Impact of policies on residential multi-energy systems for consumers and prosumers

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Introduction
In multi-energy systems (MES), different energy carriers such as electricity, heat, and gas interact with each other. When optimally designed and operated, MES can outperform energy systems without sector coupling in terms of economic, environmental, and social sustainability [1][2][3][4][5]. MES can thus contribute to the transition towards affordable, low-carbon and secure energy. Since policies affect the adoption and the design of MES, questions arise about how policy design enables the benefits of MES, and how they shape the interplay between the three sustainability dimensions. Models can be used to understand the impact of policy schemes on the transition to sustainable energy systems [6,7]. The impact of policy mechanisms on energy choices of end-users might differ depending on their ability to benefit from them. This paper aims to understand the impact of policy schemes on the design of MES and their economic, environmental, and social sustainability whilst considering differences in the ability of consumers and prosumers to benefit from subsidies.

Background and scope of the study
We build upon the state-of-the art in two directions, namely the multi-criteria optimization of multi-energy systems and the assessment of policy mechanisms within energy systems models.

Multi-criteria optimization to identify sustainability benefits of MES
Optimization techniques allow to assess different configurations of the design and operation of an energy system, which can favor one objective over others. Trade-offs are typically found when optimizing for one objective, such as CO 2 emissions, which in turn leads to a significantly worse performance in other aspects, such as the cost of energy [3]. Optimization has been applied extensively to assess potential sustainability benefits of MES, and potential trade-offs between them. Economic, environmental, and social sustainability are assessed through multi-objective optimization, where they are represented by cost, CO 2 emissions, and self-sufficiency, respectively, as the objective function [4]. Pareto fronts along the three sustainability dimensions reveal trade-offs and interdependencies between the three objective functions [8,9], thereby also assessing the compatibility of the three sustainability dimensions in MES design and operation. Other studies focus on two of the three sustainability dimensions. Some assess the compatibility of cost and emission objective functions, finding that economic and environmental sustainability are not mutually exclusive when designing energy systems [3,5,10,11]. Other studies consider social sustainability by assessing the feasibility and cost of energy autarky [12][13][14][15], mostly finding that energy autarky is achievable, albeit at a significantly higher cost than non-autonomous energy systems [16]. These findings suggest that the objectives of social and economic sustainability need to be weighed against each other.
Our work builds on the efforts of identifying trade-offs and synergies between the three sustainability dimensions and assesses which policies enable MES that combine economic, environmental, and social sustainability objectives. Thus, it aims at bridging the gap between the theoretical sustainability potential of MES, which is assessed through multi-objective optimization, and its policy-enabled realization.

Policy mechanisms in energy system modeling
The effect of policy mechanisms on the design and operation of energy systems has been the subject to various studies in energy systems modeling, as summarized in Table 1. A carbon tax is included in a considerable share of studies, and is found to be a cost-efficient policy to enable decarbonization [17][18][19], although its effect was found to be dependent on other factors such as energy prices [7]. Economic studies confirm these findings from the engineering discipline, identifying a price on carbon as the most cost-efficient policy to foster decarbonization [20,21], confirming its effect on reducing emissions via ex-post analyses [21,22], and emphasizing its importance in enabling the energy transition [23]. In addition to a carbon tax, feed-in tariffs (FiT) and investment support mechanisms for photovoltaic (PV) systems are examined in multiple studies. There is agreement about their contribution to lowering the CO 2 emissions and increase self-sufficiency [17,18]. However, these mechanisms may induce high societal costs [24,25], and require a high policy level for any effect to become evident [26]. In contrast to the aforementioned incentive-based policies, there are also regulation-based policy mechanisms. One example are carbon caps, which impose a constraint on the CO 2 emissions that can be emitted by the energy system. Carbon caps increase costs for the energy system [27], with independent energy systems in island mode experiencing higher cost increases than systems that are connected to a larger system [26].
To the best knowledge of the authors, the difference in the impact of policy mechanisms on consumers and prosumers has so far not been studied in the context of energy systems optimization. However, consumer heterogeneity has been accounted for in adjacent fields. In the field of modeling, ex-ante analysis of policies and an anticipated analysis of their effectiveness is common. For example, agent-based modeling is used to model climate change mitigation policies and allows for including heterogeneous consumers [28]. They can represent agent interactions and be used to assess energy equity implications of policies for different stakeholders [29,30]. Moreover, total cost of ownership models of electric vehicles can account for heterogeneous driving patterns of consumers [31]. In the field of economics, consumer heterogeneity can be considered in ex-post analyzes and surveys. Examples for ex-post analyzes are accounting for energy poverty when assessing energy savings due to policies [32], distinguishing between private and business users when assessing the factors fostering electric vehicle adoption [33], and considering the consumer's energy literacy when assessing energy efficiency decisions and deducing policy implications [34]. Surveys can be used to identify differences in preferences with respect to solar energy policies [35].
Our work builds on the findings of the field of energy systems modeling and assesses the impact of a feed-in tariff, a PV rebate, a carbon tax, and a carbon cap on the design and operation of MES. Moreover, we learn from adjacent fields and account for end-user heterogeneity by distinguishing between consumers and prosumers, who do not and do benefit from the subsidy schemes above (i.e. FiT and PV rebate), respectively. This enables a more detailed analysis on who bears the cost of different policy mechanisms, which is an aspect so far neglected by previous studies. Moreover, we quantify and compare the impact of these policy mechanisms in terms of economic, environmental and social sustainability.

