Thomas Reverdy
email: thomas.reverdy@grenoble-inp.fr

implementation of a market-based instrument, French tradable certificates for energy efficiency

come

Performative struggles and theory-practice decoupling in the design and the implementation of a market-based instrument, French tradable certificates for energy efficiency

Introduction

In the early 21st century, the European Commission promoted market-based instruments to support climate change mitigation, renewable energy, and energy efficiency. MBIs are an important illustration of Michel [START_REF] Callon | What does it mean to say that economics is performative[END_REF] suggestion that economic theory is no longer just a social science aimed at understanding economic behavior, but an engineering science that contributes to the creation of new economic realities [START_REF] Blok | Clash of the eco-sciences: Carbon marketization, environmental NGOs and performativity as politics[END_REF][START_REF] Callon | Civilizing markets: Carbon trading between in vitro and in vivo experiments[END_REF][START_REF] Mackenzie | Making Things the Same: Gases, Emission Rights and the Politics of Carbon Markets[END_REF].

Market-based instruments for environmental protection have been created and developed through an intellectual exploration by economics, which has been largely involved in their legitimization.

They are usually associated with Ronald Coase's theorem [START_REF] Coase | The problem of social cost[END_REF], which promotes the internalization of market externalities through the allocation of property rights and the organization of their exchange. According to Ronald Coase's theorem, the exchange of property rights between economic agents, assuming no transaction costs, allows for an efficient allocation of resources regardless of the initial distribution of property rights. This economic theory inspired John Dales to prescribe a system of transferable quotas for water pollution [START_REF] Dales | Pollution, property and prices; an essay in policy-making and economics[END_REF]). The policy instrument, formalized by [START_REF] Montgomery | Markets in licenses and efficient pollution control programs[END_REF], spread to many areas of environmental protection and spawned intense discussions in environmental economics [START_REF] Freeman | Moving to markets in environmental regulation: Lessons from twenty years of experience[END_REF]. Subsequently, it was associated with climate change mitigation in the negotiations of the Kyoto Protocol [START_REF] Meckling | Carbon coalitions: Business, climate politics, and the rise of emissions trading[END_REF]) and adopted by the European Union in its policies against greenhouse gas emissions [START_REF] Braun | The evolution of emissions trading in the European Union-The role of policy networks, knowledge and policy entrepreneurs[END_REF]Engels 2006;[START_REF] Voss | Innovation processes in governance: The development of 'emissions trading'as a new policy instrument[END_REF], in favor of renewable energy [START_REF] Lauber | The struggle over support schemes for renewable electricity in the European Union: A discursive-institutionalist analysis[END_REF] or energy efficiency [START_REF] Mallard | Can energy savings certificates help in boiler scrappage programs? Exploring the market agencing of a detachment policy[END_REF].

There is an extensive research literature in economic sociology on the controversies and readjustments of market-based instruments designed to address public concerns [START_REF] Frankel | The organization of markets for collective concerns and their failures[END_REF]). The analysis of the implementation of the European Union Emissions Trading Scheme (EU ETS) and the Clean Development Mechanism (CDM) has significantly contributed to the construction and discussion of performativity theory by highlighting the influence of economic expertise [START_REF] Callon | Civilizing markets: Carbon trading between in vitro and in vivo experiments[END_REF][START_REF] Voss | Innovation processes in governance: The development of 'emissions trading'as a new policy instrument[END_REF][START_REF] Braun | The evolution of emissions trading in the European Union-The role of policy networks, knowledge and policy entrepreneurs[END_REF]) and ideological and political forces [START_REF] Lohmann | Carbon trading, climate justice and the production of ignorance: Ten examples[END_REF][START_REF] Blok | Clash of the eco-sciences: Carbon marketization, environmental NGOs and performativity as politics[END_REF]. Implementation showed how integrating the instrument into an international agreement and European political and calculating practices produced spectacular unexpected effects [START_REF] Mackenzie | Making Things the Same: Gases, Emission Rights and the Politics of Carbon Markets[END_REF]. Numerous adjustments revealed the inertia of learning dynamics by European institutions [START_REF] Neyland | Can Markets Solve Problems?: An Empirical Inquiry into Neoliberalism in Action[END_REF], the never-ending political negotiation of the scope [START_REF] Bryant | The Politics of Carbon Market Design: Rethinking the Techno-politics and Post-politics of Climate Change[END_REF], and the limited impact on competitiveness [START_REF] Ehrenstein | Economic under-determination: Industrial competitiveness and free allowances in the European carbon market[END_REF].

However, this research rarely evaluates the extent of these adaptations with respect to the original theory: to what extent these adaptations may or may not contradict the theory. Some issues and expertise may be complementary to the initial theory and can contribute to the design by informing some key decisions of the implementation. Adjustments may also be justified by the practical considerations of implementation and be judged to be "consistent" with the theory.

Deviations may be triggered by "performative struggles" [START_REF] Callon | What does it mean to say that economics is performative[END_REF], when explicit and implicit competing theories may significantly interfere or clash with the initial theory [START_REF] D'adderio | Performing modularity: Competing rules, performative struggles and the effect of organizational theories on the organization[END_REF]. This chapter borrows from the sociology of public policy the notion of instruments (Halpern 2010; [START_REF] Lascoumes | Introduction: Understanding public policy through its instruments-from the nature of instruments to the sociology of public policy instrumentation[END_REF], which encourages the identification of intermediaries and activities through which a policy is implemented. This notion helps to explain how performative struggles can lead to a decoupling between the principles of an instrument and its design and implementation. We borrow the notion of decoupling from organizational analysis: decoupling can be a strategy for members of an organization to deal with the divergence between external legitimacy constraints (formal adoption of a method or theory) and pragmatic issues at the implementation level [START_REF] Bromley | From Smoke and Mirrors to Walking the Talk: Decoupling in the Contemporary World[END_REF][START_REF] Meyer | Institutionalized organizations: Formal structure as myth and ceremony[END_REF]. This chapter describes the design and implementation of a market-based instrument, the Energy Efficiency Certificate (EEC)1 , in France, from initial discussion in 2002 to recent modifications in 2021. The EEC is based on the distribution to energy suppliers of obligations to contribute to energy efficiency in proportion to their sales. Energy suppliers can meet this obligation by purchasing EECs from other actors that organize and finance energy efficiency activities, such as local authorities, social housing and specialized intermediaries. When they initiate an energy efficiency project, these actors receive one EEC for each MWh of gas and electricity saved. The EEC can be considered a market instrument, as it allowed a policy objective to be marketized through the buying and selling of an allowance. Through the certificate market, each energy efficiency project would be rewarded proportionally to the energy gained by selling certificates to energy suppliers.

After presenting the academic debate on performativity, the chapter reconstructs a narrative of the design and implementation of the instrument. This narrative is based on field research with interviews with the original designers of the instrument, officials directly involved in the design of the instrument and some participants in key activities, but also actors indirectly involved through the evaluation, as officials and academics. The interviews helped to identify and interpret key documents. The interviews were very consistent about the history of the instrument. In a first phase, the designers of the instrument (government advisors and civil servants) aimed at clarifying the objectives, making explicit adjustments for its practical implementation and controllability, trying to respect as much as possible the basic principles of the market mechanism. During this phase, the performativity of the instrument was affected by a competing instrument based on the tax credit. In a second phase, as the economic impact of the ECCs increased, the administration responsible for their design progressively deviated from the basic principles. The administration introduced a controversial practice of allocating bonus certificates to specific projects in order to increase support for them. This practice became systematic: coefficients were defined for each type of operation to adjust the revenue from certificates to an estimated cost of the operation. The systematic and general introduction of coefficients contradicts the principle of equivalence on which market-based instruments are constructed [START_REF] Levin | Pollution futures: Commensuration, commodification, and the market for air[END_REF][START_REF] Mackenzie | Making Things the Same: Gases, Emission Rights and the Politics of Carbon Markets[END_REF]. This fundamental deviation was motivated by a different theory of efficiency than the one supporting the market-based instrument.

From theory performativity to decoupling from theory

Market-based instruments are an excellent subject for studying the performativity of the economy. The notion of performativity is associated with how particular economic ideas, models, and instruments perform, configure, and shape the economy [START_REF] Callon | The Law of Markets[END_REF]. In this chapter, we focus on planned performativity, situations in which economic experts contribute to the design of new economic instruments and markets to manage collective concerns, such as pollution [START_REF] Mackenzie | Making Things the Same: Gases, Emission Rights and the Politics of Carbon Markets[END_REF], natural resources [START_REF] Holm | Framing fish, making markets: The construction of Individual Transferable Quotas (ITQs)[END_REF], biodiversity [START_REF] Barral | Conservation, finance, bureaucrats: Managing time and space in the production of environmental intangibles[END_REF], and other situations in which economic experts identify "market failures" [START_REF] Reverdy | The redesign of electricity markets under EU influence: The capacity mechanism in Britain and France[END_REF].

