Machine Learning-Based Evaluation of Solar Photovoltaic Panel Exergy and Efficiency Under Real Climate Conditions

Abstract

The purpose of this study article is to provide a detailed examination of the performance of exergy electric panels, exergy efficiency panels and exergy solar panels under the climatic circumstances of the Utrecht region in the Netherlands. The study explores the performance of these solar panels in terms of both their energy efficiency and their exergy efficiency. Additionally, the study investigates critical factors such as solar radiation, module internal temperature, air temperature, maximum power, and solar energy efficiency. Environmental factors have a considerable impact on panel performance; temperature has a negative impact on efficiency, whereas an increase in solar radiation leads to an increase in energy and exergy output. These findings offer significant insights that can be used to increase the utilization of solar energy in locations that have a temperate oceanic climate, particularly in the context of the climatic conditions of the Utrecht region. The usefulness of the linear regression model in machine learning was validated by performance measures such as R², RMSE, MAE, and MAPE. Furthermore, an R² value of 0.94889 was found for the parameters that were utilized. Policy makers, researchers, and industry stakeholders who seek to successfully utilize solar energy in the face of changing climatic conditions may find this research to be an important reference.

1. Introduction

As technology progresses, a significant portion of humanity’s growing energy demands continues to be met by fossil fuels. However, fossil fuels are not only depleting but also pose serious environmental challenges, making the transition to clean and renewable energy sources essential. The Paris Climate Agreement, ratified by numerous countries, including Turkey in 2015, seeks to encourage energy production from more sustainable and environmentally friendly sources, moving away from fossil fuels. Throughout history, energy has always been a fundamental necessity for human society. Providing energy in a cost-efficient and environmentally sustainable way is vital for the health of our planet and the well-being of future generations. As a result, renewable energy is gaining increasing importance. With energy resources depleting, sustainability is becoming the central issue facing humanity. Energy, being indestructible, has been studied in various forms, leading to extensive research aimed at improving energy systems. Ref. [1] asserted in his research that energy comprises more than just work; it encompasses several types including electrical, chemical, mechanical, thermal, and nuclear energy. The first law of thermodynamics is expressed through the principle of energy conservation. The second law of thermodynamics underscores the importance of both the quantity and the quality of energy, saying that energy diminishes in both aspects during transformation, a concept known as exergy. Ref. [2] contend that exergy analysis is an effective analytical approach that synthesizes the first and second laws of thermodynamics. Exergy is defined as the maximum work potential that can be derived from a system under designated reference conditions. Exergy analysis includes entropy generation inside the system, providing a comprehensive assessment of system performance through the discovery of exergy loss in its components [3]. Exergy analysis is an essential tool for evaluating the effective use of solar energy producing systems by employing the second law of thermodynamics [4]. Exergoeconomic analysis, which combines exergy analysis with economic concepts, is a methodology that evaluates the costs related to system outputs and exergy degradation, enabling the systematic examination and optimization of energy systems. The authors of [5] argue that while solar radiation maps serve as a valuable initial resource, they are not sufficient for a complete assessment of electrical energy output from photovoltaic power plants, which is necessary for estimating regional potential. This limitation arises because other environmental factors, beyond solar radiation, also play a significant role in system performance. Factors such as ambient temperature, wind speed, pollution, and others can notably influence the efficiency of solar power systems. Additionally, energy-based solar radiation maps fail to provide accurate data on photovoltaic system efficiency, as they do not account for the actual thermodynamic value of the energy source. This challenge can be effectively addressed using exergy methods, which are grounded in the first and second laws of thermodynamics. According to [6], thermoeconomics, or exergoeconomics, is an engineering field that merges exergy analysis with economic theory. It has been widely applied for over 20 years to optimize the design, synthesis, and operation of energy systems, improving exergy efficiency and reducing unit production costs. Moreover, it offers valuable insights for system designers or investors that cannot be obtained through traditional energy analysis or economic assessments. The