Numerical and experimental research of the characteristics of concentration solar cells

Abstract

The development of automatic tracking solar concentrator photovoltaic systems is currently attracting growing interest. High concentration photovoltaic systems (HCPVs) combining triple-junction InGaP/lnGaAs/Ge solar cells with a concentrator provide high conversion efficiencies. The mathematical model for triple-junction solar cells, having a higher efficiency and superior temperature characteristics, was established based on the one-diode equivalent circuit cell model. A paraboloidal concentrator with a secondary optic system and a concentration ratio in the range of 100X–150X along with a sun tracking system was developed in this study. The GaInP/GalnAs/Ge triple-junction solar cell, produced by AZUR SPACE Solar Power, was also used in this study. The solar cells produced by Shanghai Solar Youth Energy (SY) and Shenzhen Yinshengsheng Technology Co. Ltd. (YXS) were used as comparison samples in a further comparative study at different concentration ratios (200X–1000X). A detailed analysis on the factors that influence the electrical output characteristics of the InGaP/lnGaAs/Ge solar cell was conducted with a dish-style concentrating photovoltaic system. The results show that the short-circuit current (I_sc) and the open-circuit voltage (V_oc) of multi-junction solar cells increases with the increasing concentration ratio, while the cell efficiency (η_c) of the solar cells increases first and then decreases with increasing concentration ratio. With increasing solar cell temperature, I_sc increases, while V_oc and η_c decrease. A comparison of the experimental and simulation results indicate that the maximum root mean square error is less than 10%, which provides a certain theoretical basis for the study of the characteristics of triple-junction solar cell that can be applied in the analysis and discussion regarding the influence of the relevant parameters on the performance of high concentration photovoltaic systems.

Graphical abstract

Keywords

concentration / three-junction solar cell / mathematical model / electrical properties / solar energy

1 Introduction

The total global energy generated in 2013 reached 23000 TWh. The power generation using oil, natural gas, and other fossil fuels accounted for approximately 68% of this total, while renewable energy sources contributed less than 6%. To limit global warming below 2°C per year, the carbon dioxide emissions must decrease by 90% by 2050 through intense deployment of renewable energy. According to the scenarios forecast by IEA [1], renewable energy could reach a global share of 65% by 2050. The International Energy Agency (IEA) predicts that the solar process heat capacity of China’s industry will be 179 GW_th in 2020, 435 GW_th in 2030, and 1125 GW_th in 2050 [2]. Concentrating photovoltaic systems (CPVs) are able to offer numerous economic advantages by integrating concentration solar cells with inexpensive concentrators. Considerable efforts have been invested in developing high-efficiency solar cells based on III–V materials which provide excellent performance under highly concentrated light [3–7]. The usable spectrum of solar energy could be extended through the use of multi-junction solar cells, which are more efficient than single-junction cells [8,9]. CPVs can achieve significant cost reductions through advanced manufacturing for semiconductors with economically viable high concentration ratio mirrors. When the series resistance becomes a limiting factor, the efficiency of multi-junction solar cells will increase with the concentration ratio.