Scope and structure of this work
Previous work investigate the role of MES towards a sustainable energy system, and identify trade-offs and synergies with respect to the individual sustainability dimensions. However, less attention has been paid to the incentives that could enable the potential sustainability benefits of MES and their impact on heterogeneous end-users, with most studies focusing on specific policy mechanisms. This study builds on these efforts and bridges the gap between the potential sustainability benefits of MES, that have been previously identified via multi-objective optimization, and their realization through policies. Table 1 Overview of energy system modeling studies assessing the impact of policy mechanisms on energy systems. Scope refers to the geographic scope of the study. Distributed refers to small-scale application with decentralized production of heat and electricity. Regional and national refer to a wider geographic scope, which includes centralized electricity production units. Hence, our study offers insights into how different policy mechanisms impact the three sustainability dimensions, and how this impact differs for different end-user types. It combines the multi-criteria assessment of MES with a quantification of the impact of different policy mechanisms on the economic, environmental, and social sustainability of MES, while considering the uncertainty of the most relevant energy quantities. Additionally, the cost of the policies is quantified for different types of end-users as a levy on the retail energy prices, thus assessing whether they are fairly distributed. The application of the general methodology is exemplified with reference to the Swiss case study. This paper is structured as follows. Section 3 describes the policy implementation, the optimization problem, the sustainability metrics, and the sensitivity analysis developed to investigate the impact of policy mechanisms on MES design, operation and objectives. Section 4 presents and discusses the application to the Swiss case, and Section 5 draws conclusions.

Methods
Four policy mechanisms are modeled that impact the decisionmaking of end-users and hence the sustainability of the resulting energy system: (1) a feed-in-tariff for locally produced electricity, (2) an investment support mechanism called a PV rebate, (3) a carbon tax, and (4) a forced emission reduction, i.e. a carbon cap policy. The decisionmaking is approximated by an optimization approach, in which the end-users minimize their total annual cost related to their energy needs.
Thus, perfectly rational decision-making, full knowledge and perfect foresight are implicitly assumed. The study distinguishes between endusers: Those, who have the option to install PV and can therefore benefit from PV-related subsidies (prosumers), and those who cannot (consumers). The optimization model determines the technology capacities and operation, the cost of energy for both end-user types, the carbon emissions, and the self-sufficiency of prosumers. Uncertainties of input assumptions are tackled via a global sensitivity analysis. Fig. 1 provides an overview of the applied methodology.

