**3. Experimental Results**

### *3.1. Probabilistic Characteristics of the Pestrike Gap*

The prestrike phenomenon in the vacuum circuit breakers can be characterized by prestrike gap *d*, which is the instantaneous vacuum gap length between the contacts of the vacuum interrupters. It is measured at the occurrence of the first prestrike during a current-making operation. In this work, a probabilistic analysis of the prestrike gap under different applied voltage levels had been carried out. The distributions of the prestrike gaps in the test groups under each voltage level were calculated. Let *f*(*d*) be the density function (DF) of the prestrike gap distributions, which indicates the chance of the prestrike between the contacts in VIs at a vacuum gap length *d*. The corresponding complementary cumulative distribution function (CCDF) can be then calculated by:

$$F(d) = \int\_{d}^{\infty} f(u) du\tag{1}$$

which gives the probability of the prestrike between the contacts in VIs when the vacuum gap length is no less than *d*. Figures 4–6 show the calculated CCDF under applied voltage *U*s from the experimentally measured prestrike gaps in *Test* 1, *Test* 2 and *Test* 3, respectively. In *Test* 3, the total prestrike gap as well as the prestrike gaps in VI\_A (high voltage side) and in VI\_B (low voltage side) were all given.

**Figure 4.** The complementary cumulative distribution function of prestrike gaps in *Test* 1.

For the characterization of the CCDF, the Weibull distribution is used and the CCDF is then given by:

$$F(d) = \exp\left(-\left(\frac{d}{\eta}\right)^{\theta}\right) \tag{2}$$

where η > 0 is the shape parameter of the Weibull distribution [24] and β > 0 is the scale parameter of the Weibull distribution or the characteristic value of the prestrike gap. Both η and β are obtained by fitting the experimental data, as shown in Table 2.

**Figure 5.** The complementary cumulative distribution function of prestrike gaps in *Test* 2.

**Figure 6.** *Cont.*

**Figure 6.** The complementary cumulative distribution function of prestrike gaps in *Test* 3 with VI\_A and VI\_B in series: complementary cumulative distribution function (CCDF) of prestrike gaps in VI\_A (**a**), in VI\_B (**b**), and CCDF of the total prestrike gaps (**c**).


**Table 2.** Values of the Weibull parameters.

The fitted curves with the Weibull distribution given in (2) with respect to *Test* 1, *Test* 2 and *Test* 3 are shown in Figures 4–6, respectively, where a good agreemen<sup>t</sup> between the fitted curves and the experimental data can be observed. Besides the evaluation of the fit from the figures graphically, R-square is used in order to evaluate the goodness of the fit numerically, which is the square of the correlation between the predicted prestrike gaps and the experimentally measured values. R-square can take on any value between zero and one, where a value closer to one indicates a better performance of the fitted model [24–26]. The calculated values of the R-square parameter are presented in the last column of Table 2. From Table 2 it can be noted that the values of R-square are no smaller than 95% in all tests, which indicates that the prestrike gap has a Weibull distribution with parameters η and β

and that the model with the given parameters fits the prestrike gap distributions well. CCDF density function *f*(*d*) is given by

$$f(d|\eta,\beta) = \frac{\beta}{\eta} \left(\frac{d}{\eta}\right)^{\beta - 1} \exp\left(-\left(\frac{d}{\eta}\right)^{\beta}\right) \tag{3}$$

### *3.2. Scatters in the Prestrike Gap*

The scatter in the prestrike gap is of significant importance for investigating the prestrike phenomenon in vacuum interrupters. Moreover, for the phase-controlled switching application with vacuum circuit breakers, a better knowledge of the scatter in the prestrike gap could help to improve control accuracy. The scatter in the prestrike gap can be characterized by the standard deviation of the experimental measurement of the prestrike gaps, which is given by:

$$
\sigma = \sqrt{\frac{1}{N} \sum\_{i=1}^{N} \left( d\_i - \overline{d} \right)^2} \tag{4}
$$

where *N* is the number of each experimental data set; *d* is the average value of each set of the experimental data, and *di* is the prestrike gap measured during the *i*th making operation. The scatters in the prestrike gaps in all the test groups under varied applied voltages are given in Table 3. In the last column, σ indicates the scatter, where a lager value of σ indicates more scatter in the experimental data of the prestrike gaps.


**Table 3.** Prestrike gaps of 10%, 50% and 90% and scatters in prestrike gaps.

To study the probabilistic characteristics of prestrike gaps, the 10% prestrike gap *d*10, 50% prestrike gap *d*50, and 90% prestrike gap *d*90, i.e., the prestrike gap at which the value of CCDF is 10%, 50%, and 90%, respectively, are commonly used [7,16,25,26]. In this case, the influence of the approximation error between the fitted CCDF and the measured prestrike gaps that are higher than *d*90 or lower than *d*10 on the probabilistic characteristics of prestrike gaps can be neglected. The values of *d*10, *d*50, and *d*90 can be calculated by (2), which are shown in Table 3. It can be noted that with the increment of applied voltage *U*s, the *d*10, *d*50 and *d*90 are all increasing significantly. Moreover, the 50% prestrike gap *d*50 can be used as one of the most important parameters since it gives the prestrike gap at which there is a

50% chance for the vacuum gap to achieve breakdown in the VCB, i.e., *d*50 acts as the mean value of the prestrike gap in the sense of the Weibull distribution. The relationships between *d*50 and applied voltage *U*s in *Test* 1, *Test* 2, and *Test* 3 are shown in Figure 7, Figure 8, and Figure 9, respectively, where the error bar at each data point shows the corresponding scatter in the prestrike gaps. It can be noted that the value of *d*50 is approximately proportional to applied voltage *U*s.

**Figure 7.** The relationships between *d*50 and applied voltage *U*s in *Test* 1 with VI\_A.

**Figure 8.** The relationships between *d*50 and applied voltage *U*s in *Test* 2 with VI\_B.

(**c**) **Figure 9.** The relationships between *d*50 and applied voltage *U*s in *Test* 3: VI\_A (**a**), VI\_B (**b**), and the total prestrike gaps (**c**).
