During the past few years there has been a significant interest in spectrum splitting systems to increase the overall efficiency of photovoltaic solar energy systems. However, methods for comparing the performance of spectrum splitting systems and the effects of optical spectral filter design on system performance are not well developed. This paper addresses these two areas. The system conversion efficiency is examined in detail and the role of optical spectral filters with respect to the efficiency is developed. A new metric termed the Improvement over Best Bandgap is defined which expresses the efficiency gain of the spectrum splitting system with respect to a similar system that contains the highest constituent single bandgap photovoltaic cell. This parameter indicates the benefit of using the more complex spectrum splitting system with respect to a single bandgap photovoltaic system. Metrics are also provided to assess the performance of experimental spectral filters in different spectrum splitting configurations. The paper concludes by using the methodology to evaluate spectrum splitting systems with different filter configurations and indicates the overall efficiency improvement that is possible with ideal and experimental designs.
© 2014 Optical Society of America
Considerable research is being devoted to increasing the conversion efficiency of photovoltaic (PV) systems. Increased efficiency will lower the cost per energy output of solar energy systems and make them a more viable renewable energy alternative. Shockley and Queisser have shown that systems based on single junction PV cells are limited to a system efficiency of 33% . This restriction results from the mismatch between the photon energy of the incident solar illumination and the inability of a single junction device to optimally convert the broad incident spectrum.
One approach to overcome this difficulty is to incorporate multiple bandgap PV cells that are optimized to convert different parts of the incident spectrum to electrical power. Two approaches are being investigated to implement this concept: integrated multi-junction PV cells in tandem , and a spatially separated collection of single junction cells [3–5]. Tandem multi-junction cells must be made from similar materials that can be latticed matched and the cell interfaces. In addition the different junction cells are connected in series and therefore the output current is limited by the cell with the lowest output current . Finally tandem cell systems are typically used at high efficiency to increase cell conversion efficiency and to reduce the cost of the system by incorporating a large collection optic with a lower cost/area than the multi-junction PV cell.
Spectrum splitting configurations distribute incident photons onto several single bandgap PV cells that are spatially separated . This allows PV cells with dissimilar materials to be used and is not restricted by current matching effects. In addition the system can be operated at lower concentration ratios thereby requiring less stringent tracking and cooling techniques. However, the spectrum splitting optical system must be designed to spatially separate spectral bands that optimally match the spectral responsivity of the PV cells in the system. The optical system must also provide high transfer efficiency and negligible overlap between spectral bands.
In this paper, the conversion efficiency of each PV cell will be defined as a spectral quantity that is modified by the filter spectral transmittance (and different system configurations) and then integrated to determine the overall spectrum splitting system (SSS) optical to electrical conversion efficiency.
Although there have been different spectrum splitting geometries proposed to date, they have not been presented with a uniform and detailed evaluation method [4,5]. In particular, the overall advantage (or disadvantage) of using a particular SSS design has not been quantified equivalently across different systems . In this paper, the important metrics for spectrum splitting systems are developed to provide an assessment of conversion efficiency, the influence of spectral filters on system performance, and guidance on system design. It is shown that careful consideration of the wavelength dependence of the spectral filter and PV cell must be made to achieve high system conversion efficiency. A metric that compares the efficiency of the spectrum splitting system with respect to a similar system with the highest efficiency single bandgap cell from the system is also introduced. This parameter called the Improvement over Best Bandgap (IoBB) provides an indication of the benefit of the added complexity of the spectrum splitting system with respect to a simpler single cell system. Metrics are also defined that indicate the relative performance of experimental to ideal spectral filters and their impact on overall system efficiency.
The properties of different types of filters for use in spectrum splitting systems are also examined. These include spectral band separation and dispersive optical filters. The effect of the linear dependence of the spectral responsivity of PV cells on the overlap characteristics of spectrum splitting filters is also examined and is found to limit system conversion efficiency.
The paper concludes with an evaluation of ideal and experimental filters as well as ideal and experimental PV cell characteristics for 4- and 2-cell spectrum splitting systems. The metrics illustrate the conversion efficiency that is possible with ideal components and the sensitivity to different component properties.
2. Efficiency metrics for spectrum splitting systems
2.1 Spectral conversion efficiency (SCE)
The overall conversion efficiency of a PV cell or system is the ratio of electrical output power to the optical power of the incident solar illumination. For a single junction PV device, the optical-to-electrical conversion efficiency can be expressed as [2,7]:8]. In this case PIN-optical = PAM15.
The current density, JSC, is a function of both the incident spectrum and the spectral responsivity, SR(λ) [A/W], of the PV cell according to :7] and is given by: SCE(λ) = VOC⋅FF⋅SR(λ). The SCE is a useful PV cell parameter because it expresses the absorption properties of the cell (SR(λ)), the bandgap properties (VOC), and the circuit properties of the PV cell (FF).
When a PV cell is used in a spectrum splitting system the transmittance of the spectral filters used to separate different spectral bands must be included in the conversion efficiency expression. The efficiency for the kth PV cell in the system is expressed as:
For a system with K different PV cells, the total overall system efficiency is given by the following expression:5]).
2.2 Optical efficiency
The optical efficiency of a spectrum splitting system is a function of the spectral transmittance and the ability of the optical system to deliver different spectral components to their respective PV cells. However the definition is complicated by the overlap of the spectral responsivities of the different PV cells used in the system. Figure 1 above shows the response for three ideal PV cells with different bandgaps and absorption properties. Since the ideal PV cell responsivity is a linear function of wavelength there is overlap at short wavelengths and as a result it is not possible to isolate a distinct spectral band with an optical filter. In addition it is not possible to separate the optical efficiency (ηoe) as a spectrally distinct factor affecting the total efficiency as in:
2.3 Filter performance
Spectrum splitting filters separate wavelength bands for use by different PV cells with matching spectral responsivities. The process of specifying a filter is complicated by the overlap and lack of sharp transitions of PV cell spectral responsivity characteristics. As a result the filter set must be optimized for the particular set of PV cells used in the system. An ideal spectral filter will have a transmittance or reflectance value of 100% within the selected spectral band and zero outside the band. Figure 2 shows the spectral responsivities for three different PV cells: an InGaP2- Emcore [9,10], an GaAs-Alta  and Si-PERL  PV cells. The figure also shows the filter transmittance for an ideal and experimental filter implemented with a volume hologram that are designed to select the kth (GaAs) spectral band. In addition, if more than two PV cells are considered, then significant spectral overlap of the cell SCEs can occur making optimization of the filter the transmittance function more complex typically reducing the IoBB of the system.
It is possible to consider the filter performance within (In-Band) and outside (Out-of-Band) the designed spectral band of the filter. These two regions have the following characteristics:
- In-Band Filter Region: This region contains spectral components that are in the primary transmission band of the filter and overlap with the spectral range where the maximum SCE of the kth PV cell occurs. There is no appreciable overlap with adjacent PVk ± 1 cell spectral response. This is the optimized spectral range for the bandgap energy. This filter region corresponds to the maximum output from kth filter-PV cell combination.
- Out-of-Band Region: This filter region consists of two areas. The first corresponds to long wavelengths that are not absorbed by the kth PV cell. As shown in Fig. 2, SCE = 0 in this range for the kth PV cell, changes to the filter do not have any effect on the performance of the kth PV cell but affect the output of the kth + 1 PV cell. Wavelengths shorter than the in-band spectral range have significant response from both the kth and (kth-1) PV cells. Therefore in this region, the spectral range and transmittance of the filter must be optimized in conjunction with consideration of the in-band performance of the kth-1 filter/PV cell combination.
2.4 Improvement over best single bandgap (IoBB)
The total efficiency is frequently stated as a performance metric for a variety of PV systems including spectrum splitting [4,5,12]. However this parameter does not provide insight into how much of an improvement the spectrum splitting system provides with respect to using a single type of PV cell with one bandgap over the same system aperture. Comparing the performance of a spectrum splitting systems with the output when a single cell with the highest conversion efficiency is used indicates the performance gain that will be obtained with the more complex system. This metric is called the Improvement over Best Bandgap (IoBB) and is defined as:
2.5 Filter overlap metric
In order to evaluate SSS configurations and compare them with ideal filter performance, we define the filter overlap O as,Eq. (8) with practical filter performance data (measured, simulated, etc.) and IoBBIdeal is calculated using ideal filter parameters (i.e. a filter with 100% in-band performance and no spectral components in the out-of-band regions as shown in Fig. 2) and is the maximum improvement that can be obtained for a specific combination of PV cells. The overlap metric (O) indicates how close particular filter is of achieving the maximum ideal improvement.
Deviation of actual filter performance with respect to ideal filter characteristics can be quantified with:
3. Spectrum splitting system configurations
The metrics from the previous section provide a methodology to evaluate and compare different spectrum splitting systems. For clarity, the expressions below only include PV cell and filter spectral characteristics of the components used in the spectrum splitting system. Other factors (such as shadowing, reflectance, tracking error, etc.) have been omitted but can be easily included in the analysis if necessary .
3.1 Tandem spectrum splitting PV systems
In a tandem spectrum splitting configuration, the absorption of each junction effectively filters the incident spectrum for subsequent PV cells. Multi-junction PV cells are typically designed so that the top layer completely absorbs shorter wavelength bands allowing longer wavelengths to penetrate to the lower energy bandgap junction that is near the bottom of the multi-junction stack. For a three bandgap system, the total efficiency can be calculated as:
3.2 Lateral spectrum splitting PV systems
Lateral spectrum splitting can be realized in a variety of configurations. In each case spectral bands are spatially separated outside of the PV cell layer with external optical components.
3.2.1 Band pass filter spectrum splitting
Reflection type band pass filters can be formed using dielectric coating (dichroics) or with holographic reflection gratings. The mth reflected spectral band, Rm(λ), is laterally separated as the transmitted band Tm(λ) continues down the optical axis as shown in Fig. 3(a). In order to realize a multi-PV cell system the filters can be cascaded as shown in Fig. 3(b). In this case, the total efficiency of a system with M cascaded band pass filters and K PV cells can be calculated as:
3.2.2 Spatially separated spectral band systems
In a spatially separated spectral band configuration, the filters are arranged next to each other instead in a cascade. The separated band is directed to an adjacent spectrally matched PV cell as shown in Fig. 4.In this case the unfiltered spectrum is matched to the PV cell directly below the filter.
The PV cell below a filter receives the spectrally matched pass band of the filter above it and the separated band of the adjacent filter. In this case, the total system efficiency can be calculated as:Fig. 4 above.
3.2.3 Dispersive spectrum splitting
Spectral separation can also be achieved using the dispersion of a filter caused by diffraction or refraction. The magnitude of filter dispersion depends on the type of component used and can be expressed as a dispersion factor given by:Eq. (16). If the dispersion factor is uniform across the filter aperture then the situation shown in Fig. 5(a) results in which the filter aperture with a mixture of wavelengths is projected onto the receiver plane. This configuration by itself is not useful for spectrum splitting since the incident wavelengths are not completely separated at the receiver plane except at the edges of the projected aperture. In order to realize a useful spectrum splitting element, the dispersion factor must be varied across the aperture (with an element like a focusing hologram as shown in Fig. 5(b)) or a separate focusing element must be used as in .
The degree to which the spatially dispersed spectrum on the receiver plane is matched to the responsivity of a PV cell can be quantified using a spectral overlap parameter τk(λ). The spectral overlap parameter will be maximum when the PV cell aperture, spectral responsivity and the dispersed spectrum match in size and location, and decreases with mismatch of these parameters. A diagram of this function is shown in Fig. 6 and illustrates the effect of focusing on the overlap parameter as a function of wavelength.
With the above definitions, the total efficiency of a spectrum splitting system using a dispersive optical element with K different PV cells can be calculated using the following expression:
4. Examples of spectrum splitting systems
In this section the metrics and methodology discussed in the previous sections are applied to systems using ideal and experimental PV cell data and filter characteristics.
4.1 Ideal filter band pass spectrum splitting system with ideal PV cell SCE performance
Figure 7 shows the results of modeling a band pass configuration as shown in Fig. 1 with K = 4 and the highest efficiency PV cell as PV4. Each curve was obtained using: the SCE definition in Eq. (4), the bandgap wavelength [10,14] indicated in the graph for each cell, and the Current-Voltage (IV) characteristics were obtained with detailed balance analysis [1,15]. The filters were modeled with ideal spectral response (with 100% in-band and 0% out-of-band transmittance as shown overlaid on the SCE curves). The filter parameters (bandwidths: short and long wavelength cutoffs, and central wavelengths) were obtained by maximizing the total system efficiency ηSSS. The filtered efficiencies are less than the reported STC full spectrum efficiencies but the overall SSS efficiency is larger than any of the PV cells that comprise the system.
For this case, all components of the SSS have ideal characteristics and the efficiency is maximum at ηSSS = 51.89% (for the four PV cells selected). The system’s efficiency corresponds to an IoBB of 67.02% over the InGaP2-Emcore PV cell (best single junction efficiency). Since the filters are ideal, the IoBB = IoBBIdeal and the filter overlap O is equal to unity.
4.2 Ideal filter band pass spectrum splitting system with reported PV cell SCE characteristics
Figure 8 shows modeling results for the same system as the previous section using experimental PV cell performance data and an ideal filter spectral response. The IoBB shows that the SSS performance can be expected to be 47.78% more efficient that the GaAs-Alta PV cell (at 28.19%) for a total system efficiency of ηSSS = 41.65%. The filter overlap (O) is still equal to unity since ideal filters were used in the analysis.
4.2 Holographic filter spectrum splitting systems with reported PV cell SCE characteristics
In the following system experimental values for a holographic reflection filter are used with two different PV cells (GaAs-Alta  and Si-PERL  PV cells). The resulting system performance is shown in Fig. 9 as well as results for the same PV cells with ideal spectral filters. With ideal filters, a maximum system efficiency (ηSSS) of 34.38% can be achieved and the improvement over best single cell efficiency is (IoBB = IoBBIdeal) 21.99%. The overall system efficiency and the IoBB are reduced to 31.88% and 13.09% respectively when experimental holographic filter performance  is considered. From examination of Fig. 9 it can be seen that out-of-band loss is primarily due to secondary Bragg matching at a wavelength near 0.5μm. Light diffracted away from the GaAs cell by the secondary order is incident on the Si-PERL cell with lower SCE than GaAs and therefore reduces the overall system performance. The in-band losses are caused by the non-square spectral shape and limited bandwidth of the holographic filter reflectance spectrum. As a result the most efficient part of the Si-PERL cell is not utilized effectively.
The holographic filter overlap (O) for this system is 59.53% with a corresponding filter loss (Ofilter loss) of 40.47% (Eqs. (9) and (10) respectively). The in-band losses account for 96.22% of the losses while the out-of-band regions account for an additional 3.78% of the losses. Therefore the system efficiency can be significantly increased with improved hologram diffraction efficiency within the in-band spectral range at the expense of increasing the efficiency of the secondary order of the hologram and its effect on the out of band loss.
Results for a dispersive spectrum splitting system with three PV cells implemented with a holographic transmission filter and a lens  are shown in Fig. 10. In this analysis, the reported data for three PV cells (GaAs-Alta , Si-PERL , and a GaSb [14,16]) are used. When ideal filters are used, a maximum system efficiency (ηSSS) of 38.24% is achieved. The IoBB improvement of this system is 35.66%. When experimental data for holographic spectral filters are used, the overall system efficiency and the IoBB are reduced to 33.43% and 18.61% respectively. The corresponding holographic filter overlap and filter loss are O = 52.18% and Ofilter loss = 47.81% respectively. The spectral overlap parameter τk(λ) with the geometry in  has a spectral response close to ideal. However, since the diffraction efficiency of the hologram is less than 100% the in-band performance is reduced. The in-band losses are then 7.45%, 23.90%, and 16.50% for PV cells k = 1 (GaSb), 2 (Si-PERL) and 3 (GaAs-Alta) respectively. The filter overlap analysis shows that improving the filter shape for k = 2 followed by reducing the loss for k = 3 can significantly improve the performance of this configuration.
The overlap metric (O) can be used to compare systems that have different configurations and different numbers of PV cells, (i.e. the systems shown in Figs. 9 and 10). The dispersive system in Fig. 10 has a better IoBB and system efficiency than the band pass system of Fig. 9. But the larger filter overlap metric of the band pass system indicates a filter that is closer to ideal. The improved performance of the dispersive system is due to increased number of PV cells with different bandgaps offsetting the losses of the dispersive transmission grating.
In this paper a set of metrics were presented for evaluating spectrum splitting systems. These metrics were then used to evaluate the specific issues related to different types of spectral filters used in spectrum splitting systems. The system efficiency (ηsss) incorporates the spectral conversion efficiency (SCE(λ)) of the cell and the filtered Air Mass (T(λ)⋅EAM1.5(λ)) spectrum and provides a measure for comparing different configurations. Another important metric that was defined is the Improvement over Best Bandgap (IoBB) and indicates the degree to which the efficiency increases with respect to a similar system that uses a single bandgap PV cell. This parameter provides an indication of whether the improved performance is worth the greater complexity of the spectrum splitting system. Parameters specific to the spectral filter performance were also introduced and include the spectral overlap with adjacent spectral bands in the filter system and loss with respect to an ideal filter.
A description of different types of spectral filters and spectrum separation mechanisms was provided giving an overview of the capabilities and limitations of each approach. These include: absorption filtering in tandem PV cell systems, band pass filters, spatially separated spectral band filters, and dispersive filters.
An analysis was then performed of two spectrum splitting configurations: a system with ideal spectral filters and 4 ideal PV cells compared to a similar system with ideal spectral filters and 4 experimental PV cells; and a system with ideal spectral filters and two experimental PV cells compared to a similar system with experimental holographic spectral filters. The results showed: ηSSS = 51.89% and IoBB of 67.02% for the ideal filter with four ideal PV cell system compared to ηSSS = 41.65% and IoBB of 47.78% for the ideal filter with 4 experimental PV cells. For the two cell system ηSSS = 34.38% and IoBB = 21.99% for the ideal filter system and ηSSS = 31.88% and IoBB = 13.09% for the holographic filter case. Analysis of the holographic filter characteristics shows that the primary loss with respect to an ideal filter occurs due to lower in-band efficiency rather spectral overlap with adjacent PV cell spectral bands. This characteristic can be improved using different hologram exposure and processing methods.
Future improvements to the system metrics and analysis methods will incorporate the PV cell efficiency change with concentration ratio. In ideal spectrum splitting systems the concentration ratio increases with the number of PV cells. Systems with greater numbers of PV cells therefore require using cells that are more efficient with higher concentration. This will also require optical systems that can readily incorporate different focusing powers to realize correspondingly different concentration ratios.
The authors wish to acknowledge support from the NSF/DOE ERC cooperative agreement No. EEC-1041895, NSF Grant No. 0925085, State of Arizona TRIF(WEES) program and the Research Corporation.
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