Photovoltaic Arrays[LINK]
The Photovoltaics.f90 module includes three different
models referred to as “Simple”, “Equivalent One-Diode” and
“Sandia” and the choice will determine the mathematical models
(and input data) used to determine the energy produced by
solar/electric conversion panels. The EnergyPlus photovoltaic
array models are called one at a time at the HVAC system
timestep along with other electrical generation components
such as gas turbines and diesel engines.
All of the photovoltaic models share the same models for
predicting incident solar radiation that are also used for the
solar thermal calculations and are described in the section
Climate, Sky and Solar/Shading Calculations.
Note that some of the terminology used to discussed
photovoltaics overlaps with terminology used to discuss
Fortran programs. The word module may refer to a PV
panel or to a fortran90 programming entity. Model may
refer to a manufacturers production model for a specific type
of PV module or to a mathematical model used for engineering
analysis. Array may refer to a collection of PV
modules wired together or to a mathematical variable with
multiple elements.
The PV modules are assumed to always run when the total
incident solar is greater than 0.3 Watts. If the incident
solar is less than 0.3, then the modules produce no power.
PV arrays are managed by an electric load center. The load
center is a “load” with respect to generating equipment but is
a “supply center” for the rest of the building. PV arrays need
to be connected to ElectricLoadCenter:Distribution
objects that have a DC buss type.
The Generator:PV:Simple object describes about the simplest
model for predicting photovoltaic energy production. In this
model the user specifies the efficiency with which surfaces
convert incident solar radiation to electricity. (In the other
models this efficiency is determined as part of the model.)
The full geometric model for solar radiation is used,
including sky models, shading, and reflections, to determine
the incident solar resource. The model accepts arbitrary
conversion efficiencies and does not require actual production
units be tested to obtain empirical performance coefficients.
(The Energy+.idd sets the range of conversion efficiencies to
be on [0..1], but the user could alter the Energy+.idd to
extend this range if desired.)
Mathematical
Description[LINK]
Nomenclature for Simple Photovoltaic model
| Mathematical variable |
Description |
| P |
Electrical power produced by
photovoltaics [W] |
| Asurf |
Net area of surface
[m2] |
| factiv |
Fraction of surface area with
active solar cells [ ] |
| GT |
Total solar radiation incident
on PV array [W/m2] |
| cell |
Module conversion efficiency [
] |
| invert |
DC to AC conversion efficiency [
] |
The usable electrical power produced by a PV surface are
calculated using:
On the right hand side of this equation, only GT
is calculated by EnergyPlus and the rest are user inputs.
Power levels are assumed constant over the timestep to arrive
at energy production.
There are two modes that can be selected by the user that
govern how the PV system is coupled to the building surfaces.
If the integration mode is selected as ‘DECOUPLED’ then no
adjustments are made to account for energy extracted in the
form of electricity. If the integration mode is selected as
‘INTEGRATED’ then the energy extracted in the form of
electricity is removed from surface heat transfer calculations
using a sink term. This sink term is lagged from the previous
timestep.
Equivalent One-Diode
Model[LINK]
This model predicts the electrical performance of a
photovoltaic (PV) array. This model is also known as the
“TRNSYS PV” model.
Mathematically speaking, the EnergyPlus PV module employs
equations for an empirical equivalent circuit model to predict
the current-voltage characteristics of a single module. This
circuit consists of a DC current source, diode, and either one
or two resistors. The strength of the current source is
dependent on solar radiation and the IV characteristics of the
diode are temperature-dependent. The results for a single
module equivalent circuit are extrapolated to predict the
performance of a multi-module array.
The module employs a “four-parameter” equivalent circuit to
model crystalline (both mono and poly) PV modules developed at
the University of Wisconsin – Madison [2]. The values of these
parameters cannot normally be obtained directly from
manufacturers’ catalogs. However, the PV module will
automatically calculate them from commonly available data. The
PV module also includes an optional incidence angle modifier
correlation to calculate how the reflectance of the PV module
surface varies with the angle of incidence of solar
radiation.
The module determines PV current as a function of load
voltage. Other OUTPUTS include current and voltage at the
maximum power point along the IV curve, open-circuit voltage,
short circuit current as well as electrical load met and
unmet.
