Increasing Photovoltaic Systems Efficiency Through the Implementation of Statistical Methods

Abstract

The article emphasises both the advantages and disadvantages of photovoltaic power plant deployment, assessing the current stage of development as well as the deficient characteristic criteria, such as the occupied specific surface area or the associated unpredictability. The authors consider that current technologies related to photovoltaic plants provide a maximum efficiency of approximately 28%. Consequently, management methods must be applied in order to improve efficiency and eliminate the reported deficiencies. When assessing a medium- to high-power PV plant, the initial investment, projected efficiency, and parameters of the desired plant are correlated, and sometimes, a cheaper and less efficient power plant can be recommended. Although solar trackers may represent a viable solution in certain scenarios, their effectiveness is strongly influenced by various factors, including panel orientation, climatic conditions, installed capacity, and the specific technologies. These variables can significantly affect such systems’ overall efficiency and suitability. The present study proposes a statistical approach to assessing the economic efficiency of photovoltaic systems equipped with solar trackers, aiming to enhance energy production performance. The results are correlated and validated using field data obtained from existing literature studies to ensure the reliability and accuracy of the analysis. For a better analysis, the paper presents two methods, ANOVA and STEM, which are derived from quality control. The novelty aspect of this proposal consists of the combination of specific data obtained from the PVGIS platform with a new approach for optimisation of energy production in photovoltaic systems based on geographical coordinates. The STEM statistical method provides a high degree of novelty because, although it is a well-known method, it has not yet been applied to analyse the technical and economic efficiency of photovoltaic systems. One of the main advantages of this method is its ability to incorporate a wide range of technical and economic performance parameters. A case study is provided to evaluate the benefits of implementing the STEM method.

1. Introduction

When considering applications such as PV power plants, substantial increases in technical and economic performance can be achieved by improving safety in operation, location, and mounting systems, as well as optimising specific management parameters. This involves the application of statistical methods.

The first theme addressed in the article is selecting the tilt angle of the PV panels used on a PV plant to optimise the efficiency of photovoltaic systems using statistical analysis based on the ANOVA method. There are several nonparametric methods of statistical analysis, the most common being the description of the Kruskal–Wallis test in Excel [1], using the Taguchi and ANOVA Analysis for PV plants [2], a multi-objective design optimisation strategy for hybrid PV plants [3], one-way ANOVA analysis of variance [4], operating parameters and degradation of PV [5], and statistical methods and artificial neural network approaches for PV plant decisions [6].

The second approach involves a managerial and statistical analysis based on a STEM-type method. Economic efficiency analysis should be conducted during the initial phase of developing a new photovoltaic (PV) plant. This analysis must consider technical and financial factors relevant to the specific location of the proposed facility. Quality management systems began to be developed in the early 20th century as statistical sampling techniques were introduced into the quality control methodology. In the following years, there was an increasing demand for increased productivity. It became apparent that a more robust, structured, and logical approach to quality control was needed. There are numerous statistical methods for product quality control, including the impact of product quality on sales, as demonstrated by the Sales Training Evaluation Model (STEM) [7,8]. Thus, the technical level method can be used to evaluate the quality of products based on the von Neumann–Morgenstern utility principle and the Cobb–Douglas production functions [9]. A relation of the form defines the absolute technical level (Hai) [10,11].

The STEM method currently used in the education system, where STEM stands for science, technology, engineering, and mathematics, although it shares the same acronym, is distinct from the sales method STEM discussed in this context.

2. Literature Review

2.1. Review of High-Performance Technologies for PV Cells and Modules

Due to the exponential growth of industrial production, with significant implications for the development of energy sources and all types of transportation, there is an urgent need to reduce the carbon footprint. Therefore, most countries worldwide are interested in developing and optimising renewable energy sources: solar, wind, geothermal, hydropower, ocean energy, and bioenergy. The photovoltaic (PV) conversion provides the most outstanding development dynamics. According to the Clean Energy Technology Observatory [12], in 2022, the cumulative global installed photovoltaic capacity exceeded one terawatt (TW), while cumulative electricity generation from photovoltaic sources was approximately 158 terawatt-hours (TWh). Considering this dynamic, various equipment and processes use PV systems for an efficient power supply [13]. Despite certain shortcomings in solar energy conversion efficiency, it remains one of the most reliable and affordable energy sources, especially in remote areas. Its conversion can provide electricity to supply isolated households by ensuring the end-users’ energy consumption and management of other potential various generation sources [14] or different equipment for monitoring environmental parameters or biodiversity protection, such as behavioural barriers associated with river water intakes [15].

In theory, the maximum efficiency of a single junction p-n solar cell is 30% at 1.1 eV, but the maximum laboratory efficiency for a single junction is about 27%. At the same time, the efficiency of photovoltaic modules is lower than that of a PV cell due to resistive losses in the solar cells, as well as variations in individual solar cell efficiencies [16]. Although the conversion efficiency of hydroelectric power plants is around 90% and that of power plants is about 40%, both have a very low carbon footprint. Yet, a series of arguments plead for the priority of photovoltaic conversion [17,18,19]. First of all, solar energy is practically available everywhere, unlike hydroelectric power, and, relative to nuclear power plants, there are no radioactive residues. However, although PV systems have an almost zero carbon footprint, this can be significant if the manufacturing processes are taken into account [20,21]. Additionally, extending the lifespan of PV modules and addressing waste management at the end of their lifespan have recently become significant concerns in the context of the circular economy. This concern becomes more pressing as the prospect of equipping photovoltaic power plants with lithium-based batteries becomes a requirement for efficiency. Therefore, recycling these devices will involve more significant challenges than recycling PV modules.

