Assessment of the Gross, Technical and Economic Potential of Region’s Solar Energy for Photovoltaic Energetics

Abstract

A great number of factors determining the development of photovoltaics are associated with the assessment of possible volumes of solar energy use in correlation with the technical and economic characteristics of photovoltaic equipment. An appropriate assessment of solar energy potential that applies universally to any subsequent use option still remains a crucial task. This work is devoted to the assessment and analysis of the gross, technical and economic potentials of solar energy for photovoltaics. The smart analysis includes the basic program working in the context of connection to databases and to the programs used for determining required initial data or, as a limited option, in the context of full or partial initial data input by the user. Therefore, optimally, a smart network is formed, which for the purposes of obtaining the values of potentials, uses the most up-to-date values of initial data and other required information. This work sets out the tried and tested assessment program for the potentials of solar energy available in large and medium areas. The proposed approach to the analysis of solar energy potential in a region makes it possible to secure a high degree of assessment reliability which can be used for more detailed calculations, including the potentials analysis for a specific point on the ground or a specific type of PV system.

1. Introduction

This work is devoted to the assessment and analysis of the gross, technical and economic potentials of solar energy (see Figure 1) for generating electricity through photovoltaic conversion.

Constant improvement of PV equipment is not the only factor affecting the increasing use of solar energy for electrification and the expanding implementation of PV systems. A great number of factors determining the development of photovoltaics are associated with the assessment of possible volumes of solar energy use in correlation with the technical and economic characteristics of PV equipment.

The values of solar energy potentials for different regions are necessary for the development of regional, state and interstate development programs, ecological programs, short-term and long-term forecasts, legislative instruments, and so forth [1,2].

Figure 1. Three types of renewable generation potential (traditional ratio) [3].

Figure 1. Three types of renewable generation potential (traditional ratio) [3].

The assessment of potentials is important both for implementing individual PV systems and for building and operating solar power plants [4,5,6,7,8]. A correct assessment of potentials ensures maximally accurate forecasts at a macrolevel and optimal solutions on the placement, priority and financing of PV system implementation, interoperability between PV systems and other equipment, systems and facilities and choice of top-priority PV technologies [9,10].

Over recent years, much interesting research was pursued with respect to both worldwide assessment of solar energy potential and assessment of solar energy potential in individual regions [3,11,12,13,14,15,16,17,18,19,20,21,22,23]. That notwithstanding, an appropriate solar energy potentials assessment that is universal for various subsequent use options still remains a crucial task.

This is all the more so as solar energy is the only one from among the renewable energy sources for which a change in the traditional proportion between the gross, technical and economic potentials (see Figure 1) is possible. Thanks to the development of integrated PV equipment (IPV), primarily building-integrated photovoltaics (BIPV) and building-applied photovoltaics (BAPV), the area for technical potential calculation in populated regions may be considerably greater than the Earth’s surface area used for calculation of the gross potential. That is, taking into account BIPV and BAPV, the technical potential of solar energy may be greater than the gross potential (see Figure 2).

The analysis showed that even the most evident indicator, i.e., the unit gross potential (that is equal to solar irradiation), is evaluated in various ways for one and the same territory in different sources.

For the purposes of building new PV facilities, a wide variety of estimates are proposed: from very crude and rough (as, for example, in [24]) to more or less accurate [25,26,27,28]. In [24] the technical potential of solar energy in a region, if planar modules are used, is estimated as the solar energy arrival at 1/5 of the region’s area multiplied by the modules’ efficiency factor. In [25] the gross, technical and economic potentials of solar energy are determined for 26 different regions of the Earth subject to various possible restrictions on the surface.

In [26] the values of potentials are computed using the geographic information system for the Krasnodar Region. In [27] the solar energy potentials in the Greater Mekong area are considered for the purpose of forecasting the cost of energy received from any kind of renewable energy sources (including solar energy). In [28] the NASA data on the solar energy arrival and temperature are used. RETScreen software was used to calculate the economic potential for 22 different locations in Libya. The best location was found for a network solar power plant. In [29] it is proposed to use roofs for placing solar modules on them. Numerical simulation was used to compute the gross and technical potentials for 11 regions in the world. It is noted that the use of roofs for generating electric and heat energy may result in a considerable saving of heat and electric energy.

Top-down methodology [30,31] takes into account mineral reserves, land competence and other limits which are not considered by other authors. Such an approach is indeed important for global estimates for a long period of time. In case of relatively short-term estimates, the consideration, for instance, of mineral reserves makes no sense except if their development and use is fully prepared for launch. The research of the same authors [32] is one of those we based our own on when elaborating out approach. We completely agree that “many of the estimations found in the literature are hardly compatible with the rest of human activities.” However, the meaningful constraints on the use of solar energy for photovoltaic energetics, which in their opinion exist, do not relate to the solar energy.

2. General

The smart analysis includes the basic program working under the condition of connection to databases and to the programs of determining required initial data or, as a limited option, under the condition of full or partial initial data input by the user.

Thus, in the optimal variant, a smart network is formed, which for the purposes of obtaining the values of potentials uses the most up-to-date values of initial data and other required information that is received, processed and calculated in parallel with the calculation of potentials.

A program extension is envisaged which enables a more detailed analysis, both for a territory of a lesser area and for a greater amount of initial data or their variants that is taken into account, and also a comparative outcome analysis using the specified assessment criteria. In this case, the matrices of potentials values for various options and comparison results are released at the output.

This article provides the description of the basic analysis algorithm (“central steps”).

Figure 3 shows the general diagram of the calculation logic. If the gross (GP), technical (TP) and economic (EP) potentials are determined within the same task, the calculation is performed sequentially starting from GP. While calculating the subsequent potential, the calculation results for the previous potentials are used, for example the calculated values of solar irradiation on the active surface of PV modules, their operating temperatures, and so on.

A region is the total of K areas, or zones, in which the density of incoming solar energy and the Earth’s albedo, as well as geographical, climatic and weather conditions are homogenous throughout the total area of the zone. Such zones have the linear dimensions of up to ≈200 km. The number of zones in the region, their locations and areas are set out in a separate table.

The results are released as the values of potentials (kW∙h/yr), where the calculated area is determined, and/or, if more universally, as the specific values of potentials (kW∙h/m²∙yr).

The technical and economic potentials are determined considering BIPV.

The minimum scope of calculations is as follows: month, zone, the most popular type of PV modules, overall economic indicators of industrial and domestic consumers.

The calculations are performed for two ultimate options of solar irradiation on the active surfaces of PV modules based on the adopted variants of their orientation and installation: a permanent installation at the optimal tilt angle to the horizon and a continuous tracking of the sun when using solar radiation concentrators. The program assumption is that the modules operate with free air movement around them, i.e., they are installed in an open rack.

The extensions allow calculations to take into account additional factors influencing the module irradiance and temperature by connecting to other programs or values/coefficients entered by the user, e.g., to consider air pollution in the region and the pertinent change in the radiation spectrum.

3. Determining the Gross Potential of Solar Energy in a Region

3.1. General Formula for Calculating the Potential

The gross (theoretical) potential of solar energy in a region for photovoltaic energetics, hereinafter the “GP” (kW∙h/yr), is the long-time average total energy of solar radiation arriving at the region’s area in one year.

