Proposed constraint handling Traditional constraint handling
50 100 150 200 250 300
Fig. 7.4. Convergence profiles for different constraint handling methods.
that violated to the constraint were simply set to zero. With the strong tendency towards the global optimum, the GA was able to continuously escape from such local maxima and gradually converge to the optimal value of performance index. However, the proposed method is superior to the conventional method with the advantage of fast convergence speed.
An investigation into the optimization by using different time subintervals for dividing the feed rate control actions was conducted in order to develop the suitable optimization strategy. Figure 7.5 shows the best performance indices for optimizations with different number of intervals: 4, 8, 16 and 80.
As discussed above, with the stochastic nature, the GA can eventually reach the similar optimum points for different number of intervals with different convergence rates. It can be seen from Figure 7.5 that the smaller the number of intervals, the faster the convergence rate. The algorithm took about 50 generations to converge when the feed rate was divided into four piecewise control actions, while it took about 1000 generations to reach the optimal value of index when the feed rate was divided into 80 sub-feed actions. On the other hand, because the value of feed rate within the subinterval is fixed, a
No. of intervals = 4 No. of intervals = 8 No. of intervals = 16 No. of intervals = 80
Fig. 7.5. Convergence profiles for different number of intervals.
smaller number of intervals means a longer time for each interval, and it gives less flexibility to the feed rate profile. As discussed by Na et al. , the necessary conditions for optimality at different subintervals rather than a given subinterval on the profile can not be fulfilled. In this sense, a feed rate profile that is obtained using a small number of intervals (e.g. four) may be viewed in principle as a suboptimal result, although it may produce a performance index value which is very close to others. In fact, as seen in Figure 7.5, the performance index of the feed rate with four intervals is higher than those with higher number of intervals. However, the index value for four intervals could hardly improve once it converges from the generation number of 50, whereas the indices for 16 and 80 intervals evolve slowly from starting point up to the end of the optimization.
A novel feature of the proposed optimization strategy is the incremental interval number technique. In order to achieve faster convergence as well as a higher performance index, the number of intervals was initially chosen as four, then was increased to eight, 16, and 80 during the progress of the optimization procedure. Figure 7.6 shows the evolvement of the performance index when using the proposed optimization strategy. Due to the fast convergence, the four intervals was used in the first 100 generations. The end population was then transferred into the population with the feed rate represented by eight piecewise control actions, and the GA was run on the new population for 500 generations. The same procedure was carried out until the last run of the GA with 80 intervals. The fast convergence was achieve by the feed rate with four intervals, and a higher index was obtained by further dividing the feed rate profile based on the result of the previous stage. A relatively big improvement was observed for the feed rate with eight intervals. As the number of intervals increased, the improvements became smaller.
- - Interval No.
— Interval No.
= 16 "
1-11 Interval No.
Fig. 7.6. Performance index profiles for the proposed optimization strategy.
Comparisons of computation times for different number of intervals are given in Table 7.1. The optimizations were performed on a IBM compatible computer with an Intel Pentium II Celeron 633 MHz processor. The population size was chosen as 250.
The top half of Table 7.1 shows the running times of fixed number of intervals, i.e., the number of intervals was unchanged from the beginning to the end of the optimization. The bottom half of the table shows the computation times spent with the incremental number of intervals, i.e., the number of intervals was increased from four to 80 during the course of the optimization. It can be seen from the table that the times spent were the same for the optimizations when the number of intervals equalled four. However, from intervals eight to 80, the computation times of optimization with an incremental number of intervals were shorter than those with fixed number of intervals. This is due to the end population of a previous stage is subsequently converted into the initial population of the next stage when the subdivision was increased. Thus the times for initialization were saved.
The development of optimal feed rate profile using the optimization strategy is shown in Figure 7.7(a)-(d). A significant change of feed rate profile appeared from number of intervals four to eight. From number eight to 80, however, there was no big alteration in the shape of the profile. It was only "fine tuned" by the optimization procedure when it was further subdivided.
A large number and high frequency of fluctuations appeared on the the feed rate profile with 80 intervals as shown in Figure 7.7(a). This makes the profile impractical to implement using the laboratory controller. Moreover, due to the continuous behavior of the fermentation process, a highly fluctuating feed rate may cause unexpected disturbances to the system . Practically, further increase in subdivision for optimization might lead to even worse results . In the experiments of this study, 16 was chosen as the final number of intervals to divide the entire feed rate into equal length of sub-feed rates. This selection is also a compromise between running time and the performance index. Higher subdivision may result in higher yield but is more time consuming.
Subdivision = 16
Fig. 7.7. Evolution of the optimal feed rate profile using the proposed optimization strategy.
