Evaluating Solar Energy Potential Through Clear Sky Index Characterization Across Elevation Profiles in Mozambique

Abstract

The characteristics and types of the sky can greatly influence photovoltaic (PV) power generation, potentially leading to a reduction in both the lifespan and efficiency of the entire system. Driven by the challenge of addressing fluctuations in solar PV energy utilization, the aim was to assess the solar energy potential by analyzing the clear sky index $\left(K_t^{*}\right)$ across elevation profiles. To achieve this, a theoretical model for determining $K_t^{*}$ was employed, which encapsulated the solar energy analysis. Initially, solar energy data collected from approximately 16 stations in various provinces of Mozambique, as part of the solar energy measurement initiatives by INAM, FUNAE, AERONET, and Meteonorm, was processed. Subsequently, the clear sky radiation was calculated, and $K_t^{*}$ was established. The statistical findings indicate a reduction in energy contribution from the predictors, accounting for 28% of the total incident energy; however, there are progressive increases averaging around ~0.02, with $K_t^{*}$ values ranging from 0.4 to 0.9, demonstrating a strong correlation between 0.7 and 0.9 across several stations and predictor parameters. No significant climate change effects were noted. The radiation flux is directed from areas with higher $K_t^{*}$ to those with lower values, as illustrated in the heat map. The region experiences an increase in atmospheric parameter deposition, with concentrations around ~0.20, yet there remains a substantial energy flow potential of 92% for PV applications. This interaction can also be applied in other locations to assess the potential for available solar energy, as the analyzed solar energy spectrum aligns closely with the theoretical statistical calibration of energy distribution relevant to the global solar energy population process.

1. Introduction

The utilization of solar energy on the Earth’s surface is crucial and has increasingly gained significance over generations [1,2]. Even during ancient times, our ancestors harnessed solar energy for drying and preserving food as well as for daily illumination [2,3]. However, with the current energy demand reaching approximately 23,090.0 GW [4] in 2024 and projections estimating a future requirement of 34,989.0 GW [5] by 2030, the reliance on fossil fuels for energy generation remains predominant, despite their detrimental environmental impact through atmospheric pollution and the consequent reduction in solar energy availability [6]. There are ongoing initiatives to embrace clean and renewable technologies [2,7,8,9,10]. Since the 1960s, there has been a notable trend towards the implementation of technologies that convert hydropower and wind energy [8,9] through the establishment of power plants and wind turbines [6,11,12,13,14]. While these solutions require significant investment and can fulfill a substantial portion of energy needs, they also necessitate adaptation to local ecological conditions, which poses risks to the ecosystem [9,15,16]. The utilization of clean energy sources, such as solar technology, has demonstrated itself as an effective alternative due to its efficiency and versatility, allowing for the conversion of solar radiation into a steady supply of electrical power during peak sunlight hours [9,17,18]. The generation of solar energy from PV cells within a solar power plant is primarily determined by the solar radiation incident [1,5] on their surface. Various methods of solar energy production exist across different global regions, influenced by a range of atmospheric, meteorological, geospatial, temporal, and other factors [19,20,21]. These variables can be modeled to optimize the output flow of a solar facility [22]. Generally, larger areas are more conducive to consistent energy production, unlike smaller installations that may experience significant output reductions due to cloud cover [23,24,25,26,27]. It is estimated that less than 70% [22,28,29] of the solar energy emitted by the atmosphere reaches the Earth’s surface, contingent upon local climatic conditions [1,2,26,29]. The remaining energy is either absorbed or reflected back into space [2], with the primary factor being the total transmittance, which results from the interaction of various atmospheric components, including uniformly mixed gases, water vapor, particulate matter, and aerosols, which contribute to the ozone layer [30,31,32,33,34,35]. Furthermore, the geographical location and seasonal variations in solar energy availability can differ significantly [25,36,37,38,39]. The atmospheric parameters that predict solar energy deposition are crucial factors influencing the solar irradiation that reaches the Earth’s surface, which can be harnessed for PV conversions in solar power plants [1,2,28,33,39]. These contributions primarily stem from industrial activities [40,41] that can excessively release gases into the atmosphere due to the extraction and production of raw materials [2], the processing of sand in refineries and stone cutting facilities, and combustion processes that emit solid particles [42,43], particularly from uncontrolled burning in agricultural and forested areas [44,45,46,47,48]. This burning releases gases [1,2,5] that enhance the ozone layer and others that significantly diminish solar energy [49], alongside aerosol emissions, volcanic eruptions, and sea emissions that are deposited in the atmosphere and can either absorb or reflect solar energy [8,50,51,52]. Seasonal variations also play a role; during hot and rainy seasons, there is an increased incidence of solar energy as rain helps to clear some atmospheric particles [22,53,54,55]. The high energy flow during these times excites and disperses gases and particles [2,15,39,56], raising their temperatures and allowing for greater solar incidence on Earth, thus creating optimal conditions for solar energy conversion [23,57,58]. Conversely, in colder seasons, when the Earth is positioned farther from the Sun and oriented away from it [1,29,46,59], the energy flow is significantly reduced, leading to a higher accumulation of particles that contribute minimally to atmospheric energy transitions due to their low excitation levels [60,61,62]. Most particles, vapors, and gases remain suspended in a fractal cloud, and depending on the extent of their deposition, this can further diminish solar energy availability [2,30,42,47]. Tropical regions, particularly those near the equator with minimal industrial activity and lower emissions of atmospheric pollutants, experience a more consistent availability of solar energy compared to polar regions, which are characterized by glaciers and a significant presence of stationary vapors that lead to lower temperatures [32,48,63,64]. These vapors absorb and release energy as diffuse radiation, resulting in a portion of the solar energy in these areas being diminished [2,33,65]. Additionally, the presence of greenhouse gases exacerbates global warming, as the solar energy is not effectively released into the Earth’s surface but is instead trapped in the atmosphere [29,36,38,66], leading to increased particle agitation akin to the entropy observed in semi-open systems [2,19,22,38]. This process involves atmospheric particles in the emission excitation process, ultimately resulting in rising temperatures [22,28,29]. Therefore, it is essential to comprehend the actual variability in solar energy across different elevations to accurately assess its variability in relation to sea level, considering the diverse landforms present on Earth [30,56,59,67]. The greater the scale of solar energy measurement, the more accurately the true variability of solar energy can be assessed [48,68,69,70]. Although an optimal scale for observing this variability has yet to be established, current observations utilize scales of 0.001, 0.002, and 0.00001 s [3,22,30,59,67,71,72], which effectively capture the true variability without addressing sub-consistent descriptions [1,22,30]. Even the $K_t^{*}$, which eliminates variability caused by solar geometry, encompasses the predicted global energy, which does not exceed 1 [5,29,66]. For optimal measurement, however, the variation line remains inconsistent even at sub-minute scales, allowing for the observation of fluctuations caused by cloud movement and atmospheric interference [51,63,68,73]. Consequently, days can be categorized into clear sky days, characterized by full solar incidence and minimal deviation in energy; cloudy sky days, which exhibit significant deviation due to cloud cover and atmospheric conditions that reduce PV production; and intermediate sky days, which display high deviation and possess characteristics that fall between clear and cloudy days [52,74,75]. The daily variation in solar energy availability impacts the output of solar power plants [1,39,73]. This indicates that the solar energy reaching the conversion area exhibits varying power levels [76,77]. Consequently, for several minutes, multiple power curves can be observed for the same solar cell. Additionally, different solar stations experience varying levels of solar incidence, which can also lead to fluctuations in PV production [1,39,41,78]. These variations in output can occur even with the implementation of PV rectification technology, which adjusts the continuous power signal and can result in output changes over short intervals due to the movement of solar clouds [12,25,31]. To mitigate these fluctuations, it is essential to dimension the implementation of solar plants based on accurate forecasts of solar energy flow, particularly by determining the $K_t^{*}$ [22,39,59,71]. This approach allows for the operation of the PV plant to be aligned with the actual energy available, thereby reducing the likelihood of output fluctuations below projected levels and enhancing both the durability and efficiency of the system [1,29,58,79].

Recent studies on variability related to PV energy analyze how to compensate for the territorial intermittency of solar energy, and observe a reduction in the set of the most intense variations representing 5.5 to 15% of the analysis period against 1.7% taken individually [31,66,79,80]. Researchers analyzed PV systems and found that using a solar power optimizer can increase productivity [55,81,82,83,84,85]. They also measured the variability of PV power production and concluded that it can be measured by determining the amount and step of the dispersion factor for a fleet of identical PV systems [23,59,61,86,87]. In terms of short-term PV power variations, it has been shown that the intermediate sky type is the most potentially problematic due to the spatiotemporal variability of the network of increasing numbers of PV power systems [31,67,72,88,89,90]. Researchers predicted clear sky irradiance for solar resource mapping in large-scale applications and observed that uncertain predictions under ideal clear sky conditions can propagate and affect all sky predictions, resulting in potential bases in solar resources at the continental scale [6,30,66,91,92]. Researchers have discussed how temporal averaging affects short-term GHI variability in intermediate sky circumstances and concluded that values >1 tend to underestimate variability, while values <1 probably increase it [56,59,71]. According to an assessment of the behavior of the $K_t^{*}$ using in situ recorded GHI and correlation (1 h), the daily radiation is the result of the integration of a variety of conditions, with clear and cloudy conditions predominating. Temporal variability in southern Mozambique was analyzed for 1 and 10 min and was found to be dominated by clear and intermediate days [22,26,69,76,93,94,95,96]. Very-short-term GHI was predicted and solar radiation variability was found [11,18,33,92,97,98]. GHI data was estimated on a horizontal surface with the optimal inclination angle of the modeling (5 min), and it was concluded that inclined surfaces present variability [19,90,94,99,100,101,102]. Short-term PV variability based on cloud velocity and GHI was analyzed, and it was concluded that short-term ramp rates decrease with plant size [22,80,103,104,105,106].

As previously mentioned, it is evident that solar energy is influenced by atmospheric predictive parameters, which induce oscillations on the Earth’s surface. This impact varies across different types of days, exhibiting a probability density influx with consistent deviations, particularly on cloudy and clear sky days. Such fluctuations pose a significant challenge to the output stability of PV power plants, ultimately diminishing their efficiency and longevity. Furthermore, intermediate sky days characterized by high deviations and variances present additional complications. This phenomenon is corroborated by the findings of Gueymard [30], Lohlman et al. [71], and Hoff and Perez [59] that highlight the variability in solar energy across diverse climatic regions. The authors note that intermediate days with substantial deviations share traits with both cloudy and clear sky days, leading to a reduction in the solar plant’s output power curve and the emergence of bimodal production patterns. This means that on the same day, there can be two distinct peaks in power production for the same energy estimate. Consequently, there is a pressing need to assess solar energy potential across various elevations to achieve a consistent PV power output that does not compromise the performance of other solar equipment, thereby enhancing overall efficiency. This necessity is primarily driven by the goal of accurately understanding the variability in solar energy, its behavioral patterns, and the implications of climate change over the years, particularly during the 21st century, with comparative analyses across multiple years and study stations.

It was essential to (1) assess and categorize the types of days, showcasing the study area predominantly characterized by clear sky days, in comparison with nations in the Southern African region; (2) investigate the $K_t^{*}$, focusing on the attributes of clear and overcast days, with particular attention to intermediate days that exhibit potential solar instability; and (3) examine the solar energy flow description through heat map analysis in the study area, extending to other regions, while conducting a comparative analysis of solar energy flows susceptible to climate change. This necessity arose from the urgent requirement to evaluate solar energy potential by characterizing the $K_t^{*}$ in elevation profiles across Mozambique, in light of the current push for the widespread adoption of clean energy, alongside the observation of climate change and its implications for PV production and conversion, which are influenced by the growth dynamics of particulate matter in the Earth’s atmosphere. The findings are innovative as they not only classify day types for evaluating solar energy potential at various measurement stations, but also illustrate the propagation of solar energy flux through heat map assessments, incorporating 21 years of continuous measurements, including short-term sampling, while also evaluating climate change in relation to the $K_t^{*}$ across different day types. This is crucial for mitigating fluctuations in solar plant output and enhancing its efficiency and longevity, while also comparing performance in terms of output power under diverse sky conditions.

In pursuit of this objective, short-scale solar energy samples were gathered across various locations in Mozambique, southern Africa. These included interprovincial distances in the southern region at UEM-Maputo, Massangena, and Dindiza stations, as well as in the mid–east and western regions at Pomene, Marávia, Nhangau, Lugela-1 and -2, and Nhapassa 1 and 2 stations. Additionally, in the northeast and western regions, samples were collected at Massangulo-1 and -2, Nanhupo-1 and -2, and Chomba stations during global horizontal solar energy measurement (GHI) campaigns. The measurements were taken at 10 min intervals during the years 2012 and 2013, while global summary analysis results were compiled from 2004 to 2024, with intervals of 1, 10 min, and 1 day recorded in the databases of PVGIS [107], Meteonorm [108], NASA [109], and NOAA [110]. Furthermore, predictor data samples concerning aerosol concentration density, dust, uniformly mixed gases, and ozone were sourced from the National Institute of Meteorology of Mozambique (INAM) [111], Mozambique National Energy Fund’s (FUNAE) [112], and the AERONET [113] database. The sample is suitable for analysis, and its processing can be organized to eliminate systematic errors, allowing for the determination of a quantity that correlates the measured global radiation with theoretical radiation. This process begins with predicting global solar energy while considering parameters that diminish its intensity upon reaching the Earth’s surface, thereby facilitating a more accurate assessment of its actual behavior by excluding all factors that contribute to this reduction.

It is evident that in the majority of measurements, solar energy closely aligns with theoretical energy values, indicating that the chosen model is optimal for both clear and cloudy days, with a notable peak in the increments of the $K_t^{*}$. On clear days, the $K_t^{*}$ approaches 0.98, while on cloudy days, it hovers around 0.3. During intermediate sky conditions, the index exhibits characteristics between clear and cloudy. However, the south, southwest, and md–east regions, along with parts of the northwest, experience a significant flow of solar energy, primarily characterized by intermediate sky days, leading to a decoupling of energy throughout the defined analysis area. The energy flux maintains a stable $K_t^{*}$, and over the years, increases in the $K_t^{*}$ have been both compensatory and consistent, reflecting minimal changes in the solar energy pattern across the entire study area of Mozambique. Nevertheless, this region holds considerable potential for PV applications. The solar energy flux depicted in the heat map illustrates a transition from areas with high $K_t^{*}$ flux to internal regions with lower indices, attributed to the persistent strengthening of local cloud cover. This analysis framework can be applied to various regions to assess and estimate solar energy potential at different elevations, thereby characterizing and mitigating fluctuations by utilizing local variables and parameters from multiple geolocated sites worldwide.

