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A numerical model of nonuniform solar cell stability

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A numerical model of nonuniform solar cell stability
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  A numerical model of nonuniform solar cell stability A. C. Vasko  • V. G. Karpov Received: 16 July 2014/Accepted: 5 September 2014   Springer-Verlag Berlin Heidelberg 2014 Abstract  We model degradation in extended two-dimensional solar cells. The model couples electrical andthermal physics, so that voltage and temperature can varyover a device. Degradation is modeled phenomenologi-cally, such that local open circuit voltage degradation isdependent on local temperature. Initial device nonunifor-mity is introduced with an initial spatial distribution of random shunt conductances. A comparison of pre- to post-stress performance shows variations, despite the deter-ministic nature of the degradation model, which is inter-preted as a consequence of temperature variations. 1 Introduction Solar cell stability is an important factor in determiningdevice lifetime, return on investment, pricing, and warrantypolicies. Stability is often studied under harsher conditionsthan working cells are exposed to in the field, so that30 years of field stress can be inferred with 20 days of laboratory study, through increased concentrations of light,higher temperatures, or greater humidity (accelerated lifetesting, or ALT). Higher temperature seems to be a fun-damental factor. Established experimental work on devicedegradation has found that average parameter degradationin time can be fit to an Arrhenius phenomenological model[1, 2], such as d ð P = P o Þ d t    ¼  Z   exp    E kT     ð 1 Þ in which  P  could be one of many of the metrics used tocharacterize solar cell performance, such as efficiency, fillfactor, or open circuit voltage, and  P o  is the initial,unstressed value of this metric. A typical activation energy, E  , may be about 1 eV, so that the elevated temperature of 100   C can accelerate degradation, compared with roomtemperature, by a factor of over 4,000. Accelerated lifetesting has been argued to demonstrate cells may have alifetime greater than 100 years [3].Though average cell degradation may be well-con-trolled, other aspects of degradation are less understood.Various mechanisms for degradation within CdTe cellshave been proposed in terms of atomic phenomenon [4],though there is no consensus in the field as to what modesare present in certain samples, or if all possible modes areyet accounted for. A particular phenomenon that is not wellunderstood is that nominally identical cells may degradedifferently, at any phase through the degradation process.For instance, degradation form of Eq. (1) only describes thedegradation average, but has no information regardingdispersion about the degradation mean.What has not been considered prior to now is the phe-nomenon of nonuniform degradation, where differentregions of an extended cell may, at any given time, bothhave different material parameters and which may degradeat different rates. Exact analytic treatment of nonunifor-mity in extended area cells has proven difficult, butnumerical modeling has allowed understanding to progress[5].Our approach in this paper is to use a finite elementapproach to modeling solar cells. First, we discuss thegeneral numerical nature of the model, how the voltagedistribution is determined over the sample and from that NSF award No. 1066749.A. C. Vasko ( & )    V. G. KarpovDepartment of Physics and Astronomy, The Universityof Toledo, Toledo, OH 43606, USAe-mail: anthony.vasko@rockets.utoledo.edu  1 3 Appl. Phys. ADOI 10.1007/s00339-014-8772-x  the temperature distribution over the sample. Next, we willdiscuss models for solar cell degradation, which uses thepreviously obtained temperature distribution as input.Then, we will present degradation simulation data anddiscuss implications. 2 Numerical model 2.1 Voltage modelOur model divides a continuous, two-dimensional cell intomany nodes containing and connected with finite elements.We have previously used similar methods to simulateadmittance spectroscopy measurements over distributedsolar cells [6] and to model the experimentally observedphenomenon of spontaneous hotspot formation in solarmodules [7].Figure 1 shows an elemental node of the cell in themodel.The local voltage at each node point  ð  x ;  y Þ  at the frontand back contact electrodes (denoted through  V  fc  and  V  bc ,respectively) is determined through Kirchoff’s current law X a V  a bc  V  bc ð  x ;  y Þ q bc þ V  fc ð  x ;  y Þ V  bc ð  x ;  y Þ  R sh þ  I  L   I  D  ¼ 0 ð 2 Þ X a V  a fc  V  fc ð  x ;  y Þ q fc þ V  bc ð  x ;  y Þ V  fc ð  x ;  y Þ  R sh   I  L þ  I  D  ¼ 0 ð 3 Þ In this,  I  L  is the light-generated current in the activematerial (which is equal to the light-generated currentdensity multiplied by the node area).  I  D  is the currentthrough the diode.  