## 4 Test and Results
### Multistage TNEP Problem
#### 4.2.2 IEEE 24-Bus System  
For this system, the total generation capacity for stage 1 is equal to 10,215 MW, for stage 2 is equal to 10,726 MW, and for stage 3 is equal to 11,262 MW. The total demand for stage 1 is equal to 8560 MW, the total demand for stage 2 is equal to 8988 MW, and the total demand for stage 3 is equal to 9437 MW. For all tests carried out, the HEGA parameters are: \(k_{p}=20\), \(k_{s}=3\%\), \(k_{m}=5\%\) and \(k_{dr}=1\%\) (population size, selection rate, mutation rate, and diversification rate, respectively). The number of linear blocks was Y = 8. A summary of results for this system is presented in Table 6.  
Test 1: The best solution found by the HEGA for the multistage TNEP problem without SCC and active power losses has a total investment cost of US$220.286 \(\times\) 10\({}^{6}\) and no load shedding. This solution is found after solving 1585 LPs with a computational time of 0.48 min.  
Test 2: The solution found by the HEGA for the multistage TNEP with SCC has a total investment cost of US$209.243 \(\times\) 10\({}^{6}\) and no load shedding. This solution is found after solving 190,375 LPs with a computational time of 44.09 min. Test 3: The best solution found by the HEGA for the multistage TNEP problem considering power losses presents a total investment cost of US$238.35 \(\times\) 10\({}^{6}\), and no load \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Test**} & **Total Cost** & **LPs** & **Time** & **Stage Cost** & **Power Losses** & **Configuration** & **Compensated** \\  & **(10\({}^{6}\) US$)** & **(10\({}^{6}\) US$)** & **(10\({}^{6}\) US$)** & **(10\({}^{6}\) US$)** & **(10\({}^{6}\) US$)** & **(10\({}^{6}\) US$)** \\ \hline \multirow{4}{*}{1} & \multirow{4}{*}{220.286} & \multirow{4}{*}{1685} & \multirow{4}{*}{0.481} & \multirow{4}{*}{1} & \multirow{4}{*}{164.00} & \multirow{4}{*}{–} & \(n_{6-10}=1\), & \multirow{4}{*}{–} \\  & & & & & & & \(n_{7-8}=2\), & \\  & & & & & & \(n_{10-12}=1\), & \\  & & & & & & & \(n_{11-13}=1\), & \\  & & & & & & & \(n_{4-6}=1\) & \\ \cline{3-8}  & & & & 2 & 21.87 & – & \(n_{20-23}=1\), & – \\ \cline{3-8}  & & & & 3 & 14.34 & – & \(n_{1-5}=1\), & – \\  & & & & & & & \(n_{3-24}=1\) & – \\ \hline \multirow{4}{*}{2} & \multirow{4}{*}{209.243} & \multirow{4}{*}{190,375} & \multirow{4}{*}{44.09} & \multirow{4}{*}{1} & \multirow{4}{*}{152.00} & \multirow{4}{*}{–} & \(n_{6-10}=1\), & \multirow{4}{*}{–} \\  & & & & & & \(n_{7-8}=2\), & \\  & & & & & & \(n_{10-12}=1\), & \\  & & & & & & \(n_{14-16}=1\) & \\ \cline{3-8}  & & & & 2 & 48,114 & – & \(n_{12-13}=1\) & No compensated lines \\ \cline{3-8}  & & & & & & & \(n_{14-16}=1\) & \\ \cline{3-8}  & & & & & & & & \(n_{1-3}=1\) \\ \cline{3-8}  & & & & & & & (type 3), & \\ \hline \multirow{4}{*}{3} & \multirow{4}{*}{238.35} & \multirow{4}{*}{15,395} & \multirow{4}{*}{5.82} & \multirow{4}{*}{1} & \multirow{4}{*}{152.00} & \multirow{4}{*}{220.47} & \(n_{6-10}=1\), & \multirow{4}{*}{–} \\  & & & & & & \(n_{7-8}=2\), & \\  & & & & & & \(n_{10-12}=1\), & \\  & & & & & & \(n_{14-16}=1\) & \\ \cline{3-8}  & & & & 2 & 48.114 & 169.40 & \(n_{12-13}=1\) & – \\ \cline{3-8}  & & & & & & & \(n_{9-12}=1\), & \\ \cline{3-8}  & & & & & & \(n_{20-23}=1\) & – \\ \hline \multirow{4}{*}{4} & \multirow{4}{*}{227.264} & \multirow{4}{*}{179,863} & \multirow{4}{*}{60.48} & \multirow{4}{*}{1} & \multirow{4}{*}{152.07} & \multirow{4}{*}{220.47} & \(n_{6-10}=1\), & \multirow{4}{*}{–} \\  & & & & & & \(n_{7-8}=2\), & \\ \cline{3-8}  & & & & & & \(n_{10-12}=1\), & \\ \cline{3-8}  & & & & & & \(n_{14-16}=1\) & \\ \cline{3-8}  & & & & & & & \(n_{1-3}=1\) \\ \cline{3-8}  & & & & & & & (type 3), & \\ \cline{3-8}  & & & & & & & \(n_{2-6}=1\) \\ \cline{3-8}  & & & & & & & (type 1), & \\ \cline{3-8}  & & & & & & & \(n_{14-16}=1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Summary of results for the IEEE 24-bus system (multistage TNEP). shedding. This solution is found after solving 15,395 LPs with a computational time of 5.82 min.