Digital Twin of the Photoelectric Converter of the Power Transmission System over Optical Fiber

The photovoltaic converter (PVC) is a key device in a power-over-fiber system (PoFS). The aim of the work is to create a digital twin (DT) to predict the behaviour of PVC based on their specification data, which is the first step towards creating a universal DT of the entire PoFS. The paper considers the theoretical principles of operation of PVC using the zone theory of solids. A relationship has been established between the theoretical parameters and specification data of real converters. A five-element equivalent circuit of a PVC has been created and a DT in MATLAB Simulink has been developed on its basis. The dependences of the main parameters of the DT on the optical power at the converter input are calculated. Experiments were carried out on an installation with PVC of the YCH-H6424-15-FC-A brand. The optical powers at which all characteristics of the solar cell coincide with an accuracy of 2% are determined. The discrepancy between the calculated and experimental characteristics with increasing optical power is explained by the fact that the current transfer coefficient decreases at high current strength. This phenomenon is substantiated by the zone theory, but is not taken into account in the DT.


Introduction
Progress in the development of photonics as a science has led to the emergence of modern semiconductor lasers, fiber-optic communication lines and photovoltaic converters.Integration of these devices into a single system allows using their competitive advantages to create modern power supply systems with energy transmission via optical fiber [1 -3].At the same time it is possible to achieve full galvanic isolation of primary and secondary energy converters, noise immunity of power-over-fiber system line from external electromagnetic influences [4], absence of fracture [5] even at fiber destruction, possibility of laying in hard-to-reach places and integration into composite materials.The application of these power supply systems is currently of a niche nature and is used in situations where the use of traditional wire power supply lines is either unsafe or difficult [6].Their use is most effective in monitoring systems for remote or hard-to-access locations, as well as in explosion-and fire-hazardous situations [7 -10].In the foreseeable future, their application areas are likely to be significantly expanded, and in this regard [11,12], the search for new technical solutions to the main problem of all dual energy conversion systems -low efficiency -is going on all over the world.
The analysis of energy losses performed in the works [8,9,13] shows that at transmission of small to units of watts and average to tens of watts of power the main sources of losses are primary electrooptical converters made on the basis of solid-state semiconductor lasers, power-over-fiber system lines and secondary optoelectronic pre-transducers.Usually energy sources of semiconductor laser diodes are power electric networks with practically unlimited (kilowatts and more) energy resource, so the efficiency (efficiency factor) of primary converters of about 50% is quite acceptable.Energy losses in optical fiber limit the length of the laid line to tens of kilometers, which is quite sufficient for most applications.The most critical are the energy losses in the secondary photovoltaic converter, which efficiency improvement is currently the most urgent task.
In [14 -17] the current state of development of various PVC designs is described.It is noted that there are three wavelength ranges in which optimal matching of parameters of semiconductor solid-state lasers, power-over-fiber system lines, and photovoltaic converters is possible.Currently, the most commonly used is the first with a central wavelength of 850 nm for which there is a wide choice of semiconductor light sources and receivers used in power-over-fiber systems.It should be noted that such devices are usually complex multilayer structures based on gallium arsenide heterojunctions with threecomponent structures of the same substance with the addition of aluminum, indium atoms, etc. [18].Developers of power supply systems, as a rule, cannot change the internal design of devices.In order to increase efficiency (efficiency), they have several possibilities at their disposal: selection within the transparency window of the operating wavelengths of emitters and receivers, optical powers, cooling modes and, in the presence of a feedback system, optimization of operating modes when the load coprotection of the secondary converter changes.In order to reduce the development time of the system, the authors propose to create its DT, allowing to find the optimal parameters of operation before the physical manufacture of its prototype.
The first step is to develop in the MATLAB application program package a DT of a photovoltaic converter operating on active and reactive loads, both constant and dynamically changing in time [19].The purpose of this work is to create a universal DT for modeling the operation of PV converters of different ratings of one manufacturer, as well as different manufacturers, including serial and experimental samples.

