Energy Yield Modeling for Optimization and Analysis of Perovskite‐Silicon Tandem Solar Cells Under Realistic Outdoor Conditions

A comprehensive algorithm for calculating the energy yield (EY) is used as the most important figure‐of‐merit in determining the capabilities of PV devices in realistic operation. The model is based on advanced optical modeling and extensive opto‐thermo‐electrical characterization. It takes the realistic environmental and device installation data fully into account, as well as the complete set of device specifications including all relevant temperature‐induced variations. A detailed analysis and optimization of two‐terminal perovskite‐silicon tandem devices are done in terms of long‐term electrical energy production at different geographical locations from distinct Köppen–Geiger–Photovoltaic climate zones (KGPV). It is shown that the optimal perovskite bandgap (EG,PK) in a tandem device operating under realistic conditions is in all cases higher than the one optimized under STC, yet does not differ significantly from location to location, which is beneficial from the manufacturing point of view. The optimal EG,PK is influenced primarily by the spectral distribution of incident irradiance, and not so much by the operating temperature conditions. Moreover, a clear linear dependency between the optimal EG,PK and the weighted average photon energy of the incident irradiance is demonstrated, which presents an important rule for rapid tandem design.


Introduction
In just a few years, perovskite-based tandem photovoltaic (PV) devices have firmly established themselves as one of the key players in the future evolution of photovoltaics, since they enable a costeffective way to advance the efficiency of state-of-the-art silicon single-junction solar cells. [1] Successful integration of perovskite DOI: 10.1002/adts.202200931 (PK) cell with silicon (Si) cell to form a PK-Si tandem solar device has shown tremendously rapid efficiency progress in less than a decade of its development. [2,3] The progress is mostly driven by the large area of possible improvements in the perovskite solar cell research field, which have already led to the current PK-Si tandem efficiency record of 31.3%. [2] The design process of record tandem solar cells is conventionally based on studying and optimizing the device under standard test conditions (STC) established in the controlled environment inside laboratories. Studies have mostly been focused on how different perovskite bandgaps (E G,PK ) [4,5] and texturization affect the device efficiency. [6,7] The radiative efficiency limit of PK-Si tandems states the optimal perovskite bandgap to be 1.73 eV under STC conditions, [8] while optical simulations under STC indicate a lower value around 1.70 eV. [9] However, this does not consider the complete structure of the device with experimentally determined optical, thermal and electrical properties as well as optical changes evoked by lamination and module integration. Furthermore, with the transition from STC to realistic operation outdoors, not only the device long-term behavior but also the optimal perovskite bandgap may become significantly influenced by the constantly changing environmental conditions, such as irradiance and temperature. [10] Accordingly, there has recently been a pronounced tendency in the photovoltaic research field to approach not only experimental characterization, but also numerical analysis and optimization of tandem PV devices by taking the realistic operating conditions fully into account. [11][12][13][14][15][16] This is ultimately crucial for proper design and optimization of large-area tandem PV modules and eventually also large-scale systems to achieve an effective increase in the long-term performance as well as for enabling reliable assessment of the projected energy payback times and extraction of fundamental information such as degradation rates. [17][18][19] In this respect, numerous studies have been reported [13,16,[20][21][22][23][24][25][26][27][28][29][30][31] in which some form of a model evaluating the longterm production of (area-normalized) electrical energy -the energy yield (EY), given in kWh m −2 -is used to analyze the capabilities of tandems in realistic operation. Many studies employ planar or simplified device architectures to make more generalized Adv. Theory Simul. 2023, 6,2200931 www.advancedsciencenews.com www.advtheorysimul.com statements about the prediction of realistic performance limitations of perovskite-based tandem solar PV modules, [12] . [16] Others, on the other hand, focus on specific aspects, such as precise modeling of the electrical behavior of multi-junction PV modules, [13] optical response analysis to variations in spectral irradiance, [11,20,25] or the application of light-managements foils. [9] In general, it can be stated that EY modeling of tandems has come a long way since the early reports by Hörantner et al. [13] and Dupré et al., [16] especially in terms of the complexity of various EY modeling aspects (optical, electrical and thermal) that have nicely been represented by Schmager et al. [24] Multi-junction solar cells were developed to achieve a higher power conversion efficiency (PCE) by reducing the thermalization losses and ensuring a more efficient harvesting of the solar spectrum. The presence of multiple subcells in the device, however, results in a more complex structure accompanied by new sources of losses, especially those related to current mismatch [32] in monolithically integrated tandem cells. According to the literature, [10] the bandgap temperature dependencies of perovskite (top) and silicon (bottom) subcell follow opposing trends, thus shifting the two-terminal (2T) tandem devices optimized at STC away from current matching, depending on the operating temperature. Therefore, in order to predict and ultimately improve the long-term EY of solar cells and PV modules, not only optical and electrical, but also accurate thermal modeling is of high importance. [33][34][35][36][37][38] An increasing number of studies [21,24,29,31,[39][40][41] investigate the EY of multijunction solar cells by considering the temperature dependence of various cell parameters, but according to our knowledge they disregard specific aspects such as forced convective heat transfer due to wind and/or, more importantly, temperature-induced bandgap shifting that may significantly affect not only electrical properties but also the spectral absorptance and quantum efficiency characteristics of the device.