Policy implementation
Five policy levels are tested for each policy mechanism, ranging from no policy being in place to a maximum value, and where possible Swiss reference values were used to be consistent with the Swiss case study: (1) The feed-in-tariff in Switzerland varies between municipalities.
Here, a maximum value of Rp 23/kWh is assumed, which is the highest feed-in tariff currently being paid by any municipality in Switzerland [36]. One Rappen (Rp) is one hundredth of a Swiss Franc (CHF). (2) Residential PV systems in Switzerland currently receive an initial one-off rebate, and a feed-in-tariff during the operation of the PV system [37]. The PV rebate in Switzerland is designed to account for around 20% of the capital cost of new PV systems and is Overview of the methodology. The analysis consists of two different parts: A reference case, in which reference values are considered for the input data assumptions, and a sensitivity analysis, in which the uncertainty associated with the input data is accounted for.  [38,39]. To broaden the range of possible rebates, a maximum rebate of 40% is considered in this analysis. (3) The maximum carbon tax considered is CHF 336/tCO 2 , which is the maximum carbon tax considered by the Swiss authorities by 2030 [40]. (4) The highest carbon cap policy level corresponds to the lowest attainable CO 2 emissions, as determined by the optimization model described in Section 3.2. Different policy levels correspond to different values of allowed CO 2 emissions of both end-user types. To illustrate, consider a medium level of the carbon cap policy mechanism: each end-user needs to achieve 50% of their maximum CO 2 emissions reduction, which differs depending on their ability to install PV. Table 2 lists all policy mechanisms and their corresponding numerical values.
In Switzerland and Germany, the cost of support schemes for lowcarbon technologies is passed on to the end-users through a levy on retail energy prices [41,42]. The levy is designed to redistribute the cost of subsidies, and this analysis aims at quantifying how different types of end-users are affected by this mechanism. Namely, this study distinguishes between consumers and prosumers. The measure in which the two types of end-users benefit from policies, depend on the specific policy mechanism. The levy allows to assess, if the cost of policies are fairly distributed across end-users.
The levy is calculated via an iterative approach to account for nonlinear feedback effects (see Eq. (2)), which occurs due to mechanisms such as a fuel switch away from electric heating stemming from the increase in retail prices [43]. To include these non-linear feedback effects, the levy, , is designed to cover the cost of the feed-in-tariff C FiT and of the PV rebate C R through a surcharge on the retail electricity price. It is calculated by dividing the total cost of policies by the overall electricity imported U e, , by all end-users ∈  at all time steps ∈ {1, … , }: Recently, it was criticized that much of cost for the energy transition is borne by electricity consumers, which discourages investments in heating electrification [44]. To provide an alternative assessment, we are considering a cost distribution mechanism where the levy applies to all retail energy imports, i.e. gas and electricity. In that case, the levy is calculated as: The cost of the FiT is calculated as the difference between the remuneration received via the FiT and the revenue that would have been achieved on the wholesale day ahead market: where V e, , is the electricity exported by end-user at time step , FiT is the FiT remuneration, and WH, is the wholesale electricity market price at time step . The cost for the PV rebate is calculated as a fraction of the overall investment cost of the PV installations of all prosumers: where PV is the unit installation cost of PV systems, S PV, is the size of the PV system of prosumer , and s R is the rebate level. The exported electricity, the size of the PV system, and the imported electricity are decision variables of the optimization model. Therefore, the introduction of the levy introduces a non-linearity to the model (see Eq. (2)). This is handled by adopting an iterative approach, which starts from a guess value of the levy, and solves the optimization model iteratively until the value of the levy converges with a tolerance of 10 −3 . Typically, convergence is achieved in five to six iterations. Fig. 2 illustrates the approach for calculating the levy.

Fig. 2.
The iterative optimization approach, in which the levy on retail electricity price converges to a value where it covers the cost of the feed-in tariff and the PV rebate. The optimization is performed separately for both end-user types, reflecting a shift towards decentralized decision-making.

Optimization problem
The optimization problem minimizes the cost of energy supply to the heat and electricity demand of end-users, which are assumed to be single households. Households can meet their demand through installing and operating technologies and via electricity and gas imports from the distribution grids. The available technologies include electricity-driven heat pumps (edHP), natural gas boilers, hot water thermal storage tanks (HWTS), and lithium-ion batteries. Additionally, for prosumers solar photovoltaics (PV) is available. Thus, two types of end-users are defined: Those who have the option of installing PV, and those who do not have that option. An overview of the available technologies and the resulting MES is shown in Fig. 3.
The solar irradiance, energy import prices, technology characteristics and unit costs, and energy demand profiles are input parameters to the optimization model. The model results include the selection, size and operation of technologies, operational and annualized capital cost, and associated emissions. The optimization model builds upon earlier work and it is formulated as a mixed integer linear program (MILP), which determines the optimal selection, size and operation of the available technologies that minimize the total annual cost of the system while meeting the energy demands of the end users [3,45]. The approach was adapted to include the non-linear feedback from introducing a levy to finance the PV subsidies and Fig. 2 provides an overview of the resulting optimization approach. The MILP is implemented in Matlab using Yalmip [46] and solved using Gurobi [47]. In . .
where c and d correspond to the cost vectors associated with the continuous and binary decision variables x and y with the dimensions and , respectively. and are the constraint matrices for x and y, respectively. b is the known term of the inequalities. For a detailed description of the input data, decision variables, constraints, and objective function of the model please refer to Appendix A.

