Skip to main content

​
Summary

The 7-parameter model extends the 5-parameter single-diode model with a voltage-dependent current term, following the equivalent circuit proposed by Merten et al. (1998). The additional term captures carrier recombination at low irradiance. This mechanism is most relevant for thin-film technologies but can improve accuracy for any module where shunt resistance alone underestimates low-light losses. The model adds two parameters— (VbiV_{bi}) and recombination parameter (di2/μτd_i^2/\mu\tau)—to the five standard single-diode parameters. All five base parameters are scaled by Parameter Translation in the same way as for the 5-parameter model; VbiV_{bi} and di2/μτd_i^2/\mu\tau are not scaled.

​
Inputs

NameSymbolUnitsDescription
PhotocurrentIphI_{ph}ALight-generated current
Saturation CurrentI0I_0ADiode reverse saturation current
Series ResistanceRsR_sΩSeries resistance (includes module internal resistance and DC wiring resistance)
Shunt ResistanceRshR_{sh}ΩShunt resistance of module
Diode Ideality Factorγ\gamma—Diode ideality factor
Number of CellsNsN_s—Cells in series within module
Cell TemperatureTcT_c°COperating cell temperature
Built-in VoltageVbiV_{bi}VBuilt-in voltage per cell
Recombination Parameterdi2/μτd_i^2/\mu\tauVLumped recombination parameter combining recombination layer thickness and effective carrier mobility-lifetime product

​
Outputs

NameSymbolUnitsDescription
Max Power VoltageVmpV_{mp}VVoltage at maximum power point
Max Power CurrentImpI_{mp}ACurrent at maximum power point
Max PowerPmpP_{mp}WVmp×ImpV_{mp} \times I_{mp}
Open-Circuit VoltageVocV_{oc}VVoltage at open-circuit (I=0I = 0)

​
Detailed Description

​
Circuit Equation

The 7-parameter adds a recombination current to the 5-parameter circuit equation: I=Iph−I0(exp⁡ ⁣(q(V+IRs)NskTcγ)−1)−V+IRsRsh−(di2/μτ)⋅IphNsVbi−(V+IRs)I = I_{ph} - I_0 \left(\exp\!\left(\frac{q(V + IR_s)}{N_s k T_c \gamma}\right) - 1\right) - \frac{V + IR_s}{R_{sh}} - \frac{(d_i^2/\mu\tau) \cdot I_{ph}}{N_s V_{bi} - (V + IR_s)} As in the 5-parameter model, Vth=NsγkTc/qV_{th} = N_s \gamma k T_c / q is the modified thermal voltage and Vint=V+IRsV_{int} = V + IR_s is the internal voltage: I(Vint)=Iph−I0(eVint/Vth−1)−VintRsh−(di2/μτ)⋅IphNsVbi−VintI(V_{int}) = I_{ph} - I_0 (e^{V_{int}/V_{th}} - 1) - \frac{V_{int}}{R_{sh}} - \frac{(d_i^2/\mu\tau) \cdot I_{ph}}{N_s V_{bi} - V_{int}} The last term represents a voltage-dependent current loss that increases as VintV_{int} approaches NsVbiN_s V_{bi} and is proportional to . It was originally proposed by Merten et al. (1998) to model recombination in amorphous silicon p-i-n junctions, but is used more broadly as an empirical correction that improves low-irradiance accuracy beyond what shunt resistance alone provides.

​
Maximum Power Point

PlantPredict uses the same internal-voltage approach as the 5-parameter model, expressing I(Vint)I(V_{int}) and V(Vint)V(V_{int}) as explicit functions and solving dP/dVint=0dP/dV_{int} = 0 via Newton-Raphson iteration. Once the optimal internal voltage Vint,mpV_{int,mp} has converged: Imp=I(Vint,mp)I_{mp} = I(V_{int,mp}) Vmp=Vint,mp−ImpRsV_{mp} = V_{int,mp} - I_{mp} R_s Pmp=Vmp×ImpP_{mp} = V_{mp} \times I_{mp}

​
VV Given II

The recombination term’s pole at NsVbiN_s V_{bi} prevents reformulation into the Lambert W canonical form used by the 5-parameter model. PlantPredict instead solves for VintV_{int} directly via Newton-Raphson iteration on the circuit equation and recovers the terminal voltage as V=Vint−IRsV = V_{int} - IR_s. The VocV_{oc} is obtained as the special case with I=0I = 0.

​
II Given VV

When the terminal voltage is fixed—for instance, when set by the inverter at an operating point away from MPP (e.g., clipping)—the Lambert W reformulation is again not applicable. PlantPredict solves for VintV_{int} via Newton-Raphson with I=(Vint−V)/RsI = (V_{int} - V)/R_s substituted into the circuit equation, then computes II from the converged VintV_{int}.

​
References

  • Merten, J., Asensi, J. M., Voz, C., Shah, A. V., Platz, R., & Andreu, J. (1998). Improved equivalent circuit and analytical model for amorphous silicon solar cells and modules. IEEE Transactions on Electron Devices, 45(2), 423–429.