The history of photovoltaic conversion will soon complete two centuries of existence, with its beginnings marked by the French experimental physicist Alexandre-Edmond Becquerel in 1839. The progress in this field has been modest given that over the last half-century, we have witnessed an exponential increase in efficiency [16,22]. Thus, nowadays, apart from technologies of the first-generation a-Si, c-Si mono and polycrystalline, there are envisaged superior efficiencies characterising the new technologies, even in the commercial field, as presented in

Table 1. Efficiency of various PV panels [16,23,24].

PV Panel Type	Efficiency [%]
PERC (passivated emitter rear contact)	22.8
Bifacial PERC	<25
PERL (passivated emitter with rear locally diffused cells)	24.5
HIT (heterojunction with intrinsic thin layer)	26
HJ-IBC (heterojunction-interdigitated back contact)	26.7
TOPCon (tunnel oxide passivated contact)	26.1
Thin-film CIGS (copper indium gallium selenide)	22.2
Tandem solar cells based on silicon and perovskite	32

:

The table above should also include thin-layer cadmium telluride (CdTe) solar cells, which are the second most common photovoltaic PV technology after crystalline silicon, representing 21% of the U.S. market and 4% of the global market in 2022, with an efficiency of approximately 24%, and GaAs module technology with concentrators (Fresnel lenses), which can reach an efficiency of 28% [25,26].

2.2. Analysis of the PV Market in Romania

The cumulative global PV installed capacity increased from 483.1 GW in 2018 to 580.2 GW in 2019, representing a 21% relative growth [27]. It further increased from 866.02 GW in 2021 to 1055.03 GW in 2022, representing a 22% relative growth [28]. The highest contribution of currently installed PV systems is identified in Asia. Relevant examples come from China (175 GW in 2018 and 392.46 GW in 2022), Japan (55.5 GW in 2018 and 83.1 GW in 2022), and India (26.8 GW in 2018 and 62.85 GW in 2022). Europe comes in second regarding PV installed capacity, with considerable shares in Germany (45.9 GW in 2018 and 66.66 GW in 2022), while the UK has an insignificant increase (13.4 GW in 2018 and 14.66 GW in 2022).

Specific prices for PV panels decreased approximately linearly starting in 2008, by about 35% on the American market and by about 80% on the European market, likely due to cheaper imports from China [29,30]. For instance, in the USA, the mean price of PV panels varied from USD 2.51 to USD 3.31 per watt in 2020, while the average costs were USD 0.40–USD 0.60 per watt for polycrystalline panels and USD 0.60–USD 0.90 per watt for monocrystalline panels as of January 2020. Chinese production became the market leader relatively quickly, starting in 2007. Since China is the world’s largest manufacturer and consumer of PV cells and modules, Chinese production has driven down prices from USD 4.73 per watt in 2007 to USD 0.19 per watt in 2020 [29].

Considering the significant variability of the statistical data available in the literature,

Table 2. Commercial analysis of the PV market in Romania for 2024.

Panel Type	Surface [m2]	Peak Power [Wp]	Efficiency [%]	Technology	Price [USD]	Price Ratio [USD/Wp]	Power Ratio [W/m2]
Yingli Solar	1.9	330	17	Si-polycrystalline	109	0.33	174
Solaro LR5-72HPH-575M	2.5	575	22.3	Si-mono N Multi Busbar	140	0.24	230
ENF-Germany	2.6	580	22	HJT Half-Cell Bifacial	NA	0.35	223
Risen Solar RSM132 8 660	3.1	660	21.3	Mono PERC BIFACIAL	297	0.45	213
Jinko	2.27	460	20.3	Si-mono Bifac. Half-Cut	118	0.26	203
Jinko Solar JKM440N-54HL4R-V	2	440	22	TOPCon N type mono	96.8	0.22	220
Canadian Solar CS3L-375MS	1.8	375	20.8	Mono PERC	95	0.25	208
JA Solar JAM72S30	2.5	550	22	PERC	200	0.4	220
Macsun Solar Ms-Hcpv120w	0.48	133	28	GaAs Si Fresnel lens	286	2.15	277

includes several relevant commercial offers of photovoltaic modules available on the Romanian market in February 2024 based on various technologies: Si-mono and poly-crystalline, PERC and TOPCon technologies, bifacial variant, and GaAs technology with concentrators (Fresnel lenses), all with an efficiency between 17% and 28%. Table 2 is based on the prices provided by retailers’ online shops in Romania.

The efficiency shown in Column 4 is calculated by dividing the PV module’s peak power output from Column 3 by the maximum irradiation in the region (1000 W/m²) for the panel area from Column 2.

