1. IntroductionBecause of the increasing demand on energy, solar energy is attracting increasing attention due to its significant advantages. As one of the most important branches of the solar energy industry, photovoltaic (PV) electricity generation has developed faster and faster in recent years.[1–3] However, the efficiency of PV cells is low and the cost is high in spite of many years development. It still remains expensive relative to competing forms of energy sources.[4] In order to decrease the cost of PV electricity generation, light concentration is an important way to achieve power gain.[5–7] Under the guidance of this idea, the luminescent solar concentrator (LSC) appeared.
As early as 1976, the idea of an optical waveguide to collect sunlight was proposed by Lambe and Weber,[8] and it experienced rapid development in the following decades.[9–15] LSC is composed of PV cells and fluorescent dyes which are uniformly distributed in the waveguide that consists of transparent material, as shown in Fig. 1. Solar light shining in the LSC is absorbed and reemitted by dyes. The new photons reemitted by dyes whose emission angle is smaller than the critical angle will escape and the rest will be trapped in the waveguide through total internal reflection and be redirected to the PV cells attached to the edges of the waveguide. Unlike traditional solar concentrators, the LSC concentrates both diffuse light and direct light, which makes a sun tracking system unnecessary.[16–18] Due to the narrow absorption band of existing fluorescent materials and other reasons, the efficiency of LSC is still low.[11,19–21] In order to collect more photons, Mansour developed an LSC with bottom-facing cells (BMP-LSC) that can absorb both direct sunlight and wave-guided light,[22] which is considered as a more effective way for cost reduction.[23–26]
Researchers have paid a lot of attention to modeling LSCs.[27–33] Theoretical calculation is an important means to analyze and optimize LSCs. At present, two main kinds of theoretical models, the thermodynamic model and the light-tracing model, have been developed.[18] The thermodynamic model is based on the energy transfer between mesh points in the LSC, and the whole volume of LSC is integrated.[27] Light-tracing for LSC uses a basic light-tracing principle, which means that a photon is traced until it leaves the LSC throught escape or absorption. Whether a physical process has occurred or not is determined by comparing the probability of a single physical process with a randomly generated number. Finally, the fate of numerous photons was tallied. The Monte Carlo light-tracing model affords greater flexibility so that it is widely used in complex systems. Carrascosa et al. simulated monolayer and multilayer LSC films doped with Rholdamine6G and Flourol555 dyes,[28] and the results agree with the experiment. Wilton et al. simulated the PbSe quantum dot LSC by ray tracing model.[31] Leow et al. simulated the fluorescence transmission process of BMP-LSC by two-dimensional light tracing model,[32] in which the effects of optical waveguide thickness, cell width and spacing on the efficiency of the LSC cell are analyzed. In short, to better optimize the LSC parameters and increase the efficiency of LSC, the simulation of the LSC is very necessary.
Comparing with edge-mounted PV cell LSCs, more parameters are involved in the BMP-LSC, so optimizing the system experimentally is time consuming and difficult. Meanwhile, by the theoretical calculation, it can also be more convenient to analyze and calculate the loss mechanism of fluorescence. However, there has been less research on large size LSCs, although the large size LSC is what we will use eventually.[15,34] In this paper, a three-dimensional (3D) Monte Carlo program is written in Matlab and used to compute the performance of a large-size LSC. The contribution of each optical loss mechanism and the effects of PV cells area, PV cells position, LSC thickness, and dyes concentration were simulated.
2. Theoretical modelSeventy-two PV cells were arranged in a Tic-Tac-Toe layout as shown in Fig. 2. PMMA was chosen as the transparent material of LSC for its low cost as well as low extinction coefficient.[27,28,35] Single crystalline silicon solar cells, which have the band gap width of 1.1 eV, were attached to the bottom of the waveguide. A mass ratio of 2:1 fluorescent dyes Lumogen Red 305[10,31,34] and Yellow 083 were mixed and uniformly distributed in the optical waveguide. Figure 3 shows the emission and absorption spectrums of Red 305 and Yellow 083.
The Monte Carlo light-tracing model of LSCs was created using matlab to evaluate the device performance. A minimum of 100000 photons were created in each run. The light-tracing algorithm flow chart is depicted in Fig. 4.
