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Development of a computer algorithm based on a conjugate gradient approach for optimization of fed-batch fermentations

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Abstract

The problem of optimization of fed-batch fermentations using the substrate feed rate as the control variable is singular in nature. Previous approaches, including the boundary condition iteration method and transformation to a nonsingular problem using a different control variable, do not work well for solving optimization of systems governed by more than four differential equations. The applicability of a first-order conjugate gradient algorithm for optimizing fed-batch fermentations was tested for systems of varing complexity. This approach does not need any variable transformation ora priori knowledge of the control arc sequence. Constraints on the feed rate are handled in a simple and direct manner. The algorithm worked very well for three, four, and five-dimensional singular systems. The correctness of the optimal profile was judged by observing the variation in the sign of the gradient of the Hamiltonian. The gradient was found to be zero during the singular period and had the appropriate sign on the boundary arcs. The optimization method based on conjugated gradient approach can be complementary to the boundary condition iteration method for determination of the exact optimum profile.

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Abbreviations

F:

fraction of plasmid-free cells, or feed rate L/hr

g:

gradient of the Hamiltonian with respect to the control vector

H:

Hamiltonian

I:

a parameter defined in Eq. (13)

K, k:

model parameters, or weight on the penalty function

kd :

first order inactivation constant [hr1]

p:

probability of forming a plasmid-free cell upon cell division

R:

fraction of glucose that is channeled through the fermentative pathway, or metabolic flux [(g glucose)/hr. OD]

S:

substrate concentration [g/L]

s:

conjugate gradient direction

t:

time [hr]

V:

volume [L]

w:

a boundary function defined by Eq. (6), (12)

X:

cell mass concentration [g/L or OD]

x:

state vector

Yxy :

yield of respiratory pathway [OD/(g. glucose)]

Yxj :

yield of fermentative pathway [OD/(g. glucose)]

α:

a search parameter in the conjugate gradient method

β:

a search parameter in the conjugate gradient method

λ :

adjoint vector, or costate vector

μ:

specific cell growth rate [hr]

Π:

performance index, or objective function

π:

invertase formation rate [KU/hr]

ϕ:

gradient of Hamiltonian in singular problems

+:

plasmid-containing cells

-:

plasmid-free cells

*:

optimal

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Authors and Affiliations

  1. School of Chemical Engineering, Purdue University, IN 47907, West Lafayette, USA

    Anant Patkar

  2. Department of Food Science and Technology, Research Center for New-Biomaterials in Agriculture, Seoul National University, 441-744, Suwon, Korea

    Anant Patkar, Dong Hoon Lee & Jin-Ho Seo

Authors
  1. Anant Patkar
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  2. Dong Hoon Lee
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  3. Jin-Ho Seo
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Patkar, A., Lee, D.H. & Seo, JH. Development of a computer algorithm based on a conjugate gradient approach for optimization of fed-batch fermentations. Korean J. Chem. Eng. 10, 146–155 (1993). https://doi.org/10.1007/BF02705137

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  • DOI: https://doi.org/10.1007/BF02705137

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