*2.2. Problem Formulations*

⎧ ⎪⎪⎨

⎪⎪⎩

*RMSE* between the measured data and the calculated data usually serves as the objective function [54–56]:

$$\text{RMSE} = \sqrt{\frac{1}{N} \sum\_{k=1}^{N} f^2(V, I, \mathbf{x})} \tag{6}$$

where *x* represents the solution vector and *N* represents the actual data's amount, and *f*(*V*, *I*, *x*) calculates the current error in the following way.

For SDM:

$$\begin{cases} f(V, I, \mathbf{x}) = I\_{\mathrm{ph}} - \frac{V + IR\_{\mathrm{s}}}{R\_{\mathrm{sh}}} - I\_{\mathrm{ssd}} \Big[ \exp\left(\frac{q(V + IR\_{\mathrm{s}})}{nkT}\right) - 1 \right] - I\\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \begin{pmatrix} I\_{\mathrm{ph}} \ R\_{\mathrm{ssd}}, R\_{\mathrm{s}\prime} R\_{\mathrm{sh}\prime} \mathfrak{n} \end{pmatrix} \end{cases} \tag{7}$$

For DDM:

$$\begin{cases} f(V, I, \mathbf{x}) = I\_{\mathrm{pl}} - \frac{V + IR\_{\mathrm{s}}}{R\_{\mathrm{sl}}} - I\_{\mathrm{sd1}} \Big[ \exp\left(\frac{q(V + IR\_{\mathrm{s}})}{n\_{1}kT}\right) - 1\Big] - I\_{\mathrm{sd2}} \Big[ \exp\left(\frac{q(V + IR\_{\mathrm{s}})}{n\_{2}kT}\right) - 1\Big] - I\\ \mathbf{x} = \left(I\_{PV}, I\_{\mathrm{sd1}}, I\_{\mathrm{sd2}}, R\_{\mathrm{s}}, I\_{\mathrm{sl}}, n\_{1}, n\_{2}\right) \end{cases} \tag{8}$$

For TDM:

$$\begin{cases} f(V, I, \mathbf{x}) = I\_{\text{pl}} - \frac{V + IR\_{\text{sf}}}{R\_{\text{sl}}} - \sum\_{j \to 3} I\_{\text{ssd}j} \left[ \exp\left(\frac{q(V + IR\_{\text{s}})}{n\_{j}kT}\right) - 1 \right] - I\\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left( I\_{PV\_{\text{s}}} I\_{\text{ssd2}}, I\_{\text{ssd3}}, I\_{\text{ssd1}}, n\_{1}, n\_{2}, n\_{3} \right) \end{cases} \tag{9}$$

For PV module:

$$\begin{cases} f(V, I, \mathbf{x}) = I\_{\mathrm{pl}} N\_p - \frac{V + IR\_s N\_s / N\_p}{R\_{\mathrm{sl}} N\_{\mathrm{s}} / N\_p} - I\_{\mathrm{ssd}} N\_p \left[ \exp\left(\frac{q\left(V + IR\_s N\_{\mathrm{s}} / N\_p\right)}{n N\_{\mathrm{s}} \mathrm{KT}}\right) - 1\right] - I\\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \begin{cases} I\_{\mathrm{pl}} \mathrm{-T}\_{\mathrm{s}} \mathrm{-T}\_{\mathrm{s}} = \mathrm{I} \\\ \mathrm{I}\_{\mathrm{pl}} \mathrm{-}\_{\mathrm{s}} \mathrm{-}\_{\mathrm{s}} \mathrm{-}\_{\mathrm{s}} \mathrm{-}\_{\mathrm{sl}} \mathrm{n} \end{cases} \end{cases} \tag{10}$$

For the objective function RMSE, its computation requires solving methods with the ability to solve implicit functions. Commonly used are deterministic and metaheuristic methods. Several deterministic methods, including Newton Raphson [24], Lambert W function [25], Levenberg Marquardt [57], and Berndt–Hall–Hall–Hausman [58], have successfully solved the non-linear problem. However, it does not mean that deterministic methods can tackle the challenge of initial value sensitivity well. Due to challenges such as non-linearity and non-convexity, metaheuristics are considered to be the best solution for solving this issue.

### *2.3. Indicators Summary*

Varied algorithmic settings substantially affect the results of metaheuristic methods and various indicators can evaluate the results from diverse aspects. Hence, we summarize the approach and case settings and the performance evaluation indicators. Usually, the literature has drawn characteristic curves to visualize the accuracy of the extracted parameters. Nevertheless, when the parameters' difference is not very large, some general and objective indicators are used as the basis for evaluating the advantages and disadvantages of different methods. Here, we highlight the commonly used indicators to compare them:


$$\text{IAE} = |f(V, I, \mathbf{x})| \tag{11}$$

$$\text{SIAE} = \sum\_{k=1}^{N} IAE \tag{12}$$

$$\text{MIAE} = \frac{1}{N} \sum\_{k=1}^{N} IAE \tag{13}$$

• In addition, a few works in the literature also use evaluation indicators such as the sum of squares of power, current, and voltage errors (ERR) [61].
