ORNL/CP-96303
Active Power Components
of Instantaneous Phasors
(Abstract)
CEJ
Abstract
The unique property of the instantaneous phasor
method is that the phasors of one phase can be used to
represent all three phases.
The active power
components that include the fundamental-frequency,
positive-sequence,
neg ative-sequence,
and
the
harmonic power components are studied in this paper.
John S. Hsu
Senior Member
Oak Ridge National Laboratory
Post Office Box 2009, MS 8038
Oak Ridge, Tennessee 37831-8038
Phone:
Fax:
c-11,
.aura,
(423) 24 1-5045
(423)
241-6124
.
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PAPER DIGEST
Active Power Components of Instantaneous Phasors
John S. Hsu
Senior Member
Oak Ridge National Laboratory*
Post Office Box 2009, MS 8038
Oak Ridge, Tennessee 37831-8038
Key
words: Active power, Symmetrical propeq,
Instantaneous phasor method, Phasors of one phase, Three
Fundamental-frequency,
Positive-sequence,
phases,
Negative-sequence, Harmonic power.
where the zero-sequence component for the three-phase
voltages is
Abstract
The unique property of the instantaneous phasor
method is that the phasors of one phase can be used to
represent all three phases. The active power components
that include the fundamental-hquency, positive-sequence,
negative-sequence, and the harmonic power components are
studied in this paper.
The real values of the instantaneous phasors are simply the
instantaneous phase values without the zero-sequence
component as given in (3).
I. INTRODUCIlON
The instantaneous phasor method originated by
the author to solve power quality and efficiency of threephase systems with unbalanced and distorted voltages and
currents has a unique symmetrical property. The phasors
of one phase can be used to represent three phases[ 1,2].
.
The power quality of a three-phase system can be
indicated by the roundness of the trajectories of the voltage
and current phasors [ 1, 21. The fundamental-frequency,
positive-sequence components of the immediate past cycle
can be obtained to guide the compensation of the present
values. This approach for power quality improvement is
different from the recent development of the instantaneous
reactive power [3-51. The active power components of
instantaneous phasors are studied in this paper.
The instantaneous phasors of voltages and currents
derived in [l] can either be presented in a vector format or
in a complex number format. The arbitrarily chosen
complex number format of the instantaneous phasors, such
as the voltages, Vu,V, ,and V, ,are given in (1). They have
the same magnitude but are 12O-degrees apart.
_* _
,._
-
(3)
The instantaneous phasors' imaginary values
denoted as v u q , vbs ,and vCq can be obtained from (4)
vuq =-'b -
6 '
,and
The numerators of equation (4) are actually the
instantaneousline to line voltages that are not affected by
the zero-sequence component.
The instantaneous phasor magnitude, V, of
Vu,V,, and V, can be derived from (2), (31, and (4). The
result is that
Prepared by the Oak Kiage7&uOnaI &&tory.
dG &up. iennessee 3 1131,managed by Lockheed Martin Research Corp. for the U. S.
Department of Energy under contract DE-AC05-960R22464.
The submitted manuscript has been authored by a contractor of the U. S. Government under contract No. DE-AC05-960R22464.
Accordingly. the U. S. Government retans a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or ailow
othen to do so. for U. S. Government purposes.
PAPER DIGEST
Where
381.589
Alternatively,
the
instantaneous
phasor
magnitude, V , of V,, vb, and V , can be derived from a
phase, for instance, from phase-a
II. TRAJECTORIES OF INSTANTANEOUS PHASORS
The voltages and currents obtained from a field
test of a 7.5-hp, 4-pole induction motor are shown in Fig.
1. The voltages are slightly unbalanced, and the currents
are significantly more unbalanced.
-385.3 17
385.358
Fig. 2 Trajectory of instantaneous phasors, Vu,v b , and V,.
14.591
16
0
-16.061
Fig. 3
-16
15.97
Trajectory of instantaneous current phasors,
I,, Ib and I,.
I
Fig. 1 Phase voltages and currents at full load
The trajectories of instantaneous voltage phasors,
Vu,5,and
5,of (1) are shown in Fig. 2, where the three
the trajectories of instantaneous current phasors, ia, ib, and
i,, shown in Fig. 3.
Figs. 2 and 3 also show that for the polluted
voltages and currents, the instantaneous phasor trajectories
constant speed [I, 21. The six peaks shown in the current
trajectory suggest a strong fifth or seventh harmonic
content.
PAPER DIGEST
The unique symmetrical property of the
instantaneous phasors of three phases permits taking the
voltage and current phasors of one phase to calculate the
various power components of the entire motor.
The trajectories of instantaneous phasors shown in
Figs. 2 and 3 can be broken down into the phasors for
various frequency components. Subsequently, the phasors
of each fkquency component can be further broken down
into positive and negative-sequence phasor components.
Fig. 4 shows the flow chart for obtaining the
fundamental three-phase balanced components. The bottom
left of the figure is the input example of three phase
voltages, vn, vb, and v,.
