Optimising Sustainable Home Energy Systems Amid Evolving Energy Market Landscape

Abstract

The paper presents a linear optimisation model aimed at improving the design and operational efficiency of home energy systems (HESs). It focuses on integrating photovoltaic (PV) installations, hybrid heating systems, and emerging energy storage systems (ESSs). Driven by the EU climate policy and the evolution of the Polish electricity market, which have caused price volatility, the model examines the economic and technical feasibility of shifting detached and semi-detached houses towards low-emission or zero-emission energy self-sufficiency. The model simultaneously optimises the sizing and hourly operation of electricity and heat storage systems, using real-world data from PV output, electricity and gas consumption, and weather conditions. The key contributions include optimisation based on large data samples, evaluation of the synergy between a hybrid heating system with a gas boiler (GB) and a heat pump (HP), analysis of the impact of demand-side management (DSM), storage capacity decline, and comparison of commercial and emerging storage technologies such as lithium-ion batteries, redox flow batteries, and high-temperature thermal storage (HTS). Analysis of multiple scenarios based on three consecutive heating seasons and projected future conditions demonstrates that integrated PV and storage systems, when properly designed and optimally controlled, significantly lower energy costs for prosumers, enhance energy autonomy, and decrease CO₂ emissions. The results indicate that under current market conditions, Li-ion batteries and HTS provide the most economically viable storage options.

1. Introduction

For addressing climate change and reducing dependence on energy imports, the EU promotes renewable energy sources (RESs). The ambitious goal to cut EU greenhouse gas emissions by at least 55% by 2030 [1] and to set the EU on the path to achieve climate neutrality by 2050 will further boost the dynamic development of RESs across various energy vectors and a range of industrial and social activities. This path also involves decarbonising households, where electricity use, space heating, and personal transport are the main contributors to high CO₂ emissions in Poland [2].

Supported by generous investment subsidies and driven by the rising and volatile prices of energy carriers, energy consumers have already made significant efforts to reduce their risk exposure. Furthermore, new CO₂ emission regulations for buildings have notably increased the adoption of individual photovoltaic installations and expanded the deployment of heat pumps for space heating [3,4].

Figure 1 shows the progress in total PV power in residential installations and power plants in Poland.

However, this success brings several challenges for the underdeveloped distribution grid, and to the stability and controllability of the entire Polish power system.

The Polish TSO (Polskie Sieci Elektroenergetyczne S.A., PSE) reports significant issues with power system balancing during periods of peak PV production and low demand. Distribution system operators (DSOs) signal difficulties in exporting excess energy from LV to MV distribution grids. Low levels of auto-consumption in Polish households (between 22 and 28%), ageing LV networks designed for downstream energy flow, and the quality of key PV installation components often cause voltage problems and restrict PV output [5].

PSE mainly tackles this issue through economic measures, such as adjusting the settlement rules for energy exported to the grid. However, the number of delivery and settlement periods where the market approach fails is increasing, and the TSO is forced to curtail PV production of large PV plants connected to MV and HV networks. The ongoing debate suggests that these last-resort technical measures could eventually be extended to PVs below 10 kWp, typical of residential installations connected to LV networks [6].

Although this publication and the developed and validated optimisation model have been triggered by the experience gained during the transformation of the wholesale Polish power sector, electricity market, and the first steps to implementing a fully competitive retail market, the approach is not limited to these contexts. It is adaptable and consistent with a variety of EU electricity market designs, including ultimate methods for the settlement of imbalances, definition of prosumers or energy communities (with local generation oriented on self-consumption), and importantly, it aligns with the EU-wide future implementation of the extension of the emission trading system (ETS).

2. RES Balancing and Settlement

In the early days of the PV sector in Poland, the first settlement system, based on net-metering, was highly successful, prosumer-friendly, and played a crucial role in the growth of small PV installations. However, this solution did not promote prosumers’ self-consumption nor motivate customers to invest in ESSs.

Thus, in April 2022, net-metering was replaced by net-billing. The current settlement system is based on double-price settlement. Imported energy is purchased at a rate determined by the individual supply contract (either a flat rate or by time zone) and this is further increased with a distribution tariff. Exported energy is settled using a monthly price (RCEm) calculated ex post by the TSO and approved by the regulator [7]. This reflects the cost of the overall system balancing, including the financial compensations made by TSO to RES for the out-of-market curtailment of generation.

The implementation of the new settlement method led to a significant decrease in the profitability of investments in PV installations, especially when combined with increased energy use due to heat pumps in new buildings or switching from gas- or coal-fired boilers in existing buildings [8]. Prosumers view this settlement method as unpredictable and unclear due to the complexity of the algorithm and TSO’s regular adjustments to calculated prices.

Therefore, the ultimate solution is to implement dynamic pricing (time of use, ToU) based on overall system balancing trends, with a single balancing price applied in both the wholesale and retail markets. The dynamic pricing settlement is already available to Polish prosumers on a voluntary basis, as an alternative to the monthly settlement of exported energy. It is likely to become a mandatory method by 2027.

Recent amendments to the wholesale balancing market, effective from 2023, permit negative price signals, thereby increasing price risk for prosumers. This change is expected to boost self-consumption and enhance energy independence, mainly through the deployment of energy storage systems (ESSs) and the optimisation of load profiles with home energy monitoring and management systems.

Furthermore, owners of new or refurbished buildings will soon face environmental restrictions related to the use of gas boilers, and all consumers will have to bear the economic consequences of the expanded EU Emission Trading System (ETS2), coming in 2027, which will significantly impact existing individual heating systems that burn fossil fuels.

3. Underlying Research and Progress Beyond the State of the Art

The optimisation of HESs has received significant research interest recently, motivated by practical reasons, as this research demonstrates direct relevance to rapidly evolving structures and the operation of residential energy systems.

Early research mainly focused on short-term optimisation, typically covering a single day or week, and concentrated primarily on individual storage technologies. There has been limited exploration of the synergy between the two most vital home energy sub-systems: electricity and heat. In this context, the work of Beaudin and Zareipour [9] provided a comparative analysis of past research activity in this area, emphasising modelling approaches and their implications.

Referring to individual landmark papers, Ajao et al. [10] proposed one of the first comprehensive models combining residential PV installations, household appliances, EVs with battery storage, with optimisation based on mixed-integer programming (MILP).

Langer and Volling [11] used the same optimisation method, but they focused on combining electrical storage and thermal storage in a home heating system with a heat pump and dynamic electricity prices. Although the paper considered a detailed model of the heat pump and the home heating system, the optimisation horizon was limited to 24 h, focusing only on a single electricity storage technology, and thermal storage was limited to a simple water buffer model.

Huy, Dinh, and Kim [12] advanced the field by converting the MILP into a multi-objective optimisation model, comparing deterministic and stochastic models, and emphasising user thermal comfort. Similar to the authors of this publication, who explored a stochastic model for battery storage optimisation [13], the referenced paper concentrates on daily optimisation.

For a longer optimisation horizon, Song Gao et al. [14] published a paper that explored the integration of electric and thermal storage optimisation over an entire year (8760 h). The authors utilised a comprehensive cost structure that included both capital expenditure (CAPEX) and operational expenditure (OPEX), formulating the optimisation problem as an MILP model.

