# Modelling Photovoltaic Systems Using PSpice - 45271 04

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Solar Cell Arrays, PV Modules and PV Generators

Summary

This chapter describes the association of solar cells to form arrays, PV standard modules and PV generators. The properties of series and parallel connections of solar cells are first described and the role played by bypass diodes is illustrated. Conversion of a PV module standard characteristics to arbitrary conditions of irradiance and temperature is described and more general or behavioural PSpice models are used for modules and generators extending the solar cell models described in

Chapter 3.

4.1 Introduction

Single solar cells have a limited potential to provide power at high voltage levels because, as has been shown in Chapter 2, the open circuit voltage is independent of the solar cell area and is limited by the semiconductor properties. In most photovoltaic applications, voltages greater than some tens of volts are required and, even for conventional electronics, a minimum of around one volt is common nowadays. It is then mandatory to connect solar cells in series in order to scale-up the voltage produced by a PV generator. This series connection has some peculiar properties that will be described in this chapter.

PV applications range from a few watts in portable applications to megawatts in PV plants, so it is not only required to scale-up the voltage but also the current, because the maximum solar cell area is also limited due to manufacturing and assembly procedures. This means that parallel connection of PV cells and modules is the most commonly used approach to tailor the output current of a given PV installation, taking into account all the system components and losses.

78 SOLAR CELL ARRAYS, PVMODULES AND PV GENERATORS 4.2 Series Connection of Solar Cells

The schematic shown in Figure 4.1 is the circuit corresponding to the series association of two solar cells. A number of cases can be distinguished, depending on the irradiance levels or internal parameter values of the different cells.

(42) (43)

(303) L r (302) xcelll

Subcircuit

(300)

Figure 4.1 Association of two solar cells in series

4.2.1 The corresponding netlist for an association of two solar cells is given by: Association of identical solar cells

*SERIES .CIR .include cell-2. lib xcel1l454342ce1-2params:area=126.6 jO=le-1 j02=1E-9

+ jsc=O.O343 rs=le-3 rsh=lOOOOO xcell2 045 4 cell-2params:area=126.6 jO=le-1 j02=1E-9 + jsc=O.O343 rs=le-3 rsh=100000 vbias 43 0 dc 0 virradl42 45dclOOO virrad244OdclOOO

.dcvbiasO1.2 0.01

.probe .plot dc i(vbias) .end

SERIES CONNECTION OF SOLAR CELLS 79

As can be seen the two solar cells have the same value as the short circuit current due to the same irradiance value and have equal values of the series and shunt resistances. It is then expected that the total I( V) characteristic has the same value as the short circuit current of any of the two solar cells and that the total voltage is twice the voltage drop in one single solar cell. The output of the array is between nodes (43) and (0) and node (45) is the node common to the two cells. This is shown in Figure 4.2.

o I (vbias) vbias Figure 4.2 Plot of the I( V) curve of the association of two identical solar cells in series

4.2.2 Association of identical solar cells with different irradiance levels: hot spot problem

Imagine that two solar cells are connected in series but the irradiance they receive is not the same. This a common situation due, for example, to the presence of dirt in one of the solar cells. The previous netlist has been modified to consider cell number 2 illuminated with an irradiance of 700 W/m2 whereas cell number 1 receives an irradiance of lo00 W/m2. This is achieved by modifying the value of the voltage source representing the value of the irradiance in cell number 2.

The result is shown in the upper plot in Figure 4.3. As can be seen the association of the two solar cells, as could be expected by the series association, generates a short circuit current equal to the short circuit current generated by the less illuminated solar cell (namely 0.0343 x (700/1000) x 126.6 = 3.03 A). What also happens is that the voltage drop in the two cells is split unevenly for operating points at voltages smaller than the open circuit voltage. This is clearly seen in the bottom graph in Figure 4.3 where the voltage drop of the two cells is plotted separately. As can he seen, for instance, at short circuit, the voltage drop in cell number 1 is 533 mV (as measured using the cursor) whereas the drop in cell number 2 is -533 mV ensuring that the total voltage across the association is, of course, zero. This

80 SOLAR CELL ARRAYS, PVMODULES AND PVGENERATORS vbias 0 V(43)-V(45) v V(45)

Figure 4.3 I( V) curve of a two-series solar cell array with cell number one having an irradiance of 1000 W/m2 and cell number 2 an irradiance of 700 W/m2 means that under certain operating conditions one of the solar cells, the less illuminated, may be under reverse bias. This has relevant consequences, as can be seen by plotting not only the power delivered by the two solar cell series string, but also the power delivered individually by each solar cell as shown in Figure 4.4.