Multi-junction solar cells have been utilized in concentrating systems in which sunlight is focused by lenses or mirrors onto a much smaller area. By being incorporated into concentrating systems, multi-junction solar cells could remain superior in terms of efficiency while simultaneously reducing their associated costs [10–12]. Numerous studies have been conducted to investigate the performance of multi-junction cells incorporated into concentrating systems at different concentrating ratios. The effect of light intensity on the electrical properties of triple-junction solar cells, such as efficiency and power, has been studied by many researchers. The effects of light intensity and spot uniformity on the properties of triple-junction solar cells have been explored by Li et al. [13]. The experimental results indicate that the power and the efficiency are proportional to the light intensity, and reach a maximum of 3.039 W, 3.812 W, and 4.298 W for a concentration ratio of 300X, 400X, and 500X, respectively. Besides, it is shown that the efficiency and power are 3.53% and 0.819 W lower, respectively, than those of a solar cell without a homogenizer. Mi et al. [14] have developed a HCPV system consisted of 96 independent modules containing triple-junction solar cells. The primary optic is a Fresnel lens at a 500X geometric concentration ratio while the secondary optic device is a core-shaped reflective tube. Research has revealed that the direct normal irradiation determines the power of an HCPV system. Moreover, a larger performance ratio is observed for dry seasons in comparison to wet seasons, which is mainly caused by the power inversion process instead of the effect of cell temperature on performance. Yamaguchi et al. [15] have shown that the efficiency of concentration solar cell does not always increase with increasing concentration ratios. A peak efficiency of 36.5% is obtained at 200X. The solar cell efficiency decreases after this point with an increasing concentration ratio. Aiken et al. [16] have analyzed the efficiency loss factors for InGaP/lnGaAs/Ge three-junction solar cells at a concentration ratio of 520X. The results indicate that elevated operating temperatures and light intensity non-uniformity are the two main loss mechanisms. The efficiency of the module decreases by 1.4% when the light is not well homogenized. Some researchers have studied the temperature characteristics of triple-junction solar cells. The effects of temperature on the open-circuit voltage (V_oc) of InGaP/lnGaAs/Ge solar cells have been analyzed by Nishioka et al. [17]. The results indicate that V_oc exhibits a nearly linear decrease as cell temperature increases. The temperature coefficient of V_oc was –0.0060 V/K in the low-temperature range (303.15 K–373.15 K) and –0.0048 V/K in the high-temperature range (443.15 K–513.15 K). Kinsey et al. [18] have applied a simple diode model to analyze the temperature coefficient of V_oc for InGaP/lnGaAs/Ge solar cells to predict the performance of multi-junction solar cells. It is observed that the voltage temperature coefficient decreases with an increasing concentration ratio while the efficiency temperature coefficient decreases. Almonacid et al. [19] have investigated the relationship between the cell temperature and the cell efficiency (h_c). It is indicated that the efficiency of the solar cell decreases with an increasing cell temperature. For instance, the efficiency of a concentration solar cell decreases from 39% to 31% as the cell temperature increases from 298.15 K to 373.15 K at a concentration ratio of 200X. Nishioka et al. [20] have investigated the performance of InGaP/lnGaAs/Ge three-junction solar cell at concentration ratios of 1X–200X and cell temperatures of 298.15 K–413.15 K. The results indicate that the temperature coefficients of V_oc, h_c, and FF decrease as the concentration ratio increases at a cell temperature of 298.15 K, and an increasing in short-circuit current (I_sc). Kinsey et al. [21] have investigated the performance of InGaP/lnGaAs/Ge three-junction solar cells exposed to different concentration ratios (1X–1000X) and cell temperatures (273.15 K–393.15 K). The results demonstrate that both h_c and the fill factor decrease as a function of the concentration ratio after an initial increase. It is also interesting to note that h_c and the fill factor decrease with an increase in cell temperature. For a constant solar cell temperature, the fill factor approaches a maximum value at a concentration ratio of 200X, and h_c reaches a maximum value at a concentration of 500X. Various mathematical models have been implemented to predict the performance of triple-junction solar cells under varying conditions. A Random-Forest (RF) model of the temperature of a multi-junction solar cell has been established by Renno and Petito [22]. Furthermore, an artificial neural network (ANN) model and a linear regression model (LRM) have also been researched to compare with the results of the RF model. The theoretical results are found to be in accord with those of the experiment. The results show that the root mean square error (RMSE) of the LRM, ANN, and RF models are 4.29, 3.47, and 1.95, respectively, indicating that the calculation results of the RF model are more accurate than those of the other two. Based on ANNs, a new Matlab/Simulink model for a InGaP/lnGaAs/Ge solar cell has been established by Rezk and Hasaneen [23]. The simulation results reveal that the daily generation of the HCPV module could be improved from 3.37 kWh to 3.75 kWh. Based on a double diode equivalent, a mathematical model of the triple-junction solar cell has been developed by Catelani et al. [24] using the Matlab software package. The simulation results are experimentally validated which indicate that the model could be used to predict the performance of a triple-junction solar cell. The performance of InGaP/lnGaAs/Ge solar cells at a concentration ratio of 1X has been investigated by Nishioka et al. [25], which demonstrates that the error between the theoretical and measured values of the open-circuit voltage is less than 1.5% at a set temperature. Considering the effects of the irradiance and spectral properties of the incident sunlight, a mathematical model has been developed by Kurtz et al. [26], which can be used to calculate the efficiency of multi-junction solar cells. The results suggest that the use of a detailed balance model could make it possible to calculate ideal efficiencies. Gómez-Gil et al. [27] have developed a novel method in which the global horizontal irradiation and photovoltaic geographical information system database would be used to predict the power of HCPV. It is found that the monthly performance ratio varies for both silicon solar cells and three-junction solar cells, and the deviation between the theoretical and experimental output is no more than 2% for annual energy production and varies between 5.6% and 16.1% for monthly energy production. A mathematical model for the performance of the triple-junction solar cells has been developed by Wang et al. [28]. The simulation results are validated experimentally through the use of a paraboloidal concentrator with a secondary optic system. The results indicate that a power of 1.52 W/cm² and an average efficiency of 29.3% are achieved for a direct solar irradiance and cell temperature of 450 W/m² and 338.05 K, respectively. Moreover, the experimental results for V_oc and I_sc conform reasonably well with the predictions of the mathematical model.

