Référence électronique

Roffidal Emilie, "The Drawing School of Bayonne", translated by Alexandra Woolley, Les papiers d'ACA-RES, Brefs historiques, put online in March 2020

Émilie ROFFIDAL CNRS UMR 5136-FRAMESPA The Drawing School of Bayonne

On the 27th of April 1778, prompted by the painter Pierre Lagleire (1745-1813), the Bayonne city council authorized the founding of a free Academic Drawing School which officially opened on the 21 st of July of the same year. Pierre Lagleire, originally from Marmande, attended as a student the Académie de peinture et de sculpture of Toulouse* and then became secretary to the parliament's president Bordeaux Lalanne (?-?) for six years (attestation of the 24 th of November 1775). To ensure the creation of the school, Lagleire used all of his connections in Toulouse, Bordeaux and Bayonne. If the "list of people whose help was requested for the founding" (GG234) was destroyed in the fire of the council archives in 1889, other documents recount how he proceeded. On the 28 th of March 1778, a certificate was written by the Royal Academy of Toulouse which was signed by the knight of Aufréry (Toulouse, 1736(Toulouse, -1813) ) moderator, by Lamothe (1715-1785) permanent secretary and by Bardy (1709Bardy ( -1794) ) 1 . It indicates that Lagleire "attended all his classes when he was student, was a role model of dedication and wisdom to his pairs" and the institution rewarded him with four prizes 2 . Recommendations letters were also written by Nicolas Dupré de Saint-Maur (Paris, 1732-1791) when he was the intendant of Bordeaux city in December 1778 and by M. Hiriart, the Bayonne Priest, guarantying the painter's religious observance.

With such an appeal, the Bayonne city council agreed to the Drawing School and asked that Lagleire seek affiliation from the Académie royale de peinture et de sculpture** of Paris. The director of the drawing school of Poitiers and painter François Aujolest Pagès (Bordeaux, 1746-Poitiers, 1801) made the request on his behalf which did not seem to please the permanent secretary of the Parisian academy Antoine Renou (Paris, 1713-1801) who demanded that Lagleire not use a middleman. On the 24 th of October 1778, Lagleire wrote to him directly asking him for a "legal and solid scheme" for the Bayonne school. Yet, the-then director of the Royal Academy, Jean-Baptiste-Marie Pierre (Paris, 1714-1789) wrote to d'Angiviller asking him to wait until Lagleire had proved his talent before agreeing to the 1 This would be Christophe-Suzanne de Lamothe (1715-1785) and Jacques de Bardy (or Bardi) (1709-1794). Lamothe, council member of the Toulouse parliament, received the honor of being "an ordinary associate" to the Royal Academy of Paris in 1774 and was permanent secretary for 1774 to 1776 (after the Marquis of Orbessan). Il took part in the Salons of the Academy as a collector in 1759,1767, 1774, 1777 and 1784. On the Lamothe family, see Franck Florenson, L'hôtel de Lamothe-Trilhe au cloître Saint-Étienne, 8 rue Sainte-Anne, master's thesis (supervised by Yves Bruand), Université de Toulouse-Le Mirail, 1993. Bardy, was also a council member of parliament, and took part in the Academic Salons of 1768 and 1773. My thanks go to Marjorie Guillin for this information; 2 The details of the prices remain unknown. patronage and praised the aldermen of Bayonne who considered the opening of the school only on a trial-basis. Lagleire continued his search for sponsors and befriended Jean-Etienne Montucla (Lyon, 1725-Versailles, 1799), first assistant of the King's Buildings, to whom he explained his attempt to officially open the school and asked him to be his intermediary with d'Angiviller. On the 26 th of December 1778, the city council confirmed its support to the drawing school that will bring "real progress to the arts" and "benefit many".