The gross potential of the region is the sum of gross potentials of the zones forming it:

$$GP=\sum G{P}_j$$

(1)

Depending on the volume and nature of the initial data used, the calculation of the gross potential of solar energy in a zone is performed using the following three options.

3.2. Determining the Gross Potential of Solar Energy in a Zone Subject to the Volume and Nature of the Initial Data Used

3.2.1. There Is a Weather Station or Satellite-Based Measurement Data Are Used in the Zone

If meteorological data on the long-term average of incoming solar energy for each month of the year are available, the total solar irradiation is calculated according to the following formulas:

$$H_j={{\displaystyle \sum }}_{i=1}^{12}{H}_i;$$

(2)

$$H_i=H_{\mathrm{d}i}+H_{\mathrm{def}i};$$

(3)

$$H={{\displaystyle \sum }}_{j=1}^K{H}_j={{\displaystyle \sum }}_{j=1}^K{{\displaystyle \sum }}_{i=1}^{12}{H}_{ij}.$$

(4)

where H_j is the long-time average annual total solar irradiation in the j zone, kW∙h/(m²∙yr); H is the long-time average annual solar irradiation in region, kW∙h/(m²∙yr); H_i is the long-time average annual solar irradiation on horizontal surfaces in the i month of the year, kW∙h/(m²∙mo); H_d i is the long-time average annual direct solar irradiation on horizontal surfaces in the i month of the year, kW∙h/(m²∙mo); and H_def i is the long-time average annual diffuse solar irradiation on horizontal surfaces in the i month of the year, kW∙h/(m²∙mo).

The gross potential of the j zone is determined as follows:

$$G{P}_j=H_j\cdot S_j,$$

(5)

and for the region, Equation (1) has a similar form:

$$GP=H\cdot S.$$

(6)

In Equations (5) and (6), S_j and S are total areas of the j zone and region, respectively (m²).

The calculated values of H_i, H_di and H_defi, H_j, GP_j and H (if necessary) are used to create a table. In addition, the meteorological data (if any) on the root-mean-square dispersion of solar irradiation (as absolute values or percentage) are inserted into the table.

Similarly, a calculation is performed using the satellite-based data, for example, the NASA database. However, when choosing such a calculation option, it should be taken into account that these data do not involve to the full extent the peculiarities of solar radiation arriving on the active surface of PV modules.

3.2.2. There Is No Weather Station in the Zone

Where there are no meteorological data on solar irradiation but the neighboring weather station data are known and may be used for determining the average values of sunshine duration $\langle t_{\mathrm{s}i}\rangle$ for each month of the year, H_i is determined in the following manner.

To calculate the solar irradiation, an indirect method is employed through the use of the radiation data obtained at the neighboring weather stations and in the adjacent territories and with the application of the Angstrom formula [33] improved by [34]. The average monthly total solar irradiation is determined according to the following formulas:

$$H_i=H_{0i}\cdot \left(a_i+b_i\cdot \frac{\langle t_{si}\rangle }{t_{0i}}\right) ;$$

(7)

$$H_{0i}=\langle G_{Mi}\rangle \cdot \langle \cos {\theta }_i\rangle \cdot t_{0i },$$

(8)

where for the i month of the year: H_0i is the long-time average annual solar irradiation on horizontal surfaces under clear sky, kW∙h/(m²∙mo); a_i and b_i are the empirical coefficients (a_i + b_i = 1) calculated for k trapeziums by which the analyzed territory is broken down for each month (see, for example, [24]); $\langle G_{Mi}\rangle$ is the average intensity of solar radiation on the active surface of PV modules for atmospheric mass M (W/m²); θ_i is the angle between the sun vector and the zenith vector (angle of incidence on horizontal surfaces); $\langle t_{\mathrm{s}i}\rangle$ is the empirical duration of sunshine for the locality in question during the month, h/mo; and t_0i is the astronomically possible duration of sunshine for the given locality during the month, h/mo.

The factors of Equation (8) for the i month are determined using the following formulas:

$$\langle G_{Mi}\rangle =G_1\cdot {\left(\frac{G_1}{G_0}\right)}^{M_i-1}=1000\cdot {\left(\frac{1000}{1360}\right)}^{M_i-1};$$

(9)

$$M_i=\frac{2}{\sqrt{{\langle \cos {\theta }_i\rangle }^2+0.06}+\langle \cos {\theta }_i\rangle } ;$$

(10)

$$\langle \cos {\theta }_i\rangle =\sin {\delta }_i\cdot \sin \phi +\cos {\delta }_i\cdot \cos \phi \cdot \frac{\sin {\omega }_{\mathrm{s}-\mathrm{s}i}}{{\omega }_{\mathrm{s}-\mathrm{s}i}};$$

(11)

$$t_{0i}=12\cdot n_i\cdot \left(\frac{2\cdot {\omega }_{\mathrm{s}-\mathrm{s}i}}{\pi }\right) ,$$

(12)

where G₀ = 1360 W/m² is the solar constant; G₁ is the direct solar irradiance corresponding to terrestrial solar radiation in the southerly latitudes at sea level and on a clear day—these are approximately the conditions of atmospheric mass M = 1, G₁ = 1000 W/m²; δ is the mean solar declination that is determined by the Cooper formula [35] (see

Table 1. Average monthly solar declination.

Month	I	II	III	IV	V	VI	VII	VIII	IX	X	XI	XII
δ, rad	−21.1	−14.1	−2.8	9.2	18.7	+23.1	+21.3	+13.5	+2.0	−9.6	−18.7	−23.5

), rad; ω_s-si is the sunrise–sunset angle in the i month, rad; and n_i is the number of days in the i month.

The angle ω_s-si is determined by the following formula:

$$\cos {\omega }_{\mathrm{s}-\mathrm{s}}=-tg\delta \cdot tg\phi$$

(13)

The Calculation Procedure

The parameter values of φ, δ_i, a_i, b_i, n_i and t_si for each month of the year in this zone are the initial data for calculations.

The parameter values of ω_s-si, t_0i, $\langle \cos {\theta }_i\rangle$, M_i, $\langle G_{Mi}\rangle$, H_0i and H_i are consequently calculated for each month of the year in this zone using Equations (7)–(13).

Using Equations (2), (4) and (5), the calculation of H_j, H and GP_j is completed and the table of values of H_i, H_j, H and GP_j is formed.

3.2.3. The Data on Solar Irradiation on Inclined Surfaces Are Available

If there are only data on long-time average annual solar irradiation in each month on a surface inclined at the angle β to the horizontal and oriented to the south, H_ti (W∙h/(m²∙mo)), the monthly solar irradiation H_i is calculated using the following formulas:

$$H_{\mathrm{t}i}=H_{\mathrm{d}i}\cdot \left(\frac{{\omega }_{\max }}{{\omega }_{\mathrm{s}-\mathrm{s}i}}\right)\cdot \frac{\langle \cos {\xi }_i\rangle }{\langle \cos {\theta }_i\rangle }+H_{\mathrm{def}i}\cdot \frac{1+\cos {\beta }_{\mathrm{opt}i}}{2}+{\rho }_i\cdot \left(H_{\mathrm{d}i}+H_{\mathrm{def}i}\right)\cdot \frac{1-\cos {\beta }_{\mathrm{opt}i}}{2};$$

(14)

or

$$H_{\mathrm{t}i}=H_i\left(\left(1-{\epsilon }_i\right)\cdot \left(\frac{{\omega }_{\max }}{{\omega }_{\mathrm{s}-\mathrm{s}i}}\right)\cdot \frac{\langle \cos {\xi }_i\rangle }{\langle \cos {\theta }_i\rangle }+{\epsilon }_i\cdot \frac{1+\cos {\beta }_{\mathrm{opt}i}}{2}+{\rho }_i\cdot \frac{1-\cos {\beta }_{\mathrm{opt}i}}{2}\right).$$

(15)

where ω_max is the astronomically possible hour angle of maximum insolation of the inclined surface, rad; and for the i month of the year: β_{opt i} is the optimal angle of inclination of the PV module active surface to the horizontal; ξ_i is the angle between the sun vector and the normal line to the inclined surface (angle of incidence on the inclined surface) oriented to the south; ε_i = H_di/H_i is the share of diffuse radiation in the total solar radiation; and ρ_i is the ground surface albedo.

