Algorithm for Rapid Estimation of the Performance of Small Rooftop Solar PV Use by Households

Abstract

The purpose of the research presented in this paper was to develop an analytical instrument for measuring the efficiency of investing in renewable energy sources suitable for use by the average single-family homeowner. The paper develops an algorithm to quickly estimate the efficiency of small rooftop solar photovoltaic (PV) systems use by households. The algorithm is developed on the basis of the MS Excel software package. It is easy to use and allows estimation of the maximum efficiency of the installation of a photovoltaic system, taking into account the preferences of the household, the technical characteristics of the photovoltaic system, and the parameters of the building and location. The validation of the algorithm was carried out on the example of Opole Province (Poland). The selection of photovoltaic panels is based on 17 types of panels made in different technologies, which allows rational choice of modules based on consumer preferences according to the price/quality ratio. The practical use of the developed application should increase the level of informed consumer decision-making in the process of designing a photovoltaic installation and concluding a contract with the installation company.

1. Introduction

The development of the modern world is heading towards the intensification of digitalization processes in all aspects of socio-economic human life, resulting in a significant increase in the demand for electricity. According to the International Energy Agency (IEA), in 2021 alone, there was a 6% increase in global demand for electricity compared to the previous year. For the next three years, the IEA forecasts demand to increase by another 7.6%, meaning that by the beginning of 2025, global electricity consumption will have increased by nearly 14% compared to 2020 [1]. Furthermore, between 2020 and 2050, global electricity demand is expected to increase by 80% [2].

Most countries in the world have found it effective, efficient and necessary to increase the amount of electricity produced from renewable energy sources: bioenergy, geothermal energy, hydropower (gravitational energy), solar energy and wind energy. The European Union in the perspective of 2030 has recognized, as the most important goal, to achieve 40% of renewable energy sources in its energy mix [3].

The level of utilization of renewable energy sources and plans for the future depend on the specificity and structure of the national economy of each country. Thus, according to the objectives of Poland’s Energy Policy until 2040, the share of renewable energy sources in gross final energy consumption in 2030 should amount to 23% [4]. Such dynamics is also reflected in the structure of the European Union energy system by installed capacity already in 2020: 483,579 MW—energy using non-renewable sources (50.19%); 479,910 MW—energy using renewable sources (49.81%), including:

• 176,768 MW/18.35%—wind farms;
• 138,935 MW/14.42%—solar power plants;
• 128,943 MW/13.38%—hydro/sea power plants;
• 34,393 MW/3.75%—biofuel energy recovery power plants;
• 896 MW of installed capacity, of which only 0.09% are geothermal power plants [5].

In recent years, photovoltaic (PV) systems have become very popular and have been recognized as one of the most accessible sources of renewable energy, including for households and small businesses [6,7,8]. Examples of exceptions among EU countries that have not opted for solar energy recovery include:

• Sweden (50.11% hydropower, 29.46% wind energy and 16.11% bioenergy, while the installed capacity using solar energy recovery technology is 4.31%);
• Finland (37.24% hydropower, 29.84% bioenergy and 28.43 wind energy, while the installed capacity using solar energy recovery technology is 4.49%);
• Other countries where the share of installed solar capacity is less than 5% include Croatia (2.36%), Ireland (0.86%), and Latvia (0.38%). It is worth noting that in these countries the structure of renewable energy is dominated by hydropower and wind energy.

Accurate prediction of the performance of energy systems from renewable energy sources, including systems based on grid-connected photovoltaic panel technology, is crucial to increase the overall penetration of renewable energy sources in the power grid, and at the same time optimize the efficiency of the operation of such a grid [9].

In order to design a photovoltaic system, it is necessary to have expertise on the subject, which prevents or complicates the responsible and rational behavior of households when deciding to implement such a project. Therefore, installation companies may use unfair practices towards their customers by imposing on them systems that are not exactly adapted to the specific object by using more standardized and cheaper solutions.

According to Polish law, which is subordinate to the EU laws and acts, intentional or unintentional misrepresentation may be considered a misdemeanor (socially harmful act, prohibited by law) (Civil Code, Articles 84, 86, 88) [10]. However, the onus is on the consumer claiming a misrepresentation within the meaning of the legislation to prove as to the error caused (Civil Code, Article 6). This wording of the regulations, combined with the very likely lack of necessary expertise of the average single-family homeowner, does not fully protect the installation company customer from making a potentially irrational decision. An example of a factor causing an erroneous decision may be an overestimation of the efficiency of the photovoltaic system proposed by the company, failure to include the effect of shading, showing as a determining factor the peak power of the installed system compared to the needs of a particular household.

In recent years, many studies have been conducted on the estimation of the energy potential of photovoltaic systems, taking into account various techno-economic factors and local specifics. Such studies include the results of Adaramola M.S. and Vagnes E.E.T. [11], who developed a method and conducted an evaluation of the actual performance of a grid-connected photovoltaic system installed on the roof of a laboratory building at the Norwegian University of Life Science, As, Norway. The evaluation method developed by the authors takes into account the technical characteristics of the photovoltaic panels, the losses of the installed energy system, the structural characteristics of the building and its location. Khatri, R. [12] presented a methodology for the financial and environmental assessment of a photovoltaic power plant installation project for meeting the energy needs of a hostel of MNIT University in Jaipur city.

A technical-economic evaluation of the actual performance of the photovoltaic system installed on the roof of the faculty of the medicine building at An-Najah National University, Nablus, Palestine, was addressed in the work of Ibrik I.H. [13]. A similar study was conductedfour years earlier in Italy [14].

Al-Aboosi F.Y. and Al-Aboosi A.F. [15] proposed a system to analyze the efficiency of grid-connected rooftop photovoltaic systems from technical (energy efficiency and energy loss of photovoltaic panels), economic (return on investment, payback period, averaged cost of energy) and environmental (carbon emissions) perspectives. The researchers’ analysis takes into account the intensity of solar radiation, two uniaxial tracking modes, shadow effect, and the PV cell temperature impact on the performance of the system installed at Texas A&M University.

In order to evaluate the feasibility of a grid-connected PV system on the roof of a residential building in Manit, Bhopal, India, Shukla A.K., Sudhakar, K. and Baredar P. [16] conducted a computational simulation. While modeling the photovoltaic system, the building location parameters and technical characteristics of the photovoltaic system were considered. The authors used Solargis PV Planner software to analyze the performance of the PV system. Odeh S. and Nguyen T.H. [17] developed a method to estimate the potential of residential photovoltaic systems by accurately identifying the usable area of building roofs for the placement of photovoltaic panels. The authors estimated a roof suitability factor to identify the actual usable area of different types of roofs, taking into account obstacles to photovoltaic panel placement such as solar windows, satellite dishes, chimneys, and air conditioning units. For the impetus study, a sample of 400 residential houses from the suburbs included in the Sydney City Council 2030 development plan was selected. Imam A.A., Al-Turki Y.A. and R.S.R. [18] developed a technical-economic model to analyze the performance, cost-effectiveness and feasibility of residential grid-connected solar PV for a typical apartment in Saudi Arabia.

Rodríguez-Martinez Á. and Rodríguez-Monroy C. [19] presented a computational method that offers the possibility to estimate the maximum power of installed photovoltaic panels to be used in manufacturing plants, taking into account technical variables such as building location, slope and roof area, and the technology of the photovoltaic module to be installed. The study was carried out on the example of four manufacturing plants situated at different latitudes in Spain. A similar study regarding the design of the photovoltaic system at Kuantan Airport, Malaysia, was conducted by Sreenath S., Sudhakar, K. et al. [20].

Nguyen D.P.N. and Lauwaert J. [21] attempted to develop a model to calculate the total daily, monthly and annual energy yield of photovoltaic installations considering local weather conditions. The model considers the effect of solar spectrum, time of day and orientation of photovoltaic panels, which provides an estimate of the potential of using tandem solar cells in a specific location. The validation of the model has been carried out on the example of selected regions of Belgium and Vietnam.

Bazrafshan M., Yalamanchili L. et al. [22] developed a stochastic program to minimize the cost of installation and operation of photovoltaic panels and maximize the benefits of distributed energy units by optimizing the location and sizing of photovoltaic systems. In order to determine the most effective technological solution for using renewable energy sources in warm urban areas, Awan A.B. et al. [23] conducted a comparative performance analysis of ground-mounted and rooftop photovoltaic systems. The results showed that rooftop photovoltaic systems perform better from a technical and economic point of view.

