Reliability Enhancement of Photovoltaic Systems under Partial Shading through a Two-Step Module Placement Approach

: Partial shading has a negative impact on photovoltaic systems by forcing the connected modules to generate lower power, creating severe unexpected power losses. To resolve this issue, numerous solutions have been proposed, among which conﬁguration modiﬁcation has recently attracted a greater audience. The preliminary approach to module reconﬁguration was based on the alteration of electrical connections through switches, which introduces lag due to the large number of switches and sensors, complex algorithms, and impractical application. Hence, static techniques are considered to be a cost-effective, low-complexity and easy-to-adopt solution for efﬁciently reducing the losses due to shading. Hence, this paper proposes a two-step module replacement approach that is validated under multiple partial shading conditions, and the performance is compared with various conventional and hybrid conﬁgurations and a static electrical reconﬁguration technique using mathematical analysis, comparative parameters and power curves analysis. The validation was performed using the MATLAB platform for two system sizes—6 × 6 and 18 × 3—proving its applicability for arbitrary system sizes. On the basis of the depth investigation, an average power increase of 17.49%, 14.47%, and 14.12% for the two-step approach compared to the conventional, hybrid and electrical reconﬁguration was observed in the partial shading cases considered.


Introduction
Solar Photovoltaic (PV) systems have emerged as a viable source among all renewables due to their lower degree of required maintenance, location-independent installation, costless generation, and ease of implementation [1]. Additionally, recent advances in terms of efficiency and cost reduction in PV technologies have prompted installers to opt for solar PV systems [2]. In recent years, the governments of many developing countries have taken various initiatives by providing incentives and tax relief for solar PV system installation in order to reduce their nations' energy dependence on fossil fuels [3]. Hence, PV systems are turning out to be the most reliable and a major source of energy generation for industrial, residential and office sectors, facilitating a reduction in electricity consumption from the grid, and hence a reduction in environmental impact [4].
PV systems are expected to generate the maximum power throughout their operational period; however, various unavoidable phenomena affect system performance and raise reliability questions [5]. One of the main factors reducing the performance of the system is partial shading among the modules occurring due to natural phenomena such as shadows caused by nearby objects, clouds, dust, etc., creating multiple irradiance operation scenarios Reconfiguration based on the Follow The Regularized Leader (FTRL) prediction model showed better performance than other dynamic reconfigurations under all the shading scenarios, offering better shade ability [28]. Similarly, another reconfiguration based on the genetic algorithm optimization technique was proposed, and it was found that the GA-based tool was able to work under changing environmental conditions and generate higher power output under all shading cases [29]. The algorithms proposed above have the advantage of higher power output during partial shading than conventional configurations; however, the major demerits include requiring large sensors and switches, complex algorithms for the switching matrix, and real-time application, as none of these algorithms have been tested experimentally. Hence, in contrast, static reconfigurations are gaining importance due to their not requiring switches or sensors, and there being no dynamic sections, as well as their ability to generate higher power than conventional techniques. The static reconfiguration begins with the proposal of a fixed electrical connection that has been validated experimentally using a 5 × 5 array and found to have a higher performance than conventional techniques [30]. Another well-known reconfiguration technique is the Sudoku technique, where the modules of the array are physically relocated to disperse the shading and validated using a 9 × 9 array [31]. A zig-zag reconfiguration was proposed and compared with conventional TCT, OTCT and NTCT schemes and found to have a higher power output [31]. Another novel magic square-based reconfiguration technique for PV arrays was proposed and compared with TCT, and it was found to offer a power enhancement of 11.53% under partial shading [32]. Another magic square algorithm was proposed and found to have higher power output than the Sudoku or Optimal-Sudoku, offering higher power output with reduced losses during shading [33]. Most recently, a four-square Sudoku reconfiguration was proposed and evaluated in comparison with several static and dynamic reconfiguration approaches, and was found to have 38.016% higher power [34]. An Ancient Chinese Magic square reconfiguration was proposed, compared with TCT and Sudoku, and concluded to offer higher power output during partial shading on the basis of simulation and experimental validations [35]. An improved nonsymmetrical reconfiguration puzzle reconfiguration scheme was proposed that introduced an odd-even configuration and redefined the shift distance in order to obtain a static reconfiguration [36]. A new skyscraper puzzle-based one-time reconfiguration has been proposed, tested using 5 × 5 and 9 × 9 arrays, compared with TCT, Dominance Square and Sudoku technique, and found to have higher shade dispersion capability during partial shading [37]. A ken-ken puzzle pattern-based reconfiguration was proposed, tested in a 4 × 4 array and found to offer a 10.85% power improvement compared to LS, Sudoku, and OE [38]. However, these techniques are typically validated using symmetric arrays such as 9 × 9 which are not common in real-world scenarios, and their application to asymmetric arrays has not been investigated. A one-time electrical reconfiguration was proposed for PV arrays and was found to have 20% higher power output than the conventional PV array configurations [39]. However, the technique has been compared with limited configurations and the performance with respect to other techniques has not been studied. Recently, a digital image encryption-based reconfiguration was proposed and tested using 9 × 9, 7 × 7, 6 × 6, 5 × 5, 4 × 4, 3 × 5, 4 × 3, 5 × 9, and 6 × 20 array sizes and compared with 41 reconfiguration techniques under 100 distinct shading cases [40]. The proposed technique exhibited 30.81%, 36.36%, 38.15%, 33.77%, 16.62%, 21.8%, 18.42%, and 16.79% higher power output in the 9 × 9, 7 × 7, 6 × 6, 5 × 5, 4 × 4, 4 × 3, 5 × 9, and 6 × 20 PV arrays, respectively. However, the algorithm proposed in that work was based on image encryption, which is complex and needs proper understanding related to the image encryption methodology. Another 64 various reconfigurations have been reviewed in the literature, where it is stated that most static reconfiguration techniques are limited to symmetric arrays and are difficult to implement in practical scenarios [41,42]. Additionally, it is stated that dynamic reconfigurations are costly due to the requirement of switches, sensors and control algorithms, which increase the cost of the system. Hence, in this paper, a two-step approach based on the combination of two-times position replacement of modules is proposed, which is applicable for both symmetric and asymmetric arrays and enhances the reliability of the arrays during partial shading. The proposed technique is static, in that it contains no dynamic parts, switches, sensors, or complex algorithms, and can be effectively implemented in arbitrary array sizes with no additional complexity. The validation of the proposed technique is performed using two array sizes of 6 × 6 (short) and 18 × 3 (long) in the MATLAB/Simulink platform under numerous shading patterns. The power-voltage curves, maximum power output, mismatch loss, power reduction, power increment, efficiency, and performance index of the proposed approach are compared to that of conventional SP, BL, HC and TCT configurations, hybrid SP-TCT, BL-TCT, HC-TCT and BL-HC, and static electrical reconfiguration methodology under shading cases.
The remainder of this paper provides a description of the mathematical modeling of the PV modules and formulations in Section 2, a description of conventional and hybrid configurations in Section 3, an explanation of the implementation of the proposed two-step approach for the arrays in Section 4, an analysis under numerous partial shading patterns for 6 × 6 and 18 × 3 arrays along with a performance comparison in Section 5, followed by a conclusion in Section 6.

