Cédric Clastres
email: cedric.clastres@univ-grenoble-alpes.fr

Jacques Percebois
email: jacques.percebois@umontpellier.fr

Olivier Rebenaque
email: olivier.rebenaque@chaireeconomieduclimat.orgboris.solier@chaireeconomieduclimat.org

Boris Solier

RENEWABLE SELF-CONSUMPTION

Keywords: L94, Q42, L51 Self-consumption Cross-subsidies Network tariff Policy instruments

The deployment of renewable energies relies upon incentive policies to make their use profitable for owner. However, their development needs adjustments of network to manage intermittency and additional energy fed into the grid. Moreover, the Public Service Obligation Tariffs (PSOT) are increasing to fund policies that support renewable energy deployment. Therefore, some decisions are taken to promote selfconsumption by owners of renewable energy power plants, as photovoltaic prosumers. This behavior is encouraged by payment exemptions of PSOT, special tariffs dedicated to remunerate each selfconsumed energy unit or savings on the variable part of the network tariff. Thus, some cross-subsidies appear between self-consumers and other users of the network to compensate all these previous selfconsumers' gains. We show that these cross-subsidies occur but they strongly rely on self-consumption rate and on renewable energy share in the total produced or consumed energy. So, currently, the levels of cross-subsidies are not significant for consumers. We also show that regulator could fund these cross-subsidies increasing the fixed part of the network tariff for prosumers.

1 Introduction Public and environmental policies have launched the deployment of renewable Energies, as photovoltaic (PV) capacities. We do not come back on environmental policies and the obligation for all countries to develop renewable energies to achieve the 3*20 goals of the third Energy Package. Regarding public policies, we could classify them into two categories [START_REF] Zachmann | When and How to Support Renewables? Letting the Data Speak[END_REF]. The first one is the financial support for Research and Development (R&D) in a specific technology. The second one is based on subsidies to develop investments in selective generation processes, in our case in clean technologies as for example PV generation capacities. These later incentive policies are decided by public authorities to internalize high investment costs in renewable energies and to lower electricity prices (IEA PVPS, 2016). In the sector of renewables, we could cite the Feed In Tariffs, the Feed in Premium or the Net metering as public incentive policies which have improved the increase in renewable energy capacities. The PV sector will benefit from these policies and we could observe that PV capacities have increased since 2007 at a great rate in the world, from 1 GW in 2000 to 230 GW in 2015 (Figure 1). This increase in PV installed capacities has dropped the cost of PV panels. Between 1975 and2008, their costs have decreased at a 2% rate (from 65$/W to 4$/W). Since 2008, we observe an accelerate decreasing rate to achieve 1$/W today, so a decrease in cost of 80% in this period of time [START_REF] Mir-Artigues | The Economics and Policy of Solar Photovoltaic Generation[END_REF]. According to Bloomberg New Energy Finance (2016), when PV installed capacities increase twofold, the cost of panels is reduced by 23.4%. However, PV energy has not reached the grid parity in all country. The grid parity is achieved when energy prices are equal than the Levelized Cost of Electricity (LCOE) for PV technology. This could be the case in areas where solar radiations are important or in countries with high electricity prices. According to SOLAR POWER EUROPE (2016), only 1.3% of PV capacities have been developed without incentive policies. For these installations, subsidies are not useful and, because they are not included in the public budget [START_REF] Couture | A Policymaker's Guide to Feed-in Tariff Policy Design[END_REF][START_REF] Mendonça | Powering the green economy: the feed-in tariff handbook[END_REF], weight of Public Service Obligation Tariffs could be reduced for consumers. Thus, in most of studied cases, public incentive policies are always needed to develop renewable energy capacities (IEA PVPS, 2016). Large renewable energy capacities go with grid and financing difficulties. Network operators, mainly Distribution System Operator (DSO) 1 , must invest in new capacities to manage the surplus of distributed energy fed into the grid and its intermittency. Incentive policies must be compensated with revenues coming from consumption or network uses. Thus, we often observe an increase in Public Service Obligation Tariff (PSOT). For example, in France, the PSOT, called the CSPE (Compensation for Electricity Public Services) and paid by all consumers of electricity, has increased between 2010 and 2015 from 120 million € to 2.2 billion €. This increase mainly comes from the obligation for suppliers to buy renewable energies at regulated prices (Feed In Tariffs). Thus, we observe two kinds of transfers between electricity consumers. The first one is a transfer based on the PSOT. Indeed, the PSOT increases for all consumers but part of it is redistributed towards PV producers, which could also be consumers (usually called prosumers). The second one is a transfer based on network tariffs. Grid costs are increasing because of distributed energy management. This increase must be compensated by the regulated tariff which is the unique network operator's revenue. As this regulated tariff is paid by all consumers of electricity, we observe another transfer towards prosumers.

In recent years, and to internalize increasing network and intermittency costs, the deployment of self-consumption for renewable energies has been studied and implemented in several countries. Selfconsumption situation is defined as the opportunity for a prosumer to consume part or the entire amount of its own generation (Figure 2). Two kinds of values are usually used to evaluate the Self-Consumption (SC) for a prosumer (Table 1). The first one is the SC rate (τ SC) which is defined as the share of PV production that is consumed in the generation area. The second one is the Self-Generation (SG) rate (τ SG) which is defined as the share of consumed electricity that is not withdrawn from the grid.

Table 1: Self-consumption and self-generation rates

Self-consumption rate

Self-generation rate τ SC = Total PV electricity consumed for a day on site Total PV generation for a day on site τ SG =

Total PV electricity consumed for the day on site Total Electricity consumed for a day on site

These two rates are different and do not refer to the same objectives. If the objective is to maximize the SC rate τ SC, then prosumers must decrease their PV generation (and their PV installed capacity) or must shift part of their demand in hours when PV generation occurs. In this case, demand could stay at the same level; prosumers do not have incentive to reduce it. Otherwise, if the objective is to maximize the SG rate τ SC, then prosumer could have incentives to reduce their demand.

According to [START_REF] Jäger-Waldau | PV Staus Report[END_REF], the SC rate is around 30% in Europe. In some implementations, as for example in France for consumers with smart-meters, this rate was around 45% for households and 70% for industries (PV-NET, 2016). In Italy, the 2015 SC rate was 48% in the tertiary sector, 58% for the industry and 42% in the agricultural sector (GSE, 2015). In Germany, this rate is 27% for households that benefit from an incentive policy to promote self-consumption. The rate is more often below 50% for households [START_REF] Luthander | Photovoltaic self-consumption in buildings: A review[END_REF], except for Sweden [START_REF] Thygesen | Simulation and analysis of a solar assisted heat pump system with two different storage types for high levels of PV electricity self-consumption[END_REF][START_REF] Widén | Improved photovoltaic self-consumption with appliance scheduling in 200 singlefamily buildings[END_REF]. The self-consumption rate is different in Europe because it depends on solar radiation but it also depends on the self-consumption regulatory framework. For instance, in Italy, the incentive to selfconsume is based on a net-metering scheme which doesn't encourage a high self-consumption rate. However, industrial sector could have greater SC rates because of their level of consumption when PV generation occurs [START_REF] Ong | Impacts of regional electricity prices and building type on the economics of commercial photovoltaic systems[END_REF]. So, some industrial prosumers could have a SC rate between 95 and 100% (cf section 3.5). There exist some innovations that could increase this rate. If prosumers use batteries to store their produced energy to consume it at low solar radiation periods, self-consumption could increase from 10 to 24 percentage points. If some technologies, as smart technologies, are used to manage prosumers' demand, for example shifting some uses when the sun shines, self-consumption could increase from 2 to 15 percentage points [START_REF] Luthander | Photovoltaic self-consumption in buildings: A review[END_REF].

Thus, self-consumption could improve the management of the PV intermittency. However, it creates some cross-subsidies between consumers because of the incentive policies that could be applied to favor SC. In this paper, we propose to study these cross-subsidies from standard electricity consumers to prosumers that consume their own energy production. In a second section, we begin with the analysis of SC deployment and on incentive policies that could be applied to foster it. In a third section, we present some empirical illustrations of SC deployment, as several countries have decided to favor it to internalize renewable energies impacts on the electricity sector. In the fourth section, we focus on crosssubsidies between categories of consumers. We show with some implementations that cross-subsidies occur but the amount is currently low. In the fifth section, we propose two methodologies to recover the financial losses for the system operator (Distribution System Operator). We show that an increase in fixed part of the network tariff paid by self-consumers could internalize cross-subsidies.

Incentive policies and self-consumption deployment

Main features in favor of self-consumption

Investing or not in renewable energies and consuming its own production rely on several features. Firstly, the initial cost of investment in renewable installations impacts consumers' decision to realize the project. Secondly, the electricity price paid by the consumers, the solar radiations and demand curve act upon the profitability of the project. Thirdly, the network tariff components reinforce the reduction in costs for investors but create concerns on network costs recovery. Indeed, network tariffs are often based on two parts: a fixed part based on the contractual power (in MW) and a variable part based on consumed energy (in MWh). The more this later part is, the more self-consumption should be attractive because of savings on variable costs, as the electricity withdrawn from the grid is reduced [START_REF] Schittekatte | Future-proof tariff design: Recovering sunk grid costs in a world where consumers are pushing back[END_REF]. Finally, the exemption on the taxes for the electricity self-consumed creates an incentive to invest in a PV power plant. These features could be influenced by public decisions. The cost barriers could be overcome with low rates for loan or with tax fees exemptions. Electricity price levels are often influenced by taxes set by public authorities. Network recovery costs are set by the regulation scheme that could be adapt to favor self-consumption if included in the energy policy goals of the country.

Besides these economic factors, adding behavioral analyses are relevant because they also influence the decision of investment. They are based for examples on environmental concerns, consumed energy management, the origin of power (renewables or not), security and reliability, etc.… [START_REF] Boughen | Understanding the residential customer perspective to emerging electricity technologies: Informing the CSIRO Future Grid Forum[END_REF][START_REF] Bremdal | Prosumer oriented business in the energy market[END_REF][START_REF] Farhar | Public response to residential grid-tied PV systems in Colorado: A qualitative market assessment[END_REF].

Table 2 summarizes the main features to favor self-consumption.

Incentive policies and self-consumption

As the deployment of renewable energies shows it, the role of public policies is often essential to foster the adoption of new energy technologies or new behavior. Thus, we observe several decisions in favor of SC. Each of them has been implemented and, today, some are eliminated because of their low impact on the improvement of the system.

