Download Comparison of Time-Based Non-Active Power Definitions for Active

Comparison of Time-Based Non-Active Power Definitions for Active Filtering
Leon M. Tolbert, Senior Member, IEEE
Thomas G. Habetler, Senior Member, IEEE
The University of Tennessee
Electrical and Computer Engineering
Knoxville, TN 37996-2100
Tel: (865) 974-2881
Fax: (865) 974-5483
e-mail: [email protected]
Georgia Institute of Technology
Electrical and Computer Engineering
Atlanta, GA 30332-0250
Tel: (404) 894-9829
Fax: (404) 894-4641
e-mail: [email protected]
additional losses in the generation and transmission system
pay for those losses, whereas customers want utilities to
supply them with voltage that contains no distortion.
Instantaneous active power p(t) is defined as the time rate
of energy transfer or energy utilization. It is a physical
quantity and satisfies the principle of conservation of energy
[5]. For a single-phase circuit, it is defined as the
instantaneous product of voltage and current:
Abstract Many definitions have been formulated to
characterize non-active power for nonsinusoidal waveforms in
electrical systems, and no single, universally valid power theory
has been adopted as a standard for non-active power. Most of
the non-active power theories formulated thus far have had a
particular type of compensation in mind, which has influenced
the conventions used in the development of the definitions.
Because nonsinusoidal loads are expected to continue to
proliferate throughout electrical distribution systems, nonactive power theories will only grow in importance for
applications such as non-active power compensation, harmonic
load identification, voltage distortion mitigation, and metering.
This paper presents a comprehensive technical survey of the
published literature on the topic and briefly outlines the salient
points of each of the different theories as well as each one’s
applicability to active power filtering.
p(t ) = u (t )i(t ) .
(1)
Active power P is the time average of the instantaneous
power over one period of the wave p(t). For a polyphase
circuit with M phases, instantaneous active power is the sum
of the of the active powers of the individual phases:
Keywords: non-active current, non-active power, reactive
current, reactive power, active power, var compensation
M
p(t ) = ∑ um (t )im (t ) =
m =1
I. INTRODUCTION
M
∑
m =1
p m (t ) .
(2)
Non-active power is not a physical quantity but can be
thought of as the useless energy that causes increased line
losses and greater generation requirements for utilities.
Because non-active power is a conventional quantity, the
conservation of non-active power does not hold true. The
first definitions of non-active power involved rms or average
values, and not until more recently did expressions for
instantaneous non-active power appear in the literature.
Most of the theories have chosen voltage to be a reference and
to decompose current into an active component that is
responsible for the transmission of the active power and an
oscillating non-active component that yields zero average
active power. These theories are formulated with the goal of
identifying and eliminating non-active power, or non-active
current, to improve the power factor of an electrical system.
While some authors restricted their analyses to the time- or
frequency-domain, others formulated a more unifying concept
by utilizing both domains.
Frequency-based theories were generally developed for
metering or power systems that had steady-state harmonic
components instead of active filtering as their main objective;
thus, these definitions are not considered in this paper. Most
Non-active power theories for nonsinusoidal waveforms
were first formulated in the 1920’s and 1930’s by Budeanu
[1, 2] and Fryze [3, 4]. Because the vast majority of the
generation and use of electrical energy at that time involved
periodic, sinusoidal waveforms, these two theories were not
developed further by others until the 1970’s when widespread
use of power electronic converters made study of the subject
more than just an academic exercise. The nonsinusoidal,
non-periodic loads and voltage waveforms present in today’s
electrical systems have made the issue of instantaneous nonactive power a contentious subject of discussion as several
different theories and analyses have been formulated in the
last 30 years.
Closely related to the definitions of active and non-active
power are the definitions of apparent power and power factor.
Several proposals have been made for the definition of both,
but no consensus has been reached for these two conventional
quantities.
