## 5 Discussion
### Multi-Stage Co-Optimization of Microgrid Sizing and Energy Management  
In the multi-stage approach, two methodologies are predominantly discussed in the literature. The first methodology begins by addressing the component sizing dilemma, followed by resolving the energy management challenge in a subsequent phase. This sequential methodology leverages optimization-based methods, commercial software solutions, or rule-based strategies for problem formulation. Figure 9 illustrates the outline of this methodology. This methodology has been widely used by researchers. For example, the study [127] contrasts hydrogen and BESS in PV systems by employing a GA for optimal sizing and a rule-based strategy. It focuses on self-sufficiency, NPV, and the impact on the power grid. In [109], the multi-stage methodology is also applied. The stochastic fractal search (SFS) and the symbiotic organisms search (SOS) algorithms are used for sizing, while the energy management uses a rule-based strategy. Figure 8: General framework of the single stage problem. Figure 9: General framework of the multi-stage optimization problem. Another example can be found in [41; 55], where GA is used for sizing and a rule-based approach is used for energy management. The NSGA-II with several rule-based options is presented in [56]. The same approach is presented in [60], with the PSO algorithm used for sizing of the grid-connected microgrid. Shifting from heuristic approaches for sizing and rules-based approaches for energy management, some studies have implemented optimization algorithms for both stages. In [100], the model uses the Karush--Kuhn--Tucker approach to simplify the bi-level problem into a single-level one. A mix of binary PSO and QP finds the optimal BESS size and operation schedule to enhance microgrid design and reliability. In [128], the methodology applies the two-stage method, where the first stage focuses on determining the size of the microgrid. In contrast, the second stage addresses the daily energy management considerations. This study implements MILP optimization techniques in both stages. In [129], the first stage employs an evolutionary algorithm to handle the sizing task, exploring a vast solution space to identify viable configurations. In the second stage, MILP fine-tunes these configurations by meticulously scheduling microgrid assets to balance demand fulfillment with the dual objectives of cost minimization and reliability maximization. A similar approach is found in [130], where the study uses a GA to identify optimal sizing solutions that concurrently lower the lifecycle CO\({}_{2}\) emissions and total costs of the microgrid. Moreover, an EMS is developed, employing a MILP algorithm to efficiently distribute power flow while also reducing CO\({}_{2}\) emissions. In [131], a stochastic two-stage model is presented for optimizing microgrid design by addressing the uncertainties in DER. The initial stage focuses on setting capacities for PV, WT, diesel generation, and ESS. In contrast, the second stage targets operational decisions, such as diesel and ESS usage. The study in [132] introduces a method for optimizing microgrid components in \"EI Espino\" community. It uses linear programming (LP), firstly to minimize costs while meeting energy needs, and secondly to incorporate operational and reliability constraints. In [133], the paper applies a two-stage stochastic optimization approach to minimize both capital and operational costs considering uncertainties in demand. The second form of multi-stage optimization utilizes decomposition algorithms, notably the Dantzig-Wolfe decomposition [134]. This method entails an iterative process that includes several subproblems and a master problem. Each iteration introduces a new variable to the master problem, facilitating a gradual convergence towards the solution. Another key decomposition algorithm is the alternating direction method of multipliers [135], which segments the optimization challenge into multiple sub-problems without incorporating a master problem.