In several countries, net-metering or net-billing are used to manage SC (Dufo-Lopez and Bernal-Agustin, 2015). Net-metering consists in the calculation of the net prosumers' consumption as the total consumption of the site minus the renewable energy volumes fed into the grid (DGEC, 2014). Prosumers only pay the net consumption of their site. In this case, the network acts as a back-up power plant, i.e it supplies the residual demand of the site and offers security and reliability services. This mechanism assumes that the value of the energy fed into the grid is the same that the energy consumed from the grid. Net-billing is different from net-metering on this feature. Indeed, the exported energy to the grid and the imported energy from the grid have different monetary values. The exported energy is valued at market price or at the "avoided cost price", rather than the imported value is valued at a greater price, the retail price (Dufo-Lopez and Bernal-Agustin, 2015). So, prosumers' revenues could be:

• In the case of net-metering, revenues are only reductions on their bill because of the payment of the net consumption. In case of exported energy greater than consumed energy from the grid, they could receive certificates that could be used to increase the energy fed into the grid for a period in which consumed energy is greater than renewable energy production. In other words, renewable energy surpluses in one period could be used to compensate a reduction in renewable energy production in another period of consumption. • In the case of net-billing, their revenues are the monetary difference between the renewable energy fed into the grid and valued at a feed-in-tariff (market price or other valuation) and the consumed energy paid at retail price. Public authorities must set the feed-in-tariff at an incentive level to improve self-consumption.

Net-metering and net-billing stand for 16% of the incentive policies that foster the deployment of PV installation, the main driver staying the feed-in-tariff scheme (IEA PVPS, 2016c). The cost of PV generation is also an important driver of the PV deployment. In case of net-metering, as savings are the unique revenue for prosumers, costs of PV generation must be lower than retail price to develop selfconsumption. In case of net-billing, we have the same conclusion on PV costs adding the impact of the exported tariff. This one must not be too high to restore the incentive to self-consumption. Otherwise, prosumers earn more gains to feed all their production into the grid. The net-metering system has been criticized by several authors (Gauthier et al., 2017). These appreciations are mainly based on its inefficiency to induce good economic signal to the deployment of renewable energies2 and because it creates numerous transfers between prosumers and other consumers (transfers of the network costs). The European Union has recently prohibited the use of net-metering to foster the deployment of selfconsumption.

Thus, self-consumption is profitable if the grid parity is achieved for the PV generation, i.e if the levelized cost of PV electricity is lower than the retail price. In that case, PSOT should be reduced, selfconsumption being economically profitable. Some countries, as Germany or Denmark, have increased their retail prices (respectively around 0.26€ and 0.29€ per kWh) to reduce the costs of the renewable policies. Since 2009, the levelized costs of PV electricity have decreased from $0.40 per kWh to $0.06 per kWh (Solar Power Europe, 2017). Projects in some world areas have achieved lower costs, as in Argentina ($0.33 per kWh), in Mexico ($0.29 per kWh), in Chile ($0.29 per kWh), in USA ($0.27 per kWh) or in Abu Dhabi ($0.24 per kWh). Other PV projects, mainly in the south of Europe, have also significantly reduced their costs to achieve a range of 0.035 to 0.06€ per kWh. Then, self-consumption could stand for a relevant percentage of the renewable energy production (Figure 3).

Empirical experiences in Europe on self-consumption deployment

In Europe, PV capacities have grown significantly since 2006 (Figure 4). This growth is widely due to incentive policies, as Feed-In-tariffs (FIT) schemes.

Germany

Germany is the European country with the stronger increase in PV generation. Its FIT policy is the main driver of this growth. So, it has earlier known discussions on the costs of the FIT policy, which reached near than 1.2 billion euros in 2006 (BMU, 2007). Moreover, the exemption of PSOT for several firms has increased the small end-users' charges (of an amount of 420 million euros in 2006). So, since 2009, Germany has set premiums to favor self-consumption. Each self-consumed kWh was paid at this premium (Table 2). Besides this premium, FIT have decreased of about 10% to reduce the profitability of renewable energy fed into the grid. An exemption on the payment of fixed part of network tariffs and taxes was also decided for prosumers. These incentive policies have achieved their goals. Between 2009 and 2014, the percentage of sites that self-consumed their own generation has grown to achieve in 2014 a range of 70 to 95% for installations with a capacity between 0 to 1000 kW (BUNDESNETZAGENTUR, 2015). A simple comparison between revenue from FIT and self-consumption could resume this growth. Selfconsumption is profitable if the revenues from premiums and savings from retail price are greater than the revenue of FIT, as we could see in Equation 1∑ 𝐩 is the premium for self-consumption (Table 2), q AC t the self-consumed energy, 𝑝 𝑡 the energy price, b the fixed part of the network tariff (paid by the self-consumers), q x t the amount of energy fed into the grid and 𝑝 𝐹𝐼𝑇 the FIT. Some figures from Bundesnetzagentur (2015) could be used to compute the revenue from one self-consumed kWh or sold at the FIT (Figure 5). Each year, self-consumption is more profitable rather than FIT. Self-consumption rates have increased to achieve a range of 20% to 40% in 2014. During the period 2009-2013, Germany has known a significant increase in self consumption volumes, increasing the costs of incentives policies ( Source: BMU

As the costs of the policy are always important, the government of Germany has decided in 2012 several changes. Firstly, the FIT is again reduced (of 20 to 26%). Secondly, only part of PV capacities could benefit from FIT. Thirdly, the potential energy fed into the grid from PV installations is lowered (with a maximum of 70% of PV generation, reduced to 50% in 2016). If a producer found economically profitable to continue to value its production at the FIT, it had to contract with the Network System Operator (NSO) a "feed-in management" clause, i.e the NSO could cut the PV generation if necessary. In this case, the PV producer could receive a fee. Fourthly, self-consumption premiums were stopped because of the achievement of the grid parity for PV generation (Fraunhofer ISE, 2017;[START_REF] Fulton | The German Feed-in Tariff: Recent Policy Changes[END_REF]. This was due to a huge increase in retail price of electricity, the EEG amount was the main driver of this increase (from 5% of the household's retail rate in 2009 from 14% in 20124 ). All these measures created incentives for all prosumers to increase their self-consumption. For example, in 2014, 95% of residential prosumers are under self-consumption (BMU). In the same period, selfconsumption is extended to a group of consumers in the same area. Then, they benefit from EEG or network tariff exemptions (they ended up in 2016). Government also wants to develop the storage of renewable energies to improve self-consumption during all the hours of the day (and not only when PV generation occurs). It has invested between 2013 and 2015 about 50 million € per year for this policy to cover at least 30% of the costs for one installation5 .

Despite the improvement in self-consumption rate, the charges of the incentive policies have increased (Table 3). This is mainly due to the higher gap between FIT and spot price because of great renewable energy volumes in the spot market, reducing spot prices (FRAUNHOFER ISE, 2017). Thus, this difference has raised charges paid by end-consumers. They also have suffered from cross-subsidies. Their appearance is due to the exemptions in EEG or network tariffs payments for the prosumers. All network and EEG costs are passed on to users of the network or consumers of electricity from the grid. So, they subsidize self-consumers. According to l'OFAEnR, self-consumption has involved losses in EEG revenues of 100 million € in 2012 and could achieve 300 million € for the 2013-2018 period. To reduce these inequalities, self-consumers had to pay 30% of the EEG fees on self-consumed kWh in 2014. This percentage had increased to achieve 35% in 2016 and 40% in 2017 (EEG law, 2014). However, some are always excluded from taxes, as small generation sites or industrial and large consumers. These losses in EEG revenues have stood for 5.1 billion € in 2014 (Fraunhofer ISE, 2017). As costs of distribution network have known a huge increase, from 14.6 billion € in 2011 to 17.7 billion € in 2016 (OFAEnR), Distribution Network Operator could now apply a reduction in annual renewable generation (EEG law 2012, §6, 1, 2).

The implementation in Germany of self-consumption is certainly those we have the more information. We could conclude that the deployment of self-consumption goes with an increase in retail prices and a decrease in FIT. To foster this development, some incentive policies have been added, as feed-inpremium or exemptions on taxes and network tariffs. These policies have increased PSOT (EEG costs) and cross-subsidies between consumers of electricity withdrawn from the grid and prosumers. Public authorities came back with the tax on the self-consumed volumes to partially restore the equality between consumers.

The United Kingdom

The United Kingdom set Feed-in-Tariffs in April 2010 to improve the deployment of PV generation lower than 5 MW (Table 4). These tariffs are applied on the total PV generation of the site. They could be increased by an export tariff for all PV energy fed into the grid. On the period 2010 to 2016, this tariff involves in a range of 3 pence/kWh in 2010 to 4.85 pence/kWh in 2016 (OFGEM). According to this incentive policy, revenues of self-producers are divided into three components, as we could see it in Equation 2:

𝑹 𝒕 = ∑ [(𝒑 𝒕 -𝒃) * 𝒒 𝑨𝑪 𝒕 ] + ∑ 𝒒 𝒕 * 𝒑 𝑭𝑰𝑻 + ∑ 𝒒 𝒙 𝒕 * 𝒑 𝒙 𝒕 * 𝒏 𝒕=𝟏 𝒏 𝒕=𝟏 𝒏 𝒕=𝟏 (𝟐)
With 𝑝 𝑡 , 𝑏, 𝑞 𝐴𝐶 𝑡 , 𝑝 𝐹𝐼𝑇 , 𝑞 𝑥 𝑡 the same parameters as in Equation 1, 𝑞 𝑡 the total PV generation with 𝑞 𝑡 = 𝑞 𝐴𝐶 𝑡 + 𝑞 𝑥 𝑡 and 𝑝 𝑥 𝑡 * the export tariff.

The first component, [(𝑝 𝑡 -𝑏) * 𝑞 𝐴𝐶 𝑡 ], is the savings on electricity bills because they don't withdraw 𝑞 𝐴𝐶 𝑡 from the grid. The second one, (𝑞 𝑡 * 𝑝 𝐹𝐼𝑇 ), is the revenue from the FIT incentives, applied on all PV generation. The third component, (𝑞 𝑥 𝑡 * 𝑝 𝑥 𝑡 * ), is the revenue from PV energy exported to the grid.

