A national laboratory of the U.S. Department of Energy

Office of Energy Efficiency & Renewable Energy

National Renewable Energy Laboratory 

Innovation for Our Energy Future 

Comparison of Predictive 

Models for Photovoltaic 

Module Performance 

Preprint 

B. Marion 

National Renewable Energy Laboratory 

Presented at the 33rd IEEE Photovoltaic Specialists Conference 

San Diego, California 

May 11–16, 2008 

Conference Paper 

NREL/CP-520-42511 

May 2008 

NREL is operated by Midwest Research Institute ● Battelle Contract No. DE-AC36-99-GO10337 
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COMPARISON OF PREDICTIVE MODELS 

FOR PHOTOVOLTAIC MODULE PERFORMANCE 

B. Marion 

National Renewable Energy Laboratory (NREL) 

1617 Cole Boulevard, Golden CO 80401 

ABSTRACT 

This paper examines three models used to estimate the 

performance of photovoltaic (PV) modules when the 

irradiances and PV cell temperatures are known. The 

results presented here were obtained by comparing 

modeled and measured maximum power (Pm) for PV 

modules that rely on different technologies. The models 

evaluated for estimating Pm are (1) the power temperature 

coefficient model, (2) the PVFORM model, and (3) the 

bilinear interpolation model. These range from simple 

models with few input parameters to more complex 

models with more extensive PV module characterization 

procedures. NREL researchers determined modeling error 

statistics by comparing model estimates with measured 

PV performance data. A modification to the power 

temperature coefficient model was also evaluated that 

provided improved accuracy. 

INTRODUCTION 

Models are a key element for predicting the performance 

of PV systems and for use in standards that assign a PV 

module an energy rating. This paper examines three 

models used to estimate the Pm of PV modules when the 

irradiance and PV cell temperature are known: (1) the 

power temperature coefficient model, (2) the PVFORM 

model, and (3) the bilinear interpolation model. 

For modeling values of Pm, an “effective” plane-of-array 

(POA) irradiance (Ee) and the PV cell temperature (T) are 

used as model inputs. Using Ee essentially removes the 

effects of variations in solar spectrum and reflectance 

losses, and permits the influence of irradiance and 

temperature on model performance for Pm to be more 

easily studied. Eq. 1 is used to determine Ee from T and 

the PV module’s short-circuit current (Isc). The equation 

assumes that Isc is proportional to the irradiance if T and 

the spectral distribution of the irradiance are constant. 

Zero subscripts denote performance at Standard 

Reporting Conditions (SRC), which consists of an 

irradiance of 1,000 W/m2

 with a spectral distribution 

conforming to the air mass (AM) 1.5 spectrum [1] and a 

PV cell temperature of 25°C. 

[ (

where 

Isc = short-circuit current, A 

 E = POA irradiance, W/m2

Α = short-circuit current correction factor 

 for temperature, °C-1

 T = PV cell temperature, °C. 

THE POWER TEMPERATURE COEFFICIENT MODEL 

This model applies a temperature correction to Pm to 

account for departures in cell temperature from those at 

SRC. Pm is assumed to be linear with respect to the 

effective irradiance if the temperature is constant. Eq. 2 

represents this model. Zero subscripts denote 

performance at SRC. 

[ ( 0

0

1 0 P TT

E

E P m e m −⋅γ+⋅⋅= )] , (2) 

where 

γ = maximum power correction factor for 

 temperature, °C-1. 

THE PVFORM MODEL

The PVFORM model is the same as the power 

temperature coefficient model for irradiance levels greater 

than 125 W/m2

, but PVFORM uses a different formulation 

for irradiance levels of less than 125 W/m2

 to account for 

reductions in output observed for crystalline silicon 

modules [2,3]. 

For Ee ≤ 125 W/m2

, 

[ ( 0

0

2

1 0080

0 P TT

E

E. P m e m −⋅γ+⋅⋅ ⋅ = )] . (3) 

THE BILINEAR INTERPOLATION MODEL 

)] 0

0

1 0 I TT

EI E

sc

sc e −⋅α+⋅

⋅ = , (1) 

This model is based on work by Hishikawa and colleagues 

which developed translation equations for interpolating, 

with respect to the irradiance, a current-voltage (I-V) curve 

from two I-V curves at the same PV cell temperature [4]. 