General Nomenclature for the PV model
| Mathematical variable |
Description |
| |
Slope of PV array [degrees] |
| |
Empirical PV curve-fitting
parameter |
| |
Semiconductor bandgap [eV] |
| c |
Module conversion
efficiency |
| |
Temperature coefficient of
short-circuit current [A/K] |
| |
Temperature coefficient of
open-circuit voltage [V/K] |
| |
Angle of incidence for solar
radiation [degrees] |
| |
Module transmittance-absorptance
product |
| normal |
Module transmittance-absorptance
product at normal incidence |
| GT |
Total radiation incident on PV
array |
| GT,beam |
Beam component of incident
radiation |
| GT,diff |
Diffuse component of incident
radiation |
| GT,gnd |
Ground-reflected component of
incident radiation |
| GT,NOCT |
Incident radiation at NOCT
conditions |
| GT,ref |
Incident radiation at reference
conditions |
| I |
Current |
| IL |
Module photocurrent |
| IL,ref |
Module photocurrent at reference
conditions |
| Io |
Diode reverse saturation
current |
| Io,ref |
Diode reverse saturation current
at reference conditions |
| Isc |
Short-circuit current |
| Isc,ref |
Short-circuit current at
reference conditions |
| Imp |
Current at maximum power point
along IV curve |
| Imp,ref |
Current at maximum power point
along IV curve, reference conditions |
| IAM |
Dimensionless incidence angle
modifier |
| K |
Boltzmann constant [J/K] |
| NP |
Number of modules in parallel in
array |
| NS |
Number of modules in series in
array |
| Ns |
Number of individual cells in
module |
| P |
PV output power |
| Pmax |
PV output power at maximum power
point along IV curve |
| Q |
Electron charge constant |
| Rs |
Module series resistance
[] |
| Rsh |
Module shunt resistance [] |
| Tc |
Module temperature [K] |
| Tc,NOCT |
Module temperature at NOCT
conditions [K] |
| Tc,ref |
Module temperature at reference
conditions [K] |
| UL |
Array thermal loss
coefficient |
| V |
Voltage |
| Vmp |
Voltage at maximum power point
along IV curve |
| Vmp,ref |
Voltage at maximum power point
along IV curve, reference conditions |
| Voc |
Open-circuit voltage |
| Voc,ref |
Open-circuit voltage at
reference conditions [V] |
Mathematical
Description[LINK]
PV Section 1:
Four-Parameter Model[LINK]
The four-parameter equivalent circuit model was developed
largely by Townsend [1989] and is detailed by Duffie and
Beckman [1991]. The model was first incorporated into a
component for the TRNSYS simulation package by Eckstein
[1990]. The EnergyPlus module employs the Eckstein model for
crystalline PV modules, using it whenever the short-circuit IV
slope is set to zero or a positive value as modified by
Ulleberg [2000]. The four parameter model assumes that the
slope of the IV curve is zero at the short-circuit
condition:
This is a reasonable approximation for crystalline modules.
The “four parameters” in the model are
IL,ref, Io,ref, , and
Rs. These are empirical values that cannot
be determined directly through physical measurement. The
EnergyPlus model calculates these values from manufactures’
catalog data as discussed in the following section on
calculating these parameters
The four-parameter equivalent circuit is shown in the
following figure:
V is the load voltage and I is the
current flowing through the load and PV.
Determining Performance under Operating
Conditions
The IV characteristics of a PV change with both insolation
and temperature. The PV model employs these environmental
conditions along with the four module constants
IL,ref, Io,ref, , and
Rs to generate an IV curve at each
timestep.
The current-voltage equation of circuit shown in the
previous figure is as follows:
Rs and are constants. The
photocurrent IL depends linearly on
incident radiation:
The reference insolation Gref is nearly
always defined as 1000 W/m2. The diode reverse
saturation current Io is a temperature
dependent quantity:
Equation gives the current implicitly as a function of
voltage. Once Io and
IL are found from Eqs. 3 and 4, Newton’s
method is employed to calculate the PV current. In addition,
an iterative search routine finds the current
(Imp)~~ and voltage
(Vmp) at the point of maximum power along
the IV curve.
Calculating IL,ref, Io,ref,
, and Rs
The Idf specification for the PV model include several
values which must be read from manufacturers’ PV module
catalogs. The manufactures’ values are used to determine the
equivalent circuit characteristics IL,ref,
Io,ref, , and Rs.
These characteristics define an equivalent circuit that is
employed to find the PV performance at each timestep, as
described previously. This section describes the algebra and
calculation algorithms used to solve for the four equivalent
circuit characteristics.
Three of these values, IL,ref,
Io,ref, , may be isolated algebraically.
The first step is to substitute the current and voltage into
Eq. at the open-circuit, short circuit, and maximum power
conditions:
In each case the “-1” term is may be dropped to simplify
the algebra. This approximation has little influence on the
right side of the equations since because the magnitude of
Iois very small, generally on the order of
10-6 A. Some rearrangement then yields the
following three expressions which isolate
IL,ref, Io,ref, :
At this point an additional equation is needed in order to
determine the last unknown parameter. Taking the analytical
derivative of voltage with respect to temperature at the
reference open-circuit condition derives the fourth equation.