AM1.5 was used as the initial emission light and only the photons which have wavelength smaller than 1124 nm will be launched. Rest photons cannot be absorbed by cells as well as dyes, so have no effect on the LSC. The photons are given a direction perpendicular to the LSC top face, and a starting position on the LSC top surface evenly over the entire panel. The probability of reflection at the LSC top face is calculated by Fresnel’s equation of the incident photons at the air–PMMA interface
where
represents the refractive index of PMMA and
represents the refractive index of air. The probability of top face reflection is 0.04.
The probability of absorption and the absorption location is determined by the Beer–Lambert law
where
s represents the photons transmission distance and
C is the concentration of dyes.
and
, the absorption coefficients of Red 305 and Yellow 083, are measured as functions of wavelength in our laboratory with a Jobin Yvon LUOROLOG-3-TAU spectrometer.
ζ1 is a random variable that is uniformly distributed between [0, 1]. When
s is the smaller than the thickness of the waveguide, it can be considered that the photon is absorbed by dyes. Otherwise, the photon is unabsorbed. The absorption spectrums of Red 305 and Yellow 083 have been given in the previous paper. Whether the surviving (not absorbed by dyes as well as PMMA) photons reach cells is determined by their position and cells. When the photons is absorbed by dyes, if
it can be considered that the photon is absorbed by Red 305. Otherwise, the photon is absorbed by Yellow 083.
When the photon is absorbed by the dyes, it is reemitted only if
where
is the quantum yield of dyes and
ζ3 is a random variable. Next, the photon’s new wavelength is sampled randomly from the measured emission spectrum, as if the luminescence were memoryless. The remitted direction of the photons is determined isotropically, so the azimuth (
) is uniform from 0 to
, as well as the polar angle (
) according to
Whether the photon transfers or escapes is determined by comparing the critical angle of reflection (
) and θ. The critical angle is determined by the refractive index of the transparent medium:
If the photon’s polar angle is larger than the critical angle, it will be trapped in the waveguide through total internal reflections, and then be redirected to the PV cells. Whether the photons reach the cell is determined by each point of total reflection.
When the photon is determined to have reached the cell, the path length will be calculated from the azimuth, polar angles, and total reflection times by trigonometry:
At the same time, the absorption path-length (
) is determined by the Beer–Lambert law at the emission wavelength
where
α is the absorption coefficient of PMMA. When
s1 is smaller than
s2, the photon is considered to have reached the solar cell. Otherwise, the photon is considered to be absorbed.
When
the photon is considered to be absorbed by Red 305; else if
the photon is considered to be absorbed by Yellow 083; otherwise, the photon is considered to be absorbed by PMMA. If the photon is considered to be reabsorbed, the position can be calculated and a new photon will be emitted according to the PQ and emission spectrum.
Finally, the photons that reach the cell are tallied, and the optical efficiency is calculated by
where
is the photons collected by cell and
is the total launched photons.
In the terms of practicability, the power efficiency draws more attention. The power of cells can be calculated by
where the fill factor
FF and the open-circuit voltage
are assumed to be constant. The short-circuit current
is proportional to photons which reach the cells. In this model, we can get
since
represents the photons that reached the cells.
It can be considered that the power efficiency is proportional to optical efficiency
where
is the photoelectric conversion efficiency of PV cells, the specific figure of which is 17% for single crystalline silicon cells.
In order to facilitate the analysis, power gain (G) was defined by
where
is the total area of all the cells and
is the front surface area of the LSC.
3. Results and discussion3.1. Effect of dye concentrationDye concentration is a very important parameter in the process of solar light collection and fluorescence transmission. On one hand, high concentration of the dye can increase the absorption of the sunlight, thereby increasing the fluorescence. On the other hand, the high concentration of dye will increase reabsorption, which is expected to result in the decrease of the efficiency. Specific attention was given to simulate a rectangular LSC with dimension of 122 cm× 61 cm× 0.7 cm, on which 72 stripe cells with 7.8 cm× 2 cm were attached and the distance between cells is 20.3 cm, as shown in Fig. 2.
