The voltage of a line must remain fairly constant because the active power consumed by customers varies. Generally, the voltage variation from zero to full load should not exceed a certain value ranging from 5% to 10% of the nominal voltage depending on the type of system. The maximum power that a line can carry also depends on the voltage variation [23] . Here, the study is done on a compensated inductive line. We consider an inductive line of negligible resistance but which has an inductive reactance X and a capacitive reactance X_c. The voltage V measured at the end of the line decreases when the load increases [23] .

$V_{\infty }=E-X\cdot I$ (1)

with: E: the electromotive force in Volts.

The relation (1) between the voltage V at the load busbar and the load current I is called. This system which defines the equation of a straight line passing through V and having the slope as shown in Figure 1 next.

As a reminder, this model only makes sense in the case of a steady state. We also consider that this machine is connected to a so-called “infinite” network, i.e. one whose voltages and phase remain constant regardless of the disturbances imposed [23] . The operation of this system is then described by Equation (1), written in polar notation. The network here represents the synchronous reference. This results in the vector diagram in Figure 2 below.

$E-jX\cdot I=V_{\infty }$ (2)

The regulation can be improved and the transportable power increased by adding a capacitive reactance X_c at the end of the line. The voltage V can be kept constant by adding the value of X_c so that the reactive power supplied by the capacitor see expression (3) below is equal to half the reactive power absorbed by the line see expression (4) below [23] :

$P_{ab}=\frac{u^2}{x_c}$ (3)

$P_f=x_c\cdot I^2$ (4)

with: $P_f$: power supplied by the capacitor in kVAr and $P_{ab}$: reactive power absorbed by the line in kVAr.

However, we see that there is always an upper limit P_max to the active power carried by the line.

$P_{ac}=\frac{E}{x}\left(2E-V_{\infty }\right)\cdot \sin \delta$ (5)

with: $P_{ac}$: Upper limit to the active power that the line transports in MW.

Relation (5) shows that the transmissible power as a function of the phase shift angle “δ” exists between the machine and the infinite network. The curve of this power as a function of the phase shift is then half-sinusoidal see Figure 3 below [23] .

Points “S” such that “δ” is less than π/2 are stable, while points “I” for which “δ” ∈ [π/2, π] are unstable. This can be explained by the following two remarks: Let “P_m” be the mechanical power supplied to the generator, cfr Figure 3 [23] :

For “δ < π/2”, if a small disturbance occurs tending to increase or decrease the rotor angle, the electrical power produced becomes greater or less than the mechanical power rotor angle, the electrical power produced becomes greater or less than the mechanical power. Given the equation of motion of the synchronous machine, the angle δ tends to decrease because “ $\frac{{\text{d}}^2\delta }{\text{d}{t}^2}<0$”. In both cases, there is a propensity of the system to return to equilibrium.

In both cases, the system tends to return to equilibrium.

For “δ > π/2”, the behavior is the opposite: far from its equilibrium position, the rotor tends to move further away from it! Although in theory the stability zone extends from 0 to π/2, in practice we are limited to a value of the order of δ_max ~ 30˚ to 35˚, for a heavily loaded network. The stability margin thus preserved makes it possible to maintain the system in a stable state, even in the event of a transient disturbance such as a change in a modification of the load distribution during a network reconfiguration, a modification of energy production, or in the event of a temporary fault or loss of the generator set.

Note: A distinction must be made between the angle “δ” between the voltages of two neighboring network nodes and the angle between two. The value of the latter can reach 45˚ [23] .

In the equivalent diagram of Figure 4 below, we neglect, as a first approximation, the effect of the capacities of the capacitors and the inherent resistance of the lines. The latter is then represented only by an inductance X as shown in the following diagram [23] :

By a reasoning similar to the previous one, we deduce that the power transfer is guided by expression 6. These results can also be obtained from the load distribution equations.

