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Theoretical concentration limit and maximum annual optical efficiency of static/low-concentration CPV for horizontal integration to vehicle bodies

Daisuke Sato, Taizo Masuda, Ryota Tomizawa, and Noboru Yamada

Open Access Open Access

Abstract

This study proposes an extended theoretical formula to characterize the relationship between the geometrical concentration ratio Cg and arbitrary incidence-angle range to objectively understand the theoretical limit performance of vehicle-integrated concentrator photovoltaics (VICPVs), i.e., static and low-concentration CPVs installed on a horizontal surface. Theoretical analysis revealed that the maximum annual optical efficiency (maximum annual solar energy yield) can be achieved by selectively collecting the sunlight incident only from a specific incidence-angle range of θ1–θ2 (θ1 ≠ 0°), regardless of the geographical location, which is associated with the angular distribution of the annual solar energy on the concentrator aperture. Moreover, the results of the solar concentrator design based on the obtained formula clarified that the annual optical efficiencies of the optical systems with aspheric lens or dielectric-filled crossed compound-parabolic-concentrator lens were 64–89% of the theoretical limit depending on the location for Cg = 3.5×. Nevertheless, the concentrator design can be further improved for practical applications.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Concentrator photovoltaics (CPVs) use low-cost concentrator optics, such as mirrors or lenses, to collect sunlight onto small-area high-efficiency solar cells (e.g., III–V multijunction solar cells), which can achieve high electricity yields at reduced costs [1]. Originally, CPV systems were developed for large-scale power plants in geographical regions with high annual solar energy obtained from direct sunlight. Therefore, high-concentration CPV (HCPV) modules, with geometrical concentration ratios Cg of 100–1000×, are mounted on big-bonded sun-trackers [2], where Cg is defined as the ratio of the concentrator aperture area to the solar cell area. In addition to such utility-scale applications, micro-CPV designs employing miniaturized lenses (1–10 mm order) and microscale solar cells (0.1–1 mm order) have been proposed for small-scale applications with space constraints [3,4]. Additionally, the potential of various micro-CPV architectures, including III–V/Si hybrid CPVs [5–9], microtracking CPVs [10–13], and highly transparent (i.e., diffuse transmitting) CPVs [14] have been recognized. In recent times, a micro-CPV design employing static and low-concentration optics (Cg $\le $ 5×) has been studied for application on vehicle bodies, because it can provide high electricity yields at reduced costs [15]. Certain authors have designed static and low-concentration CPV (LCPV) modules and demonstrated daily-averaged module efficiencies of ∼21% (Cg = 3.5×) [16] and ∼25% (Cg = 1.76×) [17] under horizontal installation through outdoor tests using prototype modules. Furthermore, a stretchable micro-CPV module was developed, and the concept was validated via outdoor measurement of the daily-generated electricity yield using a prototype module fitted perfectly on a spherical surface with a curvature radius of 100 mm [18]. Moreover, the annual electricity yield is required to be further enhanced to improve the cost per unit of electricity. The power generation performance of the CPVs is mainly determined by the conversion efficiency of the solar cell and the optical efficiency of the concentrator, which is defined as the ratio of the solar energy collected on the solar cell to that incident on the concentrator aperture. Therefore, it is important to design concentrator optics exhibiting angular optical efficiency tailored to the angular distribution of the annual solar energy on the module installation surfaces, as well as to improve the solar cell efficiency. In this context, sun-tracked CPVs can maintain high optical efficiencies throughout the year by intensively collecting the sunlight that is incident from an approximately 0° incidence angle (i.e., perpendicular to the concentrator aperture) because a significantly large proportion of the annual solar energy on the sun-tracked surfaces is incident from low incidence angles.

Understanding the theoretical concentration limit and establishing the concentrator design techniques is vital to approach the theoretical limit performance. The theoretical concentration limit has been defined using the following formula [19]:

$$ C_{\mathrm{g}}=\frac{n_{\text {conc }}{ }^{2}}{n_{\text {air }}{ }^{2} \sin ^{2} \theta_{\mathrm{a}}} $$
where nconc and nair denote the refractive indices of the concentrator material and air, respectively, and θa [°] represents the acceptance half angle (i.e., incidence angle corresponding to the concentration limit). The above expression indicates that the optical efficiency is constant of 100% in the incidence angle range of 0°–θa [°] (depicted in Fig. 1 (a)) for optical systems having the solar cell optically coupled to the concentrator. Generally, CPV modules operate under inclined static installation (for LCPVs), with the inclination angle similar to the latitude of the installation site, or under sun-tracking (for HCPVs). In such operation modes, a larger portion of the annual solar energy incident on the concentrator aperture is obtained at a low incidence-angle range. Therefore, the theoretical concentration limit determined using Eq. (1) is considered as the ideal performance of the solar concentrator for the inclined static and sun-tracked CPVs. Moreover, several researchers have reported on various design schemes to achieve the theoretical limit performance [20–24]. Meanwhile, the vehicle-integrated CPV (VICPV) examined in this study is required to be installed on near-horizontal surfaces, such as roofs and engine hoods. Among the total annual solar energy incident on the horizontal surfaces, the amount of solar energy obtained from the midrange incidence angle accounts for a larger proportion than that coming from the low incidence-angle range [15]. In addition, the installation azimuth of the modules is not determined, as they are mounted on moving bodies. Given the angular distribution of the annual solar energy on the horizontal surfaces, the conventional theoretical limit (i.e., Eq. (1) for the targeted incidence-angle range of 0°–θa) does not represent the ideal angular optical efficiency of VICPVs. Instead, as shown in Fig. 1 (b), if the sunlight from the midrange incidence angle (i.e., θ1–θ2, θ1 ≠ 0°, θ2>θa) can be intensively collected, even if solar collection from low incidence-angle range (0°–θ1) is sacrificed, the annual solar energy yield would be enhanced. However, the theoretical concentration limit of the solar concentrator exhibiting such angular concentration characteristics has not been quantitively investigated. Therefore, a new design guideline that is distinct from the conventional theory of concentration limit is required for the optimal designing of concentrator optics for VICPVs.

 figure: Fig. 1.

Fig. 1. Comparison of targeted incidence-angle range to obtain maximum annual optical efficiency (i.e., maximum annual solar energy yield). (a) Conventional CPV (i.e., sun-tracked CPV or inclined static CPV). (b) Vehicle-integrated CPV (i.e., horizontally installed CPV).

