3rd International Conference on Energy Systems and Technologies
16 – 19 Feb. 2015, Cairo, Egypt
DEDUCTION OF TWO-DIODE MODEL PARAMETERS FOR
PHOTOVOLTAIC SYSTEM
Adel A. Elbaset1, Hamdi Ali2, Montaser Abd-El Sattar2
(1)
Department of Electrical Engineering, Minia University, El-Minia, 61517, Egypt
E-mail: Adel.Soliman@mu.edu.eg
(2)
Department of Electrical and Computer Engineering, El-Minia High Institute
for Engineering and Technology, El-Minia, Egypt
E-mail: hamdihesha@yahoo.com, mymn2013@yahoo.com
The paper presents a proposed two-diode model for PV module to describe I-V and P-V
characteristic curves at different weather conditions such as, temperature and solar
radiation. Two two-diode model parameters are estimated using Newton-Raphson
method with the aid of initial values which are derived from basic equations of an
equivalent circuit for two-diode model and manufacturing data sheet at standard test
conditions. The two-diode model parameter represent an important role in design,
manufacturing and performance of PV system at different weather conditions especially
at low radiation. Newton-Raphson method is used to describe non-linear output
characteristic curves of I-V and P-V. The proposed two-diode model is validated for
multi-crystalline solar cell PV modules. Results are compared with the manufacturer’s
data sheet curves and the proposed results of other published research works. The results
of proposed model are validated with an excellent manner with respect to data sheet and
other published research works.
Keywords: PV modules, Seven-parameter model, Two-diode model, Single diode model.
1. INTRODUCTION
Nowadays, solar photovoltaic systems become popular and have many applications
in the world that extended from remote area energy services, house appliances up to grid
utilities. The rapid growth of PV system utilizations is due to many benefits and
advantages such as availability everywhere which reduces costs and losses, free,
abundant, and pollution free. It also represents the most important available renewable
energy resources due to its permanent energy source in everywhere of the world to
generate electricity on site where it is needed, which reducing CO2 emission in
environment. Silicon is the basic material required for the production of solar cells based
on crystalline technology. Most of solar cells are based on multi-crystalline silicon
technology due to their reliability and high efficiency for manufacturing PV solar
modules [1-3].
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The PV system operation depends on many physical parameters like site latitude of
PV systems, weather conditions, the panel tilt and its azimuth angles, the air and surfaces
surrounding temperatures and finally electrical loads. Although PV systems have many
advantageous, but unfortunately, they suffered from changing of system performance due
to weather variations, high cost installation and low efficiency that is hardly reached up to
20% for module. Therefore, the modeling of PV system becomes important in design,
manufacturing, and operation of the PV based power systems to obtain optimization
performance from such system [4-8].
PV cell is the main building block of PV module which represents the main unit of
electrical solar power generation system. The PV module consists of many PV cells
connected in series/ parallel manner for each module to produce I-V and P-V curves.
These characteristic curves depend mainly on weather conditions such as solar radiation
and cell temperature. So, it is important to model a PV module to obtain accurate
manufacturing, operation, and discovering the causes of degradation of PV performance.
The PV system modeling should fulfill the following criteria [9-11]:
1- It should be simple and fast.
2- It should be predicted the I-V and P-V characteristic curves with accurate manner.
3- It should be developed and has a comprehensive tool.
4- It should be validated PV system manufacturing data sheet.
Ref. [1] described an improved five-parameter which are Rs, Rsh, a, Iph and Is for
single diode model that is capable of analytically describing the I-V characteristic curves
of a PV module for each generic condition of operative temperature and solar radiation.
The parameters of the equivalent electrical circuit parameters are solved by a system
equations based on data commonly issued by manufacturers in standard test conditions
with a trial and error process. Ref. [5] estimated the solar cell parameters of single diode
model using the hybrid genetic algorithm and Nelder-Mead simplex search method from
the given voltage - current data. Ref. [6] implemented a generalized PV model based on
single diode model using Matlab/Simulink software package for PV cell, module, and
array. Ref. [8] proposed a PV single diode model by Hybrid Genetic Algorithm and
Particle Swarm Optimization techniques. Refs. [11-20] proposed several computational
methods with different techniques for single diode model, but most of these techniques
required new additional coefficients into the model equations causing increase of their
computational burdens. The equivalent electric circuit of PV cell is represented by Wolf
which is composed of many lumped elements such as a current source, a diode and a
series resistance for each cell [21]. Wolf modeling has been simplified to a single diode
model as shown in Fig. 1. The general characteristic equation of a single diode model is
given as [1,6,17,25].
Figure 1. Single diode circuit model of PV cell.
