Introduction, evaluation and application of an energy balance model for photovoltaic modules

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Solar Energy
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Introduction, evaluation and application of an energy balance model for
photovoltaic modules
Jannik Heusingera,b,1,⁎, Ashley M. Broadbenta,b, David J. Sailora,b,c, Matei Georgescua,b,c
a School of Geographical Sciences and Urban Planning, Arizona State University, 85281 Tempe, USA
bUrban Climate Research Center, Arizona State University, 85281 Tempe, USA
cGlobal Institute of Sustainability, Arizona State University, 85281 Tempe, USA
A R T I C L E I N F O
Keywords:
Photovoltaic
Surface energy balance
Renewable energy
A B S T R A C T
Here we introduce a new energy balance model that accurately simulates the complete diurnal dynamics of
photovoltaic (PV) thermal behavior with routinely available meteorological input. The model is evaluated ex-
tensively against observed module surface temperatures (day and nighttime), electrical output, and sensible heat
flux measurements. It is demonstrated that different tracking systems have a significant effect on module tem-
peratures and sensible heat fluxes by modulating the total radiation received on the PV surface. A model in-
tercomparison study indicates significant improvements in the representation of module temperatures compared
to an earlier study, the commercial software PVsyst and the Python package PVLIB. A sensitivity study de-
monstrates a considerable effect of the PV conversion efficiency and longwave emissivity on sensible heat fluxes
emitted by the module. The model is available as a stand-alone program (UCRC-Solar) written in Python and
planned to be implemented in mesoscale meteorological models to study the geophysical impacts of PV arrays at
larger spatial and longer temporal scales.
1. Introduction
By 2040, solar photovoltaics (PV) are expected to make up the
largest share of renewable energy production worldwide (International
Energy Agency, 2017). This transition will lead to a considerable in-
crease in the deployment of PV systems in the form of solar farms and
rooftop solar installations. These efforts to mitigate global warming by
reduction of greenhouse gas emissions are associated with land cover
changes (e.g. surface roughness and albedo), which may modify the
local surface energy balance and local meteorology dependent on initial
landscape characteristics. Therefore, a comprehensive evaluation of PV
systems should consider geochemical (greenhouse gases) and geophy-
sical (surface energy balance) impacts (Broadbent et al., 2019). The
geophysical impacts of PV systems have previously been studied in
mesoscale atmospheric models (Masson et al., 2014; Millstein and
Menon, 2011; Salamanca et al., 2016; Taha, 2013) and global-scale
general circulation models (GCMs) (Hu et al., 2016). Representation of
PV has been simplistic, including changes to effective surface albedo
and simple energy balance models. Furthermore, these PV models have
not been evaluated with observed meteorological data, which are
necessary to demonstrate the merit of such models. Here we present
and evaluate a new PV surface energy balance model that can represent
the full range of geophysical processes associated with PV systems.
The geophysical impacts of PV systems are relevant to human
thermal comfort in urban areas where widespread rooftop PV pene-
tration could lead to local climate impacts. Due to the impacts of cli-
mate change, cities should implement mitigation strategies that alle-
viate heat stress during heat waves (Krayenhoff et al., 2018). In
particular, expected warming in cities arising from urban expansion
and greenhouse gas induced climate change is leading to increased
frequency of average and extreme heat over most land regions across
the US (Georgescu et al., 2014, 2013; Krayenhoff et al., 2018) and the
globe (Stocker et al., 2013). The effect of rooftop PV on the urban
sensible heat fluxes and corresponding near-surface temperature has
been studied with building energy software (Scherba et al., 2011) and
mesoscale models (Masson et al., 2014; Salamanca et al., 2016; Taha,
2013). The simplest approach used is to modify rooftop albedo to re-
present PV module effective albedo, by accounting for an assumed rate
of solar absorption (Taha, 2013). Masson et al. (2014) improved on this
approach and developed a simple PV surface energy balance model;
https://doi.org/10.1016/j.solener.2019.11.041
Received 18 February 2019; Received in revised form 8 September 2019; Accepted 12 November 2019
⁎ Corresponding author at: 975 S Myrtle Ave, Tempe, AZ, USA.
E-mail addresses: nheusing@asu.edu, j.heusinger@tu-braunschweig.de (J. Heusinger), ashley.broadbent@asu.edu (A.M. Broadbent),
David.Sailor@asu.edu (D.J. Sailor), Matei.Georgescu@asu.edu (M. Georgescu).
1 Present address: Institute of Geoecology, TU Braunschweig, Langer Kamp 19c, 38106 Braunschweig, Germany.
Solar Energy 195 (2020) 382–395
Available online 29 November 2019
0038-092X/ © 2019 International Solar Energy Society. Published by Elsevier Ltd. All rights reserved.
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mailto:nheusing@asu.edu
mailto:j.heusinger@tu-braunschweig.de
mailto:ashley.broadbent@asu.edu
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however, this approach includes unevaluated assumptions about PV
energy balance. Lastly, only Scherba et al. (2011) evaluated their PV
model with measured surface temperatures of a PV module, high-
lighting the uncertainty associated with current PV energy balance
modeling approaches.
Several studies on sensible heat flux of plates with varying shapes,
PV modules and building surfaces indicate some uncertainty regarding
the available correlations between wind velocity, temperatures and
sensible heat flux (Jürges, 1924; Kumar and Mullick, 2010; Sharples
and Charlesworth, 1998; Sparrow et al., 1979; Sparrow and Tien, 1977;
Test et al., 1981). The dependency of sensible heat flux on the angle of
attack (AOA) seems to be dependent on the shape of the plate/ PV
module (Sparrow et al., 1979; Sparrow and Tien, 1977). For square
plates almost no dependency of the AOA and yaw was found (Sparrow
and Tien, 1977). For rectangular plates the angle of attack lead to an
uncertainty of± 10% (Sparrow et al., 1979). To give an indication of
the related sensible heat flux uncertainty we implemented four different
well-known correlations (Jürges, 1924; Kumar and Mullick, 2010;
Sharples and Charlesworth, 1998; Test et al., 1981) and examined their
influence on module temperatures.
