This chapter deals with the problem of design and implementation of optimal control for a bench-scale fermentation of Saccharomyces cerevisiae. A modified GA is proposed for solving the dynamic constrained optimization problem. The optimal profiles are verified by applying them to the laboratory experiments. Among all 12 runs, the one that is controlled by the optimal feed rate profile based on the DO neural model yields the highest product. The main advantage of the approach is that the optimization can be accomplished without a priori knowledge or detailed kinetic models of the processes.

7.1 Definition of an Optimal Feed Rate Profile

The growth of Saccharomyces cerevisiae in the fermentation process can be described by the stoichiometries of three pure metabolic routes, namely, ox-idative glucose catabolism, reductive (fermentative) glucose catabolism and ethanol utilization [18]. Metabolic pathways that take place during the fermentation are expressed in the following three stoichiometric equations:

Oxidation of glucose (R1): S + a1O2 3 b1X + c1CO2 (7.1)

Reduction of glucose (R2): S b2X + c2CO2 + d2P (7.2)

Oxidation of ethanol (R3): P + a3O2 3 b3X + c3CO2 (7.3)

where, X, S, P, O2 and CO2 are the reaction components, namely, microorganisms, consumed substrate, ethanol, oxygen and carbon dioxide respectively; The parameters ai, a3, bi, b2, b3, C1, c2, c3 and d2 are the stoichiometric coefficients, which are the yields of the three reactions; The reaction rates at which three metabolic pathways take place during the fermentation are r1, r2 and r2 [114].

L. Z. Chen et al.: Modelling and Optimization of Biotechnological Processes, Studies in Computational Intelligence (SCI) 15, 91-108 (2006)

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As described in the above equations, according to different fermentation conditions and controls, three different metabolic routes may occur in the growth of microorganisms, which are governed by the respiratory capacity of the cells or so called overflow mechanism. If the substrate flux is low, and there is excess respiratory capacity of the cells, both pathways R1 and R3 are activated, but R1 is observed to have higher priority than R3. Pathway R2 is activated if the substrate flux is high and the respiratory capacity limitations of cells are exceeded.

The limited respiratory capacity can be represented by a bottleneck principle for oxidative substrate utilization [18]. As illustrated in Figure 7.1, three cases (a), (b) and (c) represent the pathways of glucose oxidation, glucose reduction and ethanol oxidation respectively. In case (a), the total amount of substrate can pass the bottleneck, thus the substrate is metabolized purely oxidatively through the pathway R1 (see Equation 7.1). Case (b) represents the substrate flux exceeding the bottleneck of substrate utilization. Part of the substrate that passes the bottleneck is metabolized through the same pathway as in case (a). However, the residual part of the substrate that can not pass the bottleneck is metabolized reductively, and ethanol is formed (see Equation 7.2). Case (c) illustrates the pathway of the oxidation of ethanol, which is described by Equation 7.3.

Restrictions of feed rates in fed-batch fermentations

Even though all three pathways can lead to biomass production, the pathway R2 should be avoided in order to prevent ethanol formation, which results in wasting of the substrate. The pathway R3 takes place only following the occurrence of the pathway R2, and the biomass produced in R3 is very low compared to that produced in the pathway R1. Also, part of the ethanol may be lost due to volatilization. Thus, in order to achieve a high production yield, some control strategies, such as those proposed in [8] and [20], were to assure that the pathway R1 was tightly maintained throughout the fermentation.

For fed-batch fermentation, the substrate concentration inside the reactor can be manipulated by adjusting the feed rate of input substrate. If the residual concentration of substrate in the reaction medium is high, then the substrate flux is high. Conversely, if the residual substrate concentration is low, the substrate flux is low.

A simple illustration of typical stoichiometric constraints on the feed rate during a fed-batch cultivation is shown in Figure 7.2 [115]. In the initial phase of fermentation, the cell density is very low and the feed rate should remain low to avoid overflow metabolism. As the cells grow and cell mass increases, the feed rate can be allowed to increase to meet the nutrient requirement of growing cells. An increased feed rate, however, leads to increased oxygen consumption and eventually the constraint from the limited respiratory capacity may be reached. The feed rate should thus be decreased in order to allow the consumption of residual glucose and ethanol.

Practically, in order to achieve high productivity, a high feed rate is usually needed. However, a bottleneck effect may become significant if the feed rate is too high. To avoid the formation of ethanol, which indicates the occurrence of overflow metabolism, the feed rate should be kept at a sufficiently low level to guarantee only the pathway R1 takes place. In such a case, however, in order to obtain a high productivity, the process may have to be carried out for a long time. This is because there exists an excess respiratory capacity, which is not fully utilized by the cells for growing and reproducing. Moreover, the cells may starve if under-feed happens. On the other hand, both the highest yield and the shortest process time are always desirable for process optimization, since they are of considerable economic importance [56]. However, there is a conflict between these two optimality criteria. A fast production rate usually results in a formation of ethanol, thus decreasing cell mass yield. To avoid ethanol formation, pathway R1 should be maintained throughout the fermentation. This may slow down the whole process. Practically, ethanol formation is always observed during fermentation processes. In past studies, ethanol formation was allowed in some optimal feeding strategies in order to achieve short process times [75]. Produced ethanol was then consumed to produce cell mass at the end of the cultivation. Maintaining the correct balance is therefore of great important to optimizing the production yield.

Maximum feed rate

Maximum respiratory capacity

Maximum respiratory capacity

starvation starvation

Time

Fig. 7.2. An example of feed rate profile under the constraints of overflow metabolism in a fed-batch fermentation.

