## 3 Experimental Results and Analysis
### Breakdown Characteristics Analysis  
As discussed in the previous section, the number of streamer branches decreased significantly in nano-liquids, which are accompanied by fewer and shorter secondary streamer branches. Further research is needed to explore the modification mechanism of nano-particles on transformer oil, therefore, we measured the breakdown voltage of transformer oil, TiO\({}_{2}\) nano-liquids and Al\({}_{2}\)O\({}_{3}\) nano-liquids with a series of volume fractions. Based on the analysis of breakdown voltage and combined with the previous empirical research, when the order of magnitudes of the volume fraction nano-liquids reaches 0.1% [42; 43], there would be an obvious modification effect, i.e., the increase in the breakdown strength of the liquid dielectric after nano-modification. Considering the modification effect on microsecond pulse, we chose the typical volume fractions ranging from 0.2% to 0.8% [41]. Twenty repeat breakdown strength measurements were applied for each group.  
There is a uniform distributed electric field between the two spherical electrodes; the breakdown strength \(E\) can be expressed as [42]  
\[E=U/d \tag{1}\]  
where \(U\) is the breakdown voltage, \(d\) is the electrode gap. The Weibull distribution has been proved to be appropriate for the estimation of average breakdown field strength associated with the probability of breakdown, and the probability of cumulative percentage of breakdowns is defined by [44; 45]  
\[P(U)=1-e^{-(U/\alpha)^{\beta}} \tag{2}\]  
where \(\alpha\) is the scale parameter representing the characteristic breakdown voltage, and \(\beta\) is the shape parameter measuring the sensitivity of the insulation system.  
Figure 6 shows the Weibull distribution of pure transformer oil, TiO\({}_{2}\) nano-liquids and Al\({}_{2}\)O\({}_{3}\) nano-liquids with the volume fractions ranging from 0.2% to 0.8%. When the volume fraction of nano-liquids is 0.2%, there is a clear intersect interference of Weibull distribution between nano-liquids and pure transformer oil, which means a similar average breakdown strength. Furthermore, the average breakdown strength of nano-liquids gradually increases with the volume fractions when the volume fraction is below 0.6%, and then it is started to fall when the volume fraction is beyond 0.6%, which indicates that 0.6% is the threshold level for positive influence and represents the optimal volume fraction of the two nano-liquids.  
Table 1 compared the average breakdown strength and modification effect of the pure transformer oil and nano-liquids. The modification effect of TiO\({}_{2}\) nano-liquids ranged from 2.3 to 18% and the modification effect of Al\({}_{2}\)O\({}_{3}\) nano-liquids ranged from 5 to 18%. The results have further proven that 0.6% is the optimal volume fraction for the two types of nano-liquid. In general, it is indicated that there is a deep connection between the breakdown strength and nano-particles.  
In order to explore how nano-particles affect the modification effect, optimal volume fraction, and streamer morphology, we discuss the electric field distribution (potential barrier) and hydromechanical properties around TiO\({}_{2}\) nano-particles and Al\({}_{2}\)O\({}_{3}\) nano-particles in the next section of this article.