A Digital-Twin and Machine-Learning Framework for the Design of Multiobjective Agrophotovoltaic Solar Farms
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Computational Mechanics https://doi.org/10.1007/s00466-021-02035-z ORIGINAL PAPER A digital-twin and machine-learning framework for the design of multiobjective agrophotovoltaic solar farms T. I. Zohdi1 Received: 2 March 2021 / Accepted: 10 May 2021 © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract This work develops a computational Digital-Twin framework to track and optimize the flow of solar power through complex, multipurpose, solar farm facilities, such as Agrophotovoltaic (APV) systems. APV systems symbiotically cohabitate power- generation facilities and agricultural production systems. In this work, solar power flow is rapidly computed with a reduced order model of Maxwell’s equations, based on a high-frequency decomposition of the irradiance into multiple rays, which are propagated forward in time to ascertain multiple reflections and absorption for various source-system configurations, varying multi-panel inclination, panel refractive indices, sizes, shapes, heights, ground refractive properties, etc. The method allows for a solar installation to be tested from multiple source directions quickly and uses a genomic-based Machine-Learning Algorithm to optimize the system. This is particularly useful for planning of complex next-generation solar farm systems involving bifacial (double-sided) panelling, which are capable of capturing ground albedo reflection, exemplified by APV systems. Numerical examples are provided to illustrate the results, with the overall goal being to provide a computational framework to rapidly design and deploy complex APV systems. Keywords Agrophotovoltaics · Digital-twin · Machine-learning 1 Introduction gies to store heat and redeploy to run steam turbines, (e) artificial photosynthesis (biomimicry of plants), etc. Photo- Theoretically, if properly harnessed, solar energy is the most voltaic systems are the most widely used solar conversion plentiful energy source available to society. A rough calcu- systems. They employ the semi-conducting materials that lation (Fig. 1) to ascertain the amount of worldwide solar exhibit the photovoltaic effect. With dramatic price drops power available combines (a) The Earth’s radius: R ≈ and readily available silicon, such systems have come to 6,400,000 m, (b) The Earth’s surface area: A ≈ 4π R2,(c) dominate solar energy conversion systems. The first prac- The peak solar power per unit area: p ≈1300 Watts/m2 and tical solar cell was built in 1954 at Bell Labs, with the space (d) The assumption that roughly 1/2 of the Earth is illu- industry being an early adopter, driven by power limitations minated at any given time and that, on average, the overall in space for satellites. In the US alone, installed solar energy surface achieves 1/2 peak illumination. This yields an esti- use has increased by a factor of 20 in the last decade. Issues = Ap ≈ mate of P 4 167,358 terawatts. The actual number is around the technology now focus on large-scale deploy- approximately 173,000 terawatts (NREL [1]), which is still ment and integration with other societal infrastructure in approximately 10,000 times the worldwide power usage in urban and rural settings (NREL [1]), such as agricultural pro- 2019. Solar power can be harnessed in a variety of ways duction systems. However, placement of such facilities has (a) photovoltaics (largest use), (b) solar heating (primarily become increasingly harder in both urban and rural areas. water heating), (c) solar-thermal technologies (using mirrors, Next generation facilities need to be blended with other concentrators and steam turbines), (d) molten salt technolo- societal structures. Such ‘blended’ systems are exemplified by Agrophotovoltaic (APV) systems, which symbiotically B T. I. Zohdi cohabitate power-generation facilities and agricultural pro- [email protected] duction systems. APV systems have become increasingly popular to help reduce tensions between pure solar farms 1 Department of Mechanical Engineering 6117 Etcheverry Hall, and large-scale agriculture over desirable land. University of California, Berkeley, CA 94720-1740, USA 123 Computational Mechanics POSSIBLE SOURCES SUN WATER MAXIMIZE ABSORPTION/MINIMIZE LOSSES SOLARSSOOLALARAR PANELPPAAANNENELEL EARTH x3 x2 AGRICULTUREA x1 TARGETED LEVEL OF GROUND ABSORPTION Fig. 1 Illumination of Earth for worldwide power estimates, an overall system and an APV unit APV systems were pioneered in the 1980s (Goetzberger tual setting. The system can then be deployed in the physical and Zastrow [2]) and have steadily grown as photovoltaic sys- world afterwards, reducing the potential costs of experiments tems have become more robust and inexpensive. We refer the and accelerating development of new technologies. By creat- reader to [2–25] for a broad survey of such systems.1 Regard- ing Digital-Twins of complex, symbiotic APV systems, one less of the exact type of blended system, there is a necessity to can safely and efficiently