# Evolutionary Design Optimization of an Alkaline Water Electrolysis Cell for Hydrogen Production
## 4 Sensitivity Study of the Results to the Evolution Parameters
### Impact of \(p_{m}\)  
The mutation probability \(p_{m}\) is important for the initial population draw, the exploration phase and the genetic stirring all along the evolution. If \(p_{m}\) is too low, no new genetic material is found, and if too high it is disruptive and prevent keeping good solutions. Here we study its impact upon the fitnesses.  
Initially, \(p_{m}\) = was set at 1.55% as recommended by [PERSON] [26]. In order to refute or affirm this choice, a sensitivity study based on \(p_{m}\) was conducted. In order to observe only the operator effect of the mutation, elitism was deactivated (i.e., the fact that the best individual is always re-injected into the population). The results of the evolutions are shown in Figure 14. The maximum fitness increases from 0 to about 0.5%, then reaches a plateau of up to 2% and then gradually decreases. For \(p_{m}\) = [0.5-2]%, there is a peak at \(p_{m}\) = 0.7%. However, the difference between the fitness value at \(p_{m}\) = 0.7% and 1.5% is very small. The choice of \(p_{m}\) recommended by [PERSON] is therefore relevant as far as we focus on the best fitness.  
Figure 14: Result of the sensitivity study of the fitness to the mutation rate. In (**a**) the whole research space has been plotted and (**b**) only the mutation rate between 0 and 2% has been plotted.  
Figure 13: Result of the sensitivity study of the fitness to the \(R_{GP}\) (**a**) and \(P_{GP}\) (**b**).  
In Figure 15, the evolution parameters are the ones just defined. We see in this 100-averaged run that the exploration phase stops at gen = 10, followed by the exploitation phase ending for gen = 90. This time, we observe a convergence phase (gen = 90 to 117) where \(f_{avg}\) stagnates just below \(f_{best}\) that increases slightly. The convergence gap between \(f_{best}\) and \(f_{avg}\) is only 3% of the \(f_{best}\) value, as expected in the canonical GA (<5%). In conclusion, we can consider these evolution parameters to be validated.  
Eventually, the evolution parameter are summarized in Table 8.  
These evolution parameters allow to get the optimum DP set summarized in Table 9. The main results are the difference in catholyte and anolyte thickness and electrode heights. Due to the presence of gases, the minimum of electrolyte thickness cannot be chosen. The electrode height is set to the minimum value available. There is twice as much gas injected in the catholyte than in anolyte. Thus, the catholyte thickness \(h_{H2}\) is bigger than the anolyte thickness \(h_{H2}\). The lower the electrode height, the lesser gases injected resulting in a lower void fraction. The pressure stays at 1 bar. It can be explained by the fact that the surface coverage sensitivity regarding the pressure has not been taken into account. Thus, at relatively low current density, the pressure effect increases the cell voltage.  
\begin{table}
\begin{tabular}{c c} \hline \hline  & **Value** \\ \hline \(k_{pop}\) & 17 \\ \hline \(k_{gen}\) & 118 \\ \hline \(p_{c}\) & 50\% \\ \hline \(p_{m}\) & 0.5\% \\ \hline \(R_{GP}\) & 7 \\ \hline \(P_{GP}\) & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Optimum evolution parameters.