ejsr_47_1_12

European Journal of Scientific Research

ISSN 1450-216X Vol.47 No.1 (2010), pp.122-134

© EuroJournals Publishing, Inc. 2010

http://www.eurojournals.com/ejsr.htm

Development of an Improved Model of SPV Cell for Partially

Shaded Solar Photovoltaic Arrays

R.Ramaprabha

Department of EEE, SSN College of Engineering, Chennai, Tamilnadu, India

Tel: +91-044-27475065; Fax: +91-044-27475063

E-mail: [email protected]

B.L.Mathur

Department of EEE, SSN College of Engineering, Chennai, Tamilnadu, India

Tel: +91-044-27475065; Fax: +91-044-27475063

E-mail: [email protected]

Abstract

A loaded illuminated Solar Photovoltaic (SPV) cell produces forward voltage and

forward current. When a number of such cells are connected as series string and some of

these cells are not illuminated then the voltage across unilluminated cells may get reversed,

though the current remains in the forward direction. In a commercial module, groups of

cells (generally 18) are shunted by a diode to limit the reverse voltage to 0.7 V but all the

cells are not shunted. Characteristics of these unilluminated cells are therefore required to

be studied in the second quadrant also. This work presents first and second quadrant model

of SPV cell. Equivalent shunt resistance (Rsh) in the model varies with environmental

parameters. The effect of change in Rsh hitherto neglected by many researchers has been

correctly modelled and incorporated in the equivalent circuit. The developed model has

been validated through experimentation using a novel simple electronic load.

Keywords: SPV Module, Improved model, Reverse Characteristics, MATLAB, Novel

Electronic load

1. Introduction Solar Photovoltaic (SPV) cells directly convert sunlight into electricity. Many SPV cells are grouped

together to form a module. Modules are normally formed by series connection of SPV cells to get the

required output voltage. Modules having large output currents are realized by increasing the surface

area of each SPV cell or by connecting several of these in parallel. A SPV array may be either a

module or group of modules connected in series/parallel configuration. Output of the SPV array may

directly feed loads or may use power electronic converter for further processing. These converters may

be used to serve different purposes like controlling the power flow in grid connected systems, track the

maximum power available from the SPV array. Model of SPV system is therefore required to study

and optimize the performance of the complete system including these converters and other connected

loads. This paper aims at developing a complete mathematical model of a SPV cell suitable for

analysis of a non-uniformly illuminated solar array. SOLKAR (Model No.3712/0507) cells and

Development of an Improved Model of SPV Cell for Partially Shaded Solar Photovoltaic Arrays 123

modules available in our laboratory were used for validation of the developed model. MATLAB-M file

coding has been used for simulation of the model.

Many analog and mathematical models have been reported in the literature. Different models

suggested by Gow and Manning (1999), Hyvarinen and Karila (2003), Pongratananukul and Kasparis

(2004), have different levels of accuracy and are suitable for different purposes. In a two diode model

proposed by Chowdhury et al (2007), the additional diode represents the effect of recombination of

carriers. A three-diode model is proposed by Nishioka et al (2007) to include the effects of factors not

considered in earlier models. The one diode model proposed by Vitorino et al (2007) offers a good

compromise between simplicity and accuracy and has been used by several researchers. The model has

been sometimes modified, but its basic structure composed of a current source and a parallel diode has

been retained. Most of the authors have neglected the influence of shunt resistance or have taken it as a

constant (Hiren Patel and Agarwal 2008, Koutroulis et al 2008, Vitorino et al 2007, Veerachary 2006,

Walker 2001). However, the reverse voltage characteristics of the module are greatly influenced by the

shunt resistance particularly in crystalline type SPV cells. The proposed model is an improvement over

the existing one diode model in respect of including the effect of varying shunt resistance with

environmental parameters. Experimental determination of voltage-current characteristic of a solar

module is required to validate the model. An attempt to determine such a characteristic by connecting a

variable resistance across the module and measuring the voltage and current by meters is not accurate

due to varying cloud conditions and temperature during the experiment. A simple and novel method

using a current source to quickly draw the characteristic in both first and second quadrant and

recording the results has been developed for this purpose and presented in this paper. Characteristic

between voltage and current with current less than the short circuit current is referred here as the first

quadrant characteristic and between voltage and current for currents more than the short circuit value is

referred here as second quadrant characteristic.

