## 2 Methodology  
### Layout of Simulated Systems  
Various configurations of heating systems, using air-to-water heat pumps, biomass boilers, and integrating both systems, were modeled to assess and contrast their impact on energy, environmental, and economic performance. Figure 1 illustrates the layout of the simulated systems. All components depicted in Figure 1 were provided within the TRNSYS libraries and custom dynamic models developed by the authors, as detailed in [30]. Figure 1: Cont. Figure 1: _Cont._  
Figure 1a depicts the monovalent systems, which serve as baselines for the energy performance analysis conducted in this paper. In these configurations, the heating system relies solely on a single heat generator, namely an air-to-water heat pump. In this scenario, the generator is sized to meet the building's heating peak load. Consequently, the energy demand of the building is fulfilled throughout the entire winter season. Another option is using a biomass boiler for the heating system in winter, which is illustrated in Figure 1b. In this configuration, a heat pump is used just for the cooling demand. To consider the hybrid system concept, which involves the parallel or series operation of heat generators, four configurations were introduced. These configurations involve coupling an air-to-water heat pump with a backup device, specifically a biomass boiler. In these cases, the heat pump size is reduced because a portion of the heating load is covered by the backup system. In configurations 1 and 2, the heat generators are linked in series to enhance the energy performance of the heat pump by utilizing a lower sink temperature. Two bypass valves are employed: a cut-off bypass valve ensures that in cut-off situations, the flow rate of the heat pump is reduced to zero. Additionally, the boiler cut-off bypass valve (configuration 2) splits the water flow between the heat pump and the boiler based on the opening of the tempering valve. In this valve, the flow rate to the boiler is regulated according to the difference between the set point and the output temperature of the heat pump, ensuring that the temperature of the tank inlet remains at 55 \({}^{\circ}\)C. Configurations 3 and 4, on the other hand, are defined as independent systems, where each system has its own separate hot water cycle. In both configurations, the mass flow rates on the supply side (boiler and heat pump) are constant, and the outlet temperature of the heat pump is monitored to control the heat pump and boiler. If the temperature falls below the set point, the boiler is activated to support the heat pump, raising the outlet temperature. Once the outlet temperature reaches a nominal difference of 5 \({}^{\circ}\)C, the controller sends a stop signal to turn off the boiler. Configuration 4 uses two valves to mix the flow rate of the two sides (boiler and heat pump). A tempering valve is used to ensure that the supply temperature to the home is always maintained at 55 \({}^{\circ}\)C. Depending on the difference between the outlet temperature of the heat pump's tank and the set point, the tempering valve regulates the flow rate through the boiler tank. Additionally, the water flow rate serving the heating demand has been determined by maintaining a nominal temperature difference of 5 \({}^{\circ}\)C.  
The heat pump utilized in the simulations is an inverter-driven unit based on a compression cycle. Its model employs a performance-map approach, detailed in [29]. The  
Figure 1: Layouts of the simulated heating systems: (**a**) configuration 0-HP, (**b**) configuration 0-Boiler, (**c**) configuration 1, (**d**) configuration 2, (**e**) configuration 3, (**f**) configuration 4.  
heating capacity and Coefficient of Performance (COP) of the heat pump are determined using a three-dimensional lookup table. This table is a function of the inverter frequency (\(\Phi\)), as well as the sink and source temperatures (i.e., supply water temperature, T\({}_{\text{wo}}\), and external air temperature, T\({}_{\text{ext}}\)).