Theory shows and experience confirms that the shape of the lightning discharge current is bi-exponential [8] and its temporal evolution is represented by the curve in Figure 1 below:

I ( t ) = I 0 [ exp ( − α t ) − exp ( − β t ) ] (1)

The numerical values of the lightning parameters α et β of the bi-exponential equation [3] made it possible to obtain the pace of the curve of the figure below confirmed in theory and experimentally.

The lightning surge in electrical grid equipment results in sudden variation d i d t in lightning current on the transmission line.

u = R i ( t ) + L d i ( t ) d t (2)

By comparing the equation below to the induced electromotive force, the current lightning model in the Laplace domain becomes;

I ( s ) = μ I 0 S 2 π r L [ β ( s + α ) − α ( s + β ) ( s + α ) ( s + β ) ( s + R L ) ] (3)

The temporal lightning current flowing to the ground through the earth electrode represented by Figure 1 reaches its peak for values greater than 2.5 × 10 kA in a few microseconds.

Following Frédéric ELIE’s publication “Lightning and pitch voltage November 2005”, the parameters of the lightning current are determined experimentally and have the following values:

α = 0.02 μs, β = 1 μs, for a current amplitude of I₀ = 30 kA. μs β μs

The rise time to reach the peak of the wave is obtained by:

t m = 1 / ( − α ) ⋅ ln ( β / α ) (4)

The order of magnitude is t_m = 2.2 s.μ

The peak of the bi-exponential lightning current shown in Figure 2 constitutes shock energy; therefore, aggression to the equipment of the electrical network and presents a danger for people located in an environment close to the station.

The guard wire located at the top of transmission line pylons shown in Figure 2

also plays the role of lightning current detection; it is also used for the transmission of the high frequency signal of telecommunication between the production centers and the distribution networks in the consumption centers.

This ground wire is connected to the ground via an L-inductor cable through the pylon and a very low-value R-series resistor.

The guard wire surge transfer function translated by Equation (2) gives:

F 1 ( s ) = I ( s ) U ( s ) = 1 R + L s (5)

F₁(s) is the transfer function of the line guard cable and the numerical values of the numerical values of R = 0.07 Ω and L = 1.01 m_H are given in the table of characteristics of the line 220 kV. These values come from the National Electricity Company, SNEL/RDC [9].

F 1 ( s ) = 1 0.07 + s (6)

For the study of the phenomenon of overvoltage transmitted to the transformer of the HV substation by coupling with the flow circuit, one limits oneself to an equivalent diagram of a winding on the high voltage side in Figure 3 below.

This equivalent diagram reflects the initial distribution of the overvoltage that propagates in the power transformer of the substation.

The application of the Kirchhoff equation to node P in Figure 3 gives.

i + i k − ( i − ∂ i ∂ x d x ) − ( i k + ∂ i ∂ x d x ) − i c = 0 (7)

i c = − ∂ ( i + i k ) ∂ x d x     and     i c = − c ∂ u ∂ t d x (8)

Expressions of Equation (8) gives

∂ ( i + i k ) ∂ x = − c ∂ u ∂ t → i k = − k d x ∂ 2 u ∂ x ∂ t d x → ∂ i k ∂ x = − c ∂ 3 u ∂ x 2 ∂ t (9)

The current in inductor of equivalent diagram

∂ i ∂ t = − 1 L ∂ u ∂ x → ∂ u ∂ x = − L ∂ i ∂ t (10)

∂ ( i + i k ) ∂ x = − 1 L ∂ u ∂ t ∂ x 2 − c ∂ 3 u ∂ x 2 ∂ t (11)

Ask i + i k = i a and ∂ X ∂ t being the propagation speed of the overvoltage in the transformer V,

We have:

∂ i a ∂ t = − 1 V [ 1 L ∂ u ∂ t + C ∂ 3 u ∂ t 3 ] → V = − 1 L ∂ u ∂ t − C ∂ 3 u ∂ t 3 (12)

V is the speed of the surge wave in the tr with almost low attenuation (V = 3 × 10⁸ m∙s⁻²),

In frequency mode, the Equation (12) becomes:

F 2 ( s ) = U ( s ) I ( s ) = − V 1 L + C S 2 (13)

The numerical values of F₂(s) are given Ru, pu = 0.01 pu; Xu,pu = 0.04 to 0.18; X_L = 0.1 to 0.2 [9].

The transformer transfer function during the overvoltage becomes:

F 2 ( s ) = U ( s ) I ( s ) = − V 1 0.2 + 0.04 S 2 (14)

The behavior of the non-linear resistance of the variance elements ZnO is represented by a current expression as a function of voltage (V = F(I)), extracted in the voltage-current characteristic of the lightning arrester.

