**1. Introduction**

Photovoltaic systems (PVSs) are the most common in the "green" energy sector. A considerable share of PVSs falls on local objects (LO) for various purposes, where hybrid solar power plants with connection to the alternating current distribution grid (DG) are used. This corresponds to the current trend of localizing consumption at the generation site [1].

Shavolkin,Shvedchykova, I.; Gerlici, J.; Kravchenko, K.; Pribilinec, F.

**Citation:**

Hybrid Photovoltaic Systems with a Storage Battery for the Remote Objects of Railway Transport Infrastructure. *Energies* **2022**, *15*, 4883. https://doi.org/10.3390/en15134883

 O.;

> Use of

Academic Editors: Larysa Neduzha and Jan Kalivoda

Received: 25 May 2022 Accepted: 30 June 2022 Published: 2 July 2022

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The issue of localization of the consumption of energy, generated by PV, can be solved when using a PVS to increase the load power of the LO above the limit on consumption from the grid. Such an application of PVS can be in demand during the development (expansion) of facilities with an increase in consumption, when the possibilities of increasing the capacity of the existing connection to the power grid are exhausted.

PVS application can be useful for railway infrastructure facilities remote from the transformer substation, including those with the seasonal nature of the load (consumption). The simplified structure of the power supply of LO with a PVS and storage battery (SB) is shown in Figure 1. The implementation of this option may be cheaper than laying a new power transmission line and replacing DG equipment. The advantage of this solution is the possibility of reducing the cost of paying for consumption from the DG during the term of operation of the PVS and the possibility of autonomous operation in the case of DG disruptions.

**Figure 1.** Simplified structure of power supply of LO load.

This also applies to objects of extensive railway transport infrastructure, including remote objects on sections of the grid that are not currently electrified. The main task when using PVS for such facilities is to provide for their own needs. At the same time, the possibility of increasing their power when using existing DG is limited by the consumption limit. The issue of energy saving remains relevant. First of all, this concerns the reduction in electricity consumption from DG. The use of storage batteries in the PVS with modern methods of energy managemen<sup>t</sup> [2] allows rationally redistributing the energy in the system in time. This also achieves a reduction in consumption from DGs and an increase in the reliability of the power supply of LO. Improving the performance of systems with renewable sources of electricity is an urgen<sup>t</sup> task, which contributes to the further development of energy with distributed sources of electricity.

#### **2. Literature Review and Problem Statement**

The demand for the use of hybrid PVS with SB is confirmed by the fact that the electrical market is widely represented by various solutions of hybrid inverters [3,4], which are designed for LO. They have all the equipment for connecting a photovoltaic battery (PV) and SB, as well as sufficiently powerful software. They are designed for self-consumption of LO while reducing consumption from DG, and they provide an uninterruptible power supply function.

With a power consumption of LO in excess of 10 kW, it is advisable to use three-phase multifunctional inverters while maintaining a power factor close to 1 at the point of common coupling (PCC) to the grid [5–12]. This allows to unload the grid from reactive power and

ensures the symmetry of loading the phases of the grid at an unbalanced load [9–12]. A previous study [13] presented a three-phase PVS with an SB with four converters with a common link of direct current (DC). In the DC link, a proportional–integral (PI) voltage controller (VC) was used. The controller set the currents of the battery and supercapacitor. The "multiconverter" provided a given active power with the proper quality of electricity, PV operation with tracking the maximum power point (MPPT), and extended battery life with hybrid storage. In [10,11], the structure of the control system of a multifunctional inverter of a PVS with an SB with voltage stabilization in the DC link with three VCs was considered. The VC controls the currents of the SB and PV, as well as current in the PCC. The structure changes in accordance with the mode of operation, and only one of the VCs is always used. Ensuring the efficiency of the use of hybrid PVS with energy storage for LO is usually associated with a reduction in the cost of consuming electricity from the grid and an increase in the reliability of the power supply [14]. With a wide range of changes in PV generation during the year, cost reduction is achieved by overestimating the power of PV relative to the load power. This allows providing acceptable indicators in cloudy weather and winter.

PV generation changes significantly during the day and year. Therefore, the increase in the efficiency of PVS energy managemen<sup>t</sup> is associated with the use of the forecast of PV generation [15–19]. The issues with obtaining an accurate forecast were considered in a number of studies, particularly [15,16]. Recently, web resources have been made available that provide a forecast with a discreteness of 0.5 h or less, including an individual one at the location [20,21]. For the application without generation to the grid, the main purpose of the forecast is load planning for the next day and the possibility of adjustment when it changes [19].

Certain opportunities to reduce the cost of paying for electricity consumed by LO from DG are provided by taking into account the tariffication of paymen<sup>t</sup> [19,22–25]. Features of the implementation of PVS in the application of static and dynamic tariffs were considered in [22].

When using PV for the needs of LO (without generating electricity to the grid), it is achievable to reduce the cost of electricity consumption from DG by up to five times at one tariff rate (up to seven times at wo rates) in the summer [19,26]. In winter, the reduction is insignificant—around 1.2 times. A general estimation of cost reduction for the year is not given. In [27], along with meeting the needs of the LO, the use of the planned generation of electricity in the grid during peak hours was considered. This allowed significantly reducing electricity costs. Overall estimation of cost reduction during the year was not given. Regardless, there was still an underutilization of PV energy in the summer.

An effective tool for assessing the efficiency of the energy managemen<sup>t</sup> of a PVS with an SB is mathematical modeling [26–29]. Modeling of a hybrid system with a supercapacitor for the PV generation period was considered in [29]. Modeling of energy processes in the daily cycle with an estimate of the cost of paying for electricity consumed from the DG was considered in [26,27]. The use of archival data on PV generation [30] allowed studying the operation of the system in different weather conditions with an estimation of the cost of electricity, consumed from the DG.

The possibilities of PVS use for power increase for infrastructure objects, remote from the transformer substation, over the limit of consumption have not been sufficiently studied. This is related to the determination of parameters and limiting capabilities in different seasons of the year, the features of the formation of the SB state of charge in the process of operation, and the realization of the principles of added power formation at maximum use of PV energy. In this case, it is possible to reduce the installed power of the PV and the battery. An important role in assessing the capabilities of the system is performed by mathematical modeling.

