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Contact grid optimization methodology for front contact concentration solar cells
ARTICLE IN PRESS
Solar Energy Materials & Solar Cells 80 (2003) 155–166
Contact grid optimization methodology for front contact concentration solar cells " F. Bizzi, A. Antonini*, M. Stefancich, D. Vincenzi, C. Malagu, A. Ronzoni, G. Martinelli University of Ferrara and INFM, Via Paradiso 12, 4410 Ferrara, Italy Received 10 March 2003; accepted 1 July 2003
Abstract The high current generated in cells under concentrated sunlight causes a voltage drop on the front contact grid. This drop, proportional to the current intensity and combined with the non-linear I=V characteristic of the diode, limits the cell efficiency in the mid- and highconcentration region. A simulation method capable of evaluating this kind of loss for general contact patterns and different concentration levels is therefore proposed. The simulated I2V curve can be employed for concentration-dependent pattern and coverage optimization. r 2003 Elsevier B.V. All rights reserved. Keywords: Solar concentration; Solar cells; Ohmic losses; Series resistance; Numerical methods
1. Introduction Photovoltaic concentrator systems offer an alternative to photovoltaic flat panels with potentially large cost savings [1]. Such a system consists of a primary light collector, based either on mirror or lenses, concentrating the sunlight over a small area where high efficiency photovoltaic cells produce the actual electric energy. The main advantage of this approach is that large amount of energy can be produced with a limited surface of photovoltaic receiver which is the most expensive system part. The cell design must cope, however, with some problems related to the high energy flux under concentration. In particular the high current density in the cells leads to collection losses important mostly in the case of concentrated solar light. At *Corresponding author. Tel.: +39-0532974329; fax: +39-0532974213. E-mail address: [email protected] (A. Antonini). 0927-0248/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.solmat.2003.07.001
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the same time the optical losses caused by the front contacts reduce proportionally the overall system performance. Possible solutions to this problem, involving the shift of contacts on the back of the cell, are actually in use [2] but they require a complicated solar cell construction process and high-quality material, increasing significantly the costs. Photolithographic technology for front contact concentrator solar cells [3] allows, on the other side, to obtain high efficiencies with lower-grade silicon and moderate process cost. The use of a front contact grid induces two kind of efficiency losses. Optical losses are caused by the shading of the cell surface by the metal grid. These losses are therefore reduced with the minimization of the contact area. On the other hand, under solar concentration, the cell front contacts carry high currents causing proportional voltage drops between the contacted silicon surface and the load. This induces the so-called ‘ohmic losses’, the precise nature of which will be discussed in this work. The reduction of these losses is commonly performed through thicker and wider contacts, increasing contact area and optimization of contact pattern. Whereas an efficient tool is needed for the determination of the optimal compromise between these contrasting issues, the techniques actually employed to calculate the effects of contact-related losses [4] are complex and impose limitations on the grid shape. We propose here an alternative approach to the problem allowing one to study and optimize, under different concentration, more general contact patterns and open the way to different contact optimization solutions.
2. Ohmic losses The fill factor (FF) is a fundamental parameter for the PV cells; it indicates the ratio between the V I value at the maximum power point and the product Voc Isc FF ¼
Vpmax Ipmax : Voc Isc
ð1Þ
Lower than ideal FFs are caused by parasitic series (Rs ), shunt (Rshunt ) resistances, ohmic losses, and non-ideal diode properties. This fact is clear if one considers, for example, the following model: qðV þ JRs Þ V þ JRs J ¼ Jph J0 exp ; ð2Þ AkT Rshunt where A is the ideality factor of the diode, k the Boltzmann constant, T the temperature in Kelvin, and q the electron charge. Moreover in this model, Rs is dependent on the current density. This is caused by the so-called ‘current-crowding’ effect, and the higher the emitter sheet resistance the more pronounced the effect. The component of ‘current-crowding’ owing to lateral minority carrier diffusion through the bulk, as discussed in Ref. [5], is not considered in this work.