Many research papers have been devoted to increasing the potential of photovoltaic systems on the basis of optimizing the location of photovoltaic panels at the appropriate inclination angle with respect to the Earth’s surface. Scientists have developed models and programs to find the inclination angle of solar panels at which photovoltaic cells could absorb maximum energy. The developed models allow the efficiency of photovoltaic systems to be increased, taking into account local specificities [24,25,26,27,28,29,30,31,32,33].

However, none of the analyzed scientific works is addressed to households that are in the process of making a decision regarding the installation of photovoltaic systems.

The main objective of the research presented in thisarticle was to develop an analytical instrument for measuring the efficiency of investing in renewable energy sources useful for use by an average owner of a single-family house.

The paper develops an algorithm for rapid estimation of the efficiency of use of small rooftop photovoltaic systems by households.

The developed algorithm allows for a quick estimation of the maximum efficiency of a photovoltaic system for a household, and comparing this efficiency with the electricity demand, disregarding technical details, but taking into account factors such as:

• Building location according to the southern direction;
• Roof angle of inclination;
• Shape of the roof (monopitch, pitched equal, pitched unequal, hipped);
• Transmission losses of electricity in the installed photovoltaic system based on consumer preferences;
• Selection of optimal photovoltaic panels from the point of view of technology based on customer preference between price and quality;
• Technical data of photovoltaic panels selected for analysis, made in 17 different technologies;
• Monthly and hourly insolation level, kWh/m² for the city of Opole Province (Poland);
• Performance of photovoltaic panels on sunny days, on days with partial cloud cover and on days with heavy cloud cover, estimated based on the adjustment factor developed by the authors;
• Photovoltaic panel performance by month based on average air temperature, estimated on the basis of an adjustment factor developed by the authors;
• The maximum roof area remaining after subtracting the necessary technical clearances between separate rows of PV panels and the clearances from the edge of the roof at the perimeter.

Based on the author’s algorithm, a simulation of photovoltaic installations was carried out in a single-family house cited in Turawa-Marszalki, Opole, Poland.Possible system losses were taken into account when calculating the efficiency of photovoltaic installations.

The main advantages of this method are the selection of the type of solar panel depending on the customer’s preferences, and taking into account basic data on atmospheric conditions in the area of the studied location. The practical use of the developed application should increase the level of informed consumer decision-making in the process of designing a photovoltaic installation, and concluding in a contract with the installation company.

2. Logic of the Developed Author’s Algorithm for Rapid Estimation of the Performance of the Use of Photovoltaic Installations by Households

The selection of photovoltaic panels is based on 17 types of panels made with different technologies, which enables rational selection of panels based on consumer preference according to price/quality ratio. The input data characterizing the selected panels for analysis are presented in

Table 1. Preliminary technical data and background information on the types of photovoltaic panels selected for analysis, made with 17 different technologies.

Type of Photovoltaic Panel Manufacturing Technology	Photovoltaic Panel Performance Indicators				Estimated Net Price Per Wp, (Polish Zloty)	Dimensions, m
	Typical Power, Wp		Typical Efficiency, %	Coefficient of Average Power Change Per 1 °C, %		a	b
	Wp/m2	Wp/Panel
Amorphous	83.33	100	8.30	0.25	1.6	1	1.2
CIGS	134.25	140	13.40	0.38	2.1	0.66	1.58
CdTe	138.89	100	13.90	0.34	2	0.6	1.2
Poly-glass-glass	169.38	275	16.90	0.44	2.2	0.99	1.64
Poly-(classic)	172.46	280	17.20	0.41	1.5	0.99	1.64
Poly-Smart (Solar Edge)	172.46	280	17.20	0.43	2.1	0.99	1.64
Poly-half-cut	176.46	290	17.60	0.38	1.6	0.99	1.66
Mono-(classic)	181.69	295	18.20	0.43	1.8	0.99	1.64
Mono-Smart (Solar Edge)	181.69	295	18.20	0.44	2.2	0.99	1.64
Mono-bifacial	182.93	300	18.30	0.38	3.2	1	1.64
Poly-MWT	184.77	300	18.50	0.36	1.7	0.99	1.64
Mono-glass-glass	184.77	300	18.50	0.45	2.4	0.99	1.64
Mono-PERC	197.09	320	19.70	0.45	1.9	0.99	1.64
Mono-HIT	197.66	330	20.10	0.29	2.8	1.05	1.59
Mono-half-cut	197.76	325	19.80	0.37	1.9	0.99	1.66
Mono-MWT	200.17	325	20.00	0.36	2.1	0.99	1.64
Mono-all back contact	205.25	333	20.50	0.33	3.3	1.04	1.56
Average value	174.18	269.88	17.43	0.38	2.14
Minimum value	83.33	100	8.30	0.25	1.50
Maximum value	205.25	333	20.50	0.45	3.30

CIGS—copper indium gallium selenide solar cell, CdTe—Cadmium telluride (CdTe) photovoltaics, MWT—Metal Wrap Through, PERC—Passivated Emitter & Rear Cell (or Contact), HIT—Heterojunction with Intrinsic Thin layer.

, according to the ranking “Typical power, W_p/m²”.

The minimum power of photovoltaic panels considered is 83.33 W_p/m², this is for panels using amorphous modules. Generally, these are the panels with low efficiency (average 8.3%), while they have advantages such as low weight, low response to temperature changes, smaller dimensions, and one of the lowest prices among the analyzed panels per W_p. The average efficiency of the modules in the set is 174.18 W_p/m² of the initial installed power. According to the list, the most expensive and also the most efficient are modules made in Mono-all back contact technology (average output thanks to this technology is 205.25 W_p/m² and efficiency is 20.5%, respectively).

The basic element necessary for the selection of photovoltaic panel technology depending on customer preference is to determine the ratio of preferred quality with respect to preferred price, using the following formula:

z = x + y = 100%; x = z – y = 100% − y; y = z – x = 100% − x

(1)

where

x: preference level of photovoltaic panel with lower prices (determined by the customer), %;

y: preference level of higher quality photovoltaic panel (determined by the customer), %;

z: overall preference level, %.

The next element of the developed algorithm is the determination by the customer of the level of preference of the system efficiency with respect to the need to install a photovoltaic system with resistance to temperature changes (if the investment is more intended for the summer period, generally in hot days for air conditioning). Proposed values for year-round use of the photovoltaic system for an efficiency preference of no less than 50%. These two preference coefficients are related to the Y-index, namely the level determining the preference of quality over price. The calculation is performed using the following formula:

f + s = 100%; f = 100% − s; s = 100% − f

(2)

where

s: level of preference for efficiency of the photovoltaic installation over resistance to high temperatures in summer (hot days), %;

f: level of preference of resistance of the photovoltaic installation to high temperatures during the summer period (hot days) over the installation efficiency factor, %.

It is also worth noting that based on the simulations performed, the impact of ambient temperature on the performance of photovoltaic panels in the analyzed location (Opole Province, Poland) is quite significant in certain months, so as noted above, it is not recommended to reduce the preference of efficiency over temperature by more than 50%. The basic formula for determining the temperature reached by a photovoltaic panel under certain conditions of solar irradiance and ambient temperature, using the NOCT (Normal Operating Cell Temperature) parameter, is as follows [34]:

$$\mathrm{Tcell}=\mathrm{Tambient}+\frac{\left(\mathrm{NOCT}-20\right)\ast \mathrm{E}}{800}$$

(3)

where

Tcell: the temperature reached by the photovoltaic cells in the photovoltaic panel under defined conditions of insolation and ambient temperature;

NOCT:Normal Operating Cell Temperaturein degrees Celsius according to the characteristics of the cell;

E: insolation level, W/m².

In modifying this formula for the purposes of the study, the following assumptions were used:

• First, given that the NOCT parameter is almost always reported for selected photovoltaic panels with a tolerance of plus or minus 20 °C, and NOCT values are in the range of 43 °C to 45 °C, the NOCT factorc_onst—45 °C was considered standard;
• Second, the insolation level E wastaken as the average insolation level in a specific i-th month (E_i);
• Third, T_ambient based on statistical data was determined at the level of T₁ = 25, T₂ = 30 oraz T₃ = 35 (ambient temperature values below 20 °C were eliminated by determining the proportion of days in the i-th month within specific temperature ranges U_ti).