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The photo current
[6,15]:
is a function of temperature and solar radiation is given as follows
The single diode saturation current as function of working PV temperature is given as
follows [6,15]:
Although single diode is more popular for PV modeling, but it has many disadvantageous
such as [22]:
1- It exhibits high deficiencies when studying PV performance with temperature
variations.
2- It neglects recombination loss in PV cell depletion region.
3- Deterioration its accuracy at low radiation levels especially at open circuit
.
The single diode model was based on the assumption that the recombination loss in
the depletion region is absent. In a real solar cell, the recombination represents a
substantial loss, which cannot be adequately modeled using a single diode. Consideration
of this loss leads to a more precise model known as the two-diode model [23]. However,
the inclusion of the additional diode increases the PV model parameters to sevenparameter (new parameters:
, a2). Also, the two-diode model is proposed to improve
the accuracy of PV module and to overcome the disadvantageous of single diode model.
The main purpose now is to estimate the values of all the model parameters within a
reasonable simulation time [22-27]. Ref. [22] employed a Matlab / Simulink to simulate a
PV system with a two-diode model. The inputs to the simulator are information available
on standard PV module datasheets. Ref. [23] estimates four-parameter of two-diode
model which are short-circuit current (Isc), saturation current (Is), the series resistance, Rs
and the parallel resistance, Rsh. The three other parameters of such model were simplified
to Is1 = Is2 = Is and arbitrarily chosen ideality factors of diodes a1 and a2.
There were many mathematical techniques of the two-diode model in Refs. [22-27]
such as, the Levenberg / Merquardt technique, an equivalent thevenin circuit technique to
estimate the model parameters, and finally, the simplification technique of the two-diode
current equations using iteration method. However, in all these techniques many new
additional coefficients are introduced into the equations and difficulty arises in
determining the initial values of the parameters. These cited methods are very sensitive to
the initial conditions and, if not properly guided by an initial estimation of the
parameters, lead to inconsistent results.
This paper proposed a novel seven-parameter model for PV modules to predict I-V
and P-V characteristic curves based on three main points such as open circuit, maximum
power and short circuit at STC which lead to accurate results. The proposed model is
based on:
1- Basic equations of two-diode model and data sheets of the PV manufacturing.
2- Newton-Raphson method for estimated the seven-parameter.
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2. METHODOLOGY
The two-diode equivalent circuits based model are shown in Figs. 2 and 3 [6], for
cells and modules. The equivalent circuit of the module consists of series and parallel
and
respectively.
cells with
Figure 2. Two-diode circuit model of PV cell model.
Figure 3. Equivalent circuit model of generalized PV.
The general equation of two-diode model is given by [25]:
The seven-parameter respectively known as follow, photo current
, saturation currents
of two diodes
series and shunt resistances
and
, and ideality factors
of two diodes
and
.
These seven-parameter can be arranged respectively in the following matrix form
[24,28]:
The photo current
follows [6,22]:
is a function of temperature and solar radiation is expressed as
The two-diode saturation currents as function of working PV temperature are written as
follow [6,22]:
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The current equation of module is given as [6,22]:
where:
Ns: is the number of series cells.
Np: is the number of parallel cells.
The derivative of output current with respect to cell voltage is given from Eq. (4) as:
To determine seven-parameter, it needs seven-equation that will be solved
simultaneously using Newton-Raphson method. The seven-equation are derived as
follow:
The first equation is obtained from open circuit condition, then I=0, V=Voc and Eq. (4)
becomes as:
The second equation is given from short circuit condition, then V=0, I=Isc and Eq. (4)
expressed as:
The third equation is obtained from the maximum power and Eq. (4) rewritten as:
Moreover the 4 th, 5 th and 6 th equations are deduced from current derivative
equation at open circuit, short circuit, and maximum power points in data sheet of PV
module as follow:
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where:
at open circuit where
at short circuit where
and
and
at STC.
at STC.
The 7th equation is obtained from diode ideality factors for which their summation is
greater than or equal to three for all PV cell types. The ideality factors of two-diode
model are given as follow [6,18,20]:
(18)
The particular solution is starting by choosing suitable initial values of two-diode
ideality factors
and . Choose the initial value of
as a partial value from
to
satisfy the following equation:
where:
is the fractional number
and is given by
is the reciprocal slope of I-V curve of cell at open circuit voltage.
and recombination
The equations of diffusion saturated current of diode, D1,
saturated current of diode, D2,
are derived from Eqs. (12-14) and (17) as follow:
where:
Finally, the diffusion saturated current of D1,
of D2,
are computed as follow:
The initial value of
and recombination saturated current
is derived from Eq. (17) as follow:
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The final initial value of photo current is estimated from Eq. (12).