The average value $\langle \cos {\xi }_i\rangle$ determines the average flow of solar radiation falling on the inclined surface in the day’s interval of hour angles of the inclined surface insolation Δω:

$$\langle \cos {\xi }_i\rangle =\cos \left(\phi -{\beta }_{\mathrm{opt}i}\right)\cdot \cos {\delta }_i\cdot \langle \cos \omega \rangle +\sin \left(\phi -{\beta }_{\mathrm{opt}i}\right)\cdot \sin {\delta }_i;$$

(16)

$$\langle \cos \omega \rangle =\frac{\sin {\omega }_{\max }}{{\omega }_{\max }},$$

(17)

where ω = πt/12 is the hour angle of solar motion that is equal to 0 at solar noon; each hour of time t corresponds to 15° of longitude, and the values of an hour angle before noon are deemed positive, while those after noon are deemed negative.

For the ‘winter’ half of the year (δ ≤ 0) ω_max = ω_s-s. For the ‘summer’ half of the year (δ ≥ 0) ω_max = ω_S, where ω_S is the hour angle of the sun meeting the condition of cosξ = 0, and

$$\cos {\omega }_{\mathrm{S}}=-tg\delta \cdot tg\left(\phi -\beta \right).$$

(18)

For calculating the optimal value of the inclination angle, β_opt is used, which is determined as follows:

$${\beta }_{\mathrm{opt}i}=\phi -{\delta }_i.$$

(19)

There are two options, depending on the chosen accuracy of calculations: β_{opt i} = β_opt = constant for all months of the year and β_{opt i} = variable.

The Calculation Procedure

The parameter values of φ, β_{opt i}, δ_i, ε_i, ρ_i and H_ti for each month of the year in this zone are the initial data for calculations.

The parameter values of ω_s-si, ω_S, $\langle \cos {\theta }_i\rangle$, $\langle \cos {\xi }_i\rangle$ and H_i are consequently calculated for each month of the year in this zone using Equations (11), (13)–(16) and (18).

Using Equations (2), (4) and (5) the calculation of H_j, H and GP_j is performed and the table of values of H_i, H_j, H and GP_j is formed.

Upon the potential of each zone being calculated, the gross potential of the region GP is calculated using Equation (1) as the sum of the gross potentials of the zones.

4. Determining the Technical Potential of Solar Energy in a Region

4.1. General Formulas for Calculating the Potential

The technical potential of solar energy in a region for photovoltaic energetics, hereinafter the “TP” (kW∙h/yr), is the long-time average electrical energy which may be generated in the region through photovoltaic conversion of solar radiation in one year with the current state of the art in science and technology and subject to environmental compliance.

The technical potential of solar energy is the sum of the technical potentials of thermal energy and electrical energy obtained through conversion of solar radiation.

The technical potential of the region is the sum of the technical potentials of the zones forming a part of it:

$$TP={\displaystyle \sum }T{P}_j.$$

(20)

The technical potential of a zone is determined as follows:

$$T{P}_j={{\displaystyle \sum }}_{i=1}^{12}T{P}_i,$$

(21)

where the technical potential of the i month in the j zone, subject to the chosen position of the surface at which solar radiation arrives, is equal to:

$$T{P}_i=H_i\cdot {\eta }_1\left(1-\chi \cdot \left(T_i-T_1\right)\right)\cdot S_{TPj};$$

or

$$T{P}_i=H_{\mathrm{t}i}\cdot {\eta }_1\left(1-\chi \cdot \left(T_i-T_1\right)\right)\cdot S_{TPj}.$$

(22)

where S_TPj is the portion of the j zone area which economically and environmentally is reckoned to be expedient for installing PV systems (without regard to BIPV, AgroPV, etc.), m²; it is equal to such portion of the total area of the zone S_j as is left after deducting the territories in which the placement of PV systems/PV modules is prohibited: T_i is the average monthly operating temperature of a PV cell, K; T₁ is the operating temperature of a PV cell in the agreed standard lighting conditions (G_M1 = 1000 W/m²), T₁ = 25 °C = 298.15 K; η₁ is the efficiency factor in the agreed standard lighting conditions; χ is the temperature gradient which depends mainly on the PV cell type and design.

The calculation takes into account that the efficiency of PV cells, PV modules and, accordingly, PV systems is a composite function of lighting intensity and spectral structure, and the operating temperature of elements which may change in a random way. The calculation logic is based on the fact that as the operating temperature T₁ rises, the efficiency factor decreases (see, for instance [36,37]). This is chiefly originated by the linear drop in the no-load voltage, V_o.c, because of the sharp exponential growth in the reverse saturation current, J₀, and the relevant decrease in the volt-ampere characteristic filling factor, FF [38]. Contemporaneously, the photocurrent, J_f, increases, but very weakly, so that the full temperature gradient of the efficiency factor turns out to be negative and the explicit temperature dependence of the efficiency factor for lightning, AM 1, is represented as follows:

$$\eta ={\eta }_1\cdot \left[1-\chi \cdot \left(T-T_1\right)\right].$$

(23)

For the purpose of uniform valuation, the values of the parameters in the agreed standard lighting conditions, AM 1 (G_M1 = 1000 W/m²), are used at an operational temperature of T₁ = 25 °C = 298.15 K.

The average monthly temperature of a PV cell [K] is equal to:

$$T_i=\frac{\frac{H_i}{t_{\mathrm{PV}i}}\cdot \left[\alpha -{\eta }_1\cdot \left(1+\chi \cdot T_1\right)\right]+\langle \lambda \rangle \cdot \langle T_{\mathrm{amb}i}\rangle }{\langle \lambda \rangle -\frac{H_i}{t_{\mathrm{PV}i}}\cdot {\eta }_1\cdot \chi },$$

(24)

where α is the integrated coefficient of solar radiation absorption by the photovoltaic elements of PV cells; λ is the coefficient of heat transmission from the surface of PV modules or the radiator of concentrator photovoltaic (CPV) modules (W/(m²∙K); $\langle T_{\mathrm{amb}i}\rangle$ is the average monthly environmental temperature during daylight (during the operation of PV systems), K; t_PVi is the average duration of PV systems operation in the i month, h; t_PVi ≤ t_si and is determined based on the value of t_si and the number of clear, overcast and partly overcast days in the zone. If there are no data on the number of clear, overcast and partly overcast days in the zone, it is taken that t_PVi = t_si.

