1. The document proposes modifying the single-diode PV module model in PVsyst to include temperature dependence for the series resistance (Rs) parameter based on analysis of temperature-dependent I-V data. 2. The analysis found no clear temperature dependence for the ideality factor (γ) but did find Rs increased linearly with temperature for different PV module types, consistent with known temperature dependence of metal resistivity. 3. Modeling with a constant γ and linear temperature-dependent Rs produced more accurate predictions of voltage, current, and power versus the standard PVsyst model, especially their dependence on irradiance and temperature.

1. Single-Diode Model with
Rs Temperature Dependence
Last Update: 2017-05-08
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 = 𝑮 𝑮 𝟎 ∙ 𝑰𝒑𝒉 𝟎 ∙ 𝟏 + 𝝁 𝑰𝒔𝒄 ∙ 𝑻 𝑪 − 𝑻 𝟎
Iph proportional to irradiance and linear with temperature
• 𝑰𝒐 = 𝑰𝒐 𝟎 ∙ 𝑻 𝑪 𝑻 𝟎
𝟑 ∙ 𝒆𝒙𝒑 𝒒 ∙ 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 ∙ 𝑮 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 = 𝑮 𝑮 𝟎 ∙ 𝑰𝒑𝒉 𝟎 ∙ 𝟏 + 𝝁 𝑰𝒔𝒄 ∙ 𝑻 𝑪 − 𝑻 𝟎
Identical to PVsyst
• 𝑰𝒐 = 𝑰𝒐 𝟎 ∙ 𝑻 𝑪 𝑻 𝟎
𝟑 ∙ 𝒆𝒙𝒑 𝒒 ∙ 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 ∙ 𝑮 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