Optimizatin of fed-batch processes: Challenges and solutions

Optimizatin of fed-batch processes : Challenges and solutions

A. W. S. HENRIQUES, A. C. COSTA, T. L. M. ALVES and E. L. LIMA

Programa de Engenharia Química/ COPPE/UFRJ

C. P. Box 68502, 21945-970, Rio de Janeiro - RJ, Brazil

e-mail: [e-mail]

(Received: January 19, 1999; Accepted: April 12, 1999)

Abstract - The optimization problem of a fed-batch alcoholic fermentation by Zymomonas mobilis, solved using Pontryagin's maximum principle and the singular control theory, is used to discuss the main challenges that usually arise during the implementation of optimal strategies. For each problem a practical solution is proposed. In order to solve the optimization problem, a hybrid neural model is developed using experimental data from batch and fed-batch fermentations. The calculated optimal solution is experimentally implemented in open-loop scheme.

Keywords: optimization, fed-batch processes, control theory.

INTRODUCTION

The stirred tank operating in fed-batch mode is a chemical reactor frequently used in the industry (Jorgensen and Jensen, l989; Schnelle Jr and Richards, 1986; McGreavy, l983). Its use is specially advantageous for systems such as bioprocesses and polimerization processes, in which products of a desired quality can be obtained by efficient (optimal) operation, (Rahman and Palanki, l986; Lim et al., l994; Embiruçu et al., l996). In these processes, different reaction rates between competitive species, inhibition phenomena etc. determine that some reactants should be added (or that the temperature should be varied) according to certain temporal laws. Although this kind of reactor is very flexible, its operation and control is usually difficult. Its transient behavior, which leads to operation under a wide range of conditions, magnifies problems associated with the inherent nonlinearity of chemical processes.

One of the main objectives of batch reactor engineering is the optimization of reactor operation, striving for higher productivity, a shorter reaction time, higher product quality, good reproduction of results from batch to batch, etc. Once a performance index is defined and some constraints are fixed, involving a dynamic model and other physical limitations, the traditional way to determine the best operational conditions is the application of nonlinear optimization techniques, which results in control actions for implementation in open-loop scheme. The practical implementation of these strategies involves many challenges that arise mainly due to the lack of knowledge of the true behavior of the process (uncertainties). Due to unforeseen uncertainties, sometimes an optimal strategy determined in a simulation study results in a complete failure when experimentally implemented. In these cases, to make implementation possible, a series of ad-hoc adaptations have to be made in the precalculated strategy, which in some cases modify it to the point that it loses its optimal characteristics. This is one of the main limitations in the use of advanced techniques for the automatic operation of batch processes that favor manual operation, wich is highly dependent on the abilities of a "specialist".

In this work, studying the optimal fed-batch operation of a bioprocess, ethanol production by Zymomonas mobilis, we faced many important challenges. The proposed solutions worked efficiently and are not restricted to the process studied.

In order to solve the proposed optimization problem, a mathematical model of the process was needed. This was the first and most important challenge. Construction of reliable models for bioprocesses is a difficult task due to the complex nature of microbial metabolism and the highly nonlinear nature of its kinetics. For these processes, the development of detailed models based on fundamental principles and intense kinetic studies is frequently expensive and time-consuming. The use of neural networks is a promising solution to overcome this kind of difficulty. In this work, a hybrid neural model was chosen, as this model is reported to perform better than "black-box" neural model and significantly less data are required for its training (Psichogios and Ungar, 1992). The idea was to use all the available knowledge of the process to aid in the development of the model. At the same time, the structure of the model was made simple enough to enable its on-line adaptation and allow the development of an adaptive optimal control strategy.

MATERIAL AND METHODS

Batch and fed-batch experiments were conducted in a two-liter bioreactor, with the temperature controlled at 30 °C and the pH at 6.0. Zymomonas mobilis strain CP4 (ATCC31821) was used. The fermentation medium consisted of glucose (concentration varied during the experiments), 5 g/l of yeast extract, 1.5 g/l of (NH₄)₂SO₄, 2 g/l of KH₂PO4 and 1 g/l of MgSO₄ (Veeramallu and Agrawal, 1988). During fed-batch operation, a solution containing glucose and nutrients was fed into the fermenter with a peristaltic pump. The experiments were conducted using a 10%v/v inoculum with the same composition as the fermentation medium. Samples were assayed for dry cell weight, glucose and ethanol. After measurement of the sample volume, it was centrifuged. The supernatant was removed for glucose and ethanol analyses by the Glucose GOD-PAP method (Merck system), using a Varian 3350 gas chromatographer. The residue was used for determination of dry cell weight.

