As described above, the \(I-V\) optimizer should be seeded with initial values that reflect the experimentally measured data. In this regard, an analytical model is derived to express each targeted parameter. Due to the nature of perovskite solar cells, a capacitance-dependent model is a must. Accordingly, the standard D.C. analytical model, such as the model in Abdulrazzaq et al. (2022), cannot be utilized. Consequently, a transient analytical model is suggested. The general form for the I-V equation, while applying the single diode model, can be written as (Abdelrazek et al. 2022):
$$I={I}_{ph}-{I}_{01}\left(\mathrm{exp}\left(\frac{q(V+{R}_{s}I)}{{n}_{1}kT}\right)-1\right)-\frac{(V+I{R}_{s})}{{R}_{sh}}-{C}_{eq}\frac{d(V+I{R}_{s})}{dt}$$
(1.a)
$$I={I}_{ph}-{I}_{01}\left(\mathrm{exp}\left(\frac{q\left(V+{R}_{s}I\right)}{{n}_{1}kT}\right)-1\right)-{I}_{02}\left(\mathrm{exp}\left(\frac{q\left(V+{R}_{s}I\right)}{{n}_{2}kT}\right)-\frac{(V+I{R}_{s})}{{R}_{sh}}-{C}_{eq}\frac{d(V+I{R}_{s})}{dt}\right)$$
(1.b)
$$I={I}_{ph}-{I}_{01}\left(\mathrm{exp}\left(\frac{q(V+{R}_{s}I)}{{n}_{1}kT}\right)-1\right)-{I}_{02}(\mathrm{exp}\left(\frac{q\left(V+{R}_{s}I\right)}{{n}_{2}kT}\right)-{I}_{03}\left(\mathrm{exp}\left(\frac{q(V+{R}_{s}I)}{{n}_{3}kT}\right)-\frac{(V+I{R}_{s})}{{R}_{sh}}-{C}_{eq}\frac{d(V+I{R}_{s})}{dt}\right)$$
(1.c)
where \({I}_{ph}\) is the photo-generated current in (A), \({I}_{\mathrm{01,2},3}\) are the first, second, and third diode reverse saturation current in (A), respectively, \({n}_{\mathrm{1,2},3}\) are the first, second, and third diode ideality factor, respectively, \({R}_{s}\) is the series resistance in (Ω), \({R}_{sh}\) is the shunt resistance in (Ω), \(I\), and \(V\) are the output PSC’s current and voltage, respectively, \(k\) is the Boltzmann’s constant (1.38065 × 10−23 J/K), \(T\) is the cell temperature in (K), \({C}_{eq}\) is the equivalent capacitance in (F), \(q\) is the unit charge (\(1.6 \times 10^{ - 19}\)C), and the differential operator − DDT indicates the rate of change for the independent variable \(t\), representing time in (sec.).
It can be observed that the three above equations represent the possible three dynamic electronic models for PSCs, considering single diode model (SDM), double diode model (DDM), and triple diode model (TDM), respectively. As discussed earlier, the PSC equivalent capacitance is a voltage-dependent parameter. Accordingly, the \(C-V\) characteristic can be a linear, quadratic, or third-order polynomial (see Fig. 1d). as given by Abdelrazek et al. (2022):
$${C}_{eq}=a+b\left(V+{R}_{s}I\right)$$
(2.a)
$${C}_{eq}=a+b\left(V+{R}_{s}I\right)+c{\left(V+{R}_{s}I\right)}^{2}$$
(2.b)
$${C}_{eq}=a+b\left(V+{R}_{s}I\right)+c{\left(V+{R}_{s}I\right)}^{2}+ d{\left(V+{R}_{s}I\right)}^{3}$$
(2.c)
where \(a\), \(b\), \(c\), and \(d\) are positive fitting parameters extracted from the experimental C-V characteristic fitting processes. Solving Eqs. (1) and (2) can result in nine different analytical models based on the selected diode model and the C-V fitting. The following sub-sections demonstrated the complete analytical solutions, seeking a final symbolic time-dependent closed form for each electronic model’s parameters.
3.1 Single-diode PV model with linear C-V fitting
In this solution, Eqs. (1.a) and (2.a) are used. The analytical solution procedure is initiated by substituting the boundary conditions values, including short-circuit, open-circuit, and maximum points. At open circuit conditions, \(I=0\) and \(V= {V}_{oc}.\) At such a condition, substituting Eq. (2.a) into Eq. (1.a), we get:
$${I}_{ph}={I}_{01}\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-1\right)+\frac{{V}_{oc}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right) \times \frac{d}{dt}({V}_{oc})\right)$$
(3)
where, \({V}_{oc}\) is the open circuit voltage. Substitute Eq. (3) into (1.a), we get:
$$I=\left({I}_{01}\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-1\right)+\frac{{V}_{0c}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)\right)-{I}_{01}\left(\mathrm{exp}\left(\frac{q(V+{R}_{s}I)}{{n}_{1}kT}\right)-1\right)-\frac{(V+I{R}_{s})}{{R}_{sh}}-(a+b\left(V+{R}_{s}I)\right)\frac{d(V+I{R}_{s})}{dt}$$
(4.a)
Applying short circuit condition, \(I = {I}_{sc}\) and \(V= 0\), one can obtain:
$${I}_{sc}=\left({I}_{01}\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-1\right)+\frac{{V}_{0c}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)\right)-{I}_{01}\left(\mathrm{exp}\left(\frac{q({I}_{sc}{R}_{s})}{{n}_{1}kT}\right)-1\right)-\frac{({I}_{sc}{R}_{s})}{{R}_{sh}}-\left(a+b\left({R}_{s}{I}_{sc}\right)\frac{d({I}_{sc}{R}_{s})}{dt}\right)$$
(4.b)
$${I}_{sc}={I}_{01}\left[\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-\mathrm{exp}\left(\frac{q{I}_{sc}{R}_{s}}{{n}_{1}kT}\right)\right]+\frac{{V}_{oc}-{I}_{sc}{R}_{s}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)- (a+b\left({R}_{s}{I}_{sc}\right)\frac{d({I}_{sc}{R}_{s})}{dt}$$
(4.c)
where \({I}_{sc}\) is the short circuit current. At maximum power point, \(I = {I}_{mp}\) and \(V= {V}_{mp}\), solving Eq. (4.a) at maximum point results:
$${I}_{mp}={I}_{01}\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-1\right)+\frac{{V}_{0c}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)\frac{d({V}_{oc})}{dt}\right)-{I}_{01}(\mathrm{exp}\left(\frac{q\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}{{n}_{1}kT}\right)-1)-\frac{{V}_{mp}+{I}_{mp}{R}_{s}}{{R}_{sh}}-(a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right))\frac{d({V}_{mp}+{I}_{mp}{R}_{s})}{dt}$$
(5.a)
$${I}_{mp}\left(1+\frac{{R}_{s}}{{R}_{sh}}\right)={I}_{01}\left(\mathrm{exp}\left(\frac{q{V}_{\mathit{oc}}}{{n}_{1}\mathit{kT}}\right)-\mathrm{exp}\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s}}{{n}_{1}kT}\right)\right)+\frac{{V}_{oc}- {V}_{mp}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)\frac{d({V}_{oc})}{dt}\right)-\left((a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right))\frac{d({V}_{mp}+ {I}_{mp}{R}_{s})}{dt}\right)$$
(5.b)
where \({I}_{mp}\) is the maximum power point current, \({V}_{mp}\) is the maximum power point voltage. The output power of PV module at each point on the \(I-V\) curve is calculated as
and its derivative with respect to voltage is given by:
$$\frac{dP}{dV}=I+V\frac{dI}{dV}$$
(6.b)
Knowing that the maximum power point is a turning point with zero slope, the power derivative can be written as:
$${\left.\frac{dP}{dV}\right|}_{P={p}_{{m}_{P}}}=0$$
(6.c)
Accordingly, Eq. (6.b) can be reformatted as:
$${\left.\frac{dI}{dV}\right|}_{P={P}_{{m}_{P}}}=-\frac{{I}_{mp}}{{V}_{mp}}$$
(6.d)
The term \(\frac{dI}{dV}\) is obtained by differentiating the Eq. (1.a) with respect to voltage, while considering voltage independent photo-generated current as resulted in Eq. (3), the \(\frac{dI}{dV}\) can be formulated as:
$$\frac{dI}{dV}=\frac{-q {I}_{01}}{{n}_{1}kT}\left(1+{R}_{s}\frac{dI}{dV}\right)\mathrm{exp}\left(\frac{q(V+I{R}_{s})}{{n}_{1}kT}\right)-\frac{1}{{R}_{sh}}\left(1+{R}_{s}\frac{dI}{dV}\right)-(a+b\left(V+{R}_{s}I)\right)\frac{d}{dt}\left[1+{R}_{s}\frac{dI}{dV}\right]-\left(\frac{d}{dt}\left(V+I{R}_{s}\right)\right) \left(b\left(1+{R}_{s}\frac{dI}{dV}\right)\right)$$
(7)
Solving Eq. (7) at the maximum point, and substituting in (6.d) we get:
$$\frac{{I}_{mp}}{{V}_{mp}}=\frac{q {I}_{01}}{{n}_{1}kT}\left(1-{R}_{s} \frac{{I}_{mp}}{{V}_{mp}}\right)\mathit{exp}\left(\frac{q({V}_{mp}+{I}_{mp}{R}_{s})}{{n}_{1}kT}\right)+\frac{1}{{R}_{sh}}\left(1-{R}_{s}\frac{{I}_{mp}}{{V}_{mp}}\right)+\left(a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right)\right)\frac{d}{dt}\left[1-{R}_{s}\frac{{I}_{mp}}{{V}_{mp}}\right]+\left(\frac{d}{dt}\left({V}_{mp}+{I}_{mp}{R}_{s}\right)\right)\left(b\left(1+{R}_{s}\frac{dI}{dV}\right)\right)$$
(8)
Alternatively, Eq. (7) can be reformulated as:
$$\frac{dI}{dV}\left(1+\frac{{R}_{s}}{{R}_{sh}}+\frac{q{I}_{01}{R}_{s}}{{n}_{1}kT}exp\frac{q\left(V+I{R}_{s}\right) }{{n}_{1}kT}+{R}_{s}b\frac{d\left(V+I{R}_{s}\right)}{dt}\right)=\left(\frac{-q{I}_{01}}{{n}_{1}kT}exp\frac{q\left(V+I{R}_{s}\right)}{{n}_{1}kT}\right)-\frac{1}{{R}_{sh}}-(a+b\left(V+{R}_{s}I)\right)\frac{d}{dt}\left({R}_{s}\frac{dI}{dV}\right)-b\frac{d\left(V+I{R}_{s}\right)}{dt}$$
(9.a)
$$\frac{dI}{dV}=\frac{\left(\frac{-q{I}_{01}}{{n}_{1}kT}\mathrm{e}xp\frac{q\left(V+I{R}_{s}\right)}{{n}_{1}kT}\right)-\frac{1}{{R}_{sh}}-(a+b\left(V+{R}_{s}I\right))\frac{d}{dt}\left({R}_{s}\frac{dI}{dV}\right)-b\frac{d\left(V+I{R}_{s}\right)}{dt}}{1+\frac{{R}_{s}}{{R}_{sh}}+\frac{q{I}_{01}{R}_{s}}{{n}_{1}kT}exp\frac{q\left(V+I{R}_{s}\right) }{{n}_{1}kT}+{R}_{s}b\frac{d\left(V+I{R}_{s}\right)}{dt}}$$
(9.b)
Considering that the shunt resistance is inversely proportional to the \(\frac{dI}{dV}\) close to the short-circuit point, then, solving Eq. (9.b) at short0circuit condition results:
$${\left.\frac{dI}{dV}\right|}_{I={I}_{sc},V=0}= \frac{\frac{-q{I}_{o1}}{{n}_{1}kT}exp\left(\frac{q\left({I}_{sc}{R}_{s}\right)}{{n}_{1}kT}\right)-\frac{1}{{R}_{sh}}-\left(a+b\left({R}_{s}{I}_{sc}\right))\frac{d}{dt}({R}_{s}{\left.\frac{dI}{dV}\right|}_{I={I}_{sc},V=0}\right)-b\frac{d\left({I}_{sc}{R}_{s}\right)}{dt} }{1+\frac{{R}_{s}}{{R}_{sh}}+\frac{q{I}_{01}{R}_{s}}{{n}_{1}kT}exp\frac{q\left({I}_{sc}{R}_{s}\right) }{{n}_{1}kT}+{R}_{s}b\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}}$$
(10)
The term \({I}_{01}\mathrm{exp}\left(\frac{q{I}_{sc}{R}_{s}}{{n}_{1}kT}\right)\) represents the diode current and this is too small compared to the short-circuit current and thus it can be neglected. In addition, since \({R}_{s}\)< < \({R}_{sh}\), the term \({R}_{s}\)/\({R}_{sh}\) may also be neglected. Therefore, the derivative can be approximated as
$${\left.\frac{dI}{dV}\right|}_{I={I}_{sc},V=0}=\frac{ -\frac{1}{{R}_{sho}}-\left(a+b\left({R}_{s}{I}_{sc}\right)\frac{d}{dt}\left({R}_{s}{\left.\frac{dI}{dV}\right|}_{I={I}_{sc},V=0}\right)-b\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}\right) }{1+{R}_{s}b\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}}$$
(11)
Substituting Eq. (11) into Eq. (7), applying short-circuit condition, one can obtain:
$$\begin{gathered} \frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}} \hfill \\ = \frac{{ - q I_{01} }}{{n_{1} kT}}\left( {1 + R_{s} \left( {\frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right)} \right)\exp \left( {\frac{{q\left( {I_{sc} R_{s} } \right)}}{{n_{1} kT}}} \right) \hfill \\ - \frac{1}{{R_{sho} }}\left( {1 + R_{s} \left( {\frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right)} \right) \hfill \\ - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left[ {1 + R_{s} \left( {\frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right)} \right] \hfill \\ - \frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}\left( {b\left( {1 + R_{s} \frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right)} \right) \hfill \\ \end{gathered}$$
(12.a)
$$\begin{gathered} \frac{{ \frac{1}{{R_{sho} }} + \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) + b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}} \hfill \\ = \left( {1 + R_{s} \left( {\frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right)} \right) \hfill \\ \left( {\frac{{q I_{01} }}{{n_{1} kT}}\exp \left( {\frac{{q\left( {I_{sc} R_{s} } \right)}}{{n_{1} kT}}} \right) + \frac{1}{{R_{sho} }} + b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}} \right) + \left( {a + b\left( {R_{s} I_{sc} } \right)} \right) \hfill \\ \frac{d}{dt}\left[ {1 + R_{s} \frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right] \hfill \\ \end{gathered}$$
(12.b)
Here the term \({R}_{sho}\) indicates the initial guess for the \({R}_{sh}\) at the short-circuit point. The same approach can be applied to estimate the series resistance by solving Eq. (9.b) under the open-circuit condition, to reach:
$${\left.\frac{dI}{dV}\right|}_{I=0,V={V}_{oc}} =\frac{\frac{-q{I}_{01}}{{n}_{1}kT}exp\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-\frac{1}{{R}_{sh}}-\left(a+b\left({V}_{oc}\right)\right)\frac{d}{dt}\left({R}_{s}{\left.\frac{dI}{dV}\right|}_{I=0,V={V}_{oc}}\right)- b\frac{d\left({V}_{oc}\right)}{dt}}{1+\frac{{R}_{s}}{{R}_{sh}}+\frac{q{I}_{01}{R}_{s}}{{n}_{1}kT}exp\frac{q\left({V}_{oc}\right) }{{n}_{1}kT}+{R}_{s}b\frac{d\left({V}_{oc}\right)}{dt}}$$
(13)
This gives a simple expression to find the initial value of the series resistance (\({R}_{so}\)) defined by the negative slope of the \(I-V\) curve near the open circuit region, while considering the same approximations concerning the diode current and the series-shunt resistance ratio, here we reach:
