A Cyclodissipativity Condition for Power Factor Improvementunder Nonsinusoidal Source with Signiﬁcant Impedance
Dunstano del PuertoFlores, Romeo Ortega, and Jacquelien M. A. Scherpen
Abstract
—The main contribution of this paper is an extension of a recent result that reformulates and solves the powerfactor compensation for nonlinear loads under nonsinusoidalregime in terms of cyclodissipativity. In the aforementionedresult the generator was assumed to be ideal, that is, withnegligible impedance. In this work, we formulate the powerfactor compensation problem in a way that explicitly accountsfor the effects of a signiﬁcant source impedance.
I. I
NTRODUCTION
Recently, in [1] it has been established that the classicalproblem in electrical engineering of optimizing energy transfer from an alternating current (ac) source to a load withnonsinusoidal (but periodic) source voltage is equivalentto imposing the property of cyclodissipativity to the sourceterminals. Since this framework is based on the cyclodissipativity property, see [2], the improvement of power factor(PF) is done independent of the reactive power deﬁnition,which is a matter of discussions in the power community,see [3].Using this framework the classical capacitor or inductorcompensators were interpreted in terms of energy equalization, see [1] for more details. We have presented anextension of the results in [4] where we considered arbitrarylossless linear timeinvariant (LTI) ﬁlters, and proved that forgeneral lossless LTI ﬁlters the PF is reduced if and only if acertain equalization condition between the weighted powersof inductors and capacitors of the load is ensured. Althoughthe aforementioned results were obtained by consideringnonlinear loads, the generator was assumed to be ideal,that is, with negligible impedance. However, in practice thesource impedance is often signiﬁcant.We consider the energy transfer from a known voltagesource
v
s
to a given ﬁxed load
i
ℓ
, see Figure 1. Thestandard approach to improve the PF is to place a losslesscompensator,
Y
c
, between the source and the load. ThePF compensation conﬁguration considered in the paper isdepicted in Figure 2.In this work, our task is to formulate the power factor compensation problem in a way that explicitly accounts for theeffects of a nonnegligible source impedance,
Z
s
, on the loadvoltage and current. We prove that cyclodissipativity providesa rigorous mathematical framework useful to analyze and
J.M.A. Scherpen and D. del PuertoFlores are with the Deparmentof Discrete Technology and Production Automation, RijksuniversiteitGroningen, Nijenborgh 4, 9747 AG, Groningen, The Netherlands. (email:
{
j.m.a.scherpen,d.del.puerto.flores
}
@rug.nl
)R. Ortega is with the Laboratoire des Signaux et Syst`emes,Plateau de Moulon, 91192, GifsurYvette, cedex, France. (email:
ortega@lss.supelec.fr
)
design power factor compensators for general nonlinear loadsoperating in nonsinusoidal regimes with signiﬁcant sourceimpedance.First, we brieﬂy review our result from [4]. Then, inSection IV we give a cyclodissipativity characterization of the lossless LTI PF Compensation problem with sourceimpedance. Next we provide a geometric characterization inSection V and we end with an example in Section VI.II. P
OWER
F
ACTOR
I
MPROVEMENT
A. Framework
i
s
Z
s
v
us
i
ℓ
v
s
Fig. 1. Power delivery system with signiﬁcant source impedance.
i
s
i
c
Z
s
Y
c
v
cs
i
ℓ
v
s
Fig. 2. Parallel load compensation in a power delivery system withsigniﬁcant source impedance.
We consider the energy transfer from an
n
phase acgenerator to a load where the source is not assumed to beideal, but has a signiﬁcant impedance, see Figure 1. Allsignals are assumed to be periodic and have ﬁnite power,that is, they belong to the space
L
n
2
=
x
: [0
,T
)
→
R
n
:
x
2
:= 1
T
T
0

x
(
τ
)

