GENERALIZED INSTANTANEOUS REACTIVE POWER THEORY IN POLY-PHASE
POWER SYSTEMS
HERRERA REYES
GENERALIZED INSTANTANEOUS REACTIVE POWER THEORY IN
POLY-PHASE POWER SYSTEMS
R. S. Herrera, P. Salmerón, J. R. Vázquez, S. P. Litrán, A. Pérez
ESCUELA POLITÉCNICA SUPERIOR. UNIVERSITY OF HUELVA
Department of Electrical Engineering
Ctra/ Palos de la Frontera s/n, Palos de la Frontera, Huelva, Spain
Tel.: +034– 959 21 75 85, Fax: +034 – 959 21 73 04
E-Mail: reyes.sanchez@dfaie@uhu.es; patricio@uhu.es; vazquez@uhu.es; salvador@uhu.es;
aperez@uhu.es
URL: http://www.uhu.es/eps
Keywords
«Instantaneous Reactive Power Theory», «Active Power Filter», «Power Quality», «Multiphase systems».
Abstract
The systematic use of Active Power Filters, APFs, to compensate nonlinear loads, has extended the use of
the instantaneous reactive power theory. Originally, the p-q theory appeared; it obtained constant
instantaneous power in the source side after compensation. Later, other formulations have been developed.
They have allowed different compensation objectives to be obtained. Nevertheless, all of them can only be
applied to three-phase systems, i.e. those formulations frameworks can not be used to establish control
strategies in poly-phase systems. This paper presents a new approach, based on geometric algebra, which
can be applied to multiphase systems. A new tensor product is introduced that allows the operative
definition of the instantaneous reactive power tensor and its derived instantaneous reactive current
component. In addition, in this paper a new concept of instantaneous power multivector is introduced
which allows the instantaneous reactive power theory to be encoded in an analog way as single-phase
systems analysis. The new expressions are applied to a practical multi-phase system. Power and current
terms are calculated and the associated waveforms are presented. Results corroborate the correspondence
with the traditional electric power theory.
Introduction
At the beginning of the eighties, the instantaneous reactive power theory was published. It had the
objective of being applied in a systematic way to the active power filters (APF) control design to
compensate nonlinear loads, [1,2]. Besides, it has been used as theoretical framework to establish the
electric properties of energy transmission between source and load in the most general asymmetry and
distortion conditions. A lot of approaches have been appeared since the publication of the Akagi et al.
original work, [3-17]. They compose an important set of formulations that may be considered into the
instantaneous reactive power theory framework. Thus, besides the original formulation, they stand out,
among others, the modified p-q or cross product formulation, [3-6], the d-q formulation, [7], (or its
alternative, the id-iq, [8]) in the rotating frame, the p-q-r formulation, [9], and the vectorial formulation,
[10-12]. All of them relate the power transfer in a three-phase system in function to the instantaneous
power (instantaneous real power) p(t) and to the instantaneous imaginary (or reactive) power, q(t),
depending on the formulation. This last quantity marks the difference between the instantaneous reactive
power theory and the rest of other possible theories about the electric power. The instantaneous imaginary
(or reactive) power, q(t), are formally defined, in most of the cases, as the voltage and current vectors
cross product. In any case, instantaneous reactive power theory reclaim the fact that to relate the energy
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POWER SYSTEMS
HERRERA REYES
transfer between source and load in a three-phase system, p(t) is not enough, but a power quantity like q(t)
is necessary. In fact, those power variables are the base of the current decomposition in its different
components, and only in the case of single-phase systems, p(t) is enough.
All the formulations proposed have been limited to three or four-wire, three-phase systems, and all of
them have a difficult generalization to poly-phase systems. Up now there have been a few attempts in the
regular papers. In [5], it is proposed the decomposition of an n-dimensional current vector in two
G
G
G
orthogonal components, i p and iq . The first one, the instantaneous power current i p ( t ) , is obtained by
G
G
means of the projection of the current vector, i ( t ) , over the voltage vector, v ( t ) . The second one, the
G
G
G
instantaneous non-active (or reactive) current iq ( t ) , is the i p ( t ) complement to obtain the current i ( t ) ,
G
and from the length iq ( t ) , the instantaneous reactive power q(t) is derived. However, in [5] it is not
G
established an operative expression to obtain the current component iq ( t ) , or the power q(t). An important
advance in the generalization to multiphase systems is presented in [13-14]. In this paper an instantaneous
n-phase power p(t) and an instantaneous n-phase reactive power tensor q are proposed, where the second
one is a second-order asymmetrical tensor. They are defined by means of formal expressions.
