Allocation of Series Capacitors Bank in Electric Power Distribution Systems

Abstract

To ensure the quality of the energy in distribution networks, configurations of voltage regulators (VRs) and Shunt Capacitor Banks (BCshunts) are traditionally used along the feeders. Series Capacitors Bank (BCseries) along the feeders can also help to improve the power quality. In this paper, a strategy is proposed based on a mathematical formulation for an optimization problem where the aim is to allocate and size BCshunts and VRs (as traditionally implemented in the electricity sector) together with BCseries in distribution networks. These devices are allocated using Genetic Algorithm together with Multiperiod Optimal Power Flow. Using the proposed strategy, it is analyzed the network performance by studying: (i) the allocation of VRs and BCshunts simultaneously, and (ii) the allocation of VRs, BCshunts and BCseries simultaneously. The objective function is based on the minimization of active power loss costs, penalties for voltage violations, and equipment costs. Results are presented for a 90-bus radial distribution system which show that the use of BCseries is suitable for feeders with abrupt load variations, and that this approach competes positively with the simultaneous allocation of VR and BCshunt devices, with shorter investment amortization times and improved quality indices.

Keywords:
series capacitor bank; shunt capacitor bank; voltage regulators; distribution networks; genetic algorithms; multiperiod optimal power flow

HIGHLIGHTS

Allocation of Capacitors in Series in electric power distribution systems

Genetic Algorithm (GA) and Non-Linear Multi-Period Optimal Power Flow (NLMOPF)

Electric power distribution systems, 90 buses, used to test the model

Analysis of impact on steady state operation

INTRODUCTION

The continuous growth in electrical load means that investment is required to maintain proper operation. This investment may include the installation of new equipment with a good cost/benefit ratio for the installation.

The costs must be carefully evaluated in addition to the benefits, as they can support decision-making regarding future investments. From this point of view, it becomes interesting to evaluate the operation of a network following the insertion of devices that can improve performance, such as series capacitors (BCseries), shunt capacitors (BCshunts) and voltage regulators (VRs).

There are several works in which the allocation of BCshunts and VRs has been optimized simultaneously, such as [¹,²], in which Genetic Algorithms (GA) were used, and [³], in which an optimization problem was solved by combining GA with a single-period Optimal Power Flow with the aim of minimizing the costs of losses, voltage profile violations, voltage drops and equipment. In [⁴], a linearized optimization problem was proposed, which was solved with conventional mixed-integer linear programming software.

In [⁵], a single-period OPF was used to solve the electrical problem and to determine the taps for the VRs by minimizing the specified voltage deviation, while GA was used to solve the simultaneous allocation problem by minimizing the power factor deviation of substation. The authors of [⁶] proposed a similar approach to solving the problem using OPF and GA, but which included voltage stability analyses.

Most of the articles that have dealt with the installation of BCseries have used heuristic techniques; for example, in [⁷], a voltage adjustment was performed with BCseries on a feeder with a voltage of 13.8 kV and 60 irrigation motors. These motors were being automatically turned off due to the occurrence of voltage sags (<10%), and the previously installed VRs and BCshunts were unable to adjust the voltage profile. A BCseries capacitive reactance designed had 117.29% greater than the inductive reactance upstream of the installation point, to overcome these profile and disconnection problems.

From a search of the literature, it was not found works that specifically dealt with the optimal allocation of BCseries in distribution networks. This article aims to address this gap by proposing an optimization method for allocating BCseries simultaneously with BCshunts and VRs, in a similar way to the approach proposed in [³] for the allocation of BCshunts and VRs.

The main differences of this article are:

• This work uses a Multi-period Optimal Power Flow (MOPF) to simulate the effect of load over a day, rather than a single-period Optimal Power Flow (OPF) used in [³]. In [³], a single-period OPF was executed three times to evaluate the allocation performance for light, medium and heavy loads. In this article, the single-period OPF is replaced by a MOPF, as proposed in [⁸] to simulate the network over 24 hours with hourly discretization, optimizing the voltage regulator taps and obtaining the network status every hour.

• BCseries are added to the traditional allocation process of BCshunts and VRs;

• In [³], the evaluation criteria for the allocations involved minimization of the costs of losses, voltage profile violations, voltage drop and equipment costs. As described in [⁶], a stability index that was developed for distribution networks will be also evaluated in this article.

The main objective of this study is to develop a computational program to allocate and size BCseries in distribution networks together with other devices such as BCshunts and VRs.

In the following section, a brief description of BCseries, BCshunts and VRs used in distribution networks is presented; followed by the proposed optimization problem formulation, results and some conclusions.

Voltage and reactive power regulator devices

In this section, it is succinctly described the most important technologies for voltage regulation and reactive power provision used in energy distribution networks, such as BCseries, BCshunts, and VRs.

The fundamental function of capacitors, whether in a series (BCseries) or shunt configuration (BCshunts), is to regulate the voltage and reactive power flows at the point where they are installed. A shunt capacitor does this by changing the power factor of the load, while a series capacitor does by compensating the upstream inductive series reactance of the circuit.

Capacitors connected in series in the distribution line have been used to a very limited extent in distribution circuits, as they represent a more specialized type of equipment with a limited range of application [⁸]. BCseries can provide an automatic and instantaneous voltage increase as the load increases, in addition to reactive compensation and voltage control when the load changes abruptly [⁹]. For example, when starting a motor, it demands a large initial current, up 8 times nominal current, with low power factor and causes a momentary voltage sag across the distribution network feeder. This voltage sag is sudden and lasts a few seconds, until the motor reaches the rated speed. This load disturbance happens not only with motors, but also with welding equipment, furnaces and other fluctuating loads. Traditional solutions to solve the voltage fluctuations, as re-conducting distribution line, construction of a new substation or installation of other compensation equipment, are not successful. However, the installation of BCseries can represent a more economical solution to this voltage problem.

The installation of BCseries in electrical distribution systems is cost-effective, as it enables a more immediate response to voltage drops and allows large loads to connect to the electrical system (such as large induction motors). Although it can bring some challenges and require studies in order to avoid electrical resonance that can cause damages to another equipment and loads along the feeder [¹⁰], besides the complexity protection systems.

The Brazilian Association of Technical Standards has issued NBR 8763 - Series Capacitors for Power Systems [¹¹] establishes requirements relating to nominal characteristics, tests, and performance, as well as instructions for installation and operation. However, this standard does not include criteria for sizing capacitors in series.