Marion and colleagues expanded this work by developing 

a method in which four I-V curves could be used to 

bilinearly interpolate an I-V curve with respect to both 

irradiance and PV cell temperature [5]. 

1
0 5 10 15 20 25

Voltage (V)

0

1

2

3

4

Current (

A)

1

2 5

7

6

3

4

Fig. 1. Numbering of I-V curves for bilinear interpolation 

model 

Figure 1 illustrates the bilinear interpolation methodology. 

I-V curves 1 through 4 are the reference I-V curves 

measured for combinations of two PV cell temperature 

and two irradiance settings. I-V curves 1 and 2 were 

measured at the same nominal irradiance, as were I-V 

curves 3 and 4. I-V curves 1 and 3 were measured at the 

same nominal PV cell temperature, as were I-V curves 2 

and 4. The term nominal is used in referring to irradiance 

and temperature settings because the method’s equations 

accommodate unintended variations in settings that might 

occur. To perform the bilinear interpolation, I-V curves 5 

and 6 are interpolated with respect to open-circuit voltage 

(Voc) from I-V curves 1 and 2, and 3 and 4, respectively, 

and I-V curve 7 is interpolated with respect to Isc from I-V 

curves 5 and 6. I-V curve 7 is the translated I-V curve for 

the desired conditions of irradiance and PV cell 

temperature. 

Equations 4 and 5 are used to determine values of Isc and 

Voc. Subscripts with a value of one denote performance for 

reference I-V curve 1. 

 [ ( 1

1

1 1 I TT

E

E I sc e sc −⋅α+⋅⋅= )]

)]

 (4) 

and 

[ ] ( ) [ ( )( 1 1 1 1 1 VV ococ ⋅+⋅+⋅−⋅β+⋅= e E/ElnbTmTT . (5) 

where 

β = open circuit voltage correction factor 

 for temperature, °C-1

 m = open circuit voltage correction factor 

 for irradiance, °C-1

 b = open circuit voltage correction factor 

 for irradiance, dimensionless. 

VALIDATION DATA 

NREL’s Performance and Energy Rating Testbed (PERT) 

supplied the data used to evaluate model performance. 

These data were collected from April 1, 2005 through 

March 31, 2006, except for the CdTe PV module, on which 

data were available only from 11 November, 2005 through 

March 31, 2006. The data include I-V curves measured at 

15-minute intervals with coincident measurements of POA 

irradiance and PV module back-surface temperatures. To 

reduce data uncertainty, pyranometer calibration data 

were used to correct irradiance measurements for cosine 

response. In addition, cell temperatures were estimated by 

adding 2.5°C per 1,000 W/m2

 irradiance to the PV module 

back-surface temperatures. 

Data were screened to eliminate data recorded under the 

following conditions (which would cause inaccurate 

measurements): (1) days with recorded snow on PV 

modules and pyranometers; (2) periods of the day during 

and after rainfall as indicated by accumulated rainfall 

measurements at the nearby Solar Radiation Research 

Laboratory (SRRL); and (3) unstable irradiance resulting 

from the presence of cloud movement in the vicinity of the 

sun. This last condition could be detected when the next 

PV module’s I-V curve measurement showed a change in 

irradiance of more than 10 W/m2

. The PERT measures I-V 

curves in sequence, with each PV module’s I-V curve 

requiring 3 to 4 seconds. Missing data and the datascreening criteria reduced the available data by about 

30%. 

PV MODULE CHARACTERIZATION 

To obtain the information required by the models, their 

performance was characterized using PERT I-V curve 

data measured on March 14, 2006, a day with clear skies 

and a fairly large temperature differential between morning 

and afternoon. These conditions afforded the opportunity 

to select four reference I-V curves that met the criteria for 

the bilinear interpolation method (two irradiance levels and 

two temperatures for an irradiance level). For the four I-V 

curves selected, the corresponding measurement times, 

nominal irradiances, and PV module temperatures were 

as follows: 7:15 a.m., 220 W/m2

 and 0°C; 10:15 a.m., 

1040 W/m2

 and 35°C; 2:00 p.m., 1040 W/m2 and 50°C; 

5:00 p.m., 220 W/m2

 and 20°C. 

