## 7 Additional STHTS Problems (as Test Cases)
### Case 2: Non-Cascaded and Non-Pumped Storage STHTS Problem While Not Considering Transmission Losses  
This problem is similar to the Section 6.2 Case 2 problem of the last section. However, in this new system, the transmission line losses are considered as zero. The thermal and hydro units have the following characteristics. Thermal system: Equations (29) and (30):  
\[F_{T}=700+4.8P_{T}+0.0005P_{T}^{2}\ \ (\$/h) \tag{29}\]  
\[200\ MW\leq P_{T}\leq 1200\ MW \tag{30}\]  
Hydro system: Equations (31)-(33):  
\[D_{\text{is}}(P_{hyd})=260+10P_{hyd}\ \ (acre-ft/h) \tag{31}\]  
for:  
\[0\ MW\leq P_{hyd}\leq 350\ MW \tag{32}\]  
\[D_{\text{is}}(P_{H,Pump})=0\ \ (acre-ft/h)\ with\ P_{hyd}=0\ MW \tag{33}\]  
The starting and ending volume of the reservoir were 16,000 (acre-ft) and 12,000 (acre-ft) with a maximum allowed volume of 18,000 (acre-ft) and a minimum allowed volume of 12,000 (acre-ft). A constant water inflow of 2000 (acre-ft/h) was to be maintained in the reservoir for each scheduling interval. It was considered that there were no electrical transmission power losses. The load demand during each interval of four hours each is given in Table 12. Table 13 shows the SPSS results of the independent sample \(t\)-test showing the comparison between simple APSO and improved APSO on the non-pumped storage non-cascaded STHTS problem without transmission losses. Table 14 presents the power flow and cost optimization with the improved APSO algorithm implementation on the selected problem. Table 15 shows the comparison of some performance parameters between APSO and improved APSO on the non-cascaded and non-pumped storage STHTS problem while not considering transmission losses. Convergence results for the simple APSO and improved APSO for this case are shown in Figures 11 and 12, respectively. compared to the simple APSO algorithm. If the tuning parameters of both the algorithms are adjusted properly, both algorithms can help in finding a good approximate of the global optimum solution. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Performance Parameter or Algorithm** & **APSO** & **Improved APSO** \\ \hline Minimum Cost & 72,662.81 & 72,658.09 \\ Average Cost & 72,684.09 & 72,682.82 \\ Maximum Cost & 72,708.60 & 22,717.87 \\ No. of acceptable convergences & 72 out of 100 trials & 80 out of 100 trials \\ Standard deviation & 10.62 & 10.44 \\ Average No. of iterations & 50 & 50 \\ Average computation time * & 0.512 s & 0.506 s \\ \hline \hline \end{tabular}  
* Average computation time is according to MATLAB 2015 on a Core i5 second-generation processor.  
\end{table} 
Table 13: SPSS results of the independent sample \(t\)-test showing the comparison between simple APSO and improved APSO on the non-pumped storage non-cascaded STHTS problem without transmission losses. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**Independent Samples \(t\)-Test**} \\ \hline
**Levene’s Test for the Equality** & \multicolumn{8}{c}{\(t\)**-Test for the Equality of Means**} \\ \multicolumn{8}{c}{**of Variances**} \\ \hline
**Comparison** & \multirow{2}{*}{**F**} & \multirow{2}{*}{**Sig. **} & \multirow{2}{*}{**t**} & \multirow{2}{*}{**df**} & \multirow{2}{*}{**Sig. ** (2-tailed)**} & \multirow{2}{*}{**Mean** (2-tailed)**} & \multirow{2}{*}{**Difference**} & \multicolumn{2}{c}{**Std. Error**} & \multicolumn{2}{c}{**95\% Confidence Interval**} \\ \cline{5-10} \cl  
Optimization of Some General Deterministic Non-Linear Test Functions Using the Proposed Improved APSO Algorithm  
To check the performance of the proposed improved APSO algorithm, some hard and general deterministic optimization problems have been solved using the APSO and improved APSO algorithms. The problem functions are given in Table 16 and are plotted in Figure 13. Table 17 presents the results of the implementations of APSO and the proposed improved APSO algorithm on these complex optimization functions, showing the capability of these two algorithms to solve such non-linear and multi-modal objective functions.