The RBF-FD and RBF-FDTD Methods for Solving Time-Domain Electrical Transient Problems in Power Systems

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Introduction
Te transient waveforms of current and voltage that occur on power system elements such as overhead power lines and grounding systems as a result of direct or indirect lightning strikes, short-circuit faults, and circuit breaker switching operations [1] propagate along power lines to line-connected electric devices. Tese transient waveforms can signifcantly damage insulators and equipment and interrupt electricity, depending on the existing time and peak value. As a result, one of the primary tasks of overhead power line operation and economic insulation design is to accurately calculate these waveforms.
PDEs can express the variation of current and voltage during the transient state in either the frequency domain or the time domain. Te latter, known as ODEs, is a commonly used form in which the variables of current and voltage depend only on time and are solved using classical methods such as Laplace or Fourier transforms [2], traveling wave methods such as the lattice-diagram method, the method of characteristics, and the state-space method [3][4][5]. Furthermore, for modeling electrical transient voltage waveforms, EMTP-ATP software, which is based on Bergeron's method, has become popular. matrix [11], wavelet transforms [12,13], and so on, directly solve transient equations. Tese methods are well known for their efciency, power, and robustness in providing numerical solutions to time-dependent diferential equations. Te FDTD method, in particular, is the most widely used mathematically simple analysis technique for timedependent electromagnetic applications. It approximates PDE derivatives in both time and space at the same time and then solves PDEs using either a matrix analysis or an iterative method.
Over the last decade, RBF-based methods have been developed and successfully applied to science and engineering problems, as well as electromagnetic problems [14][15][16][17][18]. Tese methods produce solutions that are frequently far more accurate than traditional numerical methods. As a result, they have gained increasing attention from the scientifc community interested in solving diferential equations. RBF meshfree methods (global and local RBF meshfree method) in the time domain and use a difference scheme (called RBF-FD scheme) to compute the solutions on them is presented in [19]. Tis approach is similar to the FETD method. It means that at each time step, the spatial component is solved using the RBF meshfree method. On the other hand, new numerical methods combining RBFs and traditional approaches have been developed. Tese include RBF-DQ [20] and RBF-FD [21][22][23][24][25][26][27][28][29][30], which are used in 1-D and 2-D spatial domains, respectively. Unfortunately, the application of these improved methods to time-dependent ODEs and PDEs has not been widely adopted, particularly in the context of electrical transient problems.
In this paper, we develop and apply the RBF-FD method and the RBF-FDTD method, in which central RBF-FD approximations of the frst-and second-order derivatives of the function are formulated in the matrix form (see Appendix A) using RBFs of GA, MQ, IQ, and IMQ types based on the theory presented in [22], to modeling the transient voltage waveforms of a typical electrical circuit and a 220 kV single-phase transmission line defned by time-dependent ODEs, lightning-induced voltages on a 110 kV single-phase distribution line, and lightning transient voltages along two horizontal grounding electrodes defned by time-dependent PDE systems [31][32][33][34][35][36], respectively. Furthermore, when using RBFs, it is important to note that selecting an "optimal" shape parameter is an important task because it infuences not only the accuracy of the solution but also the numerical stability; this is still an open problem that requires more research. For this purpose, Bayona's algorithm [23] for selecting "optimal" shape parameters based on the minimizing approximation error is used. Te computed results of the transient voltages of a typical electrical circuit and a 220 kV single-phase transmission line obtained using the RBF-FD method were compared to those provided by EMTP-ATP software and classical numerical methods such as fnite diference, state variable, wavelet, and TLM methods, and the computed results of lightning-induced voltages due to a nearby lightning strike on the 110 kV single-phase distribution line calculated using the RBF-FDTD method were also compared to the ones and LIOV [37]. All of these results show that our proposed numerical methods are more accurate than traditional numerical methods when it comes to selecting "optimal" shape parameters. Finally, a diferent approach to the RBF-FDTD method is presented for economics and efciency. Te solutions of lightning transient voltages along two horizontal grounding electrodes computed using this new method agree very well with those obtained by our traditional FDTD method and shown in [38].
