The Impact of Photovoltaic Installations on Changes in Voltage Levels in the Low-Voltage Network

Abstract

Due to the dynamic increase in the number of prosumer electrical installations in Poland, one may observe many negative effects of their development, including the deterioration of energy quality parameters and the reliability of the existing distribution network. The installation of solar panels in Polish homes was mainly motivated by economic reasons. One of the most important problems of the distribution network is the increase in voltage. The aim of this work was to develop a practical method for determining the maximum voltage changes caused by the connection of photovoltaic installations. To accomplish this, a representative low-voltage overhead line, typical of those found in Poland, was modeled using the NEPLAN software. More than 100 distinct simulations were conducted, exploring various locations and power capacities of photovoltaic installations and utilizing authentic annual profiles for both electrical loads and photovoltaic generation. From the analysis of the data obtained, relationships that enable the determination of voltage changes induced by photovoltaic connections at any node within the low-voltage circuit were established. The computational results derived from this simplified model demonstrate sufficient accuracy for practical applications, and the required input data is accessible to distribution system operators.

1. Introduction

The increasing number of micro-installation in the low-voltage network is the result of changes in the electricity market and growing environmental concerns about global warming. Renewable energy sources (RESs) are considered as elements contributing to increased energy security, as their operation helps to reduce the dependence on fossil fuels. Economic factors related to the costs of energy from conventional power plants, with the costs of RESs falling at the same time, are also important. Renewable energy systems, especially photovoltaic (PV) systems used in private households, are rapidly gaining popularity. When installed on a residential building or other small-scale facilities, they can significantly reduce the amount of energy drawn from the mains and thus lower the associated costs. In addition, global trends such as the development of electromobility are becoming a reality, and distribution system operators are facing new challenges in the form of increased energy demand, rapidly changing loads, and growing instability in the power grid.

The connection of a renewable energy source, even a small-scale one, to the power network is regulated by relevant legislation. The legal acts in Poland that define the conditions for the operation and connection of renewable energy sources to the network are the Act of the 10 April 1997 Energy Law, as amended [1]; the Renewable Energy Sources Act, as amended [2]; and the Regulation of the Minister of Economy of 4 May 2007 [3]. Power sources with a rated power not exceeding 40 kW may theoretically be connected directly to the low-voltage network at any point on the network. However, the conditions specified in the grid code of the relevant operator must be met. The possibility of connecting micro-installations to low-voltage distribution networks, practically without consulting the network operator (only notification), can cause problems with keeping the voltage within acceptable limits. In the past, low-voltage distribution networks did not apply voltage control. On-load tap changers are not used in MV/LV transformer substations. Voltage levels can also be affected by the distribution of sources in a given network. Clustering them at one point (one branch) may produce different effects than a uniform distribution. Furthermore, the presence of sources close to the MV/LV transformer will also cause different effects than the placement of sources at the end of the network (at points furthest from the transformer).

The demand for connecting prosumers’ electric installations to low-voltage networks is constantly growing. Distribution network operators are overwhelmed by the large number of connection requests that exceed network capacities. Furthermore, the processing of applications for micro-installations is being handled on an unprecedented scale. Hundreds of thousands of such cases have a significant impact on the operation of the distribution network. The increase in their number has produced many consequences that were not expected in long-term analyses. In addition to the positive impact, i.e., the increase in the share of renewable energy sources in Poland’s energy production, there are also negative aspects that result in economic and technical problems. The problem can arise when there are too many PV installations. When electricity generation exceeds consumption, part of the energy is fed back into the network, causing a voltage rise in the power network near the points where PV installations are connected. This process continues until the voltage reaches the nominal network voltage ($U_n$) [1] limit of 1.1, at which point the inverter shuts down automatically, interrupting electricity generation and leading to losses. An important aspect of this work is the simulation study of the impact of photovoltaic generation on the actual low-voltage network. The analysis carried out in this document presents an analytical relationship of voltage changes in the low-voltage network caused by photovoltaic generation. The main focus of this work involves the analysis of the impact of photovoltaic generation on the low-voltage network under examination using the NEPLAN software, taking into account both annual consumer load profiles, annual photovoltaic generation profiles, different penetration levels, and photovoltaic locations. The analysis was carried out for a real, rural low-voltage network of considerable length. More than 100 different variants of the photovoltaic distribution in the network were used, analysing their impact on voltage changes at individual nodes, and other connection points for sources with various capacities and profiles were considered.

The remaining part of the article is organized as follows: Section 2 provides an overview of issues related to photovoltaic generation, including a review of selected articles on distributed generation within the distribution network. Section 3 presents a case study: the analyzed low-voltage overhead network, modeled using the NEPLAN program. Section 4 discusses the research results obtained by the authors. Section 5 introduces a method for determining voltage changes in the LV network caused by connecting a PV installation, based on the active and reactive power flowing in the tested line. Section 6 details a simplified method for determining the maximum voltage changes resulting from connecting a PV installation. Finally, Section 7 summarizes the article’s conclusions.

2. Selected Problems of Photovoltaic Generation

Renewable energy and the shift towards decentralized energy generation have radically changed energy systems around the world. This change is driven by an increased demand for renewable energy generation, reduced carbon emissions, improved energy security, and better transmission and distribution systems. The reduction in the cost of photovoltaic modules and technical developments in the conversion of power electronics and semiconductor devices make photovoltaic systems one of the most promising renewable energy sources. Network-connected photovoltaic systems offer many technical advantages, such as flexibility, simple installation in any location with access to sunlight, no pollution, no noise emissions, and low maintenance [4,5,6,7,8,9]. Such systems can cover the consumer’s electricity demand and reduce electricity bills while feeding surplus energy back into the network or using the network as a backup system during periods of insufficient photovoltaic generation. However, the growing share of PV systems can lead to technical problems for the distribution network, mainly low voltage [10], such as excessive energy production in the circuit and overvoltage problems in LV networks. Problems intensify when PV energy generation is at its peak and household demand is at its lowest. Among the reported problems, the key issues include voltage control and voltage imbalances, as well as voltage increases along distribution lines due to reverse power flow.

