IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 4, JULY 2005
901
Analysis of Power Losses for Instantaneous
Compensation of Three-Phase
Four-Wire Systems
Juan-Carlos Montaño, Senior Member, IEEE, Patricio Salmerón, and Jaime Prieto Thomas
Abstract—This paper deals with the effect that the instantaneous compensation in three-phase four-wire systems, including
or not the compensation of the neutral current, has on the supply
line power losses. Thus, for three-phase circuits, the instantaneous
compensation criterion has been established based on the instantaneous power theory. According to the instantaneous value concept
the noninstantaneous power current is reduced, without altering
the instantaneous active power. Two approaches are marked in
this paper for instantaneous compensation: the first one is for
eliminating the total noninstantaneous power current but the
neutral current can still flow. The second one for eliminating the
modified noninstantaneous power current, thus the neutral current component is compensated. It demonstrates that, in common
situations of medium and low relative-values of the zero-sequence
voltage, the total losses (line and neutral losses) obtained with
the second approach are lower than those obtained with the first
approach. The same results are obtained when a criterion based
on the average value concept is used. Simulated and experimental
results are obtained to confirm the theoretical properties and to
show the compensator performance.
Index Terms—Neutral current, three-phase four-wire systems,
zero-sequence voltage.
I. INTRODUCTION
L
OAD compensation in power engineering is an important
topic to be considered nowadays. It is the procedure used
to obtain the supply currents sinusoidal and balanced or with
the same waveforms as the respective source voltages. That
is, source currents without harmonic distortion and balanced
components. At present, active power line conditioners (APLC)
make possible harmonic elimination, but no single and universally accepted electric power theory exists to obtain the control
algorithm for balanced or unbalanced-load compensation in
nonsinusoidal situations.
In general, from the compensation point of view, for
three-phase systems in distorted and/or unbalanced condition
the analysis of the line current is based on the assumption that
any current may be divided into two or more components:
active current and all other nonactive components. There are
Manuscript received July 28, 2004; revised November 8, 2004. This work was
supported by the Comision Interministerial de Ciencia y Tecnología (CICYT),
Spanish Minister of Science and Technology. Recommended by Associate Editor V. Staudt.
J.-C. Montaño is with the Spanish Research Council (CSIC), Sevilla 41080,
Spain (e-mail: montano@irnase.csic.es).
P. Salmerón and J. P. Thomas are with the Department of Electrical
Engineering, University of Huelva, Spain (e-mail: patricio@uhu.es;
jaime.prieto@die.uhu.es).
Digital Object Identifier 10.1109/TPEL.2005.850956
two main branches of power theories. The first was introduced
by Fryze [1], and is based on the average value concept. The
active current components (which are calculated using time-average measurements) are in phase with the source voltages,
and the nonactive current components are in out of phase
with the source voltages. This theory permits to speak of a
time-average compensation type to eliminate the nonactive
current. The concept presented by Fryze has resulted in the
development of much more sophisticated theories, some based
on Fourier’s analysis of voltage and current waveforms such
as the Czarnecki’s theory [2], and others developed directly
in the time domain such as the theories by Depenbrock and
co-authors [3], Kuster and Moore [4], Page [5], Filipski [6], etc.
The second main branch of power theories belongs to the
so-called “ – theory,” by Akagi et al. [7], who analyzed the
source current according to the instantaneous value concept.
The value of the instantaneous non active current is determined
at any single time instant, and hence the corresponding compensating current. This enables instantaneous compensation,
using switching devices, which theoretically requires no energy
storage elements. Later, several power theories, which are
derived of the – theory, emerged thereby that have aided to
make up the art state on electric power [8]–[16].
A detailed analysis of the – based work in comparison to
the Fryze based work, specially for four-conductor systems, is
found in [17]. Also, the subject of minimal line losses for multiwire systems at arbitrary resistances of the feeding conductors
has already been trated [18], [19]. But the present work studies
the case of the neutral current elimination as a derived process
of the nonactive current compensation according to – based
theories. The study is further extended to stationary and periodic conditions, where Fryze based theories are applicable.
