**2. Increasing Energy Production by Increasing Voltage—Improvement in the Design of Photoelectric Cells**

*2.1. Vertical High-Voltage Multi-Junction Silicon PV Cells*

Three technologies have been used to form multi-layer structures: soldering unit cells using solder plugs with diffused layers, epitaxy, and breakdown [44]. Currently, PV cells are manufactured via plug soldering with subsequent cutting into size-defined cells and the application of antireflection coating with nanoclusters to the end face (active surface) [45,46]. Figure 2a,b show the structure of a typical vertical high-voltage PV cell and a PV module composed of these cells. As one can see from Figure 2, both the cells and modules of this design are much more space-saving than those manufactured using traditional planar technology. The cell voltage is equal to the sum of the voltages of microcells (0.58–0.62 V per microcell). The efficiency achieved for the designed cells under single-fold radiation is comparable to that of conventional silicon PV cells.

**Figure 2.** (**a**) PV cell structure; (**b**) PV module based on vertical multi-junction PV cells, 40 60 × 10 mm PV cells; (**c**) *I*–*V* curve of developed PV modules, 10 suns; (**d**) dependence of efficiency on irradiance.

PV cells have universality in relation to the radiation level, i.e., in contrast to traditional silicon planar PV cells, their voltage does not decrease as concentration increases. Moreover, they have low sensitivity to the radiation incidence angle.

PV modules based on vertical high-voltage multi-junction silicon PV cells have a voltage rate of up to 1000 ± 15 V at the maximum power point. For solar radiation concentration ratios ranging from 10 to 100 suns, the specific power of such modules is up to 0.245 ± 0.01 W/cm2, and their efficiency is up to 25.3%. The modules are encapsulated with polysiloxane gels (see Section 2.3), which facilitates the extension of their expected service life up to about 50 years. The operating range of their ambient temperature is from –60 to 110 ◦C. The module efficiency, which is 18–22%, remains the same when the temperature rises up to 60 ◦C, which simplifies their cooling system while working with concentrators. The production process does not require the use of silver, screen printing, photolithography, or any other time-consuming procedure or expensive materials. Figure 2c shows the *I*–*V* curve of the developed PV module. The test conditions were as follows: direct normal irradiance of 10 × 1000 W/m2, cell temperature of 25 ◦C, and spectral irradiance distribution of direct normal AM1.5. Figure 2d shows the PV module's efficiency dependence on the level of active surface irradiance [46].

The operating voltage of 1000 ± 15 V enables the use of these PV modules with transformerless inverters and connecting them to high-voltage direct-current 110–500 kV power lines without the need for converting substations. To obtain the operating voltage of 1000 V with the use of conventional planar PV modules, it would be necessary to connect more than 1800 150 × 150 mm planar PV cells in series, in which case the total length of the modules would exceed 300 m. It is also important that the operating voltage of one module can be equal to the accepted nominal DC voltage of large photovoltaic systems (system voltage), for example, solar power plants such as AgriPV systems.

The main problem involves the reproducibility of characteristics (wide spread of parameter values). Thus far, it has been impossible to achieve parameter values in the required narrow interval with the change from batch to batch.

#### *2.2. Planar High-Voltage Multi-Junction Silicon PV Cells*

The currently developed planar high-voltage silicon PV cells are multi-junction homogeneous PV structures [25–30]. The doped layers form planar diodes *n*+–*p*–*p*<sup>+</sup> (*p*+–*n*–*n*+; *n*–*p*–*p*+; *p*–*n*–*n*+) or *n*–*p* microstructures (unit cells) connected in series in the direction of radiation propagation. Figure 3 shows the design options of planar high-voltage PV cells, while Table 1 presents the advantages, design specificity, and technological features borrowed from PV cells of various types in order to develop planar high-voltage multi-junction PV cells.

**Figure 3.** (**a**) High-voltage planar PV cells with basic option; (**b**) structure manufactured using the electric breakdown technology; (**c**) structure with nanoclusters; (**d**) bifacial structure.

**Table 1.** Advantages, specificity of design, and technologies borrowed from various types of PV cells to develop high-voltage planar multi-junction PV cells.


The small (≤10 μm) thickness of layers and, consequently, unit cells make them practically transparent considering solar radiation. Each unit cell receives radiation that consecutively passed through the previous semiconductor layers. The thickness of the base unit cell area does not exceed the diffusion length of minority charge carriers.