Another limitation of existing PV surface energy balance models
used in climatological studies is lack of representation of solar tracking
systems. All of the aforementioned models assume PV modules have a
static orientation and angle. In the U.S., the majority of utility scale
solar arrays (> 5 MW) have tracking systems installed (Bolinger and
Seel, 2018). Tracking systems change electricity production and influ-
ence the PV surface energy balance, leading to differing geophysical
impacts throughout the diurnal cycle (Broadbent et al., 2019). This
suggests that the thermal effect of PV modules with tracking systems
has not been represented realistically to date.
Here, a detailed PV energy balance model is presented that is able to
simulate the total radiation received by the panel for different types of
PV tracking systems (non-tracking flat or tilted PV, one-axis and two-
axis tracking systems) and the full range of geophysical processes (e.g.
sensible heat flux exchange) associated with PV systems. The focus on
modeling geophysical impacts makes our model unique among PV
models that are currently available and can be used for novel applica-
tions, such as assessing the thermal impactsof PV on local tempera-
tures. The model is evaluated extensively against surface temperature,
electrical output, and sensible heat flux data, previously presented in
Broadbent et al. (2019). A model inter-comparison with Masson et al.
(2014), the software PVsyst (Mermoud, 1995) (a solar industry stan-
dard) and the Python package PVLIB is performed with emphasis on
module temperatures. The effect of PV on the thermal environment is
evaluated for different tracking and non-tracking systems and its sen-
sitivity to longwave emissivity and conversion efficiency is in-
vestigated. The PV model, hereafter UCRC-Solar, can be used as a stand-
alone program (written in Python) with routinely available meteor-
ological variables. In future work, the model will be implemented and
coupled dynamically within meteorological models (e.g., the Weather
Research and Forecasting System) to analyze the impact of PV array
deployment on near surface air temperature and meteorology.
2. Model description
First, the model structure and model equations are presented, fol-
lowed by a description of the model evaluation procedure. The
minimum input variables needed to execute the model are
List of symbols
Latin symbols
a empirically derived coefficients in objective hysteresis
model (1)
A surface area (m2)
APV absorptivity of silicon (1)
b distance between rows of PV panels (m)
B vertical distance between upper and lower end of PV panel
(m)
CL specific heat capacity of module layer (J kg−1 K−1)
Cmodule heat capacity of PV panel (J K−1)
D horizontal distance between upper and lower end of PV
panel (m)
dL module layer thickness (m)
EffPV maximum electrical conversion efficiency of PV module
(1)
FCLD cloud cover (1)
FPV plan area fraction of PV modules (1)
Fshd shaded fraction of the ground surface (1)
GVFPV ground view factor (1)
hc turbulent convection coefficient (W m−2 K−1)
↓L downwelling longwave radiation (W m−2)
↑L G longwave radiation from ground surface received by the
PV panel (W m−2)
→L PV longwave radiation from adjacent PV panels (W m−2)
↑L PV upwelling longwave radiation from the PV panel (W m−2)
↓L PV downwelling longwave radiation from the PV panel (W
m−2)
↓L sky downwelling longwave radiation received by the PV panel
(W m−2)
∗LPV net longwave radiation of PV panel (W m
−2)
Pout power produced by PV panel divided by A (W m−2)
PVFPV PV view factor (1)
∗Q net radiation at ground surface (W m−2)
QG ground heat flux at the surface (W m−2)
QH_PV sensible heat flux from PV panel (W m−2)
SVFPV sky view factor (1)
↓SW global horizontal radiation (W m−2)
SWcell shortwave radiation reaching the cell surface (W m−2)
↓SW dif diffuse horizontal radiation (W m−2)
SWdir direct horizontal radiation (W m−2)
SWtot total shortwave radiation received by the PV panel (W
m−2)
TA air temperature (K)
TG average ground surface temperature (K)
TPV module temperature (K)
Tshd modeled ground surface temperature in shade (K)
Tsun sun-exposed modeled ground surface temperature (K)
u wind velocity (m s−1)
Greek symbols
αG albedo of ground surface (1)
αPV albedo of PV panel (1)
β tilt angle of the PV panel (°)
εbttm emissivity of downward PV panel surface (1)
εclear clear sky emissivity (1)
εtop emissivity of upward PV panel surface (1)
κ thermal diffusivity (m2 s−1)
ϕ view angle to calculated view fractions (1)
ρL density of module layer (kg m
−3)
σ Stefan-Boltzmann constant (W m−2 K−4)
(πα) transmissivity-absorptance product (1)
θh incidence angle (°)
θzh zenith angle (°)
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
383
meteorological variables that are routinely observed: air temperature,
relative humidity, wind velocity, atmospheric pressure or global hor-
izontal radiation (Table 1). This means that if global radiation is not
provided, it will be modeled for clear sky conditions, which needs at-
mospheric pressure as input. Other variables such as diffuse horizontal
radiation, longwave radiation from sky and ground surface tempera-
tures can be modeled as well, which is explained in more detail in the
next section. The input data can be provided as 30min or 60min
averages and is interpolated to the model time step automatically. Be-
sides that, only the location (Lat/Lon) and some geometrical input in-
formation, e.g. the panel tilt angle is needed to run the model. No
height corrections are implemented in the presented model version, i.e.
the variable heights should approximately match the height of the
modeled PV module. Additional, optional variables that may not be
routinely available for some regions of the world, but can be expected
to improve model performance, are given in Table 2.
