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Unformatted text preview: 828 IEEE 'I'RANSACI'lONS ON POWER ELECTRONICS, VOL 19, NO. 3. MAY 2004 On Some Misinterpretations
of the Instantaneous Reactive Power p—q Theory Leszek S. Czarnecki, Fellow, IEEE Abstract—The main features of the instantaneous reactive
power (IRP) pq Theory, considered as a power theory of
threephase systems, are analyzed in this paper using the theory
of the currents’ physical components (CPC).'This analysis shows
that the p and q powers are not associated with separate power
phenomena, but with multiple phenomena. Moreover, the results
of the IRP pq Theory contradict some common interpretations
of power phenomena in threephase circuits. Namely, according
to the IRP pq Theory the instantaneous reactive current can
occur even if a load has zero reactive power, Q. Similarly, the
instantaneous active current can occur even if a load has zero
active power, P. Moreover, these two currents in circuits with a
sinusoidal supply voltage can be nonsinusoidal even if there is
no source of current distortion in the load. The analysis shows
that a pair of values of instantaneous active and reactive p and
q powers does not enable us to draw any conclusion with respect
to the power properﬁes of threephase unbalanced loads even in
a sinusoidal situation. Thus, the Instantaneous Reactive Power
pq Theory does not identify power properties of such loads
instantaneously. This conclusion may have an importance for
control algorithms of active power ﬁlters. The paper reveals the
relationship between the p and q powers and the active, reactive
and unbalanced powers, P, Q, and D and speciﬁes the required
energy storage capability of activerpower ﬁlters operated under
sinusoidal unbalaneed conditions. Index Terms—Active current, active power ﬁlters, apparent
power, currents’ physical components (CPC) power theory, in
stantaneous active power, instantaneous reactive power, reactive
current, switching compensators, unbalanced power, unbalanced
systems. I. INTRODUCTION HE INSTANTANEOUS reactive power (IRP) p—q Theory,
developed by Akagi, Kanazawa and Nabae, [l], [2], pro
vides mathematical fundamentals for the control of switching
compensators, known commonly as “active power ﬁlters.”
Although there is still substantial confusion [3] with respect
to power phenomena in electrical systems and there are reports
[4], [5] on some shortcomings of the IRP p—q Theory, it seems to
be well established [6]—[9] in the electrical engineering commu
nity involved in switching compensator design. Moreover, there
are attempts [10]—[l2] to provide deeper fundamentals for this
Theory and it is becoming a theoretical tool for power proper
ties of threephase system analysis [13]—[15] and instrumenta tion [16]. Manuscript received March 14, 2003; revised October 15, 2003. Recom
mended by Associate Editor H. du T. Mouton. The author is with the Electrical and Computer Engineering Depart
ment, Louisiana State University, Baton Rouge, LA 70802 USA (email:
[email protected]; czameck®ee.lsu.edu). Digital Object Identiﬁer 10.1109/I'PEL.2004.826500 When the IRP p—q Theory is considered as a theoretical fun
damental for a control algorithm design, it is irrelevant whether
it interprets power properties of electrical circuits correctly or
not. It is enough that it enables the achievement of the control
objectives. However, when it is considered as a power theory
one could expect that it does provide a credible interpretation of
power phenomena in electrical systems. Having this expectation in mind, the following dilemma oc
curs. Power properties of threephase, threewire systems with
only sinusoidal voltages and currents, i.e., even without any har
monic distortion, are determined by three independent features
of the system. 1) permanent energy transmission and associated active
power, P; 2) presence of reactive elements in the load and associmed
reactive power, Q; 3) load imbalance that causes supply current asymmetry and
associated unbalanced power, D. Thus, how can the IRP p—q Theory, based on only two power
quantities, p and q, identify and describe'three independent
power properties? Moreover, according to Akagi and Nabae,
[2] who developed the Instantaneous Reactive Power p—q
Theory, its development was a response to “...the demand to
instantaneously compensate the reactive power. . ..” The adverb
instantaneous in the name of this Theory and deﬁnitions of p
and q powers in terms of instantaneous value of voltages and
currents, suggest the possibility of instantaneous identiﬁcation
and compensation of the reactive power of a threephase load.