$P=\frac{u_s^2}{x}\cdot \sin \delta$ (6)

Either:

$P=\frac{u_s}{x}\left(2u_s-u_R\right)\cdot \sin \delta$ (7)

with: $P$: power transmissible by the inductive line in MW; $u_s$: source voltage in kV and $u_R$: voltage at the line output or at the load terminals kV.

Disturbances in the network load pattern have a direct impact on this angle δ, which requires it to be kept, for stability reasons, below a value generally lower than 30˚ - 35˚. For a connection loaded at its natural power, this angle limits the series reactance, and therefore sets a limit length. The maximum power P_max is reached if δ = 30˚ and u_R = u_s.

$P_{\max }=\frac{u_s}{2\cdot x}\left(2u_s-u_R\right)$ (8)

Legend: $P_{\max }$: Maximum power in MW; $u_s$: Source voltage in kV; $u_R$: Load voltage in kV and $x$: Line reactance in Ohm .

The lightning channel and the elements carrying the lightning current to the ground create an electromagnetic field. Induced currents and voltages will then appear in nearby conductors. The resulting potential differences can in turn cause breakdowns in the electronic or electrical elements connected to these conductors. These breakdowns can also be of high intensity and create a risk of ignition or destruction of the same type as that created by the direct strike. Some equipment sensitive to electromagnetic disturbances can be disturbed or destroyed by the field created by a nearby lightning strike. In the presence of a thundercloud, a cloud-to-ground discharge can be initiated. This is due to a deformation of the electric field [17] [18] [21] .

To model lightning, it is considered as a time-varying current wave [23] . The oldest and simplest analytical model is a difference of two decaying exponentials. It was proposed by Bruce and Golde in 1941 [Cooray V., 2003] [24] - [27] . It is represented by the following relation:

$i\left(t\right)=I_0\left({\text{e}}^{-At}-{\text{e}}^{-Bt}\right)$ (9)

$I_0$ is the amplitude of the base current, $A$ and $B$ are time constants.

The analytical model usually adopted is that proposed by Heidler, frequently known as the “Heidler function”, where the lightning wave is represented by the base channel current by the following expression [Heidler F., 1985] [28] :

$i\left(0,t\right)=\frac{I_0}{\eta }\frac{{\left(\frac{t}{{\tau }_1}\right)}^n}{1+{\left(\frac{t}{{\tau }_1}\right)}^n}{\text{e}}^{\frac{-t}{{\tau }_2}}$ (10)

Legend:

$I_0$: amplitude of the base channel current in kA;

${\tau }_1$: Rise time constant

${\tau }_2$: Decay time constant;

$\eta$: Correction factor for wave amplitude and

$n$: is an exponent varying between 2 and 10.

The Heidler function is preferred over the classical (double exponential) form for the following reasons:

In order to reproduce a particular lightning waveform, a combination of two Heidler functions can be used. Relation (2) then becomes:

$i\left(0,t\right)=\frac{I_{01}}{{\eta }_1}\cdot \frac{{\left(\frac{t}{{\tau }_{11}}\right)}^{n_1}}{1+{\left(\frac{t}{{\tau }_{21}}\right)}^{n_1}}{\text{e}}^{\frac{-t}{{\tau }_{22}}}+\frac{I_{02}}{{\eta }_2}\cdot \frac{{\left(\frac{t}{{\tau }_{11}}\right)}^{n_2}}{1+{\left(\frac{t}{{\tau }_{21}}\right)}^{n_2}}{\text{e}}^{\frac{-t}{{\tau }_{22}}}$ (11)

A combination of the Heidler function and the double exponential (relations 1 and 3) was used by Nucci to reproduce a typical lightning wave obtained by measurements [Nucci, CA, Diendorfer G., Uman M., Rachidi F., Ianoz M., and Mazzetti C., 1990] [29] .

