To investigate the shading effects of solar arrays in agricultural settings and to understand their implications for crop production, this study employed a three-dimensional simulation approach using the Ladybug software. The research focused on analyzing the distribution of solar radiation beneath solar panels and the resulting shading rates, which are critical parameters for designing agrivoltaic systems. Experiments were conducted based on the geographical and meteorological conditions of Kunming and Pu’er cities in Yunnan Province, China. The primary objective was to quantify how variations in solar panel spacing, installation height, and tilt angle influence the shading environment, thereby providing operational guidelines for optimizing the balance between photovoltaic power generation and agricultural yield.
The study utilized a digital model of a solar panel array to simulate the shading patterns during the peak solar radiation period, specifically from 9:00 to 16:00 on the summer solstice (June 21). The simulation incorporated three key variables: the spacing between panels along the x-axis (20 cm, 50 cm, and 100 cm), the height of the panels above the ground (150 cm, 200 cm, and 250 cm), and the tilt angle of the panels (0°, 15°, and 30°). The performance metric evaluated was the shading rate, defined as the proportion of solar radiation blocked by the solar panel relative to the total available radiation in an unshaded area. The cumulative radiation received by the ground surface beneath the array was also recorded for each configuration.
Methodology and Simulation Setup
The simulation environment was built upon the Ladybug plugin, which allows for detailed analysis of solar radiation based on geographic location, sky conditions, and geometric obstructions. The meteorological data for Kunming (latitude 25°07′ N, longitude 102°44′ E, elevation 1770 m) and Pu’er (latitude 23°04′ N, longitude 101°02′ E, elevation 1302 m) were sourced from the EnergyPlus website using the Chinese Standard Weather Data (CSWD) format. This dataset provided hourly values for direct normal irradiance, diffuse horizontal irradiance, and other relevant parameters.
The solar array model was constructed as a series of rectangular panels. The ground area beneath the array was divided into a grid of 1 square meter cells for analysis. The key calculation for this study is the shading rate, which can be expressed as:
$$SR = \frac{R_{unshaded} – R_{shaded}}{R_{unshaded}} \times 100\%$$
where \( SR \) is the shading rate, \( R_{unshaded} \) is the cumulative solar radiation on a horizontal surface without any obstruction (the control condition), and \( R_{shaded} \) is the cumulative radiation measured at a specific grid point under the solar panel array.
The core of the analysis involves calculating the cumulative radiation \( R \) at each point over time. This is given by the integral of the direct and diffuse components of solar radiation over the specified time period, considering the view factor of the sky and the obstruction provided by the solar panel. For a given point \( P \) on the ground, the cumulative radiation is calculated as:
$$R(P) = \int_{t_1}^{t_2} [I_{direct}(t) \cdot f_{sky}(P, t) + I_{diffuse}(t) \cdot VF_{sky}(P)] dt$$
where \( I_{direct}(t) \) and \( I_{diffuse}(t) \) are the direct and diffuse solar irradiances at time \( t \), \( f_{sky}(P, t) \) is a binary function indicating whether the point is in direct sunlight (1 if yes, 0 if no), and \( VF_{sky}(P) \) is the sky view factor, which represents the fraction of the sky dome visible from point \( P \). The simulation period \( t_1 \) to \( t_2 \) is set to the daily window of 9:00 to 16:00.
Experimental Variables and Parameters
The study systematically varied three parameters of the solar panel array to assess their impact on the shading rate. The base configuration for comparison was set with an x-axis spacing of 20 cm, a y-axis spacing of 50 cm, an installation height of 150 cm, and a tilt angle of 0°. The variations are summarized in the table below.
| Parameter | Variable Values | Fixed Conditions |
|---|---|---|
| X-axis Spacing | 20, 50, 100 cm | Height: 150 cm, Tilt: 0°, Y-spacing: 50 cm |
| Installation Height | 150, 200, 250 cm | X-spacing: 20 cm, Tilt: 0°, Y-spacing: 50 cm |
| Tilt Angle | 0°, 15°, 30° | X-spacing: 20 cm, Height: 150 cm, Y-spacing: 50 cm |
Results and Discussion
The simulation results revealed significant differences in the shading rate based on geographic location and the configuration of the solar panel array. The first part of the analysis focused on the monthly variation of shading rates for a standard configuration (20 cm spacing, 150 cm height, 0° tilt). Data for cumulative radiation and shading rates for both Kunming and Pu’er are presented in the following table.
