Voltage Rise Mitigation in Medium-Voltage Networks with Long Underground Cables and Low Power Demand

Abstract

Medium-voltage (MV) distribution networks that are spread through larger territory and threatened by extreme weather conditions are sometimes formed by very long underground cable lines. In such circumstances, a significant amount of capacitive reactive power flow can be generated. If, concurrently, there is low power demand in the network, it can result in significant reverse reactive power flows and voltage rise issues. This paper proposes a general approach for analyzing and mitigating voltage rise issues and demonstrates it using an example of a real distribution network that operates under the described conditions. Previous studies that dealt with this problem did not include the allocation of multiple shunt reactors in a larger distribution network, modeling a high number of lines that create reverse reactive power flows, and modeling the main distribution transformers, which are the locations where voltage rise predominantly occurs. In this paper, we demonstrate that precise allocation and placement of multiple shunt reactors in a fully modeled, larger distribution system, including transformer models, can reduce reverse reactive power flows, thereby improving voltage in the distribution system. If hourly control of the power factor from the distributed generation unit is also implemented, the voltage can be further improved.

1. Introduction

Medium-voltage overhead distribution networks that are spread through a large territory are vulnerable to severe weather conditions. A study [1] showed that in some places, weather-related outage events account for 78.09% of the total, with lightning, contributing 63.26%, and wind, contributing 12.27%, as the two main weather factors. Additionally, wildfires account for 2.14%, icing for 0.16%, and geological events for 0.11%.

In conditions where overhead line damage is frequent, medium-voltage networks are sometimes formed by long underground cable lines, ensuring enhanced reliability in the supply of electrical energy [2]. Electric power distribution networks were primarily developed for power flows directed towards consumers [3]. Due to the inductive characteristics of consumption, reactive power flows are commonly directed towards consumers, with variations in the inductive component roughly corresponding to fluctuations in electrical demand, especially when the medium-voltage network is predominantly implemented using overhead lines.

Underground medium-voltage cable lines generate reverse reactive power flows [4] toward the distribution network feeding point, where the quantity of reactive power flows increases with the length of the lines. In distribution networks with very low power demand and very long cable lines, significant reverse reactive power flows may occur, leading to voltage rises in the distribution network.

Standard indicators of voltage quality are defined by the EN 50160 standard [5]. The permissible voltage deviation is typically specified as ±10% of the nominal voltage. Voltage fluctuation limits are set to ensure that short-term flicker does not exceed 1.0 and long-term flicker remains below 0.8, as outlined in the same standard. Additionally, the standard specifies that voltage asymmetry between phases (voltage unbalance) should be ≤2%, ensuring balanced distribution across three-phase systems. Limits for non-sinusoidal voltages dictate that total harmonic distortion (THD) should not exceed 8%, maintaining the integrity of voltage waveforms and minimizing interference with electrical equipment.

The problems with voltage rise can result in the need for power curtailment of DG units [6]. The voltage rise issue is typically investigated in the context of the connection of distributed generation sources [7]. Papers related to voltage rises typically address the reduction in voltage rises resulting from reverse power flows caused by the connection of distributed generation units. In Ref. [8], the author proposed a phase shifting strategy to mitigate local voltage rises and reverse power flows caused by high photovoltaic penetration. In Ref. [9], the power curtailment of distributed generation is proposed to mitigate voltage rise. In Ref. [10], the authors considered the voltage rise issue during the distributed generation allocation process. In Ref. [11], the authors proposed the dynamic operation of a voltage regulator for voltage regulation in a distribution system with a significant integration of distributed generation. In Ref. [12], the authors proposed the usage of distributed energy storage for mitigation of the voltage rise impact caused by rooftop solar PV systems. In Ref. [13], the authors proposed a decentralized control strategy using droop-based control to adjust electric vehicle (EV) charging rates in real time based on nodal voltages, voltage unbalance, and state of charge (SOC) to mitigate voltage rise from rooftop solar PV systems. The paper [14] introduces a novel reactive power control technique tailored for low-voltage distribution networks with high photovoltaic (PV) system penetration, addressing voltage rise challenges by dynamically adjusting reactive power based on both PV active power injection and real-time network voltage conditions. It demonstrates superior voltage regulation and reduced losses compared to conventional fixed power factor and voltage-based methods, highlighting its adaptability to varying scenarios despite a potentially slightly slower response time. The paper [15] addresses voltage regulation challenges arising from increased distributed PV installations in distribution systems. It introduces a PID closed-loop-based volt-var (VV) function and a mode-switching function for PV smart inverters to enhance voltage regulation effectiveness. The paper [16] introduces a control strategy for grid-connected inverters interfacing with battery storage systems (BSS) to enhance PV penetration in low-voltage (LV) grids. The strategy optimizes inverter set points for active and reactive power based on grid voltage, considering battery specifications and state of charge (SoC). The paper [17] explores leveraging electric minibus taxi (EmBT) charging alongside residential solar photovoltaic (PV) generation to mitigate voltage rises in low-voltage (LV) networks with high penetration of distributed energy resources (DERs).

