculations involve analyzing conductive, convective, and radiant heat transfer through the building envelope. In addition to heat transfer through the building envelope, heat load modeling also considers internal heat gains from occupants, lighting, and appliances. These internal gains can significantly impact the building’s overall heat load of the building, particularly in well-insulated buildings. Software tools are widely used to facilitate heat load modeling. These tools allow for the creation of detailed building models and the performance of complex heat transfer calculations. They often incorporate climate data, material databases, and calculation algorithms based on established standards, such as EN ISO 52016-1 [23] and EN ISO 13790 [24]. Accurate heat load modeling is essential for designing energy-efficient buildings that provide comfortable indoor environments while minimizing energy consumption. By using detailed modeling and simulation techniques, architects and engineers can make informed decisions about building design and HVAC system selection. This standard incorporates hourly or monthly calculation methods, accounting for both steady-state and transient heat transfer. The simulation encompasses the interaction between the air source heat pump, the PV system, and the buffer tank. The PV installation’s energy generation is modeled based on local solar irradiance data, while the heat pump’s performance is simulated considering varying ambient temperatures and load demands. The buffer tank’s role in storing thermal energy is crucial for optimizing the system’s efficiency, allowing for the decoupling of heat generation and demand. The analysis focuses on evaluating the system’s overall energy efficiency, self-consumption of PV-generated electricity, and the reduction in grid dependency. The results provide valuable insights into the potential of integrated renewable energy systems for achieving sustainable and energy-efficient household heating [25,26,27].

The approach to estimating the building’s heat load was meticulously executed, leveraging a sophisticated methodology rooted in the development of a detailed 3D building model and adherence to the EN ISO 52016-1 standard [23]. This approach provides a high degree of accuracy and reliability, essential for designing efficient heating, ventilation, and air conditioning (HVAC) systems. The process commenced with the creation of a comprehensive 3D digital model of the building. This model served as a precise virtual representation of the building’s geometry, incorporating all architectural elements, including walls, roofs, floors, windows, and doors. Each element was defined with its accurate dimensions and material properties. Crucially, thermal characteristics, such as thermal conductivity, heat capacity, and density, were assigned to each material, enabling the simulation of the building’s thermal performance under varying conditions [28,29].

Integral to the process was the utilization of the EN ISO 52016-1 standard [23], which provides detailed procedures for calculating the energy needs for heating and cooling in buildings. This standard employs advanced calculation methods that consider both steady-state and dynamic thermal behavior, accounting for the complex interactions between the building, its occupants, and the surrounding environment.

Firstly, the transmission heat losses were calculated. This involved determining the heat loss through each building element (walls, windows, roofs, floors) based on their U-values (heat transfer coefficients), surface areas, and the temperature difference between the indoor and outdoor environments. The U-values were calculated based on the thermal resistance of each layer of the building element, ensuring accuracy in the transmission heat loss calculations.

Secondly, the ventilation heat losses were calculated. This involved determining the heat loss due to air exchange between the indoor and outdoor environments. The ventilation rate, which is the amount of air exchanged per unit of time, was determined based on the building’s occupancy, ventilation system design, and air infiltration characteristics. The ventilation heat losses were then calculated based on the ventilation rate and the temperature difference between the indoor and outdoor environments.

Thirdly, the heat losses due to thermal bridges were calculated. Thermal bridges are areas of the building envelope where the thermal resistance is reduced, leading to increased heat loss. The calculations involved identifying and quantifying the heat loss through linear and point thermal bridges, ensuring that these losses were accurately accounted for in the overall heat load calculation.

Finally, the design heat load was determined by summing the transmission, ventilation, and thermal bridge heat losses. This value represents the maximum amount of heat required to maintain the desired indoor temperature under design conditions. The 3D model proved invaluable in accurately determining surface areas, identifying thermal bridges, and simulating the impact of solar radiation on different building facades. The integration of climatic data, including temperature, humidity, and solar radiation, further enhanced the accuracy of the calculations.

EN ISO 52016-1 calculates the energy needs for heating and cooling, internal temperatures, and energy loads of a building zone using a dynamic hourly (or sub-hourly) time-step method. It is based on a thermal network model representing the heat transfer paths and thermal capacities within the building zone. A common simplification, conceptually aligned with the standard, is the 5R1C model (five resistances, one capacitance) which links thermal nodes representing the external environment, internal air, and the building’s thermal mass.

The fundamental principle is that the sum of all heat flows into the zone air node (θ_air), plus the heat supplied by the heating/cooling system (Q_HC,nd), must balance the heat transferred to internal surfaces and ventilation loss.

Simplified quasi-steady state for air node

$${\mathsf{\Phi }}_{\mathrm{t}\mathrm{r},\mathrm{i}}+{\mathsf{\Phi }}_{\mathrm{v}\mathrm{e}}+{\mathsf{\Phi }}_{\mathrm{s}\mathrm{o}\mathrm{l},\mathrm{a}\mathrm{i}\mathrm{r}}+{\mathsf{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{t},\mathrm{a}\mathrm{i}\mathrm{r}}+{\mathrm{Q}}_{\mathrm{H}\mathrm{C},\mathrm{n}\mathrm{d}}=0$$

(1)

where:

• Φ_tr,i = Convective heat transfer from internal surfaces to zone air.
• Φ_ve = Heat transfer due to ventilation and infiltration.
• Φ_sol,air = Solar gains transferred directly to zone air (often minor).
• Φ_int,air = Convective part of internal heat gains.
• Q_HC,nd = Energy delivered (+) or extracted (−) by the heating/cooling system to the air node to meet the setpoint (Energy Need). Positive for cooling need, negative for heating need.