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Most of the earlier theoretical calculations [9,10] used a fixed value for the series resistance. The Rs is calculated by defining an equivalent lumped series resistance, starting from an equivalent electrical circuit with distributed series resistance under uniform illumination [6]. This resistance is calculated as 1 L 1 ¼ Rd ; Rs-lumped ¼ Rsq 3 W 3
ð3Þ
where Rsq is the sheet resistance in O/square, L the distance from the finger and W the width of the considered piece of cell. Rs-lumped gives, under one sun, the effective power dissipation across the sheet. This method is applicable under the condition 1 AkT DVd p ; 3 q
ð4Þ
where DVd is the voltage drop across the considered sheet. The lumped resistance is the zero order of a perturbation theory approach to cell modeling [6]. Under concentration, the effect of variation of the equivalent resistance with the photocurrent density and with the operating point [7] can be modeled by the higher-order terms of the theory, which contributes as ‘dynamical’ resistances [6]. Physically, this is due to the so-called ‘distributed diode effect’ [3]. Distributed resistance results in both a junction voltage and consequently generated current density that varies with position. Due to the voltage drop along the metallic finger and the front-doped cell surface, the cell region further away from the load operates at a forward voltage higher than that applied to the load. This induces, according to the non-linear I2V characteristic of the p2n junction, a higher local density of forward current flowing through the junction instead of the external load with an overall reduction of the drained current. Since the voltage drops are proportional to the current density and the forward current depends exponentially on the voltage, this effect becomes crucial under solar concentration. For a single finger solar cell like that shown in Fig. 1, the equivalent electrical circuit with distributed parameters is shown in Fig. 2. When the generated current is high, the current losses due to the bending of the I2V
Fig. 1. A single finger solar cell is a good example to evaluate the effect of series resistance on the cell fill factor. It allows to compare systems available in literature with the proposed self-convergent algorithm.
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Fig. 2. Discrete equivalent circuit of solar cell element in Fig. 1.
characteristic of the diodes become relevant and the slope of the total characteristic decreases. In this paper we evaluated the FF of a solar cell taking into account the current dependence of the series resistance with its contribution from sheet resistance and contact grid. To be as pervasive as possible and to minimize the high resistance path covered by the electrons, the contact must be designed with the aid of a quantitative analysis tool.
3. Actual method to simulate I2V characteristics Common analytical approaches to obtain the I2V characteristics, starting from the layout of the contact grid and from the sheet and metal resistivity, are based on the solution of large, non-linear electric circuits [4]. The I2V curve of a single finger solar cell, sketched in Fig. 1, is obtained as the solution of the circuit in Fig. 2. This takes into account, in a discrete form, the distributed series resistance of the front doped cells surface and of the metallic grid. A semi-linear I2V characteristic for every couple source-diode, as sketched in Fig. 3, is used in place of the real exponential curve, allowing a semi-analytical solution of the circuit. The bending point marks the beginning of forward polarization of the elementary diode. The solution of the circuit is presented in detail in Refs. [4,8] but the technique is difficult to generalize to non-trivial patterns owing to some underlying assumptions. To properly draw the equation system using the approximation of semi-linear I2V characteristic, it is necessary to know a priori
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Fig. 3. Linearized version of the I2V characteristic of an elementary diode element employed for the global cell I2V characteristic computation.
the order of diodes reaching the bending point, resulting from the increase of the voltage load. In general this is impossible in the case of high currents, very resistive contacts or complicated patterns of metallic grids. Here, a method is proposed to obtain the cell I2V characteristic without the above-mentioned limitations, useful for wide illumination range. 4. Self-convergent method In the proposed scheme the solar cell is divided into elementary ‘cells’ characterized by a tabulated I2V curve allowing to take into account their intrinsic non-linearity. These basic elements are connected to one another and to the contact network according to the physical contact layout. The cell equivalent circuit can therefore be designed. The solution strategy considers each active element as an abstract voltage controlled current generator, as shown in Fig. 5, behaving according to the elementary I=V curve and driven by the average voltage at the element boundary. The voltage distribution on the generator ensemble is univocally determined by the elementary currents injected in the resistive contact network and by the external load. The determination of the I=V curve for the elementary cell is a preliminary step and is a generalization of the approach described in Ref. [4]. The equivalent circuit for a single contact solar cell element is sketched in Fig. 4. If the sector is divided up into n slices, parallel to the electric contact, each of them can be represented by an ideal current generator with a parallel diode, and two resistances representing the relative portion of heavily doped front cell sheet. In the case of rectangular elements, these resistances can be assumed equal. The higher the number of these slices, the more precise the I2V characteristic of the element. A straightforward solution of the electric circuit in Fig. 4 is unpractical, due to the presence of the non-linear electric elements, corresponding to the slices, which generate current according to Eq. (5). A simplification is obtained by the semi-linear approximation of each slice I=V characteristic. This results in Eq. (6), along the prescription of [4]. 1 qVi ii ¼ Iph I0 exp 1 ; ð5Þ n AkT
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Fig. 4. A simple design for a solar cell with a single busbar contact connected to an external resistance load and its equivalent circuit.
