Familiaux Dans

Les Cooperatives De

L A Reforme

Agraire Du

Olivier Petit
email: olivier.petit@univ-artois.fr

Valère Martin
email: valere.mart1@gmail.com

Marcel Kuper
email: kuper@cirad.fr

; Fatah

Claire Papin Stammose
email: papinstammose.claire@yahoo.fr

WORKER, PEASANT OR ENTREPRENEUR? ANALYSIS OF THE ENTREPRENEURIAL LOGICS AND PRACTICES OF FAMILY FARMERS IN AGRARIAN REFORM COOPERATIVES IN THE SAISS (MOROCCO) TRAVAILLEUR, PAYSAN OU ENTREPRENEUR ? ANALYSE DES PRATIQUES ET LOGIQUES ENTREPRENEURIALES DES AGRICULTEURS

Keywords: groundwater economy, innovation, modes of farming, entrepreneurial practices and logics, irrigation, Saiss, Morocco

The aim of this paper is to analyze the emerging entrepreneurial practices and the underlying logic of family farms in two agrarian reform cooperatives in Morocco. These practices can be explained by the constant negotiation of multiple and sometimes even antagonistic logics (peasant, entrepreneurial, proletarian, capitalistic) within these farms in a context of rapid agrarian change and a juxtaposition of different farm types on the same territory. Five factors illustrate this emergence: (1) the access to credit, (2) the functioning of the farm (rotation of the crops, use of inputs, workforce), 3) the access to groundwater resources, (4) the marketing practices adopted by farmers and (5) the informational factors. The porosity of the peasant and entrepreneurial worlds is the main lesson we can draw from our study. There is a subtle process of hybridization between the peasant and entrepreneurial modes of farming, with a wide range of profiles, ranging from a pure 'peasant', to a pure 'entrepreneur' and in between the peasant-entrepreneur and the entrepreneur-peasant. If we only focus on the political discourse, the trend in the development of new modes of farming seems inescapable. Our study stresses the resistance of practices and logics of peasant modes of farming which can mix with a 'modern' vision of agriculture. However, the siren songs of entrepreneurship can lead to bankruptcy, an exit from agriculture, which could have a strong impact on the social cohesion of the Moroccan society, particularly in rural areas.

Introduction and analytical framew!ork

The aim of this paper is to analyze the (emerging) entrepreneurial practices in family farms in Morocco. These practices can be explained by the constant negotiation of multiple and sometimes even antagonistic logics (peasant, entrepreneurial, proletarian, capitalistic) within these farms in a context of rapid agrarian change and a juxtaposition of different farm types on the same territory. The setting of our study is the development of a "groundwater economy" [START_REF] Shah | Taming the Anarchy: Groundwater Governance in South Asia[END_REF] in the Saiss (Morocco) which has witnessed over the past 25 years a rapid growth of irrigated agriculture, mainly depending on groundwater (almost 50,000 ha of irrigated area, of which 45,000 ha directly depend on groundwater in 2012). The evolution of technology, public policies and new market opportunities (among many other factors) have profoundly transformed the relationships between farmers, land, innovation processes and their agricultural practices [START_REF] Bekkar | On the difficulty of managing an invisible resource: farmers' strategies and perceptions of groundwater use, field evidence from Morocco[END_REF][START_REF] Benouniche | Bricolage as innovation: opening the black box of drip irrigation systems[END_REF]Dugué et al., 2015). However, policy makers and development agents continue to view small-scale farming as a social activity, as if they were still following a traditional model of peasantry. In contrast, our paper will show that several forms of hybridization of logics and practices within family farms can be observed.

Three typical modes of farming

Van der Ploeg (2009) contrasts three typical models of agriculture, namely the capitalist, peasant and entrepreneurial modes of farming (fig. 1). The first corresponds to the model of a capitalist agricultural system, mostly driven by agrobusiness market exports. This model is characterized by productive capital, the massive use of hired labor, financial and technical capital, organized in an "Empire" where the consumer and the producer never meet. In contrast, the model of peasant agriculture represents a small farming structure based on the use of an essentially family labor, mainly focusing on subsistence farming and short supply and commercialization chains. The third model would be intermediate: the agricultural entrepreneur. It would evolve between the two aforementioned models according to its size and financial capacity. However, according to [START_REF] Van Der Ploeg | The new peasantries: struggle for autonomy and sustainability in an era of empire and globalization[END_REF], there is no clear dividing line between these three models : "At the interfaces there is considerable overlap and ambiguity, and "borderlines" are crossed through complex moves both backwards and forwards" (Van der Ploeg, 2009). However, these three models of agriculture follow different logics and practices in terms of networking, organization, and production.

The distinction between logics and practices in the modes of farming

The study of farming practices aims at examining the organizational structure, the modes of acquisition and use of inputs (land, water etc.), from breeding to processing and commercialization of agricultural products, information acquisition, innovation and spin-offs. The main criteria of distinction are based on the analysis developed by [START_REF] Van Der Ploeg | The new peasantries: struggle for autonomy and sustainability in an era of empire and globalization[END_REF] grounded on the differentiation between the peasant and entrepreneurial modes of farming.

From a methodological point of view, a distinction is made between practices and logics of these two modes of farming. Practices describe how farmers act and how a mode of farming is concretely implemented. Logics refer to the strategy and organization of the mode of farming. Logics do not necessarily predetermine practices, because experimental practices can also influence farmers' logics for example. Since the peasant and entrepreneurial modes of farming are largely an ideal type6 , our description of farming logics and practices of these two modes of farming are also idealized. However, as we will show below, this distinction will help us, in a second step, to identify several kinds of hybridization.

To summarize, the entrepreneurial practices aim at developing processes of production and commercialization more or less connected to the market, generating an added-value, which will enable to expand the investment and capitalization capacities. These practices are based on an increased use of wage-labor force and on a strong investment in technologies. They are also based on a dependence on markets for 1) input provision such as seeds, fertilizers and pesticides, 2) productive resources such as land and water, and 3) financial means (credit).

The entrepreneurial logics are much more difficult to characterize. However, they are grounded, according to us, on a will to generate higher profits so as to develop, expand and intensify food production. Subsidence food producing agriculture is not considered as a satisfactory end. Besides, attachment to land is weaker. Land is much more considered as an input whose profitability must be maximized. This way of thinking is intrinsically based on a short-term horizon, because the substitutability of one land compared to one another is understood only from the possibility to generate benefits. Finally, the entrepreneurial logics are based on an organization of the farm developed on a shareholder basis or according to a hierarchical logic: in all cases outside of the family sphere.

In contrast, the peasant practices are based on loose ties with the market economy, concerning the employment of workforce (the workforce used is mainly related to the extended network of the family), the use of inputs (natural fertilizers, coming from their own livestock) and financial aspects (no bank loan contracts but informal credit). Peasant logics are characterized by a mode of development aiming at subsidence food producing agriculture or turned towards the acquisition of comfort. The added value generated by agricultural production is intended neither to fund the expansion of the farm (in terms of land acquisition or investment in costly technologies), nor the degradation of productive resources such as land or water which will enable the development of farming in the medium and long run. Moreover, the peasant logics are based on the idea of land transfer towards future generations, so as to help them becoming a farmer. Thus, the organization is primarily turned towards the family.

Based on the characteristics identified by [START_REF] Schumpeter | The Theory of Economic Development[END_REF], followed by the works of [START_REF] Wennekers | Linking Entrepreneurship and Economic Growth[END_REF] and van der Ploeg (2009) among others, we can propose a definition of the entrepreneur applicable to the area of agriculture. The entrepreneur can be defined as an actor connected with capital markets (land, workforce, agricultural inputs…) and able to innovate or to use innovation to develop wealth creation in a competitive market. The modes of organization developed by the entrepreneur can be inspired from preexisting or innovating models, but the main factor explaining the innovation spin-off is linked with his capacity to mobilize capital.

The hypothesis behind the links between entrepreneurial practices and logics is based on the idea that a pro-capitalist behavior leads to the implementation of market-based strategies to develop agricultural projects. The agricultural project is the projection of a behavior and of an agricultural practice aiming at generalizing an (observed or already implemented) practice in the future. Thus, the nature of the agricultural project and the strategic means enabling its implementation will determine whether the farmers follow an entrepreneurial or a peasant mode of farming. For instance, drilling a tube-well can be understood as an entrepreneurial agricultural project if -based on a bank loan contract and on commercial inputsit aims at generating profits enabling to expand the surface of the lands cultivated, which, in turn will generate a new capital-intensive project (in the agricultural domain or in another). In all cases, this project to drill a tube-well is expected to cover far more than subsidence food producing agriculture.

Concretely, agricultural practices are not empty of meaning. The more it will be connected to the market and motivated to increase agricultural yields beyond the basic needs and search for comfort, the more it will be an obvious translation of entrepreneurial logics.

Inspired by this analytical framework, an empirical study was conducted between April and June 2013 [START_REF] Papin-Stammose | Résilience de la paysannerie marocaine dans un contexte de mutation de l'accès aux ressources foncières et aux eaux souterraines[END_REF] and between April and June 2014 [START_REF] Martin | Etude des pratiques et logiques entrepreneuriales en agriculture dans deux coopératives de la réforme agraire : plaine du Saïss (Maroc)[END_REF] in two agrarian reform cooperatives located in the Saiss plain (Morocco): Ezzahra and Iqaddar. Based on the observation of the practices and on the analysis of logics developed by the diversity of farmers established in these cooperatives, we analyze the factors explaining the emergence of entrepreneurial agricultural practices (selection of crops, irrigation vs rain-fed, livestock, land markets, land renting, access to credit, etc.). However, we contrast this trend by showing the persistence and resurgence of peasant practices (informal credit, land association, etc.). Then, we present the multiple and (sometimes) antagonistic logics inside family farms and their incarnation in agricultural practices. This case study in the Saiss plain will allow us to extend the analysis of the modes of agriculture proposed by [START_REF] Van Der Ploeg | The new peasantries: struggle for autonomy and sustainability in an era of empire and globalization[END_REF], so as to stress the various forms of hybridization currently occurring.

