# Optimizing Sustainable Energy Transitions in Small Isolated Grids Using Multi-Criteria Approaches
## 3 Materials and Methods
### Optimization Procedures of Multi-Objective Problems  
EnergyPLAN is a powerful simulation tool that enables the analysis of energy balances beyond just cost concerns. Unlike traditional approaches focused solely on minimizing expenses, EnergyPLAN allows for the examination of different future energy generation mixes, ensuring a balance between economic feasibility and environmental sustainability, particularly by achieving zero CO\({}_{2}\) emissions. It operates by conducting hourly energy balances, usually over an entire year, in which the energy demand at each time step is compared with the supply capacity of the generation systems plus the existing storage capacity [51]. Once all possible system arrangements are simulated, the software provides key performance characteristics, helping researchers assess different energy strategies. While single-objective optimization, such as cost minimization, is useful, energy system planning often concerns numerous competing goals. These can include reducing overall system costs, minimizing GHG emissions, and maximizing renewable energy integration. Multi-objective optimization techniques are employed to address these complex trade-offs. These approaches allow for the simultaneous consideration of several goals by exploring a wide range of possible solutions and ranking them based on their performance. Multi-objective optimization algorithms follow an iterative process inspired by natural selection and evolutionary strategies. The key steps include generating an initial population of solutions, evaluating them based on predefined objectives such as cost and emissions, and improving solutions through selection, crossover, and mutation. Over multiple generations, these solutions evolved to enhance their performance. The final result is often represented as a Pareto front, which illustrates the trade-offs between competing objectives. This  
Figure 7: Schematic overview presenting the key input and output data of the EnergyPLAN tool. front consists of optimal solutions where improving one objective, such as lowering costs, may compromise another, such as reducing emissions. By analyzing the Pareto front, decision-makers can select the best energy system configuration based on their priorities. EnergyPLAN's integration with optimization algorithms offers a structured approach to energy system planning, enabling decision-makers to explore diverse solutions rather than relying on a single predefined objective. As energy transitions become more complex, adopting a multi-objective optimization framework ensures that economic, environmental, and technical criteria are adequately balanced. To start, a multi-objective minimization problem generally follows a structure like the following [54]:  
\[\underset{x}{\text{\emph{minim}}}[f_{k}(x)]\qquad k=1,2,3,\ \ldots,K \tag{1}\]  
\[\underset{x_{i}}{\text{\emph{subject to constraints}}} \tag{2}\]  
where \(f_{k}(x)\) are the \(k\) objective functions to be minimized and \(x\) is the vector of the i decision variables or parameters. The decision variables' space is constrained by Lower and Upper Bounds (LB/UB, respectively) and might have n inequality constraints of type \(h(x)\). If an \(x_{i}\) solution space meets the constraints, then it is named a possible solution. The specific context of the current research, specifically the energy situation of El Hierro, must be integrated into the optimization framework. This study aims to minimize both the system cost and GHG emissions. The decision variables include the installed capacities of various technologies such as wind, solar photovoltaic, hydropower, energy storage, and diesel generators. The optimization is subject to constraints like maximum installable power for each technology and, in the case of storage systems, limits on both power capacity and total energy storage. Detailed information about these parameters and constraints specific to this research is provided in Section 4.2.