2. Materials and Methods

2.1. Data Collection and Processing

During the FUNAE [112] 2012, 2013, and 2014 solar radiation measurement campaign, the GHI data sample was gathered in eleven provinces of Mozambique, with the southern, mid, and northern locales being UEM-Maputo, Massangena, Dindiza, Pomene, Marávia, Nhapassa (two stations), Nhangau, Massangulo (two stations), Nanhupo (two stations), and Chomba. The INAM [111] measurement stations were used to measure a second GHI sample in each of the provinces around the previously mentioned stations. Measurement intervals of one to ten minutes were used for the measurements. These stations are set up with high-resolution radiometers at interprovincial and interstation lengths of 1000 km, distributing the sensors uniformly across the area at kilometer intervals. The PY 5886 pyranometer radiometer sensors (made by NRG Systems, located in Houston, TX, USA [32,114]) were used to measure the GHI and DHI radiation components. They were attached to a system that had an inserted pyrheliometer that measured the DNI component directly; Figure 1 illustrates the behavior of these sensors.

Setting Up and Running Data Collection Devices

To better collect solar energy and prevent the effects of shading, bird landings, objects obstructing the incidence of solar energy, human and animal action, and other factors, the pyranometers were placed on a mobile-phone-type antenna tower at each of the 11 measuring stations, some of which were 40 m above the ground and others 60 m. This is depicted in Figure 1.

Solar radiation emitted by the Sun, encompassing electromagnetic radiation across various spectrum bands from ultraviolet to infrared, is referred to as global solar radiation (GHI). This radiation, which includes both direct and diffuse components, is measured by pyranometers that are properly leveled and oriented to ensure accurate readings. These instruments generate electrical signals that are transmitted to a laboratory through shielded wiring systems designed to protect against electromagnetic interference and ensure proper grounding, supported by channels and insulation for enhanced safety and durability. The signals from the pyranometers are received by a data logger, where they are amplified and converted from analog to digital format. Data are recorded at intervals ranging from 1 to 10 min up to 1 h for long-term monitoring. Subsequently, these data are transmitted to a computer via USB, networks, or serial connections, as illustrated in Figure 1b. The computer, which can receive data in real time or download it from the data logger, utilizes specific software (such as Campbell Logger Net version NRD-4.0, LabVIEW version 3, or scripts in Python 3.9, R-2024 and Matlab 2025a) to visualize radiation levels in real time and generate graphs of the solar spectrum.

The pyranometer for data collection was established with protective parameters including a linearity of ±0.5%, a cosine accuracy of ±1%, a spectral range with a 1 min response time, and a location factor spanning 295–2800 nm. The pyrheliometer exhibited optimal calibration characteristics. The sensors were installed horizontally on the roof of a university building, with inter-sensor distances of approximately 15 m. The sensors necessitate biweekly maintenance, which involves cleaning the glass dome and ensuring the sensors are horizontally aligned. Each instrument underwent routine seasonal maintenance during the measurement period, which encompassed data transmission, battery replacement, cleaning, and realignment. The times for cleaning and orientation were recorded during these activities, excluding data selection times that involved turbulence and interference. Throughout the week-long campaign, the voltage angle was documented if any inaccuracies in data calibration were identified, although this was not consistently observed.

To prevent human interference, shade, and other obstacles that prevent the measurement of solar radiation, additional sensors were positioned on towers 40 to 60 m high in each of the 11 provinces. Additional readings from a reputable thermopile pyranometer location were used to confirm the accuracy of the measurements for periods that were chosen at random. The verified sample of readings made with the instrument was taken from the Meteonorm database. From 19 June 2019 to 31 December 2021, measurements were conducted using monthly averages and short-scale measurement intervals of roughly four minutes, one hour, and one day. The AERONET [113] was also used to gather a sample of atmospheric parameter data, such as the proportion of aerosol concentrations, water vapors, ozone, and uniformly mixed gases. These data were used as predictor data in sample extrapolation and interpolation of a sample of missing data in the locations indicated in

Table 1. Coordinates of the measuring stations (property of AERONET).

ID	Site Name	λ (nm)	Amplitude	Level	Long. (°)	Lat. (°)	A (m)
1A	Niassa	400–500	4″, 1 and 24 h	2.0	37.5665	−12.155	510
3A	Sofala	400–500	4″, 1 and 24 h	2.0	37.5665	−12.155	510

.

The collected data representing time series, as well as atmospheric, spatial, and geographic data, were then organized into a tab.

2.2. Sample Size

The sample was generated to aid in calculations using data from the MZ06–Tete–Chiputo, MZ11–Sofala–Vanduzi, MZ21–Manica–Choa, MZ32–Zambezia–Lugela, MZ25–Niassa–Massangulo, MZ24–Nanhupo, and MZ03–Cabo–Delgado–Ocua campaigns when the sun rose and set (between 6:00 am and 6:00 pm). From the sixth month of the year onward, the data covered only the months of June through December for the FUNAE, AERONET, Meteosat, and INAM data campaigns. The samples collected at the stations included three-year intervals of complete measurements (2012 to 2014 and 2019 to 2021). In 2020 and 2021, the data covered all months of the year of complete measurements, totaling roughly 823,478.00 daily radiation data points each in the useful study area.

2.3. Procedural Execution Order for Solar Energy Analysis

In the data processing phase, the measured sample was extracted and the location of solar energy incidence was separated, that is, full hours of solar energy in all years, approximately three years in each stations of the 16 in total. Each day was analyzed and separated into acceptable and unacceptable types of days. Unacceptable days were a typology of days in which constant failures were observed, with only approximately 10 to 30 of the total daily sample being averaged over a short measurement scale.

However, optimal characteristics refer to the acceptable typology in which there is a minimization of outliers, i.e., the energy complies with the criteria of data included in the solar constant, without major deviations outside of nature. The unacceptable ones presented major deviations in relaxation to the solar constant, theoretical irradiation, and energy according to the statistical model; however, this range presented outliers that completely distorted the results, in addition to the acceptable sample obeying the standardized normal distribution taken to proceed with the study.

As illustrated in the second stage of the statistical design shown in Figure 2, the sample was modeled and corrected for errors, extrapolation and filling in missing data using the Random Forest model and ordinary kriging. The optimal parameters were chosen and tested for the conditions mentioned above, and good behavior’ was chosen and analyzed as input in the MLM. The $K_t^{*}$ model, or the correlation between the expected value and the theoretical global solar radiation of clear skies, was examined during the model and validation phase. After that, the solar energy variability and each station’s correlation coefficients were computed and examined. Finally, the correlational relationship between the $K_t^{*}$ of several stations was analyzed using correlation coefficient intermediates.

An accurate assessment would have taken into account the regressors that offer the relative variability of energy required to totally eradicate energy variations at a solar plant’s output and fully estimate the real potential of the available energy. This focuses mostly on smaller areas that might be vulnerable to capping because of fluctuations in level, rather than bigger areas where generation might occasionally be erratic. The investigation revealed a high correlation between local predictor parameters and local energy.

2.4. Study Area

The study region (Mozambique) is located between the parallels 10°27′ and 26°52′ south latitude and between the meridians 30°12 and 40°51′ east longitude, where the stations were installed. The characteristics in terms of latitude and longitude are described in

Table 2. Location of the study stations and details of the available $K_t^{*}$ datasets.

Station	Province	Tower	Longitude (X)	Latitude (Y)	Nr. Stations
MZ03_UEM	Maputo	FUNAE	33°6′1.64″ E	25°19′18.02″ S	1
MZ03_Masangena	Gaza	MCeL	32°56′26.72″ E	21°34′59.51″ S	1
MZ03_Dindiza	Gaza	FUNAE	33°25′22.78″ E	23°27′37.09″ S	1
MZ03_Pomene	Inhambane	MCeL	35°35′35.52″ E	17°47′32.54″ S	1
MZ03_Chomba	Cabo Delgado	FUNAE	39°23′36.16″ E	11°32′57.57″ S	1
MZ06_Maravia	Tete	FUNAE	31°40′33.7″ E	14°58′28.07″ S	1
MZ11_Nhangau	Sofala	FUNAE	35°12′18.72″ E	19°43′46.64″ S	1
MZ21_Nhapassa	Manica	MCeL	33°13′79″ E	17°47′32.54″ S	2
MZ24_Nanhupo	Nampula	MCeL	39°30′46.77″ E	15°57′57.38″ S	2
MZ25_Massangulo	Niassa	TDM	35°26′12.82″ E	13°54′25.93″ S	2
MZ32_Lugela	Zambezia	MCeL	36°42′47.51″ E	16°28′44.5″ S	2

. Additionally, the topographical cut of the Mozambican region, which has high potential for solar energy each year, is presented in a dispersed manner. As shown in Figure 3, the measurement stations installed in each province have taken into consideration the lowering of parameters because they are mounted at heights of 40 to 60 m. Their location at sea becomes strategic for the implementation of clean solar energy projects in arable lands and makes it easy to manage, maintain, and reproduce energy for various purposes, all of which help the high density of biomass and other renewable resources like wind.

2.5. Experimental Procedure

The impact of regional variations in solar energy on a solar plant’s PV output power was examined. As Figure 4 illustrates, for various sky types, the output power without power behaves differently on days with bimodal characteristics, i.e., days with strong cloudiness and others with weak cloudiness. This creates instability in the operation of the remaining devices in the system and puts not only its efficiency but also its lifespan and service life at risk.

The combined solar power generated for clear sky days is at least 517,600, with a median of 1,049,700, and for cloudy sky days, it is at least 265,000 with a median of 1,049,700, at most 3,110,400 with a median of 1,292,683, deviating to 745,655; for intermediate sky days, it is at least 386,200 with a median of 1,172,600. Figure 4 illustrates the effects of the solar power of a solar cell for various sky types using measurements of 1 and 10 min. This is considered in terms of the sample collected throughout the Mozambican region, recorded for the daily course, with measurement intervals of 1 and 10 min for an amplitude of 1 day, showing a maximum in the order of 2,895,800 with an average of 1,379,974, deviating to 590,100. For the short-scale and long-term measurements, the energy density is recorded in the order above 60%, with the majority of days having an intermittent sky across the entire region. However, an attempt was made to estimate the atmospheric transmittance for the incident radiation ${\tau }_b$ based on the received and predicted solar energy flow. Equation (1) was then used to calculate the direct radiation for clear skies [1,2,114].

$$G_{cnb}=G_{on}{\tau }_b$$

(1)

where $G_{on}$ is the extraterrestrial radiation incident on a plane normal to the radiation on the same day and is expressed in references [1,87]. For the zenith angle, at an altitude above 2.5 km, the clear sky normal DNI was calculated using Equation (2) [56,88,89,115].

$$G_{cb}=G_{on}{\tau }_b\cos {\theta }_z$$

(2)

For periods of any hour, the direct horizontal radiation was estimated. The Total Theoretical Radiation $G_{Clear}=Total$, was determined, which represents the sum of the direct radiation on the horizontal surface that arrives in a time, and the diffuse radiation on the horizontal surface [1,114].

A diverse set of machine learning and statistical modeling techniques—including Simple Linear Regression (SLR), Random Forest (RF), Regression Kriging (RK), Support Vector Machine model (SVM), AutoRegressive Integrated Moving Average (ARIMA) model, Gradient Boosting Machines (GBMs), Gaussian Process regression (GPR), and Artificial Neural Networks (ANN) with integrated Long Short-Term Memory (LSTM) cells—was comprehensively evaluated through comparative performance analysis [1,2,114]. ANN were specifically targeted because they demonstrated a lower ANN error margin of order 8.0 W/m², as indicated by the relationship in Equation (3) [90,91,114].

$${GHI}_i=\sum _{i=1}^n(w_{ij}{u}_j+w_{0j})$$

(3)

The Random Forest model (RF) with ${\lambda }_i={\displaystyle \frac{d_{i0}^{-r}}{\sum _{z_i{N}_0^k}{d}_{i0}^{-r}}}$ measures relative inverse distance weights between stations, with $r$ being a parameter that amplifies the relevance of closer neighbors and reduces the weights of more distant ones, given by Equation (4) [2,56,82,115,116].

$$\overline{z}\left(N_0^k\right)=\sum _{z_i{N}_0^k}{\lambda }_i{z}_i$$

(4)

The SVM model operates by maximizing the margin between classes within a set of multidisciplinary input vectors, which incorporates a hinge loss term with regularization given in Equation (5) [1,44,76,114,116].

$$min\left({‖w‖}^2+C\left[{\displaystyle \frac{1}{n}}{\sum }_{i=1}^{n}max\left(\mathrm{0,1}-y_i\left(w^T{x}_i-b\right)\right)\right]\right)$$

(5)

The ARIMA model, employed for time forecasting, uses autoregressive and moving average components with lagged values and residuals. It general formulation is given in Equation (6), where L is the lag operator and ${\epsilon }_t$ represents a normally distributes error term with zero mean [18,117,118].

$$\left(1+\sum _{i=1}^{p^{,}}{\alpha }_i{L}^i\right)X_t=\left(1+\sum _{i=1}^q{\theta }_i{L}^i\right){\epsilon }_t$$

(6)

The RK combines a linear regression model with geostatistical kriging to account for spatial autocorrelation in residuals [114,119]. RK prediction for a target $x_0$ is calculated using estimated trend components $\widehat{m}\left(x_0\right)$ and spatially interpolated residuals $\widehat{e}\left(x_0\right)$, as given by Equation (7) [1,2,46].

$$\widehat{z}\left(x_0\right)=\sum _{k=0}^p{\widehat{\beta }}_k{q}_k\left(x_0\right)+\widehat{e}\left(x_0\right)$$

(7)

The SLR describes the lines dependency between a predictor x and a response variable y. [12,114]. It is modeled with additive gaussian noise ${\epsilon }_i,$, assuming homoscedasticity and independence across observations, given in Equation (8) [1,114,120,121].

$$y_i={\beta }_0+{\beta }_1{x}_i+{\epsilon }_i, i=1,\dots ,n$$

(8)

The Persistence model provides a naïve forecast buy assuming that future observations will replicate the most recent value. This method, often used as reference model, is expressed by Equation (9) [114,122,123].

$${\widehat{GHI}}_{t-1}={GHI}_t$$

(9)

GBMs, construct a strong predict model by sequentialy training weak learners (eg. Decision trees). The final $F_M\left(x\right)$ aggregates the output M base learners scaled by a learning rate $\eta$, starting from an initial $F_0\left(x\right)$, given in Equation (10) [115,122,124].

$${\widehat{y}}_i=F_0\left(x_i\right)+\sum _{m=1}^M\eta h_m\left(x\right)$$

(10)

The GPR is a non-parametric Bayesian regression framework, assumes underlying data-generating function $f\left(x\right)$ follows a gaussian process with mean function $m\left(x\right)$ and a function $k\left(x,x^{\prime }\right)$, given by Equation (11) [99,102,125,126].

$$f\left(x\right)~GP\left(m\left(x\right),k\left(x,x^{\prime }\right)\right)$$

(11)

LSTM units central to certain classes of recurrent neural networks, are specially designed to learn long-term dependencies in sequential data. The forget gate function, which determines which past information to discard, is mathematically defined by Equation (12), where $\sigma$ is the sigmoid activation function [14,116,124,126,127].

$$f_t=\sigma \left(W_f\left[h_{t-1},x_t\right]+b_f\right)$$

(12)

To remove the variability due to the reduction in incident solar radiation on the horizontal surface, we introduced a normalized quantity, the brightness index, $K_t^{\mathrm{*}}$ (GHI ratio for daily extraterrestrial radiation) [88,119,120]. Relating GHI and clear sky radiation, the $K_t^{\mathrm{*}}$ was defined by Equation (13) [114,121,122,126].

$$K_t^{\mathrm{*}}={\displaystyle \frac{GHI}{G_{Clear}}}$$

(13)

To quantify temporal variability, a dedicated metric was utilized to evaluate the difference between successive values of the $K_t^{*}$ [2,114,126]. A fixed time resolution of ten minutes was adopted to ensure accuracy in capturing the variation between consecutive observations, as expressed in Equation (14) [2,122,125,126,127].

$$∆K_t^{*}=K_{t+1}^{*}-K_t^{*}$$

(14)

The distribution of $K_t^{\mathrm{*}}$ (Figure 5a), its smoothed variation (Figure 5b) and the $∆K_t^{*}$ depicted in Figure 5c as a function of the time of day, for a time interval of one and ten minutes and amplitude of one day, on a clear sky day, for acceptable and unacceptable days, shows that at the UEM–Maputo station, throughout the clear sky day of February 1, 2012, at sunrise, the solar energy presents itself with low density, reaching $K_t^{\mathrm{*}}$ in the order of 0.4 and $∆K_t^{*}$ in the order of 0.0001. However, in the interval from 07:30:00 to 10:19:00 h, it presents itself with abrupt fluctuations with $K_t^{*}$ in the order of 0.300, 1.0001 and its $∆K_t^{*}$ in the order of −1.000, 0.6000. Over the rest of the day, the solar energy depletes, presenting high density in the order of 1.000, and low $∆K_t^{*}$ in the order of −1.000, transcribing throughout the day with maximum averages of $K_t^{\mathrm{*}}$ in the order of 0.8162 and $∆K_t^{*}$ in the order of −0.0473, as shown in Figure 5d.

