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Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
ISSN 1843-6188
MODELLING OF THE PV PANELS CIRCUIT PARAMETERS USING
THE 4-TERMINALS EQUATIONS AND BRUNE’S CONDITIONS
Horia ANDREI1, Costin CEPISCA2, SORIN DAN GRIGORESCU2
Traian IVANOVICI1, Paul ANDREI 2
1
2
Faculty of Electrical Engineering University Valahia Targoviste 18-20 Blv. Unirii, 130082a
Faculty of Electrical Engineering University Politehnica Bucharest 313 Splaiul Independentei, 060032
Email: [email protected]
Abstract: In this paper the performances of photovoltaic (PV)
panels has been analysed, for different connections, by using the 4terminals theory and Brune’s conditions. In order to characterize
the PV parameters and to obtain the maximum power generated, a
new set of equations is established for all the typical PV
connection. A Lab VIEW application is implemented to prove the
theoretical models and to obtain the optimum connection model in
terms of generated power. The examples presented have shown the
comparisons between all the typical connections of PV panels
functioning in the same conditions of irradiance.
2.
The parameters and performances of the PV cells are
evaluated here by using the one-exponential equivalent
model and four-terminals circuit equations.
2.1 Equivalent circuits of PV cell
Many equivalent circuits have been proposed in the
literature in order to assess the behavior of the PV cell.
One of the models often used in the so-called doubleexponential, containing a two-diode circuit [7], [8], [9].
Alternatively, a more simplified model can be used, in
which the equivalent circuit of the solar cell has been
implemented by using the one-exponential model with a
single diode.
This circuit equivalent of a PV cell can be modelled
through the circuit shown in Figure 1. The intrinsic
silicon p-n junction characteristic is simulated as a diode
in the circuit equivalent.
Keywords: PV panels, circuit parameters, four-terminal equations,
Brunes’s conditions, maximum power, LabVIEW application.
1.
PHOTOVOLTAIC CELL CIRCUIT MODEL
INTRODUCTION
Photovoltaic (PV) system has been successfully used for
over five decade [1], [2], [3], [4]. Whether in DC or AC
form, photovoltaic panel provide power for systems in
many applications on earth and space. Its principles of
operation are therefore well understood, and circuit
equivalents have been developed that accurately model
the nonlinear relationship between the current and
voltage of a photovoltaic cell.
To solve the PV system maximum power point (MPP)
problem for different type of PV connection is utterly
important in order to improve the overall system
efficiency [5], [6].
Typically, a solar panel comprises cells in series, parallel
or series-parallel connection. In order to characterize the
effects of each circuit parameters connections - current,
voltage and power - on the PV system operation and on
the PV generated power, in this paper a mathematical
model is constructed. Based on the classical 4-terminal
theory and the Brune’s conditions, a specific set of
equations is stated for each type of the PV panel
connection. It is demonstrate the optimum connection in
terms of maximum power generated, and thus of MPP.
For all types of the solar panel connection functioning in
the same conditions of irradiance, a specific Lab View
application is implemented in order to prove the
theoretical aspects demonstrated by authors.
Using the real set of measurements, it is possible to
interpret the behavior of a PV panel in different
connection and to shown which is the optimum model.
Figure 1. PV cell equivalent circuit.
The light generated current source Iph is introduced in
order to model the photoelectric effect: when the PV cell
p-n junction is exposed to light releases electrons and
these electrons are free to move across the junction due
to the built-in potential and create a current.
Rsh and Rs are the shunt (representing the effect of the
surface current dispersion), and respectively, series
resistance (representing the effect of the voltage drop of
the semiconductor and contacts) of a PV cell.
By using this equivalent circuit, from theoretical point
of view, the PV cell can be considered a four-terminals
circuit, whose input parameters are the voltage u(1), and
the current i(1). These two electric quantities can model
the photoelectric effect of silicon p-n junction. The
output parameters of the PV cell are the voltage u(2) and
the current i(2). The relationship between u(2) and i(2) is
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Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
known as the current –voltage characteristic I-U of the
PV cell, and can be defined by
i
( 2)
 I ph  I sd
q (u ( 2 )  R s i ( 2 ) )
kT
[e
 1] 
u (2)  Rs i (2)
Rsh
( 2)
pMPP

4( Rsh  Rs )
( I ph  I d ) 2
(8)
( 2)
u MPP

( Rsh  Rs )
( I ph  I d )
2
(9)
q (u ( 2 )  R s i ( 2 ) )
kT
[e
 1]
2.2 Four-terminal equations for PV cell
The four-terminal or two-port networks, shown in Figure
2, may be defined as a part of an electrical system,
having two pairs of terminals [10], [11], and [12]. One
port 1-1’accepts a source and the other port 2-2’ is
connected to a load, so that there is an input port and an
output port in any two-port network.
(2)
where Isd is diode’s saturation current, k = 1.3807 x 10-23
JK-1 is Boltzmann’s constant, q = 1.6022 x 10 -19 C is the
electronic charge, and T is the absolute temperature.
Letting, in relation (1), i(2) = 0, and considering Rsh = ∞,
yields the open-circuit voltage of the PV cell
( 2)
U oc