authors of [7] investigated the optimum limits of solar rooftop accessibility by exergy analysis for three systems: solar thermal, photovoltaic, and hybrid photovoltaic–thermal systems. The authors in [8] analyzed the thermal, electrical, and exergy efficiency of a solar panel. It was observed that the energy and exergy efficiencies fluctuate between 6–9% and 8–10%, respectively, throughout the day. The exergy efficiency of the photovoltaic module initially increases with solar radiation intensity and then decreases after reaching a maximum. Ref. [9] investigated the thermodynamic parameters of photovoltaic cells with a cumulative power output of 631.5 watts in Golden, Colorado, USA, employing an exergy model they formulated based on previous exergy models. They contended that exergy evaluations should be utilized for a more precise assessment and planning in solar cell systems. Ref. [10] evaluated the exergy efficiency of photovoltaic and photovoltaic–thermal systems using various thermodynamic approaches. They have illustrated the variances in exergy efficiency across several systems. They emphasized that exergy analysis should be utilized to provide more precise results in the evaluation of solar systems. Ref. [11] performed exergy assessments of photovoltaic and photovoltaic–thermal systems. The researchers examined the energy and exergy balances and determined the different exergy losses related to both thermal and electrical energy transfers. Ref. [12] performed an exergoeconomic analysis of a 750-watt solar system implemented at the meteorological park of Istanbul Technical University. Ref. [13] performed a performance assessment of polycrystalline photovoltaic modules by monthly energy and exergy measurements in North India. The study’s results revealed that the mean values for energy efficiency, power conversion efficiency, and exergy efficiency were 18.09%, 12.26%, and 11.17%, respectively. Ref. [14] analyzed the energy, power conversion, and exergy efficiency of monocrystalline solar modules over three days (21–23 March) in Adrar, Algeria. Ref. [15] analyzed the thermal, electrical, and exergy characteristics of a 36-watt solar module. Moreover, they established that exergy loss increases with elevated PV module temperature. In 2014, Ref. [16] performed exergoeconomic and environmental economic assessments of opaque and translucent photovoltaic modules employing several solar cell types. Translucent photovoltaic modules exhibit a higher energy loss rate and an increased exergy loss rate than opaque photovoltaic modules, regardless of the solar cell type. Ref. [17] assessed the energy and exergy performance of thin film photovoltaic modules on designated days each month in North India. Performance evaluations demonstrate that power conversion efficiency exceeds exergy efficiency. Ref. [18] investigated the energy and exergy efficiency by applying partial shadowing to a 75-watt polycrystalline solar panel in Elazig province. Ref. [19] performed comprehensive exergy and performance evaluations of solar electricity generation. They clarified the advantages and disadvantages of photovoltaic and hybrid systems (PV/Tsu or PV/Air) through the lens of exergy analysis. They established that the exergy efficiency of hybrid systems (PV/Tsu or PV/Air) is twice that of photovoltaic systems. Ref. [20] performed an exergy and energy analysis of a 12 MWp solar power facility employing polycrystalline photovoltaic modules at Cochin International Airport in India. In 2018, Ref. [21] established a compact solar energy system employing polycrystalline photovoltaic panels on the engineering building of Karabük University. They subsequently evaluated this system in line with the first and second laws of thermodynamics. In [22], the energy and exergy efficiency of a 10 MWp grid-connected solar power station situated on a water canal in Sama, India, were evaluated. The evaluation was based on two years of data from the plant. In [23], 7E assessments were carried out at seven different airports in India, where the researchers planned to develop a 5 MW solar power plant for each location, evaluating the projects using RETScreen expert software. The study concluded that photovoltaic power plants are theoretically viable at all airports, potentially contributing to the global development of solar power plants at airports. In [24], an energy and exergy analysis was performed on a 50 W solar module, grounded in the first and second laws of thermodynamics. The findings showed that the maximum energy efficiency and exergy efficiency of the photovoltaic module were 25.2% and 32.4%, respectively. Ref. [25] conducted energy and exergy evaluations of monocrystalline and amorphous solar cells in Northern Poland. The average annual energy efficiencies of monocrystalline cells at 9:00, 12:00, and 15:00 h were 8.3%, 8.0%, and 7.1%, respectively. In comparison, the corresponding values for amorphous cells were 2.1%, 2.2%, and 2.2%. The exergy efficiencies of monocrystalline cells at 9:00, 12:00, and 15:00 h were 6.8%, 4.9%, and 5.0%, respectively, while for amorphous cells, these values were 1.3%, 0.9%, and 0.7%, respectively.