Accordingly, the phasor-a
imaginary portion, vq , and the real portion, va - vo, can
be calculated through (3) and (4). Consequently, the
instantaneous phasor magnitude, V, is computed from (6)
or (7).
1
74:zrr
magnitude, V
,
imaginary
positive and negative
peaks (49apart from
zero-crossing points)
band pass
IFFT
harmonic are of interest, a low-Q filter that normally has
fast response is good enough. The positive and negative
peaks of the second harmonic are 45degrees apart from the
zero crossing points and are used to find the initial and
synchronization phasors’ positions. Alternatively, a FFT
may be used for the same purpose. The theory given in [2]
is briefly described as follows. The maximum and
minimum peaks of the second harmonic correspond to the
in-phase positions of the fundamental-kequency positive
and negative sequences [2] as shown in Fig. 5. When the
positive-sequence phasor and the negative-sequence phasor
coincide in the same direction, the instantaneous phasor has
the maximum magnitude (points B and D) and is in phase
with the positive-sequence component. When the positivesequence phasor and the negative-sequence phasor coincide
in opposite directions (points A and C), the instantaneous
phasor has the minimum magnitude and is again in phase
with the positive-sequence component.
I
1
Read filtered
‘i‘ i
1
fundamental
filtermr FFT
,average phasor
c
I
(1) initial and
synchronization
phasors positions
(2) equivalent rmr
phase currents
(3)
positive-sequence
value, and
(4)
I negative-sequence
phasor
magnitude
Fig. 5
fundamental
3-phase balancei
comuonents
Maximum and minimum magnitudes of
fundamental-frequencyinstantaneous phasors
The fundamental voltage and current phasors of
phase a are given in Figs. 6 and 7. They are not in perfect
circles because of the unbalanced voltages and currents [5].
Their positive and negative-sequence components can be
b,
P
The second harmonic of the instantaneous phasor
can be obtained through a sesond-order-harmonicband pass
filter. Since only zero crossing points of the second
+-
an example, the positive and negative-sequence trajectories
[2] of the current instantaneous phasor are illustrated in
Fig. 7. Subsequently, the positive-sequence power a d
negative-sequence power, which equals the fundamental-
PAPER DIGEST
frequency power minus the positive-sequencepower, can be
obtained.
The three-phase instantaneous active power, p ,
calculated from the real instantaneous voltages and currents
is
This is compared with the three-phase
instantaneous power, pphosorrcalculated from the active
voltages and currents projected from the instantaneous
phasors.
-360.71 1
Fig. 6
360.71 1
Trajectory of fundamental-frequency,phase-a
voltage instantaneous phasor, Vu,.
We have the three-phase instantaneous power
P = Pphasor
+ 3v0i0.
(10)
2. N o Average Power Produced from Harmonics Interacting
with Fundamental Components
The higher orden of harmonics interacting with
the fundamental components do not produce net power over
the time period of one fundamental cycle. This can be
proven as follows.
(sin wIt.sin N q t ) dolt
-1 1.883
Fig. 7
1 1.883
Trajectories of fundamental-frequency,phase-a
current, I,, , and its positive and negative-
ILI. ACTIVE POWER COMPONENTS
I . Three-phaseInstantaneousActive Power
where the first sine term, sinw,?, is the fundamental
component; the second sine term, sinNo,t, is the
harmonic component; W, is the fundamental angular
frequency; t is time in seconds; and N is an integer greater
than one.
ge and current phasors having constant
magnitudes and rotating at a constant angular fiquency.
a,are shown in Fig. 8. The average power associated
PAPER DIGEST
with these voltage and current phasors over a time period of
the angular frequency, W , is
Average power
fundamental cycle naturally covers all the higher order
harmonics.
Use N c to denote the number of data taking per
cycle, and n to denote the sequence of taking the data. The
function, arg, means argument. The average power can be
written as
1
- 1 Illu Ilcos(a)
= IlVa
2
1
=-va
.la,
2
where the magnitudes of the phasors are IIV,IIand Il1,II.
Equation (12) indicates that the average power is equal to
half of the dot product of the voltage and current phasors,
or to half of the product of their magnitudes multiplied by
the cosine of the angle, a,between the phasors.
This manner of calculating average power from
the instantaneous voltage and current phasors will be used
in the calculations for various power components.
5. Positive-Sequence and Negative-Sequence Average
Power Componem are lndependentfrorn Each Other
The fundamental-frequency, three-phase, positive
and negative-sequence, instantaneous voltage and current
phasors are shown in Fig. 9. The suffix, P, is for the
positive sequence, and the suffix, N, is for the negative
sequence.
NJN
U
IC P
real
L
ViG-
w
i 3\
trajectones
Fig. 8
Voltage and current phasors with constant
magnitudes and rotating at a constant angular
frequency, 0.
