# Learning-Based Branching Acceleration for Unit Commitment with Few Training Samples
## 5 Numerical Studies
### Application with the Branch-and-Cut Algorithm  
Branching acceleration for UC proposed in this paper is independent of branch selection, node selection, and bound-tightening policies in the B&B algorithm. As a result, it can be applied with various B&B improvement methods, such as the branch-and-cut (B&C) algorithm, to further enhance the overall performance.  
B&C is one of the most effective algorithms for solving MILP problems [36]. It combines the B&B algorithm with the cutting plane method, playing a crucial role in improving the solution efficiency for MILP problems. The key characteristic of the B&C algorithm is its dynamic incorporation of cutting planes, particularly at the root node of the B&B search tree, to achieve a better approximation of the integer programming problem. While the cutting plane method increases the constraint size of the MILP model compared to using the B&B algorithm alone, it effectively reduces the feasible domain and the number of nodes explored, which typically results in a significant improvement in overall performance. For these reasons, the B&C algorithm is widely used in practical applications.  
In this subsection, branching acceleration for UC is combined with the B&C algorithm, which utilizes strengthened lift-and-project cuts as the cutting planes [37]. Compared with the testing process of branching acceleration for UC (Algorithm 3) described in Section 4, the combined method only adds a certain number of cutting planes at the root node of the B&B tree. The maximum number of cutting planes is set to 10. Note that this study focuses solely on evaluating the effectiveness of combining branching acceleration for UC with other B&B improvement methods. Therefore, the impact of the B&C algorithm on the solution time for the UC problem is not considered. The results are shown in Table 9.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**24-Unit Standard Testing System**} & \multicolumn{3}{c}{**10-Unit Practical Testing System**} \\ \hline  & \begin{tabular}{c} **B\&B +** \\ **B\&C** \\ **Branching** \\ **Acceleration** \\ **for UC** \\ \end{tabular} & \begin{tabular}{c} **B\&C** \\ **B\&C** \\ **for UC** \\ \end{tabular} & \begin{tabular}{c} **B\&C** \\ **Branching** \\ **Acceleration** \\ **for UC** \\ \end{tabular} & \begin{tabular}{c} **B\&C** \\ **B\&C** \\ **for UC** \\ \end{tabular} & \begin{tabular}{c} **B\&C** \\ **B\&C** \\ **Acceleration** \\ **for UC** \\ \end{tabular} \\ \hline \hline \begin{tabular}{c} Average node count in B\&B tree \\ \end{tabular} & 480 & 63 & 57 & 546 & 51 & 41 \\ \
\begin{tabular}{c} Speedup \\ Optimality gap \\ \end{tabular} & \(1.43\times\) & \(10.90\times\) & \(12.01\times\) & \(1.42\times\) & \(15.58\times\) & \(18.81\times\) \\
\begin{tabular}{c} Optimality gap \\ \end{tabular} & \(0\%\) & \(0.08\%\) & \(0.07\%\) & \(0\%\) & \(0.05\%\) & \(0.02\%\) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Performance comparison of each algorithm with 50 testing daily load profiles.  
The results demonstrate that branching acceleration for UC can be effectively combined with other B&B improvement methods (e.g., the B&C algorithm) to further enhance the overall performance, leveraging their respective strengths for optimal synergy.