MPP MPP
Temperature dependence
A constant temperature of 25°C was assumed for all equations of the previous section. It was mentioned that the characteristics change with the temperature. This section describes how to modify the solar cell equations to include the temperature dependence.
The thermal voltage Vt must be calculated for a given temperature. With the Boltzmann constant k = 1.380658 • 1023 J/K, the absolute temperature T in Kelvin (T = & K/°C + 273.15 K) and the elementary charge e = 1.60217733 • 1019 A s, the temperature voltage is given by:
The temperature dependence of the saturation currents IS1 and IS2 with the coefficients cS1 and cS2 and the band gap Eg of Table 4.2 is given by the following equations (Wolf et al, 1977) for silicon devices:
The increase of the saturation currents with rising temperature explains the decrease in the open circuit voltage. The temperature dependence of the series resistance RS, the parallel resistance Rp and the diode factor can be ignored for further considerations.
Equations 4.48 and 4.49 ignore the temperature dependence of the bad gap. While it does not significantly influence the saturation currents, its temperature dependence is decisive for the photocurrent Iph. Due to the decrease in the bad gap with rising temperature, photons with lower energy can elevate electrons into the valence band, which increases the photocurrent. Using the coefficients c1 and c2, the temperature dependence of the photocurrent is given by:
TS1 >exp
Table 4.5 Parameters for the Temperature Dependence of Various Photovoltaic Modules


Parameter 


CS1 
CS2 
c1 
c2 
Unit 
A / K3 
A K52 
m2/V 
m2/(V K) 
AEG PQ 40/50 
210.4 
18.M03 
2.24^103 
2.286^106 
Siemens M50 
170.8 
18.8^103 
3.06^103 
0.179^ 106 
Kyocera LA441J59 
371.9 
12.2^103 
2.51 ^103 
1.932 • 106 
Source: University of Oldenburg, 1994
Source: University of Oldenburg, 1994
Table 4.5 shows the parameters for the calculation of the temperature dependences of various solar modules.
Figure 4.22 shows the IV characteristic with rising temperature It shows clearly that the open circuit voltage decreases significantly when the temperature rises. On the other hand, the short circuit current only increases slightly. The result is the reduction of the MPP power at decreasing temperatures.
Parameter estimation
For the simulation of a real solar cell, for instance with the simple equivalent circuit, the cell parameters (here IPh and IS) must be estimated from measured cell characteristics. To simplify the process, the photocurrent IPh can be assumed to be equal to the short circuit current ISC. The diode factor m of an ideal diode is equal to 1. Hence two parameters are already estimated (I
isc and
1). The diode saturation current IS can be calculated from equation (4.34) using the open circuit conditions:






\\\ \ 

,9  75 °C 1 
1 9 = 50 °q S = 25 °C 
1,9 = 0 °C 
Cell voltage in V
Figure 4.22 Temperature Dependence of Solar Cell Characteristics
With this, all parameters of the simple equivalent circuit are estimated. However, this model can only provide a very rough correspondence with measured characteristics. A diode factor higher than 1 (m > 1) is used for a nonideal diode. The two independent parameters m and IS could be found relatively easy using mathematical software such as Mathematica for a given solar cell characteristic in the generating region. The simple equivalent circuit with a real diode already provides a very good correspondence between simulations and measurements.
The determination of the additional parameters RS and Rp of the extended onediode solar cell model is more difficult. With higher numbers of independent parameters, even professional mathematical programs reach their limits and the best fit has to be determined iteratively. The initial values must be chosen carefully for a good iteration convergence of the parameter estimation. However, the estimation of initial values for Rp and RS is relatively simple.
The parallel resistance Rp can be estimated by the negative slope of the IV characteristic under short circuit conditions. The slope of the IV characteristic around or beyond the open circuit voltage provides the series resistance RS:
The parameters VBr, b and n for the negative diode breakdown operation can be found analogously to the other parameters, when measured values at the point of negative diode breakdown are used for parameter estimation.
Electrical Description of Photovoltaic
Modules
Series connection of solar cells
Solar cells are normally not operated individually due to their low voltage. In photovoltaic modules, cells are mostly connected in series. A connection of these modules in series, parallel or seriesparallel combinations builds up the photovoltaic system.
Many modules are designed for operation with 12V leadacid rechargeable batteries where a series connection of 3240 silicon cells is optimal. Modules for grid connection can have many more cells connected in series in order to obtain higher voltages.
Figure 4.23 Series Connection of Photovoltaic Solar Cells (left: Electrical Symbols, Currents and Voltages; right: Top View of a Part of a Module with Crystalline Cells)
Figure 4.23 Series Connection of Photovoltaic Solar Cells (left: Electrical Symbols, Currents and Voltages; right: Top View of a Part of a Module with Crystalline Cells)
The current I through all cells i of a series connection of n cells is identical, according to Kirchhof's law (see Figure 4.23). The cell voltages Vi are added to obtain the overall module voltage V:
Given that all cells are identical and experience the same irradiance and temperature, the total voltage is given as:
The characteristics of a single cell provide easily the IV characteristic for any series connection as shown in Figure 4.24.
TD O
characteristics for one cell ■ characteristics for two cells
Characteristics for 35 cells Module characteristics
characteristics for one cell ■ characteristics for two cells
Characteristics for 35 cells Module characteristics
Module voltage in v
Figure 4.24 Construction of Module Characteristics with 36 Cells (Irradiance E = 400 W/m2, T = 300 K)
Data sheets published by the module producers often give only a limited number of parameters such as open circuit voltage VOC0, short circuit current
W voltage ^mppq and current 1MPP0 at the MPP at an irradiance of E1000
1000 W/m2 and a temperature of &25 = 25°C as well as the temperature coefficients aV and aI for voltage and current. The equations:
allow fast approximate module performance parameter estimation for different temperatures & and irradiances E. With the parameters
the relation between module current I and module voltage V can be found approximately as:
Series connection under inhomogeneous conditions
During realistic operation, not all solar cells of a series connection experience the same conditions. Soiling by leaves or bird excrement, or climatic influences such as snow covering or visual obstructions by surroundings can shadow some cells. This has a high influence on the module characteristics.
Modelling modules with nonidentical IV cell characteristics is significantly more difficult. The following example assumes that 35 cells of a module with 36 seriesconnected cells are irradiated identically (Quaschning and Hanitsch, 1996). The remaining cell experiences an irradiance reduced by 75 per cent. Even in this case the current through all cells is the same. The module characteristics can be found by choosing a range of currents from zero to the unshaded short circuit current. The voltages of the fully irradiated cells VF and the shaded cell VS are determined and added:
The module characteristic can be obtained when stopping at the short circuit current of the partially shaded cell. However, this characteristic only covers a small voltage range close to the module's open circuit voltage. Lower voltage
Figure 4.25 Construction of Module Characteristics with a 75 per cent
Shaded Cell
Module voltage In V
Figure 4.25 Construction of Module Characteristics with a 75 per cent
Shaded Cell ranges of this characteristic only occur if the current in the partially shaded cell is higher than the cell short circuit current. This is only possible in the negative voltage range of the shaded cell, and this cell then operates as a load that can be described by the equivalent circuit shown in Figure 4.19.
Figure 4.25 shows the determination of one point of the module characteristic (1). The module voltage for a given current is the sum of the voltage of the partially shaded cell (1a) and 35 times the voltage of the irradiated cells (1b). The total module characteristic of the shaded case is calculated this way point by point for different currents.
It is obvious that cell shading reduces the module performance drastically. The maximum module power decreases from P1 = 20.3 W to P2 = 6.3 W, i.e. by about 70 per cent, although only 2 per cent of the module surface is shaded. The partially shaded solar cell operates as a load in this example. The dissipated power of the shaded cell is 12.7 W and is obtained when the module is short circuited.
Other shading situations at higher irradiances can increase the power dissipated in the shaded cell up to 30 W. This will heat the cell significantly and may even destroy it. Socalled hot spots, i.e. hot areas about a millimetre in size, can occur where the cell material melts or the cell encapsulation is damaged.
To protect single cells from hot spot related thermal damage, socalled bypass diodes are integrated into the solar modules in parallel to the solar cells. These diodes are not active during regular operation, but when shading occurs, a current flows through the diodes. Hence, the integration of bypass diodes eliminates the possibility of high negative voltages, and in the process eliminates the increase in cell temperature of shaded solar cells. 
 Figure 4.26 Integration of Bypass Diodes across Single Cells or Cell Strings
Bypass diodes are usually connected across strings of 1824 cells. The reason for this is mainly economic. Two bypass diodes are sufficient for a solar module with a rated power of about 50 W containing 3640 cells. The diodes can be integrated into the module frame or module junction box. However, these diodes cannot fully protect every cell; only the use of one bypass diode for every cell can provide optimal protection. Semiconductor technology can integrate bypass diodes directly into the cells (Hasyim et al, 1986). Shadingtolerant modules with cellintegrated bypass diodes, which were first manufactured by Sharp, show significantly lower losses when they are inhomogenously irradiated.