Metering non-active power and assigning
causality to the various components of non-active power is
another area that has received much attention lately. Utilities
are interested in ensuring that customers who cause
1/7
of the time-based theories had some application of non-active
power filtering in mind when they were developed, and thus
these are the focus of this paper.
This paper presents a technical survey of the published
literature on the topic of non-active power and briefly outlines
the salient points of each of the different theories. The
applicability of each of the different theories and a
comparison between them are also included in this paper.
elements or not. They defined the active current ia the same
as Fryze in (3), but subdivided the instantaneous non-active
current i - ia into two components - an inductive or capacitive
reactive current, iql or iqc, and a residual reactive current.
Their theory reasoned that the inductive (capacitive) reactive
current could be completely compensated by means of adding
a capacitor (inductor) in parallel with the load. They defined
u as the instantaneous value of the alternating component of
II. INITIAL ORTHOGONAL CURRENTS THEORY
∫u dt , or in other words, that portion of the voltage that did
not contribute to the active power transfer from the source to
the load. They then defined the inductive (capacitive) reactive
current as follows:
Fryze introduced his idea of non-active power in 1931 [3]
by using a time-based approach where he decomposed the
source current i into an active component ia, which had the
same waveform as the source voltage u, and a non-active
component in by the following equations:
ia ≡
where
P
u
2
in ≡ i − ia
u,
T
iql = u (
By multiplying (4) by
2
2
= ia + in .
u 2, Fryze obtained the following for
Q F ≡ u in = S 2 − P 2
(6)
iqcr = i − ia − iqc .
(7)
Kusters and Moore’s definition is valid for single-phase
applications where compensation by only passive elements is
considered [8]. Because their definition, like Fryze’s,
depends on RMS or average values over some time period, it
does not lend itself to real-time compensation.
(4)
non-active power QF :
S 2 = P 2 + QF2 ,
1
2
, iqc = u&( u&i dt ) / u& .
T
iqlr = i − ia − iql ,
power over one period. Because ia and in are mutually
orthogonal, the following holds true:
2
2
Because compensation by means of passive elements did not
eliminate all of the non-active current, they defined that
portion of the current which could not be compensated as the
residual reactive current, iqlr or iqcr:
(3)
u is the rms value of u and P is the average active
i
1
u i dt ) / u
T∫
0
B. Polyphase Orthogonal Currents (FBD Method)
(5)
Depenbrock extended Fryze’s definition of non-active
power and current decomposition method for a single phase
to a polyphase system with M phases [9]. Because he also
incorporated some of the apparent power theory published by
Buchholz [10], he named his power theory the FBD method
after the three contributing authors. Buchholz defined the
instantaneous collective value of currents and voltages and
the collective RMS value of the currents and voltages in a
system with M phases as
Fryze’s theory has been found to be fundamentally sound
[4], and several authors have extended his definition by
considering multiple phases or by modifying the equation for
ia in (3).
III. TIME-DOMAIN NON-ACTIVE POWER THEORIES
One of the main thrusts behind the definition of nonactive power has largely been to build compensators that will
improve the power factor to unity without incurring
additional power losses in the electrical system. Many of the
definitions outlined in the following sections have been
developed and used to design these compensators and their
controls.
i∑ =
u∑ =
M
∑i
υ =1
2
υ
,
i∑
M
∑
υ =1
uυ2 ,
u∑
2
M
= ∑ iυ
2
,
υ =1
2
M
= ∑ uυ
2
.
(8)
υ =1
A. Inductive and Capacitive Power
Depenbrock defined collective instantaneous power
Kusters and Moore [6] (and later generalized by Page [7])
extended Fryze’s decomposition theory for periodic
nonsinusoidal waveforms in 1979 based on whether the nonactive current could be compensated by means of passive
p∑ ,
collective instantaneous conductance G p (t ) , instantaneous
collective power current i ∑ p , and instantaneous phase power
currents iυ p as
2/7
M
M
υ =1
υ =1
p∑ (t ) = ∑ uυ iυ = ∑ pυ (t ) ,
G p (t ) =
p∑
u∑2
To calculate active power P and an equivalent conductance G,
they used the time-domain crosscorrelation function
,
Rui (τ ) =
i ∑ p = G p (t ) u ∑ ,
iυ p = G p (t )uυ .
