IEEE Transactions on Power Delivery, Vol. 13, No. 4, October 1998
1342
REE-PHASE SYSTEMS
I
JC . Montaiio, Member, IEEE
IRNAS (Spanish Council for ScienMic Research)
PO. Box 1052,41080-Sevilla,Spain
Abstract: Funbental results of the generalized instantaneous power
theory have been applied for power factor correction in three-phase
systems.Instantaneous and full compensation were considered to set
the differencesbetween compensation in general conditions (periodic
or non-periodic waveforms) and compensation in steady states
@eriodicwaveforms). An active power filter is simulated as reactive
power compensator to confirm the theory.
Keywords: Three-phase systems, Instantaneous active and reactive
power, Active power line conditioners.
P. Salmeron
Department of Applied Physics and ElectricalEngineering
University of Huelva, Ctra. de Palos de la Frontera s/n
2 1819 La Rabida, Huelva, Spain
generating the current references. On the other hand, full
compensation can be obtained using the same compensators
except for the computational circuits, which need adding
integrators or low pass filters. Full compensation is more accurate
than instantaneous compensation in the sense that the current
waveform can reproduce the voltage waveform exactly. It is done
at the expenses of a slower response, i.e., a time delay equal to the
fundamental period of the voltage waveform is involved in
calculations. This is necessary to complete calculationsof average
quantities defined in periodic states.
I. INTRODUCTION
Three-phase system have been described in the time domain by
Akagi, Kanazawa, Nabae [l], who analyzed the source voltage
and current based on the instantaneous value concept. Arbitrary
voltage and cment waveforms and steady-and-kansientstates are
allowed. Their development avoids complex formulation of the
frequency domain and makes possible instantaneous
compensation, i.e., reactive power compensationusing switching
devices, which requires theoretically no energy storage
components. This explains the recent emergence of many
instantaneous power theories 12-41. One of these is a complete
vectorial model developed for explaining physically the
instantmeous energy transfer between source and load [4-61. The
source of the instantaneous active and non-active currents
(powers)is completely determined and instantaneouspower due
to zero sequence currents and voltages can be deduced from the
general formulae. In this paper, this vector theory of
instantaneous power is used for verifying the processes of
instantaneous compensation and full compensation (average
compensation).Both compensation methods reduce line losses:
the first does not dter the instantaneouspower transfer, it reduces
the instantaneous reactive current without energy storage; the
second does not alter the average power transfer, it reduces
average losses due to the instantaneous reactive current and a
component of the instantaneous active current. Instantaneous
compensation uses active power line conditioners (APLC)
(current-controlled PWM inverters 17-81) as compensators driven
by an analog controller. This controller includes computational
circuits, which comprise only adders, multipliers and dividers, for
11. INSTANTANEOUSCURRENT AND POWER
COMPONENTS
For three-phase four-wire systems (Fig. 11, the instantaneous
voltages and currents in the three phases of the line are
represented by three dimensional vectors:
t: transpose.
The instantaneous zero-sequence voltage vector, v,, and the
instantaneous zero-sequence current vector, io,are defined as
where 1, is the unity vector of size 3, and
&J0=
U , + U* + U,,
&io
=
il
+
i,
+
i,
With this it is possible to decompose the voltage vector
PE-163-PWRD-0-12-1997 A paper recommended and approved by
the IEEE Transmission and Distribution Committee of the IEEE Power
Engineering Society for publication in the IEEE Transactions on Power
Delivery. Mawscript submitted March 31, 1997; made available for
printing December 12, 1997.
1Fig. 1.-Three-phasefour-wire system
0885-8977/98/$10.00 0 1997 IEEE
(3)
1343
t
u = v + v o
uq
(4)
where v represents the voltage vector without including the
instantaneous zero sequence components. The following
relationship is verified
u2
2
= Y 2 + Yo
where U denotes the instantaneous magnitude of vector U , i.e. U'=
utU; and v and v, denote the respective instantaneousmagnitude
of vectors v and v,.
A perpendicular voltage vector has been defined [5-61
U
WO
Fig. 2.- Instantaneous space vectors on xyz coordinates.
t
(6)
Uq1=
u2-u3, u p =
U3-U1,
the instantaneous power flow.