This analytical value is matched to the open-circuit
temperature coefficient, a catalog specification:
where
The “TRNSYS PV model” uses an iterative search routine in
these four equations to calculate the equivalent circuit
characteristics. The first step is to set upper and lower
bounds for the series resistance parameter
Rs: physical constraints require the
Rs~~value to lie between 0 and the value
such that = Ns. The initial guess for
Rs is midway between these bounds. and
Io,ref are found from Eq. and Eq. , while
Eq. gives a trivial solution for IL,ref.
The model then employs Eq. to compare the analytical and
catalog values for voc. When all other variables
are held constant, the analytical value for voc
increases monotonically with series resistance (Townsend
1989). If the analytical voltage coefficient is less than the
catalog value, the lower bound for Rs is
reset to the present guess value. Likewise, the upper bound is
set to the current value if the calculated voc is
too large. After resetting the upper or lower bound for
Rs, a new guess value is found by
averaging the bounds. This procedure repeats until
Rs and converge. Note that for
IL,ref, Io,ref, , and
Rs are assumed to be constant and are
calculated only on the first call in the simulation.
Alternatively, the user may enter a known series resistance by
entering a positive value in the IDF. In this
case the iterative routine described above is skipped and Eqs.
, , and find IL,ref,
Io,ref, and directly from the given
value of Rs.
PV Section
2 : Module Operating Temperature[LINK]
The PV model uses one of five methods for determining cell
temperature data. The cell temperature of a PV module is
important because the hotter the temperature of the panel, the
lower its electrical output. The cell temperature calculation
method is chosen by the user in the EnergyPlus IDF file
through a parameter choice in the IDD entry called Integration
and Cell Temperature Mode.
If the value of this parameter is “Decoupled NOCT
Conditions” then the cell temperature of the PV is
modeled using the method from the Duffie and Beckman (1991)
for estimating cell temperature. This is based upon the
standard NOCT (Nominal Operating Cell Temperature)
measurements to compute the module temperature Tc at each
timestep. The NOCT temperature (Tc,NOCT) is the operating
temperature of the module with a wind speed of 1 m/s, no
electrical load, and a certain specified insolation and
ambient temperature [Beckman and Duffie, 1991]. The values for
insolation GT,NOCT~~ and ambient temperature
Ta,NOCT are usually 800 W/m2
and 20º C. c is the convesion efficiency
of the module, which varies with ambient conditions. is a
user-defined constant.
The equation is:
If the user specifies the “Decoupled Ulleberg
Dynamic” mode for calculating cell temperature, then
a method developed by Ulleberg is used:
In other words, the cell temperature is a function of the
privious cell temperature and the thermal capacity of the PV
module material.
If the user specifies “Integrated Surface Outside
Face” for this parameter, then the temperature result
from EnergyPlus’s modeling of surfaces is used for the cell
temperature. Also the energy exported from the surface as
electricity becomes a sink in the internal source modeling for
the heat transfer surface.
If the user specifies “Integrated Transpired
Collector” for this parameter, then the temperature
result for the unglazed transpired collector surfaces is used
for the cell temperature. Also the energy exported from the
collector surface as electricity is deprecated using a source
term in the collector’s temperature modeling.
If the user specifies “Integrated Exterior Vented
Cavity” for this parameter, then the temperature
result for the exterior cavity is used for the cell
temperature. Also the energy exported from the baffle surface
as electricity is deprecated using a source term in the
baffle’s temperature modeling.
PV Section 3 :
Multi-Array Modules[LINK]
The electrical calculations discussed in the sections above
deal only with a single module. The EnergyPlus PV component
may be used to simulate arrays with any number of modules. The
IDF defines the number of modules in series (NS) and modules
in parallel (NP) for the entire array. The total number of
modules in the array is the product of NS and NP. When
simulating a single module only, both NS and NP are set to 1.
The single-module values for all currents and voltages
discussed in PV Section 1 are multiplied by NP or NS to find
values for the entire array. This approach neglects module
mismatch losses.
With the above equations, and the assumption that the
panels operate at the maximum power point, it is a direct
calculation to determine DC power production. The performance
of an array of identical modules is assumed to be linear with
the number of modules in series and parallel. The inverter
efficiency is applied linearly to derate the energy
production. The inverter capacity forms a limit for power
production from a PV generator. A ‘load’ is passed the PV
array acting as a generator and various trivial calculations
compare PV production to this load. If the PV array is
associated with a surface that is associated with a zone, then
if the zone has any multipliers associated with it,
electricity production will be multiplied accordingly.
