A new analytical solar cell I–V curve model

Renewable Energy 36 (2011) 2171e2176

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Renewable Energy journal homepage: www.elsevier.com/locate/renene

A new analytical solar cell IeV curve model Tor Oskar Saetre*, Ole-Morten Midtgård, Georgi Hristov Yordanov University of Agder, Faculty of Engineering and Science, Serviceboks 509, NO-4898 Grimstad, Norway

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 July 2010 Accepted 12 January 2011 Available online 12 February 2011

A simple mathematical equation that can represent empirical IeV curves of individual solar cells, systems of solar cells and modules has been found. The basic model is determined by four parameters: the open circuit voltage, the short circuit current and two shape parameters. With the four parameters determined, the complete currentevoltage curve, the fill factor and the maximum power point are given by simple analytical functions. The model is valid both in the positive and the negative (dark condition) voltage range. Several simple examples demonstrate some of the potential of the model. Due to its mathematical simplicity, it is suggested that the model will be suitable for analytical modeling of complex systems.
2011 Elsevier Ltd. All rights reserved.

Keywords: Solar cell Currentevoltage curve Fill factor Maximum power point

1. Introduction

present paper can be used for modeling single cells, a system of single cells or as the output IeV curve of a module.

There are many models in the literature which represent the currentevoltage characteristics of solar cells (cf. e.g. [1e10]). They vary from models with simple assumptions to advanced models incorporating many physical variables. Fig. 1 depicts the well-known circuit model of a single junction solar cell. For simplicity, a one-diode model is chosen here. The relationship between current I and the voltage V at its terminals is given by


 VþIRs V þ IRs I ¼ IL  I0 e nVT  1  Rsh

(1)

where IL is the photogenerated current of the cell, I0 is the dark saturation current of the diode (D1), n is the diode ideality factor, a number between 1 and 2, and VT is the thermal voltage. The thermal voltage equals Boltzmann’s constant times the temperature in Kelvin divided by the magnitude of the electron charge (cf. e.g. [11]). More complex models with several diodes are frequently found in the literature (cf. e.g. [12e15]. In the present paper, a new and simple model is suggested. It can be argued that if a mathematical representation of experimental data is needed, it is sometimes good practice to choose the simplest mathematical expression which can adequately represent the data. In particular, when a larger system is being represented by multiples of a basic model, a simple expression may be better for the performance of the whole model. The equations suggested in the

* Corresponding author. Tel.: þ47 372 33137. E-mail address: [email protected] (T.O. Saetre). 0960-1481/$ e see front matter
2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2011.01.012

2. The simple IeV curve model 2.1. Model description It can be argued that the following equations are suitable for modeling the empirical IeV characteristics of solar cells:


h 1x V I ¼ Isc 1  Voc

(2)

and similarly,

" V ¼ Voc 1 




I

x #1h

Isc

(3)

where V is the voltage, Voc is the open circuit voltage, I is the current and Isc is the short circuit current. h and x are shape parameters enabling the adjustment to satisfy experimental data. 2.2. The maximum power point and the fill factor Eqs. (2) and (3) lead to the following equations for current and voltage at the maximum power point, Imp and Vmp:

1
h x Imp ¼ Isc $ hþx and

(4)

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T.O. Saetre et al. / Renewable Energy 36 (2011) 2171e2176

I Rs IL

D1

Rsh

+

and Imp can be determined in a reliable manner, Eqs. (4) and (5) can be employed:

V


x Imp h  ¼ 0 hþx Isc with x given by

_

x ¼ 

Fig. 1. Standard model of a single junction solar cell.


Vmp ¼ Voc $

x

1h

(5)

hþx

(9)

h

Voc Vmp

h

(10)

1

which can be solved numerically to yield h. x can thereafter be calculated directly by Eq. (10).