$${\left.\frac{dI}{dV}\right|}_{I=0,V={V}_{oc}}=\frac{ -\left(a+b\left({V}_{oc}\right)\right)\frac{d}{dt}\left({R}_{s}{\left.\frac{dI}{dV}\right|}_{I=0,V={V}_{oc}}\right)- b\frac{d\left({V}_{oc}\right)}{dt} }{1+{R}_{s}b\frac{d\left({V}_{oc}\right)}{dt}}$$
(14)
Substitute (14) in (7), and applying open-circuit condition, one can obtain:
$$\begin{gathered} \left( {\frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right) = \frac{{ - q I_{01} }}{{n_{1} kT}}\left( {1 + R_{s} \left( {\frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right)\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{n_{1} kT}}} \right) \hfill \\ - \frac{1}{{R_{sh} }}\left( {1 + R_{s} \left( {\frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right) - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left[ {1 + R_{s} \left( {\frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right] \hfill \\ - \frac{{d\left( {V_{oc} } \right)}}{dt}\;b\left( {1 + R_{s} \left( { \frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right) \hfill \\ \end{gathered}$$
(15.a)
$$\begin{gathered} \left( {\frac{{ \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) + b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right) = \left( {1 + R_{s} \left( {\frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right) \hfill \\ \left( {\frac{{q I_{01} }}{{n_{1} kT}}\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{n_{1} kT}}} \right) + \frac{1}{{R_{sh} }} + b\frac{{d\left( {V_{oc} } \right)}}{dt}} \right) + \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left[ {1 + R_{s} \left( {\frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right] \hfill \\ \end{gathered}$$
(15.b)
In addition to the two parasitic resistances, the diode saturation current can be extracted by Eqs. (3) and (4.c), we can reach:
$${I}_{o1}=\frac{{I}_{sc}({R}_{sh}+{R}_{s})-{V}_{oc}-{R}_{sh}\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)+ {R}_{sh}(a+b\left({R}_{s}{I}_{sc})\right)\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}}{{R}_{sh}\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-\mathrm{exp}\left(\frac{\left(q{R}_{s}{I}_{sc}\right)}{{n}_{1}kT}\right)\right)}$$
(16)
The open-circuit voltage can be obtained from the experimental data. However, the ideality factor is still an unknown, that needs to be determined.
Since, \({R}_{sh}\gg {R}_{s}\), accordingly, \(1+\frac{{R}_{s}}{{R}_{sh}}\approx 1,\) and \({I}_{sc}\gg \frac{{V}_{oc}}{{R}_{sh}}\). Furthermore, \(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)\gg \mathrm{exp}\left(\frac{{I}_{sc}{R}_{s}}{{n}_{1}kT}\right)\) is a valid assumption. Therefore, Eq. (16) can be approximated to:
$${I}_{o1}={I}_{sc}\mathrm{exp}\left(\frac{-q{V}_{oc}}{{n}_{1}kT}\right)-\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)\mathrm{exp}\left(\frac{-q{V}_{oc}}{{n}_{1}kT}\right)+ \left(\left(a+b\left({R}_{s}{I}_{sc}\right)\right)\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}\right)\mathrm{exp}\left(\frac{-q{V}_{oc}}{{n}_{1}kT}\right)$$
(17)
Substituting Eqs. (17) into (3), while considering \(\frac{{V}_{0c}}{{R}_{sh}}\) tends to zero:
$${I}_{ph}={I}_{sc}\mathrm{exp}\left(\frac{-q{V}_{oc}}{{n}_{1}kT}\right)-\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)\mathrm{exp}\left(\frac{-q{V}_{oc}}{{n}_{1}kT}\right)+ (a+b\left({R}_{s}{I}_{sc})\right)\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}+\frac{{V}_{oc}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)*\frac{d}{dt}({V}_{oc})\right)$$
(18)
Equation (5.b) can be written as by substituting Eq. (17) in it:
$$\begin{gathered} I_{mp} \left( {1 + \frac{{R_{s} }}{{R_{sh} }}} \right) = \left( {I_{sc} \exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right) - \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right)\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right) + \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right)} \right) \hfill \\ \left( {\exp \left( {\frac{{qV_{oc} }}{{n_{1} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)} \right) + \frac{{V_{oc} - V_{mp} }}{{R_{sh} }} + \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right) - \left( {\left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right)\frac{{d\left( {V_{mp} + I_{mp} R_{s} } \right)}}{dt}} \right) \hfill \\ \end{gathered}$$
(19)
Assume: \(\frac{{V}_{oc}-{V}_{mp}}{{R}_{sh}} \approx 0\) and \(1+\frac{{R}_{s}}{{R}_{sh}}\approx 1\)
$$\begin{aligned} I_{mp} & = I_{sc} \left( {1 - \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)} \right) - \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right) + \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right)\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right)\exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right) \\ + \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} - \left( {\left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}\left( {\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right)} \right)} \right)\left( {\exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)} \right) \\ + \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right) - \left( {\left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right)\frac{{d\left( {V_{mp} + I_{mp} R_{s} } \right)}}{dt}} \right) \\ \end{aligned}$$
(20)
Considering \(\frac{1}{{R}_{sh}}\left(1-{R}_{s}\frac{{I}_{mp}}{{V}_{mp}}\right)\approx 0\), Eq. (8) can be written as,
$$\begin{aligned} I_{{m_{P} }} & = V_{{m_{P} }} = \frac{q }{{n_{1} kT}}\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right)\exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} } \right)}}{{n_{1} kT}}} \right)\left( {I_{sc} \exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right)} \right) \\ - \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right)\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right) + \left( {\left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right)} \right) \\ + \left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right)\frac{d}{dt}\left[ {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right] + \left( {\frac{d}{dt}\left( {V_{mp} + I_{mp} R_{s} } \right)} \right)\left( {b\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right)} \right) \\ \end{aligned}$$
(21)
It can be observed that both Eqs. (20), and (21) are function in \(R_{s}\), and \({n}_{1}\). However, the series resistance is associated with the differentiation operator. Consequently, we equalize both equations seeking for a first order differential equation (DE) in terms of \(R_{s}\). Towards simplification, and as mentioned earlier, the boundary conditions points, at short circuit, open-circuit, and maximum power point, are treated as time independent constants, extracted from experimental measurements. Following that, the first order DE can be written as:
$$\begin{gathered} I_{sc} \left( {1 - \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)} \right) + \left( {aI_{sc} + b R_{s} I_{sc}^{2} } \right)\frac{{dR_{s} }}{dt} - \left( {aI_{sc} + b R_{s} I_{sc}^{2} } \right)\frac{{dR_{s} }}{dt} \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right) \hfill \\ - aI_{mp} \frac{{dR_{s} }}{dt} - b\left( {V_{mp} + R_{s} I_{mp} } \right)I_{mp} \frac{{dR_{s} }}{dt} - I_{sc} V_{{m_{P} }} \frac{q }{{n_{1} kT}}\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right) \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right) \hfill \\ - \left( {aI_{sc} + b R_{s} I_{sc}^{2} } \right)V_{{m_{P} }} \frac{q }{{n_{1} kT}}\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right) \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)\frac{{dR_{s} }}{dt} \hfill \\ - \frac{{I_{mp} }}{{V_{mp} }}\left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right)\frac{{dR_{s} }}{dt} - bI_{mp} \left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right) \frac{{dR_{s} }}{dt} = 0. \hfill \\ \end{gathered}$$
(22)
Equation (22) can be treated as:
$${f}^{1}\left({R}_{s}\right)\frac{d{R}_{s}}{dt}+ {f}^{2}\left({R}_{s}\right)=0$$
(23.a)
where:
$$\begin{aligned} f^{1} \left( {R_{s} } \right) = & \left( {aI_{{{\text{sc}}}} + bR_{s} I_{{{\text{sc}}}}^{2} } \right) - \left( {aI_{{{\text{sc}}}} + bR_{s} I_{{{\text{sc}}}}^{2} } \right)\exp \left( {\frac{{q(V_{{{\text{mp}}}} - V_{{{\text{oc}}}} + I_{{{\text{mp}}}} R_{s} )}}{{n_{1} kT}}} \right) \\ & - aI_{{{\text{mp}}}} - b\left( {V_{{{\text{mp}}}} + R_{s} I_{{{\text{mp}}}} } \right)I_{{{\text{mp}}}} - \left( {aI_{{{\text{sc}}}} + bR_{s} I_{{{\text{sc}}}}^{2} } \right)V_{{m_{P} }} \\ & \;\;\;\frac{q}{{n_{1} kT}}\left( {1 - R_{s} \frac{{I_{{{\text{mp}}}} }}{{V_{{{\text{mp}}}} }}} \right)\exp \left( {\frac{{q(V_{{{\text{mp}}}} - V_{{{\text{oc}}}} + I_{{{\text{mp}}}} R_{s} )}}{{n_{1} kT}}} \right) \\ & - \frac{{I_{{{\text{mp}}}} }}{{V_{{{\text{mp}}}} }}\left( {a + b\left( {V_{{{\text{mp}}}} + R_{s} I_{{{\text{mp}}}} } \right)} \right) - {\text{ b}}I_{{{\text{mp}}}} \left( {1 - R_{s} \frac{{I_{{{\text{mp}}}} }}{{V_{{{\text{mp}}}} }}} \right) \\ \end{aligned}$$
(23.b)
$$f^{2} \left( {R_{s} } \right) = I_{sc} \left( {1 - { }\exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)} \right) - I_{sc} V_{{m_{P} }} { }\frac{q }{{n_{1} kT}}\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right) \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)$$
(23.c)
Equation (23) can’t be solved analytically, however, an iterative numerical solution for both Eqs. (20) and (23) can converge with both \({R}_{s} (t)\), and \({n}_{1}(t)\). Referring to Eq. (12.b), with replacing the diode saturation current as given in (16), the shunt resistance can be calculated as:
$$\begin{gathered} \frac{{ \frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} }}{{1 + R_{s} \left( t \right)bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}} = \hfill \\ \left( {1 + R_{s} \left( t \right) \left( {\frac{{ - \frac{1}{{R_{sho} }} - bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} }}{{1 + R_{s} \left( t \right)bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right)} \right)\left( {\frac{q }{{n_{1} \left( t \right)kT}}\left( {\frac{{I_{sc} \left( {R_{sho} + R_{s} \left( t \right)} \right) - V_{oc} + R_{sho} \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right)} \right)\frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}{{R_{sho} \left( {\exp \left( {\frac{{qV_{oc} }}{{n_{1} \left( t \right)kT}}} \right) - \exp \left( {\frac{{\left( {qIsc Rs\left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right)} \right)}}} \right)\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right) + \frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}} \right) \hfill \\ + \left( {a + b\left( {R_{s} \left( t \right)I_{sc} } \right)} \right)\frac{d}{dt}\left[ {R_{s} \left( t \right)\frac{{ - \frac{1}{{R_{sho} }} - bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} }}{{1 + R_{s} bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right] \hfill \\ \end{gathered}$$
(24)
Substituting (23.a) in (24) we get:
$$\begin{gathered} \frac{{ \frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} }}{{1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}} = \left( {1 + R_{s} \left( t \right)\left( {\frac{{ - \frac{1}{{R_{sho} }} + bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} }}{{1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}} \right)} \right) \hfill \\ \left( {\frac{q }{{n_{1} \left( t \right)kT}}\left( {\frac{{I_{sc} \left( {R_{sho} + R_{s} \left( t \right)} \right) - V_{oc} - R_{sho} \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}{{R_{sho} \left( {\exp \left( {\frac{{qV_{oc} }}{{n_{1} \left( t \right)kT}}} \right) - \exp \left( {\frac{{\left( {qIsc Rs\left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right)} \right)}}} \right)\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right) + \frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) \hfill \\ + \left( {a + bR_{s} \left( t \right)I_{sc} } \right) \hfill \\ \left( \begin{gathered} \left[ {R_{s} \left( t \right)\frac{{\left( { - \frac{1}{{R_{sho} }} + bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} } \right)\left( { - R_{s} bI_{sc} \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right] - bI_{sc} \left( {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} } \right) - \left( {1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)\left( { - \frac{1}{{R_{sho}^{2} }} \frac{{d\left( {R_{sho} } \right)}}{dt} + bI_{sc} \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right]} \right)}}{{\left( {1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} }}} \right] \hfill \\ + \left( {\frac{{ \frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} }}{{1 + R_{s} bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered}$$