In contrast to the literature addressing voltage rises due to distributed generation units, the literature that investigates voltage rises caused by large capacitive currents in underground cables and variations in power demand is much less common. In Ref. [18], a reactive power control method using a variable shunt reactor and the reactive power control of a wind power plant was proposed to compensate for the capacitive reactive power of the cable. In Ref. [19], the authors addressed voltage issues resulting from substantial load variations by proposing the installation of shunt capacitor banks for voltage drop issues and shunt reactors (ShRs) for voltage rise issues. In Ref. [20], the authors deployed switchable shunt reactors in combination with DGs that have the capability of absorbing reactive power for voltage rise mitigation in a low-voltage distribution network with high penetration of photovoltaic DGs. The authors concluded that DGs have limited capacity to serve as a source of reactive power and are unable to provide sufficient reactive power for full voltage rise mitigation. In Ref. [21], the authors demonstrated that the shunt reactor can mitigate the power frequency voltage rise caused by the capacitance effect of long lines under no-load or light-load conditions in high-voltage systems. The paper [22] explores the voltage rise mitigation caused by minimal loading and excessive reactive power generated by line capacitance in magnetically controlled shunt reactors.

This paper focuses on analyzing voltage rise issues that are not primarily caused by DGs, but rather by very low power demand and a significant number of long underground cable lines that create a high amount of capacitive reactive power. The analysis was conducted on a real distribution network spread over two voltage levels. The network is located in Primorsko-Goranska County, Croatia. This distribution network is specific due to the very low power demand resulting from a significant population decrease, as well as a large number of long underground cable lines installed to withstand extreme weather conditions caused by the ice storm in 2014. A method for reducing voltage rise was introduced, involving the allocation of multiple fixed shunt reactors within the distribution network and the implementation of hourly power factor control for the existing distributed generation. The distribution system model contains two voltage levels of the distribution network with two distribution transformers that are connected between the 35 kV and 20 kV distribution networks. Modeling the distribution transformers and including them in the analysis of voltage rise issues is significant, as the majority of voltage rises occur in them. The analysis includes not only the 20 kV network and 35/20 kV transformers but also the feeder lines of the 35 kV network, which are expected to contribute to voltage rise issues due to their length and characteristics. The main 110/35 kV transformer is not included in the analysis since it has an on-load tap changer so the voltage is fixed to a set value of 1.05 p.u.

The main innovations introduced in this paper are summarized in four points:

(1)

This paper proposes a novel general approach for analyzing and mitigating voltage rise issues where the entire system is accurately modeled and various elements that contribute to voltage rises are unified into a single method for the mitigation of voltage rises on multiple voltage levels of a distribution network.

(2)

One of the novel aspects of our approach lies in the inclusion of main distribution transformers in the analysis of the voltage rise issue. The voltage rise issue is analyzed with the modeling of distribution transformers. Earlier research addressing this issue overlooked the incorporation of main distribution transformers, which are pivotal sites where voltage rise predominantly occurs.

(3)

Previous studies on this topic typically incorporated just one shunt reactor and a limited number of distribution lines. In contrast, our paper introduces a method for addressing voltage rise by strategically allocating multiple shunt reactors and implementing hourly power factor control for a distributed generation unit within extensive distribution networks featuring a high volume of lines generating reactive power flows. The allocated multiple shunt reactors simultaneously influence the reduction of voltage rise both in the lines and in the transformers.

(4)

The proposed method also unifies and accurately models the components of distribution lines responsible for creating reverse flows of capacitive reactive power, particularly the parallel branches that inject capacitive reactive power. The voltage rise issue is analyzed with the modeling of lines on two voltage levels of the distribution network.

Despite the relatively small size of the higher-level network, we demonstrated the feasibility of mitigating voltage rises in distribution lines of both voltage levels of the distribution network, as well as in the transformers that serve as connections between these voltage levels, by using shunt reactors and hourly power factor control of DGs.

2. Materials and Methods

The main objective of optimization is to mitigate voltage rise issues caused by low power demand and significant reverse power flows generated by underground cables in the distribution system. This is achieved by the allocation of up to eight fixed shunt reactors and the hourly power factor control of the existing distributed generation.

2.1. Objective Function for Voltage Rise Mitigation

The mitigation of voltage rises will be achieved by defining the objective function for voltage profile improvement across all distribution network nodes and time intervals, which can be mathematically expressed as follows:

$$Minf={\sum }_{t=1}^{Nt}{\sum }_{i=1}^{Nn}{(U_n-{U_i}^t)}^2$$

(1)

In Equation (1), N_n and N_t represent the total number of nodes and time intervals, while U_i^t represents the ith node voltage at interval t. U_n represents the desired system voltage in the distribution network. The nominal voltage is as follows:

$$U_n=21000<0°V$$

(2)

2.2. Constraints

2.2.1. Positions of Shunt Reactors

The algorithm is permitted to suggest up to eight new shunt reactors, which can be connected to nodes on the secondary side of the 35/20 kV transformers; the nodes satisfy the following constraints, respectively:

(3 ≤ ShRs_position ≤ 42) ∨ (44 ≤ ShRs_position ≤ 71)

(3)

2.2.2. Installed Power of Shunt Reactors

The algorithm is permitted to suggest the following installed reactive power of one ShR unit:

$$Q_{ShR}=n·100\left[kVAr\right]$$

(4)

In Equation (4), n represents the factor that must satisfy the constraint (5):

n_min ≤ n ≤ n_max

(5)

In this paper, n_min was set to a value of 0, and n_max was set to a value of 11.