Mathematical Modeling of PV Modules and Arrays
The work begins with the modeling of PV modules based on the single-diode electrical circuit, as presented in Figure 1, which contains a current-generating source, cells connected in series (N S ) and parallel (N P ), series resistance (R S ), and parallel resistance (R P ). The current output can be established by applying Kirchhoff's current law in the circuit, and can be given as been reviewed in the literature, where it is stated that most static reconfiguration techniques are limited to symmetric arrays and are difficult to implement in practical scenarios [41,42]. Additionally, it is stated that dynamic reconfigurations are costly due to the requirement of switches, sensors and control algorithms, which increase the cost of the system. Hence, in this paper, a two-step approach based on the combination of two-times position replacement of modules is proposed, which is applicable for both symmetric and asymmetric arrays and enhances the reliability of the arrays during partial shading. The proposed technique is static, in that it contains no dynamic parts, switches, sensors, or complex algorithms, and can be effectively implemented in arbitrary array sizes with no additional complexity. The validation of the proposed technique is performed using two array sizes of 6 × 6 (short) and 18 × 3 (long) in the MATLAB/Simulink platform under numerous shading patterns. The power-voltage curves, maximum power output, mismatch loss, power reduction, power increment, efficiency, and performance index of the proposed approach are compared to that of conventional SP, BL, HC and TCT configurations, hybrid SP-TCT, BL-TCT, HC-TCT and BL-HC, and static electrical reconfiguration methodology under shading cases.
The remainder of this paper provides a description of the mathematical modeling of the PV modules and formulations in Section 2, a description of conventional and hybrid configurations in Section 3, an explanation of the implementation of the proposed twostep approach for the arrays in Section 4, an analysis under numerous partial shading patterns for 6 × 6 and 18 × 3 arrays along with a performance comparison in Section 5, followed by a conclusion in Section 6.

Mathematical Modeling of PV Modules and Arrays
The work begins with the modeling of PV modules based on the single-diode electrical circuit, as presented in Figure 1, which contains a current-generating source, cells connected in series (NS) and parallel (NP), series resistance (RS), and parallel resistance (RP). The current output can be established by applying Kirchhoff's current law in the circuit, and can be given as The cell current is represented as IPV, with maximum voltage and current outputs being denoted by VM and IM, respectively. The parameters T, q, K, IO and α represent the The cell current is represented as I PV , with maximum voltage and current outputs being denoted by V M and I M , respectively. The parameters T, q, K, I O and α represent the module temperature (in Kelvin), electron charge (1.6 × 10 −19 C), Boltzmann constant (1.3806 × 10 −23 J/K), reverse saturation current, and ideality factor, respectively. The value of I PV in the above can be calculated using Equation (2), which shows the dependence of the current on the receiving irradiance with symbols I SC , S, S S , c i , and T S are denoted as the short-circuit current, irradiance, standard irradiance (1000 W/m 2 ), coefficient of current and standard temperature (298 K), respectively.
The module ratings are determined based on the performance under standard testing conditions, with an irradiance of 1000 W/m 2 and a module temperature of 25 • C. The maximum power, maximum voltage, open-circuit voltage, maximum current, short-circuit current and the area receiving irradiance of the PV module used under standard conditions are 325 W, 37.80 V, 46.60 V, 8.60 A, 9.20 A and 1.69 m 2 , respectively.
To generate the required power, modules are connected in combination in series and parallel, forming an array that increases the terminal voltage and output current of the system. Additionally, the modules are connected in various other combinations derived from the series-parallel configuration in order to achieve higher performance under partial shading conditions, as explained in Section 3.
The performance of the PV arrays is evaluated using various mathematical parameters, including power-voltage (P-V) characteristic curves, maximum power output, mismatch loss, power reduction, power increment, efficiency, and performance index, the formulations of which are explained below.
The maximum power output (P O ) is the main parameter used to calculate other parameters such as mismatch loss, power reduction, power increment, efficiency, and performance index, and the equations involved in the calculation are given in Equation (3), where P T , P US , P OX and A denote the total available power, power with no shading, and power output for other configurations and areas, respectively.

