The Optimal Strategy for Supplying Single-Family Homes with Electricity Using Photovoltaic Installations

Abstract

In this paper, an analysis of the feasibility of photovoltaic installations for individual consumers in Poland was carried out. For this purpose, an author’s model of the feasibility of small-scale photovoltaic installations for individual consumers was developed, depending on the following: the adopted strategy for the installed capacity and the changing economic and social environment in which electricity from this installation will be produced. Based on the model built, the costs of obtaining electricity from a photovoltaic installation were determined, using the methodology of calculated costs of electricity production. The lifetime of a small photovoltaic installation is 20 years, during which time the conditions for generating electricity can change significantly. Accordingly, five states of the economic and social environment were determined, ranging from extremely unfavourable to extremely favourable. The total costs of procuring electricity in these states took into account the additional costs or benefits of changing the price of electricity additionally procured from the electricity grid, in years with less sunshine, or sold to the grid, in years with high sunshine, and the possibilities of taxing the production of that energy. Using the method of individual choice, which is an element of game theory, the optimal strategy for equipping the customer with a small photovoltaic installation was determined, taking into account the changing economic and social environment.

1. Introduction

The progressive increase in the price of electricity in Poland is forcing individual consumers of this energy to look for opportunities to reduce their electricity usage costs. One of the possibilities to reduce these costs is the installation of photovoltaic installations on residential properties, especially in single-family houses. In single-family buildings, despite the use of increasingly energy-efficient equipment, we are seeing an increasing demand for electricity, which results from the increased use of various electrical appliances in these buildings.

The continuous increase in the price of conventional energy carriers, as well as the ecological requirements regarding CO₂, are causing, according to many authors both in Poland and worldwide, a very significant increase in interest in renewable energy sources (RES), which also represents one of the opportunities to combat global climate change [1,2,3,4,5,6,7,8]. The main renewable energy carriers in Poland include the following types of energy: geothermal, wind, biogas, briquette and pellet, and photovoltaic.

As one of the devices used in single-family homes to generate renewable energy, air source heat pumps are increasingly being installed as an alternative solution to heating these homes with gas furnaces. Heat pumps have a high demand for electricity both to drive them and to reheat buildings. During periods of high temperature drops, when the heat pump’s ability to extract sufficient heat is exhausted, the reheating of the building by means of electric heating systems is switched on.

In the last three years, photovoltaic installations of up to 15 kW have been installed in single-family homes in the greatest numbers [9,10]. The lifetime of these photovoltaic installations is 20 years. During such a long lifetime of these installations, the economic and social environment may change significantly: the price of electricity obtained from grid suppliers, the purchase price of electricity from photovoltaic installations that will be supplied to the grid, these types of installations may also be taxed [11,12]. The research problem addressed in this paper concerns the possibility of using electricity from photovoltaic installations in single-family houses, depending on the installed capacity of the installation and the economic and social environment in a given region. Each single-family building handed over to its owners is equipped with an electrical installation connected to the power grid in a given locality. The use of electricity from photovoltaic installations in these residential buildings involves additional costs related to the construction of an additional photovoltaic installation. The unit costs of electricity production from photovoltaic installations depend, to a large extent, on the level of sunshine in a given region and year. Therefore, the choice of the optimal strategy for equipping single-family houses with photovoltaic installations is a multi-level decision-making factor. Therefore, the basic research problem in this work is to determine the optimal strategy for equipping single-family houses with photovoltaic installations.

Therefore, according to the authors, a thorough analysis should be made of the costs of obtaining electricity for individual consumers with a PV installation over its lifetime and the possible purchase of this energy from the supplier in the case of insufficient production of the PV installation or the sale of this energy to the grid in the case of overproduction.

The primary aim of the study is to determine the optimum strategy for obtaining electricity by individual consumers in the turbulent economic and social environment in Poland, based on an analysis of unit costs of obtaining this energy, over the entire lifetime of photovoltaic installations.

Secondary objectives will be as follows:

-

Building a model for the procurement of electricity for residential consumers from photovoltaic installations in Poland,

-

A determination of acceptable strategies for procuring electricity and the identification of economic and social ambient states in which small-scale electricity consumers can operate,

-

A determination of the unit cost of procuring electricity in a changing economic and social environment and a determination of the optimum strategy for procuring electricity for individual consumers with an installed capacity of up to 15 kW.

The production of electricity from renewable sources can also significantly contribute to the creation of new jobs and reduce dependence on imports of conventional energy carriers. One possibility for renewable energy in Poland is the production of electricity from photovoltaic installations for individual consumers, i.e., installations mounted on the roofs of houses or other buildings, and the production of electricity from large photovoltaic installations, so-called “photovoltaic farms”.

This study attempts to assess the economic efficiency of electricity production for individual consumers from small photovoltaic systems installed most often in single-family houses up to 15 kW in Poland and to determine the optimal strategy for equipping these consumers with this type of photovoltaic system.

