Optimization of Renewable-Based Multi-Energy Systems in Residential Building Design

Abstract

Electrification is a key priority of the European Union, focusing on saving energy resources and mitigating carbon emissions through enhancing restrictions on relative policies and initiatives. For such goals to be achieved, investing in renewable energy technologies on large- and small-scale projects is promoted. These efforts were implemented in the building sector too, highlighting the importance of optimal decisions in improving the energy performance of buildings, from an economic, energy and environmental perspective. In this context, this paper aims to elaborate a decision-making methodology for building thermal design, considering the optimal selection and operation of multi-energy systems focused on renewable technologies. Solar thermal collectors, photovoltaic systems and heat pumps were included in an Energy Hub for meeting the heating, cooling and domestic hot water energy demand. Optimal decisions were achieved by formulating Mathematical Programming models in GAMS, for minimizing economic, energy and environmental parameters of the systems under a life cycle perspective. The proposed methodology was implemented in a residential building case study. Results show that combining heat pumps with photovoltaics is preferable for all of the examined criteria, while a sensitivity analysis of the economic, energy and environmental parameters, influencing the energy mixture, leads to optimal solutions with the participation of different energy systems.

1. Introduction

Social development and rising economies worldwide are considered as key motivational aspects for governments to ensure independence on covering energy requirements. Additional concerns, like the increase in energy demand, rising oil prices and fossil fuel depletion, highlight the need for energy security under the perspective of affordable and sustainable energy pricing and access. This is indicative for the electricity supplies and other energy-intensive uses, as the electricity sector seems to present a rising growth in global energy consumption, increasing its share of penetration from 15% in 2000 to around 19% in 2017 [1]. In 2022, in the European Union (EU), the share of electricity in total energy consumption was 23%, second to the rating of all energy products. All above are in line with the energy goals of mitigating greenhouse gas (GHG) emissions and exploiting technologies utilizing renewable energy sources (RESs). This is characteristic in EU 2022, as renewable energy contributes to the highest share (43%) in energy consumption [2].

Among all energy consumers (industry, transport, agriculture, etc.), the building sector represents a significant portion of nearly one-third of global energy use, and a great contribution to GHG emissions. In fact, building operations account for approximately 55% of global electricity demand, although the goals of the Paris Agreement propose a decrease by 45% by 2030 of the building energy usage, which necessitates an annual rate that is five times greater than the progress made in recent years [3,4]. Furthermore, the building stock in the EU is responsible for about 40% of total energy consumption [5,6], and accounts for about 36% of total GHG emissions too [7]. This is due to estimations mentioning that almost 75% of existing buildings are considered as energy inefficient because of their high energy consumption for heating, cooling, lighting, etc. [8]. This value was higher (97% of EU buildings) when the decarbonization goals of 2050 were considered [9], even if the contribution of the residential sector towards emission reduction targets, with a 34% reduction between 2005 and 2022, was achieved [10]. Also, the current rate of deep energy renovation for achieving primary energy savings (beyond 60%) and high energy performance in buildings stands at 0.2%, which is far away from the goals of achieving 55% GHG emission reduction (compared to 1990) by 2030 and carbon neutrality by 2050 [11]. Also, deep building energy renovations, which account for 70% of the total, should increase to a rate of 3% [12].

The EU has developed a comprehensive series of initiatives and directives, considering the improvement of building energy efficiency and decarbonization, progressing toward climate neutrality by 2050. In this context, the Clean Energy Package was introduced in 2018, focusing on accelerating the energy transition in Europe by improving building energy efficiency and integrating RES, as well as setting requirements for new constructions to meet nearly zero-energy standards [13]. This package includes relevant directives and regulations, such as the Energy Efficiency Directive (EED), mentioning the reduction of energy demand through energy savings targets, and the Energy Performance of Buildings Directive (EPBD), focusing on improving building energy efficiency through retrofitting and energy renovations, while outlining a comprehensive vision of the complete electrification of residential buildings’ energy uses, as depicted in its recent recast [14]. Also, the Renewable Energy Directive (RED) promotes the integration of RES, and the Governance Regulation (GOV) establishes frameworks for national energy and climate plans (NECPs) to monitor progress toward EU targets (especially for 2030).