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In this study, the theoretical limit performance of a VICPV (i.e., horizontally installed static LCPV) was examined. Based on Eq. (1), we derived an extended theoretical formula that is effective for an arbitrary incidence-angle range based on étendue conservation (i.e., energy conservation). Accordingly, for the design guideline of the highly efficient static solar concentrator suitable for vehicle-integrated applications, angular optical efficiencies corresponding to the maximum annual optical efficiency (maximum annual solar energy yield) were clarified for various combinations of angular distribution of annual solar energy, which is determined using the latitude and annual average of the diffuse-to-global ratio of the installation site, and the geometrical concentration ratio Cg. Furthermore, the performances of optical systems with Cg = 3.5× employing an aspheric lens or compound parabolic concentrator (CPC) lens, which are commonly used for static and low-concentration, (i.e., how close they are to the theoretical performance limit) are discussed following optical analysis via ray-tracing simulation.

2. Formularization of relationship between geometrical concentration ratio and arbitrary incidence-angle range

In this section, an extended theoretical formula is proposed relating the geometrical concentration ratio and arbitrary incidence-angle range.

The relationship between the geometrical concentration ratio Cg and acceptance angle θa, as expressed in Eq. (1), is based on the theorem of étendue conservation [25–27], which is equivalent to the entropy conservation in thermodynamics [28]. The requirement for maximum concentration, based on the second law of thermodynamics, is that the temperature is invariant over the propagation path of the light [29]. Therefore, the étendue conservation (i.e., entropy conservation) at maximum concentration (i.e., invariant temperature) condition is equivalent to energy conservation [30,31]. The étendue is considered as a volume in the phase space, which is expressed as the product of the cross-sectional area and solid angle of the light beam corresponding to the direction of its propagation [27]. In case an infinitesimal surface element with area dA immerged in a medium with refractive index n is crossed by the light beam incident from the azimuth angle φ and incidence angle (zenith angle) θ, or emits the light beam in the direction of azimuth angle φ and zenith angle θ, the étendue of the propagating light beam can be defined as follows [31,32]:

$${\rm d}G = {n^2}{\rm d}A\cos \theta {\rm d}\varOmega = {n^2}{\rm d}A\cos \theta \sin \theta {\rm d}\theta {\rm d}\varphi , $$
where dΩ denotes the infinitesimal solid angle on a spherical surface with unit radius, expressed as [33]:
$${\rm d}\varOmega = \sin \theta {\rm d}\theta {\rm d}\varphi . $$

As depicted in Fig. 2, an optical system with the solar cell optically coupled to the concentrator exit is assumed herein. The étendue of the light in the concentrator optics of CPV is preserved between the concentrator entrance (area: Aent, surrounding refractive index: nair) and at the exit (area: Acell, surrounding refractive index: nconc) [29]. In particular, the étendue of the light at the concentrator entrance Gent is the integral of dGent over the azimuth angle range of φ1–φ2 and incidence angle range of θ1–θ2, expressed as:

$${G_{\textrm{ent}}} = \int_{{\varphi _1}}^{{\varphi _2}} {\int_{{\theta _1}}^{{\theta _2}} {{\rm d}{G_{\textrm{ent}}}} = } {n_{\rm{air}}}^2{A_{\textrm{ent}}}\int_{{\varphi _1}}^{{\varphi _2}} {\int_{{\theta _1}}^{{\theta _2}} {\cos \theta \sin \theta {\rm d}\theta {\rm d}\varphi } }. $$

 figure: Fig. 2.

Fig. 2. Schematic of concentrator optics of CPV for definition of étendue.

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Similarly, the étendue of the light at the concentrator exit Gext can be expressed as:

$${G_{\textrm{ext}}} = \int_{{\varphi _1}}^{{\varphi _2}} {\int_{{\beta _1}}^{{\beta _2}} {{\rm d}{G_{\textrm{ext}}}} = {n_{\rm{conc}}}^2{A_{\rm{cell}}}} \int_{{\varphi _1}}^{{\varphi _2}} {\int_{{\beta _1}}^{{\beta _2}} {\cos \beta \sin \beta {\rm d}\beta {\rm d}\varphi } }. $$

Upon applying the étendue conservation between the concentrator entrance and exit (i.e., Gent = Gext), geometrical concentration ratio Cg is expressed as follows from Eqs. (4) and (5):

$$\begin{aligned} {C_{\rm g}} & = \frac{{{A_{\textrm{ent}}}}}{{{A_{\textrm{cell}}}}} = \frac{{{n_{\textrm{conc}}}^2\int_{{\varphi _1}}^{{\varphi _2}} {\int_{{\beta _1}}^{{\beta _2}} {\cos \beta \sin \beta {\rm d}\beta } {\rm d}\varphi } }}{{{n_{\textrm{air}}}^2\int_{{\varphi _1}}^{{\varphi _2}} {\int_{{\theta _1}}^{{\theta _2}} {\cos \theta \sin \theta {\rm d}\theta } } d\varphi }} = \frac{{{n_{\textrm{conc}}}^2\int_{0^\circ }^{360^\circ } {\int_{0^\circ }^{90^\circ } {\cos \beta \sin \beta {\rm d}\beta } {\rm d}\varphi } }}{{{n_{\textrm{air}}}^2\int_{0^\circ }^{360^\circ } {\int_{{\theta _1}}^{{\theta _2}} {\cos \theta \sin \theta {\rm d}\theta } } {\rm d}\varphi }}\\ & = \frac{{{n_{\textrm{conc}}}^2}}{{{n_{\textrm{air}}}^2({{\sin }^2}{\theta _2} - {{\sin }^2}{\theta _1})}} \end{aligned}, $$
where following intervals of angular integration were applied to maximize Cg: (1) The integration interval of the azimuth angle φ was set to 0–360° (i.e., φ1 = 0°, φ2 = 360°) for both Gent and Gext considering the following condition. Because the installation azimuth of the VICPV varies randomly, the concentrator should collect the sunlight that is incident from all azimuthal directions. Thus, the integration intervals of the azimuth angle for étendue of the concentrator entrance Gent (i.e., denominator of Eq. (6)) was set to 0–360°. To maximize Cg, the integration intervals of the azimuth angle for étendue of the concentrator exit Gext (i.e., numerator of Eq. (6)) needs to be maximized. Considering that the possible maximum azimuth angle range is 0–360°, the integration interval of the azimuth angle for Gext must be set to 0–360° for maximum concentration. (2) The integration interval of angle β for Gext was set to 0–90°(i.e., β1 = 0° and β2 = 90°), which is the maximum possible angle range for β. The maximization of the integral value of β implies minimizing the area of the concentrator exit Acell, thereby yielding the maximum possible concentration [29]. Equation (6) indicates the angular optical efficiency, in which the efficiency value is 100% for the incidence angle range of θ = θ1 –θ2, whereas the efficiency value is zero at θ < θ1 and θ > θ2, that is, the sunlight incident from outside of θ1 –θ2 cannot be collected on the solar cell. It is noted that Eq. (6) attains the same form as Eq. (1) for θ1 = 0° and θ2 = θa.