2
Rsh
and
(1)
The diode current is given by
I d  I sd
ISSN 1843-6188
kT  I ph  I sd
ln
q  I sd




(3)
Applying a short-circuit condition to the cell’s output
terminals, u(2) = 0, results the short-circuit current
Figure 2. Four-terminal network.
( 2)
I sc
 I ph
(4)
The active four-terminal network can be described by
several set of equations, which indicate the voltagecurrent relationships for each port and also how the
currents and voltages are interrelated at the two ports. In
this paper, the PV cell can be considered an active fourterminal network. Two set of equations are used, in order
to make an adequate description of PV cell behaviour in
series and parallel connection.
The impedance equations of the four-terminal network
can be used for describes the series connection, and is
defined by
( 2)
( 2)
If U oc
and I sc
are measured, then the diode’s
saturation current is approximately
I sd 
I ph
q ( 2)
U oc
e kT
(5)
1
The generated (output) power of PV cell defined by
p ( 2)  u ( 2) i ( 2)
(6)
u (1)  
 i (1)   (1) 
   Z11 Z 21     U o 
 ( 2)  
  ( 2)   ( 2) 
u  Z 21 Z 22  i  U o 
can be expressed, by using (1) and (2), as
p ( 2) 


2
Rsh
u ( 2)
I ph  I d u ( 2) 
Rsh  Rs
Rsh  Rs
Z Z 
where Z   11 21  is called the matrix of transfer
Z Z 
 21 22 
impedances, and Uo(1), Uo(2) are the open-circuit voltages,
defined by the conditions i(1) = i(2) = 0.
The admittance equations of the four-terminal network
can be used for describes the parallel connection, and is
defined by
(7)
This relationship (7) between the generated power p(2)
and the voltage u(2) is known as the power-voltage
characteristic P-U of the PV cell. In order to improve the
PV system efficiency, is necessary that the operating
point of the system to be very close to the true value of
the maximum power point (MPP) of the PV cell. The
MPP value means the extreme point of the P-U
characteristic,
dp( 2)
du ( 2)
(10)
i (1)  
 u (1)   (1) 
   Y11 Y21      I sc 
 ( 2)  
  ( 2)   ( 2) 
i  Y21 Y22  u   I sc 
 0 , thus
( 2)
u MPP
64
(11)
ISSN 1843-6188
Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
Y Y 
where Y   11 21  is called the matrix of transfer
Y Y 
 21 22 
admittances, and Isc(1), Isc(2) are the short-circuit currents,
defined by the conditions u(1) = u(2) = 0.
PV cell can be characterized by a set of equations similar
to relations (8) and (9) of the four-terminal network.
Thus, using equation (1) and Kirchhoff’s laws, yields the
impedance equation
(1)
u (1)   R
I  I 
- Rsh  i 
ph
d
    sh

 Rsh 
 ( 2)  




(
2
)
u
R - ( Rsh  Rs ) i 
I I
 ph d 
   sh
 
ie(1)  i (1) 
ie      1 
ie( 2)  i ( 2) 
1 
(16)
ns
U e  U k
(17)
k 1
Thus, the equivalent matrix of transfer impedance can be
expressed as
(12)
and, the admittance equation
i (1)  
 (1) 
- 1/ Rs  u   I ph  I d 
   ( Rsh  Rs )/Rsh Rs

 (13)

 ( 2)  
 u (2)   0 
i
1/
R
1/
R


s
s
  
  
of PV cell.
From equations (10) and (11) it is possible to identify the
matrix of transfer impedance
R

- Rsh
sh

Z 
 R - ( R  R )
s 
sh
 sh
(14)
a)
respectively the matrix of transfer admittance
( R  R )/R R
s
sh s
Y   sh
1/ R
 s
- 1/ Rs 

- 1/ Rs 
(15)
b)
Figure 3. Series connection of ns PV cell a), and the
equivalent circuit b).
which contain the parameters of the PV cell.
Imposed in (10) and (11) the open-circuit and shortcircuit conditions, the calculated values of the openkT  I ph  I sd 
(1)
( 2)
 and
circuit voltage U oc
 U oc

ln 

q  I sd

Ze 
(1)
(2)
I sc
 0, I sc
 I ph (considering
Rsh » Rs ) are equal with the values indicated in (3) and
(4). This proved the correct set of equations (10) and
(11) of the PV cell.
short-circuit
ns
n R
s sh


