Introduction

Partial shading greatly limits the PV array output and will in turn leads to the development of hotspots in the PV array which further directs to dangerous fire hazards and ultimately destroys the module [1]. To safeguard the module, a bypass diode should be installed across it and these diodes bypass the current during low irradiations [2]. Nonetheless, the installation of these diodes instigates various peaks in an array characteristics. Another popular technique to reduce the impacts of PS is by installing maximum power point tracking (MPPT) controllers in the array [3]. However, these controllers necessitate complicated power converters and various control algorithms for operation. Despite its effectiveness in tracking the maximum power during PS, using these controllers increases the complexity, overall system cost. Due to the limitations of MPPT controllers, the reconfiguration methods are regarded as the best alternative to overcome the above-said limitations [4].

The reconfiguration methods are categorized as static and dynamic reconfiguration methods. Some of the notable dynamic reconfiguration techniques in the literature are presented as follows: The authors in [5] validated a completely dynamic PV array reconfiguration scheme that employs a neuro-fuzzy methodology and a switch matrix to regulate the configuration of the electrical array. The PV array reconfiguration strategy that depends on the short-circuit current evaluated by a convolutional neural network (CNN) under PS Conditions is proposed in [6]. An Irradiance Equalization (IE) concept-based reconfiguration procedure is proposed in [7] by reconfiguring the array in such a way that the summation of irradiation received by the modules in all the rows of an array must be made equal. In contrast to the principle of IE where the array output is enhanced solely by increasing the row current without regarding the array voltage, the authors in [8] proposed a 0/1 Multi knapsack problem based optimized dynamic reconfiguration approach by directly estimating the global maximum power (GMP) of the array. In [9], the authors experimentally verified an intelligent reconfiguration scheme based on the Fuzzy logic system. The optimized configuration is calculated by fuzzy system and the result is later implemented to reconfigure the modules by using a switching matrix. Nevertheless, this approach demands numerous steps and complex circuitry.

Enhancing the solar PV power generation with the metaheuristic-based grey wolf optimizer for array reconfiguration under various scenarios has been invested in [10]. The authors in [11], developed an enhanced immune genetic algorithm to reconfigure the thermoelectric power generation system under nonuniform temperature distribution conditions in order to optimize power output. By developing a modified form for the simple hunger games-search optimizer, the authors in [12] proposed a novel hunger games search optimization approach-based array reconfiguration to reach optimal diversity and robust utilization of the best solutions. These dynamic reconfiguration approaches enhance the output during PS; however, they require many complicated switches and algorithms to administer these switches, complex controlling units, efficient driver circuits and sensors, systematic monitoring units which makes the system entirely intricate and also uneconomical. For an instance, it is reported in [13] that the number of switches obliged to dynamically reconfigure a simple 9 × 9 PV array is 1384. To evade the above complexities of dynamic approaches, static reconfiguration techniques are preferred.

A static reconfiguration method based on Sudoku-puzzle is proposed in [14] to distribute the shadow in the array to lessen the shading losses. The above sudoku-puzzle is further developed by the authors in [15] and obtained an optimal-sudoku approach to alleviate the mismatch. Despite its effectiveness, the Sudoku-puzzle is implemented for only 9 × 9 PV arrays. Moreover, the Sudoku-puzzle contains numerous sets of solutions, and obtaining the optimal reconfigured pattern is a great challenge. Furthermore, in some instances, effective shade dispersal is also not assured. A Ken-Ken-square puzzle-based configuration is proposed in [16] for 6 × 6 array to yield more power and reduce wiring losses compared to Sudoku technique. To minimize the power losses caused due to PS, the authors in [17] proposed an Odd-even (OE) pattern-based reconfiguration strategy for a simple 6 × 4 PV array. An enhanced version of the OE-based reconfiguration strategy named odd-even-prime (OEP) strategy has been proposed in [18] to be employable for both symmetric and asymmetric sizing of PV arrays. In contrast to the numerous existing strategies, the OEP strategy mitigates the row current mismatch of the asymmetric PV arrays as well. An innovative Zigzag strategy-based reconfiguration has been executed on a 4 × 3 PV array to replace the modules for increased shade dispersal [19]. However, first column in an array remains unchanged.

Most of the recommended reconfiguration strategies are not implementable for both symmetrical and non-symmetrical PV arrays. To redress this, a novel reconfiguration procedure based on Skyscraper-puzzle has been executed in [20] to augment the GMP and to lessen the line losses. A Polycube puzzle-based Dancing-Links algorithm is presented for solving a 6 × 6 Sudoku-puzzle [21]. To verify the practical viability of the proposed algorithm, a shifting cloud pattern regarding the direction and speed of wind with discrete instants has been analysed. The authors in [22] introduced a reconfiguration scheme based on a Magic-Square puzzle by using a genetic algorithm (GA) toolbox in MATLAB. A one-time reconfiguration approach based on Lo-Shu magic square is suggested in [23]. An Optimized fixed reconfiguration is exercised in [24] to alleviate the impacts of mutual shading independent of factors of shading such as sun’s location, system’s latitude, and the installation viewpoints of PV array.