3. Maximum annual optical efficiency of VICPV at various geographical locations

Assuming that the LCPV modules, in which the silicone lenses encapsulate the solar cells, (such as those reported in the literature [16–18]) are installed on the vehicle roofs/engine hoods, the maximum annual optical efficiency associated with the amount of the maximum annual solar energy collected on the solar cells is clarified herein. Based on the extended theoretical formula (i.e., Eq. (6)), the optimal angular optical efficiency that can obtain the maximum annual optical efficiency was investigated for several combinations of the angular distribution of annual solar energy and geometrical concentration ratio Cg. For the static (i.e., non-sun-tracked) CPV, the influence of the incidence-angle dependence of the annual global solar energy should be considered to evaluate the annual solar collection performance, which is different point from the sun-tracked CPV. The global solar energy is the sum of the solar energies of the direct (beam) and diffuse components. The angular distribution of the annual direct solar energy varies with the annual solar orbit determined using the latitude. Furthermore, the ratio of the annual solar energy of the diffuse component to that of the global sunlight, referred to as the annual average of the diffuse-to-global ratio, influences the angular distribution of the annual global solar energy because the diffuse sunlight is incident from various directions. In this study, five cities with various combinations of latitude Φ and annual average of diffuse-to-global ratio γ were selected based on the criteria of being a potential market for PV-integrated electric vehicles; namely, these cities (a) are ranked in the top 200 populated cities in the world [34] and (b) belong to the top 100 countries of the number of registered vehicles, as reported by the World Health Organization (WHO) [35], and are presented as follows: (1) low Φ–low γ (Khartoum, Sudan), (2) low Φ–high γ (Manila, Philippines), (3) mid Φ–low γ (Phoenix, USA), (4) mid Φ–high γ (Tokyo, Japan), and (5) high Φ–high γ (Rotterdam, Netherlands). The geographical locations and irradiance conditions of the selected cities are summarized in Fig. 3 and Table 1. Accurate modeling of the angular solar energy distribution on the vehicle roofs/engine hoods requires the correct consideration of the curvature of the installed surfaces, shading effect by the surrounding objects, inclination/direction of the vehicle bodies, and angular intensity of the diffuse radiance. Therefore, such complicated factors were not considered in this study; instead, the following assumptions were made: (1) the module is installed on a horizontal surface, (2) the angular solar energy distribution is independent of the vehicle direction (installation azimuth), and (3) the angular intensity of the diffuse radiance is isotropic.

 figure: Fig. 3.

Fig. 3. Locations selected for analysis of annual solar energy. Map of annual sum of global horizontal irradiance (GHI) is obtained from: https://meteonorm.com/en/product/maps.

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Tables Icon

Table 1. Geographical coordinate and annual solar energy on horizontal surface at each location. Annual solar energies were calculated using irradiance data extracted from commercial meteorological database (METEOTEST, METEONORM 6.0). Diffuse-to-global ratio γ is defined as (GHI − DHI) / GHI.

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Figures 4–8(a) show the angular distribution of the annual global solar energy on the concentrator aperture of the VICPV installed on the horizontal surface Econc_Global(θ). The relationship between each solar energy component was defined using the angular annual solar energy of direct sunlight on the horizontal surface Eh_Direct(θ) and the annual solar energy of diffuse sunlight on the horizontal surface Eh_Diffuse, which are determined from the solar orbit (associated with azimuth and elevation angles of the sun) and irradiance data extracted from the commercial meteorological database (METEOTEST, METEONORM 6.0), which are presented as follows:

$${E_{{\rm{conc}}\_{\textrm{Global}}}} = {E_{{\textrm{conc}}\_{\textrm{Direct}}}} + {E_{{\textrm{conc}}\_{\textrm{Diffuse}}}}, $$
$${E_{{\textrm{conc}}\_{\textrm{Direct}}}} = \int_{0^\circ }^{90^\circ } {{E_{\textrm{h}\_{\textrm{Direct}}}}(\theta )\cos \theta } {\rm d}\theta = \int_{0^\circ }^{90^\circ } {{E_{{\textrm{conc}}\_{\textrm{Direct}}}}(\theta )} {\rm d}\theta, $$
$$\begin{aligned} {E_{{\textrm{conc}}\_{\textrm{Diffuse}}}} & = \int_\varOmega {\frac{{{E_{\textrm{h}\_{\textrm{Diffuse}}}}}}{{2\pi }}\cos \theta } {\rm d}\varOmega = \int_{0^\circ }^{360^\circ } {\int_{0^\circ }^{90^\circ } {\frac{{{E_{\textrm{h}\_{\textrm{Diffuse}}}}}}{{2\pi }}\cos \theta \sin \theta {\rm d}\theta {\rm d}\varphi } } \\ & = \int_{0^\circ }^{90^\circ } {{E_{\textrm{h}\_{\textrm{Diffuse}}}}\cos \theta \sin \theta {\rm d}\theta } = \int_{0^\circ }^{90^\circ } {{E_{{\textrm{conc}}\_{\textrm{Diffuse}}}}(\theta )\textrm{d}\theta } \end{aligned}, $$
where Econc and Eh represent the annual solar energy on the concentrator aperture and horizontal surface, respectively, and the subscripts Global, Direct, and Diffuse denote the global, direct, and diffuse irradiances, respectively. Subsequently, the angular distributions of Eh_Direct and Eh_Diffuse were multiplied with the concentrator aperture area directed to the incident light (i.e., cosine loss) to obtain the angular distributions of Econc_Direct and Econc_Diffuse, respectively. Additionally, the angular intensity of the diffuse radiance was assumed as isotropic over the unit hemispherical sky dome with a solid angle of 2π sr, expressed as Eh_Diffuse/2π [kWh/(m2·sr)]. The upper tiers of Figs. 4–8 (b)–(d) show angular optical efficiencies ηopt(θ) which can maximize the annual optical efficiency (defined as Eq. (11)) for Cg = 2.5×, 3.5×, and 4.5×. Their active incidence-angle ranges θ1–θ2 were determined as restricted by Eq. (6), where nconc = 1.41 (i.e., representative refractive index of transparent silicone) and nair = 1.0 (i.e., refractive index of air) were used as inputs, and were presented at the bottom right corner in the figures. In this case, the angular annual global solar energy on the concentrator aperture Econc_Global(θ) within θ1–θ2 (highlighted by the green area in the figures) is perfectly collected on the solar cell, whereas Econc_Global(θ) outside θ1–θ2 (gray area in the figures) is not collected. This case of optimal ηopt(θ) is referred to as “Case A”, and the corresponding results are highlighted in red. The annual optical efficiency ηopt_a is defined as the ratio of the annual solar energy collected on the solar cell (i.e., angular integral of Econc_Global(θ) weighted by ηopt(θ)) to the total annual global solar energy on the concentrator aperture Econc_Global, expressed as:
$${\eta _{{\textrm{opt}}\_{\rm a}}} = \frac{{\int_{0^\circ }^{90^\circ } {{\eta _{\textrm{opt}}}\textrm{(}\theta \textrm{)}{E_{{\textrm{conc}}\_{\textrm{Global}}}}\textrm{(}\theta \textrm{)}{\rm d}\theta } }}{{{E_{{\textrm{conc}}\_{\textrm{Global}}}}}} = \frac{{\int_{0^\circ }^{90^\circ } {{\eta _{\textrm{opt}}}\textrm{(}\theta \textrm{)}{E_{{\textrm{conc}}\_{\textrm{Global}}}}\textrm{(}\theta \textrm{)}{\rm d}\theta } }}{{\int_{0^\circ }^{90^\circ } {{E_{{\textrm{conc}}\_{\textrm{Global}}}}\textrm{(}\theta \textrm{)}{\rm d}\theta } }}. $$

