Agrivoltaic system design tools for managing trade-offs between energy production, crop productivity and water consumption

We approach the question of how an agrivoltaic array will affect water consumption and plant growth by determining how the array changes the illumination conditions in the field and then applying those illumination conditions to (1) a model of ET and (2) a plant productivity model (figure 1). The same calculation captures the illumination of the photovoltaic panels and hence the electricity production. The model is implemented in MATLAB (MathWorks). For simplicity, the model assumes conditions under the array are the same as the control setting in all aspects other than illumination. While multiple studies have reported differences in air temperature in the presence of photovoltaic arrays (Barron-Gafford et al 2019, Broadbent et al 2019, Jiang et al 2021), the effects are not consistent in magnitude or sign across or within studies. To examine the sensitivity of the results to this assumption, we also considered scenarios where the air temperature under the array differed from the ambient by up to ±5 °C (see supplemental).

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The first modeling step generates an array geometry based on the unit panel dimensions, the panel orientation (azimuth and tilt angle) and the panel height (figure 2(a)). The array unit cell includes the unit panel (2 m wide and 1.3 m in height, elevated 2.5 m) and variable space, tiled two directions (figure 2(b)). The analysis considered array rows of either North-South (NS) or East-West (EW) axes. We also examined three panel tilt angles, 0, 20 and 30 degrees to the South (for EW rows) or West (for NS rows). Together, these design permutations span the density and orientation of most fixed-panel solar installations and includes average tilt angles for latitudes up to approximately 45 degrees.

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Array coverage (projected area of the panels from nadir divided by the field area) ranges from 100% to 1% and is the primary basis of comparison between the designs in the subsequent analysis. To generate the shade pattern, we first calculated the solar angle at six-minute intervals for each day of the year at each location (Blanco-Muriel et al 2001). We determined the shade fraction by comparing the area of the panel shadow projected onto the ground plane to the area of the array unit cell. Any self-shading by adjacent panels was corrected in total shading and electricity production calculations.

We model the array impact on ET using the FAO formulation of the Penman–Monteith equation (equation (2)) (Allen et al 1998).

$$\begin{equation}{\text{ET}} = \frac{{\Delta \left( {{R_n} - G} \right) + {\rho _a}{c_p}\frac{{{e_s} - {e_a}}}{{{r_a}}}}}{{\Delta + \gamma \left( {1 + \frac{{{r_s}}}{{{r_a}}}} \right)}}.\end{equation} \tag{ 2 }$$

This model estimates the ET of a plant layer based on the energy balance between incoming solar radiation (G), outgoing thermal radiation (R_n ), and latent heat from ET, as governed by the local air conditions: vapor pressure deficit (e_s –e_a ), the air density (ρ_a ), specific heat (c_p ), psychrometric constant (γ) and slope of the vapor saturation pressure curve (Δ), and bulk and aerodynamic resistances (r_s and r_a ) that represent the restrictions to water transport through the soil, plant tissues and canopy.

For full sun ET estimates, G is the global horizontal irradiance, and for full shade conditions, G is the diffuse irradiance, both taken from the National Solar Radiation Database (Sengupta et al 2018). We assume that the leaf surface temperature is equal to the air temperature, the sky temperature is 273 K and the albedo is 0.7, which together determine the outgoing radiation (R_n ) (Evangelisti et al 2019). Temperature, ambient water vapor pressure and wind speed are hourly climate normals for the location, interpolated for model timesteps (Arguez et al 2012). The surface roughness/diffusivity parameters of the PM model, r_a and r_s , are the baseline values for grass (Allen et al 1998).

To analyze water savings for each array case, we weighted the baseline full sun and full shade ET rates at each timestep by the sun and shade fractions and calculated an average ET rate for the field as a whole. This instantaneous ET rate was then integrated over each day or the total year to quantify seasonal and cumulative trends.

As in many current ecosystem modeling efforts, our model avoids distinguishing among plant species, but rather collapses plant differences into two broad types: shade-tolerant plants and sun-tolerant plants. Productivity in our model refers to photosynthetic activity, which we treat as a proxy for all crop products. We assume that the plants are managed to avoid water or nutrient restriction without specifying an irrigation method, and our model generates an instantaneous crop productivity value based on illumination and temperature. We include temperature in our model, because photosynthesis has a strong dependence on temperature and because temperature has diurnal, seasonal and location variations that interact with the diurnal and seasonal illumination changes caused by the agrivoltaic array (Farquhar et al 1980, Leuning 2002, Sage and Kubien 2007).

Our plant model calculates productivity as the product of two responses. First, plant productivity increases to an asymptote with irradiance (Pickett and Myers 1966) (equation (3)):

$$\begin{align}{\text{Light productivity response}} = { }\text{tanh} \left( {1.5*\frac{I}{{{I_{{\text{sat}}}}}}} \right).\end{align} \tag{ 3 }$$

Here I is the cumulative irradiance on the plants in W m⁻² and I_sat is the saturation irradiance. We use the cumulative irradiance rather than the PAR, because PAR is well-correlated with cumulative irradiance at the level of precision required by this model (Yu et al 2015).

