## Case Study
Further, the proposed method has a much slower increasing rate of congestion hours compared to C-UC in Figure 4, indicating the improvement of system robustness to transmission congestion. To interrogate the performance of high-dimensional PCM in solving the probabilistic real-time dispatch sub-problem, the obtained real-time transmission reserves using PCM with sparse grids (denoted by PCM-SG) are compared with those of 1000 MCS and two-point estimate method [32] (denoted by 2 PEM) with regards to both accuracy and computational effort. Since these values vary for each time step and from line to line, we standardize the results by computing the average relative error and take the MCS with 20,000 scenarios as a benchmark. Figure 5 presents the average percentage error of real-time transmission reserve requirements for all the branches and cases over the whole time horizon. As can be seen, the PCM-SG consistently has the lowest percentage error for all the four cases with a maximum value of 6.3% in case 2 and a minimum value of 2.3% in case 3. The relative errors of 2 PEM in the four cases are kept almost invariant at around 10%, showing less accuracy compared to PCM-SG, but are much lower than those of 1000 MCS, especially in case 1 and 3, where the error goes up to 20%. The computational complexity of each method can be measured by the number of scenarios involved in the simulation. Though there are more scenarios needed in PCM-SG (109 scenarios) than 2 PEM (13 scenarios), the reduction ratio is dramatic compared to 1000 MCS (183 vs 20), which is efficient for the optimization problem on the day-ahead and hourly basis. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**With Correlation**} & \multicolumn{3}{c}{**Without Correlation**} \\ \cline{3-10}  & & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 1** & **Case 2** & **Case 3** & **Case 4** \\ \hline \multirow{2}{*}{Total Cost [\$/day]} & RA-SUC & 53,784 & 45,748 & 52,574 & 45,983 & 52,313 & 45,022 & 49,468 & 45,537 \\  & C-UC & 55,652 & 49,294 & 56,620 & 47,125 & 53,659 & 48,356 & 54,532 & 46,749 \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} Wind Utilization \\ Ratio [\%] \\ \end{tabular} } & RA-SUC & 88.41 & 93.74 & 90.53 & 96.93 & 89.28 & 93.82 & 90.47 & 98.10 \\  & C-UC & 87.67 & 91.93 & 89.70 & 93.71 & 87.91 & 92.13 & 89.78 & 95.58 \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Load Serve Ratio \\ [\%] \\ \end{tabular} } & RA-SUC & 95.26 & 98.21 & 94.87 & 99.83 & 95.24 & 98.21 & 95.18 & 99.85 \\  & C-UC & 95.12 & 98.26 & 94.57 & 99.36 & 95.21 & 98.15 & 94.59 & 99.69 \\ \hline \multicolumn{2}{l}{Spinning Reserves} & RA-SUC & 10,505 & 7186 & 9806 & 7458 & 11,002 & 7437 & 10,799 & 8062 \\  & C-UC & 11,284 & 8849 & 10,892 & 9280 & 11,891 & 9166 & 11,680 & 9613 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Expected economic indices of the proposed method versus conventional unit commitment (UC). C-UC: conventional UC; RA-SUC: risk-averse stochastic unit commitment. Figure 4: Comparison of total congestion hours versus wind power level. Figure 5: Average percentage error of real-time transmission reserve requirements. PCM-SG: probabilistic collocation method with sparse grids. Figure 3: Comparison of total operation cost versus wind power level. C-UC: conventional UC; RA-SUC: risk-averse stochastic unit commitment.