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SECTION I

INTRODUCTION

THE IEEE Standard 1459 [1] includes definitions for the measurement of electric power quantities under sinusoidal, nonsinusoidal, balanced, or unbalanced conditions. The analysis of nonsinusoidal and unbalanced electrical systems is still under research, as demonstrated by works in the field [2], [3], [4], [5], [6]. It is stated in the IEEE Standard 1459 introduction that “the new definitions were developed to give guidance with respect to the quantities that should be measured or monitored for revenue purposes, engineering economic decisions, and determination of major harmonic polluters. ” Some works deal with instruments for the measurement of the electric power quantities defined in IEEE Standard 1459 [7], [8], [9], [10], [11]. In [12], a new perspective for the IEEE Standard 1459 definitions is introduced by using the stationary wavelet transform. The IEEE Standard 1459 power magnitudes are also used in the detection of the major sources of waveform distortion [13] or in the definition of the reference currents of shunt active power compensators [14], [15], [16].

Some of the definitions proposed in the IEEE Standard 1459 are still under discussion. In [17], the author disagrees with the definition of nonfundamental power. In [18], the authors propose a new definition for unbalance power following an instantaneous approach based on well-established concepts used in the IEEE Standard 1459 for the definition of active and reactive power in single-phase systems. In the approach proposed in this paper, the IEEE Standard 1459 power magnitudes are calculated using the supply voltages and the load currents. After resolving Formula$S_{e}$, the different power terms are related to the power magnitudes defined in the IEEE Standard 1459.

The structure of this paper is as follows. After the introduction, a summary of the IEEE Standard 1459 power definitions used in this paper is presented. The analysis focuses on the study of unbalanced and nonlinear loads. In Section III, the IEEE Standard 1459 power magnitudes are calculated by means of the voltage and current components Formula$V_{e}$ and Formula$I_{e}$ for an unbalanced linear load and a nonlinear balanced load. The comparison of the IEEE Standard 1459 definitions with the power magnitudes obtained by means of the proposed approach in this paper permit highlighting some differences that raise doubts about the definitions of Formula$V_{e}$ and Formula$I_{e}$ included in the IEEE Standard 1459. Section IV includes the new expressions of Formula$V_{e}$ and Formula$I_{e}$ that overcome the presented problems. Section V includes a comparison between the existing and new definitions by means of a numerical example. This paper concludes with a summary of the main points developed in this paper.

SECTION II

POWER QUANTITIES IN IEEE STANDARD 1459

The IEEE Standard 1459 establishes new electric power quantities for any situation of the electric power system. The new power magnitudes are obtained by means of the resolution of the effective apparent power in three-phase systems introduced by Buchholz in [19]. The symmetrical components of the supply fundamental voltages and load fundamental currents are used in the IEEE Standard 1459 to define several fundamental power magnitudes. The importance of the fundamental positive-sequence powers Formula$P_{1}^{+}$ and Formula$Q_{1}^{+}$ are recognized in [1] and [7].

This section details the IEEE Standard 1459 power definitions applied to some electric circuits. These power definitions will be resolved in Section III of this paper by means of a new approach that uses the supply voltages and load current components. Fig. 1 shows the general three-phase four-wire electrical system analyzed in this paper. The instantaneous supply voltages (phase to neutral) are denoted by Formula$v_{a}$, Formula$v_{b}$, and Formula$v_{c}$. The ac supply can feed the load with fundamental and harmonic symmetric or asymmetric voltage components. At the same time, the three-phase load can be balanced or unbalanced and linear or nonlinear. Formula$i_{a}$, Formula$i_{b}$, and Formula$i_{c}$ are the instantaneous supply currents demanded by the load, and Formula$i_{n}$ is the neutral instantaneous current. It is considered an ideal line in the analysis, but the effect of the unbalanced and nonlinear currents in the electrical system is included in the supply voltages.

Figure 1
Fig. 1. Three-phase four-wire electrical system under analysis.

The effective voltage Formula$(V_{e})$ is defined as a function of the supply rms line-to-neutral voltages Formula$(V_{a},V_{b},V_{c})$, and rms line-to-line voltages Formula$(V_{ab},V_{bc},V_{ca})$ as follows: Formula TeX Source $$V_{e}=\sqrt{3\left(V_{a}^{2}+V_{b}^{2}+V_{c}^{2}\right)+\left(V_{ab}^{2}+V_{bc}^{2}+V_{ca}^{2}\right)\over 18}.\eqno{\hbox{(1)}}$$

The effective current Formula$(I_{e})$ is defined as a function of the phase Formula$(I_{a},I_{b},I_{c})$ and neutral Formula$(I_{n})$ rms currents as follows: Formula TeX Source $$I_{e}=\sqrt{I_{a}^{2}+I_{b}^{2}+I_{c}^{2}+I_{n}^{2}\over 3}.\eqno{\hbox{(2)}}$$

The effective apparent power Formula$(S_{e})$ is defined as follows: Formula TeX Source $$S_{e}=3\ V_{e}I_{e}.\eqno{\hbox{(3)}}$$

As indicated in [20], several concepts of the apparent power can be found in the literature. An explanation of why Formula$V_{e}$ and Formula$I_{e}$ are defined in this way is given in [21] and [22].

Power magnitudes for unbalanced linear loads and for nonlinear loads are detailed separately in the following subsections. Section II-A summarizes power definitions for a three-phase unbalanced linear load connected to a three-phase fundamental asymmetric supply. Section II-B summarizes the IEEE Standard 1459 power definitions for a three-phase nonlinear load connected to a three-phase asymmetric supply that also includes harmonic components.

