Simulation of a Low Concentrator Photovoltaic System Using COMSOL

Abstract

The use of photovoltaic (PV) systems presents a great solution to high energy demand. Many factors limit the output of PV systems. One method of increasing the output of PV systems is to employ concentrators. The function of these concentrators is to increase the amount of solar radiation falling on a PV panel using optical devices. In this work, a simulation of a low concentrated photovoltaic system (LCPV) (V-trough model) will be conducted using COMSOL Multiphysics software package. The ray-tracing technique, based on the finite-element method, was used to study the performance of a V-trough without the incorporation of a tracking system. By investigating the effect of the mirrors’ inclination angles on the performance of the system, the optimum inclination angles were determined. The simulation was done for a non-tilted concentrator photovoltaic (CPV) system if placed in different geographical locations in Saudi Arabia with the inclination of the mirrors being changed every hour of the daylight. It was found that the concentration ratio of the suggested model increased for the city of Jeddah, for example, by 171% and 131% for double and partial coverage cases, respectively. In order to reduce the operation cost, the simulation was repeated with the restriction of the mirrors’ inclination to only three positions during the day. The concentration ratio decreased in this case by not more than 14%. When mirrors were fixed throughout the day, the concentration ratio dropped to about 50%. Such simulations will assist in investigating different designs of PV systems prior to their manufacturing. In addition, it could assist in determining the best geographic location for such CPV systems.

1. Introduction

Photovoltaic cells (PV) are one of the techniques that use sunlight to produce clean and renewable energy by converting solar radiation to electricity [1]. PV cells have high public acceptance due to their safety level since there are no emissions during the production of power. Moreover, the source of fuel (sunlight) needed is vast and unlimited, which makes the cost of operation relatively low. Despite these attractive properties, PV cells at this time cannot be a substitute for fossil fuels. This is due to its low power-conversion efficiency. In addition, PV cells have high industrial manufacturing and installation costs [2]. Since the early days of terrestrial PVs, the attractiveness of focusing sunlight to dramatically decrease the price of PV systems has been foreseen. Much effort has been made to develop cost-effective systems for concentration [3,4]. The main concept of a concentrator photovoltaic (CPV) is to utilize optical components to concentrate sunlight on a small receiving PV cell; hence the concentration ratio (CR) can reduce the cell area to the focus area provided by the concentrator. Simultaneously, the irradiance on the PV cell can also be increased by the optical components’ CR [4]. There are different varieties and classes of CPV, one of which depends on its CR, which is also alternatively defined as the number of suns. According to this, the system can be classified as low (<10 suns), medium (10–100 suns), and high CPV (100–2000 suns) [5,6,7].

The main concern in this work is the low-concentration photovoltaic (LCPV) systems. These systems have the prospect to decrease the cost per kWh of electricity in comparison to standard flat-plate PV [8]. The concentrating reflectors employed in LCPV systems are simpler in design and cheaper in manufacturing than high-concentration photovoltaic (HCPV). These static concentrators possess a high acceptance angle and can focus direct and diffuse radiation without the requirement of a tracking system [9,10].

Different designs have been suggested for LCPV, one of which is a V-trough concentrator that utilizes flat-plate mirrors [8] as shown in Figure 1. For example, Fraidenraich and Almeida [11] developed and simplified a conceptual description of the optical performance of a V-trough concentrator. They found that the optical performance of diffuse radiation is strongly dependent on the CR and slightly on the vertex angle of the trough. Therefore, a cost–benefit analysis would favor large trough angles when constructing V-trough cavities. In addition, geometric modelling of LCPV and the numerical simulations were done by Hermenean et al. [12] to find optimized parameters that can lead to maximum incident radiation on the PV cell with a small system size. The authors concluded the CR growth with increasing incident angle. However, the radiation did not depend on the tracking step. More recently, Hermenean et al. [13] also investigated two cases of the geometrical design of a V-trough based on the width of the mirrors that are attached to a PV panel with an equatorial tracking system. They described the relation between the width of the mirrors and the incident angle of the rays that fall on the system in addition to the inclination angle of the mirrors. They found that the optimum mirror inclination angle was 65° and that the CR was 2.2x when the reflected rays doubly covered the whole PV surface. Furthermore, CR was 1.6x when minimizing the width of both mirrors and the reflected rays partially swept the PV surface. As a comparison between the effect of a tracking mechanism to a fixed mirror concentrator on PV arrays, Ya’acob et al. [14] studied a practical in-field method conducted in Serdang, Selangor, Malaysia, by installing, monitoring, recording, and analyzing the data for each system separately. They concluded that the contribution of the mirror was high compared to the tracking system with an increase of 14% per day. Arias-Rosales and Mejia-Gutierrez [15] developed a mathematical model, which offers a hybrid analytical-numerical 2D geometrical representation that explains the effects of direct solar radiation on a V-trough system with geometrical and tracking parameters that are highly flexible. They found that a V-trough having two mirrors of the same width could reach a CR of 2 suns while reaching 0.61 average effective suns during the daytime, when the system remained fixed. To increase energy generation and absorption of solar radiation, coupled optical, thermal, and electrical modeling was done by Bahaidarah et al. [9]. They proved that a V-trough system is suitable for elevating energy output.

