## 3 Results and Discussion
### Genetic Algorithm Optimization
About the development of the regression model for the fermentation process under pH control, the same procedure was followed, and the optimal model was found to be:  
\[X(t,pH)=-0.073394+0.55134\frac{t}{pH} \tag{12}\]  
As can be seen from Equation (12), the optimal solution of the algorithm is the same of the previous case, that is \(\{1,\,-1\}\), demonstrating the dependence of the biomass concentration on the fermentation time and on the inverse of the pH value. It is worth noting, however, that in this second case, i.e., under pH control, the pH value is kept constant throughout the entire process, so the model could be further simplified by incorporating the term pH within the constant \(c_{1}\) (i.e., 0.55134) in Equation (13):  
\[X(t)=-0.073394+0.095059t \tag{13}\]  
By graphically representing the trend obtained by the genetic algorithm optimization, shown in Figure 9(a), it can be seen that the 1-term model is not able to reproduce the growth curve during the lag phase, and that, in this case, in general, it is characterized by a strong discrepancy with the experimental results, as also underlined by the values of the normalized RMSE error reported in Figure 9(b), i.e., greater than six for the first 8 h of fermentation. Figure 10: Genetic algorithm optimization by considering a single regression term and the response variable as a function of time and pH under pH control: (**a**) comparative trends and (**b**) normalized RMSE. \[X(t,pH)=-0.063367+0.47026\frac{t}{pH} \tag{8}\]0 Genetic algorithm optimization by considering two regression terms and the response variable as a function of time, pH, and lactic acid concentration in case of uncontrolled pH: (**a**) comparative trends and (**b**) normalized RMSE. To improve the interpolation capacity of the algorithm, as in the previous case, the lactic acid concentration term (\(X_{AL}\)) was introduced defining a new regression model, described by Equation (14):  
\[X(t,pH)=-0.063367+0.47026\frac{t}{pH} \tag{8}\]1  
Unlike the optimal model obtained by implementing the data relating to the case without pH control, described by Equation (10), the second term is given by the combination of time and pH. However, given that the pH remains constant due to the control, this can be incorporated into the empirical constants and the resulting model is described by Equation (15):  
\[X(t,pH)=-0.063367+0.47026\frac{t}{pH} \tag{8}\]2  
In fact, as shown in Figure 11a, the difference between the estimated trend and the experimental one is greater, as highlighted by the values of the normalized RMSE error shown in Figure 11b. Using a sampling time shorter than 2 h is expected to reduce the RMSE. However, to verify this statement, a new experimental campaign is needed. This can be considered as a further step to carry out the investigation in the near future. Despite the small discrepancy between the estimated and the experimental results, especially when the fermentation is carried out under pH control, the genetic algorithm is proved to be a valuable solution for microbial growth modelling. In fact, it allows to model the growth trend during all the typical phases, i.e., lag, exponential, and stationary in this case. Moreover, it gives high freedom of customization allowing the definition of new and more powerful models depending on the data availability, without the need to perform characterization analyses that may require exhaustive resources.