2. Modelling of Practical SPV Module The standard five parameter model (Rauschenbach 1980; Duffie and Beckman 2006) with improved

model equations (Marcelo Gradella Villalva et al 2009) are used to model SPV cell in I quadrant.

The model equations are given below:

( )

sh

sePVPV

t

sePVPV

rphPVR

RIV

V

RIVIII

+−

−

+

−= 1exp (1)

Where

( )[ ] ref

refrefphphG

GTTII −+= α1, and refscrefph II ,, = and scref

sh

sesh

phref IR

RRI ×

+= (2)

( )( )

1exp,

,

−−+

−+=

t

refrefoc

refrefsc

r

nV

TTV

TTII

β

α and

1exp,

,

,

,

−

=

reft

refoc

refsc

refr

V

V

II (3)

ref

treftT

TVV = and

q

kTnV

refref

tref = (4)

∞=shR or a constant value (5)

GII mrefm ×= and ( ) refmrefm TTVV −+= β (6)

+

+−−+

+−= se

t

semm

r

t

ref

seref

tref

serefmrefmref

rref

terf

ref

se RV

RIV

I

V

G

GR

V

RIV

I

V

G

GR expexp (7)

124 R.Ramaprabha and B.L.Mathur

ref

refT

Tnn = (8)

Parameters of all the equations i.e. nominal open circuit voltage (Voc,ref), nominal short circuit

current (Isc,ref), voltage at the MPP (Vm,ref), current at the MPP (Im,ref), open circuit voltage/temperature

coefficient (α), short circuit current/temperature coefficient (β),and the maximum peak output power

(Pm,ref) are given by manufacturer’s data sheet. These information are always provided with reference

to the standard test conditions (STC: Gref =1000 W/m2 and Tref=25

oC) of temperature and solar

irradiation. Some manufacturers provide V-I curve for several irradiation and temperature conditions.

These curves make easier the adjustment and the validation of the desired mathematical V-I equation.

Basically, this is all the information one can get from datasheets of SPV modules. Usually, ideality

factor ‘n’ is between 1 and 1.5 and its choice depends on other parameters of the V-I model. Some

values for ‘n’ are based on empirical analyses (De Soto et al 2006). As given by Carrero et al 2007,

there are different opinions about the best way to choose ‘n’. Because ‘n’ expresses the degree of

ideality of the diode and it is totally empirical, any initial value of ‘n’ can be chosen in order to adjust

the model. The value of ‘n’ can be later modified as expressed in equation (8) in order to improve the

model fitting. The practical SPV cell has a series resistance Rse whose influence is stronger when the

device operates in the voltage source region (low current region) and a parallel resistance Rsh with

stronger influence in the current source region (high current region) of operation. The Rse resistance is

the sum of several structural resistances of the device and basically depends on the contact resistance

of the metal base. The value of Rsh is generally high and some authors (Walker 2007, Veerachary 2006,

Celik and Acikgoz 2007, Kuo et al 2001, Khouzam at al 1994, Glass 1996, Altas and Sharaf 2007)

neglect this resistance to simplify the model. The value of Rse is very low, and sometimes this

parameter is neglected too (Glass 1996, Tan et al 2004, Kajihara and Harakawa 2005, Benavides and

Chapman 2008). The V-I characteristic of the SPV cell depends on the internal characteristics of the

device (Rse, Rsh) and on external influences such as irradiation level and temperature. The reference

value of Rse is found from the V-I characteristics at reference conditions.

Equations from (1) to (8) have been taken from Marcelo Gradella Villalva et al, 2009. Equation

(5) has been replaced by a new equation (9) whose parameters have been experimentally determined as

detailed in section 4 of this paper. Its dependency on temperature ‘T’ (Fig.9) was found to be

negligible and hence neglected to reduce the complexity of the model.

086.0

6.3

−=

GRsh (9)

Equation (1) represents a single SPV cell. This equation can be used to represent a

series/parallel connected module also by suitably modifying its parameters as shown in Table I below.

The simulation results of SPV module consists of 36 cells in series for I quadrant without and with the

effect of including varying Rsh are shown in Fig.1 and Fig.2 respectively. The comparison of the

parameters with manufacturer’s datasheet (SOLKAR module) is shown in Table II.

Table I: Modification of Parameters used in modeling SPV module from SPV cell

Parameters of the SPV cell Parameters of Series array of NS cells Parameters of Parallel array of NP cells

Iph Iph NPIph

Ir Ir NPIr

Vt NsVt Vt

Rse NsRse Rse/NP

Rsh NsRsh Rsh/NP


Figure 1: Characteristics of SPV model with constant Rsh--Reference Table II

Figure 2: Characteristics of SPV model with varying Rsh --Reference Table II

Table II: Comparison of Parameters of the Proposed Model and SOLKAR datasheet values at Reference

conditions

S.No. Parameters With constant Rsh Proposed Model Datasheet

1 Maximum Power ( Pm ) 38.69 W 37.08 W 37.08 W

2 Voltage at Maximum power( Vm ) 16.69 V 16.56 V 16.56 V

3 Current at Maximum power ( Im ) 2.32 A 2.25 A 2.25 A

4 Open circuit voltage ( Voc) 21.2 V 21.24 V 21.24 V

5 Short circuit current ( Isc ) 2.55 A 2.55 A 2.55 A

6 No. of Series Cells (Ns) 36 36 36

7 Series resistance, Rse Variable Variable Not specified

8 Shunt resistance, Rsh 145.62 Ω Variable Not specified

9 Ideality Factor, n 1.5 Variable Not specified

3. Modelling of Reverse Characteristics of SPV Cell Commercially available modules consist of a number of series connected cells to produce voltage

levels of practical use. Because of the series connection, all the cells are forced to carry the same

current called module current. If one or more cells receive less illumination as compared to others,

these cells may get reverse biased leading to their heating and possible damage. Even when a group of

cells (generally 18) are shunted by a reverse biased diode, cells within the group may get reverse

biased and may get damaged. Figure 3 shows the V-I characteristics of a single cell in first and second

quadrants. Whereas the forward characteristic extends to the open circuit voltage of approximately 0.6

Volts, the reverse biased characteristic is much more extensive and limited by the breakdown voltage.

If the cell is shaded, its short circuit current is less than the module current so that it is operated at the

reverse characteristic, causing power loss.


Figure 3: V-I Characteristics of the SPV cell in the forward and reverse biased conditions

Extension of the five parameter model (Bishop 1988) is considered for this work (Figure 4 and

equation 10).

Figure 4: SPV cell reverse bias model (Bishop’s model)

( ) ( )

44444444444 344444444444 21shmI

m

br

sePVPV

sh

sePVPV

sh

sePVPV

t

sePVPV

rphPVV

RIV

R

RIVa

R

RIV

V

RIVIII

−

+−

+−

+−

−

+

−= 11exp (10)

This model includes a modified leakage current term Ishm that describes the diode breakdown at

high negative voltages. The leakage current term Ishm, which is a function of voltage and controls the

cell reverse characteristic, consists of an ohmic term (current through the shunt resistance) and a non-

linear multiplication factor (Alonso-Garcia and Ruizb 2006, Hartman et al 1980, Quaschning and

Hanitsch 1995) describing avalanche breakdown (Pace et al 1986). Where Vbr is the junction break

down voltage, ‘a’ is the fraction of ohmic current involved in avalanche breakdown and ‘m’ is the

avalanche break down exponent. The electrical behavior of the solar cell can be described by this

equation over the whole voltage range. The unknown parameters are ‘a’, ‘Vbr’ and ‘m’. These


parameters are calculated by extracting parameters in those areas of practical V-I characteristic which

are more significant.

The measured V-I characteristics of the SPV cell under reverse biased conditions for dark

condition is shown in Figure 5. Breakdown voltage is calculated by linear regression of the straight line

of voltage against the inverse of current near breakdown region from the dark characteristics (Alonso-

Garcia 2006). The breakdown voltage Vbr is found to be 13.5 V. The other two parameters are found

by tuning them in model by trial and error method so as to match with the experimental characteristics.

The values of ‘a’ and ‘m’ were found as 0.10 and -3.70 respectively. The modelled and experimental

curves of reverse biased measurements for dark and illuminated conditions (258 W/m2) are shown in

Figure 6. The single SPV cell model in the reverse biased condition is extended to simulate SPV

module consists of 36 cells in series. The complete simulated curve of the solar module in both forward

and reverse biased modes for a particular illumination is shown in Figure 7.

Figure 5: Calculation of Vbr from the dark characteristics of SPV cell

Figure 6: Experimental and modeled reverse bias characteristics of SPV cell


Figure 7: Simulated Model of the module by Equation (10)

4. Validating the Model Table II and Figure 2 show that, the developed model and the experimental data are closely matched at

the reference remarkable points of the V-I curve, and the experimental and mathematical maximum

peak powers coincide. The objective of adjusting the mathematical V-I curve at the three remarkable

points was successfully achieved (at V=0, at V=Voc and at V=Vmp). In order to test the validity of the

model, a comparison with other experimental data (different from the reference remarkable points) is

very useful.

For different insolation and temperature the practical characteristics are easily traced out using

electronic load method described in section 5 and the relevant data traced by Digital Storage

Oscilloscope (DSO) are stored in Excel spreadsheet for comparison of model parameters. GWINSTEK

GDS-1022 DSO is used for this purpose which is calibrated using Fluke 5500A Multi-Product

calibrator. Figure 8 shows the mathematical characteristic curves at different irradiations. Figure 9

shows the mathematical characteristic curves of the SOLKAR solar module plotted with the

experimental data at three different temperature conditions. The markers in the graphs show that the

experimental (V, I) points are exactly matched with the model parameters at the remarkable points. The

accuracy of the model for the other points may be slightly improved by running more iteration with

other values of the ‘nref’, without modifications in the equations. The five suggested points of the

model (Engin Karatepe 2006) against the practical characteristics is checked to know the suitability

and accuracy of the developed model. These are presented in Table III below.

Table III: Comparison of proposed model values with practical values at remarkable points

Remarkable

voltage points

Current in amperes at T=300C

G=1000 W/m2 G=750 W/m

2 G=498 W/m

2 G=245 W/m

2

Model Practical Model Practical Model Practical Model Practical

V=0 2.55 2.55 1.91 1.91 1.27 1.27 0.63 0.63

V=0.5Voc 2.49 2.49 1.86 1.86 1.24 1.24 0.62 0.62

V=Vmp 2.25 2.25 1.69 1.69 1.12 1.12 0.56 0.56

V=0.5(Voc+Vmp) 1.65 1.65 1.24 1.24 0.79 0.79 0.37 0.37

V=Voc 0 0 0 0 0 0 0 0


Remarkable

voltage points

Current in amperes at G=985 W/m2

T=340C T=42

0C T=59

0C

Model Practical Model Practical Model Practical

V=0 2.56 2.56 2.61 2.61 2.64 2.64

V=0.5Voc 2.49 2.49 2.53 2.53 2.55 2.55

V=Vmp 2.27 2.27 2.20 2.20 2.13 2.13

V=0.5(Voc+Vmp) 1.68 1.68 1.58 1.58 1.43 1.43

V=Voc 0 0 0 0 0 0

Figure 8: Characteristics of 36 series connected cells (SOLKAR) as per the proposed model and Experimental

at T=300C

Figure 9: Characteristics of 36 series connected cells (SOLKAR) as per the Proposed model and Experimental

at G=985 W/m2


5. The Proposed Electronic Load (Characteristic Plotter) Due to randomly changing field conditions, it is difficult to use voltmeter-ammeter method to draw the

characteristics of a SPV module. Several systems for measuring the V-I characteristic of solar modules

have been proposed. Most of them can plot the characteristic in the first and fourth quadrants only.

They use adjustable resistance (http://emsolar.ee.tuberlin.de/lehre/english/pv1/index.html),

programmable electronic load (http://www.pvmeas.com/ivtester.html), active load (Benson et al 2004)

or capacitors for variable load (Recart et al 2006). It is required to develop a simple, inexpensive and

automatic V-I characteristic plotting and recording system in the first and the second quadrants.

Figure 10: Schematic of the proposed Electronic Load

[The polarities in the brackets indicate the condition when the module current exceeds the short circuit

current value]

A novel method to quickly draw the characteristics of a SPV module under field conditions is

proposed in this work. The schematic of the proposed plotter is shown in Figure 10. The Op-Amp, the

MOSFET and the resistor Rsense have been so connected that the current of the solar panel is

proportional to the voltage applied to the non-inverting port of the Op-Amp. A linear MOSFET (IRF

150/IRF 460) is used. Gate-Source port of the MOSFET is driven by a low frequency triangular wave

signal. For good results, the gate signal should be large enough to cover the entire range of the panel

current from open circuit to short circuit. To plot the characteristic in the second quadrant, the panel

current should increase beyond the short circuit value. For this purpose, a variable Regulated Power

Supply (RPS) has been connected in series. While plotting the characteristics in the first quadrant, the

RPS may be removed or set to zero and while plotting the characteristic in the second quadrant,

appropriate voltage of the RPS is set and magnitude of the triangular triggering signal is also set so that

the current swings between zero to beyond short circuit value. If a general purpose Cathode Ray

Oscilloscope (CRO) is used then the voltage applied to the non-inverting port of the Op-Amp should

be repetitive to observe a steady pattern. When the panel current varies from zero to maximum, the full

characteristic is drawn and the same characteristic is retraced when the current varies from maximum

to zero. Due to large capacitance between the cells and earth, the retraced pattern does not exactly

follow the earlier pattern and therefore two characteristics are seen on the screen of the CRO. As low

frequency signal is applied at non-inverting port of the Op-Amp, the capacitance effect is minimized. A

signal frequency of 1 Hz was therefore used. For uniform intensity of the trace on the CRO screen, the

slope of the trigger should be constant. Therefore triangular wave has been used.

In this work, DSO has been used therefore repetitive trigger signal is not required and only a

slow changing ramp signal to change the current from zero to short circuit value or beyond will be

sufficient to plot the complete characteristic. For equidistant samples, a linear slope has been used. A

typical plot of first and second quadrant characteristic using this plotter is shown in Figure 11.


Figure 11: First and second quadrant characteristics with proposed characteristic plotter

To determine the parameters of equation (9), the SOLKAR panel was mounted on a tilting

stand. The insolation received depends upon the angle of tilt. The insolation was considered as

proportional to the short circuit current of the panel. A number of characteristics were plotted and

stored for different values of insolation. Rsh was determined at each insolation by fitting a straight line

on the initial portion (near short circuit point) of the characteristic. After taking all the readings, finally

a straight line was fitted between reciprocal of Rsh and the insolation ‘G’ to determine various

parameters of equation (9). The hardware set up of electronic load is shown in Figure 12.

Figure 12: Hardware set up of the proposed Electronic load


6. Conclusion In this paper, a well known mathematical model of a SPV has been improvised using an additional

equation showing the insolation dependent shunt resistance. Parameters of this equation have been

determined through experimentation. To draw the characteristic of the SPV module in I and II

quadrant, a novel electronic load circuit has been developed so that the characteristic can be drawn

quickly and data stored before any change in insolation and/or temperature occur. This electronic load

has been utilized for experimentally determining the parameters of the model. It was experimentally

found that the value of the shunt resistance Rsh prominently depend upon insolation and its variation

with temperature is very low. An empirical relation was established between Rsh and insolation by

conducting a series of experiments recording the characteristic at different insolations. The proposed

model is more accurate when applied to analyze SPV module characteristics under partial shaded

conditions. The developed model can be interfaced with power electronics circuits to see the impact of

shading and can be used to develop new methods to reduce the adverse effects of partial shading. The

proposed electronic load method is a better circuit for display and recording of the characteristic in

field conditions as compared to other known circuits.

7. Nomenclature

IPV Solar module output current (A) Voc open circuit voltage of the module respectively

VPV Solar module output voltage (V) Vm &Im Maximum power point voltage and current

respectively

Iph Photo current of the SPV module (A) Pm Maximum power

Ir Diode reverse saturation current in the equivalent

circuit (µA)

ref Additions subscripts indicate the parameters at

reference conditions

Rse Series resistance in the equivalent circuit of the

module (mΩ)

VD & ID Voltage drop across & current through the diode

Rsh Parallel resistance in the equivalent circuit of the

module (Ω)

Ish Current through the shunt resistance (A)

n Diode ideality factor (0<n<1) Ishm Modified Ish (A),

q Electron charge ( =1.602×10-19

C ) Isc Short circuit current of the module

k Boltzman’s constant ( = 1.381×10-23

J/K) Vbr Junction break down voltage

T Tempearture (Kelvin) a Fraction of ohmic current involved in avalanche

breakdown

Vt Thermal voltage (= nkT/q) m Avalanche break down exponent

G Insolation level (at reference condition G=1000

W/m2)

NP Number of cells in parallel

α Short circuit current temperature co-efficient Index

β Open circuit voltage temperature co-efficient SPV Solar Photo Voltaic

NS Number of cells in series MPP Maximum power point

Acknowledgement The authors wish to thank the management of SSN College of Engineering, Chennai for providing all

the experimental and computational facilities to carry out this work.

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