Duffie, John A. and William A. Beckman. 1991. Solar
Engineering of Thermal Processes. New York: John Wiley
& Sons, Inc..
Eckstein, Jürgen Helmut. 1990. Detailed Modeling of
Photovoltaic Components. M. S. Thesis – Solar Energy
Laboratory, University of Wisconsin, Madison: 1990.
Ulleberg, Øystein. HYDROGEMS Component Library for
TRNSYS 15 User Manual, Institute for Energy Technology,
Kjeller, Norway **
The third model available in EnergyPlus for predicting the
electricity generated by photovoltaics is referred to as the
Sandia model. This model is based on work done at Sandia
National Lab, Albuquerque, NM by David King – with the help of
many others. The model consists of a series of empirical
relationships with coefficients that are derived from actual
testing. Once the coefficients for a particular module are
available, it is straightforward matter to use the model
equations to calculate five select points on the
current-voltage curve.
The implementation in EnergyPlus is also based on work done
by Greg Barker (2003) for the National Renewable Energy Lab
who implemented the Sandia model in FORTRAN77 as a custom type
(Type101) for the TRNSYS computer program.
There are several climate and solar orientation inputs to
the model that are managed elsewhere in EnergyPlus including:
incident beam solar, incident diffuse solar, incidence angle
of beam solar, solar zenith Angle, outdoor drybulb, wind
speed, and elevation.
Mathematical
Description[LINK]
This section presents the mathematical description of the
Sandia model from a draft report by King et, al. (2003). The
core of the model predicts the performance of a single PV
module. The following nomenclature and equations summarize the
Sandia model.
Nomenclature for Sandia PV model
| Mathematical variable |
Description |
| Isc |
Short-circuit current (A) |
| Imp |
Current at the maximum-power
point (A) |
| Ix |
Current at module V = 0.5
Voc, defines 4th point on I-V curve |
| Ixx |
Current at module V = 0.5
(Voc + Vmp), defines a 5th point on the
I-V curve |
| Voc |
Open-circuit voltage (V) |
| Vmp |
Voltage at maximum-power point
(V) |
| Pmp |
Power at maximum-power point
(W) |
| fd |
Fraction of diffuse irradiance
used by module |
| Ns |
Number of cells in series in a
module’s cell-string |
| Np |
Number of cell-strings in
parallel in module |
| k |
Boltzmann’s constant,
1.38066E-23 (J/k) |
| q |
Elementary charge, 1.60218E-19
(coulomb) |
| Tc |
Cell temperature inside module
(°C) |
| δ(Tc) |
‘Thermal voltage’ per cell at
temperature Tc, approximately 1 volt for a typical
26-cell crystalline silicon module |
| Ee |
‘Effective’ solar
irradiance |
| Eb |
Beam solar irradiance |
| Ediff |
Diffuse solar irradiance |
| C0 ,
C1 |
Empirical coefficients relating
Imp to Ee , C0 +
C1= 1 (both dimensionless) |
| C2 ,
C3 |
Empirical coefficients relating
Vmp to Ee(C2 dimensionless,
C3 is 1/V) |
| C4 ,
C5 |
Empirical coefficients relating
Ix to Ee, C4 + C5
= 1 (both dimensionless) |
| C6 ,
C7 |
Empirical coefficients relating
Ixx to Ee,C6 + C7
= 1 (both dimensionless) |
| n |
Empirically determined ‘diode
factor’ for individual cells |
| AMa |
Absolute Air Mas |
| AOI |
Solar angle-of-incidence
(degrees) from normal |
| f1(AMa) |
Empirical polynomial function
used to relate short-circuit current to the solar spectrum via
air mass |
| f2(AOI) |
Empirical polynomial function
used to relate short-circuit current to the solar
angle-of-incidence |
| a0, a1,
a2, a3, a4 |
Empirical coefficients for
f1(AMa) polynomial |
| b0, b1,
b2, b3,
b4,b5,b6 |
Empirical coefficients for
f1(AOI) polynomial |
| To |
Reference cell temperature for
rating, typically fixed at 25°C |
| Isco |
Short circuit current at
reference conditions |
| Impo |
Max power point current at
reference conditions |
| Vmpo |
Voltage at max power at
reference conditions |
| Voco |
Open circuit voltage at
reference conditions |
| Ixo |
Current at V = 0.5 Voc and at
reference conditions |
| Ixxo |
Current at V = 0.5 (Vmp + Voc)
and at reference conditions |
| αIsc |
Normalized temperature
coefficient for Isc (1/°C) |
| αImp |
Normalized temperature
coefficient for Imp (1/°C) |
| βVoc(Ee) |
Temperature coefficient for
module open-circuit-voltage as function of Ee |
| βVoco |
Temperature coefficient for
module open-circuit-voltage at reference conditions |
| mβVoco |
Coefficient for irradiance
dependence of open-circuit-voltage-temperature coefficient,
often zero (V/°C) |
| βVmp(Ee) |
Temperature coefficient for
module maximum-power-voltage as a function of