Thus, the purpose of the article is to develop principles for the use of PVS with SB to increase the power of the LO above the power limit for consumption from the grid with the maximum use of PV generation.

The main objectives of the research are as follows:


#### **3. Methodology of Research**

A study of ways to improve the control mechanism of the PVS with an SB for increasing the power of the LO was carried out on the basis of analytical methods in electrical circuits. The results of processing statistical data on the PV generation for a given point of location of the object were also used. A proportional increase in power to the original load schedule was adopted. As original, the load schedule characteristic of objects with a predominance of day loads, with peak loads in the morning and evening and a decrease in the load at night, was considered. The basic schedule of power, added and provided by the PVS, was adopted in accordance with the PV generation schedule. The maximum value of the power increase factor in winter takes into account the possibility of ensuring the battery charge within the power limit for consumption from the grid. On this basis, the energy capacity of the battery was determined, followed by an assessment of the possibilities for increasing the load power, taking into account the average monthly PV generation. The choice of a fixed value of the degree of power increase was carried out while taking into account the use of PV energy and cost reduction. The control system of the PVS converter unit was implemented on the basis of a classic double-loop structure with voltage stabilization in the DC link. When forming the *SOC*(*t*) (state of charge) of the SB schedule, a limit of DOD ≤ 80% (depth of discharge) was introduced with one deep discharge per day in the spring–summer–autumn period. The technique to calculate the reference of added power for maximum use of PV energy was realized on the basis of an analysis of the average monthly PV generation by time intervals for the taken load schedule. Analysis of energy processes in the system "DG–PVS with SB–LO load" was carried out for the daily cycle without taking into account transient processes and higher harmonics in energy converters. Energy losses were accounted for through efficiency. The properties of the SB were considered in accordance with the characteristics of the manufacturer. PV generation was estimated on the basis of monthly average values for given time intervals during the day. Data of PV generation were obtained for the location point of the object when processing archival data for 5 years. The modeling of energy processes was performed using MATLAB software package using real archive graphs of PV generation. The days were chosen when PV generation by time intervals was close to the average monthly values. The model was completed on the basis of analytical expressions for steady-state operating modes, which correspond to generally accepted proven calculation methods. When the operating modes of the system changed, the corresponding calculated expressions were used at time intervals per day.

#### **4. The Results of the Research on the Use of PVS with SB to Increase the Power of LO**

The option of PVS with an SB with a grid multifunctional inverter VSI (Figure 2) was considered. The middle pin (n) of the VSI link DC was connected to the neutral connection point of the DG. This allowed ensuring the equalization of power consumption from the grid by phases under unbalanced load LO [9–11]. This also made it possible to control the active power *Pg* in the PCC. The structure of the PVS included the following elements: a DC voltage converter PV (CPV) with a transistor key for measuring the current of the PV shortcircuit [10], and a DC voltage converter battery (CSB). The converter unit was controlled by

a control unit (CU) connected to the programmable control unit (PCU). Communication with the web resource to obtain forecast data was provided by a Wi-Fi WFM module.

**Figure 2.** Structure of hybrid PVS with SB.

When solving the issue of increasing the power of the LO, the general approach changes somewhat; the DG becomes an auxiliary source of energy of limited power. As originally proposed, the use of a load schedule was considered in accordance with the standard distribution of peak loads [19,26] for objects of the utility sector and the nondomestic sector with a single-shift mode of operation. The following distribution of load intervals was accepted: night tariff zone in the period May–August (*<sup>t</sup>*7 = 24.00, *t*1 = 7.00), day tariff (*t*1 = 7.00, *t*2 = 8.00), (*<sup>t</sup>*3 = 11.00, *t*5 = 20.00), (*<sup>t</sup>*6 = 23.00, *t*7 = 24.00), peak load zones (*t*2 = 8.00, *t*3 = 11.00), and (*t*5 = 20.00, *t*6 = 23.00); in the period autumn–winter–spring (*t*1 = 7.00, *t*2 = 8.00, *t*3 = 10.00, *t*5 = 17.00 and 18.00, *t*6 = 17.00 and 18.00, *t*7 = 24.00). An additional point of time *t*4, was also used, corresponding to the transition to an evening decrease in PV generation. This time during the year varied from *t*4 = 16.00 in June to *t*4 = 14.00 in December.

Increasing the power of the LO load in the daytime assumes that the total power *PLC* of the LO load is defined as *PLC = PLg + PC* (where *PLg* is the power which is provided by consumption from the grid (*PLg* does not exceed the limit on consumption *PLIM*), and *PC* is the added power, generated by the inverter due to the energy of PV and SB). We can take *PLg = PLIM*; then, with an increase in *PL*; the possible value of *PLC* grows with constant consumption from the grid. If the actual load power is less than *PLC*, the electricity consumption from the grid is reduced.

The value of the energy generated by PV during the year varies widely (Table 1). Table 1 shows the data [30] on the average monthly generation of PV per day *WPVAVD* at power *PPVR* = 1 kW for the Kyiv location: latitude (decimal degrees)—50.451, longitude (decimal degrees)—30.524. Similar data are given for the city of Žilina *WPVAVDZ* (latitude (decimal degrees)—49.224, longitude (decimal degrees)—18.748). In winter, the PV generation in Žilina is slightly higher. In Table 1 (in parentheses), monthly energy values per day (*W\*PVAVD*) and at time intervals during the day (*WPV*23, *WPV*34, and *WPV*45) are also presented for Kyiv. The energy values without parentheses correspond to the selected days when the generation was close to the monthly average. These values were obtained from archival data of *PPV* generation in Kyiv [30] for the period 2012–2016.


**Table 1.** Average monthly PV generation during the year.

The value of the added load power *PC* was determined on the basis of the average monthly daily generation of *WPVAVD* ≈ 2500 W in the transitional seasons of the year (Table 1): October and March. In November–February, the load, provided by PV, decreased. The average value of power in the daytime ( *PAVD = WPVAVD/tD*, where *tD* is the length of the day) was about 200 W.