In order to properly understand these effects, the structure of the network must be taken into account. In Poland (Europe and many other parts of the world), low-voltage networks usually consist of three-phase, four-wire lines powered by three-phase transformers, forming a radial distribution network [11,12,13]. Most houses operate on a single-phase system. Originally, the network was designed to operate with a unidirectional power flow, and distribution systems are operated on the assumption that energy flows unidirectionally from the MV/LV transformer to the consumer. In such situations, the typical voltage profile decreases from the transformer at the source to the subsequent consumers. Photovoltaic modules are connected to the network via an electronic power converter, which is designed to obtain maximum solar energy and feed it into the network [14]. As this type of energy is variable, when its penetration is low, it does not have a negative impact on the network. However, with many prosumers, there is the possibility of significant reverse power flow during periods of high PV production and severe voltage fluctuations on cloudy days. This can cause overloading and violations of power quality standards, such as voltages above the limits [15,16,17]. The low-voltage network described has an inherently random distribution of energy consumption.

The growing number of photovoltaic systems and the fact that these are installed at random locations and have different nominal values contribute to the voltage unbalance problem [18]. Therefore, an excessive number of installed PV systems can turn their environmentally friendly advantages into disadvantages with many possible operational problems in the power system [19,20], such as the risk of overvoltage and overloaded lines due to increased active and reactive power flow [21,22]. The operation of the low-voltage network within the permissible voltages becomes a problem due to the distribution lines having too small cross-sections for the expected voltage drop/increase to be within the permissible limits [23,24]. Another factor limiting the increase in energy produced by PV systems is the long-term load capacity of distribution lines. Exceeding permissible voltage levels is a major problem in rural and suburban networks; line overloading is a bigger problem in urban networks [25,26].

In paper [27], simulation studies conducted in the PSCAD/EMTDC environment assessed, among other factors, voltage variations under high photovoltaic penetration. The impact of different PV locations on network voltage fluctuations was analytically derived and correlated with line impedance. A quadratic function was developed, using PV active power/voltage fluctuations as the independent/dependent variables, respectively. Furthermore, an objective function proposing the minimization of daily network average node voltage fluctuations was introduced, with constraints for optimizing the PV location. Paper [28] proposed an analytical methodology to investigate the static voltage stability of unbalanced distribution networks under high PV penetration, considering both single-phase and three-phase connections. The study analytically examined the loadability of general unbalanced distribution systems and how it is affected by PV power for both single- and three-phase PV integration scenarios. The analytical outcomes were studied and validated using results from the IEEE 37-bus distribution network. The IEEE 14-bus test system was used as a research case for wind power in [29]. Simulations were performed using the DigSilent PowerFactory software, and the results were analyzed using MATLAB. Three different line loadability enhancement measures were proposed to improve the permissible penetration level. These measures, combined with reactive power compensation, aimed to enhance the voltage stability of the grid. The performance of these measures was analyzed and compared. In paper [30], a local voltage control strategy based on distributed PV generation was presented. For this purpose, numerical models for network components and PV generators were implemented in the MATLABreg Simulinkreg environment. The paper proposes a local voltage control strategy based on PV generation curtailment as an alternative to the typical "on/off" operation often required by distribution operators. Paper [31] aimed to assess the effects of dispersed generators directly connected to low-voltage networks on the distribution system. Specifically, the impact of dispersed generation on the voltage profile of an LV distribution feeder was examined. With reference to different types of load distributions along the line, analytical expressions were derived to determine the limit value of power that can be injected into the distribution network without causing overvoltages. A general expression for voltage calculation at a point along a distribution line where a single generator is connected was also developed.

Other important problems of distributed photovoltaic generation are as follows:

1.

Phase load unbalance: Low-voltage photovoltaic systems often operate on a single-phase system, which creates an imbalance with respect to the other phases. This phenomenon causes an increase in the current in the neutral conductor and thus an increase in losses [32].

2.

Increased energy losses: Total losses in low-voltage networks decrease with increasing PV penetration and local consumption of the generated energy. However, if the active power generated by PV systems exceeds the load demand, losses increase because not all the energy can be consumed and is exported to the medium-voltage network [33,34,35,36].

3.

Higher harmonics: Some photovoltaic inverters can introduce harmonics into the network, which leads to increased losses in transformers, which causes the heating of their windings, as well as the heating of the protection systems, which in turn causes their failure [37].

4.

Short-circuit currents: High levels of photovoltaic penetration cause higher short-circuit currents, which can lead to greater damage to network equipment [37].

Despite numerous studies by many researchers focusing on new methods, analyses, assessments, and optimizations of low-voltage networks with PV generation, further work is needed to develop practical methods for distribution system operators to assess the impact of photovoltaic systems on the operation of low-voltage networks. The considerations proposed in this article are a continuation of the research presented in [38] in this case on the issues of voltage changes in the tested network. An important aspect of this work is the simulation study of the impact of PV generation on the actual low-voltage network. The calculations were based on the actual generation and consumption profiles of consumers at different penetration levels and PV system locations. The annual variability of the consumer load profiles and the generation of PV in Poland was taken into account.

3. Case Study

The analysis was conducted to assess the impact of energy introduced by prosumer PV installations on the voltage level of an LV line. This study utilized measurement data from an operational network located in Poland. For modeling purposes, a rural LV network was chosen. It is a radial overhead line with branches supplied from a 40 kVA MV/LV transformer consisting of 11 overhead sections made of AFL-8 cables with a cross section of 50 mm². The unit parameters of the 50 mm² AFL-8 overhead line are as follows: unit resistance R1 = 0.606 $\mathsf{\Omega }$/km, unit reactance X1 = 0.39 $\mathsf{\Omega }$/km, unit conductance B1 = 2.959 $\mathsf{\mu }$S/km, and maximum current $I_{max}$ = 150 A.