 figure: Fig. 4.

Fig. 4. Theoretical concentration limit of VICPV at Khartoum (Low Φ−Low γ). (a) Angular distribution of annual solar energy on concentrator aperture Econc_Global(θ). Econc_Direct and Econc_Diffuse represent direct (beam) and diffuse components, respectively. (b)–(d) Optimal angular optical efficiencies for maximum annual optical efficiency based on extended formula of concentration limit (i.e., Case A) and angular optical efficiency based on conventional formula (i.e., Case B). Econc_Global(θ) within θ1–θ2 (green-colored area) is perfectly collected on the solar cell whereas Econc_Global(θ) outside θ1–θ2 (gray-colored area) is not collected. (b) Cg = 2.5×, (c) Cg = 3.5×, and (d) Cg = 4.5×. (e) Comparison of annual optical efficiencies calculated using Eq. (11) with data presented in (a)–(d).

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 figure: Fig. 5.

Fig. 5. Theoretical concentration limit of VICPV at Manila (Low Φ−High γ). (a) Angular distribution of annual solar energy on concentrator aperture Econc_Global(θ). Econc_Direct and Econc_Diffuse represent direct (beam) and diffuse components, respectively. (b)–(d) Optimal angular optical efficiencies for maximum annual optical efficiency based on extended formula of concentration limit (i.e., Case A) and angular optical efficiency based on conventional formula (i.e., Case B). Econc_Global(θ) within θ1–θ2 (green-colored area) is perfectly collected on the solar cell whereas Econc_Global(θ) outside θ1–θ2 (gray-colored area) is not collected. (b) Cg = 2.5×, (c) Cg = 3.5×, and (d) Cg = 4.5×. (e) Comparison of annual optical efficiencies calculated using Eq. (11) with data presented in (a)–(d).

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 figure: Fig. 6.

Fig. 6. Theoretical concentration limit of VICPV at Phoenix (Mid Φ−Low γ). (a) Angular distribution of annual solar energy on concentrator aperture Econc_Global(θ). Econc_Direct and Econc_Diffuse represent direct (beam) and diffuse components, respectively. (b)–(d) Optimal angular optical efficiencies for maximum annual optical efficiency based on extended formula of concentration limit (i.e., Case A) and angular optical efficiency based on conventional formula (i.e., Case B). Econc_Global(θ) within θ1–θ2 (green-colored area) is perfectly collected on the solar cell whereas Econc_Global(θ) outside θ1–θ2 (gray-colored area) is not collected. (b) Cg = 2.5×, (c) Cg = 3.5×, and (d) Cg = 4.5×. (e) Comparison of annual optical efficiencies calculated using Eq. (11) with data presented in (a)–(d).

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 figure: Fig. 7.

Fig. 7. Theoretical concentration limit of VICPV at Tokyo (Mid Φ−High γ). (a) Angular distribution of annual solar energy on concentrator aperture Econc_Global(θ). Econc_Direct and Econc_Diffuse represent direct (beam) and diffuse components, respectively. (b)–(d) Optimal angular optical efficiencies for maximum annual optical efficiency based on extended formula of concentration limit (i.e., Case A) and angular optical efficiency based on conventional formula (i.e., Case B). Econc_Global(θ) within θ1–θ2 (green-colored area) is perfectly collected on the solar cell whereas Econc_Global(θ) outside θ1–θ2 (gray-colored area) is not collected. (b) Cg = 2.5×, (c) Cg = 3.5×, and (d) Cg = 4.5×. (e) Comparison of annual optical efficiencies calculated using Eq. (11) with data presented in (a)–(d).

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 figure: Fig. 8.

Fig. 8. Theoretical concentration limit of VICPV at Rotterdam (High Φ−High γ). (a) Angular distribution of annual solar energy on concentrator aperture Econc_Global(θ). Econc_Direct and Econc_Diffuse represent direct (beam) and diffuse components, respectively. (b)–(d) Optimal angular optical efficiencies for maximum annual optical efficiency based on extended formula of concentration limit (i.e., Case A) and angular optical efficiency based on conventional formula (i.e., Case B). Econc_Global(θ) within θ1–θ2 (green-colored area) is perfectly collected on the solar cell whereas Econc_Global(θ) outside θ1–θ2 (gray-colored area) is not collected. (b) Cg = 2.5×, (c) Cg = 3.5×, and (d) Cg = 4.5×. (e) Comparison of annual optical efficiencies calculated using Eq. (11) with data presented in (a)–(d).

