## 6 Results and Discussions
### Case 1: Non-Cascaded Pumped Storage Hydrothermal Scheduling [24; 27]  
This is a two-unit hydrothermal scheduling problem as given in [24]. The total number of scheduling periods was six, out of which three periods were pumping, i.e., discharged water is pumped back to the reservoir, while three periods were generating. The thermal unit characteristics are given in Equation (16):  
\[F_{T}=3877.5+3.9795P_{T}+0.00204P_{T}^{2}\ \ \$/h \tag{16}\]  
\[200 MW\leq P_{T}\leq 2500 MW \tag{17}\]  
The generating and pumping models are given by Equations (18) and (19), respectively. \[Dis(P_{hyd})=200+2P_{hyd}(acre.ft/h)\ \ for\ \ 0\leq P_{hyd}\leq 300\ MW\ \ \ (Generatingcharacteristics) \tag{18}\]  
\[Dis(P_{H,pump})=-600(acre.ft/h)\ with\ P_{H,pump}=-300\ MW\ \ \ \ (Dischargingcharacteristics) \tag{19}\]  
The starting and ending volume of the reservoir = 8000 (acre-ft/h). The load demand at every interval is given in Table 1.  
The problem is implemented using both the APSO algorithm and improved APSO algorithm. The work in [24] has already discussed the implementation of APSO on 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 APSO algorithm. Table 2 presents the comparison between the two implementations. Both algorithms have been tested on the Case 1 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 Table 3.  
The null hypothesis was that there was no difference in the performance of the two algorithms on the pumped storage STHTS problem. This hypothesis was tested on a confidence interval value of 95%, giving a significance value (2-tailed) equal to 0.000, i.e., for both sets of a hundred and two hundred trials lesser than 0.05 or 5% [35]. This value suggests that the null hypothesis is rejected, and the statistical test quantitatively suggests that the improved APSO algorithm has performed significantly differently from the simple APSO algorithm. Furthermore, [PERSON]'s test of the equality of variances suggest that for a significance value equal to 0.000, which is less than 0.05, variances are also statistically different, significantly. Table 2 further elaborates the difference in performance by \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Interval** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Demand MW & 1600 & 1800 & 1600 & 500 & 500 & 500 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Load demand for each scheduling interval of four hours each. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Performance Parameter or Algorithm**}} & \multicolumn{4}{c}{**APSO**} & \multicolumn{4}{c}{**Improved APSO**} \\ \hline Minimum Cost & 269,642.4001 & 269,642.4000 & 269,642.4000 \\ Average Cost & 269,642.5318 & 269,642.4752 & 269,642.402 \\ Maximum Cost & 269,643.2336 & 269,642.402 & \\ No. of acceptable convergences & 4 out of 100 trials & 97 out of 100 trials & \\ Standard deviation & 6.2557539 & 0.0089976 & \\ Average No. of iterations & 7 & 8 & \\ Average computation time * & 0.462 s & 0.506 s & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of some performance parameters between APSO and improved APSO on the pumped storage STHTS problem. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Levene’s Test for the Equality**}} & \multicolumn{6}{c}{**Independent Samples \(t\)-Test**} \\ \multicolumn{1}{c}{\ \begin{tabular}{c} **of Variances** \\ \end{tabular}  } & \multicolumn{6}{c}{ \begin{tabular}{c} **\(t\)-Test for the Equality of Means** \\ \end{tabular}  } \\ \hline Comparison & \multirow{2}{*}{**F**} & \multirow{2}{*}{**Sig. **} & \multirow{2}{*}{**t**} & \multirow{2}{*}{**df**} & \multicolumn{2}{c}{**Sig.