The originality of the study lies in its context-specific analysis, integration of solar exergy, assessment of exergy electricity and efficiency, investigation of various environmental factors, focus on the impact of temperature, comprehensive analysis of the importance of solar radiation, and practical implications for policy and industry. These factors jointly enhance the understanding of solar energy application in specific climatic conditions and provide valuable insights for both the scientific community and practitioners in the field. A linear regression model in machine learning will be conducted and explained.

This study is structured as follows: Section 2 comprehensively summarizes the materials and methods used to obtain the findings of this research. Section 3 presents a comprehensive review of the results and an in-depth discussion. Finally, Section 4 provides a conclusive summary.

2. Material and Methods

This research investigates exergy solar, exergy electricity, exergy efficiency, and the operational efficacy of solar panels in relation to meteorological circumstances in the Utrecht region of the Netherlands. This study employed minute-by-minute data from the Utrecht University Photovoltaic Outdoor Test site. The data utilized includes solar radiation, air temperature, module temperature, maximum power, and efficiency. The study aims to evaluate the efficiency of the measured panel, encompassing solar exergy, electric exergy, and exergy efficiency, while analyzing these variables under different climatic situations. The document begins with the definition of the variables and a description of the data collection process. The linear regression model will thereafter be executed and explained (Figure 1). The gathered data were examined by linear regression employing Python 3.9.1 version code that utilized the Scikit-learn machine learning framework within Pandas 3. The characteristics and their corresponding values are presented in

Table 1. Characteristics of the used PV panels.

Feature	Values Between
Open Circuit Voltage (Voc) [Volt]	30–36
Short Circuit Current (Isc) [A]	6–8
Maximum Power (Pmpp) [W]	0.08–1.26
Solar Irradiation Spread [m2]	0.000516–0.007284
Maximum Voltage (Vmpp) [V]	15.18–29.34
Maximum Current (Impp) [A]	0.13–0.54
FillFactor (FF) [%]	50–75
Parallel Resistance (Rp) [Ohm]	0.18–1.20
Series Resistance (Rs) [Ohm]	0.18–0.97
Module Temperature [°C]	7.72–59.89
Efficiency [%]	5–20

.

2.1. Study Area

In the Utrecht region (Figure 2), summers are mild and partly cloudy, while winters are extended, cold, windy, and mostly overcast. Temperatures vary from 0 °C to 22 °C, remaining above −6 °C and below 28 °C. Cloud cover demonstrates seasonal variability, with July recognized as the month exhibiting the most clarity, as 56% of the sky is clear or partially overcast. December exhibits the highest incidence of precipitation, averaging 9.9 days of rainfall. Rain is the primary kind of precipitation, peaking at 34% on 22 December. The length of daylight ranges from 7 h and 44 min in December to 16 h and 45 min in June. Humidity comfort levels remain rather stable year-round. Wind velocities reach their zenith from October to April, averaging around 18.1 km/h. June is the month with the peak sun radiation, averaging 6.3 kWh. The geographical area is primarily flat, with a maximum elevation variation of 25 m and an average altitude of 4 m above sea level.

2.2. Parameters Used in PV Solar Power Plant Efficiency

The efficiency of a PV solar power plant is the ratio of the total power generated (kWh/year) to the total solar radiation received (kWh/year). It is influenced by both external (ambient) and internal (module) elements [26,27,28]. The result depends on various environmental factors (solar radiation, air temperature) and module attributes (maximum power, efficiency, module temperature), which are elaborated below.

Solar Irradiation (W/m²): The intensity of sunlight received. Elevated levels enhance electricity production.

Air Temperature (°C): Influences efficiency; elevated temperatures often diminish performance.

Module Maximum Power (kW): Peak electrical output.

Module Temperature (°C): Elevated temperatures may diminish output.

Module efficiency denotes the ratio of power produced to the solar energy absorbed. It denotes the efficacy with which a photovoltaic system converts sunlight into electrical energy. Figure 3 depicts the graphics for solar irradiance, air temperature, module maximum power, and module temperature.