[
iυ a = G uυ ,
iυ v = iυ p − iυ a .
(11)
The non-active currents iυ n then could be written as
iυ n = iυ − iυ a = iυ z + iυ v .
P
. (14)
U2
in = i − ia .
(15)
]
1/ 2
,
B=
Q
,
U2
(16)
where R$ ui was the maximum value of the crosscorrelation
between voltage and current waveforms, which occurred at
the point in the time interval dT where maximum similarity
between voltage and current waveforms was present. With the
Hilbert transform of the voltage, Enslin and Van Wyk then
defined the instantaneous reactive current as
active conductance G, which depended on the mean values of
p∑ and uΣ . Expressions for these quantities are given as
,
0
Q = R$ ui2 (τ ) − Rui2 (0)
and variation currents iυ v by introducing the equivalent
u∑2
P = Rui (0) , G =
In order to decompose the non-active current into reactive
ir and deactive id components, they calculated the reactive
power Q and equivalent susceptance B as
(10)
and noted that this component of the current can theoretically
be compensated without any time delay (instantaneously). He
then decomposed the power currents into active currents iυ a
p∑
∫u(t ) i (t − τ )dt ,
ia (t ) = G u(t ) ,
Depenbrock then defined the phase zero power currents iυz as
G=
dT
They then defined the instantaneous active and non-active
current the same as Fryze
(9)
iυ z = iυ − iυ p ,
1
dT
ir (t ) = B H {u},
(12)
where H {u}=
∞
1 u(τ ) dτ
. (17)
π −∫
τ− t
∞
Complete compensation of the non-active currents is only
possible under steady state conditions because the variation
currents depend on time average values of power and voltage.
Depenbrock also developed a decomposition of the nonactive current into orthogonal symmetrical and purely
unsymmetrical components for an unsymmetrical load [11].
The FBD method can be applied to a multiphase, steady state,
periodical system for compensation purposes.
They then defined the instantaneous deactive current as
C. Nonperiodic Waveforms
Although Enslin and Van Wyk extended Fryze’s theory to
nonperiodical single-phase systems, their values of the
autocorrelation and crosscorrelation functions in (13) and
(14) depended on the length of the interval dT and the
interval start time.
id = i − i a − ir .
The mutual orthogonality of the current components leads to
the following apparent power relation
S 2 = P 2 + F 2 = P 2 + Q2 + D 2 .
In 1988, Enslin and Van Wyk [12-15] generalized Fryze’s
time-domain principles to single-phase nonperiodic waveforms by using correlation techniques and also further
decomposed the non-active power into a reactive component
Q and a deactive component F. To find the effective value of
current or voltage, they used the time-domain autocorrelation
function
dT
1
Ruu (τ ) =
u(t ) u(t − τ )dt ,
dT ∫
0
[
U = Ruu (0)
]
1/ 2
,
[
I = Ri i (0)
]
1/ 2
.
(18)
(19)
D. Instantaneous Orthogonal Currents
Rossetto and Tenti [16] proposed an extension of Fryze’s
decomposition method to multiphase systems and also
modified equation (3) by using instantaneous values as
follows:
r v
v v u T ⋅i
v p
i p = u vT v = u v 2 .
(20)
u ⋅u
u
(13)
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v
q
Using the same definition as Fryze, they then defined the
v v v
instantaneous non-active current iq = i − i p . They also
formulated expressions for the instantaneous active power p,
instantaneous non-active power q, and instantaneous apparent
power s, where s2 = p2 + q2, as follows:
v v
p = u ip ,
v v
q = u iq ,
v v
s= u i .
imaginary axis
v v
uα ×iβ
(21)
Rossetto and Tenti’s theory appears to be truly
instantaneous and not limited to periodic waveforms.