8 1 > Iph > > > > Voc Vcut >n > : 0
if Vi oVcut ; if Vcut pVi oVoc ;
ð6Þ
if Vi XVoc :
Being Vcut Vcut ¼ Voc
2AkT : q
ð7Þ
It is then possible to define a set of points j on the I=V curve where the jth diode operating voltage overcomes Vcut and the current produced by the jth slice falls below Iph =n: This results in a slope change for the otherwise linear I=V curve. The I=V curve is determined by calculating the globally produced current Cj and operating voltage Vj in these points and by using linear interpolation elsewhere.
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We thus define the currents Iij produced by the slice i at the point j: Slices with ioj operate at voltages lower than Vcut and produce therefore the maximum current. Combining the semi-linear I=V curve in Eq. (6), the linear resistive grid and the knowledge of the voltage at the jth diode, it is possible to write the following equations for the current produced by the slices: Iph I1j ¼ ; n I2j ¼
Iph ; n
^ Ijj ¼
Iph ; n
Iðjþ1Þj ½1 þ aðjþ1Þ1 þ aj2 ¼
Iph ; n
Iðjþ2Þj ½1 þ aðjþ2Þ1 þ aðjþ1Þ2 þ aðjþ1Þ1 þ aj2 þ Iðjþ1Þj ½aðjþ1Þ1 þ aj2 ¼
Iph n
^ ð8Þ Inj ½1 þ an1 þ aðn1Þ2 þ aðn1Þ1 þ ? þ aj2 þ Iðn1Þj ½aðn1Þ1 þ aðn2Þ2 Iph þaðn2Þ1 þ ? þ aj2 þ ? þ Iðjþ1Þj ½aðjþ1Þ1 þ aj2 ¼ ; n where aik are connected to the resistances Rik in the equivalent circuit, according to Eq. (9), depending, in turn, on the shape and physical parameters of the doped semiconductor sheet 1 Iph Rik : ð9Þ aik ¼ q n 2AkT Then the global current in the jth bending point is directly calculated n n X X j Cj ¼ Iij ¼ Iph þ Iij ð10Þ n i¼1 i¼jþ1 and from the solution of the linear resistive network, Vj and the complete I2V characteristic of an elementary current source are obtained. This curve is then used in the solution of the complete circuit as a map for the iterative process. The resistances at the ohmic metal–semiconductor contacts are not considered in this work, but can be easily implemented adding an appropriate contribution to the circuit of every elementary current source. The solution of the complete circuit employs a convergent iterative algorithm. An initial trial current distribution is assigned to the elementary generators placed on the nodes of the resistive network and, through the solution of the linear resistive interconnecting network, the operating voltage of each elementary generator is computed. These voltages are used, in turn, to ‘‘correct’’ the operating points of the elementary generators until a stable configuration is found.
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The algorithm operates, as represented in Fig. 5, with three maps between the multi-dimensional space of the voltages at the nodes where a generator is present and the space, with the same dimension, of the currents generated by the elementary active elements. The elementary I=V curve, as previously computed, is placed in the current source block that represents the active element and operates as a non-linear map from the space of the voltages to that of the currents. The linear metallic resistive grid is represented, in the linear block, as a multi-dimensional, linear map from the space of the currents to that of the voltages at the generators nodes. The process starts with an arbitrary initial current distribution fed to the linear block, together with the external load voltage, and the resulting voltages are fed to the current sources block as new inputs. The new current set produced by the non-linear block passes through the mixer block where, on the basis of their present and previous value, the new current vector is generated to be fed to the linear block. This process is iterated until the discrepancy between two subsequent set of voltages, defined according to Eq. (11) as their Euclidean distance, becomes smaller than a threshold value which depends on the desired precision. The process is repeated, to convergence, for every value of the external voltage load determining, for each case, the produced output current. This allows one to construct the complete device I=V curve: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X c I c1 Þ2 : discrepancyc ¼ ð11Þ ðI n n n For every value of the external load voltage the existence of a single solution is granted by physical motivations, since it represents the solar cell operating condition. While it is clear that only the physical solution can be a fixed point for the algorithm, its convergence is strictly related to the ‘‘correction criteria’’ assumed, in the mixer block, for the current. In most cases the average between the previous and present value provides satisfying results; in some complex circuital configuration and high current densities, it can however lead to spurious oscillations preventing a stable
Fig. 5. Block diagram.