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Considering that in the present analysis, ηopt(θ) is 100% (i.e., unity) for the incidence angle range of θ = θ1–θ2, whereas ηopt(θ) at θ < θ1 and θ > θ2 is zero, Eq. (10) can be rewritten as follows:

$${\eta _{{\textrm{opt}}\_{\rm a}}} = \frac{{\int_{{\theta _1}}^{{\theta _2}} {{E_{{\textrm{conc}}\_{\textrm{Global}}}}\textrm{(}\theta \textrm{)}{\rm d}\theta } }}{{\int_{0^\circ }^{90^\circ } {{E_{{\textrm{conc}}\_{\textrm{Global}}}}\textrm{(}\theta \textrm{)}{\rm d}\theta } }}. $$

The resultant maximum (theoretical limit) ηopt_a are summarized in Figs. 4–8 (e). In addition, the angular optical efficiency, based on the conventional formula of concentration limit given by Eq. (1) (i.e., optical efficiency value is constant of 100% for θ = 0°–θa, θa: acceptance angle), and the resultant annual optical efficiency, calculated from Eq. (11) by substituting θ1 = 0° and θ2 = θa, are also shown in the lower tiers of Figs. 4–8 (b)–(d) and Figs. 4–8 (e), respectively, for comparison. This case of ηopt(θ) is referred to as “Case B”, and the corresponding results are drawn in black. Comparing the restriction of Eq. (6) for Case A and Eq. (1) for Case B, the active incidence-angle range in Case A is determined by both Cg and the angular distribution of annual solar energy (i.e., location), whereas that in Case B is determined by Cg only.

From ηopt(θ) of Case A for maximum annual optical efficiency depicted in the upper tiers of Figs. 4–8 (b)–(d), it can be said that the maximum annual optical efficiency can be achieved in the case that the sunlight within a specific incidence-angle range θ1–θ2 (θ1 ≠ 0°), which includes the incidence angle corresponding to the peak of Econc_Global(θ) is collected, regardless of the geographical location (i.e., latitude Φ and annual average of diffuse-to-global ratio γ) and geometrical concentration ratio Cg. This finding cannot be clarified from the conventional formula of the concentration limit (i.e., Eq. (1)). Comparing the annual optical efficiencies ηopt_a among those shown in Figs. 4–8 (e), the following tendency was observed: At low-latitude locations (i.e., Khartoum with low γ: Fig. 4, Manila with high γ: Fig. 5), the ηopt_a values were almost identical between the two ηopt(θ) cases (i.e., Case A vs. Case B) at both locations for every Cg. Therefore, the conventional ideal angular optical efficiency represented by Eq. (1) (i.e., Case B) can be considered as the design target. At the middle-latitude locations (i.e., Phoenix with low γ: Fig. 6, Tokyo with high γ: Fig. 7), the difference in ηopt_a between the two ηopt(θ) cases (i.e., Case A vs Case B) became larger compared to that in the case of low-latitude locations, especially for higher Cg. At the high-latitude and high-γ location (i.e., Rotterdam: Fig. 8), the difference in ηopt_a resulting from the variations in ηopt(θ) became more significant, and Case B ceased to represent the suitable angular characteristics for a VICPV. Furthermore, the maximum annual optical efficiency varied with the location, specifically, for the exhibited ranges of 86.6–93.1% for Cg = 2.5×, 62.7–78.4% for Cg = 3.5×, and 48.9–65.9% for Cg = 4.5×. In particular, for every Cg, the largest and smallest values of maximum annual optical efficiencies were observed at Khartoum and Tokyo, respectively.

Summarily, the maximum annual optical efficiency can be achieved by selectively collecting the sunlight only from a specific incidence-angle range of θ1–θ2 (θ1 ≠ 0°), regardless of the geographical location (i.e., latitude and annual average of diffuse-to-global-ratio). The optimal active incidence-angle range θ1–θ2 varies with the geographical location. Moreover, the maximum annual optical efficiencies vary depending on the geographical location as well. These results cannot be evaluated from the conventional formula of the concentration limit. Thus, the appropriate evaluation of the VICPV performance exhibiting such angular optical efficiency requires modifying the conception of the acceptance angle. Specifically, the definition of the acceptance angle ought to be extended to include the effectiveness in an arbitrary incidence-angle range of θ1–θ2, as determined by Eq. (6), including the conventional acceptance angle range of 0°–θa. Hereafter, the acceptance angle for the regular CPV (i.e., θ1 = 0°) and that for the VICPV (i.e., θ1 ≠ 0°) are referred to as “on-axis” and “off-axis” acceptance angles, respectively.

4. Concentrator design and performance analysis

In Section 3, we clarified that the maximum (theoretical limit) annual optical efficiency can be achieved by selectively collecting the sunlight incident from a specific incidence-angle range of θ1–θ2 (θ1 ≠ 0°). In context, the following question should be raised: “Can a feasible concentrator be developed with such angular concentration characteristics?” In this section, we examine the solar collection performance of the concentrator optics with Cg = 3.5× employing a polynomial aspheric lens or compound parabolic concentrator (CPC) lens [36] through ray-tracing simulation. These lenses have been commonly used and extensively studied for static and low-concentration applications. Although the aspheric lenses have been examined for vehicle-integrated applications in the previous works of the authors [15–17], it has not been clarified as to how close the performance of these lenses are to the theoretical limit. Therefore, their annual solar collection performances are quantitively discussed in comparison with the theoretical limit. As described in Section 3, optimal angular optical efficiency maximizing annual optical efficiency varies depending on the location, even though Cg is identical, which indicates that “ideally” the lens geometry should be tailored for each location to obtain the maximum annual optical efficiency. However, from the viewpoint of practical use, such as fabrication cost, a common lens is desirable for application in all locations even if the performance is slightly compromised. Therefore, in this study, we examined only a single type of lens design for the aspheric and CPC lenses.

The geometry of the world-common aspheric lens was determined as follows: out of the five lenses designed to maximize the annual optical efficiency defined in Eq. (10) for each location (the details of the lens design and ray-tracing simulation are described in next paragraph), a lens with the smallest location dependence of the annual optical efficiency was adopted. The simulated annual optical efficiency for the five types of location-specific lens designs applied to all locations (i.e., total 25 cases by round robin) exhibited a range of 46.5–51.7%. Especially, the aspheric lens optimally designed for Rotterdam exhibited the smallest location dependence of the annual optical efficiency, i.e., 49.5 ± 0.5% at all locations, which can be acceptable for practical use. Therefore, the aspheric lens for Rotterdam was considered as the world-common lens in this study.

