Photovoltaic Array Performance Model
PHOTOVOLTAIC ARRAY PERFORMANCE MODEL
D. L. King, W.
E. Boyson, J. A. Kratochvil
Sandia National Laboratories
Albuquerque, New Mexico 87185-0752
SAND2004-3535
Unlimited Release
Printed August 2004
Photovoltaic Array Performance Model
David L. King, William E. Boyson, Jay A. Kratochvil
Photovoltaic System R&D Department
Sandia National Laboratories
P. O. Box 5800
Albuquerque, New Mexico 87185-0752
Abstract
This document summarizes the equations and applications associated with the photovoltaic array performance model developed at Sandia National Laboratories over the last twelve years. Electrical, thermal, and optical characteristics for photovoltaic modules are included in the model, and the model is designed to use hourly solar resource and meteorological data. The versatility and accuracy of the model has been validated for flat-plate modules (all technologies) and for concentrator modules, as well as for large arrays of modules. Applications include system design and sizing, ‘translation’ of field performance measurements to standard reporting conditions, system performance optimization, and real-time comparison of measured versus expected system performance.
ACKNOWLEDGEMENTS
The long evolution of our array performance model has greatly benefited from valuable
interactions with talented people from a large number of organizations. The authors would like to acknowledge several colleagues from the following organizations: AstroPower (Jim Rand, Michael Johnston, Howard Wenger, John Cummings), ASU/PTL (Bob Hammond, Mani
Tamizhmani), BP Solar (John Wohlgemuth, Steve Ransom), Endecon (Chuck Whitaker, Tim Townsend, Jeff Newmiller, Bill Brooks), EPV (Alan Delahoy), Entech (Mark O’Neil), First
Solar (Geoff Rich), FSEC (Gobind Atmaram, Leighton Demetrius), Kyocera Solar (Steve Allen), Maui Solar (Michael Pelosi), NIST (Hunter Fanney), NREL (Ben Kroposki, Bill Marion, Keith Emery, Carl Osterwald, Steve Rummel), Origin Energy (Pierre Verlinden, Andy Blakers),
Pacific Solar (Paul Basore), PBS Specialties (Pete Eckert), PowerLight (Dan Shugar, Adrianne Kimber, Lori Mitchell), PVI (Bill Bottenberg), RWE Schott Solar (Miles Russell, Ron
Gonsiorawski), Shell Solar (Terry Jester, Alex Mikonowicz, Paul Norum), SolarOne (Moneer Azzam), SunSet Technologies (Jerry Anderson), SWTDI (Andy Rosenthal, John Wiles), and Sandia (Michael Quintana, John Stevens, Barry Hansen, James Gee).
Annual Distribution of Array V mp vs. Power 25-kW Array, ASE-300-DG/50 Modules, Prescott, AZ 0510********Array Maximum Power (kW)
H o u r l y A v g . V m p (V )
C u m u l a t i v e P m p
D i s t r i b u t i o n
CONTENTS
INTRODUCTION 6 PERFORMANCE EQUATIONS FOR PHOTOVOLTAIC MODULES 6 Basic Equations 7 Module Parameter Definitions 8 Irradiance Dependent Parameters 9
Resource
Solar
10
Parameters
to
Related
(Reference)
Reporting
Conditions 14 Standard
Parameters
at
Temperature Dependent Parameters 15
Model) 16
(Thermal
Operating
Module
Temperature
19
ARRAYS
PERFORMANCE
EQUATIONS
FOR
Array Performance Example 20
Production 23
Energy
Grid-Connected
System
Optimization 25 Off-Grid
System
‘TRANSLATING’ ARRAY MEASUREMENTS TO STANDARD CONDITIONS 25 Translation Equations 26
Voc
Module-String
Measurements 26 of
Analysis
and
Voltage 28
Current
Array
Analysis
of
Operating
DETERMINATION OF EFFECTIVE IRRADIANCE (E E) DURING TESTING 29 Detailed Laboratory Approach 30
Module 30 Direct
Reference
Using
Measurement
Simplified Approach Using a Single Solar Irradiance Sensors 31 Using a Predetermined Array Short-Circuit Current, I sco 32 DETERMINATION OF CELL TEMPERATURE (T C) DURING TESTING 33 MODULE DATABASE 34
PERFORMANCE
MODEL 36 SANDIA’S
OF
HISTORY
CONCLUSIONS 37 REFERENCES 38
INTRODUCTION
This document provides a detailed description of the photovoltaic module and array performance model developed at Sandia National Laboratories over the last twelve years. The performance model can be used in several distinctly different ways. It can be used to design (size) a photovoltaic array for a given application based on expected power and/or energy production on an hourly, monthly, or annual basis [1]. It can be used to determine an array power ‘rating’ by ‘translating’ measured parameters to performance at a standard reference condition. It can also be used to monitor the actual versus predicted array performance over the life of the photovoltaic system, and in doing so help diagnose problems with array performance.