In recent years, several models of the return arc, with different degrees of complexity, have been developed to assess its electromagnetic radiation (e.g. [30] - [36] ). One of the major difficulties in modeling the lightning channel is that the current can only be measured at the base of the channel; however, to determine the radiated electric and magnetic fields, it is necessary to know the current distribution along the channel. The proposed return arc models differ essentially from each other in the description of the spatial and temporal distributions of the current along the lightning channel $i\left(z^{\prime },t\right)$. In [33] , the back-arc models are classified into four categories: the gas dynamic models, the electromagnetic models [37] [38] , the distributed circuit models [39] [40] and the engineering models [30] [32] [34] . A general description of the four models is given in [33] [34] . In this paper, we will consider only the five models presenting respectively the spatial and temporal distribution of the back-arc current, namely, the BG, TL, TCS, MTLL and MTLE models. which, Rakov [41] (and recently) [33] [42] , expressed the technology models by the following generalized equation:

$i\left(z^{\prime },t\right)=u\left(t-\frac{z^{\prime }}{v_f}\right)\cdot P\left(z^{\prime }\right)\cdot i\left(0,t+\frac{z^{\prime }}{c}\right)$ (12)

where: $u\left(t\right)$: is the Heaviside function $\{\begin{array}{l}t\ge \frac{z^{\prime }}{v_f}\to u\left(t\right)=1\\ aileurs\to u\left(t\right)=0\end{array}$

$P\left(z^{\prime }\right)$: is the height-dependent attenuation factor introduced by (Rakov and Dulzon) [41] , and $v_f$: is the upward return arc propagation speed, $v$: is the propagation speed of the current wave. Table 1 below summarizes the parameters $P\left(z^{\prime }\right)$, and $v_f$ for different models mentioned above:

Based on expressions (9) and (4) below, we implemented the five spatio-temporal distribution models of the lightning current wave:

$\{\begin{array}{l}i\left(t\right)=I_0\left({\text{e}}^{-At}-{\text{e}}^{-Bt}\right)\\ i\left(z^{\prime },t\right)=u\left(t-\frac{z^{\prime }}{v_f}\right)\cdot P\left(z^{\prime }\right)\cdot i\left(0,t+\frac{z^{\prime }}{v_f}\right)\end{array}$

Bruce and Golde [23] considered that the return arc propagates vertically from the ground with a velocity “v” which is less than the speed of light “c”. The current at each point of the channel can be expressed, mathematically as follows:

$i\left(z^{\prime },t\right)=i\left(0,t-\frac{z^{\prime }}{v}\right)$ (13)

By combining equation (9) and (13), we obtain below, the BG model reproducing the spatio-temporal distribution of the lightning current wave.

$i\left(z^{\prime },t\right)=I_0\left({\text{e}}^{-A\left(t-\frac{z^{\prime }}{v}\right)}-{\text{e}}^{-B\left(t-\frac{z^{\prime }}{v}\right)}\right)$ (14)

This model equates the lightning channel to a lossless transmission line where a current pulse propagates from the ground at a constant return arc velocity “v”. This model was proposed by “Uman” and “McLain” in 1969 [42] , and is widely used to this day. The current distribution is defined by:

$i\left(z^{\prime },t\right)=i\left(0,t-\frac{z^{\prime }}{v}\right)$ (14a)

By staging expression (9) and (14a), we easily find the TL model corresponding to the lightning current distribution.

$i\left(z^{\prime },t\right)=I_0\left({\text{e}}^{-A\left(t-\frac{z^{\prime }}{v}\right)}-{\text{e}}^{-B\left(t-\frac{z^{\prime }}{v}\right)}\right)$ (15)

In order to overcome the shortcomings of the TL model while maintaining its simplicity which allows easy use in coupling calculations, a modification to the TL model has been proposed by (Nucci et al.), (Rachidi and Nucci), and (Rakov and Dulzon) [43] - [45] . These two models are described below:

$i\left(z^{\prime },t\right)=\left(1-\frac{z^{\prime }}{h}\right)\cdot i\left(0,t-\frac{z^{\prime }}{v}\right)$ (16)

Combining Equation (9) and (16), we find the “modified transmission line” (MTL) model that can reproduce the spatio-temporal distribution of lightning current.

