# Optimization of Heat Pump Systems in Buildings by Minimizing Costs and CO\({}_{2}\) Emissions
Large-scale electric HPs have been found to significantly reduce CO2 emissions compared with traditional fossil fuel sources such as coal or oil [4; 5]. Additionally, the use of HPs instead of traditional heating systems can help to reduce the burning of waste fuels, including biofuels. Although biofuels were initially considered to be a neutral or low-carbon fuel source, recent studies [6; 7; 8] have shown that the production and use of biofuels in terms of overall carbon footprint is less favorable than previously thought. Thus, HPs can have a significant impact on reducing the overall carbon footprint of buildings. Integration of HPs in buildings can have a significant impact on the overall energy consumption of the building and associated CO2 emissions. As such, there is a growing interest in optimizing the selection of HPs in order to reduce their environmental impact, lower heating costs, and improve heating system efficiency. In practice, these optimization problems can be approached through various mathematical optimization methods, including linear programming (LP), mixed-integer linear programming (MILP), genetic algorithms (GA), nonlinear optimization, and many others. These optimization methods can provide a systematic approach for decision makers to select the most suitable HP for a specific building and heating system, considering factors such as energy efficiency, initial costs, operating costs, and emissions. The use of LP in heat pump design is aimed at finding the optimal solution that maximizes the performance and efficiency of the system while minimizing costs. [PERSON] et al. [9] utilized LP to minimize annual electricity consumption while determining optimum capacities for HPs using different heat sources. In this paper, the analyzed DH system consisted of HPs, an electric peak load boiler, and short-term storage. [PERSON] et al. [10] proposed an LP model to define a cost-optimized 24-h schedule for HP operation in a residential building such that electricity demand and price would be at their lowest. [PERSON] et al. [11] investigated the integration effects of an HP on large-scale thermal energy storage in district heating systems. Another commonly used approach for designing an optimal heat pump system is MILP, which seamlessly integrates multiple variables and constraints into a mathematical optimization model. In the context of heating system design, MILP is used to optimize the performance, efficiency, and cost of the system by considering variables such as operating modes, equipment selection, and control strategies [12; 13; 14; 15; 16]. [PERSON] et al. [17] used MILP for an optimization framework to find the economically optimal design for air-to-water HP systems in single-family houses. Their optimization framework considers both the design and operation of all components, taking into account system dynamics and interdependencies among them over a one-year period. The results of the optimization model include the optimal number and capacity of HPs and the use of thermal energy storage systems for domestic hot water (DHW) and space heating (SH). However, designing an optimal heating system often requires accounting for multiple objectives, which is why multi-objective optimization techniques are frequently utilized in relevant research [18; 19; 20; 21; 22]. The study by [PERSON] et al. [23] proposed a multi-objective optimization approach based on MILP to determine the most effective level of decentralization for reducing CO2 emissions and achieving high levels of self-sustainability. The model aims to minimize both total annual costs and operational carbon emissions through simulation using the epsilon-constraint method. The analysis involves 225 unique mixed-use buildings, including single-family homes, multi-family homes, offices, a grocery store, and a sports facility located in the Swiss Alps. [PERSON] et al.