(25)
Again, Eq. (25) can’t be solved analytically, however, an iterative numerical solution can be extracted for \({R}_{sho} (t)\). Equation (25) can be treated as:
$${g}^{1}\left({R}_{sho}\right)\frac{d{R}_{sho}}{dt}+ {g}^{2}\left({R}_{sho}\right)=0$$
(26.a)
where:
$${g}^{1}\left({R}_{sho}\right)=(a+b{R}_{s}\left(t\right){I}_{sc})\left(\left[{R}_{s}\left(t\right)\frac{ \left(1-{R}_{s}b{I}_{sc}\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}\right)\left(-\frac{1}{{R}_{sho}^{2}} \right)}{(1-{R}_{s}b{I}_{sc}\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}{)}^{2}}\right]\right)$$
(26.b)
$$\begin{gathered} g^{2} \left( {R_{sho} } \right) = \left( {1 + R_{s} \left( t \right)\left( {\frac{{{ } - \frac{1}{{R_{sho} }} + bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }}}{{1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}} \right)} \right) \hfill \\ \left( {\frac{q }{{n_{1} \left( t \right)kT}}\left( {\frac{{I_{sc} \left( {R_{sho} + R_{s} \left( t \right)} \right) - V_{oc} - R_{sho} \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}{{R_{sho} \left( {\exp \left( {\frac{{qV_{oc} }}{{n_{1} \left( t \right)kT}}} \right) - \exp \left( {\frac{{\left( {qIsc Rs\left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right)} \right)}}} \right)\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right) + \frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) \hfill \\ - \frac{{{ }\frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }}}{{1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}} + \left( {a + bR_{s} \left( t \right)I_{sc} } \right) \hfill \\ \left( {\left[ {R_{s} \left( t \right)\frac{{\left( { - \frac{1}{{R_{sho} }} + bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }} \right)\left( { - R_{s} bI_{sc} \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right] - bI_{sc} \left( {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} } \right) - \left( {1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)\left( {bI_{sc} \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right]} \right)}}{{\left( {1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} }}} \right] + \left( {\frac{{{ }\frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}{ }}}{{1 + R_{s} bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) \hfill \\ \end{gathered}$$
(26.c)
Solving the first order DE in (26) results with the estimated shunt resistance at the short-circuit point. Substituting back in (16) with the extracted functions from (20), (23), and (26) results:
$${I}_{o1 }(t)=\frac{{I}_{sc}({R}_{sho}(t))-{V}_{oc}+ {R}_{sh0}(t)(a-b\left({R}_{s}(t){I}_{sc}^{2}\right)\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}}{{R}_{sho}(t)\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}(t)kT}\right)-\mathrm{exp}\left(\frac{\left(q{R}_{s}(t){I}_{sc}\right)}{{n}_{1}(t)kT}\right)\right)}$$
(27)
Finally, the photo-generated current can be driven from Eq. (3) as:
$${I}_{ph}(t)={I}_{01}(t)\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}\left(t\right)kT}\right)-1\right)+\frac{{V}_{oc}}{{R}_{sho}(t)}$$
(28)
3.2 Single-diode PV model with second and third-order C-V fitting.
Herein, we proceeded in the same derivation sequence as in Sect. 3.1, utilizing Eqs. (2.c) and (2.d) for the second and third-order capacitive voltage fitting. Considering the second-order model, a differential equation concerning \({R}_{s}\) can be extracted, typically as Eq. (22) in the first-order linear model. Towards simplification, and as implemented earlier, the boundary conditions points, at short circuit, open-circuit, and maximum power point, are treated as time-independent constants extracted from experimental measurements. Following that, the first-order D.E. can be written as:
$$I_{ph} = I_{01} \left( {\exp \left( {\frac{{qV_{oc} }}{{n_{1} kT}}} \right) - 1} \right) + \frac{{V_{oc} }}{{R_{sh} }} + \left( {\left( {a + b\left( {V_{oc} } \right)} \right)x\frac{d}{dt}\left( {V_{oc} } \right)} \right)$$
(29)
where, \({V}_{oc}\) is the open circuit voltage. Substitute Eq. (3) into (1.a), we get:
$$I=(\left({I}_{01}\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-1\right)+\frac{{V}_{0c}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)\right)-{I}_{01}\left(\mathrm{exp}\left(\frac{q(V+{R}_{s}I)}{{n}_{1}kT}\right)-1\right)-\frac{(V+I{R}_{s})}{{R}_{sh}}-(a+b\left(V+{R}_{s}I)\right)\frac{d(V+I{R}_{s})}{dt}$$
(30.a)
Applying short circuit condition, \(I = {I}_{sc}\) and \(V= 0\), one can obtain:
$${I}_{sc}=\left({I}_{01}\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-1\right)+\frac{{V}_{0c}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)\right)-{I}_{01}\left(\mathrm{exp}\left(\frac{q({I}_{sc}{R}_{s})}{{n}_{1}kT}\right)-1\right)-\frac{({I}_{sc}{R}_{s})}{{R}_{sh}}-\left(a+b\left({R}_{s}{I}_{sc}\right)\frac{d({I}_{sc}{R}_{s})}{dt}\right)$$
(30.b)
$${I}_{sc}={I}_{01}\left[\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-\mathrm{exp}\left(\frac{q{I}_{sc}{R}_{s}}{{n}_{1}kT}\right)\right]+\frac{{V}_{oc}-{I}_{sc}{R}_{s}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)- (a+b\left({R}_{s}{I}_{sc}\right)\frac{d({I}_{sc}{R}_{s})}{dt}$$
(30.c)
where \({I}_{sc}\) is the short circuit current. At maximum power point, \(I = {I}_{mp}\) and \(V= {V}_{mp}\), solving Eq. (4.a) at maximum point results:
$${I}_{mp}={I}_{01}\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-1\right)+\frac{{V}_{0c}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)\frac{d({V}_{oc})}{dt}\right)-{I}_{01}\left(\mathrm{exp}\left(\frac{q\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}{{n}_{1}kT}\right)-1\right)-\frac{{V}_{mp}+{I}_{mp}{R}_{s}}{{R}_{sh}}-(a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right))\frac{d({V}_{mp}+{I}_{mp}{R}_{s})}{dt}$$
(31.a)
$${I}_{mp}\left(1+\frac{{R}_{s}}{{R}_{sh}}\right)={I}_{01}\left(\mathrm{exp}\left(\frac{q{V}_{\mathit{oc}}}{{n}_{1}\mathit{kT}}\right)-\mathrm{exp}\left(\frac{q\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}{{n}_{1}kT}\right)\right)+\frac{{V}_{oc}- {V}_{mp}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)\frac{d({V}_{oc})}{dt}\right)-\left((a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right))\frac{d({V}_{mp}+ {I}_{mp}{R}_{s})}{dt}\right)$$
(31.b)
where \({I}_{mp}\) is the maximum power point current, \({V}_{mp}\) is the maximum power point voltage. The output power of PV module at each point on the \(I-V\) curve is calculated as
and its derivative with respect to voltage is given by:
$$\frac{dP}{dV}=I+V\frac{dI}{dV}$$
(32.b)
Knowing that the maximum power point is a turning point with zero slope, the power derivative can be written as:
$${\left.\frac{dP}{dV}\right|}_{P={p}_{{m}_{P}}}=0$$
(32.c)
Accordingly, Eq. (6.b) can be reformatted as:
$${\left.\frac{dI}{dV}\right|}_{P={P}_{{m}_{P}}}=-\frac{{I}_{mp}}{{V}_{mp}}$$
(32.d)
The term \(\frac{dI}{dV}\) is obtained by differentiating the Eq. (1.a) with respect to voltage, while considering voltage independent photo-generated current as resulted in Eq. (29), the \(\frac{dI}{dV}\) can be formulated as:
$$\frac{dI}{dV}=\frac{-q {I}_{01}}{{n}_{1}kT}\left(1+{R}_{s}\frac{dI}{dV}\right)\mathrm{exp}\left(\frac{q(V+I{R}_{s})}{{n}_{1}kT}\right)-\frac{1}{{R}_{sh}}\left(1+{R}_{s}\frac{dI}{dV}\right)-(a+b\left(V+{R}_{s}I)\right)\frac{d}{dt}\left[1+{R}_{s}\frac{dI}{dV}\right]-\left(\frac{d}{dt}\left(V+I{R}_{s}\right)\right)\left(b\left(1+{R}_{s}\frac{dI}{dV}\right)\right)$$
(33)
Solving Eq. (7) at the maximum point, and substituting in (32.d) we get:
$$\frac{{I}_{mp}}{{V}_{mp}}=\frac{q {I}_{01}}{{n}_{1}kT}\left(1-{R}_{s} \frac{{I}_{mp}}{{V}_{mp}}\right)\mathrm{exp}\left(\frac{q({V}_{mp}+{I}_{mp}{R}_{s})}{{n}_{1}kT}\right)+\frac{1}{{R}_{sh}}\left(1-{R}_{s}\frac{{I}_{mp}}{{V}_{mp}}\right)+(a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right))\frac{d}{dt}\left[1-{R}_{s}\frac{{I}_{mp}}{{V}_{mp}}\right]+\left(\frac{d}{dt}\left({V}_{mp}+{I}_{mp}{R}_{s}\right)\right)\left(b\left(1+{R}_{s}\frac{dI}{dV}\right)\right)$$
(34)
Alternatively, Eq. (33) can be reformulated as:
$$\frac{dI}{dV}\left(1+\frac{{R}_{s}}{{R}_{sh}}+\frac{q{I}_{01}{R}_{s}}{{n}_{1}kT}exp\frac{q\left(V+I{R}_{s}\right) }{{n}_{1}kT}+{R}_{s}b\frac{d\left(V+I{R}_{s}\right)}{dt}\right) =\frac{-q{I}_{01}}{{n}_{1}kT}exp\frac{q\left(V+I{R}_{s}\right)}{{n}_{1}kT})-\frac{1}{{R}_{sh}}-(a+b\left(V+{R}_{s}I)\right)\frac{d}{dt}\left({R}_{s}\frac{dI}{dV}\right)-b\frac{d\left(V+I{R}_{s}\right)}{dt}$$
(35.a)
$$\frac{dI}{dV}=\frac{\left(\frac{-q{I}_{01}}{{n}_{1}kT}\mathrm{exp}\frac{q\left(V+I{R}_{s}\right)}{{n}_{1}kT}\right)-\frac{1}{{R}_{sh}}-(a+b\left(V+{R}_{s}I\right))\frac{d}{dt}\left({R}_{s}\frac{dI}{dV}\right)-b\frac{d\left(V+I{R}_{s}\right)}{dt}}{1+\frac{{R}_{s}}{{R}_{sh}}+\frac{q{I}_{01}{R}_{s}}{{n}_{1}kT}\mathrm{exp}\frac{q\left(V+I{R}_{s}\right) }{{n}_{1}kT}+{R}_{s}b\frac{d\left(V+I{R}_{s}\right)}{dt}}$$
(35.b)
Considering that the shunt resistance is inversely proportional to the \(\frac{dI}{dV}\) close to the short-circuit point, then, solving Eq. (9.b) at short0circuit condition results:
$${\left.\frac{dI}{dV}\right|}_{I={I}_{sc},V=0}= \frac{\frac{-q{I}_{o1}}{{n}_{1}kT}exp\left(\frac{q\left({I}_{sc}{R}_{s}\right)}{{n}_{1}kT}\right)-\frac{1}{{R}_{sh}}-(a+b\left({R}_{s}{I}_{sc}\right))\frac{d}{dt}\left({R}_{s}{\left.\frac{dI}{dV}\right|}_{I={I}_{sc},V=0}\right)-b\frac{d\left({I}_{sc}{R}_{s}\right)}{dt} }{1+\frac{{R}_{s}}{{R}_{sh}}+\frac{q{I}_{01}{R}_{s}}{{n}_{1}kT}exp\frac{q\left({I}_{sc}{R}_{s}\right) }{{n}_{1}kT}+{R}_{s}b\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}}$$
(36)
The term \({I}_{01}\mathrm{exp}\left(\frac{q{I}_{sc}{R}_{s}}{{n}_{1}kT}\right)\) represents the diode current and this is too small compared to the short-circuit current and thus it can be neglected. In addition, since \({R}_{s}\)< < \({R}_{sh}\), the term \({R}_{s}\)/\({R}_{sh}\) may also be neglected. Therefore, the derivative can be approximated as
$${\left.\frac{dI}{dV}\right|}_{I={I}_{sc},V=0}=\frac{ -\frac{1}{{R}_{sho}}-(a+b\left({R}_{s}{I}_{sc}\right)\frac{d}{dt}\left({R}_{s}{\left.\frac{dI}{dV}\right|}_{I={I}_{sc},V=0}\right)-b\frac{d\left({I}_{sc}{R}_{s}\right)}{dt} }{1+{R}_{s}b\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}}$$
(37)
Substituting Eq. (37) into Eq. (33), applying short-circuit condition, one can obtain:
$$\begin{gathered} \frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}} = \frac{{ - q I_{01} }}{{n_{1} kT}} \hfill \\ \left( {1 + R_{s} \left( {\frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right)} \right) \hfill \\ \exp \left( {\frac{{q\left( {I_{sc} R_{s} } \right)}}{{n_{1} kT}}} \right) - \frac{1}{{R_{sho} }}\left( {1 + R_{s} \left( {\frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right)} \right) \hfill \\ - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left[ {1 + R_{s} \left( {\frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right)} \right] \hfill \\ - \frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}\left( {b\left( {1 + R_{s} \frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right)} \right) \hfill \\ \end{gathered}$$
(38.a)
$$\begin{gathered} \frac{{ \frac{1}{{R_{sho} }} + \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) + b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}} = \left( {1 + R_{s} \left( {\frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right)} \right) \hfill \\ \left( {\frac{{q I_{01} }}{{n_{1} kT}}\exp \left( {\frac{{q\left( {I_{sc} R_{s} } \right)}}{{n_{1} kT}}} \right) + \frac{1}{{R_{sho} }} + b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}} \right) + \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt} \hfill \\ \left[ {1 + R_{s} \frac{{ - \frac{1}{{R_{sho} }} - \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = I_{sc} ,V = 0}} } \right) - b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}}}} \right] \hfill \\ \end{gathered}$$
(38.b)
Here the term \({R}_{sho}\) indicates the initial guess for the \({R}_{sh}\) at the short-circuit point. The same approach can be applied to estimate the series resistance by solving Eq. (9.b) under the open-circuit condition, to reach:
$${\left.\frac{dI}{dV}\right|}_{I=0,V={V}_{oc}} =\frac{\frac{-q{I}_{01}}{{n}_{1}kT}\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-\frac{1}{{R}_{sh}}-(a+b\left({V}_{oc}\right))\frac{d}{dt}\left({R}_{s}{\left.\frac{dI}{dV}\right|}_{I=0,V={V}_{oc}}\right)- b\frac{d\left({V}_{oc}\right)}{dt}}{1+\frac{{R}_{s}}{{R}_{sh}}+\frac{q{I}_{01}{R}_{s}}{{n}_{1}kT}\mathrm{exp}\frac{q\left({V}_{oc}\right) }{{n}_{1}kT}+{R}_{s}b\frac{d\left({V}_{oc}\right)}{dt}}$$
