Voltage Variations and Their Reduction in a Rural Low-Voltage Network with PV Sources of Energy

Renewable sources of energy (RES), especially photovoltaic (PV) micro-sources, are very popular in many countries. This way of clean power production is applied on a wide scale in Poland as well. The Polish legal regulations and tariffs specify that every prosumer in a low-voltage network may feed this network with a power not higher than the maximum declared consumed power. In power networks with RES, the voltage level changes significantly along the power line and depends on the actually generated as well as consumed power by particular prosumers. There are cases that prosumers connected to this line cannot produce and inject the full permissible power from PV sources due to the level of a voltage higher than the technically acceptable value. In consequence, it leads to the lack of profitability of investments in installations with PV sources. In this paper, voltage variations in a real rural low-voltage network with PV micro-sources are described. The possible two general solutions of voltage levels improvement are discussed—increase in the cross-sectional area of the bare conductors in the existing overhead line as well as the replacement of the overhead line with a cable line. The recommended solution for the analyzed network, giving the best reduction of voltage variations and acceptable cost, is underlined. Such a recommendation can also be utilized in other rural networks.


Introduction
In recent years, a significant increase in the number of installed photovoltaic (PV) sources has been noticed. According to the report from 2019 [1], photovoltaic energy sources produce almost 1200 MW in Poland. PV sources are the most popular renewable energy sources (RES) among prosumers [2], especially in the so-called micro-installations (up to 50 kW). In general, the weather-related instability of PV energy production is a problem for the entire power system [3]. Moreover, as the power of installed micro-sources in the distribution network increases, problems begin to appear related to maintaining the required voltage. During the peak of the power generation by PV sources, the energy consumption by prosumers is most often low. These phenomena lead to a voltage level that exceeds the permissible value. According to the national legal regulation [4] and standard EN 50160 [5], the permissible range of the network voltage is (0.9-1.1) V n , where V n is the nominal voltage of the network. Furthermore, too high a voltage in the network may cause automatic disconnection of PV sources, which disables the production of clean power and is unfavorable from the economical point of view.
The discussed state of affairs was presented in publications [6,7]. The problems most often concern rural networks, which are characterized by long circuits [8,9]. For the network analyzed in the paper [10], it was shown that only with a 20% share of power generation by PV sources installed in the low-voltage (LV) network, no voltage problems were noted. Therefore, it seems to be reasonable to apply restrictions regarding the maximum installed power of renewable sources. In Poland, this limitation is defined by the maximum declared they assume unequal treatment of consumers in terms of the possibility of generating power by renewable sources. 6. The use of electric vehicles-especially since electric vehicle charging techniques are becoming more and more advanced and user-friendly [30]. However, it will take time to implement these solutions in many countries, and the power network problems related to PV sources are already present. 7. Replacement of power lines with larger cross-section lines. This method does not involve placing additional devices in the network that require additional maintenance. Additionally, it does not require a complete reconstruction of the network in overhead lines, and it may be enough to replace the wires. It is clearly simpler than the reconfiguration mentioned in point 1. The authors' investigation is focused on this method.
This article presents an analysis of the voltage conditions in a real rural low-voltage network located in northern Poland. The analysis of the current state of this network shows that in the case of no PV generation, the voltage in some nodes may be very close to the lower permissible limit (0.9 Vn). However, when PV generation is active, the voltage may exceed the upper permissible limit (1.1 Vn). The solutions that enable a decrease in voltage variations, especially that enable keeping them within the permissible normative range, are proposed. The most suitable solution related to the voltage improvement, which can be implemented by the operator of the analyzed network, is expressed.
The rest of this paper is organized as follows. In Section 2, the description of the analyzed real low-voltage network and its computer model are presented. Section 3 discusses the effect of the increase in the cross-sectional area of the bare conductors on the voltage variations, the possibility of voltage variations reduction by the replacement of the overhead line with a cable line as well as includes economic calculations. Conclusions flowing from the investigations, along with the recommendation for the operator of the analyzed network, are included in Section 4.