The most significant benefits of series capacitors are their fast dynamic response. It stays connected all time on the network, meaning that there is a need to evaluate their influence within the classical BCshunts and VRs, in steady-state operation, along typical days.

BCshunts are widely used in distribution systems. They provide reactive power or current to neutralize the current out-of-phase component required by an inductive load, thereby improving any unbalance in load currents [¹²]. The application of a BCshunt implies that the magnitude of the source current can be reduced, the power factor can be improved, and the voltage drop between the emitter terminal and the load is also reduced.

A VR is a device that acts when the voltage varies outside the specified limits in an electrical energy distribution network. Its objective is to compensate the voltage to pre-defined level. Its robustness, efficiency and ease of use mean that this device is widely used in electrical distribution systems in which permanent voltage regulation is desired [¹²]. The positioning of regulators on distribution feeders must be carried out in such a way that the voltage profile is as close as possible to the voltage adopted as a reference.

As described in the Introduction, there are consolidated methods for allocating VRs and BCshunts, but not for the allocation of BCseries alone or for the allocation of BCseries, VRs and BCshunts simultaneously.

In the next section, it is proposed a method of allocating and sizing all these devices individually or combined. This makes it possible to analyze the operational impacts of each installation configuration on the electrical grid in a permanent regime. The issue of properly and systematically allocating and sizing these devices involves the adjustment of many discrete and continuous variables, which leads to a combinatorial explosion. Following a scheme in the literature [³], it is developed a hybrid optimization problem in which artificial intelligence techniques such as the GA are used for allocation, and traditional mathematics approaches such as MOPF are used to evaluate the solutions obtained.

Optimization problem for the allocation of capacitor banks and voltage regulators

The optimization problem proposed in this work for the allocation of capacitor banks (shunt and series) and VRs aims to minimize the cost of electrical losses, the voltage violation costs, and the equipment acquisition costs. It involves both binary and continuous variables.

The binary optimization variables are:

• The positioning of the BCshunts (i.e., which buses are used to install the BCshunts), whether they are fixed or automatic, and their dimensions. This power selection is sized from a pre-established list of commercial options that contains the dimensions and the costs of BCshunts.

• The positioning of the VRs (i.e., which lines are used to install the VRs). The sizing and costs are obtained from a network analysis carried out following the installation of the VRs. This analysis provides the nominal current that circulates through the VR post-installation, which enables suitable values of the nominal power and commercial cost to be calculated.

• The positioning of the BCseries (i.e., which lines are used to install the BCseries) and their capacitive reactance values. The capacitive reactance is selected from a pre-established list of common values that contain the dimensions and costs of the BCseries.

These binary variables are optimized through the minimization of an evaluation function, which is solved via GA. This evaluation function is composed of five criteria: the acquisition cost for each of the three devices, plus the electrical losses and voltage violations. Each individual generated by the GA is evaluated with a MOPF, which optimizes the network's electrical losses based on the solution to the nonlinear power flow equations that correspond to the network's active and reactive power balances. The electrical quantities are monitored to satisfy the operational characteristics of the network, such as the voltage magnitude limits and power flows that circulate through the network branches over 24 hours.

The MOPF is used to calculate the continuous variables of the problem, such as the voltage magnitudes of all buses, voltage regulator taps that help maintain the voltage profile within operational limits, and the injection of active and reactive power by the substation. With the values of voltage magnitudes, the electrical losses and the sizes of the voltage violations are calculated, which are used to calculate the evaluation function of GA and carry out analyses.

The MOPF formulation applied here was inspired by a nonlinear formulation developed for distribution networks that minimizes electrical losses and battery degradation costs, as described in [⁸]. This formulation was adapted to exclude the battery charging and discharging processes that appeared in the original version.

In summary, the basic premises of the work are:

• allocation of VRs, BCshunts (fixed BC and automatic BC) and BCseries;

• BC fixed are connected along all day and automatic BCs are disconnected during light level of load (from 1 am until 8 am).

• daily study horizon with hourly discretization to consider hourly modulation of load and solar generation;

• non-linear representation of the network;

• the continuous variables are obtained via MOPF that minimizes electrical losses, calculate the voltage phasor, complex power injected by the substation and adjustment of VR taps;

• the rules for voltage limits used are based on Module 8 of the Electrical Energy Distribution Procedures in the National Electric System - PRODIST [¹³];

• the binary problem is related to the positioning and size of VRs, BCshunts (fixed or automatic) and BCseries which are solved via AG.

In the next section, the formulation of the GA is described, and the details of the MOPF formulation are presented.

Evaluation function

The proposed evaluation function of GA, denoted as fitness, has four terms, as follows:

$fitness=\min \left[{\omega }_p*f_1+{\omega }_v*f_2+{\omega }_{BCshunt}*f_3+{\omega }_{RT}*f_4++{\omega }_{BCseries}*f_5\right]$ (1)

where $fitness$ is the function to be minimized by the GA; $f_1$ is the cost of active power losses; $f_2$ is the cost of violations of voltage limits (as in [³]); $f_3$ is the cost of the BCshunts; $f_4$ is the cost of the VRs; $f_5$ is the cost of the BCseries; ${\omega }_p$ is the weight of $f_1$; ${\omega }_v$ is the weight of $f_2$; ${\omega }_{BCshunt}$ is the weight of $f_3$; ${\omega }_{RT}$ is the weight of $f_4;$ and ${\omega }_{BCseries}$ is the weight of $f_{5.}$

The function $f_1$ represents the total cost of active power losses calculated over 24 hours, according to the results obtained via MOPF:

$f_1={\displaystyle \sum }_{i=1}^{np}\left(Losse{s}_i*\mathrm{Cos}tlosses\right)$ (2)

where np is the number of periods (np=24); $i$ is the period of the day; $Losses$_i represents the total active loss during period i; and$\mathrm{Cos}tlosses$ is the value of the energy tariff in $/kWh.

Module 8 of the Electrical Energy Distribution Procedures in the National Electric System - PRODIST [¹³] contains rules for voltage limits. Table 1 presents these values and shows the rules for obtaining the voltage violations used in the mathematical formulation [³].