Using the I-V curve data and their coincident 

measurements of irradiance and PV module temperature, 

the procedures outlined by Marion and coworkers were 

used to determine the PV module coefficients α, β, m, and 

b [5]. Performance at SRC was determined using these 

module coefficients and the bilinear interpolation model. 

For the power temperature coefficient model and the 

PVFORM model, the two I-V curves measured at 10:15 

a.m. and 2:00 p.m. were used to determine γ values. 

Tables 1 and 2 list the PV modules by their technology 

and serial number (S/N), their performance at SRC, and 

their derived temperature and irradiance coefficients. 

2
Table 1. PV Module Performance at SRC 

Technology 

Module 

S/N 

Isc

(A) 

Voc

(V) 

Im

(A) 

Vm

(V) 

Fill-Factor 

(%) 

Pm

(W) 

Multi-crystal Si 4978 2.831 21.57 2.496 16.80 68.7 41.94 

Multi-crystal Si 9366 2.358 20.95 2.095 16.78 71.1 35.16 

Single-crystal Si 585-2164 4.580 21.38 4.317 17.22 75.9 74.34 

Single-crystal Si 270-2301 4.113 20.87 3.759 16.37 71.7 61.54 

Single-crystal Si 0442 4.104 21.24 3.675 16.50 69.6 60.65 

a-Si/x-Si HIT 21226001 3.517 66.16 3.258 52.48 73.5 170.97 

a-Si/a-Si/a-Si:Ge 1736 2.511 21.27 1.981 14.86 55.1 29.43 

CdTe 14407 0.880 87.16 0.713 59.55 55.3 42.44 

Table 2. PV Module Temperature and Irradiance Coefficients 

Technology 

Module 

S/N 

α

(°C-1) 

β

(°C-1) 

m 

(°C-1) b γ(°C-1) 

Multi-crystal Si 4978 3.88E-04 -3.70E-03 2.40E-04 6.02E-02 -4.66E-03 

Multi-crystal Si 9366 3.65E-04 -3.48E-03 4.40E-04 5.23E-02 -4.28E-03 

Single-crystal Si 585-2164 6.50E-04 -3.56E-03 2.43E-04 3.38E-02 -4.58E-03 

Single-crystal Si 270-2301 5.00E-04 -3.69E-03 2.66E-04 3.36E-02 -4.70E-03 

Single-crystal Si 0442 6.00E-04 -3.96E-03 8.35E-05 4.98E-02 -5.20E-03 

a-Si/x-Si HIT 21226001 1.60E-04 -2.76E-03 6.91E-05 4.85E-02 -2.78E-03 

a-Si/a-Si/a-Si:Ge 1736 5.95E-04 -4.09E-03 4.96E-04 4.44E-02 -2.05E-03 

CdTe 14407 5.97E-04 -3.10E-03 2.38E-04 4.09E-02 -1.09E-03 

CALCULATING MODEL ERROR STATISTICS 

Model estimates and measured data were compared 

using root-mean-square-error (RMSE), mean-bias-error 

(MBE), and mean-absolute-error (MAE) statistics. RMSE 

provides information on the variation of the modeled 

values from the measured values, MBE provides the 

average deviation of the modeled values from the 

measured values, and MAE provides the average absolute 

deviation of the modeled values from the measured 

values. RMSE and MAE are always positive, whereas 

MBE can be either positive or negative. RMSE, MBE, and 

MAE are defined by Eqs. 6, 7, and 8, which calculate the 

errors as a percentage of the average measured value. 

Although the word “error” is used, “difference” would be 

more accurate because the true values are not known and 

because the differences between measured and modeled 

values are being reported. 

( ) ⎥

⎦

⎤ ⎢

⎣

⎡ ÷ ⎥

⎦

⎤ ⎢

⎣

⎡ ×= ∑ = − ∑ =

n

i i

/

n

i ii x

n

xy

n

RMSE % 1

21

1

1 2 1 100 (6) 

( ) ⎥

⎦

⎤ ⎢

⎣

⎡ ÷⎥

⎦

⎤ ⎢

⎣

⎡ ×= ∑ = − ∑ =

n

i i ni ii x

n

xy

n

MBE % 1 1

1 1 100 (7) 

⎥

⎦

⎤ ⎢

⎣

⎡ ÷⎥

⎦

⎤ ⎢

⎣

⎡ ×= ∑ = − ∑ =

n

i i ni ii x

n

xy

n

MAE % 1 1

1 1 100 , (8) 

where 

yi = the ith modeled value 

xi = the ith measured value 

n = the number of measured or modeled 

 values. 