Should the potential be determined for any specific types of PV modules which, for example, underwent tests under the IEC 61853-2 [39], i.e., the empirical ratios of module operating temperature to wind speed are known for them, then the average monthly temperature of a PV module is calculated as follows:

$$T_i=\frac{1}{\left(u_0+u_1\langle v_i\rangle \right)}\cdot \frac{H_i}{t_{\mathrm{PV}i}}\hspace{1em}+\langle T_{\mathrm{amb} i}\rangle ,$$

(25)

where u₀ is the coefficient reflecting the influence of irradiance on the PV module temperature; u₁ is the coefficient reflecting the influence of wind speed on the PV module temperature; $\langle v_i\rangle$ is the average value of wind speed in the i month.

With this, it is taken that the temperature difference (T_i − T_amb) actually does not depend on the environmental temperature and generally is directly proportional to irradiance at 400 W/m² or more. In general, both coefficients depend on the method of installation of a PV module.

4.2. Procedure for Calculation

Unless otherwise specified, the initial data for calculations are the values of the parameters characterizing the current technical level of the most common flat silicon PV modules: α, η₁, χ and <λ>, as well as the data for calculating the solar irradiation in respect of each zone as set out in Section 3.2, and $\langle T_{\mathrm{amb} i}\rangle$, data on the number of clear, overcast and partly overcast days in the zone.

The calculations are performed for each zone as follows.

The total solar irradiation on the active surface of PV modules, H_i or H_ti, is calculated as described in Section 3.2.3. The average monthly operating temperature in the j zone is calculated.

The technical potential of the i-month, TP_i, is calculated.

The technical potential, TP_j, of the j zone is determined by summing up all the months, and the table of values of T_i, H_i or H_ti, TP_i and TP_j, is formed.

Upon the technical potential of each zone being calculated, the technical potential of the region TP is calculated by Equation (20) as the sum of the technical potentials of the zones.

The algorithm allows several options for the initial data values to be proposed, for instance, for different technologies of PV module manufacture for all zones or different values for different zones, and then comparison and selection or valuation to be carried out, as well as taking into consideration the particularities of BIPV, BAPV and AgriPV.

5. Determining the Economic Potential of solar Energy in a Region

5.1. General Formulas for Calculating the Potential

The economic potential of solar energy in a region for photovoltaic energetics, hereinafter the “EP” (kW∙h/yr), is the amount of annual generation of electrical energy in the region through the use of photovoltaic conversion of solar radiation, the generation of which energy is economically justified for the region at the existing level of energy and fuel production, transportation and consumption prices and subject to environmental compliance.

The economic potential of the region is the sum of the economic potentials of the zones forming a part of it:

$$EP={\displaystyle \sum }E{P}_j.$$

(26)

The economic potential of the j zone is determined as the sum of the potentials of each month of the year:

$$E{P}_j={{\displaystyle \sum }}_{i=1}^{12}E{P}_i.$$

(27)

Therefore, the economic potential of the i month for each zone is determined as follows:

$$E{P}_i=W_i\cdot S_{EPJ},$$

(28)

where W_i is the amount of electric energy generated by PV systems per unit area of the active surface of PV modules or aperture of CPV modules (unit generation) in the i-month, kW∙h/(m²∙mo) [40,41]; S_EPj is the economically expedient area of PV systems placement in the j zone, m².

The generation of the PV systems installed in the territory is calculated according to the following formulas:

The j zone in the i month

$$W_i=H_{\mathrm{PV}i}\cdot {\eta }_1\cdot \left[1-\chi \cdot \left(T_i-T_1\right)\right];$$

(29)

annual generation of the j zone

$$W_j={{\displaystyle \sum }}_{i=1}^{12}{W}_i;$$

(30)

in the n year

$$W_{jn}=k_n{{\displaystyle \sum }}_{i=1}^{12}{W}_i;$$

(31)

annual generation of the region

$$W={{\displaystyle \sum }}_{j=1}^K{W}_{jn}=k_n{{\displaystyle \sum }}_{j=1}^K{{\displaystyle \sum }}_{i=1}^{12}{W}_{ijn},$$

(32)

where H_PVi is the solar irradiation on the active surface unit of PV modules or aperture of CPV modules in the i month and k_n is the maximum capacity degradation of PV or CPV modules in the n year, n = 1, 2, …, t_s.

There are two ultimate options for determining the economic potential.

In the first option, PV modules are considered to be permanently oriented at the optimal angle of inclination to the horizontal β_opt. With this, H_PVi = H_ti is determined by Equations (14) or (15) provided that the average value $\langle \cos {\xi }_i\rangle$ is determined as follows:

$${\langle \cos {\xi }_i\rangle =\cos \left(\phi -{\beta }_{\mathrm{opt}i}\right)\cdot \cos {\delta }_i\cdot \frac{\sin {\omega }_{\mathrm{m}}}{{\omega }_{\mathrm{m}}}+\sin \left(\phi -{\beta }_{\mathrm{opt}i}\right)\cdot \sin {\delta }_i|}_{{\beta }_{\mathrm{opt}i}=\phi -{\delta }_i}.$$

(33)

In the second option, a PV system is composed of CPV modules [42] with continuous tracking of the sun and:

$$H_{PVi}=H_{\mathrm{n}i}=\frac{H_{\mathrm{d}i}}{\langle \cos {\theta }_i\rangle }+\frac{H_{\mathrm{def}i}}{K},$$

(34)

where H_ni is the solar irradiation on a normally oriented surface; K is the solar radiation concentration ratio.

5.2. Determining the Economic Parameters of PV System Use

The annual economic effect of using PV systems in the region E (€ or $, etc.) is determined as the sum of economic effects of using PV systems in each zone:

$$E={{\displaystyle \sum }}_j{E}_j.$$

(35)

The annual economic effect for each zone is as follows:

$$E_j=t_{\mathrm{s}}\left(E{P}_j-S_{EPj}{W}_{\mathrm{crit}j}\right)c_{\mathrm{t}.\mathrm{s}.\mathrm{e}j}+t_{\mathrm{s}}{D}_j\cdot \left(c_{\mathrm{loss}j}-c_{\mathrm{t}.\mathrm{s}.\mathrm{e}j}\right),$$

(36)

where t_s is the service lifetime of PV systems, year; c_t.s.e j is the unit cost of energy production using traditional sources subject to the regional fuel cost factor and regional ecological factor, EUR/(kW∙h) or USD/(kW∙h); c_loss j is the unit cost of losses from energy shortage or unit cost of the valuables produced by industry in the j zone, EUR/(kW∙h) or USD/(kW∙h); D_j is the annual electrical energy deficit and/or additional demand of industrial production for electric energy in the j zone, kW∙h/yr; W_crit j is the critical value of annual generation by PV systems determining the field of economic expediency of their use in the j zone, kWt∙h/(m²∙yr).

$$c_{\mathrm{t}.\mathrm{s}.\mathrm{e}j}=r_{\mathrm{t}.\mathrm{s}.\mathrm{e}j}\cdot r_{cj}\cdot c_{\mathrm{t}.\mathrm{s}},$$

(37)

where r_t.s.ej is the regional ecological factor of traditional energy sources; r_cj is the regional factor of traditional source energy cost; c_t.s is the unit cost of energy production using conventional energy sources, EUR/(kW∙h) or USD/(kW∙h).