The thesis of the performativity of economic theory has generated numerous debates that have enriched the analytical approach and possible conclusions [START_REF] Cochoy | Performativity, economics and politics: An overview[END_REF]). First, it has been accused of confirming the main claim of economic theories, namely that the behavior of economic agents can be consistent with the theory and thus achieve the expected results in the real economy [START_REF] Miller | Turning Callon the right way up[END_REF]. Second, it has been accused of taking a positivist view of the transfer from science to practice, as if economic expertise were unified and coherent and its enactment unambiguous and disinterested (Mirowski and Nik-Khah 2007). [START_REF] Callon | What does it mean to say that economics is performative[END_REF] responds to these critics by arguing that they have misunderstood his intellectual project, explaining that economic performativity is achieved through a laborious construction of assemblages of instruments, discourses, rules, and human skills, as a process of enactment embedded in various material and social contingencies.

The introduction of market-based instruments for greenhouse gas (GHG) mitigation in Europe is a good example of the influence of economics on the design of a policy instrument: EPA economists were involved in the initial US experience and its formalization, and EU DG Competition considered a market-based instrument as the best way to integrate "environmental externalities" into the European market (Braun 2009, Skjaerseth and[START_REF] Skjaerseth | Making the EU emissions trading system: The European commission as an entrepreneurial epistemic leader[END_REF]. The increasing influence of economic analysis of law in the field of European competition policy [START_REF] Davies | The limits of neoliberalism: Authority, sovereignty and the logic of competition[END_REF]) facilitated the criticism of other approaches such as technical regulation or direct subsidies. Market-based instruments were in line with the principles of non-discrimination between projects (fair competition) and the objective of limiting market distortions. DG Competition also promoted market-based instruments for renewable energy [START_REF] Lauber | The struggle over support schemes for renewable electricity in the European Union: A discursive-institutionalist analysis[END_REF], which was not affected by international agreements as greenhouse gas mitigation.

The purpose here is to examine the extent to which the original theory associated with the instrument is preserved, enriched by complementary logics or theories [START_REF] Asdal | From climate issue to oil issue: Offices of public administration, versions of economics, and the ordinary technologies of politics[END_REF], or undermined by a competing one or by the constraints of implementation. Four modes of adaptation can be identified. First, these instruments must be configured according to policy objectives. For example, in the case of cap-and-trade instruments, economic theory assigns to the political authorities the role of defining the geographic and technical scope of obligations, the overall level of pollution, and the conditions for the initial distribution of allowances [START_REF] Schmalensee | An interim evaluation of sulfur dioxide emissions trading[END_REF]. Second, the instrument must be adapted to the practical constraints of implementation and controllability: it follows a process similar to the one observed by Breslau for the design of the electricity market, which he qualified with the expression "applied Platonism" (Breslau 2011). Regulatory design refers to an idealized model, and each new specification must be justified in relation to the model and the policy objectives associated with the instrument. For market-based instruments, the key activities are the construction of classifications and technical metrics that define equivalences between the contribution to an environmental objective and the allocation of credits [START_REF] Levin | Pollution futures: Commensuration, commodification, and the market for air[END_REF]MacKenzie 2008), and the methodology for characterizing the additional effect of a project through counterfactual reasoning based on a reference situation [START_REF] Ehrenstein | The conditional sink: Counterfactual display in the valuation of a carbon offsetting reforestation project[END_REF]. This second step is usually presented by the designers of the instrument as 'technical' in order to depoliticize their activity [START_REF] Barry | The anti-political economy[END_REF]. Sociologists have shown that the detailed design ultimately turns out to be as political as the activities identified a priori as "political" [START_REF] Frankel | The organization of markets for collective concerns and their failures[END_REF]. The detailed design of economic policies is a relational space of different and sometimes interacting versions of economics [START_REF] Asdal | From climate issue to oil issue: Offices of public administration, versions of economics, and the ordinary technologies of politics[END_REF]). In the energy sector, for example, the adaptation of the market model involves a great diversity of expertise and political issues such as security of supply and European integration [START_REF] Silvast | Energy, economics, and performativity: Reviewing theoretical advances in social studies of markets and energy[END_REF].

According to [START_REF] Callon | Civilizing markets: Carbon trading between in vitro and in vivo experiments[END_REF], the performativity of economics includes the evaluation of the effects of instruments and an exploratory market design by NGOs, scientists, economists, in a context of reflexive modernity [START_REF] Beck | Risk society: Towards a new modernity[END_REF]. Reflexivity could concern both initial parameters of the instrument (such as quota distribution rules, the scope of projects associated to the market), but also detailed rules (equivalences, aditionality...). In the EU, this reflexivity led to the creation of an additional mechanism, the quota stability reserve for the EU ETS, to stabilize the CO2 quota price [START_REF] Neyland | Can Markets Solve Problems?: An Empirical Inquiry into Neoliberalism in Action[END_REF]). However, many authors studying the GHG credit market discussed critical capacities and considered that reflexivity suffers from structural governance weaknesses [START_REF] Lohmann | Carbon trading, climate justice and the production of ignorance: Ten examples[END_REF][START_REF] Blok | Clash of the eco-sciences: Carbon marketization, environmental NGOs and performativity as politics[END_REF].

The evaluation of the failure or success of performativity raises methodological and theoretical questions. According to [START_REF] Callon | What does it mean to say that economics is performative[END_REF], failure or success "(...) becomes clear only at the end of the tests to which it is put, through the cooperation it triggers, the oppositions and controversies it generates" (p. 330). MacKenzie ( 2003) introduced the concept of counter-performativity: aggregate behavior can lead to an outcome opposite to that originally intended. According to [START_REF] Rilinger | Conceptual limits of performativity: Assessing the feasibility of market design blueprints[END_REF], the failure of performativity can be produced by conditions that deviate from some of the intrinsic hypotheses of the economic design, hypotheses that could not have been identified or made explicit. Many attempts to address collective concerns through market mechanisms have failed in the sense that they did not meet the objectives or were not politically accepted [START_REF] Frankel | The organization of markets for collective concerns and their failures[END_REF].

The thesis of performativity probably reaches a limit when the detailed design and implementation contradicts the initial principle of the instrument and does not respect the condition of "applied Platonism" [START_REF] Breslau | Designing a market-like entity: Economics in the politics of market formation[END_REF]. Decoupling occurs when the designers and intermediaries systematically introduce adjustments that do not respect the initial hypotheses, neutralize the economic mechanism, and allow the implementation of contradictory principles of efficiency on a large scale. For example, the redesign of the French electricity market [START_REF] Reverdy | Making an exception: Market design and the politics of reregulation in the French electricity sector[END_REF]) could be qualified as decoupling because the French parliament reintroduced a regulated tariff for nuclear power, bypassing the electricity wholesale market mechanism, for electricity furniture.

Inspired by organizational analysis [START_REF] Meyer | Institutionalized organizations: Formal structure as myth and ceremony[END_REF]Rowan 1977, Bromley and[START_REF] Bromley | From Smoke and Mirrors to Walking the Talk: Decoupling in the Contemporary World[END_REF] and public policy analysis [START_REF] Lascoumes | Introduction: Understanding public policy through its instruments-from the nature of instruments to the sociology of public policy instrumentation[END_REF], we open here the hypothesis that public authorities could pursue pragmatic goals in the design and implementation of instruments, to the point of gradually deviating from the original theory. The decoupling could be encouraged by various competing ideologies and pragmatic cognitive ideas [START_REF] Heidbreder | European Union governance in the shadow of contradicting ideas: The decoupling of policy ideas and policy instruments[END_REF]. Decoupling is made possible by a lack of control, the complexity of implementation, and a degree of opacity [START_REF] Bromley | From Smoke and Mirrors to Walking the Talk: Decoupling in the Contemporary World[END_REF]. Decoupling might be preferred to abandoning the instrument for reasons of legitimacy [START_REF] Meyer | Institutionalized organizations: Formal structure as myth and ceremony[END_REF] or because some related parties and intermediaries are satisfied with the benefits it brings them [START_REF] Voss | Instrument constituencies and the supply side of policy innovation: The social life of emissions trading[END_REF]. For this reason, decoupling is difficult to characterize: many actors may have a motivation to demonstrate that the mechanism is still related to the theory it inspired.

How the favorable context for market-based instruments influenced the initial design of EEC?

The creation of a market instrument for energy efficiency was first discussed in France in 2002 during the debate on energy policy organized by the government, between a technical association for energy and environment 2 (Association Technique Energie Environnement, ATEE), the public agency for the environment and energy efficiency (Agence de l'Environnement et de Maîtrise de l'Energie, ADEME), and the Direction for Energy and Raw Materials at the Ministry of Industry (Direction Générale en charge de l'Energie et des Matières Premières, DGEMP3 ).