Table 1
Service voltage classification (adapted from [¹³] and [³])

Classification	Voltage variation range	Violation
$Adequate$	$0.95\left[pu\right]\le V_{bus}\le 1.05\left[pu\right]$	$violatio{n}_k=0$
$\mathrm{Pr}ecarious$	$0.90\left[pu\right]\le V_{bus}\le 0.93\left[pu\right]$	$violatio{n}_k=0.93-V_k$
	$V_{bus}<0.90\left[pu\right]$	$violatio{n}_k=0.90-V_k$
$Critical$	$V_{bus}>1.05\left[pu\right]$	$violatio{n}_k=V_k-1.05$

For each simulation period, the buses that violate the voltage limits are identified as $violatio{n}_k$. Each $violatio{n}_k$ is calculated as the difference between the minimum limit and voltage magnitude at the time when violation occurs (as shown in Table 1).

The adequate classification was used to calculate the violations.

The function $f_2$ corresponds to the total voltage violation cost during a day. It is calculated as the product of the total violation of the voltage limits and the fixed cost, $\mathrm{Cos}tViolac,in$ $/Vh [³]:

$f_2={\displaystyle \sum }_{k=1}^{nb.np}violatio{n}_k*\mathrm{Cos}tViolac$ (3)

For each period, only the buses identified with voltage violated are computed in eq. (3).

The BCshunt costs, $f_3$, depends on the type of capacitor (fixed or automatic) and its total reactive power size [³]:

$f_3={\displaystyle \sum }_{bbc\_shunt\in \Omega \_shunt}\left(\left(Fix{o}_{bbc\_shunt}*\mathrm{Cos}{t}_{fix}\right)+\left(Aut{o}_{bbc\_shunt}*\mathrm{Cos}{t}_{auto}\right)\right)$ (4)

where $bbc\_shunt$ indicates the buses on which the BCshunts are installed; $\Omega \_shunt$ is the set of buses with BCshunt installed; $Fi{x}_{bbc\_shunt}$ indicates the installation of a fixed BCshunt at bus $bbc$; $Cust{o}_{fix}$ is the cost of the fixed BCshunt that is installed; $Aut{o}_{bbc\_shunt}$ indicates the installation of an automatic BCshunt at bus $bbc$; and $\mathrm{Cos}{t}_{auto}$ is the cost of the installed BCshunt.

The VR cost, $f_4$, depends on the current that flows through it. As these devices are installed in an open delta configuration, two single-phase VR units are required [³], meaning that the cost function for the VRs is:

$f_4=2*{\displaystyle \sum }_{lrt\in \Psi }\mathrm{Cos}{t}_{lrt}$ (5)

where $lrt$ indicates the lines where the VRs are installed; $\Psi$ is the set of buses with VRs installed; and $Cust{o}_{lrt}$ is the cost of the VR installed along line $lrt$.

The cost of BCseries, $f_5$, depends on the allocated capacitive reactance:

$f_5={\displaystyle \sum }_{bbc\_series\in \Omega \_series}\left(\mathrm{Cos}{t}_{bbc\_series}\right)$ (6)

where $bbc\_series$ indicates the line on which the BCseries is installed, and $\Omega \_series$ is the set with the BCseries installed.

The evaluation function in Eq. (1) is calculated for each of the individuals generated by the GA, which stands out for its ability to handle multi-criteria optimization problems with many local minima. The GA applies a search mechanism in which the best individuals survive, using probabilistic transition rules [¹⁴].

The codification of the individuals is presented in Table 2. The first template gives information on the location of the VRs that can be allocated, the second template gives the location, type, and dimension of the BCshunts, and the third template gives the location and capacitive reactance of the BCseries.

Table 2
General codification of the individuals

Codification of VRs	Codification of BCshunts			Codification of BCseries
Position of the vector composed of candidate lines for allocation	Position of the vector composed of candidate buses for allocation	Type of BCshunt (fixed or automatic)	Size of BCshunt	Position of the vector composed of candidate lines for allocation	Value of the capacitive reactance
Template for VRs	Template for BCshunts			Template for BCseries

The first gene of the individual represents the lines on which the VRs can be installed and is formed of a binary sequence of nbits_VR_local bits. Once converted to decimal format, this sequence represents a position in the vector of candidate lines to which VRs can be allocated. The maximum number of VRs that can be allocated is nVR, meaning that the number of bits required for the total VR template is (nbits_VR_local * nVR).

The VR is not sized directly by the AG, that is, via its decodification process, but indirectly, by the decodification of its location. Once the installation location is known, the current that circulates through the VR is calculated and this value is used to size the commercial value and thus be able to evaluate the cost of the equipment (according to Table 5).

The BCshunt coding is composed of three parts. The first is formed of a binary sequence of nbits_BCshunt bits, which is converted to decimal to represent a position in the vector of candidate buses to which BCshunts can be allocated. The second part is formed of one bit that defines the type of BCshunt that will be installed, where a value of zero represents a fixed BC and a value of one represents an automatic BC. The third part corresponds to the size of the BCshunt (nominal three-phase power). This part of the coding is formed of nbits_BCshunt_dim bits. As single-phase system modeling is used, this value is divided by three during the MOPF simulations.

The BC fixed are connected along all the day and the automatic BCs are disconnected during light level of load (from 1 am until 8 am).

The maximum number of BCshunts that can be allocated is nBCshunt. Thus, the number of bits required for the total BCshunt template is [(nbits_BCshunt_local+ nbits_BCshunt_dim +1) * nBCshunt].

The BCseries coding is composed of two parts, the first of which is formed of a binary sequence of nbits_BCseries bits. When converted to decimal format, this sequence represents a position in the vector of candidate lines to be allocated BCseries. The second part corresponds to the capacitive reactance value for the BCseries, and is formed of nbits_BCseries_dim bits. The maximum number of BCseries that can be allocated is nBCseries, meaning that the number of bits required for the total BCshunt template is [(nbits_BCseries_local+ nbits_BCseries_dim) * nBCseries].

Furthermore, since the installation of two capacitors in series can cause electromagnetic transients, it is defined a function that detects whether two or more BCseries have been allocated to the same branch. If this occurs, the fitness function is discarded; that is, only solutions with one BCseries per branch are admitted.

Each individual in the population represents a possibility of BCshunts, VRs and BCseries allocations, and requires a total template with nbits:

$nbits=[nbits\_VR\_local*nVR\left]+\right[(nbits\_BCshunt\_local+nbits\_BCshunt\_dim+1)*nBCshunt]+[(nbits\_BCseries\_local+nbits\_BCseries\_dim)*nBCseries).$ (7)

When the individuals for each generation have been created, they are decoded, i.e., the information on the devices to be allocated to selected buses and lines of the system is found. Each configuration must be simulated by the MOPF, which performs the power dispatches and voltage profile calculations, as these are used to calculate the evaluation function given in Eq. (1).

