YieldOpt, a model to predict the power output and energy yield for concentrating photovoltaic...

RESEARCH ARTICLE

YieldOpt, a model to predict the power output andenergy yield for concentrating photovoltaic modulesMarc Steiner1*, Gerald Siefer1, Thorsten Hornung1, Gerhard Peharz2 and Andreas W. Bett1

1 Materials—Solar Cells and Technologies, Fraunhofer Institute for Solar Energy Systems, Heidenhofstr. 2, 79110 Freiburg, Germany2 Joanneum Research—Materials, Franz-Pichler-Straße 30, A-8160 Weiz, Austria

ABSTRACT

In this work, we discuss three empirical models and introduce one more detailed model named YieldOpt. All models can beused to calculate the power output and energy yield of concentrating photovoltaic (CPV) modules under different ambientconditions. The YieldOpt model combines various modeling approaches: simple model of the atmospheric radiativetransfer of sunshine for the spectral irradiance, a finite element method for thermal expansion, ray tracing for the optics, anda SPICE network model for the triple-junction solar cell. YieldOpt uses a number of constant and variable input parameters,for example, the external quantum efficiency of the cells, the temperature-dependent spectral optical efficiencies of the optics,the tracking accuracy, the direct normal irradiance, the aerosol optical depth, and the temperature of the lens and the solar cell.To verify the accuracy of the models, the I-V characteristics of five CPVmodules have been measured in a 10-min interval overa period of 1 year in Freiburg, Germany. Four modules equipped with industrial-standard lattice-matched triple-junction solarcells and one module equipped with metamorphic triple-junction solar cells are investigated. The higher accuracy of YieldOptcompared with the three empirical models in predicting the power output of all five CPV modules during this period isdemonstrated. The energy yield over a period of 1 year was predicted for all five CPV modules with a maximum deviationof 5% by the three empirical models and 3% by YieldOpt. Copyright © 2014 John Wiley & Sons, Ltd.

KEYWORDS

simulation; modeling; CPV modules; ambient conditions; SPICE; SMARTS2

*Correspondence

Marc Steiner, Materials—Solar Cells and Technologies, Fraunhofer Institute for Solar Energy Systems, Heidenhofstr. 2, 79110 Freiburg,Germany.E-mail: [email protected]

Received 29 April 2013; Revised 14 November 2013; Accepted 27 November 2013

1. INTRODUCTION

The market of high concentration photovoltaics has showna strong growth since 2008 with an installed capacity ofover 40MW in the year 2012 [1]. One reason for thisstrong growth is the further increase in the efficiency of con-centrator photovoltaic (CPV)modules. For instance, in 2012,Amonix published a module efficiency of 33.5% [2]. Othermanufacturers and institutes such as Semprius [3] and Fraun-hofer Institute for Solar Energy Systems (ISE) [1] also pub-lished module efficiencies above 32%. For the continuationof the strong market growth in the future, it is important toprove the bankability of the CPV technology and to con-vince potential investors. For that reason, in addition tothe reliability, at least two other issues are of highest impor-tance. The first goal is to increase the competitiveness ofCPV systems by a further increase of the module efficiency,

and the second goal is to have techniques to predict the an-nual and short-term energy output of CPV systems with thehighest accuracy. The latter issue demands modeling ap-proaches, which calculate the power output of CPV mod-ules under realistic ambient conditions. In addition, themodeling approaches should allow the optimization of aCPV module by a modular architecture. This modular ar-chitecture allows the investigation of the newly designedcomponents of a CPV module by replacing single compo-nents in the models.

There currently already exist a number of models calculat-ing the power output of CPV systems, for instance, [4–10].Some of these models calculate the power output of CPV sys-tems under distinct standardized ambient conditions. They areused for power rating, for instance, [4–6]. Other models cal-culate the sum of power output under any ambient conditionsoccurring within 1 year of operation. They are used to predict

PROGRESS IN PHOTOVOLTAICS: RESEARCH AND APPLICATIONSProg. Photovolt: Res. Appl. (2014)

Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/pip.2458

Copyright © 2014 John Wiley & Sons, Ltd.

the annual energy yield of CPV systems with an accuracyof a few percent, for instance, [7–10]. Both power ratingand energy yield models are not necessarily able nor arethey proven to predict an instantaneous power output forany ambient condition with the same accuracy. Both powerrating and energy yield models can be distinguished in twotypes of models. The first type of models, named as empir-ical models within this paper, describes the dependency ofthe power output from several input parameters by empiri-cal functions using various fit parameters. These empiricalfunctions ignore the detailed design of CPV modules. Thesecond modeling approach considers the detailed designof CPV modules by a modular architecture. The single ele-ments of a CPV module such as optics and multijunctionsolar cell are considered independently. The advantage ofsuch a modeling approach is the possibility to optimize aCPVmodule by the usage of the model. The modular archi-tecture of such models allows an integration of the newlydesigned components of a CPV module by replacing singlecomponents in the models. In this manner, the resultingpower output using the new component can be estimated.

Such a model approach is used by Muller et al. [11]and Ekins Daukes et al. [10], for example. Ekins Daukeset al. [10] previously published such a model called Syracuse.Therein, simple model of the atmospheric radiative transferof sunshine (SMARTS2) [12] is used to calculate thespectral irradiance under which the CPV module operatesat a distinct time and location. The required meteorologicalinput parameters are taken from a database. The requiredinput parameters for Syracuse are the ambient temperature,the module temperature, the relative humidity, the averagedaily aerosol optical depth (AOD), the date, and the time.Syracuse calculates the photo and dark saturation currentsin a multijunction solar cell by a photovoltaic device model.This device model uses the absorption coefficients of thesemiconductor layers and calculates the external quantumefficiencies by solving the semiconductor diffusion equa-tion for each subcell. The dark saturation currents areapproximated by fitting the two diode equation to measuredI-V characteristics. Finally, the current voltage characteris-tic is calculated by a SPICE network model. The Syracusemodel was validated by Chan et al. by predicting the annualenergy yield of one specific CPV module at a location inJapan with an accuracy of 2% [13].

