Optimization of fed-batch bioreactors using genetic algorithm: multiple control variables
Introduction
The usual objective in optimal control of a fed-batch bioreactor is to maximize the biomass and/or the metabolite production and the optimization has been traditionally sought with respect to substrate feed rate. Determination of optimal substrate feed rate is a problem in singular control, so called because the control variable appears linearly both in the dynamic equations describing the process and/or in the performance index which is to be optimized. In many industrially important fermentation processes, microorganisms require more than one substrate for their growth and product formation (Modak & Lim, 1989). It has long been realized that the production of antibiotics and enzymes requires precise control of the nitrogen source in addition to the carbon source. The production of a desired chemical from recombinant cell cultures often involves addition of either an inducer or repressor along with the primary growth-limiting nutrient. The optimization problem for such processes involves the determination of the optimal feed rates of two nutrients: either two growth-limiting substrates such as carbon and nitrogen or one growth-limiting substrate and an inducer or a repressor. The feed rate optimization of fed-batch bioreactors involving multiple singular control variables is a numerically difficult problem. The optimization of bioreactor performances by manipulating two control variables simultaneously by use of optimal control theory has been reported in literature (Modak & Lim, 1989, Lee & Ramirez, 1994; Lee, Hong, & Lim, 1998; Rahman & Palanki, 1998). Also, there are some reports of use of direct search methods (adaptive stochastic algorithm, dynamic programming, genetic algorithms, simulated annealing etc.) that transform the optimal control problem into a nonlinear programming problem for solution of such problems (De Tremblay, Perrier, Chavarie, & Archambault, 1992; Carrsco & Banga, 1997, Tholudur & Ramirez, 1997; Roubos, van Straten, & van Boxtel, 1999; Balsa-Canto, Banga, Alonso, & Vassiliadis, 2000; Mekarapiruk & Luus, 2000; Jayaraman, Kulkarni, Gupta, Rajesh, & Kusumaker, 2001; Nguang, Chen, & Chen 2001; Sarkar & Modak, 2003a).
In a previous study, we developed an optimization technique based on Genetic Algorithm (GA) to determine optimal substrate feeding policy for fed-batch bioreactors with a single control variable (Sarkar & Modak, 2003b). A customized GA with a problem specific representation for the decision variables was used and the algorithm incorporated to some extent the information available from the use of optimal control theory. In this study, we extend this methodology to address fed-batch processes with multiple control variables. The efficiency of the proposed method is demonstrated by taking two challenging optimal control problems in fed-batch bioreactors from the open literature: (1) induced foreign protein production by recombinant bacteria (Lee & Ramirez, 1994), (2) production of monoclonal antibody (De Tremblay et al., 1992).
Section snippets
Optimization problem
For a typical fed-batch operation with two control variables, the mass balance equations for cells, substrates, product, and total mass (assuming constant density) can be written as followswhere X, S1, S2 and P are the concentrations of cells, substrate-1, substrate-2 and product, respectively. μ, σ1, σ2 and π are the specific rates of growth, substrate-1 consumption, substrate-2 consumption and metabolite production, respectively. S
Genetic algorithm for optimal feed rate determination
The important aspect of our algorithm is the representation of the decision variables to represent a substrate feeding profile to the bioreactor. In what follows, we describe briefly the main features of our GA-based algorithm for solution of optimal control problems with multiple feed. Further implementation details of the algorithm can be found in Sarkar and Modak (2003b).
According to optimal control theory, the optimal feeding pattern will consist of intervals of maximum, minimum, and
Case study I: Induced foreign protein production by recombinant bacteria
Lee and Ramirez (1994) have developed the optimal nutrient and inducer feeding strategy for the fed-batch production of induced foreign protein using recombinant bacteria. They used the optimal control theory and showed the existence of singular control arcs for this system. Since the performance index exhibits a very low sensitivity with respect to the controls, Tholudur and Ramirez (1997) constructed a modified parameter function set to increase the sensitivity to the controls and we consider
Conclusion
The feed rate optimization of fed-batch bioreactors involving multiple singular control variables is a numerically difficult problem. An optimization procedure based on genetic algorithm is developed for the determination of substrate feeding policies in fed-batch bioreactors with multiple control variables. The proposed algorithm in this study combines genetic algorithm with certain knowledge generated through the applications of the optimal control theory to search simultaneously the correct
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