To mitigate the row currents mismatch, the authors in [25] proposed an arrow-sudoku puzzle-based reconfiguration scheme for a 6 × 6 PV array. A tom-tom puzzle-based reconfiguration scheme is proposed for a small 5 × 5 PV to physically rearrange the panels for equalizing the irradiation levels of the rows in an array [26]. To examine the efficacy of Arrow-Sudoku and Tom-Tom puzzle-based schemes, they were tested under few growing dynamic PS patterns. The authors in [27] proposed an approximately balanced dispersal of shaded panels in distinct rows of a small 4 × 4 PV array by employing a diagonally-distributed-TCT reconfiguration scheme. A new column index strategy has been developed in [28] to determine the fixed configuration of the array by physically relocating the panels. Very recently, the authors in [29, 30] have employed two chaotic map-based techniques such as chaotic baker map and henon map for PV array reconfiguration to enhance the array characteristics and mitigate the multiple power peaks. However, both these approaches fail significantly under diagonal and column type shading conditions. Further, both these strategies are not scalable for asymmetric PV arrays.

It is deduced that static reconfiguration techniques are simple, cost-effective, reliable and competent. However, some of the above-mentioned reconfiguration techniques cannot rearrange the elements in first-column and hence fail to disperse the first-column shade and thus providing a sub-optimal solution under PS. Further, only a few of these techniques are compatible with the asymmetrical and large-rated arrays. As a result, the development of an optimal reconfiguration of large-rated arrays for effective shade dispersion remains as a major research gap. The primary challenges related to the present work are as follows:

  • The methodologies employed in the dynamic reconfiguration procedures [5,6,7,8,9,10,11,12,13] are complicated and include weighted-sum methodology, early convergence issues, difficult searching mechanisms, increased processing complexity, optimised weights determination, difficult parameter tuning, multiple stages to obtain solutions, etc.

  • These dynamic strategies require a significant number of sensors and switches, complicated switching matrices, driver circuits, micro-controllers, etc. to implement, which raises substantial practical and financial concerns.

  • Numerous static approaches [14,15,16, 19,20,21,22,23,24,25,26,27,28,29] now in use are inapplicable to asymmetric arrays and are primarily compatible with a small number of symmetrical PV array sizes. In reality, the majority of existing PV arrays are asymmetrical. Finding the best puzzle-based and logic-based strategy from the infinite solution sets is a very time-consuming process.

  • The existing reconfiguration algorithms yield poor shade dispersion, which underperforms in some scenarios due to uneven shade dispersion, increased mismatch, unreliable performance, low power improvement, and inconsistent performance.

  • Further, the existing techniques demonstrate several local power maxima in the array characteristics despite increasing GMP to some extent. Since the array’s global maximum power needs to be tracked, this imposes a significant strain on the MPPT controllers and ends up making expensive, advanced MPPT controllers necessary.

To rectify the above setbacks, two novel reconfiguration techniques are proposed in this paper to disperse the shading effectively over the entire PV array under PS. The major contributions of this paper are as follows:

  • In this paper, a Knight’s Tour Magic square (KTM) and Doubly Even-order Magic square (DEM) approaches are proposed to effectively disperse the shade over the entire array thereby reducing the mismatch losses.

  • To demonstrate the potency of the proposed techniques, their performance has been examined with the conventional total-cross-tied [2, 31], odd-even [17], odd-even-prime [18] configurations extensively analyzed under 20 distinct PS patterns.

  • The performance of the considered strategies is investigated with five performance parameters such as global maximum power, power mismatch, percentage losses, efficiency, and performance ratio.

  • A majority of the other contemporary strategies are compatible only for symmetrical array sizes and hence they are not validated/ tested for the unsymmetrical array sizes. However, the proposed strategies are scalable for both symmetrical and unsymmetrical arrays. The proposed KTM and DEM approaches are tested for a symmetric 8 × 8 and asymmetric 3 × 4 PV arrays.

  • Additionally, a comparative economic analysis of the proposed KTM and DEM configurations with the conventional TCT configuration has been performed to demonstrate the associated economic benefits. The electricity units generated, revenue yielded, and the payback obtained for the considered array configurations are assessed and investigated.

  • The detailed comparative analysis of the proposed KTM and DEM approaches with the other recently reported state-of-art strategies has been performed in detail.

Modelling of Solar Cell

Several approaches for modeling a PV cell are published in the literature as follows: one-diode, two-diode, and three-diode models [32]. In this work, a one-diode equivalent circuit model is employed due to its simplicity as shown in Fig. 1.

Fig. 1
figure 1

Equivalent circuit of PV cell

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By using KCL at the junction ‘N’, the PV cell output current (I) is obtained as follows,

$$\mathrm{I}={\mathrm{I}}_{\mathrm{Ph}}-{\mathrm{I}}_{\mathrm{D}}-\frac{{\mathrm{V}}_{\mathrm{PV}}+\mathrm{I}{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}}}\left[31\right]$$
(1)
$$\mathrm{where}\ {\mathrm{I}}_{\mathrm{D}}={\mathrm{I}}_0\times \left[\exp \left(\frac{{\mathrm{V}}_{\mathrm{D}}}{{\upalpha \mathrm{V}}_{\mathrm{T}}}\right)-1\right]$$
(2)

Here, ‘Iph’ is photon-current generated, ‘ID’ is diode current, ‘Rs’ and ‘Rsh’ are series and shunt resisitances, ‘Vpv’ is cell voltage and ‘I’ is cell current, ‘Io’ is module reverse saturation current, ‘α’ is diode ideality factor and ‘VT’ is thermal voltage.