PV Array Configurations: Conventional and Hybrid
The PV array configurations are the connection schemes between the modules, and are based on the addition of additional wires to the junctions of series-parallel-connected modules in order to provide an additional path through which the module currents can flow. These configurations are highly effective at mitigating the power loss in PV arrays during partial shading, and are classified into two categories, i.e., conventional and hybrid configurations. The power, voltage and current output ratings of these array configurations remain the same as the number of modules, and the initial electrical architecture remains the same.

Conventional Configurations
The conventional array configuration used thus far is series-parallel connection, whereby the modules are connected in series for voltage increment, forming a string, and similar strings are connected in parallel for the current increment of the system. Subsequently, considering the partial shading effect and the power losses in the PV array, additional configurations were formed by connecting wires to the connection junctions of the modules, which were referred to as bridge-linked, honeycomb, and total-cross-tied, denoted by BL, HC and TCT, respectively. The schematic representation of the conventional configurations for a 6 × 6 PV array is shown in Figure 2. Among these configurations, the TCT has been proved to be effective; however, other configurations have also been reported in some cases. denoted by BL, HC and TCT, respectively. The schematic representation of the conventional configurations for a 6 × 6 PV array is shown in Figure 2. Among these configurations, the TCT has been proved to be effective; however, other configurations have also been reported in some cases.

Hybrid Configurations
Despite the higher power generation capability of the conventional configurations, the requirement of additional wires adds connection complexity, mainly in the case of large-sized PV arrays. Hence, various other configurations for the array from the combination of conventional configurations have been developed, as shown in Figure 3. The configurations were named according to the configurations combined to produce them, for example, the hybrid configuration for the series-parallel and total-cross-tied combination is named SP-TCT. Similarly, the combinations of bridge-linked and total-cross-tied, honeycomb and total-cross-tied, and bridge-lined and honeycomb are termed BL-TCT, HC-TCT and BL-HC, respectively.

Two-Step Approach of Module Placement for PV Arrays
The proposed two-step approach is a module placement methodology for PV arrays in which the modules are repositioned using a certain technique to facilitate shade dispersion and increase the output power under partial shading conditions. The approach uses a fixed architecture along with a total-cross-tied electrical connection and requires no switches, sensors, complex algorithms, or dynamic parts, thus adding the advantage of easy implementation in PV arrays. The proposed technique uses a two-step module

Hybrid Configurations
Despite the higher power generation capability of the conventional configurations, the requirement of additional wires adds connection complexity, mainly in the case of largesized PV arrays. Hence, various other configurations for the array from the combination of conventional configurations have been developed, as shown in Figure 3. The configurations were named according to the configurations combined to produce them, for example, the hybrid configuration for the series-parallel and total-cross-tied combination is named SP-TCT. Similarly, the combinations of bridge-linked and total-cross-tied, honeycomb and total-cross-tied, and bridge-lined and honeycomb are termed BL-TCT, HC-TCT and BL-HC, respectively. denoted by BL, HC and TCT, respectively. The schematic representation of the conventional configurations for a 6 × 6 PV array is shown in Figure 2. Among these configurations, the TCT has been proved to be effective; however, other configurations have also been reported in some cases.

Hybrid Configurations
Despite the higher power generation capability of the conventional configurations, the requirement of additional wires adds connection complexity, mainly in the case of large-sized PV arrays. Hence, various other configurations for the array from the combination of conventional configurations have been developed, as shown in Figure 3. The configurations were named according to the configurations combined to produce them, for example, the hybrid configuration for the series-parallel and total-cross-tied combination is named SP-TCT. Similarly, the combinations of bridge-linked and total-cross-tied, honeycomb and total-cross-tied, and bridge-lined and honeycomb are termed BL-TCT, HC-TCT and BL-HC, respectively.

Two-Step Approach of Module Placement for PV Arrays
The proposed two-step approach is a module placement methodology for PV arrays in which the modules are repositioned using a certain technique to facilitate shade dispersion and increase the output power under partial shading conditions. The approach uses a fixed architecture along with a total-cross-tied electrical connection and requires no switches, sensors, complex algorithms, or dynamic parts, thus adding the advantage of easy implementation in PV arrays. The proposed technique uses a two-step module placement approach, a graphical representation of which is provided in Figure 4, where a 6 × 6

Two-Step Approach of Module Placement for PV Arrays
The proposed two-step approach is a module placement methodology for PV arrays in which the modules are repositioned using a certain technique to facilitate shade dispersion and increase the output power under partial shading conditions. The approach uses a fixed architecture along with a total-cross-tied electrical connection and requires no switches, sensors, complex algorithms, or dynamic parts, thus adding the advantage of easy implementation in PV arrays. The proposed technique uses a two-step module placement approach, a graphical representation of which is provided in Figure 4, where a 6 × 6 PV array is taken as an example to perform the two-step approach for module replacement. The technique begins with an initial consideration of the PV modules numbered according to their row and column indices; for example, 34 denotes the modules located in the third row and fourth column of the array. The steps involved in the proposed technique do not require the use of an algorithm, but can be executed using a simple mathematical procedure, as described below. Step-1 (Inter-Group Exchange): Initially, the array is divided into multiple groups with two modules in each group, with the nomenclature being A1, A2 and A3 for the first Step-1 (Inter-group Exchange) 11

Adding PV Modules into Groups
Step-1 of Rearranged Modules Step-2 (Group Exchange)  Step-1 (Inter-Group Exchange): Initially, the array is divided into multiple groups with two modules in each group, with the nomenclature being A1, A2 and A3 for the first column, B1, B2 and B3 for the second column, C1, C2 and C3 for the third column, and so on. Then, the modules of the group are interchanged; for example, in group A1, module 11 takes the place of module 21, and vice versa. Considering another example, in group D3, modules 54 and 64 are interchanged, with 64 taking the place of 54, and vice-versa. In this manner, all the modules in the groups shown in Figure 4 (under step-1 analysis) are interchanged, resulting in a final rearranged PV array.

