THREE-PHASE POSITIVE AND NEGATIVE
SEQUENCES ESTIMATOR TO GENERATE
CURRENT REFERENCE FOR
SELECTIVE ACTIVE FILTERS
F. Ronchi , A. Tilli
Dept. of Electronics, Computer Science and Systems (DEIS)
University of Bologna
Viale Risorgimento n.2, 40136 Bologna, ITALY
fax: +39 051 20 93073
e-mail:
fronchi, atilli @deis.unibo.it
Keywords: active power filters, harmonic distor- ing software parameters. Besides, shunt active filters
have the goal to generate current equal but opposite
tion, positive and negative sequences.
to the harmonic currents in the load waveform. This
leads to a cheaper design of the filter components
Abstract
with respect to series active filters [6].
This article deals with the on-line estimation of
The performance of active power filters (APF) is
three-phase three-wire signal harmonics. It is perbased on the inverter parameters, control algorithm
formed in the d-q synchronous reference frame and
and on the method of obtaining current reference. In
exploits the Luenberger observer concept to isolate conventional load current detection methods [2],[4],
positive and negative sequences of each harmonic.
the generation of the active filter current reference is
This estimator is suitable to be implemented as refbased on the harmonic detection of load currents, userence generator for selective active power filters. ing the well known instantaneous power theory [2],
Simulation and experimental results are presented.
time-domain correlation techniques [7], FFT, etc.
1 Introduction
The objective of this paper is to provide an on-line
method to estimate load current harmonics, separating positive and negative sequences. An approach
based on Luenberger observer is proposed. The estimated harmonics can be used both to simply monitor the load and to generate current references for the
APF controller, to perform selective compensation.
The use of nonlinear devices, e.g. power electronics, generates harmonics, subharmonics, and interharmonics in voltage and current mains spectra. It
is necessary to measure and decrease this harmonic
distortion, thus reducing the power losses and the
risk of equipment damage or malfunctioning.
Moreover, in order to cope with the delay of the
voltage-source
inverter current loop, each estimated
Current harmonics have been traditionally compensated with passive filters, which have several draw- harmonic can be processed inverting the transfer
backs: their operation depends on the network function of the closed-loop control system APF. The
impedance, they have to be tuned on fixed frequen- current control is usually [8] performed in a d-q syncies, etc. Active power filters [1],[5] based on digital chronous reference frame. Hence the estimation and
controllers can be more expensive respect to the pas- isolation of positive and negative sequences of each
sive ones, but they are less network-dependant and harmonic is carried out in this reference frame. This
can be tuned on different frequencies simply chang- paper is organized as follows. In section (2) some
β
concepts about positive, negative sequences and harb
monics description are presented; in section (3) the
estimation-isolation algorithm is reported. The last
two paragraphs show simulation and experimental
results. Conclusions summarize the contents ofPSfrag
the replacements
paper.
a α
c
2 Preliminaries
According to Fortescue’s theorem [3], an unbalanced set of N phasors can be resolved into N systems of phasors called the symmetrical components
Figure 1: Fixed reference frames
of the original phasors. For a three-phase systems
(i.e. N 3), the three sets are:
with kc arbitrarily chosen constant, typically
kc
2 3 2 3 .
p p p
1. Positive sequence. Three phasors Ia Ib Ic , Let consider the positive sequence
equal in magnitude, 120o apart, with the same
iap t
Re Iap e jω t
I p cos ω t
sequence (a-b-c) as the original phasors.
i t Re I e i t Re I e 2. Negative sequence. Three phasors Ian Ibn Icn ,
equal in magnitude, 120o apart, with the opposite sequence (a-c-b) of the original phasors.
p
b
p
c
2π 3
I cos ω t 2π 3 I p cos ω t
p
It can be calculated that
i t 3. Zero sequence.
Three identical phasors
Ia0 Ib0 Ic0 : equal in magnitude, with no relative
phase displacement.
p jω t
b
p jω t
c
p
iα t
p
β
2π
3
kc I p cos ω t
2
3
kc I p sin ω t
2
Defining r e j 3 the operator that rotates a phasor
of 120o , the relationships among the sequence com- that correspond to a phasor rotating with frequency
ω in the complex plain.
ponents for a-b-c are:
Consider the negative sequence
p
2
Ia
Ia
1 r r
1
ina t
Re Ian e jω t
In cos ω t
2
n
1 r r
Ia
Ib
n
n
j
ω
t
3
ib t
Re Ib e
In cos ω t 2π 3
1 1 1
Ia0
Ic
n
n jω t
ic t
Re Ic e
In cos ω t 2π 3
The system under study is three-phase three-wire:
the sum of the three a-b-c currents is identically zero It can be calculated that
due to Kirchoff’s current law and therefore there is
3
iαn t
kc In cos ω t
no zero sequence current.
