Luis Roman

Arciniega Gil

_________________________________________________________________________

A glimpse into smart cities: opportunities for the development of energy cooperatives for citizens and businesses in Mexico

Keywords: smart cities, energy cooperatives, bottom-up initiatives, local development, sustainable development goals

Smart cities are a local development model that has found success in recent years both in the scientific literature and in local policies. Although traditionally related to the use of technologies, clean energies are a key dimension of the concept and are promoted for the development of sustainable cities and communities. Mexico has a high potential for renewable energy generation, particularly solar energy, which represents an opportunity for national development and energy security. However, Mexico's history and economic performance, strongly linked to the energy sector, make private initiative in this area highly regulated by the state. This chapter explores the opportunities offered by locally determined energy projects combined with the cooperative model as a way to reconcile interests within the Mexican energy sector. In addition, it presents how they can contribute to addressing the country's energy poverty, as well as reducing carbon emissions when promoted among industries and companies.

In this way, energy cooperatives can promote the development of a democratic Mexican energy system, while diversifying the energy mix and benefiting the public, private and social sectors.

Introduction

In the last 20 years, the concept of the smart city has taken prominence both in the scientific literature and local policies, making different authors and institutions set out specific definitions and elements in order to better understand what this implies (Albino et al., 2015, pp. 3-21). Although there is no universal definition of the smart city, three documents at the international level serve as references and place the use of technologies at the heart of the concept. These include the following: the 2030 Agenda for Sustainable Development and the New Urban Agenda Habitat III (both documents from the United Nations), as well as the ISO 37122:2019 standards establishing definitions and methodologies for a set of indicators for smart cities. In Mexico, even though there is no normative text that homogeneously defines smart cities, the National Commission for the Efficient Use of Energy (CONUEE, 2017) has considered them to be a city model that connects its inhabitants through infrastructure and the efficient use of technology.

Smart cities are indeed a trend-setting model of local development, and while the use of technology is the main element of the concept, their wide scope allows that any local project applying innovative technologies can call itself a "smart city". However, critical literature has expressed concern about the growing role of the technological private sector in the development of smart city projects, as the marketing objectives and return on the capital of enterprises may indeed conflict with the general interest and efficient management of public resources and services, originally conceived to serve society unselfishly (Grossi & Pianezzi, 2017, pp. 79-85). In response, some authors propose developing projects with a bottom-up dimension, which implies first identifying local needs and problems, and then determining projects based on locally available resources (Niaros, 2016, pp. 51-61).

A literature review shows that clean energies are a key dimension of the smart city concept and are considered to be a viable solution for achieving integral sustainable development.

Therefore, although traditionally smart cities are related to the use of technologies, this paper focuses on the manner in which locally determined energy projects can be seen as a smart solution for the development of sustainable territories. Mexico has a high potential for the development of renewable energies including solar, wind, bioenergy, hydropower and geothermal energy (Elizondo et al., 2017, p. 15). Mexico's solar energy potential is particularly relevant as it has one of the top five solar irradiation levels in the world, with a daily average of 5.5 kWh/m2 and a potential to cover up to 45% of demand [START_REF] Bello | En energía, México debe apuntar al sol[END_REF]. The implementation of smart city projects, with regard to the production of renewable energies, represents an opportunity for energy transition in Mexico (IRENA, 2015) and could help to meet Sustainable Development Goal (SDG) 7 on the access to affordable, reliable, sustainable and modern energy for all, as well as SDG 11 on the creation of sustainable cities and communities.

Moreover, it is considered that the cooperative model can contribute to the sustainability of local energy projects, as it is a form of social organization motivated by the common interests of its members. In Mexico, they are defined as a "form of organisation made up of natural persons based on common interests and the principles of solidarity, self-help and mutual assistance, with the purpose of satisfying individual and collective needs through the performance of economic activities of production, distribution and consumption of goods and services" (General Law on Cooperative Societies, Article 2). In such considerations, cooperatives are seen, on the one hand, as entities that favour the democratic management of energy production and consumption according to the community's energy needs; and on the other hand, as entities that can contribute to bringing energy to marginal areas where investment may be costly for governments or unattractive for businesses (COPAC, n.d.). This chapter focuses on how locally determined energy projects, combined with the cooperative model, can be seen as a smart solution from a bottom-up perspective for Mexico's national development and energy security. It starts by briefly setting Mexico's energy industry context and the country's commitments on climate change; then, it addresses the issue of energy poverty in Mexico and the challenges for the industry and business sector as leaders of energy consumption in the country; and finally, it states a conclusion that comprises recommendations and suggestions for future research, in light of Mexico's climate commitments and the new government's energy strategy (2018)(2019)(2020)(2021)(2022)(2023)(2024).

Context of the Mexican energy industry

It is a matter of historical fact that in the 1930s, after a long labor dispute of "El Aguila" and others in the economic conflict with the Union of Oil Workers of the Mexican Republic (1938), the Supreme Court of Mexico ruled in favor of the workers of foreign oil companies, culminating in the expropriation and further nationalization of the oil industry [START_REF] Pjf -Poder | Decreto expropiatorio del 18 de Marzo de 1938[END_REF]. This made the energy sector largely linked to the economic performance of Mexico for the next 80 years [START_REF] Colmenares | Petróleo y crecimiento económico en México 1938-2006[END_REF](Colmenares, , p. 53) and, for instance, between 2007 and 2013, oil revenue represented from 31 to 38% of fiscal income and from 7.4 to 8.7% of the national GDP (Oswald, 2017, p. 155) (CEMDA, 2017, p. 19). Moreover, the cooperative model can play a significant role in the promotion of such types of projects as it is recognized by the Mexican Constitution (Article 25) as a form of social organization for the production, distribution and consumption of socially necessary goods and services.

In that sense, and considering the solar potential that Mexico has, a strategy based on the development of photovoltaic energy could bring great benefits both for the government and particulars, thanks to the development of the cooperative model in the country. Indeed, the Ministry of Energy of Mexico (SENER, 2017) estimates that the installation of solar panels, for distributed energy generation, for 1% of the total generation capacity of the country, could represent public annual savings of around MX 1.5 billion, plus 680 million liters of unused water, and 1.3 million tons of carbon dioxide (MtCO2e) not emitted. In addition, it is considered that US$20 trillion in cooperative assets can produce around US$3 trillion of annual income (Grace et al., 2014, pp. 1-2). Mexico can then take advantage of the benefits offered by this model, while diversifying its energy matrix with clean energies and increasing a culture of cooperativism in all its types, regions and social backgrounds throughout the country.

Mexico's state of play and commitments to climate change and clean energy by 2050

Mexico is one of the signatory countries of the 2015 Paris agreement (COP21) and on this basis, the country is committing itself to reduce substantially its greenhouse gas (GHG) emissions to limit the increase in global average surface temperature to below 2°C above pre-industrial levels (Bruckner et al., 2014, p. 569). In 2013, Mexico ranked 11 th in the world in terms of GHG emissions, which means equivalent emissions of 489 MtCO2e, representing 1.4% of global emissions (Rodríguez, 2017, pp. 182-191). The LGCC (Second The latest National Inventory of GHG emissions of Mexico (2018) states that the country emitted 683 MtCO2e in 2015 of which 64% corresponded to fossil fuel consumption; 10% from livestock production systems; 8% from industrial processes; 7% related to waste management; 6% from fugitive emissions from oil, gas and mining extraction; and 5% generated by agricultural activities. Conversely, 148 MtCO2e were absorbed by vegetation, mainly forests and jungles; the balance between emissions and removals for 2015 was then 535 MtCO2e, while in 1990 these were 445 MtCO2e. This means that between 1990 and 2015, Mexico's emissions increased by 54%, with an annual growth rate (AER) of 1.7%, but also AER from 2010 to 2015 decreased to 0.8% (INECC and SEMARNAT, 2015).

In such a context, Elizondo et al. (2017, p. 19) project that by 2050, industry and transportation are likely to lead Mexican energy consumption, accounting for around 80% of demand; buildings (heating, cooling and cooking) will account for 15-20%; and the remaining 3-7% will be related to lighting. While it is estimated that half of the energy generated in 2050 will come from fossil fuels (oil and gas), it is also considered that there are good opportunities for Mexico to rely primarily on clean energy for electricity supply (mostly renewable energies) essential for heating and tackling energy poverty in the country (Elizondo et al., 2017, p. 24). Furthermore, considering that the industrial sector is the absolute leader in terms of energy demand, the government's energy strategy should consider not only the development of clean energy, but also promote energy efficiency and coordinate actions with local and transport authorities in order to comprehensively address GHG reduction, whereas mitigation measures could be shared between agricultural waste management and forestry (Elizondo et al., 2017, p. 24).

The challenge of inequality: Tackling energy poverty through locally determined energy projects

Although there is no harmonized definition of energy poverty in Mexico, García-Ochoa and Graizbord (2016, pp. 289-337) consider that this concept encompasses the lack of access to energy, the insufficient supply and the lack of economic capacity of households to meet all energy needs. In this regard, Mexico has 12.4 million households living in energy poverty (43.4% of the total), of which 7.8 million reside in urban areas (27.5%) and 4.5 million in rural environments (16%); additionally, energy poverty in Mexico increases as one moves from the urban to the rural, although in absolute terms, urban energy poverty is almost double that seen in rural areas (García-Ochoa, 2014, p. 19). Locally determined energy projects could represent a smart alternative to address Mexico's energy backlog and the cooperative model can effectively support the implementation of local energy projects (ILO, 2013).

Local-scale projects also represent an area of opportunity as evidence shows that large energy infrastructures may come into conflict with local populations, who are affected by the implementation of major projects. This is the case, for instance, of socio-environmental conflicts related to the construction of hydroelectric plants in the States of Veracruz and Puebla (Casas Mendoza & Carbajal, 2017, pp. 70-93), as well as wind power plants in the states of Oaxaca and Yucatan (Zárate Toledo & Fraga, 2016, pp. 65-95). Considering such issues are particularly important in terms of investment risks since the cancellation of projects has represented losses of approximately MX 3 billion pesos (CEMDA, 2017, p. 33). Mexico's solar potential represents in those terms an opportunity to meet the energy needs of the poorest, as photovoltaic energy production can satisfy the demand for shorter lead times and reduce potential socio-environmental conflicts linked to other types of renewable energies (Vergara et al., 2016, p. 21).

According to the International Energy Agency (IEA), 70% of electricity access can be affordably met by renewable energy; for 65% of non-electrified households, the cheapest way could be through mini-grids and in 45% of households through off-grid technology (OECD/IEA, 2011 in ILO, 2013, p. IX). In this context, local and regional governments in Mexico could promote mini-grids and off-grid based solutions to address the country's energy backlog and basing projects on the cooperative model could help overcome the lack of institutions, policies, enterprises and human capacity that is characteristic of the most marginalized territories (ILO, 2013). Subnational governments can also take advantage of the distributed generation scheme, regulated by Electric Industry Law (Article 3 XXIII), as it promotes the capacity of local communities to propose, design, implement and operate their own technology (Dafermos et al., 2015, pp. 459-460).

In this respect, the Mexican legal framework provides that producers generating less than 0.5 MW are considered as "exempt generators", meaning that they do not require permission for energy production and may also participate in the Wholesale Electricity Market (WEM) through the representation of a Qualified Service Provider (LIE, Article 3 XXV). The distributed generation model in Mexico can thus enable micro-generators to meet their energy needs while facilitating the sale of surpluses in a simpler and faster manner than higher capacity sources (White & Case, 2014, p. 3). Furthermore, this type of project can activate the mechanisms considered by the Ministries of Finance and Welfare to provide clean energy to rural communities and marginalized urban areas at the lowest cost (LIE, Article 116). Projects could also benefit from the "sustainable energy financing" mechanism, considered by the Energy Transition Law (Article 55), to replace energyinefficient equipment and devices, make improvements to buildings and install economically viable equipment to enable households to use renewable energy sources to meet their energy needs.

