*3.2. Current Limitation*

In this Section, the presented analysis aims to evaluate the transient performance of the SCs in contrast with a conventional copper cable installed at the same power system. For this reason, emphasis has been given on the calculation of the current-limitation capability as a percentage of the prospective fault current flowing through a conventional copper cable, during the quenching process. In particular, a 3-Phase-to-ground fault with fault resistance of *Rf* = 0.01 Ω was applied at the 50% of the SCs length. The same fault has been repeated for the case of conventional copper cable. The fault currents captured by the SCs model during the simulations have been compared with the prospective fault currents through the conventional copper cable, highlighting the merits arising by utilizing superconductors. Figure 13 demonstrates the RMS value of the fault currents at Bus 11 during a 6-cycle 3-Phase-to-ground fault for both cases.

**Figure 13.** RMS values of the fault currents at Bus 11 during 3-Phase-to-ground solid fault at 50% of the proposed SCs and a conventional copper cable: (**a**) Phase A, (**b**) Phase B, (**c**) Phase C.

Similarly, to the previous Section, the fault is initiated at *t* = 5.06 s and cleared after 120 ms. When the fault occurs at *t* = 5.06 s, the RMS value of the current for the SCs is slightly higher compared to that of the conventional cable, as at the initial quenching state (stage 2) the resistance of the HTS tapes has not reached high values yet. It is well-established that the short-circuit magnitude is determined by the X/R ratio of the circuit. Therefore, it can be seen that the RMS values of the fault currents

start to decrease at the time instant of *t* = 5.065 s, due to the high resistance and the significant temperature increase.

To quantify the fault current limitation by adding the fault current limiting function, a current limitation percentage of the prospective current through a conventional cable has been introduced based on Equation (30). Particularly, for the case of the SCs the RMS values of the limited fault currents during the whole quenching process (stage 2 and stage 3) have been calculated and compared with the prospective current values. Figure 14 shows the current limitation percentage per phase, verifying and supporting the practical feasibility of the proposed cable design,

$$I\_{\text{current}-limitation} \left( \% \right) = \frac{I\_{\text{conv}} - I\_{\text{SC}}}{I\_{\text{conv}}} \cdot \left( 100 \% \right) \tag{30}$$

where *Iconv* is the RMS value of the fault current flowing through the conventional copper cable and *ISC* is the fault current flowing through the SCs under the same type of fault. The current limitation presents a slight difference among phases due to the difference in phase angle of each phase at the fault instant.

**Figure 14.** Current limitation percentage (%) of the SCs for phases A, B, and C compared to a conventional copper cable during a 3-Phase-to-ground fault at 50% of cable's length.

It is evident that the installation of SCs can lead to fault current reduction up to 62.5% of the prospective current flowing through a conventional copper cable, considering the same 3-Phase-to-ground fault.

#### *3.3. Simulation Analysis of Fault Resistance E*ff*ect on the Quenching Process*

In order to achieve the maximum benefit of the designed cable, its performance under a wide range of power system conditions should be comprehensively evaluated. In the available technical literature, several studies [45–50] have investigated the impact of the fault resistance *Rf* on the superconducting current limiters. However, there are no studies available assessing the impact of the fault resistance on the SCs and the fault current limitation that it provides. Therefore, in this Section the quenching process of the SCs is analyzed in accordance with the gradual increase in the fault resistance value. Simulation studies, which include 3-Phase-to-ground faults applied at the 50% of SCs length (considering different values of *Rf*), were conducted to study the relationship between *Rf* and the quenching process. Figures 15–20 show the corresponding waveforms of the quenching stage, the fault current signatures among the layers, the resistance and the temperature of the cable for *Rf* 1 = 1 Ω, *Rf* 2 = 5 Ω and *Rf* 3 = 10 Ω, respectively.

**Figure 15.** Stages of quenching process for phase A, B, and C for 3-Phase-to-ground fault at 50% of cable's length with (**a**) *Rf* 1 = 1 Ω, (**b**) *Rf* 2 = 5 Ω, (**c**) *Rf* 3 = 10 Ω.