Hybrid Neural Model

Many methods to develop models for bioprocesses have been proposed in recent years. One of them is the use of neural networks, which offers a tool for direct use of process data to generate input-output relationships (Simutis et al., 1993). The training of a neural network, however, requires a large amount of experimental data.

Another alternative is the use of hybrid neural models. As noted by Psichogios and Ungar (1992), it is straightforward to derive an approximate model of a bioreactor from simple first principle considerations such as mass balance of the process variables. The critical factor in determining the dynamic behavior of the process is the unknown kinetics. Thus, in a hybrid neural model, the aspects of a problem whose quantitative behavior is well understood are described by deterministic mathematical equations, while neural networks describe the kinetics. In this case generalization and extrapolation are confined to only the uncertain parts of the process, and the basic model is always consistent with first principles. Besides, significantly less data are required for the training.

However, even when hybrid neural models are used, a large experimental data set is required. Thus, the biggest problem in the development of such models is the large number of experiments to be performed, which usually involves high costs and is very time-consuming. In this work two approaches were used to minimize the number of experimental runs required to develop the hybrid model. One is the use of the technique proposed by Tsen et al. (1996), which generates an "augmented" data set combining experimental data with a model that predicts the relative trends in the process behavior (Henriques, 1998). The other approach was the use of a model available in the literature (Veeramallu and Agrawal, 1990) as a source of knowledge of the process. This model, with some modifications and new parameters, was used to generate new data for the training of the neural network.

The hybrid neural model is composed of the mass balance equations of the system and the modified functional link networks (FLN) (Costa et al., 1998). The advantage of this kind of neural networks is that the weights appear linearly, which simplifies the training procedure. The resulting model, due to the characteristics of the chosen neural networks, can be easily adapted to changes in the system - culture medium, mode of operation, microorganism strain - by the reestimation of the network weights, which can be done in a simple and direct way. This strategy can help to solve another major problem that usually occurs in practical implementations: the model-process mismatch.

Optimal Control

Once a model of the process is available, the optimization problem can be solved by the use of Pontryagin's maximum principle and the singular control theory. One of the difficult aspects in the application of this maximum principle is the requirement for solving a set of differential equations for state and adjoint variables with split boundary conditions. In this work the methodology proposed by Costa (1996) was used so the choice of the dilution rate as the control variable made the analytical solution possible, thus avoiding the numerical problem (Alves et al., 1998). Clearly this represents a significant gain which, together with the simplicity of the model, enables an easy implementation of an adaptive optimization strategy. The maximized performance index was the final ethanol concentration and the final time was free. The optimal temporal profile of the control variable was obtained as a function of the state variables (biomass, substrate and product concentrations), of the kinetic rates (described by the functional link networks) and of the derivatives of the kinetic rates with respect to the state variables.

RESULTS AND DISCUSSION

The experimental data obtained from four batch and two fed-batch experiments were used to develop the hybrid neural model. The technique proposed by Tsen et al. (1996) and the modified Veeramallu and Agrawal model were used to augment the experimental data set. The experimental data were previously filtered to avoid problems during network training. The resulting expressions for the specific kinetic rates of growth, glucose consumption and ethanol formation are:

where y_p1 and y_p2 are given by Equations 4 and 5, S and P are the substrate and product concentrations.

Figures ¹ to ⁴ show the results of the hybrid neural model for two batch and two fed-batch experiments. It can be seen that the hybrid model described well the experimental data under a wide range of experimental conditions. Figures ³ and ⁴ show the fed-batch results, in which conditions of high substrate and ethanol concentrations were attained. These are conditions under which the inhibition effects are strong; nevertheless the hybrid model described the experiments very well.

Figure 1: Batch experiment S₀ = 50 g/L (--) Hybrid model and () Experimental data.

Figure 2: Batch experiment S₀ = 100 g/L (--) Hybrid model and ()Experimental data.

Figure 3: Fed-batch experiment S₀ = 85 g/L and S_F = 200 g/L (--) Hybrid model and () Experimental data.

Figure 4: Fed-batch experiment S₀ = 100 g/L and S_F = 320 g/L (--) Hybrid model and () Experimental data.

In order to supply the network with inherent characteristics of the process kinetics and increase its nonlinear approximation ability, the input vector, wich was originally given by x_e =[S P] was modified, using information from the models available in the literature. Thus, the input vector for the neural network that describes specific growth rate (m) was x_e =[S P] and for the neural network that describes the specific ethanol formation rate it was x_e =[S P 1/S 1/P]. These modified vectors were selected from tests with different vectors; those selected were the ones that maximized a previously defined performance index.