The following equation then gives the fill factor (FF):


FF ¼

1h
1 h x $ hþx hþx

x

2.3. Asymmetric plateau levels

(6)

Fig. 2 shows the effect of varying x and h on the value of the fill factor as given by Eq. (6). If two IeV curves of the same shape is considered, i.e. h and x have the same values for the two curves, with two different sets of values: Vmp1, Imp1, Vmp2, Imp2, etc., it is seen that:

Vmp1 V ¼ oc1 Vmp2 Voc2

(7)


h 1x V I ¼ Isc 1  Vc

and

Imp1 I ¼ sc1 Imp2 Isc2

At lower voltages an IeV curve of a cell or module with a high shunt resistance is typically almost horizontal. This can be described as the plateau level which often continues as a plateau well into the negative voltages. If the data indicate that the breakoff of the curve occurs at different points in the positive and negative voltage ranges, an asymmetric plateau level exists. In such a case Eqs. (2) and (3) have to be somewhat modified in order to incorporate this effect. This can be achieved by the following modifications:

(8)

Finding the h and x values based on empirical data can be a challenge. Figs. 3e5 show some variations of the curves as a function of h and x. There are several approaches possible for finding the best fit with experimental data, depending on the quality and values of the data. Normally, the Voc and the Isc data should be employed in determining the shape parameters. If Vmp

(11)

and

" V ¼ Vc 1 




I

x #1h

Isc

(12)

where Vc ¼ Voc for V  0 and Vc ¼ Vk for V < 0. Vk is a new parameter enabling the adjustments necessary for the curve to fit the data when V < 0.

Fig. 2. The fill factor (FF) as a function of the two shape variables x and h as determined by Eq. (6).

T.O. Saetre et al. / Renewable Energy 36 (2011) 2171e2176

ξ=1

ξ = 1/2

a

2173

2.0 η=1

1.0 η=1

η=2

η=5

η=15

η=200

1.5

0.8

η=15

η=0.4

I/Isc

I/Isc

0.6

0.0

1.0 η=200

0.4 0.2

η=5

η=2

0.5

η=0.15

0.0

η=0.05

-1.0

η=0.01

0.0

0.2

0.4

0.6

0.8

1.0

V/Voc

0.0

0.5

1.0

V/Voc Fig. 4. It is demonstrated in this figure that even numbers of h will produce curves bending downwards toward the V/Voc-axis when the voltage is negative. Hence, h is normally used with positive odd integers if the negative voltage characteristic is required.

ξ=1

b

-0.5

1.0 η=1

0.8

I/Isc

η=2

η=5

η=15

η=200

2.4. Linearization of the model

η=0.4

It can be argued that one of the best manners to determine the goodness of fit of a model, i.e. how well it fits experimental data, is achieved by linearization. Eq. (2) can be written in the following manner:

0.6 0.4

η=0.15

0.2

η=0.05

ðIn Þx ¼ 1  ðVn Þh

η=0.01

0.0 0.0

0.2

0.4

0.6

0.8

1.0

V/Voc ξ=2

c 1.0 η=1

η=2

η=5

η=15

(13) h

x

where Vn ¼ V/Voc and In ¼ I/Isc. Hence, ðVn Þ ˛½0; 1 and ðIn Þ ˛½0; 1 for a normal single solar cell IeV curve. A plot of the experimental IV data of such a cell should yield a straight line when the data are plotted in a (Vn)h versus (In)x plot if the shape parameters h and x are correctly optimized. More extensive goodness of fit procedures can be found in [15,16].