The term ‘acquisition cost of electricity’ refers to, the cost of producing electricity through a photovoltaic installation and the eventual purchase of this energy from the supplier in the event of insufficient production of the photovoltaic installation or the sale of this energy to the grid in the event of overproduction. ‘Unit cost of production’, on the other hand, refers to the calculated unit cost of electricity production resulting from the capacity of the installed PV installation—the LCOE.

2. Materials and Methods

2.1. Calculated Unit Cost of Electricity Production

The economic efficiency of electricity generation, depending on the strategy adopted to equip buildings with small-scale photovoltaic installations and the state of the economic and social environment, depends on the magnitude of the unit production costs-LOCE and on the variable factors of the environment in which the consumer of this energy will operate. The unit production costs of photovoltaic electricity-LOCE can be determined from relation 1.

$$LCOE={\displaystyle \frac{\sum _{t+1}^n\frac{{I_t+O\&M}_t+F_t+C_t+D_t}{{(1+d)}^t}}{\sum _{t=1}^n\frac{A_t}{{(1+d)}^t}}}$$

(1)

where

I_t—capital expenditure in the year/(investment costs),

O&M_t—operations and maintenance costs in year t,

F_t—fuel costs in year t, for a photovoltaic installation; this cost is equal to 0,

D_t—CO emission costs₂, in year t (carbon costs); in the case of a photovoltaic installation, this cost is equal to 0,

C_t—decommissioning cost in year t,

A_t—electricity generated in year t,

d—discount rate,

n—service life (the ‘lifetime’ of the object),

t—the year in which the calculation is made.

To determine the unit costs resulting from an installed PV installation, the analysis should consider all components of the cumulative cost of producing that energy—the LCOE [13], among others, all financial inputs and costs should be taken into account, which should be presented by means of comparable values, although, such as a discount account [14]. These requirements are met by the levelised cost of electricity (LCOE), which takes into account the total costs of installing and operating this installation over its lifetime and the costs of disposal [15].

For the determination of the total unit costs of electricity generation in a changing economic and social environment (Y₁ to Y₅), as well as the adopted strategy for the installed capacity of the photovoltaic installation (S₁ to S₄), it is also necessary to take into account the additional costs resulting from the necessity of a possible purchase of electricity from network suppliers, the costs of a possible taxation of this production, and the possible benefits resulting from the sale of this energy to the electricity grid, according to relation 2.

K = KJCLCOE + Kst + Podt − Esp

(2)

where

K_LCOE—calculated cost of electricity,

E_st—energy purchased from the grid from suppliers of that energy in the year [EUR/year],

P_odt—tax on energy produced per year [EUR/year],

E_sp—energy sold to the grid per year [EUR/year].

2.2. Expert-Mathematical Method—Assessment of the Prioritisation of Economic and Social Factors Influencing the Costs of Obtaining Electricity from Households in Poland

The expert-mathematical method, as a scientific tool, is used to solve complex tasks [16]. The characteristics of this method are as follows: a procedure for determining the minimum number of experts and their qualitative selection, a detailed order of carrying out all stages of the expert opinion, the use of a mathematical apparatus both in the organisation of the expert opinion and in the elaboration of the results of each stage of the research [17].

In order to determine the unit acquisition costs of electricity from small-scale PV installations for individual consumers, depending on the strategy adopted for the installed capacity of the PV installation and the state of the economic and social environment, it is necessary to determine the probability of the occurrence of a given state in the environment and the additional benefits or losses resulting from the occurrence of a given state, i.e., the possible need to buy energy from the grid, the sale of any surplus electricity to the grid, or the taxation of PV energy production. The expert-mathematical method was used to determine the probability of the occurrence of a given state in the environment (level II factors from Y to Y₂₁₂₅) and the structure of unit acquisition costs in these states (level III factors from Y₃₁₁ to Y₃₅₄), and according to this method, a consequence tree was built (Figure 1), followed by an expert study in which 92 experts participated.

One of the most effective methods for determining the importance of goals (Y factors) is the so-called goal interaction graph method, called a consequence tree or Isikawa diagram [18,19]. This method consists of determining the main objective and successively breaking down (decomposing) this objective into more and more subordinate objectives located at lower levels, resulting in the so-called consequence tree (Figure 1). The basic principle for the construction of such a consequence tree is that each lower-level objective is related to only one higher-level (more general) objective and many subordinate (more specific) objectives.

In the consequence tree presented in Figure 2, according to the expert mathematical method, we distinguish three levels of the analysed process. The first level marked with the symbol S1 (main duty) is the unit costs of obtaining electricity from small photovoltaic installations for individual consumers in Poland [EUR/kWh]; the second level (S₂₁–S₂₅) contains states in the economic and social environment. The third level (S₃₃₁–S₃₅₄) is the percentage shares in the structure of unit costs of obtaining electricity from photovoltaic sources in Poland in the assumed state of the economic and social environment.