The details of the aspheric lens design and ray-tracing simulation are as follows. The angular distribution of the annual solar energy (Figs. 4–8 (a)) was considered as Econc_Global(θ) for calculation of annual optical efficiency, and the optimal lens geometries were obtained through an optical analysis based on the ray-tracing simulation conducted using commercial software (Synopsys, Inc.; LightTools 9.1.0) with an established lens design method [16]. A ray-tracing simulation model of an optical system with an aspheric lens is presented in Fig. 9. The overall lens model comprised a 3 × 3 array of unit lenses considering the cross-talk effect with the adjacent lenses. In particular, the solar cell (thickness: 0.185 mm) was placed only for the lens located at the center of the array. To date, static and low-concentration optics have been applied mainly to building-integrated applications with horizontal or vertical installations. In this application, asymmetric lens designs, such as those reported in [21,37], are the most suitable because the angular dependence of annual solar energy from both the azimuth and zenith directions are specifically determined for the specified installation surfaces. Meanwhile, in the vehicle-integrated applications, the concentrator must equally collect the sunlight incident from various azimuthal directions because installation azimuth of the VICPV modules (i.e., direction of the vehicle bodies) varies randomly. Accordingly, a four-fold symmetrical aspheric lens was designed herein, that is, the side lengths of the unit lens and solar cell were set as 1.87 mm and 1.0 mm, respectively, to simulate a micro-CPV structure for Cg = 3.5×. Moreover, the simulated optical system was placed at the center of a hemispherical light source, and the light irradiation area was irradiated by the rays emitted from this light source. The angular intensity of the emitted rays was assumed to be independent on the azimuth angle and was weighted using Econc_Global(θ) at each location, as depicted in Figs. 4–8 (a). The proportion of the emitted ray spectral intensity was assumed to be identical to that of the AM 1.5G standard spectrum (300–2000 nm). The Fresnel reflection loss at the air-lens interface and absorption loss in the lens were considered based on wavelength dispersion. In addition, the solar cell was assumed as a black body (i.e., a perfect light absorber). The geometry of the polynomial aspheric lens can be expressed as follows:

$$z = \frac{{c({x^2} + {y^2})}}{{1 + \sqrt {1 - (1 + k){c^2}({x^2} + {y^2})} }} + \sum\limits_{j = 2}^{66} {{\alpha _j}{x^m}{y^n}}, $$
where c, k, and αj represent the lens-shape parameters, i.e., the curvature, conic constant, and polynomial coefficient, respectively. Note that the value of j can be determined as {(m + n)2 + m + 3n}/2 + 1 on the satisfaction of m + n ≦ 10. Upon considering that the lens geometry is four-fold symmetric, that is, axisymmetric with respect to both the x- and y-axes, the lens geometry can be expressed as an even function; thus, terms including an odd order (x, y, xy, x3, x2y, xy2, y3, etc) are eliminated. Therefore, the second term on the right-hand side of Eq. (12) can be expanded as follows:
$$\begin{aligned} \sum\limits_{j = 2}^{66} {{\alpha _j}{x^m}{y^n}} & = ({\rm X}2){x^2} + ({\rm Y}2){y^2} + ({\rm X}4){x^4} + ({\rm X}2{\rm Y}2){x^2}{y^2} + ({\rm Y}4){y^4}\\ & + ({\rm X}6){x^6} + ({\rm X}4{\rm Y}2){x^4}{y^2} + ({\rm X}2{\rm Y}4){x^2}{y^4} + ({\rm Y}6){y^6} \end{aligned}, $$
where (XmYn) represents the polynomial coefficient corresponding to the term xmyn. Herein, the terms with more than six degrees were eliminated to reduce the number of optimized parameters as they slightly influenced the concentration performance. Thereafter, the lens-shape parameters and lens thickness maximizing the annual optical efficiency were searched. The resultant lens-geometry parameters optimized for angular distribution of annual solar energy at Rotterdam are listed in Table 2. This lens geometry is four-fold symmetric to the z-axis, therefore, certain polynomial coefficients (e.g., (X2) and (Y2)) have the same value.

 figure: Fig. 9.

Fig. 9. A ray-tracing simulation model of optical system employing aspheric lens with coordinate system and light irradiation area.

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Table 2. Optimized parameters of the aspheric lens geometries for Rotterdam.

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Meanwhile, in this study, a dielectric-filled crossed CPC (CCPC) lens with Cg = 3.5× was also designed, wherein the aperture area was the same as that of the aspheric lenses (i.e., 1.87 mm side length/unit) for a fair comparison with the aspheric lens. The lens material was assumed as transparent silicone, and the ray-tracing simulation was performed under the simulation model composition and conditions identical to those prevailing for the aspheric lens. The acceptance angle and peak optical efficiency are in a trade-off relation for a constant Cg; their optimal combination to maximize the annual optical efficiency was accordingly searched. As a result, the annual optical efficiency was the largest at all locations when the lens thickness was 2.270 mm, which is approximately twice as large as that of the aspheric lenses.

The simulated three-dimensional incidence-angle dependences of the optical efficiency of the aspheric lens optical system and CCPC lens optical system are presented in Fig. 10 (a). The on-axis acceptance angles θa, i.e., the incidence angle θ at which the optical efficiency reduces to 90% of that at θ = 0°, are indicated on the horizontal axis with corresponding colors. Unfortunately, both lenses did not exhibit the ideal angular optical efficiency (drawn in green in Fig. 10 (a), which corresponds to Case A in Fig. 8 (c), representing the selective solar collection only from a middle incidence-angle range). The aspheric lens optical system exhibited a lower optical efficiency at θ = 0° and wider on-axis acceptance angle, whereas the CCPC lens optical system showed a higher optical efficiency at θ = 0° and narrower on-axis acceptance angle. Specifically, the ηopt (θ = 0°) and θa of the aspheric lens optical system were 48.9% and 63.7°, respectively, and similarly, those of the CCPC lens optical system were 95.6% and 34.2°, respectively. The annual optical efficiencies ηopt_a achieved by these optical systems are evaluated using Eq. (10), in which the angular distribution of the annual solar energy (i.e., Figs. 4–8 (a)) and simulated ηopt(θ) shown in Fig. 10 (a) are input. The calculated ηopt_a are presented in Fig. 10 (b) with the theoretical limit values at each location (i.e., Case A for Cg = 3.5× in Figs. 4–8 (e)). As mentioned above, regardless of the location, the ηopt_a of the aspheric lens system was within 49.5 ± 0.5%, which corresponded to a relative value of 64% (Khartoum) to 79% (Tokyo) of the theoretical limits. In contrast, the ηopt_a of the CCPC lens system decreased with the increasing Φ and γ, because the solar energy amount incident from a higher incidence angle accounted for a larger proportion in Econc_Global. Nevertheless, ηopt_a achieved a relative value of 76% (Rotterdam) to 89% (Manila) of the theoretical limit values, which were higher than those of the aspheric lens system at all locations. The reason for the lower ηopt_a of the aspheric lens is due to the flat trench area that is inactive for the light concentration, which is formed between the active lens islands to minimize the shading effect on the adjacent lens islands for higher incidence-angle light. This inactive flat area corresponds to the thinner part formed at the outer circumference of the unit lenses (Fig. 10 (a)) and accounts for 22% of the lens area (i.e., 1.87 × 1.87 mm2/unit), which requires solutions to improve the annual optical efficiency.

 figure: Fig. 10.