(39)
This gives a simple expression to find the initial value of the series resistance (\({R}_{so}\)) defined by the negative slope of the \(I-V\) curve near the open circuit region, while considering the same approximations concerning the diode current and the series-shunt resistance ratio, here we reach:
$${\left.\frac{dI}{dV}\right|}_{I=0,V={V}_{oc}}=\frac{ -(a+b\left({V}_{oc}\right))\frac{d}{dt}\left({R}_{s}{\left.\frac{dI}{dV}\right|}_{I=0,V={V}_{oc}}\right)- b\frac{d\left({V}_{oc}\right)}{dt} }{1+{R}_{s}b\frac{d\left({V}_{oc}\right)}{dt}}$$
(40)
Substitute (40) in (33), and applying open-circuit condition, one can obtain:
$$\begin{gathered} \left( {\frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right) = \frac{{ - q I_{01} }}{{n_{1} kT}}\left( {1 + R_{s} \left( { \frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right)\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{n_{1} kT}}} \right) - \frac{1}{{R_{sh} }} \hfill \\ \left( {1 + R_{s} \left( {\frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right) - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt} \hfill \\ \left[ {1 + R_{s} \left( { \frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right] - \frac{{d\left( {V_{oc} } \right)}}{dt}\left( {b\left( {1 + R_{s} \left( { \frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right)} \right) \hfill \\ \end{gathered}$$
(41.a)
$$\begin{gathered} \left( {\frac{{ \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) + b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right) = \left( {1 + R_{s} \left( {\frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right) \hfill \\ \left( {\frac{{q I_{01} }}{{n_{1} kT}}\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{n_{1} kT}}} \right) + \frac{1}{{R_{sh} }} + b\frac{{d\left( {V_{oc} } \right)}}{dt}} \right) + \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt} \hfill \\ \left[ {1 + R_{s} \left( {\frac{{ - \left( {a + b\left( {V_{oc} } \right)} \right)\frac{d}{dt}\left( {R_{s} \left. {\frac{dI}{{dV}}} \right|_{{I = 0,V = V_{oc} }} } \right) - b\frac{{d\left( {V_{oc} } \right)}}{dt} }}{{1 + R_{s} b\frac{{d\left( {V_{oc} } \right)}}{dt}}}} \right)} \right] \hfill \\ \end{gathered}$$
(41.b)
In addition to the two parasitic resistances, the diode saturation current can be extracted by Eqs. (3) and (4.c), we can reach:
$${I}_{o1}=\frac{{I}_{sc}({R}_{sh}+{R}_{s})-{V}_{oc}-{R}_{sh}\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)+ {R}_{sh}(a+b\left({R}_{s}{I}_{sc})\right)\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}}{{R}_{sh}\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-\mathrm{exp}\left(\frac{\left(q{R}_{s}{I}_{sc}\right)}{{n}_{1}kT}\right)\right)}$$
(42)
The open-circuit voltage can be obtained from the experimental data. However, the ideality factor is still an unknown, that needs to be determined.
Since, \({R}_{sh}\gg {R}_{s}\), accordingly, \(1+\frac{{R}_{s}}{{R}_{sh}}\approx 1,\) and \({I}_{sc}\gg \frac{{V}_{oc}}{{R}_{sh}}\). Furthermore, \(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)\gg \mathrm{exp}\left(\frac{{I}_{sc}{R}_{s}}{{n}_{1}kT}\right)\) is a valid assumption. Therefore, Eq. (42) can be approximated to:
$${I}_{o1}={I}_{sc}\mathrm{exp}\left(\frac{-q{V}_{oc}}{{n}_{1}kT}\right)-\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)\mathrm{exp}\left(\frac{-q{V}_{oc}}{{n}_{1}kT}\right)+ (a+b\left({R}_{s}{I}_{sc})\right)\left(\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}\right)\mathrm{exp}\left(\frac{-q{V}_{oc}}{{n}_{1}kT}\right)$$
(43)
Substituting Eqs. (43) into (29), while considering \(\frac{{V}_{0c}}{{R}_{sh}}\) tends to zero:
$${I}_{ph}={I}_{sc}\mathrm{exp}\left(\frac{-q{V}_{oc}}{{n}_{1}kT}\right)-\left((a+b\left({V}_{oc})\right)\frac{d\left({V}_{oc}\right)}{dt}\right)\mathrm{exp}\left(\frac{-q{V}_{oc}}{{n}_{1}kT}\right)+ (a+b\left({R}_{s}{I}_{sc})\right)\frac{d\left({I}_{sc}{R}_{s}\right)}{dt}+\frac{{V}_{oc}}{{R}_{sh}}+\left((a+b\left({V}_{oc})\right)*\frac{d}{dt}({V}_{oc})\right)$$
(44)
Equation (31.b) can be written as by substituting Eq. (43) in it:
$$\begin{aligned} I_{mp} \left( {1 + \frac{{R_{s} }}{{R_{sh} }}} \right) & = (I_{sc} \exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right) - \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{{{\text{d}}t}}} \right)\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right) \\ \left( {\left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right)} \right)\left( {\exp \left( {\frac{{qV_{oc} }}{{n_{1} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)} \right) \\ + \frac{{V_{oc} - V_{mp} }}{{R_{sh} }} + \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right) - \left( {\left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right)\frac{{d\left( {V_{mp} + I_{mp} R_{s} } \right)}}{dt}} \right) \\ \end{aligned}$$
(45)
Assume: \(\frac{{V}_{oc}-{V}_{mp}}{{R}_{sh}} \approx 0\) and \(1+\frac{{R}_{s}}{{R}_{sh}}\approx 1\)
$$\begin{aligned} I_{mp} & = I_{sc} \left( {1 - \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)} \right) - \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right) \\ + \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right)\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right)\exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right) \\ + \left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt} - \left( {\left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}\left( {\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right)} \right)} \right) \left( {\exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)} \right) \\ + \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right) - \left( {\left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right)\frac{{d\left( {V_{mp} + I_{mp} R_{s} } \right)}}{dt}} \right) \\ \end{aligned}$$
(46)
Considering \(\frac{1}{{R}_{sh}}\left(1-{R}_{s}\frac{{I}_{mp}}{{V}_{mp}}\right)\approx 0\), Eq. (8) can be written as,
$$\begin{aligned} I_{{m_{P} }} & = V_{{m_{P} }} \frac{q }{{n_{1} kT}}\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right)\exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} } \right)}}{{n_{1} kT}}} \right)\left( {I_{sc} \exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right)} \right) \\ - \left( {\left( {a + b\left( {V_{oc} } \right)} \right)\frac{{d\left( {V_{oc} } \right)}}{dt}} \right)\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right) + \left( {\left( {a + b\left( {R_{s} I_{sc} } \right)} \right)\frac{{d\left( {I_{sc} R_{s} } \right)}}{dt}\exp \left( {\frac{{ - qV_{oc} }}{{n_{1} kT}}} \right) } \right) \\ + \left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right)\frac{d}{dt}\left[ {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right] + \left( {\frac{d}{dt}\left( {V_{mp} + I_{mp} R_{s} } \right)} \right)\left( {b\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right)} \right) \\ \end{aligned}$$
(47)
It can be observed that both Eqs. (46), and (47) are function in \({R}_{s}\), and \({n}_{1}\). However, the series resistance is associated with the differentiation operator. Consequently, we equalize both equations seeking for a first order differential equation (DE) in terms of \({R}_{s}\). Towards simplification, and as mentioned earlier, the boundary conditions points, at short circuit, open-circuit, and maximum power point, are treated as time independent constants, extracted from experimental measurements. Following that, the first order DE can be written as:
$$\begin{gathered} I_{sc} \left( {1 - \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)} \right) + \left( {aI_{sc} + b R_{s} I_{sc}^{2} } \right)\frac{{dR_{s} }}{dt} - \left( {aI_{sc} + b R_{s} I_{sc}^{2} } \right)\frac{{dR_{s} }}{dt} \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right) \hfill \\ - aI_{mp} \frac{{dR_{s} }}{dt} - b\left( {V_{mp} + R_{s} I_{mp} } \right)I_{mp} \frac{{dR_{s} }}{dt} - I_{sc} V_{{m_{P} }} \frac{q }{{n_{1} kT}}\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right) \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right) \hfill \\ - \left( {aI_{sc} + b R_{s} I_{sc}^{2} } \right)V_{{m_{P} }} \frac{q }{{n_{1} kT}}\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right) \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right)\frac{{dR_{s} }}{dt} \hfill \\ - \frac{{I_{mp} }}{{V_{mp} }}\left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right)\frac{{dR_{s} }}{dt} - bI_{mp} \left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right) \frac{{dR_{s} }}{dt} = 0. \hfill \\ \end{gathered}$$
(48)
Equation (48) can be treated as:
$${f}^{1}\left({R}_{s}\right)\frac{d{R}_{s}}{dt}+ {f}^{2}\left({R}_{s}\right)=0$$
(49.a)
where:
$$\begin{gathered} f^{1} \left( {R_{s} } \right) = { }\left( {aI_{sc} + b R_{s} I_{sc}^{2} } \right) - \left( {aI_{sc} + b R_{s} I_{sc}^{2} } \right){ }\exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right) \hfill \\ - aI_{mp} - b\left( {V_{mp} + R_{s} I_{mp} } \right)I_{mp} - { }\left( {aI_{sc} + b R_{s} I_{sc}^{2} } \right)V_{{m_{P} }} \hfill \\ \frac{q }{{n_{1} kT}}\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right) \exp \left( {\frac{{q(V_{mp} - V_{oc} + I_{mp} R_{s} )}}{{n_{1} kT}}} \right) - \frac{{I_{mp} }}{{V_{mp} }}\left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right) - {\text{ b}}I_{mp} { }\left( {1 - R_{s} \frac{{I_{mp} }}{{V_{mp} }}} \right) \hfill \\ \end{gathered}$$
(49.b)
$${f}^{2}\left({R}_{s}\right)= {I}_{sc} \left(1-\mathrm{ exp}\left(\frac{q{(V}_{mp}-{V}_{oc}+ {I}_{mp}{R}_{s})}{{n}_{1}kT}\right)\right)-{I}_{sc} {V}_{{m}_{P}}\frac{q }{{n}_{1}kT}\left(1-{R}_{s} \frac{{I}_{mp}}{{V}_{mp}}\right)\mathrm{ exp}\left(\frac{q{(V}_{mp}-{V}_{oc}+ {I}_{mp}{R}_{s})}{{n}_{1}kT}\right)$$
(49.c)
Equation (49) can’t be solved analytically, however, an iterative numerical solution for both Eqs. (20) and (23) can converge with both \({R}_{s} (t)\), and \({n}_{1}(t)\). Referring to Eq. (48.b), with replacing the diode saturation current as given in (16), the shunt resistance can be calculated as:
$$\begin{gathered} \frac{{ \frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} }}{{1 + R_{s} \left( t \right)bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}} = \hfill \\ \left( {1 + R_{s} \left( t \right) \left( {\frac{{ - \frac{1}{{R_{sho} }} - bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} }}{{1 + R_{s} \left( t \right)bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right)} \right)\left( {\frac{q }{{n_{1} \left( t \right)kT}}\left( {\frac{{I_{sc} \left( {R_{sho} + R_{s} \left( t \right)} \right) - V_{oc} + R_{sho} \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right)\frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}} \right)}}{{R_{sho} \left( {\exp \left( {\frac{{qV_{oc} }}{{n_{1} \left( t \right)kT}}} \right) - \exp \left( {\frac{{\left( {qIsc Rs\left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right)} \right)}}} \right)\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right) + \frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}} \right) \hfill \\ + \left( {a + b\left( {R_{s} \left( t \right)I_{sc} } \right)} \right)\frac{d}{dt}\left[ {R_{s} \left( t \right)\frac{{ - \frac{1}{{R_{sho} }} - bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} }}{{1 + R_{s} bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right] \hfill \\ \end{gathered}$$
(50)
Substituting (49.a) in (50) we get:
$$\begin{gathered} \frac{{ \frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} }}{{1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}} \hfill \\ = \left( {1 + R_{s} \left( t \right)\left( {\frac{{ - \frac{1}{{R_{sho} }} + bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} }}{{1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}} \right)} \right) \hfill \\ \left( {\frac{q }{{n_{1} \left( t \right)kT}}\left( {\frac{{I_{sc} \left( {R_{sho} + R_{s} \left( t \right)} \right) - V_{oc} - R_{sho} \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}{{R_{sho} \left( {\exp \left( {\frac{{qV_{oc} }}{{n_{1} \left( t \right)kT}}} \right) - \exp \left( {\frac{{\left( {qIsc Rs\left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right)} \right)}}} \right)\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right) + \frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) \hfill \\ + \left( {a + bR_{s} \left( t \right)I_{sc} } \right) \hfill \\ \left( {\left[ {R_{s} \left( t \right)\frac{{ \left( { - \frac{1}{{R_{sho} }} + bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} } \right)\left( { - R_{s} bI_{sc} \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right] - bI_{sc} \left( {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} } \right) - \left( {1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)\left( { - \frac{1}{{R_{sho}^{2} }} \frac{{d\left( {R_{sho} } \right)}}{dt} + bI_{sc} \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right]} \right)}}{{\left( {1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} }}} \right]} \right. \hfill \\ \left. { + \left( {\frac{{ \frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} }}{{1 + R_{s} bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) \hfill \\ \end{gathered}$$
