## 2 Materials and Methods
### Design of Experiments Using Taguchi Orthogonal Arrays and ANOVA Analysis
Typically, the output response of the experimental design is evaluated using the signal-to-noise ratio (S/N), which is the method most commonly used for analysing factor effects in Taguchi's orthogonal array (OA). In this study, as mentioned earlier, Taguchi OA is employed to assess how HPPC parameters influence ECM performance. The inputs are the pulse time gap, discharge pulse time, discharge pulse C-rate, and rest time. The root mean square error (RMSE) of the error between the ECM's voltage response and the measured value under each different validation test condition is to be evaluated as the control factor. The experimental data were converted into a signal-to-noise ratio (S/N) to measure the deviation of parameter quality from the desired values. The Taguchi method involves analysing three types of performance characteristics: the larger the better, the smaller the better, and the nominal-the-better [28]. Deviations from the optimal value depend on result dispersion in the optimization process. Therefore, the analysis of variance (ANOVA) is based on the mean and variance of each test. ANOVA identifies sources of dispersion within a dataset and measures the contribution of each data point to the total variance. This technique assesses the significance of effects compared to random error, also known as noise. For the variance analysis of RMSE, the \"smaller-the-better\" criterion was chosen, and the signal-to-noise ratio (\(S/N\)) was calculated using Equation (1) as follows [26]:  
\[S/N=-10\text{log}\left(\frac{1}{n}\right)\sum y^{2} \tag{1}\]  
where \(y\) represents the response variable, and n denotes the number of experiments. Following the experiments, analysis of variance (ANOVA) was carried out to calculate the variance of the errors. Furthermore, the contribution ratio of the variable factors to the output parameters was assessed. The sum of squares for both within-group and between-group variations was calculated to establish the contribution rate. Equation (2) represents the sum of squares within the group, while Equation (3) depicts the sum of squares between groups. Here, \(l\) denotes the number of parameter levels, \(j\) is the level number of the specific parameter p, and \(((S/N)_{j}^{2}\) is the sum of the S/N ratios for the \(j\) level and parameter \(p\). \(t\) denotes the number of repetitions for each level of parameter p, m signifies the total number of experiments, and \((S/N)_{j}\) refers to the S/N ratio for the \(i\)th experiment. The \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Pulse Time Gap** & **Discharge Pulse Time** & **Discharge Pulse** & **Rest Time** \\  & **(s)** & **(s)** & **C-Rate** & **(s)** \\ \hline
1 & 50 & 50 & 1C & 60 \\
2 & 50 & 125 & 2C & 120 \\
3 & 50 & 200 & 3C & 180 \\
4 & 125 & 50 & 2C & 180 \\
5 & 125 & 125 & 3C & 60 \\
6 & 125 & 200 & 1C & 120 \\
7 & 200 & 50 & 3C & 120 \\
8 & 200 & 125 & 1C & 180 \\
9 & 200 & 200 & 2C & 60 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: L9 (34) orthogonal array in this study. percentage contribution (\(Pc\)) of the factors is determined by the ratio of the sum of squares within the group to the sum of squares between the groups, as shown in Equation (4) [29]. \[SS_{p}=\sum_{j=1}^{l}\frac{\left(S/N\right)_{j}^{2}}{t}-\frac{1}{m}\left[\sum_{i =1}^{m}(S/N)_{i}\right]^{2} \tag{2}\]  
\[SS_{T}=\sum_{i=1}^{m}\left(S/N\right)_{i}^{2}-\left[\sum_{i=1}^{m}(S/N)_{i} \right]^{2} \tag{3}\]  
\[P_{C}=\frac{SS_{p}}{SS_{T}}*100 \tag{4}\]