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Thermal aspects of c-Si photovoltaic module energy rating
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Solar Energy 83 (2009) 1425–1433 www.elsevier.com/locate/solener
Thermal aspects of c-Si photovoltaic module energy rating Emmanuel Amy de la Breteque * Laboratory for Solar Systems (L2S)/Institut National d’Energie Solaire (INES), Cadarache Outdoor Measurement Platform, BP 332, 50 Avenue du Lac Le´man, 73377 Le Bourget du Lac, France Received 20 September 2007; received in revised form 10 September 2008; accepted 19 October 2008 Available online 12 November 2008 Communicated by: Associate Editor H. Gabler
Abstract Standard test conditions (STC) of photovoltaic (PV) modules are not representative of field conditions; PV module operating temperature often rises up to 30 °C above STC temperature (25 °C), causing a performance drop of 0.5%/°C for crystalline silicium modules. Normal operating cell temperature (NOCT) provides better estimates of PV module temperature rise. It has nevertheless to be measured; moreover NOCT wind speed conditions do not always fit field conditions. The purpose of this work is to model average PV module temperature at given irradiance levels as a function of meteorological parameters and PV module implementation. Thus, no empirical knowledge of PV module thermal behaviour is required for energy rating basing on irradiation distributions over irradiance levels. Ó 2008 Published by Elsevier Ltd. Keywords: PV module; Operating temperature; Site profile
1. Introduction The power rating of photovoltaic (PV) modules at a 1000 W/m2 irradiance level under spectral irradiance distribution defined by AM 1.5 and junction temperature of 25 °C is not representative of PV modules operating conditions. These conditions nevertheless are, according to IEC standard 61215, the so-called standard test conditions (STC), and deliver a reference for PV module peak performance. But as the irradiance level of 1000 W/m2 is generally reached during only a few hours around solar noon in the plane of array (POA), and as the PV module temperature often rises up to 40–50 °C rather than 25 °C, neither the peak power nor the efficiency at STC have great chance to be observed under field conditions. There is hence a great research interest for energy rating of PV modules under field conditions. The first step of energy prediction algorithms often consists in establishing
*
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0038-092X/$ - see front matter Ó 2008 Published by Elsevier Ltd. doi:10.1016/j.solener.2008.10.013
reliable performance correlations, expressed as a relationship between efficiency on the one hand and irradiance and module temperature on the other hand: gðG; T m Þ ¼ gSTC ð1 þ aðT m 25Þ þ f ðGÞÞ
ð1Þ
Temperature dependence of efficiency of crystalline silicon modules (c-Si) is in the order of magnitude of a = 0.5%/°C. The function f differs depending on authors; its determination is beyond the scope of this paper. The second step considers meteorological conditions and PV module implementation specificities: it is necessary to know at what irradiance and temperature levels the PV modules operate. In particular, several approaches have been developed to take into account the performance drop due to PV modules temperature rise. Various thermal models (Fuentes, 1985, Ingersoll, 1986; Krauter, 1993) deliver quite accurate estimates of PV module operating temperature but require irradiance, ambient temperature and wind speed profiles. They might be used in pointwise determination of electrical output (Kenny et al., 2006) or in I–V curves translation algorithms (Marion et al., 1999).
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Nomenclature temperature factor of module efficiency (standing alone) module transmittance–absorptance factor (–) emissivity factor (–) module efficiency (–) clearness index (subscript cd stands for clear day) (–) Stefan-Boltzmann constant (S.I.) Gamma function (–) solar irradiation (W h) solar global irradiance (W/m2) average module temperature (°C or K) ambient temperature (°C or K) difference between sky temperature and ambient temperature (K)
a sa e g j r C H G Tm Ta T0
A way to reduce the number of input parameters is the definition of PV modules normal operating cell temperature (NOCT) in the International Standard IEC 61215. It corresponds to open circuit PV modules temperature at POA irradiance level of 800 W/m2, 20 °C ambient temperature and wind speed of 1 m/s. Thus, the thermal behaviour of PV modules appears to be summed up in only one parameter that has nevertheless to be adjusted depending on the module implementation mode. Fuentes (1985) therefore introduces the concept of installed nominal operating conditions temperature (INOCT), which should be determined empirically. In numerous energy rating methods, PV module temperature Tm is expressed as a function of empirical NOCT or INOCT values as follows (referred to as NOCT model in the rest of the paper): g G ðNOCT 20Þ ð2Þ Tm ¼ Ta þ 1 sa 800 Tm(G) is possibly used either in explicit models (Kenny et al., 2006), which require empirical (G, Ta) maps, or in the so-called site profile approach ((Wheldon et al., 2001), similar to previous works of Siegel, Klein or Evans (Evans, 1981; Klein, 1978; Siegel et al., 1980)), consisting in integrating the efficiency as given in Eq. (1) weighted by the solar irradiation distribution over all possible POA irradiance levels. By definition, h(G) is the contribution of irradiance level G to the overall amount of incoming irradiation. The unit of h(G) is 1/(W/m2). PV module mean efficiency is eventually given by the following equality: Z gðG; T m ðGÞÞhðGÞdG ð3Þ g¼ XG
As PV modules mean efficiency might be expressed as a linear function of its energy-weighted average temperature (Bu¨cher, 1997; Gue´rin de Montgareuil, 2003) and as, at a given irradiance level G, taking energy-weighted average is the same as taking the average, the above equation
sky temperature (°C or K) contribution of irradiance level G to the total irradiation (subscript cd stands for clear day) 1/(W/m2) a, p, k empirical parameter set for forced convection (S.I.) v wind speed (vw is a parameter of the probability density function of wind speed) (m/s) h heat transfer coefficients: hcv for convection (hnat and hforc for natural, resp. for forced convection), hIR for long-wave radiation (W/m2 K) STC standard test condition (I)NOCT (installed) nominal operating cell temperature BIPV building integrated photovoltaics POA plane of array Ts h(G)
appears to be very interesting in case Eq. (2) delivers a reliable approximation of PV module mean operating temperature for each irradiance level. Nordmann and Clavadetscher (2003) underline that experimental studies achieved in the framework of Task 2 of the photovoltaic power system program (PVPS) on 18 PV systems tend to show that mean PV module temperature at given G is indeed approximately a linear function of irradiance. Some authors anyway point out that the NOCT model is not always the best possible and propose empirical linear regressions for completing thermal aspects of their energy rating method (e.g. Tm = Ta + 0.031G for a building integrated photovoltaic (BIPV) system (Mondol et al., 2005)). These approach yields satisfying results but requires knowing a priori the thermal behaviour of the modules; these expressions moreover depend on local climate and on the way the modules are mounted (racks, roof-mounted, BIPV, etc.) and may therefore not be used elsewhere without experimental validation. The following semi-empirical approach enables a physical apprehension of the heat transfer phenomena explaining the differences between two sites; it also makes short-term field performance assessments relevant for long-term performance prediction as long as both short-term and long-term site profiles are provided. This paper hence aims at proposing a convenient way for integrating PV modules thermal behaviour in performance prediction models, without referring explicitly to PV modules temperature measurements. It sets up a thermal model and proposes an empirical validation for determining the average PV module operating temperature at different irradiance levels, as a function of meteorological parameters only. 2. Experimental setup The aim of the thermal model is to deliver reliable estimates of PV modules mean operating temperature at any
E. Amy de la Breteque / Solar Energy 83 (2009) 1425–1433
possible POA irradiance level, depending on meteorological conditions. It bases on literature models for the simulation of PV modules temperature profile (Fuentes, 1985, Ingersoll, 1986; Krauter, 1993). In these models, empirical convective heat transfer laws and sky temperature determination should be adapted to local conditions. We therefore set up two specific experiments to find the optimal correlations fitting our operating conditions. 2.1. Convective heat transfer Convective heat transfer over hot plates has been the subject of lots of experimental studies, whose review is not the main object of this work. Empirical correlations provided by thermics handbooks were obtained for experiments, in which the air flow over the plates is well known, and have to be adapted according to specific PV conditions, where the wind flow regime varies quickly (Sharples and Charlesworth, 1998). A PV module was therefore simulated by a heated 2 mm thick flat plate, painted in black, and insulated with a 10 cm thick polyurethane layer on its front side. The dimensions of the plate were 1.35 m 0.65 m (Captec); it was tilted at a 40° angle, corresponding approximately to Cadarache latitude. Incoming infrared, respectively visible, radiative flux was measured by a CG1 pyrgeometer, respectively a CM21 pyranometer (Kipp & Zonen). External air temperature was measured by a PT100 near the modules. Total heat losses u were determined directly by four black tangential gradient fluxmeters with an emissivity of 0.96 (Captec), which is similar to the emissivity of the plate. They were coupled to four surface temperature measurements (K-type thermocouples, TC Direct), so that the surface-averaged convective heat transfer coefficient is eventually given by: hcv ¼ 1 4
