**3. Problem Expression**

In order to find the PV cell/panel equivalent circuit parameters, an optimization problem was formulated and solved to mitigate the RMSE between the measured current (*I*<sup>M</sup> ) and the simulated one (*I*<sup>S</sup> ). In the optimization formula, the unknown parameters are defined as design variables; the fitness value can be formulated as follows:

$$RMSE = \sqrt{\frac{1}{N\_m} \sum\_{i=1}^{N\_m} f\_{PV}(V\_{PV}, I\_{PV}, \mathbf{x})^2} = \sqrt{\frac{1}{N\_m} \sum\_{i=1}^{N\_m} \left(I\_{\mathbf{M}} - I\_{\mathbf{S}}\right)^2} \tag{8}$$

where *Nm* indicates the number of measured patterns, *fPV* refers to the PV model function, and *I*<sup>M</sup> and *I*<sup>S</sup> are the measured and simulated currents, respectively. The DDM objective function contains seven unknown parameters; it can be written as,

$$\begin{split} \left( f\_{\text{DDM}}(V\_{\text{PV}}, \text{I}\_{\text{PV}}, \mathbf{x}) \right) &= \mathbf{I}\_{\text{PV}} - \mathbf{x}\_{7} + \mathbf{x}\_{5} \left( \exp\left(\frac{q(V\_{\text{PV}} + I\_{\text{PV}} \mathbf{x}\_{3})}{\mathbf{x}\_{1} \mathbf{K} \mathbf{T}}\right) - 1 \right) + \mathbf{x}\_{6} \left( \exp\left(\frac{q(V\_{\text{PV}} + I\_{\text{PV}} \mathbf{x}\_{3})}{\mathbf{x}\_{2} \mathbf{K} \mathbf{T}}\right) - 1 \right) \\ &+ \frac{(V\_{\text{PV}} + I\_{\text{PV}} \mathbf{x}\_{3})}{\mathbf{x}\_{4}} \\ &= \mathbf{I}\_{\text{ref}} \end{split} \tag{9}$$

$$\text{where } \underset{\frown}{\text{where }} \mathfrak{x} = \left[ A\_1 A\_2 R\_s R\_{sh} I\_{d1} I\_{d2} I\_{ph} \right].$$

On the other hand, the TDM objective function comprises nine unknown parameters, which can be expressed as,

$$\begin{split} f\_{\text{TDM}}(V\_{\text{PV}}, I\_{\text{PV}}, \mathbf{x}) &= I\_{\text{PV}} - \mathbf{x}\_{\text{\ $}} + \mathbf{x}\_{\text{\$ }} \left( \exp\left(\frac{q(V\_{\text{PV}} + I\_{\text{PV}} \mathbf{x}\_{4})}{\mathbf{x}\_{1} \mathbf{K} \mathbf{T}}\right) - 1\right) + \mathbf{x}\_{7} \left( \exp\left(\frac{q(V\_{\text{PV}} + I\_{\text{PV}} \mathbf{x}\_{4})}{\mathbf{x}\_{2} \mathbf{K} \mathbf{T}}\right) - 1\right) \\ &+ \mathbf{x}\_{8} \left( \exp\left(\frac{q(V\_{\text{PV}} + I\_{\text{PV}} \mathbf{x}\_{4})}{\mathbf{x}\_{3} \mathbf{K} \mathbf{T}}\right) - 1\right) + \frac{(V\_{\text{PV}} + I\_{\text{PV}} \mathbf{x}\_{4})}{\mathbf{x}\_{5}} \end{split} \tag{10}$$

where *x* = *<sup>A</sup>*1*A*2*A*3*RsRsh Id*<sup>1</sup> *Id*<sup>2</sup> *Id*<sup>3</sup> *Iph* .

Finally, the TDM of the PV panel objective function can be written as,

$$\begin{aligned} \left[\mathbf{I}\_{PV\\_p}(V\_{PV\\_I\\_PV}, \mathbf{x})\right] &= \mathbf{I}\_{PV} - \mathbf{x}\_7 \ast \mathbf{N}\_p + \mathbf{x}\_5 \ast \mathbf{N}\_p \left[\exp\left[\frac{q\left[V\_{PV}/\mathbf{N}\_s + \mathbf{x}\_3/N\_p \ast I\_{PV}\right]}{\mathbf{x}\_1 KT}\right]}{\mathbf{x}\_2 KT}\right] - 1 \right] + \mathbf{x}\_6\\ &\ast \mathbf{N}\_P \left[\exp\left[\frac{q\left[V\_{PV}/\mathbf{N}\_s + \mathbf{x}\_3/N\_p \cdot I\_{PV}\right]}{\mathbf{x}\_2 KT}\right] - 1\right] + \left[\frac{V\_{PV}/\mathbf{N}\_b + \mathbf{x}\_3/N\_p \ast I\_{PV}}{\mathbf{x}\_5}\right] \end{aligned} \tag{11}$$