In this paper, we present a model called YieldOpt that issimilar to the Syracuse model. The first results were pub-lished in [14]. One of the novelties of YieldOpt towardSyracuse is a more detailed treatment of the concentratoroptics. The influence of the used optics on the modules’power output is considered in YieldOpt by temperature-dependent spectral optical efficiencies calculated via a finiteelement method (FEM) and ray tracing [15]. Furthermore,the novelty of the model YieldOpt is its usage on predictingthe instantaneous power output under any ambient condi-tions rather than predicting the energy yield or the calcula-tion of the power output under standardized conditions.In addition to predicting the annual energy yield withhigh accuracy, we prove that YieldOpt is also capable of

predicting the instantaneous power output of fiveFLATCON®-type CPV modules [16]. In this paper, thedata generated with the YieldOpt computer model arecompared with experimental data provided by five modulesusing silicone-on-glass Fresnel lenses. The novelty of thismodel validation is the usage of CPV modules with solarcells of different band gap combination. Four of these mod-ules use industrial-standard lattice-matched triple-junctionsolar cells [17], whereas one module is equipped with meta-morphic triple-junction cells [18].

2. MODEL DESCRIPTION

2.1. General modeling approach

The power output of a CPV module is influenced by sev-eral components. Figure 1 schematically shows the mostimportant components: sunlight, which is altered by theinfluence of the Earth’s atmosphere; the Fresnel lens,which concentrates the sunlight; the multijunction solar cellmounted on a heat sink; the interconnection of the solarcells within the module; and the sun-tracking unit. Mostof these components are already described by adequatemodeling approaches. The core of the YieldOpt model,which is introduced in this paper, is the combination ofthese modeling approaches listed in Figure 1. These modelingapproaches and the required input parameters are described inmore details in the following sections. YieldOpt calculatedthe power output of a CPV module in seven steps:

(1) The spectral irradiance reaching the CPV module iscalculated by SMARTS2 [12].

(2) The modifications to the spectral irradiance causedby the Fresnel lens are considered by spectral opticalefficiency calculated with ray tracing, taking intoaccount the temperature of the optics [15]. Thespectral optical efficiency is multiplied with thespectral irradiance.

(3) Calculation of subcell short-circuit currents usingthe external quantum efficiency.

(4) The I-V characteristics of the multijunction solarcells are simulated by a SPICE network model[20–23] based on the calculated spectral irradianceand spectral optical efficiency.

(5) The alignment of the tracker and the module to the sunis implemented in YieldOpt multiplying the short-circuit current of the subcells with a factor derivedfrom an acceptance angle function. This functiondescribes the short-circuit current in dependence oftracker alignment. This function has been measuredfor the investigated modules using a tracking accu-racy sensor (TAS).

(6) Considering the electrical interconnection of thesolar cells in the CPV module by calculation. TheI-V characteristic of the whole module is obtainedby summing up the voltages and retaining thecurrent of each single cell for series-connected cells.

YieldOpt, a model to predict power output and energy yield M. Steiner et al.

Prog. Photovolt: Res. Appl. (2014) © 2014 John Wiley & Sons, Ltd.DOI: 10.1002/pip

For parallel connections, the currents are summedup, and the voltages are retained.

(7) Finally, the maximum power of the calculated mod-ule I-V characteristic is multiplied by a fit parameterp. This fit parameter was introduced to consider sim-plifications within the YieldOpt model and to com-pensate for an offset in the power output comparedwith a real CPV module. Those simplifications are,for instance, the homogenous illumination of thesolar cell or the assumption of an identical I–V char-acteristic for all interconnected solar cells within theCPV module. The fit parameter does not affect therelative variation of the power output over the yearbut is needed for calculating the absolute values ofthe power output of the CPV modules. This fitparameter is determined from measurement data.

2.2. Input parameters for YieldOpt

YieldOpt calculates the power output of a CPV modulebased upon several constant and variable input parameters.Constant input parameters describe the characteristics ofthe module that do not change over time. Therefore, theconstant input parameters used for YieldOpt are measuredonly once. Those constant input parameters are describedin the chapters 2.3 to 2.6. The variable input parametersdescribed in chapter 2.2 are changing over time and thusmust be measured continuously by adequate methods.Table I summarizes the variable input parameters neededfor YieldOpt, the origins of these parameters, and themodeling approaches for which they are used. The modelingapproaches that are listed in Table I and used for the combined

YieldOpt model are explained in more detail in the followingsection. First, a short overview of the 10 variable input param-eters is given. The variable input parameters used forSMARTS2 are the following: (i) the direct normal irradiance(DNI), which is measured with an Eppley normal incidentpyrheliometer (THE EPPLEY LABORATORY, INC.Newport, Rhode Island, USA); (ii) the AOD at a wavelengthof 500 nm; (iii) the precipitable water (PW); and (iv) the Ang-strom exponents α1 and α2. The AOD, the PW, and the

Figure 1. Components of a FLATCON®-type concentrating photovoltaic (CPV) module modeled by YieldOpt with adequate modelingapproaches listed on the right. The sun in space is represented by the AM0 spectrum proposed by Gueymard [19]. The influence of theEarth’s atmosphere is calculated by simple model of the atmospheric radiative transfer of sunshine (SMARTS2) as spectral irradianceimpinging on the optics of the CPV module. The temperature-dependent deformation of the Fresnel lens geometry is calculated usingthe finite element method (FEM). Using the resulting lens shape, the spectral optical efficiency is calculated via ray tracing. The I-Vcharacteristic of the multijunction solar cell is calculated by a SPICE network model, and the alignment of the module and the tracker

to the sun are considered by a measured angular acceptance function.

