IEC 60891

Uncertainly analysis for temperature and irradiance correction of measured I-V characteristics in accordance with IEC 60891

The standard IEC 60891 [1] offers 4 different procedures for temperature and irradiance correction of measured I-V characteristics. Most commonly, the two algebraic correction procedures (procedures 1 and 2) are used in test laboratories. The determination of the PV module specific I-V correction parameters is described in sections 5, 6 and 7 of the standard IEC 60891.

For algebraic procedures, the translation equations for current and voltage are expressed as a function of a set of I-V correction parameters, the PV module temperature (T) and the in-plane irradiance (G). The translation equations of an I-V data point are expressed as a function of the test conditions (index 1), the target conditions (index 2) and the PV module I-V correction parameters.

[math]\displaystyle{ I_{2}=f(T_{1},G_{1},T_{2},G_{2},IV \text{ correction parameters}) }[/math]

[math]\displaystyle{ V_{2}=f(T_{1},G_{1},T_{2},G_{2},IV \text{ correction parameters, }I_{2}) }[/math]

The mean square error of a corrected I-V data point (I₂,V₂) on the current-voltage characteristic can be calculated according to the error propagation law on the mean errors of the directly measured quantities ΔI₁, ΔV₁, ΔT₁, ΔG₁and the uncertainties of the module specific I-V correction parameters.

[math]\displaystyle{ \Delta I_{2}(T_{1},G_{1},...) = \pm \sqrt{ \left ( \frac { \delta I_{2} } { \delta T_{1} } \Delta T_{1} \right )^2 + \left ( \frac { \delta I_{2} } { \delta G_{} } \Delta G_{1} \right )^2 +... } }[/math]

[math]\displaystyle{ \Delta V_{2}(T_{1},G_{1},...) = \pm \sqrt{ \left ( \frac { \delta V_{2} } { \delta T_{1} } \Delta T_{1} \right )^2 + \left ( \frac { \delta V_{2} } { \delta G_{} } \Delta G_{} \right )^2 +... } }[/math]

The relative translation errors are given by ΔI₂ / I₂ and ΔV₂ / V₂ .

With this information the relative error of translated maximum output power [math]\displaystyle{ P_{MAX}=I_{MP} \cdot V_{MP} }[/math] is

[math]\displaystyle{ \frac {\Delta P_{MAX}} {P_{MAX}} = \pm \sqrt{ \left ( \frac {\Delta I_{MP}} {I_{MP}} \right )^2 + \left ( \frac {\Delta V_{MP}} {V_{MP}} \right )^2 } }[/math]

Correction procedure 1

[math]\displaystyle{ I_{2}=I_{1}+I_{SC1}\times \left ( \frac{G_{2}}{G_{1}}-1 \right ) +\alpha \times (T_{2}-T_{1}) }[/math] (1)

[math]\displaystyle{ V_{2}=V_{1}-R_{S} \times (I_{2}-I_{1}) -\kappa \times I_{2} \times (T_{2}-T_{1}) + \beta \times (T_{2}-T_{1}) }[/math] (2)

where:

• I₁, V₁, T₁ and G₁ are the measured current, voltage, module temperature and irradiance, respectively;
• I₂ and V₂ are the corresponding pair of current and voltage values of the to target conditions corrected IV curve;
• T₂ and G₂ are the target module temperature and irradiance respectively;
• I_SC1 is the measured short-circuit current that may result from interpolation or extrapolation of I-V data points in the short-circuit range;
• R_S is the internal series resistance;
• α and β are respectively the absolute short-circuit current and open-circuit voltage temperature coefficients at the target irradiance for correction;
• κ is the curve correction factor.

The uncertainty of I-V translation is composed of the partial derivatives, which are listed in Table 6.

