## 4 Results
To better appreciate such speed-ups, we show them in Table 2, where we see that the proposed ASYN LD provides consistent speed-ups over the entire range of operating conditions considered here. The advantages of the asynchronous methods are made clearer in Figure 4, where we see that not only the asynchronous methods provide (on average) better running times but also present significantly less variation among the problems in \(\mathcal{E}\). The latter is relevant in the day-to-day operations of ISOs, since, if there are stochastic hydrothermal unit-commitment (SHTUC) cases that take significantly more time to be solved than the expected, subsequent operation steps that depend on the results of the SHTUC might be affected. Take, for instance, the case from the Midcontinent Independent System Operator reported in Reference [3], where the (deterministic) UC is reported to have solution times varying from just 50 to over 3600 s. Such variation can be problematic in the day-to-day operation of  
Figure 3: Performance profiles for the instances with machine Combination 2.  
power systems since it may disrupt tightly scheduled operations. Naturally, methods that can reduce such variance and still produce high-quality solutions in reasonable times are appealing. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Asynchronous**} & \multicolumn{2}{c}{**Synchronous**} \\ \cline{2-5}  & **Dynamic** & **Static** & **Dynamic** & **Static** \\ \hline
40 & 135 (340) & 240 (497) & 364 (569) & 338 (490) \\
50 & 130 (279) & 222 (414) & 195 (293) & 267 (404) \\
70 & 127 (249) & 139 (242) & 161 (157) & 250 (300) \\ A & 143 (280) & 226 (418) & 187 (270) & 216 (303) \\ B & 137 (340) & 192 (406) & 305 (515) & 347 (530) \\ C & 112 (247) & 184 (377) & 229 (335) & 291 (340) \\ L & 102 (195) & 157 (317) & 172 (398) & 201 (396) \\ M & 108 (179) & 112 (111) & 142 (126) & 205 (210) \\ H & 182 (425) & 333 (585) & 405 (492) & 449 (506) \\ HQSC & 105 (202) & 129 (193) & 156 (258) & 159 (118) \\ LQSC & 156 (357) & 272 (523) & 324 (473) & 411 (534) \\ Combination 1 & 120 (282) & 207 (447) & 227 (387) & 243 (416) \\ Combination 2 & 141 (301) & 194 (348) & 253 (393) & 327 (393) \\ \hline \hline \end{tabular} The rows indicate that the average elapsed times and standard deviation given in parentheses are computed considering only the instances in \(\mathcal{E}\) with the parameter given in the column 1. For example, the averages and respective standard deviations in row 3 are computed considering all experiments for which the initial useful-reservoir-volume level is 40%. Likewise, rows 4 and 7 provide the averages over instances with scenario set A and load level L, respectively. In rows 10 and 11, HQSC and LQSC stand for high-quality stability center and low-quality stability center, respectively.  
\end{table} 
Table 1: Average elapsed time and standard deviation in seconds. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Static SYN LD** & **Dynamic SYN LD** & **Static ASYN LD** \\ \hline
40 & 60 & 63 & 44 \\
70 & 49 & 21 & 9 \\ A & 34 & 23 & 37 \\ B & 61 & 55 & 29 \\ C & 61 & 51 & 39 \\ L & 49 & 41 & 35 \\ M & 47 & 24 & 4 \\ H & 59 & 55 & 45 \\ HQSC & 34 & 33 & 19 \\ LQSC & 62 & 52 & 42 \\ Combination 1 & 50 & 47 & 42 \\ Combination 2 & 57 & 44 & 27 \\ \hline \hline \end{tabular} As in Table 1, the rows indicate that the average speed-up computed considering only the instances in \(\mathcal{E}\) with the parameter given in the column 1. Moreover, the columns indicate the method the speed-up is computed for. For example, column Static SYN (synchronous computing) LD gives the speed-ups provided by the ASYN LD over instances in the first column w.r.t. to the static synchronous level decomposition.  
\end{table} 
Table 2: Speed-ups in % provided by the asynchronous computing (ASYN) with respect to the level decomposition (LD)