2
3
Positive and negative sequences correspond to two
kc In sin ω t
inβ t
2
vectors rotating in opposite directions in the complex
plane. In order to better understand the direction that correspond to a phasor rotating with frequency
of rotation of the vector, the α β fixed reference
ω in the complex plain.
frame is considered. The matrix that changes coorThe effect of changing from a fixed reference frame
dinates from a-b-c to α β is, according to Fig. 1
to the rotating d-q frame is the following. Consider the matrix that describes the change of refer1
1 2
1 2
αβ
Tabc kc
ence frame from the fixed α β one to the d-q one,
0
3 2
3 2
which rotates synchronously with the voltage mains. 3
dq
Tαβ
ω t
cos
sin ω t m
m
Estimation
Let consider the three-phase signal u, its coordinates
in the d-q reference frame are:
sin ωm t
cos ωm t
! !
"
Id0 ∑M
Harmonics rotating with frequency n ωm (indicated
n 1 Idn cos nωmt ϕdn
u
M
I
q0 ∑n 1 Iqn cos nωmt ϕqn
as PR, Positive Rotating) are shifted into frequency
n 1 ωm in the d-q reference frame.
At first, let assume that there is only one d-q harmonic in the signal u whose PR and NR compo3
idp t
kc I p cos n 1 ωmt
nents have to be estimated. Let hωm be its frequency.
2
The proposed estimator based on the Luenberger ob3
kc I p sin n 1 ωmt
iqp t
server
scheme is the following:
2
In the same way, harmonics rotating with frequency
n ωm (indicated as NR, Negative Rotating) are where
shifted into frequency n 1 ωm in the d-q reference
frame.
A linear time-invariant (LTI) state-space representation of the PR and NR sequences at frequency ω is
given by the following oscillators
Ax t ẋ t
with
A
A
Ar
Ac
0
ω
ω
0
0 ω
ω 0
0 (PR)
if ω
0 (NR)
In a digital control system, signals are sampled with
period Ts . Hence the following discrete-time model
of previous oscillators is considered
ii kk 11 A ii kk α
α
β
β
with
A
A
Ar
Ac
cs
where
ch
sh
ch
sh
h
h
sh
ch
sh
ch
if ω
if ω
cos hωm Ts
sin hωm Ts
# K $ u t #
x̂˙ t
&%
Ah x̂ t
Ah
Arh
Arh 0
0 Ach
0
hω m
Ch x̂ t
hω m
0
(1)
h0ω hω0 C 10 01 10 01 and x̂ '$ x̂ x̂ x̂ x̂ % is the state of the estimator, where:
m
Ach
m
h
if ω
dr
qr
dc
qc
T
• x̂dr is the d-component of the estimated positive
sequence;
• x̂qr is the q-component of the estimated positive
sequence;
• x̂dc is the d-component of the estimated negative sequence;
• x̂qc is the q-component of the estimated negative
sequence.
0
(PR)
0
(NR)
The system Ah Ch is completely observable, therefore all the eigenvalues of the estimator dynamic matrix can be arbitrarily chosen by means of K.
Let u be decomposed in the sum of of u1 , the part to
be estimated, and u2 , the part to be rejected, with
u1
x
ẋ
Ch x
xdr xqr xdc xqc
Ah x
$ %
T
Defining
x̃
Mh
x̂ x estimation error
Ah KCh
matrixes Ah Ch Mh of (1) and (2) have to be replaced
by the following A C M :
(2)
A
the estimation error dynamic can be described as follows
x̃˙ Mh x̃ Ku2
11
1
Ah 0
0 Aj
0
0
and then, Laplace-transforming
('$ sI M *% ) K U s
X̃ s
h
1
C
M
2
6 676 4
0
0
6676
666
.
Ch C j
A KC
..
0
0
0
322
2
(6)
0
Az
666 C 5
(7)
(8)
z
The matrix Mh must be shaped in order to ensure that where h j
z are the numbers of the harmonics
considered for estimation. The system A C is still
1. Mh is Hurwitz. This guarantees the conver- completely observable and therefore all the estimator dynamic matrix eigenvalues can be arbitrarily
gence of the harmonic h estimation.
chosen by means of K. The conditions to take in
jω I Mh 1 K
1
2.
account are:
ω nω m n h
All the harmonics different from the h one have
1. M must be Hurwitz. This guarantees the conto be greatly decreased.
vergence of harmonics h j
z estimation.
/ +$ % ) 0 ,+.- A way to satisfy these requirements is to impose
complex conjugate eigenvalues for Mh , with the peak
frequency in hωm and a very low damp. Choosing
K
11 k 3
22
k
k1
k2
k1
k2
2
666 2. +$ jω I M % ) K ,+.- 1 / ω nω n 0 h j 6766 z This guarantees that all the harmonics different
from the h j 666 z ones are greatly decreased.
