**4. Discussion**

### *4.1. Cathode and Anode Vd*

Figure 14a shows arc boundaries, the σ inside the core (yellow), Electric potential (V) contours (red and blue contours perpendicular to arc and core boundaries), arrows, and contours of *Jnorm* in logarithmic scale (in green). The numbers in black are the V-contour tags in 2.5 V steps between. A zoomed view of conductivity for the 1 mm-thick layer near the cathode is shown at the upper right corner of Figure 14a and in Figure 14b for the upper and lower sides of the moving contact. The σ*max*, *Tmax,* and consequently *Jmax* are evident in this view, and numbers in magenta are the *J*-contour tags.

The arc column gets narrow near the cathode, while it has a bezel shape near the anode. From Figure 14c,d, it is obvious that the arc core (yellow section) has a sharper tip near the cathode than the anode.

**Figure 14.** (**<sup>a</sup>**,**b**) Arc boundaries, V contours, and *Jnorm* in Log. scale and the zoomed view of the near electrode areas at (**c**) Fixed cathode, (**d**) Moving anode, (**e**) Diagram of <sup>σ</sup>*sheath* (*x*, *<sup>t</sup>*)*<sup>t</sup>*=3.75 *ms* from cathode to the core gradient line, (**f**) Diagram of the <sup>σ</sup>*sheath* (*y*, *<sup>t</sup>*)*<sup>t</sup>*=3.75 *ms* from anode to the σ*max*, and (**g**) Simulated arc and core resistance and piecewise linear equivalent circuit of *Rarc*|*<sup>t</sup>*=3.75 *ms* for 200 *Apeak*; (An, Ca, and MC in figures refer to the anode, cathode, and the moving contact).

The zoomed view of these tips is shown in Figure 14e,f, showing the sheathes. The diagrams of the σ*sheath* (*y*, *<sup>t</sup>*)*<sup>t</sup>*=3.75 *ms* from cathode to the core centerline and σ*sheath* (*x*, *<sup>t</sup>*)*<sup>t</sup>*=3.75 *ms* from anode to the σ*max* are shown in this figure. The equipotential lines between the core and the cathode wall are very dense, which means a rather uniform field and significant linear *Vd* in the cathode sheath; this finding is shown in the conductivity curve of Figure 14e and is in line with other studies [86]. The sheath effect increases the arc resistance (*Rarc*) in front of the electrodes, as it is recognized from the broken and deformed σ diagram of the cathode sheath in Figure 14e.

It shows a change in σ (blue) near the sheath layer and the constricted V contours (brown) near the electrodes. The *J* at the cathode depends on cathode material, while it is independent of the current [87]. The *Jmax* near the moving cathode was simulated to 3.5 × 10<sup>8</sup> A/m2. The relation between *J* and electron temperature is presented in [75], and the sheath thickness is estimated based on electron temperature [88].

Our simulated sheath thickness is about 130 μm at the cathode, which complies with the mentioned research. At 3.75 ms, the simulation shows about 16 V drops along the 130 μm cathode sheath (123 mV/μm) and about 2–3 V drops in 55 μm (45 mV/μm) between the core boundary and anode. Simulated anodic and cathodic voltages are similar to others [89].

The thickness of electrode sheaths for arcs at atmospheric pressure is less than 0.7 mm in total. The sheath *Vd* and the current divided between core/column imply a piecewise linear equivalent circuit of *Rarc*|*<sup>t</sup>*=3.75 *ms* for 200 *Apeak*, shown in Figure 14g with the simulated arc and core resistances.

### *4.2. E*ff*ects of Fast Elongation on the Arc Voltage*

Figure 15a compares the measured *VMeas* (dash-dotted in green) with the simulated *Vsimulated* for 200 A, considering thermionic emission modeling. It is seen that both *VMeas* and *Vsimulated* jump from

zero to 24–33 V, then step up to 40–45 V and then increase to about 75–80 V with a linear increment rate of *m* = 10 kV/s.

**Figure 15.** Measured and simulated arc voltages for 200 *Apeak* arcs in (**a**) Prototype FS with uc = 9 m/s and (**b**) Fixed contact distance of 30 mm initiated by exploding wire at the same current cycle.