The lengths of the individual sections between the nodes are shown in

Table 1. Lengths of individual sections of the 50 mm² AFL-8 overhead line.

Section	0–A	A–B	B–C	C–D	D–E	E–F
Length of line [m]	180.75	187.35	136.25	181.28	230.03	260.03
Section	F–G	B–I	D–J	J–K	K–L	F–H
Length of line [m]	167.53	117.68	65.75	163.25	184.5	294.23

and marked according to Figure 1.

Prosumers supplied by the analyzed network are measured with remote reading meters. They are billed according to the "net-metering" method, according to which for 1 kWh of energy delivered to the network, the prosumer receives 0.8 kWh. The structure of the modeled network is shown in Figure 1.

The description of the modeled line in the NEPLAN program and the introduced simulation conditions can be found in article [38].

Table 2. Annual energy consumption and power factor for individual consumers.

Label of Consumer	Annual Energy Consumption of the Consumer [MWh]	Average Consumer Power Factor—cos $\mathit{\phi }$
ECA	2404.42	−0.95
ECB	4730.04	0.92
ECC	3258.11	−0.94
ECE	3463.61	0.92
ECF	3342.26	0.91
ECG	3174.54	0.93
ECH	2179.64	0.94
ECI	2312.74	0.90
ECJ	2043.54	−0.94
ECK	4764.64	0.92
ECL	4730.01	0.90

presents the annual energy consumption and the average power factor for the individual energy consumers labeled ECA, ECB, ECC, ECE, and ECL respectively.

The analysis presented in this paper assumes the installation of the same photovoltaic modules in different configurations in a 3-phase installation with an inverter that ensures symmetrical current generation in each phase. In this analysis, both load asymmetry and inverter generation asymmetry were ignored.

Table 3. Annual energy production by the PV system.

Label of PV	Annual PV Energy Production [MWh]
PVA	2885.31
PVB	5676.05
PVC	3909.73
PVE	4156.33
PVF	4010.71
PVG	3809.44
PVH	2615.56
PVI	2775.29
PVJ	2452.24
PVK	5717.57
PVL	5676.01

presents the energy produced by the photovoltaic system labeled PVA, PVB, PVC, PVE, and PVL. It should be noted that individual installations are not permanently assigned to the consumer—during further tests they change locations.

Figure 2 shows the total energy consumed by consumers and produced by the photovoltaic system at different times of the year.

In Poland, during several winter months, the amount of energy produced by photovoltaics definitely does not cover the energy demand.

The study was based on a low-voltage network model using the NEPLAN software. The NEPLAN software uses the Newton–Raphson method to determine power distributions and calculate voltages at individual network nodes [39].

According to the data read from external files, an n-fold analysis of load flow with load profiles is performed for each variant of the active PV system configuration. Based on the loaded data from 00:00 on 1 January 2022 to 23:00 on 31 December 2022, calculations of the flows in this network are performed. At 1-hour intervals, on the basis of consumer profiles and generation profiles of photovoltaic systems, currents, voltages, and power flows in the studied network are calculated for each analyzed period, and the results are saved in the designated file. The study was carried out for a varying number of active PV systems distributed from 0 to 11 within the network according to the following number of variants (11 variants with one, two, or three PV systems active up to a variant where all PV systems are active). The number of PV variants analyzed is shown in

Table 4. The number of connected photovoltaic installations and the corresponding number of analyzed network operation variants.

The number of connected PVs	0	1	2	3	4	5	6	7	8	9	10	11
The number of analysed variants	1	11	11	11	10	10	9	8	7	5	3	1

.

The network in this case was analyzed for 87 different PV system configurations.

4. Test Results

4.1. Voltage Changes in Network Nodes for Individual PV Generation Variants

In the modeled network, an analysis of voltage levels as a function of the distance of the node from the MV/LV transformer station supplying the network was carried out for 87 variants of the PV system distribution. Figure 3 shows the voltage variation in the tested network over the course of a year in the absence of any generation. The calculations were performed in the R environment [40].

As shown in Figure 3, the voltages in the network range from 1.002$U_n$ at point 0 of the network to 0.948$U_n$ at point H of the network. Figure 4 shows the changes in voltage levels throughout the year when photovoltaic generation is switched on at each point in the network.

As expected, the minimum voltages are the same as in the absence of PV generation. This is due to the power consumption profiles of consumers. The maximum power consumption by customers occurs when the PV system is not operating, so the voltage drops during this time are unchanged. The maximum voltages are at the points furthest from the transformer, namely at H = 1.107$U_n$ and G = 1.101$U_n$. These voltages are higher than the maximum network voltage, which is 1.1$U_n$. It was determined how often and at which points of the network the exceedance of voltage occurs. At point F it occurred three times during the year, at G four times, at H six times, and at node L once. A total of 14 exceedances of the maximum permissible voltage level occurred during the year. This is not a lot, but it is important to note that the network in question is a relatively large cross-section.

4.2. Statistical Measures of Voltage Changes in Node H of the Network for Individual Variants of PV Generation

In order to determine the influence of individual systems on the voltage level in the low-voltage network, the voltages at individual network nodes were calculated when photovoltaic systems were connected in different variants. The results show the average, median, minimum, and maximum voltage values in relation to their nominal value, and the highest difference between the phase voltage with PV generation at a given grid point and the voltage without PV generation ($\Delta U_{max}$) for node H is shown in

Table 5. Basic statistical measures of voltages in node H of the network ¹.