R sh  is the resistance of the parasiticshunt conductance.The summation index  a  is over nodes adjacent to point ð  x ;  y Þ ; that is to say, the local back contact voltages  V  a bc  atnodes that are connected to  V  bc ð  x ;  y Þ  through resistors  q bc ,and the local front contact voltages  V  a fc  at nodes that areconnected to  V  fc ð  x ;  y Þ  through resistors  q fc . The edgeboundary conditions can be applied by adjusting thenumber of adjacent nodes. For instance, for the situationfor which no current flows through the edges, interiornodes will have four adjacent nodes (which is the situationdepicted in Fig. 1), edge nodes three adjacent nodes, andcorner nodes two adjacent nodes.For nodes arraigned in a square lattice, the value of theresistors  q fc  is the sheet resistance of the transparent con-ducting oxide (TCO) front contact (  10 X ), while  q bc  isthe sheet resistance of the metalized back contact ( . 1 X ).Unmetalized areas of the back contact have large ( J 1G X )sheet resistance. The cell to be simulated is a 1 cm 2 met-alized circle on a 1.7 cm 2 square of glass. The purpose of having the substrate slightly larger than the contact size isto allow the temperature perturbations caused by shuntswithin the cell to, at least potentially, extend past thecontact area, as they might in a real cell.The diode current  I  D  is given by the temperature-dependent Shockley Equation  I  D  ¼  I  O  exp  V  bc  V  fc nkT     1    ð 4 Þ with  I  O  ¼  I  OO  exp    E  g nkT     ð 5 Þ with  E  g  equal to the bandgap of the solar cell absorbermaterial (1.5 eV in the case of CdTe), generally differentfrom the value of   E   in Eq. (1).  k   is the Boltzmann constant. n  is the diode ideality factor, generally between 1.0 and2.0; we have used 1.8, a value typical for CdTe.Equations (2) and (3) are generally not completely lin- early independent (note that for a 1-node model, so noadjacent nodes exist, Eqs. (2) and (3) are simply negatives of each other), so to obtain unique voltages, it is necessaryto define at least one point to be at zero potential. When acell is connected to a load, we define the edge of the TCOregion to be at zero potential (which is in fact similar to howthe cells would be measured in the laboratory), one end of the load is also at ground, and the other is connected to thecenter of the back contact of the contact region. Cells weremodeled at open circuit and near maximum power point. Fig. 1  Elemental node used in voltage distribution modelA. C. Vasko, V. G. Karpov  1 3  Generally, modeling at the exact maximum power pointwould be more difficult. Even experimental cells requiremaximum power point tracking circuits to find the voltageor load condition where the cell puts out maximum power.These circuits find the maximum power through someperturbation procedure. A similar, simulated perturbationprocedure would be necessary for a numerical model. Tosimplify this, we have used a load  R mp  ¼ V  OC =  I  SC  as anapproximation to the maximum power point condition. Afurther justification of using a passive load in simulationrather than an actively tracked maximum power point is thatthe experimentally observed degradation, though greatlyaccelerated near open circuit, is relatively constant atmaximum power point, and particularly at lower voltages.Equations (2) and (3) can be solved in various ways, for instance, either as a collection of multivariable nonlinearequations with Newton’s method, or they may be linearizedand iteratively solved with any technique suitable forsolving systems of linear equations.2.2 Temperature modelThe temperature distribution over the device is governedby the heat equation C  o T  = o t  ¼ Q  a ð T   T  o Þþrð j r T  Þ r ð T  4  T  4o Þ ð 6 Þ In this,  C   is the heat capacity,  T   is sample temperature,  T  o  isambienttemperature, t  istime, Q isinternalheat, a istheheattransfercoefficienttotheambient, r istheStefan–Boltzmannconstant, and  j  is the sample thermal conductivity.Although it is possible to integrate Eq. (6) to find the timeevolution of the device, we make the simplifying assump-tion that light stress is a quasi-static phenomenon such thatat most times,  o T  = o t  ¼ 0. The advantage of this assumptionis that it is computationally much faster and avoids entirelypotential issues of stiffness in Eq. (6). The exception to o T  = o t  ¼ 0 is that after the rules for degradation (see below)are applied, the degradation may result in a new voltagedistribution and new temperature distribution; however,even then we assume the new temperature distribution isreached ‘‘instantly.’’ We justify this by noting, for example,that when a cell at room temperature is exposed to 2 sunslight intensity, thermal equilibrium is reached in about10 min, which is a small time scale compared to the 20-daytime period of the light stress. Moreover, we have, for a fewtest configurations, run degradation simulations using bothfull-time integration and the quasi-static assumption, andobtained indistinguishable results.When Eq. (6) is reduced to finite element form, itbecomes Q þ X a v ð T  a  T  Þþ  A a ð T  o  T  Þþ  A r ð T  4 o   T  4 Þ¼ 0 ð 7 Þ Again, the index  a  is over adjacent nodes, and the boundarycondition (a negligible amount of heat is assumed to flowout of the sides) is dealt with as with Eqs. (2) and (3). The internal heat  Q  is made of the sum of all Jouleheating terms IV for all local resistors, as