All values of the fitness are normalized to the range [0, 1], with none of them overlapping the others. The weight values have only an enabling function. For example,

• If ${\omega }_p=1$, ${\omega }_v=1$, ${\omega }_{BCshunt}=0$, ${\omega }_{RT}=0$ and ${\omega }_{BCseries}=1$, this means that only BCseries are allocated, that is, the installation cost of BC series is considered to be zero and only the coding template relating to the allocation of BCseries is formed, as illustrated by Table 3.

• If ${\omega }_p=1$, ${\omega }_v=1$, ${\omega }_{BCshunt}=1$, ${\omega }_{RT}=1$ and ${\omega }_{BCseries}=1$, this means that VRs, BCshunts and BCseries are allocated, as Table 2.

Table 3
General codification of the individuals

Codification of BCseries
Position of the vector composed of candidate lines for allocation	Value of the capacitive reactance
Template for BCseries

In the next section, it is described the MOPF formulation used.

MULTIPERIOD OPTIMAL POWER FLOW FORMULATION

MOPF defines optimal operating points for a given set of periods, which are considered simultaneously. In this work, the MOPF optimizes the losses of the electrical grid simultaneously for all 24 periods of a daily planning horizon, and satisfy the requirements for nonlinear active and reactive power balance for the network [⁸].

Input parameters

The vectors $Pd$ and $Qd$ represent the active and reactive power, respectively, for the loads demanded over np periods of the daily schedule. The vectors $Pgsun$ and $Qgsun$ represent the active and reactive power, respectively, that is generated by photovoltaic (PV) systems over np periods. It is assumed that the power factor of the solar power plant is equal to 0.92 capacitive. These vectors have dimensions (nb.np x 1), where nb is the number of buses and np is the number of periods (np = 24 for hourly programming):

$Pd=\left[\begin{array}{c}P{d}_1^1\\ ⋮\\ \begin{array}{c}P{d}_{nb}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}P{d}_1^{np}\\ ⋮\\ P{d}_{nb}^{np}\end{array}\end{array}\end{array}\end{array}\right],Qd=\left[\begin{array}{c}Q{d}_1^1\\ ⋮\\ \begin{array}{c}Q{d}_{nb}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}Q{d}_1^{np}\\ ⋮\\ Q{d}_{nb}^{np}\end{array}\end{array}\end{array}\end{array}\right],Pgsun=\left[\begin{array}{c}Pgsu{n}_1^1\\ ⋮\\ \begin{array}{c}Pgsu{n}_{nb}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}Pgsu{n}_1^{np}\\ ⋮\\ Pgsu{n}_{nb}^{np}\end{array}\end{array}\end{array}\end{array}\right],Qgsun=\left[\begin{array}{c}Qgsu{n}_1^1\\ ⋮\\ \begin{array}{c}Qgsu{n}_{nb}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}Qgsu{n}_1^{np}\\ ⋮\\ Qgsu{n}_{nb}^{np}\end{array}\end{array}\end{array}\end{array}\right]$ (8)

where $Pd$ is the vector of active power demand, with dimensions (nb.np x 1); $P{d}_i^k$ represents the active power demanded at bus i and period k; $Qd$ is the vector of the reactive power demand, with dimensions (nb.np x 1); $Q{d}_i^k$ represents the reactive power demanded at bus i and period k; $Pgsun$ is the vector of active PV power, with dimensions (nb.np x 1); $Pgsun{l}_i^k$ represents the active PV power injected at bus i and period k; $Qgsun$ is the vector of reactive PV power, with dimensions (nb.np x 1); and $Qgsu{n}_i^k$ represents the reactive PV power injected at bus i and period k.

The maximum and minimum active and reactive power limits of the transformer at the substation (connected to bus 1) are given by:

$Psu{b}_{\max }^{}=\left[\begin{array}{c}Psub\_{\max }_1^1\\ ⋮\\ \begin{array}{c}0\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}Psub\_{\max }_1^{np}\\ ⋮\\ 0\end{array}\end{array}\end{array}\end{array}\right],Qsu{b}_{\max }^{}=\left[\begin{array}{c}Qsub\_{\max }_1^1\\ ⋮\\ \begin{array}{c}0\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}Qsub\_{\max }_1^{np}\\ ⋮\\ 0\end{array}\end{array}\end{array}\end{array}\right],Psu{b}_{\min }^{}=\left[\begin{array}{c}Psub\_{\min }_1^1\\ ⋮\\ \begin{array}{c}0\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}Psub\_{\min }_1^{np}\\ ⋮\\ 0\end{array}\end{array}\end{array}\end{array}\right],Qsu{b}_{\min }^{}=\left[\begin{array}{c}Qsub\_{\min }_1^1\\ ⋮\\ \begin{array}{c}0\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}Qsub\_{\min }_1^{np}\\ ⋮\\ 0\end{array}\end{array}\end{array}\end{array}\right]$ (9)

where $Psu{b}_{\max }^{}and$ $Psu{b}_{\min }^{}$ are vectors of the maximum and minimum active power injected by the substation, respectively, with dimensions (nb.np x 1); $Psub\_{\max }_i^k$ and $Psub\_{\min }_i^k$ are the maximum and minimum limits on the active power injected by the substation at bus 1 and period k, respectively; $Qsu{b}_{\max }^{}$ and $Qsu{b}_{\min }^{}$ are vectors of the maximum and minimum reactive power injected by the substation, with dimensions (nb.np x 1); and $Qsub\_{\max }_i^k$ and $Qsub\_{\min }_i^k$ are the maximum and minimum limits on the reactive power injected by the substation at bus 1 and period k, respectively.