Uncertainty contributions	Translation uncertainty ΔI2	Translation uncertainty ΔV2	Remarks
Irradiance measurement	[math]\displaystyle{ \frac { \delta I_{2} } { \delta G_{1} } \Delta G_{1} }[/math]	[math]\displaystyle{ \frac { \delta V_{2} } { \delta G_{1} } \Delta G_{1} }[/math]	ΔG1 is the measurement uncertainty resulting from section 3.1
Temperature measurement	[math]\displaystyle{ \frac { \delta I_{2} } { \delta T_{1} } \Delta T_{1} }[/math]	[math]\displaystyle{ \frac { \delta V_{2} } { \delta T_{1} } \Delta T_{1} }[/math]	ΔT1 is the measurement uncertainty resulting from section 3.2
Isc temperature coefficient	[math]\displaystyle{ \frac { \delta I_{2} } { \delta \alpha } \Delta \alpha }[/math]		The uncertainty of is highly dependent on the spectral irradiance of the light source, with which α has been measured. A reasonable estimate is [math]\displaystyle{ \Delta \alpha = 0.5 \cdot \alpha }[/math]
Current measurement	[math]\displaystyle{ \frac { \delta I_{2} } { \delta I_{1} } \Delta I_{1} }[/math]		is the measurement uncertainty of short circuit current, which considers all uncertainties except uncertainty related to G1 (data acquisition, potential interpolation or extrapolation, etc.)
Voc temperature coefficient		[math]\displaystyle{ \frac { \delta V_{2} } { \delta \beta } \Delta \beta }[/math]	A reasonable estimate is [math]\displaystyle{ \Delta \beta = 0.1 \cdot \beta }[/math]
Translated current		[math]\displaystyle{ \frac { \delta V_{2} } { \delta I_{2} } \Delta I_{2} }[/math]	ΔI2 to be calculated first
Internal series resistance		[math]\displaystyle{ \frac { \delta V_{2} } { \delta R_{s} } \Delta R_{s} }[/math]	A reasonable estimate is [math]\displaystyle{ \Delta R_{s} = 0.5m\Omega \cdot N_{S}/N_{P} }[/math] Where NS: No. of serially connected cells NP: No. of parallel connected cell strings
Curve correction factor		[math]\displaystyle{ \frac { \delta V_{2} } { \delta \kappa } \Delta \kappa }[/math]	A reasonable estimate is [math]\displaystyle{ \Delta \kappa = 0.5 \cdot \kappa }[/math]

Table 6: Calculation of IEC 60891 I-V translation uncertainty (procedure 1)

Note:

The translated short circuit current and open circuit voltage do not directly result from the translation formulas. Both must be interpolated or extrapolated from the translated I-V curve. Corresponding uncertainties must be additionally considered in ΔI_SC,2 and ΔV_OC,2.

Correction procedure 2

The equations are as follows:

[math]\displaystyle{ I_{2} = \frac {G_{2}} {G_{1}} \times I_{1} \times \frac { (1 + \alpha_{rel} \times (T_{2} - 25^\circ C)) } { (1 + \alpha_{rel} \times (T_{1} - 25^\circ C)) } }[/math] (1)

[math]\displaystyle{ V_{2} = V_{1} - R_{S1}' \times (I_{2}-I_{1}) - \kappa' \times I_{2} \times (T_{2}-T_{1}) + V_{OC,STC} \times \left \{ \beta_{rel} \times [ f(G_{2}) \times (T_{2} - 25^\circ C) - f(G_{1}) \times (T_{1} - 25^\circ C) ] \frac {1} {f(G_{2})} - \frac {1} {f(G_{1})} \right \} }[/math] (2)

[math]\displaystyle{ R_{S1}' = R_{S}' + \kappa' \times (T_{1} - 25^\circ C) }[/math] (3)

[math]\displaystyle{ V_{OC,STC} = \frac {V_{OC1} \times f(G_{1}) } {1 + \beta_{rel} \times (T_{1} - 25^\circ C) \times f^{2}(G_{1}) } }[/math] (4)

[math]\displaystyle{ f(G) = \frac {V_{OC,STC} } {V_{OC}(G) } = B_{2} \times ln^{2} \frac {\text {1000 }W/m^{2} } {G } = B_{1} \times ln^{} \frac {\text {1000 }W/m^{2} } {G } +1 }[/math] (5)

where:

• V_OC,STC is the open-circuit voltage at STC. It can be calculated from Eq.(4);
• α _rel is the relative short-circuit temperature coefficient;
• β_rel is the relative open-circuit voltage temperature coefficient;
• R′_S is the internal series resistance determined at 25 °C;
• R′_S1 is the internal series resistance at measured temperature T₁, it can be calculated from Eq.(3);
• κ' is the temperature coefficient of the internal series resistance R′_S;
• B₁ is the irradiance linear correction factor for V_OC that is related to the diode thermal voltage of the p-n junction and the number of cells N_S serially connected in the DUT;
• B₂ is the irradiance correction factor for V_OC, which accounts for non-linearity of V_OC with irradiance.

The uncertainty of I-V translation is composed of the partial derivatives which are listed in Table 7.