2
dτ <
∞
where
·
is the rms value and
·
is the Euclidean norm.We also deﬁne the inner product in
L
n
2
as
x,y
:= 1
T
T
0
x
⊤
(
t
)
y
(
t
)
dt.
49th IEEE Conference on Decision and ControlDecember 1517, 2010Hilton Atlanta Hotel, Atlanta, GA, USA
9781424477463/10/$26.00 ©2010 IEEE3978
The process of power factor correction is an attempt toreduce the apparent voltamperes of a load to the value of theaverage power consumed. The universally accepted deﬁnitionof PF is given as [5]:
Deﬁnition 1 (Power factor):
Consider the power deliverysystems of the Figure 1. The PF of an AC electric powersystem is deﬁned by
PF
:=
P S ,
(1)where
P
:=
v
us
,i
s
, S
:=
v
us
i
s
(2)are the active (real) power,
1
and the apparent power, respectively.From (2), it follows that
P
≤
S
. Hence
PF
∈
[
−
1
,
1]
is adimensionless measure of the energytransmission efﬁciency.Cauchy–Schwartz also tells us that a necessary and sufﬁcientcondition for the apparent power to equal the active poweris that
v
s
and
i
s
are collinear. If this is not the case,
P < S
and compensation schemes are introduced to maximize thePF.The condition for unity power factor is that the inputcurrent to a systems is proportional at all times to theinstantaneous supplied voltage.
B. The Power Factor compensation problem
The PF compensation conﬁguration considered in thepaper is depicted in Figure 2, where
Y
c
:
L
n
2
→ L
n
2
is theadmittance operator of the compensator. That is,
i
c
=
Y
c
v
cs
where
i
c
∈ L
n
2
, is the compensator current. In the simplestLTI case the operator
Y
c
can be described by its admittancetransfer matrix, which we denote by
ˆ
Y
c
(
s
)
∈
R
n
×
n
(
s
)
. Wemake the following fundamental assumption.Following standard practice, we consider only losslesscompensators, that is,
i
c
,v
cs
= 0
,
∀
v
cs
∈ L
n
2
.
(3)We recall that, if
Y
c
is a passive LTI LCnetwork, losslessnessimplies
Re
{
ˆ
Y
c
(
jω
)
}
= 0
.
(4)for all
ω
∈
R
for which
jω
is not a pole of
ˆ
Y
c
and all alternatezeros and poles are simple and lie on the imaginary axis andwhere
Re
{
ˆ
Y
c
(
jω
)
}
is the real part of the admittance transfer
ˆ
Y
c
(
jω
)
, see [6].
Deﬁnition 2 (Power factor improvement):
Given a
n
phase source voltage
v
s
(
t
)
, a linear timeinvariant (
n
phase)source impedance
Z
s
:
L
n
2
→ L
n
2
, and a ﬁxed load current
i
ℓ
, as in Figure 2, powerfactor improvement is achievedwith the lossless compensator
Y
c
if and only if
PF > PF
u
:=
v
us
,i
ℓ
v
us
i
ℓ
(5)
1
Also called average power [5].
v
s
N
i
s
Zs
v
p
i
p
Fig. 3. Circuit schematic of an
n
phase nonideal generator connected toa
n
port (possible nonlinear and time varying) load
where
PF
u
denotes the uncompensated power factor, that is,the value of
PF
with
Y
c
= 0
and
i
s
=
i
ℓ
, and, by Kirchhoff’sVoltage Law (KVL), the uncompensated voltage is
v
us
=
v
s
−
Z
s
i
ℓ
.
(6)III. PF
COMPENSATION AND CYCLODISSIPATIVITY
:I
DEAL
C
ASE
The framework to be discussed carries abstract powerconnotations (as does the term itself). This is derived fromthe interpretation of the supply rate as an input power. Toformulate our results we need the following.
Deﬁnition 3 (Cyclodissipative system, [2]):
Given a mapping
w
:
L
n
2
×L
n
2
→
R
. The
n
port system of Figure 3 iscyclodissipative with respect to the supply rate
w
(
v
p
,i
p
)
if and only if
T
0
w
(
v
p
(
t
)
,i
p
(
t
))
dt >
0
.
(7)for all
(
v
p
,i
p
)
∈ L
n
2
×L
n
2
.
A. Cyclodissipative Framework for Power Factor Compensation
To place or results in context, and make the paper self–contained, we recall the following results from [1]. Assume
