Evaluating transient behaviour of large-scale photovoltaic systems during lightning events using enhanced finite difference time domain method with variable cell size approach

Photovoltaic (PV) arrays are usually installed in open areas; hence, they are vulnerable to lightning strikes that can result in cell degradation, complete damage, service disruption, and increased maintenance costs. As a result, it is imperative to develop an effective and efficient lightning protection system by evaluating the transient behaviour of PV arrays during lightning events. The aim is to evaluate the transient analysis of large ‐ scale PV systems when subjected to lightning strikes using the finite difference time domain (FDTD

Greenhouse gas emissions are increasing due to the continuous utilisation of fossil fuels, so utilities are working nowadays to replace them with clean sources of energy [1,2].Photovoltaic (PV) systems have indeed gained significant popularity nowadays because of the widespread availability of solar power and the declining cost of PV technology compared to other renewable sources of energy [3][4][5].One of the notable advantages of PV systems is their low maintenance requirements [6].Once installed, PV systems typically operate with minimal maintenance needs, so it is a promising replacement for fossil fuels in the future [7].PV systems harness the Sun's energy by using solar panels constructed from semiconductor materials to convert sunlight into electrical power.Hence, they should be located in open-air areas and rooftops to maximise their exposure to sunlight.However, the vulnerability of PV systems to lightning strikes is a concern that needs to be addressed during the installation and design process [8].
Lightning strikes result in high transient overvoltages and potential damage to PV systems [9].Also, they can cause deterioration or destruction of the PV system as well as failures in sensitive electronic components such as meters, inverters, and data networks [10,11].Furthermore, other power system components like transmission lines can also be affected [12].The switching of vacuum circuit breakers produces also transient overvoltages in offshore wind farms [13].To ensure effective lightning protection for PV systems, it is necessary to study the transient behaviour of lightning.The differences between the lightning strikes to distribution lines and ground obtained from electromagnetic analyses and measurements are discussed in ref. [14].Different modelling methods have been employed for this purpose, including the partial element equivalent circuit (PEEC) [15], the method of moments [16], the magnetic vector method [17], and the finite difference time domain (FDTD) [18].The FDTD method offers a precise analysis of the magnetic and electrical field distribution throughout the PV system.However, its computational burden is a challenge that needs to be addressed.The FDTD method provides a highly accurate model for designing an efficient lightning protection system tailored to safeguard PV systems against lightning strikes.By incorporating the FDTD method, the design process can ensure the necessary protection measures to reduce the risks of lightning-induced transients [19].
The representation of the thin wire to be used in FDTD in the 3D simulation was discussed before ref.[20], where a general program was proposed for analysing surges based on the FDTD method.A non-uniform transmission line approach was presented before ref.[21] to model the transients during lightning occurrence in grounding systems required for the protection of PV systems, where the FDTD method was used for solving telegrapher equations to analyse the transient overvoltages along the length of the earthing electrode.The transient electromagnetic fields near the lightning channel above the ground where the PV system is installed were studied by the FDTD scheme [22].The execution time and memory needed for FDTD calculations were minimised by applying the boundary conditions of the surface impedance to avoid simulating the conductive region underground.
The FDTD method was used for modelling the earthing system during the transient period during lightning occurrence [23].This work applied the singularities by which the field varies around the conductor to Faraday's contour path to reduce the computational time of the standard FDTD, but this study suffered from the limited focus on the earthing system of the PV system and did not consider other parts connected with the earthing system.
The usage of the radial bias function (RBF) to the approximated derivatives in the finite difference was presented earlier [24] for analysing the transient overvoltages resulting from lightning strikes in grounding systems; the proposed RBF-FDTD provided higher accuracy than the conventional FDTD solution if the shape of the parameter was optimal.Different applications of the FDTD for analysing the lightning transient behaviour in different systems, such as PV systems, wind turbines, transmission lines, electric and airborne vehicles, power stations, buildings, and grounding electrodes, were discussed earlier [25], and the basic concept of FDTD was discussed.Also, the advantages and disadvantages of FDTD were compared to the other electromagnetic computation methods.The transient overvoltages generated in the structure anchoring PV system were evaluated using a reduced scale model in the field [26], where the measured values were verified using the FDTD algorithm.The reduced model and FDTD theory were used to analyse the overvoltages in the power conditioner DC side when the supporting structure of the solar array was subjected to lightning strikes.However, the PV system model did not consider PV modules, metal frames, or mounting systems.
A study of the transient currents in the grounding system using the FDTD was presented earlier [27], which was optimised to reduce the computation time by predicting the ground resistance.Moreover, the foundation model of the DC distribution system in the PV system was modelled, and the current withstand capability of the surge protective device (SPD) was evaluated using the FDTD method by analysing the transient overvoltages and currents in the PV system [28].The research work presented earlier [29] proposed an FDTD model of the thin wire with a noncircular cross-section, which was used to analyse the transient behaviour during lightning.The calculation of the correction factor of magnetic and electric fields was introduced to apply the FDTD.
