## 2 Modeling and Solution Strategies
### Multi-Objective Solution Methodology  
The proposed multi-objective solution methodology (MOHGA) consists of two steps: a Genetic Algorithm routine to perform the optimization process in searching of feasible solutions (population in GA terminology) and the Maximin metric routine for dominance measure and classification of the solutions in population to form the Pareto-set. For each \(i\)-th iteration of the optimization process, a feasible solution \(X_{i}\) represented by variable vector \(x\) (called individual in the GA terminology) is evaluated and a vector of objective-functions \(F(X_{i})\) is obtained. However, interpretations of the Maximin metric depends heavily on the values of the objective functions, which have different units and magnitudes. To mitigate this problem, a normalization of vector \(F(X_{i})\) shown in Equation (8) is proposed, where \(F(X_{0})\) corresponds to the vector of objective functions evaluated in the base-individual \(X_{0}\) referring to the solution of mono-objective problem of the system loss minimization, so that no null value to any of the objectives can be obtained. \[Y_{i}=\frac{F(X_{i})}{F(X_{0})}=\left[y_{1}^{i},y_{2}^{i},\dots,y_{n_{f}}^{i} \right]\,\forall\,\,X_{i}\in Pop,i\in[1,Popsize] \tag{8}\]  
Maximin metric provides two useful informations [27; 32]: (1) negative value indicates a non-dominated solution; (2) the more negative is the value, the less crowded is the region of Pareto frontier the solution lies. Thus, Maximin metric can be used as an additional fitness criterion to evaluate a population of solutions in order to provide observation of Pareto dominance along with the degree of clustering aiming to obtain diversity of solutions in Pareto sets. Consequently, the Pareto set can be updated at each iteration with alternatives from the population of solutions with the most negative (non-dominated) Maximin metric values. Equation (9) presents the Maximin metric of a normalized vector of objective functions \(Y_{i}\). \[maxmin_{i}=\max\biggl{\{}\min_{j=1,\dots,Npop\,,\,j\
eq i}\left\{y_{1}^{i}-y_{ 1}^{j},\dots,y_{n_{f}}^{i}-y_{n_{f}}^{j}\right\}\biggr{\}}\,,i\in[1,Popsize] \tag{9}\]  
GA solution process starts with a population and applies genetic operators to search new feasible alternatives of solution in each iteration. The new population is inserted in the last Pareto set, the resulting set is evaluated by Maximin metric as ranking criterion and only solutions with negative Maximin metric are chosen to update the Pareto set. The process is repeated until the adopted maximum number of iterations is reached. Figure 2 exemplifies the adopted codification (chromosome) of alternative solutions (individuals) for four primary sources that can be inserted in two buses. In this example, the proposed solution suggests the installation of 4 DG units of type 1 at bus 13; for type 2, the installation of 5 DG units at bus 36 and 3 DG units at bus 58; for type 3, the installation of one DG unit at bus 3 and 2 DG units at bus 47; and, for type 4, the installation of 4 DG units at bus 62. The genetic operators applied in GA solution process are described as follow:  
* Elitism: best individuals from the current Pareto set are used as individuals in the next iteration. In this work, the number of best individuals to be considered in the new population is equal to 5% of the current population size. * Parental Selection: a tournament selection among three random individuals from the current iteration is applied. The individual with more negative Maximin metric wins the competition and becomes a father for the next generation. * Mutation: a fixed mutation tax of 5% is adopted. * Recombination: the process is always accomplished (recombination tax of 100%). Three decision criteria considering the concept of preference were adopted to identify a final solution among Pareto frontier setpoints: minimum losses of active power point, maximum net present value (NPV) point, and Max-Min approximation point.