Using CPC-Based Minimizing Balancing Compensation to Reduce the Budeanu Reactive Currents Described in Extended Budeanu Theory

Abstract

This article presents principles for matching reactance parameters for minimizing balancing compensation, whose mathematical origins come from the Currents’ Physical Components (CPC) theory developed by Czarnecki. The construction of minimizing balancing compensators was considered by applying it to the concept of the extended Budeanu theory. It focuses on the possibility of compensating the Budeanu reactive current, Budeanu complemented reactive current, and both currents at the same time. In addition, the compensator also has the potential to balance the load, that is, to reduce the unbalanced current. In order to precisely illustrate the difference in the effectiveness of compensation and balancing the load, each approach to minimizing balancing compensation has its equivalent in the case of ideal compensation. The analysis of the results achieved is a comparison of the three-phase RMS values of the respective components and the current of the load at the primary load and using three approaches, with each approach divided into ideal compensation and minimizing balancing compensation. For all approaches, calculations and simulations were carried out, in which the numerical values and generated waveforms of each quantity were compiled and analyzed. The Matlab/Simulink R2023a application environment was used as computational and simulation software.

1. Introduction

The first power theories appeared to take form in the 1920s and 1930s, distinguishing two main trends. The first is based on a Fourier series [1,2,3,4,5,6], describing the energy properties of a circuit in the frequency domain by decomposing electrical waveforms into components at different frequencies. Almost in parallel, a second trend was formed, which defines energy quantities, without the use of a Fourier series, as time functions of current and voltage in the time domain [7,8,9,10,11].

The problem of the inexistence of a unified and globally accepted theory describing the energy properties of circuits with nonsinusoidal waveforms is intensified with the invention of semiconductor elements and the evolution of power electronics. Although power electronic systems have numerous advantages, they also generate negative phenomena. The increase in the number of nonlinear power electronic loads has caused a significant expansion of the quantity of higher harmonics, which negatively affects the power grid, for example, an increase in the asymmetry of supply voltage [12] transfer of capacitive reactive energy into the grid, which leads to a rise in the voltage value [13], an increase in the asymmetry of currents flowing through the network, an increase in the asymmetry of the zero sequence [14], disturbances in wireless energy transmission resulting from the commutation operation of converters [15], disturbances in the stability of the electromagnetic field [16], distortions in supply voltages in lighting circuits [17], power oscillations appearing after a disturbance caused by the lack of inertia and damping of the converter [18], a rise in power and energy losses [19], an impact on active power circulation caused by the difference in parameters between different types of generation units [20], and an increase in distortion related to higher harmonics [21].

Since the 1970s, there has been expanding interest in describing the energy properties of these devices and in methods of optimizing the power factor [22,23,24,25,26,27]. Currently, there are many concepts for studying electrical systems under distorted voltage and current waveforms. The proposed power theories are subject to debates and do not always obtain widespread acceptance.

For many decades, one of the dominant power theories was the Budeanu concept [2]. However, as Czarnecki [28] has shown, this theory is flawed. Despite the errors in this theory, it is still the main concept presented to young students of electrical engineering. An important contribution to the progress of power theory was also made by Fryze’s proposal [7], which still plays a significant role.

Currently, one of the most widely used theories (according to the author of the publication, the p-q method is a control algorithm and not a power theory—however, this is beyond the scope of this manuscript), especially in the field of active filtering, is the p-q power theory proposed by Akagi’s team [8,9] and power theory concepts related to the original p-q method along with positive [9,29,30,31] as well as negative aspects, where the main objection is the increase in the third harmonic current and its injection into the compensator current and the load current [32,33,34,35]. Nevertheless, in the opinion of many experts, the most correct interpretation of the energy properties of electrical systems and a powerful mathematical instrument is offered by the power theory based on the concept of Currents’ Physical Components, formulated by Czarnecki [3].

With the CPC power theory, it is possible to apply a minimizing balancing compensation method [36,37,38], which is used in this article. It is based on the combination of reactance parameters that minimize the reactive component of the load’s current and minimize and balance the three components of unbalance, i.e., the unbalanced current of the positive sequence, the unbalanced current of the negative sequence, and the unbalanced current of the zero sequence. Such a system is less sensitive to changes in the current–voltage conditions in the circuit and is more suitable as a passive component in hybrid filters [23,39,40,41].

Extended Budeanu theory is the most recognized and most frequently applied concept of power theory in the electrical power industry and in electrical engineering programs. However, it did not have a sufficiently clear mathematical description, especially because of the problem with the components of distortion power. The problem of the description and identification of physical phenomena has been discussed in a study [4]. The next step in the improvement in Budeanu’s power theory was to demonstrate the potential for the optimal compensation of individual components of the current [27], as described in this publication in a minimized way. It is based on modifying selected components in a way that minimizes the need for full compensation, while reducing dependence on the power supply and load conditions. Moreover, it is important to consider the economic parameter of applying a minimizing balancing compensator, i.e., the lower cost of elements necessary to construct the compensator.

This article is organized into 13 sections. Section 1 describes the differences between power theories and algorithms used to control frequency converters and the possibility of using passive configurations in the power network. Section 2 is dedicated to a brief characterization of extended Budeanu theory. Section 3 presents a theoretical illustration 1 based on calculations and simulations of a 3-phase 4-wire load following extended Budeanu theory. Section 4 explains the ideal compensation approach, which is explained in detail in [27]. Section 5 is about minimizing balancing compensation, which is inspired by Currents’ Physical Components theory. Section 6 and Section 7 presents calculations and simulations of the ideal compensator and the minimizing balancing compensator for the Budeanu complemented reactive current, which was attached to the load described in Section 4. Similarly, Section 8 and Section 9 refer to the calculations and simulations of the compensators in relation to a reduction in the Budeanu reactive current. Section 10 and Section 11 describe a complex approach to compensation, i.e., the consideration of both Budeanu currents in calculations and simulations, namely, the Budeanu complemented reactive current and the Budeanu reactive current, which together form the Czarnecki’s reactive current. Section 12 presents an analysis of the results obtained with respect to the three-phase RMS values of the load current components and the power factor after compensation as well as a brief economic analysis of the costs of the reactance components needed to build the compensators. This study is summarized in Section 13, which presents the conclusions resulting from the theoretical components and the research results.

2. Description of the Load’s Currents in Extended Budeanu Theory for 3-Phase 4-Wire Systems

For the exact derivation of the formulas and a detailed description of extended Budeanu theory, see [4]. The basis for the energy description in extended Budeanu theory is the concept of power theory proposed by Czarnecki—Currents’ Physical Components theory [5].

The circuit shown in Figure 1 is a three-phase four-wire system in which the voltage vector is symmetric and also has higher-order harmonics.

The vector of fundamental and higher-order harmonic voltages is expressed by the following equation [5]:

$$\mathit{u}\left(t\right)=\left[\begin{array}{c}{u}_{\mathrm{R}}\left(t\right)\\ u_{\mathrm{S}}\left(t\right)\\ u_{\mathrm{T}}\left(t\right)\end{array}\right]=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}\left[\begin{array}{c}{\mathit{U}}_{\mathrm{R}n}\\ {\mathit{U}}_{\mathrm{S}n}\\ {\mathit{U}}_{\mathrm{T}n}\end{array}\right]{\mathrm{e}}^{jn{\omega }_{1}t}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}{\mathit{U}}_n{\mathrm{e}}^{jn{\omega }_{1}t}}$$

(1)

where ${\mathit{U}}_n$ is the vector of the CRMS (complex RMS) value of the voltage harmonics and $n$ is the harmonic order.

When a linear load is connected to the power supply, currents that correlate with the supply voltage vector are obtained:

$$\mathit{i}\left(t\right)=\left[\begin{array}{c}{i}_{\mathrm{R}}\left(t\right)\\ i_{\mathrm{S}}\left(t\right)\\ i_{\mathrm{T}}\left(t\right)\end{array}\right]=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}\left[\begin{array}{c}{\mathit{I}}_{\mathrm{R}n}\\ {\mathit{I}}_{\mathrm{S}n}\\ {\mathit{I}}_{\mathrm{T}n}\end{array}\right]e^{jn{\omega }_{1}t}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}{\mathit{I}}_n{e}^{jn{\omega }_{1}t}}$$

(2)

where ${\mathit{I}}_n$ is the vector of CRMS (complex RMS) value of the current harmonics.