PV Modules Group Swapping Final Modules Position
Step-2 (Group Exchange): In this step, the groups are now exchanged on the basis of a simple mathematical technique that allows proper dispersion of the shading. For column 1, all of the groups remain unaltered at their original position, and for column 2, the groups are shifted an additional one position, followed by an additional three positions for the groups in column 3. Again, this procedure is repeated for the next column, i.e., column 4, where the groups remain unaltered, followed by one additional shift for column 5, two additional shifts for column 6, and so on. This logic can be expressed as follows: if 'n' represents the column of the array, 'm' states the total groups count and 'i' denotes the initial positions of the groups then for n = 1, i = 1, i.e., the groups remain altered for n = 2, i = i + 1 ≤ m (if not, then i = m − i), i.e., additional one shifting for n = 3, i = i + 2 ≤ m (if not, then i = m − i), i.e., additional two shifting for n = 4, i = 1, i.e., the groups remain altered for n = 5, i = i + 1 ≤ m (if not, then i = m − i), i.e., additional one shifting and so on.
Considering the example for a 6 × 6 PV array shown in Figure 4, the groups of the first column (A1, A2 and A3) remain at their original position. For the second column, the groups are shifted to the positions i + 1, i.e., group B1 with the initial position (i) as 1 is shifted to the i + 1 = 1 + 1 = 2nd position (<3), taking the place of B2, and B2 (i = 2) moves to the third position (i + 1 = 2 + 1 = 3 ≤ 3). For B3 (i = 3), as the updated value of 'i' was calculated to be i + 2 = 3 + 2 = 5, which is greater than m = 3, the recalculation is performed, i.e., i = m -I = 3 -3 = 1, i.e., the group is shifted to the first position. Similarly, the process continues for the third column (C1, C2 and C3) and repeats from the fourth (D1, D2 and D3) column to the sixth column (F1, F2 and F3). The final array with the two-step approach of module shifting is depicted in Figure 4 (under step-2 analysis), which is considered to be the final architecture based on which the modules are replaced. Citing an example of the final 6 × 6 array, it can be observed that in the first column, the initial module 11 is shifted to the position of 21, and vice-versa; initial module 31 is shifted to the position of 41, and vice-versa; and initial module 51 is shifted to the position of 61, and vice versa. For the second column, the modules 12, 22, 32, 42, 52 and 62 are shifted to the positions of 42, 32, 62, 52, 22 and 12, respectively, and so on. It has to be noted that the electrical connection remains the same as in the cases of a total-cross-tied connection, and hence, the electrical parameters remain equal to those of other configurations. In the case of an odd row count, the last single module can be treated as a single group, and the procedure continues according to the given methodology. Additionally, the proposed methodology is not limited to symmetric arrays, but have a wide range of applications in arbitrarily sized arrays. Some examples of other array sizes based on the two-step approach of module placement are pictorially represented in Figure 5. It is to be noted that the two-step technique is a module placement technique in which the electrical connection of the modules remains equivalent to that of the total-cross-tied connection; therefore, the electrical wiring and wiring length remain the same as that of the total-cross-tied array. Additionally, in practical applications, the array can experience a loss of 1-2% due to wires, but for the purposes of simulation validation, such losses are excluded in this study.

Analysis under Partial Shading (Results and Discussion)
The proposed two-step approach was implemented in two array sizes, i.e., symmetric (6 × 6) and asymmetric (18 × 3), and tested under numerous actual and random shading scenarios. The shade dispersion ability of the proposed methodology was validated on the basis of a comparison with conventional (SP, BL, HC and TCT) and hybrid (SP-TCT, BL-TCT, HC-TCT and BL-HC), along with the static electrical reconfiguration technique. The evaluation of the performance was carried out on the basis of the power-voltage (P~V) curves, power generation, mismatch loss, power reduction, power increment, efficiency and performance index, the mathematical formulations of which are given in Equation (3).

Symmetric 6 × 6 PV Array
The 6 × 6 PV array has the maximum power output of 9.58 kW when all modules are operating under 800 W/m 2 irradiance and 50 °C module temperature, which reflects the no-shading situation. The evaluation was conducted by considering numerous partial shading situations based on actual cases and random situations. The partial shading cases based on the actual cases used in the study for the PV arrays are pictorially presented in Figure 6. It has to be noted that all the shading cases used for testing differ from each other in terms of location, area, pattern and irradiance levels.

Analysis under Partial Shading (Results and Discussion)
The proposed two-step approach was implemented in two array sizes, i.e., symmetric (6 × 6) and asymmetric (18 × 3), and tested under numerous actual and random shading scenarios. The shade dispersion ability of the proposed methodology was validated on the basis of a comparison with conventional (SP, BL, HC and TCT) and hybrid (SP-TCT, BL-TCT, HC-TCT and BL-HC), along with the static electrical reconfiguration technique. The evaluation of the performance was carried out on the basis of the power-voltage (P~V) curves, power generation, mismatch loss, power reduction, power increment, efficiency and performance index, the mathematical formulations of which are given in Equation (3).