η=200

3. Case studies

0.8 η=0.4

I/Isc

0.6

3.1. IeV curves from the literature η=0.15

0.4

η=0.05

0.2 η=0.01

0.0 0.0

0.2

0.4

0.6

0.8

1.0

V/Voc ξ=3

d 1.0

η=1

I/Isc

0.8

η=0.4

0.6

η=0.15

0.4

η=0.05

0.2

η=0.01

η=2

η=5

η=15 η=200

0.0 0.0

0.2

0.4

0.6

0.8

An example of a simple IeV curve adapted from a paper by Fraas et al. [17], is shown in Fig. 6. The curve is plotted by employing Eq. (2) with parameters x ¼ 9 and h ¼ 11. It is seen that the curve is forming a plateau level at about I ¼ Isc for a voltage interval between approx. 1 V. The curve plotted in Fig. 6 appears to be accurately reproduced from the paper by Fraas et al. Data from this work have not been available to the present authors and the accuracy of the curve fitted can therefore not be determined. An asymmetric plateau level can be obtained by employing Eqs. (11) and (12). An example chosen from the literature is a paper by King et al. [18], who in their paper showed how to decide the absolute spectral response of separate junctions in a two-terminal triple-junction GaInP/GaAs/Ge cell. The presented currentevoltage characteristics has been used as the basis for a hypothetical triplejunction cell modeled as a series connection of three basic solar cell circuits, where the reverse bias circuit elements are also included. The individual IeV curves as well as the IeV curve of the tandem device have been modeled and can be seen in Fig. 7. The corresponding power curves have been plotted in Fig. 8.

1.0

V/Voc Fig. 3. The plotted curves demonstrate the effect of varying the two shape parameters h and x of Eq. (2). Curves for several different h-values are given for x ¼ 1/2 (a), x ¼ 1 (b), x ¼ 2 (c) and x ¼ 3 (d).

3.2. IeV curves from experimental data Two experimental data sets have been employed. One set concerns data that was obtained by an IeV curve measurement on an individual 156  156 mm2 solar cell from Q-Cells employing

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T.O. Saetre et al. / Renewable Energy 36 (2011) 2171e2176

3.0

a

ξ = 1/2

2.5

Current [A]

4.0 3.5 3.0 2.5

I/Isc

2.0 1.5 η=201

η=15

1.0

η=5

η=3

0.0 -3.0 0.0

0.5

-2.5

-2.0

-1.5

1.0

V/Voc

-1.0

-0.5

0.0

0.5

1.0

1.5

Voltage [V] Fig. 6. A simple IeV curve of a cell modeled by Eq. (2) with parameters: x ¼ 9, h ¼ 11. This curve demonstrates the ability to replicate curves found in the literature (Fraas et al. [17]).

ξ=1

b

1.0 0.5

0.0 -0.5

1.5

η=1

0.5 -1.0

2.0

2.0 1.5

I/Isc

η=201

η=15

η=5

η=3

0.0025

1.0

η=1

0.0020

0.0 -1.0

-0.5

0.0

0.5

1.0

V/Voc

0.0015 0.0010

GaInP GaAs

ξ=2

c

Current [A]

0.5

0.0005

Ge Tandem Cell

2.0

0.0000 -24.0

1.5

I/Isc

-20.0

-16.0

-12.0

-8.0

-4.0

0.0

4.0

Voltage [V] η=201 η=15

η=5

η=3

1.0

η=1

Fig. 7. Individual and the composite currentevoltage curves for a triple-junction cell. The curves are modeled by Eq. (11). Adapted from experimental data of a GaInP/GaAs/ Ge cell (King et al. [18]). GaInP-curve: h ¼ 47, x ¼ 1, Vk ¼ 8.5; GaAs-curve: h ¼ 21, Vk ¼ 10.0, x ¼ 3; Ge-curve: h ¼ 7, x ¼ 1, Vk ¼ 4.6.

0.5 0.0 -1.0

-0.5

0.0

0.5

1.0

V/Voc ξ=3

d

0.0030 2.0

GaInP GaAs

I/Isc

η=201 η=15

η=5

η=3

Power [W]

1.5 1.0

η=1

0.5

Ge

0.0020

Tandem Cell Max level

0.0010

0.0 -1.0

-0.5

0.0

0.5

1.0

V/Voc Fig. 5. IeV modeling with the voltage range allowing for negative values. The plotted curves demonstrate the effect of varying the two shape parameters h and x of Eq. (2). Curves for several different h-values are given for x ¼ 1/2 (a), x ¼ 1 (b), x ¼ 2 (c) and x ¼ 3 (d).