The transient overvoltages generated from lightning strikes in the large PV systems were analysed by the virtual surge test lab (VSTL) tool [30].The VSTL tool was based on the FDTD theory and used to analyse and evaluate the expected damage to different parts of the PV system, such as DC cables, PV panels, and inverters.However, it was mandatory to simplify this large system, which required high-capability hardware and additional costs to apply the FDTD algorithm.Analysis of storm magnetic fields using the 3-D FDTD method was explained earlier [31], where the proposed scheme enabled the modelling of large and complex earthing systems with less computation burden.Furthermore, the coupling effects analysis of the insulated wire was discussed before using the FDTD theory [31].Compared with the traditional FDTD method, this proposed scheme reduced the memory required and the unknown numbers.On the other hand, adaptive processing in the space-time domain was presented in ref. [32] to increase the computation speed of the FDTD method in large systems.The proposed scheme used a quadrature structure to execute the adaptive processing to reduce the time required when the equivalence theorem was applied to connect multiple regions.The FDTD method can also be used to study the transient overvoltages on a 10-kV distribution line based on an electromagnetic return-stroke model [33].
Based on the literature survey, the FDTD research work related to lightning transients in PV systems was quite limited, and many researchers did not study the effect of some important PV system components on the transient voltage values such as metal frames and the mounting system.The work presented in ref. [34] used simple steady-state equations for the PV module and did not take into account the impact of the frame, mounting system, and high-frequency nature of lightning strikes.Although the grounding system of the PV model was represented by reasonable accuracy in ref. [35], the PV system suffered from a lack of accuracy where the highfrequency effects of lightning were not considered.Also, the PV system model was very simple and far from the practical cases in which the mounting system and metal frame exist.Although the authors in refs.[36][37][38] considered all PV system components, including the inverter and transformer, the PV module model was very simple and neglected the practical aspects of the real PV, such as the mounting system, metal frame, and connection with the ground.The effect of the grounding system configuration on the transferred voltage between the mounting system and DC cables was discussed earlier [34].However, the authors did not consider the effect of the lightning strike position, which affects the transient overvoltage values and hence the protection system design.In addition, the authors did not consider the effect of the PV metal frame existence and the effect of changing the mounting system's grounded leg.Moreover, the execution time is still long (about 3 h), which needs further reduction by improving the FDTD model.
In order to fill the research gaps and shortcomings in the previous works, this paper studies the transient behaviour of large-scale PV systems using the FDTD approach, including panels, the mounting system, and the earthing system.In addition, the execution time of FDTD is reduced by considering a variable cell size in the PV system while studying the transient behaviour.A larger cell size for the grounding system (variable cell size) has been selected, which saved much time.The cell size in the case of the PV system is small because it is more sensitive to the transient effects of lightning and may cause damage, especially to the PV module.Hence, it is crucial to ensure higher accuracy in calculating the transient voltages and currents, making the protection system more effective and efficient.The installed protection system, including the number of SPDs, complies with the IEC-62305 standard based on risk assessment calculations which depend on the probability of different types of risks, including the risks to human lives and physical damages to structures.
On the other hand, the grounding system has been divided into larger cells because it is less sensitive to the high-frequency lightning effects and can withstand them without observed damage.The cell size of the grounding system has been selected to be less than the maximum acceptable size, keeping a reasonable accuracy of the FDTD method.Moreover, the effect of changing the lightning strike location and soil resistivity in addition to the influence of metal frame existence on the transient overvoltages were discussed.Furthermore, the transient overvoltage results obtained using the FDTD approach are compared with those obtained using the PEEC modelling approach, which is widely used by researchers in performing transient analysis.This is performed to highlight the higher accuracy of the FDTD method when compared with the PEEC method.Based on the obtained precise results by FDTD, an effective and reliable protection system can be installed.Moreover, the lightning protection scheme of a multiterminal distribution system operates more efficiently if the transient overvoltages are obtained using the FDTD method [39].In addition, a laboratory experiment is conducted on a small-scale PV system where the strike is simulated by an impulse generator.Then the obtained simulation results are compared with the measured ones using the mean absolute error method to further verify the accuracy of simulation results.
In addition to Section 1, which presented a comprehensive review of different numerical electromagnetic analysis methods, the remaining part of the paper is organised as follows.Section 2 discusses the modelling of the PV system's various components.Section 3 illustrates the three-dimensional FDTD method, including the general steps and the related equations.Section 4 presents the simulation results and their associated discussions.Finally, the conclusions are presented in Section 5.

