## 4 Application Cases  
### Testing System and Modeling Methodology  
A three generator-nine bar system is used, widely found in the international literature, as a testing system to apply the proposed model. The topology of the system is shown in Figure 4, and the information of the elements is included in Appendices A and B. For the permanent regime symmetric operation condition, Table 1 shows the considered load distribution. In order to implement the unbalanced model described in the sections above, two load conditions are presented: the nominal values of the loads and the case previous to instability in the classical model. The two scenarios analyzed are presented below:  
* Scenario 1: balanced nominal charges;
* Scenario 2: charge increase by 45%. The model proposed in this paper allows analyzing an unbalanced system in its topology, such as unbalanced lines or operation points where the loads do not have the consumed power distributed in a symmetric form in its three phases. As a first approach to the unbalanced study, unbalanced load is considered, modeled as constant power and distributed in a proportional form among its three phases  
The three-phase unbalanced model with initial balanced conditions is used as a control case because it can directly be compared to the classical analysis. This case is not real since distribution systems are naturally unbalanced. Two more asymmetrical distributions of loads were studied, named Forms A and B. These are used to analyze the effect of different kinds of unbalance on the system stability. The way in which the asymmetrical load is distributed in both forms is shown in Tables 2 and 3. \begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Loads**} & \multicolumn{3}{c}{**Percentage**} \\ \cline{2-4}  & **Phase A** & **Phase B** & **Phase C** \\ \hline Load A & 33.3\% & 33.3\% & 33.3\% \\ \hline Load B & 33.3\% & 33.3\% & 33.3\% \\ \hline Load C & 33.3\% & 33.3\% & 33.3\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Distribution by the phase of the balanced loads. Figure 4: Nine-bus testing system. Table 4 displays all the cases considered in order to evaluate the model proposed in this paper. The methodology for simulation used in this work (see Figure 5) starts first (1) with gathering the information of the system, especially focused on the parameters used in the unbalanced representation. The second step (2) is to create the model of the system in the software DigSilent, including all the unbalanced parameters. The third step (3) is to modify the loads of the system according to the study cases. In the fourth step (4), the load flow is calculated for every single study case, and the results are obtained. The fifth step (5) is to use the system parameters, load flow results, and initial conditions as the inputs for a Matlab script that calculates the resulting matrices \(A,B,C\) and \(D\) for each study case. In Step (6), the matrices are used for the eigenvalue analysis of the system, and the damping ratios for these eigenvalues are also calculated. Finally, (7) the time domain simulation is carried out in DigSilent to contrast the results. The main criteria for deciding about the stability of the eigenvalues is based on eigenvalue analysis. If an eigenvalue has a real part that is positive, it is considered unstable [18]. The system must not have any unstable eigenvalue to be classified as stable. To check the validity of the results of the proposed model, a time-based simulation of the system is constructed. The simulation includes a small perturbation and contrasts the results. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Study Cases** & **Scenario 1** & **Scenario 2** \\ \hline Balanced & Case 1 & Case 5 \\ model & Case 2 & Case 6 \\ \hline Unbalanced model with & Case 3 & Case 7 \\ unbalanced initial conditions, Form A & Case 4 & Case 8 \\ \hline Unbalanced model with & Case 4 & Case 8 \\ unbalanced initial conditions, Form B & & \\ \hline \hline \end{tabular}
\end{table}