The performance model is empirically based; however, it achieves its versatility and accuracy from the fact that individual equations used in the model are derived from individual solar cell characteristics. The versatility and accuracy of the model has been demonstrated for flat-plate modules (all technologies) and for concentrator modules, as well as for large arrays of modules. Electrical, thermal, solar spectral, and optical effects for photovoltaic modules are all included in the model [2, 3]. The performance modeling approach has been well validated during the last seven years through extensive outdoor module testing, and through inter-comparison studies with other laboratories and testing organizations [4, 5, 6, 7, 8]. Recently, the performance model has also demonstrated its value during the experimental performance optimization of off-grid photovoltaic systems [9, 10].
In an attempt to make the performance model widely applicable for the photovoltaic industry, Sandia conducts detailed outdoor performance tests on commercially available modules, and a database of the associated module performance parameters is maintained on the Sandia website (https://www.wendangku.net/doc/282761132.html,/pv). These module parameters can be used directly in the performance model described in this report. The module database is now widely used by a variety of module manufacturers and system integrators during system design and field testing activities. The combination of performance model and module database has also been incorporated in commercially available system design software [11]. In addition, it is now being considered for incorporation in other building and system energy modeling programs, including DOE-2 [12], Energy-10 [13], and the DOE-sponsored PV system analysis model (PV SunVisor) that is now being developed at NREL.
PERFORMANCE EQUATIONS FOR PHOTOVOLTAIC MODULES
The objective of any testing and modeling effort is typically to quantify and then to replicate the measured phenomenon of interest. Testing and modeling photovoltaic module performance in the outdoor environment is complicated by the influences of a variety of interactive factors related to the environment and solar cell physics. In order to effectively design, implement, and monitor the performance of photovoltaic systems, a performance model must be able to separate and quantify the influence of all significant factors. This testing and modeling challenge has been a goal of our research effort for several years.
The wasp-shaped scatter plot in Figure 1 illustrates the complexity of the modeling challenge using data recorded for a recent vintage 165-W p multi-crystalline silicon module over a five day period in January 2002 during both clear sky and cloudy/overcast conditions. The vertical spread in the P mp values is primarily caused by changes in the solar irradiance level, with lesser influences from solar spectrum, module temperature, and solar cell electrical properties. The horizontal spread in the associated V mp values is primarily caused by module temperature, with lesser influences from solar irradiance and solar cell electrical properties. Our performance model effectively separates these influences so that the chaotic behavior shown in Figure 1 can be modeled with well-behaved relationships, as will be demonstrated in subsequent charts.
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303132333435363738
Maximum Power Voltage, V mp (V)M a x i m u m P o w e r , P m p (W )
Figure 1. Scatter plot of over 3300 performance measurements recorded on five different days in January in Albuquerque with both clear sky and cloudy/overcast operating conditions for a 165-W p mc-Si module.
Basic Equations
The following equations define the model used by the Solar Technologies Department at Sandia for analyzing and modeling the performance of photovoltaic modules. The equations describe the electrical performance for individual photovoltaic modules, and can be scaled for any series or parallel combination of modules in an array. The same equations apply equally well for
individual cells, for individual modules, for large arrays of modules, and for both flat-plate and concentrator modules.
The form of the model given by Equations (1) through (10) is used when calculating the expected power and energy produced by a module, assuming that predetermined module
performance coefficients and solar resource information are available. The solar resource and weather data required by the model can be obtained from tabulated databases or from direct measurements. The three classic points on a module current-voltage (I-V) curve, short-circuit current, open-circuit voltage, and the maximum-power point, are given by the first four
equations. Figure 2 illustrates these three points, along with two additional points that better define the shape of the curve.