| PV SYSTEM MODEL
For more practical analysis, the FDTD model is applied on one string consisting of nine PV modules of a large-scale PV system as shown in Figure 1 [40].For simplicity and due to the limited hardware capabilities, one PV string is simulated to study the transient overvoltages during lightning.The transient voltages are calculated at four different points (V1, V2, V3, and V4) on the mounting system of a certain PV module as shown in Figure 1.Every three panels are connected in series to form a PV string, and the three strings are connected in parallel to form a PV array which is connected to an inverter that converts DC voltage to AC voltage.The capacity of each panel is 4 kW.The mounting structure is fabricated from C profile steel, with 3 m width, 4 m length, and 3 m height, and extends 2 m above the ground and 1 m below.The thickness of the C steel profile is 3 mm with a width of 40 mm, and the DC cable diameter is 4 mm.The mounting systems are separated by 3 m.The mounting system of PV panels is connected to a mesh grounding grid implemented Using the FDTD model, the transient overvoltages at the four corner points in a certain module can be calculated.The wiring system required in the earthing system, the DC cables that connect different PV modules, and all metal parts, including the mounting system and metal frame, are considered in the FDTD model.In addition, the proposed non-uniform cell sizing in the FDTD model can greatly reduce the execution time of simulation to about 50 min instead of about 10 h in the case of the uniform cell size.This large reduction in execution time enables the researchers to study complex systems even with low hardware capabilities.