A. Power Magnitudes for a Three-Phase Unbalanced Linear Load Connected to a Three-Phase Fundamental Asymmetric Supply

For this case, the supply voltages in Fig. 1 are not equal Formula$(V_{a}\neq V_{b}\neq V_{c})$. Formula$V_{e}$ coincides with the fundamental effective voltage Formula$(V_{e1})$ because the supply voltage only contains fundamental components. Subscript “1” is used to identify the fundamental components. Formula$V_{e1}$ is calculated in [1, p. 18, Sec. 3.2.2.8] in terms of the rms fundamental positive-, negative-, and zero-sequence voltages (Formula$V_{1}^{+}$, Formula$V_{1}^{-}$, and Formula$V_{1}^{0}$, respectively) as follows: Formula TeX Source $$V_{e}=V_{e1}=\sqrt{\left(V_{1}^{+}\right)^{2}+\left(V_{1}^{-}\right)^{2}+{\left(V_{1}^{0}\right)^{2}\over 2}}.\eqno{\hbox{(4)}}$$

The currents through the unbalanced linear load are not equal (Formula$I_{a}\neq I_{b}\neq I_{c}$ and Formula$I_{n}\neq 0$). Formula$I_{e}$ coincides with the fundamental effective current Formula$(I_{e1})$. Formula$I_{e1}$ is resolved in [1, p. 18, Sec. 3.2.2.8] in terms of the rms fundamental positive-, negative-, and zero-sequence currents (Formula$I_{1}^{+}$, Formula$I_{1}^{-}$, and Formula$I_{1}^{0}$) as follows: Formula TeX Source $$I_{e}=I_{e1}=\sqrt{\left(I_{1}^{+}\right)^{2}+\left(I_{1}^{-}\right)^{2}+4\left(I_{1}^{0}\right)^{2}}.\eqno{\hbox{(5)}}$$

According to (3)(5), Formula$S_{e}$ corresponds to the fundamental effective apparent power Formula$(S_{e1})$ Formula TeX Source $$S_{e}=S_{e1}=3\ V_{e1}I_{e1}.\eqno{\hbox{(6)}}$$

The fundamental positive-sequence apparent power Formula$(S_{1}^{+})$ is obtained by the product of Formula$V_{1}^{+}$ and Formula$I_{1}^{+}$, as follows: Formula TeX Source $$S_{1}^{+}=3V_{1}^{+}I_{1}^{+}.\eqno{\hbox{(7)}}$$

Formula$S_{1}^{+}$ exists in three-phase linear and balanced electrical systems. Formula$S_{1}^{+}$ is resolved into the fundamental positive-sequence active power Formula$(P_{1}^{+})$ and the fundamental positive-sequence reactive power Formula$(Q_{1}^{+})$ Formula TeX Source $$\eqalignno{\left(S_{1}^{+}\right)^{2}=&\,\left(3V_{1}^{+}I_{1}^{+}\right)^{2}\cdot\left(\left(\cos\theta_{1}^{+}\right)^{2}+\left(\sin\theta_{1}^{+}\right)^{2}\right)\cr =&\,\left(P_{1}^{+}\right)^{2}+\left(Q_{1}^{+}\right)^{2}.&\hbox{(8)}}$$ where Formula$\theta_{1}^{+}$ is the phase angle between Formula$V_{1}^{+}$ and Formula$I_{1}^{+}$. The unbalanced power Formula$(S_{U1})$ is calculated by means of Formula$S_{e1}$ and Formula$S_{1}^{+}$ as follows: Formula TeX Source $$S_{U1}^{2}=(S_{e1})^{2}-\left(S_{1}^{+}\right)^{2}.\eqno{\hbox{(9)}}$$

Formula$S_{U1}$ is the power magnitude that quantifies the load unbalances and the voltage asymmetries, including only fundamental components of the voltages and currents.

The fundamental negative-sequence apparent power Formula$(S_{1}^{-})$ and the fundamental zero-sequence apparent power Formula$(S_{1}^{0})$ are resolved by means of their corresponding active and reactive parts as follows: Formula TeX Source $$\eqalignno{\left(S_{1}^{-}\right)^{2}=&\,\left(P_{1}^{-}\right)^{2}+\left(Q_{1}^{-}\right)^{2}=\left(3V_{1}^{-}I_{1}^{-}\right)^{2}&\hbox{(10)}\cr\left(S_{1}^{0}\right)^{2}=&\,\left(P_{1}^{0}\right)^{2}+\left(Q_{1}^{0}\right)^{2}=\left(3V_{1}^{0}I_{1}^{0}\right)^{2}.&\hbox{(11)}}$$

The fundamental positive-, negative-, and zero-sequence active powers (Formula$P_{1}^{+}$, Formula$P_{1}^{-}$, and Formula$P_{1}^{0}$, respectively) are defined in the IEEE Standard 1459 as follows: Formula TeX Source $$\eqalignno{P_{1}^{+}=&\,3V_{1}^{+}I_{1}^{+}\cos\theta_{1}^{+}&\hbox{(12)}\cr P_{1}^{-}=&\,3V_{1}^{-}I_{1}^{-}\cos\theta_{1}^{-}&\hbox{(13)}\cr P_{1}^{0}=&\,3V_{1}^{0}I_{1}^{0}\cos\theta_{1}^{0}&\hbox{(14)}}$$ where Formula$\theta_{1}^{+}$, Formula$\theta_{1}^{-}$, and Formula$\theta_{1}^{0}$ are the phase angle between the voltage and the current of the corresponding symmetrical components. Formula$P_{1}^{+}$ is the unique active power that is considered useful while Formula$P_{1}^{-}$ and Formula$P_{1}^{0}$ are considered useless active powers due to the power losses produced in the load and line wires [6], [9], [16], [20], [22], and [23]. Instead of using the term “non-active powers” to denote the power magnitudes that are not desirable in an ideal electrical system, the authors considered the term “useless powers” to be more suitable, which includes all power magnitudes that are not Formula$P_{1}^{+}$.

The fundamental active power Formula$(P_{1})$ is obtained as the sum of the three fundamental active powers that are equal to the active power Formula$(P)$, in this case Formula TeX Source $$P_{1}=P_{1}^{+}+P_{1}^{-}+P_{1}^{0}=P.\eqno{\hbox{(15)}}$$

The fundamental positive-, negative-, and zero-sequence reactive powers (Formula$Q_{1}^{+}$, Formula$Q_{1}^{-}$, and Formula$Q_{1}^{0}$, respectively) are produced by the product of the currents that are in quadrature with their corresponding voltages and are defined as follows: Formula TeX Source $$\eqalignno{Q_{1}^{+}=&\,3V_{1}^{+}I_{1}^{+}\sin\theta_{1}^{+}&\hbox{(16)}\cr Q_{1}^{-}=&\,3V_{1}^{-}I_{1}^{-}\sin\theta_{1}^{-}&\hbox{(17)}\cr Q_{1}^{0}=&\,3V_{1}^{0}I_{1}^{0}\sin\theta_{1}^{0}.&\hbox{(18)}}$$