Carrying out a simulation is an effective way to forecast the performance of any system before it is manufactured. There are many computer simulation programs that can be used in building a physical model and then studying and analyzing its data mathematically. Altuwirqi et al. [16] performed a simulation using FORTRAN to study the optical performance of a V-trough with a tracking system placed in different cities in Saudi Arabia. They investigated two cases of the geometrical design of a V-trough that was suggested by Hermenean et al. [13]. They found that the overall efficiency of the design did not depend on the tracking system. This conclusion supports the elimination of a tracking system, which will decrease the cost and complexity of the whole PV system. Moreover, it was found that turbidity played an important role in system performance. Al-Shohani et al. [17] performed a 3-D ray-tracing simulation of a V-trough using OptisWorks with four CRs (1.5x, 2x, 2.5x, and 3x) to estimate the optimum vertex angles of a V-trough, which were 30°, 30°, 22°, and 19°, respectively. To find the effect of mirrors on the uniformity of the solar ray distribution on the surface of a PV panel, Michael et al. [18] used ray-tracing analysis, employing Monte Carlo methodology, for a tilted PV module at an angle of 13°. The analysis was done in different months for two cases of concentrator: the first had one mirror attached to the PV module, and the second had two mirrors. They found that the one-mirror design gave uniform illumination for seven months, although with less concentration, while the two-mirror design did not produce uniform illumination in any month, excluding June. Increasing the width of the mirror by 2.82 times the width of the PV module assisted in minimizing the non-uniform illumination on the PV surface. Another factor that affects the efficiency of the LCPVs is the increased temperature of the PV panel, which was studied in previous work [19].

COMSOL-Multiphysics software is an advanced simulation program for modelling and simulating physics-based problems. After constructing the geometrical model of any physical system and assigning the materials used in building the model, the needed physics package is linked to investigating the physical phenomena [20]. Parupud and coworkers [21] developed three geometric LCPV for buildings and performed simulations for it using the ray-tracing package in COMSOL. They found that the CR changed from 1.40 to 1.53 according to the geometry. In addition, they studied the cell temperature impact on the output energy and found the highest energy generation was 177 kWh/m². Baig et al. [22] studied ray tracing in building-integrated concentrating photovoltaics (BICPV) using COMSOL and discussed the effect of non-uniform incident light on the optical device of a solar cell system. Su et al. [23] improved the design of a compound parabolic concentrator (CPC) by installing a lens-wall in order to reduce the damage of irregular radiation on a normal unit of CPC. Then, they simulated the optical distribution of the design and did electrical analysis using COMSOL.

In this work, a V-trough model, which is not mounted on a tracking system, will be simulated using COMSOL. The PV panel is positioned horizontally, and the attached mirrors can be inclined at different angles during the day. Two sizes for the mirrors will be assumed to supply double or partial coverage of solar radiation on the PV panel. The ray-tracing technique will be applied to calculate the solar radiation falling on a PV cell. The optimal number of mirrors’ movement will be investigated, as will its effect on the concentration ratio of the system. The model will simulate placement in different geographical locations in Saudi Arabia.