After the documents had been signed by Bayonne and Versailles approving the rules and regulations, the school was set-up quite quickly. On the 16 th March 1779, the final contents were sent to d'Angiviller who gave them to Jean-Baptiste-Marie Pierre on the 20 th of March who only found two small amendments. He requested that the institutions papers be kept in one of the rooms for "constant and immediate access for research purposes" and that the treasurer not be "permanent". D'Angiviller then transferred the approved version to Lagleire on the 15 th of May with a letter in which he expresses his desires to see the Academy be a success : "The great lengths you went to, or so I have been told, to see (the arts) flourish, and the opulence of the town of Bayonne give me hope in thinking that they will not go wasted in such a part of the Kingdom where the arts have not yet penetrated". On the 24 th of May, d'Angiviller confirmed the approbation of the school and Lagleire, fully satisfied with the successful development of his project, replied to his letter in the same vein on the 29 th of May, underlining the artistic void of the region and confirming the successes to come, not doubting that "these same arts, ignored until now, will show multiple progress in time"

The drawing school of Bayonne was said to be "in favour of the liberal and mechanical arts" (AD. Pau, C 58). To understand Lagleire pragmatic intentions, one has to read his "Réflexions sur l'établissement projetté d'une École royale académique gratuite de dessein dans la ville et cité de Bayonne, présentées à Messieurs les amateurs du bien public [s.d.] [Thoughts on the establishment of a Free Royal Academic School of Drawing in the city of Bayonne, presented to the Gentlemen Amateurs of the Public Good]". The artist begins his argument by briefly presenting Bayonne: "the town is one the furthest away from the Capital, the most pleasant and amenable to culture and fine arts of all the cities in the Kingdom". He defends the necessity of such an establishment and its universal interest: "it is only in towns that have Fine Art Academies that one can find skilled artists and good craftsmen of all kind (…) a mason, a carpenter, an ironsmith and number of other craftsmen remain ignorant of design, yet this talent is the touchstone & the compass that drives the edifice". He then argues Bayonne's uniqueness -as a town with a port and being close to Spain -capable of educating seamen and young Spaniards. His proposal of having the aldermen's portrait painted constitutes the high point of his presentation in order to win over his readers. In another document, more than likely written to respond to his detractors complaining about the cost of the school, he reassures the town council that "the city will have no part in that department", and claims that all the academy's expenses will be covered by the "associated amateurs".

According to Lagleire (in a letter of the 24 th October 1778), the School welcomed around sixty students. The educational program remains unknown but the regulations that travelled from Bayonne to Paris and Versailles give an understanding of organizational intentions of the institution (« Projet de règlemens pour l'école académique gratuite de dessein de Bayonne, 20 mars 1779 »3 ; « Projet de statuts et autorisation de l'école académique gratuite de dessein pour la ville de Bayonne, [s.d.] »4 ; « Projet de statuts et autorisation de l'école académique gratuite de dessein pour la ville de Bayonne, Versailles, 15 mai 1779 »5 ). It is revealed that :

. The School will be under the authority of the Mayor and the aldermen of the town and that of the direction of the Royal Academy of Paris.

. It will be composed of local artists, amateurs "chosen from the respectable people of the city, who for the love for the arts, will want to contribute to the expenses and maintenance of the school" and foreign artists. The project of the 20 th of May 1779 specifies that "we can welcome all foreign amateurs and artists that present themselves, divide them into two classes, with a preference for those who express a love for the Arts, as they will contribute to the expenses and ornament of the school".

. The office, qui will get together once a month, will be composed of a President (the mayor of Bayonne), a permanent Director (named by the Royal Academy of Paris), professors, six administrators and a permanent secretary.

. The permanent secretary will be in charge of all the paperwork that will be kept in one of the schoolrooms "in order to consult them if necessary and to do an inventory every two years".

. The Treasurer will give details of the expenses every six months.

. The Director will be in charge of supplies and will be helped by 2 assistants.

. The lectures will be given by a drawing professor, a geometry professor, an anatomy professor and an architect.