It is needed to define the maximum and minimum limits on the voltage magnitudes, and the voltage regulator taps, if these are allocated:

$V_{\min }^{}=\left[\begin{array}{c}v{\min }_1^1\\ ⋮\\ \begin{array}{c}v{\min }_{nb}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}v{\min }_1^{np}\\ ⋮\\ v{\min }_{nb}^{np}\end{array}\end{array}\end{array}\end{array}\right],V_{\max }^{}=\left[\begin{array}{c}v{\max }_1^1\\ ⋮\\ \begin{array}{c}v{\max }_{nb}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}v{\max }_1^{np}\\ ⋮\\ v{\max }_{nb}^{np}\end{array}\end{array}\end{array}\end{array}\right]a_{\min }^{}=\left[\begin{array}{c}areg{\min }_1^1\\ ⋮\\ \begin{array}{c}areg{\min }_{nl}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}areg{\min }_1^{np}\\ ⋮\\ areg{\min }_{nl}^{np}\end{array}\end{array}\end{array}\end{array}\right],a_{\max }^{}=\left[\begin{array}{c}areg{\max }_1^1\\ ⋮\\ \begin{array}{c}areg{\max }_{nl}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}areg{\max }_1^{np}\\ ⋮\\ areg{\max }_{nl}^{np}\end{array}\end{array}\end{array}\end{array}\right]$ (10)

where $v{\min }_i^k$ e $v{\max }_i^k$are the maximum and minimum limits on the voltage magnitude at bus i and period k; respectively; and $are{g}_{\max }{}_j^{k}and$ $are{g}_{\min }{}_j^k$are the maximum and minimum limits on the voltage regulator taps for line j and period k, respectively.

The locations of the VRs and the series and shunt capacitors are obtained by decoding each individual generated by the algorithm. If BCshunts are allocated, decoding gives information on the buses and the values of the capacitive susceptance of the BCshunts, represented as the vector $B$. If capacitors connected in series are allocated, decoding gives information on the lines on which the capacitive reactance values of the BCseries are allocated, denoted by the vector $C$, as follows:

$B=\left[\begin{array}{c}{b}_1^1\\ ⋮\\ \begin{array}{c}{b}_{nb}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}{b}_1^{np}\\ ⋮\\ b_{nb}^{np}\end{array}\end{array}\end{array}\end{array}\right],C=\left[\begin{array}{c}{c}_1^1\\ ⋮\\ \begin{array}{c}{c}_{nl}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}{c}_1^{np}\\ ⋮\\ c_{nl}^{np}\end{array}\end{array}\end{array}\end{array}\right]$ (11)

where $B$ is a vector representing the susceptance capacitive of the BCshunts allocated by the GA (obtained from decoding individuals), with dimensions (np.nb x 1), and $C$ is a vector of the reactance capacitive of the BCseries allocated by the GA, with dimensions (np.nl x 1).

The other input data are parameters of the system lines and reference bus (or substation).

Control variables

The phasor voltage is represented in rectangular form following the approach set out in [¹⁵], and is represented in a simplified form by the phasor ${\dot{V}}^{}.$ The variable related to voltage regulator taps is represented by the vector $a$.

The vector $Psub$ represents the active power injected by the substation, with dimensions [nb.np x 1], and $Qsub$ represents the reactive power injected by the substation that energizes the distribution network, with dimensions [nb.np x 1]:

${\dot{V}}^{}=\left[\begin{array}{c}{\dot{V}}_1^1\\ ⋮\\ \begin{array}{c}{\dot{V}}_{nb}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}{\dot{V}}_1^{np}\\ ⋮\\ {\dot{V}}_{nb}^{np}\end{array}\end{array}\end{array}\end{array}\right],a=\left[\begin{array}{c}are{g}_1^1\\ ⋮\\ \begin{array}{c}are{g}_{nl}^1\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}are{g}_1^{np}\\ ⋮\\ are{g}_{nl}^{np}\end{array}\end{array}\end{array}\end{array}\right]Psub=\left[\begin{array}{c}Pg{h}_{sub}^1\\ ⋮\\ \begin{array}{c}0\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}Pg{h}_{sub}^{np}\\ ⋮\\ 0\end{array}\end{array}\end{array}\end{array}\right],Qsub=\left[\begin{array}{c}Qgh{f}_{sub}^1\\ ⋮\\ \begin{array}{c}0\\ ⋮\\ \begin{array}{c}\\ ⋮\\ \begin{array}{c}Qg{h}_{sub}^{np}\\ ⋮\\ 0\end{array}\end{array}\end{array}\end{array}\right]$ (12)

where ${\dot{V}}^{}$is a vector of voltage phasors for all buses and all periods, with dimensions (nb.np x 1); and $a$ is vector of voltage regulator taps, with dimensions (nl.np x 1).

It is clarified that $Psub,Qsub,$ $Psu{b}_{\max }^{},$ $Psu{b}_{\min }^{},Qsu{b}_{\max }^{}$ and $Qsu{b}_{\min }^{}$ are vectors with values zeros, with exception of the first position of each period, which represents the power injections and limits of the substation.

The optimization criterion minimizes the total electrical loss of the distribution network, and the final formulation of the MOPF is as follows:

$\min fperdas=u´.c\left(Psub+Pgsol-Pd\right)$ (13)

subject to:

$Psub+Pgsol-Pd=real\left\{diag\left[\dot{V}.\dot{[Y}\left(a,C\right).\dot{V}{]}^{*}\right)\right\}$ (14)

$Qsub+Qgsol+diag\left(\left|{\left(\dot{V}\right)}^2\right|.B\right)-Qd=imag\left\{diag\left[\dot{V}.\dot{[Y}\left(a,C\right).\dot{V}{]}^{*}\right)\right\}$ (15)

$\mathrm{0,7}*V_{\min }^{}\le \left|\dot{V}\right|\le V_{\max }^{}$ (16)

$a_{\min }^{}\le a\le a_{\max }^{}$ (17)

$Psu{b}_{\min }^{}\le Psub\le Psu{b}_{\max }^{}$ (18)

$Qsu{b}_{\min }^{}\le Qsub\le QPsu{b}_{\max }^{}$ (19)

where u is a unitary vector; $c\left(Psub+Pgsol-Pd\right)$ is the energy cost function; and $\dot{Y}\left(a,C\right)$ is the bus admittance matrix, with dimensions (np.nb x np.nb).

Eqs. (14) and (15) are the active and reactive power balance equations, respectively. The constraints in Eqs. (16) to (19) represent the restrictions in the form of inequalities. To ensure convergence of each solution decoded by the GA, the voltage magnitude constraints in Eq. (16) are relaxed by 70%.