Formula$Q_{1}^{+}$, Formula$Q_{1}^{-}$, and Formula$Q_{1}^{0}$ are considered useless power flows because they are not converted into any other kind of energy. The fundamental reactive power Formula$(Q_{1})$ is obtained as the sum of the reactive powers that is equal to the conventional reactive power Formula$(Q)$ in this case Formula TeX Source $$Q_{1}=Q_{1}^{+}+Q_{1}^{-}+Q_{1}^{0}=Q.\eqno{\hbox{(19)}}$$

B. Power Magnitudes for a Three-Phase Nonlinear Load Connected to a Three-Phase Nonsinusoidal Asymmetric Supply

In this case, the supply voltages include harmonic components produced by the harmonic current components demanded by the nonlinear load [24]. Formula$V_{e}$ is divided in [1] into Formula$V_{e1}$ and the nonfundamental effective voltage Formula$(V_{eH})$ as follows: Formula TeX Source $$V_{e}^{2}=V_{e1}^{2}+V_{eH}^{2}\eqno{\hbox{(20)}}$$ where the subscript “Formula$H$” denotes the nonfundamental quantities. Formula$V_{e1}$ is composed by the fundamental positive-, negative-, and zero-sequence terms of the supply voltages, as in (4). Formula$V_{eH}$ is composed by the nonfundamental voltages.

Following the same approach for the currents, Formula$I_{e}$ is resolved into Formula$I_{e1}$ and the nonfundamental effective current Formula$(I_{eH})$ as follows: Formula TeX Source $$I_{e}^{2}=I_{e1}^{2}+I_{eH}^{2}.\eqno{\hbox{(21)}}$$

Formula$I_{e1}$ is composed by the fundamental positive-, negative-, and zero-sequence terms of the load currents, as in (5). Formula$I_{eH}$ is composed by the nonfundamental currents.

The nonfundamental effective apparent power Formula$(S_{eN})$ is defined as follows: Formula TeX Source $$S_{eN}^{2}=S_{e}^{2}-S_{e1}^{2}=(3\ V_{e1}I_{eH})^{2}+(3\ V_{eH}I_{e1})^{2}+(3\ V_{eH}I_{eH})^{2}.\eqno{\hbox{(22)}}$$

Formula$S_{eN}$ is resolved in three terms. The first term is the current distortion power Formula$(D_{eI})$ Formula TeX Source $$D_{eI}^{2}=(3\ V_{e1}I_{eH})^{2}.\eqno{\hbox{(23)}}$$

The second term is the voltage distortion power Formula$(D_{eV})$ Formula TeX Source $$D_{eV}^{2}=(3\ V_{eH}I_{e1})^{2}.\eqno{\hbox{(24)}}$$

The third term is the harmonic apparent power Formula$(S_{eH})$, as defined in (25). Formula$S_{eH}$ is resolved in the harmonic active power Formula$(P_{H})$ and the harmonic distortion power Formula$(D_{eH})$ as follows: Formula TeX Source $$S_{eH}^{2}=(3\ V_{eH}I_{eH})^{2}=P_{H}^{2}+D_{eH}^{2}.\eqno{\hbox{(25)}}$$

Formula$P_{H}$ is considered a useless power since it produces losses in the load and in the electrical system [6], [9], [20], and [22]. Formula$P_{H}$ is defined by means of the harmonic voltages and currents of the same harmonic order that are in phase as follows: Formula TeX Source $$P_{H}=3\sum_{h\neq 1}V_{h}I_{h}\cos\theta_{h}\eqno{\hbox{(26)}}$$ where the subscript “Formula$h$” is used to denote the harmonic order and Formula$\theta_{h}$ is the phase angle between Formula$V_{h}$ and Formula$I_{h}$.

The active power Formula$(P)$ in the electric power system is obtained as the sum (15) and (26) Formula TeX Source $$P=P_{1}+P_{H}.\eqno{\hbox{(27)}}$$

The harmonic reactive power Formula$(Q_{H})$ is defined as follows: Formula TeX Source $$Q_{H}=3\sum_{h\neq 1}V_{h}I_{h}\sin\theta_{h}.\eqno{\hbox{(28)}}$$

The conventional reactive power Formula$(Q)$ in the electric power system is obtained as the sum of (19) and (28) Formula TeX Source $$Q=Q_{1}+Q_{H}.\eqno{\hbox{(29)}}$$

In the following section, some of the power magnitudes defined previously are calculated using the supply voltages and the load current components. The analysis is performed over two different circuits and yields differences with some of the definitions included in IEEE Standard 1459.

SECTION III

EFFECTIVE APPARENT POWER RESOLUTION BY MEANS OF THE VOLTAGE AND CURRENT COMPONENTS

In this section, Formula$S_{e}$ is resolved following an approach that uses the supply voltages and load currents. The products of the different current and voltage components that appear after this resolution are related to the different power magnitudes defined in Section II. Differences appear between the definitions of some useless active powers and reactive powers included in IEEE Standard 1459 and the expressions obtained by means of the approach proposed in this paper. To highlight these disagreements, two different cases are analyzed in the following sections.

The first studied case corresponds to a three-phase unbalanced linear load connected to a three-phase asymmetric supply that only contains fundamental voltage components. The first case is focused on the fundamental active and reactive power terms that are part of Formula$S_{U1}$. The second case highlights the IEEE Standard 1459 quantification error by means the calculation of the harmonic active and reactive powers that are part of Formula$S_{eN}$ The load of case 2 includes a balanced linear part that only demands fundamental positive-sequence active currents in parallel with a balanced nonlinear load. The supply voltages include harmonic terms of the same order than the harmonic currents, corresponding to the voltages that appear between load terminals due to nonideal distribution line impedances.

These two cases do not correspond to real situations that can appear in electrical power networks. They are selected to highlight the quantification errors included in some IEEE Standard 1459 power definitions. The quantification errors can only be demonstrated by means of the comparison between the results obtained with the approach used in this paper and the well-known active and reactive power definitions included in IEEE Standard 1459. The two cases are selected due to a reduced number of voltage and current components where the quantification errors are demonstrated. A more realistic case that includes all types of power magnitudes is presented later in this paper. It highlights the differences between IEEE Standard 1459 definitions and the proposed ones.

The subscript “_m” is used to distinguish the expressions obtained in this paper that are mistaken and disagree with the definitions included in IEEE Standard 1459. Power magnitudes calculated specifically for case 1 and case 2 are identified with subscripts “_c1” and “_c2,” respectively. Expressions proposed by the authors that redefine magnitudes included in IEEE Standard 1459 are distinguished by the subscript “#.”