In fact, for three-phase four-wire systems, the instantaneous
power theories present a serious drawback from the view of
point of neutral current control when the voltage applied to the
load is unbalanced and a zero sequence component appears. In
fact, when the zero-sequence phase voltage on the load side exists, neither of the compensation types can guarantee the elimination of the neutral current in three-phase four-wire systems.
So, the authors proposed [20] a strategy for compensating the
neutral current at the expenses of eliminating a modified non instantaneous power current,1 which in general has no maximum
1The notation instantaneous power current and noninstantaneous power current have been adopted instead of the respective instantaneous active current and
instantaneous reactive current used in [20]. This is according to the technically
correct terminology used today [21].
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 4, JULY 2005
instantaneous norm. Only when the zero sequence component
of the source voltage is zero, the instantaneous norm reach its
maximum value.
Two approaches are marked in this paper. The first one is
for eliminating the noninstantaneous power current, thus neutral current can still flow. The second one proposes the modified noninstantaneous power current elimination, and then the
neutral current component is compensated. Now, the question
is which current compensation is most economical, in particular which non instantaneous-active current elimination implies
minimal transmission losses. A practical case has been developed to confirm the theoretical properties and show the performance of the control strategies in an active compensator.
In Section II, the principle of instantaneous compensation
in nonsinusoidal three-phase four-wire systems, according the
two strategies discussed in [20] is introduced. In Section III, a
comparative study of power losses resulting of instantaneous
compensation using the two strategies is obtained. In Section IV,
the comparative study is extended to stationary and periodic
conditions. Simulation results of a practical situation, showing
the losses reduction for the two approaches of paragraphs
Section III and Section IV, are given in Section V. Principal
results of an experimental prototype designed for testing these
concepts are shown at the end of Section V.
II. THEORETICAL BACKGROUND
According to proposed theories [8]–[13], the three-phase generator and resulting line currents can be described in vector form
(1)
where is the voltage vector, which contains as elements the
instantaneous phase voltages,2 and is the current vector, which
contains as elements the instantaneous line currents.
Let and be the instantaneous norm of the above vectors
and
(2)
the instantaneous active power of the three-phase circuit
Fig. 1. System configuration of nonactive current compensator.
A. First Approach
The instantaneous power current, , is defined [8]–[13] as
the vector component obtained by projecting the current vector
on the voltage vector
(6)
The vector of noninstantaneous power currents is the
complement
(7)
1) Control Strategy: An active filter, in parallel with the
load, is used [7] for instantaneous compensation. It generates
. Then, a load current vector
the compensation current
is defined (Fig. 1), which contains the line
current components circulating through the load. In this case,
the supply current vector verifies
(8)
As shown in [13], under imbalance supply-voltage the voltage
vector, , contains a zero-sequence voltage component. It implies the existence in of a zero-sequence current component
after compensation. So, a neutral current can still flow into the
source side, producing power losses.
2) Power Losses: After compensation of the noninstantaneous power current vector
(9)
(3)
and considering (6)
For three-phase four-wire systems, the power loss in the line
and neutral conductors can be expressed as
(10)
(4)
B. Second Approach
where denotes the resistance (assumed constant) of the line
the resistance of neutral conductor and
conductor,
the neutral current. If the line conductors have
the same resistance, , the above expression becomes
A theory of instantaneous power, which permits compensation of both the zero-sequence current and the noninstantaneous
power current vectors, exists [14]–[16]. The following analysis
of the voltage vector was defined:
(5)
(11)
(12)
2For simplicity, f will be used instead of f (t) for denoting instantaneous
quantities.
(13)
MONTAÑO et al.: ANALYSIS OF POWER LOSSES
903
where the zero-sequence voltage
(14)
and
(15)
The modified instantaneous power current is defined as a current vector [14]
(16)
which is free of the zero-sequence voltage
. It implies
the definition of the vector of modified noninstantaneous power
currents as the complement
(17)
1) Control Strategy: In this second approach, the modified
noninstantaneous power current was compensated using for the
control the current vector
(18)
Fig. 2. Power loss ratio versus the zero sequence voltage magnitude (u
=1).