2.2.3. Permitted Power Factor of Existing DG

Various types of DG generators have different operational constraints for the capability of reactive power generation and absorption [23]. Since the DG in the presented network is hydropower, the synchronous generator is considered. Therefore, the permitted power factor of the existing DG used in this paper ranges from 0.85 lagging to 1.0 p.u. and from 0.95 leading to 1.0 p.u. in steps of 0.005.

2.2.4. Current Magnitudes

The magnitude of the current that flows through all lines and transformers must satisfy the following constraint:

I_i^t ≤ I_n

(6)

In Equation (6), I_i^t stands for the current of the ith branch in the time interval t, and I_n is the nominal current of each branch.

2.3. Power Demand Modeling and Distribution System Model Used for Voltage Rise Issue Analysis

The analysis was conducted using a model based on a segment of the actual distribution network supplied from the transformer station 110/35 kV Delnice. From this substation, two other substations are taken for analysis. The substations are supplied through two separate feeders.

The distribution system model is shown in Figure 1. The substation TS 35/20 kV Kupjak is supplied through an underground cable, while the substation TS 35/20 kV Gerovo is supplied through a 35 kV overhead line. The transformer station 110/35 kV Delnice has a transformer installed with an on-load tap changer, which sets the secondary side voltage at 36.75 kV, corresponding to 1.05 p.u. Therefore, the voltage in the 20 kV distribution networks is also initially set to 21 kV, corresponding to 1.05 p.u. The voltage is set to 1.05 p.u. because a lower set voltage could result in excessively low voltages in some other parts of the distribution networks that are also supplied from TS 110/35 kV Delnice. Therefore, the voltage sought to be achieved by objective function (1) during the mitigation of excessive voltage rises is 1.05 p.u.

In addition to the two transformers, the model of the network includes 69 distribution lines with a total length of 252 km, the majority of which are underground cable lines.

Power demand is modeled in nodes 4–42 and 45–71, based on maximum active and reactive power in each node and real measured data for overall daily power demand in the presented network. The maximum power in each node is shown in

Table 1. Maximum active and reactive power in nodes that create power demand.

Node	Maximum Active Power (kW)	Maximum Reactive Power (kVAr)	Node	Max. Active Power (kW)	Max. Reactive Power (kVAr)	Node	Max. Active power (kW)	Max. Reactive Power (kVAr)
4	160	52.59	24	350	115.04	46	450	147.91
5	370	121.61	25	250	82.17	47	1090	358.27
6	630	207.07	26	150	49.30	48	1030	338.54
7	1250	410.86	27	210	69.02	49	850	279.38
8	650	213.64	28	630	207.07	50	150	49.30
9	350	115.04	29	850	279.38	51	150	49.30
10	460	151.19	30	650	213.64	52	150	49.30
11	350	115.04	31	150	49.30	53	50	16.43
12	630	207.07	32	150	49.30	54	300	98.61
13	630	207.07	33	250	82.17	55	100	32.87
14	1250	410.86	34	350	115.04	56	400	131.47
15	650	213.64	35	610	200.50	57	50	16.43
16	200	65.74	36	760	249.80	58	200	65.74
17	150	49.30	37	500	164.34	59	410	134.76
18	1030	338.54	38	300	98.61	60	810	266.23
19	150	49.30	39	300	98.61	61	790	259.66
20	200	65.74	40	100	32.87	62	320	105.18
21	450	147.91	41	150	49.30	63	510	167.63
22	460	151.19	42	150	49.30
23	500	164.34	45	880	289.24

.

The normalized power demand curves are shown in Figure 2. To obtain these curves, the active power demand was measured in TS Gerovo and TS Kupjak in the period from 22 January 2023 to 21 July 2023. This six-month period is well suited for analysis as it includes both high and low consumption periods, making it a sufficiently robust dataset for representing the method. The data was obtained from the SCADA system, which provides the average active power consumption for each hour for all days within the specified period.

In this paper, the average curve shown in Figure 2 was used. The working day and weekend day curves are similar. Using an average curve is sufficient to present the method effectively. However, due to fluctuations in power consumption throughout the week, it is expected that voltage levels during weekends might be slightly higher than anticipated. Consequently, if the algorithm were to base its operation on the weekend day curve, it would likely decide on a slightly higher delivery of reactive power, either through increased deployment of ShRs or increased reactive power that the DG absorbs.

For even more accurate results, further segmentation could be considered by differentiating between weekdays and weekends. This would enable a more precise representation of power demand fluctuations throughout the week.

For each hour, the average power demand for the whole distribution system was calculated using the following equation:

$${PD}^t=\frac{{P_{measured\_total}}^t}{{\sum }_{i=1}^{Ni}{P}_{i\_max}}$$

(7)

In Equation (7), P_{measured_total}^t represents the total measured power demand of the whole distribution system during time interval t (including measurements from both substations). P_{i_max} represents the maximum allowed power demand in node i, while N_i represents the number of nodes.

In this paper, the demand changes in every node on an hourly basis, according to the curve. The active and reactive power demand at each node for time interval t is calculated as follows:

$${P_i}^t={PD}^t·P_{i\_max}$$

(8)

$${Q_i}^t=\tan \phi ·{P_i}^t$$

(9)

In Equation (9), tanφ is determined based on the chosen value for cosφ, which, in this paper, was 0.95. Therefore, the maximum reactive power in Table 1 represents the maximum reactive power of that value of cosφ.