In June 2023, a fixed, grid-connected PV solar power plant was installed on the roof of the building housing the Photovoltaic Outdoor Test (UPOT) research center at Utrecht University. In June 2023, the solar modules on the roof of the UPOT building were oriented at an angle of 37° to the south. Real experimental measurements of the electrical parameters of the PV solar plant were taken. In Figure 3 we can see the graphs of the outdoor data taken in the period 2023. With these data, the integration of solar exergy, exergy electricity and efficiency were calculated. Data were collected every 5 min from the Utrecht University Photovoltaic Outdoor Test (UPOT) facility, which evaluates the performance of various PV modules under real conditions. Solar radiation was measured using a pyranometer (Kipp & Zonen CMP11). The uncertainty associated with these measurements is estimated to be ±5%, arising from calibration errors and environmental factors.

2.3. Multiple Linear Regression

Linear regression is a statistical method used to illustrate the relationship between a dependent variable and one or more independent variables. The aim of linear regression is to determine the best line (or hyperplane for many variables) that defines the relationship among these variables [27,28].

$$Y_i={\mathsf{\beta }}_0+{\mathsf{\beta }}_1{X}_1+\cdots +{\mathsf{\beta }}_i{X}_i+\cdots +{\mathsf{\beta }}_k{X}_k+\mathsf{\epsilon }, i=\mathrm{1,2},\dots ,k$$

(1)

Y: The dependent variable. X₁, X₂,..., X_i: The independent variables. β₀: The intercept term, indicating the point of intersection of the line with the Y-axis when all X variables are set to zero. β₁, β₂,..., β_i: The coefficients that measure the impact of each independent variable on the dependent variable. ε: The error term, representing the portion of Y that is not elucidated by the model.

R² (Coefficient of Determination): Evaluates the degree to which the independent variables explain the variance in the dependent variable. R² ranges from 0 to 1, with values nearing 1 indicating an excellent fit.

$${\mathrm{R}}^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}{{\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}}$$

(2)

Mean Absolute Error (MEA) measures the average deviation between the anticipated values and the actual values. It is often employed in contexts such as approximation theory or the evaluation of predictive models, particularly in fields where little errors are crucial.

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|y_i-\widehat{y_i}\right|$$

(3)

Mean Squared Error (MSE) is a metric that measures the average of the squared deviations between expected and actual values. It is widely used to evaluate regression models.

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2$$

(4)

2.4. Account Methods

Exergy efficiency in solar panels measures the effectiveness of labor capacity in converting solar energy into electrical energy. This takes into account both the volume of solar energy and the efficacy of its use. Exergy efficiency is an essential metric for a more accurate evaluation of solar panel performance.

Exergy and Solar Energy: Solar energy denotes the transmission of solar radiation (photons) that arrives at the Earth’s surface. Nonetheless, sunlight arrives at the Earth at very low temperatures and generally in a diffuse manner. As a result, many losses occur during the direct transformation of solar energy into electrical energy. The efficiency of solar panels demonstrates these losses and the degree to which they convert energy into useable work capacity [8,29]. Exergy is the theoretical maximum amount of work achieved in a reversible process if equilibrium with the environment is achieved. In the simplest sense, exergy is the part of energy that can be used and is also referred to as availability. Energy is often an uncertain state. Exergy is a concept that shows how ‘useful’ the energy in a system can be in relation to its environment. In other words, exergy determines the capacity of an energy source to do work. Energy is usually not lost, but it comes into equilibrium with the environment and loses its ability to do work in this state of equilibrium. Exergy measures the potential of energy to do work before it reaches this equilibrium. Exergy is work or the ability to produce work. In order to calculate exergy in accordance with these definitions, environmental conditions must be known. Through reversible processes, the amount of work that can be obtained when a substance is brought into thermodynamic equilibrium with the basic elements of the natural environment is equal to the exergy of that substance [30,31,32,33].

Solar Panels and Exergy: Solar panels convert sunlight into electrical energy through photovoltaic (PV) cells. Nonetheless, during this conversion process, not all solar energy is translated into electrical energy; some is converted into heat or is lost as dissipation. Exergy evaluates the quality of energy and its capacity to perform work during the conversion process [34].

Exergy efficiency: Exergy efficiency is a concept that measures how effectively an energy system works. Basically, energy has the capacity to do a certain amount of work in a system and this capacity is called ‘exergy’. Exergy efficiency shows how much of this capacity is utilized efficiently.

In an energy conversion process, energy usually undergoes some losses. For example, an engine generates heat while running or a power plant loses a certain amount of energy. Exergy efficiency calculates how much of these losses are available to do work and how much is lost through heating, friction, etc.

