Section: New Results

Applications

Modelling and control of Anaerobic Digestion processes

Participants : Boumediene Benyahia, Amine Charfi, Radhouene Fekih-Salem, Jérôme Harmand, Guilherme Pimentel, Tewfik Sari.

We consider the AM2 or AMOCO model developed in [72] and extend both the model in itself and its analysis to the following cases:

• Depending on the AM2 model parameters, the steady states were analytically characterized and their stability were analyzed [12] . Following this study, it was shown that the overloading tolerance, a parameter proposed in [81] to on-line monitoring anaerobic processes, may be not adapted under certain operating conditions and even lead to bad operating decisions.

• Within the framework of the PhD theses of Amine Charfi and Boumediene Benyahia, we have included the fouling dynamics of membranes into the AM2 and we have analyzed the resulting model (called the AM2b) [16] , [29] .

• We actually work towards two directions: (i) we are extending these results in including into the AM2 an additional process, i. e. the hydrolysis step in order to study bioprocesses treating solid waste (the resulting model being called the AM3) [36] , [37] ; (ii) we try to find links between complex models such as the ADM1 model and simple models such as the AM2b or the AM3 [40] .

Apart from this work on the modelling of anaerobic digesters and membrane bioreactors, we have developed a number of specific simple models for control design accounting for the coupling of such processes with membrane modules in the chemostat (PhD thesis of G. Pimentel). This work aims at contributing to the efficient treatment of wastewaters produced in fish production farms. The work of G. Pimentel aims at studying the coupling of simple fouling models with the model of the chemostat in order to propose new simple models for control design.

Hybrid modelling of biofilms in plug-flow reactors

Participants : Fabien Campillo, Chloé Deygout, Annick Lesne, Alain Rapaport.

We have proposed a multi-scaled modelling that combines three scales: a microscopic one for the individual bacteria, a mesoscopic or “coarse-grained” one that homogenises at an intermediate scale the quantities relevant to the attachment/detachment process, and a macroscopic one in terms of substrate concentration.

Such a “hybrid” approach allows for modelling and understanding in plug-flow reactors the interplay between

• the formation of the biofilm at a microscopic scale, that starts from a small number of bacteria (thus a stochastic individual based description),

• the limitation of the biofilm, due the carrying capacity of the wall attachment, at a mesoscopic scale,

• the consumption of nutrient along the flow at a macroscopic level, as a solution of a coupled transport-reaction partial differential equation.

The numerical computation of such a model requires a software architecture that allows the simultaneous simulation of stochastic events at the bacteria scale and the continuous evolution (in space and time) of the substrate density.

This work has been conducted within the DISCO project (see Section 7.3 ) and the postdoctoral stay of C. Deygout hired by the project, in close collaboration with A. Lesne (LPTMC, Univ. Paris VI). A first paper on the simulation model has been published [17] .

Within the DISCO project, experiments on real tubular plug-flow reactors have been simultaneously driven at IRSTEA Antony with the perspective of comparison with numerical simulations.

The multi-species case with different bacteria specialized in different environments (poor or rich in nutrient) is a work in progress.

Individual-based models for the bacterial degradation of the cellulose

Participants : Fabien Campillo, Chloé Deygout.

We propose an individual-based model for the degradation of one cellulose bead (dozens of micrometers in diameter) by cellulolytic bacteria. This model accounts for biofilm formation with minimal hypotheses: soluble substrate diffusion combined with bacterial chemotaxis-like movement in the liquid phase, lack of bacterial movement in the solid phase. The IBM results are qualitatively different from the main macroscopic degradation models previously used for cellulose degradation. It suggests that random and discrete processes could significantly impact the cellulose degradation dynamics by their effect on the spatial structuration of the colonized cellulose particles [44] .

Modelling and control of cascade biosystems to mimic batch wine making processes

Participants : Térence Bayen, Céline Casenave, Jérôme Harmand, Alain Rapaport, Matthieu Sebbah.

An experimental setup of four tanks connected in series has been designed by the research unit SPO (Montpellier) for studying four physiological stages of yeast as steady state. The manipulated variables are the flow rates $Q_i$ of each tank with the constraint $Q_i\ge Q_{i-1}\ge 0$, and the objective is to reach simultaneously four set-points in the four tanks. We are studying two kinds of control strategies:

• a linearizing feedback law that drives exponentially the dynamics to the target. This is not the fastest strategy but is has good robustness properties. Nevertheless, the inputs constraint imposes to use saturation functions that provide satisfactory convergence in simulations but that is hard to prove mathematically.

