Modelización matemática de la radiación solar fotosintéticamente activa

ANA GARCÍA RODRÍGUEZ TESIS DOCTORAL MODELIZACIÓN MATEMÁTICA DE LA RADIACIÓN SOLAR FOTOSINTÉTICAMENTE ACTIVA Dirigida por las Doctoras: MONTSERRAT DÍEZ MEDIAVILLA CRISTINA ALONSO TRISTÁN Burgos, Febrero 2022 ii Universidad De Burgos Escuela De Doctorado Montserrat Díez Mediavilla y Cristina Alonso Tristán hacen constar: Que el presente trabajo, titulado “Modelización matemática de la radiación solar fotosintéticamente activa”, que presenta Ana García Rodríguez para la obtención del título de doctor ha sido realizado bajo su supervisión, en el Programa de Doctorado Interuniversitario en Eficiencia Energética y Sostenibilidad en Ingeniería y Arquitectura por la Universidad de Burgos, la Universidad de Vigo y la Universidad del País Vasco/Euskal Herriko Unibertsitatea (5600956; RD 99/2011). Burgos, 26 de Febrero de 2022 Fdo. Montserrat Díez Mediavilla Fdo. Cristina Alonso Tristán iii Ana García Rodríguez, 2022 iv A mi familia. v vi Esta tesis doctoral se presenta como un compendio de publicaciones cumpliendo los requisitos establecidos en el Programa de Doctorado “Eficiencia Energética y Sostenibilidad en Ingeniería y Arquitectura”, programa Interuniversitario de la Universidad de Burgos, la Universidad de Vigo y la Universidad del País Vasco/Euskal Herriko Unibertsitatea (RD 99/2011). Las publicaciones recogen los resultados obtenidos en los diferentes trabajos de investigación desarrollados para alcanzar los objetivos planteados para la realización de la tesis. Las publicaciones referidas se enumeran en orden de relación con los objetivos de la tesis en la Sección Lista de Artículos y constituyen las Secciones I a IV de esta memoria de tesis. Tres de los cuatro artículos que componen la tesis han sido publicados en revistas indexadas en el Journal of Citation Reports (JCR) y el último ha sido enviado para su publicación y actualmente se encuentra en proceso de revisión. Se incluye también un quinto artículo, ya publicado, derivado de una de las líneas de investigación futuras propuestas en la tesis. Esta tesis ha sido financiada gracias a los proyectos de investigación siguientes: 1.- Valoración técnica de los niveles de exposición a radiación solar en trabajos de exterior: identificación de grupos de riesgo y medidas de prevención. (INVESTUN/19/BU/004) Junta de Castilla y León. Dirección General de Trabajo y Prevención de riesgos laborales. IP: Montserrat Díez Mediavilla. 01/01/2019-30/09/2021. 2.- Análisis Espectral de la Radiación Solar: Aplicaciones Climáticas, Energéticas y Biológicas (RTI-2018-098900-B-I00). Ministerio de Universidades e Investigación Programa Estatal De I+D+i Orientada a los Retos de la Sociedad. IP: Cristina Alonso Tristán y Montserrat Díez Mediavilla. 1/01/2019-30/09/2022. vii 3.- Metodología para la rehabilitación energética de edificios de uso público en Castilla y León mediante integración fotovoltaica (BU021G19). Junta de Castilla y León. Programa de Apoyo a los Grupos de Investigación Reconocidos de Universidades públicas de Castilla y León. 01/01/2019-31/12/2021. IP: Montserrat Díez Mediavilla. viii Agradecimientos Lo primero de todo, agradecer enormemente a mis Directoras de Tesis, Montse y Cristina, por confiar en mí, apoyarme y ayudarme en la consecución de este trabajo. Y, sobre todo a nivel personal, por su cercanía y trato, y hacerme ver que las cosas se pueden hacer de otra manera. No me puedo olvidar en estos agradecimientos al Grupo SWIFT, en especial al “Grupo G”, donde hemos llevado el compañerismo y la expresión “todos a una” a su máxima expresión. Y con quienes las “to do list” interminables y el apagar fuegos, son mucho más llevaderos. Muchas gracias por todo. No podía faltar en estos agradecimientos mi familia, que me ha apoyado y ayudado en todo momento, calmando mis ánimos cuando más lo necesitaba, y no dejándome caer nunca. Sin vosotros, no habría sido posible. Por todo ello: MUCHAS GRACIAS xv Artículo III A. García-Rodríguez, D. Granados-López, S. García-Rodríguez, M. Díez-Mediavilla, C. Alonso-Tristán (2021). Modelling Photosynthetic Active Radiation (PAR) through meteorological indices under all sky conditions. Agricultural and Forest Meteorology, 310, 108627. DOI: https://doi.org/10.1016/j.agrformet.2021.108627 Journal Citation Reports Índice impacto (2020): 5.734. Cuartil: Q1 (12/94). Categoría JCR: Metheorology and Atmospheric Sciences Scimago Journal Rank: Índice impacto (2020): 1.837. Cuartil: Q1 (15/135) Categoría SJR: Atmospheric Science Número de citas:1 Artículo IV Ana García-Rodríguez, Sol García-Rodríguez, Diego Granados-López, Montserrat Díez-Mediavilla and Cristina Alonso-Tristán (2022). Extension of local all sky conditions 𝑃𝐴𝑅 models to different climatic zones. Applied Sciences, 12, 2372. DOI: https://doi.org/10.3390/app12052372. Journal Citation Reports: Índice impacto (2020): 2.679. Cuartil: Q2 (38/90). Categoría JCR: Engineering Multidisciplinary Scimago Journal Rank: Índice impacto (2020): 0.435. Cuartil: Q2 (1222/6334) Categoría SJR: Engineering Miscellaneous xvi Artículo V Ribas, J.R; Ribas, J.S; García,A.S; Fariña, E.A; Peña,D.G.; Rodríguez, A.G (2021). A multicriteria evaluation of sustainable riparian revegetation with local fruit trees around a reservoir of a hydroelectric power plant in Central Brazil. Sustainability, 13, (14) 7849. DOI: https://doi.org/ 10.3390/su13147849. Journal Citation Reports: Índice impacto (2020): 3.251. Cuartil: Q2 (59/125). Categoría JCR: Environmental Studies. Scimago Journal Rank: Índice impacto (2020): 0.612. Cuartil: Q2 (94/341) Categoría SJR: Environmental Science (miscellaneous). xvii xviii INDICE GENERAL DE LA TESIS Listado de Figuras .......................................................................... xx Listado de Tablas ......................................................................... xxii Nomenclatura .............................................................................. xxiv Capítulo 1 ............................................................................................1 Introducción......................................................................................1 Objetivos principal y parciales ........................................................ 14 Resumen de artículos .................................................................... 16 Capítulo 2 .......................................................................................... 19 Resultados por objetivos ................................................................ 19 Capítulo 3 .......................................................................................... 45 Conclusiones .................................................................................. 45 Líneas de futuro ............................................................................. 46 Bibliografía ......................................................................................... 49 Renuncia coautores publicaciones ..................................................... 61 Artículos............................................................................................. 63 xix xx Listado de Figuras Figura 1. Condiciones de cielo estándar CIE. Imágenes de diferentes tipos de cielo grabadas con SONA201D All-Sky Camera en Burgos, España.[47] … 10 Figura 2. Flujograma con los objetivos y sus publicaciones asociadas………15 Figura 3. Ubicación del equipo experimental en la azotea de la Escuela Politécnica Superior edificio de la Universidad de Burgos, España [47]………21 Figura 4. Vista general de la estación radiométrica situada en la Escuela Politécnica Superior de la Universidad de Burgos …………………………… 21 Figura 5. Ubicación de las siete estaciones meteorológicas de SURFRAD…22 Figura 6. Frecuencia de aparición (FOC,%) de tipos de cielo estándar CIE en Burgos, España, entre abril 2019 y enero 2020 [47] ………………………… 24 Figura 7. Frecuencia de aparición (FOC,%) de la clasificación de nubosidad CIE Burgos, España, entre abril 2019 y enero 2020 [47]. …………………… 25 Figura 8. Densidad de flujo de fotones fotosintéticos, 𝑄𝑝(𝜇𝑚𝑜𝑙 · 𝑚−1 · 𝑠−2) e irradiancia solar de banda ancha, 𝑅𝑎𝐺𝐻(𝑊 · 𝑚−2)(medida en Burgos, entre abril de 2019 y enero de 2020. [47]. …………………………………………… 26 Figura 9. Diagrama de cajas y bigotes de la ratio densidad de flujo de fotones fotosintético e irradiancia solar de banda ancha (QpRaGH) ⁄ para cada tipo de cielo estandard CIE [47]. ………………………………………………………… 27 Figura 10. Resultados estadísticos de los modelos para datos de entrada de 1 minuto……………………………………………………………………………… 31 Figura 11. Frecuencia de aparición (𝐹𝑂𝐶,%) de los tipos de cielo estándar CIE en Burgos, España, entre abril de 2019 y febrero de 2021 [74]..…………… 33 Figura 12. Análisis estadístico de la cobertura de nubes (CC) antes (1) y después (2) de filtrar los datos atípicos [74] ……………………………………34 Figura 13. El coeficiente de correlación de Pearson (𝑟(𝑃𝐴𝑅, 𝑀𝐼𝑖)) calculado para los 10 𝑀𝐼 utilizados en este estudio [74] ………………………………… 35 Figura 14. Arquitectura del sistema de red neuronal artificial mediante el algoritmo de retropropagación de Levenberg-Marquardt [74]…………………37 xxi xxii Listado de Tablas Tabla 1. 15 tipos de cielo clasificados según la norma ISO/CIE………………9 Tabla 2. Marcas y modelos de los sensores instalados en la instalación……20 Tabla 3. Modelos evaluados para la estimación de la 𝑃𝐴𝑅 ……………………29 Tabla 4. Clasificación del cielo según el índice del cielo (SI) propuesto por Igawa. ……………………………………………………………………………… 30 Tabla 5. El coeficiente de correlación de Pearson o Pearson 𝑟(𝑃𝐴𝑅,𝑀𝐼𝑖) teniendo en cuenta las condiciones de cielo definidas por la clasificación del cielo estándar (cubierto, parcial y claro) [74]……………………………………35 Tabla 6. Modelos de regresión multilineal de la 𝑃𝐴𝑅 (modificado a partir de [74]) ................................................................................... ……………………36 Tabla 7. Resultados estadísticos de los modelos de regresión multilineal (modificado a partir de [74]) ………………………………………………………36 Tabla 8. Resultados estadísticos de los modelos de redes neuronales [74] 38 Tabla 9. Índices meteorológicos medidos en Burgos [76]. ……………………40 Tabla 10. Coeficiente de Pearson, 𝑟(𝑃𝐴𝑅,𝑀𝐼𝑖) basado en las condiciones de cielo definidas por la clasificación de cielo de 𝑘𝑡(claro, parcial y cubierto) [76].41 Tabla 11. Modelos de regresión multilineal de la 𝑃𝐴𝑅 [76].……………………42 Tabla 12. Condiciones climáticas de las siete estaciones americanas (modificado a partir de [76]) .......................................................................... 43 xxiii xxiv Nomenclatura Simbolos 𝐴𝑂𝐷 profundidad óptica de los aerosoles 𝑐𝑜𝑠𝑍 coseno del ángulo zenital 𝐶𝐶 cobertura nubosa (%) 𝑒 presión de vapor de agua (ℎ𝑃𝑎) 𝜀 claridad del cielo de Perez [1] 𝛥 brillo del cielo de Perez [1] 𝐺0 irradiancia extraterrestre (𝑊 · 𝑚−2) 𝑘𝑏 fracción directa 𝑘𝑑 fracción difusa de la radiación 𝑘𝑡 índice de claridad 𝑘𝑡,𝑃𝐴𝑅 índice de claridad para el rango de radiación cubierto por la 𝑃𝐴𝑅 𝐿𝐷 longitud del día 𝑚 masa óptica relativa del aire 𝑛𝑚 duración de horas de sol 𝑃 presión de aire (ℎ𝑃𝑎) 𝑄𝑝 densidad de flujo de fotones fotosintéticos (𝜇𝑚𝑜𝑙 · 𝑚−1 · 𝑠−2) 𝑄𝑝/𝑅𝑎𝐺𝐻 relación entre 𝑄𝑝 y 𝑅𝑎𝐺𝐻 𝑅2 coeficiente de determinación 𝑅𝑎𝐷𝐻 irradiancia difusa horizontal (𝑊 · 𝑚−2) 𝑅𝑎𝐺𝐻 irradiancia global horizontal (𝑊 · 𝑚−2) 𝑅𝐻 humedad relativa (%) 𝑆 insolación relativa 𝑆𝐼 índice de cielo [2] 𝑇 temperatura del aire (º𝐶) Capítulo 1: Introducción y Objetivos 6 atenuación de 𝑃𝐴𝑅 incidente) y a temperaturas más altas que los corales del arrecife cercano. También concluyeron que la materia orgánica disuelta coloreada no contribuía en gran medida a la atenuación de la radiación solar. Jackson et al. [17] indicaron que la tasa de fotosíntesis en las zooxantelas asociadas a los corales aumenta linealmente con la 𝑃𝐴𝑅 hasta que los fotosistemas se saturan y se alcanza la tasa máxima. Una vez alcanzado este umbral máximo de luz se produce la fotoinhibición y el exceso de energía lumínica se disipa en forma de calor a través de varios mecanismos fotoprotectores. Si no se disipa toda la energía, se producen daños. Cuando la tasa de fotodeterioro supera a la de reparación fotoprotectora, se acumula el fotodaño y se liberan compuestos oxidativos de oxígeno en los tejidos del coral. Si las condiciones persisten, los corales expulsarán sus zooxantelas y se blanquearán. Una alta concentración de sólidos suspendidos totales puede agravar el fotodaño al reducir aún más la capacidad de absorción de 𝑃𝐴𝑅. El actual calentamiento de los océanos, la acidificación y la disminución de la calidad del agua están afectando negativamente a los ecosistemas de los arrecifes de coral y pueden provocar un cambio de régimen ecológico, ya que la resistencia de los corales a las perturbaciones sigue disminuyendo. Esto puede llevar a un cambio en la producción de DMS y en el flujo mar-aire de los arrecifes de coral, con posibles impactos en el equilibrio radiativo local. Todos estos trabajos ponen de manifiesto la necesidad de conocer y modelar la 𝑃𝐴𝑅, debido a su implicación en diferentes aspectos de la vida. La 𝑃𝐴𝑅 es un factor clave para diferentes procesos energéticos, físicos y biológicos importantes en el medio ambiente, y está directamente relacionada con el cambio climático. Es preciso conocer la evolución de esta variable en el tiempo y en función de las condiciones atmosféricas. Capítulo 1: Introducción y Objetivos 7 Pese a su importancia, la mayor parte de las estaciones meteorológicas terrestres no disponen de sensores para la medida de la 𝑃𝐴𝑅, por lo que, en general, es necesario abordar su cálculo a través de la determinación de la relación de la 𝑃𝐴𝑅 con otras variables más habitualmente medidas. Distintos estudios publicados [18–20] en diferentes partes del mundo analizan la relación entre el flujo de fotones (𝑄𝑝), y la irradiancia global horizontal (𝑅𝑎𝐺𝐻) , 𝑄𝑝𝑅𝑎𝐺𝐻. ⁄ Los resultados ponen de manifiesto que la relación no es constante pero que, a efectos prácticos, la 𝑃𝐴𝑅 supone entre un 45 −50% de la 𝑅𝑎𝐺𝐻. Diferentes autores han establecido modelos matemáticos para obtener la ratio 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ a través de otros parámetros climáticos. De los resultados obtenidos concluyeron que apenas hay dependencias significativas de la relación 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ con factores como la localización, estacionalidad o los efectos diurnos [21]. El estudio de dependencia de la 𝑃𝐴𝑅 con las condiciones de nubosidad del cielo ha sido abordado en numerosas ocasiones, obteniendo que la ratio 𝑄𝑝𝑅𝑎𝐺𝐻, ⁄ presenta valores más elevados en condiciones de cielo cubierto y disminuye en condiciones de cielo claro [22,23]. Este hecho es atribuible a los fenómenos de absorción y dispersión de la radiación solar a través de diferentes regiones del espectro. Además de la modelización de la 𝑃𝐴𝑅 a través de su relación directa con la 𝑅𝑎𝐺𝐻 , se han desarrollado modelos que incluyen otros parámetros y variables meteorológicas. Matemáticamente, muchos de estos modelos, están basados en regresiones lineales a partir de medidas de 𝑅𝑎𝐺𝐻 [24–26], o valores de masa óptica relativa del aire (𝑚) [27], como variables únicas. Otros autores han modelado la 𝑃𝐴𝑅 considerando la influencia simultánea de diversas variables. Zempila et al. [28] desarrollaron un modelo apartir de la 𝑅𝑎𝐺𝐻, del ángulo solar zenital (𝑍), de la columna de vapor de agua precipitable (𝑊𝑉) y de la profundidad óptica de los aerosoles (𝐴𝑂𝐷). Wang et al. [29] estimaron Capítulo 1: Introducción y Objetivos 8 la 𝑃𝐴𝑅 a partir del índice de claridad (𝑘𝑡), la longitud del día (𝐿𝐷) y el ángulo zenital (𝑍). Ferrera-Cobos et al. [30] utilizaron como variables de entrada para sus modelos la 𝑅𝑎𝐺𝐻, la irradiancia extraterrestre (𝐺0), la temperatura (𝑇) y la humedad relativa (𝑅𝐻). En la mayoría de los trabajos se han analizado estas relaciones en función de las condiciones de cielo. Para cielos claros, Mottus et al. [31] encontraron que 𝑍 es la variable más importante y el índice de claridad (𝑘𝑡) permite usar el modelo en presencia de nubes. Distintos estudios [30,32–34] destacan que la 𝑃𝐴𝑅 es una variable de carácter fuertemente local de modo que los modelos desarrollados dependen de la localización geográfica donde se obtienen los datos experimentales. Como toda componente espectral de la radiación solar, la 𝑃𝐴𝑅 está relacionada con las condiciones atmosféricas, representadas por los tipos de cielo. La clasificación de cielos (cubiertos, parcialmente cubiertos y claros) es un problema complejo debido a la utilización de un concepto abstracto como es la consideración de cubierto, claro o todo el rango de cielos parciales. Tradicionalmente, se han clasificado los cielos en función de la nubosidad atendiendo a distintos índices meteorológicos o combinaciones de ellos, como el índice de claridad (𝑘𝑡) [32,35,36], la insolación relativa (𝑆) [37,38], la claridad del cielo (𝜀), o el brillo del cielo (𝛥) [39–42] . Es necesario tener en cuenta la capacidad limitada para clasificar cielos a partir de índices meteorológicos [43], demostrada en diferentes trabajos. En este sentido, Kittler et al. [44,45] propusieron un conjunto de tipos de cielo estándar que describían distintas distribuciones angulares de luminancia en la bóveda celeste. Esta propuesta se consolidó en 2004 con el CIE Standard General Sky recogido en la norma ISO 15469:2004(E)/CIE S 011/E:2003 [46]. La norma ISO/CIE establece los 15 tipos de cielo que se muestran en la Tabla 1. De ellos, los cinco primeros pertenecen al grupo de cielos cubiertos, los cinco siguientes Capítulo 1: Introducción y Objetivos 9 al de cielos intermedios y los últimos cinco al de claros. La Figura 1 muestra las principales características de cada tipo de cielo estándar según la norma ISO/CIE. Tabla 1. 