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Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions
Milano, October 13-15, 2003
A Time-Domain Approach to Power Term
Definitions under Non-Sinusoidal Conditions
Paolo Tenti*, AEI member and Paolo Mattavelli**, AEI member
*Department of Information Engineering, University of Padova, Italy
** Department of Electrical, Management, and Mechanical Engineering, University of Udine, Italy
The paper presents a time-domain approach to current and voltage decomposition and power terms definition under
periodic, non-sinusoidal conditions. The approach makes reference to basic physical quantities only, like voltages,
currents and their integrals and derivatives. Based on these quantities, instantaneous and average power terms are
introduced, which are conservative in any electrical networks and naturally extend the definitions used under
sinusoidal conditions. Since the defined power terms are additive quantities, the impact of each load connected to the
network on total reactive and harmonic absorption can easily be evaluated. Moreover, the reactive power term has a
precise physical meaning, being directly related to the energy stored in the network. An orthogonal decomposition of
currents (and voltages) is also introduced, where each current component relates to a specific power or energy term
and/or to supply voltage and load current distortion. Correspondingly, a decomposition of the apparent power into
active, reactive and distortion terms is proposed, where the influence of supply voltage distortion on the distortion
power is clearly indicated. Finally, the single-phase approach is extended to poly-phase systems. Relying on
conservative quantities, the proposed approach offers a basis for distributed reactive and harmonic compensation
under non-sinusoidal conditions.
Based on the above considerations, the theory presented
here makes use only of elementary operations in the time
domain on the above quantities. For the purpose, suitable
homo-variables are introduced, which are defined under
the assumption of periodic conditions and are
homogenous to voltage, current and power terms. After a
description of the properties of homo-variables, it is
shown that homo-powers are conservative in any
networks, which allows to introduce the concept of
conservative complex power under non-sinusoidal
conditions. The definition of reactive power follows,
which is related to the average energy storage in the
network. Based on the given power term definition, a
current decomposition is then introduced, where each
current term relates to some power/energy phenomenon,
taking also into account supply voltage and load current
distortion. Consequently, the apparent power is
decomposed into active, reactive and distortion terms,
where the influence of supply and load distortion is
apparent. Finally, the approach, initially developed for
single-phase networks, is naturally extended to polyphase systems.
Although application to metering and compensation
problems is not treated in this paper, the proposed theory
looks capable to help for the solution of these problems
too.
1. Introduction
Power theories under non-sinusoidal conditions are
mainly aimed to approach two classes of problems, the
first relating to metering, tariffs, and distorting load
identification issues, the second to harmonic and reactive
power compensation issues.
Several approaches have been developed, both in timeand frequency domain, which are suitable to solve classes
of problems under non-sinusoidal conditions, like design
and optimization of passive compensation networks
[1-6], design and control of active compensators [7-9],
identification of distorting loads [10], measuring
techniques [11]. In addition, comprehensive theories have
been developed [12], which offer the basis for a general
analysis of networks behavior under non-sinusoidal
conditions: some of them relate to the frequency domain
[13-15], some to the time domain [7-9,16], these latter
giving special emphasis to instantaneous quantities or
average quantities, depending on the aim of the work.
So far, no one of the approaches described in the
literature showed the capability to solve all classes of
problems, thus the discussion on power term definitions
and current decomposition methods under non-sinusoidal
conditions is still active and fruitful of new results.
This paper tries to give a contribution by introducing a
time-domain theory which relies on the basic observation
that, while power absorption is related to currents and
voltages across network bipoles, energy storage is related
to magnetic flux and electrical charges, i.e., to the time
integrals of voltages and currents. Moreover, elementary
bipole equations relate currents, voltages and their timeintegrals and derivatives. Finally, these latter quantities
are easily measured also under non-sinusoidal conditions,
provided that the measuring equipment has a suitable
frequency bandwidth.
2. Definitions and
homo-variables
properties
of
Consider a real, continuous and periodic variable x(t)
with zero dc value, i.e.:
x=
1
1T
∫ x ( t ) dt = 0
T0
Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions
Milano, October 13-15, 2003
where T is the period and overline means average value.
For such variable we can define the primitive function
x’(t) and the derivative function x”(t) as:
t
x ' ( t ) = ∫ x ( τ ) dτ
(2.1.a)
d
x(t )
dt
(2.1.b)
1 T df
1
) (
[f ( t )]T0 = 0
xoy+xoy =
dt =
∫
ωT 0 dt
2π
From 2.6.c we also have:
) (
x o x = −x o x = − x
0
x" ( t ) =
Let ω = 2π
T
1T
∫ x ' ( t ) dt
T0
For any set of real variables which are continuous in time
and periodic of period T, we define the following
operators.
(2.1.c)
3.1 Scalar Operators
be the average value of x’(t), we define the homointegral function of x(t) as:
)
x = ω ( x '− x ' )
(2.2.a)
and the homo-derivative function of x(t) as:
( 1
x = x"
(2.2.b)
ω
Note that both homo-integral and homo-derivative
quantities are homogeneous to x(t), i.e. they have the
same dimensional unit, which justifies their name. Thus,
)
(
x, x and x constitute a set of homo-variables.
Let’s first recall, for the sake of completeness, some
definitions already given in the previous section, i.e.,
average value:
x =x=
xoy = x⋅y =
[
so that, at any time:
(2.3.a)
(2.3.b)
1
∫ x ( t ) y( t ) dt
T0
1T
∫ x ( t ) y( t ) dt
T0
(3.2)
]
Remember that, under sinusoidal operation:
) (
x+x=0
thus:
x& α = x& β
and:
)
(
x2 + x2
x2 + x2
x& α = x& β =
=
=X
2
2
The internal product between two periodic variables
x(t) and y(t) is defined as:
xoy=
(3.1)
The complex binomial representation of a real variable
x(t) is defined as the two-elements complex vector:
)
(
 x + jx
x − jx 
x& = x& α x& β = 
(3.3)