In the consequence tree shown in Figure 1, we distinguish between local and systemic goal priorities. Local priorities are ratings of the relative importance of detailed objectives that are components of a single overarching objective. The determination of local priorities is to be carried out in such a way that the sum of the points allocated by the experts during the assessment in a given area of any overarching objective is equal to 100 (C₃₁₁ + C₃₁₂ + C₃₁₃ + C₃₁₄ + C₃₁₅ = 100). In doing so, the norming condition is strictly adhered to, i.e., the sum of the local priorities at a given level (Level II or Level III) will always equal 100.

The system priority of an objective determines its contribution to the main objective, and we determine this by multiplying the value of the lower-level objective by the value of the higher-level objectives. This will give an obvious idea of the relative contribution of each lower-level objective to the achievement of the higher-level objective (C₃₁₁ × C₂₁).

As can be seen from Figure 1, a number of factors (targets) will influence the size of the unit cost of obtaining electricity from small-scale PV installations for residential consumers.

These factors have different effects on the process under consideration. Therefore, achieving high economic efficiency in the extraction of electricity from a photovoltaic installation is a difficult multi-level decision problem [19].

The data obtained from the expert method should be processed using mathematical methods. Therefore, a new method called the expert-mathematical method was developed, which is a comprehensive method [18], and its application allows for a rational combination of an intuitive–logical process with quantitative and qualitative methods of mathematical data processing. The final scores obtained from the compilation of individual expert assessments will allow a uniform overall assessment of the level of importance of individual factors. In this case, the concordance of the subjective assessments of many experts becomes an objective assessment of the problem in question. The mathematical and statistical method used to develop the expert assessments allows reliable final results to be obtained. Such a combination allows the greatest efficiency to be achieved in solving the problem under consideration [18].

The minimum number of experts required is determined according to the following relationship:

$$N_E={\displaystyle \frac{f\beta (b-1)}{(\gamma +1)(b-1)\Theta 0}}$$

(3)

where f_β (b − 1) is the quantile of the distribution χ² corresponding to the confidence level β and the number of degrees of freedom b − 1, b is the number of factors assessed, γ is the assumed accuracy in the assessment of concordance, and Θ₀ is the critical value of the concordance coefficient.

The result of the generalisation of expert assessments is the established mean value: arithmetic mean, median, or modal value. The quantitative measures characterising the spread of the overall expert assessments are the statistical indices of these assessments: mean square deviation, coefficient of pairwise serial correlation, and others.

The quantity under study can be expressed in series (at the appropriate levels) and in scaled scores, with the latter being determined in the final determination phase and converted into a score expressed in parts of a unit, i.e., in percentages.

Where the principle of serialisation—the distribution of factors in decreasing order of importance—is applied, the dispersion coefficient of concordance is used to assess the degree of concordance of expert judgements, which, in the absence of equal ranks, is determined from the following relationship:

$$\Theta ={\displaystyle \frac{12S}{N{E}^2(b^3-b)}}$$

(4)

where S is the sum of the squares of the deviations of the actual series values:

$$S=\sum _{j=1}^b(\stackrel{-}{r}j-\stackrel{-}{r}{)}^2$$

(5)

where ${\stackrel{\_}{r}}_j$ is the sum of the ranks given by the experts to the jth factor and $\overline{r}$ is the arithmetic mean of the sum of the ranks:

$$\overline{r}={\displaystyle \frac{{\displaystyle \sum _{j=1}^b\overline{r}j}}{b}}, (1.4) \hspace{1em}\overline{r}j={\displaystyle \sum _{i=1}^{N_E}rij}$$

(6)

where ${\stackrel{\_}{r}}_{ij}$ is the the series given by the ith expert to the jth factor.

Where similar series exist, the concordance coefficient is determined according to the following formula:

$$\Theta ={\displaystyle \frac{S}{\frac{1}{12}{N_E}^2(b^3-b)-NE{\sum }_{i=1}^{N_E}Ti}}$$

(7)

where T_i is the index of similar series:

$$Ti={\displaystyle \frac{1}{12}}{\displaystyle \sum _{i=1}^p(t{i}^3-ti)}$$

(8)

where p is the number of groups of equal ranks in the evaluation of the jth expert and t_i is the number of repetitions of equal ranks in the p-e group.

The concordance coefficient is a quantitative assessment of the degree of concordance of the experts’ assessments and varies between 1 ≥ Θ ≥ 0, with Θ = 0 expressing no concordance in the experts’ opinion and Θ = 1 being total concordance in their opinion. Therefore, by adopting a critical value of Θ₀ closer to zero, we have to simultaneously reckon with an increased number of experts, but at the same time, we ensure a higher reliability of the results.