2.4. Distribution Line Modeling, Data, and Voltage Rise Calculation

In this paper, the “backward/forward sweep” algorithm described in [24] was used for modeling distribution lines and load flow calculations. Matlab, Version R2015A, a widely used software platform known for its capabilities in numerical analysis, data processing, and visualization, was employed for implementing both the load flow calculation and the optimization algorithm and for processing the experimental results.

The distribution line model is shown in Figure 3. The longitudinal branch contains the distribution line series resistance R_DLi and the series inductive reactance X_DLi. Since the distribution system model contains distribution lines of two voltage levels, all series line impedances were reduced to the 21 kV voltage level. Real and reactive power P_i, Q_i, P_i−1, and Q_i−1 are calculated using the load flow algorithm from P_Pi, Q_Pi, P_P,i−1, and Q_P,i−1, which represent the active and reactive power demand in starting node i−1 and end node i.

The model also contains parallel branches, which are introduced to model the capacitive currents that underground distribution lines create. For each line, one part of that reactive power is added to the distribution line starting node and the other half to the distribution line end node. The reactive power is calculated using Equation (10).

$$\frac{\overline{Q_{Ci}}}{2}={U_i}^2·\omega ·\frac{C_i}{2}·{(\mathit{cos}{\phi }_i+j\mathit{sin}{\phi }_i)}^2$$

(10)

In Equation (10), C_i represents the capacitance of the ith distribution line, U_i represents system voltage, and ${\phi }_i$ represents the phase angle of the voltage.

The current that flows through the distribution transformer and creates voltage rises or drops is calculated using Equation (11).

$$\overline{I_{DLi}}=\frac{P_i·\mathit{cos}{\phi }_i+(Q_i+Q_{Ci}/2)·\mathit{sin}{\phi }_i}{\sqrt{3}{U}_i}+j\frac{P_i·\mathit{sin}{\phi }_i-(Q_i+Q_{Ci}/2)·\mathit{cos}{\phi }_i}{\sqrt{3}{U}_i}$$

(11)

In Equation (10), Q_Ci represent the reactive power of the parallel branch of ith distribution line.

The voltage rise on the distribution line is calculated using Equation (12):

$$\overline{V_i}-\overline{V_{i-1}}=-(R_{DLi}+j{X}_{DLi})·\overline{I_{DLi}}$$

(12)

In Equation (12), $\overline{V_{i-1}}$ represent the voltage in the node at the beginning of the ith distribution line and $\overline{V_i}$ the voltage at the end of the distribution line. R_DLi and and X_DLi represent the serial resistance and reactance of the ith distribution line, respectively. Distribution line lengths are shown in

Table 2. Distribution lines’ length.

Line No.	Length (km)	Line No.	Length (km)	Line No.	Length (km)	Line No.	Length (km)
2	7.92	20	2.37	37	6.18	55	1.03
4	1.35	21	2.34	38	1.89	56	3.86
5	2.4	22	6.45	39	4.09	57	4.13
6	1.33	23	4.83	40	10.42	58	1.51
7	0.59	24	0.29	41	3.38	59	5.53
8	1.18	25	8.63	42	11.42	60	1.17
9	1.34	26	3.68	43	28.28	61	0.22
10	2.96	27	2.2	45	5.82	62	0.81
11	5.6	28	1.35	46	1.9	63	1.38
12	2.55	29	6.62	47	6.91	64	3.84
13	1.15	30	1.55	48	6.15	65	5.54
14	1.75	31	1.51	49	2.13	66	0.7
15	4.4	32	3.12	50	5.15	67	1.82
16	0.56	33	4.84	51	7.28	68	0.25
17	1.59	34	2.74	52	1.99	69	3.817
18	1.56	35	4.68	53	2.29	70	2.75
19	2.27	36	4.78	54	2.95	71	3.53

, and other data are shown in

Table 3. Distribution lines’ data.

Line No.	Type	R (Ω/km)	X (Ω/km)	B (µS/km)
2	35 kVcable	0.176	0.12	63
43	35 kV overhead	0.253	0.35	3.3
4, 5, 7–19, 22–30	20 kV cable	0.164	0.122	85.5
32, 33, 35–42,	20 kV cable	0.164	0.122	85.5
45–54, 56, 57, 59,	20 kV cable	0.164	0.122	85.5
61, 62, 65–71	20 kV cable	0.164	0.122	85.5
6, 21, 58, 60	20 kV overhead	0.43	0.345	3
20, 31, 34, 55, 63, 64	20 kV overhead	0.86	0.365	3

.

2.5. Distribution Transformers’ Modeling, Data, Voltage Rise Calculation, and Importance in Analysis

Two distribution transformers are integrated into the model because most voltage rises are expected to occur at the transformers. The transformer model is shown in Figure 4, and transformer data are shown in

Table 4. Distribution transformer data.

Transformer No.	From Node	To Node	Installed Power (MVA)	PTRm (W)	i0 (%)	uk (%)
1	2	3	8	47,000	0.8	7
2	43	44	4	24,800	0.8	6

.

