The test function models the maximum power of a single-diode solar cell. It was introduced by Constantine et al. (2015)[^Constantine] to demonstrate the active subspace method for (input) dimension reduction and sensitivity analysis.
The model consists of the following set of equations:
$$
\begin{aligned}
\mathcal{M}(\boldsymbol{x}; \boldsymbol{p}) & = P{\text{max}}(\boldsymbol{x}) = \max{I, V} I V \
I(\boldsymbol{x}) & = I_L - IS \left( \exp{\left(\frac{V + I R{S}}{Ns n V{\text{th}}} \right)} - 1 \right) - \frac{V + I R_S}{R_P} \
IL(\boldsymbol{x}) & = I{SC} + IS \left( \exp{\left(\frac{I{SC} R_{S}}{Ns n V{\text{th}}} \right)} - 1 \right) - \frac{I_{SC} R_S}{R_P}, \
\end{aligned}
$$
where $\boldsymbol{x} = { I_{SC}, I_S, n, R_S, R_P }$ is the vector of input variables modeled as independent random variables whose marginals are specified in the table below.
Symbol
Bounds
Description
Units
$I_{SC}$
$[0.05989, 0.23958]$
Short circuit current
$[\text{A}]$
$I_S$
$[2.2 \times 10^{-11}, 2.2 \times 10^{-7}]$
Diode reverse saturation current
$[\text{A}]$
$n$
$[1, 2]$
Ideality factor
$[-]$
$R_S$
$[0.16625, 0.66500]$
Series resistance
$[\Omega]$
$R_P$
$[93.75,375.00]$
Parallel (shunt) resistance
$[\Omega]$
Furthermore, $\boldsymbol{p} = { Ns, V{\text{th}} }$ is the vector of parameters, namely, the number of cells connected in series and the thermal voltage. In the paper, the number of cells is set to $1$ while the thermal voltage is set to $T = 25 \, [^\text{o}\text{C}]$ with the following relation:
$$
V_\text{th} = k T
$$
where $k$ is the Boltzmann constant in $[\text{eV K}^{-1}]$.
Note that the function above is implicit in the sense that the variable $I$ appears in both sides of the equation and thus must be solved numerically for a given $V$.
[^Constantine]: P. G. Constantine, B. Zaharatos, and M. Campanelli, “Discovering an active subspace in a single‐diode solar cell model,” Statistical Analysis, vol. 8, no. 5–6, pp. 264–273, Oct. 2015, doi: 10.1002/sam.11281.
The test function models the maximum power of a single-diode solar cell. It was introduced by Constantine et al. (2015)[^Constantine] to demonstrate the active subspace method for (input) dimension reduction and sensitivity analysis.
The model consists of the following set of equations:
$$ \begin{aligned} \mathcal{M}(\boldsymbol{x}; \boldsymbol{p}) & = P{\text{max}}(\boldsymbol{x}) = \max{I, V} I V \ I(\boldsymbol{x}) & = I_L - IS \left( \exp{\left(\frac{V + I R{S}}{Ns n V{\text{th}}} \right)} - 1 \right) - \frac{V + I R_S}{R_P} \ IL(\boldsymbol{x}) & = I{SC} + IS \left( \exp{\left(\frac{I{SC} R_{S}}{Ns n V{\text{th}}} \right)} - 1 \right) - \frac{I_{SC} R_S}{R_P}, \ \end{aligned} $$
where $\boldsymbol{x} = { I_{SC}, I_S, n, R_S, R_P }$ is the vector of input variables modeled as independent random variables whose marginals are specified in the table below.
Furthermore, $\boldsymbol{p} = { Ns, V{\text{th}} }$ is the vector of parameters, namely, the number of cells connected in series and the thermal voltage. In the paper, the number of cells is set to $1$ while the thermal voltage is set to $T = 25 \, [^\text{o}\text{C}]$ with the following relation:
$$ V_\text{th} = k T $$
where $k$ is the Boltzmann constant in $[\text{eV K}^{-1}]$.
Note that the function above is implicit in the sense that the variable $I$ appears in both sides of the equation and thus must be solved numerically for a given $V$.
[^Constantine]: P. G. Constantine, B. Zaharatos, and M. Campanelli, “Discovering an active subspace in a single‐diode solar cell model,” Statistical Analysis, vol. 8, no. 5–6, pp. 264–273, Oct. 2015, doi: 10.1002/sam.11281.