## 3 Results
### Voltage Sag Matrix
#### 3.1.4 Single-Phase Fault  
The last type of asymmetrical fault--the single-phase short-circuit--causes the lightest damage to the grid. For this reason, due to the shortest propagation path, fault detection is the most difficult. Voltage swells are characteristic for this type of fault, similar to the case of a two-phase-to-ground short-circuit. The fault occurs between phase A and the ground. Voltage sag matrixes are given in Tables 11 and 12. The single-phase fault pattern depends on neutral mode and electrical distance from the power source (Table 13). Voltage swells are not observed when the short-circuit occurs at the generator's busbar (Figure 50). At the fault node, the lowest voltage swell magnitudes are typical for an HV grid (if compared with either an MV or an LV). A typical fault node pattern in distribution grids (10 kV) is given in Figure 51: contrary to the HV grid, the fault does not affect phase-to-phase voltage when it occurs in either an MV or an LV grid (except Scenario No. 5). Despite being the most common fault in overhead power lines, the operator will almost always avoid legal responsibility. For a three-wire system, EN 50160:2010 requires only phase-to-phase voltage to be taken into consideration; however, it is obvious that phase-to-ground events are also harmful for grid equipment (in particular voltage swells for insulation). On the other hand, single-phase faults, which occur in an MV grid (Scenarios Nos. 5-7), do not interrupt end-users on the LV side, and for this reason, this is the longest-lasting fault. The disconnection of a short-circuited line along with other required commutations is initiated only after the discovery of the fault location by electricians (e.g., after a few hours). \begin{table}
\begin{tabular}{c c  
Single-phase faults in the TSO grid are more dangerous. In the case of Scenarios No. 1, No. 2, and No. 4, when the fault occurs in an HV grid, downstream penetration into an LV grid is stronger than in the case of Scenarios Nos. 5-7 (see Tables 11 and 12). For example, a single-phase fault at the primary winding of the 35/10 kV transformer (Scenario No. 6) does not propagate even to the 10 kV grid (Figure 52), but the fault at the 110 kV busbar (Scenario No. 4), given in Figure 53, reaches both sides--the MV grid (Figure 54) and consequently the LV grid. On the other hand, contrary to all previous types of faults, the propagation path mitigates the consequences by itself--many residual voltage gains (in particular conversion from voltage interruption to sag) can be seen in Table 12. In the case of Scenario No. 4, transition from M4 to M5 can be given as an example: the degree of phase-to-ground voltage unbalance at M5, which is caused by the fault at M4 (Figure 53b), is sufficiently enhanced (Figure 54b). Figure 51: Scenario No. 3, single-phase fault. (**a**) Phase-to-phase voltage of M2a; (**b**) Phase-to-ground voltage of M2a. Figure 50: Scenario No. 1, single-phase fault. (**a**) Phase-to-phase voltage of M2; (**b**) Phase-to-ground-voltage of M2. Upstream propagation of single-phase faults is considerably weaker. Contrary to the previous cases, the single-phase sag usually does not propagate upwards through a transformer, even if it occurs at the secondary winding. In Scenarios No. 8 and No. 9, the faults (Figure 55) not only do not propagate to the neighboring 0.4 kV line, but also do not cause any voltage unbalance at M6 and consequently M7a (Figure 56). In Scenario No. 4, despite the fact that the unbalance is decreasing upwards (across the 110 kV line), it is observed at the primary winding of the autotransformer (Figures 57 and 58). The propagation of symmetrical components from M4 to M2 is given in Figure 59: the negative sequence phase-to-phase unbalance at M2 is 4.9%, the phase-to-ground is 2.5%. These values (especially for the 330 kV grid) are high, because the limit on negative sequence unbalance in EN 50160:2010 (which is applicable up to 110 kV) is 2-3%. Figure 52: Scenario No. 6, single-phase fault.