Ee |
| βVmpo |
Temperature coefficient for
module maximum-power-voltage at reference conditions |
| mβVmpo |
Cofficient for irradiance
dependence of maximum-power-voltage-temperature coefficient,
often zero (V/°C) |
| Tm |
PV module temperature at back
suface (°C) |
| Ta |
Ambient outdoor drybulb
temperature (°C) |
| E |
Solar irradiance incident on
module surface (W/m2) |
| WS |
Wind speed at standard 10-m
height (m/s) |
| a |
Empirical coefficient relating
module temperature at low wind and high solar irradiance |
| b |
Empirical coefficient relating
module temperature decrease with increasing wind speed |
| Tc |
Temperature of solar cell inside
module (°C) |
| Eo |
Reference solar irradiance (1000
W/m2) |
| ΔT |
Temperature difference between
Tc and Tmat Eo(°C), |
The current implementation in EnergyPlus focuses on
determining performance at the maximum power-point but also
calculates, and reports, four other points on the I-V curve so
that the data are available for analyses outside of
EnergyPlus. The equations below use the module performance
parameters that are available in a database provided by Sandia
National Laboratory (see www.sandia.gov/pv). The following
equations form the basis of the Sandia model implemented in
EnergyPlus:
where,
With the above equations, and the assumption that the
panels operate at the maximum power point, it is a direct
calculation to determine DC power production. The performance
of an array of identical modules is assumed to be linear with
the number of modules in series and parallel. The inverter
efficiency is applied linearly to derate the energy
production. The inverter capacity forms a limit for power
production from a PV generator. A ‘load’ is passed the PV
array acting as a generator and various trivial calculations
compare PV production to this load. If the PV array is
associated with a surface that is associated with a zone, then
if the zone has any multipliers associated with it,
electricity production will be multiplied accordingly.
The equation above for Tm is used to predict
back-of-module temperature when the mode ‘SANDIA RACK’ is
selected. This would be appropriate for most rack mounted PV
installations. If the user selects ‘EPLUS INTEGRATED’ then the
back-of-module temperature is obtained from the outside face
surface temperature calculated by the full complement of Heat
Balance models using Conduction Transfer Functions that is
native to EnergyPlus. And energy exported from the surface is
accounted for using a source/sink term that is lagged from the
previous timestep (pingpong).
King, D.L., Boyson, W.E., Kratochvil J.A. 2003.
Photovoltaic Array Performance Model. Sandia National
Laboratories, Albuquerque, NM 87185, November 2003 currently
in DRAFT
Barker, G. 2003 (unknown). Predicting Long-Term Performance
of Photovoltaic Arrays. Submitted as deliverable for NREL
subcontract LAX-1-30480-02. Currently in DRAFT
Davis, M.W., Fanney, A.H., and Dougherty B.P. 2002.
Measured Versus Predicted Performance of Building
Integrated Photovoltaics. from the conference Solar 2002,
Sunrise on the Reliable Energy Economy, June 15-19, 2002,
Reno, NV. Available from NIST website.
King, D.L. 1996. Photovoltaic Module and Array Performance
Characterization Methods for All System Operating Conditions.
Sandia National Laboratory. Albuquerque, NM 87185
Photovoltaic Arrays[LINK]
The Photovoltaics.f90 module includes three different models referred to as “Simple”, “Equivalent One-Diode” and “Sandia” and the choice will determine the mathematical models (and input data) used to determine the energy produced by solar/electric conversion panels. The EnergyPlus photovoltaic array models are called one at a time at the HVAC system timestep along with other electrical generation components such as gas turbines and diesel engines.
All of the photovoltaic models share the same models for predicting incident solar radiation that are also used for the solar thermal calculations and are described in the section Climate, Sky and Solar/Shading Calculations.
Note that some of the terminology used to discussed photovoltaics overlaps with terminology used to discuss Fortran programs. The word module may refer to a PV panel or to a fortran90 programming entity. Model may refer to a manufacturers production model for a specific type of PV module or to a mathematical model used for engineering analysis. Array may refer to a collection of PV modules wired together or to a mathematical variable with multiple elements.
