Prosumage of solar electricity: Tariff design, capacity investments, and power sector effects

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Introduction
Decarbonizing energy supply leads to a substantial transformation of the power sector.In many countries, increasing shares of renewable electricity are generated by small-scale distributed solar photovoltaic (PV) plants (IEA, 2018).Driven by regulation and dedicated support schemes, private households account for a substantial share of PV installations.In particular, partial self-supply with PV becomes more attractive.Households' self-generation shares can further increase by the advent of PV-plus-battery systems.Batteries have recently experienced a substantial drop in costs; a trend that is expected to continue in the future (Schmidt et al., 2017).In this respect, the term prosumage emerged to describe the activity of a household that generates its own PV electricity, enhanced by battery storage, while still being connected to the grid (von Hirschhausen, 2017).
We investigate how the design of retail and feed-in tariffs (FIT) affects household decisions to invest in PV and battery storage systems.We also explore impacts on the power sector in terms of renewable energy capacities, peak PV feed-in, and the contribution of households to non-energy power sector costs, that is, costs for the electricity network infrastructure or renewable support schemes.To this end, we first provide the intuition how household incentives are shaped by retail tariffs, feed-in tariffs, and respective investment cost for PV and batteries.In a second step, we numerically explore these incentives and their effects in a computational equilibrium model applied to a German 2030 setting.In doing so, we take interactions between households' decisions and the wholesale power market into account.
Central results show that lower feed-in tariffs substantially reduce PV investments.Yet effects on battery capacity and PV self-generation are less pronounced.Higher fixed parts and lower volumetric components in retail tariffs lead to lower optimal battery capacities and self-generation.In turn, households contribute more to non-energy power sector costs such as costs for the electricity network.We further find that limiting peak feed-in, which can relief distribution grids, is possible without substantially distorting households' incentives.
While previous research features a number of analyses on the economic viability of prosumage for specific households, only few contributions consider feedback of prosumage on the power sector.Yet these are largely silent on household behavior and the impact of the regulatory setting.Combining the household, power sector, and regulatory policy perspectives, this paper aims to fill a gap in the literature.It contributes to the academic and policy debates on retail tariff design, increasing deployment of decentral PV and storage systems, and contribution to recovering network costs, renewable support charges, and other non-energy power sector costs.
The remainder of this paper is structured as follows.Section 2 reviews relevant literature on prosumage.Section 3 introduces a conceptual framework and provides some intuition.Based on this, we develop a formal equilibrium model in section 4. Section 5 presents the numerical results; section 6 discusses limitations of our approach and outlines avenues for future research.Section 7 concludes.

Literature review
This paper contributes to three overlapping strands of the literature on the economics of prosumage: the household perspective, the power sector perspective, and the policy perspective.
First, concerning the household perspective, a range of publications analyze the economic viability of prosumage systems.For Germanya country with favorable market conditions -Hoppmann et al. (2014) were among the first to argue that falling battery costs will spur the prosumage segment.Kaschub et al. (2016) come to a similar conclusion: according to their analysis, prosumage will be economical for households in the short run even in the absence of subsidies, driven by self-consumption.Using self-generated electricity for electric vehicles would support this trend.Dietrich and Weber (2018) also conclude on the profitability of PV-plus-battery systems in the near future, and highlight that economies of scale would incentivize larger installations.A comparative study for Germany and Ireland derives analogous results (Bertsch et al., 2017).Studies on the viability of residential prosumage also exist for electricity markets in other countries like Australia (Muenzel et al., 2015;Say et al., 2018Say et al., , 2020)), Brazil (Vilaça Gomes et al., 2018), France (Jin and Yu, 2018), Italy (Cucchiella et al., 2016), Spain (Prol and Steininger, 2017;Solano et al., 2018), the United Kingdom (Green and Staffell, 2017), and the United States (Khalilpour and Vassallo, 2015;Say et al., 2018;Tervo et al., 2018).
Second, concerning the power sector perspective, prosumage contrasts with the traditional supply-and demand-side division.A widespread adoption of residential PV-plus-battery systems affects both prosumage households and other electricity consumers as well as power generators.This could have broad technical, socio-economic, and political repercussions, as discussed by Agnew and Dargusch (2015), Schill et al. (2017), and Schill et al. (2019).More specifically, residential PV-plus-battery systems provide low-carbon energy and can thus help to achieve climate targets.However, depending on prosumagers' price signals and objectives, economic inefficiencies may arise.These relate to sub-optimal investment in the long run, for instance redundant storage infrastructure (Say et al., 2020), and sub-optimal dispatch in the short run, for instance a modest contribution to system peak shaving or valley filling (Green and Staffell, 2017;Schill et al., 2017).
Besides investments and dispatch, Marwitz and Elsland (2018), Moshövel et al. (2015), and Neetzow et al. (2019) discuss whether and under which regulatory circumstances prosumage may require expanding the electricity distribution grid infrastructure.In contrast, Young et al. (2019) conclude for an Australian region that benefits from reduced peak grid demand may outweigh foregone revenues from self-consumption for network operators.
Third, concerning the regulatory policy perspective, an increasing number of studies address the question how to price prosumagers' consumption and generation.Proposed policies can be broadly divided into a revision of retail rate structures, remuneration schemes for decentral generation, and other policy measures.For the German context, Ossenbrink (2017) investigates the implications of feed-in and retail tariff schemes for residential PV systems without batteries.Based on the ratios of the levelized cost of electricity (LCOE) of PV to the feed-in and retail tariffs, respectively, he derives conditions under which households act as pure consumers or engage in prosuming activities.However, storage is excluded from the analysis.More recently, Thomsen and Weber (2019) assess how the tariff design affects profitability and operation of small-scale PV-plus-battery systems in Germany, yet assuming fixed prosumage PV and battery capacities and not analyzing investment incentives.
Broadening the scope, tariff design triggers re-distributive effects in the context of prosumage.Generally, volumetric energy charges lead to a burden shift from prosumage households to pure consumers (Roulot and Raineri, 2018).Since prosumage households have a lower grid energy consumption (load defection), they pay fewer network fees and other charges although they still enjoy all grid services (Simshauser, 2016).A growing prosumage segment can induce distributive justice concerns at the consumer level and cost recovery issues for utility operators (Hinz et al., 2018;Kubli, 2018;Roulot and Raineri, 2018;Schittekatte et al., 2018), leading to what has been referred to as death spiral in the most extreme case (Costello and Hemphill, 2014;Laws et al., 2017): ever fewer customers must pay ever higher grid charges which, in turn, incentivizes further load defection.However, given that this would require major uptake of residential PV-plus-battery systems, such scenario is rather unlikely to occur in many developed countries' power sectors (Darghouth et al., 2016;Laws et al., 2017). 1  In conclusion, the literature features individual analyses that provide an in-depth treatment of specific households, yet mostly do not take interactions between prosumagers and the power sector into account.Power sector studies mostly neither incorporate prosumage households' incentives nor the tariff design.Regulatory studies generally lack a numerical underpinning.We aim to contribute to all three perspectives and provide an analysis of tariff design for prosumage, which also includes PV and storage investment incentives of households as well as interactions with the power sector.While we focus on the German context, results are of interest also for other markets where feed-in tariffs and high volumetric retail tariffs drive solar prosumage.