3. NEWTON-RAPHSON METHOD [28-30]
3.1. Estimation of Seven-parameter
Newton-Raphson method is a numerical technique established using seven previous
equations for estimation seven-parameter of two-diode model in the form of
,
where is array of the seven-parameter as in Eq.(5).
and the Jacobian matrices
are derived from previous seven-equation as follow:
=
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The main steps of computer algorithm for estimation the PV model parameters as
shown in flowchart of Fig. 4 are summarized as follow:
1- Computing the initial values of PV module parameters.
2- Forming both matrices
and
of PV system parameters.
3- Computing seven-parameter values using Newton-Raphson iteration method.
The iterative process is repeated up to the difference between
and
reaches an
acceptably small value.
TSTC , I sc , Voc
, I ,V
mp
mp
≤
Figure 4. Flowchart for estimation PV module parameters using Newton-Raphson method
3.2. Establishing I-V and P-V Characteristic Curves
I-V and P-V characteristic curves are established the model parameters of Eqs. (4,9)
using Newton-Raphson method.
where n denotes the nth iteration, and
The flowchart of computer program is shown in Fig. 5.
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I ph
, I s1 , I , R , R , a 1 , a 2
s 2
sh
s
I ph
≤
Figure 5. Flowchart for establishing I-V and P-V curves.
4.
VERIFICATION AND RESULTS
4.1. Comparing Results with Manufacturing Data Sheet
The proposed two-diode model is validated by estimated parameters of multicrystalline of two different modules, MSX-60 and KC-200GT which have data sheet as
shown in Table 1. Table 2 shows the seven-parameter values of both MSX-60 and KC200GT multi-crystalline modules as compared with the results of Ref. [22]. The results
have reasonable values as compared with previous research work and the differences are
due to assumption of
=
in Ref. [22].
Fig. 6 shows I-V and P-V characteristic curves of the proposed two-diode model
using Newton-Raphson numerical iteration method as compared with data sheet at STC
for previous two modules of multi-crystalline. This figure shows an excellent matching
between manufacturer curves and computed results at STC. Tables 3-4 show matching
points for short circuit current, open circuit voltage and maximum values for current,
voltage and power. These tables show excellent results between data sheet and the
proposed model. The percentage deviation between theses values are due to measuring
process of manufacturer which normally has accuracy within determined range.
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Table 1. Data sheet parameters of multi-crystalline modules.
Data sheet
parameter
Multi-Crystalline solar cells
BP Solar [22]
MSX-60
3.8000
21.1000
3.5000
17.1000
-80×10-3
3×10-3
36
1
Np
Kyocera [22]
KC-200GT
8.2100
32.9000
7.6100
26.3000
-123×10-3
3.1800×10-3
54
1
Table 2. MSX-60 and KC-200GT parameters as compared with results of reference [22].
BP Solar
MSX-60
Computed results
Results of
using Newtonreference
[22]
Raphson
method
3.8084
3.8000
-10
4.8723×10
4.7040×10-10
6.1528×10-10
4.7040×10-10
0.3692
0.3500
169.0471
176.4000
1.0003
1.0000
1.9997
≥1.2000
Model
parameter
9
X: 0
Y: 8.21
8.2100
4.2180×10-10
4.2180×10-10
0.3200
160.5000
1.0000
≥1.2000
Solar KC-200GT module
X: 26.1
Y: 7.646
7
Newton Raphson method
Manufacturing data
6
Current [A]
8.2237
4.1437×10-10
1.9032×10-6
0.3305
196.5000
1.0003
1.9997
V = 26.1 V
I = 7.646 A
V = 0 V , I = 8.21 A
8
Multi-Crystalline Kyocera
KC-200GT
Computed results using
Results of
Newton-Raphson
reference
[22]
method
5
V = 17.1 V
I = 3.493
V = 0 V, I = 3.8 A
4
X: 0
Y: 3.8Solar
3
MSX-60 module
X: 17.1
Y: 3.493
2
1
0
V = 21.06 V , I = 0 A
0
5
10
15
20
PV voltage in volt
V = 32.85 V
I=0 A
25
30
35
a) I-V curve
b) P-V curve
Figure 6. Comparison between manufacturing data sheet and two-diode model using NewtonRaphson method.
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Table 3. Matching points of data sheet and two-diode model for MSX-60 module.