The PV modules are assumed to always run when the total incident solar is greater than 0.3 Watts. If the incident solar is less than 0.3, then the modules produce no power.
PV arrays are managed by an electric load center. The load center is a “load” with respect to generating equipment but is a “supply center” for the rest of the building. PV arrays need to be connected to ElectricLoadCenter:Distribution objects that have a DC buss type.
Simple Model[LINK]
The Generator:PV:Simple object describes about the simplest model for predicting photovoltaic energy production. In this model the user specifies the efficiency with which surfaces convert incident solar radiation to electricity. (In the other models this efficiency is determined as part of the model.) The full geometric model for solar radiation is used, including sky models, shading, and reflections, to determine the incident solar resource. The model accepts arbitrary conversion efficiencies and does not require actual production units be tested to obtain empirical performance coefficients. (The Energy+.idd sets the range of conversion efficiencies to be on [0..1], but the user could alter the Energy+.idd to extend this range if desired.)
Mathematical Description[LINK]
The usable electrical power produced by a PV surface are calculated using:
On the right hand side of this equation, only GT is calculated by EnergyPlus and the rest are user inputs. Power levels are assumed constant over the timestep to arrive at energy production.
There are two modes that can be selected by the user that govern how the PV system is coupled to the building surfaces. If the integration mode is selected as ‘DECOUPLED’ then no adjustments are made to account for energy extracted in the form of electricity. If the integration mode is selected as ‘INTEGRATED’ then the energy extracted in the form of electricity is removed from surface heat transfer calculations using a sink term. This sink term is lagged from the previous timestep.
Equivalent One-Diode Model[LINK]
This model predicts the electrical performance of a photovoltaic (PV) array. This model is also known as the “TRNSYS PV” model.
Mathematically speaking, the EnergyPlus PV module employs equations for an empirical equivalent circuit model to predict the current-voltage characteristics of a single module. This circuit consists of a DC current source, diode, and either one or two resistors. The strength of the current source is dependent on solar radiation and the IV characteristics of the diode are temperature-dependent. The results for a single module equivalent circuit are extrapolated to predict the performance of a multi-module array.
The module employs a “four-parameter” equivalent circuit to model crystalline (both mono and poly) PV modules developed at the University of Wisconsin – Madison [2]. The values of these parameters cannot normally be obtained directly from manufacturers’ catalogs. However, the PV module will automatically calculate them from commonly available data. The PV module also includes an optional incidence angle modifier correlation to calculate how the reflectance of the PV module surface varies with the angle of incidence of solar radiation.
The module determines PV current as a function of load voltage. Other OUTPUTS include current and voltage at the maximum power point along the IV curve, open-circuit voltage, short circuit current as well as electrical load met and unmet.
Mathematical Description[LINK]
PV Section 1: Four-Parameter Model[LINK]
The four-parameter equivalent circuit model was developed largely by Townsend [1989] and is detailed by Duffie and Beckman [1991]. The model was first incorporated into a component for the TRNSYS simulation package by Eckstein [1990]. The EnergyPlus module employs the Eckstein model for crystalline PV modules, using it whenever the short-circuit IV slope is set to zero or a positive value as modified by Ulleberg [2000]. The four parameter model assumes that the slope of the IV curve is zero at the short-circuit condition:
This is a reasonable approximation for crystalline modules. The “four parameters” in the model are IL,ref, Io,ref, , and Rs. These are empirical values that cannot be determined directly through physical measurement. The EnergyPlus model calculates these values from manufactures’ catalog data as discussed in the following section on calculating these parameters
The four-parameter equivalent circuit is shown in the following figure:
V is the load voltage and I is the current flowing through the load and PV.
Determining Performance under Operating Conditions
The IV characteristics of a PV change with both insolation and temperature. The PV model employs these environmental conditions along with the four module constants IL,ref, Io,ref, , and Rs to generate an IV curve at each timestep.
The current-voltage equation of circuit shown in the previous figure is as follows:
Rs and are constants. The photocurrent IL depends linearly on incident radiation:
The reference insolation Gref is nearly always defined as 1000 W/m2. The diode reverse saturation current Io is a temperature dependent quantity:
Equation gives the current implicitly as a function of voltage. Once Io and IL are found from Eqs. 3 and 4, Newton’s method is employed to calculate the PV current. In addition, an iterative search routine finds the current (Imp)~~ and voltage (Vmp) at the point of maximum power along the IV curve.