$i\left(z^{\prime },t\right)=\left(1-\frac{z^{\prime }}{h}\right)\cdot I_0\cdot \left({\text{e}}^{-A\left(t-\frac{z^{\prime }}{v}\right)}-{\text{e}}^{-B\left(t-\frac{z^{\prime }}{v}\right)}\right)$ (17)

In this model the spatio-temporal distribution, the parameter λ represents the decay rate of the current intensity along the channel; its value, was determined by (Nucci and Rachidi) (λ = about 2 km) [46] , and by (Lin et al.) using simultaneous recordings of electromagnetic fields at several distances published in [47] [48] . The MTLE model is the most used, the compact MTLE model of the lightning current distribution $i\left(z^{\prime },t\right)$ is defined by:

$i\left(z^{\prime },t\right)={\text{e}}^{\frac{-z^{\prime }}{\lambda }}\cdot i\left(0,t-\frac{z^{\prime }}{v}\right)$ (18)

The staging of Equations (9) and (18) gives the generic MTLE model below:

$i\left(z^{\prime },t\right)=I_0\cdot \left({\text{e}}^{-A\left(t-\frac{z^{\prime }}{v}\right)}-{\text{e}}^{-B\left(t-\frac{z^{\prime }}{v}\right)}\right)\cdot {\text{e}}^{\frac{-z^{\prime }}{\lambda }}$ (19)

For the calculation of the electromagnetic field radiated by a ground-cloud lightning discharge, the commonly adopted geometry is that shown in Figure 5 above. The Lightning channel is considered as a one-dimensional vertical antenna of height H placed above a conductive plane. The return arc propagates vertically from the ground with a speed v. It is traversed by a current whose spatio-temporal distribution $i\left(z^{\prime },t\right)$ determines the electromagnetic field at any point in space.

For the case of a perfectly conductive ground, the dipole formalism offers the advantage of a mathematical model which can be written in both frequency and time, which avoids mathematical transformations (FFT or Laplace) [48] .

$\{\begin{array}{l}\text{d}{E}_r\left(r,z,t\right)=\frac{\text{d}{z}^{\prime }}{4\pi {\epsilon }_0}[\frac{3r\left(z-z^{\prime }\right)}{R^5}{\displaystyle \underset{0}{\overset{t}{\int }}i\left(z^{\prime },\tau -\frac{R}{c}\right)\text{d}\tau }+\frac{3r\left(z-z^{\prime }\right)}{c{R}^4}i\left(z^{\prime },t-\frac{R}{c}\right)\begin{array}{c}\\ \\ \stackrel{}{}\end{array}\\ -\frac{r\left(z-z^{\prime }\right)}{c^2{R}^3}\frac{\partial i\left(z^{\prime },t-\frac{R}{c}\right)}{\partial t}]\\ \text{d}{E}_z\left(r,z,t\right)=\frac{\text{d}{z}^{\prime }}{4\pi {\epsilon }_0}[\frac{2{\left(z-z^{\prime }\right)}^2-r^2}{R^5}{\displaystyle \underset{0}{\overset{t}{\int }}i\left(z^{\prime },\tau -\frac{R}{c}\right)\text{d}\tau }\begin{array}{c}\\ \\ \stackrel{}{}\end{array}\\ +\frac{2{\left(z-z^{\prime }\right)}^2-r^2}{c{R}^4}i\left(z^{\prime },t-\frac{R}{c}\right)-\frac{r^2}{c^2{R}^3}\frac{\partial i\left(z^{\prime },t-\frac{R}{c}\right)}{\partial t}]\\ \text{d}{H}_{\phi }\left(r,z,t\right)=\frac{\text{d}{z}^{\prime }}{4\pi }\left[\frac{r}{R^3}i\left(z^{\prime },t-\frac{R}{c}\right)+\frac{r}{c{R}^2}\frac{\partial i\left(z^{\prime },t-\frac{R}{c}\right)}{\partial t}\right]\\ R=\sqrt{r^2-{\left(z-z^{\prime }\right)}^2}\end{array}$ (20)

where: $i\left(z^{\prime },t\right)$: the current flowing through the dipole $\text{d}{z}^{\prime }$ at time $t$; ${\epsilon }_0$: the permittivity of the vacuum; $c$: the speed of light; $R$: the distance between the dipole and the observation point; $r$: the horizontal distance between the channel and the observation point.