In the same concept, the COVID-Related Recovery Plan for Europe provided funds for sustainable recovery. This plan includes several programs focusing on building energy renovation through initiatives, such as the EU Green Deal, the Renovation Wave and the “Fit for 55” package, setting strategic priorities for the 2019–2024 period [13]. More specifically, the EU Green Deal provides an extensive roadmap for GHG neutrality by 2050 considering several sectors (economy, environment, energy, mobility, agriculture, etc.), aiming to conserve, while also enhancing, the EU natural capital, and protecting citizens from environment-related risks and impacts too. It is recognized that the building sector is important for the decarbonization effort, setting goals such as doubling energy renovation rates, mitigating energy poverty, and generating green jobs [15]. This led to the “Fit for 55” package being released in 2021, which includes revisions of the EPBD, EED and RED, essentially aligning them with the targets of reducing GHG emission by 55% by 2030. The main provisions included in this package consider the transition from nearly zero-energy buildings (NZEBs) to zero-emission buildings (ZEBs), the introduction of Building Renovation Passports, the definition of lifecycle GHG emissions and the identification of Minimum Energy Performance Standards (MEPSs) for significant building retrofitting. In addition, the Renovation Wave initiative, which was launched alongside “Fit for 55”, was set under the perspective of the Green Deal [16]. This initiative focuses on the decarbonization of the building sector, emphasizing heating and cooling systems for public and residential buildings and addressing energy poverty too. It also provided financial and technical assistance for strengthening the incentives for large-scale renovations. The REPowerEU Plan, introduced in 2022, enhanced the beforementioned goals by reducing dependence on fossil fuels (including natural gas) and scaling up the deployment of RES and electrification, considering the energy crisis caused by geopolitical tensions [17].

Considering the energy problem in the building sector, as well as the goals and measures set into several initiatives, the necessity for implementing advanced energy-management strategies and technologies is highlighted. Interventions in building envelopes via the implementation of different energy-saving measures (ESMs), such as enhancing thermal insulation, as well as the decision-making for the energy systems (energy supply systems, ESSs) to meet energy demand (heating, cooling, air conditioning and electricity) are considered crucial parameters for improving building energy efficiency. The decisions made for ESMs and ESSs are essential for the development of efficient energy-management strategies, influencing building sustainability and performance throughout its entire lifecycle, especially when decisions were made in the early design stages. In this context, considerable efforts have been devoted to improving building energy efficiency through different strategies by examining several factors, such as indoor and outdoor climate, building fabric, energy systems, the behaviour of the occupants, etc., under the perspective of economic, energy, environmental, sustainable, technical, social and many other aspects. Such examples include the integration of renewable-based technologies into buildings [18], the use of high-performance energy systems and devices [19], the development of innovative design methods [20], like multi-energy systems and Energy Hubs (EHs) [21,22], the creation of appropriate mathematical models describing aspects from building operation [23], the assurance of indoor thermal comfort through energy-efficient solutions [24], the improvement of thermal insulation in building envelopes [25], and more.

All of the above formulate a multi-parameter decision-making problem for building design, where different alternatives should be examined under an optimization process [26]. Such processes can be described as an attempt to find the optimal values for a set of variables (decision variables), while satisfying various constraints, under the optimization of specific objectives. Different optimization methods can be combined with several dynamic simulation tools such as EnergyPlus 24.2.0 and TRNSYS^® 18.05.0001, in order to manage complex design spaces and identify optimal solutions [27,28,29]. The development of multi-energy systems, combined with the concept of EHs, provides a solution for implementing optimization methodologies in building thermal design. EHs set the coupling of different energy carriers, under the perspective of importing, producing, converting and storing different types of energy, while its respective optimization process proposes optimal decisions considering the selection of different energy systems.

The growing environmental and energy concerns mentioned above highlight the need for adopting the concept of multi-energy systems, combining multiple energy sources (conventional and renewable) with different energy uses. This is indicative in the literature, where the deployment of integrated electricity and heating systems seems to be a strategic response. For example, in [30], a novel method for providing an optimal time-scaling matching for integrated electricity and heating systems was developed, while in [31], the integration of gas-grid models into simulation tools for capturing their operation was depicted. Furthermore, the concept of multi-energy systems and their coupling with the EHs in optimization problems for the building sector is commonly used, especially under modelling, operation and planning issues. In [32], a literature review on the status of multi-energy systems, together with evaluation methods for the penetration of RESs in the building sector is presented. Moreover, ref. [33] modelled a multi-energy system to meet the thermal and electrical needs of 40 residential buildings, utilizing solar and wind energy, as well as a cogeneration unit. Furthermore, ref. [21] developed an EH with multi-energy systems for optimizing the design of a residential building. The systems examined were photovoltaic systems, a gas condensing boiler, heat pumps, photovoltaic thermal collectors and electric chillers, with the aim of minimizing investment costs and the use of non-renewable sources. A novel strategy for residential and commercial EH optimization was developed in [34] by integrating specific demand response programs and time-of-use tariffs for achieving cost minimization and improvement in resource scheduling considering uncertain conditions. Similarly, ref. [35] proposed a multi-objective optimization control strategy for EHs within multi-energy systems. They aimed at distributing thermal and electrical loads in an efficient manner, opting for balanced solutions between economic and environmental criteria. Also, ref. [36] proposes an energy management strategy for the operation of integrated electricity and heating systems in intelligent buildings (the interconnection of photovoltaic and heat pumps), for meeting conventional electrical and heating loads of buildings. The objective was the minimization of the overall system cost, considering alternative scheduling scenarios for energy supply and demand.