Data
sheet
parameter
,A
Data sheet
measuring
results
3.8000
21.1000
3.5000
17.1000
60.0000
BP Solar MSX-60
Newton-Raphson
results of Fig. 6
3.8000
21.0600
3.4930
17.1000
59.7200
Percentage of
deviation
zero
-0.1896%
-0.2000%
zero
-0.4667%
Table 4. Matching Points of data sheet and two-diode model for KC-200GT module.
Data
sheet
parameter
Multi-Crystalline Kyocera KC-200GT
Data sheet
Newton-Raphson
Percentage of
measuring
results of Fig. 6
deviation
results
8.2100
8.2100
zero
32.9000
32.8500
-0.1520%
7.6100
7.6460
0.4731%
26.3000
26.1000
-0.7605%
200.0000
199.6000
-0.2000%
4.2 Comparing Results of MSX-60 Module with Single Diode Model
Fig. 7 shows computed curves of MSX-60 module as compared with single diode
model of Ref. [18] at different radiation levels and constant temperature. Figs. 8 indicate
computed curves of MSX-60 module at different temperatures and constant radiation as
compared with manufacturing curves and a single diode model. From these figures, it can
be seen that inaccuracies and inefficiencies of single diode as compared with two-diode
results at different radiation and temperature levels.
b) P-V curve
a) I-V curve
Figure 7. Comparing between single diode of reference [18] and calculated (new model) using
Newton-Raphson method at different radiation levels.
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b) P-V curve
a) I-V curve
Figure 8. Comparing between single diode of reference [18] and calculated (new model) using
Newton-Raphson method at various temperatures and 1000 W/m2.
4.3. Comparing Results of KC-200GT Module with Manufacturing Data Sheet
In Figs. 9-10, the I-V and P-V computed curves with seven-parameter are compared
with the characteristic curves issued by manufacturer of KC-200GT module at different
radiation and temperature levels. These figures indicate a good agreement between the
provided and calculated data at different radiation and temperature levels especially at
low radiation for Newton-Raphson method.
a) I-V curve
b) P-V curve
Figure 9. Comparing between calculated (new model) using Newton-Raphson method and
manufacturing curves at different radiation levels and temperature 250 C.
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a) I-V curve
b) P-V curve
Figure 10. Comparing between calculated (new model) using Newton-Raphson method and
manufacturing curves at different temperatures and radiation 1000 W/m2.
CONCLUSION
An accurate two-diode model of PV module is proposed using Matlab software with
the aid of Newton-Raphson method. The two-diode model uses only data commonly
provided by manufacturers, which numerically solved exactly to determine I-V and P-V
curves, for different radiation and temperature levels. The results obtained are in good
agreement with those published previously. The accuracy of the proposed model is
verified using practical data from various manufacturers of multi-crystalline two different
PV modules. The results of two-diode model are compared to the popular single diode
model results. It is deduced in all comparing that the proposed model is superior than
single diode model when subjected to different radiation and temperature levels,
particularly at lower radiation conditions. The proposed model is powerful and accurate
for using solar PV modules. This paper gives an accurate representation of the I-V and PV characteristic curves of PV module, which will serve as a proposed model for
researches in the field of PV modeling.
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APPENDIX
Nomenclature
PV Photovoltaic
I
output or load current of PV model (A)
V output or load voltage of PV model (V)
Iph photo current (A)
Imp current at the maximum power point (A)
Isc short circuit current of the module (A)
Vmp voltage at the maximum power point (V)
Voc open circuit voltage of the module (V)
Is
cell reverse saturation current (A)
cell reverse saturation current (A) at standard test
conditions (STC)
Is1 diffusion saturated current of D1 (A)
Is2 recombination saturated current of D2 (A)
diffusion saturated current of D1 (A) at standard
test conditions (STC)
recombination saturated current of D2 (A) at
standard test conditions (STC)
q
electron charge (1.6 * 10-19C)
k
Boltzmann constant (1.38 * 10-23J/K)
T
cell working temperature (K)
a
a1,a2
Rs
Rso
diode ideality factor for single diode model
diode ideality factors for two-diode model
series resistance (Ω)
reciprocal slope of the I-V characteristic for
V=Voc and I=0, (Ω)
Rsh shunt resistance (Ω)
Rsho reciprocal slope of the I-V characteristic for
V=0 and I=Isc , (Ω)
VT thermal voltage (V) (VT = kT/q)
G solar radiation (kW/m2)
GSTC solar radiation at standard test conditions (STC)
[GSTC=1kW/m2]
Iph at STC photo current at standard test conditions (A)
Ki short circuit current coefficient (A/C°)
TSTC temperature of PV cell at standard test conditions
Eg band gap energy of semiconductor (eV)
Ns number of series cells
Np number of parallel cells
X
array of seven-parameter
fractional number
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