This is one of the main reasons for this theory’s attractiveness,
both as a theoretical fundamental of control algorithms and as
a power theory. Thus, the question occurs, is such an instanta
neous identiﬁcation of power properties of three—phase systems,
possible? The answer to this question is of a fundamental value
for the power theory. It could be important also for control
algorithms of active power ﬁlters. To answer such a question, this paper investigates how the
IRP p—q Theory interprets and describes power phenomena in
threephase, threewire systems. The Theory of the Currents’
Physical Components (CPC) developed in [17] by the author of
this paper is used as a tool for the study. The IRP p—q Theory was deve10ped for threephase systems
with nonsinusoidal voltages and currents. To provide credible
results for such systems, it should provide them for any subset
of such systems. Threephase, threewire systems with sinu
soidal voltages and currents form just such a subset with rela
tively simple and easy to comprehend power phenomena. There
fore, properties of the IRP pq Theory are veriﬁed using such
simple circuits. 08858993/04S20.00 © 2004 IEEE CZARNECKI: MlSlNI‘ERPRETATlONS OF THE INSTANTANFDUS REACHVE POWER pq THEORY 829 II. INSTANTANEOUS ACTIVE AND REACHVE CURRENTS The notions of the active and reactive currents have meanings
that were established in electrical engineering long ago. The
active current, deﬁned by Fryze [18] in 1932, is the smallest load
current that is necessary [10] if the load at the supply voltage
u(t) has the active power, P, and has the same waveform as the supply voltage. This current was deﬁned as . P za(t) = 8711.0) = G¢u(t) (l)
where U is the supply voltage RMS value. For threephase, threewire systems, shown in Fig. l, Fryze’s deﬁnition of the
active current is generalized [17] to the form ‘Ra P “R
in“) = 1:5“ ' W Us = G¢u(t) (2)
’Ta “T where u denotes the threephase RMS value of the supply voltage, namely
u = ‘lUﬁ + U; + U72“ (3) The reactive current is the component of the supply current
delayed by 1r / 2 with respect to the supply voltage and is deﬁned
insinglephase systems with sinusoidal voltage and current as —Qd d W) = —U—2d(wt)u(t) = B‘aot) W) (4) For three—phase, threewire systems this deﬁnition is general
ized [17], [21] to . Q d d 1,(t) — wmua) — Be d(wt)u(t). Botluhe active and reactive currents have an explicit physical
meaning. They are associated with the presence of the active and
reactive powers, P and Q, and are related to the load equivalent
conductance, Ga, and susceptance, Be. The concept of the ac
live current is also important for the design of compensators.
Because it is the smallest supply current of the load that has the
active power, P, this is the only current that should remain in the
supply lines of the load after compensation of all useless current
components. Now, let us compare the features of the active and reactive
currents, as deﬁned by (2) and (5), with the instantaneous active
and reactive currents in the IRP pq Theory. Observe however,
that the adjective instantaneous in the name of these currents
does not distinguish them from the common active and reac
tive currents, since deﬁnitions (2) and (5) specify instantaneous
values of these currents. The IRP pq Theory has evolved from the Fortescue, Park and
Clarke Transforms of voltages and currents speciﬁed in natural,
phase B, S and T coordinates, as shown in Fig. 1, into a pair
of voltages and currents in orthogonal a and ,3 coordinates. The
Clarke Transform of threephase voltages has the form (5) UR ua_21_%—% 6
[“ﬁ]— 50735—’§é :2: U Fig. l. Threephase, threewire system. For threephase, threewire systems as shown in Fig. l, with line
voltages referenced to an artiﬁcial zero, so that 11.3 + as + W E
0, the Clarke Transform of the line voltages can be simpliﬁed to theform
[:2] = [Q 35] [11:] =Cl:::l and similarly for the line currents, since in + is + i7» E 0 Jx/i: 0 l [“‘l  C [“‘l
1 . — . .
l 7; V5 25 15
The line currents can be calculated from the currents in the a
and [i coordinates with the inverse Clarke Transform [in] = ﬁ’ 0 in = 01 i0 (9)
ZS _ 713. 715 ig ig .
