## 1 Introduction  
It is now well established that the building sector significantly contributes to the primary energy demand and greenhouse gas (GHG) emissions in developed countries [1]. With the operation of buildings accounting for 28% of global CO2 emissions [2], alternative low-carbon technologies are necessary to meet buildings' heating and cooling needs. Reducing buildings' energy consumption and utilizing renewable energies are essential steps for decreasing greenhouse gas emissions. To enhance the efficiency of building heating and cooling systems, various solutions can be implemented [3, 4, 5]. The use of heat pumps is certainly a crucial measure to achieve energy savings. Heat pump designs, commonly used for space cooling or heating, efficiently transfer heat from the surrounding air (or other renewable sources) to the water, rather than using energy primarily to heat water directly. Consequently, integrating a heat pump design with a backup heater can significantly increase overall efficiency, depending on the choice of heat pump technologies. Hybrid heat pumps (HHPs) combine an electrically driven heat pump, typically an air-to-water system, with a gas condensing boiler within a single unit. Under suitable operating conditions, HHPs have been shown to achieve higher seasonal performance compared to electric-only heat pumps (EHPs). This improved performance results from the ability to operate in gas boiler (GB) mode under conditions where the EHP efficiency is lower [6]. For instance, during the coldest weather periods, when heating loads are higher and EHP performance declines, HHPs can switch to GB mode, which may be more efficient under these conditions. Moreover, since a hybrid heat pump includes a gas boiler that can be used to meet peaks in thermal demand, it allows for a lower rating of the EHP, leading to smaller EHP equipment and reduced power demand [7]. As noted by [PERSON] [8], hybrid heating systems incorporate multiple heat sources, each fulfilling energy needs under varying time periods or operational conditions. The choice of heat sources must consider local climate and policy factors. Traditionally, boilers are preferred by investors among conventional heat sources. Importantly, there are numerous instances where a combination of electric heat pumps (HPs) and gas boiler systems is implemented on a larger scale, such as in district or industrial settings [9]. Hybrid heat pumps (HHPs) can alternate between using biomass (BM), electricity, or both (HP and boiler) to fulfill heating requirements, depending on various factors affecting system efficiency. These factors include outdoor temperature, flow temperature, and the cost of gas and electricity at any given time [10] which can make it more or less favorable to use these systems and to control them according to different strategies. A study by [PERSON] et al. [11] found that over-sizing heat pumps is a critical design issue that significantly impacts the energy performance of heat pump systems, affecting both seasonal and annual energy performance. [PERSON] et al. [12] conducted an economic analysis of a hybrid heat pump system, comparing it with conventional gas-fired water heaters for residential houses in Korea. The study found that operating the hybrid system could save approximately 4% in annual energy costs. Typically, the performance of HHP systems is strongly influenced by how the hybrid system is integrated, controlled, and operated. [PERSON] and [PERSON] [13] noted that electricity and natural gas prices play a significant role in HHP performance, as these prices can vary by region. The authors found that HHP systems are more effective in residential buildings when the system is carefully integrated to optimize both gas and electricity usage. Therefore, an effective operation strategy is crucial for a hybrid heating system. [PERSON] et al. [14] investigated a configuration for both heating and cooling in buildings, utilizing a biomass-powered absorption heat pump operated with a LiBr/H2O working substance.