4. Average Power of Immntaneous Phasors
Because of the unique property of the
instantaneous phasors, any one phase, such as the phase-a
average power, represents one third of the average threephase power. In general the trajectory of an instantaneous
phasor is not a circle, and the phasor’s angular frequency is
not a constant. However, at a given time instant, the
innstantanenas uhasors can be viewed as a set of
the instantaneous averaged power of the phase-a
instantaneous phasors over a fundamental cycle. The
Fig. 9
Fundamental-frequency,positive and negativesequence instantaneous phasors
The products of currents and voltages that contain
positive and negative-sequence components yield two
independent positive-sequence and negative-sequence power
components as verified by the following derivation.
c
PAPER DIGEST
This equation can be simplified as follows.
Other harmonic powers, such as the fifth
harmonic power can also be obtained in a like manner as
demonstrated for the calculations of hndamental-frequency
power components.
IV.CONCLUSIONS
3
= - Vap
2
lap
+ 73 Vu,
0
The unique property of the instantaneous phasor
method is that the phasors of one phase can be used to
represent all three phases. The active power components
that include the fundamental-frequency, positive-sequence,
negative-sequence,and the harmonic power components m
studied in this paper.
I,,
V. ACKNOWLEDGMENT
(15)
Equation (15) shows that because of the unique
symmetrical property of the three-phase instantaneous
phasors, the power components of three-phase can be
obtained through the phasors of one phase.
Two
independent positive-sequence and negative-sequence power
components are obtained.
The summation of following terms in the
derivation of Equation (15) is zero.
Encouragement from the Power Electronics Group
headed by Mr. Donald Adams and the helpful discussions
from Dr. John McKeever are gratefully acknowledged.
VI. REFERENCES
[I] John S. Hsu, “Instantaneous Phasor Method for
Obtaining
Instantaneous
Balanced
Fundamental
Components for Power Quality Control and Continuous
Diagnostics,” Paper Number: 98WM360, Power
Engineering Society Winter Meeting, 1998, Tampa, FL.
[2] John S. Hsu, “Instantaneous Phasor Method Under
Severely Unbalanced Situations,” Paper Number:
98SM202, Power Engineering Society Summer Meeting,
1998, San Diego, CA.
[3] H. Akagi, Y. Kanazawa A. Nabae, “Instantaneous
Reactive Power Compensators Comprising Switching
Devices without Energy Storage Components,” IEEE
Trans. Ind. Appl., Vol. 20, pp. 625-630, May/June 1984.
[4] A. Nabae and T. Tanaka, “A New Definition of
Instantaneous Active-ReactiveCurrent and Power Based on
Instantaneous Space Vectors on Polar Coordinates in
Three-phase Circuits,” EWES Winter Meeting, Paper
NO. 96WM227-9 PWRD, 1996.
[5] F. 2. Peng and J. S. Lai, “Reactive Power and‘
Harmonic Compensation Based on the Generalized
%5w-F&&qp
Incem-&rs.d
Conrerence on harmorucs and Quality of Power, pp. 8389, Las Vegas, NV, October 1618,1996.
=O
r+
c '
PAPER DZGEST
VII.
APPENDIX
is derived from the fundamental-frequency, instantaneous
phasors shown in Figs. 6 and 7.
Various power components of the sample case are
calculated as examples. The number of data taking, N c , is
ar&itrariiy chosen as 84 per fundamental cycle in the
calculations.
e = 6,094 W.
The sample yields
I . Total Power Averaged Over a Cycle
The three-phase total active power given by (8) is
averaged over a fundamental cycle.
The positive-sequence, fundamend power
component refemng to (12), Zulp of Fig. 7, and Vu,, (not
shown) is
This average total power yields
eo,,,,= 6,099 W.
- arg(luIp
rill-
) - (nV1ma.x- nllmax)-l27r
N,
2. Total Power Defined by ( l o )Averaged Over a Cycle
1
N.
(17)
It results in E&tal = 6,099 W.
where n = nV1 max is the data-taking sequential number
corresponding to the maximum magnitude of fundamentalfrequency voltage phasor at point B or D shown in Fig. 5 ,
and n = nV1 min corresponds to the minimum magnitude
of voltage phasor at point A or C shown in Fig. 5. The
similar meanings of n = nll max and n = nll minare for
the current phasors.
Where from (9) and (1 3) the first summation term in (17)
For the sample data Fip = 6,067 W.
is
(18)
This term yields 6,097 W for the sample- The
instantaneous phasors, V, andl,, of the example are
shown in Figs. 2 and 3.
The second summation term of (17) is the average
zero-sequence power component,
1
Nc
- C WO,
io,).
N, n=l
This term yields 2 Watts. The zero-sequence
component, vo, is dehed by (2), and is evaluated in a
likp
~ p n n 9s
- ~qiver! i n 131
The fundamental-frequency power, 8 , that
includes positive and negative-sequence power components
The fundamental-frequency, negative-sequence
power is the difference of fundamental-ikquency power and
its positive-sequence component.
It gives 27 Watts for the sample case.
The fundamental-frequency, negative-sequence
power can also be presented in a format similar to (21).