With the voltages and currents transformed to the a and [3 co ordinates, the instantaneous power of the load can be expressed
as (8) p = uaia + 11,3753 (10) referred to as the instantaneous real or instantaneous active
power in the IRP p—q Theory. The instantaneous imaginary
power, q, is deﬁned in [2] as (11) and is usually referred to [6]——[l6] as the instantaneous reactive
power. With these two instantaneous powers, the instantaneous ac
tive and reactive currents are deﬁned in [2]. The instantaneous
active current, ip, is deﬁned in the a and [i coordinates as q = uaip —— rigid, ua _ 11.3 i = — = —
up 11% + “33177 1/31; “(2' + “g? (12)
while the instantaneous reactive current, iq, is deﬁned as
—uﬁ . “a
z = — ’l, = — .
all 11% + ugqi ﬂq 11% + ugq (13) The instantaneous active and reactive currents in the supply lines
can be calculated from these currents in the a and [i coordinates
with the inverse Clarke Transform, (9), namely erode] Worm . zSp ‘ﬁp , iSq iﬁq
Unfortunately, the reactive current has little in common with the reactive power, Q, of the load. This is shown in Illustration (14) \\ 830 82 Fig. 2. Example of a circuit with resistive unbalanced load. 1, where the IRP pq Theory is applied to a circuit with zero reactive power, Q.
Illustration 1: Let us assume that a resistive load, connected as shown in Fig. 2, is supplied from a symmetrical source of a
sinusoidal, positive sequence voltage, with it}; = x/2U cos cult,
U = 120 V. Given such assumptions, the supply voltage in the
a and ﬂ coordinates has the value an _ C ﬁUcoswlt
ug _ «EU cos(w1t — 120°) Since the line currents are equal to ﬁU cos wlt =[\/§Usinw1t]'
(15) 1,, = \/21cos(w1t + 30°) = 45, I = 103.9 A, iT o. The supply current in the a and ,8 coordinates has the value [in] :C[ in ] _ [ﬁlcos(w1t+30°)] i5 —’iR — ICOS(w1t + 30°) (16) and consequently, the instantaneous active power of such a load
is equal to p = uaia + «1,511; = \/§UI [1 + cos2(w1t + 30°)] (17)
and the instantaneous reactive power is equal to
q = uaig — 1152",, = —\/§UIsin 2(w1t + 30°). (18) Thus, the instantaneous active current of the load in the a and
[3 coordinates, can be calculated. It is equal to . in, = ﬂ}; = I [1 + c052(w1t + 30°)] coswlt, (19)
u o .
16,, = mp = [[1 + c052(w1t + 30 )]smw1t (20) which in the phasecoordinates is equal to [21m] =I[1+cos2(w1t+3°°)]c_l[ coswlt]
tsp sin out = £1 [1 + c052(w1t + 30°)] coswlt ] . (21) 0X [cos(w1t — 120°) The instantaneous reactive power, q, of the load considered in
this Illustration is not equal to zero, hence, according to the IRP
pq Theory, a reactive current has to occur in the supply lines.
Its value in the a and ,8 coordinates is equal to 1“, 2 q = Isin 2(w1t + 30°) sin cult, (22) i =—
aq u§+uﬁ IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 19. NO. 3. MAY 2004 u
° 2 q = —Isin 2(w1t + 30°)cosw1t m <23) iﬁq = and consequently, the inverse Clarke Transform results in the
instantaneous reactive current in the supply lines [i39]=1siu2(w1t+ 30°)01[ ’qu
2 .