2 
 2
where dots identify complex quantities, j is the imaginary
unit and bolded symbols refer to complex binomial
variables.
x = 2 X sin(ω t )
)
x = − 2 X cos(ω t )
(
x = 2 X cos(ω t )
T
1T
∫ x ( t ) dt
T0
and internal product:
Under sinusoidal operation (i.e., if all quantities are sinusoidal) we can
assume:
) (
x+x=0
)
(
)(
x2 + x2 = x2 + x2 = x2 − x x = 2 X2
(2.7)
3. Operators
be the angular frequency, and
x' =
2
(2.4)
(3.4.a)
(3.4.b)
where X is the rms value of x(t). Complex variables x& α and x& β are
therefore identical and are represented, in the complex plane, by a vector
of constant amplitude X rotating at angular speed ω .
and consequently the norm of x(t) is:
1 T 2
(2.5)
∫ x ( t ) dt = X
T 0
X being the rms value of x(t).
We can now enunciate three basic properties of the
homo-variables. First:
() )(
x=x=x
(2.6.a)
which derives directly from definitions 2.2. Second:
)
(
xox = xox =0
(2.6.b)
which is easily demonstrated as follows:
x = xox =
The cross product between two real variables x(t) and
y(t) is defined as the cross product between their complex
binomial representations x& and y& , in the form:
x& α y& β* + x& β y& α*
x×y =
2
(3.5.a)
where the asterisk means complex conjugate. Note that
the result of the cross product is generally a complex
number. Moreover:
T
T
1 dx
1  x 2 (t) 
( 1
xox = ∫x
dt =

 =0
T 0 ω dt
2π  2 
0
)
and similarly for x o x . Third:
) ( ( )
x o y = x o y = −x o y
(2.6.c)
In fact, let:
)
f =xy
we have:
)
df dx
) dy
) (
)(
=
= ω x y + x ω y = ω (x y + x y )
y+x
dt
dt dt
Thus:
x × y = (y × x )*
(3.5.b)
i.e., inverting the variables in the product gives a complex
conjugate result.
Developing the product we obtain:
2
Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions
Milano, October 13-15, 2003
)
(
)(
( )
x + jx y + jy x y − x y
xy+xy
⋅
=
+j
2
2
2
2
(
)
()
) (
x − jx y − jy x y − x y
xy+xy
β α*
⋅
=
−j
x& y& =
2
2
2
2
)( ()
)
)
( (
2xy − xy− xy j  xy − xy xy − xy
+ 
+
x×y =

4
2
2
2

x& α y& β* =
x = x•x =
(3.6)
(3.12)
N
n =1
and the corresponding average operators:
 x1 
1T
x = x = ∫ x ( t ) dt =  ..  average value
 
T0
 x N 
]
(3.13)
(3.14)
1T
∫ x n ( t ) y n ( t ) dt
n =1T 0
N
N
x o y = ∑ x n o yn = ∑
n =1
difference. The result of the cross product is therefore constant. The
demonstration is easily done by assuming:
(
)
x = 2 X sin(ω t ), x = − x = 2 X cos(ω t )
(
)
y = 2 Y sin(ω t − ϕ), y = − y = 2 Y cos(ω t − ϕ)
which gives:
)(
()
x y − x y = x y − x y = 2 X Y[sin(ωt ) ⋅ sin(ωt − ϕ) + cos(ωt ) ⋅ cos(ωt − ϕ)] =
x = xox =
N
∑ X 2n
internal product
(3.15)
norm
(3.16)
n =1
N
x ⊗ y = ∑ x n ⊗ yn
external product
(3.17.a)
n =1
= 2 X Y cos ϕ
( )
) (
x y + x y = −( x y + x y) = 2 X Y[sin(ωt ) ⋅ cos(ωt − ϕ) − cos(ωt ) ⋅ sin(ωt − ϕ)] =
= 2 X Y sin ϕ
Consequently:
Note that:
(
)
x⊗y = y⊗x *
(3.17.b)
and:
2
x × x = X2
x⊗x = x
(3.17.c)
We also recall the Cauchy-Schwartz inequality for the
internal product:
(3.8.b)
The external product between two real variables x(t)
and y(t) is defined as the average value of their cross
product, i.e.:
)
(
xoy−xoy
x⊗y = x×y = xoy+ j
(3.9)
2
xoy ≤ x y
[
2
(3.18.b)
The demonstration is immediate observing that, from (3.9):
[
]
ℜx⊗y = xoy
Note that, recalling Eq.2.7, application of the CauchySchwarz inequality gives:
) (
) (
2
xox = x ≤ x x
(3.10)
(3.19)
4. Homo-variables in electrical
networks. Conservation of Power
Terms
3.2 Vector operators
The above definitions can be easily extended to vectors of
real periodic and continuous variables. In fact, given
vectors x and y of size N:
 y1   y1 ( t ) 
 x1   x1 ( t ) 
 ..   .. 
 ..   .. 