2.3. Assessment of PV Power Plants’ Operating Conditions

The final product of a power plant is the electric power supplied to the grid at 50 Hz and various voltage levels, depending on the power plant’s structure. For photovoltaic plants, the voltage level varies depending on the injection points in the system, which include 11 kV, 33 kV, 110 kV, and 400 kV. Electricity is a product; like any product, it must meet its quality requirements. They are specified in the IEC 61000-4-30:2015+A1:2021 and EN 50160 standards [31,32]. A PV system that does not comply with these standards will generate problems for the transmission and distribution grids, potentially risking disconnection from the power system. These interruptions can affect the efficiency of photovoltaic plants, decreasing the installed power/electricity produced ratio.

2.4. Using Statistical Methods for the Assessment of PV Power Plants

Parametric statistical techniques begin with a series of conditions regarding the normality and homogeneity of the subjects’ results distribution. Nonparametric techniques are also referred to as statistical techniques that are independent of data distribution when these conditions are not met. Nonparametric tests can be used when there are doubts about the normality of statistical data. The advantage of nonparametric tests is that they are more robust, meaning they use fewer assumptions than parametric tests. Nonparametric tests do not need a known a priori distribution of observed data or a large volume of data.

2.5. Prediction of PV System Performance Using PVGIS Platform

A geographic information system (GIS) is a computer system used to create, store, analyse, and process global information. GIS technology can be applied in various scientific fields, including resource management, environmental impact studies, mapping, and route planning. For photovoltaic system applications, the most used are the web applications from the National Renewable Energy Laboratory (NREL) and PVGIS.

NREL is led by the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy and the Alliance for Sustainable Energy, LLC, a partnership between Battelle and MRIGlobal. NREL’s geospatial data science research utilises geographic information systems to produce maps, analyses, models, applications, and visualisations that inform energy planning and production, including the PVWatts Calculator. The analogue of this application for Europe is the PVGIS Platform.

PVGIS is a web application that provides users with data on solar radiation and energy production generated by photovoltaic systems in most parts of the world. It is entirely free to use, with no restrictions on how the results can be used, and no registration is required. The application utilises terrestrial and satellite databases that have been statistically processed over extended periods [33].

The necessary input data consider the following:

• The location on the displayed map or the analytical entry of the geographical position (latitude and longitude);
• Choosing the database for solar irradiation;
• Photovoltaic cells/panels technology (Si-mono/polycrystalline);
• Installed power of the photovoltaic power plant [kWp];
• System power losses (typical 14%);
• Orientation of the panels according to azimuth (−90° East, 0° South, and 90° West, respectively);
• Orientation of the panels according to the angle of inclination to the horizontal—in the case of the mount without tracking;
• Characteristic of the tracking system (one or two axes);
• Economic data:

  ▪

  Cost of investment (in the currency of the respective country);

  ▪

  Interest rate [%/an];

  ▪

  Estimated service life [years].

The output data include the following:

• Global horizontal irradiation (GHI)—This value is the monthly sum of the energy of solar radiation hitting one square meter from a horizontal plane, measured in kWh/m²;
• Direct average irradiation (DNI)—This value is the monthly sum of the solar radiation energy hitting one square meter always oriented in the direction of the sun, measured in kWh/m², including only the radiation coming directly from the sun;
• Global irradiation, optimal angle—This value is the monthly sum of the solar radiation energy hitting one square meter oriented toward the equator at the angle of inclination that gives the highest annual irradiation, measured in kWh/m²;
• Global irradiation, selected angle (GTI)—This value is the monthly sum of the solar radiation energy hitting one square meter from a plane oriented toward the equator at the angle of inclination chosen by the user, measured in kWh/m²;
• Ratio of diffuse radiation to global radiation—Much of the radiation that reaches the ground does not come directly from the sun but from scattering clouds and water vapour from the air (blue sky). This is known as diffuse radiation. This number represents the fraction of the total radiation that reaches the ground due to diffuse radiation.
• Electricity production per month (January–December). According to input data for photovoltaic systems, PVGIS can also calculate the cost of electricity generated by the photovoltaic system. The calculation is based on a “Levelized Cost of Energy” (LCOE) method, similar to how a fixed-rate mortgage is calculated.
• The total cost of purchasing and installing the photovoltaic system in the currency of the respective country;
• The interest rate, in % per annum, is assumed to be constant throughout the lifetime of the photovoltaic system;
• Estimated lifespan of the photovoltaic system in years.

The data provided by PVGIS are used as a precursor to investigate the possibility of increasing efficiency through statistical methods, either by optimising the structure of photovoltaic systems (ANOVA method) or by optimising specific economic parameters (STEM method). Higher-efficiency photovoltaic plants will help reduce the cost of an installed photovoltaic system, decrease the occupied area, and implicitly substantially improve the technical and economic parameters related to circular economy principles.

3. Methodology

Statistical methods can be applied to achieve the efficient operation of PV systems. Therefore, the methodology for each approach is described below.

The following outlines the theory and methodology utilised for applying statistical methods to analyse the efficiency of the photovoltaic (PV) system: ANOVA for optimising the PV module’s structure and STEM for optimising specific economic parameters.