We took the basic load schedule (average *PL* values by time intervals) taking into account peak loads in the morning and evening with a decrease in load after *t*4 until the evening peak, e.g., *PL*23*B* = 200 W ( *PLAVD*), *PL*34*B* = 180 W (0.9 *PLAVD*), *PL*45*B* = 160 W (0.8 *PLAVD*), and *PL*56*B =* 200 W ( *PLAVD*). The total night load of LO could be taken from the condition *PLg*62 *= PLIM* − *PB* (we took *PLIM* equal to the peak power *PLIM = PL23*B *=* 200 W, *PB = UBIB*—power, consumed from the grid to charge the SB ( *UB* and *IB*—voltage and current of the SB)). At the same time, *PLg*62 ≥ *PLMIN*, *PLgMIN =* 0.2 *PL*56 in summer, and *PLgMIN =* 0.3 *PL*56 in winter. The total energy transmitted by the inverter to increasing power ( *PL*) at the interval (*<sup>t</sup>*2, *t*6) was *WL*26 = 2740 Wh in summer, *WL*26 = 2580 Wh in autumn–spring, and *WL*26= 2410 Wh in winter.

The effective use of the SB's capabilities for the redistribution of energy in the system involves the formation of the S*OC*(*t*) dependency *Q*<sup>∗</sup>(*t*) ( *Q*∗ = 100 *Q*/*QR* , *Q* = *Q*0 + - *IBdt* , *QR = CB*—the rated value (100%) or capacity (Ah) of the battery, *Q*0—the initial value). Increasing the power during peak hours in the morning and evening, when the PV generation is small, implies a deep discharge of the SB. That is, we have two deep discharges per day. In these conditions, the use of lithium-ion batteries is preferable.

Two variants of implementation were considered: (1) with a maximum increase in power in accordance with PV generation, which is available for the consumers with seasonal load; (2) with a constant increase in power during the year.

For variant (1), it was assumed that the planning of load using the day-ahead forecast was possible.

For calculation of the value of SB energy capacity, in the interval (*<sup>t</sup>*4, *t*6), the PV generation *WPV*45 is small, and the increase in power is achieved mainly due to the energy of the SB. At the same time, the energy balance is determined by the following expression:

$$0.01 \cdot \Delta Q\_{46}^\* W\_B \cdot \eta\_C \cdot \eta\_B = P\_{C45}(t\_5 - t\_4) + P\_{C56}(t\_6 - t\_5) - \mathcal{W}\_{PV45} \cdot \eta\_{C} \tag{1}$$

where *WB* is the SB energy capacity (*WB = UBCB*), <sup>Δ</sup>*Q*<sup>∗</sup>46 = *Q*<sup>∗</sup>4 − *Q*<sup>∗</sup>6, *ηC* is the general efficiency of the SB voltage converter and grid inverter, and *ηB* is the efficiency of the SB.

*WB* is calculated from the condition of the functioning of the added power of the load during the evening peak hours (*<sup>t</sup>*5, *t*6) and on the intervals (*<sup>t</sup>*4, *t*5), when PV generation is significantly reduced. This is most typical in winter, with a longer evening peak (4 h). The limitation is to ensure the possibility of the battery charge and the operation of the load at night within the framework of *PLIM*. When *WPV*46 = 0, the value of the energy capacity of the battery SB is

$$\mathcal{W}\_{\rm B} = \frac{\mathcal{W}\_{\rm C46}}{0.01 \cdot \Delta Q\_{46}^\* \cdot \eta\_C \cdot \eta\_B} \, ^\prime \tag{2}$$

where *WC*46 = *WL*46 *(ρ* − 1); *ρ >* 1 indicates the degree of increase in the load power.

It was assumed that the night load in winter also proportionally increases *PLg* = *ρ* 0.3 *PLIM*. The possible value for the battery charge in the framework of limit is denoted as

$$
\Delta W\_{B62} = (t\_2 - t\_6) P\_{LIM} (1 - 0.3\rho) \,. \tag{3}
$$

Alternatively,

$$
\Delta W\_{\text{B62}} = \frac{W\_{L46} \left(\rho - 1\right) \Delta \dot{Q}^\*\_{\text{ } \text{62}}}{\Delta Q^\*\_{\text{ } 46} \left(\eta\_{\text{C}} \cdot \eta\_{\text{B}}\right)^2}. \tag{4}
$$

The maximum value *ρMAX* in the interval (*<sup>t</sup>*4, *t*6) can be determined in accordance with Equations (2) and (3). In this case, at DOD6 ≤ 80%, *ρMAX* = 1.721. Accepting *ρ* = 1.7, *WB* = 1164 Wh. At DOD6 ≤ 90%, the value is *WB* = 1034 Wh. The average value *WB* = 1099 Wh (corresponding to, for example, *CB* = 43 Ah at *UB* = 25.6 V) can be accepted. The resulting value is sufficient to ensure the added power at *ρMAX* = 1.7 and average monthly PV generation in December in the interval (*<sup>t</sup>*2, *t*6).

However, for the same value *PLC* = *ρPL* and *PLg* ≤ *PLIM*, the different variants of reference of the added power *PC* for the interval (*<sup>t</sup>*2, *t*6) (in Table 2, data are presented for *ρ* = 1.7) are possible. This allows planning *PC*(*t)* according to the conditions.


**Table 2.** Variants of the reference *PC* and *PLg* at *ρ* = 1.7.

Above the variant, *va* was considered. At minimal PV generation (in winter), variant *vb* ensured the biggest value *ρ*, whereby *PC*26*(t) = ρPL*26*(t)* − *PLIM*. In this case, *WC*46 = *ρWL*46 − *PLIM(t*6 − *t*4), and the value Δ*WB*62 can be expressed as

$$
\Delta W\_{\rm B62} = \frac{P\_{\rm LIM} \left( 6\rho - \left( t\_6 - t\_4 \right) \right) \Delta Q^\*\_{\rm 62}}{\Delta Q^\*\_{\rm 46} \left( \eta\_{\rm C} \cdot \eta\_{\rm B} \right)^2}. \tag{5}
$$

We ge<sup>t</sup> values *ρMAX* = 1.78 and *WB* = 968 Wh (corresponding to, for example, *CB* = 37.8 Ah at *UB* = 25.6 V). The advantage of this variant is the minimal value of *PC* in the interval (*<sup>t</sup>*4, *t*6) when the PV generation is minimal.

