Difference between revisions of "Bioreactor"

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The bioreactor example is an easy bioreactor with an substrate that is converted to a product by the biomass in the reactor. It has three states and a control that is describing the feed concentration of substrate. It is taken from the examples folder of the ACADO toolkit described in:
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{{Dimensions
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|nd        = 1
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|nx        = 3
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|nu        = 1
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|nc        = 2
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|nre      = 3
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}}
  
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  Houska, Boris, Hans Joachim Ferreau, and Moritz Diehl. "ACADO toolkit—An open‐source framework for automatic control and dynamic optimization."
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The bioreactor problem describes an substrate that is converted to a product by the biomass in the reactor. It has three states and a control that is describing the feed concentration of the substrate. The problem is taken from the examples folder of the ACADO toolkit described in:
 +
<bib id="Houska2011a" />
 +
 
 +
  Houska, Boris, Hans Joachim Ferreau, and Moritz Diehl.  
 +
"ACADO toolkit—An open‐source framework for automatic control and dynamic optimization."
 
  Optimal Control Applications and Methods 32.3 (2011): 298-312.
 
  Optimal Control Applications and Methods 32.3 (2011): 298-312.
  
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Originally it seems to be taken from:
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Originally the problem seems to be motivated by:
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<bib id="Versyck1999" />
  
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  VERSYCK, KARINA J., and JAN F. VAN IMPE. "Feed rate optimization for fed-batch bioreactors: From optimal process performance to optimal parameter estimation."
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  VERSYCK, KARINA J., and JAN F. VAN IMPE.  
 +
"Feed rate optimization for fed-batch bioreactors: From optimal process performance to optimal parameter estimation."
 
  Chemical Engineering Communications 172.1 (1999): 107-124.
 
  Chemical Engineering Communications 172.1 (1999): 107-124.
  
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\dot{X}&=&-DX+\mu X \\
 
\dot{X}&=&-DX+\mu X \\
 
\dot{S}&=& D(S_{f}-S)-\mu /Y_{xs} X \\
 
\dot{S}&=& D(S_{f}-S)-\mu /Y_{xs} X \\
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\dot{P}&=&-DP+ (\alpha \mu +\beta) X
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\dot{P}&=&-DP+ (\alpha \mu +\beta) X.
 
\end{array}
 
\end{array}
 
</math>
 
</math>
 
</p>
 
</p>
  
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The three states describe the concentration of the enzyme (<math>X</math>), the substrate (<math>S</math>), and the product (<math>P</math>) in the reactor. In steady state the feed and outlet are eaqual and dilute all three concentrations with a ratio <math>D</math>. The enzyme
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The right-hand side of these equations will be summed up in <math> f(x, S_f) </math>.
  
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In this section the dynamic model, that describes the physical or technical process of interest, should be described. First the differential equation should be written down. Then the states, inputs, and parameters should be declared (Dimension, Type (integer, real, binary, ...)). Finally the parameters that are not free in the optimization and the coefficients should be stated in an extra table or list. All equations and functions are defined over an interval <math>I \subset \R</math>. Initial values and free parameters are taken into account later in the Optimal Control Problem.
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The three states describe the concentration of the biomass (<math>X</math>), the substrate (<math>S</math>), and the product (<math>P</math>) in the reactor. In steady state the feed and outlet are equal and dilute all three concentrations with a ratio <math>D</math>. The biomass grows with a rate
 +
<math>\mu</math>, while it eats up the substrate with the rate <math>\mu/Y_{xs}</math> and produces product at a rate <math>(\alpha \mu +\beta)</math>. The rate <math>\mu</math> is given by:
  
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<math>\mu = \mu_{m}*(1-P/P_{m})*S/(K_m+S+S^2/K_i)</math>
  
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The parameter (coefficient) <math> p </math> could be stated in a table, if it is fixed.
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The fixed parameters (constants) of the model are as follows.
  
 
{| class="wikitable"
 
{| class="wikitable"
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|+Parameter(s)
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|+Parameters
 
|-
 
|-
 
|Name
 
|Name
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|Unit
 
|Unit
 
|-
 
|-
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|Parameter
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|Dilution
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|<math>p</math>
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|<math>D</math>
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|23
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|0.15
 
|[-]
 
|[-]
 +
|-
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|Rate coefficient
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|<math>K_i</math>
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|22
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|[-]
 +
|-
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|Rate coefficient
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|<math>K_m</math>
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|1.2
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|[-]
 +
|-
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|Rate coefficient
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|<math>P_m</math>
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|50
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|[-]
 +
|-
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|Substrate to Biomass rate
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|<math>Y_{xs}</math>
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|0.4
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|[-]
 +
|-
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|Linear slope
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|<math>\alpha</math>
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|2.2
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|[-]
 +
|-
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|Linear intercept
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|<math>\beta</math>
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|0.2
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|[-]
 +
|-
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|Maximal growth rate
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|<math>\mu_m</math>
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|0.48
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|[-]
 +
|-
 