The standard uses a system of equations based on thermal resistances (R) and conductances (H = 1/R) connecting different thermal nodes. Key nodes often include external air (θ_e), internal air (θ_air), internal surfaces (θ_si), external surfaces (θ_se), and a central node representing the effective thermal mass of the zone (θ_m) with capacitance C_m.

Heat Flow (General):

$$\mathsf{\Phi }=\mathrm{H}\cdot \left({\mathsf{\theta }}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}-{\mathsf{\theta }}_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{k}}\right)$$

(2)

Energy Balance for the Internal Air Node (θ_air): This node links ventilation, convective internal gains, convective heat transfer from internal surfaces, and the heating/cooling system.

$${\mathrm{H}}_{\mathrm{ve}}\cdot \left({\mathsf{\theta }}_{\mathrm{e}}-{\mathsf{\theta }}_{\mathrm{a}\mathrm{i}\mathrm{r}}\right)+{\mathrm{H}}_{\mathrm{i}\mathrm{s}}\cdot \left({\mathsf{\theta }}_{\mathrm{s}\mathrm{i}}-{\mathsf{\theta }}_{\mathrm{a}\mathrm{i}\mathrm{r}}\right)+{\mathsf{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{t},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}+{\mathrm{Q}}_{\mathrm{H}\mathrm{C},\mathrm{n}\mathrm{d}}={\mathrm{C}}_{\mathrm{a}\mathrm{i}\mathrm{r}}\cdot {\displaystyle \frac{\mathrm{d}{\mathsf{\theta }}_{\mathrm{a}\mathrm{i}\mathrm{r}}}{\mathrm{d}\mathrm{t}}}\approx 0$$

(3)

• H_ve = Thermal conductance for ventilation/infiltration.
• H_is = Convective heat transfer coefficient between internal surfaces and zone air.
• C_air = Thermal capacitance of the zone air (often negligible or lumped).

Energy Balance for the Central Mass Node (θ_m): This node primarily links radiative heat flows and conductive transfer through the building fabric. It represents the thermal inertia.

$${\mathrm{H}}_{\mathrm{tr},\mathrm{ms}}\cdot \left({\mathsf{\theta }}_{\mathrm{si}}-{\mathsf{\theta }}_{\mathrm{m}}\right)+{\mathrm{H}}_{\mathrm{tr},\mathrm{op}}\cdot \left({\mathsf{\theta }}_{\mathrm{e}}-{\mathsf{\theta }}_{\mathrm{m}}\right)+{\mathsf{\Phi }}_{\mathrm{int},\mathrm{rad}}+{\mathsf{\Phi }}_{\mathrm{s}\mathrm{o}\mathrm{l},\mathrm{r}\mathrm{a}\mathrm{d}}={\mathrm{C}}_{\mathrm{m}}\cdot {\displaystyle \frac{\mathrm{d}{\mathsf{\theta }}_{\mathrm{m}}}{\mathrm{d}\mathrm{t}}}$$

(4)

• H_tr,ms = Radiant/convective conductance between internal surfaces and the thermal mass node.
• H_tr,op = Conductance representing heat transfer through the opaque building fabric to the external environment (simplified path).
• Φ_int,rad = Radiant part of internal heat gains transferred to the mass.
• Φ_sol,rad = Solar gains absorbed by the internal mass/surfaces.
• C_m = Effective thermal capacitance of the zone’s thermal mass connected to this node.

At each time step, the calculation proceeds as follows:

Step 1: Calculate the internal air temperature θ_air,0 assuming no heating or cooling (Q_HC,nd = 0). This is the “free-floating” temperature.

Step 2: Compare θ_air,0 with the heating (θ_H,set) and cooling (θ_C,set) setpoint temperatures for that hour.

• If θ_air,0 < θ_H,set (Heating required): The heating need Q_H,nd is calculated as the heat flux required to raise the internal air temperature from θ_air,0 to θ_H,set. Q_H,nd is negative (heat supplied). Q_C,nd = 0. The final θ_air is set to θ_H,set.
• If θ_air,0 > θ_C,set (Cooling required): The cooling need Q_C,nd is calculated as the heat flux required to lower the internal air temperature from θ_air,0 to θ_C,set. Q_C,nd is positive (heat extracted). Q_H,nd = 0. The final θ_air is set to θ_C,set.
• If θ_H,set ≤ θ_air,0 ≤ θ_C,set (Within deadband): No heating or cooling is needed. Q_H,nd = 0, Q_C,nd = 0. The final θ_air remains θ_air,0.