MODELING ERRORS FOR PM

Table 3 gives modeling error statistics for Pm by model 

and PV module. The Number of Data Points column lists, 

by PV model, the number of measured I-V curves that 

were compared with model estimates. There are fewer I-V 

curves for the CdTe PV module because the period of 

data collection was shorter. The Ave column shows the 

average Pm of the measured data by PV module. Overall, 

the bilinear interpolation model performed best with the 

lowest errors. All models appear to adequately account for 

PV module temperature. The main differences were seen 

in the model estimates at lower irradiances, as shown 

graphically in Fig. 2. The figure uses a scatterplot to 

illustrate modeled versus measured Pm for the multicrystal Si PV module S/N 4978 when the power 

temperature coefficient model is used. This PV module is 

used for illustration purposes because it exhibited the 

largest departure from linearity with respect to irradiance. 

As a consequence, it also had the largest MBE. The 

scatterplot includes a diagonal line with a slope of one for 

ease in comparing modeled and measured values. If 

modeled and measured values are in perfect agreement, 

the data points reside on the diagonal line. Data points 

above the line indicate model overestimates and data 

points below the line indicate model underestimates. 

For the a-Si/a-Si/a-Si:Ge PV module, the data show about 

a 5% reversible degradation-recovery cycle from late 

winter (degradation) to late summer (annealing). Figure 3 

shows this as increased modeling error for September 

2005 when compared to March 2006. PV module 

performance was characterized using data measured at 

the beginning of the recovery cycle. For September, 

relative PV module performance had improved by about 

5%, and the models underestimated the performance by

3
Table 3. MBE, MAE, and RMSE Statistics for Pm by Model and PV Module 

Power Temperature 

Coefficient Model PVFORM Model Bilinear Interpolation Model 

Technology 

Module 

S/N 

Number 

of Data 

Points. 

Ave. 

(W) MBE 

(%) 

MAE 

(%) 

RMSE 

(%) 

MBE 

(%) 

MAE 

(%) 

RMSE 

(%) 

MBE 

(%) 

MAE 

(%) 

RMSE 

(%) 

Multi-crystal Si 4978 11816 19.50 3.9 4.3 5.5 2.8 3.4 4.8 -0.3 0.7 1.6 

Multi-crystal Si 9366 11830 16.66 3.2 3.4 4.0 2.1 3.0 3.6 0.1 0.7 1.2 

Single-crystal Si 585-2164 11844 36.12 0.0 1.5 1.8 -1.1 1.7 2.1 -0.4 1.4 1.7 

Single-crystal Si 270-2301 11870 29.86 0.4 1.6 1.9 -0.7 1.6 1.9 0.2 1.5 1.8 

Single-crystal Si 0442 12468 28.92 0.9 2.2 2.8 -0.2 2.1 2.7 -0.8 1.2 2.0 

a-Si/x-Si HIT 21226001 11878 84.72 1.1 1.4 1.7 0.1 1.7 2.1 0.2 0.7 1.0 

a-Si/a-Si/a-Si:Ge 1736 11878 15.31 -2.6 3.2 4.5 -3.5 3.9 4.9 -2.0 3.3 4.4 

CdTe 14407 3415 25.78 -1.6 2.3 3.0 -2.2 2.8 3.4 -0.9 1.9 2.5 

about the same amount. Accounting for this seasonal 

change in PV module efficiency would improve model 

performance when applied to a-Si technologies. 

0 20 40

Measured Power (W)

60

0

20

40

60

Modeled Power (W)

Slope equals 1

Power Temperature Coefficient Model

Fig. 2. Modeled versus measured Pm for the multi-crystal 

Si PV module S/N 4978 when using the power 

temperature coefficient model. 