Definitions
We define a residential prosumager as a grid-connected household with a PV panel and a battery (Schill et al., 2017).Fig. 1 presents a stylized representation.The household generates electricity (G PV ) that it consumes at times (G pro2pro ), feeds into the grid at other times (G pro2m ), potentially curtails (CU pro ), or stores in the battery (STO in,pro ) for future consumption (STO out,pro ).Still, the household may consume electricity from the grid, i.e. the market, at any point in time (E m2pro ).For clarity, we assume that the battery can only be used for deferring self-consumption. 2  There are two standard metrics that describe the dependency of prosumage households on power provision from the grid: the rate of selfconsumption and the autarky rate (Luthander et al., 2015;Weniger et al., 2014).The rate of self-consumption SC is the fraction of electricity generated on-site that is either directly consumed or stored in the battery for future self-consumption.
The lower the rate of self-consumption SC, the higher the revenues 1 A related debate discusses the metering design in case of residential photovoltaic (plus battery) systems.Specifically, under net metering consumers' PV electricity grid feed-in and grid-consumption are offset against each other over a longer time horizon (Hughes and Bell, 2006).This was shown to substantially raise incentives for residential standalone PV investments in different settings (e.g., Eid et al., 2014;Picciariello et al., 2015;Darghouth et al., 2016), fuelling discussions on distributional justice. 2 In principle, other uses are conceivable like smoothing grid consumption and feed-in or providing flexibility by storing in and out grid electricity.Schill et al. (2017) show that such additional battery use can lower total electricity sector costs.
generated from feeding energy into the grid.For example, a selfconsumption rate of 40% implies that 60% of the generated electricity receive a remuneration, such as the feed-in tariff.This explains why relying solely on levelized cost of electricity (LCOE) to determine the profitability of a PV-plus-battery system for a single household is inadequate.
The autarky rate A, also referred to as rate of self-generation or rate of self-sufficiency, is the share of household electricity demand covered by generation from the PV-plus-battery system.This includes directly consumed energy and energy discharged from the battery storage.
The higher a household's autarky rate A, the less it pays for energy consumed from the grid.Both the autarky and self-consumption rates are defined over a specified time interval, usually a year.

Intuition: Incentives for prosumage
For a rational household, economic incentives for prosumage are largely driven by investment costs for PV and storage systems as well as grid consumption and feed-in tariffs. 3Under the current German setting, households face a time-invariant, volumetric rate for PV grid feed-in, the feed-in tariff, and a time-invariant volumetric rate for grid consumption, the retail tariff, that, depending on the specific retailer a household chooses, may also come along with some fixed part.For convenience, and as standard in aggregate household electricity price statistics to even out varying tariff designs,4 the fixed part is distributed to the volumetric rate when reporting retail tariffs.The average German residential retail tariff for electricity was 0.21 EUR/kWh in 2008.A decade later, in 2018, it had increased to about 0.30 EUR/kWh, of which 0.23 EUR/kWh accrued for non-energy components like charges for recovering network costs, taxes, and renewable support payments; a share of 80%.Importantly, self-consumption from small-scale PV systems below 10 kW is exempt from any such cost components.At the same time, the cost degression of PV systems has led to a substantial reduction of feed-in tariffs.They have decreased from more than 0.46 EUR/kWh in 2008 to less than 0.12 EUR/kWh in 2018.Fig. 2 illustrates these developments over the last decade.
In principle, if the retail tariff exceeds the FIT, self-consumption of PV electricity becomes economical.Yet this socket parity (Bazilian et al., 2013) does not necessarily imply economic viability of prosumage because residential demand and PV supply may only partly coincide in time.Savings from further substituting grid consumption with self-consumption must over-compensate the necessary investments into a battery.
To this end, Fig. 3 provides a more comprehensive illustration of incentives for investments in residential PV and battery systems.It extends an analysis by Ossenbrink (2017) by adding the storage dimension.In the following, we go through the figure's areas A to F, using the German situation as an example.The lower left area A refers to a situation where the LCOE of a decentral PV installation exceed both the retail and feed-in tariffs.Hence, there is no financial incentive for households to invest in PV or battery systems (''Pure consumer'').This was the situation in Germany before a FIT was introduced in the year 2000.
The 45-degree line starting at the upper right corner of area A marks the points where the FIT and the retail price are equal.In the upper left area B, the FIT exceeds the levelized costs of PV, and the FIT is also higher than the retail tariff.This characterizes the market situation in Germany before 2012.Households have an incentive to feed all generated PV electricity into the grid and satisfy their demand with grid consumption. 5This means they are full grid producers and consumers.Moreover, since the PV system can generate positive revenues, households are incentivized to invest in a PV system that is as large as possible.Yet there is no incentive to install battery storage since it is more attractive to generate revenues from the FIT than to avoid consuming from the grid.
When the retail tariff exceeds both the LCOE of PV and the FIT, selfconsumption becomes attractive (''Prosumer'' areas C and D).Consider area C first.The FIT exceeds the levelized costs for PV and creates an incentive to install the maximum PV capacity.At the same time, the retail tariff is higher than the FIT, meaning that it is attractive to substitute as much grid consumption as possible with on-site generation.In area D, the FIT is not high enough to cover the LCOE of PV.In this situation, households no longer have an incentive to install the maximum possible PV capacity.The optimal PV capacity trades off the costs for the PV system with the revenues collected through the FIT and the  expenditure on grid consumption saved through self-consumption.
In both areas E and F, installing a battery storage is more profitable than a PV-stand-alone system.For this to be the case, the retail tariff must not only be high relative to the LCOE of PV, but also relative to the FIT.Since the battery is used to offset grid consumption with selfgenerated PV energy, which would otherwise be fed into the grid, the difference between the two tariffs must cover the levelized costs of storage (LCOS).The higher the storage costs, the further areas E and F are shifted to the right.Similarly, battery capacity deployed by households increases with this difference since it increases incentives for offsetting grid consumption.As before, these two prosumage situations differ with respect to the deployed PV capacity.If the FIT is above the LCOE of PV, in area F, households deploy the maximum PV capacity, while the optimal PV size is smaller in E. Households are incentivized to maximize their self-consumption in both cases.The market situations depicted in E and F have not yet been reached in Germany by 2019 since storage costs are, still, too high.