4 X ui 0:96ðrT 4i uIR uvis Þ Ti Ta i¼1
ð4Þ
This approach avoids assuming that the plate is at energy balance, yielding interesting results for transitory regimes. The wind speed is measured by a three cup vane anemometer mounted on a 2 m high mast about 2 m from the eastern side of the plate. This shows the advantage of being more representative of local wind conditions near the plate, without altering too much the wind flow over it. The disadvantage of this method is that measurements stay very location specific. The meteorological norms for wind speed measurements (10 m above ground and free wind flow) are, in our opinion, nevertheless not adapted to such kind of studies, as wind profiles near PV system will have local properties. Data were collected by a HP 34970A data logger. All measurements were achieved in Cadarache during summer 2005 at a sampling interval of 2 s. In order to reduce uncertainties, only data for which the temperature difference
1427
between the plate and the ambient was greater than 10 °C were selected, yielding 4340 exploitable data points. This experiment allows an accurate characterisation of PV implementation modes with regard to convective heat transfers. 2.2. Sky temperature PV module front side exchanges heat with the sky, whose temperature ranges from 25 °C to some degrees below ambient temperature, depending on sky conditions. It is convenient to suppose that the sky temperature, Ts, is equal to ambient temperature for overcast skies, and 20 °C below ambient temperature for clear skies. In some PV module thermal models (Krauter, 1993), Ts is expressed as an empirical function of ambient temperature. But, in this work, the purpose is to deal with average sky conditions at fixed irradiance levels. It is hence necessary to further investigate sky temperature for predicting a reliable estimate of its average value. Another challenge is to complete this with few ‘‘general purpose” input parameters to ensure good applicability of the model. We aim at expressing downward long-wave atmospheric irradiance in terms of temperature difference between sky temperature and ambient temperature: uIR ¼ rðT a T 0 Þ
4
ð5Þ
Josey et al. (2003) relate T0 to sky conditions by quantifying the cloudiness using okta measurements, that is, quantifying cloud cover. Replacing okta measurements by another unit that is commonly used in the scope of PV applications was our goal. Daily clearness index is interesting for that purpose, as it is influenced by many characteristics of atmospheric conditions; among them cloud cover and atmospheric water content, both having great influence on sky temperature. We hence monitored downward long-wave atmospheric irradiance thanks to a CG1 pyrgeometer (Kipp & Zonen), mounted on a Brusag tracker for shadowing direct solar irradiance on the sensor surface. Experimental results were averaged over a day and then related to clearness index measurements thanks to a ventilated CM21 (Kipp & Zonen) pyranometer delivering global horizontal irradiance estimates: Hh ð6Þ T s ¼ f H AM0 External temperature was measured by a PT100. Relative humidity was also measured by a TRH300 sensor (Rixen), to achieve as exact as possible the characterization of atmospheric conditions. Data were recorded in Cadarache at a 5 min scan interval from May 2004 to January 2005. 2.3. Long-term monitoring of PV modules thermal behaviour Thermal and electrical characteristics of a rack-mounted BP 585 PV module (BP Solar) were monitored during the
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whole year 2003, in Cadarache (43°390 North, 5°460 East). The temperature of its rear side was measured by a K-type thermocouple. Wind speed was measured 3 m above ground, in the neighbourhood of the module. POA irradiance and horizontal irradiance were measured by two CM21 Kipp & Zonen pyranometers. As we also proceeded to I–V curve measurements, the module was mostly at open circuit conditions. 3. Thermal model 3.1. Heat transfer in a PV module Energy balance easily delivers the differential equation governing PV modules temperature. For clarity in further arguments, we suppose that the temperature gradient between PV module surfaces and PV cells is negligible (PV module temperature is uniform). With a linear expression for radiative heat transfer, we obtain the following equation: cp
dT m ¼ ðsa gSTC ÞG hcv ðT m T a Þ ðhIR;f þ hIR;b Þ dt ðT m T a Þ hIR;f ðT a T s Þ
ð7Þ
The transmission–absorption coefficient sa of the module, respectively its efficiency g is considered to be constant and equal to its mean value, respectively to STC value. This assumption is relevant in the framework of a thermal model, as the variations of PV modules efficiency will contribute for a very small amount to the part of irradiance converted into heat. A variation of efficiency of 10% will induce a variation of approximately 1% of the total irradiance converted into heat. Heat transfer in the module eventually breaks down into three transfer modes; (i) conduction from the cells to the sides of the module, which is not taken into account here, (ii) long-wave radiation between the module sides and the ground (possibly the roof) and the sky, (iii) convective heat transfer from the module to external air. In Eq. (7), the linear expression of radiative heat transfer is supposed to be valid over the whole PV module operating temperature range. Heat transfer coefficients are given by the following equalities, and have to be adjusted according to view factors between each of PV module sides and the ground, respectively the sky: 2 hIR;b ¼ eb rðT 2m þ T a ðGÞ ÞðT m þ T a ðGÞÞ hIR;f ¼ ef rðT 2 þ T s ðGÞ2 ÞðT m þ T s ðGÞ2 Þ
ð8Þ
m
View factors are simple functions of the tilt angle of the module, the skyline and the implementation of the module (in particular, for BIPV modules, the view factor between the rear side and the roof is 1; if there is no air layer between the roof and the PV module, conductive heat transfer through the building envelope should be taken into account). Convective heat transfer is either buoyancy-driven (natural convection) or wind-induced (forced convection). The
expression of convective flux in Eq. (6) is purely phenomenological and avoids using CFD techniques. The heat transfer coefficient is given as a function of wind speed, PV modules surface temperature and ambient temperature, as proposed in previous works (Fuentes, 1985, Jones and Underwood, 2001; Krauter, 1993, Romero, 1999). Numerical combination of natural forced convection is chosen, basing on remarks by Churchill (1977). This model is only valid for hot plates (Tm > Ta) tilted at x°: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hcv ¼ 3 h3nat þ h3forc ( 1=3 ð9Þ hnat ¼ kðsin x ðT T a ÞÞ hforc ¼ avp The parameters set (a, p, k) is determined empirically. Sharples and Charlesworth propose a good overview of literature models (Sharples and Charlesworth, 1998). According to their work, this set is a function of PV module implementation; choosing a model of convective heat transfer is hence quite cumbersome when lacking of adapted similar experimental references (Notton et al., 2005). Establishing an exhaustive convective heat transfer coefficients database specific of PV applications, depending on PV modules implementation mode, would be a relevant way to avoid inaccurate determination of PV modules operating temperature. Such empirical correlations for convective heat transfer are in fact the only parameter related to the way the modules are mounted. Field performance of PV modules is hence a function of (i) electrical performance, (ii) of local climate, which is taken into account in thermal models and in spectral response of PV module and (iii) of the implementation mode, governing mainly convective heat transfer. 3.2. Two hypothesis We make a first hypothesis that, for long time periods, the left-hand term of Eq. (7) has zero average, meaning that thermal mass effects have eventually no influence on the average of PV module operating temperature. This assumption is also implicitly made by authors using NOCT model for estimating the average PV modules temperature at given irradiance levels (e.g. Kenny et al., 2006) and has been experimentally validated in previous works (Amy de la Brete`que, 2005). As a consequence, PV module temperature at given irradiances, and for sky or ambient temperature conditions fixed to their average value at the considered irradiance level, is a function of wind speed only. The fact that convection heat transfer coefficients are still expressed in terms of wind speed and PV module surface temperature does not prevent us from completing the computation by iterations. Given the probability density function of the wind speed at G, p(v|G), the above remark leads to the following equality for mean PV module temperature:
E. Amy de la Breteque / Solar Energy 83 (2009) 1425–1433
T m ðGÞ ¼
Z
1429
1
T m ðv; T a ðGÞ; T s ðGÞÞpðvjGÞdv
3.5
ð10Þ
natural convection
0
Or, for linear radiative heat transfer: T m T a ðGÞ ¼ ððsa gSTC ÞG hIR;f ðT a T s ðGÞÞÞ Z 1 1 pðvjGÞdv hIR;f þ hIR;b þ hcv;f þ hcv;b 0 ð11Þ In the computation of the above two equations, we made a second hypothesis, claiming that sky temperature, respectively ambient temperature, may be set equal to its average value at fixed irradiance. It bases on the fact that, for small temperature ranges, radiative heat transfer might be considered linear, so that Tm Ta does not depend on ambient temperature level. Further, a theoretical study shows that Tm Ta decreases with increasing Ta, so that ambient temperature levels above average Ta value at given G compensate for lower ambient temperatures. Thus, no (G, Ta) map is required anymore for characterizing local climate properties. Moreover, in this work, mean sky temperature will be supposed constant over all possible irradiance levels.
logarithm of average convective coefficient transfer
3 2.5 2 1.5 1
forced convection
0.5 0 -4
-3
-2
-1
0
1
2
logarithm of wind speed experimental data
model
Fig. 1. Average convective coefficient transfer for the rear side of a flat plate (wind speed bins width = 0.25 m/s).
