## 3 Results and Discussion
### HPES System Discharging  
While the Alpha and Beta models for the Pelton turbine and PCS are very similar as previously discussed in Sections 2.2 and 2.3.2, the Beta model includes transients and integrates a nested PID control loop. Additionally, the Pelton wheel's inertia and damping are also considered together with power loss calculations in the form of windage and friction. Figure 10 presents an example comparison of Run 3 between the typical results yielded from each numerical model. As may be observed, the plots all follow the same trends, where a minimal difference in simulation time may be noted. Similar to the comparison between the numerical models in Section 3.1, the time difference results from the transients experienced in the Beta model, where an overshoot or undershoot within the control systems impacts the time required to fully discharge the PCS. Figure 10 presents the overall efficiency of both models, where a key observation is that while the efficiency is constant across the entire simulation for the Alpha model, a slight variation in efficiency occurs in the Beta model due to the inclusion of the Friction and Windage power losses, since these are directly proportional to the Pelton Wheel speed and the cube of Pelton Wheel speed, as shown in Equations (15) and (16), respectively.  
Table 9 presents the results for the RMSE\({}_{N}\) across the five considered runs for the main parameters of the numerical models. While all error results are reasonably low (<6%) and show the codes' comparability and reliability, several parameters show a trend. The Input Power, Flow Rate, and Nozzle Position all increase with increased constant power input, while the PCS Pressure decreases with increased constant power input. The reason for a PCS Pressure decrease may be due to reduced overshoot and settling time compared to the other runs, since operating at 5 MW is the rated power of the Pelton turbine considered. The increase in error across the former three variables, however, results from their correlation through the nested PID controller, whereby a higher power requirement seems to increase error within the Beta model, while the Alpha model continues to provide a steady-state solution across all runs. Since the Beta discharging model does not include electrical parameters, the computational time between both models is negligible, only taking a maximum of 45 s to complete a full discharging cycle.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Discharging Runs**} & \multirow{2}{*}{**Input Power**} & \multicolumn{4}{c}{**Normalised RMSE (\%)**} \\  & & **Pelton Speed** & **Flow Rate** & **PCS Pressure** & **Gen. Torque** & **Nozzle Position** \\ \hline Run 1 & 3.10 & 4.52 & 2.83 & 4.27 & 4.39 & 1.51 \\ Run 2 & 3.45 & 2.78 & 3.80 & 2.67 & 3.50 & 2.26 \\ Run 3 & 3.76 & 3.53 & 4.54 & 2.13 & 3.69 & 2.84 \\ Run 4 & 3.96 & 3.98 & 5.15 & 1.87 & 4.08 & 3.38 \\ Run 5 & 4.57 & 4.46 & 5.66 & 1.72 & 4.51 & 3.91 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Summary of all the Normalised Root Mean Square Error (RMSE\({}_{N}\)) results comparing all the Alpha and Beta model for discharging.