2. Materials and Methods

2.1. The Model

The V-trough model is a simple design that consists of one PV panel with two attached flat mirrors that are inclined by an angle (θ) as illustrated in Figure 1. To simulate a V-trough, the parameters that affect the model must be determined. These parameters can be divided into two categories: the first category is related to the geographical location where the amount of direct solar radiation falling on the V-trough will be determined. The second category is related to the geometrical design parameters of the V-trough [13,16,24].

Firstly, the geographical location parameters include the latitude angle (φ), the declination angle (δ), the hour angle (ω), and the solar altitude angle (α). The latter is also known as the elevation angle, and it refers to the angle between an imaginary line in the middle of the sun in the sky and the horizontal. The elevation angle can be calculated from the following equation [25]:

α = sin⁻¹ (cos φ cos δ cos ω + sin φ sin δ)

(1)

where the hour angle is given by ω = 15° (12 − T_S); here, T_S is the solar time. From these parameters, the direct solar radiation (B_S) can be calculated from the relation [26]

B_S = B₀ exp (−T_R/(0.9 + 9.4 sin α))

(2)

where T_R is the turbidity factor that describes the condition of the sky. It is classified into ten classifications, each of which describes how the air contains particles of dust and water vapor that cause solar beam attenuation [27]. T_R plays a critical role in solar-system efficiency, as it measures the clearness of the sky. It has values greater or equal to one. For example, for polluted air T_R = 6, while for clean warm air T_R = 3, and it equals 2 for very clean cold air [16]. That is, a high value reflects an elevated content of atmospheric water vapor and dust particles [27]. The values of T_R for Riyadh, Jeddah, Dhahran, Aljawf, and Sharurah are listed in

Table 1. Geographical parameters of selected cities.

Parameter	Dhahran	Jeddah	Aljawf	Sharurah	Riyadh
Declination angle (δ) (degree) (a)	23.5	23.5	23.5	23.5	23.5
Latitude angle (φ) (degree) (a)	26.3	21.51	29.79	17.47	24.64
Longitude angle (l) (degree) (a)	50.11	39.19	39.93	47.08	46.72
Turbidity factor (TR) (b)	5.4	4.2	4.5	6	5.3

^(a) Reference [28]. ^(b) Reference [29].

. Since T_R gives an indication of the optical thickness of the atmosphere, its value affects the amount of solar radiation reaching the surface of the Earth, hence, affecting the performance of LCPV systems. B₀ is the extra-terrestrial radiation given by [13,25]

B₀ = 1367[1 + 0.0334cos(0.9856 × (N − 2.27))]

(3)

Here, N is the day number of the year.

Secondly, the parameters due to the geometrical design describe the design of the V-trough, which includes the width of both mirrors (L₂), the width of the PV panel (L₁) (which is constant), and the inclination angle for each mirror (θ). Altuwirqi et al. [16] concluded that the tracking system had no effect on the amount of total radiation that falls on the PV. Therefore, in this work, the system was placed horizontally with no tilt and without a tracking system to track the sun over the day. The PV is assumed stationary, while the attached mirrors can be tilted at different angles. The width of the mirrors was fixed at two different lengths to enable us to investigate two cases: the first when the mirror length, L₂, is twice the width of the PV panel, L₁, to ensure that the reflected solar radiation covers the whole PV surface as shown in Figure 2a (this case is called double coverage). The second case, when L₂ is the same length of L₁ so the reflected solar radiation from each mirror will partially cover the PV surface, is illustrated in Figure 2b (this case is called partial coverage).

To simulate the two cases of V-trough design, the geographical locations where the system could be assembled were selected to be Dhahran, Jeddah, Aljawf, Sharurah, and Riyadh. These cities represent different locations in Saudi Arabia, in the east, west, north, south, and center, respectively. The geographic information of these cities was determined and incorporated in Table 1. All required information was taken from [28], except T_R values were taken from Diabaté et al. [29].