4. Results and validation
persion reduces when binning the data into wind speed bins as transitory effects have no longer influence on the measurements; further computation will show that the inaccuracy of this model has little influence on the final result. The optimal set of empirical parameters (a, p, k) used in (9) for our experimental set up is (a, p, k) = (6.5, 0.58, 1.1). It was obtained by searching a minimum value of the root mean square error on the measured convective heat flux.
4.1. Convective heat transfer
4.2. Average sky temperature for all sky conditions
The prevailing wind conditions at the experiment location are west-north-west and south east, whereas the plate was south oriented. The collected data are analysed regardless of wind direction for two reasons; (i) the influence of this parameter on such a small geometry has been proven to be rather small (Sharples and Charlesworth, 1998) and (ii) in case it would not be negligible, it would probably strongly depend on the dimensions on the system (the wind still at the leeward surface increasing with the dimensions of the plate). Wind speed conditions during experiment are characterized by two parameters of the Weibull distributions fitting the empirical probability density function (see Section 4.3); for this work (aw, vw) = (2.5, 2.3 m/s). Basing on conclusions by Sharples and Charlesworth (1998), we fitted the data with a power function of wind speed for forced convection. We also aimed at fitting the natural convection parameter k at the same time, and visualising the mixed convection regime where natural and forced convections coexist. The log–log diagram of Fig. 1 is a way to achieve this for convective heat transfer coefficients averaged over 0.25 m/s wide wind speed bins. It underlines moreover the fact that the convective coefficient for forced convection is reasonably well estimated by a power function of wind speed. The accuracy of the model is not very high when simulating instantaneous convective flux; the root mean square error on the measured convective heat flux is about 65 W/ m2, representing 24% of the mean observed value. This dis-
The results of the measurements achieved over the whole year 2004 in Cadarache show a relation between day-averaged sky temperature and daily clearness index j, which might be approximated by the following linear function (Amy de la Brete`que, 2005): 0 if j < 0; 15 ð12Þ To ¼ Ta Ts ¼ 45ðj 0; 15Þ if j P 0; 15 Clear day measurements (j > 0.65) are not well fitted by the above empirical law; the deviation might be related to the average partial water pressure (deduced from relative humidity measurements through the Magnus Teten formula: pw = 6.105 HR exp(17.27 Ta/(237.7 + Ta)) as shown in Fig. 2. This improvement of the model could nevertheless not be taken into account as it is necessary to keep the relationship between T0 and j linear for applying it to longer periods. In order to use this empirical law within the global thermal model (see Section 3), we take the energy-weighted average of Eq. (12), so that an estimate of the temperature difference between the external air and the sky still will be related to the mean clearness index of a longer time period, which is, in fact, the energy-weighted average of the daily clearness indexes. The root mean square error is 2.4 °C on Ta Ts; it has to be summed together with the error of the model of Eq. (11). It eventually yields a root mean square error on downward atmospheric long-wave irradiance incident on
10
3.5
40
8
3
35
6 4 2 0 -2
0
5
10
15
20
30
2.5
25 2 20 1.5 15 1
10
0.5
5
-4
0
-6
temperature (°C)
E. Amy de la Breteque / Solar Energy 83 (2009) 1425–1433
part of incident irradiation (1000/ (W/m²) )
residuals of daily average of sky temperature model (K)
1430
0
200
-8
400
600
800
1000
0 1200
lower bound of irradiance bins (W/m²)
daily average water pressure (hPa)
experimental data
Fig. 2. Sky temperature model residuals as a function of day-average partial water pressure, for j > 0.65.
the PV module front side of up to 45 W/m2 (monthly), representing approximately 13% of its average value. For the purpose of the thermal model, the period of time over which it is eventually necessary to average sky temperature is the time, during which irradiance is equal to a given level G. Experimental measurements of T0 for a complete year (2004), shown in Table 1, validate the above hypothesis (see Section 3) claiming that that the average sky temperature dependence on irradiance level is small. Eq. (12) is then valid for longer time periods. In the framework of this paper, we will use it for monthly clearness indexes.
4.3. Site profile for Cadarache The monitoring of meteorological parameters delivers Cadarache site profile as defined by Wheldon et al. (2001). Fig. 3 shows an example of a monthly site profile in Cadarache. It appears in Cadarache that the average distribution of irradiation over POA irradiance level hm(G) is proportional to that of a clear day, hcd(G), up to a given irradiance Glim; and the ratio between both functions is the ratio of the clear day clearness index and the mean clearness index. For irradiances greater than Glim, the distribution function decreases linearly until clear day irradiance at solar noon Gmax, so that it can be possible to model it from one typical clear day irradiance profile as follows: ( hm ðGÞ ¼ jjcd hcd ðGÞ for G 6 Glim having : max jcd hm ðGÞ ¼ GGG hcd ðGlim Þ lim Gmax j ð13Þ jcd x þ 12 ðGlim GÞ jjcd hcd ðGlim Þ ¼ 1 where : j RG x ¼ 0 lim hcd ðGÞdG
model
ambient temperature
Fig. 3. Empirical and modelled site profile for average March in Cadarache (2001–2003).
Results of this model are also shown in Fig. 3 for average March over three years in Cadarache. The above expression (Eq. (13)) for the distribution of irradiation over irradiance levels fits well empirical data, but we observed, for some months, that the predicted value of Glim is not always accurate. It differs from experimental Glim from one or two irradiance level bins. The width of these bins has an influence on the accuracy of the result. This model also does not fit the highest irradiance levels, greater than clear day irradiance at solar noon, caused by reflections on the clouds, which contribute for a small part to the total irradiance (0.5% during average March in Cadarache). The above argument (see Section 3) showed that it is necessary to integrate a description of local wind conditions in the site profile in order to avoid referring directly to PV module temperature measurements. Numerous experimental works (e.g. Henessey, 1977; Takle and Brown, 1977) underline the fact that a Weibull probability density function, which is a generalisation of the Rayleigh distribution that would be expected for isotropic, Gaussian wind speeds, fits reasonably well experimental data: a 1 aw v w ðv=vw Þaw e ð14Þ pðvÞ ¼ vw vw Computing PV module operating temperature for a theoretical case study tends to show that the parameter aw, which must be determined empirically, has little influence on the final result. Table 2 shows the average difference between calculated PV module operating temperature when taking aw into account or setting it to 2. This case study is defined as follows; ambient temperature is given by Ta = 0.0075G + 293 (in K), sky temperature is 20 K below ambient temperature and the second Weibull parameter vw is set to 1.5 m/s.
Table 1 Average difference temperature between the external air and the sky at different horizontal irradiance levels in Cadarache. Horizontal irradiance (W/m2)
0
100
200
300
400
500
600
700
800
900
1000
Ta Ts (°C)
12.5
14.5
17.6
20.4
21.1
514.5
22.5
23.2
23.5
23.3
19.5
E. Amy de la Breteque / Solar Energy 83 (2009) 1425–1433
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Table 2 Difference between calculated PV module temperature with aw = 3 or aw = 1. Irradiance bins (lower bound in W/m2)
0
100
200
300
400
500
600
700
800
900
Temperature difference (°C)
1.0
0.0
0.1
0.5
1.3
1.6
1.7
0.8
0.5
1.1
clearness index in 2003 in Cadarache, and the corresponding T0 (see Eq. (9)) values for the determination of the average sky temperature; Table 4 is the description of local wind conditions in Cadarache. Table 5 shows the mean root square error on predicted PV module operating temperature over the range of all possible irradiance levels and, in comparison, the NOCT model performance (as in Eq. (2); BP 585 NOCT value was found to be 46 °C). In this table, we excluded the highest irradiance level for the months of January, April, September and December because the great deviation from the model is not representative in these irradiance bins as they account for a very small part of total irradiation (less than 0.5% of total irradiation). The cloud reflections are by definition transitory effects that can not be properly explained by the model as it does not take them into account. We also proceeded to the same computation when assuming that wind speed and irradiance are independent stochastic variables, so that it would be possible to assume that the average wind speed at any irradiance is equal to
The fact that a Dirac function might be summed to the Weibull function in order to better take into account still wind periods (see Ramirez and Carta, 2005) also has little influence on the final result, as will eventually be shown in Section 4. Both above assumptions are purely empirical, so that nothing guarantees that they would be valid for another site. In Cadarache anyway, average wind speed at given irradiance is eventually the only parameter needed for characterizing local wind conditions, as the second empirical parameter of the Weibull distribution is given by: v ð15Þ vw ¼ C 1 þ a1w where aw is set to 2 (so that we use a Rayleigh distribution). 4.4. PV module average operating temperature at given irradiance levels All elements required for the computation of the thermal model are now available. Table 3 shows the monthly
Table 3 Monthly average clearness index and average temperature difference between the external air and the sky.