Sustainability dimensions
Economic, environmental, and social sustainability, are quantified as the cost per unit of energy demand , self-sufficiency , and emissions per unit of energy demand , respectively. The cost per unit of energy demand is calculated by dividing the total annual cost of the end-user by their total electricity e, , and heat demand h, , summed over all time steps .
The cost differs between both types of end-users due to their differing ability to install PV and, therefore, produce electricity locally. Thus, this approach allows to assess the distribution of costs amongst consumers and prosumers, which is a relevant aspect of the economic sustainability dimension that is often overlooked. Increasingly, municipalities and decentralized energy systems target a balanced energy autonomy [48], which requires that over the course of one year, the electricity exports exceed the electricity imports of the system [49,50]. Thus, this definition of self-sufficiency is used for this study, and it is calculated as the fraction of energy that is produced locally and not imported in form of electricity or gas from the distribution grid. Thus, the self sufficiency is calculated as the electricity that is produced locally PV, , divided by the electricity demand including the amount of electricity covering the heat demand when electrified through heat pumps: where edHP denotes the coefficient of performance of the heat pump. Lastly, the emissions per energy demand are computed by dividing the overall CO 2 emissions of the end-user related to their electricity and gas imports by their total electricity and heat demand: where e, , and g, , denote the energy imports of electricity and gas, respectively, and e, , and g, , describes the corresponding emission factors.

Sensitivity analysis
The optimization model is solved deterministically, i.e. a perfect knowledge of all input parameters is assumed. However, the uncertainty associated with the input data is tackled via global sensitivity analysis. The parameters subject to uncertainty analysis are selected based on their high impact on the optimal design and operation of the resulting energy system [51][52][53][54].
A PERT distribution is used to model the probability density of all parameters, which requires a minimum, maximum, and most likely, mode, value. The parameter ranges are deduced from available scientific literature, as well as from commercially available products. The minimum, maximum and mode values of all parameters that are included in the sensitivity analysis and their corresponding references are listed in Table A.1 in Appendix A.5. The computational burden of the sensitivity analysis is reduced via Latin Hypercube Sampling (LHS). LHS is particularly suitable to perform sensitivity analyses of computationally intense models due to its effective stratification across the range of uncertain parameters, which allows for a smaller number of samples than other approaches [55]. For the set of input parameters = ( 1 , 2 , … , ), LHS divides the sample space of parameter ∈ into disjoint intervals with equal marginal probability 1 . For each parameter , one sample from each interval is taken, so that realizations of the parameter are sampled. The realizations of the parameters are combined to input scenarios in a random manner without replacement [56]. Here, = 300 scenarios are created.