15 tipos de cielo clasificados según la norma ISO / CIE. Tipo de cielo Descripción de la distribución de luminancia 1 Cielo cubierto estándar CIE, gradación acusada y uniformidad acimutal. 2 Cubierto, con gradación pronunciada y un ligero brillo hacia el sol. 3 Cubierto, moderadamente gradado, con uniformidad azimutal. 4 Cubierto, moderadamente graduado, con un ligero brillo hacia el sol. 5 Cubierto, brumoso o nublado, con uniformidad general. 6 Parcialmente nublado, con gradación uniforme y un ligero brillo hacia el sol. 7 Parcialmente nublado, con un efecto circunsolar brillante y gradación uniforme. 8 Parcialmente nublado, bastante uniforme, con una clara corona solar. 9 Parcialmente nublado, con sol oscurecido. 10 Parcialmente nublado, con región circunsolar más brillante. 11 Cielo blanco-azulado con una clara corona solar. 12 Cielo claro estándar CIE, turbidez de baja iluminancia. 13 Cielo claro estándar CIE, atmósfera contaminada. 14 Cielo turbio sin nubes con amplia corona solar. 15 Cielo turbio blanco-azulado con amplia corona solar. Capítulo 1: Introducción y Objetivos 10 Figura 1. Condiciones de cielo estándar CIE. Imágenes de diferentes tipos de cielo grabadas con un SONA201D SONA201D All-Sky Camera—Day en Burgos, España [47]. En relación con la temporalidad de las medidas, Frouin y Pinker [48] observaron que, en las regiones tropicales y árticas, la proporción de 𝑃𝐴𝑅 e insolación permanecía bastante constante en diferentes escalas de tiempo tanto diarias como más largas, independientemente de la Capítulo 1: Introducción y Objetivos 11 nubosidad, la composición atmosférica, el tipo de superficie, la estación o la duración del día. La inteligencia artificial se ha utilizado también para modelar la 𝑃𝐴𝑅, utilizando algoritmos supervisados, como por ejempo, las redes neuronales artificiales (𝐴𝑁𝑁). La principal ventaja de estos sistemas radica en que no es necesario, a priori, conocer las relaciones existentes entre las variables de entrada y salida, permitiendo, con el entrenamiento adecuado del sistema, generar un modelo lo suficientemente preciso para ser utilizado con datos de entrada diferentes a los del entrenamiento. Ferrera-Cobos et al. [30] modelaron la 𝑃𝐴𝑅 en climas oceánicos y mediterráneos a través del test de 22 modelos, 11 de regresión multilineal (𝑀𝐿𝑅) y 11 modelos basados en 𝐴𝑁𝑁, utilizando como variables de entrada la 𝑅𝑎𝐺𝐻, la 𝐺0, la 𝑇 y la 𝑅𝐻. Concluyeron que zonas con diferentes condiciones climáticas necesitan diferentes modelos, ajustándose mejor los modelos para climas mediterráneos, que los modelos para climas oceánicos. Estos últimos necesitan una corrección dependiente de su localización geográfica, ya que el mayor porcentaje de humedad que existe en un clima oceánico por la presencia de agua en la atmósfera absorbe más radiación en el espectro infrarrojo aumentando la relación entre la 𝑃𝐴𝑅 y 𝑅𝑎𝐺𝐻. Aunque el uso combinado de satélites geoestacionarios y de órbita polar es una solución óptima para estimar la 𝑃𝐴𝑅 , es necesario disponer de una red terrestre de medida de 𝑃𝐴𝑅 para validar los modelos [48]. Sudhakar et al. [25] obtuvieron un modelo de regresión de potencia para la estimación de la 𝑃𝐴𝑅 en la India en 6 latitudes, entre 9º y 34º, basado en el promedio horario y mensual de la radiación global diaria. El ajuste de dicho modelo tuvo en cuenta los distintos ángulos solares, nubosidad, condiciones climáticas y su relación con la 𝑅𝑎𝐺𝐻. Para desarrollar los distintos modelos de 𝑃𝐴𝑅, varios autores [26,30] han utilizado datos satelitales proporcionados por la Satellite Capítulo 1: Introducción y Objetivos 12 Application Facility on Climate Monitoring (CM-SAF), aunque para su validación ha sido necesario el uso de datos de campo. Para cubrir esa necesidad, Vindel et al. [49] presentaron una metodología para determinar ubicaciones óptimas para instalar estaciones de medición de 𝑃𝐴𝑅. Esta metodología se basa en un proceso de agrupamiento aplicado a la 𝑘𝑡,𝑃𝐴𝑅 , parámetro que se calcula dividiendo la 𝑃𝐴𝑅 recibida en la superficie de la Tierra por la parte del espectro correspondiente a la banda de la 𝑃𝐴𝑅 en la parte superior de la atmósfera (39,8% del total). Nyamsi et al. [50] presentaron y validaron un método de cielo despejado para estimar la 𝑃𝐴𝑅 a partir del contenido total de la columna de ozono (𝑇𝑂𝐶), del 𝑊𝑉 y de las propiedades ópticas de los aerosoles proporcionadas por el Copernicus Atmosphere Monitoring Service. El método se validó comparando sus resultados con el valor de 𝑃𝐴𝑅 experimental medido cada minuto, en siete estaciones de la red Surface Radiation (SURFRAD) de Estados Unidos, ubicadas en localizaciones con diferentes climatologías. En todas las estaciones el coeficiente de determinación (𝑅2) resultó mayor de 0,97, lo que demuestra que la gran mayoría de la variabilidad temporal se reproduce bien con el método propuesto. Los autores indicaron que el modelo propuesto se puede combinar con otros que tengan en cuenta la atenuación debida a las nubes para proporcionar estimaciones de 𝑃𝐴𝑅 de todo el cielo. Ferrera-Cobos et al. [30], al comparar modelos de 𝑀𝐿𝑅 y modelos basados en 𝐴𝑁𝑁, observaron que, para ambos métodos, la variable más determinante para la estimación de la 𝑃𝐴𝑅 era la 𝑅𝑎𝐺𝐻, independientemente del clima. De hecho, al realizar los modelos con 𝐴𝑁𝑁, cuando la 𝑅𝑎𝐺𝐻 no estaba incluida en ellos, dichos modelos empeoraban claramente. En este sentido Jacovides et al. [51] utilizaron como variables de entrada para la realización de los modelos basados en 𝐴𝑁𝑁 la duración de horas de sol (𝑛𝑀), la 𝑇, la 𝑅𝑎𝐺𝐻, la 𝐺0 y la 𝑅𝐻. Capítulo 1: Introducción y Objetivos 13 Obtuvieron que la duración de la luz solar juega un papel importante a la hora de obtener predicciones de modelos aceptables, y que el modelo que mejor predice los valores de 𝑃𝐴𝑅 es el que combina la 𝑛𝑚 y la 𝑅𝑎𝐺𝐻. López et al. [52] realizaron un modelo a través de datos de 𝑃𝐴𝑅 obtenidos de estaciones radiométricas mediante una 𝐴𝑁𝑁, estimando la 𝑃𝐴𝑅 a partir de la 𝑅𝑎𝐺𝐻 horaria como única variable de entrada. Con una segunda 𝐴𝑁𝑁 se desarrolló un modelo basado en mediciones de la duración de la luz solar y se demostró que es una alternativa aceptable para calcular la 𝑃𝐴𝑅. Pankaew et al. [53],a partir de un modelo de 𝐴𝑁𝑁 cuyas variables de entrada fueron diferentes parámetros atmosféricos (coseno del ángulo de cénit solar ,𝑐𝑜𝑠𝑍, índice de nubosidad,𝐶𝐶, WV, y 𝐴𝑂𝐷, estimaron la 𝑃𝐴𝑅 horaria, a partir de datos satelitales. Qin et al. [54] propusieron ocho modelos de inteligencia artificial, de los cuales, el obtenido a partir de una red neuronal de retropropagación (𝐵𝑃) fue el que presentó la mayor precisión. Wang et al. [55] propusieron tres métodos de 𝐴𝑁𝑁 mejorados, (perceptrón multicapa, 𝑀𝐿𝑃, red neuronal de regresión generalizada, 𝐺𝑅𝑁𝑁, y red neuronal de base radial, 𝑅𝐵𝑁𝑁, para la estimación de la 𝑃𝐴𝑅, a partir de observaciones horarias a largo plazo de 𝑃𝐴𝑅 , 𝑅𝑎𝐺𝐻 y diferentes variables meteorológicas ( 𝑇, 𝑅𝐻, temperatura de rocío (𝑇𝑑), presión de vapor de agua (𝑒), presión de aire (𝑃)). Concluyeron que los diferentes parámetros meteorológicos influyen de manera diferente en la estimación de la 𝑃𝐴𝑅 según los diferentes ecosistemas afectados (tierras agrícolas, tierra, bosque, bahía, pradera, desierto y lago). Capítulo 1: Introducción y Objetivos 14 1.2. Objetivos principal y parciales El objetivo principal de este trabajo es caracterizar y modelar la radiación fotosintéticamente activa (𝑷𝑨𝑹). Como se ha puesto de manifiesto en la Sección anterior, la 𝑃𝐴𝑅 es un parámetro clave para la obtención de biomasa, para la fotosíntesis y para la producción vegetal primaria. Los objetivos parciales planteados para obtener el objetivo principal son los siguientes: 1.- Análisis y estudio estadístico de la 𝑃𝐴𝑅. 2.- Revisión bibliográfica de modelos de 𝑃𝐴𝑅 y ajuste con valores experimentales en diferentes ubicaciones. 3.- Modelado local de la 𝑃𝐴𝑅 con diferentes índices meteorológicos para todo tipo de cielo y con diferentes categorías de cielo. 4.- Modelado de la 𝑃𝐴𝑅 para Burgos en función de los tipos de cielo clasificados según el índice de claridad (𝑘𝑡) y validación de los mismos para otras localizaciones. Cada uno de estos objetivos parciales que completan el objetivo principal se ha materializado con la publicación de un artículo científico en una revista bajo revisión por pares e indexada en el Journal Citation Reports, en los primeros cuartiles de distintas categorías, como se ha indicado en la Sección Listado de Artículos de esta memoria. La interrelación entre los objetivos y los resultados obtenidos se muestra en el flujograma de la Figura 2: Capítulo 1: Introducción y Objetivos 15 Figura 2. Flujograma con los objetivos y sus publicaciones asociadas Este documento se estructura de la siguiente forma: en el Capítulo 1, se realiza la introducción del tema fundamental del trabajo, la 𝑃𝐴𝑅, su importancia e influencia en la fotosíntesis, el cambio climático y la producción vegetal, a través de la revisión del estado del arte. En este mismo capítulo se describe el objetivo principal y los objetivos parciales del trabajo y se incluye un breve resumen de los artículos que constituyen el trabajo. En el Capítulo 2: Resultados por Objetivos, se describe la investigación realizada para caracterizar y modelar la 𝑃𝐴𝑅 de acuerdo con los artículos publicados, con los resultados obtenidos y que van conformando los objetivos parciales. Para finalizar, el Capítulo 3 recoge las conclusiones y las líneas futuras de trabajo. Capítulo 2: Resultados por objetivos 22 (𝑅𝐻). Otras variables utilizadas se han calculado a través de expresiones conocidas como la temperatura de rocío (𝑇𝑑), la presión de vapor de agua (𝑒) y el agua precipitable (𝑊𝑉). Figura 5. Ubicación de las siete estaciones meteorológicas de SURFRAD. La red SURFRAD somete los datos de irradiancia registrados por sus estaciones al procedimiento de control de calidad propuesto por la Baseline Surface Radiation Network (BSRN) y recogido en Long y Dutton [57]. Dicho procedimiento contempla tres niveles de control de las medidas de irradiancia , pero ninguno para la 𝑃𝐴𝑅. En este trabajo, se ha establecido un criterio de control de calidad de 𝑃𝐴𝑅 relativo a sus límites físicos: la 𝑃𝐴𝑅 global sobre el plano horizontal no debe superar el valor de 𝑃𝐴𝑅 extraterrestre sobre el plano horizontal correspondiente a ese mismo momento. Gueymard [58] obtuvo un valor de 1361.1 𝑊 ∙ 𝑚−2 de constante solar o irradiancia solar total pero sin valor estandarizado para la irradiancia en el rango espectral de la 𝑃𝐴𝑅. La curva de irradiancia entre 400 𝑦 700 𝑛𝑚 del espectro extraterrestre propuesto por Gueymard et al. [59] fue integrado, para poder estimar esta constante, resultando un valor de 530.8 𝑊 ∙ 𝑚−2. El objetivo general de la tesis, la caracterización y modelado de la 𝑃𝐴𝑅, se ha dividido en 4 tareas, que conforman cada una de ellas un objetivo Capítulo 2: Resultados por objetivos 23 parcial de la tesis, y que se completan con los cuatro artículos publicados que recogen los resultados de la investigación realizada. 2.1. Objetivo parcial 1: Análisis y estudio estadístico de la 𝑷𝑨𝑹 El Objetivo Parcial 1, como se observa en el flujograma representado en la Figura 1, radica en conocer la variabilidad temporal y la relación de esta componente de la radiación solar con las condiciones climáticas. Este objetivo parcial se concreta en el artículo Photosynthetic Active Radiation, Solar Irradiance and the CIE Standard Sky Classification, publicado en Applied Sciences. Como se ha comentado anteriormente, la escasez de datos experimentales de 𝑃𝐴𝑅 hace necesario modelarla a partir de variables medidas habitualmente, como puede ser la 𝑅𝑎𝐺𝐻. Muchos trabajos coinciden en que la relación 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ es prácticamente constante y se establece una ratio 𝑄𝑝/𝑅𝑎𝐺𝐻 que oscila entre 1,75-2,3 𝜇𝑚𝑜𝑙 · 𝐽−1 [60]. En este trabajo, se evalúa la relación 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ en Burgos y se analiza su dependencia con las características climatológicas definidas a partir de los distintos tipos de cielo y para diferentes intervalos temporales (diezminutales, horarios, diarios, mensuales, estacionales). La revisión bibliográfica ha comprobado que la mayoría de los autores trabajan para cielos claros [31,50,61,62]. En este estudio se ha utilizado el estándar ISO/CIE para la clasificación de los cielos. Para ello, se ha caracterizado la frecuencia de aparición, 𝐹𝑂𝐶,(%), de cada tipo de cielo estándar ISO/CIE en Burgos, considerando para su clasificación la ratio de normalización (NR) introducida por Littlefair [63,64]. La 𝐹𝑂𝐶 de cada tipo de cielo durante el periodo estudiado, desde abril de 2019 hasta enero de 2020, se muestra en la Figura 6. Capítulo 2: Resultados por objetivos 24 Figura 6. Frecuencia de aparición (FOC, %) de tipos de cielo estándar CIE en Burgos, España, entre abril 2019 y enero 2020 [47]. De la Figura 6 se deduce que, en la ciudad de Burgos aparecen los 15 tipos de cielo clasificados según la norma ISO/CIE. Los tipos de cielo correspondientes a las categorías 11,12, y 13, que se describen como cielos claros, presentan una frecuencia de aparición alta, entre el 10 y 14,5%, seguidos del tipo de cielo 14, con una frecuencia del 8,4%. Los tipos de cielo 1,7, 8 y 15 presentan valores de 𝐹𝑂𝐶 en torno al 7%. Los cielos 5,9 y 10 presentan una frecuencia muy baja, inferior al 3%. La utilización del estándar ISO/CIE para la clasificación de cielos es poco usual en la literatura científica, puesto que esta clasificación precisa de dispositivos sky-scanner para poder obtener la distribución de luminancia y radiancia de la cúpula celeste [45]. Habitualmente, los cielos se clasifican, atendiendo a la nubosidad, en cielos claros, parciales y cubiertos. El estándar ISO/CIE considera cielos cubiertos los tipos de cielo 1 a 5, parciales los tipos 6 a 10 y claros los tipos 11 a 15. Otras clasificaciones que emplean la misma taxonomía (cielos claros, parciales y cubiertos) utilizan como criterio de evaluación, el valor numérico de uno o varios 𝑀𝐼’𝑠 combinados, estableciendo intervalos de clasificación [43]. Atendiendo a esta clasificación, la 𝐹𝑂𝐶 de cada una de dichas categorías en Burgos se describe en la Figura 7. Durante la campaña de medidas, los cielos son predominantemente claros, con una 𝐹𝑂𝐶 cercana al 62%; mientras que las categorías de Capítulo 2: Resultados por objetivos 25 cielos cubiertos y parcialmente cubiertos tienen una frecuencia considerablemente inferior a la de los cielos claros, entorno al 20%. Figura 7. Frecuencia de aparición (FOC, %) de la clasificación de nubosidad CIE Burgos, España, entre abril 2019 y enero 2020 [47]. Los resultados obtenidos revelan que los cielos cubiertos predominan durante los meses de noviembre a enero y los cielos claros de mayo a octubre. Además, los cielos claros predominan en todas las horas del día, aunque las diferencias de 𝐹𝑂𝐶 con respecto a los cielos cubiertos y parciales, decrecen hacia el mediodía. Una vez conocidas las características de los cielos en la localización de medida, el trabajo ha analizado la dependencia de la ratio 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ en base diezminutal. Se observa (Figura 8) una relación lineal prácticamente perfecta entre la 𝑄𝑝 y la 𝑅𝑎𝐺𝐻, con un valor de 𝑅2= 0,99. La pendiente de la curva es 1,893 ± 0,001 𝜇𝑚𝑜𝑙 ∙ 𝐽−1, valor que se ajusta a los datos obtenidos por otros autores [65]. Los valores 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ se han promediado en base diezminutal, media horaria y mensual y se han analizado estadísticamente para detectar valores anómalos. Se han obtenido las gráficas de cajas y bigotes 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ para cada una de las bases temporales analizadas. Capítulo 2: Resultados por objetivos 26 Figura 8. Densidad de flujo de fotones fotosintéticos, 𝑄𝑝 (𝜇𝑚𝑜𝑙 ∙ 𝑠−1 ∙ 𝑚−2) e irradiancia solar de banda ancha, 𝑅𝑎𝐺𝐻(𝑊 ∙ 𝑚−2) medida en Burgos, entre abril de 2019 y enero de 2020 [47]. Con respecto a la variación diaria, se observa que la relación entre ambas magnitudes va aumentando en las primeras horas del día, se estabiliza en las horas centrales y tiende a disminuir a medida que se alcanza la puesta de sol. La dispersión de los valores es mayor en las primeras y últimas horas del día frente a las centrales. Los datos mensuales son prácticamente constantes durante los meses centrales del año, de mayo a octubre, oscilando la desviación estándar entre 0,11 𝑦 0,16 𝜇𝑚𝑜𝑙 ∙ 𝐽−1 y el rango intercuartílico varía entre 0,10 𝑦 0,15 𝜇𝑚𝑜𝑙 ∙ 𝐽−1. Es de destacar que noviembre es el mes con mayor dispersión con un rango intercuartílico de 0,24 𝜇𝑚𝑜𝑙 ∙ 𝐽−1, y una desviación estándar de 0,20 𝜇𝑚𝑜𝑙 ∙ 𝐽−1. El valor máximo se obtiene en abril (1,98 ± 0,15 𝜇𝑚𝑜𝑙 ∙ 𝐽−1), mientras que el mínimo se alcanza en diciembre (1,91 ± 0,17 𝜇𝑚𝑜𝑙 ∙ 𝐽−1). La media mensual de 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ es de 1,93 ± 0,025 𝜇𝑚𝑜𝑙 ∙ 𝐽−1. El siguiente estudio estadístico se realiza considerando el cociente 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ con datos diezminutales para cada tipo de cielo clasificado según la norma ISO/CIE como se puede observar en la Figura 9. En esta gráfica se deduce que los tipos de cielo claro 12,13,14 𝑦 15 muestran desviaciones estándar muy pequeñas (0,06 a 0,11 𝜇𝑚𝑜𝑙 ∙ 𝐽−1) y un menor rango intercuartílico. Los valores Capítulo 2: Resultados por objetivos 27 más altos de 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ se obtienen en los tipos de cielo CIE 1,2,3 𝑦 4 que se corresponden con las condiciones de cielo cubierto. Los tipos de cielo 5 𝑦 9 presentan un comportamiento anómalo. La norma ISO/CIE define estos dos tipos de cielo como cubierto y parcialmente