ISSN 1843-6188 Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
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
1Faculty of Electrical Engineering University Valahia Targoviste 18-20 Blv. Unirii, 130082a
2 Faculty of Electrical Engineering University Politehnica Bucharest 313 Splaiul Independentei, 060032
Email:
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ISSN 1843-6188 Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
Abstract: In this paper the performances of photovoltaic (PV) panels has been analysed, for different connections, by using the 4-terminals 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.
Keywords: PV panels, circuit parameters, four-terminal equations, Brunes’s conditions, maximum power, LabVIEW application.
1. 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.
2. photovoltaic Cell Circuit model
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 double-exponential, 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.
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 known as the current –voltage characteristic I-U of the PV cell, and can be defined by
(1)
The diode current is given by
(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
(3)
Applying a short-circuit condition to the cell’s output terminals, u(2) = 0, results the short-circuit current
(4)
If and are measured, then the diode’s saturation current is approximately
(5)
The generated (output) power of PV cell defined by
(6)
can be expressed, by using (1) and (2), as
(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, , thus
(8)
and
(9)
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.
Figure 2. Four-terminal network.
The active four-terminal network can be described by several set of equations, which indicate the voltage-current 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 four-terminal 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
(10)
where is called the matrix of transfer 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
(11)
where is called the matrix of transfer 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
(12)
and, the admittance equation
(13)
of PV cell.
From equations (10) and (11) it is possible to identify the matrix of transfer impedance
(14)
respectively the matrix of transfer admittance
(15)
which contain the parameters of the PV cell.
Imposed in (10) and (11) the open-circuit and short-circuit conditions, the calculated values of the open-circuit voltage and short-circuit current (considering ») are equal with the values indicated in (3) and (4). This proved the correct set of equations (10) and (11) of the PV cell.
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 four-terminal network.
A series connection of ns identically PV cell, shown in Figure 3, is defined by the Brune’s conditions [15], [16].
(16)
(17)
Thus, the equivalent matrix of transfer impedance can be expressed as
a)
b)
Figure 3. Series connection of ns PV cell a), and the equivalent circuit b).
(18)
If it is consider a parallel connection of np identically PV cell, shown in Figure 4, the Brune’s conditions are
(19)
(20)
According to the above conditions (15), (16) the equivalent matrix of transfer admittance is defined as
(21)
The generated power p(2) for ns series respectively np parallel PV cells is obtained by relations (7), (18), and (21)
(22)
a)
b)
Figure 4. Parallel connection of np PV cell a), and the equivalent circuit b).
(23)
On the theoretical point of view, when, and , then and
3. LabVIew application
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 open-circuit voltage is = 0,59 V, the short-circuit current is = 2,16 A, and the maximum power point occurs at = 0,48 V, =0,96 Wp. All the measurements are effected at the solar irradiance E = 1000 W/m2 and the temperature 250 Celsius.
Figures 5 and 6 have shown the current-voltage i-u characteristics for series and parallel connection of different number of PV cells.
Figure 5. I-U characteristics for series connection of PV cell.
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.
Figure 7. P-U characteristics for series connection of PV cell.
Figure 8. P-U characteristics for parallel connection of PV cell.
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.
Table 1. The MPP for series and parallel connection of PV cell
Connection / MPP1 cell / 0,883 W
2 cell in series / 1,767 W
3 cell in series / 2,651 W
4 cell in series / 3,534 W
5 cell in series / 4,419 W
2 cell in parallel / 1,723 W
3 cell in parallel / 2,585 W
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. CONCLUSIONS
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.
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