So, prosumers have an incentive to consume their own production if exported prices, 𝑝 𝑥 𝑡 * , are lower than retail price minus the fixed part of network tariff they have to pay, (𝑝 𝑡 -𝑏). This is often verified as exported prices are lower than 4.65 pence per kWh and as the fixed parts of network tariffs are respectively around 3.2% and 0.8% of the retail price for households and small industries (COMMISSION EUROPÉENNE, 2015). For example, in 2011, retail prices were, respectively for households and small firms, 13.74 pence/kWh, with 0.44 pence/kWh for fixed network fee, and 10.87 pence/kWh, with 0.085 pence/kWh for fixed network fee (Eurostat). As the exported tariff was 3.1 pence/kWh, it is easy to conclude that self-consumption was and remains profitable.

As expected, a huge increase in PV capacities has followed the FIT policy (Table 4). To avoid transferring a significant part of the FIT costs onto consumers, producers are paying their connecting costs to the network. The government has also reduced the level of FIT, the total costs being 18% above the forecasted implementation FIT costs (£94 million instead of the forecasted £80 million, [DECC, 2015]). The predictions of the Department of Energy & Climate Change (DECC) has increased to achieve a cost of £9.1 billion for the incentive policy on the 2010-2021 period (its first forecast was $8.6 billion in 2030). In the same way, the initial 2010-2030 predictions of a £8.5 increase in households' energy bill have been revised. The DECC has observed an increase of £9 in 2014, that could be of £14 in 2020 (DECC, 2015). Thus, in 2016, FIT were divided by a factor 2 with the objective to maintain these costs under £100 million per year. Beside this decision, some experimentations (with dynamic retail pricing) to increase self-consumption, or to shift loads to optimize the use of PV generation, are implemented.

To conclude, incentive policies of FIT and export tariffs have significantly contributed to the increase in PV capacities (Figure 6). However, this increase has been done at a high cost for consumers. This policy has also increases the self-consumption. We could note that there was no exemption of taxes and fees for self-consumers or prosumers. Nevertheless, cross-subsidies always exist when network costs are considered. Self-consumers do not pay a fee to compensate their decrease in consumption withdrawn from the grid. These cross-subsidies could create an increase in network costs for others consumers.

Spain

As in previous countries we have presented, Spain has known a substantial increase in costs of incentive policies to develop renewable energies. For instance in PV technology, this cost was around 194 million € in 2007, 991 million € in 2008 and it has raised to 2600 million € per year on the 2009-2012 period (del Río and Mir-Artigues, 2014). These costs are linked with a FIT policy followed by a premium scheme. Renewable producers must sell their generation at the spot price and then, they received a premium determined as the difference between renewable costs and the spot prices.

In October 2015, public authorities regulated the administrative, technique and economic conditions for self-consumption plants. Indeed, the electricity sector known a large deficit as government wanted to minimize the impact of renewable energies deployment costs on consumers' bills (del Río and Mir-Artigues, 2014). Then, to reduce this deficit, the decision for a deployment of self-consumption programs is adopted. This decision followed the implementation of a net-metering policy, decided in 2011 to increase capacities in distributed generation.

Since October 2015, all PV producers or prosumers must pay a fee for self-consumption. This fee, called backtop toll, is a two part tariff with a fixed fee for the subscribed capacity and a variable fee correlated with the produced energy. These fees are self-consumers' contributions to network costs' recovery. Only PV plants with a capacity lower than 10 kW and with a self-consumption rate of 100% were exempted of these charges.

Despite great solar radiations and some incentives to self-consume, its deployment is not a success. The development of small PV power plants remains low because of charges and a fixed part standing for 84% in the network tariff (COMMISSION EUROPÉENNE, 2015).

Denmark

Since 2006, Denmark has decided to use the net-metering policy to support the deployment of renewable energies. In 2011, the decision to increase the retail price of electricity has been taken to achieve 30c€/kWh in 2013; taxes stand for 58% of this price. Therefore, self-production and self-consumption were increasingly profitable. Investment in PV power plants has grown from around 15 MW in 2011 to 450 MW mid-2013 (EnergiNet). The cost of net-metering was fairly high, around 65 million € in 2012 (Kjaer et al, 2013). Indeed, 99% of PV capacities were under net-metering. Then, revenues from taxes were reduced because of lower volumes of energy withdrawn from the grid. To reduce this counterproductive effect, and as the grid parity was achieve for PV energy, the period of compensation in the net-metering program was reduced from one year to one hour. The residual PV generation, i,e PV generation which is not self-consumed along one hour, is valued through a premium as in Spain. The premium was in a range of 5 to 8 c€/kWh to maintain the price for PV energy fed into the grid. This reinforce the idea of the main goal: to favor the self-consumption. Small PV power plants (lower than 50 kW) are exempted from the PSOT8 for each self-consumed kWh. Others are only exempted from the renewable energy subsidies on such kWh. Currently, government is working on a network fee option paid by the self-consumers to compensate the network losses in self-consumption models (IEA PVPS, 2016).

France

As several of its neighbors, France has used FIT to foster the deployment of renewable energies. At the end of 2016, the installed PV capacities were 7 134 MW and have known an exponential growth since 2007. Despite a decrease in FIT following the reduction in costs of PV energy, PSOT has increased from 7.5 €/MWh in 2011 to 22.5€/MWh in 2016. Thus, in 2016, French government has decided to introduce a support for the self-consumption, both individual and collective. The regulator published a call for tenders standing for 50 MW for PV installations (with a capacity for each PV power plant between 100 and 500 kW, so industrial, tertiary and agricultural sectors could be interested in) 9 . Bidders must indicate in their bid the forecasted rate of self-consumption and the premium they want to receive for each selfconsumed MWh. A mark-up until 5€/MWh could be added to their premium. First results have shown that answers were on projects with a self-consumption rate of at least 97%.

As firstly the grid parity is not yet achieved in France and secondly the retail price is lower than in other countries, incentive policies must be adopted to favor self-consumption. To obtain the subsidies, selfconsumption rates must be higher than 50%. The revenue for self-consumption is set as in Equation 3:

𝑪𝑹 𝒕 = [(𝑷 * + 𝟏𝟎) * 𝒒 𝑨𝑪 𝒕 ] + [𝑷 * * 𝐪 𝐱 𝐭 ] -[𝟏𝟐 * 𝒒 𝒕 * ( 𝐏𝐨𝐰𝐞𝐫 𝐦𝐚𝐱 𝐭 𝑲 𝒕 )] (𝟑)
With 𝑞 𝐴𝐶 𝑡 , q x t 𝑎𝑛𝑑 𝑞 𝑡 the same parameters as in Equation 2, 𝑃 * the premium of the bid, Power max t the maximum of the PV generation fed into the grid and 𝐾 𝑡 the PV installed capacities. The two first terms of Equation 3 stand for revenues from self-consumption and PV energy fed into the grid. Incentives are clearly in favor of self-consumption because of the mark-up added to the premium. The third term of Equation 3 represents losses for bidders if they fed electricity into the grid.

Self-consumption and cross-subsidies to recover network costs

The different implementations have shown that to internalize the costs of renewable energies deployment, several countries have decided to promote self-consumption. They have used net-metering or net-billing to remunerate prosumers that consume their own renewable production. Nevertheless, the positive impact of self-consumption is not obvious without the achievement of the grid parity for renewable energies. Indeed, self-consumption has positive effects on the welfare. Public authorities could reduce policy supports for renewables. Thus, the evolution of PSOT could be reduced. This advancement is nevertheless linked with incentive policies to develop self-consumption. Then, households and industries increase their energy autonomy and could manage their consumption to increase their selfconsumption rate. The Distribution System Operator (DSO) decreases its costs of network management, the energy fed into the grid being lower under self-consumption, and of reinforcing networks. As the renewable energy is consumed near its production area, network losses are also improved [START_REF] Cohen | Effects of distributed PV generation on California's distribution system, Part 1: Engineering simulations[END_REF][START_REF] Mendez Quezada | Assessment of Energy Distribution Losses for Increasing Penetration of Distributed Generation[END_REF] However, self-consumption negatively impacts the volume of energy withdrawn from the grid. Network tariffs are usually two-part tariffs. A simple explanation is that the fixed part is paid for the capital expenditures (CAPEX) in €/MW and the variable part for operational expenditures (OPEX) in €/MWh. This allocation of costs could be modified (Commission Européenne, 2015;[START_REF] Schwartz | Etude comparative des tarifs d'utilisation des réseaux de distribution d'électricité en Europe[END_REF] and some CAPEX could be recovered with the variable part. As implementations have shown it, selfconsumers pay the fixed part but only a small variable part. In some cases, they could be exempted from taxes, so they benefit from incentives policy but they do not contribute to the recovering of costs of network use or PSOT. Thus, revenues from network tariffs could be reduced, and the DSO could know difficulties to recover its costs, financial flows and prices becoming unstable (Picciariello et al, 2015;Kubli, 2015;Simshauser, 2015). When the financial deficit is recovered by a increase of the variable component, the system comes in a "death spiral" (Costello et al., 2014;Felder et al., 2014). For instance, network tariffs are increasing because variable costs must be recovered on smaller electricity volumes withdrawn from the grid. This increase could become an incentive for others consumers to invest in selfconsumption, reducing their demand to the grid. The demand is again reduced, then network costs increase and so on.

Data and assumptions

In this section, we describe the set of assumptions and data used to forecast the volume of photovoltaic capacities installed by 2021 as well as the consumption and PV generation load curves for different consumer profiles

a. Assumptions on forecasted PV capacities in 2021

In October 2017, 7.7 GW of photovoltaic (PV) capacities were connected to the grid in France (Ministère de la transition écologique et solidaire10 ), representing 7% of the total electricity generating capacity. Enedis, the historical DSO which manages 95% of the grid in France, published the breakdown of PV capacities for each capacity level in its network (table 6). The ADEME (Agence pour l'Environnement et de la Maitrise de l'Energie) uses a similar breakdown capacity by sector. Residential, industrial and commercial segments represent 52% of total capacities connected to Enedis' grid. This percentage might not be similar for the PV capacities in France because PV power plants above 12 MW are connected to the transmission grid. Moreover, on January 1 st 2018, 2.6 GW were commissioned but not connected yet, including approximately 74% of utility-scale PV installations. The development of photovoltaic cpacities in France was triggered by the government who decided to extend policy supports until 2023. Indeed, France's government is hoping to reach about 10.2 GW in 2019, and 18.2 GW in the case of a "low scenario" and 20.2 GW in the case of a "high scenario" in 2023. From 2016 to the first quarter of 2019, the government has published a calendar of PV tenders in order to reach those goals. Over this period, 3 GW will be allocated to ground-mounted power plants (between 500 kW and 17 MW) and 1.35 GW to rooftop power plants (between 100 kW and 8 MW). Other tenders regarding self-consumption installations are planned. However, those calls for tenders will also deal with all categories of renewable power plants that have a capacity between 100 kW and 500 kW. The first call for tenders was launched in 2016 and, currently, 40 MW were essentially allocated to PV projects. In order to develop self-consumption in a regulative way, the government decided to renew this type of tenders until 2019, for a capacity of 150 MW per year.