Description of the Analyzed Network
The analyzed low-voltage network is an overhead rural network in which the main supply line is 4 × AL 35 mm 2 (four-wire, aluminum bare conductors of nominal crosssectional area 35 mm 2 ). The farthest prosumer is 1.15 km away from a 15/0.4 kV power transformer substation. The loads are mainly single-family or summer-holiday houses (profile 3 and 4, respectively- Figure 1). Other loads are a shop and small farms, which correspond to load profiles 2 and 1, respectively ( Figure 1). The maximum declared consumed power of individual loads is presented in Table 1. To perform the computer analysis, the profiles of the loads from Figure 1 were assumed.   The maximum permissible generated power is equal to the maximum consumed power [11]. The modelled structure of the analyzed low-voltage network is depicted in Figure 2. For the node relatively far from the transformer substation (node w_325), daily voltage variations have been measured ( Figure 3). One can see that these variations (recorded for 9 days) are within the wide range and even exceed the upper permissible limit (dashed trace). It is an example to show that voltage variations/deviations in this network may really exceed the upper limit, which is unacceptable. Therefore, this real low-voltage network was modelled to perform extended voltage analysis.  The maximum permissible generated power is equal to the maximum consumed power [11]. The modelled structure of the analyzed low-voltage network is depicted in Figure 2. For the node relatively far from the transformer substation (node w_325), daily voltage variations have been measured ( Figure 3). One can see that these variations (recorded for 9 days) are within the wide range and even exceed the upper permissible limit (dashed trace). It is an example to show that voltage variations/deviations in this network may really exceed the upper limit, which is unacceptable. Therefore, this real low-voltage network was modelled to perform extended voltage analysis.  AsXSn4: overhead line composed of 4 insulated aluminum conductors; YKY 4 × 10: cable line composed of 4 copper conductors of cross-sectional area 10 mm 2 ; O301_1, Profile 3: modelled load in node w_303_1, the prosumer has profile 3 (see Figure 1); P301_1: modelled PV generation in node w_301_1; 0.10 and 0.06-the length of the power line, 0.10 km and 0.06 km, respectively.   The main line, which parameters are changed in subsequent computer simulations, runs from the substation node to node w_325 ( Figure 2). Connections to prosumers or another intermediate node depart directly from the selected nodes. The network is a radial type. The analyzed low-voltage network has been modelled with the use of DIgSILENT Power Factory software. Voltage conditions (daily variation) in selected nodes (w_301nearest to the transformer; w_303-in the middle of the main line; w_325-at the end of the main line) for the current network status are presented in Figure 4a, whereas Figure 4b presents assumed daily profile of the PV generation.
The most favorable voltage conditions are observed in node w_301 ( Figure 4a). For the whole day (24 h), the voltage is within the permissible range-the lowest voltage is around 0.95 V n (the case without PV generation, ca. 7:00 a.m.), and the highest voltage is almost 1.06 V n (the case with PV generation, ca. 12:00 p.m.). A much wider range of voltage is noted in the case of node w_303. Moreover, during the maximum PV generation, the voltage value exceeds the upper permissible level of 1.1 V n . The worst voltage conditions are observed in node w_325. This node (w_325) is far from the transformer (1.15 km) and, due to the voltage drop, the voltage at the period 7:00 a.m.-8:00 a.m. is below permissible 0.9 V n . What is worse, during the period with the maximum PV generation (11:00 a.m.-2:00 p.m.), the voltage value significantly exceeds permissible 1.1 V n . It may lead to the damage of the prosumer's current-using equipment. Additionally, PV generators can be disconnected automatically. The voltage conditions in the analyzed network are not acceptable, so local DSO is searching for the solution, which in a relatively simple way could improve levels of voltage.

General Assumptions
To improve voltage conditions in the analyzed low-voltage network, the following main solutions are considered: another intermediate node depart directly from the selected nodes. The network is a radial type. The analyzed low-voltage network has been modelled with the use of DIgSILENT Power Factory software. Voltage conditions (daily variation) in selected nodes (w_301nearest to the transformer; w_303-in the middle of the main line; w_325-at the end of the main line) for the current network status are presented in Figure 4a, whereas Figure  4b presents assumed daily profile of the PV generation.  The most favorable voltage conditions are observed in node w_301 ( Figure 4a). For the whole day (24 h), the voltage is within the permissible range-the lowest voltage is around 0.95 Vn (the case without PV generation, ca. 7:00 a.m.), and the highest voltage is almost 1.06 Vn (the case with PV generation, ca. 12:00 p.m.). A much wider range of voltage is noted in the case of node w_303. Moreover, during the maximum PV generation, the voltage value exceeds the upper permissible level of 1.1 Vn. The worst voltage conditions are observed in node w_325. This node (w_325) is far from the transformer (1.15 km) and, due to the voltage drop, the voltage at the period 7:00 a.m.-8:00 a.m. is below permissible 0.9 Vn. What is worse, during the period with the maximum PV generation (11:00 a.m.-2:00 p.m.), the voltage value significantly exceeds permissible 1.1 Vn. It may lead to the damage of the prosumer's current-using equipment. Additionally, PV generators can be disconnected automatically. The voltage conditions in the analyzed network are not acceptable, so local DSO is searching for the solution, which in a relatively simple way could improve levels of voltage.