The literature mentioned above highlights the need to develop reliable and flexible decision-making methodologies for improving the process of building thermal design and enhancing their efficiency. Also, the concept of connecting EHs with multi-energy systems under optimization processes is used in many studies, where the use of integrated electricity and heating systems is highlighted, especially for defining management strategies for their operation. Many studies focus on obtaining cost-effective operation schemes, considering the energy production side under optimal multi-energy coordination [37,38]. Net-metering is a simpler approach for energy management, utilized in this study. More specifically, net-metering is a regulatory financial mechanism designed to promote the adoption of photovoltaic systems by enabling a bidirectional flow of electricity between the grid and consumers, under energy management. The surplus electricity produced by photovoltaics is fed back into the utility grid, while, when such production is limited, the electricity demand can be met by the grid. The net-metering strategy is based on balancing electricity production and consumption, reducing the dependency on the grid.

Although most of such studies can integrate dynamic simulations, their goal is limited to one criterion, like the minimization of the overall economic costs. However, the proposed methodology implements a decision-making process, considering several criteria, giving a life cycle perspective too. More specifically, this study is focused on renewable-based multi-energy systems, following the EU goal of increasing the use of RESs. In this context, heat pumps, combined with photovoltaic systems or the electricity grid and solar thermal collectors, were examined, giving a life cycle perspective on their construction and operation, which is an added value for such studies. In addition to this, the present analysis incorporates a novel model utilizing the f-chart method for sizing the solar thermal collectors. The main scope of this study is to provide optimal decisions for the sizing and operation of systems, in order to meet the thermal energy demand of a building (space heating and cooling and domestic hot water—DHW). The objectives of the optimization include multiple conflicting criteria, considering economic, energy and environmental parameters under a life cycle perspective (embodied energy and GHG emissions), bolstering the innovativeness of this study. The optimization process was based on the development of Mathematical Programming (MP) models in the General Algebraic Modelling System (GAMS), and the SCIP solver was used for providing the optimal solutions. The proposed methodology was implemented to a case study multi-storey residential building. A sensitivity analysis was conducted considering economic and environmental parameters, as well as different primary energy factors, depending on alternative energy mixtures constituting the electricity grid, performing an innovative investigation. Finally, the optimal decisions resulting from the GAMS were compared with the ones provided by a brute force analysis.

2. Materials and Methods

2.1. Framework of the Decision-Making Process

The overall framework of the proposed decision-making methodology considers several essential steps, as depicted in Figure 1, in order for the optimal solutions to be provided, upon the concept of improving economic, energy and environmental aspects of building energy systems. The first key parameters include identifying the geometric and thermal properties of the building envelope. The utilization of an energy simulation tool, considering the beforementioned characteristics of the building and the climatic conditions, are essential for calculating the energy demand for heating, cooling, DHW, etc. Afterwards, the architecture of the EH should be formulated, including the key parameters characterizing the thermal behaviour of the building envelope, as well as the basic aspects of the multi-energy systems involved. The concept of the EH incorporates an optimization methodology, considering alternative criteria for minimizing economic, energy and environmental aspects. The arising optimization problem was approached by developing MP models in GAMS, in order for the optimal decisions to be defined.

2.2. Building the Case Study

The case study building is a 4-storey residential building, with a total floor plan surface area of about 390 m² (Figure 2). The basic geometric characteristics are presented in

Table 1. Geometric parameters of the case study building envelope.

Geometric Parameters	Values
Height (m)	12
Floor area (m2)	390.5
Volume (m3)	1171.5
Façade surface (m2)	667.5
Window surface (m2)	41

. The building was constructed in Thessaloniki, Greece, which is considered to belong to climatic zone C, according to the Greek EPBD for climatic zone C [39]. The U-values of the basic building envelope components are presented in

Table 2. U-values of the basic building envelope components.

Components	U-Values (W/m2K)
External wall	0.6
Floor	0.85
Roof	0.55
Windows	2.4

, leading to a well-insulated building (Um = 0.74 W/m²K).

Considering the beforementioned values for the building examined in this study, an energy simulation tool, called TEE KENAK, was used for calculating heating/cooling and DHW energy demand on a monthly basis, excluding energy systems. This tool follows the methodology provided by Greek EPBD, considering simulation data and parameters as set in [39]. Figure 3 presents the results of the simulation, highlighting that cooling demand is higher than the heating demand. This is due to the climatic conditions and the building thermal properties considered, as well as due to the assumption of no shading during the year. Also, defining the power demand for sizing the energy systems for each energy use is an important factor in building energy analysis. For this reason, the worst-case scenario for the climatic conditions was considered, calculating 25.1 kW for heating and 16.3 kW for cooling. Similarly, the power for the DHW is calculated at 1.8 kW, assuming 5 h of system daily operation to fully meet the needs, as defined in the technical guidelines provided in [39].