0 10 20 30 40 5

Measured Power (W)

0

0

10

20

30

40

50

Modeled Power (W)

March 2006

September 2005

Slope equals 1

Fig. 3. Modeled versus measured Pm for the a-Si/a-Si/aSi:Ge PV module S/N 1736 showing the effect of 

annealing for September 2005 and March 2006 for the 

power temperature coefficient model. 

AN IMPROVED MODEL 

The power temperature coefficient model yielded 

surprisingly good results for some of the PV modules, 

notably the single-crystal silicon and the a-Si/x-Si HIT PV 

modules. Because these PV modules maintained their 

efficiency at low irradiance levels, the model’s assumption 

that Pm was linear with respect to the effective irradiance if 

the temperature is constant was valid. 

If, however, the PV module does not maintain its efficiency 

at lower irradiance levels, the model will overestimate 

performance. This is illustrated in Figure 4 for the multicrystal Si PV module S/N 4978, with the model error 

expressed as a percentage of the measured value and 

plotted as a function of irradiance to demonstrate how 

irradiance level affects model accuracy. 

Figure 4 should be used with caution for ascertaining the 

overall impact on energy production because a large 

percentage error at a low irradiance may have less impact 

on energy production than a smaller percentage error at a 

high irradiance. Figure 5 presents the results in units of 

the error in modeled power divided by Pm at SRC. This 

results in better information on the error in overall energy 

0 200 400 600 800 1000 1200

Irradiance (W/m2)

-20

0

20

40

60

80

Modeled Power % Error

Power Temperature Coefficient Model

Fig. 4. Percentage error in modeled Pm for the multi-crystal 

Si PV module S/N 4978 when using the power 

temperature coefficient model. 

4
0 200 400 600 800 1000 1200

Irradiance (W/m2)

-0.10

-0.05

0.00

0.05

0.10

(Modeled Pm - Measured Pm) / Pm0

k

Error = k [(Eo-Ee)/(Ee-200)]

Error = k [1-(1-Ee/200)4]

Fig. 5. Normalized error in modeled Pm for the multi-crystal 

Si PV module S/N 4978 when using the power 

temperature coefficient model, and graphical 

representation of functions to correct for non-linearity of 

Pm with respect to irradiance. 

production, and it also offers insight about how a 

correction factor might be applied to minimize the model 

error at low irradiance. This is also graphically represented 

in Figure 5. 

To correct for the modeling error as a function of 

irradiance, a model was developed that considers the 

error in two irradiance regions: ≤ 200 W/m2

 and > 200 

W/m2

. For irradiance levels ≤ 200 W/m2, the correction is a 

function of an irradiance term raised to the 4th power. For 

irradiance levels > 200 W/m2

, the correction is linear with 

respect to irradiance. The correction functions have a 

maximum value of k when Ee = 200 W/m2

, and a value of 

zero for Ee = 0 or 1,000 W/m2

. 

Based on experimental observations, the correction 

functions yield results that are closer to measured 

performance for PV modules whose Pm is nonlinear with 

respect to the effective irradiance. If a PV module’s Pm is 

linear with respect to the effective irradiance, the value of 

k is zero and—in effect—no correction is applied. 

The irradiance correction factor, k, is determined with Eq. 

9: 

m0

Lm Lmeas

P

)T,E(P)T,E(P k − = , (9) 

where 

EL = effective low irradiance, ~ 200 W/m2 

Pm(EL,T) = Pm from Eq. 2 for EL and T conditions 

Pmeas(EL,T) = measured Pm for EL and T conditions. 

Equations 10 and 11 represent the improved power 

temperature coefficient model with an irradiance 

nonlinearity correction: 

For Ee > 200 W/m2

 [ ] ( ) ⎥

⎦

⎤ ⎢

⎣

⎡

−

− ⋅−−⋅γ+⋅⋅= 200

1

0

0

0

0 0 E

EE kTT

E

E PP e e mm . (10) 

For Ee ≤ 200 W/m2

 [ ] ( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞ ⎜

⎝

⎛

−−⋅−−⋅γ+⋅⋅=

4

0

0 200

1 11 0

e e mm

E kTT

E

E PP . (11) 

A significant development for this approach is on the 

horizon. Beginning in January 2009, manufacturers will be 

furnishing data corresponding to Pmeas(EL,T) and based on 

IEC 61215 and 61646, to the California Energy 

Commission so that the manufacturers’ PV modules can 

be listed by the Commission as eligible components for 

incentive programs [6]. Consequently, this will allow the 

new method to be implemented with manufacturersupplied information without the need for further 

characterization measurements. Some manufacturer data 

sheets already contain this information as “reduction in 

efficiency under low irradiance (200 W/m2

).” For example, 

a reduction in efficiency value by 10% would correspond 

to k = 0.10 x 200/1000 = 0.020. 