Fig. 10. (a) Simulated 3D incidence-angle dependences of optical efficiency of optical system with Cg = 3.5× employing aspheric lens and that employing CCPC lens. Overview of unit cell-lens pair and on-axis acceptance angles θa, i.e., incidence angle θ at which the optical efficiency reduces to 90% of that at θ = 0°, are indicated. The ideal angular optical efficiency for the angular distribution of annual solar energy at Rotterdam is also presented. (b) Annual optical efficiencies achieved by aspheric and CCPC lenses, and comparison with theoretical limits. Theoretical limit values correspond to annual optical efficiencies of “Case A” for Cg = 3.5×, as depicted in Figs. 4–8 (e).

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Summarily, for the aspheric or CCPC lenses, a larger amount of sunlight with an incidence angle of nearly 0° was collected on the solar cells; therefore, producing the ideal angular concentration characteristics could not be achieved. Thus, further studies should be performed to approach the ideal angular optical efficiency and the annual optical efficiency limit.

5. Conclusion

The theoretical limit performance of a VICPV (i.e., static and low-concentration CPV installed on a horizontal surface) was characterized using a theoretical formula that determined the relationship between the geometrical concentration ratio Cg and arbitrary incidence-angle range in a three-dimensional space according to the étendue conservation (i.e., energy conservation). Based on the theoretical formula, the optimal angular optical efficiency yielding the maximum annual optical efficiency (maximum yearly solar energy yield) was investigated for optical systems with Cg = 2.5×, 3.5×, and 4.5× at five geographical locations, where the latitude and annual average of diffuse-to-global ratio are different. The theoretical analysis results revealed the following: the maximum annual optical efficiency can be achieved by selectively collecting the sunlight incident only from a specific incidence-angle range of θ1–θ2 (θ1 ≠ 0°), regardless of the location and Cg; the maximum annual optical efficiency and its corresponding active incidence-angle range vary with the location and Cg; the variation of annual optical efficiency with the angular optical efficiency becomes more significant with an increase in the latitude, annual average of diffuse-to-global ratio, and Cg. To properly evaluate the performance of the VICPV exhibiting such angular optical efficiencies, the off-axis acceptance angle should be defined, wherein the corresponding incidence-angle range is θ1–θ2 (θ1 ≠ 0°), apart from the conventional on-axis acceptance angle, wherein the corresponding incidence-angle range is 0°–θa.

In cases of the aspheric or dielectric-filled CCPC lenses, a larger amount of sunlight with an incidence angle of nearly 0° was inevitably collected on the solar cells; therefore, the ideal angular concentration characteristics could not be achieved. The annual optical efficiencies of the aspheric lens systems were ∼50%, regardless of the location, which corresponded to 64–79% of the theoretical limit. In contrast, the annual optical efficiency of the CCPC lens system varied in the range of 54–69% depending on the location, which corresponded to 76–89% of the theoretical limit. Therefore, the CCPC lens can exhibit better annual solar collection performance than the aspheric lens at every location; nevertheless, further improvements can be made to the concentrator design.

Thus, future studies will focus on the investigation of concentrator geometries and optical system structures to approach the ideal angular optical efficiency and the annual optical efficiency limit, eventually establishing its design technique. In addition, the effect of lens temperature variation (causing thermal deformation and variation of refractive index) on the annual solar collection performance will be also examined by accumulating dynamic (i.e., instantaneous) solar collection performance over a one-year period.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)
Fig. 1.
Fig. 1. Comparison of targeted incidence-angle range to obtain maximum annual optical efficiency (i.e., maximum annual solar energy yield). (a) Conventional CPV (i.e., sun-tracked CPV or inclined static CPV). (b) Vehicle-integrated CPV (i.e., horizontally installed CPV).

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Fig. 2.
Fig. 2. Schematic of concentrator optics of CPV for definition of étendue.

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Fig. 3.
Fig. 3. Locations selected for analysis of annual solar energy. Map of annual sum of global horizontal irradiance (GHI) is obtained from: https://meteonorm.com/en/product/maps.

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Fig. 4.
Fig. 4. Theoretical concentration limit of VICPV at Khartoum (Low Φ−Low γ). (a) Angular distribution of annual solar energy on concentrator aperture Econc_Global(θ). Econc_Direct and Econc_Diffuse represent direct (beam) and diffuse components, respectively. (b)–(d) Optimal angular optical efficiencies for maximum annual optical efficiency based on extended formula of concentration limit (i.e., Case A) and angular optical efficiency based on conventional formula (i.e., Case B). Econc_Global(θ) within θ1–θ2 (green-colored area) is perfectly collected on the solar cell whereas Econc_Global(θ) outside θ1–θ2 (gray-colored area) is not collected. (b) Cg = 2.5×, (c) Cg = 3.5×, and (d) Cg = 4.5×. (e) Comparison of annual optical efficiencies calculated using Eq. (11) with data presented in (a)–(d).

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Fig. 5.
Fig. 5. Theoretical concentration limit of VICPV at Manila (Low Φ−High γ). (a) Angular distribution of annual solar energy on concentrator aperture Econc_Global(θ). Econc_Direct and Econc_Diffuse represent direct (beam) and diffuse components, respectively. (b)–(d) Optimal angular optical efficiencies for maximum annual optical efficiency based on extended formula of concentration limit (i.e., Case A) and angular optical efficiency based on conventional formula (i.e., Case B). Econc_Global(θ) within θ1–θ2 (green-colored area) is perfectly collected on the solar cell whereas Econc_Global(θ) outside θ1–θ2 (gray-colored area) is not collected. (b) Cg = 2.5×, (c) Cg = 3.5×, and (d) Cg = 4.5×. (e) Comparison of annual optical efficiencies calculated using Eq. (11) with data presented in (a)–(d).

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Fig. 6.
Fig. 6. Theoretical concentration limit of VICPV at Phoenix (Mid Φ−Low γ). (a) Angular distribution of annual solar energy on concentrator aperture Econc_Global(θ). Econc_Direct and Econc_Diffuse represent direct (beam) and diffuse components, respectively. (b)–(d) Optimal angular optical efficiencies for maximum annual optical efficiency based on extended formula of concentration limit (i.e., Case A) and angular optical efficiency based on conventional formula (i.e., Case B). Econc_Global(θ) within θ1–θ2 (green-colored area) is perfectly collected on the solar cell whereas Econc_Global(θ) outside θ1–θ2 (gray-colored area) is not collected. (b) Cg = 2.5×, (c) Cg = 3.5×, and (d) Cg = 4.5×. (e) Comparison of annual optical efficiencies calculated using Eq. (11) with data presented in (a)–(d).