The non-linear behavior of the ZnO elements of the lightning arrester is mainly due to non-linear resistors and capacitors as represented by the figure below.

Equation (23) below reflects this nonlinearity [11].

I = K 1 V α 1 + K 2 V α 2 (15)

where:

I: Current in mA;

V: Voltage in kV;

α: Coefficient of non-linearity of lightning arrester varistors: 30 < α < 100;

K: Dielectric constant of materials specific to the lightning arrester: K₁ = K₂ = 100.

By generalizing, we obtain:

I = K V α (16)

Since the numerical values of and K is very large, we can linearize this expression by the characteristic of the lightning arrester below in the area between U_p and U_c respectively: priming voltage (protection level) and lightning arrester set voltage (maximum permanent working voltage).

In this linear area between U_c and U_p of Figure 4, the character takes the form: y = a x

I = K V (17)

where:

K = U P − U C I c − I n being the slope of the line et V a potential difference seen the characteristic in its part U_C and U_P, we can then write:

i ( t ) = K d v d t (18)

In the field of Laplace, we get

F 3 ( s ) = I ( s ) U ( s ) = K s (19)

F₃(s) is the transfer function of the zinc oxide surge arrester used in the HT substation.

Les values of U_c = 225 kV, U_p = 245 kV, I_c = 100 kA andI_n = 20 kA according to the manufacturers [11].

The coefficient K becomes:

K = U p − U c I c − I n = 245 − 225 100 − 20 = 0.25

In the field of Laplace, subject to the hypotheses mentioned above, we have:

F 3 ( s ) = I ( s ) U ( s ) = 0.25 s (20)

F₃(s) is the model of the lightning arrester.

In an electrical energy transmission network with HV lines and substations, grounding is carried out in several forms [12] [13]:

­ Simple grounding systems, simple electrodes;

­ Complex grounding systems, such as the large ground grids of the HT substation illustrated in Figure 5.

From the above, the grounding grid of the substation is assimilated to a transmission line characterized by an input quantity (the lightning current) and an output quantity (the potential difference around the ground socket).

Modeling of the propagation of the lightning current through the grounding grid must therefore use a description of the transmission line by constants distributed by the following expressions:

∂ i ∂ t = G u ( x , t ) + C ∂ u ∂ t (21)

∂ u ∂ t = R i ( x , t ) + L ∂ i ∂ t (22)

Equations (21) and (22) are called telegraph equations translating the variation in time and space of current and voltage along the overvoltage line.

Hypothesis:

The ground network of HV substations being local and of small dimensions, we can admit that the flow of current in the ground is mainly through the transverse elements; the current and the distribution of the potential around the ground socket are weakly related to the variable x.

Thus: only Equation (23) and the time variable are taken into account. The expression (21) becomes:

∂ i ∂ t = G u ( t ) + C ∂ u ∂ t

Laplace’s transform gives:

F 4 ( s ) = U ( s ) I ( s ) = s G + C s (23)

F₄(s) is the transfer function of the grounding grid of the HT station.

G and C shave the transverse parameters of the interconnected network (underground grounding grid) whose numerical values are obtained by the following assumptions:

• The soil is homogeneous and its characteristic parameters are constants; ε, μ et ρ.

• The reinforcements of the capacitor thus formed are the different layers of the ground and the grounding grid.

In the realization of the ground network of HT substations, some standard values are recommended by the standards (NFC 13 - 100 and NFC 11 - 200, CEI 61140, IEEE 80 - 2000) [14]:

• Dimensions of the grounding grid, GTC-25/4 series are: 2.50 × 0.4 m;

• Soil characteristics [8]: m; h ≤ 1   m ; ε r = 5 ; μ r = 1 et ρ = 10   Ω ⋅ m .

These parameters make it possible to define the permittivity and relative permeability (ε_r, μ_r) in relation to their values in a vacuum [15] [16]:

­ ε r = ε ε 0 is the relative permittivity and is μ r = μ μ 0 the relative magnetic permeability, ε r and μ r are the corresponding permittivity and permeability of the soil.

In the majority of cases, we μ = μ 0 consider; for μ r = 1 .

➢ For the resistance of the grounding of the station, we have:

R = ρ 4 r and r = a b π = 2.50 × 0.4 3.14 = 0.56 and R = 10 4 × 0.56 = 10 2.24 = 4.4   Ω

This value is acceptable by the standard and gives rise to high grounding performance ( R < 10   Ω ); a and b are respectively the length and width of the grid.