j T0 (K)
January
February
March
April
May
June
July
August
November
December
0.60 20.2
0.54 17.5
0.62 21.0
0.58 19.2
0.67 23.3
0.74 26.7
0.74 26.4
0.67 23.5
0.50 15.6
0.57 18.7
Table 4 Monthly average wind speed (in m/s) at given irradiance levels. Irradiance bin (W/m2)
January
February
March
April
May
June
0–100 100–200 200–300 300–400 400–500 500–600 600–700 700–800 800–900 900–1000 >1000
0.94 1.00 0.87 0.92 1.10 1.50 1.57 1.64 1.46 2.67 3.40
1.46 1.91 2.03 2.21 2.33 2.12 2.23 2.37 2.23 2.35 2.42
1.06 1.61 1.86 1.83 1.85 1.69 1.70 1.82 1.97 2.10 2.05
1.38 2.11 2.36 2.63 2.52 2.56 3.02 2.78 2.87 2.62 2.73
1.26 1.49 1.87 2.05 2.30 2.47 2.64 2.51 2.80 2.82 2.84
1.07 1.55 1.64 1.65 1.76 1.78 1.90 1.95 1.97 1.97 1.95
0–100 100–200 200–300 300–400 400–500 500–600 600–700 700–800 800–900 900–1000 >1000
July 1.57 1.94 2.02 2.20 2.22 2.27 2.38 2.49 2.60 2.68 3.19
August 1.14 1.69 1.88 1.86 1.82 1.80 1.87 1.93 1.84 1.82 2.42
September 1.12 1.61 1.76 1.58 1.76 1.68 1.70 1.72 1.64 1.70 2.07
October 1.37 1.54 1.36 1.84 1.88 1.71 2.04 2.29 2.30 1.78 2.99
November 1.14 1.21 1.35 1.52 1.44 1.35 1.25 1.34 1.31 1.30 1.08
December 1.35 1.31 1.29 1.10 1.38 1.31 1.62 1.37 1.69 2.20 2.74
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Table 5 Performance of explicit thermal model and NOCT model for the estimate of average PV module temperature at different irradiance levels. January
February
March
April
May
June
NOCT model 2nd model Explicit thermal model
2.0 2.8 0.8
3.8 2.8 1.5
4.2 3.2 1.3
4.4 1.7 0.8
5.4 2.6 1.6
2.4 1.5 2.3
NOCT model 2nd model Explicit thermal model
July 5.7 2.2 1.4
August 4.5 2.7 1.8
September 2.0 1.2 0.7
October 4.9 5.9 3.2
November 0.6 0.9 1.1
December 3.9 4.5 2.4
the overall average wind speed (‘‘2nd model” in Table 5). It is in fact the assumption made by NOCT-like models, which has the advantage that the calculation does not depend on the POA. The performance of the model decreases by 0.9 °C in average, which is negligible for energy rating as the impact on PV module efficiency would be in the order of magnitude of 0.5%, depending on irradiation distribution. This point nevertheless proves that NOCT models are relevant if (i) Eq. (16) (see below) is true and if (ii) wind speed and irradiance are uncorrelated. In particular, NOCT-like models deliver rather good estimates of PV module temperature when wind speed conditions are close to NOCT wind speed conditions as defined in IEC 61215 (v = 1 m/s), and are less reliable for windy months (e.g.
45
température (°C)
40 35 30 25 20 15 10 5 0 0
200
400
600
800
1000
irradiance (W/m²) ambient temperature experimental data
explicit thermal model NOCT model
70
temperature (°C)
60 50 40 30 20 10 0 0
200
400
600
800
1000
1200
irradiance (W/m²) ambient temperature experimental data
explicit thermal model NOCT model
Fig. 4. Explicit thermal model and NOCT model for January 2003 and May 2003.
Average 3.7 2.7 1.6
February, April, May or July). Fig. 4 shows an explicit comparison between the model presented in this paper and NOCT model for January 2003, respectively May 2003, where the wind speed conditions are close to NOCT conditions, respectively higher than NOCT conditions. Z 1 T m ðv; T a ð800Þ; T s ÞpðvÞdv ð16Þ NOCT ¼ 0
5. Conclusion Thermal models based on NOCT provide reliable estimates of PV module operating temperature when local wind conditions are close to the wind conditions as specified in the NOCT measurements standard, that is, if average wind speed is not too high in the neighbourhood of the modules and if wind and irradiance are uncorrelated in the sense of stochastic variables. These models nevertheless require empirical knowledge of the NOCT value, whose reliability depends on PV modules implementation mode. The growing importance of BIPV systems made the introduction of the concept of empirical Installed NOCT necessary. Next step consists in replacing, as proposed in this work, NOCT or INOCT empirical measurements by an algorithm whose input parameters are statistical properties of the local climate. Wind characteristics are hence also taken into account in the site profile. Two aspects of the model should nevertheless be further investigated. Firstly, if no empirical NOCT is required, empirical convection laws have to be estimated for each typical mode of PV module implementation. This point is of great importance, as the temperature module estimate strongly depends on the parameter set (a, p) used in forced convection models. Moreover, the huge number of empirical correlations in the literature often does not refer to general geometries that are typical of PV applications. The discrepancies between the proposed models are also a problem for taking into account thermal aspects of PV energy rating; a specific model database is lacking. The second point concerns empirical correlations or models proposed in this work to estimate mean sky temperature or irradiation distribution; they would be of more interest if validated for other climates. The proposed approach, consisting in estimating average PV module operating temperature at different irradi-
E. Amy de la Breteque / Solar Energy 83 (2009) 1425–1433
ance levels, eventually reduces the set of input parameters for PV module energy rating using site profiles. This set only consists in four meteorological variables; irradiation distribution, clearness index, average ambient temperature and wind speed at different irradiance levels. No empirical description of PV module behaviour is required. This approach moreover shows the advantage of distinguishing between the influence of electrical performance of the PV module on the one hand, local climate and PV module implementation mode on the other hand, thus making a link between indoor and outdoor characterization of PV module performance. Acknowledgments The author would like to thank Mr. Gue´rin de Montgareuil for all outdoor measurements that he completed, and which we exploited for this work. This work is funded in part by the Re´gion Provence Alpes Coˆte d’Azur. References Amy de la Brete`que, E., 2005. Statistical properties of Pv modules thermal behaviour. In: Proceedings of 20th EPVSEC. Bu¨cher, K., 1997. Site dependence of the energy collection of Pv modules. Solar Energy Materials and Solar Cells, 85–94. Churchill, S.W., 1977. A comprehensive correlating equation for laminar, assisting, forced and free convection. AIChE Journal 1, 10–16. Evans, D.L., 1981. Simplified method for predicting photovoltaic array output. Solar energy 6, 555–560. Fuentes, M.K., 1985. A simplified thermal model for flat-plate photovoltaic arrays, Sandia National Laboratories Technical Report. Gue´rin de Montgareuil, A., 2003. An empirical synthetic law between the modules energy output and the meteorological data. In: Proceedings of Third WCPEC. Henessey, J.P., 1977. Some aspects of wind power statistics. Journal of Applied Meteorology, 119–128. Ingersoll, J.G., 1986. Simplified calculation of solar cell temperatures in terrestrial photovoltaic arrays. Journal of Solar Energy Engineering, 95–101.