On the other hand, over the last 40 years geometric algebra has been developed. It constituted a
mathematical framework based on Clifford and Grassman algebras. The geometric algebra integrates
vectors and matrix and the operations traditionally applied to them into a unified and coherent language.
In addition, its expressions can be expanded to any number of dimensions in a easy way, [18]. Into de
geometric algebra framework, the geometric product is presented. It allows the instantaneous power
multivector to be defined. Thus, the power analysis of n-phase systems is possible in an analog way as the
single-phase systems.
In this paper, the [13-14] line is followed and, into the geometric algebra framework, there are introduced
the mathematical operations necessary to define, in an operative way, each of current components and
each of power terms in polyphase systems. The development is not only the calculation of instantaneous
reactive current as the difference between instantaneous current and instantaneous power current, but it
evolves the introduction of an instantaneous reactive current defined in function to an instantaneous
imaginary power.
Instantaneous Reactive Power Theory for Polyphase Systems
Mathematical Foundations
Based on Fig. 1, the voltage and current vectors are defined as follows:
G
G
T
T
u = u = ⎡⎣u1 u2 ... un ⎤⎦ ; i = i = [i1 i2 ... in ]
(1)
From these voltage and current vectors within n-phase systems, the instantaneous real power, p(t), and the
instantaneous reactive power tensor, q(t), can be defined. The first one is obtained from the inner product,
whose result is a scalar valor and the second one is obtained from the outer product, whose result is a
hemi-symmetrical tensor, represented by a matrix. It is:
G
G
G
G
u (t ) ⋅ i (t ) = p (t ) ; u (t ) × i (t ) = q (t )
(2)
In fact, the outer product is defined by means of the tensor product in the next way:
G
G G
G
G G
u (t ) × i (t ) = i ⊗ u − u ⊗ i
(
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HERRERA REYES
which, in explicit way, has expressions derived from now on.
i1(t)
i2(t)
i3(t)
SUPPLY
VOLTAGE
LOAD
in(t)
u1(t)
u2(t)
un(t)
Fig. 1: n-phase system where the voltage is defined with respect to an arbitrary reference
The tensor product of current vector over voltage vector is:
⎡ i1u1 i1u2
⎡ i1 ⎤
⎢
G G ⎢⎢i2 ⎥⎥
iu iu
i ⊗u =
⎡⎣u1 u2 ... un ⎤⎦ = ⎢ 2 1 2 2
⎢ ...
⎢...⎥
...
⎢
⎢ ⎥
⎣⎢in ⎦⎥
⎣⎢in u1 in u2
... i1un ⎤
... i2un ⎥⎥
... ... ⎥
⎥
... in un ⎦⎥
(4)
And the tensor product of voltage vector over current vector is:
⎡ u1i1 u1i2
⎡ u1 ⎤
⎢ ⎥
⎢
G
ui ui
u
G
u ⊗ i = ⎢ 2 ⎥ ⎡⎣i1 i2 ... in ⎤⎦ = ⎢ 2 1 2 2
⎢ ...
⎢ ... ⎥
...
⎢ ⎥
⎢
u
i
u
u
n i2
⎣⎢ n ⎦⎥
⎣⎢ n 1
... u1in ⎤
... u2in ⎥⎥
... ... ⎥
⎥
... un in ⎦⎥
(5)
... i1un − u1in ⎤
... i2un − u2in ⎥⎥
⎥
...
...
⎥
...
0
⎦
(6)
So, the instantaneous reactive power tensor gets the next form:
0
i1u2 − u1i2
⎡
⎢
G
u i −i u
0
G
q = u ×i = ⎢ 1 2 1 2
⎢
...
...
⎢
⎣u1in − i1un u2in − i2un
where all the terms above and below the main diagonal are the same and with opposed sign, and so it is a
hemi-symmetrical tensor. Its norm determines the instantaneous reactive power q(t):
q(t ) = q =
qu
(7)
(u ⋅ u )
1/ 2
Both power terms characterize the energy transfer between source and load in a multiphase system.
Moreover, the definition of an only power term is possible, which encodes all the information about the
power transfer in an n-phase system. Thus, the instantaneous power multivector is defined as the
geometric product of voltage and current vectors:
GG G G G G
s(t) = ui = u ⋅ i + u × i
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It is obtained from the combination of inner and outer products. From (2), s(t) can be expressed as:
s(t) = p(t ) + q(t)
(9)
i.e., the instantaneous power multivector combines in an only mathematical entity the scalar valor which
represents the instantaneous power and the tensor value which represents the instantaneous reactive
power. This kind of representation is usual in the geometric algebra framework, whose elements belongs
to this kind of entity are denominated multivectors.