2.3. Multi-Energy Systems Parameters

The proposed multi-energy systems configuration is focused on enhancing the RES penetration for meeting building energy demand. In fact, air–water heat pumps, solar thermal collectors and photovoltaic systems were included, as well as the interconnected electricity grid. It is noted that the analysis excludes the impact of different terminal units, as well as the temperature of the working medium, on energy recovery efficiency, assuming that the proposed solutions are capable of meeting the energy requirements.

For each system and their energy sources, the basic techno-economic, energy and environmental parameters, considering life cycle aspects as well, should be clarified. More specifically, the economic parameters include the operational costs provided by the consumption of the respective energy sources, as well as the costs of installing the proposed energy systems. In addition, the energy parameters account for the primary energy factors, considering the origin of the energy sources during energy system operation. Moreover, they take the embodied energy of the systems into account. This corresponds to the cumulative energy consumption originating from fossil fuels and RESs across several life cycle stages up to the system construction. Similarly, the environmental aspects incorporate the equivalent CO₂ emissions during the operation of the energy systems and the embodied emissions coming from the construction life cycle stages. The estimated impact values of the embodied energy and emissions of the examined energy systems resulted from a life cycle assessment (LCA) analysis conducted in [40,41]. In these studies, the construction of the inventory database was made by utilizing the “Ecoinvent” database and the Environmental Product Declaration (EPD) [42]. Also, the “CML 2 Baseline 2000” and cumulative energy demand (CED) methods were implemented for defining the results of energy and environmental impacts [43,44].

Regarding the technical characteristics of the energy systems examined, their lifetime duration and their efficiency were considered. For the calculation of the coefficient of performance (COP) of the heat pump, an empirical equation was used, as provided in [45]:

$$\mathrm{C}\mathrm{O}\mathrm{P}=0.001·\Delta {\mathrm{T}}^2-0.1534·\Delta \mathrm{T}+7.3775,$$

(1)

where ΔT represents the difference between outdoor air temperature and the outlet temperature of the working medium (water), which is assumed to be 35 °C and 15 °C for heating and cooling, respectively; underfloor heating and cooling are considered for attributing thermal energy to the building.

Also, the f-chart method, developed by Duffie and Beckman, was utilized for sizing a flat plate solar thermal collector (

Table 3. Basic design parameters of a flat plate solar thermal collector [47].

Solar Collector Parameters	Values
Thermal performance curve slope—FRUL	4 W/m2K
Thermal performance intercept—FR(τα)n	0.75
Coefficient for collector location—τα/(τα)n	0.96
Heat exchanger coefficient—F’R/FR	0.95
Storage tank volume per collector area—M	75 L/m2

). This approach is widely used to evaluate the annual thermal performance of active heating systems in buildings, as it offers reliable results and has a relatively easy implementation; the minimum temperature of energy delivery is around 20 °C [46]. In more detail, it provides a fully developed methodology for calculating the solar thermal fraction of meeting heating and DHW demand for a specific solar heating system. There are some key variables considered as essential for the f-chart method. The primary design variable seems to be the area of the solar collector, while the secondary variables include the collector types, storage capacity, fluid flow rates and collector heat exchanger sizes. Two dimensionless variables (X, the ratio of collector losses to heating loads and Y, the ratio of absorbed solar radiation to heating loads), resulting from several correlations, were introduced, in order for the fraction of the monthly heating load supplied by solar energy to be estimated.

It should be mentioned that the f-chart method primarily applies to systems designed to meet DHW demand and a small percentage of heating demands, as defined in [46]. However, in the proposed analysis, it was assumed that the f-chart could also predict heating operation on a larger scale, considering the presence of underfloor heating, setting a balance between the temperature levels of demand (load) and supply (collector operation).

Surrogate polynomial models were developed in ALAMO (Automatic Learning of Algebraic Models) [48] for simulating the operation of the solar thermal collector, considering the results of the f-chart implementation. More specifically, these models, which are essential for the optimization process, calculate the collector area as a function of the fraction of the heating load supplied by solar energy. The process of creating such surrogate models was performed to provide models that maximize R², which represents the variability explained by the model, considering the total number of deviations and minimizing the root mean square error (RMSE).

Considering all of the above, the technical, economic, energy and environmental parameters utilized in the proposed study were presented in

Table 4. Economic, energy and environmental operational costs for the energy sources [39,49,50,51].

Energy Sources	Economic Costs (EUR/kWh)	Primary Energy Factors	GHG Emission Coefficients (kg CO2/kWh)
Solar Energy	0	0	0
Electrical Energy	0.12/0.19	2.9/2.1/1.8	0.989/0.6/0.2

and

Table 5. Technical, economic, energy and environmental parameters for energy systems.

Energy System Parameters	Heat Pump	Solar Collector	Photovoltaic System
Efficiency	4/4.3 1	f-chart	0.2
Life Duration (years)	25	25	25
Installation Costs	600 EUR/kW	380 EUR/m2	250 EUR/m2
Energy Costs	142 kWh/kW	103 kWh/m2	192.5 kWh/m2
Environmental Costs	337 kg CO2/kW	1890 kg CO2/m2	1022 kg CO2/m2

¹ The values represent the COP and EER for the heat pumps, respectively.

