PVsyst 6, arguably the most popular PV modeling software in the industry, includes a temperature coefficient for the ideality factor in the single-diode model for PV modules. Multi-temperature and multi-irradiance I-V data from IEC 61853-1 tests on silicon modules, however, reveal no significant temperature dependence of the ideality factor. The series resistance shows a stronger temperature dependence, but this behavior is currently not captured by the PVsyst 6 module performance model. It is shown that, once the temperature dependence of the series resistance is included, the single-diode model can reproduce I-V curve point values Voc, Imp, and Vmp with greater accuracy.

1. Single-Diode Model with
Rs Temperature Dependence
44th IEEE Photovoltaic Specialists Conference, 2017
Kyumin Lee, PhD
Chief Engineer, CFV Solar Test Laboratory
kyumin.lee@cfvsolar.com

2. PVsyst 6 Single-Diode Model
Single-Diode Model with Rs Temperature Dependence2
• 𝑰𝒑𝒉 = Τ𝑮 𝑮 𝟎 ∙ 𝑰𝒑𝒉 𝟎 ∙ 𝟏 + 𝝁 𝑰𝒔𝒄 ∙ 𝑻 𝑪 − 𝑻 𝟎
Iph proportional to irradiance and linear with temperature
• 𝑰𝒐 = 𝑰𝒐 𝟎 ∙ Τ𝑻 𝑪 𝑻 𝟎
𝟑
∙ 𝒆𝒙𝒑 Τ𝒒 ∙ 𝑬 𝒈 𝜸 ∙ 𝒌 ∙
𝟏
𝑻 𝟎
−
𝟏
𝑻 𝑪
De Soto model
• 𝜸 = 𝜸 𝟎 ∙ 𝟏 + 𝝁 𝜸 ∙ 𝑻 𝑪 − 𝑻 𝟎
Ideality factor gamma varies linearly with temperature; Optional but used often
• 𝑹𝒔𝒉 = 𝑹𝒔𝒉 𝟎 + (𝑹𝒔𝒉 𝑮=𝟎 − 𝑹𝒔𝒉 𝟎) ∙ 𝒆𝒙𝒑[−𝑹𝒔𝒉𝑬𝒙𝒑 ∙ Τ𝑮 𝑮 𝟎 ]
Rsh varies exponentially with temperature
• 𝑹𝒔 = 𝑹𝒔 𝟎 Series resistance constant, irrespective of irradiance and temperature

3. Observations on PVsyst 6 Model
1. Ideality Factor γ Dependent on Temperature?
• No temperature dependence reported for Si devices
2. Shunt Resistance Rsh Exponentially Dependent on
Irradiance?
• No clear consensus; De Soto: Rsh = Rsh,ref * (Gref/G),
multiple reports of negative temperature coefficient
• Calculating Rsh from I-V curve is already challenging.
• All in all, not important for modern Si modules (STC Rsh high enough)
3. Series Resistance Rs Independent of Temperature?
• Multiple reports and physical arguments for T dependence.
• Rs temperature coefficient included in IEC 60891 corr. proc. 2
A lot of the reported work are on cells, or on PV modules with
old technology.
Single-Diode Model with Rs Temperature Dependence3

4. Verifying G/T Dependence of γ and Rs
Modules: 72-cell Poly 315W (η 15.9%), 60-cell PERC 295W (18.0%),
72-cell n-PERT 375W (19.1%)
1. Use IEC 61853-1 test data
to derive γ and Rs at each T.
• Ideality factor γ from regression
on Voc(G) – Voc(Go) versus ln G
(Sandia, IEC 60904-5, “Suns-Voc”)
• Rs with Swanson method
(IEC 60891-compatible)
2. Optimize PVsyst model parameters.
3. Optimize parameters for a revised model
(“Rs TempCo”; Linearly T-dependent Rs and constant γ).
4. Compare residuals for the two models.
Single-Diode Model with Rs Temperature Dependence4

5. Ideality Factor Dependent on T?
• Voc values were analyzed to
derive γ at 15, 25, 50, and 75 °C.
• Data shows no clear
T dependence of γ,
for all three Si module types.
• 72-Cell Poly 315W: -0.041 %/°C
• 60-Cell PERC 295W: +0.006 %/°C
• 72-Cell n-PERT 375W: +0.015 %/°C
Single-Diode Model with Rs Temperature Dependence5
72-Cell Poly 315W
60-Cell PERC 295W
72-Cell n-PERT 375W
Slope = γ

6. Series Resistance Independent of T?
• Swanson method was applied to
IV curves to derive Rs at
15, 25, 50, and 75°C.
• Data shows clear T dependence
of Rs, for all 3 Si module types.
• 72-Cell Poly 315W: +0.405 %/°C
• 60-Cell PERC 295W: +0.356 %/°C
• 72-Cell n-PERT 375W: +0.164 %/°C
Single-Diode Model with Rs Temperature Dependence6
72-Cell Poly 315W
60-Cell PERC 295W
72-Cell n-PERT 375W
Slope = Rs