Fig. 2 shows (24) for several values of
ranging between
extreme values 1/3 and 2. For neutral conductor with zero resistance (24) simplifies to
(25)
resulting in
(19)
which is less than one if
tance (24) simplifies to
, and for equal conductor resis-
(20)
(26)
i.e., the supply current after compensation is free of the zerosequence current component, and the instantaneous active
power generated by the compensator is zero. Thus, the neutral
current can be eliminated after instantaneous compensation.
2) Power Losses: After compensating , the power loss in
the line and neutral conductors can be expressed as
which is higher than one if
.
Fig. 2 shows the power loss ratio dependence of the instantaneous norm of voltage component with the related condition:
1. For
0, then
1, which implies the coincidence of
formulation in the two approaches and equal power losses. For
1, and
,
equal conductor resistance,
1, i.e., the total power losses (considering the line and
then
neutral conductors) compensating according the first approach
are higher than compensating according the second approach.
1/3, then
If
(21)
(27)
III. POWER LOSS COMPARISON
A comparative study of power losses after instantaneous
compensation using the approaches of the above paragraph is
performed.
So, considering (15) expression (10) can be transformed as
(22)
for comparing with the expression of the second approach
indicating that for any instantaneous value of
, less losses
will be delivered with the first approach. But note that condition
1/3, means that the neutral-wire section is three times the
phase-wire section. This is a limit case (hypothetical case) in the
distribution of electrical power, where the conductor volume of
the three-phase circuit is equal to three single-phase independent circuits.
Note also that in actual power systems, conditions of instan1 are
taneous norm values outside the shown margin of
unrealistic.
(23)
Then, the ratio of the above power loss expressions can be considered after some manipulations
(24)
where
, i.e., the instantaneous norm is taken as
reference for obtaining the power loss ratio as a function of .
IV. QUANTITIES DEFINED IN STATIONARY AND
PERIODIC CONDITIONS
The average value concept for three-phase systems, in contrast to that of the instantaneous value, has been based on timeaveraged quantities developed in both the frequency domain and
the time domain. Under stationary and periodic conditions, the
average value of the instantaneous active power and the rms
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value of voltage and current vectors, corresponding to the fundamental period , have been defined [8].
The instantaneous power (3) averaged over the observation
interval or period defines the active power
where
(37)
(28)
is the modified equivalent conductance of the load.
1) Control Strategy: The compensating current is given by
The rms of the voltages and of the currents are, respectively
(38)
and the source current after compensation
(39)
(29)
2) Power Losses: After compensation the power loss in the
line and neutral conductors can be expressed as
As in Section II, two approaches are considered in this
situation.
A. First Approach
The active current vector,
average value concept as
, was defined [8] according to the
(30)
which is an extended definition of the Fryze’s active current [1].
Equation (30) defines those line currents that would be absorbed
by an equivalent symmetric resistive load having the same average power consumption as the actual load at voltage . So, the
load presents an equivalent conductance defined by
(31)
1) Control Strategy: The compensating current
(32)
is the current vector which must be supplied by the compensator.
It implies that the source current after compensation is
(33)
Although no power dissipation exists in the compensator [20]
, can not be controlled to zero
the source side neutral current,
if the voltage zero-sequence exists, i.e.,
(34)
2) Power Losses: As in (10), after compensation of the nonactive current vector
(35)
B. Second Approach
The modified active current vector, , was defined [14] according to the average value concept as
(36)
(40)
The same considerations of Section III can be applied to the
power loss expressions (35) and (40) for obtaining the power
loss ratio (24) as a function of , the correspondent rms value
of .
V. SIMULATION AND EXPERIMENTAL RESULTS OF
PRACTICAL CASES
The three-phase four-wire system of Fig. 1 was simulated
using Mathcad. First, a sinusoidal three-phase voltage source
with balanced resistive impedance but with 5% voltage unbalance (negative sequence) and preexisting fifth harmonic voltage
of 2.5% was considered [Fig. 3(a)]. The load consists of a typical
six-pulse converter with inductor-input filter. For the first situation, instantaneous compensation following approach 1 and 2
are considered as Cases 1 and 2, respectively. Waveforms corresponding to the practical results included in [22], in particular the phase-A voltage and phase-A load current circulating
through the converter, are shown in Fig. 3(a). The zero sequence
voltage resulting of the voltage unbalance is shown in Fig. 3(b).