Compensation techniques with this theory have not yet
appeared in the literature.
v
iα
v v
uβ ×iα
In 1983, Akagi and coauthors [17-18] introduced a novel
concept, the p-q theory, by applying Park’s transformation to
a three-phase, three-wire system (a-b-c) to change it to a twophase plane (α-β) and an orthogonal reactive axis as shown
in Fig. 1. They also showed how this theory could be used for
three-phase, four-wire systems by introducing the zero-phase
sequence components. The p-q theory transforms
instantaneous voltage and current measurements by the
following matrix equations:
1 / 2
2
 1
3
0

1/ 2
1 / 2 ua 
 
− 1/ 2
− 1 / 2 ub  ,
3 / 2 − 3 / 2
u 
 c 
(22)
i0 
 
iα  =
i β 
 
1 / 2
2
 1
3
0

1/ 2
1 / 2 i a 
 
− 1 / 2 − 1 / 2 ib  .
3 / 2 − 3 / 2
i 
 c 
(23)
α
r
For symmetrical voltage sources, | q | is equal to the threephase non-active power, but for asymmetrical voltage sources,
r
| q | is not equal to the three phase reactive power. This has
introduced errors in some p-q control algorithms [19].
F. Park Power
Using Akagi’s Park transformation of phase voltages and
currents, Ferrero and Superti-Furga [20-21] defined their
instantaneous non-active power differently, which enabled
them to generalize their theory based on power definitions.
Instead of an α-β real plane with an orthogonal imaginary
axis q, they used a direct-quadrature (d-q) plane. In a three
phase system, they transformed instantaneous voltage and
currents to Park vectors (d-q only, without zero sequence) of
v
v
voltages u (t ) and currents i (t ) just as Akagi did and then
defined the Park instantaneous complex power as
v
v
v
a p (t ) = u (t ) ⋅i ∗(t ) .
(26)
By expanding (26), they defined the Park real power and Park
imaginary power as
[ ]
v
p p (t ) = Re a p ( t ) = ud id + uq iq ,
(24)
[ ]
v
q p ( t ) = Im a p ( t ) = uq id − ud iq .
They then introduced the instantaneous imaginary power
r
space vector q defined by
r v
r
r v
q = uα × i β + uβ × iα .
v
uα
Fig. 1. Instantaneous space vectors of voltage and current for pq-theory.
Akagi and coauthors then defined the Park instantaneous
active power pp and the zero phase-sequence instantaneous
power po as follows:
v v
v v v v
p0 = u0 ⋅i0 ,
p p = uα ⋅iα + uβ ⋅i β ,
p = p p + p0 = ua ia + ub ib + uc ic .
v
uβ
real plane
E. p-q Theory
u0 
 
uα  =
uβ 
 
β
v
iβ
(27)
Next, they stated that the instantaneous active power p(t)
is the algebraic sum of the zero sequence power p0(t)=u0 i0 and
the Park power in (27). The only difference between Ferrero’s
approach and Akagi’s approach is that the Park imaginary
power (instantaneous non-active power) defined by Ferrero is
a characteristic quantity of the three-phase system whereas
Akagi’s instantaneous imaginary power is not.
(25)
As seen in Fig. 1, this space vector is an imaginary axis
vector perpendicular to the α-β coordinate real plane and is
composed of the sum of the products of voltages and currents
r
in orthogonal axes. This meant that q could not be dimensioned in W, VA, or VAR.
4/7
In addition, Ferrero and Superti-Furga used Park vectors
for current and voltage to extend Fryze’s theory to three-wire,
three-phase systems. From (3), they developed the following
v
vectorial equations for active currents ia (t ) and residual
v
currents i x (t ) :
v
Pp v
ia ( t ) = 2 ⋅u( t ) ,
U
v
v
v
i x ( t ) = i (t ) − i a ( t ) .
instantaneous active current (30) and active current defined
by Fryze (3). All of the terms in (30) are instantaneous ones,
whereas those in (3) have RMS and average values over a
period T. Willems theory enables compensation without
energy storage devices whereas those of Fryze or Kusters and
Moore do not.
