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Design and Analysis of Output Filter for 3 phase
UPS
Inverter
Byoungwoo Ryu*, Jaesik
Kim**,
Jaeho Choi , and Changho Choi
Chungbuk National University*
48 Gaesindong, Cheongju, Chungbuk
Phone: +82-43-261-2425, Fax: +82-43-276-7217
E-Mail: choi@,power.chunPbuk.ac.kr
POSCON Ltd., Seoul, S. KORE A**
361-763
S.
KOREA
Abstract
This paper proposes a design method of the
L-C
output filter for three-phase
UPS
inverter: The
harmonic components
of
the output voltages are
dependent
on
the
L-C
value and the system time
constant defined by the switching frequency, the
speed o f control processo< the controller gain, and
etc., but their characteristics are complicate and
entangled each othel: In this paper: the relation
between the
L-C
value and the system time constant
are described with the closed form and the filter
values can be calculated to meet the limit of voltage
THD considering the system time constant. The
schematic design procedure is given by a single f l ow
chart and the proposed filter design algorithm is
verified through the simulation an d experiments.
Key words: output filter design, UPS, zero-output
impedance
1 Introduction
There are two kinds of harmonic sources that make
the UPS output voltage waveform be distorted: One
is caused by the PWM switching of inverter and the
other is caused by the nonlinear characteristics of the
load. If any linear loads are connected to the inverter
output side, then the output voltage harmonics due to
the PWM switching can be eliminated by inserting
the L-C filter between the inverter and the loads, and
it
is well known how to design the optimal values of
the filter components [ l , 21. But the configuration of
the power circuit with L-C filter cannot meet the zero
output impedance at all. Therefore, the capacitor
voltage is usually collapsed in case of the sudden
load change or under the nonlinear load.
By using the fast feedback controller for the output
capacitor voltage, the harmonic distortion due to the
nonlinear load characteristics, such as the capacitor
input type rectifier load or transient load change, can
be complemented [3]. But the performance of the
feedback controller is dependent on the system time
constant defined by the switching frequency, the
speed of control processor, and the controller gain.
And the harmonics suppression characteristics of the
L-C filter and the feedback controller are entangled
each other and complicate to analyze. Therefore, it is
difficult to design the L-C filter considering the
system time constant with the conventional methods.
In this paper, the harmonics characteristics due to
the L-C filter value and the system time constant are
described with the closed form and L-C filter values
can be calculated to meet the given specification of
voltage THD, considering the system time constant.
It means that the harmonic components due to the
PWM switching of inverter and the nonlinear load
characteristics are decoupled separately, and the L-C
filter value and the performance of the feedback
controller are determined to meet each allotted ratio
of THD, for the harmonics suppression [4, 51.
The design procedure of the L-C filter for three-
phase UPS inverter is described. The transfer
function of the filter output voltage to the load
current is given as the closed form and it is possible
to get the L-C values by solving this closed form.
The capacitor value can be calculated with the given
system time constant defined by the performance of
the system controller. This procedure is more
practical to design the power circuit and control
circuit of inverter for UPS than the conventional one.
And also, the effect of the load current to the voltage
distortion can be calculated from the closed form, so
it is possible to analyzed the system how much the
voltage waveform is distorted under the nonlinear
load.
Finally, the schematic design procedure is given by
a single flow chart and the proposed filter design
algorithm is verified through the simulation.
2
Three phase PWM VSI system
The three-phase voltage and curren t equations of
0-7803-7156-9/02/ 10.000 2002 IEEE 941
PCC-Osaka 2002
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P WM INVERTER
LC
FILTER LOAD
I
I
Fig.
1
Configuration of three-phase PWM-VSI power
circuit.
I
I
Fig.
2
Single-phase equivalent block diagram.
the input and output sides of the filters are transferred
to , he dq-values under the stationary reference frame
as (1) and ( 2 ) .