Symmetric 6 × 6 PV Array
The 6 × 6 PV array has the maximum power output of 9.58 kW when all modules are operating under 800 W/m 2 irradiance and 50 • C module temperature, which reflects the no-shading situation. The evaluation was conducted by considering numerous partial shading situations based on actual cases and random situations. The partial shading cases based on the actual cases used in the study for the PV arrays are pictorially presented in Figure 6. It has to be noted that all the shading cases used for testing differ from each other in terms of location, area, pattern and irradiance levels.

Evaluation under Partial Shading S1
The blocks showing the irradiance levels of the PV array during partial shading S1 ( Figure 6a) are shown in Figure 7, where the partial shading reduced the received irradiance of the modules from 800 W/m 2 (50 °C) to 100 W/m 2 (53 °C) and 150 W/m 2 (52.6 °C). As the conventional and hybrid configurations have no shading dispersion capability, the block for total-cross-tied (TCT) represents the irradiance levels for all. Irradiance received by the 6 × 6 PV array during partial shading S1 for (a) conventional, hybrid and TCT, (b) static electrical reconfiguration (shade dispersion), and (c) the two-step approach (shade dispersion).
As most of this analysis demonstrated the increased effectiveness of the TCT over the other configurations, the mathematical comparison of power output was performed with TCT. For partial shading S1 in an array with TCT, the current output by different rows based on the irradiance block in Figure 7a was calculated as follows: Shading S1 Shading S2 Shading S3 Shading S4

Evaluation under Partial Shading S1
The blocks showing the irradiance levels of the PV array during partial shading S1 ( Figure 6a) are shown in Figure 7, where the partial shading reduced the received irradiance of the modules from 800 W/m 2 (50 • C) to 100 W/m 2 (53 • C) and 150 W/m 2 (52.6 • C). As the conventional and hybrid configurations have no shading dispersion capability, the block for total-cross-tied (TCT) represents the irradiance levels for all.

Evaluation under Partial Shading S1
The blocks showing the irradiance levels of the PV array during partial shading S1 ( Figure 6a) are shown in Figure 7, where the partial shading reduced the received irradiance of the modules from 800 W/m 2 (50 °C) to 100 W/m 2 (53 °C) and 150 W/m 2 (52.6 °C). As the conventional and hybrid configurations have no shading dispersion capability, the block for total-cross-tied (TCT) represents the irradiance levels for all. Irradiance received by the 6 × 6 PV array during partial shading S1 for (a) conventional, hybrid and TCT, (b) static electrical reconfiguration (shade dispersion), and (c) the two-step approach (shade dispersion).
As most of this analysis demonstrated the increased effectiveness of the TCT over the other configurations, the mathematical comparison of power output was performed with TCT. For partial shading S1 in an array with TCT, the current output by different rows based on the irradiance block in Figure 7a was calculated as follows:  Irradiance received by the 6 × 6 PV array during partial shading S1 for (a) conventional, hybrid and TCT, (b) static electrical reconfiguration (shade dispersion), and (c) the two-step approach (shade dispersion).
As most of this analysis demonstrated the increased effectiveness of the TCT over the other configurations, the mathematical comparison of power output was performed with TCT. For partial shading S1 in an array with TCT, the current output by different rows based on the irradiance block in Figure 7a was calculated as follows: The mathematical current estimation for the static electrical reconfiguration using the irradiance block presented in Figure 7b was calculated using Equation (5).
For the two-step approach, the irradiance block in Figure 7c was used to estimate the current output of the array calculated in Equation (6).
The mathematical estimations of the current, voltage and power outputs for different rows of TCT, static electrical reconfiguration and the two-step approach are summarized in Table 1. It can be seen that TCT, static electrical reconfiguration and the two-step approach have a mathematically equal estimated power output value of 20.7 V M I M .; however, by viewing the current output values of the rows, it can be seen that there is a minimum current difference between the rows in the case of the two-step approach. The validation of the effective shade dispersion by the proposed two-step approach was further performed using P-V curve analysis and parameter comparison with other configurations.
The total available power of the array during partial shading S1 was calculated to be 8.17 kW, on the basis of which the power output of the SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT and BL-HC were determined to be 7.08 kW, 7.08 kW, 7.14 kW, 7.34 kW, 7.23 kW, 7.25 kW, 7.22 kW and 7.18 kW, respectively. Static electrical reconfiguration generated a higher power output of 7.40 kW than the conventional and hybrid configurations; however, the two-step approach presented a maximum power output of 7.52 kW, representing a 6.21%, 6.21%, 5.32%, 2.45%, 4.01%, 3.72%, 4.15%, 4.73% and 1.63% increment in power compared to SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC and static electrical reconfiguration, respectively. On the basis of the parameter calculation formulas given in Equation (3), the mismatch loss and power reduction experienced when using the two-step approach were calculated and found to have lower values of 0.65 kW and 2.05 kW, respectively. Additionally, there is an increase in the values of efficiency and performance index of 15.39% and 78.57% when using the two-step approach compared to when using the other configurations. Among all of the configurations, the SP showed the poorest performance and suffered higher mismatch loss (1.09 kW) and power reduction (2.49 kW) with lower efficiency (14.49%) and performance index (73.98%). The comparison of the P~V curve of the array with the proposed two-step approach for partial shading S1 with the conventional (SP, BL, HC and TCT), hybrid (SP-TCT, BL-TCT, HC-TCT, BL-HC) and static electrical reconfiguration is graphically presented in Figure 8a, 8b and 8c, respectively. Additionally, a comparison of the configurations in terms of different parameters was performed through bar graph analysis, as given in Figure 8d. On the basis of the graphical results, it can be observed that the proposed approach performed well during partial shading S1, and can be stated to be the best technique for power enhancement. Additionally, the arrays were operated under the same shading case by maintaining the temperature of all modules at 50 • C, and it was observed that the power generation remained nearly the same as that in the varying temperature case. The average power difference under constant temperature and varying temperature cases under partial shading was determined to be 0.019 W, which is much less than the total array rating. On the basis of this analysis, it can be stated that module temperature has a minimal effect on the power generation of the array during partial shading, and therefore, temperature variation was considered in this study.

