# Reduced Models (and control) of in-situ decontamination of large water resources

Antoine Rousseau, Alain Rapaport

► To cite this version:

Antoine Rousseau, Alain Rapaport. Reduced Models (and control) of in-situ decontamination of large water resources. Sino-French Conference on Modeling, Mathematical Analysis and Computation, Chuanju Xu; Alain Miranville, Jun 2017, Xiamen, China. pp.27. hal-02785037

HAL Id: hal-02785037

https://hal.inrae.fr/hal-02785037

Submitted on 4 Jun 2020

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<figure><img src="image_1.png" /></figure>
Reduced Models (and control) of in-situ
decontamination of large water resources

Antoine ROUSSEAU,

Inria, Team LEMON (Montpellier)

In honor of my "brother in sciences" Claudius

<figure><img src="image_2.png" /></figure>

IMAG

INSTITUT MONTPELLIER IN
ALEXANDER GROTHENDIECK

<figure><img src="image_3.png" /></figure>

June 9th, 2017, SFC2MAC, Xiamen

Joint work with Alain Rapaport (INRA)
## Outline

1. Introduction : the Taihu problem

2. A simple ODE model

3. A PDE-based model for the lake

4. Back to ODEs
Outline

1. Introduction : the Taihu problem

2. A simple ODE model

3. A PDE-based model for the lake

4. Back to ODEs
# Applicative Framework

<figure><img src="image_4.png" /><figcaption>Polluted Taihu (algae), Yangtze Delta plain, Wuxi, China.</figcaption></figure>

Courtesy Desert Research Institute

## Objectives :

* use a bioreactor to remove pollution from the lake

* do it as efficiently as possible
# Bioremediation problem

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

## Parameters and unknowns

- $V_L$ and $V_R$ the volumes of the resource and bioreactor,

- $X_R$ the biomass concentration in the bioreactor,

- $S_L$ and $S_R$ the pollutant concentrations,

- $\mu(\cdot)$ the biomass growth law,

- $Q = Q(t)$ the pump discharge, controlled by the user.
## Outline

1. Introduction : the Taihu problem

2. A simple ODE model

3. A PDE-based model for the lake

4. Back to ODEs
ODE-based model : the homogeneous case $S_L = S_L(t)$

<figure><img src="image_7.png" /></figure>

$$ \begin{cases} \dot{X}_R = \mu(S_R) X_R - \frac{Q}{V_R} X_R, \\ \dot{S}_R = -\mu(S_R) X_R + \frac{Q}{V_R} (S_L - S_R), \\ \dot{S}_L = \frac{Q}{V_L} (S_R - S_L), \end{cases} $$
ODE-based model : the homogeneous case $S_L = S_L(t)$

<figure><img src="image_8.png" /></figure>

$$ \begin{cases} \dot{X}_R = \mu(S_R) X_R - \frac{Q}{V_R} X_R, \\ \dot{S}_R = -\mu(S_R) X_R + \frac{Q}{V_R} (S_L - S_R), \\ \dot{S}_L = \frac{Q}{V_L} (S_R - S_L), \end{cases} $$

Chemostat equations :
J. Monod (1910-1976)
ODE-based model : the homogeneous case $S_L = S_L(t)$

$$
\begin{cases}
\dot{X}_R = & \mu(S_R) X_R - \frac{Q}{V_R} X_R, \\
\dot{S}_R = & -\mu(S_R) X_R + \frac{Q}{V_R} (S_L - S_R), \\
\dot{S}_L = & \frac{Q}{V_L} (S_R^\infty - S_L),
\end{cases}
$$

Slow-fast
approximation :

$$
\epsilon = V_R / V_L \ll 1
$$

Reference : Gajardo et al, Automatica 2011.
ODE-based model : the homogeneous case $S_L = S_L(t)$

$$
\left\{
\begin{aligned}
\dot{X}_R &= \mu(S_R) X_R - \frac{Q}{V_R} X_R, \\
\dot{S}_R &= -\mu(S_R) X_R + \frac{Q}{V_R} (S_L - S_R), \\
\dot{S}_L &= \frac{Q}{V_L} (S_R^\infty - S_L),
\end{aligned}
\right.
$$

Slow-fast
approximation :

$$
\epsilon = V_R / V_L \ll 1
$$

Reference : Gajardo et al, Automatica 2011.

