## 1 Introduction
In spite of it being an effective method, it is only focused on the energy of the supercapacitor neglecting the energy from the inductor, which leads to a non-optimal solution. Additionally, Ref. [13] shows an application where a DC-DC converter is used for Maximum Power Point Tracking (MPPT), which is an application that continually requires duty cycle changes in the converter, in order to perturb the actual state and observe the resultant behavior. Hence, in applications with periodic perturbations, an inefficient inductance selection may complicate the MPPT calibration, regarding unexpected or aggressive responses against the induced perturbations. Thus, addressing the issue, a novel design methodology is proposed in this paper, seeking to improve the design of DC-DC converters through the optimization of the inductor selection. As will be later detailed in this paper, the inductance selection is a multi-objective optimization problem; since, focusing only on the output voltage of the circuit may lead to unreal current peaks or impossible required behaviors, just as validated in [2]. Therefore, to achieve an efficient design with a low ripple performance and stable behavior, the proposal is to optimize the energy from the inductor, by also optimizing the output voltage in the circuit. The optimization in this proposal is made through a metaheuristic approach, which allows implementing the multi-objective cost function in order to evaluate the performance of the circuit. Therefore, to obtain a correct representation of the system to evaluate, the Small-Signal Modelingfrom [14] is proposed due to its accuracy for a dynamic representation; since according to [10], the Small-Signal approach allows modeling and designing the circuit based on its dynamic behavior. The metaheuristic algorithm implemented for this modeling approach, is the Earthquake Algorithm (EA) proposed in [15], which as explained in [16], is a geo-inspired algorithm based in the behavior of the velocities of earthquake waves. Moreover, the EA is the first metaheuristic algorithm geo-inspired, which has a great suitability for different optimization problems (as validated in [17]), provided by the fine and aggressive searching capabilities, provided by the behavior of the P and S wave velocities (as highlighted in [2]). As cost function for the optimization, the Integral of Time Multiplied by Absolute Error (ITAE) performance index is taken as inspiration; where, as explained in [18] and validated in [15], the performance index may lead to small overshoots with damped oscillations profiles, which is exactly what the optimization is looking for. In terms of the current behavior of the circuit, a dampened profile is expected, in order to avoid the undesired large current overshoots, without neglecting the speed with which the converter voltage is expected to respond, which enables better and more efficient designs, as explained in [4]. Therefore, Section 2 deepens in the inductor behavior in basic converter topologies, to better understand the relevance of a correct inductance selection. Meanwhile, Section 3 explains the proposed design methodology for DC-DC converters based on multi-objective metaheuristic optimization. Later, the Earthquake Algorithm (EA) is explained in Section 4, where the key elements for the metaheuristic optimization through the EA are also summarized. Thus, a buck converter case study is presented in Section 5, in order to exemplify the novel design methodology, which is used for the experimental testbed designed in Section 6. Finally, Section 7 presents the results of this work, followed by the final conclusions in Section 8.