## 3 Results
### Voltage Sag Matrix
#### 3.1.1 Three-Phase Fault  
Three-phase short-circuit is the rarest and the most severe fault, also the simplest to investigate due to the symmetry. In Tables 1-3, residual voltages (p.u.) are given: the monitor number (according to Figure 4) with its pre-fault voltage (according to Figure 5) are given in the first two columns, the fault location (i.e., corresponding monitor number according to Figure 4) with the scenario in the brackets (respectively) is given in the first rows. The voltages of the fault nodes are underlined. Scenario No. 8 does not have an assigned monitor. Results are also given in the figures (starting from Figure 6): voltage in volts is given in the ordinate axis, and time in seconds is given in the abscissa axis. notching ringing, since according to IEEE Std 1159-2019, notching is \"a periodic waveform disturbance caused by normal operation of power electronic devices.\"  
Since electric generators are (an effective) measure for voltage sag mitigation (as mentioned in Section 1), the first additional question arises: what is the dependence between the length of the 330 kV line and the residual voltage of M1. The results are given in Figure 9: the sag threshold (0.9 p.u.) is reached when the distance is approximately equal to 20 km, and the dependence can be approximated with a natural logarithm model (with a very high coefficient of determination). Since a three-phase fault is the most severe fault, it is the easiest fault to detect. The opposite boundary condition is a single-phase fault (see Section 3.1.4): in this case, the threshold is 10 km (Figure 10). The line is modeled as a distributed-parameter line (see Appendix A). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Monitor** & **Pre-Fault** & **1 (No. 1)** & **2 (No. 2)** & **2a (No. 3)** & **4 (No. 4)** & **5c (No. 5)** & **5 (No. 6)** & **6 (No. 7)** & **(No. 8)** & **7 (No. 9)** \\ \hline
**2a** & 1.026 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\
**2b** & 1.026 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\
**3a** & 1.020 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\
**5a** & 0.840 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\
**5b** & 0.825 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\
**5c** & 0.901 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\
**5d** & 0.715 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\
**5e** & 0.772 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\
**7a** & 0.924 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\ \hline \end{tabular}
\end{table} 
Table 2: Three-phase fault voltage sag matrix of the rest of the nodes. Figure 6: Scenario No. 2, three-phase fault. (**a**) Phase-to-phase voltage of M5c; (**b**) Phase-to-ground-voltage of M5c. Figure 8: Scenario No. 2, three-phase fault. (**a**) Phase-to-phase voltage of M2d; (**b**) Phase-to-ground-voltage of M2d. Figure 7: Scenario No. 2, three-phase fault. (**a**) Phase-to-phase voltage of M1; (**b**) Phase-to-ground-voltage of M1. Figure 9: Scenario No. 2, three-phase fault. The dependence between the length of the 330 kV line and the residual voltage of M1. The short-circuit at the autotransformer's tertiary winding (Scenario No. 3) causes a power supply interruption (0.1 p.u.) in all nodes of the left branch, including both the primary and secondary windings of the autotransformer (Figure 11). Analogously to Scenario No. 2, transients are observed in the right branch (Figure 12). At this point, the second additional question arises about the propagation path when the fault location (node) is shifted from the tertiary winding (along the same 10 kV line) by 0.5-1 km (modification of Scenario No. 3). The results are given in Figure 13: when the distance is increased by 0.5 km, the fault does not propagate against the power-flow and does not affect the autotransformer windings. The last fault of the TSO grid occurs at the 110 kV busbar (Scenario No. 4). Similarly, the sag does not propagate against the power-flow (Figure 14), i.e., from M4 to M3.