A high exergy efficiency means that energy is utilized more efficiently and the system operates with fewer losses. This means generating less waste heat and more work.

It evaluates the effectiveness of a solar panel or photovoltaic system in transforming sunshine into energy (the ability to perform labor). This concept takes into account both the amount of sunlight and the ability of this energy to perform tasks. Exergy efficiency evaluates the attainment of superior grade energy.

Exergy efficiency is defined as follows:

Exergy In: The energy contained in solar radiation. This varies according to temperature and the strength of solar radiation.

Exergy Output: The exergy associated with the electrical energy produced by the solar panel [25,35].

$${\eta }_{\mathrm{e}\mathrm{x}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}}={\displaystyle \frac{\mathrm{E}\mathrm{x}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{O}\mathrm{u}\mathrm{t}}{\mathrm{E}\mathrm{x}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{I}\mathrm{n}}}$$

(5)

Exergy Efficiency and Loss in Solar Panels: In energy conversion devices such as solar panels, the predominant energy loss is converted into low-grade heat. Examining these losses improves our understanding of the effectiveness of solar panel systems. Although solar panels predominantly convert sunlight into electrical energy, a considerable amount of sunlight is converted into heat, leading to energy loss. These losses are designated as exergy losses [36,37,38,39,40,41].

Exergy loss can be determined as follows:

$$\mathrm{E}\mathrm{x}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{L}\mathrm{o}\mathrm{s}\mathrm{s}=\mathrm{E}\mathrm{x}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{I}\mathrm{n}-\mathrm{E}\mathrm{x}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{O}\mathrm{u}\mathrm{t}$$

(6)

2.4.1. Exergy Efficiency Calculation in Solar Panel

Entered Exergy: The exergy of sunlight reaching the surface. The exergy of sunlight is usually calculated as follows [34]:

$$\mathrm{E}\mathrm{x}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}\left(\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{s}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r} \mathrm{i}\mathrm{r}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)=Q.\left(1-{\displaystyle \frac{T_0}{T_{\mathrm{e}\mathrm{n}\mathrm{v}}}}\right)$$

(7)

Q denotes the power of solar radiation. T₀ denotes the temperature of sunlight. T_env denotes the ambient temperature. Efficiency is the quotient of exergy output to exergy input.

2.4.2. Exergy Efficiency and Temperature in Photovoltaic Systems

In a solar panel, the predominant energy conversion occurs into heat. When sunlight contacts the panel’s surface, a fraction is converted into electrical energy, while the rest is transformed into heat and released. As the temperature escalates, the panel’s efficiency declines due to a spike in exergy loss. At increased temperatures, the panel’s power generation capacity declines. As a result, exergy loss is generally more pronounced in solar panels due to higher temperatures. An efficient system design depends on controlling and managing the temperature of the solar panel to minimize exergy loss. Exergy analysis of solar panels is a method utilized to evaluate energy efficiency and environmental impacts. Exergy measures the quality of energy in a system and its capacity to do tasks. This document presents the formulae and processes for the fundamental exergy analysis of solar panel systems.

Exergy Definition and Basic Formula: Exergy denotes the maximum work that an energy source may accomplish while in equilibrium with its surroundings. In solar energy systems, two principal exergies can be calculated:

Solar exergy and electrical exergy: In a solar panel, the exergy of an energy source (e.g., sunlight) is usually calculated as follows [6,25]:

$$\mathrm{E}\mathrm{x}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}=Q.\left(1-{\displaystyle \frac{T_0}{T}}\right)$$

(8)

Example: Exergy (Joule). Q: Energy transfer (watt). T₀: Ambient temperature (Kelvin). T: Temperature of the solar panel (K). This formula delineates the disparity between thermal energy and its potential to do work in relation to the environment.

Solar Exergy (Exergy of Solar Energy): The energy of the sun’s rays’ incident on the solar panel’s surface can be ascertained by calculating solar exergy. This value derives from the subsequent formula:

$${Ex}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}=I.\left(1-{\displaystyle \frac{T_0}{T_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}}}}\right)$$

(9)

I: Solar radiation (W/m²). T₀: Environmental temperature (Kelvin). T_module: Surface temperature of the panel (Kelvin).