In summary, Mexico's energy poverty could be tackled by all three levels of government by promoting cooperative photovoltaic energy projects. In fact, electricity access can be affordably met by renewable energy and the cooperative model presents a series of competitive advantages based on a democratic control that allows better decision-making in the production, supply, distribution and consumption of electricity (ILO, 2013). Likewise, this type of project can contribute to bringing energy to marginal areas where investment can be costly for governments or unattractive for businesses (COPAC, n.d.). From this perspective, local and regional governments, in partnership with the national government, could promote community energy systems to effectively address the energy backlog in the country, particularly in marginalized urban and rural areas, while empowering people and increasing local knowledge for sustainable energy generation (Walker & Simcock, 2012, p. 194).

Reconciling interests: The benefits of the cooperative model for industry and business

Although energy production is relatively easy for "exempt generators", they do not always meet the needs of industries and businesses (leaders in energy consumption by 2050) that have to resort to medium-scale production (0.5 to 10 MW); yet the installed production capacity of small, medium and large photovoltaic arrays in recent years shows a significant growth trend that represents an opportunity for cooperativism within the Mexican industrial and business sector (DGRV, 2018, p. 122). Mexican legislation considers producers of more than 0.5MW as "qualified generators", therefore subject to the regulations of the WEM (LIE, Article 3 XLVII). Plants between 0.5 and 2 MW are particularly affected by this legal framework as the installation process is more complex and the investment less profitable; however, it is considered that industries and businesses can benefit from the cooperative model by implementing an investment system that allows the installation of larger capacity generators under a collective measurement scheme for energy production and consumption (DGRV, 2018, p. 120).

In fact, Mexican law considers a model of isolated supply for the generation or import of electric energy to satisfy one's own needs (LIE, Article 22). The Energy Regulatory Commission of Mexico (CRE, 2017) has reinforced this criterion and provided that a group of persons that have similar commercial and financial interests can be considered a "group of economic interest" in order to meet personal energy needs. This concept is in line with Mexican jurisprudence on administrative matters, which considers that a "group of economic interest" can coordinate its activities to achieve a certain common objective (Thesis I.4o.A. J/66. Group of economic interest. Its concept and elements that make it up in terms of economic competition. Volume XXVIII, November 2008, p. 1244). As such, this could be seen as an opportunity for industries and businesses as it allows synchronizing their activities in order to achieve a common goal and generate more than 500kW without carrying out all the formalities for participating in the WEM (DGRV, 2018, p. 124).

Notwithstanding, it is worth noting that financing mechanisms, especially for power plants of more than 0.5MW and less than 2MW, still need to be strengthened as this is the sector with the lowest return on investment (DGRV, 2018, p. 129). Commercial banking is developing financial mechanisms to meet the needs of the different groups in the distributed generation, although, it continues to face certain obstacles including price uncertainty that increases investment risk, and loans that do not give a grace period over the construction phase necessary to increase their attractiveness (DGRV, 2018, p. 129).

Consequentially, the leadership of the State in the promotion, funding and integration of the concepts of ownership, design, effective participation and representation, with regard to energy production, is a necessary condition for the successful deployment of medium-scale generation plants in Mexico (Climate Transparency, 2019, pp. 6, 9). LTE provides for financial and investment instruments and establishes that the resources for the energy transition must come from the Federal Expenditure Budget, financial instruments available for public works and services, as well as from private contributions (LTE, Article 43). Funds are intended to attract and guide public and private financial resources, national or international, to support programs and projects that help to diversify and enrich the energy matrix (LTE, Article 48). In this context, clean energy certificates (CELs) created by the Law for the Use of Renewable Energies and Energy Transition Financing ( 2008) can be seen as a promising mechanism, as CELs are issued by the CRE and certify the production of a certain amount of electricity as coming from clean sources (LIE, Article 3 VIII).

According to the Mexican Institute for Competitiveness (IMCO), the proper regulation of CELs can help diversify the energy mixwhich had an 80% share of fossil fuels in 2015 (SENER, 2016, p. 30), while fostering a CEL's exchange market in line with the objectives of clean energy production of Mexico (IMCO, 2014, in Moreno Sánchez, 2017, p. 28). In this sense, CELs could encourage the purchase of surplus energy from medium-sized cooperative producers, thus promoting investments in energy generation, while mitigating vulnerability to supply interruptions and reducing market volatility (IPCC, 2011, p. 20).

Cooperatives can then generate their own clean energy and consider a long-term repayment plan to cover the initial investment (DGRV, 2018, p. 128) through the sale of surplus energy to a trading service provider or supplier (LIE, Article 23).

While from a business perspective the objective is to make projects profitable at a low cost per kWh by competing with the prices established by the WEM, medium-scale cooperatives can offer other types of benefits to their members. This includes a reduced technology price thanks to the economies of scale, savings in electricity consumption compared with high tariffs, collective awareness raised by actively participating in the generation of clean energy and the possibility of selling the surplus generated to the distributed generation network (DGRV, 2018). In addition, they can collaterally favor access to affordable clean energy, encourage the creation of local jobs, contribute to improved performance of the national energy grid and, in general, promote economic sustainability (ILO, 2013, p. 29) and energy security for the poorest communities, industries and businesses.

Conclusions

The smart city concept has developed in recent years as a way of providing solutions to local problems through the use of technology. Although the concept is broad and its scope diverse, the implementation of clean energy projects is shown to be a central axis of the smart city concept and can contribute to promoting sustainable development and reducing carbon emissions. In the case of Mexico, history and national economic performance are strongly linked to the energy sector, which makes the participation of the private sector in this area highly regulated by the State. Furthermore, although in recent years the Mexican energy sector has been opened to various types of investments, in particular by States, municipalities and the private sector, in practice, energy policy remains a matter of federal competence, which means that there is a degree of centralization in this area that significantly affects the development of smart energy projects at the local scale.

The cooperative model can help reconcile interests within the Mexican energy sector, thus contributing to the development of local projects, the promotion of investment, and diversification of the energy mix in favor of energy security and the creation of a democratic national energy system. Indeed, cooperatives are shown to be important contributors to the development of smart local energy projects, favoring democratic management of energy production and consumption according to the energy needs of a community, while bringing positive benefits to the State, business and society at large. In addition, based on the cooperative model, the development of locally determined smart energy projects can help address two main issues related to the energy sector in Mexico such as the country's energy backlog and poverty, and the reduction of carbon emissions from the industrial and business sectors as leaders of energy consumption by 2050.

In this way, the government could promote mini-grids and off-grid solutions to address the country's energy backlog, while overcoming the lack of institutions, policies, enterprises and human capacity of most marginalized areas. Micro-generators can then meet their energy needs and sell their surplus energy production under the country's distributed generation scheme. Moreover, projects can take advantage of the policies and strategies considered by the Ministries of Finance and Welfare, as well as of the "sustainable energy financing" mechanism considered by the Energy Transition Law, to provide clean energy to rural communities and relegated urban areas at the lowest cost. Finally, they can allow better decision-making in the production, supply and distribution of electricity, while empowering communities and increasing local knowledge for sustainable energy generation.

In the same manner, the government can encourage cooperativism among industries and businesses as "groups of economic interest" in order to produce their own energy needs and promote energy efficiency in this sector. The expected benefits can be translated into a reduction in the price of technology through economies of scale, savings in electricity consumption compared with high tariffs, collective awareness raised by actively participating in the generation of clean energy and the possibility of selling the energy surplus to the distributed generation network. In this context, CELs could be seen as a tool to promote a market for the purchase and sale of clean energy, enabling medium-scale cooperatives to meet their energy needs, while considering a repayment plan to cover the initial investment through the sale of energy surplus to a trading service provider or supplier. This can help increase interest in clean energy production, thereby diversifying and enriching the energy mix, mitigating vulnerability to supply disruptions, as well as reducing energy market volatility.

Finally, yet importantly, although the Mexican government (2019-2024) has stressed the importance of the development of smaller-scale and locally determined renewable energy projects, major State investments have been focused, until today, on increasing the exploitation of hydrocarbons and the renovation of the national hydroelectric system. Detailed monitoring of these plans is suggested for further research and their examination, in light of climate change commitments, is relevant to the assessment of Mexico's energy transition. Similarly, the development of cooperatives as a form of social entrepreneurship, empowerment and active participation of Mexican society in energy transition is also suggested as a topic of interest for future research.

NB: Many thanks to Natasa Kurucki, electrical engineer, as well as Christopher Fernandes, renewable energy engineer, for their valuable comments that helped prepare the final version of this article.

  transitional article) mandates to reduce the country's emissions by 30% by 2020 and 50% by 2050, in comparison with 2000; the National Congress has also set limits in the use of fossil fuels for electricity generation of 65% by 2024, 60% by 2035 and 50% by 2050. The Ministry of Energy of Mexico has integrated these objectives into the national development policies of the electricity sector, yet without precise quantitative targets, due to the absence of technical and economic studies on the large-scale integration of different types of clean and renewable energies (Vidal-Amaro et al., 2015, pp. 80-96).

to promote locally determined renewable energy projects as local governments, citizens and enterprises may participate directly in the generation, trade, transmission and distribution of electricity

  . The energy industry is a sensitive issue in Mexican politics and the participation of the private sector in this area is highly regulated by the State (Climate Transparency, 2019, p. 9).

	Yet, within the last decade, Mexico has entered a period of important structural reforms that
	break with this long-standing tradition of a hard state monopoly on energy matters. This
	includes the General Climate Change Law (LGCC) of 2012; the Energy Reform of 2013;
	the Electricity Industry Law (LIE) of 2014; and the Energy Transition Law (LTE) of 2015.
	Within this framework, energy policy remains the sole responsibility of the State, but the
	door is open for the government to enter into contracts with individuals to participate in
	activities related to the electricity industry (Mexican Constitution, Article 27). This
	represents an opportunity


Ibrahim Al Asmi

Roman Le Goff Latimier

Yasmine Lalau

Thomas Brian

Hamid Ben Ahmed

Co-optimization of a high temperature thermal storage as per its modeling accuracy

Coupling energy networks becomes unavoidable in order to decarbonize human usages, increase global energy efficiency and ensure flexibility in so-called ''multi-energy'' network. In such a network, high temperature thermal energy storage (HTTES) can be a relevant solution when designed and managed in an optimal way. However, the precise modeling of its physical behavior requires complex models whose computational costs are not compatible with optimal control. A fortiori, a co-optimization approach requires to select a less precise but faster model. This article proposes to study the consequences of using a panel of such lighter models, in particular by discussing the modeling of losses. To do so, two business models will be discussed on a case study composed of a heat network linking a concentrated solar power (CSP) to thermal industrial load. When losses are not a consideration, the use of very simplistic models is sufficient to determine a good estimate of storage sizing. However, if losses are included, a proper co-optimization can only be achieved by using a metamodeling approach.