 n R -n ( R  R ) 
s
s
s
sh
sh


 Z k n s Z  
k 1
-n s Rsh
(18)
If it is consider a parallel connection of np identically PV
cell, shown in Figure 4, the Brune’s conditions are
current
ue(1)  u (1) 
 1 
ue  
ue(2)  u (2) 
 1 
2.3 Connections of the PV cell
Typically, a solar panel comprises ns PV cells in series or
np PV cells in parallel [13], [14].
If these are well-matched and evenly illuminated, then
for each connection will be established further sets of
equations and equivalent transfer matrix that take into
consideration Brune’s conditions for operation as fourterminal network.
A series connection of ns identically PV cell, shown in
Figure 3, is defined by the Brune’s conditions [15], [16].
Ie 
(19)
np
 Ik
(20)
k 1
According to the above conditions (15), (16) the
equivalent matrix of transfer admittance is defined as
n ( R  R )/ R R - n /R 
np
p sh
s
p s
sh s
 (21)
Ye   Yk n pY  
 n /R
k 1
- n p /Rs 
 p s
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Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
The generated power p(2) for ns series respectively np
parallel PV cells is obtained by relations (7), (18), and
(21)
ps(2) 

ISSN 1843-6188
Figures 5 and 6 have shown the current-voltage i-u
characteristics for series and parallel connection of
different number of PV cells.
2

Rsh
u ( 2)
I ph  I d u (2) 
Rsh  Rs
ns ( Rsh  Rs )
I-U characteristics
(22)
2,500
1 cell
2,000
2 cell in series
1,500
3 cell in series
1,000
4 cell in series
I (A)
5 cell in series
0,500
0,000
0,000
0,500
1,000
1,500
2,000
2,500
3,000
3,500
U (V)
Figure 5. I-U characteristics for series connection of PV
cell.
I-U characteristics
7,000
6,000
I (A)
5,000
1 cell
4,000
2 cell in parallel
3,000
3 cell in parallel
2,000
1,000
0,000
0,000
a)
0,100
0,200
0,300
0,400
0,500
0,600
0,700
U (V)
Figure 6. I-U characteristics for parallel connection of PV
cell.
The power-voltage P-U characteristics for the same
series and parallel connection of the PV cell are
illustrated in Figure 7 and Figure 8.
b)
Figure 4. Parallel connection of np PV cell a), and the
equivalent circuit b).
P-U characteristics
p (p2) 
2
n p u ( 2)
Rsh
I ph  I d u (2) 
n p ( Rsh  Rs )
Rsh  Rs


5,000
(23)
1 cell
4,000
2 cell in series
3,000
3 cell in series
2,000
4 cell in series
P (W)
On the theoretical point of view, when ns   , and
Rsh
ps( 2) 
I ph  I d u ( 2) and
then
np   ,
Rsh  Rs

5 cell in series
1,000

0,000
0,000
1,000
1,500
2,000
2,500
3,000
3,500
U (V)
Figure 7. P-U characteristics for series connection of PV
cell.
p(p2)  .
3.
0,500
P-U characteristics
LABVIEW APPLICATION
3,000
2,500
The characteristics of the PV cells for different
connections are evaluated in this paper by using a
LabVIEW application [17], [18], [19], [20].
In order to provide a numerical example, a PV panel
with silicon monocrystalline cell Saturn has been
selected. The diameter of the cell is 100 mm, the openis
at
3 cell in parallel
0,500
0,000
0,000
V,
( 2)
p MPP
=0,96
Wp.
All
0,100
0,200
0,300
0,400
0,500
0,600
0,700
U (V)
Figure 8. P-U characteristics for parallel connection of PV cell.
A, and the maximum power point occurs
0,48
2 cell in parallel
1,000
( 2)
circuit voltage is U oc
= 0,59 V, the short-circuit current
( 2)
I sc
= 2,16
( 2)
u MPP
=
1 cell
2,000
P (V) 1,500
The important data in what concerns the MPP, are
extracted from all the P-U characteristics (Figures 7 and
8), and are summarized in Table 1.
the
measurements are effected at the solar irradiance E =
1000 W/m2 and the temperature 250 Celsius.
Table 1. The MPP for series and parallel connection of PV cell
66
ISSN 1843-6188
Connection
1 cell
2 cell in series
3 cell in series
4 cell in series
5 cell in series
2 cell in parallel
3 cell in parallel
Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
MPP
0,883 W
1,767 W
2,651 W
3,534 W
4,419 W
1,723 W
2,585 W
[10]
[11]
The table is seen as the output power (MPP) of the PV
system is higher values in the case of PV cells connected
in series.
4.
[12]
CONCLUSIONS
[13]
In this paper, the performances of PV systems has been
quantified by using the four-terminals equations and
Brune’s conditions for series and parallel connection.
A new set of matrix equations that describe the behavior
of PV cell is presented. The equivalent parameters and
the MPP of the PV system for each type of cell
connection are calculated.
The example presented has shown some comparisons
between the PV systems in different connection. The
method can be extended to multiple connections of the
PV cells to increase the efficiency of the system.
[14]
[15]
[16]
5.
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