Sustainability dimensions
The effect of the policy types and levels on the emissions and cost related to the energy supply of households is displayed in Fig. 4. Restricting the emissions, i.e. a carbon cap, is the most cost-effective policy to decarbonize the residential energy supply. However, this policy translates least directly to a mechanism that can be implemented in reality, since such an emission limit would be difficult to enforce on the individual household-level. The second most cost-effective policy to decarbonize the energy supply is the carbon tax. Here, it should be noted that the model does not account for the re-investment of the carbon tax and therefore underestimates its cost-effectiveness. If the carbon tax is used to achieve further decarbonization or if it is given back to the households who paid it in some form, the cost efficiency of the carbon tax could be improved, hence the related results in Fig. 4 can be interpreted as conservative. The subsidy mechanisms aimed at fostering PV adoption are less successful in achieving decarbonization. The feed-in-tariff increases both cost and emissions of the energy supply, and the PV rebate increases the cost while having only minimal impact on the emissions. In the following, the impact of the policies on the three sustainability dimensions is discussed in detail. Fig. 5(a) and (b) show the cost of final energy consumption for both end-user types, depending on the policy type and level. Comparing the cost for consumers and prosumers types shows that having the option to install PV results in lower total annual costs, consistently for all policy levels and types. However, incentive-based policies, i.e. a higher FiT and a higher rebate on PV installations, only decrease the cost for the prosumers, but increase the cost of the consumers. This can be explained by the levy, which increases with an increasing levels of incentives and thus makes importing electricity from the distribution network more expensive (Fig. 6(a)). Prosumers can avoid paying the levy by producing electricity locally, leading to a disproportional share of cost that is paid for by consumers. It should be noted that in Switzerland the levy is capped at 2.3 Rp/kWh, which is exceeded in case of   the high and maximum FiT level. Thus, the inequality observed in these cases is greater than what would be seen in reality. Nonetheless, also in the cases where the limit is not exceeded, an unfair distribution of cost can be observed. In contrast, the carbon tax and carbon cap increase the cost for all end-users, since they do not include a mechanism such as the levy. Fig. 5(c) reveals a trade-off between subsidizing renewable energy deployment and increasing the retail electricity price through the levy. Despite fostering PV adoption, the FiT actually increases overall emissions, and the PV rebate fails to achieve substantial decarbonization of energy supply. For low FiT levels and low to high rebate levels, the installed PV capacity increases and thus the emissions related to electricity imports decrease slightly. The retail price increases, but since the overall cost for the subsidies is moderate, this increase does not lead to a significant shift away from heating electrification, hence from heating decarbonization. However, such a shift away from heating electrification can be observed for higher subsidy levels: The high cost of the subsidies leads to a high levy on the retail electricity price, making electric heating uneconomic and, therefore, increasing the use of fossilfuel powered heating systems (Fig. 7). This increase in emissions from the use of gas boilers cannot be offset by the reduction of emissions from electricity imports due to the installation of additional PV systems. Therefore, the overall system emissions increase, if the subsidy level is not chosen adequately. In contrast, the two policy types targeting decarbonization in general, namely the carbon tax and carbon tax, increase the deployment of heat pumps and therefore achieve greater decarbonization. Fig. 7(c), (d) suggests that the shift away from electric heating due to a high levy on retail electricity might be alleviated by applying the levy to all energy imports, i.e. gas and electricity. Doing so can result in emissions savings of up to 41% as analyzed in Appendix B. Fig. 5(d) shows the self-sufficiency of prosumers as a function of the policy type and level. All policy types foster PV adoption and therefore increase self-sufficiency. However, there are significant differences in the extent of PV deployment. The FiT is the most effective policy in fostering PV adoption, with full self-sufficiency achieved already for a high policy level at a cost increase across all end-users of 6% ( Fig. 5(a) and (b)). However, the cost is distributed unevenly, with the prosumers actually paying 4.5% less and the consumers paying 17% more than in the no policy case (Fig. 5(a) and (b)). This uneven distribution of cost needs to be considered when designing policy mechanisms aimed at fostering self-sufficiency. The PV rebate also fosters PV adoption, but to a lesser extent than the FiT. The carbon cap policy achieves full self-sufficiency only when the maximum policy level is enforced (i.e. emissions are minimized). The carbon tax has the lowest impact on the self-sufficiency, even though it achieves a significant emission reduction as seen in Fig. 5(a) and (b). However, the decarbonization efforts incentivized by the carbon tax are mainly realized via electrification of heat (Fig. 7) and only to a lesser extent via the deployment of PV. This reflects the trade-off between deploying PV to decarbonize electricity and electrifying heating to decarbonize heating. In the case of the carbon tax, heat decarbonization occurs first as it is the most cost-efficient approach.

