## 6 Results and Discussions
### Case 2: Non-Cascaded Non-Pumped STHTS Considering Transmission Losses [25]  
This is a two-unit and non-pumped hydrothermal scheduling problem, as given in [25]. The total number of scheduling periods is six, and the discharging characteristics of the thermal unit are given in Equations (20) and (21); where the fuel cost is \(1.15\) $/MBTU. \[F_{T}=500+8P_{T}+0.0016P_{T}^{2}\ \ (MBTU/h) \tag{20}\]  
\[150\ MW\leq P_{T}\leq 1500\ MW \tag{21}\]  
The water discharging constraints are given in Equations (22)-(25):  
\[D_{is}(P_{hyd})=330+4.97P_{hyd}\ \ (acre-ft/h) \tag{22}\]  
for:  
\[0\ MW\leq P_{hyd}\leq 1000\ MW \tag{23}\]  
\[D_{is}(P_{hyd})=5300+12(P_{hyd}-1000)+0.05(P_{hyd}-1000)^{2}\ \ (acre-ft/h) \tag{24}\]  
for  
\[1000\ MW\leq P_{hyd}\leq 1100\ MW \tag{25}\]  
The transmission losses are given by Equation (26):  
Figure 4: Convergence characteristics of the improved APSO algorithm on the non-cascaded pumped STHTS problem. Figure 5: Convergence characteristics of the improved APSO algorithm on the non-cascaded pumped STHTS problem for five different instances. \[(P_{loss})=0.0008P_{hyd}^{2}\ \ (MW) \tag{26}\]  
The discharge rate characteristics are given in Equation (27):  
\[5300\ (acre-ft/hr)\leq D_{is}\leq 7000\ (acre-ft/hr) \tag{27}\]  
The load demand in every interval is given in \[150\ MW\leq P_{T}\leq 1500\ MW \tag{21}\]0  
The reservoir's water volume flow constraints as given in [25; 27] are:  
1. 100,000 acre-feet constitute the volume of water in the reservoir prior to the first scheduling period. 2. 60,000 acre-feet constitute the volume of water in the reservoir in the last period. 3. Volume limit constraints in the first iteration are in acre-ft: 60,000 (acre ft) \(\leq\) V \(\leq\) 120,000 (acre-ft)
4. Continuous incoming flow into the reservoir is 2000 acre-feet/hour throughout the scheduling period. 5. The continuity equation as given in Equation (15) is to be met. The problem was implemented using both APSO algorithm and the improved APSO algorithm. The work in [25] already discussed the implementation of APSO for this problem; however, this paper presents its comparison with the improved APSO algorithm on hypothetical testing. The improved APSO algorithm has shown even more promising results as compared to the APSO algorithm. Table 6 presents the comparison between the two implementations. Both algorithms have been tested on the Section 6.2 Case 2 problem for sets of a hundred and two hundred trials each to have a normally-distributed dataset. Their means were compared using the hypothesis testing independent sample \(t\)-test. The results obtained from the SPSS software run are given in \[150\ MW\leq P_{T}\leq 1500\ MW \tag{21}\]1  
For this (Section 6.2 Case 2) problem, the value of significance of two-tailed \(t\)-test showed that both algorithms were not significantly different in terms of performance because the value was greater than 5%. This means that for more than 25% of the time, the results were somewhat repeated. Therefore, at the 95% confidence level, the null hypothesis could not be rejected, and the means were not statistically different. However, the improved APSO has been able to find a better approximation to the global best optima, as given in \[150\ MW\leq P_{T}\leq 1500\ MW \tag{21}\]2 The number of iterations and computation time were very low for both of these algorithms; however, due to the dynamic search space squeezing, improved APSO took a \begin{table}
\begin{tabular}{c c c} \hline \hline
**Performance Parameter or Algorithm** & **APSO** & **Improved APSO** \\ \hline Minimum Cost & 727,870.00 & 727,855.8 \\ Average Cost & 728,422.7 & 728,539.4 \\ Maximum Cost & 730,103.3 & 732,386.8 \\ No. of acceptable convergences & 15 out of 100 trials & 15 out of 100 trials \\ Standard deviation & 413.35 & 539.34 \\ Average No. of iterations & 6 & 7 \\ Average computation time \({}^{\star}\) & 1.035 s & 1.227 s \\ \hline \hline \end{tabular} \({}^{\star}\) Average computation time is according to MATLAB 2015 on a Core i5 second-generation processor.  
\end{table}