= §Ism2(w1t+30°) X sin wlt
sin(w1t — 120°) sin wlt
— COS (4)1t (24) Equations (21) and (24) show that the names instantaneous ac
tive current and instantaneous reactive current were given in the
IRP pq Theory to currents that have nothing in common with
the notion of the active and reactive currents as used in elec
trical engineering. Also the reactive current 2}, occurs in supply
lines of the load in spite of its zero reactive power, Q. Moreover,
both the active and reactive currents in systems with sinusoidal
voltage and linear loads that do not generate harmonics, as con
sidered in the Illustration, are nonsinusoidal. For example, the
active current in the line R is equal to i3? = % [2 coswlt + cos (wlt + 60°) cos(3w1t + 60°)}
(25)
thus, its contains the third order harmonic. This conclusion ob
tained from the IRP pq Theory is in sharp contradiction to the
notion of the active current that was introduced to electrical en
gineering by Fryze. The’suggestion of the IRP pq Theory that
line currents of a linear load, that does not generate harmonics in
a system with a sinusoidal voltage, contain a nonsinusoidal com
ponent should be considered as a major misconception of elec
trical phenomena in threephase circuits. Moreover, unlike the
active current, in, deﬁned by Fryze, the active current, ip, that
results from the IRP p—q Theory is not the current that should
remain in the supply lines after the load is compensated to unity power factor. It cannot be considered the compensation goal.
Illustration 1 also shows that the Instantaneous Reactive
Power pq Theory does not identify power properties of the
load instantaneously. Both powers are timevarying quantities,
so that, a pair of their values at any single instant of time does
not identify power properties of the load. The possibility of
instantaneous identiﬁcation of the active and reactive powers,
1) and q, does not mean that power properties of the load are
identiﬁed instantaneously. For example, for t = 'r, such that
(our + 30°) = 0, the instantaneous reactive power, q = 0, as
it is in a circuit with purely resistive balanced load, while for
t = 1', such that (wrr + 30°) = 90°, both the instantaneous
active and reactive powers are equal to zero. Thus, power
properties of the load cannot be identiﬁed without observation
of the p and q powers over the entire cycle of their variability.
However, even such an observation of these powers does not
explain power properties of the load without additional analysis.
For example, the instantaneous reactive power, q, has occurred
in the circuit considered in Illustration 1 not because of the
load reactive power, Q, but because of the load imbalance.
How could this imbalance be identiﬁed having the values of the
instantaneous reactive power q? Therefore, the IRP pq Theory
has no advantages with respect to the time interval needed for CZARNECKI: MISINTERPRETATIONS OF THE INSTANTANEOUS REACTIVE POWER pq THEORY 831 Fig. 3. Example of threephase circuit with purely reactive load.
the identiﬁcation of load properties over power theories based
on the frequencydomain approach that require the system to be observed over one period, T, of its variability. Illustration 1 has demonstrated that the instantaneous reactive
current has nothing in common with the load reactive power, Q.
It also occurs that the instantaneous active current in the IRP pq
Theory has nothing in common with the load active power, P.