Although the statistical analysis yielded different results when the modeled $K_t^{*}$ values were changed, it was able to improve the analysis’s accuracy and RMSE (Root Mean Squared Error) with a 30% margin of error. The daily solar energy development is thus depicted in Figure 6a in a typical manner, with high values at midday and declining toward the end of the day. On the other hand, the $\Delta K_t^{*}$ are calculated similarly, and the $K_t^{*}$ model is part of the theoretical irradiation spectrum of the theoretical clear sky irradiation model, which improves the model and the distribution’s suitability. As depicted in Figure 6b, the propagation of daily pattern of $K_t^{*}$ and $\Delta K_t^{*}$ reflects the systematic variation of the ratio between global solar radiation at the surface and daily extraterrestrial solar radiation, allowing the allowing the visualization of cloudiness trends, atmospheric conditions, and afference of solar PV power generation.

The nominal variance was evaluated by computing the standard deviation of $K_t^{*}$ and $∆K_t^{*}$, as depicted in Equation (15) [71,123,128,129].

$$\sigma \left(∆K_t^{*}\right)=\sqrt{var\left[{∆K_t^{*}}_{∆t}\right]}$$

(15)

After estimating and obtaining solar energy based on $K_t^{*}$, it was classified into day classes with a percentage of 16 classification blocks. However, the classification of days was carried out for each year at all measured stations in order to better map the energy.

The days centered on the first qualifier; from the statistical classification, less than 25% of the total data were classified as cloudy sky days, characterized by low deviation relative to the sample norm of these days. However, days with an average $K_t^{*}$ between 75% and 25% were classified as intermediate sky days, presenting high deviation and persistent fluctuations in solar energy that can lead to potential changes in PV production. This can cause damage to apartments due to the high variability in the sample in a solar plant. Days that were above 75% of the sample, namely, the maximum of each year, were classified as clear sky days, characterized by low deviation and high intensity. The $K_t^{*}$ was higher in percentage during the hot and rainy season, but exhibited greater deviation relative to cloudy sky days.

The study analyzed 21 years of data from clear, cloudy, and intermediate sky days. Cloudy sky days had low deviation relative to the sample norm, while intermediate sky days had high deviation and persistent fluctuations in solar energy. Clear sky days had low deviation and high intensity, with a higher percentage in the hot and rainy season.

The analysis showed that cloudy sky days have a $K_t^{*}$ close to 0.2, intermediate sky days have values with peaks in region of 0.9, and intermediate sky days have values of intermediate complexity between the two. A summative analysis showed an increase in the $K_t^{*}$ from cloudy to clear sky days as we moved away from the cloudy sky. The relationship between the peak and decrease tails appeared smoother and higher on intermediate sky days, showing a smooth decrease but maintaining very high potential solar energy fluxes. On all types of days, the $K_t^{*}$ increases present a central maximum close to zero. On cloudy days, the frequency density decreases abruptly if deviated to values below 0.001. On clear days, the frequency density drops to below 0.02 and then again to values still below 0.0001, but there is a smooth decrease in the second drop of thin and long wings.

Intermediate sky days also present a central maximum close to zero, but also a gentle drop, presenting arms with long and flattened edges related to high fluxes of solar energy fluctuations prone to extreme variability. This group analysis addresses mixed characteristics, allowing for better management of the PV fleet and elimination of apparent variability. Using all data classified as intermediate sky conditions in Mozambique, from each pyranometer at each location, we calculated time series based on the highest resolution data available within a period and averaged the result. A time series obtained under intermediate sky conditions, using temporal averages and scales up to six orders of magnitude, revealed that there was high variability in the $K_t^{*}$ on a temporal scale. However, variability was apparent throughout the region on a spatial–temporal scale in the southern region shown in panel (a) with low variability, affected little by industrial action and human activity, and high fluctuation increases in the mid region shown in panel (b) and the northern region in panel (c), with high industrial activity. However, this is specifically selected as a case that presents unusual volatility time series with short-term variability at all time scales, as shown in panel (d) in Figure 7.

Taking the values of $K_t^{*}$ for different classes of days (clear, intermediate, and cloudy) [114,122], a connection was made as a function of the distance of the correlation coefficient or systematic connection of the clear sky index ${\chi }_{ij}^{K_t^{*}}$, given by Equation (16) [32,114].

$${\chi }_{ij}^{k_t^{*}}={\displaystyle \frac{cov\left(∆k_{\tau ,i}^{*}\left(t\right),∆k_{\tau ,j}^{*}\right)}{{\sigma }_{∆k_{\tau ,i}^{*}\left(t\right)}{\sigma }_{∆k_{\tau ,j}^{*}}}}$$

(16)

Statistically, $-1\le {\chi }_{ij}^{k_t^{*}}\le 1$, and spatially, considering that for a subspace station between two points $x$ and $y$, the randomized values that relate to $x+y$ are ${\sigma }_{x+y}=\sqrt{{\sigma }_x^2+{\sigma }_y^2+2{\chi }_{xy}{\sigma }_x{\sigma }_y}$ with ${\sigma }_{∆k_{\tau ,i}^{*}\left(t\right)}\ne 0 \mathrm{a}\mathrm{n}\mathrm{d} {\sigma }_{∆k_{\tau ,j}^{*}}\ne 0$, we write the formula as per Amjad et al. and Pereira and Tanaka, where $∆k_{\tau ,i}^{*}\left(t\right)$ e $∆k_{\tau }^{*},j\left(t\right)$, and $\overline{∆k_{\tau ,i}^{*}}$ e $\overline{∆k_{\tau ,j}^{*}}$ are the increments and the corresponding altimetric averages individually in the time series between two locations $i$ and $j$, and $\tau$ is the number of points between two series of measurements [32,94,114]. By analogy, the correlation or systematic connection of the increments in the clear sky index ${\chi }_{ij}^{{∆k}_t^{*}}$ was determined. The correlation arrangement was taken between two equidistant measuring stations, with Nhangau station as the first station, followed by Nhapassa-1 and Nhapassa-2, and the evaluation destination was Marávia station. A comparison was made of the systematic connection of the $\Delta K_t^{*}$ using the ${\chi }_{ij}^{{∆k}_t^{*}}$, first with the spatial correlation coefficient model proposed by Equation (17) [5,92,93,94,95,122].

$${\chi }_{ij}^{∆k_{\tau }^{*}}=exp\left({\displaystyle \frac{l_{ij}ln\left(0.2\right)}{1.5.\tau }}\right)$$

(17)

Finally, the spatial correlation coefficient model proposed by Hoff and Perez was utilized, using a range of relative cloud speed values $3.21 \mathrm{m}/\mathrm{s}<v<7.38 \mathrm{m}/\mathrm{s}$ for the region, given by Equation (18) [5,7,92,97,122].

$${\chi }^{∆k_{\tau }^{*}}={\displaystyle \frac{\tau .v_i}{\tau .v_i+l_{ij}}}$$

(18)

Inferring the regression coefficient of $K_t^{*}$ and its increments ${\beta }_{ij}^{{∆K}_t^{*}}$, with ${\alpha }_l$ as the intercept constant and ${\epsilon }_t$ as the error term, as shown in Equation (19), described the characteristics of the regression inferential analysis of $K_t^{*}$ and its increments for various classes and types of days in all years between the previously indicated seasons [71,73,95,122,123].

$$∆K_{t,j}^{*}\left(t\right)={\alpha }_l+{\beta }_{ij}^{{∆K}_t^{*}}∆K_{t,i}^{*}\left(t\right)+{\epsilon }_t$$

(19)

The normalized standard deviation, also known as the coefficient of variation (${\mathsf{\Gamma }}^{K_t^{*}}$), is calculated as the ratio of the standard deviation to the mean of the solar energy data, as per Equation (20) [38,50,95,114,122,126].

$${\mathsf{\Gamma }}^{K_t^{*}}={\displaystyle \frac{\sigma \left(∆K_t^{*}\right)}{\overline{K_t^{*}}}}$$

(20)

The accessibility of solar energy was analyzed by observing the distribution of deviations for each day of the year and the numbers of acceptable, unacceptable, and non-applicable days. In terms of the years in which the data measurement presented mainly excellent data without many failures, most occurred in 2012 for all of the obtained classes.

A quantitative diagram was drawn indicating the highest percentages. The optimal days are the clearest days of frequency in each year, as well as the cloudiest days from the observation of the peak of the probability density function estimated from the density estimation kernel of each day, in a given study area. The probability density function f(x) was evaluated through the kernel density estimation function (PDF). The GHI and Total Theoretical Radiation spectra were used chronologically to identify a maximum of 10 days in which the GHI closely matched the Total Theoretical Radiation, indicating a satisfactory performance, and the same method was used to identify unsatisfactory days.

There are differences in size and timing between solar energy properties in central and northern Mozambique. Although the flat regions are typically 20 m above sea level, there are also mountains at around 5400 m that support a range of activities, from atmospheric contributions to small-scale emissions.

Wave emission measurements are impacted by the increased concentration of aerosols in the Eastern Mediterranean region due to human and industrial activity compared to the north, with the area having more intermediate sky days and a higher proportion of upper intermediate sky and clear sky days. The hydrological cycle is compromised, and widespread electrification is required in underdeveloped areas such as Mozambique as a result of the current climate crisis, which has a detrimental effect on energy production due to climate change and harmful emissions from human and natural activities.

Using MLMs of atmospheric, geographic, climatic, and spatial variables, the study sought to model short-scale parametric predictions of solar energy in the mid–northern region of Mozambique. The findings revealed a strong relationship between the transmittances and irradiances of ozone, water vapor, aerosols, and nebulized gases, with the latter exhibiting the least amount of attenuation. The generator characteristics of clear, cloudy, and intermediate days was characterized using the current–voltage of the PV converter and its dependence on solar radiation and cell temperature. Considering the current I, with cell voltage V, the power is given by Equation (21) [1,2,114,127,128,129].

$$P=IV$$

(21)

The I–V characteristics of the model is given by Equation (22) [1,29,30,31,32,33,34,35,36,37,38,114,130,131,132,133,134,135,136].

$$I=I_L-I_0\left[e^{\left({\displaystyle \frac{V+I{R}_S}{a}}\right)}-1\right]-{\displaystyle \frac{V+I{R}_s}{R_{sh}}}$$

(22)

Five parameters need to be known: the light current $I_L$, the diode reverse saturation current $I_0$, the series resistance $R_S$, the shunt resistance, and a parameter $a$. All five parameters may be functions of cell temperature and absorbed solar radiation [1,2,7,114].

3. Results

3.1. Percentage Estimate of Different Types of Days

Under annual observations in 2012 with greater uniformity of measurements, the Nhangau and Marávia stations have the greatest potential for total availability of solar radiation, with approximately 68% and 67% of acceptable days, followed by the Nhapassa-2 and Nhapassa-1 stations with approximately 57% and 55%. This comparison illustrates the difference between theoretical radiation in clear skies and experimental horizontal global solar radiation on the Earth’s surface throughout the entire region of Mozambique. The difference between Nhapassa-1 and Nhapassa-2 is roughly 65% in 2013, 70% in 2014, 63% in 2013, and 64% in 2014. The intermediate energy potential between the Nhapassa-1 and Nhapassa-2 stations and Marávia station is potentially higher when compared to that of Nhangau due to heat flow transport phenomena where the mass flow of solar energy continues in the seasonal direction of establishing the solar energy balance.

However, in the southern region, greater experimental precision of solar energy accessibility was observed, with greater potential of total solar radiation availability around Pomene station during the year 2013, estimated at about 81.4371% of acceptable days, 17.9641% of unacceptable days, and 0.5988% of unacceptable days, as can be seen in Figure 8.

In 2014, 81.1377% of acceptable days, 18.8623% of unacceptable days, and 0.0009% of inapplicable days were observed; this was lower in 2012, estimated at 18.3099% for acceptable days, 9.0141% for unacceptable days, and 72.6761% for non-applicable days. Next, at the Dindiza station in 2012, approximately 73.0909% of acceptable days, 22.5455% of unacceptable days, and 4.3636% of non-applicable days were observed. In 2014, 69.9708% of acceptable days, 30.0292% of unacceptable days, and 0.0000% of non-applicable days were observed; during 2013, 68.3196% of acceptable days, 31.4050% of unacceptable days, and 0.2755% of non-applicable days were observed. At Massangena station, in 2012, 71.6364% of acceptable days, 23.2727% of unacceptable days, and 5.0909% of non-applicable days were observed. In 2013, 71.0811% of acceptable days, 28.6486% of unacceptable days, and 0.2703% of non-applicable days were observed. Lower levels of solar energy accessibility content were observable at the UEM–Maputo station during the year 2012, where 42.1622% of acceptable days, 29.7297% of unacceptable days, and 28.1081% of non-applicable days were observed.

Throughout the central region, a higher level of solar energy accessibility can be observed, with a greater potential for total solar radiation availability at the Nhapassa-1 station during 2014, with approximately 70.4110% of acceptable days, 29.0411% of unacceptable days, and 0.5479% of non-applicable days. In 2013, 65.1934% of acceptable days, 34.2541% of unacceptable days, and 0.5525% of non-applicable days can be observed; in addition, in 2012, 55.1402% of acceptable days, 31.3084% of unacceptable days, and 13.5514% of non-applicable days can be observed. This is followed by Nhangau station, where, in 2012, 68.2927% of acceptable days, 26.0163% of unacceptable days, and 5.6911% of non-applicable days can be observed, and in 2013, 0.0012% of acceptable days, 99.1826% of unacceptable days, and 0.8174% of non-applicable days can be observed. However, in 2014, 0.0024% of acceptable days, 38.0822% of unacceptable days, and 61.9178% of non-applicable days can be observed.