I sc = I sco?f1(AM a)?{(E b?f2(AOI)+f d?E diff) / E o}?{1+αIsc?(T c-T o)} (1)
I mp = I mpo?{C0?E e + C1?E e2}?{1 + αImp?(T c-T o)} (2)
V oc = V oco + N s?δ(T c)?ln(E e) + βVoc(E e)?(T c-T o) (3)
(4)
V mp = V mpo + C2?N s?δ(T c)?ln(E e) + C3?N s?{δ(T c)?ln(E e)}2 + βVmp(E e)?(T c-T o)
P mp = I mp?V mp (5) FF = P mp / (I sc?V oc) (6) where:
E e = I sc / [I sco?{1+αIsc?(T c-T o)}] (7)
q (8)
/
δ(T c) = n?k?(T c+273.15)
The two additional points on the I-V curve are defined by Equations (9) and (10). The fourth
point (I x) is defined at a voltage equal to one-half of the open-circuit voltage, and the fifth (I xx) at
a voltage midway between V mp and V oc. The five points provided by the performance model provide the basic shape of the I-V curve and can be used to regenerate a close approximation to
the entire I-V curve in cases where an operating voltage other than the maximum-power-voltage
is required. For example, in battery charging applications, the system’s operating voltage may
be forced by the battery’s state-of-charge to a value other than V mp.
I x = I xo?{ C4?E e + C5?E e2}?{1 + (αIsc)?(T c-T o)} (9)
I xx = I xxo?{ C6?E e + C7?E e2}?{1 + (αImp)?(T c-T o)} (10)
The following six sections of this document discuss all parameters and coefficients used in the equations above that define the performance model. These sections include discussions and definitions of parameters associated with basic electrical characteristics, irradiance dependence, solar resource, standard reporting conditions, temperature dependence, and module operating temperature.
Module Parameter Definitions
I sc = Short-circuit current (A)
I mp = Current at the maximum-power point (A)
I x = Current at module V = 0.5?V oc, defines 4th point on I-V curve for modeling curve shape
I xx = Current at module V = 0.5?(V oc +V mp), defines 5th point on I-V curve for modeling curve
shape
V oc = Open-circuit voltage (V)
V mp = Voltage at maximum-power point (V)
P mp = Power at maximum-power point (W)
FF = Fill Factor (dimensionless)
N s = Number of cells in series in a module’s cell-string
N p = Number of cell-strings in parallel in module
k = Boltzmann’s constant, 1.38066E-23 (J/K)
q = Elementary charge, 1.60218E-19 (coulomb)
T c = Cell temperature inside module (°C)
T o = Reference cell temperature, typically 25°C
E o = Reference solar irradiance, typically 1000 W/m 2
δ(T c ) = ‘Thermal voltage’ per cell at temperature T c . For diode factor of unity (n=1) and a cell temperature of 25oC, the thermal voltage is about 26 mV per cell.
Module Voltage (V)
M o d u l e C u r r e n t (A )
Figure 2. Illustration of a module I-V curve showing the five points on the curve that are provided by the Sandia performance model.
Irradiance Dependent Parameters
The following module performance parameters relate the module’s voltage and current, and as a result the shape of the I-V curve (fill factor), to the solar irradiance level.
Figure 3 illustrates how the measured values for module V mp and V oc may vary as a function of the effective irradiance. In this example, the measured values previously shown in Figure 1 were first translated to a common temperature (50oC) in order to remove temperature dependence. Then the coefficients (n, C 2, C 3) were obtained using regression analyses based on Equations (3) and (4). The coefficients were in turn used in our performance model to calculate voltage versus irradiance behavior at different operating temperatures. The validity of this modeling approach can be appreciated when it is recognized that the 3300 measured data points illustrated were
recorded during both clear and cloudy conditions on five different days with solar irradiance from 80 to 1200 W/m2 and module temperature from 6 to 45 oC.
Figure 4 illustrates how the measured values for module current (I sc, I mp, I x, I xx) may vary as a function of the effective irradiance. Similar to the voltage analysis, the measured current values were translated to a common temperature to remove temperature dependence. The performance coefficients (C0, C1, C4, C5, C6, C7) associated with I mp, I x, and I xx were then determined using regression analyses based on Equations (2), (9), and (10). Our formulation of the performance model uses the complexity associated with Equation (1) to account for any ‘non-linear’ behavior associated with I sc. As a result, the plot of I sc versus the ‘effective irradiance’ variable is always linear. The relationships for the other three current values can be nonlinear (parabolic) in order to closely match the I-V curve shape over a wide irradiance range. The formulation also takes advantage of the ‘known’ condition at an effective irradiance of zero, i.e. the currents are zero, thus helping make the model robust even at low irradiance conditions. The definitions for coefficients are as follows:
E e = The ‘effective’ solar irradiance as previously defined by Equation (7). This value
describes the fraction of the total solar irradiance incident on the module to which the cells inside actually respond. When tabulated solar resource data are used in predicting module performance, Equation (7) is used directly. When direct measurements of solar resource variables are used, then alternative procedures can be used for determining the effective irradiance, as discussed later in this document.