The problem represented by Eqs. (13) to (19) is solved by the interior point method in its primal-dual version [¹⁵]. The output variables used to calculate the GA evaluation function in Eq. (1) are the nodal voltage profile, which is used to calculate the total electrical loss of the network for the entire period, the voltage violations, and the sizing of the nominal power of the allocated VRs.

The algorithm for solving the optimization problem proposed in the previous sections can be summarized as follows:

Step 1: Input the electrical network parameters and the physical limits (impedance of lines and transformer$,Pd$,$Qd,Pgsun,Qgsun,Psu{b}_{\max }^{},Qsu{b}_{\max }^{},Psu{b}_{\min }^{},Qsu{b}_{\min }^{},V_{\min }^{},V_{\max }^{},a_{\min }^{},a_{\max }^{},B,C$ (that are input data of MOPF) and network topology.

Step 2: Define the combination of equipment to be allocated (BCshunts and/or VRs and/or BCseries), via weight qualification (${\omega }_p$, ${\omega }_v$, ${\omega }_{BCshunt}$, ${\omega }_{RT}$ and ${\omega }_{BCseries}$) , and the maximum numbers of devices to be allocated (nVR, nBCshunt and nBCseries).

Step 3: Define maximum number of generations and adjustment of the GA parameters.

The suggested options for the parameters used for the genetic operators in this work developed are listed in Table 4.

Table 4
Initial parameters used for genetic operators

Features	Parameters	Configuration
Individual	Codification	Binary
	Number of bts	$nbits$
Population	Size	20 individuals
	Initial	Random
Selection	Elitism	2 individuals
	Method	Roulette
Crossing	Type	Disperse
	Tax	80%
Mutation	Type	Uniform
	Tax	10%
Stop	Criterium	500 generations

Step 4: Simulate the system without equipment, to calculate the worst losses (Eq. 2) and voltage violations (Eq. 3).

Step 5: Execute of the GA. After decoding each individual and updating the input data (as Table 2), the MOPF is executed and the values that make up the evaluation function are obtained as in Eq. (1).

Step 6: When convergence of the process has been reached, output the allocation results.

Step 7: END.

To complement the impact analysis of equipment allocations, the results of which are presented in the next section, it is also carried out a study of the voltage instability based on the index $VS{I}_k$ proposed in [¹⁶]. This index was chosen because it is specifically applicable to radial networks. According to [¹⁶], the voltage stability index $VS{I}_k$ for each bus k is restricted to the range [0, 1], where the higher the value, the better the voltage stability of the bus.

The calculation of the instability index [¹⁶] calculated for each bus in the system is formulated as follows:

$VS{I}_k=\frac{4.\left\{P_k{}^2+Q_k{}^2\right\}\left\{r_{jj}{}^2+x_{jj}{}^2\right\}}{{\left\{\left|{\dot{V}}_m{}^2\right|-2.P_k.r_{jj}-2Q_k.x_{jj}\right\}}^2}$ (20)

where $P_k$ e $Q_{k}is$ active and reactive power injected at the bus $k$, respectively, $r_{jj}$ and $x_{jj}$ are resistance and reactance of the line connecting the buses m and $k$; $m$ is the bus that feeds bus $k$; $k$ is the bus fed by bus m; and ${\dot{V}}_m$ is the voltage phasor at bus m.

Another index to analyze the impacts of equipment allocations is the Return on Investment (ROI), which is calculated, in this work, as the time (in months) to amortize the total investment.

After the optimal allocating, the daily savings obtained are calculated using the cost reduction of losses after the optimal allocation of equipment. To do this, the daily savings ($DailySavings$) are obtained calculating the cost losses ($f_1$) when no equipment is installed ($f_{1_{withoutallocation}}$) and after the installation of the selected equipment ($f_{1_{withallocation}}$):

$DailySavings=\left[\left(f_{1_{withoutallocation}}-f_{1_{withallocation}}\right)\right]$ (21)

The total cost of the optimal allocated equipment is:

$TotalEquipment\mathrm{Cos}t=f_3+f_4+f_5$ (22)

Therefore, the time (in months) to amortize the investment is:

$ROI=\frac{TotalEquipment\mathrm{Cos}t}{DailyEconomy.30}.$ (23)

RESULTS

The system used to analyze the proposed optimization problem has 90 buses and was adapted from the 69-bus system described in [¹⁷]. Twenty buses were added and coupled to the medium-voltage network using transformers [¹⁸], and a further 16 PV generation systems were added [⁵], as shown in Figure 1.

The 90-bus system has a total nominal loading of 4.4742 MW and 3.2031 Mvar. The base apparent power value used was equal to 1 MVA, with levels of 13.8 kV for medium-voltage operation and 220 V for low-voltage operation.

In all of the simulations, the maximum number of devices that could be allocated is equal to three (nVR = nBCshunt = nBCseries = 3). The energy tariff value used to calculate the cost of losses is $\mathrm{Cos}tlosses=$ 0.932 $/kWh, and the value of the voltage violation cost was $\mathrm{Cos}tViol$= 228.32 $/Vh. All of the lines were considered as candidates for the allocation of VRs and BCseries, and all of the buses were considered as candidates for the allocation of BCshunts.

Table 5 presents the power and cost values for the VRs used, as proposed in [³], which have been updated for 2024. Table 6 shows some of the power and cost values for the BCshunts used, as proposed in [³], which have been updated to 2024 values. Table 7 shows the power and cost values obtained for the BCseries [¹⁹].

Figure 1
Single-line diagram of a 90-bus system [⁵]

The results given here were obtained for a basic load profile situation with an abrupt increase in the load on bus 76 at 3 pm, simulating the entry of a large load such as an induction motor (Figure 2). This profile was adopted because it represents a situation in which studies in the literature indicate that the installation of BCseries is required.

The analyses presented here enable to compare the results from the use of conventional equipment (VRs and BCshunts) with those from the use of BCseries together with VRs and BCseries, to evaluate whether there are any advantages from considering BCseries as an alternative in distribution networks to improve the efficiency energy.

The following cases were simulated:

• Simulation 1: The system is simulated without no allocation of devices, as a reference case;

• Simulation 2: VRs and BCshunts are allocated via GA, to evaluate the results from using conventional dispositives;

• Simulation 3: VRs, BCshunts and BCseries are allocated via GA, to evaluate the results from allocating BCseries in addition to conventional diapositives, and

• Simulation 4: VRs and BCseries are allocated via GA, to evaluate the results from allocating BCseries with VRs.