Introduction

In 2018, 76% of global greenhouse gas emissions (GHG) are emitted in the energy sector, where 42% of this energy is designed to meet heat and electricity needs in residential, service and industrial sectors [START_REF]Paris-IEA, Greenhouse Gas Emissions from Energy Data Explorer[END_REF]. To overcome this problem, governments and industries seek energy strategies and technical solutions to reduce the carbon footprint of the produced energy. In Europe, 78% of electricity, in the residential sector, has been consumed for space and water heating in 2020 [START_REF]Eurostat, Energy consumption in households -Statistics Explained[END_REF]. However, considering the coupling between heat and electricity sectors is unusual today when building networks, even though it is necessary to reach a global optimal design and management. This approach is complex, as it uses innovative technologies and requires fine optimization to reach a profitable solutions, compared to standard approaches [START_REF] Mohammadi | Optimal management of energy hubs and smart energy hubs -A review[END_REF]. The efforts made to overcome the difficulties of a multi-energy network are justified by the numerous resources and technologies that could then be exploited much more widely than today [START_REF] Mancarella | MES (multi-energy systems): An overview of concepts and evaluation models[END_REF]. For instance, 8900 TWh/year of waste heat can be exploited if considered when designing energy networks [START_REF] Forman | Estimating the global waste heat potential[END_REF]. Moreover, the share of renewable energy in the final consumption needs to be doubled by 2030, to be on track with the net zero emissions by 2050 Scenario [START_REF]Paris-IEA, Greenhouse Gas Emissions from Energy Data Explorer[END_REF]. Hence, flexible networks will be required to accommodate important amount of intermittent energy and to meet heat and electrical needs. To achieve this, conversion systems (Power-To-Heat, Heat-To-Power...) to couple networks and multiple storage technologies do exist such as Electrochemical Energy Storage (EES) and Thermal Energy Storage (TES). TES can have many uses such as storing excess of heat or even electricity, after being converted into heat by the help of Power-To-Heat (PTH) system, during production peak periods. In the other way, TES can deliver heat or electricity, coupled to a steam turbine or an Organic Rankine Cycle (ORC), during load peak periods [START_REF] Al Asmi | Impact of coupling thermal and electrical carriers on the optimal management of a multi-energy network[END_REF].

TES technologies are multiple and can be categorized based on their temperature, physical principle, geometry, storage medium (HSM) and heat transfer fluid (HTF) as discussed in [START_REF] Zinurov | Assessment of thermal storage technologies in energy sector[END_REF]. Here, a sensible thermocline high temperature thermal energy storage (HTTES), using air as HTF and rocks as HSM, will be studied [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF]. This technology has proven to be cost effective and recently undergone numerous developments as shown in [START_REF] Forsberg | Converting excess low-price electricity into high-temperature stored heat for industry and high-value electricity production[END_REF][START_REF] Lalau | Energy analysis and life cycle assessment of a thermal energy storage unit involving conventional or recycled storage materials and devoted to industrial waste heat valorisation[END_REF][START_REF] Cascetta | Thermocline vs. two-tank direct thermal storage system for concentrating solar power plants: A comparative techno-economic assessment[END_REF]. In a multi-energy network, the HTTES system represents an important flexibility to accommodate intermittent energy resources but also to meet uncontrolled demand. The investment cost of such a system (CAPEX) can be a barrier if not optimized. In other words, if operation costs (OPEX) and benefices do not payback the CAPEX E-mail address: Ibrahim.alasmi@ens-rennes.fr (I. Al Asmi). over its lifetime, the system is considered as not profitable. Therefore, it is essential to co-optimize such a system coupling its design and its management to find the best compromise between both. This requires the storage to be designed and operated optimally with the help of a predictive controller that calculates the installed capacity and the management strategy. To do so, it is decisive to have a controller Atmospheric transmission between the heliostat and the receiver v Interstitial between fluid and solid w Wall model enabling the anticipation of future events that would lead to a realistic optimal design and management. Additionally, within the control optimization, the chosen model requires accuracy and speed to reach the best performance within acceptable time cost adapted for real-time management applications. Moreover, the controller model needs to be suitable for the objective function that changes from case to another. Typically, a simple objective function would not consider losses of storage system. On the other hand, objective function in multi-energy network needs to consider storage losses and other terms that depends on local storage information. As a result, HTTES cooptimization requires appropriate models in terms of computation time, accuracy but also the ability to estimate all objective function terms. Therefore, the sensitivity of HTTES co-optimization to different modeling levels need to be evaluated for different objective functions more or less complex.

Nomenclature

State of the art

HTTES modeling have been the subject of many research and development work, mainly in the recent years. Here we will focus on time-dependent HTTES models as they allow the anticipation of future events based on the current state of the system. Most of those models solve partial differential equations (PDE) to estimate precisely the evolution of the system temperature respecting the energy conservation principle. Those models are relatively time consuming but relevant for simulation purposes along several cycles. [START_REF] Marongiu | Modeling of high temperature thermal energy storage in rock beds -Experimental comparison and parametric study[END_REF][START_REF] Hoffmann | Experimental and numerical investigation of a thermocline thermal energy storage tank[END_REF][START_REF] Calderón-Vásquez | Review on modeling approaches for packed-bed thermal storage systems[END_REF] describe some of commonly used models in the literature, that solve PDE along one or two spacial dimensions (1D and 2D). It is to be noticed that in 1D models, radial effects are neglected compared to 2D models. [START_REF] Celador | Implications of the modelling of stratified hot water storage tanks in the simulation of CHP plants[END_REF] investigated three hot water TES models of increasing complexity level. They stated that accurate modeling of losses and storage temperature has an impact on the profitability of the projects. [START_REF] Hoffmann | A thermocline thermal energy storage system with filler materials for concentrated solar power plants: Experimental data and numerical model sensitivity to different experimental tank scales[END_REF] proposes and experimentally validates 1D models based on PDE by the help of an oil/rock HTTES prototype. A similar model for air/rock HTTES has been conducted by [START_REF] Touzo | Experimental and numerical analysis of a packed-bed thermal energy storage system designed to recover high temperature waste heat: an industrial scale up[END_REF], where radial effects have been neglected to achieve 1D model. Besides, other work focused on developing models without solving PDE to avoid computation time limitation. [START_REF] Esence | Étude et modélisation des systèmes de stockage thermique de type régénératif solide/fluide[END_REF] describes a model based on an analytical formulation, used to calculate the storage temperature evolution with low computation time but only for partial cycle configurations. [START_REF] Bonanos | Analysis of thermocline thermal energy storage systems with generic initial condition algebraic model[END_REF] proposed a new algebraic formulation of the developed model in [START_REF] Votyakov | A perturbation model for stratified thermal energy storage tanks[END_REF][START_REF] Votyakov | Algebraic model for thermocline thermal storage tank with filler material[END_REF]. In this formulation, the algebraic solution for thermocline HTTES is given as polynomial function, and allowing for any initial temperature profile to predict the storage temperature evolution. [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF] suggested a metamodel based on data issued from 1D -PDE model simulations, representing a suitable trade-off between accuracy and computation time. In case of manufacturing defects or highly dimension-dependent phenomena like flow channeling, this metamodel can be constructed on data issued from experimental measures. However, researchers conclude that such an experiments based metamodel requires experimental validation.

The state of the art shows that, thermal storage optimization research are mainly conducted to calculate the optimal design or the management strategy. 0D models are often used for those purposes as they have low computation time adapted for predictive controller usages. Model Predictive Control (MPC) has been developed, to define the optimal thermal storage strategy for a heat network case study, in [START_REF] Cox | Real time optimal control of district cooling system with thermal energy storage using neural networks[END_REF][START_REF] Jørgensen | Economic MPC for a linear stochastic system of energy units[END_REF][START_REF] Verrilli | Model predictive control-based optimal operations of district heating system with thermal energy storage and flexible loads[END_REF]. Other studies such as [START_REF] Leśko | Operational optimization in district heating systems with the use of thermal energy storage[END_REF] investigate the operational optimization of a heat district network using TES and solving Mixed Integer Linear Programming (MILP) problem. Researchers agree that 0D models can by used for a quick estimation of large systems optimal design and management. It is to be mentioned that, in all those studies, simulation and optimization models are identical. Moreover, [START_REF] Limpens | EnergyScope TD: A novel opensource model for regional energy systems[END_REF][START_REF] Gabrielli | Optimal design of multienergy systems with seasonal storage[END_REF] study the design optimization, whereas [START_REF] Cao | Distributionally robust heat-andelectricity pricing for energy hub with uncertain demands[END_REF] investigates the pricing and the management, of a multi-energy system at the scale of a region. The models involved in these works are 0D in order to guarantee reasonable computation time and low complexity level. The impact of coupling management and design when optimizing a storage system had been a subject of interest of several papers in EES field. [START_REF] Goff Latimier | Preliminary sizing of a collaborative system: Photovoltaic power plant and electric vehicle fleet[END_REF] studied the impact of forecasting models on the sizing of electrical vehicle fleet, where batteries are charged by the help of a photovoltaic power plant. It appeared that the forecast model quality impact the optimal battery size but author recommend further investigations. [START_REF] Goff Latimier | Cooptimisation de l'engagement de production et de la capacite de stockage associeea une ferme photovoltaique, prenant en compte le vieillissement de la batterie[END_REF] investigated the impact of battery aging model on the management and design results. Researchers conclude that introducing an aging model, even a simplistic one, greatly improves the choice of storage capacity. Three EES models of different accuracy level, used for optimal design, have been explained in [START_REF] Rosewater | Battery energy storage models for optimal control[END_REF]. It concludes that a temperature model is necessary when it impacts the controller actions. In addition, introducing aging into controller models can have important impact on the rate of degradation. Researchers conclude that trade-offs between model complexity and model accuracy can be difficult to navigate, but offer significant benefits in terms of performance.

This review shows that multiple models of HTTES exist with increasing level of complexity and accuracy. However, most of optimization work related to HTTES uses 0D models for computation time reasons. On the other hand, a 1D metamodel based on experience was proposed but lack experimental validation. Moreover, the state of the art shows that the impact of the model choice on the HTTES management and design was not evaluated. Also, most of works study either the management or the design of the HTTES but, to our knowledge, none of them study the co-optimization of this system.

Scope of this work

The goal of the present contribution is to study the impact of the HTTES model choice on the performance of the simulator and the cooptimization controller for two case studies. Therefore, four models of increasing complexity, including models that consider losses and others that does not, are being selected from the literature and investigated. Also, the first case study is formulated with simple objective function that does not take into account losses. On the contrary, the second case study takes into account losses in the objective function. The impact of model choice is measured by the help of time, management and design cost. For this purpose, a new physical metric is formulated to evaluate management and design costs. As a result, suitable tradeoff between accuracy and computation time can be found for future real-time management and design purpose. Also, the impact of model choice, in function of its accuracy and computation time, on the cooptimization results is being analyzed for different storage capacities. Moreover, 1D metamodel based on measured data is being studied in the case of HTTES with manufacturing defects. This metamodel is constructed on the bases of four experiments and being validated over three days of continuous experimental tests by the help of an HTTES test facility in the south of France.

Section 2 presents the selected models from the literature that are used along this article. Section 3 describes the case study used to conduct the co-optimization study, the mathematical formulation as well as the optimization algorithm. Section 4 discuss simulations models and evaluate their performance. Also, in case of defected installation, metamodel based on experimental measures is discussed as well as the facility, experimental protocol and instrumentation used for its experimental validation. In Section 5, the co-optimization results are shown and analyzed. Finally, Section 6 will be dedicated for conclusions and perspectives of this work.