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Fig. 6. Traditional comb pattern.
convergence. These purely numerical effects can be avoided, and the convergence to the ‘‘true’’ solution recovered, by reducing the rate of current change at each step. The new current value can, for example, be a weighted average between the previous and new value with a strong importance toward the former. Higher weights slow down the convergence but reduce the influence of numerical instabilities.
5. Results The method is applied to the equivalent circuit in Fig. 2 and the results, obtained for different values of finger resistance, are visible in Fig. 7. Fig. 8, extracted directly from Ref. [4], describes the same equivalent circuit with the method exposed by Andreev. The corresponding FF of these curves are 0.55%, 0.50%, 0.46% and 0.42%. The excellent agreement confirms the reliability of the method for standard patterns. The influence of the finger resistance on the cell FF is apparent for the different contact grid. Convergence for more complex pattern, designed according to a fractal layout, has been demonstrated and the experimental verification is currently in progress. According to the aim of this article we give here an example of contact layout for a cell with a traditional comb pattern indicating the simulated work conditions and FF. The characteristics of the cell are the following: Cell dimensions=1 cm 1 cm; Uniform illumination with a flux density, under 1 sun=0.1 W/cm2;
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Fig. 7. (b–e) I2V curves obtained with the method described in the article for the sector in Fig. 1, where the metallic finger resistance are taken with values, respectively, of 0, 0.25, 0.5, 0.75O. They are for a cell at Iph=1 A, Rsc=1 O, Rc=0, A ¼ 1 (A is the ideality factor). (a) I2V characteristic for a branch in Fig. 2 like that shown in Fig. 4.
Fig. 8. All the curves represent the corresponding ones in the previous graphics. These are reproduced directly from the work of Andreev.
Voc ð1 sunÞ ¼ 650 mV; Contact grid with a coverage area=10% and thickness=10 mm; Rosheet ¼ 60 O/square; Rometal ¼ 2:5 10e6 O-cm.
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The current is drained out at the four points, extremities of the bus bars. In this case, with a 1 Sun FF of 82% and efficiency of 20.25%, the FF decreases to 75–76% at 150 Suns, taking into account only the sheet, finger and bus bar resistances. This decrease of FF was evaluated for a cell with the above-described layout, which appears to be optimal for a traditional comb pattern like that in Fig. 6, with 10% coverage area: Finger width=5 mm; Finger number=82; Bus bar width=280 mm. Since the resistance at the metal–semiconductor interface has not been considered; the actual FF losses under concentration may be higher than expected.
6. Conclusions The proposed method is a useful tool for comparing the performances of different contact patterns for standard solar cells. Moreover, it overcomes the limitation of conventional approaches and allows to obtain the complete voltage distribution on the front surface both for the case of concentrating solar cells and for complex contact patterns. Therefore, this allows one to numerically optimize pattern and surface treatments and to explore non-conventional contact layouts and structures. Searching the maximum for the FF, it is possible to choose automatically the best contact layout among a set of parameter-dependent possibilities. Currently, the method only requires the existence of an elementary cell to be repeated all over the structure, but the generalization to patterns with more than one kind of cell is straightforward.
References [1] R.M. Swanson, The promise of concentrators, Prog. Photovoltaic Res. Appl. 8 (2000) 93–111. [2] P. Verlinden, R.A. Sinton, R.M. Swanson, Int. J. Solar Energy 6 (1988) 347–365. [3] Araki Kenji, Yamaguchi Masafumi. An Si concentrator cell by single photolithography process, Solar Energy Mater. Solar Cells 65(1–4) (2001) 437–443. [4] V.M. Andreev, V.A. Grilikhes, V.D. Rumyantsev, Photovoltaic Conversion of Concentrated Sunlight, Wiley, New York, 1997. [5] A.G. Aberle, S.R. Wenham, M.A. Green, A New Method for accurate measurement of the lumped series resistance of solar cells, Conference record, 23rd IEEE Photovoltaic Specialists Conference, Louisvilly, KY, May, 1993, pp. 133–139. [6] L.D. Nielsen, Distributed series resistance effects in solar cells, IEEE Trans. Electron Dev. 29 (5) (1982) 821–827. [7] G. Araujo, A. Cuevas, J. Ruiz, The effect of distributed series resistance on the dark and illuminated current–voltage characteristics of solar cells, IEEE Trans. Electron Dev. 33 (3) (1986) 391–401.