  

  
x =  x n  =  x n (t) 
y =  yn  =  y n (t) 

  

  
 ..   .. 
 ..   .. 
 y N   y N ( t )
x N   x N ( t )
we define the following instantaneous operators:
dot product
]
ℜx⊗y ≤ x y
Consequently:
x ⊗ x = x o x = x = X2
X being the rms value of variable x(t).
(3.18.a)
which, for the external product, becomes:
This is easily obtained recalling that:
) (
( )
x o y = −x o y = − x o y
)
)
x o y = −x o y
(
(
x o y = −x o y
and then applying such equalities in (3.6).
N
( x& αn y& βn* + x& βn y& αn * )
2
n =1
cross product
N
Note that under sinusoidal conditions:
(3.8.a)
x × y = x& α y& β* = x& β y& α* = X Y cosϕ + j X Ysinϕ
where X and Y are the rms values of x(t) and y(t) and ϕ is their phase
x • y = ∑ x n yn
magnitude
n =1
x × y = ∑ x n × yn = ∑
The cross product of a quantity by itself is a real number,
given by:
)(
x& α x& β* + x& β x& α*
x2 − x x
x×x =
= ℜ x& α x& β* =
(3.7)
2
2
where symbol ℜ relates to the real part of the argument.
Similarly, symbol ℑ refers to the imaginary part.
[
N
∑ x 2n
Given a generic electrical network Π with L branches,
let u be the vector of a consistent set of branch voltages
(i.e., voltages which satisfy the Kirchhoff’s Law for
Voltages, KLV) and i the vector of a consistent set of
branch currents (i.e., currents which satisfy the
Kirchhoff’s Law for Currents, KLC). The Tellegen’s
Theorem states that:
u•i = 0
(4.1)
which corresponds to the principle of conservation of
the instantaneous power if u and i are simultaneous
quantities.
)
(
Considering now homo-voltages u and u , we observe
that they are consistent with network Π .
(3.11)
n =1
3
Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions
Milano, October 13-15, 2003
Thus:
Consider in fact a generic mesh
µ
L
of M branches and the corresponding
branch voltages u µ . Application of KLT gives:
M
u µm
m =1
∑
u×i =
(4.2.a)
=0
du µm
M
(
(4.2.b)
= ω ∑ u µm = 0
dt
m =1
which demonstrates that homo-derivatives of mesh voltages are
consistent with mesh equation. Since the above consideration applies to
each mesh in the network, we can affirm that homo-voltages u( are
∑
l=1
) (
L
m =1
and finally:
u×i =
( )
L
∑ u l il − ∑ u l il
l =1
l =1
4
L
)
 L)
 ∑ u l i l − ∑ u l il
j  l=1
l =1
+ 
+
2
2


Derivation of this equation gives:
M
2 ∑ u lil −
L
+
(
L
(

∑ u l il − ∑ u l i l 
l =1
2
l=1




)
( (
) ( ( )
)
2u•i − u• i − u• i j  u•i− u• i u• i − u•i

+ 
+

4
2
2
2

Since, according to Eqs.4.3, every dot product in the expression is zero,
Eq.4.4 is demonstrated.
consistent with network Π .
Integration of Eq.4.2.a gives:
M
t
µ
∫o ∑ u m ( τ) dτ = 0
m =1
5. Power Terms in single-phase
networks
since the integrated function is identically zero. Let: u'µm = ∫ t u µm ( τ) dτ ,
0
the above equation becomes:
M
(4.2.c)
∑ u'µm = 0
Given a single-phase network Π including set λ of L
branches, let assume that the network exchanges power
with the rest of the grid through a set µ of M single-
m =1
which also implies:
M
1 T M µ
∑ u' m ( t ) dt = ∑ u 'µm = 0
∫
0
T m =1
m =1
Recalling now from Eq.2.2.a that:
)
uµ
u 'µm = m + u'µm
ω
equation 4.2.c becomes:
M
1 M )µ
∑ u m + ∑ u 'µm = 0
ω m =1
m =1
Substituting Eq.4.2.d, we finally obtain:
M )
∑ u µm = 0
(4.2.d)
phase ports (exchange ports). Let uµ and iµ be the
vectors of voltages and currents measured at the
exchange ports, u λ and iλ being the vectors of network
branch voltages and currents, application of the principle
of conservation of the instantaneous power gives:
which demonstrates that also homo-integrals of mesh voltages are
consistent with mesh equation. Since this applies to each mesh in the
network, we can affirm that homo-voltages
network
⇔
uµ × iµ = u λ × y λ
⇔
(4.2.e)
m =1
)
u
u µ • iµ = u λ • y λ
M
L
m =1
l =1
∑ uµm ⋅ iµm = ∑ u λl ⋅ iλl
M
L
m =1
l =1
(5.1.a)
∑ u µm × iµm = ∑ u λl × iλl (5.1.b)
which are valid assuming as positive the power entering
the network through the exchange ports and that absorbed
by network branches. Eqs.5.1 state that there is an
instantaneous balance between the total power entering
the network and that absorbed by the network branches.
are consistent with
Π.
The above demonstration can be repeated for the currents
)
in each cut set of the network, thus homo-currents i and
(
i are consistent with network Π .
Application of Tellegen’s Theorem to every pair of
consistent sets of homo-voltages and homo-currents gives
the following equalities:
)
(
u•i = u•i = u•i = 0
) ) ) ( )
(4.3)
u• i = u• i = u• i =0
( ) ( ( (
u• i = u• i = u• i =0
which express the principle of conservation of homopowers in every electrical network.
It is interesting to note that Eqs.4.3 also imply:
u ×i = 0
(4.4)
which states the principle of conservation of
instantaneous complex power (see next section for
further analysis).
Let’s now examine the meaning and properties of
instantaneous complex power. Given voltage u and
current i measured at any network port (i.e., between any
pair of network terminals), we define instantaneous
complex power s& as:
u& α ⋅ &iβ* u& β ⋅ &i α* s&'+ s&"
+
=
(5.2.a)
s& = u × i =
2
2
2
where:
(
)(
)
ui − u i
ui+ u i
α
β
*
&
s&' = u& ⋅ i =
+j
=
(5.2.b)
2
2
= ℜ(s&' ) + j ℑ(s&' ) = r '+ j q'
) (
()
ui − u i
u i + ui
&s" = u& β ⋅ &i α* =
−j
=
(5.2.c)
2
2
= ℜ(s&" ) + j ℑ(s&" ) = r"+ j q"
where r’ and r” are the real parts of instantaneous
complex power terms s&' and s&" and are called
instantaneous real power terms. Instead, q’ and q”,
which are the imaginary parts of s&' and s&" , are called
instantaneous imaginary power terms. Substituting
(5.2.b) and (5.2.c) in (5.2.a) we obtain:
Demonstration of (4.4) can be done easily. Let L be number of branches
in the network, recalling (3.6) we have:
L
L u
& α ⋅ &i β* + u& βl ⋅ &i lα*
=
u × i = ∑ ul × il = ∑ l l
2
l =1
l=1
)
( (
) ( ( )
)
L 2u i − u
j  u i − u l il u l il − u l i l 
l il − u l il
= ∑ l l
+  l l
+