The literature review above indicates that Refs. [13–16] have focused on the electrical properties of the triple-junction solar cells, but the appropriate mathematical models are not established. References [17–21] have studied the temperature coefficient of triple-junction solar cells. However, few studies have been conducted so far on the electrical property parameters, such as I_sc, V_oc, and h_c. The studies in Refs. [22,23] have been conducted in the laboratory and thus could not accurately reflect the practical working characteristics of a triple-junction solar cell. Adequate theoretical models have been developed and validated using laboratory experimentation as described in Refs. [24–27]. Owing to the impact of numerous factors such as solar irradiance, air temperature, and wind speed, the performance of InGaP/lnGaAs/Ge three-junction solar cell varies dramatically between experiments performed outdoors and in the laboratory. In addition, although the theoretical model has been validated by outdoor experimentation in Ref. [28], the effect of temperature on I_sc, V_oc, and h_c at particular concentration ratios has not yet been investigated. It is important to note that a larger concentration ratio is capable of exerting a more significant effect. In this study, based on a dish-style concentration photovoltaic system incorporating a dual-axis tracking system, the impact of temperature and concentration ratio on the electrical properties of InGaP/lnGaAs/Ge three-junction solar cell has been thoroughly analyzed in outdoor conditions. In addition, a heat pipe is employed to cool the solar cell and a homogenizer is applied to keep the irradiance uniform and avoid bright or dim spots. Furthermore, a mathematical model of the characteristics of a three-junction solar cell is established, and the simulation results are validated using the experimental results. More importantly, a comparative study at a higher concentration ratio (200X–1000X) is performed to further verify the effectiveness of the mathematical model.

2 Experiment and simulation

2.1 Experimental setup

A schematic diagram of the experimental apparatus, composed of a concentrator, a solar cell module, and a solar tracking-system, is shown in Figs. 1 and 2. There are two stages in the concentration system used in this study. The first stage of light concentration involves a revolving parabolic mirror acting as the primary concentrator with a concentration ratio of 200X. A prismatic homogenizer is placed over the cell to ensure the uniformity of the irradiance and avoid spots reflected from the concentration stages onto the surface of the cell and to eliminate the negative effects on the photovoltaic cell associated with deviation in the angle of incidence. It thus improved the conversion efficiency. The sun tracking system which is provided with a dual-axis solar tacking system controlled the azimuth angle and the elevation angle with an accuracy of±0.5°. The heat pipe is constructed of copper and is used in the experiment to remove heat from the three-junction solar cell.

The structure of the InGaP/InGaAs/Ge triple-junction cell studied in this paper is illustrated in Fig. 3. Based on metal organic vapor phase epitaxy, the bottom cell is grown on p-type Ge. The top subcell is formed of InGaP and the middle subcell is formed of InGaAs, both of which are lattice-matched with the Ge bottom subcell. The p-AlGaAs/n-InGaP tunnel junction connects the InGaP subcell with the InGaAs subcell, and the p-GaAs/n-GaAs tunnel junction connects the InGaAs subcell and the Ge subcell. The electrodes are assembled using evaporation [29]. The experiments are conducted on a 1cm² InGaP/InGaAs/Ge three-junction solar cell equipped with an anti-reflective coating (TiO_x/Al₂O₃), and front contact metal tabs as shown in Fig. 1. Manufactured with a rated power of 25 mW at one sun (AM1.5G, 298.15 K) by Shenzhen Yin Shenxuan Energy Technology Co. Ltd., the thickness of the front contact and back contact of the three-junction solar cell is 5–7 mm and 1.5–2.5 mm respectively. Both the front contact and back contact are constructed using an Ag core covered with Au. Such cells are typically optimized to achieve maximum performance under the AM1.5D+ circumsolar spectrum. Concentration ratios of 100X–150X are applied in the experiment.