The theoretical and most of the experimental parts of this research were carried out using the basic option of the design, as shown in Figure 3a. The basic option of PV cells was fabricated through the use of epitaxy by transferring epitaxial layers to the basic unit cell (silicon wafer with a *p*–*n* junction formed via diffusion). The first basic unit cell ensures the mechanical strength of the entire structure. As the *p*<sup>+</sup> and *n*<sup>+</sup> layers are heavily doped, a conductive contact occurs between them owing to the quantum-mechanical effect of tunneling through the *p*+–*n*<sup>+</sup> junction. The dimensions of PV cells are determined by the dimensions of the wafer and correspond to the dimensions of traditional crystalline Si PV cells.

The thickness of *p, n,* and *n+* layers is from 12 nm to 9.8 μm, while that of *p*+-type layers is from 10 nm to 120 nm. The concentration of active impurity in thin heavily doped *p*+-type layers is more than 1019 cm−3. In general, the thickness of layers increases with the distance from the active surfaces, which enables the production of PV cells with the required output parameters and their nonsignificant distribution with a wide tolerance for layer identity. The number of layers is determined based on equipment capability and the specifics of the layer formation process, as well as the expediency of using solar cells with a specific number of layers in the PV modules.

To avoid losses in the circuit, each unit cell must operate under radiation at its optimal point of the volt–ampere characteristic, which means the equality of the photogenerated currents flowing through the unit cells:

$$\begin{cases} \int\_{0}^{\infty} \frac{d\Phi}{d\omega} \cdot e^{-\kappa \sum\_{k=1+2}^{N} d\_{k}} \cdot \left(1 - e^{-ad\_{i+1}}\right) \cdot d\omega = \int\_{0}^{\infty} \frac{d\Phi}{d\omega} \cdot e^{-\kappa \sum\_{k=1+1}^{N} d\_{k}} \cdot \left(1 - e^{-ad\_{i}}\right) \cdot d\omega; \; i = 2, 3, \dots, N-1\\ 0 \qquad \int\_{0}^{\infty} \frac{d\Phi}{d\omega} \cdot e^{-\kappa \sum\_{k=i+1}^{N} d\_{k}} \cdot \left(1 - e^{-ad\_{i}}\right) \cdot d\omega = \int\_{0}^{\infty} \frac{d\Phi}{d\omega} \cdot e^{-\kappa \sum\_{k=2}^{N} d\_{k}} \cdot Q\_{l}(\omega) \cdot d\omega; \quad i = 2, 3, \dots, N \end{cases} (1)$$

where Φ is the incident photon flux; *α* is the radiation absorption coefficient depending on photon frequency *ω* or wavelength *λ* and on the nature of the semiconductor; *dk* and *di* are the thickness of the *k*-th or *i*-th unit cell, respectively; *Qi*(*ω*) is the spectral charge carriers collection efficiency for the *p*–*n* junction in the *i*-th unit cell; and *N* is the number of unit cells in an entire PV cell.

The requirement of Equation (1) is met by determining the optimal values of the thickness of each unit cell as follows:

$$d\_i = \frac{1}{a} \ln \frac{1 - e^{-2a\delta} + Q\_i e^{-a\delta} \left(1 - e^{-2a\delta(i-1)}\right)}{1 - e^{-2a\delta} + Q\_i e^{-a\delta} \left(1 - e^{-2a\delta(i-2)}\right)}; \; i = 2, 3, \ldots, N\tag{2}$$

where αδ is the complex optical–technological parameter of the PV cell.

The density of the photocurrent generated in the *i*-th unit cell is subject to light absorption in the previous layers.

$$J\_{\rm phi} = q \Phi\_0 S\_i(a, \xi); \; i = 1, 2, \; \dots, N \tag{3}$$

where *q* is the absolute value of the electron's charge, Φ<sup>0</sup> is the incident photon flux density, *Si* is the spectral charge carriers collection efficiency (photoresponse) of the *i*-th unit cell, and *ξ* is the angle of radiation incidence onto the active surface.

Volt–ampere characteristic is expressed as follows:

$$V = \frac{AkT}{q} \sum\_{i=1}^{N} \ln\left(\frac{\left(\frac{q\Phi\_0 \cdot Q\_i}{1 + (N-1)\cdot Q\_i} - J\_{\text{Ph}}\right)}{J\_{0i}} + 1\right) \tag{4}$$

where *A* is the parameter of characteristic curvature, *k* is the Boltzmann constant, *T* is the cell temperature, and *J*0*<sup>i</sup>* is the reverse dark current density in the *i*-th unit cell.

