## 3 Many-Objective Marine Predators Algorithm
### Many-Objective Marine Predators Algorithm (MaMPA)  
The MPA itself cannot solve multi- or many-objective optimization problems, but it can only solve the single-objective optimization problem. To solve the MaOPF problem, where four unrelated objective functions are considered in this work and the provided solutions are a trade-off between each objective, the Pareto dominance method is adopted. This method aims to find the trade-off which is a non-dominated solution and keep it in the repository traditionally called Pareto optimal fronts. The Pareto dominance method can be formulated as follows [8]:  
\[ \begin{array}{l}\forall i=\{1,2,\ldots,M\},\ F_{i}(X_{1})\leq F_{i}(X_{2})\\ \exists j=\{1,2,\ldots,M\},\ F_{j}(X_{1})<F_{j}(X_{2})\end{array}  \tag{36}\]  
where the vector \(X_{1}\) dominates \(X_{2}\) when both conditions are satisfied; and \(M\) is the number of considered objective functions. Since the number of non-dominated solutions is uncountable and the repository has a limited size, the non-dominated sorting and crowding mechanisms are also used in this work to organize the full repository to help the algorithm find the solutions with diversity in the limited size of the repository [56]. In addition, in the optimization process of the traditional MPA, the top predator (referred to as the best solution) is found in each iteration by simply comparing each prey from each iteration with the top predator from the previous iteration, and the top predator is also used to build the Elite matrix, as in (28). So, in this work, to find the top predator in the MOOPF and MaOPF problems where the number of the non-dominated solutions is uncountable, the leader mechanism is adopted using the roulette wheel and adaptive grid operators to provide the top predator in each iteration to be able to build the Elite matrix in (28). With more than three dimensions of objectives in the MaOPF problems, the graph of Pareto fronts faces difficulty in being appropriately generated. Instead, a best-compromised solution from the optimal tradeoffs between all different objectives can be provided by using the fuzzy decision method [36] to help power the system operators to select the best solution from the various provided solutions. To apply the fuzzy decision method, fuzzy membership functions on each dimension of all non-dominated solutions are firstly computed as shown below:  
\[ \begin{array}{l}\mu_{i}^{j}=\begin{cases}\begin{array}{c}1&,f_{i}^{j}\leq f _{\min}^{j}\\ \frac{f_{\max}^{j}-f_{\min}^{j}}{f_{\max}^{j}-f_{\min}^{j}}&,f_{\min}^{j}\leq f _{i}^{j}\leq f_{\max}^{j}\\ 0&,f_{i}^{j}\geq f_{\max}^{j}\end{array}\end{array} \end{array} \tag{37}\]where \(\mu\) is the fuzzy membership function, \(1\leq i\leq N_{S}\), \(1\leq j\leq N_{\omega j}\); \(N_{S}\) is the number of non-dominated solutions; and \(f_{\max}^{j}\) and \(f_{\min}^{j}\) are the maximum and minimum values of the objective \(j\), respectively. The summation of the fuzzy membership functions on all dimensions is then calculated and normalized by the following equation:  
\[\mu_{i}^{j}=\frac{\sum\limits_{i=1}^{m}\mu_{i}^{j}}{\sum\limits_{j=1}^{NF}\sum \limits_{i=1}^{m}\mu_{i}^{j}} \tag{38}\]  
where the non-dominated solution with the greatest summation value of the normalized function is referred to as the best-compromised solution. The implementation of the proposed MaMPA for solving the MaOPF problem can be described in the following steps:  
Step 1. Initialize the system data including costs and emission coefficients, active powers of generators, voltages of generator buses, transformer tap ratios, shunt compensation capacitors, maximum and minimum limits of the constraints, maximum iteration number, size of Pareto archive, and searching populations (Prey);  
Step 2. Convert the constrained problem to an unconstrained one by (20);  
Step 3. Calculate the fitness values of all initialized preys;  
Step 4. Find the non-dominated prey by using the Pareto dominance concept as in (36) and keep them in the initial archive;  
Step 5.