Te structure of this paper is as follows: the RBF-FD approximations derived using RBFs are shown in Section 2. In Section 3, the algorithm for determining the "optimal" shape parameter is introduced. Te results of this work are presented in Section 4, where we apply the RBF-FD method, other traditional numerical methods, and EMTP-ATP software to the transient analysis of a typical electric circuit and a 220 kV single-phase transmission line, as well as the RBF-FDTD method to model lightning-induced voltages on a 110 kV single-phase overhead distribution line and lightning transient voltages along two horizontal grounding electrodes. Section 5 will include a remark and a conclusion.

RBF-FD Approximations
In general, because of its high efectiveness and simplicity, the traditional FD method, which expresses all FD approximations by using the Taylor series expansions of the frst-and second-order derivatives of the function f at an observed point [5], is recognized as the state-of-the-art numerical technique. In other words, this method is a highly efective computational tool. However, because the series is truncated with a second-order remainder, this method is usually of lower accuracy.
RBFs have received a lot of attention from the scientifc community in the last decade because of their "meshfree" or "meshless" nature. As a result, RBFs are now used to devise derivative approximations instead of Taylor's series. Some early versions of the RBF-FD method were applied [20][21][22]. We will present the formulae of the central RBF-FD approximations used for developing our proposed methods in Section 4 for solving time-dependent electrical transient problems.
Before we solve diferential equations by using the RBFs, we recall that the mth-order derivative of a function f at a point x i in the traditional fnite diferences is approximated by a linear combination of the values of the function f at some adjacent points as follows [20][21][22][23][24][25][26][27][28][29]: where N is the total number of grid points, f(x i ) is the value of f at the grid point x i , and w (m) j,i are the weighting coefcients associated with the mth-order derivative at the point x j and are computed using RBF interpolation.
Implementing (1) for the scheme of three equispaced points N � 3 of x i − ∆x, x i , and x i + ∆x in 1-D domain, the general expressions of RBF-FD approximations of the frstand second-order derivatives are written as follows: 2 International Transactions on Electrical Energy Systems dΦ dx in which the function Φ(‖ − x j ‖ 2 ) is the basis function that depends only on the distance norm from each center to some adjacent points and defned as in Table 1. Te weighting coefcients (α 1 , α 2 , α 3 ) and (β 1 , β 2 , β 3 ) are formulated as follows.
Two linear equations of (2) and (3) can be rewritten in following matrix forms as and where A is a vector [1 × 3] that contains the elements of . Terefore, the unknown weighting coefcients in (4) and (5) are given by It can be noticed that using the GA, MQ, IQ, and IMQ for the function Φ(‖ − x j ‖ 2 ) of the abovementioned procedure, we will fnd the unknown coefcients of (α GA , β GA ), (α MQ , β MQ ), (α IQ , β IQ ), and (α IMQ , β IMQ ), respectively, in detail (see Appendix A).

Choosing "Optimal" Shape Parameters
Te shape parameter c is critical in the theory and practice of the RBF-FD method, as presented in previous literature [20][21][22][23][24][25][26][27][28][29], and infuences not only the accuracy of the solution but also its numerical stability. Te accuracy of the RBF-FD method will be the same as that of the traditional FD method if this parameter c becomes large, which means c tends to infnity. Similarly, a small shape parameter causes numerical instability in the RBFs we use. Finding an "optimal" value of c that improves solution accuracy is thus an important task for researchers that is still an open issue (See [15]).
Many diferent methods for determining an "optimal" shape parameter of the RBF-FD method have been investigated and presented in numerous previous papers [21,[23][24][25]. Here, we will apply the algorithm proposed by Bayona et al. [23] for selecting "optimal" shape parameters to our RBFs, which can be described as follows.
Considering that the 1-D domain is divided by N scattered points, in which NI denotes the total number of interior points of the solution domain, thus, N-NI is two boundary points.