Modeling

In HTTES systems, a fan is used to push hot air through the porous medium as shown in Fig. 1(a) in order to charge the system with energy. The air (HTF) enters the storage through the hot section and gives up its heat to the storage materials (HSM). The heat exchange is done by convection between the fluid and the solid but also by conduction from one solid particle to another. Therefore, the storage medium heats up throughout the storage, creating what is called a thermocline that can be seen in Fig. 1(b). During the discharge process, the cold air enters the storage though the cold section and warms up thanks to the convective heat exchange with hot HSM. Here, multiple complexes heat transfer mechanisms occur such as convection, conduction and radiation between the three storage phases (solid, fluid and walls). To control the thermal charging power 𝑃 𝑠𝑡𝑜 of such a system, the flow rate and entry air temperature are accordingly calculated. During the discharge process, cold air is blown into the storage, and its flow rate is calculated based on the outlet temperature, to reach the desired thermal discharging power. In order to estimate the storage energy evolution, an energy balance can be applied using the HSM temperature. System losses can be divided into loss at walls and charging phase loss. In fact, when charging the system, the outlet temperature of the storage increases. Therefore, the charging phase air leaving the system, contains residual heat that is being lost into the atmosphere.

As said before, the predictive controller used to operate optimally the storage needs a model to estimate the system behavior. Here, the storage energy state and losses evolution in function of the power command are required for the controller to design and manage the system. This section will be dedicated for describing two 0D models, a 1D based on PDE model as well as a 1D metamodel as explained in [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF], whereas 2D models are out of the scope of this work.

Accurate model: 1D -PDE

In this work, 1D -PDE represents the most accurate model in the scope of this work. It solves PDE on the fluid (HTF) but also the two solid phases (HSM and walls) respecting the energy conservation principle. As a results, the temperature evolution can be obtained accordingly to time and to one spacial dimension (length of the storage). Here below the heat equations taking into account the convection, conduction and fluid advection phenomena :

• Fluid : 𝜀 ( 𝜌𝑐 𝑝 ) 𝑓 ( 𝜕𝑇 𝑓 𝜕𝑡 + 𝑢 𝜕𝑇 𝑓 𝜕𝑥 ) = (1a) 𝑘 eff 𝑓 𝜕 2 𝑇 𝑓 𝜕𝑥 2 + ℎ 𝑣 (𝑇 𝑠 -𝑇 𝑓 ) + ℎ 𝑤 𝐴 𝑓 ↔𝑤 𝑉 𝑓 + 𝑉 𝑠 (𝑇 𝑤 -𝑇 𝑓 )
• Solid :

(1 -𝜀) ( 𝜌𝑐 𝑝 ) 𝑠 𝜕𝑇 𝑠 𝜕𝑡 = (1b) 𝑘 eff 𝑠 𝜕 2 𝑇 𝑠 𝜕𝑥 2 + ℎ 𝑣 (𝑇 𝑓 -𝑇 𝑠 ) + ℎ 𝑤 𝐴 𝑠↔𝑤 𝑉 𝑓 + 𝑉 𝑠 (𝑇 𝑤 -𝑇 𝑠 )
• Wall :

( 𝜌𝑐 𝑝 ) 𝑤 𝜕𝑇 𝑤 𝜕𝑡 = 𝑘 𝑤 𝜕 2 𝑇 𝑤 𝜕𝑥 2 + ℎ ext 𝐴 𝑤↔ext 𝑉 𝑤 (𝑇 ext -𝑇 𝑤 )+ ℎ 𝑤 ( 𝐴 𝑓 ↔𝑤 𝑉 𝑤 (𝑇 𝑓 -𝑇 𝑤 ) + 𝐴 𝑠↔𝑤 𝑉 𝑤 (𝑇 𝑠 -𝑇 𝑤 ) ) (1c)
Physical proprieties as well as correlations for effective thermal conductivities and heat transfer coefficients are presented in Appendix. Solving numerically PDE on the three phases, using finite difference method, results in the evolution of the storage temperature. Later on, the energy evolution and the losses of the system, based on the obtained temperature, can be calculated using Eqs. (2a) and (2b). More information on how to calculate the Eqs. (1a), (1b) and (1c) parameters can be found in [START_REF] Hoffmann | Experimental and numerical investigation of a thermocline thermal energy storage tank[END_REF][START_REF] Hoffmann | A thermocline thermal energy storage system with filler materials for concentrated solar power plants: Experimental data and numerical model sensitivity to different experimental tank scales[END_REF][START_REF] Esence | Étude et modélisation des systèmes de stockage thermique de type régénératif solide/fluide[END_REF].

𝐸 𝑠𝑡𝑜 = (1 -𝜀)𝑆 𝑏𝑒𝑑 ⋅ 𝜌 s ∫ 𝐿 0 𝑐 𝑠 𝑝 (𝑥) ( 𝑇 𝑠 (𝑥) -𝑇 𝑒𝑥𝑡 ) d𝑥 (2a) 𝑃 𝑙𝑜𝑠𝑠-𝑐ℎ = { 0 if 𝑇 𝑜𝑢𝑡𝑙𝑒𝑡 = 𝑇 𝑒𝑥𝑡 ṁ ⋅ 𝑐 𝑓 𝑝 (𝑇 𝑜𝑢𝑡𝑙𝑒𝑡 -𝑇 𝑒𝑥𝑡 ) if 𝑇 𝑜𝑢𝑡𝑙𝑒𝑡 > 𝑇 𝑒𝑥𝑡 (2b)
This model has the highest computation time compared to other models in the scope of this work (approximately 1 to 10 s for the simulation of one hour of the storage operation). Hence, this model is not suitable for use in an optimization controller. Further information on models computation time can be found in Section 4.

Ideal storage: 0D -ideal

In 0D -Ideal models, the losses before saturation are completely neglected. This means that all the injected energy into the system is being stored. In other words, heat transfer between fluid and solid is supposed to be perfect and instant, and heat transfer to walls is completely neglected. As a results, the thermocline (temperature gradient) which is the zone where the HSM temperature varies is null. The dynamics of the system according to this model can be expressed as follows:

𝐸 𝑡+𝛥𝑡 𝑠𝑡𝑜 = 𝐸 𝑡 𝑠𝑡𝑜 + 𝛥𝑡 ⋅ 𝑃 𝑡 𝑠𝑡𝑜 (3a) 𝑃 𝑡 𝑙𝑜𝑠𝑠 = { 0 if 𝐸 𝑡 𝑠𝑡𝑜 < 𝐸 𝑟𝑎𝑡𝑒𝑑 𝑃 𝑡 𝑠𝑡𝑜 if 𝐸 𝑡 𝑠𝑡𝑜 = 𝐸 𝑟𝑎𝑡𝑒𝑑 (3b)
where 𝐸 𝑟𝑎𝑡𝑒𝑑 is the maximum storage capacity. Storage heat losses are zero at all times except when the system is charged even though it is already fully charged and can be calculated as expressed here above. The advantage of such a model is the low computation time that comes with rough heat transfer and losses assumptions.

Uniform temperature storage: 0D -uniform

0D -Uniform models assume the same temperature value 𝑇 𝑚𝑒𝑎𝑛 for the whole HSM. This means that the thermocline zone covers the whole storage length. This temperature can be calculated using Eq. ( 4c). In such a model, energy evolution and losses, based on 𝑇 𝑚𝑒𝑎𝑛 , are expressed in Eq. ( 4a) and (4b).

𝐸 𝑡+𝛥𝑡 𝑠𝑡𝑜 = 𝐸 𝑡 𝑠𝑡𝑜 + 𝛥𝑡 ⋅ 𝑃 𝑡 𝑠𝑡𝑜 -𝛥𝑡 ⋅ 𝑃 𝑙𝑜𝑠𝑠 (𝑇 𝑡 𝑚𝑒𝑎𝑛 , ṁ) (4a) with 𝑃 𝑡 𝑙𝑜𝑠𝑠 = ṁ ⋅ 𝑐 𝑓 𝑝 (𝑇 𝑡 𝑚𝑒𝑎𝑛 -𝑇 𝑒𝑥𝑡 ) (4b)
and

𝑇 𝑡 𝑚𝑒𝑎𝑛 = 𝐸 𝑡 𝑠𝑡𝑜 𝑐 𝑠 𝑝 𝑚 + 𝑇 𝑒𝑥𝑡 (4c)
Concerning losses, only the charging phase heat losses are being considered in this model. This model does not consider the second aspect of loss, the loss at walls, unlike 1D models. This model comes with low computation time but overestimates losses and has inaccurate representation of the storage temperature. This explains their frequent use in TES optimization work and their limitation to achieve feasible optimal design and management strategy. Consequently, a trade-off model is required to optimize design and management of TES with acceptable computation time.

Metamodel: 1D -MM

This subsection presents the metamodeling approach, based on a 1D -PDE model simulations, that has been developed in [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF]. Depending on the initial storage temperature 𝑇 𝑡 (𝑥) and the power command 𝑃 𝑠𝑡𝑜 , it is possible to calculate the evolution of the storage temperature 𝑇 𝑡+𝛥𝑡 (𝑥) after a fixed period of time 𝛥𝑡. Later on, the storage energy and losses can be deduced by applying Eq. ( 2a) and (2b). Thanks to an analytical approximation based on a pre-built information matrix, 1D -MM is able to estimate the temporal evolution of the HSM temperature along the storage length. Fig. 2 illustrated the main calculation steps in the metamodel. 1D -MM showed a trade-off between accuracy and computation time as discussed in [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF], thanks to the pre-built matrix that avoids solving PDE in the simulation stage. Therefore, it seems adequate to use 1D -MM inside a predictive controller, which will be the subject of next sections in this work.

The temperature spacial approximation in the metamodel is achieved by applying a least square algorithm to estimate the logistic function parameters described in Eq. [START_REF] Forman | Estimating the global waste heat potential[END_REF].

𝑇 (𝑥, 𝑇 𝑚𝑎𝑥 , 𝑇 𝑚𝑖𝑛 , 𝑧 𝑐 , 𝑠) = 𝑇 𝑚𝑎𝑥 + 𝑇 𝑚𝑎𝑥 -𝑇 𝑚𝑖𝑛 1 + 𝑒 (𝑥-𝑧 𝑐 )∕𝑠 (5)
The temporal approximation used to estimate the final state, starting from an initial state, is a linear interpolation based on a pre-built matrix. The construction of the information matrix use 1D -PDE simulations data as shown in algorithm 1. This step can be time consuming depending on the discretization level of the logistic function parameters (𝑇 𝑚𝑖𝑛 , 𝑇 𝑚𝑎𝑥 , 𝑧 𝑐 , 𝑠) as well as the power command 𝑃 𝑠𝑡𝑜 . On the other hand, a higher discretization level signifies higher accuracy for the metamodel as finer temporal approximation can be achieved. Therefore, it is necessary to find the best trade-off between accuracy and computation time. Here we will call 𝑖 the discretization of logistic function parameters and 𝑗 the discretization of the power command. All along this work, two metamodels will be investigated. A first one with low discretization level called 1D -MM 3,7 (𝑖 = 3 and 𝑗 = 7) and another with high discretization level called 1D -MM 7,15 (𝑖 = 7 and 𝑗 = 15). More details on the metamodel can be found in [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF]. The next subsection compares the computation time between the discussed models. , 𝑠 𝑡+𝛥𝑡 with Eq. ( 5) and least square algorithm end end