While linear and MILP optimisation techniques demonstrate strong performance in large-scale applications, they have inherent limitations in modelling the nonlinear characteristics of buildings, thermal systems, and individual electrical loads. Dinh and Kim [15] addressed these limitations in their paper by employing a nonlinear function to better capture user convenience regarding load flexibility (home appliances). As a result, they used the MINLP optimisation algorithm for 24 h scheduling. Other research on nonlinear models of flexible loads has incorporated artificial neural networks (ANNs) [16] or game theory (GA) [17]. Additionally, a separate set of studies has focused on the nonlinear optimisation of HESs, examining nonlinear thermal models of buildings and heating systems, as well as thermal comfort considerations [18,19].

In some instances, a nonlinear problem can be effectively solved through piecewise linearisation, as shown in the research conducted by Mallégol et al. [20]. In their study, the nonlinear characteristics of the efficiency performance of energy system components, especially combined heat and power (CHP), were linearised using three different methods. Artificial neural networks (ANNs) and nonlinear models have been used for the online optimisation of electric vehicle (EV) integration within HESs [21].

Moreover, recent advancements have involved the use of AI tools to model nonlinearities in the predictive control of building energy systems [22]. This research focused on nonlinear distributed model predictive control (NDMPC), where a comparison was made between the Alternating Direction Method of Multipliers (ADMM) and the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method. In this context, ANN has also served as the optimisation algorithm.

Considering the current state of the art, the most advanced research activities related to HES optimisation and their limitations in handling real-world, large-scale tasks, computational performance, and the practical benefits of advanced nonlinear models, this paper focuses on a practical application of the optimisation method for the integrated sizing and hourly scheduling of electricity and heat storage operations within the evolving Polish and EU power sectors. It also investigates new and emerging ESS technologies for individual households.

The main contributions of this work can be summarised as follows.

• Optimisation of HES structure and daily operations using metering data on PV production, energy consumption, and weather data collected over three heating seasons.
• Simultaneous optimisation of key home systems, including electrical and heating installations.
• Analysis of the economic viability of three electricity storage technologies, each with different operational parameters, investment, operation, and maintenance costs, based on the standard cost model. This includes a sensitivity analysis and the identification of break points for the emerging storage technologies.
• The optimisation model accounts for the impact of demand site management.
• The optimisation model integrates the wear-and-tear component for storage facilities.
• Analysis of trends in the Polish electricity balancing market related to the changes in the energy mix.
• Study of the consequences of the proposed implementation of ETS2 with new environmental levies imposed on residential customers.

4. Zero-Emission, Renewable-Based Households

This paper examines detached and semi-detached houses that can economically benefit from local electricity generation through a rooftop photovoltaic system designed to meet annual energy needs. The house employs a hybrid heating system that combines market signals and weather conditions, such as outdoor temperature, to effectively manage the daily operation of a heat pump and a gas boiler.

Figure 2 depicts the overall structure of the home energy system connected to the distribution grid with a 12 kW export and import limit. The building’s electrical system may include three different electricity storage options, selected and sized during the optimisation process. Thermal storage and heat pumps link the household’s electrical energy to heat production, storage, and demand.

Active heating with the heat pump involves two separate energy flows with different leverage factors (COP–hourly coefficient of performance).

The principal HES devices and energy flows between the primary energy system components and energy conversion devices denote the primary optimisation variables and input data, which are elaborated upon in Section 7.

The optimal design of an HES relies on historical data, including real-time PV output, hourly electricity consumption data, and estimated hourly demand for space heating and hot water, derived from daily gas meter readings and hourly outdoor temperature data. Hourly gas consumption during the non-heating seasons has been used to develop the hot water profile. The optimisation process also considers one year of hourly settlement prices from the Polish balancing market [23], pipeline gas prices, and relevant distribution tariffs.

The applied cost model includes both CAPEX and OPEX components. The outlays and profits distributed over the system’s entire lifespan are annualised using Formula (1). Additionally, a gradual decline in the productivity of the PV installation has been considered over the relevant 18-year lifespan, without significant additional investments.

The equivalent annual cost (EAC) denotes the yearly expense linked to owning, operating, and maintaining an asset over its entire lifespan. EAC is frequently employed in capital budgeting decisions, as it enables a company to compare the cost-effectiveness of different assets with varying lifespans, using a process known as the replacement chain method.

The calculation of EAC is carried out using the following formula:

$$EAC={\displaystyle \frac{\mathit{Asset}\mathit{Price}·\mathit{Discount}\mathit{Rate}}{1-{(1+\mathit{Discount}\mathit{Rate})}^{-n}}}$$

(1)

where n refers to the year in which expenses are incurred or income is earned, based on the effective lifespan of key components such as photovoltaic (PV) installations and storage facilities. The discount rate in the formula is determined using the Weighted Average Cost of Capital (WACC) approach, calculated assuming a 100% equity investment, with 10-year state bonds regarded as a safe alternative investment.

The optimisation period extends to 8760 settlement periods.

This approach offers a notable advantage over studies relying on standard customer demand profiles commonly used for tariff or energy contract planning.

5. EES Technology for Sustainable HES

Currently, the most common energy storage (EES) technologies for household use are lithium-ion batteries (LFP or NMC) and thermal hot water tanks. However, these solutions, which are excellent for short-term storage, do not perform well as seasonal storage. Therefore, this study explores emerging storage technologies that are not yet adapted for small energy systems.

The pre-selection of emerging storage technologies includes power-to-hydrogen (H2) systems, which utilise a combination of electrolysers and fuel cells (green hydrogen), and redox flow batteries (FB). These options were chosen primarily due to their technology readiness levels (TRLs) and the availability of commercial solutions for medium-scale electricity storage.

The technical and economic parameters of these technologies were collected from the review of several fully commercial solutions (Li-ion batteries) as well as from bibliographical research in the case of emerging technologies (H2, FB).

The NREL report [24] served as the basis for a comprehensive analysis of leading energy storage technologies. It was used as a reference source for information collected from subsequent publications, which focus on specific technologies or particular aspects of those technologies. A comparative review of the most popular EES technologies, including those discussed in this paper, is also available in the paper by Amir et al. [25]. However, it mainly concentrates on technical aspects, providing limited economic information. Recently, the techno-economic comparison of electricity storage options for renewable energy systems by Mulder and Klein [26] provided an update on the economic parameters across all well-established and emerging types of EES.

Regarding the pre-selected storage technologies, the paper by Martínez de León et al. [27] is the most valuable reference for analysing both the technical and economic aspects of hydrogen storage. Furthermore, the publication by Mylonopoulos et al. [28] offers additional information on small-scale hydrogen storage investment costs and maintenance costs.

Information on the state of the art regarding technical parameters of flow battery storage technology is detailed in the review by Shoaib et al. [29]. The techno-economic assessment by Poli et al. [30], focusing on vanadium flow batteries, provided economic parameters. Further technical details were obtained from the study by Tang et al. [31].

For PV and Li-ion batteries, basic economic and technical information was gathered from commercial products and complex solutions designed for small residential energy systems in Poland. Additional details about the degradation of battery lifespan with the number of charging and discharging cycles were obtained from work by Schade et al. [32] and from the bibliographical research conducted in another work by the authors of this publication [33].

Beyond individual research outcomes, a general review of renewable technologies, including generation and storage, provided in [34], was utilised to estimate the range of techno-economic parameters for EES.

Table 1. Charging and discharging efficiency and standby losses for the selected electricity storage technologies [24,25,26,27,28,29,30,31].