Social sustainability
Another important aspect of self-sufficiency is the share of electricity that is used locally, which is displayed in Fig. 8. This highlights the difference between policy mechanisms subsidizing PV and policy mechanisms targeting emissions reduction directly. Whilst PV subsidy mechanisms foster the adoption of PV, they do not incentivize the local utilization of the produced electricity, which can be achieved by offsetting the mismatch between energy generation and demand via battery or thermal storage. Thus, the increase in PV adoption leads to a sharp increase of exported electricity. These findings are aligned with the literature, which criticizes the aim for energy autonomy, since L. Brodnicke et al. the stress on centralized infrastructure is actually increased through extensive electricity exports and the subsidized renewables are paid for by others [57]. In contrast, the carbon tax and the carbon cap correspond to a higher deployment of thermal storage (Fig. 9), which allows shifting electricity-driven heat production to times where locally produced electricity is available, hence increasing the local electricity utilization. Fig. 9 shows the sizes of the heating technologies, namely the boiler, edHP, and HWTS, for all policy types and levels for the case that the levy applies to retail electricity imports. The carbon tax and the carbon cap schemes foster the electrification of the heating sector, whereas FiT and PV rebate lead to a shift towards fossil-fuel based heating. Moreover, much larger thermal storage installations are observed with policies schemes that target carbon reduction (i.e. carbon tax and cap), which allow to shift heat production via edHP to times where locally produced electricity is available. Moreover, the capacities of PV and battery are shown in Fig. 9 (for the prosumers only). Following the same argumentation as for the self-sufficiency, the FiT is the most efficient mechanisms to enable PV deployment, followed by carbon cap, PV rebate, and carbon tax. Generally, larger PV installations correspond to larger sizes of the battery.

Sensitivity analysis
The impact of the input uncertainty on the sustainability dimensions of MES is displayed in Fig. 10.
While ranges of cost, emissions and self-sufficiency are found, the sensitivity analysis confirms the trends of the reference case. The following considerations can be made about the three dimensions of sustainability: (1) All policies tend to increase the overall cost, with carbon cap and carbon tax increasing the cost for all end-users and therefore having the highest impact on the average cost of energy supply.
In particular, the carbon cap bears the largest uncertainty in term of ensuing energy costs. Moreover, the results suggest that the cost discrepancy due to the FiT might be greater than found in the reference case analysis, since higher costs for consumers and lower costs for prosumers are identified as likely outcomes. The sensitivity analysis hence stresses the importance to keep energy equity considerations in mind when designing policies. (2) The emissions in the reference case are at the lower end of the sensitivity analysis results. This is especially true for the FiT and the PV rebate, in which case the mean of the observed emissions in the high and maximum policy scenario are much higher than in the reference scenario. Therefore, the FiT and the PV rebate policies bears the largest uncertainties in terms of successful decarbonization. This can be explained by much higher values of the levy that can occur across the sensitivity analysis ( Fig. 6(b)). The higher levy leads to a significant shift away from electric heating, hence in much higher carbon emissions. This finding underlines that affordable electricity is a key driver for the electrification of heating and therefore for emissions reduction, and that carbon tax and carbon cap policies should be pursued when prioritizing emissions reduction. (3) Whereas the general trends are confirmed, the variance of the results is much higher for the self-sufficiency of prosumers than for the other sustainability dimensions. This effect can be attributed to the fact that both the PV price and the electricity price vary across a wide range, leading to a shift between local electricity production and imports of electricity even if all other factors are constant. Thus, the effect of policies on self-sufficiency is associated with the highest uncertainty out of considered sustainability dimensions.
L. Brodnicke et al. Fig. 10. Sustainability dimensions for all policy types and levels for the reference case (symbols) and the sensitivity analysis (shaded area) for both end-user types and the case that the levy applies to the electricity imports. Denser areas correspond to a higher frequency of results, with the lines corresponding to the 20, 60, and 80 percentiles, respectively. The lines between the symbols do not indicate a linear trend but aim to guide the eye of the reader.

Conclusions
This paper investigates the impact of policy mechanisms on the sustainability benefits of MES on different end-user types, namely consumers and prosumers. A MILP optimization model is used to model the decision-making process of consumers prosumers, which minimize the total annual cost of their energy supply. A sensitivity analysis is performed to account for the uncertainty associated with the input assumptions. Four policy mechanisms are investigated, namely a feed-in tariff, a PV rebate, a carbon tax, and a carbon cap.
Different policy mechanisms foster different system designs and thus differ in their impact on the three sustainability dimensions of MES. Hence, energy policy should be tailored to the relative importance of economic, environmental, and social sustainability. The analysis shows that subsidies for PV reduce the cost for prosumers, but increase the cost for consumers, that cannot benefit from these subsidies. Thus, energy equity considerations need to be taken into account when designing energy policy. Moreover, a trade-off is identified between wider PV deployment and electrification, hence decarbonization, of the heating sector in the MES. Subsidies that support specifically PV investments might defer money from electrification efforts and even results in higher CO 2 emissions with respect to business as usual. Similarly, when subsidies are financed through a levy on retail electricity price, they can actually have an adverse effect and increase emissions due to a shift away from electrification. Hence, high subsidies that result in high levies are found to not only lead to considerable cost increases for consumers, but also to undermine decarbonization efforts. The increase of emissions can be alleviated by applying the levy to both gas and electricity imports. Availability of affordable electricity either through grid imports or through local production is found to be a key driver of heat electrification, which is supported through carbon tax and carbon cap schemes.
Finally, the key limitation to this study is the assumption of perfect knowledge and fully rational decision making of all end-users. Future work should explore ways to account for sub-optimal decision making. Moreover, even though the study introduces some heterogeneity of endusers, there should be an assessment with a higher number of different end-users that differ in various aspects, such as geographic location, size of the household, and energy needs. Lastly, this study considers the energy demand as a fixed input parameter, without allowing for the flexibility to reduce or shift it due to refurbishments and behavior changes. These shortcomings should be addressed in further work.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability
No data was used for the research described in the article.