. It is noted that a sensitivity analysis was conducted for the parameters dealing with the electricity grid, considering the share of RES technologies in the energy mixture. In more detail, the primary energy factor of 2.1 and the GHG emission factor of 0.6 kg CO₂/kWh correspond to an energy mixture of Greece in 2019, with a share of 33.2% RESs, while the values of 1.8 and 0.2 kg CO₂/kWh, respectively, refer to an increased penetration of RESs in the energy mixture, as anticipated for the future (years 2028–2030) [49,50,51].

2.4. Energy Hub Formulation

The multi-energy systems described above were used in order to connect the energy sources with the building energy demand. This link is essential for determining the energy consumption, leading to economic, energy and environmental costs. For this reason, the EH concept was utilized, representing such connections, with its main goal being to provide optimal decisions considering the installation and operation of the energy systems under the objectives of minimizing economic, energy and environmental life cycle costs. In more detail, the formulation of the proposed EH includes the energy sources as input parameters and building energy demand as output parameters. The energy systems compose the converters in the EH concept.

This concept is illustrated in Figure 4, where electrical and solar energy are proposed to supply multi-energy systems for meeting building demand for space heating, cooling and DHW. More specifically, solar thermal collectors are proposed to cover heating or DHW demand, while heat pumps could be used for space heating, cooling and hot water. The electrical energy consumed by the heat pumps can be supplied either by photovoltaic systems or by the electricity coming from the grid. All of these decisions describe an optimization problem for defining the participation level of each of these systems in order to meet the relative building energy requirements. The optimal decisions for each criterion are made under an annual analysis, providing solutions for the systems’ participation to fully cover the energy demand of the case study building.

2.5. Optimization Problem

The formulation of such an EH, including multi-energy systems, creates an optimization problem for providing decisions on energy system installation and operation for each energy use. In this context, an MP problem was developed in order to find the optimal solutions by defining the relationship between key variables and constraints for achieving specific objectives, like minimizing the economic, energy and environmental costs of the systems. The following aspects are essential for MP model formulation, as also presented in Figure 5.

• Constant parameters include all of the fixed values that are considered to be unchanged when solving the optimization problem.
• Design variables describe the optimization factors for the decision-making process and providing optimal solutions.
• Objective functions formulated by mathematical expressions include design variables and determining the goals of the optimization problem, i.e., the optimization criteria.
• Constraints impose the boundaries or the limitations or the requirements of the design variable in the optimization problem.
• Mathematical techniques define the type of the optimization problem (linear, integer, etc.) and select the appropriate solver for finding the optimal solutions.

2.5.1. Constant Parameters

These parameters include building energy demand and power, as presented in Section 2.2, as well as all of the technical, economic, energy and environmental aspects for the energy sources and the systems included in Section 2.3. In more detail, economic costs, primary energy factors and GHG coefficients were depicted in Table 4 for electricity, defining the operation costs for each optimization criterion, combined with energy consumption. Also, Table 5 presents the values of installation costs, embodied energy and embodied GHG emissions of the examined energy systems. Such values correspond to the installation costs for each optimization criterion, combined with the size of the energy systems (in kW or m²). Such values describe the optimization goals, formulating the objective functions for the economic, energy and environmental criteria.

2.5.2. Design Variables

The optimal decisions are made according to the results provided by the design variables. The key variables in this problem provide the energy systems’ participation (${\epsilon }_i^{\mathrm{j}}$), considering their proper operation and installation for meeting building energy demand. More specifically, these variables are related to the operation of the heat pumps connected to the electricity grid, as well as to the sizing of the solar thermal collectors and the photovoltaic systems. So, the range value of these variables is (0,1). However, the sizing of the heat pumps should be defined by a binary variable (${\mathrm{b}}_{i=HP}^{\mathrm{j}}$) in order for the full demand size to be considered. The beforementioned design variables and the referring indexes are presented in

Table 6. Definition of the design variables.

Design Variables	Values	Definition
${\epsilon }_i^{\mathrm{j}}$	Non-negative [0,1]	The participation level of the energy systems’ operation
${\mathrm{b}}_{i=HP}^{\mathrm{j}}$	Binary (0 or 1)	The participation of the energy systems’ installation (HP)
i represents the energy systems examined, i.e., heat pump (HP), solar thermal collector (SC) and the photovoltaic system (PV) j represents the energy uses; thus, the building energy demand for space heating (H), cooling (C) and domestic hot water (DHW)

.