The collection efficiencies of spectral charge carriers of unit cells are determined as follows:

$$S\_1(a, \tilde{\xi}) = S\_{\mathbf{b}}(a, \tilde{\xi}) = \cos \tilde{\xi} \cdot e^{-\frac{\tilde{a}}{\cos \theta} \left(\sum\_{k=2}^{N} d\_k + 2\delta(N-1)\right)} \cdot Q\_{\mathbf{b}}(a, \theta)$$

$$S\_{\mathbf{i}}(a, \tilde{\xi}) = \cos \tilde{\xi} \cdot e^{-\frac{\tilde{a}}{\cos \theta} \left(\sum\_{k=i+1}^{N} d\_k + 2\delta(N-i) + \delta\right)} \left(1 - e^{-\frac{a}{\cos \theta} d\_i}\right); \ i = 2, \dots, N-1 \tag{5}$$

$$S\_N(a, \tilde{\xi}) = \cos \tilde{\xi} \cdot e^{-\frac{a}{\cos \theta} \delta} \cdot \left(1 - e^{-\frac{a}{\cos \theta} d\_N}\right)$$

where *Q*<sup>b</sup> is the collection efficiency of the charge carriers in the basic cell, *ϑ* is the refraction angle in a semiconductor structure, and *δ* is the thickness of the tunnel layer in the unit cells (the difference between the tunnel layers of unit cells can be neglected).

Angle *ϑ* is determined using the relation sin*ξ* = *n*·sin*ϑ*, where *n* is the radiation refraction index in the semiconductor. Angle *ξ* is equal to 0◦ when the rays are perpendicular to the active surface. The reverse dark current density in the *i*-th unit cell, given the uniform technology used in the fabrication of unit cells, does not depend on the unit cell number, i.e., *J*0*<sup>i</sup>* = *J*0, as evident from Equation (1).

The plots showing the main characteristics of the developed high-voltage multijunction PV cell with the basic structure (see Figure 3a) and different unit cell numbers are presented in Figure 4. The characteristics of the final experimental samples quite accurately correspond to the results of computer simulations and calculations.

**Figure 4.** (**a**) Open-circuit voltage of high-voltage planar PV cells under solar radiation, *a* = 0.125 μm<sup>−</sup>1, A = 2, *J0* = 10−<sup>7</sup> A/cm2, and *q*Φ = 45 mA/cm2; (**b**) dependence of the optimum unit cell thickness on it number; (**c**) dependence of efficiency on solar radiation concentration; (**d**) relative efficiency, *a* = 0.125 μm<sup>−</sup>1, A = 2, *J0* = 10−<sup>7</sup> A/cm2, and *q*Φ = 45 mA/cm2.

For the experiments, PV cells (see Figure 3a) were originally fabricated using the basic design option with the number of unit cells *N* from 1 (only the basic unit cell without epitaxy layers) to 20 (see Figure 4a). For technological reasons, more detailed research was carried out to determine the characteristics of unit cells using samples with *N* = 2, *N* = 5, and *N* = 10.

The dependence of the open-circuit voltage of PV cells on *N* is shown in Figure 4a, for various values of tunnel layer thickness *δ* for *α* = 0.125 μm<sup>−</sup>1, *Q*<sup>b</sup> = 0.8, a radiation intensity equal to that of solar radiation (*q*Φ = 45 mA/cm2), and for other parameters corresponding to the recombination current mechanism (A = 2; *J*<sup>0</sup> = 10−<sup>7</sup> A/cm2). The open-circuit voltage increases monotonically with *N* (growing almost linearly for small values of *N*), weakly depending on the values of *J*<sup>0</sup> and *Q*. In the asymptotic limit, for *N*·*Q* >> 1, it becomes dependent on *N*, to an extent that is weaker than the linear law and does not change for any value of *Q*.