The external quantum efficiency (EQE) measurements are performed based on the ReRa SpeQuest Quantum efficiency system. The data are acquired using the ReRa Photor software package. A light source consisting of Xenon and halogen is used to satisfy the primary wavelengths that make up the solar spectrum. A monochromator is used to generate quasi-monochromatic light along with a chopper for intensity modulation, which generates a test light source of variable wavelength while a continuous bias light is used to test the cell under operating conditions [30]. The EQE of the InGaP/lnGaAs/Ge three-junction solar cell is depicted in Fig. 4.

The reference patterns used in this study are the InGaP/lnGaAs/Ge three-junction solar cells produced by the AZUR SPACE Solar Power Company, Shanghai Solar Youth Energy Technology Co. Ltd., and Shenzhen Yin Shenxuan Energy Technology Co. Ltd. The particular model numbers of the solar cells used in this study are 3C35/175-100, SY-TJ10, and YS-TJ1010, respectively, whose performance parameters are listed in Table 1.

2.2 Measuring equipment

The experiments were conducted on the roof of a building of the University of Shanghai for Science and Technology (Shanghai, China). Figure 5 displays the arrangement of the test system. The temperature of the module was measured using thermocouples (T type,±0.1K) and acquired using the Key sight 34970A data logger. The direct solar radiation was measured using a pyrheliometer (modelTBS-2-2), manufactured by Jinzhou Sunshine Technology Co. Ltd. The pyrheliometer was fixed to the module to measurements the radiation every 30 s along with the weather data such as wind speed, environmental temperature, humidity, and global solar radiation. An I-V tracer (IT8500, ITECH Electronic Co. Ltd.) was used to record the output and voltage at the maximum power point of the photovoltaic module every 60 s.

2.3 Mathematical model of InGaP/lnGaAs/Ge three-junction solar cell

As shown in Fig. 6 [26], the characteristics of a three-junction InGaP/lnGaAs/Ge solar cell can be expressed as an equivalent circuit. The top subcell (InGaP), middle subcell (InGaAs), and bottom subcell (Ge) are all lattice-matched. Here J_sc and J_L are the short circuit current and the load current respectively, which are opposite in direction to the forward bias currents of the diodes “D.” V is the voltage and R_sh is the shunt resistance. The series resistance is represented by R_s and includes both the contact resistance and the resistance of the base and diffused regions.

In general, the diode current characteristics are described by [31]

(1)J_{\mathrm{L}}=J_{\mathrm{sc},i}-J_{\mathrm{o,}i}({\mathrm{e}}^{\frac{q(V_i+J_{\mathrm{L}}A{R}_{s,i})}{n_i{k}_{\mathrm{B}}T}}-\mathrm{1})-{\displaystyle \frac{V_i+J_{\mathrm{L}}A{R}_{s,i}}{A{R}_{\mathrm{sh},i}}},

where i represents the subcell number (1= top, 2= medium, and 3= bottom), J_o is the diode reverse saturation current density, n is the ideality factor, k is the Boltzmann constant, q is the electric charge, A is the cell area, and T is the absolute temperature. The reverse saturation current is given by [32]

(2)J_{\mathrm{o},i}=k_i{T}^{\left(\mathrm{3}+\raisebox{1ex}{${\gamma }_i$}\!\left/ \!\raisebox{-1ex}{$\mathrm{2}$}\right.\right)}{\mathrm{e}}^{-\frac{E_{\mathrm{g}}}{n_i{k}_{\mathrm{B}}T}},

where E_g is the band gap while k_i and g_i are constants.

As the energy band gap (E_g) decreases slightly with an increase in temperature, Eq. (3) applies [18].

(3)E_{\mathrm{g}}=E_{\mathrm{g}}(\mathrm{0})-{\displaystyle \frac{\alpha {{\rm T}}^{\mathrm{2}}}{{\rm T}+\sigma }},

where a and s are material dependent constants. I_sc increases slightly with temperature. If the shunt resistance is sufficiently large to be neglected, the voltage can be derived from Eq. (1) as

(4)V={\displaystyle \frac{k_{\mathrm{B}}T}{q}}\left[n_{\mathrm{1}}\ln \left({\displaystyle \frac{J_{\mathrm{sc,1}}-J_{\mathrm{L}}}{J_{\mathrm{o,1}}}}\mathrm{+1}\right)+n_{\mathrm{2}}\ln \left({\displaystyle \frac{J_{\mathrm{sc,2}}-J_{\mathrm{L}}}{J_{\mathrm{o,2}}}}\mathrm{+1}\right)+n_{\mathrm{3}}\ln \left({\displaystyle \frac{J_{\mathrm{sc,3}}-J_L}{J_{\mathrm{o,3}}}}\mathrm{+1}\right)\right]-J_{\mathrm{L}}A{R}_s,

and V_oc can be calculated by setting J_L = 0.