Description of case study

The goal of this work is to study the sensitivity of HTTES cooptimization results to model choice in terms of controller performance and computation time, for different business models. Here, studied Fig. 4. Power time series for thermal source and load. In blue is the thermal source power issued from 5500 m 2 CSP field in the south of France, with a total annual production of 7 500 MWh. In pink is the load power consumed by 2 MW industry in France, with a total annual consumption of 5 000 MWh. All data corresponds to 2013 and can be found in [START_REF] Ruhnau | Time series of heat demand and heat pump efficiency for energy system modeling[END_REF][START_REF]JRC Photovoltaic Geographical Information System (PVGIS) -European Commission[END_REF]. models consider storage losses more or less accurately as described in Section 2. Therefore, two business models have been formulated to compare the obtained results in case that losses are counted or not in the objective function. Indeed to calculate losses accurately, local information such as outlet temperature needs to be estimated which is the case of multi-energy networks. Here, the case study includes a thermal carrier with two sources and one load as this would be sufficient to study the HTTES co-optimization sensitivity to models. The case study used for the optimization work is presented in Fig. 3. The consumed thermal power 𝑃 𝑙𝑜𝑎𝑑 is given by [START_REF] Ruhnau | Time series of heat demand and heat pump efficiency for energy system modeling[END_REF], an aggregated estimation of consumed thermal power by heat networks in France in 2013. Based on those data, the load is supposed to be an industrial with a thermal peak load of 2 MW that consumes a total of 5000 MWh per year. As shown in Fig. 3, this case study includes a main source which is a fuel boiler 𝑃 𝑓 𝑢𝑒𝑙 to cover the load. Additionally, a renewable production source of 5500 square meters concentrated solar power (CSP) field will be connected in order to avoid fuel consumption. This source is composed of heliostat mirrors and thermal receiver tower as described in [START_REF] Ziyati | A multiphysics model of large-scale compact PV-CSP hybrid plants[END_REF], with a total production of 7500 MWh per year and a production peak of 5 MW. The solar data for the direct normal irradiation (DNI) is issued from PVGIS database developed by the European commission [START_REF]JRC Photovoltaic Geographical Information System (PVGIS) -European Commission[END_REF], and correspond to the Occitania region in the south of France in 2013. The thermal output of the central is calculated using Eq. [START_REF] Al Asmi | Impact of coupling thermal and electrical carriers on the optimal management of a multi-energy network[END_REF].

𝑃 𝑝𝑟𝑜𝑑 (𝑡) = 𝜂 ℎ𝑒𝑙 (𝑡) ⋅ 𝑆 ℎ𝑒𝑙 ⋅ 𝐷𝑁𝐼(𝑡) (6a) with 𝜂 ℎ𝑒𝑙 = 𝜌 𝑟𝑒𝑓 ⋅ 𝜂 𝑡𝑟𝑎 ⋅ 𝑐𝑜𝑠𝜃 ⋅ 𝜂 𝑠ℎ ⋅ 𝜂 𝑏𝑙𝑜 ⋅ 𝜂 𝑠𝑝𝑖𝑙 (6b)
where 𝑃 𝑝𝑟𝑜𝑑 is the CSP field thermal produced power, 𝜂 ℎ𝑒𝑙 is the instantaneous optical efficiency of the heliostat field, 𝑆 ℎ𝑒𝑙 is the reflective area (here 5500 m 2 ), 𝐷𝑁𝐼 is the direct normal irradiation, 𝜌 𝑟𝑒𝑓 is the heliostat reflectivity, 𝜂 𝑡𝑟𝑎 is the atmospheric transmission between the heliostat and the receiver, 𝑐𝑜𝑠𝜃 is the cosine of the angle formed between the normal to the heliostat surface and the incident rays, 𝜂 𝑠ℎ is the fraction of the heliostat surface shaded by adjacent heliostats, 𝜂 𝑏𝑙𝑜 is the fraction of the reflected sunlight blocked by adjacent heliostats and 𝜂 𝑠𝑝𝑖𝑙 refers to the fraction of reflected sunlight missing the receiver due to other technological errors. For simplicity, 𝜂 ℎ𝑒𝑙 is assumed constant and equal to 0.9. In addition, HTTES is considered with charge and discharge thermal power 𝑃 𝑠𝑡𝑜 . As a result, the storage flexibility would help the CSP energy to meet the considered load and so that allows further reduction in boiler fuel consumption. Here, thermal sources, load and storage system are supposed to be on high temperature of 600 • C. It is to be noticed that, the only considered power loss in this case in the heat storage losses (losses from heat network and others are not considered). Both production and load time series can be visualized over a year in Fig. 4.

The goal of this case study is to optimize the management of the thermal storage for different design points in order to minimize the objective function. Two objective functions are studied here and correspond to two different business models (BM). In the first one, only the call on the main boiler 𝑃 𝑓 𝑢𝑒𝑙 in case of overconsumption is to be minimized over a year without paying the losses related to the storage system. In the second one, thermal losses linked to storage 𝑃 𝑙𝑜𝑠𝑠 is to be minimized also with the call on the boiler which mean that the storage operator pays the losses also. This problem is deterministic and subject to physical constraints which are explained in Eq. [START_REF] Zinurov | Assessment of thermal storage technologies in energy sector[END_REF].

min 𝑛 𝑤 ∑ 𝑡=𝑖 𝐶 𝑡𝑜𝑡 (𝑡) (7a)
where

𝐶 𝑡𝑜𝑡 = { 𝑃 𝑓 𝑢𝑒𝑙 ⋅ 𝛥𝑡, for BM 𝑛 𝑜 1 (𝑃 𝑓 𝑢𝑒𝑙 + 𝑃 𝑙𝑜𝑠𝑠 ) ⋅ 𝛥𝑡, for BM 𝑛 𝑜 2 (7b)
with 𝑃 𝑚𝑖𝑠 (𝑡) -𝑃 𝑠𝑡𝑜 (𝑡) + 𝑃 𝑓 𝑢𝑒𝑙 (𝑡) -𝑃 𝑠ℎ𝑒𝑑 (𝑡) = 0 (7c)

𝑃 𝑚𝑖𝑠 (𝑡) = 𝑃 𝑝𝑟𝑜𝑑 (𝑡) -𝑃 𝑙𝑜𝑎𝑑 (𝑡) (7d) 0 ≤ 𝑃 𝑓 𝑢𝑒𝑙 (𝑡) and 0 ≤ 𝑃 𝑠ℎ𝑒𝑑 (𝑡) (7e)
-𝑃 𝑟𝑎𝑡𝑒𝑑 ≤ 𝑃 𝑠𝑡𝑜 (𝑡) ≤ 𝑃 𝑟𝑎𝑡𝑒𝑑 (7f) 0 ≤ 𝐸 𝑠𝑡𝑜 (𝑡) ≤ 𝐸 𝑟𝑎𝑡𝑒𝑑 ( 7g
)
𝑃 𝑟𝑎𝑡𝑒𝑑 = 𝐸 𝑟𝑎𝑡𝑒𝑑 (7h)
It is to be noticed that 𝑃 𝑠𝑡𝑜 follows a receptor convention: 𝑃 𝑠𝑡𝑜 has a positive sign when charging and negative when discharging. The storage dynamics reflecting storage energy 𝐸 𝑠𝑡𝑜 and heat loss 𝑃 𝑙𝑜𝑠𝑠 evolution in function of the storage command 𝑃 𝑠𝑡𝑜 are estimated using one of the four models described in the previous sections. The problem is solved numerically using Model Predictive Control (MPC) as described in algorithm 2. The time horizon 𝑁 of the resolution is 8760 h. At each time step (here 𝛥𝑡 = 1 h), an optimization problem is formulated and solved on a sliding window of size 𝑛 𝑤 . Only the first sample of the output sequence is implemented for simulation, subsequently the sliding window is shifted of 1 h. Within the present simulation framework, the new state of the system at the next time step is estimated by the accurate model 1D -PDE simulation, which is considered as a reference. Then a new optimization problem is solved using this new information. In whole work, the accurate 1D -PDE model is the only model used for simulation and estimation of the real system energy state and losses. It is to be noticed that investigated HTTES capacities are ranging from 2 to 20 MWh 𝑡ℎ . This choice has been made based on the HTTES studied technology which is adapted for daily to weekly cycles but not for seasonal storage purpose. This means that a part of fuel consumption is not replaceable as more load appears in winter period, oppositely to CSP production that is more important in summer period as shown in Fig. 4. In other words, a seasonal storage would be needed to cover the totality of the heat consumption in this case study -which is out of this work scope. Otherwise, an irreducible residual consumption is unavoidable.

Simulation models discussion

The goal of this section is to validate the studied models for simulation purposes and to evaluate their committed errors. This step is necessary before beginning the co-optimization work. Indeed, HTTES installations can be divided into two categories; installations that fit with 1D -PDE assumptions and installations presenting manufacturing defects, aging or highly dimension-dependent phenomena like flow channeling.

Standard HTTES models validation

In most designs, 1D -PDE reproduces faithfully the real system behavior, except in certain configurations as described in [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF]. In general, this is the case of large scale HTTES without manufacturing defects. An alternative could be the 1D -MM based on 1D -PDE simulations data which is accurate and divides the computation time by a 1000 factor. Also, 0D models can be used to estimate storage energy evolution with low computation time but not local variable such as storage outlet temperature. Here, we will study, on the basis of a year optimal command simulation, the evolution of the HTTES energy based on four models. 1D -PDE is supposed to reproduce faithfully the real storage energy evolution. Hence, the 1D -PDE simulation results will be compared to the four models, listed in Table 1, to calculate each model error. For this purpose, an example of 20 MWh capacity HTTES is used to conduct simulations and evaluate models errors.

Fig. 5 shows the evolution of the storage energy over a year with a time step of 1 h. Also, the relative error as well as the normalized root mean square deviation (NRMSD) (defined in [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF]), between studied models and 1D -PDE are illustrated. Indeed the NRMSD error has been selected as a metric in order to sidestep calculation problems that appear when investigated values are near from zero. It can be defined as :

√ √ √ √ 𝑛 ∑ 𝑖=0 (𝑥 𝑒𝑥𝑝 𝑖 -𝑥 𝑚𝑜𝑑𝑒𝑙 𝑖 ) 2 𝑛 ( 8b
)
where 𝑥 𝑚𝑎𝑥 and 𝑥 𝑚𝑖𝑛 are the maximum and minimum in the investigated numerical values, respectively and 𝑛 is the sample size.

Results show that for 0D models, errors are mostly committed when battery is near from saturation. This can be explained by the fact that, 0D models assumptions estimate poorly the outlet temperature. Therefore, losses during the charge phase, that are mostly present in energy levels near saturation, are underestimated in 0D -Ideal and overestimated in 0D -Uniform. It is to be noticed that 1D -MM 3,7 that is poorly discretized, commits more errors than 0D models in mean. Therefore, it is important to find the best trade-off of discretization in terms of computation time and accuracy for metamodels. Furthermore, the error committed by 1D -MM 7,15 is the lowest error as it represents faithfully the 1D -PDE. NRMSD values for 0D -Ideal, 0D -Uniform, 1D -MM 3,7 and 1D -MM 7,15 can be visualized in Table 1. 𝑁 𝑐𝑜𝑛𝑠 stands for the number of simulation iterations, and 𝑡 𝑐𝑜𝑛𝑠 for the computation time, to build the metamodel information matrix. The model computation time to simulate one hour of storage operation will be called 𝑡 𝑠𝑖𝑚 . In this example, the 1D -MM allows to reduce the computation time by a factor of 1000 for simulation purposes, compared to the 1D -PDE model. As a result, for storage systems without defects, 1D -MM 7,15 shows high accuracy with suitable computation time as well as the ability to estimate local variables of HTTES. This can be an alternative for 1D -PDE, in co-optimization work, that is accurate but time consuming.