C0, C1 = Empirically determined coefficients relating I mp to effective irradiance, E e. C0+C1 = 1, (dimensionless)
C2, C3 = Empirically determined coefficients relating V mp to effective irradiance (C2 is
dimensionless, and C3 has units of 1/V)
C4, C5 = Empirically determined coefficients relating the current (I x), to effective irradiance,
E e. C4+C5 = 1, (dimensionless)
C6, C7 = Empirically determined coefficients relating the current (I xx) to effective irradiance,
E e. C6+C7 = 1, (dimensionless)
n = Empirically determined ‘diode factor’ associated with individual cells in the module, with a value typically near unity, (dimensionless). It is determined using measurements of V oc translated to a common temperature and plotted versus the natural logarithm of effective irradiance. This relationship is typically linear over a wide range of irradiance (~0.1 to 1.4 suns).
Parameters Related to Solar Resource
For system design or sizing purposes, the solar irradiance variables required by the performance model are typically obtained from a database or meteorological model providing estimates of hourly-average values for solar resource and weather data [14, 15]. These solar irradiance data can be manipulated using different methods in order to calculate the expected solar irradiance incident on the surface of a photovoltaic module positioned in an orientation that depends on the system design and application [16, 17]. On the other hand, for field testing or for long-term
performance monitoring, the solar irradiance in the plane of the module is often a measured value and should be used directly in the performance model.
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25
30
35
40
45
50
00.20.40.60.81 1.2 1.4
Effective Irradiance, E e (suns)V o l t a g e (V )
Figure 3. Over 3300 measurements recorded on five different days with both clear sky and cloudy/overcast operating conditions for 165-W p mc-Si module. Measured values for V oc and V mp were translated to a common temperature, 50°C. Regression analyses provided coefficients used in the performance model used to predicted curves at different operating conditions.
1
2
3
45
6
7
00.20.40.60.81 1.2 1.4Effective Irradiance, E e (suns)C u r r e n t (A
)
Figure 4. Over 3300 measurements recorded on five different days with both clear sky and cloudy/overcast operating conditions for 165-W p mc-Si module. Measured values for currents were translated to a common temperature, 50°C, prior to regression analysis.
The empirical functions f 1(AM a ) and f 2(AOI) quantify the influence on module short-circuit current of variation in the solar spectrum and the optical losses due to solar angle-of-incidence. These functions are determined by a module testing laboratory using explicit outdoor test procedures [2, 8]. The intent of these two functions is to account for systematic effects that occur on a recurrent basis during the predominantly clear conditions when the majority of solar energy is collected. For example, Figure 5 illustrates how the solar spectral distribution varies as the day progresses from morning toward noon, resulting in a systematic influence on the normalized short-circuit current of a typical Si cell. For crystalline silicon modules, the normalized I sc is typically several percent higher at high air mass conditions than it is at solar noon. The effects of intermittent clouds, smoke, dust, and other meteorological occurrences can for all practical purposes be considered random influences that average out on a weekly, monthly, or annual basis. For modules from the same manufacturer, these two empirical functions can often be considered ‘generic’, as long as the cell type and module superstrate
material (e.g., glass) are the same. Figures 6 and 7 illustrate typical examples for the empirically determined functions.
It can be seen in Figure 6 that the influence of the changing solar spectrum is relatively small for air mass values between 1 and 2. In the context of annual energy production, it should also be recognized that over 90% of the solar energy available over an entire year occurs at air mass values less than 3. So, the spectral influence illustrated at air mass values higher than 3 is of somewhat academic importance for the system designer. As documented elsewhere [1], the cumulative effect of the solar spectral influence on annual energy production is typically quite small, less than 3%. Nonetheless, using our modeling approach, it is straightforward to include the systematic influence of solar spectral variation.
0.0
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0.6
0.81.01.21.41.6
1.8
200400600800100012001400160018002000220024002600
Wavelength (nm)I r r a d i a n c e (W /m 2/n m )
Figure 5. Measured solar spectral irradiance on a clear day in Davis, CA, at different air mass conditions during the day. The normalized spectral response of a typical silicon solar cell is superimposed for comparison.
