## 3 Proposed V-I Characteristic Model of Electrical Arc
### Grey Wolf Optimization (GWO) Algorithm
The position vectors for \(\alpha\), \(\beta\), or \(\delta\) are given in Equations (12a)-(12g):  
\[\overset{\rightarrow}{D_{\alpha}}=\left|\overset{\rightarrow}{C}_{1}\cdot \overset{\rightarrow}{X_{\alpha}}-\overset{\rightarrow}{X}\right| \tag{12a}\]  
Figure 7: The hierarchical structure of the GWO algorithm (top–down from strong to weak). \[\stackrel{{\rightarrow}}{{D}}_{\beta}=\left|\stackrel{{ \rightarrow}}{{\mathrm{C}}}_{2}\cdot\stackrel{{\rightarrow}}{{X}}_{ \beta}-\stackrel{{\rightarrow}}{{X}}\right| \tag{12b}\] \[\stackrel{{\rightarrow}}{{D}}_{\delta}=\left|\stackrel{{ \rightarrow}}{{\mathrm{C}}}_{3}\cdot\stackrel{{\rightarrow}}{{X}}_{ \delta}-\stackrel{{\rightarrow}}{{X}}\right|\] (12c) Figure 7.0 (12d) Figure 7.1 (12e) Figure 7.2 (12f) Figure 7.3  
The pseudo-code of the GWO algorithm is given in Algorithm 1. ```  Initialize grey wolf population:  Xi = {C1_i, C2_i, I1_i, I2_i, I3_i, I4_i, \(\eta\)_i, n_i} for i = 1 to n // C1, C2: Heidler function coefficients // I1-I4: current threshold points for Double S-curve // \(\eta\), n: slope and shape parameters  Set control parameters:  a = 2  A and C will be computed during updates  For each search agent Xi:  Compute estimated voltage V_est using:  - Modified Heidler Function (for rising arc regions)  - Double S-curves (for falling arc regions)  Calculate objective functions:  f1 = RMSE(V_measured, V_test)  \(\Omega\) = MAE(V_measured, V_test)  Identify non-dominated solutions and initialize the archive with them // Leader selection  X\(\alpha\) = SelectLeader(archive)  Temporarily remove X\(\alpha\) from the archive  X\(\beta\) = SelectLeader(archive)  Temporarily remove X\(\beta\) from the archive  X\(\delta\) = SelectLeader(archive)  Add X\(\alpha\) and X\(\beta\) back to the archive // Optimization Loop t = 1  while t < MaxIter:  for each search agent Xi:  Compute vectors A and C:  A = 2 * a * rand() - a  C = 2 * rand()  Update Xi position using Equations (12a)-(12g)  Enforce physical bounds on Xi parameters (e.g., C1 > 0, I1 < I2, etc.) ### Proposed Model

The proposed model utilizes a mathematical framework to delineate the V-I characteristic of the electric arc, incorporating double S-curves and the modified Heidler function. The polynomial equations established by the curve-fitting procedure, discussed in the previous section, serve as a reference for calculating the mathematical parameters of the model. The characteristic curve, as illustrated in Figure 6 is segmented into three distinct regions to construct the proposed model. The first region, defined by the conditions \(di/dt>0,\ i>0\), and \(V_{arc}>0\), is modeled using the modified Heidler function, whereas the reduction in current is represented by double S-curves. The parameters of the equations that yield the optimal fit for the polynomial expressions of the proposed model were identified using the GWO technique. #### 3.3.1 Region 1 and Region 4: Rising Current Phase \((\text{di}/\text{dt}>0)\)

Figure 8a illustrates the eighth-degree polynomial equation derived from the curve-fitting method to represent the V-I characteristic of the electric arc. Figure 8b illustrates the Heidler function employed to model lightning strikes in the investigation of lightning-induced overvoltages. The Heidler function describes the temporal variation of current, making it a suitable approach for modeling arc behavior. To establish the mathematical relationship between current and voltage in the arc, the polynomial expression in Figure 8a is reformulated by substituting current and voltage parameters in place of time and current in the Heidler function. The Heidler function is further adjusted, as given in Equation (13), to represent the current as a function of time as follows [53]:

Figure 7.4In Equation (14), the parameter \(\eta\) is a scaling factor and is defined as follows:

Figure 7.5

In these equations, \(i_{0}\) represents the peak current value, \(T_{1}\) represents the time constant associated with the current's rise time, and \(T_{2}\) represents the time constant related to the current's decay time.