The first principle of this instrument was to impose energy savings targets on energy distributors to be achieved with their customers. This idea, inspired by the US policy of demand management, was well received by French utilities, which had developed energy efficiency services for consumers as part of their public service mission. In the context of the liberalization of the electricity and gas sectors, the formalization of this obligation was therefore an opportunity to turn this activity into a competitive advantage.

The second principle was to encourage other economic actors to contribute to energy efficiency actions by allowing them to obtain certificates for the amount of energy saved and to sell the The Energy Act promoted the creation of a market between two types of participants. On the one hand, the electricity and gas suppliers, the so-called "obligated parties", to whom the energy administration distributes obligations according to the quantities of energy supplied and the sales revenue; on the other hand, the "eligible parties", the leaders of energy efficiency projects, of which the initial non-exhaustive list includes industrialists, service providers, municipalities, social landlords, etc. The law defines the nature of the exchange between these two types of participants by creating energy savings certificates based on a unit of measurement: the cumulative and discounted kWh, or "kWh cumac", which corresponds to the cumulative quantity of energy saved following an investment (installation of an energy-efficient appliance, insulation work, etc.)5 .

Beyond the initial idea, however, such an instrument presupposes a number of parameters inherent in its principles. First of all, any market-based instrument presupposes the existence of an overall political objective, which is then distributed among the parties involved. In this case, the political authority defines by law an amount of energy to be saved for several years6 . The definition of the total amount of energy to be saved has an economic implication for the obligated parties, who will have to finance the actions.

The detailed design of the instrument: the constraints of standardization

In the first phases of its implementation, the detailed design of the instrument was fully in line with the economic principles of market-based instruments, such as the commensurability between actions and the allocation of certificates. On the supply side, the administration in charge of the energy sector (DGEMP and, after 2008, DGEC) defines technical rules for the characterization of energy savings and their conversion into EEC. The administration defined standardized operations with a theoretical energy gain for each of them. To do this, it defined theoretical baseline conditions from which experts would be able to calculate energy gains and ensure the additional effect: the baseline condition for building insulation is the average performance of the existing building stock; for equipment, the calculation takes into account the minimum performance for equipment listed in the EU Eco-design Directive. The designers preferred a cautious interpretation of the evaluation of energy gains, implementing their own representation of household investment decisions. This representation rather referred to the behavior of a privileged household, able to invest regularly in the renewal of its heating equipment and whose home insulation was in the average range. It did not take into account a household living in a poorly insulated house with a very limited investment capacity. The choice of baseline condition had the effect of excluding households that would be most in need of investment subsidies.

The second design activity is the calculation of energy gains. The evaluation of the technical characteristics of the equipment or intervention is entrusted to the ATEE, which collaborates with experts from energy suppliers, equipment suppliers and representatives of artisans. While the adoption of a market mechanism was intended to avoid the capture of regulation by private interests, the calculation of energy gains per operation is not exempt from this type of attempt.

According to the instrument designers, obligated parties and equipment suppliers have tended to encourage an overestimation of potential savings. This allows the obligated parties to reach their targets more quickly and the equipment suppliers, who are in competition with each other, to defend the performance of their equipment. ADEME has limited expertise in detecting abuses. In the event of a conflict, the DGEC acts as arbitrator, recognizing that the mechanism is seriously vulnerable to "capture" of the regulator by private interests.

The detailed design faces similar practical constraints as the GHG credit in the CDM, such as characterizing the contribution to environmental goals through standardization of operations and counterfactual reasoning [START_REF] Chiapello | The fabrication of environmental intangibles as a questionable response to environmental problems[END_REF]. This necessary practice faces intrinsic contradictions with some promises, such as the ability to avoid capture. Capture is displaced from the calculation of costs to the evaluation of the contribution to the environmental objective.

The coexistence of instruments with different definitions of efficiency

Since its creation, the EEC suffered from the coexistence of another policy instrument, the tax credit for the purchase of energy equipment, which remained the main energy policy instrument in terms of financial support7 . It provided a direct subsidy, calculated as a percentage of the equipment purchase bill, which depended (from 15% to 50%) on the different types of equipment, with annual variations according to budgetary arbitrations and parliamentary amendments. Since the same investments were supported by both instruments, the EECs often played a marginal role in financing those that were already profitable under the tax credit. During the first and second periods, the energy efficiency targets were limited and easily achieved thanks to the tax credit, and the price of the certificates remained quite low, i.e. 2-4€/MWh cumac.

Gradually, in the following phases, both the increase of the targets and the increase of the prices of the certificates changed the nature of the contribution of the certificates. The combination of EEC and tax credit has achieved a certain balance between the operations. Insulation measures tend to be supported by the EECs, equipment such as boilers by the tax credit. According to the DGEC, the overlap of the two mechanisms should be beneficial by compensating for some of its arbitrary limits in the calculation of energy gains for the EECs. Conversely, the EECs allow a certain continuity of financing despite the budgetary uncertainties of the tax credit, which is revised each year according to the annual state budget law. Nevertheless, this system creates uncertainty and complexity for the craftsmen who have to explain the two mechanisms to their clients. It also creates numerous inconsistencies when the tax credit is modified as a result of the annual state budget without being coordinated with the EECs. Thus, from the very start, the performativity of the theory was hindered by the coexistence of the market instrument with a previously implemented instrument defined on a very different theoretical basis, whose economic effects prevented the new instrument from producing the expected effects. But at this stage, the deviation from the theory is not integrated into the instrument itself.

Diversification of evaluations and performativity struggles

At the end the second period (2014), the EEC system was also strongly contested by obligated parties and some academics, who emphasized the costs for consumers and the opacity of the system [START_REF] Giraudet | European experiences with white certificate obligations: A critical review of existing evaluations[END_REF]. The administration published the first critical reports on its effectiveness. According to these reports, the renovations in private homes did not lead to a measurable reduction in the bill, raising questions about the calculation of potential benefits, the quality of the installation, and rebound effects among consumers8 . Academics came to the same conclusion [START_REF] Blaise | Quel est l'impact des travaux de rénovation énergétique des logements sur la consommation d'énergie[END_REF]. In addition, they believed that the coexistence with the tax credit made the evaluation difficult.

Political criticism focused on the social impact: the mechanism mainly benefited a wealthy population able to invest, which combined the tax credit with access to bank loans. In response to these criticisms, the government decided to redirect part of the EECs to households in precarious energy situations. As of 2016, operations related to vulnerable households would be subject to specific certificates and would be subject to separate obligations. This segmentation of the market partly responded to the social criticism of the instrument.

Between 2015 and 2018, as a result of more ambitious energy-saving targets, the price per MWh cumac increased significantly, from 2€ to 9€. The increase soon raised questions among the market designers: the more efficient operation (low cost and high energy saving) could give a significant profit for intermediaries or beneficiaries. This issue was raised before by an administrative report written by civil servants of the Ministry of the Economy in 20149 .

The administrative report of the Ministry of Economy argued that the remuneration based on the estimated MWh saved could lead to large disparities in payback periods depending on the nature of the investment. In 2019, the ADEME adopted the same criticism on the windfall profits for industrial investments 10 . Implicitly, these reports questioned the principle of financing the project on the basis of the energy saved, considering that the public support (through the ECCs market) should be defined at the level necessary to trigger the action, according to the different technical situations and households. The DGEC was also concerned about the profits generated by the mechanism for some operations. It therefore regularly asked the ATEE to revise the standard in order to reduce the number of MWh cumac attributed to low-cost operations. In the absence of a cap, the calculation was discreetly adjusted to avoid providing windfall profits to operations that were too lucrative because they were too efficient. This strategy requires a good knowledge of operating costs and an anticipation of the EEC price. The DGEC relied on data from the renovation observatories set up by the utilities or the ADEME. These adjustments were modest, but they introduced new methods of evaluating efficiency by integrating the evaluation of the cost of actions in the allocation of EECs.

The decoupling process: the generalization of bonus calculation

New "counterperformance" of the mechanism led to stronger adjustments in allocating certificates to efficiency projects. In 2016 and 2017, obligated parties warned the Ministry of the risk of failing to meet the targets for the period, which could lead to a sharp increase in the price of certificates and, consequently, a significant increase in energy prices. In order to adjust the volume, the Ministry then introduced a bonus principle, which consists of increasing the remuneration of certain operations by adding a multiplier coefficient to the initial quantification of the MWh cumac. In this way, the operation can generate more certificates, be better financed and thus be better facilitated. This led to the creation of more certificates (as a result of the existence of more operations, each of which is more highly valued), which would allow obligated parties to meet their obligations without an excessive increase in the price of certificates. For the obligated parties, the budget remained under control. This bonus practice thus made it possible to adjust the total cost of the policy. This practice had its critics among obligated parties and within the ATEE. They felt that it distorted competition between solutions. For the concerned projects, the bonus increased the revenues: some companies were able to attract customers by offering insulation work for 1€, which was very profitable for all the intermediaries (advertisers, tradesmen...). It has led to unscrupulous interventions by companies that do mediocre work, for beneficiaries who would not be able to control it, or who would not be motivated to control it because they would not pay for it.