3.1. ANOVA Method Regarding Operational and Structural Analysis

The parametric statistical techniques begin with a series of conditions regarding the normality and homogeneity of the subjects’ results distribution. Nonparametric techniques are also referred to as statistical techniques that are independent of data distribution when these assumptions are not met. The advantages of this technique lie in its ability to process a diverse range of data, both qualitatively and quantitatively. Nonparametric tests can be used when there are doubts about the normality of statistical data. The advantage of nonparametric tests is that they are more robust and use fewer assumptions than parametric tests. Nonparametric tests do not need a known a priori distribution of observed data or a large volume of data. The main disadvantage is the lower power to detect the falsity of a null hypothesis. There are several nonparametric methods, the most common being:

• Fisher’s test technique—(χ²);
• Mann–Whitney U test (equivalent to the independent parametric t-test);
• Wilcoxon test of paired ranks (ANOVA equivalent of repeated measurements or t-dependent);
• Kruskal–Wallis ANOVA test of ranks, equivalent to simple ANOVA (data are converted to ranks).
• Analysis of variance (ANOVA) is a statistical formula that compares variances across different groups’ means (or averages). In this paper, we propose using the Kruskal–Wallis ANOVA test of ranks to determine the tilt angle of the panels based on geographical coordinates and season.

The primary phenomenon that generates the application of statistical methods is data variability. The general notion of variability refers to the measurable deviations of individual values from a central value, known as the mean (μ). By analysing statistical collectivity with the help of central tendency indicators, the manifestation of a particular phenomenon can be predicted. A collectivity has a specific internal organisation defined by how individual values are distributed around the central value. Thus, one central value may be credible, and another may not. For this reason, the central tendency analysis must be supplemented with indicators of variation and the form of distribution.

In probability theory and statistics, variance is the mean of the squared deviations of a random variable from its mean. Informally speaking, it measures the distance of individual (random) values in a crowd from their average. Variance is a crucial concept in statistics, used in both descriptive and inferential statistics, as well as in hypothesis testing and Monte Carlo methods. Statistical methods are widely used throughout science.

Analysis of variance (ANOVA) is a statistical method used to compare the means of three or more groups. Its purpose is to determine whether there are significant differences between groups and to identify which groups have these differences. ANOVA compares the variability between groups (intergroup variability) with the variability within groups (intragroup variability). If the intergroup variability is significantly greater than the intragroup variability, then we can conclude that there are significant differences between the groups.

In the article, the Kruskal–Wallis H test will be applied using Relationship (1):

$$H={\displaystyle \frac{12}{N·\left(N+1\right)}}·\sum _{i=1}^k{\displaystyle \frac{R_i^2}{n_i}}+3\left(N+1\right)$$

(1)

• N = the size of the sample analysed (total number of observations across all groups);
• k = number of groups;
• R = the sum of the ranks of a group; n_i = the number of elements in a group.

3.2. Application of STEM Method Focused on Economic Efficiency

The managerial approach based on a STEM-type statistical analysis is currently applied to product competitiveness analysis. As mentioned, this method is used for the first time in photovoltaic power plants, which are considered market-promoted products. The process was applied to five existing PV plants with six known characteristics, three of which represent the subset of characteristics whose value is directly proportional to the quality of the product, and the other three represent the subset of characteristics whose value is inversely proportional to the quality of the product, as described in the next section.

The technical level method can evaluate product quality based on the von Neuman–Morgenstern utility principle and the Cobb–Douglas production functions [9]. A form relation defines the absolute technical level H_ai [10,11].

$$H_{ai}=a·{\prod }_{j\in S_1}{\left({\displaystyle \frac{K_{ij}}{K_{1j}}}\right)}^{{\gamma }_j}·{{\prod }_{j\in S_2}\left({\displaystyle \frac{K_{ij}}{K_{1j}}}\right)}^{{-\gamma }_j}$$

(2)

where

• S₁ represents the subset of characteristics whose value is directly proportional to the quality of the product;
• S₂ represents the subset of characteristics whose value is inversely proportional to the quality of the product;
• i represents the product for which the technical level is calculated;
• 1 represents the reference product (any of the analysed products can be considered a reference);
• a is a constant that defines the technical level of the reference product (usually a = 1000, to differentiate the products sufficiently);
• j is the characteristic of the product;
• K_ij is the value of the characteristic j for product i;
• γ_j represents the weight of the characteristic j.

The relative technical level can also be calculated with the relation:

$$H_{ri}=H_{ai}·100/max\left(H_{ai}\right)$$

(3)

The product with the highest technical level has a value of 100, while the others have values below this.

The primary challenge is determining the weights of the characteristics, γ_j, to apply the above relations. Two methods are primarily known for this: the MISENIT and STEM methods [10,11]. The MISENIT method is based on the elasticity of the operating expense function, which is the property of a function that changes when its argument changes. The STEM method is less finesse than the previous one. Still, it allows, on the other hand, a faster calculation of the weights, given that the analyst must be a specialist in exploiting the respective products. To accomplish this, the characteristics are compared two by two, defining a matrix with the elements a_j1j2 established as follows:

$$a_j=\left\{\begin{array}{cc}4,\hfill & K_{j1} \gg K_{j2}\hfill \\ 2,\hfill & K_{j1}>K_{j2}\hfill \\ 1,\hfill & K_{j1} \approx K_{j2}\hfill \\ 0,\hfill & K_{j1}<K_{j2}\hfill \end{array}\right.$$

(4)

where K_j are the quality characteristics.

The signs that intervene have the following meanings:

(≫)—much more important;

(>)—more important;

(≈)—equivalent;

(<)—less important.