To evaluate Eqs. 10 and 11 for the PV modules being 

assessed, k values for each PV module were determined 

using Equation 9 and the low irradiance I-V curve data 

recorded on March 14, 2006 (and previously used to 

determine temperature and irradiance coefficients for the 

bilinear interpolation model). The effective irradiance was 

about 220 W/m2

, sufficiently close to 200 W/m2 that it 

would not impact the value of k because the change in 

error value is relatively constant near 200 W/m2

 (see Fig. 

5). This is another attribute of the improved model. Table 4 

lists the two values of k for each PV module. Both morning 

and afternoon data yielded approximately the same k 

values. Two of the PV modules have negative k values, 

meaning that module efficiency at low irradiance levels 

increases relative to their efficiencies at SRC. 

Table 4. Values of k from March 14, 2006 Data 

Technology Module S/N k 

(T~0°C) 

k 

(T~20°C) 

Multi-crystal Si 4978 0.046 0.041 

Multi-crystal Si 9366 0.029 0.029 

Single-crystal Si 585-2164 0.003 0.003 

Single-crystal Si 270-2301 0.002 0.002 

Single-crystal Si 0442 0.022 0.018 

a-Si/x-Si HIT 21226001 0.012 0.009 

a-Si/a-Si/a-Si:Ge 1736 -0.002 -0.005 

CdTe 14407 -0.016 -0.012 

For evaluating this new model, Eqs. 10 and 11 were used 

to model Pm for each of the PV modules. Then, the results 

were compared to measured Pm values using the 

methodology described previously. Values of k from the 

last column of Table 4 were used because the 

temperatures for these values were closest to the 

5
temperature for SRC. Table 5 presents the results, 

showing that the modeling errors are comparable to those 

presented in Table 3 for the bilinear interpolation model. 

Table 5. MBE, MAE, and RMSE Statistics for Pm for the 

Power Temperature Coefficient Model with Correction for 

Irradiance Nonlinearity.

Technology Module S/N MBE 

(%) 

MAE 

(%) 

RMSE 

(%) 

Multi-crystal Si 4978 -0.4 0.8 1.7 

Multi-crystal Si 9366 0.2 0.6 1.1 

Single-crystal Si 585-2164 -0.3 1.4 1.7 

Single-crystal Si 270-2301 0.2 1.5 1.9 

Single-crystal Si 0442 -0.9 1.4 2.2 

a-Si/x-Si HIT 21226001 0.2 0.7 1.1 

a-Si/a-Si/a-Si:Ge 1736 -2.0 3.3 4.4 

CdTe 14407 -0.8 1.8 2.4 

SUMMARY 

Error statistics were determined for PV models that 

estimated the performance of PV modules of various 

technologies operated at NREL from April 1, 2005 through 

March 31, 2006. For modeling Pm using an effective 

irradiance and cell temperature, error statistics were 

determined for the following models (listed in order of 

overall performance): (1) the bilinear interpolation model, 

(2) the PVFORM model, and (3) the power temperature 

coefficient model. Results for the bilinear interpolation 

model were consistent with other evaluations by NREL 

and other organizations [7, 8, 9]. Using characterization 

data at a low irradiance (~200 W/m2

) in the bilinear 

interpolation model helped it to account for the nonlinearity 

of Pm with irradiance level. 

This work also evaluated a model that addressed the 

inability of the power temperature coefficient model to 

account for nonlinearity of Pm with respect to irradiance 

that is seen in some PV modules. Though equivalent in 

accuracy to the bilinear interpolation model, this model 

proved to be simpler and require less module-specific 

characterization data. Module characteristic data are 

reduced to three parameters: Pm0

, γ, and Pmeas(EL,T). PV 

manufacturers currently provide the first two parameters, 

and manufacturers will make the last parameter available 

in the near future. Consequently, this model has 

advantages in terms of its simplicity, accuracy, and readily 

available input parameters. 