Defected HTTES models validation

As the construction of a HTTES is not an easy task, it can happen that the behavior of a real infrastructure does not correspond to the 1D -PDE model. This may result from imperfections during the construction such as non homogeneous air distribution, increased wall effects, manufacturing defects or material aging particularly in small scale installations. Such flaws could undermine the effectiveness of the whole co-optimization approach, since the behavior of the actual infrastructure could no longer be anticipated before implementation. Thus, this paragraph presents how a metamodel can be built from experimental tests on an installation with defects. The accuracy evolution of such a model can be estimated in function of the number of experimental tests and measurement frequency as described in [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF]. This subsection will investigate the construction of such a model and evaluate its performance based on real experimental data and will compare its performance to 1D -PDE in this configuration.

On the one hand, this allows for the optimal control of an installation that cannot be modeled analytically. Moreover, in an operational co-optimization approach, an analysis of the risks related to such defects could be taken into account.

Fig. 6 describes the HTTES facility studied in this work. This installation is classified as high-temperature unpressurized air/rock packedbed storage, where atmospheric air is used as heat transfer fluid (HTF) and rocks as heat storage material (HSM). Further more, the packed bed has a horizontal cylindrical geometry with a maximum storage capacity 𝐸 𝑚𝑎𝑥 of 40 kWh th (calculated for an ambient temperature of 0 • C). 𝑃 𝐶𝐻,𝑚𝑎𝑥 and 𝑃 𝐷𝐶,𝑚𝑎𝑥 are the maximum charge and discharge power, here equal to 27.6 and 37.6 kW th respectively. The presented installation is a prototype that has been developed by the company Eco-Tech CERAM for material characterization purpose and innovative thermal storage applications validation. In this work, the facility has been connected to an electrical heater as shown in Fig. 6(a). The storage is composed of a metallic cylinder that encapsulated insulation and the HSM. The HTTES is equipped with a total of 29 temperature sensors of which 9 are along the central axis of the unit distributed over the sections named A and B, as shown in Fig. 6(b). To avoid wall effects, only temperatures date from the 9 central axis sensors will be used. A summary of the main characteristics of the HTTES, such as the bed diameter 𝐷 𝑏𝑒𝑑 , length 𝐿 𝑏𝑒𝑑 , volume 𝑉 𝑏𝑒𝑑 , particle diameter 𝐷 𝑝 , storage maximum temperature 𝑇 𝑠𝑡𝑜𝑟𝑎𝑔𝑒,𝑚𝑎𝑥 can be found in Table 2. It is to be mentioned that the whole installation has been adapted to receive realtime commands remotely passing by a virtual private network. This applies for all registers, heater and fan. Also, all measures are recorded in real time on a remote data base.

Since this HTTES facility is of small capacity and has been operating under extreme conditions for more than 5 years, its behavior differs 3. Results show the evolution of the storage energy estimation and the error between investigated models and the experimental data over 78 hours experiment. from hypothesis supposed by most of literature models. This is due to important air leakage and preferential passages that become more important in small scale designs and which are not modeled neither in 1D -PDE nor in 0D models. In fact, energy balance has been calculated on the basis of the installation measures. This showed that, nearly 35% (in average) of the energy consumed by the heating system is not being transferred to the storage, during the charge phase. Although losses through the outlet section during the charging phase has been accounted for, this energy difference is still being observed. After several verification by the mean of measuring instruments and mass balance, an important hot air leakage has been confirmed and located upstream of the storage inlet. It is to be noticed that other source of errors can impact the accuracy of the 1D -PDE model. Moreover, model correlations choice can be another source of errors. The impact of HTTES model correlations choice has been discussed partially in some research work [START_REF] Esence | Étude et modélisation des systèmes de stockage thermique de type régénératif solide/fluide[END_REF][START_REF] Hoffmann | Stockage thermique pour centrale solaire thermodynamique à concentration mettant en oeuvre des matériaux céramiques naturels ou recyclés[END_REF]. In general, results show that models are poorly sensitive to heat transfer coefficient with the outside and between walls and other components. However, heat transfer coefficient between solid fluid ℎ 𝑣 can has an important impact on the model results.

Hence, 0D -Ideal, 0D -Uniform, and two 1D -PDE models have been used to simulate the behavior of the studied facility using an optimal management command over 78 h. Each 1D -PDE model use a different correlation for ℎ 𝑣 coefficient as shown in Appendix. The first 1D -PDE model uses Wakao, 1979 [36] correlation where the second one uses Coutier, 1982 one [START_REF] Coutier | Two applications of a numerical approach of heat transfer process within rock beds[END_REF]. A sample of models results, only for the two 1D -PDE models are shown in Fig. 7(a). It shows high discrepancy between both models and experimental results. Indeed, the first correlation underestimates the heat transfer efficiency, compared to experiments, whereas the second one overestimates it. In terms of NRMSD, the scores values are 12, 17, 13 and 30%, for 1D -PDE (Wakao, 1979), 1D -PDE [START_REF] Coutier | Two applications of a numerical approach of heat transfer process within rock beds[END_REF], 0D -Uniform and 0D -Ideal respectively.

In the following, we will focus on the metamodeling approach when physical defects occur and lead to important discrepancies between experience and models results, independently of the choice of the models correlations.

To address this limitation, this same installation is being used to conduct four experimental tests, with full charging and discharging cycles. Those tests results will be used to construct the 1D -MM based on experimental measured data with variable power commands and fixed measurement frequency as illustrated in Table 3. Once the experiments are conducted, the data collected and treated, and then the 1D -MM can be constructed. Afterward, this model is used to simulate an optimal management command over 78 h, which is also being applied on the same test facility in order to evaluate the performance of the constructed model. Fig. 7(b) shows the evolution of the storage energy level for both experimental data and the 1D -MM based on experiments as well as the error of the model. It is to be noticed that the NRMSD value is slightly higher than 6%, which is near from values estimated by [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF]. Also, this 1D -MM has higher accuracy than the three other investigated models studied here. As a consequence, metamodeling based on experience can replace physical equation based models in case of defected installations or specific designs. This section showed that 1D -PDE and 1D -MM can be used to simulate non defected large scale HTTES installations in the designing phase. On the other hand, experimental based metamodels can be used to simulate or optimize management of existing defected or small scale HTTES installations. In the next section, we will admit 1D -PDE as a simulation model to conduct co-optimization work, in the design phase, on a unique accurate base. All investigated installations in the next section are supposed to be without manufacturing defects.

Co-optimization

The goal of this section is to select appropriate controller model for co-optimization work based on two business models. The first BM does not count losses while the second one does. This aims to distinguish the impact of models on co-optimization when taking into account losses or not. The optimization model performance evaluation will be based on a physical metric that will be presented later in this section. All along this work, simulation and optimization models will be distinguished. In fact, 1D -PDE model will be used along this work to simulate the optimization results in order to verify the accuracy of the calculated optimal commands. The choice of the 1D -PDE is motivated by the fact that real system behavior is reproduced faithfully by this model in most designs especially for large scale storages [START_REF] Al Asmi | Thermocline thermal storage modeling towards its predictive optimal management[END_REF] -except regarding defaults as discussed Section 4.2. The studied optimization models are : 0D -Ideal, 0D -Uniform, 1D -MM 3,7 and 1D -MM 7,15 . Moreover, a study for sliding window size 𝑤 selection in function of the storage capacity will be conducted. Based on the obtained results, the minimum accepted size (the size from which the cost function converges to an asymptote) can be decided for each capacity in order to limit the calculation time. In the rest of this work, the optimization is coded in Python language and uses SciPy library [START_REF] Virtanen | SciPy 1.0 Contributors, SciPy 1.0: Fundamental algorithms for scientific computing in Python[END_REF]. A summary of all models in the scope of this study are described in Table 1. It is to be noted that Sequential Least Squares Programming (SLSQP) optimization algorithm from SciPy library has been used to solve the minimization problem along this work.

Identification of sliding window size

For this part, 0D -Uniform model is used to solve problem over 8760 h with different sliding window sizes as shown in Fig. 8. For each capacity, the smallest window size is chosen, ensuring the minimum value of the cost function with 5 % tolerance. Results shows a faster convergence of the cost function for small battery capacities. This can be explained by the fact that for small storage capacity, saturation is reached earlier. Therefore, future events anticipation has lower weight in the cost function. The sliding window size selected for the rest of this study is fixed on 24 h. This choice is made as it is the smallest window size (less time consuming) that guarantees convergence of cost function values for all studied capacities.

Results discussion and analysis

In this subsection, four models (as described in Table 1) of increasing accuracy will be used for the optimization process while only 1D -PDE will be used for the simulation of the optimization results. As a reminder, the storage system operator in this study case has two business models. The first one aims to avoid the boiler fuel consumption only and the second one aims to avoid losses as well as boiler consumption as explained in Section 3. For each storage system design, multiple optimization models have been used to calculate an optimal management command. Once the command is obtained, the simulator is launched in order to evaluate the real system behavior according to the optimal command and to calculate the system life cost in terms of primary energy. In order to evaluate the model impact on both design and management results, multiple criteria can be used for this purpose. Here we focus on criteria that are based only on system energy costs without accounting for financial aspects. A first criteria can be the sum of the system primary energy costs over a fixed number of years. Those costs are equivalent to the cumulative energy demand (CED) and being calculated by the help of the tool developed by [START_REF] Lalau | Energy analysis and life cycle assessment of a thermal energy storage unit involving conventional or recycled storage materials and devoted to industrial waste heat valorisation[END_REF]. Here 𝐸 𝑑𝑒𝑠𝑖𝑔𝑛 is the CED needed for the storage construction, 𝐸 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 is the CED needed to operate the storage system during a year of operation, 𝐸 𝑙𝑜𝑠𝑠 is the CED of generated losses by the storage system over a year of operation, and 𝐸 𝑓 𝑢𝑒𝑙 is the CED of consumed fuel that can be substituted thanks to the storage flexibility over a year of operation. It is to be noticed that losses are supposed to have the same CED value as the fuel per MWh of generated heat (here the fuel is natural gas with a CED of 1.215 MWh-𝑒𝑞 per MWh of generated heat). Also, it is reminded that a residual consumption is unavoidable in the considered case study due to the absence of a seasonal storage. Then for the sake of readability of the results, only the boiler consumption 𝐸 𝑓 𝑢𝑒𝑙 that can be substituted thanks to storage flexibility, will be visualized in the next figures as defined here below:

System life cost =𝐸 𝑑𝑒𝑠𝑖𝑔𝑛 + (9a)
(𝐸 𝑓 𝑢𝑒𝑙 + 𝐸 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 + 𝐸 𝑙𝑜𝑠𝑠 ) ⋅ 𝑁 𝑦𝑒𝑎𝑟𝑠 with 𝐸 𝑓 𝑢𝑒𝑙 =𝐸 𝑓 𝑢𝑒𝑙-𝑠𝑎𝑣𝑒𝑑 𝑚𝑎𝑥 -𝐸 𝑓 𝑢𝑒𝑙-𝑠𝑎𝑣𝑒𝑑 (9b)
where 𝑁 𝑦𝑒𝑎𝑟𝑠 is the operating period of the installation in years, 𝐸 𝑓 𝑢𝑒𝑙-𝑠𝑎𝑣𝑒𝑑 𝑚𝑎𝑥 is the maximum yearly saved fuel when installing a 20 MWh 𝑡ℎ HTTES and optimizing management with the 1D -MM 7,15 model (here equal to 1516 MWh/year) and 𝐸 𝑓 𝑢𝑒𝑙-𝑠𝑎𝑣𝑒𝑑 is the avoided fuel consumption over a year of operation of the investigated storage capacity. Fig. 9 shows the evolution of the system cost components as a function of the storage installed capacity over a year of operation and for BM n o 2 (with losses being taken into account in the objective function). This figure can be used to evaluate the performance of the controller and the obtained design if the HTTES relevance is evaluated over a one year period.