The concordance coefficient calculated according to relation (4) or (7) is an assessment of the actual value of the coefficient and therefore represents a random quantity. In order to check that the decreasing number of serialised factors is not random, an assessment of the meaningful value of the concordance coefficient, according to the concordance criterion, should be carried out using the following relation:

$${\chi }^2={\displaystyle \frac{S}{\frac{1}{12}{N}_{E}b(b+1)-\frac{1}{b-1}{\displaystyle \sum _{i=1}^{N_E}Ti}}}$$

(9)

The quantile of the distribution χ² corresponding to the confidence level β and the number of degrees of freedom b − 1 [f_β (b − 1)] are shown in

Table 1. Quantile of the distribution χ² corresponding to the confidence level β and the number of degrees of freedom b − 1.

b − 1	fβ (b − 1) at β
1	0.700	0.800	0.900	0.950	0.975	0.990	0.995	0.999
2	2.14	3.23	4.60	5.99	7.38	9.21	10.60	13.82
3	3.66	4.65	6.24	7.88	9.36	11.14	12.84	16.26
4	4.88	5.99	7.77	9.49	11.16	13.27	14.88	18.48
5	6.05	7.30	9.25	11.05	12.85	15.10	16.75	20.50

Source: Izdebski, 2003.

.

If the calculated value χ² is greater than the tabulated value χ²_tab in Table 1 and the concordance coefficient is significantly different from zero, it can be argued that the concordance of the expert assessments is not random.

Assessing the local priorities of the stated objectives by means of scaled evaluations (expressed in points or percentages) is somewhat more complicated than assessing priorities by prioritising them at appropriate levels of importance. However, it has the significant advantage that it results in direct numerical values for the priorities and their contribution to the overall structure of the factors considered. In this case, the consistency of the experts’ judgements is assessed by means of the coefficient of variance.

From the expert data, the local priority m_j of the jth factor is determined:

$$m_j=\left({\displaystyle \sum _{i=1}^{N_E}{m}_{ij}}\right)/N_E$$

(10)

where m_ij is the normalised importance coefficient of the jth factor determined by the ith expert.

The mean square deviation of the importance factor of the jth factor is then determined:

$$g_j=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^{N_E}{(mj-mij)}^2}}{N_E-1}}}, \mathrm{for} {\mathrm{N}}_{\mathrm{E}}\le 30$$

(11)

$$g_j=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^{N_E}{(\overline{m}j-mij)}^2}}{N_E}}}, \mathrm{for} {\mathrm{N}}_{\mathrm{E}}>30$$

(12)

Based on these indices, a variation factor is calculated for each factor:

$$V_j={\displaystyle \frac{gj}{\overline{m}j}}$$

(13)

It is considered that if V_j ≤ 0.25, the concordance of the experts’ designated individual validity assessments is sufficient. If, on the other hand, V_j > 0.25, the concordance is insufficient. The following gradation of variation coefficients can also be adopted: V_j ≤ 0.10—concordance high; 0.10–0.25—higher than average; 0.16–0.25—average; 0.26–0.35—lower than average; V_j > 0.35—low.

2.3. Game Theory—Determination of the Optimal Strategy for Equipping the User with a Small Photovoltaic Installation

In order to determine the optimal strategy for equipping the user with small-scale PV, it is most advisable to use individual choice criteria that are elements of game theory.

The choice of this strategy can be made using two groups of individual selection criteria. According to the first group of criteria, in which we need to know the probability of a given state in the economic and social environment, the analysis is carried out on the basis of the determined total unit costs of acquiring electricity from small-scale photovoltaic installations for individual consumers.

Using the maximum average win criterion, the optimal strategy will be the one for which the sum of the total unit acquisition costs, M(Si, Yj), is the minimum:

$$Siopt=\sum _{j=1}^{d\sum }PjM(Si, Yj)\to min.$$

(14)

where d is the number of states in the economic and social environment Y_j and Pj is the probability of occurrence ${\displaystyle \sum _{j=1}^{d}Pj}$ of a given state in the economic and social environment.

In Wald’s maximum pessimism criterion, which is based on the principals of a possible worse situation, but for which a minimum ‘win’ is guaranteed, the optimal strategy is determined in two stages. In each case, the minimum value M(Si, Rsj) is chosen, and then, from each of these, the one that is the maximum is determined:

$$Siopt=\underset{i}{min}\underset{j}{max}M(Si, Yrj)$$

(15)

For the value determined in this way, a corresponding strategy is adopted, which ensures a minimum win and, at the same time, guarantees that there is no possibility of a worse result.

According to the next criterion, ‘pessimism-optimism’ by Gurvic, which is a compromise when selecting the optimal solution between extreme pessimism and reckless risk (optimism), the optimal strategy is determined by the following relationship:

$$Siopt=\underset{i}{min}\left[\kappa \underset{j}{max}M\right(Si, Yj)+(1-\kappa \left)\underset{j}{min}M\right(Si, Yj\left)\right]$$

(16)

where κ is the coefficient indicating the amount of pessimism and optimism.