Te RBF-FD error is given by where For fnding the "optimal" shape parameter c * , the approximation error E(c) of (8) is minimized by the following expression as In the case of practical problems where we do not know the exact solution, the RBF-FD error can be obtained by the following expression as where Γ(c) is a NI × NI sparse matrix that contains the weighting coefcients of (1) and ϵ(c) is a vector of the local error of ϵ n (x i , c).
In addition, we can also fnd an approximate value of c * e to the "optimal" shape parameter of c * as where ϵ e (c) is the estimated local error. For a much more detailed presentation and discussion of this algorithm, we refer the readers to [23].

Case 1: RBF-FD Method Applied to a Typical Electric
Circuit. As a frst example of RBF-FD application to ODEs, we used this method to compute the transient voltage on the capacitor of the typical electric circuit shown in Figure 1.
Consider a switching operation in which the voltage source in the circuit of Figure 1 switched on at t � 0, the voltage across the capacitor and the current satisfy the system of the following equations as International Transactions on Electrical Energy Systems 3 We can obtain the time-dependent 2nd-order ODE of the capacitor voltage by frst diferentiating the current variable in (13) with respect to t and then substituting it into (12).
Te analytical solution of the capacitor voltage of (14) obtained using the Laplace transform method is given by To apply the RBF-FD method to (14), the unknown weighting coefcients of the RBF-FD approximations of the temporal derivatives are formulated the same as (6) and (7) by replacing ∆x with ∆t, thus the RBF-FD equation of the capacitor voltage can be written as v n+1 We divided the time interval of ten seconds into 1000 to compare numerical solutions. Figure 2 shows the numerical results of the capacitor voltage calculated using the IMQ-FD method and many diferent approaches. Tis fgure shows that all obtained solutions are well superimposed. We used two common error norms to evaluate the accuracy of all solutions: the root-mean-square error (RMS-error) and the infnitive error (‖E‖∞). Table 2 shows the results of two error norms for all methods. It is clear that the IMQ-FD method is more accurate than all the rest of the methods.
Furthermore, it is critical to understand that the accuracy of the RBF-FD method is heavily infuenced by the shape parameter c. To demonstrate this, we applied the RBF-FD method to the algorithm [23] for determining "optimal" shape parameters presented in Section 3. Te obtained results of two error norms are shown in Figure 3 and Table 3. Te accuracy of the RBF-FD method is higher than that of the traditional FD method at the values of the "optimal" shape parameters, as shown in Table 3, and in this case, the IMQ-FD realizes the smallest error when compared to the MQ-and GA-FD versions. It is especially noticeable in Table 3. When the exact solution is unknown, Table 3 shows that the values of c * e and ‖E(c * e )‖ ∞ obtained using (11) are equivalent to the values of c * and ‖E(c * )‖ ∞ . Tis algorithm will be used efectively in Case 2 and Case 4 of this section.
In general, aside from selecting the "optimal" shape parameters, the accuracy of these methods is also afected by the total number of discretized points, which means that as the total number increases, so does the error. However, in this case, the computation cost will increase signifcantly. A comparison of the infnitive error norm while increasing the total number of temporal steps is illustrated in Figure 4 to evaluate the efectiveness and economics of the RBF-FD method compared to the FD method. Tis is demonstrated by the fact that the RBF-FD method is signifcantly more efcient and cost-efective than the conventional FD method. It means that even with only a few hundred points, the RBF-FD solution has the same accuracy as the conventional FD method with a thousand points, and the RBF-FD method is still stable. As a result, the proposed method for analyzing the transient behavior of mediumvoltage lines and short transmission lines defned by the lumped-parameters model is both accurate and efcient.