		Li-ion	H2	FB
Charging efficiency	[%]	85–98	60–80	70–80
		(95)	(70)	(75)
Discharging efficiency	[%]	90–95	40–60	75–85
		(95)	(50)	(80)
Standby losses	[%]	2–3 per month	∼0	∼0
		(0.004 per hour)
Energy density	$[\mathrm{Wh}·{\mathrm{kg}}^{-1}]$	100–200	800–1000	25–30

displays charging and discharging efficiency parameters, along with standby losses, for selected electricity storage technologies considered for household system optimisation. Values in parentheses and bold are the parameters used in this research.

In the field of thermal storage technologies, the range of mature solutions (TRL 8–9) is broader. However, considering factors such as space constraints, safety concerns, and ease of daily maintenance, the authors focus on solid-state sensible heat storage materials. These materials are characterised by a high density, a wide range of operational temperatures, and low specific heat, which enhances energy density and decreases the space required for thermal storage installations.

A comprehensive overview of well-established ultra-high temperature thermal storage solutions is available in Robinson’s two publications [35,36], along with other thermal storage technologies covered in the review of sensible heat storage by Bespalko et al. [37].

Information about losses and simulation models of thermal storage facilities was derived from the work by Junga et al. [38], which focuses on a hot water storage tank. A review paper published by Tian et al. [39] served as the primary source for the economic model of HTS.

The technical specifications for the high-temperature storage solutions examined in this paper are detailed in

Table 2. Technical parameters of the pre-selected thermal storage technologies, including charging and discharging efficiency and standby losses for the range of working temperatures [35,36,37,39].

	Density	Working Temperature	Specific Heat	Thermal Conductivity	Efficiency	Standby Losses
	$\mathbf{[}\mathbf{kg}·{\left({\mathbf{m}}^{\mathbf{3}}\right)}^{\mathbf{-}\mathbf{1}}\mathbf{]}$	${\mathbf{[}}^{\mathbf{\circ }}\mathbf{C}\mathbf{]}$	$\mathbf{[}\mathbf{kJ}\mathbf{·}{\mathbf{(}\mathbf{kg}\mathbf{·}\mathbf{°}\mathbf{C}\mathbf{)}}^{\mathbf{-}\mathbf{1}}\mathbf{]}$	$\mathbf{[}\mathbf{W}\mathbf{·}{\mathbf{m}·\mathbf{K}}^{\mathbf{-}\mathbf{1}}\mathbf{]}$	$\mathsf{[}\mathsf{\%}\mathsf{]}$	$\mathbf{[}\mathbf{mW}\mathbf{·}{\mathbf{(}\mathbf{kg}\mathbf{·}\mathbf{h}\mathbf{)}}^{\mathbf{-}\mathbf{1}}\mathbf{]}$
Cast steel	7800	200–700	0.60	40	50–90	0.2–0.6
					(90)	(0.4)
Cast iron	7200	200–400	0.56	37	50–90	0.2–0.4
					(90)	(0.3)
Magnesia fire bricks	3000	200–1200	1.15	5.0	50–90	0.4-2.7
					(90)	(1.6)
Reinforced concrete	2200	200–400	0.85	1.5	50–90	0.6–1.2
					(90)	(0.9)
Hot water	997	50–90	4.18	0.6	50–90	1.7–2.9
					(90)	(2.3)

.

The efficiency of charging and discharging processes varies considerably depending on the heat exchange method employed. In cases involving passive charging through an electrical heater, along with passive, direct discharging to the heating system using a single exchanger and external flow control, the charging and discharging efficiencies are set to 100% and 90%, respectively [37]. Developing a simplified and consistent model for standby losses across thermal storage materials with differing sensitivities proves more complex. These losses depend on factors such as insulation performance (conductivity), the temperature difference between the storage medium and the ambient indoor environment containing the tank, and the surface area of the storage unit [31,34,35]. Based on selected case studies and general thermodynamic principles, the authors determined the maximum and minimum efficiency values for high-temperature storage materials, and average values were used for optimisation.

Table 3. Economic parameters of the primary household energy system components [24,25,27,28,29,30,31,34,40].

		PV	Li-Ion	H2	FB	HTS
CAPEX (power)	$[\mathrm{EUR}·{\mathrm{kW}}^{-1}]$	1000–1300	200–400	1700–3700	400–850
		(1150)	(250)	(2000)	(700)	N/A
CAPEX (capacity)	$[\mathrm{EUR}·{\mathrm{kWh}}^{-1}]$	0	250–400	5–15	100–400	1–60
			(300)	(5)	(400)	(6.60)
OPEX (fixed, power)	$[\mathrm{EUR}·{(\mathrm{kWh}·\mathrm{year})}^{-1}]$	0–40	0–10	30–100	5–15
		(5)	(5)	(30)	(10)	0
OPEX (fixed, capacity)	$[\mathrm{EUR}·{(\mathrm{kWh}·\mathrm{year})}^{-1}]$	0	0	0-10	0-10
				(3)	(2)	0
OPEX (variable, energy)	$[\mathrm{EUR}·{\mathrm{kWh}}^{-1}]$	0	0	0	0	0
Lifespan	[years]	15–25	15–25	5–30	10–20	20–30
		(20)	(20)	(30)	(20)	(30)
Number of cycles	[–]	N/A	2000–10,000	>10,000	>12,000	Unlimited
			(5000)	(10,000)	(20,000)

displays the economic parameters for the key system components [24,25,27,28,29,30,31,34,40]. Values in parentheses and in bold indicate the primary data used in the calculations.

For HTS thermal storage, the selected CAPEX value relates to magnesia fire bricks (6) and cast steel (60) [39].

In the case of Li-ion battery storage, the literature review and studies conducted by the authors of [33] revealed a wide range of values and models for the degradation of battery capacity depending on the effective number of charging and discharging cycles. These parameters are strongly influenced by factors such as the battery brand (which may involve minor adjustments in the underlying chemical storage technology), operational strategy, environmental conditions, and calendar degradation [32]. In this study, the battery operation was limited to charging and discharging between a factor of 0.3 and 0.9 of its nominal capacity. It was assumed to operate within indoor temperatures ranging from 10 to 25 °C, following the manufacturer’s recommended nominal charging and discharging power. Based on these assumptions, the effective number of cycles before the battery must be replaced due to a reduction in operational capacity to 30% of its nominal value was established at a maximum feasible limit of 10,000 cycles.

6. Market Input Data

The regulatory framework, energy market design and rules, and economic parameters significantly influence the sizing of the roof PV and EES. They include imbalance settlement rules, electricity distribution tariffs, and environmental levies.

RCE prices from July 2022 to May 2025 were utilised to calculate gains and losses related to energy exchanges with the grid. They were adjusted to include applicable taxes and relevant distribution tariffs.

The statistical chart presented in Figure 3 illustrates trends in the Polish balancing market, highlighting a rising number of settlement periods characterised by negative prices. This phenomenon is due to overproduction linked to PV generation during peak solar radiation hours. High prices occur during evening hours when more costly coal and gas power plants are needed to balance the increased power demand. These large power stations must recover their start-up costs and account for environmental charges related to carbon dioxide emissions (EUAs).

The work in this paper considers future regulatory and environmental charges. Therefore, recent price movements in the CO₂ emissions and gas markets are analysed. The market data shown in

Table 4. Electricity and gas distribution tariffs, and average annual emission allowance prices (EUAs) for January 2022–June 2025 [35,36,41,42,43].