Model, data, and scenarios
In the following, we illustrate the incentive structure by means of a numerical model applied to German 2030 scenarios.Beyond prosumage household decisions, we investigate selected effects on the power sector.

Model
The model features a prosumage household agent and a benevolent power sector operator.We assume that both agents have perfect foresight.Each solves a linear cost minimization problem such that the Karush-Kuhn-Tucker (KKT) conditions are both necessary and sufficient for global optimality of the solution.Throughout the mathematical exposition, capital letters denote variables and lower-case letters parameters.
The prosumage agent minimizes her annual electricity expenditure Z pro by deciding on optimal prosumage system investment and dispatch (2a).Retail costs for grid electricity consumption consist of a fixed component and a volumetric component for hourly electricity consumption from the grid E m2pro h .The volumetric component comprises a price for energy t ener h , which may vary from hour to hour, and a timeinvariant part t other , which represents non-energy charges like network fees, taxes or renewable support surcharges.Non-energy charges may also be raised by a fixed annual component t fix .Depending on the scenario, cost components may be zero or positive.To avoid costly grid consumption, the household can invest in PV capacity N pro pv as well as lithium-ion battery energy and power capacities, N pro,E sto and N pro,P sto , respectively.Annualized investment costs are factored in via c inv and the annual fixed costs via c fix .Investments into PV, battery power, and battery energy capacities are mutually independent.Prosumage households can also lower their annual electricity bill by generating positive revenues through selling energy G pro2m h to the grid at a, potentially timevarying, price of t prod h .
Households' inelastic electricity demand d pro h must be satisfied in each hour, either with directly consumed PV generation G pro2pro h , through energy discharged from storage STO out,pro h or with grid consumption (2b).For each constraint, the respective Lagrange multiplier, or shadow price, is given in parentheses.
The hourly available PV energy depends on the exogenous capacity factor φ avail pv,h ∈ [0, 1] and the installed capacity.It can be consumed by the household, sold to the market, curtailed CU pro h or stored STO in,pro h (2c).
The stored energy at the end of each hour STO l,pro h is equal to the storage level at the end of the previous hour, minus the energy discharged in the current hour, plus the charged energy, corrected by the battery's roundtrip efficiency η (2d).In the first period, the household starts with an empty storage (2e).Battery use is subject to the installed power and energy capacities (2f-2h).
The maximum PV capacity per household is limited to m pv to reflect space restrictions or regulatory thresholds (2i).Appendix A.2.2 gives a full account of the Lagrangian and according KKT first-order optimality conditions.
The second agent is a benevolent power sector operator.She minimizes total system costs through optimal dispatch of a given power plant and pumped-hydro storage fleet.This dispatch is equivalent to a competitive market outcome.The market clears in every hour, meaning that total generation must equal total consumption.This includes both the grid demand from the prosumage household and a non-prosumage inelastic demand representing all other consumers.The according dual variable λ enbal h can be interpreted as the wholesale market price that is passed on to the prosumage household in some scenarios.The power sector dispatch is based on the open-source model DIETER (Zerrahn and Schill, 2017).Appendix A.2.3 lists all equations.
In order to analyze prosumage households and generators jointly in an equilibrium problem, we combine the KKT conditions of the two programs in a mixed complementarity problem (MCP) (Facchinei and Pang, 2007).Fig. 4 illustrates the interaction between the wholesale generators as well as storage operators and the prosumage households.The problem is implemented in GAMS and solved with the PATH solver (Dirkse and Ferris, 1995).All model sets, decision variables, and parameters are listed in Table 3, Table 4, and Table 5 in appendix A The model is solved for every second hour of the year.This allows capturing all the important daily and seasonal demand variability as well as fluctuating generation of wind and solar plants while ensuring tolerable computation times.We provide the model code and input data under a permissive open-source license in a public repository. 6