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Jones, A.D., Underwood, C.P., 2001. A thermal model for photovoltaic systems. Solar Energy 4, 334–359. Josey, S., Pascal, R., Yelland, M., 2003. A new formula for determining the atmospheric longwave flux at the ocean surface at mid-high latitudes. Journal of Geophysical Research 108. Kenny, R.P., Dunlop, E.D., Ossenbrink, H.A., Mu¨llejans, H., 2006. A practical method for the energy rating of c-Si photovoltaic modules based on standard tests. Progress in Photovoltaics: Research and Applications, 155–166. Klein, S.A., 1978. Calculation of flat-plate collector utilizability. Solar Energy, 393–402. Krauter, S., 1993. Betriebsmodell Der Optischen, Thermischen Und Elektrischen Parameter Von Photovoltaischen Modulen. Koester Verlag. Marion, B., et al., 1999. Validation of a photovoltaic module energy rating procedure at NREL, NREL Technical Report. Mondol, J.D., Yohannis, Y.G., Smyth, M., Norton, B., 2005. Long-term validated simulation of a building integrated photovoltaic system. Solar Energy, 163–176. Nordmann, T., Clavadetscher, 2003. Understanding temperature effects on PV system performance, IEA task 2 Technical Report. Notton, G., Cristofari, C., Mattei, M., Poggi, P., Muselli, M., 2005. Modelling of the thermal behaviour of a Pv module in view to Pv/ thermal applications. In: Proceedings of 20th EPVSEC. Ramirez, P., Carta, J.A., 2005. Influence of the data sampling interval in the estimation of the parameters of the Weibull wind speed probability distribution: a case study. Energy Conversion and Management, 2419– 2438. Romero, F.C., 1999. Calculo de la temperatura de operacion de celulas solares en un panel fotovoltaico, Ciemat Technical Report. Sharples, S., Charlesworth, P.S., 1998. Full-scale measurements of windinduced convective heat transfer from mounted flat plate solar collector. Solar Energy 2, 69–77. Siegel, M.D., Klein, S.A., Beckman, W.A., 1980. A simplified method for estimating the monthly-average performance of photovoltaic systems. Solar Energy, 413–418. Takle, E.S., Brown, J.M., 1977. Note on the use of Weibull statistics to characterize wind-speed data. Journal of Applied Meteorology, 556– 559. Wheldon, A., Doptidis, G., Cherrault, J.-Y., Wheeler, S., 2001. Application of frequency distribution of irradiance and temperature to their comparison of different Pv technologies. In: Proceedings of 17th PVSEC.
Solar Energy 83 (2009) 1425–1433 www.elsevier.com/locate/solener
Thermal aspects of c-Si photovoltaic module energy rating Emmanuel Amy de la Breteque * Laboratory for Solar Systems (L2S)/Institut National d’Energie Solaire (INES), Cadarache Outdoor Measurement Platform, BP 332, 50 Avenue du Lac Le´man, 73377 Le Bourget du Lac, France Received 20 September 2007; received in revised form 10 September 2008; accepted 19 October 2008 Available online 12 November 2008 Communicated by: Associate Editor H. Gabler
Abstract Standard test conditions (STC) of photovoltaic (PV) modules are not representative of field conditions; PV module operating temperature often rises up to 30 °C above STC temperature (25 °C), causing a performance drop of 0.5%/°C for crystalline silicium modules. Normal operating cell temperature (NOCT) provides better estimates of PV module temperature rise. It has nevertheless to be measured; moreover NOCT wind speed conditions do not always fit field conditions. The purpose of this work is to model average PV module temperature at given irradiance levels as a function of meteorological parameters and PV module implementation. Thus, no empirical knowledge of PV module thermal behaviour is required for energy rating basing on irradiation distributions over irradiance levels. Ó 2008 Published by Elsevier Ltd. Keywords: PV module; Operating temperature; Site profile
1. Introduction The power rating of photovoltaic (PV) modules at a 1000 W/m2 irradiance level under spectral irradiance distribution defined by AM 1.5 and junction temperature of 25 °C is not representative of PV modules operating conditions. These conditions nevertheless are, according to IEC standard 61215, the so-called standard test conditions (STC), and deliver a reference for PV module peak performance. But as the irradiance level of 1000 W/m2 is generally reached during only a few hours around solar noon in the plane of array (POA), and as the PV module temperature often rises up to 40–50 °C rather than 25 °C, neither the peak power nor the efficiency at STC have great chance to be observed under field conditions. There is hence a great research interest for energy rating of PV modules under field conditions. The first step of energy prediction algorithms often consists in establishing
*
Tel.: +33 4 79 44 45 46; fax: +33 4 42 25 73 65. E-mail address: [email protected]
0038-092X/$ - see front matter Ó 2008 Published by Elsevier Ltd. doi:10.1016/j.solener.2008.10.013
reliable performance correlations, expressed as a relationship between efficiency on the one hand and irradiance and module temperature on the other hand: gðG; T m Þ ¼ gSTC ð1 þ aðT m 25Þ þ f ðGÞÞ
ð1Þ
Temperature dependence of efficiency of crystalline silicon modules (c-Si) is in the order of magnitude of a = 0.5%/°C. The function f differs depending on authors; its determination is beyond the scope of this paper. The second step considers meteorological conditions and PV module implementation specificities: it is necessary to know at what irradiance and temperature levels the PV modules operate. In particular, several approaches have been developed to take into account the performance drop due to PV modules temperature rise. Various thermal models (Fuentes, 1985, Ingersoll, 1986; Krauter, 1993) deliver quite accurate estimates of PV module operating temperature but require irradiance, ambient temperature and wind speed profiles. They might be used in pointwise determination of electrical output (Kenny et al., 2006) or in I–V curves translation algorithms (Marion et al., 1999).
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Nomenclature temperature factor of module efficiency (standing alone) module transmittance–absorptance factor (–) emissivity factor (–) module efficiency (–) clearness index (subscript cd stands for clear day) (–) Stefan-Boltzmann constant (S.I.) Gamma function (–) solar irradiation (W h) solar global irradiance (W/m2) average module temperature (°C or K) ambient temperature (°C or K) difference between sky temperature and ambient temperature (K)
a sa e g j r C H G Tm Ta T0
A way to reduce the number of input parameters is the definition of PV modules normal operating cell temperature (NOCT) in the International Standard IEC 61215. It corresponds to open circuit PV modules temperature at POA irradiance level of 800 W/m2, 20 °C ambient temperature and wind speed of 1 m/s. Thus, the thermal behaviour of PV modules appears to be summed up in only one parameter that has nevertheless to be adjusted depending on the module implementation mode. Fuentes (1985) therefore introduces the concept of installed nominal operating conditions temperature (INOCT), which should be determined empirically. In numerous energy rating methods, PV module temperature Tm is expressed as a function of empirical NOCT or INOCT values as follows (referred to as NOCT model in the rest of the paper): g G ðNOCT 20Þ ð2Þ Tm ¼ Ta þ 1 sa 800 Tm(G) is possibly used either in explicit models (Kenny et al., 2006), which require empirical (G, Ta) maps, or in the so-called site profile approach ((Wheldon et al., 2001), similar to previous works of Siegel, Klein or Evans (Evans, 1981; Klein, 1978; Siegel et al., 1980)), consisting in integrating the efficiency as given in Eq. (1) weighted by the solar irradiation distribution over all possible POA irradiance levels. By definition, h(G) is the contribution of irradiance level G to the overall amount of incoming irradiation. The unit of h(G) is 1/(W/m2). PV module mean efficiency is eventually given by the following equality: Z gðG; T m ðGÞÞhðGÞdG ð3Þ g¼ XG
As PV modules mean efficiency might be expressed as a linear function of its energy-weighted average temperature (Bu¨cher, 1997; Gue´rin de Montgareuil, 2003) and as, at a given irradiance level G, taking energy-weighted average is the same as taking the average, the above equation
sky temperature (°C or K) contribution of irradiance level G to the total irradiation (subscript cd stands for clear day) 1/(W/m2) a, p, k empirical parameter set for forced convection (S.I.) v wind speed (vw is a parameter of the probability density function of wind speed) (m/s) h heat transfer coefficients: hcv for convection (hnat and hforc for natural, resp. for forced convection), hIR for long-wave radiation (W/m2 K) STC standard test condition (I)NOCT (installed) nominal operating cell temperature BIPV building integrated photovoltaics POA plane of array Ts h(G)
appears to be very interesting in case Eq. (2) delivers a reliable approximation of PV module mean operating temperature for each irradiance level. Nordmann and Clavadetscher (2003) underline that experimental studies achieved in the framework of Task 2 of the photovoltaic power system program (PVPS) on 18 PV systems tend to show that mean PV module temperature at given G is indeed approximately a linear function of irradiance. Some authors anyway point out that the NOCT model is not always the best possible and propose empirical linear regressions for completing thermal aspects of their energy rating method (e.g. Tm = Ta + 0.031G for a building integrated photovoltaic (BIPV) system (Mondol et al., 2005)). These approach yields satisfying results but requires knowing a priori the thermal behaviour of the modules; these expressions moreover depend on local climate and on the way the modules are mounted (racks, roof-mounted, BIPV, etc.) and may therefore not be used elsewhere without experimental validation. The following semi-empirical approach enables a physical apprehension of the heat transfer phenomena explaining the differences between two sites; it also makes short-term field performance assessments relevant for long-term performance prediction as long as both short-term and long-term site profiles are provided. This paper hence aims at proposing a convenient way for integrating PV modules thermal behaviour in performance prediction models, without referring explicitly to PV modules temperature measurements. It sets up a thermal model and proposes an empirical validation for determining the average PV module operating temperature at different irradiance levels, as a function of meteorological parameters only. 2. Experimental setup The aim of the thermal model is to deliver reliable estimates of PV modules mean operating temperature at any
E. Amy de la Breteque / Solar Energy 83 (2009) 1425–1433