Under these conditions, the weights of the characteristics are calculated with Relation (5):

$${\gamma }_j={\displaystyle \frac{\sum _{j2}{a}_{j1j2}}{\sum _{j1}\sum _{j2}\left(a_{j1}\right)·\left(a_{j2}\right)}}$$

(5)

The STEM method algorithm flowchart shows the computing steps for the relative technical level evaluation:

According to Figure 1, the first step of the presented algorithm involves selecting parameters from sets S1 and S2, respectively. For PV power plants, the three parameters whose value is directly proportional to the electricity supplied are the panel’s efficiency, expressed in percentages, the installed power of the panel, described in Wp, and the presence of a tracking-type orientation system. The parameters whose value is inversely proportional to the efficiency of the electricity supplied are the surface area of the panel (m²), the specific cost of the panel (EUR/Wp), and the total losses of the photovoltaic system relative to inverters, transformers, and cables (typically 14%). The second step involves selecting several PV power plants for product quality analysis, whose parameters, as mentioned above, are well defined. The third step is crucial because it consists of assigning scores to the mutual influences between the six parameters of sets S1 and S2, respectively, according to Relationship (4). These influence scores are presented in a table or matrix with terms a_j, as illustrated in the next chapter for a specific case. Based on this table, the range γ_j, the exponent from Equation (2), is calculated according to Relation (5).

Finally, steps four and five only apply Relations (2) and (3) to calculate the absolute and relative technical level.

4. Results

When assessing a medium-to-high-power photovoltaic plant, the initial investment, projected efficiency, and parameters of the desired plant are correlated. In some cases, a cheaper and less efficient plant may be recommended. This paper presents two methods for enhanced analysis: ANOVA and STEM, which are derived from quality control principles.

4.1. Analysis of the ANOVA Method Results

Regarding the statistical methods, a presentation of the PVGIS platform [30] with input and output data was provided to predict electricity production for a photovoltaic plant based on geographical and meteorological parameters, as well as the technical data of the used PV panels. Based on the data provided by the PVGIS platform, an analysis is conducted to examine the impact of a panel inclination angle on electricity production. The literature shows that orientation tracking systems can increase efficiency by up to 30% (depending on latitude), as shown in Figure 2, generated by the PVGIS platform.

Series 1 (blue) represents the monthly electricity production for the photovoltaic system of 1.19 MWp in the case of fixed panels inclined at an angle of 30°, series 2 (red) and 3 (grey) for the case of single-axis tracking system—azimuth and tilt (inclination), respectively—and series 4 (orange) for the case of two-axis tracking systems. It is observed that there is no significant difference between the three tracking systems (with one axis and with two axes, respectively).

Although the increase in electricity production is significant in the case of tracking systems, they are more complex, require maintenance, and may not be cost-effective for all applications. This analysis is fundamental to deciding to what extent the investment is economically justified, depending on geographical coordinates.

In this respect, the statistical analysis carried out by the ANOVA method, corroborated by the data obtained through the PVGIS platform, shows the following: on the one hand, between three consecutive months, there is no statistically significant difference between the groups in terms of the angle of inclination; on the other hand, the panels can be segmented according to the angle of inclination.

Keeping the criterion of three consecutive months, obtained by applying the ANOVA method, as an example for the case of latitude 45°, an optimal arrangement would be the following:

• For June, July, and August, an optimum tilt angle of 30° is obtained;
• For September, October, and November, an optimal tilt angle of 50° is obtained;
• For December, January, and February, an optimal tilt angle of 70° is obtained;
• For March, April, and May, an optimal tilt angle of 40° is obtained.

The methodology remains valid for other geographical coordinates, provided the data are available from the PVGIS platform.

• Case analysis—Fixed tilt angle

The statistical analysis begins with the specific case of a photovoltaic power plant with a capacity of 1.19 MWp [34], which generates a declared amount of renewable energy of approximately 1428 MWh/year. The envisaged PV system is 26°55′ east longitude and 46°35′ north latitude in Romania.

Table 3. PVGIS predicts the annual energy production (MWh) of PV plants at seven different tilt angles.

Month	10°	20°	30°	37°	40°	50°	60°	70 °
January	39	46	52	55	57	60	61	61
February	54	61	66	68	69	71	70	69
March	97	105	110	112	112	112	110	104
April	128	133	135	135	134	130	123	113
May	150	151	149	146	144	137	126	112
June	157	156	152	148	146	136	123	107
July	166	166	163	159	157	147	134	118
August	155	159	160	159	158	152	142	129
September	119	127	132	134	135	134	130	122
October	79	89	97	101	102	105	105	102
November	44	52	59	62	63	66	68	67
December	33	39	45	48	49	52	54	54

summarises these data, obtained through the PVGIS platform (PVGIS—re.jrc.europa.eu—European Union [33]) for each month of the year, with the values provided in MWh.

The influence of the panels’ inclination on annual electricity production will be analysed statistically, applying the Kruskal–Wallis test to sets of cases with three distributions.

We propose analysing two sets of cases, each with three distributions, using July as a reference, as shown in

Table 4. Distributions per month for the data analysed.

Case	Months Under Surveillance
A	May 2024	July 2024	September 2024
B	June 2024	July 2024	August 2024

.