Based on the results depicted in Figure 16, when fault resistance is *Rf* 1 = 1 Ω, the HTS tapes quench at the first half fault cycle and enter normal state (stage 3), as it can be seen in Figure 15a. The resistance and the temperature reach their maximum values at *t* = 5.065 *s*, as shown in Figures 19a and 20a, respectively. Therefore, the current starts to flow through the stabilizer layer at *t* = 5.065 s, 5 ms after the fault occurs. For the case of *Rf* 2 = 5 Ω, HTS tapes quench after one fault cycle (at *t* = 5.082 s) and it is noticeable by Figure 15b that stage 2 lasts for a slightly longer period (few ms). Considering a fault resistance of *Rf* 1 = 1 Ω, SCs operate within stage 2 for 5.5 ms, while, for *Rf* 2 = 5 Ω, stage 2 lasts for 18 ms. Furthermore, for fault resistance *Rf* 2 = 5 Ω, the first fault current peaks depicted in Figure 18c,e,g are lower compared to the fault current peaks extracted during the fault with *Rf* 1 = 1 Ω, as a larger value of fault resistance results in lower fault currents. Regarding the case of *Rf* 3 = 10 Ω, the HTS tapes of the faulted phases quench, reaching only stage 2, without entering into normal state. Therefore, the maximum value of the SCs equivalent resistance is low, *Req* = 0.058 Ω and the fault currents flow through the HTS layers. This behaviour indicates that the increase in the fault resistance value affects the quenching degree and consequently the current sharing between the HTS and stabilizer layers. Particularly, as it has already been analyzed (also reported in [28]), temperature increase plays a key role in the resulting value of the equivalent resistance and the quenching degree, which in turn is determined by the generated resistive heat, the magnitude, and the duration of the fault current. By observing, Figure 20a, it is obvious that for a 3-Phase-to-ground fault with *Rf* 1 = 1 Ω, the temperature of the superconductor exceeds the critical value *TC*, reaching the maximum value of 250 K. When a 3-Phase-to-ground fault with *Rf* 2 = 5 Ω occurs, the temperature exceeds the critical value *TC* = 92 K, but it is noticeable from Figure 20b, that the temperature reaches the maximum value of 250 K with a delay, which affects the quenching process. In the last case of *Rf* 3 = 10 Ω, the boundary condition *If ault* > *IC* of quenching has been met. Although the temperature does not reach the critical value *TC*, resulting in "incomplete quenching". The resistance of the HTS tapes reach low values, affecting the value of the equivalent resistance and the current distribution among the layers and resulting in small percentage of fault current limitation. The fault current flows mainly through the HTS layers.

**Figure 16.** Fault current signatures for 3-Phase-to-ground fault at 50% of cable's length: with *Rf* 1 = 1 Ω (**a**) phase currents at Bus 11, (**b**) phase voltages at Bus 11, (**c**) current in HTS layer of phase A, (**d**) current in copper layer of phase A, (**e**) current in HTS layer of phase B, (**f**) current in copper layer of phase B, (**g**) current in HTS layer of phase C, (**h**) current in copper slayer of phase C.

The results revealed that the fault resistance has a considerable impact on the SCs performance for the same type of fault, considering the same fault location. For instance, further increase in the fault resistance can lead to much lower fault currents, even below the critical current *IC*, preventing SCs from quenching. This has been confirmed by Figure 18, where, during a 3-Phase-to-ground fault with *Rf* 3 = 10 Ω, the first peak of the fault current is below the critical current *IC*, and therefore there is no quenching or fault current sharing between the two layers. Consequently, low values of fault resistance result in higher fault current (with respect to the critical current *IC*), which lead to SCs quenching during the first half cycle, and therefore to greater fault current limitation capability (Figure 16). High fault resistance affects the quenching degree and jeopardizes the fault current limiting capability of the cable. This can be explained by the reduced allocation of fault current within different layers during current limitation mode, as fault current is predominately limited by the fault resistance value.

**Figure 17.** Fault current signatures for 3-Phase-to-ground fault at 50% of cable's length: with *Rf* 2 = 5 Ω (**a**) phase currents at Bus 11, (**b**) phase voltages at Bus 11, (**c**) current in HTS layer of phase A, (**d**) current in copper layer of phase A, (**e**) current in HTS layer of phase B, (**f**) current in copper layer of phase B, (**g**) current in HTS layer of phase C, (**h**) current in copper slayer of phase C.

**Figure 18.** Fault current signatures for 3-Phase-to-ground fault at 50% of cable's length: with *Rf* 3 = 10 Ω (**a**) phase currents at Bus 11, (**b**) phase voltages at Bus 11, (**c**) current in HTS layer of phase A, (**d**) current in copper layer of phase A, (**e**) current in HTS layer of phase B, (**f**) current in copper layer of phase B, (**g**) current in HTS layer of phase C, (**h**) current in copper slayer of phase C.

**Figure 19.** Equivalent resistance for phases A, B, and C for 3-Phase-to-ground fault at 50% of cable's length, with: (**a**) *Rf* 1 = 1 Ω, (**b**) *Rf* 2 = 5 Ω, (**c**) *Rf* 3 = 10 Ω.

**Figure 20.** Temperature for phases A, B, and C for 3-Phase-to-ground fault at 50% of cable's length, with: (**a**) *Rf* 1 = 1 Ω (**b**) *Rf* 2 = 5 Ω (**c**) *Rf* 3 = 10 Ω.