0.6
0.7
0.80.91.01.1
0.5 1.5 2.5 3.5 4.5 5.5 6.5
Absolute Air Mass, AM a R e l a t i v e R e s p o n s e , f 1(A M a )
Figure 6. Typical empirical relationship illustrating the influence of solar spectral variation on module short-circuit current, relative to the AM a =1.5 reference condition. Results were measured at Sandia National Laboratories for a variety of commercial modules.
0.2
0.3
0.4
0.5
0.60.70.80.91.0
1.1
0102030405060708090Angle-of-Incidence, AOI (deg)R e l a t i v e R e s p o n s e , f 2(A O I )
Figure 7. Typical empirical relationship illustrating the influence of solar angle-of-incidence in reducing a module’s short-circuit-current. Results were measured at Sandia National
Laboratories for four different module manufacturers. The effect is dominated by the reflectance characteristics of the glass surface.
Figure 7 shows that the influence of optical (reflectance) losses for flat-plate modules is typically negligible until the solar angle-of-incidence is greater than about 55 degrees. This loss is in
addition to the typical ‘cosine’ loss for a module surface that is not oriented perpendicular to the path of sunlight. The cumulative effect (loss) over the year should be considered for different
system designs and module orientations. For modules that accurately track the sun, there is no optical loss. In the case of a vertically oriented flat-plate module in the south wall of a building, the annual energy loss due to optical loss is about 5% [1].
Our performance model is also directly applicable to concentrator photovoltaic modules. In this case, the empirical functions, f1(AM a) and f2(AOI), take on somewhat greater roles. The effects of solar spectral influence, variation in optical efficiency over the day, module misalignment, and non-linear behavior of I sc versus irradiance can all be adequately accounted for in f1(AM a). As previously discussed, the intent of these empirically-determined relationships is to account for the bulk of the effect of known systematic influences, with the assumption that other uncontrollable factors result in random effects that average out over the year. For concentrator modules, the term angle-of-incidence can be considered synonymous with ‘tracking error.’ Therefore, using predetermined coefficients, the f2(AOI) function can be used to quantify the effect of tracking error on concentrator module performance. The definitions for parameters are as follows:
E b = E dni cos(AOI), beam component of solar irradiance incident on the module surface,
(W/m2)
E diff = Diffuse component of solar irradiance incident on the module surface, (W/m2)
f d = Fraction of diffuse irradiance used by module, typically assumed to be 1 for flat-plate
modules. For point-focus concentrator modules, a value of zero is typically assumed, and for low-concentration modules a value between zero and 1 can be determined.
E e = “Effective” irradiance to which the PV cells in the module respond, (dimensionless, or
“suns”)
E o = Reference solar irradiance, typically 1000 W/m2, with ASTM standard spectrum.
AM a = Absolute air mass, (dimensionless). This value is calculated from sun elevation angle and site altitude, and it provides a relative measure of the path length the sun must travel through the atmosphere, AM a=1 at sea level when the sun is directly overhead.
AOI = Solar angle-of-incidence, (degrees). AOI is the angle between a line perpendicular (normal) to the module surface and the beam component of sunlight.
T c = Temperature of cells inside module, (°C). Typically determined from module back surface temperature measurements, or from a thermal model using solar resource and
environmental data.
