# Arc Modeling and Kurtosis Detection of Fault with Arc in Power Distribution Networks
## 3 Arc Modeling  
### Mathematical Arc Modeling  
It is difficult to accurately simulate a complete arc phenomenon due to its irregularity and reproducibility. The arc behavior changes from one half-cycle to the other half-cycle, as revealed by the experimental arc characteristics shown in Figure 5c. An arc phenomenon is composed as one cycle specific to the arc, which is characterized by unsymmetrical half cycles. Therefore, the arc parameters required for arc modeling were determined by dividing the arc region based on the amplitude of the phase voltage for each half-cycle. Over the past few decades, many popular methods have been proposed for implementing the arc phenomenon. However, most of the previous methods use the phase current for calculating arc conductance (arc impedance) as an input of arc dynamics. Hence, it is difficult to accurately obtain the arc conductance (arc impedance) because the phase current is also affected by load fluctuation and load type. Therefore, in this study, we designed  
Figure 5: Experimental arc characteristics. (**a**) Phase impedance. (**b**) Phase impedance (1 cycle). (**c**) Voltage-current curve (1 cycle). (**d**) Impedance-voltage curve (1 cycle). a mathematical arc model by separating two regions (viz. positive half cycle and negative half cycle) as follows:  
* Positive half cycle After the ignition time, the arc impedance (\(Z_{arc}\)) can be determined by the following equations for positive phase voltage. The first and third equations are nonlinear functions and the second equation is a linear function. The last equation is defined as a constant value because there is no waveform distortion due to the arc in the region where the phase voltage exceeds \(V_{th3}\). \[\left\{ \begin{array}{ll}Z_{p,min}\exp(a_{p1}(V-V_{p1})),&0\leq V<V_{p1}\\ k_{p1}V+k_{p2},&V_{p1}\leq V<V_{p2}\\ (Z_{p,max}-Z_{p,const})\exp(a_{p2}(V-V_{p2}))+Z_{p,const},&V_{p2}\leq V<V_{p3}\\ Z_{p,const},&V\geq V_{p3}\end{array} \right.\] (1) where \(k_{p1}=\frac{Z_{p,max}-Z_{p,min}}{V_{p2}-V_{p1}}\), \(k_{p2}=\frac{-Z_{p,max}V_{p1}+Z_{p,min}V_{p2}}{V_{p2}-V_{p1}}\), \(a_{p1}\) and \(a_{p2}\) are positive integers. * Negative half cycle Similar to the case of the positive half cycle, the arc impedance can be determined by the following equations for negative phase voltage: \[\left\{ \begin{array}{ll}Z_{n,min}\exp(a_{n1}(-V-V_{n1})),&V_{n1}\leq V<0\\ -k_{n1}V+k_{n2},&V_{n2}\leq V<V_{n1}\\ (Z_{n,max}-Z_{n,const})\exp(a_{n2}(V+V_{n2}))+Z_{n.const},&V_{n3}\leq V<V_{n2} \\ Z_{n,const},&V<V_{n3}\end{array} \right.\] (2) where \(k_{n1}=\frac{Z_{n,max}-Z_{n,min}}{V_{p2}-V_{n1}}\), \(k_{n2}=\frac{-Z_{n,max}V_{p1}+Z_{n,min}V_{p2}}{V_{p2}-V_{n1}}\), \(a_{n1}\) and \(a_{n2}\) are positive integers. Figure 6 shows the proposed arc model, which calculates the arc impedance by using the phase voltage as the input. Since arc impedance depends on the magnitude of the phase voltage, the proposed arc model consists of the nonlinear arc impedance region ([\(V_{n3}\), \(V_{n2}\)], [\(V_{n1}\), \(V_{p1}\)], [\(V_{p2}\), \(V_{p3}\)]), linear impedance region ([\(V_{n2}\), \(V_{n1}\)], [\(V_{p1}\), \(V_{p2}\)]), constant arc impedance region ([\(-\infty\), \(V_{n3}\)], [\(V_{p3}\), \(\infty\)]). The corresponding simulated arc characteristics are illustrated in the comparison shown in Figure 7. We can observe that two V-I curves are similar and have small relative error between the experimental model and simulation model, even if the V-I curve has nonlinearity and asymmetry. The arc parameters are determined such that they match the results of V-I and Z-V curves in the arc experiment. These comparisons verify the accuracy of the arc model. As shown in the comparison results, the simulated arc model is similar to the experimental arc model. The arc parameters used in arc modeling are listed in Table 1.