At Marávia station, in 2012, 67.3913% of acceptable days, 23.9130% of unacceptable days, and 8.6957% of non-applicable days could be observed, and in 2013, 39.7196% of acceptable days, 48.1308% of unacceptable days, and 12.1495% of non-applicable days could be observed. At Nhapassa-2 station, the highest level of solar energy accessibility was observed in 2014, with 63.5616% of acceptable days, 35.8904% of unacceptable days, and 0.5479% of non-applicable days. In 2013, it was possible to observe 62.9834% of acceptable days, 36.4641% of unacceptable days, and 0.5525% of non-applicable days; however, a lower level was observed in 2012, when 57.2770% of acceptable days, 28.6385% of unacceptable days, and 14.0845% of non-applicable days were observed. The Lugela-1 station, in 2012, exhibited 44.265% of acceptable days, 67% of unacceptable days, and 56% of non-applicable days. However, in 2013, 44% of acceptable days, 36% of unacceptable days, and 25% of non-applicable days were observed, and in 2014, there were 27% of acceptable days, 24% of unacceptable days, and 46% of non-applicable days. In addition, for the Lugela-2 station in 2012, there were approximately 64% of acceptable days, 46% of unacceptable days, and 28% of non-applicable days. In 2013, 46% of acceptable days, 78% of unacceptable days, and 22% of non-applicable days can be observed; and in 2014, 12% of acceptable days, 14% of unacceptable days, and 62% of non-applicable days can be observed.

In addition to what was previously described, the north region shows a higher level of solar energy with a higher potential for the total availability of solar radiation at the Massangulo-1 station. The experimental observations during 2014 showed approximately 82.4176% of acceptable days, 15.9341% of unacceptable days, and 1.6484% of non-applicable days. In 2012, 82.1990% of acceptable days, 16.7539% of unacceptable days, and 1.0471% of non-applicable days can be observed; however, in 2013, 77.7473% of acceptable days, 22.2527% of unacceptable days, and 0.0020% of non-applicable days can be observed. Then, at Nanhupo-2 station, the highest accessibility level was observed in 2012, with approximately 81.8182% of acceptable days, 18.1818% of unacceptable days, and 0.0044% of non-applicable days. However, the comparison of the Marávia stations in 2013 and Nhangau in 2013 and 2014 was hampered by a number of circumstances, including repeated cloud reflections and obstructions to the measurement of solar radiation.

3.2. Casual Inference Factors for Determining Solar Energy

An analysis of solar energy on the Earth’s surface indicates that approximately 70% of the Sun’s radiation is absorbed, diminished, and dispersed among various atmospheric elements and physical processes, while the remaining fraction escapes into space, assuming that solar rays follow their typical path until they reach the Earth’s surface. This phenomenon is influenced by several barriers encountered by the radiation beam. A detailed examination of the primary atmospheric barriers in the southern (southwest), mid (east), and northern (north) regions during the years 2012 and 2013, as illustrated in Figure 9, reveals that transmittance is notably higher due to the ozone layer, with minimal variation in solar energy transmittance attributed to uniformly mixed gases present in the atmosphere, which arise from a range of sources including natural, domestic, human, and industrial activities. The presence of water vapor contributes even less to attenuation, resulting in lower transmittance compared to the mixed gases. Consequently, regional aerosol emissions linked to various human and environmental activities are found in greater concentrations in the central region, which experiences more industrial operations related to the extraction of heavy sands, mineral resources, and other materials. The most significant relative attenuation is attributed to the Rayleigh scattering effect.

The above-described causal effects of solar energy inference are compared to their actual behavior in a summative estimate of solar energy using the ANN model, whose estimation error is displayed in Figure 3. There is a significant chance that the estimate will be lower than the arithmetic averaging. The finding reveals irradiance over all spatially and temporally arranged points between the years 2012 and 2013; the diffuse solar energy originates from summative statistics of interaction of the irradiation beam with aerosols, the Rayleigh effect, and multiple reflections (depending mainly on the albedo). In Figure 4, it is shown that the Rayleigh scattering effect contributes significantly, followed by scattering by aerosols and, finally, the multiple reflections, which define the diffuse solar energy when incident on an area of interest. However, direct solar energy, which reaches the Earth with no obstructions, is greater relative to diffuse solar energy.

A summative statistic between the last two described energies enables the prediction of solar energy on the Earth’s surface, the relative knowledge of which is of extreme importance. For the two years observed, the estimate of the energy due to all described causal effects, geographical, atmospheric, climatic, and meteorological factors is close to the theoretical solar energy in skies with few clouds, as shown in Figure 10.

3.3. Characterization of the Variability in Intermediate Sky Days in Terms of Clear Sky Index

The distribution of the $K_t^{*}$ on intermediate sky days exhibits a bimodal pattern, characterized by a probability density concentrated around values of 0.8 and 0.2, as illustrated by histograms with a bin width of 0.05 and average measurement intervals for each location. Nevertheless, at a 6 h interval with one-minute measurements, the frequency and probability density of bimodality are reduced in the southern region, followed by a segment of the northern area, while being more pronounced in the western part of the central region, as shown in Figure 11.

In the southern region, the $K_t^{*}$ distribution showed a bimodal decline, peaking in 2012 at 0.02709, with a minimum of 0.0340 and a high of 0.9835. With 1299 mean lengths, this distribution had a mean of 0.7858 and a frequency density of 0.020. In 2012, it had a minimum of 0.0340, a maximum of 1.0099, and a maximum central centroid of 0.9835 in the state of Dindiza. The order’s frequency density was 0.020; however, it fell precipitously to 0.001 with a mean of 0.009759. The $K_t^{*}$ distribution in the state of Pomene had a mean of 1379 classes, a low frequency of 0.010, a maximum of 0.872, and a short side. In 2013, the central centroid reached a maximum of 0.9835. Over 1299 average lengths with 36 classes, the frequency density peaked at 0.020 before dropping precipitously to 0.001 and a mean of 0.7858. On unacceptable days, the $K_t^{*}$ distribution in the southern area peaked in 2012 at 0.8256, with an incremental level of 0.030655 to 0.010087 and a frequency of 0.10 to 0.25. Moreover, 33 classes with a minimum of 0.0003, a maximum of 0.9090, and a median of 0.5295 were present. In Dindiza, the $K_t^{*}$ distribution had maximums of roughly 0.3544 and low-strength washed frequency distributions below 0.010, 0.023341, 0.2366, and 1820. The frequency distribution in Pomene was 0.10 to 0.25, with an incremental level between 0.03067 and 0.0108. In 2013, there were 44 classes in the $K_t^{*}$ distribution, with frequency distributions below 0.010, a mean ranging from 0.4904 to 0.9079, a minimum of 0.0043, an incremental level of 0.023341, and maximums of around 0.3544, as depicted in

Table 3. Variability of $K_t^{*}$ in the south region.

Region	Station	Year/Day Type		${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{i}\mathit{n}}^{\mathit{*}}$	${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{a}\mathit{x}}^{\mathit{*}}$	$\overline{{\mathit{K}}_{\mathit{t}}^{\mathit{*}}}$	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{o}\mathit{n}\mathit{e}}^{\mathit{*}}$	$\mathit{\sigma }{\mathit{K}}_{\mathit{t}}^{\mathit{*}}$	Size	Classes	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{t}\mathit{w}\mathit{o}}^{\mathit{*}}$
South	UEM-Maputo	2012	Acceptable	0.1796	0.6260	0.8490	0.012	0.1752	5391	73	0.0088
			Unacceptable	0.0093	0.9160	0.5577	0.018	0.3013	10,922	105	0.019
	Dindiza	2012	Acceptable	0.0480	0.9291	0.7494	0.0252	0.2364	2101	46	0.0116
			Unacceptable	0.0397	0.9094	0.6227	0.0373	0.2196	668	26	0.0097
		2013	Acceptable	0.0705	0.9278	0.9561	0.0353	0.4146	1966	41	0.0098
			Unacceptable	0.0014	0.9894	0.7287	0.0199	0.4934	1482	54	0.0078
		2014	Acceptable	0.0045	0.9875	0.7582	0.0278	0.3584	1486	42	0.0089
			Unacceptable	0.0018	0.9785	0.8024	0.0299	0.3984	3492	52	0.0026
	Massangena	2012	Acceptable	0.0001	0.9259	0.4256	0.0157	0.29088	4155	64	0.0109
			Unacceptable	0.0001	0.9094	0.4738	0.0124	0.29088	5695	75	0.0188
		2013	Acceptable	0.1523	0.9269	0.7426	0.0224	0.2064	1531	39	0.0087
	Pomene	2012	Acceptable	0.0259	0.9192	0.8848	0.0293	0.1729	1172	34	0.0099
			Unacceptable	0.0001	0.9094	0.4738	0.01577	0.29088	4155	64	0.0109
		2013	Acceptable	0.0743	0.9456	0.9304	0.0346	0.1359	950	31	0.0176
			Unacceptable	0.0556	0.9097	0.6151	0.0307	0.2554	963	31	0.0095

.

The mid region’s $K_t^{*}$ distribution peaked in 2012 with a central centroid of 0.9835, a midday average of 0.020, and a low in the 0.0340 range. With a low frequency of 0.010, a high of 0.872, and a minimum of 0.0289, an average of 1379 of 37 classes was recorded in 2013. With a slope of 0.2212 and an incremental level of 0.0337, the distribution at Nhangau peaked in 2012 at 0.0852 before declining to 0.01. After that, the distribution became flattened with sweeping, long branches. At Nhapassa-1 station, the $K_t^{*}$ distribution peaked in the middle at values near 0.97 and then steadily declined with increments of 0.0405, peaking at 0.9095 and falling to a minimum of 0.0494. With increments of 0.04005, the distribution at the Nhapassa-2 stage likewise decreased gradually, peaking in the middle at values near 0.97. At Lugela-1 (a), the $K_t^{*}$ distribution in 2012 had a median of 0.6846, a maximum of 0.7497, a progressive reduction with increments of 0.0288, a maximum of 1.0095, and a minimum of 0.0001 using 1224 mean lengths and 35 classes. For a mean of 02501, the $K_t^{*}$ envelope at Lugela-2 (a) showed a central peak at 0.0351, gradually decreasing with increments of 0.035039, reaching a maximum of 0.9811 and a minimum of 0.0001 with a density above 0.0011.

In the mid region of Marávia, the $K_t^{*}$ distribution reached its maximum in 2012 with a frequency of 0.10 to 0.25 and a lengthy and unclean orthosis. With short and badly positioned orthoses, frequency values below 0.010, and an incremental level of 0.0233, it had a median length of 1820 with 43 classes, a median of 0.4904, a maximum of 0.9079, and maxima in the order of 0.3544 in 2013. After peaking at 0.1091 in 2012, the distribution in Nhangau decreased to 0.02 with a displacement of 0.2473 and an incremental level of 0.0447. Density data showed that the distribution’s long, awkwardly positioned branches were steadily growing. Long and shifted branches were consistently increasing in the $K_t^{*}$ distribution, with density values for 47 classes out of 2244 ranging from −0.0312 to 0.6270 and −0.9098 to 0.0001. With increments of 0.0405, the $K_t^{*}$ distribution steadily decreased in the Nhapassa-1 stage, peaking at 0.9095 and reducing to a minimum of 0.0494. The $K_t^{*}$ distribution in 2013 had a central peak with a maximum of 0.9095 and a minimum of 0.0494, with an increment level of 0.0096 and a shift of 0.1831 for a total of 575 samples with a density <0.012. At Lugela-1, the $K_t^{*}$ distribution peaks at values near 0.97 and then gradually declines with increments from 0.024117 to 0.0001, with a central maximum for values near 0.0093 and a low of 0.0001 for mean values. At Lugela-2, the distribution peaks at values near 0.97 and gradually declines with increments, producing 1764 sample sizes with a density below 0.0001 and 575 sample sizes with a density below 0.012, as shown in

Table 4. The $K_t^{*}$ variability in the mid region.

Region	Station	Year/Day Type		${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{i}\mathit{n}}^{\mathit{*}}$	${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{a}\mathit{x}}^{\mathit{*}}$	$\overline{{\mathit{K}}_{\mathit{t}}^{\mathit{*}}}$	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{o}\mathit{n}\mathit{e}}^{\mathit{*}}$	$\mathit{\sigma }{\mathit{K}}_{\mathit{t}}^{\mathit{*}}$	Size	Classes	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{t}\mathit{w}\mathit{o}}^{\mathit{*}}$
Mid	Marávia	2012	Acceptable	0.0098	0.9099	0.7589	0.0196	0.2361	2554	51	0.0123
			Unacceptable	0.0022	0.9299	0.6968	0.0388	0.2226	675	26	0.017
		2013	Acceptable	0.0013	0.9410	0.6659	0.0281	0.2443	1379	37	0.0104
			Unacceptable	0.0001	0.9079	0.4891	0.0168	0.2542	3631	60	0.0109
	Nhangau	2012	Acceptable	0.0530	0.9097	0.8634	0.0504	0.2049	373	19	0.0096
			Unacceptable	0.0116	0.9892	0.6088	0.0369	0.2757	739	27	0.0099
		2013	Acceptable	0.0568	0.927	0.8691	0.0253	0.2146	1386	38	0.0098
			Unacceptable	0.0229	0.9933	0.0850	0.0107	0.1220	8099	90	0.0097
		2014	Unacceptable	0.0512	0.9898	0.8971	0.0553	01456	5698	44	0.0085
	Nhapassa-1	2012	Acceptable	0.1157	0.9099	0.8197	0.0319	0.2051	810	28	0.0089
			Unacceptable	0.0785	0.9086	0.6319	0.0321	0.2445	823	29	0.0093
		2013	Acceptable	0.0705	0.9278	0.9561	0.0353	0.4146	1966	41	0.0098
			Unacceptable	0.0014	0.9489	0.7287	0.0199	0.4934	1482	54	0.0078
		2014	Acceptable	0.0045	0.9875	0.7582	0.0278	0.3584	1486	42	0.0089
			Unacceptable	0.0018	0.9858	0.8024	0.0299	0.3984	3492	52	0.0026
	Nhapassa-2	2012	Acceptable	0.2739	0.9245	0.8489	0.0283	0.1542	657	26	0.0074
			Unacceptable	0.0691	0.9198	0.6001	0.034	0.2181	793	28	0.0094
		2013	Acceptable	0.0741	0.9298	0.7771	0.0253	0.2146	1366	37	0.0094
		2014	Acceptable	0.0084	0.9788	0.9254	0.0289	0.3938	3485	55	0.0049
			Unacceptable	0.0024	0.8448	0.8294	0.0122	0.5935	3482	39	0.0199
	Lugela-1	2012	Acceptable	0.0058	0.9094	0.7812	0.0386	0.2592	688	26	0.0104
			Unacceptable	0.0024	0.8448	0.8294	0.0283	0.1542	657	56	0.0074
		2013	Acceptable	0.0001	0.9093	0.3748	0.0287	0.2609	1206	35	0.0109
			Unacceptable	0.0001	0.9979	0.1131	0.0169	0.2101	3446	59	0.0099
		2014	Acceptable	0.0014	0.9448	0.6294	0.0283	0.1542	657	26	0.0078
			Unacceptable	0.0001	0.9871	0.1102	0.0174	0.2047	3375	58	0.0098
	Lugela-2	2012	Acceptable	0.0014	0.9448	0.6294	0.0199	0.2934	2482	50	0.0099
			Unacceptable	0.0001	0.9066	0.3150	0.025	0.2684	1560	40	0.0099
		2013	Unacceptable	0.0001	0.9899	0.1119	0.0168	0.2106	3420	59	0.0099
		2014	Acceptable	0.0012	0.9441	0.9294	0.0289	0.2264	2492	57	0.0082
			Unacceptable	0.0001	0.4294	0.1158	0.12626	0.2683	3328	58	0.0943

.

