Parameter Translation

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Summary

Parameter Translation scales the five standard parameters—, , , , and —from reference conditions (typically : 25 °C, 1000 W/m²) to actual operating conditions using physics-based scaling relationships. PlantPredict also scales for use in downstream calculations. All parameters are scaled before solving the single-diode circuit equation. The two additional parameters of the seven-parameter model ($d_i^2/\mu\tau$ and $V_{bi}$) are not scaled, so this procedure applies similarly to both model variants.

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Inputs

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Outputs

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Detailed Description

The general of a PV cell is described by: $$I = I_{ph} - I_0 \left(\exp\!\left(\frac{q(V + IR_s)}{N_c k T_c \gamma}\right) - 1\right) - \frac{V + IR_s}{R_{sh}} - \frac{(d_i^2/\mu\tau) \cdot I_{ph}}{N_c V_{bi} - (V + IR_s)}$$ where $q = 1.602 \times 10^{-19}$ C is the elementary charge and $k = 1.381 \times 10^{-23}$ J/K is the Boltzmann constant. The first five parameters—photocurrent $I_{ph}$, saturation current $I_0$, ideality factor $\gamma$, series resistance $R_s$, and shunt resistance $R_{sh}$—define the 5-parameter model. The last term, governed by the recombination parameter $d_i^2/\mu\tau$ and $V_{bi}$, is an optional extension used by the 7-parameter model to improve accuracy at low irradiance. For the 5-parameter model, $d_i^2/\mu\tau = 0$ and this term vanishes. Module datasheets characterize these parameters at reference conditions (typically 25 °C, 1000 W/m²). Since cell temperature and irradiance vary continuously during operation, each parameter must be scaled from reference to actual conditions before solving the circuit equation. Additionally, a per-module DC equivalent series resistance is added to account for wiring losses. The scaling relationships below are applied at every simulation time step. All temperatures are converted to Kelvin before use in the equations below. Throughout, $\Delta T = T_c - T_{ref}$ denotes the temperature difference from reference conditions.

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Series Resistance

The per-module DC wiring resistance $R_{DC,module}$ is added to the reference value: $$R_s = R_{s,ref} + R_{DC,module}$$

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Shunt Resistance

Shunt resistance increases at low irradiance due to reduced minority carrier concentration. The exponential model below, aligned with PVsyst (Mermoud & Lejeune, 2010), is an empirical fit that interpolates between a finite dark shunt resistance $R_{sh,0}$ at zero irradiance and the reference value at STC: $$R_{sh} = R_{sh,ref} + (R_{sh,0} - R_{sh,ref}) \exp\left(-R_{sh,exp} \frac{G'_{POA,tot,eff}}{G_{ref}}\right)$$ where:

• $R_{sh,ref}$ is the reference shunt resistance, typically defined at STC
• $R_{sh,0}$ is the dark shunt resistance (at $G = 0$ W/m²)
• $R_{sh,exp}$ is the exponential dependency coefficient (default 5.5)

Note that PlantPredict uses $R_{sh,ref}$ directly rather than computing an intermediate base value $R_{sh,base}$. As a result, $R_{sh}$ evaluated at $G'_{POA,tot,eff} = G_{ref}$ does not exactly equal $R_{sh,ref}$—it includes a residual term $(R_{sh,0} - R_{sh,ref}) \cdot e^{-R_{sh,exp}}$. For the default $R_{sh,exp} = 5.5$, this residual is less than 0.5% of $(R_{sh,0} - R_{sh,ref})$.

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Diode Ideality Factor

PlantPredict supports two models for the temperature dependence of the diode ideality factor. The linear model is the standard approach, aligned with PVsyst. The polynomial model provides additional flexibility for technologies where the ideality factor exhibits nonlinear temperature dependence, as observed by Sauer et al. (2015). Linear model: $$\gamma = \gamma_{ref} (1 + \alpha_{\gamma} \Delta T)$$ The coefficient $\alpha_{\gamma}$ is typically derived during parameter extraction from the temperature coefficient of maximum power ($\beta_{P_{mp}}$) reported on the module datasheet. It is chosen so that the fully scaled single-diode model reproduces the correct temperature coefficient of power. Polynomial model: $$\gamma = \gamma_{ref} (1 + a_{\gamma} \Delta T + b_{\gamma} \Delta T^2 + c_{\gamma} \Delta T^3 + d_{\gamma} \Delta T^4)$$ where $a_{\gamma}, b_{\gamma}, c_{\gamma}, d_{\gamma}$ are polynomial coefficients. In practice, these are derived by fitting the single-diode model to measured I-V curves at multiple temperatures, extracting $\gamma$ at each temperature, and then fitting a 4th-degree polynomial to $\gamma$ as a function of $\Delta T$.

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Saturation Current

The saturation current $I_0$ represents the recombination current of charge carriers across the solar cell in the dark. Its temperature dependence follows from the intrinsic carrier concentration $n_i$, where $n_i^2 \propto T^3 \exp(-E_g / kT)$ and $E_g$ is the . The ideality factor $\gamma$ in the exponent accounts for non-ideal recombination: $$I_0 = I_{0,ref} \frac{T_c^3}{T_{ref}^3} \exp\left(\frac{q E_g}{k \gamma} \left(\frac{1}{T_{ref}} - \frac{1}{T_c}\right)\right)$$

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Short-Circuit Current

Short-circuit current scales linearly with irradiance and is corrected for temperature using the coefficient $\alpha_{I_{sc}}$ (typically from the module datasheet, in %/°C; converted to 1/°C before use in the equation below): $$I_{sc} = I_{sc,ref} \frac{G'_{POA,tot,eff}}{G_{ref}} (1 + \alpha_{I_{sc}} \Delta T)$$

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Photocurrent

The photocurrent $I_{ph}$ is obtained by evaluating the I-V equation at $V = 0$ and solving for $I_{ph}$ to ensure that $I(V=0) = I_{sc}$: $$I_{ph} = \frac{I_{sc}\!\left(1 + \frac{R_s}{R_{sh}}\right) + I_0 \left(\exp\!\left(\frac{q \, I_{sc} R_s}{k \, T_c \, N_c \, \gamma}\right) - 1\right)}{1 - \frac{d_i^2/\mu\tau}{N_c V_{bi} - I_{sc} R_s}}$$ For the 5-parameter model, $d_i^2/\mu\tau = 0$ and the relationship simplifies to: $$I_{ph} = I_{sc}\!\left(1 + \frac{R_s}{R_{sh}}\right) + I_0 \left(\exp\!\left(\frac{q \, I_{sc} R_s}{k \, T_c \, N_c \, \gamma}\right) - 1\right)$$

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References

• Mermoud, A., & Lejeune, T. (2010). Performance assessment of a simulation model for PV modules of any available technology. 25th European Photovoltaic Solar Energy Conference, Valencia, Spain.
• Sauer, K. J., Roessler, T., & Hansen, C. W. (2015). Modeling the irradiance and temperature dependence of photovoltaic modules in PVsyst. IEEE Journal of Photovoltaics, 5(1), 152–158. DOI: 10.1109/JPHOTOV.2014.2364402
• De Soto, W., Klein, S. A., & Beckman, W. A. (2006). Improvement and validation of a model for photovoltaic array performance. Solar Energy, 80(1), 78–88. DOI: 10.1016/j.solener.2005.06.010