Main variants of designs of photovoltaic converters of laser radiation
Nowadays there is a great variety of photo-electric converters designs.They can be divided according to several main features: 1. Spectrum of converted energy: wide, for example, solar radiation and narrow -laser radiation; 2. Energy characteristics: optical radiation flux density, output voltage, output currents, sensitivity to changes in load resistance, efficiency of conversion of radiation energy into electrical energy; 3. Design features: input of optical radiation, converter materials and semiconductor converter design.In the present work, a DT of the photovoltaic converter of laser radiation developed for the Powerover-Fiber (PoF) system will be developed.The peculiarity of the above designs is the narrow spectrum of laser radiation and Gaussian energy distribution in the beam transmitted along the optical fiber and high optical energy density.
To increase efficiency, modern FETs are made on the basis of hetero-structures.In the first window of optical fiber transparency (central wavelength 850 nm), the most suitable material is gallium arsenide (GaAs) with a bandgap width ∆W = 1.42 eV at a temperature of 300 K [20].Addition of aluminum to the AlxGa1-xAs compound allows changing the concentration of Al to x = 0.4 (40% of Ga atoms are substituted) to linearly increase the bandgap width to 1.9 eV.To reduce the width of the forbidden zone aluminum is replaced by indium and three-component mixture InxGa1-xAs allows you to vary it in a wide range from 0.354 eV in indium arsenide to 1.42 in gallium arsenide.The formula for calculating the forbidden band width [21] ∆ = 0,354 + 0,63  + 0,43 ![eV]. ( Introduction of elements of groups II and V of the Mendeleev table allows to create layers with pand n-type conductivity, respectively.In a simplified form, the basic designs of PVCs with end and side illumination are presented in Figure 1,a-d.It should be noted that in real designs there are more layers: a layer of illuminating coating is applied on top of the illuminated surface of the transducer to reduce the reflection coefficient calculated by the Fresnel formula for normal incidence  = . where n1 = 1 is the refractive index of air, and n2 ≈ 4 -of gallium arsenide and its three-component compounds.The introduction of an enlightening coating makes it possible to reduce the reflection coefficient to 1% and less for a narrow spectral band of radiation, which is characteristic of laser radiation.Also, to reduce mechanical stresses at the interface of individual layers, special buffer layers are introduced to harmonize the constant crystal lattices (the difference should not exceed 0.5% and the smaller it is, the better the harmonization, the fewer defects on the border and the less loss of electrical power during current flow).The advantage of end illumination is the uniformity of illumination of the semiconductor with a cylindrical design as shown in Figure 1, a-c.The disadvantage is the light absorption losses in the top electrode, which is partially offset by its location in the region of illumination recession at the periphery of the Gaussian distribution of incident radiation energy and the use of semitransparent electrodes.The matching of the illuminated surface with the area of the light spot is usually achieved by installing an optical radiation homogenizer in front of the crystal surface.
The advantage of side illumination of a photovoltaic converter is the uniformity of alternating semiconductor layers [14], but for efficient operation it is necessary to illuminate them uniformly, which implies the use of radiation homogenizers [22].

Effects of light on a p-n junction
Figure 2 shows a schematic diagram of the arrangement of layers of the simplest photovoltaic converter of laser radiation energy into DC energy.A thin n-type semiconductor is built up on a monocrystalline p-type semiconductor.The bottom electrode is solid and the top electrode is circular.To reduce light loss on reflection, an opaque coating is applied on top.Laser radiation has the form of a Gaussian beam and penetrates into the depth of the photovoltaic converter through the brightening coating.During the first passage, part of the optical energy is converted into electric current energy, and part of it reaches the lower electrode and reflects from it as from a mirror and passes through the semiconductor, increasing the efficiency of photovoltaic conversion.