Analysis of the entrepreneurial logics and practices of family farmers in the Saiss

Presentation of the case study and context of the Groundwater Economy in the Saiss

Our study was carried out in two agrarian reform cooperatives covering a total area of 700 ha: Ezzahra and Iqaddar (Fig. 2). These cooperatives, located in the province of El Hajeb, were part of a larger cooperative created by the State in 1978 for the production of cereals, after the dismantling of the agricultural land management society (SOGETA). The assignees of the large cooperative worked under the supervision of staff of the Ministry of Agriculture, and often compared their situation to that of labourers. The State distributed only in 1991 land use rights of 9-13 ha to 65 members of State cooperative. In the 1990's, organizing work on a cooperative basis, under the direction of the State, was mostly based on imposed crop rotation, so as to facilitate cultivation operations, inputs acquisition and product commercialisation. But the failure of this system, based on rain-fed agriculture, considered unprofitable by the farmers themselves, was progressively abandoned by the farmers who succeeded to obtain an access to groundwater resources through individual wells and progressively developed diversified family farms. In these two cooperatives, groundwater access is difficult to obtain. In 2006, only 9 wells (3 located in Ezzahra and 6 in Iqaddar) were identified, which demonstrate that the massive development of groundwater exploitation through tube-wells is very recent.

Fig. 2 : Study area

In 2006, the State privatized the land, which was considered "frozen" by the administration [START_REF] Bossenbroek | One doesn't sell one's parents' : gendered experiences of shifting tenure regimes in the agricultural plain of the Sais in Morocco[END_REF] and members of the cooperatives could obtain property of the land for quite low prices. Following the ambitious 'Green Morocco Plan' of 2008 aimed at intensifying agriculture through irrigated, high value crops and at attracting investors to agriculture by providing substantial subsidies [START_REF] Akesbi | Le Maghreb face aux nouveaux enjeux mondiaux. Les investissements verts dans l'agriculture au Maroc[END_REF][START_REF] Faysse | The rationale of the Green Morocco Plan: missing links between goals and implementation[END_REF], a lively land market emerged in these cooperatives. Land prices were soaring very high (up to 50,000€/ha) and a number of the cooperative members had debts and were forced to sell the land. In 2014, 53 % of the surface of both cooperatives are now in the hands of these investors 7 and even 72 % of the irrigated area within the cooperatives 8 . In 2014, there was a tremendous increase in the number of tube-wells (18 in Ezzahra and 20 in Iqaddar), most of whom were installed by the investors, who are mostly 7 35 % for land purchasers and 18 % for lessees. 8 30% by land purchasers and 42 % by lessees. planting fruit orchards. Their irrigation equipment is actively subsidized by the State to promote intensive irrigated agriculture. The former cooperative members, on the other hand, only irrigate 21% of their land due to lack of capital to invest in tube-wells, but also due to reduce the risk linked to volatile agricultural markets for the high value crops. They are considered small-scale family farmers not so much because of their farm size (7-13 ha), but rather because of the lack of capital on these farms. There are also a number of lessees in these cooperatives, growing horticulture, by renting in the land from the cooperative members. These lessees also make intensive use of groundwater by installing tube-wells.

The emergence of entrepreneurial practices

The aim of this section is to understand the emergence of entrepreneurial practices in the two cooperatives under study, in a context dominated, during the last decades, by a peasant mode of farming, and confronted with new entrepreneurial modes of farming of investors. We will more particularly focus on five factors (1) the financial factors (investment, credit), (2) the factors linked with the functioning of the farm: rotation of the crops, the use of inputs such as workforce, fertilizers and pesticides, seeds, etc.), (3) the factors linked to the access to water resources and especially to groundwater, (4) the commercialization practices adopted by farmers and (5) the informational factors.

First, it is important to mention that the situation is really contrasted between the two cooperatives in the domain of credit. In 2005-2006, two cooperatives of the region were drawn and selected to benefit from a cancellation of debt. Ezzahra was concerned, contrary to Iqaddar. This has influenced the way farmers have managed their (symbolic and use) relation to formal credit.

The 'Green Morocco Plan' encourages the use of credit to increase farm performance through a credit policy managed by the bank 'Credit Agricole'. In practice, the interest rates are high (7 to 10% as an average in 2014) and this generally discourages small farmers to invest and request a bank credit. Moreover, the loan with interest is sometimes presented (especially in rural areas of Morocco) as 'haram' (illicit). But their disinterest is also related to peasant logic. A small farmer from Ezzahra states: "I did not use a bank credit after the decision of the king Mohamed VI to cancel our debt. Taking a loan means that you don't work for yourself, but for the bank. I prefer to work with less surface area rather than being exhausted to work on a large surface area with the money of the bank." This is the reason why many farmers prefer the solution of an association of production, with a neighbor for instance, to divide the cost of investment. Informal credit (interest-free) is however a solution largely adopted (in 5 cases out of 9 in Iqaddar and in 5 cases out of 13 in Ezzahra).

In Iqaddar, because the cancellation of debt did not occur and because the proportion of new investors coming generally from the city is more important than in Ezzahra, resorting to formal credit is more frequent. These new investors directly benefit from the credit policy of the Moroccan State, since they generally exploit large farms (55 ha as an average for the new investors, compared to 5 ha as an average for family farmers) and because the minimum size of the farms required to benefit from this credit policy is 5 ha… To put it in a nutshell, the new agricultural policy implemented by the Moroccan State, through the 'Green Morocco Plan' can be summarized with the famous slogan: "Big is beautiful"! The functioning of the farms also reflects an evolution in the modes of farming. Historically, cooperative members have witnessed a similar trend in the two cooperatives concerning the evolution of rain-fed and irrigated total areas. Groundwater access (through wells) first developed slowly and assignees were able to apprehend and develop progressively irrigated agriculture. Groundwater access (through tube-wells) then accelerated from 2006 onwards with the arrival of the investors. The rain-fed surface area has decreased from 1995 to 2009 in both cooperatives. Since 2009 however, there is stagnation of these surface areas. This change in the access to water, coupled with the possibility to rent or sell land, has also generated a change in crop rotation and in the varieties grown in the area. If cereals are still largely used by small farmers as rotational crops with legumes such as beans, the development of high value crops for horticulture (potatoes, onions) has developed in line with the evolution of irrigation and change of land status. In parallel, the planting of fruit trees and the development of orchards has been observed in the past decades.

Concerning the wage-labor used in both cooperatives, all the farmers interviewed hire seasonal workers, during specific periods. However, a distinction can be made since the former assignees continue to employ family workforce, whereas larger farms hire a wage-labor force, especially since the landowners are generally absent from the farm. This mechanism causes a number of interactions between the different farming modes, as a lot of young men and women, descendants of the former assignees, work on the large farms, thus observing and practicing new forms of agriculture.

In the area of seeds provision and agricultural inputs (fertilizers, pesticides), most of the small/family farmers produce themselves their seeds and use natural fertilizers but none of the farmers we met are only relying on self-productionthey have a clear link with the seeds and agricultural inputs markets. In larger farms, however, the wide majority of the farmers we met (almost) exclusively buy their seeds and agricultural inputs. This situation clearly shows a strict distinction between the peasant and entrepreneurial modes of farming.

Farmers also develop a different relation to land, water and especially groundwater access. Whereas small /family farmers generally continue to rely on rain-fed agriculture with only a small portion of their land which is irrigated, larger farms cover the majority of their cultivated surface area with irrigation (especially drip irrigation). Irrigation has been highly subsidized during the past ten years by the Moroccan government, but the property rights over the lands must be clearly established to obtain the authorization to drill a tube-well and the family farmers often do not have the means to finance the associated costs (equipped tube-well, drip irrigation, high cost inputs). In this context, small farmers who have not been able to obtain the formal property rights over the land they cultivate (especially in the case of Iqaddar) cannot obtain the authorization to drill a tube-well and benefit from the subsidy for the development of drip irrigation. Thus, it is possible to identify a twotiered system: large farms, cultivating high added-valued crops and able to plant orchards, have a comfortable groundwater access and benefit from subsidies to convert their farm to drip irrigation; whereas most of the small and family farms are still in a peasant mode of farming, cultivating rain-fed crops without the possibility to benefit from the subsidies of the Moroccan government. Of course, as will be later shown, this contrasted picture does not exactly illustrate all the spectrum of the farming practices which can be observed, since in many cases, a process of hybridization occurs. In terms of the marketing all the farmers we met in Ezzahra and Iqaddar confirmed that in 2009-2010, with the massive arrival of investors in both cooperatives, the price of onion seeds has substantially increased and this situation has generated the development of onion seed beds. "The buyers were often civil servants who didn't know anything about agriculture. They all bought the seeds at high prices. I couldn't afford to pay this price, then I decided to replant onions, just like my neighbors did. The following year, a lessee proposed me to buy a portion of my seed bed. I then decided to start planting more seed beds to sell the seeds." Mohammed, small farmer in Iqaddar. This situation can help us to understand the evolution of marketing practices between the farmers in the cooperativesdepending on their status and knowledge in agricultureand the role played by economic factorssuch as the seeds price -in the development of the marketing practices. At the same time, as mentioned by Mohammed during our interview, there is clearly an imitation process which is operating in the development of seed beds and in the marketing processes. The quotation above is also interesting in relation to the importance of informational factors. Channels of information are often difficult to circumscribe, since they can hardly be observable, except if you have a regular and repeated communication with the various actors. Our study of the farming practices and logics in Iqaddar and Ezzahra has benefited from long and repeated interactions with farmers, but also with local merchants (providers of agricultural inputs, materials, etc.). One figure is especially interesting to mention in this network of actors operating in the Saiss region: the brokers. Brokers are often people with multiple activities (taxi driver and farmer; local merchant and farmer, for instance). They act as nodal points or intermediaries in the groundwater economy, collecting information all along their activities and developing their businesses thanks to their deep knowledge of the places, actors, legislations, opportunities of subsidies, etc. Family farmers who generally work in their fields are less connected with brokers, since these brokers usually stay in the small towns of the region where the economic activity of the groundwater economy is especially developing (El Hajeb, Ain Taoujdate for instance). On the contrary, investors and managers of larger farms have more time to spend in town and they also benefit from continuous interactions with these brokers. Information is really crucial to succeed in agriculture in this groundwater economy.