| Month | Unshaded Rad. (Kunming) | Unshaded Rad. (Pu’er) | Shaded Rad. (Kunming) | Shaded Rad. (Pu’er) | Shading Rate (Kunming) | Shading Rate (Pu’er) |
|---|---|---|---|---|---|---|
| Jan | 103.76 | 104.93 | 38.97 | 39.26 | 62.4% | 62.6% |
| Feb | 108.08 | 115.42 | 40.55 | 43.92 | 62.5% | 61.9% |
| Mar | 153.38 | 148.76 | 60.50 | 58.92 | 60.6% | 60.4% |
| Apr | 159.56 | 149.45 | 62.84 | 58.77 | 60.6% | 60.7% |
| May | 135.69 | 135.61 | 52.26 | 52.68 | 61.5% | 61.2% |
| Jun | 121.13 | 126.96 | 46.57 | 49.14 | 61.6% | 61.3% |
| Jul | 125.12 | 126.88 | 48.32 | 49.29 | 61.4% | 61.2% |
| Aug | 122.64 | 124.73 | 47.77 | 48.67 | 61.1% | 61.0% |
| Sep | 107.13 | 121.69 | 42.49 | 48.36 | 60.3% | 60.3% |
| Oct | 100.39 | 112.66 | 38.68 | 44.18 | 61.5% | 60.8% |
| Nov | 82.83 | 93.32 | 31.11 | 35.74 | 62.4% | 61.7% |
| Dec | 85.60 | 101.41 | 33.02 | 38.22 | 61.4% | 62.3% |
From the data, it is evident that the shading rate varies throughout the year due to changes in the solar elevation angle. The largest difference in shading rate between the two cities under identical configurations occurred in December, where Kunming showed a rate of 61.4% and Pu’er showed 62.3%, a difference of 0.9 percentage points. This indicates that the local climate, specifically the ratio of direct to diffuse radiation, significantly affects the shading efficiency of the solar panel.
Impact of Solar Panel Spacing on Shading
The effect of the distance between solar panel rows (x-axis spacing) on the shading rate was analyzed. For the simulation on the summer solstice, the cumulative radiation from 9:00 to 16:00 was calculated. The results show that increasing the spacing between solar panel rows allows more sunlight to reach the ground, thereby reducing the shading rate.
The relationship between the average shading rate and the x-axis spacing can be approximated by a logarithmic function for the given conditions. For the simulated day, the peak shading rate observed over the ground grid declined significantly as spacing increased. For example, in Pu’er, the peak shading rate fell from 64.4% at 20 cm spacing to 44.8% at 100 cm spacing. Similarly, the minimum shading rate in the area also decreased. This direct relationship can be summarized mathematically. For a given configuration, the reduction in peak shading rate \( \Delta SR_{peak} \) relative to a baseline spacing \( d_0 \) could be modeled as:
$$\Delta SR_{peak}(d) = \alpha \cdot \ln\left(\frac{d}{d_0}\right)$$
where \( d \) is the new spacing and \( \alpha \) is a coefficient dependent on geographic location and panel geometry. In this study, for the transition from 20 cm to 100 cm spacing, the value of \( \alpha \) would reflect a significant reduction in shading.
Impact of Solar Panel Installation Height on Shading
Modifying the height of the solar panel array had a non-linear and location-dependent effect on the shading rate. Increasing the height from 150 cm to 200 cm generally led to higher shading rates in the central zones of the array due to the wider shadow cast by the panels. However, at the edges of the array, higher placement could reduce shading as more diffuse light reached the ground.