Into the geometric algebra framework the reverse of an element, for example q(t), is defined as the
following expression:
q† = ( −1)
k ( k −1) / 2
q
(10)
where k is the vectors number which generate q, i.e., k=2. It allows the s(t) multivector norm to be
defined as:
(
s = ss +
where
0
)
1/ 2
0
(11)
means scalar value.
From (10) and (11) the square value of s(t) multivector norm can be obtained:
2
s =
( p + q )( p + q )
+
0
= p 2 + q 2 = s (t )
(12)
Where s(t) is the instantaneous apparent power. The introduction of the instantaneous power multivector
allows a power term for multiphase system to be obtained, according to the same approach used in
sinusoidal steady state.
Current vector decomposition
G
Considering any vector, as for example voltage vector u , its inverse, into the geometric algebra
framework, can be defined as follows:
u −1 =
u†
u†
=
u†u u 2
(13)
where u is the instantaneous norm of vector u ( t ) . Multiplying by the left side s(t) multivector the uG ( t )
inverse, next expression is obtained:
u −1s(t) = u −1ui = i = i ( t )
(14)
Thus, from (9):
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i=
u†
u
2
s(t) =
u
u
2
( p + q) =
up
u
2
+
HERRERA REYES
uq
u
(15)
2
where the fact that u† ≡ u has been considered. In this way, current vector is divided in two current
G
components: instantaneous power current ip ( t ) = i p ( t ) ,
u
ip =
u
p (t )
2
(16)
G
and instantaneous reactive current iq ( t ) = iq ( t ) ,
iq =
uq
u
(17)
2
In (17), voltage vector u and tensor q operate by means of geometrical product, i.e.:
uq = u ⋅ q + u × q = q × u
(18)
because u·q≡0.
Therefore, into the geometric algebra framework, a decomposition of current vector has been possible for
multiphase systems. This decomposition is orthogonal. In fact:
i = ii + = ( i p + i q ) ( i +p + i q+ )
2
0
= ip
2
+ iq
2
(19)
On the other hand, according to (19) and (9), current vector norm has the following expression:
2
2
2
⎛ up ⎞⎛ u + p ⎞ ⎛ uq ⎞⎛ q +u + ⎞
u p2 u q u+
q2 u
p2
p2 + q2
=
+
=
+
=
i = i i + i i = ⎜ 2 ⎟⎜ 2 ⎟ + ⎜ 2 ⎟⎜
⎟
2
2
2
2
2
2
⎜ u ⎟⎜ u ⎟ ⎜ u ⎟⎜ u 2 ⎟ u 2 u 2
u u
u
u u
u
⎝
⎠⎝
⎠ ⎝
⎠⎝
⎠
2
+
p p
+
q q
(20)
And so,
p2 + q2 = s2 (t ) = u
2
2
i = u
2
ip
2
+ u
2
iq
2
(21)
The similarity to single-phase circuits is complete. Thus, the next equation generalizes the instantaneous
power theory to n-phase systems from n=1. I. e.:
G
G
us ( t )
i (t ) =
2
(22)
u
Several compensation approaches can be achieved applying this new formulation: instantaneous
compensation or average compensation. Into the second approach framework, several objectives can be
considered: constant source current, unity power factor and balanced sinusoidal source current, [11], [14]-
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HERRERA REYES
[16]. Besides, thanks to know the mathematical expression of instantaneous reactive power, there are other
compensation objectives to impose.
Compensation Current
In previous sections, general power terms have been derived within geometric algebra framework. In
addition, those expressions have been used to decompose the current vector into two orthogonal
components. Therefore, in this section, a control strategy is established and the compensation current is
derived by means of the developed expressions. The considered system is shown in figure 2, which
presents two elements supplied by an only source. One of those elements can be considered the load and
the other the compensator.
Source
Load
Compensator
Fig. 2 General three-phase four-wire system
Thus, according to the fact that the instantaneous power multivector is additive, it is:
s S = s L + sC
(23)
where sS is the source instantaneous power multivector, sL the corresponding to the load and sC, the
corresponding to the compensator. The instantaneous compensation strategy imposes that the compensator
instantaneous power multivector is the same as the load instantaneous reactive power, that is:
s C = −q L
(24)
In addition, within this strategy, the compensation instantaneous power multivector has the following
expression:
sC = ui C = u ( −i Lq )
(25)
i C = −i Lq
(26)
and:
Thus, according to (22):
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iC =
us C
2
u
=
HERRERA REYES
u ( −q L )
u
(27)
2
On the other hand, the instantaneous current supplied by the source is the next:
i S = i L + iC =
us L
u
2
+
−uq L
u
2
=
u
u
2
(sL − q L ) =
u pL
u
2
G
≡ iLp
(28)
Therefore, applying a control strategy which compensates the load instantaneous reactive power tensor,
the instantaneous current supplied by the source sets to the instantaneous power current incoming the load.