A sharp increase in EEC prices in 2018 prompted the DGEC to once again resort to bonuses.

However, the DGEC rationalized the calculation method. It integrated the tax credit mechanism and tried to ensure that the additional support did not exceed the total cost. A principle of zero residual cost for the most vulnerable households was adopted by the DGEC, integrating a new understanding of the consumer's choice, coming from the professional representatives, according to whom it would be "impossible for the vast majority of individuals" to insulate their attics or low floors "without the one-euro offers" 11 .

The calculation method starts from the desired target in terms of minimum premium for each type of work. For example, the target for the insulation of attics and roofs is a minimum premium of 20€/m2. However, the technical calculation of energy saved gives 1.4 MWh cumac per m2 for this operation. Assuming a MWh cumac price of 5.5€, the DGEMP applies a coefficient of 2.57 to the initial calculation so that the operation receives 20€/m2. The same calculation is applied to all types of plants. The coefficients range from 1.29 to 17.3 in order to obtain the target "minimum premium" evaluated for each operation. This means that a project that saves 1 MWh cumac will receive between 1.29 and 17.3 certificates, depending on the type of operation.

There are some inconsistencies in this new strategy. The amount of revenue from EECs for each operation still depends on the market price of the certificates. For the calculation of the bonus coefficient in 2019, the DGEC had arbitrarily taken a certificate value of 5.5€/MWh cumac.

However, the value of certificates depends on unpredictable factors, including the effect of subsidies. In 2019, the price remained high at around 9€/MWh cumac, which meant that the subsidized companies received much more funding than initially calculated. The administration learned from this experience that the calculation of the coefficients must answer two contradictory objectives: stabilize the price of the EECs by controlling the volume of emitted certificates and, at the same time, avoid overpaying the operations. An official suggested this image: "The administration acts like artillerymen in the First World War: it fires the projectile, looks where it falls and readjusts the angle". 12 .

Thus, the bonus policy was rationalized in order to provide a subsidy for each operation, adjusted with other sources of financing and according to the capacities of the households, and finally to keep the EEC prices under control. Observers and officials stress the importance of the DGEC's intervention. An academic explained: "When the DGEC defines the level of the premium for each operation, we are no longer in a market mechanism". 13 . Academic papers confirm that the practice of bonuses contradicts the principles of the market mechanism [START_REF] Giraudet | Selling and saving energy: Energy efficiency obligations in liberalized energy markets[END_REF], Glachand et al. 2020, Crampes and Léautier 2021). In a market-based instrument, the equivalence between energy savings and the allocation of certificates is essential because it guarantees that funds are allocated to the most efficient projects. Bonuses break the equivalence between certificate allocation and contribution. The bonus affects the commensurability and the mechanism of allocation to more efficient operations.

Why maintain a market-based instrument that has lost its original logic and has become particularly complex and fragile in the face of cyclical fluctuations or estimates? According to the 12 Interview with a civil servant involved in the detailed calculation of coefficients 13 Interview with an academic. interviews, the maintenance of the instrument has less and less to do with economic theory and more and more to do with its institutional status and the relationship within the government.

Since it is based on a system of obligation and exchange, it escapes budgetary arbitration, parliamentary debates on state budgets and the control of the Ministry of the Economy. This instrument is entirely under the responsibility and control of the Ministry of the Environment.

The associated costs do not appear in the state budget and do not constitute public expenditure: "For the time being, the political objective is to make the system last, because it is difficult to deprive oneself of some 4-5 billion euros in extra-budgetary funds that are directed towards virtuous effects". 14 . The EEC also enjoys a positive political image because the cost of the policy is hidden in energy bills. However, this strategy is also controversial within the administration.

Considering the EEC as a mechanism that combines taxation and subsidies, officials of the Ministry of Economy and Finance are concerned about the lack of transparency and invisibility in the state budget and in the calculation of what they consider to be mandatory taxation. They are also concerned that the allocation of financial support is not optimized.

Conditions for decoupling

This type of decoupling is not limited to the French EEC. The UK Tradable Green Certificate was also about differentiating between different categories of projects (renewable energy technologies such as solar PV, wind, etc.) by allocating them different quantities of green certificates for one MWh of renewable electricity. Overly profitable projects were given fewer certificates per MWh of electricity, while less efficient projects were given more certificates, in order to adjust revenues to the costs of each technology [START_REF] Woodman | Learning from experience? The development of the Renewables Obligation in England and Wales 2002-2010[END_REF]. In contrast, we do not observe this kind of segmentation and adjustment coefficient in the case of the EU ETS.

The comparison with these two other regulatory market-based instruments allows us to suggest conditions favorable to the decoupling process. The first condition for decoupling is a critical evaluation based on alternative theories of efficiency. The original theory of economic efficiency assumed that, in the context of public authority ignorance of costs, the market-based instrument optimizes the allocation to lower-cost operations. But this economic theory of efficiency ignores the distributive effects of the market instrument associated with the inframarginal rent: since the market price of the allowance is formed by the cost of the most expensive operations (marginal cost), those less expensive, considered as inframarginal, benefit from the same allowance price.

The difference between their costs and the price produces a profit, the inframarginal rent. This problem was not completely ignored by the initial promoters of the instrument, who thought that the mechanism would optimize itself: the most attractive opportunities would be financed first, become saturated, which would lead to an increase in the price of the kWh cumac to seek more costly operations. However, experience has shown that there is no saturation because the construction sector is not necessarily able to take on one type of operation and then another.

Given the specialization of craftsmen, a diversity of operations must be encouraged. Criticism of the windfall profits generated by the instrument stems from another implicit theory of efficiency and equity associated with traditional subsidy policy. According to this logic, the level of support should be adjusted to the additional costs of operations. The support should make the operation affordable if it is deemed sufficiently efficient. This is typically the traditional efficiency logic of civil servants. The same concern motivated the abandonment of tradable green certificates [START_REF] Cointe | Feed-in tariffs in the European Union: Renewable energy policy, the internal electricity market and economic expertise[END_REF]Nadaï 2018, Lauber and[START_REF] Lauber | The struggle over support schemes for renewable electricity in the European Union: A discursive-institutionalist analysis[END_REF]. Another issue, the lack of EEC price control, was also considered as a counter-performance by civil servants. According to the theory of market-based instruments, the price increase should be considered as a natural result of the mechanism responding to the scarcity of quotas or certificates. Another principle of efficiency, the control of the budget, is gradually asserting itself as the collective awareness of the economic effects of the mechanism. The EU ETS faces the same problem of fluctuating quota prices, but adopts a different solution, the market stability reserve [START_REF] Neyland | Can Markets Solve Problems?: An Empirical Inquiry into Neoliberalism in Action[END_REF].

A second condition for decoupling is attachment to the instrument, even if the functional promises of the instrument are contested. Underperformance does not necessarily mean that the instrument is abandoned. The resistance could be explained by its structural properties [START_REF] Voss | Instrument constituencies and the supply side of policy innovation: The social life of emissions trading[END_REF], the coalition of actors that benefit from its specific design: the members of the cabinet of the Ministry of the Environment support the EEC because it finances an environmental policy without a discussion in the state budget. The EEC also benefits from the support of experts and intermediaries who contribute to its organization. These supporters of the EEC need that the instrument is not evaluated as a tax subsidy mechanism. Therefore, EEC rely on the "rationalized myth" [START_REF] Meyer | Institutionalized organizations: Formal structure as myth and ceremony[END_REF] of the efficiency of market-based instruments. This myth is preserved by the fact that the adjustments are "technical" and not visible.