Table I. Variable input parameters used for YieldOpt, thesources of the parameters, and modeling approaches for which

the parameters are needed.

Variable inputparameter Source Used for

DNI [W/m2] Pyrheliometer SMARTS2AOD (500 nm) (�) MFRSRPW (cm)Angstromexponent α1 (�)Angstromexponent α2 (�)Date and time Radio-controlled clockTOptics (°C) PT100 on lens plate FEM, ray

tracingTSolar cell (°C) PT100 on heatsink LTSpiceΔAzimuth (°) Tracking accuracy

sensorΔElevation (°)

DNI, direct normal irradiance; AOD, aerosol optical depth; PW, precipitable

water; MFRSR, multifilter rotating shadow band radiometer; SMARTS2,

simple model of the atmospheric radiative transfer of sunshine; FEM, finite

element model.

YieldOpt, a model to predict power output and energy yieldM. Steiner et al.


Angstrom exponents are obtained in Freiburg by using datataken in a 1min interval from a multifilter rotating shadowband radiometer, Prede POM-01 Sky Radiometer (PredeCo., Ltd., Tokyo, Japan). The numerical values for the AODand the PWare determined in amanner similar to the approachintroduced by Ingold et al. [24]. The Angstrom exponents α1and α2 are obtained by fitting the turbidity law to the AOD[12,25] determined at the wavelength 400 and 500nm for α1and 500, 675, 870, 840, and 1020nm for α2. Additionally,the date and the time (together with the location), which definethe air mass calculated by SMARTS2, are determined with aradio-controlled clock. The temperature of the lens is measuredby a PT100 sensor (UnithermMesstechnikGmbHBruchköbel,Germany) mounted on the lens plate and is used as an input forthe computer simulation of the optics. The solar cell tempera-ture is approximated by a PT100 sensor on the heat sink ofone solar cell within the module and is needed for thesimulation of the solar cell’s I–V curve by LTSpice (LinearTechnology, Milpitas, California, USA). The alignment of thetracking unit to the sun is taken into account through themeasurement of the orientation of the tracking unit in azimuthand elevation with a Black Photon TAS (Black PhotonInstruments GmbH Freiburg im Breisgau, Germany) [26].

2.3. Calculation of spectral irradiance withSMARTS2

In the model YieldOpt, the spectral irradiance reaching theEarth is calculated with the SMARTS2 developed byGueymard [12] based upon several input parameters. Theinput parameters of SMARTS2 are documented in [12].The SMARTS2 input parameters that have a significantinfluence on the power output of CPV modules are theAOD at 500 nm, the Angstrom coefficients α1 and α2, andthe PW. The DNI calculated by SMARTS2 is only validfor clear and cloudless days.

In this work, all of the measured data are considered, thatis, the data affected by cirrus clouds are not removed. Forthat reason, in the model YieldOpt, the resulting spectralirradiance is scaled to the DNI value that was measured onthe tracking unit by a pyrheliometer. All of the SMARTS2input parameters that have no significant influence on thepower output of CPV modules are set as defined in theAM1.5d ASTMG-173-03 reference spectrum for the calcu-lation with the model YieldOpt.

2.4. Ray tracing of the optics

The primary optics in the FLATCON-type modules is a4 × 4 cm2 silicone-on-glass Fresnel lens with a nominalfocal length of 76mm. The influence of this Fresnel lenson the spectral irradiance impinging the solar cell withinthe CPVmodule is considered by spectral optical efficienciesin the model YieldOpt. The spectral optical efficiency iscalculated by Hornung et al. using the FEM and a raytracing technique [15]. Hornung et al. considered thedeformation of the Fresnel lens structure due to the differentthermal expansions of glass and silicone as well as the

change in the refractive index with the lens temperature.By this means, the optical efficiency of the Fresnel lens iscalculated as a function of the wavelength and of the dis-tance of the solar cell from the lens. In the model YieldOpt,the calculated spectral optical efficiency is multiplied by thespectral irradiance calculated by SMARTS2 to estimate theactual spectral irradiance impinging the solar cell withinthe CPV module. Figure 2 presents the spectral optical effi-ciencies used in YieldOpt for different lens temperaturesbetween �30°C and 70°C. Figure 2 shows that for lenstemperatures above 20°C, the decrease in optical efficiencyis more prominent at longer wavelengths compared withshort wavelengths and vice versa for lens temperaturesbelow 20°C. Furthermore, Figure 3 shows the changes inthe spectral optical efficiency considered in YieldOpt fordifferent distances between the lens and solar cell. A lens-to-cell distance that is shorter than the focal length of thelens leads to a decrease of the optical efficiency in the longwavelengths range and increases the optical efficiency inthe short wavelengths range because of the chromatic aber-ration of the Fresnel lens. In contrast, a lens-to-cell distancethat is longer than the focal length of the lens leads to anincreasing optical efficiency at long wavelengths and adecreasing efficiency at short wavelengths. Note that thisbehavior is specific for each optic; thus, the presented datacannot be generalized. However, the presented methodol-ogy can be applied to any optics.