4
2
2
2
l =1


4
Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions
Milano, October 13-15, 2003
s& = ℜ(s& ) + j ℑ(s&) = r + j q =
r ' + r"
q'+q"
+j
(5.2.d)
2
2
It is noticeable that under sinusoidal conditions:
r = r ' = r" = U I cos ϕ = P
r ' + r"
q α + qβ
+j
=
2
2
)
( ( (5.7)
)(
()
)
1  ui − u i ui − u i  j  ui − u i u i − ui
+ 

= 
+
+
 2 2
2 2
2
2 


where terms:
)
)
ui − u i
qα =
(5.8.a)
2
( (
u i − ui
qβ =
(5.8.b)
2
are respectively defined as instantaneous integral
reactive power (qα) and instantaneous derivative reactive
power (qβ). Such reactive power terms are relevant for
instantaneous compensation of distorted networks.
s& = r + j q =
(5.3.a)
(5.3.b)
q = q' = q" = U I sin ϕ = Q
where U and I are the rms values of voltage u and current i and ϕ is
their phase difference. This means that real power terms r, r’, r” coincide
at each instant with active power P, while imaginary power terms q, q’,
q” coincide at each instant with reactive power Q.
Correspondingly, instantaneous complex power terms become:
(5.3.c)
s& = s&' = s&" = P + j Q
and coincide with the complex power, as usually defined for sinusoidal
operation.
Let now consider some relevant average quantities. From
(5.2.b) and (5.2.c), recalling property (2.6.c), we obtain:
(
) (
)
&S' = s& ' = r '+ j q ' = u o i − u o i + j u o i + u o i =
2
2
(5.4.a)
(
)
uoi+ uo i
= uoi+ j
= P + jQ
2
) (
( )
&S" = s&" = r"+ j q" = u o i − u o i − j u o i + u o i =
2
2
(5.4.b)
(
)
uoi+ uo i
= uoi + j
= P + jQ
2
which show that average complex power terms S& ' and
S& " coincide. Therefore:
S& '+S& "
S& = u ⊗ i =
= P + jQ
(5.5.a)
2
where:
P = uoi
(5.5.b)
is the active power and:
(
)
uoi + uo i
Q=
(5.5.c)
2
is defined as reactive power, extending the reactive
power definition under sinusoidal conditions.
Analysing the physical meaning of reactive power is the
goal of next section. Here we give some additional
definitions. Let’s write:
Q=
Q α + Qβ
2
It is immediate to verify the under sinusoidal conditions q α and qβ
coincide at each instant with reactive power Q.
Observe that all instantaneous and average power terms
defined in this section are conservative in each network,
irrespective of its topology and nature of branches.
It is also interesting to note that the Cauchy-Schwarz
inequality, in both forms (3.18.a) and (3.18.b), gives:
P ≤ u i = UI = A
(5.9)
where U and I are the rms values of voltage and current,
respectively, and their product A is the apparent power,
which, contrary to previous power terms, is a nonconservative quantity.
6. Power Terms in passive linear
networks
Consider a passive network made up of linear resistors,
inductors and capacitors. Let’s compute the power terms
defined above for each type of elementary bipole.
6.1 Resistor
The bipole equations are:
u = Ri ⇔ i = Gu
)
)
1
)
)
u = R i ⇔ i = G u with: G =
(6.1.a)
R
(
(
(
(
u = R i ⇔ i = Gu
where R is resistance and G conductance of the bipole.
Under the assumption of linearity, R and G are constant.
We have:
)(
)(
u 2 − u u i2 − i i
(6.1.b)
=
r ' = r" =
2R
2G
) )
)
)
ui − u i uu − uu
α
q =
=
=0
2
2R
( (
( (
u i − ui u u − uu
(6.1.c)
qβ =
=
=0
2
2R
(5.6.a)
where:
)
) u) o i − u o i
)
Q = u o i = −u o i =
(5.6.b)
2
is defined as integral reactive power, since its calculation
only requires consideration of homo-integral voltage and
current terms. Similarly we define:
( (
(
uo i −uoi
(
Qβ = u o i = − u o i =
(5.6.c)
2
as derivative reactive power, since its calculation only
requires consideration of homo-derivative voltage and
current terms.
α
q=
Let’s now go back to instantaneous quantities. We can
easily rewrite (5.2.d) in the form:
q α + qβ
=0
2
Instantaneous reactive power terms q, q α and qβ are
therefore zero at each instant for any resistor. It can be
5
Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions
Milano, October 13-15, 2003
shown that terms q’ and q” are not zero, although they
sum zero. Moreover, according to Eqs.5.2:
r = r ' = r" = P
(6.1.d)
∞
k =1
Qβ =
(6.2.g)
=
2ω2 L
and observe that, in general:
β
α
we
have
β
2
= −ω C U 2
(6.3.g)
2
I2
(6.3.h)
ωC
ωC
where U and I are the rms values of capacitor current and
voltage, respectively. Observe that, let
1
(6.3.i)
wC = C u2
2
be the instantaneous energy stored in the capacitor, its
average value is given by:
i
=−
1 T
1 T 2
C U2
Qα
w
(
t
)
dt
C
u
(
t
)
dt
=
=
=
−
∫ C
∫
T 0
2 0
2
2ω
(6.3.j)
which shows that the integral reactive power is related to
the average capacitor energy by the relation:
WC = w C =
(6.2.l)
Qα = −2 ω WC
(6.3.k)
which shows the physical meaning of the integral reactive
power. Let’s assume, for symmetry:
where:
L U 