Case 1: Three-Phase Unbalanced Linear Load Connected to an Asymmetric Fundamental Supply

In this case, the three-phase unbalanced linear load Formula$(Z_{a}\neq Z_{b}\neq Z_{c})$ demands fundamental positive-, negative-, and zero-sequence current components (Formula$I_{a}\neq I_{b}\neq I_{c}$ and Formula$I_{n}\neq 0$), as represented in Fig. 2. The current through the neutral wire corresponds to the sum of the currents through the three lines. The neutral current includes only the sum of the fundamental zero-sequence components because positive and negative currents are zero-sum components [25]. The three-phase supply is asymmetric, including fundamental positive-, negative-, and zero-sequence voltage components. The IEEE Standard 1459 power magnitudes for this case are detailed in Section II-A.

Figure 2
Fig. 2. Equivalent circuit of a three-phase unbalanced linear load connected to an asymmetric fundamental supply.

Replacing (4) and (5) in (6) and expanding terms [18], Formula$S_{e1}$ is expressed as follows: Formula TeX Source $$\eqalignno{S_{e1}^{2}=&\,9\cdot\left[\left(V_{1}^{+}\cdot I_{1}^{+}\right)^{2}+\left(V_{1}^{+}\cdot I_{1}^{-}\right)^{2}+4\cdot\left(V_{1}^{+}\cdot I_{1}^{0}\right)^{2}\right.\cr&\qquad+\left(V_{1}^{-}\cdot I_{1}^{+}\right)^{2}+\left(V_{1}^{-}\cdot I_{1}^{-}\right)^{2}+4\cdot\left(V_{1}^{-}\cdot I_{1}^{0}\right)^{2}\cr&\qquad+{1\over 2}\cdot\left(V_{1}^{0}\cdot I_{1}^{+}\right)^{2}+{1\over 2}\cdot\left(V_{1}^{0}\cdot I_{1}^{-}\right)^{2}\cr&\qquad\left.+2\cdot\left(V_{1}^{0}\cdot I_{1}^{0}\right)^{2}\right].&\hbox{(30)}}$$

The first term in (30) appears in the expression of Formula$S_{1}^{+}$ in (7), while the remaining terms belong to Formula$S_{U1}$, which is expressed as follows: Formula TeX Source $$\eqalignno{S_{U1}^{2}\!=\!&\,9\cdot\left[\left(V_{1}^{+}\cdot I_{1}^{-}\right)^{2}+4\cdot\left(V_{1}^{+}\cdot I_{1}^{0}\right)^{2}+\left(V_{1}^{-}\cdot I_{1}^{+}\right)^{2}\right.\cr&\qquad+\left(V_{1}^{-}\cdot I_{1}^{-}\right)^{2}+4\cdot\left(V_{1}^{-}\cdot I_{1}^{0}\right)^{2}+{1\over 2}\cdot\left(V_{1}^{0}\cdot I_{1}^{+}\right)^{2}\cr&\qquad\left.+{1\over 2}\cdot\left(V_{1}^{0}\cdot I_{1}^{-}\right)^{2}+2\cdot\left(V_{1}^{0}\cdot I_{1}^{0}\right)^{2}\right].&\hbox{(31)}}$$

The term Formula$V_{1}^{-}\cdot I_{1}^{-}$ appears in the expression of Formula$S_{1}^{-}$ in (10) while the term Formula$V_{1}^{0}\cdot I_{1}^{0}$ appears in the expression of Formula$S_{1}^{0}$ in (11). A different resolution of Formula$S_{U1}$ is reported in [26]. The fundamental zero-sequence apparent power obtained by means of (31), denoted as Formula$S_{1\_c1}^{0}$, can be written as follows: Formula TeX Source $$\left(S_{1\_c1}^{0}\right)^{2}=\left(P_{1\_m}^{0}\right)^{2}+\left(Q_{1\_m}^{0}\right)^{2}=\left(3\sqrt{2}V_{1}^{0}\cdot I_{1}^{0}\right)^{2}.\eqno{\hbox{(32)}}$$

By means of the use of the voltage and current components in the definitions included in IEEE Standard 1459, the expressions of the zero-sequence active power Formula$(P_{1\_m}^{0})$ and the zero-sequence reactive power Formula$(Q_{1\_m}^{0})$ included in (32) are as follows: Formula TeX Source $$\eqalignno{P_{1\_m}^{0}=&\,\sqrt{2}\left(3V_{1}^{0}I_{1}^{0}\cos\theta_{1}^{0}\right)&\hbox{(33)}\cr Q_{1\_m}^{0}=&\,\sqrt{2}\left(3V_{1}^{0}I_{1}^{0}\sin\theta_{1}^{0}\right).&\hbox{(34)}}$$

In the expressions of Formula$P_{1\_m}^{0}$ and Formula$Q_{1\_m}^{0}$, the factor Formula$\sqrt{2}$ disagrees with the commonly accepted expression of Formula$P_{1}^{0}$ and Formula$Q_{1}^{0}$ [(14) and (18), respectively] included in IEEE Standard 1459. To obtain (14) and (18), all of the factors that multiply the power terms between brackets in (30) must be equal to one. It is stated in [17] that “the definition of non-fundamental power Formula$S$ is flawed.” After the analysis was performed previously, the previous statement can be extended to include a fundamental power magnitude Formula$S_{U1}$.

The remaining power magnitudes defined in IEEE Standard 1459 (Formula$P_{1}^{+}$, Formula$P_{1}^{-}$, Formula$Q_{1}^{+}$, and Formula$Q_{1}^{-}$) coincide with the expressions obtained by means of the use of the voltage and current components. Nevertheless, it is necessary to modify the definitions included in (4) and (5) as they appear in (35) and (36) to reach an agreement in the definitions included in IEEE Standard 1459 with the results obtained by means of the voltage and current components Formula TeX Source $$\eqalignno{V_{e1\#}=&\,\sqrt{\left(V_{1}^{+}\right)^{2}+\left(V_{1}^{-}\right)^{2}+\left(V_{1}^{0}\right)^{2}}&\hbox{(35)}\cr\cr I_{e1\#}=&\,\sqrt{\left(I_{1}^{+}\right)^{2}+\left(I_{1}^{-}\right)^{2}+\left(I_{1}^{0}\right)^{2}}.&\hbox{(36)}}$$

The expressions obtained for this case show that the fundamental zero-sequence current is over valuated in (5) by a factor of 4 while the fundamental zero-sequence voltage is under valuated in (5) by a factor of 1/2. These factors result in a value of Formula$S_{U1}$ that does not quantify the unbalance phenomenon correctly. The product of (35) and (36) yields the expression of the new fundamental effective apparent power Formula$(S_{e1\#})$ that contains the same terms as (30), but with all factors equal to one.