(51)
Again, Eq. (51) can’t be solved analytically, however, an iterative numerical solution can be extracted for \({R}_{sho} (t)\). Equation (51) can be treated as:
$${g}^{1}\left({R}_{sho}\right)\frac{d{R}_{sho}}{dt}+ {g}^{2}\left({R}_{sho}\right)=0$$
(52.a)
where:
$${g}^{1}\left({R}_{sho}\right)=(a+b{R}_{s}\left(t\right){I}_{sc})\left(\left[{R}_{s}\left(t\right)\frac{ \left(1-{R}_{s}b{I}_{sc}\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}\right)\left(-\frac{1}{{R}_{sho}^{2}} \right)}{(1-{R}_{s}b{I}_{sc}\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}{)}^{2}}\right]\right)$$
(52.b)
$$\begin{gathered} g^{2} \left( {R_{sho} } \right) = \left( {1 + R_{s} \left( t \right)\left( {\frac{{{ } - \frac{1}{{R_{sho} }} + bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }}}{{1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}} \right)} \right) \hfill \\ \left( {\frac{q }{{n_{1} \left( t \right)kT}}\left( {\frac{{I_{sc} \left( {R_{sho} + R_{s} \left( t \right)} \right) - V_{oc} - R_{sho} \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)}}{{R_{sho} \left( {\exp \left( {\frac{{qV_{oc} }}{{n_{1} \left( t \right)kT}}} \right) - \exp \left( {\frac{{\left( {qIsc Rs\left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right)} \right)}}} \right)\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{n_{1} \left( t \right)kT}}} \right) + \frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) \hfill \\ - \frac{{{ }\frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }}}{{1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}} + \left( {a + bR_{s} \left( t \right)I_{sc} } \right) \hfill \\ \left( {\left[ {R_{s} \left( t \right)\frac{{\left( { - \frac{1}{{R_{sho} }} + bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }} \right)\left( { - R_{s} bI_{sc} \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right] - bI_{sc} \left( {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} } \right) - \left( {1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)\left( {bI_{sc} \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right]} \right)}}{{\left( {1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} }}} \right]} \right. \hfill \\ \left. { + \left( {\frac{{ \frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} }}{{1 + R_{s} bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) \hfill \\ \end{gathered}$$
(52.c)
Solving the first order DE in (26) results with the estimated shunt resistance at the short-circuit point. Substituting back in (16) with the extracted functions from (20), (23), and (26) results:
$${I}_{o1 }(t)=\frac{{I}_{sc}({R}_{sho}(t))-{V}_{oc}+ {R}_{sh0}(t)(a-b\left({R}_{s}(t){I}_{sc}^{2}\right)\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}}{{R}_{sho}(t)\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}(t)kT}\right)-\mathrm{exp}\left(\frac{\left(q{R}_{s}(t){I}_{sc}\right)}{{n}_{1}(t)kT}\right)\right)}$$
(53)
Finally, the photo-generated current can be driven from Eq. (3) as:
$${I}_{ph}(t)={I}_{01}(t)\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}\left(t\right)kT}\right)-1\right)+\frac{{V}_{oc}}{{R}_{sho}(t)}$$
(54)
3.3 Double-diode PV model with first, second, and third-order C-V fitting.
Scaling up the circuit model to a double-diode model, cf. Figure 1b directly impacted the analytical model procedure, adding two new unknowns, \({I}_{02}\), and \({n}_{2}\). Consequently, two new equations are introduced. Simple the summation of the diode ideality factors can be expressed as:
$$\sum_{i=1}^{n}{n}_{i}=n+1$$
(55)
where \(n\) represents the diode model order, for example, \(n=2\) for the double diode model. Additionally, the exact derivation procedure for the linear \(C-V\) model in Sect. 3.1 can be applied up to Eq. (15.b) by applying the general \(I-V\) equation in (1. b) instead of (1. a). Applying the same assumptions as in (16), we can reach the following:
$$I_{{sc}} = I_{{o1}} \left( {\exp \left( {\frac{{qV_{{oc}} }}{{n_{1} kT}}} \right)} \right) + I_{{o2}} \left( {\exp \left( {\frac{{qV_{{oc}} }}{{n_{2} kT}}} \right)} \right) + \left( {\left( {a + b\left( {V_{{oc}} } \right)} \right)\frac{{d\left( {V_{{oc}} } \right)}}{{dt}}} \right) - ~\left( {a + b\left( {R_{s} I_{{sc}} } \right)} \right)\frac{{d\left( {I_{{sc}} R_{s} } \right)}}{{dt}}$$
(56)
$$\begin{gathered} I_{{mp}} =\, I_{{o1}} \left( {\exp \left( {\frac{{qV_{{oc}} }}{{n_{1} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{{mp}} + I_{{mp}} R_{s} } \right)}}{{n_{1} kT}}} \right)} \right) + I_{{02}} \left( {\exp \left( {\frac{{qV_{{oc}} }}{{n_{2} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{{mp}} + I_{{mp}} R_{s} } \right)}}{{n_{2} kT}}} \right)} \right) \hfill \\ + \left( {\left( {a + b\left( {V_{{oc}} } \right)} \right)\frac{{d\left( {V_{{oc}} } \right)}}{{dt}}} \right) - \left( {\left( {a + b\left( {V_{{mp}} + R_{s} I_{{mp}} } \right)} \right)\frac{{d\left( {V_{{mp}} + ~I_{{mp}} Rs} \right)}}{{dt}}} \right) \hfill \\ \end{gathered}$$
(57)
Therefore, using above two equations, \({I}_{o1}\) can be obtained from Eq. (56) and \({I}_{o2}\) can be obtained from Eq. (57) as the following:
$$I_{{o1}} = I_{{sc}} \left( {\exp \left( { - \frac{{qV_{{oc}} }}{{{\mathbf{n}}_{{\mathbf{1}}} kT}}} \right)} \right) - I_{{o2}} \left( {\exp \left( {\frac{{ - qV_{{oc}} }}{{{\mathbf{n}}_{{\mathbf{2}}} kT}}} \right)} \right) - \frac{{\left( {a + b\left( {V_{{oc}} } \right)\frac{{d\left( {V_{{oc}} } \right)}}{{dt}}} \right)~ - ~\left( {a + b\left( {R_{s} I_{{sc}} } \right)} \right)\frac{{d\left( {I_{{sc}} R_{s} } \right)}}{{dt}}~~}}{{\left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{{\mathbf{1}}} kT}}} \right)} \right)}}$$
(58)
$$I_{{o2}} = \;\frac{\begin{gathered} I_{{mp}} - ~I_{{o1}} \left( {\left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q(V_{{mp}} + I_{{mp}} R_{s} )}}{{{\mathbf{n}}_{1} kT}}} \right)} \right)} \right)\left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{{mp}} + I_{{mp}} R_{s} } \right)}}{{{\mathbf{n}}_{1} kT}}} \right)} \right) \hfill \\ - \left( {\left( {a + b\left( {V_{{oc}} } \right)} \right)\frac{{d\left( {V_{{oc}} } \right)}}{{dt}}} \right) + \left( {\left( {a + b\left( {V_{{mp}} + R_{s} I_{{mp}} } \right)} \right)\frac{{d\left( {V_{{mp}} + ~I_{{mp}} Rs} \right)}}{{dt}}} \right)~ \hfill \\ \end{gathered} }{{\left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{{mp}} + I_{{mp}} R_{s} )}}{{{\mathbf{n}}_{2} kT}}} \right)} \right)}}$$
(59)
Utilizing Eqs. (58), and (59), one can reach a general equation for the series resistance, given by:
$$\begin{gathered} \left( {\frac{{~b\frac{{d\left( {V_{{oc}} } \right)}}{{dt}}~}}{{1 + R_{s} b\frac{{d\left( {V_{{oc}} } \right)}}{{dt}} + \left( {a + b\left( {\left( {V_{{oc}} } \right)} \right)} \right)\frac{d}{{dt}}\left( {R_{s} } \right)}}} \right) = \left( {1 + R_{s} ~\left( {\frac{{~ - ~b\frac{{d\left( {V_{{oc}} } \right)}}{{dt}}~}}{{1 + R_{s} b\frac{{d\left( {V_{{oc}} } \right)}}{{dt}} + \left( {a + b\left( {\left( {V_{{oc}} } \right)} \right)} \right)\frac{d}{{dt}}\left( {R_{s} } \right)}}} \right)} \right) \hfill \\ \left( {\frac{{q~\left( {I_{{sc}} \left( {\exp \left( { - \frac{{qV_{{oc}} }}{{{\mathbf{n}}_{{\mathbf{1}}} kT}}} \right)} \right)} \right) - I_{{o2}} \left( {\exp \left( {\frac{{ - qV_{{oc}} }}{{{\mathbf{n}}_{{\mathbf{2}}} kT}}} \right) - \frac{{\left( {\left( {a + b\left( {V_{{oc}} } \right)} \right)\frac{{d\left( {V_{{oc}} } \right)}}{{dt}}} \right)~ - ~\left( {a + b\left( {R_{s} I_{{sc}} } \right)} \right)\frac{{d\left( {I_{{sc}} R_{s} } \right)}}{{dt}}~~}}{{\left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{{\mathbf{1}}} kT}}} \right)} \right)}}} \right)~}}{{n_{1} kT}}} \right. \hfill \\ \frac{{\exp \left( {\frac{{q\left( {V_{{oc}} } \right)}}{{n_{1} kT}}} \right) + q~\left( {\frac{{I_{{mp}} - ~I_{{o1}} \left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{{\mathbf{1}}} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{{mp}} + I_{{mp}} R_{s} } \right)}}{{{\mathbf{n}}_{{\mathbf{1}}} kT}}} \right)} \right) - \left( {\left( {a + b\left( {V_{{oc}} } \right)} \right)\frac{{d\left( {V_{{oc}} } \right)}}{{dt}}} \right) + \left( {\left( {a + b\left( {V_{{mp}} + R_{s} I_{{mp}} } \right)} \right)\frac{{d\left( {V_{{mp}} + ~I_{{mp}} Rs} \right)}}{{dt}}} \right)~}}{{\left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{{\mathbf{2}}} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{{mp}} + I_{{mp}} R_{s} } \right)}}{{{\mathbf{n}}_{{\mathbf{2}}} kT}}} \right)} \right)}}} \right)}}{{{\mathbf{n}}_{{\mathbf{2}}} kT}} \hfill \\ \left. {\exp \left( {\frac{{q\left( {V_{{oc}} } \right)}}{{{\mathbf{n}}_{{\mathbf{2}}} kT}}} \right) + b\frac{{d\left( {V_{{oc}} } \right)}}{{dt}}} \right) \hfill \\ + \left( {a + b\left( {V_{{oc}} } \right)} \right)\frac{d}{{dt}}\left[ {1 + R_{s} \left( {~\frac{{~ - ~b\frac{{d\left( {V_{{oc}} } \right)}}{{dt}}~}}{{1 + R_{s} b\frac{{d\left( {V_{{oc}} } \right)}}{{dt}} + \left( {a + b\left( {\left( {V_{{oc}} } \right)} \right)} \right)\frac{d}{{dt}}\left( {R_{s} } \right)}}} \right)} \right] \hfill \\ \end{gathered}$$
(60)
It can be observed that Eq. (60) is a function in \({R}_{s}\), \({n}_{1}\), and \({n}_{2}\), but \({n}_{1}\) and \({n}_{2}\) can be estimated based on Eq. (55). However, the series resistance is associated with the differentiation operator. Consequently, we seek a first-order differential equation (D.E.) regarding \({R}_{s}\). Towards simplification, and as mentioned earlier, the boundary conditions points, at short circuit, open-circuit, and maximum power point, are treated as time-independent constants extracted from experimental measurements. Following that, the first-order D.E. can be written as:
$$\begin{gathered} \left( {\frac{q }{{n_{1} kT}}\left( {I_{sc} } \right) - I_{o2} \left( {\exp \left( {\frac{{ - qV_{oc} }}{{{\varvec{n}}_{2} kT}}} \right)\left( {\frac{{q\left( {V_{oc} } \right)}}{{n_{1} kT}}} \right)} \right) - \left( {a + b\left( {R_{s} I_{sc}^{2} } \right)} \right)\frac{{d\left( {R_{s} } \right)}}{dt}} \right) \hfill \\ + \frac{q }{{n_{2} kT}}\left( {\frac{{I_{mp} \exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\varvec{n}}_{2} kT}}} \right) - I_{o1} \left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{{\varvec{n}}_{1} kT}}} \right)} \right)\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\varvec{n}}_{2} kT}}} \right)} \right) + \left( {\left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right)I_{mp} \exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\varvec{n}}_{2} kT}}} \right)\frac{{d\left( {R_{s} } \right)}}{dt}} \right) }}{{\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{{\varvec{n}}_{2} kT}}} \right)} \right)}}} \right) = 0 \hfill \\ \end{gathered}$$
(61)
Equation (61) can be treated as:
$${f}^{1}\left({R}_{s}\right)\frac{d{R}_{s}}{dt}+ {f}^{2}\left({R}_{s}\right)=0$$
(62.a)
where:
$${f}^{1}\left({R}_{s}\right)=-\frac{q }{{n}_{1}kT}\left(a+b\left({R}_{s}{I}_{sc}^{2}\right)\right)+\frac{q }{{n}_{2}kT}\left(\frac{\left(a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right)\right){I}_{mp}\mathit{exp}\left(\frac{q\left({V}_{oc}\right)}{{{\varvec{n}}}_{2}kT}\right)\frac{d\left({R}_{s}\right)}{dt} }{\left(\mathrm{exp}\left(\frac{q{V}_{\mathit{oc}}}{{{\varvec{n}}}_{2}\mathit{kT}}\right)-\mathit{exp}\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s})}{{{\varvec{n}}}_{2}kT}\right)\right)}\right)$$
(62.b)
$${f}^{2}\left({R}_{s}\right)=\frac{q }{{n}_{1}kT}(\left({I}_{sc}\right)-{I}_{o2}\left(\mathrm{exp}\left(\frac{-q{V}_{oc}}{{{\varvec{n}}}_{2}kT}\right)\mathrm{exp}\left(\frac{q\left({V}_{oc}\right)}{{n}_{1}kT}\right)\right)+\frac{q }{{n}_{2}kT}\left(\frac{{I}_{mp}\mathrm{exp}\left(\frac{q\left({V}_{oc}\right)}{{{\varvec{n}}}_{2}kT}\right)- {I}_{o1}\left((\mathrm{exp}\left(\frac{q{V}_{\mathit{oc}}}{{{\varvec{n}}}_{1}\mathit{kT}}\right)-\mathrm{exp}\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s})}{{{\varvec{n}}}_{1}kT}\right)\right)\mathrm{exp}\left(\frac{q\left({V}_{oc}\right)}{{{\varvec{n}}}_{2}kT}\right) }{(\left(\mathrm{exp}\left(\frac{q{V}_{\mathit{oc}}}{{{\varvec{n}}}_{2}\mathit{kT}}\right)-\mathrm{exp}\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s})}{{{\varvec{n}}}_{2}kT}\right)\right)}\right)$$