2.5. Alignment of tracking unit and moduleto the sun

Imperfect alignment of the tracking unit and of the moduleto the sun causes a drop in the short-circuit current ISC ofthe solar cells in a CPV module due to the movement of

Figure 2. Spectral optical efficiency calculated by thefinite elementmethod and ray tracing for a characteristic 40×40-mm2 silicone-on-glass Fresnel lens with a focal length of 76mm using a circu-lar target with a 2.3mm diameter. The spectral optical efficiencyused in the model YieldOpt is calculated for lens temperaturesbetween �30°C and 70°C. For lens temperatures above 20°C,the optical efficiency decreases stronger at long wavelengthsthan at short wavelengths. For lens temperatures below 20°C,the optical efficiency decreases stronger at short wavelengths.



the focal spots of the Fresnel lenses off the solar cells. Themodel YieldOpt assumes that all cells and lenses arealigned in the same way and have identical I-V characteris-tics. Under this assumption, the drop in the ISC of the solarcells causes the same decrease in the ISC of the CPV mod-ule. In the model YieldOpt, the quantitative decrease of theISC of the CPV module as a function of the angle ofmisalignment of the tracker to the sun is taken from a mea-sured angular acceptance function (AAF) of the module onthe tracking unit. This AAF was measured for each indi-vidual CPV module. Figure 4 shows the characteristicAAF of the ISC of the module ISE069T measured at theFraunhofer ISE sun-tracking unit. The alignment of thesun-tracking unit in elevation and azimuth is measuredby a Black Photon TAS [26].

2.6. SPICE simulation of the solar cell’s I–Vcharacteristic

In the model YieldOpt, the multijunction solar cell is cal-culated as described by the two-diode model. In the two-diode model, the I-V curve of a multijunction solar cell isdescribed by two diodes and one current source for eachsubcell as well as one lumped series resistance. In themodel YieldOpt, the parallel resistance is neglected. Thiselectrical circuit is realized by a SPICE network modelusing the software LTSpice [20] in the model YieldOpt.LTSpice calculates the I–V curve for a given set of materialparameters needed for the network elements. For eachsubcell, the two diodes require the dark current densitiesJ01 and J02 as inputs. The current source corresponds tothe short-circuit current density JSC of the respective

subcell. J01, J02, and JSC are dependent on the temperature,the band gap, and the material quality of the solar cell. Fur-thermore, JSC depends on the spectral irradiance impingingon the CPV module, the temperature of the Fresnel lens,the external quantum efficiency (EQE) of the triple-junctionsolar cell, and the alignment of the tracker to the sun.

The subcell material parameters J01, J02, and JSC areobtained for the model YieldOpt as follows. The JSC of thesubcells are determined using the spectral irradiance calculatedby SMARTS2. This spectral irradiance is multiplied by theoptical efficiency calculated by ray tracing at a distinct lenstemperature. From the resulting modified spectral irradiance,the short-circuit current densities of the three subcells are deter-mined using the EQE of the triple-junction solar cell at thecell’s operating temperature. The EQE at the cell’s operatingtemperature is derived from the EQE measured at a tempera-ture of 298 K. The method to calculate the EQE of the cellsat any temperature is explained in Section 2.7. In the modelYieldOpt, the JSC of the subcells are then derived from thecalculated EQE at the corresponding temperature, from thespectrum impinging the solar cell, and by multiplying the fac-tor given by the AAF measured as explained in chapter 2.5.

The dark saturation current densities J01 and J02 of the threesubcells are needed as material parameters for the SPICE net-work model and thus for the model YieldOpt. These two ma-terial parameters J01 and J02 are calculated using the equationspublished by Reinhardt et al. [27] and using a procedurepresented in detail in [28]. A similar procedure to derive J01and J02 was published by Nishioka et al. in [29], for instance.

The lumped series resistance is determined by fitting thedark I-V curves calculated by LTSpice to the dark I-Vcurves measured at the triple-junction solar cells.

Figure 3. Spectral optical efficiency calculated by the finite ele-ment metho0064 and ray tracing for a characteristic 40×40-mm2

silicone-on-glass Fresnel lens and a lens temperature of 35°C usinga circular target with a 2.3-mm diameter. The spectral opticalefficiency used in the model YieldOpt is calculated for differentdistances between the lens and the target. The nominal focallength of the lens is 76mm. If the distance between the lens andthe target is longer than the focal length, the optical efficiency isdecreased at short wavelength and increased at long wavelengthand vice versa for distances between the lens and the target that

are shorter than the focal length.

Figure 4. Angular acceptance function as response of the short-circuit current of the concentrating photovoltaic module ISE069Ton the misalignment of the tracker to the sun in elevation andazimuth. Values are normalized to the maximum short-circuit cur-rent. The dependence of the short-circuit current on the alignment

of the tracker is used as a material parameter for YieldOpt.



2.7. Calculating the external quantumefficiency at any temperature

The main differences between the EQE at room tempera-ture (298 K) and the EQE at another temperature arecaused by the change in the energy gaps of the subcells.The dependence of the energy gap on the temperatureEgap(T) used in this work was introduced by Varshniet al. in [30] with the following equation:

Egap Tð Þ ¼ E0 � αT2= T þ βð Þ (1)

The required parameters E0, α, and β for the Ga0.5In0.5P,Ga0.99In0.1As, and Ge subcells of the lattice-matched triple-junction cell and for the Ga0.35In0.65P and Ga0.83In0.17Assubcells of the metamorphic triple-junction cell were takenfrom Levinshtein et al. [31,32]. The parameters for theGa0.35In0.65P subcell are obtained by interpolation of theparameters given by Levinshtein. From Equation (1), itcan be derived that for a temperature increase from T1 toT2, photons with the wavelength λ +Δλ can be additionallyabsorbed by the subcells due to the increased band gaps.Δλcan be calculated as follows:

Δλgap T1; T2ð Þ ¼ 1239:9=ΔEgap T1;T2ð Þ¼ 1239:9= Egap T2ð Þ � Egap T1ð Þ� �

(2)

The approach to calculate the EQE of the subcells at anytemperature from the EQE measured at 298 K in the modelYieldOpt is to split the EQE at a fixed wavelength λi. λi iscalculated by Equation (3), whereas λi,Min and λi,Max arethe minimum and maximum wavelength whose EQEs aregreater than 95% of the maximum EQE.