Conventional PV Array Configurations

Some of the notable conventional array configurations reported in the literature are Series (S), Parallel (P), Series-Parallel (SP), Honey-Comb (HC), Bridge-linked (BL), and Total-Cross-Tied [33]. The TCT configuration shown in Fig. 2 is preferable over other configurations due to following reasons: increased GMP, enhanced array characteristics, better output and reduced mismatch losses during PS, extensive employment in practical applications, better reliability [33]. Despite its effectiveness over other configurations under PS, TCT configuration has major setback as it doesn’t disperse the shadow and curtails the output. Therefore, there is a desperate necessity for array reconfiguration for efficient shade dispersal during PS.

Fig. 2
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TCT configurations of an 8 × 8 PV array

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Proposed Methodologies

To attain effective shade dispersal through reconfiguration, the following techniques are employed:

Knight’s Tour Magic Square (KTM)

Knight’s Tour Magic square (KTM) is a semi-magic square proposed by Leonhard Euler in 1759. The reason why it is called a semi-magic square is: Firstly, all the numbers in each row and column add up to give 260. Secondly, all the numbers in each 4 × 4 sub-matrix add up to give 130 [34]. The chess knight is enabled to travel in L-direction on a chessboard whose squares are enumerated from 1 to 64 along the knight’s pathway, i.e., the first cell is designated as “1,” the following cell it visits is designated as “2,” and so forth as shown in Fig. 3(b). Here the pathway is termed as a tour if the knight moves about the chessboard in a general way (one vertical move and two horizontal moves or the other way round).

Fig. 3
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a An 8 × 8 matrix numbered from 1 to 64, b Obtained Knight tour’s magic square matrix

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According to the knight’s tour problem, the knight is moved in such a way that it should visit all the cells in the chessboard only once. So, the main constraints of this problem are: (a) each cell must be visited only once and (b) the knight should not overpass the board. These constraints can be solved by employing a recursive backtracking algorithm. It is executed to redirect the knight’s move when there doesn’t exist any further possible moves and also if not all the cells have been toured [35]. The chess knight follows closed tour and open tour approaches while moving. In closed tour approach, the knight returns to its original starting point after visiting each cell only once. However, in an open tour approach which is employed in this work, the knight just visits all the cells but will not return to the starting point. An 8 × 8 matrix and its obtained Knight tour’s magic square matrix is shown in Figs. 3(a) and 3(b) respectively.

Doubly Even-Order Magic Square (DEM)

A magic square of order N × N contains the numbers numbered from 1 to N2 are positioned in such a way that the sum of all the numbers in a row, in a column, and in the diagonal is the same. This constant numerical sum of all the numbers in a row or column or diagonal of the magic square is termed as a magic sum or magic constant [36]. For an 8 × 8 magic square, the magic constant is 260. Generally, the magic constant of any magic square can be determined by.

where M is magic constant, and N is order of the magic square. The magic squares are majorly classified on the basis of their order as follows: odd-order magic square with order (2 N +1), then doubly even-order magic square with order (4 N), and finally singly-even magic square with order (4 N +2), where N = 1,2,3 and so on [37]. In this paper, DEM approach is employed for reconfiguration. The obtained Doubly Even-order Magic square matrix is given in Fig. 4(b).

Fig. 4
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(a) An 8 × 8 matrix numbered from 1 to 64 (b) Obtained Doubly Even-order Magic square matrix

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The step-by-step formulation of Doubly Even-order Magic square is as follows:

Step 1: Fill all the cells with numbers from 1 to N2 (i.e. 1 to 64) sequentially in a row-wise manner.

Step 2: Draw two main diagonals of the square. Swap the blue-coloured diagonal numbers that are equally distant from the centre.

Step 3: Form a diamond-shaped 4 × 4 sub-square lines which pass through 16 red-coloured numbers as shown in Fig. 3(a).

Step 4: Then, extract these blue-coloured and red-coloured numbers from the 8 × 8 square and form them in a sequence as follows: 1–4–5-8-10-11-14-15-18-19-22-23-25-28-29-32-33-36,-37–40–42-43-46-47-50-51-54–55-57-60-61-64.

Step 5: Fill up the empty cells (formed due to the extraction of coloured numbers in step 4) sequentially by reversing the order of the above-formed sequence as follows: 64–61–60-57-55-54–51--50-47-46-43-42-40-37-36-33-32-29-28-25-23-22-19-18-15-14-11-10-8-5-4-1.

The flowchart of detailed step by step procedure of the proposed KTM and DEM approaches is shown in Fig. 5(a) and (b) respectively.

Fig. 5
figure 5

Flowchart of detailed step by step procedure of (a) KTM and (b) DEM approaches

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Application of the Proposed Strategies in Reconfiguration

By using the proposed algorithms achieves uniform shade dispersion for irradiation equalisation, in contrast to the majority of existing reconfiguration solutions that distribute the shadow effect unevenly. In contrast to the current methodologies, this ensures reliable and optimal performance under any kind of shading environment. While the solar PV array is rearranged in accordance with a preconfigured rearranged matrix design pattern developed by using proposed algorithms. This ensures that the shading effect is significantly reduced but the electrical characteristics of the array are retained. The solar panels of the conventional TCT-configured array are realigned electrically or moved physically (as depicted in Fig.7) following the patterns of rearranged matrices obtained by the proposed strategies shown in Fig. 6. The obtained KTM matrix by the knight tour approach is applied for reconfiguration of the conventional TCT-configured PV array. The differences in architectural topological structure between TCT and the proposed KTM configuration is shown in Fig.7.