7. Physical Reasons for T Dependence of Rs
• Metals have positive temp. coeff. of resistivity (TCR).
• Silver (cell gridlines): +0.38%/°C, Copper (ribbon wires): +0.39%/°C
• About 80% of Rs of a PV module
is due to Ag and Cu.
• Calculated for a module with
72 poly-Si Al BSF cells, 4BB;
Total Rs = 0.310 Ω
• TCR of Si varies depending on
doping type, doping level, and
impurities present. It can even
be negative.
• Since Si contribution is only ~20%, it is reasonable to assume
a metal-like T dependence for the Rs of a PV module.
Single-Diode Model with Rs Temperature Dependence7

8. Proposal: “Rs TempCo” Model
Single-Diode Model with Rs Temperature Dependence8
• 𝑰𝒑𝒉 = Τ𝑮 𝑮 𝟎 ∙ 𝑰𝒑𝒉 𝟎 ∙ 𝟏 + 𝝁 𝑰𝒔𝒄 ∙ 𝑻 𝑪 − 𝑻 𝟎
Identical to PVsyst
• 𝑰𝒐 = 𝑰𝒐 𝟎 ∙ Τ𝑻 𝑪 𝑻 𝟎
𝟑
∙ 𝒆𝒙𝒑 Τ𝒒 ∙ 𝑬 𝒈 𝜸 ∙ 𝒌 ∙
𝟏
𝑻 𝟎
−
𝟏
𝑻 𝑪
Identical to PVsyst
• 𝜸 = 𝜸 𝟎 Ideality factor constant, irrespective of irradiance and temperature
• 𝑹𝒔𝒉 = 𝑹𝒔𝒉 𝟎 + (𝑹𝒔𝒉 𝑮=𝟎 − 𝑹𝒔𝒉 𝟎) ∙ 𝒆𝒙𝒑[−𝑹𝒔𝒉𝑬𝒙𝒑 ∙ Τ𝑮 𝑮 𝟎 ]
Identical to PVsyst
• 𝑹𝒔 = 𝑹𝒔 𝟎 ∙ 𝟏 + 𝝁 𝑹𝒔 ∙ 𝑻 𝑪 − 𝑻 𝟎
Series resistance varies linearly with temperature

9. PANOpt® Model Optimization
• Iterative solver was seeded with values from regression.
• Solver ran to get the lowest RMS error of Pmp over 61853-1.
Single-Diode Model with Rs Temperature Dependence9
Module γ μγ [10-3/C] Rs [Ω] μRs [mΩ/C] RMSE [W]
72-Cell
Poly
315W
Regression 1.162 -0.473 0.312 +1.27
PVsyst 1.000 -0.334 0.341 N/A 0.338
Rs TempCo 1.098 N/A 0.306 +1.39 0.287
60-Cell
PERC
295W
Regression 1.102 +0.069 0.274 +0.97
PVsyst 1.000 -0.421 0.283 N/A 0.232
Rs TempCo 1.112 N/A 0.247 +1.53 0.191
72-Cell
n-PERT
375W
Regression 1.072 +0.155 0.310 +0.51
PVsyst 1.000 -0.326 0.315 N/A 0.256
Rs TempCo 1.091 N/A 0.268 +1.52 0.326

10. Residuals – 72-Cell Poly 315W
PVsyst
Model:
Residuals
show clear
correlation
to G and T.
Rs TempCo
Model:
Residuals
are smaller
and less
correlated
to G and T.
Single-Diode Model with Rs Temperature Dependence10

11. Residuals – 60-Cell PERC 295W
PVsyst
Model:
Residuals
show clear
correlation
to G and T.
Rs TempCo
Model:
Residuals
are smaller
and less
correlated
to G and T.
Single-Diode Model with Rs Temperature Dependence11

12. Residuals – 72-Cell n-PERT 375W
PVsyst
Model:
Residuals
show clear
correlation
to G and T.
Rs TempCo
Model:
Residuals
are smaller
and less
correlated
to G and T.
Single-Diode Model with Rs Temperature Dependence12

13. Final Notes
• Temperature dependence of material resistivity is a well-
known phenomenon.
• There is no clear evidence of T dependence of ideality factor.
• Proposed “Rs TempCo” model, using constant ideality factor
and T-dependent Rs, predicts Voc, Imp, and Vmp with greater
accuracy than PVsyst 6 model.
• “Rs TempCo” model does not necessary improve Pmp
accuracy. After optimization, RMSE of Pmp is very small for
both models (~0.1% of STC Pmp).
• Should the single-diode model be physical or empirical?
• T dependence of Rs has more physical basis than that of γ.
• Even the “Rs TempCo” model can lose physical significance if we
enforce nameplate Isc, Voc, Imp, and Vmp values instead of the
measured ones.
Single-Diode Model with Rs Temperature Dependence13