Case 2: RBF-FD Method Applied to the Single-Phase
Transmission Line. In order to apply the RBF-FD method to the second example of ODEs, we examine the equivalent circuit diagram of the 220 kV short transmission line using  International Transactions on Electrical Energy Systems a lumped-parameter model in Vietnam, which is depicted in Figure 5. Tese parameters are assigned to this line: R � 11.42 Ω, L � 292.3 mH, C � 1.54 μF. Tis line is equivalent to a two-port model, allowing us to discover an analytical solution using the "lsim" function in MATLAB.
Te single-phase voltage source of U 0 � 220 � 2 √ sin (100πt) kV is switched on at t � 0 in the single-phase equivalent circuit of Figure 5, and the receiving-ended voltage and current must satisfy the following set of equations: First, by diferentiating the current variable in (18) with respect to t and then substituting it into (17), the timedependent 2nd-order ODE of the receiving end voltage can be obtained as follows: Ten, by applying the RBF-FD approximations of the frst-and second-order derivatives in the temporal domain to the receiving end voltage in (19), we derive the RBF-FD equation as (20).
As shown in Figure 6, the solutions of the transient voltage at the receiving end of this line calculated in the time domain using the RBF-FD, conventional FD, wavelets, state variable, and TLM techniques, as well as the EMTP-ATP software, compare favorably with the analytical solution. It is noted in this fgure that all the obtained solutions are well superimposed.
In general, the number of time steps used afects the accuracy of the solution in transient modeling. In this example, we divided the one-second time interval by 1000, which also means that the time step utilized in this case is 1.10 − 6 . To evaluate the accuracy of solutions, we use the root-mean-square error (RMS-error) and the infnitive error (‖E‖∞) error norms to all techniques. Two error norms for all approaches are displayed in Table 4 where it can be seen that the IMQ-FD solution most closely matches the analytical one, while the EMTP-ATP error norm is the lowest. Tere is a very high level of agreement about accuracy and applicability to the RBF-FD method in practical application.
It is extremely important to be aware of the fact that the accuracy of RBF-FDM is signifcantly dependent on the shape parameter c that is selected, as is described in Section 3. In order to demonstrate this, we apply the algorithm for fnding the "optimal" shape parameters of three RBFs (IMQ, MQ, and GA) that are presented in Section 3 to (20)     International Transactions on Electrical Energy Systems 5 220 kV single-phase transmission line. Te comparison between RMS-error and ‖E‖ ∞ norms of RBF-FD and FD methods is presented in Figure 7 and Table 5. It has come to our attention that the RBF-FDM is, in comparison to the conventional FDM, more accurate. In general, it is important to take note of the fact that there is widespread consensus regarding the range of the shape parameter values for which we always obtain RBF solutions that are more precise than those obtained by the conventional FDM for this practical transient problem in power systems. Figure 8 illustrates the behavior of the RMS-error and (‖E‖∞) norms of the RBF-FD and FD methods for the transient voltage solutions while increasing the total number of time steps in order to demonstrate that the RBF-FDM is more efcient than the traditional FDM while maintaining similar accuracy. Again, using a few hundred points of time steps in the RBF-FDM, it is possible to fnd a solution that has the same level of accuracy as traditional numerical methods using more than a thousand points.
in which L and C are the per-unit length inductance and capacitance, V s (x, t) and I(x, t) are the scattered voltage and the total current along the line, respectively, and E e x (x, h, t) is the incident horizontal electric feld along the x axis at the phase-conductor height.
Te total induced voltage at any observation point along the line can be calculated using the Agrawal et al. model. Tis voltage can be derived from the scattered voltage and the fnite integral of the incident vertical electric feld, which is also referred to as the incident voltage [32][33][34][35]39] using the following expression: where is the incident voltage and the incident vertical electric feld that does not vary in the height range of the line 0 < z < h.
In this study, we determined the scattered voltage on the overhead distribution line by applying Taylor's series expansions with the second-order truncating term that was proposed by Paolone et al. [33] in the time domain. Hence, both the variables of the distributed voltage and the total current can be expressed by (23) and (24).