Variant *vc* is included in Table 2. This variant is more tied to the PV generation schedule during the day.

There are days in winter when *WPVD* ≤ 60 Wh. In this case, it is possible to use a night charge of the battery up to 100%, followed by a discharge during the day (from 8:00 a.m. to 9:00 p.m.) at DOD6 = 10%. This allows using the energy Δ *WB*26 = 768 Wh (at *WB* = 968 Wh) in the load. Accordingly, *ρ* = ( Δ *WB*26 + *WL*26*LIM*)*/WL*26*LIM =* 1.3. At small values of *WPVD* (for example, at *WPVD* ≤ 300 Wh), there is the possibility to redistribute the energy of SB by intervals. For example, the degree of increase in power in peak hours can be saved without increasing from 10:00 a.m. to 5:00 p.m. Then, for example, 0.3 Δ *WB*26 can be used in the morning peak, and, taking into account the duration, 0.6 Δ *WB*26 can be used in the evening peak. This allows ensuring *ρ* ≈ 1.6 in intervals (*<sup>t</sup>*2, *t*3) and (*<sup>t</sup>*5, *t*6).

In the period spring–autumn, there is the possibility to increase *ρ*. Determination of *ρ* is carried out in the accordance with forecast data of PV generation on the next day on the basis of the balance of energy in the intervals (*<sup>t</sup>*2, *t*6) and (*<sup>t</sup>*4, *t*6).

It is possible to calculate *ρ* for the interval (*<sup>t</sup>*2, *t*6) as follows:

$$\rho\_{26} = \frac{\mathcal{W}\_{\text{PV26}} \eta\_{\text{C}} + \Delta \mathcal{W}\_{\text{B26}} - \mathcal{W}\_{\text{g}} \eta\_{\text{E26}} + P\_{\text{LIM26}}}{\mathcal{W}\_{\text{L26}}},\tag{6}$$

where *WgR*26 indicates a decrease in energy consumption from the grid, and *WLIM*26 = *PLIM*(*<sup>t</sup>*6 − *<sup>t</sup>*2).

Depending on the PV generation, the implementation is possible (a) with the discharge of the SB without a decrease in consumption from the grid, (b) without the discharge of the SB and a decrease in consumption from the grid, (c) with minimal discharge of the SB and a decrease in consumption from the grid, or (d) without the discharge of the SB and a decrease in consumption from the grid. Variant (a) is specific to winter with small PV generation, when *ρMAX* = 1.7 is accepted.

For interval (*<sup>t</sup>*4, *t*6),

$$\rho\_{46} = \frac{k\mathcal{W}\_{PV46}\eta\_C + \Delta\mathcal{W}\_{B46} - \mathcal{W}\_{\mathcal{g}R46} + P\_{LIM46}}{W\_{L46}},\tag{7}$$

where *k* is a coefficient, taking into account the use of PV energy for consumption, and Δ *WB*46 corresponds to DOD = 80%.

The value of *k* takes into account that, when the SB is fully charged by the time *t*4, only part of the PV energy is used for consumption by the load. The rest of the energy provides a reduction in consumption from the grid. Thus, *k =* 1 is accepted if *WPV*45 · *ηC* < *WC*45, while *k =* 0.5 is accepted if *WPV*45 · *ηC* ≥ *WC*45.

Furthermore, *ρ*46 can be defined at the decrease in consumption in Equation (6) when *WgR*46 = 0. For March (Table 1), *ρ*46 *=* 1.8. Compared with the value *ρ*26 = 2.02 according to Equation (6) for option (b), we accept the smaller value. In this case, with *ρ* = 1.8, it is possible to implement option (c). Thus, for *ρ* = 1.8, we find the following value:

$$
\Delta Q^\*\_{.23} = \frac{W\_{PV23} \eta\_C - W\_{C23}}{0.01 W\_B \eta\_C \eta\_B}. \tag{8}
$$

For March Δ*Q\**23 = 17% > 0, i.e., the SB state of charge increases, and the night battery charge is not needed. Then, in accordance with Equation (6), we have a decrease in consumption on the order of 572 Wh. With the average monthly values of PV generation for the summer period, *ρ* = 1.95 is achievable.

A graph of the added power *PC(t)* and the state of charge of the SB *Q\*(t)* is presented in Figure 3 in accordance with the obtained value of *ρ*. We consider options *vb* and *vc* when using the PV energy *WPV* forecast data over time intervals. At the same time, it is desirable to (1) use the lowest possible power consumption at night (until 8:00 a.m.) per battery charge, and (2) ensure the condition *Q\**4 → 100%, taking into account the reduction in PV generation in the evening. A prerequisite is the maximum use of PV energy.

**Figure 3.** Formation of schedule of state of the charge of SB.

Reference to the initial value *Q\**2*R* is carried out in accordance with values Δ*Q\**23 and Δ*Q\**24 (calculated similarly to Equation (8)). If Δ*Q\**24 ≤ 0, then *Q\**2*R =* 100% (curves 1, 5, and 6 in Figure 3). In this case, when Δ*Q\**24 > 0 and Δ*Q\**23 ≤ 0, then *Q\**2*R* = (100 − Δ*Q\**24) ≥ 40% (curve 2 in Figure 3). If Δ*Q\**24 > 0, Δ*Q\**23 > 0, and *WPV*23 · *<sup>η</sup>C*/*W*1*C*23 < 1.5 (*W*1*C*23 is the value for the basic schedule with given *ρ*), then *Q\**2*R* = (100 − Δ*Q\**24*)* ≥ *(Q\**6 *+ δ)* (curve 3 in Figure 3,*δ =* 10–15%). In all cases, the reference of the added power is *PC*23 *= <sup>P</sup>*1*C*23.

If Δ*Q\**24 > 0, Δ*Q\**23 > 0, and *WPV*23 · *<sup>η</sup>C*/*W*1*C*23 ≥ 1.5*,* then *PLC*23 ≥ (*PC*23 *= PPV*23 · *ηC)* ≥ *<sup>P</sup>*1*C*23, which is given in accordance with PV generation. In this case, the SB charge (curve 4 in Figure 3) is not carried out. It also does not use the battery charge at night, but there may be some battery charge before 8:00 a.m. (sunny morning).

