The social profitability of photovoltaics in Germany

Abstract While Germany has led the market in photovoltaic (PV) implementation throughout the last decade, there has been increasing criticism of PV support policies due to their high cost. Although declining, the levelized cost of electricity (LCOE) from PV is still above the German wholesale electricity price. However, using LCOE as an evaluation yardstick falls short in at least 2 respects: It neither takes into account integration costs rising with PV penetration (ie, undervaluing its actual cost) nor avoided externalities of replacing conventional for renewable generation (social cost overvaluation). We thus calculate the social profitability of PV in Germany by including not only private costs and benefits but also integration costs to the electricity system and avoided environmental externalities, using the internal rate of return and the profitability index as indicators. Our results show that when these factors are considered, the social profitability of PV in Germany is higher than 10% at the lower bound of the social cost of carbon (150€/tCO2) up to a penetration level of at least 15% and positive up to a penetration level of at least 25%. Results also show the level of private profitability if all externalities were internalized and assert that subsidies are justified to align private and social profitability. The proposed method could be used as a complementary indicator to private profitability by public institutions, development banks, and companies with social responsibility values.


| INTRODUCTION
Germany has led the photovoltaics (PV) implementation market throughout the last decade, with the highest installed capacity per capita and the second highest in absolute value after China. 1 However, although PV costs have been declining during the last 3 decades at a learning rate (ie, cost decline per each doubling in installed capacity) between 20% and 24%, 2-4 PV support policies have been under increasing criticism due to their high costs.
While PV is quickly achieving grid parity in many parts of the world, 5 its levelized cost (LCOE) is still above wholesale electricity prices in Germany. 4 Additionally, increasing PV penetration causes rising integration costs to the electricity system-due to its variability, uncertainty, and location specificity. 6,7 Finally, all generation technologies also cause external costs not included in their market prices, 8 causing social damages, which are usually ignored in traditional cost-benefit analyses. Therefore, any comprehensive cost-benefit analysis of generation technologies should take into account not only private costs and benefits but also the integration costs caused to the electricity system and the avoided or additional environmental externalities of switching between technologies.
We develop a method to calculate the "social profitability" of PV by including integration costs and avoided externalities alongside the assessment of private costs and benefits, applying our approach to Germany. By including social costs and benefits, we perform a more comprehensive analysis to inform policy and investment decisions.
First, our method demonstrates the competitiveness of the technology when social costs and benefits are taken into account. Second, it can justify taxes/subsidies when social profitability is lower/higher than private profitability to arrive at welfare-optimal investment decisions.
Third, it indicates private profitability levels if all externalities were to be internalized. Finally, it is a useful indicator to complement private We calculate "social profitability" through 2 widely used indicators: the internal rate of return (IRR) and the profitability index (PI), and according to the 2 approaches available in the literature: considering integration costs either as an additional cost or as a lower value of PV electricity. 7 Aside from this calculation, we present a wide range of results regarding the main parameters to cope with the uncertainty concerning actual values of these parameters and their future evolution.
The remaining of the paper is structured as follows: Section 2 frames the discussion regarding the profitability and competitiveness of variable renewables. Section 3 presents the method for the calculation of the social profitability of PV and the input data to the different approaches. Section 4 presents the main results, first as a function of the social cost of carbon (SCC) and then, as a function of PV installation cost and electricity yield. Section 5 summarizes insights derived from the previous sections' analysis and its relevance.

| PROFITABILITY AND COMPETITIVENESS OF VARIABLE RENEWABLES
Competitiveness of different electricity-generating technologies is usually assessed and compared through the LCOE, which measures the life cycle costs of a technology per kilowatt hour of electricity generated during the lifetime of the system, discounted at a determined discount rate. [9][10][11] However, this indicator has increasingly been criticized for its inability to capture a range of features divergent across technologies, and other indicators have been proposed, such as the levelized avoided cost of electricity, which accounts for how much it would cost the grid to generate the electricity otherwise displaced by the new generation project, 12 and the System LCOE, which includes the cost of integrating the new generation into the existing electricity system, moving beyond the traditional LCOE. 6 From a levelized costs perspective, competitiveness or "grid parity" is achieved when a generation technology reaches the costs of conventional technologies. 5,13 Equivalently, competitiveness can be assessed from the point of view of profitability. In this sense, competitiveness would be achieved when a technology is profitable in the market without subsidies. 14,15 However, the existence of market failures may entail the departure of private profitability from social profitability, which would not be the case in the presence of perfect and complete markets.
Only in the latter case would private and social profitability be equal and the amount of investment be welfare optimal. In the presence of incomplete and/or imperfect markets, however, private investment decisions may yield socially suboptimal outcomes. In this case, the government can subsidize/tax activities with positive/negative externalities such that private and social profitability align again. 16 Photovoltaic profitability has been widely studied regarding evaluating the impact and evolution of feed-in tariffs and other incentives. [17][18][19][20][21][22] However, on the one hand, private profitability does not capture the integration costs caused to the electricity system in situations with higher PV penetration and, on the other hand, the avoided environmental externalities when PV displaces conventional generation. On this basis, we calculate the social profitability of PV by taking into account not only private costs and benefits but also integration costs imposed on the electricity system due to higher PV penetration as well as net avoided environmental external costs. Since we want to evaluate the social profitability of the technology to displace conventional generation, we do not include corrective incentives such as feed-in tariffs or investment subsidies in our calculations, since they are interventions designed to correct for externalities themselves and would therefore result in double counting. Positive social profitability would entail that the technology is competitive when all factors are accounted for. Likewise, social profitability above/below private profitability would justify subsidies/taxes to that technology to adjust for social benefits/costs. Finally, social profitability shows how much private profitability would be if all externalities were internalized.