Following [4], the current of a load in a three-phase four-wire system with distorted waveforms can have 7 mutually orthogonal components. Each of the components can be defined in the time waveform or a three-phase RMS value. In addition, each of the currents, in combination with the voltage, results in the appropriate power.

Table 1. Summary of the currents and component powers based on the extended Budeanu theory.

Components	Quantity	Time Waveform	Three-Phase RMS Value
Active	Current	${\mathit{i}}_{\mathrm{a}}={\mathrm{G}}_{\mathrm{e}}\mathit{u}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}{G}_{\mathrm{e}}{\mathit{U}}_n{e}^{jn{\omega }_{1}t}}$	$\Vert {\mathit{i}}_{\mathrm{a}}\Vert =G_{\mathrm{e}}\Vert \mathit{u}\Vert ={\displaystyle \frac{P}{\Vert \mathit{u}\Vert }}$
	Power	-	$P=\Vert \mathit{u}\Vert \Vert {\mathit{i}}_{\mathrm{a}}\Vert$
Scattered	Current	${\mathit{i}}_{\mathrm{s}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}\left(G_n-G_{\mathrm{e}}\right){\mathit{U}}_n{e}^{jn{\omega }_{1}t}}$	$\Vert {\mathit{i}}_{\mathrm{s}}\Vert =\sqrt{{\displaystyle \sum _{n\in N}\left[{\left(G_n-G_{\mathrm{e}}\right)}^2{\Vert {\mathit{u}}_n\Vert }^2\right]}}$
	Power	-	$D_{\mathrm{s}}=\Vert \mathit{u}\Vert \Vert {\mathit{i}}_{\mathrm{s}}\Vert$
Budeanu reactive	Current	${\mathit{i}}_{\mathrm{rB}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}j{B}_{\mathrm{eB}}{\mathit{U}}_n{e}^{jn{\omega }_{1}t}}$	$\Vert {\mathit{i}}_{\mathrm{rB}}\Vert =\left|B_{\mathrm{eB}}\right|\Vert \mathit{u}\Vert ={\displaystyle \frac{\left|Q_{\mathrm{B}}\right|}{\Vert \mathit{u}\Vert }}$
	Power	-	$Q_{\mathrm{B}}=\Vert \mathit{u}\Vert \Vert {\mathit{i}}_{\mathrm{rB}}\Vert$
Budeanu complemented reactive	Current	${\mathit{i}}_{\mathrm{crB}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}\left[j\left(B_n-B_{\mathrm{eB}}\right)\right]{\mathit{U}}_n{e}^{jn{\omega }_{1}t}}$	$\Vert {\mathit{i}}_{\mathrm{crB}}\Vert =\sqrt{{\displaystyle \sum _{n\in N}\left[{\left(\left(B_n-B_{\mathrm{eB}}\right)\Vert {\mathit{u}}_n\Vert \right)}^2\right]}}$
	Power	-	$Q_{\mathrm{crB}}=\Vert \mathit{u}\Vert \Vert {\mathit{i}}_{\mathrm{crB}}\Vert$
Positive-sequence unbalanced	Current	${\mathit{i}}_{\mathrm{u}}^{\mathrm{p}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}{\mathit{I}}_{\mathrm{u}n}^{\mathrm{p}}{e}^{jn{\omega }_{1}t}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}{Y}_{\mathrm{u}n}^{\mathrm{p}}{\mathit{1}}^{\mathrm{p}}{U}_{\mathrm{R}n}{e}^{jn{\omega }_{1}t}}$	$\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{p}}\Vert =\sqrt{3}{\displaystyle \sum _{n\in N}{Y}_{\mathrm{u}n}^{\mathrm{p}}{U}_{\mathrm{R}n}}$
	Power	-	$D_{\mathrm{u}}^{\mathrm{p}}=\Vert \mathit{u}\Vert \Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{p}}\Vert$
Negative-sequence unbalanced	Current	${\mathit{i}}_{\mathrm{u}}^{\mathrm{n}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}{\mathit{I}}_{\mathrm{u}n}^{\mathrm{n}}{e}^{jn{\omega }_{1}t}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}{Y}_{\mathrm{u}n}^{\mathrm{n}}{\mathit{1}}^{\mathrm{n}}{U}_{\mathrm{R}n}{e}^{jn{\omega }_{1}t}}$	$\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{n}}\Vert =\sqrt{3}{\displaystyle \sum _{n\in N}{Y}_{\mathrm{u}n}^{\mathrm{n}}{U}_{\mathrm{R}n}}$
	Power	-	$D_{\mathrm{u}}^{\mathrm{n}}=\Vert \mathit{u}\Vert \Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{n}}\Vert$
Zero-sequence unbalanced	Current	${\mathit{i}}_{\mathrm{u}}^{\mathrm{z}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}{\mathit{I}}_{\mathrm{u}n}^{\mathrm{z}}{e}^{jn{\omega }_{1}t}}=\sqrt{2}\mathrm{Re}{\displaystyle \sum _{n\in N}{Y}_{\mathrm{u}n}^{\mathrm{z}}{\mathit{1}}^{\mathrm{z}}{U}_{\mathrm{R}n}{e}^{jn{\omega }_{1}t}}$	$\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{z}}\Vert =\sqrt{3}{\displaystyle \sum _{n\in N}{Y}_{\mathrm{u}n}^{\mathrm{z}}{U}_{\mathrm{R}n}}$
	Power	-	$D_{\mathrm{u}}^{\mathrm{z}}=\Vert \mathit{u}\Vert \Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{z}}\Vert$

summarizes the current components and power of a linear load as given in the extended Budeanu theory.

All the current components of the load and the power components of the load form the entire current’s value $\Vert \mathit{i}\Vert$:

$${\Vert \mathit{i}\Vert }^2={\Vert {\mathit{i}}_{\mathrm{a}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{s}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{rB}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{crB}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{p}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{n}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{z}}\Vert }^2$$

(3)

and three-phase apparent power $S$ is as follows:

$$S=\Vert \mathit{u}\Vert \Vert \mathit{i}\Vert$$

(4)

The $\lambda$ power factor of a system described in the extended Budeanu theory is adequately expressed in the CPC theory, i.e., it can be described using the component powers:

$$\lambda ={\displaystyle \frac{P}{S}}={\displaystyle \frac{P}{\sqrt{P^2+D_{\mathrm{s}}^2+Q_{\mathrm{B}}^2+Q_{\mathrm{crB}}^2+D_{\mathrm{u}}^{\mathrm{p}2}+D_{\mathrm{u}}^{\mathrm{n}2}+D_{\mathrm{u}}^{\mathrm{z}2}}}}$$

(5)

or the currents’ components:

$$\lambda ={\displaystyle \frac{\Vert {\mathit{i}}_{\mathrm{a}}\Vert }{\Vert \mathit{i}\Vert }}={\displaystyle \frac{\Vert {\mathit{i}}_{\mathrm{a}}\Vert }{\sqrt{{\Vert {\mathit{i}}_{\mathrm{a}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{s}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{B}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{crB}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{p}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{n}}\Vert }^2+{\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{z}}\Vert }^2}}}$$

(6)

To find the parameters of the Budeanu compensator for minimizing the Budeanu reactive current and/or the Budeanu complemented reactive current, the components of the current of the load have to be determined, as presented above. The specification of the parameters of a minimizing balancing compensator can be achieved in three steps, namely:

• Step 1—the determination of equivalent parameters and load currents.
• Step 2—the determination of the parameters of an ideal compensator based on the equivalent parameters of the load.
• Step 3—the determination of the parameters of the minimizing balancing compensator based on the parameters of the ideal compensator and the equivalent parameters of the load.