manipulate, improve, and optimize optimize these complex multiobjective systems so that they agriculture, careful water use, and integrated solar energy operate properly. If configurations are properly optimized in virtual settings, before deploying them in the physical the approach can yield the best of both worlds, yielding world. We formulate a multiobjective optimization problem energy and abundant agriculture. The approach allows for whose goal is to maximize power absorbed by the panels and agricultural use of land in areas that would otherwise have which also simultaneously matches the power absorbed by been impossible to utilize. A common theme in these new the ground for agriculture. The approach can yield the best of paradigms is optimal system deployment. For a next gen- both worlds, delivering energy and abundant agriculture. The eration solar farm designer, there are multiple surfaces to approach allows planners to utilize and optimize land use in track, such the panel front, panel back, panel sides and the complex regions that would otherwise have been impossible. ground, and therefore a tool that keeps track of the absorption by each is advantageous. Accordingly, this work develops a computational Digital-Twin framework to track and optimize 2 Digital-twin structure and reduced-order the flow of solar power through complex multipurpose solar model farm facilities, such APV systems. Specifically, solar power flow is rapidly computed with a reduced order model of 2.1 Assumptions for reduced order model Maxwell’s equations, based on a high-frequency decomposi- tion of the irradiance into multiple rays, which are propagated The interest here is on the absorption of an initially (test) forward in time to ascertain multiple reflections and absorp- coherent pulse (Fig. 2), represented by multiple collimated tion for various system configurations, varying multi-panel (parallel) rays (initially forming a planar wave front), where inclination, panel refractive indices, sizes, shapes, heights, each ray is a vector in the direction of the flow of power (the ground refractive properties, etc. The method allows for a rays are parallel to the initial wave’s propagation vector). We solar installation to be tested from multiple source direc- make the following observations: tions quickly and uses a genomic-based Machine-Learning Algorithm to optimize the system. This is particularly useful • It is assumed that the features of the surface to be irra- for planning of complex next-generation solar farm systems diated are at least an order of magnitude larger than the involving bifacial (double-sided) panelling, which are capa- wavelength of the incident radiation (essentially specular ble of capturing ground albedo reflection, exemplified by (non-diffusive) optical surfaces), therefore “geometri- APV systems (Fig. 1). A key to the presented approach is cal” ray tracing theory is applicable, and is well-suited the ‘Digital-Twin’ paradigm of physical reality, i.e., digital for the systems of interest. It is important to empha- replicas of complex systems that can then be inexpensively size the regimes of validity of such a model are where and safely manipulated, improved and optimized in a vir- the surface features are larger than the optical wave- lengths. For example, if we were to use rays (10−8 m ≤ λ ≤ × −7 1 APV systems can involve a variety of aspects, even utilizing pollinat- 4 10 m), the features in this analysis would ing insects, such as bees, to ‘solar grazing’ systems. be assumed to possess scales larger than approximately 123 Computational Mechanics BEAM DECOMPOSITION VARIOUS SHAPES SOURCE INCOMING RAYS N I i FRONT Θ np R2 incident i θ θ θ PANEL Θ 1 2 3 r R1 Ir h R3 GROUND ng reflected Θ I SOLAR PANEL SURFACE a a absorbed Fig. 2 Left: an overall system. Right: an electromagnetic pulse applied to a surface and a reduced order ray model 4×10−6 m. For systems containing features smaller than • The power for a ray in the pulse/beam is then given by ¯ r ¯ b this, one can simply use the model as a qualitative guide. I = IA = IA /Nr , where Nr is the number of rays • Ray-tracing is a method that is employed to produce in the beam (Fig. 2) and Ar can be considered the area rapid approximate solutions to wave-equations for high- associated with a ray. frequency/small-wavelength applications where the pri- • Fresnel reflection relations can then be used to compute mary interest is in the overall propagation of power.2 changes in the magnitude of the reflected rays (and the • Ray-tracing methods proceed by initially representing amount absorbed), with directional changes given by the wave fronts by an array of discrete rays. Thereafter, the laws of reflection. problem becomes one of a primarily geometric char- acter, where one tracks the changing trajectories and magnitudes of individual rays which are dictated by the Essentially, rays are a mathematical construction/discreti- reflectivity and the Fresnel conditions (if a ray encounters zation of a pulse/beam. We refer the reader to Gross [26] and a material interface). Zohdi [28–34] for details.