Figure 2. Layout of the photovoltaic converter layers
Figure 3 shows the zone diagram of the heterojunction and the processes of generation, recombination and movement of the main charge carriers -electrons [16].The figure shows a highly doped broad-zone n+-type emitter on the right, and a low-alloyed narrow-zone p-type base on the left, between which there is a p-n junction (the solid vertical line indicates the metallurgical boundary, and the dashed lines indicate the boundaries of the junction on the emitter and base sides).The electric field  2⃗ resulting from the formation of the p-n junction is inside its boundaries.In the upper part of the figure is the conduction zone, the lower boundary of which -the bottom of the conduction zone is indicated by a thick horizontal line.At the metallurgical boundary, the conduction zone has a break like a small "peach".At the bottom is located valence zone gap which has the form of a "wall", which is a barrier to the movement of holes.Thus, in the heterojunction conduction is carried out due to the movement of one main type of charge carrier in our case -electrons.Practically, holes do not contribute to conduction and their motion will not be considered further.A quantum of light with energy hυ falls from the side of the emitter with forbidden zone width ∆Wn > hυ.The energy of the quantum is not enough to form an electron-hole pair, so the emitter is transparent to light.The light quantum is absorbed in the p-type narrow-gap base, with the formation of an electron-hole light pair (photovoltaic effect).Conversion efficiency is characterized by the value of quantum yield  = where Ne is the number of electron-hole pairs formed during the absorption of Nq quanta of light.
In the depth of the base the intensity of the light flux exponentially decreases according to the Bouguer-Lambert law [17] where Ф0 is the light flux incident on the emitter, k = k(λ) is the wavelength-dependent absorption coefficient, x is the distance to the base depth.At the distance x = 1/k, the intensity of the light flux decreases by e = 2.71 times, this distance is called the average electron diffusion length in the p-type base Ln.If the base thickness lb < Ln, it is a diode with a "thin" base, if lb > Ln, it is a diode with a "thick" base.Usually diodes with "thin" base are used.
Near the red boundary of the photovoltaic effect, absorption decreases, so the energy of light quanta must be greater than the width of the forbidden zone Wp.In this case, as shown in the figure, the electron first moves to energy levels above the bottom of the conduction zone, and then as a result of collisions with the atoms of the crystal lattice gives the latter part of its energy and falls to a free energy level near the bottom of the conduction zone.The released energy is converted into heat, which heats the device.Designers are forced to find a compromise between the absorption and heating processes by selecting laser radiation sources with a wavelength not much longer than the red limit of the photovoltaic effect.
To create electric photocurrent in the generator mode of PVC operation, electrons from the depth of the base must reach the boundary of the p-n junction and drifting towards the electric field  2⃗ be injected into the emitter of n + -type.In the base electrons are non-main charge carriers (they are several orders of magnitude smaller than holes) and after a short time -lifetime they can combine with holes.To prevent this from happening, the base can be made multilayer consisting of several heterojunctions with different width of the forbidden zone, in which the absorption region will be thin, and the buffer layers (necessary for coordination of constant crystal lattices) and the single crystal (substrate) on which the other layers are built up -wide-area, transparent to laser radiation.Other options are also possible: swap the base and emitter (see Figure 2) or create an electric field "embedded" in the base by doping it unevenly.Then, in addition to diffusion (which is present in any case), the second -drift mechanism of electron transfer is added and the efficiency of electric current generation increases.
When electrons are transferred within the base, electrons can recombine with holes.To account for recombination, a transfer coefficient is introduced where Ne is the number of electron-hole pairs formed as a result of light generation, Np is the number of electrons that did not recombine during transfer through the base.The transfer coefficient depends on several factors: the ratio of the base thickness to the average electron diffusion length in the base, the mobility of charge carriers, the presence of an electric field "embedded" in the base, and the ratio of the concentration of electrons Ne and holes Np in the p-type base.For gallium arsenide at T = 300 K the mobility of electrons  е = 8500 ?( ! )•+ @ is 14.2 times greater than the mobility of holes  д = 600 ?( ! )•+ @, so when creating PVC always use electronic conductivity.The typical dependence of the transfer coefficient at lb = Ln on the photocurrent strength is shown in Figure 3.The dependence starts from the origin, since in the absence of current there is no electron transfer and χ = 0.In this case, in a p-type semiconductor there is an equilibrium concentration of non-basic charge carriers -electrons, given by the ratio: where the concentration of main charge carriers -holes pp ≈ Na at T = 300 K, here Na is the concentration of acceptor impurities in the base, np is the concentration of non-main charge carriers -electrons,  .=  .-concentrations of "thermal" holes and electrons where ∆W is the width of the forbidden zone in electron volts, Boltzmann constant, and the density of quantum states (the number of allowed energy levels per unit energy, i.e., reduced to 1 eV) at the bottom of the conduction band is here  9 * is the effective mass of the electron, and the double in front of the bracket appears taking into account the spin (two electrons with different spins can be at the same energy level).
With increasing current strength, χ first increases, then goes to a constant level and finally at high currents slightly decreases due to a decrease in the lifetime of electrons at their high non-equilibrium concentration (see Figure 4).
The greater the concentration of electrons, the more likely they are to recombine with holes, and hence the lifetime is shorter.
where kλ is the absorption coefficient at the laser wavelength λ, S is the area of the illuminated PVC surface, the number of photons incident per unit area per unit time  ; = Ф 8= , here the edge frequency  = / > .Formula (9) will be further used to develop a DT of the PVC.The voltage generator circuit (Figure 5, a) consists of a source of electromotive force (EMF) Ɛ with internal resistance rp.The load is a resistor Rn, through which current flows The current generator (Fig. 5, b) consists of a current source I with a parallel internal resistance rsh and an external load resistance Rn, through which the current flows  ?=  −  CD (11) by setting the load voltage PVC can be represented as a combination of current and voltage generators.Then the equivalent circuit of the photovoltaic converter (Figure 5, c) contains five elements: generators of photocurrent Iph and current of the directly biased p-n junction of the diode Id, as well as resistances: internal shunt rш and series rp and external load resistance Rn.This most commonly used five-element equivalent circuit, in most practically important situations adequately describes the behavior of PVCs operating in the quasi-stationary mode at slowly varying light flux and active load.In this mode of operation, the influence of PVC capacitances -barrier and diffusion, as well as parasitic capacitances of PVC, load and connecting wires -are not taken into account.There are more complicated equivalent schemes of PVCs [24].In the present work, a five-component scheme is used.In this case, the load current [25] It should be noted that for a heterojunction, the magnitude of the diode reverse current I0 depends on the bias voltage and temperature.