Fig. 6
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Original and rearranged matrices obtained by KTM and DEM approaches

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Fig. 7
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(a) Conventional TCT and (b) proposed KTM configuration

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The Fig.7 shows the TCT-based configuration of the PV array where the panel number “27” is positioned in the fifth row and third column (between Node-E and Node-F). However, based on the rearranged matrices that were acquired, this panel is now oriented in the sixth row and the fourth column, as shown in Fig. 7. (b). Similar to this, all panels are effectively reconfigured using the rearranged matrices to disperse the shadow equitably. As an illustration, if the PV panels (PV1, PV30, PV33, PV16, PV3, PV24) within the first row of the newly configured array are shadowed, this row-wise shadow effect is spread to all rows, maintaining the balanced irradiation throughout the rows and minimising the mismatch since these panels are positioned physically in the first row but are electrically connected to multiple rows.

Various Performance Parameters

To verify the effectiveness of the array configurations, the system is tested with the following performance parameters.

Mismatch Power (MMP)

In a PV array, mismatch losses occur due to the interconnection of PV cells having different characteristics [38, 39] and it is the variation between GMP obtained under unshaded (GMPunshaded) and shaded conditions (GMPshaded).

$${\mathrm{MM}}_{\mathrm{p}}\left(\mathrm{W}\right)={\mathrm{GMP}}_{\mathrm{unshaded}}-{\mathrm{GMP}}_{\mathrm{shaded}}$$

Percentage Power Loss (PL)

It is the ratio of difference between GMP obtained at unshaded and shaded conditions to the GMP obtained under unshaded conditions.

$${\mathrm{P}}_{\mathrm{L}}\left(\%\right)=\frac{{\mathrm{GMP}}_{\mathrm{unshaded}}-\mathrm{GMP}{\mathrm{I}}_{\mathrm{shaded}}}{{\mathrm{GMP}}_{\mathrm{unshaded}}}$$

Efficiency (η)

It is the ratio of PV array energy output to the amount of incident solar irradiation delivered by the sun. It is computed as follows:

$$\upeta\ \left(\%\right)=\frac{{\mathrm{V}}_{\mathrm{mpp}}\times {\mathrm{I}}_{\mathrm{mpp}}}{\mathrm{G}\times \mathrm{A}}$$

where, ‘G’ is amount of solar irradiation (W/m2) and A is PV panel area (A = 1425 mm × 990 mm considered for analysis).

Performance Ratio (PR)

It is the ratio of GMP obtained under PS conditions to the GMP obtained under standard test conditions (STC). PR gives what factor of the generated solar energy is utilized efficiently.

$$\mathrm{PR}\left(\%\right)=\frac{{\mathrm{GMP}}_{\mathrm{PS}}}{{\mathrm{GMP}}_{\mathrm{STC}}}$$

Results and Discussion

To attain effective shade dispersal, the PV array is reconfigured by using the rearranged matrices obtained by KTM and DEM approaches as shown in Figs. 3(b) and 4(b) respectively. Further, to substantiate the effectiveness of both the proposed techniques over the notable benchmark TCT configuration and the existing odd-even (OE) and odd-even-prime (OEP) strategies, an 8 × 8 PV array configured in TCT [7], OE [17], OEP [18], KTM and DEM arrangements has been tested using MATLAB software. The efficacy of the proposed techniques is tested under four non-uniform and uniform-shading patterns with the aforementioned five performance indices. The performance parameter values of various array configurations under different shading conditions are given in Fig. 22 and Table. 4. The performance investigation under various shading conditions is performed as follows: The GMP obtained by TCT configuration under STC is 11,582.1 W.

During the non-uniform shading conditions, all the un-shaded modules are considered to receive irradiation of 900 W/m2 whereas shaded modules receive the irradiation of 600 W/m2, 400 W/m2, and 200 W/m2.

Under Short-Broad (SB) Type Shading

During SB shading, the shading pattern considered is short and broad typed as shown in Fig. 8(a) and the shade dispersion witch KTM and DEM techniques are shown in Figs. 8(b) and 8(c) respectively.

Fig. 8
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Short-Broad shading pattern: (a) TCT configuration, shade dispersion with (b) KTM and (c) DEM techniques respectively

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In an array, the number of variations in the row currents indicates the number of bypasses. Table 1 gives the comparative analysis of theoretical computations of various configurations under SB pattern. It is noted that for TCT configuration, the row currents vary from 3.8Im to 7.2Im and this large variation is significantly reduced by employing KTM and DEM techniques which vary from 5.7Im to 6.9Im and 5.5Im to 7.2Im respectively. This is clearly manifested in the significant enhancement in power-voltage characteristics as shown in Fig. 9 which has very fewer multiple peaks and is closer to the ideal characteristics. Both the proposed KTM and DEM techniques generate the highest GMP of 44VmIm compared to TCT which is only 36VmIm. The improved power-voltage (PV) and current-voltage (IV) characteristics obtained by the proposed KTM, and DEM facilitates the efficient tracking of the global maximum power. Further, due to the mitigated row current mismatch through shade dispersion by the KTM and DEM approaches, the global maximum power has been augmented significantly.