Calculating IL,ref, Io,ref, , and Rs
The Idf specification for the PV model include several values which must be read from manufacturers’ PV module catalogs. The manufactures’ values are used to determine the equivalent circuit characteristics IL,ref, Io,ref, , and Rs. These characteristics define an equivalent circuit that is employed to find the PV performance at each timestep, as described previously. This section describes the algebra and calculation algorithms used to solve for the four equivalent circuit characteristics.
Three of these values, IL,ref, Io,ref, , may be isolated algebraically. The first step is to substitute the current and voltage into Eq. at the open-circuit, short circuit, and maximum power conditions:
In each case the “-1” term is may be dropped to simplify the algebra. This approximation has little influence on the right side of the equations since because the magnitude of Iois very small, generally on the order of 10-6 A. Some rearrangement then yields the following three expressions which isolate IL,ref, Io,ref, :
At this point an additional equation is needed in order to determine the last unknown parameter. Taking the analytical derivative of voltage with respect to temperature at the reference open-circuit condition derives the fourth equation. This analytical value is matched to the open-circuit temperature coefficient, a catalog specification:
where
The “TRNSYS PV model” uses an iterative search routine in these four equations to calculate the equivalent circuit characteristics. The first step is to set upper and lower bounds for the series resistance parameter Rs: physical constraints require the Rs~~value to lie between 0 and the value such that = Ns. The initial guess for Rs is midway between these bounds. and Io,ref are found from Eq. and Eq. , while Eq. gives a trivial solution for IL,ref. The model then employs Eq. to compare the analytical and catalog values for voc. When all other variables are held constant, the analytical value for voc increases monotonically with series resistance (Townsend 1989). If the analytical voltage coefficient is less than the catalog value, the lower bound for Rs is reset to the present guess value. Likewise, the upper bound is set to the current value if the calculated voc is too large. After resetting the upper or lower bound for Rs, a new guess value is found by averaging the bounds. This procedure repeats until Rs and converge. Note that for IL,ref, Io,ref, , and Rs are assumed to be constant and are calculated only on the first call in the simulation. Alternatively, the user may enter a known series resistance by entering a positive value in the IDF. In this case the iterative routine described above is skipped and Eqs. , , and find IL,ref, Io,ref, and directly from the given value of Rs.
PV Section 2 : Module Operating Temperature[LINK]
The PV model uses one of five methods for determining cell temperature data. The cell temperature of a PV module is important because the hotter the temperature of the panel, the lower its electrical output. The cell temperature calculation method is chosen by the user in the EnergyPlus IDF file through a parameter choice in the IDD entry called Integration and Cell Temperature Mode.
If the value of this parameter is “Decoupled NOCT Conditions” then the cell temperature of the PV is modeled using the method from the Duffie and Beckman (1991) for estimating cell temperature. This is based upon the standard NOCT (Nominal Operating Cell Temperature) measurements to compute the module temperature Tc at each timestep. The NOCT temperature (Tc,NOCT) is the operating temperature of the module with a wind speed of 1 m/s, no electrical load, and a certain specified insolation and ambient temperature [Beckman and Duffie, 1991]. The values for insolation GT,NOCT~~ and ambient temperature Ta,NOCT are usually 800 W/m2 and 20º C. c is the convesion efficiency of the module, which varies with ambient conditions. is a user-defined constant.
The equation is:
If the user specifies the “Decoupled Ulleberg Dynamic” mode for calculating cell temperature, then a method developed by Ulleberg is used:
In other words, the cell temperature is a function of the privious cell temperature and the thermal capacity of the PV module material.
If the user specifies “Integrated Surface Outside Face” for this parameter, then the temperature result from EnergyPlus’s modeling of surfaces is used for the cell temperature. Also the energy exported from the surface as electricity becomes a sink in the internal source modeling for the heat transfer surface.
If the user specifies “Integrated Transpired Collector” for this parameter, then the temperature result for the unglazed transpired collector surfaces is used for the cell temperature. Also the energy exported from the collector surface as electricity is deprecated using a source term in the collector’s temperature modeling.
If the user specifies “Integrated Exterior Vented Cavity” for this parameter, then the temperature result for the exterior cavity is used for the cell temperature. Also the energy exported from the baffle surface as electricity is deprecated using a source term in the baffle’s temperature modeling.
PV Section 3 : Multi-Array Modules[LINK]
The electrical calculations discussed in the sections above deal only with a single module. The EnergyPlus PV component may be used to simulate arrays with any number of modules. The IDF defines the number of modules in series (NS) and modules in parallel (NP) for the entire array. The total number of modules in the array is the product of NS and NP. When simulating a single module only, both NS and NP are set to 1. The single-module values for all currents and voltages discussed in PV Section 1 are multiplied by NP or NS to find values for the entire array. This approach neglects module mismatch losses.