. Correspondence will be established with the Royal Academy of Paris "informing the institution of its principal events, or to consult with the Academy when in difficulty".

If the beginnings of the School seem straightforward in comparison with other institutions, the reality tells a different story. From 1781, the School's soundness is questioned due to its lack of financial support from the city council. Lagleire, already in financial predicament, asked the aldermen to be exonerated from his capitation. He reminded them of his dedication to the "Public Good" (Letter of the 9 th June 1781) and that he was "an artist who gives his all and even his own money to promote the arts", declaring "that it would not be unjust and not without prejudice to the law, to broaden the grasp of taxation to that of the general public to the advantage of those who for free". This letter stresses the difficulties that the artist had to face, and he also argued that he supported "the Free Academic School of Drawing" "out of his own pocket over thirty months". If the subject of his letter is tax exemption, or at least the capitation, he also expresses his regrets and desires for the school: "Any other town, would have, without a doubt your Highness, rushed to take part in such an interesting cause for the Public Good and would have at least found the necessary housing for the education of its students as the elite of the Kingdom with commercial interest and consequence, that would be too poor to give a sum of five or six pounds per year and by doing so, would have given an certain advantage to their ill-fated fellow citizens. Unfortunately, the taste for arts is not a dominant taste in this country and this being the one and only reason I can fathom for the lack of assistance I have found in Bayonne". Lagleire's request like all the others, that are kept in the town's archives, were all denied. The aldermen's response was just as straightforward, replying that the artist "attacks the lack of taste for the arts in this country: one could say that he has yet to be educated". They also justify their refusal for financial aid by reminding the artist of his own personal wealth, stressing that he can rely on the lucrative income of his second wife, Marguerite Costes, clockmaker in the one of main lanes of the town.

Having not found the help he had hoped for, Lagleire left Bayonne in 1783 for Poitiers, where he more than likely was welcomed by Aujolest Pagès. He offered his services as a portraitist and as a painter of religious scenes for churches and also gave drawing lessons. The painter gave the Academy one last shot in 1784, as indicated by a letter he sent d'Angiviller, with the new founded support of the Bishop of Evreux. In it, he asks for housing in Bayonne by suggesting the Chateau-Vieux (or Old Castle) and requests an income to "promote fine arts in this remote part of the Kingdom where they are unknown". However, his requests remain unanswered: Montucla wrote in the superior part of the letter "The Count can do nothing in regards to what is asked and I replied to M. Lagleire at the time when he wrote to me, giving details". The short-lived school of Bayonne came to an end.

Archival Sources

National Archives

O/1/1933/A . AN, O/1/1933/A, dossier 10, doc. 1, "Réflexions sur l'établissement projetté d'une École royale académique gratuite de dessein dans la ville et cité de Bayonne, présentées