On the basis of this criterion, it can be concluded that, given the unknown probability of a situation occurring, the choice of an appropriate solution should be made on the basis of one’s own experience or that of experts and common sense. In this case, the optimal strategy should be determined from the following condition: for κ = 1 the criterion of pessimism–optimism transforms into a criterion of maximum pessimism, and at κ = 0, we obtain a solution devoid of all prudence such as for the criterion of extreme optimism, for which the largest win in a given row is the maximum value resulting from relation 17:

$$Si=\underset{i}{min}\underset{j}{max}M(Si, Yj)$$

(17)

The ratio κ is determined either analytically or by means of a complex expert-mathematical method. The more dangerous the situation, the lower the risk should be and the κ should be closer to 1. In this study the coefficient κ was determined using the expert-mathematical method, and its value was 0.60, meaning that 60% of the strategy is pessimistic and 40% optimistic.

In the second group (without knowledge of the probability of a given condition), we include two criteria: minimum average risk and minimum risk. For these two criteria, the previously determined matrix of the unit total costs of obtaining electricity from small-scale PV installations for individual consumers has to be transformed into a risk matrix. For this purpose, the difference between the minimum value of M_min (Si, Yj) in the jth column and all its other values should be calculated, for the adopted strategy S and the same economic and political environment, that is, when Yj = const:

R(Si, Yj) = M(Si, Yj) − Mmin(Si, Yj)

(18)

where R(Si, Yj) is the risk-loss from a possibly incorrectly adopted strategy and R(Si, Yj) ≥ 0.

Once the amount of risk has been determined in this way, a corresponding strategy is adopted, using an appropriate criterion that ensures the least amount of risk in the worst-case situations.

When applying the criterion of minimum average risk, we consider the aspirational summed values in the individual rows as the optimum:

$$Siopt=\sum _{j=1}^{d\sum }PjR(Si,Yj)\to min$$

(19)

If the choice of strategy is made without taking into account the probability of a given state in the environment, the so-called insufficient Laplace’ basis principle is applied, the essence of which is that all possible states in the environment are equally likely, and therefore, the probability of each state occurring is determined by an overall number d:

P = 1/d = const

(20)

Since the multiplier is the same for all ambient states and does not affect the magnitude of the minimum in expressions (21) and (22), this expression can be written in the following form:

$$Siopt=\sum _{j=1}^{d\sum }M(Si, Yj)\to min$$

(21)

$$Siopt=\sum _{j=1}^{d\sum }R(Si, Yj)\to min$$

(22)

When determining the optimal strategy on the basis of Sevige’s minimum risk criterion, it is determined not by ‘win-win’ but by the amount of risk. According to this criterion, the optimal strategy is also determined in two steps. Sequentially, the maximum risk value R(Si,Ysj) is calculated in each row, and then, the one that is the minimum is determined from these (23):

$$Siopt=\underset{i}{min}\underset{j}{max}R(Si, YSj)$$

(23)

3. Research Results

3.1. Model for Obtaining Electricity from Small Photovoltaic Installations for Individual Consumers in Poland

The large increase in energy demand due to rapid economic development and the limited and steadily decreasing supply of traditional energy sources, as well as excessive environmental pollution due to increasing concentrations of dust and gases in the atmosphere, are the main factors driving the growing interest in renewable energy sources [20]. Photovoltaic electricity is considered the most important and promising renewable energy source [21,22]. Developing such a model for future projects can ensure optimal decisions or make new small-scale PV projects viable.

The author’s model of the possibility of obtaining electricity from small photovoltaic installations for individual consumers in Poland is based on a strategy of economic efficiency of small photovoltaic installations up to 15 kW, installed in single-family houses or other small residential buildings, which depends on the installed capacity of the installation. The following strategies, which are feasible for single-family houses, are assumed in this study (Figure 2):

S₁—the production of electricity from a photovoltaic installation is such that there will be no shortage of energy even with a minimum solar insolation of 950 kWh/m² year,

S₂—the production of electricity from a photovoltaic installation is such that there will be no shortage of energy with average or better-than-average insolation over the year 1050 kWh/m² year,

S₃—the production of electricity from a photovoltaic installation is such that this energy will not be lacking only with high insolation during the year, 1150 kWh/m² year,

S₄—We are not setting up a photovoltaic installation, and we buy our electricity from the grid supplier.

Depending on the strategy adopted, it can be assumed that the unit cost of procuring electricity will result from the unit cost of own production from photovoltaic installations, the purchase of electricity from suppliers of this energy, the possibility of taxing this production, or the sale of this energy to a supplier in the event of excess own production.

The lifetime of a photovoltaic installation is approximately 20 years. During this period, the economic and social environment may change significantly. Within this environment, the price of electricity provided by grid suppliers may change, the production of electricity from a photovoltaic installation may be taxed through legislation, and the price of energy purchased by grid suppliers may also change. Therefore, before choosing an appropriate strategy to equip a residential building or other structure with a small-scale PV installation, possible changes in the states of the economic and social environment should be considered. The states of the economic and social environment can change, from possibly favourable to unfavourable. In this way, we obtain an interval of considered states from unfavourable Y₂ via average Y₃ to favourable Y₄. In order to ensure full orthogonality of the considered interval, it was extended by 21.5% of the assumed size of the interval, counting from average to unfavourable very unfavourable Y₁ and from average to favourable Y₅—very favourable [18,23].

