## 2 Design Selection of an Autonomous Photovoltaic Installation  
A system with solar-battery (SB) orientation is bulky and requires a complex automated electric drive that tracks the Sun. In our case, a fixed roof panel with a tilt angle \(\beta=45^{\circ}\) was suitable [7, 34], to which the accumulator battery (AB) was connected in parallel. Users required a standard voltage of 220 V with a frequency of 50 Hz and used a stabilized sinusoidal voltage converter. In the dark hours of the day, the power supply of the converter (inverter) was provided by a battery that was charged during the day. The selection of the voltage of the panels was carried out by taking into account the safety of the power supply and the battery, which decreased with increasing voltage, and the reliability of the AB and SB. In high-voltage circuits (220 V), reliability decreases; a high-voltage AB has a large voltage between its elements and requires a complex balancing system to prevent damage; the best-developed design of ABs is automotive. A sealed lead-acid battery operates at a voltage of between 10 and 14 V. A hypothetical structure of an autonomous photovoltaic system is shown in Figure 1 and contains a current source, PV1, imitating a solar cell, a battery, and an inverter with a step-up transformer. The battery starts charging when the voltage of the solar battery increases to the minimum battery voltage level due to an increase in illumination, which is \(10\div 12\) V.  
Figure 2 shows the possibilities for matching the energy characteristics of the elements of PV tubes, taking into account external conditions with the following typical characteristics: SB current ISB (A), SB voltage USB (V), change in ambient temperature T (\({}^{\circ}\)C), solar radiation power depending on the time of day PC (W/m\({}^{2}\)) [35; 36; 37]. It is clear that the energy deficit from the SB occurs from 6 to 7 in the morning and from 17 to 18 in the evening when USB < UAB, and energy losses are proportional to the area of triangles A and B. We calculate the maximum charge theoretically separated from the \(Q_{SB}\) over the time of illumination as follows:  
\[Q_{SB}=I_{SB}*t_{c} \tag{1}\]  
We calculate the charge losses \(\Delta\)Q when USB < UAB during the day. This happens twice every hour:  
\[Q_{m}=\frac{I_{max\Delta}*t_{\Delta}}{2}=Q_{e}, \tag{2}\]  
Figure 1: Equivalent circuit of a PV with a model SB in the form of a controlled current source PV1 and a converter to increase the voltage of the solar battery. Figure 2: Matching power supply characteristics to battery voltage depending on sunlight. where \(Q_{m}\) is the morning charge, \(Q_{e}\) is the evening charge, \(I_{max\lambda}\) is the maximum charge of the SB one hour after sunrise, and \(t_{\Delta}\) is the time when USB < UAB. Let us calculate the total charge loss due to USB < UAB:  
\[\sum_{\Delta}Q=\Delta Q_{m}+\Delta Q_{e}, \tag{3}\]  
Let us calculate the relative value of the charge loss without a converter:  
\[\zeta=\frac{\sum_{\Delta}Q}{Q_{SB}}=2\%, \tag{4}\]  
where \(\zeta\) is part of the capacity loss. Due to these 2% losses, there was no great need to increase the voltage. We proved that we could create a PV without a voltage converter between the power supply and the battery. It is relevant for southern regions. The converter on L2, VT3, and VD6 (Figure 1), which increases the voltage, had an efficiency of \(\cong\) 0.8. Its inclusion allowed the use of energy to charge the battery from 6 to 7 in the morning and from 17 to 18 h (Figure 2). The extra energy, on the other hand, was equal to the losses in the boost converter L2, VT3, and VD6. This was because the areas of triangles A and B were equal, so we used a diode VD6 to connect SB and AB in parallel. The study aimed to determine the optimal voltage on AB and SB at fixed load parameters. We chose the PV structure with the simplest circuit for battery protection and control.