Figure 3
Fig. 3. Power system with zero-sequence harmonic components.

Case 2: Three-Phase Balanced Nonlinear Load Connected to a Three-Phase Nonsinusoidal Symmetric Supply

The circuit analyzed in this case is represented in Fig. 3. The load is balanced and includes a linear part (represented by the resistance R) in parallel with a nonlinear distorting load (denoted as D). Harmonic orders of voltage and current components in balanced systems can be classified according to the rotation of the corresponding phasors [14], [27]. Harmonic current components of an order equal to Formula$3n+3$ (with Formula${\rm n}=0,1,\ldots,\infty$) are in phase (with any rotation sequence) and are denoted as zero-sequence components. To simplify the analysis, it is assumed for this case that the nonlinear load demands only zero-sequence current components Formula$(i_{ha}^{0},i_{hb}^{0},i_{hc}^{0})$, with an rms value equal to Formula$I_{h}^{0}$ in the three phases. The resistances demand only fundamental positive-sequence active current components Formula$(i_{1a}^{+a},i_{1b}^{+a},i_{1c}^{+a})$, with an rms value equal to Formula$I_{1}^{+a}$ in the three phases. The neutral current is equal to the sum of the three line zero-sequence current components Formula$(i_{ha}^{0}+i_{hb}^{0}+i_{hc}^{0})$, with an rms value equal to Formula$3\cdot I_{h}^{0}$.

The supply voltage includes the fundamental positive-sequence voltages Formula$(v_{1a}^{+},v_{1b}^{+},v_{1c}^{+})$ plus some harmonic zero-sequence components Formula$(v_{ha}^{0},v_{hb}^{0},v_{hc}^{0})$ that appear due to the flow of the harmonic zero-sequence current components through the distribution lines. Under this condition, the harmonic order of the current and voltage harmonic components is the same. The rms value of the fundamental positive-sequence voltages is Formula$V_{1}^{+}$, while the rms value of the zero-sequence voltage components is Formula$V_{h}^{0}$. For this case, Formula$V_{e}$ is calculated by means of (20) as follows: Formula TeX Source $$V_{e}=\sqrt{\left(V_{1}^{+}\right)^{2}+{1\over 2}\left(V_{h}^{0}\right)^{2}}.\eqno{\hbox{(37)}}$$

Replacing the values of the load currents in (21) defined previously, Formula$I_{e}$ is equal to Formula TeX Source $$I_{e}=\sqrt{\left(I_{1}^{+a}\right)^{2}+4\left(I_{h}^{0}\right)^{2}}.\eqno{\hbox{(38)}}$$

Substituting (37) and (38) into (3), and expanding the terms, Formula$S_{e}$ is written as follows: Formula TeX Source $$S_{e}^{2}\!=\!9\!\left[\left(V_{1}^{+}I_{1}^{+a}\right)^{2}\!+\!4\left(V_{1}^{+}I_{h}^{0}\right)^{2}\!+\!{1\over 2}\left(V_{h}^{0}I_{1}^{+a}\right)^{2}\!+\!2\left(V_{h}^{0}I_{h}^{0}\right)^{2}\right].\eqno{\lower12pt\hbox{(39)}}$$

The first term in (39) corresponds to Formula$P_{1}^{+}$, and is obtained by the product of the fundamental positive-sequence voltage with the fundamental positive-sequence active current. The remaining terms in (39) are part of Formula$S_{eN}$ Formula TeX Source $$S_{eN}^{2}=9\left[4\left(V_{1}^{+}I_{h}^{0}\right)^{2}+{1\over 2}\left(V_{h}^{0}I_{1}^{+a}\right)^{2}+2\left(V_{h}^{0}I_{h}^{0}\right)^{2}\right].\eqno{\hbox{(40)}}$$

It is possible to identify the power magnitudes in (40) and detailed in (23)(25). The first term in (40) includes the product of Formula$V_{1}^{+}$ and Formula$I_{h}^{0}$ and corresponds to Formula$D_{eI}$. By rearranging the terms, Formula$D_{eI}$ can be expressed as follows: Formula TeX Source $$D_{eI}^{2}=\left(3\ V_{1}^{+}\left(2I_{h}^{0}\right)\right)^{2}.\eqno{\hbox{(41)}}$$

The expression of Formula$I_{eH}$ in (23) is equal to Formula$2\cdot I_{h}^{0}$ in this case. The second term in (40) includes the product of Formula$V_{h}^{0}$ and Formula$I_{1}^{+}$ and corresponds to Formula$D_{eV}$. By rearranging the terms, Formula$D_{eV}$ can be expressed as follows: Formula TeX Source $$D_{eV}^{2}=\left(3\ \left({1\over\sqrt{2}}V_{h}^{0}\right)I_{1}^{+}\right)^{2}.\eqno{\hbox{(42)}}$$

The expression of Formula$V_{eH}$ in (24) is equal to Formula$((1/\sqrt{2})\cdot V_{h}^{0})$. The last term in (40) includes the product of Formula$V_{h}^{0}$ and Formula$I_{h}^{0}$ and corresponds to Formula$S_{eH}$. By rearranging the terms, Formula$S_{eH}$ can be expressed as follows: Formula TeX Source $$S_{eH}^{2}=(3\ V_{eH}I_{eH})^{2}=2\left(3\ V_{h}^{0}I_{h}^{0}\right)^{2}.\eqno{\hbox{(43)}}$$

As detailed in (25), Formula$S_{eH}$ can be resolved by means of Formula$P_{H}$ and Formula$D_{eH}$. Since (43) includes harmonic voltage and current components of the same order, Formula$S_{eH}$ is resolved as follows: Formula TeX Source $$S_{eH}^{2}=2\cdot\left(3\ V_{h}^{0}I_{h}^{0}\right)^{2}\left(\left(\cos\theta_{h}^{0}\right)^{2}+\left(\sin\theta_{h}^{0}\right)^{2}\right).\eqno{\hbox{(44)}}$$