PV Location	Mean	Median	Max	$\Delta$Umax [V]	PV Location	Mean	Median	Max	$\Delta$Umax [V]
A	0.982	0.985	0.995	1.066	ABCGI	0.987	0.987	1.031	10.489
B	0.983	0.985	1.001	2.980	ABEHK	0.989	0.988	1.041	12.064
C	0.983	0.985	0.999	2.400	ACGJL	0.988	0.987	1.041	12.831
E	0.984	0.986	1.005	3.771	AEHIJ	0.987	0.987	1.029	9.134
F	0.984	0.986	1.007	4.123	AGHIL	0.989	0.988	1.042	12.956
G	0.984	0.986	1.006	3.559	BCEFK	0.99	0.988	1.049	14.565
H	0.984	0.986	1.003	2.886	BCEKL	0.99	0.988	1.053	15.564
I	0.982	0.985	0.996	0.987	BEFIJ	0.989	0.988	1.039	12.253
J	0.983	0.985	0.997	1.363	CGHIL	0.99	0.988	1.047	14.144
K	0.984	0.986	1.004	3.077	ABCEFG	0.991	0.988	1.059	16.896
L	0.984	0.986	1.009	5.308	ABFHJL	0.991	0.988	1.056	16.316
AB	0.983	0.986	1.005	4.017	ABFIKL	0.991	0.988	1.053	15.494
AG	0.984	0.986	1.009	4.601	ABGJKL	0.991	0.988	1.051	15.131
BE	0.985	0.986	1.015	6.664	ACEGIJ	0.989	0.988	1.039	12.158
BI	0.984	0.986	1.005	3.785	ADCEFG	0.991	0.988	1.059	16.896
CE	0.985	0.986	1.013	6.095	AEGHIL	0.991	0.988	1.056	16.324
CG	0.985	0.987	1.014	5.894	BCEFGK	0.993	0.988	1.062	17.703
FG	0.986	0.987	1.021	7.530	BCEFJK	0.991	0.988	1.054	15.459
GH	0.986	0.987	1.018	6.309	BEGIJK	0.991	0.988	1.05	13.752
JK	0.985	0.987	1.01	4.391	ABCEFGH	0.993	0.988	1.069	19.242
KL	0.986	0.987	1.019	7.182	ABCGJKL	0.992	0.988	1.06	17.229
ABC	0.985	0.986	1.014	6.327	ACEGIJL	0.991	0.988	1.059	16.928
ACF	0.986	0.987	1.019	7.466	ACEHIKL	0.992	0.988	1.059	17.009
AGK	0.986	0.987	1.022	7.210	ACGHIKL	0.992	0.988	1.059	16.843
BFK	0.987	0.987	1.029	8.920	BCEFGJK	0.993	0.989	1.067	18.576
CEG	0.987	0.987	1.027	9.469	BCFGJKL	0.994	0.988	1.072	19.892
CEL	0.987	0.987	1.034	11.111	BEFHIJK	0.993	0.988	1.062	16.590
CFG	0.987	0.987	1.029	9.796	ABCEFGHI	0.993	0.988	1.072	19.956
CHJ	0.986	0.987	1.017	6.094	ABCFGIJK	0.993	0.988	1.062	16.955
FGH	0.988	0.988	1.033	10.031	ABCFHIJL	0.992	0.988	1.068	19.118
HIL	0.986	0.987	1.024	8.667	ABEFGIKL	0.995	0.989	1.08	21.839
IKL	0.987	0.987	1.023	11.111	ACEGHIJL	0.993	0.988	1.069	19.283
ABCE	0.987	0.987	1.029	9.916	AEFGHJKL	0.995	0.989	1.08	21.700
ABKL	0.988	0.987	1.033	10.971	BCEFGJKL	0.997	0.989	1.095	25.309
ACIJ	0.985	0.986	1.012	5.234	ABCEFGHIJ	0.994	0.989	1.076	20.821
AEIJ	0.986	0.987	1.017	6.582	ABCGHIJKL	0.994	0.989	1.074	20.297
BCEF	0.988	0.987	1.041	12.760	ACEGHIJKL	0.995	0.989	1.076	20.951
BEGK	0.989	0.988	1.041	11.881	BCEFGHJKL	0.997	0.989	1.095	25.309
CEFG	0.989	0.988	1.043	13.247	CEFGHIJKL	0.997	0.989	1.087	23.496
CFIL	0.988	0.987	1.039	12.224	ABCEFGHIJK	0.996	0.989	1.085	22.514
EHIJ	0.987	0.987	1.026	8.272	ABCEFGHJKL	0.998	0.989	1.099	26.207
GHIL	0.988	0.988	1.038	11.973	BCEFGHIJKL	0.998	0.989	1.098	25.989
ABCEF	0.989	0.988	1.045	13.730	ABCEFGHIJKL	0.998	0.989	1.102	26.883

¹ In each analyzed case, the minimum voltage value was 0.945$U_n$. This is because the minimum voltage drop occurs when there is no photovoltaic generation.

. Node H was selected because of the highest voltage occurring in this node, as shown in Figure 4. The voltage at the supply node of the analyzed circuit is not equal to the nominal voltage of the network; its average value is 0.982$U_n$; the minimum is 0.945$U_n$, and the maximum is 0.994$U_n$. The analysis of the results obtained showed that the distribution of sources in the network affects the formation of voltage levels in the network. Thus, it also has significance in terms of the occurrence of their extreme values. The largest annual energy production from PV systems can be found at prosumer node K, and the additional voltage increase caused by PV generation for this node is 3.08 V, which shows that both the connection point of the generation and its power have an impact on the voltage in the network. The highest voltage increase of 26.21 V was observed when the PV systems were in nodes A, B, C, E, F, G, H, J, K, and L, but then the voltage did not exceed the permissible value and was 1.099$U_n$. A voltage above the permissible value of 1.102$U_n$ was observed when the PV systems were operating for all consumers and the voltage increase caused by PV operation was less than 10% of the rated voltage. The minimum voltage value is 0.945$U_n$ and is constant for all variants. This is because it occurs at a time when there is no generation. The voltage for different generations at a specific time was also examined. The results of the calculations for point H on 15 July at 3 p.m. are shown in Figure 5.

The analysis of the results showed that the voltage will vary depending on the number of generations, the connection point of the PV system, and the energy generated by the PV system. If the PV system is connected to node I, the voltage at node H is the lowest at 0.993$U_n$; it is highest when the generation is connected to node L at 1.013$U_n$. It can be seen that the lowest voltage at a given point does not mean that the generation was the lowest; the node (nodes) where the generation occurred is also important.