  à Messieurs les amateurs du bien public [s.d.]," collationné et mis en ligne dans Les ressources d'ACAcollationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/61. . AN, O/1/1933/A, dossier 10, doc. 4, "[Lettre de D'Angiviller à Pierre au sujet de l'école de Bayonne, Versailles, 24 mai 1779]," collationné et mis en ligne dans Les ressources d'ACA-/1933/A, dossier 10, doc. 5, "[Lettre de Laglaire aux maires et échevins de la ville de Bayonne, et réponse, 1779]," collationné et mis en ligne dans Les ressources d'ACA-RES, collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/64. . AN, O/1/1933/A, dossier 10, doc. 7, "[Lettre de Laglaire à Montucla, Bayonne, 26 décembre 1778]," collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/65. . AN, O/1/1933/A, dossier 10, doc. 8, "[Lettre de d'Angiviller à Pierre, Versailles, 20 mars 1779]," collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/66. . AN, O/1/1933/A, dossier 10, doc. 9, "[Lettre officielle du maire et des échevins de Bayonne, Bayonne, 26 décembre 1778]," collationné et mis en ligne dans Les ressources d'ACA-RES, /1933/A, dossier 10, doc. 10, "[Lettre de Laglaire, Bayonne, 24 octobre 1778]," collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/68. . AN, O/1/1933/A, dossier 10, doc. 11, "[Lettre de Laglaire à Montucla, Bayonne, 13 février 1779]," collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/69. . AN, O/1/1933/A, dossier 10, doc. 12, "Projet de règlemens pour l'école académique gratuite de dessein de Bayonne, 20 mars 1779," collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acaresarchives.nakalona.fr/items/show/70. . AN, O/1/1933/A, dossier 10, doc. 13, "Projet de statuts et autorisation de l'école académique gratuite de dessein pour la ville de Bayonne [s.d.] ," collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acaresarchives.nakalona.fr/items/show/71. . AN, O/1/1933/A, dossier 10, doc. 14, "Lettre de d'Angiviller à Laglaire, Versailles, décembre 1778]," collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/72. . AN, O/1/1933/A, dossier 10, doc. 15, "[Lettre de Laglaire à Dupré de Saint-Maur, Bayonne, [s.d.] ; réponse de ce dernier, Bordeaux, 8 décembre 1778]," collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acaresarchives.nakalona.fr/items/show/73. . AN, O/1/1933/A, dossier 10, doc. 16, "Nottes sur la demande d'une École académique à Bayonne, [s.d.]," collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/74.

	RES, archives.nakalona.fr/items/show/59. fonds d'archives, consulté . AN, O/1/1933/A, dossier 10, doc. 2, "[Extrait des registres de l'Académie royale de peinture le 20 octobre 2019, https://acares-et de sculpture, concernant Bayonne, du 26 septembre 1778] ," collationné et mis en ligne dans Les ressources d'ACA-RES, fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/60. . AN, O/1/1933/A, dossier 10, doc. 3, "[Lettre de Laglaire, Bayonne, 29 mai fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/62. . AN, O/1fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/63 . AN, 0/1/1933/A, dossier 10, doc. 6, "[Lettre de Pierre à d'Angiviller, Paris, 7 novembre 1778]," fonds d'archives, consulté le 20 octobre 2019, https://acares-archives.nakalona.fr/items/show/67. 1779]," RES, . AN, O/1

AN, O/1/1933/A, dossier 10, doc. 12, "Projet de règlemens pour l'école académique gratuite de dessein de Bayonne, 20 mars 1779," collated and put online in Les ressources d'ACA-RES, fonds d'archives, consulted on the 20 of October 2019, https://acares-archives.nakalona.fr/items/show/70.

AN, O/1/1933/A, dossier 10, doc. 13, "Projet de statuts et autorisation de l'école académique gratuite de dessein pour la ville deBayonne [s.d.], collated and put online in Les ressources d'ACA-RES, fonds d'archives, consulted on the 20 of October 2019, https://acares-archives.nakalona.fr/items/show/71.

AN, O/1/1933/A, dossier 10, doc. 21, "Projet de statuts et autorisation de l'école académique gratuite de dessein pour la ville de Bayonne, Versailles, 15 mai 1779," collated and put online in Les ressources d'ACA-RES, fonds d'archives, consulted on the 20 of October 2019, https://acares-archives.nakalona.fr/items/show/79.


Lucas José Da

Silva Moreira

Gildas Besançon
email: gildas.besancon@grenoble-inp.fr

Francesco Ferrante
email: francesco.ferrante@grenoble-inp.fr

Mirko Fiacchini
email: mirko.fiacchini@grenoble-inp.fr

Hervé Roustan
email: herve.roustan@riotinto.com

L J Da

Model Based Approach for Online Monitoring of Aluminum Production Process

Keywords: Aluminum electrolysis, state estimation, Kalman filter

In Hall-Heroult process for aluminium production, estimating the alumina concentration and the anode-cathode distance (ACD) remains a challenge. One of the difficulties arises from the fact that it is not possible to measure those quantities continuously during the pot operation. This article presents a novel approach for on line estimating alumina concentration and ACD in a regular aluminum-reduction pot cell using a Linear Kalman Filter. This is done by using an appropriate dynamical model for the pot, which is obtained by combining the first principle modeling and experimental identification of alumina concentration behavior from irregularly sampled data. Moreover, a dynamical model for the pot resistance is identified as a function of the alumina concentration and ACD data. The proposed approach is validated on an industrial platform.