The useless harmonic active power included in (44) is denoted as Formula$P_{H\_m}$ and is expressed as follows: Formula TeX Source $$P_{H\_m}=\sqrt{2}\left(3V_{h}^{0}I_{h}^{0}\cos\theta_{h}^{0}\right).\eqno{\hbox{(45)}}$$

The other term in (44) corresponds to Formula$D_{eH}$ and in the case under analysis, it corresponds to a harmonic reactive power that is expressed as follows: Formula TeX Source $$D_{eH}=Q_{H\_m}=\sqrt{2}\left(3V_{h}^{0}I_{h}^{0}\sin\theta_{h}^{0}\right).\eqno{\hbox{(46)}}$$

Expressions (45) and (46) do not coincide with the commonly accepted expressions of Formula$P_{H}$ and Formula$Q_{H}$ written in (26) and (28), respectively. The differences are due to the factor Formula$\sqrt{2}$ that multiplies the remaining terms that appear in Formula$P_{H}$ and Formula$Q_{H}$. This factor is responsible for the increase in the nonfundamental power quoted in [17]. To correct these errors, it is necessary to modify (37) and (38) as follows: Formula TeX Source $$\eqalignno{V_{e\_c2}=&\,\sqrt{\left(V_{1}^{+}\right)^{2}+\left(V_{h}^{0}\right)^{2}}&\hbox{(47)}\cr\cr I_{e\_c2}=&\,\sqrt{\left(I_{1}^{+a}\right)^{2}+\left(I_{h}^{0}\right)^{2}}.&\hbox{(48)}}$$

These expressions yield to new expressions of the effective apparent power Formula$(S_{e\_c2})$ and the nonfundamental effective apparent power Formula$(S_{eN\_c2})$ Formula TeX Source $$\eqalignno{S_{e\_c2}^{2}=&\,9\left[\left(V_{1}^{+}I_{1}^{+a}\right)^{2}+\left(V_{1}^{+}I_{h}^{0}\right)^{2}+\left(V_{h}^{0}I_{1}^{+a}\right)^{2}\right. \left.+\left(V_{h}^{0}I_{h}^{0}\right)^{2}\right]\cr&&\hbox{(49)}\cr\noalign{\vskip 4pt}S_{eN\_c2}^{2}=&\,9\left[\left(V_{1}^{+}I_{h}^{0}\right)^{2}+\left(V_{h}^{0}I_{1}^{+a}\right)^{2}+\left(V_{h}^{0}I_{h}^{0}\right)^{2}\right].&\hbox{(50)}}$$

The expressions of Formula$S_{e\_c2}$ and Formula$S_{eN\_c2}$ contain the same terms as that of (39) and (40), but all of the factors that multiply the terms between brackets are equal to one. The last term in (50) includes Formula$P_{H}$ and Formula$Q_{H}$ as defined in (26) and (28), respectively, avoiding the erroneous factors found in (45) and (46).

The analysis of these new expressions yields new definitions of the effective voltage and current valid for all kinds of situations in the electrical system.

SECTION IV

EFFECTIVE QUANTITIES IN UNBALANCED AND NONLINEAR SYSTEMS

The IEEE Standard 1459 states in its introduction that “This trial-use standard is meant to provide definitions extended from the well-established concepts.” Two different approaches are used in the IEEE Standard 1459 to obtain new definitions. The first approach uses some voltage and current components to define power magnitudes. For example, the fundamental positive-sequence voltage and current are used to define Formula$P_{1}^{+}$ and Formula$Q_{1}^{+}$, as seen in (12) and (16). The second approach uses the subtraction of some fundamental power magnitudes to some effective power to obtain a new power magnitude. Examples of this second approach are the definitions of Formula$S_{U1}$, as in (9), and Formula$S_{eN}$, as in (22).

The approach used in this paper to calculate Formula$S_{U1}$ and Formula$S_{eN}$ in the studied cases is based on the use of the voltage and current components in the expressions of Formula$S_{e}$. The number of voltage and current components in the analysis of Formula$S_{U1}$ is limited to six, corresponding to the fundamental positive-, negative-, and zero-sequence voltage and current components. The case of a balanced nonlinear load that demands harmonic zero-sequence current is selected to limit the number of voltage and current components when Formula$S_{eN}$ is analyzed. The flow of current through the neutral wire is the common feature between the two studied cases.

After resolving Formula$S_{e}$ by means of the voltage and current components, problems arise with some commonly accepted power magnitudes such as the harmonic active power and the fundamental zero-sequence active power. The problems come up when zero-sequence components of voltage and current exist in the power system. In all of these cases, the expressions of the power magnitudes include a factor that yields to expressions of the power magnitudes that do not coincide with the commonly accepted power magnitudes.

After the results presented in cases 1 and 2 for two different three-phase four-wire electrical systems, it is necessary to modify the definitions of Formula$V_{e}$ and Formula$I_{e}$ included in IEEE Standard 1459. To remove the erroneous factors, the proposed definitions of the effective voltage Formula$V_{e\#}$ and the effective current Formula$I_{e\#}$ are as follows: Formula TeX Source $$\eqalignno{V_{e\#}=&\,\sqrt{V_{a}^{2}+V_{b}^{2}+V_{c}^{2}\over 3}&\hbox{(51)}\cr\noalign{\vskip 4pt} I_{e\#}=&\,\sqrt{I_{a}^{2}+I_{b}^{2}+I_{c}^{2}\over 3}.&\hbox{(52)}}$$

With (51) and (52), a new expression of the effective apparent power Formula$(S_{e\#})$ is obtained Formula TeX Source $$S_{e\#}^{2}=9V_{e\#}^{2}I_{e\#}^{2}=\left(V_{a}^{2}+V_{b}^{2}+V_{c}^{2}\right)\left(I_{a}^{2}+I_{b}^{2}+I_{c}^{2}\right)\eqno{\hbox{(53)}}$$ which yields to a new expression of the fundamental effective apparent power Formula$(S_{e1\#})$ Formula TeX Source $$S_{e1\#}^{2}=9V_{e1\#}^{2}I_{e1\#}^{2}=\left(V_{a1}^{2}+V_{b1}^{2}+V_{c1}^{2}\right)\left(I_{a1}^{2}+I_{b1}^{2}+I_{c1}^{2}\right).\eqno{\lower12pt\hbox{(54)}}$$