In order to study the interaction between a lightning wave and a wire structure (power transmission line, cable, pylon, etc.), several works have been published in the literature and various mathematical models are proposed [49] [50] .

The most rigorous modeling is that based on antenna theory which consists of solving an integral equation in electric or magnetic field by the numerical method called moments [51] . However, the threadlike nature of power lines has led to a more flexible modeling, it is the modeling by transmission line theory which is established under certain conditions [52] . In this case it is a question of solving the equations of the lines with or without second member in time or frequency.

For this purpose, we present the modeling based on the resolution of the equations of the lines with or without second member by FDTD of the interaction between a lightning wave (direct and indirect impact) and a power transmission line. The field theory developed from Maxwell’s equations [53] , is a rigorous modeling but not always easy to apply and allows to establish the second theory that of electric circuits. At any point in space, which is not located on a separation surface between two media, that is to say, in a linear, homogeneous, and isotopic medium (LHI), the general equations of Maxwell specify that [54] :

$\nabla \times E\left(t,r\right)=-\frac{\partial B\left(t,r\right)}{\partial t}$ (21)

$\nabla \times H\left(t,r\right)=J\left(t,r\right)+{\epsilon }_0\frac{\partial E\left(t,r\right)}{\partial t}$ (22)

$\nabla \cdot D\left(t,r\right)=\rho \left(t,r\right)$ (23)

$\nabla \cdot B\left(t,r\right)=0$ (24)

The basic variables of these equations are:

$B$: Magnetic induction [Tesla: T]

$H$: Magnetic field intensity [Ampere/meter²: Am⁻²]

$D$: Electric flux density [Dove/meter: Cm⁻²]

$E$: Electric field density [Volt/meter: Vm⁻¹]

$J$: Electric current density [ampere/meter²: Am⁻²]

$\rho$: Electric charge density [Coulomb/meter³: Cm⁻³]

With:

$B=\mu H$ (25)

$D=\epsilon E$ (26)

$J=\sigma E$ (27)

And:

$\mu$: Magnetic permeability

$\epsilon$: Electrical permeability

$\sigma$: Electrical conductivity

Consider a transmission line with n conductors parallel to the x-axis and located at a height h above a ground of finite conductivity and subjected to an electromagnetic disturbance and $\left[E^e\right]$ as $\left[B^e\right]$ shown in Figure 6 .

From equations (22) and (23) [54] , we can model the coupling between an external electromagnetic field and a transmission line via the following system of equations [55] :

$\{\begin{array}{l}\frac{\text{d}\left[U\left(x\right)\right]}{\text{d}x}+\left[Z\right]\left[I\left(x\right)\right]=j\omega \left[{\displaystyle \underset{0}{\overset{h}{\int }}{B}_y^e\left(x,z\right)\text{d}z}\right]+\left[E_x^e\left(x,0\right)\right]\\ \frac{\text{d}\left[I\left(x\right)\right]}{\text{d}x}+\left[Y\right]\left[U\left(x\right)\right]=-\left[Y\right]\left[{\displaystyle \underset{0}{\overset{h}{\int }}{E}_z^e\left(x,z\right)\text{d}z}\right]\end{array}$ (28)

with: $\left[Z\right]$ and $\left[Y\right]$: respectively denote the longitudinal linear impedance and the transverse admittance of the line.