• a minimal time feedback. Due to lack of local controllability imposed by the constraint on the inputs, the optimal synthesis is not smooth with the presence of “barriers”. The input constraint $Q_i\ge Q_{i-1}\ge 0$ is unusual in optimal control problems that are linear w.r.t. to the control. The optimality of candidate singular arcs is still open for this problem.

This summer, some experiments have been made to test the first feedback law on the experimental setup. The control law seems to perform work, but other experiments should be made next year with more reliable input flow pumps.

This work was conducted as a part of the European CAFE project (Computer-Aided Food processes for control Engineering) described in Section 7.1 .

Modelling and control of an ice cream crystallization process

Participants : Céline Casenave, Denis Dochain.

In the ice cream industry, the type of final desired product (large cartons or ice creams on a stick) determine the viscosity at which the ice cream has to be produced. The control the viscosity of the ice cream at the outlet of a continuous crystallizer is therefore an important challenge. The problem has been studied in two steps. First, we have completed and validated on experimental data the reduced order model of the system. This model has been obtained by application of the method of moments on a population balance equation describing the evolution of the crystal size distribution. Then, we have proposed a nonlinear control strategy to control of the viscosity of the ice cream with the temperature of the refrigerant fluid of the crystallizer. This control strategy is based on a linearizing control law coupled with a Smith predictor to account for the measurement delay. The control has been validated on an experimental pilot plant located at IRSTEA (Antony, France).

This work was conducted as a part of the European CAFE project (Computer-Aided Food processes for control Engineering) described in Section 7.1 .

Bioremediation of natural resources

Participants : Sébastien Barbier, Jérôme Harmand, Alain Rapaport, Antoine Rousseau.

The objective of this work is to provide efficient strategies for the bioremediation of natural water resources. The originality of the approach is to couple minimal time strategies that are determined on a simplified model with a faithful numerical model for the hydrodynamics. This work has been carried out in close cooperation with A. Rousseau. Based on a previous paper that deals with an implicit representation of the spatial inhomogeneity of the resource with a small number of homogeneous compartments (with a system of ODEs), we have implemented a coupled ODE-PDE system that accounts for the spatial non-homogeneity of pollution in natural resources. The main idea is to implement a Navier-Stokes model in the resource (such as a lake), with boundary conditions that correspond to the output feedback that has been determined to be optimal for the simple ODEs model of a (small) bioreactor. A first mathematical model has been introduced and numerical simulations have been performed in academic situations. During the internship of S. Barbier (co-advised by A. Rousseau and A. Rapaport) we built a reduced model that approximates the reference PDE model thanks to a set of ODEs with parameters. Numerical optimization is performed on these parameters in order to better fit the reference model. This will lead to a publication.

The study of the minimal time strategies on the system of ODEs has been mainly achieved in cooperation with Chilean researchers (P. Gajardo, Universidad Tecnica Federico Santa Maria, and H. Ramirez, Centro de Modelamiento Matemático) and a Chilean PhD student (V. Riquelme, Depto. Ingenieria Matematica, Universidad de Chile) within the associated team DYMECOS [57] .

Modelling and simulating terrestrial plant ecological dynamics

Participant : Fabien Campillo.

This study is part of the ANR Syscomm MODECOL that is done in collaboration particularly with the University of Rennes I, the University of La Rochelle and Inria. The first semester of 2012 was the last part of the project. We propose a very original individual-based model for clonal plant dynamics in continuous time and space that focuses on the effects of the network structure of the plants on the reproductive strategy of ramets. The model is coupled with a PDE dynamics for resources. The basic structure of the IBM encompass a population of “ramets” (the individuals) connected by “stolons or rhizomes” (the network) [13] , [22] . See http://www-sop.inria.fr/members/Fabien.Campillo/software/ibm-clonal/ for more details.

Modelling and inferring agricultural dynamics

Participants : Fabien Campillo, Angelo Raherinirina.

The International Laboratory LIRMA supports this work that is done in collaboration with the University of Fianarantsoa in Madagascar and with Dominique Hervé (IRD, Fianarantsoa, Madagascar). The aim is to study the dynamics of agricultural plots on the edge of primary forest. In [32] a simple Markov model has been successfully confronted to a first data set with the help of maximum likelihood and Bayesian approaches. On a new data set developed by IRD, the Markov hypothesis has been rejected and we proposed to use semi-Makov models: for this new dataset the law of the sojourn time on certain states will depend on the next state visited, which is incompatible with the Markov hypothesis and which is consistent with the semi-Markov hypothesis.