2.5.3. Objective Functions

The optimization criteria investigated in this study focus on minimizing the economic, energy and environmental parameters, as analysed in Section 2.3, considering heating, cooling and hot water energy demand. The form of the objective functions used in this study is presented in Equation (2).

$$\begin{gathered} Min\left[\begin{array}{c}Cost\\ Ener\\ Envir\end{array}\right]={\sum }_{i=PV}^j\left({\left[\begin{array}{c}InstCost\\ EmbEner\\ GHGEm\end{array}\right]}_i·{\displaystyle \frac{1}{L{D}_i}}·{\displaystyle \frac{\frac{Q_{dem}^j}{CO{P}^j}}{n_{PV}·H_{sol}}}·{\epsilon }_i^j\right)+ \\ {\sum }_{i=HP}^j\left({\left[\begin{array}{c}OpCost\\ PrEner\\ OpGHGEm\end{array}\right]}_i·{\displaystyle \frac{Q_{dem}^j}{CO{P}^j}}·{\epsilon }_i^j\right)+{\sum }_{i=HP}^j\left({\left[\begin{array}{c}InstCost\\ EmbEner\\ GHGEm\end{array}\right]}_i·{\displaystyle \frac{1}{L{D}_i}}·P_{dem}^j·b_i^j\right)+ \\ {\sum }_{i=SC}^{j=H,HW}\left({\left[\begin{array}{c}InstCost\\ EmbEner\\ GHGEm\end{array}\right]}_i·{\displaystyle \frac{1}{L{D}_i}}·f\left({\epsilon }_i^j\right)\right) , \end{gathered}$$

(2)

where

• Cost (EUR): The total annual economic costs including systems’ operation and installation, which are set for minimization.
• Ener (kWh): The total annual energy costs including systems’ primary energy consumption and embodied energy, which are set for minimization.
• Envir (kg CO₂): The total annual environmental costs including systems’ GHG emission during operation and the embodied emissions, which are set for minimization.
• InstCost (EUR/kW or m²): The installation costs of the systems (Table 5).
• EmbEner (kWh/kW or m²): The embodied energy of the systems (Table 5).
• GHGEm (kg CO₂/kW or m²): The embodied GHG emissions of the systems (Table 5).
• OpCost (EUR/kWh): The economic costs resulting from the systems’ operation (Table 4).
• PrEner (-): The primary energy factors for the grid electricity (Table 4).
• OpGHGEm (kg CO₂/kWh): The environmental costs, i.e., the GHG emitted during systems operation (Table 4).
• LD_i (years): The life duration of the energy systems examined (Table 5).
• Q^j_dem (kWh): The building energy demand for each energy use (j).
• P^j_dem (kW): The power demand for each energy use (j).
• H_sol (kWh/m²): The incident solar irradiation.
• n_PV, COP^j: The efficiency of the photovoltaic systems and the coefficient of performance of the heat pumps (Table 5).

2.5.4. Constraints

For finding feasible solutions, the design variables should be constrained, considering the following aspects.

• The design variable defining the participation level of the energy systems should be limited to a lower bound of 0% and an upper bound of 100%.

$$0\le {\epsilon }_i^{\mathrm{j}}\le 1 \forall i,j,$$

(3)

• The energy demand for each energy use should be fully met by at least one energy system.

$${\sum }_i{\epsilon }_i^j=1 \forall j,$$

(4)

• The installation of the heat pump should be considered only when its participation is preferable by the optimization criteria.

$${\epsilon }_{i=HP}^j\le b_{i=HP}^j \forall j,$$

(5)

2.5.5. Mathematical Techniques

GAMS is a computational tool specialized in formulating MP problems, while it incorporates several solvers for providing optimal solutions. The proposed MP problem was modelled in GAMS, as a Mixed Integer Non-Linear Problem, and the SCIP solver was used to find optimal solutions [52].

In addition, scripts in Python 3.10.6 were developed, conducting a brute force analysis of the proposed problem. The process starts with the calculation of all of the possible combinations of the energy systems examined, with an analysis level of 0.1%. This results in almost half a million combinations, which are used for calculating economic, energy and environmental costs, describing the optimization criteria. The above brute force process of investigating the optimal solutions is directly associated with the level of analysis, which specifies the number of the alternative scenarios examined. This is the reason for making brute force analysis intractable and time consuming. However, the process described above was implemented for the optimization problem of this analysis, in order to compare the optimal solutions provided by GAMS. With the analysis level at 0.1%, the brute force analysis leads to results similar to the ones provided by GAMS.

3. Results

3.1. Optimization Results of the Proposed EH

The optimization results show that the criteria examined are not conflicting with each other. In more detail, the minimization of economic and energy costs leads to the same optimal solution, while in the environmental criterion, a small differentiation was depicted. This leads to smaller GHG emissions, but higher economic and energy costs, as presented in Figure 6.

Building energy demand for heating, cooling and DHW are fully covered by 27 kW HP combined with 25.5 m² PV, under net metering. This is the optimal decision considering the economic and energy criteria. When the mitigation of GHG emissions is set as the optimization objective, the previous solution of 27 kW HP is now connected with 25 m² PV to fully meet the heating and cooling demand, while 1 m² SC is preferable for meeting the DHW demand. The participation level of the SC is 30% for the DHW demand, and the rest, 70%, is covered by the HP.

