Hybrid Models for Estimating 5-Minute Global Solar Irradiance on an Inclined Surface: A Case Study on Two Regions in Algeria

The spectral distribution of radiation reaching the Earth's surface is influenced by both the dispersal of radiation from outer space and the atmospheric constituents. The distribution of this phenomenon on land is of utmost importance in a wide range of applications, including earth-based solar power systems, the Earth's reflectivity, and photochemical reactions [1]. These applications require precise knowledge of solar resource availability in different regions [2]. In Algeria, the national energy plan emphasizes the rapid expansion of solar energy, particularly through the development of solar photovoltaic (PV) and solar power technologies. The government aims to launch multiple solar PV projects with a combined capacity of 800 MWp by 2020, with plans for additional projects with an annual capacity of 200 MWp from 2021 to 2030. However, accurately determining the distribution of solar radiation (SR) remains a challenge [3].

In regions where observed data is unavailable, it is customary to estimate SR using models, often classified into radiative transfer models [4], remote sensing retrievals [5], machine learning models [6, 7], and empirical models (EMs) [8, 9]. The EMs are widely used for SR estimation due to their low computational cost and their readily available inputs [10]. Among them are the models developed by Liu & Jordan, Perrin de Brichambaut, and Capderou. Various studies have utilized these models to estimate global, beam, and diffuse radiation across several regions in Algeria. For instance, Hamdani et al. [11] applied the Capderou, Perrin de Brichambaut, and R. Sun models to calculate hourly SR in Ghardaia, a city in the northern central region of the Sahara Desert of Algeria. Similarly, S-Koussa et al. [12] used the Bird & Hulstrom model to determine the three components of SR per hour in Ghardaia. Further, Benkaciali and Gairaa [13] applied both Liu & Jordan, and Brichambaut models to estimate daily and hourly SR at a specific location. Mesri-Merad et al. utilize several models, including Lacis & Hansen, Bird & Hulstrom, Davies & Hay, and Atwater, to compute hourly global SR (GSR) at two locations: Bouzareah in northern Algiers and Ghardaia [14]. Nia et al. [15] conducted a study using the Angstrom & Prescott model to assess monthly GSR across four cities in Algeria: Algiers, Oran, Bechar, and Tamanrasset. Bouramdane et al. [16] applied two semi-empirical methods to estimate daily inclined SR in Ghardaia. Their findings showed that the Liu & Jordan model produced more accurate results compared to experimental data, particularly at dawn and sunset. In addition, the Perrin de Brichambaut model proved to be the most suitable around solar noon. In another study, Lantri et al. [17] used available meteorological records to estimate different components of SR on a horizontal surface, concluding that Model 3 was the most accurate for estimating the global component based on relative humidity and water vapor tension. Soulouknga et al. [18] evaluated six EMs over 26 years of meteorological data from the Abeche site. Their results showed that the Sabbagh model provided the most accurate estimation, with excellent precision based on statistical indicators. EMs in the literature typically represent clear-sky conditions, leading to notable discrepancies compared to measured values, especially on cloudy days. Therefore, three models, Liu & Jordan, Capderou, and Perrin de Brichambaut, develop semi-EMs to estimate solar irradiance on horizontal surfaces. These models use consistent equations to convert data from horizontal to inclined planes, based on the well-known Liu & Jordan model from 1968. Since then, other transposition models (TMs) have emerged, offering enhancements to facilitate the transformation from horizontal to inclined planes, primarily differing in their treatment of the diffuse component.

From this survey, it can be observed that despite the availability of numerous estimation methods (EMs), only a few have been evaluated for short-term (5-minute) GSR predictions at Algerian sites. Given the inherently dynamic nature of SR, which can fluctuate significantly within minutes, and the critical need for real-time control and optimization in PV systems, 5-minute GSR estimation is crucial for ensuring their efficient and stable operation. Forecasts from a few seconds to a few minutes are necessary to manage rapid fluctuations and ramp rates, leading to smoother power production. This allows PV inverters, energy storage systems, and grid control equipment to adjust preemptively before cloud-induced power drops occur. A 5-minute resolution aligns with the response time of automatic grid control systems and helps prevent costly power quality issues, spinning reserve activations, and voltage fluctuations that can impact grid stability. For large PV installations, this results in measurable improvements in capacity factor, reduced curtailment losses, and enhanced grid integration capabilities, making it an essential tool for modern grid-scale PV systems [19, 20]. On the other hand, we note that EMs typically demonstrate the highest accuracy under clear-sky conditions; however, their precision diminishes significantly during cloudy periods, when SR exhibits the most variability. Furthermore, the performance investigation of hybrid models combining EMs with TMs has not yet been adequately explored in the literature. To address this research gap, this study investigates the performance of innovative hybrid models that merge five EMs with five TMs for predicting 5-minute GSR on inclined surfaces at two climatically diverse locations in Algeria: Ghardaia and Bouzareah. The main aim of the proposed method is to ensure an accurate estimation of the GSR on inclined surfaces. The paper’s key contributions are:

