1. Introduction

Currently, very high efficiency solar cells are available on the market. These cells, which are based on III-V semiconductors, can reach efficiencies higher than 40% [2]. Since they are very expensive, they need to be operated at high concentration ratios to reduce semiconductor area. The idea that motivates a CPV system is to concentrate light by using low-cost optical elements so that the final product can be cost-effective.

Cell impact in total system cost is divided by the concentration ratio. Hence, high concentration ratios can be necessary to make CPV system based on III-V cells cost competitive. However, this is not the only requirement the optical system must fulfill to constitute a good photovoltaic concentrator. Acceptance angle of CPV system is barely a few times the angle subtended by the sun and its impact is often underestimated: wide acceptance angles can greatly reduce assembly and alignment requirements. Assembling the different elementary units that form a module in factories is done by assembly-line robots, so wider acceptance angles allow larger machinery tolerances. Acceptance angle is also dramatically important in field installation, where alignment and assembly of different modules in the tracker can become very difficult if acceptance angle is very narrow. Tracker stiffness and performance are also enormously influenced by acceptance angle. Wider acceptance angles allow less stiff trackers which translate into less material-intensive trackers and, as consequence, cheaper ones. Because tracker cost is an important factor in system total cost, the $/Wp figure can be significatively reduced by increasing acceptance angle. In addition, acceptance angle has a great impact in annual energy generation, so it is directly responsible for the cost of KWh of electricity generated, that is, it can affect whether the energy generated by the CPV system is competitive or not.

Another consequence of the optical system is that the irradiance distribution at the optics exit, that is, over the cell, is not uniform. Many designs lead to huge irradiance peaks compared to average irradiance over the cell. This lack of irradiance uniformity can put long term reliability of the cell at a risk. Concentration peaks can cause thermal stresses which could damage the cell. In addition, it has not yet been shown what maximum local current density can be handle by the tunnel diode in a multijunction cell. Moreover, lack of uniformity can increase the effective series resistance and decrease the Fill Factor. All these problems are being deeply studied at the moment by simulation [3] and experimental measurement [4].

The solution usually proposed for rising concentration, increasing the acceptance angle and/or equalizing irradiance over the cell consist of using a Secondary Optical Element (SOE) in addition to the Primary Optical Element (POE). It is obvious that energy generation enhancement and other aspects mentioned above should overcome the cost of adding an extra element to the system. In the past, few different designs have been studied, designed and manufactured. But actually, most of the CPV system in the market include a SOE. In this paper, the most common SOE designs are optimized and their performances are compared under the same primary lens. An aspheric lens, which is described later, is used as a POE. This comparison is made in terms, not only of the concentration level achievable by each SOE, but also of the acceptance angle and exit irradiation distribution.

2. Optical System description

The optical systems under study are composed of two elements. The primary, which would be the same in all cases under study, is an aspheric plano-convex lens and the secondaries are either refractive or reflexive. System geometric concentration is defined as the ratio of the entrance aperture (determined by primary lens diameter) and the optics exit area (determined by the cell diameter). For comparison, both magnitudes are kept equal in all designs, resulting in a geometric concentration ratio of 1000X. Most of CPV modules designs based on III-V cells have a concentration ratio higher than 300X [5], [6], [7], to become cost-effective. A concentration limit of 1000X would be desirable [8] if a good optical efficiency (over 70%) and wide acceptance angle (beyond ±1) were attained. Under these conditions the level of 1000X seems to be the limit for optics system based on refractive POE so this figure has been considered for the cases under study. Once the geometric concentration and primary are fixed, the secondary will be the only influence on the system optical efficiency. It can be understood that the SOE determines the effective system concentration X_effective = η_{optical,system} · X_geometric.

The idea of using an aspheric lens to achieve high concentration by eliminating spherical aberration is widely known and its profile can be easily calculated by using Fermat principle. It is also well known that this profile can be transformed into a Fresnel lens to reduce the amount of material necessary and light absorbance. When using an aspherical lens (or its equivalent Fresnel lens) attainable concentration is limited not only by the sun angular aperture but also by chromatic aberration and by Fresnel losses in surfaces where refractive index changes. These effects are inherent to refractive optics and cannot be avoided. To carry out this study the primary lens f-number is fixed to 1.1. This f-number is chosen high enough so that Fresnel losses in the second air-lens interface do not become very significant.

Three families of secondary are the most widely taken into account in practical applications: those based on reflexive surface (truncated inverted cone or truncated inverted pyramid), compound parabolic concentrator (CPC) and dome-shape lens. A simple idea to increase concentration and improve acceptance angle consist of adding an inverted reflexive cone over the cell. Some families of rays that would not impact the cell are redirected to it by the cone. This secondary can be manufactured easily and inexpensively by rolling a metal sheet. A high reflectivity value for all wavelengths useful for the cell must be assured in order to maintain a good optical efficiency. A simplified version of the cone, the truncated inverted pyramid, is even easier to manufacture by bending a metal sheet. It is an intelligent solution to couple a circular primary lens with a square cell. In addition, it homogenizes the irradiance distribution over the cell. For ray-tracing simulation in this case a square cell of equivalent area has been considered.

The compound parabolic concentrator is one of the most studied CPV systems. The CPC is designed using the edge-ray principle [9] and it is a perfect example of how to use nonimaging optics to achieve high concentration and optical efficiency simultaneously. CPC is an ideal concentrator, that is, it is able to maximize concentration as much as predicted by theory. It also provides very wide acceptance angle and, since rays are redirected to the cell by using totally internal reflection, no losses due to metallic reflectivity take place. One of the main disadvantages of this design is that it provides a deep non-uniform irradiation distribution over the cell. In addition it is usually manufactured from glass, resulting usually very expensive for CPV applications since it requires large amounts of material and optical surfaces usually need to be polished after manufacturing. Plastic injected CPCs could also be used if long-term reliability was assured which involves meeting challenges such as proper thermal cooling and UV protection (or rejection at primary optics).

The 2D dome design process consists of imposing the condition that all rays coming from one of the exterior points of the primary lens reaches the opposite extreme point of the cell. An asymmetric caustic curve is obtained, to transform it into three dimensional dome, caustic curve must be rotated around optical axis. Either rotating one of the caustic part or the other, two different domes (A, B) can be obtained. This refractive secondary can be advantageous: it is less material-intensive than the CPC and its shape (similar to a semi-sphere) can be easier manufactured not only from plastic but also glass. This approach has been thoroughly studied by Sandia Lab (Concept-90) [10]. Another similar design with a different approach was proposed by Davies [11].

3. Results

Chromatic aberration and solar aperture angle limit attainable concentration for refractive lenses. In fact, a single stage optic system based on a Fresnel lens would provide very poor optical efficiency and acceptance angle at the 1000X concentration level. In the case under study, the primary lens creates a spot at its focal distance bigger than the cell size. That means that if no secondary were used part of the light would not reach the cell. Hence, adding a secondary will automatically increase the amount of light collected by the cell and as a consequence will improve the optical efficiency. Optical efficiency is commonly defined for normal incidence, i.e., light transmission when the normal to the aperture area of the optical system is aiming to the centre of the source. The capability to transmit to the receiver deviated beams from normal will be analyzed with the angular transmission curves (see Fig. 2). Therefore, optical efficiency corresponds to the angular transmission value for null deviation. From transmission curves, two angles, θ _90% and θ _80% are obtained as those giving a relative light transmission respectively of 90% and 80% of the maximum obtained at normal incidence in each case.

 

Fig. 1. Profiles for the different secondaries under study: truncated reflexive cone (green) and pyramid (red), CPC (yellow), dome A (blue), dome B (orange). Dimensions in millimeters.

Download Full Size | PDF

With all the system components on axis, the optical efficiency is slightly higher when using reflexive secondaries than when using refractive ones (see Table 1). Despite an assumed reflectivity of 85%, with normal incidence, most of light reaches the cell without bouncing off the secondary walls. On the other hand, in case of using refractive secondaries a new index-changing interface is added, which increases Fresnel losses, leading to a slight lower optical efficiency for null angular deviation. Consequently, higher optical efficiency is obtained by using the reflexive cone and pyramid. The reflexive cone performs better because of its revolution symmetry. However, the reflexive pyramid would offer higher performance if a square cell is used.

Table 1. Optical efficiency comparison for different secondaries. θ _90% and θ _80% are deviation angles for an optical efficiency of 90% and 80% of the maximum. The first one (θ _90%) is usually known as acceptance angle.

View Table

As it is shown in Table 1, in the case of an inverted reflexive cone or pyramid, the acceptance angle is slightly increased with respect to the system without secondary. However, much better acceptance angle enhancement is obtained when choosing the CPC, providing ±1.4 for 90% relative transmission. Although dome A design has a poor acceptance angle (±0.58), dome B design has a significantly wider one (±0.94). As was explained in the introduction, acceptance angle defines tracking, assembly and alignment requirements and its impact can be decisive in the annual energy generation figure. It is clear the advantage of any of the studied secondary with high concentration optical system. Although CPC and dome B provides better performance, reflexive truncated secondaries are currently widely used [6] because of their low cost. Nevertheless, if the benefit of wide acceptance angle is considered, which can relax other system components and consequently reduce manufacturing and installing cost, refractive solutions such as dome or CPC may be justified.

 

Fig. 2. Angular transmission curves for the different secondaries studied. Circular marks over the lines indicates the deviation angle where optical efficiency becomes 90% and 80% of the maximum.

Download Full Size | PDF

Another figure of merit of a CPV system is irradiance uniformity in the cell. Irradiance distribution at the optics exit for each secondary are represented in Fig. 3 for the case of system on-axis. In relation to concentration peaks which could damage the cell, CPC irradiance distribution has the most significant peak. However, it is important to remark here that these results are predicted from simulations, considering ideal optical surfaces, and concentration peak observed is caused by the sum of radiation reflected by the entire secondary surface. That means that any manufacturing surface defect, assembly error, dust presence, scattering or other non-ideal factor will diminish this concentration peak. Instead of considering POE and SOE surfaces ideally specular, scattering can be applied to them to made simulation more realistic. In Fig. 4 gaussian scattering is applied to all optical surfaces. Gaussian scattering is defined by a radiance distribution R(α) that varies with angle α by the following function $R\left(\alpha \right)=R_0\exp \left[\right(-\frac{1}{2}\left){\left(\frac{\alpha }{\sigma }\right)}^2\right]$ where R ₀ is radiance in the specular direction and σ is standard deviation of Gaussian distribution in degrees, in this case σ = 0.2. Surface representation of irradiance distribution can be sometimes misunderstood, so the encircled energy figure can also be used in order to compare irradiance distribution. Encircled energy is defined as the percentage of total energy contained in a circle of a given radius. Because surface is quadratic with radius, a perfectly uniform irradiance will have a perfect convex parabolic representation. In other words, the more concave the encircle irradiance representation seems, the less uniform the irradiance distribution is. As it can be observed in Fig. 3, dome A has the most uniform irradiance distribution, as a consequence of the requirements imposed on the dome A optical design. CPC, although having the highest irradiance peak, has a nearly uniform irradiance distribution over the rest of the cell.

 

Fig. 3. Irradiance distribution over the cell and encircled energy for the different secondaries studied at deviation angle θ = 0.

Download Full Size | PDF

 

Fig. 4. Irradiance distribution at CPC exit considering (a) perfect specular optical surfaces and (b) optical surfaces where gaussian scattering takes place with σ = 0.2.

Download Full Size | PDF

4. Conclusion

Different secondaries under a reference primary lens have been compared. Reflexive secondaries (inverted cone or pyramid) show the best optical efficiency. CPC (±1.4) and dome B (±0.94) designs allow the wider acceptance angles. Dome A design creates the most uniform irradiance distribution over the cell. The intention of this work is not to declare the best secondary design in all cases, as these performance figures must be analyzed together with the resulting material and manufacturing costs before choosing the most appropriate secondary for a particular CPV system. However, the results of this study may prove useful in guiding this selection process.