## 5 Numerical Studies
### Comparative Analysis Of Approaches  
To illustrate the advantages of the proposed approach, several traditional methods are selected for comparative analysis. The cases are set as follows:  
1. Case 1: Assume that the output of PV&WPP has no prediction error, thereby transforming the problem into deterministic optimization. The results of Case 1 can be used as the benchmark for the uncertain optimization.
2. Case 2: The problem is formulated in the traditional RO form, using an ellipsoidal shape to define the uncertainty set. The case is equivalent to having a single cluster.
3. Case 3: Using the DRO approach to model the problem, where the ambiguity set is constructed based on the Wasserstein distance. The case is equivalent to the formulation where the number of clusters equals the number of uncertain parameter samples.
4. Case 4: This is the proposed approach, and the dataset of uncertain parameters is clustered into K subsets.  
Firstly, the optimization results are compared across the aforementioned cases, where the result of Case 4 varies with the number of clusters. Figure 13 shows the total cost values for \(K\)= 0\(\sim\)120. The dashed lines of different colors in the figure represent the objective function values of Case 2, Case 3 and Case 1 from top to bottom. They are independent of the number of clusters and are therefore a straight line. From Figure 13, it can be observed that Case 1, which represents deterministic optimization, yields the lowest total cost and serves as the benchmark for other cases. Case 2 results in the highest total cost, indicating that the outcome of RO is the most conservative. Additionally, the result of Case 2 is identical to that of Case 4 when \(K=0\). The optimization result of Case 4 fluctuates between those of Case 2 and Case 3. On the one hand, it shows that the choice of K significantly influences the optimization performance of Case 4. On the other hand, it also demonstrates that Case 4 can reduce the conservativeness of RO and result in better outcomes.  
Figure 12: Total costs under different carbon prices.  
Secondly, we also compared the computation times under the different cases, and the results are presented in Figure 14. The dashed lines corresponding to the bars of Case 4 represent the upper and lower bounds of the computation time for different values of K, denoted by t.max and t.min, respectively. From Figure 14, it is evident that the computation time for Case 1 is much smaller than that of the other cases, while Case 3 has the largest computation time, which is attributed to its complexity of methodology. The computation time for Case 2 is the same as the minimum computation time for Case 4, but the maximum computation time for Case 4 is still smaller than that of Case 3. This indicates that the clustering process effectively reduces the computation burden of DRO, thereby decreasing the computation time.  
In summary, compared to the benchmark represented by Case 1, Case 2 yields the most conservative optimization results. Case 3 brings the longest computation time, which  
Figure 14: Computation times under different cases.  
Figure 13: Operation results under different cases.  
represents DRO. The metrics for Case 4 fall between those of Case 2 and Case 3, which means that the proposed MRO approach can achieve a balance between conservativeness and computation time based on the choice of the clustering number K.