Table 1 Comparison of theoretical computations of TCT, KTM and DEM configurations under SB pattern
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Fig. 9
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Power-Voltage characteristics of TCT, KTM and DEM under SB shading pattern

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The GMP obtained by TCT configuration is 7611 W. The concentrated SB shading has been effectively dispersed by KTM technique which yields the GMP of 9521 W which is 25.1% more than TCT. Similarly, DEM technique also yields the highest GMP of 9542 W which is 25.4% more than TCT. Further, the existing OE and OEP configurations despite scalable for all array sizes, fail to disperse the shade uniformly thereby yielding more mismatch losses. The GMP obtained by OE and OEP arrangements are 8321 W and 9078 W which is only 9.32% and 19.27% respectively. The respective MMP obtained by the proposed KTM, and DEM configurations is also reduced significantly. It is noted that the proposed techniques reduce MMP nearly by 50% compared to conventional TCT.

The respective PL computed for KTM, and DEM techniques is 17.79% and 17.61% that again confirms that the proposed techniques are preferable over conventional ones. The KTM and DEM techniques have the maximum generated efficiency of 13.6% and 13.63% respectively when compared with TCT which is only 10.87%. It is remarkably manifested in the fact that the values of PR, and MMP are more superior to conventional TCT.

Under Short-Narrow (SN) Type Shading

The shading pattern considered is short and narrow in which the lower right part of the array has been shaded as shown in Fig. 10(a) and the shade dispersion witch KTM and DEM techniques - are shown in Figs. 10(b) and 10(c) respectively. The theoretical computations for TCT, KTM, and DEM techniques explicates that DEM technique yields the lowest row current variation with only two bypasses as shown in Table 2 and thus providing much smoother power-voltage characteristics as shown in Fig. 11. By employing the DEM technique, a very narrow range of row current variations is achieved i.e., from 6Im to 6.2Im. Followed by DEM, the KTM technique also provides smoother characteristics compared to TCT. The GMP obtained by DEM and KTM techniques are 48VmIm and 40VmIm which is highest compared to TCT which yields only 38.4VmIm.

Fig. 10
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Short-Narrow shading pattern: (a) TCT configuration, shade dispersion with (b) KTM and (c) DEM techniques respectively

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Table 2 Comparison of theoretical computations of TCT, KTM and DEM configurations under SN pattern
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Fig. 11
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Power-Voltage characteristics of TCT, KTM and DEM under SN shading pattern

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Under, this condition, the GMP obtained by TCT, KTM and DEM techniques is 8578.5 W, 8871.8 W, 9878.5 W respectively. The respective DEM and KTM techniques distribute the shade effectively thereby enhancing the GMP by 1300 W and 293.3 W which is 15.16% and 3.42% more than TCT. During this case, the existing odd-even strategy yields very lower power (−2.4%) even compared to the standard benchmark TCT configuration. Besides, the improved OE strategy named as odd-even-prime strategy also underperforms in this case maximizing the GMP only by 1.72%. The MMP obtained by TCT, KTM, and DEM configuration is 3003.5 W, 2710.2 W, and 1703.5 W. The proposed DEM and KTM techniques reduce the MMP by significant percentage when compared to TCT. The PL computed for TCT is 25.93%, whereas the respective DEM and KTM techniques reduce it to 14.7% and 23.4%. The DEM technique yields the highest efficiency, which is 14.23%. Followed by KTM (12.78%) and TCT (12.35%) configurations. The efficacy of the respective DEM and KTM techniques is evinced with a greater PR of 0.852 and 0.765. Further, both DEM and KTM techniques provides an extremely better performance which is also manifested in the enhancement of the array characteristics.

Under Long-Broad (LB) Type Shading

The shading pattern considered is long and broad typed as shown in Fig. 12(a) and the shade dispersion witch KTM and DEM techniques are shown in Figs. 12(b) and 12(c) respectively. The maximum power output obtained by TCT, KTM, and DEM techniques is 31.2VmIm, 36VmIm, and 40.8VmIm. It is noticed that from Table. 3 that by employing the DEM and KTM techniques, a narrow range of row current variation is obtained which is from 5.2Im to 6.6Im and 5.1Im to 7.2Im respectively. This narrow range has led to the reduction of multiple peaks thereby obtaining smooth characteristics of an array as shown in Fig. 13. However, the conventional TCT yields a comparati - ly broad range of row currents varying from 4.5Im to 7.2Im, thus exhibiting many bypasses. The number of bypasses obtained by TCT is four and these bypasses are significantly reduced by half by employing the proposed KTM and DEM techniques. By executing the proposed approaches, the smoother characteristics are obtained exhibiting only one global and local power peak, whereas the conventional TCT exhibits four multiple power peaks. This causes the misleading of tracking the global maximum power from the conventional TCT array configuration and the MPPT controllers stuck at a local optimum extracting sub-optimal output from the array.