The underlying logics of farmers

Farmers logics, as mentioned above, are much more complex to grasp. However, in-depth interviews with the farmers (assignees, lessees) and land investors can help to understand the main elements which can influence the logics behind the various modes of farming. These elements can be identified when questioning these actors on their future projects. According to our understanding of farmers' logics, these elements deal basically with (1) the logic of capital use, (2), the relationship to the land, (3) the capacity to expand the size of the farm (inside and outside of the cooperative), and finally (4), the farm organization.

In the case of family farmers, the profits associated with production are mostly dedicated to satisfy family needs and to related projects. Entrepreneurs, on the contrary, use the profits associated with production as a primary source of productive investment, which represent a strategy of intensification and development of the farm. The way farmers project themselves in the future is a good indicator of their intrinsic logic.

Another important dividing line between the peasant and entrepreneurial logics holds in the relationship established between farmers and the land they cultivate. Assignees and their families, generally have a very strong relationship with the land, since the land ownership has been very hard to obtain. They have come a long way, from being laborers to family farmers. However, attachment to the land has a different meaning in Ezzahra and Iqaddar. In Ezzahra, attachment to the land is more deeply rooted for assignees and their families (especially assignees' sons) than in Iqaddar. In Ezzahra, the sons of 3 assignees only (out of 13) have stated their readiness to sell (part of) their lands 9 . As reported by Hichem, heir of an assignee "Why should I sell my land?

If I sell what allows me to live, what should I do with this money? I'm a farmer, I know how to grow crops and take care of my livestock. Doing another job could be a pleasant activity but I don't know what I could leave to my children afterwards. Shame on those who sell their lands to buy a car and don't do anything

intelligent with this money. The land sustains us, I will never sell it, except if the life of my children is at stake". In Iqaddar, the price of the lands has grown very quickly in the past ten years and assignees have been more tempted to sell their lands to investors. Lessees and investors have of course a completely different relationship with the land they cultivate. Lessees are sometimes only looking for short-term incomes. They take out the riches of the land without any consideration for soil deterioration or pollution. Investors generally do not cultivate themselves their lands but they hire a manager who is in charge to develop irrigated intensive farming.

The differences identified in the relationship to land and in the capacity to mobilize the profits associated with production also explain the contrasted trends concerning the capacity to expand the size of the farm. Since most peasant modes of farming do not generate enough profit to buy new lands, family farmers generally prefer the solution of a local production association or to rent a parcel to a neighbor. In contrast, lessees coming from outside of the cooperative and new land owners investors do not only cultivate and grow up crops in one location, but have several parcels disseminated in the Saiss region. This strategy is explicitly considered, since the very beginning, as a model of expansion and intensification, coupled with the use of technologies aiming at generating short-term profits.

The last distinctive element identified in our study area concerning farmers logics is based on the structural organization of the farms. As already mentioned, assignees rely most of the time on their own families as a workforce. The patriarchal organization mode of farming is also connected with inheritance issues. The sons of the assignees share the work in the farm, in relation with their own future interest in the farm production. In the majority of the cases, family farms mode of organization is based on a strict division of labor. The father is the central decision-maker, even if he is no longer able to work for reasons of age or disability. The sons work in the fields and take care of livestock breeding. In Iqaddar, an unusual situation was observed, since the decision-making process is in the hands of the father and the oldest son. The latter has invested a large part of his savings in the farm and has been able to renegotiate his status in the decisionmaking process. Another interesting situation occurred on a family farm in Ezzahra, where the youngest son has developed new production techniques (drip irrigation) and has negotiated with his father and with his six older brothers a contract enabling him to obtain 25% of the net profit of the farm. However, beyond these specific cases, it is interesting to mention that in most cases, the decisional weight of the father is decreasing with the ability to work effectively in the fields. In several cases, informal association with neighbors has been observed, without knowledge and consent of the father. In this patriarchal mode of organization, gender inequalities are obvious. However, mothers sometimes play a hidden role as reported in another cooperative of the same area (Ait Ali) by [START_REF] Bossenbroek | One doesn't sell one's parents' : gendered experiences of shifting tenure regimes in the agricultural plain of the Sais in Morocco[END_REF]. The attraction to an entrepreneurial mode of farming is however generating, according to the same authors, a 'professionalization' of farming "with farming identities increasingly becoming reserved for, and actively taken up by, some ambitious young male farmers whose fathers or mothers used to be members of the state cooperative. Their positive appreciation of the opportunities offered by mechanization, higher value crops and drip irrigation is based on labelling of these changes as 'new' and 'modern'. They mark a new era, and allow male youngsters to positively distinguish themselves from their old-fashioned 'peasant' parents, and to become new, clean and entrepreneurial farmers." Bossenbroek and Zwarteveen (2015: 163).

In contrast with this patriarchal mode of farming, the organization of production in large farms owned by private investors is different. These private investors do not generally have the knowledge to conduct their farms and in most cases, when they arrived in the late 2000 in Ezzahra, but probably more often in Iqaddar, they searched for an association with the assignees who were already cultivating the lands since 20 or 30 years of time. The private investorsmost of them working as civil servantssought to maximize their investment and oriented their farm towards arboriculture, primarily plums or apples, considered as high-value crops. Their association with small farmers was considered as an opportunity to develop projects, based on the mastery of agricultureand especially horticultureof these small farmers. However, the concentration of investors developing horticulture in the same area of production and their lack of knowledge of the marketing processes generated financial worries according to five testimonies gathered in Iqaddar during our fieldwork. This situation helps us to stress the high dependence of the new investors on a variety of local actors who can have a crucial role from an organizational point of view.

Discussion and conclusion -New peasantries? Hybridization in peasant and entrepreneurial practices and logics

The understanding of peasant and entreprenaurial logics and practices, based on the fieldworks conducted in 2013 and 2014 in Ezzahra and Iqaddar and presented above, is only partial if we consider these two modes of farming as hermetic.

The porosity of the worlds, idealized in our presentation following the methodology we chose to adopt, is the main lesson we can draw from our study. What happens in reality is a subtle process of hybridization between the peasant and entrepreneurial modes of farming, with a wide range of profiles, ranging from a pure 'peasant', to a pure 'entrepreneur' and integrating, in between, the figures of the peasant-entrepreneur and of the entrepreneur-peasant.

The figure of the entrepreneur, as idealized by the Moroccan agricultural policy (Green Morocco Plan of 2008) seems to operate as a powerful and strange magnet to small farmers, especially young men, who continue to develop their activities in a structure and organizational mode of farming inherited from their ancestors (the Peasant mode of farming), but try, on a trial and error basis, to progressively integrate the knowledge, techniques and innovations developed in the entrepreneurial mode of farming. However, this world of entrepreneurship is not really 'their' world, and many resistances continue to operate in the way they conduct their farms. This is the reason why, many small farmers accept to rent (part of) their lands but refuse to sell. Selling their lands would mean turning away from agriculture to develop another economic activity. Several farmers we met in Iqaddar and Ezzahra returned from their brief attempt to develop a new mode of farming. Indebted and unable to continue on the trend of 'modern' agriculture, they decided to renew their livestockconsidered traditionally as the best savings method. But there are also several examples of success. For instance, one of the families decided to start to irrigate his lands in the late 1990's, only for horticulture. They currently generate profit and have adopted the techniques of the entrepreneurial mode of farming. However, the organizational structure of the farm is still embedded in the family.

If we only focus on the political discourse in the field of agriculture, the trend in the development of new modes of farming seems inescapable. Our study stresses the persistence and resistance of practices and logics of ancestral modes of farming which can mix with a 'modern' vision of agriculture. However, in several cases, the siren songs of entrepreneurship can lead to bankruptcy, an exit from agriculture, which could also have a strong impact on the social cohesion of the Moroccan society, particularly in rural areas such as the one we have studied.

Fig. 1 .

 1 Fig. 1. Three typical modes of farming (source: Van der Ploeg, 2009)

According toWeber (1949 :90) "It is not a description of reality but it aims to give unambiguous means of expression to such a description".

THEME 3 : WHAT GOVERNANCE FOR GROUNDWATER AND SURFACE WATER USE IN AGRICULTURE?

Worker, Peasant or Entrepreneur? Analysis of the entrepreneurial logics and practices of family farmers in agrarian reform cooperatives in the Saiss (Morocco)


Mathieu ! Paquier
email: mathieu.paquier@univ@brest.fr!

Vincent ! Koehl

Brice ! Jantzem

Effect of headphone position on absolute threshold measurements

Keywords: audiometry, !absolute!thresholds, !headphone!position!

teaching and research institutions in France or abroad, or from public or private research centers.

1.4.%Variability%specifically%due%to%headphone%positionning%

Most!of!test@retest!(with!systematic!removing!of!headphone)!experiments!cited!above! have! mixed! the! two! contributions! "threshold! variance"! and! "fitting! variance".! Indeed,! when! only! two! audiograms! are! realized,! the! effect! of! headphone! repositioning! is! merged! with! cognitive!factors,!procedure!validity,!etc.!! In! order! to! separate! between! "threshold! variance"! and! "fitting! variance",! three! audiograms! have! to! be! measured:! two! with! the! same! headphone! position! (the! difference! between!these!measurements!involves!only!"threshold!variance"),!and!a!third!with!an!other! headphone!position!(the!difference!between!this!third!measurement!and!the!two!other!ones! involves!both!"threshold!variance"!and!"fitting!variance").! Few! studies! have! separated! the! two! contributions! by! repeating! the! threshold! measurement! both! with! and! without! headphone! replacement.! Two! of! them,! using! insert! earphones,! found! an! effect! of! earphone! replacement! [11,! 24].! Hickling! [5]! showed! that! the! removal! and! replacement! of! TDH39! supra@aural! headphone! significantly! reduced! the! reliability!of!6!and!8!kHz!thresholds!in!comparison!to!repeated!threshold!measurements!for! the! same! headphone! position! (but! no! effect! at! 1! and! 2! kHz).! ! However,! only! these! four! frequencies! were! under! test.! Erlandsson! et! al.! [START_REF] Erlandsson | ! The! reliability! of! Bekesy! sweep! audiometry! recording! and! effects! of! the! earphone! position[END_REF]! and! Gauz! et! al.! [START_REF] Gauz | !High@frequency!Bekesy!audiometry:! III.!Reliability!and!validity!revisited.!The)[END_REF]! both! compared!!! auditory! thresholds! obtained! when! a! circumaural! headphone! was! repeatedly! replaced! to! thresholds!obtained!when!the!headphone!position!was!fixed.!Erlandsson!et!al.! [START_REF] Erlandsson | ! The! reliability! of! Bekesy! sweep! audiometry! recording! and! effects! of! the! earphone! position[END_REF]!reported!a! significant! effect! of! circum@aural! headphone! replacement! on! threshold! measurements! but! Gauz!et!al.! [START_REF] Gauz | !High@frequency!Bekesy!audiometry:! III.!Reliability!and!validity!revisited.!The)[END_REF]!reported!no!such!significant!effect.! ! ! ! ! ! 4!

