## 7 Additional STHTS Problems (as Test Cases)
### Case 1: Non-Cascaded Pumped-Storage STHTS Problem Considering Valve Point Loading  
In steam turbine-based thermal generators, valves control the steam entering the turbine through separate nozzle groups. Each nozzle group achieves best efficiency when operated at full output. Therefore, when increasing the output, valves are opened in sequence in order to achieve the highest possible efficiency for a given output. This causes a rippled cost curve, which is usually a sinusoidal wave riding on the quadratic cost function [19]. A valve point loading-based STHTS problem was given to test the effectiveness of the proposed improved APSO algorithm in comparison with the already tested simple APSO algorithm in the literature. The system taken was that of Case 1 with a valve point loading-based cost function. The cost function is now as given in (28):  
\[F_{T}=3877.5+3.9795P_{T}+0.00204P_{T}^{2}+500\sin(0.085(200-P_{T}))\ \ \$/h \tag{28}\]  
The rest of the problem data are the same as Section 6.1 Case 1 (Equations (17)-(19)) of the previous section. The statistical results of both the simple APSO and improved APSO algorithms are shown in Table 9. Table 10 shows the results of the SPSS results of the independent sample \(t\)-test performed for the comparison of the simple APSO and improved APSO algorithms on the non-cascaded pumped-storage STHTS problem considering valve point loading. Table 11 shows the results of power flow and cost optimization with the improved APSO algorithm implementation on the non-cascaded pumped-storage STHTS problem considering valve point loading. Convergence results for the simple APSO and improved APSO are shown in Figures 9 and 10, respectively.  
The results show that there is no significant difference between the performances of these two algorithms, statistically; however, the improved APSO algorithm has reached acceptable solutions more quickly than the simple APSO algorithm. If the tuning parameters of both algorithms are adjusted properly, both algorithms can help in finding a good approximate of the global optimum solution.  
Figure 9: Convergence characteristics of the simple APSO algorithm on the non-cascaded pumped-storage STHTS problem considering valve point loading effect  
Figure 10: Convergence characteristics of the improved APSO algorithm on the non-cascaded pumped-storage STHTS problem considering valve point loading effect  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Performance Parameter or Algorithm** & **APSO** & **Improved APSO** \\ \hline Minimum Cost & 265,200.2082 & 265,200.2082 \\ Average Cost & 265,433.4813 & 265,459.0192 \\ Maximum Cost & 265,781.3508 & 269,539.6085 \\ No. of acceptable convergences & 53 out of 100 trials & 59 out of 100 trials \\ Standard deviation & 174.6004 & 454.7182 \\ Average No. of iterations & 7 & 7 \\ Average computation time \({}^{*}\) & 0.5 s & 0.6 s \\ \hline \hline \end{tabular} \({}^{*}\) Average computation time is according to MATLAB 2015 on a Core i5 second-generation processor.  
\end{table}
Table 9: Comparison of some performance parameters between APSO and improved APSO on the pumped storage STHTS problem considering valve point loading.