It can be seen that the power delivered by solar cell number 2, which is the less illuminated, may be negative if the total association works at an operating point below some 0.5 V. This indicates that some of the power produced by solar cell number 1 is dissipated by

solar cell number 2 thereby reducing the available output power and increasing the temperature locally at cell number 2.

0 I (vbias)*v(45) v I (vbias)*v(43) A I (vbias)*(v(43)-v(45)) vbias

Figure 4.4 Power delivered (positive) or consumed (negative) on the two solar cells unevenly illuminated of Figure 4.3 and total power, as a function of the total voltage across the series association

SERIES CONNECTION OF SOLAR CELLS 81

This effect is called the ‘hot spot’ problem, which may be important in PV modules where only one of the series string of solar cells is less illuminated and which then has to dissipate some of the power generated by the rest of the cells. The analysis of this problem is extended to a PV module later in this chapter.

4.2.3 Bypass diode in series strings of solar cells

As shown above, the current available in a series connection of solar cells is limited by the current of the solar cell which is less illuminated. The extension of this behaviour for a situation in which one of the solar cells is completely in the dark, or has a catastrophic failure, converts this solar cell to an open circuit, and hence all the series string will be in open circuit. This can be avoided by the use of bypass diodes which can be placed across every solar cell or across part of the series string. This is illustrated in the following example.

Eample 4.1

Assume a series string of 12 identical solar cells. Assume that cell number six is completely shadowed. To avoid the complete loss of power generation by this string, a diode is connected across the faulty device in reverse direction as shown in Figure 4.5.

Plot the final I( V) characteristic and the voltage drop at the bypass diode and the power dissipated by the bypass diode.

The corresponding netlist is shown in the file ‘bypass.cir’ listed in Annex 4. The bypass diode is connected between the nodes (53) and (5) of the string. The results are shown in Figure 4.6 where a comparison is made between the I( V) characterisitic of the solar cell array with all cells illuminated at 1000 W/m2 and the same array with cell number 6 totally shadowed and bypassed by a diode. It can be seen that the total maximum voltage is 5.5 V instead of approximately 7 V.

Figure 4.7 shows the bypass diode voltage, ranging from 0.871 V to -0.532 V in the range explored, along with the power dissipation, which takes a value of 3.74 W in most of the range of voltages. These two curves allow a proper sizing of the diode.

It may be concluded that a bypass diode will save the operation of the array when a cell is in darkness, at the price of a reduced voltage.

1211109 87654321 n nnr-innn

I- +I Figure 4.5 Series array of 12 solar cells with a bypass diode connected across cell number 6

82 SOLAR CELL ARRAYS, PVMODULES AND PVGENERAlORS

Figure 4.6 Effect of a bypass diode across a shadowed solar cell in a series array

OV 1.OV 2.0V 3.0V 4.0V 5.0V 6.0V 7.0V 8.OV

0 I (dbypass)*(v(5)-v(53)) vibas

Figure 4.7 Diagram of the voltage across the bypass diode (top) and of the power dissipated (bottom)

4.3 Shunt Connection of Solar Cells

We have seen in preceding sections that the scaling-up of the voltage can be performed in PV arrays by connecting solar cells in series. Scaling of current can be achieved by scaling-up the solar cell area, or by parallel association of solar cells of a given area or, more generally, by parallel association of series strings of solar cells. Such is the case in large arrays of solar cells for outer-space applications or for terrestrial PV modules and plants. The netlist for the parallel association of two identical solar cells is shown below where nodes (43) and (0) are the common nodes to the two cells and the respective irradiance values are set at nodes (42) and (4).

SHUNT CONNECTION OF SOLAR CELLS 83 ov 100mV 200mV 300mV 400mV 500mV 600rnV 0 I (vbias) 0 I (xcelll.rs) v I (xcell2.r~) vbias

Figure 4.8 Association of two solar cells in parallel with different imadiance

*shunt .cir .include cell-2.lib xcelll 0 43 42 ce1-2params:area=126.6 jO=le-1 j02=1E-9 + jsc=O.O343 rs=le-2 rsh=lOOO xcell2 0 43 4 ce1-2params:area=126.6 jO=le-1 j02=1E-9 + jsc =O. 0343 rs = le - 2 rsh= 1000

vbias 43 0 dc 0 virradl42 0 dc 1000 virrad2 4 0 dc 700 .plot dc i(vbias)

.dcvbiasOO.60.01 .probe .end

Figure 4.8 shows the I( V) characteristics of the two solar cells which are not subject to the same value of the irradiance, namely 1000 W/m2 and 700 W/m2 respectively. As can be seen the short circuit current is the addition of the two short circuit currents.