λi ¼ λi;Min þ λi;Max

� �=2∀EQE > 0:95�EQEi;Max (3)

The EQE measured at T1 = 298 K is adjusted for thewavelengths above and below λi as defined in Equation (4).All of the measured EQE(298 K) values of a subcell aboveλi are shifted from λ to λ +Δλ. All of the EQE(298 K) valuesbelow λi are subtracted by the difference between the EQE(T1) and the EQE(T2) of the subcell with the higher bandgap, where i specifies the subcell. The modeled EQE at T2

between λi and λi +Δλ is calculated by linear interpolation.

Figure 5 shows an EQE measured at 298 K and at 343 Kfor the top, middle, and bottom subcells of a metamorphictriple-junction solar cell in comparison with a calculatedEQE at 343 K with the model YieldOpt. The calculatedtop cell EQE is obtained by shifting the EQE measured ata wavelength λ to a wavelength λ +Δλ for all of the top cell

EQEs at wavelengths above λ1 presented in Figure 5. Thetop cell EQEs at wavelengths below λ1 remains unchanged.The EQE between λ1 and λ1 +Δλ is calculated by linearinterpolation. Figure 6 shows the measured EQEs of theGaInAs middle subcell at 298 and 343 K compared withthe calculated EQE at 343 K that is also shown in Figure 5.For the calculation of the middle cell EQE at 343 K, thecalculated difference of the top cell EQE at 298 K to thecalculated EQE at 343 K is subtracted from the EQE ofthe middle cell measured at 298 K up to λ2. Furthermore,the middle cell EQE measured at a wavelength λ is shiftedto a wavelength λ +Δλ for wavelength above λ2. For thecalculation of the bottom cell EQE at 343 K, the procedureexplained for themiddle cell is used, replacing λ2 with λ3. Theproof of this approach is performed by the calculation of theshort-circuit current densities of the three subcells of onelattice-matched and of one metamorphic triple-junction

solar cell using the AM1.5d ASTM G-137-03 referencespectrum and the EQE. The short-circuit current densitiesdetermined with the calculated EQEs have an averageddifference of 1.0% from the short-circuit current densitydetermined by the measured EQEs at the temperatures of283, 313, 328, and 343 K.

EQEi λ; T2ð Þ ¼∀λ < λi : EQEi λ;T1ð Þ � EQEi�1 λ; T2ð Þ � EQEi�1 λ;T1ð Þð Þ∀λ > λi þ Δλ : EQEi λþ Δλ; T1ð Þ∀λi < λ < λi þ Δλ : linear interpolation

8><>: (4)

Figure 5. Measured external quantum efficiency (EQE) of ametamorphic Ga0.35In0.65P top, a Ga0.83In0.17As middle, and a Gebottom subcell at temperatures of 298 and 343 K. The measuredGaInP, GaInAs, and Ge EQEs at 298 K are used to calculate theEQEs at 343 K with the model YieldOpt. EQEs at wavelengthsabove λ1 for the top, λ2 for the middle, and λ3 for the bottom cellare shifted to the wavelength λ+Δλ. EQEs of the middle cellbelow λ2 (λ3 for the bottom cell) are subtracted by the differenceof top cell EQE at T=298 K and the calculated top cell EQE at343 K. EQE of the bottom cell below λ3 are subtracted by thedifference of middle cell EQE at T=298 K and the calculated

middle cell EQE at 343 K.



3. EXPERIMENTAL SETUP,VALIDATION OF THE POWER, ANDENERGY YIELD PREDICTION

3.1. Description of empirical models as abenchmark for YieldOpt

As a benchmark for the agreement of YieldOpt with themeasured data, three empirical multilinear regressionmodels called DNI-model, Z-model and ZMPP-model areused [33]. It is assumed that no correlation between theinput parameters of the models exists and that the superpo-sition principle is applicable. The first model is a modifica-tion of the model introduced by Araki et al. [7] defined byEquation (5) with DNI as single input parameter and a, b,and c as fit parameters. This model only considers theinfluence of different spectral conditions on the poweroutput of CPVmodules indirectly by second and third orderDNI effects.

P DNIð Þ ¼ a�DNI1 þ b�DNI2 þ c�DNI3 (5)

One approach to consider the spectral conditions directly isthe spectral parameter Z. The spectral parameter Z quantifiesthe impact of the spectrum on the solar cell parametersand is defined by Equation (6) following Peharz et al. [34].

Z ¼ 1� 2

1þ Itop=Itop; AM1:5d

Imid=Imid; AM1:5d

(6)

Itop and Imid are the currents generated by a Ga0.5In0.5P topcomponent cell and by a Ga0.99In0.01As middle componentcell under the actual measurement conditions, and Itop,AM1.5d and Imid,AM1.5d are the corresponding currents underthe reference conditions AM1.5d ASTM G-173-03. Aspectral parameter Z of zero indicates spectral conditionswith current matching of the subcells similar to AM1.5d.Spectral conditions with a Z below zero are more red-rich

(higher middle cell current), and conditions with a Z abovezero are more blue-rich (higher top cell current) comparedwith the AM1.5d spectrum.

The spectral parameter Z as well as the DNI and theambient temperature T are used as inputs for the Z-modeland ZMPP-model. The idea of the Z-model and of theZMPP-model is to consider the impact of the three inputparameters on the short-circuit current (ISC), the open-circuit voltage (VOC), the fill factor (FF), the maximumpower current (IMPP), and the maximum power voltage(VMPP) independently in separate equations. The Z-modeluses ISC, VOC, and FF as defined in Equations (7), (8),and (9). The maximum power output (PMPP) of a CPVmodule is calculated in the Z-model by multiplying ISC,VOC, and FF as defined in Equation (10).