| Social profitability index and social rate of return
We calculate social profitability by computing the internal rate of return and its equivalent profitability index, including not only private costs and benefits but also integration costs and avoided environmental externalities. We assume that new PV generation displaces nonrenewable generation, and we do not consider other spill-over effects or broader macro policy objectives such as energy security or job creation.
We calculate the internal rate of return (IRR) as the discount rate Both indicators are equivalent such that a breakeven 0% IRR corresponds to a PI of 0 at zero discount rate. Likewise, 2 SPI(0) is equivalent to 10% SRR (see Figure A1 in the supplementary materials for a detailed equivalence between both indicators and across different discount rates).
Finally, we can calculate the breakeven SCC, breakeven installation cost (PV in ) and breakeven annual electricity yield (or equivalent solar irradiation) (EPV) at a certain discount rate by simply clearing the interest parameter in Equation 1 as shown, respectively, in Equations 9 to 11. In other words, we calculate the value of SCC, installation cost, and annual electricity yield necessary to achieve a certain level of social profitability (eg, 0%, 5%, or 10% as shown in Figures 4-6).

| Cost vs value approach to integration costs
Increasing the penetration of variable renewable energy (VRE) technologies causes integration costs to the electricity system. [25][26][27] Integration costs arise because of the uncertainty, variability, and location specificity of this type of technologies, which are usually higher than those of conventional dispatchable generation alternatives. Integration costs can be measured and conceptualized in a "cost approach" as the marginal cost of increasing VRE penetration or in a "value approach" as the declining marginal value of VRE electricity at higher VRE penetration. 7 We use both approaches to calculate the social profitability of PV in Germany.
Integration costs can be decomposed into 3 subcategories according to the respective feature of the VRE causing them. 6 The input parameters of our model-and therefore our results-are a rough estimation of shape and order of magnitude at different penetration rates, rather than a final and exact calculation. Input data are extracted from the literature and refers specifically to PV in European (profile and grid costs) and American (balancing costs) thermal systems.
We assume balancing costs to be a linear function of penetration starting from 2€/MWh at 0% 28 up to 6€/MWh at 30% penetration. 29 According to Hirth et al, 7 grid costs are within the single-digit range in terms of € per megawatt hour. Therefore, we consider the most pessimistic shape of grid costs suggested by MIT 30 (an inverted U curve peaking at 20% penetration) at an assumed 10€/MWh to be in the high end of the range. Profile costs, derived from the variability of the PV generation, entail the most important share of integration costs ( Figure 1). Within profile costs, backup costs are most prominent at low penetration levels, reduction of load hours for conventional generators prevails at penetrations of 5% to 20%, and overproduction costs rocket to dominance beyond that point. 6 Figure 1 illustrates the input data for the cost model, which is derived from the aforementioned literature and constitutes a quantification of the shape and order of magnitude rather than an exact calculation. 6 Likewise, increasing penetration of almost-zero marginal cost electricity technologies cause the decline of wholesale electricity prices by shifting the supply curve to the right in the so-called merit-order effect. [31][32][33][34] This effect is particularly relevant for PV since the price drop is stronger at times of high PV generation. Therefore, the market value of PV electricity declines as its penetration in the market increases. [35][36][37][38][39] In the presence of perfect and complete markets, the decline in the value of PV electricity would be equal to the decrease of its value factor in the wholesale electricity market, where the value factor represents the ratio of the unit revenue of PV (ie, the generation-weighted average price) and the time-weighted average wholesale electricity price, 39,40 in other words, the average remuneration of PV producers relative to the average wholesale price. Therefore, integration costs can also be understood as the decline in the market value of PV electricity, this approach being equivalent to the cost perspective. 6,7 The market value of PV electricity can be estimated empirically expost from market data or theoretically ex-ante through dispatch or investment and dispatch models. Hirth 39  to the German electricity market. Among these 2, the empirical would be more suitable for our analysis than the review estimates as it relates to the country of our concern. However, only the ex-ante approach allows for the flexibility necessary to accommodate future changes in the electricity mix. While the ex-post approach relies on an extrapolation of the current electricity system, in the investment and dispatch model, the electricity system adapts capacity and  We use the model benchmark estimates at 0€/tCO 2 SCC (since we account for the SCC separately) and its respective power extrapolation (see Figure A4 in the supplementary materials for more details on extrapolations) as input data. The estimates at 0€/tCO 2 show a higher solar value at low penetration but a stronger decline as penetration increases. These estimates are derived from the partial equilibrium EMMA model, which is a dispatch and investment model representing the Northwestern European power system (Germany, Belgium, The Netherlands, France, and Poland) and does not account for internal grid constraints, so it only partially captures grid integration costs, the total of which are relatively small according to Ueckerdt et al 6 (see Figure 1).