A third condition of decoupling is the introduction of new principles of action in the adjustment of the mechanism, which contradict the original theory. The economic rationale, calculation methods, and professional values of civil servants have deeply influenced the design of the instrument, leading to the differentiation in the allocation of allowances. By rehabilitating their ability to calculate project costs, civil servants have recovered a role that was contested by the New Economy of Law [START_REF] Stigler | The theory of economic regulation[END_REF], considering that the administration faces information asymmetry and strategic capture, and only the market mechanism can allocate resources efficiently. The ability of the public authority to compensate for distributive effects makes the difference between "positive allocation", such as CDM credits, TGC and EEC, where allowances are allocated to economic actors who have no obligation, and "negative" allocation, based on an initial allocation to polluters, such as the EU ETS quota [START_REF] Chiapello | The fabrication of environmental intangibles as a questionable response to environmental problems[END_REF]. In the case of positive allocation, the entire positive contribution is converted into certificates (or credits) that are remunerated by the exchange mechanism. The difference between the project cost and the market price leads to a significant inframarginal rent for low-cost projects [START_REF] Blok | Clash of the eco-sciences: Carbon marketization, environmental NGOs and performativity as politics[END_REF]). In the case of negative allocation, the initial allocation of allowances can become a strategy to reduce windfall profits [START_REF] Wettestad | Smokescreen politics? Ratcheting up EU emissions trading in 2017[END_REF]. For example, European authorities have introduced discrimination between electricity production and industrial installations, as well as between industrial sectors [START_REF] Ehrenstein | Economic under-determination: Industrial competitiveness and free allowances in the European carbon market[END_REF]. This possibility to intervene in the parameterization of the instrument probably reduces the motivation to violate the principle of equivalence between emission reductions and quotas. It could explain why only the French EEC and the UK TGC were affected by the practices of project discrimination and adjustment coefficients, and not the EU ETS.

Finally, decoupling is facilitated by the lack of control over compliance with the original principles of the instrument. The initial formalization left implicit many of the rules and principles that justified its existence and guaranteed its effectiveness, such as the rules for allocating certificates according to performance in terms of energy savings. The decision to entrust the detailed design to the Cabinet of the Ministry and the DGEC, which is directly answerable to the government, and not to an independent authority, has allowed the implementation of an alternative definition of efficiency. Thus, officials of the Ministry of Finance tried to denounce the hypocrisy of qualifying the EEC as a market mechanism in order to requalify it as a tax and subsidy mechanism in order to obtain the right to control it. But they did not defend the original principles of the mechanism.

Taking into account the organizational and institutional context in which a public policy instrument is designed and implemented helps to explain the dynamics that promote the decoupling of theory and practice. Indeed, this organizational context helps to explain how actors appropriate different complementary or contradictory theories, develop their autonomy of action, and translate theories more or less faithfully to the situation at hand. It explains how a definition of efficiency justifying a market instrument (allocation to the more efficient project in a context of ignorance of costs) could be abandoned by French officials involved in the design and implementation of the EEC, in favor of another definition of efficiency (adaptation of subsidies to costs). The concept of the instrument of public action makes it possible to take into account both the initial influence of the theory and its social roots and permanence, even if it departs from the initial theory that gave rise to it. The instrument is both a resource and a constraint for action, but it could also be malleable to integrate conflicting expectations.

  certificates to energy suppliers to help them meet their obligations. This possibility of exchange could create a market. The justification for the exchange of certificates comes from a favorable institutional context in Europe for market-based instruments for environmental concerns. The Energy Law published in 2005, which defines the EEC mechanism, clearly refers to the EU Emissions Trading Scheme and the economic arguments defending market-based instruments in terms of efficient allocation 4 . The exchange of certificates was also defended by the members of demand. It relies on a small team of permanent staff and brings together more than 2,200 members who participate voluntarily in the production of technical expertise. the ATEE, who advocated the development of energy services to compete with the dominant historical suppliers. ADEME was also convinced by the positive feedback from the previous implementation of a market-based instrument for the same purpose in the United Kingdom, launched on April 1, 2002.

I will use this expression as it is the most usual in the English language. The French name is "certificat d'économie d'énergie", or energy saving certificates.

ATEE was created in 1978 in the context of the oil crisis and the beginning of an energy policy. Financed by major energy producers, consumers and public authorities, its mission is to promote technical solutions to control energy

In 2008, the issue of energy was moved from DGEMP to a new Direction for Energy and Climate (Direction Générale de l'Energie et du Climat, DGEC) attached to the Ministry of the Environment.

Law n° 2005-781 of 13 July 2005 of the program setting the energy policy guidelines

The energy savings achieved in each year following the first year are discounted by dividing the previous year's savings by 1.04, which results in a discount rate of 4%. The discount rate assures the commensurability between energy saving over the years.

Since its creation in 2006, the system has gone through four periods, with increasingly ambitious objectives, from 54 TWh cumac for the first period to 2133 TWh cumac for the last one.

The French tax credit is a reduction in personal income tax and takes the form of a subsidy for households that pay no tax.

CGEDD, CGEiet, IGF, Les certificats d'économies d'énergie : efficacité énergétique et analyse économique (Energy-saving certificates : energy efficiency and economic analysis). 2014

CGEDD, CGEiet, IGF, idem.

ADEME, Evaluation du dispositif des certificats d'économie d'énergie -Rapport synthétique. (Evaluation of the energy-saving certificate device -Summary report) 2019


Carlos Granero-Belinchón
email: carlos.granero-belinchon@imt-atlantique.fr

Neural network based generation of a 1-dimensional stochastic field with turbulent velocity statistics

We define and study a fully-convolutional neural network stochastic model, NN-Turb, which generates a 1-dimensional field with some turbulent velocity statistics. In particular, the generated process satisfies the Kolmogorov 2/3 law for the second order structure function. It also presents negative skewness across scales (i.e. Kolmogorov 4/5 law) and exhibits intermittency as characterized by skewness and flatness. Furthermore, our model is never in contact with turbulent data and only needs the desired statistical behavior of the structure functions across scales for training.

I. INTRODUCTION

Turbulence is characterized by non-linear, multiscale and non-local interactions [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]. Moreover, it presents long-range dependencies and intermittency [START_REF] Frisch | On the singularity structure of fully developed turbulence[END_REF] making it a very interesting subject of study in the field of complex systems [START_REF] She | Towards a complex system approach for the study of turbulence[END_REF] and multifractals [START_REF] Dubrulle | A correspondence between the multifractal model of turbulence and the Navier-Stokes equations[END_REF].

The generation of stochastic fields sharing the statistical behavior of turbulence has been matter of study during the last century. Thus, several stochastic models [START_REF] Mandelbrot | Fractional brownian motions fractional noises and applications[END_REF][START_REF] Flandrin | On the spectrum of fractional Brownian motions[END_REF][START_REF] Nawroth | Multiscale reconstruction of time series[END_REF][START_REF] Gontier | A comparison of fatigue loads of wind turbine resulting from a non-Gaussian turbulence model vs. standard ones[END_REF][START_REF] Robert | Hydrodynamic turbulence and inter-mittent random fields[END_REF][START_REF] Chevillard | A stochastic representation of the local structure of turbulence[END_REF][START_REF] Chevillard | Regularized fractional Ornstein-Uhlenbeck processes and their relevance to the modeling of fluid turbulence[END_REF][START_REF] Peinke | The FokkerPlanck approach to complex spatiotemporal disordered systems[END_REF] have been proposed to recover the very known energy distribution and energy cascade, in the sense of the Kolmogorov's 2/3 and 4/5 laws respectively [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF], as well as intermittency [START_REF] Kolmogorov | A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number[END_REF][START_REF] Obukhov | Some specific features of atmospheric turbulence[END_REF]. From the first fractional Brownian motion [START_REF] Mandelbrot | Fractional brownian motions fractional noises and applications[END_REF][START_REF] Flandrin | On the spectrum of fractional Brownian motions[END_REF], only recovering the energy distribution, the modelling of turbulence evolved towards more complex fields introducing also intermittency [START_REF] Bacry | Multifractal random walk[END_REF] or both intermittency and energy cascade [START_REF] Chevillard | A stochastic representation of the local structure of turbulence[END_REF][START_REF] Chevillard | On a skewed and multifractal unidimensional random field, as a probabilistic representation of Kolmogorov's views on turbulence[END_REF]. However, the generation of stochastic fields matching the statistical properties of turbulence is still challenging since while modelling the energy distribution only requires second order statistics, modelling the energy cascade and intermittency requires high order ones. Most of the models of the state of the art only focus on second order statistics and so do not exhibit intermittency nor do they respect Kolmogorov's 4/5 law [START_REF] Peinke | On chaos, fractals and turbulence[END_REF][START_REF] Mann | Wind field simulation[END_REF][START_REF] Klein | A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations[END_REF][START_REF] Hoepffner | Realizing turbulent statistics[END_REF].