If monthly data are available, then:

$$c_{\mathrm{loss}j}={{\displaystyle \sum }}_{i=1}^{12}{c}_{\mathrm{loss}i} ,D_j={{\displaystyle \sum }}_{i=1}^{12}{D}_i \mathrm{and} W_{\mathrm{crit}j}={{\displaystyle \sum }}_{i=1}^{12}{W}_{\mathrm{crit}i}.$$

(38)

The critical value of specific generation by PV systems W_critj is determined as follows:

$$W_{\mathrm{crit}j}=r_{\mathrm{e}j}\left(1-\gamma t_{\mathrm{s}}\right)\frac{c_{\mathrm{PV}j}}{t_{\mathrm{s}}-c_{\mathrm{t}.\mathrm{s}.\mathrm{e} j}},$$

(39)

where r_ej is the territorial ecological factor of using PV systems; c_PVj is the unit final cost of PV systems (before the date of putting them into operation), EUR/m² or USD/m² and γ is the standard cost of operation, 1/year.

The payback time for PV systems t_pb (year) is calculated according to the following formula:

$$t_{\mathrm{pb}j}=\frac{r_{\mathrm{e}j}{c}_{\mathrm{PV}j}}{W_j{c}_{\mathrm{t}.\mathrm{s}.\mathrm{e} j}-\gamma r_{\mathrm{e}j}{c}_{\mathrm{PV}j}}.$$

(40)

5.3. Economic Potential Assessment with Different Relationships of Service Lifetime and Payback Time

In accordance with the definition, the economic potential of solar energy in a region for photovoltaic energetics is the energy which may be generated during the year by the PV systems installed in the territory, provided that their economic effect is positive or equal to zero:

$$E\ge 0.$$

(41)

The feasibility analysis of this condition provides for two options.

(1)

Where the service lifetime of PV systems is longer than, or equal to, their payback time:

$$t_{\mathrm{s}}\ge t_{\mathrm{pb}},$$

(42)

i.e., if the specific generation by PV systems is greater than, or equal to, its critical value:

$$W\ge W_{\mathrm{crit}},$$

(43)

then, by virtue of a usual condition:

$$c_{\mathrm{loss}}>c_{\mathrm{t}.\mathrm{s}.\mathrm{e}}.$$

(44)

The economic potential of using PV systems is positive with any number of installed PV systems, no matter how much area fit for this they occupy. This means that in this case it is expedient to use the greatest possible number of PV systems, i.e., the active surfaces of PV modules will occupy the greatest possible area and the economic potential of solar energy in the region for photovoltaic energetics turns out to coincide with its technical potential:

$$EP=TP.$$

(45)

(2)

Where the service lifetime of PV systems is shorter than their payback time:

$$t_{\mathrm{s}}<t_{\mathrm{pb}},$$

(46)

i.e., if the specific generation by PV systems is greater than, or equal to, its critical value:

$$W\le W_{\mathrm{crit}},$$

(47)

then the fulfillment of the condition of Equation (41) is in line with the following limitation of the PV system full capacity:

$$\left(W_{\mathrm{crit}}-W\right)S_{\mathrm{EP}}\cdot c_{\mathrm{t}.\mathrm{s}}\le D\left(c_{\mathrm{loss}}-c_{\mathrm{t}.\mathrm{s}.\mathrm{e}}\right),$$

(48)

and concurrently

$$EP>D.$$

(49)

Provided that the service lifetime is close to the payback time or, more precisely, if:

$$t_{\mathrm{pb}}-t_{\mathrm{s}}\le t_{\mathrm{s}}\frac{D}{TP}\cdot \left(\frac{c_{\mathrm{loss}}}{c_{\mathrm{t}.\mathrm{s}.\mathrm{e}}}-1\right),$$

(50)

then the economic potential, as in case 1, is equal to the technical potential, Equation (45).

If the condition of Equation (49) is fulfilled, i.e., when:

$$W\ge W_{\mathrm{crit}}\frac{c_{\mathrm{t}.\mathrm{s}}}{c_{\mathrm{loss}}},$$

(51)

the economic potential of solar energy is equal to:

$$EP=D{\left(\frac{W_{\mathrm{crit}}}{W}-1\right)}^{-1}\left(\frac{c_{\mathrm{loss}}}{c_{\mathrm{t}.\mathrm{s}}}-1\right).$$

(52)

And finally, in the realm of:

$$W<W_{\mathrm{crit}}\frac{c_{\mathrm{t}.\mathrm{s}}}{c_{\mathrm{loss}}},$$

(53)

the economic potential is equal to zero:

$$EP=0.$$

(54)

where the condition of Equation (53) is fulfilled, the cost of the energy produced by PV systems is so high that the products of industry manufactured with their use do not cover, in terms of cost, the electric power costs, i.e., the use of this power plant is inexpedient.

Even for different zones of the region we can envisage different scenarios of economic potential assessment of the same type of PV systems, which is subject to climate and energy consumption features. Accordingly, the values of zone economic potentials are determined by Equations (45), (52) or (54).

The economic potential of solar energy in the region, EP, is increasingly dependent on the annual amount of energy obtained from the active surface unit of PV modules, W, which is determined by three parameters: critical value of energy generation, W_crit, demand of industry and domestic consumers in the region for electric energy, D/TP, and price, c_loss/c_t.s.e. The typical correlation is shown in Figure 4.

For the purpose of assessment, the notions have been introduced of surplus of economic potential of solar electric energy in the region and zone, ΔEP_j and ΔEP, constituting the difference between the economic potential and the total demand for electric energy in the region/zone:

$$\Delta EP=EP-Q,$$

(55)

and

$$\Delta E{P}_j=E{P}_j-Q_j,$$

(56)

where Q is the region’s total demand for electrical energy, kW∙h/yr, which is the sum of demands for electrical energy on the part of production, Q_p, and domestic consumers, Q_d:

$$Q=Q_{\mathrm{p}}+Q_{\mathrm{d}}.$$

(57)

The zone’s total demand for electric power, Q_j, is determined in a similar way.

If ΔEP > 0 or ΔEP_j > 0, a region or zone, respectively, is an economically justified potential donor of electrical energy.

5.4. Procedure for Calculation

Option 1: PV systems are composed of flat PV modules permanently oriented at an angle of inclination to the horizontal that is β_opt.

The initial data for calculations are the initial data set out in Section 3: service lifetime of PV systems, t_s, maximum capacity degradation, k_n, for a year of service, as well as the values for each zone of c_PV, factors r_t.s.e, r_c, r_e and c_t.s.e, c_loss, D (or c_loss.i, D_i in each month of the year for each zone), values of Q_p, Q_d and Q_pj, Q_dj for the region and each zone, respectively.

The calculations are performed for each zone in the following manner.

The solar irradiation on the active surface of PV modules placed at the angle of inclination to the horizontal, β_opt and H_ti, and the average monthly operating temperature in the zone j, are calculated as described in Section 2 subject to Equation (33). Alternatively, the values which were obtained when calculating the technical potential are used if the calculations of both potentials are performed together.

Consecutive calculations are performed for the generation by PV systems, W_i, for each month of the year (using Equation (29)), annual generation in the j zone, W_j, (using Equation (30)), and generation in the region, W (using Equation (32)) and W_crit (using Equation (39)).