1.5.%Summary% %

From!all!these!previous!works!it!can!be!seen!that!:! @ Studies! exhibited! large! differences! in! their! measurement! methods! and! most! of! them! consider! a! 5! or! 10dB! precision.! As! a! consequence,! results! are! often! contradictory! and! do! not! enable! to! really! know! if! circumaural! headphones! are! more!or!less!reliable!than!supra@aural!headphones.! @ Most! of! the! studies! used! only! one! retest! (with! headphone! repositioning),! and! did! not!enable!to!draw!a!global!conclusion!about!the!effect!of!the!headphone!placement! independently!of!the!"threshold!variance".!! @ Most! of! the! studies! tested! frequencies! up! to! 8kHz! (however! few! papers! tend! to! indicate! that! positioning! reliability! would! be! more! problematic! in! higher! frequencies,!but!counter@arguments!might!be!found!among!this!literature! [START_REF] Zhou | ! Reliability! of! pure@tone! thresholds! at! high! frequencies.! The![END_REF]).!!! @ In! addition,! the! circumaural! headphone! HD600! is! nowadays! often! used! in!

  psychoacoustics!experiments!and!a!recent!study![23]!has!shown!that!modifications! of! the! position! of! this! headphone! led! to! audible! changes.! However! no! study! has! specifically!studied!the!threshold!reliability!with!this!headphone.! ! Taking! all! these! considerations! into! account,! the! present! study! aimed! at! measuring! auditory!thresholds!with!high!precision!(1!dB!step)!on!a!large!frequency!range!(from!125!Hz! up!to!14!kHz)!using!both!Sennheiser!HD600!(circum@aural)!and!Telephonics!TDH39!(supra@ aural)! headphones,! with! and! without! headphone! repositioning.! Threshold! variability! was! then! investigated! by! comparing! the! audiograms! obtained! on! both! identical! and! different! headphone! positions,! and! enabled! to! separate! between! "threshold! variance"! and! "fitting! variance"!ANSI!procedure![26]!was!used.!The!audiometry!included!measurements!at! octave!intervals!from!250!Hz!through!8!kHz,!and!additional!thresholds!were!tested!at!125!Hz,! 6!kHz,!11!kHz,!and!14!kHz.!We!used!the!octave!intervals!from!250!Hz!through!8!kHz!because! they!are!used!in!clinical!audiometry!(according!to!ANSI).!Several!frequencies!were!added!to! expand!the!measurement!range!towards!the!low!and!high!frequencies!(125!Hz,!11!and!14!kHz! respectively

9 ! 11 !

 911 Figure! 1! (for! HD600)! and! Figure! 2! (for! TDH39)! show! the! absolute! value! of! the! differences! between! different! or! identical! positions,! for! each! frequency! and! for! all! subjects.!Note! that! for! different) positions,! the! two! differences! for! consecutive! and! non@consecutive! measurements! (|B@A|! and! |C@A|,! or! |C@B|! and! |C@A|)! were! merged.! As! the! distribution! of! differences!did!not!fit!a!normal!distribution!according!to!a!Kolmogorov!-Smirnov!test,!means! and!standard!deviations!were!not!suitable,!so!boxplots!were!displayed!!instead:!on!each!box,! the!central!mark!is!the!median,!the!edges!of!the!box!are!the!25th!and!75th!percentiles,!and!the! crosses!indicate!the!subjects!out!of!this!range.!


Rafael Muñoz-Tamayo

Pierre Martinon
email: pierre.martinon@inria.fr

Gaël Bougaran
email: gael.bougaran@ifremer.fr

Francis Mairet
email: francis.mairet@inria.fr

Olivier Bernard
email: olivier.bernard@inria.fr

Getting the most out of it: optimal experiments for parameter estimation of microalgae growth models

Keywords: biofuel, biological processes, modelling, parameter identification, optimal experiment design

Mathematical models are expected to play a pivotal role for driving microalgal production towards a profitable process of renewable energy generation. To render models of microalgae growth useful tools for prediction and process optimization, reliable parameters need to be provided. This reliability implies a careful design of experiments that can be exploited for parameter estimation. In this paper, we provide guidelines for the design of experiments with high informative content based on optimal experiment techniques to attain an accurate parameter estimation. We study a real experimental device devoted to evaluate the effect of temperature and light on microalgae growth. On the basis of a mathematical model of the experimental system, the optimal experiment design problem was formulated and solved with both static (constant light and temperature) and dynamic (time varying light and temperature) approaches. Simulation results indicated that the optimal experiment design allows for a more accurate parameter estimation than that provided by the existing experimental protocol. For its efficacy in terms of the maximum likelihood properties and its practical aspects of implementation, the dynamic approach is recommended over the static approach.

Introduction

Microalgae have received a specific attention in the framework of renewable energy generation [START_REF] Williams | Microalgae as biodiesel & biomass feedstocks: Review & analysis of the biochemistry, energetics & economics[END_REF]. However, optimizing productivity in large scale systems is a difficult task since microalgae growth is driven by multiple factors including light intensity, temperature and pH [START_REF] Posten | Design principles of photo-bioreactors for cultivation of microalgae[END_REF]. Mathematical modelling is thus required for quantifying the effect of environmental factors on microalgae dynamics.

In order to obtain reliable models that can be used in prediction and optimization of large scale systems, the model calibration stage requires carefully designed experiments with high informative content. Providing accurate parameters is indeed crucial since model-based optimality might be sensitive to parameters values as shown in [START_REF] Muñoz-Tamayo | Optimizing microalgal production in raceway systems[END_REF]. Moreover, assessing the effect of operational factors via sensitivity analysis can provide useful information for improving configuration design of photobioreactors [START_REF] Pereira | Parametric sensitivity analysis for temperature control in outdoor photobioreactors[END_REF].

Looking for high informative experiments is the objective of optimal experiment design (OED) for parameter estimation. Extensive work has been done for tackling the OED problem for dynamical systems (see, e.g., [START_REF] Goodwin | Dynamic System Identification: Experiment Design and Data Analysis[END_REF][START_REF] Walter | Identification of Parametric Models from Experimental Data[END_REF][START_REF] Keesman | System Identification: an Introduction[END_REF][START_REF] Franceschini | Model-based design of experiments for parameter precision: state of the art[END_REF]). The OED problem can be formulated as an optimal control problem. For low dimension models, analytical solutions may be obtained by the application of Pontryagin's Maximum Principle, which provides necessary conditions to be satisfied by the optimal inputs (see,e.g. [START_REF] Stigter | Optimal parametric sensitivity control of a fed-batch reactor[END_REF]). When model complexity increases, analytical solutions are arduous to obtain and thus the solution of the OED problem relies on numerical optimization techniques (see,e.g. [START_REF] Bauer | Numerical methods for optimum experimental design in DAE systems[END_REF][START_REF] Balsa-Canto | Computational procedures for optimal experimental design in biological systems[END_REF]).

When dealing with biological systems, OED approaches are based either on static or dynamic experiments (see, e.g., [START_REF] Bernaerts | On the design of optimal dynamic experiments for parameter estimation of a Ratkowsky-type growth kinetics at suboptimal temperatures[END_REF][START_REF] Bernaerts | Optimal experiment design for cardinal values estimation: guidelines for data collection[END_REF]). In this work, we analyze these two strategies and capitalize the available tools for OED to provide guidelines for the design of optimal experiments that allow an efficient assessment of the effect of temperature and light on microalgae growth. The model under investigation represents a real experimental device used to assess optimal growth conditions under batch mode. This device is operated at IFREMER Nantes, France.

The paper is organized as follows. Section 2 presents the system under study and its mathematical description, which corresponds to a simplified model of microalgae growth.

The OED framework based on this model is detailed in Section 3. In Section 4 we show the results of solving the OED problem. Two strategies are analyzed, namely static and dynamic approaches. Furthermore, we discuss about the relevance of OED for modeldriven decisions on raceway performance. For that, we make use of a local sensitivity analysis of a more complex model describing microalgae growth in an outdoor pond. In Appendix A, we discuss about the structural and practical identifiability of the model.

The main conclusions of the study are summarized in Section 5.

Modelling

We focus our study on the effect of temperature and light on the growth of microalgae.

More precisely, we aim at designing efficient experimental protocols for a real experimental system that allow an accurate estimation of the model parameters. The experimental apparatus, named the TIP (Fig. 1), consists of 18 batch photobioreactors located inside an incubator (see [START_REF] Marchetti | Optimizing conditions for the continuous culture of Isochrysis affinis galbana relevant to commercial hatcheries[END_REF] for more details). In each photobioreactor, it is possible to regulate the temperature, pH and light intensity.

Following the models developed for microalgae growth [START_REF] Bernard | Hurdles and challenges for modelling and control of microalgae for CO2 mitigation and biofuel production[END_REF][START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF], we study here a simplified model of microalgae growth under the hypotheses that the experiment is carried out at low cellular concentrations and under conditions of non-limiting nutrients. The first hypothesis implies that light is homogeneous along the depth of the photobioreactor.