Data
The size of the prosumage segment is set to one million households, in accordance with prosumage growth projections for Germany (BNetzA, 2018b).This number represents approximately ten percent of all single-family and two-family houses that are potentially suitable for PV-plus-battery systems (Prognos, 2016).That is, the prosumage agent represents the aggregate of all prosumage households.In line with the current threshold for being exempt from paying the renewable surcharge on self-consumed electricity, we set an upper PV investment limit of 10 kW per household.The annual load of each prosumage household is assumed to be 5 MWh, the average value of a German single-family household (Statistisches Bundesamt Destatis, 2018a).
Table 1 lists the relevant parameters for the prosumage segment, mainly drawing on the European Energy Technology Reference Indicator projection (ETRI) for 2030 (JRC, 2014).The expected LCOE of small-scale PV for a household range between 0.07 and 0.08 EUR/kWh, including expenses for value-added tax (VAT).The assumed storage costs imply a substantial reduction compared to current levels in Germany.Though this projection is optimistic regarding storage cost decline, it lies within the expected range for 2030 reported by Schmidt et al. (2017).
Demand of prosumage households follows the standard load profile of the German Association of Energy and Water Industries (BDEW, 2015) and is scaled up to represent one million households.The hourly PV capacity factor is a capacity-normalized, country-aggregated time series taken from Pfenninger and Staffell (2016) and Staffell and Pfenninger (2016).These sources are based on reanalysis data of the year 2012, which constitutes an average weather year.
The power sector is calibrated in line with official projections for Germany in 2030.Generation and storage capacities shown in     (ENTSOE, 2018).This projection of future demand is supposed to be representative of a normal weather year.We subtract the demand of the prosumage segment from the national demand curve.As for renewable energy sources, hourly capacity factors are taken from Pfenninger and Staffell (2016) and Staffell and Pfenninger (2016).

Scenarios
We devise a number of scenarios of the year 2030 that differ with respect to which price signals prosumage households receive (Table 2).Some scenarios also include other, non-price-based policies.
Scenario Retail 30 FIT 8 is the baseline scenario.The levels of the retail tariff and the FIT are comparable to the situation of German households by 2019.To clearly isolate incentives, the retail tariff for electricity consumption is purely volumetric and comprises a timeconstant, volumetric energy charge t ener h and volumetric other charges t other .The energy charge of 0.05 EUR/kWh is calibrated endogenously according to the average energy wholesale market price λ enbal in an initial model calibration run without prosumage.The tariff component t other of 0.25 EUR/kWh broadly reflects the level of non-energy charges contained in an average German purely volumetric electricity retail price by 2019.Specifically, these other charges comprise charges for network-related costs, taxes, the renewable energy surcharge, and several smaller surcharges targeting, for instance, combined heat and power plants or the connection of offshore wind farms.Among the other charges, the network-related components account for the biggest proportion. 7The overall volumetric consumption price for households sums up to 0.30 EUR/kWh.For a pure consumer household with an annual load of 5 MWh, this results in an annual expenditure of 1500 EUR on the electricity bill.The FIT is assumed to be 0.08 EUR/kWh, which is slightly above the projected LCOE of PV.
The first group of scenarios keep the volumetric retail tariff constant and vary the level of the feed-in tariff.Scenario Retail 30 FIT 0 represents an extreme case in which grid feed-in of PV electricity by households is prohibited.Scenario Retail 30 FIT 8 Cap is the same as the baseline, but additionally restricts the hourly maximum in-feed of prosumage PV energy into the grid to 50% of the installed PV capacity.This is in line with the requirements for preferential loans in Germany by 2019.
The second group of scenarios implement a greater fixed part for retail tariffs to capture the network costs, renewable surcharges, and other non-energy price components.Households pay an annual fixed fee t fix and, in turn, a lower volumetric charge t other .This reflects a more capacity-oriented tariff design.For a pure consumer household with an annual load of 5 MWh, all scenarios with a fixed-part retail tariff result in the same annual bill of 1500 EUR as under the baseline and volumetric scenarios.
The third group of scenarios represent dynamic real-time pricing (RTP) schemes.Households pay a volumetric retail tariff of 0.30 EUR/ kWh in scenario Retail 30 FIT RTP and sell their electricity at the current wholesale market price, represented by the dual of the hourly power sector energy balance λ enbal h in the model.Scenario Retail RTP FIT 5 assumes a time-varying energy price component of the retail tariff, in addition to a fixed volumetric component of 0.25 EUR/ kWh accounting for non-energy charges for network costs, renewable support, and others.The feed-in tariff, in turn, is fixed at 0.05 EUR/kWh.The other two dynamic pricing scenarios impose a real-time price on both the retail and the feed-in sides, with an additional market premium of 0.03 EUR/kWh in scenario Retail RTP FIT RTP + 3.This is motivated by the idea that the mean market value of PV energy is typically low in hours with high PV feed-in.The market premium may help to cover the cost difference between LCOE of PV and the wholesale market price.

Results
We show and interpret the model results with respect to household investments, how households use their prosumage systems, and several implications for the power sector.In doing so, we relate our findings to the prosumage incentives that households receive through price signals, as illustrated in Fig. 3.