		2022	2023	2024	2025
Exchange rate	[PLN/€]	4.68	4.54	4.31	4.23
Network gas fuel tariff (W-2.1)	[€/MWh]	43.55	107.57–144.01	68.47	57.58
Network gas distribution tariff (W-2.1WA)	[€/MWh]	5.70	6.08	7.98	8.43
Electricity distribution tariff (G11)	[€/MWh]	49.53	82.16	88.56	89.60
Average annual price of EUAs	[€/${\mathrm{t}}_{{\mathrm{CO}}_2}$]	84.01	83.85	65.13	69.58

demonstrates a significant rise in electricity tariffs, reflecting increased costs associated with system balancing and the easing of transmission and distribution constraints.

The figures in Table 4 represent net values and require further comment. The substantial rise in energy costs for households between 2021 and 2023 led the Polish government to introduce various economic measures to help families facing rising energy bills. These temporary measures included tax exemptions and social tariffs for low-income families and low-consumption households. Such social benefits are not included in our studies.

7. Optimisation Model and Implementation

The optimisation problem involves the simultaneous sizing of the PV installation, the selection of storage technology, the specification of the ESS volume, and the efficient utilisation of these resources. The aim is to minimise the cost of supplying energy carriers to the household within the fully competitive retail energy markets. The optimisation covers 8760 h and determines both the optimal capacity for each of the pre-selected electrical and thermal storage technologies and the optimal utilisation programmes for the storage facilities to minimise the total cost of energy carriers, including annualised investment costs and operational expenses.

Given the scale of the optimisation task, characterised by a large number of optimisation variables (148,932), the range of optimisation methods available is limited. MINLP methods were excluded as they struggle with large-scale tasks, and the problem was modelled using linear programming (LP). The necessary simplifications were implemented to address the nonlinearity introduced by storage facility degradation, as well as the nonlinearities associated with charging, discharging, and standby losses in the storage systems. However, the nonlinear characteristic of the coefficient of performance (COP) for the air-to-water heat pump is fully modelled, with hourly COP values calculated based on the difference between outdoor temperature and the temperature of the building’s heating system water, as determined by the heating curve set by the building’s HVAC regulator (or the hot water tank, in the case of hot water preparation).

The optimisation model’s cost function covers expenses related to buying and selling energy from the grid ($C{F}_e$), the cost of pipeline gas purchase for heating and hot water preparation ($C{F}_g$), and both investment and operational costs for the PV generation plant, high-temperature thermal storage ($C{F}_{hts}$), battery electricity storage ($C{F}_{bat}$), hydrogen storage ($C{F}_{h2}$), and flow battery storage ($C{F}_{fb}$).

$$CF={CF}_e+{CF}_g+{CF}_{pv}+{CF}_{hts}+{CF}_{bat}+{CF}_{h2}+{CF}_{fb}$$

(2)

The individual elements of the above formula are described below (for variable symbols, see Appendix A):

$${CF}_e=\sum _{h=1}^{8760}{E}_{en.im}^h·(C_{eb}^h+C_{ed})-\sum _{h=1}^{8760}{E}_{en.ex}^h·\left(C_{es}^h\right)$$

(3)

$${CF}_g=\sum _{h=1}^{8760}{E}_{hg}^h·(C_{gb}^h+C_{gd})$$

(4)

$$\begin{array}{cc}\hfill {CF}_{pv}=& {\displaystyle \frac{pv·{CPXp}_{eg.pv}}{{lft}_{eg.pv}}}+{\displaystyle \frac{pv·{CPXe}_{eg.pv}}{{lft}_{eg.pv}}}+\hfill \\ \hfill & pv·{OPXf}_{eg.pv}+{OPXv}_{eg.pv}·\sum _{h=1}^{8760}pv·E_{eg.pv}^h\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill C{F}_{hts}=& {\displaystyle \frac{m_{hts}·{CPXp}_{hs.hts}}{{lft}_{hs.hts}}}+{\displaystyle \frac{m_{hts}·{CPXe}_{hs.hts}}{{lft}_{hs.hts}}}+\hfill \\ \hfill & m_{hts}·{OPXf}_{hs.hts}+{OPXv}_{hs.hts}\sum _{h=1}^{8760}d{E}_{hs.hts.Dw}^h\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {CF}_{bat}=& {\displaystyle \frac{E_{es.bat}^{max}·{CPXe}_{es.bat}}{{lft}_{es.bat}}}+\hfill \\ \hfill & k_{bat}·{\displaystyle \frac{{CPXe}_{es.bat}·{\displaystyle \sum _{h=1}^{8760}}{dE}_{es.bat}^h}{{nnoc}_{es.bat}}}·({lft}_{bat}-1)+{\displaystyle \frac{P_{es.bat}^{max}·{CPXp}_{es.bat}}{{lft}_{es.bat}}}+\hfill \\ \hfill & E_{es.bat}^{max}·{OPXf}_{es.bat}+{OPXv}_{es.bat}·\sum _{h=1}^{8760}{dE}_{es.bat.Dw}^h\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {CF}_{h2}=& {\displaystyle \frac{E_{es.h2}^{max}·{CPXe}_{es.h2}}{{lft}_{es.h2}}}+k_{h2}·{\displaystyle \frac{{CPXe}_{es.h2}·{\displaystyle \sum _{h=1}^{8760}}{dE}_{es.h2}^h}{{nnoc}_{es.h2}}}·({lft}_{h2}-1)+\hfill \\ \hfill & {\displaystyle \frac{P_{es.h2}^{max}·{CPXp}_{es.h2}}{{lft}_{es.h2}}}+E_{es.h2}^{max}·{OPXf}_{es.h2}+{OPXv}_{es.h2}·\sum _{h=1}^{8760}{dE}_{es.h2.Dw}^h\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {CF}_{fb}=& {\displaystyle \frac{E_{es.fb}^{max}·{CPXe}_{es.fb}}{{lft}_{es.fb}}}+k_{fb}·{\displaystyle \frac{{CPXe}_{es.fb}·{\displaystyle \sum _{h=1}^{8760}}{dE}_{es.fb}^h}{{nnoc}_{es.fb}}}·({lft}_{fb}-1)+\hfill \\ \hfill & {\displaystyle \frac{P_{es.fb}^{max}·{CPXp}_{es.fb}}{{lft}_{es.fb}}}+E_{es.fb}^{max}·{OPXf}_{es.fb}+{OPXv}_{es.fb}·\sum _{h=1}^{8760}{dE}_{es.fb.Dw}^h\hfill \end{array}$$

(9)

The second component of the above formulas represents a simplified model of the reduction in the effective lifespan of the considered storage technologies as the number of charging and discharging cycles per year increases. For hydrogen storage and flow batteries, the degradation impact factors ($k_{fb},k_{h2}$) take very small values, making the wear-and-tear component negligible. Whereas, for Li-ion batteries, the number of charging and discharging cycles per year influences the choice of technology and the capacity sizing ($k_{bat}\cong 1$).