(62.c)
Extracting the series resistance numerically, and solving (44) and (45) simultaneously results:
$$I_{{o1}} \left( t \right) = \frac{\begin{gathered} ~I_{{mp}} \left( {\exp \left( { - \frac{{qV_{{oc}} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right) - \left( {\left( {a + b\left( {V_{{mp}} + R_{s} \left( t \right)I_{{mp}} } \right)} \right)I_{{mp}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}~} \right) - \left( {\left( {\exp \left( {\frac{{qV_{{oc}} }}{{\user2{n}_{2} kT}}} \right)} \right) - \exp \left( {\frac{{q\left( {V_{{mp}} + I_{{mp}} R_{s} \left( t \right)} \right)}}{{\user2{n}_{2} kT}}} \right)} \right) \hfill \\ \left( {\left( {I_{{sc}} \left( {\exp \left( { - \frac{{qV_{{oc}} }}{{\user2{n}_{1} kT}}} \right)} \right)} \right) + ~\frac{{~\left( {a + b\left( {R_{s} \left( t \right)I_{{sc}}^{2} } \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}~~~}}{{\left( {\exp \left( {\frac{{qV_{{oc}} }}{{\user2{n}_{1} kT}}} \right)} \right)}}} \right) \hfill \\ \end{gathered} }{{\left( {\exp \left( {\frac{{ - qV_{{oc}} }}{{{\mathbf{n}}_{2} kT}}} \right)\left( {\exp \left( {\frac{{qV_{{oc}} }}{{\user2{n}_{1} kT}}} \right)} \right) - \exp \left( {\frac{{q\left( {V_{{mp}} + I_{{mp}} R_{s} \left( t \right)} \right)}}{{\user2{n}_{1} kT}}} \right)} \right){\text{~}} + {\text{~}}\left( {\exp \left( {\frac{{q\left( {V_{{mp}} + I_{{mp}} R_{s} \left( t \right)} \right) - V_{{oc}} }}{{\user2{n}_{2} kT}}} \right)} \right)}}$$
(63.a)
$$I_{o2} \left( t \right) = \frac{{ - \left( {\left( {a + b\left( {V_{mp} + R_{s} \left( t \right)I_{mp} } \right)} \right)I_{mp} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }} \right) + I_{mp } - I_{sc} + I_{sc} {\text{e}}\left( {{\text{xp}}\left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right))}}{{{\mathbf{n}}_{1} kT}}} \right)} \right) + \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{\text{exp}}\left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right) - V_{oc} } \right)}}{{{\mathbf{n}}_{1} kT}}} \right)}}{{\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right)}}{{{\mathbf{n}}_{2} kT}}} \right)} \right) - \left( {\exp \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right)} \right) + \left( {\left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right){\text{exp}}\left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right))}}{{{\mathbf{n}}_{1} kT}}} \right)} \right)}}$$
(63.b)
Again, following the same procedure as in Sect. 3.1, a DE for the shunt resistance can be determined by:
$${g}^{1}\left({R}_{sho}\right)\frac{d{R}_{sho}}{dt}+ {g}^{2}\left({R}_{sho}\right)=0$$
(64.a)
where:
$${g}^{1}\left({R}_{sho}\right)=(a+b{R}_{s}\left(t\right){I}_{sc})\left(\left[{R}_{s}\left(t\right)\frac{ \left(1-{R}_{s}b{I}_{sc}\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}-(a+b\left(\left({I}_{sc}{R}_{s}(t)\right)\right)\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}\right)\left(-\frac{1}{{R}_{sho}^{2}} \right)}{(1-{R}_{s}b{I}_{sc}\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}{-(a+b\left(\left({I}_{sc}{R}_{s}(t)\right)\right)\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)})}^{2}}\right]\right)$$
(64.b)
$$\begin{aligned} g^{2} \left( {R_{sho} } \right) & = \left( {1 + R_{s} \left( t \right)\left( {\frac{{{ } - \frac{1}{{R_{sho} }} + bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }}}{{1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - (a + b\left( {\left( {I_{sc} R_{s} \left( t \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}} \right)} \right) \\ \left( {\frac{{q I_{o1} \left( t \right)}}{{n_{1} kT}}\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{n_{1} kT}}} \right) + + \frac{{q I_{o2} \left( t \right)}}{{{\mathbf{n}}_{2} kT}}\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{2} kT}}} \right) + \frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) \\ - \frac{{{ }\frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }}}{{1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {\left( {I_{sc} R_{s} \left( t \right)} \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}} + \left( {a + bR_{s} \left( t \right)I_{sc} } \right) \\ \left( {\left[ {\begin{array}{*{20}c} {R_{s} \left( t \right)\frac{{\left( { - \frac{1}{{R_{sho} }} + bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }} \right)\left( { - R_{s} bI_{sc} \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right] - bI_{sc} \left( {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} - \left( {a + b\left( {\left( {I_{sc} R_{s} \left( t \right)} \right)} \right)} \right) \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right] - bI_{sc} \left( {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} } \right)}}{{\left( {1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {\left( {I_{sc} R_{s} \left( t \right)} \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} }}} \\ { - R_{s} \left( t \right)\frac{{{ }\left( {1 - R_{s} \left( t \right)bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {\left( {I_{sc} R_{s} \left( t \right)} \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)\left( {bI_{sc} \frac{d}{dt}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right]} \right)}}{{\left( {1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) - \left( {a + b\left( {\left( {I_{sc} R_{s} \left( t \right)} \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}})^{2} }}} \\ { + \left( {\frac{{{ }\frac{1}{{R_{sho} }} - bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }}}{{1 - R_{s} bI_{sc} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {\left( {I_{sc} R_{s} \left( t \right)} \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \\ \end{array} } \right]} \right) \\ \end{aligned}$$
(64.c)
Finally, the photo-generated current can be driven as:
$${I}_{ph}\left(t\right)={I}_{01}\left(t\right)\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-1\right)+{I}_{02}\left(t\right)\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{2}kT}\right)-1\right)+\frac{{V}_{oc}}{{R}_{sho}(t)}$$
(65)
Following the same procedure as from Eq. (56) to (65), a similar analytical solution for the double-diode second order C-V fitting can be obtained. The DE representing the series resistance, as in (48), can be driven as:
$${f}^{1}\left({R}_{s}\right)\frac{d{R}_{s}}{dt}+ {f}^{2}\left({R}_{s}\right)=0$$
(66.a)
where:
$${f}^{1}\left({R}_{s}\right)=-\frac{q }{{n}_{1}kT}\left(a+b\left({R}_{s}{I}_{sc}^{2}\right)+c({I}_{sc}^{3}{R}_{s}^{2})\right)+\frac{q }{{n}_{2}kT}\left(\frac{\left(a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right)+c{\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}^{2}\right){I}_{mp}\mathit{exp}\left(\frac{q\left({V}_{oc}\right)}{{{\varvec{n}}}_{2}kT}\right)\frac{d\left({R}_{s}\right)}{dt} }{((\mathit{exp}\left(\frac{q{V}_{\mathit{oc}}}{{{\varvec{n}}}_{2}\mathit{kT}}\right)-\mathit{exp}\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s})}{{{\varvec{n}}}_{2}kT}\right))}\right)$$
(66.b)
$${f}^{2}\left({R}_{s}\right)=\frac{q }{{n}_{1}kT}(\left({I}_{sc}\right)-{I}_{o2}(\mathit{exp}\left(\frac{-q{V}_{oc}}{{{\varvec{n}}}_{2}kT}\right)\mathit{exp}\left(\frac{q\left({V}_{oc}\right)}{{n}_{1}kT}\right)))+\frac{q }{{n}_{2}kT}\left(\frac{{I}_{mp}\mathit{exp}\left(\frac{q\left({V}_{oc}\right)}{{{\varvec{n}}}_{2}kT}\right)- {I}_{o1}((\mathit{exp}\left(\frac{q{V}_{\mathit{oc}}}{{{\varvec{n}}}_{1}\mathit{kT}}\right)-\mathit{exp}\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s})}{{{\varvec{n}}}_{1}kT}\right))\mathit{exp}\left(\frac{q\left({V}_{oc}\right)}{{{\varvec{n}}}_{2}kT}\right) }{((\mathit{exp}\left(\frac{q{V}_{\mathit{oc}}}{{{\varvec{n}}}_{2}\mathit{kT}}\right)-\mathit{exp}\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s})}{{{\varvec{n}}}_{2}kT}\right))}\right)$$
(66.c)
Accordingly, the dark saturation current can be extracted as:
$$I_{o1} \left( t \right) = \frac{\begin{gathered} \frac{{ I_{mp} \left( {\exp \left( { - \frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right) - \left( {\left( {a + b\left( {V_{mp} + R_{s} \left( t \right)I_{mp} } \right) + c\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)^{2} } \right)I_{mp} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} } \right){ } }}{{\left( {\left( {{ }\exp \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right)\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right))}}{{{\varvec{n}}_{1} kT}}} \right)} \right)} \right){ } + { }\left( {\exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right) - V_{oc} }}{{{\varvec{n}}_{2} kT}}} \right)} \right)}} - \hfill \\ \left( {\left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right)}}{{{\varvec{n}}_{2} kT}}} \right)} \right)} \right)\left( {\left( {I_{sc} \left( {\exp \left( { - \frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right)} \right)} \right) + \frac{{ \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right) + c\left( {I_{sc}^{3} R_{s}^{2} \left( t \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} }}{{\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right)} \right)}}} \right)} \right){ } \hfill \\ \end{gathered} }{{\left( {{ }\left( {\exp \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right)\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)}}{{{\varvec{n}}_{1} kT}}} \right)} \right)} \right) + { }\left( {\exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right) - V_{oc} } \right)}}{{{\varvec{n}}_{2} kT}}} \right)} \right)}}$$
(67.a)
$$I_{o2} \left( t \right) = \frac{{\frac{{\left( {\left( {a + b\left( {V_{mp} + R_{s} \left( t \right)I_{mp} } \right) + c\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)^{2} } \right)I_{mp} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }} \right) + I_{mp } - I_{sc} + I_{sc} \left( {{\text{exp}}\left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{1} kT}}} \right)} \right)}}{{\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right) - \exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{2} kT}}} \right) - \left( {\exp \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right) + \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right){\text{exp}}\left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{1} kT}}} \right)} \right)}} + }}{{\frac{{\left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right) + c\left( {I_{sc}^{3} R_{s}^{2} \left( t \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} + c\left( {I_{sc}^{3} R_{s}^{2} \left( t \right)} \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{\text{exp}}\left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right) - V_{oc} } \right)}}{{{\mathbf{n}}_{1} kT}}} \right)}}{{\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right)}}{{{\mathbf{n}}_{2} kT}}} \right)} \right) - \left( {\exp \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right) + \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right){\text{exp}}\left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{1} kT}}} \right)} \right)}}}}$$
(67.b)
The DE in terms of \({R}_{sho} (t)\) is obtained as:
$${g}^{1}\left({R}_{sho}\right)\frac{d{R}_{sho}}{dt}+ {g}^{2}\left({R}_{sho}\right)=0$$
(68.a)
where:
$${g}^{1}\left({R}_{sho}\right)=(a+b{R}_{s}\left(t\right){I}_{sc}+c{\left({I}_{sc}{R}_{s}(t)\right)}^{2})\left(\left[{R}_{s}\left(t\right)\frac{ \left(1-{R}_{s}b{I}_{sc}\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}-2\mathrm{c}{R}_{s}{I}_{sc}^{2}\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}-(a+b\left(\left({I}_{sc}{R}_{s}(t)\right)+c{\left({I}_{sc}{R}_{s}(t)\right)}^{2}\right)\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}\right)\left(-\frac{1}{{R}_{sho}^{2}} \right)}{(1-{R}_{s}b{I}_{sc}\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}-2{R}_{s}^{2}(t)c{I}_{sc}^{2}\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)}{-(a+b\left(\left({I}_{sc}{R}_{s}(t)\right)+c{\left({I}_{sc}{R}_{s}(t)\right)}^{2}\right)\frac{{f}^{2}\left({R}_{s}\left(t\right)\right)}{{f}^{1}\left({R}_{s}\left(t\right)\right)})}^{2}}\right]\right)$$
(68.b)