Electrical Exergy (Exergy for Electricity Generation): Electrical exergy. The following formula is used to calculate the exergy of the electricity generated by solar panels:

$${Ex}_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}=P_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(1-{\displaystyle \frac{T_0}{T_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}}}}\right)$$

(10)

Electrical Exergy (Joule). P_electric: Electric power (watt). Electrical exergy reflects the panel’s efficiency and its ability to execute work under varying environmental conditions.

Exergy Efficiency: Exergy efficiency is calculated to determine how much work a system can do. The exergy efficiency for a solar panel can be calculated as follows:

$${\eta }_{\mathrm{e}\mathrm{x}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}}={\displaystyle \frac{{Ex}_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}}{{Ex}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}}}$$

(11)

Here, η_exergy represents the exergy efficiency of the solar panel, Ex_eletric represents the electrical exergy, and Ex_solar represents the exergy of solar energy.

Heat Loss and Exergy of Heat Energy: The heat loss of the solar panel can also be calculated by exergy analysis. This is used to calculate the exergy of the heat energy emitted by the solar panel to its surroundings:

$${Ex}_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}=Q_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\left(1-{\displaystyle \frac{T_0}{T_{\mathrm{p}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{l}}}}\right)$$

(12)

Q_{heat loss}: Thermal energy emitted by the solar panel to the environment (watts). T_panel: Surface temperature of the panel (in Kelvin). Exergy analysis of solar panels is utilized to assess energy efficiency.

The exergy of solar energy is calculated. The electrical exergy is calculated based on the electricity generated by the panel. Exergy efficiency is defined as the ratio of the electrical exergy generated to the exergy of the solar energy. Exergy analysis is an essential tool for measuring the efficiency of solar panels and examining the environmental impacts of the system.

3. Results and Discussion

3.1. Applying Multiple Linear Regression

This study first performs a multiple linear regression analysis. The commonly employed module and environmental factors affecting solar panel efficiency are identified and analyzed. This study seeks to analyze the impact of ambient and module variables on panel power efficiency by linear regression analysis. The principal objective of our research is to analyze the efficacy of solar panels in correlation with diverse climate data. The exergy of solar energy, electricity, and panel efficiency are examined concerning module internal temperature, ambient air temperature, efficiency, maximum power, and solar radiation. The energy efficiency of solar panels, an essential measure for converting solar radiation into useful energy, is influenced by various factors.

Multiple linear regression models were evaluated using data obtained from measurements of the Photovoltaic (PV) Solar Power Plant of the Utrecht University Photovoltaic Outdoor Test (UPOT) facility. Prior to importing the data into the models, several pre-processing steps were undertaken to ensure its quality and compatibility. These steps included data cleaning, normalization, and partitioning the dataset into training and test sets using 10-fold cross-validation. The dataset, which contains 5 min interval measurements from the Utrecht University Photovoltaic Outdoor Test (UPOT) facility for the city of Utrecht, covers data from the month of June. A total of 477 5 min samples were collected from the solar panel, and the dataset was split using 10-fold cross-validation, where the data were divided into 10 folds. Nine of these folds were used for training, while the remaining fold was used for testing. This process was repeated 10 times to ensure consistent results and minimize variance.

The results demonstrate that the R² value was 0.94889 in the multiple linear regression study. The results for mean squared error (MSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) are displayed in

Table 2. Training and test sets to validate the results of the datasets according to the internal parameters.

Model	R2	RMSE	MAE
RF	0.94889	0.0180499	0.0983381

. The linear regression model demonstrated robust performance in the assessment criterion.

Table 3. Correlation coefficients between solar PV module efficiency and solar PV module efficiency parameters.

Input Internal Parameters	PV Solar Power Efficiency
Maximum Power	+0.314
Module Temperature	+0.106
Air Temperature	+0.980
Irradiation	+0.318

shows the correlation coefficients between PV solar energy plant efficiency and other variables. The results show that there is a significant positive correlation between solar PV module efficiency and solar irradiance and solar PV module power. These correlations are shown in Figure 4.

Figure 5 illustrates that the distribution of residuals closely resembles a Gaussian curve, signifying that errors are primarily concentrated around zero. The error distribution approaches an ideal condition in data science experiments.

The correlation graph (Figure 6) and scatter plot (Figure 7) of data variables exhibit the parameters together with the correlations of internal and external parameter estimations detailed in

Table 4. Correlations of external parameter estimates.