2.1. Model structure
The energy balance of the PV module (Fig. 1) is modeled after Jones
and Underwood, (2001):
= + − −∗C dT
dt
L SW Q Pmodule PV PV tot H PV out_ (1)
where Cmodule is the heat capacity (J K−1) of the module, TPV is the
module temperature (K), ∗LPV is the net longwave radiation of the PV (W
m−2), SWtot the total radiation (W m−2),QH PV_ the sensible heat flux (W
m−2) and Pout the electrical power produced by the panel (W m−2). This
approach implies that we calculate the spatially-averaged temperature
of the module which represents a 0D approach. Since PV modules have
very low thermal mass, this is considered a reasonable assumption.
The heat capacity of the module is defined as the sum of the heat
capacity of its individual layers:
∑= ∗ ∗ ∗
=
C A d ρ Cmodule
L
n
L L L
1 (2)
where A is the surface area (m2), dL the thickness (m), ρL the density (kg
m−3) and CL the specific heat capacity of the layer (J kg−1 K−1).
The total radiation incident on the module is calculated after Duffie
and Beckman (2013), where in-plane diffuse radiation is calculated
after Perez et al. (1987) and assuming isotropic reflected radiation from
the ground:
⎜ ⎟= ∗ ⎛
⎝
⎞
⎠
∗ − + + ↓ ∗
∗ ⎛
⎝
− ⎞
⎠
SW SW θ
θ
α SW SW α
β
cos
cos
(1 )
1 cos
2
tot dir
h
zh
PV dif PV G_
(3)
where ↓SW is global horizontal radiation (W m−2), SWdir is direct ra-
diation (W m−2), SWdif PV_ is in-plane diffuse radiation (W m−2), θh and
θzh are the incidence angle (°) and zenith angle (°), respectively and β is
the tilt angle of the module (°), calculated after (Braun and Mitchell,
1983). The contribution of shortwave radiation to the rear side was
assumed to be negligible since monofacial PV modules typically have a
highly reflective, white backsheet. The equations for the calculation of
θh and θzh are given in Stackhouse and Whitlock (2009). θh depends on
θzh, the solar azimuth, the azimuth of the PV module and β. The azi-
muth of the PV module and β in turn depend on the tracking system. All
tracking systems presented in Braun and Mitchell (1983) haven been
implemented, and additionally flat and tilted non-tracking systems
(Fig. 2, Table 3). We assigned IDs to each tracking system for clar-
ification purposes. The reflectivity of the glazing was calculated after
Fresnel’s law. When SW↓ data is not provided the user can chose to
simulate SW↓ for clear sky conditions with the Bird and Hulstrom
(1981) model or the Perez and Ineichen model (Ineichen and Perez,
2002, Perez et al., 2002). In the latter case, the model code from the
Python package PVLIB is used. The diffuse horizontal radiation was
calculated after Ridley et al. (2010), which estimates the diffuse ra-
diation fraction based on the clearness index. The αPV is modified due to
angle of incidence losses after De Soto et al. (2006).
The sensible heat flux is calculated as:
= ∗ ∗ −Q h T T2 ( )H PV c PV A_ (4)
where, hc (W m−2 K−1) is the turbulent convection coefficient and TA is
the air temperature (K). The turbulent convection coefficient is calcu-
lated after four well known correlations (Table 4). The characteristic
length Lchar is calculated for each study and used to normalize the re-
sults based on
hc ∝ Lchar-0.2, as pointed out by Kumar and Mullick (2010):
=L A C4 /char (5)
where A is the surface area and C the circumference of the studied
plate. The resultingsensible heat fluxes from each equation are aver-
aged in the model.
The incoming longwave radiation received at the PV module is
weighted by the view factors of ground (GVFPV), sky (SVFPV) and other
PV modules adjacent to the PV module of interest (PVFPV):
=PVF ϕ/180PV
= −SVF ϕ(180 )/180PV (7)
= −GVF ϕ(180 )/180PV
where ϕ is a view angle at the center of the PV module, which is cal-
culated as the arithmetic average of two view angles ϕ1 and ϕ2 (c.f.
Fig. 1). ϕ1 and ϕ2 are calculated as the sum of ϕ1.1, ϕ1.2 and ϕ2.1, ϕ2.2,
respectively:
=
+
ϕ B
b D
1.1 arctan /2
/2
=
−
ϕ B
b D
1.2 arctan /2
/2 (8)
=
+
ϕ B
c D
2.1 arctan /2
/2
=
−
ϕ B
c D
2.2 arctan /2
/2
where distances B, D, b and c are calculated from trigonometric theory,
as demonstrated in Fig. 3. The fact that SVFPV and GVFPV are calculated
in the same way is explained by symmetry.
The net longwave exchange of a PV module in an array is then
calculated by:
= ↓ + → + ↑ + − ∗ ↓ − ↑ − ↓∗L L L L ε L L L(1 )PV sky PV G G sky PV PV (9)
where
↓ = ↓ ∗L L SVFsky PV (10)
→ = ∗ ∗ ∗ + ∗ ∗ ∗L PVF σ ε T PVF σ ε T
2 2PV
PV
top PV
PV
bttm PV
4 4
↑ = ∗ ∗ ∗L σ ε T GVFG G G PV4
↑ = ∗ ∗L σ ε TPV top PV 4
Table 1
Minimum input variables needed to run the model.
Acronym Variable
TA Air temperature (°C)
RH Relative humidity (%)
u Wind speed (m s−1)
p or ↓SW Atmospheric pressure (hPa)/ Global horizontal radiation (W m−2)
Datetime Date and time in one string
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
384
↓ = ∗ ∗L σ ε TPV bttm PV 4
where ↓L sky (W m−2) is downwelling longwave radiation from the sky
derived from the sky view factor of the PV module, →L PV (W m−2) is
the longwave radiation from adjacent PV modules, εtop and εbttm are the
emissivities of the upward and downward facing sides of the PV
module, respectively. ↑L G is the fraction of the longwave radiation from
the ground surface that is received by the panel, TG is the average
ground surface temperature (K) and TPV is the average module tem-
perature (K). ↑L PV is the upwelling longwave radiation and ↓L PV is
downwelling longwave radiation from the PV module, respectively.