The selected cases were chosen so that we have two sets of data with different ranges of variation (5 months and 3 months, respectively) to apply the presented statistical method.

Table 5. Case A: Monthly energy production and ranks related to the panel’s angle (in MWh).

Orientation	May	July	September
10°	150	166	119
20°	151	166	127
30°	149	163	132
37°	146	159	134
40°	144	157	135
50°	137	147	134
60°	126	134	130
70°	112	118	122

shows the characteristic values that express the amount of electrical energy obtained depending on the inclination angle.

These values were selected from Table 3 above. Column 37° means the optimum value of the tilt angle of the panel for which the maximum annual electric energy is obtained according to the PVGIS platform for the geographical and climatological coordinates considered.

Next,

Table 6. Case A: Table of values with explanation of ranks.

Rank	1	2	3	4	5	6	7	8	9	10	11	12
Value	112	118	119	122	126	127	130	132	134	134	134	135
Avg rank									10
Rank	13	14	15	16	17	18	19	20	21	22	23	24
Value	137	144	146	147	149	150	151	157	159	163	166	166
Avg rank											23.5

was built with 24 columns corresponding to the 24 values from Table 5, representing the ranks under Relation (2). The values are placed in ascending order. Where we have identical values, the corresponding rank is the arithmetic mean of the ranks corresponding to similar values. Ranks are relative numbers to 100, which represents the minimum value.

Next, the table of ranks was built for the three analysed months: May, July, and September (

Table 7. Case A: The analysis (table of ranks) for May, July, and September.

Orientation	May		July		September
10°	150	18	166	23.5	119	3
20°	151	19	166	23.5	127	6
30°	149	17	163	22	132	8
37°	146	15	159	21	134	10
40°	144	14	157	29	135	12
50°	137	13	147	16	134	10
60°	126	5	134	10	130	7
70°	112	1	118	2	122	4
Rang sum		102		147		60
(Rang sum)2		10,404		21,609		3600

).

The last two rows in Table 7 represent the sum of the ranks and the square of the sum of the ranks for the analysed cases, respectively. Next, the equation characteristic of the H_STAT (Kruskal–Wallis test) was applied:

$$H_{STAT}=14.03$$

(6)

Relationship (6) represents the value of the test applied in the present case, called the H statistic (H_STAT).

The value thus obtained is compared with χ² obtained from the Chi-Square Distribution Table for 2 degrees of freedom (df = 2) and alpha = 0.05 (statistical confidence interval of 95%), and the value obtained is

$${\chi }^2=5.991$$

(7)

According to the theory, hypothesis H₀, for which the three distribution probabilities are equivalent, is denoted by H_STAT > χ², meaning that hypothesis H₀ is rejected due to a significant difference between the groups.

As a conclusion, for case A, the H₀ hypothesis is rejected because the value of H_STAT is greater than the χ². There is a significant difference between the analysed groups.

Next, the same algorithm was applied for case B, as presented in

Table 8. Case B: Monthly energy production (in MWh) and ranks related to the panel’s angle for June, July, and August.

Orientation	June 2024	July 2024	August 2024
10°	157	166	155
20°	156	166	159
30°	152	163	160
37°	148	159	159
40°	146	157	158
50°	136	147	152
60°	123	134	142
70°	107	118	129

.

Table 8 is similar to the above-presented Table 5, with the May, July, and September triad replaced by June, July, and August.

Table 9. Case B: A table of values with explanation of ranks.

Rank	1	2	3	4	5	6	7	8	9	10	11	12
Value	107	118	123	129	134	136	142	146	147	148	152	152
Avg rank											11.5
Rank	13	14	15	16	17	18	19	20	21	22	23	24
Value	155	156	157	157	158	159	159	159	160	163	166	166
Avg rank			157				159				23.5

presents the table of values for Case B, similar to Table 6.

Table 10. Case B: The analysis (table of ranks) for June, July, and August.

Orientation	June		July		August
10°	157	15.5	166	23.5	155	13
20°	156	14	166	23.5	159	19
30°	152	11.5	163	22	160	21
37°	148	10	159	19	159	19
40°	146	8	157	15.5	158	17
50°	136	6	147	9	152	11.5
60°	123	3	134	5	142	7
70°	107	1	118	2	129	4
Rang sum		69		99.5		111.5
(Rang sum)2		4761		9900.25		12,432.25

presents the table of ranks for Case B, which is identical to Table 7.

Table 9 and Table 10 resemble Table 6 and Table 7 above, built for June, July, and August.

$$H_{STAT}=-7.3$$

(8)

For case B, the H₀ hypothesis cannot be rejected because the value of H_STAT is smaller than the χ², as presented in Equation (8). In this case, with the analysed months having similar energy production values, there is no significant difference between the analysed groups, as indicated by the Kruskal–Wallis H test.

4.2. Applying the STEM Method Results

According to the flowchart in Figure 1, the following steps were considered:

• Step 1

The characteristics j of the PV power plants belonging to the sets S1 and S2, respectively, are chosen:

• Efficiency of photovoltaic panels (E), the greater, the better;
• Nominal power of a panel (P), the greater, the better;
• Tracking systems (T), the greater, the better;
• Panel surface (S), the smaller, the better;
• The specific cost of a panel (C), the smaller, the better;
• Overall losses (L), the smaller, the better.

