## 4 Design Example and Simulation Results  
This section illustrates the design process of the proposed solution, where the performance of the proposed HESS is evaluated using detailed circuital simulations. The test-bed consists of a DC microgrid connected to a 12 V battery and a supercapacitor with a rated voltage of 16 V. This proof-of-concept HESS has a rated power of 40 W, where the Sepic/Zeta converter has a switching frequency \(F=40\) kHz, with inductors  
Figure 9: Structure of the adaptive LQG controller.  
\(L_{1}=L_{2}=680\)\(\upmu\)H and an intermediate capacitor \(C_{i}=330\)\(\upmu\)F. Finally, this design example considers load (bus) current transients up to 3 A, and hence the \(i_{dc}\) range is \([-3,3]\) A.  
The supercapacitor design is preformed using Equations (29) and (30) with a maximum bus current change \(\Delta i_{dcmax}=6\) A. Moreover, the supercapacitor is designed to ensure a maximum voltage change of 1% and a cut frequency \(f_{c}=0.5\) Hz of the HPF, which results in a minimum value of the supercapacitor equal to 34.15 F. Due to the commercial availability and cost, the supercapacitor BMOD0058E016C02 [34] is used, which has a capacitance equal to 58 F, thus fulfilling expression (30). Table 1 summarizes the parameters of this design example.  
The matrices \(A_{m}\) and \(B_{m}\) of the space-state model are obtained offline by using Equation (14). The parameters of these matrices are calculated for different operating points resulting from variations on the supercapacitor voltage (\(v_{sc}\)). Then, the \(K\) and \(\ell\) parameters of the controller are calculated using the LQR function from Matlab 2024b(r) for all the space-state systems. For any other operating point, the adaptive gain-scheduling strategy determines the control parameters using a linear interpolation from the parameters previously calculated.  
Figure 10 shows a pole-zero map to verify the close-loop stability of the system for different voltage (and SoC) conditions of both battery and supercapacitor. In this analysis, multiple operating points are simulated considering variations of 2 V for \(v_{dc}\) and \(v_{sc}\), thus covering all the operation range. It is observed that the pole-zero points for all operating points are in the left side of the S-plane, which confirms the system stability.  
On the other hand, Figure 11 shows the pole-zero map considering parametric variations on the converter elements, e.g., due to parametric tolerances. This analysis considers a tolerance of 15% for the inductors and 20% for the capacitors, and the results demonstrate that tolerances in the circuit components do not represent a risk for the system stability.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(v_{dc}\) & \(12\) V \\ \(v_{sc}\) & \(6-16\) V \\ \(F\) & \(40\) kHz \\ \(i_{dc}\) & \([-3,3]\) A \\ \(L_{1}=L_{2}\) & \(680\)\(\upmu\)H \\ \(C_{i}\) & \(330\)\(\upmu\)F \\ \(C_{sc}\) & \(58\) F \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters for the design example.  
Figure 10: Pole-zero map of the adaptive LQG controller.  
Then, the performance of both the HESS and the adaptive control system are evaluated in terms of current reference tracking and capacitor voltage regulation. The HESS is implemented using detailed circuital models in Simulink(r) from Matlab(r). The main objective of the HESS is to provide the high-frequency current perturbations of the DC bus, which could be produced by power variations among loads and sources connected to the bus. In this way, the battery only provides the low-frequency current variations. Moreover, in real operation conditions, the supercapacitor voltage could be lower, equal to, or higher than the DC bus voltage, which implies that the power converter must operate in buck mode, boost mode, or at unitary gain. The following subsections report the simulation results in these different operating conditions.