| Climate change and other environmental externalities
Externalities can be defined as the costs or benefits caused by an economic activity to a third party not involved in the transaction. Since external costs or benefits are not included in market prices, they lead to a welfare-suboptimal allocation of resources. 16   Hirt 2013 and 2015 39,40 Abbreviation: PV, photovoltaic.
*Extrapolation (see Figure A4 in the supplementary materials for details).
In this sense, van den Bergh and Botzen 49 argue that the average estimates from IAMs are gross underestimates: first, because they include both low and high social discount rates, which undermine the present value of future damages, and second, because they ignore factors such as (1)  itive CC feedback loops, which is also likely to be a source of undervaluation, in addition to the latest developments in earth and climate sciences, which point at higher climate sensitivity than expected, 50 permafrost tipping point risks, 51 higher than expected sea level rise, 52,53 and even climate-driven polar motion, 54 lead us to expect that the SCC might be even higher than the lower bound of 150€ 2015 /tCO 2 suggested by van den Bergh and Botzen. 49 We still use the value of 150€/tCO 2 as a benchmark lower bound of the SCC but report a wide range between 0 and 300€/tCO 2 to account for the uncertainty surrounding its actual value and the (more likely than not) upwards evolution of future estimations acknowledging risk aversion.

| Other data and assumptions
Since we calculate our model in real terms and assume that nominal operation and maintenance costs, electricity prices, and integration costs (the latter over time but at identical penetration levels) rise at the same rate as inflation, their respective real escalation rates are equal to 0. The SCC increases over time because GHG concentrations increase over time, with future emissions thus producing larger incremental damages. 41 Although its real escalation rate is also uncer-  Table 2 for a summary of the input data).

| RESULTS AND DISCUSSION
We now present our model results, comparing the cost and the value approaches. While their interpretation up to 15% penetration is the same, it differs beyond that point. Both approaches include endogenous adaptation of the electricity system to higher VRE generation.
However, while the cost approach includes overproduction costs arising beyond 15% penetration, these are not captured in the value approach, since input data are extrapolated beyond the point at which overproduction start to arise. Therefore, the cost and value illustrations below can be considered as worst and best cases, respectively, considering either no additional adaptation measures beyond endogenous electricity system capacity optimization (cost approach) or perfect adaptation to remove all overproduction costs beyond 15% penetration (value approach). 5% in the last 10 years, so the white area between the solid and the dashed lines in Figures 4-6, representing 0% and 5% social profitability levels, respectively, is a plausible range for covering the "social cost of capital." Figure 4 shows the results for the 2 approaches as a function of PV penetration and the SCC. The color gradient represents the social profitability index (SPI) at 0% discount rate, and the solid line shows the social profitability breakeven (ie, social rate of return (SRR) equal to 0%). The dashed and the dotted lines show SRR of 5% and 10%, respectively (see Figure A1 in the supplementary materials for a detailed relation between the SPI and the SRR and between the SPI at different discount rates). Both cost and value approaches agree that PV profitability at current penetration (8%) 57 is above 10% at the lower bound of the SCC (150€/tCO 2 ), and around 5% at a SCC as low as 50€/tCO 2 . Indeed, PV would still be socially profitable at that level of the SCC and double the current penetration (50€/tCO 2 and 16% penetration).

