# Optimizing Sustainable Energy Transitions in Small Isolated Grids Using Multi-Criteria Approaches
## 4 Results and Discussion
### Multi-Objective Optimization
These limits are typically in the form of maximum power to be installed, and in the case of storage systems, are also usually given by the total energy that can be stored by the system. Then, in the current case, the optimization decision function to minimize the total annual costs is:  
\[Total\ Annual\ Cost\ [M\&]=\left[\sum_{i}\left(C_{i}\cdot x_{i}\right)\cdot \left(\frac{1}{L_{i}}+O_{i}\right)\right]+C_{Diesel}\cdot F_{Diesel} \tag{5}\]  
where \(x_{i}\) and \(C_{i}\), respectively, represent the installed power (MW) and the cost per unit of installed power (EUR/MW) of each subsystem (solar PV, wind, hydro turbination, hydro pumping and diesel), except \(x_{\text{Storage}}\) and \(C_{\text{Storage}}\) which represent the total installed capacity (MWh) and its unitary cost of storage (EUR/MWh) for the hydro storage subsystem (dam-related equipment). While \(L_{i}\) are the lifetimes of each technology, in the diesel contribution, this term has not been considered since this subsystem is considered to be amortized. In the same manner, \(O_{i}\) represents the annual O&M costs of each technology (usually a percentage of the \(C_{i}/100\)) per technology, but in this case, diesel generation has to be considered as the rest of the contributions. Diesel generation also has fuel costs, the contribution which is accounted for in the total diesel generation, \(G_{Diesel}\) (MWh) and the unitary cost of this diesel fuel, \(F_{Diesel}\) (EUR/MWh). All these costs are summarized in Table 2.  
On the other hand, there is the expression associated with the second function to be optimized, the one related to GHG emissions. This is given by the following expression:  
\[CO_{2}\ Life\ Cycle\ Emissions\ (tons)=\sum_{i}G_{i}\cdot E_{i} \tag{6}\]  
where \(E_{i}\) is the life cycle emissions of each generation technology (tCO\({}_{2}\)eq/MWhe) and, as in the previous equation, \(G_{i}\) is the total generation of each technology. In this case, the contributions are diesel, solar PV, wind and storage (where the emissions of the life cycle of the complete installation, i.e., pumping, turbination, dams and other components of the reverse pumping system, have been considered). The data on the life cycle emissions for each technology is shown in Table 3.  
These variables are subject to the following group of constraints:  
\[ \begin{array}{c}CurrentInstalledPower\leq x_{i}\leq PotentialInstalledPower\ (MW)\\ Constraints:\\ 11.5\leq x_{Wind}\leq 25.3\\ 0\leq x_{PV}\leq 25\\ 11.25\leq x_{Turb.}\leq 15.0\\ 6.0\leq x_{Pump}\leq 12.0\\ \end{array}  \tag{7}\]  
\[ \begin{array}{c}CurrentInstalledCapacity\leq x_{i}\leq PotentialInstalledCapacity\ (MWh)\\ Constraint:\\ 225\leq x_{Storage}\leq 500\\ \end{array}  \tag{8}\]  
The inferior limits for wind power and reverse pumping storage (which are 11.5 MW and 6 MW, respectively) are established based on existing installed powers, ensuring that optimization does not propose lowering the current arrangement. The superior constraints for wind (25.3 MW) and solar PV (25 MW) are determined considering land availability and resources in El Hierro. For example, reports suggest that rooftop solar capacity could reach up to 85 MW [57]. However, such an elevated upper limit would create an excessively large solution space, making analysis and visualization difficult. It would also be unrealistic due to visual impact and potential technical-electrical concerns. The upper bound for reverse pumping storage (12 MW) is established based on expected maximum demand and system flexibility needs. Turbine power remains constant at its actual value, as the peak demand of the island is around 8 MW, indicating no need for an increase. However, it is maintained to ensure it can meet future demand growth. For the capacity of the hydro storage system, the inferior limit (225 MWh) denotes the present existing size, while the upper bound (500 MWh) reflects the maximum possible expansion of existing facilities, as no other suitable locations are available for new installations on the island.