$$\begin{gathered} g^{2} \left( {R_{{sho}} } \right) = ~\left( {1 + R_{s} \left( t \right)\left( {\frac{{ - ~\frac{1}{{R_{{sho}} }} + bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} + 2cR_{s} I_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}{{1 - R_{s} \left( t \right)bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - 2R_{s}^{2} \left( t \right)cI_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {I_{{sc}} R_{s} } \right) + c\left( {R_{s} I_{{sc}} } \right)^{2} } \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}} \right)} \right) \hfill \\ \left( {\left( {\frac{{q~I_{{o1}} \left( t \right)}}{{n_{1} kT}}\exp \left( {\frac{{q\left( {I_{{sc}} R_{s} \left( t \right)} \right)}}{{n_{1} kT}}} \right) + + \frac{{q~I_{{o2}} \left( t \right)}}{{{\mathbf{n}}_{2} kT}}\exp \left( {\frac{{q\left( {I_{{sc}} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{2} kT}}} \right) + \frac{1}{{R_{{sho}} }} - bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) - 2cR_{s} I_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) \hfill \\ - \frac{{{\text{~}}\frac{1}{{R_{{sho}} }} - bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - 2cR_{s} I_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{\text{~}}}}{{1 - R_{s} \left( t \right)bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - 2R_{s}^{2} \left( t \right)cI_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {\left( {I_{{sc}} R_{s} \left( t \right)} \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}~ \hfill \\ + \left( {a + bR_{s} \left( t \right)I_{{sc}} + c\left( {I_{{sc}} R_{s} \left( t \right)} \right)^{2} } \right) \hfill \\ \left( {\left[ {R_{s} \left( t \right)\frac{{{\text{~}}\left( { - \frac{1}{{R_{{sho}} }} + bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} + 2cR_{s} I_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}~} \right)\left( { - R_{s} bI_{{sc}} ~\frac{d}{{dt}}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right] - bI_{{sc}} \left( {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} - 2R_{s}^{2} \left( t \right)cI_{{sc}}^{2} ~\frac{d}{{dt}}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right] - 4R_{s} cI_{{sc}}^{2} \left( {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} - \left( {a + b\left( {\left( {I_{{sc}} R_{s} \left( t \right)} \right) + c\left( {I_{{sc}} R_{s} \left( t \right)} \right)^{2} } \right)} \right)~\frac{d}{{dt}}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right] - \left( {bI_{{sc}} - 2cR_{s} I_{{sc}}^{2} } \right)\left( {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} } \right)}}{{\left( {1 - R_{s} bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - 2R_{s}^{2} \left( t \right)cI_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {\left( {I_{{sc}} R_{s} \left( t \right)} \right)} \right) + c\left( {I_{{sc}} R_{s} \left( t \right)} \right)^{2} } \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} }}} \right.} \right. \hfill \\ \left. { - R_{s} \left( t \right)\frac{{\left( {\left( {1 - R_{s} bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) - 2R_{s}^{2} \left( t \right)cI_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {I_{{sc}} R_{s} } \right) + c\left( {R_{s} I_{{sc}} } \right)^{2} } \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)\left( {bI_{{sc}} \frac{d}{{dt}}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} + 2cR_{s} I_{{sc}}^{2} \frac{d}{{dt}}\left[ {\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right]} \right]} \right)}}{{\left( {1 - R_{s} bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - 2R_{s}^{2} \left( t \right)cI_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {\left( {I_{{sc}} R_{s} \left( t \right)} \right)} \right) + c\left( {I_{{sc}} R_{s} \left( t \right)} \right)^{2} } \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right)^{2} }}} \right] \hfill \\ \left. {\left( {\frac{{{\text{~}}\frac{1}{{R_{{sho}} }} - bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - 2cR_{s} I_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{\text{~}}}}{{1 - R_{s} bI_{{sc}} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - 2R_{s}^{2} \left( t \right)cI_{{sc}}^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {\left( {I_{{sc}} R_{s} \left( t \right)} \right)} \right)} \right) + c\left( {I_{{sc}} R_{s} \left( t \right)} \right)^{2} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}}}} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}} \right) \hfill \\ \end{gathered}$$
(68.c)
Finally, the photo-generated current can be driven with the same general form as in Eq. (65). Finally, a typical model is found to the double diode model with third order C-V fitting, as follows:
$${f}^{1}\left({R}_{s}\right)\frac{d{R}_{s}}{dt}+ {f}^{2}\left({R}_{s}\right)=0$$
(69.a)
where:
$${f}^{1}\left({R}_{s}\right)=-\frac{q }{{n}_{1}kT}\left(a+b\left({R}_{s}{I}_{sc}^{2}\right)+c\left({I}_{sc}^{3}{R}_{s}^{2}\right)+d({I}_{sc}^{4}{R}_{s}^{3}\right)+\frac{q }{{n}_{2}kT}\left(\frac{\left(a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right)+c{\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}^{2}+d{\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}^{3}\right){I}_{mp}\mathit{exp}\left(\frac{q\left({V}_{oc}\right)}{{{\varvec{n}}}_{2}kT}\right)\frac{d\left({R}_{s}\right)}{dt} }{((\mathit{exp}\left(\frac{q{V}_{\mathit{oc}}}{{{\varvec{n}}}_{2}\mathit{kT}}\right)-\mathit{exp}\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s})}{{{\varvec{n}}}_{2}kT}\right))}\right)$$
(69.b)
$$f^{2} \left( {R_{s} } \right) = \frac{{q~}}{{\user2{n}_{1} kT}}(\left( {I_{{sc}} } \right) - I_{{o2}} (\exp \left( {\frac{{ - qV_{{oc}} }}{{\user2{n}_{2} kT}}} \right)\exp \left( {\frac{{q\left( {V_{{oc}} } \right)}}{{n_{1} kT}}} \right))) + \frac{{q~}}{{n_{2} kT}}\left( {\frac{{I_{{mp}} \exp \left( {\frac{{q\left( {V_{{oc}} } \right)}}{{n_{2} kT}}} \right) - ~I_{{o1}} ((\exp \left( {\frac{{qV_{{oc}} }}{{n_{1} kT}}} \right) - \exp \left( {\frac{{q(V_{{mp}} + I_{{mp}} R_{s} )}}{{n_{1} kT}}} \right))\exp \left( {\frac{{q\left( {V_{{oc}} } \right)}}{{\user2{n}_{2} kT}}} \right)~}}{{((\exp \left( {\frac{{qV_{{oc}} }}{{\user2{n}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{{mp}} + I_{{mp}} R_{s} )}}{{n_{2} kT}}} \right))}}} \right)$$
(69.c)
$$\begin{gathered} I_{o1} \left( t \right) \hfill \\ = \frac{{I_{mp} \left( {\exp \left( { - \frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right) - \left( {\left( {a + b\left( {V_{mp} + R_{s} \left( t \right)I_{mp} } \right) + c\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)^{2} + d\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)^{3} } \right)I_{mp} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} } \right)}}{{\left( {\left( {{ }\exp \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right))\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right))}}{{{\varvec{n}}_{1} kT}}} \right)} \right)} \right){ } + { }(\exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right) - V_{oc} }}{{{\varvec{n}}_{2} kT}}} \right)} \right)}} \hfill \\ - \frac{{\left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right)}}{{{\varvec{n}}_{2} kT}}} \right)} \right)\left( {\left( {I_{sc} \left( {\exp \left( { - \frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right)} \right)} \right) + \frac{{ \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right) + c(I_{sc}^{3} R_{s}^{2} \left( t \right) + d(I_{sc}^{4} R_{s}^{3} \left( t \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} }}{{\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right)} \right)}}} \right)} \right)}}{{\left( {\left( {\left( {{ }\exp \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right)\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)}}{{{\varvec{n}}_{1} kT}}} \right)} \right)} \right){ } + { }\left( {\exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right) - V_{oc} }}{{{\varvec{n}}_{2} kT}}} \right)} \right)} \right)}} \hfill \\ \end{gathered}$$
(70.a)
$$\begin{gathered} I_{o2} \left( t \right) = \frac{{ - \left( {\left( {a + b\left( {V_{mp} + R_{s} \left( t \right)I_{mp} } \right) + c\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)^{2} + d\left( {V_{mp} + I_{mp} R_{s} } \right)^{3} } \right)I_{mp} \frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{ }} \right) + I_{mp } - I_{sc} + I_{sc} \left( {{\text{exp}}\left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{1} kT}}} \right)} \right)}}{{\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right)}}{{{\mathbf{n}}_{2} kT}}} \right)} \right) - \left( {\exp \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right) + \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right){\text{exp}}\left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{1} kT}}} \right)} \right)}} \hfill \\ + \frac{{\left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} } \right) + c\left( {I_{sc}^{3} R_{s}^{2} \left( t \right) + d\left( {I_{sc}^{4} R_{s}^{3} \left( t \right)} \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}} - \left( {a + b\left( {R_{s} \left( t \right)I_{sc}^{2} + c\left( {I_{sc}^{3} R_{s}^{2} \left( t \right)} \right)} \right) + d\left( {I_{sc}^{4} R_{s}^{3} \left( t \right)} \right)} \right)\frac{{f^{2} \left( {R_{s} \left( t \right)} \right)}}{{f^{1} \left( {R_{s} \left( t \right)} \right)}}{\text{exp}}\left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right) - V_{oc} } \right)}}{{{\mathbf{n}}_{1} kT}}} \right)}}{{\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} \left( t \right)}}{{{\mathbf{n}}_{2} kT}}} \right)} \right) - \left( {\exp \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right) + \left( {\frac{{ - qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right){\text{exp}}\left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{1} kT}}} \right)} \right)}}. \hfill \\ \end{gathered}$$
(70.b)
The shunt resistance can be calculated as
$$\begin{gathered} \frac{{ \frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + 2cR_{s} \left( t \right)I_{sc}^{2} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + 3d\left( {I_{sc}^{3} R_{s}^{2} \left( t \right)} \right)^{2} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} }}{{1 + R_{s} \left( t \right)bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + 2R_{s}^{2} \left( t \right)cI_{sc}^{2} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + 3R_{s}^{3} \left( t \right)dI_{sc}^{3} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + \left( {a + b\left( {I_{sc} R_{s} } \right) + c\left( {R_{s} I_{sc} } \right)^{2} + d\left( {I_{sc} R_{s} } \right)^{3} } \right)\frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}} \hfill \\ = \left( {1 + R_{s} \left( {\frac{{ - \frac{1}{{R_{sho} }} - bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} - 2cR_{s} \left( t \right)I_{sc}^{2} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} - 3d\left( {I_{sc}^{3} R_{s}^{2} \left( t \right)} \right)^{2} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}{{1 + R_{s} \left( t \right)bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + 2R_{s}^{2} \left( t \right)cI_{sc}^{2} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + 3R_{s}^{3} \left( t \right)dI_{sc}^{3} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + \left( {a + b\left( {I_{sc} R_{s} } \right) + c\left( {R_{s} I_{sc} } \right)^{2} + d\left( {I_{sc} R_{s} } \right)^{3} } \right)\frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right)} \right) \hfill \\ \left( {\frac{{q I_{01} }}{{n_{1} kT}}\exp \left( {\frac{{q\left( {I_{sc} R_{s} (t} \right)}}{{n_{1} kT}}} \right) + \frac{{q I_{o2} }}{{{\mathbf{n}}_{{\mathbf{2}}} kT}}\exp \left( {\frac{{q\left( {I_{sc} R_{s} (t} \right)}}{{{\mathbf{n}}_{{\mathbf{2}}} kT}}} \right) + \frac{1}{{R_{sho} }} + b\frac{{d\left( {I_{sc} R_{s} (t} \right)}}{dt} + 2c\left( {(I_{sc} R_{s} } \right)\frac{{d\left( {I_{sc} R_{s} (t} \right)}}{dt} + 3d\left( {I_{sc} R_{s} } \right)^{2} \frac{{d\left( {I_{sc} R_{s} (t} \right)}}{dt}} \right) \hfill \\ + \left( {a + b\left( {R_{s} I_{sc} ) + c\left( {I_{sc} R_{s} } \right)^{2} + d\left( {I_{sc} R_{s} } \right)^{3} } \right)} \right)\frac{d}{dt}\left[ {1 + R_{s} \frac{{ - \frac{1}{{R_{sho} }} - bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} - 2cR_{s} \left( t \right)I_{sc}^{2} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} - 3d\left( {I_{sc}^{3} R_{s}^{2} \left( t \right)} \right)^{2} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}{{1 + R_{s} \left( t \right)bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + 2R_{s}^{2} \left( t \right)cI_{sc}^{2} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + 3R_{s}^{3} \left( t \right)dI_{sc}^{3} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + \left( {a + b\left( {I_{sc} R_{s} } \right) + c\left( {R_{s} I_{sc} } \right)^{2} + d\left( {I_{sc} R_{s} } \right)^{3} } \right)\frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right] \hfill \\ \end{gathered}$$
(71)
