## 2 Materials and Methods
### Principles of IDE Algorithm and Multi-Objective Optimization
#### 2.4.2 Improved Self-Adaptive Differential Evolutionary Algorithm
Limitation-Handling Technique  
Because optimization issues in the real-world are often connected to various limitations, the alpha (\(\alpha\))-finite approach has been utilized to address limitations. The \(\alpha\)-finite technique treats several infeasible solutions as feasible approaches whenever such non-applicable solutions are under a specific level. In order to control the level \(\alpha\), Equation (28) is presented. \[\alpha(0)=\varnothing(\mathrm{x}_{\varnothing}) \tag{28}\]  
\[\alpha(\mathrm{t})=\left\{ \begin{array}{cc}\alpha(0)\left(1-\frac{\mathrm{t} }{\mathrm{T}_{\mathrm{c}}}\right)^{\mathrm{cp}},&0<t<\mathrm{T}_{\mathrm{c}}, \\ 0,&t\geq\mathrm{T}_{\mathrm{c}}\end{array} \right. \tag{29}\]  
where \(x_{\varnothing}\) represents the top \(\theta th\) solution at the initialization phase; \(cp\) denotes a parameter for controlling the speed of limitations reducing relaxation; \(\theta\) has been suggested to be \(0.2*NP\); also \(cp\in[2,10]\). For having an efficiently using of the information performed by the infeasible solutions, the concept of pseudo feasible has been described. Generally, the pseudo feasible solution gives the overall solution whose general limit violation becomes less than \(\alpha\). A detailed study about the limited-dominate connection between individuals consisting of pseudo feasible and pseudo infeasible solutions has been presented in the reference [48]. After the phase of sorting operation based on the limited-dominate connection criteria, exactly \(NP\) solutions should be selected from combined population \(X^{t}\cup U^{t}\) to new population \(X^{t+1}\). By combining the self-adaptive evolutionary method and immune operation, the step by step pseudo code of the new improved self-adaptive evolutionary is presented as below:* [1] Set the parameters' values and initialize the external archive. Set the generation number G = 0 randomly initialize a population of NP individuals uniformly distributed in the range [Xmin,Xmax]
* [2] Evaluate the fitness of initial solutions
* [3] WHILE the termination criterion is not satisfied, DO
* [4] Generating offspring solutions by DE's trial vector generation operators
* [5] Mutation step FOR \(i=1\) to NP Generate a mutated vector according to the self-adaptive mutation strategy END FOR
* [6] Crossover step FOR \(i=1\) to NP Generate a trial vector for each target vector using binomial crossover operator. END FOR
* [7] Divide the combination of parent and offspring population into two subpopulations according to pseudo feasible concept, SubP1 (pseudo feasible solutions) and SubP2 (pseudo infeasible solutions)
* [8] Order the SubP1 by non-dominated sorting procedure
* [9] Order the SubP2 according to overall violation degree
* [10] Add all the non-dominated solutions into external archive and remove the inferior ones. * [11] Select solutions for the main population of the next generation
* [12] Update the parameters
* [13] Increment the generation count G = G + 1. END WHILE  
All simulations are performed in a typical system that has MT resources, PVs, WTs, FCs, ES units with having the capability of upstream grid power exchange.