possible POA irradiance level, depending on meteorological conditions. It bases on literature models for the simulation of PV modules temperature profile (Fuentes, 1985, Ingersoll, 1986; Krauter, 1993). In these models, empirical convective heat transfer laws and sky temperature determination should be adapted to local conditions. We therefore set up two specific experiments to find the optimal correlations fitting our operating conditions. 2.1. Convective heat transfer Convective heat transfer over hot plates has been the subject of lots of experimental studies, whose review is not the main object of this work. Empirical correlations provided by thermics handbooks were obtained for experiments, in which the air flow over the plates is well known, and have to be adapted according to specific PV conditions, where the wind flow regime varies quickly (Sharples and Charlesworth, 1998). A PV module was therefore simulated by a heated 2 mm thick flat plate, painted in black, and insulated with a 10 cm thick polyurethane layer on its front side. The dimensions of the plate were 1.35 m 0.65 m (Captec); it was tilted at a 40° angle, corresponding approximately to Cadarache latitude. Incoming infrared, respectively visible, radiative flux was measured by a CG1 pyrgeometer, respectively a CM21 pyranometer (Kipp & Zonen). External air temperature was measured by a PT100 near the modules. Total heat losses u were determined directly by four black tangential gradient fluxmeters with an emissivity of 0.96 (Captec), which is similar to the emissivity of the plate. They were coupled to four surface temperature measurements (K-type thermocouples, TC Direct), so that the surface-averaged convective heat transfer coefficient is eventually given by: hcv ¼ 1 4
4 X ui 0:96ðrT 4i uIR uvis Þ Ti Ta i¼1
ð4Þ
This approach avoids assuming that the plate is at energy balance, yielding interesting results for transitory regimes. The wind speed is measured by a three cup vane anemometer mounted on a 2 m high mast about 2 m from the eastern side of the plate. This shows the advantage of being more representative of local wind conditions near the plate, without altering too much the wind flow over it. The disadvantage of this method is that measurements stay very location specific. The meteorological norms for wind speed measurements (10 m above ground and free wind flow) are, in our opinion, nevertheless not adapted to such kind of studies, as wind profiles near PV system will have local properties. Data were collected by a HP 34970A data logger. All measurements were achieved in Cadarache during summer 2005 at a sampling interval of 2 s. In order to reduce uncertainties, only data for which the temperature difference
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between the plate and the ambient was greater than 10 °C were selected, yielding 4340 exploitable data points. This experiment allows an accurate characterisation of PV implementation modes with regard to convective heat transfers. 2.2. Sky temperature PV module front side exchanges heat with the sky, whose temperature ranges from 25 °C to some degrees below ambient temperature, depending on sky conditions. It is convenient to suppose that the sky temperature, Ts, is equal to ambient temperature for overcast skies, and 20 °C below ambient temperature for clear skies. In some PV module thermal models (Krauter, 1993), Ts is expressed as an empirical function of ambient temperature. But, in this work, the purpose is to deal with average sky conditions at fixed irradiance levels. It is hence necessary to further investigate sky temperature for predicting a reliable estimate of its average value. Another challenge is to complete this with few ‘‘general purpose” input parameters to ensure good applicability of the model. We aim at expressing downward long-wave atmospheric irradiance in terms of temperature difference between sky temperature and ambient temperature: uIR ¼ rðT a T 0 Þ
4
ð5Þ
Josey et al. (2003) relate T0 to sky conditions by quantifying the cloudiness using okta measurements, that is, quantifying cloud cover. Replacing okta measurements by another unit that is commonly used in the scope of PV applications was our goal. Daily clearness index is interesting for that purpose, as it is influenced by many characteristics of atmospheric conditions; among them cloud cover and atmospheric water content, both having great influence on sky temperature. We hence monitored downward long-wave atmospheric irradiance thanks to a CG1 pyrgeometer (Kipp & Zonen), mounted on a Brusag tracker for shadowing direct solar irradiance on the sensor surface. Experimental results were averaged over a day and then related to clearness index measurements thanks to a ventilated CM21 (Kipp & Zonen) pyranometer delivering global horizontal irradiance estimates: Hh ð6Þ T s ¼ f H AM0 External temperature was measured by a PT100. Relative humidity was also measured by a TRH300 sensor (Rixen), to achieve as exact as possible the characterization of atmospheric conditions. Data were recorded in Cadarache at a 5 min scan interval from May 2004 to January 2005. 2.3. Long-term monitoring of PV modules thermal behaviour Thermal and electrical characteristics of a rack-mounted BP 585 PV module (BP Solar) were monitored during the
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whole year 2003, in Cadarache (43°390 North, 5°460 East). The temperature of its rear side was measured by a K-type thermocouple. Wind speed was measured 3 m above ground, in the neighbourhood of the module. POA irradiance and horizontal irradiance were measured by two CM21 Kipp & Zonen pyranometers. As we also proceeded to I–V curve measurements, the module was mostly at open circuit conditions. 3. Thermal model 3.1. Heat transfer in a PV module Energy balance easily delivers the differential equation governing PV modules temperature. For clarity in further arguments, we suppose that the temperature gradient between PV module surfaces and PV cells is negligible (PV module temperature is uniform). With a linear expression for radiative heat transfer, we obtain the following equation: cp
dT m ¼ ðsa gSTC ÞG hcv ðT m T a Þ ðhIR;f þ hIR;b Þ dt ðT m T a Þ hIR;f ðT a T s Þ
ð7Þ
The transmission–absorption coefficient sa of the module, respectively its efficiency g is considered to be constant and equal to its mean value, respectively to STC value. This assumption is relevant in the framework of a thermal model, as the variations of PV modules efficiency will contribute for a very small amount to the part of irradiance converted into heat. A variation of efficiency of 10% will induce a variation of approximately 1% of the total irradiance converted into heat. Heat transfer in the module eventually breaks down into three transfer modes; (i) conduction from the cells to the sides of the module, which is not taken into account here, (ii) long-wave radiation between the module sides and the ground (possibly the roof) and the sky, (iii) convective heat transfer from the module to external air. In Eq. (7), the linear expression of radiative heat transfer is supposed to be valid over the whole PV module operating temperature range. Heat transfer coefficients are given by the following equalities, and have to be adjusted according to view factors between each of PV module sides and the ground, respectively the sky: 2 hIR;b ¼ eb rðT 2m þ T a ðGÞ ÞðT m þ T a ðGÞÞ hIR;f ¼ ef rðT 2 þ T s ðGÞ2 ÞðT m þ T s ðGÞ2 Þ
ð8Þ
m
View factors are simple functions of the tilt angle of the module, the skyline and the implementation of the module (in particular, for BIPV modules, the view factor between the rear side and the roof is 1; if there is no air layer between the roof and the PV module, conductive heat transfer through the building envelope should be taken into account). Convective heat transfer is either buoyancy-driven (natural convection) or wind-induced (forced convection). The
expression of convective flux in Eq. (6) is purely phenomenological and avoids using CFD techniques. The heat transfer coefficient is given as a function of wind speed, PV modules surface temperature and ambient temperature, as proposed in previous works (Fuentes, 1985, Jones and Underwood, 2001; Krauter, 1993, Romero, 1999). Numerical combination of natural forced convection is chosen, basing on remarks by Churchill (1977). This model is only valid for hot plates (Tm > Ta) tilted at x°: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hcv ¼ 3 h3nat þ h3forc ( 1=3 ð9Þ hnat ¼ kðsin x ðT T a ÞÞ hforc ¼ avp The parameters set (a, p, k) is determined empirically. Sharples and Charlesworth propose a good overview of literature models (Sharples and Charlesworth, 1998). According to their work, this set is a function of PV module implementation; choosing a model of convective heat transfer is hence quite cumbersome when lacking of adapted similar experimental references (Notton et al., 2005). Establishing an exhaustive convective heat transfer coefficients database specific of PV applications, depending on PV modules implementation mode, would be a relevant way to avoid inaccurate determination of PV modules operating temperature. Such empirical correlations for convective heat transfer are in fact the only parameter related to the way the modules are mounted. Field performance of PV modules is hence a function of (i) electrical performance, (ii) of local climate, which is taken into account in thermal models and in spectral response of PV module and (iii) of the implementation mode, governing mainly convective heat transfer. 3.2. Two hypothesis We make a first hypothesis that, for long time periods, the left-hand term of Eq. (7) has zero average, meaning that thermal mass effects have eventually no influence on the average of PV module operating temperature. This assumption is also implicitly made by authors using NOCT model for estimating the average PV modules temperature at given irradiance levels (e.g. Kenny et al., 2006) and has been experimentally validated in previous works (Amy de la Brete`que, 2005). As a consequence, PV module temperature at given irradiances, and for sky or ambient temperature conditions fixed to their average value at the considered irradiance level, is a function of wind speed only. The fact that convection heat transfer coefficients are still expressed in terms of wind speed and PV module surface temperature does not prevent us from completing the computation by iterations. Given the probability density function of the wind speed at G, p(v|G), the above remark leads to the following equality for mean PV module temperature:
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T m ðGÞ ¼
Z
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1
T m ðv; T a ðGÞ; T s ðGÞÞpðvjGÞdv
3.5
ð10Þ
natural convection
0
Or, for linear radiative heat transfer: T m T a ðGÞ ¼ ððsa gSTC ÞG hIR;f ðT a T s ðGÞÞÞ Z 1 1 pðvjGÞdv hIR;f þ hIR;b þ hcv;f þ hcv;b 0 ð11Þ In the computation of the above two equations, we made a second hypothesis, claiming that sky temperature, respectively ambient temperature, may be set equal to its average value at fixed irradiance. It bases on the fact that, for small temperature ranges, radiative heat transfer might be considered linear, so that Tm Ta does not depend on ambient temperature level. Further, a theoretical study shows that Tm Ta decreases with increasing Ta, so that ambient temperature levels above average Ta value at given G compensate for lower ambient temperatures. Thus, no (G, Ta) map is required anymore for characterizing local climate properties. Moreover, in this work, mean sky temperature will be supposed constant over all possible irradiance levels.