General Assumptions
To improve voltage conditions in the analyzed low-voltage network, the following main solutions are considered:  The aforementioned first solution makes that the reactance of the line practically does not change, but the resistance decreases significantly. In the second solution, the line resistance is almost constant, but the reactance decreases around 4 times. As an extension, the effect of cables YAKY 4 × 50 mm 2 and YAKY 4 × 70 mm 2 is also analyzed. Table 2 presents the nominal parameters of the overhead lines and the cable lines. Voltage conditions along a line mainly depend on the voltage drop in it. Figure 5 presents the general relation between voltages in two nodes. Voltage V fB at node B depends on the geometric difference of the voltage phasor V fA at node A and the voltage drop ∆V AB across the A-B section. Due to the fact that for MV and LV networks, the angle β reaches small values (V fA · cos(β) ≈ V fA ), the following formula can be used to calculate the voltage at node B: X L -reactance of the section A-B; P-active power flow (consumed at node B); Q-reactive power flow (consumed at node B). Based on the aforementioned description ( Figure 5 and Equation (1)), the computer simulation assumes that PV sources generate only active power (it refers to practice), so the voltage drop in the line is affected by its resistance. During consumption of energy, the assumed load reactance-to-resistance ratio is 0.2. Voltage conditions along a line mainly depend on the voltage drop in it. Figure 5 presents the general relation between voltages in two nodes. Voltage VfB at node B depends on the geometric difference of the voltage phasor VfA at node A and the voltage drop ΔVAB across the A-B section. Due to the fact that for MV and LV networks, the angle β reaches small values (VfA • cos(β) ≈ VfA), the following formula can be used to calculate the voltage at node B: XL-reactance of the section A-B; P-active power flow (consumed at node B); Q-reactive power flow (consumed at node B). Based on the aforementioned description ( Figure 5 and Equation (1)), the computer simulation assumes that PV sources generate only active power (it refers to practice), so the voltage drop in the line is affected by its resistance. During consumption of energy, the assumed load reactance-to-resistance ratio is 0.2.              In the case of the nodes w_325 (Figure 8), w_305 and w_305_3 (Figure 9), the voltage exceeds the permissible range (in selected periods of the day) when the cross-sectional area of the conductors is equal to 35 mm 2 . For other cross-sections (50 mm 2 , 70 mm 2 ), the voltage varies in the permissible range.
Aggregated results of the voltage variations for the cross-sections 35 mm 2 , 50 mm 2 and 70 mm 2 are presented in Figure 10. The highest voltage variations are at node w_325-the farthest analyzed node from the power transformer substation. In the case of the nodes w_325 (Figure 8), w_305 and w_305_3 (Figure 9), the voltage exceeds the permissible range (in selected periods of the day) when the cross-sectional area of the conductors is equal to 35 mm 2 . For other cross-sections (50 mm 2 , 70 mm 2 ), the voltage varies in the permissible range.
Aggregated results of the voltage variations for the cross-sections 35 mm 2 , 50 mm 2 and 70 mm 2 are presented in Figure 10. The highest voltage variations are at node w_325the farthest analyzed node from the power transformer substation. The voltage variations in the main line are the critical point of concern because, for radial power networks, they are of the greatest importance from the voltage stability point of view. If there are no significant voltage variations in the main line, the branches can be analyzed then. Figure 10a shows that the highest voltage variations (especially changes from node to node) are in the main line-compare relative changes in the main line (e.g., node w_301 vs. node w_302) and relative changes in the branch line (e.g., node w_301 vs. nodes w_301_1/w_301_2/w_301_3). Therefore, it is reasonable to reduce the voltage variations/deviations by the main line modernization.
It is clearly seen that replacing currently installed conductors 4 × AL 35 mm 2 with conductors 4 × AL 70 mm 2 gives a positive result, and voltage variations are limited to the acceptable range (with some excess) in every node. Thus, taking into account the obtained results, one can conclude that the decrease in the line resistance results in a significant decrease in the voltage variation/deviations in the considered network.
In the case of the nodes w_325 (Figure 8), w_305 and w_305_3 (Figure 9), the voltage exceeds the permissible range (in selected periods of the day) when the cross-sectional area of the conductors is equal to 35 mm 2 . For other cross-sections (50 mm 2 , 70 mm 2 ), the voltage varies in the permissible range.
Aggregated results of the voltage variations for the cross-sections 35 mm 2 , 50 mm 2 and 70 mm 2 are presented in Figure 10. The highest voltage variations are at node w_325the farthest analyzed node from the power transformer substation.
(a)  The voltage variations in the main line are the critical point of concern because, for radial power networks, they are of the greatest importance from the voltage stability point of view. If there are no significant voltage variations in the main line, the branches can be analyzed then. Figure 10a shows that the highest voltage variations (especially changes from node to node) are in the main line-compare relative changes in the main line (e.g., substation  w_301  w_301_1  w_301_2  w_301_3  w_301_3A  w_301_3B  w_301_4  w_301_4A  w_301_4B  w_302  w_302_1  w_303  w_303_1  w_303_2  w_304  w_304_1  w_304_2  w_304_3  w_304_4  w_305  w_305_1  w_305_2  w_305_3 w_301  w_301_1  w_301_2  w_301_3  w_301_3A  w_301_3B  w_301_4  w_301_4A  w_301_4B  w_302  w_302_1  w_303  w_303_1  w_303_2  w_304  w_304_1  w_304_2  w_304_3  w_304_4  w_305  w_305_1  w_305_2  w_305_3