(5)V_{\mathrm{oc}}={\displaystyle \frac{k_{\mathrm{B}}T}{q}}\left[n_{\mathrm{1}}\ln \left({\displaystyle \frac{J_{\mathrm{sc,1}}}{J_{\mathrm{o,1}}}}\mathrm{+1}\right)+n_{\mathrm{2}}\ln \left({\displaystyle \frac{J_{\mathrm{sc,2}}}{J_{\mathrm{o,2}}}}\mathrm{+1}\right)+n_{\mathrm{3}}\ln \left({\displaystyle \frac{J_{\mathrm{sc,3}}}{J_{\mathrm{o,3}}}}\mathrm{+1}\right)\right].

According to Eq. (4), h_c can be described as

(6)P=J_{\mathrm{L}}VA.

When dP/dJ_L = 0, the mathematical expression of the best power point is expressed as

(7)\begin{array}{l}{\displaystyle \frac{k_{\mathrm{B}}T}{q}}\left[n_{\mathrm{1}}\ln \left({\displaystyle \frac{J_{\mathrm{sc,1}}-J_{\mathrm{L}}}{J_{\mathrm{o,1}}}}\mathrm{+1}\right)+n_{\mathrm{2}}\ln \left({\displaystyle \frac{J_{\mathrm{sc,2}}-J_{\mathrm{L}}}{J_{\mathrm{o,2}}}}\mathrm{+1}\right)+n_{\mathrm{3}}\ln \left({\displaystyle \frac{J_{\mathrm{sc,3}}-J_{\mathrm{L}}}{J_{\mathrm{o,3}}}}\mathrm{+1}\right)\right]\\ -J_L\left[{\displaystyle \frac{k_{\mathrm{B}}T}{q}}\left({\displaystyle \frac{n_{\mathrm{1}}}{J_{\mathrm{sc,1}}-J_{\mathrm{L}}+J_{\mathrm{o,}\mathrm{1}}}}+{\displaystyle \frac{n_{\mathrm{2}}}{J_{\mathrm{sc,}\mathrm{2}}-J_{\mathrm{L}}+J_{\mathrm{o,2}}}}+{\displaystyle \frac{n_{\mathrm{3}}}{J_{\mathrm{sc,}\mathrm{3}}-J_{\mathrm{L}}+J_{\mathrm{o,}\mathrm{3}}}}\right)+\mathrm{2A}{R}_{\mathrm{s}}\right]\mathrm{=0}.\end{array}

According to Eq. (7), the current density J_m can be calculated. The voltage of the maximum power point can be obtained by substituting J_m into Eq. (4).

The selection of materials for each sub-cell was based on lattice-matching, current-matching, and optoelectronic property matching.

As each the sub-cell was connected in series, the same current passed through each junction. The materials were arranged in the order of decreasing band gap (E_g), which allowed sub-bandgap light to be transmitted to lower sub-cells. Therefore, adequate band-gaps had to be selected to ensure that the current generation in each of the sub-cells allow them to achieve current matching.

According to Eq. (3), the resulting derivative can be expressed as [18]

(8){\displaystyle \frac{\partial E_{\mathrm{g}}}{\partial T}}=-{\displaystyle \frac{\mathrm{2}\alpha T}{T+\sigma }}+{\displaystyle \frac{\alpha T^{\mathrm{2}}}{{(T+\sigma )}^{\mathrm{2}}}}.

Figure 7 plots the change in band gap for each sub-cells vs. temperature using Eq. (3). The band gap for each of the sub-cells decreased with increasing temperature. The variation in band gap with temperature indicated by Eq. (8) was higher for InGaP than for InGaAs. For instance, performing a calculation using Eq. (8) at 343 K yielded a value of –3.74 × 10^-4 eV/K for InGaP and –4.64 × 10^-4 eV/K for InGaAs. As the temperature increased, the change in the band gap of the top subcell was smaller than that of the middle subcell, causing the current in the top subcell to approximate that of the middle subcell.