• Step 2

It chose five representative PV power plants (i = 1 … 5), with the respective general characteristics:

• PV power plant Borzesti (Romania), 1.19 MWp—considered the reference [34], using monocrystalline PV panels;
• PV power plant Stupini–Brasov (Romania), 20 MWp, Ref. [35] monocrystalline panels;
• PV power plant, Ouarzazate (Morocco), 20 kW, Refs. [36,37,38] using HCPV technology, including tracking;
• PV power plant Elazig (Turkey), 1 MWp, Ref. [30], using polycrystalline panels;
• PV power plant Bisha (Saudi Arabia), 10 MWp, Ref. [39] using monocrystalline panels and a single-axis tracking system.

Furthermore, the characteristics table identified in Step 1 was constructed using data extracted from the bibliography for the proposed PV plants (

Table 11. Characteristics of the chosen PV power plants.

PV Power Plant	1. (Ref.)	2. (A)	3. (B)	4. (C)	5. (D)
E [%]	20	15	40	16.6	15.7
P [Wp]	370	280	150	270	255
T (yes/no)	1.10 (no)	1.10 (no)	1.30 (yes)	1.10 (no)	1.30 (yes)
S [m2]	1.77	1.94	0.512	1.62	1.6
C [EUR/Wp]	0.34	0.21	8.68 × 1.1	0.51	0.41 × 1.1
L [%]	14	14	14	14	14

). Since PV Power Plant 1 was considered the reference, the following notations were assigned: 2 ≡ A, 3 ≡ B, 4 ≡ C, and 5 ≡ D.

The PV builders provide the values for efficiency (E), the rated power of the used PV panel (P), panel surface (S), and the specific cost of a PV panel (C) in Table 11 [27,31,32,33,34,35,36].

In the above table for the tracking systems (T) parameter, a coefficient of 1.10 (+10%) was assigned for photovoltaic systems without tracking and 1.30 (+30%) for those with tracking because systems oriented according to the sun improve yield by approximately 30%. At the same time, a tracking system increases costs by about 10% compared to the specific cost (C), as specified in the fifth row of the table above.

For the overall losses (L) parameter from Table 11, it is currently considered that these losses represent approximately 14% of the electricity produced by a PV power plant, which is equal to all the PV plants analysed. In the future, if a plant diminishes its own losses, this parameter can be adjusted accordingly.

In

Table 12. Table of characteristics, K_ij.

(j)\(i)	Unit	1. (ref.)	A.	B.	C.	D.
		K1j	K2j (A)	K3j (B)	K4j (C)	K5j (D)
E +	%	20	15	40	16.6	15.7
P +	Wp	370	280	150	270	255
T +	%	10	10	13	10	13
S −	m2	1.77	1.94	0.512	1.62	1.6
C −	EUR/Wp	0.34	0.21	9.55	0.51	0.45
L −	%	14	14	14	14	14

, the matrix of characteristics K_ij was calculated from the data in Table 11.

• Step 3

The coefficients a_ij (0, 1, 2, 4) from

Table 13. The weight of the characteristics γ_j.

(j) ↓ →(i)	E +	P +	T +	S −	C −	L −	$\sum _{\mathit{j}2}{\mathit{a}}_{\mathit{j}1\mathit{j}2}$ (∑)	γj
E +		0	4	1	0	0	5	0.125
P +	2		0	1	0	0	3	0.075
T +	0	2		0	0	0	2	0.05
S −	1	0	4		1	1	7	(−) 0.175
C −	4	2	2	1		1	10	(−) 0.25
L −	4	4	2	1	2		13	(−) 0.325
					${\sum }_{\mathit{j}1}{\sum }_{\mathit{j}2}\left({\mathit{a}}_{\mathit{j}1}\right)\left({\mathit{a}}_{\mathit{j}2}\right)=40$

have been assigned, according to Relationship (6), from the weight we attribute to the influence of the quality characteristics K_ij on the product compared two by two, as presented in Table 12.

In Table 13, the matrix/table a_j was built. The weight of the characteristics γ_j was calculated according to Relation (7):

• Step 4

In the end, the absolute technical level, Ha_i, was calculated according to Relationships (2,3), considering the H_REF = 1000:

Ha_A ≈ 1045

(9)

Ha_B ≈ 549

(10)

Ha_C ≈ 865

(11)

Ha_D ≈ 904

(12)

• Step 5

The relative technical level, Hr_i, shall be calculated in [%] from H_REF = 1000, using Relation (4):

Hr_A = 104.5%

(13)

Hr_B = 54.9%

(14)

Hr_C = 86.5%

(15)

Hr_D = 90.4%

(16)

Result: Hr_A > H_REF > Hr_D > Hr_C > Hr_B.

Following this analysis, the best-performing PV power plant is Stupini–Brasov (Id 2), followed by Id 1 (reference), Id 5, Id 4, and Id 3, respectively.

5. Discussion

When assessing a medium-to-high-power PV plant, the initial investment, projected efficiency, and parameters of the desired plant are correlated, and sometimes, a cheaper and less efficient power plant can be recommended. For a more comprehensive analysis, the paper presents two methods, ANOVA and STEM, which are derived from quality control.