Instantaneous current vector i is decomposed according to its
orthogonalprojections on coordinate axes of Fig. 2; i.e.,
Uq3= U1-U2
. . lpv
. + 1.
a = a0 +
as long as
t
l3U
4
=
0
(7)
Thus, we can define a coordinate system with rectangular unit
vectors x,, yh z, having the respective directions of the voltage
vectors v,, v and U,:
VO
X0"
VP
V
Yo=
V
=
Q
o +y p +
x P
O Yo
z
9
OV
io, i,and i, were determined mathematically by the Lagrange
multipliers procedure [6],
t.
lo = vo- v O I o
-- v _P_o=
0
VO VO
Po
-
.#
2
0
YO
YO
U
zo=
4
U
Q
(Fii.2) to explain the power exchange between source and load.
Thus, the scalar product of the voltage and current vectors,
i,
=
uii
uq-=
4
"",
t
(I
defines the instantaneous power delivered to the load (sum of
three-phase instantaneouspowers). It includes two components:
p = v'i, the instantaneouspower exchanged between the source
and the load when one (v,) 01 both (v, and io) zero-sequence
signals are absent.
p o = v,'.i, the instantaneous zero-sequence power exchanged
between the source and the load when both signals (v, and io)are
present.
The scalar product of the perpendicular-voltage vector and
cment vector,
t.
q= U4 a
They are, respectively,the currents of minimum 'magnitude' for
&ansferringto the load the total instantaneous powerp, at voltage
v,,, the total instantaneous power p at voltage v and the total
instantaneous power q at voltage uq.
The orthogonalprojection of vector i on the plane determined by
v and v,, is the instantaneous active current 4. It is responsible
for the instautane~us
power transfer p + p , between the generator
and the load, and is obtained by adding ioand ip,
. .
1P = l o +
*
Po
P
apv= x0- + y 0-
Yo
defies the instantaneous non-active power that is supplied by
line-to-line voltages. This type of power does not contribute to
Current component i, is obtained by projecting current vector i
on the perpendicular voltage vector ug.In three-phase systems
this component can be named instantaneous reactive current.
1344
Since i, satisfiesthe condition of orthogonality
i pt i ,
=
"9
0
4
the relationship between the instantaneous norms of the current
components can be established,
Thus, the breakdown proposed for the current is not arbitrary.
It is imposed physically by those components necessary for the
complete description of the process of instantaneoustransfer of
energy between somce and load.
All the above equations can be considered valid for n-phase
systems, changing the vector from three to n 15-61, For singlephase systems only scalar quantities are Considered and zero
sequence components do not exist. Thus, the instantaneous
cment, i ip- ip, for this configurationis then
i
Fig. 3.- Space vectors components of i, for periodical conditions.
Or, considering (4), (15) and (20),
. P
1-
V
Hence, the instantaneous current vector for the case of periodic
signals can be analyzed according to
where v = v,.
III. CASE OF PERIODIC WAVEFORMS
. .
.
i = i ea+ +
In periodic waveforms and polyphase asymmetrical loads, the
active current component
i,
=
pr
4
Each of these current components is associated with the
corresponding components of the instantaneous power,
Ggv
U
t.i
=
P
is defied as the current that circulates through a resistive
symmetricalload that has the same average power P at voltage U.
In this definition a direct generalizationof Fryze's active current
[9] for three-phase systems has been followed. The equivalent
conductance G, is defined as follows:
i
U
t.ia +
11
t.i
Pr
P = P a +Pf
(25)
(26)
The first component of the instantaneous power is the
instantaneous active power. It has an average value equal to the
active power P.The second component is termed thefluctuating
power and has an average value of nil.
IV. INSTANTANEOUS COMPENSATION
where V is the average norm of the voltage vector v,
correspondingto the period T,
The component of residual current (Fig. 3), which has been
denominatedfluctuatingcurrent, is the difference:
To reduce the line losses as much as possible without alterhg
the instantaneous power transmission, the instantaneous
imaginary power 4 should be eliminated. The compensator can
be built theoretically with switching devices without reactive
elements; in thisway its incoming instantaneous power is null and
it does not interfere in the instantaneous transfer of energy
between source and load.