0.0000 0.0

0.5

1.0

1.5

2.0

2.5

Voltage [V] Fig. 8. Power curves of the individual and composite devices of Fig. 7. The sum of the maximum power of the three individual junctions can be calculated by Eqs. (4) and (5) and the result is seen as an unreachable horizontal level at the top (broken line).

T.O. Saetre et al. / Renewable Energy 36 (2011) 2171e2176

2175

0.8

Current [A]

0.6

0.4

0.2

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Voltage [V] Fig. 9. Modeling an experimental data set of an individual solar cell. x: Experimental data points. Solid red line: Generated by employing Eq. (2) with x ¼ 0.9 and h ¼ 12. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Elkem Solar polycrystalline-Si. Isc and Voc are approximately 7.8 A and 0.62 V respectively. The cell was measured under weak sunlight coming through a window. No direct irradiance measurements were taken, but the measured short circuit current indicates an irradiance of approximately 0.08 kW/m2. For this reason the cell’s apparent performance, shown in Fig. 9, was lower than it would have been in an optimized test. However, the purpose of the measurement was to generate the actual response of a real solar cell and to investigate whether the model proposed in this paper could be utilized to represent the actual recorded data. The experimental data has been plotted in the linearized way described in Section 2.4 in Fig. 10. The shape parameters h and x can be fine-tuned if standard methods of calculating the deviation from a straight line are employed, e.g. by calculating the correlation coefficient. The other set of data consists of an outdoor IeV curve measurement of a polycrystalline-Si PV module from Q-Cells. The data was recorded in January 2010 in Southern Norway (latitude N58 020 ). The module consists of 60 large area cells connected in series. The IeV curve sweep time used was 280 ms, and the variation of irradiance during the sweep was within 1%. Because of the non-zero minimum resistance of the electronic load that was used (of about 0.15 Ohm), a complete short circuit condition was not obtained. The short circuit current was therefore determined from linear extrapolation of the lowest-voltage part of the IeV curve. 1.0

Fig. 11. Modeling an experimental data set of a module of solar cells. x: Experimental data points. Solid red line: Generated by employing Eq. (2) with x ¼ 0.7 and h ¼ 14. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9 shows experimental data from the single cell marked as crosses and the corresponding continuous curve given by Eq. (2) with x ¼ 0.9 and h ¼ 12. The suitability of applying the model for the module can be seen in Fig. 11. The experimental data are marked as crosses and the solid line is determined by Eq. (2). x ¼ 0.7 and h ¼ 14 in this case. 4. Discussion and conclusion In the present paper, a new analytical mathematical model for representing empirical currentevoltage curves of both single PV cells and modules has been introduced. It has been demonstrated in several example cases that the equations suggested can model IeV curves quite well and that a linearization is possible. Moreover, it has been shown that the model is capable of representing PV cells also in the negative voltage range. It is suggested that further research may find useful functions for the x and h shape parameters based on physical conditions such as effective irradiance to the solar cells. Thus, the equations given in this paper may in the future be employed in predicting the IeV curve for devices based on the energy input. In conclusion, it has been demonstrated that Eqs. (2) and (3) can be used as simple mathematical models to represent the IeV curves of individual solar cells and modules. The equations are not derived from physical principles but are simple to use and therefore have a potential to be useful in analytical analysis and design of systems.

0.8

Acknowledgment 0.6

(In)ξ

The authors would like to thank Elkem Solar for financial support and for supplying solar cells.

0.4

References 0.2

0.0 0.0

0.2

0.4

0.6

(Vn)

0.8

1.0

η

Fig. 10. The experimental data points in Fig. 9 is plotted in a linearized plot as (Vn)h versus (In)x. Plotted in this manner, the experimental data should yield a straight line if the model is accurate. In this manner, the values of the shape variables h and x can also be determined.

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