Te RBF-FDTD approach was utilized in this study in order to solve the equation system consisting of (23) and (24). Tis method requires that the 1-D approximations described in Section 2 be applied to each individual variable of t and x. Hence, the RBF-FDTD scheme of the second order can be formulated as (25) and (26).
We denote here.
iii) k is the kth-spatial step (iv) n is the nth-time step (v) ∆x, ∆t are the size of spatial and time steps, respectively (vi) Nx, NT are the number of spatial and time steps, respectively Te boundary conditions in terms of the scattered voltage and the total current with the loads at two line terminations are given by where Z A and Z B are chosen equal to the characteristic impedance of the line to avoid refected waves at two line terminations.
We consider an example of the 110 kV single-phase distribution line as shown in Figure 9. Te height of the line is 10 meters, the radius of the line is 5 millimeters, the length of the line is assumed to be one kilometer, the characteristic impedance is 500 Ω, and the location of the lightning stroke is y 0 � 50 meters from the center of the line. Te lightning current that is being utilized in this study is the subsequent return stroke. Te peak value of the channel base current is 12 kA, the maximum time-derivative is going to be 50 kA/μs, and the velocity of the return stroke is equal to 1.3.10 8 m/s. In this particular case study, all of the parameters are the same as those in [32][33][34][35][36].
Te MQ, IMQ, and GA RBF-FDTD methods are utilized for the numerical implementation in order to calculate the lightning-induced voltage waveform on the line [36] that is described by the equation system consisting of (25) and (26). Figure 10 presents the lightning-induced voltage on the entire line over time that was obtained by utilizing the MQ-FDTD approach.
In this work, we can assume the solution of LIOV [37] as a benchmark one so that we can determine the "optimal" shape parameter and evaluate how accurate the results are. Figure 11 displays the RMS error that was produced by the MQ-FDTD technique inside the value ranges that were specifed for both c x and c t . From this fnding, we are able to fnd the pair of "optimal" shape parameter values (c * x , c * t ) that corresponds to the minimum RMS-error norm, and the lightning-induced voltage at the center of the line is illustrated in Figure 12. When compared to the FDTD solution, it has been demonstrated that the MQ-FDTD solution can be well superimposed with the LIOV one. In order to achieve these outcomes, the number of temporal steps selected for both approaches is 1000 and the number of spatial steps selected for both approaches is 200.
On the basis of this optimal algorithm of (c * x , c * t ), the comparison of the RMS-error and infnitive error norms between the RBF-FDTD and FDTD methods is illustrated as in Table 6. Furthermore, Figure 13 is an illustration of the   International Transactions on Electrical Energy Systems RMS-error norm versus c x . Te fndings indicate that the RBF-FDTD method produces highly accurate solutions when the "optimal" shape parameters are chosen, and these solutions will converge to the conventional FDTD solution corresponding to larger values of (c x , c t ).    International Transactions on Electrical Energy Systems 9 A comparison of the RMS-error norm while increasing the total number of spatial steps in the MQ-FDTD and FDTD methods is illustrated in Figure 14 to evaluate the efectiveness of using the RBF-FDTD method for the practical problem of power systems defned by timedependent PDEs. Tis is demonstrated by the fact that the RBF-FDTD method is signifcantly more efcient than the traditional FDTD method. Tis means that we can obtain the RBF-FDTD solution with only a few dozen points, which are still more accurate than the conventional FDTD solution with hundreds or thousands of points. As a result, the computational cost of the RBF-FDTD method is signifcantly reduced, particularly when applied to the transient analysis of long power lines or large electric networks.

Case 4: RBF-FDTD Method Applied to Two Horizontal
Grounding Electrodes. In this example, we showed the use of the RBF-FDTD approach for simulating the lightning transient voltages along two 20 m and 100 m horizontal grounding electrodes buried at the depth of 0.5 m in the soil with ϵ r � 50, ρ � 100, and μ r � 1 as in Figure 15. Every electrode is a copper conductor that is 15 millimeters in diameter.