Acknowledgments
This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie (MSC) grant agreement No. 847585 -RESPONSE.

Appendix A. Data, decision variables, constraints, and objective function of the optimization problem
In the following, all aspects of the optimization model, namely input data, decision variables, constraints, and the objective functions, are briefly described. A more detailed description of the optimization model is reported in [3].

A.1. Input data
The time-dependent input data to the optimization model are obtained for a representative household in the city of Zurich, Switzerland. We do not consider different climate conditions as we focus on a specific geographic case study. Input data to the optimization model are: • The heat D h ∈ R × and electricity D e ∈ R × demand profiles, which are based on one of the typical archetypes of a single family house in Zurich, Switzerland, as identified by Petkov et al. [100]. The demand profiles have an hourly resolution and are created using the CESAR-P modeling tool for the year 2020 [101]. • The solar irradiance profile I ∈ R , which is obtained through the work of Pfenninger and Staffel (Renewables.ninja platform) [102][103][104], based on the CM-SAF SARAH and the MERRA-2 data sets [105,106]. • The export price of electricity v e ∈ R , which depends on whether a FiT v FiT ∈ R is available or the electricity is exported and sold to the wholesale electricity price v WH ∈ R . The 2020 wholesale electricity price is used [107].
• The import price of gas u g ∈ R and electricity u e ∈ R , which are assumed to be constant at all time steps. They are subject to the sensitivity analysis and taken as 8.71 Rp/kWh and 21.78 Rp/kWh in the reference case, respectively. • The carbon intensity of natural gas, which accounts for its life cycle emissions and it is taken equal to g = 237 gCO 2 ∕kWh [3]. The carbon intensity of imported electricity e is assumed to be 58.8 gCO 2 ∕kWh in the reference case. • The parameters describing the technical, economic, and environmental performance of the available technologies. More specifically, for all technologies within the set of available technologies , such parameters are the technology cost ∈ R, efficiency ∈ R, and lifetime ∈ R. These assumptions are subject to the sensitivity analysis. Reference case values are listed in Table A.1 in Section 3.4.
• The discount rate ∈ R for annualizing the technology capital cost, which is subject to the sensitivity analysis. In the reference case, a discount rate of 5.13% is assumed.

A.2. Decision variables
The following decision variables are included in the optimization, i.e. those variables that are determined by the solution optimizer: • The size of all technologies S ∈ R × . • The input power F ∈ R × × and output power P ∈ R × × of all technologies. • The stored energy E ∈ R s × × of storage technologies, with s being the number of available storage technologies. • The imported electrical and gas power U e ∈ R × and U g ∈ R × , respectively.
• The exported electrical power V e ∈ R × . • The annual capital cost c ∈ R × and maintenance cost m ∈ R × of all technologies.
• The operational cost c ∈ R × and the carbon tax cost CO 2 ∈ R × resulting from the imports and exports of natural gas and electricity.

A.3. Constraints
The constraints of the optimization model can be generally grouped in two categories, namely energy balances and individual technology operations. The former enforce that the sum of imported and converted energy must equal the sum of exported and used energy at all times. The latter enforce real-world technology performance and behavior. The constraints describing the technology design and operation are described in [3]. In addition, the maximum PV size for each prosumer is restricted to a maximum size, which allows to cover the entire electrified energy demand of the prosumer, i.e. the electricity demand if the heat demand is covered via a heat pump: Moreover, the maximum size of the residential battery is restricted to 13.5 kWh, which is the maximum size commercial solutions currently available at residential level [76]. This is done to avoid unrealistically sized battery installations when enforcing the maximum possible emissions reduction, i.e. a carbon cap.