In this contribution, we use energy yield modeling to extract information about the behavior of 2T PK-Si tandem devices in realistic operation. Our main goal is to qualitatively and quantitatively evaluate the long-term electrical energy production of PK-Si PV devices operating under realistic conditions and to optimize their structures with respect to the bandgap of the top perovskite subcell. For that purpose, certain knowledge of the device's opto-thermo-electrical behavior is required to perform accurate EY modeling that combines: i) the long-term measurements of meteorological data, ii) detailed optical simulations that take the exact 3D geometry of the tandem device fully into account (including the microtextured surface(s) of the layers), and iii) experimentally determined J-V characteristics of individual single-junction solar cells composing the investigated tandem PV device. To our best knowledge, however, such a comprehensive EY framework equipped with the models describing optothermo-electrical mechanisms that interplay in tandems has not yet been reported, which compelled us to develop and employ our own energy yield modeling algorithm.
We first present the procedure of the developed EY model in which each aspect of device operation is handled in a comprehensive way, including all relevant temperature-induced variations. Then, we analyze the PK-Si tandem devices in terms of the energy yield obtained under realistic conditions at different geographical locations from distinct Köppen-Geiger-photovoltaic climate zones (KGPV [42] ) and in different plane-of-array orientations. The optimal bandgap of the perovskite subcell is determined for each case, and the extent of relative EY losses attributed to the deviations of E G,PK from the optimal value is analyzed and discussed in detail. Finally, we demonstrate how different environmental aspects affect tandem device operation, and above all we draw a close correlation between the optimal E G,PK and the spectral distribution of incident irradiance, which is expressed in terms of the average photon energy weighted by the incident irradiance and averaged over a given time period. With the presented results we aim to draw conclusions that would provide clear design guidelines for perovskite-silicon tandem devices intended for outdoor operation.

Energy Yield Modeling
For the purpose of this work, we developed an intricate EY calculation algorithm that is based on three modeling approaches -optical, electrical, and thermal, which are coupled to determine the current density-voltage (J-V) characteristics of the investigated tandem solar cell at any given time and conditions. The algorithm is based on an experimentally verified modeling framework developed for single-junction solar cells in our previous work, [19,17] which has now been significantly expanded and improved in order to handle any kind of tandem device in an arbitrary orientation and/or electrical configuration. To obtain realistic EY results that would lead to trustworthy conclusions with directly applicable design guidelines, we strived to take all key aspects of real world operation of tandem solar cells fully into account, which lead to the development of our own energy yield model.
The analysis presented in this study is focused on 2T PK-Si tandem devices. The bottom cell is a double-side textured rearemitter n-type silicon heterojunction (SHJ) cell, which has been optimized for the tandem architecture as reported previously. [9,43] For the top perovskite cell we selected the "inverted" design (pi-n), of which the absorber layer is comprised of a multiple cation, multiple halide composition. The complete nonencapsulated monolithic tandem PV device is structured as follows: IZO contact (80 nm) / SnO 2 interfacial layer (10 nm) / C 60 electron transport layer (10 nm) / perovskite absorber (1000 nm) / MeO-2PACz hole transport layer (monolayer) / ITO contact (20 nm) / nc-SiOx:H electron transport layer (95 nm) / a-Si:H (i) passivation layer (5 nm) / c-Si n-doped absorber (250 000 nm) / a-Si:H (i) passivation layer (5 nm) / a-Si:H (p) hole transport layer (8 nm) / {ZnO:Al (70 nm) / Ag (500 nm)} contact. Additionally, an encapsulation stack -glass (3.2 mm) / ethylene-vinyl acetate-EVA (500 µm) -is placed above the described structure. For the perovskite absorber, we have selected a fixed thickness of 1000 nm, which is able to absorb virtually all incident irradiance up to the bandgap and ensures maximal power (as demonstrated also in Figure S4 of the Supporting Information), whereas the bandgap itself was optimized with respect to the energy yield.