| 3D FINITE DIFFERENCE TIME DOMAIN METHOD
The FDTD is based on investigating the electric and magnetic fields in the system according to Maxwell's equations which gives highly accurate results to determine the transient voltages and currents.The FDTD calculates the electric and magnetic fields at discrete time intervals where Maxwell's differential equations are converted to finite differences in the time domain.The electric and magnetic field values are then determined at each time instant to give a full description of the fields through time.The main challenge facing the FDTD method is its complexity and time-consuming execution.This paper applies the FDTD method to the PV system, considering all system components.The execution time in this work is improved by increasing the cell size of the earthing system to the maximum stability limit and using smaller cell sizes for the panel, mounting system, and DC cables to provide proper protection for the PV system components.The electric and magnetic fields inside any studied system are described by the following differential equations as follows [20]: where E and H are the electric and magnetic fields, respectively.Moreover, μ, ε, and σ represent the permeability, permittivity, and conductivity, respectively.The PV system is divided in the space into cubic cells, and the values of electric and magnetic fields in the cell are updated at each time-step (dt).The magnetic field (H) is computed at 0.5dt, 1.5dt, …, (n þ 0.5)dt while the electric field (E) is computed at 0, dt, 2dt, …, ndt instants.The size of the cell (dx, dy, and dz) should be less than 1/10 of the wavelength of the highest frequency in the transient period [25].
The time-step (dt) required to apply FDTD to study the electric and magnetic fields distribution must be set to fulfil the stability condition stated by Equation (3) [25].It is clear that the time-step mainly depends on the cell dimensions, where the small cell dimension will require a smaller step size and more complexity, but the gain will be a higher accuracy of results.
dt ≤ 1 c ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi The flowchart summarising the steps to execute FDTD is shown in Figure 2 [41].The FDTD main steps are described as follows: (1) Setting the system parameters, the electric and magnetic field's initial conditions, and the values of conductivity, permeability, permittivity, and resistivity.(2) Updating the electric field and magnetic field values at each time-step based on the previous values of the magnetic and electric fields in the previous iteration, where the equations used to calculate the updated values will be discussed in the next section.(3) Checking if the time-steps have finished according to the simulation time, and the value of the time-step is adjusted to achieve the algorithm's stability.(4) The obtained output data are extracted according to the final updated values of magnetic and electric fields distribution at all points in the studied PV system based on the system dimensions and the cell dimensions.The initial conditions of the electric and magnetic fields are set to zero, where their updated values in x-direction (E x and H x ) at each time-step (n) can be calculated in the 3D space coordinates (i, j, k) as follows [41]: where ζ ix and M ix represent the impressed electric and magnetic current densities in the x-direction, respectively.The updated values of the magnetic and electric fields are obtained based on the values of the electric and magnetic fields at the previous time instants.The execution of these calculations continues till the end of the total time.In the studies of lightning transient analysis, there are many representations of the lightning channel, one of these representations is the phased current source array in air.The lightning channel is composed of discrete, linear current segments that facilitate the transfer of electrical charge.These charge-carrying segments are interconnected and are surrounded by corona sheath segments, which share a similar geometric structure.The corona sheath segments serve to insulate the charge-carrying segments from the surrounding environment, allowing the lightning channel to maintain its integrity and effectively transport electrical charge.The lightning current is represented by a lumped current source at the bottom of the lightning channel.The lumped current source is connected at the striking point and expressed by a mathematical equation such as Heidler or double exponential functions.The lightning current formula was proved by Heidler [42] based on converting the real values of lightning current in the time domain to its corresponding equation in the time domain.The Heidler function is expressed using Equation ( 6) as follows [42]: where I max is the lightning peak current.K is the Peak current correction factor.τ 1 and τ 2 are the time constants.The value of peak current is taken as 100 kA, K = 0.93, τ 1 = 19 μs and τ 2 = 485 μs [42].This current is applied in the z-direction, which is added to the updated equations of the cell.
If the cell contains a thin wire, its relative permittivity, effective conductivity, and relative permeability will be changed, which will change the propagating electric and magnetic field values in these cells.The change in the electric and magnetic field will definitely affect the values of the produced overvoltages and currents.The updated values of the relative permittivity, effective conductivity, and relative permeability are determined as follows [41]: where m is the correction factor, a is the thin wire radius, s is the cell side length, and μ r means relative permeability.The PV model and the earthing system are shown in Figure 1, which are modelled using the FDTD method by the appropriate time and space sizes according to the previous subsection.The PV and grounding system were divided in the space into cubic cells of sides dx, dy, and dz.The value of dx, dy, and dz is 0.005 m for each cell inside the PV system and 0.05 m for the grounding system.The number of cells in x, y, and z directions is 3600, 3000, and 600 inside the PV system grid and 360, 300, and 60 inside the grounding system grid, respectively.This non-uniform division of cells in the space reduces the execution time by software from 1 h in the case of the uniform division of cells to nearly half an hour in the case of the non-uniform division of cells.Also, the flop counts, including multiplication/division and addition/ subtraction for the grounding system analysis in the proposed scheme, were reduced to about 55% of the flops in the uniform grid.The PC utilised for the transient voltage calculation includes a processor manufactured by Intel, specifically the Intel (R) Core (TM) i5-2450M CPU, which operates at a clock speed of 2.5 GHz.The processor is complemented by 4 GB of installed memory, providing sufficient computational capacity for the simulation.The MATLAB software platform is employed as the environment for conducting the analysis.The constants for the analysis by the FDTD method are summarised in Table 1.
In addition, the magnetic and electric field distribution at each time-step is calculated and updated at all system points.The analysis results focus on the four corner points in the mounting system according to Figure 1.The voltage values at these points are calculated considering the change of the striking point with change in the soil resistivity, existence of metal frame, and change in the grounding leg in the mounting system.Based on the impulse current value in (kA) and the calculated transient overvoltages in (kV), the appropriate SPD can be selected by matching its impulse current value and the required voltage protection level in (kV) based on the calculated transient overvoltages from the simulation model.The percentage difference (%D) in the simulation results between the FDTD and PEEC methods is calculated based on Equation (11) as follows:

