# Evolutionary Design Optimization of an Alkaline Water Electrolysis Cell for Hydrogen Production
## 2 Physics and Economy of the Alkaline Water Electrolysis Cell
### Artificial Neural Network (ANN)  
These four dimensionless numbers (Equations (21)-(24)) have been used to correlate the void fraction using an artificial neural network (ANN). Appendix B shows the design of experiments used to train the ANN. The numerical search field is summarized in Table 1. This numerical search field allows the finding an optimum design for an average current density \(j\) in [10\({}^{3}\)-10\({}^{4}\)] A m\({}^{-2}\), an electrode height \(H_{dec}\) in [0.01-0.1] m, a temperature \(T\) in [293-350] K, a pressure \(p\) in [1-30] bar, a mass fraction \(Y\) in [0.2-0.4], a electrolyte thickness \(h\) in [\(1.5\times 10^{-4}\)-\(1.5\times 10^{-3}\)] m.  
The dimensionless numbers have been reduced using Equation (21). \[\overline{X}=\frac{x\,-\,\left(\frac{x_{max}+x_{min}}{2}\right)}{\frac{x_{max }-x_{min}}{2}} \tag{21}\]  
With \(\overline{X}\) the reduced dimensionless number, \(x\) the dimensionless number  
The weights of the ANN are obtained using the software NeurOne(r). Because the evolutions of the void fraction according to the four dimensionless parameters are non-linear, the learning is done by a Levenberg-Marquardt (L-M) algorithm [24] with parameters initialized according to a normal law N(0, 0.1) over about 70 iterations. The L-M algorithm consists in reducing a Sum Square Error (SSE) defined by Equation (22). \[SSE\left(\left( \begin{array}{c}\overline{\overline{Re}}_{V_{G}}\\ \overline{\overline{Fr}}_{V_{G}}\\ \overline{h}\\ \overline{r}\end{array} \right)\right)=\frac{1}{2}\sum_{i}^{i=O}{e_{i}^{2}}= \frac{1}{2}\sum_{i}^{i=O}{\left({\varepsilon_{fluent,i}}\,-\,\,{\varepsilon_{ANN,i}}\right)}^{2} \tag{22}\]  
\(\stackrel{{\rightarrow}}{{w}}_{N,M}\) the weight vector with the subscript \(N\) the number of layer (from 0 to \(N\)) and \(M\) the total number of weights per layer and \(O\) the number of outputs. \begin{table}
\begin{tabular}{c c c c c} \hline
**Dimensionless Parameter** & \(x_{min}\) & \(x_{max}\) & \(\frac{x_{max}+x_{min}}{2}\) & \(\frac{x_{min}-x_{min}}{2}\) \\ \hline \(Re_{V_{G}}\) & \(1\times 10^{-2}\) & 300 & 150.005 & 149.995 \\ \(Fr_{V_{G}}\) & \(5.85\times 10^{4}\) & \(5.30\times 10^{10}\) & \(2.65\times 10^{10}\) & \(2.65\times 10^{10}\) \\ \(r^{*}\) & \(1.14\times 10^{-5}\) & \(1.50\times 10^{-3}\) & \(7.75\times 10^{-2}\) & \(7.25\times 10^{-2}\) \\ \(h^{*}\) & \(5\times 10^{-3}\) & \(1.50\times 10^{-1}\) & \(7.557\times 10^{-4}\) & \(7.442\times 10^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Numerical search field for every dimensionless parameters. The SSE is reduced by a combination of two algorithm: the steepest descent algorithm and the Gauss-Newton algorithm [24]. Thus, the weight vector is updated following the Equation (23). \[J_{k}=\left[ \begin{array}{cccc}\frac{\partial\epsilon}{\partial u_{0}u_{0}}& \frac{\partial\epsilon}{\partial u_{1}u_{0}}&\cdots&\frac{\partial\epsilon}{ \partial u_{0}u_{0}}\\ \frac{\partial\epsilon}{\partial u_{0}u_{1}}&\frac{\partial\epsilon}{ \partial u_{1,1}}&\cdots&\frac{\partial\epsilon}{\partial u_{N,1}}\\ \vdots&\vdots&\cdots&\vdots\\ \frac{\partial\epsilon}{\partial u_{0}M}&\frac{\partial\epsilon}{\partial u_ {1,M}}&\cdots&\frac{\partial\epsilon}{\partial u_{N,M}}\\ \end{array} \right] \tag{23}\]  
\[\stackrel{{\rightarrow}}{{w}}_{N,Mk\ +\ 1}=\stackrel{{ \rightarrow}}{{w}}_{N,Mk}-\left(J_{k}^{T}I_{k}\ +\ \lambda I\right)^{-1}\!J_{k}\!\!\left(\stackrel{{ \rightarrow}}{{\varepsilon}}_{fluent,k}-\stackrel{{ \rightarrow}}{{\varepsilon}}_{ANN,k}\right) \tag{24}\]  
With \(J_{k}\) the Jacobian matrix, the \(k\) subscript for the number of update and \(\lambda\) the combination coefficient. If \(\lambda\) is close to 0 then Gauss-Newton algorithm is used whereas when \(\lambda\) is very large the L-M algorithm switches to the steepest descent algorithm. The Figure 2 gives the architecture of the artificial neural network used to estimate the void fraction, which correspond to the Equations (25)-(29).