## 3 Differential Evolution-Based Parameter Selection  
Aiming at the selection of optimal parameters for the classical wireless energy transfer topologies, an algorithm developed from the Differential Evolution (DE) algorithm was used. DE is a meta-heuristic based on the process of species evolution, which consists of generating an initial random population, mutating, crossing characteristics between individuals, and selecting the best individuals at the end of each generation [13]. For the proposed application, some changes were made in the pseudocode of the presented algorithm. First, an initial population is defined, composed of NP vectors of randomly chosen parameters, _pop(i, 1)_ and _pop(i, 2)_, in the interval between the minimum of 0.01 uH and the maximum of 100 uH, which correspond to the values of the primary and secondary inductances. The population is created from a uniform probability distribution and follows a natural evolution with the number of individuals fixed during the optimization process. It is noteworthy that the population must meet the non-bifurcation criteria. That is, if the generated population presents bifurcation, it will be discarded, and a new population will be formed. As shown in Figure 11, the generation of the initial population is illustrated. In one generation, three individuals are selected, \(X(1)\), \(X(2)\), and \(X(3)\), obtained from a randomly established population, in an interval from 1 to NP. Another vector, named the target vector _pop(i, j)_, is also selected randomly. In the first generation, therefore, the target vector and the vectors \(X(1)\), \(X(2)\) and \(X(3)\) are selected to be part of the mutation, crossover,  
Figure 11: Initial population generation. Figure 10: Impedance angle as a function of \(\kappa\). (**a**) SS. (**b**) SP. (**c**) PS. (**d**) PP. and selection steps. The four individuals must be distinct from each other, meeting the condition that \(X(1)\
eq X(2)\
eq X(3)\
eq pop(i,:)\), as illustrated in Figure 12.  
In the context of evolutionary computation, mutation can be defined as a change or perturbation with a random element [23]. From the random selection of three individuals of a generation G, \(X(1)\), \(X(2)\), and \(X(3)\), a vector of mutated parameters (\(V\)) is obtained. The differential mutation operation is performed according to (1). \[V=X(3)+F\times(X(2)-X(1)) \tag{1}\]  
In accordance with the operation, \(X(2)-X(1)\) is defined as the difference between the parameter vectors. Let \(F\) be a real constant \(\in\) [0, 2], called the mutation factor. This factor is responsible for controlling the size of the step to be performed, that is orienting the vector amplitude. The weighted vector difference is added to the third individual \(X(3)\) as illustrated in Figure 13.  
The mutated vector \(V\) is combined with the target vector, \(pop(i,j)\), i.e., an arbitrarily chosen target vector, and the combination results in the trial vector. This process is called crossover, increasing the diversity of mutated individuals. The algorithm performs the combination respecting the real crossover constant \(cr\in\) [0, 1]. Let rand be a random scalar \(\in\) [0, 1]; if rand is less than the \(cr\), the muted vector parameters are selected; otherwise, the target vector is selected. The crossover operation is exemplified in Figure 14. It is noteworthy that, in this paper, the mutation and crossover operations were performed concurrently. After the mutation and crossover of the individuals, the selection of the best individuals is carried out. Initially, the value of the characteristics was analyzed, and it was verified if the value was within the range of 0.01 \(\mu\)H to 100 \(\mu\)H. If it was within the range, the characteristic was maintained; otherwise, it was replaced by the characteristic of the previous individual. Subsequently, the bifurcation criterion was analyzed.