In this work, the analysis was restricted to studying the 

accuracy of the models when using inputs of measured 

PV temperatures and of irradiances calculated from 

measured Isc. This is analogous to measuring the 

irradiance with a calibrated reference cell that matched the 

optical and spectral characteristics of the PV module. 

Modeling errors would have been increased if (1) the 

irradiance had been measured with a pyranometer or 

modeled; (2) the PV cell temperature had been modeled; 

or (3) Pm at SRC had been determined independently of 

the PERT measurements or by another laboratory 

(because of uncertainties associated with the 

reproducibility of measurements among laboratories). 

ACKNOWLEDGEMENTS 

The author acknowledges Dave Trudell and Joe del Cueto 

at NREL for maintaining and operating the PERT systems. 

Peer review comments and suggestions were provided by 

NREL staff Keith Emery and Joe del Cueto. This work was 

supported by the U.S. Department of Energy under 

Contract No. DE-AC36-99GO10337 with the National 

Renewable Energy Laboratory. 

REFERENCES 

[1] ASTM E 892-92. Standard Tables for Terrestrial 

Solar Spectral Irradiances at AM 1.5 for a 37° Tilted 

Surface, West Conshohocken, PA: ASTM International, 

1992. 

[2] D. Menicucci and J. Fernandez. User’s Manual for 

PVFORM: A Photovoltaic System Simulation Program for 

Stand-Alone and Grid-Interactive Applications, SAND85-

0376, Albuquerque, NM: Sandia National Laboratories, 

1988. 

[3] D. Menicucci. “Photovoltaic Array Performance 

Simulation Models,” Solar Cells 18, 1986, pp. 383-392. 

[4] Y. Hishikawa, Y. Imura, and T. Oshiro. “IrradianceDependence And Translation of the I-V Characteristics of 

Crystalline Silicon Solar Cells.” Proceedings of the 28th

IEEE PV Specialist Conference, Anchorage, AK, 2000, 

pp.1464–1467. 

[5] B. Marion, S. Rummel, and A. Anderberg. “CurrentVoltage Curve Translation by Bilinear Interpolation,” 

Progress in Photovoltaics: Research and Applications 

12, 2004, pp. 593-607. 

[6] California Energy Commission. Draft Guidelines for 

California’s Solar Electric Incentive Programs Pursuant to 

Senate Bill 1, CEC-300-2007-012-D, Sacramento, CA: 

California Energy Commission, September, 2007. 

[7] B. Marion, J. del Cueto, and B. Sekulic. “Modeling 

Current-Voltage Curves Using Bilinear Interpolation,” 

Proceedings of the World Renewable Energy Congress 

VIII (WREC 2004), Denver, CO, 2004. 

[8] Y. Tsuno, Y. Hishikawa, and K. Kurokawa. 

“Temperature and Irradiance Dependence of the I-V 

Curves of Various Kinds of Solar Cells,” Proceedings of 

the 15th International Photovoltaic Science & Engineering 

Conference (PVSEC-15), Shanghai, China, 2005. 

[9] J. Tsutsui, Y. Sato, and K. Kurokawa. “Modeling the 

Performance of Several Photovoltaic Modules,” 

Presented at the IEEE 4th World Conference on 

Photovoltaic Solar Energy Conversion, Waikoloa, HI, 

2006.

6
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Comparison of Predictive Models for Photovoltaic Module 

Performance: Preprint 

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This paper examines three models used to estimate the performance of photovoltaic (PV) modules when the 

irradiances and PV cell temperatures are known. The results presented here were obtained by comparing modeled 

and measured maximum power (Pm) for PV modules that rely on different technologies. The models evaluated for 

estimating Pm are (1) the power temperature coefficient model, (2) the PVFORM model, and (3) the bilinear 

interpolation model. These range from simple models with few input parameters to more complex models with more 

extensive PV module characterization procedures. NREL researchers determined modeling error statistics by 

comparing model estimates with measured PV performance data. A modification to the power temperature coefficient 

model was also evaluated that provided improved accuracy. 

15. SUBJECT TERMS

PV; module; measured maximum power; power temperature coefficient; bilinear interpolation; PVFORM model; 

irradiance; short-circuit current; 

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