The actual calculation of Q_H,nd or Q_C,nd is implicitly part of solving the nodal balance equations with the respective setpoint temperature enforced as the value for θ_air. This description provides a conceptual overview using simplified equations. The actual implementation within EN ISO 52016-1 involves detailed procedures for calculating each conductance and heat flow component, defining the specific thermal network model, and numerically solving the resulting system of equations. Refer to the official standard document for the complete and precise methodology [23]. The output of this process was a detailed and reliable estimate of the building’s heat load, providing essential data for the design of efficient HVAC systems. The rigor inherent in combining a precise 3D model with the EN ISO 52016-1 standard [23,29] ensured that the heat load calculations were both accurate and relevant.

The model was created with Audytor OZC 7.0Pro software. Calculations of the heat load followed the EN ISO 13790 [24].

Below we present the calculation methodology according to the monthly quasi-steady state method described in the EN ISO 13790:2008 standard “Energy performance of buildings—Calculation of energy use for heating and cooling” together with the key equations.

EN ISO 13790 has been largely superseded by the EN ISO 52000 family of standards (especially EN ISO 52016-1 [23] for hourly calculations), but its monthly method remains relevant for understanding certain calculation approaches and is still referenced or used in some contexts.

The core principle of the monthly method is an energy balance calculated for each month, determining the energy needed for heating (Q_H,nd) and cooling (Q_C,nd).

Calculation of Total Heat Transfer (Losses) for the Month (Q_ht).

This combines heat transfer through transmission (Q_tr) and ventilation (Q_ve). Heat Transfer Coefficient by Transmission (H_tr): Represents heat loss through the building envelope.

$${\mathrm{H}}_{\left\{\mathrm{t}\mathrm{r}\right\}}={\mathsf{\Sigma }}_{\mathrm{i}}\left({\mathrm{A}}_{\mathrm{i}}\cdot {\mathrm{U}}_{\mathrm{i}}\right)+{\mathsf{\Sigma }}_{\mathrm{k}}\left({\mathrm{l}}_{\mathrm{k}}\cdot {\mathsf{\Psi }}_{\mathrm{k}}\right)+{\mathsf{\Sigma }}_{\mathrm{j}}\left({\mathsf{\chi }}_{\mathrm{j}}\right)$$

(5)

where:

• A_i = Area of building element i (m²)
• U_i = Thermal transmittance of building element i (W/(m²·K))
• l_k = Length of thermal bridge k (m)
• Ψ_k = Linear thermal transmittance of thermal bridge k (W/(m·K))
• χ_j = Point thermal transmittance of thermal bridge j (W/K)

Heat Transfer Coefficient by Ventilation (H_ve): Represents heat loss due to air exchange.

$${\mathrm{H}}_{\left\{\mathrm{v}\mathrm{e}\right\}}={\mathsf{\rho }}_{\mathrm{a}}\cdot {\mathrm{c}}_{\mathrm{a}}\cdot {\mathrm{q}}_{\mathrm{ve}}$$

(6)

where:

• ρ_a = Density of air (≈1.2 kg/m³)
• c_a = Specific heat capacity of air (≈1000 J/(kg·K))
• q_ve = Average ventilation volume flow rate for the month (m³/s)

Total Monthly Heat Transfer (Q_ht): Calculated based on the average monthly temperature difference between the internal setpoint and the external environment.

$${\mathrm{Q}}_{\mathrm{ht}}=\left({\mathrm{H}}_{\mathrm{t}\mathrm{r}}+{\mathrm{H}}_{\mathrm{v}\mathrm{e}}\right)\cdot \left({\mathsf{\theta }}_{\mathrm{i}\mathrm{n}\mathrm{t},\mathrm{s},\mathrm{H}}-{\mathsf{\theta }}_{\mathrm{e},\mathrm{m}\mathrm{o}\mathrm{n}}\right)\cdot \mathsf{\Delta }{\mathrm{t}}_{\mathrm{m}\mathrm{o}\mathrm{n}}$$

(7)

(for heating mode calculation)

$${\mathrm{Q}}_{\mathrm{ht}}=\left({\mathrm{H}}_{\mathrm{t}\mathrm{r}}+{\mathrm{H}}_{\mathrm{v}\mathrm{e}}\right)\cdot \left({\mathsf{\theta }}_{\mathrm{e},\mathrm{m}\mathrm{o}\mathrm{n}}-{\mathsf{\theta }}_{\mathrm{int},\mathrm{s},\mathrm{c}}\right)\cdot \mathsf{\Delta }{\mathrm{t}}_{\mathrm{mon}}$$

(8)

(for cooling mode calculation—using relevant setpoints)

Simplified representation often uses a single Q_ht calculation based on heating setpoint difference, as gains/losses determine the mode. Let’s use the heating mode definition for the energy balance:

$${\mathrm{Q}}_{\mathrm{ht}}=\left({\mathrm{H}}_{\mathrm{t}\mathrm{r}}+{\mathrm{H}}_{\mathrm{v}\mathrm{e}}\right)\cdot {\left[{\mathsf{\theta }}_{\mathrm{i}\mathrm{n}\mathrm{t},\mathrm{s},\mathrm{H}}-{\mathsf{\theta }}_{\mathrm{e},\mathrm{m}\mathrm{o}\mathrm{n}}\right]}^{+}\cdot \mathsf{\Delta }{\mathrm{t}}_{\mathrm{m}\mathrm{o}\mathrm{n}}$$

(9)

where:

• θ_int,s,H = Internal setpoint temperature for heating (°C)
• θ_e,mon = Average monthly external air temperature (°C)
• Δt_mon = Duration of the month (in seconds)
• [x]⁺ means max (x, 0)—heat transfer considered only when the internal setpoint is higher than the external temp. (This is a simplification; the standard calculates based on total difference and uses utilization factors).