| Effect of changing striking point
The transient overvoltages of the four corner points (V1, V2, V3, and V4) are changed by changing the lightning striking point, as shown in Figure 3a,b.The distance between points 1 and 4 and points 2 and 3 is 3 m.Also, the distance between points 1 and 2 and points 3 and 4 is 4 m. Figure 3a shows the transient overvoltages when lightning strikes point 1.It could be noted that the overvoltage value at point 1 is the largest when lightning strikes point 1 because the magnetic and electric fields are at their maximum value at the striking point.
On the other hand, the overvoltage values decrease by increasing the distance from the striking point because the electric and magnetic fields associated with the lightning current decrease when moving away from the strike location.The minimum voltage is obtained at point 3 since it had the least lightning effect, where it is located at the farthest position from the striking point.Furthermore, Figure 3b shows the overvoltage values when lightning strikes point 4. It is clear that point 4 (the striking point) has the largest overvoltage value compared with the other points.In addition, the transient overvoltages decreased by moving away from the striking point due to the decreasing of the magnetic fields when moving away from the striking point.The results show that the point of the lightning strike has the maximum voltage value and the transient voltages decrease when moving away from the striking point towards the grounding system.This occurs according to Maxwell's theory, which is related to the propagation of the electric and magnetic fields in the space and time domains as travelling waves, where these fields become weaker when moving away from the source location (striking point).
T A B L E 1 Constant values used with FDTD method.
Constant Value dt 1 � 10 −8 s for PV and mounting system and 1 � 10 −6 s for the grounding system dx 0.005 m for the PV system and 0.05 m for the grounding system dy 0.005 m for the PV system and 0.05 m for the grounding system dz 0.005 m for the PV system and 0.05 m for the grounding system

6
- In addition, the obtained FDTD results are compared with PEEC results of the same PV system [25].Then, the overvoltage results of the FDTD model and the PEEC method at points 1 and 4 are compared to highlight the higher accuracy of the FDTD method compared with the PEEC method, which is used widely in performing the transient analysis, as shown in Figure 4a,b, respectively.It could be noted that there is a small difference between the results obtained by the two methods for the voltages at points 1 and 4. The difference between the two curves is about 9% when lightning strikes point 1 and 12% when lightning strikes point 4. The higher values of transient voltages calculated by the FDTD method help to design a more reliable lightning protection system.

| Effect of changing soil resistivity when lightning strikes point 1
Changing the soil resistivity is considered one of the most influencing factors on the transient overvoltages at different points.In this case, lightning strikes point 1 in the mounting system.The resistivity of the soil was increased by replacing the clay soil of low resistivity with clayey sand soil of large resistivity, as illustrated by Figure 5.When the value of soil resistivity rises, this leads to a higher value of grounding resistance and consequently increases the overvoltage at different points.We can observe an increase in the transient overvoltages from 8 MV to about 17 MV when the clayey sand soil is used instead of clay soil.Hence, the soil type is an essential factor in analysing the transient overvoltages, where selecting low-resistivity soil is important to obtain lower transient overvoltages at different points.As a result, the requirements for additional protection systems during lightning strikes are reduced.On the other hand, Figure 6 compares the overvoltage values at point 1 obtained by FDTD and PEEC methods in the case of clayey sand soil.It could be noted that the percentage difference between the two methods is about 15%.Based on the obtained results, it is important to make the grounding soil resistivity as small as possible since it greatly affects the transient overvoltages in the PV system when subjected to lightning strikes.HETITA ET AL. -7

| Effect of metal frame existence when lightning strikes point 1
According to the manufacturer, PV modules may have a metal frame, while others do not.The metal frame is important to protect the PV module during the transportation process.In this work, the effect of the absence of the metal frame on the transient overvoltages is illustrated in Figure 7.It is clear that the absence of the metal frame causes a decrease in the total system resistance, which results in a decrease in the transient overvoltages at different points during lightning.The obtained results show that the existence of the metal frame causes higher overvoltages than the modules without any metal frame, which requires more protection for the PV modules.The results show that the existence of the metal frame will cause higher transient overvoltages due to the mutual effect, which has a considerable effect on lightning protection system design requirements and cost.Moreover, a comparison between the results obtained using FDTD and PEEC methods is illustrated in Figure 8.It could be noted that the percentage difference between the two methods is around 5%.