2  ω L 
k =1
qβ = Qβ = −
Qα = 2 ω WL
(6.2.k)
which shows the physical meaning of the integral reactive
power. Let’s assume, for symmetry:
=
k =1
operation
q α = Qα = −ω C u
1 T
1 T 2
L I 2 Qα
=
=
=
w
(
t
)
dt
L
i
(
t
)
dt
∫ L
∫
T 0
2 0
2
2ω
(6.2.j)
which shows that the integral reactive power is related to
the average inductor energy by the relation:
U2
∞
The bipole equations are:
)
du
i
(
i=C
(6.3.a)
= ωC u ⇔ u =
dt
ωC
where C is the capacitance of the bipole, which is
constant under the assumption of linearity. The power
terms are:
)(
ui − u i
r' =
≠0
(6.3.b)
2
)
(
( (
ui − u i
uu − uu
r" =
= ωC
=0
(6.3.c)
2
2
)
)
)(
u u − u2
ui − u i
(6.3.d)
= ωC
qα =
2
2
( (
)(
u i − u i i i − i2
β
q =
(6.3.e)
=
2
2 ωC
In average terms:
r' = P = 0
(6.3.f)
since the capacitor does not absorb active power.
Moreover:
WL = w L =
WLβ =
∞
= ω L ∑ k 2 I 2k ≥ Q α = ω L ∑ I 2k
6.3 Capacitor
2
Qβ = 2 ω WLβ
k =1
WLβ = WL .
U2
(6.2.h)
ωL ωL
where I and U are the rms values of inductor current and
voltage, respectively. Observe that, let
1
(6.2.i)
w L = L i2
2
be the instantaneous energy stored in the inductor, its
average value is given by:
u
k =1
and
r" = r ' = 0
2.
Moreover
q = q = q' = q" = q = Q = Q = Q' = Q" = Q = ω L I
The bipole equations are:
)
(
u
di
(6.2.a)
u = L = ωL i ⇔ i =
ωL
dt
where L is the inductance of the bipole, which is constant
under the assumption of linearity. The power terms are:
)(
) )
ui − u i uu − uu
r' =
=
=0
(6.2.b)
2
2ωL
()
ui − u i
r" =
≠0
(6.2.c)
2
)
)(
)
i2 − i i
ui − u i
(6.2.d)
= ωL
qα =
2
2
( (
)(
u i − u i u2 − u u
β
q =
(6.2.e)
=
2
2 ωL
In average terms:
r" = P = 0
(6.2.f)
since the inductor does not absorb active power.
Moreover:
q β = Qβ =
∑ U 2k
ωL
sinusoidal
Under
6.2 Inductor
= ω L I2
∞
∑ U 2k with: U k = k ω L I k .
K
α
2
U2 =
Thus:
Under sinusoidal operation, owing to (2.3.b), r = r ' = r" = P at each
instant, P being active power absorbed by the resistor. Moreover
q’=q”=0 at each instant.
q α = Qα = ω L i
and:
I 2 = ∑ I 2k
2
(6.2.m)
Qβ = −2 ω WCβ
(6.3.l)
where:
Qβ ≥ Qα
(6.2.n)
the equality being valid only under sinusoidal operation.
WCβ =
In fact, let Uk and Ik be rms value of the k-th harmonic component of
voltage and current, respectively, we have:
6
I2
2ω2C
=
C I 


2  ω C 
2
(6.3.m)
Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions
Milano, October 13-15, 2003
following decomposition for the current (or,
symmetrically, for the voltage):
i = ia + i n = i a + i r + i v = ia + iq + is + i v
(7.1)
Under
sinusoidal
operation
we
have
and
r" = r ' = 0
q α = qβ = q' = q" = q = Q α = Qβ = Q' = Q" = Q = −ω C U 2 . Moreover
WCβ = WC .
where the terms are named as follows:
ia
active current
in
non-active current
ir
reactive current
iv
void current
iq
main reactive current
is
secondary reactive current
In (7.1) all current terms are orthogonal, i.e.:
ia ⊥ i n ia ⊥ i r ia ⊥ i v ia ⊥ iq ia ⊥ is
Similarly to the case of inductive bipoles, it can be
demonstrated that:
Qβ ≤ Qα
(6.3.n)
both terms being negative. In a more general form, which
is valid for both inductive and capacitive bipoles, we can
write:
Qβ ≥ Qα
(6.4)
the equal sign being valid only for sinusoidal operation.
ir ⊥ i v
L
N
P = ∑ u l o i l = ∑ PR n = PR tot
l =1
n =1
L
M
K
l =1
m =1
k =1
K
 M