(28)
H. Polar Coordinates
Recently, Nabae and Tanaka [23] proposed a new method
of calculating active and non-active current and power in
three-phase, three-wire circuits based on instantaneous space
vectors with polar coordinates. Their technique is similar to
the p-q theory transformation except that active and nonactive currents can be calculated directly from the
v
v
instantaneous voltage u and current i space vectors instead
of calculating powers first. They defined the instantaneous
voltage and current space vectors as
They then noted that non-active power compensation based
on these vector definitions would be more effective than
separate time-domain decomposition for each of the
individual phases.
The p-q theory has been widely used for active
compensation in three phase systems. Drawbacks of the
original theory is that it does not lend itself to compensation
by passive energy storage components and could not be used
for single phase or multiphase systems. Also, zero sequence
components are completely attributed to active power without
consideration that they may also contribute to non-active
power.
v
u=
r
i =
G. Multiphase Generalization of the p-q Theory
In 1992 Willems [22] generalized Akagi’s and Ferrero’s
p-q theories for three-phase systems to polyphase systems.
For an electrical system with m phases, Willems represented
the instantaneous currents and voltages by m-dimensional
v
v
vectors i (t ) and u( t ) . The instantaneous active power
transmitted is the scalar product of these two vectors:
v
v
(29)
p(t ) = u (t ) T i (t ) .
v
j
2
(i a + ib ε
3
2π
3
+ uc ε
+ ic ε
δ
j
j
4π
3
4π
3
) = u ( t ) ε jθ ( t ) ,
) = i (t )ε j {θ( t ) + φ( t )},
v
ip
v
u
u
(30)
(33)
γ
φ
v v
where | u |2 = u T u . Instantaneous non-active current is then
v
v
v
i q ( t ) = i (t ) − i p ( t ) .
(31)
v
iq
v
i
Fig. 2. Space vectors of current and voltage in γ
-δcoordinates.
Willems then defined the magnitude of the instantaneous
imaginary power as
v
v
q ( t ) = u (t ) i q ( t ) .
2π
3
where ua, ub, and uc are instantaneous phase voltages and ia,
ib, and ic are instantaneous line currents. The angle φ(t)
v
v
between u and i is also an instantaneous value. The authors
v
v
then applied a rotating-coordinate transformation to u and i
v
to fix the vectors on a γ
-δcoordinate plane with u as the
reference vector as shown in Fig. 2. This transformation led
v
v
to expressing u and i as
He then defined the instantaneous active current by projecting
v
v
the vector i (t ) onto the vector u (t ) :
v
p( t ) v
i p (t ) = v 2 u (t ) ,
u (t )
j
2
(ua + ub ε
3
v
u = u( t ) ,
(32)
v
i = i (t )ε jφ( t ) .
(34)
v
Nabae and Tanaka then decomposed i into instantaneous
active and non-active components as follows:
A positive or negative sign can be associated with q(t) only
for three-phase systems without zero sequence components.
In more general cases, this is not possible.
One should note the difference in Willems definition of
i p (t ) = i (t ) cos φ(t ) ,
5/7
iq (t ) = i (t ) sin φ(t ) .
(35)
Then, the instantaneous active power p(t) and instantaneous
non-active power q(t) are defined as
J. Generalized Non-Active Power
Peng [26] extended Fryze’s theory to multiphase and/or
non-periodic systems. He defined the active current and nonactive current using the following equations:
p(t ) = u(t ) i p (t ) = u(t ) i (t ) cos φ(t ) ,
q (t ) = u(t ) i q (t ) = u(t ) i (t ) sin φ(t ) .