In the northern region, the average daily stomatographic density was 0.9999 in 2012; it dropped to 0.02948 with 1124 classes found. The median of the 2014 mean $K_t^{*}$ curve was 0.0060, the maximum was 0.0099, the minimum was 0.0061, and the steep decrease was 0.009933. An average of 0.5283, a mean of 0.9964, a mean of 0.9984, and a minimum of 0.0110 were recorded for the Massangulo-2 stage. There was a steady and continuous decline in the frequency of 14 classes of 181 and 317 m. At Nanhupo-1 in 2012, the density and frequency values were 0.1, reaching their maximums at 0.4021 and 0.0014, respectively, for $K_t^{*}$. At Nanhupo-2, the highest density in 2013 was 0.2989, while the mean was 0.5283, the maximum was 0.9964, and the minimum was 0.0110. The 14,181-m classes’ frequency steadily and progressively declined. At Chomba in 2012, the intermediate center between clear days had a maximum of 0.5887, a minimum of 0.5433, a maximum of 0.5401, and a deviation between 0.0095 and 0.0099.

The northern region’s clear-cutting season displayed notable fluctuations in mean values, with a drop below 0.05 and a center maximum of 0.4060 in 2012. The mean values dropped below 0.05 in 2014, falling as low as 0.0040 and as high as 0.9975. In the Massangulo-2 stage, the frequency density values dropped sharply from 0.002 to 0.001, reaching a minimum of 0.0001 and a maximum of 0.9994. The median at Nanhupo-1 was 0.4449, with a total length of 661 and 26 classes. In 2012, the central maximum was 0.5424, the minimum was 0.12, and the maximum was 0.9975. The height progressively declined from a maximum of 0.9994 to a minimum of 0.01 with a median of 0.1073, but the density value plummeted sharply to 0.022. The height steadily declined to a maximum of 0.1947 in 2014, but the density number fell sharply to 0.09. A progressive decrease in the long arms and a central peak of 0.2218 in the order were noted in Nanhupo-2. The 33 classes’ average height was 1057 m, with a minimum of 0.0303 in the incremental level and a variance of 0.0024952 in the order. In 2012, the center frequency density of Chomba was 0.0392, but it thereafter dropped to 0.15. The 519-hectare area has 23 classes, each having an incidence of 0.0433 and a variance of 0.24589. The central frequency density peaked at 0.0988 in 2013 and then began to drop to 0.015, as shown in

Table 5. The $K_t^{*}$ variability in the north region.

Region	Station	Year/Day Type		${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{i}\mathit{n}}^{\mathit{*}}$	${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{a}\mathit{x}}^{\mathit{*}}$	$\overline{{\mathit{K}}_{\mathit{t}}^{\mathit{*}}}$	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{o}\mathit{n}\mathit{e}}^{\mathit{*}}$	$\mathit{\sigma }{\mathit{K}}_{\mathit{t}}^{\mathit{*}}$	Size	Classes	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{t}\mathit{w}\mathit{o}}^{\mathit{*}}$
North	Chomba	2012	Acceptable	0.0071	0.9958	0.6019	0.0762	0.2695	180	13	0.0099
			Unacceptable	0.0012	0.9959	0.2555	0.0285	0.2799	1249	35	0.0024
		2013	Unacceptable	0.0014	0.9998	0.6294	0.0199	0.5689	2482	50	0.0085
	Massangulo-1	2012	Acceptable	0.0001	0.9966	0.3150	0.0251	0.2684	1560	40	0.0086
			Unacceptable	0.0024	0.9998	0.6294	0.0199	0.4454	2482	50	0.0075
		2013	Acceptable	0.0041	0.9966	0.3150	0.0225	0.2684	1560	60	0.0045
			Unacceptable	0.0046	0.9998	0.6294	0.0199	0.2934	2482	50	0.0076
		2014	Acceptable	0.0005	0.9966	0.3150	0.0446	0.3784	1630	30	0.0045
			Unacceptable	0.0029	0.9998	0.6294	0.0199	0.2934	2482	50	0.0028
	Massangulo-2	2012	Acceptable	0.0042	0.9977	0.3150	0.0775	0.4984	1125	60	0.0092
			Unacceptable	0.0018	0.9998	0.6294	0.0199	0.6934	2482	50	0.0071
		2013	Acceptable	0.0007	0.9944	0.3150	0.0248	0.4478	1560	70	0.0039
			Unacceptable	0.0018	0.9998	0.6294	0.0199	0.1327	2482	50	0.0089
		2014	Acceptable	0.0027	0.9928	0.3150	0.0369	0.3582	1489	80	0.0072
			Unacceptable	0.0019	0.9979	0.6294	0.0199	0.3568	2369	60	0.0092

.

3.4. Variability of Intermediate Sky Days in Terms of Clear Sky Index Increments

The global maxima in the generated distributions are near zero. The broad, flattened arms have a tendency to diminish to lower probability densities over time. The probability densities, however, have a tendency to drop off quickly as the positive and negative increment values rise and fall. The tails drastically diminish when secondary maxima are ignored. The distributions are more rounded around clear central peaks for longer time averages of 10 min and 1 h. Strong changes between clear and cloudy states, however, are common during intermediate sky days; these probability density shifts are typically covered by a time step of the order of 0.01 s. Clear sky indices are low on cloudy days, high on clear sky days, and in the middle of the range between the two are intermediate days with deviating fluctuations. This brings the characteristics of clear days closer to those of cloudy days on a daily basis, making them susceptible to fluctuations, as shown in Figure 12.

A maximum density of 0.010, a reduction of 0.017569, and an average length of 1299 were observed in the UEM-Maputo region in 2012. In Dindiza state, the greatest central point of the Δ$K_t^{*}$ distribution was at 0.01256, the frequency density was at 0.010 with a slope of 0.0488, and the slew was at 0.001 with a slope of 0.0176. With a mean center point of 0.0831 in 2014 and a skew of 0.0175, the ∆$K_t^{*}$ distribution’s greatest central point in 2013 was 0.0831. In Massangena state, the $K_t^{*}$ distribution had a mean length of 1299, a central maximum of 0.0126, a frequency density of 0.010, a slope of 0.0488, and a slope of 0.001 with a slope of 0.0176 for 36 classes. The distribution of ∆$K_t^{*}$ has a frequency density of 0.001, a central maximum of 0.0831, and a skew of 0.0175 for 1378 mean lengths and 37 classes. In the 2012 Pomene midday averages, the mean ∆$K_t^{*}$ was 0.01256, with the maximum values for frequency density and slope deviation being 0.01256 and 0.0488, respectively. The southern region recorded the highest mean density of ∆$K_t^{*}$ in 2012, with a mean duration of 11,007, a maximum of 0.020, and a minimum of 0.002. With a low of 0.002 and a maximum of 0.1312, the average daily density in Dindiza state was 0.1312. The density in 33 classes had a mean of 1107 and a minimum of 0.001636, while the mean value of ∆$K_t^{*}$ dropped to 0.002. After declining to 0.0106 with an incremental level of 0.2199 in 2013, the mean value of ∆$K_t^{*}$ then dropped to 0.001 with an incremental level of 0.0167. The average number of hours per day in the state of Massangena was 0.1312, with a low of 0.002 and a maximum of 0.0202. With a mean height of 1820, a mean difference of 0.0489, a maximum of 0.0108, and a minimum of 0.001, there were 43 classes in 2013. In Pomene state, the ∆$K_t^{*}$ distribution had a maximum center point of 0.1312, a frequency density of 0.020, and a slope of 0.0495, as shown in

Table 6. Variability of Δ$K_t^{*}$ in the Southern region.

Region	Station	Year/Day Type		${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{i}\mathit{n}}^{\mathit{*}}$	${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{a}\mathit{x}}^{\mathit{*}}$	$\overline{{\mathit{K}}_{\mathit{t}}^{\mathit{*}}}$	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{o}\mathit{n}\mathit{e}}^{\mathit{*}}$	$\mathit{\sigma }{\mathit{K}}_{\mathit{t}}^{\mathit{*}}$	Size	Classes	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{t}\mathit{w}\mathit{o}}^{\mathit{*}}$
South	UEM-Maputo	2012	Acceptable	0.1796	0.9626	0.8490	0.012	0.1752	5391	73	0.0088
			Unacceptable	0.0093	0.9916	0.5577	0.018	0.3013	10,922	105	0.019
	Dindiza	2012	Acceptable	0.0480	0.9291	0.7494	0.0252	0.2364	2101	46	0.0116
			Unacceptable	0.0397	0.9094	0.6227	0.0373	0.2196	668	26	0.0097
		2013	Acceptable	0.0397	0.9125	0.7287	0.0373	0.2196	268	26	0.0027
			Unacceptable	0.0397	0.9022	0.8244	0.0373	0.2126	778	44	0.0044
		2014	Acceptable	0.0397	0.9444	0.9222	0.04423	0.2196	6548	29	0.0025
			Unacceptable	0.0397	0.9584	0.7527	0.0225	0.2426	6148	77	0.0014
	Massangena	2012	Acceptable	0.0397	0.9334	0.8027	0.0563	0.2145	4568	56	0.0047
			Unacceptable	0.0397	0.9124	0.8244	0.048	0.2596	7878	28	0.0046
		2013	Acceptable	0.1523	0.9269	0.7426	0.0224	0.2064	1531	39	0.0087
			Unacceptable	0.0397	0.9994	0.92321	0.0373	0.2196	668	26	0.0089
	Pomene	2012	Acceptable	0.0259	0.9692	0.8848	0.0293	0.1729	1172	34	0.0099
			Unacceptable	0.0001	0.7924	0.4738	0.01577	0.29088	4155	64	0.0109
		2013	Acceptable	0.0743	0.9456	0.9304	0.0346	0.1359	950	31	0.017
			Unacceptable	0.0556	0.9097	0.6151	0.0307	0.2554	963	31	0.0095

.

The central density of ∆$K_t^{*}$ in 2012 was 0.0126, with increases of 0.010 and 0.0488, according to experimental data from the central region. It then dropped to 0.001 with an increase of 0.0175 of 36 classes and an average height of 1299. In 2013, the central density of ∆$K_t^{*}$ peaked at 0.0831, and then decreased to 0.001 with a 0.047364 increment. After reaching a central maximum of 0.01949 in 2012, the Δ$K_t^{*}$ distribution at Nhangau dropped to 0.02 with a slope of 0.2485 and an incremental level of 0.0495. The distribution displayed lengthy, sweeping branches for 31,985 classes, with maximum and minimum density values of −0.0802 and 0.0002, respectively. The distribution’s highest value in 2014 was 0.0001. At Nhapassa-1 stadium, the ∆$K_t^{*}$ distribution steadily decreased in steps of 0.04005, peaking at 1.0095 and down to a low of 0.0494. For values near 0.0481, the Δ$K_t^{*}$ distribution had a central maximum at the Lugela-1 (a) and Lugela-2 (b) stages in 2012. The mean values showed a steady fall with increments of 0.0518, a maximum of 0.7497, and a minimum of −0.9041.

The mid region’s $K_t^{*}$ distribution had a mean density of less than 0.0001, a mean deviation of 0.263587, and a mean of 0.0508. In 2012, the frequency density was 0.0202, the slope was 0.0495, and the greatest mean value of Δ$K_t^{*}$ was 0.1312. In 2013, with 43 classes, an average height of 1820, a frequency density of 0.0105, and a slope of 0.0015, the greatest mean value of ∆$K_t^{*}$ was 0.0489. The 2012 central maximum of the ∆$K_t^{*}$ distribution at Nhangau was 0.4816; it decreased to 0.02 with an incremental level of 0.064795 and an offset of 0.158242. Long and offset branches were consistently increasing, according to density statistics for the distribution, with 0.0001 for 22,476 m classes and −0.9438 for 2013. The 2013 ∆$K_t^{*}$ distribution had a central maximum of 0.0098, an offset of 0.052248, and an incremental level of 0.023872. When values rise, the Nhapassa-1 state gradually decreases, with a central peak at values near 0.97. The ∆$K_t^{*}$ distribution at Nhapassa-2 station has a central peak at values close to 0.97 and gradually decreases with increments of 0.0405, reaching a maximum of 0.9095 and a minimum of 0.0494. The total area is 575 square meters, with a density below 0.012, a mean of 0.0096, and a deviation of 0.1833, as depicted in

Table 7. Variability of Δ$K_t^{*}$ in the central region.

Region	Station	Year/Day Type		${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{i}\mathit{n}}^{\mathit{*}}$	${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{a}\mathit{x}}^{\mathit{*}}$	$\overline{{\mathit{K}}_{\mathit{t}}^{\mathit{*}}}$	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{o}\mathit{n}\mathit{e}}^{\mathit{*}}$	$\mathit{\sigma }{\mathit{K}}_{\mathit{t}}^{\mathit{*}}$	Size	Classes	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{t}\mathit{w}\mathit{o}}^{\mathit{*}}$
Mid	Marávia	2012	Acceptable	0.0099	0.7471	0.0754	0.0345	0.2964	2554	51	0.0175
			Unacceptable	0.0098	0.6175	0.0247	0.0626	0.2099	675	26	0.0163
		2013	Acceptable	0.0091	0.7433	0.0939	0.0474	0.332	1378	37	0.0175
			Unacceptable	0.0057	0.6739	0.0162	0.0279	0.1848	3629	60	0.0167
	Nhangau	2012	Acceptable	0.0097	0.4092	0.0727	0.075	0.2263	373	19	0.0142
			Unacceptable	0.0085	0.5670	0.0467	0.05835	0.20732	739	27	0.0158
		2013	Unacceptable	0.0933	0.4829	0.0057	0.0164	0.0931	8098	90	0.0147
		2014	Acceptable	0.0089	0.8563	0.0896	0.0444	0.3325	1278	39	0.0186
			Unacceptable	0.0068	0.7899	0.0569	0.0339	0.1889	2629	62	0.0197
	Nhapassa-1	2012	Acceptable	0.0083	0.7992	0.0621	0.0645	0.2557	810	28	0.018
			Unacceptable	0.0856	0.7219	0.0185	0.0588	0.2063	823	29	0.0171
		2013	Acceptable	0.0082	0.7533	0.0944	0.0294	0.4332	1378	29	0.0278
			Unacceptable	0.0044	0.9737	0.0158	0.04589	0.1948	3629	44	0.0289
		2014	Acceptable	0.0048	0.4483	0.0949	0.0589	0.4532	1298	29	0.0148
			Unacceptable	0.0055	0.8769	0.0178	0.0279	0.1948	3599	69	0.0189
	Nhapassa-2	2012	Acceptable	0.0869	0.2839	0.2351	0.0489	0.07223	657	26	0.0127
			Unacceptable	0.0098	0.7323	0.0255	0.06221	0.2199	793	28	0.0174
		2013	Acceptable	0.0098	0.7572	0.0627	0.0478	0.2788	1366	37	0.0177
			Unacceptable	0.0089	0.8453	0.0456	0.0474	0.332	1378	37	0.0175
		2014	Acceptable	0.0058	0.5939	0.0459	0.0569	0.1895	3896	62	0.0157
			Unacceptable	0.0049	0.8929	0.0259	0.0248	0.1588	3656	58	0.0189
	Lugela-1	2012	Acceptable	0.0072	0.7926	0.0508	0.069	0.1874	688	26	0.0179
			Unacceptable	0.0908	0.6863	0.0363	0.0479	0.2037	1206	35	0.0168
		2013	Acceptable	0.0058	0.6896	0.0177	0.0289	0.1895	4628	65	0.0144
			Unacceptable	0.0978	0.9791	0.0642	0.0335	0.2592	3445	59	0.0198
		2014	Acceptable	0.0099	0.6369	0.0256	0.0288	0.1856	3628	65	0.0178
			Unacceptable	0.0888	0.9598	0.0702	0.0335	0.2393	3375	58	0.0195
	Lugela-2	2012	Acceptable	0.0994	0.8033	0.0847	0.03605	0.2604	2482	50	0.0182
			Unacceptable	0.0909	0.9172	0.0321	0.049	0.1923	1560	39	0.0198
		2013	Acceptable	0.0089	0.6756	0.0178	0.0289	0.1856	3628	68	0.0178
			Unacceptable	0.0917	0.9788	0.0615	0.0339	0.2629	3418	58	0.0197
		2014	Acceptable	0.0089	0.6745	0.0135	0.0278	0.1856	5689	25	0.0148
			Unacceptable	0.0044	0.8787	0.0248	0.0598	0.1878	3628	72	0.0148

.