Fig. 9 shows high total life cost for the system controlled by a 0D -Ideal caused by important loss generation even though the fuel cost is well minimized compared to other models. This can be explained by the poor capacity of this model to estimate losses as it considers no loss before storage saturation. On the other hand, 0D -Uniform succeeds to limit losses, as this model overestimates the storage losses, but achieve a higher fuel consumption compared to metamodels results particularly when compared to 1D -MM 7,15 . Optimal designs and total life cost values over a year of operating system for different models are illustrated on Fig. 9 with 𝑥 symbols. Reductions of optimal costs are 30%, 20% and 14% when comparing 1D -MM 7,15 to 0D -Ideal, 0D -Uniform and 1D -MM 3,7 respectively. It is to be noticed that all models find near optimal designs around 10 MWh 𝑡ℎ , except of 0D -Ideal that situates optimal design around 4 MWh 𝑡ℎ storage.

To investigate the impact of the facility's operating time, Fig. 10 shows the evolution of the sum of system life costs after multiple year operation for BM n o 2 where 1D -MM 7,15 has been used in the controller. The red dots shows the optimal designs that minimize the sum of system life costs, which are 12, 16 and 18 MWh 𝑡ℎ after 1, 2 and 4 years of operation respectively. It is be noticed that the optimal design changes in function of number of years of operation. In fact, the longer the length of operation, the higher the saved fuel cost, whereas the design cost is constant and independent of the operation length. This means that, for long term projections, optimal designs can have higher storage capacities compared to short term projections, which is the case here. Indeed, higher storage capacities (tending towards seasonal storage systems) are more profitable when storage operator is engaged for longer years of exploitation, but comes with higher risk (uncertainties linked to far future). Fig. 9. Evolution of the system life cost components, after a year of exploitation, according to thermal storage installed capacity. Each sub figure refers to a system that has been operated with a controller using one of the four investigated models. 𝑥 symbols on the plots indicates the minimum values of the system life cost after a year of operation and correspond to the optimal design. 10. Evolution of the sum of system life costs as a function of the installed storage capacity after different years of operation. Results are based on BM n o 2 and issued from a controller that uses 1D -MM 7,15 for co-optimization.

Comparison between BM n o 1 and BM n o 2

In order to discuss the difference between the two business models in a way that is independent of the length of operation, the Energy Payback Time (EPBT) is introduced in this section, as proposed in [START_REF] Lalau | Energy analysis and life cycle assessment of a thermal energy storage unit involving conventional or recycled storage materials and devoted to industrial waste heat valorisation[END_REF]. This criteria is used to calculate the time needed to payback the initial investment (here 𝐸 𝑑𝑒𝑠𝑖𝑔𝑛 ), which allows to determine the storage capacity to install with shortest payback time as follows:

𝐸𝑃 𝐵𝑇 = 𝐸 𝑑𝑒𝑠𝑖𝑔𝑛
𝐸 𝑓 𝑢𝑒𝑙-𝑠𝑎𝑣𝑒𝑑 -𝐸 𝑙𝑜𝑠𝑠 -𝐸 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 [START_REF] Lalau | Energy analysis and life cycle assessment of a thermal energy storage unit involving conventional or recycled storage materials and devoted to industrial waste heat valorisation[END_REF] Here, for each design the EPBT criteria is evaluated as defined in Eq. [START_REF] Lalau | Energy analysis and life cycle assessment of a thermal energy storage unit involving conventional or recycled storage materials and devoted to industrial waste heat valorisation[END_REF]. This criteria is calculated without taking into account the loss cost 𝐸 𝑙𝑜𝑠𝑠 in BM n o 1 and with loss cost in BM n o 2. It is to be noticed that EPBT criteria, unlike the previous criteria, is adapted for high risk projects where the storage operator is engaged on short amount of time.

Figs. 11(a) and 11(b) show the evolution of EPBT in function of the storage installed capacity for each optimization model and for the two defined BM. Fig. 11(a) for BM n o 1 shows that all models, except 1D -MM 3,7 , have similar controller performances and objective function minimized value. In mean, for all investigated storage capacities, 1D -MM 3,7 increases the EPBT criteria by 5% as shown in Table 4. This is an illustration of the consequences of insufficient discretization when building a metamodel. Also, degraded model performance decreases (compared to other models) for large storage capacities. On the other hand, 0D models have similar management performances as 1D -MM 7,15 for BM n o 1, where objective function accounts only for fuel consumption without finer terms such as losses that depends on local system temperature values. Although 1D -MM 3,7 has degraded performance, all models give the same optimal design situated around 4 MWh 𝑡ℎ storage. This shows that in case of BM n o 1, where only fuel consumption is being taken into account in the objective function, simple and degraded models can be used to find the optimal design. However, degraded models do not succeed to reach other models optimal management performance.

As a comparison, regarding BM n o 2, Fig. 11(b) shows that an increasing error can be observed for larger storage capacities when investigating the 0D -Ideal results. This can be explained by the poor estimation of generated losses, in 0D -Ideal, that is being taken into account in BM n o 2. Those errors become more important as more losses are generated in large storage capacities. It is to be noticed that 1D -MM 7,15 has the best optimal management score that other models could not reach. In mean, for all investigated storage capacities, the increase in EPBT criteria is 100, 25 and 14% for 0D -Ideal, 0D -Uniform and 1D -MM 3,7 respectively compared to 1D -MM 7,15 . On the other hand, the increase in optimal EPBT criteria is 35, 32 and 15% for 0D -Ideal, 0D -Uniform and 1D -MM 3,7 respectively compared to 1D -MM 7,15 as shown in Table 4. However, the decrease in models management performances, except for 0D -Ideal, does not prevent them from reaching the optimal design as they commit similar errors for all storage capacities. All models except 0D -Ideal reach similar optimal design around 6 MWh 𝑡ℎ . Only 0D -Ideal reaches lower optimal design capacity around 4 MWh 𝑡ℎ . The results show that for BM n o 2, where losses are being taken into account, only models that consider global losses can reach the optimal design point. On the contrary, 0D -Ideal that only considers losses after storage saturation, fails to reach the optimal design. It is to be mentioned that metamodels results show improvements when they are more discretized (as described in Table 1) achieving better results for 1D -MM 7,15 . In fact, larger state discretization vectors in metamodel lead to higher accuracy and lower approximations during the interpolation process. The limit of the discretization is governed by the choice of the logistic function used to approximate the storage temperature curve in the metamodel. Finally, it is to be mentioned that, the EPBT criteria for the optimal design does not increase significantly by accounting the storage losses. In effect, it is possible to apply an optimal management avoiding losses without decreasing significantly the saved gas amount thanks to storage flexibility.

Simulation vs co-optimization performances

As a reminder, Section 4 showed that 1D -MM 7,15 have the best simulation performance score with an NRMSD lower than 3% when compared to 1D -PDE as shown in Table 1. Hence, the optimization scores achieved by 1D -MM 7,15 are the best achieved scores in both BM n o 1 and BM n o 2. In BM n o 1, 0D -Ideal as well as 0D -Uniform optimized EPBT remain very close from 1D -MM 7,15 EPBT for almost all investigated capacities. This occurs although both 0D models commit non neglected errors in simulation process. On the other hand, 1D -MM 3,7 commits the highest EPBT and NRMSD simulation errors. It seems that models which are able to estimate approximately the objective function terms (here storage energy level only for BM n o 1) can be used for optimal design purposes. However, the more models are accurate in the storage energy simulation, the more management performance is improved. In BM n o 2, 0D -Ideal fails to reach the optimal design as it does not consider storage losses before saturation.

Table 4

Summary of different models optimization performances. The increase in EPBT is calculated relatively to 1D -MM 7,15 EPBT that reaches the lowest values. 𝑡 𝑜𝑝𝑡 is the computation time to solve the optimization problem over one hour using MPC algorithm as described in Section 3.

Models

Loss An important increase in EPBT can also be observed in the management performance, although it has an acceptable NRMSD simulation score. This shows that models which does not consider one of the objective function terms, are not able to find the optimal design and scores poor management performance. Other models that consider losses succeed to reach the optimal design and acceptable management scores even though those models commit important NRMSD simulation errors. In terms of simulation and optimization computation time, in average, 0D models are 10 times faster than metamodels, which are 100 to 1000 times faster than 1D -PDE as described in Tables 1 and4. In other words, for objective functions such as BM n o 1 where the goal is to optimize system-wide quantities, 0D models reach satisfying optimization scores compared to accurate 1D models with lower computation time. However, for finer objective functions such as BM n o 2, 1D -MM 7,15 allows to reach a more optimal design and management score with greater but acceptable computation time.

Conclusions and perspectives

In this work, 5 models of increasing complexity level (0D -Ideal, 0D -Uniform, 2 metamodels and 1D -PDE), have been investigated. Those models have been confronted using a HTTES co-optimization problem, on the basis of a CSP energy source and industrial load over a year, where boiler fuel is to be avoided. Two BM have been studied, with and without considering losses generated by the HTTES. A simulation over a year has been carried out, where NRMSD has been calculated between each investigated model results and those of 1D -PDE (the latter is the most accurate model considered in this study). Results show that the metamodel that has high discretization level achieves the best NRMSD score lower than 3%.

Later on, a defected experimental setup is used to validate the models on the basis of real measurements. In this case it has been shown that all previously studied models fail to predict the behavior of such installation with NRMSD greater than 12%. However, a new metamodel constructed on the basis of real measurements, succeeds to predict the experimental setup behavior with NRMSD lower than 6%. Furthermore, a co-optimization study has been conducted using the available models and for the two BM. A first optimal design evaluation using CED criteria has been conducted, over multiple operating periods. Results show that the longer the HTTES operating period, the greater is the optimal design capacity, with 12 and 18 MWh optimal capacities after 1 and 4 years of operation respectively. Therefore, it is more profitable to invest in greater storage capacities if storage operator is engaged over longer period, but this come with higher risk linked to unknown future events. Finally, a criteria called EPBT has been established for models performance comparison. BM n 𝑜 1 shows that, when losses are not considered, all models succeed to reach the optimal design, but the most degraded model fails to reach the optimal management performance with an increase of 3% in the EPBT criteria for the latter. BM n 𝑜 2 shows that, when losses are considered, models which consider losses succeed to predict the optimal design point but fails to achieve the same management performance as the highly discretized metamodel. In average over all investigated designs, an increase of EPBT criteria of 100, 25 and 14% for 0D -Ideal, 0D -Uniform and 1D -MM 3,7 respectively, can be observed compared to 1D -MM 7,15 EPBT values. Finally, the comparison between the 2 BM shows that, considering losses increases slightly the EPBT from 4.3 to 5.5 months, if co-optimized with appropriate model.

To summarize, we can distinguish three categories of models.

• First, models that consider all objective function terms with high accuracy. They reach the optimal design but also the optimal management score. This comes with considerable calculation time, in average 10 s per MPC iteration. • Second, models that consider approximately all objective function terms. They reach the optimal design but score less optimal management. Generally, they have low computation time, in average 1 s per MPC iteration. • Finally, models that do not consider all objective function terms.