logarithm of average convective coefficient transfer
3 2.5 2 1.5 1
forced convection
0.5 0 -4
-3
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-1
0
1
2
logarithm of wind speed experimental data
model
Fig. 1. Average convective coefficient transfer for the rear side of a flat plate (wind speed bins width = 0.25 m/s).
4. Results and validation
persion reduces when binning the data into wind speed bins as transitory effects have no longer influence on the measurements; further computation will show that the inaccuracy of this model has little influence on the final result. The optimal set of empirical parameters (a, p, k) used in (9) for our experimental set up is (a, p, k) = (6.5, 0.58, 1.1). It was obtained by searching a minimum value of the root mean square error on the measured convective heat flux.
4.1. Convective heat transfer
4.2. Average sky temperature for all sky conditions
The prevailing wind conditions at the experiment location are west-north-west and south east, whereas the plate was south oriented. The collected data are analysed regardless of wind direction for two reasons; (i) the influence of this parameter on such a small geometry has been proven to be rather small (Sharples and Charlesworth, 1998) and (ii) in case it would not be negligible, it would probably strongly depend on the dimensions on the system (the wind still at the leeward surface increasing with the dimensions of the plate). Wind speed conditions during experiment are characterized by two parameters of the Weibull distributions fitting the empirical probability density function (see Section 4.3); for this work (aw, vw) = (2.5, 2.3 m/s). Basing on conclusions by Sharples and Charlesworth (1998), we fitted the data with a power function of wind speed for forced convection. We also aimed at fitting the natural convection parameter k at the same time, and visualising the mixed convection regime where natural and forced convections coexist. The log–log diagram of Fig. 1 is a way to achieve this for convective heat transfer coefficients averaged over 0.25 m/s wide wind speed bins. It underlines moreover the fact that the convective coefficient for forced convection is reasonably well estimated by a power function of wind speed. The accuracy of the model is not very high when simulating instantaneous convective flux; the root mean square error on the measured convective heat flux is about 65 W/ m2, representing 24% of the mean observed value. This dis-
The results of the measurements achieved over the whole year 2004 in Cadarache show a relation between day-averaged sky temperature and daily clearness index j, which might be approximated by the following linear function (Amy de la Brete`que, 2005): 0 if j < 0; 15 ð12Þ To ¼ Ta Ts ¼ 45ðj 0; 15Þ if j P 0; 15 Clear day measurements (j > 0.65) are not well fitted by the above empirical law; the deviation might be related to the average partial water pressure (deduced from relative humidity measurements through the Magnus Teten formula: pw = 6.105 HR exp(17.27 Ta/(237.7 + Ta)) as shown in Fig. 2. This improvement of the model could nevertheless not be taken into account as it is necessary to keep the relationship between T0 and j linear for applying it to longer periods. In order to use this empirical law within the global thermal model (see Section 3), we take the energy-weighted average of Eq. (12), so that an estimate of the temperature difference between the external air and the sky still will be related to the mean clearness index of a longer time period, which is, in fact, the energy-weighted average of the daily clearness indexes. The root mean square error is 2.4 °C on Ta Ts; it has to be summed together with the error of the model of Eq. (11). It eventually yields a root mean square error on downward atmospheric long-wave irradiance incident on
10
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35
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30
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25 2 20 1.5 15 1
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part of incident irradiation (1000/ (W/m²) )
residuals of daily average of sky temperature model (K)
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0
200
-8
400
600
800
1000
0 1200
lower bound of irradiance bins (W/m²)
daily average water pressure (hPa)
experimental data
Fig. 2. Sky temperature model residuals as a function of day-average partial water pressure, for j > 0.65.
the PV module front side of up to 45 W/m2 (monthly), representing approximately 13% of its average value. For the purpose of the thermal model, the period of time over which it is eventually necessary to average sky temperature is the time, during which irradiance is equal to a given level G. Experimental measurements of T0 for a complete year (2004), shown in Table 1, validate the above hypothesis (see Section 3) claiming that that the average sky temperature dependence on irradiance level is small. Eq. (12) is then valid for longer time periods. In the framework of this paper, we will use it for monthly clearness indexes.
4.3. Site profile for Cadarache The monitoring of meteorological parameters delivers Cadarache site profile as defined by Wheldon et al. (2001). Fig. 3 shows an example of a monthly site profile in Cadarache. It appears in Cadarache that the average distribution of irradiation over POA irradiance level hm(G) is proportional to that of a clear day, hcd(G), up to a given irradiance Glim; and the ratio between both functions is the ratio of the clear day clearness index and the mean clearness index. For irradiances greater than Glim, the distribution function decreases linearly until clear day irradiance at solar noon Gmax, so that it can be possible to model it from one typical clear day irradiance profile as follows: ( hm ðGÞ ¼ jjcd hcd ðGÞ for G 6 Glim having : max jcd hm ðGÞ ¼ GGG hcd ðGlim Þ lim Gmax j ð13Þ jcd x þ 12 ðGlim GÞ jjcd hcd ðGlim Þ ¼ 1 where : j RG x ¼ 0 lim hcd ðGÞdG
model
ambient temperature
Fig. 3. Empirical and modelled site profile for average March in Cadarache (2001–2003).
Results of this model are also shown in Fig. 3 for average March over three years in Cadarache. The above expression (Eq. (13)) for the distribution of irradiation over irradiance levels fits well empirical data, but we observed, for some months, that the predicted value of Glim is not always accurate. It differs from experimental Glim from one or two irradiance level bins. The width of these bins has an influence on the accuracy of the result. This model also does not fit the highest irradiance levels, greater than clear day irradiance at solar noon, caused by reflections on the clouds, which contribute for a small part to the total irradiance (0.5% during average March in Cadarache). The above argument (see Section 3) showed that it is necessary to integrate a description of local wind conditions in the site profile in order to avoid referring directly to PV module temperature measurements. Numerous experimental works (e.g. Henessey, 1977; Takle and Brown, 1977) underline the fact that a Weibull probability density function, which is a generalisation of the Rayleigh distribution that would be expected for isotropic, Gaussian wind speeds, fits reasonably well experimental data: a 1 aw v w ðv=vw Þaw e ð14Þ pðvÞ ¼ vw vw Computing PV module operating temperature for a theoretical case study tends to show that the parameter aw, which must be determined empirically, has little influence on the final result. Table 2 shows the average difference between calculated PV module operating temperature when taking aw into account or setting it to 2. This case study is defined as follows; ambient temperature is given by Ta = 0.0075G + 293 (in K), sky temperature is 20 K below ambient temperature and the second Weibull parameter vw is set to 1.5 m/s.