Optimal household investments into PV and storage
In the baseline scenario Retail 30 FIT 8, prosumage households install a PV capacity of 10 kW, storage energy capacity of 5.7 kWh, and storage power capacity of 1.2 kW (Fig. 6).The FIT is above the levelized cost of PV and grid feed-in never constitutes a net loss.The storage helps to substitute grid energy, priced at the volumetric retail tariff, by selfgenerated energy, at the levelized cost of PV and storage.Consequently, prosumage is profitable, and the baseline scenario refers to area F in Fig. 3.
Panel 6a shows the results for the first group of scenarios.At a volumetric retail tariff of 0.30 EUR/kWh, a FIT below the LCOE of PV of 0.08 EUR/kWh yields lower optimal PV investments, referring to area E in Fig. 3. Yet optimal battery energy capacities are relatively stable because the storage is built to optimize the amount of self-consumed PV generation given the retail price.Beyond a storage capacity of 5-6 kWh, the costs of additional storage become too high compared to the small increase in self-consumption and avoidance of grid consumption that can be reached.This finding also holds true in the absence of remuneration for grid feed-in.A feed-in cap yields somewhat higher optimal storage capacities to accommodate a greater share of self-consumption.
Results for the second group of scenarios follow a similar line of reasoning (panel 6b).A high feed-in tariff of 0.08 EUR/kWh, above the LCOE of PV, incentivizes maximum solar capacities.Without any grid feed-in, PV capacities are optimized only to serve self-consumption.Optimal battery capacities are largely governed by the design of the retail tariff and, accordingly, the profitability of self-consumption.Greater fixed parts, together with lower volumetric price components, trigger smaller optimal storage capacities.Eventually, the difference between the volumetric retail tariff and FIT is too small to profitably cover expenditure on the storage battery.Consequently, scenarios Retail 15 FIT 8 and Retail 15 FIT 0 refer to areas C and D in Fig. 3, respectively.
If PV feed-in is remunerated by the real-time price, optimal PV capacities are below the maximum, at around 5.5 kW per household (panel 6c).In fact, the average price for which households sell electricity to the market is slightly above 0.04 EUR/kWh, rendering results comparable to scenario Retail 30 FIT 4. Thus, a FIT of 0.05 EUR/kWh, or a market premium of 0.03 EUR/kWh, yield greater optimal PV investments.Average real-time retail prices at which households buy electricity from the market are slightly below 0.05 EUR/kWh.With the volumetric component of 0.25 EUR/kWh, the eventual retail rate is around 0.30 EUR/kWh.Accordingly, optimal storage energy investments are comparable to the first group of scenarios and range between 5 and 6 kWh.

Optimal household dispatch: self-generation, self-consumption, and expenditure
We start with some intuition.Fig. 7 shows the dispatch behavior of a prosumage household for three sunny days at the end of April, taken from the baseline scenario.During the day, the available solar energy exceeds the demand of the prosumage household, and it can consume only a part of PV energy directly.Most of the PV energy is sent to the grid, peaking in hours of high solar radiation.Prosumage households charge their battery fully in the morning and discharge it in the evening when available PV energy declines.Both the feed-in and retail tariffs are time-invariant.Therefore, the household is not incentivized to schedule grid feed-in to hours with higher prices, for instance in the morning, and grid demand to hours with lower prices, for instance at night.Fig. 8 shows how, depending on the tariff design, households use their PV electricity and from which sources they satisfy their electricity demand.In the baseline scenario, households satisfy 4.0 MWh of their annual electricity demand of 5.0 MWh with self-generated electricity (panel 8a).This corresponds to an autarky rate of 80%.Almost half of the annual demand (2.4 MWh) is directly satisfied with PV energy.Around 30% (1.6 MWh) of demand is met with energy from the battery, while only one fifth (1.0 MWh) is obtained from the grid (panel 8a).The overall PV generation of 10.9 MWh exceeds annual demand, yielding a self-consumption rate slightly below 40%.More than half of the generated PV electricity is sent to the grid.This does not change much if the baseline setting is combined with a cap on maximum PV feed-in.
For a lower FIT, the composition of sources that satisfy electricity demand stays relatively stablethe autarky rate is between 64% and 78% -with a somewhat increasing share of grid electricity (panel 8a).Likewise, the absolute volumes of self-consumption, direct or facilitated by the battery, only decrease slightly.Yet optimal PV panels are smaller and generate overall less electricity, yielding a higher relative proportion of self-consumption.If grid feed-in is prohibited, households consume 80% of their PV electricity themselves, with the remaining energy curtailed.
For greater fixed parts in retail tariffs, and accordingly lower volumetric parts, panel 8b shows a stronger dependency on the grid.In this group of scenarios, it is less attractive to substitute grid energy with selfgenerated energy.For illustration, compare the baseline Retail 30 FIT 8 with scenario Retail 15 FIT 8.With a halved volumetric retail price, a prosumage household satisfies 2.6 MWh of her annual electricity demand from the grid, compared to only 1.0 MWh in the baseline.Accordingly, the autarky rate drops from 80% to below 50%.In the most Fig. 6.Optimal PV and storage energy capacities of prosumage households.extreme scenario Retail 15 FIT 0, prosumage households satisfy only 30% (1.5 MWh) of their demand on-site and source approximately 70% (3.5 MWh) from the grid.Lower volumetric retail tariffs also decrease the volume of self-consumed energy.In the baseline, 4.1 MWh out of 10.9 MWh PV generation are consumed on-site, that is, around 38%.In scenario Retail 15 FIT 8, only 2.4 MWh are consumed on-site, amounting to about 22%, and 8.5 MWh are fed into the grid.Batteryfacilitated self-consumption does not take place.
Results for real-time pricing, with a mean price of around 0.05 EUR/ kWh for energy consumption plus a volumetric non-energy charge of 0.25 EUR/kWh, are comparable to the fixed retail rate of 0.30 EUR/kWh (panel 8c).Likewise, feed-in remuneration at real-time market prices of, on average, 0.04 EUR/kWh results in a similar use pattern for PV electricity as under the scenario with a comparable fixed FIT.A market premium of 0.03 EUR/kWh increases the mean real-time feed-in price, with an accordingly greater grid feed-in.
Tariff design also affects the electricity bill of prosumage households (Fig. 9).Prosumage households profit the most in scenarios where they offset large parts of their grid consumption.In the baseline Retail 30 FIT 8, their annual bill on electricity expenditure is 785 EUR (panel 9a).It includes annualized costs of the PV and storage systems, expenses for grid consumption, and revenues from PV power feed-in.Compared to a pure consumer, the bill almost halves.If the FIT is lower, total net expenditures rise slightly, with a pronounced shift toward expenses for grid consumption.If grid feed-in is capped at 50% of the installed PV capacity, household expenditures are largely the same as under the baseline.This means that households can effectively adjust their energy feed-in without incurring curtailment losses.
Lower volumetric retail tariffs with higher fixed parts influence expenditures of prosumage households to a greater extent (panel 9b).Since they must pay a fixed network charge in any case, the saving potential of prosumage compared to pure consumer behavior deteriorates.In any of the scenarios, non-energy payments for grid consumption constitute a substantial part of household expenditures.
The real-time pricing scenarios (panel 9c) only have moderate effects on the electricity bill when compared to the baseline.Prosumage households can still reduce their annual expenditure below 1000 EUR, representing a cut of more than a third over pure consumers.Differences  C. Günther et al. between the real-time pricing scenarios are rather small because the assumed inelastic demand limits households' options to respond to price signals on the consumption side.