With the above equations, and the assumption that the panels operate at the maximum power point, it is a direct calculation to determine DC power production. The performance of an array of identical modules is assumed to be linear with the number of modules in series and parallel. The inverter efficiency is applied linearly to derate the energy production. The inverter capacity forms a limit for power production from a PV generator. A ‘load’ is passed the PV array acting as a generator and various trivial calculations compare PV production to this load. If the PV array is associated with a surface that is associated with a zone, then if the zone has any multipliers associated with it, electricity production will be multiplied accordingly.
References[LINK]
Duffie, John A. and William A. Beckman. 1991. Solar Engineering of Thermal Processes. New York: John Wiley & Sons, Inc..
Eckstein, Jürgen Helmut. 1990. Detailed Modeling of Photovoltaic Components. M. S. Thesis – Solar Energy Laboratory, University of Wisconsin, Madison: 1990.
Ulleberg, Øystein. HYDROGEMS Component Library for TRNSYS 15 User Manual, Institute for Energy Technology, Kjeller, Norway **
Sandia Photovoltaic Performance Model[LINK]
The third model available in EnergyPlus for predicting the electricity generated by photovoltaics is referred to as the Sandia model. This model is based on work done at Sandia National Lab, Albuquerque, NM by David King – with the help of many others. The model consists of a series of empirical relationships with coefficients that are derived from actual testing. Once the coefficients for a particular module are available, it is straightforward matter to use the model equations to calculate five select points on the current-voltage curve.
The implementation in EnergyPlus is also based on work done by Greg Barker (2003) for the National Renewable Energy Lab who implemented the Sandia model in FORTRAN77 as a custom type (Type101) for the TRNSYS computer program.
There are several climate and solar orientation inputs to the model that are managed elsewhere in EnergyPlus including: incident beam solar, incident diffuse solar, incidence angle of beam solar, solar zenith Angle, outdoor drybulb, wind speed, and elevation.
Mathematical Description[LINK]
This section presents the mathematical description of the Sandia model from a draft report by King et, al. (2003). The core of the model predicts the performance of a single PV module. The following nomenclature and equations summarize the Sandia model.
The current implementation in EnergyPlus focuses on determining performance at the maximum power-point but also calculates, and reports, four other points on the I-V curve so that the data are available for analyses outside of EnergyPlus. The equations below use the module performance parameters that are available in a database provided by Sandia National Laboratory (see www.sandia.gov/pv). The following equations form the basis of the Sandia model implemented in EnergyPlus:
where,
With the above equations, and the assumption that the panels operate at the maximum power point, it is a direct calculation to determine DC power production. The performance of an array of identical modules is assumed to be linear with the number of modules in series and parallel. The inverter efficiency is applied linearly to derate the energy production. The inverter capacity forms a limit for power production from a PV generator. A ‘load’ is passed the PV array acting as a generator and various trivial calculations compare PV production to this load. If the PV array is associated with a surface that is associated with a zone, then if the zone has any multipliers associated with it, electricity production will be multiplied accordingly.
The equation above for Tm is used to predict back-of-module temperature when the mode ‘SANDIA RACK’ is selected. This would be appropriate for most rack mounted PV installations. If the user selects ‘EPLUS INTEGRATED’ then the back-of-module temperature is obtained from the outside face surface temperature calculated by the full complement of Heat Balance models using Conduction Transfer Functions that is native to EnergyPlus. And energy exported from the surface is accounted for using a source/sink term that is lagged from the previous timestep (pingpong).
References[LINK]
King, D.L., Boyson, W.E., Kratochvil J.A. 2003. Photovoltaic Array Performance Model. Sandia National Laboratories, Albuquerque, NM 87185, November 2003 currently in DRAFT
Barker, G. 2003 (unknown). Predicting Long-Term Performance of Photovoltaic Arrays. Submitted as deliverable for NREL subcontract LAX-1-30480-02. Currently in DRAFT
Davis, M.W., Fanney, A.H., and Dougherty B.P. 2002. Measured Versus Predicted Performance of Building Integrated Photovoltaics. from the conference Solar 2002, Sunrise on the Reliable Energy Economy, June 15-19, 2002, Reno, NV. Available from NIST website.
King, D.L. 1996. Photovoltaic Module and Array Performance Characterization Methods for All System Operating Conditions. Sandia National Laboratory. Albuquerque, NM 87185
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