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[8] Kh.K. Aripov, V.D. Rumyantsev, L. Hernandez, An account of distributed nature of front layer and contact grid resistance at calculation of solar cell current–voltage characteristics, Appl. Solar Energy No 5 1986. [9] M.A. Green, SOLAR CELLS Operating Principles, Technology, and System Applications, PrenticeHall, Englewood Cliffs, NJ, 1982. [10] R.J. Handy, Theoretical analysis of the series resistance of a solar cell, Solid State Electron 10 (7) (1967) 765–775.
Solar Energy Materials & Solar Cells 80 (2003) 155–166
Contact grid optimization methodology for front contact concentration solar cells " F. Bizzi, A. Antonini*, M. Stefancich, D. Vincenzi, C. Malagu, A. Ronzoni, G. Martinelli University of Ferrara and INFM, Via Paradiso 12, 4410 Ferrara, Italy Received 10 March 2003; accepted 1 July 2003
Abstract The high current generated in cells under concentrated sunlight causes a voltage drop on the front contact grid. This drop, proportional to the current intensity and combined with the non-linear I=V characteristic of the diode, limits the cell efficiency in the mid- and highconcentration region. A simulation method capable of evaluating this kind of loss for general contact patterns and different concentration levels is therefore proposed. The simulated I2V curve can be employed for concentration-dependent pattern and coverage optimization. r 2003 Elsevier B.V. All rights reserved. Keywords: Solar concentration; Solar cells; Ohmic losses; Series resistance; Numerical methods
1. Introduction Photovoltaic concentrator systems offer an alternative to photovoltaic flat panels with potentially large cost savings [1]. Such a system consists of a primary light collector, based either on mirror or lenses, concentrating the sunlight over a small area where high efficiency photovoltaic cells produce the actual electric energy. The main advantage of this approach is that large amount of energy can be produced with a limited surface of photovoltaic receiver which is the most expensive system part. The cell design must cope, however, with some problems related to the high energy flux under concentration. In particular the high current density in the cells leads to collection losses important mostly in the case of concentrated solar light. At *Corresponding author. Tel.: +39-0532974329; fax: +39-0532974213. E-mail address: [email protected] (A. Antonini). 0927-0248/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.solmat.2003.07.001
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the same time the optical losses caused by the front contacts reduce proportionally the overall system performance. Possible solutions to this problem, involving the shift of contacts on the back of the cell, are actually in use [2] but they require a complicated solar cell construction process and high-quality material, increasing significantly the costs. Photolithographic technology for front contact concentrator solar cells [3] allows, on the other side, to obtain high efficiencies with lower-grade silicon and moderate process cost. The use of a front contact grid induces two kind of efficiency losses. Optical losses are caused by the shading of the cell surface by the metal grid. These losses are therefore reduced with the minimization of the contact area. On the other hand, under solar concentration, the cell front contacts carry high currents causing proportional voltage drops between the contacted silicon surface and the load. This induces the so-called ‘ohmic losses’, the precise nature of which will be discussed in this work. The reduction of these losses is commonly performed through thicker and wider contacts, increasing contact area and optimization of contact pattern. Whereas an efficient tool is needed for the determination of the optimal compromise between these contrasting issues, the techniques actually employed to calculate the effects of contact-related losses [4] are complex and impose limitations on the grid shape. We propose here an alternative approach to the problem allowing one to study and optimize, under different concentration, more general contact patterns and open the way to different contact optimization solutions.
2. Ohmic losses The fill factor (FF) is a fundamental parameter for the PV cells; it indicates the ratio between the V I value at the maximum power point and the product Voc Isc FF ¼
Vpmax Ipmax : Voc Isc
ð1Þ
Lower than ideal FFs are caused by parasitic series (Rs ), shunt (Rshunt ) resistances, ohmic losses, and non-ideal diode properties. This fact is clear if one considers, for example, the following model: qðV þ JRs Þ V þ JRs J ¼ Jph J0 exp ; ð2Þ AkT Rshunt where A is the ideality factor of the diode, k the Boltzmann constant, T the temperature in Kelvin, and q the electron charge. Moreover in this model, Rs is dependent on the current density. This is caused by the so-called ‘current-crowding’ effect, and the higher the emitter sheet resistance the more pronounced the effect. The component of ‘current-crowding’ owing to lateral minority carrier diffusion through the bulk, as discussed in Ref. [5], is not considered in this work.