Minimising the above cost function is subject to the following constraints:

• C.1. Home thermal balance

$$\begin{array}{cc}\hfill & E_{hdh}^h+E_{hdw}^h-{dE}_{hs.hts.Dw}^h·{\eta }_{hs.hts}-\hfill \\ \hfill & E_{ehh}^h·{COP}_h^h-E_{ehw}^h·{COP}_w^h-E_{hg}^h=0,\forall h=1,\dots ,8760.\hfill \end{array}$$

(10)

• C.2. Heat pump output limit

$$E_{hdh}^h-E_{ehh}^h·{COP}_h^h\ge 0,\forall h=1,\dots ,8760.$$

(11)

• C.3. Home power balance

$$\begin{array}{cc}pv·E_{eg.pv}^h+E_{en.im}^h+{dE}_{es.bat.Dw}^h·{\eta }_{es.bat.D}+{dE}_{es.h2.Dw}^h·& {\eta }_{es.h2.D}+\hfill \\ {dE}_{es.fb.Dw}^h·{\eta }_{es.fb.D}-E_{ed}^h·{dsm}^h-E_{ehh}^h-E_{ehw}^h-E_{ed}^h-& {dE}_{hs.hts.Up}^h=0,\hfill \\ & \forall h=1,\dots ,8760.\hfill \end{array}$$

(12)

• C.4. HTS energy balance

$$\begin{array}{cc}\hfill & E_{hs.hts}^1-{sbl}_{hs.hts}·m_{hts}-{\eta }_{hs.hts}·d{E}_{hs.hts.Up}^1+d{E}_{hs.hts.Dw}^1=0,\mathrm{for}h=1\hfill \\ \hfill & E_{hs.hts}^h-{sbl}_{hs.hts}·m_{hts}-{\eta }_{hs.hts}·d{E}_{hs.hts.Up}^h+d{E}_{hs.hts.Dw}^h=0,\forall h=2,\dots ,8760.\hfill \end{array}$$

(13)

• C.5. Li-ion energy balance

$$\begin{array}{cc}\hfill & E_{es.bat}^1-{sbl}_{es.bat}·E_{es.bat}^0-{\eta }_{es.bat}·d{E}_{es.bat.Up}^1+d{E}_{es.bat.Dw}^1=0,\mathrm{for}h=1.\hfill \\ \hfill & E_{es.bat}^h-{sbl}_{es.bat}·E_{es.bat}^{h-1}-{\eta }_{es.bat}·d{E}_{es.bat.Up}^h+d{E}_{es.bat.Dw}^h=0,\forall h=2,\dots ,8760.\hfill \end{array}$$

(14)

• C.6. H2 energy balance

$$\begin{array}{cc}\hfill & E_{es.h2}^1-{sbl}_{es.h2}·E_{es.h2}^0-{\eta }_{es.h2}·d{E}_{es.h2.Up}^1+d{E}_{es.h2.Dw}^1=0,\mathrm{for}h=1.\hfill \\ \hfill & E_{es.h2}^h-{sbl}_{es.h2}·E_{es.h2}^{h-1}-{\eta }_{es.h2}·d{E}_{es.h2.Up}^h+d{E}_{es.h2.Dw}^h=0,\forall h=2,\dots ,8760.\hfill \end{array}$$

(15)

• C.7. FB energy balance

$$\begin{array}{cc}\hfill & E_{es.fb}^1-{sbl}_{es.fb}·E_{es.fb}^0-{\eta }_{es.fb}·d{E}_{es.fb.Up}^1+d{E}_{es.fb.Dw}^1=0,\mathrm{for}h=1.\hfill \\ \hfill & E_{es.fb}^h-{sbl}_{es.fb}·E_{es.fb}^{h-1}-{\eta }_{es.fb}·d{E}_{es.fb.Up}^h+d{E}_{es.fb.Dw}^h=0,\forall h=2,\dots ,8760.\hfill \end{array}$$

(16)

• C.8. Li-ion and discharging applied capacity limits

$$\begin{array}{cc}\hfill & E_{es.bat}^h-E_{es.bat}^{max}·0.9\le 0\hfill \\ \hfill & E_{es.bat}^h-E_{es.bat}^{max}·0.3\ge 0,\forall h=1,\dots ,8760.\hfill \end{array}$$

(17)

• C.9. H2 charging and discharging applied capacity limits

$$\begin{array}{cc}\hfill & E_{es.h2}^h-E_{es.h2}^{max}·1\le 0\hfill \\ \hfill & E_{es.h2}^h-E_{es.h2}^{max}·0.01\ge 0,\forall h=1,\dots ,8760.\hfill \end{array}$$

(18)

• C.10. FB charging and discharging applied capacity limits

$$\begin{array}{cc}\hfill & E_{es.fb}^h-E_{es.fb}^{max}·1\le 0\hfill \\ \hfill & E_{es.fb}^h-E_{es.fb}^{max}·0.01\ge 0,\forall h=1,\dots ,8760.\hfill \end{array}$$

(19)

• C.11. Li-ion charging and discharging power limits

$$\begin{array}{cc}\hfill & {dE}_{es.bat.Up}^h-P_{es.bat}^{max}\le 0\hfill \\ \hfill & {dE}_{es.bat.Dw}^h-P_{es.bat}^{max}\le 0,\forall h=1,\dots ,8760.\hfill \end{array}$$

(20)

• C.12. H2 storage charging and discharging power limits

$$\begin{array}{cc}\hfill & {dE}_{es.h2.Up}^h-P_{es.h2}^{max}\le 0\hfill \\ \hfill & {dE}_{es.h2.Dw}^h-P_{es.h2}^{max}\le 0,\forall h=1,\dots ,8760.\hfill \end{array}$$

(21)

• C.13. FB charging and discharging power limits

$$\begin{array}{cc}\hfill & {dE}_{es.fb.Up}^h-P_{es.fb}^{max}\le 0\hfill \\ \hfill & {dE}_{es.fb.Dw}^h-P_{es.fb}^{max}\le 0,\forall h=1,\dots ,8760\hfill \end{array}$$

(22)

C.14.

The upper limit of PV generation per year, which, in the case of prosumers, should correspond to their yearly electricity consumption

$$\sum _{h=1}^{8760}\left(pv·E_{eg.pv}^h-E_{ed}^h-E_{eh}^h-E_{ew}^h\right)\le 0$$

(23)

• C.15. Maximum and minimum temperature limits for HTS

$$\begin{array}{cc}\hfill & E_{hs.hts}^h-(T_{hs.hts}^{max}+273)·m_{hts}·{cw}_{hts}\le 0\hfill \\ \hfill & E_{hs.hts}^h-(T_{tkh}^h+273)·{m_{hts}·cw}_{hts}\ge 0,\forall h=1,\dots ,8760.\hfill \end{array}$$

(24)

• C.16. Load flexibility constraint

$$\begin{array}{cc}\hfill & \sum _{i=0}^{23}{E_{ed}^{h+i}·dsm}^{h+i}=\sum _{i=0}^{23}{E}_{ed}^{h+i},\forall h\in \left\{1,25,49,73\dots ,8737\right\}.\hfill \end{array}$$

(25)

C.17.

