## 5 Case Study
### Model Validation  
In order to ensure a correct optimization through the metaheuristic algorithm, the objective functions in Equations (10) and (11) have to be validated. Therefore, as explained above in Section 3, the _MATLAB/Simulink_ simulation through its _Simscape_ library is used for the model validation, since allows validating the dynamic performance of the converter.  
It is important to highlight that the Simulink model is taken as reference for the mathematical equations validation, since it allows evaluating how the system dynamically reacts by also considering the ripple of the circuit. Nevertheless, the Simulink model is not taken as the simulating model for the optimization methodology regarding the computational cost of the simulation (which is higher than the transfer functions simulation), which would make it less efficient for the metaheuristic algorithm.  
Figure 11: Cascade controller scheme for voltage and current in a buck converter, using the small-signal model transfer functions.  
Thus, the simulations regarding the transfer functions and the circuit model, were performed using the parameters from Table 2. Then, Figure 12a shows the resultant curves for the output voltage, where it is clear the correct behavior of the transfer function compared against the circuit model.  
To demonstrate a correct comparison against the real behavior of the system, the highlighted circled area is zoomed in, showing the performance of \(G_{vd}\) against the ripple in the circuit.  
Additionally, Figure 12b shows the validation of the \(G_{id}\) transfer function, where it can be analyzed that the equation correctly represents the behavior of the current through the inductor. As in the previous case, a certain area, is highlighted and zoomed in, showing the behavior of the transfer function compared to the ripple of the current in the circuit.  
As explained when the equations were obtained, \(G_{vd}\) and \(G_{id}\) represent the an average approach of the variables to be analyzed, therefore it is important to highlight from Figure 12, that the obtained curves represent the average behavior of the ripple in the real circuit model (exemplified by the Simscape model), validating then the behavior of both equations.  
Even so, to quantify the model validation, the statistical mean can be computed to obtain a statistical comparison and therefore, to measure the accuracy of the models. Table 3 shows the results for the \(G_{id}\) objective function. Meanwhile, Table 4 shows the accuracy results for the \(G_{vd}\) transfer function.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Simscape Model & 4.8104 [V] \\ \(G_{vd}\) & 4.8105 [V] \\ \hline Accuracy & 99.9990 [\%] \\ \hline \hline \end{tabular}
\end{table}
Table 4: Statistic mean of the output voltage.  
Figure 12: Validation of the output voltage (**a**) and current (**b**) transfer functions.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Simscape Model & 0.1669 [A] \\ \(G_{id}\) & 0.1678 [A] \\ \hline Accuracy & 99.4636 [\%] \\ \hline \hline \end{tabular}
\end{table}
Table 3: Statistic mean of the inductor current.  
Finally, as a last double-check measure for the model validation, the ITAE performance index was computed for both voltage and current curves. The obtained \(ITAE=0.0004\) between the _Simscape vs \(G_{vd}\)_ voltage curves, and the \(ITAE=0.0017\) between the _Simscape vs \(G_{id}\)_ current curves, completely validated the objective functions.