2.2. COMSOL Set Up

This work focuses on studying the optical performance of a V-trough under solar radiation using COMSOL. This program relies on the finite element method (FEM). FEM is a method to solve complex partial differential equations (PDEs) that describe a physical problem by subdividing the system into smaller parts and approximating the PDEs with simple analytical equations [20]. In order to simulate the V-trough, we followed essential steps. Firstly, the parameters that affect the system were defined as global parameters, either geographical or geometrical parameters. The length and the width of the PV panel were fixed to 1.0 m and 0.5 m, respectively, which are the typical measurement of a PV panel present in the market. The day number in the year N was taken to be 172, as it represents summer solstice. This day has the longest daylight period and would give the maximum output for the LCPV system. In addition, for comparison, N was also taken to be 356, for the shortest daylight, to show the minimum output of such systems. Furthermore, B_S was defined as a global parameter and used to calculate the intensity of the solar radiation that falls on the system for each city. The geometry of the V-trough for both cases (double and partial coverage) was then built. In this study, the simulation was performed for the ideal case of having 100% reflecting mirrors and the material of the PV panel 100% absorbent. Air and the materials of the system are considered homogeneous; hence, the refractive index is constant within each domain.

The COMSOL physics module that was applied to the system was the geometrical optics interface, as it contains all the required physical laws to study the optical rays’ interaction with the system. The process that was applied to the boundaries of the geometry was mainly the reflection of the rays off the mirrors’ surface. The reflection type was set as a specular reflection with a reflectivity equal to one. The solar radiation option was chosen to enable the calculation of the amount of solar radiation falling on the geographical location under investigation. The amount of radiation falling on any specific location will depend on the day of the year, the time of day (which is defined as the solar time over the daylight interval), and the turbidity T_R.

Two types of mesh generator were used. The first type is free triangular, which was created for the top surface of a PV, with a maximum element size equal to 0.0095 m. This size is more suitable with the number of incident rays that hit the PV surface satisfying the condition of one ray at each mesh element. The second type is free tetrahedral, which was used for the entire geometry with extremely fine element size.

For the evaluation steps in the study, the ray-tracing technique was used to trace the solar radiation falling on the system under the conditions of each geographical location. By knowing the amount of power each ray holds, the amount of irradiance, B_PV, in W/m² falling on the PV panel can be calculated. Figure 3 shows the COMSOL calculation method structure.

3. Results and Discussion

The irradiance on the PV panel was calculated for a simple PV module with the two cases of V-trough design: the double and partial coverage. The concentration ratio for all cases was calculated. In addition, the optimum inclination angles of both mirrors were obtained at each hour of daylight. The results are presented below.

3.1. Optimum Inclination Angles of Mirrors

3.1.1. Double Coverage

In order to find the optimum inclination angles of the mirrors, the daylight was divided into three periods: morning (sunrise to 10:00), noon (10:00–14:00), and afternoon (14:00–sunset). The angles for the noon period were determined first, since the rays of the sun are almost normal to the PV panel; hence, θ for both mirrors will be the same. Figure 4 shows the amount of B_PV falling on the PV panel vs. inclination angles at each hour for the selected cities. It can be seen from Figure 4 that, for all chosen cities, the most significant amount of B_PV was at 12:00, where α is almost 90°, as expected. Moreover, the optimum angle for all cities at 12:00 was the same. This value was 70° for both mirrors. Although the amount of B_PV is maximum at 70° inclination for both mirrors exactly at noon, its value drops at times before and after 12:00 at this angle. In fact, the amount of B_PV can become a minimum at 70° inclination at the beginning or end of the noon period, and other angles show better performance. However, regardless of this fluctuation at the 70° inclination, this angle remains the optimum angle for the whole noon period, since the average B_PV value over the whole length of this period is the highest among all inclination angles. Moreover, the behavior of B_PV away from sharp noon was found to be different for the different cities. This is attributed to the effect of the location of each city and its longitude angle. For examples, cities that have comparable longitude angles, such as Jeddah and Aljawf (l ≈ 39°), exhibited similar B_PV behavior throughout the noon period. The other three cities, Dhahran (l ≈ 50°), Sharurah (l ≈ 47°), and Riyadh (l ≈ 46.7°), showed similar behavior since their longitude angles are similar.