Annual calendar cycle constraint for all considered types of storage facilities

$$\begin{array}{cc}\hfill & E_{hs.hts}^{8760}-E_{hs.hts}^0·1.1\le 0\hfill \\ \hfill & {0.9·E}_{hs.hts}^0-E_{hs.hts}^{8760}\le 0\hfill \\ \hfill & E_{hs.bat}^{8760}-E_{hs.bat}^0·1.1\le 0\hfill \\ \hfill & {0.9·E}_{hs.bat}^0-E_{hs.bat}^{8760}\le 0\hfill \\ \hfill & E_{hs.h2}^{8760}-E_{hs.h2}^0·1.1\le 0\hfill \\ \hfill & {0.9·E}_{hs.h2}^0-E_{hs.h2}^{8760}\le 0\hfill \\ \hfill & E_{hs.fb}^{8760}-E_{hs.fb}^0·1.1\le 0\hfill \\ \hfill & {0.9·E}_{hs.fb}^0-E_{hs.fb}^{8760}\le 0\hfill \end{array}$$

(26)

Beyond the constraint equations C.1–C.17, individual optimisation variables are subject to physical or grid connection code-imposed limits, which include the following: grid import and export limited to 12 kW (as specified by the supply contract and the protection device at the PCC), and charging and discharging limits for all considered energy storage technologies (also limited to 12 kW).

The optimisation model was developed in the MATLAB^® environment using the Optimisation Toolbox. The model is solved with MATLAB^®’s internal linprog function. The dual simplex optimisation algorithm was used with the standard constraint tolerance set to 1e-07. The computation time varies between 150 and 250 s, which is acceptable for practical application in HES design.

8. Scenarios

The optimisation model presented was employed to analyse past, present, and potential future scenarios.

Initially, the reference scenarios for each heating season (2022/2023, 2023/2024, and 2024/2025) were assessed. The scenarios S22A00, S23A00, and S24A00 describe the household without PV installation or HP heating. HES relies solely on grid-imported electricity and GB for heating and hot water preparation. These scenarios include the original hourly electricity demand profiles. In contrast, the scenarios S22B00, S23B00, and S24B00 explore the effects of potential load flexibility, applying a flexibility factor of 10% (demand-side management) and a load shift period within 24 h, which involves demand reduction and rebound. These assumptions regarding load flexibility were consistently used in all subsequent scenarios where DSM is included (constraint C.16).

The scenarios S22A02–S24B02 examine the effects of installing a heat pump and operating a hybrid heating system, where the use of HP or GB depends on outdoor temperature (COP) and electricity and gas prices. The scenarios S22A01–S24B01 reflect the modifications of S22A02–S24B02 with the addition of PV.

In the next stage, we examined the basic scenarios (S22A, S23A, and S24A) for each heating season to select and size electrical and thermal storage technologies, considering real energy market prices, distribution service fees, and environmental taxes that prosumers must pay. Like in the reference scenarios, the hourly load profiles were initially applied without considering load flexibility.

In the scenarios S22B, S23B, and S24B, the basic scenarios above were re-optimised to reflect the potential effects of load profile adjustments. The scenarios S22Ad–S24Bd include additional modifications that account for the impact of storage capacity decline.

Finally, we prepared and analysed future scenarios (SF0–SFds) using the most adverse market prices for pipeline gas (2023), along with EUAs from 2022 and RCE prices for the heating season of 2024/2025. These conditions present the best opportunities for price arbitrage, and with increasing expenses for energy carrier acquisition and rising risks in the balancing market, they promote greater household energy independence and HES decarbonisation.

Load flexibility is considered a standard solution in future scenarios, reflecting advancements in home automation and smart home appliances.

New environmental levies related to ETS2 are included in the cost of pipeline gas purchased. In Poland, the carbon dioxide emissions from a typical gas boiler range between 198 and 215 g/kWh of consumed pipeline natural gas, with the higher figure used in this paper.

In the reference scenario SF0, only GB is used for heating and hot water production, and PV, ESS, and load flexibility are not considered.

The scenario SF adopts a hybrid heating system that incorporates PV, HP, and ESS but does not account for storage capacity degradation. The wear-and-tear effects on all storage technologies are included in the scenario SFd.

Based on the future scenarios described above, a zero-emission household use case was developed, featuring a heating system that utilises only HP and HTS-based electrical heating (SFdCO2). It is important to recognise that achieving zero local emissions does not necessarily mean a household is carbon-neutral, as this depends on the overall power system’s electricity generation mix and the balance between energy exported and imported from the grid. In the next section, we report an analysis of household energy autarky (independence) separately across all future scenarios.

Finally, the scenarios SF and SFd, which use magnesia firebricks for HTS, were modified to investigate the use of cast steel. This raises the per-unit cost of storage, reduces the operational temperature range, and therefore requires more space for the same capacity. These are the scenarios SFs and SFds.

A summary of all the developed, analysed, and comprehensively detailed scenarios is presented in

Table 5. Summary of all analysed scenarios for the optimal modernisation and decarbonisation of the home energy system.

	GB	GB + HP	GB + HP + PV	GB + HP + PV + ES
BaU	S22A00, S23A00, S24A00	S22A02, S23A02, S24A02	S22A01, S23A01, S24A01
DSM	S22B00, S23B00, S24B00, SF0	S22B02, S23B02, S24B02	S22B01, S23B01, S24B01
ES				S22A, S23A, S24A
ES + DSM				S22B, S23B, S24B, SF, SFs
ESw&t				S22Ad, S23Ad, S24Ad
ESw&t + DSM				S22Bd, S23Bd, S24Bd, SFd, SFds

. In this context, the term BaU (business as usual) refers to reference scenarios that illustrate a typical Polish household energy system (HES) without local electricity generation or energy storage, and with a heating system dependent on a gas boiler. The acronym DSM signifies the incorporation of load flexibility into the BaU scenario. The designation ES indicates the HES equipped with an energy storage facility, while excluding the effects of degradation, which are addressed in the ESw&t scenarios. The column headings in the table reflect the incremental upgrades of the HES, commencing with the basic variant (GB, gas boiler), followed by the hybrid heating system (HP, heat pump), the inclusion of a roof-mounted photovoltaic installation (PV), and concluding with the implementation of various energy storage solutions (ES).

9. Analysis of the Optimisation Results

The core reference scenarios, which depend solely on grid-imported electricity and use a gas boiler for heating and hot water, show similar energy flows as depicted in Figure 4 for all three analysed heating seasons. In these basic reference scenarios, the two energy systems operate independently, and the possible synergy between electricity demand and heat needs is not explored.

The results for scenarios S22A01–S24B01, which feature PV generation without ESS, show that the optimised generation capacity ranges from 6.18 to 7.24 kWh across all the analysed scenarios. This relatively narrow range, as illustrated in Figure 5, is mainly determined by constraint C.14. This constraint limits the total output of the PV installation to the amount of electrical energy consumed by the household over a year. Polish distribution system operators (DSOs) now carefully assess this limitation before granting connection permits or approvals for expanding existing PV installations.

Assuming there is no direct curtailment by the DSO or TSO, the PV output in Figure 6 is directly based on the installed capacity.

For the SFd and SFdCO2 scenarios, it is important to highlight a case where both the peak PV power and electricity production could decrease, which is explained further in this section.

The optimisation results for technology selection and the sizing of the storage facility are shown in Figure 7 and Figure 8.

Based on the technical and economic parameters outlined in Table 1, Table 2, Table 3 and Table 4, along with the energy market data, only two ESS technologies are economically viable. The storage capacity of the Li-ion battery varies from 0 kWh in the scenarios S23Ad and S23Bd to 31.91 kWh in the scenarios S24A and S24B. The difference between the outcomes for the use cases S23A and S23B demonstrates that the impact of load flexibility is minimal (6.24 kWh versus 5.80 kWh).