A more direct representation aligned with the standard’s balanced approach is just:

$${\mathrm{Q}}_{\mathrm{ht},\mathrm{mon}}=\left({\mathrm{H}}_{\mathrm{t}\mathrm{r}}+{\mathrm{H}}_{\mathrm{v}\mathrm{e}}\right)\cdot \left({\mathsf{\theta }}_{\mathrm{i}\mathrm{n}\mathrm{t},\mathrm{r}\mathrm{e}\mathrm{f}}-{\mathsf{\theta }}_{\mathrm{e},\mathrm{m}\mathrm{o}\mathrm{n}}\right)\cdot \mathsf{\Delta }{\mathrm{t}}_{\mathrm{m}\mathrm{o}\mathrm{n}}$$

(10)

(Using an internal reference temperature θ_int,ref)

The final energy use requires accounting for the efficiencies of the heating and cooling generation, distribution, emission, and control systems (η_H,tot, η_C,tot).

$${\mathrm{Q}}_{\mathrm{H},\mathrm{u}\mathrm{s}\mathrm{e}}={\displaystyle \frac{{\mathrm{Q}}_{\mathrm{H},\mathrm{n}\mathrm{d},\mathrm{a}\mathrm{n}}}{{\mathsf{\eta }}_{\mathrm{H},\mathrm{tot}}}}$$

(11)

$${\mathrm{Q}}_{\mathrm{C},\mathrm{use}}={\displaystyle \frac{{\mathrm{Q}}_{\mathrm{C},\mathrm{n}\mathrm{d},\mathrm{a}\mathrm{n}}}{{\mathsf{\eta }}_{\mathrm{C},\mathrm{t}\mathrm{o}\mathrm{t}}}}$$

(12)

η_C,tot would incorporate the system’s Coefficient of Performance (COP) or Energy Efficiency Ratio (EER)).

This provides a structured overview of the monthly calculation methodology in EN ISO 13790 using key equations. The standard itself contains much more detail on calculating the input parameters (H_tr, H_ve, Q_int, Q_sol, a_H, a_C, system efficiencies, etc.).

To precisely estimate the heat load of the studied building, the use of advanced virtual modeling proved crucial. The application of a digital model enabled a detailed analysis of the building’s energy characteristics, considering its structural specifics, orientation to the cardinal directions, and climatic conditions. The virtual building model was created using Audytor OZC software, widely recognized as a reference tool in energy auditing and the design of heating and ventilation systems [28,29]. Audytor OZC, with its versatility and precision, facilitated the accurate representation of the building’s geometry, including all structural elements such as walls, windows, doors, roofs, and floors. Each of these elements was defined with its thermal properties, such as the heat transfer coefficient (U-value), heat capacity, and solar radiation absorption coefficient. As a result, the virtual model faithfully reflected the building’s actual thermal parameters (Figure 2).

A key element of the calculation process was the application of EN ISO 13790, which is the European standard for calculating the energy demand for heating and cooling buildings. This standard specifies calculation methods that account for both static and dynamic aspects of heat exchange in a building. The heat load calculations were performed using the monthly method, which considers the variability of climatic conditions throughout the year. The calculation process in Audytor OZC, based on EN ISO 13790, comprised several key stages. The first of these was defining the input data, such as the building’s location, climatic conditions (external temperature, solar radiation, humidity), technical parameters of building materials, and internal parameters (internal temperature, humidity, heat gains from occupants and equipment). The software then calculated heat losses through building partitions, ventilation heat losses, and heat gains from solar radiation and internal sources. This process begins with the meticulous input of building geometry and material properties. Users define the building’s dimensions, wall, roof, floor, and window specifications, ensuring that the model accurately reflects the physical structure. Each material is assigned its thermal properties, including thermal conductivity, specific heat capacity, and density, which are critical for calculating heat transfer. Following the geometric and material input, the software requires the definition of boundary conditions. This includes the building’s location, which is crucial for accessing climate data. Audytor OZC utilizes integrated climate databases or allows users to import custom data, ensuring that calculations are based on local weather patterns. The software then processes monthly average temperatures, solar radiation, humidity, and wind speed, all of which influence heat transfer.

The EN ISO 13790 and EN ISO 52016-1 standard [23,24] provide various calculation methods, and Audytor OZC implements the monthly calculation method, which is well-suited for assessing long-term energy performance. This method involves calculating monthly heat losses and gains, considering both static and dynamic thermal behavior.