Introduction

Aluminum manufacturing is a challenging industrial area based on the Hall-Heroult process [START_REF] Grjotheim | Aluminium Electrolysis: Fundamentals of the Hall-Héroult Process[END_REF][START_REF] Grjotheim | Aluminium Smelter Technology: A Pure and Applied Approach[END_REF]. In this setting, model and control challenges arise from the limitations in the process information continuously available, which can lead to inaccurate results [START_REF] Jakobsen | Estimating alumina concentration distribution in aluminium electrolysis cells[END_REF]. Moreover, this restrains the process production efficiency since it is risky to work with uncertainties or imprecise data [START_REF] Yao | State and parameter estimation in Hall-Héroult cells using iterated extended Kalman filter[END_REF].

During the electrolysis reaction which is the basis of the process, the anodecathode distance (ACD) is continuously affected by liquid aluminum production, carbon consumption. Other aspects affect the ACD as the electric perturbations and changes in the composition of the bath. An inappropriate ACD can impact in the efficiency of the process and safety conditions. In fact, a large ACD decreases the pot cell production and a small value can cause a short-circuit between the produced aluminum and the anode [START_REF] Keniry | Digital processing of anode current signals: an opportunity for improved cell diagnosis and control[END_REF]. Unfortunately, it is not possible to measure it during the process operation, and its dynamical behaviors is very difficult to represent in full details. See [START_REF] Jakobsen | Estimating alumina concentration distribution in aluminium electrolysis cells[END_REF][START_REF] Yao | Estimation of spatial alumina concentration in an aluminum reduction cell using a multilevel state observer[END_REF] for more details.

Another essential variable in aluminum production is the dissolved alumina concentration (w Al 2 O 3 ). Generally, it is required to keep it in a specific range to avoid the formation of sludge or the onset of anode effect [START_REF] Bearne | The development of aluminum reduction cell process control[END_REF]. Although monitoring importance, measurements this quantity continuously is costly. In common operation, just a few samples per week are manually taken, which makes it difficult to obtain an experimental model.

In practice, to monitor and regulate aluminum production, an indirect measurement is used: the pseudo-resistance. This signal is computed using the line current and pot voltage direct measurements. Other important variables for the Hall-Heroult process like aluminum fluoride concentration, bath temperature, metal height, etc. are only sporadically measured. The pseudo-resistance is used to determine the alumina feed rate injected in the system. The alumina feeding rate is based on the alternation of slower (underfeeding) and faster (overfeeding) periods than the rate corresponding to normal consumption of the cell [START_REF] Homsi | Overview of process control in reduction cells and potlines[END_REF]. The feeding periods alternation occurs when the rate of variation of pseudo-resistance exceeds a set value. Moreover, it is usually defined a setpoint resistance and thermal balance to ensure control stability and obtain a good current efficiency. This value is adjusted by the ACD regulation [START_REF] Homsi | Overview of process control in reduction cells and potlines[END_REF]. Many authors indicate that the pseudo-resistance could not be a good indicator for the operation [START_REF] Yao | Monitoring local alumina dissolution in aluminum reduction cells using state estimation[END_REF][START_REF] Haupin | Aiming for zero anode effects[END_REF].