Uncertainty contributions	Translation uncertainty ΔI2	Translation uncertainty ΔV2	Translation uncertainty ΔVOC,2	Remarks
Irradiance measurement	[math]\displaystyle{ \frac { \delta I_{2} } { \delta G_{1} } \Delta G_{1} }[/math]	[math]\displaystyle{ \frac { \delta V_{2} } { \delta G_{1} } \Delta G_{1} }[/math]	[math]\displaystyle{ \frac { \delta V_{2} } { \delta G_{1} } \Delta G_{1} }[/math]	ΔG1 is the measurement uncertainty resulting from section 3.1
Temperature measurement	[math]\displaystyle{ \frac { \delta I_{2} } { \delta T_{1} } \Delta T_{1} }[/math]	[math]\displaystyle{ \frac { \delta V_{2} } { \delta T_{1} } \Delta T_{1} }[/math]	[math]\displaystyle{ \frac { \delta V_{2} } { \delta T_{1} } \Delta T_{1} }[/math]	ΔT1 is the measurement uncertainty resulting from section 3.2
Isc temperature coefficient	[math]\displaystyle{ \frac { \delta I_{2} } { \delta \alpha } \Delta \alpha_{rel} }[/math]			The uncertainty of  is highly dependent on the spectral irradiance of the light source, with which α has been measured. A reasonable estimate is [math]\displaystyle{ \Delta \alpha _{rel} = 0.5 \cdot \alpha _{rel} }[/math]
Voc temperature coefficient		[math]\displaystyle{ \frac { \delta V_{2} } { \delta \beta } \Delta \beta_{rel} }[/math]	[math]\displaystyle{ \frac { \delta V_{2} } { \delta \beta } \Delta \beta_{rel} }[/math]	A reasonable estimate is [math]\displaystyle{ \Delta \beta _{rel} = 0.1 \cdot \beta _{rel} }[/math]
Translated current		[math]\displaystyle{ \frac { \delta V_{2} } { \delta I_{2} } \Delta I_{2} }[/math]		ΔI2 to be calculated first
Internal series resistance		[math]\displaystyle{ \frac { \delta V_{2} } { \delta R_{s} } \Delta R_{s}' }[/math]		The internal series resistance is subject to production tolerance. If not measured, reasonable estimates are [math]\displaystyle{ R_{s}' = 5 m \Omega \cdot N_{s} / N_{p} }[/math] [math]\displaystyle{ \Delta R_{s}' = 5 m \Omega \cdot N_{s} / N_{p} }[/math] where NS : No. of serially connected cells NP : No. of parallel connected cell strings
Curve correction factor		[math]\displaystyle{ \frac { \delta V_{2} } { \delta \kappa } \Delta \kappa' }[/math]		A reasonable estimate is [math]\displaystyle{ \Delta \kappa ' = 0.5 \cdot \Delta \kappa ' }[/math]
Irradiance linear correction factor for VOC		[math]\displaystyle{ \frac { \delta V_{2} } { \delta B_{1} } \Delta B_{1} }[/math]	[math]\displaystyle{ \frac { \delta V_{2} } { \delta B_{1} } \Delta B_{1} }[/math]	A reasonable estimate is [math]\displaystyle{ \Delta B_{1} = 0.1 \cdot B_{1} }[/math]
Irradiance non-linear correction factor for VOC		[math]\displaystyle{ \frac { \delta V_{2} } { \delta B_{2} } \Delta B_{2} }[/math]	[math]\displaystyle{ \frac { \delta V_{2} } { \delta B_{2} } \Delta B_{2} }[/math]	A reasonable estimate is [math]\displaystyle{ \Delta B_{2} = 0.5 \cdot B_{2} }[/math]

Table 7: Calculation of IEC 60891 I-V translation uncertainty (procedure 2)

Note: The translated short circuit current does not directly result from the translation formulas. It must be interpolated or extrapolated from the translated I-V curve. The corresponding uncertainty must be additionally considered in ΔI_SC,2.

For procedure 1 the translated V_oc does not result directly from the formula and must be interpolated or extrapolated. The associated uncertainty depends on how far the I-V curve was measured in the negative current range (beyond V_OC). In case of extrapolation, a quadratic extrapolation is recommended where the I-V data points should cover a range of at least 2 V.

References

[1] IEC 60891:2021 “Photovoltaic devices - Procedures for temperature and irradiance corrections to measured I-V characteristics”