Theorem (case $\mu(X) = \mu X$)

For any $X_R(0) > 0$ and $S_R(0) = S_L > 0$, our system has a unique global
solution. In addition, there exists a critical flow rate $Q_c > 0$, depending on $S_L$,
such that asymptotically :

* if $Q > Q_c$, then $(S_R(s), X_R(s))$ converges towards $(S_L, 0)$,

* if $0 < Q < Q_c$, then $(S_R(s), X_R(s))$ converges towards $(S_R^\infty, X_R^\infty)$ with
  $S_R^\infty = \frac{Q}{\mu V_R} < S_L$.
# Optimal flow rate for the bioremediation (linear $\mu$)

Back to the lake equations...

$$ \dot{S}_L = \frac{Q}{V_L} \left( \frac{Q}{\mu V_R} - S_L \right), $$

What is the best flow rate $Q$?
# Optimal flow rate for the bioremediation (linear $\mu$)

Back to the lake equations...

$$ \dot{S}_L = \frac{Q}{V_L} \left( \frac{Q}{\mu V_R} - S_L \right), $$

What is the best flow rate $Q$?

$$ Q(t) = \frac{\mu V_R}{2} S_L(t) $$
# Optimal flow rate for the bioremediation (linear $\mu$)

Back to the lake equations...

$$ \dot{S}_L = \frac{Q}{V_L} \left( \frac{Q}{\mu V_R} - S_L \right), $$

What is the best flow rate $Q$?

$$ Q(t) = \frac{\mu V_R}{2} S_L(t) $$

<figure><img src="image_9.png" /></figure>
# Outline

1. Introduction : the Taihu problem

2. A simple ODE model

3. A PDE-based model for the lake

4. Back to ODEs
The non-homogeneous case : $S_L = S_L(t, x, y)$

Flow rate Q

Lake

Bioreactor

Biomass

<figure><img src="image_10.png" /></figure>

Navier Stokes & transport
equations in the resource

$$
\left\{
\begin{array}{l}
\displaystyle \frac{\partial u}{\partial t} + u \cdot \nabla u - \nu_u \Delta u + \nabla p = 0, \\
\\
\displaystyle \nabla \cdot u = 0, \\
\\
\displaystyle \frac{\partial S_L}{\partial t} + u \cdot \nabla S_L - \nu_s \Delta S_L = 0.
\end{array}
\right.
$$

Boundary conditions

$$
\left\{
\begin{array}{l}
u_{in}(t) = \mathcal{A} \times \mathcal{Q}(t), \\
u_{out}(t) = \mathcal{A} \times \mathcal{Q}(t), \\
\text{B.C on } S_L
\end{array}
\right.
$$
<figure><img src="image_11.png" /></figure>