The second component of the plant model is a temperature response that rises with temperature to a peak value and then declines. The function is based on the maximum catalytic rate as a function of leaf temperature, as reported in (Leuning 2002), which captures the asymmetric rise and fall of activity of the enzyme Rubisco with temperature (equation (4)):

$$\begin{align}&{\text{Temperature productivity response}} \nonumber\\ & \quad = C*\frac{{{e^{\left( {{\raise0.7ex\hbox{${{H_a}}$} \!\mathord{\left/ {}\right.} \!\lower0.7ex\hbox{${R{T_0}}$}}} \right)\left( {1 - {\raise0.7ex\hbox{${{T_0}}$} \!\mathord{\left/ {}\right.} \!\lower0.7ex\hbox{$T$}}} \right)}}}}{{1 + {e^{{\raise0.7ex\hbox{${\left( {{S_v}T - {H_d}} \right)}$} \!\mathord{\left/ {}\right.} \!\lower0.7ex\hbox{${\left( {RT} \right)}$}}}}}}\quad\end{align} \tag{ 4 }$$

H_a is the activation energy and H_d the deactivation energy, R is the gas constant, T₀ is a reference temperature (298 K), T is the leaf temperature, and S_v is an entropic term. For this model we use the referenced tabulated values for H_a and S_v corresponding to Brassica rapa, while we adjust the value of H_d to set the desired peak temperature and C is a scaling constant that adjusts the peak value to one. The full plant model productivity (equation (5)) is the product of the light response (LPR) and temperature response (TPR) for a given set of conditions:

$$\begin{align}{\text{Instantaneous Crop Prod}}\left( {T,{ }I} \right) = {\text{LPR}}\left( I \right)*{\text{TPR}}\left( T \right).\end{align} \tag{ 5 }$$

The model output value range lies between zero and one, where one represents the maximum possible instantaneous productivity.

Sun tolerant plants use the same model parameters for both full sun and full shade conditions, specifically I_sat is 300 W m⁻² and the peak response temperature is 26 °C. For the shade-tolerant plants, the model uses the same parameters in the shade, but in full sun the peak response temperature is 22 °C, simulating plants that are more sensitive to high temperatures in full sun. The values for the peak response temperatures and I_sat parameters are chosen to match the aggregate photosynthetic performance of a variety of plants in agrivoltaic experiments under full sun and full shade conditions (Barron-Gafford et al n.d.).

We calculate the plant response for each six-minute time step under full sun and full shade conditions. Agrivoltaic productivity for the field for each time step is the average of the full sun and full shade productivity values, weighted by the instantaneous shade fraction of the field. The instantaneous productivity values are integrated over each day and over a growing season from day 120 to day 275 of the year. We also include an integration over the whole year to obtain cumulative values of the crop productivity potential for settings that may grow crops over multiple seasons. All productivity values are compared to an unshaded baseline case.

The analysis examines the impact of agrivoltaic designs on water consumption and crop and electricity productivity for an agrivoltaic system in Tucson, AZ (mean temperature 21 °C, mean annual precipitation 269 mm) (Arguez et al 2012). The analysis is further extended to Stockton, CA (mean temperature 17 °C, mean annual precipitation 342 mm) and Aurora, CO (mean temperature 11 °C, mean annual precipitation 367 mm) to explore how water consumption, crop productivity and LER depend on system design, location and crop selection. The three locations span the climatic space for the majority of the Southwestern United States, a region that has large agricultural production, arid or semi-arid climate, and water resources under increasing stress due to climate change and demand growth (Overpeck and Udall 2020).

For each location we calculated the shade pattern, total water consumption and crop productivity for arrays with both EW and NS orientation, panel coverage ranging from 1% to 100% and tilt angles of 0, 20 and 30 degrees to the South (for EW rows) or West (for NS rows). We simultaneously calculated the electricity production, with panel illumination calculated as the sum of the direct irradiance projected onto the panel area and the diffuse irradiance. Electricity production of select array configurations at each location were compared to projections from the PVWatts (Dobos 2014) model for validation. Portions of the panel shaded by adjacent panels are assumed to produce zero electricity. For all scenarios, we compare the production of crops and electricity for the agrivoltaic system to a baseline case consisting of a full sun conventional agriculture field and a second photovoltaic-only field with a baseline photovoltaic array consisting of EW rows of panels tilted 30 degrees to the south and 50% area coverage. For both conventional and agrivoltaic crop productivity we consider a summer growing season from day 120 to day 275. By comparing the agrivoltaic crop and energy productivity to the baseline, we obtain an LER value for each design permutation.

There is a clear correlation between array coverage and water savings, due to shading reducing the incoming irradiation and therefore the ET rate (figure 3(a)). The NS arrays show a linear trend until roughly 70% coverage for all tilt orientations. The EW arrays follow three slightly different linear trends corresponding to different panel tilts, with larger tilt values experiencing larger water savings. At very high coverage fractions, the linear trend for all array types shifts to a lower rate of increase with area, due to increasing self-shading of the array. If the air temperature were warmer under the array, water savings decrease by roughly 3% per degree of temperature difference. Conversely, savings increase by 3% per degree if the air is cooler (see supplemental).