After taking into account the above modifications, a formula was proposed to determine an adjustment factor for the effect of temperature on the performance of photovoltaic panels in each month, which, among other things, takes into account the average temperature power factor P_max:

$${\mathrm{TEPM}}_{\mathrm{i}}=25\frac{(45-{20)\mathrm{E}}_{\mathrm{i}}}{800}{\mathrm{U}}_{1\mathrm{i}}{\mathrm{P}}_{\max }+30\frac{(45-{20)\mathrm{E}}_{\mathrm{i}}}{800}{\mathrm{U}}_{2\mathrm{i}}{\mathrm{P}}_{\max }+35\frac{(45-{20)\mathrm{E}}_{\mathrm{i}}}{800}{\mathrm{U}}_{3\mathrm{i}}{\mathrm{P}}_{\max }$$

(4)

In selecting the least efficient photovoltaic panels (of those selected for analysis in this study) from the point of view of the effect of ambient temperature and insolation level on the performance of these panels, namely K5 (Mono-PERC technology) and K7 (Mono-glass-glass technology), for which the temperature power index P_max is 0.45, the adjustment factor of the ambient temperature effect according to each month is shown in

Table 2. Adjustment factor for the effect of temperature in each month with P_max = 0.45.

Month	Proportion of Days in Specific Temperature Ranges, %			Average Level of Insolation (Ei), W/m2	Proportion of Days in Specific Temperature Ranges, %
	U1 (Ambient Temperature above 35 °C)	U2 (Ambient Temperature from 30 °C to 35 °C)	U3 (Ambient Temperature from 25 °C to 30 °C)
January	0.000	0.000	0.000	190.6	1.00000
February	0.000	0.000	0.000	304.9	1.00000
March	0.000	0.000	0.645	374.4	0.99893
April	0.000	1.667	13.667	536.6	0.97080
May	0.323	11.935	33.871	539.3	0.91029
June	4.333	21.333	40.333	548.7	0.86808
July	11.401	34.853	36.482	558.3	0.82899
August	12.052	35.505	37.459	511.6	0.82978
September	0.333	15.000	32.667	440.0	0.91278
October	0.000	1.929	18.328	338.4	0.96714
November	0.000	0.000	0.333	250.8	0.99951
December	0.000	0.000	0.000	192.8	1.00000

. From the results of the calculations, it can be seen that the negative effect of ambient temperature occurs in the period from March to November (while in March and November this effect is not significant and amounts to a reduction in the performance of photovoltaic panels of less than 1%). On the other hand, more significant changes in performance occur in the months of July–August, when only due to high temperatures a decrease in the performance of photovoltaic panels by almost 18% is predicted. Conducted author’s simulation results give reason to conclude that, in the case of the most efficient, from this point of view, with photovoltaic panel K17 (Amorphous) it is possible to reduce losses in electricity production resulting from heating the photovoltaic panel by almost two times (in the months of July–August, losses resulting from the influence of ambient temperature should not exceed 10% of power) (

Table 3. Adjustment factor for the effect of temperature in each month with P_max = 0.25.

Month	Proportion of Days in Specific Temperature Ranges, %			Average Level of Insolation, W/m2	Adjustment Factor
	U1 (Ambient Temperature above 35 °C)	U2 (Ambient Temperature from 30 °C to 35 °C)	U3 (Ambient Temperature from 25 °C to 30 °C)
January	0.000	0.000	0.000	190.6	1.00000
February	0.000	0.000	0.000	304.9	1.00000
March	0.000	0.000	0.645	374.4	0.99941
April	0.000	1.667	13.667	536.6	0.98378
May	0.323	11.935	33.871	539.3	0.95016
June	4.333	21.333	40.333	548.7	0.92671
July	11.401	34.853	36.482	558.3	0.90499
August	12.052	35.505	37.459	511.6	0.90543
September	0.333	15.000	32.667	440.0	0.95154
October	0.000	1.929	18.328	338.4	0.98174
November	0.000	0.000	0.333	250.8	0.99973
December	0.000	0.000	0.000	192.8	1.00000

).

Thus, the overall matching factor of a photovoltaic panel to customer preference can be written as follows:

$${\mathrm{w}}_{\mathrm{i}}= \mathrm{x}\ast \frac{{\mathrm{a}}_{\mathrm{i}}}{{\mathrm{a}}_{\max }}+\mathrm{ys}\ast \frac{{\mathrm{b}}_{\mathrm{i}}}{{\mathrm{b}}_{\max }}+\mathrm{yf}\ast \frac{{\mathrm{c}}_{\mathrm{i}}}{{\mathrm{c}}_{\max }}$$

(5)

where

w_i: the matching factor of the i-th technology to customer preferences;

x: lower priced photovoltaic panel preference level (determined by customer), %;

a_i: number of points granted according to the ranking in the “Price” category for the i-th type of photovoltaic panels;

a_max: maximum number of points granted in the “Price” category;

y: the level of preference for a higher quality photovoltaic panel (determined by the customer), %;

b_i: number of points granted according to the ranking in the category “Temperature” for the i-th type of photovoltaic panels;

b_max: maximum number of points granted for the “Temperature” category;

f: the level of preference for the resistance of the photovoltaic installation to high temperatures in summer (hot days) over the efficiency of the installation, %;

c_i: number of points granted according to the ranking in the “Efficiency” category for the i-th type of photovoltaic panels;

c_max: the maximum number of points granted in the “ Efficiency” category.

After entering the data specifying the preferences of a particular customer, we can proceed to the procedure of selecting a specific photovoltaic panel for further analysis. At this stage, we analyze the technical data of photovoltaic panels in terms of their ranking (the better the rated panel looks in comparison with competing panels, the more points are granted). The comparison of panels according to selected parameters and their scores are presented in the matrices shown in Figure 1, while the abbreviations shown in Figure 1 are given in

Table 4. Codification of types of photovoltaic (PV) panel manufacturing technologies.

Abbreviation	Type of Photovoltaic Panel Manufacturing Technology
K1	Mono-all back contact
K2	Mono-HIT
K3	Mono-MWT
K4	Mono-half-cut
K5	Mono-PERC
K6	Poly-MWT
K7	Mono-glass-glass
K8	Mono-bifacial
K9	Mono-(classic)
K10	Mono-Smart (Solar Edge)
K11	Poly-half-cut
K12	Poly-(classic)
K13	Poly-Smart (Solar Edge)
K14	Poly-glass-glass
K15	CdTe
K16	CIGS
K17	Amorphous

.

According to the results of the study, three EPT (“Efficiency/Price/Temperature”) matrices were constructed. The first EP (“Efficiency/Price”) matrix shows the competitive positioning of the individual photovoltaic panel technologies (Figure 1A) with respect to price and efficiency.

In general, the efficient option can be considered technologies where, for a relatively low price of W_p generation, one can get more or less good efficiency on the level of some more expensive models. Indeed, we can tentatively consider the technologies in the upper right part of the matrix, namely K6 (Poly-MWT), K5 (Mono-MWT) and K4 (Mono-half-cut) technologies as the most favorable option. The most unprofitable from this point of view is the technology lying within the lower left part of the matrix: K14 (Poly-glass-glass), which, with relatively high costs, is not characterized by competitive efficiency. Such a matrix gives an opportunity to quickly compare the technologies selected for analysis and, if necessary, make a rational decision. Therefore, if the decision is between K8 and K7, the rational choice is K7, which is more efficient and cheaper than K8.

When analyzing the PT (“Price/Temperature”) matrix, the more temperature tolerant technologies have a price above average (upper left part of the matrix) (Figure 1B). A relatively safe solution with a limited budget is to choose K15 (CdTe) or K6 (Poly-MWT) technology. In the case of a very limited budget and use of the photovoltaic system, mostly in the summer during hot days, it is worth paying attention to the K17 technology, namely amorphous photovoltaic panels, which are characterized by high resistance to changes in ambient temperature and a relatively low price, which consequently translates into very low efficiency of photovoltaic panels of this type.

When analyzing the ET (“Efficiency/Temperature”) matrix (Figure 1C), the technologies located in the upper right part of the matrix—K1 (Mono-all back contact), and K2 (Mono-HIT)—look best compared to their competitors. In the case of year-round use of the designed system and taking into account the geographical location of Poland, it is worth paying attention also to all technologies located in the upper right part of the matrix, which are characterized by good efficiency, while their resistance to temperature is above average.