The series resistance R_TRi and the series inductive reactance of the distribution transformer X_TRi have the largest impact on voltage rises and voltage drops that occur. The series resistance and reactance could be calculated as follows:

$$R_{TRi}=P_{Cu}·\frac{{U_n}^2}{{S_n}^2}$$

(13)

$$Z_{TRi}=\frac{u_k}{100}·\frac{{U_n}^2}{{S_n}^2}$$

(14)

$$X_{TRi}=\sqrt{{Z_t}^2-{R_t}^2}$$

(15)

The aforementioned impedance values of the transformers were calculated for the nominal voltage value of 21 kV because this voltage is the voltage on the secondary side of the two transformers in the actual network. Additionally, all the line impedances were recalculated for the same voltage. In Equations (13)–(15), S_n represents the transformer installed power, P_Cu denotes transformer copper loss, and u_k is the transformer short-circuit voltage.

The current that flows through the distribution transformer and creates voltage rises or drops is calculated using Equation (16):

$$\overline{I_{TRi}}=\frac{\left(P_i+P_{TRm,i}\right)·\mathit{cos}{\phi }_i+\left(Q_i+Q_{TRm,i}\right)·\mathit{sin}{\phi }_i}{\sqrt{3}{U}_i}+j\frac{\left(P_i+P_{TRm,i}\right)·\mathit{sin}{\phi }_i-(Q_i+Q_{TRm,i})·\mathit{cos}{\phi }_i}{\sqrt{3}{U}_i}$$

(16)

P_TRm,i and Q_TRm,i represent active and reactive power in the parallel branch of the transformer, where P_TRm,i is obtained from the transformer open-circuit test and Q_TRm,i is calculated as follows:

$$Q_{TRm,i}=\frac{{U_n}^2}{{X_{mi}}^2}$$

(17)

$$X_{m,i}=\frac{U_n}{\begin{array}{c}\sqrt{3}·\frac{i_{0\%}}{100}·I_n\\ \end{array}}$$

(18)

In Equation (18), i_0% stands for the transformer open-circuit current, and I_n denotes the transformer nominal current.

The voltage rise in the power transformer is calculated using Equation (19):

$$\overline{V_i}-\overline{V_{i-1}}=-(R_{TRi}+j{X}_{TRi})·\overline{I_{TRi}}$$

(19)

Based on Equations (13) and (15) and the distribution transformer data shown in Table 4, the serial impedance of the distribution transformer TS 35/20 kV Kupjak can be calculated to be 0.324 + j3.845 [Ω] and the impedance of the distribution transformer TS 35/20 kV Gerovo to be 0.684 + j6.58 [Ω]. If we compare these impedances with the impedances of the distribution lines given in Table 3, we can conclude that the transformer reactances are larger. We can also conclude that the reverse power flow through distribution transformers is larger because it contains the sum of the reverse power flows from all distribution lines after the transformer. Therefore, from Equation (19), used for the calculation of the voltage rise in the transformer, we can conclude that the voltage rise in the transformer will have a significant impact on the overall voltage rise in the distribution system at all nodes located after the distribution transformer.

From Equation (19), we can see that voltage rises could be decreased by decreasing the reverse reactive part of the current $\overline{I_{TRi}}$ by using shunt reactors and the power factor control of DGs.

2.6. Distributed Generation Unit Data

In this paper, one existing DG with a constant power output is considered with a connection at node 47. The installed power of this DG is 2500 kW.

2.7. Concise Theory and Specific Advantages of Genetic Algorithms

The genetic algorithm (GA) was chosen for the optimization. GA is a computational method inspired by natural selection and genetics, initially proposed by John Holland in the 1960s and further developed by researchers such as David Goldberg. GA simulates the process of natural evolution to efficiently solve optimization and search problems.

At the core of GA is a population of potential solutions, represented as chromosomes encoding candidate solutions. The algorithm iteratively evolves this population over generations, guided by principles akin to biological evolution.

The iterative process of the genetic algorithm begins with an initial population of chromosomes:

$$P^{\left(0\right)}=\{{x_1}^{\left(0\right)},{x_2}^{\left(0\right)},\dots ,{x_N}^{\left(0\right)}\}$$

(20)

Each chromosome represents a potential solution to the problem at hand. These chromosomes undergo evaluation, where their fitness is assessed based on an objective function defined by the problem domain. This fitness evaluation determines how well each chromosome solves the problem and guides the selection process. During selection, chromosomes with higher fitness are more likely to be chosen as parents for the next generation. Various selection methods, such as roulette wheel selection or tournament selection, are employed to determine which chromosomes proceed to the next stage. Selected parent chromosomes undergo crossover, where genetic material (genes) is exchanged to create offspring. Crossover points are randomly selected to maintain genetic diversity within the population. Additionally, GA introduces occasional mutations, where random changes are applied to offspring chromosomes. This mutation process promotes the exploration of new solutions and prevents premature convergence to suboptimal solutions. After generating offspring through crossover and mutation, the algorithm replaces less fit individuals in the current population with these new offspring. Elitism strategies may be employed to ensure that the best-performing individuals survive across generations, maintaining the population’s overall fitness. The iterative process continues until a termination criterion is met, such as reaching a maximum number of generations or achieving convergence where further improvement is unlikely.

GA was chosen because each solution of optimization in this research has 40 optimization variables, and GA is particularly effective in tackling intricate combinatorial problems with a large number of optimization variables [25]. Therefore, one of the main advantages is that genetic algorithms excel at exploring solution spaces and finding global optimal solutions, which is crucial in complex optimization problems where the search space is large or uneven. Another benefit is that by combining global exploration through selection and crossover with local exploration through mutation, genetic algorithms strike a balance between intensive and extensive search, often yielding superior results compared to metaheuristics focused solely on one aspect of search. Genetic algorithms can be easily adapted to meet specific optimization problem requirements, such as different types of constraints or multi-objective optimization, ensuring their wide applicability.