Practical Application
In this section, definitions presented in section II are applied to the system shown in figure 3. It is a 12
pulses rectifier. A three-phase source supplies a three-winding transformer YY, YΔ. Secondary winding
supplies two rectifiers connected in series and with an inductive load in the dc side. The voltage and
source current presented in this section have been measured in the transformer secondary winding, which
constitute a six-phase system, section M in figure 3. This system has been chosen to illustrate the
definitions proposed in previous sections because it corresponds to a practical case where six-phase
waveforms can be identified. Besides, in the system usual configuration, the six-phase voltage is
composed of two three-phase voltage systems delayed 30º. Thus, the six-phase voltage system is
unbalanced sinusoidal, as can be seen in table Ia). This is a poly-phase system with general configuration
derived from a practical circuit frequently used in power electronics. The application of the approach
developed along the paper to a system of this kind reveals that the approach allows the analysis of a
practical and general case.
M
Transf
Source
Load
Fig. 3 Twelve pulse rectifier, used as load in the power system
Besides, the phase 1 of the current components in M has been presented in table I. They have been
calculated according to definitions presented in section II.
Table Id) shows the phase 1 of the six-phase waveform corresponding to the current required by the load.
It corresponds to a strongly non linear load. Table Ib) presents the phase 1 of instantaneous power current
waveform obtained applying (16). The complete waveform corresponds to two balanced three-phase
systems with an argument difference of 30º between both. This current includes the complete
instantaneous real power, which can be seen in table Ie). This waveform is the corresponding to the inner
product uT ⋅ i and to the expression uT ⋅ i p where ip has been calculated according to the equation (16).
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Table I: Simulation results
a) Voltage in M
d) Phase 1 of current required by the load
300
10
8
200
6
100
4
2
0
0
0,005
0,01
0,015
0,02
0,025
0,03
0,035
0
-2
0
0,005
0,01
0,015
0,02
0,025
0,03
0,035
-100
-4
-6
-200
-8
-10
-300
P has e1
P hase2
P hase3
P has e4
P has e5
P hase6
b) Phase 1 of instantaneous power current
10
e) Instantaneous real power
6,E+07
8
6,E+07
6
4
6,E+07
2
6,E+07
0
-2
0
0,005
0,01
0,015
0,02
0,025
0,03
0,035
6,E+07
6,E+07
-4
-6
6,E+07
-8
6,E+07
-10
0
c) Phase 1 of instantaneous reactive current
5
0,005
0,01
0,015
0,02
0,025
0,03
0,035
f) Instantaneous reactive power
2500
4
3
2000
2
1
1500
0
-1
0
0,005
0,01
0,015
0,02
0,025
0,03
0,035
1000
-2
-3
500
-4
-5
0
0
0,005
0,01
0,015
0,02
0,025
0,03
0,035
These two waveforms are superimposed. So, effectively, the instantaneous power current carries the
whole instantaneous real power. Table Ic) presents the phase 1 of the six-phase waveform corresponding
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to the instantaneous reactive current. It transfers the whole instantaneous reactive power. To corroborate
it, table If) presents two waveforms. One of them has been calculated as follows:
q (t ) = s2 (t ) − p2 (t )
(29)
where s(t) is the instantaneous apparent power obtained according to the next expression:
G
G
s (t ) = u (t ) i (t )
(30)
Another graph has been calculated as the norm (according to (12)) of the second-order skew-symmetric
tensor defined in (6).
The waveforms presented in table I third column, second row are superimposed, too. So, the instantaneous
reactive current transfers the whole instantaneous reactive power.
Conclusions
In this paper, a generalized instantaneous reactive power theory formulation has been presented. This new
formulation proposes a way of calculating the instantaneous reactive power which can be applied not only
to three-phase power systems, but to n-phase power systems. It has been developed into the geometric
algebra framework. Thus, it is introduced a new tensor product which allows the determination of the
instantaneous reactive power tensor and the instantaneous reactive current component. In addition, the
instantaneous power multivector is introduced which allows the instantaneous reactive power theory to be
encoded in an analog way as single-phase systems analysis. This new approach allows the definitions of
different compensation strategies in n-phase systems. The power and current concepts presented in the
paper are used to calculate the compensation current in a general m-phase system. Moreover, the new
expressions are applied to a practical multi-phase system. Power and current terms are calculated and the
associated waveforms are presented. Results corroborate the correspondence with the traditional electric
power theory.
References
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[10] P. Salmerón, J.C. Montaño, Instantaneous Power Components in Polyphase Systems Under Nonsinusoidal
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