(b) Total-cross-tied configuration

The TCT electrical arrangement is an extended modification of the SP configuration by means of fixing ties across the parallel strings. This cross-tied-based modification is responsible for enhancing the current through each parallel string and constant voltage during PSCs. Mathematically; voltage analysis is expressed for the PV array in Equation (2).

$$\mathbf{V}\_{\text{array}} = \sum\_{\mathbf{k}=1}^{6} \mathbf{V}\_{\text{mk}} \tag{2}$$

where Vmk refers to the voltage (maximum) at the kth row. Each string of the PV modules is linked in parallel; therefore, the total current drawn by the PV array is the sum of the individual currents drawn by each module in the array. Furthermore, a mathematical approach is applied to each node using Kirchhoff's current law. As a consequence, the array current *Iarray* can be expressed in Equation (3) as

$$I\_{array} = \sum\_{q=1}^{6} \left( I\_{kq} - I\_{(k+1)q} \right) = 0, \ p = 1, 2, 3, \dots 9 \tag{3}$$

where *k* and *q* are the number of rows and columns in the considered 6 × 6 size PV array. Figure 3 depicts the TCT array's electrical configuration of PV modules.


**Figure 3.** PV module arrangements in TCT configuration (6 × 6 size). (**a**) Nomenclature, (**b**) placement of PV module.

#### *2.3. Game-Theory-Based PV Array Configurations*

(a) SDK and I-SDK configurations

As the shadow is diffused over the array, the game-theory-based SDK layout guarantees that mismatch losses are minimized. The wiring connections are completed once the panels have been properly arranged and they stay unmodified. This decreases computing difficulties while also preventing the overuse of sensors and switches [33].

The SDK change in columns 2–6 is known as the I-SDK puzzle. By repositioning the PV modules without disturbing the electrical arrangement, the suggested I-SDK layout is implemented in the TCT PV array. In this game puzzle, higher dispersion is found based on the optimal placement of all the integer numbers in an array. This integer placement modification is responsible for higher shade dispersion. The SDK and I-SDK puzzles and approach are depicted in Figure 4a–d as


**Figure 4.** *Cont*.


**Figure 4.** (**a**) Number placement for I-SDK; (**b**) number placement for SDK; (**c**) electrical connections for SDK (**d**) methodology to achieve SDK game theory.

#### (b) Symmetric matrix-based configurations

Cyclic arrangement of integer numbers from 1 to 6 is carried out to establish the SM game-theory-based arrangement. The summation of the considered integer numbers in each row and column is found to be equal as per SM development guidelines. In addition to that, either of the diagonal elements keeps repeating within it. Figure 5a depicts all of the assets of the 6 × 6 size SM as

**Figure 5.** (**a**) Row, column and single diagonal property; (**b**) repeated submatrix elements.

The summation of all the items in each particular row/column, according to SM characteristics, is 21. Furthermore, in Figure 5b, there is a repetition of 3 × 3 size square submatrices.

To represent the row-column summation rules, mathematical assumptions are made. In this context, the SM size is considered in the order of *<sup>p</sup>* × *<sup>q</sup>*. Moreover, the *nth* element can be placed corresponding to the *pth* and *qth* row-column, respectively. So, the location

of the PV module (*npq*) in an array can be written in a generalized way and expressed in Equation (4) as

$$n\_{pq}, where \left\{ \begin{array}{ll} p = no. of row \\ q = no. of column \end{array} \middle| \begin{array}{l} (p = 1, 2, \dots, 6) \\ (q = 1, 2, \dots, 6) \end{array} \right. \tag{4}$$

In Figure 5a,b, we see the mathematical equations for four distinct cases of row-wise summing and that are accomplished using Equation (5) as follows:

$$\sum\_{p=1}^{6} n\_{pq}(\text{Summation for } p^{\text{th}} \text{row}) = \sum\_{q=1}^{6} n\_{pq}(\text{Summation for } q^{\text{th}} \text{column}) \tag{5}$$

The above Equations (4) and (5) are involved in the guidelines to achieve the 6 × 6 size SM setup, and the scientific method to establish the SM is shown in Figure 6 as

**Figure 6.** Methodology to achieve SM game theory.

In SM, there are six rows and columns in a 6 × 6 PV array, respectively. As per the nomenclature shown in Figure 7a, the first digit of each individual PV module depicts the row count, while the second digit depicts the column count. It is an easier nomenclaturebased methodology to understand the electrical arrangements of PV modules in an array. In Figure 7b, the PV module locations are migrated using the recommended SM structure but the electrical contacts of the PV panels within PSCs remain unchanged.


**Figure 7.** (**a**) Nomenclature of PV modules; (**b**) PV module arrangements for SM configuration.