v
s
is ideal, i.e.,
Z
s
= 0
.
v
s
i
s
i
c
i
l
Y
c
Y
l
Fig. 4. Parallel load compensation in a power delivery system withnegligible source impedance.
Proposition 4 ([1]):
Consider the system of Figure 4 andﬁxed
Y
ℓ
. The compensator
Y
c
improves the PF if and onlyif the system is cyclodissipative with respect to the supplyrate
w
(
v
s
,i
s
) := (
Y
ℓ
v
s
+
i
s
)
⊤
(
Y
ℓ
v
s
−
i
s
)
.
(8)The next result follows from Proposition 4 and it characterizes the set of all compensators
Y
c
that improve the powerfactor for a given
Y
ℓ
.
3979
Corollary 5 ([1]):
Consider the system of Figure 4 Then
Y
c
improves the PF for a given
Y
ℓ
if and only if
Y
c
satisﬁes
2
Y
ℓ
v
s
,Y
c
v
s
+
Y
c
v
s
2
<
0
,
∀
v
s
∈ L
n
2
.
(9)Dually, given
Y
c
, the PF is improved for all
Y
ℓ
that satisfy(9).
B. Compensation with Lossless LTI Compensator equatesWeighted Power Equalization.
In this section we review the results presented in [4], whichextended the results in [1]. Similarly to [1], the class of RLC circuits that we consider as load models consists of possibly nonlinear lumped dynamic elements (
n
L
inductors,
n
C
capacitors) and nonlinear static elements (
n
R
resistors).Capacitors and inductors are deﬁned by the physical lawsand constitutive relations [7]:
i
C
= ˙
q
C
, v
C
=
∇
H
C
(
q
C
)
,
(10)
v
L
= ˙
φ
L
, i
L
=
∇
H
L
(
φ
L
)
,
(11)respectively, where
i
C
,
v
C
,
q
C
∈
R
n
C
are the capacitorscurrents, voltages and charges, and
i
L
,
v
L
,
φ
L
∈
R
n
L
are theinductors currents, voltages and ﬂux–linkages,
H
L
:
R
n
L
→
R
is the magnetic energy stored in the inductors,
H
C
:
R
n
C
→
R
is the electric energy stored in the capacitors,and
∇
is the gradient operator. We assume that the energyfunctions are twice differentiable. For linear capacitors andinductors
H
C
(
q
C
) = 12
q
⊤
C
C
−
1
q
C
, H
L
(
φ
L
) = 12
φ
⊤
L
L
−
1
φ
L
,
respectively, with
L
∈
R
n
L
×
n
L
,
C
∈
R
n
C
×
n
C
. To avoidcluttering the notation we assume
L,C
are diagonal matrices.Finally, we distinguish between two sets of nonlinear staticresistors:
n
R
i
current–controlled resistors and
n
R
v
voltage–controlled resistors, for which the characteristic are given by
v
R
i
= ˆ
v
R
i
(
i
R
i
)
,
and
i
R
v
=ˆ
i
R
v
(
v
R
v
)
,
respectively, where
i
R
i
,
v
R
i
∈
R
n
Ri
are the currents, voltages of the currentcontrolled resistors, and
i
R
v
,
v
R
v
∈
R
n
Rv
are the currents, voltages of the voltagecontrolled resistors,with
n
R
=
n
R
i
+
n
R
v
.Recalling the deﬁnition of real power (2) we introduce thefollowing.
Deﬁnition 6 (Weighted (real) power):
Given a compensator admittance
Y
c
the weighted (real) power of a single–phase circuit with port variables
(
v,i
)
∈ L
2
×L
2
is givenby
P
w
:=
Y
c
v,i
.
(12)For instance, if
Y
c
is LTI
P
w
=
∞
k
=
−∞
ˆ
Y
c
[
k
]ˆ
V
[
k
]ˆ
I
∗
[
k
]
(13)where
ˆ
V
[
k
]
,
ˆ
I
[
k
]
are the
k
th spectral lines of
v
and
i
,respectively, and
ˆ
Y
c
[
k
] := ˆ
Y
c
(
kω
0
)
, with
ω
0
:=
2
πT
. Thatis,
P
w
is the sum of the power components of the circuitmodulated by the frequency response of
Y
c
—hence the useof the “weighted” qualiﬁer.
2
The aforementioned deﬁnitionmotivates the next result.
Proposition 7:
Consider the system of Figure 2 with
n
=1
,
3
a full nonlinear RLC load and a ﬁxed LTI losslesscompensator
Y
c
with admittance transfer function
ˆ
Y
c
(
jω
)
which has a zero at the srcin.i) PF is improved if and only if
12
V
ws
+
n
L
q
=1
P
wL
q
+
n
C
q
=1
P
wC
q
<
0
(14)where
V
ws
is the rms value of the ﬁltered voltagesource, that is,
V
ws
:=