1
m
1
(3) These conditions can be satisfied forcing complex
conjugate eigenvalues for M, with the peak frequencies at the harmonics that have to be estimated, and
it can be found that the characteristic polynomial of a very low damp. In particular, if all the harmonics
from 1 to hM have to be estimated, then the rejection
Mh is
must be ensured only for the harmonics higher than
2
2
2
hM ; hence j ω I
M 1
K can be shaped
s 2k1 s hωm
2 hω m k 2
as a low pass filter that greatly attenuates harmonics
Therefore, k1 and k2 can be chosen to im- higher than hM .
pose polynomial damp δ and natural frequency In order to implement the proposed estimator on
ωn
1
2 δ 2 1 h ωm
a DSP board, its time-discrete version must be investigated. All the considerations made about the
k1
δ ωn
(4) continuous-time estimators can be repeated for the
ωn2 ω 2
discrete-time ones. In particular, in the case of one
(5)
k2
d-q harmonic PR and NR components to be isolated,
2ω
the discrete-time solution is
Starting from the previous result, let consider the
x̂ k 1
general case of many harmonics to be estimated. The
Ah x̂ k K u k Ch x̂ k
(9)
4
k2
k1
5
)
$
%) 8
# $ 9
:%
( with
uβ t
Ah
Arh
Arh 0
0 Ach
ch
sh
sh
ch
cs cs 10 01 10 01 Ach
Ch
h
h
h
h
10 sin
10 sin
20 sin
10 sin
10 sin
10 sin
10 sin
20 sin
10 sin
4ωmt
5ω m t
7ω m t π 5
8ω m t π 5
10ωmt
11ωmt
12ωmt
13ωmt π 5
14ωmt π 5
with ωm 2π 50 rad s. Assume that the goal is to
estimate harmonics 5 7 11 13. The 5th harmonic
has NR sequency only, while the 7th one has the PR
one only. Hence both are mapped as a 6th harmonic
in the d-q reference frame. The 11th harmonic is
completely NR, the 13th one is PR. Hence both are
2
mapped as a 12th harmonic into the d-q reference
z2 2 cos ωn 1 δ 2 Ts e δ ωn Ts z e 2δ ωn Ts
frame. A continuous-time estimator having dynamic
the same gain matrix K reported in (3) can be con- matrix (8) is considered.
sidered, with
Ar6 0
0
0
The considerations about the placement of the eigenvalues are the same seen for the continuous-time estimators. In particular, if there is only one d-q harmonic to divide into its PR and NR components, and
the following characteristic polynomial is desired
;
k1
k2
)
c cos ω 1 e) 2s2k c h
n
δ ωn Ts
1 h
<
)
)
δ 2 Ts e
1
δ ωn Ts
h
11
A
C
0
0
0
22
Ac6 0
0
0 Ar12 0
0
0 Ac12
1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1
If the harmonics to estimate are several, then the matrixes to consider have the same structure of (6), (7) In order to ensure high rejection to the other harmonand (8).
ics, the matrix K is chosen to obtain for M the following characteristic polynomial:
s 4 Simulation results
2
In order to verify the performances of the estimawith
tor, the following simulation has been executed. Let
u α uβ T ,
consider the three-phase signal u
uα t
m
m
m
m
m
'$ %
10 cos 4ω t 10 cos 5ω t 20 cos 7ω t π 5 10 cos 8ω t π 5 10 cos 10ω t 10 cos 11ω t 10 cos 12ω t 20 cos 13ω t π 5 10 cos 14ω t π 5 m
m
m
m
2δ ωn6 s
2
ωn6
δ
ωn6
ωn12
s 2
2
2δ ωn12 s
0 6 60ω01
1 2δ
112 ω2δ
2
ωn12
2
m
2
m
2
All the estimator state variables have initial values
equal to zero. Each of the resulting estimated harmonics is converted from d-q to α β coordinates
and compared with the original ones. The α component of the resulting estimation errors is presented in
Fig. 2. The β component is similar and therefore is
omitted.