	Maximum Power	Module Temperature	Efficiency	Air Temperature	Irradiation
Maximum Power	1
Module Temperature	0.553477	1
Efficiency	0.377574	0.593858	1
Air Temperature	0.367222	0.800951	0.354029	1
Irradiation	0.013694	0.028378	−0.013180	0.073490	1

.

To alleviate the risk of overfitting in data mining algorithms, datasets are commonly partitioned into training and test sets for the purpose of result validation. The linear regression curves for all data are depicted in Figure 8a,b.

The ideal sub-model was determined to be the one exhibiting the highest cross-validation R² value in Figure 9. This part evaluates the goodness of fit criterion for the models, demonstrating that linear regression is the most efficient model for both the training and test datasets. The performance gap between the training and test sets indicates that the model is suffering from overfitting. Thus, the generalization of the concept poses difficulties. However, for all models, even if the difference is not as dramatic as in linear regression, the same situation applies.

3.2. Applying Exergy Solar, Exergy Electric and Exergy Efficiency

This section analyzes the chronological development of energy efficiency and investigates its evolution. Figure 10 depicts the temporal variations in energy efficiency observed for a certain panel. Energy efficiency is a vital factor in evaluating the overall performance of photovoltaic (PV) technology due to its substantial direct impact.

This study examines how ambient temperature, module temperature, solar radiation, and maximum power affect the efficiency of the panel technology. Figure 11 illustrates the impact of these factors on the energy efficiency of the selected panel type. The data presented in the curves reveal a clear and consistent pattern: As the ambient temperature increases, the energy efficiency of the panels decreases. This negative correlation between ambient temperature and energy efficiency indicates that the photovoltaic (PV) panels are highly sensitive to temperature fluctuations. This is a well-known characteristic of solar panels, making it crucial to understand their energy efficiency. The reduction in efficiency with rising temperature can be explained by various factors. The primary reason is the temperature sensitivity of the semiconductor materials used in the panels, particularly silicon. As the temperature increases, silicon’s conductivity typically rises, resulting in greater internal losses in the solar cells, ultimately leading to a decrease in the panels’ overall efficiency.

Additionally, the effect of temperature on panel efficiency is closely related to the performance of the semiconductor bandgap. As temperatures rise, the bandgap tends to decrease, impairing the material’s capacity to effectively absorb and convert sunlight into electricity.

Figure 12 presents key insights regarding how solar radiation influences the energy efficiency of the specified solar panel characteristics. The data shown in the curves reveal a positive relationship between irradiance levels and panel performance. As irradiance or solar energy increases, the energy efficiency of the panels improves proportionally, owing to the higher energy received per unit area. This relationship is fundamental to the operation of photovoltaic systems. Solar panels are designed to convert sunlight into electricity, and higher irradiance levels provide more energy for conversion, effectively maximizing power output.

The observed increase in energy efficiency with rising irradiance levels underscores the ability of these solar panels to effectively harness solar exergy, as shown in Figure 12. This aligns with the fundamental principle that higher solar input results in greater electrical output. As irradiation levels rise, both the quality and quantity of energy produced by the solar panel improve. This enhancement is due to the panels greater capacity to generate energy in conditions of higher sunlight intensity. This study has tangible implications for the design and execution of solar energy systems. A thorough understanding of the connection between irradiance and energy efficiency is essential for enhancing the performance of solar installations. System designers and operators can utilize these data to forecast energy output under different lighting conditions, account for fluctuations in energy production, and make well-informed decisions regarding the selection and placement of solar panels.

Figure 13 illustrates the variations in solar exergy, electric exergy, and exergy efficiency for the analyzed silicon panels. The importance of solar exergy, electric exergy, and exergy efficiency lies in their ability to transform solar radiation into useful solar exergy, electric exergy, and exergy output. It is crucial to understand that these efficiencies are likely dependent on specific conditions and influenced by factors such as ambient temperature, irradiance levels, and other environmental variables.