cubierto, pero en este caso para Burgos presentan valores medios de 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ más próximos a los valores de las categorías de cielo claro. Figura 9. Diagrama de cajas y bigotes de la ratio densidad de flujo de fotones fotosintético e irradiancia solar de banda ancha (𝑄𝑝𝑅𝑎𝐺𝐻) ⁄ para cada tipo de cielo estandard CIE [47]. Del estudio de esta clasificación se desprende que el promedio de la relación 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ para cielos claros y para cielos cubiertos es de 1,90 ± 0,11 𝜇𝑚𝑜𝑙 ∙ 𝐽−1 y 1,98 ± 0,16 𝜇𝑚𝑜𝑙 ∙ 𝐽−1, respectivamente. Los cielos parcialmente cubiertos muestran la mayor dispersión de los datos con valores de desviación estándar de 0,17 y un rango intercuartílico de 0,19 𝜇𝑚𝑜𝑙 ∙ 𝐽−1 , siendo el valor medio 1,93 𝜇𝑚𝑜𝑙 ∙ 𝐽−1. Sin embargo, el conjunto de datos de cielos claros presenta el mayor número de valores atípicos. Para concluir el análisis estadístico, se realiza el cálculo del coeficiente de correlación de Spearman, que determina la fuerza y la dirección de la correlación entre 2 variables. También en este caso se analiza la relación 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ con la clasificación CIE, y se obtiene una correlación moderada de −0,23 y un 𝑝 − 𝑣𝑎𝑙𝑜𝑟 < 0,001. Como el 𝑝 − 𝑣𝑎𝑙𝑜𝑟 es inferior a 0,05 en todas las pruebas y este parámetro es el que Capítulo 2: Resultados por objetivos 28 determina la significancia de los resultados con la hipótesis nula (es decir, los resultados son debidos al azar), podemos concluir que hay una diferencia estadísticamente significativa entre los promedios de la relación para cada tipo de cielo CIE, con un nivel de significancia del 5%. Cuando se analiza la correlación Spearman para la clasificación en tres tipos de cielo se obtiene un valor de 0,52 lo que califica como débil la relación entre el valor de la relación 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄y la clasificación del cielo estándar ISO/CIE en tres categorías. Dado que los valores de la relación son más altos en condiciones de cielo cubierto y menores para cielo claro, se concluye la importancia de la presencia o ausencia de nubes en el valor del ratio 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄. Analizada la 𝑃𝐴𝑅, su variación y su relación con la irradiancia global horizontal y el tipo de cielo, se ha conseguido cumplir el primer objetivo parcial de la tesis doctoral: Análisis y estudio estadístico de la 𝑷𝑨𝑹. 2.2. OBJETIVO PARCIAL 2: Revisión bibliográfica de modelos de 𝑷𝑨𝑹 y ajuste con valores experimentales en diferentes ubicaciones. El Objetivo Parcial 2 consiste en determinar los diferentes modelos de 𝑃𝐴𝑅 desarrollados hasta el momento, a través de una rigurosa revisión bibliográfica. Este objetivo se completa con un análisis pormenorizado de la bondad estos modelos mediante su aplicación a los datos experimentales obtenidos de 7 estaciones pertenecientes a la red SURFRAD, que incluye estaciones situadas en localizaciones con diferentes condiciones climáticas según la clasificación Köppen [66]. Este objetivo parcial se concreta en el artículo Validation and Calibration of Models to Estimate Photosynthetically Active Radiation Considering Different Time Scales and Sky Conditions, trabajo actualmente en proceso de revisión en la revista Advances in Space Research. Capítulo 2: Resultados por objetivos 29 La revisión bibliográfica realizada ha permitido seleccionar 21 modelos de 𝑃𝐴𝑅 desarrollados en distintas ubicaciones, con diferentes escalas temporales. La selección de los modelos se ha realizado atendiendo a las variables disponibles en las estaciones meteorológicas seleccionadas. Estos modelos se describen en la Tabla 3, donde se muestra la expresión matemática del modelo y la referencia bibliográfica de la que se ha obtenido. Distintos autores manifiestan que la validez de los modelos depende de las características meteorológicas estacionales y geográficas del lugar para el que se desarrolla el modelo [30,32–34]. Los modelos seleccionados se desarrollaron originalmente para datos con diferente resolución temporal, en concreto, para valores diezminutales y mensuales. Tabla 3. Modelos evaluados para la estimación de la 𝑃𝐴𝑅. Modelo EXPRESIÓN REF. M1 𝑃𝐴𝑅 = 𝐺(𝑎1𝐿𝑛𝜖 + 𝑏1𝐿𝑛Δ + 𝑐1𝑇𝑑+ 𝑑1𝑐𝑜𝑠2𝜃𝑧+ 𝑓1) [42] M2 𝑃𝐴𝑅 = 𝐺(𝑎2ln𝜀 + 𝑏2ln∆ + c2cos2𝜃𝑧+ 𝑑2) [42] M3 𝑃𝐴𝑅 = 𝐺(𝑎3ln𝑘𝑇+ 𝑏3𝑇𝑑+ 𝑐3cos𝜃𝑧+ 𝑑3) [42] M4 𝑃𝐴𝑅 = 𝐺(𝑎4ln𝑘𝑇𝑏4cos𝜃𝑧+ 𝑐4) [42] M5 𝑃𝐴𝑅 = 𝐺(𝑎5ln𝑘𝑇+ 𝑏5) [50] M6 𝑃𝐴𝑅 = 𝐺[𝑎6(ln𝑘𝑇)2+ 𝑏6ln𝑘𝑇+ 𝑐6] [67] M7 𝑃𝐴𝑅 = 𝐺(𝑎7ln𝑘𝑑+ 𝑏7ln∆ + 𝑐7) [67] M8 𝑃𝐴𝑅 = 𝐺[𝑎8(ln𝑘𝑇)2+ 𝑏8ln𝑘𝑇+ 𝑐8𝑇𝑑 + d8] [67] M9 𝑃𝐴𝑅 = 𝐺[𝑎9(ln𝑘𝑇)2+ 𝑏9ln𝑘𝑇+ 𝑐9cos𝜃𝑧+ 𝑑9] [67] M10 𝑃𝐴𝑅 = 𝐺[𝑎10(ln𝑘𝑇)2+ 𝑏10 ln𝑘𝑇+ 𝑐10𝑇𝑑+ 𝑑10 cos𝜃𝑧+ 𝑓10] [67] M11 𝑃𝐴𝑅 = 𝐺(𝑎11 ln𝜀 + 𝑏11 ln∆ + 𝑐11 ln𝑤 + 𝑑11 cos𝜃𝑧+ 𝑓11) [68] M12 𝑃𝐴𝑅 = 𝐺(𝑎12 ln𝜀 + 𝑏12 ln∆ + 𝑐12 cos𝜃𝑧+ 𝑑12) [68] M13 𝑃𝐴𝑅 = 𝐺(𝑎13 ln𝑘𝑇+ 𝑏13𝑤 + 𝑐13 cos𝜃𝑧+ 𝑑13) [68] M14 𝑃𝐴𝑅 = (𝑎14𝑘𝑇+ 𝑏14𝑘𝑇2+ 𝑐14𝑘𝑇3+ 𝑑14) (cos𝜃𝑧)𝑓 14 [29] M15 𝑃𝐴𝑅 = 𝑎15𝑘𝑇cos𝜃𝑧 [65] M16 𝑃𝐴𝑅 = 𝑎16𝐺 + 𝑏16 [69] M17 𝑃𝐴𝑅 = 𝑎17𝐺 + 𝑏17𝑘𝑇+ 𝑐17 [69] M18 𝑃𝐴𝑅 = 𝑎18𝐺 + 𝑏18𝑘𝑇+ 𝑐18 · 𝑒 + 𝑑18 [69] M19 𝑃𝐴𝑅 = 𝐺(𝑎19 ln𝑘𝑇+ 𝑏19 ln𝑚 + 𝑐19 · 𝑒 + 𝑑19) [70] M20 𝑃𝐴𝑅 = 𝑎20 · 𝐺 + 𝑏20𝐺cos𝜃𝑧+ 𝑐20𝐺𝑘𝑇+ 𝑑20 [71] M21 𝑃𝐴𝑅 = 𝑎23𝑘𝑇cos𝜃𝑧+ 𝑏23 Variación M15 Las primeras mediciones de radiación de la red SURFRAD se tomaron en 1995 y las últimas, en 2003. Con el objetivo de comparar series temporales paralelas, en este trabajo se ha considerado la serie de registros comprendida entre 2004 y 2018. Capítulo 2: Resultados por objetivos 30 Los datos de las estaciones radiométricas se registran minutalmente y se clasifican según el tipo de cielo utilizando para la clasificación el índice de cielo propuesto por Igawa [2]. Este índice considera 5 tipos de cielo atendiendo a los intervalos mostrados en la Tabla 4. Tabla 4. Clasificación del cielo según el índice del cielo (SI) propuesto por Igawa Clasificación de cielos Índice de Cielo (SI) Cielo Claro (Claro) < 0.15 Cielo Intermedio-Claro (I-Claro) 0.15 ≤ SI <0.3 Cielo Intermedio (Intermedio) 0.3 ≤ SI <0.9 Cielo Intermedio-Cubierto (I-Cubierto) 0.9 ≤ SI < 1.15 Cielo Cubierto (Cubierto) SI ≥ 1.15 Para conseguir homogeneidad, todos los modelos se ajustan para valores de 𝑃𝐴𝑅, medidos en 𝑊 ∙ 𝑚−2, adaptando los modelos originales que en algunos casos proponen el cálculo de la relación 𝑃𝐴𝑅 𝑅𝑎𝐺𝐻. ⁄ El estudio se realiza tanto en base minutal como en base horaria. Los modelos se ajustan de forma global con los datos de todas las estaciones, empleando el 50% de los datos experimentales para calibrar el modelo y determinar los coeficientes propios del modelo en cada localización y el 50% de datos restante para la comprobación de la bondad del modelo. La distribución del conjunto de datos experimentales entre el conjunto de calibración y el de prueba se realiza arbitrariamente. Los indicadores estadísticos tradicionales, coeficiente de determinación (𝑅2), error cuadrático medio (𝑅𝑆𝑀𝐸), el error cuadrático medio relativo (𝑅𝑀𝑆𝐸𝑟 ), la desviación relativa estándar (𝑅𝑆𝐷) y error de porcentaje medio (𝑀𝑃𝐸) se emplean como indicadores de la calidad del ajuste. La Figura 10 muestra todos los parámetros estadísticos obtenidos para los 21 modelos para el caso del ajuste en base minutal. En la figura se observa que, en general, todos los modelos presentan muy buen ajuste, con valores de 𝑅2 mayores del 0,99 y con unas desviaciones relativas estándar siempre inferiores al 12%. Los valores de 𝑅𝑀𝑆𝐸𝑟, son bajos, comprendidos entre 3,9 y 5 %. Los modelos 𝑀15 𝑦 𝑀21 presentan los Capítulo 2: Resultados por objetivos 31 mayores valores de 𝑅𝑀𝑆𝐸𝑟, por encima del 4%, y el valor mayor de desviación, próximo al 12%, se obtiene para el modelo 𝑀15. Figura 10. Resultados estadísticos de los modelos para datos de entrada de 1 minuto. A continuación, se evalúa el comportamiento de los modelos calibrados a partir de los registros minutales, al utilizarlos con datos horarios. Los valores obtenidos del 𝑅𝑀𝑆𝐸𝑟 están comprendidos entre 3,6 y 4,7% y como ocurre con los datos minutales, solo los modelos 𝑀15 y 𝑀21 presentan valores ligeramente superiores. Por último, se ha analizado la bondad de los distintos modelos cuando se tienen en cuenta las características de cielo. En este trabajo, la clasificación del cielo distingue las 5 categorías mostradas en la Tabla 4. El ajuste de los diferentes modelos mejora cuando los datos están previamente clasificados, especialmente en el caso de cielos claros, intermedios-claros e intermedios, en el que los estadísticos de error presentan valores ligeramente inferiores. Es significativo que, en el caso de cielos intermedios-claros, los modelos 𝑀15 y 𝑀21 muestran un peor comportamiento, siendo los valores de 𝑅𝑆𝐷 próximos al 8%, y particularmente el modelo 𝑀21 alcanza un valor cercano al 12% para cielos intermedios. Para cielos intermedios-cubiertos y cubiertos, los modelos obtienen valores de 𝑅𝑆𝐷 y de 𝑀𝑃𝐸 superiores al resto de tipos de cielo. Para cielos intermedios-cubiertos los valores de 𝑅𝑆𝐷 que se obtienen para la desviación de los modelos 𝑀14, 𝑀16, 𝑀17, 𝑀18 y 𝑀21 son relativamente altos, comprendidos entre el 15 y 30%. En el caso de Capítulo 2: Resultados por objetivos 38 valores de 𝑛𝑀𝐵𝐸 son prácticamente insignificantes en todos los casos, aunque su valor negativo muestra que todos los modelos tienden a subestimar los valores de 𝑃𝐴𝑅. Tabla 8. Resultados estadísticos de los modelos de redes neuronales [74]. Se puede concluir que ambos procedimientos para estimar la 𝑃𝐴𝑅, mediante 𝑀𝐿𝑅 y mediante 𝐴𝑁𝑁, se ajustan muy bien para las 3 categorías de cielo establecidas: claro, parcial y cubierto. Es de destacar que la categoría de cielos para la que mejor se ajustan los modelos, tanto las 𝑀𝐿𝑅𝑠 como las 𝐴𝑁𝑁, es la de cielos claros con valores de 𝑅2 mayores del 0.99 y de 𝑛𝑅𝑆𝑀𝐸 menores del 4% en ambos casos. Una vez obtenidos diferentes modelos para diferentes tipos de cielo que nos permiten obtener valores de 𝑃𝐴𝑅 con un buen ajuste para Burgos, se ha conseguido el tercer objetivo parcial de la tesis: Modelado local de la 𝑷𝑨𝑹 con diferentes índices meteorológicos para todo tipo de cielo y con diferentes categorías de cielo. 2.4. Objetivo parcial 4: Modelado de la 𝑷𝑨𝑹 para Burgos en función de los tipos de cielo clasificados según el índice de claridad (𝒌𝒕) y validación de los mismos para otras localizaciones. El último Objetivo Parcial de la tesis incide en el estudio del carácter local de los modelos de 𝑃𝐴𝑅. Este objetivo se concreta en el artículo Extension of local all sky conditions 𝑃𝐴𝑅 models to different climatic zones, publicado en Applied Sciences. Se desarrollan nuevos modelos de 𝑃𝐴𝑅 basados en 𝑀𝐼’𝑠 a través de 𝐴𝑁𝑁𝑠 y 𝑀𝐿𝑅𝑠, que se calibran y Condiciones del cielo 𝑹𝟐 nRMSE (%) nMAE (%) nMBE (%) ANN1 (claros) 0.992 3.846 2.59 -2.10∙10-6 ANN2 (parciales) 0.976 7.795 5.00 -3.10∙10-6 ANN3 (cubiertos) 0.987 6.466 4.34 -5.10∙10-6 Capítulo 2: Resultados por objetivos 39 validan en Burgos gracias a los datos experimentales disponibles. Los modelos desarrollados tienen en cuenta las condiciones del cielo utilizando como parámetro para su clasificación el índice de claridad, 𝑘𝑡, parámetro tradicionalmente empleado para esta caracterización [43]. La ventaja de usar el índice de claridad es que se puede obtener en la mayoría de las estaciones meteorológicas y se permite la clasificación de los cielos en tres categorías. Los modelos se desarrollan utilizando tanto 𝑀𝐿𝑅𝑠 como 𝐴𝑁𝑁𝑠 , siguiendo los procedimientos ya descritos en la Sección 2.3. La diferencia radica en los 𝑀𝐼’𝑠 seleccionados como entradas a los modelos. Los modelos desarrollados para Burgos, se aplican utilizando los datos experimentales obtenidos de las estaciones de la red SURFRAD, red que se ha utilizado también en la Sección 2.2. En este trabajo, se van a utilizar los 𝑀𝐼’𝑠 disponibles en esta red que coinciden con los que se registran o se pueden calcular en Burgos, y que se resumen en la Tabla 9. Todas las variables y datos meteorológicos se utilizan en base diezminutal. Los datos de la estación de Burgos se obtienen directamente en esta base temporal, mientras que los datos de las estaciones de la red SURFRAD, registrados minutalmente, se promedian cada diez minutos para realizar la transformación de la base temporal. Capítulo 2: Resultados por objetivos 40 Tabla 9. Índices meteorológicos medidos en Burgos [76]. 𝑴𝑰 𝑴𝑰 Expresión 𝑅𝑎𝐺𝐻 Radiación global horizontal Medida directamente 𝑘𝑑 Fracción horizontal difusa 𝑘𝑑=𝑅𝑎𝐷𝐻 𝑅𝑎𝐺𝐻 𝑄𝑝 Densidad de flujo de fotones fotosintéticos Medida directamente 𝑃𝐴𝑅 Radiación fotosintéticamente activa 𝑃𝐴𝑅 = 𝑄𝑝/4.57𝜇𝑚𝑜𝑙 ∙ 𝐽−1 𝑘𝑡 Índice de claridad 𝑘𝑡=𝑅𝑎𝐺𝐻 𝐵𝑠𝑐 ∙ 𝜀0∙ 𝑐𝑜𝑠𝑍𝑠 𝑇 Temperatura del aire Medida directamente 𝑃 Presión Medida directamente 𝑇𝑑 Temperatura del punto de rocío 𝑇𝑑=35.859 ∙ 𝑙𝑜𝑔𝑃 𝑣−21.48.496 𝑙𝑜𝑔𝑃 𝑣−10.2858 𝑐𝑜𝑠𝑍 Coseno azimuth solar 𝑐𝑜𝑠𝑍 = 𝑠𝑖𝑛𝛿 ∙ 𝑠𝑖𝑛𝜙 + 𝑐𝑜𝑠𝛿 ∙ 𝑐𝑜𝑠𝜙 ∙ 𝑐𝑜𝑠𝜔 𝜀 Índice de claridad 𝜀 = 𝑅𝑎𝐷𝐻 + 𝑅𝑎𝐵 𝑅𝑎𝐷𝐻 + 1.04 ∙ 𝑍𝑠 3 1 + 1.04 ∙ 𝑍𝑠 3 𝛥 Factor de brillo Δ = 𝑚 ∙ 𝑅𝑎𝐷𝐻 𝐵𝑠𝑐 ∙ 𝜀0∙ 𝑐𝑜𝑠𝑍𝑠 La selección de variables a través del coeficiente de correlación de Pearson, con mayor relación con la 𝑃𝐴𝑅, se realiza con los datos experimentales de Burgos incluyendo todos los tipos de cielo. Un segundo análisis se realiza clasificando previamente los datos en función de los tipos de cielo atendiendo al valor del 𝑘𝑡. Los resultados del análisis se muestran en la Tabla 10. Capítulo 2: Resultados por objetivos 41 Tabla 10. Coeficiente de Pearson, 𝑟 (𝑃𝐴𝑅,𝑀𝐼𝑖) basado en las condiciones de cielo definidas por la clasificación de cielo de 𝑘𝑡 (claro, parcial y cubierto) [76]. | 𝑟 (𝑃𝐴𝑅,𝑀𝐼𝑖) | Intervalos (𝒌𝒕) [1-0.9] (0.9-0.7] (0.7-0.5] (0.5-0.3] (0.3,0] Todo tipo de cielo 𝑅𝑎𝐺𝐻 𝑐𝑜𝑠𝑍, 𝑘𝑡 𝑘𝑑, 𝜀 𝑇 𝛥, 𝑃, 𝑇𝑑 Claro 𝑅𝑎𝐺𝐻,𝑐𝑜𝑠𝑍 𝑘𝑡, 𝜀 𝑇 𝑘𝑑, 𝛥,𝑃, 𝑇𝑑 Parcial 𝑅𝑎𝐺𝐻,𝑐𝑜𝑠𝑍 𝑇 𝑘𝑡, 𝑘𝑑, 𝛥,𝜀, 𝑃, 𝑇𝑑 Cubierto 𝑅𝑎𝐺𝐻 𝑐𝑜𝑠𝑍 𝑘𝑡, 𝛥 𝑘𝑑, 𝜀,𝑃,𝑇, 𝑇𝑑 Como es previsible, se observa que la 𝑅𝑎𝐺𝐻 es el parámetro que más influye, independientemente del tipo de cielo. 𝑐𝑜𝑠𝑍 tiene una muy fuerte relación para cielos claros y parciales y fuerte para todo tipo de cielo y para cielos cubiertos. Los índices 𝑘𝑡, 𝑘𝑑, 𝜀 y Δ, presentan una relación más débil en general, salvo para los cielos parciales en los que no hay ningún efecto sobre el valor de 𝑃𝐴𝑅. A partir de los valores experimentales recogidos en la estación de Burgos se han desarrollado modelos de medida de la 𝑃𝐴𝑅 mediante 𝑀𝐿𝑅𝑠 y 𝐴𝑁𝑁 . Para calibrar los modelos basados en regresiones multilineales, se utilizan el 85% de los datos lo que permite calcular los coeficientes de cada modelo. Para comprobar la bondad de los modelos se utiliza el 15% restantes de los valores experimentales. La bondad de los modelos se determina a través de los siguientes parámetros estadísticos: 𝑅2, 𝑛𝑀𝐵𝐸 y 𝑛𝑅𝑀𝑆𝐸. Los resultados del modelo, aplicados al conjunto de datos experimentales de test disponibles, se muestran en la Tabla 11. Todos los modelos presentan valores de 𝑅2 entre 0,97 y 0,99. Los valores de 𝑛𝑅𝑀𝑆𝐸 en todas las categorías de cielo están por debajo del 7,5%. Capítulo 2: Resultados por objetivos 42 Tabla 11. Modelos de regresión multilineal de la 𝑃𝐴𝑅 [76]. Condiciones del cielo 𝑹𝟐 nRMSE (%) nMBE (%) 𝑀𝐿𝑅1 (Todos los tipos) 0.994 4.37 -2.74∙10-13 𝑀𝐿𝑅2 (Cielos claros) 0.990 3.27 -1.45∙10-14 𝑀𝐿𝑅3 (Cielos parciales) 0.977 6.80 -5.32∙10-13 𝑀𝐿𝑅4 (Cielos cubiertos) 0.978 7.33 1.87∙10-13 Los modelos basados en 𝐴𝑁𝑁 se han entrenado mediante el algoritmo Levenberg-Marquardt Back-Propagation (𝐿𝑀𝐵𝑃) [75] para cada categoría: todo tipo de cielos, claros, parciales y cubiertos, usando los índices que más influencia han demostrado en la 𝑃𝐴𝑅 a través del coeficiente de Pearson (Tabla 10). En este caso, se divide en dos grupos el 85% de los datos experimentales que conformaban en conjunto de calibración para los modelos 𝑀𝐿𝑅. Uno de estos grupos, que comprende el 70% de los datos, se usa para el entrenamiento de la 𝐴𝑁𝑁, mientras que el 15% restante se emplea para la validación del modelo. El mismo conjunto de datos empleado anteriormente para la determinación de la bondad del modelo, se emplea para testar la fiabilidad del modelo 𝐴𝑁𝑁. Los resultados de estas pruebas, arrojan valores de 𝑅2 superiores al 0,97 y valores de 𝑛𝑅𝑀𝑆𝐸 inferiores al 8%. Una vez obtenidos los modelos en Burgos, cuya climatología corresponde con un tipo Csb (clima Mediterraneo oceánico), según la clasificación climática Köppen [66], se evalúan con los datos de las siete estaciones meteorológicas de la red SURFRAD. En la Tabla 12 se muestra la clasificación Köppen de cada una de las localizaciones analizadas. Capítulo 2: Resultados por objetivos 43 Tabla 12. Condiciones climáticas de las siete estaciones americanas (modificado a partir de [76]). Estaciones Clima Bondville, Ilinois Dfa Table Mountain, Boulder, Colorado Bsk Desert Rock, Nevada Bwh Fort Peck, Montana Bsk Goodwin Creek, Missisippi Cfa Penn State, Univ. Pennsylvania Dfb Sioux Falls, South Dakota Dfa (Dfa:continental húmedo de verano cálido; Bsk: semiárido frío; Bwh: árido desértico; Cfa: subtropical húmedo; Dfb: continental húmedo de verano moderado) Los resultados obtenidos por los modelos, tanto 𝑀𝐿𝑅 como 𝐴𝑁𝑁 en las estaciones meteorológicas de la red SURFRAD, presentan valores de 𝑅2> 0,99 y valores de 𝑛𝑅𝑀𝑆𝐸 inferiores al 7%. Se observa que, para cielos claros, para ambos procedimientos se obtienen valores de 𝑅2> 0,98 y 𝑛𝑅𝑀𝑆𝐸𝑠 aún más bajos, ya que en ninguna estación se obtienen valores de 𝑛𝑅𝑀𝑆𝐸 > 6%. Cuando se analizan los cielos parciales, el valor de 𝑅2 está también por encima del 0,99 y los errores (𝑛𝑅𝑀𝑆𝐸 𝑦 𝑛𝑀𝐵𝐸) son menores del 9,5%. Por último, con los cielos cubiertos siguen ajustándose muy bien los modelos con valores de 𝑅2> 0,98. Sin embargo, hay que destacar que los 𝑛𝑅𝑀𝑆𝐸𝑠 obtenidos con los modelos 𝑀𝐿𝑅 son menores del 8%, pero con los modelos 𝐴𝑁𝑁 en la estación Desert Rock, de clima árido, se alcanza un 𝑛𝑅𝑀𝑆𝐸 de 11,07%, relativamente grande frente al resto. Al analizar los resultados según zonas