## 7 Additional STHTS Problems (as Test Cases)
### Case 2: Non-Cascaded and Non-Pumped Storage STHTS Problem While Not Considering Transmission Losses
Figure 11: Convergence characteristics of the simple APSO algorithm on the non-cascaded non-pumped STHTS problem without transmission loss. Figure 12: Convergence characteristics of the improved APSO algorithm on the non-cascaded non-pumped STHTS problem without transmission loss. It can be seen that both the APSO and improved APSO algorithms have shown promising results to find the good approximates to the determined or known global minima of the highly non-linear and multi-modal test functions. The performances of both APSO and improved APSO were good; however, the improved APSO had the capability to find closer results to the determined or known values of the minima of these functions. The average value of the set of minima obtained for the case of the egg-crate function was high, as well as the value of standard deviation. The reason is that the egg-crate function had many local minima quite near the global minimum, as can be seen in Figure 13. Therefore, due to the dependency of the particle updating process, only on the g* value, and due to the presence of fewer tuning parameters, both the simple APSO and improved APSO algorithms sometimes became stuck in the local optima. If some chaotic maps are introduced, the stochastic nature of APSO and improved APSO can be increased to allow the particles to escape from local peaks. However, the improved APSO algorithm has still achieved a higher number of successful results for the egg-crate function. Similarly, for the Michaelwicz 2D function, both improved APSO and APSO sometimes become stuck in the local minima. However, for most of the time, good and robust solutions were achieved. If the neighborhood topologies of particles and some stochastic term are dominantly added in the particle update equation, the number of successful achievements of global optimum solution can be increased to more than 90% of the time. \begin{table}
\begin{tabular}{l l} \hline \hline
**Optimization Function** & Michaelwics 2D function \\
**Mathematical form** & \(f(x,y)=-sin(x)\sin^{20}{(\frac{x^{2}}{\pi})}+\sin(y)\sin^{20}(\frac{2y^{2}}{ \pi})\) \\  & \((x,y)=[0,4]\times[0,4]\) \\
**Nature of the problem** & Highly non-linear and multi-modal \\
**and its global optimum value** & **Global optimum approximately at** \\  & \((x,y,z)=(2.20319,1.57049,-1.801)\) \\ \hline
**Optimization Function** & Rosenbrock 2D function (Banana function) \\
**Mathematical form** & \(f(x,y)=(1-x)^{2}+100\times(y-x^{2})^{2}\) \\  & \((x,y)=[-5,5]\times[-5,5]\) \\
**Nature of the problem** & Non-linear and convex \\
**and its global optimum value** & **Global optimum at** \\  & \((x,y,z)=(1,1,0)\) \\ \hline
**Optimization Function** & De Jong 2D function \\
**Mathematical form** & \(f(x,y)=x^{2}+y^{2}\) \\  & \((x,y)=[-5,5]\times[-5,5]\) \\
**Nature of the problem** & Non-linear and convex \\
**and its global optimum value** & **Global optimum at** \\  & \((x,y,z)=(0,0,0)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Selected functions for the validation of the proposed improved APSO. The results obtained from the implementation of the improved APSO and the results taken from [1] of the implementation of the simple APSO algorithm on the deterministic, non-linear, and multi-modal functions show that both of these algorithms are a brilliant choice to obtain robust solutions of the optimization problems. Moreover, the elegance of these APSO variants in terms of a few tuning parameters and a single-step particle update equation make them a preferred choice as optimization algorithms.