It can be seen that the lowest voltage at a given point does not mean that the generation was the lowest; it is also important at which node (nodes) the generation occurred. One might notice that the closer the generation was to the MV/LV transformer, the lesser the voltage increases in the circuit.

Taking the generation in four nodes as an example, one might notice that when the generation is in the ACIJ nodes, the voltage is the lowest and amounts to 1.013$U_n$. The total generation for this case is 12,022.57 MWh; the lowest generation is 11,999.42 MWh and occurs when the PV systems are in nodes EHIJ, and the voltage is 1.025$U_n$. It is not the lowest because the PV sources are located at consumers connected in the final sections of the circuit.

4.3. Statistical Measures of Voltage Changes in Node H of the Network for Different Line Cross-Sections

Low-voltage distribution networks are networks with open and closed structures but operating in open configurations (open connectors at points). For each distribution network node, the voltage deviation shall be within the limits specified in the Ordinance of the Minister of Climate and Environment of 22 March 2023 on the detailed conditions of electricity system operation, i.e. ±10%. In order to determine the influence of the line cross-section on the maximum changes in voltage levels in the network, their variability distributions were determined when each consumer has a PV installation. In the NEPLAN program, three equal networks were modeled—identical in terms of structure and connected prosumers; they only differed in terms of the cross-section of the lines. In addition to the network described in the Research Model section, two more networks were modeled: the first with a cross-section of 35 mm² and the second with a cross-section of 70 mm². The unit resistance of the AFL-8 35 mm² overhead line with a rated voltage of 0.4 kV was assumed to be R1 = 0.852 $\mathsf{\Omega }$/km. For the AFL-8 70 mm² line, R1 = 0.441 $\mathsf{\Omega }$/km was assumed. The lengths of the individual sections between the nodes marked in accordance with Figure 1 were identical. As it has been shown before that the greatest voltage changes occurred in node H, the further analysis concerns only this node. The results of the research are presented in charts combining histograms, box plots, and probability density functions calculated using kernel density estimators (KDEs) with the use of a normal kernel [41]. The most common histogram presents a fully visual assessment of the distribution of the analyzed data. A box plot is a form of graphical interpretation of the distribution of statistical features of a variable. These plots show the modal value directly visible on the PDF plots, as well as the skewness of the distributions and outliers. In the description of some of the figures, selected statistical measures of the analyzed distributions are provided in tabular form, namely the minimum and maximum values, the first quantile, the third quantile, the median, and the average voltage value. The results of the tests for lines with a cross-section of 35 mm² are shown in Figure 6. The calculations were performed in the R environment [40].

Figure 6. Histogram, box plot, and probability density function determined using the KDE (red line) of voltage changes at node H over the course of a year for a line cross-section of 35 mm² with all photovoltaic systems active.

Figure 6. Histogram, box plot, and probability density function determined using the KDE (red line) of voltage changes at node H over the course of a year for a line cross-section of 35 mm² with all photovoltaic systems active.

Min.	1st Qu.	Median	Mean	3rd Qu.	Max.
0.97	0.97	0.99	0.99	1.01	1.13

For the 35 mm² line cross-section, the voltage increased to a value of 1.13$U_n$ during the simulation tests. Obviously, under real network conditions, above a value of 1.1$U_n$, an emergency shutdown of the inverter and the interruption of the power generation process should occur. In node H, the voltage exceeded 1.1$U_n$ as much as 160 times. The test results for a 50 mm² line are shown in Figure 7.

Figure 7. Histogram, box plot, and probability density function determined using the KDE (red line) of voltage changes at node H over the course of a year for a line cross-section of 50 mm² with all photovoltaic systems active.

Figure 7. Histogram, box plot, and probability density function determined using the KDE (red line) of voltage changes at node H over the course of a year for a line cross-section of 50 mm² with all photovoltaic systems active.

Min.	1st Qu.	Median	Mean	3rd Qu.	Max.
0.94	0.98	0.99	0.99	1.01	1.10

For the 50 mm² line cross section, by far the most common voltage at node H is below the value of $U_n$; occurring relatively less often when it exceeds the nominal value. The resulting distribution of voltage variations at the node is right-skewed. As shown earlier, the voltage of 1.1$U_n$ ascended at node H only six times. The test results for a 70 mm² line are shown in Figure 8.

Figure 8. Histogram, box plot, and probability density function determined using the KDE (red line) of voltage changes at node H over the course of a year for a line cross-section of 70 mm² with all photovoltaic systems active.

Figure 8. Histogram, box plot, and probability density function determined using the KDE (red line) of voltage changes at node H over the course of a year for a line cross-section of 70 mm² with all photovoltaic systems active.

Min.	1st Qu.	Median	Mean	3rd Qu.	Max.
0.92	0.98	0.99	0.99	1.01	1.08

In the case of the 70 mm² line cross section, the maximum permissible voltage threshold was not exceeded in any of the simulations carried out. Despite the fact that the distributions of voltage changes for the network with each cross-section have an identical median of 0.99$U_n$, most importantly, they differ in their maximum values, which are 1.13$U_n$, 1.1$U_n$, and 1.08$U_n$, respectively.

5. Determination of the Voltage Changes Caused by the Connection of a Photovoltaic Installation

The expected level of voltage fluctuations at a distribution network node can be determined from the balance of voltage dips and deviations according to the following equation.

$$\Delta U_{WLV\%}=\delta U_{NMV}-\Delta U_{MW\%}+\delta U_{\theta \%}+\delta U_{PZ\%}-\Delta U_{T\%}-\Delta U_{LV\%}.$$

(1)

where $\Delta U_{WLV\%}$—voltage changes in the low-voltage distribution network, $\delta U_{NMV}$—voltage deviation from the nominal value of the network on the MV buses at the HV/MV substation, $\Delta U_{MW\%}$—voltage changes in the medium-voltage distribution network, $\delta U_{\theta \%}$—voltage deviation associated with the HV/MV substation, $\delta U_{PZ\%}$—voltage deviation due to the MV/LV transformer’s tap changer position changer, $\Delta U_{T\%}$—voltage changes at the MV/LV transformer, and $\Delta U_{LV\%}$—the voltage changes in the lines of the LV network.