= 2 ω ∑ WL m − ∑ WC k  = 2 ω WL tot − WC tot
k =1
 m =1

L
M
K
(
Qβ = ∑ u l o il = ∑ QβL + ∑ QβC =
(
m
m =1

= 2 ω ∑ WLβ − ∑ WCβ
k
 m =1 m k =1
K
(7.2)
k =1
k
(

 = WLβ − WCβ
tot
tot

where upper-case symbols refer to rms value of variables.
Let’s now derive the various current terms.
Active current ia is defined as the projection of current i
on voltage u:
iou
P
ia = 2 u = 2 u = G a u
(7.4)
U
u
and represents the minimum current (i.e. the current with
minimum norm) capable to convey the average power P
absorbed by the port. This current coincides with the
Fryze current and Ga is the equivalent (or Fryze)
conductance seen from the port terminals. Note that:
ia o u = i o u = P
(7.4.a)
Moreover:
P
(7.4.b)
Ia = ia = Ga U =
U
(6.4.a)
)
∑ u l o i l = ∑ Q αL m + ∑ Q Cα k =
M
is ⊥ i v
I 2 = Ia2 + I 2n = Ia2 + I 2r + I 2v = Ia2 + I q2 + Is2 + I 2v (7.3)
Given a linear passive network including L branches,
with N resistors, M inductors and K capacitors, we can
compute the total power absorption by simply adding the
power absorbed by each bipole of the network. Thus:
l =1
iq ⊥ i v
so that:
6.4 Network absorption
Qα =
i q ⊥ is
)
)
(6.4.b)
(6.4.c)
In fact, only resistors absorb active power, while reactive
power terms are absorbed by reactive elements.
It’s very important to note that total average integral
reactive power Q α is proportional to the difference
between total average inductive energy and total
average capacitive energy in the network. This relates
directly the absorption of integral reactive power to the
average energy stored in the network
Eqs. (6.4.b) and (6.4.c) are relevant for the analysis of
reactive compensation by means of controlled reactors or
capacitors, since they define the amount of inductive or
capacitive energy which must be stored in the
compensating equipment in order to compensate for the
reactive power absorbed by the loads.
It is also noticeable that terms WL and WC are
tot
2
P
ia o i = G a u o i = G a P =   = i a
U
and, recalling (2.6.b):
)
)
ia o u = G a u o u = 0
(
(
ia o u = G a u o u = 0
The power terms associated to ia are:
Pa = P
Q aα = 0
2
= Ia2 (7.4.c)
(7.4.d)
(7.4.e)
Qβa = 0
(7.4.f)
Non-active current in is defined as the current portion
exceeding ia:
i n = i − ia
(7.5)
and is orthogonal to ia . In fact, recalling (7.4.c):
tot
simply computed by adding the amounts of average
energy stored in each inductor or capacitor of the
network, respectively.
i n o i a = (i − i a ) o ia = i o i a − i a
Moreover:
2
2
=0
(7.5.a)
i n o i = i n o (ia + i n ) = i n
(7.5.b)
The power terms associated to in are:
Pn = P − Pa = 0
Q αn = Q α − Q aα = Q α
(7.5.c)
Qβn = Qβ − Qβa = Qβ
which show that there is no active power related to
current in , which justifies its name.
7. Current decomposition in singlephase networks
In this section we discuss an orthogonal decomposition of
the current into active, reactive and void terms. In
particular, given voltage u and current i measured at any
single-phase port of generic network Π , we derive the
7
Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions
Milano, October 13-15, 2003
while imposing unity norm gives the value of ∆ :
) (
)
(
) U2U2 − U4
2
ν = ν o ν = ∆2 U 2 + δ 2 U 2 − 2 δ U 2 = ∆2 U 2
=1
U4
Reactive current ir can generally be expressed as a linear
)
(
combination of voltages u and u :
)) ((
ir = B u + B u
(7.6)
)
(
where B and B are the equivalent inductive and
capacitive susceptances seen from the port. According to
such definition, reactive current ir does not involve active
power absorption, while it fully accounts for integral and
derivative reactive power absorption. This requires that:
)
)
u o i r = u o i = Qα
(7.6.a)
(
(
u o i r = u o i = −Qβ
)
Substitution of (7.6) in (7.6.a) gives coefficients B and
(
B:
(
) Q α U 2 − Qβ U 2
B= )2(2
U U − U4
(7.6.b)
)
( Q α U 2 − Qβ U 2
B= )2(2
U U − U4
The power terms associated to ir are, by definition:
(7.6.c)
Pr = 0
Q αr = Q α
Qβr = Qβ
(
U2
⇒ ∆= ) ) (
2 2
U U U − U4
(7.8.b)
Assume now current is as the projection of current i on
ν:
) ) ( (
i s = (i o ν ) ν = I s ν = Bs u + Bs u
(7.9)
)
(
where Bs and Bs are the equivalent secondary
susceptances.
Developing the internal product in (7.9) gives the rms