Selected effects on the power sector
Beyond households, the prosumage tariffs implemented in the different scenarios also affect the power sector.We discuss implications for peak feed-in, PV generation, and recovery of non-energy costs.

Peak feed-in
While we do not model the electricity distribution grid explicitly and idiosyncratic configurations prevent general conclusions, peak demand and peak feed-in are suitable indicators: higher peaks incur higher stress.Fig. 10 shows residual load duration curves of a prosumage households for selected scenarios.A residual load duration curve is a graphical representation of residual load, that is, household net demand from the grid or net feed-in to the grid.It is ordered for all hours of a year in a descending fashion.
In the baseline scenario Retail 30 FIT 8, prosumage households consume electricity from the grid in 20% of all hours (panel 10a).Their residual load is zero during 46% of all hours, meaning that they satisfy their own electricity demand without sending surplus energy to the grid.In the remaining third of the year, prosumage households feed PV energy to the grid.For comparison, the upper dark gray lines indicate the residual load duration curve of a pure consumer.Her residual load is positive throughout the year because she consumes power from the grid at all times.
Both the sizes of the PV panel and battery shape the residual load duration curve.To this end, compare the baseline with scenario Retail 15 FIT 8, in which households do not invest in storage, but have the same PV capacities (panel 10b).In that case, there are no hours of zero residual load.Households consume grid energy 60% of the time; the remaining 40% of hours they feed surplus energy into the grid.The effect of PV capacities on the residual load duration curve can be inferred from comparing the baseline with scenario Retail 30 FIT 2, where prosumage households have a battery capacity of comparable size yet less than half of the PV capacity (panel 10a).This results in more hours of grid consumption, less hours of energy feed-in, and a lower absolute feed-in of surplus energy.
The left-hand sides of all panels in Fig. 10 indicate that none of the pricing schemes help to reduce peak residual demand.In all scenarios, it is around 1.3 kW.Thus, prosumage households do not shift their consumption patterns in a way that alleviates potential stress on the distribution grid.The right-hand sides of the residual load duration curves show the feed-in peaks.The peak is the higher the larger the PV capacity.In the baseline, it amounts to 6.3 kW; a similar order of magnitude emerges in the other scenarios with maximum PV investment.With 3.5 kW, it is about half this size if maximum feed-in is capped at 50%.Note that this reduction is reached with only little financial disadvantage for prosumage households.
In the scenarios with real-time pricing on the generation side (panel 10c), the maximum feed-in is also somewhat lower, around 2 kW, compared to the baseline.Beyond the smaller PV size, this is also driven by low market prices in hours with high solar radiation.Thus, households have an incentive to avoid those hours for feeding into the grid.Comparing scenarios Retail 30 FIT 4 and Retail 30 FIT RTP illustrates this point.While PV and storage capacities are about the same size, the maximum feed-in is 1 kW higher for the time-invariant FIT.A similar rationale applies to the market premium scenario compared to the fixed FIT scenarios.Also here, maximum feed-in is lower by about 1 kW at similar capacities.

Contribution to PV expansion and non-energy power sector costs
Finally, we provide results how prosumage tariffs can impact the expansion of PV capacities on the one hand and recovery of non-energy costs of the power sector on the other.These non-energy power sector costs may account for network costs, surcharges for financing renewables, and other fees.To this end, we quantify a prosumage household's contribution by summing up expenses on the volumetric and fixed charges t other and t fix .We contrast this figure with households' PV investments.Fig. 11 shows the power sector non-energy cost contribution in EUR on the vertical axis.The PV capacities in kilowatts, as contribution to providing renewable energy, are on the horizontal axis.The further to the north-east, the better a scenario addresses both dimensions.
In the baseline scenario, prosumage households have an autarky rate of 80% and contribute to non-energy power sector costs with about 245 EUR/year (panel 11a).For a pure consumer, this number amounts to 1250 EUR/year.Thus, the baseline tariff scheme may hamper cost recovery of energy infrastructures such as the electricity network.It incentivizes households to lower grid consumption and save on the retail tariff with the according volumetric surcharges.Also in the other scenarios with dominant volumetric retail pricing, prosumage households  contribute modestly to network costs, renewable subsidies, and other non-energy power sector costs.In general, the higher the autarky rate and the less a household pays for volumetric price components, the less it contributes.Concerning the contribution to expanding renewable energy, the baseline features maximum PV investments of 10 kW.As discussed, lower FITs lead to lower PV investments and, thus, a lower contribution to expanding renewables.
In contrast, if households face a fixed part in their retail tariff independent of their annual consumption level, as in the second group of scenarios, greater autarky does not necessarily go along with a lower contribution to non-energy power sector costs (panel 11b).The fixed payments allow households to save on volumetric retail expenses for energy, yet make sure that they contribute to network costs, the renewable surcharge, and other non-energy power sector costs.As in all scenarios, a FIT above the LCOE of PV triggers maximum investments.Thus, these tariff designs appear most suitable to involve prosumage households in the recovery of fixed power sector costs and incentivize large PV capacities.Results for the real-time pricing scenarios (panel 11c) do not differ much from the scenarios with a purely volumetric retail tariff.