Clear sky days were the focus of the study’s analysis of data on days deemed inappropriate in the northern region. The shortest and longest arms dropped sharply to −0.50 in increments of −0.0495 during 2012, when the central maximum of −0.0058 was discovered. The results showed a median of −0.1218; a maximum of −0.685; and a minimum of −0.9999. A tiny central maximum was found in 2013 with lengthy, aching arms and a value of −0.0381. Incidence ranged from −0.0356 to −0.0175, with a mean of −0.1346. Targeting the left, long-distance steel purchases in Massangulo-2 state demonstrated a steady reduction, peaking at 0.0775 in the middle. In 2012, Nanhupo-1’s density of ∆$K_t^{*}$ had a steep decline, reaching a peak of −0.0219 before gradually decreasing to a frequency density of 0.25. Following the collapse of the center maximum of ∆$K_t^{*}$, the lengthy and asymmetrically displaced sand softening in Nanhupo-2 was described with a frequency of 0.022 in 2012.

Experimental analysis was conducted to identify the aesthetic category for clear days in the northern region. The Massangulo-1 stage in 2012 had a maximum central with watery arms and a length of 0.0304, with a mean of −0.0262. The highest central with a mean of −0.0262 was observed in 2014. When using lengthy and rejected keys, frequency disturbances in the range of 0.02000 peaked in 2012 and then slightly decreased to 0.056. With a rising level from 0.0475 to 0.0998, the sample displayed 42 classes of departure from the mean of 0.0244. With long and discarded keys, frequency disruptions in the order of 0.056 thereafter declined marginally after reaching a peak of 0.0951 in 2014. The highest central peak of Nanhupo-1 in 2012 was 0.0514, which was followed by a frequency density of 0.0402 and a slow decline to 0.0101. The 2013 peak at 0.0951 was burst to a frequency density of 0.0205, after which it decreased gradually. After peaking at 0.0198 in 2014, the frequency density progressively dropped to 0.0252 and 0.0101. The Nanhupo-2 state showed three peak values of ∆$K_t^{*}$ in 2012, 2013, and 2014, with a total of 1056 classes and 33 lateral deviation classes. In the Nanhupo-2 region, the frequency of density progressively declined, as shown in

Table 8. Variability of Δ$K_t^{*}$ in the north region.

Region	Station	Year/Day Type	${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{i}\mathit{n}}^{\mathit{*}}$	${\mathit{K}}_{\mathit{t}\mathit{m}\mathit{a}\mathit{x}}^{\mathit{*}}$	$\overline{{\mathit{K}}_{\mathit{t}}^{\mathit{*}}}$	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{o}\mathit{n}\mathit{e}}^{\mathit{*}}$	$\mathit{\sigma }{\mathit{K}}_{\mathit{t}}^{\mathit{*}}$	Size	Classes	$\Delta {\mathit{K}}_{\mathit{t}-\mathit{t}\mathit{w}\mathit{o}}^{\mathit{*}}$
North	Chomba	Acceptable	0.0781	0.5479	0.0825	0.1174	0.2428	180	13	0.0153
		Unacceptable	0.0988	0.9815	0.0566	0.0563	0.2597	1241	35	0.0197
		Acceptable	0.0088	0.9823	0.0355	0.0821	0.399	689	28	0.0174
		Unacceptable	0.0599	0.2844	0.3561	0.0569	0.0823	745	256	0.0227
	Massangulo-1	Acceptable	0.0072	0.7926	0.0508	0.069	0.1874	688	26	0.0179
		Unacceptable	0.0908	0.6863	0.0363	0.0479	0.2037	1206	35	0.0168
		Acceptable	0.0869	0.2839	0.2351	0.0489	0.07223	657	26	0.0127
		Unacceptable	0.0098	0.7323	0.0255	0.06221	0.2199	793	28	0.0174
		Acceptable	0.0869	0.2839	0.2351	0.0489	0.07223	657	26	0.0127
		Unacceptable	0.0098	0.7323	0.0255	0.06221	0.2199	793	28	0.0174
	Massangulo-2	Acceptable	0.0072	0.7926	0.0508	0.069	0.1874	688	26	0.0179
		Unacceptable	0.0908	0.6863	0.0363	0.0479	0.2037	1206	35	0.0168
		Acceptable	0.0869	0.2839	0.2351	0.0489	0.07223	657	26	0.0127
		Unacceptable	0.0098	0.7323	0.0255	0.06221	0.2199	793	28	0.0174
		Acceptable	0.0869	0.2839	0.2351	0.0489	0.07223	657	26	0.0127
		Unacceptable	0.0098	0.7323	0.0255	0.06221	0.2199	793	28	0.0174

.

3.5. Standard Deviation of the Variational of Clear Sky Index Coefficient Variation (${\Gamma }^{K_t^{*}}$)

On the intermediate sky day, the frequency density of observation values close to 1 (about 0.120) is lower than that of observation values close to 0.66 (about 0.060), but frequency density values lower than 0.020 are reached for values from $K_t^{*}$ to 0. These values are within the theoretical radiation spectrum, ranging from –1 to 1, with the measurement observations. However, the standard deviation of the ${\mathsf{\Gamma }}^{K_t^{*}}$ decreases as a function of the averaging time, in the order of 6 to 18 h of the measurement period. The differences between the measurement sites and sample interpolation are largely removed by the $K_t^{*}$ normalization. Although the cloud stages are different at different sites, the statistical structures are very similar at all locations, as shown in Figure 13.

In

Table 9. Variability of ${\mathsf{\Gamma }}^{K_t^{*}}$ in Mozambique.

Region	Station	2012		2013		2014
		Acceptable	Unacceptable	Acceptable	Unacceptable	Acceptable	Unacceptable
South	UEM-Maputo	0.0098	0.5295	---	---	---	---
	Dindiza	0.0097	0.5295	0.2416	0.2329	0.2426	---
	Massangena	0.0095	0.2561	0.2421	0.2326	---	---
	Pomene	0.0099	0.0187	0.2426	0.5632	0.2426	0.2329
Mid	Marávia	0.0095	0.5295	0.2421	0.23269	---	---
	Nhangau	0.1833	0.2477	---	0.1243	0.2221	0.2221
	Nhapassa-2	0.1830	0.1831	0.1813	0.1831	0.1833	0.1813
	Lugela-1	0.2791	0.2658	---	0.2159	---	0.2073
	Lugela-2	0.2117	0.1833	---	0.2067	---	0.2148
North	Chomba	0.2458	0.2459	0.2131	0.2191	---	0.2194
	Massangulo-1	0.2773	0.2293	---	---	0.6006	0.2274
	Massangulo-2	0.2655	0.33256	0.2876	0.1995	---	---
	Nanhupo-1	0.2766	0.2657	0.2855	---	---	0.1897
	Nanhupo-2	0.2657	0.2252	0.2874	0.2065	---	0.2718

, the southern region of Mozambique shows normalized deviations from the $K_t^{*}$ in terms of intermediate sky days, as observed at various stations. These deviations range from 0.009759 to 0.24216 in 2012, 2013, and 2014, respectively. The southern region also showed deviations from the $K_t^{*}$ at various stations, including UEM-Maputo, Dindiza, Massangena, and Pomene. The average deviations from the $K_t^{*}$ were 0.5295 in 2012, 0.23269 in 2013, and 0.23269 in 2014. These deviations highlight the need for improved visibility and statistical characterization of acceptable days in the southern region.

Figure 3 shows that the acceptable days in the central region vary across stations. In Marávia, it was 0.009759 in 2012, 0.24216 in 2013, 0.22103 in 2014, 0.183013 in 2012, 0.183013 in 2013, and 0.183013 in 2014. In the Nhapassa-1, Nhapassa-2, Lugela-1, and Lugela-2 stations, the acceptable days were 0.5295, 0.23269, 0.247297, 0.124327, 0.222103, 0.183013, 0.18301, 0.183013, 0.260958, 0.210359, 0.20027, 0.206917, and 0.21458. In Lugela-1, it was 0.260958, 0.210359, 0.20027, 0.206917, and 0.21458. In terms of unacceptable days, Marávia had 0.5295 in 2012, 0.23269 in 2013, 0.247297 in 2012, 0.247297 in 2013, 0.183013, 0.18301, 0.183013, 0.260958, 0.210359 in 2013, 0.20027 in 2014, and 0.206917 in 2015.

The north region has seen a range of acceptable days for different stations, with Massangulo-1 (a) experiencing the highest average daily deviation of 0.277384 in 2012. Massangulo-2 also experienced a high average daily deviation of 0.265507 in 2012, followed by 0.276806 in 2013, 0.285945 in 2014, and 0.265507 in 2013. Chomba station had the lowest daily deviation of 0.049366 in 2012. Massangulo-1 station had the highest daily deviation of 0.220393 in 2012, followed by 0.227344 in 2014, 0.33256 in 2012, 0.199765, 0.199765, 0.26057 in 2013, 0.199765, 0.271468 in 2012, and 0.0.211394 in 2013.

3.6. Standard Deviation of the Variational ${\Delta K}_t^{*}$ Coefficient

The increase in the $K_t^{*}$ in the southern region was observed to have variational coefficients. In 2012, the index showed a normalized deviation of 0.0012, while in 2013 it was 0.3366 and in 2014 it was 0.3316. The average deviation was 0.2564; at Massangena, it was 0.2564, and at Pomene, it was 0.0015. In terms of unacceptable days, the index showed a deviation of 0.2196; at UEM-Maputo, it was 0.5638, and at Dindiza, it was 0.002. The average deviation was 0.2196; at Massangena, it was 0.002, and at Pomene, it was 0.002, as presented in

Table 10. Variabilidade de ${\mathsf{\Gamma }}^{{\Delta K}_t^{*}}$ in Mozambique.

Region	Station	2012		2013		2014
		Acceptable	Unacceptable	Acceptable	Unacceptable	Acceptable	Unacceptable
South	UEM-Maputo	0.0015	0.5638	---	---	---	0.2196
	Dindiza	0.0015	0.0025	0.3336	0.2166	0.3366	0.2456
	Massangena	0.2564	0.0025	0.33166	0.2166	0.4562	0.1564
	Pomene	0.0016	0.0025	0.3336	0.2196	0.3336	0.2196
Mid	Marávia	0.0014	0.0024	0.3336	0.2196	---	---
	Nhangau	0. 2481	0.1542	0.2564	0.0528	0.2203	0.2103
	Nhapassa-1	0.1803	0.1833	0.1813	0.1833	0.1813	0.0494
	Nhapassa-2	0.1813	0.1013	0.1813	0.1803	0.1813	0.2398
	Lugela-1	0.2587	0.2031	0.2546	---	0.2398	---
	Lugela-2	0.3721	0.1888	---	0.2609	---	0.2479
North	Chomba	0.2632	0.2844	---	0.2056	---	0.2624
	Massangulo-1	0.3157	0.2439	0.3369	---	---	0.1846
	Massangulo-2	0.2429	0.2479	0.2179	0.2479	0.2479	0.2479
	Nanhupo-1	0.2345	0.2208	0.2479	0.2479	---	0.2624
	Nanhupo-2	0.3267	0.5689	0.5236	0.2656	---	0.2624

.

The central region showed varying acceptable days in terms of days spent at stations. Marávia had the highest acceptable days in 2012 at 0.001, followed by Nhangau at 0.2485 and then 0.2223 in 2014. Nhapassa-1 and Nhapassa-2 had the lowest acceptable days at 0.1833 and 0.18103, respectively. Lugela-1 and Lugela-2 had the lowest acceptable days at 0.2637 and 0.3721, respectively. Marávia had the highest acceptable days at 0.0025, followed by Nhangau at 0.1542, Nhapassa-1 at 0.1813, Nhapassa-2 at 0.1833, Lugela-1 at 0.2031, Lugela-2 at 0.1888, and Lugela-2 at 0.2398.

The north region saw a rise in unacceptable days, with stations experiencing varying levels of unacceptable days. Massangulo-1 experienced 0.3157 in 2012, followed by 0.3369 in 2013, 0.2479 in 2014, 0.2863 in 2012, 0.2479 in 2013, 0.3267 in 2013, and 0.5236, respectively. Nanhupo-1 experienced 0.2427 in 2012, 0.3267 in 2013, and 0.5236, and Chomba experienced 0.2639 in 2012. Similarly, Nanhupo-1 experienced 0.2398 in 2012, 0.2579 in 2013, and 0.2624 in 2014. Chomba experienced 0.2284 in 2012, 0.2656 in 2013, and 0.2624 in 2013.