Modeling of photovoltaic converter in MATLAB
In MATLAB Simulink modeling environment, a simulation model of a photovoltaic converter [26,27], shown in Figure 6, has been developed.Figure 6.Simulation model of a photovoltaic converter.Blocks: "P_opt" -output optical power of the optical radiation source; "Energy converter" -converter of optical power into photocurrent; "PV cell"equivalent circuit of the photovoltaic converter; "Load" -load resistance; "oscilloscope" -oscilloscope In the block "P_opt" sets the output optical power of the source of optical radiation in watts.MATLAB allows modeling the processes occurring in the PVC in a wide range of power changes (for example, from 0 to 45 W as in this work).In the "Energy converter" block (see Figure 7) the conversion of optical power "P" into photocurrent "Iph" takes place.To perform the conversion it is necessary to enter the parameters: illumination "IL", area of the sensitive area of the photovoltaic converter "S", obtained experimentally.

Set parameters:
• "P" -output optical power of the optical radiation source; • "IL" -illumination; • "S" -area of the sensitive area of the photovoltaic converter; • "Iph" -result of current generation.
The correspondence of the input parameters to the parameters in formula ( 1) is given below: • optical power "P" [W]; • illuminance "IL"  = (1 − ) Ф N ,, where R -reflection coefficient, α -angle of beam incidence (i.e. between the direction of the wave vector and the normal to the area of the sensitive surface of the photodetector).Due to the small reflection coefficient (not more than 1%) and normal incidence of light on the PVC surface, the formula is simplified as follows where "S" is the area of the sensitive area of the photovoltaic converter; • the coefficient of the generated current (highlighted by a rectangular area in Figure 8) corresponds to the product  " 5 8= in formula (1).In the "PV cell" block, the equivalent circuit of the PV cell is modeled as shown in Figure 8.It corresponds to the circuit in Figure 5, b, without load resistor.• "Forward voltage" -the forward voltage that must be applied to the diode to start the photovoltaic conversion process.It has a value of 19 volts; • "On resistance" -resistance of the short-circuited diode.It has a value of 1 ohm; • "Off conductance" -the reverse bias conductance of the diode.It has a value of 1/ohm=10-9 [Cm], which corresponds to a resistance of 1 megohm; • "Series resistance" -series resistance of the sensitive zone of the PVC.It has a value of 7 Ohm.The above parameters of the modeled substitution scheme are chosen on the basis of expert evaluation and can be specified in further studies.
Load" block -load resistance.As the load resistance, a variable resistor R is selected, which allows to change the resistance within the range from zero to 1 GOhm.