Thus, the basic scheme of the instantaneous compensation [I]
includes a compensatorwhose instantaneous power components
verify pc=0, and qC=q, where 4 is the instantaneous imaginary
power on the load side (Fig. 4). The compensating current,
corresponding to the command current for the compensator, is
given by
1345
\
-
.iI
\
i2
\
LOAD
I3
'c2
id
pc = o qc= -4
c
3
CO MPENSA TO1
Fig. 6.- Block diagram for calculatingthe full compensating current in
periodic conditions.
instautaneouspower to obtain unity PF. Under this condition, the
voltage source supplies the active current i, defined by Fryze [9]
i
ia
. I.,
= i -
(30)
where i,eJ is the residual current
Fig. 5.- Block diagram for the calculation of the instantaneous compensating
currentfor three-phasethree-wire (io= 0) or four-wiresystems (io+ 0).
with phase components, derived from (3);
Fig. 5 shows the calculation of compensating-currentreference
values for three-phase three-wire or three-phase four-wire
systems. Calculation is performed instantaneously,i.e., without
time delay, by directly using the instantaneousvoltages
and currents sensed on the load side. For three-phase three-wire
system,
icI
+
ic2 + ic3
=
0
(29)
and the compensating current does not include the zero-phase
sequence current io. Thus, the dashed line of figure 5 includes
elements not necessary in this case.
V.FULL COMPENSATION
In the steady states the possibility of full compensationmay be
of interest, i.e., the elimination of the non-active terms of the
In steady or transient states, we refer to full compensationas a
compensation with or without energy storage elements for
modifying ip or p without altering the average power transfer,P.
The compensating current eliminates the residual component of
the source current (31) and, according to (19) and (24), the source
current waveform reproduces the source voltage wavefonn.
Fig. 6 shows the calculation to obtain the compensatingcurrent
in the case. of periodic waveforms. The diagram includes two low
pass fdters (LPF) for detamining average power P and squared
average-norm U.Here, the compensator response involves a
minimum time delay due to the settling time of fdters. For
sinusoidalvoltage source U'= CP is verified, thus fdtering u2 is not
necessary.
VI. SIMULATION OF PRACTICAL CASES
Instantaneous and full compensation of a three-phase four-wire
system is simulated in two practical cases using Mathcad 5.0 Plus
package. Fig. 7 shows the configurationof the simulated system.
A compensator consisting of a conventionalthree-leg converter
[8] is connected in parallel with a non-linear load (a three-phase
two-way convertor). The switching elements SCR of the circuit
are controlled by a gating angle of 40 deg referred to the zerocrossing instant of the phase voltage.
Case 1 considers a balanced sinusoidal supply voltage. Fig. 8
shows the injection of the instantaneous compensating current
after one period of the phase voltage. The reactive power
compensationis reached instantaneously and the source supplies
a distorted current waveform to the load but verifying the
principle of mini" instantaneous power transfer between
source and load.
1346
10
(4
i
cl
0
1
-10
/
I
I
I
41
a:*
-2
I
1
I
I
10
20
30
40
I
t (ms>
Fig. 9.-Waveformsofthe full compensation system of fig. 7 (phase-1) for
balanced voltage source.
Fig. 7.- Simulated c u e of instantaneous compensationof a non-linear load.
Instantaneouspower changes after compensation and maintains
the expected constantvalue P (its average value).
Case 2 consists of the same circuit as case 1but considering an
unbalanced supply voltage, i.e., the voltage amplitude of phase 2
is now 20% lower than in case 1. Thus, zero-sequence
components of voltage and current now exist. Fig. 10 shows
(from top to bottom) three cycles of the zero-sequence voltage
and simulationresultsof instantaneous compensationfor phase-1.
Instantaneouspower waveform differs from that in case 1 due to
the unbalanced voltage but the power exchange is not altered by
the instantaneous compensation process. For full compensation
(fig. l l ) , waveforms are similar to case 1 except for the
instantaneouspower, which is not constant after compensation.
It is now sinusoidal, flowing from source to load at a frequency
twice the supply frequency. As is shown, the zero-sequence
current is eliminated after compensation.
- IO
4
0
2
0
0
I
I
I
I
10
20
30
40
I
I
VII. CONCLUSIONS
t (ms)
Fig. 8.-Waveformsof the instantaneous compensationsystem of fig. 7 (phase1) for balanced voltage source.
Instantaneous power remains unchanged before and after the
instantaneouscompensation.
Full compensationcan be obtained with the same compensator
but changing the conk01 algorithm,i.e., using as reference values
the output currents of figure 6. Fig. 9 shows, for phase 1,
waveforms of the full compensation system before and after
starting compensation. When compensation starts, the source
current changes its waveform from square to sinusoidal,
indicating the existence of an equivalent purely resistive load.