The fulfillment of the conditions for Equations (43), (50), (51), (53) is analyzed and the region of definition of economic potential is determined. The potential value, EP_j, is calculated.

Q_j and the surplus potential ΔEPj are calculated.

Option 2: PV systems are composed of CPV modules which are oriented to, and continuously keep track of, the sun.

The initial values of the parameters are the same as for option 1, with the exception of the PV system cost, c_PV, which, in accordance with the practical data for option 2, is taken to be equal to C/3 with K = 13, and the temperature coefficient of the efficiency factor, which is theoretically and practically less and is taken to be equal to χ = 0.003 K⁻¹.

With regard to each zone, the values of H_ni are calculated by Equation (34) for i = 1, 2, …, 12. Then, calculations are performed similarly to option 1, and EP_j and ΔEPj are determined.

After comparing the values of economic potential under the two calculation options, the greater value is chosen which is taken as the economic potential of solar energy in the zone EP_j.

Upon the economic potential of each zone being determined, the economic potential of the region is calculated by Equation (26) as the sum of economic potentials of the zones.

6. Example of Solar Energy Potential Assessment for Photovoltaic Conversion in the Olkhon District of the Irkutsk Region (52° N and 105° E)

6.1. Climatic Conditions

The Island of Olkhon is situated in Lake Baikal and its geographical coordinates are 52° north latitude and 105° east longitude (see Figure 5). The island area is S = 730 km².

There are weather stations in the settlements of Khuzhir and Uzur, and there was a weather station in Cape Tashkai before 1964. The entire territory of the island is considered a single zone, and the potentials of the Island of Olkhon are calculated as the potentials of a zone.

Some characteristics of solar irradiation on horizontal surfaces and the reflecting power of the earth’s surface are set forth in

Table 2. Direct and diffuse solar irradiation on horizontal surfaces during a 10 h period (from 07 a.m. to 05 p.m.) on clear and partly overcast days, and surface albedo, ρ, by month.

Characteristics		Month
		3	4	5	6	7	8	9
Clear days	Hdi, kW∙h/m2	4.58	5.62	7.25	7.17	6.68	5.82	4.22
	Hdef.i, kW∙h/m2	0.39	1.38	1.01	1.21	1.32	0.96	0.66
Partly overcast days	Hdi, kW∙h/m2	2.15	2.52	3.48	3.34	3.21	2.80	1.96
	Hdef.i, kW∙h/m2	1.35	2.24	2.40	2.48	2.53	2.02	1.41
Surface albedo ρ		0.60	0.20	0.15	0.17	0.17	0.17	0.18

and

Table 3. Number of clear, partly overcast and overcast days, average duration of sunshine $\langle t_{\mathrm{s}i}\rangle$, solar irradiation on horizontal surfaces, H_i, during a 10 h period (from 07 a.m. to 05 p.m.), and share of diffuse radiation, ε, in the sunny months of the year.

Characteristics	Month
	3	4	5	6	7	8	9
Number of clear days	25	15	13	12	13	13	11
Number of partly overcast days	5.9	14.5	17.3	16.1	15.9	15.0	17.0
Number of overcast days	0.1	0.5	0.7	1.9	2.1	3.0	2.0
Average duration of sunshine $\langle t_{\mathrm{s}i}\rangle$, h/mo	356.0	400.8	467.4	461.2	462.0	402.8	345.0
Total solar irradiation Hi, kW∙h/m2∙mo	144.9	74,0.	209.1	194.3	195.3	160.4	94.0
Share of diffuse radiation ε	0.12	0.31	0.26	0.28	0.29	0.27	0.33

. All of the calculations are performed in terms of local time.

The air temperature during daylight hours for different months of the year, to the exclusion of the ‘darkest’ months, November, December and January, is shown in

Table 4. Average monthly air temperature (°C) for each hour of day-time and average monthly ambient temperature (°C) during daylight hours for the time of day 08 a.m. to 08 p.m.

Time of Day	Month
	2	3	4	5	6	7	8	9	10
8 a.m.	−20.1	−12.7	−2.3	4.9	11.1	14.5	14.3	9.5	0.9
9 a.m.	−18.8	−11.1	−1,3	5.5	11.7	14.9	14.7	9.4	1.0
10 a.m.	−17.4	−9.9	−0.6	6.1	12.2	15.2	15.0	9.7	2.8
11 a.m.	−16.1	−8.9	0.1	6.6	12.5	15.6	15.4	10.1	3.3
12 noon	−15.5	−8.1	0.6	6.9	12.9	15.8	15.7	10.3	3.6
01 p.m.	−14.6	−7.4	1.0	7.4	13.3	16.2	16.0	10.6	3.9
02 p.m.	−14.3	−7.0	1.2	7.6	13.6	16.5	16.1	10.7	3.9
03 p.m.	−14.2	−6.8	1.3	7.7	13.7	16.7	16.3	10.8	3.9
04 p.m.	−14.7	−7.0	1.2	7.4	13.5	16.6	16.2	10.7	3.6
05 p.m.	−15.7	−7.6	0.7	7.0	13.2	16.4	16.1	10.5	2.9
06 p.m.	−16,.8	−9.0	−0.2	6.4	12.7	15.9	15.7	9.9	2.3
07 p.m.	−17.3	−10.2	−1.4	5.4	12.1	15.4	15.1	9.4	1.9
08 p.m.	−17.7	−10.9	−2.1	4.4	11.1	14.8	14.6	9.2	1.6
08 a.m. to 08 p.m.	−16.4	−9.0	−0.1	6.4	12.6	15.7	15.5	10.1	2.7

.

For November, December and January there are average temperature data during daylight hours based on the average daily temperature in the pertinent month: $\langle T_{\mathrm{amb}11}\rangle$ = −5 °C; $\langle T_{\mathrm{amb}12}\rangle$ = −11 °C; $\langle T_{\mathrm{amb}1}\rangle$ = −17 °C.

The total solar radiation on horizontal surfaces was calculated using the data from Table 4 (by the number of sunny days) and the data from Table 3. For example, for March the calculation was as follows:

H₃ = (4.58 + 0.39) 25 + (2.15 + 1.35) 5.9 = 144.9 [kW∙h/m²∙mo]

The share of diffused radiation, ε, for each month specified in Table 4 is determined as the ratio of diffuse irradiation to total irradiation for the relevant month and calculated using the data from Table 3 and Table 4. For instance, for March this is:

ε = (0.39 ∙ 25 + 1.35 ∙ 5.9)/144.9 = 0.122

Hence, during the time of year that is the most efficient for using solar energy the average share of diffuse radiation is about 0.3 of the total irradiance. It should be noted that in the autumn and winter period (October to February), the share of diffuse radiation grows sharply, so that the average yearly share of diffuse radiation in the total solar radiation is about 0.4.

For comparison and supplementation of Table 4,

Table 5. Total monthly irradiation, H_ti, and yearly irradiation, H_t, on surfaces tilted at an angle of 45° to the horizontal, Irkutsk.

	Month												Ht, kW∙h/(m2∙yr)
	1	2	3	4	5	6	7	8	9	10	11	12
Hti, kW∙h/ (m2∙mo)	64.7	108.0	160.5	167.1	167.8	170.8	160.1	149.0	135.7	106.5	63.4	46.8	1500.5

provides data on monthly and annual solar radiation on surfaces inclined at 45° to the horizontal in the weather station nearest to the village of Khuzhir, situated in Irkutsk.