• Develop HMs by combining EMs and TMs intended for estimating the 5-minute GSR on inclined surfaces. Such a combination has two main benefits: i) it enables the use of TMs where actual measurements of horizontal solar irradiance are unavailable, a common situation in many parts of Algeria, particularly the southern regions, and ii) it aims to improve modeling outcomes from EMs, particularly for cloudy days.
• Applying the developed HMs for two regions, Ghardaia and Bouzareah, located in the south and north of Algeria, each characterized by unique atmospheric conditions.
• Investigating and comparing the performance of the developed HMs to EMs-based GSR estimation, considering four key statistical metrics.
• Improving the GSR estimation accuracy, especially under cloudy conditions, in which an NRMSE with a reduction of up to 90% is achieved for some cases.

The remainder of the paper is structured as follows: Section 2 describes the proposed method and provides a comprehensive evaluation of five EMs and five TMs. In Section 3, the results of a comparative study evaluating the performance of the developed models are presented and discussed. Finally, Section 4 offers the main conclusions.

The global expansion of solar energy has underscored the imperative for an accurate and meticulous assessment of SR. Using only traditional models, such as empirical ones, may not be enough to achieve accurate estimation with high performance. Hybridizing these models with other models is a compelling approach to overcome the limitations of individual estimating models and enhance estimation accuracy. Figure 2 shows the developed HMs, which merge five empirical and five transposition models, resulting in 25 combinations, to predict GSR on inclined surfaces. Each HM consists of one EM and one TM. The EM estimates the horizontal GSR, G_h, from the input data, while the TM facilitates the conversion from a flat to an inclined surface and achieves the titled GSR, G. These HMs offer two significant benefits. First, it enables the application of TMs in cases where horizontal radiation data are unavailable, a common occurrence across many locations in Algeria, particularly in the expansive southern area. Second, it enhances the performance of EMs, leading to more accurate estimation. Note that the five EM and TM are selected due to their: i) simplicity and proven estimation accuracy, ii) extensive validation in prior literature for SR estimation in similar climates/regions [11].

The mathematical formulas of each EM and TM involved are given in the next subsections.

2.png

Figure 2. Schematic diagram of the HMs designed for GSR estimation on inclined surfaces

3.1 Empirical models

3.1.1 Capderou model

As Figure 3 depicts, the total radiation, G, incident on a surface with any given orientation is expressed as the sum of two components as follows [3]:

3.png

Figure 3. Solar radiation distribution

$G=I+D$     (1)

where, I and D denote the direct and diffuse radiations, which are defined below.

a) Direct radiation: The direct radiation on the inclined surface is expressed by:

$I=I_{S C} exp \left[-T_l\left(0.9+\frac{9.4}{(0.89)^z} \sin (h)\right)^{-1}\right] \cos (i)$     (2)

where, I_SC is the solar constant, h is the sun height, z is the altitude, T_l is the link trouble factor, and i is the angle of incidence, for a horizontal plane is given by $\cos (i)=\sin (h)$.

b) Diffuse radiation: The diffuse radiation consists of three components as given in Eq. (3):

$D=D_1+D_2+D_3$     (3)

with:

• D₁ is the diffuse radiation on behalf of the sky expressed by:

$D_1=\delta_d \cos (i)+\delta_i \frac{1+\sin (\omega)}{2}+\delta_h \cos (\omega)$     (4)

where, the terms $\delta_d, \delta_i$, and $\delta_h$ refer to the direct or circumsolar, isotropic, and horizon circles, respectively, and $\omega$ denotes the hour angle.

• D₂ is the diffuse radiation from the ground or the so-called reflected radiation and is defined as:

$D_2=\delta_a \frac{1-\sin (\omega)}{2}$     (5)

where, $\delta_a$ defines the soil diffusion coefficient, and for a horizontal plane, the diffuse radiation from the ground is zero.