In the last decades, neural network (NN) models evidenced their potential to deal with non-linear complex problems [START_REF] Lguensat | Learning generalized quasi-geostrophic models using deep neural numerical models[END_REF][START_REF] Ouala | Learning latent dynamics for partially observed chaotic systems[END_REF][START_REF] Di Carlo | Post processing sparse and instantaneous 2D velocity fields using physics-informed neural networks[END_REF]. Specifically, generative NN models have been recently developed [START_REF] Ruthotto | An introduction to deep generative modeling[END_REF]. These models aim to learn the statistical distribution of some data to then create new data matching the underlying distribution [START_REF] Goodfellow | Generative Adversarial Nets[END_REF][START_REF] Roy | Using generative adversarial networks (GAN) to simulate central-place foraging trajectories[END_REF]. Some NN models of turbulence already appeared in the last years [START_REF] Geneva | Multi-fidelity generative deep learning turbulent flows[END_REF][START_REF] Kim | Deep unsupervised learning of turbulence for inflow generation at various Reynolds numbers[END_REF][START_REF] Buzzicotti | Reconstruction of turbulent data with deep generative models for semantic inpainting from TURB-Rot database[END_REF][START_REF] Drygala | Generative modeling of turbulence[END_REF]. However, these NN models of turbulence use to present training and validation approaches that only focus on second order statistics [START_REF] Kim | Deep unsupervised learning of turbulence for inflow generation at various Reynolds numbers[END_REF][START_REF] Liu | Deep learning methods for super-resolution reconstruction of turbulent flows[END_REF][START_REF] Yousif | High-fidelity reconstruction of turbulent flow from spatially limited data using enhanced super-resolution generative adversarial network[END_REF][START_REF] Yousif | Super-resolution reconstruction of turbulent flow fields at various Reynolds numbers based on generative adversarial networks[END_REF] do not modelling then the energy cascade nor intermittency. In addition, all these works learn from experimental or numerical turbulent data, sometimes helped by additional physic information [START_REF] Kim | Deep fluids: Agenerative network for parameterized fluid simulations[END_REF][START_REF] Wang | Towards physics-informed deep learning forturbulent flow prediction[END_REF]. This introduces a strong dependence on databases that are not always easily available. However, a NN model is just a non-linear function Ψ θ completely defined by the weights θ of its neurons, and so, we can formulate an optimization problem of Ψ θ with respect to a given criterion [START_REF] Villarrubia | Artificial neural networks used in optimization problems[END_REF][START_REF] Abdolrasol | Artificial neural networks based optimization techniques: a review[END_REF]. From this viewpoint, we don't need to feed our model with data.

We propose to avoid learning from data and directly impose the multiscale statistical behavior described by Kolmogorov and Obukhov theories. The proposed approach is a multiscale generalization of the Generative Moment Matching Networks from Li et al. [START_REF] Li | Generative Moment Matching Networks[END_REF] and grounds on the Kolmogorov multiscale descriptions of second, third and fourth order structure functions of turbulent velocity across scales [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF][START_REF] Kolmogorov | A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number[END_REF][START_REF] Obukhov | Some specific features of atmospheric turbulence[END_REF][START_REF] Chevillard | A phenomenological theory of eulerian and lagrangian velocity fluctuations in turbulent flows[END_REF]. We focus on these three functions since they are representative of the energy distribution, the energy cascade and intermittency respectively [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF]. Since our model do not impose the behavior across scales of higher order structure functions, the full probability density function (pdf) across scales of the generated stochastic field will be different from the turbulent velocity one. Only second, third, and fourth order structure functions will be recovered. So, our model takes a Gaussian white noise as input and only needs the desired evolution across scales of these three structure functions of turbulent velocity for learning. These functions can be defined from experimental or numerical data, but also from empirical or theoretical models.

The main originality of this work is the proposed approach, which directly imposes the evolution across scales of the structure functions of the stochastic field and presents several advantages compared to the current state of the art: it is not limited to second order statistics, it is easily generalizable to consider higher and higher order statistics and it does not need data for training nor synthesis. The main disadvantage of this approach is the current lack of explainability of Neural Networks, even if progresses are being done in the field [START_REF] Subel | Explaining the physics of transfer learning in datadriventurbulence modeling[END_REF].

In section II, we describe the multiscale statistical behavior of turbulent velocity as illustrated by the Kolmogorov-Obukhov theories. Then, section III presents our NN model, that we named NN-Turb, as well as the optimization approach used to train it. Finally, in sections IV and V we show the statistical multiscale behavior of the NN-Turb stochastic field, we compare it with experimental turbulent velocity and with a regularized Multifractal Random Walk (rMRW) [START_REF] Robert | Hydrodynamic turbulence and inter-mittent random fields[END_REF], and we give some conclusions and perspectives.

II. ISOTROPIC AND HOMOGENEOUS FULLY DEVELOPPED TURBULENCE

The Kolmogorov 1941 (K41) statistical multiscale theory of turbulence prescribes the existence of three domains of scales with different statistical behaviors: integral, inertial and dissipative domains, where the energy is respectively injected in the flow, transferred across scales and dissipated [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF][START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF]. The integral scale L separates the integral and inertial domains, while the Kolmogorov scale η divides the inertial and dissipative ones.

The Kolmogorov 1941 theory enuntiates two statistical relationships for the longitudinal turbulent velocity in the inertial domain of scales: the 2/3 and the 4/5 laws, that illustrate respectively the energy distribution and cascade across scales [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF][START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF]:

δ l v(x) = v(x + l) -v(x) (1)
S p (l) = (δ l v(x)) p (2)
S 2 (l) ∝ l 2/3 (3) S 3 (l) ∝ - 4 5 l (4)
where x is the spatial dimension, v(x) is the longitudinal turbulent velocity, δ l v(x) is the velocity increment of size l and S p (l) is the p-th order structure function. Thus, in the inertial domain the Kolmogorov 2/3 law implies that S 2 (l) ∝ l 2/3 , and the Kolmogorov 4/5 law imposes S 3 (l) ∝ -l. The 4/5 law was directly derived by Kolmogorov from the Navier-Stokes equations and must be respected by any model of turbulence [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF]. Moreover, the Kolmogorov and Obukhov 1962 theory (KO62) corrected K41 by introducing intermittency: the energy dissipation rate is inhomogeneous and should be considered locally [START_REF] Kolmogorov | A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number[END_REF][START_REF] Obukhov | Some specific features of atmospheric turbulence[END_REF]. Moreover, this correction leads to the emergence of extreme values of the velocity increments both in the inertial and dissipative domains, and so, to a deformation of the pdf of the velocity increments when the scale decreases, from Gaussian in the integral domain to strongly non-Gaussian in the dissipative one [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF][START_REF] Frisch | On the singularity structure of fully developed turbulence[END_REF].

For practical purposes we define the skewness and flatness as:

S(l) = S 3 (l) S 2 (l) 3/2
(5)

F(l) = S 4 (l) S 2 (l) 2 . ( 6
)
In these measures the dominant effects of the energy distribution across scales (S 2 (l)) are compensated and so they allow us to finely study high-order statistics. On the one hand, the skewness characterizes the degree of asymmetry of the distributions of the velocity increments and from the Kolmogorov 4/5 law it is a signature of the energy cascade. On the other hand, the Flatness describes the significance of the tails of the distribution of the increments and its evolution across scales characterizes intermittency.

In this work, we will focus on four main points of the Kolmogorov-Obukhov theories that we will impose to the process generated by our NN-Turb model: a) turbulence presents three domain of scales with different statistical behaviors, b) in the inertial domain S 2 (l) matches the 2/3 Kolmogorov law, c) in the dissipative and inertial domaines the skewness should be negative (4/5 Kolmogorov law) and d) in the integral domain the flow statistics are close to Gaussian and become non-Gaussian at small scales, i.e. the flatness of the velocity increments increases when the scale decreases. Thus, our NN-Turb model will generate intermittent processes with the desired energy distribution and cascade as prescribed by Kolmogorov and Obukhov theories. Note that all along the paper and following the Kolmogorov 1941 theory, the 2/3 and 4/5 laws of Kolmogorov are respectively identified as signatures of the distribution and cascade of energy across scales [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF][START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF]. Note also that the structure functions of order higher than four are not imposed by the model and so, the generated process will not reproduce the full pdf of turbulent velocity increments. It focuses on second, third and fourth order statistics, contrary to most of the existing literature only focusing on second order ones [6, 18-21, 29, 32-34] or second and fourth order ones [START_REF] Bacry | Multifractal random walk[END_REF][START_REF] Friedrich | Superstatistical wind fields from pointwise atmospheric turbulence measurements[END_REF].

III. TURBULENT VELOCITY NEURAL NETWORK BASED GENERATION

This section describes the proposed deep learning approach, referred to as NN-Turb, for modelling a 1-dimensional stochastic field with some turbulent velocity statistics. We introduce our neural network model and the optimization setup.

