Mermoud one-diode

Method

The module is a current source in parallel with a diode and a shunt resistance, followed by a series resistance in series with the output (1). The photocurrent, the saturation current and the ideality factor at standard conditions are solved from the short-circuit current, the open-circuit voltage and a third point that the One-diode STC point step sets, with the series and shunt resistances from the module file. The photocurrent follows the irradiance and the temperature coefficient of the short-circuit current (2). The shunt resistance rises as the irradiance falls (3). The ideality factor drifts with the temperature (4), and the saturation current follows a temperature law with the bandgap, with a floor of 2.07 × 10⁻¹⁴ A (5). The maximum-power point is found exactly on the curve at every timestep.

The math

1. Circuit
2. Photocurrent
3. Shunt resistance
4. Ideality factor
5. Saturation current

$$I_m = I_{ph} - I_0\left[\exp\!\left(\frac{V_m + I_m R_s}{N_{cs}\,\gamma_d\,k_B T / q}\right) - 1\right] - \frac{V_m + I_m R_s}{R_{sh}(G)} \tag{1}$$ $$I_{ph} = \frac{G}{G_{ref}}\left[I_{ph,ref} + \mu_{I_{sc}}\,(T - T_{ref})\right] \tag{2}$$ $$R_{sh}(G) = R_{sh,b} + \left(R_{sh,0} - R_{sh,b}\right)e^{-c_{sh}\,G/G_{ref}} \tag{3}$$ $$\gamma_d = \gamma_{d,ref} + \mu_{\gamma_d}\,(T - T_{ref}) \tag{4}$$ $$I_0 = \max\!\left(2.07 \times 10^{-14}\ \text{A},\ I_{0,ref}\left(\frac{T}{T_{ref}}\right)^3 \exp\!\left[\frac{E_g}{\gamma_d\,k_B}\left(\frac{1}{T_{ref}} - \frac{1}{T}\right)\right]\right) \tag{5}$$

where

Symbol	Meaning	Unit
$I_m$	the module current	A
$V_m$	the module voltage	V
$I_{ph}$	the photocurrent	A
$I_0$	the diode saturation current	A
$R_s$	the series resistance	Ω
$N_{cs}$	the number of cells in series	—
$\gamma_d$	the diode ideality factor	—
$k_B$	Boltzmann's constant	J/K
$T$	the cell temperature	K
$q$	the elementary charge	C
$R_{sh}(G)$	the shunt resistance at irradiance $G$	Ω
$G$	the effective irradiance	W/m²
$G_{ref}$	the irradiance of standard test conditions, 1000 W/m²	W/m²
$I_{ph,ref}$	the photocurrent at standard test conditions	A
$\mu_{I_{sc}}$	the temperature coefficient of the short-circuit current	A/K
$T_{ref}$	the cell temperature of standard test conditions, 25 °C	K
$R_{sh,0}$	the shunt resistance in the dark, from the module file, else 4 times the shunt resistance at standard conditions	Ω
$R_{sh,b}$	the shunt resistance at high irradiance, set so that $R_{sh}(G_{ref})$ equals the shunt resistance at standard conditions	Ω
$c_{sh}$	the exponent of the law, from the module file, else 5.5	—
$\gamma_{d,ref}$	the ideality factor at standard conditions	—
$\mu_{\gamma_d}$	the temperature coefficient of the ideality factor, from the module file or fitted to the power temperature coefficient	1/K
$I_{0,ref}$	the saturation current at standard conditions	A
$E_g$	the bandgap, 1.121 eV (1.796 × 10⁻¹⁹ J) for every cell technology	J

References

1. De Soto, Klein & Beckman 2006, Improvement and validation of a model for photovoltaic array performance, Solar Energy 80(1) 78–88
2. Mermoud & Lejeune 2010, Performance assessment of a simulation model for PV modules of any available technology, 25th European Photovoltaic Solar Energy Conference
3. PVsyst 8 help, Standard one-diode model

API

module.model = mermoud, or options.module_model = mermoud

Symbols follow the symbol registry.