Table 5
Nominal currents and VR costs (adapted from [³])

Nominal current for VR [A]	Cost of VRs [$]
50	94,000.00
100	95,000.00
150	112,000.00
200	129,000.00
250	145,250.00
300	161,750.00
350	175,750.00

Table 6
Power and prices for fixed and automatic BCshunts (adapted from [³])

Power of BCshunt [kvar]	Cost of BCshunt [$]
	$Cust{o}_{fix}$ [$]	$Aut{o}_{bbc\_shunt}$ [$]
150	7,250.00	44,708.00
300	14,500.00	89,417.00
450	16,433.00	105,447.00
600	18,125.00	106,618.00
700	19,333.00	105,689.00
900	20,542.00	107,844.00
1200	22,958.00	112,241.00
1500	25,375.00	116,725.00
1650	26,583.00	118,992.00
1800	27,792.00	121,273.00
1950	28,517.00	120,762.00
2100	29,242.00	122,419.00

Table 7
Capacitive reactance and BCseries costs (adapted from [¹⁹])

Capacitive reactance [ohms]	Cost of BCseries [$]
36	85,134.00
42	72,972.00
48	63,851.00
49	62,547.00
50	61,297.00
54	56,756.00
55	55,724.00
59	51,946.00
64	47,888.00
69	44,418.00
70	43,783.00
72	42,567.00
75	40,864.00
76	40,327.00

Figure 2
Active load curves of all the buses with a peak load on bus 76 at 3 pm

Simulation 1: No allocation of control devices

Figure 3 shows the voltage profiles of all buses in the 90-bus system over 24 hours, without the installation of VRs, BCshunts or BCseries. The buses with voltages violations are 16-28, 62-66, 72-76, 87 and 88.

Table 8 presents the values of the optimization criteria calculated for this base case, i.e., the cost of electrical losses, cost due to voltage violations, and voltage stability index (VSI) value.

Figure 3
Voltage profiles without allocation of equipment.

Table 8
Evaluation criteria with no allocation of control devices

Evaluation criteria	Cost [$]	Normalized value [pu]
Cost due to electrical losses	1,603.50	1.00
Cost due to voltage violation	166.47	1.09
Total cost	1,769.97	1.00
Average voltage stability index	0.84 pu	0.84

Simulation 2: Allocation of voltage regulators and shunt capacitors via GA

Figure 4 shows the voltage profiles for all buses of the 90-bus system over 24 hours, after the allocation of VRs and BCshunts. No buses were found to have voltage violations.

Figure 4
Voltage profile over 24 hours with allocation of VR and BCshunt via GA

Table 9 presents the results for the location, size and cost of the VRs and BCshunts allocated by the optimization problem (Simulation 2). Table 10 presents the values of the optimization criteria calculated cost for Simulation 2. In relation to the base case (Simulation 1), the value of the total loss cost decreased from R$1,603.50 to R$1,250.00 (around 22%). The total average voltage stability index value increased from 0.84 (without devices) to 0.96.

Table 9
Allocation of voltage regulators and shunt capacitors via GA

Voltage regulators allocated
Line with VR	Current [A]	Power [kVA]	Cost [$]
9	100	126.6	190,000.00
Total Cost of VR			190,000.00
Shunt capacitors allocated
Bus with BCshunt	Type	Power [kvar]	Cost [R$]
5	Fix	700	29,242.00
23	Fix	150	16,433.00
58	Fix	700	29,242.00
Total cost of BCshunt			74,917.00
Total cost of VR and BCshunt			264,917.00

Table 10
Evaluation criteria without of VRs and BCshunts via GA

Evaluation criteria	Cost [$]	Normalized value [pu]
Cost due to electrical losses	1,250.00	0.78
Cost due to voltage violation	0	0
Cost of BCshunts	74.917.00	0.08
Cost of VRs	190,000.00	0.19
Total cost	264,917.00	0.27
Average voltage stability index	0.96 pu	0.96

Simulation 3: Allocation of voltage regulators, shunt capacitors and capacitors in series through GA

Figure 5 shows the voltage profiles for all buses of the 90-bus system over 24 hours, after the allocation of VRs, BCshunts and BCseries. No buses showed voltage violations, although abrupt variations in the voltage magnitudes appeared at 3 pm, when a voltage drop was seen at bus 76.

Figure 5
Voltage profile along 24 h with VR, BCshunt and BCseries allocation via GA

Table 11 presents the locations, sizes and costs of the VRs, BCshunts and BCseries allocated as a result of solving the optimization problem (Simulation 3).

Table 11
Allocation of voltage regulators, shunt capacitors and series capacitors via GA

Voltage regulators allocated
Line with VR	Current [A]	Power [kVA]	Cost [$]
9	100	126.6	190,000.00
Total cost of VR			190,000.00
Shunt capacitors allocated
Bus with BCshunt	Type	Bus with BCshunt	Type
12	Fix	150	16,433.00
64	Fix	400	22,958.00
75	Fix	150	16,433.00
Total cost of BCshunt			55,824.00
BCseries allocated
Line with BCseries	Capacitive reactance [Ω]		Cost [$]
30	82		43,500.00
17	80		43,500.00
Total cost of BCseries			87,000.00
Total cost of VR, BCshunts and BCseries			332,824.00

Table 12 presents the values of the optimization criteria calculated cost for Simulation 3. In relation to the base case (Simulation 1), the value of the total electrical loss decreased from R$1,603.50 to R$1,120.00, (around of 30%). The total average index value increased from 0.84 (without devices) to 0.95.

Table 12
Evaluation criteria with allocation of VRs, BCshunts and BCseries via GA

Evaluation criteria	Cost [$]	Normalized value [pu]
Cost due to electrical losses	1,120.00	0.70
Cost due to voltage violation	0	0
Cost of BCshunts	55,824.00	0.06
Cost of RTs	190,000.00	0.19
Cost of BCseries	87,000.00	0.09
Total cost	332,824.00	0.33
Average voltage stability index	0.95 pu	0.95

Simulation 4: Allocation of voltage regulators and capacitors in series through GA

Figure 6 shows the voltage profiles for all buses of the 90-bus system over 24 hours, after the allocation of VRs and BCseries. No buses showed voltage violations and there were no abrupt variations in voltage magnitudes at 3pm, when a voltage drop was observed at bus 76.

Figure 6
Voltage profile along 24 h with VR and BCseries allocation via GA

Table 13 presents the locations, sizes and costs of the VRs and BCseries allocated as a result of solving the optimization problem (Simulation 4).

Table 13
Allocation of voltage regulators and series capacitors via GA

Voltage regulators allocated
Line with VR	Current [A]	Power [kVA]	Cost [$]
12	50	63.30	188,000.00
55	50	63.30	188,000.00
Total Cost of VR			376,000.00
Serie capacitors allocated
Bus with BCserie	Capacitive reactance [Ω]		Cost [R$]
63	66		43,500.00
12	82		43,500.00
Total cost of BCserie			87,000.00
Total cost of VR and BCserie			463,000.00

Table 14 presents the values of the optimization criteria calculated cost for Simulation 4. In relation to the base case (Simulation 1), the value of the total electrical loss decreased from R$1,603.50 to R$1,400.00, (around of 33%). The total average index value increased from 0.84 (without devices) to 0.88.

Table 14
Allocation of voltage regulator and series capacitor via GA

Evaluation criteria	Cost [$]	Normalized value [pu]
Cost due to electrical losses	1,400.00	0.87
Cost due to voltage violation	0.00	0.00
Total cost	1,400.00	0.87
Average voltage stability index	0.88 pu	0.88

Table 15 presents a summary of results for all of the simulations performed here.

Table 15
Electrical losses, total cost and average IET for all simulations

Simulation		Losses		Voltage violation	Total cost	VSI	Amortization
		$	pu	$	$	pu	Months
1	No allocation	1,603.00	1.00	166.00	1,769.00	0.84	-
2	VR and BCshunt allocated	1,250.00	0.78	0	264,917.00	0.96	25
3	VR, BCshunt and BCseries allocated	1,120.00	0.70	0	333,944.00	0.95	23
4	VR and BCseries allocated	1,400.00	0.87	0	463,000.00	0.88	76

From a comparison of the values of the loss costs, total cost and VSI obtained from Simulations 2, 3 and 4, as shown in Table 15, it can be observed that the lowest loss cost occurs when BCseries are inserted in conjunction with VRs and BCshunts. This reduction allows for faster amortization of total costs over the useful life of the equipment, which is an average of two years.

Some other observations that can be made are as follows:

• - The allocated VRs had the same dimensions of 126.6 kVA and were allocated on the same line (line 9) in Simulations 2 and 3;

• - BCshunts were allocated to the same branches but different buses. In Simulation 3, the BCshunt values had a reduced value for the power, as shown in Table 16;

• - The two BCseries allocated in Simulation 3 were installed on the branches with the greatest voltage drops;

• - In the simulation 4, due to the allocation of two VR, the total cost were higher.

Table 16
Equipment allocated in Simulations 2, 3 and 4

	Line with VR / Power [kVA]	Buses with BCshunt / Power [kvar]	Lines with BCseries / Capacitive reactance [Ω]
Simulation 2	9 / 126.6	5, 23 and 58 / 700, 150 and 700	No allocation
Simulation 3	9 / 126.6	12, 64 and 75 / 150, 400 and 150	30 and 17 / 82 and 80
Simulation 4	12 and 15 / 63.3 and 63.3	No allocation	63 and 12 / 66 and 82

The electrical losses and voltage violations obtained from Simulations 2, 3 and 4 were zero; in other words, all allocations were able to adjust the voltage curves over 24 hours to values above the lower voltage limit of 0.95 pu, which was found to be violated for the system without device allocation (Simulation 1).

In the cases analyzed here, with abrupt load changes, the BCseries operated adequately as voltage control devices, making corrections to the voltage drops. However, when the aim was to correct the voltage levels of an entire radial distribution system, classical control devices gave a better result, as they were able to adjust the voltages of all buses, with a slight improvement in the voltage stability indices and a lower cost. Despite the higher initial cost for the BCseries allocation of around 20% higher than for the scheme without these devices, the savings generated by the reduction in electrical losses (around 30%) shortened the time for the return on investment from 25 months to 23 months, in simulations 2 and 3.

The computational processing of the simulations was performed on a computer with an 11th generation Intel Core i7 processor, 2.8GHz, 16GB of RAM, using Windows 11 operating system and Matlab software, with the processing time values ​​shown in Table 17.

Table 17
Simulation processing times

Simulation	Configuration	Processing time
1	None allocation	≃1 min
2	VR and BCshunt allocated	4.86 h
3	VR, BCshunt and BCserie allocated	11.53 h
4	VR and BCserie allocated	2.73 h

CONCLUSION

In this article, it was formulated an optimization problem that was used to allocate BCseries to solve the traditional problem of improving the voltage profiles and reducing the electrical losses in a distribution network. This problem is traditionally solved by allocating VRs and/or BCshunts. The proposed optimization problem makes the following contributions to existing work in this area:

• - describe a new branch of study involving the optimal allocation of BCseries, as an improvement to the heuristic methods used in the sector.

• - model an optimization problem for the allocation of BCseries in addition to the usual BCshunts and VRs, which will enable studies of the impact of using BCseries in the network together with the typical VRs and BCshunts.

• - use voltage stability indices to evaluate the steady state allocations of VRs, BCshunts and BCseries.

Although the approach described here can be used to analyze allocations based on any combinations of VRs, BCshunts and BCseries, a configuration was chosen where all types of devices were allowed to be allocated. In this case, it was observed that the use of BCseries together with VRs and BCshunts led to a reduction in electrical losses that was 8% greater than that obtained when only VRs and BCshunts were used. This decrease resulted in a competitive return on investment compared with the adoption of only VRs and BCseries, thereby opening the way for further study of these devices regarding the planning of distribution networks and suggesting a greater range of possibilities for their implementation as a solution for improving energy efficiency.

The methodology proposed in this work enabled us to analyze the operational impacts of installing a combination of BCshunts, VRs and BCseries on the electrical network in the permanent regime; however, future studies of the electromagnetic transients must be carried out to finalize the planning of the network.

Acknowledgments

This research was funded by CAPES - Brazilian Federal Agency for Support and Evaluation of Graduate Education within the Ministry of Education of Brazil.

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Edited by

• Editor-in-Chief:
  Alexandre Rasi Aoki

• Associate Editor:
  Alexandre Rasi Aoki

Publication Dates

• Publication in this collection
  03 Feb 2025
• Date of issue
  2025

History

• Received
  28 May 2024
• Accepted
  30 Sept 2024