A. NN-Turb model

We propose a fully convolutional model for the generation of a 1-dimensional field with some turbulent velocity statistics:

δ ls u(x) = Ψ(w(x)) (7)
where l s is the sampling distance of the generated fields, i.e. the smallest available scale of analysis. Ψ is our model which takes as input a Gaussian white noise w(x) with zero mean and unit variance, and produces the field corresponding to the velocity increment δ ls u(x). Finally, the generated turbulent velocity field u(x) is defined as the cumulative sum of δ ls u(x):

u(x) = x -∞ δ ls u(y)dy (8)
So, our NN-Turb model performs a double operation on the input noise w(x). On the one hand it deformates the Gaussian pdf of w(x) to a pdf in agreement with the turbulent velocity statistics at the sampling scale l s . On the other hand, our model introduces a structure of dependencies (multi-point correlations) in the initial white noise used as input. This approach implies the existence of a small scale process δ ls u(x) with a given distribution and a given structure of dependencies whose cumulative sum statistically behaves as turbulent velocity up to fourth order statistics. Physically the scales in the dissipative domain take information from larger scales and this information is encoded in the structure of dependencies and pdf of δ ls u(x) [START_REF] Yakhot | Probability densities in strong turbulence[END_REF]. Moreover, directly generating a small scale increment in the dissipative domain and constructing the full process u(x) by eq.( 8) facilitates the continuous deformation of the pdf of the increments across scales from the dissipative to the integral domain.

The operator Ψ follows a U-net architecture performing a multi-resolution processing of the input Gaussian white noise based on convolutional kernels of different sizes [START_REF] Ronneberger | U-Net: convolutional networks for biomedical image segmentation[END_REF]. This generative approach consists of a non-linear filtering of the input noise, thus generalizing the linear filtering version of [START_REF] Klein | A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations[END_REF][START_REF] Hoepffner | Realizing turbulent statistics[END_REF]. See A for more details on the NN-Turb architecture.

B. Optimization setup

From eq.( 7) the only input of our model is a Gaussian white noise w(x). In addition, during training, we need to introduce the desired statistics across scales of the generated process u(x). Indeed, we impose the evolution across scales of S 2 (l), S(l) and F(l), see section II.

In this work, the reference curves of the second order structure function, skewness and flatness, that we note S r 2 (l), S r (l) and F r (l), are obtained from statistical measures on the Modane wind tunnel dataset [START_REF] Kahalerras | Intermittency and Reynolds number[END_REF]. It consists on Eulerian longitudinal velocity measurements v(x) obtained from a grid turbulence setup. The sampling frequency of the setup was f s = 25 kHz, the mean velocity of the flow is v = 20.5 m/s, and the Taylor-scale based Reynolds number of the flow is R λ = 2500. Thus the flow is considered as exhibiting fully developed turbulence. For this dataset, we use the Taylor frozen turbulence hypothesis [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF] in order to interpret temporal variations as spatial ones. Thus, the sampling distance can be expressed as l s = v /f s . Detailed multiscale statistical analyses of Modane turbulent velocity signals have been previously provided in [START_REF] Chevillard | A phenomenological theory of eulerian and lagrangian velocity fluctuations in turbulent flows[END_REF][START_REF] Kahalerras | Intermittency and Reynolds number[END_REF][START_REF] Gagne | New trends in nonlinear dynamics and pattern-forming phenomena[END_REF][START_REF] Arneodo | Revealing a lognormal cascading process in turbulent velocity statistics with wavelet analysis[END_REF]. Furthermore, from previous studies the integral and Kolmogorov scales for this flow are respectively L = 2350 l s and η = 5 l s [START_REF] Granero-Belinchon | Scaling of information in turbulence[END_REF]. Figures 2 a),b) and c) provide respectively the evolution across scales of log(S r

2 ), S r and log(F r /3) in blue. Here and all along the paper, the natural logarithm is used and noted log.

We want to point out that even if in this work the reference curves are obtained from statistical measures on real data, the used learning approach allows us to use empirical or theoretical laws to define them. Our model never sees turbulent data neither during training nor during fields generation.

We consider the optimization of our NN-Turb model according to the following four losses:

1. the mean squared error between the reference log(S r 2 (l)), S r (l) and F r (l), and log(S 2 (l)), S(l) and F(l) of the generated field u(x).

L S2 = E l log(S r 2 (l)) -log(S 2 (l)) 2 (9)
L S = E l S r (l) -S(l) 2 (10)
L F = E l F r (l) -F(l) 2 (11)
where E l is the expected value operator. The logarithm used only in eq.( 9) allows to homogenize across scales the magnitude of the standard deviation of S 2 and, more importantly, to highlight the differences across scales of S 2 for scales in the dissipative and inertial ranges. This is an advantage for training the network. The logarithm is not used on the skewness because this statistic can take negative values, and it is not used on the flatness because the logarithm produces the inverse effect on this statistic: it reduces the differences of F across scales for scales in the dissipative and inertial ranges.

a regularization loss L

C = 1 -r(u (x), w(x))
where r is the cross-correlation function between u (x), which is the centered and standardized version of u(x), and w(x) which is the Gaussian noise used as input. This loss aims at increasing the correlation between these two time-series.

The optimization criterion L is a weighted sum of these losses:

L = α • (L S2 + L S + L F ) + β • L C (12)
We set empirically the weights of the three first losses to α = α S2 = α S = α F = 1 and the weight of the regularization loss to β = 0.1. These weights were chosen by grid searching with a grid step of 0.1 for β ∈ [0, 1) and with α S ∈ {0.5, 1}, α F ∈ {0.5, 1} and fixed α S2 = 1. We decided to impose α S2 ≥ α S and α S2 ≥ α F since S 2 directly intervenes in the definition of S and F. We also imposed α S > β and α F > β. Changes in α S and α F impact very slightly the results, on the other hand the value of β was important for converging to an adequate field. Consequently, our optimization approach completely grounds on the Kolmogorov theories as well as on previous descriptions of turbulent velocity fields. The three mean squared error losses impose the desired 2/3 and 4/5 laws of Kolmogorov, i.e. the distribution and cascade of energy across scales, as well as intermittency. The regularization loss is used to impose stationarity at large scales as desired for turbulent velocity.

Using Pytorch, our learning setup relies on Adam optimizer with a learning rate of 2e-3 for the first 100 epochs, 1e-3 for epochs between 100 and 1000 and 5e-4 for remaining epochs up to epoch 2000. The open source code is available at https://github.com/cgranerob/NN-Turb.

IV. RESULTS AND DISCUSSION

In this section, we study the process u(x) generated by our NN-Turb model Ψ. For this purpose, we generate 256 realizations of u(x), each one of size N = 2 15 samples. To avoid border effects due to convolutions, we first generate realizations of size N + N b samples, where N b is the number of samples impacted by border effects. In our case of study N b = 2 13 . Then, we only consider the N samples far from the borders to define u(x). We analyse the second order structure function, skewness and flatness of u(x) across scales for scales going from the dissipative domain to the integral one. We perform the same analysis on Modane experimental turbulent velocity for comparison. To compare NN-Turb to current state of the art models, we also study a regularized Multifractal Random Walk, z(x), [START_REF] Robert | Hydrodynamic turbulence and inter-mittent random fields[END_REF] parameterized to simulate Modane experimental turbulent velocity: the small and large scales of regularization are = 1 and R = e 8 respectively, and the long-range dependence and intermittency parameters are respectively c 1 = 1/3 and c 2 = 0.025. The used large scale regularization function is Gaussian. A single realization of size 2 23 was generated and sliced to provide 256 realizations of size N .

Figure 1 illustrates three realizations of u(x) among the 256 generated. We observe dynamics at very different scales, from very small scales of the order of the sampling distance l s to scales of the order of the integral scale L of the process (to facilitate visualization a red box of width L is shown).

Figure 2 a), b) and c) show respectively log(S 2 (l)), S(l) and log(F(l)/3) of the NN-Turb (black), Modane (blue) and rMRW (green) fields in function of log(l/L). The vertical dashed black lines indicate the integral and Kolmogorov scales as obtained for Modane turbulent data in previous studies [START_REF] Granero-Belinchon | Scaling of information in turbulence[END_REF]. Thus, we observe different behaviors of the studied statistics depending on the domain of scales.

In figure 2 a) we observe, independently of the process, a plateau of log(S 2 (l)) for scales l larger than the integral scale L as expected for turbulent velocity. These are the more energetic scales. Then, when the scale decreases through the inertial region, log(S 2 (l)) also decreases following the 2/3 law of Kolmogorov. In the dissipative domain, log(S 2 (l)) decreases faster with the scale than in the inertial one, and its slope is close to 2 as described by the Batchelor model [START_REF] Batchelor | Pressure fluctuations in isotropic turbulence[END_REF]. This behavior in the dissipative domain is a signature of the smoothness of the field at these scales. While the NN-Turb field perfectly matches the Modane behavior within the errorbars for any scale of analysis, the rMRW field matches the Modane behavior in the dissipative and inertial ranges but the impact of the large scales is more localized and does not propagate through the inertial range. This leads, in the case of the rMRW process, to a linear behavior with a slope of 2/3 remaining at larger scales compared to Modane.