The *PC*34 value in the interval (*<sup>t</sup>*3, *t*4) is *PC*34 = *WPV*34*ηC*−0.01Δ*Q*<sup>∗</sup>34*WBηCη<sup>B</sup>* (*<sup>t</sup>*4−*t*3) ≤ *PLC*34, where Δ*Q\**34 is defined using values *Q\**2*R* and Δ*Q\**23, as described above.

In the interval (*<sup>t</sup>*4, *t*5) with a large PV generation, the state of SB charge can increase (curve 7 in Figure 3), be unchanged (≈100%), or decrease. The possibility of an increase is excluded by increasing the *PC*45, which is carried out automatically. The formation of the degree of discharge in the interval (*<sup>t</sup>*5, *t*6) is carried out in the mode of regulation of the SB current.

The battery charge when *Q\** ≥ *Q\*d* = 90–92% is reached at a constant value of the voltage [31]. In this case, the battery current is determined by the charge curve and is significantly reduced. Thus, the ability of the SB to receive energy is limited. Therefore, when *Q\** ≥ *Q\*d*, the value of *PC* is set according to the actual PV generation as *PLC* ≥ *PC* = (*PPV* · *ηC* − *PB)* ≥ *<sup>P</sup>*1*C*. The limitation *PLC* ≥ *PC* is implemented by reducing the PV generation. The introduction of this mode allows ensuring more complete use of the PV energy, particularly when the calculated value of *PC*34 is less than required.

Figure 3 also shows a graph for the case when the PV energy is not enough to charge the battery up to 100% (curve 5) and for the case when the PV generation is close to 0 (curve 6). In these cases, the task is realized to support the load during peak hours.

Consider the variant with the constant increase of power during the year. For the choice of value *ρ*, consider the option *vb* with reference to the added power when the value *WB* is minimal. According to Equations (2) and (5), when *ρ* increases, *WB* also increases. An important issue is the underutilization of the installed PV power at high solar activity in the summer. To estimate, we introduce the PV energy utilization factor,

$$k\_{PV} = \frac{W\_{C26} + W\_{\text{g}R26} - \Delta W\_{\text{R26}}}{m\_{\text{P}}W\_{\text{PV26}}\eta\_{\text{C}}},\tag{9}$$

where *WgR*26 indicates a decrease in energy consumption from the grid, *WC*26 is the energy consumed by the added load, and *mP* is a coefficient of PV power recalculation relatively installed power *PPVR* = 1 kW.

The value of *WgR*26 can take place during hours of high daytime solar activity *tda*, when *PPV*·*η<sup>C</sup> > PC + PB* (*PB = UBIB* is the power of SB charging). To exclude the generation of energy into the grid, the restriction *PgR* ≤ *PLIM* is used, which is achieved by regulating (reducing) the PV generation. In the limited case, *WgR*26 *= PLIM* · *tda* (*tda =* 6 h, from 10:00 a.m. to 4:00 p.m.).

The value of Δ *WB*26 in the summer period with the average monthly PV generation is taken as equal to Δ *WB*26 = 0 (nighttime battery charging is not required).

To assess the efficiency, we use the coefficient of cost reduction for electricity consumed from the grid (at one tariff rate and full use of PV energy). For the winter period (December), when the power increase in the interval (*<sup>t</sup>*2, *t*6) is achieved while maintaining the consumption from the grid within the limit,

$$k\_E = \frac{\mathcal{W}\_{\rm LC}}{\mathcal{W}\_{\mathcal{S}}} = \frac{\mathcal{W}\_{\rm LC}}{\mathcal{W}\_{\rm LC} + \Delta \mathcal{W}\_{\rm B26} / \left(\eta\_{\rm C} \eta\_{\rm B}\right)^2 - m\_P \mathcal{W}\_{\rm PV} \eta\_{\rm C}} \,\tag{10}$$

where *WLC* = *WLC*62 + *WC*26 + *PLIM*(*<sup>t</sup>*6 − *<sup>t</sup>*2) is the total energy consumed by the load.

At *mP* = 1, with an increase in *ρ* and, accordingly, *WC*26, the value of *kE* decreases, and the value of *kP* increases. If the installed power ( *mP* < 1) is reduced, then there is a reverse change in the coefficients.

Consider the option with *ρ* = 1.6 in comparison with the variant for *ρ* = 1.7 discussed above. This allows reducing the *WB* value from 968 Wh to 800 Wh (by 21%). Almost the same value of *kE* for December, in this case, can be obtained with *mP* = 0.86 and an increase in *kP* to 0.71 instead of 0.679 (in June). If we recalculate to the same load power, we ge<sup>t</sup> a decrease in the installed PV power by 9.4% and in the energy capacity *WB* of the SB by 14%. At the same time, it remains possible to increase the power to *ρ* = 1.8.

The reference of the value of added power and the formation of a graph of the state of charge of the SB are carried out according to the method discussed above.

A simplified structure of the PVS control system is shown in Figure 4. The system of automatic control of the CU converter unit is implemented according to well-known principles with voltage stabilization in the DC link *Ud* at the VSI input [10,13]. This is provided by three proportional–integral (PI) voltage controllers (VCs): VCIPV forms the PV current reference; VCIB forms the SB current reference; VCIg forms the reference of the current at the point of common coupling to the grid. The system (Figure 4) contains three channels:


**Figure 4.** Structure of the control system of PVS.

The control of switches in the structure and the formation of current references are carried out by the PCU. The PCU also processes the prediction data according to the WFM block signal and the specified time intervals (TH). The phase lock loop (PLL) and offline control PVS channel when the mains voltage is turned off in Figure 4 are not shown (current reference of inverter phases *<sup>i</sup>*1*Ca,b,c*; in this case, a separate load voltage controller is set [10]).

