A new methodology for early BMP assessment using a mathematical model

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A new methodology for early BMP assessment using a mathematical model

Sabrina Gu érin ² St éphane Mottelet ¹ Sam Azimi ² Jean Bernier ² Laura Andr é ³ Thierry Ribeiro ³ Andr é Pauss ¹ Vincent Rocher ²

1

Universit ´e de Technologie de Compi `egne, FRANCE

2

SIAAP Direction D ´eveloppement et Prospective, Colombes, FRANCE

3

UniLaSalle Beauvais, FRANCE

15 ^th Anaerobic Digestion 2017 Conference

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MOCOPEE Program (Modeling, Control and Optimization of Wastewater Treatment Processes, www.mocopee.com)

The Mocopee research program aims to build the metrological and mathematical tools (signal processing, treatment processes modeling, regulation) required to improve the control and the optimization of water and sludge treatment plants.

R&D actions on the AD process :

I

validate at industrial scale sewage sludge BMP estimation methods

I

build predictive models of AD process

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The different substrates considered in the study

Primary, biologic (nitrification, denitrification), floated, mixed and thickened sludge, from different plants of SIAAP (Seine centre, Seine aval, Seine Gr ´esillons)

Inoculum sampled at the output of the digester

preliminary work :

S. Gu érin et al. (2016), Cartographie des boues de STEP et r éduction du temps de mesure du potentiel m éthanog ène :

couplage exp érimentation en r éacteur / mod élisation

, L’eau, l’industrie, les

nuisances, n

^o

397

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Experimental device

500 ml reactors,I/S ratio=3 CO ₂ trapping

Mean flow measurement by ≈ 10ml throttles

AMPTS

Full compliance with experts recommendations :

C. Holliger et al. (43 auteurs) (2016), Towards a standardization of biomethane potential tests, Water

Science &Technology

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Experimental study

36 batchs in triplicates MS, MV, DCO, DBO measurement

BMP obtained after 20 days No evident correlation between BMP and a priori measurements !

Could a model of AD digestion help to make an early assessment of BMP ?

0 2 4 6 8 10

t (day) 0

1 2 3

CH4 flow (g COD/L/day)

0 2 4 6 8 10

t (day) 0

1 2 3

0 2 4 6 8 10

t (day) 0

1 2 3

0 2 4 6 8 10

t (day) 0

1 2 3

0 2 4 6 8 10

t (day) 0

1 2 3

0 2 4 6 8 10

t (day) 0

1 2 3

0 2 4 6 8 10

t (day) 0

1 2 3

0 2 4 6 8 10

t (day) 0

1 2 3

0 2 4 6 8 10

t (day) 0

1 2 3

0 2 4 6 8 10

t (day) 0

1 2 3

0 2 4 6 8 10

t (day) 0

1 2 3

0 2 4 6 8 10

t (day) 0

1 2 3

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Modified AM2 model

O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi, J. P. Steyer (2001), Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnology and

bioengineering

R. Fekih Salem, N. Abdellatif, T. Sari, and J. Harmand (2012), On a three step model of anaerobic digestion including the hydrolysis of particulate matter, MATHMOD 2012

A. Donoso-Bravo, S. P ´erez-Elvira and F. Fdz-Polanco (2014), Simplified mechanistic model for the two-stage anaerobic degradation of sewage sludge, Environmental Technology

S ₀ −−−−→ ^r

⁰

S ₁ , (Hydrolysis)

S ₁ −−−−→ ^r

¹

Y _X

₁

X ₁ + (1 − Y _X

₁

)S ₂ + k ₄ CO ₂ , (Acidification) S ₂ −−−−→ ^r

²

Y _X

₂

X ₂ + (1 − Y _X

₂

)CH ₄ + k ₅ CO ₂ , (Methanogenesis) S ₀ : insoluble organic molecules, S ₁ : simple compounds (fatty acids, peptides, amino acids, . . .), S ₂ : volatile fatty acids

Warning : we use the

batch

version of this model !

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Modified AM2 model

Differential equations system

Perfectly mixed batch reactor, states of the sytem : S ₀ , S ₁ , S ₂ , X ₁ , X ₂ S ₀ ⁰ = −r ₀ , S ₁ ⁰ = r ₀ − r ₁ , S ₂ ⁰ = (1 − Y _X

₁

)r ₁ − r ₂ , X ₁ ⁰ = Y _X

₁

r ₁ , X ₂ ⁰ = Y _X

₂

r ₂ , CH ₄ ⁰ = (1 − Y _X

₂

)r ₂ initial conditions : S ₀ (0) = S ₀ ⁰ , S ₁ (0) = S ₁ ⁰ , X ₁ (0) = X ₁ ⁰ , X ₂ (0) = X ₂ ⁰ . reaction rates :

r ₀ = µ ₀ S ₀ , r ₁ = µ ^max ₁ S ₁ X ₁

S ₁ + K _S

₁

, r ₂ = µ ^max ₂ S ₂ X ₂

S ₂ + K _S

₂

+ S ₂ ² /K _I · Parameters θ = (Y _X

₁

, Y _X

₂

, µ ₀ , µ ^max ₁ , µ ^max ₂ , K _S

₁

, K _S

₂

, K _I

| {z }

θ

c

: kinetic parameters

, X ₁ ⁰ , X ₂ ⁰ , S ₀ ⁰ , S ₁ ⁰ , S ₂ ⁰

| {z }

θ

b

: batch parameters

)

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Modified AM2 model

Identifiability

Knowing CH ₄ (t), ∀t > 0 can we uniquely identify parameters ? I O. Bernard et al. (2001), R. Fekih Salem et al. (2012) model : no

I A. Donoso et al. (2014) : a priori no , but certain algebraic expressions are identifiable :

CH4(∞) = (1 − Y _X

₂

)

(1 − Y _X

₁

)(S ₀ ⁰ + S ₁ ⁰ ) + S ₂ ⁰

Other interesting expressions (relative proportions of substrates) : S ₀ ⁰

P 2

i=0 S _i ⁰ , S ₁ ⁰ P 2

i=0 S ⁰ _i , S ₂ ⁰ P 2

i=0 S _i ⁰

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Modified AM2 model

Practical identification

Goals :

1

obtain a mathematical model allowing to reproduce the methane rate of our 108 experiences, without necessarily uniquely describe state variables

(X ₁ , X ₂ , S ₁ , S ₂ , S ₀ ).

