*2.1. CAD Approach*

The GIS metallic enclosure was modeled through six parallel aluminum bars interconnected in such a manner that the resultant geometry constitutes the GIS shell equivalent structure (hexagon-shape geometry) (see Figure 2). The thickness of the aluminum bars was considered to be equal with the distance between the inner and outer walls of the GIS enclosure, accounting also for the thickness of both metallic walls. The metallic flanges located at each GIS equipment enclosure ends were modeled through aluminum conductors with a similar diameter as the aluminum bars, in order to ensure the galvanic connection between every two consecutive bars.

**Figure 2.** CAD representation of GIS pipe (**a**) octagon (**b**) pentagon.

To understand and to endorse the adopted hexagon-based geometry approach, three simplified CAD geometries (pentagon, hexagon, and octagon layouts) were designed and tested in similar modeling conditions for a single GIS bus section. The endorsement and validation of the geometric approach were performed considering lightning surge scenario, considering a voltage breakdown fault. The purpose of the study was to quantify how the di fferent number of parallel aluminum bars representing the solid metallic pipe will impact the transient response of the enclosure during voltage breakdown fault, taking into account very-high-frequency transients.

From a geometric point of view, each CAD model was built following similar design procedures. The radius of geometric circle of the enclosure (located at a half distance between the inner and the

outer GIS walls), r = 0.248 m, was taken as reference, with all three investigated geometries being inscribed in a circle with the corresponding circle.

The average errors computed analyzing the numerical values of maximum and minimum amplitudes measured for the first through fourth periods of the time domain waveform were between acceptable limits, between 2–5%. Due to the fact that the spatial distribution of the aluminum bars surrounding the phase conductor was slightly di fferent for each particular model, dissimilarities between numerical values were to be expected. However, the neglectable di fferences between recorded TGPR values considering di fferent models were obtained as a result of conductive coupling, which dominates the transient response of the system during voltage breakdown fault. According to graphical as well as numerical results (see Figure 3), there were negligible di fferences between the primary parameters of the TGPR when di fferent enclosures were considered during the computational process. Due to geometric symmetry reasons as well as from a computational time point of view, the hexagon-based geometry was developed and further used for the real GIS model analysis.

**Figure 3.** Transient ground potential rise during lightning surge scenario, pentagon-, hexagon-, and octagon-based geometries 10 μs.

Figure 4 illustrates the equivalent CAD model representing the metallic enclosure of a single GIS Bus configuration.

**Figure 4.** CAD model representing one GIS bus section enclosure.

#### *2.2. Adopted Electromagnetic Field Theory Approach: Partial Equivalent Element Circuit Method*

The partial element equivalent circuit (PEEC) method is derived from Maxwell's equations and provides a full-wave solution to them [41]. The PEEC method interprets Maxwell's equations to a circuit domain. The theoretical derivation of the PEEC method for the thin wire structures proposed by Yutthagowith and Ametani in [42] starts from a total electric field on a wire surface:

$$\overline{s} \cdot \overline{E^i}(\overline{r}) = \overline{s} \left( \overline{E^i}(\overline{r}) + \overline{E^s}(\overline{r}) \right) = \overline{s} \cdot \frac{\overline{f}(\overline{r})}{\sigma} \tag{1}$$

Whole system equations in the frequency domain can be written in a matrix form corresponding to a modified nodal analysis (MNA) formulation, as shown in Equation (2):

$$
\begin{vmatrix} j\omega P^{-1} + Y\_a & A^T \\ -A & R + j\omega L \end{vmatrix} \begin{vmatrix} \phi \\ \end{vmatrix} \begin{vmatrix} \phi \\ I \end{vmatrix} = \begin{vmatrix} I\_S \\ \end{vmatrix} \tag{2}
$$

where *A* is an incident matrix that expresses the cell connectivity, *R* is a matrix of series resistances of current cells, *L* is a matrix of partial inductances of current cells including the retardation effect, *P* is a matrix of partial potential coefficients of the potential cells including the retardation effect, *Y* is a vector of potentials on the potential cells, *I* is a vector of currents along with current cells, *US* is a vector of voltage sources, *IS* is a vector of external current sources, and *Ya* is an additional admittance matrix of linear and nonlinear elements [25].

The purpose of the computer-based solvers is to accurately expend analytical methods elaborated for basic structures to real-case scenarios considering metallic towers, different geometries of the grounding grid, gas-insulated substation metallic enclosure, and the electromagnetic interaction between such structures. Although, initially, the PEEC numerical approach was intended to be applied on inductivity analysis across integrated circuit boards, improvements and modifications have been made in recent years in order to be successfully applied on transient regime analysis. As was presented in previous sections, PEEC approach has several advantages compared to DE form-based methods, which require the discretization of the entire computational domain's volume, thus imposing limitations regarding the analysis of complex geometries. Moreover, the outcome provided by PEEC approach does not require post-processing procedures in order to obtain the potential and currents along a circuit as FDTD, MoM, and FEM methods. Regarding the application toward gas-insulated substation transient behavior analysis, there are barely a few papers throughout public literature focused on this topic.

The substation is energized by two 220/110-kV autotransformers, which are modeled through equivalent impedances, *Z* = 0.1 Ω. Pure resistive loads were assumed during the computational process, modeled as transversal impedances to Earth connected at each particular phase conductor *Z* = 1 Ω. The VFTO source was modeled through an ideal electromotive force (EMF) generator situated at the tripped disconnector switch location. Beside the transient source, each metallic element became an individual electromagnetic source, which generates a particular electric and magnetic field, affecting the metallic structures located in nearby surroundings, the electromagnetic chain reaction. The transient waveform is described by a double exponential function, presented in [12] with the following parameters α = 2.31049 · <sup>10</sup>5s−1, β = 8.17350372 · <sup>10</sup>9s−1, and *Vm* = 1 p.u. in order to obtain per unit results considering fault severity. The following relation describes the time domain distribution of the voltage source in the XGSLab software package:

$$V(t) = \frac{V\_m}{k} \Big(e^{\frac{\pi t}{\tau\_1}} - e^{\frac{\pi t}{\tau\_2}}\Big) \tag{3}$$

$$\tau\_1 = \frac{1}{\beta} \text{ and } \tau\_2 = \frac{1}{\alpha}$$

where