The second hypothesis implies that the cells grow in exponential phase. The resulting mass balance equation on the TIP system reads

ẋ = f (x, θ, I, T, t) = µ(θ, I, T )x, x(0) = x 0 , (1)
with x the biomass concentration, I the light intensity and T the temperature in the reactor, θ the parameter vector and µ(•) the specific growth rate µ(•) defined by

µ(θ, I, T ) = µ max φ I φ T . ( 2
)
with µ max the maximal specific growth rate. The factors φ I , φ T , detailed below, represent the effects of light and temperature on microalgae growth.

Temperature has a homogeneous effect on uptake and growth rates [START_REF] Geider | Light and temperature dependence of the carbon to chlorophyll a ratio in microalgae and cyanobacteria: implications for physiology and growth of phytoplankton[END_REF]. The effect of temperature is described by the cardinal model developed for bacteria [START_REF] Rosso | An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model[END_REF] and validated for microalgae [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF].

φ T =              0, T < T min (T -Tmax)(T -T min ) 2 (Topt-T min )[(Topt-T min )(T -Topt)-(Topt-Tmax)(Topt+T min -2T )] , T ∈ [T min , T max ] 0, T > T max .
(

) 3
The effect of light (φ I ) on microalgae growth is often represented by a Haldane type kinetics that accounts for photoinhibition [START_REF] Peeters | The relationship between light intensity and photosynthesis-A simple mathematical model[END_REF]. The following parameterization of the standard Haldane equation is used [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF] φ

I = I I + µmax α I Iopt -1 2 , ( 4
)
where α is the initial slope of the growth response curve w.r.t. light.

In terms of practical identifiability properties, Eq. ( 4) excels the standard Haldane kinetics. For a brief discussion, the reader is referred to Appendix A.

The above equations implies that microalgae exhibit a maximal growth rate at optimal conditions of light (I opt ) and temperature (T opt ).

The model is then determined by the parameter vector θ θ = [µ max , α, I opt , T min , T max , T opt ].

In the next Section, we tackle the OED problem locally, that is the design of optimal experiments is carried out on the basis of nominal values θ. Table 1 shows the nominal values of the model parameters used in this study. They correspond to the microalgae Isochrysis aff. galbana, currently named as Tisochrysis lutea [START_REF] Bendif | On the description of Tisochrysis lutea gen. nov. sp. nov. and Isochrysis nuda sp. nov. in the Isochrysidales, and the transfer of Dicrateria to the Prymnesiales (Haptophyta)[END_REF]. Parameter values were mainly obtained from [START_REF] Muñoz-Tamayo | Optimizing microalgal production in raceway systems[END_REF] and [START_REF] Mairet | Modelling neutral lipid production by the microalga Isochrysis aff. galbana under nitrogen limitation[END_REF]. The temperature parameters are those of Nannochloropsis oceanica [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgal growth[END_REF] whose maximal and optimal temperatures are close to those of Tisochrysis lutea [START_REF] Renaud | Effect of temperature on growth, chemical composition and fatty acid composition of tropical Australian microalgae grown in batch cultures[END_REF].

OED problem for parameter estimation

The problem of OED for parameter estimation consists in designing an experimental protocol that provides data with high informative content to allow an accurate identification of the model parameters, that is to provide estimates with small confidence intervals.

Classical approaches of OED for parameter estimation rely on the optimization of a scalar function of the Fisher information matrix (FIM), since this matrix is the core for the calculation of the confidence intervals of the parameter estimates (see, e.g., [START_REF] Walter | Identification of Parametric Models from Experimental Data[END_REF], [START_REF] Franceschini | Model-based design of experiments for parameter precision: state of the art[END_REF]). Recent approaches such as the Sigma Point Method have been proposed to estimate parameter uncertainty without the explicit calculation of the FIM [START_REF] Schenkendorf | Optimal experimental design with the sigma point method[END_REF]. Here, we will focus on the classical approach.

Let us recall some basic principles of parameter identification. We consider here a local design approach. Our aim is to design optimal experiments on the basis of the nominal parameter vector θ. We first assume that the ith measurement (observation) y i of our experiment is modelled as:

y i = y m i (θ * ) + ε, ( 5
)
where y m i (θ * ) is the deterministic output of the model and θ * the true value of the parameter vector. The measurement error ε is here assumed to follow a normal distribution ε ∼ N(0, σ 2 ). Note that [START_REF] Goodwin | Dynamic System Identification: Experiment Design and Data Analysis[END_REF] implies that a deterministic model is available and represents adequately the system. Moreover, the model structure must be structurally identifiable.

In Appendix A, structural identifiability of the model is checked.

The maximum likelihood (ML) estimate θ of θ minimizes the cost function

J(θ) = 1 σ 2 s n i=1 (y i -y m i (θ)) 2 , ( 6
)
with n the number of data points.

The covariance matrix P of θ can be approximated to

P = F -1 ( θ), ( 7
)
where F is the Fisher information matrix. An estimate of the standard deviation of θ j is given by

η j = P jj . ( 8
)
We will be then interested in designing an experiment that render η j small.

In our case study, we aim at determining optimal profiles (or levels) of temperature (T ) and light intensity (I) for attaining an accurate estimation of parameters. Optimal experiments are built w.r.t. the D-optimality criterion, which maximizes the determinant of the FIM. Maximizing the determinant implies minimizing the volume of the confidence ellipsoids for the parameters [START_REF] Walter | Identification of Parametric Models from Experimental Data[END_REF].

By means of simulations, we tested also other optimality criteria, namely E-optimality (maximization of the smallest eigenvalue of the FIM) and modified E-optimality ((minimization of the condition number of the FIM)). D-optimality provided the best results in terms of the volume of the confidence ellipsoids. Therefore, we chose D-optimality as criterion of optimal design. Interestingly, the modified E-optimality criterion resulted in large confidence intervals. Indeed, it has been noted that since the modified E-optimality criterion is a criterion of shape of the ellipsoids, it is possible to obtain circular confidence regions with large volumes [START_REF] Pauw | Optimal experimental design for calibration of bioprocess models: a validated software toolbox[END_REF].

It should be noted that the performance of the obtained optimal experiment strongly depend on the nominal values of the estimates of the parameter vector. Ideally, θ should be as close as possible to θ * . In our case study, the nominal values of the parameters used are expected to be close to the true values, since the selection of priors was based on published experimental studies.

The OED problem is tackled by means of two strategies, namely dynamic and static approaches, which are detailed in the following.

Dynamic approach

The OED by the dynamic approach is directly applied on the dynamic (primary) model [START_REF] Williams | Microalgae as biodiesel & biomass feedstocks: Review & analysis of the biochemistry, energetics & economics[END_REF]. Here, the temperature and light intensity can be set to vary in time.

For the dynamic approach, the FIM reads as follows

F d ( θ) = 2 σ 2 d ne k=1 nt i=1 ∂y m k,i ∂θ T θ ∂y m k,i ∂θ θ = 2 σ 2 d M d , (9)
with n t the number of sampling times. Here, y m k,i is the biomass concentration predicted by the model (1) for the kth experiment at the ith time and σ 2 d is the noise variance of the measurement of biomass concentration. M d is the matrix resulting from the summation term. This formulation is made for facilitating further discussion. The terms in brackets in [START_REF] Stigter | Optimal parametric sensitivity control of a fed-batch reactor[END_REF] contains the local sensitivities of the model output w.r.t. the parameter vector θ.

The sensitivity functions were computed automatically with the Matlab Toolbox IDEAS [START_REF] Muñoz-Tamayo | IDEAS: a Parameter Identification Toolbox with Symbolic Analysis of Uncertainty and its Application to Biological Modelling[END_REF]. The toolbox is devoted to estimate parameters of ODE models. It uses symbolic differentiation to calculate the sensitivity functions for the evaluation of the FIM.

An approximate noise variance σ 2 d = 9.31 was calculated from the data reported in [START_REF] Lacour | Diel variations of carbohydrates and neutral lipids in nitrogen-sufficient and nitrogen-starved cyclostat cultures of Isochrysis sp[END_REF] and the mathematical model developed in [START_REF] Mairet | Modelling microalgae growth in nitrogen limited photobiorector for estimating biomass, carbohydrate and neutral lipid productivities[END_REF].

The OED problem is defined as

min ϕ d Det (F) , ( 10
)
with ϕ d the design vector

ϕ d = [T 1 (t), I 1 (t), . . . , T (t) ne , I(t) ne ] ,
such that

T L = 12 ≤ T k (t) ≤ T U = 33.2 • C I L = 20 ≤ I k (t) ≤ I U = 1200 µE m -2 s -1 ṪL = -5 ≤ Ṫk (t) ≤ ṪU = 15 • C, (11)
with n e the number of distinct experiments. We set n e = 9 with duplicate experiments.

The boundaries in [START_REF] Balsa-Canto | Computational procedures for optimal experimental design in biological systems[END_REF] correspond to the physical boundaries of the TIP system. Note that the rate of temperature change ( Ṫ ) is imposed. This constraint is bounded by the thermal dynamics of the equipment but also it must account for the potential thermal stress induced to the microalgal cells.

No boundaries were imposed to the rate of change of light, since it can be changed instantaneously. We assumed that microalgae respond instantaneously to light changes.

However, it is known that microalgae can adapt its photosynthetic system to changes of light [START_REF] Geider | Light and temperature dependence of the carbon to chlorophyll a ratio in microalgae and cyanobacteria: implications for physiology and growth of phytoplankton[END_REF]. Here, we consider time scales larger than the photosynthesis response time (in the range of minutes for photoinhibition). In this case study, we neglect however photoacclimation (adaptation of the pigment content to light intensity, at the scale of weeks). Further experiments will be needed to assess the dynamics of such an adaptation.

Note that ϕ d is of infinite dimension. However, ϕ d will be further transformed into a finite dimension vector to solve the optimization problem numerically.

Before attempting to solve the full OED problem, we first partitioned the original OED problem into simpler subproblems in which we study the effect of either temperature or light. This strategy was for instance used in [START_REF] Van Derlinden | Accurate estimation of cardinal growth temperatures of Escherichia coli from optimal dynamic experiments[END_REF] to estimate the cardinal temperatures for E. coli.