The technologies at the bottom left of the matrix look the worst compared to the competition, respectively. These include: K16 (CIGS), K14 (Poly-glass-glass), K12 (Poly-classic), K13 (Poly-Smart/Solar Edge) and K11 (Poly-half-cut).

3. Case Study

3.1. Simulation of Photovoltaic Panel Technology Selection Based on Different Customer (Prosumer) Preferences

Option 1: Prosumer’s choice: x = 0, y = 100, f = 100, s = 0.

In this case, our prosumer prefers only the quality of the components used in the design and implementation of the photovoltaic system. Based on the following ranking, the developed proprietary algorithm to quickly estimate the performance of photovoltaic panel installation will suggest the prosumer to use panels with “Mono-all back contact” technology. The whole ranking under suchassumptions is provided in Figure 2.

Option 2: Prosumer’s choice: x = 50, y = 50, f = 50, s = 50.

Such assumptions can be made when one prefers an effective system from the price/quality point of view and plans to use the installation as a year-round renewable energy source. According to the calculation results, the developed algorithm will advise the prosumer to use poly-MWT photovoltaic cells (Figure 3).

Option 3: Prosumer’s choice: x = 0, y = 100, 0 ≤ f ≤ 100, 0 ≤ s ≤ 100.

Based on the results of possible simulations, in a situation where the customer focuses only on price and makes it the only deciding factor, the developed algorithm after calculations will indicate as the best rational option photovoltaic panels with Poly(classic) technology; the second place in the ranking will be for photovoltaic panels with Poly technology with cells cut in half (Figure 4).

3.2. Gathering Technical Data about the Property Necessary to Estimate the Maximum Possible Electricity Generation from the Photovoltaic System

In addition to the discussed preferences, the user of the application is asked to provide:

• Necessary technical data of the property (

  Table 5. Basic technical data of the property.

  Roof: Monopitch *
  	Length, m	Roof slope
  		Does the roof slope in this direction? (Yes/No)	Provide the angle of inclination?
  Front of the construction
  Right side of the construction
  Left side of the construction
  Rear wall
  Roof: pitched equal *
  	Length, m	Roof slope
  		Does the roof slope in this direction?	Provide the angle of inclination?
  Front of the construction
  Right side of the construction
  Left side of the construction
  Rear wall
  Roof: pitched unequal *
  	Does the roof slope in this direction? (Yes/No)	Provide the angle of inclination?	Indicate the length of the wall to which the roof is sloped, m
  Front of the construction
  Right side of the construction
  Left side of the construction
  Rear wall
  Indicate the corresponding lengths on the left or front wall of the building, m
  from the right corner of the wall to the ridge		from the left corner to the ridge
  Roof: hipped
  		Indicate the length, m	Provide the angle of inclination?
  Front of the construction
  Right side of the construction
  Left side of the construction
  Rear wall

  * Fill in only the parts of the table filled in with color.

  );
• Data regarding the location of the front wall of the property in relation to the southern direction;
• Annual electricity demand in kWh.

The second element is an indication of the location of the front wall of the property with respect to the southern direction (z).

In order to describe the mechanism of calculating the angle of inflection from the southern direction of particular walls of the building and at the same time the roof surface, the following variables were introduced: a—angle of inflection from the south of the front wall of the building; b—angle of inflection from the south of the right wall of the building; c—angle of inflection from the south of the left front wall: d—angle of inflection from the south of the rear wall of the building; z—direction of the front wall position.

With these findings, z coefficient should be in the next range:

$$0\le \mathrm{z}\le 360$$

(6)

Therefore,

Front wall (a)

$$\left\{\begin{array}{c}z\ge 0\\ z\le 180\end{array}\to a=z,\right. \mathrm{if} \left\{\begin{array}{c}z\le 360\\ z>180\end{array}\to a=360-z\right.$$

(7)

In other words, in our case we can restrict ourselves to the following conditions:

$$\mathrm{z}\le 180\to \mathrm{a}=\mathrm{z}\vee \mathrm{z}>180\to \mathrm{a}=360-\mathrm{z}$$

(8)

Right wall (b)

$$\mathrm{if} \mathrm{z}>180\wedge \mathrm{a}-90\ge 0\vee \mathrm{z}\le 180\wedge \mathrm{a}+90\ge 0,$$

then if

$$\mathrm{a}>180\Rightarrow \mathrm{b}=\mathrm{a}-90 \mathrm{and} \mathrm{a}\le 180\Rightarrow \mathrm{b}=\mathrm{a}+90$$

(9)

if

$$\mathrm{z}>180\wedge \mathrm{a}-90<0 \vee \mathrm{z}\le 180\wedge \mathrm{a}+90<0.$$

Then if

$$\mathrm{a}>180⟹\mathrm{b}=\left(\mathrm{a}-90\right)\ast -1,$$

(10)

if

$$\mathrm{a}\le 180⟹\mathrm{b}=\left(\mathrm{a}+90\right)\ast -1$$

Left wall (c)

If

$$\mathrm{z}>180\wedge \mathrm{a}+90\ge 0\vee \mathrm{z}\le 180\wedge \mathrm{a}-90\ge 0$$

Then if

$$\mathrm{a}>180⟹\mathrm{c}=\mathrm{a}+90$$

(11)

if

$$\mathrm{a}\le 180⟹\mathrm{c}=\mathrm{a}-90$$

if

$$\mathrm{z}>180\wedge \mathrm{a}+90<0 \vee \mathrm{z}\le 180\wedge \mathrm{a}-90<0.$$

Then if

$$\mathrm{a}>180⟹\mathrm{c}=\left(\mathrm{a}+90\right)\ast -1$$

(12)

if

$$\mathrm{a}\le 180⟹\mathrm{c}=\left(\mathrm{a}-90\right)\ast -1$$

Rear wall (d)

$$\mathrm{d}=180-\mathrm{a}$$

(13)

The annual demand for electricity in kWh is the third element necessary for further calculations. After providing the above data, the algorithm starts the calculation in order to estimate the maximum possible electricity generation from the photovoltaic installation.

3.3. Main Computations of the Algorithm

Step one. Extracting from the database the average daily insolation level of the horizontal surface (A) provided in W/m²/hour in i-th months (

Table 6. Average daily insolation level of the horizontal surface (A), W/m²/h *.

Months	Time
	0–5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20–24
January						141	165	165	139
February					170	225	255	257	230	178	108
March			110	199	277	335	335	334	333	274	194	105
April			123	220	313	391	446	472	736	430	336	282	186
May		104	197	294	385	460	511	534	527	490	427	344	249	152
June		129	222	317	406	480	531	556	551	517	458	378	286	190	100
July		108	199	295	385	461	515	542	540	508	451	372	280	183
August			145	242	333	411	466	493	490	457	396	315	221	124
September				169	258	332	383	406	398	360	295	212	120
October					168	235	279	297	284	243	197	102
November					139	178	193	181	145
December						126	144	137	107

* Own elaboration based on the report developed by ScanTheSun application on android smartphone. Link to the application: https://play.google.com/store/apps/details?id=com.scanthesun&hl=pl&gl=US (accessed on 9 March 2022).

).

Step two. Correction of insolation level according to prevailing weather conditions based on the number of sunny days, partly cloudy days, and highly cloudy days. The author’s approach is to try to combine this adjustment factor with customer (prosumer) preferences.

Statistical data about the average number of days and the level of cloudiness in particular months for the towns of Opole Province, Poland, are given in

Table 7. Historical data of average number of days by cloud cover level and months, days *.

Month	Sunny (xi), Days	Partly Cloudy (vi), Days	Highly Cloudy (zi), Days
January	3.2	10.3	17.5
February	2.4	10.7	15.2
March	4.0	14.5	12.5
April	6.4	15.1	8.5
May	6.8	16.7	7.5
June	5.4	17.9	6.7
July	8.4	17.5	5.1
August	8.8	16.5	5.6
September	7.6	13.9	8.5
October	7.9	13.7	9.4
November	4.8	11.6	13.5
December	3.3	10.6	17.0

* Own elaboration based on data from Institute of Meteorology and Water Management National Research Institute. Link to website: https://klimat.imgw.pl/pl/climate-normals/TSR_AVE (accessed on 11 April 2022).

.

