## 4 Case Study
### Parameter Attack Scenario II  
1. A measurement cyber-attack of magnitude 5 \(\sigma\) is added to reactive power flow from bus 31 to bus 17 (\(Q_{31-17}\)). 2. An unbalanced parameter FDI attack is injected to the series and shunt parameter of the line 47-69 (13% on parameter \(g\), \(-7\%\) on parameter \(b\), 8% on parameter \(b^{sh}\)). In this scenario, the attack is detected and the result is shown in Table 3, where the objective function is higher than the C value for this scenario (\(C=\chi^{2}=775.1861\)). For identification step, a descending list of \(CME^{N}\) is built and shown in the same table. The largest \(CME^{N}\) values (absolute values) are associated with reactive power injection \(Q_{47}\), real power injection \(P_{47}\), real power flow \(P_{69-47}\) and reactive power flow \(Q_{69-47}\). This scenario characterizes a parameter cyber-attack on line 47-69 [19]. After identification, the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Processing Measurement Cyber-Attack Step 1**} \\ \hline \multicolumn{4}{c}{\(J\) (x) = 1246.587 \(>\) C = 775.1861 Attack Detected!} \\ \hline \multicolumn{4}{c}{\(CME^{N}\) Descending List} \\ \hline
**Measurement** & \(II\) & \(CME^{N}\) \\ \hline \(Q_{23}\) & 0.0395 & \(-8.4156\) \\ \(Q_{32-23}\) & 3.9464 & 6.9948 \\ \(Q_{23-25}\) & 0.1380 & 5.9432 \\ \(P_{23-22}\) & 0.3557 & 5.5826 \\ \(Q_{24}\) & 0.4774 & \(-4.4724\) \\ \(Q_{32}\) & 3.9464 & 3.9948 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Processing cyber-attacks. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{4}{c}{**Parameter Correction**} \\ \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 \(g_{23-32}\) & 2.2169 & 1.8179 & 2.2285 (0.525\%) & 2.2385 (0.974\%) \\ \(b_{23-32}^{2}\) & \(-8.0635\) & \(-9.0311\) & \(-8.0782\) (0.183\%) & \(-8.0854\) (0.271\%) \\ \(b_{23-32}^{44}\) & 0.0587 & 0.0551 & 0.0586(0.170\%) & 0.0589(0.340\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Corrected Parameters using the updated parameter correction Jacobian matrix \(\tau_{up}\). Figure 3: Correction for line 23–32 using presented model (18). net system parameter are corrected using the presented model in (18). Still, only flow \(P_{69-47}\) and \(Q_{69-47}\) are provided in current measurements configuration. The lack of missing the real power flow measurement \(P_{47-69}\) limits the possibility of performing state of the art model in (7), since incomplete information prevent this model to calculate real power loss mentioned in (7). However, with synthetic measurement \(Q_{47-69}\) and \(P_{47-69}\) provided, one will be able to use presented overdetermined model in (18) to perform parameter correction. Correction converges after 16 iterations, and corrected values and comparable results are presented in Table 4. System net parameters \(g_{47-69}\), \(b_{47-69}\), \(b_{47-69}\) have approximation error \((0.499\%)\), \((0.241\%)\) and \((0.092\%)\) after convergence. After correction, a new state estimation is performed, objective function value \(837.6015\) is obtained which is still higher than threshold C in Table 5. As seen, the only \(CME^{N}\) value (absolute value) above the threshold is the reactive power flow \(Q_{31-17}\). Therefore, the measurement \(Q_{31-17}\) is in error. The correction of measurements as shown in the flowchart is performed using their \(CNE\) values. The corrected measurement is shown in Table 6. After re-running the state estimator, the \(\chi^{2}\) is smaller than C, thus no further FDI attack detected. To further evaluate the robustness of presented model, different measurement noise levels are simulated.