The control principle (Table 3) is based on the control of active power, consumed from the grid (in PCC) *Pg* = *PLC* − *<sup>P</sup>*1*C* (*P*1*C* is the value of the added load power at the current time interval, and *PLCi* is the value of the active load power according to the measured values of currents and phase voltages) at *PLIM* ≥ *Pg* ≥ 0. The initial parameters are the recommended schedule (maximum average value) *PLCR(t)* and the base schedule *PCB(t)* for the accepted value of *ρ,* and the calculated schedule *<sup>P</sup>*<sup>1</sup>*C(t)*. There is a possible situation when *PLC* = *PCR*. With this in mind, deviation compensation Δ*PLC* = *(PLC* − *PLCR)* ≥ 0 is used in the reference *<sup>P</sup>*1*C* = *<sup>P</sup>*1*C +* Δ*PLC*. The operating modes of the control system by intervals and the used controllers are given in Table 3. Calculation expressions for steady modes are given in the description of the model of energy processes.


**Table 3.** Operating modes of the PVS with SB.

Consider the functioning of the system, starting from the intervals (*<sup>t</sup>*2, *t*3). PV operates in maximum power mode (MPPT). The SB current is set by the controller VC*IB* → *<sup>I</sup>*1*B*. Then, *<sup>I</sup>*1*gm =* √2*Pgi*/3*Ugph* (*Ugph*—phase voltage of the grid) is defined according to *Pg* = *PLC* − *<sup>P</sup>*1*C*23 and is supplied to the input of the current set unit RCU + CCL. When the load is reduced, *Pg* decreases; when the load is increased, *Pg* also increases. If *WPV*<sup>23</sup>·*ηC/WC*<sup>23</sup> ≥ 1.5, then the value

of *PC* is set by PV generation under the condition *PCL* ≥ (*PC = PPV* · *ηC*) ≥ *<sup>P</sup>*1*C*23. In this case, at *PPV*·*η<sup>C</sup>* < *<sup>P</sup>*1*C*23, there is *PC = <sup>P</sup>*1*C*23. At *PPV* · *ηC* ≥ *<sup>P</sup>*1*C*23, there is *PPVηC* = *PC* and the current of SB charge decreases to 0; when *PPVηC* ≥ *PLC*, the SB charge is restored.

When switching to the interval (*<sup>t</sup>*3, *t*5) and *Q\** < *Q\*d*, the operating mode is saved. When *Q\** ≥ *Q\*d* = 90–92%, the SB current is determined by the charging characteristic *IB(Q\*)*; if the set value *<sup>I</sup>*1*B* > *IB(Q\*)*, then VCIB goes into saturation. The battery cannot consume all the energy. This leads to an increase in the voltage *Ud* on the capacitors in the DC link of the inverter. If *PLC* > *PC* ≥ *PCB*, then, upon reaching the switching threshold (*Ud* + Δ*U*), the VCI**g** controller (*g4* mode) is activated and reduces *<sup>I</sup>*1*gm* (*Pg*). If *PPVηC* ≥ *PLC*, then the VCI**PV** controller (*g3* mode) is activated and *<sup>I</sup>*1*PV* (*PPV*) is reduced. Thus, we have two conditions for switching of controllers (excluding the influence of transients).

In the interval (*<sup>t</sup>*5*, t*6), the discharge of SB is carried out with current (given by controller VCI**B**).

$$I\_{B56R} = \frac{0.01C\_B(Q^\*\_{.5} - Q^\*\_{.6})}{(t\_6 - t\_5)}$$

The added power is set in accordance with an average value of SB voltage *UBAV* as *PC*56 = *IB*56*R*·*UBAV +* Δ*PLC*, if Δ*PLC* = *(PLC* − *PLCR) >* 0.

For the night period in the interval (*<sup>t</sup>*6*, t*2), the reference of the current of SB charge is

$$I\_{B62R} = \frac{0.01C\_B(O^\*\_2 - Q^\*\_t)}{(t - t\_2)}\rho$$

where *Q\*t* is the current measured value, for example, with a resolution of 1 h.

In the period spring-autumn, the SB charge in the morning is possible from PV at *PPV*·*η<sup>C</sup>* ≥ 0.

Referencing of the value *Q\**2*R* and planning of the recommended (maximum) load *PLCR* is carried out according to the forecast for the next day. The given values of the added power in the intervals are specified at the beginning of the day in accordance with the current forecast of PV generation. Subsequently, with an interval of 1 h, the adjustment is carried out.

#### **5. Modeling of Energy Processes in the Daily Cycle**

The initial data for modeling were from an archive of PV generation on the mode of maximum power *PPVM(t).* Accordingly, the values of *ρ*, *Q\**2*R*, the base *PCB(t)* schedule for the accepted value of *ρ,* and the calculated *P*<sup>1</sup>*C(t)* schedule were calculated. The load power values of the recommended *PLCR(t)* and the actual values of *PLC(t)*, *PCB(t)*, *P*<sup>1</sup>*C(t)* are given in tabular form. The time intervals are given by the variables *t*12, *t*23*, t*34*, t*45*, t*56*, t*67*,* and *t*71, which take on the value 1 at the corresponding time. The following auxiliary variables are also used:

$$\eta = \begin{cases} 1, \text{ if } Q^\* \ge Q^\*\_{\
u} \\ 0, \text{ if } Q^\* < Q^\*\_{\
u} \end{cases}, \text{ } p\\v = \begin{cases} 1, \text{ if } P\_{\text{PV}M} \eta\_{\text{C}} \ge P^1\_{\text{C}} \\ 0, \text{ if } P\_{\text{PV}M} \eta\_{\text{C}} < P^1\_{\text{C}} \end{cases}, \text{ } lc = \begin{cases} 1, \text{ if } P\_{\text{C}} \ge P\_{\text{LC}} \\ 0, \text{ if } P\_{\text{C}} < P\_{\text{LC}} \end{cases}, \text{ } \epsilon$$

*c* = 1, if PLC ≥ *PPVMηC* ≥ *<sup>P</sup>*1*C*23 0, if PLC < *PPVMηC* < *<sup>P</sup>*1*C*23 , *s* = 1, if *PPVMηC* > 0 0, if *PPVMηC* ≤ 0 , *h* = 1, if (*PLC* − *PLCR*) > 0 0, if (*PLC* − *PLCR*) ≤ 0 ,

$$\eta c = \begin{cases} 1, \text{ if } P\_{\text{LC}} > P\_{\text{C}} \\ 0, \text{ if } P\_{\text{C}} \ge P\_{\text{LC}} \end{cases}, w = \begin{cases} 1, \text{ if } W\_{\text{PV23}} \cdot \eta\_{\text{C}}/W\_{\text{CB23}} \ge 1.5 \\\ 0, \text{ if } W\_{\text{PV23}} \cdot \eta\_{\text{C}}/W\_{\text{CB23}} < 1.5 \end{cases}$$

The variable *w* is precalculated. To measure the values of *Q\*t*, *Q\**5, sample-and-hold schemes are used.

