## 3 Results and Discussion
### HPES System Charging
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**PID Regulation Quality Indicators**} \\
**Charging Runs** & **Maximum Overshoot (\%)** & \multicolumn{4}{c}{**Settling Time (s)**} & \multicolumn{4}{c}{**Steady State Error (MW)**} \\  & **Alpha Model** & **Beta Model** & **Alpha Model** & **Beta Model** & **Alpha Model** & **Beta Model** \\ \hline Run 1 & 0 & 0 & 8 & 366 & 0.001 & 0.011 \\ Run 2 & 0 & 0 & 21 & 85 & 0.001 & 0.004 \\ Run 3 & 0 & 8 & 19 & 257 & 0.002 & 0.003 \\ Run 4 & 0 & 0 & 21 & 150 & 0.002 & 0.002 \\ Run 5 &  
Figure 9 presents more results comparing the same runs within the Alpha and Beta models. The two numerical models follow the same trend in each plot. Table 8 presents the RMSE\({}_{N}\) results to provide a more reliable representation on the models' comparative accuracy. The results obtained clearly illustrate the Alpha model's capabilities of reproducing the Beta model simulation predictions for the same prescribed operational conditions. A particularly important trend is that, despite the maximum RMSE\({}_{N}\) across all runs reaching only 13.9%, Runs 6, 8, and 10 experienced the highest errors in the majority of parameters. While variables such as pump speed and flow rate are expected to differ due to the different factors considered in the numerical calculations, such a result also coincides with the fact that the three mentioned runs consisted of the highest constant power, implying that RMSE\({}_{N}\) has a direct proportionality with input power. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Charging Runs** & \multicolumn{4}{c}{**Normalised RMSE (\%)**} \\  & **Input Power** & **Pump Speed** & **Flow Rate** & **PCS Pressure** & **Pump Efficiency** \\ \hline Run 1 & 6.94 & 3.51 & 10.50 & 2.51 & 8.31 \\ Run 2 & 5.76 & 4.26 & 6.38 & 1.73 & 7.29 \\ Run 3 & 4.18 & 3.93 & 4.99 & 2.56 & 2.13 \\ Run 4 & 6.79 & 5.66 & 7.74 & 1.99 & 8.83 \\ Run 5 & 5.74 & 5.67 & 6.01 & 1.29 & 6.36 \\ Run 6 & 9.11 & 7.73 & 10.30 & 1.68 & 12.20 \\ Run 7 & 4.18 & 4.59 & 6.53 & 1.12 & 7.51 \\ Run 8 & 9.32 & 8.29 & 10.30 & 2.57 & 11.30 \\ Run 9 & 5.61 & 4.84 & 8.64 & 2.33 & 8.55 \\ Run 10 & 4.81 & 5.75 & 6.37 & 2.37 & 5.43 \\ Run 11 & 4.52 & 4.43 & 9.07 & 3.57 & 13.90 \\ Run 12 & 6.36 & 4.36 & 11.70 & 2.97 & 11.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Summary of all results for the Normalised Root Mean Square Error (RMSE\({}_{N}\)) comparing the predictions for the Alpha and Beta model PCS charging process. Figure 9: Plots showing (**a**) Pump Speed variation for Run 4, (**b**) Pump Speed variation for Run 10, (**c**) Pump Efficiency variation for Run 4, and (**d**) Pump Efficiency variation for Run 10. Although the power was a system input, the RMSE\({}_{N}\) was still calculated since minor variations occurred which were regulated by the speed controller to maintain power as close to the desired (input) power as possible. The calculation was thus also used to prove that the Beta model controller gain variations across all runs all provided a stable system, since the power difference between the desired and shaft power was the controller's input value.