## 3 Results
### Voltage Sag Matrix
#### 3.1.5 Case Studies  
Some practical examples will be given in order to support the validity and adequacy of the results in real conditions. Siemens PSS/E software tool is used for calculations. The first example is given in Figure 60: three scenarios are created in the 330 kV grid of the Baltic power system operating in the 2017 mode.  
The voltage sag matrix is given in Table 14 and it can clearly be seen that the results are not symmetrical. Latvia's node is the most robust, and Estonia's node is in second place: if a three-phase fault occurs at Jelgava, then the residual voltage at Tartu is 0.84 p.u., meanwhile at Telsiai it is only 0.31 p.u. If the fault occurs at Telsiai, the residual voltage at Jelgava is 0.63; but otherwise, when the fault occurs at Jelgava, the residual voltage at Telsiai is 0.31 p.u. In both cases, when the fault location is in either Latvia or Estonia, the respective voltage sag is not mitigated at Lithuania's 110 kV node in Tryksiai; however, when the fault occurs at Lithuania's node (Telsiai), the residual voltage of Tryksiai is increased by 0.26 p.u. As has been explained before, the result depends on the power-flow direction, the number of power supply lines (or generally speaking on topology) and distances from power plants, also reactive power generated by both reactive power compensation devices and transmission lines, which jointly determine the grid node's voltage stability, i.e., reactive power-voltage characteristic. Particularly in the case of a three-phase fault, downward  
Figure 57: Scenario No. 4, single-phase fault. (**a**  
propagation does not depend on line length, and upward propagation has a logarithmic nature (see Section 3.1.1).  
Figure 59: Scenario No. 4, single-phase fault. (**a**) Symmetrical magnitudes of the phase-to-phase voltage of M4; (**b**) Symmetrical magnitudes of the phase-to-ground voltage of M4; (**c**) symmetrical magnitudes of the phase-to-phase voltage of M3; (**d**) Symmetrical magnitudes of the phase-to-ground voltage of M3; (**e**) Symmetrical magnitudes of the phase-to-phase voltage of M2; (**f**) Symmetrical magnitudes of the phase-to-ground voltage of M2.  
order to avoid both successive voltage sag and the triggering of overcurrent relays. In this particular case, the highest economic losses can be suffered by interrupting the following processes: both atmospheric and reduced pressure fractional distillations (separation of a mixture into its component parts), hydrodesulfurization (sulfur removal by converting it to hydrogen sulfide H\({}_{2}\)S, i.e., by adding hydrogen), and catalytic cracking (conversion of high-molecular weight hydrocarbons to lighter products). In spite of that, other critical systems must also be taken into consideration (particularly for safety reasons), for example, fire-fighting systems, the cooling loop, the nitrogen loop [5].  
Please note that a three-phase electric motor behaves like a generator upon the occurrence of a three-phase short-circuit (i.e., begins to supply the fault node); thus, the given example is slightly different from the previous cases. The process depends particularly on motor type. For example, the generation of conventional synchronous motors is longer than induction motors; hence, a higher residual voltage (and consequently higher probability of successful self-starting) can be expected [58]. If a motor is equipped with a variable-frequency drive, the drive limits short-circuit current and immediately disconnect the motor; thus, interruption of a manufacturing process can be expected [60].