3.7. Connection of the $K_t^{*}$ Between Two Measurement Stations

The spatial autocorrelation ${\chi }_{ij}^{K_t^{*}}$ in relation to the distances between pyranometers for both acceptable and unacceptable days exhibits a decline as the distance increases. This phenomenon has also been documented in the works of Hoff and Perez [59], Mucomole et al. [114], and G. M. Lohmann et al. [67], who assessed the influence of 10 to 16 and 50 to 99 pyranometers over extensive distances, yielding numerous correlation points. The correlation structures of $K_t^{\mathrm{*}}$ among interprovincial stations, across various sky types, align with the corresponding cloud patterns, demonstrating that both cloudy and clear sky conditions exhibit greater homogeneity, which is indicative of the presence or absence of cloud layers, occasionally influenced by localized shading factors affecting the pyranometers. The increasing correlation of $K_t^{\mathrm{*}}$ is also encapsulated in a decorrelation pattern relative to distance, suggesting that energy is characterized by its occurrence from the commencement to the conclusion of the study period, diminishing from cloudy to clear skies and subsequently to intermediate skies. The days classified as intermediate skies are depicted in the decorrelation curve as having the highest potential for fluctuations, as both types of clouds can be observed, interfering with the solar energy reaching the horizontal surface. The correlation comparison with the model proposed by Marcos et al. (2011) [126], utilizing Pearson’s coefficients, and the findings of Hoff and Perez [59] regarding local fraternal clouds indicates a reduction in correlation associated with intermediate conditions, which may also reflect disturbances under such circumstances. This can potentially be analyzed further using the fraternal cloud model to account for all variability introduced by clouds. When employing the fraternal cloud model, the intermediate sky days are represented with a decorrelation that combines the two types of days, as illustrated in Figure 14.

3.8. Incremental Analysis of High Fluctuations in the $K_t^{*}$

It is evident from the years under study that the hot and wet season has the highest radiation levels at the Maputo-1 station, while the cold and dry season has the lowest radiation levels. The mean $K_t^{*}$ does, however, noticeably rise in 2012, at 0.6781. A trend deviation of 0.1867 and a negative increment of −0.088 in $K_t^{*}$ accompany this increase, suggesting a shift to 0.3093. According to the PDF function, $K_t^{*}$ is 0.0459 for every year. The Gaza-1 province has different climate patterns across the studied time. It is routinely observed that radiation levels are highest during the hot and wet season and lowest during the cold and dry season.

However, there is a noteworthy rise in the average radiation index in 2012, 2013, and 2014, indicating an intriguing pattern. The radiation received in cloudless conditions is measured by $K_t^{*}$, which has values of 0.6979, 0.6612, and 0.6667, respectively. There is a trend towards 0.2014, indicating a modest rise in radiation levels. Additionally, it is evident that $∆K_t^{*}$ is trending upward, with values of 0.3152, 0.30925, and 0.3145, respectively. This strengthens the trend in the direction of 0.3093. Remarkably, Figure 15 illustrates that the PDF function calculates the radiation index to be 1.0001 for every year.

Contrasting climate variations are evident in Gaza across the examined time. High radiation levels are experienced during the hot and rainy season, whereas the lowest radiation levels are experienced during the cold and dry season. In 2012 and 2013, however, the mean $K_t^{*}$ increased significantly, reaching 0.7595 and 0.6574, respectively. Accompanying this increase were increases in $∆K_t^{*}$ of −0.1935 and −0.1367 as well as a trend towards 0.1959. Furthermore, a trend in the direction of 0.3854 was observed. The PDF function calculates the consistency of 0.1957 throughout all years. Across the years analyzed, it is shown that the Inhambane station consistently exhibits the highest radiation levels during the hot and rainy season, while the lowest radiation levels are observed during the cold and dry season. However, in 2013 and 2014, there was a notable increase in the mean $K_t^{*}$. Specifically, the $K_t^{*}$ values for these years are 0.6962 and 0.7019, respectively. Furthermore, there is a trend of deviation towards 0.2454, indicating a deviation from the mean. Figure 14 shows that there is an increase in $∆K_t^{*}$ of −0.1142 and −0.1979 for the respective years, along with a trend of deviation towards 0.2982. The PDF function estimates whether the value of 0.1214 holds for all years.

The Tete station continuously shows the highest radiation levels during the hot and wet season, while the lowest radiation levels are noted during the cold and dry season, according to the years examined. However, the average $K_t^{*}$ increased significantly in 2012 and 2013, reaching 0.7131 and 0.5877, respectively. A tendency towards 0.2323 and a $∆K_t^{*}$ of −0.0785 and −0.0851, respectively, accompanied this development. Furthermore, there was a tendency in the direction of 0.2839. For every year, the PDF function predicts a value of 0.0504. Radiation levels at Sofala station were lower during the cold and dry seasons and higher during the hot and wet ones. However, the average $K_t^{*}$ increased in 2012, 2013, and 2014. These years have respective values of 0.6839, 0.0902, and 0.0882. Also, there is a propensity for a deviation of 0.2205. Furthermore, it is demonstrated that $∆K_t^{*}$ is increasing, with values of −0.0761, −0.007, and −0.0069, as well as a tendency toward a deviation of 0.2089. For every year, the PDF function calculates a value of 0.0547. Radiation levels at another Manica site during the hot and shallow season are higher than those at the trials conducted during the cold and dry season. On the other hand, the average $K_t^{*}$ increased significantly in 2012, 2013, and 2014. In particular, these years’ $K_t^{*}$ values are 0.6154, 0.6015, and 0.6042, respectively. Furthermore, a variation with a value of 0.2366 tended to occur.

It is demonstrated that, with a tendency of deviation towards 0.3093, $∆K_t^{*}$ increases by −0.0898, −0.1113, and −0.0182 for the corresponding years $K_t^{*}$ is estimated by the PDF function to be 0.1026 for every year. Lastly, it is demonstrated that radiation levels at the Manica station are higher during the hot and dry season and lower during the cold and rainy season. However, there was a noticeable rise in the average $K_t^{*}$ in 2012, 2013, and 2014. In particular, these years’ $K_t^{*}$ values are 0.6026, 0.5823, and 0.5902 correspondingly. Additionally, a rising pattern is shown by a trend of deviation towards 0.2322. Additionally, it is demonstrated that the $∆K_t^{*}$ for these years has a bias towards 0.3093 and is −0.0755, −0.0609, and −0.0484 correspondingly. For every year, the $K_t^{*}$ is estimated by the PDF function to be 0.03269.

3.9. Solar Energy Potential Through $K_t^{*}$, $∆K_t^{*}$ and Its Characterization Across Elevation Profiles in Mozambique

The long-term spatial and temporal trends identified from 2004 to 2024, measured over short intervals of 1, 10 min, and 1 h, reveal a reverse trend from south to north. Specifically, the southern provinces of Maputo and Gaza exhibit higher $K_t^{*}$, whereas the northern provinces, including Nampula, Niassa, and Cabo Delgado, display the lowest indices during the observed period. Notably, Tete in the mid–west region emerges as the area with the highest $K_t^{*}$ in the country, highlighting its potential as a key region for solar energy initiatives. Conversely, coastal provinces, particularly those near the northern and mid–northern coasts such as Cabo Delgado and Nampula, experience increased cloud cover, likely due to moisture from the Indian Ocean. In contrast, the interior regions, especially Tete and Manica in the mid–west, are characterized by drier and sunnier conditions, as illustrated in Figure 16.

The long-term spatial–temporal trends observed—that is, during the period 2004 to 2024, on a short measurement scale of 1, 10 min, and 1 h—show a geospatial trend in the north and mid–north under greater impact; that is, the northern provinces (Nampula, Nassa, Cabo Delgado) and central provinces (Manica, Sofala) had the largest reductions in the $K_t^{*}$. This may be related to regional climate change, increased atmospheric humidity, or seasonal events (such as cyclones). Throughout the south of the country, the $K_t^{*}$ is more stable; that is, in provinces such as Maputo, Inhambane and Tete, they show little or almost no reduction, maintaining high potential for solar generation, as depicted in Figure 17.

However, there is potential for generation throughout the country, being influenced by the mining industry that is being developed in the central and northern regions. This activity has introduced particles such as dust, aerosols, and gases into the atmosphere that enrich the ozone layer and contribute to the formation of cloud that absorbs some of the solar energy, especially in the regions in question, presenting $∆K_t^{*}$ as depicted in Figure 17.

3.10. Global Solar Energy Potential Through $K_t^{*}$ and Its Characterization Across Elevation Profiles

From 2004 to 2024, various regions exhibited unique geographic patterns regarding solar radiation availability, primarily influenced by atmospheric conditions. Arid or tropical areas with minimal cloud cover year-round, such as Saudi Arabia, Egypt, Kuwait, Qatar, the Seychelles, Nauru, and other tropical island nations, show high clear $K_t^{*}$ values ranging from 0.75 to 0.989, indicating significant potential for solar energy generation. Conversely, regions characterized by substantial cloud cover and precipitation demonstrate a moderate $K_t^{*}$, classified as medium, typically between 0.6 and 0.7, including countries like Brazil, India, Mexico, and Vietnam, which experience a mix of intermediate days and varying trends. In contrast, areas with temperate climates and higher latitudes where cloud cover is more prevalent, such as Germany, Canada, the United Kingdom, Ireland, Russia, Denmark, and Finland, exhibit lower solar energy potential, reflected in a low clear sky index of ≤0.5, as illustrated in Figure 2.

Figure 18 illustrates the global variation in clear sky frequency, which directly correlates with the potential for solar energy incidence. Notably, arid and tropical regions such as the Sahara, the Middle East, tropical islands, and the northeast exhibit a higher prevalence of clear sky indices, characterized by enhanced efficiency and high saturation levels, indicating clear sky indices ranging from 0.75 to 0.9. The heat map in Figure 17 highlights areas marked by cold, dark, and yellowish tones, indicative of low indices (0.4 to 0.55), typical of regions with frequent cloud cover, precipitation, or fog, particularly in temperate climates and high latitude areas like Western Europe, North America, and Russia. A latitudinal gradient emerges near the equator, promoting a tendency for elevated solar energy flow due to the consistent and direct exposure to solar radiation throughout the year. Conversely, as one moves towards the poles, the $K_t^{*}$ diminishes, reflecting the characteristics of polar regions, which are often marked by snow and dry climates that impede energy flow. Interestingly, tropical regions with persistent high cloudiness, such as Indonesia and the Amazon, demonstrate average solar energy levels despite their proximity to the equator.

3.11. Analysis of the Annual Course of the Development and Trend of the Clear Sky Index

The solar energy potential in the examined region is assessed to have a high density for effective photovoltaic performance, with an average clear sky index of 0.67. This is in contrast to neighboring areas, which exhibit higher solar energy incidence with $K_t^{*}$ values around 0.7 and 0.8. These values fluctuate based on geospatial and temporal factors throughout the study area. Nevertheless, during the analyzed years, the hot and rainy season consistently shows elevated $K_t^{*}$ ranging from 0.6 to 0.98. This phenomenon is primarily attributed to the Earth’s orientation concerning direct solar radiation in the southern hemisphere, which minimizes the scattering of particles that typically diminish solar energy flow. As a result, there are more days characterized by clear to partially clear skies. Conversely, the hot and rainy season also experiences lower $K_t^{*}$ values between 0.1 and 0.4, leading to an increase in cloudy days, particularly notable in June and July. Figure 2 illustrates that over the study period, the south, mid–west, and parts of the northeast regions generally exhibit a robust solar energy flow with an average clear sky index of 0.86, while other regions affected by cloud dynamics have seen a decline in their solar energy availability.

Figure 19 further demonstrates that there is no significant increase or decrease in the $K_t^{*}$ throughout the years analyzed across all seasons. This indicates that there are no notable changes or climate variations in the $K_t^{*}$ for this location during the examined period. Nevertheless, the region shows promise for solar resource utilization, as variations in atmospheric parameters predicting solar energy from both natural and anthropogenic sources, along with the effects of El Niño/La Niña in certain years, enhance the reliability of solar resources as stable and predictable for energy applications in the area. While differences between years are noted, they are gradual, lacking substantial spikes or abrupt shifts, which supports the implementation of solar energy forecasts across the three regions analyzed to optimize solar energy utilization and size systems according to the actual solar energy flow available. The solar energy potential in the region is notably high, consistently averaging between 0.60 and 0.85, with significant fluctuations occurring over short temporal intervals, yet minimal on hourly measurement scales. These fluctuations are particularly evident on days with intermediate sky conditions, exhibiting traits between clear and cloudy days, indicating potential for diversified power generation that may contribute to the reduced efficiency of solar systems due to energy oscillation, thereby raising questions about the maximum power point observed.

3.12. Analysis of the Annual Course of Development and Trend of Increases in the Clear Sky Index in Mozambique

The analysis of data from the AERONET database at the stations located in Niassa, Gorongoza, UEM-Maputo, and Inhaca reveals a significant contribution of solar energy from uniformly mixed gases, with the ozone layer exhibiting an average transmittance of 0.9825. This is followed by contributions from water vapor, aerosol absorption, and notable effects from Rayleigh scattering and MIE. During both the dry and rainy seasons, there is a reduced incidence of solar energy, as predictive parameters are densely concentrated in a stationary manner within the Earth’s atmosphere. After the winter equinox on March 23 in the Southern Hemisphere, the region’s orientation results in a diminished direct solar energy projection, leading to a lower intensity of solar energy received in the area under study. Consequently, fewer particles are dispersed due to the solar intensity compared to the hot and rainy season. Nevertheless, the particles that do predict solar energy are concentrated in the atmospheric layer and absorb significantly, resulting in a considerable decrease in solar energy. Over the years analyzed, the north, mid–south, and southeast regions demonstrate consistent increases, indicating a greater resistance to the transmission of solar energy.

The primary reason for this is the significant emission of particles that influence atmospheric conditions, which accumulate in the atmosphere, thereby intensifying cloud formation. Additionally, activities such as mining and agricultural burning contribute to the reduction in solar energy. Nevertheless, areas including the mid–east, south, southwest, and northwest exhibit considerably less obstruction to solar energy across all evaluated years, indicating a high potential for solar energy generation. The predicted energy output is greater and more stable, which helps mitigate fluctuations in solar energy production at solar facilities. The analysis across all years reveals a consistent trend, particularly in the months of January, March, and December, which experience more volatile increases and sharp positive or negative variations from year to year. However, it is more common to observe frequent negative variations during the dry and cold season, particularly between May and July, with a peak occurring in July, depending on cloud displacement. The period from June to August typically shows smaller interannual variations, suggesting that winter conditions are more stable regarding the $K_t^{*}$, making it ideal for solar applications. It is essential to size solar plants based on the actual potential of available solar energy to completely eliminate fluctuations in the plant’s output power and to maximize the operational efficiency of photovoltaic solar systems. Furthermore, Figure 20 illustrates that the rates fluctuate significantly in an extensive annual assessment from 2004 to 2024, ranging from a minimum of −0.35 to a maximum of 0.39, which are considered optimal for the performance of solar energy systems based on their estimates, taking into account the actual flow of available solar energy as described.

The random fluctuations illustrated in Figure 6, which occur sporadically throughout the months of the year and alternate between positive and negative values of the $K_t^{*}$, distinctly reveal a subtle trend of gradual increase or decrease in energy from 2004 to 2024. This trend showcases an unexpected stochastic behavior, characteristic of climate systems that exhibit spatial–temporal variations across the region of interest for solar energy utilization, accompanied by minor natural fluctuations. This may occasionally suggest more variable atmospheric conditions during these periods. Additionally, Figure 20 indicates that most annual increases across the analyzed months are nearly zero, as shown by the horizontal reference line, suggesting that interannual variability is not significant, and the clear sky index remains relatively stable from one year to the next.