The source of climate data for the Audytor OZC is a typical meteorological year according to EN ISO 15927-4:2005—ISO [30]. The values are published by the Polish Institute of Meteorology and Water Management [31]. In this case, for the localization of the city of Tarnów.

The calculation of heat losses through building partitions considered the heat transfer coefficient (U-value) for each partition, the temperature difference between the interior and the environment, and the partition’s surface area. Ventilation heat losses were calculated based on the ventilation air flow and temperature difference. Heat gains from solar radiation are considered the building’s orientation, solar radiation, and the solar radiation absorption coefficient of the external surfaces. Heat gains from internal sources are considered the heat generated by occupants and electrical equipment.

• Key stages of the calculation process include:
  • Heat Loss Calculation:

    ✓

    Transmission Losses: These are calculated for each building element (walls, windows, roof, floor) using the U-value (heat transfer coefficient), the surface area of the element, and the temperature difference between the interior and exterior. Audytor OZC accurately calculates U-values by considering the thermal resistance of each layer of the building element.

    ✓

    Ventilation Losses: These are determined by the air exchange rate, the volume of the building, and the temperature difference between the interior and exterior. The software allows for the input of natural and mechanical ventilation rates.

    ✓

    Thermal Bridges: Audytor OZC allows for the input of linear and point thermal bridges. Those are places with an increased heat transfer, which are calculated separately.

  • Heat Gain Calculation:

    ✓

    Solar Gains: These are calculated based on the building’s orientation, window properties (solar heat gain coefficient), and solar radiation data. The software accounts for shading from adjacent buildings or landscape features.

    ✓

    Internal Gains: These include heat generated by occupants, lighting, and appliances. Users can define occupancy schedules and appliance usage patterns to accurately model internal heat gains.

  • Dynamic Thermal Behavior:

    ✓

    Audytor OZC incorporates dynamic thermal behavior by considering the thermal mass of building elements. Thermal mass influences the building’s ability to store and release heat, which affects its temperature stability and energy consumption.

    ✓

    The software calculates the influence of the capacity of the building elements to store heat.

  • Monthly Energy Balance:

    ✓

    The software calculates a monthly energy balance by summing heat losses and subtracting heat gains. This provides a detailed overview of the building’s heating and cooling requirements throughout the year.

    ✓

    The monthly method allows for determining the seasonal energy demand.

  • Results and Reporting:

    ✓

    Audytor OZC generates detailed reports that include monthly heat loads, energy consumption, and other relevant metrics. These reports can be used to assess the building’s energy performance and identify opportunities for improvement.

    ✓

    The software provides graphical representations of the results, facilitating the interpretation of the data.

The main construction parameters of the building used in the simulation are gathered in

Table 1. Main construction parameters of the analyzed building.

Type of Building Element	Construction	Heat Transfer Coefficient U [W/m2K]	Total Partition Area in the Building [m2]
External walls	Cement-lime render (inside), cinder-ash blocks of varied thickness (12.5/25 cm), polystyrene with lambda = 0.040 W/mK (17/19 cm) silicone-silicate render (outside)	0.194/0.199	40.30/214.86
Internal walls	cinder-ash blocks of varied thickness (12.5/25 cm with cement-lime render (inside)	1.553/2.159	71.67/23.15
Floor	Terracotta floor finish, concrete screed, asphalt felt insulation, concrete, gravel	0.482/0.495	67.84/43.76
Ceilings	12 cm reinforced concrete slab, 25 cm rockwool insulation Lambda = 0.039 W/mK, concrete screed.	0.127	121.04
Roof	Sheet metal roofing with wooden truss	6.913	147.79
Windows	PCV window frame, heat transfer coefficient provided by the manufacturer	0.900/1.300	8.68/42.20
Doors	MDF, timber frame with wood core	1.300	3.22

. The provided partitions were used to create a virtual model in Figure 2.

The analyzed building is oriented according to Figure 2, located in an area unobstructed by other buildings. The internal heat gains are calculated based on EN ISO 52016-1, assuming 6.8 W/m² gains for the floor surface area. The building design temperature is 24 °C for bathrooms and 20 °C for other heated rooms. The ventilation is provided by a natural system. The air infiltrates the building through leaks in windows and doors and is discharged through air vents. The ventilation airflow is calculated to provide the minimum air change per hour of 0.5 1/h for the cubic capacity of the rooms, resulting in an airflow of 338.6 m³/h.

The result of the calculations was the monthly heat load, which considered both heat losses and heat gains. These data were crucial for the optimal selection of the heat pump, energy storage, and photovoltaic installation. Precise estimation of the heat load allowed for accurate matching of the heating device and energy storage capacities, contributing to increased energy efficiency and minimized operating costs. Virtual building modeling, combined with ISO 13790 [24], allowed for the acquisition of reliable and precise results, which provide a solid foundation for further analysis and optimization of the heating system. This made it possible to design a system that not only meets thermal comfort requirements but also is characterized by high energy efficiency and minimal environmental impact.

2.2. Transient Heat Pump Model

The transient system model was developed in Ebsilon software. The heat load was correlated to temperature and characterized by a linear function of ambient temperature. However, the required heat load was reduced by the solar gains of the building.