Evaluation under Partial Shading S2
The irradiance block for the TCT, conventional and hybrid configurations, static electrical reconfiguration and the two-step approach for partial shading S2 (Figure 6b) are shown in Figure 9a,b,c, respectively. It can be observed that the partially shaded modules have four irradiance values-200 W/m 2 , 250 W/m 2 , 300 W/m 2 and 350 W/m 2 -along with 800 W/m 2 for the unshaded modules. The summarized mathematical estimation values for currents, voltages, and powers of rows for TCT, static electrical reconfiguration and the proposed two-step approach are tabulated in Table 2. From the mathematical calculations, it can be observed that the two-step approach has a higher mathematical power value of 21. temperature of all modules at 50 °C, and it was observed that the power generation remained nearly the same as that in the varying temperature case. The average power difference under constant temperature and varying temperature cases under partial shading was determined to be 0.019 W, which is much less than the total array rating. On the basis of this analysis, it can be stated that module temperature has a minimal effect on the power generation of the array during partial shading, and therefore, temperature variation was considered in this study.

Evaluation under Partial Shading S2
The irradiance block for the TCT, conventional and hybrid configurations, static electrical reconfiguration and the two-step approach for partial shading S2 (Figure 6b) are shown in Figure 9a,b,c, respectively. It can be observed that the partially shaded modules have four irradiance values-200 W/m 2 , 250 W/m 2 , 300 W/m 2 and 350 W/m 2 -along with 800 W/m 2 for the unshaded modules. The summarized mathematical estimation values for currents, voltages, and powers of rows for TCT, static electrical reconfiguration and the proposed two-step approach are tabulated in Table 2. From the mathematical calculations, it can be observed that the two-step approach has a higher mathematical power value of 21.6 VMIM, whereas the static reconfiguration and TCT have 18.90 VMIM and 18 VMIM, respectively. The total available power output of the array was theoretically calculated to be 7.91 kW.

Evaluation under Partial Shading S2
The irradiance block for the TCT, conventional and hybrid configurations, static electrical reconfiguration and the two-step approach for partial shading S2 (Figure 6b) are shown in Figure 9a,b,c, respectively. It can be observed that the partially shaded modules have four irradiance values-200 W/m 2 , 250 W/m 2 , 300 W/m 2 and 350 W/m 2 -along with 800 W/m 2 for the unshaded modules. The summarized mathematical estimation values for currents, voltages, and powers of rows for TCT, static electrical reconfiguration and the proposed two-step approach are tabulated in Table 2. From the mathematical calculations, it can be observed that the two-step approach has a higher mathematical power value of 21.6 VMIM, whereas the static reconfiguration and TCT have 18.90 VMIM and 18 VMIM, respectively. The total available power output of the array was theoretically calculated to be 7.91 kW.   The power output achieved when using the two-step approach was notably higher, with a value equal to 7.78 kW, whereas the SP has a value of 6.19 kW, along with 6.39 kW for BL, 6.49 kW for HC, 6.63 kW for TCT, 6.57 kW for SP-TCT, 6.59 kW for BL-TCT, 6.57 kW for HC-TCT, 6.51 kW for BL-HC and 6.90 kW for static reconfiguration.

Evaluation under Partial Shading S3
The total power available in the PV array was calculated to be 8.48 kW for shading S3 (Figure 11) with the higher mathematical power generation of 24 VMIM by static reconfiguration and two-step approach (Table 3). A graphical comparison of the two-step approach with the conventional and hybrid configurations along with static electrical reconfiguration in terms of P-V curves and parameters is presented in Figure 12. The maximum power output of SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC, static electrical reconfiguration and two-step approach was determined to be 7.50 kW, 7.52 kW, 7.55 kW, 7.73 kW, 7.70 kW, 7.70 kW, 7.72 kW, 7.59 kW, 8.30 kW and 8.38 kW, respectively. The twostep approach had the lowest mismatch loss and power reduction, at 0.06 kW and 1.15 kW, respectively, along with a higher efficiency and performance index, at 17.24% and 87.98%, respectively. The power increment achieved when using the two-step approach was calculated to be 13.06%, 12.76%, 12.31%, 9.70%, 10.12%, 10.12%, 9.84%, 11.72% and 2.16%, respectively.

