## 3 Proposed V-I Characteristic Model of Electrical Arc
### Grey Wolf Optimization (GWO) Algorithm
The parameter \(n\) is the gradient coefficient that determines the steepness of the current variation. In Equation (13), when the current variable is replaced with voltage and the time variable is designated as current, the arc voltage can be expressed as a current-dependent voltage function, as defined in Equation (15):

Figure 7.6

The negative alternans of arc current and voltage is defined by multiplying Equation (15) by the sign function [52]. In Equation (15), \(i_{0}\) and \(i_{1}\) are the current threshold value and the peak value of the current, respectively. The function parameters in Equation (15) can be efficiently determined using the GWO algorithm when the eighth-degree polynomial curve is serving as a reference. The values derived from experimental measurements were used to apply the constraints defined for the GWO algorithm. * \(0<V<V_{max}\) (the min and max values of the arc voltage obtained in the experimental results)
* \(0<i<80,000\) (the min and max values of the arc current obtained in the experimental results)

In Equation (15), applying the GWO algorithm, as detailed in the pseudo code in Algorithm 1, yields the equation parameters for the optimal polynomial curve solution as follows: \(V_{ex}\) = 314 V, \(\alpha\) = 1.505, \(n\) = 1012, \(i_{1}\) = 220 kA, and \(i_{2}\) = 1850 A.

Figure 7.7 Proposed Model: (**a**) the eighth-degree polynomial formula derived from the curve-fitting method; (**b**) the time-current waveform of the Heidler function. Figure 9 illustrates the V-I characteristic of the arc, represented by an eighth-degree polynomial expression derived through curve fitting and the modified Heidler function, optimized via the GWO algorithm. Additionally, Figure 9 demonstrates that the proposed method effectively characterizes the arc's V-I behavior during the rising current phase. #### 3.3.2. Region 2 and Region 3 (\(di/dt<0\))

The V-I characteristic of the arc can be modeled using a third-degree polynomial equation when the derivative of the current negative is \((di/dt<0)\), indicating a decreasing current phase. This representation is obtained through the curve-fitting method. To enhance accuracy, the cubic polynomial expression is divided into two regions, as shown in Figure 10a. In this case, the solution of the polynomial expression can be more accurately modeled using double S-curves, as shown in Figure 10b. The S-curve's approach is illustrated in Figure 10c [54]. The mathematical representation of the S-curve is calculated by Equation (16) as follows:

Figure 7.8

where:

\(V_{m}\): is the maximum saturation voltage on the S-curve. \(HG_{min}\): is the present value in the minimum hyper growth state, \(HG_{max}\): is the present value in the maximum hyper growth state, \(i\): is the arc current. The experimental data indicate that a minimum threshold voltage of 20 V is required for arc formation [55]. The third-degree polynomial curve exhibits a smooth transition at 30 kA. Consequently, a value of 30 kA has been established as the threshold for differentiating the S-curves. Figure 7.9 The eighth-degree polynomial curve from curve fitting for the V–I characteristic and modified Heidler function with the GWO algorithm. In Region 2, the peak voltage of the initial S-curve is denoted as \(V_{1}\), whereas in Region 3, the peak voltage of the subsequent S-curve is denoted as \(V_{2}\).