Primarily, the Ebsilon software is used in the simulation of conventional [32] or more alternative approaches [33] to power plants. However, the software has a transient module for time-series calculation of renewable systems [34,35], allowing the development of a time-dependent model of the heating system.

The meteorological data were provided by the PVGIS model [36,37]. The location in Kraków, Poland, was selected using the meteorological data from 2023 to 2024. The time span in the analysis was October 2023 to April 2024, covering a single heating season. The data consisted of solar irradiance, outdoor temperature, and wind speed with an hourly time step. Next, the data were modified to a 10 min time step, interpolating the data with the mean value.

The meteorological parameters were fed to the developed model in Ebsilon software (Figure 3), allowing calculation of the dynamic process.

The model (Figure 3) consists of several connections:

• The main cycle of the heat pump is where the heat pump heat sink is connected to the buffer tank. The mass flow is controlled to achieve a 5 K overheat compared to the tank temperature.
• The second cycle is the central heating system of a building. The heat load is calculated based on results from OZC software for different temperature levels, reduced by solar gains dependent on the solar irradiance value. The temperature difference in the heat distribution system for a specific load is set as 12 K, corresponding to ISO 13790:2008 [24] for a floor heating system. Mass flow in this cycle is controlled to fit the heat load and temperature difference. For the buffer tank calculations, an additional 200 L of water volume is added for a load of water in the floor heating system.
• The heat pump power supply is connected to the PV system and the external power source, representing the power grid. The PV system component is connected to the time series and its effectiveness is based on the global irradiance value and the sun angle calculated with date and time. The PV system component’s current–voltage relationship is based on a parameter model [38]. From this relation, the maximum power point is derived (the pair of voltage and current that produces the maximum power). The model comprises parameters provided by the manufacturer.
• An additional element in the main cycle is the electric heater. The heater is controlled by the main controller, providing the missing heat in case the heat pump’s maximum efficiency drops below its required load.
• The model uses additional connections representing logic values of the heat pump heat load (black lines in Figure 2). The heat pump is controlled based on buffer tank temperature, building demand, and available PV energy. During PV energy production, the heat pump stores the heat in a buffer tank up to a temperature of 50 °C. When the tank is full, the heat pump provides only the heat for the current load. When the PV system production drops, the system uses tank storage up to a temperature of 35 °C. When the temperature drops below that level heat pump provides just the required heat load, maintaining the 35 °C in the buffer tank. Additionally, the algorithm for heat pump control (Figure 3) features additional steering in boundary values. To compensate for storage heat losses, in case the buffer tank temperature exceeds the boundaries, the heat load is slightly modified. This modification allows the heat pump to work continuously without sudden stops in case of reaching the boundaries.
• The controlling component in the model is based on EBSscript, a programming language based on PASCAL syntax. The algorithm for controlling heat pump load is presented in Figure 4.

The heat pump in the system uses specification matrices for the COP value. The effectiveness depends on the temperature of the heat source—in this case, outdoor air temperature and the temperature of the heat sink outlet—water filling the buffer tank. The characteristic is based on a commercially available device (GalmetAirmax² 6 GT, Głubczyce, Poland [39]). The heat pump product fiche gives information on efficiency at different temperatures according to EU regulation No 913/2013 [40]. The fiche has COP values for heat outlet temperatures of 35 and 55 °C and four levels of air temperature: −7, 2, 7, and 12 °C. In the characteristic, the interpolation was made with a bicubic algorithm, and values outside the matrix were extrapolated with the nearest border method. The resulting characteristic is visualized in Figure 5.

Part of the COP characteristic, the heat pump efficiency falls with lower ambient temperature. This dependency is described in the heat pump product fiche [39]. The heat load of a building increases with lower ambient temperature; however, the heat pump power decreases with lower ambient temperature. In the lowest temperatures, an additional electric heater is used to compensate for this effect, acting as a peak heat source.

In calculation, the non-linear equations are set to reach convergence when the error drops below 10⁻⁶. A single calculation step uses 1000–10,000 iterations and requires 500–2000 ms of time. The calculation speed is strongly dependent on the dampening of controllers in the model; however, decreasing the strength of damping can lead to failures in calculations. Considering the calculations are based on a step-by-step method, such failure renders the results incorrect, providing the damping necessary.

The model was calculated in three variants, for different sizes of the buffer tank:

• 200 l—representing no buffer tank, just the load of water in the floor heating system.
• 700 l—representing system load and 500 l buffer tank.
• 1200 l—representing system load and 1000 l buffer tank.

The sizes of the buffer tank variants were chosen according to commercially available tanks for households.

3. Results and Discussion

The model produced results in 10 min time step, in tabulated form for each analyzed date and time. During calculations, the results can be generated from any of the components used in the model (Figure 3). The specification values and their respective descriptions are presented in

Table 2. Parameters used in the simulated heating system model.