| Effect of changing the grounded leg
The change of the leg connected with the earthing system causes a change in the transient overvoltages at different points.Figure 9 illustrates the effect of changing the grounded leg in the mounting system.When the grounded leg is changed from leg 1 to leg 2, this leads to a minimum voltage at the grounded leg (leg 2).When lightning strikes the PV system, the current tends to flow through the shortest path to earth (the path with lower resistance), making the propagated electric and magnetic fields larger.Hence, the transient overvoltage at the point connected to the grounded leg decreases.Based on the previous observations, the grounded leg location change is another factor that changes the transient overvoltages, which should be considered in the lightning protection system design.The results obtained by the FDTD method are compared with those obtained by the PEEC method, as shown in Figure 10.It could be noted that there is about 0.6 MV difference between the PEEC and FDTD methods, representing a 12% difference.This difference appeared due to the different concepts of applying the two methods, where the FDTD uses the differential form of Maxwell's equations while the PEEC uses the integral form.Also, the mutual couplings cannot be represented accurately with the PEEC method.On the other hand, the FDTD depends on the propagation of the electric and magnetic fields at each point in the system, which ensures higher accuracy.The results show that the grounded leg has a lower transient voltage value due to its direct connection to the grounding system, which makes the impedance of the current path lower than the other legs.

| Experimental validation of FDTD theory results
In this part, the FDTD simulation results are further verified in the laboratory using a small-scale PV system and an impulse generator.The impulse generator strikes the PV module, and the generated voltage is monitored through a capacitive divider connected to the digital oscilloscope to display and store the transient voltage.The oscilloscope meets the IEC-11802 standard, which is applicable to the test equipment in dielectric tests with impulse voltage.Also, the installation of the small-scale PV system under test complies with the IEC-60364 standard.The experimental setup arrangement is shown in Figure 11.
The small-scale PV system, shown in Figure 12, consists of one PV module with a maximum power of 285 W, an open -9 circuit voltage is 38.7 V, and a short circuit is 9.42 A. Moreover, the voltage at maximum power is 31.7 V, the current at maximum power is 9 A, and the irradiance E = 1000 W/m 2 .The maximum PV system voltage, which represents the maximum safety limit in sizing the PV module, equals 1000 V, so the maximum clamping voltage of the SPD varistor should not exceed 1000 V.The PV panel mounting system is connected to the grounding system through the grounding electrode to discharge lightning current to the ground, as shown in Figure 12.The mounting system consists of four girders, two front, and two rear legs, all made of Aluminium.The mounting system front leg height is 0.25 m, the rear leg height is 0.6 m, and the girders length is 1 m.
The impulse generator has a maximum energy of 2.5 kJ and a maximum generated voltage of 100 kV.The impulse generator charges a capacitor with a capacitance value of 0.5 μF when the striking voltage is 10 kV.The front time resistor of the impulse generator is 72 Ω.
The impulse generator is connected at point 1 in the mounting system through a copper sheet.The mounting system of the PV module is connected to the grounding electrode by a copper sheet.The transient overvoltages at points 2 and 3 obtained using PEEC, FDTD, and laboratory measurements are illustrated in Figures 13 and 14.Moreover, the overvoltage peak values at points 2 and 3 obtained by PEEC, FDTD, and lab measurements are shown in Figure 15.It is clear that the voltages at point 3 are lower than that of point 2 because it is closer to the striking point (point 1).In addition, there is a large agreement between the results of FDTD, PEEC, and those obtained in the laboratory.However, the FDTD method gives higher accurate results since it is closer to the laboratory measurements.
The laboratory measurements verify the correctness of the obtained simulation results.The error between the results of laboratory and FDTD and PEEC methods is calculated using the mean absolute error (M AE ) method, expressed using Equation (12) as follows [43]: where V Mi and V Si are the measured and simulated voltage values, respectively.n is the number of observations, which is taken as 10 observations.The value of M AE is 5% in the case of the FDTD method and 11% in the case of the PEEC method.
It could be noted that the voltage values obtained using FDTD are closer to those obtained with measurements, confirming that the FDTD is more accurate than PEEC.