PR, 6th harmonic
20
magnitude
1.5
0
1
-20
0
0.5
10
1
1.5
2
2.5
NR, 6th harmonic
3
3.5
0.5
0
200
0
220
240
260
280
300
320
340
360
380
400
220
240
260
280
300
320
340
360
380
400
-10
0
0.5
20
frag replacements
1
1.5
2
2.5
3
PR, 12th harmonic
150
3.5
phase
100
PSfrag replacements
50
0
0
-50
-20
0
0.5
1
1.5
2
2.5
3
-100
3.5
NR, 12th harmonic
10
-150
200
0
-10
0
0.5
1
1.5
2
2.5
3
3.5
Figure 4: FFT of current signal to be processed,
phase a
Figure 2: Estimation errors, α axis
vma
400
200
vma
400
0
-200
-400
3.966 3.968
200
3.97
3.972 3.974 3.976 3.978
0
1
400
3.97
3.975
3.98
3.985
3.99
3.995
3.98
3.982 3.984 3.986
3.98
3.982 3.984 3.986
0
10
5
0
-5
-10
PSfrag replacements
-0.5
-1
3.966 3.968
3.97
3.972 3.974 3.976 3.978
NR
2
1
3.975
3.98
frag replacements
3.985
3.99
3.995
4
0
-1
-2
ilb
3.966 3.968
10
5
0
-5
-10
3.97
3.982 3.984 3.986
0.5
4
ila
3.97
3.98
PR
200
3.975
3.98
3.985
3.99
3.995
4
3.97
3.972 3.974 3.976 3.978
Figure 5: harmonics estimated in simulation, d axis
Figure 3: Phase voltage vma and current signal to be K is chosen to obtain for M6 a characteristical polynomial with damp δ 0 001 and natural frequency
processed
ωn 6ωm 1 2δ 2 . The d-components of the estimator outputs are shown in Fig. 5. They match the
In order to make comparisons among simulation amplitudes in Fig. 4. In order to verify if also the
and experimental results, the three-phase signal of phases are well estimated, they have been converted
Fig. 3 is considered. In a fixed reference frame, into a-b-c coordinates and subtracted from the origithis signal is characterized by the following harmon- nal signals. The results have 5th and 7th near to zero,
with the fundamental at therefore estimations are good.
ics: 1 5 7 11 13 17 19
fm 50 Hz. The goal is to estimate the 5th and
the 7th harmonics, that are shown in Fig. 4. They 5 Experimental results
both are shifted into 6th harmonic on the d-q reference frame. The three-phase signal is sampled The current signal to be processed is the one dewith frequency fs 7 kHz and then converted from scribed in the previous section and is shown in Fig. 3.
the fixed a-b-c reference frame to the rotating d-q The three-phase signals are acquired and sampled
one. A time-discrete estimator having the structure with frequency fs 7kHz. The DSP on-line exeof (9) is considered, with h 6. The gain matrix cutes all the calculus to change reference frame from
6
666
vma
400
200
0
-200
-400
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
0.02
PR
1
0.5
0
frag replacements
-0.5
-1
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
0.02
0.012 0.014 0.016 0.018
0.02
NR
2
1
[4] L. Malesani, L. Rossetto, P. Tenti. “Active filter power filter with hybrid energy storage’,
IEEE Trans.Power Electron., volume 6, pp.
392–397, (July 1997).
[5] P. Mattavelli. “A closed-loop selective harmonic compensation for active filters”, IEEE
Trans.Ind.Applicat., volume 37, pp. 81–89,
(January/February 2001).
0
-1
-2
0
0.002 0.004 0.006 0.008
0.01
Figure 6: harmonics estimated in laboratory
a-b-c to d-q and implements a time-discrete estimator having the structure of (9), with h 6. The dcomponents of estimated harmonics are shown in
Fig. 6. They match the ones resulting from simulation.
6 Conclusions
A positive and negative sequence on-line estimator
has been presented. It allows to separate the positive
from the negative sequence, that hence can be separately processed by the active power filter controller.
Simulation and experimental results have been presented.
References
[1] H. Akagi.
“New trends in active filters for power conditioning”,
IEEE
Trans.Ind.Applicat., volume 32, pp. 1312–
1332, (Nov./Dec.1996).
[2] H. Akagi, A. Nabae. “Control strategy of active power filters using multiple voltage source
PWM converters”, IEEE Trans.Ind.Applicat.,
volume IA-22, pp. 460–465, (May/June 1986).
[3] C.L.Fortescue newblock “Method of symmetrical coordinates applied to the solution of
polyphase networks”, Trans.AIEE volume 37,
pp. 1027–1140, (1918).
[6] F. Ronchi, A.Tilli “Design methodology for
shunt active filters”, Proc. 10th EPE-PEMC
2002, (Cavtat & Dubrovnik, Croatia, September 2002), to be published, .
[7] G. L. Van Harmelen, J.H.R. Enslin. “Realtime dynamic control of dynamic power filters
in supplies with high contamination”, IEEE
Trans. Power Electron., volume 8, pp. 301–
308, (July 1993).
[8] P. Verdelho, G. D. Marques. “An active
power filter and unbalanced current compensator”, IEEE Trans.Ind.Applicat., volume 44,
pp. 321–328, (June 1997).