Parameter Name	Parameter Type	Description
Ambient temperature	controlling	Ambient air temperature based on meteorological data at a height of 2 m.
Global solar irradiation	controlling	Global solar irradiance in the horizontal plane
Building heat load	Result	Current heat load of building, calculated based on ambient air temperature
Heat pump load	Result	Required load of the heat pump system, calculated based on control algorithm
PV produced power	Result	Power produced by PV system, calculated on Sun. DNI and geometric sun height.
Grid power load	Result	Calculated based on required power of the heat pump, electric heater and PV production balance
Heat pump COP value	Result	COP value for the heat pump, based on ambient air temperature and heat pump heat sink temperature
Buffer tank temperature	Result	Current temperature in the buffer tank at the beginning of calculation period.

.

The model was set to produce results for 24 different parameters. The most relevant parameters are presented in Table 2.

The results were validated with norm heat load calculations. The building heat load for 10 min time step was used to calculate the heating season heat demand. The heat demand in the simulation was 41.29 GJ. This value consists of the load of the building envelope and the ventilation system load reduced by the solar gains of the building. The respective heat load in norm calculations equals 45.60 GJ. The discrepancy between the values is 9.5%.

Similar verification was performed on meteorological data to check the discrepancy with the monthly mean method used in OZC software. That analysis was possible with the use of the degree-day parameter. It is defined as the difference between the designed building temperature, in this case, 20 °C, and the daily mean temperature. The OZC software calculates the mean monthly temperature and number of heating days per month. That value is used to calculate degree days based on the equation:

$$S_d=\sum (20-T_{mm})·n_h$$

(13)

where:

$T_{mm}=monthlymeantemperature$

$n_h=amountofmontlhyheatingdays$

The values were published by the local weather station. In the case of historical data, the degree days were calculated based on the daily mean temperature. By definition, the degree days are counted only when the mean temperature is lower than 15 °C. In case of higher temperature potential sun and domestic heat gains nullify the need for heating. The meteorological data is based on the equation:

$$S_d=\sum _{T_{dm}>0}(20-T_{dm})$$

(14)

where:

$T_{dm}=dailymeantemperature$

The results of the degree days calculation are compared in

Table 3. Comparison of degree-days calculation for meteorological data and monthly mean value.

Degree-Days Calculation Method	Monthly Mean—Audytor OZC Software	Daily Mean—Meteorological Data
	3441 °C days	3431 °C days

. The difference between the values is lower than 0.3%, which allows one to conclude that the input values for both methods are comparable without leading to additional error.

The three variants were summarized and had their seasonal energy balance calculated. The seasonal heat demand was the same for each variant, just as the energy produced by the PV system. There is a small mismatch in the case of heat produced by the heat pump—the value varies in the range of 1.1% (

Table 4. Results are constant for each buffer tank variant.

Building Heat Demand	Heat Provided by the Heat Pump	Power Produced by the PV System
41.29 GJ	41.64 ± 0.44 GJ	2857 kWh

). The cause of this is a different buffer tank temperature level at the end calculation period. Storing more heat at the end of the calculation period causes the produced heat value to be disrupted.

The values that varied for the variant are shown in

Table 5. Seasonal results of energy balance for all analyzed variants.

Buffer Tank Variant	Grid Power Consumption	Power Fed to Grid	PV Power Used for Heat Pump	Heat Pump Total Power Use	Electric Heater Power Use	Mean COP	Power Self Consumption	Share of PV Power Used in Heat Pump
	[kWh]	[kWh]	[kWh]	[kWh]	[kWh]	[-]	[-]	[-]
No buffer	2540	1907	950	3376	114	3.69	0.333	0.272
500 L	2187	1534	1323	3397	112	3.63	0.463	0.378
1000 L	1906	1265	1592	3385	112	3.62	0.557	0.455

. The values were calculated as fixed heat or power production for each of the 10 min time-steps analyzed in the study. Total heat pump power use was calculated as the sum of grid power consumption and PV power used in the heat pump.

The results show that increasing the size of the buffer decreases the consumption of power from the grid, simultaneously increasing the usage of PV. Adding a 500 L tank decreased the grid power consumption by 14%, and the 1000 L size decreased the consumption by 25%.

The mean COP value shows little difference between the buffer tank sizes. Adding the buffer tank causes the COP to drop due to the heat pump working with a higher heat sink temperature. However, during those moments, the heat pump is using power provided by the PV system, not causing an increase in energy costs.

The power self-consumption is based on the proportion of PV energy used to power the heat pump and the total production. For the basing variant, the self-consumption was at the level of 33%. That level is higher than annual values found in the literature [41], due to the study considering only the heating season portion of the year, between October and April—7 months. During that period, the heat loads led to high daily power consumption and considerably low PV output—increasing self-consumption in comparison with a whole year of study. The variant that used a 500 L buffer tank had a self-consumption of 46%, and the 1000 L sized buffer self-consumption reached 56%.

The last calculated parameter was the share of PV in heat pump power, defined as the ratio of PV power used in the heat pump to total power demand. The value reached 27% for the basic variant, 38% for 500 l, and 46% for 1000 L.

Adding the buffer tanks proves to be an efficient solution for higher PV usage and could therefore lead to savings on energy costs.

The results can also be featured in the form of a graph. By sorting the power flow, for each time step, results in the chart with total energy use as the area under each curve (Figure 6).