Discussion of limitations
Our analysis is subject to several limitations.First, the model does not endogenously capture all power sector costs related to prosumage.Specifically, we can only give an indication whether a certain tariff setting may be detrimental to distribution grids; a deeper analysis would require explicitly modeling the underlying power flows.We also take on a static perspective.We set the non-energy cost component of retail tariffs to recover network expenses, renewable support payments, and other non-energy power sector costs exogenously and abstract from their potential adaption over time.Hence, concerning utilities, the death spiral effect is not explicitly accounted for.Concerning households, this hampers the analysis of rebound effects.Thus, determining an optimal tariff design is eventually not possible, and not aimed for, within the chosen modeling framework.
Second, we apply several simplifications to the power sector dispatch problem.The model abstracts from inter-temporal dispatch restrictions like ramping constraints or minimum up and down times of thermal generators.This tends to overestimate the flexibility of conventional generators.Beyond pumped-hydro storage, we abstract from further flexibility options in the power sector level like flexible sector coupling.Together with our focus on Germany only, this tends to under-estimate flexibility and, accordingly, overestimate price volatility in the dynamic pricing scenarios.
Third, we apply several simplifications to the household side.Modelwise, we abstract from the influence of uncertainty.However, we expect this to have little impact on results.It is plausible that necessary information on demand, solar radiation, and wholesale market prices is readily available from smart forecasting tools, and prosumage scheduling is close to optimal in shorter time frames.Data-wise, the standard load profile and national PV capacity factors are likely smoother than actual households' profiles.This tends to overestimate the match between household demand and PV generation, yielding higher autarky and self-consumption rates than observed in reality (Weniger et al., 2014).The German profiles also represent a temperate-climate country; in other countries, for instance with relevant household air conditioning, load peaks may rather coincide with PV supply peaks, thus implying different incentives for household investments.On a behavioral note, imperfect information or transaction costs may drive away household decisions from optimal values identified in our research.However, it is reasonable to assume households do, on average, choose a system according to their consumption needs and market incentives.When addressing these limitations, we expect our findings to be preserved, though potentially less pronounced than suggested by the model.Most importantly, we abstract from heterogeneous households.Therefore, we cannot explicitly derive distributional implications between different prosuming households as well as prosuming and non-prosuming households.
Last, cost assumptions for residential PV and storage batteries are important input parameters.If storage costs decline to a lesser extent than assumed or if interest rates increase, substituting grid energy with self-generated power will be less profitable.However, given the already high retail price in Germany, storage will still become profitable even if investment costs are considerably higher than in the model scenarios.As a consequence, optimal storage capacity would be smaller, but general findings would still apply since all scenario results would be affected in a fairly similar manner. 8 Possible directions for future research could address these limitations.Specifically, a more detailed, endogenous representation of retail and network tariffs would enable a dynamic perspective on tariff Fig. 11.Tradeoff between the annual contribution to non-energy power sector costs and PV capacities. 8In a not reported sensitivity with double storage costs, prosumage households invest in a storage energy capacity of 4 kWh and a PV capacity of 10 kW.Only if we calibrate storage costs in line with the most conservative cost decrease prediction of Schmidt et al. (2017), self-consumption without batteries will be more profitable than prosumage in the model scenarios.Model results are less sensitive regarding the costs of PV systems.
C. Günther et al. formation and cross-subsidies between consumers and prosumage households.Also, analyses for other countries may provide complementary insights for different household load and PV supply profiles.Moreover, future research could analyze the effects of different tariff designs when including demand-side management, heat pumps, and electric vehicles on the household side.It is likely that this additional flexibility and increased demand will intensify the observed differences between scenarios.

Conclusion and policy implications
Solar prosumage is still a niche phenomenon in most power markets as of 2019.However, as storage costs further decline and selfconsumption of electricity becomes increasingly attractive, solar prosumage has the potential to unfold disruptive effects on the power sector.Whether it is attractive for households to invest in batteries and increase their level of PV self-generation vitally depends on the design of retail and feed-in tariffs.
Our model-based analysis for German 2030 scenarios shows that prosumage becomes profitable in many settings, even in absence of purchasing subsidies for batteries or a FIT for PV energy.Given established cost projections for PV and battery storage as well as a high volumetric retail price, households face strong incentives to substitute large parts of their grid energy consumption with self-generated energy.Departing from the current electricity tariff design, and introducing a greater fixed tariff part for households, would deteriorate incentives to invest in storage systems.
At high feed-in tariffs, households invest in large PV capacities, reach considerable shares of self-generation, and feed a good amount of surplus PV energy into the grid.In contrast, households opt for smaller PV capacities at low feed-in remunerations, since they then face a trade-off regarding the optimal PV system size.Yet the optimal storage capacity for a prosumage household is less sensitive to the feed-in tariff schemes.This is because the storage capacity is driven by the time profiles of residual residential electricity demand and PV generation: beyond a certain storage capacity, the costs of increasing self-generation through additional storage become prohibitive.
From an energy (transition) policy perspective, prosumage is not desirable or detrimental as such.Potentially unintended consequences can arise if high volumetric retail and FIT pricing schemes prevail.High volumetric retail tariffs give prosumage households the incentive to reach high levels of autarky.While these households save costs through self-generation, they contribute less to the national total of grid expenses, renewable surcharges or taxes.Especially concerning grid costs, such revenue shortfalls eventually must be covered by other, nonprivileged electricity consumers, giving rise to distributional issues.This is aggravated by the conjuncture that prosumage households could induce distribution grid overloads because of peak PV feed-in and thus over-proportionately contribute to network instability and costs.
While a high feed-in tariff increases the renewable energy capacity provided by the residential sector, it does not convey incentives for energy-market-oriented dispatch of PV-plus-battery systems.Real-time pricing could incentivize prosumage households to better align their self-consumption patterns with the electricity wholesale market.A maximum feed-in policy, which caps the peak energy fed into the grid, seems to be suitable to achieve distribution grid reliefs, without causing a large financial disadvantage for prosumage households.
Taken together, none of the tariff design options examined here seem to dominate in the investigated respects.A greater contribution to nonenergy power sector costs can generally be achieved with a greater role for fixed non-energy charges.This can be combined with a FIT, and potentially also with a feed-in cap.Given the FIT is high enough, it incentivizes high investments into PV capacity.At the same time, it is relatively easy to implement from an administrative and ICT perspective.Such a combination of FITs and higher fixed retail tariff parts may thus help to foster the transition to renewable energies and, at the same time, to keep up households' contribution to recover non-energy power sector costs.
small-scale PV plants by 2030(OPSD, 2018;;Wiese et al., 2019).A great share of this capacity can be expected to be still operational (Schill et al., 2017) and could be upgraded with a battery storage to engage in prosumage (DIHK, 2018).