The above analysis should not lead to the conclusion that the output power generated by a parallel string of two solar cells illuminated at a intensity of 50% of total irradiance is exactly the same as the power generated by just one solar cell illuminated by the full irradiance. This is due to the power losses by series and shunt resistances. The following example illustrates this case.

84 SOLAR CELL ARRAYS, PVMODULES AND PVGENERATORS

Example 4.2 Assume a parallel connection of two identical solar cells with the following parameters:

Rab = 100 12, Rs = 0.5 0, area = 8cm2, Jo = 1 x lo-”, J~c = 0.0343A/cm2

Compare the output maximum power of this mini PV module in the two following cases: Case A, the two solar cells are half shadowed receiving an irradiance of 500W/m2, and Case B when one of the cells is completely shadowed and the other receives full irradiance of 1000 W/m2.

The solution uses the netlist above and replaces the irradiance by the values of cases A and

B in sequence. The files are listed in Annex 4 under the names ‘example4-2.cir’ and ‘example4-2b.cir’. The result is shown in Figure 4.9.

150mW lOOmW

50mW

ow ov lOOmV 200mV 300mV 400mV 500mV 6DomV v I (vbias) *V(43) vbias

Figure 4.9 Solution of Example 4.2

As the values of the shunt resistance have been selected deliberately low to exagerate the differences, it can be seen that neither the open circuit voltage nor the maximum power are. the same thereby indicating that the assumption that a shadowed solar cell can be simply eliminated does not produce in general accurate results because the associated losses to the parasitic resistances are not taken into account.

4.4 The Terrestrial PV Module

The most popular photovoltaic module is a particular case of a series string of solar cells. In terrestrial applications the PV standard modules are composed of a number of solar cells connected in series. The number is usually 3 to 36 but different associations are also available. The connections between cells are made using metal stripes. The PV module characteristic is the result of the voltage scaling of the Z(V) characteristic of a single solar

THE TERRESTRIAL PVMODULE 85 cell. In PSpice it would be easy to scale-up a model of series string devices extending what has been illustrated in Example 4.1.

There are, however, two main reasons why a more compact formulation of a PV mduk is required. The first reason is that as the number of solar cells in series increases. so du;, the number of nodes of the circuit. Generally, educational and evaluation versions of FSpce da not allow the simulation of a circuit with more than a certain number of nodes. 'Ibe SEC& reason is that as the scaling rules of current and voltage are known and hold in general, ik is simpler and more useful to develop a more compact model, based on these des, which could be used, as a model for a single PV module, and then scaled-up to build the mould af a PV plant. Consider the I( V) characteristic of a single solar cell:

Let us consider some simplifying assumptions, in particular thal the shunt resistance, of a solar cell is large and its effects can be neglected, and that the effects of the secand &ode are also negligible. So, assuming I,, = 0 and Rsh = co, equation (4.1) becomes where I,, = IL has also been assumed.

is considered are the following: The scaling rules of voltages, currents and resistances when a matrix of N, x Np dar C&S

where subscript M stands for 'Module' and subscripts without M stand for a single mlar ed6. The scaling rule of the series resistance is the same as that of a N, x Np da€km of resistors:

Substituting in equation (4.2),

86 SOLAR CELL ARRAYS, PVMODULES AND PV GENERAlORS Moreover, from equation (4.2) in open circuit, I0 can be written as:

I0 = (& - 1) (4.10) using now equations (4.4) and (4.6)

(4.1 I) I0 = N,(e* - 1)

Equation (4.1 1) is very useful as, in the general case, the electrical PV parameters values of a PV module are known (such as ZScM and VocM) rather than physical parameters such as Zo. In fact the PSpice code can be written in such a way that the value of I0 is internally computed from the data of the open circuit voltage and short circuit current as shown below. Substituting equation (4.1 1) in equation (4.9)

(4.12)

Neglecting the unity in the two expressions between brackets,

(4.13)

Equation (4.13) is a very compact expression of the I(V) characteristics of a PV module, useful for hand calculations, in particular.

On the other hand, the value of the PV module series resistance is not normally given in the commercial technical sheets. However, the maximum power is either directly given or can be easily calculated from the conversion efficiency value. Most often the value of P,, is available at standard conditions. From this information the value of the module series resis- tance can be calculated using the same approach used in Section 3.13 and in Example 3.5. To do this, the value of the fill factor of the PV module when the series resistance is zero is required. We will assume that the fill factor of a PV module of a string of identical solar cells equals that of a single solar cell. This comes from the scaling rules shown in equations (4.3) to (4.7)

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