## 1 Introduction
The heating and cooling demands throughout the entire year were met by a dual-purpose biomass-driven heat pump, occasionally supplemented by a biomass boiler. The study examined the energy, economic, and environmental benefits of the proposed design. [PERSON] [15] proposed an innovative economic-based control logic for a parallel loop hybrid system consisting of an air-to-water heat pump (AWHP) and a gas-fired water heater. This control logic allows the optimal working point of the system (including the mass flow rate and supply water set-point temperature for each device) to be determined. By using this approach, significant economic benefits of up to 60% can be achieved compared to traditional solutions. Another critical factor for maximizing the energy savings attainable with the adoption of HHP systems is the sizing of the heating system components (such as the boiler, heat pump, and thermal storage) and its impact on the control system algorithm determining when to activate or deactivate the heaters. [PERSON] et al. [16] aimed to assess the applicability of a compact hybrid heat pump (HHP) as a retrofit solution for an Irish bungalow. The system utilized in the study was a commercially available HHP comprising an 8 kW air-source heat pump (ASHP) and a 33 kW gas boiler. They found that hybrid systems achieve greater primary energy savings compared to all-electric heat pumps because hybrid systems can adjust delivery temperatures flexibly, whereas all-electric heat pump systems typically operate with a fixed delivery temperature. [PERSON] et al. [17] conducted an analysis using dynamic simulations to investigate how the control system of heaters in a hybrid system (comprising a heat pump and gas boiler) influences the seasonal efficiency of the system. They utilized numerical results to determine the optimal values of control settings for HHP systems. [PERSON] et al. [18] explored the implications of hybrid heating on the residential heating sector, comparing it to both traditional technology (gas boilers) and advanced low-carbon technology (heat pumps). They discovered that a compact hybrid system featuring a 2.5 kW heat pump could potentially achieve a 50% reduction in energy demand at the national stock level. [PERSON] et al. [19] investigated the optimal sizing of the main elements in a hybrid system, including PV panels, air-source heat pump, electric boiler, and thermal storage. They considered factors such as electricity price, space heating load profiles, domestic hot water demand, and the size of the photovoltaic field. Their results revealed that the thermal load profile has the most significant influence on system sizing. Furthermore, [PERSON] et al. [20] compared the energy performance of monovalent and bivalent heating systems, examining various types of heat generators (such as oil and gas boilers, micro CHP, and heat pumps) along with backup units (including electric heaters and gas boilers). Their findings indicated that electrical backup systems are not a financially advantageous solution. In research conducted by [PERSON] et al. [21], multiple buildings in Great Britain, each fitted with a hybrid heating system comprising a gas boiler and a heat pump, were examined. The study investigated the performance of these systems when operated based on electricity price variations. The results indicated that running the heat pump primarily during periods of low electricity prices and the gas boiler during peak electricity price times can lead to a 5-10% increase in the coefficient of performance (COP). [PERSON] et al. [22] analyzed the cost-optimal sizing and hourly control strategy of a hybrid heat pump system for heating applications. Thermal storage is employed to reduce both operating costs and primary energy consumption by scheduling the operation of the heat pump during periods when energy costs are lower, thereby optimizing profitability. The findings indicated an energy cost reduction of up to 8% compared to a baseline scenario lacking storage capacity.