climáticas, apenas se observa diferencias en los valores de 𝑅2, encontrándose todos por encima de 0,980. El mayor error 𝑛𝑅𝑀𝑆𝐸 se obtiene para cielos cubiertos y se corresponde con la zona desértica (Bwh), zona climática muy diferente a la de Burgos, mientras que el menor 𝑛𝑅𝑆𝑀𝐸 se da en zonas continentales (Dfb y Dfa) con cielos claros. En resumen, las Capítulo 2: Resultados por objetivos 44 características locales de la 𝑃𝐴𝑅, se atenúan cuando los modelos desarrollados incluyen las variables adecuadas. Con este trabajo, se se ha conseguido el cuarto objetivo parcial de la tesis: Análisis de la localidad de los modelos de 𝑷𝑨𝑹. Capítulo 3: Conclusiones y líneas de futuro 45 Capítulo 3: Conclusiones y líneas de futuro 3.1. Conclusiones El objetivo general de esta Tesis es modelar la radiación fotosintéticamente activa (𝑷𝑨𝑹). Este objetivo se ha conseguido a través de cuatro objetivos parciales materializados en 4 artículos publicados en revistas científicas indexadas en JCR. Como primera conclusión, se ha demostrado una dependencia representativa entre la relación 𝑄𝑝𝑅𝑎𝐺𝐻 ⁄ y las condiciones de tipo de cielo clasificadas según la propuesta de la norma ISO/CIE. Los valores más altos de esta relación corresponden a cielos cubiertos, mientras que los valores más bajos y dispersos corresponden a cielos claros. La 𝑃𝐴𝑅 no presenta ninguna tendencia diaria o estacional característica, siendo las condiciones de cielo y la presencia de nubes el parámetro que más influye en su ratio con respecto a la irradiancia global horizontal. Se ha realizado una exhaustiva revisión de los modelos de 𝑃𝐴𝑅 global existentes. Todos los autores indican un fuerte carácter local de esta componente de la radiación solar. Todos los modelos, existentes y los desarrollados en este trabajo, presentan los mejores resultados para predecir el valor de la 𝑃𝐴𝑅 a través de otras magnitudes, cuando la irradiancia global horizontal es variable de entrada en el modelo, y esto para todos los tipos de cielo independientemente de la metodología de clasificación de cielos utilizada. La base temporal de los datos experimentales utilizada para la obtención de los modelos no tiene excesiva influencia en la bondad de los mismos, aunque cuanto mayor es la resolución temporal de los Capítulo 3: Conclusiones y líneas de futuro 46 datos, los modelos presentan, en general, valores inferiores de los parámetros estadísticos característicos, obteniendo mejores ajustes. Cuando los modelos se calibran utilizando datos agregados de diferentes localizaciones, estos arrojan resultados satisfactorios en localizaciones de diferentes características climáticas de la clasificación de Köppen. Generalmente, los modelos calculados para diferentes tipos de cielo presentan mejor ajuste en el caso de cielos claros y errores más elevados en el caso de cielos cubiertos. Y esto independientemente del procedimiento de clasificación de cielos utilizado. Este hecho se puede explicar por las diferencias en el espesor y las características de las nubes y la presencia de aerosoles, que dan lugar a distintos efectos de dispersión y absorción de la radiación en las distintas bandas del espectro. Las redes neuronales artificiales y los métodos tradicionales basados en regresiones multilineales son procedimientos equivalentes para modelar la radiación fotosintéticamente activa cuando se realiza una selección de variables de entrada previa adecuada que incluye, al menos la irradiancia global horizontal como índice meteorológico con mayor influencia. A partir de datos experimentales medidos en Burgos (España) y clasificados según los tipos de cielo propuestos por la norma ISO/CIE, se ha modelizado matemáticamente la 𝑃𝐴𝑅 mediante regresión multilineal y redes neuronales, obteniendo muy buenos ajustes con ambos procedimientos para las 3 categorías de cielo (claro, parcial y cubierto). Todos los valores de 𝑅2 son mayores de 0,97 y los errores (𝑛𝑅𝑀𝑆𝐸) menores del 8%; destacando que para cielos claros, es la categoría de cielo donde se obtiene menores 𝑛𝑅𝑀𝑆𝐸. La modelización de datos de 𝑃𝐴𝑅 con datos experimentales de Burgos, clasificados según tipos de cielo a partir del índice de claridad (𝑘𝑡), Capítulo 3: Conclusiones y líneas de futuro 47 mediante regresiones multilineales y redes neuronales, permite obtener modelos con muy buenos ajustes para las 3 categorías de cielo. Además, dichos modelos se pueden aplicar a diferentes localizaciones con distinto clima según la clasificación de Köppen con un error razonablemente bajo. Los modelos presentan los errores cuadráticos mayores para cada tipo de cielo cuando se aplican a localizaciones con clima desértico cálido. En definitiva, el trabajo ha conseguido obtener un conocimiento representativo de la radiación fotosintéticamente activa en su componente global relacionándola con índices meteorológicos habitualmente registrados en estaciones meteorológicas lo que permite su modelado a través de estos índices y para diferentes condiciones de cielo y localizaciones geográficas con distintos climas característicos. 3.2 Líneas de futuro Una vez modelada la 𝑃𝐴𝑅 , se pueden abrir nuevos ámbitos de investigación en los que poner en práctica estos modelos, ya que puede resultar de gran utilidad. 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García‐Rodríguez, S.. García‐Rodríguez, D.. Granados‐ López, M.. Díez‐Mediavilla, C. Alonso‐Tristán, Extension of PAR Models under Local All Sky Conditions to Different Climatic Zones, Appl. Siences. 12 (2022) 1–15. https://doi.org/https://doi.org/10.3390/app12052372. Bibliografía 60 61 Los Investigadores SOL GARCÍA RODRÍGUEZ, con DNI 71278921N y DIEGO GRANADOS LÓPEZ, con DNI 71292836N como coautores de los siguientes trabajos: • García-Rodríguez, S. García-Rodríguez, M. Díez-Mediavilla and C. Alonso-Tristán (2020). Photosynthetic Active Radiation, Solar Irradiance and the CIE Standard Sky Classification. Applied Sciences 10 (22) ,8007. DOI: https://doi.org/10.3390/app10228007 • García-Rodríguez, D. Granados-López, S. García-Rodríguez, M. Díez-Mediavilla, C. Alonso-Tristán (2021). Modelling Photosynthetic Active Radiation (PAR) through meteorological indices under all sky conditions. Agricultural and Forest Meteorology, 310, 108627. DOI: https://doi.org/10.1016/j.agrformet.2021.108627 • García-Rodríguez, S. García-Rodríguez, D. Granados-López, M. Díez-Mediavilla and C. Alonso-Tristán (2022). Extension of local all sky conditions PAR models to different climatic zones. Applied Sciences, 12, 2372. DOI: https://doi.org/10.3390/app12052372. expresamos mediante el presente escrito nuestra renuncia a la utilización de las anteriores publicaciones para la presentación de una tesis por compendio de artículos. En Burgos, a 26 de febrero de 2022 Fdo. Diego Granados López Fdo. Sol García Rodríguez 62 63 Artículos Appl. Sci. 2020,10, 8007 7 of 14 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 15 Figure 5. Monthly frequency of occurrence of clear, partial, and overcast sky conditions in Burgos between April 2019 and January 2020. Figure 6. Daily frequency of occurrence of clear, partial, and overcast sky conditions in Burgos between April 2019 and January 2020. 4. Temporal Variability of the 𝑸𝒑𝑹𝒔 ⁄ Ratio The seasonal characteristics of the 𝑄𝑅 ⁄ ratio were studied at differing intervals: tenminute, hourly, daily and monthly. Figure 7 shows the high positive correlation between 𝑄 and 𝑅 at ten-minute intervals (𝑅 0.992) with a slope of 1.893  0.001 𝜇𝑚𝑜𝑙 ∙ 𝐽. This value is close to the mean value 1.93  0.15 𝜇𝑚𝑜𝑙 ∙ 𝐽, with a standard deviation of 0.15 𝜇𝑚𝑜𝑙 ∙ 𝐽. The 𝑄𝑅 ⁄ ratio had similar values to those reported by other researchers [15], ranging between 1.21 and 2.84 𝜇𝑚𝑜𝑙 ∙ 𝐽. Figure 6. Daily frequency of occurrence of clear, partial, and overcast sky conditions in Burgos between April 2019 and January 2020. 4. Temporal Variability of the Qp/RsRatio The seasonal characteristics of the Qp/Rs ratio were studied at differing intervals: ten-minute, hourly, daily and monthly. Figure 7shows the high positive correlation between Qp and Rs at ten-minute intervals ( R2= 0.992) with a slope of 1.893 ± 0.001 µ mol · J −1 . This value is close to the mean value 1.93 ± 0.15 µ mol · J −1 , with a standard deviation of ± 0.15 µ mol · J −1 . The Qp/Rs ratio had similar values to those reported by other researchers [15], ranging between 1.21 and 2.84 µmol·J−1. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 15 Figure 7. Photosynthetic photon flux density, 𝑄  󰇛𝜇𝑚𝑜𝑙 ∙ 𝑠  ∙𝑚  ), and broadband solar irradiance, 𝑅  󰇛𝑊∙𝑚  ), measured in Burgos, between April 2019 and January 2020. Figure 8 shows the box-plot of the mean hourly values of the 𝑄𝑅 ⁄ ratio, calculated from the average of the ten-minute data, from sunrise to sunset, using the whole data base. The graph represents the mean value, the median, the three quartiles and both the maximum and the minimum values of the data, as well as the outlier values. Rising in the early hours of the day, 𝑄𝑅 ⁄ stabilized in the central hours and tended to decrease at sun set. Higher dispersion of the values in the first and last hours than in the central hours of the day may be observed, as the interquartile range shows. The standard deviation ranged from 0.11 𝜇𝑚𝑜𝑙 ∙ 𝐽 within the hourly interval starting at 7:00 to 0.17 𝜇𝑚𝑜𝑙 ∙ 𝐽, within the hourly interval starting at 14:00. The average values were higher than the median values in all hours of the day except for the hourly interval starting at 06:00. The hourly average of the ratio was 1.910  0.016 𝜇𝑚𝑜𝑙 ∙ 𝐽, with maximum and minimum values of 1.98  0.11 𝜇𝑚𝑜𝑙 ∙ 𝐽 and 1.75  0.16 𝜇𝑚𝑜𝑙 ∙ 𝐽, respectively, at 07:00 and at 19:00 h. Figure 8. Box-plot of the hourly average of the ratio of photosynthetic photon flux density to broadband solar irradiance, 𝑄  𝑅  ⁄. Blue crosses indicate the mean and blue lines inside the box, the median. The limits of the boxes give the 1st, 2nd, and 3rd quartiles, while the extreme whiskers are the minimum and the maximum points. Blue circles represent outliers. Figure 9 presents the monthly values of the 𝑄𝑅 ⁄ ratio calculated with the average of the tenminute data, using the whole data base. Figure 8 shows that the monthly data were almost constant throughout the central months of the year, from May to October, with a standard Figure 7. Photosynthetic photon flux density, Qp ( µ mol · s −1· m −2 ), and broadband solar irradiance, Rs(W·m−2), measured in Burgos, between April 2019 and January 2020. Figure 8shows the box-plot of the mean hourly values of the Qp/Rs ratio, calculated from the average of the ten-minute data, from sunrise to sunset, using the whole data base. The graph represents the mean value, the median, the three quartiles and both the maximum and the minimum values of the data, as well as the outlier values. Rising in the early hours of the day, Qp/Rs stabilized in the central hours and tended to decrease at sun set. Higher dispersion of the values in the first and last hours than in the central hours of the day may be observed, as the interquartile range shows. The standard deviation ranged from 0.11 µ mol · J −1 within the hourly interval starting at 7:00 to 0.17 µ mol · J −1 , within the hourly interval starting at 14:00. The average values were higher than the median values in all hours of the day except for the hourly interval starting at 06:00. The hourly average of the Appl. Sci. 2020,10, 8007 8 of 14 ratio was 1.910 ± 0.016 µ mol · J −1 , with maximum and minimum values of 1.98 ± 0.11 µ mol · J −1 and 1.75 ±0.16 µmol·J−1, respectively, at 07:00 and at 19:00 h. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 15 Figure 7. Photosynthetic photon flux density, 𝑄  󰇛𝜇𝑚𝑜𝑙 ∙ 𝑠  ∙𝑚  ), and broadband solar irradiance, 𝑅  󰇛𝑊∙𝑚  ), measured in Burgos, between April 2019 and January 2020. Figure 8 shows the box-plot of the mean hourly values of the 𝑄𝑅 ⁄ ratio, calculated from the average of the ten-minute data, from sunrise to sunset, using the whole data base. The graph represents the mean value, the median, the three quartiles and both the maximum and the minimum values of the data, as well as the outlier values. Rising in the early hours of the day, 𝑄𝑅 ⁄ stabilized in the central hours and tended to decrease at sun set. Higher dispersion of the values in the first and last hours than in the central hours of the day may be observed, as the interquartile range shows. The standard deviation ranged from 0.11 𝜇𝑚𝑜𝑙 ∙ 𝐽 within the hourly interval starting at 7:00 to 0.17 𝜇𝑚𝑜𝑙 ∙ 𝐽, within the hourly interval starting at 14:00. The average values were higher than the median values in all hours of the day except for the hourly interval starting at 06:00. The hourly average of the ratio was 1.910  0.016 𝜇𝑚𝑜𝑙 ∙ 𝐽, with maximum and minimum values of 1.98  0.11 𝜇𝑚𝑜𝑙 ∙ 𝐽 and 1.75  0.16 𝜇𝑚𝑜𝑙 ∙ 𝐽, respectively, at 07:00 and at 19:00 h. Figure 8. Box-plot of the hourly average of the ratio of photosynthetic photon flux density to broadband solar irradiance, 𝑄  𝑅  ⁄. Blue crosses indicate the mean and blue lines inside the box, the median. The limits of the boxes give the 1st, 2nd, and 3rd quartiles, while the extreme whiskers are the minimum and the maximum points. Blue circles represent outliers. Figure 9 presents the monthly values of the 𝑄𝑅 ⁄ ratio calculated with the average of the tenminute data, using the whole data base. Figure 8 shows that the monthly data were almost constant throughout the central months of the year, from May to October, with a standard Figure 8. Box-plot of the hourly average of the ratio of photosynthetic photon flux density to broadband solar irradiance, Qp/Rs . Blue crosses indicate the mean and blue lines inside the box, the median. The limits of the boxes give the 1st, 2nd, and 3rd quartiles, while the extreme whiskers are the minimum and the maximum points. Blue circles represent outliers. Figure 9presents the monthly values of the Qp/Rs ratio calculated with the average of the tenminute data, using the whole data base. Figure 8shows that the monthly data were almost constant throughout the central months of the year, from May to October, with a standard deviation between 0.11 and 0.16 µ mol · J −1 , and an interquartile range between 0.10 and 0.15 µ mol · J −1 . November was the measurement campaign month with the highest data dispersion: the interquartile range was 0.24 µ mol · J −1 , with a standard deviation of 0.20 µ mol · J −1 . The maximum value was recorded in April (1.98 ± 0.15 µ mol · J −1 ), while the minimum was reached in December (1.91 ± 0.17 µ mol · J −1 ). The monthly average of Qp/Rs was 1.930 ± 0.025 µ mol · J −1 . Based on the results of the monthly average of Qp/Rs , some authors have suggested the existence of a seasonal dependence of this term. Alados et al. [ 4 ] recorded higher values in the summer months and lower values from November to January, in Granada, Spain. However, in Greece, Proutsos et al. [ 47 ] recorded the highest values for autumn (September) while the lowest averages (March) were recorded in spring, with intermediate values for summer and winter. In Midwestern US [ 48 ], the lower values with smaller deviations were recorded in the summer months while the winter months showed higher values of the ratio with larger deviations. In Lhasa (Tibetan Plateau), the ratio of photosynthetically-active to broadband solar radiation increased almost linearly from January to June and decreased until the end of the year in the same way [ 49 ]. In this study, slightly higher values were recorded in spring and autumn. The monthly average of the Qp/Rs ratio was always above the median and the number of outliers above the maximum value was greater and had a higher absolute value than those below the minimum value. Appl. Sci. 2020,10, 8007 9 of 14 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 15 deviation between 0.11 and 0.16 𝜇𝑚𝑜𝑙 ∙ 𝐽, and an interquartile range between 0.10 and 0.15 𝜇𝑚𝑜𝑙 ∙ 𝐽. November was the measurement campaign month with the highest data dispersion: the interquartile range was 0.24 𝜇𝑚𝑜𝑙 ∙ 𝐽 , with a standard deviation of 0.20 𝜇𝑚𝑜𝑙 ∙ 𝐽 . The maximum value was recorded in April (1.98  0.15 𝜇𝑚𝑜𝑙 ∙ 𝐽), while the minimum was reached in December (1.91  0.17 𝜇𝑚𝑜𝑙 ∙ 𝐽 ). The monthly average of 𝑄𝑅 ⁄ was 1.930  0.025 𝜇𝑚𝑜𝑙 ∙ 𝐽. Based on the results of the monthly average of 𝑄𝑅 ⁄, some authors have suggested the existence of a seasonal dependence of this term. Alados et al. [4] recorded higher values in the summer months and lower values from November to January, in Granada, Spain. However, in Greece, Proutsos et al. [47] recorded the highest values for autumn (September) while the lowest averages (March) were recorded in spring, with intermediate values for summer and winter. In Midwestern US [48], the lower values with smaller deviations were recorded in the summer months while the winter months showed higher values of the ratio with larger deviations. In Lhasa (Tibetan Plateau), the ratio of photosynthetically-active to broadband solar radiation increased almost linearly from January to June and decreased until the end of the year in the same way [49]. In this study, slightly higher values were recorded in spring and autumn. The monthly average of the 𝑄𝑅 ⁄ ratio was always above the median and the number of outliers above the maximum value was greater and had a higher absolute value than those below the minimum value. Figure 9. Box-plot of the monthly average of the photosynthetic photon flux density to broadband solar irradiance, 𝑄  𝑅  ⁄, ratio. A deeper analysis was conducted with the hourly values for the two months and the extreme values of the monthly averages of 𝑄𝑅 ⁄. Figure 10 shows the daily pattern of the hourly averages of the 𝑄𝑅 ⁄ ratio, for April and November, including information on mean, median and variability through the quartile range. April presented higher and constant values throughout the day, decreasing in the last hours of the day, while November showed lower values of the ratio and more variability throughout the day. Both months presented similar patterns, with constant values of the 𝑄𝑅 ⁄ ratio around noon, in a trend that decreased over the last few hours of the day. Apart from the differences in daily means, shown previously, there is also evidence of greater variability during November, mainly in the first hours of the day. Figure 9. Box-plot of the monthly average of the photosynthetic photon flux density to broadband solar irradiance, Qp/Rs, ratio. A deeper analysis was conducted with the hourly values for the two months and the extreme values of the monthly averages of Qp/Rs . Figure 10 shows the daily pattern of the hourly averages of the Qp/Rs ratio, for April and November, including information on mean, median and variability through the quartile range. April presented higher and constant values throughout the day, decreasing in the last hours of the day, while November showed lower values of the ratio and more variability throughout the day. Both months presented similar patterns, with constant values of the Qp/Rs ratio around noon, in a trend that decreased over the last few hours of the day. Apart from the differences in daily means, shown previously, there is also evidence of greater variability during November, mainly in the first hours of the day. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 15 (a) April (b) November Figure 10. Daily pattern of the ratio photosynthetically-active photon flux density to broadband solar irradiance. 5. Variability of the 𝑸𝒑𝑹𝒔 ⁄ Ratio with the CIE Standard Sky Types Figure 11 shows the average photosynthetic photon flux density to broadband solar irradiance, 𝑄𝑅 ⁄, ratio, calculated on a ten minutes basis, for each CIE standard sky type. Clear sky types 12, 13, 14, and 15 showed smaller standard deviations (from 0.06 to 0.11 𝜇𝑚𝑜𝑙 ∙ 𝐽) and a smaller interquartile range (from 0.05 to 0.09 𝜇𝑚𝑜𝑙 ∙ 𝐽). However, the numbers of outliers were important for all these categories. The dispersion of the data within the categories corresponding to overcast and partial overcast skies was similar, with standard deviations between 0.15 𝜇𝑚𝑜𝑙 ∙ 𝐽 (sky type 4) and 0.21 𝜇𝑚𝑜𝑙 ∙ 𝐽−1 (sky type 9) and an interquartile range between 0.16 𝜇𝑚𝑜𝑙 ∙ 𝐽 (sky types 3 and 4) and 0.21 𝜇𝑚𝑜𝑙 ∙ 𝐽 (sky types 8 and 9). The highest values of the 𝑄𝑅 ⁄ ratio were under CIE standard sky types 1, 2, 3, and 4; all categories of overcast sky conditions. The categories of clear skies (CIE standard sky types from 11 to 15) showed the smallest values of the ratio. Sky types 5 and 9 presented an anomalous behavior. They are categorized as overcast and partial overcast categories, but the average values of the 𝑄𝑅 ⁄ ratios for both sky types were closer to the values of clear sky categories. Furthermore, sky type 9 showed the smallest average value: 1.89  0.21 𝜇𝑚𝑜𝑙 ∙ 𝐽. The FOCs of both sky types, 5 and 9, were very scarce in the measurement campaign: 1.65% and 0.64%, respectively. CIE standard sky type 5, described as “overcast, foggy or cloudy, with overall uniformity”, presented high uniformity in terms of illuminance and broadband solar irradiance, as well sky type 9, described as “partly cloudy with a shaded sun position”. Figure 10. Daily pattern of the ratio photosynthetically-active photon flux density to broadband solar irradiance. Appl. Sci. 2020,10, 8007 10 of 14 5. Variability of the Qp/RsRatio with the CIE Standard Sky Types Figure 11 shows the average photosynthetic photon flux density to broadband solar irradiance, Qp/Rs , ratio, calculated on a ten minutes basis, for each CIE standard sky type. Clear sky types 12, 13, 14, and 15 showed smaller standard deviations (from 0.06 to 0.11 µ mol · J −1 ) and a smaller interquartile range (from 0.05 to 0.09 µ mol · J −1 ). However, the numbers of outliers were important for all these categories. The dispersion of the data within the categories corresponding to overcast and partial overcast skies was similar, with standard deviations between 0.15 µ mol · J −1 (sky type 4) and 0.21 µ mol · J −1 (sky type 9) and an interquartile range between 0.16 µ mol · J −1 (sky types 3 and 4) and 0.21 µ mol · J −1 (sky types 8 and 9). The highest values of the Qp/Rs ratio were under CIE standard sky types 1, 2, 3, and 4; all categories of overcast sky conditions. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 15 Figure 11. Box-plot ratio of photosynthetic photon flux density to broadband solar irradiance, 𝑄  𝑅  ⁄, for each CIE standard sky type. When the 15 CIE standard sky types were reduced to three, (overcast, from sky types 1 to 5, partial, from sky type 6 to 10, and clear sky, from sky types 11 to 15), the average photosynthetic photon flux density to broadband solar irradiance ratio, 𝑄𝑅 ⁄, increased when sky cloudiness increased, as shown in Figure 12. The average 𝑄𝑅 ⁄ ratio for clear skies and for overcast skies was 1.90  0.11 𝜇𝑚𝑜𝑙 ∙ 𝐽 and 1.98  0.16 𝜇𝑚𝑜𝑙 ∙ 𝐽, respectively. Partial cloudy skies showed the highest dispersion of the data, with standard deviation and interquartile range values of 0.17 and 0.19 𝜇𝑚𝑜𝑙 ∙ 𝐽 , respectively. However, the clear skies dataset had the largest number of outliers. The Spearman correlation coefficient is a non-parametric measure of rank correlation, to determine the strength and direction of the relationship between two variables. If two datasets X and X’ are strongly correlated, then the Spearman coefficient will be 1 (direct correlation) or if otherwise –1 (inverse correlation). Although 𝑟󰇛𝑄,𝐶𝐼𝐸)=0.56 (p-value < 0.001) and 𝑟󰇛𝑅,𝐶𝐼𝐸) = 0.57 (p-value < 0.001) both imply a moderate correlation, 󰇛𝑄𝑅 ⁄,𝐶𝐼𝐸) = −0.23 (p-value < 0.001). The p-value determine the significance of the results in relation to the null hypothesis (the results are due to random chance). The lower the p-value, the greater the statistical significance of the test and greater the confirmation of the hypothesis [50]. As the p-value was lower than 0.05 in all the tests, there was a statistically significant difference between the average 𝑄𝑅 ⁄ ratios for each type of CIE sky, at a significance level of 5%. Figure 12. Box-plot of the ratio of photosynthetic photon flux density to broadband solar irradiance, 𝑄  𝑅  ⁄, by CIE cloudiness sky classifications. Figure 11. Box-plot ratio of photosynthetic photon flux density to broadband solar irradiance, Qp/Rs , for each CIE standard sky type. The categories of clear skies (CIE standard sky types from 11 to 15) showed the smallest values of the ratio. Sky types 5 and 9 presented an anomalous behavior. They are categorized as overcast and partial overcast categories, but the average values of the Qp/Rs ratios for both sky types were closer to the values of clear sky categories. Furthermore, sky type 9 showed the smallest average value: 1.89 ± 0.21 µ mol · J −1 . The FOCs of both sky types, 5 and 9, were very scarce in the measurement campaign: 1.65% and 0.64%, respectively. CIE standard sky type 5, described as “overcast, foggy or cloudy, with overall uniformity”, presented high uniformity in terms of illuminance and broadband solar irradiance, as well sky type 9, described as “partly cloudy with a shaded sun position”. When the 15 CIE standard sky types were reduced to three, (overcast, from sky types 1 to 5, partial, from sky type 6 to 10, and clear sky, from sky types 11 to 15), the average photosynthetic photon flux density to broadband solar irradiance ratio, Qp/Rs , increased when sky cloudiness increased, as shown in Figure 12. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 15 Figure 11. Box-plot ratio of photosynthetic photon flux density to broadband solar irradiance, 𝑄  𝑅  ⁄, for each CIE standard sky type. When the 15 CIE standard sky types were reduced to three, (overcast, from sky types 1 to 5, partial, from sky type 6 to 10, and clear sky, from sky types 11 to 15), the average photosynthetic photon flux density to broadband solar irradiance ratio, 𝑄𝑅 ⁄, increased when sky cloudiness increased, as shown in Figure 12. The average 𝑄𝑅 ⁄ ratio for clear skies and for overcast skies was 1.90  0.11 𝜇𝑚𝑜𝑙 ∙ 𝐽 and 1.98  0.16 𝜇𝑚𝑜𝑙 ∙ 𝐽, respectively. Partial cloudy skies showed the highest dispersion of the data, with standard deviation and interquartile range values of 0.17 and 0.19 𝜇𝑚𝑜𝑙 ∙ 𝐽 , respectively. However, the clear skies dataset had the largest number of outliers. The Spearman correlation coefficient is a non-parametric measure of rank correlation, to determine the strength and direction of the relationship between two variables. If two datasets X and X’ are strongly correlated, then the Spearman coefficient will be 1 (direct correlation) or if otherwise –1 (inverse correlation). Although 𝑟󰇛𝑄,𝐶𝐼𝐸)=0.56 (p-value < 0.001) and 𝑟󰇛𝑅,𝐶𝐼𝐸) = 0.57 (p-value < 0.001) both imply a moderate correlation, 󰇛𝑄𝑅 ⁄,𝐶𝐼𝐸) = −0.23 (p-value < 0.001). The p-value determine the significance of the results in relation to the null hypothesis (the results are due to random chance). The lower the p-value, the greater the statistical significance of the test and greater the confirmation of the hypothesis [50]. As the p-value was lower than 0.05 in all the tests, there was a statistically significant difference between the average 𝑄𝑅 ⁄ ratios for each type of CIE sky, at a significance level of 5%. Figure 12. Box-plot of the ratio of photosynthetic photon flux density to broadband solar irradiance, 𝑄  𝑅  ⁄, by CIE cloudiness sky classifications. Figure 12. Box-plot of the ratio of photosynthetic photon flux density to broadband solar irradiance, Qp/Rs, by CIE cloudiness sky classifications. Appl. Sci. 2020,10, 8007 11 of 14 The average Qp/Rs ratio for clear skies and for overcast skies was 1.90 ± 0.11 µ mol · J −1 and 1.98 ± 0.16 µ mol · J −1 , respectively. Partial cloudy skies showed the highest dispersion of the data, with standard deviation and interquartile range values of 0.17 and 0.19 µ mol · J −1 , respectively. However, the clear skies dataset had the largest number of outliers. The Spearman correlation coefficient is a non-parametric measure of rank correlation, to determine the strength and direction of the relationship between two variables. If two datasets X and X’ are strongly correlated, then the Spearman coefficient will be 1 (direct correlation) or if otherwise –1 (inverse correlation). Although r(QP,CIE) =0.56 (p-value <0.001) and r(Rs,CIE) =0.57 (p-value <0.001) both imply a moderate correlation, (QP/Rs,CIE) = − 0.23 (p-value <0.001). The p-value determine the significance of the results in relation to the null hypothesis (the results are due to random chance). The lower the p-value, the greater the statistical significance of the test and greater the confirmation of the hypothesis [ 50 ]. As the p-value was lower than 0.05 in all the tests, there was a statistically significant difference between the average Qp/Rs ratios for each type of CIE sky, at a significance level of 5%. When only three CIE categories were considered (clear =3, partial =2, and overcast =1), r(QP,CIEcloudiness) =0.52 (p-value <0.001) and r(Rs,CIEcloudiness) =0.52 (p-value <0.001), and (QP/Rs,CIEcloudiness) = − 0.22 (p-value <0.001). From these results, a weak relationship can be established between the value of the QP/Rs ratio and the CIE standard sky classification. However, previous results were confirmed in this research, in so far as the ratio of photosynthetic photon flux density to broadband solar irradiance, Qp/Rs , presented its highest values over cloudy conditions and decreased with clear sky conditions, following the CIE Standard Sky classification as a reference for defining the sky conditions. 6. Conclusions The analysis of photosynthetic photon flux density to broadband solar radiation ratios registered between April 2019 and January 2020 in Burgos, Spain, at ten-minute intervals, has shown a representative dependency on the sky type conditions classified by CIE taxonomy. The higher values of the Qp/Rs ratio were for overcast sky types, while the values were lower and more dispersed under clear sky conditions. Statistically significant differences have been found in the Qp/Rs ratios for each CIE standard sky type. The overcast sky types presented the highest values of the ratio, with the clear sky categories presenting the lowest and most dispersed values. During the experimental campaign, there were only two exceptions to the expected behavior: sky types 5 and 9. They belonged to covered and partial covered sky type categories, respectively, presenting values closer to the clear sky categories according to the CIE standard. The main characteristic of both categories was a high uniformity in terms of illumination. The higher dispersion of data corresponding to clear skies categories could be explained by the presence of aerosols or atmospheric turbidity, characteristics of clear sky types 13, 14 and 15, with FOC’s 15%, 8% and 7% (Figure 3). No seasonal dependency of Qp/Rs can be established, as highlighted in this and the other studies that were reviewed, due mainly to the different sky conditions recorded during the experimental campaigns. As shown in Figure 5, between April 2019 to January 2020 in Burgos, Spain, clear skies predominated in summer, while in winter the overcast conditions presented the highest frequency of occurrence. The analysis of the hourly values also revealed a daily pattern with higher and more stable values of Qp/Rs in the first hours of the day that tended to stabilize around noon and to decrease around sunset. The study of the daily pattern of the sky types (clear, overcast and partial) show that, although clear skies are predominant at all hours of the day, the differences of the frequency of occurrence with respect to overcast and partially overcast skies decreases towards noon, as Figure 6showed. Appl. Sci. 2020,10, 8007 12 of 14 As has been demonstrated in this study and in others, the most influential factor in the Qp/Rs value was the presence of overcast sky conditions. Although other authors used different climatological parameters for sky classification, the CIE Standard sky classification has proven itself to offer a good overall framework that can represent the sky conditions, covering the complete spectrum of sky categories. Author Contributions: Conceptualization, M.D.-M. and C.A.-T.; methodology, A.G.-R. and S.G.-R.; software, A.G.-R.; validation, A.G.-R., M.D.-M.; formal analysis, A.G.-R. and S.G.-R.; investigation, A.G.-R. and S.G.-R.; original draft preparation, M.D.-M.; writing—review and editing, C.A.-T.; visualization, A.G.R. and S.G.-R.; supervision, M.D.-M. and C.A.-T.; project administration, M.D.-M.; funding acquisition, C.A.-T. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Spanish Ministry of Science and Innovation, grant number RTI2018-098900-B-I00 and Consejería de Educación, Junta de Castilla y León, grant number BU021G19. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Advances in Space Research Validation and Calibration of Models to Estimate Photosynthetically Active Radiation Considering Different Time Scales and Sky Conditions --Manuscript Draft-- Manuscript Number: AISR-D-22-00024 Article Type: ES - Earth Sciences Keywords: Photosynthetically Active Radiation; Solar irradiance; sky types; Model calibration Corresponding Author: Ignacio García Ruiz, Ph.D. Public University of Navarre: Universidad Publica de Navarra Pamplona, Navarra SPAIN First Author: Marian de Blas, PhD Order of Authors: Marian de Blas, PhD Ignacio García Ruiz, Ph.D. Ana García-Rodríguez José Luis Torres, PhD Abstract: Photosynthetically Active Radiation (PAR) is a fundamental parameter for developing plant productivity models. Nevertheless, instrumentation for measuring PAR and to record it is scarce at conventional meteorological stations. Several procedures have therefore been proposed for PAR indirect estimation. In this work, 21 previously published analytical models that correlate PAR with easily available meteorological parameters are collected. Although longer time scales were considered in the original publications, a minute range was applied in this work to calibrate the PAR models. In total, more than 10 million input records were gathered from the SURFRAD station network from a 10-year long time series with data frequencies recorded every 1 minute. After calibration, the models were validated for minute and hourly data, obtaining in all cases low fitting errors. The model for the estimation of PAR should then be selected, solely considering the meteorological variables available at the specific location. When applying the calibrated models to input data classified according to sky conditions (from clear to overcast), the PAR models continued to perform satisfactorily, although the error statistics of some models for overcast skies that establish relative values worsened. Suggested Reviewers: Harry D. Kambezidis, PhD Research Director, National Observatory of Athens: Ethnikon Asteroskopeion Athenon [email protected] Jesús Polo, PhD Senior Researcher, CIEMAT: Centro de Investigaciones Energeticas Medioambientales y Tecnologicas [email protected] Luis Manuel Navas, PhD Professor, Universidad de Valladolid [email protected] Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Validation and Calibration of Models to Estimate Photosynthetically Active Radiation Considering Di↵erent Time Scales and Sky Conditions Marian de Blasa, Ana Garc´ıa-Rodr´ıguezb,IgnacioGarc´ıa a,⇤, Jos´e Luis Torresa aInstitute of Smart Cities (ISC), Department of Engineering, Public University of Navarre, Campus Arrosad´ıa, 