## 4 Methodology
### Optimization Framework: JADE  
In parameter estimation problems, evolutionary algorithms are often utilized. In this paper, adaptive differential evolution with an external archive (JADE) [29] is employed as the optimization framework. JADE has advantages in its usage of crossover and mutation  
Figure 6: Summary of the proposed method. functions similar to other evolutionary algorithms [28] and adaptation for the parameters of mutation and crossover factors in an 'evolution of the evolution' concept similar to other differential evolutionary algorithms [37]. Furthermore, since the best candidate to be used in mutated offspring generation is selected from the percentage of the best population members instead of the best one itself, premature convergence is avoided. Moreover, it benefits from an external archive whose members are used in differential mutation, which is a further precaution against impoverishment. _Mutation:_ In JADE with an external archive, in addition to the population set, denoted by \(\mathbf{P}\), an archive of solutions, denoted by \(\mathbf{A}\), is preserved. A mutation vector is generated:  
\[\mathbf{v}_{i,g}=\mathbf{x}_{i,g}+F_{i}\Big{(}\mathbf{x}_{\text{best},g}^{p}- \mathbf{x}_{i,g}\Big{)}+F_{i}\big{(}\mathbf{x}_{\text{r}1,g}-\mathbf{\hat{x}}_ {\text{r}2,g}\big{)} \tag{14}\]  
where \(g\) indicates the generation number, \(i\) indicates the ID in the population, \(\mathbf{x}_{i,g}\) is the vector in \(\mathbf{P}\) going through the mutation operation, \(F_{i}\) is the mutation factor associated with the vector \(\mathbf{x}_{i,g}\), \(\mathbf{x}_{\text{best},g}^{p}\) is the vector randomly chosen as one of the top \(100p\%\) individuals in \(\mathbf{P}\) with \(p\in(0,1]\), \(\mathbf{x}_{\text{r}1,g}\) is a vector randomly chosen from \(\mathbf{P}\) satisfying \(\mathbf{x}_{\text{r}1,g}\
eq\mathbf{x}_{i,g}\), and finally, \(\mathbf{\hat{x}}_{\text{r}2,g}\) is a vector randomly chosen from \(\mathbf{P}\cup\mathbf{A}\) satisfying \(\mathbf{\hat{x}}_{\text{r}2,g}\
eq\mathbf{x}_{i,g}\) and \(\mathbf{\hat{x}}_{\text{r}2,g}\
eq\mathbf{x}_{\text{r}1,g}\). The mutation factor \(F_{i}\) is selected for each individual \(\mathbf{x}_{i,g}\) of generation \(g\) according to a Cauchy distribution with location parameter \(\mu_{F}\) and scale parameter 0.1. If \(F_{i}>1\), then it is truncated to \(F_{i}=1\), and if \(F_{i}<0\), then it is regenerated. The value \(\mu_{F}\) is an adaptive parameter (see below for parameter adaptation). \[F_{i}=randc_{i}(\mu_{F},0.1) \tag{15}\]  
_Crossover:_ This is performed after mutation and a binomial selection are done:  
\[u_{j,i,g}=\left\{ \begin{array}{ll}v_{j,g},&\text{if }rand(0,1)\leq CR_{i} \text{ or }j=j_{rand}\\ x_{j,g},&\text{otherwise}\end{array} \right. \tag{16}\]  
where \(rand(0,1)\) is a uniform random number generator outputting within the interval of \([0,1]\) independently for each generation \(g\), individual \(i\) and component \(j\). The parameter \(j_{rand}\) is a random integer generator with the same size as the individual length, and \(CR_{i}\) is the crossover probability generated independently for each individual \(\mathbf{x}_{i}\) according to a normal distribution of mean \(\mu_{CR}\) and a standard deviation of 0.1. It is truncated to \([0,1]\). The variable \(\mu_{CR}\) is an adaptive parameter  
\[CR_{i}=randn_{i}(\mu_{CR},0.1) \tag{17}\]  
_Selection and Parameter Updates:_ The next spring \(\mathbf{x}_{i,g+1}\) is set equal to the one with the minimum cost function between \(\mathbf{x}_{i,g}\) and \(\mathbf{u}_{i,g}\). If \(\mathbf{u}_{i,g}\) is selected, then \(\mathbf{x}_{i,g}\) is moved to the archive set \(\mathbf{A}\), and \(\mu_{CR}\) and \(\mu_{F}\) are labelled as successful parameters. At each generation \(g\), randomly selected solutions in the archive are removed to keep the size of \(\mathbf{A}\) less than or equal to the population size, if necessary. For each generation \(g\), denote \(S_{CR}\) as the set of all successful crossover probabilities and denote \(S_{F}\) as the set of all successful mutation factors.