## 6 Simulation Results
### First Stage: One Day Evaluation
#### 6.1.7 Proposed Predictive Dynamic Smoothing (PDS)  
The PDS was evaluated by varying the prediction window size W (5 and 10). However, since this method relies on the accuracy of the predictions, its robustness against prediction inaccuracies was also evaluated. For this purpose, the assumed prediction was contaminated with Gaussian noise as follows:  
\[\overset{\sim}{\mathrm{P}}_{\mathrm{P}\mathrm{v}}[\mathrm{k}+\mathrm{i}]=\mathrm{ P}_{\mathrm{P}\mathrm{v}}[\mathrm{k}+\mathrm{i}]\bigg{(}1+\frac{\mu(\%)\times\mathrm{rand}}{100 \%}\bigg{)} \tag{25}\]  
Figure 16: Simulation results for the ELES method. (**a**) One-day data results showing the total output power (\(\mathrm{P}_{\mathrm{out}}\)), the ESS output power (\(\mathrm{P}_{\mathrm{ess}}\)), the PRR at the output (\(\mathrm{PRR}_{\mathrm{out}}\)), and the state of charge (SOC). (**b**) Zoomed figures of the corresponding black squares.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Measurement** & \(\mathbf{\alpha}\) = 0.05 & \(\mathbf{\alpha}\) = 0.06 & \(\mathbf{\alpha}\) = 0.07 \\ \hline D(\%) & 0.0072 & 0.0057 & 0.0046 \\ \hline NC(80\%) & 0.7206 & 0.5709 & 0.4599 \\ \hline Violations (mins) & 0 & 0 & 1 \\ \hline SOC (end of the day) -SOC(ref) & \(-31.949\) & \(-22.593\) & \(-20.975\) \\ \hline \(\sigma^{2}(\mathrm{PRR}_{\mathrm{out}})\) & 3.9163 & 5.3242 & 6.8792 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Summary of the simulation results for the ELES method.  
where rand is a random scalar with standard normal distribution and \(\mu(\%)\) is the variable noise parameter in the percent of the contracted output power. This means that a \(\mu=20\%\) incurred in an estimated power estimation error of approximately 2 MW.  
Simulation results for the PDS method with W = 5 are shown in Figure 17. It is noted that, compared to other methods, the SOC variations got reduced and there were no PRR violations. It is also noted that the SOC variations are critically affected by the \(\mu\) parameter. A summary of these results is presented in Table 9. The PDS shows to be robust against an estimation noise of at least 20%. The battery degradation is also small compared to other methods.  
Simulation Results of PRRC-Proposed with W=5  
Figure 17: Simulation results for the proposed PDS method. (**a**) One day data results showing the total output power (\(P_{\text{out}}\)), the ESS output power (\(P_{\text{ess}}\)), the PRR at the output (\(PRR_{\text{out}}\)), and the state of charge (SOC). (**b**) Zoomed figures of the corresponding black squares.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Measurement** & \(\mu\) = 0\% & \(\mu\) = 10\% & \(\mu\) = 20\% \\ \hline D(\%) & 0.0016 & 0.0020 & 0.0026 \\ \hline NC(80\%) & 0.1591 & 0.1990 & 0.2621 \\ \hline Violations (mins) & 0 & 0 & 0 \\ \hline SOC (end of the day) -SOC(ref) & \(-1.5788\) & \(14.4696\) & \(29.9847\) \\ \hline \(\sigma^{2}(PRR_{\text{out}})\) & 14.2955 & 14.1632 & 15.9081 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Summary of the simulation results for the proposed PDS method with W = 5.  
Simulation results for the PDS method with W = 10 are shown in Figure 18. It is noted that this window size significantly reduced the PRR and SOC variations. It is concluded that, as the window size increases, the PDS is more robust against uncertainties in the predictions. It is also noted that the SOC variations are critically affected by the \(\upmu\) parameter. A summary of these results is presented in Table 10. Although none of the tests showed PRR violations and the PRR variance was significantly reduced, a PDS method with a W = 10 showed twice the battery degradation compared with a window size of W = 5. Figure 19. Comparison of the effect of selecting the window size using the PDS method. Figure 19 shows the main effects of selecting the window size.