The development of small and medium PV power plants was mainly driven by a feed-in tariff scheme. But willingness to self-consume increased since last year. Indeed, self-consumers represent 36% of total photovoltaic investments in 2016 (8 000 setups) and 70% of the first semester in 2017 (6 000 setups12 ). Moreover, France's energy transition law, which has been passed on February 24 th , 2017 sets specific supports for self-consumers. Now, it has become a reality and all plants with a capacity lower than 100 kW connected after August 2017 are assumed to self-consume a part of their own PV production. In the following study, only these installations and self-consumption tenders are considered to estimate crosssubsidies between standard consumers and self-consumers. Self-consumption tenders are assumed to be allocated for photovoltaic projects and we assume that they will be renewed until 2021. For capacities lower than 100 kW, the government sets a goal of 350 MW a year. The same target is used until 2021 in the present article and photovoltaic deployment will be the same for households and firms, i,e 175 MW for each sector (table 7). Regarding self-consumption tenders, the selected firms aren't connected to their installations yet. We assume that it will be the case in 2018 and the selected firms for future tenders will have connected their PV installation one year after their designation. According to the calendar of tenders13 , PV capacities from tenders are following the trend in the table below. In August 2021, 1.89 GW of self-consumption PV capacities are expected.

b. Consumer characteristics

As the consumer profiles are heterogeneous, several assumptions are made to characterize them. Consumer profiles are based on Enedis' typology which are used by the French regulator to set network tariffs. The consumer profiles are characterized according to their load consumption and the voltage level which they are connected to. Households are split in two categories according to their load profiles. The first one represents consumers with a single flat tariff who have usually a low electric load and no electric heater. The second one corresponds to consumers with a Time-Of-Use (TOU) tariff. These households have a flatter load curve than the first category because most of them have a hot water tank which turns on during off-peak hours. According to the sample provided by the French regulator, the first profile called "RES1" represents 60% of the household consumers and the other one called "RES2" represents 40% of them. Both of them have a contractual power of 6 kVa. Firms with PV panels under 100 kW are assumed to be connected at a voltage level between 36 kVa and 250 kVa which corresponds to a specific interval between low and medium voltage. For this specific voltage interval, Enedis has made two different profiles which differ according to the gap between the average for a profile 𝑖 (𝐴𝐶 𝑖 ) is computed by dividing the annual consumption (𝑇𝐶 𝑖 ) with the number of halfhours in a year:

𝐴𝐶 𝑖 = 𝑇𝐶 𝑖 ∑ 𝐻𝐻 (4)
Then, consumption for each half-hour 𝐶 𝑖,𝐻𝐻 is computed by multiplying the average consumption with each half-hour coefficient (and 𝑛 corresponds to the total of half-hours in a year):

𝐶 𝑖,𝐻𝐻 = ∑ 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑖,𝐻𝐻 * 𝐴𝐶 𝑖 𝑛 𝐻𝐻=1

(5)

In order to compute the volume of PV generation for each profile per half-hour, we estimate first the annual average load factor. We rely on data provided by Enedis regarding the volume of photovoltaic generating capacities on a yearly basis from 2012 to 2017. Based on the data, we deduce the average load factor of solar capacities (L𝐹) using the following relation:

L𝐹 = 𝐺 𝑃𝑉 𝐶 𝑃𝑉 * ∑ 𝐻𝐻 (6)
Then, annual photovoltaic generation for the profile 𝑖 (𝐺 𝑃𝑉 𝑖 ) is computed from equation 3 :

𝐺 𝑃𝑉 𝑖 = 𝐾 𝑃𝑉 𝑖 * 𝐿𝐹 * ∑ 𝐻𝐻 (7)
Where, 𝐾 𝑃𝑉 is the total capacity, 𝐿𝐹 is the load factor as expressed in relation (3) and ∑ 𝐻𝐻 is the number of half-hours in a year. By comparing the volume of generation and consumption for each halfhour, we can assess the volume of self-consumption and generation injected into the grid. Typical daily load curves are depicted in figure 7. For some hours, we can observe both consumption during a peak and an off-peak tariff because the peak hours are set locally by the DSO. So, in average, for a same hour, we can observe a certain percentage of peak and off-peak consumption.

Network tariff

The network tariff presents two different components: a fixed charge, representing both metering and administrative management costs and also the contractual capacity subscribed by the consumer (fixed network costs), and a variable charge depending on the volume of electricity withdrawals. As we have seen it before, self-consumers save money only on the variable charge. In august 2017, the French energy regulatory authority has issued new network tariff. The evolution of these rates is set until 2020 and they have known several changes from the previous one. Specific tariffs by season (winter and summer) are introduced for consumers fitted of a smart meter (Linky). As this smart meter will be deployed for all consumers in 2021, all of them will have this specific tariff14 . Variable daily rate is still applied for some consumer types (short and medium uses). Self-consumers have a specific metering and administrative management charge which is above normal consumer's one. Every year, network tariffs are slightly adjusted, in order to account for inflation and a clearance rate. The regulator (CRE) published expectations of network rates until 2020 for residential consumers. However, concerning firms, these figures are only available for the years 2017 to 2018. For our representative consumers, network tariffs are done in table 9. The network tariffs for the year 2017 are illustrated in table 10. Variable part of the tariff stands for between 80-85% except for large users (45%). Fixed part of the tariff change between 20-25% except for large users (55%).

The simulation model to estimate cross-subsidies

Here, we present the model framework used to estimate the impact of self-consumption of electricity on the revenue losses for the distribution network operator and the corresponding increase in network tariff applied to all consumers. This methodology is based on the one set up by [START_REF] Athawale | Residential Rate Design and Death Spiral for Electric Utilities: Efficiency and Equity Considerations[END_REF]. Consumers pay the variable part of network tariff on electricity withdrawals. Self-generators save money on the volume of self-consumed electricity. However, it creates a deficit for the network operator. In this case, feed-in generation is assumed to have no impact on network costs. However, in order to balance their financial accounts, network operators increase the fixed part of the rate. This pathway induces cross-subsidies from standard consumers to self-consumers.

Network operator deficit

To estimate cross-subsidies across years, we estimate the number of self-consumers first. We set different self-consumer profiles which install a specific PV size as explained in the previous section. By dividing the cumulative capacities for each profile by the average PV size for the self-consumer profile, we computed the number of self-consumers for each year (table 11). Network financial losses are computed by multiplying the volume of self-consumption by the corresponding network tariff for each period and consumer profile. Network operators face a deficit with the decrease of the electricity withdrawal but self-consumption decreases the electric losses on the grid. Indeed, electric losses are due to the resistance of the line. When electricity pass through the lines, a fraction of the electricity is transformed into heat. This process depends on the specificity of the lines, the current flowing and also the length of the electric lines. Between 2012 et 2017, the average of electric losses in Enedis' grid was about 6.8%. The electric losses (𝐿 𝑝 ) on the French grid are given by:

𝐿 = 𝑎𝑃 2 + 𝑏𝑃 + 𝑐 (9)
With 𝑃 as the amount of power, 𝑎 a parameter applied to the weeks, 𝑏 a parameter applied to the weekends and 𝑐 a non-technical parameter which depends on counting error and fraud. The estimated load curves allow us to compute the volume of self-consumption for all weeks and week-ends. So, we multiplied the parameter 𝑎 and 𝑏 respectively by the total amount of self-consumption during weeks and week-ends. The parameter 𝑐 is independent of the volume of self-consumption because it depends on counting error and fraud. So, we don't take into account this parameter to compute the volume of electric losses avoided. To estimate the benefit from the self-consumption on the electric losses, the average future prices15 for each year is multiplied by the volume of electric losses avoided.

Cross-subsidies estimation

To cover its financial deficit, the DSO might increase the network tariff for all consumers. In order to limit the death spiral phenomenon induced by the increase of the variable charge (Eid, 2014;Picciariello, 2015, Simshauser, 2016), we assume that the DSO will increase the fixed charge for every consumer:

𝐶𝑆 = 𝐹𝐷 𝐶𝑃 ( 10
)
Where 𝐶𝑆 is the amount of cross-subsidies per kilowatt, 𝐹𝐷 is the DSO's financial deficit and 𝐶𝑃 the total contractual power. The last one is provided by the French regulator until 2021 (CRE, 2016). Then, we assess the impact of the development of self-consumption by multiplying the amount of crosssubsidies with the contractual power for each representative consumer.

Finally, we compare these results with the recovery of DSO's financial losses by the increase in the fixed charge only for the self-consumers.

The results: small values of cross-subsidies between consumers (CC)

For each representative consumer, self-consumption rates are presented in table 12 below. Obviously, they rely on the consumption levels for each consumer, the consumers RES 1 being the lower consumers, so with lower self-consumption rates (31 and 55% respectively for winter and summer periods). Large firms could consume the major part of their PV generation (around 98 to 100% respectively for low rate summer period and high rate winter period). These results are consistent with profiles of each consumer we presented before (figures 7). They are also consistent with the intuition that self-consumption rates are higher for firms rather than for residential consumers, the activity of firms being more often in accordance with hours where the PV generation occurs. The gain per consumer is higher for large industries, according to the self-consumed volumes and the smaller number of self-consumers. Indeed, according to consumption profiles, each residential consumer could save 48€ per year and firms between 1813€ or 3819€ per year respectively for small and large industries 17 . The global amount of savings is increasing from one period of time to another because of an increase in the number of self-consumers and PV generation capacities. As these increases are linear ones, savings are also increasing linearly. As we have seen it before, these savings are also losses for the DSO. Self-consumers do not pay the variable part of the network tariffs which create imbalances in DSO's revenues. However, self-consumption also leads to cost savings for the DSO in the management of network losses. Using an average price in electricity future markets of 40,87 €/MWh for 2018, 40,24 €/MWh for 2019, 40,5 €/MWh for 2020 and 2021, and the equation 9, we could compute avoided network losses per year. They are presented in table 14. Savings in network losses could stand for 8,4% of DSO's revenue losses in 2021. They reduce financial losses that the DSO must recover through its network charge when self-consumption occurs. As we have seen before, self-consumption creates cross-subsidies between consumers. Indeed, regulator could decide to increase the fixed or variable part of the network tariff paid by each consumer to fund DSO losses in revenue. The increase in the variable part of the tariff will be very low, because of the lower self-consumption volumes compared to the overall consumption (self-consumption stands for 0,07% in 2018 to 0,35% in 2021 of the total forecasted consumption). Thus cross-subsidies will represent between 0,02€/MWh in 2018 to 0,07€/MWh in 2021. The increase in the fixed part should be higher because of the less volumes contracted in MW or kW but they remain very low if we consider the fixed part paid by consumers (Table 8). Cross-subsidies are between 0,022€/kW in 2018 to 0,09€/kW in 2021. The increase in the fixed part of the network tariff for each consumer is presented in Figure 9. For each consumer, cross-subsidies only marginally increase its fixed part of network costs with a maximum of a 2% for consumers RES 1 in 2021. This increase remains below 1% for large consumers and below 1.7% for consumers with Time of Use pricing or small firms on the four studied periods.