The adjustment factor for cloud cover and user preference is as follows:

for sunny days:

β = const = 1

(14)

number of sunny days after correction (x_ij) will amount to:

x_ij = βx_i = x_i

(15)

for days with partial cloud cover:

$$\mathsf{\gamma }=5+20\left(1-\mathrm{y}\right)$$

(16)

where $\mathsf{\gamma }$ is the preference level of the higher quality photovoltaic panel (determined by the customer), therefore:

$${\mathrm{v}}_{\mathrm{ij}}=\frac{\mathsf{\gamma }}{100}{\mathrm{v}}_{\mathrm{i}}$$

(17)

for days with high cloud cover:

$$\mathsf{\alpha }=20+80\left(1-\mathrm{y}\right)$$

(18)

where y is the preference level of the higher quality photovoltaic panel (determined by the customer), therefore:

$${\mathrm{z}}_{\mathrm{ij}}=\frac{\mathrm{a}}{100}{\mathrm{z}}_{\mathrm{i}}$$

(19)

Step three. Correction of the amount of energy produced by the photovoltaic system due to the ambient temperature as a factor that changes the performance and efficiency of photovoltaic panels. In order to perform the calculations, the historical data on the number of days in consecutive months when the temperature (T, °C) is within certain ranges (nine temperature ranges) were taken into account (

Table 8. Number of days with specific temperature ranges occurring in each month, days.

	Range, °C	30 ≤ T	25 ≤ T < 30	20 ≤ T < 25	15 ≤ T < 20	10 ≤ T < 15	5 ≤ T < 10	0 ≤ T < 5	−5 ≤ T < 0	T < −5
Months
January						1.2	7.1	12.8	7.8	1.8
February					0.2	2.8	7.7	10.3	5.8	1.2
March				0.2	2.9	8.6	10.7	6.5	2.0	0.1
April			0.5	4.1	10.2	9.7	4.5	1.0
May		0.1	3.7	10.5	11.6	4.7	0.4
June		1.3	6.4	12.1	8.7	1.5
July		3.5	10.7	11.2	5.2	0.1
August		3.7	10.9	11.5	4.3	0.3
September		0.1	4.5	9.8	11.1	4.2	0.3
October			0.6	5.7	9.2	10.6	4.6	0.4
November				0.1	2.3	8.5	11.2	6.9	0.9	0.1
December						1.4	8.8	14.4	5.6	0.7
Total, days		8.7	37.3	65.2	65.7	53.6	55.3	52.3	22.1	3.9
Share of days per year, %		2.39%	10.24%	17.91%	18.04%	14.72%	15.19%	14.36%	6.07%	1.07%

).

According to the historical data, it can be stated that in the analyzed territory (Opole Province, Poland), on average, 2.39% or on 8.7 days there is heat above 30 °C; on average, 10.24% or on 37.3 days the temperature is kept within 25 °C to 30 °C; and 17.91% or on 65.2 days the temperature is within 20–25 °C. Considering that the technical characteristics of the photovoltaic panel are provided according to the international STC standard for an insolation level of 1000 W/m² and at a temperature of 25 °C, it can be concluded that 28.15% of the time the installed photovoltaic system will operate with a lower efficiency level due to heating of the photovoltaic panel elements (it is also important that this temperature falls in the spring-autumn season (from March to November). It is also worth noting that the calculations take into account not only the characteristics of the panel according to STC standard, but also the characteristics of the photovoltaic panel according to NOCT standard, which should increase the level of efficiency and reliability of the calculations performed.

The effect of temperature on the performance of the entire installed photovoltaic system is determined for each of the analyzed technology types and depends on the temperature coefficient that determines the power loss in percentage for a temperature deviation from 25 °C by each 1 °C (temperature coefficient of the open circuit voltage (Voc)). The analysis of each technology type relative to the competing technology is illustrated in Figure 1B,C.

Example adjustment factors for each month and by selected PV panels using different technologies will be as follows (

Table 9. Adjustment factors for the amount of energy generated by photovoltaic panels depending on changes in ambient temperature.

Months	Type of Photovoltaic Panel Manufacturing Technology
	K1 “Mono-All Back Contact”	K3 “Mono-MWT”	K6 “Poly-MWT”	K4 “Mono-Half-Cut”	K12 “Poly-(Classic)”
January	0.743	0.720	0.720	0.712	0.681
February	0.779	0.759	0.759	0.753	0.726
March	0.809	0.791	0.791	0.786	0.762
April	0.887	0.877	0.877	0.873	0.860
May	0.938	0.932	0.932	0.931	0.923
June	0.968	0.965	0.965	0.964	0.960
July	0.995	0.995	0.995	0.994	0.994
August	0.997	0.997	0.997	0.997	0.996
September	0.944	0.939	0.939	0.937	0.930
October	0.889	0.879	0.879	0.875	0.862
November	0.816	0.799	0.799	0.793	0.771
December	0.754	0.731	0.731	0.724	0.694
Monthly average	0.876	0.865	0.865	0.862	0.847

).

The data in Table 9 show that among the five types of technologies selected for comparison from the 17 technologies analyzed in this application, the best coping with changes in ambient temperature in the conditions prevailing on the territory of Opole Province are the panels made in K1 technology, and the worst, respectively, are the panels made in K12 technology.

Step four. Correction by efficiency of photovoltaic panels based on technical data and considered as average performance (typical performance) of analyzed technologies. The adjustment factor is given in Table 1 (column “Typical efficiency, %”).

Step five. Correction of the amount of generated energy taking into account losses in the photovoltaic system. According to the conducted research, the minimum losses in the energy system are 10%, and the acceptable limit with cheaper components (inverter, wires, batteries (if any) and others) can cause losses up to 30% and more.

In order to simplify the procedure for estimating the performance of photovoltaic installations by households, an adjustment factor for the amount of losses was assumed in the algorithm depending on the determination of the level of user preference for quality over price. This is justified by the fact that, theoretically, the higher the preference for quality over price, the more precisely the whole photovoltaic system will be designed, and the better-quality components will be selected. The intra-system loss adjustment factor is calculated based on the following formula:

$$\begin{gathered} W_{\mathrm{adjustment}}=1-a-b\frac{100-y}{100} \\ \mathrm{a}=0.1 \\ \mathrm{b}=0.2 \end{gathered}$$

(20)

where

W_adjustment: the factor that corrects the amount of generated electricity by the amount of losses associated with the transmission of energy within the photovoltaic system and with the losses that depend on the efficiency of the selected components;

a: minimum amount of losses in the system;

b: the part of losses depending on the quality of selected elements;

y: level of customer preference for higher quality over price (determined by the customer), %.

Step six refers to the correction of the amount of energy recovered by the photovoltaic system depending on the inclination angle of the installation and the angle of its inflection from the south direction. The adjustment factors used are presented in

Table 10. Adjustment factors for the amount of energy recovered by a photovoltaic system as a function of the inclination angle of the installation and the angle of its inflection from the south. The color marks the height of the correction factor from “green” the smallest to “red” the greatest.