For the morning and afternoon periods, θ has to be different for both mirrors in order to avoid the effect of shading from the opposite mirror. Figure 5 illustrates the performance of the model in the morning period for each city when θ₁ is fixed at specific angles equal to, greater than, or smaller than α, while θ₂ varies from 70° to 125° (measured from the horizontal of the opposite mirror). It can be seen from Figure 5 how the amount of B_PV is greatly influenced by the inclination angles of the two mirrors. At 6:00 a.m., the highest amount of B_PV was when θ₁ = 5° and θ₂ = 125° for all cities, while α varied between 5° to 14°. As the day progresses, the optimum angle θ₂ shifts toward smaller inclination angles, reaching a value of between θ₂ = 95° and 105° at 9:00 a.m. Furthermore, the amount of B_PV is affected by the relation between θ₁ and α. As θ₁ becomes comparable to α, the amount of B_PV decreases. This is due to the fact that, in this case more rays were blocked by mirror M1. Hence, the maximum values for B_PV occurs when θ₁ is much lesser than α. Due to the symmetrical shape of the CPV, the same results were found when fixing the inclination angle of mirror M2 and varying that of M1. The optimum angles for the afternoon period were similar to the morning period, since the altitude angles are the same for the corresponding solar hour.

3.1.2. Partial Coverage

One method of reducing the cost of an LCPV system is to reduce its size through the reduction of the size of its attached mirrors. A major drawback of this reduction in the overall size of the system is that only partial coverage of the PV panel occurs. To investigate the performance of a partial coverage case, the city of Jeddah was chosen to conduct a partial coverage simulation, which can be later compared with the results of this city with double coverage, reported above.

Figure 6 illustrates the irradiance falling on the PV panel vs. the inclination angles for the noon period. As can be seen, the highest amount of irradiance occurs in the solar hours 12:00 and 13:00, when the mirrors M1 and M2 were inclined at the angles θ₁ = θ₂ = 65°. These inclinations are nearly the same inclinations that gave maximum B_PV in the noon period with double coverage, in Jeddah. However, it should be noted that the value of B_PV at these optimum angles for partial coverage was lower by 200 W/m² than for double coverage.

As for the morning period, Figure 7 illustrates the amount of B_PV falling with partial coverage vs. mirror inclination angles during the morning hours. It can be seen from Figure 7, when compared with Figure 5, that the optimum angles for partial coverage differed slightly from those of double coverage.

As was mentioned previously, due to the symmetrical shape of the CPV, the optimum angles for the afternoon period were the same as that for the morning period.

3.2. Concentration Ratio

The concentration ratio (CR) can be calculated by taking the ratio of the total amount of B_PV falling on the panel to the total amount of B_S. By using the optimum inclination angles found above, we were able to simulate the performance of the model throughout the longest day in the year, when the system is located at different geographical locations. These simulations allowed the calculation of the CR of the V-trough model.

The CR was calculated for the V-trough model for two geometrical designs and two operating methods. The two designs are, firstly, the large mirrors, which guarantee double coverage of the PV panel and, secondly, the smaller mirrors that reduce the size of the system but render the coverage only partial. The operating methods involve the number of times the inclinations of the mirrors are changed during the daylight period. The first case is when the inclination of the mirrors is changed every hour and set at the optimum angle for that hour. The second case is limited to three inclinations per day; one during the morning interval, the second during the noon period, and lastly a position during the afternoon interval. The limitation of positions in the second operation method is aimed at reducing the energy required for operating the system. For comparison, a third case will also be investigated wherein the mirrors remain fixed throughout the day at an angle of 65°.

Figure 8a,b show the performance of a PV system when placed in the city of Jeddah for the longest and shortest days of the year. Both figures illustrate the amount of B_PV falling on the system in seven different situations compared to the amount of B_S. The first situation is when the PV panel is not attached to any mirrors; therefore, no concentration of radiation took place. The second and third situations are double and partial coverage, while changing the mirrors’ inclination every hour of the daylight. The fourth and fifth situations are double and partial coverage when the mirrors’ inclinations are changed only three times during the day; once in the morning period, the second during the noon interval, and the third during the afternoon. The values of the inclination angles that the mirrors were fixed at during these three periods were chosen to give the greatest amount of B_PV during those intervals. These angles can be deducted from Figure 4, Figure 5, Figure 6 and Figure 7.