The current PV generation takes into account the following regulation:

$$P\_{PV}\eta\_{\mathbb{C}} = P\_{PVM} \cdot \eta\_{\mathbb{C}} \cdot (\overline{q} + q\emptyset) + (P\_{\mathbb{C}} + P\_{\mathbb{B}})q \cdot l\mathbb{c}\_{\prime}$$

where *PB = UBIB*. The value of added power is

$$P\_{\mathbb{C}} = P\_{\mathbb{LC}} \cdot t\mathfrak{c}2 + P\_{\mathbb{C}}^{1}(t\_{23} \cdot \overline{w} + t\_{34} \cdot \overline{q} + t\_{45} \cdot \overline{p}\overline{v}) + P\_{\mathbb{C56}} \cdot t\mathfrak{s}\mathfrak{c} + (P\_{\mathbb{C23}} \cdot \overline{c} + 1)(P\_{\mathbb{C23}} \cdot \overline{v} + P\_{\mathbb{C56}} \cdot \overline{v}) + ((P\_{\mathbb{P}VM} \cdot \eta\_{\mathbb{C}} - P\_{\mathbb{B}})q \cdot \overline{lc} + P\_{\mathbb{C5}} \cdot q \cdot lc)(t\_{34} + t\_{45}),$$

where *PC*56 = *IB*56*R*·*UBAV + h(PLC* − *PLCR).*

> The power consumed from the grid is *Pg* = (*PLC* − *PC*)*<sup>t</sup>*26 + (*PLC* + *PB*)*<sup>t</sup>*62 . The current of the SB is

$$I\_B = t\_{62} \cdot (I\_{B62R} \cdot \overline{\mathfrak{s}} + \frac{P\_{PV} \eta\_C}{l I\_B} \overline{s}) + (\overline{q} + q\overline{c}) \cdot t\_{25} \cdot \frac{P\_{PV} \eta\_C - P\_C}{l I\_B} + I\_B (Q^\*) \cdot q \cdot \overline{q\overline{c}} \cdot t\_{25} + t\_{56} \frac{P\_{C56}}{l I\_B} \overline{s}$$

The SB model is constructed according to the principles set in [26,27]. Data sheets given by the manufacturer were used [31]: charge characteristics *IBC*(*Q*∗) and *UBC(Q*∗) at *IB* ≥ 0 and discharge characteristic *UBR*(*Q*∗) at *IB < 0,* set in tabular form. The current can be calculated as 

$$I\_B = \begin{cases} \ I\_{B\prime} \text{ if } Q^\* < Q^\*\_{\,d} \\\ I\_B(Q^\*)\_{\prime} \text{ if } Q^\* \ge Q^\*\_{\,d} \end{cases}$$

.

The SB state of charge (*SOC*) taking into account energy losses is

$$Q = Q\_0 + \int I^1\_{\;B} dt\_\prime$$

where *<sup>I</sup>*1*B* = *IB* · *ηB* if *IB* ≥ 0, and *<sup>I</sup>*1*B* = *IB*/*ηB* if *IB* < 0.

To estimate the reduction in electricity costs at a single tariff rate (taking its value equal to 1), the *kE = WL*/*Wg* coefficient was used (*WL* = - 24 0 *PLCdt* is the energy consumed by the LO load per day (without taking into account the energy on the SB charge at night), and *Wg* = - 24 *Pgdt* is the energy consumed by LO from the grid).

#### **6. Simulation Results**

0

To set the PV generation, archival data were used [30] with the selection of days when the generation by intervals was close to the average monthly generation (Table 1). These days correspond to the energy values without parentheses in Table 1. Values *Q*<sup>∗</sup>6, *kE*, and *ρ* for the case when *PLC* ≤ 2*PLIM* and the actual value of total load power *PLC = PLCR* are given in Table 4.


**Table 4.** Simulation results.

Oscillograms *PLC*, *PLCR*, *Pc*, *PPVM*, *PPV*, *Q*<sup>∗</sup>, *IB*, and *Pg* (for clarity, *Pg* is shown as negative) are given as described below.

• In Figure 5 for the December day with a total generation twice below the average (*WPV* = 500 Wh). In this case, *kE* = 1.04, *p* = 1.5 (*PPVR* = 0.86 kW, *WB* = 800 Wh);

**Figure 5.** Oscillograms *PLC*, *PC*, *PPV*, *Q*<sup>∗</sup>, *IB*, and *Pg* for a December day with general generation *WPV* = 500 Wh (*Q*∗ and *IB* values are shown at scale 2 and 10, respectively).

• In Figure 6 for the July day with a total generation in 3.3 times below the average (*WPV* = 1320 Wh). In this case, *kE* = 1.22*,p=* 1.6 (*PPVR* = 0.86 kW, *WB* = 800 Wh);

**Figure 6.** Oscillograms *PLC*, *PC*, *PPV*, *Q*<sup>∗</sup>, *IB*, and *Pg* for a July day with general generation *WPV* = 1320 Wh (*Q*∗ and *IB* values are shown at scale 2 and 10, respectively).

• In Figure 7a for the May day with the generation, corresponding to the average monthly values at *PLC* = *PLCR*, *PPVR* = 1 kW, *WB* = 968 Wh, and at limit values *ρ =* 1.95 with *kE* = 2.618;

**Figure 7.** Oscillograms *PLC*, *PLCR*, *PC*, *PPVM*, *PPV*, *Q*<sup>∗</sup>, *IB*, and *Pg* for the May day with a generation corresponding to the monthly average (*Q*∗ and *IB* values are shown at scale 2 and 10, respectively): (**a**) *PLC* = *PLCR*, *PPVR* = 1 kW, *WB* = 968 Wh, and at limit values *ρ =* 1.95 with *kE* = 2.618; (**b**) *PLC* = *PLCR*, *PPVR* = 0.86 kW, *WB* = 800 Wh, and at limit values *p =* 1.8 with *kE* = 2.356; (**c**) *PLC* = *PLCR*, *PPVR* = 0.86 kW, *WB* = 800 Wh, and *p =* 1.6 with *kE* = 2.795.