Replacement of the Overhead Line with a Cable Line
As an alternative to the low-voltage overhead line, a cable line can be installed. Such a solution is favorable from the point of view of the reliability of supply but requires that the power line is completely rebuilt. In the computer simulations, a cable line, as an alternative to the overhead line with the same cross-section, is considered. The investigation aims to verify the effect of the power line reactance reduction on voltage variations.
The results of the simulations for two selected characteristic nodes: w_301 and w_325, are presented in Figures 11-13. In each case, both compared power lines (overhead AL vs. cable YAKY) have the same nominal cross-sectional areas of the conductors (almost the same resistance), but the cable line reactance is around 4 times lower than the overhead line (see Table 2). On the basis of these simulations, one can conclude that a significant decrease in the line reactance (the use of the cable line instead of the overhead line) does not give the expected result.

Replacement of the Overhead Line with a Cable Line
As an alternative to the low-voltage overhead line, a cable line can be installed. Such a solution is favorable from the point of view of the reliability of supply but requires that the power line is completely rebuilt. In the computer simulations, a cable line, as an alternative to the overhead line with the same cross-section, is considered. The investigation aims to verify the effect of the power line reactance reduction on voltage variations.
The results of the simulations for two selected characteristic nodes: w_301 and w_325, are presented in Figures 11-13. In each case, both compared power lines (overhead AL vs. cable YAKY) have the same nominal cross-sectional areas of the conductors (almost the same resistance), but the cable line reactance is around 4 times lower than the overhead line (see Table 2). On the basis of these simulations, one can conclude that a significant decrease in the line reactance (the use of the cable line instead of the overhead line) does not give the expected result.   Generally, the voltage is slightly higher than for the overhead line, but in the case of high PV generation, this further aggravates the voltage conditions in the network. In the case of node w_325, during the highest PV generation (around 12:00 p.m.), power cables YAKY 4 × 35 mm 2 ( Figure 11b) and YAKY 4 × 50 mm 2 (Figure 12b) give higher voltage excess (beyond the upper permissible limit 1.1 V n ) than solutions based on overhead AL lines. Generally, the voltage is slightly higher than for the overhead line, but in the case of high PV generation, this further aggravates the voltage conditions in the network. In the case of node w_325, during the highest PV generation (around 12:00 p.m.), power cables YAKY 4 × 35 mm 2 ( Figure 11b) and YAKY 4 × 50 mm 2 (Figure 12b) give higher voltage excess (beyond the upper permissible limit 1.1 Vn) than solutions based on overhead AL lines.