The shape of *Varc* complies with other researches [90], but the first jump of the measured voltage is higher because of higher simulated temperature in laminar modeling. The jump at the end of *Varc* is missed in measurement and simulation with a sampling rate of 50 μs, but it is detected with a sampling rate of 2 μs (in purple) of simulated output and is apparent in other experiments [37,91,92]. The slope of the measured and simulated voltage falls close to zero with the arc mode change. Figure 15b shows the simulated *Vsim* for 200 *Apeak* arc between a fixed contact distance of 30 mm (stationary arc) initiated by exploding wire at the same current cycle [93]. *Varc* is higher in the elongated state.

Figure 16a compares the measured (Dashed Line) and simulated (Solid Line) *Varc* for 200 A arc. The higher the *uc*, the faster the elongation and the higher the voltage for the similar arc currents. By increasing *uc* to 22.5 m/s, *Varc* reaches six times its value in the fixed contact distance at 200 A. Figure 16b shows the influence of elongating speed on *Varc* for 200 A arc. The *m* [kV/s] is related to the *uc*, thus it increases in higher *uc*. The *Varc* is an essential measure for a successful commutation.

**Figure 16.** (**a**) *VMeas* and *Vsimulated* for *uc* = 7, 9, 13.5, and 22.5 m/s, (**b**) *Vsimulated* in first 1.5 ms for *uc* = 5–80 m/s, for 200 *Apeak* arcs.

The failure of fast switches results in current commutation failure, and consequently failure in HVDC breakers and FCLs. The relation between the contact velocity and current is a matter of interruption performance and needs the simulation of transient recovery voltage (TRV) slopes. It is already studied through this validated model and is shown in Figure 6 of [94]. The relationship between the contact velocity and Thomson coil current or the variation in the actuator parameter was already published in [13]. Arc behavior in the failure of FS due to high currents, or insufficient *uc*

through data-driven decision making (DDDM), is the target of future studies. DDDM is based on actual data rather than intuition or observation alone. The hard truth is that simulation output alone is not enough. So, making organizational decisions for failure Prediction can be utilized through di fferent modeling techniques [95] like Gaussian Process Regression Models [96] in the failure study.

### *4.3. E*ff*ects of Fast Elongation on the Convective Cooling and Vd*/*mm*

Figure 17a shows the maximum convection flux for 200 A arc at *uc* of 5, 7, 9, 13.5, and 27 m/s. It is also evident that convective cooling is related to *uc*. Besides, the maximum of the convection is dependent on the maximum of the *U (Umax)*, and the *Umax* is dependent on the *uc* again [97]. Figure 17b shows the voltage per length for the 200A arc for *uc* of 5, 7, 9, 13.5, and 27 m/s within the first 3 ms. Dividing 50 V reported simulated voltages [98] to 8 mm distance between parallel rails [99] and it is already known that elongation increases the cooling and the *Varc, total*, but it decreases the arc cross-section. All these changes shall increase the resistance per unit length but, according to Figure 14g, the resistance remains almost fixed and therefore the *Vd* per unit length inside the arc is decreased.

**Figure 17.** (**a**) Max. convection flux for 200 A arc at *uc* = 5, 7, 9, 13.5, 27 m/s. Voltage per length of arc for (**b**) the first 3 ms for 200 A arcs and *uc* of 5, 7, 9, 13.5 and 27 m/s, and (**c**) the last 2 ms for 200 A arcs and *uc* = 5, 7, 9 m/s.

The physical reason behind this observation is that the elongation makes the arc narrower, so in a fixed length, the plasma volume is reduced, and therefore, lower energy is needed to keep the plasma in the previous condition considering fixed energy per volume for the plasma. As the current is supplied through the current source, the voltage per length is reduced. Except for the first 1 ms in the rest of this period, the voltage per length of arc remains between 2.5–4.5 V/m, which is in line with other measurements [37].

Figure 17c shows the voltage per length of the arc in the last 2 ms before CZ for 200 A arcs at *uc* of 5, 7, and 9 m/s. In this period, the voltage per length of arc remains between 1.7–2.3 V/m. The physical reason is that convective cooling is reduced, as is shown in Figure 16b. Therefore, lower input energy is needed to keep the plasma in the previous condition as the loss is reduced. Thus, the voltage per length is reduced again.