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Most of the earlier theoretical calculations [9,10] used a fixed value for the series resistance. The Rs is calculated by defining an equivalent lumped series resistance, starting from an equivalent electrical circuit with distributed series resistance under uniform illumination [6]. This resistance is calculated as 1 L 1 ¼ Rd ; Rs-lumped ¼ Rsq 3 W 3
ð3Þ
where Rsq is the sheet resistance in O/square, L the distance from the finger and W the width of the considered piece of cell. Rs-lumped gives, under one sun, the effective power dissipation across the sheet. This method is applicable under the condition 1 AkT DVd p ; 3 q
ð4Þ
where DVd is the voltage drop across the considered sheet. The lumped resistance is the zero order of a perturbation theory approach to cell modeling [6]. Under concentration, the effect of variation of the equivalent resistance with the photocurrent density and with the operating point [7] can be modeled by the higher-order terms of the theory, which contributes as ‘dynamical’ resistances [6]. Physically, this is due to the so-called ‘distributed diode effect’ [3]. Distributed resistance results in both a junction voltage and consequently generated current density that varies with position. Due to the voltage drop along the metallic finger and the front-doped cell surface, the cell region further away from the load operates at a forward voltage higher than that applied to the load. This induces, according to the non-linear I2V characteristic of the p2n junction, a higher local density of forward current flowing through the junction instead of the external load with an overall reduction of the drained current. Since the voltage drops are proportional to the current density and the forward current depends exponentially on the voltage, this effect becomes crucial under solar concentration. For a single finger solar cell like that shown in Fig. 1, the equivalent electrical circuit with distributed parameters is shown in Fig. 2. When the generated current is high, the current losses due to the bending of the I2V
Fig. 1. A single finger solar cell is a good example to evaluate the effect of series resistance on the cell fill factor. It allows to compare systems available in literature with the proposed self-convergent algorithm.
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Fig. 2. Discrete equivalent circuit of solar cell element in Fig. 1.
characteristic of the diodes become relevant and the slope of the total characteristic decreases. In this paper we evaluated the FF of a solar cell taking into account the current dependence of the series resistance with its contribution from sheet resistance and contact grid. To be as pervasive as possible and to minimize the high resistance path covered by the electrons, the contact must be designed with the aid of a quantitative analysis tool.
3. Actual method to simulate I2V characteristics Common analytical approaches to obtain the I2V characteristics, starting from the layout of the contact grid and from the sheet and metal resistivity, are based on the solution of large, non-linear electric circuits [4]. The I2V curve of a single finger solar cell, sketched in Fig. 1, is obtained as the solution of the circuit in Fig. 2. This takes into account, in a discrete form, the distributed series resistance of the front doped cells surface and of the metallic grid. A semi-linear I2V characteristic for every couple source-diode, as sketched in Fig. 3, is used in place of the real exponential curve, allowing a semi-analytical solution of the circuit. The bending point marks the beginning of forward polarization of the elementary diode. The solution of the circuit is presented in detail in Refs. [4,8] but the technique is difficult to generalize to non-trivial patterns owing to some underlying assumptions. To properly draw the equation system using the approximation of semi-linear I2V characteristic, it is necessary to know a priori
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Fig. 3. Linearized version of the I2V characteristic of an elementary diode element employed for the global cell I2V characteristic computation.
the order of diodes reaching the bending point, resulting from the increase of the voltage load. In general this is impossible in the case of high currents, very resistive contacts or complicated patterns of metallic grids. Here, a method is proposed to obtain the cell I2V characteristic without the above-mentioned limitations, useful for wide illumination range. 4. Self-convergent method In the proposed scheme the solar cell is divided into elementary ‘cells’ characterized by a tabulated I2V curve allowing to take into account their intrinsic non-linearity. These basic elements are connected to one another and to the contact network according to the physical contact layout. The cell equivalent circuit can therefore be designed. The solution strategy considers each active element as an abstract voltage controlled current generator, as shown in Fig. 5, behaving according to the elementary I=V curve and driven by the average voltage at the element boundary. The voltage distribution on the generator ensemble is univocally determined by the elementary currents injected in the resistive contact network and by the external load. The determination of the I=V curve for the elementary cell is a preliminary step and is a generalization of the approach described in Ref. [4]. The equivalent circuit for a single contact solar cell element is sketched in Fig. 4. If the sector is divided up into n slices, parallel to the electric contact, each of them can be represented by an ideal current generator with a parallel diode, and two resistances representing the relative portion of heavily doped front cell sheet. In the case of rectangular elements, these resistances can be assumed equal. The higher the number of these slices, the more precise the I2V characteristic of the element. A straightforward solution of the electric circuit in Fig. 4 is unpractical, due to the presence of the non-linear electric elements, corresponding to the slices, which generate current according to Eq. (5). A simplification is obtained by the semi-linear approximation of each slice I=V characteristic. This results in Eq. (6), along the prescription of [4]. 1 qVi ii ¼ Iph I0 exp 1 ; ð5Þ n AkT
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Fig. 4. A simple design for a solar cell with a single busbar contact connected to an external resistance load and its equivalent circuit.