31006 Pamplona, Spain bResearch Group Solar and Wind Feasibility Technologies (SWIFT), Electromechanical Engineering Department, University of Burgos, 09006 Burgos, Spain Abstract Photosynthetically Active Radiation (P AR) is a fundamental parameter for developing plant productivity models. Nevertheless, instrumentation for measuring PAR and to record it is scarce at conventional meteorological stations. Several procedures have therefore been proposed for PAR indirect estimation. In this work, 21 previously published analytical models that correlate P AR with easily available meteorological parameters are collected. Although longer time scales were considered in the original publications, a minute range was applied in this work to calibrate the PAR models. In total, more than 10 million input records were gathered from the SURFRAD station network from a 10-year long time series with data frequencies recorded every 1 minute. After calibration, the models were validated for minute and hourly data, obtaining in all cases low fitting errors. The model for the estimation of PAR should then be selected, solely considering the meteorological variables available at the specific location. When applying the calibrated models to input data classified according to sky conditions (from clear to overcast), the PAR models continued to perform satisfactorily, although the error statistics of some models for overcast skies that establish relative values worsened. Keywords: Photosynthetically active radiation, solar irradiance, sky types, model calibration ⇤Corresponding author. Email address: [email protected] (Ignacio Garc´ıa) Preprint submitted to Advances in Space Research January 3, 2022 Manuscript Click here to view linked References When the calibration site and the location of interest have comparable134 atmospheric conditions, it is expected that the atmospheric conditions in135 both places will cause a similar attenuation of solar radiation in the PAR136 wavelength range. Therefore, the model for PAR determination should pro-137 duce accurate results. Di↵erent coefficients in similar equations have been138 proposed, to take into account the sky conditions (Ferrera-Cobos et al.,2020;139 Jacovides et al.,2007;Wang et al.,2013;Yu et al.,2015).140 In this work, 21 models for PAR estimation were calibrated and evalu-141 ated. For this purpose, a 10-year long series of 1-minute resolution data was142 configured from data recorded at seven weather stations within the SUFRAD143 network, covering a wide range of climatic conditions in the USA. The main144 objective of this work is to present models that can be generally applied and145 that provide accurate PAR values using meteorological variables measured146 at ground stations and considering time intervals between 1 minute and 1147 hour. An objective that was achieved with a data set of more than 10 mil-148 lion PAR and meteorological measurements that are representative of a large149 variety of regional and weather conditions.150 The expected errors in PAR estimation when the 21 models under anal-151 ysis were applied were firstly quantified for all sky conditions together and152 then considering the prevailing sky conditions. By doing so, users who need153 to know the PAR at a given location are provided with a set of easy-to-154 implement equations and they can select the most appropriate model, de-155 pending on the available meteorological input data.156 This paper is organized into five sections. The meteorological data and its157 classification in sky types are detailed in Section 2.InSection3,thePAR158 8 models and the calibration procedure is described. The calibrated models159 for the minute-time scale and the statistical analysis of the results when the160 models were applied to two-time scales (minute and hourly) and to di↵erent161 sky types are presented in Section 4. Finally, the conclusions are detailed in162 Section 5.163 2. Weather data164 We used high quality datasets retrieved from the Surface Radiation Bud-165 get Network (SURFRAD) FTP Server based in the United States: PAR166 on the horizontal plane, global and di↵use solar irradiance on the horizontal167 plane, air temperature and relative humidity.168 Global PAR was measured with an LI-COR Quantum sensor within the169 400 to 700 nm broadband range. PAR values are given in W ·m2.TheG170 values were obtained from a Spectrolab SR-75 pyranometer and di↵use irradi-171 ance was measured on the horizontal plane with an Eppley 8-48 pyranometer.172 Both pyranometers measure data within the 280-3000 nm broadband range.173 The seven stations belonging to the SURFRAD network are distributed174 throughout di↵erent climatic areas in the United States, as shown in Figure175 1. The ground properties of these stations were described in Wandji Nyamsi176 et al. (2019). Table 1shows the geographical location and the basic climatic177 data from the seven weather stations.178 All the SURFRAD stations never started to take measures at the same179 time. The earliest radiation measurements were taken in 1995 and the latest,180 in 2003. Notwithstanding, with the aim of comparing similar series of data for181 model calibration purposes, we considered a common time range of records182 9 40°N 30°N 120°W 110°W 100°W 90°W 80°W Figure 1. Locations of the seven SURFRAD weather stations. Table 1. Geographical location and basic climatic data from the seven SURFRAD weather stations. Station Code Latitude (N) Longitude (W) Elevation (m) Average temperature (oC) Average relative humidity (%) Average G (W ·m2) Average PAR (W ·m2) Bondville, Illinois BON 40.0519 88.3731 230 11.53 72.65 170.46 73.82 Table Mountain, Boulder, Colorado TBL 40.1250 105.237 1689 12.25 46.92 193.71 84.29 Desert Rock, Nevada DRA 36.6237 116.019 1007 18.81 28.04 235.18 104.22 Fort Peck, Montana FPK 48.3078 105.102 634 5.84 67.24 161.89 72.66 Goodwin Creek, Mississippi GWN 34.2547 89.8729 98 16.73 71.89 182.14 79.07 Penn. State Univ., Pennsylvania PSU 40.7201 77.9309 376 10.52 70.01 152.83 65.73 Sioux Falls, South Dakota SXF 43.734 96.6233 473 7.96 73.13 166.50 74.40 between 1/1/2009 and 31/12/2018 at every station, with a temporal resolu-183 tion of 1 minute. Irradiance data collected at the stations within the network184 had previously passed the Baseline Surface Radiation Network (BSRN) qual-185 ity control checks, as explained in Long and Dutton (2002). Unfortunately,186 criteria for Quality Control (QC) of PAR are neither covered in the BSRN187 10 procedures or in any assessment standards we have consulted in the bibliogra-188 phy available to us. Therefore, an in-house quality-control method described189 below was set in accordance with the physical limits of PAR. Global PAR190 on the horizontal plane cannot be higher than extraterrestrial PAR on the191 horizontal plane corresponding to the same time. Extraterrestrial PAR on192 the horizontal plane (P0)iscalculatedwithacorrectionfactor(fc) applied193 to the PAR solar constant (Psc)thattakesintoaccounttheeccentricityof194 the Earth’s orbit and then multiplies the result by the cosine of the solar195 zenith angle, Equation (1).196 P0=fcPsc cos ✓z.(1) Gueymard (2018) obtained a value of 1361.1 W ·m2for the solar con-197 stant or total solar irradiance, but no standardized value for the irradiance198 in the PAR spectral range. The irradiance curve between 400 and 700 nm199 of the extraterrestrial spectrum proposed by Gueymard et al. (2002)was200 integrated, in order to estimate this constant, resulting in a value of 530.8201 W·m2. Gueymard’s extraterrestrial spectrum is proposed to determine202 the reference AM1.5 spectrum specified in standard IEC 60904-3:2019 (IEC,203 2019). Data corresponding to solar elevation angles lower than 5were dis-204 carded to avoid the cosine response problems of global irradiance and PAR205 measuring instruments.206 Table 2shows the number of available measurements at each station to207 calibrate and to validate PAR estimation models after having passed the QC.208 Calibration of a model implies using 50% of the data, alternatively chosen,209 for the model calibration and 50% to validate the resulting expressions. Data210 11 are classified into five sky types according to the Sky Index (Si)proposed211 by Igawa (2014)thatcanbeseeninTable3. This classification analyzes the212 influence of a given atmospheric condition, represented by the sky type, on213 the behavior of the models to be evaluated.214 Table 2. Total data and data classified according to the sky type that have passed the quality control and the solar minimum elevation parameter for each station. Time series includes data from 2009 to 2018. Station All sky Clear I-Clear Intermediate I-Overcast Overcast BON 1371257 636401 259743 352698 105381 17034 % 100 46.41 18.94 25.72 7.68 1.24 DRA 1852600 1399386 185258 214100 49715 4141 % 100 75.54 10 11.56 2.68 0.22 FPK 1363076 682583 232362 323409 108049 16673 % 100 50.08 17.05 23.73 7.93 1.22 GWN 1478690 700298 265390 340367 138794 33841 % 100 47.36 17.95 23.02 9.39 2.29 PSU 1310735 490228 216272 348225 200990 55020 % 100 37.4 16.5 26.57 15.33 4.2 SXF 1413606 701128 229540 327663 130533 24742 % 100 49.6 16.24 23.18 9.23 1.75 TBL 1545048 883883 182998 314297 138336 25534 % 100 57.21 11.84 20.34 8.95 1.65 Total 10335012 5493907 1571563 2220759 871798 176985 % 100 53.16 15.21 21.49 8.44 1.71 12 Table 3. Sky classification according to the Sky Index (Si) proposed by Igawa (2014) and PAR/G mean values and standard deviation obtained for each sky type. Sky classification Sky Index (Si) Clear Sky (Clear) <0.15 Intermediate Clear Sky (I-Clear) 0.15 Si < 0.3 Intermediate Sky (Intermediate) 0.3 Si < 0.9 Intermediate Overcast Sky (I-Overcast) 0.9 Si < 1.15 Overcast Sky (Overcast) Si 1.15 3. Methodology215 Table 4shows the mathematical expressions corresponding to the 21 mod-216 els considered in this work for estimating PAR.217 SURFRAD stations register a large number of meteorological variables,218 but only those necessary for the application of the models are considered in219 this project: global PAR on the horizontal plane, global and di↵use solar220 irradiance on the horizontal plane, and air temperature and relative humid-221 ity. Various models are used to estimate atmospheric water vapor content222 that include as a variable either the dew point, Td(C) (e.g., M1, M3; M8;223 M10), or the precipitable water content, w(cm) (e.g.M11, M13), or the va-224 por pressure, e(hPa) (e.g., M18, M19). These parameters are not directly225 registered in the stations; hence they are estimated on the basis of either226 air temperature or relative humidity. Notice that M21 is a variation of M15227 which includes an independent term.228 Models M1 to M10 for the PAR/G fraction were previously calibrated229 by Yu et al. (2015), considering daily values obtained from the SURFRAD230 13 Table 4. Models evaluated for PAR estimation. Model Expression Variables Reference M1 PAR =G(a1ln "+b1ln +c1Td+d1cos2✓z+f1)G,",,Td,cos✓zAlados et al. (1996) M2 PAR =G(a2ln "+b2ln +c2cos2✓z+d2)G,kd,,cos✓zAlados et al. (1996) M3 PAR =G(a3ln kT+b3Td+c3cos ✓z+d3)G,kT,Td,cos✓zAlados et al. (1996) M4 PAR =G(a4ln kTb4cos ✓z+c4)G,kT,cos✓zAlados et al. (1996) M5 PAR =G(a5ln kT+b5)G,kTWandji Nyamsi et al. (2019) M6 PAR =G⇥a6(ln kT)2+b6ln kT+c6⇤G,kTYu et al. (2015) M7 PAR =G(a7ln kd+b7ln +c7)G,,kdYu et al. (2015) M8 PAR =G⇥a8(ln kT)2+b8ln kT+c8Td+d8⇤G,kT,TdYu et al. (2015) M9 PAR =G⇥a9(ln kT)2+b9ln kT+c9cos ✓z+d9⇤G,kT,cos✓zYu et al. (2015) M10 PAR =G⇥a10 (ln kT)2+b10 ln kT+c10Td+d10 cos ✓z+f10⇤G,kT,Td,cos✓zYu et al. (2015) M11 PAR =G(a11 ln "+b11 ln +c11 ln w+d11 cos ✓z+f11)G,",,w,cos✓zHu et al. (2007) M12 PAR =G(a12 ln "+b12 ln +c12 cos ✓z+d12)G,",,cos✓zHu et al. (2007) M13 PAR =G(a13 ln kT+b13w+c13 cos ✓z+d13)G,kT,w,cos✓zHu et al. (2007) M14 PAR =a14kT+b14kT2+c14kT3+d14(cos ✓z)f14 kT,cos✓zWang et al. (2013) M15 PAR =a15kTcos ✓zkT,cos✓zFoyo-Moreno et al. (2017) M16 PAR =a16G+b16 GAguiar et al. (2011) M17 PAR =a17G+b17kT+c17 G,kTAguiar et al. (2011) M18 PAR =a18G+b18kT+c18e+d18 G,kT,eAguiar et al. (2011) M19 PAR =G(a19 ln kT+b19 ln m+c19e+d19)G,kT,m(*),eMizoguchi et al. (2013) M20 PAR =a20G+b20Gcos ✓z+c20GkT+d20 G,kT,cos✓zGe et al. (2011) M21 PAR =a23kTcos ✓z+b23 kT,cos✓zVariation of M15 (*) Relative optical air mass (m) calculated according to Kasten and Young (1989). station network over the period between 2009 and 2010 and for all sites and231 all sky conditions together. The resulting R2values ranged from 0.4599 (M5)232 to 0.5475 (M10), which are low values for this parameter, and RMSE values233 (in MJ ·m2) from 0.02131 (M9) to 0.02326 (M7). It should be noted that234 Yu et al. (2015) attributed models M5 to M10 to various authors, but on235 consulting the referenced publications, it was found that the models and236 the citations could not be cross-referenced. In this work, we have therefore237 attributed the models whose original reference is uncertain to Yu et al. (2015).238 14 We downloaded 10335012 data items (Table 2) from all stations that had239 passed the QC to conduct the calibration process described in this work240 and to fit the results of models M1 to M21. All the models are fitted for241 PAR (in W ·m2) and not for the PAR/G fraction, in order to achieve242 homogeneity. However, models M1 to M13 and M19 were originally published243 considering PAR/G as the dependent variable. A minute time base was used244 to adjust the models, because the PAR values obtained with this frequency245 were thought to provide users with useful input data for later studies that246 might require in-depth analysis. The behavior of the models obtained from247 the adjustment considering an hourly basis will also be evaluated, since data248 from the input parameters required in the models is often available at this249 frequency. Although the models could be tested for larger time bases (daily250 PAR values), PAR values of that frequency are not considered to be of251 practical interest. In each case, the models that o↵ered the best results were252 selected.253 The following statistics, described by Equations (2)to(6), were used for254 the validation of the models:255 •Determination coefficient:256 R2=PN i=1(EiEavg)(MiMavg) nhPN i=1(EiEavg)2ihPN i=1(MiMavg)2io0.5.(2) •Root Mean Squared Error:257 RMSE ="1 N N X i=1 (EiMi)2#0.5 .(3) 15 •Relative Root Mean Squared Error:258 RMSEr(%) = 100 Mavg "1 N N X i=1 (EiMi)2#0.5 .(4) •Relative Standard Deviation:259 RSD(%) = "1 N N X i=1 ✓EiMi Mi◆2#0.5 100.(5) •Mean Percentage Error:260 MPE(%) = 1 N N X i=1 EiMi Mi!100,(6) where Eiis the calculated value, Eavg is the mean of the calculated261 values, Miis the measured value, Mavg is the mean of the measured262 values and Nis the total number of data items.263 4. Results and Discussion264 Table 5shows the results of the coefficients for models M1 to M21, cali-265 brated for all sites and all sky types together.266 Figure 2a shows the statistical results derived from the fit of the experi-267 mental data obtained at minute frequency with the theoretical models. All268 the models yielded very satisfactory fitted results, with high values of the co-269 efficient of determination R2over 0.99 and RMSE values ranging from 8.416270 W·m2in M10 to 10.93 W ·m2in M15. As can be seen, relative deviations271 are low, although in some of the models some of the statistics were some-272 what higher. Based on these results of the models under study, it cannot be273 16 Table 5. Calibrated models for PAR using 1-minute input data. Model Calibrated expression M1 PAR =G(0.0270876"0.028342 ln + 0.000152279Td0.0206671 cos2✓z+0.425525) M2 PAR =G(0.0280839 ln "0.0286075 ln 0.0182345 cos2✓z+0.425987) M3 PAR =G(0.0278875 ln kT+0.000129279Td0.028706 cos ✓z+0.437906) M4 PAR =G(0.0295837 ln kT0.0258378 cos ✓z+0.435928) M5 PAR =G(0.0350336 ln kT+0.415212) M6 PAR =G⇥0.0340535 (ln kT)20.0757318 ln kT+0.406944⇤ M7 PAR =G(0.0336608 ln kd0.0349165 ln + 0.413051) M8 PAR =⇥0.0349876 (ln kT)20.0774526 ln kT0.0000716119Td+0.406896⇤(36) M9 PAR =G⇥0.0197733 (ln kT)20.0538075 ln kT0.02303 cos ✓z+0.428876⇤ M10 PAR =G⇥0.0169287 (ln kT)20.0490527 ln kT+0.0000967929Td0.0255814 cos ✓z+0.431371⇤ M11 PAR =G(0.0271005 ln "0.0284931 ln + 0.0022675 ln w0.0297581 cos ✓z+0.435327) M12 PAR =G(0.028097 ln "0.0287232 ln 0.0264467 cos ✓z+0.434845) M13 PAR =G(0.0281807 ln kT+0.000938004w0.0279869 cos ✓z+0.436261) M14 PAR =481.162kT+202.545kT2167.836kT3+16.7594(cos ✓z)0.928878 M15 PAR =574.278kTcos ✓z M16 PAR =0.413286G+8.38447 M17 PAR =0.414235G1.94159kT+9.13333 M18 PAR =0.41312G0.0993371kT+0.0885944e+7.6279 M19 PAR =G(0.0277219 ln kT+0.0176706 ln m+0.000155103e+0.409838) M20 PAR =0.474298G0.0196574Gcos ✓z0.0483035GkT+1.403 M21 PAR =550.309kTcos ✓z+11.5173 said that there are one or more models that stand out for their good or bad274 quality. 