(36)
v
P (t ) v
i p (t ) = 2 u p (t ) ,
U p (t )
Nabae and Tanaka also noted that ip(t), iq(t), p(t), and q(t)
could be decomposed into currents and powers originating
from fundamental frequency symmetrical conditions and from
harmonic or unsymmetric conditions, but did not elaborate on
how to do this decomposition or what its application may be.
Their theory, like the original p-q theory, is restricted to three
phase systems.
where U p (t ) =
1
Tc
and P (t ) =
I. Generalized Instantaneous Reactive Power
t
v
vT
∫u p (τ ) ⋅u p (τ )dτ
(38)
(39)
t − Tc
1
Tc
t
∫p(τ)dτ
(40)
t − Tc
v
u p (t ) is the reference voltage, which can be chosen as the
v
v
voltage itself, u p (t ) = u (t ) ; or as the fundamental component
v
v
v
v
v
v
of u (t ) , where u (t ) = u f (t ) + u h (t ) and u p (t ) = u f (t ) ; or as
Peng [24-25] has proposed a generalized instantaneous
reactive power theory for three-phase systems by considering
zero sequence components to contribute to non-active power
and active power whereas the pq theory erroneously limits
their contribution to active power only. Instead of first
decomposing the current into orthogonal components and
then calculating powers like Willems or Rossetto, they define
power components first and then decompose the current.
Peng expressed voltage and current for a three-phase
system as instantaneous space vectors, and defined the
instantaneous active power to be the inner product of these
vectors just like Akagi in (24). He also defined the
v
instantaneous non-active power vector q to be the cross
product of the voltage vector and current vector and its
magnitude as the instantaneous non-active power q:
ub uc 


i i
qa   b c 
v v v
 uc ua 
q = u ×i = 
qb  =  i i  ,
c a

qc 
 u u 
a
b


i
i
a
b


v
v
v
i q (t ) = i (t ) − i p (t )
something else depending on the compensation objectives.
Peng showed that this definition was valid for singlephase or polyphase circuits as well as for nonsinusoidal and
nonperiodic waveforms. He also showed that many of the
previous definitions can be deducted from (38 – 40).
IV. CONCLUSIONS
A nonsinusoidal electrical system has numerous degrees of
freedom and the number of parameters required to completely
describe the system could theoretically be infinite. For this
reason, no single, universally valid power theory has been
formulated for nonsinusoidal conditions. Most of the timebased non-active power theories formulated thus far have had
a particular type of compensation in mind, which has
influenced the conventions used in the development of the
definitions. It should also be noted that there are also just as
many definitions formulated using some form of a frequency
harmonic decomposition basis.
Table I summarizes the applicability of each of the
theories discussed in the preceding sections. The table shows
if each definition is applicable to single-, three-, or multiplephase systems and whether the definition has a periodic or
averaging basis or whether the definition uses instantaneous
measurements.
Because nonsinusoidal loads are expected to continue to
proliferate throughout electrical distribution systems, nonactive power theories will only grow in importance for
applications such as non-active power compensation,
harmonic load identification, voltage distortion mitigation,
and metering.
v
q = q = qa2 + qb2 + qc2
(37)
v
He then defined the instantaneous active current vector, i p
the same as Rossetto in (20), the instantaneous non-active
v qv ×uv
current vector iq = v 2 , and the instantaneous apparent
u
power s the same as Rossetto in (21).
Peng’s generalized instantaneous power theory for threephase systems is also applicable to unbalanced systems and
three-phase four-wire systems which have zero sequence
components.
6/7
[16] L. Rossetto, P. Tenti, “Evaluation of Instantaneous Power Terms in MultiPhase Systems: Techniques and Applications to Power-Conditioning
Equipment,” ETEP, vol. 4, Nov. 1994, pp. 469-475.
[17] H. Akagi, Y. Kanazawa, A. Nabae, “Instantaneous Reactive Power
Compensators Comprising Switching Devices without Energy Storage
Components,” IEEE Trans. Ind. Applicat., vol. IA-20, no. 3, May 1984,
pp. 625-631.