Figure 2 b) illustrates that the generated field u(x) is negative skewed as requested by the Kolmogorov 4/5 law. Moreover, due to intermittency effects in the dissipative domain, the skewness decreases whith the scale for scales l < η. However, this decrease is much more steep than the one from Modane experiment. On the other hand, the increments of the rMRW field present, by construction, null skewness indicating that their pdfs are symmetrical.

In figure 2 c), independently of the process, log(F(l)/3) goes from zero in the integral domain of scales to larger values when the scale decreases. This is a signature of intermittency in both the inertial and dissipative domains [START_REF] Kolmogorov | A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number[END_REF][START_REF] Chevillard | A phenomenological theory of eulerian and lagrangian velocity fluctuations in turbulent flows[END_REF][START_REF] Anselmet | High-order velocity structure functions in turbulent shear flows[END_REF]. Moreover, the flatness of Modane and NN-Turb shows a linear behavior in the inertial domain with slope -0.1 (red dashed line). This behavior is representative of homogeneous and isotropic turbulence [START_REF] Chevillard | A phenomenological theory of eulerian and lagrangian velocity fluctuations in turbulent flows[END_REF]. Furthermore, for these two processes, the intermittency is stronger in the dissipative domain and so the increase of flatness when the scale decreases becomes steeper [START_REF] Chevillard | On the rapid increase of intermittency in the near-dissipationrange of fully developed turbulence[END_REF]. The flatness of the rMRW process behaves otherwise and does not capture the steeper increase in the disipative domain. The slope -0.1 in the inertial range is reproduced but in a smaller domain of scales.

Our stochastic field recovers the correct behavior of S 2 (l) in all the domains of scales. In addition, it recovers the good behavior of skewness and flatness in the integral and inertial domains. However, in the dissipative domain the decrease of skewness is to steep and the increase of flatness is not enough compared to Modane statistics. This illustrates the complexity of correctly recovering high-order statistics in this region.

Finally, figure 3 shows the logarithm of the pdf of the standardized increments of a) the Modane turbulent velocity signal (δ l v(x))/σ δ l v ), b) the NN-Turb field (δ l u(x))/σ δ l u ) and c) the rMRW field (δ l z(x))/σ δ l z ), for different scales l. Scales from the dissipative domain to the integral one are considered. For Modane and NN-Turb processes we observe an evolution from non-Gaussian pdfs at small scales: long-tailed and asymmetric, to close to Gaussian when approaching the integral domain (a Gaussian pdf with zero mean and unit variance is illustrated in red dashed line). So, the generated field u(x) presents intermittency and becomes Gaussian at large scales. However, the evolution of the pdf of the increments of u(x) is not exactly the one expected for a turbulent field [START_REF] Chevillard | A phenomenological theory of eulerian and lagrangian velocity fluctuations in turbulent flows[END_REF]: the extreme values at small scales are only partially recovered while the asymmetry of the distribution of the increments at these scales is overvalued. On the other hand, the rMRW process do not present asymmetries in the pdfs of its increments but better recover the broad shape of the pdfs and both the positive and negative extreme events.

V. CONCLUSIONS

We propose a fully-convolutional NN model, NN-Turb, to generate a 1-dimensional field with some turbulent velocity statistics. Our stochastic model takes as input a Gaussian white noise and perform a double operation on it: 1) it introduces the desired structure of dependencies and 2) it deformates the Gaussian pdf of the input to a long-tailed and skewed one. Very importantly, our model only needs the aimed evolution across scales of the second order structure function, skewness and flatness for learning, and so, it does not require turbulent data.

The generated 1-dimensional field u(x) correctly recovers the 2/3 and 4/5 laws of Kolmogorov as well as the flatness behavior in the inertial domain described in [START_REF] Chevillard | A phenomenological theory of eulerian and lagrangian velocity fluctuations in turbulent flows[END_REF][START_REF] Chevillard | On the rapid increase of intermittency in the near-dissipationrange of fully developed turbulence[END_REF]. From this perspective, u(x) models adequately the energy distribution, energy cascade and intermittency of turbulence while remaining close to Gaussian at large scales. However, we also illustrated that the pdfs of the increments of the generated fields do not necessarily match the expected behavior of turbulent ones. So, our stochastic field can reproduce the statistical behavior of the second, third and fourth order structure functions of turbulent velocity without matching the exact pdf deformation across scales.

Three main future perspectives are considered. First, the application of the proposed learning approach for generating 2D images of homogeneous and isotropic turbulent velocity. Second, the empirical definition of the structure functions S p (l) according to a limited number of parameters c m . This will allow us to completely avoid the use of experimental data. More interestingly this will allow us to introduce these parameters c m as inputs of our model, and so, to generate different types of processes with diverse multifractal properties. Thus, we aim to generalize this NN optimization approach, which does not need data, to other kind of non-linear physical systems. Finally, the definition of a learning setup in which we don't impose the evolution across scales of some structure functions but the evolution of the pdfs of the increments directly.

FIG. 1 .

 1 FIG. 1. Illustration three realizations of process u(x) generated with NN-Turb in function of the spatial variable x/L. The red box corresponds to the length of an integral scale L.

FIG. 2 FIG. 3 .

 23 FIG.2. a) Logarithm of the second order structure function log(S2(l)), b) skewness S(l) and c) logarithm of the flatness log(F(l)/3) in function of the logarithm of the scale of analysis log(l/L) for the NN-Turb generated field (black), Modane (blue) and rMRW (green). Curves represent the mean value and errorbars the standard deviations calculated on 256 realizations. Red dashed lines in a) have a slope 2 in the dissipative domain and 2/3 in the inertial one describing respectively the behaviors of the Batchelor model[START_REF] Batchelor | Pressure fluctuations in isotropic turbulence[END_REF] and the 2/3 Kolmogorov law. Red dashed line in c) has a slope -0.1 previously described for the log(F(l)/3) in the inertial domain[START_REF] Chevillard | A phenomenological theory of eulerian and lagrangian velocity fluctuations in turbulent flows[END_REF]. The vertical black dashed lines correspond to the Kolmogorov and integral scales, η and L of Modane.

ACKNOWLEDGMENT

The author wish to thank Stéphane G. Roux for providing the code for generating rMRW processes. This work was supported by the French National Research Agency (ANR-21-CE46-0011-01), within the program Appel à projets générique 2021.

Appendix A: NN-Turb architecture Figure 4 illustrates the architecture of NN-Turb. It is a fully convolutional U-net architecture grounding on multiscale decomposition to modify the Gaussian white noise used as input. Thus, it is composed of an encoder, and a decoder that are connected by a convolutional bridge. Because of the symmetry of the architecture, both the Gaussian white noise used as input and the output field have the same size, i.e. the same number N of samples.

The encoder blocks are the combination of 1d convolution layer with batch normalization, non-linear ReLU activation function and average pooling. The decoder blocks are the combination of 1d convolution transpose layer with batch normalization, non-linear ReLU activation and upsampling layer. A bridge with a 1d convolution layer with batch normalization and non-linear ReLU activation followed by a 1d convolution transpose layer with batch normalization and non-linear ReLU activation is used to connect the encoder and the decoder of the U-net. The number of channels evolves as follows: 1 → 16 → 32 → 64 → 128 → 256 → 128 → 64 → 32 → 16 → 1) and the kernel sizes are [START_REF] Frisch | Turbulence: the legacy of A.N. Kolmogorov[END_REF][START_REF] Frisch | On the singularity structure of fully developed turbulence[END_REF][START_REF] Dubrulle | A correspondence between the multifractal model of turbulence and the Navier-Stokes equations[END_REF][START_REF] Gontier | A comparison of fatigue loads of wind turbine resulting from a non-Gaussian turbulence model vs. standard ones[END_REF][START_REF] Bacry | Multifractal random walk[END_REF][START_REF] Liu | Deep learning methods for super-resolution reconstruction of turbulent flows[END_REF]64]. Furthermore, we introduced additive long-skip connections between the encoder and the decoder layers [START_REF] Ronneberger | U-Net: convolutional networks for biomedical image segmentation[END_REF].

We want to point out that in both the encoder and the decoder the kernel sizes of the different layers increase exponentially in order to rapidly and completely sample the dissipative, inertial and integral domains of turbulence. This is specially important to recover the expected multi-scale behavior of each domain. Furthermore, the average pooling and upscaling layers also facilitate to process the whole domain of scales of interest without increasing dramatically the computational cost of learning.

The additive long-skip connections have been introduced to facilitate the learning [START_REF] Ronneberger | U-Net: convolutional networks for biomedical image segmentation[END_REF]. Moreover, the longest skip connection, the one connecting the second and next-to-last layers, appeared as crutial to impose Gaussianity at large scales.

Furthermore, we decided to generate the smallest available velocity increment δ ls u(x) instead of directly the turbulent velocity u(x) since the long-range dependencies of the velocity could complicate the learning process. The range of dependencies of the velocity increments are shorter.