## 5 Experimental Results
### Results and Analyses  
We report in Tables 2 and 3 the computational results of the instances produced by our tabu search algorithm. In each table, the first column gives the names of the test instances. Each block corresponds to a certain number of iterations. For example, the values of \"Ratio (%)\" and \"Time (s)\" under the block \"1000 iterations\" is the ratio of the total power imbalance obtained at the end of the 1000 th iteration to the total generation power and the corresponding computation time. The block \"0 iteration\" shows the results of the initial solutions generated by the two-stage approach (see Section 4.2). In the last row of each table, we present the average \"Ratio (%)\" and \"Time (s)\" of these instances. To more clearly demonstrate the converging behavior of our algorithm, we graphically present the values of \"Ratio (%)\" in Figures 10-13, where the \(x\)-axis indicates the number of iterations. As revealed in these figures, our tabu search algorithm improved the initial solutions significantly. We can get the same conclusion from the last rows of Tables 2 and 3. At the end of 500 iterations, the algorithm reduced the average ratio from 16.04% down to 0.87%; this quality improvement reflects the effectiveness of our algorithm. Although we allow the algorithm to run 10,000 iterations, the results at the end of 500 iterations are very close to the final results, which can be easily observed from the above tables and figures. The ratios of these instances are very small. At the end of 500 iterations, the biggest ratio when \(L=7\) (respectively, \(L=15\)) is only 6.86% (respectively, 6.73%); most of the ratios are less than one percent; and many ratios are equal to zero. The average and greatest computation times consumed by 500 iterations are 0.169 and 0.312 s, respectively. Moreover, at the end of 1000 iterations, our tabu search algorithm consumed less than 0.7 s for each instance. Therefore, we can say that our tabu search algorithm is able to deal with the instances with up to 3120 nodes well within 0.7 s. These observations imply that our tabu search converged quickly and is able to efficiently produce high-quality solutions. \begin{table}
\begin{tabular}{c c c c c c} \hline Name & \# Edges & \# Generation nodes & \# Load nodes & Total generation power (Mega Watt) & Total load power (Mega Watts) \\ \hline _ieee118_ & 186 & 19 & 99 & 3,966.93 & \(-\)3966.93 \\ _sop2737_ & 3506 & 185 & 2552 & 10,391.16 & \(-\)10,391.16 \\ _wop2746_ & 3514 & 336 & 2410 & 17,037.25 & \(-\)17,037.25 \\ _wp3012_ & 3572 & 292 & 2720 & 24,487.14 & \(-\)24,487.14 \\ _sp3120_ & 3693 & 241 & 2879 & 19,713.23 & \(-\)19,713.23 \\ \hline \end{tabular}
\end{table} 
Table 1: The information of the five power systems. There is no clear difference between the algorithms with \(L=7\) and \(L=15\), since their solution quality and computation time are almost the same. We pictorially present the final solutions of instances _ieee118-2_, _ieee118-3-a_ and _ieee118-3-b_ in Figures 14-17, where the black circles denote the generator nodes, the white circles are load nodes and the names of the tripped edges are shown at the bottom of each figure. The algorithms with \(L=7\) and \(L=15\) produced the same solutions for instances _ieee118-2_ and _ieee118-3-a_, which are shown in Figures 14 and 1  
Figure 14: The pictorial result of instance _ieee118-2_. Figure 13: The convergence curves of the remaining nine instances generated by the algorithm with \(L=15\). Figure 15: The pictorial result of instance _ieee118-3-a_. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Method** & **Separation Boundaries** & **Imbalance Power (MW)** & **Imbalance Power (Ratio \%)** & **Time** \\ \hline Tabu search & 25-26, 17-18, 17-16, 14-4, 14-13, 12-13, 12-13, 12-13, 4-5, 1-2 & 1026.8 & 16.40\% & 0.001 s \\ Ant search & 25-26, 17-18, 17-16, 16-15, 10-13, 13-12, 4-5, 1-2 & 1670.3 & 26.60\% & 1 s \\ \hline \hline \end{tabular}
\end{table}