As the concentration of dyes increases from 5 ppm to 160 ppm, the power gain raises steeper and then declines gently (Fig. 5(a)). The maximum gain of 1.2 appears at 80 ppm, which is equivalent to the energy efficiency of 3.2%. The fluorescence and direct light contributions to the power gain are shown in Fig. 5(b). The fluorescence contribution increases with the concentration at first and then saturates, while the direct light contribution varies in a reverse way. The direct light is the main source of the power gain, as the photons absorbed by dye are very limited although two kinds of dyes were used. It is why the bottom cell is important as it can absorb direct light. Only little photons are absorbed by Yellow 083 (1/6 of Red 305) (Fig. 5(c)), however it still contributes a little to the power gain since the absorption range of Yellow 083 is mainly at the absorption gap of Red 305. In the whole fluorescence transmission of LSC, the reabsorption is a problem that attracts a lot of attention.[19,36] It is also a very complex and complicated physical process. In order to facilitate the analysis, only the relationship between the time of iteration reabsorption and fluorescence concentration is discussed. As shown in Fig. 5(d), reabsorption increases as concentration increases. When the concentration is larger than 80 ppm, the photons absorbed by dye change slowly while the reabsorption is still increasing fast. This is mainly due to the tailing of Red 305 absorption spectrum which extends to the emission spectrum. Although the tailing part of the absorption spectrum is weak, with the increase of the dye concentration, the reabsorption capacity increases and causes strong reabsorption.
The contribution of each optical loss mechanism in LSCs can be calculated using the Monte Carlo simulation via dividing lost photons into different types based on when the loss event occurred. In principle, optical losses in a BMP-LSC can be attributed to six mechanisms: unabsorbed loss, escaping cone loss, non-radiative, interface reflection, PMMA absorption, and escape from sides. Incident photons reflecting off the top face of the LSC is always 4% and irrespective of dye concentration. The loss ratio of the other mechanisms is given in Fig. 6. Unabsorbed losses is the biggest part. This is attributed to the narrow absorption bandwidth of dyes. With the increase of concentration, more absorption, reabsorption, and thereby more reemission occurred. The latter causes more photons to leave the LSC through an escape cone. Extending Stokes shift of dyes and increasing the refractive index of the waveguide are solutions to this issue. The optical losses caused by PMMA absorption and non-radiative contribute only a small part to the total loss, which is mainly thanks to the low absorption coefficient of PMMA and high quantum efficiency of Red 305 and Yellow 083. It is worth noticing that less fluorescence escapes from the side, which indicates that the cells arrangement is reasonable.
3.2. Effect of LSC thicknessBy increasing the thickness of LSC while keeping the total of dyes in the LSC, the concentration of dye decreases so that the reabsorption is suppressed, and the fluorescence launched by the dye does not change. The power gain under different thickness of the LSC is given in the Fig. 7.
From Fig. 7, we found that the power gain increases with the increase of the thickness. However, the power gain increases slowly and eventually decreases when the thickness increases to 12 mm. In the process of fluorescence propagating along the waveguide, the probability of photon arrival to the solar cell is related with the distance of two landing positions and the width of cell. The thicker the LSC is, the larger the distance is, which leads to the increase of optical-path before reaching a cell and therefore raises the loss. The optimal thickness is 12 mm.
3.3. Effect of cell widthA simple and effective way to improve efficiency is increasing the width of the cell, which however causes the increase of cost and decrease of power gain. Shown in Fig. 8 are the power efficiency, power gain of direct light and fluorescence change as the increase of cell width.
We found that the power efficiency increases almost linearly with the width while the power gain decreases sharply as width increases. This is due to the fact that the direct light as well as the fluorescence both increase with the increase of cell width, as shown in Figs. 8(c) and 8(d). As the direct light is the main source of the output power, the power efficiency shows a nearly linear increase with the increase of cell width. Compared with direct light, the fluorescence reaching the cells changes slightly. For fluorescence, with the increase of cell width, the reabsorption is suppressed by reducing the number of reflections and the loss from side is also suppressed (Fig. 9). However, the reabsorption decreases slightly since large reabsorption occurred in a quite small optical path (Fig. 10). With increasing width, fluorescence average to a single cell area decreases, causing decrease of the power gain.
3.4. Effect of LSC sizeTo enhance the output power of LSC, cell distance is also a controllable parameter. A series of LSC with increasing size as well as cell distance were simulated using Monte Carlo ray tracing and the result is shown in Fig. 11.
With the size increase, the output power of LSC as well as power gain increase (Fig. 11(a)), but the power efficiency decreases rapidly (Fig. 11(b)). Larger area of LSC increases the amount of photon captured by dyes which in turn raises the LSC output power. With the increase of LSC size and the corresponding decrease of cells coverage, the ratio of unabsorbed loss rises rapidly and thereby the power efficiency decreases. More than that, photons captured at the position further away from the PV cells experience higher probabilities of reabsorption, which is also a disadvantage.