Those models fail to find the optimal design and management strategy.

In future work, multi-energies network could be considered to evaluate the model impact in such environment. Additionally, more Adapted for cylindrical packed bed [START_REF] Hoffmann | Stockage thermique pour centrale solaire thermodynamique à concentration mettant en oeuvre des matériaux céramiques naturels ou recyclés[END_REF] ℎ 𝑎𝑖𝑟 [W m -2 K -1 ] Heat transfer coefficient between walls and outside air (0.664𝑅𝑒 0.5 𝑎𝑖𝑟 𝑃 𝑟 0.5 𝑎𝑖𝑟 )𝑘 𝑎𝑖𝑟 ∕𝐿 𝑏𝑒𝑑 Adapted for forced convection of an incompressible laminar air flow on a plane wall [START_REF] Hoffmann | Stockage thermique pour centrale solaire thermodynamique à concentration mettant en oeuvre des matériaux céramiques naturels ou recyclés[END_REF] sophisticated objective functions and business cases can be investigated for further comprehension of modeling impact on controller performance. Finally, the study of HTTES co-optimization under uncertainties is essential as it corresponds to real time management characteristics.

CRediT authorship contribution statement

Declaration of competing interest

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

Fig. 1 .

 1 Fig. 1. HTTES simplified illustration.

Fig. 2 .

 2 Fig. 2. Illustration of the inputs, outputs and the main activated tasks when running the metamodel. 𝐿𝐹 : Logistic Function, 𝐿𝑆 : Least Square algorithm.

Fig. 3 .

 3 Fig. 3. Synopsis of the reference case study.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Simulation results of an optimal management issued from four investigated models and compared to 1D -PDE for a 20 MWh storage. Results show the evolution of the storage energy estimation and the error between the investigated model and the accurate 1D -PDE over a year.

Fig. 7 .

 7 Fig.7. Simulation results of an optimal management, issued from two 1D -PDE models and 1D -MM, compared to experimental test data for a 40 kWh test facility. 1D -PDE models use different correlations for ℎ 𝑣 coefficient as described in Appendix. The 1D -MM used in this figure is a metamodel built on experimental results measured for previous experiments explained in Table3. Results show the evolution of the storage energy estimation and the error between investigated models and the experimental data over 78 hours experiment.

Fig. 8 .

 8 Fig. 8. Evolution of the cost function, with 0D -Uniform as optimization model and 1D -PDE as simulation model, according to increasing sliding window size. Curves correspond to investigated storage capacities.

Fig. 11 .

 11 Fig. 11. Evolution of the Energy Pay Back Time (EPBT) according to battery capacity for BM n o 1 and BM n o 2. The optimization uses different models as described in the legend. The simulation of results is carried out by the physical model 1D -PDE.

  Al Asmi: Conceptualization, Methodology, Data curation, Software, Validation, Writing -original draft, Visualization. Roman Le Goff Latimier: Conceptualization, Writing -review & editing, Supervision, Funding acquisition. Yasmine Lalau: Life-cycle assessment tools, data and review. Thomas Brian: Experimental setup and testing. Hamid Ben Ahmed: Writing -review & editing.

  𝑛 𝑤 by solving Eq. (7) using a model in Table 1 ; simulate 𝑃 𝑠𝑡𝑜 [0] with 1D -PDE model ; calculate 𝑃 𝑓 𝑢𝑒𝑙 (𝑡) + 𝑃 𝑙𝑜𝑠𝑠 (𝑡) and energy balance with Eq. (2a) end

	Algorithm 2: MPC description
	Data: 𝑛 𝑤 , 𝑁
		𝑁	𝑖+𝑛 𝑤
	Result: min	∑ 𝑖=0	∑ 𝑡=𝑖	𝐶 𝑡𝑜𝑡 (𝑡)
	initialization;
	for i in N do
	calculate ⟨𝑷 𝒔𝒕𝒐 ⟩

Table 1

 1 Summary of studied models simulation performance in terms of computation time and NRMSD error. The NRMSD concerns the storage energy evolution over a year of operation, and has been calculated relatively to 1D -PDE results.

	Model	𝑁 𝑐𝑜𝑛𝑠	𝑡 𝑐𝑜𝑛𝑠 [s]	𝑡 𝑠𝑖𝑚 [s] a	NRMSD [%]
	0D -Ideal	-	-	< 0.001	8
	0D -Uniform	-	-	< 0.001	13
	1D -MM 3,7	567	600	≈ 0.001	15
	1D -MM 7,15	36 015	36 000	≈ 0.001	3
	1D -PDE	-	-	1 -10	-

a Intel ® Core i7-6820HQ CPU @ 2.70 GHz.

Table 2

 2 Summary of the HTTES prototype characteristics used for conducting experimental tests.

	Geometry	Flow	HTF	HSM	𝐷 𝑏𝑒𝑑	𝐿 𝑏𝑒𝑑	𝑉 𝑏𝑒𝑑	𝐷 𝑝	𝐸 𝑚𝑎𝑥	𝑇 𝑠𝑡𝑜𝑟𝑎𝑔𝑒,𝑚𝑎𝑥	𝑃 𝐶𝐻,𝑚𝑎𝑥	𝑃 𝐷𝐶,𝑚𝑎𝑥	𝜀	Sensors
					[m]	[m]	[m 3 ]	[mm]	[kWh th ]	[ • C]	[kW]	[kW]		used
	Cylindrical	Horizontal	Air	rocks	0.35	1.5	0.144	30	40	575	27.6	37.6	0.425	9

Table 3

 3 Summary of the different conducted experimental tests for the metamodel construction.

	Test number	1	2	3	4
	Charge power [kW]	13.4	18.15	22.87	27.6
	Discharge power [kW]	20	25.87	31.74	37.6
	Test duration [h:m]	8:15	6:32	5:38	5:00
	Measurement period [s]	60	60	60	60

Table A . 1

 A1 Summary of all correlations and physical proprieties used in this study 1D -PDE model. l 𝑤𝑎𝑙𝑙 is the wall thickness, l 𝑖𝑠𝑜 is the insulation layer thickness, 𝑅𝑒 and 𝑃 𝑟 are Reynolds and Prandtl numbers. 𝑏𝑒𝑑 +2 l 𝑤𝑎𝑙𝑙 ) ⋅L 𝑏𝑒𝑑 +(2 𝜋⋅ (D 𝑏𝑒𝑑 +2 ⋅ l 𝑤𝑎𝑙𝑙 ) 2 )/4 𝑃 𝑟 0.33 )𝑘 𝑓 ∕𝐷 𝑝 (700(𝜌 𝑓 ⋅ 𝑢/𝐷 𝑝 ) 0.76 203𝑅𝑒 0.33 𝑃 𝑟 0.33 +0.220𝑅𝑒 0.8 𝑃 𝑟 0.4 )𝑘 𝑓 ∕𝐷 𝑝

	Parameter	Signification	Correlation/value		Choice justification	Reference
	𝜌 𝑓 [kg m -3 ]	Air density	6.75E-18 T 6 𝑓 -2.429E-14 T 5 𝑓 +	Experimental measurements	[17]
				3.561E-11 T 4 𝑓 -2.799E-8 T 3 𝑓 +
				1.343E-5 T 2 𝑓 -0.004509 T 𝑓 + 1.274
	c 𝑝𝑓 [J kg -1 K -1 ]	Air heat capacity	2.42E-10 T 4 𝑓 -7.131E-7 T 3 𝑓 +	Experimental measurements	[17]
				6.581E-4 T 2 𝑓 -8.615E-3 T 𝑓 + 1006
	k 𝑓 [W m -1 K -1 ]	Air thermal	9.38E-12 T 3 𝑓 -2.59E-8 T 2 𝑓 +	Experimental measurements	[17]
			conductivity	7.30E-5 T 𝑓 + 2.477E-02
	k 𝑒𝑓 𝑓 𝑓	[W m -1 K -1 ]	Air effective thermal	𝜀 ⋅ 𝑘 𝑓				HTF conductivity in porous	[17]
			conductivity					medium
	𝜌 𝑠 [kg m -3 ]	Bauxite density	3005				Experimental measurements	[17]
	c 𝑝𝑠 [J kg -1 K -1 ]	Bauxite heat capacity	8.890E-10 T 3 𝑠 -1.850E-6 T 2 𝑠 +	Experimental measurements	[17]
				1.531E-3 T 1 𝑠 + 7.527E-1
	k 𝑠 [W m -1 K -1 ]	Bauxite thermal	-2.518E-9 T 3 𝑠 + 5.423E-6 T 2 𝑠 -	Experimental measurements	[17]
			conductivity	4.95E-3 T 1 𝑠 + 5.070
	k 𝑒𝑓 𝑓 𝑠	[W m -1 K -1 ]	Bauxite effective	(1 -𝜀) ⋅ 𝑘 𝑠			HSM conductivity in porous	[17]
			thermal conductivity					medium
	𝜌 𝑤 [kg m -3 ]	Wall density	8070				Steel density	-
	c 𝑝𝑤 [J kg -1 K -1 ]	Wall heat capacity	500				Steel heat capacity	-
	k 𝑤 [W m -1 K -1 ]	Wall thermal	18-45				Experimental measurements	[18]
			conductivity
	𝐴 𝑓 ↔𝑤 [m 2 ]	Exchange surface	(𝜀)(𝜋⋅ D 𝑏𝑒𝑑 ⋅L 𝑏𝑒𝑑 +(2 𝜋⋅ D 2 𝑏𝑒𝑑 )/4)	Exchange surface in porous	[35]
			between fluid and walls					medium
	𝐴 𝑠↔𝑤 [m 2 ]	Exchange surface	(1-𝜀)(𝜋⋅ D 𝑏𝑒𝑑 ⋅L 𝑏𝑒𝑑 +(2 𝜋⋅ D 2 𝑏𝑒𝑑 )/4)	Exchange surface in porous	[35]
			between solid and walls					medium
	𝐴 𝑤↔ext [m 2 ]	Exchange surface	𝜋⋅ (D Exchange surface in porous	[35]
			between walls and					medium
			outside
	ℎ 𝑣 [W m -3 K -1 ]	Heat transfer coefficient	(2+1.1𝑅𝑒 0.6 Models that takes into account	Wakao and
			between fluid and solid					inter-particle diffusion and	Funazkri, 1979
								adapted for air-solid bed	[36] Coutier
								and Farber,
								1982 [37]
	ℎ 𝑤 [W m -2 K -1 ]	Heat transfer coefficient	(0.Adapted for near-wall zone	Beek, 1962
			between walls and bed					air-solid bed with spherical	[18]
								particles and with 𝑅𝑒>40
	ℎ 𝑒𝑥𝑡 [W m -2 K -1 ]	Global heat transfer coefficient between walls and outside	1 𝐴 𝑤↔ext	⋅	1 𝐷 𝑏𝑒𝑑 +𝑙 𝑤𝑎𝑙𝑙 +𝑙 𝑖𝑠𝑜 𝐷 𝑏𝑒𝑑 +𝑙 𝑤𝑎𝑙𝑙 2𝜋𝑘 𝑖𝑠𝑜 𝐿 𝑏𝑒𝑑 log	+	1 ℎ 𝑎𝑖𝑟 ⋅ 𝐴 𝑤↔ext

Acknowledgments

The authors would like to express their gratitude to the Association Nationale de la Recherche et de la Technologie (ANRT France 2018/1534) and the Danish Energy Technology Development and Demonstration Program (EUDP 64016-0027) for funding this work. Appendix See Table A.1.

Data availability

The data that has been used is confidential.