As seen from Equation (2), the optimal thickness *di* decreases in inverse proportion to the radiation absorption coefficient α, logarithmically reduces as the unit cell number *i* increases, and increases as the collection efficiency of charge carriers *Q*<sup>b</sup> increases in the first basic unity cell. The dependence of the optimal thickness of a unit cell on its number for various values of *δ* is shown in Figure 4b. PV cells may operate under concentrated radiation without a noticeable decrease in voltage, and they have low sensitivity to the solar radiation incidence angle. The efficiency of PV cells does not exceed the maximum efficiency of the basic unit cell only for small values of the tunnel layer thickness *δ*. For a given value of radiation absorption coefficient *α*, this thickness, layer number *N*, and the relevant value of open-circuit voltage must not exceed the corresponding limiting values. For example, for *<sup>α</sup>* = 0.125 <sup>μ</sup>m−1, there must be *<sup>δ</sup>* ≤ 0.16 <sup>μ</sup>m, *<sup>N</sup>* ≤ 9, and *<sup>V</sup>*o.c ≤ 4.5 V. A sharp drop in PV cell efficiency was observed if *δ* ≥ 0.32 μm.

Experimental samples of PV modules were manufactured from 60 PV cells with a size of 156.75 × 156.75 mm with *N* = 10. The PV cells were connected in a series–parallel comprising two strings of 30 cells each or in series. The modules were encapsulated using polysiloxane gels (see Section 2.3). Figure 5a shows the module during testing. Additionally, Figure 5b shows the averaged *I*–*V* curve of the developed modules under the standard test conditions (STC) (cell temperature of 25 ◦C, irradiance of 1 sun, i.e., 1000 W/m2, and spectral irradiance distribution of AM1.5).

**Figure 5.** (**a**) PV module with planar high-voltage multi-junction silicon PV cells (*N* = 10) on the test bench; (**b**) *I*–*V* curve of the developed modules at STC, *N* = 10 (curve 1—two strings of 30 cells each and curve 2—1 strings of 60 cells).

#### *2.3. Performance Deterioration Reduction and External Impact Hardening—Improvement in PV Module Manufacturing Technology*

Planar crystalline Si terrestrial PV modules manufactured using the conventional lamination technology lose up to 15–20% of their maximum power at STC after 20 years of operation in tropical climates and after 25 years of operation in temperate climates. One of the reasons for it is the degradation of optical polymeric encapsulants such as ethylene vinyl acetate (EVA) and other plastics induced by ultraviolet radiation and high temperatures [47].

High-voltage cell modules considered in Sections 2.1 and 2.2 were manufactured using polysiloxane gels instead of conventional EVA as the filling material [46,48]. The replacement of the standard encapsulant with EVA and the conventional lamination process with a silicon composition that is poured as liquid and then hardens and turns into polysiloxane gels considerably slows down the performance deterioration of such modules in time. The process reaction is as follows:

According to the laboratory test data (tests in compliance with requirements of IEC 61215-2:2021, IEC 62716:2013, IEC 61701:2020, and IEC/TS 63126:2020 [49–52]), the module's ultimate power degradation at STC was approximately 15% for 50 years. Such a result was achieved thanks to a considerably lower corrosiveness of silicon gels. Moreover, the initial output power of the modules increased thanks to higher gel transparency and a decrease in the operating temperature of PV cells. PV modules are fire-safe and have increased resistance to operation in environments with a high concentration of ammonia and in tropical and marine climates.

The comparative characteristics of modules encapsulated with polysiloxane gels and conventional silicon modules filled with EVA are shown in Table 2.


**Table 2.** Comparison of the characteristics of PV modules with two types of filling materials [48].

The encapsulation technology ensures a reduction in costs thanks to higher operational characteristics of modules (higher resistance to aggressive environments, e.g., in environments with the presence of ammonia in the ambient air, high humidity, etc., which are characteristic of various agricultural facilities), along with a considerable decrease in the degradation of these characteristics.

#### **3. Increasing Energy Production by Increasing Voltage and Current per Unit Area of Cell Active Surface—Concentrator Design Improvement**

In the course of our studies, we sought to optimize the form of the RS of both focontype and compound parabolic concentrators. In the case of focon-type concentrators, it was important to solve the problem of nonuniform radiation distribution over the RR's operating surface, while in the case of compound parabolic concentrators, we had to minimize Fresnel losses. As a result, two types of effective CPV modules were built: the first based on focon-type concentrators and planar high-voltage cells, and the second based on compound parabolic concentrators and vertical high-voltage cells. The designs were based on the conditions that the concentrators should have such dimensions that would lead to comparable dimensions of the resulting CPV modules to those of conventional crystalline silicon PV modules, and that CPV modules could be used in individual power supply systems. Additionally, the design of CPV modules should match the components of mass-produced PV systems as much as possible.