A
1
vc i
o
c / s
The three-phase system can be described by the
two single-phase systems of 'd' and 'q' as shown in
Fig. 2 without the coupling terms. The dq single-
phase equivalent blocks are shown as the M I S 0
system with two inputs of the inverter output voltage
and the load current in Fig. 2 . We will deal with the
equivalent system of 'd' axis only in the equations
below and it can be applied to
q
axis equally.
From Fig. 2 , the transfer function of the capacitor
output voltage to the inverter output voltage and the
load current can be described in (3) neglecting the
resistance of inductor [ 5 ]
In conventional method of the filter design, the
second term is treated as the disturbance and only the
first term is considered to determine the L-C values.
Once the product of L,Cj is calculated from the
expecting cut-off frequency of the L-C filter, the
exact values of Lf nd Cf are determined at values that
minimize the cost function defined by the summation
of the reactive powers. This design procedure can be
applied to a linear load well. But under the nonlinear
load or transient load change, we cannot neglect the
harmonics components due to the second term.
If
the inductor value is minimized at almost zero in
(3),
the capacitor voltage can be independent on the
load current characteristics and it means that the UPS
system works as an ideal voltage source. To minimize
the inductor value, the capacitor value should be
maximized. But as the increase of the capacitor value,
the power rating of an inverter will be increased due
to the increase of the reactive power.
So
it is
permitted to increase the inductor value for the
decrease of the capacitor value as far as the harmonic
components of the capacitor voltage is controlled
with the help of fast feedback voltage controller. It is
clear from the above that the filter design value
should be compromised as the performance of the
system controller. But they are coupled each other
and the filter values are adjusted as the performance
of the controller by the try-and-error method in the
conventional methods.
3 Proposed filter design method
3.1
Transfer function of system controller
Figure 3 shows the system block diagram with
controller. The control system is designed based on
double regulation loops: the main feature is the inner
control loop of the current flowing through the filter
capacitor can be limited within a given range and a
second feedback has been performed using an outer
voltage loop which makes it possible to improve the
dynamics of the entire control system. The transfer
function is driven as
(4).
I 1
Fig. 3 System block diagram with controller
In (4), let's consider the input-output transfer
characteristics responding to the voltage command
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and the load current separately. For the analysis of
the effect due to the capacitor com mand voltage, the
effect due to the load current can be neglected by
putting zero, I,,=O. Then th e controller gains, Kp, Ki,
and
K ,
can be designed to m eet-the ideal controller.
For this, the gain of the transfer function satisfies unit
to make all the values of the nominator and the
denominator are same within the passing bandwidth.
From the above, (4) is simplified as (5).
If the inverter output L-C filter is designed well to
eliminate the harmonics included in the inverter
output voltage, then the command voltage of
5),
Vcd*
is substituted by the fundamental component of the
inverter output voltage, Vadl s in (6).
For the analysis of the effect due to the change of
load current only, the first term o f the right side of (6)
can be neglected and the transfer characteristics of
the capacitor voltage to the load current is given as
7).
7 )
c d ( j n o o ) L P ~ W , ~
IOd ( i n ) K , K ,
3.2
Design of con troller gain
For the design of controller gain, CDM(Coefficient
Diagram Method) controller with observer canonical
form is used [6]. The characteristic equation of (4) is
defined in
CDM
controller by (8) to meet the
robustness.
a
r 2
3 5 s 3 0 s
ao no
Y2YI
YI
With the comparison of the coefficients of two
equations, the transfer function of the capacitor
voltage to the load current given by (7) is described
as
(9).
( 9 )
If the specification of the capacitor voltage
harmonics caused by the nonlinear load current is
given arbitrary, then the relation between the system
time constant and the capacitor value
C
is given
as the closed form of (9). In this equation, the
stability factors of are used here as y , =2.5 and
y 2
2
which satisfy the Manabes recommendation
[5,61.
And also, from the same methods of comparison,
all the values of controller gains,
Kp,
Ki,nd
K ,
are
given as like the following equations.