Because of this lack of continuous measurements, nonlinear estimators have been developed by a few researchers [START_REF] Kolås | State estimation is the real challenge in NMPC[END_REF][START_REF] Jakobsen | Estimating alumina concentration distribution in aluminium electrolysis cells[END_REF][START_REF] Hestetun | Detecting abnormal feed rate in aluminium electrolysis using extended kalman filter[END_REF]. All of them are focused on a two-step procedure: Obtain a nonlinear model and apply an estimator strategy. However, an imprecise model can generate inaccurate results. As the Hall-Heroult process has a complex dynamics, it is not easy to find a proper model that reproduces its behavior. Moreover, it is necessary to have an initial ACD measurement which is not possible during operation. To overcome these limitations, some authors have been working on iterative filtering procedures yet, these require more computation effort [START_REF] Yao | State and parameter estimation in Hall-Héroult cells using iterated extended Kalman filter[END_REF].

In this paper, an online monitoring technique based on Linear Kalman filtering state-estimation is proposed. It uses a model that combines physical-chemical preliminary knowledge and experimental identified aspects. This system representation enables a real-time overall alumina concentration and ACD monitoring which can be used to avoid anode effects and improve the production. This procedure is validated with operational collected data and tested for different conditions in APXe50 pot cell localized in Laboratoire de Recherches de Fabrications (LRF) in Saint Jean de Maurienne, France.

The paper is organized as follows: The modeling approach and validation are presented in Section 2. The online monitoring and experimental results are explained in Section 3. The conclusions are discussed in Section 4.

Modeling

The process description of this paper is based on dynamical equations using the manipulated and process variables available in modern aluminum reduction cells. Initially, the ACD modeling is explained based on a chemical balance. Moreover, some simplified models for the alumina concentration and the pot resistance are developed. At the end of this section, an overall state-space model for the aluminum process is given. It is assumed that the pot is always in regular operation. In particular, it is assumed that line current and alumina pot feeding do not undergo on a significant overshoot.

Anode-Cathode Distance

The ACD is defined as the difference between the heights of aluminum produced (Al H eight ) and carbon consumed (C H eight ). Its dynamics depends on the anodes busbar beam movement (BM) and the initial distance (ACD ini ) as follows:

ACD(t) = Al H eight (t) -C H eight (t) + BM(t) + ACD ini (1)
The beam movement derivative is considered as one of the system inputs (u 1 ):

u 1 (t) = d dt BM(t) (2)
Using the electrochemical relations of chemical balance it is possible to compute the ACD and its variation in discrete-time for computer implementation by the following equation structure:

ACD[n + 1] = ACD[n] + T s (u 1 [n] + βI[n]) (3)
where T s is the sampling time, β is a combination of pot physical-chemical parameters and I is the line current applied to the system. As it is not possible to measure the ACD during the operation, the value of β is computed using theoretical values for a regular pot operation. Moreover, during the metal tapping, it is not considered that the beam is moving.

Alumina Concentration

The variation of the alumina concentration ( d dt w Al 2 O 3 ) is given by the difference between the concentration amount of dissolved alumina (w Al 2 O 3in ) that is resulted from the alumina injected in the pot by the feeders and the amount of alumina that is consumed by the chemical equation (w Al 2 O 3cons ). Hence the following chemical balance can be considered:

d dt w Al 2 O 3 (t) = w Al 2 O 3in (t) -w Al 2 O 3cons (t) (4)
It can be assumed a quick alumina solving [START_REF] Biedler | Modeling of an aluminum reduction cell for the development of a state estimator[END_REF]. This approach does not consider any agglomeration of the alumina powder since the dissolution constant is 0.099s -1 and the sampling rate is 1 minute. The frequency of the feeders (F) is considered as one of the system inputs (u 2 ):

u 2 [n] = F[n -D] (5)
where D is a constant time-delay due to alumina diffusion lag. Therefore, equation ( 4) can be rewritten in discrete-time as:

w Al 2 O 3 [n + 1] = w Al 2 O 3 [n] + T s (α 1 u 2 [n] -α 2 I[n]) (6)
where α 1 and α 2 are a combination of pot physical-chemical parameters.