Degrees		Angle of Inflection from the South
		90	85	80	75	70	65	60	55	50	45	40	35	30	25	20	15	10	5	0
Angle of inclination	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
	5	1	1	1	1.01	1.01	1.01	1.02	1.02	1.02	1.03	1.03	1.03	1.03	1.03	1.03	1.03	1.04	1.04	1.04
	10	0.99	1	1.01	1.01	1.02	1.02	1.03	1.04	1.04	1.05	1.05	1.05	1.06	1.06	1.06	1.06	1.07	1.07	1.07
	15	0.98	0.99	1	1.01	1.02	1.03	1.04	1.05	1.05	1.06	1.07	1.07	1.08	1.08	1.09	1.09	1.09	1.09	1.1
	20	0.97	0.98	1	1.01	1.02	1.03	1.04	1.05	1.06	1.07	1.08	1.09	1.09	1.1	1.1	1.11	1.1	1.11	1.11
	25	0.96	0.97	0.99	1	1.02	1.03	1.05	1.06	1.07	1.08	1.09	1.1	1.1	1.11	1.12	1.12	1.12	1.12	1.13
	30	0.94	0.96	0.98	1	1.01	1.03	1.04	1.06	1.07	1.08	1.09	1.1	1.11	1.12	1.12	1.13	1.13	1.13	1.14
	35	0.93	0.95	0.97	0.99	1	1.02	1.04	1.05	1.07	1.08	1.09	1.1	1.11	1.12	1.12	1.13	1.13	1.14	1.15
	40	0.91	0.93	0.95	0.97	0.99	1.01	1.03	1.04	1.06	1.07	1.09	1.1	1.1	1.11	1.12	1.12	1.13	1.13	1.14
	45	0.88	0.91	0.93	0.96	0.98	1	1.01	1.03	1.05	1.06	1.07	1.09	1.1	1.1	1.11	1.11	1.12	1.12	1.12
	50	0.87	0.89	0.92	0.94	0.96	0.98	1	1.01	1.03	1.04	1.06	1.07	1.08	1.09	1.09	1.1	1.1	1.1	1.11
	55	0.85	0.87	0.89	0.92	0.94	0.96	0.97	0.99	1.01	1.02	1.04	1.05	1.06	1.07	1.07	1.08	1.08	1.08	1.08
	60	0.82	0.85	0.87	0.89	0.91	0.93	0.95	1	0.98	1	1.01	1.02	1.03	1.04	1.05	1.05	1.05	1.06	1.06
	65	0.8	0.82	0.84	0.86	0.88	0.9	0.92	0.94	0.95	0.97	0.98	0.99	1	1.01	1.02	1.02	1.02	1.02	1.02
	70	0.77	0.79	0.81	0.83	0.85	0.87	0.89	0.91	0.92	0.93	0.95	0.96	0.97	0.97	0.98	0.98	0.99	0.99	0.99
	75	0.74	0.76	0.78	0.8	0.82	0.84	0.86	0.87	0.89	0.9	0.91	0.92	0.93	0.93	0.94	0.94	0.94	0.95	0.95
	80	0.71	0.73	0.75	0.77	0.79	0.8	0.82	0.83	0.85	0.86	0.87	0.88	0.89	0.89	0.89	0.9	0.9	0.9	0.9
	85	0.67	0.69	0.71	0.73	0.75	0.77	0.78	0.79	0.81	0.82	0.83	0.83	0.84	0.84	0.85	0.85	0.85	0.85	0.85
	90	0.64	0.66	0.68	0.69	0.71	0.72	0.74	0.75	0.76	0.77	0.78	0.79	0.79	0.79	0.8	0.8	0.8	0.8	0.8

.

The results of the analysis show that the most optimal location of the installation is its inflection with respect to the southern direction from 0 to 30 degrees and the inclination of the photovoltaic panels at an angle from 25 to 45 degrees (the values in the table are marked with a gradient from yellow to red, where yellow color means low performance and red color, respectively, higher performance in the process of recovering electricity from solar radiation).

Step seven. The algorithm itself selects the optimal positioning of the panels depending on the dimensions of the recommended photovoltaic panel, and calculates the largest possible number of panels to be installed(without taking into account roof windows, chimney and other structural elements that reduce the favorable surface for mounting photovoltaic panels). However, the ventilation/technical spaces between separate photovoltaic panels and their rows, the mounting and mandatory spacing from the roof edge necessary to compensate for wind effects on the roof structure are taken into account. The following technical spacing sizes are used in the developed algorithm:

• 20 cm on each side along the side edges of the roof;
• 35 cm from the bottom and 35 cm from the top edge of the roof;
• 1 cm of mandatory technical distance between individual rows of photovoltaic panels vertically and horizontally as the necessary ventilation space.

Step eight. The algorithm provides the results of its calculations in the form of a graph of the maximum possible amount of energy recovered with the user’s preferences specified at the outset, the required property data entered into the system by the user, and all the adjustment factors described above.

3.4. Analysis of the Performance of Photovoltaic Installation for Single-Family House Construction in Opole Province, Poland

The analysis concerns a building plot with registration number 30/7 in Turawa-Marszalki (Opole Province, Poland) with area of 603 m². User preferences are as follows (Figure 5).

Building technical data: type of roof—pitched equal.

Table 11. Technical data of the planned single-family house in Turawa-Marszalki.

	Length, m	Roof Slope
		Does the Roof Slope in This Direction?	Provide the Angle of Inclination?
Front of the construction	10	No	0
Right side of the construction	8	Yes	35
Left side of the construction	8	Yes	35
Rear wall	10	No	0

provides the lengths of the building walls (the length of the front wall—10 m, the length of the right wall and 8 m of the left wall, respectively), the side to which the roof will be sloped (in the analyzed case equally to the left and to the right side of the building), and the roof inclination angle of 35 degrees. According to the location of the analyzed property, the position of the front wall and its inflection from the southern direction are given in Figure 6. The demand for electricity is expected to be 4000 kWh per year, or 333.3 kWh per month on average.

The developed algorithm based on MS Excel and taking into account the assumptions given above will give us the following results (

Table 12. Photovoltaic panel selection report by consumer preference.

Photovoltaic Panel Technology Selected	K2-Mono-HIT
Maximum number of panels installed	20
including:
front wall of the building	0
right wall of the building	20
left wall of the building	0
rear wall of the building	0
Typical power, Wp	330
Typical efficiency, %	20.1
Installed power, kWp	6.6
Energy demand, kWh	4000
Energy production, first year, kWh	4802

). Based on the results in Table 12, we can conclude that the only effective roof area suitable for installing a photovoltaic system is the part of the roof sloping on the right side of the building (front projection). Therefore, even at the building design stage, it is worth taking into account the results of the calculations performed on the basis of the algorithm for rapid estimation of the performance of small rooftop photovoltaic systems. Such calculations will allow minimizing additional technical obstacles (for example, roof windows).

Based on the prosumer’s preferences, the system advises installation of photovoltaic panels using Mono-HIT technology (maximum number of panels—20 pieces, typical power—330 W_p, typical efficiency—20.1%, installed system power—6.6 kW_p).

Calculations show that in the first year of operation of the photovoltaic panels, the system is expected to generate 4802 kWh of electricity, which should be sufficient compared to the expected demand.

The predicted accumulated electricity production by consecutive months is given in Figure 7. According to the calculations, the most energy will be recovered in the period May–July (in May—695 kWh; in June—717 kWh; in July—699 kWh). In the months of January, February, October, November and December, the electricity production (not including degradation) will not be higher than the average monthly demand, which means that there will be a shortage of electricity during the heating period.

The share of electricity production per month per year is provided in Figure 8.

4. Results of the Analyses and Discussion

The performed calculations show that, taking into account only the projected amount of electricity produced by the PV system, the installation of photovoltaic panels can be favorably evaluated by the owners of the analyzed property in Marszalki-Turawa. However, we cannot evaluate the effectiveness of the photovoltaic system based only on one indicator. First, the problem of autonomy of the PV system due to the amount of electricity produced in particular hours and months is a debatable question.

According to the projected electricity demand of the analyzed property (4000 kWh per year or 333.3 kWh average per month), the developed proprietary algorithm allows us to estimate the percentage of electricity generated by the PV system with respect to the demand of the analyzed property (

Table 13. Forecasting the electricity production of a photovoltaic system for a period of 25 years considering the PV degradation factor in kW and the percentage of recovered energy from energy demand.

Year of PV System Use	Electricity Production Projection, kW				Projection of Percent Recovered Energy from Demand, %
	Building Wall
	Front of the Construction	Right Side of the Construction	Left of the Construction	Rear Wall	Front of the Construction	Right Side of the Construction	Left Side of the Construction	Rear Wall
1		4763.3				119.1
2		4724.9				118.1
3		4686.4				117.2
4		4648				116.2
5		4609.6				115.2
6		4571.2				114.3
7		4532.8				113.3
8		4494.4				112.4
9		4456				111.4
10		4417.5				110.4
11		4379.1				109.5
12		4340.7				108.5
13		4302.3				107.6
14		4263.9				106.6
15		4225.5				105.6
16		4187.1				104.7
17		4148.7				103.7
18		4110.2				102.8
19		4071.8				101.8
20		4033.4				100.8
21		3995				99.9
22		3956.6				98.9
23		3918.2				98.0
24		3879.8				97.0
25		3841.3				96.0

). The data presented in Table 13 shows that the installed system is able to generate the necessary amount of electricity from the first until the twentieth year of operation of photovoltaic panels. Actually, the system with twenty PV panels is not able to meet all the energy needs of the household due to the lack of constant access to a source of electricity (which is due to the specific nature of PV technology). An alternative solution may be to use an off-grid system with the installation of an energy store (the decision depends on the investor and the specific proposals of the installation company).However, after 21 years of operation of photovoltaic panels, the designed system, even theoretically, will not be able to achieve full autonomy.

The developed algorithm for rapid estimation of the efficiency of small rooftop photovoltaic installations allows visualization of the photovoltaic system activities in terms of household electricity use (insert “working hours”). In simple terms “What can we plug in and when?”.