Though a number of developed feeding strategies and proposed optimal feed rates can be found in the literature [43,116], most of them are either setpoint or mathematical model-based control. Set-point control or tracking was employed in [8,20]. In these cases, control strategies were to keep some important variables at their critical values in order to achieve the highest possible productivity. However, a considerable process knowledge is required to make such methods successful. A similar requirement has to be met for mathematical model-based optimization approaches [75,117]. Recently, a neural network model-based process optimization has drawn considerable attention due to the successful application to highly nonlinear dynamic systems [56] because less a priori knowledge is needed to build neural network models than conventional methods. This advantage makes neural network approaches more flexible and transformable from one process to another, thus increasing the development efficiency and benefit-cost ratio.

The performance of fed-batch fermentation of baker's yeast is characterized by the yield obtained at the end of the process. The optimal operation of the fed-batch fermentor can be expressed as: given operational constraints, determine the feeding strategy that optimizes a productivity objective [118].

Thus the optimization problem can be formulated as follows [114]:

where, tf is the final process time; Xtf and Vtf are the final biomass concentrations and the final volume, respectively; F(t) is the feed flow rate, which is equally divided into Nsub constant control actions:

where, Nsub is the number of intervals of the feed rate profile.

The optimization is subject to the following constraints:

The initial conditions and the glucose concentration of feeding solution are given in Table 6.1 as shown in Chapter 6.

In order to avoid the optimization procedure being stuck in local maximum, a simple strategy is used in the search procedure. The feed rate profile is firstly divided into four piecewise of control variables with equal time length, which are used as the variables of the GA to find the optimal solution. After the convergency of the GA, the final four piecewise feed rate are then divided into eight equal lengths of sub-feed rates. The GA is then run again, the feed rate is then divided again, and so on. It terminates when the improvement on performance index is less than a predefined value over a certain number of iterations or when a predefined maximum number of intervals, Nmax, has been reached. The flowchart of this strategy is illustrated in Figure 7.3.

When the subinterval of the process horizon is decreased (i.e., the subdivision of control actions is increased), the final (best) population obtained from the preceding run of the GA should be divided according to the new subdivisions. A subdivisional operation is thus required to divide the population into smaller subintervals. The time steps in the evaluation function should be also updated corresponding to the change of time intervals in the population.

The constraint on the final volume given in Equation 7.6 is implemented as a penalty function. Usually, a validity checking procedure, through which each candidate solution has to pass, is adopted to isolate the solution that does not hold the constraint. If the final volume produced by a feed rate solution is not two liters, the fitness value of this solution is set to be zero, which means this individual has less chance to survive. Otherwise, the fitness value of the candidate solution is evaluated using the objective function given

in Equation 7.4. However, an invalid feed rate solution may possess some properties which are close to those in the optimal solution, especially for those with high fitness values (before set to be zero) and final volume that is less than two liters. The fitness values being simply set to zero will deprive them of the chance to evolve into the optimum. On the other hand, a slow convergency may be observed due to the fitness values of a large number of candidate solutions being set to be zero.

Instead of giving zero to the fitness value, a penalty form of objective function was employed to calculate the fitness value for the solution that violated the constraint [45]. A penalized fitness value was assigned to the solution depending on how far it deviated from the limit:

J = X. x V. - r ■ [Vmax - V(tf )]2 If V(tf) > Vmax J = Xt. X V, If V (tf) < Vmax ( )

where r is the penalty coefficient, which was assigned to the value of 100.

However, the constraint violation still remains after applying the penalty form of objective function. This may also reduce the convergence rate as it takes a long time to find a solution with the highest fitness value as well as holding the constraint.

To solve the above problem, a new method proposed in this study is to amend or repair the conflicting candidate solutions. There are three cases for each candidate solution being evaluated. In case one, the final volume is the same as the constraint value, so no modifications are needed for both the solution and its fitness value. In case two, the final volume produced by the solution is higher than the constraint value. The new fitness of this candidate is simply set to zero because it is usually not a promising solution. Even though a high fitness value appears, it is normally caused by a high feed rate rather than an optimal feed rate. In case three, the final volume produced is lower than the constraint value. This solution is rectified to meet the constraint requirement by adding the deficiency uniformly in order to produce the final volume of Vmax. In other words, the feed rate trajectory of this solution is moved up in parallel to produce the required final volume, where the shape of the trajectory is preserved. The new procedure can be summarized as follows:

1. Evaluate the fitness value J according to Equation 7.4 for the candidate solution Fi, i = 0,1, • • • ,n, calculate the final volume V that produced by

c) If V < V {tf), Finew — Fi + {V{tf) - V)/tf, re-evaluate the fitness value Jnew according to Equation 7.4.

2. Run the GA using the new fitness value Jnew.

3. Repeat steps 1-2, until a termination criterion is reached.

7.4 Optimization Results and Discussion

The constraint handling method proposed in this work was tested in comparison with the conventional one. Figure 7.4 shows the convergence profiles of the GA using two different constraint handling methods. The best performance indices of each generation were plotted against the generation count. Neural network model I was employed in the tests and a fixed number of intervals, eight, was used to divide the feed rate control profiles.

It can be seen from Figure 7.4 that both indices climbed up to 24 at the starting points of the optimization runs. However, from the range of 24.5-26, a rapid increase of the fitness value was observed for the proposed constraint handling method. The maximum performance index was achieved within 50 generations, while the conventional method took 250 generations to reached the maximum performance index. Several temporary stallings appeared in the performance index of the conventional method. This was due to the algorithm being stuck in some local maxima because the fitness values of the solutions

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