2.8. Implementation of an Optimization Algorithm for Allocation of Shunt Reactors and Hourly DG Power Factor Recommendation

The optimization variables for one solution of the optimization algorithm are shown in

Table 5. Genetic algorithm optimization variables.

Proposed DG Power Factor for All Hours	Position of 8 Shunt Reactors	Sizes of 8 Shunt Reactors
Gene 1–Gene 24	Gene 25–Gene 32	Gene 33–Gene 40

. The process of optimization is shown in Figure 5, and the optimization process is similar to the GA optimization process described in [26].

The algorithm starts with inputting data about distribution network topology, network element impedances and susceptances, maximum power at each node, power demand, constraints, and genetic algorithm parameters such as population size, number of generations, and operator factors. The value of the chosen crossover factor is 0.9, and the chosen mutation factor is 0.1.

The first step is the creation of an initial population of chromosomes with data about proposed positions and sizes for shunt reactors and optimal power factors of DGs, if applicable. All proposed solutions for all chromosomes must adhere to the constraints described in Section 2.2.1, Section 2.2.2, Section 2.2.3 and Section 2.2.4 of this paper. After that, the first generation of the algorithm is initiated. For the proposed solution for the ShRs and power factors of DGs, the new data about active and reactive power in each node is calculated. Then, load flow calculations are made using a backward–forward sweep algorithm for the whole distribution network at all time intervals. After that, voltage rises are calculated for all distribution lines and transformers using Equations (12) and (19). Then, all chromosomes are evaluated using the calculation of the objective function given in Equation (1), which calculates the overall sum of voltage rises in the whole distribution network in all time intervals. After that, genetic algorithms apply crossover and mutation operators if the probability of each operator is met. All proposed solutions for all chromosomes after applying crossover and mutation factors must adhere to the constraints described in Section 2.2.1, Section 2.2.2, Section 2.2.3 and Section 2.2.4 of this paper.

The process repeats until the last generation of the algorithm is executed. After that, the algorithm finds the best solution by calculating the objective function given in Equation (1).

2.9. Comparision with Other Established Methods for Voltage Rise Mitigation

This study proposes a novel approach for addressing voltage rise issues in distribution networks through a comprehensive methodology that integrates various contributing factors into a unified solution. By accurately modeling the entire system, including medium-voltage lines, distribution transformers, and the effects of low power demand and long underground cable lines, the method aims to mitigate voltage rises effectively across multiple voltage levels. Existing methods typically focus on mitigating voltage rises at single voltage levels, whereas our approach uniquely addresses these challenges across multiple voltage levels of the distribution network.

Unlike traditional approaches that may focus on singular aspects such as shunt reactors or power factor control alone [19], this method combines the allocation of fixed shunt reactors with hourly power factor control of distributed generators (DGs). This integrated approach ensures a more balanced reduction in voltage rises by effectively managing reactive power flows at critical nodes. The study emphasizes the role of distribution transformers in voltage rise occurrences, showing that significant voltage rises often occur at these points due to reverse reactive power flows. This insight underscores the importance of modeling transformers to effectively tackle voltage rise challenges, a perspective that distinguishes this study from the existing literature.

2.10. Cost Analysis of Shunt Reactor Deployment

The existing studies on the installation of shunt reactors [18,19,20] have excluded a comprehensive cost–benefit analysis. Shunt reactors are typically installed in power grids, facing issues with excessive reverse flows of reactive power. Installing compensating shunt reactors is a necessary solution to mitigate these reverse flows. An alternative would be to deactivate the cables, generating these reverse flows and connecting existing consumers via new overhead lines. However, such an approach is often not economically viable compared to the installation of shunt reactors.

According to [27], an MV aluminum conductor with a cross-sectional area of 185 mm², which is the most prevalent in the network in our study, generates a capacitive reactive power of 34.2 kVAr/km. This means that to reduce the capacitive reactive power by 1 MVAr, approximately 29.24 km of the mentioned cable would need to be removed and replaced with overhead lines. According to [28], the construction cost of overhead lines would amount to at least USD 62,150 per kilometer. Therefore, the construction of a 29.24 km overhead line needed to remove a 29.24 km cable line could cost around USD 1.8 million.

According to [29], the cost of installing shunt reactors is USD 54,000/MVAr. Therefore, installing 1 MVAr of shunt reactors at an approximate cost of USD 54,000 would present a much more cost-effective solution compared to alternatives, such as replacing a 29.24 km cable with overhead lines, which could cost around USD 1.8 million.

3. Results

3.1. Cases Studied

The simulations were made to demonstrate the effectiveness of the proposed method for mitigating voltage rises by allocating shunt reactions and using the hourly recommended power factors of the DG unit. The simulation was made for a single objective function for voltage profile improvement based on the fitness function given in Equation (1).

To compare the results, three simulation cases were considered. Before applying the method, load flow calculations were made for the initial network state without connected shunt reactors and with the existing DG unit working at unity power factor. Then, in Case 1, the algorithm recommended the hourly power factor of the existing DG without connecting new shunt reactors. In Case 2, the algorithm allocated and sized up to eight shunt reactors without hourly power factor control of the existing DG. Finally, in Case 4, the algorithm used both new shunt reactors and hourly power factor control of the DG.