As an example of such visualization, tenof the most necessary devices in a single-family house were taken for analysis, i.e.,:

Appliance #1 “Refrigerator”. Samsung RB38T603CS9 refrigerator model was chosen for the analysis—energy class “A+++” according to the new system “C”; annual energy consumption—196 kWh; considering that there are 365 days in a year with 24 h each, the hourly energy consumption can be determined as 0.0193 kWh [35];

Appliance #2 “Washing Machine”. The washing machine Amica WA2S612BKiSJD was chosen for the analysis—energy class according to the new system “B”; energy consumption—48 kWh for 100 cycles, taking into account that the minimum time of the standard program is one hour, the hourly energy consumption can be determined as 0.48 kWh [36];

Appliance #3 “Oven”. The oven Whirlpool W6OM44S1PBSSW Collection was chosen for the analysis—energy class “A+”; energy consumption in traditional mode is 0.91 kWh; connected power—not less than 3 kW [37];

Appliance #4 “Induction cooktop”. Three models were selected for analysis: Whirlpool WL S3160 BF [38], Samsung NZ64T3707A1 [39] and Electrolux Slim-fit CIV654 [40]. Due to the specifics of the device, we are more interested in, not the electricity consumption, but rather the connection power, which for the models listed is 7.2 kW. This means that we cannot use this type of appliance without the voltage in the network at the level of 7.2 kW;

Appliance #5 “Electric Kettle”. A 1.7 L Tefal brand kettle, model Display KO851 with 1.8 kW [41], was selected for analysis;

Appliance #6 “Iron”. A Tefal brand iron, model Tefal Smart Protect Plus FV6870 with 2.8 kW [42], was selected for analysis;

Device #7 “Television”. A TV from Samsung, model QLED QE65!80AAT DVB-T2/HEVC [43], was selected for the analysis. Energy consumption: in SDR mode 135 kWh/1000h (0.135 kWh/h); in HDR mode 256 kWh/1000h (0.256 kWh/h).

Appliance #8 “Heater”. A 1 kW NEO 90-090 convector heater [44] was selected for analysis. According to the description on the website, the heater is able to heat areas up to 15 square meters. As such, we can assume that for every 15 square meters of the area of a single-family house, 1 kW of electricity is necessary. The minimum area to be heated is the area occupied by the building.

Appliance #9 “Water heater”. Electric water heater ZELMECH ZL-DV80 with rated power of 2 kW and capacity of 80 L [45] was selected for the analysis.

Appliance #10 “Reserve-XXX”—2.5 kW.

Preliminary utility analysis of the designed photovoltaic system gives reason to conclude that the recovered electricity will not be sufficient for some basic single-family home equipment (

Table 14. Basic results of the energy performance evaluation of the designed photovoltaic system for the analyzed single-family house without considering the degradation factor.

#	Appliance	Connection Power, kW	Is It Theoretically Possible to Turn on the Device?	To Include into Analysis?	Demand for Energy, kWh
1	Refrigerator	0.0193	Yes	Yes	0.02
2	Washing Machine	0.48	Yes	Yes	0.48
3	Oven	0.91	Yes	Yes	3.00
4	Induction cooktop	7.2	No	Yes	7.20
5	Kettle	1.8	Yes	Yes	1.80
6	Iron	2.8	Yes	Yes	2.80
7	TV, SDR mode	0.135	Yes	Yes	0.14
8	Radiator	5.33	No	Yes	5.33
9	Water heater	2	Yes	Yes	2.00
10	Reserve–XXX	2.5	Yes	Yes	2.50
				Total, kWh	25.27

).

Such items are:

First, the induction cooktop, which cannot be activated due to the lack of adequate voltage generated by the photovoltaic system. According to the technical data, the connection power of the analyzed model is 7.2 kW; at this moment, the highest planned power in the photovoltaic system without considering the degradation of the system is 4.25 kW in the month of April from 13:00 to 14:00.

Second, it is not possible to use the photovoltaic system alone to heat the building. According to the calculations, the necessary minimum heating demand is 5.33 kWh. Indeed, such power is needed to heat the whole building at once and you can turn on the heaters one by one directly in the rooms where the occupants arrive at a certain moment, but such a solution is not optimal.

Thirdly, since the oven requires at least 3 kW of energy to be turned on, it will also be possible to turn it on during the same hours as the induction cooktop only for one hour in April, assuming that everything else will be turned off (then we will be left with 1.25 kW of reserve power).

Table 15. Hours during which the power generated by the designed photovoltaic system will be sufficient for effective use of the refrigerator (without taking into account system degradation).

Months	Daily Production, kWh by Hour
	Hours
	0–5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20–24
January						0.83	0.97	0.97	0.82
February					1.00	1.32	1.50	1.51	1.35	1.05	0.64
March			0.65	1.17	1.63	1.97	1.97	1.96	1.96	1.61	1.14	0.62
April			0.71	1.27	1.81	2.26	2.58	2.73	4.25	2.48	1.94	1.63	1.07
May		0.58	1.09	1.63	2.14	2.55	2.83	2.96	2.92	2.72	2.37	1.91	1.38	0.84
June		0.69	1.20	1.71	2.19	2.59	2.86	3.00	2.97	2.78	2.47	2.04	1.54	1.02	0.54
July		0.57	1.04	1.55	2.02	2.41	2.70	2.84	2.83	2.66	2.36	1.95	1.47	0.96
August			0.76	1.27	1.75	2.15	2.44	2.58	2.57	2.40	2.08	1.65	1.16	0.65
September				0.94	1.43	1.84	2.13	2.26	2.21	2.00	1.64	1.18	0.67
October					0.97	1.35	1.61	1.71	1.64	1.40	1.14	0.59
November					0.82	1.05	1.14	1.07	0.85
December						0.74	0.85	0.81	0.63

,

Table 16. Hours during which the power generated by the designed photovoltaic system will be sufficient for the effective operation of the unit “Water heater” (without taking into account the degradation of the system). The hours when there is not enough electricity produced by the photovoltaic system are marked in color.

Months	Daily Production, kWh by Hour
	Hours
	0–5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20–24
January		0.00	0.00	0.00	0.00	0.83	0.97	0.97	0.82	0.00	0.00	0.00	0.00	0.00	0.00
February		0.00	0.00	0.00	1.00	1.32	1.50	1.51	1.35	1.05	0.64	0.00	0.00	0.00	0.00
March		0.00	0.65	1.17	1.63	1.97	1.97	1.96	1.96	1.61	1.14	0.62	0.00	0.00	0.00
April		0.00	0.71	1.27	1.81	2.26	2.58	2.73	4.25	2.48	1.94	1.63	1.07	0.00	0.00
May		0.58	1.09	1.63	2.14	2.55	2.83	2.96	2.92	2.72	2.37	1.91	1.38	0.84	0.00
June		0.69	1.20	1.71	2.19	2.59	2.86	3.00	2.97	2.78	2.47	2.04	1.54	1.02	0.54
July		0.57	1.04	1.55	2.02	2.41	2.70	2.84	2.83	2.66	2.36	1.95	1.47	0.96	0.00
August		0.00	0.76	1.27	1.75	2.15	2.44	2.58	2.57	2.40	2.08	1.65	1.16	0.65	0.00
September		0.00	0.00	0.94	1.43	1.84	2.13	2.26	2.21	2.00	1.64	1.18	0.67	0.00	0.00
October		0.00	0.00	0.00	0.97	1.35	1.61	1.71	1.64	1.40	1.14	0.59	0.00	0.00	0.00
November		0.00	0.00	0.00	0.82	1.05	1.14	1.07	0.85	0.00	0.00	0.00	0.00	0.00	0.00
December		0.00	0.00	0.00	0.00	0.74	0.85	0.81	0.63	0.00	0.00	0.00	0.00	0.00	0.00

,

Table 17. Hours during which the power generated by the designed photovoltaic system will be sufficient for effective operation of the iron (without taking into account the degradation of the system). The hours when there is not enough electricity produced by the photovoltaic system are marked in color.