In order to select the optimum strategy for retrofitting single-family houses with photovoltaic installations (from the aforementioned to S₁ to S₅), it is necessary to carry out a thorough analysis of the factors affecting the unit cost of obtaining electricity from the installation (using an expert-mathematical method) and then to select the optimum strategy for retrofitting these structures with photovoltaic installations (selection based on individual choice theory) [24].

3.2. Costs of Obtaining Electricity Depending on the Equipment Strategy Adopted for the Photovoltaic Installation and the Economic and Social Environment for This Production

From an economic point of view, small photovoltaic installations that have the possibility of selling surplus electricity are more attractive than self-consumption installations, because the payback period is about 12 years [25]. The change in accounting, also under EU law, has contributed to holding back the market for PV installations due to concerns about energy prices and payback time [26]. Nevertheless, the very high increase in electricity prices is resulting in further consumer interest in this type of energy. Therefore, the magnitude of the costs of producing electricity from small photovoltaic installations should be carefully analysed, depending on the strategy adopted, to equip the user with this installation in Poland. Therefore, it is necessary to determine, using an expert-mathematical method, the amount of the costs of obtaining this energy, depending on the strategy adopted and the state of the economic and social environment in which the user of this installation may find himself.

The minimum number of experts determined by relation No. 3 was 42, in order to have a higher probability of compatibility of the obtained research results in the first stage; therefore, 92 experts participated in the research. The results of the tests carried out using the expert-mathematical method of level II factors are shown in

Table 2. Level II factors. Probability of occurrence of a condition in the environment from very unfavourable Y₁ to very favourable Y₅.

Factor	Status of the Economic and Social Environment for Obtaining Eclectic Energy from Small Photovoltaic Installations for Individual Consumers in Poland	Value of Local Priority [Probability of Occurrence of a Condition in the Environment Y1–Y5 (%)].	Coefficient of Variance Vj
C21	Very unfavourable Y1	4.1	0.14
C22	Unfavourable Y2	10.3	0.12
C23	Average Y3	71.6	0.16
C24	Favourable Y4	10.1	0.19
C25	Very favourable Y5	4.3	0.15
Concordance ratio Θ0		0.624
Criterion χ2		29.78

Source: own research, 2023.

, while the third level is shown in Figure 3.

The value of the local priority of the factors of the second level of Table 2 determined on the basis of the relation No. 10, with the experts’ distribution of 100 points (100%), means the probability of the occurrence of the respective state in the economic and social environment from Y₁ to Y₅ during the whole lifetime of the small photovoltaic installation. As can be seen from Table 2, the probability of the occurrence of the extremely unfavourable Y₁ and extremely favourable Y₅ states is 4.1% and 4.3% respectively, and the unfavourable Y₂ and favourable Y₄ states is 10.3% and 10.1%. The most likely occurrence of average states Y₃ is 71.6%. Figure 3 shows the local priorities for all (five) states in the economic and social environment for obtaining this type of energy.

The figure above shows the local priorities (e.g., C₃₁₁–C₃₁₄ and the other four) representing the percentage contribution of each of the four factors to the cost of acquiring electricity from a PV installation in Poland. As can be seen from Figure 3, the production costs resulting from the installed capacity and insolation of the installation have a significant share in the structure of the total unit cost of acquiring electricity from small PV installations and range from 87% in the average economic and social environment Y₁ to 62% in the extremely unfavourable environment Y₃ and represents a very significant share in each environment. The other three factors analysed have varying shares; in some ambient states, their share is zero.

Table 3. Assumptions for calculating the cost of obtaining electricity from a 10 kW photovoltaic installation (discount rate 6.5%, Q3 2022).

Data to Determine Production Costs	Unit	Assumed Photovoltaic Retrofit Strategies
		S1	S2	S3	S4
Investment expenditure	EUR/kW	1350.0	2210.0	1957.0	0.0
Capacity utilisation rate	%	13.0
O&M costs	EUR/kW year	26.9	25.4	23.91	0.0
Installation disposal costs	EUR/year	74.5	89.4	104.2	0.00
Service life	Years	20	20	20	0.0

Source: own research, 2023.

shows the assumptions for determining the total cost of electricity production from small-scale PV installations for residential consumers. The following assumptions were made for this calculation: capital expenditure, O&M costs, installation disposal costs, operating time, and capacity utilisation rate.

Based on relation 1, we can determine the amount of the specific costs of electricity production resulting from the installed capacity and insolation of the installation, the amount of which does not change depending on the economic and social environment, but will vary depending on the strategy adopted for the power of the photovoltaic installation installed (from S₁ to S₄), Figure 1.