Analysis of the graph can show the change in heat pump behavior. Most visible is the middle part of the graph, where for the largest buffer tank, the zero-power level has the largest plateau. This is caused by an increase in PV energy use. When PV energy is completely used to power the heat pump, the grid balance remains at zero; also, larger storage capabilities lead to prolonged discharge time when the heat pump remains turned off. The highest power levels, exceeding 1 kW, remain similar for all variants, the highest heat load is correlated with the lowest temperature and low irradiance. In such conditions, the buffer tank becomes obsolete. A sharp rise in power demand is indistinguishable in all variants and corresponds to the use of an additional heater. With decreasing power balance, between 1 and 0 kW, increasing the buffer size causes a quicker “drop” to zero. The heat pump works in periods of high solar irradiance using the PV power, rather than keeping the partial load. The negative values show when the PV power is fed to the grid. Increasing the buffer tank causes a smaller amount of energy to be sold to the grid.

Transient simulation modeling is widely used in the research of heat pump systems. Such an approach is used for dynamic modeling of heat pump cycle calculations [42]. For seasonal calculations, the results can be produced for evaluation of hybrid heating systems [43,44] or the effect of exploitation on ground heat sources [45]

The studies found in the available literature prove that this trend results in similar conditions. Research on the effect of buffer tanks in the optimization of heating systems [46] has proven that increasing the size of the buffer leads to a decrease in power consumption (up to 12%). However, this study only analyzed the effect of the buffer tank on heating system optimization without the storage effect of PV energy. The advantage, however, is the verification of results based on 6 test projects. A study on a buffer tank and control system of a heat pump [47] proved that a larger storage tank decreases energy costs. The study, however, does not use a PV system; the profits are gained by using the heat pump during higher ambient temperatures.

The results provide both the total seasonal values that allow comparison of effectiveness during the analyzed period, and a charted analysis of the behavior of the device, leading to conclusions on the use of the proposed heat pump simulation method.

4. Conclusions

The transient calculation of a household heating system is a topic of many scientific works. However, in this study, the normative method for the calculation of house heat load was linked with thermodynamic simulations with time-series calculation in Ebsilon software. The method provides new possibilities for calculation with a more complicated and extended thermodynamic model. The transient calculation based on Ebsilon is not a widely used approach. However, it allows more complicated cycle design, including dynamic simulation of the heat pump refrigerant cycle. This approach can be used in further studies to evaluate the transient evaluation of different refrigerants and cycle designs. One such possibility is the simulation of a complete heat pump cycle connected with a transient heat load.

The model was verified by comparison of the calculated provided heat with the normative heat load. The discrepancy between norm calculations and modeling was lower than 10%, proving the method can be used for such calculations.

The results of heat pump power for different-sized buffer tanks prove that heat storage can efficiently increase the self-consumption of PV power. The addition of a 1000 L buffer tank caused the self-consumption share in a heating season to increase by 63%.

The provided methodology offers a robust foundation that can be further utilized in more advanced calculations and simulations, extending beyond the analysis of the complete heat pump cycle itself. Its capability allows for modeling the complete household heating system, moving beyond treating the building as a universal heat receiver. Instead, it enables simulation as a dynamic system possessing heat capacity and inertia, critically dependent on the specific design of the heating distribution system (e.g., radiators, underfloor heating) and the building’s thermal loops and overall structure. Engineers and HVAC designers can use this approach to more accurately size components (heat pump, storage, emitters) and predict the real-world dynamic performance of integrated systems within specific building types, such as those typical in Poland. It allows for comparing the impact of different emitter types (e.g., low-inertia radiators vs. high-inertia underfloor heating) on heat pump efficiency, cycling, and overall comfort. Researchers and energy consultants can incorporate these detailed heating system models into whole-building energy simulations. This leads to more accurate predictions of energy consumption, thermal comfort, and the effectiveness of energy efficiency measures, as the model accounts for the thermal inertia of both the building and the heating system itself. The methodology’s control aspect, implemented via EBS-script with its Pascal-based syntax, provides a flexible environment for creating and evaluating sophisticated control strategies. This is crucial for:

• Optimizing Performance: Developing controllers that adapt to variable electricity prices, grid signals (demand response), or maximize self-consumption from local PV generation.
• Improving Comfort: Designing predictive controls that account for building inertia to minimize temperature fluctuations and anticipate heating needs.
• Research and Prototyping: Testing novel control concepts (e.g., Model Predictive Control, AI-based algorithms) in a simulated environment before real-world implementation.

Detailed models can potentially be used to establish baseline performance expectations, aiding in the virtual commissioning of control strategies or the development of algorithms for fault detection and diagnostics in real systems. The platform can serve as a powerful tool for teaching students and professionals about the complex interactions within integrated building energy systems and the development of advanced control logic. In summary, the methodology provides a versatile framework not only for analyzing the core heat pump operation but also for the holistic simulation, optimization, and control design of complete residential heating systems interacting dynamically with the building structure.

In conclusion, the study provided immediate results for the buffer tank and a methodology that can be further used in differently targeted applications. The methodology incorporating transient simulation of the thermodynamic cycle in Ebsilon is an innovative tool that can be used for dynamic modeling of the heat pump cycle in the heating system.