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Field Crops Research 107, 29–42. doi:10.1016/j.fcr.2007.12.571 014.572 32 Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Conflict of Interest Agricultural and Forest Meteorology 310 (2021) 108627 5 through simple mathematical expressions (Li et al., 2013). Therefore, the CIE standard sky taxonomy was selected for atmospheric conditions reference. A complete description of the CIE standard sky classification and the procedure to obtain the CIE standard sky classification from the sky scanner measurements can be found in previous works (Granados-L´ opez et al., 2021; Su´ arez-García et al., 2020; Su´ arez-García et al., 2018). Fig. 2 shows the Frequency Of Occurrency (FOC) of the different CIE standard sky types in Burgos during the experimental campaign, which extended from April, 2019 to February, 2021. As can be seen, in Burgos, the clear sky conditions were prevalent, as sky types 11 to 15 were the most frequent over the period under study. Sky type 13 (Cloudless polluted with a broader solar corona) had the highest FOC (17.4%) followed by sky types 7 (Partly cloudy with a uniform gradation and a brighter circumsolar effect) and 8 (Partly cloudy, rather uniform with a clear solar corona). Once the reference sky conditions had been established, The MIs included in the study were analyzed from the sky conditions to identify the correlation between the MIs and the CIE standard sky conditions. The box-plot of the MIs, with respect to the CIE standard sky type, gave the following conclusions: air temperature, pressure, and cosZ were not influenced by the sky type, while CC, kt, kd, kb, ε ,and Δ were greatly affected by the sky conditions. A new quality test of the experimental data was introduced, discarding any simultaneous data corresponding to outliers in the box-plots. Fig. 3 shows the results for CC and T, where the influence of the sky type on the MI value is almost negligible in the case of T and very important in the case of CC. The discarded data reached 10%. 4. Feature selection Feature Selection (FS) is the identification of related features within a set of data and the removal of irrelevant or less important features that contribute little or nothing to the definition of the target variable, so as to achieve models of greater accuracy. FS is one of the core concepts of ML that will impact on the performance of the developed model, improving its precision and reducing its complexity and overfitting as well as its runtime. In this work, the Pearson criterion was used to analyze the degree of correlation of each of the selected MIs to PAR. The Pearson criterion is based on the Pearson correlation coefficient, r. For the case under study, if PAR and one selected MI are strongly correlated, the Pearson coefficient is 1 (direct correlation) or -1 (inverse correlation). However, a Pearson coefficient near 0 implies a weak or null correlation. The Thumb rule (Mukaka, 2012) established five r intervals for the correlation: direct (1 ≥ |r(PAR,MIi)| ≥ 0.9), strong (0.9 >|r(PAR,MIi)| ≥ 0.7), moderate (0.7 >|r(PAR,MIi)| ≥ 0.5), weak (0.5 >|r(PAR,MIi)| ≥ 0.3), and negligible (|r(PAR,MIi)| <0.3). Fig. 4 shows the Pearson coefficient, r(PAR,MIi),calculated for the 10 MIs used in this study, regardless of the sky conditions. A direct correlation was obtained between PAR and RaGH and a strong correlation between PAR and ε , kt, and cosZ.The r coefficient with the other MIs was moderate, weak or negligible, so, these MIs were discarded as inputs for modelling PAR. As Fig. 4 highlights, RaGH, the geometric parameter solar azimuth cosine, cosZ, Perez’s clearness index, ε ,and the clearness index, kt, showed the strongest correlation to PAR (|r(PAR,MIi)| >0.7). These results agree with the literature, in that RaGH (or alternatively kt)is the most widely used parameter for modelling PAR (Nwokolo and Amadi, 2018). Air pressure, P, the brightness factor, Δ, and cloud cover, CC, presented weak correlations with PAR. The data were clustered, taking into account the CIE standard sky Table 5 Multilinear regression models of PAR. Sky conditions Multilinear regression model R2 nRMSE (%) nMAE (%) nMBE (%) Clear skies (MLR1) PAR =0.3806⋅ RaGH+0.524⋅ ε + 33.247⋅cosZ−2.646 0.991 3.93 2.72 -2⋅10 −6 Partial skies (MLR2) PAR =0.3958⋅ RaGH+18.282⋅cosZ − 2.508 0.976 7.81 5.04 -2⋅10 −6 Overcast skies (MLR3) PAR =0.4335⋅ RaGH−7.726⋅kt− 9.078⋅Δ +4.065 0.985 6.62 4.44 -5⋅10 −6 Fig. 5. Correlogram of the multilinear regression model of PAR: a) Clear skies; b) Partial skies; c) Overcast skies. A. García-Rodríguez et al. Agricultural and Forest Meteorology 310 (2021) 108627 6 classification for clear sky (CIE sky types 1 to 5), and partial (CIE sky types 6 to 10) and overcast sky conditions. The Pearson correlation coefficient or Pearson’s r(PAR,MIi)was calculated taking into account the sky classification and the results are shown in Table 4. RaGH was the MI with the strongest correlation to PAR, regardless of sky conditions. cosZ only showed a very strong correlation to PAR for clear skies. MIs with strong correlations to PAR were ε for clear skies,cosZ,for partial skies, and kt,and Δ in overcast sky conditions. The brightness factor, Δ, that presented weak correlations to PAR when data were not clustered, highlighted its strong correlation in overcast sky conditions. In the following sections, the PAR is modelled by following different procedures for each of the CIE standard sky conditions that were developed using the MIs with the Pearson correlation coefficient |r(PAR,MIi)|>0.7. 5. Multilinear regression models Three multilinear regression models, one for each sky type, were formulated to estimate PAR, based on the previous feature selection process shown in Table 4. The experimental data-set, taking into account the CIE standard classification, was randomly divided into two groups: the first one, comprising 85% of the data, was used to fit the models. The other group, with the remaining 15% of the data, was used to validate the models. The statistics used for this validation were as follows: the corresponding determination coefficient (R2), the normalized mean bias error (nMBE), the normalized mean absolute error, nMAE, and the root mean square error (nRMSE). The expressions for these last three statistical terms are: nMBE =1 PARexp ∑n i=1(PARmod −PARexp) n×100 (%)(7) nRMSE =1 PARexp  ∑n i=1(PARmod −PARexp)2 n √×100 (%)(8) nMAE =1 PARexp ∑n i=1PARmod −PARexp n×100 (%)(9) where n is the number of the experimental data used for fitting the models; PARmod are the modelled values of PAR; and PARexp is the experimental value of PAR.As Table 5 shows, the results of fitting the models presented good correlation with the experimental data, given that R2>0.97 and nRMSE was lower than 8% and nMAE lower than 5%. The small and negative values of nMBE indicate that the models present a good fit although tend to underestimate PAR values. The graph in Fig. 5 shows the multilinear regression models for each of the sky Fig. 6. ANN system architecture using the Levenberg-Marquardt Backpropagation algorithm. Fig. 7. Scheme of a neuron with k inputs. The weighting matrix and the bias are used for calculating the neuron output thought the activation function. Table 6 Statistical results of the ANN models. Sky conditions R2 nRMSE (%) nMAE (%) nMBE (%) Clear skies (ANN1) 0.992 3.846 2.59 -2⋅10 −6 Partial skies (ANN2) 0.977 7.795 5.00 -3⋅10 −6 Overcast skies (ANN3) 0.987 6.466 4.34 -5⋅10 −6 A. García-Rodríguez et al. Agricultural and Forest Meteorology 310 (2021) 108627 7 conditions. 6. Artificial neural network for modelling PAR This section proposes an Artificial Neural Network (ANN) trained with the Levenberg-Marquardt Back-Propagation (LMBP) algorithm (Du and Stephanus, 2018) that is used to model PAR for each of the CIE standard sky conditions (clear, partial and overcast skies). ANN training uses an iterative process to assign the correct weight that each neuron must set to each input to obtain the desired output value, PAR data in this case. Including the dependency between the input and output variables is not necessary, because the ANN is training to learn this dependency. ANN works as a ‘black box’ and only the final result is known and not the intermediate processes that are followed. It is therefore not possible to obtain the relationship between the input variables and their relative weight in the final result. Fig. 6 illustrates the system architecture used in this work. A three-layer configuration has been chosen: the input layer, where the selected MIs for the model were introduced in the system; the hidden layer, or the information processing centers and the output layer, where the result is obtained. Each processing center (neuron) adjusts to the other neurons in an interactive process. The fit of the weights (weighting matrix, W) in each iteration is done using the Levenberg Marquardt algorithm (Lv et al., 2017). Fig. 7 shows the structure of a neuron where the input information is calculated from eq. 10: n=∑ K j=1 WjPj+b(10) Wj are the components of the weighting matrix, Pjare the input variables and b is the bias. The neuron generates an output, a, in Fig. 7, through the activation function (f(n)) given, in this work, by Eq. 11: f(n) = 1 1+e−n(11) The input data-set of an ANN is divided into three groups for its training process: a training group (with 70% of data), a validation dataset (15% of data) and a test data-set (15% of data). The training and validation groups have been established by dividing the fitting dataset used in the MLR modelling. The training group is used to determine the weighted matrix and the bias in an iterative process. The training is over when the results of the performance of the resulting model, calculated using the validation set, reach the desired quality. The test data group is used to calculate the performance of the model. The test data-set matched the one used for the test of the multilinear regression models fitted in the previous section. Three ANNs were developed and tested in this work, one for each sky type, based on the feature selection previously performed and shown in Table 4. The statistic used for the validation of the models were the corresponding determination coefficient (R2), the normalized mean bias error (nMBE), and the normalized root mean square error (nRMSE), previously defined. Table 6 summarizes the statistical results of the ANNs, and Fig. 8 shows a graph of these results. The slope and intercept (W⋅m−2)of the correlograms shown in Figs. 5 and 8 give a practically perfect fit for all sky conditions, as the results show in Table 7. Fig. 8. Correlogram of the ANN model of PAR: a) Clear skies; b) Partial skies; c) Overcast skies. Table 7 Results of the correlograms of the different models. Sky conditions R2 Slope Intercept (W⋅m−2)Sky conditions R2 Slope Intercept(W⋅m−2) Clear skies (MLR1) 0.992 1.0009 -0.6753 Clear skies (ANN1) 0.992 1.0004 -0.5484 Partial skies (MLR2) 0.978 0.9922 0.6845 Partial skies (ANN2) 0.978 0.9924 0.6078 Overcast skies (MLR3) 0.987 0.9990 -0.2836 Overcast skies (ANN3) 0.988 1.0002 -0.3550 A. García-Rodríguez et al. Agricultural and Forest Meteorology 310 (2021) 108627 8 7. Conclusions Two different procedures, multilinear regression and ANNs, have been used to develop PAR models using Meteorological Indices (MIs). A previous feature selection procedure has been performed with the Pearson’s correlation coefficient that pointed to the most influential variables, so that the irrelevant ones could be discarded. As its main novelty, sky conditions were included in the modelling of PAR, using the CIE standard sky classification as the criterion for the definition of clear, partial, and overcast skies. The experimental data-set collected at tenminute intervals, was extended for twenty-two months. In addition to the traditional quality filters for solar radiation measurements, a second filtering procedure has been applied in statistical terms and according to the clustering of the data following the CIE standard sky classification, which guarantees the quality and homogeneity of the experimental data for each of the established sky conditions. A feature selection procedure has been applied before the modelling of PAR, for selecting the most influential variables and discarding the most redundant ones. In the feature selection procedure, a maximum of three MIs were selected as input for the models. RaGH was the common MI used by all models and for all sky conditions. The additional variables were the geometrical parameter, cosZ, and three variables related to the sky conditions, kt, ε , and Δ.These variables have been used in other works for modelling PAR (Aguiar et al., 2012; Alados-Arboledas et al., 2000; Alados et al., 1996; L´ opez et al., 2001; Wang et al., 2014) with different time intervals and following different strategies, obtaining comparable results to this work. Both modelling methods, multilinear regression and ANN, have obtained very high determination coefficients (R2) with very close results in the models for each of the different sky conditions. Slight improvements were observed in the ANN models. The lowest nRMSE values were obtained for clear skies models while modelling of partial conditions yielded the highest values. nMBE values were practically insignificant in all cases, although its negative value showed than all models tend to underestimate PAR values. Regarding the ANN models of PAR, the use of Levenberg-Marquardt Back-Propagation (LMBP) algorithm with three layers (input, hidden layer and output) was the chosen configuration, based on literature data (Ferrera-Cobos et al., 2020) and previous experience. In this study, the equivalence of multilinear regression models and ANN models of PAR following a feature selection procedure has been highlighted. The main advantage of the multilinear regression models is knowledge of the relationship between the variables: in the case of PAR, the RaGH coefficient indicated that PAR value is higher under overcast sky conditions than in partial and clear skies, as previous works have demonstrated (García-Rodríguez et al., 2020). The transferability of the models to other locations and its local adaptation will be discussed in future works. CRediT authorship contribution statement A. García-Rodríguez: Investigation, Methodology, Formal analysis, Validation. D. Granados-L´ opez: Investigation, Methodology, Formal analysis, Validation. S. García-Rodríguez: Investigation, Methodology, Software, Visualization. M. Díez-Mediavilla: Conceptualization, Supervision, Funding acquisition, Project administration. C. AlonsoTrist´ an: Conceptualization, Supervision, Writing – review & editing, Funding acquisition, Project administration. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors gratefully acknowledge the financial support provided by the Regional Government of Castilla y Le´ on, under projects BU021G19 and INVESTUN/19/BU/0004 and the Spanish Ministry of Science & Innovation under the I+D +i state program “Challenges Research Projects” (Ref. RTI2018-098900-B-I00). 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García-Rodríguez et al.   Citation: García-Rodríguez, A.; García-Rodríguez, S.; Granados-López, D.; Díez-Mediavilla, M.; Alonso-Tristán, C. Extension of PAR Models under Local All-Sky Conditions to Different Climatic Zones. Appl. Sci. 2022,12, 2372. https://doi.org/10.3390/ app12052372 Academic Editors: Harry D. Kambezidis and Basil Psiloglou Received: 12 January 2022 Accepted: 22 February 2022 Published: 24 February 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article Extension of PAR Models under Local All-Sky Conditions to Different Climatic Zones Ana García-Rodríguez , Sol García-Rodríguez, Diego Granados-López, Montserrat Díez-Mediavilla and Cristina Alonso-Tristán * Research Group Solar and Wind Feasibility Technologies (SWIFT), Electromechanical Engineering Department, University of Burgos, 09006 Burgos, Spain; [email protected] (A.G.-R.); [email protected] (S.G.-R.); [email protected] (D.G.-L.); [email protected] (M.D.