well as theinternal heat within the active solar material. For resistorsspanning two nodes, half of the heat generated within theresistor is assigned to each node. For the active solarmaterial, if light energy  H   falls on the node, the internalheat generated in the node is  H  þ  I  ð V  bc  V  fc Þ , with  I  the net current through the current source and Shockleydiode [8], with the convention that the light-generatedcurrent is negative. Note that when the solar materialgenerates useful power, the product  IV   is negative, so thatthe heat is reduced. For 1 sun light intensity,  H   = 0.1 W/ cm 2 . If the light is concentrated by a factor of   c  (eg.,  c  =2, for 2 suns), we use  H   ¼ c  0 : 1 W/cm 2 and alsoincrease the light-generated current within the cell by thesame factor.As a simplification, we treat the substrate as a two-dimensional sheet, so that  v  is a thermal sheet conductance,with  v ¼ d   j , with  d   the thickness of the substrate(3 mm, typical for a CdTe glass substrate) and  j  thematerial thermal conductivity (0.85 W/m/K for glass).  A  isthe area of a node within the model (which decreases as thenode resolution increases).  a  is the air heat transfer coef-ficient; we have used a typical value of 10 W/K/m 2 .  r  isthe Stefan–Boltzmann constant, 5 : 67    10  8 W/K  4  /m 2 .For a uniform cell (with no shunts), within this model usingthese constants, the equilibrium temperature for a cell at 1sun is   350 K   and at 2 suns is   400 K  , which is con-sistent with experiment.Since the voltage distribution will affect local heatgeneration and thus temperature, and temperature affectsvoltage distribution, the system of equations defined byEqs. (2), (3), and (7) is linked. This can be dealt with numerically by iterating solutions of them alternately, untilthe solutions converge self-consistently. As the majority of the heat in the system is the voltage-independent  H  , thisprocess tends to converge in only a few iterations.Figure 2 shows an example of the result of these cal-culation. In the upper image, a cell with 4 randomly gen-erated shunts (  140 X  each) is held at room temperatureand open circuit under 1 sun (testing conditions). What isshown is the voltage distribution over the TCO. For anideal cell, without shunts, the entire TCO would be zerovoltage; the presence of the shunts perturbs this voltage. Inthe lower image, the same cell is near maximum powerpoint and stressed under 2 suns and allowed to reach itsequilibrium temperature. The location of the shunts cor-responds to spots of localized heating. Moreover, as theseshunts (as well as the load) drain power from the rest of the A numerical model of nonuniform solar cell stability  1 3  conductive area, there is some relative cooling in somesections of the device.2.3 Degradation model and resultsHere, we take a generic and phenomenological approach.We assume that Eq. (1) applies to the local open circuitvoltage of a cell and that this open circuit voltage is alteredby changing the saturation current  I  OO  in Eq. (5). To beprecise, we assume that after some simulated time D t   in themodel, the local open circuit voltage experiences a change D V  OC  ¼ D tB exp    E kT  stress    ð 8 Þ We have used a thermally activated form for voltagedegradation as this is a form which experimental groupshave found classifies degradation in accelerated life testing[1, 2]. Moreover, other groups have found that, of the various secondary metrics that degrade in ALT, open cir-cuit voltage is most highly correlated to overall efficiencydegradation [3]. The energy  E   in Eq. (8) is chosen as 1 eV,as that is the value obtained from experimental studies. Theprecise physical meaning of this energy is not known for acertainty (and it does not equal the bandgap of the absorbermaterial), but it does correspond to the characteristicatomic energy (such as in the creation of point defects)which appears in some physical theories of device degra-dation [4].For stress at open circuit, we have used  B  = 10 6 V/s,while for stress near maximum power point, we have used  B ¼ 2 : 5    10 5 V/s. The reason for this difference is that itcorresponds to the observed degradation rates[1] for cellsstressed at different biasing conditions. A possible expla-nation for the difference in degradation rate is that a sourceof degradation may be copper diffusion, which is biasdependent [9].The temperature  T  stress  is the local temperature when thecell is at 2 suns light soak.The initial  I  OO  is found through  I  OO  ¼  j L  A =  exp  V  OC nkT  RT    1    ð 9 Þ In Eq. (9), the temperature  T  RT  is the room temperature,295 K, as typically during light stress experiments celldegradation is monitored by temporarily removing the cellsfrom stress and measuring them at 1 sun and room tem-perature (which may include cooling to prevent cells fromheating above room temperature).  