Detailed description of the device structure in conjunction with the information about the incident zenith ( ) and azimuth ( ) angle of incident irradiance provide the first set of data required to optically simulate the device to obtain the wavelengthdependent absorptances (A) within the active layers of each subcell in the structure. In this study, the optical modeling of the tandems is performed by coupled wave-ray optics methods in the simulator CROWM, [44][45][46] which can take the complete geometry, the material properties as well as the realistic illumination conditions of the device fully into account. The optical model can handle any kind of textures (not only random but also regular, such as in the case of light-management foils applied to the front surface of the device [17] ), as well as the exact configuration of rear metal contacts and other laterally varying features (for example if an interdigitated back contact design would be selected for the silicon subcell [47] ). In the present study, however, we have focused only on the random pyramidal texturization of the silicon subcell, since such design of tandem devices is also appealing for the industry.
In general, an optical simulation needs to be performed for each unique combination of the incident angles ( and ) and the operating device temperature (T cell ), which affects the bandgap of the perovskite and silicon subcell within the tandem. Since the aforementioned combinations are different for each geographical location, installation, and orientation of the device, this may lead to a large number of simulations to be performed during EY modeling. Therefore, to keep computational time within reasonable limits, optical simulations were executed in advance for selected discrete combinations of , , and E G of both subcells. [17] During the EY calculation, we used these simulation results to obtain the exact A in each subcell for each unique combination of the actual calculated incident angles and T cell by means of interpolation.
In order to best emulate the device's operation outdoors, the realistic, constantly changing irradiance and temperature data need to be collected and linked with the optical simulation results to assess the short-circuit current density (J sc ) of each subcell in the tandem device. The second set of input data is, therefore, the long-term measurements of meteorological data acquired at the geographical location of the solar cell operation, namely: the position of the Sun, the spectrally resolved global tilted irradiance (GTI), and direct normal irradiance (DNI). Based on these data and the orientation of the device, the spectrally resolved direct (G dir ), diffuse (G diff ), and albedo (G albedo ) irradiance in the plane of the device are calculated and finally applied to the simulated A to obtain the photo-generated J sc of each subcell (assuming Lambertian angular intensity distribution for the diffuse and albedo irradiance). In this study, we employ actual measured meteoro-logical data -ambient temperature (T a ), wind speed (WS), spectrally resolved GTI and DNI -for selected geographical locations. They were acquired from the National Solar Radiation Database [48] in hourly intervals, given in a total of 8760 data points over the entire typical meteorological year (TMY [49] ).
In the scope of this work, the operating device temperature was calculated empirically based on Duffie-Beckman (DB) model [50,51] as a function of the total irradiance (G tot = G dir + G diff + G albedo ), ambient temperature and wind speed. It provides an extension to the Ross model [52] by using additional empirical term to take the wind speed into account. The operating device temperature is required both for electrical modeling (explained in the following paragraph) as well as for including the effects of temperature-induced bandgap variations of the perovskite and silicon subcell (E G,PK (T cell ) and E G,Si (T cell )), which significantly affect their spectral absorptance and quantum efficiency characteristics and, consequently, also the photo-generated J sc values. These bandgap variations were determined experimentally and are presented in Figure 1a) together with the values obtained from the literature. [32] Approximately linear E G (T cell ) dependencies of PK and Si subcell with distinct opposing trends can clearly be observed.
The electrical model embraces measured J-V characteristics of individual single-junction solar (sub-)cells that comprise the studied tandem PV device. We measured the J-V curves of both a nonencapsulated single-junction perovskite cell [19] as well as a nonencapsulated single-junction silicon cell (shown in Figure  S3 of the Supporting Information) inside the laboratory under controlled conditions: incident irradiance (AM1.5G) and device temperature were varied from 10 to 120 mW cm −2 in intervals of 10 mW cm −2 (corresponding to relative irradiance G rel = 10-120%) and from 25 to 85°C in intervals of 15°C, respectively. The STC PCE values were 18.5% for the PK single-junction device, and 22.8% for the Si single-junction device. From this set of measured J-V curves, appropriate interpolation/extrapolation methods were employed to determine the ones that correspond to the exact conditions of each specific time data point during EY calculation.
It is important to note, however, that charge carrier generation (and thus the electrical parameters) inside a tandem device with additional textured interfaces and encapsulation layers differs from that of the single-junction cell, even if the same irradiance is applied. Therefore, to determine from the measured set the correct pair of J-V curves according to any given illumination conditions, it is necessary to first establish a merit of equivalent relative irradiance (G rel,eq ) that is applicable for different device structures (assuming that no additional electrical losses occur by changing the texturization of the device or by adding encapsulation). In our EY model, G rel,eq is defined for each time data point and for each subcell in the tandem as the ratio of the simulated subcell J sc to the measured J sc of the nonencapsulated singlejunction cell under total irradiance of 100 mW cm −2 and T cell corresponding to the data point (using interpolation/extrapolation for temperatures not on the measured grid). The combination of G rel,eq and T cell , by which any given time data point in TMY is characterized, is therefore used to determine the correct J-V curve of each subcell from those measured at different G rel and T cell conditions.
In the final part of the energy yield algorithm, the two subcell J-V characteristics are graphically summed (summation of voltage values at constant current values) to account for their series connection, and the J-V curve of the entire tandem is thus obtained. The electrical energy is calculated by multiplying the power at the maximum power point (MPP) of the corresponding tandem J-V curve by the duration of the data point (one hour in our case). The long-term generated electrical energy -energy yield (EY) -of the investigated device is finally obtained by summation of the calculated energy over the desired time period (one year in our case).
An example of the process described above is presented in Figure 1b). The goal in this particular example is to determine the J-V curve (and the MPP) of a tandem device located in Atlanta, USA, for one specific date and time: May 13, 2014, at 11:00. We first calculate the cell temperature (42°C) and, by means of optical simulations, the equivalent relative irradiance for each of the two subcells (85% for PK, 40% for Si). Then, the J-V curves measured for both single-junction cells at different G rel /T cell combinations (only a few are shown by dotted and dashed curves) are interpolated to match the G rel,eq /T cell combination of the given data point (full red and blue curves). Finally, the two curves are graphically summed to construct the J-V curve of the tandem de-vice (green curve), from which the maximum power point can be extracted.

E G,PK Optimization-Selected Location
The investigated encapsulated PK-Si tandem device was first analyzed under STC conditions (25°C; AM1.5G spectrum, 100 mW cm −2 irradiance, perpendicular incidence) under which laboratory devices are normally measured, evaluated and optimized. Under these conditions, the developed EY calculation algorithm was employed to determine the optimal perovskite bandgap E G,PK = 1.684 eV, at which the tandem device exhibits a high PCE of 28.26%. The determined optimal bandgap fits with the reports from the literature, while the obtained PCE represents a state-ofthe-art encapsulated device, solidifying our findings.
Next, the developed energy yield model was used to determine the optimal E G,PK of the device mounted in the optimal open-rack configuration in Atlanta, Georgia, USA. We selected this location (latitude N 33.73°) since it represents the DH KGPV [42] climate zone that covers a large area of the USA as well as Southern Europe. In this realistic, field operation case with total annual irradiation G tot = 1847 kWh m −2 , the largest annual EY max of 497 kWh m −2 was obtained at E G,PK = 1.719 eV, resulting in the overall conversion efficiency EY/G tot = 26.9%. The optimal E G,PK for realistic operation in Atlanta is therefore by almost 35 meV higher than that the one obtained under STC; later on we will show, and study in detail, that this is a general conclusion applicable for any location.
The simulation results of EY and three main PV parameters (J sc , V oc , FF) are presented in Figure 2 for a broader range of E G,PK from 1.65 to 1.80 eV. For STC, all these values were calculated at a single set of conditions (25°C, AM1.5G spectrum, 100 mW cm −2 irradiance, perpendicular incidence), whereas in the case of realistic operation in Atlanta, the entire year needed to be taken into account: in this case, therefore, J sc values represent the annual integrals, J sc = ∑ . Weighted averaging was selected to promote data points in which the device is generating more energy, and is used also for other variables in later sections. Further on, full lines in Figure 2a) depict EY losses relative to EY max achieved at the optimal E G,PK , whereas dashed lines represent J sc losses relative to the maximal J sc,max achieved at ideal current matching conditions. As expected, the obtained convex shape of EY losses is driven primarily by the current mismatch, whose impact increases when moving away from the optimal bandgap (indicated by the peaks of the curves), reducing both J sc as well as the energy output. Additionally, we can observe that EY max can in general be obtained at a slightly different E G,PK (by a few meV) compared to J sc,max , and also that the convex shape of EY losses is wider and asymmetric compared to J sc losses. Both observations can be explained by the corresponding V oc and FF curves shown in Figure 2b,c). The V oc linearly increases with increasing E G,PK , whereas the FF versus E G,PK profile exhibits a distinct "V" shape [53] with the minimum close to ideal current matching conditions. Consequently, in certain cases, a slight current mismatch is favorable for boosting FF at the expense of a minimal drop in J sc to render an overall larger EY. [53] Moving away from the optimal values, relative J sc losses increase more rapidly than EY losses. In other words, EY is less sensitive to the change in E G,PK compared to J sc . The reason for this lies in the increase of FF when either subcell begins to limit the current of the complete tandem device as demonstrated in Figure 2c). Due to nonequal shunt resistances of the two subcells, this increase is especially pronounced at lower E G,PK values when J sc is limited by the silicon subcell with a higher shunt resistance. At higher bandgaps, on the other hand, EY is increased additionally by the increasing V oc , causing asymmetry of the EY curve.
Finally, based on the results in Figure 2, we can conclude that certain deviations in the perovskite bandgap can be tolerated without a significant drop of EY, as also noted in refs. [13,53]. Our model confirms this observation also when taking complete temperature-induced bandgap variations into account. For the case of our studied device, we quantify that the perovskite bandgap variations up to 30 meV from the optimum are allowed to still keep EY within 99% of its maximal attainable value. This presents a major advantage from the industry perspective, since processing-based deviations that might result in E G,PK being different from the target value would not visibly affect device operation.
While the analysis so far was focused on typical open-rack installation, we further investigated how the optimal E G,PK of the tandem solar cell may be affected by any other chosen orientation of the device. This information is especially important for the building-applied (BAPV) and building-integrated (BIPV) schemes in which the device is following the orientation of the mounting surface and can face an arbitrary direction away from the south, and also for tracking installations. For this purpose, we again considered Atlanta as the target location. Several EY calculations were executed for different device orientations, which are in the horizontal coordinate system given by the module azimuth (A M ) and the inclination angle ( ). [50] In our work, the azimuth angle was varied from 90°(module facing East) to 270°(module facing West) in intervals of 10°, and the inclination angle was var-ied from 0°to 90°in intervals of 5°. For each combination of A M and , we determined the orientation-specific optimal E G,PK that rendered the maximal energy yield EY max .
The results presented in Figure 3a) illustrate that the optimal E G,PK values are very robust to most orientations of the tandem device, and they never deviate for more than 35 meV away from the value obtained for the optimal open-rack orientation ( = 32°a nd A M = 175°for Atlanta). Indeed, if the open-rack optimal E G,PK (1.719 eV) would be used for all other orientations, the EY losses relative to the maximal achievable EY max (rendered by E G,PK that is optimized for each orientation separately) would be only minor. This is demonstrated in Figure 3b), especially if the device is facing south (A M = 180°) and only rotates along the horizontal axis (which is typically the case in seasonal axis tracking systems). But even in more extreme cases, for example if we applied the open-rack optimal E G,PK to an east-or west-facing façade, the relative EY losses attributed to the nonoptimal perovskite bandgap, in this case, would still be limited to 1% (see top right and left corner in Figure 3b)). The results show that precise tuning of the perovskite bandgap does not need to be considered when designing tandems for different static (e.g., BAPV) or dynamic (tracking) orientations, which from the industry perspective presents yet another important advantage.
Finally, it should be noted that the apparent robustness of E G,PK conveyed through the above discussion does not mean that exact optimization for certain specific operating conditions is without value. On the contrary, while the observations above teach us that it is feasible for the industry to mass-produce identical tandem PV modules for different locations and for different orientations without substantial losses, the developed EY modeling algorithm nevertheless allows us, if required, to optimize the device specifically for some target set of operating conditions, thus achieving optimal solar harvesting with potentially substantial absolute EY gains, especially for large PV systems.

Loss Analysis
While energy yield modeling can be used efficiently to perform device optimization as demonstrated above, it is also a powerful tool for isolating and studying the influence of various physical mechanisms that define the behavior of the investigated device. Thus, it is often possible to obtain critical information with a high impact on design guidelines, which otherwise could not be measured experimentally.
In this respect, our EY algorithm was employed to perform an analysis of energy losses in the tandem device, in dependence on the perovskite bandgap. The results are presented in Figure 4a). The red lines represent thermal losses, which are calculated as the reduction of the realistic EY compared to the fictive EY that would be obtained if the cell temperature was fixed to 25°C in all cases (with all other operating conditions unchanged). Two cases were considered in the scope of this study, for two specific instances of device operation. For open-rack utility-scale configuration, where the upper and bottom side of the PV module are efficiently subjected to convective cooling due to wind, temperature losses of the device located in Atlanta (weighted average T cell = 31.2°C and maximal T cell,max = 61.0°C) are almost negligible, at around 1% for the optimal E G,PK (full red line). However, if  . a) Relative energy losses related to temperature (red lines), and excess, nonextracted energy (blue). b) temperature coefficient as a function of E G,PK for STC and Atlanta-specific realistic conditions. For the device located in Atlanta we show weighted average of the by considering temperatureinduced variations of E G,PK and E G,Si (solid line) as well as by treating the bandgaps of both subcells as temperature independent (dashed line). c) The difference between the maximal currents that would be generated in the two subcells (with no limitation, e.g., in a 4T configuration) for three different perovskite bandgaps. Number of transitions, N t , from one to another subcell limitation is stated, as well as, number of hours when perovskite (N PK ) or silicon (N Si ) contain excess carriers.
rooftop or BIPV installations with little to no wind cooling at the back side of the modules were considered, the predicted energy losses increase up to 4% (dashed red line) due to the higher temperatures (T cell = 41.9°C, T cell,max = 68.1°C). Interestingly, the losses are not bandgap independent but increase with increasing E G,PK . With higher perovskite bandgap, it is the perovskite subcell that limits the J sc of the tandem device. And since E G,PK widens with increasing temperature, the tandem current decreases, resulting in increased losses. With lower perovskite bandgap, on the other hand, it is the silicon subcell that limits the tandem J sc .
And since E G,Si narrows with increasing temperature, the tandem current increases, reducing the losses, as indicated by the figure.
The above observation lead us further to consider the temperature coefficient ( ) of the investigated tandem device and its dependence on E G,PK . The temperature coefficient was simulated for different perovskite bandgaps under STC, and also under Atlanta-specific realistic conditions. In the latter case, we also calculated values that would be obtained if E G,PK and E G,Si were not temperature dependent. Realistic conditions again required weighted averaging to obtain a single representative annual value. The results plotted with solid lines in Figure 4b) demonstrate that in both cases we can achieve different temperature coefficients depending on current mismatch conditions, as has been speculated by Liu et al. [10] and Aydin et al. [30] With the energy yield model, we are able to evaluate the change in temperature coefficient, which is ranging from −0.17 and 20.13% K −1 (strong silicon subcell limitation) to −0.26 and -0.23% K −1 (strong perovskite limitation) for STC and Atlanta-specific conditions, respectively. This confirms that for accurate reporting of tandem temperature coefficients also mismatch between the subcells has to be reported. Additionally, the results also show that realistic operating conditions are expected to produce lower values compared to STC. Furthermore, while solid lines show that temperature coefficient increases with increasing E G,PK , it can be observed (dashed line) that this is not the case if bandgap temperature variations are not included in the EY modeling of tandem PV devices. This observation confirms the importance of considering the T cell dependencies of PK and Si bandgap to accurately extract the device's vital parameters such as temperature coefficients.
Returning to the analysis of energy losses, the remaining blue curve in Figure 4a) represents the losses associated with twoterminal (2T) electrical configuration of the device, in which the total short-circuit current density is limited by the smaller of the two subcells. The losses are calculated as the drop of EY in a 2T tandem solar cell relative to EY that would be obtained in an otherwise identical 4T device. As expected and also evident from the figure, the losses quickly increase when moving away from the optimal conditions. The fact that not all charge carriers can be extracted from the nonlimiting subcell of the tandem device in 2T configuration, however, might have additional long-term stability implications as the excess carriers ("V oc " condition) can speed-up the degradation. [54] With that in mind, it might prove to be beneficial for long-term high performance tandem operation to select E G,PK at which the perovskite subcell either overly dominates or limits the performance of the tandem, instead of the optimal E G,PK . Furthermore, constant alternations between the two limiting situations could also be regarded as another degradation mechanism. Figure 4c) shows the difference between the maximal currents that would be generated in the two subcells (with no limitation, e.g., in a 4T configuration) for three different perovskite bandgaps. With E G,PK 30 meV below the optimum (E G,PK = 1.689 eV), the device is predominantly limited by the silicon subcell. Throughout the year, there are 3714 hours (N PK ) in which the perovskite subcell is dominating charge carrier generation, and only 461 hours (N Si ) in which the roles are reversed with the silicon subcell. Altogether, there are 551 transitions (N t ) between the two situations, which are caused by temperature changes and, mostly, daily spectral variations. The conditions are reversed, yet otherwise similar, with E G,PK 30 meV above the optimum (E G,PK = 1.749 eV). At the optimal perovskite bandgap, the energy losses are the lowest and the limitation times between the two subcells are more evenly distributed, however, the transitions occur almost twice as often, 952 times per year.
While further investigation of both these potential degradation mechanisms on fabricated devices is beyond the scope of this contribution, energy yield modeling offers insight into their occurrence and can help evaluate their effect, which is otherwise challenging using typical measurement techniques. [19] And finally, once the exact behavior of different degradation sources is known, their rates can be fed back into the EY modeling algorithm in order to optimize the design of tandem devices for an extended long-term operation.

E G,PK Optimization-Different KGPV Climate Zones
In the last stage of our work, we generalized our findings by extending our study to cover a broader range of different geographical locations, focusing in particular on distinct Köppen-Geigerphotovoltaic (KGPV) climate zones. [42] We analyzed nine different Central and North America-based geographical locations, all of which are listed in Table 1 together with the corresponding KGPV climate zone designations (in parentheses) and geographical latitudes. For each of the locations, we repeated the E G,PK optimization procedure that has been already performed for the case of Atlanta. The maximal achievable energy yield values (EY max ) and the corresponding optimal E G,PK values were extracted from the calculations for each location and are summarized in the last two columns of Table 1.
The EY(E G,PK ) results for five selected locations -Panama, Denver, Atlanta, Minneapolis, and Juneau -are presented in Figure 5a). As expected, the main driver behind the annual energy yield is G tot -higher the irradiance, the more energy is generated by the tandem device. Additionally, we again calculated the relative difference of EY obtained at each E G,PK in the given range compared to EY max obtained at the optimal E G,PK . These results are presented in Figure 5b). The optimal E G,PK for each of the five selected locations as well as the E G,PK value optimized under STC are indicated in both figures by vertical lines.
As already observed in the case of Atlanta, the optimal E G,PK values determined for all nine geographical locations differ from the optimal value obtained under STC, and are indeed higher in all cases. This observation is also in line with previous studies. [16,27,55] However, while the geographical location does influence the value of the optimal perovskite bandgap, its impact is only minor despite the strong variety in climate conditions. The highest E G,PK was determined for Panama (1.724 eV), and the lowest for Denver (1.704 eV), with only 20 meV difference between them. Therefore, even if an arbitrary E G,PK value within this range (1.704-1.724 eV) would be selected for all geographical locations, the annual energy yield would remain within 99.5% of EY max in all cases. Furthermore, if we averaged the optimal E G,PK values in Table 1 and used this average (1.716 eV) for all locations, the energy losses would be even smaller, less than 0.2% in all cases.
This indicates that the climate zone in which the tandem device is planned to operate does not need to be strictly Table 1. Summary of numerical results for each of the nine geographical locations from distinct KGPV zones. [42] . Latitude [N]  included among the requirements for optimization of the perovskite bandgap. Additionally, the impact of any processingbased deviations is also expected to be minor and can be tolerated to some extent, as was already discussed before. These conclusions are again key for manufacturing purposes, since they indicate that a single perovskite bandgap within a generous tolerance range can be selected at which the PV module would operate with close to 100% capability, no matter the location nor the orientation of its installation. From the results obtained so far it can be observed that the tandem device installed at Panama (AH) exhibits the highest optimal E G,PK , slightly exceeding Minneapolis (EM) and Juneau (EL). That was not expected at first sight, since the device at Panama is subjected to much higher operating temperatures that should render the highest blue and red shift of the perovskite and silicon bandgap, respectively. This interesting observation can be explained by considering the weighted averages of the device temperature T cell = and the average photon energy (APE i designates the average photon energy that characterizes the spectrum of G tot,i and is calculated in the wavelength range of 300-1200 nm). The annual values of T cell and APE for all locations are given in Table 1. The T cell in Panama is almost 14°C higher compared to Minneapolis (and almost 20°C higher compared to Juneau), which would intuitively call for a lower optimal E G,PK . At the same time, however, Panama represents a climate zone with the highest annual APE (blue rich) compared to all other locations, which compensates or even exceeds the effect of higher T cell and, thus, requires an increase in the optimal E G,PK . In contrast, the device installed at Denver (CH) exhibits the smallest optimal E G,PK , since it is subjected to G tot with the lowest APE (red rich) and relatively moderate T cell values. These results demonstrate that spectral distribution of incident irradiance as well as the device temperature both play indispensable roles in determining the annual energy yield of tandem solar cells and, therefore, need to be taken fully into account.
To study these inter-dependencies in a greater detail, with the ultimate goal of indicating the key parameter that defines the optimal E G,PK value, we move from optimization performed on a yearly basis to a monthly based optimization. The monthly averaged values of APE and T cell (hourly APE and T cell values weighted by the incident irradiance and averaged over the given time period) for each of the five selected geographical locations are plotted in Figures 6a,b). Significant variations both in absolute values and in month-to-month profiles can be observed, demonstrating clearly how meteorological conditions vary across different KGPV climate zones. The realistic environmental conditions of each month were then used to determine the month-specific optimal E G,PK values, which are presented in Figure 6c). The results show that the optimal E G,PK values may indeed vary notably from Figure 6. a) Monthly weighted average APE (calculated in the wavelength range 300-1200 nm), b) monthly weighted average T cell and c) month-specific optimal E G,PK values for each of the selected locations. d) The dependence between the optimal E G,PK and APE values for different optimization cases. month to month, especially for locations with pronounced seasonal variations. More importantly, however, the results reveal that there is a distinct correlation between the optimal E G,PK values and the values of APE - Figures 6a,c) show virtually identical profiles.
The results of our EY modeling, which is based on realistic input parameters including complete temperature-induced bandgap variations of both subcells, therefore confirm the qualitatively observed trend [16,27,31,56,57] that the optimal perovskite bandgap in the tandem device is mostly influenced by the spectral distribution of the incident irradiance. However, the main challenge was to find a quantitative correlation that could serve as a design guideline. For this purpose, we plotted the dependence between the optimal E G,PK and APE values in Figure 6d), specifically for different cases of perovskite bandgap optimization performed under location-specific realistic environmental conditions over different time periods: i) over each month (circles), ii) over each season (triangles), and iii) over the entire year (squares). The four seasons in season-based optimization were defined as: winter (December-February), spring (March-May), summer (June-August), and autumn (September-November). The data point corresponding to STC conditions is also included (asterisk); it can be observed that APE is among the lowest in this case, which explains the lower optimal E G,PK value obtained when optimizing the tandem device under STC, as observed previously. The results in Figure 6d reveal an important trend that practically all different E G,PK (APE) combinations are tightly clustered along a distinct linear dependency (0.61 + 0.60⋅APE; approximated with a black line), regardless of the conditions used in the optimization process, be it STC or location and time periodbased realistic operating conditions. By knowing APE, therefore, the optimal perovskite bandgap can be determined without any need for additional location-specific simulations. This is a powerful conclusion that can be applied directly for rapid design of tandem devices for any target application. Nevertheless, the complete energy yield model such as ours is still needed to accurately determine the energy output, since this cannot simply be calculated from APE.

Conclusions
We developed a comprehensive numerical model that enables accurate estimation of the long-term energy yield of two-terminal perovskite-silicon tandem PV devices operating under realistic outdoor conditions. Measured meteorological data in conjunction with advanced optical modeling is employed to determine the J sc of the device at each data point of the time interval, whereas the measured illumination-and temperature-dependent J-V characteristics of each subcell combined with J sc and T cell values ultimately provide the output electrical energy by means of interpolation and graphical summation. Complete temperatureinduced bandgap variations of both subcells is also taken into account at all stages of the EY modeling algorithm.
Using the model, we calculated the annual energy yield of the encapsulated perovskite-silicon tandem device under STC and when located at different geographical locations from distinct KGPV climate zones. For each case, we determined the optimal bandgap of the perovskite subcell. We demonstrated that the optimal perovskite bandgap in a tandem device operating under realistic conditions is in all cases higher than the one optimized under STC, which is primarily the influence of the spectral distribution of incident irradiance. We observed a clear linear dependency that can be drawn between the optimal E G,PK and the weighted average of the average photon energy (APE) contained in the spectrum, which presents an important design guideline for device fabrication. Therefore, just knowing the APE is enough to determine the optimal perovskite bandgap.
Furthermore, the results showed that certain deviations from the optimal E G,PK can be tolerated without a significant impact on the performance. We showed that a single appropriately chosen E G,PK can be used for all locations and even for all possible orientations of the tandem device, with the annual energy yield in all cases reaching at least 99.8% of the maximal obtainable value that could be reached by case-specific bandgap optimization. This observation presents a major advantage for manufacturing of tandem PV modules, since both the location as well as the orientation of the device can be excluded from design requirements.
The developed energy yield model was also employed to study the various sources of energy losses. Through the analysis of thermal losses, we observed that the losses are not bandgap independent. The temperature coefficient and consequently the losses are both influenced by the opposing temperature dependency of the subcell bandgaps; in general, larger values are expected when the current is primarily being limited by the perovskite subcell. In addition, also including the wind has proven to be an important feature of our EY calculations as it can alter the thermal losses by several percentages. Finally, observing the current limitation conditions on the hourly basis, information about potential degradation mechanisms can also be discerned. One is the duration in which any of the two subcells is being limited by the other, and the second is the number of transitions between the two situations; they are both influenced by the daily temperature and spectral variations.
With the analysis presented in this contribution, we show the benefits of detailed energy yield modeling and offer an insight into the data that can be obtained by using it. Through energy yield modeling we can reach beyond the limits of typical measurement techniques in order to obtain hidden information that can be vital for further evolution of the technology. It is crucial for predicting the generated energy of PV devices at different locations, determining the optimal perovskite bandgap for highest energy generation, monitoring degradation of devices during testing, tracking transitions between subcell limitations and quantifying the excess charge carrier generation in individual subcells, and potentially even including degradation rates for long-term analysis, all of which could in the end serve as a base for estimating the levelized-cost of electricity (LCOE).

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.