Also, it is important to mention that these optimal solutions are the same when the electricity costs are reduced to 0.12 EUR/kWh, as well as when the values of primary energy and GHG coefficients are lower (2.1/0.6 or 1.8/0.2), considering a more renewable-based energy mixture for electricity [50,51].

Also, a more detailed analysis of the optimal economic values was conducted by inserting constraints in the optimization problem. Such analysis examines the basic alternative scenarios for meeting the energy demand of the building (heating/cooling/DHW), considering two values for the electricity cost. In this context, Figure 7 presents the distribution of the economic costs of these scenarios. More specifically, it appears that the minimum costs arise with the PV dominance, while the participation of SC (mainly in heating) causes an increase in costs, highlighting a great potential for economic savings. Also, economic solutions are provided when HP–PV are combined for meeting heating and cooling demand, and with SC for the DHW demand. This is due to the lower energy requirements for DHW, compared to the other two energy uses. Last but not least, the solution of HP-ElGrid seems to be economically advantageous in several cases, compared to renewable-based scenarios, especially when the price of electricity is lower.

Considering the other two criteria (energy and environmental), the distribution order for the 18 basic scenarios is different from the one in the economic criterion, especially when minimizing the GHG emissions (Figure 8). The primary energy factors and the GHG coefficient considered in this analysis are 2.1 and 0.6 kg/kWh CO₂, respectively. Results show that for the environmental criterion, the scenarios with the highest GHG emissions are those with the highest participation of the HP-ElGrid systems. While the renewable-based technologies (PV and SC) dominate, the environmental footprint decreases. The distribution order of these scenarios in the energy criterion is in line with the one in the economic criterion, apart from some exceptions of HP-ElGrid dominance that seem to be preferable.

3.2. Optimization Results of the EH Without PVs

The analysis was expanded to cases where PVs are totally excluded from the proposed EH, so the energy demand can be met either by SC or by HP (connected to the electricity grid) or by a combination of these two alternatives. This is important in order for a more comprehensive analysis to be carried out, providing multiple optimal results for the different criteria examined. For this reason, implementing a sensitivity analysis for the economic, energy and environmental parameters for the electricity from the grid (ElGrid), seems to be essential for the decision-making process.

In

Table 7. Optimal results of energy system participation for the EH excluding PVs, considering all of the energy uses and optimization criteria.

Optimization Criteria	Economic	Energy	Environmental
Heating	100% HP-ElGrid	51.6% HP-ElGrid 48.4% SC (36 m2)	12.6% HP-ElGrid 87.4% SC (140.5 m2)
Cooling	100% HP-ElGrid	100% HP-ElGrid	100% HP-ElGrid
DHW	40% HP-ElGrid 60% SC (2.5 m2)	18.8% HP-ElGrid 81.2% SC (5.5 m2)	4.6% HP-ElGrid 95.4% SC (14.5 m2)

, the optimal decisions of the participation level of each energy system are presented, considering all of the criteria examined. The economic, energy and environmental parameters of the ElGrid are set as 0.19 EUR/kWh, 2.9 and 0.989 kg CO₂/kWh. Results show that the HP-ElGrid (27 kW) is dominant for the economic criterion, while when minimizing the energy and environmental costs, the participation of the SC is increased, especially for heating and DHW uses. This leads to higher economic costs, as depicted in Figure 9, which displays high conflict between the optimization criteria.

In addition, when decreasing the cost of electricity (0.12 EUR/kWh) in the economic criterion, the participation of SC for meeting the DHW demand drops to 41.9% (1.5 m²). This leads to lower economic costs (23% reduction), but an increase is depicted in the energy and environmental costs (1.5–2% increase). The optimal results for the other criteria remain the same, as there is no change in energy or environmental parameters.

Considering a sensitivity analysis of the values of primary energy factors and environmental costs for the electricity, the optimization results propose minimum dependence on the SC when the energy mixture is based on RESs. This is indicative for the energy and environmental optimization criteria, leading to high participation rates of the EL-Grid. Such results are presented in

Table 8. Optimal annual results of energy system participation for the EH excluding PVs, through a sensitivity analysis of primary energy factors (2.9/2.1/1.8) and environmental costs (0.989/0.6/0.2 kg CO₂/kWh), considering the (a) economic, (b) energy and (c) environmental optimization criteria.