Fig. 12
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Long-Broad shading pattern: (a) TCT configuration, shade dispersion with (b) KTM and (c) DEM techniques respectively

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Table 3 Comparison of theoretical computations of TCT, KTM and DEM configurations under LB pattern
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Fig. 13
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Power-Voltage characteristics of TCT, KTM and DEM under LB shading pattern

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The conventional TCT generates the GMP of 6770.4 W whereas the proposed DEM configuration yields 8559.6 W which is 26.43% more than TCT. The proposed KTM technique also yields 15.97% more than TCT configuration which is 1080.74 W. The MATLAB computational results show that the proposed DEM and KTM techniques yield superior performance with the reduced mismatch power of 7623 W and 9683 W respectively which is 36.59% and 19.45% less when compared to TCT (12,021 W). Following the proposed strategies, the existing OE and OEP configurations yield the enhancement of only 6870.7 W and 7752.4 W which is 1.48% and 14.5% more respectively. It is observed that the proposed DEM technique holds the first leading place giving the reduced PL of 26.09%, efficiency of 11.29%. Furthermore, this technique gives a peerless performance with a PR of 0.739. Followed by DEM, the KTM technique holds the next leading place by proffering the reduced PL of 32.21%, efficiency of 10.37%. Besides these factors, this technique also yields better performance with a PR of 0.677.

To further, demonstrate the employability of the proposed techniques, various cases of following uniform shading patterns are also considered. Here, the un-shaded and shaded modules are considered to receive irradiation of 900 W/m2 and 500 W/m2 respectively.

Under L-Shaped Shading

In this condition, the pattern of the shading is L-shaped as shown in Fig. 14(a) and the shade dispersion witch KTM and DEM techniques are shown in Figs. 14(b) and 14(c) respectively. During this condition, TCT yields the GMP of 9200 W whereas the proposed KTM technique yields the GMP of 9808 W which is 608 W (6.06%) more than the TCT configuration. The DEM configuration yields the GMP of 10,175 W which is 975 W (10.61%) more than that of TCT. The proposed DEM technique yields extremely superior performance over TCT by yielding the reduced MMP of 1407 W, PL of 12.14%, efficiency of 14.09%, and PR of 0878. The proposed DEM and -KTM techniques yield smoother array power-voltage characteristics compared to TCT configurations which is shown in Fig. 15.

Fig. 14
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L-shaped shading pattern: (a) TCT configuration, shade dispersion with (b) KTM and (c) DEM techniques respectively

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Fig. 15
figure 15

Power-Voltage characteristics of TCT, KTM and DEM under L-shaped pattern

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Under Triangle-Shaped Shading

Here, the lower triangular part of the 8 × 8 PV array is shaded as shown in Fig. 16(a) and the shade dispersion witch KTM and DEM techniques are shown in Figs. 16(b) and 16(c) respectively. The concentrated shading subjected over the corner portion of the PV array is dispersed into various rows of the array through the KTM and DEM configurations. The conventional TCT array configuration exhibits five bypasses (five power peaks) in the array characteristics. Whereas the proposed strategies mitigate these peaks to a significant extent as shown in the Fig. 16 of power-voltage characteristics of the array. The respective DEM and KTM techniques yield the GMP of 10,659 W and 10,343 W which is 7.19% and 4.02% than TCT.

Fig. 16
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Triangle-shaped shading pattern: (a) TCT configuration, shade dispersion with (b) KTM and (c) DEM techniques respectively

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The proposed DEM and KTM techniques performed better than TCT in delivering outstanding performance with additional parameters like MMP, PL, efficiency, and PR. The respective values obtained with KTM configurations are 1239 W, 10.69%, 13.67%, and 0.893 respectively whereas the respective values obtained with DEM technique are 923 W, 10.69%, 14.09%, and 0.920 respectively. When compared to TCT, the array characteristics are improved by employing proposed techniques as shown in Fig. 17

Fig. 17
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Power-Voltage characteristics of TCT, KTM and DEM under triangle-shaped pattern

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Fig. 18
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Rhombus-shaped shading pattern: (a) TCT configuration, shade dispersion with (b) KTM and (c) DEM techniques respectively

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Under Rhombus-Shaped Shading

A special shading pattern in the form of a rhombus has been considered as shown in Fig. 18(a) and the shade dispersion witch KTM and DEM techniques are shown in Figs. 20(b) and 20(c) respectively. The concentrated shadow covering the significant portion of the PV array leads to the mismatch between the distinct rows of the array. Due to the mismatch, the irradiation levels in all the rows are unbalanced by a considerable amount. Further, this concentrated shade has been dispersed to various rows mitigating the mismatch and improving the array characteristics. The TCT configuration generates the GMP of 9368 W. The proposed KTM and DEM configuration yields the GMP of 10,338 W and 10,518 W which is 990 W (10.35%) and 1150 W (12.27%) more compared to the TCT configuration. The performance of the DEM technique is on par with the KTM technique under this shading condition. The proposed KTM technique yields the lowest MMP, lowest PL, highest efficiency, and highest PR with the values of 1239 W, 10.69%, 13.67%, and 0.893 respectively. The respective power-voltage characteristics are shown in Fig. 19.

Fig. 19
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Power-Voltage characteristics of TCT, KTM and DEM under rhombus-shaped pattern

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Under Square-Shaped Shading

The small portion of the PV array is considered to be subjected to a square-shaped shading pattern at the upper left corner of the array as shown in Fig. 20(a). The shade dispersion with the KTM and DEM techniques are shown in Figs. 20(b) and 20(c) respectively. The GMP obtained for TCT, KTM and DEM configurations are 10,282 W, 10696 W, and 10,696 W respectively. By employing the DEM and KTM techniques, the respective enhancement in GMP is 4.02% and 4.03% compared to that of the conventional TCT configuration. The proposed DEM technique yields the lowest MMP, lowest % power loss, highest efficiency, and highest PR with the values of 886 W, 7.64%, 14.05%, and 0.923 respectively. The respective power-voltage characteristics are shown in Fig. 20. It is observed from the figure that the array PV characteristics (Fig. 21) are enhanced significantly due to the mitigation of row-current mismatch in the KTM and DEM configurations of the PV array. The power enhancement after the reconfiguration is less which is approximately 4% since the shaded area of the PV array is less. Besides, both the proposed configurations effectively disperse the shade over the array to mitigate the losses.