Investigation of the main characteristics of the photovoltaic converter
The photograph of the photovoltaic converter YCH-H6424-15-FC-A with the top part (transition from the end of the optical fiber) removed is shown in Figure 9.The figure shows metal electrodes of contacts connected by gold-coated conductors with electrical lead contacts insulated from the case.The surface of the converter shows longitudinal strips of alternating layers of semiconductors connected in series with individual photovoltaic converters, which allows to obtain sufficiently high (up to 20 V) output voltages, with the same design of individual converters.The design provides low leakage losses (large value of resistance Rsh, of the order of 1 MOhm), small series resistance Rp, of the order of 10 ohms and less).At the same time, the maximum achievable currents at properly selected load resistance are close to 700 mA.
The block diagram of the investigation of the main characteristics of the YCH-H6424-15-FC-A photovoltaic pre-inverter is shown in Figure 10.The source of laser radiation was a solid-state laser diode (K976DA5RN-70.00W) 1 with a power supply unit (GPD 74303S) 2, adjustable in power from 0.25 to 70 W at a wavelength of 980 nm.A multimode optical cable 3, 2 meters long, served to supply optical power to the input of a photovoltaic converter (YCH-H6424-15-FC-A) 4, the load of which was a resistance store 5. Ammeters and voltmeters A1 and V1 and A2 and V2, respectively, were used to control the power at the input of the primary and output of the secondary converters.The experiment was conducted under laboratory conditions at an ambient temperature of 23 ℃ and humidity of about 50%.To remove excessive heat, the photovoltaic converter was mounted on a specially designed heat pipe cooling system.
Table 1 shows the nameplate characteristics of the YCH-H6424-15-FC-A photovoltaic pre-inverter, which are taken as reference in this paper.In the course of work the load characteristic of the photovoltaic converter YCH-H6424-15-FC-A was investigated and the main characteristics given in Table 2 were determined.These characteristics coincided with the specification characteristics given in Table 1 within the limits of the experiment error not exceeding 2%.
The main characteristics of the converter YCH-H6424-15-FC-A obtained experimentally are given in Table 2.
It should be noted that the optical powers used in the experiments differed slightly from the specification ones.This is due to the fact that during calibration of the optical radiation source, the indicated powers were set with an accuracy of at least 0.1%.The reference points presented in Table 2 are the closest to the specification ones (see Table 1).They were used in modeling.From the analysis (Tables 1 and 2) it follows that the experimentally obtained values coincided with the specification values with an accuracy of 2%.

Functionality check of the digital twin of the photovoltaic converter
For three values of optical power Popt: 10.38, 28.69 and 44.67 W, the developed DT was used to calculate: maximum electrical power Pmax (W), maximum output voltage Vmax (V), maximum load current Imax (mA) and converter Efficiency (%).It should be noted that each calculation was performed for optimal load impedances, different at different optical powers, at which maximum performance is achievable.Only in this case they can be compared with the specification values.For convenience of analysis, the dependencies are presented in the graphs of Figure 11, a-d   and maximum efficiency (η) These effects compensate each other, therefore, as can be seen from Figure 11, c and d, the calculated and nameplate values of power in the load and efficiency coincided for all optical powers coincided with an accuracy of 2%.
The differences in the voltage and current graphs are probably due to the fact that the DT does not take into account all the physical processes occurring in the real PVC, so Figure 4 shows that at high currents generated by the PVC, the transfer coefficient χ due to the temperature increase decreases, which according to the formula (1) leads to a decrease in the current Iph and, accordingly, the maximum load current Imax.
PVC warming up with increasing current leads to a decrease in the no-load voltage UOC (Figure 5, b) and, accordingly, in the maximum output voltage Vmax, which is observed in Figure 11, а.However, as mentioned above, the calculated dependence decreases faster.This can be explained by the fact that when the current increases, the higher voltage falls on the internal resistance of the generator Rn, which remains constant in the DT, but in the real generator decreases, as in all semiconductors with increasing temperature.Therefore, the calculated dependence falls faster, and the drop in Vmax is proportional to the increase in Imax.