The impact of self-consumption on network access pricing: a way to reform the network tariff

The promotion of self-generation / self-consumption of renewable electricity is done in France thanks to two strong incentives: a partial exemption (40%) of the costs of connection to the distribution network, on the one hand, an exemption from the CSPE for the self-produced and self-consumed part of electricity, on the other. In the latter case, the balance sheet is neutral for the self-producer since what the producer saves by not paying the CSPE is largely compensated by the loss of profit that would be obtained if he sold his production instead of consuming it and thereby benefiting from the guaranteed purchase price (FIT for feed-in tariffs). The CSPE essentially covers the difference between the guaranteed price (FIT) and the market price, which is assumed not to be very far from the production cost of self-produced kWh when one approaches the market parity.

But the self-producer remains connected to the network because it has to withdraw a part of its consumption when its demand for electricity is not covered by its production due to intermittence of solar. It therefore has to pay the network access tariff for the kWh withdrawn from the network.

As mentioned in the previous section, the tariff for access to electricity network is generally a two-part tariff which includes a share proportional to the capacity subscribed by the final consumer in kW and a share proportional to the amount of electricity withdrawn in kWh. There is also sometimes an injection rate, at least for high voltage levels. In countries like France which maintain the principle of spatial equalization of tariffs in the name of equal treatment of all consumers, this tariff is the same independently of the location of the consumer, ie its distance from the network. The regulatory commission generally authorizes a horo-seasonal differentiation of tariffs to take account of the period and time during which the kWh is withdrawn. Because this differentiation is the standard adopted by the supplier, it is normal to apply it to the distributor due to the observation that the networks are congested at the busiest times of the year at the demand peaks.

The important issue for the network operator is that of sharing between fixed and variable tariff parts.

In Europe, and in particular in France, the fixed share for households is in the order of 20 to 30% of the all taxes included tariff (TURPE) compared with 70 to 80% for the variable part. Only a few countries, such as the Netherlands, have a 100% based on the subscribed power. The choice between the fixed and the variable parts is not neutral: a high fixed share penalizes consumers who consume little kWh and therefore have a low utilization of the subscribed power. A low fixed share penalizes large consumers since the fixed costs of the network are largely recovered according to the number of kWh withdrawn.

The large-scale development of self-consumption of electricity (by households as well as small and medium-sized enterprises) puts the question of the optimal structure of the tariff on the agenda. With the current structure, which relies on a low fixed component, a large producer of photovoltaic or wind power that consumes almost all of its production and only uses the network on a marginal basis will pay the variable costs associated with which is fair, but will only marginally contribute to the coverage of fixed network costs, which will lead to "cross-subsidization" to the detriment of other customers who rely heavily on the network since the coverage of fixed costs is mainly on a pro rata basis. For reasons of equity, the structure of the tariff for access to the transmission and distribution networks must therefore be reviewed by increasing the fixed part of the tariff. Indeed, regulator, defining network regulated tariffs, could use several tools to compensate losses for DSO:

• An increase in the variable part of the network tariff for all consumers or only selfconsumers. Each kWh withdrawn from the network compensates the DSO's revenue losses.

Cross-subsidies always exist in the first case and incentives for self-consumption increase in the second case, with the risk of "death spiral". According to these difficulties, this solution is not consistent with the internalization of cross-subsidies. • An increase in the fixed part of the network tariff for all consumers (but we keep the existence of cross-subsidies and the "free-riding" behavior) or only for self-consumers (but the incentive to self-consume and to invest in own generation could be reduced). As the incentive to invest in PV generation does not only rely on self-consumption gains, this decision could improve the payment of network costs by self-consumers 18 . Moreover, all consumers connected to the grid must pay a fee to DSO to avoid shortfalls. In fact, the selfproducer who remains connected to the distribution network considers this network as a back-up and must therefore be given a pricing that takes account of this specific behavior.

The interest of the network operator, supported by the regulator, is to recover its costs while guaranteeing the fairness of the tariff between consumers. Self-consumers have to support the increase in costs they create, but also they must benefit for savings linked with the decrease in consumption from the grid, as we saw with the improvement in physical network losses.

In other words, with self-consumption, networks must be able to cope with the same power but with a lower utilization rate and with a part of the users that de facto behave like a « free rider ». It is therefore necessary to restore equity by charging the self-consumer for this guarantee of access to the network. Two approaches are possible to determine the fair price of access to the network of a self-consumer: either one sets the tariff according to the willingness to pay, or it is fixed by taking account of the network operator's shortfall and charging that customer with what would have been its fixed-cost bill if it had not been a self-consumer.

A tariff established according to the willingness to pay of the self-consumer.

In this case, recourse to the network can be considered as a relief which makes it possible to avoid a loss of well-being for the customer at the time of the intermittent occurrence of his wind or photovoltaic production. Thus, the "cost of failure" approach can be used. From the point of view of a mesh network, it is generally considered that the failure cost of one kWh (Value Of Lost Load -VOLL) is a parabolic function of the "depth of failure" defined as the share of energy not supplied as a percentage of the electricity demand (Hansen & Percebois) i.e.:

𝒀 = 𝒂𝒇 𝟐 + 𝒃𝒇 + 𝒄 ( 11
)
where c represents the marginal cost of the last plant called just before the failure, f the depth of the failure, and a and b of the parameters to be estimated through a survey of consumers who have suffered losses as a result of the interruption of the supply. It is often estimates of insurance companies that are used nationally for these estimates. As an example, RTE carried out a survey of 1600 customers in France during 2010-2011 (households and companies). The economic cost of a power outage of more than 3 minutes, according to this survey, is 26 euros per kWh, which is very high (recall that the price inclusive of one kWh purchased by a household is of the order of 0.17 euro). The cost is higher for businesses than for households. But this is a macroeconomic cost that takes into account the economic losses of various kinds that will have an impact on the evolution of the GDP in case of blackout.

For a household, the survey reveals that the average cost of a 1-hour cut is 25 euros, but households would be willing to pay only 17 euros to avoid it and they would ask 49 euros for the network operators in return of a voluntary load-shedding (source: RTE survey). These figures include both the cost of production, the cost of transmission and distribution of the kWh and the corresponding taxes. As selfconsumers also save expenditures in energy purchases and in PSOT, it could pay a fixed fee to the network operator to avoid shortfall. Which fee consumers are willing to pay to maintain this service? It all depends on their aversion to the risk and the alternative solutions available to them in case of unavailability of their production (electricity storage for example). So, consumers have to manage between the maximization of savings with self-consumption and the probability to be cut if the DSO does not cover all its network costs, yet facing to VOLL for each shedded kWh. According to its overall savings with self-consumption, and its WTP to avoid shortfall (between 17€/kWh and 25€/kWh), consumer would be willing to pay the DSO's revenue losses. Thus, RES 1 and RES 2 could agree to pay a maximum fixed fee of 48€, ENT 1 and ENT 3 respectively fees of 1813€ and 3819€. These fees are consistent with a shortfall of few kWh compared to the consumption of end users (a shortfall between 0.32 and 2.8 kWh for residential consumers, 72.5 and 106.6 kWh for small firms, 152.6 and 224.6kWh for large firms). According to these fees, the DSO could recover its costs to continue to serve consumers. We may think that the more the share of self-production increases, the more the consumers' willingness to pay decreases. Indeed, consumers reduce the use of network so they could have fewer concerns on the reliability and security of the network they marginally use.

Obviously, the consumer will waive this guarantee if the cost of an individual storage allows him to no longer connect to the network. The cost of storage therefore constitutes the ceiling price of the emergency tariff. As the cost of battery storage is downward, it is likely that self-producers will be less and less willing to pay to maintain a guarantee of network access.

A tariff established on revenue losses of the network operator

This time, we are talking about the "damage" suffered in terms of revenue by the network operator as a result of self-consumption and we are trying to see how to recover this shortfall by modifying the structure of the network access tariff.

As we have seen before, an increase in variable part of the network tariff does not reduce cross-subsidies and could maintain or make the "death spiral phenomenon" worse. Therefore, changes in fixed part of the network tariff seem to be preferred. DSO must recover the financial amount of self-consumers' savings from the variable part of the network tariff. Indeed, the network tariff paid without and with self-consumption is presented in Table 15.

As expected, the network tariffs paid by consumers are lower with self-consumption as withdrawals from the grid are lower. The decrease in network tariffs is lower for greater firms, variable network tariffs being smaller rather than for other consumers. Moreover, residual consumption staying large as the self-consumption stands for 17,6% of their overall consumption (against 43,6% and 28.6% of respectively RES1 and RES2 overall consumption, 38.7% of small firm overall consumption).

As we seen before, for each profile of consumers, DSO must recover 48€ on each residential selfconsumer, 1813€ and 3819€ respectively on each small or large firms self-consumer. Each year, the fixed part of the network tariff for self-consumers must increase of these amounts to anticipate and internalize losses in DSO's revenues. Simple calculations lead to new tariffs for subscribed power (Table 16) 19 . The increase in fixed part must be significant for households and small firm to internalize the negative effect of self-consumption on DSO's revenues. The intuition relies on their network contracts features, as they pay lower fees for subscribed power and greater variable tariffs, and on lower residual demand.

Using this new fixed part, network tariffs paid by self-consumers are the following (Table 17).

The overall network tariff is the same as the one without self-consumption but its structure differs, the weight of the fixed part has increased. Indeed, fixed part of the tariff stands for 38 to 62% of the costs recovery whereas it stood for 16 to 55% in the initial structure of network tariff. A large part of network costs is now recovered on the fixed part. Moreover, the overall network tariff for self-consumers has also increase to compensate losses in DSO's revenue. It has increased from 8% for large firms (from 47 310€ to 51 129€), from 42% for small firms (from 4 285€ to 6 097€), and from 54% and 28% respectively for consumers with fixed retail rate or Time of Use contracts (from 88€ to 136€ and from 173€ to 222€).