value of is:
(
)
u o iq = Q α
(7.7.a)
)
Qα
Bq = ) 2
U
(7.7.b)
reactive power Qsα associated to is is zero. From (7.8) and
(7.9) we easily find the derivative reactive power term
Qβs associated to is. In conclusion, the power terms
associated to is are:
Ps = 0
Qqα + Qsα = Qα
which confirms that current ir accounts for full reactive
power absorption.
Void current iv is defined as the residual term:
i v = i − ia − i r
(7.11)
Note that:
i v o u = (i − i a − i r ) o u = i o u − ia o u = 0 (7.11.a)
)
)
)
)
i v o u = (i − i a − i r ) o u = i o u o −i r o u = 0 (7.11.b)
(
(
(
(
i v o u = (i − i a − i r ) o u = i o u − i r o u = 0 (7.11.c)
Thus iv is orthogonal to ia, ir, iq, is. This also implies that:
(7.7.c)
i v o i = i v o (i a + i r + i v ) = i v
Q αv = Q α − Q aα − Q αr = 0
Qβv
(7.11.d)
β
− Qβa
(7.11.e)
− Qβr
=Q
=0
This justifies the name of such current, since no power
terms are associated to it.
Under sinusoidal operation, i.e. with sinusoidal voltages
and currents, current iv would be zero, similar to current
is. However, while current is vanishes for sinusoidal
voltages, this is not generally true for current iv, which
depends on current distortion too.
Finally, observe that:
(7.8)
ν =1
∆ and δ being suitable coefficients. Recalling (2.7), the orthogonality
condition gives the value of δ :
)
)
U2
)
( )
(7.8.a)
∆ (u + δ u ) o u = ∆ U 2 − δ U 2 = 0 ⇒ δ = 2
U
(
2
The associated power terms are:
Pv = P − Pa − Pr = 0
(7.7.d)
Qα ( ) U 2
(
Qβq = − u o i q = − ) 2 u o u = ) 2 Q α
U
U
In order to derive secondary reactive current is, let’s first
)
(
define variable ν as a linear combination of u and u
)
which has unity norm and is orthogonal to u .
)
(
ν = ∆ (u + δ u )
)
νou = 0
(7.10)
Qβq + Qβs = Qβ
which gives:
Thus:
Q sα = 0
(7.9.b)
U2
(
( )
(
Qβs = − u o i s = − ∆ I s u o ( u + δ u ) = Qβ − ) 2 Q α
U
Note that:
Thus:
Qα
Iq = iq = )
U
The power terms associated to current iq are:
Pq = 0
Q qα = Q α
)
Is = ∆ Qα − δ Qβ
(7.9.a)
)
By definition, is is orthogonal to iq and u , thus integral
Let’s now split current ir into two orthogonal terms iq
(main reactive current) and is (secondary reactive
current), where current iq is defined as the minimum
current accounting for reactive power Qα, which relates
to the energy stored in the network. Current iq can
therefore be determined by projecting current i on voltage
)
u:
)
iou ) ) )
(7.7)
i q = ) 2 u = Bq u
u
)
where Bq is the equivalent main susceptance, and
imposing:
)
)
i r o i = i r o (ia + i r + i v ) = i r
8
2
(7.12.a)
Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions
Milano, October 13-15, 2003
i q o i = i q o ( i a + i q + is + i v ) = i q
is o i = i s o ( i a + i q + i s + i v ) = i s
2
2
and Qβ. Moreover, since Qα and Qβ are conservative
quantities, the compensating elements can be connected
at every port (or multiple ports) of the network. This
makes possible the application of distributed reactive
compensation techniques.
Distortion power term Dv is not directly depending on Qα
and Qβ, and generally cannot be compensated by reactive
compensation means. Its compensation may therefore
require use of active power filters.
(7.12.b)
(7.12.c)
A similar treatment can symmetrically be developed for
voltage decomposition into active, reactive and void
terms.
8. Apparent power decomposition in
single-phase networks
The apparent power decomposition follows naturally
from the above current decomposition. In fact, let:
A == u i = U I
(8.1)
9. Extension to poly-phase networks
The extension of the above theory to poly-phase systems
is straightforward in basic terms, however it opens
several interesting questions regarding, for example,
analysis of unbalance terms, symmetrical components,
distributed compensation, etc.. These aspects will be
covered in a later paper. He we will simply observe that
all definitions given above are easily transposed in the
poly-phase domain. In fact, given vectors u and i of the
voltages and currents measured at any N-phase port of
network Π , we define the instantaneous complex power
as:
β*
α*
u& α ⋅ &i
u& β ⋅ &i
s&'+ js&"
+
=
s& = u × i =
(9.1)
2
2
2
where:
)
(
 u1 + ju1 
 u1 − ju1 
β
 u& 1α  