Y
c
v
s

2
=
∞
k
=1

ˆ
Y
c
(
k
)ˆ
V
s
(
k
)

2
and
P
wC
q
:=
∞
k
=
−∞
ˆ
Y
c
[
k
]ˆ
V
C
q
[
k
]ˆ
I
∗
C
q
[
k
]
P
wL
q
:=
∞
k
=
−∞
ˆ
Y
c
[
k
]ˆ
V
L
q
[
k
]ˆ
I
∗
L
q
[
k
]
,
are the weighted powers of the
q
–th inductor andcapacitor, respectively.ii) Condition (14) may be equivalently expressed as
(
1
p
Y
c
)
v
L
,
∇
2
H
L
v
L
−
i
C
,
(
1
p
Y
c
)
∇
2
H
C
i
C
>
12
V
ws
(15)where
p
:=
ddt
.iii) If the inductors and capacitors are linear their weightedpowers become
P
wC
q
:= 2
ω
0
∞
k
=1
k
I
m
{
ˆ
Y
c
[
k
]
}
n
C
q
=1
C
q

ˆ
V
C
q
[
k
]

2
P
wL
q
:=
−
2
ω
0
∞
k
=1
k
I
m
{
ˆ
Y
c
[
k
]
}
n
L
q
=1
L
q

ˆ
I
L
q
[
k
]

2
.
iv) Furthermore, the results iiii can be extended for ageneral LTI lossless compensator, if the resistors of the load are linear timeinvariants.Condition 14 indicates that the power factor improvementif and only if a certain equalization condition between theweighted powers of compensator and load is ensured. Wenow continue with extending these result to the nonidealsource case.IV. A C
YCLODISSIPATIVITY
C
ONDITION FOR
P
OWER
F
ACTOR
I
MPROVEMENT
: N
ON