4 Filter design procedure
Before proceeding of the filter design, the UPS
specification and the system parameters are given as
followings:
1) Total harmonic distortion factor of voltage,
THD,
is below 5% in case of both
0.8
linear
lagging load and nonlinear rectifier load with the
crest factor,
CF=1.
2 ) Modulation index,
my
is given as
0.7
considering
the variation range of DC link voltage and load.
The flowchart of filter design procedure is shown
in Fig.
4.
In the design process, if the calculated total
harmonic distortion factor of the output voltage THD,
doesnt meet the specification, then it repeats the
routine after decreasing the rate of voltage harmonics
due to the dominant load current harmon ics [4,5].
4.1
Define the harmonics
of SPWM VSI
voltage
and
load
current
The voltage harmonics of
SPWM-VSI
is defined
ideally as (13) and the fundamental component and
dominant harmonics of inverter output voltage
normalized by DC input voltage
is
given as Table
I.
Under the assumption of ideal voltage waveform,
the nonlinear load current harmonics of capacitor
input type rectifier with the crest factor, CF=3 is
analyzed using the computer simulation. Table I
shows the Fourier coefficients of load current
normalized by the fundamental component.
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Table I. Normalized inverter voltage harmonics,
v a d ~ v d ~s the change of mo dulation index, mp
mf
Order
1
F S W
f7 2
1 0.9 0.8 0.7 0.6 0.5
1.00 0.90 0.80
0.70
0.60
0.50
0.60 0.71 0.82 0.92 1.01 1.08
0.32 0.27 0.22 0.17 0.13
0.09
n
Mag.
4.2 Determine the impedance ratio of inductor
and capacitor
The impedance ratio
of
inductor and capacitor is
determined to satisfy the condition that the ratio of
output voltage to input voltage at the specific
harmonic order shown
in (14)
can be smaller than
3[ ]
of the fundamental component of the output
voltage as shown in (15). The specific harmonic
order will be given as the same value of switching
frequency.
1 3
5
7 9 11 13
1.00 0.73
0.35
0.07 0.04
0.02 0.01
(14)
4.3 Determine the capacitance value based on the
cost function
The upper limit of the capacitance is determined
at the value that minimizes the cost function defined
by the total kVA
of
the filter L-C components as
shown in (16).
In (16),
W
is the w eighting coefficient determined
due to the relative cost of inductor to the capacitor.
4.4 Determine the system time constant and
capacitance value
The system time constant zi s selected 3 times
longer than the switching time considering the
system stability and the dynamics empirically. Then
the capacitance value is calculated using the given
values of system time constant and the rate of
permitting voltage harmonics due to the load current
from (8). The permission level of the harmonics due
to the nonlinear load current is initially selected 1 A]
considering the remaining harmonics after
eliminating with
L-C
filter. If the THD, canno t meet
the specification finally, then the calculation of the
capacitance value should be repeated after decreasing
this level.
0.01 y2y,2 1
xc
~
n 2 w i I o n
5
4.5
Determine the gains of controller
Using
1
0) to (1 2), all values of controller gains
K,,,
Ki,
nd K, are determined with the given value of
the stability factor and the determined values of the
system time constant and capacitance.
4.6
Calculation of the output voltage harmonics
The total harmonic distortion factor of the output
voltage THD,is calculated from (4) as described in
(1 8). The determined values of controller gains and
the L-C parameters are used for this calculation.
After the calculation of THD,, if it does not meet
the specification, then the above process should be
repeated from the top after modifying the condition
as followings: 1 ) In case o f linear load, repeat the
calculation process after decreasing the impedance
ratio
of
inductance and conductance from
3[ ].
2 ) In
case of nonlinear load, repeat it after decreasing the
harmonics ratio caused by the load current from
1
?I.
