# Analyzing the Impact of Variability and Uncertainty on Power System Flexibility
## 4 Case Study
### Results for the Scenarios  
Figure 7 represents the result of the sensitivity analysis of the increased variability. The \(RSP_{t}\) values differ with the level of the net load; the largest \(RSP_{t}\) appears at the highest level of net load, i.e., S1. The largest difference in \(RSP_{t}\) according to the net load is between S5 (i.e., the net load is 100%) and S3 (i.e., the net load is 110%) when the increased variability is 18%. In S7 and S9, the values of \(RSP_{t}\) are almost unchanged with the uncertainty compared to the other scenarios. In the other scenarios, \(RSP_{t}\) increases in a step-wise fashion. This may be related to the failure cases of the generating units.  
Figure 8 shows the results of the sensitivity analysis for uncertainty. As in the case of increased variability, \(RSP_{t}\) is heavily influenced by the net load. The largest value of \(RSP_{t}\) is found when the level of the net load is highest (i.e., S2). S2 (i.e., 120% of the net load) and S4 (i.e., 110% of the net load), when the uncertainty is 4%, make the largest difference in \(RSP_{t}\). In most scenarios, the value of \(RSP_{t}\) does not increase much further after it increases sharply. This is related to the characteristics of the probability distribution of the uncertainty, which is modeled as a normal random variable. The left and right sides of the normal distribution are symmetrical with respect to its average, and an increase in the left side (right side) decreases (increases) the \(RSP_{t}\) at the same rate. The effects of both sides on the value of \(RSP_{t}\) cancel each other out.  
When comparing S1, S3, and S5 with S2, S4, and S6, respectively, the values of \(RSP_{t}\) and the maximum increments (i.e., the change in the \(RSP_{t}\) divided by the change in the uncertainty) in the former scenarios (i.e., S1, S3, and S5) are greater than those in the latter scenarios (i.e., S2, S4, and S6). S7 and S9 are compared to S8 and S10, respectively. The maximum increments in S8 and S10 are greater than those in S7 and S9. The value of \(RSP_{t}\) in S8 (S10) is larger than that in S7 (S9) from 10% (12%) to 21% for each parameter. Regarding flexibility, increased variability is more effective than  
\begin{table}
\begin{tabular}{c c c c} \hline
**Scenario, S\#** & **Net Load** & **Increased Variability of the Net Load at 19 h** & **Uncertainty of the Net Load at 19 h** \\ \hline S1 & 120\% & 0\% to 20\% & Fixed, i.e., 5\% \\ S2 & 120\% & Fixed, 0\% & 0\% to 20\% \\ S3 & 110\% & 0\% to 20\% & Fixed, i.e., 5\% \\ S4 & 110\% & Fixed, 0\% & 0\% to 20\% \\ S5 & 100\% & 0\% to 20\% & Fixed, i.e., 5\% \\ S6 & 100\% & Fixed, 0\% & 0\% to 20\% \\ S7 & 90\% & 0\% to 20\% & Fixed, i.e., 5\% \\ S8 & 90\%uncertainty when the level of the net load is equal to or greater than 100% (i.e., high and medium types). For the low type, uncertainty is more influential in terms of the maximum increment; however, the more influential parameter in terms of \(RSP_{t}\) differs by section.  
If the \(RSP_{t}\) criterion is set the same as the value of \(RSP_{t}\) of the given system (i.e., \(2.0982\times 10^{-4}\)), the allowed ranges of increased variability and uncertainty are then determined as shown in Table 3. S1, S2, S3, S4, and S5 do not have allowed ranges under the given conditions, while S6, S7, S8, S9, and S10 have allowed ranges. For reference, Figures 9 and 10 represent the enlarged graphs of Figures 7 and 8, respectively. On the graphs, each point having the value of \(RSP_{t}\) smaller than the \(RSP_{t}\) criterion is included in the allowed ranges.  
Figure 8: Results of sensitivity analysis for uncertainty.  
Figure 7: Results of sensitivity analysis for increased variability.