When incoming sky longwave radiation is not available, it is cal-
culated after Loridan et al. (2011):
↓ = + − ∗ ∗ ∗L ε ε F σ T[ (1 ) ]clear clear CLD A4 (6)
where ↓L is the sky longwave radiation (W m−2), εclear is the clear sky
emissivity, FCLD is estimated cloud cover (c.f. Loridan et al., 2011 for
further details) and σ is the Stefan-Boltzmann constant (W m−2 K−4).
The average ground surface temperature is represented by:
= ⎧
⎨⎩
∗ + ∗ − ↓ >
∗ + ∗ −
T
T F T F if SW
T F T F otherwise
(1 ) 0
(1 ),G
shd shd sun shd
shd PV sun PV (11)
where Tshd and Tsun are the ground surface temperature under shaded
and sun-exposed conditions, respectively. The Fshd is the shaded fraction
of the ground surface and FPV is the plan area fraction of the PV. If data
for Tshd and Tsun are not available, they can be modeled with an explicit
finite difference solution of the 1D heat diffusion equation:
∂
∂
= ∂
∂
T
t
κ T
x
( )
2
2 (12)
where κ is the thermal diffusivity of the ground. A heat flux upper and
lower boundary condition is chosen; the lower boundary condition is
assumed to equal 0 and the upper boundary condition is calculated by
the Objective Hysteresis Model (OHM) (Grimmond and Oke, 1999):
= ∗ + ∗ ∂
∂
+∗
∗
Q a Q a Q
t
aG 1 2 3 (13)
and:
∂
∂
= −
∗
+
∗
−
∗Q
t
Q Q0.5( )t t1 1 (14)
where QG (W m−2) is the ground heat flux at the surface, ∗Q is net
radiation (W m−2) and a1, a2 and a3 are empirically derived coeffi-
cients, which are available for different impervious and vegetated
surfaces (Grimmond and Oke, 1999). ∗Q is calculated for sun exposed
and shaded conditions, respectively. This approach of modeling surface
temperatures implies that varying soil moisture conditions and the in-
fluence of evaporation cannot be represented in UCRC-Solar (in offline
mode), when measured ground surface temperatures are not available.
However, planned future work to couple our PV model to a land surface
model that includes hydrological processes will resolve this issue.
The net radiation of the ground surface ∗QG is calculated by:
Table 2
Optional input variables.
Acronym Variable
↓SW dif Diffuse horizontal radiation (W m−2)
↓L Longwave radiation from sky (W m−2)
TG Ground surface temperature (°C)
Fig. 1. Schematic visualization of the PV radiation and energy balance.
Fig. 2. Schematic visualization of implemented tracking systems in UCRC-Solar. The arrows indicate the orientation of the rotation axes of the different systems.
Table 3
Tracking systems implemented in UCRC-Solar.
Tracking system ID Tracking system
1 flat, non-tracking
2 tilted, non-tracking
3 1-axis tracking system with horizontal rotation axis
4 1-axis tracking system with sloped rotation axis
5 vertical 1-axis tracking system, with fixed sloped PV
module
6 2-axis tracking system
Table 4
Equations for calculating the turbulent convection coefficient hc and char-
acteristic lengths Lchar.
Study Equation Lchar
Jürges (1924) ∗ + ∗− ∗e u4.6 6.137u0.6 0.78 not defined
Test et al. (1981) ∗ +u2.56 8.55 0.976
Sharples and Charlesworth (1998) ∗ +u3.3 6.5 1.193
Kumar and Mullick (2010) ∗ +u3.87 6.9 3.45
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
385
= ↓ − ∗ ∗ + ↓ ∗ −∗Q L ε σ T SW α(1 )G G sun shd G/ 4 (15)
where αG is the albedo of the ground surface. The electrical output of
the module is calculated with slight modifications after Masson et al.
(2014):
= ∗ ∗ − ∗ −P SW Eff Tmin[1, 1 0.005 ( 298.15)]out cell PV PV (16)
where EffPV is the maximum electrical energy conversion efficiency of
the PV module at a reference solar radiation of 1000Wm−2 and a re-
ference temperature of 25 °C and SWcell is the shortwave radiation that is
transmitted through the glazing and absorbed by the cell surface:
⎜ ⎟= ∗ ∗ ∗ ⎛
⎝
⎞
⎠
+ ∗
∗ ⎛
⎝
+ ⎞
⎠
+ ↓ ∗ ∗ ∗ ⎛
⎝
− ⎞
⎠
SW M SW πα θ
θ
SW πα
β
SW πα α
β
( ( ) cos
cos
( )
1 cos
2
( )
1 cos
2
)
cell dir dir
h
zh
dif dif
G G
(17)
where πα( ) is the transmissivity-absorptance product of the glazing for
direct (dir), diffuse (dif) and ground reflected (G) diffuse radiation
calculated after Duffie and Beckman (2013). The air mass modifier (M)
accounts for changes in the spectral distribution of the incident radia-
tion and is calculated after King et al. (2004).
2.2. Model evaluation
2.2.1. Module and site characteristics
To evaluate our PV model, we use meteorological and eddy covar-
iance data collected from a utility-scale PV power station in Southern
Arizona (32°33′16.6″N, 111°17′03.7″W) managed by Arizona Public
Service (APS) over a period of nine months (Broadbent et al., 2019).
Eddy covariance is a state-of-the art method to measure turbulent
fluxes, such as the sensible heat flux of specific land cover types through
high frequency (> 10Hz) measurements of 3-dimensional wind fields,
temperature, and humidity. For a more detailed explanation and further
information about the instruments including their accuracy, we refer to
Broadbent et al. (2019). The 40MW facility has ~180,000 Trina Solar
TSM-315PD14 modules in operation and due to its size, is an excellent
site for studying the dynamic thermal impacts of PV systems. The
evaluation site has a horizontal 1-axis type tracking system (Tracking
system #3, Table 3). The height of the horizontal axis is 1.33m, the
length of the module perpendicular to the tilt axis is 1.96m and the
width between PV rows is 5.64m. This results in a plan area fraction of
35% when the PV panels are horizontal. The tilt angle was set to 30° at
night at the validation site, therefore we set β =30° for θzh < 0.
