# Optimal Solution of Environmental Economic Dispatch Problems Using QPGPSO-\(\omega\)
## 4 Results and Discussion
### Case 4 of Ten Thermal Units (CEED + VPL + Losses)  
The test scenario involved ten thermal-based generating units, taking into account the VPL and power losses, as well as the combined emission economic dispatch (CEED). The specifications and parameters of the thermal TPPs, as well as their corresponding emissions, were obtained from [49]. The mathematical expressions used in the analysis are given by (11) and (12). Furthermore, Table 8 presents the power production and corresponding costs for each of the considered plants when simulated using the QPGPSO-\(\upomega\) algorithm.  
Figure 6 illustrates the cost convergence curve for the CEED case, showing the progression of cost values as the algorithm iteratively updated and refined the solutions.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Plant No.** & **QPGPSO-\(\upomega\) (MW)** & **QPGPSO-\(\upomega\) (S)** & **QPGPSO-\(\upomega\) (lb)** \\ \hline P\({}_{1}\) & 55 & 3645.188 & 307.474 \\ \hline P\({}_{2}\) & 80 & 4837.056 & 332.4252 \\ \hline P\({}_{3}\) & 94.1025 & 5460.377 & 302.58 \\ \hline P\({}_{4}\) & 111.10 & 6752.469 & 314.458 \\ \hline P\({}_{5}\) & 81.09 & 4897.367 & 57.87 \\ \hline P\({}_{6}\) & 82.215 & 5158.461 & 60.396 \\ \hline P\({}_{7}\) & 300 & 15,936.28 & 466.1098 \\ \hline P\({}_{8}\) & 340 & 18,034.35 & 655.8089 \\ \hline P\({}_{9}\) & 470 & 23,455.1 & 905.7544 \\ \hline P\({}_{10}\) & 470 & 23,356.94 & 902.882 \\ \hline P\({}_{\text{T}}\) & 2083.5 & - & - \\ \hline C\({}_{\text{T}}\) & - & 111,533.59 & - \\ \hline P\({}_{\text{loss}}\) & 83.402 & - & - \\ \hline E (lb) & - & - & 4305.76 \\ \hline \(PPF\) & 25.903 & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 8: Produced powers and their corresponding costs of the individual plants for case 4.  
Figure 5: Cost convergence curve of RES-based EDPs with RLSD.  
Overall, the combination of the quasi-population-based approach and the inertial weight parameter in QPGPSO-\(\omega\) results in an effective exploration of the search space, a balanced trade-off between exploration and exploitation, and convergence to favorable solutions. Comparisons with other techniques validated the superior performance of QPGPSO-\(\omega\) in terms of cost and emissions reduction, with slightly improved computational times. Therefore, QPGPSO-\(\omega\) stood out as a promising optimization approach for similar problems, offering efficient and effective solutions.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & **Min. Cost (USD)** & **Average Cost (USD)** & **E (lb)** & **CPU Time (s)** \\ \hline EMOCA [49] & 113,444.8500 & - & 4113.9800 & 2.90 \\ \hline MSA [51] & 113,350.0000 & - & 4102.4000 & \\ \hline QPSO [53] & 119,005.3000 & 121,621.7500 & 4041.9171 & - \\ \hline GQPSO [53] & 112,429.7444 & 113,102.4627 & 4032.9320 & - \\ \hline LAPO [67] & 113,412.3500 & - & 4115.5730 & 3.109 \\ \hline SMA [68] & 113,490.0000 & - & 4108.6000 & - \\ \hline PDE [68] & 113,510.0000 & - & 4111.4000 & 4.23 \\ \hline ABC-PSO [68] & 113,420.0000 & - & 4120.1000 & - \\ \hline MODE [68] & 113,484.0000 & - & 4124.9000 & 3.82 \\ \hline NSGAII [68] & 113,539.0000 & - & 4130.2000 & 6.02 \\ \hline SPEA-2 [68] & 113,520.0000 & - & 4109.1000 & 7.53 \\ \hline BSA-WSM [69] & 112,559.8061 & - & 4223.7248 & - \\ \hline GWO [70] & 111,729.4238 & - & 4247.5200 & - \\ \hline MHPSO-BAAC [66] & 106,278.0500 & 109,728.0500 & 4318.8900 & 2.242 \\ \hline MHPSO-BAAC-\(\chi\)[66] & 106,236.1500 & 109,632.1500 & 4310.9500 & 2.362 \\ \hline QPGPSO-\(\omega\) & 111,533.5900 & 111,649.8210 & 4305.7600 & 2.358 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Comparison of QPSPSO-\(\omega\) for CEED case.  
Figure 6: Cost convergence curve of EDP problem.