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Agrivoltaic arrays with different azimuth orientation but equivalent coverage both decrease water consumption by a similar amount relative to the full sun baseline (figure 3(b)). The NS array has a higher water consumption in the winter months, while the EW array is higher in the early summer, suggesting seasonal variations in the shading pattern for the two classes of array.

The daily and seasonal trends in shading fraction for these two arrays (see supplemental) illuminate these seasonal differences in water consumption. For south-tilted EW array, the shade fraction is higher in winter months when the sun's average zenith angle is larger, but the shade fraction does not vary over the course of the day. By contrast, the NS array has a consistent shade fraction over the course of the year, but the tilt to the west causes a higher shade fraction in the afternoon, with most light reaching the ground in the morning. The shade fraction averaged over the day for a NS array is independent of the panel tilt angle, only the distribution of light within the day is affected.

For the sun-tolerant crop (figure 4(a)), productivity in the shade is lower than productivity in the sun for the entirety of both days considered. By contrast, the shade-tolerant model (figure 4(b)) shows lower in-shade productivity on day 50, but higher in-shade productivity the majority of day 200. On day 200, the temperature rapidly rises to a level above the peak activity temperature for the model in full-sun, and productivity is suppressed. In the shade, the model has a higher peak activity temperature and suffers a less severe mid-day depression in productivity. Over the course of the year, the shade-tolerant daily plant productivity in the agrivoltaic setting shows a large variation, ranging from 40% of the control productivity in the winter to over 110% of the control setting productivity for days in the mid to late summer. The sun-tolerant plant model also exhibits lower productivity in winter, particularly for EW arrays at roughly 50%–60% of the control setting, however in summer the productivity is only 80% of the control setting value for both EW and NS arrays.

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Comparing the performance of both crop models over the full year (figure 5(a)), as the field coverage fraction increases, plant productivity suffers a larger penalty relative to the control case. The shade-tolerant crop model shows a slightly higher crop productivity compared to the sun-tolerant model for arrays with NS rows. For field coverage fractions of 40% or higher, plant productivity varies by roughly 10 to 20 percentage points, depending on the combination of crop and array design.

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Reanalysis for the summer growing season alone (figure 5(b)) shows a large separation in the behavior of the two plant models. The sun tolerant crop productivity still declines with array coverage, but the shade-tolerant crop under NS arrays with a western tilt has a productivity up to 8% higher than the full-sun baseline. This effect comes from the uneven distribution of shading over the course of the day for west-tilted NS arrays, which allows the shade-tolerant crop to take advantage of morning sun while temperatures are lower. The crop productivity curves (figure 4(b)) illustrate this behavior in detail. The shade tolerant crop under EW arrays shows a productivity penalty of 10% or less compared to the baseline, but it receives no net benefit from the shading.

The water consumption over the summer growing season (day 120–day 275) versus the field coverage fraction (figure 5(c)) is almost identical to the full-year trend (figure 3(a)). For both NS and EW arrays, water consumption declines by roughly 45% of the array coverage fraction. This indicates that for NS arrays with 40% or higher coverage, the agrivoltaic field would have 20%–45% water savings over the full-sun control depending on the coverage level, with productivity enhanced above the baseline for shade-tolerant crops (figure 5(b)).

For Tucson (figure 6(a)) the shade-tolerant crop model productivity and water saving trade-off is small to positive. However, for the sun-tolerant model, the water savings are close to the loss in crop performance regardless of design. In the results for Stockton (figure 6(b)) the shade-tolerant model performs well but does not show productivity benefits. Tilted NS arrays lose less than 5% relative to the baseline. This performance difference compared to Tucson is due to the cooler temperatures in Stockton, which also result in lower water savings. For Aurora, CO (figure 6(c)), the behavior of both crop models and both array orientations collapse into a single behavior trend. Based on these simulations, an agrivoltaic system in Aurora would suffer a plant productivity penalty of roughly two thirds of the water savings over the summer growing season regardless of the coverage level when compared to a full-sun field. The water savings are lowest for this location among the three, at roughly 1/3 of the coverage level. The effect of warmer or cooler temperatures under the array also depends on location, crop type and season considered. In Tucson warmer temperatures reduce summer shade tolerant crop productivity to a level comparable to that in Stockton, while crops benefit from warmer temperatures in Aurora (see supplemental).

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All agrivoltaic scenarios considered had LER values greater than one, indicating that even low-coverage designs increase the land use efficiency. While LER increases with area coverage, there is a large degree of variation in the LER at comparable coverage levels (figure 7(a)). At coverage of 60% and higher, the choice of array orientation and tilt and the shade tolerant vs sun tolerant crop can change the LER value by up to 0.6. In all cases, the range of potential LER values increases with the coverage, however the Aurora location LER values are much more closely bounded for a given coverage level. As the narrow green region indicates, this is due to the Aurora location having productivity independent of crop type (figure 7(c)).

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