3.2. Results and Comments

The results for the locations and sizes of new shunt reactors for various cases are shown in

Table 6. Allocation of shunt reactors.

	Case 1		Case 2		Case 3
New ShR	Size (kVAr)	Location	Size (kVAr)	Location	Size (kVAr)	Location
1	-	-	1000	49	700	48
2	-	-	900	40	600	30
3	-	-	800	3	600	42
4	-	-	100	37	800	49
5	-	-	200	25	700	4
6	-	-	300	57	700	48
7	-	-	800	49	1000	47
8	-	-	1100	24	1100	18
Overall	-	-	5200	-	6200	-

. The results for recommended DG power factors for previously installed DG units are shown in

Table 7. Recommended DG power factors.

Hour	Power Factor (Case 1)	Power Factor (Case 2)	Power Factor (Case 3)
1–4	0.95cap	1	0.96, 0.97, 0.975, 0.975
5–8	0.95cap	1	0.97, 0.96, 0.895, 0.865
9–12	0.95cap	1	0.86, 0.85, 0.875, 0.865
13–16	0.95cap	1	0.855, 0.865, 0.895, 0.915
17–20	0.95cap	1	0.91, 0.91, 0.91, 0.905
21–24	0.95cap	1	0.915, 0.915, 0.93, 0.955

.

In Case 1, voltage mitigation is achieved only by controlling the power factor of one DG installed in part of the distribution network powered by TS 35/20 kV Gerovo in nodes 45 to 71. The algorithm in all parts of the day proposed the maximum leading power factor of 0.95. With this power factor, the existing DG behaves as a consumer of excessive reactive power. However, since the DG produces 2.5 MW of real power, with a leading power factor of 0.95, the DG unit can consume only around 0.84 MVAr of reactive power. Figure 6 shows the overall reactive power in branch 44 of the power transformer in TS 35/20 kV Gerovo. The large reverse reactive power flow is partially reduced, but is still high.

Therefore, the voltage rise is slightly mitigated, as shown in Figure 7, which shows the voltage in the whole distribution network at 3 AM. In nodes 45 to 71, which are powered from TS 35/20 kV Gerovo, the maximum voltage rise is reduced, but only from 1.13 p.u. to 1.11 p.u. The voltage at the end of MV line 43 is around 1.07 p.u., which shows that even the voltage rise issue in that MV line is not fixed. The voltage in nodes 2 to 42 is not affected because changing the power factor of the DG unit in node 47 does not have an impact on that part of the distribution network.

In Case 2, the algorithm proposed the locations and sizes of new shunt reactors, but without power factor control of the existing DG. The algorithm recommended the allocation of shunt reactors with a total installed power of 5.2 MVAr. Figure 7 shows that the voltage magnitudes at all nodes are significantly reduced. The maximum voltage rise is 1.07 p.u. and the voltage at all nodes is near the desired magnitude of 1.05 p.u. Figure 8 shows the voltage in node 44 on the secondary side of the transformer in TS 35/20 kV Gerovo. The voltage in this case is within the range of 1.04 p.u. to slightly above 1.06 p.u., which is much better than in Case 1.

In Case 3, the algorithm proposed the locations and sizes of new shunt reactors and also the optimal power factor of the existing DG, as shown on Figure 9. The algorithm recommended the allocation of shunt reactors with a total installed power of 6.2 MVAr. The algorithm proposed a lagging power factor of the existing DG in the range of 0.85 to 0.975 p.u., as shown in Figure 8.

The combination of new shunt reactors and DG power factor control achieved the best results in voltage rise mitigation. Figure 7 shows that the voltage in all nodes in Case 3 is very close to the desired 1.05 p.u. at 3 AM. Figure 8 shows that the voltage in node 44, on the secondary side of the transformer in TS 35/20 kV Gerovo, is very close to the desired value of 1.05 p.u at all times of the day.

To summarize, Figure 6 shows the reactive power flow in branch 44 at the TS 35/20 kV Gerovo transformer. In Case 1, voltage rise mitigation adjusted the DG power factor, resulting in a modest reduction in the reverse reactive power flow. Cases 2 and 3 achieved significant reductions in reactive power flows through the deployment of SHRs, with Case 3 also optimizing DG power factor control. Figure 7 displays the voltage profile of the whole network at 3 A.M. In Case 1, adjusting DG power factors slightly reduced the voltage rises to around 1.11 p.u. Case 2, using SHRs, saw voltages peak at 1.07 p.u. Case 3, with SHRs and optimized DG factors, maintained stable voltages of around 1.05 p.u. across all nodes. Figure 8 shows the voltage at node 44 on the secondary side of TS 35/20 kV Gerovo at 3 AM. Case 1 limits voltage to 1.1 p.u., Case 2 ranges between 1.04 to slightly above 1.06 p.u., and Case 3 is near the desired 1.05 p.u. Figure 9 illustrates the proposed DG power factor for Case 3, ranging from 0.85 to 0.975 p.u., as recommended by the algorithm.

3.3. The Importance of Modeling Distribution Transformers in Voltage Rise Analysis

Figure 10 shows part of the distribution network, which includes one 35 kV overhead line between nodes 1 and 43, one 35/20 kV distribution transformer between nodes 43 and 44, and five 20 kV distribution lines from node 44 to node 49.