Each subproblem is dedicated to improve the accuracy of the estimation of a couple of parameters, while the other parameters are assumed to be known. In this case, F d is a square matrix of dimension 2 × 2 for each subproblem (for the full OED problem, the FIM is of dimension 6 × 6). The initial concentration of biomass was set to x 0 = 10 mg/L and. The duration of the experiment to t f = 4 d with ten equidistant sampling times.

When studying the temperature parameters, the light was set to

I = 547 µE m -2 s -1 ,
and when studying the light, the temperature was set to T = 26.7 • C. These constant values correspond to the optimal values for growth obtained from the model parameters (Table 1). This choice is supported by the fact that the FIM of each subproblem only involves either parameters related to the effect of light or to the effect of temperature, therefore the other experimental input only affects relatively the calculation of the sensitivity functions. By setting the experimental inputs to their optimal values, we favor growth.

A total number of nine subproblems was obtained. Table 2 shows the combination of parameters and the experiment input (T or I) for each subproblem. In practice, the nine solutions will be implemented in duplicates in the TIP.

The resulting subproblems were solved numerically via two discretization methods, namely sequential and simultaneous. The discretization allows to convert the original infinite dimensional optimization problem into a finite dimension problem. In the sequential approach (also called control vector parametrization (CVP)), the control variables are approximated by a set of basis functions that depend on a finite number of real parameters.

In the simultaneous approach, all state and control variables are discretized w.r.t. time.

Hence, this method is also known as total discretization. In this case, the dimension of the optimization problem depends on the number of discretization steps [START_REF] Chachuat | Global Methods for Dynamic Optimization and Mixed-Integer Dynamic Optimization[END_REF][START_REF] Biegler | Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes[END_REF].

The simultaneous method was implemented with the open source toolbox Bocop [START_REF] Bonnans | Bocop -A collection of examples[END_REF](http://bocop.org), based on the Ipopt solver [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF]. The simultaneous method used a Midpoint discretization with 1000 steps, with a 10 -14 tolerance for solving the discretized problem. All state and control variables were initialized with constant values.

Numerical solutions of the CVP approach were obtained with the SSmGo toolbox (http://www.iim.csic.es/ gingproc/ssmGO.html), with the parameterization depicted in Fig. 2. SSmGo performs global optimization by using a scatter search method [START_REF] Rodríguez-Fernández | Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems[END_REF][START_REF] Egea | Martí, Scatter search for chemical and bio-process optimization[END_REF].

The experiment inputs are thus defined by four parameters, namely u 1 , u 2 , t 1 , t 2 . The dimension of the optimization problem is therefore 9 × 4 with the decision vector

ϕ d = [u 1 (1), u 2 (1), t 1 (1), t 2 (1) . . . , u 1 (n e ), u 2 (n e ), t 1 (n e ), t 2 (n e )] . (12)

Static approach

The OED by the static approach is based on the secondary model of growth (here represented in (2)). In this approach, one experiment is characterized by a constant environment (T ,I in our case). The dynamic data of the biomass evolution for a given experiment is first used to calculate the maximal growth. Once different growth rates are calculated at different conditions of temperature and light intensity, the parameter estimation procedure is applied on the growth model [START_REF] Posten | Design principles of photo-bioreactors for cultivation of microalgae[END_REF].

Since the TIP system allows to run 18 experiments simultaneously, a parallel design procedure is here used. Hence, the following OED strategies aim at finding the nine best experiment conditions to account for duplicate experiments.

For the static approach, the FIM is computed as

F s ( θ) = 2 σ 2 s ne k=1 ∂y m k ∂θ T θ ∂y m k ∂θ θ = 2 σ 2 s M s , ( 13
)
where y m k is the maximal growth predicted by the model ( 2 The OED problem is then defined as

min ϕs Det (F) , ( 14
)
with ϕ s the design vector

ϕ s = [T 1 , I 1 , . . . , T ne , I ne ] ,
such that

T L = 12 ≤ T k ≤ T U = 33.2 • C I L = 20 ≤ I k ≤ I U = 1200 µE m -2 s -1 . ( 15
)
The design vector ϕ s ∈ R ne .

To solve the OED problem of the static approach, the Matlab optimization toolbox SSmGo was used.

Results and Discussion

Before presenting the resulting optimal experiments for both static and dynamic approaches, we should keep in mind that in our case study the D-optimal experiments do not depend on the value of the noise variance σ 2 , given that we assumed that the measurement errors are homoscedastic. Indeed, the optimal experiment inputs depend only on the matrix M, defined previously in [START_REF] Stigter | Optimal parametric sensitivity control of a fed-batch reactor[END_REF][START_REF] Bernaerts | Optimal experiment design for cardinal values estimation: guidelines for data collection[END_REF]. On the other hand, the confidence intervals of the estimates do depend on the actual value of σ since the estimate of the standard deviation of the parameter θ j is given by

η j = σ √ 2 ( M jj ) -1 . ( 16
)

OED by the static approach

The nine D-optimal experiments are given in Table 3. These experiments are defined by six levels of light intensity and five levels of temperature (if the decimal digits are omitted). Note that the nine experiments include the repetition of two experimental conditions (experiments 1,2 and experiments 5,6), which results in seven distinct experimental conditions. This result is not surprising since D-optimal often calls for the repetition of a small number of experimental conditions [START_REF] Walter | Identification of Parametric Models from Experimental Data[END_REF]. Simulated data resulted from the D-optimal experiments are illustrated in Fig 5A .     The performance of the D-optimal experiments was compared by means of simulation with a equidistant full 3 2 factorial design including duplicates and with the central composite design currently used in the TIP device [START_REF] Marchetti | Optimizing conditions for the continuous culture of Isochrysis affinis galbana relevant to commercial hatcheries[END_REF]. This composite design involved 17 experiments with five levels for the environmental variables temperature, pH and light intensity. Since in our study the effect of the pH is not considered in the OED problem, we only took into account the levels for temperature and light of the 17 experiments. The maximal level of temperature used in [START_REF] Marchetti | Optimizing conditions for the continuous culture of Isochrysis affinis galbana relevant to commercial hatcheries[END_REF] was 33.7 • C. We set the maximal temperature of culture to 33.2 • C, which is lower than the nominal value of the upper temperature for algae growth (T max ).

Table 4 illustrates the advantage of the D-optimal experiments over the factorial designs. Firstly, we notice that with the equidistant full factorial design the determinant of the FIM is zero, implying that the FIM is singular. Indeed, the inverse of the condition number of the FIM (defined as ratio of the largest eigenvalue to the smallest one) is smaller than the precision of floating point format (2•10 -16 ). In this case, confidence intervals for the parameter estimates can not be computed on the basis of the density of the estimator. To identify alternatives for guaranteeing a non-singular FIM for a full factorial design, a series of computations was performed. From the computations, it is concluded that a miminum number of four levels need to be considered in a full factorial design to provide a well-conditioned FIM. Other option to avoid an ill-conditioned FIM is to reduce the dimension of the matrix by splitting the full problem into subproblems (as we did for the dynamic OED). Our computations indicated that for combinations of five parameters

The combination that resulted in a singular FIM was [µ max , α, T min , T max , T opt ]. For combinations of four parameters (FIM has dimension 4×4), the FIM was well-conditioned for all the fifteen combinations.

It should be noted that in a simulation study performed in [START_REF] Van Derlinden | The impact of experiment design on the parameter estimation of cardinal parameter models in predictive microbiology[END_REF], full factorial design was applied for a cardinal model describing the effects of temperature, pH and water activity on the microbial growth rate, and the estimated values were close to the nominal values used in the simulation. However, we should not be content only with this result, since the actual values need to be supported by their corresponding confidence intervals in order to identify practical identifiability problems and to provide a quantitative measurement of the accuracy of the estimation.

We note that the composite factorial design does provide a well-conditioned FIM.

However the determinant of the FIM for this design is much lower than that obtained with the D-optimal design, and this is actually reflected on the accuracy of the estimates.

The second raw of Table 4 shows the ratio of the standard deviations of the parameters obtained with the D-optimal design to the standard deviations obtained with the composite design. D-optimal design provides lower standard deviations, 36% better in average.

This result establishes the benefit of designing optimal experiments with OED techniques for obtaining accurate parameter estimates.

OED by the dynamic approach

As it was mentioned in Section 3, the dynamic OED problem (10-11) was solved via the simultaneous and CVP approaches. While the CVP method reduces substantially the dimension of the original optimization problem, the simultaneous approach allows to find solutions without restricting the shape of the controls. These solutions potentially give better objective values, but may not be fit for practical use, if the controls have a complicated shape. Comparing the two methods also give a hint at what we lose by restricting the control shape to simple functions.

The CVP and the simultaneous approaches were compared for the case when the full problem was partitioned into nine subproblems devoted to improve the accuracy of the estimation of a couple of parameters. Overall, the CVP and the simultaneous methods find very similar solutions with the exception of the experiment 7 (see Fig. 3). The controls found with the simultaneous approach are often quite close in shape to piecewise linear functions. This confirms that our choice of shape for the controls in the CVP was a sensible one.

Table 5 compares the optimality cost functions provided by the simultaneous (J Sim )

and the CVP (J cvp ) methods. For all the nine experiments, the simultaneous approach converges to better solutions than the CVP ones. However, the CVP approach provides optimality cost functions very close to those obtained with the simultaneous approach.

In average, the cost functions obtained by the CVP approach are 95% of those obtained with the simultaneous approach.