Te lightning impulse current source that is injected at the terminal end of the grounding electrode is expressed for the lightning transient analysis in the form of a double exponential function as where I 0 � 12.935 kA, θ 1 � 190099 1/s, and θ 2 � 2922879 1/s. It notices that all these parameters are chosen the same as that in [38] for comparing our RBF-FDTD and FDTD solutions with that shown in [38].
Te nonuniform transmission line model that was used for modeling the transient voltage and current along the grounding electrode is illustrated as seen in Figure 16. Te equations for the transient voltage v(x, t) and current i(x, t) distributions are expressed using the telegrapher's equation system based on the transient electromagnetic theory as follows: in which, R e is the per-unit length resistance. L e (x, t), G e (x, t), and C e (x, t) are the per-unit length inductance, conductance, and capacitance, respectively, and are the functions of the position variable x and the temporal variable t. For a much more detailed formulation and calculation of these parameters, we refer the readers to [38].
In this study, we proposed a diferent approach of the RBF-FDTD method for solving the equation system of (30) and (31), which is to apply the conventional FD approximation to the temporal derivative term and the RBF-FD approximation to the spatial derivative term. Tus, the RBF-FDTD equation systems for the lightning transient voltage and current are denoted as (32) and (33), respectively. Due to the fact that we only need to fnd the c x shape parameter of the spatial derivative term, this approach is noticeably simpler and more cost-efective than the RBF-FDTD approach that was used in Case 3 and [40].
To select the c x shape parameter for this engineering problem for which there is no an exact solution, we used expression (11) to compute the values of c * e and ‖E(c * e )‖ ∞ , and the algorithm for this calculation is provided in Appendix B. Te result for the 100 m grounding electrode is depicted in Figure 17. It has been demonstrated that the errors of four RBFs are minimized at four diferent c * xe values. However, as c xe approaches 10 2 , these errors will converge to the same constant value. It is worth noting that when c xe is chosen in the range of [5,100], the accuracy of the solution obtained by the RBF-FDTD method is always greater than that of the traditional FDTD method.
Based on the preceding discussion, the shape parameter of the RBFs is fxed at c x � 50 in Case 4, the 20 m and 100 m long grounding electrodes are divided into 20 and 100 segments, respectively, and the number of time steps chosen is 900. Te computed lightning transient voltages at three A, B, and C points along two horizontal grounding electrodes, with A corresponding to the left terminal end, B corresponding to the middle, and C corresponding to the right terminal end as shown in Figure 16, are presented in Figures 18 and 19, respectively. Te transient waveforms simulated by the RBF-FDTD method outperform those obtained by our FDTD method and shown in [38]. Te fndings indicate that there is a good agreement, and the RBF-FDTD method works best with near-optimal shape parameters. Tese numerical results once again demonstrate the feasibility and efciency of the proposed RBF-FDTD method.  [38]. Figure 16: Equivalent circuit model of nonuniformly lumped parameters of the electrode for Case 4 [38].

Remarks and Conclusion
Here, we can remark some main points as follows: (i) Generally, the RBF-based FD methods are more accurate than the conventional FD methods in the small value range of the shape parameter. Terefore, fnding the "optimal" shape parameter is particularly critical for the application of RBF-based FD algorithms. (ii) Applying the RBF-FD and RBF-FDTD methods to the time-dependent ODEs or PDEs, the coefcients of (α 1t , α 2t , α 3t ) and (β 1t , β 2t , β 3t ) of the RBF-FD approximations in the temporal domain can be easily formulated the same as that in the spatial domain presented in Section 2 by replacing ∆x with ∆t. (iii) Tere are two RBF-FDTD approaches, the frst one is to use the RBF-FD approximation for both the temporal and spatial derivatives, and therefore, we must fnd an optimal value pair of c t and c x as shown in Figure 11. Te second approach is to use the FD approximation for the temporal derivative and the RBF-FD approximation for the spatial derivative; thus, all that remains is to fnd the optimal value of c x . Te second approach is simpler and more efective than the frst, but it still has a high level of accuracy. (iv) In some cases, if problems do not require too high accuracy, we can reduce the total number of temporal and spatial steps while still meeting the accuracy requirement. Terefore, the computational cost of the RBF-FD and RBF-FDTD methods will be signifcantly reduced (see Figures 4,8,and 14). Tus, the RBF-based FD methods are more costefective than traditional FD methods. (v) In particular, for every practical problem for which we do not have the exact solution, we frst use the traditional FD method to solve it. Te estimated shape parameter for the RBF-FD technique may then be found using the algorithm for determining the "optimal" shape parameter presented in Section 3 and [22,23]. In our experience, the shape parameter may be chosen c e ∈ [1, 100] while still ensuring that the RBF-FD approach is more stable and performs better than the standard FD method (see Figure 17 and Tables 3 and 5).