The Costs of Not Maintaining the Required Voltage Level in the Power Network
In order to estimate the financial losses related to the inability to inject power/energy into the grid (due to too high a voltage level in the network), a simplified economic analysis was performed. The economic analysis assumes average electricity prices for the first quarter of 2021 [31]. The cost Ev referred to the inability to inject the power to the network by prosumers can be presented in the following form: where: Ev-is the cost related to the inability to inject the power to the network by prosumers due to voltage that exceeds the acceptable value; Pv-is the power produced by prosumers, which cannot be injected into the network;

The Costs of Not Maintaining the Required Voltage Level in the Power Network
In order to estimate the financial losses related to the inability to inject power/energy into the grid (due to too high a voltage level in the network), a simplified economic analysis was performed. The economic analysis assumes average electricity prices for the first quarter of 2021 [31]. The cost E v referred to the inability to inject the power to the network by prosumers can be presented in the following form: where: E v -is the cost related to the inability to inject the power to the network by prosumers due to voltage that exceeds the acceptable value; P v -is the power produced by prosumers, which cannot be injected into the network; c v -is the unit cost of energy; 0.18 EUR/kWh; t v -is the time in which the level of voltage in the network is beyond the acceptable range.
Results of calculations according to (2) are presented in Table 3. It was estimated that production of energy by PV sources is possible in (365/2) sunny days per year (0.5 years) according to the profile of PV generation (see Figure 4b). The volume of energy that cannot be produced by prosumers in the analyzed network (main overhead line 4 × AL 35 mm 2 ) is assumed to be equal to 155 kWh/day (cost 27.9 EUR/day) or 28 288 kWh/year (cost 5 092 EUR/year). This volume has been calculated with the use of DIgSILENT Power Factory software, based on the aforementioned profiles of generation and levels of voltages in the network. The power P v , which cannot be injected into the network, is calculated based on the maximum declared consumed power by loads (it is equal to the maximum power of PV sources-the power list is given in Table 1) and the profile of generation (it is presented in Figure 4b). In nodes where the voltage exceeded the value of 1.1V n in a given hour, it was impossible to inject the produced power/energy into the power network. The P v value is the sum of the expected generation only for nodes where the voltage is higher than 1.1 V n . A higher cost of E v in case no. 4 than in case no. 1 (Table 3) is due to an even worse voltage level (a higher amount of energy cannot potentially be injected into the grid). In case no. 5, the voltage level is clearly improved, but a certain amount of energy still cannot be injected into the network. It is assumed in calculations that the cost E v referred to the production of power/energy from PV sources has a negative value if the voltage is within the acceptable range (after modernization of the network) and the production of clean energy is possible. The negative value of the cost can be treated as the "avoided cost"-the DSO does not contribute to the inability of the production energy from PV sources. Otherwise, it is called the "incurred cost"-the DSO is responsible for the inability of the production of clean energy, and prosumers may demand compensation.
The costs of replacement of the AL 4 × 35 mm 2 overhead line with the other lines considered in the paper have been evaluated as well. Table 4 presents the specification of costs as well as total costs referring to each considered solution. One case sees that the previously recommended solution (based on the technical aspect), i.e., replacing conductors 4 × AL 35 mm 2 with conductors 4 × AL 70 mm 2 , is associated with a relatively low total cost (11,800 EUR).

Conclusions
In low-voltage networks without PV generation, the highest voltage is at the supplying node (with transformer MV/LV), and the lowest voltage is at the end of the line. When distributed PV generation is applied, the voltage profile along the distribution line can be unexpected. Voltage can be the highest far from the transformer. In some cases, the voltage may exceed the permissible range, and, as a consequence, the automatic protection of the PV installation disconnects the installation from the public network. The prosumer cannot inject the power into the network, which leads to the lack of profitability of investments in installations with PV sources. The considerations presented in this paper show that a decrease in the line resistance (the increase in the cross-sectional area of the conductors) gives effective limitation of the voltage variations/deviations when distributed PV generation is installed. It is a relatively simple way of voltage profile improvement. The other solution-the line reactance decrease by modernization to the cable line-requires a complete reconstruction of the network (overhead line to underground cable line) and, in addition, does not give a positive effect to the voltage profiles. Thus, the recommended solution in the considered case is to replace bare conductors 4 × AL 35 mm 2 with conductors 4 × AL 70 mm 2 . The DSO is now considering modernization of the network according to the recommendation indicated by the authors.