From the study of the subproblems, we can conclude that the CVP approach with a simple piecewise parametrization seems well suited to design highly informative experiments. We now apply the CVP approach to the full OED problem, with the FIM of dimension 6 × 6. The optimal experiment inputs obtained are displayed in Fig. 4. The simulation of the nine D-optimal experiments is displayed in Fig 5B . Note that in the experiment 9, the biomass concentration exhibits, for a certain time interval, a behavior close to the steady state. This is due to the fact that the temperature reaches a very close value to T max and thus the growth rate becomes close zero. When performing the experiments, caution should be made for the selection of the maximal operational temperature. Indeed, an erroneous a priori on T max with a higher value than the real maximal temperature would lead to cell inactivation [START_REF] Bernaerts | Optimal experiment design for cardinal values estimation: guidelines for data collection[END_REF]. For microalgae cultures, T max must be well characterized to avoid operations that can be detrimental for attaining maximal productivities [START_REF] Ras | Temperature effect on microalgae: a crucial factor for outdoor production[END_REF]. In our case study, we were conservative in the selection of the prior of T max . By setting the prior lower than the maximal value reported in [START_REF] Marchetti | Optimizing conditions for the continuous culture of Isochrysis affinis galbana relevant to commercial hatcheries[END_REF], we assured that the temperature will allow growth in all the experiments.

Additionally, we wanted to assess the accuracy of the estimates when applying the optimal solutions obtained from the nine small subproblems to the full OED. Table 6 shows the ratio of the standard deviations of the estimates obtained from the full OED solutions to those obtained from the solutions of the OED subproblems. We observe that the standard deviations obtained when solving the full OED problem are usually smaller than those for the partitioned subproblems, 70% in average. For T max the estimated standard deviations of the two approaches are very close. For the light associated parameters, in particular, the accuracy of the estimation provided by the solution of the full OED problem is substantially better. A higher determinant of the FIM (two orders of magnitude) is obtained with the full OED solution while the condition numbers for the partitioned and the full OED are of the same magnitude. On the other hand, we also see that partitioning the full OED problem into small subproblems gives satisfactory results. This strategy of simplification could be easier to implement when dealing with the full model, even if a better accuracy is achieved by considering the full FIM.

Static vs Dynamic OED

A complete comparison between the static and the dynamic approaches for OED would require the knowledge on the noise variance for the measurements of the maximal growth rate (σ 2 s ) and the biomass concentration (σ 2 s ). However, even if this information is unknown a priori we can still draw a comparative analysis of the performance of these methods, assuming that the data is generated by [START_REF] Goodwin | Dynamic System Identification: Experiment Design and Data Analysis[END_REF].

The unbiased estimator of the noise variance reads as

σ 2 = 1 n -n p n i=1 [y i -y m i (θ * )] 2 , ( 17
)
with n the total number of data measurements and n p the number of parameters . Since the ML estimator is efficient asymptotically (as n → ∞), we can infer that the dynamic approach provides a more efficient estimator than the static approach. Indeed, for our case study, while the number of data points in the static approach is only 3 times the number of parameters, when applying the dynamic approach we get a number of experimental points that is 30 times the number of parameters.

To allow for a quantitative comparison, we used the approximated noise variances previously estimated σ 2 d = 9.31 and σ 2 s = 3.8 • 10 -3 to generate random simulated data for tackling the parameter estimation problem for both methods. Table 7 shows the estimated values and their confidence intervals for both approaches. The standard deviations of the parameters obtained with the dynamic approach are in average 42% lower than those given by the static approach. For the parametes µ max , I opt the dynamic approach excels substantially the static approach by providing standard deviations 13% lower. Finally, it is worth noting that for the static approach to equal in average the dynamic approach, it is required to reduce significantly the value of σ s , which is only possible for n >> n p .

The correlation matrix of the estimates for the dynamic approach was The condition numbers of both approaches are comparable (see Table 4 and Table 6).

The correlation matrices for both approaches indicated a low correlation between the parameters despite the high condition numbers . This is indeed thanks to the practical identifiability properties of the cardinal model as discussed in Appendix A.

For the dynamic approach the mean squared error (MSE) of the estimated parameters w.r.t the nominal parameters was 1.42 while for the static approach MSE = 2.30•10 3 , indicating the dynamic approach provides closest estimates to the nominal values in comparison to the static approach. Only for T max , both approaches perform equally.

Practical aspects as the labor of performing a two-step identification [START_REF] Bernaerts | On the design of optimal dynamic experiments for parameter estimation of a Ratkowsky-type growth kinetics at suboptimal temperatures[END_REF] place the static approach in disadvantage compared to the dynamic approach. These reasons lead us to favor the dynamic approach. Another benefit is that the sampling times could be further optimized within the experimental protocol, giving additional degrees of freedom.

From the mechanistic point of view, by stimulating the system with time-varying inputs, the dynamic approach allows a better characterization of the system behavior. On the opposite, the static approach can hide the relevance of certain important phenonema.

This factor is critical to our case study where microalgae are meant to grow in a dynamic environment that is periodically forced by daily variations of light and temperature. However, we should keep in mind that to take advantage fully of the dynamic approach, a step forward in the mathematical description of the process needs to be done for accounting important phenomena such as acclimation to light and temperature [START_REF] Geider | Light and temperature dependence of the carbon to chlorophyll a ratio in microalgae and cyanobacteria: implications for physiology and growth of phytoplankton[END_REF][START_REF] Zou | Thermal acclimation of respiration and photosynthesis in the marine macroalga Gracilaria lemaneiformis (Gracilariales, Rhodophyta)[END_REF] and cell inactivation due to high temperatures. For the sake of generality, a further study should be done to include the impact of the initial conditions and the physiological state of the cells on the determination of optimal experiment inputs. We also recommend to perform a preliminary experiment for which the cells get acclimated to their light and temperature growth conditions. This experiment will allow a dynamic characterization of the adaptation phenomena.

Relevance of accurate estimation on model-driven optimization

One of the ultimate goals of developing microalgae growth models is to provide a platform for model predictions and for the design of optimal control strategies for systems operated at large scale. Following this aim, we wanted to assess the relevance of providing accurate parameter estimates on the quality of the predictions for a more complex model representing the continuous cultivation of microalgae on an outdoor pond. For that, we used the raceway model described in [START_REF] Muñoz-Tamayo | Optimizing microalgal production in raceway systems[END_REF]. The model takes the configuration of a pilot-scale raceway (Algotron) located at INRA LBE, Narbonne (France). The model is described by

ṡ =D(s in -s) -ρ(•)x, ( 18
) qn =ρ(•) -(µ(•) -R(•))q n , (19) ẋ
=(µ(•) -D -R(•))x, (20)
where s (mg N/L) is the extracellular nitrogen concentration and x (mg C/L) is the concentration of carbon biomass. The term q n (g N/g C) denotes the intracellular nitrogen quota, that is the concentration of nitrogen per biomass unit. D is the dilution rate, µ(•)

is the specific growth rate, ρ is the nitrogen uptake rate and R(•) the respiration rate. For more details, the reader is referred to [START_REF] Muñoz-Tamayo | Optimizing microalgal production in raceway systems[END_REF].

Firstly, we evaluated the sensitivity of the biomass concentration with respect to the model parameters along a year of operation. Meteorological data was used for the location of Narbonne to calculate the temperature and light intensities for each month. The normalized sensitivity matrix S y was computed for each month. The element (k, j) of S y is calculated as [START_REF] Turányi | Sensitivity analysis of complex kinetic systems. Tools and applications[END_REF] S y (k, j)

= nt i=1 s k j (t i , θ) , ( 21
)
where s k j is the normalized sensitivity of the model output y m k w.r.t. θ j ,

s k j (t i , θ) = θ j y m k (t i , θ) ∂y m k ∂θ j (t i , θ) . ( 22
)
Figure 6 shows a graphical representation of the sensitivity matrices for four months.

January is the coldest month in Narbonne, while August is the warmest. October is an intermediate month. The sensitivities of June are also presented for illustration. It is interesting to observe that the influence of the parameters on the model response is modulated by the environmental conditions. Indeed, we can see a specific pattern of parameter influence for each month. In terms of the tuning importance, that is the importance of parameter changes around their nominal value for the model output [START_REF] Turányi | Sensitivity analysis of complex kinetic systems. Tools and applications[END_REF],

we observe that, overall, the most dominant parameter is T opt . In August the most dominant parameter is µ max . This month exhibits the most homogeneous distribution of the influence of parameters in comparison with months like January where the influence of two parameters (T max , T opt ) exceeds substantially the influence of the rest of parameters.

In cold months (e.g. January-March), the influence of T max is higher than the influence of µ max . This pattern is switched in warm months (e.g. June-August). In the figure, T min appears as the less influential parameter. This effect may be inverted in cold regions.

Indeed, the minimum average temperature in Narbonne used in our simulation is 4.76 • C, which is very high compared to the nominal value of T min .

Figure 7 shows the dramatic effect of an uncertainty of 5% on the nominal value of T opt on the quality of model predictions for the month of January. Both overestimation (1.05T opt ) and underestimation (0.95T opt ) of the optimal temperature results in important discrepancies between the response of the model with the nomical value of T opt and those obtained with a small perturbation of 5% on the nominal value. Hence the importance of providing accurate estimates since small changes on the parameter values can induce large changes on the biomass dynamics. Model-driven decisions are thus strongly dependent on the accuracy of the parameter estimates.

The previous result strengthens the relevance of the temperature effect for outdoor cultivation as discussed in [START_REF] Ras | Temperature effect on microalgae: a crucial factor for outdoor production[END_REF]. It should be noted that with the meteorological data used here, the temperature of the culture (T ) never exceeded T max , so the effect of temperature φ T was always higher than zero. We recalled that an overestimation on T max will have a strong impact on model predictions and system operation. In particular, when the temperature exceeds T max phenonema as cell inactivation and mortality take place. These phenomena, detrimental for attaining maximal productivities, need to be characterized by an approach combining both experiments and modelling in order to provide guidelines to mitigate negative effects.

Conclusions

We solved the OED problem for a simplified model of microalgae growth. We have determined optimal experiment conditions to provide an accurate estimation of the parameters that drive microalgae growth by modulating the effects of light and temperature.

Both static and dynamic approaches were evaluated to find D-optimal experiments. From our results, we recommend the use of the dynamic approach in virtue of the efficacy in terms of the maximum likelihood properties of the estimator. The protocol of experiment inputs determined in this study will be further implemented in the TIP system used at Ifremer Institute.

