## 4 Case Studies
### Probabilistic Forecasting of ALSC
#### 4.2.2 The Effectiveness of LHS-MCS  
In order to evaluate the performance of LHS-MCS, the results of the probabilistic indices for ALSC (expectation, standard deviation and variation coefficient) of RS-MCS are taken, with a sample size of 20,000 as the benchmark, the results of which are expressed as \(l_{accurate}\). Likewise, the results of probabilistic indices obtained from the test sample size of LHS-MCS and RS-MCS are expressed as \(l_{simulate}\). The error of indices is introduced to evaluate the accuracy of the probabilistic methods, as follows:  
\[\varepsilon_{I}=\left|\frac{I_{simulate}-I_{accurate}}{I_{accurate}}\right|\times 1 00\% \tag{52}\]  
Due to the random factors in the probabilistic evaluation of ALSC, the error of probabilistic indices of both methods are obtained by the results of 50 simulations with the given sample size. The mean error of 50 simulations, which is expressed as \(\varepsilon_{I,50}^{mean}\), is taken to demonstrate the accuracy of the tested method. In addition, to explain the robustness of the tested method, the standard deviation of the error of 50 simulations is taken, which can be expressed as \(\varepsilon_{I,50}^{std}\).  
The curves of \(\varepsilon_{\text{ALSC},50}^{mean}\), \(\varepsilon_{\text{ALSC},50}^{mean}\), \(\varepsilon_{\text{ALSC},50}^{mean}\) and \(\varepsilon_{\text{ALSC},50}^{std}\), \(\varepsilon_{\text{ALSC},50}^{std}\), and \(\varepsilon_{\text{ALSC},50}^{std}\) with different sample sizes are shown in Figures 8 and 9, respectively. It can be seen that the mean and standard deviation of the error of LHS-MCS are smaller than those of RS-MCS  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Generator** & **30** & **31** & **32** & **33** & **34** & **35** & **36** & **37** & **38** & **39** \\ \hline \(P_{G\text{max}}/\text{MW}\) & 1040 & 969 & 725 & 652 & 508 & 1030.5 & 870 & 846 & 1297.5 & 1650 \\ \hline \hline \end{tabular}
\end{table}
Table 10: The maximum active power of each generator.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Branch** & **1–2** & **1–39** & **2–3** & **2-25** & **2–30** & **3–4** & **3–18** & **4–5** & **4–14** & **5–6** & **5–8** & **6–7** & **6–11** \\ \hline \(S_{\text{jmax}}/\text{p.u.}\) & 3.75 & 3 & 6 & 6 & 6 & 3 & 3 & 4.5 & 4.5 & 9 & 6 & 7.5 & 6 \\ \hline
**Branch** & **6–31** & **7–8** & **8–9** & **9–39** & **10–11** & **10–13** & **10–32** & **12–11** & **12–13** & **13–14** & **14–15** & **15–16** & **16–17** \\ \hline \(S_{\text{jmax}}/\text{p.u.}\) & 12 & 4.5 & 3 & 3 & 6 & 6 & 10.5 & 1.5 & 1.5 & 6 & 1.5 & 6 & 5.25 \\ \hline
**Branch** & **16–19** & **16–21** & **16–24** & **17–18** & **17–27** & **19–20** & **19–33** & **20–34** & **21–22** & **22–23** & **22–35** & **23–24** & **23–36** \\ \hline \(S_{\text{jmax}}/\text{p.u.}\) & 7.5 & 6.75 & 3 & 4.5 & 1.5 & 4.5 & 10.5 & 9 & 10.5 & 1.5 & 12 &under different sample sizes, demonstrating that LHS-MCS is more accurate and stable than RS-MCS. In addition, the mean and standard deviation of the error of LHS-MCS converge faster, indicating that LHS-MCS needs a smaller sample size to achieve good accuracy. In contrast, RS-MCS requires a large sample size to achieve an acceptable accuracy.