It can be seen that none of the steps can be ignored because they are all mutually connected.

3. Theoretical Illustration 1—Original System with Three-Phase Four-Wire Linear Load

The electrical circuit in Figure 1 is powered by a symmetric nonsinusoidal voltage waveform. In addition to the fundamental harmonic, the source of nonsinusoidal voltage also produces higher-order harmonics, i.e., n = 3, 5, and 7, which form a symmetrical waveform in each phase in the following form: $u(t)=\left\{230e^{j\omega t}+5e^{j3\omega t}+30e^{j5\omega t}+5e^{j7\omega t}\right\}\mathrm{V}$. The frequency of the primary harmonic $n_1$ is 50 Hz. A three-phase load has linear elements in the components of resistance, inductive reactance, and capacitive reactance, with a load asymmetry. The electrical circuit is visualized in Figure 2.

Figure 3 shows the waveform of the supply voltage at connections points R, S, T of an imbalanced linear load.

The three-phase RMS value of the discussed voltage is as follows:

$$\Vert \mathit{u}\Vert =401.93\mathrm{V}$$

The receiver in Figure 3 consists of the elements R, L, and C. The resistance, inductive reactance, and capacitive reactance values for fundamental frequency are compiled in

Table 2. Summary of resistance, capacitive reactance, and inductive reactance values for the fundamental harmonics.

Parameter in $\mathsf{\Omega }$	R-Line	S-Line	T-Line
Resistance	4	4	4
Inductive reactance	3.142	6.283	-
Capacitive reactance	-	-	1.592

.

Using the data presented in Table 2 and the details regarding the supply voltage, the line currents for each harmonic were obtained. The resulting current waveform of the analyzed load is shown in Figure 4.

The three-phase RMS value of the load current, the waveform of which is shown in Figure 4, is as follows (3):

$$\Vert \mathit{i}\Vert =76.918\mathrm{A}$$

Based on the information in [2,4,5] regarding Budeanu’s power theory [2] and extended Budeanu theory [4], the active powers P and reactive Budeanu powers Q_B are summarized in

Table 3. Summary of three-phase active power and reactive power values of specific harmonics and the overall three-phase active power and reactive power of the entire system.

Harmonic Order	Active Power P in [W]	Reactive Power QB in [VAr]
n = 1	23,411	7872.5
n = 3	7.4	2.7
n = 5	240.9	64.2
n = 7	6.5	1.3
SUM	23,665.8	7940.7

. Based on extended Budeanu theory, the powers of the specific harmonics were summed up, even though the powers of the respective harmonic are capacitive or inductive.

The apparent power S is as follows (4):

$$S=30,916\mathrm{VA}$$

and the Budeanu distortion power D_B [2,4] is as follows:

$$D_{\mathrm{B}}=18,239.5\mathrm{VA}$$

According to the publication [4], the components of the load current described in the extended Budeanu theory are summarized in

Table 4. Summary of three-phase RMS values of component currents for individual harmonics described in the extended Budeanu theory.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{a}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{rB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$
n = 1	58.358	0.409	19.581	0.180	0	39.270	21.599
n = 3	1.269	0.418	0.426	0.114	0.822	0.587	0
n = 5	7.612	2.976	2.554	1.318	3.832	0	4.745
n = 7	1.269	0.520	0.426	0.274	0	0.774	0.663
RMS	58.880	3.077	19.756	1.363	3.920	39.282	22.124

.

The three-phase RMS value in the extended Budeanu theory is $\Vert \mathit{i}\Vert =76.918\mathrm{A}$.

In the case under study, 5 components of the current are responsible for the Budeanu distortion current. The waveform of the Budeanu distortion current described in the extended Budeanu theory is shown in Figure 5.

Table 5. Components of the distortion current described in extended Budeanu theory.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{dB}}\Vert$
n = 1	0.409	0.180	0	39.270	21.599
n = 3	0.418	0.114	0.822	0.587	0
n = 5	2.976	1.318	3.832	0	4.745
n = 7	0.520	0.274	0	0.774	0.663
RMS	3.077	1.363	3.920	39.282	22.124	45.379

summarizes the three-phase RMS values of the components of the currents, which can be used to determine the distortion current and, thus, the distortion power described by Budeanu.

Based on Equation (6), the power factor $\lambda$ is as follows:

$$\lambda =0.766$$

This requires that the receiver in Figure 2 follows the ideal compensation or minimization of the balancing compensation in order to achieve the highest possible power factor.

4. Optimal Compensation of the Budeanu Complemented Reactive Current and the Unbalanced Current

The increase in the power factor through ideal compensation is described in [27]. At this point, it should be emphasized that the calculated parameters of an ideal compensator, as the name suggests, are lossless and do not cause any other interferences (e.g., no additional resistance). This means that after finding the parameters of the reactance elements, connecting them in the required configuration, and tuning them to a given harmonic, they do not influence the current value of the rest of the harmonics.

The formulas that make this possible are demonstrated below. Extended Budeanu theory describes two reactive components of the load current, namely, the Budeanu reactive current and the Budeanu complemented reactive current. It should be pointed out, however, that after summing up both reactive currents, the reactive current described by Czarnecki in Currents’ Physical Components theory is formed. Considering the above conclusions and the basic idea in [22], the result of the two expressions concerning the compensation of the chosen reactive current component and the unbalanced current of the zero sequence for the sequence of harmonics n = 3 k + 1 is as follows:

$$\begin{array}{l}{T}_{\mathrm{R}n}=-2\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{z}}-\left(M_x\right)\\ T_{\mathrm{S}n}=-\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{z}}+\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{z}}-\left(M_x\right)\\ T_{\mathrm{T}n}=\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{z}}+\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{z}}-\left(M_x\right)\end{array}$$

(7)

where the parameter $M_x$ can be described as follows, respectively:

1.

$M_x=\left(B_n-B_{\mathrm{e}}\right)$—Budeanu complemented reactive current compensation.

2.

$M_x=B_{\mathrm{e}}$—Budeanu reactive current compensation.

3.

$M_x=\left[\left(B_n-B_{\mathrm{e}}\right)+B_{\mathrm{e}}\right]$—Budeanu complemented reactive current and Budeanu reactive current compensations, i.e., the Czarnecki reactive current.

For the positive sequence n = 3 k + 1, a delta-structure compensator is described by the following group of equations:

$$\begin{array}{l}{T}_{\mathrm{RS}n}={\displaystyle \frac{1}{3}}\left(\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{n}\text{\#}}-\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{n}\text{\#}}\right)\\ T_{\mathrm{ST}n}={\displaystyle \frac{1}{3}}\left(2\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{n}\text{\#}}\right)\\ T_{\mathrm{TR}n}={\displaystyle \frac{1}{3}}\left(-\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{n}\text{\#}}-\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{n}\text{\#}}\right)\end{array}$$

(8)

If the supply voltage has negative-sequence harmonics n = 3 k − 1, then, the answer to Equation (7) is as follows:

$$\begin{array}{l}{T}_{\mathrm{R}n}=-2\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{z}}-\left(M_x\right)\\ T_{\mathrm{S}n}=\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{z}}+\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{z}}-\left(M_x\right)\\ T_{\mathrm{T}n}=-\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{z}}+\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{z}}-\left(M_x\right)\end{array}$$

(9)

Then, the solution of Equation (8) changes. For the negative-sequence harmonics n = 3 k − 1, it is as follows:

$$\begin{array}{l}{T}_{\mathrm{RS}n}={\displaystyle \frac{1}{3}}\left(-\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{p}\text{\#}}-\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{p}\text{\#}}\right)\\ T_{\mathrm{ST}n}={\displaystyle \frac{1}{3}}\left(2\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{p}\text{\#}}\right)\\ T_{\mathrm{TR}n}={\displaystyle \frac{1}{3}}\left(\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{p}\text{\#}}-\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{p}\text{\#}}\right)\end{array}$$

(10)

For zero-sequence voltage harmonics n = 3 k, only the configuration of a star-structure compensator can be used. When choosing the compensation for the positive-sequence unbalanced current, the solution to the equation is as follows:

$$\begin{array}{l}{T}_{\mathrm{R}n}=-2\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{p}}-\left(M_x\right)\\ T_{\mathrm{S}n}=\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{p}}+\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{p}}-\left(M_x\right)\\ T_{\mathrm{T}n}=-\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{p}}+\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{p}}-\left(M_x\right)\end{array}$$

(11)

If the unbalanced current of the negative sequence is chosen for compensation, the equation has the following solution:

$$\begin{array}{l}{T}_{\mathrm{R}n}=-2\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{n}}-\left(M_x\right)\\ T_{\mathrm{S}n}=-\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{n}}+\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{n}}-\left(M_x\right)\\ T_{\mathrm{T}n}=\sqrt{3}\mathrm{Re}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{n}}+\mathrm{Im}{\mathit{Y}}_{\mathrm{u}n}^{\mathrm{n}}-\left(M_x\right)\end{array}$$

(12)

In this article, the unbalanced component of the current of the positive sequence was taken into account when setting the parameters of the compensator for harmonics n = 3 k.