• D₃ represents the retro-diffused radiation expressed as follows:

$D_2=\delta_a \frac{1-\sin (\omega)}{2}$     (6)

3.1.2 Liu & Jordan model

Liu & Jordan model define the general formula for GSR, G, received by a tilted surface comprising three components, as follows [15]:

$G=I_h \cdot R_b+D_h \cdot\left(\frac{1+\cos (\beta)}{2}\right)+\left(\frac{1-\cos (\beta)}{2}\right) \cdot G_h \cdot \rho$     (7)

In this formula, I_h, D_h, and G_h represent the direct, diffuse, and global radiation on a horizontal surface, respectively. The parameter β denotes the tilt angle, ρ is the ground albedo (reflectance of the ground), and R_b is the ratio of beam radiation incident on an inclined plane to that on a horizontal plane. For surfaces in the northern hemisphere that are south-facing, R_b is defined as:

$R_b=\frac{\cos (\varphi-\beta) \cos (\delta) \cos (\omega)+\sin (\varphi-\beta) \sin (\delta)}{\cos (\varphi) \cos (\delta) \cos (\omega)+\sin (\varphi) \sin (\delta)}$     (8)

where, $\varphi$ and $\delta$ are the latitude of the location and the declination angle of the sun, respectively.

For a horizontal plane, the GSR can be formulated as:

$G=G_h=I_h+D_h$     (9)

a) Direct radiation: The direct radiation on a horizontal surface $I_h$ is expressed as:

$I_h=A \sin (h) \exp \left(\frac{-1}{c \sin (h+2)}\right)=\frac{I}{R_b}$     (10)

where, I is the direct radiation on a tilted surface at an angle β.

b) Diffuse radiation: The diffuse radiation D_h is determined by:

$D_h=B(\sin (h))^{0.4}$     (11)

In Eqs. (10) and (11), A, B, and c are constants that take into account the nature of the sky [16].

The reflected radiation on a tilted surface, R, is given by:

$R=\left(I_h+D_h\right)\left(\frac{1-\cos (\beta)}{2}\right) \rho$     (12)

For a horizontal plane, the value of the reflected radiation is zero [16].

3.1.3 Bird & Hulstrom model

Bird & Hulstrom model developed an alternative method to determine diffuse D and direct I radiation and the total SR, G, received on a tilted surface, which is the sum of these two components [12]:

$G=I+D$     (13)

a) Direct radiation: The direct radiation, I, is calculated as follows:

$I=0.9751 * \cos \left(\theta_z\right) * I_{s c} * \tau_0 * \tau_r * \tau_\omega * \tau_g * \tau_a$     (14)

where, $\theta_z$ is the zenith angle, $\tau_0, \tau_r, \tau_\omega, \tau_g$, and $\tau_a$ indicate the ozone, Rayleigh, water, gas, and aerosols scattering transmittances, respectively, which are defined in study [21].

b) Diffuse radiation: The diffuse radiation on a tilted plane is composed of three components: $D_r$ the diffuse radiation, $D_a$ the aerosols scattering after the first pass through the atmosphere, and $D_m$, the multiply reflected diffuse radiation, which are defined as follows:

$D_r=0.395 * I_{s c} * \cos \left(\theta_z\right) * \tau_0 * \tau_g * \tau_\omega * \tau_{a a} * \frac{\left(1-\tau_r\right)}{\left(1-m_{a+} m_a^{1.02}\right)}$     (15)

$D_a=0.79 * I_{s c} * \cos \left(\theta_z\right) * \tau_0 * \tau_g * \tau_\omega * \tau_{a a} * F_c * \frac{\left(1-\tau_{a s}\right)}{\left(1-m_a+m_a^{1.02}\right)}$     (16)

$D_m=\left[\left(1+D_a+D_r\right) \cdot \rho \cdot \rho_a^{\prime}\right] /\left[1-\rho \cdot \rho_a^{\prime}\right]$     (17)

where, $\tau_{a a}$ represents the transmittance of direct radiation due to aerosol absorption, $\tau_{a s}$ is the atmospheric transmittance due to aerosol scattering, $F_c$ is the atmospheric dispersion coefficient [22]. $m_a$ defines the air mass at a specified pressure [21], and $\rho_a^{\prime}$ is the Albedo of the cloudless-sky atmosphere.