Evaluation under Partial Shading S3
The total power available in the PV array was calculated to be 8.48 kW for shading S3 (Figure 11) with the higher mathematical power generation of 24 VMIM by static reconfiguration and two-step approach (Table 3). A graphical comparison of the two-step approach with the conventional and hybrid configurations along with static electrical reconfiguration in terms of P-V curves and parameters is presented in Figure 12. The maximum power output of SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC, static electrical reconfiguration and two-step approach was determined to be 7.50 kW, 7.52 kW, 7.55 kW, 7.73 kW, 7.70 kW, 7.70 kW, 7.72 kW, 7.59 kW, 8.30 kW and 8.38 kW, respectively. The twostep approach had the lowest mismatch loss and power reduction, at 0.06 kW and 1.15 kW, respectively, along with a higher efficiency and performance index, at 17.24% and 87.98%, respectively. The power increment achieved when using the two-step approach was calculated to be 13.06%, 12.76%, 12.31%, 9.70%, 10.12%, 10.12%, 9.84%, 11.72% and 2.16%, respectively.

Evaluation under Partial Shading S4
The irradiance blocks of the array are shown in Figure 13, and the mathematical power output of the two-step approach, static reconfiguration and TCT were calculated to be 25. .85 kW, respectively. The mismatch loss and power reduction in the twostep approach were calculated to be 0.56 kW and 0 kW, which is lower than for the other techniques. Comparisons of the P~V curves and the parameters of all of the configurations, static reconfiguration, and the two-step approach are shown in Figure 15a, from which it can be observed that the two-step approach has the higher values. The increment in power when using the two-step approach was calculated to be 12.30%, 10.63%, 8.99%, 4.11%, 8.85%, 9.80%, 8.58%, 9.25% and 7.79% for SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC and static reconfiguration, respectively.

Evaluation under Partial Shading S5
The irradiance blocks of the array with conventional, hybrid and TCT, static reconfiguration, and the two-step approach are presented in Figure 14, and the maximum mathematical power outputs have been calculated to be 16.2 V M I M , 11 V M I M and 16.8 V M I M , respectively. The total power available in the array was calculated to be 5.86 kW, and the power output from SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC, static reconfiguration and the two-step approach was determined to be 4.42 kW, 3.70 kW, 3.87 kW, 5.71 kW, 4.78 kW, 4.78 kW, 4.77 kW, 4.77 kW, 4.02 kW and 5.85 kW, respectively. The performance of the two-step approach can be observed from the graphical comparison given in Figure 15b, in which the generated power output was 32.35%, 58.10%, 51.16%, 2.45%, 22.38%, 22.38%, 22.64%, 22.38% and 45.52% higher than that of SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC and static reconfiguration, respectively.

Evaluation under Partial Shading S4
The irradiance blocks of the array are shown in Figure 13, and the mathematical power output of the two-step approach, static reconfiguration and TCT were calculated to be 25.8 VMIM, 22.8 VMIM and 24 VMIM, respectively. The total power available in the array was calculated to be 8.85 kW, on the basis of which the power output by SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC, static reconfiguration and the two-step approach were determined to be 7.88 kW, 8  .85 kW, respectively. The mismatch loss and power reduction in the twostep approach were calculated to be 0.56 kW and 0 kW, which is lower than for the other techniques. Comparisons of the P~V curves and the parameters of all of the configurations, static reconfiguration, and the two-step approach are shown in Figure 15a, from which it can be observed that the two-step approach has the higher values. The increment in power when using the two-step approach was calculated to be 12.

Evaluation under Partial Shading S5
The irradiance blocks of the array with conventional, hybrid and TCT, static reconfiguration, and the two-step approach are presented in Figure 14, and the maximum mathematical power outputs have been calculated to be 16.2 VMIM, 11 VMIM and 16.8 VMIM, respectively. The total power available in the array was calculated to be 5.86 kW, and the power output from SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC, static reconfiguration and the two-step approach was determined to be 4.42 kW, 3.70 kW, 3.87 kW, 5.71 kW, 4.78 kW, 4.78 kW, 4.77 kW, 4.77 kW, 4.02 kW and 5.85 kW, respectively. The performance of the two-step approach can be observed from the graphical comparison given in Figure 15b, in which the generated power output was 32.35%, 58.10%, 51.16%, 2.45%, 22.38%, 22.38%, 22.64%, 22.38% and 45.52% higher than that of SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC and static reconfiguration, respectively.

Evaluation under Partial Shading S5
The irradiance blocks of the array with conventional, hybrid and TCT, static reconfiguration, and the two-step approach are presented in Figure 14, and the maximum mathematical power outputs have been calculated to be 16.2 VMIM, 11 VMIM and 16.8 VMIM, respectively. The total power available in the array was calculated to be 5.86 kW, and the power output from SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC, static reconfiguration and the two-step approach was determined to be 4.42 kW, 3.70 kW, 3.87 kW, 5.71 kW, 4.78 kW, 4.78 kW, 4.77 kW, 4.77 kW, 4.02 kW and 5.85 kW, respectively. The performance of the two-step approach can be observed from the graphical comparison given in Figure 15b, in which the generated power output was 32.35%, 58.10%, 51.16%, 2.   The summarized data for the different parameters of the conventional and hybrid configuration, static electrical reconfiguration, and the two-step approach for the five partial shading cases are given in Table 4.  The summarized data for the different parameters of the conventional and hybrid configuration, static electrical reconfiguration, and the two-step approach for the five partial shading cases are given in Table 4.
Furthermore, the reliability of the proposed two-step approach was tested by applying various additional random partial shading patterns (as shown in Figure 16) in the 6 × 6 array and comparing the power generation with the conventional, hybrid and static electrical reconfigurations. The P~V curves of the array for all the random partial shading cases are shown in Figure 17, from which it can be observed that the two-step approach has the highest power output. The detailed power output data of all the configurations for all the cases of partial shading are given in Table 5.