Table 2. The mirror inclination angles when fixed at three positions for double and partial coverage (for the city of Jeddah).

	Double Coverage		Partial Coverage
	θ1	θ2	θ1	θ2
Morning	20°	115°	25°	115°
Noon	70°	70°	60°	60°
Afternoon	115°	20°	115°	25°

lists these angles for both double and partial coverages when the mirrors are fixed at three positions. Lastly, as shown in Figure 8, the sixth and seventh situations are double and partial coverage for a fixed mirror’s inclinations.

The calculated B_PV values for a simple PV panel followed the same pattern, as B_S but did not show agreements in the early and later hours of the day. This is due to the large time intervals used during the calculation, in contrast with the empirical continuous function of B_S.

When mirrors were attached to the PV panel to form a V-trough, the amount of B_PV far exceeded that of B_S, reaching a maximum of more than 2000 W/m² for the longest day and 1400 W/m² for the shortest day during midday, both with double and partial coverage, when the position of the mirrors changed every hour. When the inclination of the mirrors changed only three times during the day, the maximum irradiance reaching the PV panel was also high compared to B_S. It is worth noting, as can be seen from Figure 8b, that, during the shortest day, there was no significant changes between the performance of the LCPV in the double and partial coverage cases, nor when the mirrors’ position changed from 13 to 3 position per day. This can be attributed to the fact that the direction of solar radiation on the December solstice falls at an oblique angle at noon. Therefore, the same portion of the reflected rays reaches the PV panel for the 13-positions and 3-positions. In contrast for the longest day, the B_PV for 3-positions slightly differs from 13-positions, as the direction of the solar radiation is perpendicular to the surface of the system at noon. Hence, tilting the mirror at the optimum angle for each hour ensures that more incident radiation reflects inside the V-trough cavity.

The same pattern was observed when the simulation was performed for different geographical locations, with differences in the maximum value of B_PV reached.

To visualize the values of CR when the system is placed in different geographical locations, Figure 9 illustrates the results of the configurations for the highest irradiances reached. That is, the case of double coverage, with the three operation schemes of changing the mirrors’ inclinations every hour or limiting it to three moves during the day or fixing it at one position throughout the day. It can be seen from Figure 9 that the effect of limiting the number of movements of the mirrors to three movements did not affect significantly the CR, although this restriction can help in cutting down operation costs. The reduction in CR values was as low as 1.2% for Sharurah and as high as 14% for Dhahran, while other cities varied between these two values. However, with fixed mirrors, the CR was reduced, approximately, to half its original value for all of the cities under study. Hence, simulating the different operational schemes can assist in balancing between maintaining the high efficiency of the LCPV system and reducing the cost and effort. The results in this work indicated that the optimum performance came when moving the mirrors to three positions per day.

4. Conclusions

Simulation of a low concentrator photovoltaic (LCPV) system (V-trough) was performed for two geometrical designs: double and partial coverage, using the COMSOL Multiphysics package of ray optics. The LCPV system was presumed to be non-tilted, to be suitable for flat top surfaces, and with no tracking system. After investigating the effect of the inclination angles of both mirrors on a CPV system, the optimum angles of the mirrors for obtaining maximum incident irradiance on the PV panel were determined. The concentration ratio (CR) to test the performance of the LCPV was calculated, and it was found that attaching mirrors in a V-trough shape increased the amount of irradiance falling on the PV panel. For instance, in Jeddah, the CR improved by 171% for double coverage and 131% for partial coverage. Furthermore, in this work, the effect of the number of movements on changing the mirrors’ inclination on the total amount of irradiance falling on the PV panel was studied. A comparison between fixed mirrors; between 13 and 3 movements per day; and between the morning, noon, and afternoon periods was conducted. The reduction in CR values was found to be between 1.2% and 14%, depending on the geographical location of the LCPV, when movements were reduced to three per day. Such analysis will assist in conducting a cost-effectiveness analysis of LCPV systems. The use of COMSOL as a design tool integrated with the laws of physics can allow for the improvement of other forms of CPV systems and for the measurement of its CR. This will further assist in designing and testing models prior to manufacturing.