IDEAL
C
ASE
Although the problem at hand is posed as a problem innetworks, it can be equally well interpreted as a feedback problem; the circuit of Fig. 2 is represented by the systemof Fig. 5, which consists of two systems in a feedback loop.
2
Since the spectral lines of real signals satisfy
ˆ
F
[
−
k
] = ˆ
F
∗
[
k
]
, theweighted power is a real number.
3
This condition is imposed, without loss of generality, to simplify thepresentation of the result.
3980
Speciﬁcally, the inputs are
v
s
and
i
ℓ
and the outputs are
i
s
and
v
cs
, and products are related with the instantaneousdelivered power by the source
i
⊤
s
v
s
and the instantaneousinput power into the load,
i
⊤
ℓ
v
cs
.
Z
s
Y
c
v
s c
v
s
i
c
i
s
i
l
v
zs
Fig. 5. Feedback conﬁguration
Proposition 8:
Consider the system of Figure 2. Givena
n
phase source voltage
v
s
(
t
)
, a linear timeinvariant (
n
phase) source impedance
Z
s
and a ﬁxed load current
i
ℓ
. Thecompensator
Y
c
improves the PF if and only if the system hasﬁnite gain and is cyclodissipative with respect to the supplyrate
w
(
i
s
,i
ℓ
) :=
δ
2
i
⊤
ℓ
i
ℓ
−
i
⊤
s
i
s
.
(16)for all
(
i
s
,i
ℓ
)
∈ L
n
2
×L
n
2
, where
δ
is the upper gain bound
4
and is given by
δ
=
v
cs
,i
ℓ
v
cs
(
I
+
Z
s
Y
c
)
v
cs
(
I
+
Z
s
Y
c
)
v
cs
,i
ℓ
,
(17)with
1
< δ <
∞
.
Proof:
From Kirchhoff’s Current Law (KCL), we have
i
s
=
i
c
+
i
ℓ
, and using KVL,
v
cs
=
v
s
−
Z
s
(
i
c
+
i
ℓ
)
,
(18)Substituting (19) into (18), we obtain
v
us
=
v
cs
+
Z
s
i
c
(19)From the deﬁnition of power factor and the lossless conditionof the compensator, we have
PF
:=
v
cs
,i
ℓ
v
cs
i
s
(20)And, we deﬁne
˜
α
:=
v
cs
,i
ℓ
v
cs
.
(21)
PF
u
is given by
PF
u
:=
v
us
,i
ℓ
v
us
i
ℓ
and by using (19) and
i
c
=
Y
c
v
cs
,
v
us
=
v
cs
+
Z
s
Y
c
v
cs
= (
I
+
Z
s
Y
c
)
v
cs
,
then,
PF
u
=
(
I
+
Z
s
Y
c
)
v
cs
,i
ℓ
(
I
+
Z
s
Y
c
)
v
cs
i
ℓ
,
(22)
4
See Table 1 of [8], Deﬁnition 2.1 of [9], and Deﬁnition 2 of [10].
and we deﬁne:
α
:=
(
I
+
Z
s
Y
c
)
v
cs
,i
ℓ
(
I
+
Z
s
Y
c
)
v
cs
.
(23)From Deﬁnition (5), we conclude that
PF > PF
u
if andonly if
v
cs
,i
ℓ
v
cs
i
s
>
(
I
+
Z
s
Y
c
)
v
cs
,i
ℓ
(
I
+
Z
s
Y
c
)
v
cs
i
ℓ
,
or, by (21) and (23), the inequality becomes
i
s
<
˜
αα
i
ℓ
,
(24)with
δ
:=
˜
αα
. If
Y
c
= 0
, i.e., the uncompensated case, from(17) we have that
δ >
1
and because the fact that
δ
dependsonly on bounded signals,
i
ℓ
and
v
cs
, and the operators
Z
s
and
Y
c
are LTI, we can conclude that
δ <
∞
.
Remark 9:
The results of Proposition 5 in [1] are aparticular case of Proposition 8 assuming an ideal source,i.e., the source impedance
Z
s
= 0
and, from (17), we have
δ
= 1
.The next corollary of this result is the characterization of allcompensators that improve the power factor.
Corollary 10:
Consider the system of Figure 2. Then
Y
c
improves the PF for a given
i
ℓ
if and only if
Y
c
satisﬁes
i
c
2
+ 2
i
c
,i
ℓ
<
˜
αα
2
−
1
i
ℓ
2
,
∀
v
s
,i
ℓ
∈ L
n
2
.
(25)
Proof:
Using the fact
i
s
=
i
c
+
i
ℓ
, and the square of (24),
i
s
2
<
˜
αα
2
i
ℓ
2
i
c
2
+ 2
i
c
,i
ℓ
+
i
ℓ
2
<
˜
αα
2
i
ℓ
2
i
c
2
+ 2
i
c
,i
ℓ
<
˜
αα
2
−
1
i
ℓ
2
.
and, from
i
c
=
Y
c
v
cs
, then we have
Y
c
v
cs
2
+ 2
Y
c
v
cs
,i
ℓ
<
˜
αα
2
−
1
i
ℓ
2
(26)
Remark 11:
From the feedback conﬁguration under consideration, see Fig. 5. The interconnected system is cyclodissipative with respect to (16) if the compensator
Y
c
iscyclodissipative with respect to the supply rate function