Formula$S_{e1\#}$ can be expressed by means of the fundamental symmetrical components [28] as follows: Formula TeX Source $$S_{e1\#}^{2}\!=\!\!\left(\!\left(V_{1}^{+}\right)^{2}\!+\!\left(V_{1}^{-}\right)^{2}\!+\!\left(V_{1}^{0}\right)^{2}\right)\!\cdot\!\left(\!\left(I_{1}^{+}\right)^{2}\!+\!\left(I_{1}^{-}\right)^{2}\!+\!\left(I_{1}^{0}\right)^{2}\right).\eqno{\lower12pt\hbox{(55)}}$$

With the proposed expressions of Formula$V_{e\#}$, Formula$I_{e\#}$, and Formula$S_{e\#}$, the definitions of the new unbalance power Formula$(S_{U1\#})$ and the new nonfundamental effective apparent power Formula$(S_{eN\#})$ are as follows: Formula TeX Source $$\eqalignno{S_{U1\#}^{2}=&\,S_{e1\#}^{2}-\left(S_{1}^{+}\right)^{2}&\hbox{(56)}\cr S_{eN\#}^{2}=&\,S_{e\#}^{2}-S_{e1\#}^{2}\cr=&\,9\left[(V_{e1\#}I_{eH\#})^{2}+(V_{eH\#}I_{e1\#})^{2}\right.\cr&\quad\ \left.+(V_{eH\#}I_{eH\#})^{2}\right].&\hbox{(57)}}$$

All of these power quantities obtained from (51) and (52) provide a numerical result that is smaller than that obtained using the definitions included in IEEE Standard 1459. The different active and reactive power terms calculated by means Formula$V_{e\#}$ and Formula$I_{e\#}$ agree, for any case, with the well-known definitions included in IEEE Standard 1459.

SECTION V

RESULT AND DISCUSSION

A three-phase unbalanced nonsinusoidal system is analyzed in IEEE Standard 1459 [1, p. 33, Annex A.3]. The Appendix includes a summary of the voltages and currents in the electrical system under analysis. The values represent a distribution system that includes all types of useless powers as, in fact, occurs in real electrical systems.

Using the equations defined in IEEE Standard 1459 and the Stokvis–Fortescue transformation, some calculation errors found in Annex A.3 in IEEE Standard 1459 are reported in the Appendix and in Table I. The two main errors Formula$V_{1}^{+}$ and Formula$V_{1}^{0}$ are highlighted.

Table 1
TABLE I MAGNITUDES OF THE THREE-PHASE UNBALANCED NONSINUSOIDAL SYSTEM UNDER ANALYSIS

Table I shows a comparison between the main electrical magnitudes in the system under the IEEE Standard 1459 approach and the approach proposed in this paper. Magnitudes that give the same value under both approaches are in the middle of the two columns. The current and power magnitudes calculated by the IEEE Standard 1459 definitions present, in the example, an overvaluation that varies between 13% for Formula$S_{eH}$ to 83% for Formula$S_{U1}$, with a 47% increase for Formula$S_{e}$ with respect to Formula$S_{e\#}$. Table II presents the variation of the different magnitudes according to the approach used in their calculation. Effective voltages with the new approach are slightly higher than those calculated by means of IEEE Standard 1459. The factors 1/2 in (4) and Formula$1/\sqrt{2}$ in (42) are responsible for an undervaluation of the zero-sequence fundamental and harmonic voltages in the IEEE Standard 1459 definitions. It results in a Formula$D_{eV\#}$ that is greater by the proposed approach in this paper than by the means of IEEE Standard 1459.

Table 2
TABLE II FACTORS OF VARIATION BETWEEN MAGNITUDES FOR THE THREE-PHASE UNBALANCED NONSINUSOIDAL SYSTEM

The power factor and total harmonic distortion are merit factors of the electrical system. IEEE Standard 1459 defines the effective power factor Formula$(P_{Fe})$, the fundamental positive-sequence power factor Formula$(P_{F1}^{+})$, and the equivalent total harmonic distortion (Formula${\rm THD}_{eV}$ and Formula${\rm THD}_{eI}$) as follows: Formula TeX Source $$\eqalignno{P_{Fe}=&\,{P\over S_{e}}&\hbox{(58)}\cr P_{F1}^{+}=&\,{P_{1}^{+}\over S_{1}^{+}}&\hbox{(59)}\cr {\rm THD}_{eV}=&\,{V_{eH}\over V_{e1}}&\hbox{(60)}\cr {\rm THD}_{eI}=&\,{I_{eH}\over I_{e1}}.&\hbox{(61)}}$$

A new merit factor introduced in [6] and [20] is denoted here as the total power factor Formula$(P_{FT})$. Formula$P_{FT}$ measures the relationship between the active power under ideal operating conditions Formula$(P_{1}^{+})$ and Formula$S_{e}$ Formula TeX Source $$P_{FT}={P_{1}^{+}\over S_{e}}.\eqno{\hbox{(62)}}$$

The merit factors also present some variations between both approaches. By means of the IEEE Standard 1459 definitions, the Formula${\rm THD}_{eI}$ is 1.17 times higher than Formula${\rm THD}_{eI\#}$. The Formula${\rm THD}_{eV}$ is 0.71 times smaller than Formula${\rm THD}_{eV\#}$, the power factors Formula$P_{F}$ and Formula$P_{FT}$ are the worst per the IEEE Standard 1459 approach, and only Formula$P_{F1}^{+}$ is equal in both approaches.

SECTION VI

CONCLUSION

The resolution of Formula$S_{e}$ by means of the electrical system voltages and currents permits calculating some power quantities that are a function of other power quantities in IEEE Standard 1459.

Table 3
TABLE III VOLTAGES AND CURRENTS FOR THE THREE-PHASE UNBALANCED NONSINUSOIDAL SYSTEM UNDER ANALYSIS

Power magnitudes that include fundamental and harmonic zero-sequence components yield expressions of active and reactive powers that are in disagreement with well-established definitions. The problems arise as factors that multiply the commonly accepted definitions.

If the expressions of the effective voltage and current are modified to eliminate these factors, new expressions of the effective voltage and current are defined. Some power magnitudes included in IEEE Standard 1459 are redefined in this paper while others are kept equal. The results obtained with the new definitions are compared with the existing ones by means of one of the examples included in IEEE Standard 1459. Current and power quantities following IEEE Standard 1459 definitions always produce higher results, with a 47% of increase for Formula$S_{e}$ and an 83% increase for Formula$S_{U1}$ with respect to the quantities obtained by means of the proposed definitions. Only Formula$P_{F1}^{+}$ remains equal under both approaches while the remaining power factors result in the worst values when IEEE Standard 1459 definitions are used.