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Harmonics analysis o f
PWM voltage and no nlinear load current
Calculating frequency transfer function of
filter input to filter output vo ltage
1
Determine the impedance ratio of
filter inductor and capacitor
,
output
Power
output
Voltage
Considered
Controller
Making closed form of filter capacitance and system equivalent time
constant using transfer function with contvoller
using closed form for load current harmonics
of the amplitude of dominant
30[kVA] Switching 9.54[kHz]
Frequency
22O[V] Controller 2OO[us]
time constant
f
4o[uH]
Considered L /
165[uH]
Cost function
c/ =100[uF]
c 25[uF]
R/
0 05f 0.05
calculated by cost function
Retune to the system
1
the impedance ratio of
I
equivalent time constant
Selecting controller gains
satisfying the system response
4
Analyzing output voltage harmonics under the linear and nonlinear load
using transfer function considering the system performance
of
controller
Calculating DC link voltage
I
Fig. 4 Flowchart o f filter design procedure
5 Simulation results
Table I shows the system parameters and the
design values of
L-C
filter and the controller gains
using the filter design flowchart as show n in Fig.
4.
In
case of the linear R-L load, the THD, of
capacitor output voltage is
3.2[ ]
when the
L-C
filter
is
designed by the conventional method based
on
the
cost function and 3.7[ ] when it is designed by the
proposed method, respectively. From the results, it is
verified that both methods are us ef d under the linear
load.
From Fig. 5 to Fig.
8,
the output voltage and
current waveforms and the harmonic spectra of the
output voltage waveform under the capacitor input
type three-phase rectifier with
CF=
re shown
Table 111. System parameters for simulation
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300
300
I I
0.00
0 01 0 02
0 03
T i m sec]
-300
0.00 0 01 0.02 0.03
Tims [sec]
Fig.
5
Output voltage and current waveforms with L-
C filter designed by the conventional method.
Fig. 7 Output voltage and current waveforms with L-
C filter designed by the proposed method.
.
2 4 6 8 1 12 14 16
Order
..
_ _ _ .
h
2
4 6
8 1 12 14
16
Order
Fig.
6
Harmonic spectrum of output voltage in
conventional method. proposed method.
Fig.
8
Harmonic spectrum of output voltage in
When the filter is designed by the conventional
method, the voltage waveform is distorted a little and
the its
THD,
is 8.5[ ] with the given system time
constant. Therefore, to meet the specification of
THD,, the inductor value should be reduced and the
capacitor value should be larger. And the inverter
power rating will be increased due to the increase of
the reactive power.
On the other hand, if the filter is designed by the
proposed method, then the output voltage waveform
is almost sinusoidal and the THD, comes within the
specification with 4.6[%].
6.
Conclusion
References
[ l ] S .
Dewan and
P.
Ziogas, Optimum filter design for a
single phase solid-state UPS system,
IEEE Trans. Ind.
Appl.,
vol.
IA-15, no. 6, pp. 664-669, 1979.
[2] T. G. Habetler
and
D. M. Divan, Rectified Inverter
reactive component minimization, in Conf. Rec. of
IEEE PESC87, pp. 648-657, 1987.
[3] A.
Kusko, D. Galler, and
N.
Medora, Output
impedance of PWM UPS inverter-feedback vs. filters,
in Con Rec. of IEEE IAS90, p p .
1044-1048,
1990.
[4]
. Kim,
J.
Choi, and
F.
Blaabjerg, Design and analysis
of
output filter for UPS, in ConJ: Rec. of EPE2001,
DS3.1, 2001.
This paper proposes a filter design procedure for
three-phase
UPS
inverter. By using the proposed
closed form under the given system time Constant.
Therefore, the L-C filter value can be determined
using any try and error method.
[51
J Kim, Design of output filter and controller fo r UPS
inverter, Ph. D thesis in Chungbuk National University,
2001.
[6] S. Manabe, C oefficient diagram m ethod, in ConJ Rec.
o IFAC symposium
on
Automatic Control in Aerosp ace,
method, the filter values can be calculated from the
quickly through the systematic procedure without
pp. 199-210, 1998.
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