The conductance G becomes: G = 1 R = 1 3.5 = 0.28   Ω − 1

➢ For the capacity of the ground circuit

C = ε S a and ε = ε 0 ε r = 5 × 8.85 × 10 − 12 × 10 − 12

S = surface of the reinforcements = 2.50 × 0.4 = 1 m²

C = ε S a = 44.25   F = 11   pF × 10 − 12 × 1 4 = 11 × 10 − 12

The numerical values of G and C of the transfer function of the terrestrial network being determined, we can write:

F 4 ( s ) = s 0.28 + 0.01 s (24)

The nettle of the system: V p s ( s ) desired pitch voltage during the flow of lightning current on the ground.

After obtaining all the transfer functions of the components of the surge wave flow circuit of the HV substation coupled with the power transformer and, using the servo system approach, the functional block diagram of the circuit of Lightning shock wave flow is given in Figure 6 below:

The transfer function F(s) of the components of the direct cascade flow circuit becomes:

F ( s ) = F 1 F 3 F 4 (25)

F(s) represents the transient impedance of the circuit which must act on the flow velocity of the lightning current on the ground; this lightning current is considered a disturbance to the system. Below is the pace of the index response of F(s) under the action of the disturbance.

The observation in Figure 7 below, the curve shows that the corrected system is stable. Nevertheless, it has a static error of the order of 7%, the system is therefore imprecise; it is, therefore, necessary to correct it so that the output is the image of the input regardless of the presence of the disturbance.

Considering that the aggressiveness of lightning can have an impact directly on the flow circuit and the transformer of the substation, the block diagram of the closed-loop circuit with unit return gives them to the block diagram below in Figure 8.

The output of the closed-loop system taking into account the disturbance and the overvoltage induced in the transformer makes it possible to write:

S ( B ) = C ( s ) F ( s ) 1 + C ( s ) F ( s ) + F ( s ) 1 + C ( s ) F ( s ) + F 2 ( s ) P ( s ) 1 + C ( s ) F ( s ) (26)

Faced with the imprecision presented by F(s), it is essential to synthesize a corrector whose role is to reject the disturbance P(s).

Summary of the corrector C(s)

A closed-loop, F(s) although of the second order, has two transmittances of the first order cascading with a dominant pole of the form:

F ( s ) ≅ 0.25 ( s + 1 ) ( s + 27.1 ) (27)

Experience and theory support that the response time for a first-order system is t r = 3 τ since, τ = 1   s then

T r 5 % T m = 3 τ 2.2 = 3 2.2 = 1.36       et       t r = 1.36 × 2.2 ≅ 3   μ s

The transfer function of the PID regulator has the form:

C ( s ) = P + I s + D N 1 + N s (28)

Thanks to the digital tool Matlab, Simulink, the toolbox “PID turner” below, we can determine the parameters of the correct Figure 9: Modeling of the flow circuit of the shock wave environment Matlab Simulink.

The simulation results are recorded in Table 1.

The corrector settings are: P = 4.3347 ; I = 3.303 ; D = − 0.30048 ; N = − 14.4257 .

N is softening pole, which means, a correction pole.

C ( s ) = U ( s ) ε ( s ) = 4.3347 + 3.3103 s − − 0.030048 × 14.4257 1 + 14.4257 s

C ( s ) = U ( s ) ε ( s ) = 4.3347 + 3.3103 s − − 4.3346 s s + 14.4257 = − 62.2 s s 2 + 14.43 s

C ( s ) = U ( s ) ε ( s ) = − 62.2 s + 14.43 (29)

The expression (34) represents the transfer function of the corrector sought.

The simulation consists in determining the parameters of the digital PID regulator that will have to meet certain specifications of the optimal specifications according to the standard which should be exceeded; 20% gain margin of 6 to 20 dB and phase margin between 45˚ to 65˚; these two parameters reflect the robustness of the system.

The effects of the correction after simulation are given by the following curves:

The look of the curves in Figure 9, obtained from Simulink’s PID Turner subprogram, shows that the system response meets the requirements of the specifications.

Dotted, uncorrected system response and continuous, corrected system response that shows stability in a very short time after reaching its peak.

Thanks again to the program PID Turner, it can be seen that the rejection of the disturbance by the corrector is the reverse corrected response

If the curves in Figure 10(a) and Figure 10(b) reflect the response of the system in time, the figures below reflect the same response of the system in frequency regime under the action of the corrector.

The effect of the rejection of the lightning current is also observed in Figure 11, this time in the frequency domain, translated by the Bode curve according to the phase margin and the gain margin by the specifications defined below.

The effect of the rejection of the lightning current is also observed in Figure 11, this time in the frequency domain, translated by the Bode curve according to the phase margin and the gain margin by the specifications defined above.