The following values are typically utilized to justify the selection of materials for three-junction solar cells: InGaP for the top sub-cell (E_g = 1.8eV), InGaAs for the middle sub-cell (E_g = 1.4 eV), and Ge for the bottom sub-cell (E_g = 0.67 eV). The selection of Ge is mainly due to its lattice constant, robustness, low cost, abundance, and ease of production [30].

3 Results and analysis

3.1 Parameter definition

The concentration ratio is defined as the ratio of I_sc after and before concentration, and can be expressed as

(9)C={\displaystyle \frac{I_{\mathrm{scx}}}{I_{\mathrm{sc}}}}.

I_sc increased as the cell temperature increased. Assuming that the temperature coefficient I_sc increased linearly with increasing concentration ratio, the calculated value of I_cal could be modified as

(10)I_{\mathrm{cal}}=C\times I_{\mathrm{sc}}+C\times {\displaystyle \frac{\mathrm{d}{I}_{\mathrm{sc}}}{\mathrm{d}T}}\times (T_{\mathrm{c}}-\mathrm{298}),

where {\displaystyle \frac{\mathrm{d}{I}_{\mathrm{sc}}}{\mathrm{d}T}} is the temperature coefficient of I_sc at 1X.

The efficiency of photovoltaic cells is calculated as the maximum power divided by direct radiation intensity as

(11){\eta }_{\mathrm{c}}={\displaystyle \frac{J_{\mathrm{m}}{V}_{\mathrm{m}}}{C{E}_{\mathrm{d}}}},

where C is the concentration ratio and E_d is the direct solar radiation intensity.

The error between the theoretical values and the measured values in this study was analyzed in terms of RMSE [28] as

(12)e_{\mathrm{RMS}}=\sqrt{{\displaystyle \frac{\mathrm{1}}{nm}}{\displaystyle \sum }_{i\mathrm{=1}}^n{\displaystyle \sum }_{j\mathrm{=1}}^m{\left({\displaystyle \frac{X_{ij}-{\tilde{X}}_{ij}}{{\tilde{X}}_{ij}}}\right)}^{\mathrm{2}}},

where n and m are the concentration ratio and the cell temperature, respectively; X_{ij} is the measured value, and {\tilde{X}}_{ij} is the theoretical value.

3.2 Results and discussion

3.2.1 Experiment results

Figures 6 to 8 show the daily performance of the InGaP/InGaAs/Ge three-junction solar cells used in the field test. According to the experimental data available, the ranges for the temperature and concentration ratio were 298 K≤T≤345 K and 120≤C≤150, respectively. In addition, the measured values were compared with the theoretical ones.

It is observed from Fig. 8 that I_sc increased with the rising concentration ratio and was closely associated with the increase in cell temperature, which was consistent with the theoretical predications of Eq. (2). This was mainly caused by the difference in the photon flux density which increases with the increase in concentration ratio. In addition, the increase in I_sc is somewhat influenced by the reduced band gap. The part of the photon energy that exceeds the band gap was dissipated as heat, increasing the cell temperature. According to Eq. (3), the band gap of the sub-cell decreased as the cell temperature rose. The temperature coefficient of E_g was negative, agreeing well with the simulated results of Eq. (8). The band gap of the solar cell was narrowed by increasing cell temperature, which produced more photons having sufficient energy. This caused more electron-hole pairs to be generated. Consequently, I_sc increased gradually, but in a nonlinear fashion. Both the measured and calculated I_sc values presented a similar trend, but the calculated values are larger. This was mainly attributed to the fact that the temperature coefficient of I_sc used in Eq. (10) was assumed to vary linearly with the concentration ratio. However, this assumption proved inaccurate, especially for conditions of high concentration. Discrepancies were thus noted between the theoretical and experimental values. Nevertheless, the differences were small at low concentration ratios but increased significantly at high concentration ratios, coinciding with an increase in the cell temperature. The RMSE was 4.83% for a concentration of 120X, and 9.49% for a concentration of 150X. The error between the measured and theoretical values for the same concentration ratio also demonstrated an increasing trend. At a concentration ratio of 150X, the measured values of I_sc rose from 1.239 A at 302.83 K to 1.53 A at 340.02 K, while the theoretical values of I_sc varied from 1.269 A to 1.826 A. The increasing error was mainly caused by the cell temperature increasing at high concentration ratio. The process of recombination of electron-hole pairs changes with cell temperature diversification, causing J_o,I to vary, thus producing the error.