f1(AM a) = Empirically determined polynomial relating the solar spectral influence on I sc to air mass variation over the day, where:
f1(AM a) = a0 + a1·AM a + a2·(AM a)2 + a3·(AM a)3 + a4·(AM a)4 f2(AOI) = Empirically determined polynomial relating optical influences on I sc to solar
angle-of-incidence (AOI), where:
f2(AOI) = b0 + b1·AOI + b2·(AOI)2 + b3·(AOI)3 + b4·(AOI)4 + b5·(AOI)5 Parameters at Standard Reporting (Reference) Conditions
Standard Reporting Conditions are used by the photovoltaic industry to ‘rate’ or ‘specify’ the performance of the module. This rating is provided at a single standardized (reference) operating condition [18, 19]. The associated performance parameters are typically either manufacturer’s nameplate ratings (specifications) or test results obtained from a module testing laboratory. The accuracy of these performance specifications is critical to the design of photovoltaic arrays and systems because they provide the reference point from which performance at all other operating conditions is derived. The consequence of a 10% error in the module performance rating will be a 10% effect on the annual energy production from the photovoltaic system. System integrators and module manufacturers should make every effort to ensure the accuracy of module performance ratings. The performance parameters and conditions associated with the standard reporting condition are defined as follows:
T o = Reference cell temperature for rating performance, typically 25°C
E o = Reference solar irradiance, typically 1000 W/m2
I sco = I sc(E = E o W/m2, AM a = 1.5, T c = T o°C, AOI = 0°) (A)
I mpo = I mp(E e =1, T c = T o) (A)
V oco = V oc(E e =1, T c = T o ) (V)
V mpo = V mp(E e =1, T c = T o ) (V)
I xo = I x(E e =1, T c = T o) (A)
I xxo = I xx(E e =1, T c = T o) (A)
Temperature Dependent Parameters
Although not universally recognized or standardized, the use of four separate temperature coefficients is instrumental in making our performance model versatile enough to apply equally well for all photovoltaic technologies over the full range of operating conditions. Currently standardized procedures erroneously assume that the temperature coefficient for V oc is applicable for V mp and the temperature coefficient for I sc is applicable for I mp [18]. If not available from the module manufacturer, the required parameters are available from the module database or can be measured during outdoor tests in actual operating conditions [3]. In addition, our performance model allows the temperature coefficients for voltage (V oc and V mp) to vary with solar irradiance, if necessary. For example, a concentrator silicon cell may have a V oc temperature coefficient of –2.0 mV/°C at 1X concentration, but at 200X concentration the value may drop to –1.7 mV/°C. However, for non-concentrator flat-plate modules, constant values for the voltage temperature coefficients are generally adequate.
The definitions for parameters are as follows, and when used in the performance model defined in this document, the engineering units for the temperature coefficients must be as specified below in order to be consistent with the equations.
αIsc = Normalized temperature coefficient for I sc, (1/°C). This parameter is ‘normalized’ by dividing the temperature dependence (A/°C) measured for a particular standard solar
spectrum and irradiance level by the module short-circuit current at the standard reference condition, I sco. Using these (1/°C) units makes the same value applicable for both individual modules and for parallel strings of modules.
αImp = Normalized temperature coefficient for I mp, (1/°C). Normalized in the same manner as αIsc.
βVoc(E e) = βVoco + mβVoc?(1-E e), (V/°C) Temperature coefficient for module open-circuit-voltage as a function of the effective irradiance, E e. Usually, the irradiance dependence can be neglected and βVoc is assumed to be a constant value.
βVoco = Temperature coefficient for module V oc at a 1000 W/m2 irradiance level, (V/°C) mβVoc = Coefficient providing the irradiance dependence for the V oc temperature coefficient, typically assumed to be zero, (V/°C).
βVmp(E e) = βVmpo +mβVmp?(1-E e), (V/°C) Temperature coefficient for module maximum-
power-voltage as a function of effective irradiance, E e. Usually, the irradiance dependence can be neglected and βVmp is assumed to be a constant value.
βVmpo = Temperature coefficient for module V mp at a 1000 W/m2 irradiance level, (V/°C) mβVmp = Coefficient providing the irradiance dependence for the V mp temperature coefficient, typically assumed to be zero, (V/°C).
Module Operating Temperature (Thermal Model)
When designing a photovoltaic system it is necessary to predict its expected annual energy production. To do so, a thermal model is required to estimate module operating temperature based on the local environmental conditions; solar irradiance, ambient temperature, wind speed, and perhaps wind direction. Site-dependent solar resource and meteorological data from recognized databases [14] or from meteorological models [15] are typically used to provide the environmental information required in the array design analysis. Estimates of hourly-average values for solar irradiance, ambient temperature, and wind speed are used in the thermal model to predict the associated operating temperature of the photovoltaic module. There is uncertainty associated with both the tabulated environmental data and the thermal model, but this approach has proven adequate for system design purposes.
After a system has been installed, the solar irradiance and module temperature can be measured directly and the results used in the performance model. The measured values avoid the inherent uncertainty associated with estimating module temperature based on environmental parameters, and improve the accuracy of the performance model for continuously predicting expected system performance.
In the mid-1980s, a thermal model was developed at Sandia for system engineering and performance modeling purposes [20]. Although rigorous, this early model has proven to be unnecessarily complex, not applicable to all module technologies, and not easily adaptable to site dependent influences.
A simpler empirically-based thermal model, described by Equation (11), was more recently developed at Sandia. The model has been applied successfully for flat-plate modules mounted in an open rack, for flat-plate modules with insulated back surfaces simulating building integrated situations, and for concentrator modules with finned heat sinks. The simple model has proven to be very adaptable and entirely adequate for system engineering and design purposes by providing the expected module operating temperature with an accuracy of about ±5°C.