(a) Economic Criterion
Sensitivity Analysis Parameters	Systems	Economic	Energy	Environmental
0.19/2.9/0.989 1	27 kW HP-ELGrid 2.5 m2 SC	EUR 1794	17,514.5 kWh	5947.6 kg CO2
0.19/2.1/0.6 1			12,835.3 kWh	3672.4 kg CO2
0.19/1.8/0.2 1			11,080.6 kWh	1332.7 kg CO2
(b) Energy Criterion
Sensitivity Analysis Parameters	Systems	Economic	Energy	Environmental
0.19/2.9/0.989 1	27 kW HP-ELGrid 41.5 m2 SC	EUR 2082	15,870.3 kWh	4549.9 kg CO2
0.19/2.1/0.6 1	27 kW HP-ELGrid 20 m2 SC	EUR 1892	12,293.4 kWh	3209.8 kg CO2
0.19/1.8/0.2 1	27 kW HP-ELGrid 15.5 m2 SC	EUR 1855	10,776.3 kWh	1243.7 kg CO2
(c) Environmental Criterion
Sensitivity Analysis Parameters	Systems	Economic	Energy	Environmental
0.19/2.9/0.989 1	27 kW HP-ELGrid 155 m2 SC	EUR 3569	20,796.4 kWh	3766.7 kg CO2
0.19/2.1/0.6 1	27 kW HP-ELGrid 120.5 m2 SC	EUR 3084	16,180.6 kWh	2563.1 kg CO2
0.19/1.8/0.2 1	27 kW HP-ELGrid 49.5 m2 SC	EUR 2172	11,448.6 kWh	1172 kg CO2

¹ Electricity cost (EUR/kWh)/primary energy factor (-)/environmental cost (kg CO₂/kWh).

, including the economic, energy and environmental costs, as well as the sizing of the energy systems installed. These are the optimal results for each optimization criterion, considering a constant value of electricity cost (0.19 EUR/kWh) and different values for primary energy factors and environmental costs. More specifically, the optimal decisions for the economic criterion are the same for the three scenarios of energy mixture; nevertheless, the energy and environmental values decrease in a renewable-based energy mixture. This pattern corresponds to the energy and environmental criteria too, but the decisions provide the mitigation of SC in renewable-based energy mixtures, which leads to lower economic costs.

Comparing the alternative optimal solutions of Table 7 and the solution with the participation of PVs (Section 3.1), it is important to mention that when minimizing the economic costs, the optimal solution without PVs is the one provided by renewable-based energy mixtures. This leads to low values of energy and GHG emissions.

4. Conclusions

In this study, a decision-making methodology has been proposed, considering building energy design. More specifically, the EH concept was implemented in a building in order to optimize multi-energy system installation and operation in the scope of building energy demand satisfaction. The optimization was conducted under the principles of MP, considering economic, energy and environmental criteria with a life cycle perspective.

The analysis showed that the application of the optimization process in the design of multi-energy systems highlights the investigation of multiple parameters, such as technical, economic, energy and environmental ones. Also, different limitations upon the energy sources can be easily considered, providing the possibility of evaluating the optimal results too. Last but not least, the concept of EH is important for incorporating different simulation models, like f-chart, in order to provide a more accurate analysis. Additionally, the optimization with MP seems to be less time consuming than brute force investigations, where a huge number of alternative scenarios should be considered.

The proposed RES-based EH, which is considered as a tool of energy management, highlights the ability of buildings to be prosumers, thus consumers and producers concurrently. This is indicative for the optimal results, as the combination of heat pumps and photovoltaic systems under net metering seems to be preferable for all of the examined criteria. However, in cases in which the economic costs or the primary energy factors of the electricity from the grid were decreased, the optimal decisions of HP and PV remained. This is a useful conclusion for highlighting the importance of buildings’ self-energy production, rather than transforming the energy mixture for electricity, which is essential in cases where the installation of PVs on buildings is forbidden. More specifically, the sensitivity analysis for the primary energy factors in an EH where the PVs were excluded showed that with more RESs in the energy mix, there were lower energy and environmental costs.

All in all, optimization models can enhance and streamline the building design process by assessing the advantages and drawbacks of different energy systems, enabling a comparative analysis of the available alternatives too. The proposed methodology can easily be implemented to different building types and climatic condition scenarios (future research), affecting the values of the economic, energy and environmental costs, upon the same pattern of optimal decisions. The current analysis was limited to an annual analysis for decision-making, considering data on a monthly basis. Future research could extend the proposed methodology to a shorter time-step analysis (daily and hourly), enabling the prediction of the dynamic operational behaviour of the energy systems, considering their efficiency degradation, especially for PV systems, and a more comprehensive strategy for their operation. In this context, it is important to use relevant approaches dedicated to space heating, especially for solar collector operation, that are based on validated simulation results. This is a meaningful task for future work, indicating a suitable form for integrating such approaches into the energy hub optimization concept. The incorporation of different energy storage technologies, such as batteries, in the energy hub, seems to be crucial too. Moreover, scenarios incorporating the fluctuation in energy demand and considering the indoor temperature setting could enhance the resilience of the proposed analysis. Lastly, thermal comfort parameters could also be examined in more detail, enhancing the importance of implementing multi-objective optimization methodologies through conflicting criteria scenarios.