j L  is the 1 sun light-generated current density  ð 0 : 025A/cm 2 Þ ;  A  is the nodearea, and  V  OC  is the initial open circuit voltage (0.9 V).To introduce an element of randomness and nonunifor-mity into the deterministic degradation model, each cell hasbetween 1 and 4 random shunts, of random conductance andposition, introduced. The reasoning behind this choice isthat the same total shunt conductance, spread over a numberof shunts, will have similar effect on initial cell perfor-mance; however, the same conductance confined to a single,small shunt will result in the highest heating power densityat the shunt, and the greatest temperature variations over thecell. Additional shunts would be expected to reduce thetemperature variation. An additional simulation was runwith up to 10 shunts to verify that exploration with 4 shuntswas sufficient; the results were consistent. To be clear, thismethod of introducing shunts is not intended to model theactual statistical nature of shunts in real manufactured cells.In fact, this method of introducing shunts results in somecells with open circuit voltages much lower than would beobtained from an optimized fabrication process. Rather, thissimulation is intended to be a means to explore the space of what outcomes are possible given that random shunts exist, 0.00.20.40.60.81.01.2 0.000.010.020.030.040.05 0.00.20.40.60.81.01.2    P  o   t  e  n   t   i  a   l   (   V   )   W  i d  t  h  (  c m  ) L e n g t h  ( c m  )  0.00.20.40.60.81.01.2 393.0393.5394.0394.5 0.00.20.40.60.81.01.2    T  e   m  p  e  r  a   t  u  r  e   (   K   )   W  i d  t  h  (  c m  ) L e n g t h  ( c m  )  Fig. 2  Top , example voltage distribution at open circuit over TCO of cell with random shunts.  Bottom , temperature distribution over samecell at 2 suns and near max power pointA. C. Vasko, V. G. Karpov  1 3  and whether different spatial configurations of randomshunts, which may produce the same initial open circuitvoltage, may produce such different temperature distribu-tions that the final, stressed open circuit voltages might bedifferent. An alternative interpretation of this approach isthat it may be a reasonable approximation to a situation inwhich degradation has two modes, one which infrequentlyproduces ohmic shunts and one which steadily degrades thediode quality.The degradation procedure is as follows: After the cell(with possible shunts) has been generated, its open circuitvoltage is determined at 1 sun and 295 K. For these 1 sundeterminations, the cell is not allowed to heat above 295 K;an experimental analogue would require either cooling orfast measurements. Then, the cells are stressed at 2 suns for20 days, either at open circuit or near maximum powerpoint. Throughout this time, the temperature and voltagedistributions are maintained, as discussed in the previoussections, and during which time the degradation rule Eq.(8) is repeatedly applied, by increasing  I  OO  through Eq. (9).Finally, the cell open circuit voltage is determined again, at1 sun and 295K, after the stress.The results are shown in Fig. 3. Each simulated cellcontributes one data point to the scatter plot. There is rel-atively little dispersion in the stress results of cells stressednear maximum power point, with the initial open circuitvoltage strongly predicting the after-stress voltage. More-over, all data points are reasonably described ( r  2 ¼ 0 : 97)by an average voltage loss of 14 %. The low dispersion isattributable primarily to the low degradation rate, asopposed to the biasing condition; if the degradation coef-ficient B is replaced with the larger coefficient used foropen circuit biasing, the results (not shown) would besimilar to those shown for the open circuit case.In contrast, the results of cells stressed at open circuitexhibit both a nonlinear relationship between initial andafter-stress voltage and variability in the after-stress volt-age as a function of initial voltage.We attribute the observed variability to variations intemperature and hence variation in degradation, which wewill justify in the following section.2.4 1 diode modelIn an attempt to better understand the dispersion shown inthe previous section, we use a simplified one diode mode of a solar cell, as shown in Fig. 4. We assume that the singlediode shown in Fig. 4 degrades uniformly in accordancewith the degradation rules, Eqs. (8) and (9). Solving for the shunt resistance given the open circuitvoltage is straightforward.  R sh  ¼ V  oc   I  L   I  O  exp  V  oc nkT     1     ð 10 Þ Solving for the open circuit voltage given the shunt resis-tance requires use of the Lambert W function [10], W  ð  x Þ exp ð W  ð  x ÞÞ¼  x , V  oc  ¼  I  L  R sh  nkT q  W qI O R sh nkT exp qI L R sh nkT     ð 11 Þ A procedure, then, to use this model would be: (1) Firstdeduce  R sh  from the initial  V  oc  using Eq. (10); (2) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.350.400.450.500.55       V    O   C  ,   A   f   t  e  r   S   t  r  e  s  s V OC , Initial 0.3 0.4 0.5 0.6 0.7 0.8 0.90.30.40.50.60.70.8       V    O   C  ,   A   f   t  e  r   S   t  r  e  s  s V OC , Initial Fig. 3  Comparison of open circuit voltage before and after 20 daysof stress at 2 suns. Top , stress at open circuit voltage condition.  Bottom ,stress near max power point Fig. 4  Simple 1 diode model, with shunt resistance, at open circuit.The diode degrades through increasing reverse saturation currentA numerical model of nonuniform solar cell stability  1 3
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