The analysis performed in this paper gives an explanation for the overvaluation of some of the IEEE Standard 1459 power quantities.

Figure 4
Fig. 4. Phase-to-neutral voltages (top), line currents (middle), and neutral current (bottom) waveforms in the electrical system under analysis.
Figure 5
Fig. 5. Harmonic spectrum of voltages (top) and currents (bottom) in the system under analysis.

The modifications performed in Formula$S_{e\#}$ maintain the definition of the fundamental positive-sequence powers Formula$P_{1}^{+}$ and Formula$Q_{1}^{+}$ as they appear in IEEE Standard 1459. The resolution of Formula$S_{e\#}$ for unbalanced and nonlinear loads also contains an unbalance power Formula$(S_{U1\#})$ and a nonfundamental effective apparent power Formula$(S_{eN\#})$. With the new definitions, the useless active powers (Formula$P_{1}^{-}$, Formula$P_{1}^{0}$, and Formula$P_{\rm H}$), obtained after resolving the power definitions, agree with the commonly accepted expression for these magnitudes.

APPENDIX

Table III shows the voltages and currents phasors in the system under analysis, both expressed in rms values. The values correspond to the three-phase unbalanced nonsinusoidal system analyzed in IEEE Standard 1459 [1, p. 35, Annex A.3]. The load in phase Formula$c$ is disconnected, thus increasing the load current unbalance. The voltages are measured between phase and neutral.

The phase-to-neutral voltages are represented in the top plot at Fig. 4. The order of the waveforms is Formula$a-b-c$. The origin of angles is Formula$\omega t=$ 0, with a fundamental frequency of 60 Hz. The load-current waveforms are represented in the midplot, and neutral current is represented in the bottom plot. Current through line Formula$c$ is equal to zero at any instant.

Harmonic content of the phase-to-neutral supply voltages (top plot) and the supply currents (bottom plot) are represented in Fig. 5. For each harmonic order, the bars represent phases Formula$a-b-c$ (from the left). Since the current through line Formula$c$ is equal to zero, only two bars appear in the current spectrum.

Some of the values included in Table III are different from their corresponding values in [1, Table A.5] because some mistakes exist in the example presented in IEEE Standard 1459. The erroneous terms are the following: the phase angle for the fifth harmonic component Formula$(\beta_{b5})$ is equal to −65.09° and Formula$I_{n1}$ (%) is equal to 126% (equivalent to 125.93 A). Also, some values included in Table I are modified with respect to the values included in Annex A.3 of IEEE Standard 1459 because there are some errors in the calculations. The corrected terms are as follows. Formula$V_{1}^{+}$ is equal to 278.41 V instead of 288.49 V, and Formula$V_{1}^{0}$ is equal to 7.48 V instead of 2.98 V. Small calculation errors also exist in Formula$V_{1}^{-}$, Formula$S_{e}$, and all of the active powers.

Footnotes

This work was supported by the Generalitat Valenciana under Grant GVPRE/2008/343. Paper no. TPWRD-00427-2009.

S. Orts-Grau, F. J. Gimeno-Sales, and S. Seguí-Chilet are with the Electronics Engineering Department, Universidad Politécnica de Valencia, Valencia 46022, Spain (e-mail: sorts@eln.upv.es; fjgimeno@eln.upv.es; ssegui@eln.upv.es).

N. Muñoz-Galeano is with the Electrical Engineering Department, Universidad de Antioquia, Medellín, Colombia (e-mail: nicolasm@udea.edu.co).

J. C. Alfonso-Gil is with the Department of Engineering of Industrial Systems and Design, Universidad Jaume I de Castellón, Castelló de la Plana 12071, Spain (e-mail: jalfonso@esid.uji.es).

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Authors

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Salvador Orts-Grau

Salvador Orts-Grau (M'06) was born in Valencia, Spain, in 1972. He received the M.E. degree in automation and industrial electronics engineering from the Polytechnic University of Valencia (UPVLC), Valencia, Spain.

Since 2001, he has been Assistant Professor of the Electronics Engineering Department with the UPVLC. His research work focuses on the improvement of the electrical power network quality by means of digitally controlled shunt active power compensators.

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Nicolás Muñoz-Galeano

Nicolás Muñoz-Galeano was born in Medellín, Colombia, in 1981. He received the B.E. degree in electric engineering at the Universidad de Antioquia (UdeA) in 2004.

Since 2005, he has been Professor in the Electric Engineering Department at the UdeA, Colombia, and a member of the research group GIMEL. His research work focuses on the predictive maintenance of electrical machines, the analysis of the electrical power network behavior, and the effects of nonefficient powers and its reduction by means of active compensators.

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José Carlos Alfonso-Gil

José Carlos Alfonso-Gil was born in La Cañiza, Spain, in 1974. He received the M.E. degree in automation and industrial electronics engineering from the Polytechnic University of Valencia, Valencia, Spain, where he is currently pursuing the Ph.D. degree in electronic engineering.

From 2005 to 2007, he was a Researcher with the UPVLC and since 2007, he has been a Visiting Teacher in the Department of Industrial Systems Engineering and Design, Universitat Jaume I (Castellón). His research work focuses on digital signal processors applied to the control of power converters and the design of instrumentation that measures electric quantities.

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Francisco J. Gimeno-Sales

Francisco J. Gimeno-Sales was born in Valencia, Spain, in 1958. He received the Ph.D. degree in electronics engineering from the Polytechnic University of Valencia, Valencia, Spain in 2004.

From 1986 to 1993, he was with several R&D departments, developing industrial products (hardware and software). Since 1993, he has been teaching power electronics and Formula$\mu{\rm C}$ applied to power converter control at the Electronics Engineering Department, UPVLC. His research interest is the control of power-electronic converters and grid quality.

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Salvador Seguí-Chilet

Salvador Seguí-Chilet (M'04) was born in Valencia, Spain, in 1962. He received the B.E. degree in industrial electronics from Polytechnic University of Valencia (UPVLC), Valenica, in 1986, the M.E. degree in electronic engineering from the University of Valencia in 1999, and the Ph.D. degree in electronics engineering from UPVLC in 2004.

Since 1990, he has been lecturing in the Electronics Engineering Department, UPVLC. His major fields of interest are power electronics, renewable energy systems, and active power compensators.

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