Circular lake : 1 pump
Comparing models

<figure><img src="image_12.png" /></figure>

Pollution concentration in the lake
with ODE (black) and PDE (red)
## Outline

1. Introduction : the Taihu problem

2. A simple ODE model

3. A PDE-based model for the lake

4. Back to ODEs
# Active and dead zones model

## New ODE model

$$ \dot{X} = \mu(S_R)X - \frac{Q}{V_R}X, $$

$$ \dot{S}_R = -\mu(S_R)X + \frac{Q}{V_R}(S_1 - S_R), $$

$$ \dot{S}_1 = \frac{Q}{V_1}(S_R - S_1) + \frac{d}{V_1}(S_2 - S_1), $$

$$ \dot{S}_2 = \frac{d}{V_2}(S_1 - S_2). $$

<figure><img src="image_13.png" /></figure>
# Active and dead zones model

## New ODE model

$$ \dot{X} = \mu(S_R)X - \frac{Q}{V_R}X, $$

$$ \dot{S}_R = -\mu(S_R)X + \frac{Q}{V_R}(S_1 - S_R), $$

$$ \dot{S}_1 = \frac{Q}{V_1}(S_R - S_1) + \frac{d}{V_1}(S_2 - S_1), $$

$$ \dot{S}_2 = \frac{d}{V_2}(S_1 - S_2). $$

<figure><img src="image_14.png" /></figure>

### Existence, uniqueness, optimal control for two zones

Gajardo, Ramírez, Rapaport, Riquelme. Bioremediation of natural water resources via optimal control techniques.

BIOMAT 2011, 178-190, World Sci. Publ., Hackensack, NJ, 2012.

Two parameters we can play with: $\frac{V_1}{V_1 + V_2}$ and $d$.
# Comparing models (with A. Rapaport and S. Barbier)

Optimization on the parameters $\frac{V_1}{V_1 + V_2}$ and $d$

<figure><img src="image_15.png" /><figcaption>Pollution concentration in the lake
1 ODE (black)
2 ODEs (blue)
PDE (red)</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Best parameter $d$ vs viscosity $\nu_S$ (dashed blue)
Best meansquare fit (plain red)</figcaption></figure>
In summary...

<figure><img src="image_17.png" /></figure>
Conclusion

The “take home” message is...

* we have designed a first model and questioned it,

* we proposed a more complicated (too much ?) one,

* we took an intermediary model, which seems satisfactory, on which
optimal control can be done.
Conclusion

The “take home” message is...

* we have designed a first model and questioned it,

* we proposed a more complicated (too much ?) one,

* we took an intermediary model, which seems satisfactory, on which
optimal control can be done.

BUT : is the reference (PDE) model a realistic model ?
# Conclusion

The “take home” message is...

* we have designed a first model and questioned it,

* we proposed a more complicated (too much ?) one,

* we took an intermediary model, which seems satisfactory, on which optimal control can be done.

BUT : is the reference (PDE) model a realistic model?

Certainly not...

## Sino-French project ANSWER:

<figure><img src="image_18.png" /></figure>

Analysis and Numerical Simulation of Water Ecosystems in Response to anthropogenic environmental changes
Conclusion

The “take home” message is...

* we have designed a first model and questioned it,

* we proposed a more complicated (too much ?) one,

* we took an intermediary model, which seems satisfactory, on which
optimal control can be done.

BUT : is the reference (PDE) model a realistic model?

Certainly not...

Sino-French project
ANSWER:

<figure><img src="image_19.png" /></figure>

Analysis and Numerical Simulation of Water
Ecosystems in Response to anthropogenic
environmental changes

The more we fail, the more chances we have to succeed
Thank you for your attention

A few references...

* P. Gajardo, J. Harmand, H. Ramírez, A. Rapaport and V. Riquelme
  Minimal time bioremediation of natural water resources. Automatica 2011

* P. Gajardo, H. Ramírez, A. Rapaport and V. Riquelme
  Bioremediation of natural water resources via optimal control techniques.
  BIOMAT 2012

* A. Rousseau
  Texte public de l'agrégation externe de mathématiques.
  http://agreg.org/Textes/public2016-B4.pdf, 2012.

* A. Rapaport, A. Rousseau and J. Harmand.
  Procédé de traitement d'une ressource fluide, programme d'ordinateur et module
de traitement associés. Patent No : FA 78 4546 - FR 13 55129, 2014.

* S. Barbier, A. Rapaport et A. Rousseau.
  Modelling of biological decontamination strategies of a water resource in natural
  environment. Journ. Sci. Comp, 2016.

* A. Rousseau, A. Rapaport, A. Pacholik and C. Leininger
  Action Dépollution. Serious game. https://depollution.inria.fr/

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