Temperature uncertainties of this magnitude result in less than a 3% effect on the power output from the module.
The empirically determined coefficients (a, b) used in the model are determined using thousands of temperature measurements recorded over several different days with the module operating in a near thermal-equilibrium condition (nominally clear sky conditions without temperature
transients due to intermittent cloud cover). The coefficients determined are influenced by the module construction, the mounting configuration, and the location and height where wind speed is measured.
The standard meteorological practice for recording wind speed and direction locates the
measurement device (anemometer) at a height of 10 meters in an area with a minimum number of buildings or structures obstructing air movement. The tabulated wind speed and direction data provided in meteorological databases were recorded under these conditions. However, it should be noted that by analyzing data recorded after system installation, the thermal model can be ‘fine tuned’ by determining new coefficients (a,b) that compensate for site dependent influences and anemometer installations different from standard meteorological practice.
{}a WS b a m T e E T +?=?+
(11)
where: T m = Back-surface module temperature, (°C).
T a = Ambient air temperature, (°C)
E = Solar irradiance incident on module surface, (W/m 2)
WS = Wind speed measured at standard 10-m height, (m/s)
a = Empirically-determined coefficient establishing the upper limit for module temperature at low wind speeds and high solar irradiance
b = Empirically-determined coefficient establishing the rate at which module temperature drops as wind speed increases
Figure 8 illustrates typical measured data recorded on six different days with nominally clear conditions and a wide range of irradiance, wind speed, and wind direction. The module in this case was a large-area 300-W model with tempered-glass front and back surfaces. The effect of non-equilibrium ‘heat up’ periods (~ 30-min duration) is illustrated for two mornings when the sun first illuminated the module. A linear fit to the measured data provided the intercept and slope (a, b) coefficients required in the model. After the coefficients have been determined for a specific module then it is also possible to calculate the nominal operating cell temperature (NOCT) specified by ASTM [18], as well as the module temperature associated with the commonly used PVUSA Test Condition (PTC) [19].
Wind direction can also have a small but noticeable influence on module operating temperature. However, incorporating the effect of wind direction in the thermal model is believed to be
unnecessarily complex. Therefore, in our approach the influence of wind direction on operating temperature is regarded as a random influence adding some uncertainty to the thermal model, but also tending to average out on an annual basis. Similarly, thermal transients caused by clouds
and the module’s heat capacitance can introduce random influences on module temperature, but again these random effects average out on a daily or annual basis.
Double-Glass Module (Open Rack Mount)
-6.0
-5.5-5.0
-4.5-4.0-3.5
-3.0
05101520
Wind Speed (m/s)
(At 10-m Height)l n [(T m o d -T a m b ) / E ]
Figure 8. Experimentally determined relationship for back surface temperature of a flat-plate module in an open-rack mounting configuration as a function of solar irradiance, ambient
temperature and wind speed. A linear regression fit to the data provides the coefficients (a, b) for the thermal model.
Cell temperature and back-surface module temperature can be distinctly different, particularly for concentrator modules. The temperature of cells inside the module can be related to the module back surface temperature through a simple relationship. The relationship given in
Equation (12) is based on an assumption of one-dimensional thermal heat conduction through the module materials behind the cell (encapsulant and polymer layers for flat-plate modules, ceramic dielectric and aluminum heat sink for concentrator modules). The cell temperature inside the module is then calculated using a measured back-surface temperature and a predetermined temperature difference between the back surface and the cell.
T E E T T o
m c ??+= (12) where:
T c = Cell temperature inside module, (°C)
T m = Measured back-surface module temperature, (°C).
E = Measured solar irradiance on module, (W/m 2)
E o = Reference solar irradiance on module, (1000 W/m 2)
?T = Temperature difference between the cell and the module back surface at an irradiance level of 1000 W/m 2. This temperature difference is typically 2 to 3 °C for flat-plate modules in an open-rack mount. For flat-plate modules with a thermally insulated back surface, this temperature difference can be assumed to be zero. For concentrator modules, this
temperature difference is typically determined between the cell and the root of a finned heat exchanger (heat sink) on the back of the module.
Table 1 provides empirically-determined coefficients found to be representative of different module types and mounting configurations. The cases in the table can be considered generic for typical flat-plate photovoltaic modules from different manufacturers. However, the thermal behavior of concentrator modules can vary significantly, depending on the module design. Therefore, coefficients for concentrators must be empirically determined for each module design. One example, for a 1994-vintage linear-focus concentrator module, is given in the table.
