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Research Papers

An Empiric-Stochastic Approach, Based on Normalization Parameters, to Simulate Solar Irradiance

[+] Author and Article Information
Edith Osorio de la Rosa

CONACYT,
Universidad de Quintana Roo,
Blv. Bahía s/n esq. Ignacio Comonfort,
C.P. 77019, Quintana Roo, México
e-mail: osoriodelarosa@gmail.com

Guillermo Becerra Nuñez

CONACYT,
Universidad de Quintana Roo,
Blv. Bahía s/n esq. Ignacio Comonfort,
C.P. 77019, Quintana Roo, México
e-mail: guillermobec@gmail.com

Alfredo Omar Palafox Roca

Departamento de Actuaría, Física y Matemáticas,
Escuela de Ciencias,
Universidad de las Américas Puebla, Ex Hacienda
Sta. Catarina Mártir S/N.,
San Andrés Cholula, Puebla, C.P. 72810, México
e-mail: aocontreras@gmail.com

René Ledesma-Alonso

Departamento de Ingeniería Industrial y Mecánica, Escuela de Ingeniería,
Universidad de las Américas Puebla, Ex Hacienda
Sta. Catarina Mártir S/N.,
San Andrés Cholula, Puebla, C.P. 72810, México
e-mail: rledesmaalonso@gmail.com

1Corresponding author.

Contributed by the Solar Energy Division of ASME for publication in the Journal of Solar Energy Engineering: Including Wind Energy and Building Energy Conservation. Manuscript received February 25, 2019; final manuscript received May 23, 2019; published online June 11, 2019. Assoc. Editor: M. Keith Sharp.

J. Sol. Energy Eng 141(6), 061011 (Jun 11, 2019) (9 pages) Paper No: SOL-19-1069; doi: 10.1115/1.4043863 History: Received February 25, 2019; Accepted May 25, 2019

This paper presents a methodology to estimate solar irradiance using an empiric-stochastic approach, which is based on the computation of normalization parameters from the solar irradiance data. For this study, the solar irradiance data were collected in a weather station during a year. Posttreatment included a trimmed moving average to smooth the data, the performance of a fitting procedure using a simple model to recover normalization parameters, and the estimation of a probability density, which evolves along the daytime, by means of a kernel density estimation method. The normalization parameters correspond to characteristic physical variables that allow us to decouple the short- and long-term behaviors of solar irradiance and to describe their average trends with simple equations. The normalization parameters and the probability densities allowed us to build an empiric-stochastic methodology that generates an estimate of the solar irradiance. Finally, in order to validate our method, we had run simulations of solar irradiance and afterward computed the theoretical generation of solar power, which in turn had been compared with the experimental data retrieved from a commercial photovoltaic system. Since the simulation results show a good agreement with the experimental data, this simple methodology can generate the synthetic data of solar power production and may help to design and test a photovoltaic system before installation.

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Figures

Grahic Jump Location
Fig. 1

(a) Experimental data of solar irradiance (GHI), denoted by the variable E, as a function of the short-term variable m, for the 2017–2018 seasons described in Table 1. (b) Irradiance data shown in (a) after the application of the TMA, as described by Eq. (1) with N = 5 and L = 4, denoted by the variable , as a function of the short-term variable m, for the 2017–2018 seasons described in Table 1.

Grahic Jump Location
Fig. 2

(Top) Evolution of three fitting parameters A, B, and C (dots) and their general trends , , and (solid lines), in terms of the long-term variable d. The parameters were introduced in Eq. (2), the general trends in Eq. (12), and their physical interpretation is given in text. (Bottom) Histograms of the residuals x, the difference between each fitting parameter and its corresponding trend, and the proposed probability distribution function (solid lines), which are scaled by the bins width for comparison purposes. (Color version online.)

Grahic Jump Location
Fig. 3

(a) Normalized irradiance E* as a function of the normalized short-term variable m*, defined in Eq. (3a). The results from the experimental data are shown as diamonds (red) if m* lies within the daytime limits m, whereas dots (blue) are employed for the data outside the aforesaid boundaries. The irradiance estimator E (black line), given in Eq. (4), is presented as well. (b) Discrete probability densities, in terms of the discrete short-term variable mj* (horizontal axis) and the corresponding to the rate of change of the residuals r*(mj*). The coefficient 2/J = 0.0313 and the definition of J is given in Eq. (7). (Color version online.)

Grahic Jump Location
Fig. 4

(a) Scaled probability densities (scaled PDs), in terms of the discrete short-term variable mj* (horizontal axis) and the corresponding to the rate of change of the residuals r*(mj*) A bandwith value of h = 0.0364 has been employed, and its justification is provided in the text. The coefficient 2/J = 0.0313 and the definition of J are given in Eq. (7). (b) Irradiance profiles: measured (taken from the data presented in Fig. 1), expected (E[E(d, m)] computed with Eq. (21b), corresponding to the statistics classical definition) and simulated (according to the procedure detailed in Sec. 4), for four different dates of a year, as described in Table 1.

Grahic Jump Location
Fig. 5

(a) Daily radiant exposure: measured (integration according to Eq. (15) with the data presented in Fig. 1), expected (E[I(d)] computed with Eq. (22b), corresponding to the statistics classical definition) and simulated (according to the procedure detailed in Sec. 4), along the year described in Table 1. The expected behavior is given by Eq. (22b). (b) Comparison of the measurements obtained from a photovoltaic system (black circles and line) with the statistics obtained from the simulation of a year, with 100 samples. The (blue) boxes, the (red) dashes, the (red) crosses, and the (dashed) whiskers indicate the quantiles, the median, the outliers, and the full range of the data, respectively. (Color version online.)

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