|}
 
|}
  
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This would result in an right hand side function <math>f(x,u)</math> in this case, because there are no free parameters. Of course one could also specify an implicit DAE system (<math>F(\dot{x},x,u,p,t)=0</math>), where it just hast to be said which states are differential and which are algebraic.
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== Mathematical formulation ==
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Writing shortly for the states in vector notation <math>x=(X,S,P)^T</math> the OCP reads:
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 +
<p>
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<math>
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\begin{array}{clcl}
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\displaystyle \min_{x,S_f} & J(x,S_f)\\[1.5ex]
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\mbox{s.t.}
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& \dot{x} & = & f(x,S_f)\\
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& x(0) & = & (6.5,12,22)^T \\
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& S_f & \in  &[28.7,40].
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\end{array}
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</math>
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</p>
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=== Objective ===
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<p>
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<math>
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J(x,S_f)=\int_0^{48}D(S_f-P)^2dt
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</math>
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</p>
 +
 
 +
== Reference Solution ==
 +
 
 +
Here we present the reference solution of the reimplemented example in the ACADO code generation with matlab. The source code is given in the next section.
 +
 
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<gallery caption="Reference solution" widths="551px" heights="390px" perrow="1">
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Image:ACADO_bioreactor.png| Optimal solution for the ACADO example.
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</gallery>
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== Source Code ==
 +
 
 +
Model descriptions are available in
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* [[:Category:ACADO | ACADO code]] at [[Bioreactor (ACADO)]]
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<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
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[[Category:MIOCP]] [[Category: ODE model]] [[Category:Chemical engineering]]
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[[Category:Bang bang]]

Latest revision as of 10:27, 27 July 2016

Bioreactor
State dimension: 1
Differential states: 3
Continuous control functions: 1
Path constraints: 2
Interior point equalities: 3


The bioreactor problem describes an substrate that is converted to a product by the biomass in the reactor. It has three states and a control that is describing the feed concentration of the substrate. The problem is taken from the examples folder of the ACADO toolkit described in: [Houska2011a]The entry doesn't exist yet. 

Houska, Boris, Hans Joachim Ferreau, and Moritz Diehl. 
"ACADO toolkit—An open‐source framework for automatic control and dynamic optimization."
Optimal Control Applications and Methods 32.3 (2011): 298-312.

Originally the problem seems to be motivated by: [Versyck1999]The entry doesn't exist yet. 

VERSYCK, KARINA J., and JAN F. VAN IMPE. 
"Feed rate optimization for fed-batch bioreactors: From optimal process performance to optimal parameter estimation."
Chemical Engineering Communications 172.1 (1999): 107-124.

Model Formulation

The dynamic model is an ODE model:

\begin{array}{rcl} \dot{X}&=&-DX+\mu X \\ \dot{S}&=& D(S_{f}-S)-\mu /Y_{xs} X \\ \dot{P}&=&-DP+ (\alpha \mu +\beta) X. \end{array}

The right-hand side of these equations will be summed up in f(x, S_f) .

The three states describe the concentration of the biomass (X), the substrate (S), and the product (P) in the reactor. In steady state the feed and outlet are equal and dilute all three concentrations with a ratio D. The biomass grows with a rate \mu, while it eats up the substrate with the rate \mu/Y_{xs} and produces product at a rate (\alpha \mu +\beta). The rate \mu is given by:

\mu = \mu_{m}*(1-P/P_{m})*S/(K_m+S+S^2/K_i)

The fixed parameters (constants) of the model are as follows.

Parameters
Name Symbol Value Unit
Dilution D 0.15 [-]
Rate coefficient K_i 22 [-]
Rate coefficient K_m 1.2 [-]
Rate coefficient P_m 50 [-]
Substrate to Biomass rate Y_{xs} 0.4 [-]
Linear slope \alpha 2.2 [-]
Linear intercept \beta 0.2 [-]
Maximal growth rate \mu_m 0.48 [-]

Mathematical formulation

Writing shortly for the states in vector notation x=(X,S,P)^T the OCP reads:

\begin{array}{clcl} \displaystyle \min_{x,S_f} & J(x,S_f)\\[1.5ex] \mbox{s.t.} & \dot{x} & = & f(x,S_f)\\ & x(0) & = & (6.5,12,22)^T \\ & S_f & \in &[28.7,40]. \end{array}

Objective

J(x,S_f)=\int_0^{48}D(S_f-P)^2dt

Reference Solution

Here we present the reference solution of the reimplemented example in the ACADO code generation with matlab. The source code is given in the next section.

  • Reference solution

  • Optimal solution for the ACADO example.

Source Code

Model descriptions are available in

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