For the dynamic case, we showed that a parameterization of the control input by piecewise linear functions (CVP approach) provides efficient results as compared as the simultaneous approach. Moreover, the strategy of partitioning the full OED problem into subproblems dedicated to improve the accuracy of the estimation of a couple of parameters was shown to be satisfactory. The CVP method and the partitioning of the full OED into subproblems are suitable approaches for solving the OED problem in microalgae growth models by reducing, additionally, the problem complexity. Finally, with the use of sensitivity analysis of a more complex model describing the cultivation of microalgae in a raceway, we showed the relevance of providing accurate parameters for enabling reliable model-driven decisions.

Acknowledgment

This work benefited from the support of the Facteur 4 research project founded by the French National Research Agency (ANR). Figure 3: Optimal experiment inputs given by the CVP approach (solid lines) and the simultaneous approach (dashed lines) for the partitioned OED problem. The responses of the identified models (solid and dotted lines) for both static and dynamic approaches described satisfactorily the simulated data. identification of µ max . Such property allows to weaken the correlation between µ max and α.

For illustration, we performed a D-optimal protocol consisted of five experiments with ten equidistant sampling times for both standard Haldane and the parameterized kinetics.

For the Haldane kinetics, the correlation matrix of the parameters was: As observed, the Haldane kinetics exhibits a stronger parameter correlation than the parameterized kinetics. Therefore, in terms of practical identifiability, the parameterized kinetics is preferred over the standard Haldane kinetics.

  ) for the kth experiment, n e is the number of distinct experiments (n e = 9) and σ 2 s is the noise variance associated to measurement of the maximal growth. To provide an approximate value of the noise variance, the dynamic model was simulated for nine experiments. Each of them characterized by a level of temperature and light intensity. Normal distributed data of biomass concentration was further generated by taking the value of noise variance of biomass σ 2 d . The generated noisy data was used to calculate the variance of specific growth. An approximate value of σ 2 s = 3.8 • 10 -3 was obtained.

IT

  opt -0.08 -0.36 1.0 T min 0.34 -0.04 -0.07 1.0 T max -0.19 0.10 -0.04 0.33 1.0 T opt -0.14 0.01 0.05 -0.35 -0.34 1.0For the static approach, The correlation matrix of the estimates was opt -0.21 0.04 -0.02 -0.39 -0.33 1.0

Figure 1 :Figure 2 :

 12 Figure 1: The TIP system. The device has 18 batch photobioreactors for microalgae cultivation.

Figure 4 :

 4 Figure 4: Optimal experiment inputs obtained for the full OED problem.

Figure 5 :

 5 Figure 5: Simulated data resulted from D-optimal experiments including duplicates and responses of the identified models. A. Measurements of specific growth (circles) for the static approach. B. Measurements of biomass concentrations (circles) for the dynamic approach obtained for the full OED problem.

Figure 6 :

 6 Figure 6: Overall sensitivity of the biomass concentration to the parameters in the complete raceway model developed in [3]. Results are shown for four months illustrating how the influence of the parameters on the model output is modulated by the environmental conditions.

KI

  iI -0.98 -0.92 1.0 For the parameterized kinetics, the correlation matrix of the parameters was: opt -0.25 -0.10 1.0

Table 1 :

 1 Nominal values of the model parameters.

	Parameter	Definition	Value
	α	Initial slope of the growth	0.008 (µE m -2 s -1 d) -1
		response curve w.r.t. light
	µ max Maximal specific growth rate	0.76 d -1
	I opt	Optimal light intensity	548 µE m -2 s -1
	T min	Lower temperature for	-0.20 • C
			microalgae growth
	T max	Upper temperature for	33.30 • C
			microalgae growth
	T opt	Temperature at which	26.70 • C
		growth rate is maximal

Table 2 :

 2 Subproblems of the dynamic OED strategy. I opt )I (µ max , T min ) T (µ max , T max ) T (µ max , T opt ) T (T min , T max ) T (T min , T opt ) T (T max , T opt ) T

	Experiment Couple of parameters Experiment input
	(µ max , α)	I
	(µ max , I opt )	I
	(α,

Table 3 :

 3 Experimental conditions for the static approach. Temperature 12.10 12.10 24.30 24.60 26.70 26.70 30.60 30.70 33.20

	Full factorial design
					Experiment
		1	2	3	4	5	6	7	8	9
	Temperature	12.0 22.60 33.20 12.0 22.60 33.20 12.0 22.60 33.20
	( • C)
	Light intensity	20	20	20	610	610	610	1200 1200 1200
	(µE m -2 s -1 )
	Composite factorial design						[14]
					Experiment
		1	2	3	4	5	6	7	8	9
	Temperature 15.30 19.0	19.0	19.0	19.0	24.5	24.5	24.5	24.5
	( • C)
	Light intensity 560	863	257	257	863	560	560	560	560
	(µE m -2 s -1 )
		10	11	12	13	14	15	16	17
	Temperature 24.50 24.50 24.50 30.0	30.0	30.0	30.0 33.20
	( • C)
	Light intensity 560	50	1070	863	257	257	863	560
	(µE m -2 s -1 )
	D-optimal design
					Experiment
		1	2	3	4	5	6	7	8	9
	( • C)
	Light intensity 536	536	1200	409	74	74	1200	395	547
	(µE m -2 s -1 )

Table 4 :

 4 Comparison of D-optimal design with factorial design for the static approach.

		µ max	α	I opt T min T max T opt
	η j D-optimal η j Composite factorial design	0.50 0.48 0.76 0.16 0.02 0.25
			Det(F s )	λ max /λ min
	Full factorial design		0	5.30•10 20
	Composite factorial design		381.74	1.89•10 9
	D-optimal	7.90 • 10 6	3.85 • 10 9

Table 5 :

 5 Comparison of the CVP and sequential strategies for the partitioned OED problem in the dynamic approach.

	Experiment	-log Det (F d )
		Simultaneous (J Sim ) CVP (J cvp ) J cvp /J Sim
		-26.1949	-25.8114	0.9854
		-5.37552	-5.0026	0.9306
		-10.2537	-9.8398	0.9596
		-11.3121	-10.8970	0.9633
		-16.9941	-16.5543	0.9741
		-14.856	-14.6943	0.9891
		-5.0363	-4.4793	0.8894
		-4.5046	-4.0965	0.9094
		-9.9769	-9.4997	0.9522

Table 6 :

 6 Comparison of the accuracy of the estimation obtained with the solutions of the full and partitioned OED problems in the dynamic approach.

		µ max	α	I opt T min T max T opt
	η j Full η j Partitioned	0.85 0.57 0.41 0.76 1.05 0.59
			Det(F d )	λ max /λ min
	Full OED	1.76•10 12	1.10•10 9
	Partitioned OED	4.20•10 10	4.44•10 9

Table 7 :

 7 Estimated parameters with their approximate confidence intervals for the static and dynamic OED approaches. The parameter estimation was performed with simulated noisy data.

	θ ± 2η j

, five out of six possible combinations of parameters led to a well-conditioned FIM.

Figure 7: A small uncertainty of 5% on the value of T opt leads to important mismatches on model predictions. The dynamic of the biomass concentration of the month of January with the nominal value of T opt (solid blue line) is compared to the response of the model with 0.95T opt (dotted green line) and 1.05T opt (dashed red line) .

Appendix A. Comments on structural and practical identifiability of the model Appendix A.1. Structural identifiability

To check the structural identifiability of the model, we used the time power series method [START_REF] Pohjanpalo | System Identifiability Based on the Power Series Expansion of the Solution[END_REF], briefly described below. Since all the time derivatives of the outputs are unique, it follows that a sufficient condition for the identifiability of the model is that set of equations

have a unique solution for θ [START_REF] Pohjanpalo | System Identifiability Based on the Power Series Expansion of the Solution[END_REF].

For our case study, let us consider first the identifiability of the temperature parameters of the cardinal model T min , T max , T opt . At constant light, the model is given by

with µ I = µ max φ I and x 0 a known initial concentration of biomass. The effects of light φ I and temperature φ T on microalgae growth are here recalled:

By simple inspection of (A.8) and given the biological meaning of the parameters of the cardinal model, we can infer that a series of adequate experiments running at different temperature conditions in the interval [T min , T max ] will allow to identify uniquely the temperature parameters.

Coming back to the time power series method, let us consider the case of a specific input T (t) that is infinitely differentiable w.r.t. to time. For simplicity, we chose

The Taylor series coefficients are thus a 0 =x 0 , (A.9) a 1 =µ I φ T (0)x 0 , (A.10)

Given the shaphe of φ T and applying the first-optimality condition, the following cases provide the parameters to be uniquely identifiable: a 1 = 0, and, a 2 ≥ 0, ⇒ T min = T (0), (A.12) a 1 = 0, and, a 2 < 0, ⇒ T max = T (0), (A.13)

The previous conditions can be reached by making T (t) vary along the interval [T min , T max ].

Following the same procedure, we can now check the identifiability of the light parameters I opt , α and the maximal specific growth rate µ max . Since T opt is structurally identifiable, let us consider a constant temperature T = T opt . The model is thus

By considering the case of a specific input I(t) = c 1 t+c 2 , the series expansion provides the following coefficients a 0 = x 0 , (A.17)

By applying the first-order optimality condition on φ I , we get

Injecting I opt in (A.18) provides µ max .

Once I opt is identified, evaluating (A.18) at I(0

The model is therefore structurally identifiable.

Appendix A.2. Practical identifiability

Parameter estimation of Haldane and Monod type kinetics is known to be hampered by practical identifiability problems due to strong correlation between its parameters. To represent the effect of light on microalgae growth, the Haldane kinetics is often used

where μ is the specific growth rate, K sI is the light affinity constant and K iI is the inhibition constant. By applying the first-order optimality condition, the optimal light intensity for growth is In this work, instead of the standard Haldane kinetics, we made use of the parameterized kinetics (A.7), which has the same shape than the Haldane kinetics but offers certain advantages in terms of practical identifiability properties. On the basis of good quality nominal parameters, it was previously shown that adequate inputs allow to identify the optimal conditions for growth I opt and T opt , which derive automatically on the