3.4 Triple-diode PV model with first, second, and third-order C-V fitting.
Finally, the triple-diode model, cf. Figure 1c and Eq. (1.c) are presented in this section. The updated analytical model is performed by adding two new unknowns, \({I}_{03}\), and \({n}_{3}\). The ideality factor summation in Eq. (41) can still be valid, where the ideality factors are assumed accordingly. The linear C-V fitting showed:
$${f}^{1}\left({R}_{s}\right)\frac{d{R}_{s}}{dt}+ {f}^{2}\left({R}_{s}\right)=0$$
(72.a)
where:
$${f}^{1}\left({R}_{s}\right)=-\frac{q }{{n}_{1}kT}\left(a+b\left({R}_{s}{I}_{sc}^{2}\right)\right)+\frac{q }{{n}_{2}kT}\left(\frac{\left(a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right)\right){I}_{mp}\mathit{exp}\left(\frac{q\left({V}_{oc}\right)}{{{\varvec{n}}}_{2}kT}\right)\frac{d\left({R}_{s}\right)}{dt} }{\left(\mathit{exp}\left(\frac{q{V}_{oc}}{{{\varvec{n}}}_{2}kT}\right)-exp\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s})}{{{\varvec{n}}}_{2}kT}\right)\right)}\right)$$
(72.b)
$$\begin{gathered} f^{2} \left( {R_{s} } \right) = \frac{q }{{n_{1} kT}}(\left( {I_{sc} } \right) - I_{o2} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{2} kT}}} \right)} \right) - I_{o3} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{3} kT}}} \right)} \right) + \frac{q }{{n_{2} kT}} \hfill \\ \left( {\frac{{I_{mp} \exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\varvec{n}}_{2} kT}}} \right) - I_{o1} \left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{{\varvec{n}}_{1} kT}}} \right)} \right)} \right)\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\varvec{n}}_{2} kT}}} \right) - I_{o3} \left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{3} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} } \right)}}{{{\mathbf{n}}_{3} kT}}} \right)} \right)\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\mathbf{n}}_{2} kT}}} \right)} \right) }}{{\left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} } \right)}}{{{\varvec{n}}_{2} kT}}} \right)} \right)} \right)}}} \right) \hfill \\ + \frac{q }{{n_{3} kT}}\left( {I_{ph} - I_{o1} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right) - I_{o2} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right)} \right) \hfill \\ \end{gathered}$$
(72.c)
$$I_{{o1}} = I_{{sc}} \left( {\exp - \frac{{qV_{{oc}} }}{{{\mathbf{n}}_{1} kT}}} \right) - I_{{o2}} \left( {~\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right)\left( {\exp \left( { - \frac{{qV_{{oc}} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right)~ - I_{{o3}} \left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{3} kT}}} \right)} \right)\left( {\exp \left( { - \frac{{qV_{{oc}} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right) + \left( {\frac{{~~\left( {a + b\left( {R_{s} I_{{sc}} } \right)} \right)\frac{{d\left( {I_{{sc}} R_{s} } \right)}}{{dt}}~~}}{{\left( {\exp \left( {\frac{{qV_{{oc}} }}{{kT}}} \right)} \right)}}} \right)$$
(73.a)
$$I_{o2} = \;\frac{{I_{mp} - I_{o1} \left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} } \right)}}{{{\mathbf{n}}_{1} kT}}} \right)} \right)} \right) - I_{o3} \left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{3} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} } \right)}}{{{\mathbf{n}}_{3} kT}}} \right)} \right)} \right) + \left( {\left( {a + b\left( {V_{mp} + R_{s} I_{mp} } \right)} \right)\frac{{d\left( {V_{mp} + I_{mp} Rs} \right)}}{{{\text{d}}t}}} \right){ }}}{{\left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{{\mathbf{n}}_{2} kT}}} \right)} \right)} \right)}}$$
(73.b)
$$I_{{o3}} = \frac{{I_{{ph}} - I_{{o1}} \left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right) - I_{{o2}} \left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right)~~}}{{\left( {\exp \left( {\frac{{qV_{{oc}} }}{{{\mathbf{n}}_{3} kT}}} \right)} \right)}}$$
(73.c)
$$\begin{gathered} \frac{{{ }\frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}{ }}}{{1 + R_{s} \left( t \right)bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + \left( {a + b\left( {\left( {I_{sc} R_{s} \left( t \right)} \right)} \right)} \right)\frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}} \hfill \\ = \left( {1 + R_{s} \left( {\frac{{ - \frac{1}{{R_{sho} }} - bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}{ }}}{{1 + R_{s} \left( t \right)bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + \left( {a + b\left( {\left( {I_{sc} R_{s} \left( t \right)} \right)} \right)} \right)\frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right)} \right) \hfill \\ \left( {\left( {\frac{{q I_{o1} \left( t \right)}}{{n_{1} kT}}\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{n_{1} kT}}} \right) + \frac{{q I_{o2} \left( t \right)}}{{{\mathbf{n}}_{2} kT}}\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{2} kT}}} \right) + \frac{{q I_{o3} \left( t \right)}}{{{\mathbf{n}}_{3} kT}}\exp \left( {\frac{{q\left( {I_{sc} R_{s} \left( t \right)} \right)}}{{{\mathbf{n}}_{3} kT}}} \right)} \right) + \frac{1}{{R_{sho} }} + bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}} \right) \hfill \\ + \left( {a + b\left( {R_{s} \left( t \right)I_{sc} } \right)} \right)\frac{d}{dt}\left[ {R_{s} \left( t \right)\frac{{ - \frac{1}{{R_{sho} }} - bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}{ }}}{{1 + R_{s} \left( t \right)bI_{sc} \frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt} + \left( {a + b\left( {\left( {I_{sc} R_{s} \left( t \right)} \right)} \right)} \right)\frac{{d\left( {R_{s} \left( t \right)} \right)}}{dt}}}} \right] \hfill \\ \end{gathered}$$
(74)
then, the photo-generated current can be driven as:
$${I}_{ph}\left(t\right)={I}_{01}\left(t\right)\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{1}kT}\right)-1\right)+{I}_{02}\left(t\right)\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{2}kT}\right)-1\right)+{I}_{03}\left(t\right)\left(\mathrm{exp}\left(\frac{q{V}_{oc}}{{n}_{3}kT}\right)-1\right)+\frac{{V}_{oc}}{{R}_{sho}(t)}$$
(75)
Similar outputs were reached for the triple-diode second, and third order C-V fitting, respectively, summarized as:
$${f}^{1}\left({R}_{s}\right)\frac{d{R}_{s}}{dt}+ {f}^{2}\left({R}_{s}\right)=0$$
(76.a)
where:
$${f}^{1}\left({R}_{s}\right)=-\frac{q }{{n}_{1}kT}\left(a+b\left({R}_{s}{I}_{sc}^{2}\right)+c({I}_{sc}^{3}{R}_{s}^{2})\right)+\frac{q }{{n}_{2}kT}\left(\frac{\left(a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right)+c{\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}^{2}\right){I}_{mp}\mathit{exp}\left(\frac{q\left({V}_{oc}\right)}{{{\varvec{n}}}_{2}kT}\right)\frac{d\left({R}_{s}\right)}{dt} }{\left((\mathit{exp}\left(\frac{q{V}_{\mathit{oc}}}{{{\varvec{n}}}_{2}\mathit{kT}}\right)-\mathit{exp}\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s})}{{{\varvec{n}}}_{2}kT}\right)\right)}\right)$$
(76.b)
$$\begin{gathered} f^{2} \left( {R_{s} } \right) = \frac{q }{{n_{1} kT}}\left( {\left( {I_{sc} } \right) - I_{o2} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{2} kT}}} \right)} \right) - I_{o3} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{3} kT}}} \right)} \right)} \right) + \frac{q }{{n_{2} kT}} \hfill \\ \left( { \frac{{I_{mp} \exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\varvec{n}}_{2} kT}}} \right) - I_{o1} \left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{{\varvec{n}}_{1} kT}}} \right)} \right)\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\varvec{n}}_{2} kT}}} \right)} \right) - I_{o3} \left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{3} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{{\mathbf{n}}_{3} kT}}} \right)} \right)\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\mathbf{n}}_{2} kT}}} \right)} \right) }}{{\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q\left( {V_{mp} + I_{mp} R_{s} } \right)}}{{{\varvec{n}}_{2} kT}}} \right)} \right)}}} \right) \hfill \\ + \frac{q }{{n_{3} kT}}\left( {I_{ph} - I_{o1} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right) - I_{o2} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right)} \right) \hfill \\ \end{gathered}$$
(76.c)
$${f}^{1}\left({R}_{s}\right)\frac{d{R}_{s}}{dt}+ {f}^{2}\left({R}_{s}\right)=0$$
(77.a)
where:
$${f}^{1}\left({R}_{s}\right)=-\frac{q }{{n}_{1}kT}\left(a+b\left({R}_{s}{I}_{sc}^{2}\right)+c\left({I}_{sc}^{3}{R}_{s}^{2}\right)+d({I}_{sc}^{4}{R}_{s}^{3}\right)+\frac{q }{{n}_{2}kT}\left(\frac{\left(a+b\left({V}_{mp}+{R}_{s}{I}_{mp}\right)+c{\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}^{2}+d{\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}^{3}\right){I}_{mp}\mathit{exp}\left(\frac{q\left({V}_{oc}\right)}{{{\varvec{n}}}_{2}kT}\right)\frac{d\left({R}_{s}\right)}{dt} }{((\mathit{exp}\left(\frac{q{V}_{\mathit{oc}}}{{{\varvec{n}}}_{2}\mathit{kT}}\right)-\mathit{exp}\left(\frac{q{(V}_{mp}+{I}_{mp}{R}_{s})}{{{\varvec{n}}}_{2}kT}\right))}\right)$$
(77.b)
$$\begin{gathered} f^{2} \left( {R_{s} } \right) = \frac{q }{{n_{1} kT}}\left( {I_{sc} } \right) - I_{o2} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{2} kT}}} \right)} \right) - I_{o3} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{3} kT}}} \right)} \right) + \frac{q }{{n_{2} kT}} \hfill \\ \left( {\frac{{I_{mp} \exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\varvec{n}}_{2} kT}}} \right) - I_{o1} \left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{1} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{{\varvec{n}}_{1} kT}}} \right)} \right)\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\varvec{n}}_{2} kT}}} \right)} \right) - I_{o3} \left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{3} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{{\mathbf{n}}_{3} kT}}} \right)} \right)\exp \left( {\frac{{q\left( {V_{oc} } \right)}}{{{\mathbf{n}}_{2} kT}}} \right)} \right) }}{{\left( {\left( {\exp \left( {\frac{{qV_{oc} }}{{{\varvec{n}}_{2} kT}}} \right) - \exp \left( {\frac{{q(V_{mp} + I_{mp} R_{s} )}}{{{\varvec{n}}_{2} kT}}} \right)} \right)} \right)}}} \right) \hfill \\ \frac{q }{{n_{3} kT}}\left( {I_{ph} - I_{o1} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{1} kT}}} \right)} \right) - I_{o2} \left( {\exp \left( {\frac{{qV_{oc} }}{{{\mathbf{n}}_{2} kT}}} \right)} \right)} \right) \hfill \\ \end{gathered}$$
(77.c)
Substituting Eq. (76) or (77) in Eq. (70) to (74) results in the final closed forms for the second and third-order \(C-V\) fitting in the triple-diode model. The nine analytical solutions are attached to this paper in supplementary material S1. The nine derived analytical models seed the equilibrium optimizer (E.O.) and the measured \(I-V\) data, seeking optimum parameters extraction with minimum RMS error. The E.O. algorithm along with all previously mentioned numerically solved differential equations are scripted using MATLAB. This is demonstrated in the next section.