Knowing the magnitude of unit production costs resulting from the installed capacity (unit calculated cost of electricity) calculated on the basis of relation 1 for the adopted strategy S_1–3 and the structure of unit costs of obtaining electricity from small photovoltaic installations, Figure 3 (local priorities level III), we can determine, on the basis of relation 2, the total unit costs of obtaining electricity from small photovoltaic installations depending on the changing economic and political environment Y_1–5 (

Table 4. Calculation of electricity procurement costs depending on the procurement strategy adopted and the state of the economic and production environment in Poland.

	Unit	State of the Economic and Social Environment
		Y1	Y2	Y3	Y4	Y5
Sunshine	[kWh/m2 year]	900.0	950.0	1050.0	1150.0	1200.0
S1—the production of electricity from a photovoltaic installation is such that this energy will not be in short supply with a minimum insolation of 950 kWh/m2 per year.
Unit total acquisition costs	[EUR/kWh]	0.26	0.22	0.14	0.04	−0.06
S2—the production of electricity from a photovoltaic installation is such that there is no shortage of this energy with an average annual insolation of 1050 kWh/m2 year.
Unit total acquisition costs	[EUR/kWh]	0.25	0.23	0.19	0.09	0.05
S3—the production of electricity from a photovoltaic installation is such that this energy will not be in short supply with a high annual insolation of 1150 kWh/m2 year.
Taxation of production	[EUR/kWh]	0.02	0.02	0.02	0.00	0.00
Unit total acquisition costs	[EUR/kWh]	0.24	0.21	0.14	0.03	0.00
S4—all electricity is purchased from a network supplier
Price of etheric energy from supplier	[EUR/kWh]	0.41	0.36	0.20	0.16	0.14
Probability of state in the economic and social environment (Table 2 Local Priorities Level II)		4.1	10.3	71.6	10.1	4.3

Source: own research, 2023.

). The spectacular development of PV power generation in Poland is due to hitting the right time window and lowering the costs of the technology, but above all, it is based on stakeholder cooperation and trust in the regulatory environment [11].

Optimal allocation models developed by other authors have established a strategy in which carbon emissions have an advantage over the total cost of PV electricity production. A comparative analysis was carried out using a strategy that considers the advantage of total cost over carbon emissions. The case study presented in the literature shows that the cost of PV electricity production decreases as a result of the lack of carbon dioxide emissions decreasing, compared to a conventional electricity-production strategy. Thus, the carbon emissions of the system are significantly reduced, and the energy efficiency is improved [2,27].

Therefore, in order to optimally equip residential consumers with PV, the optimal strategy for equipping these users with small-scale PV installations should be identified.

3.3. Optimal Strategy for Equipping Users with Small-Scale Photovoltaic Installations

As indicated by studies conducted in various countries around the world, there is currently a need to search for optimal strategies to equip users with energy installations that reduce environmental pollution and improve the economic efficiency of the use of energy installations [27,28].

Therefore, the authors decided to determine the optimal procurement strategy for individual electricity consumers, who have the option of obtaining electricity from grid suppliers or photovoltaic installations, based on a cost analysis.

As previously signalled in Figure 2, four possible strategies for equipping the user with photovoltaic installations S-S₁₄ were assembled in this analysis. The determination of the optimal strategy for equipping the user with small-scale PV was carried out on the basis of an analysis of the total unit cost of acquiring this energy in a varying economic and social environment, from the extremely unfavourable environment Y₁ to the extremely favourable environment Y₅ (Table 4, columns 3–7).

One of these criteria, i.e., the maximum average win, takes into account the expert-determined probability of a given state in the economic and social environment (Table 4), second-level local priorities from C₂₁ to C₂₅), and the other two criteria in this group, i.e., maximum pessimism and the pessimism–optimism criterion, do not take this probability into account (Equations (15) and (16)).

If one and the same strategy is the most favourable for all criteria considered (most of the minimum values fall on this strategy), then, this is the optimal strategy for equipping the user with a small PV plant [23]. In another situation, the one on which the majority of the minimum values fall is taken as the optimal one. If the investor, on the basis of his own reflections or calculations (he likes risks or does not like risks), chooses one of the proposed strategies, he is of course entitled to do so and therefore bears the corresponding risk of incurring certain losses or additional benefits depending on the economic and social environment in which the installation will operate.

4. Discussion

After determining the most advantageous strategy for supplying single-family homes with electricity, according to five individual selection criteria, i.e., maximum average win, minimum average risk, maximum pessimism, minimum risk, and pessimism–optimism, and comparing them with each other and then comparing the individual strategies with each other, the final strategy should be chosen, which will be the optimal strategy in terms of the size of the total unit cost of obtaining electricity from photovoltaic installations and the size of the risk associated with the adopted strategy for equipping the user with a small photovoltaic installation, Figure 4.