For example, in the Pu’er simulation, the peak shading rate increased from 64.4% at a 150 cm height to 70.0% at a 200 cm height. At a height of 250 cm, the peak shading rate further increased to 71.8%. The minimum shading rate, however, showed a more complex trend. The data suggests that the total shaded area expands as the panels are lifted higher, leading to a more uniform but higher overall shading rate in the center of the array. This can be explained by the geometric relationship between the solar panel, the sun’s altitude, and the ground plane. The shadow length on the ground \( L_{shadow} \) can be approximated by:
$$L_{shadow} = H \cdot \cot(\theta)$$
where \( H \) is the height of the bottom edge of the panel and \( \theta \) is the solar altitude angle. As \( H \) increases, the shadow length increases, covering more ground area and leading to higher shading rates in those zones.
| City | Installation Height (cm) | Peak Shading Rate (%) | Minimum Shading Rate (%) |
|---|---|---|---|
| Kunming | 150 | 65.1 | 41.7 |
| Kunming | 200 | 69.3 | 40.8 |
| Kunming | 250 | 66.8 | 31.5 |
| Pu’er | 150 | 64.4 | 50.5 |
| Pu’er | 200 | 70.0 | 55.7 |
| Pu’er | 250 | 71.8 | 45.0 |
Impact of Solar Panel Tilt Angle on Shading
The angle at which the solar panels are tilted also plays a critical role. Increasing the tilt angle from 0° to 15° and then to 30° reduced the shading rate across the entire test area. A steeper tilt angle directs the shadow away from the immediate vicinity of the panel row, allowing more light to reach the ground directly beneath and between the rows.
The simulation for the summer solstice showed a consistent decline in both the peak and minimum shading rates as the tilt angle increased. In Kunming, the peak shading rate dropped from 65.1% at a 0° tilt to 56.9% at a 30° tilt. The decrease in shading rate \( \Delta SR_{tilt} \) can be modeled by a function of the tilt angle \( \phi \) and the solar declination angle \( \delta \). The net effect on the ground is represented by the decrease in the projected area of the solar panel onto the ground. The effective length of the shadow cast by a tilted panel is affected by the angle of the panel relative to the sun’s rays. The relationship can be expressed as:
$$\Delta SR_{tilt} \propto \sin(\phi – \delta)$$
For the summer solstice, when the solar altitude is high, increasing the tilt angle of the solar panel effectively moves the panel’s normal vector closer to the sun’s zenith, thereby narrowing the shadow footprint and reducing the shading rate on the ground.
| City | Tilt Angle (deg) | Peak Shading Rate (%) | Minimum Shading Rate (%) |
|---|---|---|---|
| Kunming | 0 | 65.1 | 41.7 |
| Kunming | 15 | 61.7 | 38.6 |
| Kunming | 30 | 56.9 | 36.3 |
| Pu’er | 0 | 64.4 | 50.5 |
| Pu’er | 15 | 61.0 | 48.1 |
| Pu’er | 30 | 57.3 | 46.5 |
Conclusion
This study, based on the Ladybug simulation, has quantified how the shading effect of a solar panel array is influenced by its design parameters and geographic location. The key findings are as follows:
1. The shading rate is location-dependent, with variations observed between Kunming and Pu’er due to differences in solar radiation composition (direct vs. diffuse). The maximal difference in shading rate between the two cities under the same conditions was 0.9 percentage points in December.
2. Increasing the spacing between rows of solar panels is an effective strategy to reduce shading rates, thereby improving light conditions for crops. The shading rate showed a logarithmic decrease with increased spacing.
3. The installation height of solar panels has a complex influence. Higher placement generally leads to a larger shaded area and higher peak shading rates, which can be detrimental to crops in the center of the array, although edge effects may be reduced.
4. Increasing the tilt angle of solar panels on the summer solstice significantly reduces the shading rate, as it directs shadows away from the area directly beneath the panels.
These findings provide valuable quantitative data for the design of agrivoltaic systems. By carefully selecting the spacing, height, and tilt angle of solar panels, it is possible to achieve a desired level of shading that balances the needs of crop production with the efficiency of photovoltaic power generation.