As in LRF pots the aluminum concentration is measured a few times a day and in the same pot position, it is possible to identify an experimental model from the collected data and obtain values of α 1 and α 2 for regular operation.

Initially, the time delay D is estimated using unconstrained nonlinear optimization techniques and validated for different data set. Equation ( 6) needs a previous value of w Al 2 O 3 to calculate the next concentration. Then, the model is initialized with a measurement and the simulation is started using the signals of u 2 and I. Every time that there is a new alumina concentration data collection, the model is reinitialized to improve the simulation results.

In Fig. 1, it is shown a comparison between the model simulation using parameters α 1 and α 2 estimated with the collected alumina concentration and the respective inputs. It is possible to verify that the model provides a very good prediction of the plant behavior. The mean absolute relative error between the model and the measurements is 3.5176 %. Hence, this model can be considered validated and used to simulate and predict alumina concentration over long time horizons.

Pot Resistance

The pot resistance is an indirect measurement calculated via the pot voltage and line current. The pot voltage is modeled by a complex relation that uses ACD and alumina concentration [START_REF] Haupin | Interpreting the components of cell voltage[END_REF]. Besides that, it is taken into account other system operational parameters that cannot be measured continuously or that are not measured. Therefore, this makes it difficult to obtain a precise model. Aiming to obtain a simpler resistance model, it is assumed that all these unknown factors are constant during the pot operation . Moreover, it is not necessary to know any information about them, only the ACD and w Al 2 O 3 values are required.

It is known that the resistance model generates curves as shown in Fig. 2 for different ACD and w Al 2 O 3 . During the regular operational specification, the low and high alumina concentrations plot regions i.e. the anode effect and sludge formation areas are avoided. Then, the curves shown in Fig. 2 can be parameterized by the following equation:

(7) R(t) = c × w Al 2 O 2 3 (t) + (d + e × ACD(t))w Al 2 O 3 (t) + ( f + g × ACD(t))
where c, d, e, f and g are constant parameters to be determined. Equation ( 7) can be derived to compute the resistance variation in discrete-time by the following relation:

R[n + 1] = R[n] + (e × ∆w Al 2 O 3 [n]ACD[n] + (2c × ∆w Al 2 O 3 [n] + e × ∆ACD[n])w Al 2 O 3 [n] + d × ∆w Al 2 O 3 [n] + g × ∆ACD[n]) (8)
where:

∆ACD[n] = ACD[n] -ACD[n -1] T s (9) ∆w Al 2 O 3 [n] = w Al 2 O 3 [n] -w Al 2 O 3 [n -1] T s (10)
Notice that equation ( 8) needs an initial ACD value and this values is not measurable. Only the ACD variation is calculated by equation ( 3). This problem can 8) to set as the identifiable parameter the initial ACD combined by the other unknown variables.

In Fig. 3, a comparison between the model output simulation and real measurements is shown. The mean absolute error index computed for this data set is 2.2054%. However, it can be seen a small drift in Fig. 3. This happens because the model structure from equation ( 8) reproduces the resistance variation and needs an initial value.

State-Space Model

From the results described in previous sections, it is possible to obtain a discrete-time state-space model of the plant, with state vector defined by:

x[n] =       x 1 [n] x 2 [n] x 3 [n]       =       R[n] ACD[n] w Al 2 O 3 [n]       (11)
The current intensity is not considered as a control input because, due to the operational constraints, it cannot be manipulated. Hence, this signal is considered as a measured disturbance in the system.