19 For another empirical example on modifications of the structure of network tariffs, see CREDEN / OSE study, WP 2017. They found that the fixed part must increase of 75% for a firm with a subscribe power of 110 kW.

Conclusion

The deployment of self-consumption is used to reduce network costs and PSOT. Public authorities have decided to use net billing or net-metering to foster this deployment. Some of them have also exempted from taxes and PSOT self-consumers to increase their gains when they do not feed into the grid own PV production. This system has achieved its goals. Self-consumption rates are between 35 and 45% for households and between 70 and 100% for industrials. The success of these policies is also linked with an increase in energy prices and a decrease in FIT; thus, the profitability of self-consumption rises.

According to these exemptions and the reduction in energy withdrawn from the grid, the recovery of network costs and the funding of public services obligations are not guaranteed. Indeed, cross-subsidies exist between standard energy consumers and self-consumers. To avoid these cross-subsidies, exemptions are reduced and self-consumers will pay a share of PSOT or of network tariff. These changes usually come after a period of profitable incentive policies for self-consumers. Thus, the large-scale development of self-consumption (and self-production) therefore requires thinking about a redesign of network access tariffs if existing networks are to be financed other than by cross-subsidies, i,e deferring charges to consumers who do not opt for self-consumption. The fixed share of the network tariff must be increased for self-consumer, as they wish to continue to benefit from access to the interconnected network to serve their residual demand.

As the model shows it, cross-subsidies are very low for the next 5 years in France. Forecasted PV capacities and the share of PV production in total electricity generation or consumption are very low. So, the increase in network costs or in PSOT exist but the annual increase for consumers are not significant, only few euros on their annual bill. However, losses for Distribution System Operator should be 7.7 million € in 2018, rising until 32,6 million € in 2021. The DSO is responsible for security, public service obligations and network deployment. Thus, these costs must be recovered. As its revenue comes only from the regulated network tariff, losses in this revenue could induce lower services or shortages.

Several economic tools could be used to recover theses costs, reducing cross-subsidies. Using the consumers' willingness to pay for security could lead to high costs self-consumers have to support. The self-consumption could be reduced. Regulator could also modify the share of fixed costs in the network tariff, reducing the weight of the variable part that is not paid by self-consumers. A significant increase in fixed part paid by self-consumers reduces network system operator's losses. This increase must be of 14% for large firms, in a range of 135% to 184 % for others consumers. Nevertheless, this increase must keep incentives to self-consumption. Thus, an injection tariff could be decided to complete the policy.

This new tariff is additional revenue for the network operator and further incentives for prosumers, an opportunity cost, to consume their own production to minimize injection costs.

The important point is to introduce a special access tariff for the self-consumer, which obliges the distributor to maintain the same installed power, regardless the amount of self-consumption or the use of the PV generation. This policy could also internalize the intermittence of renewable energy, the network being used as a back-up technology to serve goods as reliability and security of supply.

As the risk on network withdrawals is increasing, with self-consumption behavior or demand side management objectives, regulators have to deal with the recovering of a greater part of network costs on the fixed part of the network tariff. This study shows that a first step could be a special network tariff for self-consumers, modifying the share of variable and fixed parts of the tariff to recover all network costs. Internalizing the risk on volumes could lead to network tariff only based on subscribed power, as in Netherlands. However, some concerns remain in the allocation of costs between consumers.

Let us recall that we only take into account the cost of access to the network. Further researches on this field could complement these findings. Obviously thanks to its photovoltaic installation the consumer will save on the invoice sent by his supplier since he will buy less electricity. We do not also take into consideration the surplus of electricity that could be sold on the market or injected into the network with a FIT. It is also necessary to raise the question of the introduction or not of an injection tariff on the network, because the distributor is entitled to request compensation for the use of its network. This could be analyzed in another research as it must impact the incentive to increase self-consumption, the surplus of energy fed into the grid being taxed. It should also be noted that the self-producer may have an interest in subscribing to a lower draw-off capacity than would be the case if he did not self-consume part of its PV generation; these are fixed costs saved. So, a network tariff heavily based on the subscribed power must not be efficient or costly to implement.

Figure 1 :

 1 Figure 1 : PV installed capacities in the world (Solar Power Europe, 2017)

Figure 2 :

 2 Figure 2 : Consumption and PV generation daily profiles for a household "prosumer"

Figure 3 :

 3 Figure 3 : Share of self-consumption for some representative prosumers (IEA PVPS, 2016c)

Figure 4 :

 4 Figure 4 : Growth in installed PV capacities in Europe (Solar Power Europe, 2017).

Figure 5 :

 5 Figure 5: Revenue from one kWh under self-consumption or FIT

Figure 6 :

 6 Figure 6 : Total PV generation, exported volumes and costs of FIT policy

Figure 7 :

 7 Figure 7: Typical consumption and PV Load curves for each profile (Monday April 3rd)

Figure 8 :

 8 Figure 8: Scheme of cross-subsidies

Figure 9 :

 9 Figure 9: Increase in fixed part of network tariffs for each consumer 2018-2021

Table 2 : Main factors impacting self-consumption deployment

 2

	Economical features	Technological features	Behavioral features
	• Investment costs in	• Energy management	• Market maturity
	renewables	tools	• Long term visibility
	• Solar radiations	• Storage	• Environmental
	• Electricity prices		concerns
	• Network tariff		• Security and reliability
	components

Table 2 : Premiums for self-consumption in Germany (c€/kWh) Capacities 100 to 500 kW 30 to 100 kW Lower than or equal to 30 kW Self-consumption rate

 2

		< 30%	> 30%	< 30%	> 30%	< 30%	> 30%
	2009	0	0	0	0	25,01	25,01
	January 2010	0	0	0	0	22,76	22,76
	July 2010	14,27	18,65	16,01	20,39	17,67	22,05
	October 2010	13,35	17,73	15,04	19,42	16,65	21,03
	January 2011	9,48	13,86	10,95	15,33	12,36	16,74
	January 2012	8,63	12,61	9,96	13,95	11,25	15,23
	January 2013	7,85	11,48	9,06	12,69	10,24	13,86
	January 2014	7,14	10,44	8,25	11,55	9,31	12,61
	Source: Sarasa-Maestro C, 2013

Table 3

 3

		50				Revenue from self
	c€/kWh	30 35 45 40				Recette autoconsommation consumption (SC rate (Taux d'autoconsommation lower than 30%) <= 30%) Revenue from self Recette autoconsommation consumption (SC rate (Taux d'autoconsommation > 30%) higher than 30%)
		25				Feed-in tariff Revenue from FIT
		20
		2009	2010	juin-10 oct-10	2011	2012
				) and their part in the PSOT to stands 44% of the Erneuerbare Energien
	Gesetz 3 (EEG) contribution in 2012 (BMU, 2016)
	3 "Act on Granting Priority to Renewable Energy Sources (Renewable Energy Sources Act, EEG)" (FRAUNHOFER
	ISE, 2017).

Table 3 : Volumes and costs of the German self-consumption program

 3

		2009	2010	2011	2012	2013	2014	2015
	Self-consumed volumes (GWh)	5	46	259	734	821
	Costs of self-consumption premium (millions €)	1	11	45	102	111
	PSOT (EEG) to PV generation (c€/kWh)	0.67	1.1	1.66	2.03
	Total of PSOT (EEG) for PV generation (Billions €)	3,2	5	7,8	9,2	9,5	10,4	10,8

Table 4 : Feed-in-Tariffs and installed capacities for PV generation (OFGEM) Size of equipments FIT (Pence/kWh) in 2010 Installed capacities (MW) 6 2010 2011 2012 2013 2014 2015

 4

	<= 4 kW	44,19 7 /50,67	86	729 273 312 214 443
	4 < kW <= 10	44,19	6	46	14	19	18	29
	10 < kW <= 100	38,50	8	178	81	115	99	180
	100 < kW <= 5 000	35,95	6	206	92	117 168 470
	Source: OFGEM

Table 6 : Cumulative capacities by sector in 01/2018 kW Sector Total plants Cumulative capacities (MW) ] 0 -9]

 6

		Residential	339 378	1 167
	] 9 -250 11 ]	Firms -buildings	31 379	2 237
	] 250 -12 000]	Ground-mounted	1 082	2 237
	Total		371 839	6 528
	Source: Enedis

Table 7 : Evolution of PV self-consumption until 08/2021 (MW)

 7

	MW	08/18	08/19	08/20	08/21	Total
	Residential	175	175	175	175	700
	Firms	175	175	175	175	700
	Tenders	90	150	150	100	490

Table 9 : Variable part of network tariff for end users

 9

	Consumers	Households/Residential consumers	Firms/Industrial consumers
	profiles	Periods	RES1	RES2	Periods	ENT1	ENT2
	Variable part of the network tariffs (€/kWh)	High rate winter Low rate winter High rate summer Low rate summer	0.0367	0.0563 0.0325 0.0131 0.0098	High rate winter Low rate winter High rate summer Low rate summer	0.0481 0.0295 0.0218 0.0179	0.0192 0.0120 0.0088 0.0077
	Fixed part of the network tariffs (€/kW)	2017-2018 2018-2019 2019-2020 2020-2021	4.36 4.8 5.24 5.68	5.89 6.33 6.77 7.21	High rate winter Low rate winter High rate summer Low rate summer	9.99 5.13 3.74 1.13	14.91
	Source: Enedis and CRE

Table 10 : Variable and fixed part of network tariff for end users (2017)

 10

	End users	Variable part	Fixed part	Network tariff
	RES1	110 €	26 €	136 €
	RES2	187 €	35 €	222 €
	ENT1	5 002 €	1 095 €	6 097 €
	ENT2	23122 €	28 007 €	51 129 €
	Source: Authors (data from Enedis and CRE)

Table 11 : Evolution of the number of self-consumers

 11 Then, we estimate the volume of PV self-consumption by each time period (𝑆𝐶 𝑝 ) by multiplying the volume of self-consumption for each consumer profile (𝑆𝐶 𝑖,𝑝 ) by the corresponding number of selfconsumers (𝑃𝑉 𝑆𝐶 𝑖 ).