During daytime, the maximum tilt angle was 60°. Based on the geo-
metry described above shading losses were assumed to be negligible.
The ground is bare soil with an albedo of 0.3. Further details about the
validation site are provided in Broadbent et al. (2019).
The PV modules consist of a glazing, cell material and a backingmaterial layer (Table 5). The heat capacities of the individual layers of
the PV modules at the validation site were not available. Instead, we
used generic values for the polycrystalline PV type (Davis et al., 2002)
(Table 5). The module surface temperatures were measured with two
Campbell Scientific CS220-L/CS240 thermocouples which were at-
tached to the backside of two PV modules. The emissivities of the upper
(glazing) and lower (backing) PV module surface were estimated to be
0.82 and 0.97, respectively by comparing the thermocouple measure-
ments with measurements conducted with an infrared temperature
camera, adjusting the emissivity setting. EffPV was set to 0.16 based on
the data sheet of the manufacturer (Trina Solar).
2.2.2. Data for model evaluation
The PV energy balance model was evaluated against measurements
from two thermocouples which were attached at the backside of two PV
modules. The model was further evaluated by comparison of modeled
against measured sensible heat fluxes by the eddy-covariance method at
the PV array and a reference site. At the PV array site, the energy
balance was not closed, which is a common phenomenon of EC mea-
surements (Broadbent et al., 2019). For the model evaluation, the
missing energy was allocated to sensible heat since latent heat was
assumed to be 0 at this desert site. The reference site is a nearby un-
modified desert site where measurements were taken for comparison.
Sensible heat fluxes at the PV array site contain the signal of the PV
modules as well as the ground surface. Therefore, a qualitative com-
parison was conducted and the following hypothesis was tested: mea-
sured sensible heat fluxes from the PV site should be between the
modeled fluxes from the PV module and the measured fluxes from the
reference site. Since ground surface temperatures were not available,
the finite difference scheme of the 1D heat diffusion equation (Eq. (12))
was employed with soil properties for sandy soil given in Table 6.
The data for the total electrical power output from the PV array
(MW) was kindly provided by the electrical service company, Arizona
Public Service (APS) in 5min time steps. The data was aggregated to
30min time steps and then divided by the total area of the PV modules
for comparison with model output which is calculated in W m−2. The
model output is the power generated by an individual PV module.
System losses e.g. due to the inverters were not accounted for.
The model error was evaluated by linear regression between mea-
sured and modeled module temperature data, the root mean square
error (RMSE) and mean bias (MB). The evaluation was conducted for
clear sky and cloudy conditions separately based on the mean daily
Fig 3. (a) Top view on two PV panel rows and (b) side view of two PV panels along axis b and (c) side view of two PV panels along axis c.
Table 5
Material properties for PV module layers for the polycrystalline type (Davis
et al., 2002).
Layer (material) dL (m) ρL (kg m−3) CL (J kg−1 K−1)
Glazing (Glass) 0.006 2500 840
Cell (Silicon) 0.00038 2330 712
Backing (Tedlar/Mylar) 0.00017 1475 1130
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
386
clearness index between 10:00 and 16:00 MST. Clear sky and cloudy
conditions were defined for a mean daily clearness index ≥0.7 and<
0.7, respectively.
2.3. Model intercomparison
We compared the results of UCRC-Solar against the software PVsyst
as a solar industry standard, the Python package PVLIB and a model
presented in Masson et al. (2014) which was used in mesoscale me-
teorological simulations.
In PVsyst a 1-axis horizontal tracking system was simulated with the
database values for the Trina Solar TSM-315PD14 module. Otherwise,
default values were used, e.g. the recommended loss factors for free
mounted systems, Uc=29W/(m2k) and Uv=0.0W/(m2k)/m/s.
PVsyst was run with the same data that was used for UCRC-Solar by
importing it into the PVsyst meteo database.
In PVLIB, module temperatures are calculated after King et al.
(2004) with the Sandia PV Array Performance Model, with in-plane
solar irradiance, wind speed (10m height) and ambient air temperature
as input variables. Measured wind speed at 1.5m was calculated for a
height of 10m assuming a logarithmic wind profile. In-plane solar ir-
radiance was modeled in PVLIB by using the single axis tracking
module, calculating direct normal irradiance with the DIRINT module
(Perez et al., 1992) and modeling diffuse radiation after Perez et al.
(1987).
The required meteorological input for the Masson model was pro-
vided by measured data (TA, ↓ ↓SW L, sky) and incoming longwave ra-
diation from the ground was provided by UCRC-Solar to provide similar
boundary conditions. In contrast to the original study, where a Ross
parameter value of 0.05 for sloped roofs was chosen, we used a value of
0.02 for free standing systems to ensure a fair comparison (Skoplaki
et al., 2008).
3. Results
3.1. Model evaluation
The model was evaluated by comparing measured and modeled PV
module temperatures, sensible heat fluxes and the electrical power
output. The evaluation periods were June 1–17, 2018 and December
1–31, 2017 and classified into clear sky and cloudy conditions (c.f. Fig.
A1). December is typically the coolest month in Southern Arizona and is
characterized by sporadic precipitation events. By contrast, June typi-
cally precedes the onset of the North America Monsoon System (Adams
and Comrie, 1997) with daily maximum temperatures routinely ex-
ceeding 40 °C and low precipitation. These periods were selected so that
different seasonal conditions (i.e. summer vs winter) and solar angles
could be compared and contrasted. Weather statistics collected for both
study periods are provided in Table 7. The precipitation data was not
available directly at the validation site and was instead collected from
Tucson International Airport. Both periods had two days with pre-
cipitation, respectively.