Fig. 20
figure 20

Square-shaped shading pattern: (a) TCT configuration, shade dispersion with (b) KTM and (c) DEM techniques respectively

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Fig. 21
figure 21

Power-Voltage characteristics of TCT, KTM and DEM under square-shaped pattern

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A comparison of the global maximum power of PV array configured in various configurations has been shown in Table 4. Further, the other additional performance parameters such as mismatch power, and PR under various PS patterns are given in Table 4 respectively. It is noted from Fig. 22 that the proposed configurations offer superior and consistent performance under all the shading conditions. Further, when compared to the KTM approach, the DEM configuration exhibited superior performance due to its lowest correlation between the adjoining panels of the reconfigured array Fig. 23.

Table 4 Comparison of GMP enhancement, mismatch power, and performance ratio under various non-uniform PS patterns
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Fig. 22
figure 22

Comparison of global maximum power under various non-uniform and uniform shading conditions

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Fig. 23
figure 23

Various progressive dynamic shading cases considered for analysis

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Under Distinct Progressive Shading Conditions

The proposed KTM and DEM configurations, the existent OE [17], the existing OEP [18], and the conventional TCT [7], are examined and tested under various shading scenarios of distinct shading groups, such as progressive left-to-right and triangular shading scenarios as shown in Fig.5. The most common manifestations of these shadings include adjacent panel shading, high structures close by, moving clouds, raised poles, skyscrapers, chimneys, roof accessories, etc., whose density goes up over time. When held as independent shading cases, these shading cases have the advantage of acting as both dynamic and static shades. The system is evaluated using the global maximum power performance indicator to confirm the superiority of the recommended configurations.

The performance of the proposed configurations has been verified under static shading conditions as detailed in Section.6.1 to 6.8. Further, it is also noted from Table 5 that the proposed configurations also perform efficiently even under moving cloud/shade conditions. It is evident from the table that there exists a significant difference between the GMP obtained by the proposed and the conventional TCT configuration

Table 5 Comparison of GMP for various configurations under progressive shading conditions
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Validation of Proposed Approaches for Asymmetrical PV Arrays

In the practical real-time scenario, the PV arrays could be symmetric or asymmetrical. Numerous static reconfiguration methods [14,15,16, 19,20,21,22,23,24,25,26,27,28,29] mentioned in the literature cannot be used for the asymmetrically sized PV arrays such as 4 × 5, 7 × 9, 11 × 8, etc. Consequently, their range of applications is very limited for employment. Whereas, the conventional array configurations, despite being scalable for all array sizes, have zero shadow dispersion ability and cannot reduce the mismatch between the rows of an array. So, very recently, two methodologies based on the odd-even (OE) and odd-even-prime (OEP) patterns are reported to overcome the scalability issues for the asymmetrical arrays. Nevertheless, as mentioned and proved in [4], that the existing OE and OEP approaches, despite being adaptable to any array size, have very poor shade dispersal capability because of their inefficient reconfiguration process involved.

To verify the scalability and compatibility of the proposed techniques for asymmetrical PV array sizing, a 3 × 4 PV array is considered for the analysis. Additionally, to demonstrate the effectiveness of the proposed configurations, their results have been compared with that of the recently reported OE and OEP-based strategies under various shading conditions shown in Fig.24. It is evident from the figure that the proposed strategies distribute the shade effectively and uniformly over the array when compared to the existing OE and OEP strategies. Besides, it is also clearly observed from the array characteristics as shown in Fig. 25, that the proposed strategies yield the highest GMP compared to the OE, OEP, and TCT configurations. Moreover, the proposed configurations exhibit comparatively better characteristics that are close to the ideal ones. Further, it is noted from Fig. 25 and Fig. 26, that the existing OE and OEP exhibits poor shade dispersion under the diagonal case yielding 10.31% lowered power even compared to the conventional TCT configuration. Hence, due to the highly inconsistent performance of the OE and OEP, the proposed strategies are preferred over them. The GMP obtained by various configurations of the asymmetrical arrays are given in Table 6.

Fig. 24
figure 24

Various shading cases and the corresponding shade dispersion obtained for various approaches

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Fig. 25
figure 25

PV array characteristics obtained for unsymmetrical PV arrays under distinct cases

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Fig. 26
figure 26

(a) Number of units generated per day and (b) per year

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Table 6 GMP (in Watt) obtained for a 4 × 3 PV array under various shading patterns
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Economic Analysis

During unshaded conditions, the PV array generates 69.492 units/day. However, under various shadowing conditions, the output is limited as the mismatch losses are increased significantly [40]. These drawbacks can be reduced to the maximum extent by employing the proposed shadow dispensing techniques. To demonstrate the effectiveness, feasibility, and usefulness of the suggested techniques, the generated electricity units and the revenue growth are assessed and investigated. During the normal days, the PV array functions effectively during the peak sun hours, so for analysis, three aforementioned shading patterns (SB, SN and LB) are considered to take place over the array each hour from 10 a.m. to 4 p.m. The solar power tariff of ₹3 (INR) per unit is considered for assessment. A comparative analysis of the respective units and revenue generated is shown in Figs. 27 and 28.