| Social profitability as a function of the SCC
While the primary axes provide the basic information to interpret

| Social profitability as a function of the installation cost
We now extend our analysis to cover a wide range of possibilities regarding PV installation costs and PV yield per kilowatt peak, which deepen the dynamic interpretation of the results and allow the reader to evaluate different scenarios according to assumptions on the evolution of different parameters. Since at the lower bound of the SCC FIGURE 4 Social profitability index at 0% discount rate [SPI(0)] and social rate of return (SRR) as a function of PV penetration [%] and the social cost of carbon (SCC, €/tCO 2 ). Note: The right-side axis illustrates the year, in which the corresponding SCC is reached according to the "highimpact" scenario of the Interagency Working Group. 41 The upper axis depicts the evolution of PV penetration in Germany 57

| Social profitability as a function of the PV yield per watt peak
Finally, we present the social profitability of PV as a function of its penetration and its potential yield per kilowatt peak, again with a SCC at only half its lower bound (75€/tCO 2 ). The potential PV yield per kilowatt peak is taken from the Photovoltaic Geographical Information System (PVGIS), 55 which assumes a 75% performance rate. Figure 6 illustrates the position of several European countries as a function of their respective PV penetration and average PV potential. Due to the lack of country-specific integration cost estimations, these results can provide a useful illustration about their approximate position. FIGURE 5 Social profitability index at 0% discount rate [SPI(0)] and social rate of return (SRR) at 75€/tCO 2 social cost of carbon as a function of photovoltaic (PV) penetration (%) and installation cost (PV in [€/kWp]). Note: Ranges labelled by years represent the forecasted evolution of PV installation costs by the Fraunhofer Institute 11 with their respective uncertainty ranges, arbitrarily located along the 10% SRR line for a clear visualization. Input data used for calculations to the right of the vertical dotted line are extrapolations FIGURE 6 Social profitability index at 0% discount rate [SPI (0) Figure 6A.

| CONCLUSIONS
We have calculated the social profitability of PV in Germany by including not only private costs and benefits but also integration costs of higher PV penetration in the electricity system and avoided external costs of displacing nonrenewable generation. We have computed social profitability from a cost and value perspective. Both approaches agree that at the lower bound of the SCC (150€/tCO 2 ), PV social profitability is above 10% at current German penetration level (8%), and it is still positive up to at least double the current penetration level even at only 50€/tCO 2 .
This entails that PV is competitive when both integration costs and externalities are included in the analysis and shows the level of private profitability if all the external costs and benefits were internalized.
Therefore, subsidies to PV are economically justified in these ranges to align private and social profitability and reach a welfare-optimal allocation of resources.
Although uncertainties are large for penetration levels beyond 15%, the comparison between cost and value approaches suggests the high potential of exogenous adaptation (such as storage, demand side adjustment, and intercontinental interconnections) to boost PV social profitability also in situations beyond 20% penetration. In the worst case, if no additional adaptation measures are implemented, PV will only be competitive beyond 20% penetration to abate high CC damages at exponentially increasing costs due to overproduction.
On the contrary, if all overproduction costs could be totally removed at zero cost, PV would be socially profitable at half the lower bound of the SCC, independently of its penetration.
The method presented in this article could be used as a comple- ENDNOTES † Indeed, the value approach results beyond 15% penetration show a similar shape as produced by the cost approach when overproduction costs are excluded (see Figure A2 in the supplementary materials). ‡ More specifically, this results in a profitability index or IRR for a specific penetration level. If penetration increases further in the future, this may induce changes to the integration costs or value factor, also for all earlier investments. As we neglect this potential impact, our results are valid for the respective point in time and a specific penetration level, but profitability and SRR could decline for such investment, when penetration is further enhanced later-in the same way as the new profitability index at the higher penetration level will decline. § Originally reported in $ per ton of carbon 46 : 151$ per ton of carbon is equivalent to 41.2$ per ton of CO 2 .
** van den Bergh and Botzen 53 derive the lower bound of the SCC of 125$ 1995 /tCO 2 by applying a 206% surcharge on the average value indicated by Tol 46 (personal communication to authors). We apply the US GDP deflator (Federal Reserve database) for 1995 to 2015 (1.46) and the 2015 average US$ to € exchange rate (1.11) and perform a sensitivity analysis with the long-term (1990-2015) average exchange rates (1.11-1.23) to account for the real depreciation of the US$ in the last years, together resulting in a range between 148€ 2015 /tCO 2 and 164€ 2015 /tCO 2 , out of which we consider a conservative value of 150€ 2015 /tCO 2 as a benchmark for the lower bound of the SCC.