## 4 Characterization of the Discharge of Insulation Defects in Switchgear
### Inversion of Partial Discharge by Characteristic Gas Volume Fraction  
Through the previous analysis, it can be found that different discharge types correspond to different gas volume fraction change rates of characteristic gases, which are unique. Therefore, this paper uses the change rate of a characteristic gas volume fraction to select the regression equation to invert the partial discharge of defect propagation in discharge. Use gas sensors to detect the volume fraction changes of characteristic gases O\({}_{3}\), CO, and NO\({}_{2}\), and then calculate the current partial discharge of the air switch equipment  
Figure 17: Fitted curves of CO, NO\({}_{2}\), and O\({}_{3}\) volume fractions versus partial discharge volume in Metal Particle Discharge. (**a**) CO gas volume fraction fitting curve. (**b**) NO\({}_{2}\) gas volume fraction fitting curve. (**c**) O\({}_{3}\) gas volume fraction fitting curve. based on the inversion method designed in the previous section. The calculation process of partial discharge characteristic gas inversion can be summarized as shown in Figure 18. The detection of internal insulation defects in switchgear can be achieved through online monitoring based on characteristic gases according to this process. The equations for calculating the first-order and second-order derivatives of the characteristic gases are shown in the following equations:  
\[C_{i}^{\prime}=\frac{C_{i}^{t+1}-C_{i}^{t}}{\Delta t} \tag{15}\]  
\[C_{i}^{\prime\prime}=\frac{C_{i}^{t+2}-2C_{i}^{t+1}+C_{i}^{t}}{\Delta t^{2}} \tag{16}\]  
where denotes the volume fraction of the \(i\)th characteristic gas in the layer at time \(t\), \(C_{i}^{t+1}\) denotes the volume fraction of the \(i\)th characteristic gas in the layer at time \(t+1\), and \(C_{i}^{t+2}\) denotes the volume fraction of the \(i\)th characteristic gas in the layer at time \(t+2\). The time-domain regression equation for the characteristic gas under the defects of metallic protrusions is shown in the following equation. Here, this paper only provides regression equations for four voltages: 11 kV, 13 kV, 15 kV, and 17 kV. However, the method proposed in this paper can fully re-estimate the situation for other voltages. \[y_{CO}=\left\{ \begin{array}{llll}&-0.0014t^{2}+1.2191t-0.4938&R^{2}=0.99&11 \;\text{kV}\\ &0.0009t^{2}+1.3635t+9.7560&R^{2}=0.99&13\;\text{kV}\\ &-0.0179t^{2}+2.5341t-2.3939&R^{2}=0.99&15\;\text{kV}\\ &-0.0089t^{2}+3.3238t+1.1509&R^{2}=0.99&17\;\text{kV}\\ \end{array} \right. \tag{17}\]  
\[y_{NO_{2}}=\left\{ \begin{array}{llll}-0.0480t^{2}+7.5267t+23.9337&R^{2}=0.99 \\ -0.0797t^{2}+12.9247t+41.0222&R^{2}=0.99\\ -0.0601t^{2}+10.7539t+32.8268&R^{2}=0.99\\ -0.0840t^{2}+13.5991t+42.3192&R^{2}=0.99\\ \end{array} \right. \tag{18}\]  
\[y_{O_{3}}=\left\{ \begin{array}{llll}-0.0301t^{2}+4.0320t+74.5742&R^{2}=0.998 \\ -0.0497t^{2}+6.6555t+122.2203&R^{2}=0.998\\ -0.0526t^{2}+7.0292t+129.0536&R^{2}=0.998\\ -0.0602t^{2}+8.0500t+148.5663&R^{2}=0.998\\ \end{array} \right. \tag{19}\]  
The time-domain regression equation for the characteristic gas under air gap defects is shown in the following equation. Air gap discharge requires higher voltage. Here,  
Figure 18: Flowchart of inverse partial discharge volume calculation. this paper only provides regression equations for four voltages: 18 kV, 29 kV, 22 kV, and 24 kV. The method proposed in this paper can also be used to re-estimate the situation at other voltages. \[y_{CO}=\left\{ \begin{array}{ccc}0.0005t^{2}+0.9834t+9.7644&R^{2}=0.99&18\text{ kV}\\ -0.0011t^{2}+4.0732t+7.6492&R^{2}=0.99&20\text{ kV}\\ -0.0030t^{2}+5.3297t+2.1726&R^{2}=0.99&22\text{ kV}\\ -0.0006t^{2}+5.6624t+8.5664&R^{2}=0.99&24\text{ kV}\end{array} \right. \tag{20}\]  
\[y_{NO_{2}}=\left\{ \begin{array}{ccc}-0.0480t^{2}+7.5267t+23.9337&R^{2}=0.99\\ -0.0797t^{2}+12.9247t+41.0222&R^{2}=0.99\\ -0.0601t^{2}+10.7539t+32.8268&R^{2}=0.99\\ -0.0840t^{2}+13.5991t+42.3192&R^{2}=0.99\end{array} \right. \tag{21}\]  
\[y_{O_{3}}=\left\{