u&

2 
2 
 
 1 
β
&u α =  ...  = 
&
... ) 
u =  ...  = 
... ( 
 u& α   u N + ju N 
 u& β   u N − ju N 
 N 
 N 
2 
2 


(9.2)
α
&
and similarly for binomial complex current terms i and
be the apparent power, we can split the current into its
orthogonal components, obtaining:
A 2 = U 2 I 2 = U 2 I a2 + U 2 I 2q + U 2 Is2 + U 2 I 2v
(8.2)
Note that:
U 2 I 2a = P 2
(8.3)
and:
( )2 =
U2
U 2 I q2 = ) 2 Q α
U
(8.4)
)2
2
2
U −U
= Qα +
Q α = Q 2q + D 2q
)2
U
where we have named Qq the main reactive power, which
coincides with integral reactive power Qα, and Dq the
distortion power term associated to Iq . This term can be
)
defined since U 2 is always greater than U 2 , and
vanishes in absence of voltage distortion. Similarly we
define:
( )
2
( )
Ds2 = U 2 Is2
(8.5)
which is another distortion power term which vanishes in
absence of voltage distortion, and:
&iβ . Developing the expressions, we find:
(
) (
)
u•i − u• i
u•i+ u• i
α &β*
s&' = u& • i =
+j
=
(9.3.a)
2
2
= ℜ(s&' ) + j ℑ(s&' ) = r '+ j q'
) (
( )
u•i − u• i
u• i + u•i
β &α *
s&" = u& • i =
−j
=
(9.3.b)
2
2
= ℜ(s&" ) + j ℑ(s&" ) = r"+ j q"
In average terms we have:
(
) (
)
uoi − uo i
uoi + uo i
S& ' = s& ' = r '+ j q ' =
+j
=
2
2
(9.4.a)
(
)
uoi + uo i
= uoi + j
= P + jQ
2
) (
( )
&S" = s&" = r"+ j q" = u o i − u o i − j u o i + u o i =
2
2
(9.4.b)
(
)
uoi+ uo i
= uoi + j
= P + jQ
2
and:
S& '+S& "
= P + jQ
(9.4.c)
S& = u ⊗ i =
2
where:
D2v = U 2 I 2v
(8.6)
which is the last distortion power term, generally
depending on both voltage and current distortion.
We can finally write:
A 2 = P 2 + Q2q + D2
(8.7)
where P is the active power, representing the average
power absorbed through the port, Qq is the main reactive
power, related to the average energy stored in the
network (inductive energy terms being considered with
positive sign, capacitive terms with negative sign), and D
is the distortion power, which vanishes under sinusoidal
conditions.
In (8.7) we have set:
D2 = Dq2 + Ds2 + D 2v
(8.8)
the first two terms being dependent on voltage distortion,
the last on both voltage and current distortion.
It is worthwhile to note that power terms Qq , Dq and Ds
can be compensated by means of usual reactive
compensation means (i.e., controlled reactors and
capacitors). In fact, these power terms depend on Qα and
Qβ and would decrease if suitable reactive elements are
connected to the network to reduce total absorption of Qα
9
Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions
Milano, October 13-15, 2003
P = uoi
is active power. Moreover:
(
)
uoi + uo i
Q=
2
is reactive power, and:
Q=
specific power and energy terms in conjunction with
supply and load distortion coefficients. As a result, the
apparent power is decomposed into active, reactive and
distortion terms, each one having a precise physical
meaning. The case of poly-phase systems has also been
considered as a natural extension of the single-phase
approach.
Further developments will relate to the analysis of supply
and load unbalance in poly-phase systems, application of
symmetrical components under non-sinusoidal conditions
and distributed reactive and harmonic compensation, both
using controllable reactors and capacitors and active
power filters.
(9.5)
(9.6.a)
Q α + Qβ
2
(9.6.b)
where:
)
) u) o i − u o i
)
Qα = u o i = − u o i =
2
is integral reactive power, and:
( (
(
uo i −uoi
(
β
Q = u o i = −u o i =
2
is derivative reactive power.
We define the current decomposition as follows:
iou
P
ia = 2 u = 2 u
u
u
is active current vector;
)
i o u ) Qα )
iq = ) 2 u = ) 2 u
u
u
is main reactive current vector;
is = (i o ν ) ν
is secondary reactive current vector, where:
)2
u
1
)
(
ν = ∆ (u + δ u )
δ= 2
∆= )
)
u
u
u
(9.6.c)
(9.6.d)
BIBLIOGRAPHY
[1] Shepherd, W., Zakikhani, P.: Suggested definition of reactive
power for nonsinusoidal systems. “Proc. Inst. Elec. Eng.“, vol. 119,
pp.1361-1362, Sept. 1972.
[2] Kuster , N.L, Moore, W.J.M.: On the Definition of Reactive Power
under Non-Sinusoidal Condition. “IEEE Trans. on Power Apparatus
and Systems”, PAS-99 (1980), pp.1845-1854.
[3] Page, C.H.: Reactive Power in Non-Sinusoidal Situations. “IEEE
Trans. of Instrumentation and Measuremen”t, Vol IM-29, No.4, Dec
1990, pp 420-423.
[4] Czarnecki, L.S.: Power factor improvement of three-phase
unbalanced loads with nonsinusoidal supply voltages, “European
Trans. on Electrical Power Engineering (ETEP)”, Vol. 3, No. 1,
Jan./Feb. 1993, pp. 67-72.
[5] Czarnecki, L.S.: Minimization of Reactive Power under
Nonsinusoidal Conditions. “IEEE Trans. on Instrumentation and
Measurements”, vol.IM-36 (1987), no.1, pp. 18-22.
[6] Willems, J.L.: Power factor correction for distorted bus voltages.
“Electr. Mach. a. Power Syst“, 13 (1987), pp.207-218.
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(9.7.a)
(9.7.b)
(9.7.c)
2
2
u
(2
u − u
4
(9.7.d)
Finally:
i r = iq + is
(9.7.e)
is reactive current vector, and:
i v = i − ia − i r
(9.7.f)
is void current vector.
All current terms are orthogonal, thus:
i
2
= ia
2
+ ir
2
+ iv
2
= ia
2
+ iq
2
+ is
2
+ iv
2
(9.7)
The active power decomposition assumes the same form
given in (8.7) and (8.8).
10. Conclusions
A general approach to current, voltage and power terms
definition under non-sinusoidal conditions has been
presented, which is entirely developed in the time domain
and is valid for every network. The approach is valid for
periodic operation, and makes reference to physical
quantities which are easily expressed in terms of
voltages, currents and their integrals and derivatives.
Instantaneous and average power terms have been
identified, which are conservative and can be applied to
analyze the impact of each load on reactive and distortion
power absorption. A reactive power definition has also
been introduced, which is related to the energy stored in
the network.
An orthogonal decomposition of currents (and voltages)
has then been described, where each component relates to
10