Conclusion
In the course of performing the work: 1.The main variants of designs of photovoltaic converters of laser radiation are considered.2. The effect of light on the p-n junction using the basic provisions of the zone theory of solids is considered.
3. The connection of the main parameters used in the description of the processes of conversion of optical energy into electric current energy with the main specification characteristics of PVC is shown.4. The choice of the equivalent circuit of the PVC consisting of five elements is justified.5.The DT of PVC in MATLAB Simulink modeling environment is developed.6.The choice of model parameters closest to the main characteristics of the PVC brand YCH-H6424-15-FC-A is justified.7. The dependences of the main parameters of the DT are calculated: maximum output voltage Vmax; maximum load current Imax; maximum electrical power Pmax; PVC energy conversion efficiency (%) on the optical power at the converter input in the large signal mode.8.An experimental setup has been assembled, on which the above-mentioned main characteristics of the PVC have been studied.9.It was found that at an optical power of 10 W, all the characteristics (calculated, experimentally obtained and specification) coincided with an accuracy of up to 2%, which indicates the operability of the digital PVC double; 10.It is noted that with an increase in optical power, the parameters of the DT -the maximum output voltage Vmax and the maximum load current Imax differ from the experimental ones.The dependencies are of a multidirectional nature and change approximately the same in percentage terms of absolute value.An explanation of this phenomenon is given from the point of view of the theory of a solid body.11.The main characteristics (maximum electrical power Pmax and PVC energy conversion efficiency (%)) of the DT coincide with the characteristics of the real converter.This indicates an adequate description of real processes in the DT program.12.The proposed model of the DT of the PVC requires testing for versatility (suitable for photovoltaic converters of different designs from different manufacturers).The universal model of the DT, developed on the basis of the results of the present work will allow using it for predicting the behavior of other PVCs based on their specification data without conducting additional experiments.This significantly reduces the development time of the system.

Funding
The research was funded by the Ministry of Science and Higher Education of the Russian Federation (Project No. FSNM-2023-0005).

Figure. 1 .
Figure. 1. Variants of designs of photovoltaic converters with end illumination: asingle PV power converter with vertical structure; b -several horizontally interconnected multiple PV laser power converters; c -several vertically interconnected multiple PV laser power converters; d -several horizontally interconnected single PV laser power converters with side illumination

Figure 3 .
Figure 3. Zone diagram of the heterojunction

Figure 4 .
Figure 4. Dependence of the transfer coefficient on the current strength

4 .
Equivalent circuit of PVC Electrical generators are divided into voltage and current generators, to analyze their operation use equivalent circuits (substitution diagrams), graphical representations of which are shown in Figure 5, a, b [23].

Figure 5 .
Figure 5. Equivalent schemes of generators: a -voltage, b -current, cphotovoltaic converter with load

Figure 8 .
Figure 8. Equivalent circuit of photocell substitution in MATLAB Simulink modeling environment

Figure 10 .
Figure 10.Block diagram of the study of the load characteristic of the photovoltaic converter , respectively.

Figure 11 .
Figure 11.Graphs of dependences on optical power Ropt: a -maximum output voltage Vmax; b -maximum load current Imax; c -maximum electric power Pmax; d -converter efficiency Efficiency (%).

Figure 11 ,
Figure11, a-d shows that at an optical power of 10.38 W all the main characteristics of the calculated, experimental and specification characteristics coincided with an accuracy of 2%, which indicates the performance of the DT PVC.When increasing up to 28.69 and further 44.67W, divergences are observed in Figure11, a and b.The calculated value of Vmax decreases faster than the specification value and at Popt = 44.67W the error reaches 20%.In turn, the calculated value Imax increases faster than the specification value and at Popt = 44.67W the discrepancy is also 20%.When calculating the maximum power Pmax= Vmax• Imax(15) ?=  D −  FD +  GHI =  &  ?