	Profiles	2017-18	2018-19	2019-20	2020-21
	RES1	35 000	70 000	105 000	140 000
	RES2	23 333	46 667	70 000	93 333
	ENT1	2 500	5 000	7 500	10 000
	ENT3	300	800	1 300	1 633
	Total	61 133	122 467	183 800	244 967

Table 12 : Self-consumption rates for each representative consumer Periods of consumption

 12 Using self-consumption volumes and network tariffs presented in table 9, we could easily compute the variable network charges self-consumers saved. They are presented in table 13.

		Consumer with fixed retail rate (RES 1)	Consumer with Time of Use (RES 2) pricing	Small firms (ENT 1)	Large firms (ENT 3)
	High rate winter	55%	84%	91%	100%
	Low rate winter	-	80%	90%	99%
	High rate summer	31%	34%	69%	98%
	Low rate summer	-	30%	64%	98%
	Overall self-consumption rate	37%	45%	74%	98%
	Source: Authors (data from Enedis and CRE)

Table 13 : Savings (€) in network charges for self-consumers 2018-2021

 13

	Periods of					Savings per
	consumption 16 / Consumer's	2018	2019	2020	2021	self-consumers
	profiles
	RES 1	1 683 194 € 3 366 387 €	5 049 581 €	6 732 775 €	48 €
	RES 2	1 130 852 € 2 261 704 €	3 392 556 €	4 523 408 €	48 €
	ENT 1	4 531 478 € 9 062 957 € 13 594 435 € 18 125 913 €	1 813 €
	ENT 3	1 145 848 € 3 055 595 €	4 965 343 €	6 238 507 €	3 819 €
	Total	8 491 372 € 17 746 644 € 27 001 915 € 35 620 604 €
	Sources: Authors (data from Enedis and CRE)

Table 14 : Savings in network losses for the DSO 2018-2021

 14

	Periods of consumption	2018	2019	2020	2021
	Savings in network losses in MWh	16312 MWh 35992 MWh 55672 MWh 72546 MWh
	Savings in network losses in €	754 995 €	1 487 638 €	2 244 203 € 2 992 271 €
	Sources: Authors (data from Enedis and CRE)

Table 15 : Network tariffs with and without self-consumption Consumer's profile Network tariffs without self- consumption in €

 15

					Network tariffs with self-
						consumption in €	Variations
		Variable part	Fixed Part	Overall network tariff	Variable part	Fixed Part	Overall network tariff	in network tariffs (%)
	RES1	110	26	136	62	26	88	35
	RES2	186	35	222	138	35	173	22
	ENT1	5 002	1 095	6 097	3 189	1 095	4 285	30
	ENT3	23 122	28 007	51 129	19 303	28 007	47 310	7,5
	Sources: Authors (data from Enedis and CRE)

Table 16 : Increases in fixed part of network tariffs Consumer's profile

 16

		Average subscribed power (kW)	Tariffs increase in € per kW	New fixed part in network tariff (€)	Increase of the fixed part (%)
	RES1	6	8,02	74	184
	RES2	6	8,08	84	137
	ENT1	51	35,54	2 908	165
	ENT3	470	8,13	31 826	14
	Sources: Authors (data from Enedis and CRE)

Table 17 : New structure of network tariffs for self-consumers Consumer's profile Network tariffs with self-consumption and increasing fixed part in €

 17

						Overall
		Variable part		Fixed part		network
						tariff
	RES1	62 €	46%	74 €	54%	136
	RES2	138 €	62%	84 €	38%	222
	ENT1	3 189 €	52%	2 908 €	48%	6 097
	ENT3	19 303 €	38%	31 826 €	62%	51 129

The distributed generation is mainly connected to distribution network. For example, in France, 93% of distributed generation is connected to distribution network.

The imported and exported energy from and to the grid are valued at the same price, the retail price that is higher than the wholesale price.

At the same period, the retail rate evolved from 22.75 c€/kWh in 2009 to 26.06 c€/kWh in 2012.

This proportion has decreased to 22% in 2016, to 19% in 2017 and will be at 10% in 2018.

These figures on installed capacities are for the period April of year "n" to April year "n+1".

The FIT is lower if the PV capacity is setup on a house under construction.

It includes all network costs linked with renewable energy deployment, as renewable energies subsidies, environmental and R&D costs.

Another call for tenders has been made in march 2017 to develop self-consumption. It applies on 450 MW until 2020.

Available at: http://www.statistiques.developpement-durable.gouv.fr/publicationweb/61

Some firms have photovoltaic power plants above 250 kW but they are only a few of them and we assume that firms will not invest beyond 500 kW. Indeed, they will prefer to participate at a call for tenders for PV power plants between 100 kW and 500 kW.

Available at: https://www.ecologique-solidaire.gouv.fr/nicolas-hulot-presente-grands-axes-accelererdeploiement-des-energies-renouvelables-electriques

 13 Available at: http://www.cre.fr/documents/appels-d-offres

Moreover, this smart meter is installed when someone invests in a photovoltaic power plant.

Available at: http://www.eex.com/en/market-data/power/futures/french-futures#!/2017/10/24

Consumption periods run from September year Y-1 to august year Y.

As consumers' profiles are constant along periods of time, savings per consumers are the same ones for each period.

Incentives to invest in PV generation and to self-consumption are also in the decrease of energy purchases at the retail price, the FIT that remunerates PV surplus fed into the grid, the exemption of PSOT, the fee that remunerates each self-consumed kWh, etc. The study of these incentives policies is out of the scope of this research.

consumption and the contractual power. When the gap is high, firms have a short use and conversely, a long use. We apply the short use profile for the firms (called ENT1) because they represent 87% of the total firms in this voltage interval. For firms elected to tenders, we assume that they are connected to the medium voltage (called ENT3). This assumption is based on the fact that the first laureates of tenders announced an average self-consumption rate of about 95%. Given that, firms have to have a big amount of electricity consumption in order to absorb the PV generation. Based on the sample provided by the French regulator, average consumption and contractual power by period of time for each profile are depicted in table 8.

c. PV size

We assume that households install a PV capacity of 3 kW. It corresponds to the median of cumulative PV panels connected in France to low voltage (Rebenaque O, 2017). The average consumption peak for firms connected to a voltage level between 36 kVa and 250 kVa is 74 kW. We assume that firms won't install a PV installation with a capacity above their peak consumption, so we applied a PV capacity of about 70 kW for a representative firm which allows an average self-consumption rate of 74%. As it has been said in the previous section, self-consumption tenders concern PV capacities between 100 et 500 kW, so we assume that these firms will install a PV capacity of 300 kW on average.

Load curves

Enedis provides coefficient of profiles for different categories of customers. A profile represents the average behavior for a customer group according to the consumption variation of every week, day and half-hour.


Jean Marçais
email: jean.marcais@gmail.com.

Jean-Raynald De Dreuzy
email: jr.dreuzy@gmail.com.

Prospective interest of deep learning for hydrological inference

Keywords: Deep learning, Explicit process-based models, Implicit data-driven models, Calibration, Models structural adequacy. Impact

Deep learning may assist hydrological inference through its increased capacities

Introduction

Decision making relative to groundwater resources requires the characterization, modeling and prediction of complex and dynamical systems with many degrees of freedom.

Nonetheless, these systems have large-scale structure that emerges from hierarchical properties based on conservation principles applied to fundamental physical quantities (e.g. mass, momentum, energy). Difficulties arise as hydrologic systems are inherently heterogeneous and sometimes chaotic. This is not specific to hydrology, it is generic to natural or man-made complex systems. Two approaches have been proposed to handle these complex systems. Whether they are called model-driven or data-driven, explicit or implicit, they differ widely by methodology. In the explicit model-driven approach, processes are physically modeled at each characteristic scale and progressively scaled up. Patterns, equivalent properties and effective laws emerge progressively through upscaling. This has been done extensively in stochastic hydrology to derive equivalent permeability and in percolation theory to identify universal scaling laws (Stauffer and Aharony 1992). Large-scale observations are integrated by adapting the model parameters through the classic, but generally ill-posed, inverse problem [START_REF] Zimmerman | A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow[END_REF]). In the implicit data-driven approach, minimal assumptions are made on the structure of the models developed [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction[END_REF][START_REF] Montgomery | Design and Analysis of Experiments[END_REF]. Rather, this approach relies on generic data-driven analysis based on statistics and artificial intelligence. Among numerous methods, supervised-learning algorithms and, especially, artificial neural networks became popular in the 90s in hydrology. One well-cited example is their use for prediction and understanding of rainfall runoff processes [START_REF] Hsu | Artificial neural-network modeling of the rainfall-runoff process[END_REF][START_REF] Hsu | Self-organizing linear output map (SOLO): An artificial neural network suitable for hydrologic modeling and analysis[END_REF].

Implicit data-driven methods have been less active for the past decade (a period called the Artificial Intelligence Winter). But, there is renewed interest in machine learning methods due to the recent successes of deep neural networks in several Artificial Intelligence benchmarks including the first-time computer win at the game of Go [START_REF] Silver | Mastering the game of Go with deep neural networks and tree search[END_REF]. In general, deep network successes have been attributed to their ability to provide efficient high-dimensional interpolators that cope with multiple scales and heterogeneous information [START_REF] Lecun | Deep learning[END_REF][START_REF] Mallat | Understanding deep convolutional networks[END_REF]. This suggests a natural opportunity for the use of deep networks for hydrological sciences. We discuss these possibilities with particular focus on their complementary and combined use with well-established model-driven approaches.

Generalization, classification and hierarchical combination abilities in deep learning

Deep learning is a new class of machine learning algorithms that recognizes high level abstractions in data. They are called deep because they are made up of some tens of hierarchically arranged layers, compared to classic neural networks that had only very few.

Abstraction is achieved through the processing of the data by the internal layers to automatically identify patterns of increasing complexity. We review three essential properties for their potential interest in hydrological sciences.

Deep learning can be used for calibration. Classic neural networks have been shown to be universal estimators [START_REF] Hornik | Multilayer feedforward networks are universal approximators[END_REF]. Deep learning can significantly improve their efficiency and accuracy [START_REF] Bengio | On the Expressive Power of Deep Architectures[END_REF][START_REF] Mhaskar | Deep vs. shallow networks: An approximation theory perspective[END_REF].

Deep learning can find robust invariants from large, high dimensional datasets, leading to improved interpolation and generalization [START_REF] Hinton | Reducing the dimensionality of data with neural networks[END_REF]. For example, a deep convolutional generative adversarial network [START_REF] Goodfellow | Generative Adversarial Networks[END_REF]) was able to generate new bedroom images having all the same elements (bed, lamp, bedside table…) as the bedroom pictures used for training [START_REF] Radford | Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks[END_REF].