For three-phase systems and non-linear loads two dZferent
procedures of load compensation were considered:instantaneous
compensation, for eliminatingthe instantaneous reactive current
and zero-sequence current, andjdl compensation, for eliminating
all the non-active current components circulating through the
load. The first procedure is able to operate in transient states
while the second procedure is conceived only in steady states
(where the compensator needs a fundamental period for
calculating average quantities P and U ) . Both can be
implementedby using the same power inverter as reactive power
compensator, but controlled by different algorithms. These
algorithms have been simulated by a mathematics package to
present compensating results that confjrm the instantaneous
power theory for three-phase systems in two practical cases.
1347
_-
I
10,
U
-10
I
,
I
I
I
20
30
I
40
I
J
t (-1
Fig. 10.-Waveforms ofthe instantaneous compensation system offig. 7
(phase-I) for unbalanced voltage source.
I
I
I
I
2,
Fig. 11.-Waveforms of the full compensation system of fig. 7 (phase-1) for
unbalanced voltage source.
VIII. REFERENCES
[l] H. Akagi, Y. Kanazawa, and A. Nabae, "Instantaneous Reactive Power
Compensators Comprising Switching Devices Without Energy Storage
Components,"lEEETrans. Znd. Appl., vol.I.4-20, no.3, May 1984, pp. 625630.
[2] L. Rossetto, P. Tenti, "Using AGFed PWM Converters as Instantaneous
Reactive Power Compensators," LEEE trans Pow. Elect., vo1.7, No.1, Jan.
1992, pp. 224-229.
[3] J. L. Willems, "A New Interpretation of The Akagi-Nabae Power
Components for Nonsinusoidal Three-phase Situations", IETE Trans.
Instnnn.Meas., vol. 41, No. 4, .4ugust 1992, pp. 523-527.
[4] F.Z. Peng and J.-S. Lai,"Generalized Instantaneous Reactive Power Theory
for Three-Phase Power Systems", E E E Trans. Instntm. Meas., Vol. 45, no.
1, Feb. 1996, pp. 293-297.
[5] P. Saimehn and J. C. Montaiio, "Simulaci6n de las Componentes Reactivas
Instantineas de Intensidad en Cargas Trifhicas no Lineales," 111Jornadas
Hispano-Lusas de Ingenieria Electrica, vol. IV, Jul. 1993, pp. 1375-1384.
[6] P. Salmeron and J. C. Montaiio, "Instantaneous power components in
polyphase systems under nonsinusoidal conditions", IEEProc.-Sei. Meas.
TechnoL,vol. 143, no. 2, Mar. 1996, pp. 151-155.
[7] H. Alcag~,"Trendsin Active Power Line Conditioners," lEl?E Trans. Power
Electronics,vol. 9, no. 3, May 1994, pp.263-268.
[SI M. Aedes, J. Hafner and K. Heumann, "Three-Phase Four-Wire Shunt
Active Filter Control Strategies," EEL? Trans. PowerElecironics, vol. 12,
no. 2, March 1997,pp. 311-318.
[9] S. Fryze, "Effective, Wattless and Apparent Power in Electrical Circuits for
the Case of NonSinusoidal Wave-Form of Current and Voltage,"
Elekfroiechnische ZeitschriJi, Vol. 53, pp. 596,1932.
VIII. BIOGRAPHIES
Jnancarlos Montaiio (M' 1980) was bom
in Sanlucar de Barrameda (Cadiz), Spain in
1940. He received the Ph.D. degree in
physics fmmthe University of Seville, Spain,
in 1972. From 1973 to 1978 he was a
Researcher at the Instituto de .4utomatica
Industrial (C.S.1.C.- Higher Council of
Scientific Research), Madrid, Spain,
working on analog signal processing,
electrical measurements in power systems,
and control of industrial processes. Since
1978 he has been rwponsible for various
projects in connection with research in
power theory of nonsinusoidal systems and
Reactive Power Control in electrical systems
at the CSIC. At present he is leading a
project for monitoring Power Quality.
Patricio Salmeron was born in Huelva
(Spain) in 1958. He received the Ph. D. in
physics from the University of Seville
(Spain) in 1993. From 1983 to 1993 he has
been with the Department of Electrical
Engineering of University of Sevilla, Spain,
and since 1993 as Professor of Electric
Circuits and Power Electronics with the
Escuela Polithcnica Superior in the
Department of Applied Physics and
ElectricalEngineering, University of Huelva,
Spain. He has joined various projects in
connectionwiththe research in power theory
ofnonsinusoidal systems and power control
in electrical systems. At present his research
includes active power filters and artificial
neural networks.