## 5 Numerical Results
### Comparison for Results from the 10-Unit System with Different Cost Function Characteristics
#### 5.1.3 Case 3: 10-Unit System with MFs, VPEs, and Other Complexity Constraints  
Next, we added several complex constraints such as spinning reserve, generating capacity, POZs, and ramp rate into the 10-unit system with MFs and VPEs. Table 3 lists the obtained results for a 2700 MW load demand and reports \(N_{fes}\) and \(ic\) for better comparisons. The IALO method was better than RCGA, FOA, IFOA, and ALO methods because the best cost of the IALO method was the smallest. The highest and the lowest improvement of the suggested method over these methods was, respectively, 7.304% and 0.005%. The values of \(N_{fes}\) from all methods indicates that the proposed method was one of the fastest methods with 3000 fitness evaluations, whereas the slowest method used 6000 fitness evaluations. However, the best cost of the IALO is slightly higher than that of the  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**Best Cost (\$/h)/Power Load (MW)**} & \multirow{2}{*}{\(N_{p}\)} & \multirow{2}{*}{\(G_{max}\)} & \multirow{2}{*}{\(N_{fes}\)} \\ \cline{2-2} \cline{6-9}  & **2400 (MW)** & **2500 (MW)** & & & & & \\ \hline HDE-DP [16] & 481.7226 & 526.2388 & 574.3808 & 623.8092 & 20 & 200 & 4000 \\ AISA [24] & 481.7230 & 526.2004 & 574.381 & 623.8090 & - & - & 3000 \\ CSA [27] & 481.7270 & 526.2522 & 574.4100 & 623.8343 & 10 & 200 & 2000 \\ RCSA [29] & 481.7227 & 526.2392 & 574.3813 & 623.8096 & 10 & 200 & 2000 \\ SDE [17] & 481.7226 & 526.2388 & 574.3808 & 623.8092 & 100 & 200 & 20,000 \\ EALHN [9] & 481.7230 & 526.2390 & 574.3810 & 623.8090 & - & - & - \\ ALO & 485.0665 & 528.0273 & 579.7788 & 625.8732 & 15 & 50 & 750 \\ IALO & 481.7230 & 526.2391 & 574.3814 & 623.8092 & 15 & 50 & 750 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the best fuel cost (\$/h) of case 1 of the 10-unit system.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & **Best Cost** & **Mean Cost** & **Worst Cost** & \multirow{2}{*}{\(S_{td}\)} & \multirow{2}{*}{\(CPU\) (s)} & \multirow{2}{*}{\(N_{p}\)} & \multirow{2}{*}{\(G_{max}\)} & \multirow{2}{*}{\(N_{fes}\)} & \(ic\) (\%) \\  & **(\$/h)** & **(\$/h)** & **(\$/h)** & **(\$/h)** & & & & & \\ \hline PSO [18] & 624.3045 & 624.5054 & 625.9252 & - & - & - & - & 3000 & 0.075 \\ FOA [37] & 664.5306 & 675.5344 & 679.4260 & - & - & - & - & 2000 & 6.124 \\ IFOA [37] & 623.8768 & 625.2704 & 629.2765 & - & - & - & - & 1000 & 0.007 \\ CSA [27] & 623.8684 & 623.9495 & 626.3666 & 0.2438 & 1.58 & 10 & 500 & 10,000 & 0.005 \\ RCSA [29] & 623.8608 & 623.8963 & 623.9353 & 0.0154 & 1.54 & 12 & 100 & 2400 & 0.004 \\ ALO & 623.9214 & 626.0822 & 643.8231 & 3.8731 & 2.8331 & 40 & 200 & 8000 & 0.014 \\ IALO & 623.8347 & 623.9930 & 626.4434 & 0.4232 & 0.8992 & 40 & 200 & 8000 & \\ \hline \hline \end{tabular}
\end{table}
Table 2: The comparison of results for case 2 of the 10-unit system.  
NRCGA method with \(-0.001\%\). This number shows that the best performance was not achieved by the proposed method, but our method is still a very efficient one. In fact, NRCGA reached the lowest cost, but the application of that method was only for the study cases with multiple fuels and VPEs, whereas the proposed method was implemented for different constraint types, different fuel types, and very large scale systems with 80 units and 90 units.