3.1.1. Module temperature
A linear regression between modeled and measured TPV indicates a
good agreement for both June and December (Fig. 4). The comparison
was conducted for clear and cloudy sky conditions. In both cases, the
deviation is small, characterized by small RMSE and mean bias as well
as a slope of the linear regression close to 1. The high agreement in-
cludes maximum daytime temperatures as well as minimum night-time
temperatures as demonstrated by Fig. 5. We studied the differences/
uncertainty in TPV caused by the maximum and minimum QH from the
four different convective heat transfer coefficients calculations (Fig. 5).
It is demonstrated that the temperature differences are mostly< 3 K.
The sensible heat fluxes vary on average by 13% at daytime and larger
differences of about 25% at night-time.
The dependence of the model error on time stepping was analyzed
for dt= 10 sec, 60 sec, 600 sec and 1800 sec, respectively (Table 8).
The error estimates were calculated by comparing the 30min averages
of the model output and the measurements. The error estimates vary
slightly between dt= 10 sec and dt= 600 sec. For dt= 1800 sec the
model became unstable, so that error estimates could not be calculated.
In this case the user is notified to reduce the time step. From this
analysis, it is demonstrated that the model can be run with up to a
10min time step and that the model error is insensitive to the time step
length until the model becomes numerically unstable. Therefore, we
recommend using a model time step of ≤600 sec.
The evaluation demonstrates that UCRC-Solar can capture diurnal
variations in the timing and amplitude of TPV under cloudy and clear
sky, summer and winter conditions.
3.1.2. Sensible heat flux
The sensible heat fluxes were compared for clear sky and cloudy
conditions (Fig. 6). The PV model calculates QH from the module (Eq.
(4)) but not the flux for surrounding ground surfaces. However, the
eddy-covariance measurements at the PV array site represent the
combined sensible heat fluxes for both the ground andPV modules. As
such, a qualitative comparison of QH is done below.
The comparison against measurements from the PV array and a
nearby reference site (without PV) demonstrates that the measured QH
from the PV array (red line, Fig. 6) is mostly between the modeled PV
module QH (blue line, Fig. 6) and the measured reference values (green
line, Fig. 6). This implies good model skill since the QH at the PV array
contains a mixed signal from ground and PV modules.
3.1.3. Power output
The mean diurnal variation of modeled and measured Pout was
compared for Jun 2018 and Dec 2017 for clear sky and cloudy condi-
tions, respectively (Fig. 7). The error estimates underline a good
agreement with a low systematic error (Table 9).
3.2. Model intercomparison
We compared the results of UCRC-Solar with the model of Masson
et al. (2014) which was used to evaluate the thermal effect of PV in
Paris, France and implemented in the Weather Research and Fore-
casting Model (WRF) to evaluate the thermal effect of PV in Phoenix,
Table 6
Ground thermal and modeling domain properties.
Parameter Value
a1 0.35
a2 0.43
a3 −36.5
heat conductivity k (W m−1 K−1) 0.3
specific heat capacity Cp (J kg−1 K−1) 921
density ρ (kg m−3) 1520
depth L (m) 1
number of grid points nz 51
Table 7
Meteorological characterization for the two study periods in June 1–17, 2018
and December 1–31, 2017.
Meteorological variables Jun Dec
P (mm) 24.2 12.7
TA_max (°C) 45.0 28.5
TA_min (°C) 16.3 −3
TA_mean (°C) 31.8 12.6
umean (m s−1) 1.5 1.3
GHImean (MJ m−2 day−1) 27.27 11.17
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
387
AZ (Salamanca et al., 2016), PVsyst as a standard software in the solar
industry and the python package PVLIB (Fig. 8). Since PVsyst and
PVLIB are focusing on the correct prediction of the power output, TPV is
calculated for daytime only in these models. Therefore, the same time
period was chosen for UCRC-Solar and the Masson model in this in-
tercomparison study.
Overall, the coefficient of determination indicates a high correlation
with measured TPV for all models, but the systematic errors (indicated
by the slope of the linear regression) in both the Masson et al. and the
PVsyst model are higher compared to UCRC-Solar. Parameter estima-
tion of the thermal loss factors Uc and Uv in PVsyst and the parameters
a, b in PVLIB could potentially improve the agreement. However, this
was not attempted since it would not represent a typical use case of the
software. Instead, the default values of Uc and Uv for free-standing
systems in PVsyst and default values for an open rack mount with a
Glass/cell/polymer sheet module type in PVLIB were chosen. Also, all
models but UCRC-Solar have RMSE values ≥4 K. Nevertheless, PVLIB
has a very low systematic error, which underlines the validity of the
approach by King et al. (2004).
While nighttime TPV is defined being equal to the ambient air
temperature in the Masson model, it erroneously predicts strong ne-
gative sensible heat fluxes at night; this is likely because QH is calcu-
lated as a residual in this approach (Fig. 9). The large negative heat flux
calculated by the Masson’s approach may explain why Masson et al.
(2014) and Salamanca et al. (2016) concluded that PV expansion can
lead to significant urban nighttime cooling.
The results of the geo-solar parameters, e.g. solar altitude and the
modeled PV tilt angles and the incident angles were compared between
Fig. 4. Scatterplot of modeled vs measured module
temperatures for clear sky (a, c) and cloudy condi-
tions (b, d) for June 2018 (a, b) and December 2017
(c, d). The blue line indicates a linear regression fit
and the dashed line shows the 1:1 line. (For inter-
pretation of the references to colour in this figure
legend, the reader is referred to the web version of
this article.)
Fig. 5. Comparison between modeled and measured module surface temperatures (TPV) for June 2018 (b) and the TPV differences caused by the differences between
maximum and minimum sensible heat fluxes (QH) (a).
Table 8
Module surface temperature (TPV) error estimates in dependence on time step
for June 2018.
dt (sec) RMSE (K) Mean bias (K)
10 2.14 1.31
60 2.12 1.31
600 2.20 1.31
1800 NaN NaN
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
388
UCRC-Solar and PVsyst and were almost identical (c.f. Fig. A2), which
means that the causes of the TPV differences between PVsyst and UCRC-
Solar are to be found in the TPV parameterization and/or in the diffuse
radiation parameterization.