Table 1. Empirically determined coefficients used to predict module back surface temperature as a function of irradiance, ambient temperature, and wind speed. Wind speed was measured at the standard meteorological height of 10 meters.
Module Type Mount a b ?T (°C)
Glass/cell/glass Open rack -3.47 -.0594 3
Glass/cell/glass Close roof mount -2.98 -.0471 1
Glass/cell/polymer sheet Open rack -3.56 -.0750 3
Glass/cell/polymer sheet Insulated back -2.81 -.0455 0
Polymer/thin-film/steel Open
rack
-3.58
-.113
3 22X Linear Concentrator Tracker -3.23 -.130 13
PERFORMANCE EQUATIONS FOR ARRAYS
Equations (1) through (10) can also be used for arrays composed of many modules by simply accounting for the series and parallel combinations of modules in the array. If the number of modules connected in series in a module-string is M s, then multiply the voltages calculated using Equations (3) and (4) by M s. If the number of module-strings connected in parallel in the array
is M p, then multiply the currents calculated using Equations (1), (2), (9), and(10) by M p. The calculated array performance using this approach is based on the expected performance of the individual modules, and as a result may be slightly optimistic because other array-level losses such as module mismatch and wiring resistance are not included.
Ideally, performance (I-V) measurements at the array level are available, in which case the accuracy of the performance model can be further improved. Array measurements can provide
the four basic performance parameters (I sco, I mpo, V oco, V mpo) at the standard reporting (reference) condition, as well as the eight other coefficients (C0, C1, …, C7). The spectral influence,
f1(AM a), the optical losses, f2(AOI), and the temperature coefficients for the array are assumed
to be available from test results on individual modules. Using array measurements, the electrical performance of the entire array can be modeled completely, in which case the model directly includes the array-level losses associated with module mismatch and wiring resistance that are
difficult to predict or determine explicitly. In essence, the array is modeled as if it were a very large module. Generally, the effect of mismatch and resistance losses is small (<5%) relative to performance expected from individual module nameplate ratings. Sandia’s module database includes several arrays of modules that were characterized in this manner.
To illustrate the procedure used to determine array performance coefficients, as well as their subsequent use in modeling the expected energy production, results for a 3.4-kW p system located in Albuquerque, New Mexico, will be presented.
Array Performance Example
The 3.4-kW p array evaluated was composed of two parallel module-strings, each with 24 crystalline silicon modules (70 W p) connected in series. There were 864 silicon cells in series in each module-string. The array was connected to a 2.5-kW inverter, and the system was connected to the local utility grid. Field performance measurements (I-V curves) were recorded on one clear morning in July using a portable curve tracer and two different solar irradiance sensors (pyranometers).
The first, most important, and perhaps most difficult challenge during array performance characterization is to determine an accurate value for the array short-circuit current (I sco) at a desired reference condition. After I sco has been determined, the remainder of the array performance analysis becomes self consistent and straight forward. The most commonly used reference condition is defined by the ASTM [18]. Nominally, the ASTM condition represents a clear sky condition with the sun at a mid-elevation angle in the sky and a module temperature of 25°C. The actual array operating condition during testing is determined by four factors: solar irradiance composed of a beam and a diffuse component, solar spectrum, solar angle-of-incidence, and array temperature. The typical effects of solar spectrum and angle-of-incidence on module performance were previously illustrated in Figures 6 and 7. The influence of these factors on the response of different pyranometers is documented elsewhere [23].
Figure 9 illustrates measured values for array I sc translated to a common temperature and plotted as a function of plane-of-array solar irradiance, as measured by two different pyranometers. The Kipp & Zonen pyranometer had a thermopile sensor and the LICOR pyranometer had a silicon photodiode sensor. The response of these pyranometers was influenced by the same factors affecting the array short-circuit current; but the magnitude of the effect for each factor differed between pyranometers and both pyranometers differed from the array. Therefore, even though the data illustrated were recorded during a calm perfectly clear day, the results indicate systematic trends rather than nice linear behavior versus measured irradiance. The most practical way to approach these field measurements is to first recognize there are a variety of interacting factors present in the measured data and then select the time period during the day when the combined effect of the factors is minimized, as illustrated in Figure 9.
Figure 10 shows the measured current values (I sc, I mp, I x, I xx) translated to a common temperature and plotted as a function of the effective irradiance, E e. The effective irradiance for each measurement was calculated using the measured I sco value and Equation (7). Alternative