Conversely, the decline in the capacity of electrical energy storage greatly influences capacity sizing. This effect is apparent across all scenarios using historical market data (S22A–S24Bd), as shown in Figure 7. The decrease in storage capacity is significant, especially when comparing the use cases S24A and S24B with S24Ad and S24Bd.

In situations where a decline in electricity storage capacity is considered, the reduction or removal of Li-ion storage volume is offset by employing high-temperature thermal storage with magnesium bricks. This HTS functions across a broad temperature range (200–1200 °C) and satisfies reasonable weight (36.1 kg) and space requirements (Table 2).

The optimised charging and discharging power for the Li-ion battery storage, as shown in Figure 8, generally aligns with the storage capacity. The highest values occur in scenarios S24A and S24B, where significant electricity exchange with the grid is driven by profits from price arbitrage opportunities.

The HTS charging and discharging power limit does not appear in Figure 8 because the grid export/import and the home electrical installation restrict direct electrical heating. Instead, the discharge capacity is determined by the thermal constraint (C.15) in the optimisation model described in Section 7.

The generation, flow, and usage of energy carriers for the scenario S24Ad are illustrated in Figure 9. The total output from GB has decreased to 2.06 MWh/year and is now limited to periods characterised by the lowest COP values, particularly during times of negative outdoor temperatures.

In the Sankey energy flow diagram in Figure 9 (and the subsequent diagrams of this type), the energy in and out balance at some nodes is not maintained. This approach is used to illustrate charge and discharge losses and to highlight the leverage of the heat pump operation (COP).

Figure 10 shows the total cost of the energy supply, including both CAPEX and OPEX expenditures.

A comparison of the reference scenarios A00 and B00, which represent the current standard household energy model relying on grid electricity and pipeline natural gas, with scenarios A01–B02, highlights the economic benefits of installing PV and modernising the heating system with an air-to-water heat pump on total energy provision costs.

Notably, the most significant example relates to future scenarios, highlighting the difference between the total costs of a business-as-usual approach (household without PV, HP, ESS) and a fully optimised renewable-powered home with storage facilities (SF, SFd, and SFdCO2).

It is important to note that the optimised energy vector and household energy system for the scenario SFd continue to utilise a minimal amount of pipeline natural gas (9.8 kWh) for heating purposes (Figure 11). In the complete decarbonisation scenario (SFdCO2), phasing out the gas boiler results in only a marginal increase in total costs, amounting to EUR 1370.82 compared to EUR 1370.65.

Furthermore, replacing magnesia fire bricks with cast steel for HTS results in a small increase in costs, amounting to EUR 1178.04 compared to EUR 1177.75 for the scenarios SFs and SF, respectively, and EUR 1396.03 versus EUR 1370.65 for the scenarios SFds and SFd. This demonstrates that, despite significant differences in the technical and economic parameters outlined in Table 2, both technologies provide comparable benefits.

The detailed breakdown of yearly expenses related to energy carriers and the maintenance of energy systems, as well as the annualised costs to upgrade HES with the installation of PV and ESS, is shown in Figure 12. Due to price fluctuations in the balancing market, between the heating seasons 2022/2023 and 2024/2025, the income from exporting energy to the grid has decreased significantly (comparing the scenarios S22Ad–S22Bd with S24Ad–S24Bd). The results from scenarios S24A–S24B, which do not account for the decline in battery storage, emphasise the importance of accurately modelling the battery wear-and-tear factor.

In the baseline future scenario SF0, the cost of burning natural gas rises significantly due to expected new environmental levies.

OPEX expenses are small and consistent across scenarios, as costs for PV and Li-ion batteries are minimal. In contrast, investment outlays are significant. In this context, the capacity decline effect included in Equations (5)–(9) of the objective function is considered part of CAPEX.

With minor adjustments to the PV installed power, differences in investment costs are mainly driven by the storage capacity.

Figure 13 shows the average cost per unit of energy used in households, including all energy carriers, distribution fees, and additional environmental costs (relevant to the scenarios SF0–SFds). Excluding scenarios that do not consider storage volume degradation, the equivalent energy price ranges from 0.072 EUR/kWh for S23Bd to 0.210 EUR/kWh for SF0. When comparing the two main reference scenarios, A00 and SF0, where households rely solely on grid electricity and pipeline natural gas, there is a projected increase in the cost of energy carriers by 76.4% (scenario S23A00 with a per unit cost of 0.119 EUR/kWh versus SF0).

The energy supply mix for the heating system, shown in Figure 14, displays a gradual shift from using a gas boiler in the basic reference scenarios to a heat pump and direct electric heating (S24Ad), which also incorporates HTS to benefit from favourable electricity prices. Naturally, with the introduction of additional environmental levies (ETS2), dependence on pipeline gas is almost eliminated.

Future scenarios (e.g., SFd, Figure 11) indicate minimal use of gas boilers, mainly during hours when outdoor temperatures drop below zero. This shows the advantages of a hybrid heating system. A more flexible EU regulatory framework, which encourages the phase-out of gas boilers through market signals and environmental taxes rather than imposing a legal ban, appears to be a more effective decarbonisation path from the prosumer’s perspective.

Figure 15 shows the use of the optimised ESS volumes. Out of two economically viable and competitive options, the use of the Li-ion battery is influenced by the volatility and hourly fluctuations in electricity prices. HTS storage only becomes economically viable when considering the wear-and-tear costs in the objective function.

Replacing magnesia fire bricks with cast steel effectively eliminates thermal storage capacity (SFd versus SFds) due to higher investment costs.

Using the results obtained for scenario S24Bd, an analysis of the differences in hourly and daily utilisation of electricity storage and thermal storage was conducted.

Table 6. Basic statistics for the state of charge of storage facilities, and volumes of charging and discharging activities.

	Li-Ion	HTS
Optimised capacity [kWh]	5.53	11.53 1
Number of accomplished cycles 2	195	96
Number of charging periods	2744	682
Number of discharging periods	1732	1353
Correlation, RCE versus charging volumes	−0.39	−0.42
Correlation, RCE versus discharging volumes	0.55	0.17
Standard deviation, SOC 3	1.30	3.61
Standard deviation, charging	0.49	1.41
Standard deviation, discharging	0.33	0.53
Mean SOC	2.98	5.48
Mean charging	0.39	1.63
Mean discharging	0.59	0.55

¹ For thermal storage, the optimised storage capacity is derived from the optimised mass and the range of the working temperatures. ² The number of cycles is calculated here as the volume of energy flowing through the tank during the year divided by the optimized, nominal capacity of the tank. ³ State of charge of the storage facility.

summarises the performance of the two most competitive storage technologies.

The key differences are highlighted in bold. These differences illustrate the effective utilisation of storage volumes, which is measured by the number of charging and discharging periods, the correlation between discharging activities and RCE prices, the state of charge (SOC), and the volumes of charging and discharging.

Electricity storage exhibits a high frequency of charging and discharging activities, typically involving small amounts, driven by hourly price arbitrage and consistent use throughout the year. In contrast, HTS is mainly used during high-PV-output seasons, with winter charging primarily triggered by especially low or negative market prices. During these short periods of low or negative electricity prices, HTS charges large volumes, followed by gradual discharging until the next price opportunity. Therefore, lithium-ion batteries act as short-term storage solutions, with their charging cycles aligned with hourly and daily PV and load balancing. Meanwhile, HTS storage functions more as medium-term storage, characterised by irregular charging cycles that go beyond daily or even weekly patterns, particularly in winter, gradually discharging to meet the hourly demands of the heating system.