The equation can form the basis of an analysis aimed at optimal (from the point of view of the criteria adopted) voltage regulation in the distribution network. The voltage level at the customer will depend on the voltage changes in the medium-voltage network, the tap setting in the MV/LV transformer, and the load in the low-voltage network.

The line voltage drop can be calculated in two ways:

I. Calculating the sum of the voltage drops for individual sections of the analyzed current path starting from the rails of the LV switchboard to the last connection of the radial network according to the following equation:

$$\Delta U_{RM}=\sum _{j=1}^{m-1}\Delta u_{j,j+1}$$

(2)

where $\Delta u_{j,j+1}$—section voltage drop $j, j+1$. For any section, a drop that determines the voltage drop across the section under study l is calculated, with conductivity $\gamma$ and cross-sections for

-

one phase

$$\Delta U_{j,j+1}=I_{j,j+1}\left(R_{j,j+1}cos{\phi }_{j,j+1}+X_{j,j+1}sin{\phi }_{j,j+1}\right)$$

(3)

-

three-phase line

$$\Delta U_{j,j+1}=\sqrt{3}{I}_{j,j+1}\left(R_{j,j+1}cos{\phi }_{j,j+1}+X_{j,j+1}sin{\phi }_{j,j+1}\right)$$

(4)

where $I_{j,j+1}$—load current in the section $j, j+1$, $cos{\phi }_{j,j+1}$—power factor of the section $j, j+1$, and $R_{j,j+1}, X_{j,j+1}$—resistance and reactance of the conductor.

II. Calculating the voltage drop caused by the flow of currents drawn at successive consumption points and the impedance of the sections from the power supply to a given consumption point according to the following equation:

$$\Delta U_{zk}=\sum _{i=1}^n\Delta u_{z,i}$$

(5)

where $\Delta u_{z,i}$—the voltage drop across the section from the supply to the i-th consumer point. For an arbitrary load, the voltage drop caused by the current of the i-th load on the section $l_{z,i}$ of conductivity $\gamma$ and cross-section s is calculated for

-

one phase

$$\Delta U_{z,i}=I_i\left(R_{z,i}cos{\phi }_i+X_{z,i}sin{\phi }_i\right)$$

(6)

-

three-phase line

$$\Delta U_{z,i}=\sqrt{3}{I}_i\left(R_{z,i}cos{\phi }_i+X_{z,i}sin{\phi }_i\right)$$

(7)

where $I_i$—the load current of the i-th consumer, $cos{\phi }_i$—the power factor of the i-th consumer, $R_{z,i}, X_{z,i}$—the resistance and reactance of the conductor from the supply to the i-th consumer.

Subsequent connections of PV installations in the distribution network can lead to a reverse power flow, causing the voltage to rise at the consumers connected at the end of the circuit. Distribution system operators are primarily interested in information on what power and where connected PV installations will not cause the voltage to rise above the limit value. To calculate the voltage changes caused by the connection of a PV installation at a specific point on the circuit, Equations (5) and (6) should be used. By substituting the PV-generated current $I_i$ in Equation (6) with $P_{PVi}/U$, the equation for the voltage changes caused by the connection of the PV installation is obtained:

$$\Delta U_{PV}=\sum _{i=1}^n{\displaystyle \frac{P_{PVi}{R}_{Z,i}+Q_{PVi}{X}_{Z,i}}{U}}$$

(8)

where $P_{PVi}$—the active power of the connected PV installation at node i, $Q_{PVi}$—the reactive power of the connected PV installation at node i, $R_{Z,i}$—the resistance of the line from the supply to the i-th PV, $X_{Z,i}$—the reactance of the line from the supply to the i-th PV, and U—the rated voltage. For photovoltaic installations, usually only the active power is taken into account; for PV installations it can be assumed that $cos{\phi }_i=1$.

This equation can be used when the active and reactive powers are known in each section of the analyzed line. Distribution system operators do not have such detailed data on the active and reactive power flows (in specific hours of the year).

6. Simplified Method for Determining the Maximum Voltage Changes

The voltage in any node of an LV circuit with PV generation depends on the following, among other things:

• The voltage at the circuit supply. It is defined by Equation (1);
• The voltage drop caused by the flow of power drawn by consumers;
• The voltage increase caused by the flow of power generated by PV installations.

The following part of the study deals with considerations for determining the maximum voltage increase caused by connecting a photovoltaic installation to the low-voltage network at high saturation of PV installations. During the maximum generation in the network, it has to be taken into account that the flows are bidirectional (both deep into the grid and in the feed direction). This fact must be taken into account in the calculations. Connecting a photovoltaic installation increases the voltage at each point in the network. Therefore, a formula that can be used to determine how much the voltage will increase at a given point in the network when PV installations are connected to the network was developed. In the following analysis, it is simplified that the flow in the direction to the MV/LV transformer will depend on the sum of the PV power and the sum of the energy consumed by the consumers.

For i (this PV installation), an important question is how far generation PV_i feeds the consumers deep into the grid. Figure 9 shows a condition reflecting generation $P_{PVi}$ at node i that transmits part of its energy to the MV/LV transformer (resistance $R_{z,i}$), and part of its energy is transmitted deep into the grid, further feeding the grid (resistance $R_{i,x}$).

For i (this PV installation), an important question is how far the generation $P_{PVi}$ feeds the consumers deep into the grid. For this purpose, the length of the line $l_P{V}_{i,x}$ onto which the energy generated by the generation $P_{PVi}$ penetrates will be determined from the following relation:

$$l_{P{V}_{i,x}}={\displaystyle \frac{P_{PVi}{T}_{s}L}{E_{odb}}}$$

(9)

where $T_s$—the peak load duration for PV, L—the total length of the main circuit, and $E_{odb}$—the energy consumed during the year by the consumers connected to the circuit under analysis. The peak load duration for photovoltaic installations was assumed to be 1000–1100 h.