6.2. Gross Potential of Solar Energy

In accordance with Equations (1), (2) and (5), for determining the gross potential, it is necessary to obtain the data on monthly solar irradiation on horizontal surfaces, H_i (i = 1, 2, …, 12).

The values of H_i, calculated and set out in Table 4, relate to the seven sunniest months, with i = 3–9. In order to calculate the solar irradiation in other months (i = 1, 2, 10, 11, 12), it is expedient to use the data on monthly irradiation on inclined surfaces, presented in Table 5, in accordance with the methodology described in Section 3.2.3.

Table 6. Average monthly parameters of solar irradiation in wintertime for Khuzhir.

Month	1	2	10	11	12
Solar declination angle δ, rad (Table 1)	−21.1	−14.1	−9.6	−18.7	−23.5
cosωS	0.494	0.322	0.217	0.433	0.557
<cosω>	0.825	0.762	0.722	0.803	0.847
<cosθ>	0.190	0.263	0.307	0.216	0.164
<cosξ>	0.726	0.703	0.684	0.716	0.723
Hi, kW∙h/(m2∙mo) 1	39.7	78.4	83.3	41.6	26.4

¹ In calculation the following values were used: ε = 0.8 and ρ = 0.7.

contains the average monthly parameters for these months, calculated using coefficients from Equations (15)–(18) for the Khuzhir latitude, φ = 52°.

The results of calculating the mean angle of inclination of direct solar radiation to the normal line, <cosθ_i>, and monthly direct solar irradiation on normally oriented surfaces, H_d.ni, during a 10 h period (from 07 a.m. to 05 p.m.) for each month of the year are represented in

Table 7. Average monthly angle of inclination and monthly direct solar irradiation on normally oriented surfaces from 07 a.m. to 05 p.m. of each day in Khuzhir.

Month	1	2	3	4	5	6	7	8	9	10	11	12
<cosθi>	0.190	0.263	0.415	0.574	0.683	0.727	0.710	0.626	0.482	0.307	0.216	0.164
Hd.ni, kW∙h/(m2∙mo)	41.8	59.6	304.6	210.3	226.2	192.4	194.3	188.0	130.4	54.3	38.6	32.2

. The values of H_d.ni were calculated using Equation (34) using the following formula:

$$H_{\mathrm{d}.\mathrm{n}i}=\frac{H_{\mathrm{d}i}}{\langle \cos {\theta }_i\rangle }=\left(1-\epsilon \right)\frac{H_i}{\langle \cos {\theta }_i\rangle }.$$

(58)

Considering that the territory under analysis was not divided into zones, the solar irradiation on horizontal surfaces unit per annum is determined using Equation (2) as follows:

$$H={H_j|}_{j=1}={{\displaystyle \sum }}_{i=1}^{12}{H}_i=1441.1 [\mathrm{kW}\cdot \mathrm{h}/({\mathrm{m}}^2\cdot \mathrm{yr})]$$

and the gross potential of Olkhon is:

$$GP=G{P}_j=H_j\cdot S=1052.222 [\mathrm{MW}\cdot \mathrm{h}/\mathrm{yr}]$$

6.3. Technical Potential of Solar Energy

The initial data are the initial data in Section 6.2, as well as the parameter values characterizing the current state of the art for the most commonly encountered flat silicon PV modules:

α = 0.97; η₁ = 0.20; χ = 0.004 K⁻¹; T₁ = 298 K; <λ> = 40 W/(m²∙K); <k_n> = 0.99

The values of the average monthly operating temperature, T_i, calculated by Equation (24) using the values of T_ambi (Table 2), H_i (Table 4 and Table 6) and t_si (Table 4) are set out in

Table 8. Average monthly operation time of PV systems, t_PVi, and operating temperature of PV cells, T_i, and unit monthly technical potentials of solar energy for photovoltaic energetics, TP_i/S_TP.

Month	1	2	3	4	5	6	7	8	9	10	11	12
Ti, K	259.6 *	263.3	273.7	285.2	293.9	300.1	302.9	300.6	290.1	281.7	271.7 *	264.6 *
tPVi, h/mo	256.3	256.9	264.3	273.2	279.7	285.9	289.0	288.8	283.4	276.0	268.3	262.3
TPi/STP, kW h/(m2∙mo)	9.26	18.26	32.89	36.27	44.88	40.74	40.46	33.26	19.90	18.13	9.31	6.03

* The following estimated values of air temperature during daylight hours were used: T_amb1 = −17 °C; T_amb11 = −5 °C; T_amb12 = −11 °C.

. The average PV system operation time, t_PVi, from March to September was determined using the values of t_si and the number of clear, partly overcast and overcast days in the month (Table 4). For autumn and winter months (i = 1, 2, 10, 11, 12), which make a small contribution to the potential, the following approximate relationship was used: t_PVi = t_si = 0.9 t_0i.

Table 8 represents the values of the monthly technical potentials per surface unit of the Island of Olkhon where PV systems may be installed. These potentials were calculated using Equation (22) by the following formula:

$${T{P}_i/S_{TP}=H_i\cdot {\eta }_1\left(1-\chi \cdot \left(T_i-T_1\right)\right)|}_{S_{TP}=S_{TPj}}.$$

(59)

The potential TP under Equations (20) and (21) is determined by summing up all of the months, taking into account that the territory of the island is deemed one zone and, therefore, $TP=T{P}_j={{\displaystyle \sum }}_{i=1}^{12}T{P}_i$ with j = 1:

TP = 288.3 S_TP [MW h/yr]

6.4. Economic Potential of Solar Energy

6.4.1. Initial Data

The initial data are the initial data in Section 6.3 or/and calculation results in Section 6.3, as well as the characteristics of PV systems: unit final cost subject to the ecological factor, c_PV.e (c_PV.e = c_PV ∙r_e = 510 USD/m² = 30,600 RUB/m²).

Furthermore, the following apply to the levels of power supply to people and industry in the Island of Olkhon:

• Unit cost of traditional fuel subject to the regional and ecological factors for 2022 in renominated prices c_t.s.e = 15.25 RUB/kW∙h;
• Demand of industry for electric energy D = 818 MW∙h/yr;
• Unit cost of losses from electric power shortage (unit cost of industrial products) (assessment by the order of values) c_loss = 100 RUB/kW∙h;
• Population N = 1744 people;
• Domestic demand of population for electric energy Q_d = 1159.18 MW∙h/yr.

6.4.2. Parameters of PV Systems Equipped with Traditional Planar Silicon PV Modules

The solar irradiation on the active surface unit of PV modules, H_PVi, is equal to the values of solar irradiation on surfaces tilted at the angle β_opt (Equation (19)) to the horizontal, H_ti. Using the values of H_ti and ambient temperature, T_i, determined as specified in Section 6.2 and Section 6.3, the generation by PV systems installed on the island area unit, W_i, is calculated by Equation (29) for each month of the year and for a 10-h interval (from 07 a.m. to 05 p.m.). The results are set out in

Table 9. Total monthly solar irradiation H_ti, operating temperature of PV cells T_i and generation W_i by PV systems based on traditional planar silicon PV modules with inclination of module active surfaces at the angle β_opt.