3.3. Model application
3.3.1. Tracking/Non-tracking systems
The model can be run with four different tracking systems and two
Fig. 6. Comparison between modeled and measured QH at the PV array and a reference site without PV (ref site) for Jun 2018 (a, b) and Dec 2017 (c, d) for clear sky
(a, c) and cloudy conditions (b, d), respectively. The standard deviation is indicated by shaded areas.
Fig 7. Mean diurnal modeled and measured electrical power output for Jun 2018 (a, b) and Dec 2017 (c, d) for clear sky (a, c) and cloudy conditions (b, d),
respectively. The standard deviation is indicated by shaded areas.
Table 9
Error estimates for Pout.
Jun Dec
RMSE (W m−2) 13.8 9.1
MB (W m−2) −2.6 0.82
R2 0.93 0.93
Slope 1.04 0.95
Intercept 0.60 0.44
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
389
different non-tracking systems (Fig. 10). The tracking systems modulate
the total radiation received by the module during the day and thereby
influence TPV as well as Pout. The tilt angle of the PV system 2 and the
slope of the axis of PV system 4 were set to 15° and the azimuth was set
to 0°, i.e. south oriented. The magnitude of the effects on TPV and Pout
are dependent on the sun’s path, which is demonstrated by using input
data from June 2018 versus December 2017 from the evaluation site. In
June, the maximum difference between the two systems is 8 K
(Fig. 10a). However, in December the TPV differences between the 2
axes system and the flat, horizontal module reach a maximum of 9 K in
the mean diurnal course (Fig. 10b). Similarly, for Pout higher differences
between the systems occur in Dec compared to Jun. Assuming no lim-
itations imposed by PV array infrastructure, our model demonstrates
that a 2-axis tracking system can increase Pout by ~50Wm−2 during
the afternoon solar maximum. These findings illustrate that the impacts
of tracking systems in lower latitudes are more apparent during the
winter season where solar angles are lower.
3.3.2. Sensitivity of QH to longwave emissivity and conversion efficiency
Given the concern that PV arrays could lead to a local heating im-
pact (Barron-Gafford et al., 2016; Broadbent et al., 2019), we con-
ducted a series of sensitivity tests to assess how PV module QH varies
with module emissivity and maximum conversion efficiency. The effect
of the longwave emissivity of the upper side of the PV module on QH
was studied by increasing the emissivity in 0.05 steps from 0.85 to 0.95
(lower side emissivity being constant at 0.95). Furthermore, the effect
of the conversion efficiency was studied in three different scenarios
with 20%, 30% and 40% conversion efficiency. The effects were studied
for Jun 4, 2018, a clear sky day at the validation site location for a
horizontal tracking system (Table 10). It is demonstrated that the cu-
mulative daily sensible heat flux can be substantially reduced by in-
creasing the conversion efficiency, while the emissivity has a non-
Fig. 8. Module temperatures of UCRC-Solar (a), Masson et al. (b), PVsyst (c) and PVLIB (d) compared against measured TPV for the June 2018 daytime validation
data.
Fig. 9. Modeled sensible heat fluxes by Masson et al. model and UCRC-Solar.
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
390
negligible effect as well (Fig. 11). According to the model, the cumu-
lative sensible heat flux is linearly dependent on both parameters (c.f.
Fig. A3and Fig. A4).
The cumulative QH at the reference site (no PV influence, sandy soil,
α~0.3) for thesame day was 5.4 MJ m−2. The results indicate that a
substantial increase in conversion efficiency is needed to potentially
remove the heating effect of PV arrays on the near surface atmosphere
observed by Barron-Gafford et al. (2016) and Broadbent et al. (2019)
4. Discussion
The presented PV surface energy balance model accurately re-
produces the complete diurnal variation of TPV as demonstrated by low
RMSE < 3 K and mean bias values (−2 K < MB < 2K) under clear
sky and cloudy conditions, in different seasons, and over all hours of a
day. A qualitative evaluation indicates that the sensible heat fluxes are
plausible and in good agreement with measured sensible heat fluxes by
eddy-covariance. The model results are not sensitive to the time step as
long as a time step is chosen, which is below the numerical instability
threshold. The model can be run with a minimum of just 4 meteor-
ological variables (TA, RH, u, p or SW↓), which are routinely available.
For example, all the input variables including optional variables are
available from the National Solar Radiation Database (NSRDB) for the
whole contiguous United States. UCRC-Solar can be used to model PV
modules above different ground surfaces, such as dry soil or concrete
surfaces by changing the ground thermal properties.
Current limitations of the model are: (1) the effects of precipitation
and evaporation are not reflected in the model; (2) solar panels which
are directly attached to the surface (i.e. without having an air layer
between solar panel and roof/ground surface) are also not represented
in the current version; (3) the sensible heat flux is not calculated in
dependence on the tilt angle, i.e. in dependence on the angle of attack;
(4) a more detailed, physical representation of the electrical output is
desirable. Improvements on these limitations are planned in upcoming
Fig. 10. Module temperatures and power output for different tracking systems for June 2018 (a, c), and December 2017 (b, d).
Table 10
Meteorological description for Jun 4, 2018.
Meteorological variable Value
TAmax (°C) 44.1
TAmin (°C) 21.2
RHmax (%) 15.4
RHmin (%) 4.2
umax (m s−1) 3.1
umin (m s−1) 0.4
SW↓max (W m−2) 1010
Pmean (hPa) 1011
Fig. 11. Modeled sensible heat flux in dependence on conversion efficiency and
emissivity.
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
391
releases of UCRC-Solar. Detailed empirical studies on the influence of
the angle of attack on the sensible heat flux emitted by PV modules
would be desirable for model improvements. An earlier study indicated
that different angles of attack lead to a QH uncertainty of± 10% with
rectangular plates (Sparrow et al., 1979).