Having determined the length of the line onto which the energy penetrates during the peak generation of the installation $P_{PVi}$, we determine the corresponding resistance $R_{i,x}$:

$$R_{i,x}={\displaystyle \frac{l_{P{V}_{i,x}}}{\gamma s}}$$

(10)

Next

$$\Delta U_{PV}=\sum _{I=1}^N{\displaystyle \frac{P_{PVi}(R_{Z,i}+R_{i,X})}{U}}·k_{PV}$$

(11)

where $k_{PV}$—the correction factor taking into account the PV connection. The $k_{PV}$ factor takes into account the divergence of PV peaks: the lower-than-peak energy consumed by the customer at the time of peak PV generation and the higher-than-rated voltage at the PV connection point.

It is largely dependent on the ratio of the energy generated by the PV to the energy consumed by consumers $(E_{PV}/E_{odb})$. If value $E_{PV}/E_{odb}⩽1$, the ratio $k_{PV}$ is 0.95; for $1<E_{PV}/E_{odb}<1.2$ the ratio is 0.9; for $E_{PV}/E_{odb}⩾1.2$, the ratio is 0.85.

The derived formula can be used when the condition $E_{PV}/E_{odb}>0.5$ is met, and the power consumption and generation profiles are similar to the Polish ones.

In order to check what discrepancies exist between the voltage changes calculated by the NEPLAN program and the voltage changes from Relation (11), calculations were carried out for three variants: Variant III—There is no relationship between the energy consumed by the consumer and the energy produced by the PV.

• Variant I—The prosumer has a photovoltaic installation with an energy of 1.2 of the energy consumed by the prosumer;
• Variant II—The prosumer has a PV installation with an energy of 1.73 of their energy input;
• Variant III—There is no relationship between the energy consumed by the consumer and the energy produced by the PV.

In order to determine the impact of the PV installation on the voltage levels at the grid nodes, the voltage changes caused by the connected PV installations were calculated. The results of the calculations are presented for node H.

Table 6. Variant I: PV location, $E_{PV}/E_{odb}$, NEPLAN, Equation (11), difference, and error.

PV Location	${\mathit{E}}_{\mathit{PV}}/{\mathit{E}}_{\mathit{odb}}$	NEPLAN	Equation (11)	Difference [V]	Error [%]
ABEFGIKL	0.95	21.84	22.41	−0.57	−2.61
ACEGHIJL	0.78	19.28	18.70	0.58	3.02
AEFGHJKL	0.86	21.70	23.65	−1.95	−8.99
BCEFGJKL	0.97	23.06	24.18	−1.11	−4.83
ABCEFGHIJ	0.89	20.82	21.31	−0.49	−2.36
ABCGHIJKL	0.98	20.30	21.37	−1.07	−5.28
ACEGHIJKL	0.93	20.95	22.43	−1.48	−7.06
BCEFGHJKL	1.04	25.31	23.18	2.13	8.42
CEFGHIJKL	0.96	23.50	25.89	−2.39	−10.17
ABCEFGHIJK	1.04	22.51	23.72	−1.21	−5.37
ABCEFGHJKL	1.12	26.21	26.32	−0.11	−0.42
BCEFGHIJKL	1.12	25.99	26.68	−0.69	−2.66
ABCEFGHIJKL	1.20	26.88	27.22	−0.34	−1.27
ABCEFGH	0.74	19.24	18.91	0.33	1.71
ABCGJKL	0.83	17.23	17.38	−0.15	−0.89
ACEGIJL	0.70	16.93	15.67	1.26	7.42
ACEHIKL	0.76	17.01	17.17	−0.17	−0.97
ACGHIKL	0.75	16.84	17.65	−0.81	−4.79
BCEFGJK	0.82	18.58	20.48	−1.90	−10.25
BCFGJKL	0.86	19.89	20.84	−0.95	−4.77
BEFHIJK	0.75	16.59	18.80	−2.21	−13.33
ABCEFGHI	0.82	19.96	19.87	0.09	0.43
ABCFGIJK	0.86	16.96	18.67	−1.72	−10.13
ABCFHIJL	0.82	19.12	17.86	1.26	6.59

shows the results of the voltage change calculations for one phase resulting from the PV connection at different locations in Variant I. For Variant I, the difference between the voltage changes calculated by the NEPLAN program and that calculated according to Equation 9 ranges from −2.39 V to +2.13 V, with an average of −0.53 V and a standard deviation of 1.53. The error, expressed in relation to the voltage changes calculated with the NEPLAN program, ranges from −13.36% to 8.42%, with an average of minus 2.76%.

Table 7. Variant II: PV location, $E_{PV}/E_{odb}$, NEPLAN, Equation (11), difference, and error.

PV Location	${\mathit{E}}_{\mathit{PV}}/{\mathit{E}}_{\mathit{odb}}$	NEPLAN	Equation (11)	Difference [V]	Error [%]
ABCEFGH	1.07	15.37	17.33	−1.96	−12.76
ABCGJKL	1.19	15.48	15.67	−0.19	−1.23
ACEGJIL	0.91	14.47	15.07	−0.60	−4.14
ACEHIKL	1.10	15.63	16.09	−0.46	−2.85
BCEFGJK	1.08	16.09	16.15	−0.06	−0.37
BCFGJKL	1.18	17.76	17.85	−0.09	−0.52
BEFHJIK	1.24	16.71	17.18	−0.46	−2.68
ABCEFGHI	1.18	16.20	18.19	−1.99	−12.30
ABCFGJK	1.24	16.69	16.25	0.44	2.71
ABCFHJIL	1.19	16.86	17.14	−0.28	−1.64
ABEFGIKL	1.12	16.86	20.09	−3.23	−16.08
ACEGHJIL	1.37	18.19	20.48	−2.29	−11.19
AEFGHJIK	1.24	20.09	20.28	−0.19	−0.94
BCEFGJKL	1.40	20.74	18.44	2.30	12.47
ABCEFGHIJ	1.28	18.86	18.44	0.42	2.22
ABCGHJKL	1.41	18.67	18.32	0.35	1.86
ACEGHJKL	1.35	19.43	19.35	0.08	0.41
BCEFGHJKL	1.40	22.56	24.08	−1.51	−6.70
CEFGHJKL	1.51	21.40	21.60	−0.20	−0.78
ABCEFGHJK	1.62	21.33	23.92	−2.59	−10.82
ABCEFGHIJK	1.62	23.35	24.27	−0.92	−3.93
BCEFGHJKL	1.73	23.89	24.74	−0.84	−3.53