[18] H. Akagi, A. Nabae, “The p-q Theory in Three-Phase Systems under NonSinusoidal Conditions,” ETEP, vol. 3, Jan. 1993, pp. 27-31.
[19] M. Kohata, et. al., “A Novel Compensator Using Static Induction
Thyristors for Reactive Power and Harmonics,” Conference Record –
Power Electronics Specialists Conference, 1988, pp. 843-849.
[20] A. Ferrero, G. Superti-Furga, “A New Approach to the Definition of Power
Components in Three-Phase Systems under Nonsinusoidal Conditions,”
IEEE Trans. Instrum. Meas., vol. 40, June 1991, pp. 568-577.
[21] A. Ferrero, A. P. Morando, R. Ottoboni, G. Superti-Furga, “On the
Meaning of the Park Power Components in Three-Phase Systems under
Non-Sinusoidal Conditions,” ETEP, vol. 3, Jan. 1993, pp. 33-43.
[22] J. L. Willems, “A New Interpretation of the Akagi-Nabae Power
Components for Nonsinusoidal Three-Phase Situations,” IEEE Trans.
Instrum. Meas., vol. 41, Aug. 1992, pp. 523-527.
[23] A. Nabae, T. Tanaka, “A New Definition of Instantaneous Active-Reactive
Current and Power Based on Instantaneous Space Vectors on Polar
Coordinates in Three-Phase Circuits,” Conference Record - IEEE/PES
1996 Winter Meeting, Baltimore, Md.
[24] F. Z. Peng, J. S. Lai, “Generalized Instantaneous Reactive Power Theory
for Three-Phase Power Systems,” IEEE Trans. Instrum. Meas., vol. 45,
no. 1, Feb. 1996, pp. 293-297.
[25] F. Z. Peng, G. W. Ott, Jr., D. J. Adams, “Harmonic and Reactive Power
Compensation Based on the Generalized Reactive Power Theory for
Three-Phase Four-Wire Systems,” IEEE Trans. Power Electronics, vol.
13, no. 6, Nov. 1998, pp. 1174-1181.
[26] F. Z. Peng, L. M. Tolbert, “Compensation of Nonactive Current in Power
Systems - Definitions from a Compensation Standpoint,” IEEE Power
Engineering Society Summer Meeting, July 15-20, 2000, Seattle, WA,
pp. 983-987.
Table I. Time-Based Non-Active Power Definition Comparison
Time
Period
Phases
Single Three Multiple Avg. Inst.
Theory [Reference]
Orthogonal Currents [3]
Inductive and Capacitive Power [6]
X
X
Polyphase Orthogonal Currents [9]
Nonperiodic Waveforms [12]
X
X
Instantaneous Orthogonal Currents [16]
X
X
X
X
X
p-q Theory [17]
X
X
X
X
X
X
X
X
Park Power [19]
Multiphase p-q Theory [22]
X
Polar Coordinates [23]
Generalized Instantaneous Reactive
Power [24]
Generalized Non-active Power [26]
X
X
X
X
X
X
X
X
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
C. I. Budeanu, “Reactive and Fictitious Powers,” Inst. Romain de
I’Energie, Bucharest, Rumania, 1927. (In Romanian)
L. S. Czarnecki, “What Is Wrong with the Budeanu Concept of Reactive
and Distortion Power and Why It Should Be Abandoned,” IEEE Trans.
Instrum. Meas., vol. 36, Sept. 1987, pp. 834-837.
S. Fryze, “Active, Reactive, and Apparent Power in Non-Sinusoidal
Systems,” Przeglad Elektrot., no. 7, 1931, pp. 193-203. (In Polish)
H. Supronowicz, J. Janczak, “Compensation of the Reactive Power Drawn
from the Line by Nonlinear Consumers,” Conference Record - IEEE IAS
Annual Meeting, 1990, pp. 1093-1098.