ISC DNI; Z;Tð Þ ¼ ISC;0 þ i1DNI

þ Max ISC=DNI½ � þ i2jZ-ZMatchjð Þ�DNIþi3Tþ i4T

2 (7)

VOC DNI; Z; Tð Þ ¼ VOC;0 þ v1DNI þ v2Log DNI½ � þ v3T

(8)

FF DNI; Z; Tð Þ ¼ FF0 þ f1DNIþ f2DNI2 þ f3jZ-ZMatchj

þf4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ-ZMatchj j

pþ f5T

(9)

PMPP DNI; Z;Tð Þ ¼ ISC DNI;Z;Tð Þ�VOC DNI; Z; Tð Þ�FF DNI;Z;Tð Þ(10)

For the ZMPP-model, IMPP and VMPP are calculatedusing Equations (11) and (12). The maximum power outputPMPP is then calculated by multiplying IMPP and VMPP

according to Equation (13).

IMPP DNI;Z; Tð Þ ¼ IMPP;0 þ j1DNI

þDNI Max IMPP=DNI½ � þ j2jZ-ZMatchjð Þþj3Tþ j4T

2 (11)

VMPP DNI;Z; Tð Þ ¼ VMPP;0 þ w1DNI2 þ w2jZ-ZMatchj

þw3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ-ZMatchj j

pþ w4T (12)

PMPP DNI; Z; Tð Þ ¼ IMPP DNI; Z; Tð Þ�VMPP DNI; Z; Tð Þ (13)

The difference of the ZMPP-model compared with theZ-model is that no full I-V curve of the module must bemeasured. For the ZMPP-model, the maximum power pointdata are sufficient to determine the fit parameters.

The input and fit parameters used by the three empiricalmodels and by YieldOpt are summarized in Table II. Theempirical models require several fit parameters, whereasthe YieldOpt model has only one fit parameter, which ismultiplied by the PMPP calculated by the SPICE networkmodel. All fit parameters must be obtained from the mea-sured data in a period called the fit period.

Figure 6. Measured external quantum efficiency (EQE) of theGa0.65In0.65As middle subcell of a metamorphic triple-junctioncell at temperatures of 298 and 343 °K . The dotted line shows

the EQE calculated with YieldOpt for 343°K.



ZMatch represents the spectral parameter Z for which theratio of ISC to DNI and IMPP to DNI is maximum. AtZMatch, the top and middle cells of the triple-junction cellwithin the CPV module operate in a current-matchedmode. The measurement period in which the fit parametersof the models are determined should include ZMatch toobtain reasonable fit parameters for the Z-model andZMPP-model. Therefore, the 1-month periods in whichthe fit parameters are obtained are the months of April,May, July or August in the year 2012; in June 2012, thesun-tracking unit was not operational. For the determinationof the fit parameter of YieldOpt, all other months could alsobe used as fit periods with the same agreement betweenmeasurement and model.

3.2. Statistical approach for the validationprocess

To validate the introduced YieldOpt model, the instanta-neous power output and the yearly energy yield of the fiveFLATCON-type modules listed in Table V are predictedand compared with the measured data for a period of 1 year.The agreement between the measured and calculated data isquantified by the deviation of the predicted to experimentalenergy yield and by the normalized root mean square errors(NRMSE) of the instantaneous power output. The deviationof the energy yield as defined in Equation (14) quantifies theaccuracy in the prediction of the sum of the power outputduring a specific period.

Δpredicted energy yield ¼ 1�∑n

i¼1PModel;i

∑n

i¼1PMeasurement;i

(14)

PMeasurement,i and PModel,i in Equation (14) represent themeasured and modeled power outputs, respectively, of onemodule at a distinct time. The total number of measured I-Vcurves is denoted by n. Between 7000 and 12000 I–V curveswere measured per module. However, an accurate predictionof the yearly energy yield is much easier to achieve than aprediction of the instantaneous power, simply because ofstatistical reasons. The NRMSE defined in Equation (15)serves as a benchmark for how accurately the power outputof a CPV module at a specific time and under certainambient conditions is predicted.

NRMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑n

i¼1PMeasurement;i � PModel;i

� �2n

vuuut,PMax � PMinð Þ

(15)

PMax and PMin are the maximum and minimum poweroutputs within the measurement period. In this work, theagreement between the measurement and the model isclassified by NRMSE as listed in Table III.

3.3. Experimental setup and filtering themeasurement data

The measurement of the I-V curves of the CPV modules atFraunhofer ISE uses a fully automated measurementsystem [35]. This system measures a CPV module’s I-Vcurve every 10min. The single I-V curve is measured within30 s. In addition to every measured I-V curve, the ambientconditions are automatically logged. The DNI is measuredbefore and after the measurement of an I-V curve. Themodules and sensors are cleaned on regular bases. Becauseof the fully automated nature of the measurement, some ofthe I-V characteristics are faulty because of the shading ofthe modules, the shading of sensors or a strong misalignmentof the tracker. Other sources of errors are rapid changes in theambient conditions, for instance, due to clouds during themeasurement of a single I-V curve. Therefore, an importantissue is the elimination of faulty data. This task is performedby the application of data filters before the simulated data canbe compared with the measured data. In this work, the filtersused are listed in the succeeding text. These filters are basedupon experiences gathered from over 20 different types ofmodules using 2 years of measurement data. The applicationof these filters to the measurement data eliminates

Table II. Input and fit parameters used by the three empirical models and by YieldOpt.

Model Input parameter Fit parameter

DNI-model DNI a,b,cZ-model DNI, Z, Tambient ISC,0, i1, i2, i3, i4, VOC,0, v1, v2, v3, FF0, f1, f2, f3, f4, f5, ZMatch

ZMPP-model DNI, Z, Tambient IMPP,0, j1, j2, j3, j4, VMPP,0, w1, w2, w3, ZMatch

YieldOpt DNI, AOD@500nm, PW, α1, α2, date, time,TOptics, TSolar cell, ΔAzimuth, ΔElevation

p

DNI, direct normal irradiance; AOD, aerosol optical depth; PW, precipitable water.