| CONCLUSIONS
In this work, the 3-D FDTD has been used to model all PV system components of the large-scale system, and the transient overvoltages are calculated considering the change of the striking point and the soil resistivity, the effect of the metal frame existence, and the change of the grounded leg.It is observed that the transient voltages are highest at the striking point and decrease by moving away from it.Moreover, the transient overvoltages are increased by changing the soil from clay with resistivity to clayey sand with higher resistivity.In addition, the metal frame's existence increases the transient overvoltages at different points in the PV system.Furthermore, the connection of the mounting leg decreases the overvoltages at this point during the occurrence of lightning.The results obtained using the FDTD method as a differential base method are compared with the PEEC method as an integral base method.The results obtained using the FDTD method are more accurate because it considers all mutual effects between conductors at all points in space.Moreover, it considers the type of medium in which the magnetic and electric fields propagate rather than simply representing each conductor segment by its equivalent circuit as in the PEEC method.
A laboratory experiment has been conducted on a smallscale PV system using an impulse generator to verify the accuracy of simulation results.It is concluded that the measured overvoltage values at different points are very close to the PEEC and FDTD results, proving the simulation model's accuracy.By comparing the results obtained using FDTD and PEEC methods, the FDTD method gives higher accuracy results close to the laboratory measurements.The mean absolute error of the results obtained using FDTD and PEEC methods compared with the laboratory measurements is 5% and 11%, respectively.The comparison of simulation results with experimental ones confirms the accuracy and reliability of the FDTD simulation model for large-scale PV systems.This allows for the calculation of transient overvoltages at different points on the PV system, enabling the design of effective lightning protection systems to protect the system from damage and interruption of service, ensuring safe operation.
The main challenge of this study is obtaining detailed data on the parameters of large-scale PV systems from manufacturers, which may not always be available.Future research can be extended to study the transient analysis using the FDTD method during lightning occurrence on DC cable connecting PV modules and the inverter to install a suitable protection system for the inverter to prevent its damage during lightning.Moreover, the transients can be studied in hybrid renewable energy systems, including wind turbines and PV systems or any other combination of renewable sources.

HETITA ET AL. - 3 23977264, 0 ,
Downloaded from https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/hve2.12440 by Aalto University, Wiley Online Library on [15/05/2024].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensewith stranded copper conductors of 95 mm 2 buried at 1 m depth, and this grid connects the mounting structures of PV panels to the ground.The grounding wires of all panels are connected to the earthing system.There are nine individual meshes separated by 3 m distance with dimensions of 3 m width and 4 m length, where each 3 meshes are connected together.Finally, the combination of the nine individual meshes is connected to the grounding point of the inverter.The grounding conductors are buried in clay soil or sand soil to study the effect of changing the soil resistivity on the resultant transient overvoltages.The DC cables connect the DC output of each module in series and parallel connections, and finally, the total DC voltage is connected to the inverter.All wires are represented in the simulation, including PV mounting conductors, DC cables, grounding rods, and PV panel wires, which are usually neglected by previous research works.

F I G U R E 1
Schematic diagram of a large-scale photovoltaic system.

F I G U R E 2
Flowchart describing finite difference time domain steps.DISCUSSIONS

F I G U R E 3
Transient overvoltages at the corner points (V1, V2, V3, and V4) when lightning (a) strikes point 1 and (b) point 4. F I G U R E 4 Point 1 voltage for FDTD and PEEC methods when lightning strikes (a) point 1 with clay soil and (b) point 4 with clay soil.FDTD, finite difference time domain; PEEC, partial element equivalent circuit.

F I G R E 6 F I G U R E 7 F I G U R E 8 F I G U R E 5
Point 1 voltage for FDTD and PEEC methods with clayey sand soil.FDTD, finite difference time domain; PEEC, partial element equivalent circuit.Transient overvoltages under the absence of a metal frame.Point 1 voltage for FDTD and PEEC methods with a frameless PV module.FDTD, finite difference time domain; PEEC, partial element equivalent circuit; PV, photovoltaic.Transient overvoltages when the soil resistivity increased with clayey sand soil when lightning strikes point 1.

F I G U R E 1 0
Point 1 voltage for FDTD and PEEC methods when changing the grounding leg.FDTD, finite difference time domain; PEEC, partial element equivalent circuit.F I G U R E 9 Transient overvoltages under the change of the grounded leg.F I G U R E 1 1 Circuit diagram of the measurement setup.F I G U R E 1 2 Small-scale PV system in the laboratory.PV, photovoltaic.HETITA ET AL.

F I G U R E 1 3 F I G U R E 1 4 F G U R E 1 5 10 -
Transient overvoltages at point 2 obtained using PEEC, FDTD, and LAB results.FDTD, finite difference time domain; PEEC, partial element equivalent circuit.Transient overvoltages at point 3 obtained using PEEC, FDTD, and LAB results.FDTD, finite difference time domain; PEEC, partial element equivalent circuit.Comparison between overvoltage peak values at points 2 and 3 obtained using PEEC, FDTD, and LAB results.FDTD, finite difference time domain; PEEC, partial element equivalent circuit.HETITA ET AL.