w
c
(
i
ℓ
,i
c
) :=
−
i
⊤
c
i
c
−
2
i
⊤
c
i
ℓ
−
(1
−
δ
)(1 +
δ
)
i
⊤
ℓ
i
ℓ
.
where the input is
i
ℓ
and
i
c
is the output.V. G
EOMETRICAL INTERPRETATION
Referring to Fig. 6 we have a geometric interpretation of power factor compensation. Fig. 6 depicts the vector
v
cs
,
v
us
,
i
s
,
i
c
and
i
ℓ
. The angles
β
and
β
u
are deﬁned as
β
:= cos
−
1
PF, β
u
:= cos
−
1
PF
u
,
3981
or,
β
=
∠
(
v
cs
,i
s
)
and
β
u
=
∠
(
v
us
,i
ℓ
)
. Then, it is clear fromFig. 6 that
PF > PF
u
if only if
β < β
u
.From (17), by assuming that
v
cs
,i
ℓ
>
0
and
(
I
+
Z
s
Y
c
)
v
cs
,i
ℓ
>
0
, then we have that
1
< δ <
∞
.
v
s c
v
s u
Z
s
v
s u
Y
C
i
c
i
s
~
u
l
i
Fig. 6. Geometric interpretation of power factor compensation.
Remark 12:
The projection of
i
ℓ
onto
v
cs
is the vectordenoted and deﬁned by
proj(
i
ℓ
,v
cs
) :=
v
cs
,i
ℓ
v
cs
2
v
cs
,
with magnitude
˜
α
:=
v
cs
,i
ℓ
v
cs
.
Remark 13:
Consider the projection of
i
ℓ
onto
v
us
is thevector denoted and deﬁned by
proj(
i
ℓ
,v
us
) :=
v
us
,i
ℓ
v
us
2
v
us
,
with magnitude
α
:=
v
us
,i
ℓ
v
us
,
and with
v
us
= (
I
+
Z
s
Y
c
)
v
cs
, we obtain
α
:=
(
I
+
Z
s
Y
c
)
v
cs
,i
ℓ
(
I
+
Z
s
Y
c
)
v
cs
.
VI. E
XAMPLE
In this section we present an example that illustrate someof the points discussed in the paper.Consider the threephase distribution system of Fig 7.A linear, RL load per phase, which admittance
Y
ℓ
i
with
i
∈ {
1
,
2
,
3
}
, is starconnected with a set of starconnected,compensated capacitors
ˆ
Y
c
i
(
s
) =
C
c
i
s
, with
i
∈ {
1
,
2
,
3
}
,per phase. Assuming balanced, threephase operation, withno d.c. component of current, the problem can be representedby the singlephase equivalent circuit in the Fig. 8, see[11].
v
s
(
t
)
,
v
o
(
t
)
are the supply and load instantaneousvoltages per phase. The RMS voltage
ˆ
V
s
is maintainedconstant at
33
kV. The distribution line has the followingdata: 3phase, 20 miles of 336.4 MCM
5
, 26/7 ACSR
6
with14 ft. conductor spacing,
R
= 0
.
278 Ω
/mile/conductor,
X
L
= 0
.
516 Ω
/mile/conductor. The RL load per phase isassumed to be lumped resistance in series with lumped, pureinductance with 10 MVA, 0.65 PF lagging at
50
Hz.
n r
s1
r
s2
r
s3
L
s1
L
s2
L
s3
Y
l 1
Y
l 2
Y
l 3
n v
s1
v
s2
v
s3
C
1
C
2
C
3
n
Fig. 7. Threephase distribution system in parallel with a linear, balancedRL load.
v
s
r
s
R
l
L
s
L
l
C v
o
i
s
i
c
i
l
Fig. 8. Singlephase equivalent circuit of the threephase system of Figure7.
Now, consider the perphase equivalent circuit, Fig. 8. Theimpedance line is
Z
s
= 20(0
.
278 +
j
0
.
516) Ω
, the linearload is
Y
ℓ
= 1
/Z
ℓ
with
Z
ℓ
= 20(0
.
278 +
j
0
.
516) Ω
, andthe uncompensated total impedance is
Z
uT
=
Z
s
+
Z
ℓ
. Theimpedance of the compensator is
Z
c
(
C
) =
1
jωC
, then thecompensated total impedance
Z
cT
(
C
) =
Z
s
+
Z
c
Z
ℓ
Z
c
+
Z
ℓ
.
Condition (24) helps us to obtain the parameters for a givencompensator
Y
c
, i.e., the capacitance for this example, suchthat the power factor is improved. Where, the bounded gainis
δ
(
C
) =
ℜ{
Z
cℓ
}
ˆ
V
s
√
31
√

Z
cT

2
ℜ{
Z
ℓ
}
ˆ
V
s
√
31
√

Z
uT

2ˆ
V
s
√
3

Z
ℓ

Z
uT

ˆ
V
s
√
3

Z
cℓ

Z
cT

,
(27)the rms value of the input current is
i
s
(
C
)
=ˆ
V
s
√
31

Z
cT

,
5
The equivalent cross sectional area is 336,400 circular mils.
6
An Aluminum Conductor Steel Reinforced (or ACSR) cable with 26aluminum conductors and a core of 7 steel conductors.
3982