It can be observed in Equations 1 to 3 that only two neural networks were trained. As it is known from the models available in the literature that the substrate consumption rate (s) can be described as the ratio of the product formation rate (p) and a yield factor (Y_P/S), Equation 3 was used to describe this rate. This decision was taken because the experimental data for glucose concentration are extremely noisy, due to the experimental errors associated with the analytical technique used. Equation 3 is a function of only ethanol concentration, for which experimental measurements are more reliable and less noisy than glucose concentration measurements. Naturally, this problem wouldn't exist if a more accurate analytical technique were used for the glucose concentration measurements.

Once the model is obtained, the optimal control problem can be solved. In order to solve this problem, the derivatives of the kinetic rates with respect to the state variables are required. This could be another drawback to implementing the optimal strategy, but the results from many tests showed that these derivatives can be approximated by the derivatives of the corresponding neural networks. The simple structure of the chosen networks facilitated calculation of these derivatives, but the high sensitivity of the derivative to noisy data demands special attention in any new case studied.

The optimal temporal profile of the dilution rate calculated based on the hybrid neural model is shown in Figure ⁵. Figure ⁶ shows the results predicted by the model and the experimental data obtained when the optimal profile from Figure ¹ was implemented in open-loop scheme. The operational conditions are initial biomass concentration X₀ = 0.12 g/L, initial glucose concentration S₀ = 105 g/L, initial ethanol concentration P₀ = 3.2 g/L, initial volume V₀ = 1 L, feed glucose concentration S_F=270 g/L and final volume V_f = 2 L.

Figure 5: Optimal temporal profile of the dilution rate calculated based on the hybrid neural model.

Figure 6: Experimental open-loop implementation of the optimal control. (--) hybrid neural model and () experimental data.

As can be seen in Figure ⁵, initially the bioreactor is operated in batch mode. This mode of operation lasts until the process reaches the singular arc. At this point, the dilution rate assumes its singular value until the reactor is full. Then, the reactor is operated in batch mode until the final condition is attained, in this case, a substrate concentration equal to zero. The time in which the singular arc is reached and the expression that describes the singular dilution rate are determined by the methodology proposed by Costa (1996) and described in Alves et al. (1998). In Figure ⁶ it can be seen that the results predicted by the hybrid neural model are very similar to the experimental data.

The final ethanol concentration attained, P_f=95.6 g/L, is near the theoretical value (approximately 96 g/L). The mass of glucose consumed in this fed-batch experiment is equivalent to that which would be consumed in a batch with an initial glucose concentration of S₀=190 g/L. Veeramallu and Agrawal (1990) presented the results for a batch experiment with an initial glucose concentration of S₀=200 g/L. The fermentation time required for all the glucose to be consumed was 50 h while in the optimized fed-batch performed in the present work, it was less than 20 h. The long duration of the batch experiment is due to the strong inhibition caused by high glucose concentrations, and it evidences the importance of the use not only of the fed-batch mode but also of optimal control techniques in order to achieve high productivities.

In this work the process studied was seen to be robust for implemented control strategies. This robustness must be carefully analyzed in each new project.

CONCLUSIONS

During the design and implementation of an optimal control strategy in the ethanol production process many practical difficulties were met. These difficulties were mainly related to the development of a mathematical model to represent the process. Implementation of an optimal strategy requires a model that describes the process under a wide range of operational conditions but that has a structure which is simple enough to facilitate some of the mathematical calculations involved.

The proposed hybrid neural model, using modified functional link networks to describe the kinetic rates, modelled the dynamic behavior of the process well and adjusted the batch and fed-batch experiments. The simple structure chosen for the neural networks facilitated implementation of optimal control.

The use of techniques to augment the experimental data set permitted the development of a reliable mathematical model of the process from a small number of experimental runs. Use of these techniques significantly reduces a very common problem in process identification: the small amount of available experimental data.

The kinetic rate expressions in the hybrid model developed are very simple and can be easily updated, as their parameters appear linearly due to the structure chosen for the neural networks (FLN). Thus, the hybrid model can be easily adapted to system changes. Use of an adaptive strategy enables the model to monitor the real behavior of the process, which can change significantly during fermentation.

Results obtained from the experimental implementation of the calculated optimal profile showed the importance of using optimization techniques to maximize the productivity, mainly in process in which high substrate concentrations inhibits fermentation. However, this study also showed that the calculated optimal strategy could only be implemented after appropriate solutions were found for a series of practical problems. These solutions are part of a wide set of tools available in process engineering, and some of them need to be further studied to achieve a better definition of their application domain.

ACKNOWLEDGMENTS

The authors acknowledge the financial support received from CNPq.

REFERENCES

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Publication Dates

• Publication in this collection
  15 Sept 1999
• Date of issue
  June 1999

History

• Received
  19 Jan 1999
• Accepted
  12 Apr 1999