 Figure 4.27 Simulation of Module Characteristics with Bypass Diodes across Different Numbers of Cells (E = 1000 W/m2, T = 300 K)
MPP unshaded
MPP unshaded
Figure 4.28 PV Characteristic of a Module with 36 Cells and Two Bypass Diodes. A Single Cell is Shaded to Different Degrees; All Other Cells Are Fully Irradiated (E = 574 W/m2, T = 300 K)
Module voltage in V
Figure 4.28 PV Characteristic of a Module with 36 Cells and Two Bypass Diodes. A Single Cell is Shaded to Different Degrees; All Other Cells Are Fully Irradiated (E = 574 W/m2, T = 300 K)
Figure 4.26 illustrates bypass diode integration across cells and strings of cells. The bypass diode switches as soon as a small negative voltage of about 0.7 V is applied, depending on the type of diode. This negative voltage occurs if the voltage of the shaded cell is equal to the sum of the voltages of the irradiated cells plus that of the bypass diode.
Figure 4.27 shows the shape of IV characteristics with bypass diodes across a varying number of cells. In this example, one cell is 75 per cent shaded. It is obvious, that the significant drop in the IV characteristics moves towards higher voltages with decreasing number of cells per bypass diode. This occurs because the bypass diode switches earlier. It also reduces the power loss and the strain on singles cells.
Figure 4.28 shows the powervoltage characteristics of a module with two bypass diodes across 18 cells for different shading situations. Depending on the degree of shading, the MPP shifts and high losses occur although bypass diodes are integrated.
Parallel connection of solar cells
A parallel connection of solar cells is also possible. Parallel connections are less often used than series connections because the associated current increase results in higher transmission losses. Therefore, this section gives only a rough overview on parallel connection.
Parallelconnected solar cells as shown in Figure 4.29 all have the same voltage V. The cell currents Ii are added to obtain the overall current I: 
 Figure 4.29 Parallel Connection of n Solar Cells
A parallel connection of cells is significantly less susceptible to partial shading, and problems associated with cell damage are much less likely.
Large solar generators often use modules with parallel cell strings of multiple seriesconnected solar cells. However, parallel bypass diodes must also secure cell protection for these modules. Socalled serialconnected blocking diodes will protect cell strings; however, because of their high permanent diode losses and low protective action, they are used in only very few cases.
Technical data of solar modules
Table 4.6 summarizes technical data for selected solar modules. To point out differences in module technology, some modules are included that are no longer available.
This table contains three modules with monocrystalline cells, one with polycrystalline, one with amorphous and one with copper indium diselenide (CIS) solar cells. All modules have only seriesconnected cells. MPP power and efficiency are given for standard testing conditions (see section on 'Further electrical solar cell parameters', p136). The module efficiency is always lower than the theoretical cell efficiency since the module contains inactive areas between the cells. The module efficiency of the BP module is 13.5 per cent, whereas the efficiency of a single cell is 16 per cent. Almost all modules only have two bypass diodes; only a few modules contain a higher number of bypass diodes.
Table 4.6 Technical Data for Various Photovoltaic Modules
Name 

SM 55 
BP5S5 NT51AS5E50ALF UPM SSO 
ST40 
Manufacturer 

Siemens 
BP Solar 
Sharp 
ASE 
Unlsolar Siemens 
Number of cells 

36 (312) 
36 (49) 
36 (49) 
36 (49) 
 
 
Cell type 

monoSi 
monoSi 
monoSi 
polySi 
aSi 
CIS 
MPP power Pmpp 
Wp 
55 
B5 
B5.5 
50 
22 
3B 
Rated current /MPP 
A 
3.15 
4.72 
4.91 
2.9 
1.4 
2.29 
Rated voltage VMPP 
V 
17.4 
1B.0 
17.4 
17.2 
15.6 
16.6 
Short circuit cur. /SC 
A 
3.45 
5.00 
5.5 
3.2 
1.B 
2.59 
Open circuit volt. VOC 
V 
21.7 
22.03 
22.0 
20.7 
22.0 
22.2 
Temp. coeff. aISC 
%/oC 
+0.04 
+0.03 
+0.05 
+0.09 
* 
+0.01 
Temp. coeff. aVOC 
%/oC 
0.34 
0.34 
0.35 
0.3B 
* 
0.60 
Temp. coeff aPMPP 
%/oC 
* 
* 
0.53 
0.47 
* 
* 
Module efficiency 
% 
12.9 
13.5 
13.4 
11.5 
5.4 
B.9 
Length 
mm 
1293 
11BB 
1200 
965 
1194 
1293 
Width 
mm 
329 
530 
530 
452 
343 
329 
Weight 
kg 
5.5 
7.5 
B.5 
6.1 
3.6 
7.0 
Bypass diodes 

2 
2 
36 
2 
13 
1 
Note: * = no information
Note: * = no information
Solar Generator with Load Resistive load
The previous sections described characteristics of solar cells, modules and generators only. In reality, photovoltaic modules should provide electricity that is used by the consumer with an electric load.
The simplest load is an electric resistance R (Figure 4.30). A straight line describes the resistance characteristic. Ohm's law describes the relation between current and voltage:

 Figure 4.30 Solar Generator with Resistive Load

 Figure 4.31 Solar Module with Resistive Load at Different Operating Conditions
If the current I through the resistance is set equal to the current of the solar cell [see Equation (4.34)], the common voltage and the operation point can be found by solving the equation for the voltage V. However, numerical methods are needed to obtain the solution.
For a graphical estimation of the operating point, IV characteristics of the resistance and solar cell characteristics are drawn into the same diagram. The intersection of both characteristics then provides the operating point.
Figure 4.31 illustrates that the operating point of a solar module varies strongly with the operating conditions. Here, the module is operated close to the MPP at an irradiance of 400 W/m2 and a temperature of 25°C. At other irradiances and temperatures, the module is operated suboptimally and the output power is much less than the possible maximum power. Voltage and power at the resistive load vary significantly.
DCDC converter
The power output of the solar module can be increased if a DC (direct current)DC converter is connected between solar generator and load as shown in Figure 4.32.
The converter generates a voltage at the load that is different from that of the solar generator. Taking up the previous resistance example, Figure 4.33 shows that the power output of the module increases at higher irradiances if the solar generator is operated at a constant voltage. The power output can be increased even more if the solar generator voltage varies with temperature, i.e. if the voltage increases with falling temperatures.
Good DCDC converters have efficiencies of more than 90 per cent. Only a small part of the generated power is dissipated as heat. Input power P1 and
output power P2 are identical for an ideal converter with an efficiency of 100 per cent:
Since the voltages are different, the currents I1 and I2 are also different.
Buck converters
If the load voltage is always lower than the solar generator voltage, a socalled buck converter, as shown in Figure 4.34, can be used.
Switches and diodes are considered as ideal in the following calculations. The switch S is closed for the time period TE and the current i2 through the inductance L creates a magnetic field that stores energy. The voltage vL at the inductance is:
£=1000 W/m2 9 = 75 °C 

9 = 25 °C T —\ 






e =400 W/m2 9 = 25 °C 











Module voltage in V
10 15
Module voltage in V
Figure 4.33 Solar Module with Constant Voltage Load for Three Different Operating Conditions 
 Figure 4.34 Circuit of a Buck Converter with Resistive Load
The switch is then opened for a time period TA and the magnetic field of the inductance collapses and drives a current through the resistance R and the diode D. When neglecting the forward voltage drop at the diode, the output voltage v2 at the contacts becomes:
After the period TS = TE + TA the procedure repeats. The mean voltage vD with the duty cycle 8 = TE/TS is:
With IN = V1/R and t = L/R, the current i2 through inductance and load becomes:
The current i2 is between the maximum current (Equation 4.73) and the minimum current (Equation 4.74):
The minimum current is:
Using Imin and Imax leads to the current ^(Michel, 1992):
/max 



\ 
jS*"1^ 77 
\i 2 


f ^inin 


7R 

r 
f 
7 
„ T* U 


Figure 4.35 Current i2 and Voltage vD for a Buck Converter TJr)\
The average of the current i 2 is:
In practice, the output voltage should be relatively constant. Therefore, the capacitors C1 and C2 shown in Figure 4.36 are inserted. Capacitor C1 buffers the solar generator energy when the switch is open.
For an ideal inductance L, the mean output voltage is:
with the input voltage V1 and the duty cycle 6. Assuming ideal electronic components, the mean output current i 2 is given by:
Cr vi
Figure 4.36 Buck Converter with Capacitors
This is obtained from the mean input current and the reciprocal of the duty cycle. If the mean output current I2 is lower than:
the current through the inductance during the switch blocking phase decreases until it reaches zero. The diode will then block the current and the voltage at the inductance becomes zero. The voltage and current thus go into a discontinuous operation mode. A suitable sizing of components can avoid this. If the limiting lower current I2 lim is known, the inductance can then be sized by: ,
Switching frequencies f = 1/T between 20 kHz and 200 kHz seems to be a good compromise. The output capacity C2 is obtained by:
using the ripple of the output voltage V2, i.e. the maximum desired voltage fluctuation AV2.
Semiconductor components such as power fieldeffect bipolar transistors, or thyristors, for higher powers are mainly used for the switch S. Some integrated circuits (IC) exist that can directly control the duty period. For small power applications, the transistors are already integrated into some ICs.
Boost converters
Boost converters are suitable for applications with higher output voltages than input voltages. The principle boost converter structure is similar to that of the buck converter, except that the diode, switch and inductance change positions. Figure 4.37 shows the boost converter circuit.
If the switch S is closed, a magnetic field is created in the inductance L with the voltage vL = V1 (vL > 0). When the switch opens, the voltage v2 = V1  vL (vL < 0) is applied to the load. This voltage is higher than the input voltage V1. In this case, the voltage drop at the diode has been neglected. When the switch closes, the capacitor C2 retains the load voltage. The diode D avoids the discharging of the capacitor through the switch S.
Generator f i
Load

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