Regarding statistical methods, the PVGIS platform, which contains input and output data, was presented to predict electricity generation for a PV power plant based on geographical and meteorological parameters, as well as the technical data of the panels used. Based on the data provided by the PVGIS platform, an analysis is conducted to examine the impact of panel tilt angle on electricity production.

Cases A and B prove that using a fixed-tilt panel for generating maximum annual energy amount (as provided through PVGIS) is about 37°, but this is not the best solution because there are substantial variations in the monthly production.

By looking for the criterion of three consecutive months, for the case of latitude 45°, an optimal seasonal tilt angle would be the following:

• 30° for June, July, and August;
• 50° for September, October, and November;
• 70° for December, January, and February;
• 40° for March, April, and May.

From Table 3, the annual energy production for the optimum tilt angle of 37° is 1327 MWh, while the sum of energy with seasonal tilt angle is 1354 MWh. Seasonal activities to modify the tilt angle are not economically justified unless they can be incorporated into periodic maintenance procedures.

Solar tracking systems, such as single-axis or dual-axis trackers, can increase energy production by 28% or 31% compared to fixed panels. However, they are more complex, require maintenance, and may not be cost-effective for all applications.

Next, the economic efficiency of five different PV power plants was analysed using the STEM statistical method, based on the data presented in the bibliography. The technical and economic parameters considered in this analysis are the declared efficiency of the photovoltaic panels, the power and panel surface according to the catalogue data, the specific cost of a panel (EUR/Wp), the estimated losses, and the presence or absence of a tracking system. The data analysis showed that the most efficient are those with panels with monocrystalline Si technology without a tracking system. Although from a technical point of view, the PV power plant with HCPV technology (utilising a concentrator and tracking system) is the most efficient, from an economic perspective, it is surpassed by the other four systems. Thus, as a perspective for applying the STEM statistical method, an additional parameter should be introduced regarding the effective cost of the land rented or purchased for a PV power plant to highlight the adequate savings obtained by applying HCPV technology.

Monocrystalline silicon panels have been considered the most economically efficient solution, especially for medium-to-small-scale projects, due to their high efficiency, long lifespan (25–30 years), and high performance per square meter.

Although tracking systems can enhance energy output, various studies have identified key factors that may limit their overall financial viability. These include the significantly higher initial investment costs (estimated to increase total system expenditures by 25–40% due to the addition of motors, gears, sensors, and control software), as well as increased risk of mechanical failure, particularly in environments with ice, sand, or heavy precipitation that can impair tracking functionality. Furthermore, the system’s energy consumption, while relatively low, also contributes to operational costs.

In addition, tracking the sun is minimally beneficial in high-latitude regions or areas with frequent cloud cover. For example, during Romania’s cold and cloudy months, photovoltaic energy production can drop below 10%, even when using tracking systems, as only diffuse light is available.

Trackers require more space between rows to avoid shadowing as they move, reducing land-use efficiency.

Considering all the mentioned factors, the conclusion was that adding extra production capacity to a photovoltaic system (more panels) was more cost-effective than increasing efficiency with trackers.

The value of the STEM method lies in its flexibility to incorporate various technical and economic parameters. For instance, S1 represents a value directly proportional to the quality of the product, while S2 denotes a value inversely proportional to the product’s quality.

6. Conclusions

The statistical analysis, presented using the ANOVA method and the Kruskal–Wallis test, aims to determine the influence of an optimal tilt angle on the season. From the study of the data presented in Table 5, it was found that for May, July, and September (as examples), there is a significant difference in the influence of the tilt angle of the panels. If the time parameter analysed is restricted to the summer months of June, July, and August (Table 8), it was found that there is no significant difference regarding the tilt angle of the panels. Thus, it can be concluded that an alternative method could be to abandon tracking systems and build PV power plants with groups of panels oriented from the beginning under the optimal angle corresponding to each season for systems in temperate areas with four seasons.

The novel aspect of this proposal is the combination of specific data obtained from the PVGIS platform with a new approach to optimising electricity production in photovoltaic systems based on geographical coordinates. Instead of using costly tracking orientation systems that require elaborate maintenance, the proposed method suggests dividing the photovoltaic modules into groups. Each group would have a different tilt angle optimised for a specific season.

The STEM statistical method presents a high degree of novelty because, although it is a well-known method, it has not yet been applied to analyse photovoltaic systems’ technical and economic efficiency. One of the main advantages of this method is its ability to incorporate a wide range of technical and economic performance parameters from both the S1 and S2 categories, as illustrated. By applying this method, we can significantly expand the scope for statistical analysis of the investment efficiency of photovoltaic systems.

The statistical analysis leads to a conclusion consistent with the known situation worldwide. Data analysis indicated that, under the conditions considered in this study, the most economically efficient PV power plants use monocrystalline silicon modules and do not employ tracking systems. By correlating data from the PVGIS platform with findings from existing literature, this study aims to provide a clearer perspective to support investment decision-making, particularly by minimising uncertainties related to the strong dependency on various external factors. The analysis revealed that, in most cases, increasing the installed capacity of a photovoltaic system (by sizing a larger array) is more cost-effective than improving efficiency through the use of tracking systems. Nevertheless, solar trackers remain a viable option in specific contexts. This article also discusses the limitations encountered, particularly those associated with the availability and reliability of validation data.