Finally, two RBF-FD and RBF-FDTD approaches for simulating the transient voltage of a typical electric circuit, lightning-induced voltages on the power distribution line, and lightning transient voltages along two horizontal grounding electrodes are presented. In addition, parameters such as the type of RBFs, the number of temporal and spatial steps, and the technique for selecting "optimal" shape parameters are used to increase the accuracy of RBF-FD and RBF-FDTD solutions. Te numerical results obtained by the proposed methods are compared with the numerical results obtained by many other existing numerical methods, and it can be determined that (i) the frst advantage is that, using the same number of discretization points, our proposed methods are more accurate than other numerical methods when choosing the "optimal" shape parameter. It should be noted that the computation cost of the RBF-based FD method, in general, is much higher than that of the traditional methods; (ii) the second advantage is that we can reduce the number of discretization steps in the RBF-based FD approach while still obtaining the same level of accuracy as with the conventional FD method that requires more discretization steps. In this case, the computation cost can be  signifcantly reduced; (iii) our proposed methods can be applied to any practical transient problem without the need for an exact solution, such as lossless, uniform, nonuniform lumped, and distributed parameter lines, as well as complex electrical networks in power systems, by using the algorithm for estimating the shape parameter. In the near future, this will all be done.

A. The Weighting Coefficients of RBF Approximations
We consider the GA RBF as presented in Table 1, the frstand second-order derivatives with respect to the variable x are presented as Substituting (A.1) into (2) and (A.2) into (3) centered at three points of x i − ∆x, x i and x i + ∆x, we obtain two systems of linear equations presented in the following matrix forms as It can be noticed that applying the abovementioned procedure of (A.1) to (A.8) to the MQ and IMQ RBFs in the same way, we can also fnd the unknown weighting coefcients of (α MQ , β MQ ), (α IQ , β IQ ), and (α IMQ , β IMQ ), respectively.

B. The Optimal Estimated Shape Parameter
We consider the infnitive error norm (11) of the numerical solution of the transient voltage v(x, t) in Case 4, and the estimated local error is formulated as Applying the RBF approximation of (2) to (B.1), we get It is easy to see that v(x i − ∆x) � − v(x i + ∆x) and α 1x � − α 3x . So, (B.2) can be rewritten as follows: International Transactions on Electrical Energy Systems ϵ e (c) � 2α 3x v x i + ∆x − v ′ x i .

(B.3)
Expanding Taylor's series to the transient voltage, we

(B.4)
Substituting (B.4) to (B.3), we obtain the following: It can be noticed that three derivatives of v ′ (x i ), v ‴ (x i ), and v (V) (x i ) are calculated based on the equation system of (30) and (31) where that is solved using the conventional FDTD method.
Substituting (B.5) to (11), we obtain the RBF-FD error norm and the optimal value c * e that is presented as in Figure 17.
Applying the abovementioned procedure of (B.1) to (B.5) to the transient voltage v c (t) in Case 1 and Case 2 in the same way by replacing x with t, we can also fnd the values of c * e and ‖E(c * e )‖ ∞ as presented in Tables 3 and 5.

Data Availability
Te parameter data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.