4. PV Generator Characteristics on Clear, Cloudy, and Intermediate Days in Mozambique Region

Under standard conditions, the average data presented in Figure 21 illustrate the current–voltage (I-V) characteristics of a typical PV module. The current axis indicates that at $V=0$, the short-circuit current $\left(I_{sc}\right)$ is observed, while the intersection with the voltage axis at $I=0$ represents the open-circuit voltage $\left(V_{oc}\right)$. For this module, the current gradually declines to approximately 15 A before experiencing a rapid decrease, reaching open-circuit conditions at around 21.4 V. In comparison, a single silicon cell measuring 0.5 cm², under a $K_t^{*}$ radiation forecast of $1000 \mathrm{W}/{\mathrm{m}}^2$, exhibits an open-circuit voltage of approximately 0.6 V and a short-circuit current ranging from 20 to 30 mA. The power as a function of voltage can be derived from the rectangle representing the maximum power point; however, in practice, cells operate at a point on the I-V curve that aligns with the I-V characteristics of the load.

Under current conditions, the voltage gradually decreases to approximately 0.5 V before rapidly declining to open-circuit conditions at around 0.6 V, with a short-circuit current ranging from 5.5 to 6.0 A. The maximum power point operates at 2.98 W, aligning with the I-V characteristics of the load, as illustrated in Figure 21a.

Conversely, during intermediate conditions, the current similarly decreases to about 0.5 V before rapidly dropping to open-circuit conditions at around 0.6 V, exhibiting bimodal behavior in short-circuit current between 3.4 and 3.6 A, and 4.5 to 4.9 A. The maximum power point in these conditions operates at 1.7 W, consistent with the I-V characteristics of the load, as shown in Figure 21b. These intermediate days pose significant challenges for PV utilization. Nevertheless, the implementation of energy systems based on this technology enhances efficiency and prolongs the lifespan of PV systems, as they function with the actual energy available.

Figure 21c indicates that PV power production is limited on cloudy days, with maximum power point operations between 1.50 and 1.71 W, which also correspond to the I-V characteristics of the load. It is noted that the current decreases slowly to about 0.5 V and then sharply to open-circuit conditions at approximately 0.6 V, with a short-circuit current between 3.3 and 3.9 A.

5. Discussion

The region experiences a significant prevalence of solar energy, with potential for PV utilization under clear and partially cloudy conditions. This is marked by a steady rise in the maximum power point (MPP), which poses risks to grid reliability, particularly during overcast conditions. To address the temporary shortfall in PV generation—due to high fluctuations, hot spots, and power interruptions—hybridization of PV plants may be necessary. Similar findings are reported by Wansah JF et al. (2015) [27], Wang et al. (2006) [125], and Mucomole et al. (2025) [22], who designed PV systems for standalone applications, utilizing days with peak radiation as a generation benchmark (both clear and cloudy days with reduced radiation) to tailor the site according to daily energy needs and the energy that can be fed into the grid. Furthermore, additional studies with comparable outcomes are documented by Marcos et al. [126], Hoff and Perez [59], and G. M. Lohmann et al. [127], who, despite conducting analyses in a limited scope for private power plant implementation, noted that clear and mixed sky conditions can influence electrical quality levels. The evaluation of solar energy potential through the analysis of the $K_t^{*}$ in elevation profiles throughout the year indicates that the hot and dry season, characterized by clear and intermediate days, along with the cold and dry season featuring cloudy days, is crucial for the dependable forecasting of PV systems in the region. This aligns with the findings of Perpiñán et al. [116], Lave et al. [128], and Hoff and Perez [59], who observed similar results when examining the temporal variability of soil energy, based on the $K_t^{*}$ determined at sub-second intervals, further highlighting the intensity of fluctuations at this scale and the characterization of each day type, while also acknowledging the potential variability on intermediate days and the increased deviation observed during these periods.

The dispersion of acceptable days presents a uniform distribution in relation to unacceptable days, which result from days with measurement failures and/or the presence of external interferences, such as reflections by clouds and the absorption of solar radiation in the atmosphere, among others, verified at the measurement stations. Additionally, similar results were obtained by Lave et al. [128], Beyer et al. [129], Wienold and Christoffersen [63], and Trapero et al. [39], agreeing more with temporal variations that present with incremental values with a central tendency in all samples, here classified as acceptable and unacceptable. The transmittances and irradiances in all the measurement stations analyzed agree very well, as do the transmittances of the individual atmospheric constituents. Similar evidence is observed in Sarver et al. (2013) [49] when they evaluate solar energy on the Earth’s surface and report a substantial contribution of dust from construction sites, which is heavily deposited in the atmosphere, strengthening the possible population of dust in high and low layers of the atmosphere, as well as aerosols. These are directly linked to the reduction in solar energy through the absorption of these particles in regions of exploration and emission of solid sands and volcanic activity, among others. However, Iqbal (1983) [2] and Mucomole et al. (2025) [22] present similar results, considering the multiple transmittance that encompasses all solar parameters that participate in the reduction in solar energy, with greater emphasis on the transmittance by uniformly mixed gases of greater representation in the order of 0.98.

The variations in low and high albedo are affected by numerous reflections occurring between the Earth’s surface and the clear sky atmosphere. It is evident that a low albedo value of 0.25 corresponds to reduced diffuse irradiation, which is a result of these multiple reflections [2,102,122]. In 2012, the warm and rainy season accounted for approximately 9.2790% of clear sky days, 2.0620% of cloudy sky days, and 9.2790% of intermediate sky days, which includes 8.2480% of lower intermediate sky days and 1.0310% of upper intermediate sky days. Conversely, the cool and dry season contributed around 6.1860% of clear sky days, 13.4030% of cloudy sky days, and 21.6510% of intermediate sky days, with 8.2480% of lower intermediate sky days and 13.4030% of upper intermediate sky days. Notably, of the total 38.14% of unacceptable days, the hot and rainy season contributed approximately 2.0616% of clear sky days, 3.0924% of cloudy sky days, and 4.2492% of intermediate sky days, which included 2.0611% of lower intermediate sky days and 2.0611% of upper intermediate sky days. In contrast, the cold and dry season contributed about 7.2157% of clear sky days, 7.2157% of cloudy sky days, and 14.8723% of intermediate sky days, comprising 8.2444% of lower intermediate sky days and 6.1833% of upper intermediate sky days. The coefficients of the $K_t^{*}$, which serve as distance correlation functions over time, are highest at the initial point in Nhangau and lowest along the arrival path in Marávia. Some studies have reported similar findings, such as those by R. Perez et al. [72], who examined clarity distances under 10 km and time lags under 15 min, utilizing virtual pyranometer networks with single-point complexity temporal resolutions as low as 20 s. Furthermore, G. M. Lohmann et al. [127] indicated that the spectacle distances for the linear distance scale and time lag may not be applicable to observed multipoint samples of $K_t^{*}$ fields at extremely high spatiotemporal resolutions. Together with the brightness distances, Hoff and Perez [59] provide evidence for this relationship by presenting a linear scale of decorrelation distances based on satellite data.

Intermediate sky conditions exhibit smoother transitions with a greater likelihood of significant fluctuations compared to both cloudy and clear skies. Furthermore, they demonstrate a bimodal distribution in the development of the clear sky index, a phenomenon also noted by Gueymard [30] and G. M. Lohmann and Monahan [71], who reached comparable conclusions through high-resolution GHI data across various climatic settings and limited spatial dimensions. Additionally, Woyte et al. [130], Quoc Hung and Mishra [131], Mucomole et al. [22], and Lorenz et al. [34] indicate that these strong fluctuations may be amplified when analyzing tilted photovoltaic panels. The normalized deviation patterns of the clear sky index in the southern region reveal a significant decrease in variability for time increments of 0.01 s, 1 min, and 10 min. However, when the original high-resolution data are averaged over a 10 min interval, larger normalized deviation increments are observed in the western region, corroborating findings from Gueymard [30], G. M. Lohmann and Monahan [71], Marquez and Coimbra [33], Hoff and Perez [59], Perez et al. [56], and Marcos et al. [126], which suggest an energy correlation behavior at scales down to sub-minute and sub-second across all-day classifications. Excluding winter data, the deviation increment pattern reduces to approximately 0.96 of its initial value, significantly lower than the values associated with 1- and 10-min intervals during a 6 h solar energy assessment. This trend is consistent across regions with similar characteristics nationwide, as they share the same climatic conditions with maximum temperatures ranging from 37° to 41°, exhibiting correlations of coefficients around 0.86 in the south, 0.87 in the center, and 0.79 in the north, thereby providing an optimal model estimate, as also observed by Gueymard [132], Bódis et al. [133], and Widén et al. [15]. Nonetheless, there remains considerable variability linked to emissions. However, there is a strong variability associated with emissions throughout the region, mainly of industrial origin, which provide the space cloud that deposits and absorbs solar radiation, especially in the summer when the hemispheric orientation is opposite to that of the sun, and the incoming radiation does not greatly disturb the thermal state and the particles remain in an almost stationary flow in the region of high dust in the atmosphere, also observed similarly in Bokoye et al. [134], Duffie and Beckman [1]; Klein et al. [120], and Mucomole et al. [22].

Regions such as the mid–east, south, and northwest present considerably less obstruction to solar energy in all years evaluated, being estimated with high potential, but the forecasted energy is greater and in a better portion to eliminate fluctuations in solar energy in a solar plant. However, the increases vary in an annual assessment of wide observation from 2004 to 2024, from minimums of −0.35 to 0.42, which are considerable for the optimal performance of solar energy systems based on their estimate, taking into account the real flow of available solar energy, estimated in the description. This is observed accordingly in Rumbayan et al. (2012) [19], Mucomole et al. (2025) [22] and Lorenz et al. (2011) [135].

In the generalized evaluation of diverse surfaces, it is observed that the regions of the study site close to the equator appear a latitudinal gradient, which influences the trends of high solar energy flux due to the full and direct incidence of solar energy at all seasons of the year. However, as the clear sky index gradient moves towards the poles, the indices gradually decrease until reaching the polar regions characterized by the presence of snow, dry climates, and other factors that inhibit the passage of the energy flux. This is also observed in diaspora regions, as described in Santos et al. [55], Kapica et al. [57], Hamad et al. [53], Mucomole et al. [22], and Iqbal [136], which additionally describe that tropical climate locations with high cloudiness constants (Indonesia and the Amazon) have average solar energy despite being located close to the equator.

6. Conclusions

In order to better establish projects with knowledge of the relative estimate of the solar energy resources between different elevations, it is necessary to understand the relationship between the various points between the provinces of a region. This is achieved in conjunction with the search for mechanisms to electrify rural areas and the use of clean sources, such as PV solar energy (efficient, clean, and without oscillation).

The country of Mozambique has limited access to energy, and the PV systems currently in operation have a limited useful life due to a series of problems that affect their operation. Based on data measured during the FUNAE and INAM campaign and collected from Meteonorm and AERONET, the relationship between solar energy and the clear sky index was determined on various elevation surfaces, with a view to the entire region, with special attention paid to intermediate sky days, which have characteristics similar to clear and cloudy days, but present the possibility of fluctuations in the production of a solar plant, which places great emphasis on the efficiency of the plant. After analyzing the probability distributions of the $K_t^{*}$ and its short-term increments on a series of time scales from 1 min to 10 min, and studying the standard deviation of each time series as a function of the average time scale and increment step—not only with a view to intermediate sky days described with potential for fluctuations, but also with a view to all classes—the following conclusions can be drawn.

The assessment of the solar energy potential through the characterization of the $K_t^{*}$ in elevation profiles in Mozambique reveals significant regional disparities. The northern and central provinces, particularly Niassa (−0.1109), Gaza (−0.0943), and Manica (−0.0853), recorded the largest reductions in clear sky conditions, indicating a decrease in the potential for solar irradiance. In contrast, southern provinces such as Inhambane (−0.0035) and Tete (−0.026) showed minimal deviations from historical averages, maintaining strong potential for solar energy capture, which is essential to optimize the deployment and design of solar infrastructure, especially in variable climate conditions.

Correlation coefficients at interprovincial distances generally appear to decrease with distance and increase with increasing drift time.

The existence of several factors, such as multiple reflection by clouds and the blocking of solar energy measurements by various factors, inhibited the comparative evaluation of stations, also due to the heat flux transport factor, where the mass flux of solar energy proceeds in a seasonal direction to establish the solar energy balance.

Correlation coefficients generally increase with drift time scale and decrease with distance; a greater bias towards lower intermediate sky days is observed in the predicted intermediate sky days, which explains most of the expected variability and in situ clear sky index.

The solar energy exhibits considerable decorrelation over the spatiotemporal range examined in the intermediate sky days and in the predicted GHI sample. The simple scattering albedo encourages the molecular absorber transmittances to remain largely independent of declination for all sites examined, in accordance with the projected solar energy. A random alternation is observed between the days and months of the year in terms of the positive and negative values of the $K_t^{*}$, both for the short-scale sample between 2012 and 2014, and for the period from 2004 to 2024, clearly depicting a weak trend of consistency of increase or decrease in energy over the period of 21 years, implying a development resulting from a surprising stochastic dynamic representative of climate systems that present themselves with spatio-temporal development throughout the region of interest for solar use with small natural fluctuations. In regions close to the equator, a latitudinal gradient appears, which influences the tendency for high solar energy flux due to the full and direct incidence of solar energy in all seasons of the year. However, as the $K_t^{*}$ gradient moves towards the poles, the indices gradually decrease towards the polar regions characterized by the presence of snow, dry climates, and other factors that inhibit the passage of the energy flux.

The evaluation of the solar energy potential through the characterization of the $K_t^{*}$ in elevation profiles reveals that tropical climates with high constant cloudiness have average solar energy even though they are located close to the equator.

Intermediate sky day conditions with high decay have been shown to present arms of PDFs of increased transmission with flatter tails and more probabilities of strong fluctuations in relation to cloudy and overcast skies. On intermediate sky days, the development of the $K_t^{*}$ presents a bimodal behavior, with a focus on low variability in the southern region and its interior, with greater variability in the central and western regions, as well as in the northern region. On cloudy days, it presents characteristics similar to both cloudy days and clear sky days, with average values close to 1. The wings present a central maximum close to zero and decrease as the $K_t^{*}$, presenting flat and long arms with thin wings and a distribution structure faithful to the normal distribution integrated in the determined probability interval.

It was concluded that, despite the increase in precision obtained by the parameterization, by evaluating the solar energy potential through the characterization of the clear sky index in elevation profiles, the area still offers potential for solar application, with average clear sky and intermediate potential values of 25% and 51%, respectively.

The estimated solar energy allows the model to be evaluated in any reality, since it is within the theoretical spectrum of irradiation in clear skies, with results also applicable in parallel to all available intermediate sky data, where the distributions of the $K_t^{*}$ are strongly bimodal on average time scales. The peaks are separated by a $K_t^{*}$ difference of approximately 0.9 and represent, respectively, cloud-covered and cloud-free states. For the clear sky index, an average time is obtained.

The analysis of the clear sky indices by province in Mozambique shows a clear spatial distribution of the availability of solar radiation. The southern and central–western regions, especially Tete (0.9345) and Maputo (0.849), have the highest $K_t^{*}$ values, standing out as areas with excellent potential for solar energy generation. In contrast, northern provinces such as Cabo Delgado (0.4434) and Nampula (0.5649) have lower frequency of clear skies, which may impact the performance levels of photovoltaic systems. This information is crucial for the strategic planning of solar infrastructure, energy policies, and environmental sustainability projects in Mozambique.