V_oc shows an opposing trend in Fig. 9, where V_oc decreases with increasing of cell temperature. Such a phenomenon resulted from the decrease in the band gaps of the solar cell. The band gaps generally decreased with an increase in the cell temperature, and the increase in minority-carrier diffusion length, as well as the shift in the optic absorption edge energy, causing V_oc to gradually become small. The calculated and measured values of V_oc demonstrated the same trend, but the calculated values were larger than the measured ones. As the cell temperature increased, the difference between the calculated and measured values decreased. As the concentration ratio started to increase, however, the errors between the calculated and measured values increased. The RMSE was 0.8% for a concentration of 120X and 1.9% for a concentration of 150X. However, the measured values of V_oc decreased from 2.84 V at 302.83 K to 2.76 V at 340.02 K for a concentration of 150X while the theoretical values of V_oc dropped from 2.96 V to 2.74 V. This might be caused by the auger recombination becoming the major recombination processes as the temperature increased along with high concentration conditions. This could have generated a larger variety in the cell carrier concentration. Thus, the error between the theoretical and experimental data became larger as the concentration ratio increased.

Figure 10 shows the h_c of a three-junction solar cell as a function of cell temperature for different concentration ratios. h_c decreased with rising temperature. This was mainly caused by the band gap of the sub-cell narrowing as the cell temperature increased. With the cell temperature rising, J_o,i increased exponentially according to Eq. (2). This was significantly greater than the rate of rise of I_sc and the maximum power point current (I_m) as shown in Eq. (4). Consequently, the maximum power point voltage (V_m) decreased, thereby decreasing h_c with increasing cell temperature according to Eq. (6). It also became clear that a high concentration ratio was preferable to increase the efficiency of multi-junction PV cells. This was due to the fact that I_sc linearly grows and V_oc logarithmically increases according to Eqs. (5) and (8) with the rising concentration ratio. The calculated and measured values of h_c demonstrated the same trend. However, the calculated values proved larger than the measured values. The difference between the calculated and measured values grew as the cell temperature and concentration ratio increased. The RMSE was 0.8% for a concentration of 120X and 3.3% for a concentration of 150X. However, the measured values of h_c reduced from 34.56% at 302.83 K to 30.19% at 340.02 K for a concentration of 150X, while the theoretical values of h_c changed from 35.25% to 32.46%. This could be caused by some preliminary assumptions proposed in this study, such as ignoring the effect of changing the spectrum, assuming a linear variation of temperature coefficient of I_sc with the concentration ratio, and assuming an infinite shunt resistance. An idealized value for the calculated values was provided by these assumptions. Especially, for the conditions of high concentration ratio, the errors became larger.

3.2.2 Parameters of triple-junction solar cell

This study also used the GaInP/GalnAs/Ge triple-junction solar cell produced by AZUR SPACE Solar Power, Shanghai Solar Youth Energy (SY), and Shenzhen Yinshengsheng Technology Co. Ltd. (YXS) as comparison samples. The conditions under which the parameters were assessed were as follows: the incident spectrum was AM1.5, i.e., the radiation intensity was 1000 W/m² and the cell temperature was 298.15 K. The electrical characteristics of the triple-junction solar cell were calculated for varying concentration ratios (200X–1000X), and the calculated values was compared with the measured values, as exhibited in Figs. 11 to 14.

The I_sc of the triple-junction solar cells changed with the concentration ratio as shown in Fig. 11. The I_sc of the triple-junction solar cells increased as the concentration ratio increased and demonstrated a linear relationship. When the concentration ratio increased from 200X to 1000X, the measured value of the I_sc of the three kinds of cells (AZUR SPACE, YXS, and SY) increased from 2.7 A to 13.7 A, 2.7 A to 13.6 A, and 2.6 A to 13 A, respectively, while the theoretical value increased from 2.8 A to 14 A. The reason for this was that the photon flux density was increasing as the concentration ratio increased. The measured values of I_sc for the cells were less than the theoretical values and the errors increased with the increasing concentration ratio. The RMSE between the measured values and the theoretical values for the short circuit current of the three kinds of cells (AZUR SPACE, YXS, and SY) were 2.31%, 2.77%, and 7.43%, respectively.