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Fig. 7.
Fig. 7. Theoretical concentration limit of VICPV at Tokyo (Mid Φ−High γ). (a) Angular distribution of annual solar energy on concentrator aperture Econc_Global(θ). Econc_Direct and Econc_Diffuse represent direct (beam) and diffuse components, respectively. (b)–(d) Optimal angular optical efficiencies for maximum annual optical efficiency based on extended formula of concentration limit (i.e., Case A) and angular optical efficiency based on conventional formula (i.e., Case B). Econc_Global(θ) within θ1–θ2 (green-colored area) is perfectly collected on the solar cell whereas Econc_Global(θ) outside θ1–θ2 (gray-colored area) is not collected. (b) Cg = 2.5×, (c) Cg = 3.5×, and (d) Cg = 4.5×. (e) Comparison of annual optical efficiencies calculated using Eq. (11) with data presented in (a)–(d).

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Fig. 8.
Fig. 8. Theoretical concentration limit of VICPV at Rotterdam (High Φ−High γ). (a) Angular distribution of annual solar energy on concentrator aperture Econc_Global(θ). Econc_Direct and Econc_Diffuse represent direct (beam) and diffuse components, respectively. (b)–(d) Optimal angular optical efficiencies for maximum annual optical efficiency based on extended formula of concentration limit (i.e., Case A) and angular optical efficiency based on conventional formula (i.e., Case B). Econc_Global(θ) within θ1–θ2 (green-colored area) is perfectly collected on the solar cell whereas Econc_Global(θ) outside θ1–θ2 (gray-colored area) is not collected. (b) Cg = 2.5×, (c) Cg = 3.5×, and (d) Cg = 4.5×. (e) Comparison of annual optical efficiencies calculated using Eq. (11) with data presented in (a)–(d).

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Fig. 9.
Fig. 9. A ray-tracing simulation model of optical system employing aspheric lens with coordinate system and light irradiation area.

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Fig. 10.
Fig. 10. (a) Simulated 3D incidence-angle dependences of optical efficiency of optical system with Cg = 3.5× employing aspheric lens and that employing CCPC lens. Overview of unit cell-lens pair and on-axis acceptance angles θa, i.e., incidence angle θ at which the optical efficiency reduces to 90% of that at θ = 0°, are indicated. The ideal angular optical efficiency for the angular distribution of annual solar energy at Rotterdam is also presented. (b) Annual optical efficiencies achieved by aspheric and CCPC lenses, and comparison with theoretical limits. Theoretical limit values correspond to annual optical efficiencies of “Case A” for Cg = 3.5×, as depicted in Figs. 4–8 (e).

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Tables (2)

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Table 1. Geographical coordinate and annual solar energy on horizontal surface at each location. Annual solar energies were calculated using irradiance data extracted from commercial meteorological database (METEOTEST, METEONORM 6.0). Diffuse-to-global ratio γ is defined as (GHI − DHI) / GHI.

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Table 2. Optimized parameters of the aspheric lens geometries for Rotterdam.

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Equations (13)

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C g = n conc  2 n air  2 sin 2 ⁡ θ a
d G = n 2 d A cos ⁡ θ d Ω = n 2 d A cos ⁡ θ sin ⁡ θ d θ d φ ,
d Ω = sin ⁡ θ d θ d φ .
G ent = ∫ φ 1 φ 2 ∫ θ 1 θ 2 d G ent = n a i r 2 A ent ∫ φ 1 φ 2 ∫ θ 1 θ 2 cos ⁡ θ sin ⁡ θ d θ d φ .
G ext = ∫ φ 1 φ 2 ∫ β 1 β 2 d G ext = n c o n c 2 A c e l l ∫ φ 1 φ 2 ∫ β 1 β 2 cos ⁡ β sin ⁡ β d β d φ .
C g = A ent A cell = n conc 2 ∫ φ 1 φ 2 ∫ β 1 β 2 cos ⁡ β sin ⁡ β d β d φ n air 2 ∫ φ 1 φ 2 ∫ θ 1 θ 2 cos ⁡ θ sin ⁡ θ d θ d φ = n conc 2 ∫ 0 ∘ 360 ∘ ∫ 0 ∘ 90 ∘ cos ⁡ β sin ⁡ β d β d φ n air 2 ∫ 0 ∘ 360 ∘ ∫ θ 1 θ 2 cos ⁡ θ sin ⁡ θ d θ d φ = n conc 2 n air 2 ( sin 2 θ 2 − sin 2 θ 1 ) ,
E c o n c _ Global = E conc _ Direct + E conc _ Diffuse ,
E conc _ Direct = ∫ 0 ∘ 90 ∘ E h _ Direct ( θ ) cos ⁡ θ d θ = ∫ 0 ∘ 90 ∘ E conc _ Direct ( θ ) d θ ,
E conc _ Diffuse = ∫ Ω E h _ Diffuse 2 π cos ⁡ θ d Ω = ∫ 0 ∘ 360 ∘ ∫ 0 ∘ 90 ∘ E h _ Diffuse 2 π cos ⁡ θ sin ⁡ θ d θ d φ = ∫ 0 ∘ 90 ∘ E h _ Diffuse cos ⁡ θ sin ⁡ θ d θ = ∫ 0 ∘ 90 ∘ E conc _ Diffuse ( θ ) d θ ,
η opt _ a = ∫ 0 ∘ 90 ∘ η opt ( θ ) E conc _ Global ( θ ) d θ E conc _ Global = ∫ 0 ∘ 90 ∘ η opt ( θ ) E conc _ Global ( θ ) d θ ∫ 0 ∘ 90 ∘ E conc _ Global ( θ ) d θ .
η opt _ a = ∫ θ 1 θ 2 E conc _ Global ( θ ) d θ ∫ 0 ∘ 90 ∘ E conc _ Global ( θ ) d θ .
z = c ( x 2 + y 2 ) 1 + 1 − ( 1 + k ) c 2 ( x 2 + y 2 ) + ∑ j = 2 66 α j x m y n ,
∑ j = 2 66 α j x m y n = ( X 2 ) x 2 + ( Y 2 ) y 2 + ( X 4 ) x 4 + ( X 2 Y 2 ) x 2 y 2 + ( Y 4 ) y 4 + ( X 6 ) x 6 + ( X 4 Y 2 ) x 4 y 2 + ( X 2 Y 4 ) x 2 y 4 + ( Y 6 ) y 6 ,
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