Fig. 27
figure 27

(a) Revenue generated per day and (b) per year

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Fig. 28
figure 28

Performance comparison of various strategies using different indices (A) Scalability, (B) Consistency, (C) Maximum Power, (D) Complexity, (E) Array Characteristics Enhancement, (F) Sensors and Switches Requirement, (G) Payback Period

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The units generated per day by TCT, KTM, and DEM techniques are 51.47kWh, 56.7kWh, and 59.12kWh respectively. The respective units generated per year 154.4kWh, 170.126kWh, and 177.361kWh. So, for a year, the proposed DEM and KTM techniques generates 1812.65 and 2795.9 units more compared to TCT configuration respectively. By employing the proposed DEM and KTM techniques, the respective revenue generation is increased by ₹5735.99 (INR) and ₹8379.31 (INR) which is 14.9% and 10.18% and more compared to the benchmark TCT. From the comparative economic analysis as given in Table 7, it is deserving to acknowledge that both proposed DEM and KTM techniques effectively disperses the shadow over the array and are best suitable to be employed under the partial shading conditions. By increasing the ratings of PV panels, the output power is maximized thereby maximizing the revenue generated.

Table 7 A comparative economic analysis of TCT, KTM and DEM configurations
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Comparison of the Proposed Approaches in Comparison to the State-of-the-Art Strategies

The advantages of the proposed approaches in comparison to the current state-of-the-art strategies are listed as follows:

  • The proposed approaches do not use switching matrices, choose switching matrices at the best possible rate, and furthermore do not have memory-related switching matrix operation problems, in contrast to AI approaches [5, 6].

  • The EAR-based approaches [7, 8] demand for a controller to feed the switching pulses for the switches that correspond to the appropriate shade dispersion. In addition, a large number of switching patterns are required to determine the best PV array reconfiguration strategy. On the contrary, the suggested techniques eliminate the necessity of these operations.

  • The proposed methods circumvent the shortcomings of population-based meta - heuristic optimization algorithms [10,11,12,13] such as high dimension search space, convergence & parameter selection issues, methodology challenges, being stuck at local optimum, weighting factor shortlisting difficulties, gigantic computations, heavy computational time, etc.

  • The proposed approaches, in contrast with the current logic- and puzzle-based methodologies [14,15,16, 19,20,21, 23,24,25,26], are fully scalable and adaptable with all array sizes. Additionally, the proposed configurations evenly and effectively spread the shade over the array.

  • The proposed reconfiguration strategies effectively address a number of shortcomings of the current magic square-based [22], shift-based [27], and index-based [28] methods, including inadequate scalability, a high association between the diagonal arrangement of panels, inconsistency, etc.

  • The recent OE [17] and OEP [18] approaches, while adaptable to any array sizing, produce performance that is extremely inconsistent since the panels’ poor placement after reconfiguration has a strong correlation between the next shaded panels row-wise. The proposed ones, in contrast, are consistently more efficient in shaded environments and are also generally compatible.

The overall comparison of the proposed techniques with the state of art techniques are shown in Fig. 28.

Conclusions

The GMP obtained by the PV array is mainly restrained due to PS. The proposed KTM and DEM reconfiguration techniques effectively disperses the shade thereby enhancing the GMP and lessening the mismatch losses. For the various shading conditions considered for symmetrical 8 × 8 array such as SB, SN, LB, L-shaped, triangle-shaped, rhombus-shaped, and square-shaped, the enhancement in GMP from TCT to DEM configuration is 25.38%, 15.16%, 26.43%, 10.59%, 7.19%, 10.02%, and 4.03% respectively, and the enhancement in GMP from TCT to KTM configuration is 25.09%, 3.42%, 15.97%, 6.61%, 4.03%, 10.09%, 4.03% respectively. Moreover, there is a significant improvement in the PV and IV characteristics as the number of bypasses are significantly reduced by the proposed techniques. By employing the proposed DEM and KTM techniques, the respective annual generated revenue is increased by 14.9% and 10.18% compared to benchmark TCT. Further, for the unsymmetrical 4 × 3 array, the enhancement in GMP of the KTM configuration wrt TCT is 25.52%, 10.64%, 13.85%, 5.55%, 3.92% and the enhancement in GMP of the DEM configuration wrt TCT is 26.06%, 25.11%, 16.82%, 10.80%, 10.46%, 7.57%, 4.24% respectively. The existing OE and OEP configurations offer inconsistent and inferior performance compared to the proposed ones.

Hence, from the technical and economic analysis, it is exemplary to accept that both proposed techniques efficiently distribute the shadow over the entire array and are quite fitting to be exercised for both symmetrical and unsymmetrical PV arrays under PS. By increasing the ratings of PV panels, the output power is maximized thereby maximizing the revenue generated. Both the proposed DEM and KTM techniques outperformed the conventional TCT for all performance indices. Further, when compared to KTM, the DEM technique renders superior performance under most shading conditions and proves to be an effective solution for PS issues.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.