The coupling equations of the system (28), Taylor model, can be expressed in the form of three equivalent formulations, in which the source terms are functions:

Remaining within the fundamental assumptions of transmission lines [52] , for this article, we treat the coupling of the lightning wave with a wire structure from the model established by Agrawal [57] . This model is numerically more interesting than the other two, because it only involves a single source term in one of the two equations of the system (28); this source term does not contain any differentiation with respect to time and space. In the formulation established by Agrawal, the coupling equations are in terms of so-called “Scattered Voltage” voltages and total currents. For the case of a line with linear parameters independent of frequency, the coupling equations are:

$\{\begin{array}{l}\frac{\partial \left[u^s\left(x,t\right)\right]}{\partial x}+\left[R\right]\left[i\left(x,t\right)\right]+\left[L\right]\frac{\partial \left[i\left(x,t\right)\right]}{\partial t}=\left[E_x^e\left(x,h,t\right)\right]\\ \frac{\partial \left[i\left(x,t\right)\right]}{\partial x}+\left[G\right]\left[u^s\left(x,t\right)\right]+\left[C\right]\frac{\partial \left[u^s\left(x,t\right)\right]}{\partial t}=\left[0\right]\end{array}$ (29)

where: $\left[E_x^s\left(x,h,t\right)\right]$: is the vector containing the tangential component of the exciting electric field on each conductor; $\left[u^s\left(x,t\right)\right]$ is the vector of diffracted voltages in the time domain; $\left[i\left(x,t\right)\right]$: is the vector of line currents in the time domain; $\left[R\right]$, $\left[L\right]$, $\left[G\right]$ and $\left[C\right]$: are respectively the matrices of the resistances, inductances, conductances and linear capacitances of the line.

In the case of a direct lightning strike on an overhead line, injection of a quantity of energy into the line. This is a localized injection at a point, which is the case considered by our article ( Figure 7 below).

Then, the system of equations (28) is written as follows:

$\{\begin{array}{l}\frac{\partial \left[u\left(x,t\right)\right]}{\partial x}+\left[R\right]\left[i\left(x,t\right)\right]+\left[L\right]\frac{\partial \left[i\left(x,t\right)\right]}{\partial t}=\left[0\right]\\ \frac{\partial \left[i\left(x,t\right)\right]}{\partial x}+\left[G\right]\left[u\left(x,t\right)\right]+\left[C\right]\frac{\partial \left[u\left(x,t\right)\right]}{\partial t}=\left[0\right]\end{array}$ (30)

Equation (1) of system (30) can be written in the following form:

$\frac{\partial t}{\partial t}\frac{\partial \left[u\left(x,t\right)\right]}{\partial x}+\left[R\right]\left[i\left(x,t\right)\right]+\left[R\right]\frac{\partial \left[i\left(x,t\right)\right]}{\partial x}=0$ (31)

$\frac{\partial t}{\partial x}\frac{\partial \left[u\left(x,t\right)\right]}{\partial t}+\left[R\right]\left[i\left(x,t\right)\right]+\left[R\right]\frac{\partial \left[i\left(x,t\right)\right]}{\partial x}=0$ (32)

Or: $\frac{\partial t}{\partial x}=v$ propagation speed of the lightning shock wave in m/sec. For this purpose, let us inject this expression into (32). We therefore find:

$\frac{1}{v}\frac{\partial \left[u\left(x,t\right)\right]}{\partial t}+\left[R\right]\left[i\left(x,t\right)\right]+\left[R\right]\frac{\partial \left[i\left(x,t\right)\right]}{\partial x}=0$ (33)

where: $\left[u\left(x,t\right)\right]$ and $\left[i\left(x,t\right)\right]$: vectors of nodal voltages and currents in transient lightning regime.

For the coupling analysis by solving the line equations by FDTD, the conductor is subdivided alternately into current and voltage nodes ( Figure 8 ).

Two consecutive nodes of the same type are separated by an interval Δx. Note that the end nodes are voltage nodes and that the exciting electric field tangential to the conductor is calculated on the current nodes [55] .

As in space, the current and voltage are offset by half a time step. More precisely, the current samples are Δt/2 ahead of the voltage. Figure 9 and Equation (34) illustrate the interleaving of voltages and currents in space and time.