IEEE Standard Dictionary of Electrical and Electronics Terms, 1993,
pp. 988-989.
N. L. Kusters, W. J. M. Moore, “On the Definition of Reactive Power
under Non-Sinusoidal Conditions,” IEEE Trans. Power Apparatus Sys.,
Vol. PAS-99, Sept. 1980.
C. H. Page, “Reactive Power in Nonsinusoidal Situations,” IEEE Trans.
Instrum. Meas., vol. IM-29, Dec. 1980, pp. 420-423.
L. S. Czarnecki, “A Time-Domain Approach to Reactive Current
Minimization in Nonsinusoidal Situations,” IEEE Trans. Instrum. Meas.,
vol. 39, Oct. 1990, pp. 698-703.
M. Depenbrock, “The FBD Method, a Generally Applicable Tool for
Analyzing Power Relations,” IEEE Trans. Power Systems, vol. 8, May
1993, pp. 381-387.
F. Buchholz, “Das Begriffsystem Rechtleistung, Wirkleistung, Totale
Blindleistung,” Selbstverlag Munchen, 1950.
M. Depenbrock, “Some Remarks to Active and Fictitious Power in
Polyphase and Single-Phase Systems,” ETEP, vol. 3, Jan. 1993, pp. 15-19.
J. H. R. Enslin, J. D. Van Wyk, “Measurement and Compensation of
Fictitious Power Under Nonsinusoidal Voltage and Current Conditions,”
IEEE Trans. Instrum. Meas., vol. 37, no. 4, Sept. 1988, pp. 403-408.
J. H. R. Enslin, J. D. Van Wyk, “A New Control Philosophy for Power
Electronic Converters as Fictitious Power Compensators,” IEEE Trans.
Power Electronics, vol. 5, no. 1, Jan. 1990, pp. 88-97. Discussion, vol. 5,
no. 4, Oct. 1990, pp. 503-504.
J. H. R. Enslin, J. D. Van Wyk, M. Naude, “Adaptive, Closed-Loop
Control of Dynamic Power Filters as Fictitious Power Compensators,”
IEEE Trans. Industrial Electronics, vol. 37, June 1990, pp. 203-211.
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Trans. Power Delivery, vol. 6, Oct. 1991, pp. 1774-1780.
V. BIOGRAPHIES
Leon M. Tolbert (S’88-M’91-SM’98) received the B.E.E., M.S., and Ph.D.
degrees in electrical engineering from the Georgia Institute of Technology,
Atlanta.
He joined the Engineering Division of Lockheed Martin Energy Systems in
1991 and worked on several electrical distribution projects at the three U.S.
Department of Energy plants in Oak Ridge, Tennessee. In 1997, he became a
research engineer in the Power Electronics and Electric Machine Research
Center at the Oak Ridge National Laboratory. In 1999, he was appointed as an
assistant professor in the Department of Electrical and Computer Engineering at
The University of Tennessee, Knoxville.
Dr. Tolbert is a Registered Professional Engineer in the state of Tennessee.
Thomas G. Habetler (S’82-M’83-S’85-M’89-SM’92) received the B.S.E.E.
degree in 1981 and the M.S. degree in 1984, both in electrical engineering, from
Marquette University, Milwaukee, WI, and the Ph.D. degree from the University
of Wisconsin-Madison, in 1989.
From 1983-1985 he was employed by the Electro-Motive Division of
General Motors as a Project Engineer. While there, he was involved in the
design of switching power supplies and voltage regulators for locomotive
applications. He is currently an Associate Professor of Electrical Engineering at
the Georgia Institute of Technology, Atlanta, USA. His research interests are in
electric machine protection and condition monitoring, switching converter
technology, and drives.
Dr. Habetler serves as Vice President of Operations for the IEEE Power
Electronics Society, and Chair of the Industrial Power Converter Committee of
the Industry Applications Society.
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