A.2 Model details
This appendix section provides more details on the numerical model.

A.2.1 Technical model details
The following tables collect the sets, variables, and parameters contained in the model formulation.

A.2.2 Prosumage household optimization
The cost minimization problem of the prosumage household is given by equation ( 2).The corresponding Lagrangian is: The first order KKT conditions corresponding to the household's problem are:

A.2.3 Power sector dispatch optimization
The generation side of the power sector is represented by a benevolent system operator who minimizes total system costs Z sys through optimal dispatch.This dispatch is equivalent to a competitive equilibrium outcome and results in dispatching generators and storage based on the short-term variable costs or opportunity costs, respectively.The system costs for power generation comprise the short-term variable costs of conventional generators c m con only, as shown in the objective function A.5a. , λ sto sto,h1 free ∀sto (A.6m) Equations A.4 and A.6 are combined to form the MCP that it solved to determine numerical results.

A.2.4 Power sector data
Table 6 compiles numerical assumptions on technologies in the central power sector.

Fig. 2 .
Fig. 2. Development of the feed-in tariff for small-scale PV and the average residential electricity retail tariff in Germany.Source: Own illustration with data from BNetzA (2018a).

Fig. 3 .
Fig. 3. Illustration of incentives for investments in residential PV and battery systems as a function of feed-in tariffs (FIT), retail tariffs, and the LCOE of PV.Source: Own illustration based on Ossenbrink (2017).
Fig. correspond to the middle scenario B in the latest release of the approved German Grid Development Plan 2030 (NEP 2030) of the Bundesnetzagentur, the German network agency (BNetzA, 2018b).Costs and technical parameters for operation of power plants and storage are taken from the Grid Development Plan whenever possible and completed with information from Schröder et al. (2013) and Pape et al. (2014).

Fig. 4 .
Fig. 4. Illustration of model set-up as equilibrium problem between prosumage households and the power sector featuring wholesale generators and storage operators.

Fig. 5 .
Fig. 5. Assumed power sector generation and storage capacities in GW based on the middle scenario B of the German Grid Development Plan 2030 (BNetzA, 2018b).

Fig. 7 .
Fig. 7. Exemplary dispatch plot of a prosumage household in the baseline scenario.

Fig. 9 .
Fig. 9.Annual expenditures of a prosumage household on electricity, broken down into annualized investments into PV and storage capacities, costs for grid electricity, and revenues from PV grid feed-in.
Table in Appendix A.2.4 lists the complete cost parameters for the power sector dispatch.

Table 1
Selected parameters for the prosumage segment.The German load profile for pure consumers follows the Ten-Year Network Development Plan 2030 of the European Network of Transmission System Operators for Electricity Note: Referenced values exclude VAT as stated by the source.Own calculations of annualized costs include VAT. 6https://doi.org/10.5281/zenodo.3345784.
7See BDEW (2020) (in German) for a complete breakdown for a typical German household.C.Günther et al.

Table 3
Sets included in the model

Table 4
Variables included in the model sto Annualized specific investment costs into storage energy (continued on next page) C. Günther et al.

Table 5
(continued ) Renewable generation technologies and storage are assumed to incur no variable operating costs.Equation (A.5b) shows the energy balance, i.e., the market clearing condition, which must hold in each hour.The sum of non-prosumage consumer demand d h , prosumage grid energy demand E m2pro h , and storage intake STO in sto,h must equal generation of conventional power plants G con,h , renewable generation G res,h , storage discharging STO out sto,h , and prosumage energy fed to the grid G pro2m h .The dual variable to this constraint λ enbal his interpreted as the wholesale electricity price.It indicates the marginal costs associated with a marginal increase on the demand side.As is the case for prosumage households, energy generation of each renewable technology res at the power sector level depends on the hourly capacity factor φ avail res,h and the exogenous capacity n res .Equation A.5c prescribes that hourly available renewable energy can either be curtailed CU res,h or consumed on the market G res,h :φ avail res,h *n res = G res,h + CU res,h ∀res, hConventional generation is perfectly dispatchable and only constrained by the installed capacity n con as shown in equation A.5d.Besides conventional and renewable generation technologies, pumped-hydro power storage is also included for power supply on the power sector level.Equation (A5e) describes the change of the energy level of the storage over time.It is the sum of the energy level of the prior period STO l sto,h− 1 and current storage charging STO in sto,h , minus the energy discharged STO out sto,h .Both charging and discharging activities account for storage energy losses.Furthermore, the storage energy level is constrained by its energy capacity as given by equation A.5g.Likewise, storage charging and discharging cannot exceed the installed power capacity.The first order KKT conditions corresponding to the power sector dispatch are given by: * n res − G res,h − CU res,h , λ resgen res,h free ∀res, h