8 1 > Iph > > >
if Vi oVcut ; if Vcut pVi oVoc ;
ð6Þ
if Vi XVoc :
Being Vcut Vcut ¼ Voc
2AkT : q
ð7Þ
It is then possible to define a set of points j on the I=V curve where the jth diode operating voltage overcomes Vcut and the current produced by the jth slice falls below Iph =n: This results in a slope change for the otherwise linear I=V curve. The I=V curve is determined by calculating the globally produced current Cj and operating voltage Vj in these points and by using linear interpolation elsewhere.
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We thus define the currents Iij produced by the slice i at the point j: Slices with ioj operate at voltages lower than Vcut and produce therefore the maximum current. Combining the semi-linear I=V curve in Eq. (6), the linear resistive grid and the knowledge of the voltage at the jth diode, it is possible to write the following equations for the current produced by the slices: Iph I1j ¼ ; n I2j ¼
Iph ; n
^ Ijj ¼
Iph ; n
Iðjþ1Þj ½1 þ aðjþ1Þ1 þ aj2 ¼
Iph ; n
Iðjþ2Þj ½1 þ aðjþ2Þ1 þ aðjþ1Þ2 þ aðjþ1Þ1 þ aj2 þ Iðjþ1Þj ½aðjþ1Þ1 þ aj2 ¼
Iph n
^ ð8Þ Inj ½1 þ an1 þ aðn1Þ2 þ aðn1Þ1 þ ? þ aj2 þ Iðn1Þj ½aðn1Þ1 þ aðn2Þ2 Iph þaðn2Þ1 þ ? þ aj2 þ ? þ Iðjþ1Þj ½aðjþ1Þ1 þ aj2 ¼ ; n where aik are connected to the resistances Rik in the equivalent circuit, according to Eq. (9), depending, in turn, on the shape and physical parameters of the doped semiconductor sheet 1 Iph Rik : ð9Þ aik ¼ q n 2AkT Then the global current in the jth bending point is directly calculated n n X X j Cj ¼ Iij ¼ Iph þ Iij ð10Þ n i¼1 i¼jþ1 and from the solution of the linear resistive network, Vj and the complete I2V characteristic of an elementary current source are obtained. This curve is then used in the solution of the complete circuit as a map for the iterative process. The resistances at the ohmic metal–semiconductor contacts are not considered in this work, but can be easily implemented adding an appropriate contribution to the circuit of every elementary current source. The solution of the complete circuit employs a convergent iterative algorithm. An initial trial current distribution is assigned to the elementary generators placed on the nodes of the resistive network and, through the solution of the linear resistive interconnecting network, the operating voltage of each elementary generator is computed. These voltages are used, in turn, to ‘‘correct’’ the operating points of the elementary generators until a stable configuration is found.
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The algorithm operates, as represented in Fig. 5, with three maps between the multi-dimensional space of the voltages at the nodes where a generator is present and the space, with the same dimension, of the currents generated by the elementary active elements. The elementary I=V curve, as previously computed, is placed in the current source block that represents the active element and operates as a non-linear map from the space of the voltages to that of the currents. The linear metallic resistive grid is represented, in the linear block, as a multi-dimensional, linear map from the space of the currents to that of the voltages at the generators nodes. The process starts with an arbitrary initial current distribution fed to the linear block, together with the external load voltage, and the resulting voltages are fed to the current sources block as new inputs. The new current set produced by the non-linear block passes through the mixer block where, on the basis of their present and previous value, the new current vector is generated to be fed to the linear block. This process is iterated until the discrepancy between two subsequent set of voltages, defined according to Eq. (11) as their Euclidean distance, becomes smaller than a threshold value which depends on the desired precision. The process is repeated, to convergence, for every value of the external voltage load determining, for each case, the produced output current. This allows one to construct the complete device I=V curve: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X c I c1 Þ2 : discrepancyc ¼ ð11Þ ðI n n n For every value of the external load voltage the existence of a single solution is granted by physical motivations, since it represents the solar cell operating condition. While it is clear that only the physical solution can be a fixed point for the algorithm, its convergence is strictly related to the ‘‘correction criteria’’ assumed, in the mixer block, for the current. In most cases the average between the previous and present value provides satisfying results; in some complex circuital configuration and high current densities, it can however lead to spurious oscillations preventing a stable
Fig. 5. Block diagram.
