## 3 Renewable Energy Resource Modeling
### Fast Scenario Reduction Method  
The data used in this work, including load levels, solar irradiance, and wind speed, are presented in Figures 4-6 and are sourced from Reference [18]. This dataset contains hourly values for an entire year, resulting in 8760 data points for each variable. Figure 7 presents a focused view of the data, illustrating the variations recorded over a single day. Handling such a large volume of data presents numerous challenges [18], and scenario reduction is one of the techniques employed to deal with this complexity. Among the various reduction methods available, the fast scenario reduction (FSR) method [19] is chosen in this paper for its efficiency, reliability, and ability to select a smaller set of representative scenarios from a large set while preserving the original set's statistical properties. The procedure of the FSR technique can be expressed in terms of the following steps, with the goal of reducing the number of scenarios from \(N_{s}\) to \(N_{s}^{*}\).  
**Step 1:** Scenario Generation--Start by generating a large set of scenarios using methods: like Monte Carlo simulations, historical data, or other suitable techniques. Organizes these scenarios into a matrix, denoted as \(S\), where each row \(s_{i}\) represents a scenario containing the loading level \(LL_{i}\), the irradiance \(G_{i}\), and the wind speed \(v_{i}\), as follows:  
Figure 4: Hourly wind speed over a year.  
Figure 5: Hourly solar irradiation over a year.  
Figure 6: Hourly loading level over a year.  
\[S=\left[\begin{array}{ccc}LL_{1}&G_{1}&v_{1}\\ \vdots&\vdots&\vdots\\ LL_{i}&G_{i}&v_{i}\\ \vdots&\vdots&\vdots\\ LL_{Ns}&G_{Ns}&v_{Ns}\end{array}\right]\gets s_{1} \tag{27}\]  
**Step 2:**: Additionally, initialize the probability of each scenario to \(\tau_{i}=1/N_{s}\), where \(N_{s}\) represents the total number of scenarios.
**Step 3:**: Distance Calculation--Calculate distances between each pair of scenarios \(s_{i}\) and \(s_{j}\), using an appropriate measure to form a distance matrix. In this paper, the Euclidean distance with the 2-norm is adopted, and can be expressed as follows:  
\[d(s_{i},s_{j})=\sqrt{\left(L_{i}-L_{j}\right)^{2}+\left(G_{i}-G_{j}\right)^{2 }+\left(G_{i}-G_{j}\right)^{2}} \tag{28}\]
**Step 4:**: Scenarios Merging--Identify the pair of scenarios \(s_{i}\) and \(s_{j}\) that have the smallest Euclidean distance, as calculated in Step 2. Merge these two scenarios into a single representative scenario, often by taking the weighted average, based on their probabilities. The new scenario's values for loading, irradiance, and wind speed can be computed as:  
\[\left\{\begin{array}{l}LL_{new}=\left(LL_{i}+LL_{j}\right)/2\\ G_{new}=\left(G_{i}+G_{j}\right)/2\\ v_{new}=\left(v_{i}+v_{j}\right)/2\end{array}\right. \tag{29}\]  
Also, update the probability of the merged scenario \(\tau_{new}=\tau_{i}+\tau_{j}\) and decrease the number of scenarios \(N_{s}\) by 1.  
**Step 5:**: Termination Check--Determine if the stopping criterion has been met (e.g., by reaching a predefined tolerance level or desired number of scenarios \(N_{s}^{*}\)); otherwise, return to Step 2.  
In this paper, the scenario reduction process is applied to a timescale representing the total number of hours in a year, resulting in an initial set of \(N_{s}=8760\) scenarios. The chosen number of reduced scenarios, after applying the FSR method, is \(N_{s}^{*}=15\). Table 1 presents the obtained scenarios, including specific details for each scenario \(s\) such as the loading level, solar irradiance, wind speed, and their corresponding probabilities (\(\tau_{s}\)).  
Figure 7: (a) Focused variations over a day regarding loading level; (b) focused variations over a day regarding wind speed and irradiation.