The input parameters of YieldOpt are described in detail in chapter 2.

Table III. Classification of agreement between themeasurementand the model by the normalized root mean square errors value.

Classification NRMSE

Bad >7.0%Adequate 5.0–7.0%Satisfying 3.5–5.0%Good 2.0–3.5%Very good 0.0–2.0%

NRMSE, normalized root mean square errors.



approximately 10% to 20% of the measured data of a singlemodule. The filter 2 removes physically false values. Thefilters 3 and 9 are necessary because of changing weatherconditions, and the filters 4, 5, 6, and 7 are needed becauseof sensor and module shading as well as soiling and faultyalignment of the modules. Filter 9 is only applied for cloudydays with strongly varying weather conditions. First, thefilters 1, 2, and 3 are applied to all measurement data. Next,the filters 4, 5, and 6 are used, followed by filter 7. Finally,filter 8 and 9 are applied to the remaining data.

(1) The DNI> 100W/m2.(2) The open-circuit voltage VOC, the FF, the module

efficiency, the short-circuit current, the current atthe maximum power point IMPP, and the voltage atthe maximum power point VMPP are all greater thanzero. Additionally, IMPP< ISC and VMPP<VOC.

(3) The deviation in DNI before and after the measure-ment is less than 5%.

(4) The IMPP/ISC is higher than the median of the mea-sured IMPP/ISC values times 0.85.

(5) The VMPP/VOC is higher than the median of themeasured VMPP/VOC values times 0.85.

(6) ISC/DNI is greater than or equal to the median of themeasured ISC/DNI values divided by 3.

(7) All values of ISC/DNI of one module in the wholemeasurement period are grouped into bins. The sizeof the bins is 20% of the absolute median deviationof all values of ISC/DNI. The number of values inthe bins must be higher than 0.2% of the total num-ber of measurement values of ISC/DNI. This filter isadditionally applied to IMPP/ISC and VMPP/VOC. ForISC/DNI, this filter is only applied to values abovethe median of the measured ISC/DNI values.

(8) The alignment of the tracker to the sun is within 1°in tracking in azimuth and elevation.

(9) Days with less than 20 measured I-V curves permodule are dropped.

3.4. Ambient conditions and investigatedmodules

Over the period of 1 year (4 October 2011 until 3 October2012), five FLATCON-type modules [16] were measuredin Freiburg, Germany (48°N, 7.9°E). Ambient conditionsof the test period are listed as minimum, mean, and maxi-mum in Table IV. The investigated five FLATCON-typemodules [16] are listed in Table V. Four of the investigatedmodules consist of six triple-junction solar cells connectedin series, whereas one module (ISE069T) comprises onlyone triple-junction solar cell. The solar cells in the modulesare connected in parallel to bypass diodes. Four of themodules use lattice-matched triple-junction cells with bandgaps of 1.9, 1.4, and 0.7 eV [17], and one module(ISE064T) uses metamorphic triple-junction cells withband gaps of 1.7, 1.2, and 0.7 eV [18]. One of the modules(ISE049T) uses a reflective secondary optic [36]. Thesingle-cell module is equipped with temperature sensors

mounted on the lens plate and on the solar cell’s heat sink.The lens and cell temperatures measured on this moduleare used as input parameters for the calculation withYieldOpt for all five modules. The implementation of thesefive modules in the YieldOpt model differs in the distanceof lens to solar cell, in the acceptance angle function, andin the fit parameter of the modules. For the modulesISE047T, ISE049T, ISE059T, and ISE069T, the identicalEQE measured at 298 K is used. For module ISE064T,an EQE measured from a metamorphic triple-junction solarcell with the same epitaxial layers as the solar cells withinthe module is used.

3.5. Results and discussion

The power output and the energy yield are calculated forthe test period of 4 October 2011 to 3 October 2012. April,May, July or August are used as four different periods inwhich the fit parameters of the four models are determined.The fit period of August is missing for the moduleISE069T because of a defect of the module. In theremaining time of the test period without the 1-month fitperiod, the power output and energy yield are predicted.The agreement of the predicted to the measured poweroutput of the five FLATCON-type CPV modules listed inTable V using the NRMSE defined in Equation (15) arepresented in Figure 7 for the four investigated models. The

Table IV. Ambient conditions of the test period listed asminimum,mean, and maximum values.

Variable input parameter Min Mean Max

DNI (W/m2) 100 730 950AOD (500 nm) (�) 0 0.2 0.9PW (cm) 0.5 2.5 5.1Angstrom exponent α1 (�) 0.27 1.5 2.8Angstrom exponent α2 (�) 0.17 1.7 3Tambient (°C) �7 23 38Z (�) �0.47 �0.01 0.14

DNI, direct normal irradiance; AOD, aerosol optical depth; PW, precipitable

water.

Table V. Investigated concentrating photovoltaic modules andthe used cell type, the number of series-connected cells, and

the distance of the lens to the solar cell.

Name ofmodule

Celltype

Numberof cells

Distance lensto cell Comment

ISE047T 3JLM 6 75.4mm —

ISE049T 3JLM 6 74.8mm Secondary opticISE059T 3JLM 6 75.8mm —

ISE064T 3JMM 6 75.2mm —

ISE069T 3JLM 1 74.4mm PT100 at lens plateand heatsink

The module ISE049T uses a reflective secondary optics. The single cell

module ISE069T has a PT100 temperature sensor at the lens plate and

at the heatsink.



four modules ISE047T, ISE049T, ISE059T, and ISE069Tusing lattice-matched triple-junction cells [17] and themodule ISE064T using metamorphic triple-junction solarcells [18] show NRMSE values for the YieldOpt modelbetween 2.6% and 3.9%, which is a good-to-satisfactoryagreement of the measured to the calculated data. The DNImodel shows the highest values for NRMSE between 4%and 8%, corresponding to a satisfying-to-bad agreement.The NRMSE values of 2.6–5% are determined for theZ-model and ZMPP-model, showing a good-to-satisfyingagreement. The Z-model and ZMPP-model show a strongervariation in the agreement between the measurement andthe simulation while using different months as fit periodscompared with the DNI-model and YieldOpt. The fitparameter of the model YieldOpt can be determined in thewinter month as well with a similar agreement betweenmeasurement and simulation. The restriction of the fitperiod to April, May, July, and August is only because ofthe three empirical models.