A.4. Objective function
For all individual end-users, the optimization model minimizes the total annualized cost under consideration of policy incentives and constraints. The cost is calculated and optimized for each end-user separately. To improve readability, the end-user index is omitted in the following. The annualized cost comprises four components, namely the capital c , operational o , maintenance m , and carbon tax CO 2 costs: The capital cost is calculated as where the capital cost for each technology is multiplied with the corresponding annuity factor accounting for the lifetime of the technology and the discount rate . The operational cost is calculated as the sum of the costs of importing and exporting electricity and gas throughout the where the export price of electricity e, either corresponds to the feed-in tariff or the wholesale electricity price and the export price of electricity includes the levy . The annual maintenance cost is calculated as a fraction of the annualized capital cost: The carbon tax cost is calculated as the sum of emissions associated with the electricity and gas imports, which are priced with the carbon tax CO 2 : ( e, e, + g, g, A.5. Input data used in the sensitivity analysis Table A.1 includes the input data used for the sensitivity analysis. The minimum ( min ) and maximum values ( max ) correspond to the smallest and the largest values found in literature, respectively. The mode value ( mode ) is calculated by taking the average value of the average minimum and maximum values. The PERT distribution is a special case of the beta distribution, whose shape parameters are calculated according to [108]. A default value of = 4 is used for the height of the distribution. For the reference case results, the mode value of all input parameters is used.

Appendix B. Sustainability dimensions for a levy that applies to all energy imports
To assess different cost allocation strategies, the reference case analysis is performed for a levy that applies to all energy imports, i.e. gas and electricity. Therefore, the levy per kWh of energy is significantly lower than in the case that the levy only applies to the electricity imports, because it applies to a larger quantity of imported energy (Fig. B.1).  illustrate that applying the levy to both the gas and electricity imports instead of on the electricity imports alone results in only marginal differences in the cost for the end-users of up to 2%. Hence, applying the levy to both the gas and electricity imports only marginally improves the energy equity considerations of the subsidies. Similarly, the difference in cost allocation only marginally impacts the self sufficiency of prosumers (Fig. B.2(d)). However, applying the levy to all energy imports reduces the overall system emissions by 41% and 11% compared to the reference case analysis for the maximum FiT and PV rebate level, respectively (Fig. B.2(c)). These difference can be explained by a higher electric heating quota, which is incentivized due to the higher gas cost and the lower electricity cost compared to a levy that only applies to electricity imports (Fig. 7). Hence, a levy that applies to all energy imports could provide a mechanism to avoid a shift away from electrification due to higher electricity prices. Fig. B.3 shows the results of the sensitivity analysis, performed for the case that the levy is applied to both gas and electricity imports. Note that only the results for the FiT and PV rebate are displayed, since the carbon tax and carbon cap policies are not affected by the levy. The sensitivity analysis suggests that the reference case results are relatively robust with respect to the costs for both end-user types. However, there is a wide range of results with respect to the emissions and the self sufficiency. The reference case seems to yield optimistic results with respect to emissions and rather pessimistic results with regards to self sufficiency. Nonetheless, the trends that we are seeing in the reference case analysis are confirmed: Even for high subsidy levels and therefore high levies, we do not see an increase in emissions since there is no shift away from electrification due to high electricity prices. Moreover, the positive effect of the subsidies on PV deployment and therefore self sufficiency remains unchanged. Thus, the sensitivity analysis confirms that applying the levy to both gas and electricity imports might be able  Fig. B.3. Sustainability dimensions for the FiT and PV rebate for the reference case (symbols) and the sensitivity analysis (shaded area) for both end-user types and the case that the levy applies to both the gas and electricity imports. Denser areas correspond to a higher frequency of results, with the lines corresponding to the 20, 60, and 80 percentiles, respectively. The lines between the symbols do not indicate a linear trend but aim to guide the eye of the reader. to avoid the shift away from electrification. However, the energy equity considerations remain unchanged, since some end-users are positively affected by the subsidies, whilst others end up having to pay more.