Months	Daily Production, kWh by Hour
	Hours
	0–5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20–24
January		0.00	0.00	0.00	0.00	0.83	0.97	0.97	0.82	0.00	0.00	0.00	0.00	0.00	0.00
February		0.00	0.00	0.00	1.00	1.32	1.50	1.51	1.35	1.05	0.64	0.00	0.00	0.00	0.00
March		0.00	0.65	1.17	1.63	1.97	1.97	1.96	1.96	1.61	1.14	0.62	0.00	0.00	0.00
April		0.00	0.71	1.27	1.81	2.26	2.58	2.73	4.25	2.48	1.94	1.63	1.07	0.00	0.00
May		0.58	1.09	1.63	2.14	2.55	2.83	2.96	2.92	2.72	2.37	1.91	1.38	0.84	0.00
June		0.69	1.20	1.71	2.19	2.59	2.86	3.00	2.97	2.78	2.47	2.04	1.54	1.02	0.54
July		0.57	1.04	1.55	2.02	2.41	2.70	2.84	2.83	2.66	2.36	1.95	1.47	0.96	0.00
August		0.00	0.76	1.27	1.75	2.15	2.44	2.58	2.57	2.40	2.08	1.65	1.16	0.65	0.00
September		0.00	0.00	0.94	1.43	1.84	2.13	2.26	2.21	2.00	1.64	1.18	0.67	0.00	0.00
October		0.00	0.00	0.00	0.97	1.35	1.61	1.71	1.64	1.40	1.14	0.59	0.00	0.00	0.00
November		0.00	0.00	0.00	0.82	1.05	1.14	1.07	0.85	0.00	0.00	0.00	0.00	0.00	0.00
December		0.00	0.00	0.00	0.00	0.74	0.85	0.81	0.63	0.00	0.00	0.00	0.00	0.00	0.00

,

Table 18. Hours during which the power generated by the designed photovoltaic system will be sufficient for effective use of the kettle (not including system degradation). The hours when there is not enough electricity produced by the photovoltaic system are marked in color.

Months	Daily Production, kWh by Hour
	Hours
	0–5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20–24
January		0.00	0.00	0.00	0.00	0.83	0.97	0.97	0.82	0.00	0.00	0.00	0.00	0.00	0.00
February		0.00	0.00	0.00	1.00	1.32	1.50	1.51	1.35	1.05	0.64	0.00	0.00	0.00	0.00
March		0.00	0.65	1.17	1.63	1.97	1.97	1.96	1.96	1.61	1.14	0.62	0.00	0.00	0.00
April		0.00	0.71	1.27	1.81	2.26	2.58	2.73	4.25	2.48	1.94	1.63	1.07	0.00	0.00
May		0.58	1.09	1.63	2.14	2.55	2.83	2.96	2.92	2.72	2.37	1.91	1.38	0.84	0.00
June		0.69	1.20	1.71	2.19	2.59	2.86	3.00	2.97	2.78	2.47	2.04	1.54	1.02	0.54
July		0.57	1.04	1.55	2.02	2.41	2.70	2.84	2.83	2.66	2.36	1.95	1.47	0.96	0.00
August		0.00	0.76	1.27	1.75	2.15	2.44	2.58	2.57	2.40	2.08	1.65	1.16	0.65	0.00
September		0.00	0.00	0.94	1.43	1.84	2.13	2.26	2.21	2.00	1.64	1.18	0.67	0.00	0.00
October		0.00	0.00	0.00	0.97	1.35	1.61	1.71	1.64	1.40	1.14	0.59	0.00	0.00	0.00
November		0.00	0.00	0.00	0.82	1.05	1.14	1.07	0.85	0.00	0.00	0.00	0.00	0.00	0.00
December		0.00	0.00	0.00	0.00	0.74	0.85	0.81	0.63	0.00	0.00	0.00	0.00	0.00	0.00

and

Table 19. Hours during which the power generated by the designed photovoltaic system will be sufficient to effectively operate simultaneously a refrigerator, a TV in SDR mode and a water heater (without taking into account system degradation). The hours when there is not enough electricity produced by the photovoltaic system are marked in color.

Months	Daily Production, kWh by Hour
	Hours
	0–5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20–24
January		0.00	0.00	0.00	0.00	0.83	0.97	0.97	0.82	0.00	0.00	0.00	0.00	0.00	0.00
February		0.00	0.00	0.00	1.00	1.32	1.50	1.51	1.35	1.05	0.64	0.00	0.00	0.00	0.00
March		0.00	0.65	1.17	1.63	1.97	1.97	1.96	1.96	1.61	1.14	0.62	0.00	0.00	0.00
April		0.00	0.71	1.27	1.81	2.26	2.58	2.73	4.25	2.48	1.94	1.63	1.07	0.00	0.00
May		0.58	1.09	1.63	2.14	2.55	2.83	2.96	2.92	2.72	2.37	1.91	1.38	0.84	0.00
June		0.69	1.20	1.71	2.19	2.59	2.86	3.00	2.97	2.78	2.47	2.04	1.54	1.02	0.54
July		0.57	1.04	1.55	2.02	2.41	2.70	2.84	2.83	2.66	2.36	1.95	1.47	0.96	0.00
August		0.00	0.76	1.27	1.75	2.15	2.44	2.58	2.57	2.40	2.08	1.65	1.16	0.65	0.00
September		0.00	0.00	0.94	1.43	1.84	2.13	2.26	2.21	2.00	1.64	1.18	0.67	0.00	0.00
October		0.00	0.00	0.00	0.97	1.35	1.61	1.71	1.64	1.40	1.14	0.59	0.00	0.00	0.00
November		0.00	0.00	0.00	0.82	1.05	1.14	1.07	0.85	0.00	0.00	0.00	0.00	0.00	0.00
December		0.00	0.00	0.00	0.00	0.74	0.85	0.81	0.63	0.00	0.00	0.00	0.00	0.00	0.00

present the simulated hours of operation of the different appliances in the case of autonomous operation of the photovoltaic system of the analyzed single-family house:

• The hours when the refrigerator could work are shown in Table 15;
• The hours when the water heater could work are shown in Table 16;
• The hours when the iron could be turned on are shown in Table 17;
• the hours when the kettle will most likely be able to be turned on are shown in Table 18;
• The hours when the power generated by the designed photovoltaic system will be sufficient to efficiently use simultaneously a refrigerator, a TV in SDR mode, and a water heater (without taking into account system degradation) are shown in Table 19.

Summarizing the conducted research, we can state the following:

• In the case of the analyzed property, the photovoltaic system cannot function autonomously and needs a connection to the power grid, but it can become an additional source of electricity and secure a certain level of independence from external energy sources for the property owners;
• Only a comprehensive analysis of the energy potential of photovoltaic panels, taking into account technical, climatic and weather factors, as well as personal preferences of consumers, will make it possible to take a correct decision on the installation of PV systems in households. Such an analysis is made possible by the algorithm developed by the authors for the rapid estimation of the effectiveness of small rooftop PV systems;
• The proposed algorithm gives the probable amount of covering the own demand for electricity over a period of 25 years (most manufacturers predict a linear degradation of photovoltaic panels for this period), which in turn can approximate the estimated savings. However, due to the fact that the price of 1 KW/h of electricity supplied and the subscription price depends on the specific electricity supplier, connection capacity, photovoltaic system power, tariff chosen by the investor, according to which the amount due for the electricity consumed is calculated, the issue of state subsidies for this type of investment project, the possibility of disconnecting a specific installation in the event of a load on the supplier’s energy system, as well as various variants and technical issues, it is impossible in one article to estimate the profitability of such an undertaking that would take into account all the above-mentioned conditions. Due to the fact that the decision to install a solar panel system should not be imposed on the investor, the proposed system leaves the final decision in the hands of the investor (household). Therefore, the investor must conduct simulations of various technical variants of photovoltaic installations before making the final decision.

5. Conclusions

The paper develops an algorithm for rapid estimation of the efficiency of use of small rooftop photovoltaic systems by households. The algorithm was developed in such a way as to estimate the maximum possible production of electricity by the photovoltaic system, taking into account the roof surface of a single-family house, its location by the ratio of the south direction and the angle of inclination. The algorithm does not provide for the minimum number of photovoltaic panels to be installed, but optimizes their maximum number, taking into account installation breaks, optimal distance from the roof borders to reduce wind resistance and other technical data of the selected photovoltaic panel (for example: width, height).

The validation of the algorithm was carried out on the example of Opole Province (Poland).

The proposed proprietary algorithm is in the first place an attempt to create a tool for a potential investor. Currently, analytical systems aimed at installation companies are available on the market and not always fully comply with all customer requirements. As such, the proposed proprietary algorithm is not an alternative to the existing ones, but rather an attempt to inclusively take into account the majority of consumer preferences and important technical issues in order to support the decision of a potential investor and prepare them for negotiations with the installation company.