$\{\begin{array}{l}{\left(u^s\right)}_k^n=u^s\left(\left(k-1\right)\Delta x,n\Delta t\right)\\ {\left(i\right)}_k^{n+1/2}=i\left(\left(k-1/2\right)\Delta x,\left(n+1/2\right)\Delta t\right)\\ {\left(E_x^e\right)}_k^{n+1/2}=E_x^e\left(\left(k-1/2\right)\Delta x,\left(n+1/2\right)\Delta t\right)\end{array}$ (34)

With:

$\{\begin{array}{l}{t}_{\max }=n_{\max }\Delta t\\ l=\left(k_{\max }-1\right)\Delta x\end{array}$ (35)

More precise approximations known as centered differences are found by expanding Equation (30) in Taylor series. In the case of a direct lightning strike on an overhead line, the FDTD discretization of Equations (30) allows us to write the following system of recurrence equations, respectively for $k=1,\cdots ,k_{\max }-1$ and $k=2,\cdots ,k_{\max }-1$:

$\{\begin{array}{l}\left[i_k^{n+3/2}\right]={\left[\left[L\right]\frac{\Delta x}{\Delta t}+\frac{\left[R\right]}{2}\Delta x\right]}^{-1}\left\{\left[\left[L\right]\frac{\Delta x}{\Delta t}-\frac{\left[R\right]}{2}\Delta x\right]\left[i_k^{n+1/2}\right]-\left(\left[{\left(u\right)}_{k+1}^{n+1}\right]-\left[{\left(u\right)}_k^{n+1}\right]\right)\right\}\\ \left[{\left(u\right)}_k^{n+1}\right]={\left[\left[C\right]\frac{\Delta x}{\Delta t}+\frac{\left[G\right]}{2}\Delta x\right]}^{-1}\left\{\left[\left[C\right]\frac{\Delta x}{\Delta t}-\frac{\left[G\right]}{2}\Delta x\right]\left[{\left(u\right)}_k^n\right]-\left(\left[i_k^{n+1/2}\right]-\left[i_{k-1}^{n+1/2}\right]\right)\right\}\end{array}$(36)

Since we consider the case of direct lightning strike, we assume that the propagation of the induced voltage and current occurs through the longitudinal elements (R and L). For this purpose, the equation considered for the simulation is the first of the system (36). We can therefore derive the following recurrence equation:

$u_k^{n+1}=u_k^{n-1}+\left[L\frac{\Delta x}{\Delta t}+R\frac{\Delta x}{2}\right]i_k^{n+3/2}+\left[R\frac{\Delta x}{2}-L\frac{\Delta x}{\Delta t}\right]i_k^{n+1/2}$ (37)

We directly obtain the FDTD model of the power at any point along each line conductor of the HV electrical network in lightning regime, by replacing $u_R=u_k^{n+1}$ in Equation (8). We therefore obtain:

$P_k^{n+1}=\frac{u_s}{2X}\left[2u_s-\left\{u_k^{n-1}+\left[L\frac{\Delta x}{\Delta t}+R\frac{\Delta x}{2}\right]i_k^{n+3/2}+\left[R\frac{\Delta x}{2}-L\frac{\Delta x}{\Delta t}\right]i_k^{n+1/2}\right\}\right]$ (38)

with: $P_k^{n+1}$: FDTD model of power in lightning surge regime.

The parameters of the current at the base of the channel for the different models are presented in Table 2 below [40] : Propagation speed : v = 1.5 × 10⁸ m/s; Lightning impact assessment point : z’ = 0 km, 2 km and 4 km, vertical axis z = 7 km; distance from the impact observation point r = 500 m or 0.5 km; the height of the lightning channel was set to a value of 7.5 km; the vacuum magnetic permeability ${\mu }_0=4\pi \times {10}^{-7}$ H/m and the vacuum electrical permittivity ${\epsilon }_0=\frac{{10}^{-9}}{36\pi }$ F/m.

The numerical values of R = 0.07 Ω/km, L = 0.011 mH/km and the length of the line 262 km are given in the table of characteristics of the 220 kV line of the SNEL/DRC West interconnected network, Inga-kimwenza section [58] .

See Figures 10-15 below: