## Abstract

In this study, an evolutionary algorithm (EA), which consists of genetic and immune algorithms, is introduced to design the optical geometry of a nonimaging Fresnel lens; this lens generates the uniform flux concentration required for a photovoltaic cell. Herein, a design procedure that incorporates a ray-tracing technique in the EA is described, and the validity of the design is demonstrated. The results show that the EA automatically generated a unique geometry of the Fresnel lens; the use of this geometry resulted in better uniform flux concentration with high optical efficiency.

©2010 Optical Society of America

## 1. Introduction

Currently, concentrating photovoltaics (CPV) is one of the key technologies being used to promote the use of photovoltaic electricity. The main purpose of CPV is to enable the use of a low-cost concentrator for concentrating solar radiation, which is then supplied to a highly efficient (high-cost) photovoltaic (PV) cell with a small cell area. Selecting the best concentrator and designing an optimal optical system is still an important issue, and hence, concentrators including mirrors and/or lenses have been studied [1–5]. A nonimaging Fresnel lens is one of the most frequently used optical elements in CPV-based concentrators because of its high optical efficiency, small thickness, and low cost of fabrication using plastics. However, it is fairly difficult for designers to optimize the geometry of a nonimaging Fresnel lens. The optimization is generally achieved by adopting a time-consuming trial-and-error method.

With this background, the objective of this study was to introduce a versatile optimization method to overcome the abovementioned problem. The evolutionary algorithm (EA), by performing numerous calculations, automatically generates the best possible optical geometry that would yield the desired performance. The advantage of the present method is demonstrated by presenting an example of the optimization of a linear nonimaging Fresnel lens for achieving uniform flux concentration.

## 2. Evolutionary algorithm

The EA was applied in the form of a hybrid algorithm consisting of a genetic algorithm (GA) and an immune algorithm (IA). The GA has recently been used to design the optical geometry of a diffraction lens to achieve microwave concentration [6] and to design an efficient Fresnel lens for lighting device with multiple LEDs [7,8]. One of the present authors has developed a GA-IA hybrid algorithm for a solar concentrator [9], which accelerates the evolution speed of the algorithm, i.e., reduces the CPU time. A flow chart of the GA-IA hybrid algorithm is shown in Fig. 1 . The major steps of the algorithm are as follows. (1) Generate initial geometries. (2) Evaluate optical performance using an evaluation index, which is defined before the calculation as per requirements. (3) GA manipulations: Generate new geometries by crossover and mutation using two geometries from the previous generation selected by following a predefined rule. (4) Select geometries for the next generation. A typical GA process involves repetition of steps (1)–(4); each iteration automatically increases the optical performance of geometry. However, the GA process tends to prematurely converge to a local solution; i.e., it does not always generate the best geometry. To mitigate this problem, the IA process (steps (5)–(7)) is combined with the GA process. The IA has been developed to accelerate the GA calculation [10]. The IA process followed in the present method can be described as follows. (5) Find similar geometries by calculating the differences among vertex coordinates of geometries. (6) For the next generation, eliminate all elements except one from each group of similar geometries. (7) Generate new geometries to keep the total number of geometries constant. Thus, the IA process helps to maintain a variety of geometries and consequently accelerate the evolution speed of the algorithm. Alternatively, the IA process can be called instead of the GA process, as shown in Fig. 1.

## 3. Optimization of nonimaging Fresnel lens by EA

#### 3.1 Model and method

One of the problems associated with the nonimaging Fresnel lens is insufficient uniformity of energy flux distribution on a PV cell because of chromatic aberration [11]. This problem often necessitates the addition of a secondary optical element such as a light-guiding glass plate, compound parabolic concentrator, or homogenizer to the system by coupling the element with the PV cell [12,13]. In this study, the authors attempted to optimize the geometry so as to achieve uniform concentration without using a secondary optical element. The proposed model of the nonimaging Fresnel lens is shown in Fig. 2
. The specifications of the model are listed in Table 1
. Only a cross section of the lens is optimized to elucidate the potential of the present method. Furthermore, the calculation is simplified by assuming that no volumetric absorption loss occurs inside the lens. The size of the model was decided to be similar to that of the Fresnel lens found in the literature [5]. The ray-tracing method was used to evaluate the optical performance of geometries, i.e., to calculate the evaluation index of geometries. The propagation of rays follows Snell’s law and Fresnel equations. In the proposed model, the incident angle range is set to *θ _{in}* = ± 0.25°, which is the same as that in a solar disk. It should be noted that the present optimization was carried out by taking into account the effects of three representative refractive indices:

*n*= 1.4790 (

*λ*= 1.20 μm), 1.4835 (0.80 μm), and 1.5095 (0.40 μm); these indices were selected on the basis of the optical data for PMMA, which is a dominant material used in the fabrication of the Fresnel lens.

An outline of GA manipulations, namely, crossover and mutation, is illustrated in Fig. 3
. During crossover, vertex coordinates are exchanged between two independent Fresnel lens geometries according to a predefined probability ( = 0.5 in the present optimization). During mutation, vertex coordinates are randomly changed according to a predefined probability ( = 0.3). The lens geometry consists of 50 prism segments, each with a width of *w*. The total number of geometries is 200. GA manipulations start from the center segment of the Fresnel lens to the outer segment one by one. Each segment consists of three vertices (vertex 1, 2, and 3 as shown in Fig. 3). To keep curve and slope continuity, outer side vertex 3 of the previous segment was directly used as center side vertex 1 of the next segment. In addition, interval between center side and outer side vertices of each segment was kept to be the prism width *w*. Middle vertex 2 of each segment was kept within vertex 1 and 3 in the *y*-direction. Vertical range allowed to all vertices was ± 0.1 mm from *y* = 0 level.

#### 3.2 Evaluation index

The evaluation index, which determines the optical performance of the geometry, plays an important role in EA optimization because geometries that are retained for the next generation are selected on the basis of the value of the evaluation index. The evaluation index *E* used in the present optimization can be represented as follows:

*I*is total incident energy over Fresnel lens aperture.

_{incident}*I*and

_{t}*I*denote the total energy over the PV cell surface and the energy at each finite element of the PV cell surface, respectively.

_{i}*I*and

_{t}*I*are summation for the three representative refractive indices. The PV cell surface is equally divided into 100 finite elements, i.e.,

_{i}*N*= 100.

*F*, which denotes the weighting factor, is primary set to 0.3 and 0.5; for verification of the effect of the weighting value on the optical performance of the geometry,

*F*is varied in the range 0.1 - 0.9. The first term of Eq. (1) is the optical efficiency, and

*D*of the second term represents the degree of nonuniformity of flux concentration on the PV cell surface. Therefore, the value of index

*E*will increase as the uniformity and optical efficiency increase. Although

*D*was defined as normalized sum of absolute deviation from an average energy, one can use variance instead of

*D*because final result may not be much affected, whereas speed of evolution may be affected in some cases.

## 4. Results

EA optimization was carried out by using a standard personal computer. For comparison, the conventional nonimaging Fresnel lens designed using the methodology proposed by Leutz [6] was included in the generation of the initial geometries. The relationship between *E* and the generation (iteration) step for *F* = 0.3 is shown in Fig. 4
. The calculation was terminated at the 100th generation. The value of *E* for the EA-generated Fresnel lens before the first generation is same as that for the conventional Fresnel lens (*E* = 0.60). As expected, the value of *E* increased with the generation step and became *E* = 0.75 at the 100th generation.

Figure 5
shows the energy distribution on the PV cell surface (between points A and B shown in Fig. 2). The concentration for three wavelengths, i.e., for the three refractive indices, is plotted; the average concentration for the three wavelengths is also determined. Figure 5(a) shows the energy distribution for the conventional Fresnel lens; this distribution represents a typical energy distribution influenced by chromatic aberration. In contrast, Fig. 5(b) and (c) shows the energy distribution for the EA-generated Fresnel lens at the 100th generation for *F* = 0.3 and *F* = 0.5, respectively. It can be clearly observed that the average energy distribution in the EA-generated Fresnel lens is more uniform than that in the conventional one. The uniformity of the energy distribution at each wavelength in the EA-generated lens is also better than that in the conventional one. For *F* = 0.5, a highly uniform concentration is observed. The optical efficiency of the conventional Fresnel lens is *η* = 84.0%, whereas that of the EA-generated Fresnel lens is *η* = 82.7% (*F* = 0.3) and 77.4% (*F* = 0.5). Optical efficiency-uniformity trade-off relation was examined by tuning *F* value and is clearly represented in Fig. 7
. Additional result for a wider incident angle *θ _{in}* = ± 0.5° was also plotted in Fig. 7. It is obvious that the wider acceptance angle resulted in the lower optical efficiency. Although a trade-off between uniformity and optical efficiency is unavoidable, one can obtain the best possible geometry with a minimal decline of optical efficiency by choosing an appropriate value of

*F*. It is unlikely that there exists a geometry that has an optical efficiency much higher than that of the present EA-generated geometry for the same uniformity; however, it is difficult to definitively confirm this fact. Because the actual amount of electricity generated depends on the characteristics of the installed PV cell, the value of

*F*should be selected according to the cell characteristics.

Figure 6
shows the geometry of the conventional and EA-generated Fresnel lenses corresponding to the energy distribution shown in Fig. 5. The geometry illustrated in this figure is a top-ranked geometry among 200 EA-generated geometries rated by index *E* at the 100th generation. The top 20 geometries differ slightly from each other. It should be noted that the *y*-scale in Fig. 6 is reduced to 1/10th of the original scale. The actual slopes of these geometries are gentler than those shown in Fig. 6. The EA-generated Fresnel lens has an irregular geometry; however, the authors confirmed that a Fresnel lens can still be fabricated using this geometry because limits can be imposed on the prism angles of the EA-generated Fresnel lens such that the lens can be practically fabricated using techniques such as pressing. By using EA optimization, such fabrication limits can easily be set on the automatically generated geometry.

## 5. Conclusion

The EA optimization method developed in this study can be used to automatically generate a Fresnel lens geometry with uniform flux concentration, without adopting trial-and-error procedures and without relying on considerably difficult theoretical approaches. The authors conclude that the EA is one of the powerful and versatile methods that can be used to select the best possible geometry according to one’s requirements. For example, one can apply this method to determine the best possible geometry that yields not only uniform concentration but also a wider acceptance angle. Furthermore, more complicated systems with multiple optical elements, e.g., a Fresnel lens combined with a secondary or higher optical element, can also be optimized by using this method. The generation of a three-dimensional geometry would enable the design of a wide range of practical solar concentrators. The effectiveness of the method used for the generation of such a three-dimensional geometry can be increased with the use of supercomputing.

## Acknowledgments

This work was partially supported by the Japan Science and Technology Agency under the FY2009 program Research for Promoting Technological Seeds (No.05-B09).

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