Implementation of the optimal controls The optimal control based on neural model I
Three experimental runs controlled by different optimal feed rate profiles were carried out using BioFlo 3000 bench-scale fermentors. The detailed experimental setup and procedures are described in Chapter 6. The result of optimization based on the proposed neural network model I and experimental validation are given in Figure 7.8. The optimal feed rate profile is plotted in Figure 7.8(a). This shows the substrate was fed into the fermentor at a middle level of feed rate (0.18 L/h) during the first hour. The feed rate then rose rapidly to its maximum value (0.2988 L/h) to allow a fast feeding. In the second hour, the feed rate was decreased from the maximum value to a lower value (0.16 L/h). After half an hour, it was further decreased to 0.15 L/h, which was the minimum feed rate during the whole fermentation process. In the third hour, the feed rate was increased gradually with a small extent (< 0.01 L/h) every half an hour until the end of the process.
This feed rate profile conforms to the overflow metabolism of the growth of Saccharomyces cerevisiae. At the beginning of the fermentation, the seed of microorganisms is just inoculated into the reactor. The feed rate is not permitted to be high in order to allow the cells to adapt to the new condition of growth. A gentle middle level of feed rate allows the cells to get into the exponential growth phase as quickly as possible, without the overflow metabolism taking place. As the cells are brought to the active state, a high feed rate is required to supply sufficient nutrients to the cells and allows the cells to grow adequately. However, as explained in Section 7.1, a continuous high feed rate leads to increased oxygen consumption and overflow metabolic pathways. Thus, a switch to a low feed rate is necessary to avoid the bottleneck effect and in the same time, to allow the consumption of residual glucose and ethanol.
The biomass concentration produced by the optimal feed rate is shown in Figure 7.8(b). The final biomass concentration of the experimental result was 11.02 g/L, which is the highest among all of those obtained from experimental runs in this research, which are summarized in Table 7.2. From Figure 7.8(b)-(c), one can see a close prediction of biomass growth under optimal condition was also achieved. The final biomass predicted was 10.6698 g/L, which was in a good agreement with the experimental result (11.02 g/L). The prediction percentage error was less than 10% during the whole fermentation period.
The optimal control based on neural model II
The optimization result based on neural network model II is given in Figure 7.9(a)-(c). As shown in Figure 7.9(a), the feeding started with a very high feed rate (0.24 L/h) and lasted for four hours. A low feed rate was then used from the fourth to fifth hour followed by another high feed rate until the end of the fermentation. Obviously, this feed rate profile was in conflict with the
Fig. 7.8. Optimization result based on the cascade recurrent network model I
overflow metabolism and the neural network model II failed to predict the optimal trajectory. The produced biomass concentration was the lowest one among the three optimization runs as can be seen from Table 7.2. The main reason for this may be due to the neural network being trained with insufficient glucose data, which were measured at a long sampling time (30 minutes) in this work. However, as can be found from the error plot in Figure 7.9(c), the neural network surprisingly estimated the biomass concentration with similar accuracy to that of neural model I, even though it is not the optimal trajectory. The most likely explanation for this result is that this neural model can predict well in the "experimental space", which is the space spanned by the measurement data used for training the network. However, its extrapolation property is poor . It is expected that similar results can be achieved as those obtained from neural model I, if an on-line glucose sensor, which can detect glucose concentration more frequently (e.g., every five minutes), is available.
The optimal control based on the mathematical model
For a comparison, the results of optimization based on the unstructured mathematical model (Equations 6.4 - 6.5) is presented in Figure 7.10(a)-(c). The feed rate profile, as plotted in Figure 7.10(a), has a similar pattern as that shown in Figure 7.8(a). The difference between these two profiles is that the highest feed rate (about 0.25 L/h) lasted for two hours, while Figure 7.8(a) shows that the highest feed rate (0.2988 L/h) lasted for just one hour. The final biomass produced was 9.53 g/L as shown in Figure 7.8(b), which is less than that obtained using neural model I, but higher than that obtained from neural model II. From the error plot in Figure 7.8(c), it is found that the prediction accuracy is not as good as the neural models. However, a very accurate prediction was made on the final biomass concentration. From the above results, it is obvious that the mathematical model is less accurate in prediction during the initial and the middle phase of the fed-batch fermentation process, but it has a good extrapolation ability to predict the final biomass concentration . The performance of the mathematical model for optimization is better than that of the neural model trained with insufficient information, but is worse than that of the neural model trained with adequate data sets.
The final biomass concentrations and total reaction times of the experiments that were carried out in this research are summarized in Table 7.2. Run 1 to run 9 are correspondingly the experiments that used the feed rate f1 to f9 shown in Figure 6.10. Run op1 to op3 were the optimization experiments one to three as shown in Figure 7.8, 7.9 and 7.10, respectively. Table 7.2 shows that the highest biomass concentration and shortest reaction time was achieved in the run op1, which is the optimization experiment based on the neural network model I.
Optimization experiment 2
Optimization experiment 2
7.4 Optimization Results and Discussion 107 Optimization experiment 3
0 Measured data — Mathematical model prediction ~L-
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