-M.) *Correspondence: [email protected] or [email protected] Abstract: Four models for predicting Photosynthetically Active Radiation (PAR) were obtained through MultiLinear Regression (MLR) and an Artificial Neural Network (ANN) based on 10 meteorological indices previously selected from a feature selection algorithm. One model was developed for all sky conditions and the other three for clear, partial, and overcast skies, using a sky classification based on the clearness index (k t ). The experimental data were recorded in Burgos (Spain) at ten-minute intervals over 23 months between 2019 and 2021. Fits above 0.97 and Root Mean Square Error (RMSE) values below 7.5% were observed. The models developed for clear and overcast sky conditions yielded better results. Application of the models to the seven experimental ground stations that constitute the Surface Radiation Budget Network (SURFRAD) located in different Köppen climatic zones of the USA yielded fitted values higher than 0.98 and RMSE values less than 11% in all cases regardless of the sky type. Keywords: photosynthetically active radiation; k t sky classification; ANN; multilinear regression models 1. Introduction Photosynthetically Active Radiation (PAR) is a key factor for photosynthesis, vegetation growth, and climate change. The importance of its measurement and modelling is widely recognized [ 1 – 3 ], as PAR has a major influence on plant canopy growth, agricultural yields, and other environmental variables. Measurement of the PAR band is needed for sky-modelling of biomass growth and forestry production [ 3 – 6 ] and in natural greenhouse illumination [ 7 ]. Moreover, PAR affects the relationship between atmospheric systems and plants, so much so that its availability is a regulatory factor in the natural carbon cycle and in CO2, water, and energy exchanges within the atmosphere [8]. Within the solar radiation spectrum, PAR is the portion with a wavelength between 400 and 700 nm [ 9 ]. Unfortunately, PAR sensors are not commonly found at ground meteorological stations [ 10 ]. It is therefore usually measured on the basis of other meteorological parameters and variables. The relationship between two different variables and PAR has been analyzed in different works, obtaining mathematical models of greater or lesser accuracy. Many of these models described in the literature for modelling PAR are based on linear regressions of global horizontal radiation [ 3 , 11 , 12 ], optical air mass [ 1 ], and the clearness index [ 13 ]. Other authors modelled PAR by simultaneously taking several variables into account: solar radiation, solar zenith angle, columnar perceptible water vapour, and aerosol optical depth [ 14 ]. Wang et al. [ 13 ] estimated PAR with the clearness index, day length, and zenith angle. Ferrera-Cobos et al. [ 15 ] used Global Horizontal Irradiance (RaGH), Global extraterrestrial irradiance (G 0 ), atmospheric Temperature (T), and Relative Humidity (RH) as input variables for their models. These relationships have been analysed in most studies as a function of sky conditions [ 16 ] to estimate PAR for clear skies where the most important Appl. Sci. 2022,12, 2372. https://doi.org/10.3390/app12052372 https://www.mdpi.com/journal/applsci Appl. Sci. 2022,12, 2372 2 of 14 variable is the solar zenith angle. The brightness index indicates whether the model is used in the presence of clouds. In several studies, PAR has been treated as a variable that depends on the location where it is measured or estimated [ 17 ]. As regards measurement temporality, the ratio of PAR and insolation in both tropical and arctic regions has been observed to remain fairly constant on different days and over longer time scales, regardless of cloud cover, atmospheric composition, surface type, season, and day length [18]. Ferrera-Cobos et al. [ 15 ] modeled PAR in oceanic and Mediterranean climates, testing 22 (11 multi-linear regression and 11 artificial neural network) models, using RaGH,G 0 , T, and RH as their input variables. They concluded that areas with different climatic conditions needed different models. The Mediterranean climatic models showed better fits, and the models for oceanic climates needed some corrections depending on their geographical location, as the higher humidity of an oceanic climate, due to higher levels of atmospheric water vapour, means more radiation in the infrared spectrum, increasing the PAR to G 0 ratio. Although the combination of using geostationary and polar-orbiting satellites is an optimal solution for estimating PAR, a network of PAR sensors is necessary for better validation [ 18 ]. Sudhakar et al. proposed a PAR estimation model for India at six latitudes between 9 ◦ to 34 ◦ , based on hourly and monthly averages of daily global radiation and a power regression model that accounted for different solar angles, cloud cover, and climatic conditions, which they linked to RaGH [3]. Several researchers have used data from different satellites, supplied from CM-SAF, to develop their PAR models, although field data had to be used for validation in each study [ 12 , 15 ]—a need that Vindel et al. [ 19 ] considered when presenting methodology to determine optimal locations for PAR measurement stations. This methodology is based on a clustering process applied to the k t ,PAR, calculated by dividing the PAR at the Earth’s surface by the part of the spectrum corresponding to the PAR band at the top of the atmosphere (39.8% of the total). In recent years, machine learning techniques have been used to develop algorithms for estimating PAR. The most decisive variable for PAR estimation was RaGH when comparing MLR and ANN models, regardless of climate [ 15 ]. Even when ANN models were run, their results clearly worsened without RaGH. In this sense, Jacovides [ 20 ] used the sunshine fraction (nN), T,RaGH,G 0 , and RH as input variables for the ANN models. They found that sunshine duration plays an important role in obtaining acceptable model predictions and that the model that best predicted PAR values combined sunshine duration and RaGH. Lopez et al. [ 10 ] presented a model using PAR data collected at radiometric stations using a neural network. They estimated PAR with RaGH as the only measured variable. A second ANN model based on sunshine duration measurements was shown to be an acceptable alternative for calculating PAR. In contrast, Pankaew et al. [ 21 ] developed an ANN model for estimating hourly PAR data using seven atmospheric parameters (cosine of solar zenith angle, cloud index, precipitable water content, and aerosol optical depth) as the input collected from satellite data. They concluded that PAR estimation with an ANN model presented a good fit with a root mean square difference of 10.2%. Qin et al. [2] tested eight artificial intelligence models, among which the BackPropagation neural network model yielded the highest accuracy. Wang et al. [ 22 ] proposed three improved ANN models: MLP, Generalized Regression Neural Network (GRNN), and Radial Basis Neural Network (RBNN) for PAR estimation, from long-term hourly observations of RaGH and meteorological variables (air temperature, relative humidity, dew point temperature, water vapour pressure, air pressure). They found that different meteorological parameters influenced PAR estimation in accordance with each particular (agricultural land, soil, forest, bay, prairie, desert, and lake) ecosystem. Yu et al. [ 23 ] studied the relationship between hourly PAR and RaGH from data collected over three years at the Bondville, IL, and Sioux Falls, SD, ground weather stations (United States). From these data, they determined the temporal variability of the PAR fraction and its dependence on different sky conditions (defined by the clearness index (k t )). Furthermore, the results in terms of the normalized Root Mean Square Error (nRMSE), the Appl. Sci. 2022,12, 2372 3 of 14 (R 2 ) coefficient of determination, the Mean Percentage Error (MPE), and Relative Standard Error (RSE) from the ANN-based models were compared with the same results from four existing conventional regression models. The authors found that the ANN model could accurately predict hourly PAR, especially under cloudy and clear sky conditions. In general, as the literature review has highlighted, research on the modelling of different components of solar radiation has focused on obtaining models at specific locations, which can rarely be applied directly to other locations and usually requires local recalibration to achieve adequate results. In this study, our aim is to extend locally obtained models for PAR estimations as a function of the sky type to other locations. The sky conditions were determined from the clearness index (k t ). Three different PAR models, one for each sky type (clear, overcast, and partial), were developed from experimental data collected over ten-minute intervals at Burgos, Spain. The meteorological indices had previously been selected as the model inputs. Multi-Linear Regression (MLR) and ANN were the two modelling procedures. Both models were applied to experimental PAR data from all seven ground stations that form the SURFRAD network, corresponding to various Köppen–Geiger climate classification types [24]. The structure of this paper is as follows: after the Introduction Section, the databases, including the meteorological measurement stations, are described in Section 2. This section also includes the definition and description of the meteorological indices used for modelling PAR. In Section 3, the feature selection algorithm is described and PAR modelling is introduced using MLR and ANN models. In Section 4, the adjustment of the models obtained in Burgos for their application to the seven SURFRAD meteorological stations located in the United States is described. Finally, the main results and conclusions are presented in Section 5. 2. Description of the Experimental Data Figure 1shows the location of the weather station in Burgos, Spain (42 ◦ 21 0 04 00 N, 3 ◦ 41 0 20 00 W, 856 metres Above Mean Sea Level) (AMSL) where the data were collected for this study. The prevailing climate in Burgos, both oceanic and Mediterranean, is classified as Csb in the Köppen climate classification system This weather station, described in previous work [ 25 ], is situated on the flat roof of a building at the University of Burgos, with no external obstructions or reflections from other surfaces. In addition, data were collected from all seven stations within the Surface Radiation Budget Network (SURFRAD), dependent on the National Oceanic and Atmospheric Administration (NOAA): Bondville station, Sioux Falls station, Boulder station, Desert Rock station, Fort Peck station, Goodwin Creek Station, and Penn State station. Figure 2shows the location of the weather station in Burgos and the location of the seven SURFRAD weather stations. In Burgos, RaGH and Diffuse Horizontal Irradiation (RaDH) were measured in W · m −2 using a Hukseflux pyranometer, model SR11, and a Hukseflux pyrheliometer, model DR01, respectively. PAR was measured from Photosynthetic Photon Flux Density, Q p ( µ mol · s −1· m −2 ), data and was then converted into PAR data (W · m −2 ) using McCree’s conversion factor (4.57 µ mol · J −1 ) [ 26 ] using a EKO quantum sensor, model ML-020P. All meteorological and radiometric data were recorded every ten minutes (averages from 30 s). The experimental campaign took place from April 2019 to February 2021. Experimental data were analyzed and then filtered using conventional quality criteria [ 27 ]. If a dataset failed to pass the quality criteria, then all simultaneous datasets were rejected. The original dataset counted 71,600 datums (ten-minute datasets), 36% of which were eliminated after the filtering procedure. In the USA, PAR, global, and diffuse irradiance values are given in W · m −2 . The PAR values were measured employing a LI-COR Quantum sensor, while the RaGH values were measured using a pyranometer Spectrolab SR-75, and RaDH was measured with an Eppley, Model 8–48 pyranometer. The experimental campaign spanned 10 years, from January 2009 to December 2018 at each station, with a temporal resolution of 1 min, moving to ten-minute values by calculating the average of the values within that interval. Appl. Sci. 2022,12, 2372 4 of 14 Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 15 measured with an Eppley, Model 8-48 pyranometer. The experimental campaign spanned 10 years, from January 2009 to December 2018 at each station, with a temporal resolution of 1 min, moving to ten-minute values by calculating the average of the values within that interval. Figure 1. Experimental facility on the roof of the Higher Polytechnic School of. Burgos University, Spain. The Q p , RaGH, and RaDH sensors are shown in the detailed pictures. Figure 2. Location of the weather stations (Burgos and USA). Each Meteorological Index (MI) shown in Table 1 was determined at the single Spanish station and the seven meteorological stations in the USA. The following meteorological indices were directly obtained from the experimental measurements: T, P, RaGH, and PAR, obtained from Q p , which were measured with pyranometers and a quantum sensor, respectively. The solar azimuth cosine (cosZ) was calculated from the geometrical data of the location, using well-established mathematical relationships [28]. Finally, the dew point temperature () was calculated from the vapour water pressure [29] and RH. The other indices, k t [28], the horizontal diffuse fraction, k d [30], and Perez’s clearness index, ε, Figure 1. Experimental facility on the roof of the Higher Polytechnic School of. Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 15 measured with an Eppley, Model 8-48 pyranometer. The experimental campaign spanned 10 years, from January 2009 to December 2018 at each station, with a temporal resolution of 1 min, moving to ten-minute values by calculating the average of the values within that interval. Figure 1. Experimental facility on the roof of the Higher Polytechnic School of. Burgos University, Spain. The Q p , RaGH, and RaDH sensors are shown in the detailed pictures. Figure 2. Location of the weather stations (Burgos and USA). Each Meteorological Index (MI) shown in Table 1 was determined at the single Spanish station and the seven meteorological stations in the USA. The following meteorological indices were directly obtained from the experimental measurements: T, P, RaGH, and PAR, obtained from Q p , which were measured with pyranometers and a quantum sensor, respectively. The solar azimuth cosine (cosZ) was calculated from the geometrical data of the location, using well-established mathematical relationships [28]. Finally, the dew point temperature () was calculated from the vapour water pressure [29] and RH. The other indices, k t [28], the horizontal diffuse fraction, k d [30], and Perez’s clearness index, ε, Figure 2. Location of the weather stations (Burgos and USA). Burgos University, Spain. The Q p ,RaGH, and RaDH sensors are shown in the detailed pictures. Each Meteorological Index (MI) shown in Table 1was determined at the single Spanish station and the seven meteorological stations in the USA. The following meteorological indices were directly obtained from the experimental measurements: T,P,RaGH, and PAR, obtained from Q p , which were measured with pyranometers and a quantum sensor, respectively. The solar azimuth cosine (cosZ) was calculated from the geometrical data of the location, using well-established mathematical relationships [ 28 ]. Finally, the dew point temperature ( Td ) was calculated from the vapour water pressure [ 29 ] and RH. The other indices, k t [ 28 ], the horizontal diffuse fraction, k d [ 30 ], and Perez’s clearness index, ε ,and Perez’s brightness factor, ∆ [ 31 ], were calculated using equations described elsewhere [ 32 ]. Appl. Sci. 2022,12, 2372 5 of 14 Table 1. Meteorological indices (MIs) measured in Burgos. MI MI Expression Ref. RaGH Global Horizontal Irradiance recorded kdHorizontal diffuse fraction kd=RaDH RaGH [30] QpPhotosynthetic photon flux density recorded PAR Photosynthetically active radiation PAR =Qp/4.57 µmol·J−1[26] ktClearness index kt=RaGH Bsc·ε0·cosZs[28] TAir temperature recorded PPressure recorded TdDew point temperature Td=35.859·logPv−21.48.496 logPv−10.2858 [29] cosZ Solar azimuth cosine cosZ =sinδ·sinφ+cosδ·cosφ·cosω[28] εPerez’s clearness index ε=RaDH+RaB RaDH +k·Z3 s 1+k·Z3 s [31] ∆Perez’s Brightness factor ∆=m·RaDH Bsc·ε0·cosZs[31] ε0= 1 + 0.033 ·cos[2·π·dn/365] is the average value of the orbital eccentricity of the Earth. dn is the day of the year. Bsc is the extraterrestrial irradiance constant (1361.1 W · m −2 [ 33 ]). k= 1.04 (or 5.56 × 10 −6 if Zs is expressed in degrees). Zs is the angle between the sky zenith and sun. δ , φ , ω are the respective declination, hour angle, and geographic latitude of the specific location. 3. Methodology The dataset was distributed into three categories of sky conditions based on the clearness index, k t , [ 28 ] and the values adapted by Suarez-García [ 34 ] considering clear [0.65, 1), partial (0.35, 0.65), and overcast (0 < 0.35] skies. Figure 3shows the Frequency Of Occurrence (FOC) of the different sky types in Burgos during the experimental campaign, which extended from April 2019 to February 2021. As can be seen, in Burgos, the clear sky conditions were prevalent except from November to January, when cloudy sky conditions occur more frequently. Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 15 Figure 3. Monthly Frequency Of Occurrence (FOC, %) of clear, partial, and cloudy sky conditions based on the clearness index, k t , in Burgos, Spain, between April 2019 and February 2021. 3.1. Feature Selection The meteorological indices with the greatest influence on the PAR estimation models were selected to improve the precision and reduce the complexity of obtaining the models in Burgos, regardless of whether all sky types that can appear in Burgos were considered or whether the data were classified into clear, partial, and overcast skies. The first step determines the relationship of the different MIs to the PAR. The Pearson criterion, based on the Pearson correlation coefficient (r), was used to determine the influence or weight that each of index has on the PAR component. The criteria are as follows: if r is close to 0, the corresponding MI has a very weak relationship with PAR, whereas if it is close to 1, or −1, the relationship is very strong. The Thumb rule [35] established five r intervals for the correlation: direct (1 ≥ |r (PAR, MI i )| ≥ 0.9), strong (0.9 > |r (PAR, MI i )| ≥ 0.7), moderate (0.7 > |r (PAR, MI i )| ≥ 0.5), weak (0.5 > |r (PAR, MI i )| ≥ 0.3), and negligible (|r(PAR, MI i )| < 0.3). Table 2 shows the different intervals of Pearson’s coefficients for the different MIs according to the classification of skies over Burgos with the k t sky classification (clear, partial, and overcast). Table 2. Pearson’s r (PAR, MI i ) based on the sky conditions according to the k t sky classification (clear, partial, and overcast). |r (PAR, MI i )| k t Sky Type [1–0.9] (0.9–0.7] (0.7–0.5] (0.5–0.3] (0.3,0] All sky conditions RaGH cosZ , k t k d , ε T Δ, P, T d Clear RaGH, cosZ k t , ε T k d , Δ, P, T d Partial RaGH, cosZ T k t , k d , Δ, ε, P, T d Overcast RaGH cosZ k t , Δ k d , ε, P, T, T d From the results shown in Table 2, it can be observed that RaGH is the MI that has a very strong and direct influence on PAR for all types of skies, coinciding with the results obtained by Ferrera-Cobos et al. [15]. Likewise, for all-sky types, cosZ and k t also have a strong relationship, and k d and ε have a moderate relationship with PAR. In the case of analyzing clear skies, there is a direct and strong relation with RaGH and a moderate relation with, k t , cosZ, and, ε. For partial skies, there is also a direct and strong relation, in addition to RaGH, with cosZ. For overcast skies, there is also a direct and strong relation Figure 3. Monthly Frequency Of Occurrence (FOC, %) of clear, partial, and cloudy sky conditions based on the clearness index, kt, in Burgos, Spain, between April 2019 and February 2021.