This is shown in Illustration 2. Illustration 2: Let us consider a circuit with a purely reactive
load as shown in Fig. 3, supplied like in Illustration 1, thus, with
the voltage in the a and ﬂ coordinates speciﬁed by (15). The line
currents in such a circuit are equal to in = ﬂl cos(w1t — 60°),
I = 103.92 A is = '43, ir = 0. (26) thus, the line currents in the a and ﬂ coordinates are equal to
z“, _ 0 ﬁt cos(w1t — 60°)
i5 — \/§Icos(w1t— 60°) _ ﬁl cos(w1t — 60°)
_ [ —Isin(w1t— 60°) ] ' (27) Consequently, the instantaneous real and_imaginary powers
have the values p = uaia + um = JEUI cos(2w1t — 30°), (28)
q = uaip — um}, = t/ﬁUI [1 + sin (2w1t — 30°)]. (29)
In spite of the zero active power of the load, there is a nonzero
active current in the circuit. Its value in the a and ﬂ coordinates
is equal to
"a = u2+ufep = g [cos(w1t — 30°) + cos (3w1t — 30°)], (30) "ﬂ i =———
5? uﬁ+u§p = — g[sin(w1t — 30°)  sin (3w1t — 30°)] (31) and in the phase coordinates is equal to in m
7:5? iﬁp
— 1 f 0 l
" I 1
'75’ 75
% [cos (wlt — 30°) + cos (3w1t — 30°)]
X ~12:[sin(w1t—30°)—sin(3w1t—30°)] '
(32) In particular, the instantaneous active current in the line B is
equal to tap: %[cos(w1t + 30°) + cos(3w1t — 30°)]. (33) This means that according to the IRP pq Theory an active cur—
rent occurs even in purely reactive circuits. This current is non
sinusoidal even if there is no source of harmonics in the supply
source and the load. Illustrations 1 and 2 show some features of the IRP pq
Theory when it is applied to systems with sinusoidal voltages
and currents without any explanation of these features. They
can be explained using the Theory of the Currents’ Physical
Components. HI. APPARENT POWER RELATED AMBIGUI’I‘Y Power theory provides deﬁnitions of various powers in elec
trical circuits along with relations between them. Unfortunately,
there is a major ambiguity with respect to one of the most com
monly used powers namely the apparent power, S, in three
phase systems. This ambiguity exists even at sinusoidal voltages
and currents [3]. Namely, according to the IEEE Standard Dic
tionary of Electrical and Electronics Terms [19], the apparent
power is deﬁned as SA = URIR + UsIs + UTIT (34) and this power is referred to as arithmetical apparent power or
as SG=W referred to as geometrical apparent power. There exists a third
deﬁnition of the apparent power for threephase threewire sys
tems introduced for such systems under nonsinusoidal condi
tions in [17], but not referenced by the IEEE Standard Dictio
nary, namely (35) S = llull ' Hill (36) For systems with sinusoidal voltages and currents this apparent
power is equal to 5:53=‘,/U,2,+U§+U%,/I},+U§+I; and was suggested [20] by Buchholz in 1922. In balanced three
phase systems with sinusoidal voltages and currents these three
apparent powers have the same numerical value. The difference
becomes visible when the load is unbalanced or waveforms are
nonsinusoidal. This difference in a circuit with an unbalanced
load is shown in Illustration 3. Illustration 3: For the system shown in Fig. 2, deﬁnitions
(34), (35), and (37) result in (37) SA = 24.9 kVA, $0 = 21.6 kVA, $3 = 30.5 kVA and consequently, the power factor, /\ = P/ S, that means the
raﬁo of the active and apparent power depends on the selection
of the apparent power deﬁnition. Since the active power of the
load in the Illustration considered is equal to P = 21.6 kW, 832 Fig. 4. Circuit with balanced resistive load and load active power,
P = 100 kW. then, depending on the selection of the apparent power defini
tion, different values of the power factor are obtained, namely AA = 0.87, /\G = 1, AB = 0.71. Thus, it seems to be unclear what is the true value of the power
factor. It is unclear as well, what is the power rating of a com—
pensator, Sc, needed for the power factor improvement to unity value, that means
Sc = \/ 52 — P2. The result depends on the selection of the apparent power deﬁni
tion. Unfortunately, any relation between powers in threephase
circuits cannot be established without due clariﬁcation of this
ambiguity related to the apparent power deﬁnition. (3 8) IV. SELECTION OF THE APPARENT POWER DEFINITION The apparent power is notassociated with a particular power
phenomenon in electrical circuits. It is a conventional quantity, a
figure of merit that describes the supply—equipment with respect
to the voltage and current RMS values the supply is capable to
provide customers. The same is with the power factor, A. It is
the ratio of the active and apparent powers, P/ S. Its decline at
a speciﬁed load active power, P, is associated with an increase
in the RMS value of the supply current and consequently, with
an increase...
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