**Figure 8.** Oscillograms *PLC*, *PLCR*, *PC*, *PPVM*, *PPV*, *Q*<sup>∗</sup>, *IB*, and *Pg* for the September day with the generation, corresponding to the average monthly values at *PLC* = *PLCR*, *PPVR* = 0.86 kW, *WB* = 800 Wh, and *ρ =* 1.6 with *kE* =2(*Q*∗ and *IB* values are shown at scale 2 and 10, respectively).

Changing the reference of the added power *PC*23 in the interval (*<sup>t</sup>*2*, t*3) from a fixed calculated value to the value corresponding to the actual PV schedule in the spring–autumn period allows increasing *kE* by 2–4% with the exclusion of SB discharge.

#### **7. Discussion of the Results of the Study on Increasing the Power of the LO Using PVS with SB**

Increasing the load power of the LO above the limit of power consumption from the grid while reducing the installed power of PV and SB and decreasing the cost of paying for electricity, consumed by the LO, from the DG is possible due to the following:


This article is a development of previous studies [19,26], which considered increasing the efficiency of hybrid PVS with SB for the needs of local facilities. This was achieved by reducing the cost of electricity consumed from the grid when using the PV generation forecast. A common problem is the significant overestimation of the PV power in relation to the load power of the LO, which is necessary for use in conditions of low PV generation.

As a result, even with medium PV generation, there is a significant underutilization of PV energy with the need for regulation of PV generation. A feature of the proposed solutions is a change in the general approach to the use of PV and SB energy when the load power, added to the limit, is formed. Directed formation of the graph of added power allows reducing the installed power of PV and SB.

There are certain limitations regarding the use of the results of the work, as described below.


Further development of this work is connected with the optimization of system parameters. It is also required to study the possibilities of correction of deviations in the values of the actual PV generation according to the forecast data and possible changes in the current forecast during the day.

## **8. Conclusions**

With the selected parameters of the PVS, it is possible to increase the power of the LO over the limit for consumption from the grid up to 1.7–1.95 times. This depends on the average monthly PV generation of during the year. Limiting the degree of power increase to a value of 1.6, when choosing the parameters, allows reducing the installed power of the PV and SB. The possibility of increasing the degree of power increase to a value of 1.8 (if necessary) remains in this case.

It is advisable to select the parameters of the PVS on the basis of the data on the average monthly PV generation for the taken schedule of the load. It is possible to use archival data from web resources with open access at any point location of the object. The ratio of PV power and added load power is determined on the basis of PV generation in the transitional seasons (in this case, March and October). The energy capacity of the SB is determined by the power of the added load from the condition of the sufficiency of its operation in the pre-evening hours and evening peak hours with a given limitation of the DOD battery. The use of the basic schedule of the added power with reference to solar activity, while maintaining the resulting power of the LO, makes it possible to reduce the energy capacity of the SB. Therefore, when setting the degree of power increase *ρ* = 1.7 for winter, the energy capacity can be reduced by 13.5%. Limiting the value of *ρ* at the level of *ρ* = 1.6 allows reducing the installed power of the PV and the SB. When converted to the same total load power, the reduction for *PPVR* is 9.4%, and that for *WBR* is 14%.

The principle for controlling the power consumed by the LO from the grid in accordance with the reference value of the added load power and the total load power of the LO was developed. The reference of the value of the added power can be determined from the condition for the formation of the *SOC*(*t*) graph of the SB according to the PV generation forecast data.

At the same time, at certain time intervals, reference of the value of the added power can be carried out according to the schedule of the PV generation. This contributes to the maximum use of its energy. With the average monthly PV generation in the spring–autumn period, the SB discharge during the hours of the morning load peak is not used. Accordingly, the nighttime SB charge from the grid is not used. This helps to reduce consumption from the grid and reduce the number of deep discharge cycles, which contributes to longer battery life.

On the basis of a formalized description of energy processes in steady-state conditions for the daily cycle of the system's functioning, the simulation of the system was completed. The simulation results showed that choosing the PVS parameters on the basis of *ρ* = 1.7 by setting *ρ* = 1.7–1.95 depending on the average monthly PV generation reduced the cost of electricity, consumed from the grid, for one rate of paymen<sup>t</sup> by 1.1 to 2.68 times. When choosing the PVS parameters on the basis of *ρ* = 1.6 and a constant degree of power increase during the year, it is possible to reduce electricity costs by up to 7% in the summer.

**Author Contributions:** Conceptualization, O.S., I.S. and J.G.; methodology, O.S., I.S. and J.G.; validation, O.S. and I.S.; formal analysis, I.S. and K.K.; investigation, O.S., I.S., K.K. and F.P.; resources, I.S. and K.K.; data curation, O.S. and I.S.; writing—original draft preparation, O.S., I.S. and F.P.; writing—review and editing, J.G.; visualization, I.S. and K.K.; supervision, J.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This publication was issued thanks to support from the Cultural and Educational Grant Agency of the Ministry of Education of the Slovak Republic through the projects "Implementation of modern methods of computer and experimental analysis of properties of vehicle components in the education of future vehicle designers" (Project No. KEGA 036ŽU-4/2021) and "Development of advanced virtual models for studying and investigation of transport means operation characteristics" (Project No. KEGA 023ŽU-4/2020). This research was also supported by the Slovak Research and Development Agency of the Ministry of Education, Science, Research, and Sport of the Slovak Republic in Educational Grant Agency of the Ministry of Education of the Slovak Republic through the project "Investigation of the properties of railway brake components in simulated operating conditions on a flywheel brake stand" (Project No. VEGA 1/0513/22).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This work was supported by the Ministry of Education and Science of Ukraine in the joint Ukrainian–Slovak R&D project "Energy managemen<sup>t</sup> improvement of hybrid photovoltaic systems of local objects with storage batteries" (0122U002588).

**Conflicts of Interest:** The authors declare no conflict of interest.