Without the implementation of the presented method, it is notable (Figure 10b) that the overall voltage rise from the start to the end node of this part of the distribution network is approximately 0.08 p.u. The voltage rise in the 35 kV line between nodes 1 and 43 is approximately 0.03 p.u. The voltage rise in the 35/20 kV distribution transformer between nodes 43 and 44 is 0.04 p.u. The voltage rise in the 20 kV part of the distribution network between nodes 44 and 49 is 0.01 p.u. It is clear that the voltage rise in the distribution transformer represents approximately half of the overall voltage rise. It is also important to note that the voltage rise in the distribution transformer is caused not only by reverse power flows from the distribution network between nodes 44 and 49 but also by reverse power flows from all other distribution lines that are modeled after that distribution transformer. The results for Cases 1 to 3 show that with the implementation of the presented method, the voltage rises decreased. In Case 3, the algorithm even recommended a small voltage drop in the distribution transformer in order to keep the voltage in the remaining part of the network close to the desired one. Therefore, modeling distribution transformers is very important in voltage rise analysis.

4. Conclusions

This paper proposed a novel general approach for analyzing and mitigating voltage rise issues in distribution networks where the entire system is accurately modeled and various elements that contribute to voltage rises are unified into a single method for the mitigation of voltage rises on multiple voltage levels of a distribution network. The proposed method may prove beneficial in the analysis and mitigation of voltage rises caused by low power demand and very long underground cable lines, which create a significant amount of reverse reactive power flow. The primary objective of this paper was to reduce voltage rises. The proposed method has been demonstrated in the analysis of voltage rise issues within a real distribution network operating under these conditions, with a focus on exploring potential solutions.

The distribution system model used in this paper contains MV lines at two voltage levels and distribution transformers that are connections between these two voltage levels. Through simulations, it was established that by implementing hourly power factor control of the DG unit, voltage rise issues could be mitigated. The maximum voltage rise was reduced, but only from 1.13 p.u. to 1.11 p.u., during a critical part of the day when the demand is lowest. Another case showed that precise allocation and sizing of fixed shunt reactors can also decrease reactive power flows and reduce voltage rises. The maximum voltage rise during a critical part of the day was reduced from 1.13 p.u. to 1.07 p.u.

The best results in the mitigation of voltage rises are achieved by combining the allocation of new shunt reactors and the control of the power factor of the existing DG. In this case, the voltage in all nodes in the part of the network with the installed DG is held very close to the set value of 1.05 p.u. The fixed shunt reactor consumes a large amount of reactive power, while changing the DG power factor precisely injects the remaining reactive power of the DG unit, through which the transformer voltage is controlled during different time intervals.

The scientific innovations of this paper are summarized as follows:

(1)

In this paper, we proposed the use of a new method for an improved approach to evaluating voltage rise, where the entire system is more accurately modeled and various elements that contribute to voltage rise are unified into a single method for the mitigation of voltage rises. In addition to the usual modeling of distribution lines, the method precisely models transformers due to their significance for voltage rises. It also models distribution lines at two voltage levels, not just at one. The method also unifies and accurately models the components of distribution lines responsible for creating reverse flows of capacitive reactive power, particularly the parallel branches that inject capacitive reactive power.

(2)

Unlike traditional methods that often address voltage rise issues in isolation or at a single voltage level, our method provides an integrated approach that considers multiple voltage levels and unifies various mitigation elements into a coherent strategy. This holistic modeling significantly enhances the accuracy and effectiveness of voltage rise mitigation.

(3)

In the analysis of voltage rises caused by low power demand and long underground cable lines that generate reverse reactive power flows, it is important to model distribution transformers because the transformers are the location where voltage rise predominantly occurs. The research showed that in the selected network example, which is based on a real distribution network, in the moments of the highest voltage increase, half of the voltage rise occurs in the transformer.

(4)

It is possible to mitigate voltage rises with the use of shunt reactors or DG power factor control. The research results based on a real example showed that the voltage at 3 a.m. could be decreased from 1.13 to 1.11 p.u. with DG power factor control and from 1.13 to 1.07 p.u. with shunt reactors. However, the best results are achieved by combining the allocation of new shunt reactors and hourly control of the power factor of DGs. In this case, the highest voltage is decreased very close to the desired value of 1.05 p.u. The best results in mitigation of voltage rises are achieved by using a fixed shunt reactor that consumes a large amount of reactive power and changing the DG power factor to precisely inject the remaining reactive power of the DG unit.

Future research should include testing the obtained results using other optimization techniques and focus on influencing voltage rise prevention during the development planning of distribution networks. Strategies such as reducing the cross-sectional area of cable conductors to minimize transverse capacitance and planning overhead lines can effectively decrease reverse capacitive currents, thereby mitigating voltage rises without the need for additional investments in ShRs or DG power factor control. Moreover, evaluating the impact of integrating new DGs on voltage conditions and optimizing their integration into existing networks are important areas for further investigation. Exploring network reconfiguration possibilities also offers promising avenues for reducing voltage rises in distribution systems. Additionally, it is necessary to investigate how excess reverse reactive power affects higher-level transmission networks. Given that during the analysis of this study, the authors noted frequent instances of elevated input voltages from the transmission network, voltage rise represents a problem that extends across multiple voltage levels and should thus be addressed accordingly. This study provides a solution for analyzing the voltage rise issue on multiple voltage levels of a network.