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Fig. 6. Traditional comb pattern.
convergence. These purely numerical effects can be avoided, and the convergence to the ‘‘true’’ solution recovered, by reducing the rate of current change at each step. The new current value can, for example, be a weighted average between the previous and new value with a strong importance toward the former. Higher weights slow down the convergence but reduce the influence of numerical instabilities.
5. Results The method is applied to the equivalent circuit in Fig. 2 and the results, obtained for different values of finger resistance, are visible in Fig. 7. Fig. 8, extracted directly from Ref. [4], describes the same equivalent circuit with the method exposed by Andreev. The corresponding FF of these curves are 0.55%, 0.50%, 0.46% and 0.42%. The excellent agreement confirms the reliability of the method for standard patterns. The influence of the finger resistance on the cell FF is apparent for the different contact grid. Convergence for more complex pattern, designed according to a fractal layout, has been demonstrated and the experimental verification is currently in progress. According to the aim of this article we give here an example of contact layout for a cell with a traditional comb pattern indicating the simulated work conditions and FF. The characteristics of the cell are the following: Cell dimensions=1 cm 1 cm; Uniform illumination with a flux density, under 1 sun=0.1 W/cm2;
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Fig. 7. (b–e) I2V curves obtained with the method described in the article for the sector in Fig. 1, where the metallic finger resistance are taken with values, respectively, of 0, 0.25, 0.5, 0.75O. They are for a cell at Iph=1 A, Rsc=1 O, Rc=0, A ¼ 1 (A is the ideality factor). (a) I2V characteristic for a branch in Fig. 2 like that shown in Fig. 4.
Fig. 8. All the curves represent the corresponding ones in the previous graphics. These are reproduced directly from the work of Andreev.
Voc ð1 sunÞ ¼ 650 mV; Contact grid with a coverage area=10% and thickness=10 mm; Rosheet ¼ 60 O/square; Rometal ¼ 2:5 10e6 O-cm.
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The current is drained out at the four points, extremities of the bus bars. In this case, with a 1 Sun FF of 82% and efficiency of 20.25%, the FF decreases to 75–76% at 150 Suns, taking into account only the sheet, finger and bus bar resistances. This decrease of FF was evaluated for a cell with the above-described layout, which appears to be optimal for a traditional comb pattern like that in Fig. 6, with 10% coverage area: Finger width=5 mm; Finger number=82; Bus bar width=280 mm. Since the resistance at the metal–semiconductor interface has not been considered; the actual FF losses under concentration may be higher than expected.
6. Conclusions The proposed method is a useful tool for comparing the performances of different contact patterns for standard solar cells. Moreover, it overcomes the limitation of conventional approaches and allows to obtain the complete voltage distribution on the front surface both for the case of concentrating solar cells and for complex contact patterns. Therefore, this allows one to numerically optimize pattern and surface treatments and to explore non-conventional contact layouts and structures. Searching the maximum for the FF, it is possible to choose automatically the best contact layout among a set of parameter-dependent possibilities. Currently, the method only requires the existence of an elementary cell to be repeated all over the structure, but the generalization to patterns with more than one kind of cell is straightforward.
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[8] Kh.K. Aripov, V.D. Rumyantsev, L. Hernandez, An account of distributed nature of front layer and contact grid resistance at calculation of solar cell current–voltage characteristics, Appl. Solar Energy No 5 1986. [9] M.A. Green, SOLAR CELLS Operating Principles, Technology, and System Applications, PrenticeHall, Englewood Cliffs, NJ, 1982. [10] R.J. Handy, Theoretical analysis of the series resistance of a solar cell, Solid State Electron 10 (7) (1967) 765–775.