Month	1	2	3	4	5	6	7	8	9	10	11	12
HPVi, kW∙h/(m2∙mo)	44.2	64.4	306.0	214.4	230.4	196.6	198.7	191.3	132.8	59.4	41.2	33.8
Ti, K	260.1	262.0	284.7	288.1	295.4	300.3	303.2	302.9	293.0	280.0	271.7 *	265.4 *
Wi, kW∙h/(m2∙mo)	6.40	9.28	41.36	28.70	30.84	25.38	25.43	24.50	17.52	8.14	5.81	4.82

* The following estimated values of air temperature during daylight hours were used: T_amb1 = −17 °C; T_amb11 = −5 °C; T_amb12 = −11 °C.

.

The annual generation in the year of systems installation (k_n = 1), W₁, is determined by summing up W_i for all the months of the year using Equations (30) or (31):

W₁ = 228.2 kW∙h/(m²∙yr).

6.4.3. Parameters of PV Systems Equipped with CPV Modules with Continuous Tracking of the Sun

The initial parameter values are the same as in Section 6.4.2, except for the PV systems cost, which in accordance with the practical data is taken to be equal to c_PVe/3 with K = 13 and the temperature coefficient of efficiency factor, which is theoretically and practically lower than expected and considered to be equal to χ = 0.003 K⁻¹.

The values of H_PVi are calculated by Equation (34) for i = 1, 2, …, 12 using the values of monthly solar irradiation on normally oriented surfaces, H_d.ni, and values of <cosθ_i> (see the calculation in Table 7). The other calculations are performed similarly to those for PV systems fitted with planar silicon PV modules. The results are set out in

Table 10. Monthly solar irradiation H_PVi, operating temperature of PV cells T_i and generation W_i by PV systems based on CPV modules with continuous tracking of the sun.

Month	1	2	3	4	5	6	7	8	9	10	11	12
Hti, kW∙h/(m2∙ mo)	50.5	86.4	281.3	209.7	226.6	203.4	206.5	183.8	127.1	87.5	49.8	36.5
Ti, K	259.6 *	263.3	273.7	285.2	293.9	300.1	302.9	300.6	290.1	281.7	271.7 *	264.6 *
Wi, kW∙h/(m2∙ mo)	7.59	12.76	39.30	28.39	29.81	26.15	26.23	23.49	16.88	12.11	7.14	5.36

* The following estimated values of air temperature during daylight hours were used: T_amb1 = −17 °C; T_amb11 = −5 °C; T_amb12 = −11 °C.

.

The annual generation in the first year of systems operation is:

W₁ = 235.2 kW∙h/(m²∙^∙yr).

Hence, the unit energy generation by CPV modules with continuous tracking of the sun turned out to be approximately equal to the unit energy generation by ordinary planar PV modules, so the conclusion on expediency of using modules of any design depends on economic indicators.

6.5. Economic Potential Subject to Different Service Lifetimes of PV Systems

The calculation results for PV systems based on traditional planar silicon PV modules are represented in

Table 11. Critical values of annual generation W_crit by PV systems based on planar silicon PV modules with inclination of module active surfaces at the angle β_opt, and relevant values of economic potential EP and economic surplus potential ΔEP of solar energy for the photovoltaic energetics of the Island of Olkhon, subject to different service lifetimes of PV systems t_s.

ts, Years	γ = 0.02/yr; tpb − n/a; te2 = 13.5 Years		γ = 0; tpb = 59.1 Years; te2 = 10.6 Years
	Wcrit, kW∙h/(m2∙yr)	EP, GW∙h/yr	Wcrit, kW∙h/(m2∙yr)	EP, GW∙h/yr
10	1667	0	1389	0
12	1435	0	1157	0.95
14	1270	0.85	992.1	1.16
16	1146	0.96	868.1	1.39
18	1049	1.08	771.6	1.63
20	972.2	1.19	694.4	1.91
22	909.1	1.30	631.3	2.21
24	856.5	1.41	578.7	2.55
26	812.0	1.52	534.2	2.93
28	773.8	1.63	496.0	3.36
30	740.7	1.73	463.0	3.85

.

The calculation results for PV systems based on CPV modules with continuous tracking of the sun are set forth in

Table 12. Critical values of annual generation W_crit by PV systems based on CPV modules with continuous tracking of the sun and relevant values of economic potential EP and economic surplus potential ΔEP of solar energy for the photovoltaic energetics of the Island of Olkhon, subject to different service lifetimes of PV systems t_s.

ts, Years	γ = 0.05/yr; tpb − n/a; te2 = 4.5 Years		γ = 0; tpb = 20.3 Years; te2 = 3.7 Years
	Wcrit, kW∙h/(m2∙yr)	EP, GW∙h/yr	Wcrit, kW∙h/(m2∙yr)	EP, GW∙h/yr
4	1389	0	1157	0.92
6	1003	1.10	771.6	1.57
8	810.2	1.46	578.7	2.43
10	694.4	1.82	463.0	3.62
12	617.3	2.19	385.8	5.40
14	562.2	2.55	330.7	8.30
16	520.8	2.91	289.4	13.90
18	488.7	3.26	257.2	29.32
20	463.0	3.62	231.5	89.20
22	441.9	3.98	210.4	156.18
24	424.4	4.33	192.9	TP
26	409.5	4.69	178.1	TP

.

The received results show that PV systems based on CPV modules with continuous tracking of the sun have an advantage over PV systems based on traditional planar silicon PV modules in terms of both plant payback times and assessed economic potentials of electric energy from solar radiation [43].

The total demand of the region for electric energy, Q, in accordance with Equation (57) is:

Q = 3178 GW∙h/yr

The calculation results concerning the economic surplus potential of solar energy for the photovoltaic energetics of the Island of Olkhon, ΔEP, (Equation (55)) are also set out in Table 11 and Table 12.

The surplus ΔEP for both types of PV systems for the entire service lifetime turns out to be positive, and the region is an economically justified potential donor of electric energy from the sun with the service lifetime of PV systems exceeding 7–10 years for the range of standard operation cost γ = 0–0.05/yr.

7. Conclusions

In spite of a large body of research, the development of a universally appropriate general algorithm for the assessment of solar energy potentials that is clearly correlated with the calculations for specific operation conditions continues to be a crucial task. The best results are given by the option of overall calculation of all potentials by one and the same algorithm and using the same initial data.

Solar energy is the only one from among the renewable energy sources for which a change in the traditional proportion between the gross, technical and economic potentials is possible. Thanks to the development of IPV, primarily BIPV and BAPV, the area for technical potential calculation in settled lands may be considerably greater than the Earth’s surface area used to calculate the gross potential. That means that subject to BIPV, the technical potential of solar energy, and even the economic potential, may be greater than the gross potential.

The proposed approach to the analysis of solar energy potential in the region makes it possible to secure a high degree of assessment reliability and use it for more detailed calculations, including the potentials analysis for a specific point on the ground or a specific type of PV system. This work sets out the tried and tested assessment program for the potentials of solar energy arriving at large and medium areas (up to ~200 km²). An example of calculations for Olkhon Island is given.

In the longer term, the program may be applied with any software involving the values of potentials, e.g., the software for analyzing the options of industrial and other project planning, region development programs, regional economy development planning and assessment programs, etc., as well as building construction and reconstruction programs, for more accurate determination of S_IPV and extended feasibility of IPV implementation.