Organizing equations ( 3), ( 6) and ( 8) results in the following description:

(12)                x[n + 1] =        1 a 12 [n] a 13 [n] 0 1 0 0 0 1        x[n] +        b 1 [n] b 2 [n] b 3 [n]        R[n] = 1 0 0 x[n]
where:

a 12 [n] = e(α 1 u 2 [n] -α 2 I[n]) (13)
a 13 [n] = (2c(α 1 u 2 [n] -α 2 I[n]) + e(u 1 [n] + βI[n]) (14)
b

1 [n] = (d(α 1 u 2 [n] -α 2 I[n]) + g(u 1 [n] + βI[n])) (15)
b 2 [n] = (u 1 [n] + βI[n]) (16)
b 3 [n] = (α 1 u 2 [n] -α 2 I[n]) (17)
which all the parameters known. This model representation permits to design a Linear Kalman filter usage [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF], even with a nonlinear model. This is justified because equation ( 12) is affine in the state. Hence, it is possible to recover the state vector for a given input set. This makes it possible to use the valid model for online ACD and w Al 2 O 3 monitoring.

Online Monitoring

As the model is valid and it is possible also to organize it in a state-space form, the states can be estimated using a linear observer fed with the plant inputs and outputs. The schematic implementation is shown in Fig. 4. The estimation of the state signals for pot resistance ( R), alumina concentration ( ŵ Al 2 O 3 ), and anode-cathode distance ( ÂCD) are calculated in real-time from inputs and output signals. Therefore, this makes it possible an online state monitoring. The observer selected is a Kalman Filter. It requires first to tune the noise matrices to perform the estimations. As the chosen sampling time is 1 minute, the noise covariance level is set in small value for the process and measurement matrices. The covariance matrix has to be tuned as well. According to the chosen matrix values, the convergence can be faster or slower. In this case, it is recommended a higher value for the ACD covariance than other states since it is not possible to measure it. Further details in Kalman Filter can be found in [START_REF] Simon | Optimal state estimation: Kalman, H infinity, and nonlinear approaches[END_REF].

To ensure the performance, the filter was tested using different initial values of ACD and w Al 2 O 3 . The resulting estimations for each of them, labeled as 1, 2, 3 and 4, are compared with real measurements of R and values computed by the model for all states, as shown in Fig. 5. Notice the fast convergence for all state estimations, less than 1 hour, even with different initial conditions. As R is measured, it converges faster than other states variables. The other signals estimations take more time to converge to a value closer to those computed by the model. Therefore, the efficiency of the state estimations can be guaranteed and they can be used for online monitoring of a regular pot cell.

This online monitoring can be used to predict local anode effects as well, around the alumina collection area. This is done by monitoring a small concentration of alumina. In Fig. 6, it is shown the alumina concentration model simulation between to measurements and the surrounding anodes currents measurements labeled with the respective position number. It is possible to verify that after alumina concentration reaches small values, the currents start in anode 21 to have a disturbance behavior. After some time, it is verified in the individual current anodes measurements many peaks. This characterizes an anode effect in the pot cell that was noticed by model based monitoring model based system.

Conclusion

In this article, a novel approach for online monitoring of aluminum reduction cells based on the model is proposed. This approach combines the available signals in the pot cell, physical-chemical knowledge, and system identification methods to obtain a state-space model. This representation captures only the regular operation dynamics. Then, it is possible to obtain a simple model that captures the desired operational dynamics, neglecting complex relations. This model structure can be applied to a Linear Kalman Filter and used it to estimate two important signals for the pot operation: ACD and w Al 2 O 3 .

The estimation results were tested using experimental data, which showed good accuracy for ACD, and alumina concentration estimation, as well as resistance prediction. Moreover, it can allow to detect local anode effects based on the alumina concentration monitoring. In conclusion, the proposed method seems to provide promisingly efficient results without knowing much about the pot dynamics. Then, it can be used for pot monitoring during regular operation.

Fig. 1

 1 Fig. 1 Alumina Concentration Model Validation

Fig. 2

 2 Fig. 2 Typical pot resistance curve as function of alumina concentration and ACD

Fig. 3

 3 Fig. 3 Resistance Model Validation

Fig. 4

 4 Fig. 4 Schematic Observer Implementation

Fig. 5

 5 Fig. 5 States Estimations

Acknowledgments

The authors wish to acknowledge project PIANO funded by the French Fonds Unique Interministériel (FUI).