Figure 16 compares on-site energy generation (PV), imported and exported electricity, and household consumption. It shows that, in most scenarios, local generation closely matches local consumption. However, this balance shifts in scenarios S24Ad and S24Bd, where the household’s reliance on imported electricity increases again. This situation happens because ESS explores price arbitrage opportunities, which boost storage capacity but reduce household energy autonomy.

In scenarios that heavily utilise high-temperature thermal storage, such as S24Ad, S24Bd, SFd, and SFdCO2, Figure 16 shows an increase in overall electricity consumption while exports to the grid decrease. The surplus energy generated on-site is used for heating and hot water preparation via the HTS buffer. As a result, the gap between imported and exported electricity widens, since the optimised system benefits from very low or negative prices in the balancing market, making it economically viable to charge the HTS through passive electrical heating.

A more detailed analysis of household energy autarky, considering all energy carriers, is presented in Figure 17, which illustrates both absolute and relative energy autonomy.

Absolute autonomy refers to the proportion of energy generated by PVs that is either directly used on-site or stored in an ESS for later use, without interacting with the grid, as well as the energy consumed on-site. Relative autonomy describes the relationship between the total energy produced on-site, including energy exported to the grid, and local consumption. In this context, the heat pump’s contribution, measured by its COP, is regarded as a local zero-emission energy source that provides additional thermal energy.

The chart indicates that even without PVs and ESSs (as seen in scenarios S22A02, S22B02, S23A02, S23B02, S24A02, and S24B02), HP helps to achieve relative energy autonomy levels ranging from 21.37% to 28.22%. Local electricity generation from PVs boosts this relative energy autonomy to between 80.79% (scenarios S24A01, S24A02) and 93.40% (scenarios S23A01, S23A02), with absolute autonomy achieved at 77.78% (scenario S23B01).

Excluding unrealistic scenarios that do not account for storage capacity decline, the deployment of electrical and thermal storage facilities raises the relative autonomy to 93.21% and achieves an absolute independence of 77.57% (S23Bd). The results for the season 2024/2025 are slightly less favourable due to decreased PV capacity and a growing gap between energy imported from and exported to the grid. In future scenarios considering the impacts of new emission fees linked to ETS2 implementation, relative autonomy nearly reaches 99%, while absolute independence reaches 94.32% (SFds), despite minimal pipeline gas use in the heating system. The results for the SFdCO2 scenario, where gas boilers for heating are banned, show marginally lower figures (93.92% for relative autonomy and 86.52% for absolute autonomy). This pattern arises from increased electrical energy demand and the widening gap between grid energy imports and exports, as previously discussed.

CAPEX expenditures are shown in Figure 18. Excluding all unrealistic scenarios, the investment ranges from EUR 7107.98 for the smallest PV installation without ESS (S24A01, S24B01) to EUR 15918.42 for the scenario with the largest lithium-ion battery (SFds), as illustrated in Figure 6. The difference in investment costs between SFd and SFds (EUR 13,214.56 versus EUR 15,918.42) indicates that a comparable economic benefit for the customer, measured by the average energy price of 0.103 EUR/kWh versus 0.100 EUR/kWh (Figure 13), can be achieved with significantly lower expenditures by engaging an HTS buffer to explore the synergy between two household energy systems.

Up to this point, the analysis was conducted using selected values of economic parameters outlined in Table 3, which, according to the reviewed reference publications, may vary significantly. Therefore, the authors performed a cost sensitivity analysis to evaluate the profitability and competitiveness of targeted storage technologies across a comprehensive range of capital costs ($CPXe$). The analysis is based on scenario S24Bd.

The results, shown in Figure 19, indicate that three storage technologies can be cost-effective: Li-ion batteries, HTS using magnesia bricks, and redox flow batteries. The crossover cost of Li-ion batteries compared to other storage options is 250 EUR/kWh, while the entry point for redox flow batteries occurs when the $CPXe$ is priced below 300 EUR/kWh.

Compared to electricity storage, which can be used for both balancing electrical loads and supporting heating systems, thermal storage facilities are limited to domestic heating and hot water systems. Thermal storage becomes economically viable and competitive with electricity storage when $CPXe$ exceeds 250 EUR/kWh for redox flow batteries and 300 EUR/kWh for Li-ion batteries, respectively.

10. Conclusions

This publication presents findings from ongoing research focused on optimising the design and operation of energy systems in detached or semi-detached homes. The authors have developed a linear optimisation model for the combined sizing of the key HES components: PV and ESS.

• Economic feasibility

Among the four optimised storage technologies—Li-ion batteries, hydrogen storage using electrolysers and fuel cells, redox flow batteries, and two high-temperature thermal storage methods—only two proved to be economically viable and competitive within the current market and regulatory framework without the need for investment subsidies.

• Electricity market trend analysis

With PV installed capacity and generation rising, electricity market prices are becoming more volatile and increasingly influenced by weather conditions. Consequently, the growing number of hours with negative prices encourages the adoption of alternative heating methods and emerging thermal storage solutions.

The analysis of the three heating seasons confirms that price arbitrage opportunities and the implementation of ToU prices drive the feasibility and profitability of storage facilities.

The notable differences in the results for the two consecutive seasons, 2023/2024 and 2024/2025, emphasise the importance of a proper market design and regulatory framework (settlement rules) for the broader adoption of storage facilities in residential installations.

• Home energy system decarbonisation

Beyond economic benefits for prosumers and technical advantages for grid and system operators, choosing the appropriate size and type of electricity storage technology enhances household energy independence and reduces environmental impact. CO₂ emissions from burning natural gas and those linked to electricity imported from the grid—in Poland mainly produced by large coal-fired power stations—are nearly eliminated in the likely future scenarios. The analysis of future scenarios, which include additional environmental levies on pipeline gas used for heating and hot water preparation, was carried out using the most adverse EUA prices (84.01 EUR/${\mathrm{t}}_{{\mathrm{CO}}_2}$, 2022).

This analysis indicates that replacing gas boilers with the widely adopted air-to-water heat pumps in Poland could be economically feasible. However, for existing buildings with the necessary infrastructure already in place, maintaining a hybrid heating system may still offer modest benefits for consumers during periods when gas boiler heating is activated due to low outdoor temperatures.

• Optimisation model efficiency and further improvements

Regarding the proposed optimisation model, the selection and sizing of storage technologies largely depend on modelling the decline in storage capacity caused by increasing charging and discharging cycles, especially with Li-ion battery storage. A simple linear decline model has been used, but this aspect requires further research and refinement. Particular attention should be given to decommissioning and recycling costs, which, at the time of working on this publication, were difficult to assess.

The impact of limited load flexibility is minor, but it is doubtful that the flexibility factor could be increased above 10% without decomposition of the load profiles into individual devices. Unfortunately, metering data for the most energy-intensive household appliances has only recently become available from several monitored households, and a more detailed study of demand-side management (DSM) will be included in future publications. Considering the limited economic impact, this area is not a priority for future research aimed at practical applications.

On the other hand, it is important to model standby losses for high-temperature thermal storage in greater detail. This must be achieved without compromising the linearity of the optimisation model or its computational efficiency, which presents significant research challenges. Progress in this area will rely on gathering the necessary measurement data from medium- and long-term thermal storage systems, as this data will be crucial for validating the improved linear model.