In this situation, the simulation conditions were chosen to compare the two criteria of the above paragraph for equal conductor
resistance.
Under compensation, the source side neutral current of Case
2, shown in Fig. 3(c), is perfectly controlled at zero using the
second approach. As shown in Fig. 3(d), the defined power
losses rating changes between maximum and minimum peak
. Thus equal
values in coincidence with the peak points of
1) of both approaches is only verified
losses condition (
before compensation and when the zero sequence voltage, ,
, indicating higher power
reaches zero value. Otherwise
losses using the first-approach compensation method.
Comparison of the two criteria for equal conductor resistance,
Cases 1 and 2, is shown in Table I where the simulation results
. As expected, line losses
are measured at the peak point of
are the lowest using the first approach, which implies maximum
noninstantaneous power current compensation. On the contrary,
MONTAÑO et al.: ANALYSIS OF POWER LOSSES
905
Fig. 3. Simulation results in Case 1 and Case 2: (a) voltage and load current of phase-A, Y axis units correspond to both 500 V/div and 500 A/div, (b) zero-sequence
voltage, (c) neutral currents on the source side Case 1 and Case 2 (thick line), and (d) power loss rating of the uncompensated and compensated system.
total losses adding line and neutral losses are the lowest using
the second approach. Note that at the considered time instant,
the neutral losses after compensating with the algorithm of the
first approach are higher than the existing neutral losses without
compensation.
A second situation was considered, the simulation includes
now a voltage source with preexisting third and fifth harmonic
voltages of 5% and 2.5%, respectively, applied to the same arrangement of the first situation. Instantaneous compensation following approach 1 and 2 are now considered as Cases 3 and 4,
respectively. Waveforms corresponding to the practical results
are shown in Fig. 4. The influence of the third harmonic (3.78%)
is evidenced in both the zero sequence voltage and the neutral
currents [Fig. 4(b) and (c), respectively].
Under compensation, the source side neutral current of Case
4, shown in Fig. 4(c), is perfectly controlled at zero. As shown in
Fig. 4(d), the defined power losses rating changes between maximum and minimum peak values in coincidence with the peak
. As in the first situation, equal losses condition
points of
of both approaches is only verified before compensation and when the zero sequence voltage reaches zero value.
Comparison of the two criteria for equal conductor resistance,
Cases 3 and 4, is shown in Table II where the simulation results
. Results are similar to those
measured at the peak point of
of Cases 1 and 2 (Table I): line losses are the lowest using the
first approach but total losses adding line and neutral losses are
the lowest using the second approach. As above, neutral losses
compensating with the algorithm of the first approach are higher
than neutral losses without compensation.
Finally, an experimental model power system has been developed in the laboratory. It consists of a voltage set with 10%
,
)
of voltage unbalance (
applied throughout a short line, implemented with equivalent
circuits [23], to a four-wire three-phase unbalanced nonac regulinear load. The load consists of three single-phase
lators in star-connection with accessible neutral wire. The asymmetry of the load has been made using different values of
TABLE I
SIMULATION RESULTS FOR EQUAL CONDUCTOR RESISTANCES
loads and switching angles. Table III shows the main parameter
values of the experimental case, for the load and line impedances. Note that the results shown in Table I (Simulation Results) and Table III (Experimental Results) are different, they
correspond to two different cases showing the generality of the
theoretical properties. Simulation results are due to a realistic
load of high power consumption. Experimental results show
an extreme case of unbalanced voltages and asymmetric load,
which can be measured clearly with conventional meters.
The control system of the active conditioner was realized
through the dSPACE data acquisition and control board ds1103,
and the monitoring program ControlDesk. The line losses have
been measured using two power quality analyzers. The three
line losses were measured by the T//Teamware three-phase
Electrical QUality Analyzer (EQUA); and the neutral line
losses by the single-phase FLUKE 43 Power Quality Analyzer.
Line losses measurements, in this situation, are summarized
in Table IV, where equal line losses for the first and the second
approach are shown. However, total losses are lowered using the
second approach for compensation. Figs. 5–8 show results of
the tests captured from the screen of the ControlDesk program.
Applied voltages before compensation are shown in Fig. 5.