2

being able to use this model to predict the BMP from new data measured after 4 days Pitfalls to bypass :

I No identifiability of parameters = ⇒ numerical problems !

I Important mass of data (108 experiences to assimilate)

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Identification of parameters

Available measurements, simulations

For each batch #i we have

I (t _k ⁱ ) _k=1...m

_i

the times of throttle switchs

I (D _k ⁱ ) k=2...m

_i

the mean CH ₄ flow rate measured at t = t _k ⁱ , k = 1 . . . m _i For θ = (θ c , θ b ) we can simulate the mean flow of CH ₄ :

d _k ⁱ (θ _c , θ _b ) =

CH ₄ (t _k ⁱ ) − CH ₄ (t _k−1 ⁱ )

/(t _k ⁱ − t _k−1 ⁱ ),

and the function

J _i (θ c , θ _b ) =

m

i

X

k=2

(t _k ⁱ − t _k−1 ⁱ )(D _k ⁱ − d _k ⁱ (θ c , θ _b )) ²

evaluates the misfit between measurements of batch #i and the simulation with

parameters θ = (θ c , θ _b )

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Identification of parameters

Strategy

1

Learning phase : minimize with respect to ξ = (θ c , θ ¹ _b . . . , θ _b ⁶⁹ ) ∈ R ³⁵³ ξ ˆ = arg min

ξ J (ξ) = X

i=1...69

J _i (θ c , θ _b ⁱ ) + λkξk ² ,

We obtain θ ˆ c ∈ R ⁸ which is used for the prediction

I

Optimization : interior points method (fminc, MATLAB)

I

Moderate computation time (computer with 20 processors Xeon E5-2660-v2)

2

Prediction/validation phase at T = 4 days : minimize with respect to θ ⁱ _b ∈ R ⁵ θ ˆ ⁱ _b = arg min

θ

b

J _i ^T ( θ ˆ _c , θ _b ), i = 70 . . . 108

J _i ^T ( θ ˆ c , θ _b ) = X

k=2

t

k

≤T

(t _k ⁱ − t _k−1 ⁱ )(D _k ⁱ − d _i ^k ( θ ˆ c , θ _b )) ²

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Results

Learning on batchs #1 to #69

Kinetic parameters θ ˆ c

Y _X

₁

0, 58 Y _X

₂

0, 12

µ 0 0, 29

µ ^max ₁ 3, 63 µ ^max ₂ 2, 67 K _S

₁

1, 02 K _S

₂

3, 45 K _I 1, 44

0 200 400 600 800 1000 1200 1400

true BMP (NmL) 0

200 400 600 800 1000 1200

predicted BMP (NmL)

Model-3

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Results

Learning, primary sludge

0 5 10 15

t (days) 0

100 200 300 400

flow (NmL/d)

Model-3 - batch n°5 (Manip 20130717, cell=8)

0 5 10 15

t (days) 0

500 1000

Vol (NmL))

0 5 10 15

t (days) 0

5 10

g.COD/L

S0 S₁ S₂

0 5 10 15

t (days) 0

5 10 15

g.COD/L

Model-3 - batch n°5 (Manip 20130717, cell=8) X₁ X₂

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Results

Learning, nitrification sludge

0 5 10 15

t (days) 0

100 200 300 400

flow (NmL/d)

0 5 10 15

t (days) 0

500 1000

Vol (NmL))

0 5 10 15

t (days) 0

5 10

g.COD/L

S0 S₁ S₂

0 5 10 15

t (days) 0

2 4 6 8

g.COD/L

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Results

Prediction/validation on batchs #70 to #108 at T = 4 days

0 200 400 600 800 1000 1200 1400

true BMP (NmL) 0

200 400 600 800 1000 1200

predicted BMP (NmL)

Model-3

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Results

Prediction, floated sludge

0 5 10 15

t (days) 0

100 200 300 400

flow (NmL/d)

0 5 10 15

t (days) 0

500 1000

Vol (NmL))

0 5 10 15

t (days) 0

5 10

g.COD/L

S0 S₁ S₂

0 5 10 15

t (days) 0

5 10

g.COD/L

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Results

Prediction, thickened sludge

0 5 10 15

t (days) 0

100 200 300 400

flow (NmL/d)

0 5 10 15

t (days) 0

500 1000

Vol (NmL))

0 5 10 15

t (days) 0

5 10

g.COD/L

S0 S₁ S₂

0 5 10 15

t (days) 0

5 10

g.COD/L

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Trends and conclusions

Results :

I

Well fitted kinetics in learning phase and good prediction of BMP at 4 days

I

Ratios of S

₀

, S

₁

, S

₂

seem to be interpretable

Planned improvements :

I

Theoretical study of identifiability of parameters in learning phase

I

DBO and VSS measurements should be taken into account

I

Coupling between triplicates has to be considered

I

Confidence intervals should be computed for the predicted BMP

I

Actual model should be simplified and compared with other models

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A new methodology for early BMP assessment using a mathematical model