## 4 Case Study
### Parameter Attack Scenario II
A combined parameter error metric is presented to illustrate the total parameter error after correction, while \(e^{p}=\left\|\frac{\left[g_{\text{true}};b_{\text{true}};b_{\text{true}}^{ \text{th}}\right]-\left[g_{\text{true}}(base);b_{\text{true}}(base);b_{\text {true}}^{\text{th}}(base)\right]}{\left[g_{\text{true}}(base);b_{\text{true }}(base)\right];b_{\text{true}}^{\text{th}}(base)}\right\|_{2}\cdot e^{p}\) represents the weighted norm of the sum of all parameter errors. A 100 Monte-Carlo simulation is performed and average value is presented. Figure 4 shows a comparison between state-of-the-art solution presented in (7) and proposed model (18). One can see in Figure 5 that the error increases with noise level when using state-of-the-art solution model. However, the proposed model under different noise level provides error below 0.07. To further illustrate the robustness of proposed solution under different noise level, comparison result is presented in Figure 6. In Figure 6, the highest error \(e^{p}\) of line 47-69 reaches 0.5 after system convergence when the noise level increase to 1 standard deviation using state-of-the-art solution (7), while using presented model (18) all values of \(e^{p}\), under different noise level, are lower than 0.06. Figure 4: Correction for line 47–69 using presented model (18). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Processing Measurement Correction**} \\ \hline \hline
**Measurement** & **Database** & **Erroneous** &
\begin{tabular}{c} **Correction Using CNE** \\ **(Approximation Error) [30]** \\ \end{tabular} \\ \hline \(Q_{\text{31-17}}\) & \(-0.1754\) & \(-0.1643\) & \(-0.1761\) (\(0.399\%\)) \\ \hline \end{tabular}
\end{table} 
Table 6: Corrected measurement using the CNE. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{4}{c}{**Processing Measurement Cyber-Attack Step 1**} \\ \hline \hline
**Parameter** & **Database** & **Erroneous** & \
\begin{tabular}{c} **Presented Correction** \\ **(Approximation Error)** \\ \end{tabular} &
\begin{tabular}{c} **State-of-the-art Correction** \\ **(Approximation Error) [31]** \\ \end{tabular} \\ \hline \(\delta_{\text{47-69}}\) & \(1.0012\) & \(1.1314\) & \(1.0062\) (\(0.499\%\)) & \(1.0088\) (\(0.759\%\)) \\ \(b_{\text{47-69}}^{\text{49}}\) & \(-3.2955\) & \(-3.0648\) & \(-3.2876\) (\(0.241\%\)) & \(-3.2826\) (\(0.361\%\)) \\ \(b_{\text{47-69}}^{\text{49}}\) & \(0.0355\) & \(0.0383\) & \(0.035532\)(\(0.092\%\)) & \(0.0357\)(\(0.563\%\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Corrected parameters using the updated parameter correction Jacobian matrix \(\tau_{up}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Processing Measurement Cyber-Attack Step 1**} \\ \hline \hline
**Parameter** & **Database** & **Erroneous** &
\begin{tabular}{c} **Correction Using CNE** \\ **(Approximation Error) [30]** \\ \end{tabular} \\ \hline \(Q_{\text{31-17}}\) & \(-0.1754\) & \(-0.1643\) & \(-0.1761\) (\(0.399\%\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Processing Cyber-attacks. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{4}{c}{**Processing Measurement Cyber-Attack Step 1**} \\ \hline \hline
**Parameter** & **Database** & **Erroneous** &
\begin{tabular}{c} **Presented Correction** \\ **(Approximation Error)** [31]** \\ \end{tabular} \\ \hline \(\delta_{\text{47-69}}\) & \(1.0012\) & \(1.1314\) & \(1.0062\) (\(0.499\%\)) & \(1.0088\) (\(0.759\%\)) \\ \(b_{\text{47-69}}^{\text{49}}\) & \(-3.2955\) & \(-3.0648\) & \(-3.2876\) (\(0.241\%\)) & \(-3.2826\) (\(0.361\%\)) \\ \(b_{\text{47-69}}^{\text{49}}\) & \(0.0355\) & \(0.0383\) & \(0.035532\)(\(0.092\%\)) & \(0.0357\)(\(0.563\%\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Processing Cyber-attacks. Figure 5: Correction for line 23–32. Figure 6: Correction for line 47–69.