The variation in V_oc of the triple-junction solar cells as a function of concentration ratio is shown in Fig. 12. With the change in I_sc, the V_oc of the solar cells increased with the increasing concentration ratio. The difference was that the V_oc and the concentration ratio show a logarithmic relationship. When the concentration ratio increased from 200X to 1000X, the measured values for the V_oc of the three kinds of solar cells (AZUR SPACE, YXS, and SY) increased from 3.05 V to 3.18 V, 3.03 V to 3.16 V, and 3.08 V to 3.27 V, respectively, while the theoretical value increased from 3.09 V to 3.29 V. According to Eq. (5), the V_oc of a solar cell is determined by the change in I_sc and the dark saturation current. The increasing concentration ratio led to a linear increase in I_sc. Therefore, V_oc increased logarithmically with the increasing concentration ratio. The measured values of V_oc for the solar cells were less than the theoretical values, and the calculation error increased with the increasing concentration ratio. The RMSE between the measured and theoretical values of V_oc for the three kinds of cells (AZUR SPACE, YXS, and SY) were 2.81%, 3.53%, and 0.77%, respectively.

The variation in h_c for the triple-junction solar cells with concentration ratio is shown in Fig. 13. Unlike the variation in V_oc and I_sc, the h_c of the solar cells increased first and then decreased with the increasing concentration ratio. When the concentration ratio is 200X, the measured values of the h_c for the three kinds of solar cells (AZUR SPACE, YXS, and SY) are 35.91%, 35.33%, and 34.79%, respectively, while the theoretical value is 36.56%. The measured values reach the peaks of 36.13%, 35.58% and 35.06%, at a concentration ratio of 400X, respectively, while the theoretical value reach the peak of 36.91% at a concentration ratio of 400X. As the concentration ratio is 1000X, the measured values of the h_c are 34.52%, 34.01%, and 33.59%, respectively, while the theoretical value is 35.93%. As can be seen from the equivalent circuit of the solar cell, the series resistance can reduce the I_sc and the voltage at both ends of the load, resulting in a drop in cell conversion efficiency. However, the effect of the series resistance became increasingly apparent with the increasing concentration ratio. As the series resistance increased quickly, the conversion efficiency decreased quickly. Therefore, the accuracy of the theoretical calculation results could be improved by accurately calculating the series resistance of solar cells under high-concentration conditions. In addition, the measured values of the h_c for the solar cells were less than the theoretical values, and the errors increased with the increasing concentration ratio. The RMSE between the measured and theoretical values of the three kinds of solar cells (AZUR SPACE, YXS, and EMCORE) were 2.94%, 4.35%, and 5.62%, respectively.

4 Conclusions

Based on the equivalent circuit of a single-diode model, a mathematical model for the GaInP/GalnAs/Ge triple-junction solar cell was established. Detailed analysis of the characteristics of three-junction solar cells for high concentrations was performed based on the dish-style concentrating photovoltaic system. Besides, the GaInP/GalnAs/Ge triple-junction solar cells which produced by AZUR SPACE Solar Power, Shanghai Solar Youth Energy (SY), and Shenzhen Yinshengsheng Technology Co. Ltd. (YXS) were used as comparison samples for a further comparative study under varying concentrations (200X–1000X).

The theoretical values were compared with the measured values. The outdoor experiments showed that the direct solar radiation and the solar cell temperature were two main influencing factors on the performance of the solar cell. The field measured and simulated results for both I_sc, V_oc, and h_c at different concentrations agreed reasonably well. The difference was small at low radiation rates but increased significantly at high radiation rates, coinciding with sharp increase in the cell temperature. In addition, the selection of assumed conditions, which were employed in order to complete the calculations, affected the errors between the theoretical and measured values. For example, the temperature coefficient of I_sc was assumed to demonstrate a nonlinear variation with the concentration ratio, proving that the accuracy of these assumptions could improve the calculation accuracy of the model.

Under the condition of outdoor experiments, there was a certain error between the theoretical values and the measured values of the electrical characteristics of the solar cell. The error associated with I_sc proved large while the errors associated with V_oc and h_c were small. The theoretical values were larger than the measured values, and the error increased with the increasing concentration ratio. The RMSE of I_sc was not more than 10%, the RMSE of the V_oc was not more than 2%, and the RMSE of the h_c was not more than 4%.

In a further comparative study at high concentrations (200X–1000X), the I_sc of the GaInP/GalnAs/Ge triple-junction solar cell increased linearly with the increasing concentration ratio, while V_oc increased logarithmically. In addition, h_c increased initially and then decreased.

The maximum error in the modeled electrical characteristics of the solar cells was no more than 10%, indicating that this method could be used for theoretical calculation in high concentration photovoltaic system.

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