Deep learning successes might indicate fundamental capacities to replicate multiscale modeling of physical principles. Generalization properties of deep convolutional networks have been related to the locality and upscaling principles of wavelets [START_REF] Mallat | Understanding deep convolutional networks[END_REF]. The multilayer convolutive structure of deep networks is well adapted to let hierarchical patterns emerge through the combination of smaller scale elements. Potential consistency with physical processes may be further linked to the compositional nature of some physical processes, which might be derived from advanced combinations of symmetry, locality and compositional principles [START_REF] Lin | Why does deep and cheap learning work so well?[END_REF]. As an illustration, deep networks have prospectively been used to infer quantum energy of complex molecules, reaching state of the art quality of evaluation alternatively derived from Schrödinger equations [START_REF] Hirn | Quantum energy regression using scattering transforms[END_REF]. This inherent upscaling opens new opportunities to apply deep learning to real physical systems [START_REF] Baldi | Searching for exotic particles in high-energy physics with deep learning[END_REF][START_REF] Denil | Learning to Perform Physics Experiments via Deep Reinforcement Learning[END_REF].

Deep learning prospects for hydrological inference

Among the different interests of deep learning for hydrology, the first is its potential contribution to calibration, following the use of artificial neural networks in the 90s. Deep networks are especially designed for handling large data sets. They might be considered for prediction issues (Figure 1, blue arrows), provided that the following conditions are met. Hydrological data are highly diverse and would require some extensive pre-processing to produce uniform training sets for deep networks. Basic prerequisite of deep learning methods should also be checked in terms of the density and nature of the training information.

Nevertheless, deep neural networks have already been used for predicting precipitation from satellite clouds images and reduce significantly the bias compared to traditional neural networks [START_REF] Tao | A Deep Neural Network Modeling Framework to Reduce Bias in Satellite Precipitation Products[END_REF].

Second, deep networks may contribute to the initial choice of the structure of a physical model, which strongly conditions eventual predictions. For geological storage of high-level nuclear wastes, this issue has been handled by requesting independent models (models ensemble) from different research groups [START_REF] Zimmerman | A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow[END_REF][START_REF] Bodin | Predictive modelling of hydraulic head responses to dipole flow experiments in a fractured/karstified limestone aquifer: Insights from a comparison of five modelling approaches to real-field experiments[END_REF]). Such benchmarks are, however, possible only for highly sensitive applications with significant costs. On the other hand, modeling as well as computational capacities have critically progressed to the point such that it is feasible to simulate a broad range of model configurations [START_REF] Kumar | Hydrocomplexity: Addressing water security and emergent environmental risks[END_REF]. Models also become more faithful to the point where synthetic catchments, synthetic aquifers and virtual observatories no longer seem out of reach [START_REF] Thomas | Constitution of a catchment virtual observatory for sharing flow and transport models outputs[END_REF][START_REF] Yu | Cyber-Innovated Watershed Research at the Shale Hills Critical Zone Observatory[END_REF]. The limiting factor increasingly comes from the exploration and analysis of the resulting simulations [START_REF] Hermans | TITLE TO BE FINALIZED[END_REF]. Deep networks might contribute to more systematic interpretation through interpolation and category classification. They could be considered for interpolating explicit models as has been done in surrogate modeling (Figure 1, green arrows) [START_REF] Razavi | Review of surrogate modeling in water resources[END_REF][START_REF] Asher | A review of surrogate models and their application to groundwater modeling[END_REF]. They could also be trained on extensive databases of model realizations that support uncertainty analyses (de Pasquale 2017). The advantage of deep networks over other interpolators is their ability to cope with many simulations while inherently complying with different levels of organization of the hydrological systems [START_REF] Sposito | Scale Dependence and Scale Invariance in Hydrology[END_REF]. Practically, this will require careful consideration of methods to execute many models, manage errors, systematize formatting of inputs and outputs, and ensure easy access to simulation results.

Using deep networks as a possible high-dimensional interpolator is motivated by some reduction in computational costs through the use of existing simulations. But, current interest is also directed to model reduction, category classification and uncertainty quantification through the constitution of extensive databases of simulations (Figure 1, green arrows) [START_REF] Clark | A unified approach for process-based hydrologic modeling: 1. Modeling concept[END_REF]. Model reduction is an active field of research in applied mathematics and computational science [START_REF] Schilders | Introduction to Model Order Reduction[END_REF]) designed to retain only the key structural parameters of the models for the phenomena or predictions of interest [START_REF] Castelletti | A general framework for Dynamic Emulation Modelling in environmental problems[END_REF]. While the first target would be to refine our understanding of hydrological processes, several other issues could be handled. For example, the ability to identify model classes would help to address the question of the model structural adequacy beyond the identification of optimal parametrization [START_REF] Gupta | Towards a comprehensive assessment of model structural adequacy[END_REF][START_REF] Guthke | TITLE TO BE FINALIZED[END_REF]. It might also contribute to identify classes of acceptable models at the core of null-space identification [START_REF] Gallagher | Parameter estimation and uncertainty analysis for a watershed model[END_REF], equifinality analysis [START_REF] Beven | A manifesto for the equifinality thesis[END_REF], and, more generally, uncertainty quantification. Even more prospectively, we might investigate its potentialities in proposing alternative model structural assumptions, especially for investigating the likelihood of diverse hydrological scenarios of critical importance for stakeholders [START_REF] Ferre | Revisiting the Relationship between Data, Models, and Decision-Making[END_REF][START_REF] Marshall | TITLE TO BE FINALIZED[END_REF] and in assisting the design of hydrological experiments [START_REF] Kikuchi | On the optimal design of experiments for conceptual and predictive discrimination of hydrologic system models[END_REF] (Figure 1). Deep networks might be used to assess expert knowledge in a more systematic way [START_REF] Seibert | On the dialog between experimentalist and modeler in catchment hydrology: Use of soft data for multicriteria model calibration[END_REF][START_REF] Hrachowitz | Process consistency in models: The importance of system signatures, expert knowledge, and process complexity[END_REF]. The real strength of deep networks, and implicit data-driven models in general, is that emergent system properties can be identified that cannot be uncovered by explicit models that are imposed on an analysis (Figure 1, orange arrows). This may automatically adapt our models to conform to natural processes and to the data available [START_REF] Kirchner | Getting the right answers for the right reasons: Linking measurements, analyses, and models to advance the science of hydrology[END_REF][START_REF] Troch | Dealing with Landscape Heterogeneity in Watershed Hydrology: A Review of Recent Progress toward New Hydrological Theory[END_REF]), while exploring model space more completely.

Despite the potential contributions of deep learning, we do not suggest that they should replace classic approaches in hydrology. Rather, they are attractive as a complementary tool having, at least conceptually, some properties highly relevant to hydrological systems like their built-in hierarchical combination. This could similarly benefit ongoing automated model-building efforts that also require some definition of the range of model structures to consider and the ways in which they should be combined [START_REF] Clark | A unified approach for process-based hydrologic modeling: 1. Modeling concept[END_REF]. In addition to these enhanced capacities, deep networks will find particular use in integrating larger data sets and explicit model results produced by rapid technological progress in automated and remote sensing. In this sense, we see deep learning as a loose coupling between growing data-driven and model-driven approaches (Figure 1). It is not designed as an effective calibration technique applied on explicit physical models [START_REF] Certes | Application of the pilot point method to the identification of aquifer transmissivities[END_REF][START_REF] Doherty | Ground water model calibration using pilot points and regularization[END_REF], but rather as an implicit data-driven approach to correct errors from explicit physical based models [START_REF] Szidarovszky | A Hybrid Artificial Neural Network-Numerical Model for Ground Water Problems[END_REF][START_REF] Demissie | Integrating a calibrated groundwater flow model with error-correcting data-driven models to improve predictions[END_REF][START_REF] Gusyev | Use of Nested Flow Models and Interpolation Techniques for Science-Based Management of the Sheyenne National Grassland[END_REF]. Prospective testing and interactions are needed with the deep learning community to understand how information content between synthetic models and field data should be balanced in the training sequence [START_REF] Nearing | The quantity and quality of information in hydrologic models[END_REF].

A roadmap to investigate the relevance of deep learning to hydrology

We propose three steps toward integrating deep learning into hydrologic science: testing on data; testing on benchmarks; and collaboration with the wider deep learning community.

Testing on hydrological numerical data will require the collection and organization of systematic and well-organized databases. Availability of such database has been one of the main reasons for the success of deep learning techniques [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF]. This could include measured data, but should also rely heavily on synthetic data sets. Such synthetic hydrological databases are achievable, for example by building extensive stochastic simulations performed on heterogeneous media [START_REF] Pirot | TITLE TO BE FINALIZED[END_REF]. Constitution of systematic and standardized field databases is also under way following the dynamics of critical zone observatories, open international normalization initiatives [START_REF] Ames | HydroDesktop: Web services-based software for hydrologic data discovery, download, visualization, and analysis[END_REF]) and regulatory incentives.

Testing on hydrological benchmarks can begin with target synthetic benchmarks described above. More complicated benchmarks can be developed according to their consistency with existing deep learning benchmarks, i.e. based on images or succession of images to describe processes [START_REF] Mathieu | Deep multi-scale video prediction beyond mean square error[END_REF]. Hydrologic insight will be necessary to ensure that the benchmarks are relevant to important hydrological complexities, especially those related to the multi-scale nature of the hydrologic systems.

Cross-disciplinary discussions with the deep learning community will be required to take full advantage of deep learning methods and to introduce hydrology as a topic of interest to that community. Data heterogeneity, in terms of quantities measured and scales, have already been mentioned and are critical to improve our ability to integrate diverse information sources.

Hydrology offers an opportunity for the deep learning community to expand their interest beyond controlled and deterministic systems like the double pendulum [START_REF] Schmidt | Distilling Free-Form Natural Laws from Experimental Data[END_REF] to natural systems that are characterized by epistemic or aleatory uncertainties. The heterogeneous and sometimes chaotic nature of the hydrologic system makes it difficult to directly transpose existing results obtained on deterministic experiments but is an opportunity to assess deep learning potentialities. . More prospectively, it might be applied to discover patterns in data, find trends in synthetic versus real data misfits to assist hydrologists in formulating testable hypotheses (orange arrows). This last possibility is seen as an unsupervised learning task as opposed to supervised learning for the other ones with real or synthetic datasets.

FigureFigure 1 :

 1 Figure

Acknowledgements

We thank Ty Ferre for inviting us to contribute to this issue, for his constructive and encouraging feedback. We thank Guillaume Pirot for his review.