Table 1 Average difference temperature between the external air and the sky at different horizontal irradiance levels in Cadarache. Horizontal irradiance (W/m2)
0
100
200
300
400
500
600
700
800
900
1000
Ta Ts (°C)
12.5
14.5
17.6
20.4
21.1
514.5
22.5
23.2
23.5
23.3
19.5
E. Amy de la Breteque / Solar Energy 83 (2009) 1425–1433
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Table 2 Difference between calculated PV module temperature with aw = 3 or aw = 1. Irradiance bins (lower bound in W/m2)
0
100
200
300
400
500
600
700
800
900
Temperature difference (°C)
1.0
0.0
0.1
0.5
1.3
1.6
1.7
0.8
0.5
1.1
clearness index in 2003 in Cadarache, and the corresponding T0 (see Eq. (9)) values for the determination of the average sky temperature; Table 4 is the description of local wind conditions in Cadarache. Table 5 shows the mean root square error on predicted PV module operating temperature over the range of all possible irradiance levels and, in comparison, the NOCT model performance (as in Eq. (2); BP 585 NOCT value was found to be 46 °C). In this table, we excluded the highest irradiance level for the months of January, April, September and December because the great deviation from the model is not representative in these irradiance bins as they account for a very small part of total irradiation (less than 0.5% of total irradiation). The cloud reflections are by definition transitory effects that can not be properly explained by the model as it does not take them into account. We also proceeded to the same computation when assuming that wind speed and irradiance are independent stochastic variables, so that it would be possible to assume that the average wind speed at any irradiance is equal to
The fact that a Dirac function might be summed to the Weibull function in order to better take into account still wind periods (see Ramirez and Carta, 2005) also has little influence on the final result, as will eventually be shown in Section 4. Both above assumptions are purely empirical, so that nothing guarantees that they would be valid for another site. In Cadarache anyway, average wind speed at given irradiance is eventually the only parameter needed for characterizing local wind conditions, as the second empirical parameter of the Weibull distribution is given by: v ð15Þ vw ¼ C 1 þ a1w where aw is set to 2 (so that we use a Rayleigh distribution). 4.4. PV module average operating temperature at given irradiance levels All elements required for the computation of the thermal model are now available. Table 3 shows the monthly
Table 3 Monthly average clearness index and average temperature difference between the external air and the sky.
j T0 (K)
January
February
March
April
May
June
July
August
November
December
0.60 20.2
0.54 17.5
0.62 21.0
0.58 19.2
0.67 23.3
0.74 26.7
0.74 26.4
0.67 23.5
0.50 15.6
0.57 18.7
Table 4 Monthly average wind speed (in m/s) at given irradiance levels. Irradiance bin (W/m2)
January
February
March
April
May
June
0–100 100–200 200–300 300–400 400–500 500–600 600–700 700–800 800–900 900–1000 >1000
0.94 1.00 0.87 0.92 1.10 1.50 1.57 1.64 1.46 2.67 3.40
1.46 1.91 2.03 2.21 2.33 2.12 2.23 2.37 2.23 2.35 2.42
1.06 1.61 1.86 1.83 1.85 1.69 1.70 1.82 1.97 2.10 2.05
1.38 2.11 2.36 2.63 2.52 2.56 3.02 2.78 2.87 2.62 2.73
1.26 1.49 1.87 2.05 2.30 2.47 2.64 2.51 2.80 2.82 2.84
1.07 1.55 1.64 1.65 1.76 1.78 1.90 1.95 1.97 1.97 1.95
0–100 100–200 200–300 300–400 400–500 500–600 600–700 700–800 800–900 900–1000 >1000
July 1.57 1.94 2.02 2.20 2.22 2.27 2.38 2.49 2.60 2.68 3.19
August 1.14 1.69 1.88 1.86 1.82 1.80 1.87 1.93 1.84 1.82 2.42
September 1.12 1.61 1.76 1.58 1.76 1.68 1.70 1.72 1.64 1.70 2.07
October 1.37 1.54 1.36 1.84 1.88 1.71 2.04 2.29 2.30 1.78 2.99
November 1.14 1.21 1.35 1.52 1.44 1.35 1.25 1.34 1.31 1.30 1.08
December 1.35 1.31 1.29 1.10 1.38 1.31 1.62 1.37 1.69 2.20 2.74
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Table 5 Performance of explicit thermal model and NOCT model for the estimate of average PV module temperature at different irradiance levels. January
February
March
April
May
June
NOCT model 2nd model Explicit thermal model
2.0 2.8 0.8
3.8 2.8 1.5
4.2 3.2 1.3
4.4 1.7 0.8
5.4 2.6 1.6
2.4 1.5 2.3
NOCT model 2nd model Explicit thermal model
July 5.7 2.2 1.4
August 4.5 2.7 1.8
September 2.0 1.2 0.7
October 4.9 5.9 3.2
November 0.6 0.9 1.1
December 3.9 4.5 2.4
the overall average wind speed (‘‘2nd model” in Table 5). It is in fact the assumption made by NOCT-like models, which has the advantage that the calculation does not depend on the POA. The performance of the model decreases by 0.9 °C in average, which is negligible for energy rating as the impact on PV module efficiency would be in the order of magnitude of 0.5%, depending on irradiation distribution. This point nevertheless proves that NOCT models are relevant if (i) Eq. (16) (see below) is true and if (ii) wind speed and irradiance are uncorrelated. In particular, NOCT-like models deliver rather good estimates of PV module temperature when wind speed conditions are close to NOCT wind speed conditions as defined in IEC 61215 (v = 1 m/s), and are less reliable for windy months (e.g.
45
température (°C)
40 35 30 25 20 15 10 5 0 0
200
400
600
800
1000
irradiance (W/m²) ambient temperature experimental data
explicit thermal model NOCT model
70
temperature (°C)
60 50 40 30 20 10 0 0
200
400
600
800
1000
1200
irradiance (W/m²) ambient temperature experimental data
explicit thermal model NOCT model
Fig. 4. Explicit thermal model and NOCT model for January 2003 and May 2003.
Average 3.7 2.7 1.6
February, April, May or July). Fig. 4 shows an explicit comparison between the model presented in this paper and NOCT model for January 2003, respectively May 2003, where the wind speed conditions are close to NOCT conditions, respectively higher than NOCT conditions. Z 1 T m ðv; T a ð800Þ; T s ÞpðvÞdv ð16Þ NOCT ¼ 0
5. Conclusion Thermal models based on NOCT provide reliable estimates of PV module operating temperature when local wind conditions are close to the wind conditions as specified in the NOCT measurements standard, that is, if average wind speed is not too high in the neighbourhood of the modules and if wind and irradiance are uncorrelated in the sense of stochastic variables. These models nevertheless require empirical knowledge of the NOCT value, whose reliability depends on PV modules implementation mode. The growing importance of BIPV systems made the introduction of the concept of empirical Installed NOCT necessary. Next step consists in replacing, as proposed in this work, NOCT or INOCT empirical measurements by an algorithm whose input parameters are statistical properties of the local climate. Wind characteristics are hence also taken into account in the site profile. Two aspects of the model should nevertheless be further investigated. Firstly, if no empirical NOCT is required, empirical convection laws have to be estimated for each typical mode of PV module implementation. This point is of great importance, as the temperature module estimate strongly depends on the parameter set (a, p) used in forced convection models. Moreover, the huge number of empirical correlations in the literature often does not refer to general geometries that are typical of PV applications. The discrepancies between the proposed models are also a problem for taking into account thermal aspects of PV energy rating; a specific model database is lacking. The second point concerns empirical correlations or models proposed in this work to estimate mean sky temperature or irradiation distribution; they would be of more interest if validated for other climates. The proposed approach, consisting in estimating average PV module operating temperature at different irradi-
E. Amy de la Breteque / Solar Energy 83 (2009) 1425–1433
ance levels, eventually reduces the set of input parameters for PV module energy rating using site profiles. This set only consists in four meteorological variables; irradiation distribution, clearness index, average ambient temperature and wind speed at different irradiance levels. No empirical description of PV module behaviour is required. This approach moreover shows the advantage of distinguishing between the influence of electrical performance of the PV module on the one hand, local climate and PV module implementation mode on the other hand, thus making a link between indoor and outdoor characterization of PV module performance. Acknowledgments The author would like to thank Mr. Gue´rin de Montgareuil for all outdoor measurements that he completed, and which we exploited for this work. This work is funded in part by the Re´gion Provence Alpes Coˆte d’Azur. References Amy de la Brete`que, E., 2005. Statistical properties of Pv modules thermal behaviour. In: Proceedings of 20th EPVSEC. Bu¨cher, K., 1997. Site dependence of the energy collection of Pv modules. Solar Energy Materials and Solar Cells, 85–94. Churchill, S.W., 1977. A comprehensive correlating equation for laminar, assisting, forced and free convection. AIChE Journal 1, 10–16. Evans, D.L., 1981. Simplified method for predicting photovoltaic array output. Solar energy 6, 555–560. Fuentes, M.K., 1985. A simplified thermal model for flat-plate photovoltaic arrays, Sandia National Laboratories Technical Report. Gue´rin de Montgareuil, A., 2003. An empirical synthetic law between the modules energy output and the meteorological data. In: Proceedings of Third WCPEC. Henessey, J.P., 1977. Some aspects of wind power statistics. Journal of Applied Meteorology, 119–128. Ingersoll, J.G., 1986. Simplified calculation of solar cell temperatures in terrestrial photovoltaic arrays. Journal of Solar Energy Engineering, 95–101.
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