3.1.4 Davies & Hay's model

The general formula proposed by Davies & Hay for calculating the GSR, G_h, received on a horizontal plane is:

$G_h=I_h+D_h$     (18)

where, I_h and D_h are the direct and diffuse radiations on a horizontal surface. To convert to SR on a tilted surface, the Liu & Jordan formula (Eq. (8)) is applied.

a) Direct radiation: In this model, I_h is given by:

$I_h=I_{s c} \cos \left(\theta_z\right)\left[\left(1-\alpha_0\right) \tau_r-\alpha_\omega\right] \tau_a$     (19)

where, $\alpha_0$ and $\alpha_\omega$ represent the fractions of incident energy absorbed by the ozone layer and water vapor, respectively.

b) Diffuse radiation: The diffuse radiation, $D_h$, is composed of three components:

$D_h=D_r+D_a+D_m$     (20)

where, $D_r$ is the diffuse radiation after Rayleigh scattering, $D_a$ is the diffuse radiation after scattering and absorption by aerosols, and $D_m$ represents the multiply reflected diffuse radiation. $D_m$ is a function of the albedo of the cloudless-sky atmosphere $\rho_a^{\prime}$ and the ground albedo $\rho$.

3.1.5 Perrin de Brichambaut model

The GSR on a tilted surface, G, as presented by Perrin de Brichambaut, can be calculated for any location and time using (21):

$G=I_n * R_b+D+D_s$     (21)

where, I_nis the direct normal irradiance, which is the irradiance received by a surface perpendicular to the sun rays, R_b is the inclination factor, defined in Eq. (8), D denotes the scattered or diffuse radiation on a tilted surface, while D_s is the diffuse radiation reflected from the ground and received on a horizontal plane.

3.2 Transposition models

The TMs aim to convert horizontal irradiance to in-plane irradiance. All the TMs proposed by several authors differ only in terms of the diffuse transposition factor, Rd, formulation. Diffuse radiation models for inclined surfaces are generally categorized into two types: isotropic and anisotropic. The key difference between them lies in how they partition the sky based on the intensity of diffuse radiation. Isotropic models assume a uniform distribution of diffuse radiation across the entire sky [1], while anisotropic models include modules that depict regions with higher levels of diffuse radiation.

Various mathematical models have been developed in the literature to calculate the factor R_d. In this study, we will focus on five specific models: Hay, Willmot, Temps and Coulson, Klucher, and Steven and Unsworth. The R_d formulation of these models is defined as follows.

3.2.1 Hay model [1]

$R_d^{ {Hay }}=\left(1-A_4\right) * \cos ^2\left(\frac{\beta}{2}\right)+A_4 * R_b$     (22)

where, the anisotropy index; $A_4=\frac{I_h}{I_0}$ and $R_b=\frac{\cos (\theta)}{\cos \left(\theta_z\right)}$ with $\theta$ is the incident angle.

3.2.2 Willmott model

$R_d^{{WILLMOTT }}=A_4^{\prime} \cdot R_b+C_\beta\left(1-A_4^{\prime}\right)$     (23)

where, $C_\beta=1.0115-0.20293 \beta-0.080823 \beta^2$

3.2.3 Temps and Coulson’s model

$R_d^{T E P M S}=cos ^2\left(\frac{\beta}{2}\right) \cdot\left(1+sin ^3\left(\frac{\beta}{2}\right)\right) \cdot\left(1+cos ^2 \theta sin ^3 \theta_z\right)$     (24)

3.2.4 Klucher’s model

$R_d^{K L U C H E R}=\cos ^2\left(\frac{\beta}{2}\right)\left[1+f_k \cos ^2 \theta\left(\sin ^3 \theta_z\right)\right]\left[1+f_k \sin ^3\left(\frac{\beta}{2}\right)\right]$     (25)

where, f_kdenotes the modulation function [16].

3.2.5 Steven and Unsworth’s model

$R_d^{S T E V E N}=S * \frac{\cos (\theta)}{\cos \left(\theta_z\right)}+(-S)$$\left[cos ^2\left(\frac{\beta}{2}\right)+\frac{2 * b}{\pi(3+2 * b)} \cdot\left(sin (\beta)-\beta \cos (\beta)-\pi sin ^2\left(\frac{\beta}{2}\right)\right)\right]$     (26)

where, S denotes the anisotropy index, and b is a constant.