Asymmetric 18 × 3 PV Array
The application of the proposed two-step approach was validated using the 16 × 3 (long) array, which was placed in the 8 × 6 architecture in an irradiance block, i.e., the first two columns contain a single string of 16 modules, with eight modules in a column. Similarly, columns 2 and 3 contain another string with eight modules in a single column, and so on. The array was evaluated by considering the partial shading condition that starts from the top of the array, moves towards the bottom, and ends with different irradiance levels, as shown in Figure 18. The array operates under the no shading condition in case A, whereas shading develops in case B, with the first row receiving 600 W/m 2 , before progressively moving downward for case C, with irradiance levels of 600 W/m 2 and 550 W/m 2 . The partial shading progressively moves in a downward direction until case Q, where the partial shading ends. The P~V curves of the respective partial shading cases are shown in Figure 19, where it can be observed that under these types of shading conditions, all the configurations under comparison have equal power output, with the two-step approach generating higher power for all cases.  11.99 kW and 13.41 kW, respectively. The comparison of the power output for all the configurations and the two-step approach is graphically represented in Figure 20, from which it can be observed that for an increasing level of partial shading from case A to I, and subsequently decreasing shading from case I to Q, the two-step approach generated the highest power output. Additionally, the comparison of the efficiency and performance index for all partial shading cases is shown in Figure 21, from which it can be observed that the two-step approach presents higher values than the others.   Hence, the analysis conducted in this paper shows the efficacy of the proposed technique compared to conventional and hybrid configurations and static reconfiguration under a variety of partial shading cases. Additionally, the two-step approach overcomes the limitations of different configurations with regard to symmetric application by exhibiting wide applicability for arbitrary array sizes. For the 6 × 6 array, the average power enhancements of the two-step approach compared to the SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC and static electrical reconfiguration were noted to be 15.35%, 20.15%, 18.21%, 6.46%, 11.74%, 11.90%, 11.73%, 12.44% and 14.60%, respectively. Similarly, for random partial shading cases for a 6 × 6 array, the average power enhancement of the two-step approach was calculated to be 22.73%, 21.09%, 19.65%, 16.25%, 16.67%, 16.68%, 16.61%, 17.94% and 13.64% compared to SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC and static reconfiguration, respectively. Similarly, for the 16 × 3 array, the two-step approach had an average power 22.41% higher than that of other configurations during partial shading. Therefore, on the basis of these results, it can be stated that the two-step approach has a higher power output and performance under all partial shading conditions.  Figure 19. P~V curves for the partial shading cases in an 18 × 3 PV array with conventional and hybrid configurations using the two-step approach.  Figure 19. P~V curves for the partial shading cases in an 18 × 3 PV array with conventional and hybrid configurations using the two-step approach. Hence, the analysis conducted in this paper shows the efficacy of the proposed technique compared to conventional and hybrid configurations and static reconfiguration under a variety of partial shading cases. Additionally, the two-step approach overcomes the limitations of different configurations with regard to symmetric application by exhibiting wide applicability for arbitrary array sizes. For the 6 × 6 array, the average power enhancements of the two-step approach compared to the SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC and static electrical reconfiguration were noted to be 15.35%, 20.15%, 18.21%, 6.46%, 11.74%, 11.90%, 11.73%, 12.44% and 14.60%, respectively. Similarly, for random partial shading cases for a 6 × 6 array, the average power enhancement of the two-step approach was calculated to be 22.73%, 21.09%, 19.65%, 16.25%, 16.67%, 16.68%, 16.61%, 17.94% and 13.64% compared to SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC and static reconfiguration, respectively. Similarly, for the 16 × 3 array, the two-step approach had an average power 22.41% higher than that of other configurations during partial Increasing Shading Strength Decreasing Shading Strength Figure 21. Comparison of (a) efficiency and (b) performance index of the 8 × 3 configurations using the proposed two-step approach for partial shading cases.

Conclusions
A two-step approach for module placement with the aim of increasing the power output of the PV arrays during partial shading was proposed in this paper. The approach was validated under numerous partial shading cases and its performance compared with that of conventional configurations such as SP, BL, HC and TCT, hybrid configurations such as SP-TCT, BL-TCT, HC-TCT and BL-HC, and a static electrical reconfiguration technique. The application of the two-step approach in symmetrical and asymmetrical array sizes was verified for two array sizes-6 × 6 (short) and 16 × 3 (long)-under numerous partial shading cases. On the basis of the analysis conducted in this paper, the following conclusions regarding the advantages of the proposed two-step approach were drawn: Output value 5.35%, 20.15%, 18.21%, 6.46%, 11.74%, 11.90%, 11.73%, 12.44% and 14.60% higher than in the SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC and static reconfiguration in the 6 × 6 array. Power output 22.73%, 21.09%, 19.65%, 16.25%, 16.67%, 16.68%, 16.61%, 17.94% and 13.64% higher than in the SP, BL, HC, TCT, SP-TCT, BL-TCT, HC-TCT, BL-HC and static reconfiguration in the 6 × 6 array under random shading. Power output 22.14% higher than in all other configurations in the 16 × 3 array. Reduced losses, higher performance and higher efficiency compared to other configurations in all partial shading cases. Applicable for both symmetric and asymmetric arrays. Requires no switches, sensors, or complex algorithms. Wide practical application. Easy to implement in PV arrays.
Hence, it can be stated that implementation of the proposed two-step approach in PV arrays can increase their reliability and reduce the chances of system failure under partial shading conditions.