shows the results of the voltage change calculations for one phase resulting from the PV connection at different locations in Variant II.

For variant II, the difference between the voltage changes varies from −1.99 V to 0.69 V, an average of −0.56 V, with a standard deviation of 0.72. The error is on average minus 3.02%.

Table 8. Variant III: PV location, in this location other PVs, $E_{PV}/E_{odb}$, NEPLAN, Equation (11), difference, and error.

Place with PV Location	In This Location Other PVs	${\mathit{E}}_{\mathit{PV}}/{\mathit{E}}_{\mathit{odb}}$	NEPLAN	Equation (11)	Difference [V]	Error [%]
ABEFGIKL	BKALCEIF	0.96	23.46	21.46	1.99	8.49
ACEGHIJL	KLBACEFI	0.96	24.51	22.40	2.11	8.60
AEFGHJKL	LIFAKEBC	0.96	24.28	26.41	−2.13	−8.77
BCEFGJKL	IKAGHBEC	0.87	20.44	21.17	−0.73	−3.59
ABCEFGHIJ	CEFBKALGI	1.06	28.41	26.57	1.84	6.47
ABCGHIJKL	IGFBAJKLC	1.01	24.98	22.64	2.34	9.37
ACEGHIJKL	EIFAGHBCJ	0.89	21.81	21.37	0.44	2.02
BCEFGHJKL	JKLAIBCHG	0.98	27.85	27.53	0.32	1.14
CEFGHIJKL	HIACLJBEG	0.93	29.09	26.74	2.35	8.08
ABCEFGHIJK	BKIACEGHLJF	1.20	28.08	27.05	1.03	3.68
ABCEFGHJKL	FJLBKIAGEC	1.13	27.40	27.31	0.10	0.35
BCEFGHIJKL	EFGHABCIJL	1.04	28.52	25.87	2.65	9.30
ABCEFGHIJKL	KLIEGCBAFHJ	1.20	29.65	28.16	1.48	5.00
ABCEFGH	JFBGIKA	0.75	17.76	20.03	−2.27	−12.79
ABCGJKL	GIKLACB	0.84	20.47	18.41	2.06	10.05
ACEGIJK	GHIBCKA	0.75	16.64	16.92	−0.29	−1.74
ACEHIKL	JKIACLF	0.75	17.41	16.65	0.76	4.38
ACGHIKL	BGKEHAJ	0.75	16.22	18.38	−2.16	−13.34
BCEFGJK	IKLJACB	0.80	20.94	19.93	1.02	4.86
BCFGJKL	ALJBCEG	0.78	21.37	19.50	1.87	8.74

shows the results of the voltage change calculations for one phase resulting from the PV connection at different locations in Variant II.

For Variant III, the difference in voltage changes varies from minus 2.27 V to 2.65 V, giving an average error of 0.74% with a standard deviation of 1.535. It can be seen that the higher the ratio of energy generated by PV to energy consumed by consumers, the lower the errors. It is possible to reduce the difference between the exact calculation of the voltage changes and the approximation from Equation (11) by replacing the average energy density consumed by consumers with the actual density. However, there is a problem with the data, distribution system operators do not always have it. Equation (11) will allow them, with a minimum amount of data, to calculate by how much the voltages in the low-voltage grid will change when PV installations are switched on. It will also allow DSOs to effectively designate those grids where they install transformers with load-controlled taps where they will gain the greatest benefit. What remains to be determined is the maximum allowable voltage changes resulting from the connection of PV installations. As can be seen from Equation (1), the voltage on the power supply of a given circuit is dependent on the operation of higher voltage networks (MV, 110 kV); it will usually be lower than the rated voltage of the LV network. Therefore, it can be assumed that a phase voltage changes caused by the connection of PV installations less than 23 V will not cause a voltage overshoot of more than 10%, which is confirmed by the calculation results summarized in Table 6, Table 7 and Table 8.

To verify the obtained results, calculations were performed for several different low-voltage network circuits where the condition $E_{PV}/E_{odb}>0.5$ was met. Verifications considered varying photovoltaic generation powers and different generation points along the line and not exclusively the end points. The errors expressed in volts are presented in Figure 10.

Figure 10 shows that the differences in voltage changes, expressed in volts, between the exact calculations made with the NEPLAN program and Equation (11), which was made for several low-voltage network circuits, range from 2.5 V to −1.5 V. The error ranges from 11.40% to minus 9.10%. The results obtained are consistent with the previous analysis.

7. Conclusions

The distribution of the generation in the LV network influences the voltage levels that can be observed in such a network during its operation. As shown in this study, the distribution of the same number of sources with different powers gives voltage rises different ranges, with the size of the generation and the distance of the generation source from the MV/LV transformer being important. The voltages were raised to the least extent in systems where the generation was concentrated close to the MV/LV transformer, and to the greatest extent in those systems where it was concentrated at customers connected at the ends of the main line and/or the ends of the branches. The formula has been derived whereby the voltage increase in network nodes caused by the PV connection can be calculated. This will give distribution system operators the opportunity to decide where to install MV/LV transformers with on-load tap changers.