Figure 8 presents the deviation of the predicted to themeasured energy yield as defined in Equation (14) for thefour investigated models. The energy yield of the fiveCPV modules is predicted with a deviation between �4%and +5%. The most accurate predictions are made byYieldOpt with deviations between �3.1% and +2.7%.The Z-based models have a similar deviation in the pre-dicted energy as the DNI-model. However, the Z-basedmodels have a more accurate prediction compared to theDNI model for the module ISE064T with metamorphic

triple-junction cells. The deviation from the measuredenergy yield for the empirical models shows a strongdependency of the used fit period. The accuracy of theenergy yield prediction of YieldOpt shows a notably lowerdependence on the used fit period.

Figure 9 shows the comparison of the maximum powerpoint PMPP of the measured data with the data predicted byYieldOpt on 11 August 2012. The calculated PMPP shows agood agreement with the measurement. Approximately50% of the considered days show a similar or better agree-ment between the measurement and the data predicted byYieldOpt for all modules and fit periods. Moreover, in

Figure 7. Normalized root mean square errors (NRMSE) as abenchmark for the agreement between the predicted and themeasured instantaneous power outputs for the five FLATCON®-type modules listed in Table V using four different predictionmodels. Calculations of the bars of one module with same hatch-ing use, from left to right, April,May, July or August as the 1-monthperiods to determine the fit parameters of the four models. In therespective complementary period between October 2011 andSeptember 2012, the power output of the five modules ispredicted. The NRMSE is referred to the whole year excludingthe respective 1-month fit period. The YieldOpt model shows agood-to-satisfying agreement for the four modules using lattice-matched triple-junction solar cells and for the module ISE064T

using metamorphic solar cells.

Figure 8. Deviation of the predicted to the measured energyyields of the five FLATCON®-type modules using four differentpredictionmodels. Bars of onemodulewith samecolor andhatchinguse, from left to right, April, May, July or August as 1-month periodsto determine the fit parameters of the fourmodels. In the respectivecomplementary period between October 2011 and September2012, the energy yield of the five modules is predicted. Thedeviation in energy yield is referred to the whole year excluding therespective 1-month fit period. The YieldOpt model has the highestaccuracy between -3.1% and +2.7% to predict the energy yield.

Figure 9. Measured maximum power point (PMPP) of the moduleISE047T in comparison with the data predicted by YieldOpt on 11August 2012. A normalized root mean square errors of 2.6%wascalculated for this day, showing a good agreement between the

measurement and the model.



approximately 80% of the days, the agreement is betterthan satisfying, showing a NRMSE of 5.0% and better.This result underlines the ability of YieldOpt to predictthe instantaneous power output of CPV modules undervarious ambient conditions with a high accuracy. Further-more, Figure 10 presents a comparison of the measuredand predicted short-circuit currents and open-circuitvoltages. The ISC is slightly underestimated by YieldOptbetween 11:00 and 15:00. YieldOpt predicts the currentmatching of the top and middle subcells to occur at approx-imately 11:00. Before 11:00, the top cell limits the current;after 11:00, the middle cell limits the current in the simula-tion. However, the middle cell current is calculated slightlytoo low, which explains the underestimation of the PMPP

by the YieldOpt model at 11 August.

4. CONCLUSIONS

In this work, we present a model called YieldOpt, whichcan be used for the prediction of the power output andenergy yields of CPV modules. YieldOpt models theimportant elements of a CPV module such as the spectralirradiance, the concentrating optics, the multijunction solarcell, and the tracking and alignment of the CPV moduleseparately by adequate modeling approaches. The spectralirradiance is calculated by SMARTS2, the optics via FEMand ray tracing methods, and the solar cell by a SPICEnetwork model. The alignment of the module and of thetracker to the sun is considered by measured AAFs. As abenchmark for the YieldOpt model, three empirical modelsare compared. The empirical models use the DNI, theambient temperature, and the spectral parameter Z as inputparameters. The empirical models require between 3 and16 fit parameters. In contrast, YieldOpt uses over 20

constant and variable input parameters but only one fitparameter. Nevertheless, as proven in this paper, YieldOptallows a high accuracy in predicting the energy yield inFreiburg, Germany, for a period of 1 year. Five FLATCON-type CPVmodules were used as test specimens. Four of thesemodules are using lattice-matched triple-junction solar cells[17], whereas one module uses metamorphic triple-junctionsolar cells [18]. Using YieldOpt, the energy yields of all fivemodules could be predicted with a maximum deviation ofapproximately 3%, whereas the three empirical modelsshowed a maximum deviation of approximately 5%. Fur-thermore, YieldOpt showed a good agreement in predictingthe instantaneous power output in Freiburg, Germany, forthe five CPV modules throughout the whole year. Thisgood agreement was quantified by NRMSE between 2.6%and 3.9% compared with 2.6% to 8.0% for the threeempirical models.

ACKNOWLEDGEMENTS

The authors would like to thank Armin Bösch and all thecolleagues of the ‘III-V Epitaxy and Solar Cell’ group fortheir contributions to this paper. This work has been partlysupported by the Federal Ministry for the Environment, theNature Conservation and Nuclear Safety (BMU) under theKoMGen project, contract number 0327567A. The authorsare responsible for the content of this paper.

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