Figure 14 illustrates the changes in electrical exergy as a function of variations in ambient temperature for the specified panel. Upon reviewing the curves in Figure 15, a clear trend emerges: as temperature rises, electrical exergy decreases accordingly. This reduction in exergy efficiency with increasing ambient temperature is a well-known occurrence in solar systems. Elevated temperatures can negatively impact solar panel performance, resulting in lower efficiency. The connection between temperature and electrical exergy is closely tied to the semiconductor properties of the materials used in solar cells. As the ambient temperature increases, the semiconductor materials in the panels experience greater electron excitation and mobility, which leads to higher resistance and a decrease in energy conversion efficiency. Maximum power is directly related to electrical exergy. Figure 14 further illustrates the temporal changes in the solar energy and exergy efficiency of the panel. It highlights the relationship between exergy and energy efficiency over time. The direct correlation between exergy efficiency and energy efficiency indicates that any improvement or reduction in one will similarly affect the other. It is important to note that exergy efficiency values consistently exceed those of energy efficiency.

This phenomenon is illustrated by the declining trend in exergy efficiency shown in Figure 15. Effectively managing and minimizing the temperature’s impact on solar panel performance is essential for improving both their efficiency and lifespan. The observed variations in exergy efficiency at different ambient temperatures offer valuable insights for system designers and operators. Examining measures like the incorporation of advanced cooling systems to alleviate the negative effects of high temperatures or the application of materials with enhanced thermal properties might improve the overall exergy efficiency of solar panels under diverse environmental conditions. Figure 15 depicts the fluctuations in exergy efficiency according to alterations in irradiance for the designated panel parameters. A clear pattern emerges through careful examination of the curves: As irradiance levels rise, the performance of both solar panels improves, as reflected in the higher exergy efficiency values. The positive relationship between irradiance, maximum power, and exergy efficiency is key to understanding the behavior of solar systems. Irradiance, which represents the amount of solar energy hitting the panels, plays a vital role in power generation.

Figure 15 shows that higher irradiance levels enhance exergy efficiency, indicating an increase in both the quality and quantity of energy generated by the solar panels. As irradiance increases, more photons strike the solar cells, leading to greater electron excitation and, consequently, better energy conversion efficiency. The increase in irradiation levels is evidenced by the upward trend in exergy efficiency. The documented data highlight the need to optimize solar panel installation in areas with abundant sunlight or to utilize monitoring technology to improve solar radiation exposure. Grasping and leveraging the relationship between irradiance and exergy efficiency is essential for designing effective and reliable photovoltaic systems that can efficiently harness solar energy across various environmental conditions. Fluctuations in exergy efficiency contribute directly to the improvement of solar panel performance and overall energy output.

The disparity between exergy and energy efficiency metrics stems from the extra insights that exergy efficiency offers. Unlike energy efficiency, which only evaluates the quantity of energy output, exergy efficiency delivers a more holistic view of exergy utilization and overall system performance. High exergy efficiency values indicate the superior performance of the analyzed panel, notwithstanding the inherent losses and inefficiencies in the system. This comprehension is essential for improving and refining the design and operating specifications of solar panels, considering both the energy yield and the thermodynamic quality of that energy.

4. Conclusions

This study investigated exergy solar, exergy electricity, exergy efficiency, and the performance of solar panels under meteorological conditions in the Utrecht region of the Netherlands. This study employed minute-by-minute data from the Utrecht University Photovoltaic Outdoor Test (UPOT) facility. The aim of the study is to evaluate the efficiency of the measured panel, encompassing solar exergy, electric exergy, and exergy efficiency, while analyzing these variables under certain climatic circumstances. The linear regression model is then employed and explained. The primary goal of this study was to conduct a thorough assessment and comparison of energy efficiency and exergy efficiency for these solar panels, with careful consideration of the impacts of ambient temperature, solar radiation, module internal temperature, and maximum power. Our findings and conclusions demonstrate that the described panel displays comparable energy and exergy efficiency under the existing climatic conditions in the Utrecht region. The results highlight the significance of the specified panel technology for the effective utilization of solar energy in the Utrecht region, hence indicating prospective opportunities for renewable energy generation in the area. The research underscores the substantial influence of environmental factors on panel efficacy. Ambient temperature plays a crucial role, as higher temperatures notably decrease both energy and exergy efficiency. On the other hand, elevated solar radiation has a beneficial effect, boosting both energy and exergy output. This research offers essential data and insights that will inform key decisions on the judicious use of solar energy in climate zones similar to the Utrecht region of the Netherlands. It highlights the adaptability and efficacy of ambient photovoltaic panels in harnessing solar energy.