Figs. 6–8 show the calculated line losses in the considered
situations of Table IV. Fig. 6 shows the instantaneous and
average values of the total power losses before compensation.
Figs. 7 and 8 show the instantaneous and average values of
the total power losses during compensation using the first and
second approaches respectively. With the second approach
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 4, JULY 2005
Fig. 4. Simulation results in Case 3 and Case 4: (a) voltage of phase-A with preexisting third and fifth harmonic voltages of 5% and 2.5%, respectively, and load
current of phase-A, (b) zero-sequence voltage, (c) neutral currents on the source side Case 3 and Case 4 (thick line), and (d) power loss rating of the compensated
system.
TABLE II
SIMULATION RESULTS FOR EQUAL CONDUCTOR RESISTANCES
TABLE III
PARAMETERS OF THE TEST SYSTEM
Fig. 6. Instantaneous and average values of the total power losses, before
compensation.
TABLE IV
EXPERIMENTAL RESULTS OF LINE LOSSES
Fig. 7. Instantaneous and average values of the total power losses, during
compensation with the first approach.
Fig. 5. Supply voltages used in the experimental model power system,
showing 10% of voltage unbalance.
Fig. 8. Instantaneous and average values of the total power losses, during
compensation with the second approach.
lower average value (Table IV) and spikes of instantaneous
total power losses are obtained. In these figures, the effect of
the high frequency switching, and power ratio values always
higher than one, can be appreciated.
MONTAÑO et al.: ANALYSIS OF POWER LOSSES
VI. CONCLUSION
The present discussion is focused on instantaneous compensation of three-phase four-wire systems when the zero-sequence
component of the source voltage or load current exists. Power
losses corresponding to two compensation criteria, based on the
instantaneous value concept, are shown for.
1) Losses resulting of the elimination of the noninstanta. Neutral losses due
neous power current component
to the existence of the zero-sequence current may still
exist.
2) Losses resulting of the elimination of the instantaneous
modified noninstantaneous power current component
and the zero-sequence current
. Neutral losses
do not exist due to the compensation of the zero-sequence
current.
The power loss resulted of the compensation is in general different with the two approaches. We can conclude saying that
the instantaneous power loss, summing power losses in the four
conductors, for the actual values of the zero-sequence voltage
and equal conductor resistance, is lower using the compensating
current defined in the second approach than using that defined
in the first approach. The same conclusions are obtained for stationary and periodic conditions where theories based on the average value concept are applicable.
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Juan-Carlos Montaño (M’80–SM’00) was born in
Sanlúcar (Cádiz), Spain. He received the Ph.D. degree in physics from the University of Seville, Spain,
in 1972.
From 1973 to 1978, he was a Researcher at the
Instituto de Automática Industrial (CSIC-Spanish
Research Council), Madrid, Spain, working on
analog signal processing, electrical measurements,
and control of industrial processes. Since 1978, he
has been responsible for various projects in connection with research in power theory of nonsinusoidal
systems, reactive power control, and power quality at the IRNAS (CSIC).
Patricio Salmerón was born in Huelva, Spain.
He received the Ph.D. in physics from the University
of Seville, Seville, Spain, in 1993.
From 1983 to 1993, he was with the Department
of Electrical Engineering, University of Seville, and
since 1993 as Professor of electric circuits and power
electronics with the Escuela Politécnica Superior,
Department of Electrical Engineering, University
of Huelva, Huelva, Spain, where he is currently
the head of the department. He has joined various
projects in connection with research in power theory
of nonsinusoidal systems and power control in electrical systems. At present,
He is the head of the Escuela Politécnica Superior, University of Huelva.
His research includes electrical power theory, active power filters, and artificial
neural networks.
Jaime Prieto Thomas was born in Madrid, Spain,
in 1969. He received the M.Sc. degree in electrical
engineering from the University of Seville, Seville,
Spain, in 1994 and is currently pursuing the Ph.D.
degree at the University of Huelva, Huelva, Spain.
After three years of engineering practice, he joined
the Electrical Engineering Department, University
of Huelva, in 1997 as Assistant Professor. His main
research interests are active power conditioning,
power quality, and power converter analysis, design,
and control.