5. Minimizing Balancing Compensation in Extended Budeanu Theory

It is clear that the RMS value of the first voltage harmonic, whether phase-to-phase or phase, is significantly higher than the RMS value of any additional harmonic [36,37,38].

The principle is that if the susceptance T_XY1 or T_X1 is negative, the delta-structure compensator between the T_XY connection points may have an electrical line with a choke as its susceptance is also below zero, or for the star-structure compensator between the phase connection point and the neutral connection point with susceptance T_X, it should also have a branch with a choke. The inductance of that choke, regardless of whether it is connected between the phases or between the phase and the neutral point, should minimize the value [36,37,38]:

$${\displaystyle \sum _{n\in N}{\left(T_{kn}+\frac{1}{n{\omega }_1{L}_k}\right)}^2{U}_{kn}^2=\mathrm{minimum}}$$

(13)

If the phase-to-phase T_XY or phase T_X susceptance has a “plus” sign, the compensator may have a capacitor in the electrical line, but it needs to be put together in series with an inductor; so, for harmonic distortion, this electrical line has an inductive character, and for the basic frequency, it is still positive:

$$d_{k1}={\displaystyle \frac{{\omega }_1{C}_k}{1-{\omega }_1^2{L}_k{C}_k}}$$

(14)

This demands that the resonant frequency of that branch is higher than the fundamental harmonic frequency, as follows:

$${\omega }_{\mathrm{r}}={\displaystyle \frac{1}{\sqrt{L_k{C}_k}}}$$

(15)

The inductance and capacitance of such a branch should follow to minimize the expression below:

$${\displaystyle \sum _{n\in N}{\left(T_{kn}-\frac{n{\omega }_1{C}_k}{1-n^2{\omega }_1^2{L}_k{C}_k}\right)}^2{U}_{kn}^2=\mathrm{minimum}}$$

(16)

Inductance $L_k$ fulfills Equation (15) and can be obtained as follows:

$${\displaystyle \frac{d}{d{L}_k}}\left\{{\displaystyle \sum _{n\in N}{\left(T_{kn}+\frac{1}{n{\omega }_1{L}_k}\right)}^2{U}_{kn}^2}\right\}=0$$

(17)

By satisfying expression (17), i.e., calculating the inductance in a determined branch in this procedure, we can interpret it as the optimal inductance of this branch, as follows:

$$L_{k,\mathrm{opt}}=-{\displaystyle \frac{{\displaystyle \sum _{n\in N}\frac{U_{kn}^2}{n^2}}}{{\omega }_1{\displaystyle \sum _{n\in N}\frac{T_{kn}{U}_{kn}^2}{n}}}}$$

(18)

The left side of expression (16) represents a function with two parameters, i.e., inductance $L_k$ and capacitance $C_k$. Inductance $L_k$ is a permanently decreasing function, which means that there is no minimum for any specific value. Its specific choice depends on the person making the calculations [36,37,38]. When the inductance has been determined, the capacitance $C_k$ can be found using Equation (16). When solving Equation (16), we must satisfy the following condition:

$${\displaystyle \frac{d}{d{C}_k}}\left\{{\displaystyle \sum _{n\in N}{\left(T_{kn}-\frac{n{\omega }_1{C}_k}{1-n^2{\omega }_1^2{L}_k{C}_k}\right)}^2{U}_{kn}^2}\right\}=0$$

(19)

The distance determined by expression (19) for any branch has a minimum related to the optimal capacitor capacitance when:

$${\displaystyle \sum _{n\in N}\frac{n{T}_{kn}{U}_{kn}^2}{{\left(1-n^2{\omega }_1^2{L}_{k,\mathrm{opt}}{C}_{k,\mathrm{opt}}\right)}^2}-{\displaystyle \sum _{n\in N}\frac{n^2{\omega }_1{C}_{k,\mathrm{opt}}{U}_{kn}^2}{{\left(1-n^2{\omega }_1^2{L}_{k,\mathrm{opt}}{C}_{k,\mathrm{opt}}\right)}^3}=0}}$$

(20)

Because of the optimum capacity $C_{k,\mathrm{opt}}$ present in expression (20), this is an implicit equation. For this reason, calculating this capacity requires numerical methods, especially in an iterative process. The iterative equation for finding the optimal capacity is as follows:

$$C_{k,i+1}={\displaystyle \frac{{\displaystyle \sum _{n\in N}\frac{n{T}_{kn}{U}_{kn}^2}{{\left(1-n^2{\omega }_1^2{L}_{k,\mathrm{opt}}{C}_{k,i}\right)}^2}}}{{\omega }_1{\displaystyle \sum _{n\in N}\frac{n^2{U}_{kn}^2}{{\left(1-n^2{\omega }_1^2{L}_{k,\mathrm{opt}}{C}_{k,i}\right)}^3}}}}$$

(21)

which, after the i-th iteration, will lead to some capacities that should come together to reach the optimal capacity in the final stage of calculations.

6. Theoretical Illustration of the 2-Ideal Compensation of the Budeanu Complemented Reactive Current

The receiver in Figure 2 was connected to the Budeanu complemented reactive current (parameter $M_x=\left(B_n-B_{\mathrm{e}}\right)$) and the compensator of the unbalanced current, which is designed to compensate for the susceptances responsible for the Budeanu complemented reactive current for all the harmonics present in the voltage. Moreover, the system is able to balance the load by compensating for the unbalanced current components.

Table 6. Summary of capacitance and inductance values required to compensate for the Budeanu complemented reactive current and the unbalanced current for a compensator with a star structure.

Line	Harmonic Order
	n = 1		n = 3		n = 5		n = 7
	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]
R	-	297.58	187.1	-	96.61	-	-	3.08
S	316.53	-	-	8.98	-	4.18	51.22	-
T	-	36.43	-	10.88	-	8.44	-	7.59

and

Table 7. Summary of capacitance and inductance values required to compensate for the Budeanu complemented reactive current and the unbalanced current for a delta-structure compensator.

Line	Harmonic Order
	n = 1		n = 3		n = 5		n = 7
	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]
RS	101.10	-	-	-	-	147.2	1.04	-
ST	-	57.54	-	-	59.90	-	-	4.78
TR	74.99	-	-	-	-	7.09	42.21	-

show the capacitance and inductance values needed to compensate for the Budeanu complemented reactive current and the unbalanced current for the respective harmonics. Figure 6 shows the way in which the discussed compensator is connected (star and delta configurations). Figure 7 shows the waveforms of the line currents obtained by using the Budeanu complemented reactive current and the unbalanced current compensator.

After perfect compensation of the Budeanu complemented reactive current and the unbalanced current, the value of the line current is as follows:

$$\Vert \mathit{i}\Vert =62.195\mathrm{A}$$

The installation of the Budeanu complemented reactive current and the unbalanced current compensator changes the power values related to passive components.