As can be seen in Figure 4, the optimal strategy for equipping users with small photovoltaic installations is strategy S₁, in which electricity production from photovoltaic installations is so high that there will be no shortage of energy even with minimum sunlight of 950 kWh/m² per year. In this strategy, the function has minimum values for the three criteria considered, while for the other two, i.e., the maximum average value and minimum risk criteria, it is equal to strategy S₃. It should be emphasised here that this is the optimal long-term strategy (over the entire lifetime of the photovoltaic installation, i.e., 20 years). However, with greater sunlight, it will produce more electricity than the energy users will need. The rational use of electricity in single-family homes using the S₁ strategy can be achieved, for example, through the use of air conditioners (if they have not been used before). Air conditioners can heat the house when outside temperatures are low and, conversely, when temperatures are very high, they can cool the rooms in the house. The cooling function of air conditioners will be particularly important, as this is usually when there is high sunlight and correspondingly high electricity production by photovoltaic installations.

Therefore, opportunities for selling it or finding a rational use for it by the users of these single-family houses should be sought.

The least favourable strategy for equipping single-family houses with small photovoltaic installations is strategy S₄. In this strategy, single-family houses are not additionally equipped with photovoltaic installations and use only electricity supplied by the network provider. This is mainly due to constant increases in electricity prices. It can be assumed here that electricity prices will continue to rise in the coming years (as indicated by forecasts from both research centres and other institutions involved in price forecasting). Therefore, the electricity supply options for single-family homes should be diversified as much as possible, which can be achieved by installing photovoltaic systems on these homes [28]. Diversifying the electricity supply for single-family homes involves potential additional costs related to the installation of photovoltaic systems. Another potential risk resulting from this diversification is the volatility of the law regarding the possibility of installing such a system and, equally, using it. Users of these installations must also take into account the possibility of taxation.

Photovoltaics is one of the most effective renewable sources of energy. However, it has a significant disadvantage. Large amounts of electricity are produced by these installations on days when there is a lot of sunshine, but at night and on days with little sunshine, these installations do not produce any energy or produce very little. Therefore, energy storage is a very important issue in this type of production. These can be various types of storage facilities, from the smallest ones installed in single-family homes to very large ones organised by large economic entities. Therefore, in the near future, the authors plan to address the economic efficiency of different sizes of energy-storage facilities for photovoltaic installations.

5. Summary and Conclusions

In this study, an attempt was made to determine the optimum strategy for equipping individual consumers with small photovoltaic installations with an installed capacity of up to 15 kWp. For this purpose, a model of the possibility of producing this energy was built for four possible strategies for equipping individual consumers with PV installations. In each of the strategies adopted for the analysis, the following is true: S₁—the production of electricity from a photovoltaic installation is large enough that this energy will not be lacking even at a minimum solar insolation of 950 kWh/m² year, S₂—the production of electricity from a photovoltaic installation is large enough that this energy will not be lacking at an average solar insolation of 1050 kWh/m² year, S₃—the production of electricity from a photovoltaic installation is large enough, and S₄—we do not install a photovoltaic installation and buy electricity from a grid supplier, and we may encounter a changing economic and social environment from extremely unfavourable-Y₁, through unfavourable-Y₂, medium-Y₃, favourable-Y₄, to very favourable-Y₅.

The probabilities of the states in the economic and social environment over the lifetime of the PV installations (20 years) are, respectively, as follows: Y₁—4.1%, Y₂—10.3%, Y₃—71.6%, Y₄—10.1%, Y₅—4.3%. In the structure of the total costs of obtaining electricity, depending on the strategy adopted and the state in the economic and social environment, the share of the unit cost of production of LCOE represents from 87% to 55%, the sale of energy to the grid from 63% to 0.0%, the purchase of energy from the grid from 27% to 0.0%, and the taxation of production from 13% to 0.0%.

The total unit electricity acquisition costs for individual consumers equipped with photovoltaic installations with an installed capacity of up to 15 kW in Poland, depending on the state in the economic and social environment from Y₁ to Y₅ for the assumed electricity-acquisition strategies are, respectively, as follows: S₁ from 0.26 to gain 0, 06, S₂ from 0.25 to 0.05, S₃ from 0.24 to 0.00, S₄ from 0.41 to 0.14 EUR/kWh.

The optimum strategy for obtaining electricity determined using the theory of individual choice (which is an element of game theory) for individual consumers in Poland is strategy S₁. This strategy generates the lowest costs of obtaining electricity, and in an extremely favourable economic and social environment, Y₅ may also bring profits, and it generates the lowest risk of obtaining this energy, according to the criteria analysed. Therefore, small electricity consumers should install photovoltaic installations that produce enough electricity from the installation that there is no shortage of this energy even with a minimum insolation of 950 kWh/m² per year. The S₁ strategy is the least favourable for users, in which the consumer only uses electricity from the grid supplier. This is due to the very unstable price situation for consumers of this energy from network suppliers and the price forecasts for this energy for the coming years.