The URCR-Solar model is focused on capturing PV surface energy
balance and not PV electricity production. However, accurately pre-
dicting Pout is necessary to model TPV and QH_PV. We like to emphasize
that UCRC-Solar does not represent the true energy yield of a PV array,
because the systems losses such as wiring resistance and inverter effi-
ciency are not represented within the model. In its current form, UCRC-
Solar is only concerned with processes which directly affect TPV.
However, an accurate calculation of the energy yield of PV modules
which are connected within an array could be added in future work.
Limitations in the model evaluation are: (1) UCRC-Solar results are
sensitive to the ground surface temperatures. However, the exact soil
thermal properties were not known at the validation site. Typical values
for sandy soil were chosen therefore, introducing some uncertainty. (2)
The exact thermal properties of the PV modules at the validation site
were also not known and generic values for polycrystalline modules
have been used. Despite that, the evaluation of the module tempera-
tures showed high agreement both under clear and cloudy conditions.
We therefore conclude that the model is robust and generalizes well.
Compared to the model presented from Jones and Underwood
(2001), adaptations have been made for the sensible heat flux calcu-
lation and the longwave exchange. Jones and Underwood (2001) for
example used a constant convection coefficient for two different wind
velocity ranges, respectively, whereas it is dynamically calculated in
the present model. Furthermore, different tracking systems can be re-
presented in accordance with Braun and Mitchell (1983). In this ap-
proach, the optimal tilt angle of the PV module is calculated by trigo-
nometrical calculations based on the sun’s path.
The model intercomparison study indicated that the most accurate
estimation of TPV is done by UCRC-Solar (without parameter estima-
tion). TPV simulated by PVLIB was also in high agreement with mea-
sured TPV. Nevertheless, the regression approach implemented in PVLIB
e.g. does not allow to study the effects of different emissivities on TPV.
UCRC-Solar can be used to simulate the thermal effect of PV, i.e.
warming/cooling impacts by evaluating the module temperatures and
sensible heat fluxes. It was demonstrated that different tracking systems
Fig. A1. Meteorological conditions during the December 2017 and June 2018 validation periods including model output for diffuse and total shortwave radiation.
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
392
have a non-negligible effect on module temperatures with mean diurnal
differences of up to 9 K in December and consequently sensible heat
fluxes. This effect was not represented in earlier mesoscale studies on
the air temperature effect of large PV deployments (Masson et al., 2014;
Millstein and Menon, 2011; Salamanca et al., 2016; Taha, 2013). For
urban studies, this might be of lower significance since mostly tilted,
fixed systems are used. But also in the urban context a differentiation
between tilted and flat, horizontal systems has to be made. For studies
on the effect of large scale, rural PV deployments the representation of
tracking systems is of high significance, since the majority of utility
scale solar arrays (> 5 MW) is reported to be installed with tracking
systems in the U.S. (Bolinger and Seel, 2018). The model application
demonstrated that the results are highly sensitive to conversion effi-
ciency and longwave emissivity of the PV module. Both parameters
might increase in future PV module developments. Therefore, studies
on the impact of PV arrays in future climates have to take scenarios into
account in which these parameters are increased compared to today’s
average values. The model intercomparison results further indicate a
necessity to re-assess the meso-scale thermal impacts of PV array de-
ployment in urban as well as rural environments. A global scale PV
impact re-evaluation might as well be advisable. We plan to integrate
UCRC-Solar in WRF to study the impact of PV arrays on near-surface air
temperatures.
5. Conclusions
The model accurately reproduces the dynamic, physical behavior of
PV modules as demonstrated by a detailed validation with measured
module temperatures, electrical output and sensible heat fluxes. The
qualitative evaluation of QH indicated that modeled sensible heat fluxes
are reliable. The intercomparison with other well-known PV models
revealed that UCRC-Solar has the lowest deviations from measured TPV.
Fig. A2. Comparison of geo-solar parameters, PV tilt angle and diffuse radiation model results between PVsyst and UCRC-Solar.
Fig. A3. Dependency of sensible heat flux (QH) on emissivity for different constant conversion efficiencies.
J. Heusinger, et al. Solar Energy 195 (2020) 382–395
393
In contrast to other PV modeling software it can be used to calculate
night-time TPV, which is important for meteorological studies, such as
studies on the effect of large scale PV deployment on the urban thermal
environment. It was demonstrated that different types of tracking and
non-tracking PV systems have a considerable effect on TPV. The impacts
of tracking systems need to be considered when the meteorological
effects of PV panels are studied, and UCRC-Solaris able to represent
these effects. Further, UCRC-Solar can be run with meteorological data
that is routinely available making it easily accessible to those interested
in PV modeling. Therefore, we conclude that the model is suitable for
studies using the stand-alone Python program (UCRC-Solar) and sui-
table to be implemented in meteorological models, e.g. the Weather
Research and Forecasting system (WRF) to enable studies on the effects
of PV on larger, spatial scales. UCRC-Solar is available upon request to
the first author.
Acknowledgements
We would like to thank Arizona Public Service (APS) for providing
access to the PV array at Red Rock, AZ. Ashley Broadbent and Matei
Georgescu acknowledge support from the National Science Foundation
Sustainability Research Network (SRN) Cooperative Agreement
1444758 and SES-1520803.
Appendix A
See Figs. A1–A4.
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	Introduction, evaluation and application of an energy balance model for photovoltaic modules
	Introduction
	Model description
	Model structure
	Model evaluation
	Module and site characteristics
	Data for model evaluation
	Model intercomparison
	Results
	Model evaluation
	Module temperature
	Sensible heat flux
	Power output
	Model intercomparison
	Model application
	Tracking/Non-tracking systems
	Sensitivity of QH to longwave emissivity and conversion efficiency
	Discussion
	Conclusions
	Acknowledgements
	mk:H1_20
	References