Table 8. Summary of three-phase active power and reactive power values of individual harmonics and total three-phase active power and reactive power of the whole system after compensation of the Budeanu complemented reactive current and the unbalanced current.

Harmonic Order	Active Power P in [W]	Reactive Power QB in [VAr]
n = 1	23,411	7800.6
n = 3	7.4	3.7
n = 5	240.9	132.7
n = 7	6.5	3.7
SUM	23,665.8	7940.7

summarizes the powers of the individual harmonics and the entire system.

The apparent power S is as follows:

$$S=24,998.4\mathrm{VA}$$

and the Budeanu distortion power D_B is as follows:

$$D_{\mathrm{B}}=1340.4\mathrm{VA}$$

Based on the equations in Table 1, the three-phase RMS values of the currents’ components after the connection of the perfect Budeanu complemented reactive current and the unbalanced current compensator were calculated. The three-phase RMS values of the components of the currents for the individual harmonics described in the extended Budeanu theory are summarized in

Table 9. Summary of the three-phase RMS values of the currents’ components for the individual harmonics described in the extended Budeanu theory after compensation of the Budeanu complemented reactive current and the unbalanced current.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{a}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{rB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$
n = 1	58.358	0.409	19.581	0	0	0	0
n = 3	1.269	0.418	0.426	0	0	1.285	0
n = 5	7.612	2.976	2.554	0	0	0	0
n = 7	1.269	0.520	0.426	0	0	0	0
RMS	58.880	3.077	19.756	0	0	1.285	0

.

After perfect compensation of the Budeanu complemented reactive current and the unbalanced current, the value of the line current is $\Vert \mathit{i}\Vert =62.195\mathrm{A}$.

After including the ideal compensator of the Budeanu complemented reactive current and the unbalanced current, the Budeanu distortion current has two components. The waveform of the Budeanu distortion current after the ideal compensation of the Budeanu complemented reactive current and the unbalanced current is shown in Figure 8.

Table 10. Components of the distortion current defined in extended Budeanu theory.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{dB}}\Vert$
n = 1	0.409	0	0	0	0
n = 3	0.418	0	0	1.285	0
n = 5	2.976	0	0	0	0
n = 7	0.520	0	0	0	0
RMS	3.077	0	0	1.285	0	3.335

summarizes the three-phase RMS values of the currents’ components based on the perfect compensation of the Budeanu complemented reactive current and the unbalanced current, which allow for the description of the Budeanu distortion current.

The power factor of a circuit with an optimally compensated Budeanu complemented reactive current and the unbalanced current is as follows:

λ = 0.947

This means that the receiver in Figure 2, after connecting the optimal compensator of the Budeanu complemented reactive current and the unbalanced current, achieves the highest possible value. In the next section—Section 7, based on the obtained parameters of the ideal compensator, the parameters of the minimizing balancing compensator of the Budeanu complemented reactive current and the unbalanced current are calculated.

7. Theoretical Illustration 3—Minimizing Balancing Compensation of the Budeanu Complemented Reactive Current

A compensator was connected to the receiver in Figure 2, the main function of which is to minimize the 3-phase RMS value of the Budeanu complemented reactive current and the unbalanced current. The design and choice of parameters of the minimizing balancing compensator causes it to impact all components associated with reactance elements. Figure 9 shows how to install the abovementioned compensator. Figure 10 illustrates the current flow of the load caused by the connection of a minimizing balancing compensator.

Table 11. Summary of capacitance and inductance values necessary to minimize the Budeanu complemented reactive current and the unbalanced current for a compensator with a star structure and delta structure.

Parameter	Configuration
	Star			Delta
	R	S	T	RS	ST	TR
L [mH]	313.46	72.75	36.34	227.76	57.91	307.07
C [µF]	-	96.72	-	30.89	-	22.92

summarizes the inductance and capacitance values obtained from the parameter calculation for a compensator minimizing balancing compensator of the Budeanu complemented reactive current and the unbalanced current.

After the minimizing balancing compensation of the Budeanu complemented reactive current and the unbalanced current, the value of line current is as follows:

$$\Vert \mathit{i}\Vert =62.452\mathrm{A}$$

Table 12. Summary of three-phase active power and reactive power values of individual harmonics and total three-phase active power and reactive power of the whole system after compensation of the Budeanu complemented reactive current and the unbalanced current.

Harmonic Order	Active Power P in [W]	Reactive Power QB in [VAr]
n = 1	23,411	7724.5
n = 3	7.4	4
n = 5	240.9	133.8
n = 7	6.5	2.7
SUM	23,665.8	7865

summarizes the power values of the respective harmonics and the entire system.

The apparent power S is as follows:

$$S=25,101.5\mathrm{VA}$$

and the Budeanu distortion power D_B is as follows:

$$D_{\mathrm{B}}=2856.9\mathrm{VA}$$

Based on the relationships in Table 1, the three-phase RMS values of the currents’ components were calculated after including a minimizing balancing compensator of the Budeanu complemented reactive current and the unbalanced current. The 3-phase RMS values of the load currents for the specific harmonics described in extended Budeanu theory are summarized in

Table 13. Summary of the 3-phase RMS values of the currents’ components for the individual harmonics described in the extended Budeanu theory after minimizing the Budeanu complemented reactive current and the unbalanced current.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{a}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{rB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$
n = 1	58.358	0.409	19.394	0.004	0	0.058	0.089
n = 3	1.269	0.418	0.422	0.035	0.767	0.601	0
n = 5	7.612	2.976	2.530	0.045	4.252	0	4.586
n = 7	1.269	0.520	0.422	0.112	0	0.706	0.671
RMS	58.880	3.077	19.568	0.126	4.321	0.930	4.637

.

The load current after minimizing balancing compensation is $\Vert \mathit{i}\Vert =62.452\mathrm{A}$.

After adding the minimizing balancing compensator of the Budeanu complemented reactive current and the unbalanced current, five currents’ components represent the Budeanu distortion current. The waveform of the Budeanu distortion current after minimizing balancing compensation is given in Figure 11.

Table 14. Components of the distortion current defined in extended Budeanu theory.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{S}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{dB}}\Vert$
n = 1	0.409	0.004	0	0.058	0.089
n = 3	0.418	0.035	0.767	0.601	0
n = 5	2.976	0.045	4.252	0	4.586
n = 7	0.520	0.112	0	0.706	0.671
RMS	3.077	0.126	4.321	0.930	4.637	7.108

summarizes the three-phase RMS values of the currents’ components caused by the minimizing balancing compensation of the Budeanu complemented reactive current and the unbalanced current, which describe the distortion current defined by Budeanu.

Power factor $\lambda$ is equal to:

$$\lambda =0.943$$

This signifies that the difference between the connection of an ideal compensator to a minimizing balancing compensator of the Budeanu complemented reactive current and the unbalanced current is only 0.004.

8. Theoretical Illustration 4—Ideal Compensation of the Budeanu Reactive Current

The receiver in Figure 2 was connected to the Budeanu reactive current (parameter $M_x=B_{\mathrm{e}}$) and the unbalanced current compensator, which is designed to compensate for the susceptances responsible for the Budeanu reactive current for all the harmonics present in the voltage. Moreover, the system is able to balance the load by compensating for the unbalanced current components.

Table 15. Summary of capacitance and inductance values required to compensate for the Budeanu reactive current and the unbalanced current for a compensator with a star structure.

Line	Harmonic Order
	n = 1		n = 3		n = 5		n = 7
	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]
R	120.97	-	253.2	-	144.06	-	-	6.79
S	471.5	-	-	18.97	-	8.17	87.98	-
T	-	82.28	-	30.08	-	721.02	95.31	-

and

Table 16. Summary of capacitance and inductance values required to compensate for the Budeanu reactive current and the unbalanced current for a delta-structure compensator.

Line	Harmonic Order
	n = 1		n = 3		n = 5		n = 7
	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]
RS	101.10	-	-	-	-	147.20	1.04	-
ST	-	57.54	-	-	59.90	-	-	4.78
TR	74.99	-	-	-	-	7.09	42.21	-

show the capacitance and inductance values needed to compensate for the Budeanu reactive current and the unbalanced current for respective harmonics. Figure 12 illustrates the way in which the discussed compensator is connected (star and delta configurations). Figure 13 illustrates the waveforms of the line currents obtained by using the Budeanu reactive current and the unbalanced current compensator.

After perfect compensation of the Budeanu reactive current and the unbalanced current, the value of the line currents is as follows:

$$\Vert \mathit{i}\Vert =58.990\mathrm{A}$$

The compensator’s connection of the Budeanu reactive current and the unbalanced current changes the power values associated with passive elements.

Table 17. Summary of three-phase power values of individual harmonics and total three-phase power values of the whole system after ideal compensation of the Budeanu reactive current and the unbalanced current.

Harmonic Order	Active Power P in [W]	Reactive Power QB in [VAr]
n = 1	23,411	71.9
n = 3	7.4	−1
n = 5	240.9	−68.5
n = 7	6.5	−2.4
SUM	23,665.8	0

contains a summary of power values for the specific harmonics and the total system.

The apparent power S is as follows:

$$S=23,710\mathrm{VA}$$

and the Budeanu distortion power D_B is as follows:

$$D_{\mathrm{B}}=1448.1\mathrm{VA}$$

Calculations based on the relationships in Table 1 were used to determine the three-phase RMS values of the currents’ components after the application of an ideal compensator of the Budeanu reactive current and the unbalanced current. The three-phase RMS values of the load currents for the individual harmonics described in the extended Budeanu theory are presented in

Table 18. Components of the currents for individual harmonics described in the extended Budeanu theory after ideal compensation of the Budeanu reactive current and the unbalanced current.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{a}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{rB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$
n = 1	58.358	0.409	0	0.180	0	0	0
n = 3	1.269	0.418	0	0.114	0	1.285	0
n = 5	7.612	2.976	0	1.318	0	0	0
n = 7	1.269	0.520	0	0.274	0	0	0
RMS	58.880	3.077	0	1.363	0	1.285	0

.

The load current after optimal compensation of the Budeanu reactive current and the unbalanced current is equal to $\Vert \mathit{i}\Vert =58.990\mathrm{A}$.

When the ideal compensator of the Budeanu reactive and the unbalanced current are included, the Budeanu distortion current depends on three currents’ components. The waveform of the Budeanu distortion current after the perfect compensation of the Budeanu reactive current and the unbalanced current is demonstrated in Figure 14.

Table 19. Summary of the three-phase RMS values of the components of the distortion current defined in extended Budeanu theory.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{dB}}\Vert$
n = 1	0.409	0.180	0	0	0
n = 3	0.418	0.114	0	1.285	0
n = 5	2.976	1.318	0	0	0
n = 7	0.520	0.274	0	0	0
RMS	3.077	1.363	0	1.285	0	3.602

presents the 3-phase RMS values of the currents’ components caused by the total compensation of the Budeanu reactive current and the unbalanced current, which makes it possible to describe the Budeanu distortion current.

The value of the power factor $\lambda$ after ideal compensation of the Budeanu reactive current and the unbalanced current is as follows:

$$\lambda =0.998$$

This signifies that the receiver in Figure 2 obtained the highest possible value after connecting the ideal compensator of the Budeanu reactive current and the unbalanced current. In the next section—Section 9, the parameters of the minimizing balancing compensator of the Budeanu reactive current and the unbalanced current are calculated based on the parameters of the perfect compensator obtained.

9. Theoretical Illustration 5—Minimizing Balancing Compensation of the Budeanu Reactive Current

A compensator was connected to the system in Figure 2, the main function of which is to minimize the three-phase RMS value of the Budeanu reactive current and the unbalanced current. The design and choice of parameters of the minimizing balancing compensator causes it to impact all components associated with reactance elements. Figure 15 shows how to install the abovementioned compensator. Figure 16 illustrates the current flow of the load caused by the connection of a minimizing balancing compensator.

Table 20. Summary of capacitance and inductance values necessary to minimize the Budeanu reactive current and the unbalanced current for a compensator with a star structure and a delta structure.

Parameter	Configuration
	Star			Delta
	R	S	T	RS	ST	TR
L [mH]	190.36	48.83	82.33	227.76	57.91	307.07
C [µF]	36.95	144.09	-	30.89	-	22.92

contains a summary of the inductance and capacitance values based on the selection of parameters for the minimizing balancing compensation of the Budeanu reactive current and the unbalanced current.

After the minimizing balancing compensation of the Budeanu reactive current and the unbalanced current, the value of the line current is as follows:

$$\Vert \mathit{i}\Vert =59.370\mathrm{A}$$

The compensator’s connection of the Budeanu reactive current and the unbalanced current changes the power values related to the passive elements.

Table 21. Summary of three-phase power values of individual harmonics and total three-phase power values of the whole system after minimizing balancing compensation of the Budeanu reactive current and the unbalanced current.

Harmonic Order	Active Power P in [W]	Reactive Power QB in [VAr]
n = 1	23,411	14.5
n = 3	7.4	3.8
n = 5	240.9	130.4
n = 7	6.5	2.6
SUM	23,665.8	151.4

shows the three-phase power values of specific harmonics and the overall system.

The apparent power S is as follows:

$$S=23,863\mathrm{VA}$$

and the Budeanu distortion power D_B is as follows:

$$D_{\mathrm{B}}=3057.4\mathrm{VA}$$

Based on the equations in Table 1, the three-phase RMS values of the currents’ components were calculated after connecting a minimizing balancing compensator of the Budeanu reactive current and the unbalanced current. The three-phase RMS values of the load currents for the respective harmonics, as described in the extended Budeanu theory, are compiled in

Table 22. Summary of the three-phase RMS values of the currents’ components for each harmonic described in the extended Budeanu theory after minimizing balancing compensation of the Budeanu reactive current and the unbalanced current.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{a}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{rB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$
n = 1	58.358	0.409	0.373	0.337	0	0.144	0.007
n = 3	1.269	0.418	0.008	0.435	0.784	0.635	0
n = 5	7.612	2.976	0.049	2.462	4.332	0	4.602
n = 7	1.269	0.520	0.008	0.293	0	0.705	0.680
RMS	58.880	3.077	0.377	2.539	4.403	0.960	4.652

.

The line current after minimizing balancing compensation is $\Vert \mathit{i}\Vert =59.370\mathrm{A}$.

After adding the minimizing balancing compensator of the Budeanu reactive current and the unbalanced current, the Budeanu distortion current consists of 5 currents’ components. The waveform of the Budeanu distortion current after minimizing balancing compensation is given in Figure 17.

Table 23. Components of the distortion current defined in extended Budeanu theory.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{dB}}\Vert$
n = 1	0.409	0.004	0	0.058	0.089
n = 3	0.418	0.035	0.767	0.601	0
n = 5	2.976	0.045	4.252	0	4.586
n = 7	0.520	0.112	0	0.706	0.671
RMS	3.077	0.126	4.321	0.930	4.637	7.607

includes the three-phase RMS values of the currents’ components from the minimizing balancing compensation of the Budeanu reactive current and the unbalanced current, describing the distortion current defined by Budeanu.

Based on Equation (7), the power factor $\lambda$ is as follows:

$$\lambda =0.992$$

This means that the difference between the effect of an ideal compensation and a minimizing balancing compensation is only 0.006.

10. Theoretical Illustration 6—Ideal Compensation of the Budeanu Reactive Current and Budeanu Complemented Reactive Current—Czarnecki’s Reactive Current

The receiver in Figure 2 was connected to the Budeanu reactive current, the Budeanu complemented reactive current (parameter $M_x=\left[\left(B_n-B_{\mathrm{e}}\right)+B_{\mathrm{e}}\right]$), and the unbalanced current compensator, which is designed to compensate for the susceptances responsible for Budeanu reactive current and the Budeanu complemented reactive current for all harmonics present in the voltage. Moreover, the system is able to balance the load.

Table 24. Summary of capacitance and inductance values required for ideal compensation of the Budeanu reactive currents and the unbalanced current for a compensator with a star structure.

Line	Harmonic Order
	n = 1		n = 3		n = 5		n = 7
	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]
R	122.41	-	239.30	-	127.90	-	-	4.61
S	472.99	-	-	15.37	-	6.16	73.58	-
T	-	83.26	-	21.92	-	24.25	-	42.39

and

Table 25. Summary of capacitance and inductance values required for perfect compensation of the Budeanu reactive currents and the unbalanced current for a delta-structure compensator.

Line	Harmonic Order
	n = 1		n = 3		n = 5		n = 7
	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]	C [µF]	L [mH]
RS	101.10	-	-	-	-	147.20	1.04	-
ST	-	57.54	-	-	59.90	-	-	4.78
TR	74.99	-	-	-	-	7.09	42.21	-

show the capacitance and inductance values needed to compensate for the Budeanu reactive current, the Budeanu complemented reactive current, and the unbalanced current for the respective harmonics. Figure 18 illustrates the way in which the discussed compensator is connected (star and delta configurations). Figure 19 presents the waveforms of the line currents obtained by using the Budeanu reactive currents and the unbalanced current compensator.

After ideal compensation of Budeanu reactive currents and the unbalanced current, the value of the line currents is as follows:

$$\Vert \mathit{i}\Vert =58.974\mathrm{A}$$

The connection of a perfect compensator of the Budeanu reactive currents and the unbalanced current changes the power values associated with passive elements.

Table 26. Summary of three-phase power values of specific harmonics and total three-phase power values of the whole system after perfect compensation of the Budeanu reactive currents and the unbalanced current.

Harmonic Order	Active Power P in [W]	Reactive Power QB in [VAr]
n = 1	23,411	0
n = 3	7.4	0
n = 5	240.9	0
n = 7	6.5	0
SUM	23,665.8	0

compares the three-phase power values of individual harmonics and the complete system.

The apparent power S is as follows:

$$S=23,703.7\mathrm{VA}$$

and the Budeanu distortion power D_B is as follows:

$$D_{\mathrm{B}}=1340.4\mathrm{VA}$$

Based on the equations in Table 1, the three-phase RMS values of the currents’ components were achieved after the connection of an ideal compensator of the Budeanu reactive currents and the unbalanced current. The three-phase RMS values of the load currents’ components for the harmonics, as described in the extended Budeanu theory, are presented in

Table 27. Values of the components for individual harmonics described in the extended Budeanu theory after ideal compensation of the Budeanu reactive currents and the unbalanced current.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{a}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{rB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$
n = 1	58.358	0.409	0	0	0	0	0
n = 3	1.269	0.418	0	0	0	1.285	0
n = 5	7.612	2.976	0	0	0	0	0
n = 7	1.269	0.520	0	0	0	0	0
RMS	58.880	3.077	0	0	0	1.285	0

.

After optimal compensation of the Budeanu reactive currents and the unbalanced current, the value of the line currents is $\Vert \mathit{i}\Vert =58.974\mathrm{A}$.

After a perfect compensation of the Budeanu reactive currents and the unbalanced current, the Budeanu distortion current is formed by two components. The waveform of the Budeanu distortion current following the ideal compensation of the Budeanu reactive currents and the unbalanced current is depicted in Figure 20.

Table 28. Components of the distortion current defined in extended Budeanu theory.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{dB}}\Vert$
n = 1	0.409	0	0	0	0
n = 3	0.418	0	0	1.285	0
n = 5	2.976	0	0	0	0
n = 7	0.520	0	0	0	0
RMS	3.077	0	0	1.285	0	3.335

presents the three-phase RMS values of the currents’ components caused by the ideal compensation of the Budeanu reactive currents and the unbalanced current, which makes it possible to identify the Budeanu distortion current.

The power factor $\lambda$ after an ideal compensator of the Budeanu reactive currents and the unbalanced current is as follows:

$$\lambda =0.998$$

This means that the load in Figure 2 has reached the highest possible value after the connection of the ideal compensator of the Budeanu reactive currents and the unbalanced current. In the next section—Section 11, based on the parameters determined for the optimal compensator, the parameters of the minimizing balancing compensator of the Budeanu reactive currents and the unbalanced current are specified.

11. Theoretical Illustration 7—Minimizing Balancing Compensation of the Budeanu Reactive Current and Budeanu Complemented Reactive Current—Czarnecki’s Reactive Current

A compensator was connected to the system in Figure 2, the main function of which is to minimize the 3-phase RMS value of the Budeanu reactive current, the Budeanu complemented reactive current, and the unbalanced current. The design and choice of parameters of the minimizing balancing compensator causes it to impact all components associated with the reactance elements. Figure 21 shows how to install the abovementioned compensator. Figure 22 illustrates the current flow of the load caused by the connection of a minimizing balancing compensator.

Table 29. Summary of capacitance and inductance values necessary to minimize the Budeanu reactive currents and the unbalanced current for a compensator with a star structure and a delta structure.

Parameter	Configuration
	Star			Delta
	R	S	T	RS	ST	TR
L [mH]	188.12	48.69	83.11	227.76	57.91	307.07
C [µF]	37.39	144.53	-	30.89	-	22.92

presents the inductance and capacitance values from the parameter identification of a minimizing balancing compensator of the Budeanu reactive currents and the unbalanced current.

After the minimizing balancing compensation of the Budeanu reactive currents and the unbalanced current, the value of the line currents is as follows:

$$\Vert \mathit{i}\Vert =59.371\mathrm{A}$$

The connection of the compensator of the Budeanu reactive currents and the unbalanced current changes the power values caused by the passive elements.

Table 30. Summary of three-phase power values of respective harmonics and total three-phase power values of the whole system after minimization of the Budeanu reactive currents and the unbalanced current.

Harmonic Order	Active Power P in [W]	Reactive Power QB in [VAr]
n = 1	23,411	−52.9
n = 3	7.4	3.8
n = 5	240.9	130.5
n = 7	6.5	2.6
SUM	23,665.8	84

lists the three-phase power values of individual harmonics and the full system.

The apparent power S is as follows:

$$S=23,863\mathrm{VA}$$

and the Budeanu distortion power D_B is as follows:

$$D_{\mathrm{B}}=3060.6\mathrm{VA}$$

Based on the equations in Table 1, the three-phase RMS values of the currents’ components were found after connecting a minimizing balancing compensator of the Budeanu reactive currents and the unbalanced current. The 3-phase RMS values of the load currents for the harmonics described in the extended Budeanu theory are listed in

Table 31. Currents’ components for respective harmonics described in the extended Budeanu theory after minimization of the Budeanu reactive currents and the unbalanced current.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{a}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{rB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$
n = 1	58.358	0.409	0.207	0.340	0	0.138	0.009
n = 3	1.269	0.418	0.005	0.439	0.784	0.635	0
n = 5	7.612	2.976	0.027	2.484	4.333	0	4.602
n = 7	1.269	0.520	0.005	0.297	0	0.705	0.680
RMS	58.880	3.077	0.209	2.562	4.403	0.959	4.652

.

After minimizing balancing compensation, the value of the line currents is $\Vert \mathit{i}\Vert =59.371\mathrm{A}$.

After connecting the minimizing balancing compensator of the Budeanu reactive currents and the unbalanced current, the Budeanu distortion current consists of five currents’ components. The waveform of the Budeanu distortion current after minimizing balancing compensation is presented in Figure 23.

Table 32. Components of the distortion current defined in extended Budeanu theory.

Harmonic Order	$\Vert {\mathit{i}}_{\mathbf{s}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{crB}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{p}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{n}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{u}}^{\mathbf{z}}\Vert$	$\Vert {\mathit{i}}_{\mathbf{dB}}\Vert$
n = 1	0.409	0.207	0.340	0	0.138
n = 3	0.418	0.005	0.439	0.784	0.635
n = 5	2.976	0.027	2.484	4.333	0
n = 7	0.520	0.005	0.297	0	0.705
RMS	3.077	0.209	2.562	4.403	0.959	7.615

presents the three-phase RMS values of the currents’ components obtained from the minimizing balancing compensation of the Budeanu reactive currents and the unbalanced current, which describe the distortion current defined by Budeanu.

The power factor $\lambda$ is as follows:

$$\lambda =0.992$$

This signifies that the difference between the connection of an optimal compensator and a minimizing balancing compensator of the Budeanu reactive currents and the unbalanced current is barely 0.006.

12. Analysis of the Results

This article presents one method of minimizing and balancing the currents of a three-phase four-wire load, which has been classified into three reactive currents that can be compensated, as follows:

Approach I—the minimization of the Budeanu complement reactive current and the minimization and balancing of the three unbalanced currents.

Approach II—the minimization of the Budeanu reactive current and the minimization and balancing of the three unbalanced currents.

Approach III—the minimization of the Budeanu complemented reactive current and the Budeanu reactive current (the sum of these two currents gives Czarnecki’s reactive current—CPC theory) and the minimization and balancing of the three unbalanced currents.

It is evident that each of the approaches is based on the three-phase RMS value of the individual currents’ components of the receiver being compared to the optimal compensation of the currents’ component and to the results after minimization and balancing.

Table 33. Summary of the three-phase RMS values for the currents’ components for the primary circuit and the circuits after the balancing minimization and perfect compensation are performed.

Component	Original System	Approach
		I		II		III
		Ideal	Mini	Ideal	Mini	Ideal	Mini
$\Vert {\mathit{i}}_{\mathrm{a}}\Vert$	58.880	58.880	58.880	58.880	58.880	58.880	58.880
$\Vert {\mathit{i}}_{\mathrm{s}}\Vert$	3.077	3.077	3.077	3.077	3.077	3.077	3.077
$\Vert {\mathit{i}}_{\mathrm{rB}}\Vert$	19.756	19.756	19.568	0	0.377	0	0.209
$\Vert {\mathit{i}}_{\mathrm{crB}}\Vert$	1.363	0	0.126	1.363	2.539	0	2.562
$\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{p}}\Vert$	3.920	0	4.321	0	4.403	0	4.403
$\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{n}}\Vert$	39.282	1.285	0.930	1.285	0.960	1.285	0.959
$\Vert {\mathit{i}}_{\mathrm{u}}^{\mathrm{z}}\Vert$	22.124	0	4.637	0	4.652	0	4.652
$\Vert {\mathit{i}}_{\mathrm{dB}}\Vert$	45.379	3.335	7.108	3.602	7.607	3.335	7.615
$\Vert \mathit{i}\Vert$	76.918	62.195	62.452	58.990	59.370	58.974	59.371

presents the three-phase RMS values of the currents’ components of a three-phase four-wire load for the main system and three approaches to minimizing and balancing the currents’ components, together with the three-phase RMS values of the load current in a situation of complete compensation.

As observed in the comparison in Table 33, the optimal compensation in each of the three approaches gives a lower three-phase RMS value of the load current compared to its analog in the case of minimizing balancing compensation. However, it should be noted that the difference is up to 0.5 A. The minimum three-phase RMS value of the current was determined using the approach of ideal compensation of the Budeanu reactive current (Czarnecki’s reactive current). However, the difference between the second-best optimal compensation of the Budeanu reactive current is 0.016 A. Looking at the minimization approach to reactive compensation, the lowest three-phase RMS current value of the load was obtained in the second approach based on the limitation of the Budeanu reactive current. The delta between the lowest and second-lowest of the three-phase RMS values is only 0.001 A.

Based on the analysis of the Budeanu distortion current, the lowest value is obtained in two approaches, i.e., compensation of the Budeanu complemented reactive current and compensation of both Budeanu reactive currents. This is explained by the components of the distortion current, namely, in both situations of total compensation, the Budeanu complemented reactive current and the two components of the unbalanced current (i.e., the positive sequence and the zero sequence) were used. The remaining components of the Budeanu distortion current are the scattered current and the unbalanced current of the negative sequence, whose three-phase RMS value is equal in both situations.

Considering that this publication mainly focuses on minimizing balancing compensation, the analysis should concentrate on the values obtained through this approach to compensation. In all three approaches to minimization, it is possible to observe that after compensation, each of the reactance components of the load current is present. This is in line with the idea that the minimization approach focuses on limiting the three-phase RMS values of the specific current components of the load while also reducing the number of reactance elements used.

Table 34. Summary of power factors for all considered approaches with ideal compensation and minimizing balancing compensation.

Component	Original System	Approach
		I		II		III
		Ideal	Mini	Ideal	Mini	Ideal	Mini
$\lambda$	0.766	0.947	0.943	0.998	0.992	0.998	0.992

summarizes the power factors obtained by connecting an optimal compensator and a minimizing balancing compensator.

In Table 34, the two highest power factors corresponding to ideal compensation are highlighted in green. Equal values are obtained for both factors after rounding to 3 decimal places. When both power factors are expanded to 10 decimal places, the difference between the two values is 0.0002667250, benefitting the ideal compensation of the Budeanu reactive current.

Similarly, when focusing on the power factors highlighted in blue, i.e., the values obtained after minimizing balancing compensation, equal values are reached when rounded to 3 decimal places. Expanding both power factors to 10 decimal places, the difference between the two values is 0.0000035122, in favor of the minimizing balancing compensation of the Budeanu reactive currents (the Budeanu reactive current and Budeanu complemented reactive current—Czarnecki’s reactive current).

However, it is important to note that if the power factor were to be measured in a real system using, for example, an electrical power quality analyzer, the power factor would be displayed to three decimal places for most devices. In other words, it can be accepted that both ideal compensation and minimizing balancing compensation, in the two cases described above, lead to identical results when analyzed with respect to the power factor.

For this example, a general economic analysis of the proposed solution can also be performed. Assuming that the price of a choke is 1 u [unit] and the price of a capacitor is 0.5 u,

Table 35. Summary of generalized costs spent on the parts for the construction of a compensator.

Component	Original System	Cost of Approach [u]
		I		II		III
		Ideal	Mini	Ideal	Mini	Ideal	Mini
choke	-	12	6	10	6	11	6
capacitor	-	4.5	1.5	5.5	2	5	2
SUM	-	16.5	7.5	15.5	8	16	8

shows the estimated costs for the different approaches to designing reactance compensators.

As shown in Table 35, the lowest costs occurred when elements for the construction of a compensator were determined for the minimizing balancing compensation of the Budeanu complemented reactive current and the unbalanced current. However, this method (based on Table 34) has the worst power factor of $\lambda =0.943$. In the considered circuit, it has been clearly demonstrated that the method of minimizing balancing compensation leads to a marginal difference in power factors ($\mathsf{\Delta }\lambda =0.006$) compared to optimal compensators as well as identical generalized costs. It must also be emphasized that the cost of minimizing balancing compensation is approximately 50% lower than that of perfect compensation—the assumption of the buying cost of reactance elements.

13. Conclusions

As demonstrated in this article, the minimizing balancing compensation satisfies the basic points because it improves the power factor with only a slight reduction compared to the optimal compensation. The advantage of minimization compared to full optimum compensation is the lower buying cost of the components used in the balancing compensator.

As explained in this paper, in the case of Budeanu reactive currents and the unbalanced current minimization, the systems are not perfectly compensated like in an ideal compensation. It should also be mentioned that the systems discussed can adapt to the load fluctuations and the fluctuations in the power supply conditions. As might be expected, a system whose principal function is to limit the currents of the fundamental harmonic will be less affected by the abovementioned fluctuations than a group of systems tuned to the frequencies of the specific harmonics.

The mathematical description, in which the minimization of the Budeanu reactive currents is found, leads to the next stage in compensation, which is the development of a control algorithm for a frequency converter (active power filter) to eliminate the scattered current and the other currents’ components that have been initially limited by the passive system.